All-reflective optical systems are under consideration for future gravitational wave detector topologies. One approach in proposed designs is to use diffraction gratings as input couplers for Fabry–Perot cavities. We present an experimental demonstration of a fully suspended diffractively coupled cavity and investigate the use of conventional Pound–Drever–Hall length sensing and control techniques to maintain the required operating condition

Barr, B.W.

Britzger, M.

Bruckner, F.

Burmeister, O.

Clausnitzer, T.

Danzmann, K.

Edgar, M.P.

Kley, E.B.

Nelson, J.

Plissi, M.V.

Schnabel, R.

Strain, K.A.

Tunnermann, A.

English

All-reflective optical systems are under consideration for future gravitational wave detector topologies. One approach in proposed designs is to use diffraction gratings as input couplers for Fabry-Perot cavities. We present an experimental demonstration of a fully suspended diffractively coupled cavity and investigate the use of conventional Pound-Drever-Hall length sensing and control techniques to maintain the required operating condition

Edgar, M.

Barr, B.

Plissi, M.

Strain, K.

Brueckner, F.

Kley, E.

Tuennermann, A.

MPG.PuRe

Experimental demonstration of a suspended diffractively coupled optical cavity

3184 OPTICS LETTERS / Vol. 34, No. 20 / October 15, 2009Experimental demonstration of a suspendeddiffractively coupled optical cavityM. P. Edgar,1,* B. W. Barr,1 J. Nelson,1 M. V. Plissi,1 K. A. Strain,1 O. Burmeister,2 M. Britzger,2K. Danzmann,2 R. Schnabel,2 T. Clausnitzer,3 F. Brückner,3 E.-B. Kley,3 and A. Tünnermann31Department of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, UK2Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut) and Institut für Gravitationsphysik,Leibniz Universität Hannover, Callinstrasse 38, 30167 Hannover, Germany3Institut für Angewandte Physik, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany*Corresponding author: m.edgar@physics.gla.ac.ukReceived July 10, 2009; revised September 3, 2009; accepted September 3, 2009;posted September 16, 2009 (Doc. ID 113921); published October 12, 2009All-reflective optical systems are under consideration for future gravitational wave detector topologies. Oneapproach in proposed designs is to use diffraction gratings as input couplers for Fabry–Perot cavities. Wepresent an experimental demonstration of a fully suspended diffractively coupled cavity and investigate theuse of conventional Pound–Drever–Hall length sensing and control techniques to maintain the required op-erating condition. © 2009 Optical Society of AmericaOCIS codes: 050.1970, 050.2230, 120.2230, 120.3180, 120.5700, 230.1950.A network of first-generation laser interferometricgravitational wave detectors (LIGO, GEO600,VIRGO, and TAMA) is currently operating aroundthe world [1]. All of these instruments are based onenhanced Michelson topologies utilizing several tech-niques to increase the stored laser power and hencethe detector sensitivity, such as power recycling andoptical cavities. Light is coupled in via partiallytransmissive mirrors, and the power in the sub-strates of some of the optical components such as cou-pling mirrors or beam splitters reaches or exceeds 1kW at a wavelength of 1064 nm. Proposed designs fornext generation detectors require even higher circu-lating light levels to improve the detector sensitivity.Increasing the circulating power is troublesome,since a constant small fraction is absorbed in opticalsubstrates. This causes localized heating that be-comes progressively more serious because ever stron-ger thermal lenses are produced through the changein the refractive index with temperature [2]. A poten-tial resolution to this problem is to use all-reflectiveoptical components such as diffraction gratings tosplit and recombine light fields [3]. These devices arealso of interest in other fields, for example, nonlinearoptical cavity enhancement. A second motivation forthe use of gratings in future interferometers arises ifcryogenic techniques are required to reduce thermalnoise. It is likely that materials that are opaque at1064 nm, such as silicon, will offer the best perfor-mance at cryogenic temperatures, and therefore ei-ther a nontransmissive mechanism must be found orthe laser wavelength would need to be changed.Although the study of interferometer topologies isessentially straightforward, dealing only with thesimple relationships among linear light fields, nu-merical techniques are routinely used to compute thesignals obtained at various photodetectors in re-sponse to changes in the relative positions of the op-tics. This is a requirement for the development oflength sensing and control schemes required to keepthe interferometers at the desired operating point0146-9592/09/203184-3/$15.00 ©and to read out the signals, including the gravita-tional wave signal. It is therefore important to under-stand how to model diffractive couplers and to vali-date such simulations by experiment.Several proof-of-principle [4] and bench-top [5] ex-periments have been carried out to demonstrate theprinciples of operation for diffractive couplers in op-tical cavities. To examine the applicability of suchsystems to suspended instruments, one of the armcavities of the Glasgow 10 m prototype interferom-eter was commissioned as a diffractively coupled op-tical cavity (see Fig. 1). Each of the optics in this sys-tem is suspended as a multistage pendulum toprovide seismic isolation and allow freedom of mo-tion. The diffraction grating under investigation wasmanufactured by etching a binary structure into asilica substrate, then coating with multiple alternat-ing layers of Ta2O5 and SiO2 to give an ultra–low-loss–low-efficiency grating with a period of d=1450 nm [6]. The grating was mounted in the sec-ond order Littrow configuration, and illuminated atan angle of 47.2° with s-polarized light at 1064 nmwavelength, from a Nd:YAG laser.Fig. 1. (Color online) Simplified schematic of the three-port second-order Littrow mount grating used as the inputcoupler for a diffractive Fabry–Perot cavity. Tuned photo-diodes (PDs) are positioned at all three output ports to de-tect the DC power and rf component for derivation of thecontrol signals.2009 Optical Society of AmericaOctober 15, 2009 / Vol. 34, No. 20 / OPTICS LETTERS 3185The optical cavities within a gravitational wave de-tector must be maintained at the correct operatingpoint, usually with the light from the laser resonant.To this carrier light, various modulation sidebandsare added, e.g., as prescribed in the Pound–Drever–Hall (PDH) technique [7]. The light leaving the cavityis detected at output ports and information on the in-teraction between the frequency components canthen be extracted by demodulating the detected sig-nals with a local oscillator at the modulation fre-quency. As seen in Fig. 1 our system has three detec-tion ports. To enable signal extraction from all threeoutput ports, two different rf modulation frequencieswere used. As with the normal PDH method, thechoice of frequency for the forward-reflected andbackreflected ports is not critical, provided that thesidebands are off resonance in the cavity when thecarrier is resonant. A frequency of 10 MHz was cho-sen for convenience. To obtain a nonvanishing signalfrom the transmitted port required the use of side-bands just off resonance, at 15.24 MHz, close to thefree spectral range (FSR) of the cavity. The transmit-ted light can subsequently be detected on a tunedphotodetector and demodulated to produce a sym-metrical length sensing signal. The FSR of our cavityis 15.27 MHz and, by monitoring the DC signal fromthe transmitted port, the corresponding FWHM wasdetermined to be 13.80±0.64 kHz.The operation of a traditional two-port Fabry–Perot cavity is determined by the reflection/transmission efficiencies of the mirrors. However,utilizing a three-port diffraction grating as an inputcoupler to a Fabry–Perot cavity, the optical field in-teractions become more complex. The scattering ma-trix of a three-port grating can be represented byS3p = 2ei2 1ei1 0ei01ei1 0ei0 1ei10ei0 1ei1 2ei2 , 1where 0,1,2 and 0,1,2 are the amplitude diffractionefficiencies and the phase changes on the diffractionfor zeroth, first, and second orders, respectively, and0 is the amplitude reflectivity at normal incidence. Ifthese values are known along with the reflectivity/transmittivity efficiencies 1 /1 of the end mirror,then all properties of the cavity can be determined.The input–output relations for a three-port diffrac-tion grating coupled Fabry–Perot cavity have beeninvestigated [8] and yield amplitudes for the fields ateach port described byc1 = 2ei2 + 12e2i1+d, 2c2t = i11ei1+d, 3c3 = 0 + 12e2i1+d, 4where  represents the phase acquired after oneround trip of the cavity and the resonance factor isdefined by d= 1−01e2i−1.Based on these field equations a numerical simula-tion of the diffractive cavity was built using MATLABto allow a comparison between modeled predictionsand experimental results. Using grating parametersprovided by the Albert-Einstein-Institute in Han-nover and the specifications of the end mirror fromthe manufacturer, the light field amplitudes and thepower at each port could be calculated. Direct mea-surement of the average light power at each outputport showed agreement to within 12% of the model.The calculated finesse of the cavity is then 1177±27,which is in good agreement with the measured fi-nesse of 1107±51.One way to establish whether the sensing signalspredicted by the model agree with the experiment isto sweep the length of the cavity by at least oneround-trip wavelength (i.e., one FSR). Then the de-modulated signal from each port can be comparedwith the prediction. The slope of this signal at the op-erating point gives the response for that output port.The rf signal from each detection port can be de-modulated in two orthogonal phases, called “inphase” and “quadrature phase” for maximum andminimum responses, respectively. The demodulatedsignals obtained in the experiment are shown in Fig.2 and are seen to be in good qualitative agreementwith modeled predictions. The discrepancies betweenthe experiment and the modeled signals can be at-tributed to the finite rate of sweep in the experiment.To calibrate the demodulated signals detected atall three ports, the relative size of the in-phase slopeswas compared with the slope of the signal from theFig. 2. (Color online) Normalized rf power for all threeoutput ports C1 (top), C2 transmitted through the end mir-ror (middle), and C3 (bottom). The solid (blue) trace indi-cates in-phase measurements, and the dashed (green) traceindicates quadrature phase. The absolute scaling betweenmodeled ports 1, 2, and 3 is 2:1:163.3186 OPTICS LETTERS / Vol. 34, No. 20 / October 15, 2009transmitted port. The ratios of the signal responsesare presented in Table 1, indicating a good quantita-tive agreement with the model.Having demonstrated the validity of the model forthis experimental configuration, the numerical simu-lation can be extended to investigate a more generalcase. In particular, previous experimental work withtabletop cavities [5] had demonstrated an asymmet-ric behavior of the reflected carrier power levels. Theextent of the asymmetry was seen to be determinedby the values of the 0 and 2 diffraction efficiencies.With this in mind, we adapted the numerical simula-tion to investigate the effects of asymmetry on the de-modulated output signals with grating parameterschosen to accentuate the observable effects.It is clear from the modeled traces (see Fig. 3) thatthe demodulated signals from the forward-reflectedand backreflected ports do exhibit asymmetry as ex-pected. However, through a careful choice of demodu-lation phase, we can extract signals from each ofthese ports, which sum together to exactly recon-struct a traditional PDH locking signal. Additionally,while the asymmetry is clearly observable for our ex-aggerated case, the demodulated signals from theforward-reflected and backreflected ports will alwaysexhibit some asymmetry—this is limited by the factthat 0 and 2 have a lower bound owing to energyconservation [8].In our experimental configuration the cavity waslocked using the transmitted error signal, which ex-hibits symmetrical behavior around the center ofresonance. The ability to lock at this known operat-ing point means that we can use the diffractivelycoupled cavity to investigate the dynamic effects as-sociated with suspended optical systems. In particu-lar, when dealing with diffractive couplers, phaseshifts owing to translational motion of the couplerrelative to the laser beam are predicted to introduceadditional noise when compared with an equivalenttraditional cavity configuration [9]. Future experi-ments on this suspended diffractive system willprobe the effect of translational grating motion andassociated effects owing to cavity misalignment.In conclusion, our experimental results confirm thetheoretical foundations supporting a diffractivelycoupled Fabry–Perot cavity and successfully demon-strate the use of all-reflective optics in a fully sus-pended environment. We have developed a numericalmodel for length sensing and control signal extrac-tion from a diffractively coupled cavity based on theconventional rf sideband modulation and demodula-Table 1. Measured and Modeled Signal ResponsesParameterMeasuredValue(dB)ModeledValue(dB)Backreflected/transmitted −26.93 −22.51Transmitted/transmitted 0 0Forward-reflected/transmitted 43.07 44.74tion techniques, and we have shown that the systemis well understood through both quantitative andqualitative experimental verifications. Based onthese findings we have also observed the effect ofgrating parameters to signal extraction from a dif-fractively coupled system by utilizing the principlesof energy conservation. The framework for signal ex-traction from a suspended diffractive system is nowwell understood.This work was supported by the University of Glas-gow and the Science and Technology Facilities Coun-cil. The authors would also like to acknowledge theSonderforschungsbereich TR7 of the DeutscheForschungsgemeinschaft (DFG) for financial support.References1. P. Aufmuth and K. Danzmann, New J. Phys. 7, 202(2005).2. K. A. Strain, K. Danzmann, J. Mizuno, P. G. Nelson, A.Rüdiger, R. Schilling, and W. Winkler, Phys. Lett. A194, 124 (1994).3. A. Bunkowski, O. Burmeister, P. Beyersdorf, K.Danzmann, R. Schnabel, T. Clausnitzer, E.-B. Kley,and A. Tünnermann, Opt. Lett. 29, 2342 (2004).4. K.-X. Sun and R. L. Byer, Opt. Lett. 23, 567 (1998).5. A. Bunkowski, O. Burmeister, K. Danzmann, R.Schnabel, T. Clausnitzer, E.-B. Kley, and A.Tünnermann, Opt. Lett. 31, 2384 (2006).6. T. Clausnitzer, E.-B. Kley, A. Tünnermann, A.Bunkowski, O. Burmeister, K. Danzmann, R.Schnabel, S. Gliech, and A. Duparré, Opt. Express 13,4370 (2005).7. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G.M. Ford, A. J. Munley, and H. Ward, Appl. Phys. B 31,97 (1983).8. A. Bunkowski, O. Burmeister, K. Danzmann, and R.Schnabel, Opt. Lett. 30, 1183 (2005).9. A. Freise, A. Bunkowski, and R. Schnabel, New J.Fig. 3. (Color online) Simulated reflected field, demodu-lated signals C1 (red dashed trace) and C3 (blue dashed–dotted trace), and the combined demodulated signal C1+C3 (black solid trace). The grating parameters chosenhere are for an ideal (lossless) grating with 0=0.99663,1=0.0407, and 0=2=0.7065.Phys. 9, 433 (2007).

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        Edgar, M.P., Barr, B.W., Nelson, J., Plissi, M.V., Strain, K.A., Burmeister, O., Britzger, M., Danzmann, K., Schnabel, R., Clausnitzer, T., Bruckner, F., Kley, E.B., and Tunnermann, A. (2009) Experimental demonstration of a suspended diffractively coupled optical cavity. Optics Letters, 34 (20). pp. 3184-3186. ISSN 0146-9592   Copyright © 2009 Optical Society of America.   A copy can be downloaded for personal non-commercial research or study, without prior permission or charge   The content must not be changed in any way or reproduced in any format or medium without the formal permission of the copyright holder(s)   When referring to this work, full bibliographic details must be given      http://eprints.gla.ac.uk/62346/        Deposited on:  27 Jan 2014          Enlighten – Research publications by members of the University of Glasgow http://eprints.gla.ac.uk 3184 OPTICS LETTERS / Vol. 34, No. 20 / October 15, 2009Experimental demonstration of a suspendeddiffractively coupled optical cavityM. P. Edgar,1,* B. W. Barr,1 J. Nelson,1 M. V. Plissi,1 K. A. Strain,1 O. Burmeister,2 M. Britzger,2K. Danzmann,2 R. Schnabel,2 T. Clausnitzer,3 F. Brückner,3 E.-B. Kley,3 and A. Tünnermann31Department of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, UK2Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut) and Institut für Gravitationsphysik,Leibniz Universität Hannover, Callinstrasse 38, 30167 Hannover, Germany3Institut für Angewandte Physik, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany*Corresponding author: m.edgar@physics.gla.ac.ukReceived July 10, 2009; revised September 3, 2009; accepted September 3, 2009;posted September 16, 2009 (Doc. ID 113921); published October 12, 2009All-reflective optical systems are under consideration for future gravitational wave detector topologies. Oneapproach in proposed designs is to use diffraction gratings as input couplers for Fabry–Perot cavities. Wepresent an experimental demonstration of a fully suspended diffractively coupled cavity and investigate theuse of conventional Pound–Drever–Hall length sensing and control techniques to maintain the required op-erating condition. © 2009 Optical Society of AmericaOCIS codes: 050.1970, 050.2230, 120.2230, 120.3180, 120.5700, 230.1950.A network of first-generation laser interferometricgravitational wave detectors (LIGO, GEO600,VIRGO, and TAMA) is currently operating aroundthe world [1]. All of these instruments are based onenhanced Michelson topologies utilizing several tech-niques to increase the stored laser power and hencethe detector sensitivity, such as power recycling andoptical cavities. Light is coupled in via partiallytransmissive mirrors, and the power in the sub-strates of some of the optical components such as cou-pling mirrors or beam splitters reaches or exceeds 1kW at a wavelength of 1064 nm. Proposed designs fornext generation detectors require even higher circu-lating light levels to improve the detector sensitivity.Increasing the circulating power is troublesome,since a constant small fraction is absorbed in opticalsubstrates. This causes localized heating that be-comes progressively more serious because ever stron-ger thermal lenses are produced through the changein the refractive index with temperature [2]. A poten-tial resolution to this problem is to use all-reflectiveoptical components such as diffraction gratings tosplit and recombine light fields [3]. These devices arealso of interest in other fields, for example, nonlinearoptical cavity enhancement. A second motivation forthe use of gratings in future interferometers arises ifcryogenic techniques are required to reduce thermalnoise. It is likely that materials that are opaque at1064 nm, such as silicon, will offer the best perfor-mance at cryogenic temperatures, and therefore ei-ther a nontransmissive mechanism must be found orthe laser wavelength would need to be changed.Although the study of interferometer topologies isessentially straightforward, dealing only with thesimple relationships among linear light fields, nu-merical techniques are routinely used to compute thesignals obtained at various photodetectors in re-sponse to changes in the relative positions of the op-tics. This is a requirement for the development oflength sensing and control schemes required to keepthe interferometers at the desired operating point0146-9592/09/203184-3/$15.00 ©and to read out the signals, including the gravita-tional wave signal. It is therefore important to under-stand how to model diffractive couplers and to vali-date such simulations by experiment.Several proof-of-principle [4] and bench-top [5] ex-periments have been carried out to demonstrate theprinciples of operation for diffractive couplers in op-tical cavities. To examine the applicability of suchsystems to suspended instruments, one of the armcavities of the Glasgow 10 m prototype interferom-eter was commissioned as a diffractively coupled op-tical cavity (see Fig. 1). Each of the optics in this sys-tem is suspended as a multistage pendulum toprovide seismic isolation and allow freedom of mo-tion. The diffraction grating under investigation wasmanufactured by etching a binary structure into asilica substrate, then coating with multiple alternat-ing layers of Ta2O5 and SiO2 to give an ultra–low-loss–low-efficiency grating with a period of d=1450 nm [6]. The grating was mounted in the sec-ond order Littrow configuration, and illuminated atan angle of 47.2° with s-polarized light at 1064 nmwavelength, from a Nd:YAG laser.Fig. 1. (Color online) Simplified schematic of the three-port second-order Littrow mount grating used as the inputcoupler for a diffractive Fabry–Perot cavity. Tuned photo-diodes (PDs) are positioned at all three output ports to de-tect the DC power and rf component for derivation of thecontrol signals.2009 Optical Society of AmericaOctober 15, 2009 / Vol. 34, No. 20 / OPTICS LETTERS 3185The optical cavities within a gravitational wave de-tector must be maintained at the correct operatingpoint, usually with the light from the laser resonant.To this carrier light, various modulation sidebandsare added, e.g., as prescribed in the Pound–Drever–Hall (PDH) technique [7]. The light leaving the cavityis detected at output ports and information on the in-teraction between the frequency components canthen be extracted by demodulating the detected sig-nals with a local oscillator at the modulation fre-quency. As seen in Fig. 1 our system has three detec-tion ports. To enable signal extraction from all threeoutput ports, two different rf modulation frequencieswere used. As with the normal PDH method, thechoice of frequency for the forward-reflected andbackreflected ports is not critical, provided that thesidebands are off resonance in the cavity when thecarrier is resonant. A frequency of 10 MHz was cho-sen for convenience. To obtain a nonvanishing signalfrom the transmitted port required the use of side-bands just off resonance, at 15.24 MHz, close to thefree spectral range (FSR) of the cavity. The transmit-ted light can subsequently be detected on a tunedphotodetector and demodulated to produce a sym-metrical length sensing signal. The FSR of our cavityis 15.27 MHz and, by monitoring the DC signal fromthe transmitted port, the corresponding FWHM wasdetermined to be 13.80±0.64 kHz.The operation of a traditional two-port Fabry–Perot cavity is determined by the reflection/transmission efficiencies of the mirrors. However,utilizing a three-port diffraction grating as an inputcoupler to a Fabry–Perot cavity, the optical field in-teractions become more complex. The scattering ma-trix of a three-port grating can be represented byS3p = 2ei2 1ei1 0ei01ei1 0ei0 1ei10ei0 1ei1 2ei2 , 1where 0,1,2 and 0,1,2 are the amplitude diffractionefficiencies and the phase changes on the diffractionfor zeroth, first, and second orders, respectively, and0 is the amplitude reflectivity at normal incidence. Ifthese values are known along with the reflectivity/transmittivity efficiencies 1 /1 of the end mirror,then all properties of the cavity can be determined.The input–output relations for a three-port diffrac-tion grating coupled Fabry–Perot cavity have beeninvestigated [8] and yield amplitudes for the fields ateach port described byc1 = 2ei2 + 12e2i1+d, 2c2t = i11ei1+d, 3c3 = 0 + 12e2i1+d, 4where  represents the phase acquired after oneround trip of the cavity and the resonance factor isdefined by d= 1−01e2i−1.Based on these field equations a numerical simula-tion of the diffractive cavity was built using MATLABto allow a comparison between modeled predictionsand experimental results. Using grating parametersprovided by the Albert-Einstein-Institute in Han-nover and the specifications of the end mirror fromthe manufacturer, the light field amplitudes and thepower at each port could be calculated. Direct mea-surement of the average light power at each outputport showed agreement to within 12% of the model.The calculated finesse of the cavity is then 1177±27,which is in good agreement with the measured fi-nesse of 1107±51.One way to establish whether the sensing signalspredicted by the model agree with the experiment isto sweep the length of the cavity by at least oneround-trip wavelength (i.e., one FSR). Then the de-modulated signal from each port can be comparedwith the prediction. The slope of this signal at the op-erating point gives the response for that output port.The rf signal from each detection port can be de-modulated in two orthogonal phases, called “inphase” and “quadrature phase” for maximum andminimum responses, respectively. The demodulatedsignals obtained in the experiment are shown in Fig.2 and are seen to be in good qualitative agreementwith modeled predictions. The discrepancies betweenthe experiment and the modeled signals can be at-tributed to the finite rate of sweep in the experiment.To calibrate the demodulated signals detected atall three ports, the relative size of the in-phase slopeswas compared with the slope of the signal from theFig. 2. (Color online) Normalized rf power for all threeoutput ports C1 (top), C2 transmitted through the end mir-ror (middle), and C3 (bottom). The solid (blue) trace indi-cates in-phase measurements, and the dashed (green) traceindicates quadrature phase. The absolute scaling betweenmodeled ports 1, 2, and 3 is 2:1:163.3186 OPTICS LETTERS / Vol. 34, No. 20 / October 15, 2009transmitted port. The ratios of the signal responsesare presented in Table 1, indicating a good quantita-tive agreement with the model.Having demonstrated the validity of the model forthis experimental configuration, the numerical simu-lation can be extended to investigate a more generalcase. In particular, previous experimental work withtabletop cavities [5] had demonstrated an asymmet-ric behavior of the reflected carrier power levels. Theextent of the asymmetry was seen to be determinedby the values of the 0 and 2 diffraction efficiencies.With this in mind, we adapted the numerical simula-tion to investigate the effects of asymmetry on the de-modulated output signals with grating parameterschosen to accentuate the observable effects.It is clear from the modeled traces (see Fig. 3) thatthe demodulated signals from the forward-reflectedand backreflected ports do exhibit asymmetry as ex-pected. However, through a careful choice of demodu-lation phase, we can extract signals from each ofthese ports, which sum together to exactly recon-struct a traditional PDH locking signal. Additionally,while the asymmetry is clearly observable for our ex-aggerated case, the demodulated signals from theforward-reflected and backreflected ports will alwaysexhibit some asymmetry—this is limited by the factthat 0 and 2 have a lower bound owing to energyconservation [8].In our experimental configuration the cavity waslocked using the transmitted error signal, which ex-hibits symmetrical behavior around the center ofresonance. The ability to lock at this known operat-ing point means that we can use the diffractivelycoupled cavity to investigate the dynamic effects as-sociated with suspended optical systems. In particu-lar, when dealing with diffractive couplers, phaseshifts owing to translational motion of the couplerrelative to the laser beam are predicted to introduceadditional noise when compared with an equivalenttraditional cavity configuration [9]. Future experi-ments on this suspended diffractive system willprobe the effect of translational grating motion andassociated effects owing to cavity misalignment.In conclusion, our experimental results confirm thetheoretical foundations supporting a diffractivelycoupled Fabry–Perot cavity and successfully demon-strate the use of all-reflective optics in a fully sus-pended environment. We have developed a numericalmodel for length sensing and control signal extrac-tion from a diffractively coupled cavity based on theconventional rf sideband modulation and demodula-Table 1. Measured and Modeled Signal ResponsesParameterMeasuredValue(dB)ModeledValue(dB)Backreflected/transmitted −26.93 −22.51Transmitted/transmitted 0 0Forward-reflected/transmitted 43.07 44.74tion techniques, and we have shown that the systemis well understood through both quantitative andqualitative experimental verifications. Based onthese findings we have also observed the effect ofgrating parameters to signal extraction from a dif-fractively coupled system by utilizing the principlesof energy conservation. The framework for signal ex-traction from a suspended diffractive system is nowwell understood.This work was supported by the University of Glas-gow and the Science and Technology Facilities Coun-cil. The authors would also like to acknowledge theSonderforschungsbereich TR7 of the DeutscheForschungsgemeinschaft (DFG) for financial support.References1. P. Aufmuth and K. Danzmann, New J. Phys. 7, 202(2005).2. K. A. Strain, K. Danzmann, J. Mizuno, P. G. Nelson, A.Rüdiger, R. Schilling, and W. Winkler, Phys. Lett. A194, 124 (1994).3. A. Bunkowski, O. Burmeister, P. Beyersdorf, K.Danzmann, R. Schnabel, T. Clausnitzer, E.-B. Kley,and A. Tünnermann, Opt. Lett. 29, 2342 (2004).4. K.-X. Sun and R. L. Byer, Opt. Lett. 23, 567 (1998).5. A. Bunkowski, O. Burmeister, K. Danzmann, R.Schnabel, T. Clausnitzer, E.-B. Kley, and A.Tünnermann, Opt. Lett. 31, 2384 (2006).6. T. Clausnitzer, E.-B. Kley, A. Tünnermann, A.Bunkowski, O. Burmeister, K. Danzmann, R.Schnabel, S. Gliech, and A. Duparré, Opt. Express 13,4370 (2005).7. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G.M. Ford, A. J. Munley, and H. Ward, Appl. Phys. B 31,97 (1983).8. A. Bunkowski, O. Burmeister, K. Danzmann, and R.Schnabel, Opt. Lett. 30, 1183 (2005).9. A. Freise, A. Bunkowski, and R. Schnabel, New J.Fig. 3. (Color online) Simulated reflected field, demodu-lated signals C1 (red dashed trace) and C3 (blue dashed–dotted trace), and the combined demodulated signal C1+C3 (black solid trace). The grating parameters chosenhere are for an ideal (lossless) grating with 0=0.99663,1=0.0407, and 0=2=0.7065.Phys. 9, 433 (2007).

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Experimental demonstration of a suspended diffractively coupled optical cavity

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