C. Ruilier and F. Cassaing Vol. 18, No. 1 /January 2001 /J. Opt. Soc. Am. A 143

Coupling of large telescopes andsingle-mode waveguides:

application to stellar interferometry

Cyril Ruilier*

Observatoire de Paris, Departement de Recherche Spatiale 5, place Jules Janssen, F-92195 Meudon, France,and Departement d’Optique Theorique et Appliquee, Office National d’Etudes et de Recherches Aerospatiales,

BP 72, F-92322 Chatillon Cedex, France

Frederic Cassaing

Departement d’Optique Theorique et Appliquee, Office National d’Etudes et de Recherches Aerospatiales,BP 72, F-92322 Chatillon Cedex, France

Received January 4, 2000; revised manuscript received July 5, 2000; accepted July 31, 2000

The coupling between a turbulence-distorted optical beam and a single-mode waveguide is addressed. Thecoupling efficiency and the coupled phase are derived, both without aberrations and with small aberrations.These analytical expressions are validated by numerical simulations. Correction with adaptive optics is in-vestigated. In the general case, the Strehl ratio is a pessimistic estimator, and the coupled phase is differentand has a smaller variance than the classical phase averaged over the pupil. Application fields are hetero-dyne detection and stellar interferometry, for which spatial and modal filtering are distinguished. © 2001 Op-tical Society of America

OCIS codes: 060.2430, 120.3180, 350.1260, 010.1330, 010.1080.

1. INTRODUCTIONAfter propagation through the atmosphere, optical beamsundergo random phase disturbances induced by atmo-spheric turbulence.1 On the other hand, a beam is unal-tered when propagating in a single-mode waveguide(SMW) such as a single-mode fiber.2 Therefore injectinga turbulent beam into a SMW produces spatial filtering.

This filtering property has been used successfully instellar interferometry. Because of atmospheric turbu-lence, the visibility measured by interference betweentelescopes undergoes random fluctuations, reducing accu-racy. Since observed objects are usually unresolved bysingle apertures, coupling each beam in a SMW convertsturbulence into photometric variations that can be moni-tored by dedicated photometric channels.3 This concepthas been demonstrated by the Fiber Linked Unit for Op-tical Recombination (FLUOR) instrument,4 now produc-ing scientific results on the Infrared and Optical Tele-scope Array (IOTA) interferometer. Spatial filtering andphotometric calibration provided with single-mode fibersroutinely lead to an accuracy better than 0.5% on visibil-ity measurements.5

Spatial filtering is now widely used and will be a keycomponent for the next generation of instruments.6 Onthe ground, the limiting magnitude will be considerablyincreased with very large telescopes corrected by adaptiveoptics (AO). However, the need for spatial filtering in-creases with the telescope diameter since correction is al-ways partial.7 In space, aberrations in the optics mustbe filtered out for high rejection in interferometriccoronography.8,9 Moreover, future instruments with

0740-3232/2001/010143-07$15.00 ©

modulation, combination, and detection integrated on asingle chip by planar SMW’s will also include modalfiltering.10

The coupling of aberrated beams in SMW’s is thus acritical issue. This has been addressed without AOcorrection11 and with tip–tilt correction12 or higherorders.13 The present paper emphasizes analytical re-sults. First, modal filtering is introduced in Section 2.The coupling efficiency between a telescope without phasedistortion and a Gaussian SMW is given. The effect ofaberrations is addressed in Section 3, where analyticallaws for the coupling loss and the coupled phase are de-rived for small aberrations. Section 4 is dedicated to thecoupling of ground-based telescopes with AO correction inGaussian SMW’s: Laws are given for coupling fluctua-tions and coupled energy and are confirmed by numericalsimulations for various levels of correction.

2. MODAL FILTERINGA. NotationA telescope with a pupil diameter D and a linear centralobstruction a is considered. In the telescope pupil, as-suming a constant intensity, the incoming beam is char-acterized by a turbulent phase f(r) that can be expandedon the Zernike polynomials Zi (Ref. 14):

f~r! 5 (i51

`

aiZiS 2r

D D . (1)

The telescope’s complex pupil can also be expanded ona basis of modes Mi :

2001 Optical Society of America

144 J. Opt. Soc. Am. A/Vol. 18, No. 1 /January 2001 C. Ruilier and F. Cassaing

Pf 5 P0 exp~if! 5 (i50

`

pi~ f!Mi , (2)

where i2 5 21 and P0(r) is the pupil transmittance de-fined as

P0~r! 5 H 1 if a <2uru

D< 1

0 otherwise

. (3)

We will assume in the following that ^u&W denotes a sca-lar product with weight W, defined by

^XuY&W 5 EER2

W~r!X~r!Y* ~r!d2r. (4)

We will also use the spatial average and variance of Xwith a weight W, normalized so that ^1&W 5 1:

^X&W 5^Xu1&W

^1u1&W, (5)

sW2 ~X ! 5 ^X2&W 2 ^X&W

2 . (6)

For example, orthonormality of the Zernike polynomi-als over a full circular pupil can be written as

^ZiuZj&P05 d ij , (7)

where d ij is the Kronecker notation, and the phase vari-ance over the pupil is

sP0

2 ~ f! 5 (i52

`

ai2. (8)

B. Interferometric FilteringLet us consider as a first step the pupil-plane interferencebetween the telescope and a reference beam R, with coef-ficients ri on the modes Mi . If $Mi% is an orthonormalbasis, the signal measured by a single-pixel detector is

I 5 ^uPf 1 Ru2&P05 2 Re (

i50

`

pi~ f!ri* 1 incoh. terms.

(9)

This signal results from the superposition of interfero-grams given by each mode. Because of atmospheric tur-bulence, each pi( f) is complex with random amplitudeand phase. Information in the multimode interferencesignal is thus degraded. To compensate for atmosphericturbulence, a first solution is to correct the phase f withAO. However, to expect full correction is unrealistic.Another solution is to measure a single component of Pf .For example, if the reference beam is R 5 M0 , interfer-ence with the turbulent beam gives access to p0( f)5 ^PfuM0&P0

while rejecting incoherent photons onhigher modes.

This solution is used in heterodyne detection with a la-ser reference beam R. A similar filtering occurs in cou-pling a turbulent beam into a SMW: The coupled beamhas a perfectly defined profile M0 . Turbulence is con-verted into random fluctuations of a single complex com-ponent p0( f) and thus can be overcome by photometriccalibration and fringe tracking or short exposures.

Thus, modal filtering with a SMW must be distin-guished from the classical spatial filtering with a focal-plane pinhole, as proposed by Prasad and Loos15 forground-based interferometres. With a pinhole, the en-ergy outside the central core of the point-spread functionis removed while the amplitude and the phase passingthrough the pinhole are unaffected. The transmittedbeam is still multimode. Therefore the interferogram re-sulting from the combination of pinhole-filtered beamsfrom different telescopes would also be multimode. Onlyan infinitely small pinhole or a long succession of focalpinholes and pupil stops would realize true modal filter-ing.

Modal filtering is thus best suited for high-accuracymeasurements. Most stellar interferometers now useSMW’s instead of pinholes. We therefore address in thefollowing the coupling of an aberrated beam into a single-mode beam R 5 M0 .

C. Coupling EstimatorsThe matching between the distorted beam Pf and a SMWM0 can be characterized in the pupil plane by the normal-ized overlap integral2:

Vf 5^PfuM0&P0

~^PfuPf&P03 ^M0uM0&P0

!1/2 , (10)

where M0 is the mode profile in the pupil plane. Thanksto the Parseval theorem, a similar equation holds in theimage plane. The pupil-plane integrand,

Wf 5 PfM0* , (11)

will be used extensively since P0 , F, and M0 are knownand simple functions. We will mostly use the squaredmodulus (coupling efficiency) and the phase of Vf :

rf 5 uVfu2, (12)

Cf 5 arg Vf . (13)

Because of the Schwartz inequality, 0 < rf < 1, andsince W0 is real without aberrations, C0 5 0 and rf

< r0 . Otherwise, the exact value of these estimatorsdepends on the phase aberration f and the mode profileM0 .

D. Coupling with a Gaussian BeamWe consider in this section a single-mode fiber at the fo-cus of a telescope (focal length F, diameter D, linear cen-tral obstruction a). In the usual Gaussian approxima-tion, the LP01 mode is characterized at a wavelength l bythe radius v at 1/e (Ref. 2). Since the SMW mode is stillGaussian after propagation to the pupil plane, an analyti-cal form of the coupling efficiency can easily be derivedwithout aberrations.13 From Eqs. (10) and (12),

r0~a, b! 5 2Fexp~2b2! 2 exp~2b2a2!

b~1 2 a2!1/2 G2

, (14)

where

b 5p

2

D

l

v

F. (15)

C. Ruilier and F. Cassaing Vol. 18, No. 1 /January 2001 /J. Opt. Soc. Am. A 145

Even in the case of a plane wave front, the diffractionspot mismatches the Gaussian focal mode. Equation (14)gives a maximum coupling of ;80% when a 5 0 andv/F 5 0.71l/D. This value is used to match the tele-scope and SMW numerical apertures.12 To best of ourknowledge, Eq. (14) was published for the first time inRef. 13.

The coupling efficiency decreases while a increases,since sidelobes of the diffraction spot, in phase opposition,are reinforced. The plot of r(a, bopt) versus a is shown inRef. 13: The coupling loss is small when a < 20% and ismuch more severe above.

3. EFFECT OF SMALL ABERRATIONSWith phase disturbances over the pupil, the argument Wf

of the overlap integral of Eq. (10) is complex. Appendix Ashows that the integral encounters a modulus reductionand a phase shift. We first derive analytical expressionsof the coupling loss and the coupled phase for small aber-rations. The effects of some specific aberrations are thenemphasized.

A. Approximate Expression for Coupling EfficiencyTo estimate the loss induced by aberrations with the ana-lytical expression derived in Appendix A, we first expressVf given by Eq. (10) as a normalized phasor average us-ing the notation of Eqs. (4)–(6) and (11):

Vf

V05

^PfuM0&P0

^P0uM0&P0

5 ^exp~if!&W0. (16)

Therefore, using Eqs. (A3) and (12)–(13),

rf /r0 ' exp@2sW0

2 ~ f!#, (17)

Cf ' ^ f&W0. (18)

A special case is obtained for an infinitely small focal-plane filter. Then Cf 5 ^ f&P0

, and approximation (17)turns into the well-known Strehl ratio approximationwith the spatial phase variance over the unweightedpupil16:

M0 5 1 ⇒rf

r0' exp@2sP0

2 ~ f!#. (19)

In the other cases, the average over the pupil should beweighted by the mode profile in the pupil plane to derivethe coupling efficiency and the coupled phase. This af-fects, for example, the design of AO systems: Since thequality criterion is the residual phase variance weightedby W0 , it is more efficient to correct the center of the pu-pil than the periphery. Another consequence is that theZernike polynomials are no longer orthonormal over theweighted pupil, as detailed in Subsection 3.B.

B. Application to Zernike PolynomialsReporting the Zernike phase expansion of Eq. (1) in ap-proximation (17) shows that with modal filtering, thequantity of interest is the weighted scalar product of theZernike polynomials. Equation (7) then turns into

g ij 5 ^ZiuZj&W0Þ d ij . (20)

The absolute values of the weighted scalar-product ma-trix g ij are shown in Fig. 1 for a full circular pupil and thefirst Npol 5 91 Zernike polynomials (i.e., radial order 12included). The matrix is nearly diagonal, but some termswith the same azimuthal frequency are correlated, suchas tip–tilt and comas or piston, defocus, and spherical ab-erration. These cross-coupled terms are reinforced whena Þ 0. Figure 2 shows the covariance terms and themean cross-coupled energy. After the twentieth polyno-mial, covariance terms vary with nearly constant ampli-tude around the mean value 0.8, and the mean cross-coupled energy quickly tends toward 0. For a largenumber of polynomials with similar amplitude, one canuse the following approximation:

g i,j ' 0.8d i,j . (21)

Fig. 1. Weighted scalar product matrix of the first 91 Zernikepolynomials.

Fig. 2. Covariance g i,i (left scale, solid symbols) and meancross-coupled energy S iÞjug iju2/Npol (right scale, open symbols)for the first 91 Zernike polynomials.

146 J. Opt. Soc. Am. A/Vol. 18, No. 1 /January 2001 C. Ruilier and F. Cassaing

Figure 3 shows the evolution of the coupling efficiencyfor the first polynomials corresponding to the classical op-tical aberrations. For each mode i, the normalized cou-pling efficiency has been computed by numerical simula-tion and plotted versus the amplitude ai ranging from 0to 0.5 rad. In addition, analytical curves calculated fromrelations (17) and (20) are plotted for each aberration,with g i,i values deduced from the scalar-product matrixgiven in Table 1. Analytical curves match the numericalcurves with a relative accuracy of better than 1% forsmall ai values. An accuracy of 5% is still expected forai 5 0.8, and then accuracy rapidly decreases beyondthat value. A more extended scale can be found in Ref.13. These curves clearly show that the coupling effi-ciency depends on the strength and kind of aberrations,whereas the Strehl ratio depends only on their strengthfor a full circular pupil. Asymmetric modes and center-to-periphery modes are more penalizing. Defocus andcoma are of primary importance in the coupling loss.The Strehl ratio is thus a rather pessimistic approxima-tion. This is confirmed for higher orders, for clarity notrepresented on Fig. 3.

C. Tilt CompensationA basic correction consists of stabilizing the diffractionspot at the focus of the telescope. Considering only onedirection, the residual phase after tilt correction by an ac-tive mirror with amplitude a28 is

fres 5 f 2 a28Z2 . (22)

For classical imaging, to minimize the phase varianceover the full circular unweighted pupil, the best correc-tion is to ensure that a28 5 a2 . With modal filtering, cor-rection should differ. The optimum coupling is reachedwhen the weighted variance of fres is minimum:

Fig. 3. Coupling loss rf /r0 for the first Zernike modes, com-puted by simulation (symbols) and approximation (17) (dottedcurves). Solid curve, Strehl ratio approximation given by ap-proximation (19) and Eq. (8).

Table 1. gi,i Terms for the First ZernikePolynomials

Zi Z1 Z2,3 Z4 Z5,6 Z7,8 Z9,10 Z11

g i,i 1 0.796 1.051 0.707 0.980 0.658 1.037

a2opt8 5^ fuZ2&W0

^Z2uZ2&W0

5 (i51

`

ai

g2,i

g2,2. (23)

Figure 1 shows that some modes with i Þ 2 but thesame azimuthal frequency m 5 1 contribute to a28 . Fora full circular pupil and an optimized Gaussian mode, rel-evant terms are g2,2 ' 0.8, g2,8 ' 20.25 and g2,16' 0.05. The correction to be applied is thus closer to theangle of arrival (the first derivative of the phase over thepupil, including contributions from the same high-ordermodes and measured by classical means such as focal-plane centroiding or quad-cell detectors) than to theZernike tilt.17

D. Coupled PistonAuthors usually consider only the coupling efficiency rf ,but the argument of the complex overlap integral Cf is amajor issue in interferometry. Approximation (18) andFig. 1 show that the main term contributing to the phaseCf is of course the piston a1 with weight 1. But all otherZernike modes with azimuthal symmetry (m 5 0), suchas defocus (a4) and spherical aberration (a11), also con-tribute to the piston with respective weights 21/3 and1/10. A more detailed analysis of the piston is presentedin Subsection 4.C.

4. COUPLING WITH ADAPTIVE OPTICSPerformance of AO systems has already been examinedwith use of the Strehl ratio.18 Since the Strehl ratio hasbeen shown to be a pessimistic estimator for modal filter-ing, this section evaluates AO performance on the basis ofthe coupling efficiency in a SMW. Analytical expressionsare derived and are confirmed by numerical simulationswith the first 861 Zernike polynomials and 2048 phasescreens (so that the variance of the coupling efficiency iscalculated with a precision of ;2%) as proposed by Nico-las Roddier.19 Turbulent wave fronts are characterizedby the Fried parameter r0 (Ref. 1). Parameters of thesimulation are a 5 0, D/r0 ranging from 0 to 32, and par-tial correction with a perfect AO system, for which allZernike coefficients are forced to zero up to a radial ordern (included). Radial orders 0, 1, 3, 5, 7, 9, and 16 havebeen successively considered, respectively correspondingto a raw turbulence or correction of the first i0 Zernikepolynomials with i0 5 3, 10, 21, 36, 55, and 153. Aftertotal correction of the first n radial orders, the residualphase variance over the unweighted pupil is7,14

^fP0

2 ~ f!& t 5 (i5i011

`

^ai2& ' 0.46~n 1 1 !25/3S D

t0D 5/3

. (24)

In the following, temporal variation is omitted but as-sumed and recalled with the temporal average ^ • & t .

A. Relative FluctuationsInjection stability in the SMW is of prime importance forefficient modal filtering. Assuming that the residualphase variance over the weighted pupil is small, a first-order expansion of approximation (17) leads to

^rf& t /r0 ' 1 2 ^ sW0

2 ~ f!& t . (25)

C. Ruilier and F. Cassaing Vol. 18, No. 1 /January 2001 /J. Opt. Soc. Am. A 147

The residual phase variance over the weighted pupilcan be approximated for large i0 by approximation (21)and Eq. (24):

^ sW0

2 ~ f!& t ' (i5i011

`

g ij^ai2& t ' 0.8sP0

2 ~ fres!. (26)

Coupling fluctuations can be expressed as the ratio be-tween the root mean square and the average of the cou-pling efficiency. A similar calculation from Eq. (17) gives

s t~rf!

r0' s t@ s W0

2 ~ f!#. (27)

The right-hand term involves the fourth-order momentof f. With Eq. (25), calculation leads to7

s t~rf!

^rf& t' 0.816

sP0

2 ~ fres!

n 1 1. (28)

Figure 4 shows coupling fluctuations obtained by nu-merical simulations and by Eq. (28), for each level of cor-rection. Three regimes are clearly evidenced. For smallphase residues, coupling fluctuations obey a log-linearlaw versus D/r0 , confirming analytical expression (28).The difference between numerical simulations and ex-pression (28) decreases with the level of correction n, re-sulting from asymptotic approximation (21). In this re-gime, coupling fluctuations are due essentially to the spotdisplacement over the SMW. The standard deviation ofthe coupling efficiency is smaller than the average but in-creases more rapidly than the average decreases whenthe turbulence becomes stronger. Large values of D/r0correspond to the well-known speckle regime wherebright and dark speckles alternatively pass over theSMW.1 The coupling average balances the coupling stan-dard deviation. This balance leads to a saturation re-gime. The knee between the linear regime and the satu-ration regime occurs for D/r0 values increasing with thecorrection level i0 . The intermediate regime corre-sponds to a transition between the spot displacement re-gime and the speckle regime. This regime appears in anobvious way in the raw-turbulence case and becomesmore discreet as the low orders are corrected.

B. Coupled EnergyThe relevant figure of merit for AO is the amount of en-ergy injected into the SMW, defined as

E 5 ~D/r0!2^rf& t . (29)

Figure 5 plots E, the mean coupling efficiency being de-duced from approximations (24)–(26). The small-aberration approximation fits the simulation well on aseeing range that increases with the level of correction.The raw-turbulence case puts in evidence the competitionbetween the increase in size of the telescope and the cor-responding number of speckles over the SMW. A satura-tion regime clearly appears as soon as D/r0 > 1. For thecorrected cases, a monotonic increase in the total coupledenergy precedes a sharp decrease, as had been noted forthe tip–tilt correction in Refs. 12 and 20. The optimumcoupling for each level of correction is given by

~D/r0!opt ' 2.037~n 1 1 ! (30)

Eopt ' 1.251r0~n 1 1 !2. (31)

Table 2 shows the good agreement between approxima-tions (30) and (31) and optimum values given by simula-tions.

These laws show that for maximum injection, the orderof correction has to be adapted to the local atmosphericconditions. The same calculation with the Strehl ratiochosen as coupling estimator gives an optimum that is 5%smaller for a D/r0 value that is 13% smaller. This meansthat the gain in coupling still increases with the telescopediameter when the Strehl ratio begins to decrease. This

Fig. 4. Relative coupling fluctuations s t(rf)/^rf& t given by ap-proximation (28) (dotted curve) and by simulation (symbols).

Fig. 5. Coupled energy E in arbitrary units given by Eq. (29)(dotted curve) and by simulation (symbols).

Table 2. Coordinates of the Maxima of Fig. 5Given by Approximations (30) and (31) (top)

and Simulation (bottom)

Order n 1 3 5 7 9 16

SD

r0D

opt

4.1 8.1 12.2 16.3 20.3 34.64.9 8.4 12.4 16.3 20.2 35.0

Eopt 4.0 16.0 36.0 64 100 2894.4 15.6 33.8 60 95 318

148 J. Opt. Soc. Am. A/Vol. 18, No. 1 /January 2001 C. Ruilier and F. Cassaing

confirms that modal filtering is less sensitive to the tur-bulence than the Strehl ratio is.

The difference between correction of the optimum tiltdefined by Eq. (23) and the Zernike tilt can be evaluatedwith coupled energy. The abscissa of the optimum,D/r0 ' 4.5, is the same for the two corrections, but the in-jection is increased by 3%. This difference can be largerif a static coma exists, which can be due, for example, to amisalignment of the off-axis parabola used for the injec-tion into the fiber.

C. Fluctuation of the Coupled PistonIn stellar interferometry, the main perturbation term inamplitude is the differential piston between the pupils.This is even more the case with an interferometer thathas modal filtering and photometric calibration, when thepiston effect is the only uncontrolled parameter. A fringetracker must then be used to reduce the amount of re-sidual piston.

We have seen in Subsection 3.D that the piston injectedinto each fiber is not the classical piston a1 5 ^ f&P0

butcan be approximated by Cf 5 ^ f&W0

. In a stellar inter-ferometer with modal filtering and AO residues, the dif-ferential piston between the pupils therefore includescontributions from other modes. For example, in thecase of an 8-m telescope in the K band (D/r0 . 19) with a55-mode AO correction (n 5 9), the standard deviation ofthe coupled piston is ;27 mrad (10 nm) with a1 5 0.This optimistic value, since the simulation is made for a5 0 and does not consider AO residues for i < i0 , isabove the requirement for some applications.9 Thefringe tracker should measure the same piston as the sci-entific instrument to be effective: The two systemsshould therefore use similar spatial filtering. This filter-ing similarity has a significant effect on the design offringe trackers for stellar interferometers when signifi-cant AO residues remain.

Moreover, it could be thought that the coupled pistonCf is larger than a1 since other modes contribute. Allthe contributions resulting from the same phase screenare in fact correlated. Assuming a Kolmogorov phasescreen and using the Taylor hypothesis, one can derivethe temporal spectrum of the differential piston. Adapt-ing calculations of Ref. 21 to a weighted pupil, we canshow that the coupled piston has a lower amplitude and afaster high-frequency decrease since the weighted pupil issmoother.22

5. CONCLUSIONSFiltering is now widely used in stellar interferometry tofilter out turbulence residues. With modal filtering bySMW’s, turbulence can be fully calibrated by photometriccalibration and fringe tracking or short exposures. Withspatial filtering by pinholes, interferograms undergo ran-dom visibility attenuations since transmitted beams aremultimode. We have investigated the coupling betweenlarge telescopes and SMW’s and have derived the maxi-mum coupling efficiency without aberrations for classicalGaussian SMW’s.

The coupling loss for small aberrations is approximatedby the exponential of the opposite of the phase variance

over the pupil. This generalizes the classical Strehl-ratioapproximation, corresponding to an infinitely small guide.For a real guide, the variance should be weighted by theamplitude of the reference beam over the pupil. The cou-pling loss depends not only on the aberration strength(the classical unweighted phase variance over the pupil)but also on its kind. The formula for the coupling losshas been validated by numerical simulations for a Gauss-ian mode. It turns out that for a large number of Zernikepolynomials, the weighted variance is approximately 80%smaller than the unweighted variance: The Strehl ratiois thus a pessimistic estimator.

Likewise, the coupled phase is the mean phase of theaberrated beam weighted by the amplitude of the refer-ence beam. Therefore, assuming azimuthal symmetry ofthe reference beam, all the components with azimuthalsymmetry in the aberrated phase are taken into account.These include piston but also higher-order aberrationssuch as defocus and spherical aberration.

Modal filtering can thus considerably enhance stellarinterferometers. Future space-borne instruments23 orlarge ground-based telescopes with AO24 will use SMW’s.For better results, modal filtering must be taken into ac-count in the design of the instrument, particularly the AOsystem and the fringe tracker.25

APPENDIX A: AVERAGE OF PHASORSWITH SMALL DISPERSIONIn this appendix, an average of phasors ^exp(if)& is ap-proximated from the two first moments of the phase f.One can always write

exp~if! 5 exp~i^ f&! 3 exp@i~ f 2 ^ f&!#. (A1)

For small f 2 ^ f&, if the average operator ^•& is linearand normalized (^1& 5 1), a Taylor expansion gives

^exp~if!& 5 exp~i^ f&!F1 21

2s 2~ f! 1 ...G (A2)

' exp~i^ f&! 3 exp@ 2 s2~ f!/2#. (A3)

Expressions (A2) and (A3) have the same second-orderbehavior, or third-order when the probability densityfunction of f 2 ^ f& is even. But approximation (A3) is abetter approximation than Eq. (A2) since the fourth-orderterm, truncated in Eq. (A2), is present with the correctsign in approximation (A3). This approximation is goodfor s 2( f) < 1. Moreover approximation (A3) strictlyholds when f has a Gaussian probability density func-tion.

ACKNOWLEDGMENTSThe distinction between modal and spatial filtering is theresult of discussions within the FLUOR (Fiber LinkedUnit for Optical Recombination) and AMBER (Astronomi-cal Multiple BEam Recombiner) groups and with peoplefrom IRCOM (Institut de Recherche en CommunicationsOptiques et Micro-ondes).

Please send reprint requests to Cyril.Ruilier@alcatel.fror cassaing@onera.fr.

C. Ruilier and F. Cassaing Vol. 18, No. 1 /January 2001 /J. Opt. Soc. Am. A 149

*Now at Alcatel Corporate Research Center, Route deNozay, F-91461 Marcoussis Cedex, France.

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