Electrooptic Modulators for Cavity Locking Case Study Physics 208, Electro-optics Peter Beyersdorf
Document info 1 Class Outline
Overview of cavity locking
LIGO input spectrum
The role of modulation sidebands
Methods to generate various input spectra
Advanced LIGO modulators 67 and test ideas and analytical formulas relating to signal extraction schemes. It was also extensively used in the tabletop prototype experiment of Chapter 4 to predict the state of the interferometer as well as design and debug the experiment. It is available through the STAIC website.[49]
3Ad.2 vTheancRSEed LIGOInterfero mCoetnfiguer ration Figure 3.6 shows the RSE interferometer, along with the labels relevant to this section. There are five longitudinal degrees of freedom in RSE. “Longitudinal” indicates that ETM2 Φ+ = φ3 + φ4 Φ = φ3 φ4 − − φ+ = φ0 + (φ1 + φ2)/2 φ = (φ1 φ2)/2 φ4 − − φs = φ5 + (φ1 + φ2)/2
ITM2 rac2
φ2 PRM ITM1 ETM1 φ r + 0 + bs rac1 Laser − φ φ rprm 1 3 − φ5
PD1 PD2 + rsem SEM − PD3
Figure 3.6: The optical layout of the RSE interferometer. rprm, rbs, and rsem are the reflectivities of the power recycling mirror, the beamsplitter, and the signal mirror, respectively. rac1 and rac2 are the reflectivities of the inline and perpendicular arm cavities. Sign conventions for the mirrors are indicated. the degree of freedom is along the axis of the laser beam, as opposed to the angular degrees of freedom, which are related to the tilting motions of the mirrors. The five degrees of freedom are comprised of four cavity phases plus the dark fringe Michelson condition. The four cavities are the power recycling cavity and signal extraction cavity, as well as the two arm cavities. The two arm cavities are represented in terms Cavity Locking
Laser field must resonate in the cavities for interferometer to operate as intended
External disturbances cause cavity length and laser frequency to fluctuate, thus active sensing and control is required to keep laser resonant in the cavities
Microwave technique for locking cavities developed by Pound was adapted to optical cavities by Drever and Hall, and used by Jan Hall for advances in laser stabilization that won he and Ted Hansch the 2005 Nobel prize in Physics
Pound-Drever-Hall technique for cavity locking (alternatively laser frequency stabilization) is widely used Conceptual Model
Consider a Fabry-Perot cavity illuminated by a laser of frequency f, slightly detuned from the cavity resonance by δf=f-fres
Ein Ec Et
Er L
Treat the input mirror as having a reflectivity of r1>0 when seen from inside the cavity, and the end mirror as having a reflectivity of r2>0 when seen from inside the cavity.
t2r e2ikL r (f) = r 1 2 The cavity reflectance is cav − 1 − 1 r r e2ikL − 1 2 Conceptual Model
2 2ikL t1r2e Cavity Reflectance rcav(f) = r1 − − 1 r r e2ikL − 1 2
1
0.5
-1.5 -1 -0.5 0 0.5 1 1.5
Modulation of the round trip phase (by modulating laser frequency or cavity length) at a frequency fm produces a modulation on the reflected power at fm if cavity is off-resonance
Fig. 2. The reflected light intensity from a Fabry–Perot cavity as a function Fig. 1. Transmission of a Fabry–Perot cavity vs frequency of the incident of laser frequency, near resonance. If you modulate the laser frequency, you light. This cavity has a fairly low finesse, about 12, to make the structure of can tell which side of resonance you are on by how the reflected power the transmission lines easy to see. changes. resonance with the cavity, you can’t tell just by looking at compared with the local oscillator’s signal via a mixer. We the reflected intensity whether the frequency needs to be in- can think of a mixer as a device whose output is the product creased or decreased to bring it back onto resonance. The of its inputs, so this output will contain signals at both dc ͑or derivative of the reflected intensity, however, is antisymmet- very low frequency͒ and twice the modulation frequency. It ric about resonance. If we were to measure this derivative, is the low frequency signal that we are interested in, since we would have an error signal that we can use to lock the that is what will tell us the derivative of the reflected inten- laser. Fortunately, this is not too hard to do: We can just vary sity. A low-pass filter on the output of the mixer isolates this the frequency a little bit and see how the reflected beam low frequency signal, which then goes through a servo am- responds. plifier and into the tuning port on the laser, locking the laser Above resonance, the derivative of the reflected intensity to the cavity. with respect to laser frequency is positive. If we vary the The Faraday isolator shown in Fig. 3 keeps the reflected laser’s frequency sinusoidally over a small range, then the beam from getting back into the laser and destabilizing it. reflected intensity will also vary sinusoidally, in phase with This isolator is not necessary for understanding the tech- the variation in frequency. ͑See Fig. 2.͒ nique, but it is essential in a real system. In practice, the Below resonance, this derivative is negative. Here the re- small amount of reflected beam that gets through the optical flected intensity will vary 180° out of phase from the fre- isolator is usually enough to destabilize the laser. Similarly, quency. On resonance the reflected intensity is at a mini- the phase shifter is not essential in an ideal system but is mum, and a small frequency variation will produce no useful in practice to compensate for unequal delays in the change in the reflected intensity. two signal paths. ͑In our example, it could just as easily go By comparing the variation in the reflected intensity with between the local oscillator and the Pockels cell.͒ the frequency variation we can tell which side of resonance This conceptual model is really only valid if you are dith- we are on. Once we have a measure of the derivative of the ering the laser frequency slowly. If you dither the frequency reflected intensity with respect to frequency, we can feed this too fast, the light resonating inside the cavity won’t have measurement back to the laser to hold it on resonance. The time to completely build up or settle down, and the output purpose of the Pound–Drever–Hall method is to do just this. will not follow the curve shown in Fig. 2. However, the Figure 3 shows a basic setup. Here the frequency is modu- technique still works at higher modulation frequencies, and lated with a Pockels cell,20 driven by some local oscillator. both the noise performance and bandwidth of the servo are The reflected beam is picked off with an optical isolator ͑a typically improved. Before we address a conceptual picture polarizing beamsplitter and a quarter-wave plate makes a that does apply to the high-frequency regime, we must estab- good isolator͒ and sent into a photodetector, whose output is lish a quantitative model.
Fig. 3. The basic layout for locking a cavity to a laser. Solid lines are optical paths and dashed lines are signal paths. The signal going to the laser controls its frequency.
80 Am. J. Phys., Vol. 69, No. 1, January 2001 Eric D. Black 80 Carrier-Sideband Picture
1 R(f)
0.5
-1.5 -1 -0.5 0 0.5 f/ffsr 1 1.5
sideband carrier sideband P(f) f0-fm f0 f0+fm A (phase) modulated beam has a carrier and sidebands spaced by frequency the modulation frequency fm, as long as fm≠nffsr, the modulation frequency is not an integer multiple of the cavity’s free spectral range (ffsr=c/2L), the sidebands will reflect off the cavity with high efficiency, while the carrier will “see” a much lower reflection coefficient due ot the frequency dependence of the reflection coefficient Carrier-Sideband Picture
2!
φr(f)
!
f/ffsr -1.5 -1 -0.5 0 0.5 1 1.5
sideband carrier sideband P(f) f0-fm f0 f0+fm The phase shift upon reflection from the cavity is essentially 0 for the sidebands, but the carrier has a phase shift that is a strong function of the detuning of the cavity (i.e. the difference between the cavities resonant frequency and the carrier frequency) when the carrier is near resonance (within a linewidth Δf=ffsr/F). Near resonance it has a slope of 2πF/ffsr, when measured as a function of frequency Carrier-Sideband Picture
Phase shift of carrier transforms phase modulation into amplitude modulation
negative phase shift no phase shift positive phase shift
Fig. 2. The reflected light intensity from a Fabry–Perot cavity as a function Fig. 1. Transmission of a Fabry–Perot cavity vs frequency of the incident of laser frequency, near resonance. If you modulate the laser frequency, you light. This cavity has a fairly low finesse, about 12, to make the structure of can tell which side of resonance you are on by how the reflected power the transmission lines easy to see. changes.
resonance with the cavity, you can’t tell just by looking at compared with the local oscillator’s signal via a mixer. We the reflected intensity whether the frequency needs to be in- can think of a mixer as a device whose output is the product creased or decreased to bring it back onto resonance. The of its inputs, so this output will contain signals at both dc ͑or derivative of the reflected intensity, however, is antisymmet- very low frequency͒ and twice the modulation frequency. It ric about resonance. If we were to measure this derivative, is the low frequency signal that we are interested in, since we would have an error signal that we can use to lock the that is what will tell us the derivative of the reflected inten- laser. Fortunately, this is not too hard to do: We can just vary sity. A low-pass filter on the output of the mixer isolates this the frequency a little bit and see how the reflected beam low frequency signal, which then goes through a servo am- responds. plifier and into the tuning port on the laser, locking the laser Above resonance, the derivative of the reflected intensity to the cavity. with respect to laser frequency is positive. If we vary the The Faraday isolator shown in Fig. 3 keeps the reflected laser’s frequency sinusoidally over a small range, then the beam from getting back into the laser and destabilizing it. reflected intensity will also vary sinusoidally, in phase with This isolator is not necessary for understanding the tech- the variation in frequency. ͑See Fig. 2.͒ nique, but it is essential in a real system. In practice, the Below resonance, this derivative is negative. Here the re- small amount of reflected beam that gets through the optical flected intensity will vary 180° out of phase from the fre- isolator is usually enough to destabilize the laser. Similarly, quency. On resonance the reflected intensity is at a mini- the phase shifter is not essential in an ideal system but is mum, and a small frequency variation will produce no useful in practice to compensate for unequal delays in the change in the reflected intensity. two signal paths. ͑In our example, it could just as easily go By comparing the variation in the reflected intensity with between the local oscillator and the Pockels cell.͒ the frequency variation we can tell which side of resonance This conceptual model is really only valid if you are dith- we are on. Once we have a measure of the derivative of the ering the laser frequency slowly. If you dither the frequency reflected intensity with respect to frequency, we can feed this too fast, the light resonating inside the cavity won’t have measurement back to the laser to hold it on resonance. The time to completely build up or settle down, and the output purpose of the Pound–Drever–Hall method is to do just this. will not follow the curve shown in Fig. 2. However, the Figure 3 shows a basic setup. Here the frequency is modu- technique still works at higher modulation frequencies, and lated with a Pockels cell,20 driven by some local oscillator. both the noise performance and bandwidth of the servo are The reflected beam is picked off with an optical isolator ͑a typically improved. Before we address a conceptual picture polarizing beamsplitter and a quarter-wave plate makes a that does apply to the high-frequency regime, we must estab- good isolator͒ and sent into a photodetector, whose output is lish a quantitative model.
Fig. 3. The basic layout for locking a cavity to a laser. Solid lines are optical paths and dashed lines are signal paths. The signal going to the laser controls its frequency.
80 Am. J. Phys., Vol. 69, No. 1, January 2001 Eric D. Black 80 Fig. 2. The reflected light intensity from a Fabry–Perot cavity as a function Fig. 1. Transmission of a Fabry–Perot cavity vs frequency of the incident of laser frequency, near resonance. If you modulate the laser frequency, you light. This cavity has a fairly low finesse, about 12, to make the structure of can tell which side of resonance you are on by how the reflected power the transmission lines easy to see. changes.
resonance with the cavity, you can’t tell just by looking at compared with the local oscillator’s signal via a mixer. We the reflected intensity whether the frequency needs to be in- can think of a mixer as a device whose output is the product creased or decreased to bring it back onto resonance. The of its inputs, so this output will contain signals at both dc ͑or derivative of the reflected intensity, however, is antisymmet- very low frequency͒ and twice the modulation frequency. It ric about resonance. If we were to measure this derivative, is the low frequency signal that we are interested in, since we would have an error signal that we can use to lock the that is what will tell us the derivative of the reflected inten- laser. Fortunately, this is not too hard to do: We can just vary sity. A low-pass filter on the output of the mixer isolates this Experimentathe lfrequency Sca little bithemaand see how the reflectedtibeamc low frequency signal, which then goes through a servo am- responds. plifier and into the tuning port on the laser, locking the laser Above resonance, the derivative of the reflected intensity to the cavity. with respect to laser frequency is positive. If we vary the The Faraday isolator shown in Fig. 3 keeps the reflected laser’s frequency sinusoidally over a small range, then the beam from getting back into the laser and destabilizing it. reflected intensity will also vary sinusoidally, in phase with This isolator is not necessary for understanding the tech- the variation in frequency. ͑See Fig. 2.͒ Actuatornique, but it is essential in a real system. In practice, the Laser Pockels Cell Cavity Below resonance, (PC)this derivative is negative. Here the re- small amount of reflected beam that gets through the optical flected intensity will vary 180° out of phase from the fre- isolator is usually enough to destabilize the laser. Similarly, Modulation of the laser frequency quency. On resonance the reflected intensity is at a mini- the phase shifter is not essential in an ideal system but is mum, and a small frequency variation will produce no useful in practice to compensate for unequal delays in the is accomplished by a pockels cell change in the reflected intensity. two signal paths. ͑In our example, it could just as easily go By comparing the variation in the reflected intensity with between the local oscillator and the Pockels cell.͒ Photodetector the frequency variation we can tell which side of resonance This conceptual model is really only valid if you are dith- driven by a sinusoidal oscillator. we are on. Once we have a measure of the derivative of the ering the laser frequency slowly. If you dither the frequency reflected intensity with respect to frequency, we can feed this too fast, the light resonating inside the cavity won’t have measurement back to the laser to hold it on resonance. The time to completely build up or settle down, and the output purpose of the Pound–Drever–Hall method is to do just this. will not follow the curve shown in Fig. 2. However, the Any modulation at fm on the light Figure 3 shows a basic setup. Here the frequency is modu- technique still works at higher modulation frequencies, and lated with a Pockels cell,20 driven by some local oscillator. both the noise performance and bandwidth of the servo are The reflected Localbeam Oscillatoris picked off withMixeran opticallow passisolatorServo͑a Amp typically improved. Before we address a conceptual picture reflected from the cavity is polarizing beamsplitter (LO) and a quarter-wave filterplate makes a that does apply to the high-frequency regime, we must estab- good isolator͒ and sent into a photodetector, whose output is lish a quantitative model. detected (by mixing with the local PDH schematic for locking a cavity onto a laser Figure 1: The basic layout for locking a cavity to a laser. Solid lines are optical paths, and dashed oscillator) and used to provide lines are signal paths. The signal going to the far mirror of the cavity controls its position. feedback to the cavity (or laser). 2 A conceptual model
Fig. 1 shows a basic Pound-Drever-Hall setup. This arrangement is for locking a cavity to a laser, and Fig. 3. The basic layout for locking a 1 it is the setup you would use to measure the length noise in the cavity. You send the beam into the cavity to a laser. Solid lines are optical cavity; a photodetector looks at the reflected beam; and its output goes to an actuator that controls the paths and dashed lines are signal paths. length of the cavity. If you have set up the feedback correctly, the system will automatically adjust The signal going to the laser controls the length of the cavity until the light is resonant and then hold it there. The feedback circuit will its frequency. compensate for any disturbance (within reason) that tries to bump the system out of resonance. If you keep a record of how much force the feedback circuit supplies, you have a measurement of the noise in the cavity. Setting up the right kind of feedback is a little tricky. The system has to have some way of telling which way it should push to bring the system back on resonance. It can’t tell just by looking at the
1Locking the laser to the cavity follows essentially the same design, the only difference being that you would feed back to the laser, rather than the cavity. 80 PDHAm. sJ.chemaPhys., Vol.t69,ic No.fo1,rJanuary lockin2001 g a laser onto a cavity Eric D. Black 80
3 Quantitative Model
Recall that the phasor amplitude for phase modulated light can be written as
iωt i(ωt Ωt) i(ωt+Ωt) E E J (m)e + iJ (m) e − + e in ≈ 0 0 1 ! ! "" where m is the modulation depth, Ω=2πfm is the modulation frequency and ω=2πf0 is the carrier frequency. The field reflected from the cavity is
iωt i(ωt Ωt) i(ωt+Ωt) E E r(ω)J (m)e + iJ (m) r(ω Ω)e − + r(ω + Ω)e r ≈ 0 0 1 − which can be! approximated by ! ""
i(ωt+2π δf/f ) i(ωt Ωt) i(ωt+Ωt) E E r J (m)e F fsr + iJ (m) e − + e ≈ 0 − 0 0 1 ! ! "" when the carrier is near resonance, the sidebands are far off- resonance and r1,r2≈1. Here r0 is the magnitude of the cavity reflection on resonance and δf is the detuning of the laser from the cavity Quantitative Model
The relative intensity detected by a photodetector is proportional to the magnitude of the field squared i(ωt+2π δf/f ) i(ωt Ωt) i(ωt+Ωt) E E r J (m)e F fsr + iJ (m) e − + e ≈ 0 − 0 0 1 ! ! "" giving 2 2 2 2 I E∗E = E0 r0J0 (m) + 2J1 (m) ∝ !
i(2π δf/ffsr +Ωt) i(2π δf/ffsr +Ωt) ir0J0(m)J1(m) e F e− F − ! − " # i(2π δf/f Ωt) i(2π δf/f Ωt) + e F fsr − e− F fsr − − $ " # 2 2iΩt 2iΩt + J1(m) e + e− or $ E∗E = DC terms + 2ω% terms & + 2E2r J (m)J (m) sin (2π δf/f + Ωt) + sin (2π δf/f Ωt) 0 0 0 1 F fsr F fsr − ! " E∗E = DC terms + 2ω terms + 4E2r J (m)J (m) sin(2π δf/f ) cos(Ωt) 0 0 0 1 F fsr This is the amplitude of the modulated power at the modulation frequency, near resonance (δf< Actuator Laser Pockels Cell Cavity (PC) Consider a detected photocurrent Photodetector of the form Vdet Vlo Vdet = V0 + VΩ cos(Ωt) + V2Ω cos(2Ωt) Vmix Vlpf Local Oscillator Mixer low pass Servo Amp it gets “mixed” with a “local oscillator” (LO) filter of the form Vlo=cos(Ωt) which Fisigu re equi1: The basivc laayoluent for ltock intg aoca vimty toua llatser.iplSolidiclineas atreioopticnal paths, and dashed lines are signal paths. The signal going to the far mirror of the cavity controls its position. 2 Vmix = VloVdet = V0 cos(Ωt) + VΩ cos (Ωt) + V2Ω cos(2Ωt) cos(Ωt) 2 A conceptual model Fig. 1 shows a basic Pound-Drever-Hall setup. This arrangement is for locking a cavity to a laser, and it is the setup you would use to measure the length noise in the cavity.1 You send the beam into the cavity; a photodetector looks at the reflected beam; and its output goes to an actuator that controls the the mixed signal is then low palengssth of thfile cavtitye. Ifryoeu hdave stet oup thegfeeedbtack correctly, the system will automatically adjust the length of the cavity until the light is resonant and then hold it there. The feedback circuit will t compensate for any disturbance (within reason) that tries to bump the system out of resonance. If you keep a record of how much force the fe2edback circuit supplies, you have a measurement of the noise 1 1 This is 4 E0 r0J0(m)J1(m)δf Vlpf lim Vmixdτ = VinΩthe cavity. ≈ T T 2 Setting up theanrightdki npd orfofeviedbdeack sis aalit tlme trieacky.suTherseys teofm ha stheto hav edesomtue wany oinf tegll inofg →∞ t T which way it should push to bring the system back on resonance. It can’t tell just by looking at the ! − the laser from the cavity (or vice versa) 1Locking the laser to the cavity follows essentially the same design, the only difference being that you would feed back to the laser, rather than the cavity. 3 In this case, if ⍀ϭ⍀Ј our dc signal vanishes! If we want to F͑͒F*͑ϩ⍀͒ϪF*͑͒F͑Ϫ⍀͒ In this case, if ⍀ϭ⍀Ј our dc signal vanishes! If we want to F͑͒F*͑ϩ⍀͒ϪF*͑͒F͑Ϫ⍀͒ measure the error signal when the modulation frequency is 2 measure the error signal when the modulation frequency is d d͉F͉ 2 low we mustlow wematchmustthematchphasestheofphasesthe twoof thesignalstwo signalsgoing intogoing into d d͉F͉ Ϸ2 ReϷF2͑Re͒ F͑F͒*͑͒F*⍀͑Ϸ͒ ⍀Ϸ ⍀, ⍀, the mixer.theTurningmixer. Turninga sine intoa sinea cosineinto aiscosinea simpleis a mattersimple ofmatter of ͭ ͭd d ͮ ͮ d d introducing a 90° phase shift, which we can do with a phase introducing a 90° phase shift, which we can do with a phasewhich is purely real. Of the ⍀ terms, only the cosine term in shifter ͑or delay line͒, as shown in Fig. 3. which is purely real. Of the ⍀ terms, only the cosine term in shifter ͑or delay line͒, as shown in Fig. 3. Eq. ͑3.3͒ survives. In practice, you need a phase shifter even when the modu- Eq. ͑3.3͒ survives. In practice, you need a phase shifter even when the modu-If we approximate ͱP P ϷP /2, the reflected power lation frequency is high. There are almost always unequal If we approximatec sͱPcP0 sϷP0/2, the reflected power lation frequency is high. There are almost always unequalfrom Eq. ͑3.3͒ becomes delays indelaysthe twoin thesignaltwopathssignalthatpathsneedthattoneedbe compensatedto be compensated from Eq. ͑3.3͒ becomes 2 for to producefor to producetwo puretwosinepuretermssineattermsthe inputsat theofinputsthe mixer.of the mixer. d͉F͉ d͉F͉2 The outputTheofoutputthe mixerof thewhenmixerthewhenphasestheofphasesits twoof inputsits twoareinputs are PrefϷ͑PconstantϷ͑constantterms͒termsϩP0͒ϩP ⍀ cos⍀⍀cost ⍀t ref d 0 d not matchednot matchedcan producecan producesome odd-lookingsome odd-lookingerror signalserror signals 7 7 ͑see Bjorklund͑see Bjorklund͒, and when͒, andsettingwhenupsettinga Poundup a–PoundDrever––DreverHall –Hall ϩ͑2⍀ϩterms͑2⍀ ͒terms, ͒, lock youlockusuallyyou scanusuallythescanlaserthefrequencylaser frequencyand empiricallyand empirically adjust theadjustphasetheinphaseone signalin onepathsignaluntilpathyouuntilgetyouan geterroran errorin agreementin agreementwith ourwithexpectationour expectationfrom fromthe conceptualthe conceptual signal thatsignallooksthatlikelooksFig.like7. Fig. 7. model.model. The mixerThe willmixerfilterwilloutfiltereverythingout everythingbut thebuttermthe thattermvar-that var- ies as cosies ⍀ast.cos͑We⍀tmay. ͑Wehavemaytohaveadjustto adjustthe phasethe phaseof theofsignalthe signal before beforewe feedweitfeedintoittheintomixer.the mixer.͒ The͒PoundThe Pound–Drever–Drever–Hall–Hall IV. UNDERSTANDING THE QUANTITATIVE IV. UNDERSTANDING THE QUANTITATIVE error signalerror signalis thenis then MODEL MODEL 2 2 d͉F͉2 d͉F͉ d͉F͉2 d͉F͉ ⑀ϭP ⑀ϭP0⍀Ϸ2⍀ͱPϷP2ͱPcPs ⍀. ⍀. 0 d d c s d d A. Slow A.modulation:Slow modulation:QuantifyingQuantifyingthe conceptualthe conceptualmodel model Figure Figure6 shows6 showsa plot aofplotthisoferrorthis signal.error signal. Let’s seeLet’showseethehowquantitativethe quantitativemodel comparesmodel compareswith ourwith our conceptualconceptualmodel, wheremodel,wewhereslowlywe ditheredslowly ditheredthe laserthefre-laser fre- quency and looked at the reflected power. For our phaseB. FastB.modulationFast modulationnear resonance:near resonance:PoundPound–Drever–Drever– – quency and looked at the reflected power. For our phase Hall in practice modulatedmodulatedbeam, thebeam,instantaneousthe instantaneousfrequencyfrequencyis is Hall in practice d When Whenthe carrierthe carrieris nearis resonancenear resonanceand theandmodulationthe modulation d͑t͒ϭ ͑tϩ sin ⍀t͒ϭϩ⍀ cos ⍀t. ͑t͒ϭ ͑tϩdt sin ⍀t͒ϭϩ⍀ cos ⍀t. frequencyfrequencyis highisenoughhigh enoughthat thethatsidebandsthe sidebandsare not,arewenot,canwe can dt Ϯ⍀ assumeassumethat thethatsidebandsthe sidebandsare totallyare totallyreflected,reflected,F(ϮF⍀( ) ) The reflected power is just P ϭP ͉F()͉2, and we The reflected power is just P ϭP ͉Fref()͉02, and we ϷϪ1. Then the expression might expect it to vary overreftime0as ϷϪ1. Then the expression might expect it to vary over time as F͑͒F*͑ϩ⍀͒ϪF*͑͒F͑Ϫ⍀͒ϷϪi2 Im͕F͖͑͒, dP F͑͒F*͑ϩ⍀͒ϪF*͑͒F͑Ϫ⍀͒ϷϪi2 Im͕F͖͑͒, dP ref ͑4.1͒ Pref͑ϩ⍀ cos ⍀t͒ϷPref͑ref͒ϩ Er⍀rcoso⍀rt Signal ͑4.1͒ Pref͑ϩ⍀ cos ⍀t͒ϷPref͑͒ϩ ⍀dcos ⍀t is purely imaginary. In this regime, the cosine term in Eq. d is purely imaginary. In this regime, the cosine term in Eq. 2 ͑3.3͒ is negligible, and our error signal becomes 2 d͉F͉ ͑3.3͒ is negligible, and our error signal becomes ϷP ͑d͉͒Fϩ͉P ⍀ cos ⍀t. ref 0 ⑀ϭϪ2ͱP P Im͕F͑͒F*͑ϩ⍀͒ϪF*͑͒F͑Ϫ⍀͖͒. ϷPref͑͒ϩP0 ⍀dcos ⍀t. c s d ⑀ϭϪ2ͱPcPs Im͕F͑͒F*͑ϩ⍀͒ϪF*͑͒F͑Ϫ⍀͖͒. In the conceptual model, we dithered the frequency of the Figure 7 shows a plot of this error signal. In the laserconceptualadiabatically,model,Theslowlywe demoditheredenoughduthethatfrequencylathetestandingd of(anthewaved in-Figurelow 7paNearshowsssresonance a filplottoferthethisereflectedderror sigsignal.napowerl) essentiallyis vanishes, laser adiabatically, slowly enough that the standing wave in- Nearsinceresonance͉F()͉2theϷ0.reflectedWe do wantpowerto retainessentiallyterms tovanishes,first order in side the cavity was always in equilibrium with the incident 2 side the beam.cavity Wewascanalwaysaexpress mineathisequilibriumsuin therequantitative ofwith the incidentmodel labysemak-r’sinces de͉FF((tu))͉, nϷhowever,in0. Weg dofrto wantapproximateomto theretainthe termserrortosignal,first order in beam. Weingcan⍀ expressvery small.this Inin thethisquantitativeregime the expressionmodel by mak- F(), however, to approximate the error signal, ing ⍀ very small. In thiscaregimevity thereexpressionsonance and is callePdref Ϸan2Ps Ϫ“e4ͱrPrcPos rIm ͕sigF͑͒na͖sinl⍀”tϩ͑2⍀ terms͒. PrefϷ2PsϪ4ͱPcPs Im͕F͖͑͒sin ⍀tϩ͑2⍀ terms͒. Fig. 7. The Pound–Drever–Hall error signal, ⑀/2ͱPcPs vs /⌬fsr , when Fig. 6. The Pound–Drever–Hall error signal, ⑀/2ͱPcPs vs /⌬fsr , when the modulation frequency is high. Here, the modulation frequency is about the modulation efrequencyrror issiglow.naThel amodulations a fufrequencynctionis ofabout halfFig.a 7. The20 linewidths:Pound–Dereverrroughlyro–rH all4%sigerrorofnaasignal,freel aspectrals ⑀/2aͱ Pfurange,cPncs vstwithio/⌬na cavityfsrof, w henfinesse of Fig. 6. Thelinewidth:Pound–Daboutrever–10HϪall3 oferrora freesignal,spectral⑀/2ͱrange,PcPs withvs a/⌬cavityfsr , finessewhen of 500.the modulation500. frequency is high. Here, the modulation frequency is about the modulation frequencydetuis low.ninTheg modulation(for smafrequencyll tunisinaboutg rhalfanga e) 20 linewidths: roughlydetu4%ninofga free(fospectralr larrange,ge tuwithnina cavityg ranfinessege)of linewidth: about 10Ϫ3 of a free spectral range, with a cavity finesse of 500. 500. 83 Am. J. Phys., Vol. 69, No. 1, January 2001 Eric D. Black 83 83 Am. J. Phys., Vol. 69, No. 1, January 2001 Eric D. Black 83 Negative Feedback noise δx x1 system -Gx1 G Gx1 inv In the steady state we must have δx-Gx1=x1 giving δx x = 1 1 + G meaning the noise is suppressed by a factor of 1+G. The “noise” can be laser frequency or cavity length fluctuations 77 67 range. This ensures the carrier frequency is at an anti-resonant node, such that and test ideas and analytical formularesos rnaelnceatiings satiosfiedsigwnahenl exthetarramctcaiovnitiesschemes.resonate asItwelwl.as f also extensively used in the tabletop prototype experiment of Cf ha= pt(2ner+41)tofsrprprcedict the (3.40) 1 1 2 state of the interferometer as well as design and debug the experiment. It is available where n1 is an integer, and ffsr = c/(2lprc). This argument applies to each of through the STAIC website.[49] prc the modulation frequencies, with a different integer, n2 for the second RF sideband. Clearly the difference of the two frequencies will be some integer multiple of the free Caspviectratl raienge. s in LIGO 3.2 The RSE InterferoItm’s desietrederthat one of the RF sideband frequencies resonates in the signal cavity, while the other does not. The non-resonant RF sideband then acts as a local oscillator Figure 3.6 shows the RSE interferometer, along with the labels relevant to this section. for the resonant RF sideband, thus providing a signal extraction port for the φs There are five longitudinal degrees ofdegfrreedoee of mfreedoinm.RSE.An ex“aLomplnge soitludiutionnaisl”shoiwndin ican Fitesguretha3.1t0, where f2 = 3f1. For ETM2 Φ+ = φ3 + φ4 Φ = φ φ Input spectrum 3 4 f − − f2 f1 f0 +f1 +f2 φ+ = φ0 + (φ1 + φ2)/2 − − φ = (φ1 φ2)/2 φ4 − − Power recycling cavity φs = φ5 + (φ1 + φ2)/2 f spectral schematic Signal cavity ITM2 f spectral schematic rac2 Figure 3.10: Power and signal cavity spectral schematic for broadband RSE. The RF sidebandsφ2 are related by f2 = 3f1. PRM ITM1 ETM1 φ r + 0 + bs rac1 Laser − φ φ rprm broadband 1RSE, some care must3 be taken as to which frequency is resonant in the − signal cavity. For example, in the f = 3f case, clearly if f were resonant in the φ5 2 1 1 signal cavity, f2 would be resonant as well. PD1 PD2 + rsemThe coSnstEMraint on which frequency is resonant in the signal cavity doesn’t usually − 16 apply to a detuned RSE interferometer. Since both the frequency of one of the RF sidebaPD3nds, and the frequency of the detuning must be resonant in the signal cavity,4 4Note that this is not the detuned frequency for the gravitational wave. Figure 3.6: The optical layout of the RSE interferometer. rprm, rbs, and rsem are the reflectivities of the power recycling mirror, the beamsplitter, and the signal mirror, respectively. rac1 and rac2 are the reflectivities of the inline and perpendicular arm cavities. Sign conventions for the mirrors are indicated. the degree of freedom is along the axis of the laser beam, as opposed to the angular degrees of freedom, which are related to the tilting motions of the mirrors. The five degrees of freedom are comprised of four cavity phases plus the dark fringe Michelson condition. The four cavities are the power recycling cavity and signal extraction cavity, as well as the two arm cavities. The two arm cavities are represented in terms LIGO Input Spectrum Various frequency components are necessary for length and alignment sensing of the many cavities and interferometers in the LIGO detector phase modulation sidebands for locking “power recycling cavity” and Michelson interferometer (9 MHz) phase modulation sidebands for locking the arms (180 MHz) carrier P(f) 9 MHz sidebands f 180 MHZ sidebands 17 LIGO Modulators Initial LIGO LiNbO3 slabs with 10mm x 10mm clear aperture for Initial LIGO (operates with up to 10W of power) Transverse modulation resonant circuit geometry 9MHz phase modulation sidebands for interferometer sensing Advanced LIGO RTP (RbTiOPO4) (operates with up to 300W of power) 9 MHz and 180 MHz PM sidebands for interferometer Sensing 18 Role of Modulation Sidebands t2 Cavity Reflectance rcav(f) = r − 1 r2ei2πf/ffsr − |r(f)| -1.5 -1 -0.5 0 0.5 1 1.5 f/ffsr carrier P(f) 9 MHz sidebands f 180 MHZ sidebands Sideband frequencies are such that only one component is resonant in the cavity of interest 19 Role of Modulation Sidebands t2 Cavity Reflectance rcav(f) = r − 1 r2ei2πf/ffsr − φr=arg(r(f)) 2! f/ffsr -1.5 -1 -0.5 0 0.5 1 1.5 carrier P(f) 9 MHz sidebands f 180 MHZ sidebands Component that resonates in the cavity acquires a phase shift upon reflection that is a function of the cavities detuning. 20 Higher Order Sidebands Higher order modulation harmonics P(f) f P(f) sidebands of sidebands f Demodulation gives the sum of all frequency components spaced by flo. Higher order modulation harmonics and sidebands-of-sidebands can produce unwanted contributions 21 Unintended Sidebands Higher order (for example 18 MHZ sidebands from the 9 MHz modulator)sidebands are problematic because the intermodulation they produce can obscure intended signals Solutions: Low modulation depth Sideband cancellation Parallel modulation 22 Serial Modulation For small modulation depth J1(m)≈m/2 and J2(m)≈m/24 so intermodulation depth is m1m2/48 9 MHz 180 MHz Power spectrum after Power spectrum after 9 MHz modulator 9 MHz and 180 MHz modulators 23 Harmonic Compensation Through proper adjustment of amplitude and phase of 189 and 171 MHz signals, the intermodulation harmonics can be cancelled 9 MHz 180 MHz x - = Power spectrum after Power spectrum after Modulation spectrum 9 MHz and 180 MHz 9 MHz and 180 MHz from third modulator modulators modulators Parallel Modulation A Mach-Zehnder interferometer can 180 MHz be used to combine modulation sidebands from two independent modulators Drawbacks include reduction in 9 MHz effective modulatin depth by a factor of 2 due to sidebands lost to the unused port and increased complexity of maintaining Mach-Zehnder interference condition 25 Single Sideband Generation Some control and readout schemes require a “Single sideband” rather than a pair of phase modulated sidebands or amplitude modulated sidebands Example: mapping out the frequency response of a detuned interferometer 26 Single Sideband Generation Phase modulation produces a pair of sidebands at the modulation frequency that are each in phase with the carrier (at some instant in time) Amplitude modulation produces a pair of sidebands a the modulation frequency with one in phase with the carrier while the other is π out-of-phase with the carrier. Combining amplitude and phase modulation at the same frequency, one sideband in a pair can be enhanced while the other is suppressed. Ad been p of a g l si t An a o o g de c f v k tw se i Sub-Ca anc v b en l en fr t t an . e e r e r o l d ist r o T d LIGO na a p h t equenc a o is ha t e asin se se i v o pha e wa r d f s w d o y r gl r i y th se e r ie r Gen Single SidebandProduction e r a t io n 28 Advanced LIGO modulators Three sets of modulation electrodes on one crystal 29 Modulator Crystals To avoid interference effects from reflections off the modulator faces, the crystal has a 2x2.8° wedge The wedge leads to polarization dependent transmission angle, and thus the modulator also acts like a polarizer 30 Resonant Circuit The resonant circuit is designed to have 50Ω imp186 edance at theAP PErNDeIXsoB. ELnanECTRONItCS frequency, but high impedance at DC and low frequencies. Figure B.10: Alternative resonant transformer circuit with a capacitive voltage divider. the example, the capacitor C needs to have about 1.6 nF. It behaves differently from the previously discussed circuits in so far as there are now two resonances close to each other (a parallel resonance and a series resonance). Figure B.11 shows the input impedance of this circuit. The presence of the two resonances makes the adjustment more difficult. On the other hand, it may be an advantage that at low frequencies the input impedance is very high (instead of a short circuit as in Figure B.9). 100 ] ! 10 Impedance [ 1 9.4 9.6 9.8 10 10.2 10.4 10.6 Frequency [MHz] 31 Figure B.11: Input impedance of the alternative resonant transformer circuit with a capacitive voltage divider. B.3 The automatic lock acquisition circuit Overview of the function: The function of the automatic lock acquisition circuit can best be explained with Figure B.12. The topmost trace (labelled ‘PD5’) represents the light power that is transmitted through the reference cavity, which is detected by photodiode PD5 in Figure A.1. References Guido Mueller, David Reitze, Haisheng Rong, David Tanner, Sany Yoshida, and Jordan Camp “Reference Design Document for the Advanced LIGO Input Optics”, LIGO internal document T010002-00 (2000) James Mason, “Signal Extraction and Optical Design for an Advanced Gravitational Wave Interferometer” Ph.D. Thesis, CIT (2001) M Gray, D Shaddock, C Mow-Lowry, D. McClelland “Tunable Power Recycled RSE Michelson for LIGO II”, Internal LIGO document G000227-00-D G. Heinzel “Advanced optical techniques for laser-interferometric gravitational-wave detectors” Ph.D. Thesis MPQ (1999)OSA E Black “An introduction to Pound – Drever – Hall laser frequency stabilization”, Am. J. Phys. 69 (2001)