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4.9 Laserresonator

1. zZ do one uses a trial solution for 11 of the form DE y 2 exp lertn F is 85 w 2q After some calculation one finds ae 0 where L is a generalized Laguerre polynomial and p and are the radial and angular mode numbers L x obeys the differential aion BL Er MESEDE 2 ee 37 T Some polynomials of low order are Loa La l 1 2 Lea 4U Y 2 14 22 ha 38 As in the case of beams with a rectangular geometry the beam parameters w z and R z are the same for all cylin drical modes The phase shift is again dependent on the mode numbers and is given by p l 2 Lp l 1 are tanQz2 rw0 39 3 4 Beam Transformation by a Lens A lens can be used to focus a laser beam to a small spot or to produce a beam of suitable diameter and phase front curvature for injection into a given optical structure An ideal lens leaves the transverse field distribution of a beam mode unchanged i e an incoming fundamental Gaussian beam will emerge from the lens as a funda mental beam and a higher order mode remains a mode of the same order after passing through the lens However a lens does change the beam parameters R z and w z As these two parameters are the same for modes of all orders the following discussion is valid for all orders the relationship between the parameters of an incoming beam labeled here with the index 1 and the parameters of the a outgoing
2. BEAM CONTOUR Ri Ra Mode parameters of interest for a resonator with Fig 11 mirrors of unequal curvature where c is the velocity of light After some algebraic manipulations one obtains from 49 the following for mula for the resonant De of a mode v vo q D m n 1 are cos 1 d R 51 For the special case of the confocal resonator d R b the above relations become w Ab r Ab 27 vivo q 1 3 m n 1 52 The parameter b is known as the confocal parameter Resonators with mirrors of unequal curvature can be treated in a similar manner The geometry of such a resonator where the radii of curvature of the mirrors are R and Ra is shown in Fig 11 The diameters of the beam at the mirrors of a stable resonator 2w and 2ws are given by d wii AR a i ae E WI ae 1 d d 53 wo ARa r E Rs d Band The diameter of the beam waist 2w which is formed either inside or outside the resonator is given by d R d ts d R R d wo a gt eS TT Ri Ro 2d oe The distances f and t between the waist and the mirrors measured positive as shown in the figure are d Re d 1 NE ES UR d ee Aare 55 R Ro 2d The resonant condition is 1 v v I m mn 1 are cosy 1 dRD d Ra 56 where the square root should be given the sign of 1 d R which is equal to the sign of 1
3. If we wish we can define normalized dimensionless variables ra r A n A and pa p A so that the normalized fringe radii are given by pnt Pp 4enPn Pn m Ze Al where is the fractional order of interference on axis The spot size has a normalized radius ps n rn F A2 With these equations it is straightforward to convert the calculated curves to normalized curves which apply to any set of system parameters As an alternative we list below recipes for converting the calculated curves so that they may be used with different values of r and or A The distances ro and X are 10 em and 6 25 X 105 em respectively Figure 3 If the mirror radii are each r aro and if A bdo then i change each value of p to p pa and ii change each value of m the order of interference relative to the order on axis to m m ab iii The e axis remains unchanged Note that the curve defining p corresponds to m 1 F Figure 4 Ifr ary and A baa then change p to p a b p Figure 5 Ifr ary and A bdo then i change pto p aib p and ii change v vo to v v vo Q Figure 8 Same as Fig 4 References P Connes Rev Opt 35 37 1956 P Connes J Phys Radium 19 262 1958 P Connes in Quantum Electronics and Coherent Lighi P A Miles Ed Academic Press Inc New York 1964 p 198ff 4 W H Steel Interferometry
4. PicoScope oder PicoLog meldet Fehlercode 4 5 oder 6 Dieser Fehler wird gemeldet wenn es ein Problem mit dem Oszilloskop selbst gibt Solche Probleme k nnen daraus entstehen dass die Konfigurationseinstellungen besch digt sind ein Hardware Fehler vorliegt oder ein spezielles Programm nicht richtig funktioniert Ziehen Sie den Stecker des Oszilloskops warten Sie ein paar Sekunden und stecken Sie ihn wieder in die USB Buchse Wenn der Fehler immer noch gemeldet wird fragen Sie den Technischen Support von Pico PicoScope oder PicoLog meldet Fehlercode 7 Dieser Fehler wird gemeldet wenn das Betriebssystem zu alt ist um das Oszilloskop der Serie PicoScope 3000 zu unterst tzen Weitere Informationen finden Sie im Abschnitt Systemanforderungen Copyright 2005 7 Pico Technology Limited All rights reserved PS3000049 2 10 Series PicoScope 3000 Handbuch 3 2 Stromversorgung des PicoScope 3204 5 6 Die PC gest tzten Oszilloskope PicoScope 3204 5 6 werden normalerweise ber die USB Schnittstelle des Computers mit Strom versorgt Liegen jedoch der Computer und der Pr fling beide in Bezug zur gleichen Masse kann sich eine Masseschleife aufbauen Dies kann die Gleichstromgenauigkeit und St rfestigkeit beim Messen von Signalen mit niedrigen Pegeln verschlechtern Eine Masseschleife baut sich normalerweise auf wenn das PicoScope an einen vom Netz gespeisten Computer angeschlossen ist und zum Messen eines Signals an ei
5. Transmitted Intensity R 25 FR 2 R 65 Fr27 R 90 Fe 30 Order Number Figure 3 Fabry Perot transmission for different mirror reflectivities Choosing the Right Fabry Perot The best Fabry Perot interferometer for a particular application depends on many factors including size stability tunability free spectral range resolution light gathering power and price Distinguishing features of the various types of Fabry Perot systems are outlined below Note The word etalon is usually used for small Fabry Perots that might serve as wavelength selective filters inside the laser cavity The following discussion uses the terms etalon and interferometer according to common practice for some devices they are practically interchangeable Solid etalons and fixed air gap etalons are stable and compact making them ideal for wavelength filtering frequency calibration coherence extension and intracav ity mode selection in lasers Solid etalons are made from a piece of optically homogeneous material such as fused quartz Opposite faces are polished flat and parallel and coated to any desired reflectivity In a fixed air gap etalon two mirrors are bonded to a solid spacer element Both types are highly stable mechanically but solid etalons are more sensitive to temperature changes A solid etalon is best used in a thermally controlled housing where it can be temperature tuned or stablized Fixed air gap etalons are
6. 6 85221 Dachau Munich Germany Tel 49 8131 59560 Fax 49 8131 595699 Web http www thorlabs com Mail europe thorlabs com Doc 18356 S01 doc Rev A 1 Oct 08 Quick Reference Thorlabs Instrumentation Optical Power and Energy Meter PM100D 2009 e LABS Safety d Attention 4 All statements regarding safety of operation and technical data in the instruction manual will only apply when the unit is operated corectly The power meter PM100D must not be operated in explosion endangered environments Sensor photodiode and control inputs and outputs must only be connected with duly shielded connection cables Only with written consent from Thorlabs may changes to single components be carried out or components not supplied by Thorlabs be used Do not remove covers Refer servicing to qualified personal Table of Content 1 General Information 2 Getting Started 2 1 Unpacking 2 2Preparation 2 3Physical Overview 3 Operating the PM100D 3 1 Connecting a Power or Energy Sensor 3 2 Controlling the PM100D 3 2 1 Navigating the Menus 3 2 2 Power and Energy Measurement in the Numeric Display 8 3 2 3 Display Options 10 3 2 4 Analog Output 11 3 2 5 Battery Charging 11 3 3 Computer Interface 11 4 Addresses 12 DDP WWW W ALAS 1 General Information The PM100D Handheld Optical Power and Energy Meter is designed to measure the optical power of laser light or other monochromatic or near monochromatic light sources
7. Multipass operation of Fabry Perot interferometers is now widely used especially for experiments involving spectral analysis of light scattered from surfaces thin films or opaque materials Throughput of a properly designed multipass system is very good With 93 reflectivity mirrors the actual throughput compared to single pass with the same entrance aperture is approxi mately 50 for three pass and 30 for five pass Mirror flatness is very important for good multipass operation so that the mirror spacing can be made identical for all passes therefore only large frame Fabry Perot interferometers with distortion free mirror Suspensions are recommended Special retroreflector assemblies for three pass or five pass operation are com mercially available The use of two or more interferometers in tandem to alleviate the problems of overlapping spectral orders has often been proposed One scheme simply utilizes a low resolution Fabry Perot interferometer with a fixed spacing to prefilter a portion of the spectrum followed by a high resolution scanning interferometer with free spectal range equal to or greater than the bandwidth of the prefilter The transmission profile is that of the high resolution interfer ometer with a throughput modified by the prefilter that serves to reject adjacent spectral orders Often an interfer ence filter is added to such a system to provide complete blockage of unwanted spectral orders beyond the free s
8. Page 1 of 5 THORLABS INC Confocal Cavity Configuration Mirrors shown below are AR coated on the outer surfaces and HR coated on the inner surfaces cl RE Smm FSR c 4d Other mirror sets are available for this instrument 350nm 535nm 535nm 820nm 820nm 1275nm 1275nm 2000nm 1800nm 2500nm SETUP e Knowing the FSR of the SA210 allows the time base of an oscilloscope to be calibrated to facilitate quantitative measurements of laser line shape With a resolution of 67MHz the fine structure resulting from multiple longitudinal modes of a laser line can be resolved Note A saw tooth wave 0 20V would provide approximately 2 Free Spectral Ranges e The SA210 should be mounted so that it can be easily adjusted It is recommended that Thorlabs 1 inch Kinematic Mount KM100 be used to mount the interferometer at the 1 inch diameter flange e The apparent beam size should be approximately 1mm It is recommended that a fold mirror be used to direct the beam into the Fabry Perot interferometer A lens with focal length of 100mm can be used with the focus set roughly at the center of the housing approximately 25mm in from the flange e The maximum voltage on the piezo ramp in is not to exceed 150V e If the detector is connected directly to the scope a 5kQ terminator is needed Offset adjustment SA201 is used to center the output on the scope OPERATION To set up the SA210 Series Fabry Perot you should first
9. Phone 86 0 21 32513486 Fax 86 0 21 32513480 Email chinasales thorlabs com Web www thorlabs com Please call our hotlines send an Email to ask for your nearest distributor or just visit our homepage http www thorlabs com Copyright 2009 Thorlabs Germany 17654 D02 Rev D M0009 510 612 Series PicoScope 3000 PC Oszilloskopen Handbuch PS3000049 2 Copyright 2005 Pico Technology Limited All rights reserved I Series PicoScope 3000 Handbuch Inhaltsverzeichnis ILEINTUNFUNE HE 1 MU e e Do al 1 2 S cherheitszeichen iaa 1 3 S cherheitshinweise a ba o sale 2 ACE Hinweis ii ee ee ta era cece 2 WEE HINWEIS Se II a eee ae ae 3 6 Ga r nte ars AA a e e aL 3 7 Rechtliche Hlinwelise a Ei 4 8 War nzeichen a ne in inne 4 9 Firmenanschrift 0u 00a a a a a aa aaa 5 2 PROCURE RON tl ON ooo ona 6 1 Systemantorderunden AAA AA AAA AAA iRERaRS 6 Z Installationsanleitung aa Aa aw dancin exes 6 3 Technische Daten aa Hs Haie 8 3 FENIErDENANALUNG ari ii AA ARA AAA AAA 9 1 Fehle rcodes ura a a ed el ol add eo ln a Wai 9 2 Stromversorgung des PicoScope 3204 5 6 uuuunnunnunnunnunnunnunnunnuunnunnunnnnnunnunnunnnnnn 10 IAS AA 11 PS3000049 2 Copyright 2005 7 Pico Technology Limited All rights reserved Einf hrung 1 1 1 1 1 2 Einf hrung Ubersicht Die Serie PicoScope 3000 ist eine Reihe von High Speed PC Oszilloskopen die USB 2 0 vollst n
10. These include mirror reflec tivity mirror figure diffraction losses and alignment One of the great advantages of the spherical Fabry Perot interferometer over its plane parallel counterpart is the relative ease with which reflectivity limited resolution can be realized in practice Neglecting all but transmission losses at the mirrors the instrumental profile of an FPS is given by Eq 4 The resulting value for Avm the width of the instrumental profile is given by Avm c 1 R 4rrR 10 At this point it is useful to introduce a quantity called the finesse F of the interferometer which we can define as the ratio of the free spectral range to the instrumental width F Avs Avm 11 In terms of the finesse F the instrumental width and spectral resolving power are given by Arm C 4rF 12 and 4rF X 13 Also the expression for the interference pattern can be written as T To oke 5 1 2F m sin2 5 2 14 Here we have simply made the substitution F TR 1 R in Eq 4 Note that this expression re 954 APPLIED OPTICS Vol 7 No 5 May 1968 mains valid regardless of whether the finesse is de termined by the mirror reflectivity or by other factors When the finesse is limited by a mirror reflectivity whose value is close to unity we have Fr rR AM R 7 21 R 15 The fringe pattern described by Eq 14 is shown in Fig 4 for representative values of the
11. and the capability of the instrument to be used to display spectral information in the form of a multiple beam interference fringe pattern The FPS is clearly superior to a nonconfocal resonator for use with spatially incoherent sources and with fast pulsed light sources It is also very much easier to use with cw laser sources and permits the spectral analysis of lasers operating in a number of different transverse modes Section II deals with the theory of the FPS and includes subsections on the localized fringe pattern spectral resolution and instrument profile with finite apertures light gathering power and mode matching considerations Section III contains descriptions of prototype scanning and static FPS spectrum analyzers and prac tical procedures for their optimum use We have been able to achieve finesses well in excess of 150 with both 5 cm and 10 cm mirror spacings both instruments are thermally compensated and mechanically stable and the 5 cm FPS incorporates a piezoelectric scanning device which permits its use in either a static or rapid scan mode Table I lists the symbols used May 1968 Vol 7 No 5 APPLIED OPTICS 951 Table l List of Symbols A 5 aaa A area of FPS entrance aperture c velocity of light d FPP mirror separation D aperture diameter FPS or FPP F finesse rR 1 R FPS L cavity loss per transit M fringe pattern magnification factor r FPS mirror radii and con
12. d R for a stable resona tor There are more complicated resonator structures than the ones discussed above In part cular one can insert a lens or several lenses between the mirrors But in every case the unfolded resonator is equivalent to a periodic sequence of identical optical systems as shown in Fig 2 The elements of the ABCD matrix of this system can be used to calculate the mode parameters of the resonator One uses the ABCD law 43 and postulates self con sistency by putting q q2 Gq The roots of the resulting quadratic equation are A Ci a 35 er A DP 57 which yields for the corresponding beam radius w i ma mn 24B 7 V4 A D 58 3 6 Mode Matching It was shown in the preceding section that the modes of laser resonators can be characterized by light beams with certain properties and parameters which are defined by the resonator geometry These beams are often injected into other optical structures with different sets of beam parameters These optical structures can assume various physical forms such as resonators used in scanning Fabry Perot interferometers or regenerative amplifiers sequences of dielectric or gas lenses used as optical trans mission lines or crystals of nonlinear dielectric material employed in parametric optics experiments To match the modes of one structure to those of another one must transform a given Gaussian beam or higher order mode into another beam with
13. lt P gt keys 3 2 2 3 Readout Configuration Select Units of Main Display OK d IAE e Sub Left Sub Right EXT Depending on the connected sensor the units of measure for the large display can be configured to Watt dBm Joule Volt or Ampere For additional information two small displays are arranged below the main display Hide left Sub Display OK lo o pp Em Min Val Minimum level until MAX RESET is pressed Same unit as main display 00 DRUAN ES Frequency Displays the frequency of a power signal or the repetition rate for pulsed laser sources Temperature Shows the sensor head temperature Off Hides the left sub display Ratio MaxfMin OK CC ee Pe Max Val Maximum level until MAX RESET is pressed in main display unit Max Min Ratio between maximum and minimum Area Calculates power and energy density from the set beam diameter Alt Unit Shows the measurement in its alternate unit e g W dBm Off Hides the right sub display 3 2 2 4 Measurement Configuration Menu Photodiode Sensor Exit Menu OK Att 6 0048 a Pp MEN Thermal Sensor Exit Menu OK BE Pyroelectric Sensor Edit AY Cursor dF Save OK att 0048 frig 10 0 10 00m o po of em Att Sets an attenuation or gain factor BW Sets the bandwidth of the photodiode input stage to High or Low Acc Switches the acceleration circuit for the thermal sensor input stage Trig Sets the trigger level for lase
14. sreater than 150 over a spectral range of about 1200 A and with an instrumental transmission of 40 to 45 compared with the maximum of 50 in the nonmode matched configuration With careful mode matching the instrumental transmission goes to 80 90 the free spectral range becomes 3000 MHz and the finesse is doubled leaving the resolving power unchanged With broadband multilayer mirrors a finesse of 150 can be maintained from about 4900 A to 7000 A with an instrumental transmission of approximately 25 or 50 when mode matched With these mirrors this instrument has been used for high resolution spectros copy with argon lasers 4880 A and 5145 A He Ne lasers 6328 A and Q switched ruby lasers 6943 A A variety of experimental data is shown in Sec III E In the scanning mode the instrument is very easy to align to the incident beam of light First of all the instrument is located so that the incident light beam preferably but not necessarily more or less colli mated falls close to the center of the lens The axis of the instrument is then roughly aligned with the incident beam This can be done most conveniently by observing the light that has been reflected from the FPS back towards the source the reflected beam should be adjusted to lie fairly close to the incident beam At this point a high resolution scan display can be ob served by monitoring the detector output at an oscillo scope The display is peaked by fur
15. these mirrors may be flat or spherical and the distance between them can range anywhere from micrometers to meters All Fabry Perot designs share some common features but there are important differences which determine the right choice of interferometer for a partic ular application The author is manager of research and development at Burleigh Instruments Inc Fishers NY 14453 How It Works The Fabry Perot mirrors form an optical cavity in which successive reflections create multiple beam intefer ence fringes The simplest and most versatile design is the flat mirror cavity As shown in Figure 1 illumination by an extended monochromatic light source produces bright fringes of equal inclination in the focal plane of L producing a characteristic bull s eye pattern At the angle O where a bright fringe is observed the relationship between the source wavelength A and the mirror spacing d is mA 2nd cos O 1 where n is the refractive index of the medium between the mirrors and m is an integer identifying the order of interference A pinhole aperture on the optical axis at the focal point of L limits the light transmitted through the pinhole to that passing through the Fabry Perot parallel Bull s Eye Interference Pattern Extended Monochromatic Light Source jea Fabry Perot Interferometer Figure 1 Diagram of Fabry Perot spectrometer Lasers amp Applications July 1983 47 Figure 2
16. 4 6 10 20 N a2 A d 0 1 0 2 Fig 23 Diffraction loss per transit in decibels for the TEMa mode of a stable resonator with circular mirrors The diffraction losses for the two lowest order TEMoo and TEM modes of a stable resonator with a pair of identical circular mirrors a as gi g2 g are given in Figs 22 and 23 as functions of the Fresnel number N and for various values of g The curves are obtained by solving 77 numerically using the method of successive approximations 31 Corresponding curves for the phase shifts are shown in Figs 24 and 25 The horizontal por tions of the phase shift curves can be calculated from the formula B 29 l 1 are cos Vg1go 2p 1 1 arc cosg for gi 92 85 which is equal to the phase shift for the beam modes derived in Section 3 5 It is to be noted that the loss curves are applicable to both positive and negative values of g PHASE SHIFT DEGREES gt 0 6 TEMoo MODE 1 1 0 4 06 10 2 4 6 10 20 40 60 100 N a2 Ad Fig 24 Phase shift per transit for the TEMo mode of a stable resonator with circular mirrors PHASE SHIFT DEGREES 40 60 100 6 10 20 0 4 06 10 2 4 N a2 Ad Phase shift per transit for the TEM mode of a stable resonator with circular mirrors 0 1 0 2 Fig 25 while the phase shift curves are for positive g only the phase shift for negative g is equal to 180 degrees minus that for positive g Analytical expressions for the di
17. An oscilloscope is typically used to view the spectrum and make quantified measurements of spectral features Recommended Set up OSCILLOSCOPE O Ne O 1 CHANNEL ONE SA200 SERIES FABRY PEROT MANUFACTURED BY le S N fF MAX POWER 15W R FIN Q ge q WIE O ia 0 0 O DETECTOR EN N el 9 pea 1 Z30VAC Horia SIMA TYPE T FUSE LASER IN gt SA201 ANALYZER REAR PANEL we TRANSDUCER STACK CHANNEL TWO BSPECTRUM ANALYZER CONTROL SCAN CONTROLS SA201 ANALYZER FRONT PANEL TRIGGER Figure III Figure Ill This figure shows a schematic diagram of a typical setup that is used to measure the spectrum of a laser source Please note that for this device to be useful the linewidth of the source must be less than the FSR of the interferometer SPECTRUM ANALYZER CONTROLLER AND OTHER ACCESSORIES e The SA201 controller generates a voltage ramp which is used to scan the separation between the two cavity mirrors A photodiode is used to monitor transmission of the cavity Using the output sync signal from the controller an oscilloscope ca
18. Bevollm chtigten sind wenn nicht anders festgelegt nicht haftbar f r Verluste Sch den oder Verletzungen wie auch immer verursacht die durch die Nutzung von Ger ten oder Software von Pico Technology entstehen Zweckdienlichkeit Zwei Anwendungen sind nie ganz gleich Pico Technology kann keine Gew hr bernehmen dass Ger te oder Software f r einen bestimmten Zweck geeignet sind Daher liegt es in Ihrer Verantwortung festzustellen ob das Produkt f r Ihre Anwendung geeignet ist Kritische Anwendungen Diese Software ist f r die Ausf hrung auf Rechnern vorgesehen auf denen auch andere Software Produkte ausgef hrt werden Aus diesem Grund ist es eine Bedingung dieser Lizenz dass die Benutzung bei kritischen Anwendungen wie beispielsweise Lebenserhaltungssystemen ausgeschlossen wird Viren Bei der Erstellung wurde diese Software st ndig auf Viren berpr ft Wenn sie einmal installiert ist sind Sie jedoch selbst f r die berpr fung auf Viren verantwortlich Unterst tzung Sind Sie mit der Leistung dieser Software nicht zufrieden nehmen Sie Kontakt mit unserer technischen Abteilung auf Sie wird versuchen die Probleme innerhalb eines angemessenen Zeitraums zu beheben Sind Sie danach immer noch unzufrieden geben Sie Produkt und Software innerhalb von 28 Tagen an Ihren H ndler zur ck Upgrades Sie k nnen Upgrades kostenlos von unserer Internetseite www picotech com herunterladen Wir behalten uns jedoch das Recht v
19. Cambridge University Press Cambridge 1967 p 123 5 K M Baird and G R Hanes in Applied Optics and Optical Engineering R Kingslake Ed Academic Press Inc New York 1967 Vols 4 and 5 p 350 6 R L Fork D R Herriott and H Kogelnik Appl Opt 3 1471 1964 7 G D Boyd and J P Gordon Bell Sys Tech J 40 453 1961 8 P Laures Appl Opt 6 747 1967 ANA 966 APPLIED OPTICS Vol 7 No 5 May 1968 John S Gailey is sales manager technical optics in the Optical Products Department of Corning Glass Works Laser Beams and Resonators H KOGELNIK AND T LI Abstract This paper is a review of the theory of laser beams and resonators It is meant to be tutorial in nature and useful in scope No attempt is made to be exhaustive in the treatment Rather emphasis is placed on formulations and derivations which lead to basic understand ing and on results which bear practical significance l INTRODUCTION HE COHERENT radiation generated by lasers or gn operating in the optical or infrared wave length regions usually appears as a beam whose transverse extent is large compared to the wavelength The resonant properties of such a beam in the resonator structure its propagation characteristics in free space and its interaction behavior with various optical elements and devices have been studied extensively in recent years This paper is a review of the theory of laser beams and resonator
20. Figure 6 shows the expansion of the beam according to 20 The beam contour w z is a hyperbola with asymp totes inclined to the axis at an angle A 0 22 TWO and This is the far field diffraction angle of the fundamental mode Dividing 21 by 20 one obtains the useful relation Az qu 23 TW AR October 1966 Vol 5 No 10 APPLIED OPTICS 1553 PHASE FRONT Fig 6 Contour of a Gaussian beam which can be used to express wy and z in terms of wand R IS we w 2 j NAAR AR e k 1 25 Aru To calculate the complex phase shift a distance z away from the waist one inserts 19 into 15 to get E E A 26 q z rwo A Ns Integration of 26 yields the result jP z nfi Q2 rw Inv 1 Az rwy j arc tan Az rwo 27 The real part of P represents a phase shift difference amp be tween the Gaussian beam and an ideal plane wave while the imaginary part produces an amplitude factor wo w which gives the expected intensity decrease on the axis due to the expansion of the beam With these results for the fundamental Gaussian beam ey can be written in the form im at ur hs u r 2 4 a e e 2 28 X Ed exp j kz y a where are ENG 29 It will be seen in Section 3 5 that Cesi n Be of this kind are produced by many lasers that oscillate in the fundamental mode 3 3 Higher Order Modes
21. Fragen zur Vorbereitung oder den Vorbereitungsaufgaben aufkommen k nnen Sie sich bis einschlie lich freitags vor dem Versuch per Mail an den Betreuer wenden berlegen Sie sich vor Versuchsbeginn welche Gr en gemessen werden m ssen und erstellen Sie einen Messplan der s mtliche zu messenden Gr en inkl Fehlerangaben jedes Aufgabenteils enth lt Literatur e W Demtr der Laserspektroskopie Grundlagen und Techniken Springer 2007 e F K Kneub hl M W Sigrist Laser Vieweg Teubner 2008 e J Eichler H J Eichler Laser Bauformen Strahlf hrung Anwendungen Springer 2006 e M Hercher The Spherical Mirror Fabry Perot Interferometer Applied Optics 7 951 1968 siehe Anhang e H Kogelnik T Li Laser Beams and Resonators Applied Optics 5 1550 1966 siehe Anhang e W S Gornall Ihe World of Fabry Perots Lasers amp Applications 47 July 1983 siehe Anhang Einleitung Seit der Erfindung des Lasers light amplification by stimulated emission of radiation in den 1960er Jahren hat dieser weitreichende Anwendungen gefunden Diese Anwendungen beinhalten bspw hochaufgel ste Spektroskopie zeitlich aufgel ste Studien molekularer Dynamik mittel Erzeugung ultrakurzer Lichtpulse Fangen und K hlen von Atomen zur Erzeugung von Bose Einstein Kondensaten Medizin Chirurgie z B Laserskalpell Messtechnik z B Abstandsmessung Materialbearbeitung in der Industrie un
22. Laserresonators Benutzen Sie den Justierlaser sowie die Irisblende um eine optische Achse zu definieren Richten Sie dann das Laserrohr und die Resonatorspiegel bzgl dieser Achse aus Achten Sie hierbei darauf dass der Strahl des Justierlasers mittig durch das Laserrohr l uft und die Resonatorspiegel zentrisch trifft Der Kriimmungsradius der Spiegel betr gt R 450 mm und der Spiegeldurchmesser betr gt dm 7 75 mm Der Durchmesser des Laserrohrs betr gt ca dk 1 0 mm und seine L nge ca L 20 cm Fuhren Sie nun die Aufgaben 2 4 fur 10 Resonatorl ngen jeweils nacheinander durch Das Lasermedium soll sich bei jeder Messung in der Mitte des Resonators befinden Beginnen Sie mit einem Spiegelabstand von 25 cm und vergr ern Sie diesen dann schrittweise bis 2 cm unterhalb des gr tm gli chen Abstands Stabilit tserenze 2 Ausgangsleistung des Lasers in Abh ngigkeit von der Resonatorl nge Aufgabe zur Vorbereitung Erstellen Sie eine Messwert Tabelle mit den 11 zu untersuchenden Resonatorl ngen Die Tabelle sollte neben Feldern f r die Messwerte und den Fehler auch zwei Spalten f r die Spiegelpositionen nebst Fehler sowie eine Spalte f r Kommentare beinhalten Messen Sie die Ausgangsleistung des Lasers in Abh ngigkeit von der Resonatorl nge und bestimmen Sie so die Stabilit tserenze des Resonators Nehmen Sie zus tzlich zu den normalen Messwerten mindestens einen weiteren Messwert f r eine Resonatorl nge auf welc
23. Laserrohres erreicht werden Die Bildaufnahme erfolgt ber das Programm Beamscope Die Messdaten befinden sich im Ordner D Measurement Data Beamscope VGA Das Programm erstellt automatisch einen horizontalen und einen vertikalen Schnitt durch das Maximum der Intensit tsverteilung Tragen Sie die aus einer nichtlinearen Regression erhaltenen Strahlbreiten w in einem Diagramm ber der Resonatorl nge auf Die Betrachtung eines vertikalen und horizontalen Schnitts erm glicht einen R ckschluss auf die Genauigkeit der Messung b Vergleich mit berechneten Werten Der Auskoppelspiegel besitzt eine gew lbte Au enfl che und kollimiert den Gau schen Laserstrahl mit dem Strahlradius w z2 Das hei t es bildet sich ein neuer Gau scher Strahl mit der Strahltaille wo w 22 aus F r kleine Abst nde d k nnen Sie daher zun chst davon ausgehen dass w o w d Tragen Sie die berechneten Werte in das in 2 a erzeugte Diagramm ein Diskutieren Sie das Ergebnis 4 Longitudinale Modenstruktur in Abhangigkeit von der Resonatorlange Beobachten Sie die longitudinale Modenstruktur des HeNe Resonators in Abh ngigkeit der Resonatorl nge und vergleichen Sie die Ergebnisse mit der Theorie Die longitudinale Modenstruktur wird mit dem Fabry Perot Interferometer Thorlabs SA210 bzw SA200 amp Steuerger t SA201 gemessen Das Ausgangss gnal des Interferometers kann mit einem Digitaloszilloskop Pico Modell 3204 auf den PC mittels des Programms Pi
24. Pr fung zu vers umen kann zu Sch den am PC und zu Verletzungen bei Ihnen und anderen f hren Grunds tzlich sollte man davon ausgehen dass ein Produkt keine Schutzerdung hat Reparaturen Das Oszilloskop enth lt keine zu wartenden Teile Reparatur und Kalibrierung erfordern spezielle Pr fger te und d rfen nur von Pico Technology durchgef hrt werden CE Hinweis Die PC Oszilloskope der Serie PicoScope 3000 entsprechen den Zielen der EMC Richtlinie 89 336 EWG und die folgende Norm wurde angewendet EN61326 1 1997 Klasse B Emissionen und St rfestigkeit Produkte aus der Reihe PicoScope 3000 entsprechen auch den Zielen der Niederspannungsrichtlinie und die folgende Norm wurde angewendet BS EN 61010 1 2001 IEC 61010 1 2001 Sicherheitsanforderungen f r elektrische Ger te Bedienelemente und Laboreinsatz PS3000049 2 Copyright 2005 7 Pico Technology Limited All rights reserved Einf hrung 3 1 5 1 6 FCC Hinweis Dieses Ger t entspricht den FCC Vorschriften Teil 15 FCC US Beh rde f r Telekommunikation Der Betrieb erfordert die Einhaltung der folgenden beiden Bedingungen 1 Dieses Ger t darf keine sch dlichen St rungen verursachen 2 Dieses Ger t muss jede empfangene St rung aufnehmen einschlie lich St rungen die unerw nschte Effekte verursachen Dieses Ger t wurde gem Teil 15 der FCC Vorschriften gepr ft und entspricht den Grenzwerten f r ein digitales Ger t der Klass
25. The associated pinhole finesse is 4 PO pa where D is the pinhole diameter and fis the focal length of lens L To compute the total instrumental finesse of a Fabry Perot spectrometer the contribution from E should be included with Fp and F 1 l 1 Fi Fe Fe FP Throughput and Etendue An advantage of Fabry Perot interferometers over other types of high resolution spectrometers is their efficiency both in transmission and tendue or light gathering power For small apertures or perfectly flat and parallel mirrors the transmission on the peak of a fringe To 1 75 10 depends on A the scattering and absorption loss at the mirrors For modern multilayer dielectric mirrors A lt 0 2 Consequently mirror reflectivities as high as 98 can yield throughput close to 80 over a small aperture The tendue for a plane Fabry Perot interfer ometer is 9 x DA U 0 A 4d F 11 Ali the radiation at wavelength A within a solid angle Q subtended at a mirror aperture Am can be transmitted in the bandpass defined by the instrumental finesse F The above formulae apply to Fabry Perot interferome ters using plane mirrors Similar formulae exist for spherical mirror interferometers The most common interferometer of this type is the confocal design where identical concave mirrors are spaced by precisely their radius of curvature For this case the free spectral range 1 4nd or half of a plane mirror system
26. _ In the preceding section only one solution of 11 was discussed i e a light beam with the property that its intensity profile i In every beam cross section is given by the same function namely a Gaussian The width of this Gaussian distribution changes as the beam propagates along its axis There are other solutions of 11 with sim 1554 APPLIED OPTICS Vol 5 No 10 October 1966 ilar properties and they are discussed in this section These solutions form a complete and orthogonal set of functions and are called the modes of propagation Every arbitrary distribution of monochromatic light can be expanded in terms of these modes Because of space limitations the derivation of these modes can only be sketched here me 4 a Modes in Cartesian Coordinates For a system with a rectangular x y z geometry one can try a solution for 11 of the form ZORO exp i p Po ae a 30 where g is a function of x and z and h is a function of y and z For real g and h this postulates mode beams whose intensity patterns scale according to the width 2w z of a Gaussian beam After inserting this trial solution into 11 one arrives at differential equations for g and h of the form AH m dH m da 31 This is the differential equation for the Hermite poly toma Hm X of order m Equation 11 is satisfied Th ge vn e where m and n are the transverse mode numbers Note that the same p
27. a graphic solu tion of the mode matching problem The circle diagrams for beams are similar to the impedance charts such as the Smith chart In fact there is a close analogy between transmission line and laser beam problems and there are analog electric networks for every optical system 17 The first circle diagram for beams was proposed by Collins 18 A dual chart was discussed in 19 The basis for the derivation of these charts are the beam prop agation laws discussed in Section 3 2 One combines 17 and 19 and eliminates g to obtain FJ ES 2 1 69 o erg c Tw R A l This relation contains the four quantities w R wo and z which were used to describe the propagation of Gaussian beams in Section 3 2 Each pair of these quantities can be expressed in complex variables W and Z W A pe 1 rw R TW f Vien A jz b 2 Jz 70 where b is the confocal parameter of the beam For these variables 69 defines a conformal transformation W 1 2 71 The two dual circle diagrams are plotted in the complex planes of W and Z respectively The W plane diagram 18 is shown in Fig 14 where the variables A rw and 1 R are plotted as axes In this plane the lines of constant b 2 ww0 A and the lines of constant z of the Z plane appear as circles through the origin A beam is represented by a circle of constant b and the beam parameters w and R at a distance z from the beam waist can be easily read October 1
28. and the energy of pulsed light sources The space saving battery powered design and compatibility to all Thorlabs C Series Photodiode Thermal Pyroelectric sensors and custom Photodiode Thermal and Pyroelectric detectors combined with a fast USB device interface open a wide range of applications in Manufacturing Quality Control Quality Assurance and R amp D for stationary and field use Please refer to the user manual on the data carrier supplied with the unit for detailed function description 2 Getting Started 2 1 Unpacking Inspect the shipping container for damage If the shipping container seems to be damaged keep it until you have inspected the contents and you have inspected the PM100D mechanically and electrically Verify that you have received the following items within the hard case 1 PM100D power energy meter console 2 1GB SD memory card installed in PM100D 3 Plug In power supply with Interchangeable primary plug for USA UK Europe and Australia 4 USB cable type A to mini B 5 Quick start guide 6 USB memory stick with instrument drivers user application and operation manual 7 Certificate of Calibration 2 2 Preparation Configure the plug in power supply with the primary plug for your local power supply Connect a suitable power or energy sensor The sensors have a self fixing mechanism To plug or unplug a sensor slightly press from both sides on the pins in the connector housi
29. be stored to the built in SD memory card on the selected file until STOP is pressed With START the selected file will be overwritten 10 DR UES 3 2 4 Analog Output The analog output provides the amplified photo diode current or the amplified thermal or pyroelectric sensor voltage The signals from the analog outputs are not wavelength and zero corrected The analog output voltage is range dependent and can be calculated to Uanalog out 2V full scale range value x measurement value The analog output voltage can range from 0 3V to 2 3V 3 2 9 Battery Charging The PM100D is powered by a 1 cell LiPo battery that needs to be recharged intermittently by plugging the AC adapter or plugging the USB cable to a computer To fully charge the battery it takes approximately 3 4 hours A built in charging circuit automatically regulates and terminates the charging Following battery icons in the display header show the charging state from empty to full battery Oo Wa A AN The empty battery symbol starts blinking for one minute until the unit shuts off When an external power supply is plugged the symbols above change sequent until the battery is fully charged 3 3 Computer Interface The PM100D optical power meter contains a USB 2 0 interface When connecting the PM100D to the PC first time a new hardware will be found For proper installing the PM100D it requires a NI VISA runtime version on the PC available on the National
30. beam index 2 is studied in detail An ideal thin lens of focal length f transforms an incom ing spherical wave with a radius R immediately to the left of the lens into a spherical wave with the radius Ra immediately to the right of it where 1 1 1 A 40 Figure 8 illustrates this situation The radius of curvature is taken to be positive if the wavefront is convex as viewed from z The lens transforms the phase fronts of laser be is in eactly the same way as those of spherical waves As the diameter of a beam is the same immediately to the left and to the right of a thin lens the g parameters of the incoming and outgoing beams are related by gt 41 do q f where the g s are measured at the lens If q and qz are measured at distances d and d from the lens as indicated in Fig 9 the relation between them becomes ie G de fgs di d d2 dyd gt f n f A di f This formula is derived using 16 and 41 More complicated optical structures such as gas lenses combinations of lenses or thick lenses can be thought of as composed of a series of thin lenses at various spacings Repeated application of 16 and 41 is therefore suffi cient to calculate the effect of complicated structures on the propagation of laser beams If the ABCD matrix for the transfer of paraxial rays through the structure is known the g parameter of the output beam can be cal culated fr
31. called axial mode number giving the number of axial modes in the intracavity standing wave For the general case of a resonator made up of two mirrors radii b and bs separated by a distance d Boyd and Kogelnik have shown that the resonant frequency associated with a given TEM mna mode is given by Yang 2d q 1 MA m n x cost 1 d b 1 d b2 1 36 We interpret this as follows an arbitrary quasi mono chromatic field of frequency vo which is incident on a curved mirror resonator is decomposed into a large number of transverse modes TEMmn Each of these transverse modes will be resonant i e will be trans mitted by the high Q cavity for mirror separations satisfying d c 2m q l m n x cos 1 d bi 1 d b 37 In general therefore a given quasi monochromatic field will be decomposed into a large number of trans verse modes each of which is resonant for a different mirror separation In order for a general curved mirror cavity to be useful as a scanning spectrum analyzer or static filter the input field must be reduced to a single transverse mode of the cavity ie mode matched so that the transmission of the cavity as a function of mirror separation can be unambiguously interpreted in terms of the frequency content of the in put field This can be accomplished without consider able loss of light only in the case of a laser light source operating in a single transverse mod
32. control allows the user to continuously adjust the scan rate from 0 01ms to 0 1ms using a 10 turn trimpot Note the risetime setting may be scaled by the sweep expansion setting For example If the scan rate is set to 0 05s and the sweep expansion is adjusted from 1x to 100x then the scan rate will adjust to 5s The scaling error is typically less than 0 5 providing excellent measurement capabilities Trigger Output BNC 9 This trigger output signal may be used to externally trigger the oscilloscope The trigger is capable of driving 509 terminated cables as well as Hi Z loads such as oscilloscopes The trigger will provide an edge on the beginning and middle of the scanning ramp See Figure 2 below di 5 ms a 242590006 p 4 TDS2012 CH1 10 Y 25 MS 2 TDS2012 CH2 1 VW 25 ms Figure 2 Trigger Logic Output BNC 10 The output BNC is used to drive the SA200 scanning piezos from 1V to 45V The output is capable of driving 0 6uF piezo loads at a ramp rate of 1ms over the full voltage range The output current is internally limited to prevent damage to the output drive Note the output performance specifications assume a Thorlabs Fabry Perot Interferometer module is connected gt 1 TDS2012 CH1 10 4 S mS 2 TDS2012 CH2 2 W 5m5 11 1057042 C0H1 10 Y 10m3 2 TDS2012 CH2 2 W 10 ms Figure 3 Sawtooth Waveform Figure 4 Triangle Waveform 6679 D02 Rev E 8 08 16 05 Section 4 0 Descriptions conti
33. equivalent to calculating the transient behavior of the resonator when it is excited initially with a wave of arbi trary distribution This wave is assumed to travel back and forth between the mirrors of the resonator under going changes from transit to transit and losing energy by diffraction After many transits a quasi steady state con dition is attained where the fields for successive transits differ only by a constant multiplicative factor This steady state relative field distribution is then an eigenfunction of the integral equations and is in fact the field distribu tion of the mode that has the lowest diffraction loss for the symmetry assumed e g for even or odd symmetry in the case of infinite strip mirrors or for a given azimuthal mode index number in the case of circular mirrors the constant multiplicative factor is the eigenvalue associated with the eigenfunction and gives the diffraction loss and the phase shift of the mode Although this simple form of the iterative method gives only the lower order solu tions it can nevertheless be modified to yield higher order ones 24 35 The method of kernel expansion however is capable of yielding both low order and high order solutions Figures 19 and 20 show the relative field distributions of the TEMo and TEM modes for a resonator with a pair of identical circular mirrors N 1 a a g1 22 g as obtained by the numerical iterative method Several curves ar
34. frequency increment Avm The actual instrumental profile that is obtained when using a finite detector aperture is given by I v vo 2r IN I xdz 18 0 where g vy vo 1 2 4r and IE EUT 1 RYL 1 2F r sin 2 r c 4r and where is the frequency that would be recorded using a vanishingly small aperture Note that with a finite aperture the instrumental profile is no longer centered on v and is asymetric Figure 5 shows com puted instrumental profiles for various values of F and p the aperture radius cf Fig 14 in Sec ITI Other Factors Affecting Instrumental Finesse We have seen Eq 15 that in the absence of other losses the instrumental finesse is limited by the re flectivity of the mirrors to a value of approximately T 2 1 R In this section we consider the manner in which the finesse is degraded by other factors namely irregularities in the surfaces of the mirrors and diffraction If we wish we can associate with each loss mechanism e g mirror transmission or diffraction a contribution to the lifetime of the resonant cavity The finesse F associated with the th loss mechanism is related to the corresponding contribution to the cavity lifetime 7 by F rer 2r 19 Hence it is clear that the net instrumental finesse F is related to the individual contributions F by ui 20 so that it is useful and meaningful to consider the in dividual contribut
35. gilt Im Umgang mit Lasern st der gesunde Menschenverstand nicht zu ersetzen Einige spezielle Hinweise werden m Folgenden angef hrt l Die Laserschutzvorschriften sind immer zu beachten 2 Halten Sie Ihren Kopf n emals auf Strahlh he 3 Die Justierbrille immer aufsetzen 4 Schauen Sie nie direkt n Strahl auch nicht mit Justierbrille 5 Achtung praktisch alle Laser f r Laboranwendungen sind mindestens Klasse 3 also von vornherein f r die Augen gef hrlich ggf auch f r die Haut evil auch hierf r Schutzma nahmen ergreifen Zur Justage kann der Laserstrahl mittels einem St ck Papier sichtbar gemacht werden 6 Auch Kameras besitzen eine Zerst rschwelle 7 Spiegel und sonstige Komponenten nie in den ungeblockten Laserstrahl einbauen Vor Einbau immer berlegen n welche Richtung der Reflex geht Diese Richtung zun chst blocken bevor der Strahl wieder frei gegeben wird 8 Nie mit reflektierenden Werkzeugen im Strahlengang hantieren Unkontrollierbare Reflexe Vorsicht st z B auch mit BNC Kabeln geboten die n den Strahlengang gelangen k nnten Gleiches gilt auch f r Uhren und Ringe Diese vorsichtshalber ausziehen wenn S e mit den H nden m Strahlengang arbeiten 9 Auch Leistungsmessger te k nnen Reflexe verursachen Unbeschichtete Silizium Fotodioden reflektieren ber 30 des Lichtes 10 Achtung im Umgang mit Strahlteilerw rfeln Diese haben immer einen zweiten Ausgang Ggf abblocken
36. good spectral coverage typically with pass bands 100 nm or more in the visible with losses less than 0 2 and minimal flatness error Broader band coatings are available but they require a greater number of dielectric layers that may introduce flatness errors A 100 and higher absorption losses 0 3 to 0 4 Also available are hard coatings which are applied hot and may warp the mirror sub strates so are not advised for Fabry Perot mirrors unless necessary for resistance to high power laserbeams Ramp Generators Piezoelectric tuning of an interferometer requires a special electronic controller Its function is to sweep the Lasers amp Applications July 1983 mirror spacing in a repetitive scan by applying an adjustable ramp voltage to the piezoelectric elements so it is often called a ramp generator Modern Fabry Perot ramp generators include many additional features A common bias allows manual tuning of the mirror spacing while other bias controls permit changing the voltage on individual piezoelectric elements to tilt the movable mirror These controls also make it possible to interface automatic cavity and alignment stablization systems Generally the elements in a piezoelectric assembly are not identical but have slightly different voltage sensitivities As a result the mirror supported by that assembly will tilt as it is translated Tilt free translation can be restored by trim controls on the ram
37. integral equations First it is obvious that the mirrors can be interchanged without affecting the results that is the subscripts and superscripts one and two can be interchanged Second the diffraction loss and the intensity pattern of the mode remain invariant if both g and g are reversed in sign the eigenfunctions E and the eigenvalues y merely take on complex conjugate values An example of such equivalent systems is that of parallel plane g g2 1 and concentric g g2 1 resonator systems The third equivalence property involves the Fresnel number N and the stability factors G and Gy where N a Ad a Gi q az a Gag 79 ay If these three parameters are the same for any two resona tors then they would have the same diffraction loss the 1562 APPLIED OPTICS Vol 5 No 10 October 1966 same resonant frequency and mode patterns that are scaled versions of each other Thus the equivalence rela tions reduce greatly the number of calculations which are necessary for obtaining the solutions for the various resonator geometries 4 5 Stability Condition and Diagram Stability of optical resonators has been discussed in Section 2 in terms of geometrical optics The stability condition is given by 8 In terms of the stability factors G and Go it is 0 lt GG lt 1 or 0 lt giga lt 1 80 Resonators are stable if this condition is satisfied and unstable otherwise A stability diagram
38. is provided to allow the spectrum displayed on the oscilloscope to be shifted right or left Another convenient feature of the controller is a zoom capability that provides a 1X 2X 5X 10X 20X 50X and 100X increase in the spectral display resolution The output TTL level trigger allows the user to externally trigger an oscilloscope on either the beginning or midpoint of the ramp waveform The SA201 also includes a high precision photodetector amplifier circuit used to monitor the transmission of the cavity The amplifier provides an adjustable transimpedance gain of 10K 100K and 1M V A when driving a high impedance load such as an oscilloscope Using the output sync signal from the controller an oscilloscope can be used to display the spectrum of the input laser The detector circuitry incorporates a blanking circuit which disables the photodiode response during the falling edge of the sawtooth waveform Section 3 0 Parts List Below is a list of all components shipped with the SA201 Spectrum Analyzer Controller O SA201 Spectrum Analyzer Controller O Operating Manual O US Power Supply Line Cord O 63mA Fuse for use at 230VAC operation 125mA fuse installed in unit The following items are sold separately O SA200 5A 525 650nm Scanning Fabry Perot with Silicon Photodetector O SA200 6A 650 800nm Scanning Fabry Perot with Silicon Photodetector O SA200 7A 780 930nm Scanning Fabry Perot with Silicon Photodetector O SA200 9A 900 110
39. jedoch darauf dass nicht verschiedene Transversalmoden zu einem Bild akkumuliert werden b R umliche Intensit tsverteilung der Moden Legen Sie s nnvoll ausgerichtete Schnitte entlang der Symmetrieachsen durch die Intensit tsverteilung der Moden Hierzu dient das Programm SliceBMP Die Position der Schnitte l sst sich durch die Schieberegler rechts und oben am Bild anpassen Die Drehung durch Eingabe eines Winkels und Best tigung mit Enter Mit Klick auf Schnitt erstellen Werden sowohl das gedrehte Bild als auch die beiden Intensit tsprofile im angegebenen Ordner gespeichert c Vergleich der gemessenen Werte mit berechneten Intensit tsverteilungen Plotten Sie die berechneten Intensit tsverteilungen im jeweils zugeh rigen Graph der gemessenen Verteilung aus 6 a Passen Sie f r die Berechnung die Amplitude und die Strahlbreite w der theoretischen Verteilung an die experimentellen Daten an Stellen Sie in der Auswertung links neben dem Graph die zweidimensionale Intensit tsverteilung mit den Schnittgeraden dar welche die Lage des benutzten Schnittes innerhalb der Intensit tsverteilung aufzeigt Diskutieren S e die Qualit t der Graphenanpassung an die Messdaten Vergleichen sie den erhaltenen Radius mit dem Radius der Grundmode bei gleicher Resonatorl nge aus Aufgabe 3 a Diskutieren Sie die Ergebnisse im Hinblick auf die theoretische Beschreibung der Transversalmoden Wichtige Punkte zum Laserschutz Ganz allgemein
40. more stable thermally and unlike solid etalons they can be pressure tuned Both types allow no mechanical variability in spacing the right spacing must be preselected for a specific application Thesimplest way totuneeither etalonis by tilting This is a good technique provided tilt angles are not so large as to degrade the finesse These etalons are difficult to manufac ture with very fiat and parallel surfaces especially with large mirror spacings They are best suited for optical systems with small diameter laserbeams Variable spacing air gap etalons are similar to fixed air gap etalons except the spacing is established by a mechanically adjustable frame in which the etalon plates are mounted While adjustable mirror spacing is an advantage this design is less stable both mechani cally and thermally than the bonded etalons Applica tions are similar Piezoelectric mirror control is available for both fixed air gap and variable spacing etalons In the former the piezoelectric elements are carefully matched in length and cemented directly to Fabry Perot mirrors The latter consists of a housing with a built in piezoelec 49 A a a a A i 90 92 94 96 98 100 Reflectivity Figure 4 Finesse versus reflectivity for different plate flatnesses tric assembly that supports the Fabry Perot mirrors Some mechanical means of adjusting the mirror spacing and aligning the mirrors is provided although it is genera
41. one digit number with a blinking underline With the lt gt buttons navigate the digit to change with the AV keys increment or decrement the digit Confirm with OK JAN oS Button Appearance Focus on Button BERIN Edit Mode Blinking frame All sub menus can be left by navigating to the EXIT button at the downright soft button location or to any empty button and pressing the OK key System Settings PEGE The following sub menus will appear Measurement Settings Remote state PM100D can be switched back to local mode Line filter Sets the unit to the local line frequency 50HZ 60Hz to avoid aliasing effects Default sensor Sets the PM100D in a mode to measure photo current thermal voltage or peak voltage from a pyro electric detector See user manual Console Settings Language User interface language setting Backlight Switches the LCD and key backlight on and off Same function as the key Brightness Sets the brightness of the LCD and key backlight The setting range is 0 100 in 1 increments LCD Contrast Adjusts the LCD contrast The setting range is 0 100 in 1 increments Sound Switches on and off the key and warning sounds Shutdown The unit automatically powers off in battery operation after a certain time without user action This feature can be set to 10 minutes 1 hour or switched off and is not active when an external power supply AC adapter or USB is connec
42. operation only Thorlabs supplies a 63mA fuse with all of its SA201 units and must be installed when operating at 230VAC e 125mA Type T Slow Blow Fuse The 125mA fuse is installed from the factory This must be installed when operating the unit at 100 115VAC e Phillips Head Screwdriver 2 Preferred We do not recommend using electrically powered screwdrivers Fuse Replacement 1 Important Disconnect the scanning head or any piezo device from the SA201 output Important Disconnect the power cord Do not open the unit if the power cord is connected Remove the two screws securing the enclosure cover with a Phillips head screwdriver The screws are located on the bottom side rear corners of the unit Do not lose the screws Carefully remove the cover by sliding toward the rear of the unit Locate the fuse box between the input line voltage connector and the transformer Remove the fuse cover and slide the old fuse out Install the new fuse into the fuse cover and place back into the fuse box 125mA 100 115V c and 63mA 230Vac 8 Replace the enclosure cover and secure with the enclosure screws 6679 D02 Rev E 10 08 16 05 Section 5 0 Maintenance amp Troubleshooting continued Selecting the Line Voltage 1 Important Replace the line fuse as described above 2 Locate the voltage selector switch on the rear panel See Figure 5 leader 15 3 Switch to the appropriate line voltage 4 Install the app
43. pp 3009 3057 November 1964 40 J C Heurtley Hyperspheroidal functions optical resonators with circular mirrors in Proc Symposium on Quasi Optics New York Polytechnic Press 1964 pp 367 375 41 S R Barone and M C Newstein Fabry Perot resonances at small Fresnel numbers Appl Opt vol 3 p 1194 October 1964 42 L Bergstein and H Schachter Resonant modes of optic cavi ties of small Fresnel numbers J Opt Soc Am vol 55 pp 1226 1233 October 1965 43 A G Fox and T Li Modes in a maser interferometer with curved mirrors in Proc Third International Congress on Quantum Electronics New York Columbia University Press 1964 pp 1263 1270 44 H Kogelnik Modes in optical resonators A K Levine Ed New York Dekker 1966 in Te Modes Phase Shifts and Losses of Flat Roof Open Resonators P F CHECCACCI ANNA CONSORTINL AND ANNAMARIA SCHEGGI Abstract The integral equation of a flat roof resonator is solved by the Fox and Li method of iteration in a number of particular cases Mode patterns phase shifts and power losses are derived A good overall agreement is found with the approximate theory previously developed by Toraldo di Francia Some experimental tests carried out on a microwave model give a further confirmation of the theoretical predictions I INTRODUCTION PARTICULAR type of open resonator terminated A by roof reflectors with very s
44. referencing the scan of both instruments to the laser line frequency Tandem operation of piezoelectrically scanned interferometers is not difficult when certain simple opti cal alignment procedures are followed Synchronous frequency scanning is achieved by driving the two inter ferometers with simultaneous ramp voltages proportional to their respective mirror spacings Stabilization Techniques Often the passive thermal stability of a well designed Fabry Perot interferometer is not enough These situations include collection of weak spectra where data accumulation over long periods is necessary multi pass operation where mirror alignment is critical and tandem operation where accurate correlation in frequen cy scanning must be maintained Such cases call for some form of active stabilization One technique uses capacitance displacement trans ducers on the circumference of the Fabry Perot mirrors to monitor changes in spacing or alignment from a preset position Extremely stable and reliable instruments have been built using this technique they are particularly suited for observational astronomy where no prominent spectral features are available for optical stabilization 52 Fabry Perot Photodetector Light Beamsplitter Shutter Reference i Laser optional Figure 8 Stabilized Fabry Perot spectrometer The one drawback to capacitance transducers is that their limited displacement range typically 10
45. side rear corners of the unit Do not lose the screws Carefully remove the cover by sliding toward the rear of the unit Locate the JP3 header It is positioned in front of the heat sink and will have a shorting jumper on pin 1 Remove the shorting jumper and place across Shorting the JP3 pins to disable the blanking circuit The default setting will be blanking enabled The jumper will not be shorting the pins 7 Replace the enclosure cover and secure with the enclosure screws Section 5 0 Maintenance amp Troubleshooting DANGER The Thorlabs Spectrum Analyzer Controller SA201 must be powered off unplugged from the AC input source and disconnected from any piezo elements prior to replacing the fuse or removal of the cover Failure to do so may cause SERIOUS INJURY to the user since high voltages exist within the unit WARNUNG Sicherheitsanweisungen f r den Thorlabs Spectrum Analyzer Controller SA201 Bevor die Sicherung gewechselt oder die Geh useabdeckung entfernt werden darf mu das Ger t abgeschaltet und das Spannungsversorgungskabel von der Netzspannung getrennt werden Das nicht befolgen dieser Anweisungen kann zu lebensgef hrlichen Verletzungen f hren da das Ger t intern unter Hochspannung steht Materials Needed e SA201 Operating Manual The most recent version of this operating manual will be available on Thorlabs web site e 63mA Type T Slow Blow Fuse The 63mA fuse is required for 230V
46. superposed When the two types of ray leave the interferometer at a small angle e g if the entering beam is approximately collimated but at an angle to the axis they will form an additional interference pattern made up of equally spaced straight fringes whose sepa rations are determined by the angle at which the two beams are brought to focus This two beam i e sin interference pattern modulates the multiple beam pat tern of circular fringes and of course arises only when the two beams are coherent Examples of this inci dental two beam pattern are shown in Sec II From Eg 4 we see that bright fringes are formed in the central plane of the interferometer when p X satisfies 6 p 2mr or p r 4ep r ma 6 where m is a positive or negative integer giving the order of interference relative to the order on axis which is assumed for convenience to be an integer Fringes thus have radii given by Pm 2er 4e r mAr 7 For e gt 0 pm is single valued and m gt 0 When e lt 0 pm is two valued for m lt 0 and single valued for m gt 0 Figure 3 shows this fringe pattern in cross section for different values of e with r and A fixed Appendix I shows how to transform this curve as well as curves in later figures so that it corresponds to other values of r and or A The maximum radial dispersion in the fringe pattern dp dA is obtained in the vicinity of the fringe corre spondin
47. the EC e still complete not disassembled and not contaminated As the WEEE directive applies to self contained operational electrical and electronic products this end of life take back service does not refer to other Thorlabs products such as e pure OEM products that means assemblies to be built into a unit by the user e g OEM laser driver cards e components e mechanics and optics e left over parts of units disassembled by the user PCB s housings etc If you wish to return a Thorlabs unit for waste recovery please contact Thorlabs or your nearest dealer for further information Waste treatment on your own responsibility If you do not return an end of life unit to Thorlabs you must hand it to a company specialized in waste recovery Do not dispose of the unit in a litter bin or at a public waste disposal site Ecological background It is well known that WEEE pollutes the environment by releasing toxic products during decomposition The aim of the European RoHS directive is to reduce the content of toxic substances in electronic products in the future The intent of the WEEE directive is to enforce the recycling of WEEE A controlled recycling of end of live products will thereby avoid negative impacts on the environment al a X h O Crossed out wheelie bin symbol US PO Box 366 435 Route 206N Newton NJ 07860 Tel 1 973 579 7227 Fax 1 973 3600 4 Europe Hans Boeckler Str
48. the figure 1s equal to the phase shift experienced by the beam as given by 29 this is easily shown using 23 The dual diagram 19 is plotted in the Z plane The sets of circles in both diagrams have the same form and only the labeling of the axes and circles is different In Fig 15 both diagrams are unified in one chart The labels in parentheses correspond to the Z plane diagram and 6 is a normalizing parameter which can be arbitrarily chosen for convenience One can plot various other circle diagrams which are related to the above by conformal transformations One 1560 APPLIED OPTICS Vol 5 No 10 October 1966 such transformation makes it possible to use the Smith chart for determining complex mismatch coefficients for Gaussian beams 20 Other circle diagrams include those for optical resonators 21 which allow the graphic deter mination of certain parameters of the resonator modes 4 LASER RESONATORS FINITE APERTURE 4 1 General Mathematical Formulation In this section aperture diffraction effects due to the finite size of the mirrors are taken into account these effects were neglected in the preceding sections There it was mentioned that resonators used in laser oscillators usually take the form of an open structure consisting of a pair of mirrors facing each other Such a structure with finite mirror apertures is intrinsically lossy and unless energy is supplied to it continuously the electromagnetic field i
49. the geometrical construction for gaussian mode propagation described by Laures Thus for a confocal laser the problem of mode matching is largely a problem of aligning the optical axis of the FPS Le the line joining the centers of curvature of the two mirrors to that of the incoming light beam Stated in another way the basic problem is to locate the source or its image and both centers of curvature on a single straight line The mode matching alignment problem as just stated obviously requires both angular and lateral dis placements of the FPS regarding the source as fixed When the source is at infinity collimated however it is clear that only angular adjustments are required This simplifies the problem considerably Using a collimated beam the procedure is as follows First set up the FPS in the same manner used for observing multiple beam interference fringes Fig 10 c Then while viewing the fringe pattern adjust the angular alignment of the FPS until a two beam interference pattern of straight fringes can be seen superimposed on the circular fringes cf Fig 9 a Next make fine adjustments to increase the straight fringe spacing until it is large compared to the spot size ps or at least greater 964 APPLIED OPTICS Vol 7 No 5 May 1968 than the detector aperture at the center of an image of the circular fringe pattern Finally observe the scan display and touch up the alignment as required The pro
50. the resonant wavelength of the interferometer is a linear function of the mirror spacing it is possible to obtain a linear plot of the source spec trum simply by recording the output from the detector as a function of the mirror separation A change of 4 in the mirror separation scans through a free spectral range of c 4 r e Hz The spectral resolving power R of a spectroscopic instrument is defined by Q v Avm M ANn 9 where Avm is the minimum resolvable frequency incre ment in the vicinity of a frequency v The classical criterion for defining what is meant by minimum resolvable increment is an extension of the criter ion used by Rayleigh in discussing diffraction pat May 1968 Vol 7 No 5 APPLIED OPTICS 953 00062 00027 F 2100 00037 TRANSMITTED INTENSITY 14 A 0 30 0 40 0 50 0 60 0 70 0 80 RADIUS P IN FRINGE PLANE cm Fig 4 Calculated distribution of light in an FPS fringe pattern for a monochromatic source and various values for F the finesse Note the broad central fringe e 0 7 10 cm terns For convenience we depart from this definition slightly and define the minimum resolvable frequency in crement as the apparent spectral width full width at half maximum of a monochromatic line This is of course just the width of the instrumental profile In practice a large number of factors enter into the de termination of the instrumental profile of a Fabry Perot interferometer
51. which are discussed in Section 2 that is the field distributions are given approxi mately by Hermite Gaussian functions for rectangular mirrors 5 6 34 and by Laguerre Gaussian func tions for circular mirrors 6 7 The designation of the resonator modes is given in Section 3 5 The modes are designated as TEMmn for rectangular mirrors and TEM for circular mirrors Figure 7 shows photo graphs of some of the rectangular mode patterns of a TEMoo TEMio TEM29 TEMoo TEMo TEMo2 A ERT Teme eee aire el TEM TEM TEM TEMig TEM TEM ere pote ar 4 Ser ee TEMo2 TEMx2 TEMa22 TEM2zo TEM 21 TEM22 SQUARE MIRRORS CIRCULAR MIRRORS Fig 17 Linearly polarized resonator mode configurations for square and circular mirrors O DE Cd O Fig 18 Synthesis of different polarization configurations from the linearly polarized TEMy mode OJO laser Linearly polarized mode configurations for square mirrors and for circular mirrors are shown in Fig 17 By combining two orthogonally polarized modes of the same order it is possible to synthesize other polarization configurations this is shown in Fig 18 for the TEMo mode _ Field distributions of the resonator modes for any value of G could be obtained numerically by solving the integral equations either by the method of successive ap proximations 4 24 31 or by the method of kernel expansion 30 32 The former method of solution is
52. which goes to zero for infinitely small wave lengths 61 resembles Newton s imaging formula of geometrical optics Any lens with a focal length f gt can be used to per form the matching transformation Once f is chosen the distances d and d have to be adjusted to satisfy the matching formulas 10 Wy A o d fit VP fo page yro 63 These relations are derived by combining 60 and 61 In 63 one can choose either both plus signs or both minus signs for matching It is often useful to introduce the confocal parameters b and bs into the matching formulas They are defined by the waist diameters of the two systems to be matched by 2rw2 A Da Zrwa X 64 Using these parameters one gets for the characteristic length fo fo b1b 65 and for the matching distances d f ti V P f 1 de f tbe V F 1 66 Note that in this form of the matching formulas the wavelength does not appear explicitly Table II lists for quick reference formulas for the two important parameters of beams that emerge from various 1558 APPLIED OPTICS Vol 5 No 10 October 1966 TABLE II FORMULAS FOR THE CONFOCAL PARAMETER AND THE LOCATION OF BEAM WAIST FOR VARIOUS OPTICAL STRUCTURES d Ri d Ro d RitRe d R R2 2d Ri R2 2d R d 2R d WA 2R d n 1 2R d n 2 1 nR d 2R d Er 2n2R d n2 1 2n2R d n 1 optical structures commonly encountered They are the confocal parameter
53. 0 microme ters precludes changing the mirror spacing once a particular set of transducer elements has been attached to the mirrors The most versatile technique for active Fabry Perot stabilization is one that makes use of the light trans mitted through the interferometer to control the cavity spacing and optimize the mirror alignment A prominent spectral feature such as the laser line in a light scattering experiment is chosen as a reference frequency for cavity and alignment stabilization During setup this feature is centered at some designated position in the frequency scan By monitoring the relative intensity in two win dows on either side of this position any drift of the interferometer relative to the reference frequency can be detected and corrected through the cavity bias control on the ramp generator Parallel mirror alignment can be optimized by applying tiny angular changes to the piezoelectrically driven mirror on successive scans and detecting the resultant change in intensity of the refer ence peak After each test a correction is applied to the alignment bias voltages on the ramp generator in the direction that produced the greater throughput The main advantage of the optical stabilization technique is that it may be used with any piezoelectrical ly controlled Fabry Perot system because it makes use of the transmitted signal rather than relying on separate transducers In effect it continually corrects for
54. 0nm Scanning Fabry Perot with InGaAs Photodetector O SA200 12A 1250 1400nm Scanning Fabry Perot with InGaAs Photodetector O SA200 14A 1450 1625nm Scanning Fabry Perot with InGaAs Photodetector 6679 D02 Rev E 6 08 16 05 Section 4 0 Descriptions su TOX 20 x Sox EXPANSION CALIBRATION AMPLITUDE RISETIME 1 30V 10 100ms 1x S O a a E 4 Figure 1 SA201 Front Panel Detector Gain Adjustment 1 The SA201 includes a built in photodiode amplifier circuit This amplifier is designed specifically to operate with the detector provided with the SA200 series Fabry Perot Interferometer allowing the user to monitor the transmission of the cavity While any photodetector may be connected to the amplifier the specifica tions listed in Section 1 apply only to detectors supplied with the SA200 series The amplifier provides a transimpedance gain current to voltage gain of 10K 100K and 1M V A while driving a Hi Z load such as an oscilloscope For better noise and performance characteristics it is recommended that a 50W coax cable with a 509 terminating resistor be used The Photodetector input and output BNC s are located on the rear panel DC Offset Control 2 The DC Offset provides a continuously adjustable offset voltage over the range of O to 15V using a 10 turn potentiometer This offset adds directly to the ramp signal The DC offset control is used to adjust the waveform from left to right across an oscilloscope viewi
55. 11 Warnlampen bei Betrieb des Lasers anschalten und nach Beendigung der Arbeit wieder ausschalten 12 Daf r sorgen dass auch Dritte 1m Labor die richtigen Schutzbrillen tragen oder sich auberhalb des Laserschutzbereiches befinden 13 Filterglaser in Laserschutzbrillen d rfen grunds tzlich nicht aus oder umgebaut werden 14 In besonderem Ma e auf Beistehende achten 15 Optiken Linsen Spiegel etc nicht direkt mit den Fingern ber hren Hiermit erkl re ch dass ich die vorstehenden Punkte gelesen und verstanden habe Ich best tige dass ich eine Einf hrung in den Umgang mit Lasern sowie eine arbeitsplatzbezogene Unterweisung erhalten habe Name Unterschrift Datum The World of Fabry Perots These Elegant Instruments Are Versatile High Resolution Tunable Wavelength Filters by William S Gornall The Fabry Perot interferometer was invented by two French opticians Charles Fabry and Alfred Perot in 1897 For decades it received limited use even in spectro scopic research because few emission sources were suffi ciently monochromatic to take advantage of its high resolving power The advent of lasers in the early 1960s produced a renaissance of interest in Fabry Perot inter ferometry that continues to grow as new applications and techniques are found The Fabry Perot is the simplest of all interferome ters consisting of two partially transmitting mirrors facing each other Depending on the application
56. 1X 2X 5X 10X 20X 50X 100X 0 5 1mVnams 6 6MVpp 6679 D02 Rev E 4 08 16 05 continued Section 1 0 Specifications Trigger Characteristics Trigger Output Voltage VOH RL 50W VOL RL 50W Trigger Load Impedance Trigger Rising Edge Trigger Falling Edge Photo Diode Amplifier Characteristics Gain Steps Transimpedance Gain Hi Z Transimpedance Gain 502 7 Gain Error Output Impedance Load Impedance Output Voltage Hi Z load Output Voltage 509 load Max Output Current Bandwidth Noise RMS 8 Offset 8 TTL levels 2V min 0 5V max SOW Hi Z Ramp Start Ramp Midpoint 0 10 20dB 10K 100K 1M V A 5K 50K 500K V A 0 1 O 10K 0 12 0 12 O 100K 0 15 0 14 1M 0 3 502 5009 Hi Z 0 10V min 0 5V min 100mA 250KHz lt 0 1mV 10K 0 2mV 100K 1 5mV 1M 1mV O 10K 5mV O 100K 20mV O 1M SA200 5A 6A and 7A Detector Characteristics Detector Active Area Spectral Range Junction Cap OV Bias NEP 980nm Silicon 13mm 3 6 x 3 6mm 400 to 1100nm 110pF typ 1 2 x 10 14 W VHz SA200 9A 12A and 14A Detector Characteristics Detector Active Diameter Spectral Range Junction Cap OV Bias NEP 980nm InGaAs 1mm 700 to 1700nm 120pF 80pF typ 1x 10 14 W VHz 6679 D02 Rev E 5 08 16 05 Section 1 0 Specifications continued Notes 1 Achie
57. 4 9 Laserresonator Versuchsanleitung zum Fortgeschrittenen Praktikum Abteilung A Version 2 2 TECHNISCHE UNIVERSITAT DARMSTADT Fachbereich Physik Institut f r Angewandte Physik AG Nichtlineare Optik Quantenoptik Inhaltsverzeichnis Versuchsanleitung Anhang The World of Fabry Perots The Spherical Mirror Fabry Perot Interferometer Laser Beams and Resonators SA200 Series Scanning Fabry Perot Interferometer Datenblatt SA210 Series Scanning Fabry Perot Interferometer Datenblatt SA201 Spectrum Analyzer Controller Anleitung S120C Leistungsmesskopf Datenblatt PM100D Leistungsmessger t Kurzanleitung PicoScope 3000 PC Oszilloskop Anleitung 10 16 32 50 51 56 67 71 83 Vorbereitung e Laserprinzip Besetzungsinversion Anregungsmechanismen 3 und 4 Niveau System Einwegverst rkung e Laseroszillatoren Verst rkung durch R ckkopplung Laseroszillator Moden spektrum von Laseroszillatoren Bandbreite Eigenschaften der Laserstrahlung He Ne Gaslaser e Resonatortheorie Optische Resonatoren Resonatorgeometrie planparallel konfokal hemisph risch und deren Eigenschaften Stabilit tsdiagramm Verluste optischer Resonatoren Moden transversal und longitudinal Fabry Perot Interferometer Gau Optik e Gefahren durch Laserstrahlung siehe z B Wikipedia e Vorbereitende Aufgaben Gehen Sie die einzelnen Versuchsteile durch und bearbeiten Sie die drei Vorbereitungsaufgaben Sollten
58. 6 24 for the various resonator geometries is shown in Fig 4 where g and g are the co ordinate axes and each point on the diagram represents a particular resonator geometry The boundaries between stable and unstable shaded regions are determined by 80 which is based on geometrical optics The fields of the modes in stable resonators are more concentrated near the resonator axes than those in unstable resonators and therefore the diffraction losses of unstable resona tors are much higher than those of stable resonators The transition which occurs near the boundaries is gradual for resonators with small Fresnel numbers and more abrupt for those with large Fresnel numbers The origin of the diagram represents the confocal system with mirrors of equal curvature Ri R d and is a point of lowest diffraction loss for a given Fresnel number The fact that a system with minor deviations from the ideal confocal system may become unstable should be borne in mind when designing laser resonators 4 6 Modes of the Resonator The transverse field distributions of the resonator modes are given by the eigenfunctions of the integral equations As yet no exact analytical solution has been found for the general case of arbitrary G and G but approximate analytical expressions have been obtained to describe the fields in stable spherical mirror resonators 5 6 These approximate eigenfunctions are the same as those of the optical beam modes
59. 966 Vol 5 No 10 APPLIED OPTICS 1559 i A A cme J 2 mwg Fig 14 Geometry for the W plane circle diagram GAUSSIAN BEAM CHART 2b tw Lon b b o 02 04 06 os re 12 14 16 1 8 20 PHASE SNIFT IN DEGREE as o _ SI ety gt 20 S 5 as 3 LEE a i RR lt gt N es S gt os ips O RSS NY ER 0 4 STALK CBSE SSS SO ISS SL SN IAS AN 0 4 e CE EEI IIIS RS OS AO HETTIE N AA os PR ARAS SA 03 DEAR RITA N AAA ASS AA AA OSTIAS RIAS AS ANI AAN AO 02 a Upp Ue AU At CCOCEO EST Oo AAA Ge A YAA A Gs az PR SEU ELLE NEE ES A AA O ESA SE o O A o E E oo RN nun ILANI TO gt N ER A EEES coor e Fo gE ee Eee mare APR Bees TTT IE 0 Sasse SEE BO RTT SOK RK III TUE Ors OY LESS ZOO A ESOO SS SS a 02 BERR RRR OES 02 ESTI SS SREE n OR ESE OO ASS NAS 0 3 a gt ay Y s ay o e SOD Q N 0 3 SH CSCS RR ROR oO EPR 5 gt SCS RORY oa TI SEN 04 RITA A RS 05 SAS 0 5 FL TRITT AI Ss 0 RIIIICH AAA 0 6 SSS RH Ys 0 6 ROOT TEE or g SS SL e yA 0 7 12 14 1 6 1 8 2 0 26 hwl OR b b Fig 15 The Gaussian beam chart Both W plane and Z plane circle diagram are combined into one from the diagram When the beam passes through a lens the phase front is changed according to 40 and a new beam is formed which implies that the incoming and outgoing beams are connected in the diagram by a vertical line of length 1 f The angle shown in
60. ALLELZ PLANE Z Fig 4 Stability diagram Unstable resonator systems lie in shaded regions 3 WAVE ANALYSIS OF BEAMS AND RESONATORS In this section the wave nature of laser beams is taken into account but diffraction effects due to the finite size of apertures are neglected The latter will be discussed in Section 4 The results derived here are applicable to optical systems with large apertures i e with apertures that intercept only a negligible portion of the beam power A theory of light beams or beam waves of this kind was first given by Boyd and Gordon 5 and by Goubau and Schwering 7 The present discussion follows an analysis given in 11 3 1 Approximate Solution of the Wave Equation Laser beams are similar in many respects to plane waves however their intensity distributions are not uni form but are concentrated near the axis of propagation and their phase fronts are slightly curved A field com ponent or potential u of the coherent light satisfies the scalar wave equation V u ku 0 9 where k 27 is the propagation constant in the medium For light traveling in the z direction one writes u a y 2 exp jkz 10 where y is a slowly varying complex function which represents the differences between a laser beam and a plane wave namely a nonuniform intensity distribu tion expansion of the beam with distance of propagation curvature of the phase front and other differenc
61. ARANTHESES ARE MILLIMETERS SM1 SERIES THREAD EM TAP FOR IMPERIAL POSTS Me ADAPTER INCLUDED IR VIEWING TARGET A 0 85 216MM MD FILTER 7 JA VIEWING TARGET DETECTOR SURFACE a w Br 15M CABLE TO OBS MALE CONNECTOR SECTION A A Sensor Connector SLIGHTLY PRESS FROM f BOTH SIDES FOR PLUGGING AND UNPLUGGING Pin 3 Photodiode Anode Pin 4 Photodiode Cathode US PO Box 366 435 Route 206N Newton NJ 07860 Tel 1 973 579 7227 Fax 1 973 3600 2 Europe Hans Boeckler Str 6 85221 Dachau Munich Germany Tel 49 8131 59560 Fax 49 8131 595699 Web http www thorlabs com Mail europe thorlabs com Doc 18356 S01 doc Rev A 1 Oct 08 Spec Sheet Typical Spectral Response Graph 6 0E 02 5 0E 02 4 0E 02 3 0E 02 2 0E 02 gt Responsivity A W 1 0E 02 gt 0 0E 00 9 9 o 9 O 9 o gt NO aN oe gt AS NI nS Wavelength nm Available Accessories 120 FC FC fiber adapter 120 SMA SMA fiber adapter 120 SC SC fiber adapter 120 LC LC fiber adpter 120 ST ST fiber adapter AS4M8E M4 to 8 32 Adapter included SM1CP1 Dust cap included The S120C is also compatible to the Thorlabs imperial and metric post and post holder series and Thorlabs SM1 mechanics Cleaning and Maintenance There are no serviceable parts in the S720C head The housing may be cleaned by wiping with a soft damp cloth When cleaning the aperture f
62. C gemeinsam mit externem Trigger Festfrequenz 1kHz 5 V Rechteckwelle 600 Q Impedanz 2 1 BNC gemeinsam mit externem Trigger Variable Frequenz 100 Hz bis 1 MHZ 5 V Rechteckwelle 1 V Sinuswelle und Dreiecksfunktionen Wobbelwiederholungsfunktion Dual Slope Funktion 600 Q Impedanz PS3000049 2 Copyright 2005 7 Pico Technology Limited All rights reserved Fehlerbehandlung 9 3 Fehlerbehandlung 3 1 Fehlercodes Sehen Sie in diesem Abschnitt nach wenn Sie PicoScope oder PicoLog verwenden PicoLog meldet Fehlercode 1 Dieser Fehler wird gemeldet wenn mehr als 4 Oszilloskope auf einem PC ge ffnet werden Mit PicoLog k nnen nicht mehr als 4 Oszilloskope zugleich verwendet werden PicoScope oder PicoLog meldet Fehlercode gt Dieser Fehler wird gemeldet wenn dem Treiber nicht ausreichend Speicher zur Verf gung steht um das Oszilloskop zu betreiben Weitere Informationen finden Sie im Abschnitt Systemanforderungen PicoScope oder PicoLog meldet Fehlercode 3 Dieser Fehler zeigt an dass auf Ihrem System kein Oszilloskop aus der Serie PicoScope 3000 gefunden wurde Stellen Sie sicher dass die Software installiert ist bevor das Oszilloskop mit der USB Buchse verbunden wird und starten Sie Ihren Computer neu Stellen Sie sicher dass das Oszilloskop im Ger te Manager unter dem Eintrag USB aufgef hrt wird Wenn das Oszilloskop hier nicht aufgef hrt wird wenden Sie sich an den Technischen Support von Pico
63. C housing includes a threaded input in axis with the light input aperture that is compatible with any number of Thorlabs 1 threaded accessories This allows convenient mounting of external optics fiber adapters light filters and apertures A 8 32 threaded mounting hole is provided to accommodate posts and post holders a M4 adapter for metric posts is included The S120C is compatible with the new Thorlabs PM100D and PM100A consoles A non volatile memory in the sensor connector contains sensor information data and the NIST and PTB traceable calibration data Technical Specifications 5 400 450 nm 1001 1100 nm Diode Diode Arrays He Ne Dye Ion Lasers Ar Kr 15 Weight ooo 0 07 kg Pot SS 8 32 thread M4 Adapter included Aperture Thread SM1 outer thread Fiber Adapters optional FC SC LC SMA ST 1 2 Measured with PM100D console in bandwidth low setting Including uniformity failure Please note that the S120C power meter head is not compatible with older Thorlabs power meter consoles PM100 PM30 PM300 PM300E S100 US PO Box 366 485 Route 206N Newton NJ 07860 Tel 1 973 579 7227 Fax 1 973 3600 1 Europe Hans Boeckler Str 6 85221 Dachau Munich Germany Tel 49 8131 59560 Fax 49 8131 595699 Web http www thorlabs com Mail europe thorlabs com Doc 18356 S01 doc RevA 1 Oct 08 Spec Sheet Mechanical Drawing INFORMATION ONLY NOT FOR MANUFACTURING ALL DIMENSIONS IN P
64. Detector Aperture As mentioned earlier the diameter of the detector aperture for equivalently the diameter of the incident beam see Fig 10 a serves to define both the actual instrumental bandwidth and the tendue Asa rule one seeks a compromise in which a considerable tendue can be attained with only slight sacrifice in resolving power The curves in Fig 8 d show that the optimum compromise occurs when the radius of the detector aperture assuming it to be located adjacent to the rear interferometer mirror is just under ps Figure 14 shows experimental data for the instrument profile of a 10 cm FPS with a nominal finesse of just over 100 for various detector apertures In this figure the desira bility of using the optimum detector aperture is obvious For larger apertures the peak transmitted power goes up very little and the resolution goes down Note however that at larger apertures more total light is transmitted this may be an important consideration in applications involving weak light sources It s worth pointing out that the finite time constant of a high finesse FPS precludes the use of very high sweep speeds without suffering a loss in resolution in this case too it would make sense to use a somewhat larger detector aperture than normal 3 Alignment for Mode Matching There are undoubtedly applications where it might prove useful to extend the free spectral range of an FPS by mode matching The gain in
65. Fork et al have analyzed spherical mirror interferometers in general terms and have demon strated the extraordinarily high resolutions that can be obtained particularly when the interferometer has optical gain as in a subthreshold laser While they recognized that confocal resonators or spherical mirror Fabry Perot interferometers the two terms are inter changeable offer certain distinct advantages over non confocal arrangements the tendency to date has been to use nonconfocal cavities for high spectral resolution with laser light sources The advantages of a non The author is with the Institute of Optics University of Roches ter Rochester New York 14627 Received 2 January 1968 This work supported in part by the Air Force Cambridge Re search Laboratories confocal arrangement are a the relatively loose tolerance on the mirror separation and b the ability to select various free spectral ranges with a given pair of mirrors Its disadvantages are a the necessity to match the input radiation field to a transverse mode of the cavity and b the relatively low light gathering power of the resonator with spatially incoherent sources The FPS on the other hand requires a relatively precise control of the mirror separation with a resulting fixed free spectral range This disadvantage is largely offset by the high light gathering power of the FPS even at very high resolution freedom from mode matching con siderations
66. I C The 5 cm FPS had broadband mirrors 4800 A to 6900 A while the 10 cm FPS had con ventional narrowband mirrors peaked at approximately 6500 A The finesse in both cases was about 150 The scanning voltage was supplied directly from the oscillo scope s horizontal sweep and the duration of each sweep was 10 msec Also shown in Fig 15 is the gain profile of the single mode He Ne laser clearly showing Lamb s dip and thereby the collision broadened homogeneous linewidth in He Ne this gain profile was obtained by slowly changing the laser cavity length and integrating the displayed spectrum with a storage oscilloscope We also found that it was a simple matter to in directly determine the instrumental linewidth by mak ing a direct measurement of the FPS cavity lifetime r r 2rF rc 27 AD 43 This lifetime could be observed with fair precision by using a 10 nsec Q switched laser pulse as a realizable approximation to a temporal delta function and de tecting the transmission of the FPS as a function of time with a fast phototube and scope The resultant trace was very nearly an exponential decay and gave results in good argreement with other measurements Note that it is not necessary that the Q switched pulse have a narrow spectrum although the alignment be tween the laser beam and FPS must be good Axial amp transverse modes HE TE MULTIMODE GAIN PROFILE Fig 16 Spectra of an adjustable multimode H
67. IER CHARACTERISTICS 5 SA200 5A 6A AND 7A DETECTOR CHARACTERISTICS 5 SA200 9A 12A AND 14A DETECTOR CHARACTERISTICS 5 SECTION 2 OVERVIEW os 6 SECTION 3 PARTS LIST rs 6 SECTION 4 DESCRIPTION os 7 DETECTOR GAIN ADJUSTMENT 1 c0cceceeeeeeseeueeees 7 DC OFFSET CONTROL DB naar 7 SWEEP EXPANSION CONTROL 3 2222 7 WAVEFORM CONTROL 4 2er 7 POWER SWITCH Bo 7 POWER ON INDICATOR 6 2222er 7 AMPLITUDE CONTROL 7 mamwnmwnwa nanma mwana 8 RISETIME CONTROL 8 2222er 8 TRIGGER OUTPUT BNC 9 2222er 8 OUTPUT ENC IO cure daros noo 8 GROUND PLUG 11 oc 9 AC INPUT CONNECTOR 12 cc 22er 9 PD AMPLIFIER INPUT BNC 13 onua con 9 PD AMPLIFIER OUTPUT BNC 14 2 222 9 VOLTAGE SELECTOR SWITCH 15 cc ce 9 PD BLANKING CIRCUIT ooo 10 SECTION 5 MAINTENANCE AND TROUBLESHOOTING 10 MATERIALS NEEDED rs 10 FUSE REPLACEMENT os 10 SELECTING THE LINE VOLTAGE c0cceeeceeeeeeeuee 11 CLEANING coo 1 TECHNICAL SUPPORT 11 APPENDIX A RECOMMENDED SETUP ooo 11 6679 D02 Rev E 08 16 05 Waste Electrical and Electronic Equipment Directive As required by the WEEE Waste Electrical and Electronic Equipment Directive of the European Community and the corresponding national laws Thorlabs offers all end users in the EC the possibility to return end of life units without incurring disposal charges This offer is valid for Thorlabs electrica
68. Instruments website www ni com or from the data carrier that comes with the instrument Allow installing and follow the dialog instructions The PM100D comes with a utility software that easily enables remotely operating the PM100D also PM100A and PM100USB and visualizing and logging measurement data Perform the setup exe and follow the installing instructions The LabVIEW source code of this application is included on the data carrier as well and can be used to build own applications or to modify the utility program to specific requirements LabVIEW 8 5 1 or higher required 11 A UNES 4 Addresses Our Company is represented by several distributors and sales offices throughout the world Europe Thorlabs GmbH Hans Bockler Str 6 85221 Dachau Germany Sales and Support Phone 49 0 8131 5956 0 Fax 49 0 8131 5956 99 Email europe thorlabs com Web www thorlabs com Japan Thorlabs Japan Inc Higashi Ikebukuro Q Building 1st floor 2 23 2 Toshima ku Tokyo 170 0013 Japan Sales and Support Phone 81 3 5979 8889 Fax 81 3 5979 7285 Email sales thorlabs jp Web www thorlabs p USA Thorlabs Inc 435 Route 206 North Newton NJ 07860 USA Sales and Support Phone 1 973 579 7227 Fax 1 973 300 3600 Email sales thorlabs com techsupport thorlabs com Web www thorlabs com China Thorlabs China Oasis Middlering Centre 3 Building 712 Room 915 Zhen Bei Road Shanghai China Sales and Support
69. Sie kein Oszilloskop der Serie PicoScope 3000 mit dem PC bevor die Software installiert ist Installieren Sie die Software gem der im Installationshandbuch Ihres Oszilloskops beschriebenen Vorgehensweise Das Oszilloskop wird ber das mitgelieferte USB Kabel mit dem PC verbunden Eine zus tzliche Stromversorgung ist nicht erforderlich da der Strom aus dem USB Port bezogen wird berpr fen der Installation Nachdem die Software installiert wurde stellen Sie sicher dass das Oszilloskop mit dem PC verbunden ist und starten sie dann das Programm PicoScope oder PicoLog Das Programm sollte nun die angeschlossene Spannung anzeigen Wenn Sie eine Messsonde und PicoScope verwenden sollten Sie ein schwaches Netzspannungssignal von 50 oder 60 Hz im Oszilloskop Fenster sehen sobald Sie die Spitze der Sonde mit dem Finger ber hren Standard Oszilloskopanschl sse Oszilloskope der Serie PicoScope 3000 haben Standardanschl sse Auch der Eingangswiderstand ist standardm ig daher funktioniert die x10 Funktion mit Messsonden einwandfrei Der im folgenden Diagramm mit E bezeichnete BNC Anschluss und im Oszilloskop hat zwei Funktionen Bei normaler Verwendung ist es der Eingang f r ein externes Triggersignal mit einstellbarem Schwellwert Alternativ kann dieser Anschluss bei manchen Oszilloskopen auch verwendet werden um Sinus Rechteck und Dreieckwellen auszugeben die mit einer benutzerdefinierten Frequenz vor und zur ck laufen k
70. Super Invar Fabry Perot interferometer to the optical axis O 0 and therefore satisfying the condition ma 2nd 2 The Fabry Perot spectrometer so formed may be tuned to transmit any wavelength by varying either n or d If the medium between the mirrors is air or some other gas n can be varied by changing the gas pressure Such pressure scanning is rarely used now because it is cumbersome and slow 1f the medium is a solid n can be changed by adjusting the temperature but this too is slow and difficult to control Modern interferometers are more often tuned by changing the mirror spacing d The optical path length between the mirrors can be altered by rotating the Fabry Perot interferometer but rotation results in non linear tuning and must be limited to small angles or the resolving power of the interferometer is degraded A more versatile technique for changing d is to mount one mirror on three piezoelectric elements and translate that mirror in a direction perpendicular to its surface while the other remains fixed As Equation 2 shows any wavelength can be trans mitted through the interferometer if d changes by at least 1X For visible wavelengths it is possible to construct piezoelectric assemblies that will move several wave lengths providing ample tunability A modern Fabry Perot interferometer with piezo electric tuning is pictured in Figure 2 The main structure is heavy super Invar for mechanical rigidit
71. adjustments are required The only alignment required is that of the entire FPS etalon relative to the light source and this alignment is not critical c High tendue light gathering power without sacrifice in spectral resolution d Large acceptance angle when used as an optical filter Q Ar A rP sr e The transverse mode degeneracy of the confocal cavity obviates the necessity for mode matching f Versatility a single FPS instrument can be de signed to serve in three modes of operation 1 e scan ning fringe display and optical filtering Using readily available mirror coatings passive FPS interferometers of reasonable length 50 em can easily resolve down to a fraction of a megahertz and 1f prop erly stabilized and thermally compensated can pro vide convenient and portable secondary optical wave length standards In addition to spectroscopic appli cations we plan to use FPS instruments as optical FM discriminators and as passive elements in laser fre queney stabilization servo loops Appendix I Normalized Equations for FPS In illustrating the performance of an FPS with the calculated curves shown in various figures we chose to use specific values for the pertinent parameters r e A and p This was done rather than using dimensionless normalized parameters so the reader could get an idea of the actual values of these parameters in a typical case namely r 10 em and A 6250 A v 4 8 X 101 Hz
72. advertently Mode matching of an FPS can also be considered from a more direct point of view We have already men tioned the additional interference pattern arising from an angular misalignment of the two types of rays trans mitted by the FPS It is easy to show that when these two types of transmitted beams are aligned they are in phase on axis assuming r m 2 just out of phase at the first fringe off axis in phase again for the second fringe and so on Thus the superposition of the two transmitted beams results in constructive interference in every other free spectral range and destructive in terference in the remaining orders thereby effectively doubling the free spectral range and at the same time doubling the amount of light transmitted at a given resonance when constructive interference occurs for the transmitted beams destructive interference occurs for the two reflected beams The great advantage of the confocal FPS over a general curved mirror resonator is the freedom from the necessity to mode match in order to observe a clean spectrum This as we have just seen is due to the frequency degeneracy of even and odd transverse modes of a confocal cavity It can readily be shown from Eq 36 that the maximum value of el the departure from exact confocal mirror spacing which can be tolerated without allowing the TEMmn transverse mode to resonate at an observably different mirror spacing from that at which the TEM o m
73. al optical systems is schematically indicated in Fig 2 A single element of the sequence is characterized by its ABCD matrix The ray transfer through n consecutive elements of the sequence is described by the nth power of this matrix This can be evaluated by means of Sylves ter s theorem 2 1 sin ee C D 5 A sin n sin n DO BsinnO a sin nO D sin nO sin n 1 0 October 1966 Vol 5 No 10 APPLIED OPTICS 1551 whefe cos O 3 A D 6 Periodic sequences can be classified as either stable or unstable Sequences are stable when the trace A D obeys the inequality l1 lt 3 A D lt 1 7 Inspection of 5 shows that rays passing through a stable sequence are periodically refocused For unstable sys tems the trigonometric functions in that equation be come hyperbolic functions which indicates that the rays become more and more dispersed the further they pass through the sequence A B A B ABP AB AB N l i i Xo KI Xn Xo X an Fig 2 Periodic seguence of identical systems each characterized by its ABCD matrix 2 3 Stability of Laser Resonators A laser resonator with spherical mirrors of unequal curvature is a typical example of a periodic sequence that can be either stable or unstable 6 In Fig 3 such a resonator is shown together with its dual which is a sequence of lenses The ray paths through the two struc tures are the same except that
74. an optical material with curved end surfaces where the beam passing through it is as sumed to have phase fronts that coincide with these sur faces When one designs a matching system it is useful to know the accuracy required of the distance adjustments The discussion below indicates how the parameters bx and d change when b and fare fixed and the lens spacing d to the waist of the input beam is varied Equations 60 and 61 can be solved for b with the result 9 bi j SA e 1 zu d f T b1 2f 67 This means that the parameter b of the beam emerging from the lens changes with d according to a Lorentzian functional form as shown in Fig 12 The Lorentzian is centered at d f and has a width of b The maximum value of bs is 4f b If one inserts 67 into 60 one gets dd 1 dy f b1 21 which shows the change of d with d The change is reminiscent of a dispersion curve associated with a Lorentzian as shown in Fig 13 The extrema of this curve occur at the halfpower points of the Lorentzian The slope of the curve at d f is 2 b1 The dashed curves in the figure correspond to the geometrical optics imaging re lation between dj do and f 20 1 da f 68 3 7 Circle Diagrams The propagation of Gaussian laser beams can be repre sented graphically on a circle diagram On such a diagram one can follow a beam as it propagates in free space or passes through lenses thereby affording
75. at the center of the interferometer Note that the path of light within the interferometer is just the reverse of the case where the light incident on the interferometer is collimated All of the light falling within a central circle of radius 2p f r on lens La will pass through the detector aperture of radius p and will thus be filtered by the instrumental passband In this mode of operation the FPS can be used as a Static or tunable filter or as a scanning spec trum analyzer If the incident light is not collimated there is no significant loss in resolution but there may be a reduction in the amount of light received by the detector This holds even for gross departures from collimation in the incident beam of light The entire system should be free to rotate about the center of Ly In this way the incoming beam can be directed at the center of L and then the entire interferometer system can be rotated about this point in order to attain align ment between the incoming beam of light and the axis of the FPS The detector aperture which limits the actual instrumental bandpass for incident beams with a large diameter is located just behind the interferom eter DETECTOR HS APERTURE Li RADIUS p A H E INCIDENT LIGHT Fig 10 A versatile FPS instrument a Optical layout show ing the FPS etalon lens Li and detector aperture b arrange ment for scanning c arrangement for observing and r
76. attern scaling Penn w z applies to modes of all orders Some Hermite polynomials of oy order are Hola 1 H x 7 H x nd Hale 82 12 783 Expression 28 can be used as a mathematical descrip tion of higher order light beams if one inserts the product gh as a factor on the right hand side The intensity pat tern in a cross section of a higher order beam is thus de scribed by the product of Hermite and Gaussian functions Photographs of such mode patterns are shown in Fig 7 They were produced as modes of oscillation in a gas laser oscillator 16 Note that the number of zeros in a mode pattern is equal to the corresponding mode number and that the area occupied by a mode increases with the mode number The parameter R z in 28 is the same for all des implying that the phase front curvature is the same and changes in the same way for modes of all orders The phase shift however is a function of the mode numbers One obtains m n 2 m n 1 are tanQz2 rwo 34 _ TEMoo TEM o TEM20 TEM30 TEM TEMso TEM 60 TEM 44 TEMay TEM 33 Fig Mode patterns of a gas laser oscil lator rectangular symmetry TEMaa This means that the phase velocity increases with increas ing mode number In resonators this leads to differences in the resonant frequencies ot the various modes of oscil lation b Modes in Cylindrical Coordinates For a system with a cylindrical 7
77. b and the distance which gives the waist location of the emerging beam System No lisa resonator formed by a flat mirror and a spherical mirror of radius R System No 2 is a resonator formed by two equal spherical mirrors System No 3 is a resonator formed by mirrors of unequal curvature System No 4 d f Fig 12 The confocal parameter b as a func tion of the lens waist spacing dh 1 a f Fig 13 The waist spacing dz as a function of A lens waist spacing d is again a resonator T rmed by two equal spherical mir rors but with the reflecting surfaces deposited on plano concave optical plates of index n These plates act as negative lenses and change the characteristics of the emerging beam This lens effect is assumed not present in Systems Nos 2 and 3 System No 5 is a sequence of thin lenses of equal focal lengths f System No 6 is a system of two irises with equal apertures spaced at a distance d Shown are the parameters of a beam that will pass through both irises with the least possible beam diameter This is a beam which is confocal over the distance d This beam will also pass through a tube of length d with the optimum clearance The tube is also indicated in the figure A similar situation is shown in System No 7 which corresponds to a beam that is confocal over the length d of optical material of index n System No 8 is a spherical mirror resonator filled with material of index n or
78. beam back onto itself greatly simplifies the alignment of the cavity just align your input to within a few tenths of a millimeter of the center of the mirror set and restrict your input angles to less than a few degrees The SA200 series interferometer has two iris diaphragms that simplify this alignment requirement Finesse The finesse of the Scanning Fabry Perot interferometer is a quantity which characterizes the ability of the interferometer to resolve closely spaced spectral features it defines the resolution of the instrument For an infinitely narrow input spectrum the finesse determines the width of the measured spectrum High finesse means high resolution capability high finesse is obtained by increasing the reflectivity of the cavity mirrors However high reflective mirrors reduce the transmission of the interferometer Page 3 of 5 THORLABS INC In a typical application the SA210 Interferometer is used in conjunction with a signal generator and an oscilloscope as shown below in figure Ill A signal generator Thorlabs SA201 Fabry Perot Controller is used for generating the required scan signals for obtaining the data in this document that can produce either a triangle or saw tooth wave with an adjustable frequency 5 to 50 Hz an adjustable amplitude 15 to 40 volts and an adjustable offset The signal generator is used to repetitively scan the length of the cavity by 1 4 in order to sweep through one FSR of the interferometer
79. cal system is characterized by its distance x from the optic z axis and by its angle or slope x with respect to that axis A typical ray path through an optical structure is shown in Fig 1 The slope x of paraxial rays is assumed to be small The ray path through a given structure de pends on the optical properties of the structure and on the input conditions i e the position x and the slope x of the ray in the input plane of the system For paraxial rays the corresponding output quantities xs and xy are linearly dependent on the input quantities This is conveniently written in the matrix form AB le D Ta vo V1 1 vy TABLE I RAY TRANSFER MATRICES OF SIX ELEMENTARY OPTICAL STRUCTURES OPTICAL SYSTEM RAY TRANSFER MATRIX F d da d d gt ti de ds ddz fi fe fa fife where the slopes are measured positive as indicated in the figure The ABCD matrix is called the ray transfer matrix Its determinant is generally unity AD BC 1 2 The matrix elements are related to the focal length f of the system and to the location of the principal planes by f 1 Cc D l hi T 3 A 1 N C where h and ha are the distances of the principal planes from the input and output planes as shown in Fig 1 In Table I there are listed the ray transfer matrices of six elementary optical structures The matrix of No 1 describes the ray t
80. cavity and alignment drift the same way one would do manual ly so that any system that can be aligned can be maintained in alignment with this technique If an appropriate reference frequency does not exist in the observed spectrum one can be introduced as shown in Figure 8 Since the reference frequency is only used for a small fraction of the scan it may be blocked by a shutter the rest of the time if it would otherwise interfere with the collection of spectral intensity from the light source e References 1 C Fabry and A Perot Ann Chem Phys 16 115 1899 2 M Hercher Appl Opt 7 951 1968 3 T R Hicks N K Reay and R J Scaddan J Phys E Sci Instrum 7 27 1974 4 J R Sandercock in Light Scattering in Solids M Bal kanski Ed Flammarion Press Paris 1971 p 9 5 For further details consult Burleigh Instruments Tech Memo entitled Multipass Fabry Perot 6 J R Sandercock U S Patent 4225236 assigned to RCA Corp 1980 7 J G Dil N C J A van Hijningen F van Dorst and R M Aarts Appl Opt 20 1374 1981 Lasers amp Applications July 1983 The Spherical Mirror Fabry Perot Interferometer Michael Hercher The theory design and use of the confocal spherical mirror Fabry Perot interferometer FPS is described in detail Topics covered include performance of an FPS for small departures from the confocal mirror separation optimization of the resolution X light gather
81. cedure is somewhat inefficient in that much of the light in the collimated beam fails to reach the detector Note that the alignment precision is approximately A p and thus requires interferometric stability between the source and the FPS As an alternate procedure one can use the technique described by Fork et al for mode matching to a general curved mirror cavity Also we have found that with practice one can set up the FPS in the normal scanning mode and then hunt for the mode matched condition by making small lateral and angular adjustments while observing the scan display As the proper alignment is approached the spectral display in every other free spectral range is slightly increased in amplitude while the remaining portion of the display is decreased in amplitude This hunting procedure is not very reliable and generally takes longer than the alignment described above E Some Experimental Results One of the first characteristics of an FPS instrument which one would like to determine experimentally is its instrumental bandpass or equivalently its finesse This is conveniently accomplished by observing the spectrum of a relatively stable gas laser whose indi vidual spectral components are generally orders of magnitude narrower than one could hope to observe directly Figure 15 shows spectra of a stable single mode He Ne laser which were obtained with 5 cm and 10 cm scanning FPS instruments of the type described in Sec II
82. coScope bertragen werden Bestimmen Sie nun den Modenabstand in Abh ngigkeit der Resonatorl nge Vergleichen Sie in einer Tabelle die gemessenen mit berechneten Werten Achten Sie darauf w hrend der Messung nur die TEMoo Mode anzuregen Kalibrieren Sie zun chst die Zeitskala des Oszilloskops mit Hilfe des freien Spektralbereichs des Interferometers zur sp teren Umrechnung von der Zeitbas s in den in Frequenzraum Achten Sie bei jeder Messung darauf dass auf dem Oszilloskop klar separierbare symmetrische Lorenz Linien erkennbar sind Justieren Sie gegebenenfalls die Verkippung des Interferometers im Bezug auf den Laserstrahl und stellen sie sicher dass der Laserstrahl die Eingangs Iris des Detektors mittig trifft Stellen Sie die Iris zur Messung auf den kleinsten Durchmesser ein Speichern Sie die Daten jeweils im txt und png Format zur sp teren Kontrolle Entfernen Sie bei der Justage des Fabry Perot Interferometers nicht den Detektor wie in dessen Anleitung beschrieben wird 5 Verst rkungsbandbreite des HeNe Lasers a Aufnahme der Messdaten mithilfe der Persistenz Funktion Untersuchen Sie die Verst rkungsbandbreite des Lasers f r mindestens eine Resonatorl nge Die Persistenz Funktion des Oszilloskops eignet sich aufgrund des vorhandenen Moden Jitters zur Aufnahme des Verst rkungsprofils Es kann nur ein PNG Bild der Oszilloskop Anzeige gespeichert werden die grafisch ausgewertet werden muss Es empfiehlt sich
83. d Unterhaltungselektronik CD DVD Spieler Das Grundprinzip des Lasers l sst sich kurz folgenderma en zusammenfassen Ein Laser besteht im Wesentlichen aus drei Komponenten einem verst rkenden Medium in das von einer Energiepumpe selektiv Energie hineingepumpt wird und einem Resonator der einen Teil dieser Energie in Form elektromagnetischer Wellen in wenigen Resonatormoden speichert Die Energiepumpe erzeugt im Lasermedium eine vom thermischen Gleichgewicht extrem abweichende Besetzung eines oder mehrerer Energieniveaus Bei gen gend gro er Pumpleistung wird zumindest f r ein Energieniveau Ex die Besetzungsdichte Ny Ex gr er als die Besetzungsdichte N E f r ein energetisch tiefer liegendes Niveau das mit Ex durch einen erlaubten bergang verbunden ist Inversion Da in einem solchen Fall die induzierte Emissionsrate auf dem bergang Ex gt E gr er wird als die Absorptionsrate kann Licht beim Durchgang durch das aktive Medium verst rkt werden Die Aufgabe des Resonators ist es nun Licht das von den durch die Pumpe aktivierten Atomen des Lasermediums emittiert wird durch selektive optische R ckkopplung wieder durch das verst rkende Medium zu schicken und dadurch aus dem Laserverst rker einen selbstschwingenden Oszillator zu machen Mit anderen Worten Der Resonator speichert das Licht in wenigen Resonatormoden so dass in diesen Moden die Strahlungsdichte gro wird und damit die induzierte Emission
84. d plane and parallel to within a hundredth of a wavelength per cm of aperture so that A a is 100 and a finesse of 25 is desired then an FPP offers the greater tendue up to a mirror spacing of 4em Rather than increasing the FPP mirror spacing beyond 4 em one should in principle switch to an FPS in order to obtain higher resolution and maximum tendue D Mode Analysis of an FPS The equations derived earlier in this section have been based primarily on a geometrical analysis of a confocal resonator or FPS This as it turns out 1s adequate for most purposes A more rigorous treat ment however would involve a decomposition of the incident radiation field into eigenmodes of the resona tor as defined in the curved mirror case by Boyd and Gordon In this section we outline an analysis of this sort and to some extent justify the simpler geometrical approach We should point out that the aberrations of a confocal spherical mirror resonator which give rise to the multiple beam interference fringes described earlier can be conveniently analyzed only by the geometrical approach As shown by Boyd and Gordon the eigenmodes TEM mng of a confocal resonator are closely approxi mated by Gaussian Hermite functions Ref 7 Eq 20 The first two subscripts m and n denote the amplitude distribution of the eigenmode on a surface of constant phase and the third subscript q is the so 958 APPLIED OPTICS Vol 7 No 5 May 1968
85. diese Aufgabe bei einer Resonatorl nge von 60 81cm durchzuf hren Nehmen Sie das Verst rkungsprofil bei maximaler Ausgangsleistung der TEMoo Mode auf Bestimmen Sie anschlie end die Ausgangsleistung Reduzieren Sie nun die Ausgangsleistung durch Erh hen der Beugungsverluste auf die H lfte des Ausgangswertes Wiederholen Sie d e Messung b Bestimmen der Verst rkungsbandbreite Benutzen Sie die in Aufgabe 4 gemachte Kalibrierung der Zeitskala des Oszilloskops zur Bestimmung der Frequenzbandbreite des Verst rkungsprofils Erl utern Sie den Einfluss der Verluste auf die Verst rkungsbandbreite anhand der Messung Diskutieren Sie die erhaltenen Werte m Hinblick auf die theoretisch zu erwartende Verst rkungsbandbreite SSeS 6 Beobachtung hoherer transversaler Moden a Aufnahme der Messdaten mittels CCD Kamera Nehmen Sie mindestens vier Bilder der Intensit tsverteilung unterschiedlicher TEM Moden mit dem Programm Beamscope auf Der Schalter LIVE erm glicht es das Bild einer Intensit tsverteilung zum Speichern einzufrieren Wahlen Sie m glichst Transversal deren Verteilung klar erkennbar ist und die Sie identifizieren k nnen Speichern Sie die Bilder als BMP Dateien Notieren sie fiir jede Mode auf welche Art und Weise sie erzeugt wurde und welcher Resonatorl nge genutzt wurde Achten Sie auch hier auf eine gute Sensorbelichtung Sie k nnen durch Nutzung der Mittelungsfunktion das Bildrauschen verringern Achten Sie dabei
86. dig unterst tzen und abw rtskompatibel zu USB 1 1 sind Mit der Software von PicoScope k nnen Oszilloskope der Serie PicoScope 3000 als PC Oszilloskope und Spektrumanalysatoren verwendet werden Mit dem Programm PicoLog k nnen Oszilloskope der Serie PicoScope 3000 als Datenerfassungsger te eingesetzt werden Vielleicht sind Sie auch an der Alternative interessiert die verf gbaren API Funktionen f r die Entwicklung eigener Programme zum Erfassen und Analysieren von Oszilloskopdaten zu verwenden Ein typisches PicoScope 3000 Oszilloskop wird mit folgendem Zubeh r geliefert USB Kabel passend zu beiden USB Arten 2 Software CD Installationshandbuch Sicherheitszeichen Warnzeichen 1 Warndreieck Dieses Sicherheitszeichen gibt an dass an den angegebenen Anschl ssen eine Sicherheitsgefahr vorliegt wenn die vorgeschriebenen Sicherheitsma nahmen nicht getroffen werden Stellen Sie sicher dass alle Sicherheitsunterlagen im Zusammenhang mit dem Produkt gr ndlich gelesen werden bevor das Produkt verwendet wird Warnzeichen 2 quipotential Dieses Sicherheitszeichen gibt an dass die Au engeh use der angezeigten BNC Stecker das gleiche Potential haben d h kurzgeschlossen sind Der Benutzer muss daher die notwendigen Vorsichtsma nahmen ergreifen um zu vermeiden dass ein Potential zwischen den Au engeh useanschl ssen der BNC Anschl sse angelegt wird da dies zum Flie en eines hohen Stroms und damit zu Besch digu
87. e For the special case of a confocal or near confocal resonator such as an FPS Eqs 86 and 37 become Vang c 4 r e 2g 1 m mm 36a and r e c 4v 12q A m n 37b where the mirrors have radii r and are separated by r r Thus all transverse modes will resonate at cavity lengths of either r c 4voJ 21 1 l an integer m n even 38a or r e c 4v 21 m n odd 38b If we assume that an arbitrary input field of frequency vo is made up of an approximately equal number of even and odd transverse modes a good approximation in any instance where mode matching is not inten tionally accomplished then the cavity will be resonant for r e cl 4w lan integer 39 and the multimode free spectral range will be Avy multimode c 4r 40 If on the other hand the input field exactly matches a single mode of the cavity the free spectral range is Avys single transverse mode c 2r 41 The transition from multimode to single mode excita tion can be observed without undue difficulty as illus trated in Fig 9 Note that the free spectral range in creases by a factor of two for both scanning and static fringe modes of use of the FPS The tolerance on the alignment of the light beam relative to the axis of the FPS that is required for mode matched operation is on the order of p so that it is highly unlikely that this situation would be encountered in
88. e B Diese Grenzwerte sind darauf ausgelegt einen angemessenen Schutz vor gesundheitsgef hrdenden St rungen in Wohngeb uden sicherzustellen Dieses Ger t erzeugt und verwendet hochfrequente Spannungen und kann diese ausstrahlen wenn es nicht anweisungsgem betrieben wird kann es erhebliche St rungen des Funkverkehrs verursachen Es gibt jedoch keine Garantie dass bei einer bestimmten Einrichtung keinerlei St rungen auftreten Wenn dieses Ger t den Radio oder Fernsehempfang beeintr chtigt was durch das Aus und Einschalten des Ger ts berpr ft werden kann wird dem Benutzer der Versuch empfohlen diese St rungen durch eine oder mehrere der folgenden Ma nahmen zu beseitigen Richten Sie die Empfangsantenne neu aus oder platzieren Sie diese an einer anderen Stelle a Vergr ern Sie den Abstand zwischen dem Ger t und dem Empf nger Verbinden Sie das Ger t mit einer Steckdose die zu einem anderen Stromkreis geh rt als zu dem an den das Empfangsger t angeschlossen ist W Fragen Sie Ihren H ndler oder einen erfahrenen Radio Fernsehtechniker um Rat weitere Informationen ber Sicherheit und Wartung finden Sie in den Sicherheitshinweisen Garantie Pico Technology garantiert f r einen Zeitraum von 24 Monaten ab Auslieferdatum wenn nicht anders angegeben dass die Waren bei Lieferung frei von Material und Verarbeitungsfehlern sind Pico Technology bernimmt keine Haftung f r eine Garantieverletzung wenn d
89. e Ne gas laser Various numbers of modes were excited by adjusting the laser mirrors The 5 cm FPS shown in Fig 12 was used 330 MHz cm Figure 16 shows the spectra which were obtained us ing an inexpensive commercially available He Ne laser which could be operated in one or many transverse and axial modes by adjusting the mirror alignment This figure clearly shows the ability of the FPS to reeord the spectra of higher order transverse modes only by ob serving the laser spectrum can very weak higher order transverse modes be detected The gain profile shows a raggedness due to competition effects between dif ferent modes and an asymetry due to the presence of more than one isotope of neon in the He Ne mixture The spectra shown in Fig 16 were all obtained with a 5 cm FPS with a free spectral range of 1500 MHz or approximately 0 02 A at 6328 A The fringe patterns shown in Fig 17 a and b show the spectra of a 60 cm He Ne laser operating in three axial modes with just the THM transverse mode see Fig 17 a and with both THM and THM transverse modes see Fig 17 b These spectra were obtained with the same instrument used in obtaining the spectra shown in Fig 16 although a different laser was used The mirror separation was approximately 50 u in excess of confocal resulting in a lower radial dispersion near the center of the pattern than would otherwise be obtained Figure 17 c shows the spectrum of a Q switched laser o
90. e lowest order transverse mode of the confocal resonator the diffraction loss per pass Lp is approxi mately given by Lp 107 Bord 1 22 where po is the radius of the mirror aperture In any case of practical interest diffraction losses are com pletely negligible in comparison to other losses so that diffraction plays no significant role in determining the over all finesse For a plane parallel Fabry Perot etalon the diffraction limited finesse is approximately given by Fo FPP D 2ad 23 where d is the separation of the plane mirrors and D is the aperture diameter Other types of loss such as scattering at the mirror surface which is of course taken into account in Fp can be treated separately very easily If a small frac tion Z of the radiation incident on the mirror or making a transit of the resonator is lost then by analogy to May 1968 Vol 7 No 5 APPLIED OPTICS 955 R FILTE N y rep WIDTH TRANSMITTED LIGHT BAND WIDTH Av BROADBAND LIGHT SOURCE Fig 6 Generalized picture of a spectrometer or monochro meter Eq 15 the corresponding contribution to the finesse is given by Fy w 2L 24 To summarize the implications of this section we can say that for a spherical Fabry Perot interferometer in confocal adjustment the significant factors in de termining the finesse and resolving power are the re flectivity of the mirrors and their surface figure This
91. e pressure chamber can be used for pressure scanning or it can be partially evacuated and sealed to eliminate effects due to changes in atmospheric pressure Applications for this type of interferometer include ultrahigh resolution Spectroscopy and use as a passive feedback component in frequency stabilizing lasers C A Piezo electrically Scanned FPS System This instrument is designed along the lines illustrated in Fig 10 and can be used as a scanning spectrum analyzer as a tunable narrow bandpass filter with zero Optical power or for direct observation of multiple beam interference fringes The key element in the instru ment is the FPS etalon which is comprised of a fixed and an adjustable mirror cell and a thermally compen sated rigid spacer tube The spacer tube includes as an integral component a piezo electric ceramic section which increases in length by about 1 5 em X 103 em with the application of 50 V across the inner and outer surfaces of the tube sufficient to scan a complete free spectral range in the visible The adjustable mirror cell permits the mirror spacing to be easily set to within a fraction of a micron Further fine adjustment can be accomplished by applying a de voltage to the scan ning voltage terminal this is the method of tuning in the bandpass filter mode of operation Figure 12 shows a cut away view of the entire instrument The FPS etalon is mechanically isolated from the case by a mounting techniqu
92. e shown for different values of g ranging from zero confocal through one parallel plane to 1 2 convex unstable By virtue of the equivalence property discussed in Section 4 4 the curves are also applicable to resonators with their g values reversed in sign provided the sign of the ordinate for the phase distribution is also reversed It is seen that the field is most concentrated near the resonator axis for g 0 and tends to spread out as gl increases Therefore the diffraction loss is ex pected to be the least for confocal resonators Figure 21 shows the relative field distributions of some of the low order modes of a Fabry Perot resonator with parallel plane circular mirrors N 10 aj a2 g1 g2 1 as obtained by a modified numerical iterative method 35 It is interesting to note that these curves are not very smooth but have small wiggles on them the number of which are related to the Fresnel number These wiggles are entirely absent for the confocal resonator and appear when the resonator geometry is unstable or nearly un stable Approximate expressions for the field distribu tions of the Fabry Perot resonator modes have also been obtained by various analytical techniques 36 37 They are represented to first order by sine and cosine func tions for infinite strip mirrors and by Bessel functions for circular mirrors For the special case of the confocal resonator g1 g gt 0 the eigenfunctions are self reciproca
93. e using two large silicone rubber O rings which cushion the etalon from mechani cal shocks see Fig 12 The outer case can be struck sharply from any direction without noticeably affecting the resonant frequency of the etalon As shown in Fig DEBO I AY po AO AY PS arg ERA S 72 HASSA A e x ava e 40 DM EHRT Aa ANNEE SA are N y LA Y XA KA ATE cee AAA ES RAN Fig 12 An FPS spectrum analyzer for scanning or static mode of operation 1 removable detector photodiode 2 soft O ring for mounting FPS etalon 3 quartz mirrors r 5 cm 4 piezo electric transducer etalon spacer 5 scanning voltage terminal 6 auxiliary lens focal point is between mirrors 7 mounting flange 8 9 adjustable mirror cell 10 fixed mirror cell 11 outer case The mirror cells are designed to compensate for the thermal expansion of the etalon spacer May 1968 Vol 7 No 5 APPLIED OPTICS 961 c Fig 13 Typical fringe patterns in the vicinity of confocal separation for a 10 cm FPS In each case the source is a single mode He Ne laser Variations in the fringe patterns in each horizontal row were obtained by making small changes lt A 4 in the mirror separation a e 70 u b e 0 c e 70 p 10 the lens at one end of the instrument serves either of two functions to direct an incident beam of radia tion into the center of the FPS etalon scanning or filte
94. ecording fringe pattern detector removed x li NIN 7 Fa LA N n 2 zi A 4 o EEE h NE N FA EH L m N Y E N N uh sa mi 42 e O RUIN E a UZIZA DEN SEN m kT y P N a a Fa S7 E E a m ZA N y Fig 11 Schematic of a highly stable fixed mirror FPS etalon 1 outer case Al alloy 2 end plates with windows 3 Cer Vit etalon spacer 4 fused quartz mirrors 5 fixed mirror cell Invar 6 7 adjustable mirror cell Invar 8 ports in etalon spacer 9 phosphor bronze springs holding etalon 10 fixture for evacuating chamber and pressure scanning This design facilitates the observation of the static fringe pattern If the detector aperture is removed and a quasi collimated beam of light is incident on what was the rear of the system then the fringe pattern will be formed in the focal plane of lens Li Thus an observer or camera focused on infinity can readily view the fringe pattern If the incident beam is diverging rather than collimated then the plane of the fringe pat tern will be slightly displaced towards the lens and vice versa A low power focusing telescope is useful both for observing the fringe pattern and for photographing it In this mode of operation the system should be mounted so that it can be rotated about a point near the right hand interferometer mirror Fig 10 c To use the interferome
95. ed 1 Establishing the Confocal Mirror Separation We have pointed out that optimum performance of an FPS depends critically upon the proper spacing of the mirrors Connes has described an imaging technique for approximating this adjustment which requires only a small incoherent light souree If a He Ne laser is available the alignment can be made with high pre cision as follows First adjust the mirror separation to within a millimeter or so from knowledge of the mirror radi Then set up the FPS etalon so that a quasi collimated beam from the gas laser is incident on one mirror and arrange to view the interference fringes which are formed in the vicinity of the central plane of the etalon The laser beam diameter should be large enough to allow several fringes to be seen Next make a fine adjustment of the mirror separation to bring the mirrors closer together by a fraction of a wavelength and observe the resulting change in the central fringe radius This fine adjustment can usually be made by manually squeezing the etalon If the mirror separa tion is greater than the confocal separation the central fringe will become smaller in diameter as the mirrors are _ moved towards each other If the mirror separation is less than the confocal spacing the central fringe diam eter will ncrease as the mirrors are moved towards one another Figure 13 shows the appearance of the central fringes on either side of exact confocal spacing a
96. eir red connector housing The console will not recognize sensors from the A and B series Please contact Thorlabs for upgrading of old sensors with C Series connectors To plug or remove a sensor slightly press the two bolts in the connector housing Sensors can be hot swapped to the console after recognizing a new valid sensor the type and calibration data will be downloaded to the console in approximately 2 seconds and the unit is ready to operate with the new sensor 3 2 Controlling the PM100D 3 2 1 Navigating the Menus Each measurement screen contains of eight soft buttons that are arranged in 2 rows in the bottom of the graphics display These can be controlled by the four navigation A Y lt P and the enter edit OK key An interactive help text above describes shortly the function of each selected button The soft buttons may be configured with the following functions Type Indicator _ Function when pressing OK ha Shows a sub menu by rearranging the soft Meas button labels Config gt gt Ring Scrolls up and down the ring with the up Range A Control and down navigation buttons The changes are valid immediately A blinking button frame indicates that the control is active Confirm with OK tion key capitals button The active key gets the checked mark y Numeric Key label The button goes in the edit mode This is 1 550um contains a__ indicated by a blinking frame and
97. electromagnetic TEM So long as those assumptions are valid the Fresnel Kirchhoff formulation of Huygens principle can be invoked to obtain a pair of integral equations which relate the fields of the two opposing mirrors Further more if the mirror separation is large compared to mirror dimensions and if the mirrors are only slightly curved the two orthogonal Cartesian components of the vector field are essentially uncoupled so that separate scalar equations can be written for each component The solu tions of these scalar equations yield resonator modes which are uniformly polarized in one direction Other polarization configurations can be constructed from the uniformly polarized modes by linear superposition MIRROR je MIRROR 2 OPAQUE ABSORBING SCREENS d LENS A yon re ci P ee y Fig 16 Geometry of a spherical mirror resonator with finite mirror apertures and the equivalent sequence of lenses set in opaque absorbing screens In deriving the integral equations it is assumed that a traveling TEM wave is reflected back and forth between the mirrors The resonator is thus analogous to a trans mission medium consisting of apertures or lenses set in opaque absorbing screens see Fig 16 The fields at the two mirrors are related by the equations 24 K s1 S2 EW s2 dS2 So y VED s K s syEW sydS 72 S1 y DED s3 where the integrations are taken over the mirror surfaces S and S
98. er Defekt durch angemessenen Verschlei absichtliche Besch digung Fahrl ssigkeit Missbrauch abnormale Arbeitsbedingungen oder Nichtbeachtung von Pico Technologys m ndlichen oder schriftlichen Hinweisen zu Lagerung Installation Inbetriebnahme Gebrauch oder Wartung der Waren oder falls keine Hinweise vorliegen gutem Handelsbrauch oder falls der Kunde diese Waren ohne schriftliche Zustimmung von Pico Technology ndert oder repariert Copyright 2005 7 Pico Technology Limited All rights reserved PS3000049 2 Series PicoScope 3000 Handbuch 1 7 1 8 Rechtliche Hinweise Das in dieser Version enthaltene Material wird nur lizenziert und nicht verkauft Pico Technology Limited gew hrt der Person die das Programm installiert eine Lizenz miz den folgenden Bedingungen Zugriff Der Lizenznehmer stimmt zu nur Personen Zugriff zur Software zu gew hren die ber diese Bedingungen informiert wurden und diesen zugestimmt haben Verwendung Diese Programmversion darf nur mit Pico Produkten oder mit Daten die mit Hilfe von Pico Produkten erstellt wurden verwendet werden Copyright Pico Technology Limited beansprucht das Copyright und beh lt sich alle Rechte an den Materialien Software Dokumentationen usw dieser Version vor Sie k nnen diese Version in ihrem Originalzustand kopieren und weitergeben d rfen aber einzelne Teile der Version nur zu Sicherungszwecken kopieren Haftung Pico Technology und seine
99. es dis cussed below By inserting 10 into 9 one obtains 2 2 a dx y dz where it has been assumed that y varies so slowly with z that its second derivative 0 y dz can be neglected The differential equation 11 for y has a form similar to the time dependent Schr dinger equation It is easy to see that k Y exp q SJ 12 29 is a solution of 11 where r g y 13 The parameter P z represents a complex phase shift which is associated with the propagation of the light beam and q z is a complex beam parameter which describes the Gaussian variation in beam intensity with the distance r from the optic axis as well as the curvature of the phase front which is spherical near the axis After insertion of 12 into 11 and comparing terms of equal powers in r one obtains the relations ql 1 14 and P 15 where the prime indicates differentiation with respect to z The integration of 14 yields dl q Hz 16 which relates the beam parameter q in one plane output plane to the parameter q in a second plane input plane separated from the first by a distance z 3 2 Propagation Laws for the Fundamental Mode A coherent light beam with a Gaussian intensity pro file as obtained above is not the only solution of 11 but is perhaps the most important one This beam is often called the fundamental mode as compared to the higher order modes to be discussed later Because of its impo
100. f the mir rors of an FPP have irregularities on the order of 1 2 the resultant pattern at infinity will be completely washed out However since the fringe pattern ob tained with an FPS is localized relatively close to the surfaces of the mirrors a similar mirror figure irregu larity will not wash out the fringe pattern but will in stead distort 1t so that the fringes are no longer circular These distorted fringes tend to define coutours of equal path difference As implied above diffraction losses are much less in the case of a spherical Fabry Perot etalon than for its plane mirror counterpart The rigorous justification of this statement lies in the analytical treatment of confocal resonators given by Boyd and Gordon in which they show that for any case of practical interest to us 1 e those cases where D 4r gt X D being the diameter of the mirror aperture the diffraction losses for a confocal resonator are orders of magnitude less than for the corresponding plane parallel resonator The calculation of the exact diffraction loss in a confocal resonator requires a fairly complex analysis in which the incoming radiation field is decomposed into eigen modes of the cavity each of which has a different diffraction loss Absolute minimization of the diffrac tion losses requires proper mode matching see Sec II D In this case when the incoming radiation field has a curvature and amplitude distribution identical to that of th
101. ffraction loss and the phase shift have been obtained for the special cases of parallel plane g 1 0 and confocal g 0 geometries when the Fresnel number is either very large small dif fraction loss or very small large diffraction loss 36 38 39 41 42 In the case of the parallel plane resonator with circular mirrors the approximate expres sions valid for large N as derived by Vainshtein 36 are SM 6 A ct A 86 AL 8 _ A 87 n e October 1966 Vol 5 No 10 APPLIED OPTICS 1565 where 0 824 M 8rN and x is the p Dth zero of the Bessel function of order For the confocal resona tor with circular mirrors the corresponding expressions are 39 2r 8rN 2p l 1p 47 N 1 ol ya pip 1 1 2rN T B Br 89 Similar expressions exist for resonators with infinite strip or rectangular mirrors 36 39 The agreement be tween the values obtained from the above formulas and those from numerical methods is excellent The loss of the lowest order TEMoo mode of an unstable resonator is to first order independent of the mirror size or shape The formula for the loss which is based on geometrical optics is 12 1 vi gg en V 9192 1 Vi gg where the plus sign in front of the fraction applies for g values lying in the first and third quadrants of the stability diagram and the minus sign applies in the other two quad rants Loss curves plotted vs N ob
102. finesse In order to record the ultimate instrumental profile in the scanning mode of operation the detector aperture would be vanishingly small and the resulting instru mental profile would be given by Iw vo DIT R ToC 2 w F x r 0 4 it a If the detector aperture were increased there would initially be an increase in the amount of light collected from a finite source with little decrease in resolving power assuming perfectly spherical mirrors and con focal spacing As the aperture was opened further the amount of light collected would increase less rapidly and the resolving power would begin to decrease be coming approximately 70 of the resolving power given by Eq 16 when the radius of the detector aperture attained a value ps given by ps A F 17 We will refer to p as the spot size or spot radius ps 18 simply the radius of the mirror zone whose resonant frequency is displaced from the axial resonance by an TRANSMITTED POWER P Arbitrary units 50 40 30 20 I0 Vo 10 20 30 40 50 FREQUENCY MHz TRANSMITTED POWER P Arbitrary units 50 40 30 20 10 Vo 10 20 30 40 50 FREQUENCY MHz Fig 5 Calculated FPS instrumental profiles for two different detector aperture radii These correspond to the spectra which would be recorded using a monochromatic source in the scanning mode of operation e 0 r 10cm a p 0 05 b p 0 2 amount equal to the minimum resolvable
103. focal separation R mirror reflectivity R spectral resolving power T mirror transmission To FPS instrumental transmission U tendue QA 6 phase increment 2rA A A path difference difference between FPS mirror separation and confocal spacing r A wavelength optical frequency Avm minimum resolvable frequency difference or instrumental bandpass Avy freespectral range c dr FPS p fringe radius p radiusof central spot Ar F Q solid angle subtended by source Il Theory of Operation A Interference Fringes A spherical mirror Fabry Perot interferometer is comprised of two identical spherical mirrors separated by a distance very nearly equal to their common radius of curvature When light from a source lying close to the axis is incident on the FPS a multiple beam interfer ence pattern is produced in the vicinity of the central plane of the interferometer To see how this interfer ence pattern arises consider an entering ray which intersects the two mirrors at points P and Pz which are located at distances p and pz from the axis As shown in Fig 1 0 is the skew angle of the entering ray According to paraxial optics each mirror serves to image the other mirror back upon itself so that a paraxial ray is reentrant i e falls back upon itself after traversing the interferometer four times Fig 2 a Owing to aberration however a general ray is not reentrant but follow
104. free spectral range is accompanied by a twofold increase in both finesse and instrumental transmission so that there is no loss in spectral resolution There is however a very great reduction in the tolerance of the alignment between the FPS axis and the incident light beam and there is also the restriction that the incoming beam match a lowest order transverse mode of the cavity This latter restriction is less severe than it might be due to the degeneracy of a confocal cavity Unlike a general curved mirror cavity the position of the beam waist of the lowest order transverse mode is not uniquely de 24 MHz cm Fig 14 Observed instrumental profiles for different detector aperture diameters D Light source was a 1 cm wide collimated single mode laser beam e 0 r 10cm The 0 3 cm aper ture is clearly the best compromise between signal amplitude and resolution May 1968 Vol 7 No 5 APPLIED OPTICS 963 330 MHz cm IL u 330MHz cm Fig 15 Spectra of a single mode laser obtained with 5 cm and 10 cm scanning FPS instruments Top 10 cm FPS measured finesse 7 154 middle 5 cm broadband mirrors F 148 bottom single mode He Ne gain profile showing Lamb s dip see text termined but can be located anywhere between the two mirrors the diffraction losses are minimized however when the beam waist is at the center of the cavity This type of confocal cavity degeneracy is particularly clear from
105. g back and forth between the mirrors the beam parameters must be the same after one complete return trip of the beam This condition is used to calculate the mode parameters As the beam that repre sents a mode travels in both directions between the mirrors it forms the axial standing wave pattern that is expected for a resonator mode A laser resonator with mirrors of equal curvature is shown in Fig 10 together with the equivalent unfolded system a sequence of lenses For this symmetrical struc ture it is sufficient to postulate self consistency for one transit of the resonator which is equivalent to one full period of the lens sequence instead of a complete return 1556 APPLIED OPTICS Vol 5 No 10 October 1966 trip If the complex beam parameter is given by g1 im mediately to the right of a particular lens the beam parameter q immediately to the right of the next lens can be calculated by means of 16 and 41 as 44 Self consistency requires that g g2 g which leads to a quadratic equation for the beam parameter g at the lenses or at the mirrors of the resonator 1 1 1 0 45 a fa fd The roots of this equation are 1 1 E 1 1 46 ET fa ap where only the root that yields a real beamwidth is used Note that one gets a real beamwidth for stable resonators only From 46 one obtains immediately the real beam parameters defined in 17 One sees that R is equal to the radi
106. g to the lowest order of interference For any given value of e this fringe occurs at the value of p which corresponds to the zone of best focus for the spherical mirror By Fermat s principle this is just the value of p where dA dp is an extremum or p 2er No zone of best focus is defined for e gt 0 In the special case very nearly approximated in most applications where e 0 the fringes have radii given by Pm m gt ar 4 8 where lt 1 and 4 r e A is the exact order of inter ference on the axis It is obvious from Eq 8 that the radial dispersion in the fringe pattern is markedly nonlinear near the axis when the interferometer is precisely confocal This is of course no real disadvantage and provides the basis for the high tendue of which this type of instrument is capable If desired the dispersion may be made more nearly linear by slightly decreasing or increasing the mirror separation This is evident from Fig 3 and 1s illustrated in Sec TIT B Spectral Resolving Power In discussing spectral resolving power in this section we assume that the interferometer is set at the confocal spacing le lt X and is used in the scanning mode with a collimated light source More specifically we assume that the central fringe pattern is imaged 1 to 1 onto a plane containing an axial aperture coincident with the center of the fringe pattern behind which is located a linear detector Since
107. gative impacts on the environment Crossed out wheelie bin symbol 6679 DO2 Rev E 3 08 16 05 Section 1 0 Specifications Physical Features Dimensions W x H x D Input and Output Connectors Offset Control Amplitude Control Risetime Control Sweep Expansion Control Photodiode Gain Control Waveform Select PD Amp Features Operating Temperature Storage Temperature Power Supply Supply Type Voltage Selection Input Voltage Line Frequency Input Power Fuse Ratings Fuse Type Output Characteristics Waveform Default Waveform Sawtooth Fall Time Output Voltage Range Max Supply Current 1 Short Circuit Current 2 Short Circuit Duration Offset Adj Range Amplitude Adj Range Risetime Adj Range 3 Sweep Expansion Settings Sweep Scale Error Output Noise 5 8 x 2 8 x 12 5 147mm x 71mm x 317 5mm BNC s 10 turn Potentiometer 10 turn Trim pot 10 turn Trim pot 7 Position Rotary Switch 3 Position Rotary Switch Pushbutton w illuminated indicators Blanking with Sawtooth Waveform Falling Edge 10 C to 40 C 0 C to 85 C Linear Switch Selectable between 115 230Vac 100 115 230Vac 50 60Hz 15W max 125mA O 100 115Vac 63mA 230Vac Slow Blow Type T Sawtooth Triangle Sawtooth 1ms typ 1 to 45V offset amplitude 15mA 26mA max Continuous O to 15Voc 1 to 30V 0 01 to 0 1s 1X Sweep Exp 1 to 10s 100X Sweep Exp
108. he weniger als 2 cm von der theoretisch berechneten Stabilit tserenze entfernt ist Tragen Sie die Ergebnisse graphisch auf Diskutieren Sie die Ergebnisse Hinweise Maximieren Sie die Ausgangsleistung f r jeden Messpunkt durch Justage der Spiegel und des Laserrohrs CCD Sensor Auskoppler l HR Spiegel Wee W2 Wo en 3 Strahlbreite der Grundmode in Abh ngigkeit von der Resonatorl nge Aufgaben zur Vorbereitung Berechnen Sie f r die beiden Resonatorl ngen von 25cm und 88cm den Strahldurchmesser der TEMoo Mode am Auskoppelspiegel Bestimmen Sie f r die beiden Resonatorl ngen 25cm und 88cm den Fehler in Prozent vom erwarteten Messwert der sich aus der Vereinfachung f r einen Abstand d 8cm ergibt Diskutieren S e sp ter in der Auswertung gegebenenfalls dessen Relevanz a Aufnahme der Messdaten Bestimmen Sie Strahlbreite w am Ort des Auskoppelspiegels in Abh ngigkeit von der Resonatorl nge Nehmen Sie hierzu die Intensit tsverteillung der ausgekoppelten Laserstrahlung mit Hilfe einer CCD Kamera in festem und m glichst geringem Abstand d vom Auskoppelspiegel auf siehe Abb 1 Die Ma e der aktiven Sensorfl che betragen 4 8mm hor zontal und 3 6mm vertikal Achten S e bei der Aufnahme der Bilder darauf dass der Chip nicht bers ttigt ist F r jede Resonatorl nge soll die TEMoo Mode des Lasers angeregt werden Dies kann z B durch verkippen des
109. hown in Figure 6 Extremely high contrast can be obtained in this way Theoretically a Fabry Perot with 93 reflectivity mir rors can have a contrast of 10 in three pass operation 10 in five pass operation Although other factors such as stray reflections and mirror flatness limit the ultimate contrast performance approaching theoretical can be obtained with careful design Multipassing was made practical and popularized by the application of corner cube retroreflectors Corner cube retroreflectors displace the reflected beams lateral ly greatly simplifying the separation of input beams They also have the all important feature of producing a reflected beam accurately parallel to the input beam even if the corner cube is tilted This feature greatly simplifies Lasers amp Applications July 1983 3 Invar Mirror Cell Mirror 1st surface protrudes about 0 1mm Mounting Holes V pad ey Glass Tab Ball Invar Figure 5 Distortion free technique for mounting mirrors in a large frame Fabry Perot the optical alignment so that only the Fabry Perot mirror alignment remains critical Figure 7 shows a scheme utilizing modified corner cube retroreflectors for five pass operation of a Fabry Perot interferometer Individual beams are well sepa rated on the mirror surfaces so that cross transmission between beam paths due to stray reflections and scatter ing can be minimized by inserting light baffles
110. ilter treat it as any other fine optic Gently blow off any debris using compressed air and wipe gently with an optic tissue wetted with propanol If you suspect a problem with your S720C please call Thorlabs and an engineer will be happy to assist you As long as the sensor has not been exposed to excessive optical power please pay attention to the maximum ratings in the technical specifications the calibration should be very stable over long periods of time well over a year To keep the accuracy and performance of the S120C Thorlabs recommends a yearly recalibration starting one year after purchase US PO Box 366 435 Route 206N Newton NJ 07860 Tel 1 973 579 7227 Fax 1 973 3600 3 Europe Hans Boeckler Str 6 85221 Dachau Munich Germany Tel 49 8131 59560 Fax 49 8131 595699 Web http www thorlabs com Mail europe thorlabs com Doc 18356 S01 doc Rev A 1 Oct 08 Spec Sheet WEEE As required by the WEEE Waste Electrical and Electronic Equipment Directive of the European Community and the corresponding national laws Thorlabs offers all end users in the EC the possibility to return end of life units without incurring disposal charges This offer is valid for Thorlabs electrical and electronic equipment e sold after August 13 2005 e marked correspondingly with the crossed out wheelie bin logo see fig 1 e sold to a company or institute within the EC e currently owned by a company or institute within
111. ing power product factors limiting realizable finesse mode matching considerations alignment procedures and general design considerations Two specific instruments are described One is a versatile spectrum analyzer with piezo electric scanning the other is a highly stable etalon with fixed spacing Examples of the performance of these instruments are given l Introduction The spherical mirror Fabry Perot interferometer FPS was first described by Connes over ten years ago 8 Although this instrument is mentioned in some recent texts Connes papers contain the only detailed descriptions of the spherical mirror Fabry Perot inter ferometer This paper is intended to review and extend Connes treatments of the theory of operation of the FPS to describe specific instrument designs and to outline practical procedures for using this instrument in both static and scanning modes I have drawn freely from the results obtained by Connes particularly those contained in Ref 2 In those cases where our results differ it is generally because I consider only relatively high reflection mirrors with uniform transmission whereas Connes described interferometers in which the mirrors had zero transmission and nearly complete reflection over half of their apertures Following the introduction of curved mirror resona tors as laser cavities it was found that with little modi fication they could effectively be used as spectrum analyzers
112. ion of the Invar body with the negative coefficient of thermal expansion of the piezo actuators Page 1 of 5 THORLABS INC OR SA 435 Route 206 P O Box 366 PH 973 579 7227 Newton NJ 07860 0366 FAX 973 300 3600 www thorlabs com technicalsupport thorlabs com SA210 Series Scanning Fabry Perot Interferometer DESCRIPTION The SA210 is a high finesse Spectrum Analyzer used to examine the fine structures of the spectral characteristics of CW lasers The spectrum analyzer consists of a confocal cavity that contains two high reflectivity mirrors by varying the mirror separation with a piezoelectric transducer the cavity acts as a very narrow band pass filter Knowing the free spectral range of the SA210 allows the time base of an oscilloscope to be calibrated to facilitate quantitative measurements of a laser line shape SPECIFICATIONS Free Spectral Range FSR Measured in milliseconds FWHM Measured in microseconds FSR FWHM EAS PEA Maximum Input Voltage 150V Minimumfinesse gt 150 SS Low thermal expansion Invar 6 Dimensions 1 Flange Total Length 2 93 FSR is set by the length of the confocal cavity and is given by FSR c 4d Where d the radius of curvature of the mirrors in this case d 7 5mm see drawing on next page A thermal design balances the small coefficient of thermal expansion of the Invar body with the negative coefficient of thermal expansion of the piezo actuators
113. ions to the finesse independently First we consider the effect of irregularities in the figure of the mirror on the finesse Without knowledge of the specific nature of these irregularities it is im possible to be precise in predicting their effect on the finesse Generally however if the mirrors have a smooth irregularity on the order of A m across the aperture being used then the figure limited finesse F will be approximately Obviously by reducing the aperture or diameter of the incident light beam it is possible to minimize the re duction in instrumental finesse due to plate irregulari ties This is indeed a practical expedient in the case of If the irregularity is not smooth the loss un is more appropriately treated as a scattering loss a spherical Fabry Perot etalon in the case of a plane mirror Fabry Perot etalon however the significantly increased diffraction losses that accompany the re duction of the etalon aperture set a limit to the improve ment in finesse that can be realized by this technique Note also that in the case of the plane mirror Fabry Perot an angular misalignment of the plates is equiva lent to a corresponding plate imperfection For the spherical Fabry Perot this is not the case an angular misalignment merely redefines the optical axis of the system With regard to plate irregularities it is worthwhile pointing out another contrast between the plate mirror and spherical mirror etalons I
114. is in contrast to the case of a plane parallel Fabry Perot where diffraction and alignment can make significant contributions to the degradation of finesse and resolving power C Light Gathering Power 1 Introduction One of the major factors to consider in evaluating any spectrometer is its ability to effectively gather light from an incoherent extended source filter it with the instrumental bandpass and transmit it to some radia tion detector In general the situation can be repre sented by Fig 6 Here the spectrometer is depicted as a bandpass filter all of the radiation emanating from within a solid angle Q subtended at an aperture of area A can be transmitted within the bandpass Avm of the spectrometer If the transmission of the spectrometer at the center of the bandpass is To and the spectral radiance of the source is N then the radiant power per unit bandwidth P transmitted by the spectrometer is given by P N AQT 25 The product NA has come to be known as tendue U of the spectrometer Thus the easily remembered ex pression P NUT 26 Of course if the light source under investigation is a laser it is obvious that most of the emitted power can be put into a beam with a small cross sectional area and a small divergence In this case the tendue of the spectrometer provides a measure of the alignment tolerance between the laser beam and spectrometer rather than being a measure of the spectromete
115. ise when operating with a 50 coax cable For best results a 509 load resistor is recommended at the oscilloscope Note the amplifier gain will be halved with a 509 load connected Voltage Selector Switch 15 The voltage selector switch allows the user to select the input line voltage they will be operating the system at the factory default setting is 100 115V as shown in figure 2 To operate at 230V c this switch will have to be moved to the 230V position The line fuse will also need to be changed to properly protect the unit See section 5 for detailed instructions 6679 D02 Rev E 9 08 16 05 Section 4 0 Descriptions continued PD Blanking Circuit The detector amplifier includes a blanking circuit which blocks any photo detector response during the falling edge of the sawtooth waveform This is very useful when triggering on the photo diode spectral response because unwanted signals while the cavity resets will be removed The blanking is not available when using the triangle waveform since it is useful to see the rising and falling response overlapped during system alignment This feature may be disabled as described below 1 Important Disconnect the scanning head or any piezo device from the SA201 output 2 Important Disconnect the power cord Do not open the unit if the power cord is connected 3 Remove the two screws securing the enclosure cover with a Phillips head screwdriver The screws are located on the bottom
116. l and electronic equipment e sold after August 13th 2005 marked correspondingly with the crossed out wheelie bin logo see fig 1 sold to a company or institute within the EC currently owned by a company or institute within the EC still complete not disassembled and not contaminated As the WEEE directive applies to self contained operational electrical and electronic products this end of life take back service does not refer to other Thorlabs products such as e pure OEM products that means assemblies to be built into a unit by the user e g OEM laser driver cards e components e mechanics and optics e left over parts of units disassembled by the user PCB s housings etc If you wish to return a Thorlabs unit for waste recovery please contact Thorlabs or your nearest dealer for further information Waste treatment on your own responsibility If you do not return an end of life unit to Thorlabs you must hand it to a company specialized in waste recovery Do not dispose of the unit in a litter bin or at a public waste disposal site Ecological background It is well known that WEEE pollutes the environment by releasing toxic products during decomposition The aim of the European RoHS directive is to reduce the content of toxic substances in electronic products in the future The intent of the WEEE directive is to enforce the recycling of WEEE A controlled recycling of end of live products will thereby avoid ne
117. l under the finite Fourier infinite strip mirrors or Hankel circular mirrors transformation and exact analytical solutions exist 5 38 40 The eigenfunctions for infinite strip mirrors are given by the prolate spheroidal wave func tions and for circular mirrors by the generalized prolate spheroidal or hyperspheroidal wave functions For large Fresnel numbers these functions can be closely approxi mated by Hermite Gaussian and Laguerre Gaussian functions which are the eigenfunctions for the beam modes October 1966 Vol 5 No 10 APPLIED OPTICS 1563 0 8 RELATIVE AMPLITUDE TEMoo MODE N 1 0 0 2 F RELATIVE PHASE DEGREES o 0 2 0 4 0 6 0 8 1 0 r a Fig 19 Relative field distributions of the TEMo mode for a resonator with circular mirrors V 1 TEMo MODE N 1 0 g 0 RELATIVE AMPLITUDE RELATIVE PHASE DEGREES 1 J 0 0 2 0 4 0 6 0 8 1 0 r a Fig 20 Relative field distributions of the TEM mode for a resonator with circular mirrors V 1 1564 APPLIED OPTICS Vol 5 No 10 October 1966 TEMoy MODE TEM y MODE 0 8 0 6 0 4 0 2 Q DE 9 ams AMPLITUDE PHASE DEG o 0 2 0 4 06 08 10 0 0 2 9 4 0 6 0 8 1 0 TEMo MODE TEM MODE o s 0 6 0 4 9 2 o ras a al 9 AMPLITUDE 100 PHASE DEG 200 9 02 04 06 08 100 02 04 06 08 1 0 r a rja Fig 21 Relative field distributions of four of the low order modes of a Fab
118. lieved by rotation of the ball in the V block and does not warp the mirror surface Mirrors as large as 70 mm in diameter are routinely mounted this way without altering their surface figure These mirrors attach to the fixed and piezoelectri cally driven mirror mounts The mounting permits easy mirror changes as necessary The piezoelectric assembly provides sufficient align ment and scanning control so that once mechanically aligned the instrument can be thermally isolated in an insulated box and operated completely by remote control This is particularly advantageous for stability as the lab temperature fluctuates In extreme environments active temperature control may be added inside the thermal box Experiments that require long periods of stable operation would benefit from an electronic stabiliza tion system that actively corrects for spacing and alignment Mirrors and Coatings With the exception of mirrors used in the far infrared most Fabry Perot mirrors are made of a high quality fused silica such as Spectrosil B Plane mirror substrates are wedged at an angle of about 10 arc minutes to prevent secondary interference fringes gener ated by the back surfaces of the mirrors Also the back surfaces are antireflection coated to reduce reflections and to increase throughput High quality low loss multilayer dielectric coatings are available from the ultraviolet to the infrared These so called soft coatings give
119. lly less manageable than large scale designs In many cases this difficulty in initial alignment is not important because once the adjustment is set the range of piezoelectric control is sufficient to subsequently opti mize the interferometer mirror alignment The small size and simple integrity of piezoelectric etalons enhances thermal and mechanical stability Because these etalons are electronically tunable they can be used with active stabilization systems Fixed air gap etalons with piezoelectric control have been built with capacitance displacement transducers that can be used for automatic alignment and cavity stabilization Applications for piezoelectrically tunable etalons include spectral analysis laser tuning active optical filtering and spectroscopy all examples where it s not essential to have large spacing with full cavity adjust ment high finesse or high tendue Confocal etalons have two identical concave mirrors spaeed precisely at their common radius of curvature Each mirror images the other back upon itself so that any paraxial ray entering the interferometer is superimposed upon itself after four reflections resulting in a very high tendue Because the mirrors are spherical the require ment for parallel alignment is greatly reduced and only axial piezoelectric tuning is necessary A typical confocal interferometer has cavity spacing of 50 cm and can resolve 1 megahertz Confocal interferometers are commo
120. ly possible to obtain finesses of up to 500 with commer cially available coatings and a passive interferometer This limitation is not set by available reflectivity but by attainable mirror figure or diffraction losses in the case of very short cavities 3 Optical Layout By optical layout we mean the optical system which brings the light into the interferometer and determines the path of light leaving the interferometer For a number of reasons we have preferred to make the inter ferometer mirror blanks of very nearly zero optical It is now generally recognized that essentially scatter free surfaces of excellent figure can be obtained by continuous and ex tended 12 36 h final polishing of fused quartz blanks 960 APPLIED OPTICS Vol 7 No 5 May 1968 power This permits the use of mirrors with concentric surfaces which simplifies fabrication More impor tantly though it means that the interferometer can be used to spectrally filter a narrow collimated beam of light without appreciably affecting the collimation of the beam If a further optical system is added to the FPS it should serve to facilitate the alignment of the FPS with the light source and to get light efficiently through the interferometer and within the instrumental passband Figure 10 illustrates an optical system which has proved to be convenient and versatile Con sider first an incident collimated beam lens L brings the incident radiation to focus
121. ly small axial aperture In this case of course the tendue is also infinitesimal A reasonable compromise between spectral resolving power and tendue can be reached by increasing the mirror aperture until the resolving power is reduced to a value of approximately 0 7 Ro This as we have seen occurs when the mirror apertures have radii of approximately ps Under this condition the tendue is given by U rp rp 12 rr F 29 nN vi p INSTRUMENT TRANSMISSION To 0 0 2 3 4 5 6 7 RATIO OF MIRROR ABSORPTION TO TRANSMISSION A T Fig 7 FPS instrumental transmission as a function of the absorption transmission ratio of the mirror coatings o N O Xi 30 a 20 O Spectral resolving power R oO Instrumental bandwidth AV MHz A 2 0 2 Aperture radius plem Aperture radius plem a b Pmax R X P max Arbitrary units Arbitrary units Peak power transmitied 0 A 2 Aperture radius P cm Aperture radius p cm c d Fig 8 Computed FPS characteristics as a function of detector aperture radius p for different values of F the finesse Arrows indicate the value of ps in each case e 0 7 10 cm The maxima in the curves shown in d define the aperture radius giving the best compromise between resolving power and peak transmitted power where F is the finesse which determines the value of ps according to Eq 17 Figure 8 shows c
122. mall angles the so called flat roof resonator Fig 1 was recently described by Toraldo di Francia 1 The mathematical approach consisted in considering the solutions of the wave equation for the electric or magnetic field in the two halves of a complete diamond cavity whose normal cross section is shown in Fig 2 ignoring the fact that the reflectors are finite The two half cavities were referred to cylindrical co ordinates centered at G and H respectively and solutions were given in terms of high order cylindrical waves The field in the two half cavities was matched over the median Manuscript received May 4 1966 The research reported here was supported in part by the Air Force Cambridge Research Labo ratories through the European Office of Aerospace Research OAR U S Air Force under Contract AF 61 052 871 The authors are with the Centro Microonde Consiglio Na zionale delle Ricerche Florence Italy DN Fig 1 The flat roof resonator Fig 2 The diamond cavity plane BE by simply requiring that this plane coincide with a node or an antinode Obviously the angle of the roof must be so small that the curvature of the nodal or antinodal surfaces can be neglected Due to the high order of the cylindrical waves the field in the central region of the cavity approaches the form of a standing wave be tween the two roof reflectors while it decays so rapidly from the central region toward the ver
123. me point at which the FPP will become the better choice in terms of tendue The specific value of r the mirror spacing of the FPS at which this transition occurs depends upon both the desired finesse and the accuracy with which the mirrors of the FPP can be figured and aligned If we define an angle to represent the figure plus align ment accuracy required to maintain a finesse F with a plate diameter D a A FD 33 then the ratio Upps Urpp may be written Urrs Urpp 4rdF 0 M 34 where d is the plate separation of the FPP and where we tacitly assume that the spot size p on the FPS is small enough so that there is no problem in maintaining the necessary figure of A F If we now require that both the FPP and FPS have the same free spectral range so that d 2r then Eq 34 can be used to find the value of 7 at which the FPP becomes the better choice with regard to tendue r A a 2F 85 May 1968 Vol 7 No 5 APPLIED OPTICS 957 b Fig 9 a Scan and fringe displays of an FPS in normal opera tion Note the secondary fringe pattern b Scan and fringe displays of a very nearly mode matched FPS The alignment of the FPS relative to the source has been adjusted to eliminate the secondary fringe pattern resulting in a doubling of both the free spectral range and the instrumental transmission e 20 u r 5 cm three mode laser source For example if the mirrors of an FPP can be maintaine
124. mount the unit into a tip tilt mirror mount Thorlabs part KM100 Attach all of the connection according the drawing on page 4 Next you should remove the detector from the back of the unit and mount it in it s own mount this will aid in the initial alignment Then close the input iris and center your beam on the iris opening Leave the back iris completely open and start to scan the unit Now using the tip tilt adjustment until the beam is center through the body of the SA210 Adjust the scope gain to maximum sensitivity position the detector close to the rear opening and slowly close the back iris as you correct the 2 angular adjustments on the mirror mount Once the beam is centered you can the replace the detector on the main body and start to use the unit for measurements Page 2 of 5 THORLABS INC OVER VIEW Free Spectral Range To scan the spectra of the laser beam entering the Scanning Fabry Perot interferometer small displacement is applied to one of the cavity mirror mounted on piezoelectric transducers This operation is done by fine tuning the ramp voltage applied to the Piezoelectric elements using the controller SA201 When the mirror spacing becomes equal to an integral number of half the wavelength of the laser constructive interferences occur That spectral response of the signal can be visualized with a scope A series of periodical peaks appear on the screen of the scope The distance between consecutive peaks is called the f
125. n Da die Software die Sie ben tigen bereits auf Ihrem Computer installiert ist m ssen Sie die Pico Software CD nicht erneut einlegen Copyright 2005 7 Pico Technology Limited All rights reserved PS3000049 2 8 Series PicoScope 3000 Handbuch 2 3 Technische Daten o 8204 3205 3206 3224 3424 Maximale abtastrate 1 Kanal 50 MS s 100 MS s 200 MS s 20 MS s 20 MS s 50 MS s 100 MS s 100 MS s 3 4 Kanal Repetitive Signale 2 5 GS s 5 GS s 10 GS s 256 K 512 K 1M 512K 512K 128 K 256 K 512 K 256K 256K l 128 K ber BNC AC DC Kopplung 20 pF Eingangskapazit t en I ao Signalgenerator 1 2 1 BNC Ausgang gemeinsam mit Signalgenerator Variabler Triggerschwellwert 20 V ansteigend abfallend 12 2 mV Aufl sung 1 MQ Impedanz Spannungsbereiche 100 mV bis 20 V Bereiche in 1 2 5 3 Spannung 1 Yo Spannung 50 pom Zeit 50 pom Zeit Umgebungsbedingungen Temperaturbereich 20 mV bis 20 V 0 C bis 70 C 0 C bis 70 C 25 C f r die angegebene Genauigkeit I 20 C bis 30 C f r die angegebene Genauigkeit 25 bis 75 relative Luftfeuchtigkeit 25 bis 75 relative Luftfeuchtigkeit Feuchtigkeit berlastungsschutz Kan le 50 V Externer Trigger 30 V PC Verbindung USB 2 0 Kompatibel mit USB 1 1 100 V Stromversorgung Aus USB Port 4 6 bis 5 25 V 500 mA Aus USB Port Externe Stromversorgung ist nicht erforderlich 1 1 BN
126. n New York Benjamin 1964 E L O Neill Introduction to Sta tistical Optics Reading Mass Addison Wesley 1963 14 M Bertolotti Matrix representation of geometrical proper ties of laser cavities Nuovo Cimento vol 32 pp 1242 1257 June 1964 V P Bykov and L A Vainshtein Geometrical optics of open resonators JETP USSR vol 47 pp 508 517 August 1964 B Macke Laser cavities in geometrical optics approximation J Phys Paris vol 26 pp 104A 112A March 1965 W K Kahn Geometric optical deriva tion of formula for the variation of the spot size in a spherical mirror resonator Appl Opt vol 4 pp 758 759 June 1965 15 J R Pierce Theory and Design of Electron Beams New York Van Nostrand 1954 p 194 16 H Kogelnik and W W Rigrod Visual display of isolated optical resonator modes Proc IRE Correspondence vol 50 p 220 February 1962 17 G A Deschamps and P E Mast Beam tracing and applica tions in Proc Symposium on Quasi Optics New York Poly technic Press 1964 pp 379 395 18 S A Collins Analysis of optical resonators involving focus ing elements Appl Opt vol 3 pp 1263 1275 November 1964 19 T Li Dual forms of the Gaussian beam chart Appl Opt vol 3 pp 1315 1317 November 1964 20 T S Chu Geometrical representation of Gaussian beam propagation Bell Sys Tech J vol 45 pp 287 299 Febr
127. n be used to display the spectrum of the input laser The controller provides adjustment of the ramp voltage 0 to 20V and scan time 1ms to 5s to allow the user to choose Page 4 of 5 THORLABS INC the scan range and speed An offset control is provided to allow the spectrum displayed on the oscilloscope to be shifted right or left zoom capability provides up to 100X increase in spectral resolution e Thorlabs KM100 1 kinematic mount can be used to mount the SA210 Scanning Fabry Perot Interferometer TECHNICAL SUPPORT For further questions or if you suspect a problem with your SA210 please contact Tech Support An Applications Engineer will gladly assist you Page 5 of 5 THORLABS INC Ph 973 579 7227 gt Fax 973 300 3600 SA201 Spectrum Analyzer Controller Operating Manual Related Products ap Model Description SA200 5A 525 650nm Scanning Fabry Perot SA200 6A 650 800nm Scanning Fabry Perot SA200 7A 780 930 nm Scanning Fabry Perot SA200 9A 900 1100nm Scanning Fabry Perot SA200 12A 1250 1400nm Scanning Fabry Perot SA200 14A 1450 1625nm Scanning Fabry Perot 6679 D02 Rev E 08 16 05 Table of Contents WEEE STATEMENT re 3 SECTION 1 0 SPECIFICATIONS 222 can 4 PHYSICAL FEATURES wwwanwnma nanma wanan 4 POWER SUPPLY oo 4 OUTPUT CHARACTERISTICS 2 c ce 4 TRIGGER CHARACTERISTICS 5 PHOTO DIODE AMPLIF
128. n it will decay In this case a mode of the resonator is a slowly decaying field configuration whose relative distribution does not change with time 4 In a laser oscillator the active medium supplies enough energy to overcome the losses so that a steady state field can exist However because of nonlinear gain saturation the me dium will exhibit less gain in those regions where the field is high than in those where the field is low and so the oscillating modes of an active resonator are expected to be somewhat different from the decaying modes of the passive resonator The problem of an active resonator filled with a saturable gain medium has been solved re cently 22 23 and the computed results show that if the gain is not too large the resonator modes are essen tially unperturbed by saturation effects This is fortunate as the results which have been obtained for the passive resonator can also be used to describe the active modes of laser oscillators The problem of the open resonator is a difficult one and a rigorous solution is yet to be found However if certain simplifying assumptions are made the problem becomes tractable and physically meaningful results can be obtained The simplifying assumptions involve essen tially the quasi optic nature of the problem specifically they are 1 that the dimensions of the resonator are large compared to the wavelength and 2 that the field in the resonator is substantially transverse
129. nem anderen netzgespeisten Ger t verwendet wird In diesem Fall baut sich die Masseschleife ber die Netzerde auf wie nachstehend abgebildet Netzerde Die meisten Laptop Netzteile Ladeger te sind erdfrei und haben keinen Massebezug Verursacht jedoch der Anschluss Ihres geerdeten Laptop Netzteils St rpegel Versetzungsfehler k nnen Sie das Oszilloskop entweder verwenden w hrend der Laptop im Akkubetrieb ist oder das Oszilloskop ber den Netzadapter im Lieferumfang mit Strom versorgen Bei Bedarf sollte der Netzadapter in die Buchse hinten am Oszilloskop neben der USB Buchse eingesteckt werden Sie k nnen den Netzadapter im Betrieb ohne weiteres anschlie en und abziehen ohne eine Besch digung des Oszilloskops zu riskieren PS3000049 2 Copyright 2005 7 Pico Technology Limited All rights reserved Index 11 Index R Rechteckwelle 6 A Reparatur API 1 S API Funktionen 1 Serie PicoScope 3000 2 2 6 B Sicherheitshinweis 3 Signalgenerator 6 BNC Anschluss 6 Sonde Spektrumanalysator 1 D Systemanforderungen 6 Datenerfassungsgerat 1 T F Technische Daten 8 Externer Trigger 6 U F USB 1 6 USB 1 1 1 6 USB 2 0 1 6 Firmenanschrift 5 H Hochgeschwindigkeit 1 K Kalibration 2 M Maximaler Eingangsspannungsbereich 2 Messsonde 6 P PC Oszilloskope 1 2 PicoLog Programm 1 PicoScope 3000 Serie 1 PicoScope Software 1 6 Pr fger te 2 Copyright 2005 7 Pic
130. ng Turn the unit on by pressing the power button in the side panel After switching on the unit the graphics display will show the device status and then jump to the last measurement screen before power down The PM100D is immediately ready to use after turning on ORLAES 2 3 Physical Overview Function Keys Navigation VA lt gt Enter Edit OK Wavelength A Relative Measure A Backlight ux Figure 1 Front Panel On Off Switch USB Connector Sensor Connector DB9 female USB ANALOG Af O O S a WA 5 VDC 1A o O SENSOR DC Input Charger Analog Output SMA Figure 2 Side Panel Mounting Thread 1 4 20 SD Card Slot Figure 3 Bottom View ALAS Pull here to lift the support Removable protective Rubber boot Figure 4 Rear View Header Line with Sensor Information Date Time and Battery state N Status Line with 03 12 2008 10 20 47 CU ad Anime or REMOTE 1 E 8 9 g rouan Configurable Configurable a UW BEN A Sub Display Sub Display Hin 130 8 Max 171 5 Orts BETE L Bar Graph e Enter Measurement Range Menu OK Menu Soft Lts faster Buttons Figure 5 Numeric Display Setup Tool Tip Text ALAS 3 Operating the PM100D 3 1 Connecting a Power or Energy Sensor The PM100D supports all Thorlabs C Series photodiode thermal and pyroelectric sensors These can easily identified against older versions of Thorlabs power or energy sensors by th
131. ng am Produkt und oder angeschlossenen Ger ten f hren kann Copyright 2005 7 Pico Technology Limited All rights reserved PS3000049 2 Series PicoScope 3000 Handbuch 1 3 1 4 Sicherheitshinweise Wir empfehlen dringend vor dem ersten Verwenden des Oszilloskops die allgemeinen Sicherheitsinformationen zu lesen Die in das Ger t eingebauten Schutzvorrichtungen k nnen au er Kraft gesetzt werden wenn das Ger t nicht richtig benutzt wird Dies kann zu einem Schaden am Computer oder zu Verletzungen bei Ihnen oder anderen f hren Maximaler Eingangsspannungsbereich Oszilloskope aus der PicoScope 3000 Palette sind auf die Messung von Spannungen im Bereich von 20 V bis 20 V ausgelegt Spannungen ber 100 V k nnen physische Sch den verursachen Netzspannung Produkte von Pico Technology sind nicht f r den Gebrauch mit Netzspannung ausgelegt Verwenden Sie zum Messen von Netzspannung eine isolierende Differenzsonde die speziell f r hohe Quellspannungen ausgelegt ist Sicherheitserdung Oszilloskope aus der Serie PicoSope 3000 werden ber das mitgelieferte Kabel direkt mit der Masse des Computers verbunden Dadurch werden Interferenzst rungen minimiert Wie bei den meisten Oszilloskopen sollte es vermieden werden den Erdungsanschluss mit etwas anderem als Erde oder Masse zu verbinden Verifizieren Sie im Zweifelsfall mit einem Multimeter dass keine bedeutende Gleich oder Wechselspannung anliegt Diese
132. ng window without affecting the calibration of the cavity Sweep Expansion Control 3 The sweep expansion provides a zoom capability to increase the spectral display resolution by a factor of 1x 2x 5x 10x 20x 50x and 100x This is achieved by scaling the ramp rise time be the sweep expansion Waveform Control 4 The SA201 allows the user to select between a sawtooth and triangular waveform The sawtooth waveform is desirable for most applications however the triangle waveform is useful for cavity alignment The SA201 will default to the sawtooth waveform during the system power up To change the waveform simply press the WAVEFORM SEL button The selected waveform is indicated by the illuminated symbol to the right of the waveform select button Power Switch 5 The power switch is used to toggle the unit on and off Power On Indicator 6 The power on LED will light when the unit is powered up 6679 D02 Rev E r 08 16 05 Section 4 0 Descriptions continued Amplitude Control 7 The amplitude control allows the user to adjust the ramp amplitude from 1 to 30V peak to peak using a 10 turn trimpot Note the ramp signal is added to the DC offset This means that when the offset is set to OV the ramp will start a OV and increase to the amplitude limit setting The amplitude is used to determine how far the mir ror will be scanned or to set the spectral range of the optical head Risetime Control 8 The risetime
133. nly called spec trum analyzers when used for laser mode analysis Because mirror alignment is not critical they are easier to temperature stabilize than other high resolution Fabry Perot interferometers Thus they are often used as passive reference cavities to stabilize laser frequency and to calibrate frequency of tunable lasers The high tendue also offers an advantage for high resolution spectroscopy of diffused light sources 50 Large frame Fabry Perot interferometers are useful in spectroscopy and other research where the highest performance is desired A large frame design such as that shown in Figure 2 combines a rigid thermally stable mechanical structure with a full mirror spacing adjust ment and mirror alignment capability not found in small instruments The greatest advantage of the large frame Fabry Perotisthelight gathering power or tendue providedbythe large mirror aperture Large mirrors however are only feasible if they can be supported without distortion so that the instrument finesse is not degraded by reduced surface flatness A technique developed at Burleigh Instruments uses three Invar tabs cemented to the rim of each mirror Figure 5 Glass balls cemented in these tabs align with three V pads in the mirror cell and are secured by a spring ring that circles the mirror substrate This kinematic suspension allows the mirror to self locate so that any stress or tortional force produced by the spring ring is re
134. nnen Der integrierte Signalgenerator kann ber das Programm PicoScope oder ber API Aufrufe gesteuert werden Der Signalgenerator kann auch verwendet werden um bei eingestellter Ausgabe von Rechteckwellen die x10 Messsonden zu kompensieren PS3000049 2 Copyright 2005 7 Pico Technology Limited All rights reserved Produktinformationen 7 Anschlussdiagramm 1 A 3204 E 3205 ie 8 8 0 7 3206 Cole y 3224 ws y y u 1 USB Anschluss 2 LED Zeigt durch Aufleuchten an dass das Oszilloskop der Serie PicoScope 3000 Daten abtastet 3 12 V DC 500 mA Eingang A D Eingangskan le A D E Externer Trigger Signalgenerator Masseschleifen Bei starkem Rauschen oder Fehlspannungen bei Verwendung der Varianten PicoScope 3204 5 6 k nnte ein Masseschleifenfehler vorliegen Zu Ratschl gen zur Beseitigung dieses Problems siehe bitte Stromversorgung des 3204 5 6 Verlegung Ihres PicoScope Oszilloskops an einen anderen USB Anschluss Wenn Sie das Oszilloskop der PicoScope Serie 3000 installieren indem sie es in einen USB Anschluss stecken verkn pft Windows die Pico Treibersoftware mit diesem Anschluss Wenn Sie nun das Oszilloskop sp ter an einen anderen USB Anschluss anschlie en zeigt Windows erneut den Hardware Assistent an Folgen Sie in diesem Fall einfach den Schritten in der Kurzanleitung unter der Anweisung Schlie en Sie das PC gest tzte Oszilloskop der Serie PicoScope 3000 an den PC a
135. nued MANUFACTURED BY Model SA201 WWW THORLABS COM O MAX POWER 15VA 115 230VAC 50 60Hz PD AMPLIFIER INPUT OUTPUT O O Soo operating manua E Figure 5 SA201 Rear Panel Ground Plug 11 This ground plug is for use as a general purpose ground connection It is connected directly to the earth ground connection of the input power plug AC Input Connector 12 This is the line voltage input connection IMPORTANT The unit is configured for 100 115VAC 50 60Hz from the factory To operate at 230VAC see Section 5 PD Amplifier Input BNC 13 This input BNC is used to interface the photodetector provided with the SA200 scanning heads to the amplifier circuit The photodiode amplifier is configured to operate with the Thorlabs supplied photo detec tors however it is possible to operate user supplied photo detectors To do so the BNC center contact must be connected to the photo detector cathode and the BNC shell must be connected to the photodiode anode unbiased operation If a biased detector is to be used the BNC shell must be connected to the bias ground and the bias voltage must be negative for the circuit to operate properly PD Amplifier Output BNC 14 This BNC is the amplifier output and may be connected directly to an oscilloscope to view the cavity spectrum The amplifier gain will be set using the front panel DETECTOR control knob The amplifier output includes a 500 series resistor to minimize no
136. o Technology Limited All rights reserved PS3000049 2 12 Series PicoScope 3000 Handbuch PS3000049 2 Copyright 2005 7 Pico Technology Limited All rights reserved Pico Technology Ltd The Mill House Cambridge Street St Neots PE19 1QB Gro britannien Tel 44 0 1480 396 395 Fax 44 0 1480 396 296 Web www picotech com PS3000049 2 24 4 07 Copyright 2005 Pico Technology Limited All rights reserved
137. ode is resonant is given by held ar 2 1 m n F 42 Thus an FPS with r 10 cm and a finesse of 100 will begin to suffer a loss in effective finesse for m n 100 when je becomes greater than about 15 u Thus the variation in the mirror separation which occurs during direct scanning less than a wavelength is too small to affect the transverse mode degeneracy Of course if pressure scanning is employed it is the wave length of the light which is changed not the mirror separation Ill Experimental Work with the FPS This section is concerned with the practical aspects of the FPS including its design and fabrication align ment procedures and various modes of operation A number of applications are illustrated in the latter parts of this section A General Design Considerations There are three separate aspects of the design problem 1 the mechanical design 2 the attain ment of high finesse and 3 the optical layout Each of these is briefly discussed in the following paragraphs 1 Mechanical Design The key mechanical requirements are that the two mirrors be accurately and rigidly fixed relative to each other that there be a provision for making fine adjustments to the mirror separation either during or after fabrication that the optical separation of the mirrors be insensitive to temperature and or pressure variation and that the interferometer assembly be mechanically isolated from vibration and ac
138. om 42 October 1966 Vol 5 No 10 APPLIED OPTICS 1555 777 PHASE FRONTS a t Fig 9 Distances and parameters for a beam transformed by a thin lens Agar Bs Cq D This is a generalized form of 42 and has been called the ABCD law 10 The matrices of several optical structures are given in Section Il The ABCD law follows from the analogy between the laws for laser beams and the laws obeyed by the spherical waves in geometrical optics The radius of the spherical waves R obeys laws of the same form as 16 and 41 for the complex beam parameter q A more detailed discussion of this analogy is given in 11 q2 43 3 5 Laser Resonators Infinite Aperture The most commonly used laser resonators are com posed of two spherical or flat mirrors facing each other The stability of such open resonators has been discussed in Section 2 in terms of paraxial rays To study the modes of laser resonators one has to take account of their wave nature and this is done here by studying wave beams of the kind discussed above as they propagate back and forth between the mirrors As aperture diffraction effects are neglected throughout this section the present discussion applies only to stable resonators with mirror apertures that are large compared to the spot size of the beams A mode of a resonator is defined as a self consistent field configuragion If a mode can be represented by a wave beam propagatin
139. omputed curves for the instrumental bandwidth spectral resolving power peak transmitted power and the product peak transmitted power X resolving power all plotted as a function of aperture radius for finesses of 30 100 and 300 These curves show that the resolution drops to about 0 7R when the aperture is opened up to a value of ps M F Figure 8d however shows that the value of p which maximizes the product of the spectral resolving power and power transmitted is approximately 0 8p In practice the most convenient way to quickly attain a useful compromise between resolution and transmitted power is to start with a relatively large aperture and while observing the scanned spectrum of a narrow band source to reduce the aperture size until the transmitted power at the peak of a displayed spectral line is reduced by between 20 and 30 For this technique to be useful of course the mirror spacing must be very nearly confocal This can be accomplished without much difficulty as deseribed in Sec III 38 Comparison of FPS and FPP As Connes has pointed out it is a unique characteris tic of the FPS amongst all other types of spectrometers that as the resolving power at constant finesse is in creased by increasing the mirror radii and separation so also is the tendue constant k A U res 0 7 2F rAY 30 In writing this expression we accept the 20 to 30 loss in resolution which accompanies the reali
140. on can easily be obtained using a well made threaded mount for one mirror cell For example we found that a 12 7 mm diam cell with 16 threads cm could be manually ad justed with a precision of at least a tenth of a wave length providing that the mirror separation could be monitored by observing the spectrum from a stable gas laser Most gas lasers are far less stable both mechanically and thermally than the interferometers described here We were pleasantly surprised to find that the same precision of motion could also be obtained with a relatively loose serew fit when the slop was taken up with a thin Teflon tape commercially available as a pipe dope 2 Attainment of High Finesse The attainment of high finesse requires that the mirrors be of excellent optical quality and that they be May 1968 Vol 7 No 5 APPLIED OPTICS 959 Table Il Characteristics of Some Multilayer Reflective Coatings LS Region of high Type reflectivity F To R nenn a ur na CPE 2 A 6200 to 7000 225 0 45 0 993 B 4800 A to 6900 A 150 0 35 0 99 C 4900 to6800A 180 200 0 02 0 992 al coated with high reflectivity multilayer dielectric films For scanning applications only the central spot of radius p need satisfy these requirements If the interferometer is used to observe fringes however the mirror figure should be good over a somewhat larger area to ensure that the fringes will be circular Mirror blanks should be tested against the
141. or Updates oder Ersatzlieferungen die Ihnen zugesandt werden zu berechnen Warenzeichen Windows und Exel sind Schutzmarken der Microsoft Corporation Pico Technology Limited DrDAQ und PicoScope sind international registrierte Handelsnamen PS3000049 2 Copyright 2005 7 Pico Technology Limited All rights reserved Einf hrung 5 1 9 Firmenanschrift Adresse Pico Technology Limited The Mill House Cambridge Street St Neots Cambridgeshire PE19 1QB Gro britannien Telefon 44 1480 396395 Fax 44 1480 396296 E Mail Technischer Support support picotech com Vertrieb sales picotech com Website www picotech com Copyright 2005 7 Pico Technology Limited All rights reserved PS3000049 2 Series PicoScope 3000 Handbuch 2 2 1 2 2 Produktintormationen Systemanforderungen F r den Betrieb von Oszilloskopen der Serie PicoScope 3000 ist ein Computer erforderlich auf dem Windows oder eines der folgenden Betriebssysteme ausgef hrt wird es gilt die jeweils h here Anforderung Prozessor Mindestanforderung Pentium Prozessor oder vergleichbare Leistungsklasse Festplatte 10 MB Minimum Betriebssystem Microsoft Windows XP SP2 oder Vista Mindestens kompatibel zu USB 1 1 Kompatibilit t zu USB 2 0 wird empfohlen Muss direkt mit dem Port oder einem Aktiv HUB erbunden werden Funktioniert nicht mit einem Passiv Hub Installationsanleitung Achtung Verbinden
142. oustic pickup The requirements for rigidity and freedom from vibration dictate that the mirrors be held in a common structure rather than mounted for example on a lens bench This in turn means that any trans ducer used for varying the mechanical separation of the mirrors must be an integral part of the interferometer Mechanical isolation of the rigid interferometer as sembly is readily accomplished by mounting it in an outer case using a soft suspension in the specific de signs described later in this section the outer case could be sharply struck without producing a detectable change in the observed spectrum indicating a sta bility in the length of the optical cavity on the order of a one hundredth of a wavelength or better Insensitivity to pressure variations can be accom plished only by sealing the container holding the inter ferometer This must be done in any event 1f the interferometer is to be pressure scanned Insensitivity to temperature variations can be achieved both by con ventional compensation in which the expansions of dissimilar materials compensate for one another and by the use of very low expansion materials In an interferometer intended to serve as a passive frequency standard we have combined both methods As long as this interferometer is at a uniform temperature 1ts length can be maintained to within one part in 10 over a range of a few degrees centigrade The fine adjustment of mirror separati
143. p generator that adjust the ratio of ramp voltages applied to the separate piezoelectric elements Piezoelectric materials do not extend perfectly lin early with applied voltage One way to linearize the motion is to produce a nonlinear voltage ramp that counteracts the piezoelectric nonlinearity In Burleigh ramp generators this feature improves scan linearity tenfold Fabry Perot Systems The Fabry Perot interferometer is actually an opti cal filter passing some frequencies and rejecting others When tuned to transmit one frequency of light the greatest rejection occurs for frequencies that are dis placed by one haif of the free spectral range The ratio of maximum transmission to maximum rejection contrast is related to the finesse as shown by the transmission profiles in Figure 3 A reflectivity of 93 will typically produce a finesse of 40 and a contrast of 600 For some applications a much higher contrast ratio or larger free spectral range is necessary For this purpose combinations of interferometers that constitute Fabry Perot systems have been devised Just as with other identical filters when two or more Fabry Perot interferometers are placed in series the transmission functions multiply to improve both resolu tion and contrast In practice it is much easier more stable and less expensive to pass the light beam through different sections of the same interferometer several times A simple three pass configuration is s
144. pectral range of the prefilter interferometer Another way to eliminate overlapping spectral orders is to increase the free spectral range This is possible without reducing the resolving power by using two interferometers in tandem with slightly different a io Retroreflector Near Corner Cube Far Corner Cube Fabry Perot Plate Aperture m in past near corner cube to far corner cube 2 4 Beam from far corner cube to near corner cu Beam from near corner cube to far corner cube Beam out of near corner cube Figure 7 Retroreflector design tor a five pass interferometer mirror spacings When each is tuned to transmit a particular frequency the effective free spectral range of the pair is increased because adjacent orders of one interferometer do not coincide in frequency with those of the other The major difficulty in using such a tandem system is maintaining the two interferometers frequency locked to each other and scanning both synchronously A special mechanical design allows two interferometers to be scanned simultaneously by mounting one mirror of each on a common piezoelectrically driven mount The interferometers are set at an angle related to the ratio of the interferometer mirror spacings to achieve synchro nous frequency scanning Two separate Fabry Perot interferometers of different mirror spacing can also be operated in synchronous fashion using electronic stabili zation circuitry that couples them by
145. perating in a single mode TEM o and was obtained with a 10 cm FPS in exact confocal adjustment The anomalously large recorded linewidth gt 200 MHz is due to a continuous frequency drift during the evolution of the 10 nsec Q switched laser pulse The origin of the frequency drift has not been clearly established but it is power dependent and approaches zero near threshold for the Q switched ruby laser In many instances it is desirable to make a precise determination of the separation of two spectral lines or of the width of a single line from a photograph of an FPS fringe pattern This requires an exact knowledge of both the magnification M of the fringe pattern in the photograph and the departure from confocal separation e If the two spectral lines in question give rise to fringes of radii p and pz in the photograph of the fringe pattern both in the same free spectral range then it is straightforward to show that their frequencies differ by Av ve 11 5 4r pot pit M4 4re p pi2 M 44 where 7 is the mean optical frequency By the same token if a single spectral component gives rise to fringes of radii pi pa and pz in adjacent free spectral ranges corresponding to spectral lines of known frequency dif ference c 4r then the equation above can be used to solve for both M and e in terms of pi pz ps and r the mirror radius IV Summary We have found the spherical Fabry Perot interfer ome
146. prescribed properties This trans formation is usually accomplished with a thin lens but other more complex optical systems can be used Although the present discussion is devoted to the simple case of the thin lens it is also applicable to more complex systems provided one measures the distances from the principal planes and uses the combined focal length f of the more complex system The location of the waists of the two beams to be transformed into each other and the beam diameters at the waists are usually known or can be computed To match the beams one has to choose a lens of a focal length October 1966 Vol 5 No 10 APPLIED OPTICS 1557 f that is larger than a characteristic length fo defined by the two beams and one has to adjust the distances be tween the lens and the two beam waists according to rules derived below In Fig 9 the two beam waists are assumed to be located at distances d and dz from the lens The complex beam parameters at the waists are purely imaginary they are q Ja wa YA 59 where 2w and 2ws are the diameters of the two beams at their waists If one inserts these expressions for q and q into 42 and equates the imaginary parts one obtains qi jrw ZA FO 0 Equating the real parts results in d de f f fo 61 where fo rw ws X 62 Note that the characteristic length fy is defined by the waist diameters of the beams to be matched Except for the term fo
147. r s light gathering power 956 APPLIED OPTICS Vol 7 No 5 May 1968 2 Transmission and Etendue of a Spherical Fabry Perot Interferometer In Sec II A it was pointed out that a single beam of light incident on an FPS gives rise to two transmitted beams which are generally at a small angle to one another When both of these transmitted beams are taken into account the net transmission 7 at the center of the instrument profile is found from Eq 4a and 4b To 1 RYXT 1 22 3 7 01 R for R 1 27 When the two transmitted beams are precisely aligned the situation is somewhat different as discussed in Sec II D If we define A to be the sum of the absorption and scattering at the mirrors then 1 R T 4 so that the peak transmission may be written as To HL A T 28 This function is plotted in Fig 7 which clearly illus trates the drastic loss in net transmission whenever the absorption plus scattering losses become comparable with or exceed the transmission loss at the mirror As a rule very high reflectivities can be attained only at the expense of increased values of A T so that it is often necessary in practice to make a compromise be tween finesse and transmission This type of compro mise is discussed further in Sec III In the last section we found that the ultimate instru mental resolution which we now call Ro could be ob tained only with an infinitesimal
148. r tance it is discussed here in greater detail For convenience one introduces two real beam param eters R and w related to the complex parameter q by 1 1 A gt j 17 I E I 17 Fig 5 Amplitude distribution of the fundamental beam When 17 is inserted in 12 the physical meaning of these two parameters becomes clear One sees that R z is the radius of curvature of the wavefront that intersects the axis at z and u z is a measure of the decrease of the field amplitude E with the distance from the axis This decrease is Gaussian in form as indicated in Fig 5 and w is the distance at which the amplitude is 1 e times that on the axis Note that the intensity distribution is Gaus sian in every beam cross section and that the width of that Gaussian intensity profile changes along the axis The parameter w is often called the beam radius or spot size and 2w the beam diameter The Gaussian beam contracts to a minimum diameter 2wo at the beam waist where the phase front is plane If one measures z from this waist the expansion laws for the beam assume a simple form The complex beam parameter at the waist is purely imaginary TWO 18 oe er 18 and a distance z away from the waist the parameter is TW q qQqtz j N 19 2 After combining 19 and 17 one equates the real and imaginary parts to obtain Az wi weli 5 20 TWO TW 0 2 R 2 2 1 21 AZ
149. r modes of operation or to image the interference fringe pattern at infinity when light is incident on the other end of the instrument visual or photographic modes of operation In the scanning mode a small photode teetor is mounted directly behind the FPS We have used both a silicon photodiode and a photomultiplier as the detector the majority of the scans shown in Sec III E were obtained with a photodiode operating in the photovoltaic mode Since it is desirable for ease in alignment to rotate the entire instrument about the entrance aperture the mounting flange can be attached at either end of the instrument This flange then serves to attach the instrument to a mirror mount which is adjustable in angle The mirror radius in the FPS is 5 cm with a cor responding free spectral range of 1500 MHz Using Note added in proof By replacing the 5 cm radius mirrors with 1 em mirrors and modifying one mirror cell to provide for a l cm mirror separation we were recently able to extend the free spectral range of this instrument to 7 5 GHz or 0 09 A at 6000 A Using narrowband mirrors a finesse in excess of 100 was easily attained with a 0 5 mm detector aperture This type of instrument has proved useful in examining ion laser spectra Comparable finesse in an FPP with the same free spectral range is very difficult to obtain 962 APPLIED OPTICS Vol 7 No 5 May 1968 multilayer dielectric mirrors we can obtain a finesse of
150. r pulse detection x xxmm Set the input beam diameter for power or energy density calculation ZERO Performs a zeroing for thermal sensors and dark current adjustment for photodiode sensors 3 2 2 5 Max Reset Button Sets back the Min Max and Max Min displays 3 2 2 6 Tuning Sound Switches on and off an audible tone for laser tuning support e LABS 3 2 2 Relative Measurements A Switches on and off the relative measurement mode The main display will set to zero the offset and the absolute power or energy value will be displayed in the sub displays The bar graph and needle display will change to a measurement range from 10 to 10 of the set range 3 2 3 Display Options Select Measurement Representation OK o poo fp er 3 2 3 1 Needle Display Max value indicator 33 151 Reset Min and Max Yalues OK Rog AUTO p Meas Config Meas View Me RESET 3 2 3 2 Data Logging Screens 03 12 2008 10 24 48 C9 Y 5144C 03 12 2008 21 11 06 GD ACT Value 1695 pW 471 0BM 70m Min Value 33 16 UW 10 31 dBm Max Value 169 6 pW 7 71 dem Mean Value 162 7 PW 7 89 dem std Deviation 15 02 UW Ratio Max Min 1 820 1 2 60 dB Sample No 100 Time 0 00 29 o a za Start Logging To File OK Stop Logging To File OK l Fie soss Tue Graph Heas View Fie a058 Statstes aen System new OO MEN 1 After pressing the START button the data that are sampled in these screens will
151. ransfer over a distance d No 2 de scribes the transfer of rays through a thin lens of focal length f Here the input and output planes are immediately to the left and right of the lens No 3 is a combination of the first two It governs rays passing first over a dis tance d and then through a thin lens If the sequence is reversed the diagonal elements are interchanged The matrix of No 4 describes the rays passing through two structures of the No 3 type It is obtained by matrix multiplication The ray transfer matrix for a lenslike medium of length d is given in No 5 In this medium the refractive index varies quadratically with the distance r from the optic axis n No Ngr 4 An index variation of this kind can occur in laser crystals and in gas lenses The matrix of a dielectric material of index n and length d is given in No 6 The matrix is referred to the surrounding medium of index 1 and is computed by means of Snell s law Comparison with No 1 shows that for paraxial rays the effective distance is shortened by the optically denser material while as is well known the optical distance is lengthened 2 2 Periodic Sequences Light rays that bounce back and forth between the spherical mirrors of a laser resonator experience a periodic focusing action The effect on the rays is the same as in a periodic sequence of lenses 15 which can be used as an optical transmission line A periodic sequence of identic
152. ree spectral range FSR of the instrument From a users perspective a confocal cavity has a FSR that is given by c 4d instead of c 2d as would be the case for a plano plano cavity the factor of 2 in the denominator can be understood by inspecting the ray trace shown below in Figure Il Note that a ray entering the cavity at a height h parallel to the optical axis of the cavity makes a triangular figure eight pattern as it traverses the cavity From this pattern it is clear that the ray makes four reflections from the cavity mirrors instead of the two that would result in a plano plano cavity Hence the total round trip path through the cavity is given as 4d instead of 2d a ern ha gt ite A ani A ss HA ee ee u Bw Figure II Figure Il This figure shows a simplified ray trace for a ray entering the cavity at height h The curvature of the mirrors R and the separation being set precisely to R ensures that the input ray is imaged back onto itself after traveling a distance of approximately AR Additionally in this configuration if a paraxial ray is traced through the system as shown in figure Il it is apparent that in the confocal configuration each mirror serves to image the other mirror back onto itself so that a ray entering the cavity will after four traverses of the cavity fall back onto itself note that the focal length of a spherical mirror is R 2 This imaging of the
153. respectively In the above equations the sub scripts and superscripts one and two denote mirrors one and two s and ss are symbolic notations for transverse coordinates on the mirror surface e g s X yi and so xo Yo OF s 11 Pa and so 12 da ES and E are the relative field distribution functions over the mir rors y and y give the attenuation and phase shift suffered by the wave in transit from one mirror to the other the kernels KW and K are functions of the dis tance between s and s and therefore depend on the mirror geometry they are equal KW s s K As So but in general are not symmetric KW s s1 K s1 52 K As s AKY S9 s The integral equations given by 72 express the field at each mirror in terms of the reflected field at the other that is they are single transit equations By substituting one into the other one obtains the double transit or round trip equations which state that the field at each mirror must reproduce itself after a round trip Since the kernel for each of the double transit equations is sym metric 24 it follows 25 that the field distribution functions corresponding to the different mode orders are orthogonal over their respective mirror surfaces that is Em s En s dS 0 mn S1 J where m and n denote different mode orders It is to be noted that the orthogonality relation is non Hermitian and is the one that is generally applicable to loss
154. roc IRE Correspondence vol 48 pp 1904 1905 November 1960 Resonant modes in a maser interferometer Bell Sys Tech J vol 40 pp 453 488 March 1961 5 G D Boyd and J P Gordon Confocal multimode resonator for millimeter through optical wavelength masers Bell Sys Tech J vol 40 pp 489 508 March 1961 6 G D Boyd and H Kogelnik Generalized confocal resonator theory Bell Sys Tech J vol 41 pp 1347 1369 July 1962 1566 APPLIED OPTICS Vol 5 No 10 October 1966 7 G Goubau and F Schwering On the guided propagation of electromagnetic wave beams IRE Trans on Antennas and Propagation vol AP 9 pp 248 256 May 1961 8 J R Pierce Modes in sequences of lenses Proc Nat Acad Sci vol 47 pp 1808 1813 November 1961 9 G Goubau Optical relations for coherent wave beams in Electromagnetic Theory and Antennas New York Macmillan 1963 pp 907 918 10 H Kogelnik Imaging of optical mode Resonators with internal lenses Bell Sys Tech J vol 44 pp 455 494 March 1965 11 On the propagation of Gaussian beams of light through lenslike media including those with a loss or gain variation Appl Opt vol 4 pp 1562 1569 December 1965 12 A E Siegman Unstable optical resonators for laser applica tions Proc IEEE vol 53 pp 277 287 March 1965 13 W Brower Matrix Methods in Optical Instrument Desig
155. ropriate line cord and turn the unit on Cleaning The SA201 should only be cleaned with a soft cloth and a mild soap detergent or isopropyl alcohol Do not use a solvent based cleaner Technical Support You may use any of the following methods to contact Thorlabs in case of difficulty or if you have questions regarding the SA201 www thorlabs com Thorlabs web site will have up to date application notes and frequently asked questions regarding our products TechsupportOthorlabs com Send a detailed email message and one of our application engineers will respond promptly within 1 business day Mail Thorlabs Inc 435 Route 206N Newton NJ 07860 Phone 973 579 7227 Fax 973 300 3600 Appendix A Recommended Setup OSCILLOSCOPE SA200 SERIES FABRY PEROT SA201 ANALYZER FRONT PANEL 6679 D02 Rev E 11 08 16 05 Spec Sheet Or lt S120C Compact Photodiode Power Head with Silicon Detector The S120C power head is designed for general purpose optical power measurements The head is optimized for small thickness to fit in tight spaces The high sensitive photodiode with large active area in combination with an absorptive ND filter enables power measurements up to 50 mW in free space and fiber based applications A removable annular IR viewing target allows conveniently centering the measured beam to the active area of the photo diode The target absorbs light from 400 to 640nm and 800 to 1700nm The S7120
156. ry Perot resonator with parallel plane circular mirrors N 10 4 7 Diffraction Losses and Phase Shifts The diffraction loss a and the phase shift 8 for a par ticular mode are important quantities in that they deter mine the O and the resonant frequency of the resonator for that mode The diffraction loss is given by a 1 yl 81 which is the fractional energy lost per transit due to dif fraction effects at the mirrors The phase shift is given by 6 angle of y 82 which is the phase shift suffered or enjoyed by the wave in transit from one mirror to the other in addition to the geometrical phase shift which is given by 2rd The eigenvalue y in 81 and 82 is the appropriate y for the mode under consideration If the total resonator loss is small the Q of the resonator can be approximated by 2rd En 83 AQ t where the total resonator loss includes losses due to diffraction output coupling absorption scattering and other effects The resonant frequency is given by v vo q 1 8 7 84 where q the longitudinal mode order and vo the funda mental beat frequency are defined in Section 3 5 TEMoo MODE POWER LOSS dB i r iJ 0 4 0 6 1 0 2 4 6 10 20 40 60 100 N a nd Fig 22 Diffraction loss per transit in decibels for the TEMoo mode of a stable resonator with circular mirrors 0 1 0 2 2 TEMo MODE o o o POWER LOSS dB e ND 40 60 100 0 4 06 1 0 2
157. s Emphasis is placed on formulations and derivations which lead to basic understanding and on results which are of practical value Historically the subject of laser resonators had its origin when Dicke 1 Prokhorov 2 and Schawlow and Townes 3 independently proposed to use the Fabry Perot interferometer as a laser resonator The modes in such a structure as determined by diffraction effects were first calculated by Fox and Li 4 Boyd and Gordon 5 and Boyd and Kogelnik 6 developed a theory for resonators with spherical mirrors and approximated the modes by wave beams The concept of electromagnetic wave beams was also introduced by Goubau and Schwe ring 7 who investigated the properties of sequences of lenses for the guided transmission of electromagnetic waves Another treatment of wave beams was given by Pierce 8 The behavior of Gaussian laser beams as they interact with various optical structures has been analyzed by Goubau 9 Kogelnik 10 11 and others The present paper summarizes the various theories and is divided into three parts The first part treats the passage of paraxial rays through optical structures and is based on geometrical optics The second part is an analysis of laser beams and resonators taking into account the wave nature of the beams but ignoring diffraction effects due to the finite size of the apertures The third part treats the resonator modes taking into account aperture diffrac
158. s a path such as that shown in Fig 2 b Even though it is not reentrant if the incident ray is not at too great an angle to the axis it will continue to intersect itself in the vicinity of a point P located in the central plane of the FPS at a distance p from the axis The position of the points at which rays continue to intersect themselves determines the position of the fringe pattern If the axial mirror spacing is precisely 7 the mirror radius it is straightforward to show that the four transit path i e the path taken between successive intersections at the point P exceeds the paraxial path 4r by an amount Ao pi2p2 cos20 1 higher order terms 1 952 APPLIED OPTICS Vol 7 No 5 May 1968 More generally if the mirror spacing is r e the four transit ray path exceeds the corresponding paraxial ray path 4 r e by an amount A p p cos20 r 2elp p2 7 higher order terms 2 If we now restriet our attention to a small and distant source close to the axis of the interferometer we may write for Eq 2 Alp p 1 4ep r 3 where p is the height at which an entering ray crosses the central plane of the FPS Refering to Fig 2 b we see that for each entering ray there are two sets of trans mitted rays those which have been reflected 4m times type 1 and those which have been reflected 4m 2 times type 2 where mis an integer The interference Fig 1 General ray path in a
159. s well as at the confocal spacing As shown in Fig 13 a when the mirrors are spaced by slightly less than con focal spacing there is a zone radius 2er of high dispersion toward which all fringes gravitate as the mirror spacing is slightly reduced or equivalently as the wavelength of the quasi monochromatie light source is increased This zone approaches the center of the fringe pattern as the mirror separation is in creased and disappears through the center of the pat tern when the confocal spacing is exceeded The loca tion of this zone is an extremely sensitive indication of the mirror separation and when the mirrors are exactly confocal it is at the center of the fringe pattern The radius of this zone as a function of mirror separation is shown by the dotted line in Fig 3 If the FPS is set up in the scanning mode it is possible to peak up the adjustment of the mirror separation merely by maximizing the amplitude of a displayed laser spectrum while at the same time minimizing the apparent width of individual spectral components With very little practice it becomes a simple matter to establish confocal separation to within a mieron or so using this technique As the optimum mirror separa tion is approached one should also make fine adjust ments to the angular alignment unless a very small diameter beam is used in which case the spectral dis play is relatively insensitive to alignment 2 Selecting the Optimum
160. same master they generally require final polishing by hand if they are to have the necessary figure that is to say if they are to be spherical to within A F where F is the desired finesse Final evaluation of the mirror figure can only be made after the mirror has been coated and tested as an interferometer component Not only must the mirror figure be excellent but the polished surface must be free of microscopic scattering sites if the ulti mate in reflectivity is to be realized This is within the present state of the art and scattering losses of less than 0 3 are attainable with fused quartz blanks As we suggested in the last section one is usually forced to make some sort of compromise between finesse instrumental transmission and possibly the spectral bandwidth within which the mirrors have high re flectivity For example using commercially available multilayer coatings referred to as types A B and C we were able to obtain the performance summarized in Table II Type A is a narrow band coating covering a rather restricted spectral range but offers high finesse combined with excellent transmission Type B covers a considerably broader portion of the spectrum with fairly high instrumental transmission but has a some what reduced finesse Type C on the other hand has broad band coverage and relatively high finesse but rather low transmission By making fairly drastic sacrifices in instrumental transmission it is probab
161. spherical mirror Fabry Perot interferometer a TYPE TYPE 2 Fig 2 a Ray path in an FPS in the paraxial approximation reentrant rays b aberrated ray path showing intersection of rays at point P NEAR CONFOCAL FABRY PEROT i INN el Fee ZA INN E Cs Be III me gt V CENTRAL FRINGE WIDTH FOR FINESSE OF 100 FRINGE RADIUS p cm x N Tee wm en 04 02 02 04 06 08 0 1 DEPARTURE FROM CONFOCAL POSITION cm Fig 3 Near confocal FPS fringe patterns At each value of the solid curves give the radii of the circular interference fringes for the case of a monochromatic source and a bright fringe on axis The dashed line shows the spot size radius ps for a finesse of 100 and the dotted line defines the zone of best focus as a function of e Appendix I shows how to change the scales for different wavelengths and mirror separations patterns produced in the central plane of the inter ferometer are described by Type 1 Lo IlT 1 RI AI 2R 1 R2 x sin 0 1 2 1 4a or Type 2 l2 9 A RICA 4b where 5 p A 2r M A 0 4 r e l 5 See Table 1 for a list of symbols The derivation of these equations exactly follows the usual derivation for a plane mirror Fabry Perot interferometer FPP When the mirror reflectivity R is close to unity the interference patterns for both types of rays are the same and are
162. tained by solving the integral equations numerically have a ripply behavior which is attributable to diffraction effects 24 43 How ever the average values agree well with those obtained from 90 90 5 CONCLUDING REMARKS Space limitations made it necessary to concentrate the discussion of this article on the basic aspects of laser beams and resonators It was not possible to include such interesting topics as perturbations of resonators resona tors with tilted mirrors or to consider in detail the effect of nonlinear saturating host media Also omitted was a discussion of various resonator structures other than those formed of spherical mirrors e g resonators with corner cube reflectors resonators with output holes or fiber resonators Another important but omitted field is that of mode selection where much research work is cur rently in progress A brief survey of some of these topics is given in 44 REFERENCES 1 R H Dicke Molecular amplification and generation systems and methods U S Patent 2 851 652 September 9 1958 2 A M Prokhorov Molecular amplifier and generator for sub millimeter waves JETP USSR vol 34 pp 1658 1659 June 1958 Sov Phys JETP vol 7 pp 1140 1141 December 1958 3 A L Schawlow and C H Townes Infrared and optical masers Phys Rev vol 29 pp 1940 1949 December 1958 4 A G Fox and T Li Resonant modes in an optical maser P
163. ted Firmware Upload Needs to be enabled before uploading a new firmware version The function will automatically reset to disabled when powering down Date and Time Enters a submenu to set date and time It is possible to chose various date and time formats Console Info Shows the console related information sensor Info Shows the related information of the currently connected power or energy sensor ORS 3 2 2 Power and Energy Measurement in the Numeric Display The numeric display contains a large configurable measurement value two small sub displays for additional measurement information and a bar graph that shows the saturation degree of the chosen measurement range To control and configure the numeric display the soft buttons in the top level are arranged as following System Menu gt 3 2 2 1 Range Control Set Measurement Range AY Exit OK Roy ass Amor gt ooo fp en Up to 6 power corresponding current and 4 power energy corresponding voltage ranges can be chosen manually with the A or VW keys For power measurements an auto ranging function is available 3 2 2 2 Wavelength Correction Select Wavelength OK Edit Wavelength Hold OK Exit A A 1064 nm a 1310 om PROMESA gt 10600 nm va 1552 m The menu offers 8 individually configurable sensor independent wavelength settings To edit a wavelength keep the OK key pressed for 1 second Set the desired wavelength with the AW
164. ter as a narrow bandpass filter with zero Optical power the lens is simply removed In order to eliminate ghost fringe patterns the rear surfaces Of the FPS mirrors and both surfaces of In should be antireflection coated B Design of a Static or Pressure Scanned FPS This design shown in Fig 11 is intended primarily to eliminate variation of the axial mirror separation with temperature This FPS is thus well suited for use as a secondary frequency standard with a long term stability of better than 1 MHz with temperature con trol of 1 C This thermal stability is obtained by making the mirror spacer of Cer Vit thermal expan sion coefficient no greater than 0 1 X 10 20 and using the Invar mirror cells thermally to compensate for the residual expansion of the Cer Vit this can easily be done since the manufacturers of Cer Vit routinely supply accurate expansion data for each blank The quartz mirrors change radius by about a wavelength for a 1 C temperature change but this in troduces a negligible change in the mirror separation which determines the resonant frequency of the FPS Cer Vit is a low expansion semitransparent glasslike ceramic manufactured by Owens Illinois We have not yet established the lower limit on the frequency stability of this system The interferometer assembly is held inside a pressure chamber by means of phosphor bronze finger stock Springs which provide adequate mechanical isolation Th
165. ter to be an extremely versatile high resolution spectroscopic tool It is particularly well suited be cause of ts high resolution and limited free spectral range to the study of laser and laser derived e g stimulated and spontaneous scattering of laser light light sources It can readily be adapted for pressure or o Axial and 3 Axial modes i tr verse modes Single mode Q Switched ruby Fig 17 Top left spectrum of a He Ne gas laser operating in three axial modes and the TEM transverse mode e 20 u r 5 cm free spectral range 1500 MHz Top right same as above but with an additional THM mode in oscillation Bot tom spectrum of a 10 nsec single mode pulse from a Q switched ruby laser see text e 0 r 10 cm free spectral range 750 MHz May 1968 Vol 7 No 5 APPLIED OPTICS 965 mechanical scanning fringe display or tunable narrow band filtering The important limitations of an FPS are lts relatively narrow free spectral range generally less than a few thousand megahertz and the inability to vary the free spectral range attainable with a given pair of mirrors Some of the advantages of an FPS over other types of optical spectrum analyzers are listed below a Ease in attaining high finesse thereby taking advantage of the high reflectivities now available with commercially available dielectric coatings b Ease of alignment once the initial mirror separa tion is set no further
166. the ray pattern is folded in the resonator and unfolded in the lens sequence The focal lengths f and f of the lenses are the same as the focal lengths of the mirrors i e they are determined by the radii of curvature R and Ra of the mirrors i R 2 f2 R 2 The lens spacings are the same as the mirror spacing d One can choose as an element of the peri odic sequence a spacing followed by one lens plus another spacing followed by the second lens The ABCD matrix of such an element is given in No 4 of Table I From this one can obtain the trace and write the stability condition 7 in the form lt a e RIN R S 8 To show graphically which type of resonator is stable and which is unstable it is useful to plot a stability dia gram on which each resonator type is represented by a point This is shown in Fig 4 where the parameters d R and d Rs are drawn as the coordinate axes unstable systems are represented by points in the shaded areas Various resonator types as characterized by the relative positions of the centers of curvature of the mirrors are indicated in the appropriate regions of the diagram Also entered as alternate coordinate axes are the parameters 21 and g which play an important role in the diffraction theory of resonators see Section 4 1592 APPLIED OPTICS Vol 5 No 10 October 1966 Ri 2f Fig 3 Ra 2f2 Spherical mirror resonator and the equivalent sequence of lenses ILLAT PAR
167. ther small adjust ments to the alignment The scanning voltage is con veniently provided by the sawtooth or horizontal scan voltage from the oscilloscope This voltage will typically scan two or three free spectral ranges and assures synchronism between the applied scanning voltage and the oscilloscope sweep Moreover this technique provides a highly linear display of the optical spectrum at the oscilloscope To observe interference fringes the detector is re moved and the incident beam is directed towards the end of the instrument at which the detector was located The fringe pattern can then be viewed directly through the lens or an auxiliary telescope can be used The latter is particularly useful when the incident beam is not collimated resulting in a displacement of the plane of the fringe pattern away from the focal plane of the lens The proper alignment of the instrument axis relative to the incident beam of light can be made by observing fringe pattern and making angular adjust ments to make the fringes circular As mentioned earlier it may be desirable to change the mirror spacing slightly to obtain a more nearly linear display of frequency in the fringe pattern although this entails a reduction in the realizable spectral resolution Fig 14 D Alignment Procedures In assembling and using FPS interferometers only a few alignments or adjustments are required and in this section procedures for these are briefly describ
168. tices G and H that the absence of the complete metal walls of the dia mond outside the resonator will have very little impor tance This treatment although approximate allowed the author to understand how the resonator actually worked October 1966 Vol 5 No 10 APPLIED OPTICS 1567 435 Route 206 P O Box 366 PH 973 579 7227 Newton NJ 07860 0366 FAX 973 300 3600 www thorlabs com technicalsupport thorlabs com SA200 Series Scanning Fabry Perot Interferometer DESCRIPTION The SA200 is a high finesse Spectrum Analyzer used to examine the fine structures of the spectral characteristics of CW lasers The spectrum analyzer consists of a confocal cavity that contains two high reflectivity mirrors by varying the mirror separation with a piezoelectric transducer the cavity acts as a very narrow band pass filter Knowing the free spectral range of the SA200 allows the time base of an oscilloscope to be calibrated to facilitate quantitative measurements of a laser line shape SPECIFICATIONS Measured in milliseconds Measured in microseconds a HENAO FSR FWHM Actual Calculated Finesse 150V Minimum Finesse gt 200 o S Dimensions 2 Flange Total Length 5 85 FSR is set by the length of the confocal cavity and is given by FSR c 4d Where d the radius of curvature of the mirrors in this case d 50mm see drawing on next page A thermal design balances the small coefficient of thermal expans
169. tion effects Whenever applicable useful results are pre sented in the forms of formulas tables charts and graphs Manuscript received July 12 1966 H Kogelnik is with Bell Telephone Laboratories Inc Murray Hill N J T Li is with Bell Telephone Laboratories Inc Holmdel N J 1550 APPLIED OPTICS Vol 5 No 10 October 1966 2 PARAXIAL RAY ANALYSIS A study of the passage of paraxial rays through optical resonators transmission lines and similar structures can reveal many important properties of these systems One such geometrical property is the stability of the struc ture 6 another is the loss of unstable resonators 12 The propagation of paraxial rays through various optical structures can be described by ray transfer matrices Knowledge of these matrices is particularly useful as they also describe the propagation of Gaussian beams through these structures this will be discussed in Section 3 The present section describes briefly some ray concepts which are useful in understanding laser beams and resonators and lists the ray matrices of several optical systems of interest A more detailed treatment of ray propagation can be found in textbooks 13 and in the literature on laser resonators 14 INPUT OUTPUT PLANE PLANE Fig 1 Reference planes of an optical system A typical ray path is indicated 2 1 Ray Transfer Matrix A paraxial ray in a given cross section z const of an opti
170. tors d mals E 76 e Ra For the case of circular mirrors 4 31 32 the equa tions are reduced to the one dimensional form by using October 1966 Vol 5 No 10 APPLIED OPTICS 1561 cylindrical coordinates and by assuming a sinusoidal azimuthal variation of the field that is E r p R Ne e The radial distribution functions R and R satisfy the one dimensional integral equations a2 WORM r Vr f Kifr re Rr ro Y ra dre 0 al yP RO ra VT f Kim VI dr 77 0 where the kernel K is given by r ra d girs a K fr Ya ne Vr ra jk gir n 78 e Fr erg exp Od g r g and J is a Bessel function of the first kind and th order In 77 as and a are the radii of the mirror apertures and d is the mirror spacing the factors g and go are given by 76 Except for the special case of the confocal resonator 5 g1 g2 0 no exact analytical solution has been found for either 74 or 77 but approximate methods and numerical techniques have been employed with suc cess for their solutions Before presenting results it is appropriate to discuss two important properties which apply in general to resonators with spherical mirrors these are the properties of equivalence and stability 4 4 Equivalent Resonator Systems The equivalence properties 24 33 of spherical mirror resonators are obtained by simple algebraic manip ulations of the
171. treifer Optical resonator modes rectangular reflectors of spherical curvature J Opt Soc Am vol 55 pp 868 877 July 1965 31 T Li Diffraction loss and selection of modes in maser reso nators with circular mirrors Bell Sys Tech J vol 44 pp 917 932 May June 1965 32 J C Heurtley and W Streifer Optical resonator modes circular reflectors of spherical curvature J Opt Soc Am vol 55 pp 1472 1479 November 1965 33 J P Gordon and H Kogelnik Equivalence relations among spherical mirror optical resonators Bell Sys Tech J vol 43 pp 2873 2886 November 1964 34 F Schwering Reiterative wave beams of rectangular sym metry Arch Elect Ubertrag vol 15 pp 555 564 Decem ber 1961 35 A G Fox and T Li to be published 36 L A Vainshtein Open resonators for lasers JETP USSR vol 44 pp 1050 1067 March 1963 Sov Phys JETP vol 17 pp 709 719 September 1963 37 S R Barone Resonances of the Fabry Perot laser J Appl Phys vol 34 pp 831 843 April 1963 38 D Slepian and H O Pollak Prolate spheroidal wave func tions Fourier analysis and uncertainty I Bell Sys Tech J vol 40 pp 43 64 January 1961 39 D Slepian Prolate spheroidal wave functions Fourier anal ysis and uncertainty IV Extensions to many dimensions generalized prolate spheroidal functions Bell Sys Tech J vol 43
172. u ary 1966 21 J P Gordon A circle diagram for optical resonators Bell Sys Tech J vol 43 pp 1826 1827 July 1964 M J Offer haus Geometry of the radiation field for a laser interferom eter Philips Res Rept vol 19 pp 520 523 December 1964 22 H Statz and C L Tang Problem of mode deformation in optical masers J Appl Phys vol 36 pp 1816 1819 June 1965 23 A G Fox and T Li Effect of gain saturation on the oscillat ing modes of optical masers EEE J of Quantum Electronics vol QE 2 p Ixti April 1966 24 Modes in a maser interferometer with curved and tilted mirrors Proc IEEE vol 51 pp 80 89 January 1963 25 F B Hildebrand Methods of Applied Mathematics Englewood Cliffs N J Prentice Hall 1952 pp 412 413 26 D J Newman and S P Morgan Existence of eigenvalues of a class of integral equations arising in laser theory Bell Sys Tech J vol 43 pp 113 126 January 1964 27 J A Cochran The existence of eigenvalues for the integral equations of laser theory Bell Sys Tech J vol 44 pp 77 88 January 1965 28 H Hochstadt On the eigenvalue of a class of integral equa tions arising in laser theory SIAM Rev vol 8 pp 62 65 January 1966 29 D Gloge Calculations of Fabry Perot laser resonators by scattering matrices Arch Elect Ubertrag vol 18 pp 197 203 March 1964 30 W S
173. us of curvature of the mirrors which means that the mirror surfaces are coincident with the phase fronts of the resonator modes The width 2w of the fundamental mode is given by w 4 2 2 1 47 To calculate the beam radius w in the center of the reso nator where the phase front is plane one uses 23 with z d 2 and gets N ae we Vd2R d 48 27 The beam parameters R and w describe the modes of all orders But the phase velocities are different for the different orders so that the resonant conditions depend on the mode numbers Resonance occurs when the phase shift from one mirror to the other is a multiple of r Using 28 and 34 this condition can be written as kd 2 m n 1 arc tan Ad 2rw0 r q 1 49 where q is the number of nodes of the axial standing wave pattern the number of half wavelengths is g 1 and m and n are the rectangular mode numbers defined in Sec tion 3 3 For the modes of circular geometry one obtains a similar condition where 2p 1 1 replaces m n 1 The fundamental beat frequency vo ie the frequency spacing between successive longitudinal resonances is given by vo c 2d 50 This g is not to be confused with the complex beam parameter R 2ef R 2ef ee d ee u y viii veh 1 l 1 it ft N i A ES Me 1 1 Fig 10 Symmetrical laser resonator and the equivalent sequence of lenses The beam parameters m and gs are indicated
174. ve number of interfering beams involved in forming the interference fringes The factors that limit finesse are those which reduce the strength of the interference as the number of reflections increases Important examples are mirror reflectivity less than 100 and lack of parallelism or flatness of the mirror surfaces A separate finesse is associated with each of these factors Lasers amp Applications July 1983 The reflectivity finesse for a plane mirror interfer ometer with mirror reflectivities R is 1 R Typical intensity contours of Fabry Perot fringes for different mirror reflectivities are shown in Figure 3 The flatness finesse is Fe M 2 6 where M is the fractional wavelength deviation from true flatness or parallelism across the mirror aperture Mirror flatness is commonly specified as A M at a standard wavelength of 546 nanometers The net finesse due to flatness and reflectivity is called the instrument finesse F where 1 Ff 1 Fp 1 Fp 7 A plot of F is shown in Figure 4 for mirrors with a spherical error amounting to A 100 and A 200 over their aperture When the Fabry Perot interferometer is used in a spectrometer as shown in Figure 1 the pinhole size determines the degree of collimation of light passing through the interferometer that reaches the detector If the pinhole is too large rays passing through the Fabry Perot at different angles are accepted thus broadening the instrumental linewidth
175. ved during the sawtooth waveform fall time This is calculated by mA Cpiezo UF X AV max Atar 2 The output drive amplifier will current limit the load to 26mA max Although the unit may operate continuously under these conditions it is not recommended since the unit will heat up causing stress to the electronics The risetime adjustment range for each sweep setting is as follows Risetime Adj Range 0 01 x sweep expansion setting to 0 1 x sweep expansion setting Defined as the scaling error between 1X and any other gain settings ex 2X 0 5 Measures with SA200 series scanning head connected to output Ramp refers to the rising or scanning edge of the Output waveform The gain error does not apply when using a 50 load since the user installed output terminator will probably have a resistance tolerance greater than the gain errors above Also note that the 50W output series resistance is 49 9W 1 This will also factor into gain error when using a 50 load 8 Test performed with a 50 terminator and a 6 1 8m 50 coax cable e IA Section 2 0 Overview The SA201 is specifically designed to control Thorlabs SA200 Series Fabry Perot Interferometers The controller generates a voltage ramp which is used to scan the separation between the two cavity mirrors The controller provides adjustment of the ramp voltage and scan time allowing the user to choose the scan range and speed An offset control
176. wesentlich gr er als die spontane Emission werden kann W hrend alle Laser auf diesem Prinzip basieren ist die technische Realisierung der drei Komponenten Resonator Pumpe Medium recht vielf ltig Die Pumpe l sst sich z B durch Blitzlampen Gasentladungen Strom oder auch andere Laser implementieren Aktive Medien reichen von Gasen dotierten Festk rperkristallen Halbleitern bis zu in Fl ssigkeiten gel sten Farbstoffen In diesem Versuch soll das Laserprinzip anhand eines Helium Neon Gaslasers veranschaulicht werden Durch Aufbau und Justage eines Resonators um das akt ve Medium soll zuerst die Laseroszillation erreicht und dann die im folgenden aufgef hrten Aufgaben bearbeitet werden Versuchsdurchf hrung Auswertung WICHTIG Dokumentieren Sie immer alle Messergebnisse der einzelnen Aufgaben Die Messdaten sind am Ende des Versuchs vom Betreuer unterzeichnen zu lassen WICHTIG Die Gasentladung im Lasermedium wird ber eine Hochspannung von mehreren kV gez ndet Ber hren Sie nicht die Anschl sse Das Lasermedium inkl Halterung darf nicht von der Schiene genommen und nur vom Betreuer bewegt werden WICHTIG Die w hrend der Versuchsdurchf hrung aufgenommenen Messwerte sind im Original in die Auswertung einzuf gen Trennen Sie die Auswertung der Messwerte von der Versuchsdurchf hrung Es muss nachvollziehbar sein wie die Auswertungsergebnisse aus den Messdaten erhalten wurden Aufgaben 1 Inbetriebnahme des
177. y and low thermal sensitivity The cavity spacing can be set any where between 0 and 15 centimeters by adjusting the movable mirror mount One mirror can be mechanically aligned parallel to the other with the large alignment screws Once these are set fine alignment and tuning or 48 scanning can be performed by remote control of voltages applied to the piezoelectric assembly that supports the opposite mirror The Meaning of Finesse Before discussing further the criteria for choosing a Fabry Perot interferometer it is necessary to define some important terms A scanning Fabry Perot spectrometer illuminated with monochromatic light transmits a peak in intensity every time the wavelength satisfies Equation 2 The range of wavelengths that can be displayed in the same spectral order m without overlapping adjacent orders is called the free spectral range FSR For a plane mirror Fabry Perot with spacing 4 FSR and 3 If d is in cm FSR is in wavenumbers cm The resolvable bandwidth or instrumental resolution is the full width at half maximum of the spectral profile that would be observed from a perfectly monochromatic source It is defined arbitrarily as Ay FSR F 4 where F is called the finesse and Ay is resolvable bandwidth measured in wavenumbers Finesse is a measure of the interferometer s ability to resolve closely spaced lines the higher the finesse the better Finesse can be thought of as the effecti
178. y sys tems En 82 En S2 dS2 0 MAN 73 2 4 2 Existence of Solutions The question of the existence of solutions to the resonator integral equations has been the subject of investigation by several authors 26 28 They have given rigorous proofs of the existence of eigenvalues and eigenfunctions for kernels which belong to resonator geometries commonly encountered such as those with parallel plane and spherically curved mirrors 4 3 Integral Equations for Resonators with Spherical Mirrors When the mirrors are spherical and have rectangular or circular apertures the two dimensional integral equations can be separated and reduced to one dimensional equa tions which are amenable to solution by either analytical or numerical methods Thus in the case of rectangular mirrors 4 6 24 29 30 the one dimensional equations in Cartesian coordinates are the same as those for infinite strip mirrors for the x coordinate they are a2 ys dur x1 f K z v2 u a2 dae ai ys Du L K x vou V aida 74 where the kernel K is given by J K x1 to exp d gix Jot 222 75 Similar equations can be written for the y coordinate so that E x y u x v y and y YzY In the above equa tion a and a are the half widths of the mirrors in the x direction d is the mirror spacing k is 2r X and A is the wavelength The radii of curvature of the mirrors R and R are contained in the fac
179. zation of the tendue U We also assume that as the mirror radius 1s increased to realize higher resolving powers we are able to maintain the required figure of F across the central part of the mirror having a radius ps It isin teresting to compare this behavior with that of an FPP We assume that for an FPP a net surface figure in cluding alignment error of F can be maintained across plates of diameter D which are separated by a distance d For this FPP the tendue is TD 1 4dP and the resolving power is 2df A so that it is the product of the resolving power and tendue which re mains constant In fact the quotient U is a UR rpp 0 7 1D 2 31 The factor of 0 7 again represents the loss in resolving power associated with a useful tendue Thus an in crease in light gathering power must be paid for by a loss in resolution and vice versa The corresponding product for an FPS is given by UR rps 0 7 4r r 32 From this expression we see that by increasing the mirror radius of an FPS the tendue resolution product may be increased indefinitely as long as the mirror figure can be maintained to within A F across an aperture of diameter 2p 2 dr F At this point it should be quite clear that at high resolution and correspondingly small free spectral range the FPS excells over the FPP both in terms of tendue and resolution As the free spectral range is increased however there will be so

4.9 Laserresonator

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