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Measurement of ultralow supermirror birefringenceby use of the polarimetric differential cavityringdown technique

Jae Yong Lee, Hai-Woong Lee, Jae Wan Kim, Yong Shim Yoo, and Jae Won Hahn

We demonstrate a novel technique for measuring ultralow linear birefringence of supermirrors ~high-reflectivity dielectric mirror coatings!. The polarimetric cavity ringdown technique is used in conjunc-tion with the differential detection scheme with circular polarization to enhance the measurementsensitivity. The technique could, in principle, provide the convenience and reliability of linear detectionsignals and a reasonable tolerance to experimental imperfections. Phase retardation and orientation ofeach cavity mirror can be determined separately without the influence of the other mirror. The mini-mum detectable phase retardation achieved experimentally with this technique is ;6 3 1028 rad.© 2000 Optical Society of America

OCIS codes: 260.1440, 230.4040, 120.2230, 120.5410.

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1. Introduction

The accurate characterization of high-reflectivitymirrors is of central importance for many sensitivemeasurements that employ the multipass optical con-figuration. Along with the mirror reflectivity, theresidual phase anisotropy of a dielectric mirror coat-ing becomes all the more crucial when the detectioninvolves the minute change of a probe light polariza-tion as in the experiments that include parity non-conservation measurement,1,2 gravitational-wave

etection,3 magnetic rotation spectroscopy,4,5 mate-ial anisotropy characterization,6–8 gyrometry,9 etc.

This is because the extreme multipass sensitivity am-plifies not only the physical quantity that one wishesto measure but also the unwanted systematic errorsthat arise from the mirror phase anisotropy itself tonegate the delicate interferometric measurements.

Previously, several attempts were made to mea-sure mirror birefringence by use of ellipsometry,10

J. Y. Lee and H-W. Lee are with the Department of Physics,Korea Advanced Institute of Science and Technology, 373-1Kusong-dong Yusong-gu, Taejon 305-701, Korea. J. W. Kim, Y. S.Yoo, and J. W. Hahn ~jwhahn@kriss.re.kr! are with the Optical

igh Temperature Measurement Group, Korea Research Institutef Standards and Science, P.O. Box 102, Yusong, Taejon 305-600,orea.Received 11 January 2000; revised manuscript received 10 Feb-

uary 2000.0003-6935y00y121941-05$15.00y0© 2000 Optical Society of America

the detection of accumulated polarization distortionin a cavity,11 and the polarization oscillating cavitydecay method,12 all based on oblique incidence mea-surements that are effective only for phase retarda-tion greater than 1025 rad. With the conventionalchoice of linear polarization analyzers, the polarimet-ric cavity decay techniques reported were restrictedto the measurement of large anisotropic phase retar-dation ~.0.05°! at off-normal incidence12 or to the

etection of polarization-dependent dichroism ~lossifference!.13,14 It was not until 1995 when Jacob et

al.15 successfully characterized purely intrinsic mir-ror birefringence at normal incidence that one couldmeasure ultralow supermirror birefringence. Theyachieved the phase retardation sensitivity of &1026

rad in a high-finesse Fabry–Perot cavity configura-tion employing a linear polarization analyzer to de-tect depolarized output intensity. However, thetechnique must cope carefully with uncertainties con-tributed from the variation of cavity finesse, the angleerror in birefringence orientation, intensity fluctua-tion in the dual-detector system, and the performanceof polarizer–analyzer extinction. The ultimate sen-sitivity of their technique was set by the extinctionratio of the polarizer used in the experimental appa-ratus.

Here we present a new simple approach to charac-terize the ultralow birefringence of supermirrorsbased on a polarimetric differential ~PD! cavity ring-down ~CRD! technique.13,14 To overcome the exist-ing technical limitations as well as to achieve high

20 April 2000 y Vol. 39, No. 12 y APPLIED OPTICS 1941

1

sensitivity, we combined the advantages inherentfrom the CRD measurement and the balanced detec-tion scheme. The extreme multipass sensitivity of ahigh-finesse Fabry–Perot cavity is exploited in theCRD setup, and the polarimetric detection in con-junction with a circular polarization analyzer couldprovide the convenience of linearity in a birefringencesignal and reasonable noise immunity.

2. Principle and Theory of the Polarimetric DifferentialCavity Ringdown Technique for BirefringenceMeasurement

The foundation for our method is the CRDtechnique16–18 with which the intracavity loss is mea-sured from the time rate of intensity decay of a probelight coupled to a resonant Fabry–Perot cavity thatcontains an absorbing sample. The light signal atthe cavity exit, referred to as a ringdown signal, de-cays exponentially in time with a time constant of t 5dyc@~1 2 R! 1 +#, where d is the cavity length, c is thevelocity of light, R is the reflectivity of the cavitymirrors, and + is the single-pass fractional loss thatis due to the sample inside. Provided that the ring-down time for an empty cavity is known as t0 5dyc~1 2 R!, one can extract sample loss + from therelation + 5 ~1 2 R!~t0 2 t!yt.

In the polarization-dependent CRD method,13,14

the ringdown signals have dual ringdown times fortwo different polarizations of light given by

t6 5 SdcD 1

~1 2 R! 1 +6 , (1)

where the 1 and 2 superscripts designate the twodistinct polarization states e1 and e2, respectively.The differential loss ~+1 2 +2! can be readily deter-mined in the PD CRD measurement by

+1 2 +2 5 2 SdcD t1 2 t2

t1t2 . (2)

In Fig. 1, which shows the optical layout of the PDCRD technique, light field E0 linearly polarized alongthe x axis is coupled to a Fabry–Perot cavity com-posed of two birefringent mirrors, M1 and M2, thatproduce the eigenpolarizations with nondegenerateresonance frequencies. The intracavity light fieldgradually depolarizes into a lower degree of ellipticalpolarization during cavity round trips. On the basisof circular polarizations, this implies that the initialbalance E0

1 5 E02 between the right circularly po-

larized ~RCP! state, e1, and the left circularly polar-ized ~LCP! state, e2, is destroyed by the energytransfer from one polarization to another. The po-larization distortion caused by the energy transferapparently results in polarization-dependent gain orloss of the cavity, providing the difference in the ring-down times of the two polarizations. The resultingdifference in ringdown times between the two polar-izations is a direct indication of the rate of change inpolarization states of the intracavity light, allowingfor the corresponding birefringence characteristicsthat are involved.

942 APPLIED OPTICS y Vol. 39, No. 12 y 20 April 2000

Each birefringent mirror Mi~i 5 1, 2! can be mod-eled as a combination of an ideal mirror with reflec-tivity Ri and a wave plate of phase retardance fi withits fast axis oriented at ui from the x axis. The cor-responding Jones matrices for such mirrors are rep-resented as Mfi

~ui!, which could decompose as R~ui! zMfi

z R21~ui! with a rotation matrix R~ui! and ananisotropy matrix Mfi

for the fast axis lying on the xaxis. The intracavity optical field that undergoes Nround trips is then described by the Jones matrix,which is represented by circular polarization bases e1

and e2 as

CN 5 @Mf1~u1! z Mf2

~u2!#N

5 ~R1 R2!Ny2F 1 a 1 ib

2a 1 ib 1 GN

, (3)

where a 5 f1 sin 2u1 1 f2 sin 2u2 and b 5 f1~2 sin2

u1 2 1! 1 f2~2 sin2 u1 2 1!. If the mirror birefrin-gence is small enough to satisfy a rough criterionFfiy2p ,, 1 with cavity finesse F 5 py~1 2 R!, theaccumulated anisotropic phase retardation in an op-tical field after cavity round trips should also beminute. Under the assumption of weak birefrin-gence in cavity mirrors guaranteeing that Nfi ,, 1,the cavity Jones matrix CN is approximated as

CN < ~R1 R2!Ny2F 1 ~a 1 ib!N

~2a 1 ib!N 1 G . (4)

For a balanced input field ~E01 5 E0

2 5 1!, theresultant field component in each circular polariza-tion becomes EN

6 5 RN@~1 6 Na! 1 iNb# with R 5=R1R2. Thus the constant fractional loss that oc-curs during each round trip yields ~IN

6 2 IN116!y

IN6 ' 2@~1 2 R! 7 a#. With the corresponding per-

pass polarization loss +6 5 7a in Eq. ~2!, we can

Fig. 1. Principle and optical layout for mirror birefringence mea-surements.

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finally characterize the birefringence of the mirrorsby using the relation

z 5 f1 sin 2u1 1 f2 sin 2u2 5 S d2cD t1 2 t2

t1t2

, (5)

where the polarization-dependent ringdown times, t1

in the RCP state and t2 in the LCP state of a ring-down signal, can be measured at the two output portsof a circular polarization analyzer. As depicted inFig. 1, the circular polarization analyzer comprises aquarter wave plate ~QWP! with its fast axis at 245°rom the x axis to convert the two circular polariza-ion components in the RCP and LCP states, respec-ively, into x- and y-polarized lights and a polarizingeam splitter ~PBS! for beam separation.According to Eq. ~5!, the rotation of one cavity mir-

ror ~say, M1! while keeping the other ~say, M2! fixedat an arbitrary angle would produce a sinusoidal os-cillation in the experimentally measurable value z,

hich carries the information of the rotating mirror.he oscillation amplitude is equal to phase retarda-

ion f1 of M1, and the maximum occurs when the fastxis of M1 is oriented at angle u1 5 45°. The phasenisotropy measured in this setup is definitely thentrinsic mirror birefringence at normal incidence in-smuch as the off-normal birefringence is presum-bly rejected in the strict high-finesse cavitylignment.

3. Experimental Setup and Results

For the experimental demonstration, we used a CRDsetup with CW laser excitation based on the imple-mentation done by Rempe et al.,19 as illustrated inFig. 2. The Fabry–Perot cavity is constructed with asuper-Invar interferometer ~Burleigh RC110! thatconsists of two supermirrors ~Research Electro Optic,Inc.! on mechanical rotation stages separated by d 517 cm. Both supermirrors are spherical with a 4-mradius of curvature and a 0.5-in. (1.3-cm) diameter.A single-frequency cw ring dye laser ~Coherent 899-21!, stabilized to within approximately 1 MHz, wasused at 575-nm wavelength with less than 0.1-mWlaser power at the cavity entrance. The laser beamwas mode matched by lens L to the TEM00 mode ofthe cavity such that any higher-order mode excitationwas suppressed to 1% of the fundamental, and thenlinearly polarized along the x axis by a Glan laserpolarizer ~GLP!. The rear mirror that rests on thehollow cylindrical piezoelectric transducer ~PZT! was

Fig. 2. Schematic of the PD CRD experiment.

modulated to 150 Hz to couple the laser into thecavity. The scanning cavity was made resonantwith the laser frequency at the center of each modu-lation period by use of the bias tracking servo. At apreset threshold of on-resonance light intensity cou-pling, the acousto-optic modulator ~AOM! from NeosTechnologies, Model N23080, was triggered to switchoff the laser beam to produce ringdown signals. Thepolarization-dependent signals were analyzedthrough a QWP with its fast axis at 245° from the xxis followed by a Glan laser PBS, allowing the twoircular polarization components in the RCP and theCP states, respectively, at x- and y-polarization

ports of the PBS where two identical Hamamatsuphotomultiplier tubes ~PMT’s!, Model R955, areplaced. The detected ringdown signals were digi-tized into two channels by the 12-bit oscilloscope with10-MHz bandwidth from Gage Applied Sciences,CompuScope 1012.

The ringdown signals were recorded for the twocircular polarizations by accumulating 128 ringdownevents to achieve the desirable signal-to-noise ratio.We obtained signal-to-noise ratios with higher than30 dB and an excellent fit of the ringdown signals tosingle exponential decays. A typical result is shownin Fig. 3 as an example that corresponds to the bire-fringence signal of z 5 21.4 3 1026 rad. The singleexponential fit performed with nearly flat residualssupports the fact that our birefringence measure-ment satisfies the validity condition ~Nfi ' Ffiy2p,, 1! for the weak birefringence approximation weintroduced. The average ringdown time for the en-tire experiment was 15.6 ms, which implies a mirrorreflectivity of R 5 0.9999637 and a correspondingcavity finesse of F 5 py~1 2 R! 5 86500.

By rotating the orientation of the cavity mirrors,we separately measured the birefringence signals zfor the two individual mirrors. The linearity of abirefringence signal z on the magnitude of mirrorbirefringence, as indicated by Eq. ~5!, facilitates both

Fig. 3. Typical time trace of polarization-dependent ringdownsignals: ~a! normalized signals in different circular polarizationsnd ~b! corresponding residuals of the exponential fit.

20 April 2000 y Vol. 39, No. 12 y APPLIED OPTICS 1943

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1

the measurement and the error analysis. The char-acterization procedure for either mirror is isolatedfrom the influence of the other fixed mirror becausephase retardation f and orientation u of a fixed mir-ror contribute nothing, but they do bias a dc offset tothe oscillation of z that can be readily eliminated afterthe sinusoidal fit of z. Experimentally measureddata for z are shown in Fig. 4, where each pointrepresents the statistical average over 100 ringdowntime measurements. The dc offset zoffset is sub-tracted from the original data z, and the mirror ori-entation is assigned in terms of the fast axis angleufast based on the result of a least-squares fit of z to asinusoidal function. From the fit we have the phaseretardance in the two mirrors as f1 5 0.8 3 1026 radand f2 5 2.2 3 1026 rad. The experimental data inFig. 4 are in good agreement with the theoreticalcurve to almost within the standard deviation of zmeasured as sz ; 6 3 1028 rad.

4. Discussion of Measurement Sensitivity

Uncertainty sf in the phase retardance estimatedfrom the sinusoidal fit is theoretically warranted tobe less than sz governed by the uncertainty in ring-down time measurement styt. The accuracy of themeasurement is, however, not affected by the shot-to-shot fluctuation of the baseline ringdown timest0 5 dyc~1 2 R! or, equivalently, the cavity finesseF 5 py~1 2 R!. There is no need to specify cavityfinesse F because it is explicitly absent from Eq. ~5!,whereas cavity finesse F is implicitly taken into ac-count in the polarization-dependent ringdown timest6 for each ringdown measurement, and the commonfluctuation of F is canceled by the differential char-cter of the measurement. This feature wouldreatly enhance the sensitivity and reliability of theeasurement, especially in noisy environments.he fluctuation of the differential ringdown time

t1 2 t2!y2 in this experiment was bounded within.15% of the mean ringdown time t#, whereas the

ringdown time itself fluctuated by 0.45% in the shortterm for 100 ringdown signals and by 7% in the longterm required for the acquisition of a full z curve.

The proposed technique is inherently immune to

Fig. 4. Birefringence signal z measured with a cavity mirror pair.The symbols represent the experimental data and the curves rep-resent the sinusoidal fits.

944 APPLIED OPTICS y Vol. 39, No. 12 y 20 April 2000

intensity fluctuations of a probe light and unequaldetector gains, as is the case for a typical CRD tech-nique.16,19 Furthermore, reasonable experimentaltolerance is allowed, in principle, for the input polar-ization purity, the anisotropy of mirror substrates,and the overall analyzer performance determined bythe accuracy of retardance and alignment of a waveplate and the extinction ratio of a polarizer–analyzerconfiguration. Such technical errors introduced ei-ther before or after cavity traversal could result inlittle difference because the method takes only therate of polarization change that occurs in consecutiveround trips rather than the amount of depolarizationitself. It is theoretically predictable that the afore-mentioned imperfections represented by a smallphase error of df would cause erroneous polarizationlosses +6 ' 7a~1 6 df!, which are subsequently can-celed by the differential measurement. In addition,the finite extinction error dext of a PBS could in factleave nonvanishing errors as +6 ' 7a~1 2 2dext!, buthey would result in only a small fractional error ini, by a factor of 22dext.As to the ultimate sensitivity of our technique, it

should be noted that the major limitation is not im-posed by the extinction ratio of polarizers used in theexperimental apparatus but rather associates withthe deteriorated ringdown waveform that resultsfrom the technical noises. Theoretical estimationmade for the uncertainty styt of this experimentleads us to the shot-noise limit of the phase retarda-tion sensitivity sfushot 5 ~1 2 R!~styt!ushot as 1.8 31028 rad for 128-averaged shots. Experimentally,however, the phase retardation sensitivity of ;6 31028 rad has been achieved with the ringdown sig-nals with 230-dB technical noise that gives rise tothe ringdown time uncertainty styt of 1.5 3 1023.The inferior sensitivity of the experiment could beinferred from the additional technical noises includ-ing nonexponential decay components. Withoutsuch nonideal technical sources, one can further en-hance the measurement sensitivity of this techniqueto the lower shot-noise limit by using photodetectorsof higher responsivity and by taking an appropriatesampling rate and duration of the ringdown signal.20

The higher finesse of a cavity also permits the possi-bility of a higher measurement sensitivity in princi-ple, but this is true only when the involved technicalnoises or fluctuations are retained and do not over-whelm the gain in sensitivity that was achieved.This means that noise suppression and elimination offluctuation are the prerequisites to enhance measure-ment sensitivity with a higher cavity finesse. Ourproposed technique is thus expected to be more prom-ising for the higher finesse cavity because the ex-treme multipass feature can still be exploited withoutpainstakingly locking the cavity to the probe laserand the measurement is free from the influence offinesse fluctuation.

5. Conclusions

To summarize, the polarimetric differential cavityringdown technique with a circular polarization an-

activities by use of helicoidal waves,” Opt. Lett. 17, 360–362

alyzer has been successfully demonstrated to mea-sure the ultralow residual linear birefringence ofhigh-reflectivity mirrors. The ultimate phase re-tardation sensitivity as low as 6 3 1028 rad hasbeen achieved, and the directions of the mirror bi-refringence axes have been easily assigned. Thebalanced detection scheme based on the two circu-lar polarizations, resulting in the linear sensitivityto the linear birefringence of a sample, has facili-tated, in turn, both the measurement procedure andthe data analysis. After characterization of themirror birefringence has been carefully done, high-finesse cavities can be constructed so as to cancel orminimize the influence of the mirror birefringencefor use in the polarization-sensitive interferometricmeasurement.

This research was supported by the Korea Ministryof Science and Technology.

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