multiple-order radial-grating shearing interferometer

Multiple-order radial-grating shearing interferometer

Philip D. Henshaw and Scott K. Manlief

A shearing interferometer suitable for coherent phasing of multiple-aperture optical systems is described.The interferometer uses the properties of a rotating radial square wave grating to simplify interaperturephase measurements. A theoretical development and the results of experiments conducted at a wavelengthof 10.6 ptm are presented.

1. Introduction

Multiple-aperture optical systems have been used toincrease the resolution of an optical system.'-5 Whensuch arrays of apertures are used in a coherent systemfor both transmit and receive, the transmitted wavefronts must be adjusted so that they appear to be sec-tions of a larger coherent wave front. One commonmethod of comparing sections of wave fronts is with aradial-grating shearing interferometerY65 Here amodification of this technique is described which allowsit to be used to sense both the tilt and relative phase ofadjacent optical channels in such a way that the re-quired wave front corrections can be generated with aminimum of computation.

II. Analysis

We will consider here the requirements which mustbe satisfied for a two-channel system to supply a planewave front. As can be seen from Fig. 1, the two chan-nels must each have the proper tilt, so that a, = 2, andthe proper phase, so that 01 - 02 = 2n-7r (n integer). Ashearing interferometer which provides a shear S anda detector which can measure the phase AO, betweentwo wave fronts are necessary. (Exactly how this canbe done will be described below.) The tilt of the wavefront a relative to the direction of shear is related to SIand AO, by

sina = XAoi/2rS1 (1)

When this work was done both authors were with MIT LincolnLaboratory, P.O. Box 73, Lexington, Massachusetts 02173; P. Hen-shaw is now with SPARTA, 1844B Massachusetts Avenue, Lexington,Massachusetts 02173.

Received 19 June 1981.0003-6935/82/101772-06$01.00/0.© 1982 Optical Society of America.

as shown in Fig. 2. Now if the same shear can be ap-plied uniformly over the entire wave front it will bepossible to measure the tilt of each channel. There aretwo points to note here. First, even if the shear is notknown exactly, the tilts of each channel can be madeequal as long as the shear is uniform. Second, it is de-sirable to keep the value of the shear quite small, sincethere is a 2 ambiguity in the phase measurement.Using a small shear, larger unambiguous values of tiltcan be measured. Let us now turn to the problem ofmeasuring the relative phase of each channel.

Suppose a larger shear Sn can be applied to the two-channel system. (The reason for this notation will bemade clear below.) Now by the same reasoning as wasapplied to the single-channel measurement, the phasemeasured should be

AO,, = 27rSn sina/X, (2)

and the actual measured phase could be adjusted to thisvalue, thus assuring zero relative phase between the twochannels. However, as can be seen from Eq. (2), thedesired value of AO,, is a very sensitive function of S,and sina due to the presence of X in the denominator.Furthermore, some computation is required to producethe numerical value of ,A desired. Substituting Eq.(1) for sina into Eq. (2)

(3)

we see that this computation can be reduced consider-ably if a two-part measurement is made. In part 1 thephases corresponding to the tilt values of each channelAOk1 and AO'1 are set equal. Part 2 is a similar mea-surement of AOn in which the relationship of Sn to Siis known as accurately as possible. The relative phaseof the two channels is then adjusted to satisfy Eq. (3).The relationship between Sn and SI can be fixed exactlyto nS1 = Sn by using a square wave grating. The useof this method to compare the tilts and phases of twoadjacent channels is shown in Fig. 3 for S5 = 5 Si. Inthis figure the two 0th order wave fronts, and the dif-

1772 APPLIED OPTICS / Vol. 21, No. 10 / 15 May 1982

AO.n = (Sn/S1)i\0le

\#2

Fig. 1. Wave front requirements for a two-channel coherentsystem.

CHANNEL~~171CANEL TING X

RADIAL Of

LENS LENS 2

- -'H--- f 'i f-±-- fAPERTURE GRATING IMAGE

PLANE PLANE PLANE(x,y) (X0,yo) Cx1,y1 )

Fig. 4. Shearing interferometer for producing multiple shears.

SHEAR 1

X Asin a = 2vS1

Fig. 2. Local wave front tilt measurement using shearinginterferometer.

Consider the arrangement shown in Fig. 4. The wavefront is to be measured in the aperture plane (x,y).Lens 1 is arranged so that the wave front is focused ontothe rotating radial grating in the (xo,yo) plane. Byplacing both the aperture plane and the grating planea distance f from the lens, the Fourier transform of theaperture plane will be projected onto the grating. Thistransform is multiplied by the grating function and thentransformed by lens 2 so that the transform of the(xo,yo) plane appears in the (x1 ,yl) plane. This planeis also the image of the aperture plane.

Now let the wave front in the aperture plane beU(x,y) and the grating be G(xo,t). The field in theimage plane will then be

U(xi,yi,t) = FfF[U(x,y)] Gxo,t)l, (4)

SIGNAL AT MEASURES A CHANNEL N PHASE F 5A

SIGNAL AT 5f0 MEASURES Jf J C 1

DETECTOR

A05 1 S5 'CHANNEL 2

CHANNEL 2 Si / TH ORDER5TH ORDER JC CHANNEL 1 WAEFRONTWAVEFRONT 0 rTH ORDER

__W VAVFRONT DIRECTION OF SHEARCHANNEL 11 ST ORDERWAVEFRONT

Fig. 3. Comparison of tilts and phases of two adjacent channels.

where F is the Fourier transform operator.A useful expression can now be derived by letting the

wave front be represented by

U(x,y) = expUj0(x,y)], (5)

where 4(x ,y) is the phase of the wave front, and we havedropped the amplitude for simplicity. The grating canbe represented in general form by8

G(xo,t) = C expUj2nir(xo - ut)/P],n=--

(6)

where v is the grating velocity and P is the period.Substituting Eqs. (5) and (6) into Eq. (4) and using theFourier transforming relationship of a lens of focallength f, we get

U(xiyt) = Cn expji(x) - f/P,y]n=--

fracted wave fronts which overlap with channel 1 areshown. The detector measures AO1 corresponding tothe tilt of channel 1 and A05 corresponding to the phasedifference between channels 1 and 2. The measure-ments can be separated because they are associated withsine waves at different frequencies. This approachallows phase measurements to be made of channelswhich have a center-to-center separation much largerthan their individual diameters. The manner in whichthis can be done depends on details of the shearing in-terferometer which will now be discussed.

X exp(-j2nrvt/P). (7)

At this point it is convenient to introduce the shear

Si = Xf/P, (8)

which is the amount of sideways displacement given tothe first-order wave fronts diffracted by the grating.The intensity in the image plane is given by

I(x,y 1,t) = IU(xl,ylt)1 2.

By substituting Eqs. (7) and (8) into Eq. (9) we get(9)

15 May 1982 / Vol. 21, No. 10 / APPLIED OPTICS 1773

5Xf

l01

zk,� k� j

4"'41011�

Fig. 5. Schematic of two-channel phasing experiment.

I(x,yt) = fi CC,n=-- m=-X

X expUjP(x - nSy) - jq5(x - mSy)]

X exp[-j2gr(n - m)vtlPJ. (10)

Grouping the complex conjugate terms together, ap-propriately separating the terms for m equal to n, andincluding the phase angle of Cn into the exponentgive

I(x,y,t) = i icr2+ fi E Cnl lC.n =-X n=-- m=--

X cos[(x-nS,y)-4(x-mSy)

- 27r(n - m)vtIP + arg(Cn) - arg(Cm)]. (11)

The important part of Eq. (11) is the argument of thecosine function which contains the phase differencebetween the two wave fronts sheared by differentamounts. These cosine terms may be separated byappropriate filtering since they are modulated atfrequencies

fn-r = (n - m)vIP. (12)

This modulation may be taken advantage of to detecttwo values of shear which are exact multiples [as re-quired by Eq. (3)] by appropriate choice of gratingfunction. A good choice, which is relatively easy tofabricate, is a square wave grating with values of zeroand one, which has coefficients given by

[(-J)(n-l)/2/n n odd,

Cn = n even, (13)

n = 0.

Due to the absence of even harmonics, odd modulationfrequencies can only be obtained by interference withthe zeroth order. Furthermore, since the values of Cnare all real, arg(Cn) = 0 or 7r. Thus using bandpassfiltering at a frequency fn = nv/P and taking into ac-count contributions from both positive and negativefrequencies, the detected intensity at the point (xy) inthe image plane will be

I(x,y,t) = CnIcos[x - nSy) - (xy)]+ cos[p(x + nS,y) - (xy)]]. (14)

For small shears and plane wave fronts

k(x - nSy) -(x,y) = (x,y) - (x + nS,y), (15)

while for large shears, one of the terms in Eq. (14) willbe zero for a suitably located detector near the edge ofthe aperture. Thus the two terms in Eq. (14) may beseparated.

The conclusions which follow directly from the pre-ceding analysis are summarized:

(1) Using suitably located detectors and a squarewave grating, two shears can be measured simulta-neously using bandpass filtering at appropriatefrequencies.

(2) The measured shears are exactly related by anodd integer multiple.

(3) The ratio of shears is limited by the amplitude ofthe diffracted wave front, which drops off as 1/n.(Grating accuracy tolerances have not been consideredhere.)

(4) Positive and negative shears appear at the samefrequency. These need not be separated in the case of


F\-''S' 0 D00D;'- ' --i''''.'-1 'V, '

TH EORY

(a) PHASE ERROR A 0,

EXPERIMENT

TILT ERROR 0

; i,

NS A: 110 : y F : :

TH EORY

(b) PHASE ERROR ;A 0.5, TILT ERROR 0

Fig. 6. Comparison of experimental and theoretical far-field profiles (phase error in wave fraction).

small shear measurements on plane wave fronts, and inthe case of interchannel measurements, separation canbe achieved by an interpolation procedure starting atthe edge of the array.

Ill. Experimental Results

The goal of the two-channel phasing experiment wasto combine the output of two Fabry-Perot wedgeswitches' coherently. The essential features of ascheme to do this are shown in Fig. 5. The two channelsare derived from a single beam by means of a beamsplitter. The relative phase of the two beams can becontrolled by a piezoelectric actuator on one mirror, andthe tilts can be controlled by the small angle deflectiondevice used to fill in between digital switch positions.The two Fabry-Perot switches are arranged so that thedigital beam positions are parallel to the center-to-center line between the apertures. The beams are splitafter leaving the two switches with part going to the farfield (in this case a detector at the focus to look at thefar-field pattern) and part going into a shearing inter-ferometer. To operate over a wide field of view, agrating which produces many orders is necessary. Inthis fashion, at least 1 order will always be directed intothe shearing interferometer if the grating spacing isproperly chosen. In our experiment it was necessaryto use a beam splitter in place of the wire grating due tothe low power of the CO2 laser. The shearing inter-ferometer, which is the heart of the experiment, consistsof a rotating radial grating placed at the focus of thebeams and two detectors placed in the image plane ofthe Fabry-Perot switches. The operation of the ro-tating radial grating is described in detail in the nextsection.

The experimental setup is basically identical to thatshown in Fig. 5 except that a beam splitter was used toseparate the main beam from the diagnostic beams.

Two identical Fabry-Perot switches were used, eachcapable of producing four beam positions separated by2.5°. A 7.5-m focal length mirror was used to bring thebeams to focus where a rotating radial square wavegrating was placed. The grating consisted of 2.5-cmlong slits with a period of 4 mm at a radius of 11.4 cmfrom the center of the grating wheel. The grating wasrotated so that the grating lines moved past focus at afrequency of 1 kHz. A 4.08-m focal length mirror wasthen used to complete the relay optics train which im-aged the switch apertures onto the detectors. Theoutput signals from these detectors were filtered toobtain a first harmonic (at 1 kHz) from each and a fifthharmonic (at 5 kHz) from one. The phases of these.harmonics were then used to adjust the tilt and phaseof each aperture to achieve the desired far-field pattern.The far-field pattern was monitored by a detector witha narrow slit aperture placed at the focus of the mainbeam. A galvanometer mirror was used to scan thebeam past the slit, thus sensing the beam profile as afunction of angle, in a direction parallel to the center-to-center separation of the two channels. A comparisonof the predicted beam profile and a typical profile ob-tained during the experiment is shown in Fig. 6 for twocases: (a) when the relative phase between the twowave fronts is zero; and (b) when the wave fronts are,180O out of phase. The angular spacing of the side-lobes 0 is determined by the center-to-center separationof the apertures D and the wavelength X so that 0 = X/D= 0.1 mrad. The envelope width is given by the wave-length over the subaperture diameter and is -0.24mrad. Thus of the order of three large peaks shouldappear under the envelope at any time.

To achieve the desired relationship between theharmonic phases and the far-field pattern, a specialalignment procedure described below was followed.First, the nominal position of focus was determined by


=N -

EXPERIMENT

measuring 7.5 m from the first focusing mirror andplacing both the far-field diagnostic and the rotatinggrating in equivalent positions at this distance from themirror. Second, the beam collimation was determinedby maximizing the first harmonic signal using the focusadjustment on the collimator. Third, the beams werecarefully overlapped at focus using the far-field diag-nostic. Fourth, the detectors in the image plane of theswitch apertures were adjusted in lateral position tobring the first harmonics into an equal phase position.This effectively placed each detector in the image planeat a place where the wave fronts had equal slope. (If thewave fronts were perfectly plane, this step would not benecessary.) Diffraction was found to be an importantsource of phase error if the two channels did not haveequivalent limiting apertures (in our case an apertureplaced before the beam splitter). When this procedurewas followed carefully, the first and fifth harmonicsfrom the shearing interferometer were always found tohave the proper relationship. Care was also taken toavoid electronic phase shifts in the filters used to sep-arate the first and fifth harmonics in the shearing in-terferometer output. Three important tests of theoperation of this shearing interferometer were per-formed and are described below.

The relationship between the far-field pattern andthe output of the shearing interferometer is shown forthree different cases in Fig. 7. The left photograph

4.'0

-G.1

I-

0

I I I I I I I I

OUT-OF-PHASE -

I+ + +

.LT X/50

'HASE X/8

I I I I I I I I-a.2 0

PHASE (#-5A#,)(Fractions of a Wavelength)

0.5 0.7

Fig. 8. Loci of shearing interferometer error signals for the in-phaseand out-of-phase cases. (Units are fractions of a period of corre-

sponding modulation frequency.)

(a) PHASE ERROR = 0.037, TILT ERROR 0.015

(a) PHASE ERROR = 0.067, TILT ERROR =-0.008

(b) PHASE ERROR = -0.178, TILT ERROR = -0.037

(b) PHASE ERROR = -0.369, TILT ERROR = -0.026

A

(C) PHASE ERROR = -0.308, TILT ERROR = -0.227

Fig. 7. Far-field profiles and associated shearing interferometeroutputs.

(C) PHASE ERROR = 0194, TILT ERROR 0.0

Fig. 9. Illustration of invariance of shearing interferometer outputas input beam direction is varied. (Units are fractions of a period of

corresponding modulation frequency.)


_ .-- .

IN-PHASE

+.

shows the four outputs of the shearing interferometer.The top oscilloscope trace is a reference waveform de-rived from the transmission of an He-Ne laser beamfocused on the rotating grating. The two low frequencysine waves are at the first harmonic of the grating fre-quency and measure Ak0 and Ak01, respectively. Thehigh frequency sine wave is a fifth harmonic and is usedto measure Ap5e. (The phase units are fractions of aperiod.) In the results in Fig. 7(a) the tilts of each beamare equal, and the relative phase has been adjusted toachieve the desired far-field pattern, one peak andsymmetrical smaller sidelobes. This is the case for thetwo channels being equivalent to two parts of a largerplane wave front. In the results in Fig. 7(b) the tilts areagain equal, but the relative phase has been adjusted sothat the two channels are 0.369 wavelengths out ofphase. In this case the far-field pattern consists of twopeaks of equal magnitude with smaller symmetricalsidelobes. The results obtained with both tilt andphase errors are shown in Fig. 7(c). Here the firstharmonics are 90° out of phase (0.227 wavelengths). Inthis case not only is the desired beam shape (one centralipeak) destroyed, but the envelope of the sidelobes isbroader because the beams no longer overlap exactly inthe far field.

To obtain a quantitative measure of the accuracy ofthe shearing interferometer outputs, a number ofmeasurements were taken for the in-phase and out-of-phase conditions for equal tilts. The far-field pat-terns in each case indicated agreement with the desiredphase relation. The two measurements which wouldbe used to generate correction signals were then plotted:the phase error (A05 - 5A41 ) on the horizontal axis andthe tilt error (O, - Ap0) on the vertical axis. (Thephase units are fractions of a period.) The results areshown in Fig. 8. For the thirteen in-phase cases, indi-cated by the solid circles, the points clustered about theorigin with a standard deviation indicated by the box.The four out-of-phase cases indicated by the crossesclustered around (0.5,0) (measured in fractions of awavelength) with a 1-a variation indicated by the box.All seventeen cases had a tilt a of X/50 and a phase a ofX/8. This seems intuitively correct since it should bepossible to measure the phases of the 1-kHz sine wavesabout 5 times more accurately than the 5-kHz sine wave.Apparently the in-phase condition was easier to adjustexperimentally than the out-of-phase condition. It isalso important to note that these measurements are

made by measuring the time between an arbitrary ref-erence and the peak of each sine wave. Several suchmeasurements could be made in a short time, thus im-proving the accuracy considerably.

A second important result is to show that the desiredrelationship between the far-field pattern and theshearing interferometer outputs exists when the angleof the beams entering the shearing interferometer ischanged. Three different beam positions and the si-multaneous shearing interferometer outputs are shownin Fig. 9. The format of the photographs is identical tothat shown in Fig. 7. Only the angles of the beams en-tering the shearing interferometer were changed be-tween the examples, and no adjustments were made tothe positions of the shearing interferometer compo-nents. The asymmetry shown on the two outside beampositions can be accounted for by the spatial envelopeof the focal plane diagnostic detector response.

IV. Conclusions

In addition to verifying the predicted behavior of themultiple-order shearing interferometer, as discussed atthe end of the analysis section, two conclusions havebeen drawn from the experimental work:

(1) Large shears, which correspond to higher dif-fraction orders and thus to higher frequencies, are moredifficult to measure experimentally. An accuracy of X/8was achieved with a shear corresponding to the fifthorder.

(2) The desired relationship between the far-fieldpattern and the shearing interferometer outputs is in-variant with respect to beam input angle.

References1. P. D. Henshaw, A. Sanchez, and R. B. McSheehy, Appl. Opt. 19,

884 (1980).2. Woods Hole Summer Study, Synthetic-Aperture Optics, Vol 1,

Aug. 1967 (National Academy of Sciences, Washington, D.C.,1967).

3. J. W. Goodman, "Synthetic Aperture Optics," in Progress in Op-tics, Vol. 8, E. Wolf, Ed. (American Elsevier, New York, 1970), pp.1-150.

4. G. K. O'Neill, Science 160, 843 (1968).5. N. P. Carlton and W. F. Hoffman, Phys. Today 31, No. 9, 30

(1978).6. J. W. Hardy, J. E. Lefebvre, and C. L. Koliopoulos, J. Opt. Soc. Am.

67, 360 (1977).7. J. W. Hardy, Proc. IEEE 66, 651 (1978).8. C. A. Primmerman, "Analysis of the Radial-Grating A. C. Heter-

odyne Shearing Interferometer" (27 Dec. 1977), unpublished.

We would like to thank Antonio Sanchez and RichMcSheehy for many useful discussions of this material,Pete Cole for his help with the beam propagation sim-ulations, and Seymour Edelberg for his guidance andencouragement during this work.

This work was sponsored by the Ballistic MissileDefense Advanced Technology Center Radar Direc-torate, Department of the Army.


multiple-order radial-grating shearing interferometer

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