interferometers: equivalent sine condition

Interferometers: equivalent sine condition

Juan M. Simon and Silvia A. Comastri

The parallelism we have previously drawn between an interferometer and an equivalent optical systemenables us to study both systems similarly. The constant contrast condition along the localization surface isequivalent to the condition of isoplanatism at the system image. As the sine condition is necessary for asystem to be isoplanatic, a relationship we call the equivalent sine condition must hold if the contrast isconstant. Some classic interferometers do not satisfy this condition and we here analyze the Michelsoninterferometer. Moreover, we find that, for the record of interference fringes to be a pseudohologram, thiscondition must be unfulfilled.

1. Introduction

In an earlier paper' we drew a parallel between aninterferometer and an equivalent optical system; thisenables us to study the behavior of both systems simi-larly. The interferometers we consider are amplitudedivision and simple interference. We use a spatiallyincoherent extended source.

Moreover, in that paper,1 we mentioned that theconstant contrast condition along the localization sur-face is equivalent to the condition of local isoplanatismat the image of the system; here, in Sec. II, we show thisin more detail. On the other hand, in another previouspaper,2 we showed that the extended sine conditionmust be fulfilled in isoplanatic systems. Similarlyhere, in Sec. III, we find that a relationship, which wecall the equivalent sine condition, must hold in inter-ferometers that verify the constant contrast condition.

Furthermore, in Sec. IV, we show that if in anyobservation plane (not necessarily the localizationone) the equivalent sine condition does not hold, thefringe spacing is different for each source point.

The equivalent sine condition is unfulfilled in someclassic interferometers. As an example we consider, inSec. V, the Michelson interferometer with one of themirrors tilted so that the localized fringes may be seenon a screen. If one of the interferometer branches ismuch longer than the other, the equivalent sine condi-tion does not hold. In this case the contrast is nolonger constant along the localization surface and thenumber of visible fringes is reduced.

The authors are with University of Buenos Aires, Physics Depart-ment, 1428 Buenos Aires, Argentina.

Received 13 November 1987.0003-6935/88/224725-06$02.00/0.© 1988 Optical Society of America.

Finally, in Sec. VI, we show that the record of inter-ference fringes is a pseudohologram3 provided that theequivalent sine condition is unfulfilled on the recordedplane. In this case, taking Sec. IV into account, wefind that the permanent record of the interferencepattern is a superposition of cosinusoidal gratings,each of which retains the information of the position ofthe corresponding source point.

11. Isoplanatism in the Equivalent Optical System andContrast on the Localization Surface

In a previous paper we drew a parallel between theinterferometer and an equivalent optical system. Ac-cording to that paper (Sec. III) the extended sourcethat illuminates the interferometer corresponds to theexit pupil of the system. Moreover a perfect interfer-ometer was defined1 by analogy of a perfect system.As we know, a perfect optical system is that which isfree from aberrations and then for each field point, thepeak of the Airy pattern is located at the correspondinggeometric image. Similarly, a perfect interferometeris one which verifies that the equivalent aberrationfunction cancels and then the peak of the coherencepattern is located on the localization surface and theinterfering beams are completely coherent on that sur-face. Therefore the source that illuminates the inter-ferometer (or its images through both branches) andthe localization surface correspond respectively to theexit pupil and the ideal geometric image of the equiva-lent optical system.

Furthermore, as is well known, for real optical sys-tems illuminated by an extended object, the postulateof local isoplanatism is necessary to obtain images ofreasonably good quality. Likewise1 for real interfer-ometers the condition of constant contrast on a certainregion of the localization surface that contains enoughfringes is necessary to obtain good fringe visibility inthat region.

15 November 1988 / Vol. 27, No. 22 / APPLIED OPTICS 4725

Source

Observationplane

Interferometer

Fig. 1. Coordinate system at the interferometer: L(P), L02 (P),optical path lengths from O to Pthrough branches 1 and 2; L1(xy,P),L2(X,Y,P), optical path lengths from Q to P through branches 1 and 2;source of area a, total intensity IT, and wavelength Xo = 2r/ko; n,n',

refractive indices.

To analyze mathematically what the constant con-trast condition means we first rewrite some of theequations given in our earlier paper. 1

The complex optical disturbance at the image of theequivalent optical system is1

= E 1GO(',') exp[ik0W(i',y')II

X exp[i ko (4' + dx'dy" (1)

where W(V'j,') is the aberration function and (t',iY) arethe image coordinates.

On the other hand we found' that the total intensityat an arbitrary point P of the interferometer exit is

I(P) = I 1(P) + I 2(P) + 2 11(P) 12(P)

X Re(G12 (P) expli[512 - ao(P)]D), (2)

where 11(P) and 12(P) are the intensities at P resultingfrom branches 1 and 2; 12 is the phase difference inreflections, and we have (see Fig. 1)

a00 (P) = h 0 [L02 (P) - L01 (P)J, (3)

=12(P = JJ .I(xy) exp[-icI(xy,P)IiIT a

X expJ-i[a 10 (P)x + a0,(P)y]}dxdy, (4)

where 12(P) is the complex degree of coherence.Moreover, we have

/aL2 OL,al0(P) = ho -

( ax Jx=O ax x=0/(5

%(P) fi, - - yO( y Y=0 ay |Y=()

and the equivalent aberration function is

4'(x,y,P) = -ko[LO2(P) - L01(P)I - [L2(xyP) -L(xyP)]}

- [a1 0(P)x + a0 1(P)y]. (6)

We now require the contrast to be constant at acertain region of the localization surface which con-tains enough fringes. As the contrast is directly relat-ed to the degree of coherence, 4 from Eq. (2) this meansthat ,u12(P) must be a slowly varying function of P.

Optical system

Object Image

Fig. 2. Sine condition in optical systems for an axial object; t and ',

object and image coordinates; a and a', angles subtended by theconsidered ray in object and image spaces, respectively.

That is, we can compare the interference pattern withthe perfect one and write

All(P) = I 12(P)I exp[i012 (P)]. (7)

If A12(P) depends strongly on P, there is a quicklyvarying envelope given by 112(P)l and a fringe dis-placement given by 012 (P). If, on the contrary, A12(P)is locally constant, the interference pattern is not ap-preciably distorted over the observation region.

Then let us postulate that g12(P) is independent of Pon a certain region of a surface which we first assume tobe arbitrary, that is, it may or may not be the localiza-tion surface. If we consider two points, let us sayP andS, in this region of coordinates i' and ' + It' and, forsimplicity, -q' = 0, from Eqs. (4) and (6) we have

L02(0- L01Q(') - [L2(x,y,')- ')]

= L02(' + 6 - Lo,( ' + at/)-[L2(xyX' + ') -L1 (x,y,' + air)I. (8)

Moreover, if the surface considered is the localiza-tion one we have1 alo(P) = a0l(P) = alo(S) = a01(S) = 0and, taking Eq. (6) into account, Eq. (8) can be writtenas

F(x,y,P) = (x,y,S). (9)

From Eq. (9) we find that, if we consider a certainregion of the localization surface, the condition of con-stant contrast [ 2(P) independent of P] requires theequivalent aberration function 4(x,y,P) to be indepen-dent of P. This result is similar to that obtained foroptical systems, namely, that the condition of localisoplanatism requires the aberration function to beindependent of the image point position for a certainregion of the field.

111. Equivalent Sine Condition in Interferometers

As is known, for an optical system to be isoplanaticthe sine condition must be verified. If we first consid-er the well-known case in which there is an axial object 5

and the system is free from spherical aberration, wehave (see Fig. 2)

aW a'- sina' + sina did c ° ea ao X

(10)

4726 APPLIED OPTICS / Vol. 27, No. 22 / 15 November 1988

I taiWave

?2+iof,Source Interferometer surface

Fig. 3. Equivalent sine condition in interferometers: 0, cepoint of the source; Q, arbitrary point of the source; 02, imagethrough branch 2 in the interferometer image space; Q2, imagethrough branch 2 in the interferometer image space; P and Spoints at the localization surface; P2 and S, images of P athrough branch 2 in the interferometer object space; U2 an,points of intersection of Q2S with the wavefront and its tanrespectively; a2 and a20, angles subtended by Q2 P and 0 2 P wit.normal to the localization surface. Similar considerations are

for branch 1.

where ao and ae are paraxial angles. An equivalresult 2 is obtained for extraaxial objects. If, for plicity, we consider a meridian ray and we assumetransverse aberrations to be negligible, from our prous paper2 we get

awn -n'(sine' - sina') + - (sina - sina).

In Eq. (11) a and a' are the angles subtended byconsidered ray with the optical axis in the objectimage spaces, respectively. Similarly, ap and azthe angles subtended by the principal ray. The qutity m is the lateral magnification, that is, m = '

On the other hand, according to Sec. II, the cortion of local isoplanatism in optical systems co]sponds to the condition of constant contrast in in-ferometers. We now show that, similar to whappens in optical systems, in interferometers we fthat the validity of the constant contrast conditalong the localization surface requires the fulfillmof a relationship we call the equivalent sine conditiTo show this we consider Fig. 3. We set

G2 = L 2(') - L2(xIy, - L02( + at)

+ L2(xyX' + 6'),

the same for every ray originated at Q. Similar consid-erations are valid for points 0 and 02.

Furthermore we consider two points P and S on thelocalization surface whose images through branch 2 byinverse ray propagation are P2 and S2. Then in Eq.(12) after a little algebra we have

G2 = ([Q2S] - [Q2P]) - ([02 S] - [02P]), (14)

agent where the brackets denote optical path length.front To obtain G2 in terms of trigonometric sine func-

tions we consider Fig. 3. Let U2 and T2 be the intersec-tion points of the ray Q2S with the wavefront coming

an from Q2 and with its tangential plane, respectively.Then

[Q2S] = [Q2P] + [U2 T2] + [T2 S]. (15)

If the localization surface is sufficiently far awayfrom Q2, the approximations PT2S 7r/2 and U2T2 <<Q2P hold and we get

___ (4' CoSa2)2

L~I" 2Q2P

T 2S t' sina2 , (16)

where a 2 is the angle subtended by the source image Q2at point P.

If, for simplicity, we consider n' = 1 from Eqs. (16)and (15) we obtain

evi- [Q2S] - [Q2P] = 64' sina2 + (64/)2 cos a 2

2Q2P

(11) and a similar equation holds for [02S] - [02 P]-From Eqs. (17) and (14), G2 is given by

(17)

(~/2 Fcos2 2C1G2 = at'(sina 2 - sina 20) + (60 a2 COS 20 (18)

2LQ2P Q02JAs G1 is obtained in a way similar to G2 (see Fig. 4) fromEqs. (18) and (13), we have

(64/)2 COS2a2 cos 2a 20 -i cos2 al CoS2 alol2 iL Q2P 02P Q1 P °1P

= wt'-[sina2 - sin 20] + [sinal - sinak]}. (19)

As Eq. (19) must be valid for any value of 6t', bothterms in curved brackets must cancel. Moreover theterm on the right hand side of this equation is impor-tant when the fringes are straight and the term on theleft-hand side when they are rings. For straightfringes we only have

(12)

and we define G1 in a similar manner. Then Eq. (8)states that

G = G2 . (13)

To find 2 we assume a source point Q whose imagethrough branch 2 is Q2. If we assume the opticalsystem of branch 2 to be free from aberrations we findthat Q2 gives rise to a spherical wavefront which ar-rives at ' (Fig. 3). Owing to the properties of imageforming systems the optical path length from Q to Q2 is

sina 2 - sina2 0 = sine, - sinalo. (20)

Therefore if the contrast is constant along the local-ization surface Eq. (20) must hold. Returning to theparallelism between interferometers and optical sys-tems we note that, according to Sec. II, in the interfer-ometer the angle 2 subtended by the source imagepoint Q2 at observation point P of the localizationsurface is equivalent in the optical system (neglectinghere transverse aberrations) to the angle a' subtendedby an exit pupil point at the image point [see Eq. (11)and Ref. 2]. Similarly, the angle 20, which corre-


_.__- I -calii

sponds in the interferometer to the image of the centralpoint 0 of the source, is equivalent to the angle apsubtended in the optical system by the principal ray(central point of the pupil). Therefore there is astrong relationship between the extended sine condi-tion obtained2 for extraaxial objects in optical systems[see Eq. (11)] and the relationship given by Eq. (20).For this reason we call Eq. (20) the equivalent sinecondition.

IV. Fringe Spacing Dependence on the Source Pointand the Equivalent Sine Condition

We show now that, for straight fringes, the localfulfillment of the equivalent sine condition is a neces-sary and sufficient condition for the fringe spacing tobe independent of the position of the source pointalong a certain region of the source.

To see this let us consider one point alone, Q, of theextended source and an observation surface (whichmay or may not be the localization one). Let A be thedifference in optical path length at a point P betweentwo rays originated at Q, one of which travels throughbranch 1 and the other through branch 2. Then fromFig. 4 we obtain

= [Q Q1I + [Q1P - I[Q Q2] + Q2PII.

Source Interferometer Observationsurface

Fig. 4. Fringe spacing dependence on the source point: 0 and Q,central and arbitrary points of the source; Q1 and Q2, images of Qthrough branches 1 and 2 in the interferometer image space; P,position of maximum of order m; S, position of maximum of order m- 1; a, and a2, angles subtended by Q1P and Q2P with the normal to

the observation surface.

02 Q2

(21),

If we now assume that the maxima of order m and m- 1 are located at P and S, respectively, we may calcu-late the fringe spacing dQ corresponding to point Q.Since the optical path lengths from Q to Q, and from Qto Q2 are independent of the rays considered, takinginto account Eq. (21) we have

([Q1S] - [Q1 P]) - ([Q2S] - [Q2P]) = Xow

Ola

(22)

With a reasoning similar to that used to obtain Eq.(17), assuming straight fringes and considering that, inthis case R' = dQ, we have

[Q2S] - [Q2P] = dQ sin 2. (23)

As a similar result is obtained for [QiS] - [Q1P] fromEqs. (22) and (23) we get

1 sine 1 - sina2

dQ X0

Fig. 5. Michelson interferometer. Images of the source throughboth branches. Parameters: D = OF; 11 = FH1 ; 12 = FH 2 ; z, normalto the source; c, angle subtended by Ml; t,h, thickness and index of A.

(24)

If we take any other point of the source, for example,point 0, and we require the fringe spacing to be inde-pendent of the source point, that is, dQ = do, we findthat the equivalent sine condition [Eq. (20)] must hold.Taking Eq. (24) into account we see that the reciprocalargument is also verified. Moreover, when the obser-vation surface is the localization one, we find (fromSec. III) that, if the constant contrast condition isfulfilled, the equivalent sine condition is verified andtherefore [from Eq. (24)] the fringe spacing does notvary when the source point changes.

V. Michelson Interferometer

To see that the equivalent sine condition is not al-ways satisfied let us consider the Michelson interfer-ometer of Fig. 5. We consider, for simplicity, a linearsource of monochromatic light and thus employ only

one plate, A. There are two mirrors, M1 and M2, andM1 is tilted an-angle e so that the fringes may be seen ona screen.

Let us assume that plate A is of thickness t andrefractive index h and that it is at a distance D = OFfrom the central point, 0, of the source. Moreover let11 and 12 give the positions of mirrors M1 and M 2,respectively, relative to the plate, that is, 11 = FH1 and12 = FH2 . Then, from Fig. 5 and after a little algebra,we find that the positions of images O1 and 02 of thecentral source point 0 through both branches are

01Y1 = 21, cos2 + D(1 - 2 sin2r) +

[2 2 (sine + 3cose) ][2 2 (sinc + cosc)j

(25)


02 02\

00

Source f B

Fig. 6. Michelson interferometer. Localization surface: , anglesubtended by an arbitrary incident ray with the z axis; fl and 2,angles subtended by the emergent rays through branches 1 and 2with the normal to the z axis; P, point on the localization surface.

0 2 Y2 = 212 +D + 2 2 ]

Furthermore, if we consider a point Q at a distance xfrom 0 and if Qi and Q2 are the images of Q throughboth branches, according to paraxial theory and fromFig. 5 we have

01Q1 = 02 Q2 = OQ = x. (26)

Moreover, let us consider a ray coming from 0, whichsubtends an angle ,3 with the z axis, and which isincident on the interferometer. Then from Snell'slaws and Fig. 6 we find that the angles Ali and 2subtended by the emergent rays are

QI

Y1 2 // C

Normal to /0localization surface

Fig. 7. Michelson interferometer. Equivalent sine condition: z,axis normal to the source; P, point on the localization surface; 4', axisalong the localization surface; O,Q, source points; 01,02, images of 0through branches 1 and 2; Q1,Q2, images of Q through branches 1 and2; 01K, and 02K 2, distances from 1_and 02 to the normal to thelocalization surface at P; Q1 ,J and Q2J 2 distances from Qi and Q2 to

the normal to the localization surface at P.

_P 02Y2-PCcos# (30)

where 02 Y2 is given in Eq. (25) and PC is the distanceof P from the z axis.

Replacing Eq. (30) in Eq. (29) we obtain

x coso cos3snce2 - snce2 0 =-2 - (31

Using a similar reasoning for branch 1, if the approxi-mation Q1P O1P holds, we get

01 = + 2, [2 = - (27)

To study the fulfillment of the equivalent sine con-dition we consider Fig. 7. Here the notation is similarto that used in Fig. 3, that is, (' is one of the rectangularcoordinates along the localization surface. The anglesa2 and a2o are subtended with the normal to the local-ization surface by the emergent rays through branch 2,which were originated at the source points Q and 0,respectively (similarly, a and alo for branch 1). Fur-thermore, we assume that the localization surface sub-tends an angle with the z axis. From Fig. 7 we get

since 02 K 2 -0 2 Q 2 cosOsina2=Q2P

02K2 (28)sine2 0 =

If we now assume that Q2P 02P, from Eqs. (28) and(26) we have

sine 2- sina20 =X coso (29)

and from Fig. 7 and Eq. (27)

since - sinalo = -x cos(O + 2 ) cos(3 + 2)

01Y - PC(32)

From Eqs. (25), (31), and (32) we see that the equiva-lent sine condition [Eq. (20)] is not always verified inthe Michelson interferometer. In particular if 12 >> 11and if is not too big, sina 2 - sina201 << Isina - sinalol.Hence if 12 >> 11, the Michelson interferometer does notverify the equivalent sine condition [Eq. (20)] on thelocalization surface.

VI. Nonfulfillment of the Equivalent Sine Condition andPseudoholography

In this section we study the possibility of using therecord of the interference pattern, obtained with acertain interferometer, as a pseudohologram.

A pseudohologram 3 is a modulated interferencegrating for image reconstruction in which the ampli-tude distribution is that of the object but the recordedphase difference does not correspond to the phasedistribution along the object. The reason for choosingthe term "pseudohologram" is that it does not containall information (amplitude and phase) present in theobject, as is the case with holograms.


(31)

If the record of the interference pattern is to be usedas a pseudohologram, it must contain information con-cerning the geometrical distribution of points at thesource. As we have shown in Sec. IV, the local fulfill-ment of the equivalent sine condition is necessary andsufficient for the fringe spacing to be independent ofthe source point at a certain region that containsenough fringes. Hence, if the record of the interfer-ence pattern is to be used as a pseudohologram theinterferometer must be such that the equivalent sinecondition is not verified on the recorded plane. This isso, for example, 3 in the modified Linnik interferometerand in the triangular Cochran interferometer.

To understand the result expressed above, let usfirst consider only one point Q of the extended source.The record of the interference pattern correspondingto Q is a cosinusoidal grating of period dQ. As weknow,3 when a beam of light is incident on the gratingthere are only three diffracted beams; the directions oftwo of these beams depend on dQ and the third beam isthe directly transmitted wave.

If we now take into account the whole extendedsource we find that the record of the interference pat-tern in a plane where the equivalent sine condition isnot verified is a superposition of cosinusoidal gratingswhose spatial frequencies, 1/dQ depend on point Q ofthe source. Then when a beam illuminates this record,the directions of the emergent beams are related to the

position of points in the source. Therefore those re-cords are pseudoholograms.

V1I. Conclusions

We have shown that, if the contrast is constant alongthe localization surface, the equivalent sine conditionmust be fulfilled. We have also shown that this condi-tion does not hold in the Michelson interferometerwhen one of the branches is longer than the other.Finally, we stated that the above condition must benonfulfilled if the record of the interference pattern isto be used as a pseudohologram.

This work was supported by CONICET and theUniversity of Buenos Aires.

References1. J. M. Simon and Silvia A. Comastri, "Fringe Localization Depth,"

Appl. Opt. 26, 5125 (1987)2. S. A. Comastri and J. M. Simon, "Ray Tracing, Aberration Func-

tion and Spatial Frequencies," Optik 66, 175 (1984).3. J. M. Simon and J. 0. Ratto, "Modulated Interference Gratings

for Image Reconstruction: A Classification for HolographicMethods and Other Techniques," Opt. Pura Apl. 19, 93 (1986).

4. M. Born and E. Wolf, Principles of Optics (Pergamon, New York,1964).

5. H. H. Hopkins, "Canonical Pupil Coordinates in Geometrical andDiffraction Image Theory," Jpn. J. Appl. Phys. Suppl. 1 4, 31(1965).

Books continued from page 4700

and 1975. Itis interesting to note that Applied Optics was easily thejournal of first choice of all those early workers: of about 150literature citations given in this book one-third (52) are to articles inApplied Optics, just under another third are to patents, and the finalthird scattered among a dozen other books, journals, reports, theses,and private communications. In fact, Beiser says that the very firstarticle on holographic scanning-by I. Cindrich at the Universtiy ofMichigan-was received by Applied Optics in February 1967.

We had always known, deep down inside, that Applied Optics wasa sturdy and respectable journal for useful applications of optics,and it is comforting to see that the holographic scanning communityshares this view.

JOHN N. HOWARD

Kalman Filtering Theory. ByA. V. BALAKRISHNAN. Opti-mization Software, Inc., New York, 1987. 253 pp. $48.00.

The state-space descriptions are widely used in modeling a dy-namic system whose characteristic features, called states, are mea-sured (or observed) by means of some device described by a linearequation. If the system equation is also linear (or linearized) butotherwise contaminated with noise, the least-squares estimationwith optimal weights determined by the noise variances can be usedto yield an unbiased optimal estimate of the states from all past andpresent measurement data. Kalman filtering provides a real-timealgorithm for obtaining the unbiased optimal estimates. Based on aprediction-correction procedure, only the present measurement in-formation is used in the filtering process. This procedure is ex-tremely useful since, although the optimality criterion is governedby all the data information, there is no storage problem when onlythe present measurement is used. Of course, the so-called Kalman

gain matrices must be computed recursively in real time for thepurpose of adjusting the correction of the predicted estimates usingthe incoming data. Computation of the Kalman gain matrices,however, requires solution of certain (nonlinear) matrix Riccatiequations. Hence the difficulty in storage is now replaced by thedifficulty in real-time computation.

In understanding the theory of Kalman filtering, it is essential tohave a basic knowledge of the theory of linear systems. To appreci-ate the power of the Kalman filtering procedure, one must see manyexamples of real-time use. Furthermore, since Kalman filtering isan optimal estimation procedure based on the statistical informa-tion of both the system and measurement noise processes, a goodunderstanding of the theory of statistical estimation is most helpful.The first three chapters of the text under review are designed for theabove mentioned reasons, namely, a short review of linear systemtheory is given is Chap. 1, signal theory is briefly reviewed in Chap. 2,and an extended study of statistical estimation theory is covered inChap. 3. The fourth chapter is devoted to the study of the Kalmanfiltering theory itself. In addition to the standard topics such asKalman filtering with uncorrelated white noise, correlated signaland noise, and colored noise, a fairly detailed discussion on thesteady state theory is included. As mentioned above, to carry outthe correction process, the Kalman gain matrices must be computedin real time. It should be noted that even for time-invariant sys-tems, the Kalman gain matrices are still time-varying. Hence thedifficulty in computation is eliminated if the Kalman gain matricesare replaced by a constant gain matrix. Under very mild conditions,it can be shown that indeed for time-invariant systems the sequenceof Kalman gain matrices does converge to a constant matrix. Hencereplacement of the Kalman gain matrices by the limiting constantmatrix in the correction procedure produces a suboptimal filter.

continued on page 4770


interferometers: equivalent sine condition

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