We present a scheme for locking optical cavities with arbitrary detuning by many linewidths from resonance using an electro-optic modulator that can provide arbitrary ratios of amplitude-to-phase modulation. We demonstrate our scheme on a Fabry–Perot cavity, and show that a well-behaved linear error signal can be obtained by demodulating the reflected light from a cavity that is detuned by several linewidths.National Science Foundation (U.S.) (PHY-0757058

Ackley, Sarah F

Davis, Emily J.

Evans, Matthew J

Mavalvala, Nergis

Yam, William

Optics Letters

arXiv

W. Yam

E. Davis

S. Ackley

M. Evans

N. Mavalvala

Crossref

Continuously tunable modulation scheme for precision control of optical cavities with variable detuning

We present a scheme for locking optical cavities with arbitrary detuning many
line widths from resonance using an electro-optic modulator that can provide
arbitrary ratios of amplitude to phase modulation. We demonstrate our scheme on
a Fabry-Perot cavity, and show that a well-behaved linear error signal can be
obtained by demodulating the reflected light from a cavity that is detuned by
several line widths.Comment: 5 pages, 4 figure

Davis, Emily

Ackley, Sarah

Evans, Matthew

arXiv.org e-Print Archive

English

A continuously tunable modulation scheme for precision control of
  optical cavities with variable detuning

DSpace@MIT

arXiv:1505.05445v1  [physics.optics]  20 May 2015A continuously tunable modulation scheme for precision control of optical cavitieswith variable detuningW. Yam,1 E. Davis,2 S. Ackley,3 M. Evans,1 and N. Mavalvala11LIGO Laboratory, Massachusetts Institute of Technology, Cambridge, MA 02139.2Department of Physics, University of California, Berkeley, CA 09125.3University of California, San Francisco, CA 94110.We present a scheme for locking optical cavities with arbitrary detuning many line widths fromresonance using an electro-optic modulator that can provide arbitrary ratios of amplitude to phasemodulation. We demonstrate our scheme on a Fabry-Perot cavity, and show that a well-behavedlinear error signal can be obtained by demodulating the reflected light from a cavity that is detunedby several line widths.I. INTRODUCTIONHigh finesse optical cavities that are operated detunedfrom resonance are ubiquitously present in a variety ofapplications that include cavity optomechanics [1], laserinterferometer gravitational wave detectors [2–6], cavityQED experiments [7, 8], cold atom microscopy [9], andion trapping [10], to name a few.Acquiring low noise, high bandwidth signals for lock-ing optical cavities that are detuned from resonance by anarbitrary amount is a challenge. The Pound-Drever-Hall(PDH) locking method [11] is perhaps the most commontechnique used for locking cavities on resonance. PDH er-ror signals are generated by phase modulating the lightincident on the cavity at RF frequencies that typicallyplace the modulation sidebands outside the line widthof the cavity, and demodulating the light reflected fromthe cavity. This is an extremely efficient and powerfultechnique for cavities operating on resonance, but the er-ror signal becomes nonlinear and even changes sign whenthe cavity is detuned from resonance, making it unsuit-able for use at detunings exceeding a small fraction ofa line width. Conversely, linear error signals for cavitiesthat are detuned by half a line width can be obtained byamplitude modulating the incident light. An alternativemethod of locking a detuned cavity using the approxi-mately linear regime of the intensity of the transmittedlight – “side of fringe” or “DC” locking – is simple to im-plement, but is prone to noise couplings and offset drifts.Again the linear regime of the signal is a small fractionof a cavity line width, and the signal vanishes on reso-nance. Certainly, neither PDH nor side of fringe methodsare useful for cavities that are detuned from resonance byseveral line widths. We demonstrate a technique wherean admixture of amplitude modulation (AM) and phasemodulation (PM) yields useful error signals for cavitiesthat are on resonance or continuously detuned by manyline widths.Cusack et al. proposed the “universally tunable mod-ulator (UTM)” where an electro-optic amplitude modu-lator was modified to generate an admixture of AM andPM [12]. They connected the inputs of the two crystals ofthe electro-optic modulator to separate electronic drivesand showed that arbitrary ratios of AM and PM couldbe generated by controlling the relative phase differencebetween the two drives. Here we use a similar implemen-tation of a dual input electrooptic modulator to generatesmoothly varying error signals for a cavity operated onresonance or continuously detuned by several line widths.II. THEORYTo determine the error signals from an optical cavitydriven with an arbitrary combination of phase and ampli-tude modulated light, we review the theory of modulatedlight from the UTM, and then derive expressions for theerror signals for arbitrary detuning of the cavity.The total field of a light beam modulated at a singlefrequency, ωm, can be written asE = E0[1 + Re[A˜ exp (iωmt)]]exp(iRe[P˜ eiωmt])(1)where we have A˜ = AeiφA and P˜ = PeiφP . If we assumesmall modulation depths for both AM (A≪ 1) and PM(P ≪ 1), we haveE = E0(1 + Re[A˜ exp (iωmt)] + iRe[P˜ exp (iωmt)])(2)Regrouping terms corresponding to the sidebands andcarrier:E = E0 + E+ exp (iωmt) + E− exp (−iωmt) (3)where we have E+ =A˜+ iP˜2E0 and E− =A˜∗ + iP˜ ∗2E0,respectively.A. The Universal Tunable ModulatorThe UTM of Cusack et al consists of a modified electro-optic amplitude modulator, shown in Figure 1. An am-plitude modulator consists of two identical birefringentcrystals with their optic axes aligned orthogonal to eachother, and aligned such that they are left and right di-agonal with respect to a vertical polarizer at the outputof the modulator (see Figure 1). When the two crystals2are driven exactly out of phase, the electric field alongone direction is phase advanced while the orthogonal po-larization is phase retarded. The average phase is zero.There is no net phase modulation from elliptical to circu-lar polarization and back again. Filtering this oscillatingpolarization with a vertical polarizer produces an ampli-tude modulated beam.When the crystals are driven by separate phasors δ˜1and δ˜2, any modulation state within the space of AMand PM can be achieved. The optical field exiting theUTM is given by:~E = E˜LeiRe[δ˜1eiωmt]Lˆ+ E˜ReiRe[δ˜2eiωmt]Rˆ (4)where E˜L = ELeiσL and E˜R = EReiσR are the electricfield components along the left and right diagonal of theUTM vertical axis, and δ˜1 = δ1eiσ1 and δ˜2 = δ2eiσ2 arethe two drive phasors. Assuming left and right diagonalfield components are equal, i.e., EL = ER = Ein/√2,and the signal drives δ1,2 are small, we have:~E =Ein√2[(1 + iRe[δ˜1eiωmt])Lˆ+ eiσ(1 + iRe[δ˜2eiωmt])Rˆ](5)where σ = σR − σL. Passing through the vertical polar-izer filters along the direction of Lˆ+Rˆ√2, giving:Eout = E0[1 +12tanσ2Re[(δ˜1 − δ˜2)eiωmt]+i(12Re[(δ˜1 + δ˜2)eiωmt])] (6)where E0 =Ein2|1 + eiσ | = Ein cos(σ/2), discarding aglobal phase factor. By comparing equations (2) and(6), we see that the transfer function from UTM inputsignals to the output modulation is:P˜ =12(δ˜1 + δ˜2) (7)A˜ =12tan(σ2)(δ˜1 − δ˜2) (8)Here we have assumed that left and right diagonal fieldcomponents entering the UTM are equal in magnitude.This transfer function differs from that derived by Cu-sack et al. by a factor of cos(σ/2), where the carriersuppression has been factored out. If the input field iscircularly polarized, there is a symmetry between the twomodulation phasors and the modulation depths are equalwhen driving each input separately.B. Generating a cavity error signal withmodulated lightFor a Fabry-Perot cavity of length L with input andend mirror reflectivities r1 and r2, respectively, and cor-responding transmittances of t1 and t2, the reflection co-efficient, defined as the ratio of the cavity reflected fieldFIG. 1. An electro-optic amplitude modulator in normal op-eration (upper) and as a UTM (lower). In normal AM opera-tion, a single input voltage source drives two crystals orientedin orthogonal directions. A UTM consists of the same twocrystals wired to two independent sources. The incident lightis circularly polarized.to the incident field, isrcav(φ) = r1 − t21r2eiφ1− r1r2eiφ (9)where the round-trip phase shift is given by φ = 2ωL/c.Rewriting this in terms of line widths of detuning fromresonance, we have:rcav(nlw) = r1 − t21r2einlwφlw1− r1r2einlwφlw (10)where nlw is the number of line widths of detuning andφlw = 2ωlwL/c is the phase accumulated per line widthof detuning, for line width ωlw.To calculate the error signal, we apply the cavity re-flection coefficient to the carrier and sideband fields ofthe form of equation (3), giving a total reflected field:Eref (nlw) = rcav(nlw)E0+ rcav(nlw + nm)E+eiωmt+ rcav(nlw − nm)E−e−iωmt (11)where nm = φm/φlw is the line widths of detuning accu-mulated by the sidebands.The reflected power is now:Pref (nlw) = |Eref (nlw)|2 ={[rcav(nlw)E0]∗ rcav(nlw + nm)E++ rcav(nlw)E0 [rcav(nlw − nm)E−]∗}eiωmt+{[rcav(nlw)E0]∗ rcav(nlw − nm)E−+ rcav(nlw)E0 [rcav(nlw + nm)E+]∗} e−iωmt+ (DC Terms) + (2ωmTerms) (12)3Separating out sinωmt and cosωmt terms and writingexplicitly in terms of A˜ and P˜ , we have:Pref (nlw) = P0[Re[E(nlw)] cosωmt−Im[E(nlw)] sinωmt]+ (DC Terms) + (2ωmTerms) (13)where P0 = E20 and E(nlw) is defined as:E(nlw) = [rcav(nlw)]∗rcav(nlw + nm)(A˜+ iP˜ )+ rcav(nlw)[rcav(nlw − nm)]∗(A˜− iP˜ ) (14)We define the cosωmt term as being the in phase errorsignal, or I phase, and the sinωmt term as being the inquadrature phase, or Q phase.For high finesse cavity and a modulation frequency welloutside of the cavity line width and larger than the de-sired detuning,rcav(nlw ± nm) ∼ 1, (15)such thatE(nlw) ≃ 2(Re[rcav(nlw)]A˜+ Im[rcav(nlw)]P˜)(16)from which it is clear that the demodulated error signalfrom the cavity can be made to cross zero at any detuningnlw by setting the modulation phases such thatIm[rcav(nlw)]Re[rcav(nlw)]=−A˜P˜(17)III. EXPERIMENTAL SETUPThe experimental setup is shown in Figure 2. Lightfrom a 1064nm ND:YaG laser passes through a half-waveplate, a polarizing beam splitter (PBS), a quarter-waveplate, and through the UTM. Our UTM is a modifiedNew Focus 4140 amplitude modulator. The optical trainleading to the UTM ensures that the light incident onthe UTM is circularly polarized. A sample beam is in-cident on PD1 to directly detect amplitude modulationfor diagnostic purposes. The main beam is incident on afree space cavity that is 10 cm in length. The input andend mirrors of the cavity have amplitude reflectivity ofr1 = 0.98 and r2 = 0.997, respectively, which gives thecavity a finesse of 270. The cavity end mirror is attachedto a PZT, enabling actuation of the cavity length.The cavity length was scanned through resonances bydriving the PZT; the resulting cavity error signal wasmeasured by demodulating the signal on PDref (New Fo-cus 1811) with a local oscillator (LO). The LO was de-rived from one of the two UTM drives using a powersplitter.The inputs to the UTM were driven separately usinga signal generator that outputs two signals with an ar-bitrary phase between them. This ability to control therelative phases of the two signals going into the UTM is12PBS PBS42PDreflPDtrans PZTCavityUTM4PD1LaserFIG. 2. Schematic of the experimental setup. The path ofthe laser beam is shown in red and electronic wiring is shownin grey. Laser light in an optimal polarization is modulatedby the UTM and incident a linear Fabry-Perot cavity. Thecavity length is actuated by a PZT stack on the end mirror ofthe cavity. The two inputs to the UTM are driven by the twoseparate outputs of a two-channel signal generator. The LOport of the mixer is fed with a 25.23 MHz signal from derivedfrom one channel of a 2-channel signal generator. The RFport of the mixer is fed with the signal coming from PDref .necessary to explore the entire modulation space. Therelative phases of the drive signals going into the UTMwere adjusted in the range from 0 to 180 degrees in 30degree intervals. The phase of the signal with respect tothe LO was adjusted to measure both quadratures of thedemodulated signal.IV. RESULTSFigure 3 shows the error signals as a function of cavitydetuning for different modulation states, along with thetheoretical model corresponding to equation (14) above.As we detune the cavity, we are able to generate errorsignals based on different modulation states ranging frompure PM to pure AM. We emphasize that only a subsetof the possible modulation states are shown here. Forexample, one could drive one crystal only, varying polar-ization of the light instead to create different modulationdepths of AM and PM. This would correspond to the AMand PM modulations being in phase.The theoretical model was fit using five parameters:two parameters accounted for the difference in the slopeand offset of the amplitude of the error signal, two otherparameters accounted for the transformation from thesweep time to cavity length actuation (using a secondorder polynomial fit), and another parameter accountedfor the zero point offset.An important question to ask is: How far can the cav-ity be detuned while still yielding a useable error signal?It turns out that the if we allow the phases of the UTMdrive signal vary relative to the LO phase, the cavity can4Linewidths-2 -1 0 1 2-2.5-2-1.5-1-0.500.511.522.5DataModelFIG. 3. Cavity error signals in phase (I phase) as a function ofcavity length detuning from resonance. The different curvescorrespond to different relative phase differences of the inputdrives of 0◦ (red), 60◦ (green), 120◦ (yellow), 180◦ (blue).Dashed and solid lines correspond to the model and data, re-spectively, for each phase difference. The cavity transmissionfor a single fit is also overlaid at the top. The units on they-axis are arbitrary.be detuned past the nominal half line width of detuningusing pure AM. In Figure 4, we plot the detuning (inline widths) as a function of input drive phases. We em-phasize that this phase has a slightly different definition.Previously we assumed that the LO phase was locked toone of the UTM drives, hence only the phase differencewas important. Here the phase is set by first locking thephase of one of the drives to the LO, and varying theother drive from 0 to 180◦. Next, the drive is locked at180◦ and the other drive is varied in the same manner.The phase in Figure 4 is defined as the sum of the phasesof the two drives, ranging from both drives locked to theLO to both drives 180◦ out of phase with the LO. Wesee that this method provides a way to detune the cavityby multiple line widths. Thus, the error signal gener-ated using UTM provides a convenient way to controlthe amount of detuning of a cavity.V. CONCLUDING REMARKSWe have shown that we can use the vast modulationspace of AM and PM provided by a universal tunablemodulator to generate error signals for a cavity that canbe on resonance or many line widths detuned from res-onance. By varying the phases of the drive signals tothe UTM we can keep the cavity locked at different de-tunings. Our theoretical model for the cavity error sig-nal agrees well with our measurements of the error sig-nals derived by fixing the drive amplitudes and varyingthe phase differences of the drives. This is a fraction ofLinewidths012345Phase 1 (Rad)012345Total Phase (Rad)0 1 2 3 4 5 6Phase 2 (Rad)012345FIG. 4. The upper panel shows the number of line widthsof detuning as a function of the phases of input drives. Thephase here is defined as the sum of the phases of the inputdrives relative to the phase of the LO. As shown in the middleand lower panels, the manner in which the phase is increasedis important. The phase of one drive is allowed to increase to180◦ before increasing the other drive to the same value.the modulation space available, but has proved power-ful technique for locking optical cavities with arbitrarydetuning from resonance.ACKNOWLEDGMENTSWe thank our current and former colleagues at theMIT LIGO Lab. In particular, we are grateful to TimBodiya, Thomas Corbitt, Tomoki Isogai, Eric Oelker,Shannon Sankar, Denis Martynov, and Sam Waldmanfor their many contributions to this work. The authorsgratefully acknowledge the support of the National Sci-ence Foundation and the LIGO Laboratory, operatingunder cooperative Agreement No. PHY-0757058Appendix A: Derivation of the UTM transferfunction5[1] Kippenberg, Aspelmeyer and Marquardt.http://arxiv.org/abs/1303.0733.[2] A. Abramovici et al., LIGO: The Laser Interferome-ter Gravitational-Wave Observatory. Science 256, 325(1992).[3] The LIGO Scientific Collaboration, LIGO: The Laser In-terferometer Gravitational-Wave Observatory.Rep. Prog.Phys. 72, 076901 (2009).[4] The LIGO Scientific Collaboration, Advanced LIGO.Classical Quant. Grav. 32, 074001 (2015).[5] The Virgo Collaboration, Virgo Status. Classical Quant.Grav. 25, 184001 (2008).[6] H. Grote and the LIGO Scientific Collaboration, TheGEO 600 Status. Classical Quant. Grav. 27, 084003(2010).[7] H. Walther et al., Cavity quantum electrodynamics. Rep.Prog. Phys. 69, 1325 (2006).[8] H. Rohde, J. Eschner, F. Schmidt-Kaler, and R. Blatt,Optical decay from a Fabry-Perot cavity faster than thedecay time. J. Opt. Soc. Am. B 19, 1425 (2002).[9] Nathan Brahms, T. P. Purdy, D. W. C. Brooks, T. Bot-ter, D. M. Stamper-Kurn, Cavity-aided magnetic reso-nance microscopy of atomic transport in optical lattices.Nature Physics 7, 604 (2011).[10] H. Haffner, C. F. Roos, R. Blatt, Quantum computingwith trapped ions. Physics Reports 469, 155 (2008).[11] R. W. P. Drever et al., Laser phase and frequency stabi-lization using an optical resonator. Appl. Phys. B 31, 97(1983).[12] B. Cusack, et al., Electrooptic modulator capable of gen-erating simultaneous amplitude and phase modulations.Applied Optics, 43, 5079 (2004).

Continuously tunable modulation scheme for precision control of optical cavities with variable detuning

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