Robert M. Manning

Glenn Research Center, Cleveland, Ohio

Application of an Extended Parabolic Equation

to the Calculation of the Mean Field and the

Transverse and Longitudinal Mutual Coherence

Functions Within Atmospheric Turbulence

NASA/TM—2005-213841

August 2005
The NASA STI Program Office . . . in Profile

Since its founding, NASA has been dedicated to

the advancement of aeronautics and space

science. The NASA Scientific and Technical

Information (STI) Program Office plays a key part

in helping NASA maintain this important role.

The NASA STI Program Office is operated by

Langley Research Center, the Lead Center for

NASA’s scientific and technical information. The

NASA STI Program Office provides access to the

NASA STI Database, the largest collection of

aeronautical and space science STI in the world.

The Program Office is also NASA’s institutional

mechanism for disseminating the results of its

research and development activities. These results

are published by NASA in the NASA STI Report

Series, which includes the following report types:

• TECHNICAL PUBLICATION. Reports of

completed research or a major significant

phase of research that present the results of

NASA programs and include extensive data

or theoretical analysis. Includes compilations

of significant scientific and technical data and

information deemed to be of continuing

reference value. NASA’s counterpart of peerreviewed formal professional papers but

has less stringent limitations on manuscript

length and extent of graphic presentations.

• TECHNICAL MEMORANDUM. Scientific

and technical findings that are preliminary or

of specialized interest, e.g., quick release

reports, working papers, and bibliographies

that contain minimal annotation. Does not

contain extensive analysis.

• CONTRACTOR REPORT. Scientific and

technical findings by NASA-sponsored

contractors and grantees.

• CONFERENCE PUBLICATION. Collected

papers from scientific and technical

conferences, symposia, seminars, or other

meetings sponsored or cosponsored by

NASA.

• SPECIAL PUBLICATION. Scientific,

technical, or historical information from

NASA programs, projects, and missions,

often concerned with subjects having

substantial public interest.

• TECHNICAL TRANSLATION. Englishlanguage translations of foreign scientific

and technical material pertinent to NASA’s

mission.

Specialized services that complement the STI

Program Office’s diverse offerings include

creating custom thesauri, building customized

databases, organizing and publishing research

results . . . even providing videos.

For more information about the NASA STI

Program Office, see the following:

• Access the NASA STI Program Home Page

at http://www.sti.nasa.gov

• E-mail your question via the Internet to

help@sti.nasa.gov

• Fax your question to the NASA Access

Help Desk at 301–621–0134

• Telephone the NASA Access Help Desk at

301–621–0390

• Write to:

 NASA Access Help Desk

 NASA Center for AeroSpace Information

 7121 Standard Drive

 Hanover, MD 21076
Robert M. Manning

Glenn Research Center, Cleveland, Ohio

Application of an Extended Parabolic Equation

to the Calculation of the Mean Field and the

Transverse and Longitudinal Mutual Coherence

Functions Within Atmospheric Turbulence

NASA/TM—2005-213841

August 2005

National Aeronautics and

Space Administration

Glenn Research Center
Available from

NASA Center for Aerospace Information

7121 Standard Drive

Hanover, MD 21076

National Technical Information Service

5285 Port Royal Road

Springfield, VA 22100

Available electronically at http://gltrs.grc.nasa.gov
NASA/TM—2005-213841 1

Application of an Extended Parabolic Equation to the Calculation of 

the Mean Field and the Transverse and Longitudinal Mutual 

Coherence Functions within Atmospheric Turbulence 

Robert M. Manning 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

1. Introduction 

The parabolic equation, originally introduced by Fock and Leontovich (ref. 1), has found successful 

application in several subject areas that deal with wave propagation issues. In the case of stochastic 

electromagnetic wave propagation through random media, the use of the parabolic equation was adopted 

by Klyatskin and Tatarskii 36 years ago (refs. 2 and 3) and has since found much success in describing 

propagation phenomena from laser beams to optical images. This type of parabolic equation is accurate 

within the paraxial approximation and is therefore suited to treat small-angle wave scattering about the 

preferential direction of propagation. The paraxial approximation holds when the wavelength λ and the 

characteristic size l0 of the smallest inhomogeniety of the random medium is such that λ<<l0. However, as 

shown by Klyatskin and Tatarskii, (ref. 4), this parabolic equation in the paraxial approximation is 

sufficient in describing optical propagation within the Markov approximation (i.e., where the 

inhomogeneities which compose the random medium are taken to have a δ-function correlation in the 

direction of wave propagation). There are cases, however, in which the wavelength may be on the order 

of the size of the scattering inhomogenieties λ∼l0 thus potentially violating the paraxial approximation 

and requiring an ‘extended’ parabolic equation that still possess a preferential direction of propagation but 

is not bound by the paraxial approximation, i.e., can describe scattering at large angles out of the direction 

of propagation. Such an extended equation can only be derived from the fundamental Helmholtz equation. 

It is the purpose of this work to analytically derive solutions for the generalized mutual coherence 

function (MCF), i.e., the second order moment, of a random wave field propagating through a random 

medium within the context of the extended parabolic equation. Here, ‘generalized’ connotes the 

consideration of both the transverse as well as the longitudinal second order moments (with respect to the 

direction of propagation). Such solutions will afford a comparison between the results of the parabolic 

equation within the pararaxial approximation and those of the wide-angle extended theory. In section 2, 

the extended parabolic equation for electromagnetic wave propagation is given which, of course, derives 

from the stochastic Helmholtz equation. Since this will be an operator equation in the random electric 

field, a statistical operator method is developed in section 3 which will give a general equation for an 

arbitrary spatial statistical moment of the wave field. The generality of the operator method allows one to 

obtain an expression for the second order field moment in the direction longitudinal to the direction of 

propagation. From this, expressions are obtained as a special case for the MCF of the field, both in the 

transverse and longitudinal directions. Analytical solutions to these equations are derived for the 

Kolmogorov and Tatarskii spectra of atmospheric permittivity fluctuations within the Markov 

approximation. 
NASA/TM—2005-213841 2

2. The Wide-Angle Extended Parabolic Equation 

Consider the scalar stochastic Helmholtz equation for an electric field ( ρ) G E x, propagating principally 

along an x-axis and perpendicular to the ρ

G

-plane of an otherwise arbitrary coordinate system 

( ) + ∇ ( ) ( ) ( )( ) ρ + ρ = ε ρ ρ ∂

∂ ρ

ρ

G G G G G

, , ~ , , , 2 2 2

2

2

E x k E x k x E x

x

E x (1) 

where ε( ) ρ

G

, ~ x is the random part of the total permittivity ε( ρ) = + ε( ρ) G G, ~ x, 1 x of the propagation medium. 

Using the decomposition of the total field into a forward propagating field ( ) ρ + G E x, and a backward 

propagating field ( ) ρ − G E x, , i.e., 

() () () ( ) ( ) ( )

x

E x

x

E x

x

E x E x E x E x

∂

∂ ρ +

∂

∂ ρ = ∂∂ ρ ρ = ρ + ρ + − + −

G G G G G G , , , , (2) , , ,

with the expansion into inhomogeneous plane waves 

E ( ) () x e q iq i( ) k q x d q2 1 2 2 2 , exp ⎥⎦

⎤ ⎢⎣⎡ ρ = ⋅ρ ± − ±

∞

−∞

∞

−∞

± ∫ ∫ G G G G (3) 

one can obtain the following operator expressions for the component fields 

( ) ± ( ) + ∇ ( ) ρ = ± ( ) + ∇ ε()() () ρ [ ] ρ + ρ ∂

∂ ρ + − −

ρ

±

ρ

± G G G G G

, , , ~ 2 , , 2 1 2 2 2 2 1 2 2 2 k E x k k x E x E x

x

E x i (4) 

such an equation (or its Fourier transform) was considered by Malakov and Saichev (ref. 5) as well as by 

Klyatskin (ref. 6, p. 169) and Frankenthal and Beran (ref. 7). 

In the event that the backscattered field is insignificant with respect to the forward scattered field, i.e., 

() () ρ >> ρ + − G G E x, E x, , one can formally set ( ,ρ) = 0 − G E x ; equation (4) thus becomes a single equation for 

the forward propagating field 

( ) + ( ) + ∇ ( ) ρ = ( ) + ∇ ε( )( ) ρ ρ ∂

∂ ρ + −

ρ + ρ

+ G G G G

, , ~ 2 , , 2 1 2 2 2 2 1 2 2 2 k E x k k x E x

x

E x i (5) 

This equation, which can be called the “extended parabolic equation” (ref. 2, p. 169), was previously 

analyzed by Saichev (ref. 8) using a method different from the operator formulation in what is to follow. 

Its integral formulation is also known as the method of multiple forward scatter (refs. 9 and 10). However, 

in (ref. 10) it was pointed out that this method is applicable to situations where the wavelength λ and the 

smallest scale of inhomogeneity l0 is such that λ<l0 (ref. 11); The classical parabolic equation in the 

paraxial approximation holds for cases where λ<<l0. Hence, although equation (5) is capable of 

describing propagation situations in which the wave is scattered at angles up to π/2 with respect to the 

x-axis, its application is limited to cases where λ<l0. 
NASA/TM—2005-213841 3

The form of equation (5) can be simplified by defining the differential operator 

1 2

2

2

2 2 1 ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ ∇ ≡ + ρ

ρ k

U k (6) 

and the corresponding integral operator 

1 2

2

2

2 1

2

1 −

ρ

ρ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ ∇ ≡ +

k k

V (7) 

Equation (5) then becomes 

( ) ( ) ( )( ) , , 0 ~ , 2 , 2 4 + ρ − ε ρ ρ = ∂

∂ ρ

ρ ρ

G G G G

U E x k V x E x

x

E x ik (8) 

Hereafter, the superscript ‘+’ will be dropped. This equation can be reduced to the well-known 

stochastic parabolic equation in the paraxial approximation by expanding the operators to give 

2 2 Uρ ≈ 2k + ∇ρ and 2 Vρ ≈1 2k and transforming the field via E(x,ρ) =W (x,ρ) () exp ikx G G . In what is to 

follow, this relation will be employed to yield an operator equation for the generalized statistical moments 

of the field. Here, the term ‘generalized’ connotes moments at differing transverse coordinates ρ

G

 as well 

as differing longitudinal coordinates x. 

3. Operator Solutions for the Generalized Field Moments 

3.1 An Expression for the Generalized nmth Field Moments 

Defining the stochastic operator 

− + ε( ) ρ ∂

∂

ρ ≡ ρ ρ

G

, ~ 2 2 4 , U k V x

x

Dx ik (9) 

equation (8) simply becomes 

,ρ ( ,ρ) = 0 G Dx E x (10) 

which is easily amenable to further statistical analysis. To this end, one defines the generalized moment of 

the electric field 

( ) Γnm x ρ x ρ xn ρn xn+ ρn+ xn+m ρn+m ≡ gnm

G " G G " G G , ; , ; , ; , ; , 1 1 2 2 1 1 (11) 

∏ ( ) ∏ ( ) +

= = +

≡ ρ ρ

n m

l n

l l

n

j

gnm E x j j E x

1

*

1

, , G G (12) 
NASA/TM—2005-213841 4

Using a modification of the prescription set fourth in, e.g., (refs. 12 and 13), one can employ 

equation (10) for each of the field points ( ρ) G x, and obtain for the product of fields 

Lnmgnm ( ) x1,ρ1; x2,ρ2; xn ,ρn; xn+1,ρn+1; xn+m,ρn+m = 0 G " G G " G G (13) 

where 

+ ( ) + ε( ) ρ − ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

+

∂

∂ ≡ ρ ρ

= =

+

= +

∑ ∑ ∑ j j

n

j

n

j

n m

l n j l nm U k V x

x x

L ik j j , ~ 2 2 4

1 1 1

( ) ( ) l l

n m

l n

U k V x l l − ρ + ρ ε ρ

+

= +

∑ , ~ 2 * 4 * *

1

(14) 

In order to isolate the quantity 〈gnm〉 = Γnm from this relation, it is expedient to adopt the methods of 

(refs. 14 and 15) and decompose the operator Lnm into its average and random parts, i.e., 

*

1 1 1 1

2 U j U l x xL ik

n m

l n

n

j

n

j

n m

l n i j nm ρ

+

= +

ρ

= =

+

= +

∑ ∑ ⎟

+∑ − ∑ ⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

+

∂

∂ ≡ (15) 

and 

( ) ( ) 0 ~ , , ~ , ~ 2 ~ * *

1 1

4 = ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

≡ ε ρ − ρ ε ρ

+

= +

ρ

=

∑ ∑ l l nm

n m

l n

j j

n

j

Lnm k V x V x L l j (16) 

Hence, equation (13) becomes 

( ) 0 ~ Lnm + Lnm gnm = (17) 

Ensemble averaging this relation yields 

0 ~ Lnm Γnm + Lnmgnm = (18) 

Similarly writing 

0 ~ , ~ gnm = Γnm + gnm gnm = (19) 

and substituting into equation (18) gives 

Lnm nm L  ̃ 

nmg  ̃ nm 0 (20) 

Remembering that it is the goal of this development to obtain an expression for the general field 

moment Γnm, one follows the development given in reference 15 and subtracts equation (20) from 

equation (13) and using equation (19) obtains 
NASA/TM—2005-213841 5

0 ~ ~ ~ LnmΓnm + Lnmgnm − Lnm Γnm − Lnmgnm = (21) 

Combining the first and third members of this equation using the fact that 

[ ] Lnm − Lnm Γnm = LnmΓnm

~ gives 

0 ~ ~ ~ ~ Lnmgnm + LnmΓnm − Lnmgnm = (22) 

One must now isolate the random quantity gnm

~ by defining an operator −1 Lnm inverse to Lnm, i.e., 

1 1 = −LnmLnm . Thus, operating on equation (22) with −1 Lnm yields 

0 ~ 1 ~ 1 ~ ~ + Γ − = − − gnm LnmLnm nm Lnm Lnmgnm (23) 

Finally, operating on this relation with Lnm

~ , ensemble averaging and solving the resulting expression 

for Lnmgnm

~ ~ gives 

Lnm g nm = −[ − Lnm Lnm ] Lnm Lnm Lnm Γnm − − − ~ ~ ~ 1 ~ ~ 1 1 1 (24) 

Substituting this result back into equation (20), one obtains for the equation governing Γnm

[ ] 0 ~ ~ ~ 1 1 1 1 Γ = ⎭

⎬

⎫

⎩

⎨

⎧ − − − − − Lnm LnmLnm LnmLnmLnm nm (25) 

The solution of this operator equation gives an exact solution for the arbitrary field moments for 

wide-angle propagation through a random medium characterized by the stochastic permittivity ε( ρ) G

, ~ x

and the assumption that 0 ~

Lnm = . Just as in the case of the application of the local method of small 

perturbations (ref. 12), an explicit assumption governing the statistics of the fluctuations ε( ) ρ

G

, ~ x does not 

need to be made in the evaluation of the ensemble averages, as will now be demonstrated. 

The general relation given by equation (25) can be reduced to the parabolic equation for the field 

moments in the paraxial approximation in the case where λ<<l. In this instance, one employs the 

approximations for the operators 2 2 Uρ ≈ 2k + ∇ρ and 2 Vρ ≈1 2k used earlier. In addition, the classical 

parabolic equation considers statistical moments in the same transverse plane, i.e., x j = xl = x . Thus, the 

partial differential operators in equation (15) collapse into the single operator ∂/∂x. Equations (15) and 

(16) then become 

∑ ∑

+

= +

ρ

=

ρ

+ ∇ − ∇

∂

∂ ≈

n m

l n

n

j

nm j l x

L ik

1

2

1

2 2 (26) 

and 
NASA/TM—2005-213841 6

( ) ( )⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

≡ +∑ε ρ − ∑ε ρ

+

= = +

n m

l n

l

n

j

Lnm k x j x

1

*

1

2 , ~ , ~ ~ G G (27) 

Two related palatable approximations must now be made; since Lnm Lnm Lnm

~ ≡ + , and it is usually 

assumed that 1 ~

Lnm << , one has 

[ ] [ ] 1 ~ , 1 ~ 1 1 1 1 1 = + ≈ − ≈ − − − − −Lnm Lnm Lnm Lnm LnmLnm (28) 

where 

1

1

2

1

2

1

2

− +

= +

ρ

=

ρ

−

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

+ ∇ − ∇

∂

∂ = ∑ ∑

n m

l n

n

j

nm j l x

L ik

1

2

−

⎥

⎦

⎤ ⎢

⎣

⎡

∂

∂ ≈ x

ik

∫ = ′

x

dx

ik

0

2

1 (29) 

Equation (25) then becomes 

0 ~ ~ 2 1

1

2

1

2 Γ = ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

+ ∇ − ∇ − ∂

∂ −

+

= +

ρ

=

∑ ρ ∑ nm nm nm nm

n m

l n

n

j

L L L

x

ik

j l (30) 

where 

ε( )( ) ρ ε ′ ρ′ − ε( )( ) ρ ε ′ ρ′ −

⎢

⎢

⎣

⎡ ≈ ∫ ∑∑ ∑ ∑

=

+

= = = +

− i l

n

i

n m

l n

i k

n

i

n

k

x

nm nm nm x x x x

i

k L L L G G G G

, ~ , ~ , ~ , ~

2

~ ~ *

0 1 1 1 1

3

1

( ) x ( ) x ( ) x j ( ) x l dx

n m

j n

n m

l n

j k

n m

j n

n

k

′ ⎥

⎥

⎦

⎤

− ∑ ∑ ε ρ ε ′ ρ′ + ∑ ∑ ε ρ ε ′ ρ′

+

= +

+

= +

+

= = +

G G G G

, ~ , ~ , ~ , ~ * *

1 1

*

1 1

(31) 

which is the well-known classical paraxial form for the problem (ref. 13). It is interesting to note that the 

‘geometrical optics’ approximation made in equation (29) leads to the parabolic equation in the paraxial 

approximation. Thus, one can envision a substantial extension of this development beyond that of the 

classical treatment if one is to use the entire form of the operator 〈Lnm〉

–1, i.e., use the inverse of the 

operator 〈Lnm〉 as solved in the paraxial approximation rather than in the geometrical optics approximation 

which was used above. This will form the subject of the next section in which the first order statistical 

moment and the generalized second moment (i.e., the mutual coherence function) are derived using the 

above formalism. 
NASA/TM—2005-213841 7

3.2 Solution for the First Moment 

The first order moment of the random electric field in a plane transverse to the direction of 

propagation is defined through equation (11) to be given by 

( ) ( ) 10 1 1 Γ ;ρ = ;ρ G G x E x (32) 

which is a solution of the operator relation of equation (25), viz, 

[ ] 0 ~ ~ ~ 1 10 10 1 10 10

1 1 10 10 10 Γ = ⎭

⎬

⎫

⎩

⎨

⎧ − − − − − L L L L L L (33) 

where, from equations (15) and (16), 

( ) 1 4 10 10 , , ~ 2 ~ 2 , ˆ 1 + ≡ ε ρ ρ ≡ ρ ∂

∂ ≡ ρ ρ

G G G U L k V x

x

L ik (34) 

The ability to proceed in an analytical fashion is dependent upon the simplifying approximations used 

earlier, viz, one has that 

[ ] [ ] 1 ~ ,1 ~ 1 1 10 10

1

10

1

10 10 1

10 = + ≈ − ≈ − − − − −L L L L L L (35) 

Using this result, equation (33) becomes 

0 ~ ~

10 10

1

10 10 10 Γ = ⎭

⎬

⎫

⎩

⎨

⎧ − − L L L L (36) 

Employing the appropriate definitions of the operators, equation (36) gives 

( ) ( ) ( ) ( ) , ( ) , 0 ~ , 2 2 ~ 2 2 10 4

1

4 Γ ρ = ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

ε ρ ⎭

⎬

⎫

⎩

⎨

⎧ +

∂

∂

+ − ε ρ ∂

∂

ρ

−

ρ ρ ρ

G G G U k V x x

x

U k V x ik

x

ik (37) 

This differential equation in the operators Uρ and Vρ must now be simplified and solved for the firstorder moment Γ ( ) ρ

G

, 10 x . 

To this end, one must first deal with the factor 

≡ ( ) ρ ⎭

⎬

⎫

⎩

⎨

⎧ +

∂

∂ −

ρ

G 2 ,

1

U G x

x

ik (38) 

which is the Green function of the operators ik ∂ ∂x +Uρ

ˆ 2 , defined by 

( ) ( )( ) ρ = δ − ′ δ ρ − ρ′

⎪⎭

⎪

⎬

⎫

⎪⎩

⎪

⎨

⎧

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ ∇

+ +

∂

∂ ρ G G G G x x x

k

k

x

2ik 2 1 ,

1 2

2

2

2 (39) 
NASA/TM—2005-213841 8

where the definition of Uρ is used. Applying the approximation 2 2 Uρ ≈ 2k + ∇ρ and solving for the green 

function ( ) ρ

G G x, yields, 

() ( ) [ ] ( ) [ ( )( )]

x x

ik x x G x G x x ik x x − ′

− ρ − ρ′ − ′ ⎟ − − ′ ⎠

⎞ ⎜

⎝

⎛

π

ρ = ρ ′ ρ′ = exp 2exp 4

1 , , ; ,

G 2 G G G (40) 

Thus, the third term within the brackets of equation (37) can be written 

( ) ( ) ( ) ε( ) ρ = ⎭

⎬

⎫

⎩

⎨

⎧ +

∂

∂ ≡ ε ρ ρ

−

ρ ρ

G G " , ~ , 2 2 ~ 2 4

1

4 U k V x

x

k V x ik

( ) k G( ) () ( ) x x V x V x d dx

x

= ρ ′ ρ′ ε ρ ε ρ′ ρ′ ′ ρ ρ′

∞

−∞

∫ ∫ 2

0

2 4 , ~ , ~ 2 , ; , G G G G (41) 

Proceeding further, one now must deal with the operator products 

( ) ε( ) ρ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ ∇ ⎟ + ⎠

⎞ ⎜

⎝

⎛ ε ρ =

−

ρ

ρ

G G

, ~ 1

2

1 , ~

1 2

2

2

2 x

k k

V x (42) 

Since ε( ) ρ

G

, ~ x is a random function, it can be represented in the form of a Fourier-Stieltjes integral 

(ref. 16), i.e., 

ε( ρ) = ( κ ⋅ρ) ( κ) ∫ G G G G , exp , ~ x i dZ x (43) 

in which the spectral amplitude ( ) ρ

G dZ x, is endowed with the same statistical properties as is the random 

function ε( ) ρ

G

, ~ x as will be shown in what is to follow. Applying equation (43) to equation (42) gives 

( ) ( )( ) κ ⋅ρ κ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ κ ⎟ − ⎠

⎞ ⎜

⎝

⎛ ε ρ = ∫

−

ρ

G G G G 1 exp , 2

1 , ~ 1 2

2

2

2 i dZ x

k k

V x (44) 

Thus, the ensemble averaged product appearing in right side of equation (41) becomes, 

() ( ) ∫∫

− −

ρ ρ′ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ κ′ − ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ κ ⎟ − ⎠

⎞ ⎜

⎝

⎛ ε ρ ε ′ ρ′ =

1 2

2

2 1 2

2

2 2

2 1 1

2

1 , ~ , ~

k k k

V x V x G G

⋅ ( ) κ ⋅ρ + κ′⋅ρ′ ( κ) ( ′ κ′) G G G G G G exp i i dZ x, dZ x , (45) 

One now makes use of the fact that the atmospheric permittivity fluctuation field ε( ) ρ

G

, ~ x is taken to 

be statistically homogeneous, characterized by a power spectral density Φ ( κ) ε

G x, in the transverse plane, 

and δ-correlated in the longitudinal direction; these circumstances allow one to write (ref. 16) 

( ) κ ( ′ κ′) = δ(κ + κ′) ( − ′ κ) κ κ′ ε 2 2 dZ x, dZ x , F x x , d d G G G G G (46a) 
NASA/TM—2005-213841 9

where for δ-correlated fluctuations in the x direction, the two-dimensional spectrum ε ( ) − ′ κ

G F x x , is given 

by 

( − ′ κ) = πδ( − ′)Φ (κ) ε ε

G G F x x , 2 x x (46b) 

in which Φ( ) κ

G is the three-dimensional spectrum of permittivity fluctuations. Using these relations in 

equation (45) and performing the integrations where possible yields 

() ( ) ( ) [ ] κ ⋅ρ Φ ( ) κ κ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ κ ⎟ δ − ′ − ⎠

⎞ ⎜

⎝

⎛ ε ρ ε ′ ρ′ = π ε

∞ −

−∞

ρ ρ′ ∫ 2

1

2

2 2

2 1 exp 2

1 , 2 ~ , ~ i d

k

x x

k

V x V x d

G G G G G (47) 

where ρ ≡ ρ −ρ′ G G G d is the difference coordinate. 

Taking the statistics governing the random field ε( ρ) G

, ~ x to be also isotropic, i.e.,Φ ( ) κ = Φ ( κ) ε ε x, x, G , 

equation (41) can now finally be evaluated by substituting into it equations (40) and (47); converting the 

integration in the ρd

G -plane into one in plane polar coordinates and performing the associated integrations 

gives 

Φ ( ) κ κ κ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ κ = − π − ε

∞ −

∫ d

k

i k

1

2

2

0

2 3 " 1 (48) 

where the δ-function relation 

( ) 2

1

0

δ − ′ ′ = ∫ x x dx

x

(49) 

is employed. 

Returning to equation (37) and, substituting equation (48) into equation (37) gives 

2 2 1 1 () ( ) 10 , 0

1

2

2

0

2 3

1 2

2

2

2 Γ ρ = ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

Φ κ κ κ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ κ

+ π − ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ ∇

+ +

∂

∂ ε

∞ −

ρ

∫ G d x

k

i k

k

k

x

ik (50) 

In the plane-wave case, one has that 

( ) () () x x

k

x

k 10 10

1 2

2

2

10

1 2

2

2

1 , 1 Γ = Γ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ ∇

Γ ρ = + ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ ∇

+ ρ G ρ (51) 

since the plane wave will not possess any transverse variations. In this special case, equation (50) 

becomes 

2 2 1 () () 10 0

1

2

2

0

2 2 3 Γ = ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

Φ κ κ κ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ κ

+ + π − ∂

∂ ε

∞ −

∫ d x

k

k i k

x

ik (52) 
NASA/TM—2005-213841 10

the solution of which is 

() () ( ) ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

Φ κ κ κ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ κ − π Γ = Γ − ε

∞ −

∫ d

k

x ikx k x

1

2

2

0

2

2

10 10 1

2

0 exp (53) 

This result differs by the factor (1–κ2/κ2

)

–1 from that of the parabolic equation in the paraxial 

approximation. Of course, the later is obtained from the former by retaining the first term in the series 

expansion of the factor. The presence of this factor tends to accentuate the spatial frequencies near the 

value of the wave number k. Since this theory is applicable to those situations in which λ<l0, most of the 

contribution of this factor to wave scattering will occur at the largest spatial frequencies of the 

inhomogeneities. Hence, in the case of atmospheric turbulence, one can employ the von Karman spectral 

density 

( ) ( ) 11 6 2

0

2 2 0.033 − Φε κ = Cε κ + K (54) 

which is not bounded at the high spatial frequencies, to compare the result of equation (53) with that of 

the paraxial approximation. Substituting equation (54) into equation (53) and evaluating the integral 

(ref. 13) yields 

() () ( ) ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎪⎭

⎪

⎬

⎫

⎪⎩

⎪

⎨

⎧

⎥

⎦

⎤ ⎢

⎣

⎡ Γ = Γ − + + − − − C k x

k

K

x ikx i k K F n

2 2

2

2

0

2 1

5 3

0

5 3 10 10 ;

6

1 0 exp 1.772 1.023 0.3908 1,1; (55) 

where the structure parameters for the permittivity and refractive index are related by 2 2 Cε = 4Cn . In the 

case of the open atmosphere, one always has within the bounds of the extended parabolic equation, 

k>>K0; thus, the hypergeometric function reduces to unity and k

–5/3<<K0

–5/3, allowing equation (55) to be 

approximated by 

() () x [ikx K Cn k x] 5 3 2 2 10 10 0 0 exp 0.3908 − Γ = Γ − (56) 

which is the result of the parabolic equation in the paraxial approximation. Hence, the use of the extended 

parabolic equation only makes negligible amplitude and phase corrections to the first order moment of the 

wave field. This result establishes the accuracy of the parabolic equation in the paraxial approximation for 

the first order moment (mean field) as it applies to atmospheric turbulence. The next section will consider 

the calculation of the generalized second-order moment (generalized mutual coherence function) of the 

wave field from the extended parabolic equation and compare its result to that of the paraxial 

approximation. In addition, due to the completeness of the operator analysis, one naturally obtains 

expressions for the MCF along the longitudinal axis. 

3.3 Solution for the Generalized MCF (second moment) 

The generalized MCF of the random electric field in two planes transverse to the direction of 

propagation is, from equation (11), given by 

( ) ( ) ( ) 2 2 * 11 1 1 2 2 1 1 Γ ,ρ ; ,ρ ≡ ;ρ ;ρ G G G G x x E x E x (57) 
NASA/TM—2005-213841 11

which is a solution of equation (25), in this case given by 

[ ] 0 ~ ~ ~ 1 11 11 1 11 11

1 1 11 11 11 Γ = ⎭

⎬

⎫

⎩

⎨

⎧ − − − − − L L L L L L (58) 

where, from the definitions of equations (15) and (16), 

*

1 2

11 1 2 2 2 + ρ − ρ ∂

∂

+

∂

∂ ≡ U U

x

ik

x

L ik (59) 

and 

[ ] ( ) ( ) 0 ~ , , ~ , ~ 2 ~ 2 2 11 * * 1 1 4 11 1 2 L ≡ k Vρ ε x ρ −Vρ ε x ρ L = G G (60) 

As with the case for the first order moment, two related approximations must be made at the outset to 

render the problem analytically tractable. In particular, so long as 1 ~

L11 << , 

[ ] [ ] 1 ~ , 1 ~ 1 1 11 11

1

11

1

11 11 1

11 = + ≈ − ≈ − − − − −L L L L L L (61) 

which allows equation (58) to become 

0 ~ ~

11 11

1

11 11 11 Γ = ⎭

⎬

⎫

⎩

⎨

⎧ − − L L L L (62) 

At this point, it is suggested to connect the longitudinal coordinates x1 and x2 to the related centroid xc

and difference xd coordinates, 

1 2

1 2 , 2

x x x

x x

xc d ≡ − + ≡ (63) 

the operator expressions of equations (59) and (60) then become 

* 11 1 2 2 + ρ − ρ ∂

∂ ≡ U U

x

L ik c (64) 

0 ~ , , 2

~ , 2~ 2 ~ 2 11 * * 1 4 11 1 2 = ⎥

⎦

⎤ ⎢

⎣

⎡ ⎟

⎠

⎞ ⎜

⎝

⎛ ⎟ − ε − ρ ⎠⎞ ⎜

⎝

⎛ ≡ ρ ε + ρ ρ L x V x

x L k V x d c d c

G G (65) 

Hence, equation (62) can be written as 

⎢

⎢

⎣

⎡

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎭

⎬

⎫

⎩

⎨

⎧ ⎟

⎠

⎞ ⎜

⎝

⎛ ⎟ − ε − ρ ⎠⎞ ⎜

⎝

⎛ + − − ε + ρ ∂

∂

ρ ρ ρ ρ 2 * * 1 * 4 , 2

~ , 2

~ 2 2 1 2 1 2

G G d c d c

c

x V xx U U k V x

x

ik

, 0

2

~ , 2

~ . 2 2 2 11 * * 1 4

1

* 1 2 1 2 Γ = ⎥

⎥

⎦

⎤

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎭

⎬

⎫

⎩

⎨

⎧ ⎟

⎠

⎞ ⎜

⎝

⎛ ⎟ − ε ′ − ρ ⎠⎞ ⎜

⎝

⎛

ε ′ + ρ ⎭

⎬

⎫

⎩

⎨

⎧

+ − ∂

∂

ρ ρ

−

ρ ρ

G G d c d c

c

x V xx U U k V x

x

ik (66) 
NASA/TM—2005-213841 12

The solution of this equation commences with obtaining an expression for the Green function 

( ) 1 2

1

* 2 , , 1 2 ≡ ρ ρ

⎭

⎬

⎫

⎩

⎨

⎧

+ − ∂

∂ −

ρ ρ

G G c

c

U U G x

x

ik (67) 

Proceeding as in the last section and using the approximation 2 2 2 ˆUρ ≈ k + ∇ρ , this requires the 

solution of 

( ) ( )( )( ) 1 2 1 1 2 2 * 2 , , 1 2 ρ ρ = δ − ′ δ ρ − ρ′ δ ρ − ρ′

⎭

⎬

⎫

⎩

⎨

⎧

+ − ∂

∂

ρ ρ

G G G G G G c c

c

U U G x x x

x

ik (68) 

which is given by 

( ) [ ] ( )( ) ( )( ) c c

c c

c ik x x

x x

ik G x ρ − ρ′ − ρ − ρ′ − ′ ⎟

⎟

⎠

⎞ ⎜

⎜

⎝

⎛

− ′ ⎟

⎠

⎞ ⎜

⎝

⎛

π

⎟

⎠

⎞ ⎜

⎝

⎛ ρ ρ = − exp 2 1

2

1

2 , , 1 1 2 2

2 2

1 2

G G G G G G

( ) 1 1 2 2 = − ′ ,ρ − ρ′ ,ρ − ρ′ G G G G G xc xc (69) 

Therefore, the fourth term in equation (66) is given by 

( ) ( ) ⎟ −

⎠

⎞ ⎜

⎝

⎛ ⎟ε ′ + ρ′ ⎠⎞ ⎜

⎝

⎛ = ρ ρ ′ ρ′ ρ′ ε − ρ ρ ρ′

∞

−∞

∞

−∞

∫ ∫ ∫ 1 * 1 2 1 2

0

2 4 , 2

~ , 1

2

~ 2 , , ; , , 1 1

G G G G G G " d c d c

x x

x

x k G x x V V x

⎟ +

⎠

⎞ ⎜

⎝

⎛ ⎟ε ′ + ρ′ ⎠⎞ ⎜

⎝

⎛ ⎟ − ε − ρ ⎠⎞ ⎜

⎝

⎛ ⎟ε ′ − ρ′ ⎠⎞ ⎜

⎝

⎛ − ε + ρ ρ ρ ρ′ ρ ′ 2 1 * * 2 * 1 * , 2

~ , 2

~ , 2

~ , 2

~ 1 1 2 2

G G G G d c d c d c d c

x

x

x V V xx

x

x V V x

c d c d c d d dx

x

x

x V V x ⎟ ρ′ ρ′ ′ ⎠

⎞ ⎜

⎝

⎛ ⎟ε ′ − ρ′ ⎠⎞ ⎜

⎝

⎛ + ρ ρ′ ε − ρ 2 2 1 2 2 * 2 * * * , 2

~ , 2

~ 2 2

G G (70) 

One now employs the Fourier-Stieltjes transform as before to represent the products of Vρε, i.e., 

( ) ⎟

⎠

⎞ ⎜

⎝

⎛ κ ⋅ρ ± κ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ κ ⎟ = − ⎠

⎞ ⎜

⎝

⎛

ε ± ρ

−

ρ ∫ 1,2 1,2 1,2

1 2

2

2

1,2

2 1,2 , 2

1 exp 2

1 , 2

~ G G G G d c d c

x i dZ x

k k

x V x , etc., (71) 

and obtains the following relations (functional arguments have been suppressed but the correspondence to 

those in equation (70) follows) 

[ ] ( ) ( ) 1 2 1 1 1 1

1

2

2

1

2

2 1 exp , 2

~~ 1 1 1 κ ⋅ ρ − ρ′ − ′ κ κ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ κ ⎟ − ⎠

⎞ ⎜

⎝

⎛ ε ε = ε

−

ρ ρ′ ∫ i F x x d

k k

V V c c

G G G G (72a) 

[ ] ( ) ( ) 1 2 1 1 2 1

1

2

2

1

2

2 * * 1 exp , 2

~~ 1 1 2 κ ⋅ ρ − ρ′ − ′ + κ κ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ κ ⎟ − ⎠

⎞ ⎜

⎝

⎛ ε ε = ε

−

ρ ρ′ ∫ i F x x x d

k k

V V c c d

G G G G (72b) 
NASA/TM—2005-213841 13

[ ] ( ) ( ) 2 2 2 2 1 2

1

2

2

2

2

2

* * 1 exp , 2

~ ~ 1

1 2 − κ ⋅ ρ − ρ′ − ′ − κ κ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ κ ⎟ − ⎠

⎞ ⎜

⎝

⎛ ε ε = ε

−

ρ ρ′ ∫ i F x x x d

k k

V V c c d

G G G G (72c) 

[ ] ( ) ( ) 2 2 2 2 2 2

1

2

2

2

2

2 * * * * 1 exp , 2

~ ~ 1 2 2 − κ ⋅ ρ − ρ′ − ′ κ κ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ κ ⎟ − ⎠

⎞ ⎜

⎝

⎛ ε ε = ε

−

ρ ρ′ ∫ i F x x d

k k

V V c c

G G G G (72d) 

Substituting equations (69) and (72a) to (72d) into equation (70), performing the required integrals 

over ρ ′

1

G and ρ ′

2

G in plane polar coordinates, and taking all the spectra ( ) ( ) 1,2 1,2 Fε x,κ = Fε x,κ G , i.e., to be 

isotropic in the frequency κ

G , one obtains 

( ) ( ) − ′ κ −

⎩

⎨

⎧

⎥

⎦

⎤ ⎢

⎣

⎡ κ − ′ − ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ κ = − π − ε

− ∞

∫∫ , 2

1 exp

2

1

0 0

2

2

3 c c c c

x

F x x

k

x x i

k

i k

c

"

( ) ( )( ) κρ − ′ + κ − ⎥

⎦

⎤ ⎢

⎣

⎡ κ − ′ − ε , 2 exp 0

2

d c c d c c J F x x x

k

x x i

( ) ( )( ) κρ − ′ − κ + ⎥

⎦

⎤ ⎢

⎣

⎡ κ − ′ − − ε , 2 exp 0

2

d c c d c c J F x x x

k

x x i

( ) ( )( ) d c c c c c J F x x d dx

k

x x i κ κ

⎭

⎬

⎫

κρ − ′ κ ⎥

⎦

⎤ ⎢

⎣

⎡ κ − ′ + ε , 2 exp 0

2

(73) 

where ρ ≡ ρ1 − ρ2

G G G d is the difference coordinate. Using equation (46b) in equation (73) and performing 

the xc integration, remembering equation (49), finally gives 

( ) () Φ κ κ κ

⎭

⎬

⎫

⎩

⎨

⎧

κρ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ κ − − ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ κ = − π − ε

− ∞

∫ J d

k

i x

k

ik d

d

0

2

1

0

2

2

2 3

2 " 2 1 1 exp (74) 

Hence, equation (66) becomes 

( ) () 0

2

2 2 1 1 exp 0 11

2

1

0

2

2 * 2 3 1 2 Γ =

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

Φ κ κ κ

⎭

⎬

⎫

⎩

⎨

⎧

κρ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ κ − − ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ κ

+ − + π − ∂

∂ ε

− ∞

ρ ρ ∫ J d

k

i x

k

U U ik

x

ik d

d

c

(75) 

where Γ11 = Γ11(xc,xd,ρd). Making the plane wave approximation of equation (75), analogous to that done 

earlier for equation (50), the resulting differential equation has as a solution 

( )( ) ( ) ()

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

Φ κ κ κ

⎭

⎬

⎫

⎩

⎨

⎧

κρ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ κ − − ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ κ Γ ρ = Γ ρ − π − ε

− ∞

∫ J d

k

i x

k

x x k x d

d c d d d c 0

2

1

0

2

2

2 2 11 11 2 , , 0,0, exp 1 1 exp (76) 

(More explicitly, one has in equation (75) 
NASA/TM—2005-213841 14

( ) ( ) ( ) ( ) 11 1 1 2 2 2 2 11 1 1 2 2 * , ; , , ; , 1 2 1 2 − Γ ρ ρ ≈ ∇ − ∇ Γ ρ ρ ρ ρ ρ ρ

G G G G G G U U x x x x

( ) R ∇ d Γ x x ρd = ∇ ⋅ ρ

G G G 2 , , 11 1 2

= 0

where ≡ ( ) ρ1 + ρ2 2 G G G R is the centroid coordinate.) It is important to note the initial condition Γ(0,0.ρd); 

since one necessarily must take xc, one must also have xd = 0 by equation (63) since x1,x2≥0. 

This result cannot be analytically studied in its entirety and hence will be considered in two special 

cases. The first case is defined by xd = 0 in which one deals with the transverse MCF 

Γ(xc,ρd) ≡ Γ(xc,0,ρd). Thus, the extended parabolic equation solution for the MCF in a transverse plane at 

a distance xc from the source is, from equation (76) 

( )( ) { } () ( )

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎣

⎡

− κρ Φ κ κ κ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ κ Γ ρ = Γ ρ − π − ε

− ∞

∫ J d

k

xc d d k xc 0 d

1

0

2

2

2 2 11 , 11 0,0, exp 1 1 (77) 

As with the case of the first-order moment, this result differs from that of the paraxial approximation 

in the presence of the factor (1-κ2

/k

2

)

–1 which serves to accentuate spectral contributions for frequencies 

near k. The integral indicated in this expression cannot be analytically evaluated using the von Karman 

spectrum, equation (54). However, since the integrand is such that no singularities exist in the use of the 

unbounded Kolmogorov spectrum, viz., equation (54) with K0 = 0, one can use such a spectrum in 

equation (77) and, upon evaluating the integral, obtain 

( ) () ( ) ⎢ ( ) ( ) +

⎣

⎡

⎩

⎨

⎧ Γ ρ = Γ ρ − − − ρ − c d d k J k d x i 0 5 3 11 , 11 0, exp 3.544 2.048 1

⎥

⎥

⎦

⎤

⎪⎭

⎪

⎬

⎫

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ ρ + ρ − n c d k C x

k

k F 2 2

2 2

1 2 2 11 3

4 ; 6

17

, 6

17 0.108 1; (78) 

where 1F2 ( ) " is a generalized hypergeometric function and, as noted earlier, 2 2 Cε = 4Cn . At the outset, 

since λ<<ρd in most applications, the first term within the braces of equation (78), although quite 

interesting in structure, is negligible with respect to the second term. The hypergeometric function of the 

second term is most easily dealt with by first converting it to a Lommel function (") μ,0 s

(ref. 17, p. 986), i.e., 

( ) ( ) d d

d k s k

k

F ⎟ ρ ρ ⎠

⎞ ⎜

⎝

⎛ =⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ ρ − −

,0 3

8

11 3

2 2 2

1 2 3

11

4 ; 6

17

, 6

17 1; (79) 

The Lommel function reduces (ref. 17, p. 985), in the case where kρd>>1, to the simple approximate 

result ( )( )5 3

,0 3

8 k d k d s ρ ≈ ρ . Using this in equation (79) and (78) becomes the paraxial result 

( ) ( ) [ c ] c d d n d x k C x 2 2 5 3 Γ11 ,ρ = Γ11 0,ρ exp −1.457 ρ . (80) 
NASA/TM—2005-213841 15

Again, as with the first-order moment, corrections to the second moment afforded by the extended 

parabolic equation are negligible in the case of atmospheric turbulence. This is due to the fact that the 

Kolmogorov spectral density level near the inner scale of turbulence is much smaller than it is at larger 

scale sizes. If another spectral density were considered, e.g., one describing propagation through an 

aerosol medium, this may not be the case and significant corrections to the transverse MCF may prevail. 

The second special case in which equation (76) will be examined is in the instance where the factor 

(1-κ2

/k

2

)

–1 can be neglected; as shown above, this is a good approximation for atmospheric turbulence. 

Thus, one is now dealing with the solution for the generalized MCF for field locations at different 

transverse and longitudinal points, 

( )( ) ( ) () ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

Φ κ κ κ

⎭

⎬

⎫

⎩

⎨

⎧

κρ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ κ Γ ρ = Γ ρ − π − − ε

∞

∫ J d

k

i x

x x k x d

d c d d d c 0

2

0

2 2 11 11 2 , , 0,0, exp 1 exp (81) 

Longitudinal correlations of the wave field have been previously considered in (refs. 18 and 19) using 

different methods and obtaining different results. Setting xd = 0 in equation (81) gives the well known 

paraxial result for the transverse MCF Γ(xc,0,ρd). Unlike the cases studied above, the form of 

equation (81) suggests the use of the Kolmogorov spectrum as modified by Tatarskii, which incorporates 

a cutoff at high spatial frequencies by allowing the introduction of the inner scale of turbulence l0, viz., 

( )

0 2

2

2 11 3 5.92 0.033 exp , l

C m

m

κ ≡ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

κ

κ

Φ κ = κ − − ε ε (82) 

Substituting equation (82) into equation (81) and evaluating the resulting integral yields for the 

generalized MCF 

( )( ) ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎪⎭

⎪

⎬

⎫

⎪⎩

⎪

⎨

⎧ − κ ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ ρ Γ ρ = Γ ρ − − − −5 3

2

1 1 2 2 5 6 11 11 4 ;1; 6

5 , , 0,0, exp 4.352 m d c d d d n c B

x x k C x B F (83) 

where 

k

x B i d

m 2

1

2 +

κ ≡ (84) 

Thus, as first noted in (ref. 19), the presence of the diffraction factor on equation (81) modifies the 

effect of the cutoff frequency κm. In the case where 1 2 2 ρd > 1 κm + i xd 2k , equation (83) reduces to 

equation (80) upon employing the asymptotic representation of the confluent hypergeometric function 

1F1(...). When ρd = 0, equation (83) becomes 

( )( ) ⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

⎪⎭

⎪

⎬

⎫

⎪⎩

⎪

⎨

⎧

− ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛ κ Γ = Γ ρ − κ + − 1

2 , ,0 0,0, exp 4.352 1

5 6 2

2 2 5 3

11 11 k

ix

x x k C x d m c d d n c m (85) 

In the appropriate limits, this expression gives 
NASA/TM—2005-213841 16

( )( )

[ ]

[ ] ( ) ⎪

⎩

⎪

⎨

⎧

− + λ >>

− κ λ <<

Γ = Γ ρ

2

0

7 6 2 5 6

2

0

2 1 3

11 11

exp 0.5631 2.102 ,

exp 1.8133 ,

, ,0 0,0,

i k C x x x l

ikC x x x l

x x

n c d d

n m c d d

c d d (86) 

Hence, there is a phase variation in the longitudinal direction, as expected, with an attendant 

attenuation as the longitudinal separation is increased. It must be noted, however, that one needs to realize 

the condition xd>l0 in all cases so as to satisfy the assumption of δ-correlation of the fluctuations along the 

longitudinal axis (ref. 19). Figures 1 and 2 show plots of equation (85) with Γ11(0,0,0) = 1 for a typical 

atmospheric propagation scenario at λ = 0.63 Hm for two values of inner scale size, l0 = 1.0 mm and 

l0 = 1.0 cm, respectively. 

References 

1. Leontovich, M.A. and Fock, V.A., “Solution of the problem of propagation of electromagnetic waves 

along the earth’s surface by the method of parabolic equations,” J. Phys U.S.S.R. 10, p. 13 (1946). 

Also appears as Chapter 11 in V.A. Fock, Electromagnetic Diffraction and Propagation Problems, 

Pergamon Press Ltd, 1965. 

2. Tatarskii, V.I., “Light propagation in a medium with random refractive index inhomogeneities in the 

Markov random process approximation,” Sov. Phys. JETP 29 (6), pp. 1133–1138 (1969). 

3. Klyatskin, V.I., “Applicability of the approximation of a Markov random process in problems relating 

to the propagation of light in a medium with random inhomogeneities,” Sov. Phys. JETP 30 (3), 

pp. 520–523 (1969). 

4. Klyatskin, V.I. and Tatarskii, V.I., ”The parabolic equation approximation for propagation of waves 

in a medium with random inhomogeneities,” Sov. Phys. JETP 31 (2), pp. 335–339 (1970). 

5. Malakov, A.N., and Saichev, A.I., “Representation of wave reflected from a randomly 

inhomogeneous layer in the form of a series satisfying the causality condition,” Radiophys. Quantum 

Electron. 22 (11), pp. 922–928 (1979). 

6. Klyatskin, V.I., The Embedding Method in the Theory of Wave Propagation, Nauka, Moscow, 1986. 

In Russian. 

7. Frankenthal, S., and Beran, M.J., “Transverse spectra of waves in random media,” Waves in Random 

Media 7, pp. 257–282 (1997). 

8. Saichev, A.I., “Allowance for some corrections to the parabolic quasi-optical approximation in the 

statistical description of waves propagating in randomly inhomogeneous media,” Radiophys. 

Quantum Electron. 22 (11), pp. 980–982 (1979). 

9. Ostashev, V.E., and Tatarskii, V.I., “Multiple forward scattering method in media with random 

inhomogeneities,” Radiophys. Quantum Electron. 22, pp. 116–122 (1979). 

10. Ostashev, V.E., and Tatarskii, V.I., “Representation of the Helmholtz equation solution in the form of 

a series based on backscattering multiplicity,” Waves in Random Media, 5, pp. 125–135 (1995). 

11. Grosse, R., “Stochastic solutions of a wide-angle parabolic wave equation,” in Proc 5th Int. Symp. 

Long-Range Sound Propagation, pp. 221–227 (1992). 

12. Strohbehn, J.W., “Modern Theories in the Propagation of Optical Waves in a Turbulent Medium,” in 

Laser Beam Propagation in the Atmosphere, J.W. Strohbehn, ed. Topics in Applied Physics, vol. 25, 

(Springer-Verlag) (1978). 

13. Gurvich, A.S., and Tatarskii, V.I., “Coherence and intensity fluctuations of light in the turbulent 

atmosphere,” Radio Sci. 10 (1), pp. 3–14 (1975). 

14. Tatarskii, V.I., and Gertsenshtein, M.E., “Propagation of waves in a medium with strong fluctuation 

of the refractive index,” Sov. Phys. JETP 17 (2), pp. 458–463 (1963). 
NASA/TM—2005-213841 17

15. Manning, R.M., “Radiative transfer theory for inhomogeneous media with random extinction and 

scattering coefficients,” J. Math. Phys. 30 (10), pp. 2432–2440 (1989). 

16. Tatarskii, V.I., The Effects of the Turbulent Atmosphere on Wave Propagation, National Technical 

Information Service, Springfield, VA, 1971. 

17. Gradshteyn, I.S., and Ryzhik, I.M., Table of Integrals, Series, and Products, Corrected and Enlarged 

Edition, Academic Press, 1980. 

18. Klyatskin, V.I., “On longitudinal correlations of the field of a light wave propagating in a medium 

having random inhomogeneities,” Radiophys. Quantum Electron. 13, pp. 834–835 (1970). 

19. Nalbandyan, O.G., and Tatarskii, V.I., “Spatial correlation of the wave field in a randomly 

inhomogeneous medium,” Radiophys. Quantum Electron. 20, pp. 155–160 (1977). 

Figure 1.—Γ11(xc,xd,0) versus xd (in meters) 

for λ = 0.63 mm, l0 = 1.0 mm, xc = 5 km, 

and 2 12 10− Cn = m–2/3 
NASA/TM—2005-213841 18

Figure 2.—Γ11(xc,xd,0) versus xd (in meters) 

for λ = 0.63 mm, l0 = 1.0 cm, xc = 5 km, 

and 2 12 10− Cn =
This publication is available from the NASA Center for AeroSpace Information, 301–621–0390.

REPORT DOCUMENTATION PAGE

2. REPORT DATE

19. SECURITY CLASSIFICATION

 OF ABSTRACT

18. SECURITY CLASSIFICATION

 OF THIS PAGE

Public reporting burden for this collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources,

gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this

collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson

Davis Highway, Suite 1204, Arlington, VA 22202-4302, and to the Office of Management and Budget, Paperwork Reduction Project (0704-0188), Washington, DC 20503.

NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89)

Prescribed by ANSI Std. Z39-18

298-102

Form Approved

OMB No. 0704-0188

12b. DISTRIBUTION CODE

8. PERFORMING ORGANIZATION

 REPORT NUMBER

5. FUNDING NUMBERS

3. REPORT TYPE AND DATES COVERED

4. TITLE AND SUBTITLE

6. AUTHOR(S)

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

11. SUPPLEMENTARY NOTES

12a. DISTRIBUTION/AVAILABILITY STATEMENT

13. ABSTRACT (Maximum 200 words)

14. SUBJECT TERMS

17. SECURITY CLASSIFICATION

 OF REPORT

16. PRICE CODE

15. NUMBER OF PAGES

20. LIMITATION OF ABSTRACT

Unclassified Unclassified

Technical Memorandum

Unclassified

National Aeronautics and Space Administration

John H. Glenn Research Center at Lewis Field

Cleveland, Ohio 44135–3191

1. AGENCY USE ONLY (Leave blank)

10. SPONSORING/MONITORING

 AGENCY REPORT NUMBER

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546–0001

Available electronically at http://gltrs.grc.nasa.gov

August 2005

NASA TM—2005–213841

E–15211

WBS–22–612–30–81–03

24

Application of an Extended Parabolic Equation to the Calculation of the Mean

Field and the Transverse and Longitudinal Mutual Coherence Functions Within

Atmospheric Turbulence

Robert M. Manning

Diffraction propagation; Electromagnetic wave transmission; Optical communication

Unclassified -Unlimited

Subject Categories: 32, 59, and 74

Responsible person, Robert M. Manning, organization code RCA, 216–433–6750.

Solutions are derived for the generalized mutual coherence function (MCF), i.e., the second order moment, of a random

wave field propagating through a random medium within the context of the extended parabolic equation. Here, "generalized" connotes the consideration of both the transverse as well as the longitudinal second order moments (with respect

to the direction of propagation). Such solutions will afford a comparison between the results of the parabolic equation

within the pararaxial approximation and those of the wide-angle extended theory. To this end, a statistical operator

method is developed which gives a general equation for an arbitrary spatial statistical moment of the wave field. The

generality of the operator method allows one to obtain an expression for the second order field moment in the direction

longitudinal to the direction of propagation. Analytical solutions to these equations are derived for the Kolmogorov and

Tatarskii spectra of atmospheric permittivity fluctuations within the Markov approximation.