Application of geometricphase techniques to stellar interferometry

William J. Tango and John Davis

In stellar interferometry the fringe visibility can be measured by modulating the optical path differencebetween the two arms of an interferometer. This approach yields accurate estimates of the fringevisibility only if the bandwidth is small, and this limits the sensitivity of the technique. We proposeusing a geometric phase modulator that is achromatic and does not suffer from bandwidth restrictions.Fringe detectors using geometric phase modulation have the potential of greatly increasing thesensitivity of optical stellar interferometers. r 1996 Optical Society of America

1. Introduction

In stellar interferometry starlight collected by aper-tures separated by baseline vector b is coherentlycombined to produce an interference pattern. If thefringe visibility is measured as a function of b, itfollows from the van Cittert–Zernike theorem thatthe data can be used to obtain information about theangular structure of the star being observed.The measurement of fringe visibility is greatly

complicated by atmospheric turbulence that distortsthe arriving wave fronts by introducing randomoptical path fluctuations along the propagation paththrough the atmosphere. While techniques such asintensity interferometry and heterodyne interferom-etry are largely unaffected by these effects, theyhave limited use at optical wavelengths, and all themajor optical interferometers currently operating orunder development are amplitude interferometers inwhich the two light signals are combined directly ata dielectric beam splitter. The two beams thatemerge from the beam combiner ideally have irradi-ances given by

I61z2 5 I031 6 R5S1z264, 112

where S1z2 is the complex fringe signal expressed as afunction of optical path difference 1OPD2 z betweenthe two beams in the interferometer. The fringe

The authors are with the Chatterton Astronomy Department,School of Physics, University of Sydney, New South Wales 2006,Australia.Received 28 June 1995.0003-6935@96@040621-03$06.00@0r 1996 Optical Society of America

signal is

S1z2 5 g1z2 0g 0exp52piz@l 1 if6, 122

where g1z2 is the modulus of the Fourier transform ofthe spectral transmission of the interferometer, nor-malized to unity at z 5 0, 0g 0 is the modulus of thecomplex degree of coherence averaged over the band-width, and f includes all the phase differences thatare not due to the OPD. We define the fringevisibility to be

V1z2 5 g1z2 0g 0 . 132

The principal aim of the fringe detection system isto determine V102 5 0g 0 . In general, however, theOPD is the difference between two large terms: theexternal or astrometric OPD and an internal compen-sating OPD. The astrometric term can be of theorder of the baseline length and varies because of theEarth’s rotation. It is not practicable to computethe compensating OPD with sufficient accuracy toensure that z 5 0 and, consequently, a fringe track-ing servo system is essential. Basically the fringetracker seeks to maximize V1z2 by making smalladjustments to the compensating OPD. Because ofatmospheric turbulence the experimentally mea-sured visibility at z5 0 differs from 0g 0 ; the reductionand calibration of the losses that are due to turbu-lence are important elements in the design of practi-cal stellar interferometers, but they do not affect thisdiscussion.Formally, V1z2 can be found from the complex

fringe signal S1z2 using the identities

V 21z2 5 R25S1z26 1 I25S1z26 5 R25S1z26 1 R252iS1z26. 142

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Experimentally, Eq. 142 can be implemented by mea-suring the square of the real fringe signal, R 25S1z26,during a short sample period, then switching thephase of S1z2 by p@2 and making another measure-ment. This process is then repeated as often as isnecessary to obtain a satisfactory signal-to-noiseratio for the fringe visibility.One can approximate a p@2 phase modulation by

modulating the OPD by l@4 in alternating samples1there are technical difficulties associated with rap-idly switching the OPD, and in practice other OPDmodulation techniques are used to achieve the sameresult2. Modulating the OPD, however, is valid onlyin the quasi-monochromatic limit, and, when finitebandwidths are used, the measured visibility differsfrom V1z2. In order to minimize errors from thissource the bandwidth must be kept small, and thisrestricts the sensitivity of the interferometer.Stepping the OPD by an amount Dz is equivalent

to shifting the various frequency components by thephase 2pDz@l, and it is because this phase variesacross the bandwidth that OPD modulation intro-duces errors. The desired modulation can beachieved only across a wide bandwidth with anachromatic phase modulator that shifts all the fre-quency components in the signal by the same phaseincrement.A modulator that changes the geometric phase of

the signal is, in principle, achromatic. The geomet-ric phase, also known as the Pancharatnam or Berryphase, is the phase associated with a physical sys-tem as it is transported through parameter space.1In particular, if light in an initial polarization stateis split into two beams that are transformed into thesamefinal polarization state by two different arrange-ments of polarizing optics, theremay be a phase shiftbetween the two output light beams, even though theoptical paths for the two beams are the same. If thetwo paths are represented as trajectories on thePoincare sphere, the phase shift will be V@2, whereV is the solid angle subtended by the closed path onthe sphere.Hariharan et al.2 have shown that the geometric

phase is indeed achromatic by demonstrating awhite-light interferometer that uses geometric phasemodulation. A remarkable feature of the geometricphase shifter is that it can be placed between thebeam combiner and the detector after the two beamshave been combined.3 This means that the polariz-ing optics in the modulator do not have to be ofinterferometric quality.Berry’s analysis of geometric phase1 was based on

quantum mechanical arguments, and, consequently,one would expect that geometric phase shifts shouldpersist in the single-photon limit. Hariharan et al.have shown both theoretically and experimentallythat this is the case.4 The fact that geometric phasemodulation is achromatic and can be used withsingle photons suggests that it could be usefullyapplied to stellar interferometry.

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2. Geometric Phase Modulator

Figure 1 shows the beam-combining optics for astellar interferometer utilizing geometric phasemodulation. For simplicity the input optics and theoptics associatedwith the optical path length compen-sating system have been omitted. We also assumethat these optics have no effect on the polarizationstate of the light and that dispersive effects arenegligible. The polarizing beam splitters, PBSA,B,can be used to select a well-defined linear polariza-tion state for the input beams. The other polariza-tion state may be used for other purposes. In theSydney University Stellar Interferometer, for ex-ample, this light is used for a tip–tilt servo systemthat reduces the effects of atmospheric turbulence onthe interferometer.5 We choose the x axis to beparallel to the plane of incidence at the main beamcombiner BC, and we assume that the plane ofpolarization of the light transmitted by the twoPBS’s is also parallel to x.Half-wave plate HA rotates the plane of polariza-

tion so that it is perpendicular to x. The secondhalf-wave plate, HB, has its fast axis parallel to x andservesmerely as a dispersion-compensating element.The beams incident on main beam combiner BC areorthogonally polarized with their planes of polariza-tion parallel and perpendicular to the plane ofincidence. We assume that the transmittances andreflectances are the same for the two polarizationstates. The beams that emerge from the beamcombiner are, therefore, linear superpositions of thetwo polarization states. Two identical modulatorsare used in the output beams, and only one isdescribed in detail.The quarter-wave plate, Q1, has its fast axis

aligned at an angle of p@4 with respect to x and canbe used to convert the two linear polarizations tooppositely handed circular polarizations. Half-wave plate H, which can be rotated, is oriented sothat the angle between its fast axis and the fast axisof Q1 is u. It advances the phase of one of thecircular polarization states by 2u and retards the

Fig. 1. Optical layout of the beam-combining optics and geomet-ric phase modulators for a stellar interferometer. See text for anexplanation of the symbols.

phase of the other component by the same amount.A second quarter-wave plate, Q2, oriented with itsfast axis parallel to that ofQ1, converts the light backto linear polarization, and the PBS resolves this intox and y linearly polarized components that aredetected by the two photon-counting detectors D1and D2. The irradiances at the detectors are

I11z, u2 5 I031 1 V1z2cos14u 1 2pz@l 1 f24, 152

I21z, u2 5 I031 2 V1z2cos14u 1 2pz@l 1 f24, 162

where I0 is the irradiance that would be detected inthe absence of any interference.The output of the second modulator is the same,

apart from an interchange of the signs of the interfer-ence terms because of the p phase shift at BC. Thephase can therefore be modulated by mechanicallyrotating H. For stellar interferometry this is notpractical because sample times as short as 1 ms arerequired. Retarding plates based on ferroelectricliquid-crystal devices6 may be suitable for high-speed geometric phase switching. We also note thatthe polarizing optics, in particular the fixed retard-ing plates, are in general not achromatic. Hariha-ran and Ciddor7 have described polarizing opticsthat, when used in conjunction with ferroelectricmodulators, provide nearly ideal achromatic perfor-mance over a wide bandwidth.Experimentally, the visibility is determined differ-

entially.8 In existing interferometers the quantity1I1 2 I222 is used 3see Eq. 1124. With geometric phasemodulation one has two differential signals that canbe used. Differential measurements are essentialbecause of the scintillation noise associated with theconstant term appearing in Eq. 112 or in Eqs. 152 and162. If I02 is averaged over many sample times itsvalue will be I02 1 sI

2, where sI2 is the scintillation

noise term. This results from the fluctuations inirradiance that are due to atmospheric turbulenceand will bias measurements of the fringe visibility.By making differential measurements this source ofbias can be eliminated, although scintillation willstill have an effect on the measured visibility.In existing interferometers, the fact that the two

output beams must be used differentially limits theflexibility of the instrument. The geometric phasemodulation technique has two independent differen-tial outputs, and this means it is feasible to use twoquite different detection systems. For example, onemodulator could be used for the measurement offringe visibility in one or more narrow-wavelengthbands, whereas the other modulator could be usedfor a high-speed fringe tracker that operates withessentially white light.

3. Conclusion

One of the main reasons for the current interest ingeometric phase modulation is that it can be used to

acquire and lock onto the white-light fringe. Aparticular application is in interferometric surfaceprofiling; to avoid phase ambiguities it is necessaryto use white light and a fringe tracker is essential.3The problem of locating and tracking the white-lightfringe position in a stellar interferometer is analo-gous, and it is clear that geometric phase modulationmay greatly improve fringe tracking techniques,thereby enhancing the sensitivity of the interferome-ter so that fainter objects can be observed. Itshould also improve the accuracy of the measuredvisibility, since V 2 is affected by fluctuations in theOPD, and the magnitude of these fluctuations de-pends directly on the performance of the fringetracker.We have commenced a laboratory-based project to

confirm the operation of geometric phase modulationfor this purpose and to evaluate performance charac-teristics, including transmission losses, switchingspeed, and usable bandwidth. The systemwill thenbe installed in the SydneyUniversity Stellar Interfer-ometer9 so that it can be tested under astronomicalobserving conditions.

This research was supported by a grant from theAustralian Research Council. The authors thank P.Hariharan, M. Roy, and other members of the Physi-cal Optics Department of the School of Physics fortheir helpful discussions.

References1. M. V. Berry, ‘‘The adiabatic phase and Pancharatnam’s phase

for polarized light,’’ J. Mod. Opt. 34, 1401–1407 119872.2. P. Hariharan, K. G. Larkin, and M. Roy, ‘‘The geometric phase:

interferometric observations with white light,’’ J. Mod. Opt. 41,663–667 119942.

3. P. Hariharan andM. Roy, ‘‘White-light phase-stepping interfer-ometry for surface profiling,’’ J. Mod. Opt. 41, 2197–2201119942.

4. P. Hariharan, M. Roy, P. A. Robinson, and J. W. O’Byrne, ‘‘Thegeometric phase: observations at the single-photon level,’’ J.Mod. Opt. 40, 871–877 119932.

5. T. A. ten Brummelaar and W. J. Tango, ‘‘A wavefront tiltcorrection servo for the Sydney University Stellar Interferome-ter,’’ Exp. Astron. 4, 297–315 119942.

6. M. O. Freeman, T. A. Brown, and D. M. Walba, ‘‘Quantizedcomplex ferroelectric liquid crystal spatial light modulators,’’Appl. Opt. 31, 3917–3929 119922.

7. P. Hariharan and P. E. Ciddor, ‘‘An achromatic phase-shifteroperating on the geometric phase,’’ Opt. Commun. 110, 13–17119942.

8. W. J. Tango and R. Q. Twiss, ‘‘Michelson stellar interferom-etry,’’ in Progress in Optics, E. Wolf, ed. 1North-Holland,Amsterdam, 19802, Vol. 17, pp. 241–276.

9. J. Davis, ‘‘The Sydney University Stellar Interferometer,’’ inIAU Symposium No. 158: Very High Angular ResolutionImaging, J. G. Robertson and W. J. Tango, eds. 1KluwerAcademic, Boston, Mass., 19942, pp. 135–142.

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