Experimental Optics
Contents
1. Using o 2zo r the phase function in 12 is in general given as 0 i arctan gt 13 1 amp 2 1 amp 2 1 6 r 5a E u m n In case of resonance 7 ga with q N is required Since r equals the mirror distance d and amp 1 we find on the optical axis p 0 the eigenfrequencies from c A Vgmn such that by 2 for 2gt m n 1eN 14 Yam zg l q RDE in contrast to plane mirror cavities which are characterized by the twofold free spectral range as described by Eq 6 It should be noted that the values v from Eq 14 are degenerated since the numbers q m n are not unique Fig 6 illustrates some modes Figure 6 Amplitude distributions Amn P in the cavity center i e z 0 for the fandamental axial mode TEMoo and higher transversal modes TEM mn with 1 lt m n lt 3 in the confocal resonator Fabry P rot Interferometer 4 Setup and equipment We restrict the description to the most important parts of the kit which are essential for the experiments Any additional information can be obtained from the original literature 3 e With an output power of 2 5 mW the He Ne laser is classified as a class 3B product The laser provides a temperature dependent doublet emission line around 632 8 nm whose components are linearly polarized in perpendicular directions e Three pairs of mirrors are provided two spherical sets with radii of curvature ROC of 75 mm and2. Fabry and Alfred P rot In contrast to other more conventional types like the Michelson or Mach Zehnder inter ferometer the Fabry P rot arrangement acts as an optical resonator which may result in an extremely high spectral resolving power A A up to 10 for optical wavelengths A In this way state of the art Fabry P rot cavities may exceed the resolution of classical diffraction gratings by a factor of 100 and provide an irreplaceable tool in particular for studies of the hyperfine structure in atomic spectra Figure 1 Charles Fabry 1867 1945 left and Alfred Perot 1863 1925 right were the first French physicists to construct an optical cavity for interferometry Figure is taken from http photonics usask ca photos 2 Safety Issues 2 1 Eye hazard The HeNe laser used in this lab provides a cw output of 2 5 mW Please use appropriate laser safety goggles in order to avoid damage to your eyes It is recommended to discard any reflecting accessories like watches and jewelry Do not look directly into the laser beam Use the key switch at the laser power supply due to high voltage risks 20 kV 2 2 Electrical hazard The piezo actuator is operated at 150 V its power supply may cause an electric shock Do not remove the associated BNC cable from the back of the control unit PTC 1000 Fabry P rot Interferometer 3 Theoretical Background Fabry P rot cavities make use of curved and plane mirrors as well depending on
3. Fig 1 Basic setup of a Mach Zehnder interferometer 3 At the Screen A t s possible to observe a bright spot or f we place a diverging lens between the laser and the first beam splitter a interference ring pattern with a bright spot in the center while the screen B remains dark or as before it 1s illuminated with a interference ring pattern with a dark spot in the center The situation is illustrated in Fig 2 22 A je l a Fig 2 Complementary interference ring patterns on the two screens A and B 1 To understand the reason of this effect it is necessary to think about how the phase of a wave changes after reflection or refraction at the interface between two media Suppose to have a light beam which propagates in a medium with refraction index n and reaches the interfaces with another medium with refraction index n gt If we consider the reflected beam two different cases have to be taken into account 1 Ifnj lt ny the phase will change by n 2 Ifn gt n the phase will not change Instead the phase of the transmitted beam does not change independently on the values of n and no Being aware of that we can explain the origin of the constructive and the destructive interference at the screens A and B respectively Consider at first the formation of the interference ring pattern with a central dark spot at the screen B The phase of the beam which travels along the path 1 changes by m after
4. by means of 6 again using the value for the finesse F from above Its error should be calculated in a similar way as for dy Compare your results with theoretical predictions from the mirror s reflectivity and their ROC Use 5 and 6 once more now for the nominal data provided by the manufacturer Determine the wavelength separation of the two adjacent laser emission lines and discuss their polarization properties Note that you need to convert the directly measurable time difference from the oscilloscope into wavelengths There is no linear relation as in case of the finesse Why The polarization features should be documented by a couple of printed screen shots for various angular positions of the polarizer Calculate the piezo expansion rate in units of nm V from the oscilloscope screen shot Consider what happens during piezo expansion How are the observed resonance peaks of the Fabry P rot related to certain mechanical positions of the piezo Use an oscilloscope display similar to the right in Fig 10 5 2 Characterization of confocal and concentric setup with ROC 100 mm Determine the finesse using the oscilloscope data Calculate the free spectral range and the spectral resolution of this configuration as before and discuss your results Obviously the procedure is the same as for the 75 mm case Check if your data yield reasonable results Do the same for the concentric arrangement d 2r Compare the performance
5. d equals to the radius of curvature r The parallel input beam is effec tively transmitted without divergence Figure is taken from 5 Right Amplitude of the electric field within the confocal resonator in the case of resonance Figure is taken from http www nano physik uni muenchen de confocal cavity with an off axis parameter o gt 0 would experience an optical path difference of o 4r where the radius of curvature is denoted by r 1 A general expression for the total finesse F of confocal cavities may be thus written as 2 2 2 3 ni where mirror irregularities F have been neglected Obviously the reflective term Fp Fabry Perot Interferometer limits the total finesse for perfectly aligned resonators and on axis rays F lt Fr Figure 5 right gives an imagination of the electric field amplitude within the resonator Note the strong field enhancement due to multiple back and forth reflections A more detailed analysis considers the mode spectrum inside of the cavity Following 2 the spatial amplitude function can be written by means of the Hermitian polynomials 7 as Amn P Hm W2w x H V2w y with m neNo 12 where the Gaussian beam width w z is given as w z wo 1 2z dy with wo Ad 2r The polar coordinates are defined as 7 p cos 0 p sin 0 z assuming the origin F 0 in the center of the cavity Obviously the Rayleigh length zo TWe A coincides with the axial mirror position d 2
6. line http www thorlabs de thorcat MTN EDU QE1_M EnglishManual pdf Quantum eraser 2013 2 J D Jackson Classical electrodynamics Wiley 3th Edition 1998 3 K P Zetie et al Phys Educ 35 1 January 2000 4 R Shankar Principles of Quantum Mechanics Plenum Press 2th edition 1994 27
7. that means if we interact with it the system will reduce itself to only one of these states 5 QUESTIONS After a description of the interferometer setup the working principle and the experimental results try to answer the following questions 1 Why in the frame of the classical electromagnetism is the interference pattern on both screens destroyed when the two polarizers along the path 1 and 2 are crossed polarized Why do you restore the original interference pattern by placing the third polarizer in front of the screen B with intermediate polarization 2 Consider the situation of the single photon beam Why do you observe a complementary interference ring pattern on the two screens dark rings on one screen where you have bright rings on the other one Keeping in mind this experimental evidence do you think that a single photon can be considered divisible 3 What happens again in the single photon beam regime if the two polarizers along path 1 and path 2 are crossed polarized Do you expect to observe an interference pattern 4 Consider the situation introduced in the previous question Why by placing the third polarizer with intermediate polarization in front of the screen B do you get again an 26 interference pattern This polarizer is usually called eraser from the verb to erase Do you have any possible explanation for this expression References 1 Please copy the web address to the command
8. 100mm respectively and one set of plane mirrors The reflectivity of the mirrors is 96 for each set Please do not touch the mirror surfaces The coatings are extremely damage able and expensive Wear single use gloves whenever you mount and replace the mirrors For cleaning even from dust etc special one way lens tissues must be used You should exercise yourself in lens cleaning with some dummy optics like a piece of window glass In each mirror set one of them is mounted on an axial piezo translator The piezo ceramic actuator is driven by a maximum voltage amplitude of 150 V and provides an open loop sensitivity of 0 05 nm for 5mV noise and a maximum force generation of 5500 N The following list overviews other technical features of the device Type max stroke length L capacitance stiffness resonance HPSt 500 15 8 7 13 8 um 26 mm 140 nF 550 N um 30kHz e The piezo actuator may be controlled with respect to the amplitude an offset and the frequency when moved back and forth according to a periodical delta voltage Figure 7 illustrates the front panel of the control unit PTC 1000 The high voltage connector BNC for the piezo is located on the back as well as the piezo voltage monitor and the trigger signal output e The output signal of the silicon photo diode Siemens BPX 61 is amplified using the control unit PTC 1000 The device has an active area of 2 65 x 2 65 mm which is sufficient to collect th
9. 140 Fabry Perot Resonator miCos user manual Eschbach 2009 Tektronix Inc TDS1000B and TDS2000B Series Digital Storage Oscilloscopes User Manual Beaverton Oregon 2006 Dr W Luhs MEOS GmbH Fabry Perot Resonator Eschbach 2003 Thorlabs GmbH Operation Manual High Resolution USB2 0 CMOS and CCD Cam eras 85221 Dachau Germany 2009 20 Extra part The knowledge presented here is not mandatory but can improve your grade Quantum Eraser with MACH ZEHNDER INTERFEROMETER 1 INTRODUCTION The Mach Zehnder interferometer s a device used to determine the relative phase shift variation between two collimated beams derived by splitting the light from a single source The light beam is first split into two parts by a beam splitter and then recombined by a second beam splitter Depending on the relative phase acquired along the paths the collimated beams will form constructive or destructive interference patterns on two screens placed after the second beam splitter 21 2 SETUP AND THEORETICAL BACKGROUND The basic setup of a Mach Zehnder interferometer is illustrated in Fig 1 It consists of a light source tipically a laser two beam splitters two mirrors and two screens The light beam is divided by the first beam splitter giving origin to two different beams which travels Path 1 and Path 2 Path 20 reflector 1000 reflector Source of Path photons 7
10. _ Experimental Optics Abbe School JENA of Photonics Contact Davide Cammi e mail davide cammi uni jena de Peter Rentschler e mail peter rentschler gmx de Last edition Davide Cammi October 2013 Fabry Perot Interferometer Contents KOVENI 2 orca kee ea aS ee UR a Ca Ren eee ek 3 2 Salely ISSUES anni 3 ZA Eye ara se ee een se sel Bess ZIEL CHIC BA re ee ee es ss esse 3 Lheoretical BACK Grounds nenne 4 3 1 Ihe Plane mirror resonalor aii seite ie ia 2 3 2 Ihe COMTOC al resona Or serere 4344 rare 4 SELUP ANG e QuIpmen eek 9 4 1 Practical procedure of the experiments 0 0 0 cette cece eee etteteeeeeeeeeeee L 4 1 1 Confocal Arrangement for ROC 75mm 2 2 nasse cece cece eee e eee eee ee en eens 11 4 1 2 Stable Configurations for ROC 100mm 2 sesessssesesesessesesesseseseee L3 Ade Plane Miror Cavil 2220 are eek ie gt Goals of the experimental work 2 16 5 1 Characterization of confocal setup with ROC 75mm ines PAE EEEE ERRE E 5 2 Characterization of confocal and concentric setup with ROC 100mm u OTN 16 5 3 Characterization of plane mirror setup TEE EEEE A A ET A Preliminary questions wssisceccoscasahcsenseseneneueesdussewahnsenssss saeueencuaseabaceecesseaneees 18 B Final GUCSUIONS sus 18 C Pr paration ot he Te pore are 20 Extra part Olantum Eraser aan Ra 21 Fabry P rot Interferometer 1 Overview The Fabry P rot interferometer was invented in 1897 by Charles
11. ause unwanted phase shifts at each reflection may be denoted by An elementary theory of these surface imperfections is discussed in Appendix A From an experimental point of view the finesse describes the ratio between the free spectral range FSR v and the spectral resolution Av of the instrument in the frequency domain F v Av with dv lt c 2d 6 where c denotes the vacuum velocity of light and the equality v c 2d holds for plane mirrors Ay is usually defined as the full width FWHM of the resonance The finesse F also indicates the effective number of interfering beams within the cavity Since all light rays slightly diverge from their source due to the limited spatial coherence plane mirror cavities always produce concentric Haidinger interference rings rather than single on axis spots Such rings are shown on the left of Fig 4 From their radial intensity ring diameter 1 2 3 4 0 _ intensity _ gt 1 ring number Figure 4 Concentric interference rings of a Fabry P rot cavity with plane mirrors for two closely spaced spectral lines left and the functional dependence of the squared diameters on the ring number right The left figure is taken from http commons wikimedia org distribution characteristic parameters of the setup may be obtained 2 Following Fig 2 the condition 2dcos a ma with p 1 2 3 7 yields interference maxima for cer
12. e difference and measured in units of nm V Check if it is independent of the piezo frequency and amplitude 4 1 2 Stable Configurations for ROC 100mm According to Section 3 the free spectral range FSR and the spectral resolution depend on the mirror distance d Replace the spherical mirrors with r 75 mm by those with an ROC of 100 mm and adjust the system again as described above The distance between the resonance peaks and their width FWHM are measured again using the CURSOR Store appropriate screen shots on your memory stick and estimate the real mirror distance using the scale on the optical bench The stability of an optical resonator depends on the geometry of the system Aside from confocal arrangements several other configurations allow stable operation The general condition for a stable cavity is given as 0 lt 1 d r 1 d n lt 1l 15 for two mirrors with radii of curvature r and r2 This equation is illustrated in Fig 11 For example the concentric cavity is defined by d 2r Expand the mirror distance according to this condition and optimize the interference contrast It might be helpful to use an additional focusing lens e g f 60mm in front of the photo diode Record all required data and screen shots for the determination of the finesse as before 13 Fabry P rot Interferometer Figure 11 Stability of optical resonators made of two mirrors with radii of curvature r and ro separated by a dis
13. e unfocused laser light FWHM 1 mm without excessive losses Its spectral range of sensitivity is given as 400nm lt A lt 1100nm with a maximum around 850 nm Although all measurements can be performed under daylight conditions no di rect sunlight or artificial room illumination should hit the photo diode in order to ensure an optimized contrast Fabry P rot Interferometer Gain Factor Function Amplitude l u O Gain Variation pe E rc p a ur r Frequency Offset a Photo Diode Coupling Output Figure 7 Front panel of the control unit PTC 1000 In the photo amplifier unit the gain may be varied within 0 1 and 2500 The frequency of the piezo triangle voltage can be set between 50 Hz and 100 Hz Figure is taken from 3 We use a Tektronix TDS 2012B oscilloscope with a bandwidth of 100 MHz The amplified signal of the photo diode should be displayed on one channel the second channel is reserved for the piezo triangle voltage The oscilloscope is triggered exter nally by the control unit PTC 1000 The front panel is shown in Fig 8 An USB Flash drive with a capacity lt 2 GB may be used for quick storage of screen shots in the bmp format The procedure is as follows 1 Put your USB memory stick into the USB Flash Drive socket and wait 2 Press the SAVE RECALL button 3 Select Action Save Image and File format BMP on the scre
14. en 4 Select the function Save TEKxxxx BMP on the screen and wait Alternatively you may use the oscilloscope software on the PC and get the screen This procedure is probably more comfortable The complete printed manual of the instrument is available in the lab room 4 Please don t remove it from the lab For the plane mirror arrangement the laser beam should be expanded One ad justable divergent lens with f 5 mm and two focusing ones with f 20mm achromatic and f 150mm are available The output of the resonator is focused onto the photo diode using a convex lens with f 60 mm 10 Fabry Perot Interferometer Tektronix TDS HORIZONTAL POSITION Oo amp VOLTS DIV m SE ER CH 2 Figure 8 Front panel of the TDS 2012B An USB port for connection with a PC remote control is located on the back of the instrument Figure is taken from 4 e A rotation stage equipped with a polarizer is used for measurements of the mode spectrum of the HeNe laser 4 1 Practical procedure of the experiments An appropriate adjustment of the resonator components along the optical axis is essential for successful measurements In a first step the confocal cavity with an ROC of 75mm should be configured 4 1 1 Confocal Arrangement for ROC 75 mm This standard setup may be adjusted quickly and is used for investigations of the mode spec trum of the laser and the determination of the piez
15. er you may use the built in function from the CMOS camera software and take a screen shot as described in Sect 4 1 3 Alternatively mathematical software tools extract the corresponding data row in the gray level encoded matrix of the bmp file Once plotted the peaks and the minima in between should be marked and measured both in radius and intensity An example is illustrated in Fig 13 The visibility is defined as V r Lmnax r Imin r Umax r Imin r 17 Fabry Perot Interferometer 200 10 gt 150 2 08 D x Z 06 Q 100 5 ab E N 04 O O D P O 02 C 0 0 0 50 100 150 200 250 300 200 100 0 100 200 absolute pixel column normalized pixel number Figure 13 Analysis of the plane mirror fringe pattern On the left the bmp pixel image is shown On the right a cross section through the center of the ring system reveals the peaks and minima A Preliminary questions Please read up at home on the following keywords in common textbooks and or the inter net You should be able to explain each of them with a few sentences and or formulae HeNe laser multiple beam interference confocal optical resonator piezo actuator coherence length silicon photo diode hyperfine structure spectrometry dichroic filter etalon polarization of electromagnetic fields B Final questions In addition you should complete your lab report b
16. fore u rE n 0 n 0 re where the initial amplitude E is transmitted twice Ey t E We use the relations T t and R r for the intensity transmission and reflection respectively If there is no absorption we have T R 1 and the transmitted intensity is given as 1 R lrSl Z 1 R 4R sin A 2 3 for an incident intensity J E Obviously the phase difference 1 determines the transmitted wavelengths Am in the m order mA 2d with m gt 1 4 for an incidence angle a 0 This condition defines an optical resonator tuned by an adjustable mirror distance d We may write Eq 3 in an alternative form 1 VR ba e a ee 5 Ag Qt sin Ab 2 uae en normalized to the incident intensity This function is plotted in Fig 3 LO free spectral range gt 0 8 spectral 8 al resolution 5 0 4 0 2 0 0 2m 2 m 1 a phase difference Ad Figure 3 Normalized transmission of a Fabry P rot resonator for various values of the finesse F lt TI as a function of the phase difference Ab A monochromatic light source is presumed Fabry P rot Interferometer The finesse Fr quantifies the optical quality of the resonator made of perfectly polished and adjusted plane mirrors However even well aligned resonators with plane mirrors are severely restricted to F lt 50 unless high precision mirrors are used The contribution of mirror irregularities which c
17. functional dependence V on the finesse e Why did we use the polarizer and a CMOS camera for the plane mirror experiment Based on the measured wavelength distance for the two mode spectrum estimate if they could have been resolved by your plane mirror setup Reason your statement 19 Fabry P rot Interferometer C Preparation of the report Please keep the theory part as short as possible about 1 2 pages with the most important concepts and equations You should not just repeat the manual Work through Sect 5 step by step and try to answer all questions Please include your answers to the preliminary A and final B questions too Describe in detail what you are measuring 1 e all original plots and data must be pro vided When you calculate your results describe the way you found them derivation of formulae Make sure that all results are given with correct units That is for instance to use Hz or s7 for frequencies not s or something else Discuss your results in particular in case of mismatching or contradictory results If you were not able to do certain parts of the work program please explain References 1 http www thorlabs com Scanning Fabry P rot Interferometers 650 ff 2009 2 W Demtr der Laser Spectroscopy Vol 1 Basic Principles Springer Verlag 3 4 5 6 Berlin Heidelberg 4 edition 144 156 2008 MICOS GmbH Laser Education Kit CA 1
18. gh the fringe pattern has to be extracted Plot your values for D vs the ring number p and add a linear least squares fit to the data If possible take an appropriate software Matlab Mathematica Maple or an advanced scientific pocket calculator again for this task for the sake of accuracy Note that an excess lt 0 indicates an error in your analysis So you should consider carefully the nature of the central intensity distribution 1 e 1f it is indeed a single spot or a local minimum If you have recorded fringe patterns for both polarizations you might even perform the procedure for those two adjacent modes and try to find their wavelength separation Since the difference in the slope is very small this operation works probably only for a sufficiently large number of detected rings and thus a small statistical uncertainty in the linear fit Calculate the interference order for this setup and deduce the FSR in units of Hz As described by Eq 4 and Fig 5 the interference order is related to the wavelength 4 and the mirror distance d A modified version of this relation is valid for diffraction gratings too Why If you want to measure the absolute wavelength of an unknown light source would you either use a Fabry P rot resonator or that grating for spectroscopy Plot the radial cross section of the intensity distribution r and estimate the visibility as a function of the radius There are two options for drawing r Eith
19. nel of the oscilloscope e Find an efficient method for fine tuning of the mirror distance until the interference contrast 1s maximized For an optimized alignment the finesse should approach the theoretical maximum Using the CURSOR function of the oscilloscope the distance between the resonance peaks and their width FWHM are measured now Store appropriate screen shots on your memory stick for reference Measure the mirror distance including its estimated error 12 Fabry P rot Interferometer 1 controller switch on 3 highest finesse i photo diode i KL photo diode Figure 10 Adjustment of the confocal setup The amplitude of the resonances should be scaled down by reducing the gain at the control unit and or the volts div knob of the oscilloscope Figures are adopted from 3 During the warm up time of the laser tube the mode spectrum of the HeNe emission around 632 8 nm varies In particular the relative intensities of the closely neighbored lines oscillate on time scales of several seconds You should observe these variations until the equilibrium is reached Now the polarization of the final peak intensities is measured for relevant angular positions 0 lt gy lt 360 Store the associated screen shots on your stick The piezo expansion rate is defined as the stroke in relation to the voltag
20. ng of the cavity Exploit the concen tric fringe pattern which 1s shown on the PC screen as on the left of Fig 4 14 Fabry P rot Interferometer to computer Ph P A6 to oscilloscope Figure 12 Setup of the Fabry P rot resonator with plane mirrors 1 optical rail 2 laser with mounting and power supply 3 4 beam expander 5 fixed mirror 6 piezo driven mirror and control unit 7 focusing lens 8 CMOS camera Figure is adopted from 3 e Set the polarizer into the laser beam and try to optimize the symmetry sharpness and interference contrast of these so called Haidinger rings Exploit the polarizer for investigations of the mode spectrum if applicable e Store at least one adequate image in the bmp format on your memory stick you may also record a cross section of the radial intensity distribution through the center of the ring pattern The camera software offers an option for visualizing such one dimensional cross sections in horizontal and or vertical direction You should get a screenshot by the combination ALT GR PRINT and copy it from the clipboard into an empty work sheet of an appropriate software and store the file on your stick e Note your individual mirror separation d and estimate its uncertainty 1 e d Ad Download the CMOS camera manual 6 from the lab PC to your memory stick You will need this reference for information on the sensor and pixel size
21. o expansion Figure 9 gives an overview of the configuration For first explorations the photo amplifier should be set to DC coupling for a gain of about 50 Gain Variation center It is reasonable to start with an ordinary triangle function for the piezo movement rather than the sawtooth profile Choose an am plitude which is equivalent to a stroke near the maximum and medium values for frequency and offset The recommended adjustment procedure is as follows see Figure 10 e Remove all tabs from the optical rail 1 except for the laser 2 3 Align it as well as possible using the cross on the wall e Mount the tab with the photo diode on the rail and put the amplified 6 signal of the photo diode on one channel of the oscilloscope 11 Fabry P rot Interferometer to oscilloscope Figure 9 Setup of the Fabry P rot resonator with spherical mirrors in the confocal mode 1 optical rail 2 laser with mounting 3 laser power supply 4 fixed resonator mirror 5 piezo driven resonator mirror 6 control unit 7 photo diode Figure is adopted from 3 e Adjust both mirrors ROC 75 mm separately using back reflections and concentric interference rings e The mirror distance should be coarsely set to the confocal condition d r using the scale on the optical rail e Switch on the piezo translator using initially a medium amplitude and frequency The piezo voltage is put on the second chan
22. of the camera The model in use is called DCC1545M 15 Fabry P rot Interferometer 5 Goals of the experimental work In general all calculations must be documented by original measurement data i e tables figures and appropriate print outs or oscilloscope screen shots Please note important setup data like mirror distances and settings of the control unit PTC 1000 as well 5 1 Characterization of confocal setup with ROC 75 mm Using the measured data for the peak width and their distance from each other calculate the finesse as follows Take an appropriate screen shot on the oscilloscope similar to the right picture in Fig 10 By means of the CURSOR function measure the time distance T between two adjacent peaks of the same laser mode corresponding to a phase shift Ad 2x In the same way determine the full width at half maximum FHWM Ar for at least one of them or better take the mean FHWM of both peaks The ratio 67 At yields the dimensionless finesse 7 though the latter one is defined by inverse units 1 e frequencies Why Give results for the free spectral range and the spectral resolution of your setup based on the measured mirror distance d in units of Hz and estimate their errors The FSR y is obtained from 6 and its follow up comments Roughly approximated its experimental error originates from the uncertainty of d and should be given as v ov A v The spectral resolution is calculated now
23. tain incidence angles and integer numbers of m Let the innermost ring be denoted by the index 0 Obviously we get from Equation 7 2d cos ap 2d cos o m my A 8 The parallel transmitted light is focused to the image plane by a lens with a focal length f Fabry P rot Interferometer Thus we get the diameter D of the p interference ring in the paraxial approximation D 2ftan a 2fa for a 0 9 Combining the Equations 8 and 9 we get a linear dependence of the squared diameter Dy on the ring number p A d PER 2 EIERN D 4f WP F E with ar 10 right in Figure 4 On the other hand the slope tan y of the function 10 yields the ratio A d The quantity 0 lt e lt 1 is called the excess and determines the axis intercept b on the 3 2 The confocal resonator The finesse as defined in 5 1s valid for a constant cavity spacing d across its lateral dimen sions On axis beams would thus receive the same optical path length as parallel off axis rays when propagating through the resonator In contrast easily aligned confocal arrange ments as shown on the left of Fig 5 are made of spherical mirrors and cause various path lengths depending on the distance of the ray from the optical axis Beams which enter the DD detector mirror distance d r mirror mirror Figure 5 Left Geometry of confocal cavities made of two identical spherical mirrors The mirror distance
24. tance d The stable regions are gray underlaid and include the central point 4 1 3 Plane Mirror Cavity This experiment uses two plane resonator mirrors instead of spherical ones An optimized detection of the interference rings requires a new experimental approach Remove all optical mounts except for the laser In a first step the laser beam should be expanded with a two lens telescope It is recommended to use the f 150mm lens rather than the device with f 20mm in conjunction with the divergent lens f 5 mm The output beam must have a constant diameter which is independent of the distance from the beam expander Place the piezo mounted mirror on the optical bench and adjust the back reflected beam as accurate as possible Switch off the control unit PTC 1000 Think about a reasonable mirror distance d On the one hand the sensitivity of the setup on misaligned mirror surfaces increases with the distance d Thus the adjust ment is less challenging for a short mirror distance On the other hand the spectral resolution becomes better for large values of d Place the second mirror on the bench and adjust it coarsely using back reflections Install the CMOS camera at the end of the optical bench and the convex lens f 60 mm in front of it somewhat different from Fig 12 Display the transmitted spot on the PC screen and center the CMOS camera to the optical axis Think about an efficient algorithm for fine tuni
25. the reflection at the first beam splitter and by a after the reflection at the mirror The reflection at the second beam splitter does not induce any additional change of phase so this beam will reach the screen B with a total change of phase of 2a The beam which travels along the path 2 undergoes instead to a total change of m due to the reflection at the mirror and so will interfere destructively with the other beam at the center of the screen B 23 The situation at the screen A can be explained in a similar way As before the beam which travels along the path 1 reaches the screen A with a total change of phase of 27 But in this case also the second beam undergoes to a total change of phase of 2n and this will result in a positive interference and a corresponding bright spot on the center of the screen Fig 3 Disappearance of the interference patterns with two crossed polarizers along the paths and 2 1 3 EFFECT OF THE BEAM POLARIZATION The first objective of this experience is to investigate how the polarization state of the two beams affects the interference ring patterns on the two screens The starting configuration that we want to consider is the one with two parallel orientated polarizers placed along path 1 and path 2 between the first beam splitter and the following mirrors What we should observe is that compared to the basic setup nothing changes This means that placing as before one diverging lens in before
26. the first beam splitter on the screen B there will be an interference ring pattern with a central dark spot while on the screen A the central spot will be bright If the two polarizers are instead perpendicularly the interference patterns disappear and a bright spot appears on both screens as it is illustrated in Fig 3 24 Fig 4 Recover of the interference pattern on one screen due to the third polarizer with intermediate polarization respect to the two crossed polarizers along the paths 1 and 2 1 When a third polarizer is placed in front of the screen B with intermediate polarization between the two crossed polarizers the original interference pattern will appear again on the screen B see Fig 4 4 WHICH PATH INFORMATION IN QUANTUM PHYSICS Until now we have considered a classical electromagnetism explanation in order to describe the formation of the interference ring patterns on the screens A and B It is anyway possible to introduce a quantum mechanical description of the process if we assume to reduce the intensity of the light so that the original beam consists only of single photons emitted with a measurable time separation between each other Although we will not try to repeat the experience in such low intensity regime it is interesting to consider what could happen experimentally Consider at first the configuration in which the two polarizers are placed along the paths and 2 parallel orientated What we
27. their field of application Whereas spherical mirrors are commonly employed for confocal schemes especially the very basics of optical resonators are better studied by means of elementary plane surface reflections We thus start with a brief description of that case and switch over afterwards to peculiarities of confocal cavities 3 1 The plane mirror resonator We consider two plane mirrors with equal reflectivities 0 lt R lt 1 ina parallel arrangement separated by a distance d from each other A plane wave with an amplitude E falls onto a mirror with an angle of incidence a At each mirror surface the beam is partially reflected amplitude coefficient r lt 1 and transmitted The scheme is sketched in Fig 2 The phase Figure 2 Multiple beam interference at two parallel surfaces G and G2 separated by a distance d and filled with a dielectric medium D Figure is taken from http www loitz79 de Physik Praktikum Interferometrie htm difference Ad between two adjacent beams E and E depends on the mirror distance d and the refractive index n of the dielectric medium Ar Ad ee 1 We assume n for an air filled resonator from now on The output may be written as the coherent superposition of all amplitudes E with O lt i lt since an effectively infinite number of interfering waves is assumed for mirror reflectivities near 1 The total transmitted Fabry P rot Interferometer amplitude E results as fore
28. us wavefront errors after multiple back and forth reflections since even tiny phase shifts induced by small deviations from the ideal plane surface add up and smear out the interference contrast The surface roughness is often measured in fractions of the wavelength Ah A n Values Ah lt A 100 cannot be realized without an extraordinary technical effort As a consequence the finesse is usually limited to F lt 50 even for high reflecting mirrors Fig 14 compares both limits 80 z FR Ah 2 n N N 60 V D 40 D E Ss S E 20 C ma 0 0 0 2 0 4 0 6 0 8 1 0 0 20 40 60 80 100 mirror reflectivity R mirror surface error in units ofn Figure 14 The physical and technical finesse for plane mirrors On the left the reflection based finesse F is shown The technical limit is given in units of the wavelength fraction n right Give an explanation for the functional dependence on the right of Fig 14 Following this elementary theory you can estimate the required surface quality of the mirrors used in our lab for an optimized finesse near the theoretical limit e Reconsider the concentric ring pattern in the plane mirror arrangement Should the mirror distance d be enlarged or diminished in order to contract the ring diameters e Following its definition in Sect 5 3 the visibility V describes the interference con trast Find the
29. with 16 Fabry Perot Interferometer the confocal case Usually the concentric resonator performs worse than the confocal one Why Be careful to get a regular resonance pattern as shown on the right in Fig 10 Copy the stability plot Fig 11 into your lab report and mark the corresponding points for the confocal and the concentric case Calculate the stability criterion for all other com binations of mirrors used in this lab 1 e the spherical ones with an ROC of 75 mm and 100 mm respectively and the plane mirrors r oo In this way you will obtain 6 possi ble resonator designs Identify and discuss their stability in your plot 5 3 Characterization of plane mirror setup Calculate the mirror distance and its standard deviation using the concentric ring system and compare your result with the directly measured value Give the value of the excess e The procedure is described in Sect 3 in particular by Fig 4 and Eq 10 At first you should determine the diameters of at least three rings don t confuse about the two laser modes using a print out of the CMOS image or by image processing of its bmp version For this purpose the three dimensional RGB data of the bmp file must be first converted to one dimensional gray scale values Depending on your software this is done by a single command as in Mathematica for instance or alternatively in Photoshop or other software tools Afterwards the central row or column throu
30. would observe are the previously described interference ring patters on the screens with a dark or bright central spot on the screen B or A respectively From the classical physics point of view the result is not intuitive and is actually not predictable To get a real comprehension of the effect it 1s necessary to describe it in the framework of quantum mechanics 25 The quantum mechanical interpretation takes into account that when the two polarizers are parallel polarized the paths 1 and 2 are undistinguishable In quantum mechanics when two states are not distinguishable the physical system under interest in this case the photon is described as a superposition of both states As long as the two paths are not distinguishable we do not have any information about the actual path the photon travels along The consequence is that when the photon reaches the screens we have to treat it as a linear combination of two states the photon which travels along the path 1 and the photon which travels along the path 2 The interference pattern that we observe is thus the result of the so called interference of the photon with itself What we have presented in this section is a typical quantum mechanical effect as long as the observer does not interact with the physical system the last one is described as a superposition of different states for example the energy levels of an atom But as soon as we measure any physical observable of the system
31. y solving the following problems Their intention is to get an overview on the subject and a deeper understanding e In Sect 3 the high spectral resolution of a Fabry P rot device was explained by multiple beam interference Estimate the number of rays for an interferometer with a finesse F Compare your result with the number N of lines for a typical grating whose resolving power in the m order is given as A AA mN How do Fabry P rot resonators achieve their spectral resolution e Give a reason why the free spectral range of a plane mirror cavity 1s always c 2d whereas the FSR for confocal geometries may be just one half i e c 4d for pro nounced off axis beams What is the FSR for concentric arrangements d 2r e Calculate and plot the function F o from Equation 11 for a mirror reflectivity of 96 Why we don t use the beam expander for the confocal geometry 18 Fabry P rot Interferometer e Lasers often use spherical mirrors in the confocal distance d r Why is this arrangement so important e An important feature of optical resonators is their stability Scan the literature and or the web in order to find out how the criterion from Section 4 1 2 is derived at least in principle Is it a yes no criterion or may the stability be quantified e The plane mirror mode gives reason for simple investigations of the mirror surface quality Unlike spherical ones imperfect plane mirrors cause serio
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