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348

J. Opt. Soc. Am. A/Vol. 7, No. 3/March 1990

Mavroidis et

al.

Imaging of coherently illuminated objects through

turbulence: plane-wave illumination

T. Mavroidis, J. C. Dainty, and M. J. Northcott

Blackett Laboratory, Imperial College, London SW7 2BZ, UK Received April 17, 1989; accepted October 23, 1989

The average spectrum and energy spectrum of the short-exposure image intensity are evaluated for the case of an

object illuminated by a monochromatic plane wave and viewed through atmospheric turbulence, assuming a complex Gaussian model for the turbulence. In general, the average image energy spectrum contains diffraction-

limited information about the complex amplitude reflectivity of the object, but, unlike in the case of incoherent imaging,it is generally not possible to extract this information in a straightforward manner. This is illustrated by a computer simulation example of imaging a double-point object in incoherent and coherent illumination. When the

object is optically rough and it is physically possible to average over an object ensemble (e.g., the object rotates

slightly), then the image energy spectrum is more simply related to the object energy spectrum. The possibilities for diffraction-limited object reconstruction are discussed,and it is pointed out that the image bispectrum does not

provide diffraction-limited imaging, in contrast to the case of incoherent imaging.

1.

INTRODUCTION

There is an extensive literature on the imaging of self-lumi-

nous or incoherently illuminated objects through turbulence. In particular, the invention by Labeyrie' of the technique of astronomical speckle interferometry demonstrated that it is possible to obtain diffraction-limited information from short-exposure images. References 2 and 3 describe the basic theory of speckle interferometry and related tech7 niques such as triple correlation and bispectrum imaging.4In contrast, relatively little research appears to have been carried out on the imaging of coherently illuminated objects through turbulence. In this case it is necessary to specify precisely the mode of illumination: this may be direct (e.g., by a plane or spherical wave) or indirect (e.g., by illumination through either the same or different turbulence). Some work has been done on the scattering of light by a diffuse object illuminated through turbulence,8 which gives rise to a form of speckled speckle. The most general study is that of Fante,9 who evaluates the average image intensity and its variance for illumination of arbitrary spatial coherence. 0 Vildanov et al.1 describe an experiment on the imaging of a coherently illuminated object through turbulence for the case in which there is a point source next to the object. Kravtsov and Saichev1- 3 and Jakeman et al. 4 "15 examine

the average image intensity for the case when the viewing is

analogous (but not necessarily identical) to those used for the incoherent case. We take the basic data to be shortexposure (approximately 10-msec integration time) image intensities and evaluate the average Fourier transform and energy spectrum of these data. These averages are found over two ensembles-those of the atmospheric turbulence and of the object. The latter ensemble arises if the object is optically rough and thus generates a speckle pattern that may sweep across the detection aperture owing to slight object rotation. The results for the average Fourier transgeneral analysis by Fante9 ; the energy spectrum and bispectrum results have not been presented elsewhere. Figure 1 shows the basic geometry and coordinate notation. A number of simplifying assumptions are made in the followinganalysis: The object is assumed to be illuminated normally by a unit amplitude monochromatic plane wave (any obliquity effects are ignored). Isoplanatic imaging is assumed; the telescope is assumed to be diffraction limited. The most important assumption relates to the statistics of the complex amplitude of light at the pupil plane due to a point scatterer in the object: it is taken to be a complex Gaussian process. Using the complex Gaussian model greatly simplifies the mathematical analysis since it allows higher-order moments to be expressed as sums of products of second-order moments. It has been used extensively for the imaging of self-luminous or incoherently illuminated objects through turbulence and has been shown to include all the essential physics of the imaging process.2 20 When an ensemble of object statistics is considered, it is assumed.that the spectrum of the object amplitude is also a complex Gaussian process-this implies that the object is rough compared with the optical wavelength and thus gives rise to the usual Gaussian speckle pattern. 2 1 In Section 2 a review of the principal results for the incoherent imaging case is given; this is quite detailed, since the form of the image intensity agree with special cases of the

through the same turbulence as the illumination: in this case there may be a degree of partial wave-front reversal if the object contains specularly reflecting areas. There is also a growing interest in correlography,16-9 i.e., in using intensity correlations in the pupil plane to determine object structure. The overall aim of the present study is to determine whether it is possible to obtain diffraction-limited information, preferably in the form of an estimate of the object map, for coherently illuminated objects viewed through atmospheric turbulence by using image-plane detection methods

0740-3232/90/030348-08$02.00

analysis for the coherent case uses the same methodology,

© 1990 Optical Society of America

Mavroidis et al.

Vol. 7, No. 3/March 1990/J. Opt. Soc. Am. A

349

xI

OBJECT PLANE

OA IX')

J

T(u)=

Atmospheric Turbulence

I.

HA(u )HA*(ul + u)d2 u,

(2.A.2)

I I. IHA(u)l'd'u

where HA(u) is the Fourier transform of hA(x) and is the

PUPIL PLANE X PLANE I A(x)I

Fig. 1. Geometry of coherent imaging through turbulence.

pupil function of the atmosphere-plus-telescope systemstrictly speaking, the pupil function depends on distance t in the pupil, but for notational convenience it is written as a function of the angular frequency variable u = /X. 2 It can be shown 2 that the Fourier transform of the coherent PSF of the telescope is the pupil function Ho(u). Since a point source is now imaged through the atmosphere, one has to use a new pupil function, namely,

HA(u) = A(u) X Ho(u), (2.A.3)

and it is important to understand the similarities and differences between the two cases. In Section 3 we evaluate the average image intensity, its Fourier transform, and the average energy spectrum for the coherent case for which the

where A(u) is the complex amplitude just before the telescope pupil plane due to a point source and again is written as a function of the angular frequency variable u. Combining Eq. (2.A.2) with Eq. (2.A.3), we obtain the following

average is taken over the atmospheric statistics. In Section 4 these results are extended to the case when the averages are also taken over the object statistics. In Section 5 a computer simulation example of imaging a double-point source in incoherent and coherent illumination is presented, to illustrate the analytical results. The prospects for diffraction-limited imaging of objects illuminated by a coher-

expression for the long-exposure transfer function: (T(u))

=

Ca(u) X To(u),

(2.A.4)

ent plane wave are discussed in Section 5. Using the concept of closure phase, we show that, in contrast to the incoherent case, the bispectrum of the image intensity for

coherent imaging will not yield a diffraction-limited image.

where Ca is the normalized autocorrelation function of the atmospheric turbulence and T(u) is the optical transfer function of the imaging system. Equation (2.A.4) shows that the atmospheric turbulence plays the role of a low-pass filter. The particular properties

of Ca(u) can be found in the work of Korff, 2 3 Roddier, 3 and Fried.24 For a Kolmogorov spectrum of turbulence,

Ca(u) = exp[-3.44( 2. REVIEW OF INCOHERENT ISOPLANATIC IMAGING THROUGH TURBULENCE Although the analysis of coherent imagingis not so straightforward as that of incoherent imaging through turbulence, the methods used in the incoherent case form the basis of those in the coherent case. It is therefore useful to summarize the most important aspects of incoherent imaging through turbulence.

A. The Average Image Isoplanatic incoherent imaging is governed by the following equation2 2 :

r,

ii]

(2.A.5)

From the Eq. (2.A.5)we can see that the atmospheric contribution severely limits the resolution of the imaging system. From a diffraction-limited resolution angle of the order of X/ D, where D is the pupil diameter, one is restricted to what is termed the seeing angle, X/r,, where r is Fried's parameter. 24 The isoplanatic imaging Eq. (2.A.1)in Fourier space is

I(u) = T(u) X 0(u), (2.A.6)

and the long-exposure image is, in Fourier space, (I(u)) = (T(u)) X 0(u)

or in real space, (2.A.7)

i(x) =

or

J

IhA(x - x') 2 o(x')d 2 x'

(2.A.1)

(i(X)) = Ca(X) ® jDL(X),

(2.A.8)

i(x) = hA(x)12

O(X),

where o(x) and i(x) are the object and image intensities, respectively, hA(x) is the instantaneous amplitude pointspread function (PSF), represents convolution, and x is a two-dimensional angular coordinate. To investigate the effect of imaging through turbulence, one has to evaluate the optical transfer function of the composite atmospheric-imaging system. The normalized optical transfer function of the system is2 2

where c(x) is the long-exposure PSF of the atmosphere alone [i.e.,the inverse Fourier transform of Eq. (2.A.5)]and iDL(X)is the intensity of the incoherent, diffraction-limited image in the absence of the atmosphere. B. The Average Energy Spectrum In speckle interferometry one evaluates the average energy spectrum given by (II(U)12). In this case,

(II(U)12) = (IT(u)1 2 ) X 0(U)12,

where

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2 . =*

(IT(u)J )

2

f

(A(u,)A*(u, +

u)A*(U 2 )A(U 2 J

+ u))H(u1)H,*(u, + u)H*(u 2)H (u2 + u)d2 uId2 u 2 0

2

=

(2.B.1)

EI

( IA(u)J2)IHo(U)J2 (I(u) )a =

d2U]

The fourth-order moment in Eq. (2.B.1)can be simplified by using various assumptions for the statistics A(u). Because in this paper A(u) is taken to be complex Gaussian for the

coherent imaging case, the same property for A(u) will be

(JJ HA*(ul)OA*(ud)HA(u1 + u)OA(ul + u)d2u)

used here for the incoherent case. By using the complex Gaussian assumption, the fourthorder moment can be expressed as25 (A(u)A*(u

=

2 )A*(ul

JJ

fE

(HA*(u)HA(u1+

u))aOA*(u1)OA(u1

+ u)d2uJ.

+ u)A(U 2 + U))

With Eq. (2.A.3)the above equation becomes

(I(U))a = (2.B.2) Since

.

(A(uI)A*(u 2 )) (A*(ul + u)A(u 2 + U))

+ (A(u,)A*(ul + u)) (A*(u2 )A(u2 + u))

= ICa(U

-

JJ (A*(u,)A(ul

2 2 2 )1 d u

+ U))Ao*(U1)0A*(U1) (3.A.4)

2)1

2

2 + Ca(U)1

X H(ul + U)OA(Ul + u)d 2 U.

and Eq. (2.B.1) becomes

2 2 2 (I1 = ICa(U)1X 1T (U)1 + ) 0 (IT(u)I2 ~~~2XIOUI

J

HJ(U)H"*(U1 + U)H*(u

J

j

2 )H 0

(u2 + U)Ca(u

-

d 2 u2

oU)J2 2U 2

(2.B.3)

Following the manipulation becomes

shown in Ref. 2, Eq. (2.B.3) we have

Ca(u)= (A*(u,)A(ul + u))as

2 (IT(u)1 ) = I(T(u) )12 +

2

ITDL(U),

(2.B.4)

(I(u)a = Ca(u)

where N5 p = 2.3(D/r0 ) denotes the average number of speckles in the image and TDL(U)is the diffraction-limited optical

JJ

=

IA,DL*(ul)IA,DL(ul

+ u)d2uJ,

(3.A.5)

transfer function. The important results (for comparison with the coherent case) are that (a) a transfer function of energy spectra exists and (b) diffraction-limited transfer of

information is allowed.

where IADL(U) = HO(u)OA(U)is the Fourier transform of the

diffraction-limited image amplitude. Equation (3.A.5)can

be written as (I(u))a Ca(u)[IADL(U) * IA,DL(U)] (3.A.6a)

3. COHERENT IMAGING: AVERAGE OVER THE ATMOSPHERIC STATISTICS A. The Average Image Coherent isoplanatic imaging is governed by the equation

'A(x)

or in real space,

(l(X))a = Ca(x) liADL(X) 2 ,

(3.A.6b)

= JhA(x

-

')oA(x')d

2

x',

(3.A.1)

where A(x) and OA(x) denote the complex amplitude of the

image and the object, respectively. Image intensity is the measurable quantity, and, taking

the long-exposure average, (i(x))a = (hA(x) ® OA(X)12)a, (3.A.2)

where ca(x) is the long-exposure intensity PSF of the atmosphere and iA,DL(x)2 the intensity of the coherent, diffracis tion-limited image in the absence of the atmosphere. There are some conclusions that can be drawn from Eqs. (3.A.6a)and (3.A.6b)regarding the nature of coherent imaging through turbulence. First, by looking at (3.A.6a) one

can clearly see that no linear transfer function exists be-

tween the image intensity and any property (amplitude or intensity) of the object-this an expected result for coherent

imaging. However, the atmosphere alone does act as a linear filter on the image intensity [Eq. (3.A.6b)] in the same way as in the incoherent imaging case; comparing Eq. (2.A.8)

where (...) a denotes the ensemble average over atmospheric turbulence. By Fourier transforming both sides of Eq. (3.A.2), we get

the following expression for the long-exposure image spec-

trum:

(I(U))a = ([HA(U)OA(U)] * HA(U)OA(U)I)a, (3.A.3)

with Eq. (3.A.6b)shows that the atmospheric-intensity PSF ca(x) acts on the incoherent and coherent diffraction-limited image intensities, respectively. As in the case of incoherent imaging, the atmosphere limits resolution of the average

image in the coherent case.

where * denotes an autocorrelation integral, or, if the integral is written explicitly,

Following the incoherent imaging methodology, to overcome the filtering effect of the atmosphere one can investigate the utility of speckle imaging techniques.

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351

B. The Average Energy Spectrum

The average energy spectrum of the image intensity is found,

using Eq. (3.A.1), to be

(II(u)12)a = (I[HA(u)OA(u) * [HA(u)OA(u)]12 )

(3.B.1)

Equation (3.B.5) is the basic result of this section, and, in contrast to the incoherent case,I2 cannot be reduced further, at least in general. For the special limiting case in which the atmospheric correlation function is a delta function, i.e.,

Ca(u) = 6(U),

or, explicitly, (II(U)I2)a =

K! J

.

HA(Ul)O,(Ul)HA*(U

2 )OA*(U 2)HA*(ul

+ U)OA*(ul + u)HA(U

2

+ U)OA(U2 + u)d2uld2U

2

)

=

J (HA(Ul)HA*(U )HA*(ul + u)HA(U + u)) aOA(ul)OA*(u )OA(u + U)OA*(ul + u)d uld u

2 2 2 2 2 2

2

With Eq. (2.A.3)the above equation becomes

2 (II(u)1 )a =

Eq. (3.B.5) becomes

2

J|* J (A(ui)A*(u )A*(ul + u)A(u *

2

+ u))a

I2 =

JJ

IA,DL(U2)IA,DL*(U2)IADL(U2+ U)IADL*(u2 + u)d 2U2

X HO(ul)OA(ul)HO*(u 2 )OA*(U 2 )Ho*(ul + u) X OA*(Ul + u)H 0 (U2 + U)OA(u 2 + u)d 2 Uld 2 U2. (3.B.2)

or

2 I2 = [IIADL(U)I * IIADL(U)I].

(3.B.7)

The fourth-order moment of a random complex Gaussian variable can be reexpressed by using Eq. (2.B.2),and, substituting the given expression into Eq. (3.B.2),we can split the integral in two additive parts:

(II(U)1 2 ) = 1 + 2,

Clearly, in this limit, I2 and hence the average energy spectrum of the image (II(U)12) contain diffraction-limited information about the object. However, this information is limited to the modulus of the Fourier spectrum-furthermore, this modulus appears in an autocorrelation in Eq.

(3.B.7). However, inspection of Eq. (3.B.6) shows that it is essentially the weighted autocorrelation of IA,DL* (u 2 ) IA,DL(U2 u), the weighting factor being ICa(u)12.Thus, in + the general case, Eqs. (3.B.5) and (3.B.6), the average image

where

J1 =

Ca(u)1 2

J

**

+

J_

IA,DL(U1)IA,DL*(U2)IA,DL*(ul + U)

X IADL(U2

u)d uld 2 u 2,

- U2)I IA,DL(U1)IA,DL*(u2)

(3.B.3a)

I2 =

J

X IADL*(U

=1

IC(U 1

+ U)IADL(U2 +

u)d uld2 u 2.

(3.B.3b)

Equation (3.B.3a) can be rewritten as

ICa(u)I 2 [IADL(U)* IADL(U)] 12

energy spectrum does contain information on the diffraction-limited image spectrum and in principle it should be possible to recover the diffraction-limited image amplitude from this information. In Section 5 we show by means of a computer simulation that diffraction-limited information of the Fourier modulus is present in the image energy spectrum, although, as stated in Eq. (3.B.5), it is not easily possible to extract a quantitative value for it. 4. COHERENT IMAGING: AVERAGING DOUBLE

or, with Eq. (3.A.5a), as J1 = I (I(u)) 12.

(3.B.4)

A. The Average Image

Until now we have assumed a stationary, deterministic ob-

That is, as in incoherent speckle interferometry, there is a term equal to the squared modulus of the Fourier transform of the long-exposure image intensity, although in this case it is the coherent image intensity. To simplify I2 we make the substitution u = 1 - u 2.

Then Eq. (3.B.3b) becomes

I2 =

J.

=

.

.

2 ICa(ul)IIA,DL(U1 U2)IA,DL*(u2) +

ject in space. In this section it is assumed that the surface of the object is randomly rough compared with the wavelength (i.e., the object complex amplitude is a statistical function) and that it is possible to average the energy spectrum of the image intensity over the object ensemble (e.g., because the object rotates slowly in time). Performing this average, we find that Eq. (3.A.4)for the average image intensity becomes (((u))a)o =

X IA,DL(U2 + U)IA,DL*(U + U2 +

u)d2 uld 2 u 2

(3.B.5)

J

(A*(u)A(ul + u))aHo*(u1)Ho(u + u)

(4.A.1)

J| ICa(ui)12[J J

IA,DL*(U2)IADL(u2 + u)

X (OA*(U1)OA(U1 u))0 d u l , +

X IA,DL(U +

2)IA,DL (U +

2

+

u)d 2u 2 ]d2ui.

(3.B.6)

where ( ... ), denotes the average over the object ensemble and is assumed to be statistically independent of the atmospheric ensemble. Rearranging gives

((I(u))a)o

=

Ca(u)Co(u)[Ho(u)* HO(u)],

(4.A.2a)

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Mavroidis et al.

where CO(u) the autocorrelation of the Fourier transform is of the object amplitude-this is equal to the Fourier transform of the average intensity of the object.2 ' In real space,

Eq. (4.A.2a) becomes

(('(X))a)o = C(X) 0 (IOA(X)I )0 0

((II(U)I1)a)o = I((I(u)a)1 + ICa(u)J | * * |

X H,(ul

2 + U2)HJ(U 2 )1 d 2U d2 U2

7

C0(u')1

Ih.(x) 12,

(4.A.2b)

+ IC"(U)2

ff

ICa(ul)I 2d2 ui

where ho(x)12 the intensity PSF of the optical system. is When this is compared with Eq. (2.A.8)it can be seen that the average image intensity is equal to that given by incoherent imaging.

B. The Average Energy Spectrum

X

f

f7

IH(U 2)H 0 (U2 + u)12 d2 u 2 ICo(u,)I2 ICa(u,)1

2 2

+

d u,

Performing the average over the object statistics on Eq.

(3.B.2) gives ((II(U)12 )a)o =

Xf |

2 IH,(u 2 )H,(u 2 + u)1 d2 u 2 .

f

.

.

.

7 (OA(Ul)OA*(U2)OA*(UlU)OA(U+ U))o +

2

2 2

For some of the terms the following integrals can be represented by constants:

=

X (A(u,)A*(u 2 )A*(u1 + u)A(U2 + U))AH (Ul) 0 X H 0,*(u2 )H,*(ul + u)H0 (u 2 + u)d uId u2. (4.B.1)

f

...

f

2 IC7(U1 )1 1H(U

+ u2 )H1*(u 2 )

2

d ud

2

2

u2

72 =

Assuming, as in Section 3, complex Gaussian statistics for

2 2 )A*(u1 + u)A(U 2 + U))a = ICa(u)1

73

f

_

ICa(ui)I2 d2 u,

ICa(U1)Co(Ui)I 2 d2 ui.

=

(A(u,)A*(u

I7

+ ICa(ul

-

2 )1

2

.

Therefore as a final expression we have

2 ((II(u)I2)a)o = I((I(U))a)o)J2+ ,YICa(U)1

2 + IC (u)1 y2 0

Provided that the surface of the object is randomly rough compared with the wavelength, the spectrum of the object amplitude, OA(U),will also be complex Gaussian, and thus

(OA(Ul)OA*(U2)OA*(U1 + U)OA(U2 + U))O

=

2 1C (U)1 0

f J| IH,(U

2)H 0

(U2 + u)1 2d2 u 2

+ C(u

Equation (4.B.1) can then be split into four parts:

2 ((II(U)J 2 )a)o = ICo(u)12ICa(u)1 f

-

2 2)1 .

+ 73

-

f

IH(U)H(U + u)1 2 d2 u2 . 2 0 2

(4.B.3)

. . |

J

2 H0 (Ul)H0 *(U2 )H0 *(Ul + u)H0 (u2 + u)d u d u 2

2

2 + ICa(u)1

. f

.

IC 0(ul

-

U2)12H0 (u,)H,*(u2)H,*(ul + u)H(u2 + u)d2u d2 u 2

U 2 ) 2H(u)H

0 *(u 2 )H 0 *(u1

7 +| * * 1C u)IICa(u, 0 2 0 +u)H+u)d22. (4.B.2) * 0 - 2 2 -U)1 *(u*(ul 0(u 2u u (U, H(ul)H )H d 22 2

+ IC"(u)12

f

.

|

ICa(u

-

+ u)H0 (U2 + u)d2 u d 2 u 2

7

The first term of Eq. (4.B.2) can be seen, using Eq. (4.A.2a), to be equal to I( (I(u)) a)J2. To simplify Eq. (4.B.2) further, consider the case in which, because of the limited extent of the function IC(u)I2,it is possible to approximate H0(ul + u) by H0 (ul) within the integrals. With this approximation, and with the substitution ul = u1 - U2,

The third term is simply the energy spectrum of the average intensity of the object modulated by the diffraction-limited transfer function of the optical system (for an unapodized pupil). In real space it is proportional to

2 [o(x) * o(x)] @ [h0 (x) * h0 (x)1 .

(4.B.4)

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Vol. 7, No. 3/March 1990/J. Opt. Soc. Am. A

353

From the two expressions above it is clear that there is a term equal to that obtained in incoherent imaging. However, there are three additional terms, in contrast to the single additional term of incoherent imaging. These arise because, in the coherent case, the average over the object ensemble is carried out after the detection of the short-exposure data. For quantitative evaluation of the energy spectrum of the object, it would be desirable that the second and fourth terms in Eq. (4.B.3) be small. However, these terms are both object dependent, as can be seen from the expressions

for y, and Y3,and in general these terms are not small. The

problem of extracting a quantitative estimate of the energy spectrum of the object is therefore difficult. In the limiting case of

Ca(u) = 6(u),

object in which each point was equally bright. The procedure for incoherent imaging was identical to that described in Ref. 6, whereas for the coherent case the program was modified as required. In each case, the simulation was appropriate to visible light and D/r0 = 10; 1000 frames each with 2000 detected photons per frame were used. In brief, the simulation involved (a) calculating a single short-exposure image intensity, (b) generating photon events, (c) computing the energy spectrum (free of photon noise bias), and (d) repeating steps (a)-(c) for 1000images. The whole procedure was then repeated for a point source (to get the speckle transfer function) and carried out for both incoherent and coherent imaging. For the coherent case, the phase difference between the two points of the object was 7r/2. The results are shown in Fig. 2 for the incoherent case and

Fig. 3 for the coherent case. The upper part of Fig. 2 shows

Eq. (4.B.3) takes the form

((II(u)12)a)o

=

IC (U)[H(U) * H0 (u)]12=0 6(u) + 0 0

2 + 7 21C 0 (U)1

,6(u)

+ u)1

2

ff

IH,(U 2)Ho(U 2

d

2

u

2

+

3

17

the Fourier modulus of the double-point object for incoherent imaging; it was obtained by dividing the average energy spectrum of the image of the object by that of the image of the point source and taking the square root of the result. As expected, cosinusoidal fringes are obtained up to the diffraction limit and, as is shown in the lower part of Fig. 2, these are of unit contrast. In order to display the result for the

coherent case, we also corrected the average energy spec-

IH 0 (U2 )H 0 (U2 + u)1 2 d2 u 2 .

(4.B.5)

If the object is a point source, then

C(u) = 1. In this case Eq. (4.B.2) becomes

((II(u)1

2

)a)o = 2{ICa(u)12

f

J*

-

H0 (u)H 0 'H*(u2 )

X H0 ,*(u + u)H0 (u2 + u)d2 uld 2 u2

+f

.

..

7 ICa(Ul

2 2)1H(u)H,*(u 2 )

X H 0 "*(ul + u)H0 (u2 + u)d2Uld2U2} = 2(1I(u)12), (4.B.6)

where (II(U)12) is the average energy spectrum of a point

source for incoherent imaging. This is the expected result, the factor of 2 being due to the assumption of Gaussian statistics of the point source.

rn

1.0

0.8I

0.7I 0.3

5.

AN EXAMPLE:

DOUBLE-POINT

OBJECT

It is difficult to interpret physically the results derived in Sections 3 and 4 because of the inherent nonlinearity between object amplitude and image intensity in coherent imaging; this is compounded by the fact that we are evaluating the energy spectrum of the image intensity, and the overall result is that there is no simple transfer function for energy spectra, as there is in the incoherent case. To gain some insight into the analytical results, a Monte Carlo computer simulation of incoherent and coherent imaging through a phase screen was carried out for a double-point

0.2I

0.0 -146.0 -97.0 -49.0 0.0 49.0 98.0 146.0 1.0 x10

Relative spatial frequency

Fig. 2. Reconstructed Fourier modulus for an incoherently illuminated double-point object: top, pictorial representation; bottom, one-dimensional section.

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ent case, imaging through the pupil plane is literally an

interferometric process. In the case of the average energy

spectrum this is evident with the appearance of a fourthorder interference integral between IA,DL(u)at different positions. Thus the average energy spectrum contains diffraction-limited information about O(u) for the coherent imaging case, in contrast to the incoherent case, in which the overall result of interference is lO(u)12, the energy spectrum.

The problem is that this information is contained in a weighted integral expression, and that makes information retrieval nontrivial. In the double-average case with coherent illumination, there is a close connection with the results for an incoherent or self-luminous object. Using the concept of a pseudothermal source,26 one can model an incoherent object as an ensemble of randomly phased coherent objects, identical to those considered in the double-averaging case. However, in

-2.0 x10 1.0 0.8 m 0.7

the incoherent case one averages before detection, whereas in the double-averaging coherent case one averages after

detection. This makes no difference to the average image spectrum [i.e., Eq. (4.B.2a) exactly equals Eq. (2.A.7)] but does affect higher moments such as the energy spectrum.

Since the average of the square of a random variable is

6

0.5

0.3 0.2 0.0 -146.0 I -97.0

always greater than or equal to the square of the average, it followsthat Eq. (4.B.2) is greater than Eq. (2.B.4) multiplied by lO(u)12 has two additional terms). For a point source, (it

Eq. (4.B.6), the average energy spectrum is exactly twice

that of the incoherent case, as expected.

4

-49.0 0.0 49.0 98.0 1446.0

In the case of the imaging of incoherent objects through

Relative spatial frequency Fig. 3.

1.0 x10

turbulence, it has been shown that the average bispectrum of the image can yield the Fourier phase of the object and hence a diffraction-limited object map.f 7 For the coherent case,

Reconstructed Fourier modulus for a coherently illkiminated double-point object; top, pictorial representation; botto m, onedimensional section.

trum by that for the point source (although, of course , there is no linear transfer function in this case), and the reesult is displayed pictorially in the upper part of Fig. 3. Again,

fringes are visible, verifying the analytical result

Df Eq.

(3.B.5) that diffraction-limited resolution is obt;ained. fringes However, as is shown in the lower part of Fig. 3, the I are no longer of unit contrast and are modulated as a function of frequency; this is because there is no transfer r func-

tion of energy spectra for the coherent case.

6. DISCUSSION

Atmospheric turbulence degrades the long-exposure image

of an object illuminated by a spatially coherent plane wave in the same way as for the incoherent or self-luminous case.

An optical transfer function of the atmospheric degradation

exists and is equal to C 0(u) in all cases. This transfer func-

tion acts on the Fourier transform of the diffraction-limited image intensity, this intensity being that of the coherent

image for coherent illumination (no object averaging) and

that of the incoherent image for incoherent illumination and

coherent illumination with object averaging.

An important characteristic of coherent imaging is that

the frequency

pecti'um of the image intengity I(u) i ex-

pressed as the autocorrelation of the frequency spectrum of the image amplitude IA(u). This means that, for the coher-

Fig. 4. Fourier phase reconstructed from the average bispectrum for (top) an incoherently illuminated double-point object and for (bottom) a coherently illuminated double-point object.

Mavroidiset al.

Vol. 7, No. 3/March 1990/J. Opt. Soc. Am. A

355

an analytical evaluation of the average image bispectrum can be done for both single and double averaging, but the expressions are complicated and difficult to interpret physically. However, by using the phase closure viewpoint of the bispectrum 2 7 it can be shown that the average bispectrum of

discrete-spectrum polychromatic speckle field after propagation through the turbulent atmosphere," J. Opt. Soc. Am. 71,

1176-1179 (1981).

9. R. L. Fante, "Imaging of an object behind a random phase

screen using light of arbitrary coherence," J. Opt. Soc. Am. A 2, 2318-2328 (1985). 10. R. R. Vildanov, V. N. Kurashov, A. T. Mizzaev, and A. N.

the image intensity for the coherent case does not contain the required Fourier phase information.

As was stressed recently by Cornwall et al. 2 8 in another

Yakubov, "Formation of high-resolution images of coherently illuminated objects in a turbulent atmosphere," Opt. Spectrosc.

(USSR) 60, 513-516 (1986). 11. Yu. A. Kravtsov and A. I. Saichev, "Effects of double passage of waves in randomly inhomogeneous media," Sov. Phys. Usp. 25, 494-508 (1983). 12. Yu. A. Kravtsov and A. I. Saichev, "Effects of partial wavefront reversal during the reflection of waves in randomly inhomogeneous media," Sov. Phys. JETP 56, 291-294 (1982). 13. Yu. A. Kravtsov and A. I. Saichev, "Properties of coherent waves reflected in a turbulent medium," J. Opt. Soc. Am. A 2, 2100-2105 (1985).

context, phase closure imagingin the presence of aberrations depends on the fact that the coherence function being measured is a function only of coordinate differences in the pupil

plane. For a spatially incoherent object, the van CittertZernicke theorem guarantees that the coherence function will have such a dependance. But, for a spatially coherent object, the coherence function in the pupil is a separable function, and therefore phase closure cannot distinguish the Fourier phase of the object from the aberrations. Figure 4 illustrates this point for the computer simulation example described in Section 5. The upper part of Fig. 4 shows the Fourier phase of the incoherent double-point object reconstructed from the average image bispectrum by using standard techniques 46 ; the regions of r phase difference are clearly visible (the noise at higher frequencies is present ber of frames used in the simulation).

14. E. Jakeman, "Enhanced backscattering through a random

phase screen using light of arbitrary coherence," J. Opt. Soc. Am. A 5, 1638-1648 (1988).

15. E. Jakeman, P. R. Tapster, and A. R. Weeks, "Enhanced backscattering through a deep random phase screen," J. Phys. D 21,

S32-S36 (1988).

16. P. S. Idell, J. R. Fienup, and R. S. Goodman,"Image synthesis from nonimaged laser speckle patterns," Opt. Lett. 12, 858-860

(1987).

because of the finite number of photons per frame and numThe lower part of Fig.

17. J. R. Fienup and P. S. Idell, "Imaging correlographywith sparse

arrays of detectors," Opt. Eng. 27, 778-784 (1988). 18. D. G. Voelz, J. D. Gonglewski, and P. S. Idell, "Imaging correlog-

4 shows an attempt to reconstruct the Fourier phase for the coherent case; there is no Fourier phase information, as predicted by the phase closure argument. ACKNOWLEDGMENT This research was supported by the Royal Signals and Radar Establishment, UK. REFERENCES

1. A. Labeyrie, "Attainment of diffraction-limited resolution in large telescopes by Fourier analysing speckle patterns in star images," Astron. Astrophys. 6, 85-87 (1970).

raphy: experimental results and performance evaluation based on signal-to-noise ratio of the power spectrum estimate," in

Statistical Optics, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 976, 77-92 (1988). 19. P. S. Idell, J. D. Gonglewski, D. G. Voelz, and J. Knopp, "Image synthesis from nonimaged laser-speckle patterns: experimen-

tal verification," Opt. Lett. 14, 154-156(1989). 20. D. L. Fried, "Angular dependence of the atmospheric turbulence effect in speckle interferometry," Opt. Acta 26, 597-613 (1979). 21. J. W. Goodman, "Statistical properties of laser speckle pat-

terns," in Laser Speckle and Related Phenomena, 2nd ed., J. C.

Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 9-75. 22. J. W. Goodman, in Introduction to Fourier Optics, 1st ed., H. Hether and A. E. Siegman, eds. (McGraw-Hill, New York, 1968), Chap. 6. 23. D. Korff, "Analysis of a method for obtaining near-diffraction-

2. J. C. Dainty, "Stellar speckle interferometry," in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (SpringerVerlag, Berlin, 1984), pp. 253-315. 3. F. Roddier, "The effects of atmospheric turbulence in optical

limited information in the presence of atmospheric turbulence,"

J. Opt. Soc. Am. 63, 971-980 (1973). 24. D. L. Fried, "Optical resolution through a randomly inhomogeneous medium for very long and very short exposures," J. Opt. Soc. Am. 56, 1372-1379 (1966). 25. I. S. Reed, "On a moment theorem for complex Gaussian process-

astronomy," in Progress in Optics XIX, E. Wolf, ed. (NorthHolland, Amsterdam, 1981),pp. 281-376.

4. A. W. Lohmann, G. Weigelt, and B. Wirnitzer, "Speckle mask-

ing in astronomy: triple correlation theory and applications,"

Appl. Opt. 22, 4028-4037 (1983).

es," IRE Trans. Inf. Theory IT-8,194-195 (1962);see also J. W.

Goodman, Statistical Optics, 1st ed. (Wiley, New York, 1985), Chap. 2. 26. J. W. Goodman, "Role of coherence concepts in the study of speckle," Proc. Soc. Photo-Opt. Instrum. Eng. 194, 86-94 (1979).

5. A. W. Lohmann and B. Wirnitzer, "Triple correlations," Proc.

IEEE 72, 889-901 (1984). 6. G. R. Ayers, M. J. Northcott, and J. C. Dainty, "Knox-Thomp-

son and triple correlation imaging through atmospheric turbulence," J. Opt. Soc. Am. A 5, 963-985 (1988). 7. T. Nakajima, "Signal-to-noise ratio of the bispectral analysis of speckle interferometry," J. Opt. Soc. Am. A 5,1477-1491 (1988). 8. J. F. Holmes and V. Rao Gudimetla, "Variance of intensity for a

27. F. Roddier, "Triple correlation as a phase closure technique,"

Opt. Commun. 60, 145-149 (1986).

28. T. J. Cornwall,K. R. Anantharamaiah, and R. Narayan, "Propagation of coherence in scattering: an experiment using interplanetary scintillation," J. Opt. Soc. Am. A 6, 977-985 (1989).

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