Second-Order Statistics for Wave Propagation through ... - DTIC

Second-Order Statistics for Wave N [I

Propagation Through Complex Optical Systems

0

Prepared by H. T. YURA Electronics Research Laboratory Laboratory Operations The Aerospace Corporation El Segundo, CA 90245 and

S. G. HANSON

Ris# National Laboratory DK-4000 Roskilde

Denmark I September 1988

Prepared for SPACE DIVISION

AIR FORCE SYSTEMS COMMAND

Los Angeles Air Force Base P.O. Box 92960, Worldway Postal Center Los Angeles, CA 90009-2960

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OCT 2 0 1988

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022

This report was submitted by The Aerospace Corporation, El Segundo, CA 90245, under Contract No. F04701-85-C-0086-POO019 with the Space Division, P.O. Box 92960, Worldway Postal Center, Los Angeles, CA 90009-2960.

It was

reviewed and approved for The Aerospace Corporation by M. J. Daugherty, Director, Electronics Research Laboratory. Lt Scott W. Levinson/CNID was the project officer for the MissionOriented Investigation and Experimentation (MOIE) Program. This report has been reviewed by the Public Affairs Office (PAS) and is releasable to the National Technical Information Service (NTIS).

At NTIS, it

will be available to the general public, including foreign nationals. This technical reporL has been reviewed and is approved for publication. Publication of this report does not constitute Air Force approval of the report's findings or conclusions.

It is published only for the exchange and

stimulation of ideas.

Lt, USAFN, MOIE Project Officer SD/CNID

RAYMO M. LENG, Maj, USAF Deputy Director, AFSTC West Coast Office AFSTC/WCO OL-AB

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Second-Order

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42

16. SUPPLEMENTARY NOTATION

1?.

COSATI CODES

FIELD

GROUP

....

j

18 SUBJECT TERMS (Continue on reverse if necessary and identify by block number)

SUB-GROUP

-Gaussian

VOptical

Beams,

Statistical Optis

Propagation,

Turbulence,

'} ):

19 ABSTRACT (Continue o(reverse if necessary and identify by block number)

In this report, closed form expressions are derived for various statistical functions tha arise in optical propagatlon tnrough arbitrary optical systems that can be characterized by

complex ABCD matrix in distributed random inhomogeneities along the optical path Specifically, within the second-order Rytov approximation, explicit general expressins ntr presented for the mutual coherence function, the log-amplitude and phase correlatio function, and the mean square irradiance that is obtained in propagation through an arbitrar paraxial ABCD optical system containing Gaussian-shaped limiting apertures. Additionally, w consider the performance of adaptive-optics systems through arbitrary real paraxial ABC optical systems and derive an expression for the mean irradiance of an adaptive-optics lase transmitter through such systems.

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UNCLASSI F IED

CONTENTS

1.

INTRODUCTION ........................................................

3

2.

GENERAL CONSIDERATIONS FOR THE SCATTERED FIELD ......................

7

3.

MEAN FIELD AND IRRADIANCE OF A POINT SOURCE ........................

15

4.

GENERALIZED SECOND-ORDER MOMENTS ....................................

21

4.1

Correlation Functions .......................................... Mutual Coherence Function ......................................

21

IRRADIANCE STATISTICS ...............................................

25

5.1

Image Plane Variance or Irradiance .............................

27

5.2

Fourier Transform Plane Variance of Irradiance .................

29

6.

ADAPTIVE OPTICS AND ABCD SYSTEMS ....................................

33

7.

CONCLUSIONS .........................................................

39

4.2 5.

APPENDIX:

DERIVATION OF FI, F2 , AND F3...................................

REFERENCES ...............................................................

23

41 45

FIGURES

1.

2.

3.

Schematic Representation of Propagatiop through an Arbitrary ABCD Optical System .......................................

8

Propagation Geometry of a Point Source Located a Distance z I to the Left of a "shin Gaussian" Lens of Real Focal Length f and I/e Transmission Radius a ...............

16

Normalized Variancec

f Trradiance as a Function of

"a" for Image Plane and Fourier tranform Plan:

Propagation Geometries..............................................

30

I

! A-.ailat iilty 1Dst

... illilil~l ."--lilIli ilFi

-

4Peclal

7

¢ e

1.

INTRODUCTION

Recently, a useful generalized form of paraxial wave optics has been developed that car. treat paraxial wave propagation through any optical system that can be characterized by a complex ABCD ray matrix. 1,2

It has been shown

how Huygens' integral for propagation through a cascaded series of conventional elements, inciuding Gaussian-shaped limiting apertures located arbitrarily along the optical oath, can be accomplished in one step, using the appropriate (complex) ABCD matrix elements for that system.

This approach has

been useful for handling complicated multielement optical resonators, as well as resonators with the useful variable-reflectivity mirrors that are now being developed.1 In Ref. 2, this approach has been generalized to the case of propagation of partially coherent light and link propagation, including the effects of tilt and random jitter of the optical elements and distributed random inhomogeneities along the optical path (e.g., aerosols).

clear air turbulence and

However, the results presented in Ref. 2 for propagation in the

presence of random inhomogeneities were limited to a real ABCD system (i.e., where all the ray matrix elements are real).

Therefore, the results of Ref. 2

are limited to propagation through random media where no significant apertures or stops are located between the input and output planes of the optical system.

Additionally, for propagation between conjugate image planes, the

overall B matrix element is identically zero (for real ABCD systems), and some of the expressions in Section 7 and Appendix C of Ref. 2 must be applied carefully because of the apparent singularities that result. that

We will show

if the diffractive effects of the optical elements are included in the

analysis, no such singularities occur, and all the expressions yield finite results. Thus, the major thrust of this report is to consider propagation through a rand-m medium in a general complex ABCD paraxial optical system. y"

J,

.~

ar'-

)r-wvp

the complex optical

tor te "sutirg

field.

Explicit

"curio-uroer sLaListic.i moments of

In particular, we derive a general expression for

the mean irradiance of a point source, the mutual coherence function, the log-

3

amplitude, the phase and wave structure function, and the variance of irradiance for weak scintillation conditions.

In Section 2, we ui.cuss some

general features of the scattered field through terms of second order in the fluctuating part of the index of refraotion.

In Section 3, we discuss th'

propagation of a point source and obtain an explicit expression for the mean irradiance of a point source propagating through a complex ABCD system.

In

Section 4, we present explicit results for various second-order moments of the complex field.

In Section 5, we present some specific results for the mean

square irradiance of a spherical wave for the Kolmogorov spectrum.

We also

give explicit results for the conjugate image plane and Fourier transform plane propagation geometries.

Finally, in Section 6, we consider adaptive

optics through real ABCD systems and derive an expression for the mean irradiance of an adaptive-optics laser transmitter through such systems. The analysis in this report and in Ref. 2 applies primarily to optical systems that contain apertures of Gaussian-shaped variations in amplitude transmission across the optic axis.

However, the analysis also applies, at

least as a first approximation, to any transversely varying system whose 1 transmission has a quadratic variation to first order near the optic axis.

Thus, any "soft" aperture whose transmission varies with transverse coordinate, at least near the axis, in the approximate form

T(x)

= To(1 - a2 x 2 /2)

where T o is the on-axis transmission and the coefficient a 2 is given by a

ILd 2 T(x)1 o dx 2 x=O

can be approximated to first order by a Gaussian aperture and thus by a complex ABCD matrix.

As a result, the complex paraxial analysis is a good

approximation as long as the resulting beam wave remains sufficiently close to the axis so that la2 x2/21

(5.4)

Effects that cause beam broadening in

the output plan, (e.g., wave front tilt) are contained identically in ,I2 > and 1> 2.

Thus, the normalized variance o2 is independent of such effects and

reflects the focusing/defocusing properties of the random inhomogeneities only along the optical

path.

For example, consider a point source located at distance z, to the left of a lens of real focal length f and Gaussian radius a.

We will determine the

on-axis irradiance variance at a distance z2 to the right of the lens.

We

will assume that the intervening space between the lens and point source is filled with a turbulent medium, which is characterized by the Kolmogorov spectrum.

25

For p = 0 and the Kolmogorov spectrum [Eq. (4.10)], we obtain

2K-8/32

2L

f

2

1, Eq. (5.7) yields W(z) - - (5/12)r(-5:') 7/6 2.78 -7/6 Equations (5.6)-(5.8) are the general expressions for the on-axis logamplitude variance for the Kolmogorov spectrum.

These equations are valid for

an arbitrary complex ABCD system, as specified by O(z).

The quantity W(z) is

an additional weighting function that results from the optical system. E

In

contrast to the results for real ABCD systems, 2 no singularities result for complex ABCD systems. To be specific, we consider two well-known examples of practical interest: imaging and Fourier transform propagation geometries.

26

.... mi

l

i m

m ml lmmmmm, l ~

m

I

IMAGE PLANE VARIANCE OR IRRADIANCE

5.1

Here we have

1

1 22

1 21

For simplicity, we assume that C 2 1 0 only between the point source and the n It is straightforward to show that 2 lens. 2izlz B

and for z

1 ka 2

(5.9a)

zI

(5.9b)

BOz =z

and zL B

k 2 2 2iZl1Z

ZZ 2 zz

2(

B

1

(59c) (5-90z

)

Thus, for z ! z I

gr :

I-Z 2_

(5.10a)

•

and z2k2 2 k2 2z 2

. i

(5. lob)

1 Hence, from Eq.

(5.8), we obtain a(z/z 1 ) 1 21

where k2

a -

(5.12)

2z 1

27

...... i

l

il

-Nilmi~a i l

li

~

a

10

Over the range of integration in Eq. (5.6), where the integrand is non-zero, z -zl, and, hence, p -a. Thus, when the point source is in the far field of the lens (z, >>k a

),

i >> I over the range of integration.

near-field conditions (z


> 1 over the range of integration.

As a

result, the asymptotic limits for the on-axis log-amplitude variance, for constant

2

Cn, are given by

0.564 k7 /6 C2

n

= 0.124 k7 /6 C2

n

0

f1

5 dz 6(z) r

z11/6

Z

1

>

1

/6

'

k

a2

(5.13)

and 2> (xk = 0.906 k/6 C2 n

f

(z)5/6 f~) (z)]-7/6 rz82)

S11/6(2z 1 /k

0.612 k7 /6 Cnz 1

1z

)7/6 1



(6.7)

and

)

4c(rP;P2 = *i(r,p)

(6.8)

iSB(r,p 2 )

In obtaining Eq. (6.6)-(6.8), we have used the fact that for real ABCD systems

1 2Re + (0 i2 = 0

which can be obtained directly from Eqs. (2.15) and (2.16) for p 40

P = 0. We have

also used the fact that from statistical stationarity, ,

and = .

Upon substituting Eqs. (6.7) and (6.8) into Eq. (6.6), neglecting the correlation between phase and log-amplitude, we can be show that 1 r c = exp [-Dw(r)

1 - Ps B(r) + d(r,p)]

(6.9)

where

- x(r 2 ,pi)] > +

(6.10)

is the wave structure function of a point source located at p, in the output plane.

3

DSB(r) =

(6.11)

is the phase structure function of the beacon point source located at P2 in the output plane, and'

35

-

I

UI

d(r,p)



(2)1

(6.12)

where DSsB (r,p) =

(6.13)

is the structure function for point sources located at p, and 22 in the output plane evaluated at the points r, and r2 in the input plane [in Eq. (6.12), DSSB () DSSB (O,p)]. The quantity d is the correlation function between the residual phase difference of the object wave across the input plane and the beacon wave across the input plane.

Because they are statistically

stationary, Dw and DsB are independent of the location of the point source, in

contrast to d, which is a function of r and p. If the beacon is collocated with the object, and the measured beacon phase is a perfect estimate of the object phase, rc = function.

exp(-! DA), where DA is the log-amplitude structure

This results in the best possible performance that can be obtained

with conventional phase conjugation adaptive-optics. The structure functions that appear in Eqs. (6.10)-(6.13) can be obtained directly from the results of Sections 2 and 4.

By substituting Eqs. (2.16)

and (2.17) into Eqs. (4.2) and (4.3), we can show that for isotropic power spectra DW(r) = 8,2 k2

f dz 5 dK 0

0

DSB(r) = 412 k2 0dz

I1

f

KOn(K;z) 11

dK K n(K;z) 11

Cos(

-

J0 (Ka1 r)]

(6.14)

J0 (KaL1 r)]

(6.15)

36

and L

DSS B(r) =

412 k2

k2B

0J dz 0fdK Kon(K;z)I 1

.-2-21)

[1 - J0 (KlaxE

where E

+

cos(- )

(6.16)

]

1 - L21 P= P112

al

z)

BzL(z)

=

B

(6.17a)

BOZW B

(6.17b)

BOz(z) BzL(z) BZ

(6.17c)

and

8(z) =

Equations (6.14)-(6.77) and Eq. (6.9) represent the general solution for the corrected MCF in a real ABCD system for use in calculating the adaptive-optics mean-irradiance profile when correlations between phase and log-amplitude are neglected (the usual case considered in the literature).

If this latter

r contains an additional

restriction is relaxed, it can be shown that

c

multiplicative factor of

cos BS X( 1l2;21P22)

where BS

,

-

B3 x(2,l;L1,P2)]

the correlation function of the beacon phase and object point log-

amplitude, is given by BSBX(r I r 2 ; 1, B, r1E ;-'-

2)

:

2

-

2k2;Ld dz

0

d GKz dK KO n(K;z)

0

sin ()J

(KcaL

37

+

0221)

Within the framework of the present theory,

the inclusion of the above term

in r C provides the coupling between phase-amplitude and log-amplitude in adaptive-optics systems.

For certain applications (e.g., propagation

conditions where the interaction between atmospheric turbulence and thermal blooming is important), the inclusion of this term is necessary to obtain a self-consistent analysis. As an example, we consider the Kolmogorov spectrum in the inertial subrange 3 for propagation conditions where geometrical optics is valid.

The

cosine terms appearing on the right-hand side of Eqs. (6.15) and (6.16) can be replaced by unity. As a result, we obtain rc(r) = exp[-S c(r)]

(6.18)

where

S (r)

=

5 /3 + (alp) 5 /3 2.91 k2 f dz C2 (z) [(2r)

n

c-0 1 l2 -

2L

+

21

15/3

+ a1

For line-of-sight propagation, a I 1

1

5/3] (6.19)

102L - Q121

-

- z/L and a2 = z/L, which, when

substituted into Eq. (6.19), yields

S (r) c0

2.19 k f dx C2(z){[r(1

2 Ir(1 - z/L) + ezi

5/3

-

z/L)

5/3

+ (ez)

Ir(1 - z/L)

-

3

ezl 5 / 3 }

(6.20)

where 0 = p/L is the vector angular separation between the beacon and the object point.

Equation 6.20 is identical to previous results found in the

8 literature for common or angular anisoplanatism.

38

7.

CONCLUSIONS

Within the framework of the paraxial and the second-order Rytov approximation, we have derived explicit general expressions for calculating all second-order statistical moments of an optical wave propagating in a random media through an arbitrary complex paraxial ABCD system.

These

expressions are extensions of the corresponding results that apply both to line-of-sight propagation3 and propagation through real ABCD systems. 2 approach in this report (i.e., complex ABCD matrices.

The

Huygens-Fresnel formulation of the field) uses

This approach provides the most general form for

analyzing propagation of any generalized paraxial wave through a complex paraxial optical system in the presence of random inhomogeneities distributed arbitrarily along the optical path.

According to the Huygen's integral

approach, an arbitrary optical wave propagated through an arbitrary complex paraxial system in a random medium, including all diffraction effects (even those resulting from the "Gaussian" limiting apertures), can be accomplished in one step, with knowledge only of the ABCD matrix elements of the system. Even though Huygen's integral has been used extensively in the literature, its general range of validity is perhaps not as well understood as it ought to be.

39

r

- um.

emm ,,,,m. nu l~w mRm mm

n

u nmU

APPENDIX:

DERIVATION OF F1 , F2 , AND F3

In this appendix, we outline the derivation of the basic statistical The method used is similar

quantities F1 , F2 , and F3 introduced in Section 2.

The basic difference is that the zeroth-order

to that given in Ref. 9.

paraxial Green function, rather than the free space function, is given by Eq.

(2.8). When the optical field in the absence of the inhomogenelties is point sources, the field U0 , as obtained from Eq. (2.4), is given by

Ue(rZ) o( 1,

=

Dor 2 - 2r r" + Ao2)

BCexp e BOD)r

(A-I)

1

where C 1 = -ik/2m and the point source is located at (r,O). A.1

DERIVATION OF F1 If we substitute Eqs. (2.8) and (A-i) into Eq. (2.10), taking the

statistical average and rearranging terms, we obtain

u22

'L)> -

C1 k4

-

(4-0)

ikL

[

d

r

dyr22 B E

2 (Aozr

i1 2BOz k

SH(1,r 2 ) exp

'2 )(B Oz2 B zlL

x exp I2B-k (Alr 2 2 -£ " 2BL IL 1 2r1

~

z

2

2rr

+

DZ+L)

) -

2

212 1 + DOz2 r)

(A-2)

L

where

B

2




(r1)

is the correlation function of the fluctuations of the dielectric function [note that Bn

=



4B I and

41

•

m

Nmmwmm ms/!

|

-

I I t

(I

-

d2 K F

r2) :

azAI

opti7al elements).

(A-4)

-

S scale length of the

the smallest distance between

Consider the integration over z2 .

contribution to this integral for Izl - z2 1 - 0. 0.

1

We assume that the largest scale length is much smaller than

any propagation distance of interest (i.e.,

Bz

1A-3) 2r2"-1 + Dz2z1r

(K,z1 - z2 ) expl-iK.(r

The quantity F (K,Az) is non-zero only for inhomogeneities.

-

3 in terms of its two-dimensional Fourier transform

Next, we expand B

B

2)i

2

(Az2zl1r

2B

exp

z2 z-

,2

~A

ik

1

H r.(

Hence, the function H (rr

2

We only get a non-zero

Thus, Az1 z2

1, and

Dz2

) is given by

IZ2

H 1im 2B

H

0Eexp(

-

2i-6(r

r1 )

-ik(r2 2B

(A-5)

L2 )

To obtain this limit, we

where 6(r) is the two-dimensional delta function.

assume that k has a small imaginary part, i.e., the wave propagates in a slightly absorbing medium. Substituting Eqs. (A-4) and (A-5) into Eq. (A-2) and noting that the z-dependence of the exponential factor is slowly varying with respect to that of FC1 we obtain

(U2 (pL)>

C 1k4(2 ri/k)e - i kL (4,) 22

- (B0

dz I f d r

0 f d K 0

1 exp[

B OZ12 Io

2

L 2

Oz

(A 1

r2

-

-

2)1 2r.re+pDorl0]

2 (A 1 2 _ 2r1 .p + DzLp2) exp[- ik 2BzL 1 42

0i

u

i

u

I

f dz F (K,z)

0

(A-6)

where z2

z I in the exponential terms on the left-hand side of Eq. (A-6),

and, because F C

0 for z

scale length of the fluctuations,

on the integral over z can be extended to infinity.

0

f

FE (K,z)dz = N 4

where

the upper limit

From Ref. 3, we have

(K) n(K)

(A-7)

n(K) is the three-dimensional spectral density of the index of refrac-

tion fluctuations evaluated at K_

= 0.

Substituting Eq. (A-7) into Eq. (A-6), we obtain



L d dz

C 1 k4(2wi/k)e ik L

j d2K n (K)I

(A-8)

where , ,Bo~zA ep-kAOz

1=( B

B ) zzL

2 L( 2

exp-

BOz

r

DzL +

-zL

2

p

iJ

(A-9)

and

f2

Because MOL

MozMzL, where Mz

Oz

d2 r

-

exp

where B is given by Eq. (2.18). have again assumed that Imk < 0.

A-)

zL

k 2 1-r2 1

BOL and, thus ik

+

BOz

BzL

Oz

zL

L

-Ell

2

r

fBzzEk

BL

1z

2

is the ABCD matrix for propagation from z I

lz 2

to z 2 , we can show that AzLBOz + BzLDOz J =

r

2 zL

R

To ensure convergence of this integral, we Combining Eqs. (A-l0), (A-11), and (A-8) and

using the relations

43

9

BzL -A Oz BOL -B Oz AOL and BOz

BOLDzL

BzLDOL

we finally obtain

L