JOSA LETTERS

Coherence properties in the image of a partially coherent object David J. Carpenter and Colin Pask

Department of Applied Mathematics, Institute of Advanced Studies, Australian National University, Canberra, ACT, Australia (Received 9 August 1976; revision received 26 October 1976)

An account is given of the derivation and validity of the following law relating coherence radius Rt in the image of an object with coherence radius Ro:R i = MRo unless MRo < rA, in which case Rj=rA, where M and rA are the magnification and Airy disc radius, respectively.

I. INTRODUCTION

The degree of coherence in the image of a partially co­herent source is an important parameter in systems using that image as input to another component, e„g. , in microscopy1 and the excitation of optical fibers°

2 We first relate source and image coherence properties when a simple lens is the optical system, and then dis­cuss extensions of this work.

We consider a coherence radius R defining an area over which the light fields are strongly correlated, i. e . , coherent. The fields at points separated by distance > 2R are weakly correlated, and become more indepen­dent as the distance increases. (We give a precise mathematical definition of R l a te r ° ) In the image plane of a lens the smallest patches of coherent light pro­duced have Ri given by Ri the Airy disc radius,1 and this represents the minimum image coherence radius attainable° When the object is coherent enough to pro­duce an image with Ri > rA,, we know that for systems with small diffraction effects "the coherence is propa­gated according to the laws of geometric opt ics ." 3 Thus, noting that the object and image planes are conjugate, and applying the results in Ref. 3, we obtain Ri = MRo, where M is the magnification, and the subscripts i and o refer to image and object, respectively. We then a r ­rive at Fig. 1, and mathematically we have

We must now examine the proof and validity of (1), and this requires wave concepts° In Sec. II we move towards wave theory with a heuristic proof, and then in Sec. III we give the full Fourier optics treatment in­volving coherence functions. In each case we must build in the lens a numerical aperture property which serves to control the information flowing from object to image.

angle approximation and assume that no = nì.) Similarly, for the image,

There are now two cases to consider which follow from the geometry in Fig. 2. When θco< θlo, the lens angle shown in Fig. 2, a lens with magnification M gives

and then (2), (3), and (4) combine to give

When θco>θ10,

But the lens angle θli corresponds to the Airy disc in the image plane, so that

Thus we see the relation between (1) and the geometric acceptance properties of the lens system°

III. FOURIER OPTICS APPROACH

We now sketch the essentials of the proof of (1) using the methods of Fourier optics.6 We use the standard quasimonochromatic approach1 ,6 and ignore polarization effects so that our equations involve the amplitude U of the electric field, which may then be taken as referring to one polarization component° The system is assumed to be isoplanatic and we follow the notations and develop­ment of Ref° 6 (see pp° 95–196). Using subscripts o and ì for object and image variables as before, we then have

II. HEURISTIC ARGUMENT

There is a connection between coherence and radia­tion directionality (e .g . , Refs. 4 and 5) which we intro­duce here in the following simplistic manner. On the basis of diffraction theory, a coherent area of radius R radiates mostly into a cone of semiangle θ ≈ 1/kR, where the wave number k = 2πn/λ, ivith λ the wavelength and n the refractive index. Thus, for the object (see Fig. 2) we assign coherence angle θco by

(For the purposes of this argument we take the small-

115 J. Opt. Soc. Am., Vol. 67, No. 1, January 1977

FIG. 1. Image coherence radius Rl vs object coherence radius R0 for a simple lens system. M is the magnification and rA is the lens aperture Airy disc radius in the image plane.

Copyright© 1977 by the Optical Society of America 115

FIG. 2. Geometry for lens system. M=di /do is the magnifi­cation. The axis of the optical system is taken as the z-coor-dinate axis.

where the object coordinates xo, yo give

in the image plane, and h is the lens point spread func­tion,

where P is the pupil function and di is defined in Fig. 2.

The mutual coherence function Γ is defined for points Rol and Ro2 in the object by

where * denotes complex conjugate and ( ) the usual time average.1 With these ideas we now obtain, in an obvious vector notation,

Now let us consider an unpolarized, homogeneous, isotropic, and statistically stationary object, so that Γo

is a function of the separation distance between points 1 and 2, |Ro12l:

Further, using the fact that the optical system is i so-planatic, we know that Γl is also uniform and is a func­tion of Rl 12 = Rl 1– R i2. Then

where we have introduced ® to indicate convolution, R012 = R 0 1 – R 0 2 and X=Ro1R12. This equation is now solved using Fourier transforms6; the transform of Γi

involves the product of the transforms of h*, ht and Γo°

The transform of h, and hence h*, is already known

from the definition (10), and that of Γo depends on the form of the coherence function applicable.

If we now consider a system with circular symmetry, and use the commonly occurring functional form of Γo,1

which with our notation becomes

where α0(≃ 3° 83) is the first zero of the Bessel function J1 and Ro is the coherence radius, then the transform of r ( – R o l 2 / M ) required in (14) is the circ function,6

with cutoff frequency

The lens pupil function is taken to be aberration free for the present and, for a lens of radius L we see from (10) that we again have a circ function, for the t rans­form of h, with cutoff frequency

where we have used the standard formula for the Airy disc radius rA. The convolution theorem6 applied to (14) now tells us that Γl has a transform that is a circ function with cutoff frequency given by the smallest of the cutoff frequencies in (16) and (17)° Hence the in­verse transformation finally gives us

where

otherwise.

This is exactly the required result, (1).

We now see the conditions under which (1) is exactly correct: the optical system must be circularly sym­metr ic and isoplanatic; the object must be uniform and large compared to R0 [and theoretically infinite, since Γo in (15) extends over all space]; and the coherence function must have the standard 2J1 (aor/R)/(a0r/R) form. We have also obtained a precise mathematical meaning for the coherence radius R: the coherence function drops from its central maximum to zero over the distance R. [The conventional definition of a radius of coherence, Rcoh,

1 differs from this by the constant α0(≃3.83).]

In the above example, the form for Γ remained un­changed by the mapping from object to image; only the coherence length parameter varied. In general this will not be so, and only occurred above because the ob­ject coherence function and the lens point spread func­tion had the same form, both having the circ function for the transform, the product of these being again a circ function°

If Γo has a frequency cutoff, vco, then Γi will always have the same form as Γowhen vco< vcl , the lens point spread function cutoff frequency. However, for vco > vcl, the transform frequency domain is controlled by vcl, and we only get the point spread function for Γf when

116 J. Opt. Soc. Am., Vol. 67, No. 1, January 1977 JOSA Letters 116

FIG. 3. Image coherence length Ry vs object coherence Length R0 for object coherence function of the form in Eq. (19). Ry =MRo applies when Ro ≥ 1. 34 γA /M.

the transform of Γ° is a circ function. For example, when

where bo ≃5° 14 is the first zero of the Bessel function J2, we find vco= bo/(2πMR0), and so for Rob0 γA/aoM = 1° 34r A/M, Γ. has the same form as Γ0 and Ri = MRo° For smaller values of Rσ, Ri≃rA as shown in Fig, 3, and we see that the lens aperture is controlling the image coherence propert ies . This example serves to illustrate the general features which make the law (1) roughly true.

IV. DISCUSSION

The results given here can be trivially extended to the case where the object and image spaces have differ­ent refractive indices, n, e . g . , k in (2) and (3) becomes ko= 2πno/λ and ki = 2πni/λ, etc. The fact that h and h* are involved in (14) means that phase terms in the pupil function have no effect on coherence and so we deduce the old result7 that aberrations are not important in the present problem. The results may also be extended to the polychromatic case; Γ now becomes the c ross -spec­tral density function for a single monochromatic com­ponent of the field, and the average denoted by < > is an ensemble average.8

The result (1) may be used with a system of lenses each using the image of the previous one as object. For example, for two lenses with Airy radii γA1 and γA2 and magnifications M1 and M2, the final image following lens 2 has

where Ro is the coherence length in the original object. This result is exact under the conditions for which (1) is exact.

It is interesting to note that when the lens Airy disc no longer controls the image coherence length, i . e . , when Ri = MRo, we have the following simple scaling law:

where Li and Lo are image and object s izes . Thus the relative correlation across the image is the same as that in the object.

We conclude that result (1) quantifies the general principle that for highly incoherent situations the lens diffraction properties exert a major influence on the image state of coherence, while for more coherent s i t ­uations the lens only enters via the scaling or magnifi­cation effect. The simple law embodied in (7) is exact only when the Fourier optics techniques used in its derivation are applicable.6

ACKNOWLEDGMENTS

Our thanks are due to Professor A. W. Snyder for stimulating discussions, and to Telecom Australia for financial assistance.

1M. Born and E. Wolf, Principles of Optics, 4th ed. (Perga-mon, Oxford, 1970).

2D. J. Carpenter and C. Pask, "Optical fibre excitation by partially coherent sources , " Opt. Quant. Elect ron. , in press (1976).

3E. Wolf, "A macroscopic theory of interference and diffrac­tion of light from finite sources I , " Proc. R. Soc. London Ser. A 225, 96-111 (1954).

4E. Wolf and W. H. Car ter , "Angular distribution of radiant intensity from sources of different degrees of spatial coher­ence ," Opt. Commun. 13, 205-209 (1975).

5D. J. Carpenter and C. Pask, "Fraunhofer diffraction of partially coherent light by a circular aper ture ," Opt. Acta 23, 279-286 (1976).

6J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

7E. H. Linfoot, Recent Advances in Optics (Clarendon, Ox­ford, 1955).

8M. J. Beron and G. B. Parrent , J r . , Theory of Partial Co­herence (Prentice-Hall, Englewodd Cliffs, N. J. , 1964).

117 . J. Opt. Soc. Am., Vol. 67, No. 1, January 1977 JOSA Letters 117