Frequency Modulation Saturation Spectroscopy Laser Lock of An
Contents
1. 0 4 19 4 5 Linewidth And Noise 2 20 AO Sidebands iria aes ag Ra ee Be i 22 5 What could be improved 40 22 6 Conclusions sessa 3 Sad r Bera ei ee Ee Eei 23 1 Motivation Summary Many labs all over the world are working with Ultra Cold Atoms to study the quantum behaviour of atoms and molecules and to test or improve theoretical models that are important to understand popular and interesting phenomena like high temperature superconductivity or how to build a quantum computer that could have a great impact on how we live in the future In the Ultra Cold Atoms lab at University of Toronto atoms are loaded into a lattice potential with tunable interactions between the atoms to simulate atoms in condensed matter system and to test and study the Hubbard model that exactly describes this system but is not solvable analytically One major step of bringing the atoms to these very low temperatures is laser cooling that was simultaneously proposed by Wineland and Dehmelt 2 as well as H nsch and Schawlow 3 which is still used widely to cool the atoms to temperatures about 100 uK To reach temperatures that are needed to create BECs or degenerate Fermi gases and that are about two magnitudes of order lower evaporative cooling is used The lasers that are used in laser cooling need high frequency stability and small linewidth lt 1 MHz Several techniques are established to stabiliz2. Research Internship at University of Toronto under Prof Joseph H Thywissen on the project Frequency Modulation Saturation Spectroscopy Laser Lock of An Interference Filter Stabilized External Cavity Diode Laser Felix Stubenrauch September 16 2010 Abstract This document gives a short theoretical introduction to Frequency Mod ulation Saturation Laser Lock Moreover the experimental realization of a lock for an external cavity diode laser is described and characterized and a short manual of how to start the laser lock setup and to optimize the error signal is given It continues the work of Matthias Scholl which is described in 1 Contents 1 Motivation Summary e a dee ee ee eS 3 Qf AENEON 2 iit mas een es eek are eee oe legen Eble Das 3 2 1 Saturation Absorption Spectroscopy 3 22s baser Lock 2 2 a gh ak a oe ae SR oa cee Bak 5 3 Experiment Setup cosi pira to Pa Sa as E 11 3 1 Optical Tgolator s 0 000 a Oa N ae adn p 4 11 3 2 Electro optical Modulator EOM 11 3 352 Vapor Gellner oase Grn lect AB en aA bo e e EE 13 3 4 Photo Diode o 14 3 5 Getting started o o a 14 3 6 Beat Measurement o o 16 4 Perfomance of the System o o 18 Al Error signal 3 eaat i aaa i ee ei E t 18 4 2 Frequency Stability 2 02 ee 18 4 3 Recapture Range o 18 4 4 Long time drift
3. and dispersion at the carrier frequency and all the sidebands according to 00 E t Evexpliwot Y T wn Jn M exp inwmt 4 n 00 The even and odd order sidebands are respectively in and out of phase as the Bessel functions of negative integer order have the symmetry property J_n M 1 J M Therefore the amplitude modulations caused by two symmetric sidebands are out of phase and exactly cancel out leaving pure frequency modulation Because of the frequency dependence of the attenua tion however the sidebands have different amplitudes after the transmission through the probe leading to amplitude modulation at the modulation fre quency wm that is linear in the difference of the attenuation of the two first order sidebands figure 6 au IN power a u sideband power a u h absoprtion a u o absoprtion a u W Wm W W Wm Wy sideband power Utd Y WO sideband S wtw Y 0 Un M frequency a u frequency a u frequency a u absoprtion a u Wo error voltage a u O 0 frequency a u Figure 6 The three upper graphs show the sideband attenuation for different fre quencies of the main peak with the corresponding DC error signal underneath A DC error signal that is proportional to the amplitude modulation depth may be extracted by mixing the detected signal with the modulation sig nal This signal has to be
4. you should not put to much weight on the laser cavities and squeeze the rubber because it could take time until the alignment stays stable again The cover for the cavity that protects the inside from air flows is also acoustically isolated through rubber on top of the cavity 20 250 200 drift MHz 0 1 2 3 4 time hours Figure 10 An exemplary measurement of the frequency drift of the unlocked laser 3 with the spectrum analyzer in a beat measurement against a locked laser from another experiment 21 This and the strong error signal make the lock stable against metallic objects hitting the table like a falling pedestal Even hitting the table with a screw driver next to the laser doesn t unlock the laser After applying the rubber to the cavity the noise measurement that is described in 1 was repeated on the linear slope of the error signal No peaks could be found but only flat white noise The linewidth was measured by translating the noise of the error sig nal of the piezo feedback locked laser to frequency fluctuations leading to a linewidth around 1 MHz with 30 high errors This measurement overes timates the linewidth however because we took the maximum error signal fluctuations on the scope to calculate the frequency fluctuations Measure ments with the SA lead to similar results using Aw FG Awbeat for two beams with same linewidth Aw Since the piezo feedback is very slow in the order of 1k Hz t
5. output Moreover the oszilloscope only shows the error signal If you can t generate an error signal following this list you should try to get the saturation signal first Probably the electronics don t work correctly the photo diode is saturated the phase between the saturation signal and the modulation signal at the mixer might be wrong the alignment is bad or the vapor cell isn t heated up enough Moreover the saturation signal is easier to find because of the broad doppler peak that is suppressed in the error signal e Switch on the power suppliy that heats the vapor cells Make sure that it runs at around 2 0 A or 5 8 V Fan will be turned on with ramp lock box e Switch on laser to around 70 mA All temperature controls should show values between 10 kQ and 15kQ Make sure the beam isn t blocked 14 Wait 10 minutes until cell temperature reaches equilibrium and laser stabilizes Switch on photo diodes and oscilloscope Switch on power supply for lock box fans and amplifiers Unplug and replug the 24V bananas to start fan Make sure all fans are running Turn ramp on lock off and ramp up to maximum Start piezo mod ulation function generator Center piezo bias on lock box and piezo controller to around 70V to get maximum tuning range Check bias tuning range on lock box by turning it all the way up and down and watch the voltage on the piezo controller Change bias to the middle of the tuning range and turn piezo cont
6. splitter acousto optic modulator electro optic modulator 10 20mW potassium vapor cell pump 350u4W f 10cm f 40cm M2 Figure 7 Laser Lock Setup For reasons of simplicity most mirrors that are necessary for beam alignment are not shown in this schematic Electronics from minicircuits are shown with part numbers 12 amplifier again However this reduced the peak to peak amplitude and the signal to noise ration of the error signal significantly With this setup we achieve modulation amplitudes of around Vo 40 V The EOM has a half wave voltage voltage that has to be applied to cre ate a 180 degree phase shift at 767 nm of 173 V leading to M 0 7 As shown in figure 5 the second order sideband is still negligible with a rela tiv intensity of J2 0 7 Jo 0 7 0 4 where as Jo 0 7 75 9 and J1 0 7 Jo 0 7 15 3 1 Make sure that you start the amplifier with low modulation powers of lt 30dBm before turning up to avoid high peak power in the amplifier If you are not sure if the function generator is turned down to low powers from the previous run switch off the amplifier by unplugging the 24V bananas and check the power on the function generator The power input for the amplifier should always stay under 10dBm The amplifier starts getting non linear for power input 1dB Compr see datasheet over 4dBm The beam through the EOM should be al
7. upper limit for the drift we take the linewidth of the beat signal which is about 1 M Hz 4 3 Recapture Range The recapture range of the laser lock is illustrated in figure 9 The frequency scale is calibrated with the SA The laser frequency was detuned by changing the piezo voltage Than the time shift of the error signal was compared to the 18 Figure 8 Error signal of the F 2 transition frequency shift on the SA This calibration has to be done each time because it depends on the sweep voltage The large recapture range of 227 MHz can hardly be realized since the error signal is slightly shifted horizontally over time leading to zero crossings between the peaks Therefore the factual recapture range is about 70 MHz The servo is usually fast enough to correct the laser frequency before it changes its frequency more than this 4 4 Long time drift It is important to consider long time drifts as a property of the laser when talking about a laser lock because it leads to losing the lock when the drifts get to large The frequency of the unlocked laser drifts over time due to temperature changes and other influences even though the temperature is controlled close to the photo diode and on the bottom of the aluminium housing Since these drifts are very slow on a timescale of minutes or hours as shown in figure 10 it is easy for the integrator in the servo loop to compensate them by changing the piezo voltage over time However t
8. artment of Physics Advanced Physics Laboratory PHY4803L 8 P W Milonni and J H Eberly Lasers Wiley New York 1988 9 G E Hall and S W North Annu Rev Phys Chem 51 243 2000 24
9. e lasers to fulfill these requirements This report describes the realization and characterization of a Frequency Modulation Spectroscopy laser lock It concentrates on the creation and the optimization of the error signal and doesn t talk about the control servo needed to create the signal that is fed back to the laser At the time when this report was written the laser were only locked with slow piezo feedback However the performance has already been sufficient to be used in the experiment but will be further improved by using fast current feeback 2 Theory 2 1 Saturation Absorption Spectroscopy One of the major applications of lasers is the investigation of the energy structure of atoms and molecules with laser spectroscopy With a single laser it is possible to reach resolutions that are only limited by the Doppler width of the atomic transitions as long as the laser linewidth is smaller than the Doppler width Our laser has a free running linewidth of Avfree 1 MHz which is much smaller than the Avgopnier 1 25G Hz Doppler width of the Potassium D2 line at 766 7 nm 4 that is used as an atomic reference in our setup However it is possible to resolve energy levels within the Doppler broadening if the atomic vapor is saturated by a so called pump laser as it is the case in saturation absorption spectroscopy 1 a b Signal without pump beam Signal with pump beam Figure 1 A saturated absorption spectro
10. e velocity com ponents in beam direction that are close to zero The deep hole burned by the pump beam reduces the number of atoms available for excitation through the probe beam and therefore reduces the absorption of the latter This results in the sub doppler resolution peak at resonance shown in inset b So far we only looked at simple two level systems In systems with more energy levels additional peaks crossover peak appear exactly halfways be tween the resonance of two transitions in the SAS signal due to hole burning and optical pumping 6 7 Figure 2 shows the 9 K D2 line signal with F 1 F 2 and crossover peak see potassium energy level structure showing the D1 and D2 line in figure 3 2 2 Laser Lock As mentioned before our homebuilt ECDLaser system has to work at a very stable frequency 1 MHz with a narrow frequency distribution There are several techniques to lock stabilize a laser to a certain frequency that ensure this Single beam methods like dichroic atomic vapour laser lock DAVLL have doppler broadened features which results in large recap ture ranges of several hundred MHz However the sub doppler resolution of saturation spectroscopy methods like frequency modulation saturation spec troscopy FMS which is applied in this project leads to steeper gradients of the error signal ensuring higher stability Moreover the atomic vapor that is used in FMS gives an absolute frequency refere
11. ent Figure 4 Principle of a laser lock setup A small part of the laser intensity is used to create an optical frequency dependend signal that is detected and manipulated electronically to create a linear frequency depend error signal that is fed back to the laser to correct for the frequency aberration ing sections This signal is fed back to the laser to correct the aberration for example by changing the voltage of a piezo crystal that changes the length of the laser cavity or by controlling the current of the laser diode Frequency Modulation Spectroscopy FMS As in SAS a strong pump beam is used to saturate the excitation of atoms in a vapor cell The transmission of the weak counterpropagating probe beam is detected In FMS however the probe beam frequency is modulated before passing the vapor cell The error signal is extracted from the magnitude of the frequency component of the signal that oszillates with the modulation frequency This is described in the following section An Electro Optical Modulator EOM is used to modulate the phase of the probe beam The phase shift of light on the output of the EOM crystal is proportional to the electric field E applied inside the birefringent crystal see also Pockels effect e g 8 _ bwo C D Eal 1 where is the length of the crystal in beam direction 8 is the Pockels coef ficient and w is the frequency of the beam The polarization of light travelling through the crys
12. he lock without current feedback doesn t have any mea surable influence on the linewidth of the laser compared to the case where it is free running However it will be interesting to measure the linewidth of the current locked laser 4 6 Sidebands Since we spent a lot of thoughts on sidebands caused by longitudinal external cavity modes this short section is talking about them The free spectral range of the external cavity is 1 7GHz as shown and explained in 1 In the beginning these sidebands showed up on the beat signals between our two ECDLs with minimal relativ intensities from 30 dB Minimal means that they where sometimes even bigger depending on the laser diode currents Also Matthias Scholl already observed them Even after several internal alignments of the lenses and the interference filter they could only be decreased to relative intensities of 40dB After rearranging the setups on the optical table and reoptimizing the lasers the sidebands dissappeared from the SA signal This means that they have smaller relative intensities than 50dB now 5 What could be improved e Get power splitter after EOM amplifier with higher max power input e Built fixed frequency modulator for sweep and EOM modulation and phase shifter to optimize error signals independendly 22 e Measure linewidth and short time stability when lasers are locked with fast current feedback and piezo feedback to correct slower frequency drifts 6 C
13. he servo can only change the piezo voltage over 25V if bias on servo box is perfectly centered Because of the frequency tunability of the piezos 53 2 MHz V and 41 2 MHz V the servo can compensate for long time drifts in frequency of maximal 1 3GHz and 1 0GHz respectively Therefore the lock only 19 Figure 9 Frequency depend error signal with frequency scale The frequencies refer to the distance between the tips of the arrows The two peaks correspond to the F 2 transition left and the crossover right between F 1 and F 2 works for several hours the laser has never been seen to stay locked for more than 12 hours over night The time can be extended if the piezo voltage controller is used to decrease the servo feedback voltage from time to time which can be realized by looking at the servo output voltage on a additional scope The piezo controller has a tuning range from 0 to 140V which makes it possible to lock theoretically over maximal frequency drifts of 3 7 GHz for the left setup R 22 or laser 3 in 1 or 2 9 GHz for the right setup R 17 or laser 4 in 1 4 5 Linewidth And Noise To isolate the laser cavities from acoustic noise from the table the lasers are mounted on absorbing rubber sorbothane The bolts that are connected to the table don t touch the cavity either They only touch the rubber We had to wait several days until the output beam of the lasers stayed aligned Therefore
14. igned by increasing the trans mitted intensity first When the error signal is obtained the alignment may be further improved by optimizing the error signal 3 3 Vapor Cell We heat the potassium vapor cell to increase the absorption rate with a current of 2A through a thin wire that is coiled around the cell on both ends but not in the middle Thus the potassium atoms do not condense on the outer surfaces but in the colder middle of the cell which would lead to reflections of the beams The current of two adjacent heating wires is always running in opposite directions to avoid strong magnetic fields A fan on top of the power supply which produces a lot of heat is used for cooling We first tried to align the two counterpropagating beams through the vapor cell on top of each other and separating them with a polarizing beam splitter PBS reflecting the probe beam into the photo diode This means that probe and pump must have perpendicular polarization in the cell which is no problem as long as there is no magnetic fields that lead to Zeeman splitting of the absorption lines However the magnetic fields created by the heating wires of the vapor cell are strong enough to distort the error signal due to the splitting even though the current of two adjacent heating wires is always running in opposite directions Therefore we switched to the alignment that is shown in the schematic using a A 2 waveplate to get parallel polarization for pump and p
15. is proportional to the square of the voltage Vpp from the photo diode The voltage itself is proportional to the intensity of the beat signal 1 Hence we get VSsa x y Pa x Vpp x Ie 10 Therefore the ratio of the SA signals of the sidebands gate and the main main frequency Sg4 is proportional to the ratio of the intensities of the sidebands I and the main frequency Ip 2 side beat B 2 L 35A y ese Ss 2 ee 11 Smam peot Bo Lo This means that if the SA signal of the sidebands is 40dB smaller than the main peak on the SA the intensity of the sidebands in the experiment are also 40 dB smaller than the intensity of the main frequency 4 Perfomance of the System 4 1 Error Signal Figure 8 shows exemplarily the frequency dependend error signal around the F 2 transition We get error signals with 600mV to 800 mV peak to peak amplitudes and signal peak to peak voltage to noise ratios of about 100 The noise is measured in the horizontal parts of the error signal 4 2 Frequency Stability The frequency stability of the piezo int locked laser was measured in a beat measurement against a locked laser from another experiment As long as the laser stayed locked no frequency drift could be observed The resolution of this measurement is limited because it is hard to locate the center of the frequency distribution The peak has several local maximums caused by noise that change position continuously As an
16. low pass filtered below the modulation frequency because it contains contributions of integer multiples of the modulation fre quency In the limit of weak modulation M lt 1 only the first order sidebands contribute to this signal leading to an intensity Irm Eo exp 260 M d_1 641 cos 0 M p 1 Y41 240 sin 0 5 where 6 and Y are the attenuation and the dispersion of the ith order sidebands and 0 is the phase shift between the modulation signal and the saturation signal in the mixer The dispersion term can usually be neglected 10 Especially for a symmetric dispersion profile around the resonance frequency Equation 5 shows that we have to match the phase 0 improve the absorption signal or increase the modualtion amplitude M to get a strong error signal The treatment using fixed frequencies only makes sense when the fre quency is modulated fast in the timescale of absorption wm Z T where T is the linewidth of the transition For slow modulation the time dependent transmitted intensity just follows the absorption spectrum at the instanta neous frequency without a phase lag The natural linewidth of the 39K D2 line is about 6 MHz which is broadened in the saturation spectroscopy signal because of insufficient resolution to resolve transitions to different hy perfine states in the excited P3 2level and because of power broadening For very high modulation frequencies the sidebands showed in figure 6 lie far away from
17. n caused by random noise as a gaufian distributed error 2 2 2 OWbeat 2 OWbeat Obeat 4 01 PT i 02 ae a 6 Ow Owe where peat 01 and o2 are the standard deviations of the frequencies Wbeat w1 w2 w1 and wa we get AwWbeat V Aw Aw 7 because o x Aw has no correlation with Aw 2 If one knows the linewidth of one laser or a relation between the linewidth of both lasers one can calculate the linewidth of the other laser from the linewidth of the beat signal Intensity The intensity 1 of the beat signal can easily be derived from pleat Fy t E t A cos wt bcos wet 124 cos 2w1t 2B cos 2w t 4AB cos w 1 w2 t cos w wa t 8 AB cos w wa t The low pass filtered detected intensity at the frequency w1 wa is therefore proportional to the product of the amplitudes AB of the electromagnetic fields of the two beams that are superimposed Sidebands The intensity of the main peak is ie x AyBo with Ay and Bo being the amplitudes of the Fourier components of the main frequency w4 and wg of the two beams We assume that the second beam has first order sidebands with field amplitudes B 1 For the intensity of the beat signal of the first order sidebands 13 of the second beam and the main frequency of the first beam with field amplitude Ay we get Bs jat cAoB 1 iets 17 The signal on the SA Sya is proportional to the electronic power Pa which
18. nce Error Signal The underlying principle of all lock in methods is to create a so called error signal that is proportional to the difference between the actual possibly perturbed frequency of the laser and the reference frequency that is often obtained by using an atomic transition see fig 4 more details in the follow transmission a u frequency a u Figure 2 Saturation absorption spectroscopy signal of our setup F 1 F 2 and crossover peaks of the K D2 line are labeled see potassium energy levels D1 and D2 line in figure 3 41K isotope peaks are barely visible We dont substract the single beam doppler peak because it doesn t contribute significantly to the error signal see section 2 2 which is proportional to the first derivative of the saturation signal 93 26 6 73 39 41 F 5 2 55 2 F 7 2 31 0 2p P 3 8 4 on F 2 5 0 Pas Risin ic PR discos ARM le os NN Peale P22 87 P 16 1 P 0 419 4 F 11 2 45 4 D2 F 7 2 86 3 766 701 nm 3p F 2 11 4 F 2 20 8 2 sa 3p aii Pra 235 5 FA x penta E dl 1 19 1 125 8 P 1 34 7 F 9 2 69 0 D1 770 108 nm F 7 2 714 3 F 2 173 1 Se Si So Falsa 461 7 4 254 0 F 1 158 8 F 1 288 6 F 9 2 571 5 Figure 3 Energy level structure showing the D1 and D2 line of the potassium isotopes K 40K and 4K from 4 electronics PID to create controller error signal optical error signal experim
19. on frequencies between 10 MHz and 30 MHz For some frequencies the noise decreases signifi cantly Try to find a good trade off between noise and amplitude of the signal The slope of the linear area of the transition that you want to lock to has to be positive You can also change the cable length to optimize the phase since the wavelength is in the order of 10m This is more effort and less elegant but would be necessary if a cheap fixed frequency modulator would be used An alternative would be a homebuilt phase shifter If you unplug ca bles always switch off the amplifiers to avoid high peak currents and power reflections If both EOMs are driven by one function generator the phase for one of them has to be optimized by the cable length or a homebuilt phase shifter Ramp down to see only F 2 peak see figure 2 until ramp sweeps over the linear area of the error signal Switch off the ramp and turn on the lock The laser is locked To see beat signal AOM has to be turned on 5dBm at function generator because only the shifted first order beam of the left laser setup is coupled 1 Tf you shut down the setup always shut down the function generators continiously and before switching off the amplifiers Beat Measurement This section describes how to interpret the beat signal on the spectrum analyzer SA of two beams with different frequencies and amplitudes 16 Linewidth If we treat the random frequency distributio
20. onclusion We have been able to build an interference filter stabilized external cavity laser in the lab and lock it to the 39K D2 line transitions via frequency modulation spectroscpy Even with a slow integrator feedback to the piezo we achieve favorable linewidth and frequency stability of less than 1M Hz The error signal and the isolation of the cavity from the optical table using rubber pads are good enough to keep the laser locked even when people are moving metallic objects over the optical table The laser normally stays locked over several hours If the current error signal is used with a fast current feedback a significant improvement in linewidth and short time stability is expected The laser is ready to be used in cold atoms experiments in the lab 23 Bibliography 1 M Scholl and J H Thywissen Interference Filter Stabilized External Cavity Diode Laser Ultra Cold Atoms Lab Department of Physics University of Toronto Canada 2010 2 D Wineland and H Dehmelt Bull Am Phys Soc 20 637 1975 3 T W Hansch and A L Schawlow Optics Communications 13 68 1975 4 T G Tiecke Properties of Potassium Diploma thesis van der Waals Zeeman institute University of Amsterdam 2010 5 C J Foot Atomic Physics Oxford University Press New York 2005 6 W Demtroeder Laser Spectroscopy 5th ed Springer Verlag Berlin Heidelberg 2007 7 University of Florida Saturated Absorption Spectroscopy Dep
21. robe beam The beams lie in one hori zontal plane crossing each other in the middle of the vapor cell and having 13 a horizontal distance of 5mm on the vertical input and output surfaces of the cell 3 4 Photo Diode In the current setup a PDA8A photo diode from Thorlabs is used It has a fixed gain and a large bandwith from DC to 50 MHz The detected and DC blocked since only the modulated components of the signal contribute to the error signal signal has to be amplified by an external amplifier minicircuits ZLN 500BN LN To decrease the noise produced by the amplifier and the mixer an attenuator is used in front of the amplifier leading to a smaller peak to peak error signal A good trade off leading to an optimized ratio of signal to noise could be achieved by using a 6dBm attenuator minicircuits HAT 6 We previously used the slower PDA36A photo diode from Thorlabs with adjustable gain Since the bandwith of 17 MHz 0dB gain 12 5 MHz for 10 dB gain was only about enough for no gain two external amplifiers had to be used leading to higher noise 3 5 Getting started In this section it is described how to start both lasers The list refers to the setup when the two EOMs are driven by a single frequency modulator SRS model DS345 with double tee output modulation signal for both mixers and signal piezo ramp and amplified by a single amplifier minicircuits ZHL 1 2W with a power splitter minicircuits ZX10 12 2 at the
22. roller to 70V Start and turn up EOM modulation on function generator to around OdBm read section about EOM first to avoid destroying the amplifier Tune laser diode current slowly between 60mA and 100mA until a signal appears on the scope Make sure the scope diplays the whole piezo tuning range You expect a signal with 20 to 200mV on the scope Center signal around t 0s with piezo controller This proce dure sometimes requires a bit of patience If you loose the mode try again Turn up EOM modulation to 5dBm This should be enough to see strong error signals Optimize error signal Change pump probe relation by turning the waveplate before the second polarizing beam splitter PBS Make sure you stay in the limit of a strong pump beam realizing at least Ipump gt Iprobe Change intensity in saturation arm by turning waveplate before the first PBS Use beam intensities close to Ipump Y 350 mW and Iprobe 50mW to avoid power broadening of the saturation peaks and saturation of photo diode signal amplifier or mixer Realign EOM by optimizing the error signal Just look if you can increase the peak to peak signal by changing the mirror orienta tion 15 3 6 Change probe and pump alignment through vapor cell and realign probe to photo diode Optimize phase between the two mixed signals see theory by changing the modulation frequency Maximum peak to peak sig nal should be found for modulati
23. scopy SAS experiment 5 a Basic SAS setup b Shape of photo diode signal without and with pump beam c Velocity only component in beam direction distribution of atoms in a two level system with increasing beam frequencies from left to right hitting the resonance on the picture in the center Positive velocities are defined in probe beam direction The thick arrows are pointing out the higher intensity of the pump beam The picture on the bottom of figure 1 shows the velocity distribution of the atoms in the ground and the excited state If the beam with frequency f is for example red detuned from resonance at fo to lower frequencies by 6 f fo atoms have to counterpropagate the beam with a velocity v i E to get a doppler shift that brings them back to resonance Only atoms with a velocity in a narrow interval around this velocity will be excited Therefore atoms that are resonant with the probe beam thin arrow have negative velocities because positive velocities are defined in probe beam direction Since probe and pump have the exact same frequency atoms that are resonant with the pump beam must have the same value but opposite sign of velocity Both pump and probe beam excite resonant atoms from the ground into the excited state leaving holes in the velocity distribution of the ground state When the laser beam is in resonance with the atomic transition 6 0 both pump and probe beam get resonant with atoms that hav
24. tal stays linear if the light is polarized parallel or perpendicular to the optical axis A periodically varied voltage V Vosin wmt is applied over the height d of the crystal with modulation frequency wm creating a maximal phase shift of Pmar M Bo Ly The modulated light can than be written as E t Evexpli wot M sin wmt 2 00 Eo exp iwot y Jn M exp inwmt 3 The phase modulated light may also be written as the Fourier sum of equally spaced fixed frequencies with amplitudes that are given by the nth order Bessel functions Jn as in 3 The first three order Bessel functions are shown in figure 5 Jo 0 7 0 87 0 7 0 34 ae 7 0 06 5 0 7 0 jo oO Q O gej 2 D 0 phase shift rad Figure 5 First three order Bessel functions Vertical line shows modulation depth in our setup at M 0 7 rad The following mathematical derivation is taken from Hall s and North s review 2000 9 The effect of a sample of length L with an absorption coefficient a w and index of refraction n w can be written in terms of a complex frequency dependent transmission function T w exp d iw with amplitude attenuation and phase shift Yy where w a w L 2 and p w n w Lw c The frequency dependence of is the absorption line shape and the frequency dependence of is the dispersion line shape The electric field transmitted through a sample then depends on the absorption
25. the absorption peak and the difference between them and hence the error signal stay small The optimal modulation frequencies for our setup therefore lie between 10 MHz and 35 MHz 3 Experiment Setup Figure 7 shows our realization of the Frequency Modulation Saturation Spec troscopy Laser Lock In the following paragraphs it will be explained how to start the laser setup in practice and how to get the optimized error signal 3 1 Optical Isolator The alignment of the optical isolator should be done as described in the user s manual 3 2 Electro optical Modulator EOM We have chosen a commercial broadband electro optic phase modulator EO PM NR C1 from Thorlabs to be able to tune the modulation frequency to optimize the phase between the photo diode signal and the modulation sig nal in the mixer Resonant mode modulators have the advantage of higher modulation depth M when used with a tank circuit The modulation signal for the broadband EOM in our setup however has to be amplified by a 2W amplifier minicircuits ZHL 1 2W Because of the poor impedance match ing most of the power is reflected back from the crystal into the amplifier which has to be cooled by an additional fan over the pre mounted heat sink We also considered attenuating the amplified signal before the EOM because the reflected power would be attenuated another time before reaching the 11 lens mirror waveplate optical isolator experiment polarizing beam
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