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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: planewave 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 shortexposure 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 doublepoint 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 diffractionlimited object reconstruction are discussed,and it is pointed out that the image bispectrum does not
provide diffractionlimited imaging, in contrast to the case of incoherent imaging.
1.
INTRODUCTION
There is an extensive literature on the imaging of selflumi
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 diffractionlimited information from shortexposure 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 10msec integration time) image intensities and evaluate the average Fourier transform and energy spectrum of these data. These averages are found over two ensemblesthose 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 higherorder moments to be expressed as sums of products of secondorder moments. It has been used extensively for the imaging of selfluminous 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 processthis 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 wavefront reversal if the object contains specularly reflecting areas. There is also a growing interest in correlography,169 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 diffractionlimited information, preferably in the form of an estimate of the object map, for coherently illuminated objects viewed through atmospheric turbulence by using imageplane detection methods
07403232/90/03034808$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 atmosphereplustelescope 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 doublepoint source in incoherent and coherent illumination is presented, to illustrate the analytical results. The prospects for diffractionlimited imaging of objects illuminated by a coher
expression for the longexposure 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 diffractionlimited 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 lowpass 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 diffractionlimited 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 longexposure 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 twodimensional angular coordinate. To investigate the effect of imaging through turbulence, one has to evaluate the optical transfer function of the composite atmosphericimaging system. The normalized optical transfer function of the system is2 2
where c(x) is the longexposure 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, diffractionlimited 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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J. Opt. Soc. Am. A/Vol. 7, No. 3/March 1990
Mavroidis et al.
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 fourthorder 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 diffractionlimited 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) diffractionlimited transfer of
information is allowed.
where IADL(U) = HO(u)OA(U)is the Fourier transform of the
diffractionlimited 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 longexposure average, (i(x))a = (hA(x) ® OA(X)12)a, (3.A.2)
where ca(x) is the longexposure intensity PSF of the atmosphere and iA,DL(x)2 the intensity of the coherent, diffracis tionlimited 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 objectthis 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 longexposure 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 atmosphericintensity PSF ca(x) acts on the incoherent and coherent diffractionlimited 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.
Mavroidis et al.
Vol. 7, No. 3/March 1990/J. Opt. Soc. Am. A
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 fourthorder 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 diffractionlimited information about the object. However, this information is limited to the modulus of the Fourier spectrumfurthermore, 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 diffractionlimited image spectrum and in principle it should be possible to recover the diffractionlimited image amplitude from this information. In Section 5 we show by means of a computer simulation that diffractionlimited 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 longexposure 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 amplitudethis 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 diffractionlimited 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)
Mavroidis et al.
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 shortexposure 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 shortexposure 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 doublepoint 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:
DOUBLEPOINT
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 doublepoint
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 doublepoint object: top, pictorial representation; bottom, onedimensional section.
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Mavroidis et al.
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 diffractionlimited 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 doubleaverage case with coherent illumination, there is a close connection with the results for an incoherent or selfluminous 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 doubleaveraging case. However, in
2.0 x10 1.0 0.8 m 0.7
the incoherent case one averages before detection, whereas in the doubleaveraging 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 diffractionlimited object map.f 7 For the coherent case,
Reconstructed Fourier modulus for a coherently illkiminated doublepoint 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 diffractionlimited 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 longexposure image
of an object illuminated by a spatially coherent plane wave in the same way as for the incoherent or selfluminous 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 diffractionlimited 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 doublepoint object and for (bottom) a coherently illuminated doublepoint 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
discretespectrum polychromatic speckle field after propagation through the turbulent atmosphere," J. Opt. Soc. Am. 71,
11761179 (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, 23182328 (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 highresolution images of coherently illuminated objects in a turbulent atmosphere," Opt. Spectrosc.
(USSR) 60, 513516 (1986). 11. Yu. A. Kravtsov and A. I. Saichev, "Effects of double passage of waves in randomly inhomogeneous media," Sov. Phys. Usp. 25, 494508 (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, 291294 (1982). 13. Yu. A. Kravtsov and A. I. Saichev, "Properties of coherent waves reflected in a turbulent medium," J. Opt. Soc. Am. A 2, 21002105 (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 doublepoint 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, 16381648 (1988).
15. E. Jakeman, P. R. Tapster, and A. R. Weeks, "Enhanced backscattering through a deep random phase screen," J. Phys. D 21,
S32S36 (1988).
16. P. S. Idell, J. R. Fienup, and R. S. Goodman,"Image synthesis from nonimaged laser speckle patterns," Opt. Lett. 12, 858860
(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, 778784 (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
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