Lohmann et al. Vol. 16, No. 2 /February 1999 /J. Opt. Soc. Am. A 359

Coherence waves

Adolf W. Lohmann*

Department of Physics of Complex Systems, Weizmann Institute of Science, 76100 Rehovot, Israel

David Mendlovic† and Gal Shabtay

Faculty of Engineering, Tel-Aviv University, 69978 Tel-Aviv, Israel

Received March 19, 1998; revised manuscript received September 21, 1998; accepted September 25, 1998

In 1955 Wolf noticed that the mutual coherence function G obeys two wave equations [Proc. R. Soc. London230, 246 (1955)]. The physical optics of this finding is thoroughly presented in Mandel and Wolf’s recentmonograph [Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK 1995)]. We discussthose coherence waves in the spirit of optical signal processing. The term coherence waves is justified if oneaccepts the following paradigm: A wave is whatever obeys the wave equation. © 1999 Optical Society ofAmerica [S0740-3232(99)00902-3]

OCIS codes: 030.1640, 070.6020, 070.2580, 350.7420.

1. BACKGROUND, MOTIVATION,AND PLANOne of the parameters for characterizing an optical waveis its spatial coherence. We assume a light beam with Exas its electric-field component. By using essentially theterminology of Mandel and Wolf’s monograph on coher-ence theory,1 one gets

Ex~R, t ! 5 Re@V~R, t !#, R 5 ~x, y, z !, (1)

G~R1 , R2 , t1 , t2! 5 ^V~R1 , t1!V* ~R2 , t2!&. (2)

V is the associated analytical signal. G, the coherencefunction, is the correlation of two V values at two space–time points. The ^ & brackets in Eq. (2) imply an en-semble average over many realizations of the fluctuatingV field.

Wolf explored the coherence properties of light1–6 andshowed that G obeys two wave equations1,2:

D1G 2 ~1/c2!]2G

]t12 5 0, (3a)

D2G 2 ~1/c2!]2G

]t22 5 0. (3b)

D1 is the Laplace operator in the R1 5 (x1 , y1 , z1) space:

D1 5]2

]x12 1

]2

]y12 1

]2

]z12 . (4)

Wolf’s two equations [Eqs. (3a) and (3b)] are the core ofour study. Therefore we will present their derivationshortly.

Since laboratory experiments are nowadays usuallydone with a laser light beam, we will concentrate on thequasi-monochromatic special case. The V field maytherefore be written in terms of a slowly varying envelopeU(R, t) [see Eq. (5)], with consequences for G [Eqs. (6)and (7)]:

0740-3232/99/020359-05$15.00 ©

V~R, t ! 5 U~R, t !exp~2ivt !, (5)

G~R1 , R2 , t1 , t2! 5 ^U~R1 , t1!U* ~R2 , t2!&

3 exp@2iv~t1 2 t2!#, (6)

G~R1 , R2 , t, t ! 5 ^U~R1 , t !U* ~R2 , t !&. (7)

We will also use the symbol G and the term coherencefunction when special conditions prevail, such as quasi-monochromatic light.

Much of what follows in our study is closely related toChap. 4 on ‘‘Second order coherence theory of scalar wavefields’’ in Mandel and Wolf’s monograph.1 However, themotivations are different. Mandel and Wolf present thephysical optics of coherence theory. The terms informa-tion optics and optical signal processing characterize ourviewpoint. Such a viewpoint is best illustrated by recall-ing an interview that Gabor gave when his 1971 NobelPrize was announced. Upon being asked the question,‘‘What started you to work on holography?’’ Gabor replied:‘‘As a high-school student I was puzzled that a photo-graphic plate midway between object and camera wouldshow no structure or pattern after being exposed and de-veloped (see our Fig. 1). Yet, the information of the ob-ject must travel through that plane on its way to the cam-era. How does the light carry information from theobject to the proper image plane?’’

Looking at Gabor’s conceptual setup (Fig. 1), it is obvi-ous how the information is traveling: by means of the co-herence function G. So G is indeed a significant featurein information optics. Furthermore, G is a wave becauseit obeys a set of two wave equations, which will be derivedin Section 2. Next, in Section 3, we will couple the twoseparated four-dimensional (4D) wave equations. Eachwave equation corresponds to an Ewald’s surface in Fou-rier space (Section 4).

Recently, the synthesis and the propagation of the mu-tual intensity were also investigated in the context of thefractional Fourier transform.7,8 From Fourier optics we

1999 Optical Society of America

360 J. Opt. Soc. Am. A/Vol. 16, No. 2 /February 1999 Lohmann et al.

know about the significance of other operations such asshifting, tilting, multiplying, convolving, and spatial fil-tering. In Section 5 we will show how those operationsact upon G.

2. WAVE EQUATIONS FOR THE MUTUALCOHERENCE FUNCTION G

We start with the wave equation for V(R, t). We write itdown twice, for the space–time domains (R1 , t1) and(R2 , t2):

D1V~R1 , t1! 2 ~1/c2!] 2V~R1 , t1!

]t12 5 0, (8)

D2V~R2 , t2! 2 ~1/c2!] 2V~R2 , t2!

]t22 5 0. (9)

The Laplace operator (D) was defined in Eq. (4). We mul-tiply both terms of Eq. (8) by V* (R2 , t2), which is a con-stant for the operators D1 and ]2/]t1

2. Hence we may po-sition V* (R2 , t2) behind those operators:

D1@V~R1 , t1!V* ~R2 , t2!#

2 ~1/c2!]2@V~R1 , t1!V* ~R2 , t2!#

]t12 5 0. (10)

Now we take the ensemble average of Eq. (10) and obtainthe first of the two Wolf’s equations [Eq. (3a)]. The otherWolf’s equation [Eq. (3b)] can be arrived at by the sameprocedure.

For the quasi-monochromatic case [Eqs. (5)–(7)], we ob-tain the generalization of the Helmholtz equation:

D1G 1 k2G 5 0, (11a)

D2G 1 k2G 5 0, (11b)

where k2 5 (w/c)2.In most optical instruments and experimental setups,

the light propagates roughly along the optical z axis(paraxial propagation). Hence it is convenient to split afast-varying plane-wave term from the general field am-plitude:

V~R, t ! 5 v~R !exp@i~kz 2 vt !#. (12)

We obtain the paraxial equivalent of the Helmholtz equa-tion for monochromatic signals:

Fig. 1. Gabor’s puzzle: Why doesn’t the photo capture the in-formation traveling from the object to the camera?

Dv~R ! 1 2ik]v~R !

]z5 0. (13)

It is customary to approximate this wave equation by dis-regarding the axial part of the Laplacian:

U]2v

]z2U ! uDvu, D 5]2

]x2 1]2

]y2 ; (14)

so we obtain the approximate paraxial wave equation:

Dv~R ! 1 2ik]v~R !

]z5 0. (15)

The elementary solutions of this wave equation are

v~R ! 5 ~1/z !expS ipx2 1 y2

lz D , (16)

v~R ! 5 exp@i2p~xnx 1 yny! 2 iplz~nx2 1 ny

2!#. (17)

We now call R 5 R1 and z 5 z1 in Eq. (15), multiply byv* (R2), and finally perform an ensemble average. Wearrive at the paraxial monochromatic wave equation forG:

D 1G 1 2ik]G

]z15 0, (18a)

D 2G 1 2ik]G

]z25 0, (18b)

where now

G~R1 , R2! 5 ^v~R1!v* ~R2!&. (19)

Note that Eqs. (18) hold for the quasi-monochromatic caseas well.

3. COUPLING TO THE TWO FOUR-DIMENSIONAL WAVE EQUATIONSSo far the two Wolf equations [Eqs. (3a) and (3b)] occupytwo disjoint 4D spaces (x1 , y1 , z1 , t1) and(x2 , y2 , z2 , t2) in an eight-dimensional (8D) space withall eight coordinates. It is possible to replace the firsttwo wave equations by two other but equivalent waveequations whose 4D subspaces overlap in the 8D space(R1 , R2 , t1 , t2).

We couple the two Wolf equations in two ways, by ad-dition and by cascading the operators9:

~D1 1 D2!G 2 ~1/c2!S ]2

]t12 1

]2

]t22DG 5 0, (20)

D1D2G 2 ~1/c4!]4

]t12]t2

2 G 5 0. (21)

An elementary solution for both equations is

exp@i~k1 • R1 2 k2 • R2 2 v1t1 1 v2t2!#. (22)

This constitutes a planar G wave. The two quasi-monochromatic special cases are

~D1 1 D2!G 1 ~k12 1 k2

2!G 5 0, (23)

D1D2G 2 k12k2

2G 5 0. (24)

The paraxial counterparts to Eqs. (23) and (24) are

Lohmann et al. Vol. 16, No. 2 /February 1999 /J. Opt. Soc. Am. A 361

~D1 1 D2!G 1 2iS k1]

]z11 k2

]

]z2DG 5 0, (25)

D1D2G 1 4k1k2]2

]z1]z2G 5 0. (26)

One may wish to have a common effective k value for bothequations [Eqs. (23) and (24)]:

k12 1 k2

2 5 2k2, (27)

k12k2

2 5 k4. (28)

These conditions are compatible if

k12 5 k2

2 5 k2. (29)

This section on the coupling of the two wave equations isadmittedly speculative. There are related papers on thespecial case of z1 5 z2 and k1 5 k2 ; see Refs. 10 and 11,for example. More insight might be gained if we nowmake a transition into a higher-dimensional Fourierspace, where Ewald’s surfaces correspond to wave equa-tions.

4. EWALD’S SURFACESThe Ewald’s surfaces12 are an alternative way to describedifferential equations and especially wave equations.These surfaces express the realizable physical relation be-tween the different spectral coordinates. We introducethe concept of Ewald’s surfaces in the context of theHelmholtz equation:

Du~R ! 1 k2u~R ! 5 0. (30)

Upon inserting a three-dimensional (3D) Fourier repre-sentation

u~x, y, z ! 5 EEE u~n!exp~2pin • R !d3n, (31)

we find that the 3D spatial frequency spectrum u can benonzero only if the 3D frequency vector

n 5 ~nx , ny , nz! (32)

has a length of 1/l, i.e.,

u~n! Þ 0 only if n • n 5 1/l2. (33)

In other words, the 3D frequency spectrum is confined tothe surface of a sphere, the so-called Ewald’s sphere (seeFig. 2).

If u(R) is factorized for the paraxial situation

u~R ! 5 v~R !exp~ikz !, (34)

the spatial frequency spectrum of v(R) is now nonzero ona shifted Ewald’s sphere (see Fig. 3):

v~n! Þ 0 only if nx2 1 ny

2 1 ~nz 1 1/l!2 5 1/l2.(35)

This sphere is deformed into a paraboloid (see Fig. 4),

nx2 1 ny

2 1 2nz /l 5 0, (36)

if the paraxial approximation

]2v

]z2 ' 0, nz2 ! 1/l2 (37)

is used. In the polychromatic case the length of the spa-tial frequency vector is tied to the temporal frequencyv/2p:

n • n 5 ~v/2pc !2. (38)

Apparently, a second-order linear differential equationwith constant coefficients is equivalent to a bilinear con-dition in the associated Fourier space. This is also truefor Wolf’s equations [Eqs. (3)], the Helmholtz–Wolf equa-tions [Eqs. (11)], the paraxial G wave equation [Eqs. (18)],and the coupled G wave equations [Eqs. (20), (21), and(23)–(26)]. The general Fourier space corresponding toG(R1 , R2 , t1 , t2) has eight dimensions, with variablesn1 , n2 , v1 , and v2 . The spatial frequency vectors n1 andn2 have three dimensions each. Each wave equation re-duces the dimensionality by 1, just as linear equations doin Euclidean space, and if the condition v1 5 v2 5 vholds, the solution space is reduced by two orders. An

Fig. 2. Ewald’s sphere corresponding to the Helmholtz equa-tion.

Fig. 3. Ewald’s sphere related to the rigorous paraxial waveequation.

Fig. 4. Ewald’s paraboloid for the approximated paraxial waveequation.

362 J. Opt. Soc. Am. A/Vol. 16, No. 2 /February 1999 Lohmann et al.

uncoupled pair of differential equations, such as theWolf’s equations [Eqs. (3)], has separable conditions inthe Fourier domain:

n1 • n1 2 ~v1/2pc !2 5 0, (39a)

n2 • n2 2 ~v2/2pc !2 5 0. (39b)

By contrast, the coupled wave equations [Eqs. (20) and(21)] correspond to two conditions in the Fourier space,which do not split the eight dimensions into two indepen-dent 4D subspaces:

n1 • n1 1 n2 • n2 2 ~v1/2pc !2 2 ~v2/2pc !2 5 0, (40a)

~n1 • n1!~n2 • n2! 2 ~v1v2/2pc !4 5 0. (40b)

The various Fourier subspaces are spheres, cylinders, pa-raboloids, cones, etc., and cross sections thereof. Crosssections will especially occur in the context of the coupledequations. However, the two seven-dimensional ‘‘sur-faces’’ in 8D space [Eqs. (39)] are parallel-like concentricspheres and hyperboloids.

5. SOME PROPERTIES OF THE COHERENCEFUNCTIONIn this section we will state what happens to G if the Vwave has been modified somehow. The modifications areelementary operations for optical signal processing. Forexample, if V(R, t) is shifted in space and time, similartransformations happen to G:

V~R, t !⇒V~R 1 R, t 1 t !, (41a)

G~R1 , R2 , t1 , t2!⇒G~R1 1 R, R2 1 R, t1 1 t , t2 1 t !.

(41b)Next we consider modulation:

V~R, t !⇒V~R, t !exp~2pi n • R 2 ivt !, (42a)

V~n, v!⇒V~n 2 n, v 2 v !, (42b)

G~n, v!⇒G~n1 2 n, n2 1 n, v1 2 v, v2 1 v !, (42c)

where

V~n, v! 5 EE V~R, t !exp~22pinR 1 ivt !dRdt.

(43)

Now we explore multiplication:

V~R, t !⇒V~R, t !P~R, t !, (44a)

G~R1 , R2 , t1 , t2!⇒G~R1 , R2 , t1 , t2!

3 P~R1 , t1!P* ~R2 , t2!. (44b)

The P(R, t) are assumed to be deterministic. If we mul-tiply two independent random processes, we obtain

V~R, t !⇒VA~R, t !VB~R, t !, (45a)

G~R1 , R2 , t1 , t2!⇒GA~R1 , R2 , t1 , t2!

3 GB~R1 , R2 , t1 , t2!. (45b)

Now we treat a Fourier transformation:

V~R, t !⇒V~n, v!, (46a)

G~R1 , R2 , t1 , t2!⇒G~n1 , n2 , v1 , v2!. (46b)

Next follows an important three-step process, which is aspace–time filtering operation:

V0⇒V0⇒V0P 5 VB⇒VB , (47a)

G0⇒G0⇒G0GP 5 GB⇒GB , (47b)

where

GP~n1 , n2 , v1 , v2! 5 P~n1 , v1!P* ~2n2 , 2v2!. (48)

The process of relation (48) can also be formulated as aconvolution:

G0⇒G0 ^ GP 5 GB , (49)

where

GP~x1 , x2 , t1 , t2! 5 P~x1 , t1!P* ~x2 , t2!. (50)

Now we want to study free-space propagation for the spe-cial case

t1 5 t2 5 t, z1 5 z2 5 z, (51)

which represents a typical laboratory situation.1,6,9,13

We employ the plane-wave decomposition, and for sim-plicity we drop the y dependence, which is structured likethe x dependence:

u~x, z ! 5 E u0~n!exp$2pi@xn 1 ~z/l!A1 2 l2n2#%dn,

(52a)

u0~n! 5 E u0~x !exp~2i2pnx !dx,

u0~x ! 5 u~x, z 5 0 !. (52b)

The corresponding G operations are

G0~x1 , x2! 5 ^u0~x1!u0* ~x2!&⇒G~x1 , x2 , z !, (53)

G~x1 , x2 , z ! 5 EE G0~n1 , n2!exp$2pi@n1x1 2 n2x2

1 ~z/l!~A1 2 l2n12

2 A1 2 l2n22!#%dn1dn2 , (54)

where G0(n1,n2) 5 **G0(x1, x2)exp@22pi(n1x1 2 n2x2)#dx1dx2. Alternatively, the free-space propagation can bewritten as the following convolution:

G~x1 , x2 , z ! 5 G0~x1 , x2! ^ P~x1 , x2 , z !, (55a)

P~x1 , x2 , z ! 5 EE exp$2pi@n1x1 2 n2x2

1 ~z/l!~A1 2 l2n12

2 A1 2 l2n22!#%dn1dn2 . (55b)

Finally, we describe the same propagation process as thatabove, but now we use the paraxial approximation:

Lohmann et al. Vol. 16, No. 2 /February 1999 /J. Opt. Soc. Am. A 363

v~x, z ! 5exp~ikz !

ilz E v0~x8!exp@ip~x 2 x8!2/lz#dx8,

(56)

v~n, z ! 5 v0~n!exp~ikz !exp~2iplzn2!. (57)

The expression for the coherence function G that is due tofree-space propagation from G0 to G is now

G0~x1 , x2! 5 ^v0~x1!v0* ~x2!&⇒G~x1 , x2 , z !, (58)

G~x1 , x2 , z ! 51

~lz !2 EEG0~x18 , x28 !

3 expH S ip

lz D @~x1 2 x18 !2

2 ~x2 2 x28 !2#J dx18dx28 . (59)

In the angular spectrum domain,13,14 we have

G~n1 , n2 , z ! 5 G0~n1 , n2!exp@2ipz~n12 2 n2

2!#.(60)

Equation (59) is a special case of the convolution theorem[relation (49)], and Eq. (60) is the corresponding spatialfiltering statement.

6. CONCLUSIONSOur paradigm throughout this study has been the follow-ing: If a certain feature obeys a wave equation, then thatfeature constitutes a wave. Owing to Wolf, it has beenknown since 1955 that the coherence function G does obeytwo four-dimensional wave equations.

We have shown how the coherence function behaveslike a wave, similar to what the complex amplitude doesin so-called Fourier optics. On that basis we have takenthe first steps toward optical signal processing, with thecoherence function as a carrier of information. Amongother things we have considered the concept of spatial fil-tering of coherence function signals.

G-spatial filtering14 could be the central part of aG-signal processing system. In addition to filtering, oneneeds to know how to launch a certain G as input and howto evaluate the G output at the exit of the system. Sev-eral recent publications15–17 seem to converge with ourprogram, which should also include comparisons with co-herent and incoherent processing. We expect each ofthose processing schemes to have its own specific domainof preference.

*Permanent address, Erlangen-Nurnberg University,Physikalisches Institut, Erwin Rommelstrasse 1,91056 Erlangen, Germany.

†Send all correspondence to David Mendlovic at the ad-dress on the title page or e-mail, mend@eng.tau.ac.il.

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3. L. Mandel and E. Wolf, ‘‘Coherence properties of opticalfields,’’ Rev. Mod. Phys. 37, 231–287 (1965).

4. L. Mandel and E. Wolf, ‘‘Spectral coherence and the conceptof cross-spectral purity,’’ J. Opt. Soc. Am. 66, 529–535(1976).

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9. M. J. Beran and G. R. Parrent, Jr., Theory of Partial Coher-ence (Prentice-Hall, Englewood Cliffs, N.J., 1964), Chaps. 3and 4, pp. 36–64.

10. J. J. McCoy and M. J. Beran, ‘‘Propagation of beamed sig-nals through inhomogeneous media: a diffraction theory,’’J. Acoust. Soc. Am. 59, 1142–1149 (1976).

11. M. J. Beran, ‘‘Propagation of a finite beam in a random me-dium,’’ J. Opt. Soc. Am. 60, 518–521 (1970).

12. A. W. Lohmann, D. Mendlovic, Z. Zalevsky, and G. Shabtay,‘‘The use of Ewald’s surfaces in triple correlation optics,’’Opt. Commun. 144, 170–172 (1997).

13. A. S. Marathay, Elements of Optical Coherence Theory(Wiley, New York, 1982), Chaps. 4 and 5, pp. 60–181.

14. A. S. Marathay, ‘‘Noncoherent-object hologram: its recon-struction and optical processing,’’ J. Opt. Soc. Am. A 4,1861–1868 (1987).

15. C. Iaconis and I. A. Walmsley, ‘‘Direct measurement of thetwo-point field correlation function,’’ Opt. Lett. 21, 1783–1785 (1996).

16. J. Tu and S. Tamura, ‘‘Analytic relation for recovering themutual intensity by means of intensity information,’’ J.Opt. Soc. Am. A 15, 202–206 (1998).

17. D. Mendlovic, G. Shabtay, A. W. Lohmann, and N. Konforti,‘‘Display of spatial coherence,’’ Opt. Lett. 23, 1084–1086(1998).