Robert M. Manning

Glenn Research Center, Cleveland, Ohio

The Atmospheric Mutual Coherence Function From 

the First and Second Rytov Approximations and Its 

Comparison to That of Strong Fluctuation Theory

NASA/TM—2011-216974

January 2011
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Robert M. Manning

Glenn Research Center, Cleveland, Ohio

The Atmospheric Mutual Coherence Function From 

the First and Second Rytov Approximations and Its 

Comparison to That of Strong Fluctuation Theory

NASA/TM—2011-216974

January 2011

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
Acknowledgments

The author would like to thank Dr. Albert Wheelon for his continuing support and inspiration during this work.

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Level of Review: This material has been technically reviewed by technical management. 

This report is a formal draft or working 

paper, intended to solicit comments and 

ideas from a technical peer group.
NASA/TM—2011-216974 1 

The Atmospheric Mutual Coherence Function From the First and 

Second Rytov Approximations and Its Comparison to 

That of Strong Fluctuation Theory 

Robert M. Manning 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Abstract 

An expression for the mutual coherence function (MCF) of an electromagnetic beam wave 

propagating through atmospheric turbulence is derived within the confines of the Rytov approximation. It 

is shown that both the first and second Rytov approximations are required. The Rytov MCF is then 

compared to that which issues from the parabolic equation method of strong fluctuation theory. The 

agreement is found to be quite good in the weak fluctuation case. However, an instability is observed for 

the special case of beam wave intensities. The source of the instabilities is identified to be the 

characteristic way beam wave amplitudes are treated within the Rytov method. 

1.0 Introduction 

In the early studies in the late 1950s of electromagnetic wave propagation through atmospheric 

turbulence (Ref. 1 and 2), the weak fluctuation theory known as the Rytov approximation made 

considerable strides in the understanding of the scattering mechanisms inherent in the random permittivity 

field of the troposphere. Within the structure of the theory, propagation quantities such as the logamplitude and phase fluctuations took precedence as well as their characterizing statistical parameters, 

i.e., the corresponding correlation and structure functions of these quantities. Only in the special cases of 

plane and spherical waves were these quantities related to the prevailing statistical parameters of the 

electric field of the wave, i.e., the second order statistical moment known as the mutual coherence 

function (MCF). The case of beam wave propagation was examined (Ref. 3) but an expression for the 

MCF was never given. The theory, however, had its limitations and was replaced with the strongfluctuation theories based on the parabolic equation (Ref. 2). In this treatment, the relevant propagation 

quantities were necessarily the statistical moments of the electric field of the propagating electromagnetic 

wave; the concepts of log-amplitude and phase fluctuations did not, nor needed not, enter into the 

structure of strong fluctuation theory. Furthermore, since the MCF was the statistically most significant 

quantity for applications and experiments, strong fluctuation theory considered not only this quantity for 

the special cases of plane and spherical wave propagation but also for the more general beam wave case. 

It was found, without explanation, that the results for the MCF of the plane and spherical wave cases as 

calculated from the parabolic equation method were identical to those from the Rytov approximation. Due 

to the success of the parabolic equation method, Rytov theory fell by the wayside and the calculation of 

the MCF for the beam wave case was never carried-out within weak-fluctuation theory and a comparison 

was never made to that of strong-fluctuation theory. 

It is the purpose of the present study is to essentially finish bridging the gulf between the Rytov 

approximation and the parabolic equation method by considering the MCF of the beam wave case within 

the Rytov approximation and comparing it to the results of strong fluctuation theory. By definition, the 

weak fluctuation theoretical results should directly follow from those of the strong fluctuation results in 

the limit of weak fluctuations. This will be shown for the beam wave scenario. In the interim, it will also 

be shown that the traditional form of Rytov theory used in the classical studies is incomplete (Ref. 4); the 

subtleties between the order of the Rytov approximation and the order of magnitude of the permittivity 
NASA/TM—2011-216974 2 

fluctuations was overlooked for a complete and unified application of the theory. Finally, a surprising 

limitation of the beam wave modeling within Rytov theory is identified and studied. 

2.0 Recapitulation of the Rytov Approximations 

Using the field decomposition for the electric field of a wave propagating predominantly along the 

x-axis of a coordinate system situated in a random medium, viz., 

 

E 

( ) r = U ( ) r exp( ) ikx (1) 

in the stochastic Helmholtz equation, characterized by the stochastic permittivity field 

ε ̃ 

( ) r , 

 

∇2E 

( ) r + k 2E ( ) r + k 2ε ̃ ( ) r E ( ) r = 0 (2) 

one has for the complex amplitude 

 

∇ρ

2U 

( ) r +

∂2U 

( ) r 

∂x 2 + 2ik

∂U 

( ) r 

∂x

+ k 2ε ̃ 

( ) r U ( ) r = 0 . (3) 

In the event that the wavelength λ of the propagating wave and the size of the smallest inhomogeneity l0

characterizing the stochastic permittivity field is such that λ << l0, one has that 

 

∂2U 

( ) r 

∂x2 << 2ik

∂U 

( ) r 

∂x

 (4) 

allowing Equation (3) to be approximately written 

 

∇ρ

2U 

( ) r + 2ik

∂U 

( ) r 

∂x

+ k 2ε ̃ 

( ) r U ( ) r = 0 (5) 

which is a parabolic differential equation of the diffusion type. 

The most straightforward way of dealing with this scenario is to employ Tatarskii’s method of 

solution employing the Rytov transformation (Ref. 2). The idea is to decouple the stochastic factor 

ε ̃ 

( ) r 

from the resulting stochastic field 

U 

( ) r in Equation (5). To this end, one can employ the transformation 

(the Rytov transformation) 

 

U 

( ) r = exp ψ  ( ) ( ) r (6) 

in Equation (5) and obtain 

 

2ik ∂ψ 

( ) r 

∂x

+ ∇ρ

2ψ 

( ) r + 

∇ρψ  ( ) ( ) r 

2

+ k 2ε ̃ 

( ) r = 0 (7) 

which now a non-parametric relation but also a nonlinear one. Solving this equation via a perturbation 

expansion in the quantity ν = ε ̃ 2 , one has 

 

U 

( ) r = U0



( ) r exp ψ1



( ) r + ψ2

( ) ( ) r + (8) 

in which 

U0



( ) r is the initial field distribution and 
NASA/TM—2011-216974 3 

 

ψ1



( ) r = −k 2 Gp



r ,

 ( ) r ′

−∞

∞

∫ ε ̃ 

( ) r ′ U0



( ) r ′

U0



( ) r 

⎛

⎝

⎜

⎞

⎠

⎟ d 3r′ (9) 

is the first Rytov approximation and 

 

ψ2



( ) r = − Gp



r ,

 ( ) r ′

−∞

∞

∫ 

∇ρψ1

 [ ] ( ) r ′ 2 U0



( ) r ′

U0



( ) r 

⎛

⎝

⎜

⎞

⎠

⎟ d 3r′ (10) 

is the second Rytov approximation. Here, the parabolic equation Green function is 

 

Gp



r ,

 ( ) r ′ = − 1

4π

⎛

⎝

⎜ ⎞

⎠

⎟

exp ik 

ρ−  ( ) ρ′ 2 ( ) 2( ) x − ′ x ( )

x − ′ x 

 (11) 

Writing the complex amplitude 

U 

( ) r in terms of an amplitude 

A 

( ) r and a phase 

S 

( ) r ,i.e., 

 

U0



( ) r = A0



( ) r exp iS0

 ( ) ( ) r , U 

( ) r = A ( ) r exp iS  ( ) ( ) r (12) 

and using these representations in Equation (8) gives 

 

ψ1



( ) r + ψ2



( ) r += ln

A 

( ) r 

A0



( ) r 

⎡

⎣

⎢ ⎤

⎦

⎥ − i S0



( ) r − S ( ) ( ) r 

= χ 

( ) r + iS1



( ) r 

 (13) 

in which 

χ 

( ) r ≡ ln A ( ) r A0

 [ ] ( ) r is the log-amplitude and 

S1



( ) r ≡ S ( ) r − S0



( ) r is the phase fluctuation. 

Hence, letting 

Ψ 

( ) r ≡ ψ1



( ) r + ψ2



( ) r +, one can write for these propagation parameters 

 

χ 

( ) r = Re ψ1



( ) r + ψ2

{ } ( ) r + = 1

2

Ψ 

( ) r + Ψ* ( ) ( ) r (14) 

and 

 

S1



( ) r = Im ψ1



( ) r + ψ2

{ } ( ) r + = 1

2i

Ψ 

( ) r − Ψ* ( ) ( ) r (15) 

These random quantities can only be described by statistical functions such as averages, correlation 

functions, etc. For example, the spatial log-amplitude correlation function is given by 

 

Bχ



r 1,

 ( ) r 2 ≡ 1

4

Ψ 

( ) r 1 + Ψ*  ( ) ( ) r 1 Ψ ( ) r 2 + Ψ*  ( ) ( ) r 2

= 1

4

Ψ 

( ) r 1 Ψ ( ) r 2 + Ψ ( ) r 1 Ψ* ( ) r 2 + Ψ* ( ) r 1 Ψ ( ) r 2 + Ψ* ( ) r 1 Ψ* ( ) r 2

= 1

2

Re Ψ r ( ) 1 Ψ r ( ) 2 + Ψ r ( ) 1 Ψ* r { } ( ) 2

 (16) 

Similarly, for the spatial phase correlation and log-amplitude/phase cross correlation 

 

BS



r 1,

 ( ) r 2 = 1

2

Re Ψ 

( ) r 1 Ψ ( ) r 2 − Ψ ( ) r 1 Ψ*  { } ( ) r 2 (17) 
NASA/TM—2011-216974 4 

and 

 

BχS



r 1,

 ( ) r 2 = 1

2

Im Ψ 

( ) r 1 Ψ ( ) r 2 − Ψ ( ) r 1 Ψ*  { } ( ) r 2 (18) 

However, at this point, it is necessary to consider the relationship between the order in ε ̃ of the statistics 

of 

ψi



( ) r and the order of the Rytov approximation. 

3.0 Connection Between the First and Second Order Rytov 

Approximations—The Necessity of Using Both 

It is assumed, without loss of generality, that the average or statistical mean 

ε ̃ 

( ) r = 0 . Thus, by 

Equation (9), 

ψ1



( ) r = 0. However, by Equation (10), 

ψ2



( ) r ~ ε ̃  ( ) ( ) r 

2

 and its mean is non-zero. That is, 

one must consider the second order Rytov approximation, which is of second order in the random 

permittivity, for a non-zero mean field. Hence, one has from Equations (14) and (15), to within the second 

order Rytov approximation (i.e., of second order in the permittivity fluctuations) 

 

χ 

( ) r = Re ψ2

 { } ( ) r , S1

( ) r = Im ψ2

 { } ( ) r (19) 

Now the correlation functions formed using 

ψ1



( ) r as required by Equations (16) to (18), e.g., 

 

ψ1



( ) r 1 ψ1



( ) r 2 are also of second order in the permittivity fluctuations by Equation (9). Hence, again, to 

within second order in the permittivity fluctuations, only the first order Rytov approximation need be 

used for these quantities. Thus, for a solution to within second order in ε ̃ , by Equations (14) and (15), 

one need only keep the 

ψ1



( ) r in the evaluation of the spatial correlation functions. Hence, Equations (14) 

and (15) reduce to 

 

χ 

( ) r = Re ψ1

 { } ( ) r = 1

2 ψ1



( ) r + ψ1

*  ( ) ( ) r , S1



( ) r = Im ψ1

 { } ( ) r = 1

2i ψ1



( ) r − ψ1

*  ( ) ( ) r (20) 

where 

 

Bχ



r 1,

 ( ) r 2 = 1

2

Re ψ1



( ) r 1 ψ1

* 

( ) r 2 + ψ1



( ) r 1 ψ1

 { } ( ) r 2 (21) 

and 

 

BS



r 1,

 ( ) r 2 = 1

2

Re ψ1



( ) r 1 ψ1

* 

( ) r 2 − ψ1



( ) r 1 ψ1

 { } ( ) r 2 (22) 

Similarly, for the cross correlation of log-amplitude and phase fluctuations 

 

BχS



r 1,

 ( ) r 2 = 1

2

Im ψ1



( ) r 1 ψ1



( ) r 2 − ψ1



( ) r 1 ψ1

*  { } ( ) r 2 (23) 

Therefore, only the product averages 

ψ1



r ( ) 1 ψ1



r ( ) 2 and 

ψ1



( ) r 1 ψ1

* 

( ) r 2 need to be evaluated for these 

correlation functions. Of course, from these correlation functions come the associated variances 

 

σχ

2 

( ) r ≡ Bχ



r ,

 ( ) r , 

σS

2 

( ) r ≡ BS



r ,

 ( ) r , and 

σχS



( ) r ≡ BχS



r ,

 ( ) r . Finally, the structure functions 

 

Dχ



r 1,

 ( ) r 2 ≡ χ 

( ) r 1 − χ  ( ) ( ) r 2

2 , etc., can be calculated using Equation (20). Both the mean values given by 
NASA/TM—2011-216974 5 

Equation (19) and the correlation functions of Equations (20) to (23) are second order quantities in the 

permittivity fluctuations ε ̃ . 

What is important in the foregoing is the salient fact that the first moment given by the second order 

Rytov approximation, being 

~ ε ̃  ( ) ( ) r 

2

, is just as statistically significant as those correlation and structure 

functions found using the first order approximation (Refs. 4 and 5). Any application requiring the 

correlation and structure functions is incomplete without the consideration of the first order moments χ

and S1 . This circumstance will be demonstrated by the application of Rytov theory for the calculation 

of the MCF of the propagating wave field. 

Complete expressions for the log-amplitude and phase fluctuation statistics of a beam wave are given 

in Appendix A. 

4.0 The Mutual Coherence Function Within the Rytov Approximation 

In the original applications of Rytov theory to stochastic wave propagation, correlation and structure 

functions were calculated and compared with experimental results. The MCF, i.e., the second order 

moment of the associated electric field was relegated to strong fluctuation theory and the parabolic 

equation method. However, with the proper accounting of the second Rytov approximation, an MCF 

expression can be derived based on Rytov theory. 

Using Equation (1), the MCF for a beam wave is defined by 

 

Γ2 L,



ρ1,

 ( ) ρ2 ≡ E L, ( ) ρ1 E* L, ( ) ρ2 = U L, ( ) ρ1 U* L, ( ) ρ2 (24) 

Substituting Equation (13) into Equation (8) and using this intermediate result in Equation (24) yields 

 

Γ2 L,



ρ1,

 ( ) ρ2 = U0 L, ( ) ρ1 U0

* L,

 ( ) ρ2 exp χ L, ( ) ρ1 + χ L, ( ) ρ2 + i S1 L, ( ) ρ1 − S1 L, [ ] ( ) ( ) ρ1 (25) 

At this point, it is advantageous to recognize that the ensemble average as indicated in Equation (25) 

can be written in terms of the corresponding characteristic functional 

 

exp χ L,

 ( ) ρ1 + χ L, ( ) ρ2 + i S1 L, ( ) ρ1 − S1 L, [ ] ( ) ( ) ρ1

= exp iq χ L,

 ( ) ρ1 + χ L, ( ) ρ2 + i S1 L, ( ) ρ1 − S1 L, [ ] { } ( ) ( ) ρ1

q=−i

 (26) 

At this point, it is convenient for notational purposes to let 

χ L,

 ( ) ρ1 ≡ χ( ) 1 , etc., and to write 

χ(1) = χ1(1) + χ2(1) where χ1(1) is the first order Rytov approximation for χ(1) and χ2(1) for the second 

order approximation of the quantity; similarly for S( ) 1 = S1( ) 1 + S2 ( ) 1 . Since, as mentioned above, 

quantities up to the second order in the fluctuations ε ̃ are only being considered, one, can perform a 

cumulant expansion of the right side of Equation (26) up to second order in the parameter q and obtain 

(Ref. 6), 

 

exp iq χ L,

( ) ρ1 + χ L,( ) ρ2 + i S1 L,( ) ρ1 − S1 L,[ ] { } ( ) ( ) ρ1

q=−i

= exp iqK1 − 1

2

q2K2 + ⎡

⎣

⎢ ⎤

⎦

⎥

q=−i

= exp K1 +

1

2

K2

⎡

⎣

⎢ ⎤

⎦

⎥

 (27) 

where the first cumulant K1 is given by 
NASA/TM—2011-216974 6 

K1 = χ1( ) 1 + χ2 ( ) 1 + χ1( ) 2 +χ2 ( ) 2 + i S{ } 1( ) 1 + S2 ( ) 1 − i S{ } 1( ) 2 + S2 ( ) 2 (28) 

and the second by 

K2 = (χ1( ) 1 + χ2 ( ) 1 + χ1( ) 2 +χ2 ( ) 2 + i S{ } 1( ) 1 + S2 ( ) 1 − i S{ } 1( ) 2 + S2 ( ) 2 )

2

− χ1( ) 1 + χ2 ( ) 1 + χ1( ) 2 +χ2 ( ) 2 + i S{ } 1( ) 1 + S2 ( ) 1 − i S{ } 1( ) 2 + S2 ( ) 2

2

 (29) 

From the discussion in Section 3.0, Equation (28) reduces to the simple result 

K1 = 2 χ2 (30) 

upon remembering that χ2 and S2 are independent of transverse position (as indicated by 

Equation (A13)). Expanding the square terms in Equation (29) and ignoring terms of order higher than 

ε ̃ 2

 gives 

K2 = ( ) χ( ) 1 + χ( ) 2 2+ 2i ( ) χ( ) 1 + χ( ) 2 ( ) S( ) 1 − S( ) 2 − ( ) S( ) 1 − S( ) 2 2 (31) 

Finally, substituting Equations (30) and (31) into Equation (27) allows ensemble average in Equation (25) 

to be written 

 

exp[ ]  = exp

1

2 ( ) χ( ) 1 + χ( ) 2 2

+ i ( ) χ( ) 1 + χ( ) 2 ( ) S( ) 1 − S( ) 2 − 1

2 ( ) S( ) 1 − S( ) 2 2

+ 2 χ ⎡

⎣

⎢ ⎤

⎦

⎥ (32) 

By definition, the phase structure function is 

DS ( ) 1,2 ≡ ( ) S( ) 1 − S( ) 2 2 (33) 

But 

( ) χ( ) 1 + χ( ) 2 2 = σχ

2 ( ) 1 + 2Bχ ( ) 1,2 + σχ2 ( ) 2 (34) 

found by expanding the square within the average. However, again by definition the log-amplitude 

structure function is 

Dχ ( ) 1,2 ≡ χ( ) ( ) 1 − χ( ) 2 2 = σχ

2 ( ) 1 − 2Bχ ( ) 1,2 + σχ2 ( ) 2 (35) 

so 

( ) χ( ) 1 + χ( ) 2 2 = 2σχ

2 ( ) 1 + 2σχ2 ( ) 2 − Dχ ( ) 1,2 (36) 

Finally, 

( ) χ( ) 1 + χ( ) 2 ( ) S( ) 1 − S( ) 2 = σχS ( ) 1 − BχS ( ) 1,2 + BχS ( ) 2,1 − σχS ( ) 2 (37) 
NASA/TM—2011-216974 7 

where it is noted that, in the general case, according to Equations (A9) and (A11), BχS ( ) 1,2 ≠ BχS ( ) 2,1

and σχS ( ) 1 ≠ σχS ( ) 2 . Using Equations (26), and (27) in Equation (32) gives 

 

exp[ ]  = exp − 1

2

DW ( ) 1,2 + σχ

2 ( ) 1 + σχ2 ( ) 2 + 2 χ + i{ } σχs( ) 1 − σχs( ) 2 − BχS ( ) 1,2 + BχS ( ) 2,1 ⎡

⎣

⎢ ⎤

⎦

⎥ (38) 

where DW ( ) 1,2 = Dχ ( ) 1,2 + DS ( ) 1,2 is the wave structure function. 

An interesting occurrence of a linear combination of the cross log-amplitude/phase correlations 

appears as an imaginary term in Equation (38). Again, for the general beam wave case, this term does not 

vanish. Using Equations (A9) and (A11), one has for this linear combination 

ΔχS ≡ σχs( ) 1 − σχs( ) 2 − BχS ( ) 1,2 + BχS ( ) 2,1

= −( ) 2π

2 k 2

8

⎛

⎝

⎜ ⎞

⎠

⎟

0

L

∫ Im{ } J0 ( ) κP( ) 2,1 − J0 ( ) κP( ) 1,2

0

∞

∫ ⋅ exp − κ2

k ( ) L − x γ I

⎡

⎣

⎢ ⎤

⎦

⎥Φε ( ) κ κdκdx (39) 

where 

P i( ) , j ≡ γρi − γ*ρj . This statistical parameter as well as all the other relevant parameters are 

evaluated in Appendix B for the Kolmogorov spectrum of turbulent fluctuations. Substituting these 

results into Equation (38) gives 

 

exp[ ]  = exp −4.352k 2Cn

2 L − x

k γ I

⎛

⎝

⎜ ⎞

⎠

⎟

5/6

Re 1F1 − 5

6

,1;− kP12

2

4( ) L − x γ I

⎛

⎝

⎜

⎞

⎠

⎟

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

⎛

⎝

⎜

⎞

⎠

⎟

0

L

∫ ⎡

⎣

⎢

⎢ dx

+i( ) 2.176 k 2Cn

2 L − x

k γ I

⎛

⎝

⎜ ⎞

⎠

⎟

5/6

Im 1F1 − 5

6

,1;− kP21

2

4( ) L − x γ I

⎛

⎝

⎜

⎞

⎠

⎟

⎧

⎨

⎪

⎩

⎪ 0

L

∫

−1F1 − 5

6

;1;− kP12

2

4( ) L − x γ I

⎛

⎝

⎜

⎞

⎠

⎟

⎫

⎬

⎪

⎭

⎪

dx

⎤

⎦

⎥

⎥

 (40) 

Equation (40) can be simplified considerably. To this end, one notes that 

P2 ( ) 1,2 ≡ P12

2 = P212* (41) 

In addition to this, the integral representation of the confluent hypergeometric function 

1F1( ) a,b;z = 21−b exp( ) z 2

B a( ) ,b− a

( ) 1− t

b−a−1

( ) 1+ t a−1

exp( ) zt 2 dt

−1

1

∫ (42) 

yields the fact that 

1F1 a,b;z* ( )=1F1

*( ) a,b;z (43) 

for real a and b. Adopting the notation 

1F1 P12

2 ( )≡1F1 − 5

6

,1;− kP12

2

4γ I ( ) L − x

⎛

⎝

⎜

⎞

⎠

⎟
NASA/TM—2011-216974 8 

one has from the above considerations, 

1F1 P12

2 ( )=1F1

* P21

2 ( ) (44) 

Thus, it is found that (using Im{ }z = ( ) 1 2i z − z* [ ]) 

Im 1F1 P21

2 ( )−1F1 P122 { } ( ) = 1

i 1F1

* P12

2 ( )−1F1 P122 [ ] ( ) (45) 

and 

Re 1F1 P12

2 { } ( ) = 1

2 1F1 P12

2 ( )+1F1

* P12

2 [ ] ( ) (46) 

Substituting Equations (45) and (46) into Equation (40) finally yields the simplified result 

 

exp[ ]  = exp −4.352k 2Cn

2 γ I ( ) x

L − x

k

⎛

⎝

⎜ ⎞

⎠

⎟

0

L

∫

5 6

1F1 − 5

6

, 1;− kP21

2

4γ I ( ) x ( ) L − x

⎛

⎝

⎜

⎞

⎠

⎟ dx

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

 (47) 

As required by Equation (25), one now needs to consider the product of the initial fields of a beam 

wave; using Equation (A1), one has 

 

U0 L,

 ( ) ρ1 U0

* L,

 ( ) ρ2 = W0

2

W 2 exp − k

2

W0

2

W 2

⎛

⎝

⎜

⎞

⎠

⎟ α1 ρ1

2 + ρ22 { ( ) + i α2 − α12 + α22 [ ] ( )L ρ12 − ρ22 ( )} ⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

 (48) 

where W 2 = W0

2 ( ) 1− α2L 2

+ α1

2L2 [ ] is the beam radius at a distance L from the output aperture. Finally, 

using equations (47) and (48) in Equation (25) gives for the MCF for a beam wave, to within the second 

Rytov approximation, propagating through Kolmogorov turbulence 

 

Γ2 L,



ρ1,

 ( ) ρ2 = W0

2

W 2 exp − k

2

W0

2

W 2

⎛

⎝

⎜

⎞

⎠

⎟ α1 ρ1

2 + ρ22 { ( ) + i α2 − α12 + α22 [ ] ( )L ρ12 − ρ22 ( )} ⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

⋅exp −4.352k 2Cn

2 γ I ( ) x

L − x

k

⎛

⎝

⎜ ⎞

⎠

⎟

0

L

∫

5 6

1F1 − 5

6

,1;− kP21

2

4γ I ( ) x ( ) L − x

⎛

⎝

⎜

⎞

⎠

⎟ dx

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

 (49) 

To make contact with other formulations in what is to follow, it is advantageous to place this result in 

terms of the sum and difference coordinates of 



ρ1 and 



ρ2 , viz., 

 



ρc ≡



ρ1 + ρ2

2 , 

ρd ≡ ρ2 − ρ1 (50) 

Solving these for 



ρ1 and 



ρ2 and substituting into Equation (49) and using the definition for P21

2 gives for 

the beam wave MCF to within the second order Rytov approximation 
NASA/TM—2011-216974 9 

 

Γ2 L,



ρc ,

 ( ) ρd = W0

2

W 2 exp − k2

W0

2

W 2

⎛

⎝

⎜

⎞

⎠

⎟ 2α1 ρc

2 + ρd

2

4

⎛

⎝

⎜

⎞

⎠

⎟

⎧

⎨

⎪

⎩

⎪ − 2i α2 − α1

2 + α22 [ ] ( )L 

ρc ⋅



ρ ( ) d } ⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

⋅exp −4.352k 2Cn

2 γ I ( ) x

L − x

k

⎛

⎝

⎜ ⎞

⎠

⎟

0

L

∫

5 6

1F1 − 5

6

,1;− k γ R



ρd − 2iγ I



ρc

2

4γ I ( ) x ( ) L − x

⎛

⎝

⎜

⎞

⎠

⎟ dx

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

 (51) 

5.0 Relationship of the Rytov MCF With That of Strong Fluctuation 

Theory 

It will now be demonstrated that the result of Equation (51) agrees with the weak fluctuation 

approximation of the similar result obtained using the parabolic equation method of strong fluctuation 

theory. First, however, the plane and spherical wave cases will be considered. In the plane wave case one 

has W0 → ∞ and R0 → ∞. Thus, α1 = 0, α2 = 0, γ1(η) = 1 and γ2(η) → 0. In this limit, the asymptotic 

expansion of the confluent hypergeometric function yields 

lim

γ 2→0

1F1 − 5

6

,1; −kρd

2

4γ 2 ( ) η ( ) L − η

⎛

⎝

⎜

⎞

⎠

⎟ = 1

Γ( ) 11 6

kρd

2

4γ 2 ( ) η ( ) L − η

⎛

⎝

⎜

⎞

⎠

⎟

5 6

 (52) 

Using this expression in Equation (51) gives for a plane wave 

 

Γ2,pw



ρc ,



ρd ( ) ,L = Γ2,pw



ρd ( ) ,L = exp −1.457k 2Cn

2ρd

5 3 [ ] L (53) 

The spherical wave case is defined by W0 → 0 and R0 → ∞. Hence, α1 → ∞, α2 = 0, γ1(η) = η/L and 

γ2(η) → 0. The same asymptotic form of Equation (52) applies and Equation (51) in this case gives 

 

Γ2,sw



ρc ,



ρd ( ) ,L = kW0

2

2L

⎛

⎝

⎜

⎞

⎠

⎟

2

exp − ik

L



ρc ⋅



ρd

⎡

⎣

⎢ ⎤

⎦

⎥ exp −1.457k 2Cn

2ρd

5/3 ( ) η L 5 3 dη

0

L

∫ ⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

= kW0

2

2L

⎛

⎝

⎜

⎞

⎠

⎟

2

exp − ik

L



ρc ⋅



ρd

⎡

⎣

⎢ ⎤

⎦

⎥ exp −0.546k 2Cn

2ρd

5/3 [ ] L

 (54) 

Considering the case in which ρc = 0 (i.e., the MCF about the propagation axis) and normalizing this 

result with respect to Γ2,sw(0,0,L) gives 

 

Γ2,sw,norm 0, ( ) ρd ,L = exp −0.546k 2Cn

2ρd

5/3 [ ] L (55) 

Thus, Equation (51) gives the same results as does strong fluctuation theory in these limits. 

Now for the comparison to the general beam wave case. Consider the general solution for the beam 

wave MCF as calculated within strong-fluctuation theory, given by Equation (20) to (73) of (Ref. 7), i.e., 

 

Γ2



ρc ,

 ( ) ρd ,L = W0

2

8π

exp −aρd

2 − bκd2 [ −∞

∞

∫ + c



ρd ⋅



κd + i



ρc ⋅



κd − H ρd ,

 ( ) κd ,L ]d 2κd (56) 

where 
NASA/TM—2011-216974 10 

a ≡ 1

2W0

2 1+

α2

2

α1

2

⎛

⎝

⎜

⎞

⎠

⎟, b ≡ W 2

8 = W0

2

8 ( ) α1L 2

+ ( ) 1− α2L 2

[ ]

c ≡ 1

2

α1L − α2

α1

( ) 1− α2L ⎡

⎣

⎢ ⎤

⎦

⎥

 (57) 

 

H ρd ,

 ( ) κd ,L ≡ 2πk 2

0

L

∫ 1− exp −i



κ⋅ ρd +



κdη

k

⎛

⎝

⎜ ⎞

⎠

⎟ ⎧

⎨

⎩

⎫

⎬

⎭

⎤

⎦

⎥Φn



( ) κ

⎡

⎣

⎢

⎢ −∞

∞

∫ d 2κdη (58) 

Here, 

Φn



( ) κ is the spatial spectrum of refractive index fluctuations. (The definition of H has been 

slightly modified from that of Ishimaru to facilitate some calculations later in this development.) The 

problem now reduces to showing that Equation (56) reduces to Equation (51) in the limit of weak 

fluctuations. 

5.1 Average Intensity ( 



ρ 

d = 0) 

At the outset, it is instructive to consider the slightly simpler case of the average intensity of a beam 

wave given in strong fluctuation theory by 

 

I ρc ( ) ,L = Γ2



ρc ( ) ,0,L = W0

2

8π

exp −bκd

2 [ −∞

∞

∫ +i



ρc ⋅



κd − H 0,  ( ) κd ,L ]d 2κd (59) 

From Equation (58), one has for an isotropic turbulent spectral density 

 

H 

( ) κd ,L = 2πk 2

0

L

∫ 1− exp −i



κ⋅



κdη

k

⎛

⎝

⎜ ⎞

⎠

⎟ ⎧

⎨

⎩

⎫

⎬

⎭

⎤

⎦

⎥Φn



( ) κ

⎡

⎣

⎢

⎢ −∞

∞

∫ d 2κdη

= ( ) 2π

2

k 2

0

L

∫ 1− J0 κ

κdη

k

⎛

⎝

⎜ ⎞

⎠

⎟

⎤

⎦

⎥Φn ( ) κ κ

⎡

⎣

⎢

0

∞

∫ dκdη

 (60) 

Using the Kolmogorov spectral density for refractive index fluctuations 

Φn ( ) κ = 0.033Cn

2κ−11/3 (61) 

in Equation (60) and evaluating the resulting the κ-integral using analytic continuation yields 

H κd ( ) ,L = 0.547k1/3Cn

2L8/3κd

5/3 (62) 

Since, in the isotropic case, 

H κd ( ) ,L = H κd ( ) ,L , Equation (59) becomes in plane polar coordinates 

 

I ρc ( ) ,L = W0

2

8π

0

2π

∫ exp −bκd

2 [

0

∞

∫ +iρcκd cosθ − H( ) 0,κd ,L ]dθ κd dκd

= W0

2

4 exp −bκd

2 − H κd [ ] ( ) ,L

0

∞

∫ J0 ( ) κdρc κd dκd

 (63) 
NASA/TM—2011-216974 11 

The use of Equation (62) in Equation (63) results in an integral that cannot be analytically evaluated. 

However, in the case of weak fluctuations, i.e., H κd ( ) ,L <<1, the exponential in Equation (63) can be 

expanded to yield 

 

I ρc ( ) ,L = W0

2

4 exp −bκd

2 [ ]

0

∞

∫ ( ) 1− H( ) κd ,L + J0 ( ) κdρc κd dκd (64) 

Using Equation (62) in this result allows for an analytic evaluation and gives 

 

I ρc ( ) ,L = W0

2

8

1

b exp − ρc

2

4b

⎛

⎝

⎜ ⎞

⎠

⎟

⎧

⎨

⎩

−0.547 Γ 11

6

⎛

⎝

⎜ ⎞

⎠

⎟ k1/3Cn

2L8/3 1

b

⎛

⎝

⎜ ⎞

⎠

⎟

5/6

1F1

11

6 ,1;− ρc

2

4b

⎛

⎝

⎜ ⎞

⎠

⎟ +

⎫

⎬

⎪

⎭

⎪

 (65) 

Finally, applying the Kummer transformation 

1F1

11

6 ,1;− ρc

2

4b

⎛

⎝

⎜ ⎞

⎠

⎟ = exp − ρc

2

4b

⎛

⎝

⎜ ⎞

⎠

⎟1F1 − 5

6

,1; ρc

2

4b

⎛

⎝

⎜ ⎞

⎠

⎟ (66) 

and using the definition of b from Equation (57) in Equation (65) results in the average intensity in 

strong-fluctuation theory in the weak-fluctuation limit 

 

I ρc ( ) ,L = W0

2

W 2 exp − 2ρc

2

W 2

⎛

⎝

⎜ ⎞

⎠

⎟ 1− 2.91k1/3Cn

2L8/3 1

W

⎛

⎝

⎜ ⎞

⎠

⎟

5/3

1F1 − 5

6

,1; 2ρc

2

W 2

⎛

⎝

⎜ ⎞

⎠

⎟ + ⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

 (67) 

This result should now be compared with that of Equation (51) from Rytov theory for 



ρd = 0; in this 

instance, Equation (51) becomes 

 

I ρc ( ) ,L = W0

2

W 2 exp − 2ρc

2

W 2

⎛

⎝

⎜ ⎞

⎠

⎟

⋅exp −4.352k 2Cn

2 γ 2 ( ) η

L − η

k

⎛

⎝

⎜ ⎞

⎠

⎟

0

L

∫

5 6

1F1 − 5

6

,1;

kγ 2 ( ) η ρc

2

( ) L − η

⎛

⎝

⎜

⎞

⎠

⎟ dη

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

 (68) 

Using the definitions of γ2(η) and α1, the argument of the confluent hypergeometric function gives 

kγ 2 ( ) η ρc

2

( ) L − η = 2ρc

2

W0

2 α12L2 + ( ) 1− α2L 2

[ ] = 2ρc

2

W 2 (69) 

where the last result comes from using a relationship in Equation (57). Thus, this function becomes 

independent of η. Equation (68) is then reduced to 
NASA/TM—2011-216974 12 

 

I ρc ( ) ,L = W0

2

W 2 exp − 2ρc

2

W 2

⎛

⎝

⎜ ⎞

⎠

⎟

⋅exp −4.352k 2Cn

2

1F1 − 5

6

,1; 2ρc

2

W 2

⎛

⎝

⎜ ⎞

⎠

⎟ γ 2 ( ) η

L − η

k

⎛

⎝

⎜ ⎞

⎠

⎟

0

L

∫

5 6

dη

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

 (70) 

within which the η-integration can now be performed upon using, once again, the definitions of γ2(η) and 

α1 as well as Equation (57). This gives 

γ 2 ( ) η

L − η

k

⎛

⎝

⎜ ⎞

⎠

⎟

0

L

∫

5 6

dη = 25/6 3

8

⎛

⎝

⎜ ⎞

⎠

⎟ k−5/3L8/3 1

W

⎛

⎝

⎜ ⎞

⎠

⎟

5/3

 (71) 

Substituting this into Equation (70) yields the average intensity in the weak-fluctuation limit of Rytov 

theory 

 

I ρc ( ) ,L = W0

2

W 2 exp − 2ρc

2

W 2

⎛

⎝

⎜ ⎞

⎠

⎟ exp −2.91k1/3Cn

2L8/3 1

W

⎛

⎝

⎜ ⎞

⎠

⎟

5/3

1F1 − 5

6

,1; 2ρc

2

W 2

⎛

⎝

⎜ ⎞

⎠

⎟

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

 (72) 

the series expansion of which agrees with the result of Equation (67) (this fact was first noted in (Ref. 4)). 

Hence, the average intensity of a beam wave as calculated from strong fluctuation theory agrees with that 

of Rytov theory in the limit of weak fluctuations. 

A digression must now be made with regard to the limits of applicability of the first Rytov 

approximation before graphical depictions of the comparison of general results is given. As derived in 

(Ref. 4), the requirement that the second Rytov approximation can be neglected for correlation functions 

of log-amplitude and phase fluctuations is given by the condition χ <<1; for a general beam wave this 

is 

χ =1.187Cn

2k 7/6L11/6 Re ( )i 5/6

2F1 − 5

6

,

11

6 ;

17

6 ;

iαL

1+ iαL

⎛

⎝

⎜ ⎞

⎠

⎟ ⎧

⎨

⎩

⎫

⎬

⎭

<<1 (73) 

Hence, the limits of applicability are seen to be functions of the parameters of the particular beam wave 

being considered. Take for example, the propagation conditions given by L = 2500 m, Cn

2 = 5×10−15 m2/3, 

and λ = 0.63×10–6 m. In the case of a collimated beam wave (R0 → ∞), the behavior of Equation (73) as a 

function of W0 is displayed in Figure 1. The plot begins at the spherical wave limit and asymptotically 

reaches the plane wave limit. The function χ peaks at the value of W0 ~ Lλ = .04 m, i.e., that size of 

the first Fresnel zone. As can be seen, the condition of Equation (73) qualitatively breaks down around this 

value. 

Graphical depictions and comparisons of the general results of Equation (51), with ρd = 0, and 

Equation (63) are shown in Figures 2 to 5. Here, the normalized intensity of the collimated beam wave 

cases are shown for initial waist radii W0 of 0.0005 m, 0.005 m, 0.05 m, and 0.5 m; the first value is 

essentially for a spherical wave case and the last value effectively represents the plane wave case. In all 

cases, the following propagation parameters prevail: L = 2500 m, λ = 0.63 μ, and Cn

2 = 5×10−15 m-2/3. As 

can be seen from Figure 3, the results of Rytov theory begin to diverge at ρc = 0.2 m. This phenomenon is 

at its extreme in Figure 4. The results once again coalesce in Figure 5. 
NASA/TM—2011-216974 13 

Figure 1.—Average log-amplitude for a collimated beam wave. 

Figure 2.—Normalized intensity for a collimated beam wave of waist size=0.0005 m. 

W0 (meters)

L=2500 m

λ=0.63 μm

Cn

2=5.0x10-15 m-2/3

Collimated Beam Wave

χ

ρd (meters)

Γ2(L,0,ρd)/ 

Γ2(L,0,0)

L=2500 m

λ=0.63 μm

Cn

2=5.0x10-15 m-2/3

Collimated Beam Wave

 W0=0.0005 m

Rytov Theory

Strong Fluctuation 

Theory

χ = 0.155
NASA/TM—2011-216974 14 

Figure 3.—Normalized intensity for a collimated beam wave of waist size=0.005 m. 

Figure 4.—Normalized intensity for a collimated beam wave of waist size=0.05 m. 

ρd (meters)

Γ2(L,0,ρd)/ 

Γ2(L,0,0)

L=2500 m

λ=0.63 μm

Cn

2=5.0x10-15 m-2/3

Collimated Beam Wave

 W0=0.005 m

Rytov Theory

Strong Fluctuation 

Theory

χ = 0.206

ρd (meters)

Γ2(L,0,ρd)/ 

Γ2(L,0,0)

L=2500 m

λ=0.63 μm

Cn

2=5.0x10-15 m-2/3

Collimated Beam Wave

 W0=0.05 m

Rytov Theory

Strong Fluctuation 

Theory

χ = 0.521
NASA/TM—2011-216974 15 

Figure 5.—Normalized intensity for a collimated beam wave of waist size=0.5 m. 

Since the fundamental foundation of Rytov theory and its various approximations are based on the 

parabolic equation, Equation (5), such intensity divergences cannot arise from this fact since they are not 

observed within strong fluctuation theory, which is also based on the parabolic equation. As will be 

shown in Section 5.2, stable results are obtained for the MCF across all ranges of beam wave waist radii. 

In addition, since palatable results for the intensity are obtained for the limiting cases of plane and 

spherical waves, it is thus doubtful that such divergence phenomena are occurring due to the neglect or 

mistreatment of some propagation related mechanism or use of the paraxial approximation. Therefore, the 

beam structure becomes suspect. But again, such is not the case in strong fluctuation theory so whatever 

the source of the intensity divergences are, it is only peculiar to the way the beam wave structure is 

treated within Rytov theory. The study and identification of this intensity divergence behavior will form 

the subject of Section 6.0. 

One can now proceed to examine the complete expression for the MCF as given by Equation (56) and 

compare it to that of the Rytov theory, Equation (51). 

5.2 MCF 

From what has been gleaned from the examination of the intensity, one should consider the weak 

fluctuation limit of Equation (56) from the outset, i.e., 

 

Γ2



ρc ,

( ) ρd ,L = W0

2

8π

exp −aρd

2 − bκd2 [ −∞

∞

∫ + c



ρd ⋅



κd + i



ρc ⋅



κd ]

⋅ 1− H { } ( ) ρd ,κd ,L + d 2κd

= Γ2

(1) 

ρc ,



ρd ( ) ,L − Γ2

(2 ) 

ρc ,



ρd ( ) ,L +

 (74) 

ρd (meters)

Γ2(L,0,ρd)/ 

Γ2(L,0,0)

L=2500 m

λ=0.63 μm

Cn

2=5.0x10-15 m-2/3

Collimated Beam Wave

 W0=0.5 m

Rytov Theory

Strong Fluctuation 

Theory

χ = 0.382
NASA/TM—2011-216974 16 

where 

 

Γ2

(1) 

ρc ,



ρd ( ) ,L ≡ W0

2

8π

exp −aρd

2 − bκd2 [ −∞

∞

∫ + c



ρd ⋅



κd + i



ρc ⋅



κd ]d 2κd (75) 

and 

 

Γ2

(2 ) 

ρc ,

 ( ) ρd ,L ≡ W0

2

8π

exp −aρd

2 − bκd2 [ −∞

∞

∫ + c



ρd ⋅



κd + i



ρc ⋅



κd ]H ρd ,

 ( ) κd ,L d 2κd (76) 

The evaluation of Equation (75) is straightforward: 

 

Γ2

(1) 

ρc ,



ρd ( ) ,L = W0

2

8π

exp −a

 ( ) ρd exp i 

( ) ρc − icρd ⋅

 [ ] κd

−∞

∞

∫ exp −bκd

2 ( )d 2κd \

= W0

2

4 exp −aρd

2 ( ) J0



ρc − ic ( ) ρd κd

0

∞

∫ exp −bκd

2 ( )κd dκd

= W0

2

4

⎛

⎝

⎜

⎞

⎠

⎟

1

2b

⎛

⎝

⎜ ⎞

⎠

⎟ exp −



ρc − icρd

2

+ 4abρd

2

4b

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

 (77) 

Equation (76) becomes, with the use of Equation (58), 

 

Γ2

(2 ) 

ρc ,

 ( ) ρd ,L ≡ W0

2

8π

2πk 2 ( )−∞

∞

∫

0

L

∫ exp −aρd

2 − bκd2 [ −∞

∞

∫ + c



ρd ⋅



κd + i



ρc ⋅



κd ]

⋅ 1− exp −i



κ⋅ ρd +



κdη

k

⎛

⎝

⎜ ⎞

⎠

⎟ ⎧

⎨

⎩

⎫

⎬

⎭

⎤

⎦

⎥Φn



( ) κ

⎡

⎣

⎢

⎢

d 2κdηd 2κd

 (78) 

Here, the κd -integration will be performed first followed by the κ-integration. To this end, one has 

 

Γ2

2 

ρc ,

 ( ) ρd ,L = W0

2

4

k 2

−∞

∞

∫ { } T1 + T2

0

L

∫ Φn



( ) κ dηd 2κ (79) 

where 

 

T1 ≡ exp −aρd

2 − bκd2 + i 

ρc − ic ( ) ρd ⋅

 [ ] κd

−∞

∞

∫ d 2κd (80) 

and 

 

T2 ≡ exp −aρd

2 − bκd2 + i 

ρc − ic ( ) ρd ⋅

 [ ] κd

−∞

∞

∫ exp −i



κ⋅



ρd − i



κη

k



κd

⎡

⎣

⎢ ⎤

⎦

⎥d 2κd (81) 

The evaluation of Equation (80) follows that of Equation (77): 
NASA/TM—2011-216974 17 

 

T1 = 2π

1

2b

⎛

⎝

⎜ ⎞

⎠

⎟ exp −



ρc − icρd

2

+ 4abρd

2

4b

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

 (82) 

Similarly, Equation (81) becomes 

 

T2 = 2π

1

2b

⎛

⎝

⎜ ⎞

⎠

⎟ exp −aρd

2 − i



κ⋅

 ( ) ρd exp −



ρc − icρd −



κη

k

2

4b

⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

= 2π

1

2b

⎛

⎝

⎜ ⎞

⎠

⎟ exp −



ρc − icρd

2

+ 4abρd

2

4b

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

exp −i



S ⋅



κ − η2

4bk 2 κ2 ⎡

⎣

⎢ ⎤

⎦

⎥

 (83) 

where 

 



S ≡

−2i

η

k



ρc − ic ( ) ρd + 4b



ρd

4b (84) 

Substituting Equations (82) and (83) into Equation (79) gives 

 

Γ2

2 

ρc ,

 ( ) ρd ,L = W0

2

4

1

2b

⎛

⎝

⎜ ⎞

⎠

⎟2πk 2 exp −



ρc − icρd

2

+ 4abρd

2

4b

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

⋅

0

L

∫ −∞

∞

∫ 1− exp −i



S ⋅



κ − η2

4bk 2 κ2 ⎛

⎝

⎜ ⎞

⎠

⎟

⎡

⎣

⎢ ⎤

⎦

⎥Φn



( ) κ d 2κdη

 (85) 

Using the isotropic Kolmogorov spectral density given by Equation (61) yields for the κ-integration 

 

−∞

∞

∫ 1− exp −i



S ⋅



κ − η2

4bk 2 κ2 ⎛

⎝

⎜ ⎞

⎠

⎟

⎡

⎣

⎢ ⎤

⎦

⎥Φn



( ) κ d 2κ = 0.033 2( ) π Cn

2

⋅ 1− J0



( ) S κ exp − η2

4bk 2 κ2 ⎛

⎝

⎜ ⎞

⎠

⎟

⎡

⎣

⎢ ⎤

⎦

⎥

0

∞

∫ κ−8/3dκ

= ( ) 2π ( ) 0.033 Cn

2 1

2

⎛

⎝

⎜ ⎞

⎠

⎟ −Γ − 5

6

⎛

⎝

⎜ ⎞

⎠

⎟ ⎛

⎝

⎜ ⎞

⎠

⎟ η2

4bk 2

⎛

⎝

⎜ ⎞

⎠

⎟

5/6

1F1 − 5

6

,1;



S 

2

k 2b

η2

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟

 (86) 

where analytic continuation was used to obtain the final result. Using this in Equation (85) and 

substituting Equations (77) and (85) into Equation (74) finally yields 
NASA/TM—2011-216974 18 

 

Γ2



ρc ,



ρd ( ) ,L = W0

2

4

1

2b

⎛

⎝

⎜ ⎞

⎠

⎟ exp −



ρc − icρd

2

+ 4abρd

2

4b

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

⋅ 1− 4.351k 2Cn

2

0

L

∫ η2

4bk 2

⎛

⎝

⎜ ⎞

⎠

⎟

5/6

1F1 − 5

6

,1;



S 

2

k 2b

η2

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟

dη+

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟

 (87) 

Using the definitions for a, b, c, and 



S , and employing the transcription η → L − η, one obtains, after 

some rather involved algebra, 

 

Γ2



ρc ,



ρd ( ) ,L = 1

( ) α1L 2

+ ( ) 1− α2

2

⎛

⎝

⎜

⎞

⎠

⎟ exp −

α1 ρc

2 + ρd

2

4

⎛

⎝

⎜

⎞

⎠

⎟ + i α2 − α1

2 + α22 ( ) ( )L 

ρc ⋅



ρd

( ) α1L 2

+ ( ) 1− α2

2

⎡

⎣

⎢

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

⎥

⋅ 1− 4.351k 2Cn

2

0

L

∫ −γ 2 ( ) η

L − η

k

⎛

⎝

⎜ ⎞

⎠

⎟

5/6

1F1 − 5

6

,1;

k γ1



ρd − 2iγ 2



ρc

4γ 2 ( ) L − η

⎛

⎝

⎜

⎞

⎠

⎟ dη+

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟

 (88) 

The first two factors of this expression describe the overall beam wave structure; what is important to 

note here, however, is that the expansion given by the third term agrees with the corresponding weak 

fluctuation expansion of Equation (51). 

Graphs of the normalized MCF’s of the entire expression of Equation (51) with that of Equation (56) 

are shown in Figures 6 to 8 for a collimated beam wave subtending the spherical to plane wave limits. 

The largest discrepancy between these results occurs at W0 = 0.05 m; however, the Rytov results do not 

diverge as they do for the associated intensity of Figure 4. 

Figure 6.—Normalized MCF for a collimated beam wave of waist size=0.005 m. 

ρd (meters)

Γ2(L,0,ρd)/ 

Γ2(L,0,0)

L=5000 m

λ=0.63 μm

Cn

2=1.0x10-14 m-2/3

Collimated Beam Wave

 W0=0.005 m

Rytov Theory

Strong Fluctuation 

Theory

χ = 0.155
NASA/TM—2011-216974 19 

Figure 7.—Normalized MCF for a collimated beam wave of waist size=0.05 m. 

Figure 8.—Normalized MCF for a collimated beam wave of waist size=0.5 m. 

ρd (meters)

Γ2(L,0,ρd)/ 

Γ2(L,0,0)

L=5000 m

λ=0.63 μm

Cn

2=1.0x10-14 m-2/3

Collimated Beam Wave

 W0=0.05 m

Rytov Theory

Strong Fluctuation 

Theory

χ = 0.521

ρd (meters)

Γ2(L,0,ρd)/ 

Γ2(L,0,0)

L=5000 m

λ=0.63 μm

Cn

2=1.0x10-14 m-2/3

Collimated Beam Wave

 W0=0.5 m

Rytov Theory

Strong Fluctuation 

Theory

χ = 0.382
NASA/TM—2011-216974 20 

6.0 Limits of Beam Wave Parameters Imposed by Their Use in the Rytov 

Approximation

As shown in the last section, the calculation within the first and second Rytov approximations of the 

intensity of a beam wave propagating through turbulence revealed a divergent instability in the 

predictions for beam waves of waist radius on the order of the prevailing Fresnel zone. The MCF, 

however, remained stable in this situation. It thus becomes important to identify the source of this 

divergence for such beam waves. 

As shown in previous treatments, all orders of approximation of Rytov theory possess the 

combination of initial field distributions 

 

R L,



ρ; x,

 ( ) ρ′ ≡ U0 x,

 ( ) ρ′

U0 L,

 ( ) ρ

 (89) 

where for a Gaussian beam wave, one has Equations (A1) and (A2), viz. 

 

U0 x,

 ( ) ρ′ = 1

1+ iαx

exp − kα

2

⎛

⎝

⎜ ⎞

⎠

⎟ ρ′2

1+ iαx

⎡

⎣

⎢ ⎤

⎦

⎥ (90) 

with 

α≡α1 + iα2 , α1 ≡ 2

kW0

2 , α2 ≡ 1

R0

 (91) 

For simplicity, and without loss of generality, consider the collimated beam wave case in which R0 → ∞, 

i.e., α2 = 0. Substituting Equation (90) into Equation (89) gives 

 

R L,



ρ; x,

 ( ) ρ′ = 1+ iα1L

1+ iα1x

exp − kα1

2

ρ′2

1+ iα1x

− ρ2

1+ iα1L

⎧

⎨

⎩

⎫

⎬

⎭

⎡

⎣

⎢ ⎤

⎦

⎥

= 1+ iα1L

1+ iα1x

exp − k

2

ρ′2

1

α1

+ ix − ρ2

1

α1

+ iL

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

= 1+ iα1L

1+ iα1x

exp − k

2

ρ′2 1

α1

− ix ⎛

⎝ ⎞⎠

1

α1

⎛

⎝ ⎞⎠

2

+ x 2

−

ρ2 1

α1

− iL ⎛

⎝ ⎞⎠

1

α1

⎛

⎝ ⎞⎠

2

+ L2

⎧

⎨

⎪⎪

⎩

⎪

⎪

⎫

⎬

⎪⎪

⎭

⎪

⎪

⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

= 1+ iα1L

1+ iα1x

exp A +

ik

2

S ⎡

⎣

⎢ ⎤

⎦

⎥

 (92) 

where 

A ≡ − k

2

1

α1

1

α1

⎛

⎝ ⎞⎠

2

+ x 2

ρ′2 −

1

α1

1

α1

⎛

⎝ ⎞⎠

2

+ L2

ρ2

⎡

⎣

⎢

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

⎥

 (93) 

is the amplitude factor describing the evolution of the wave amplitude and 
NASA/TM—2011-216974 21 

S ≡ x

1

α1

⎛

⎝ ⎞⎠

2

+ x 2

ρ′2 − L

1

α1

⎛

⎝ ⎞⎠

2

+ L2

ρ2 (94) 

is the phase factor. 

An important condition for the convergence of any integral expression in which Equation (92) finds 

itself is A < 0. Thus, consider the amplitude term within the exponential of Equation (92) more closely, 

i.e., 

A ≡ − k

2

K x( )ρ′2 +

k

2

K L( )ρ2 , K z( ) ≡

1

α1

⎛

⎝ ⎞⎠

1

α1

⎛

⎝ ⎞⎠

2

+ z 2

 (95) 

Look now at the first term of on the right side of Equation (95) and consider the two cases ( ) 1 α1

2

>> x 2

and ( ) 1 α1

2

<< x 2, i.e., using the definition of α1, W0 > λx and W0 < λx . 

6.1 Case (i) (1 α1 )

2

>> x2

Take the worst case in which x = L. Thus, K x( ) ~ K L( ) = α1 and one has for the first term on 

Equation (95) 

k

2

K x( )ρ′2 ~ k

2

K L( )ρ′2 = k

2

α1ρ′2 = ρ′2

W0

2 (96) 

For any integral over ρ′ in which the expression of Equation (92) occurs, most contributions over the 

integration range come from ρ′2 W0

2 ≤1. Taking the equality to hold in the worst case and using this 

condition in Equation (96) yields 

k

2

K x( )ρ′2 ~ k

2

K L( )ρ′2 ~ 1 (97) 

Substituting this intermediate result into Equation (95) gives for the condition of convergence of any 

integral involving Equation (92), i.e., A < 0, 

A = −1+

k

2

K L( )ρ2 < 0 (98) 

6.2 Case (ii) (1 α1 )

2

<< x2

Once again, taking x = L, one has K x( ) = ( ) 1 α1 L2 . Hence, for the first term on the right of 

Equation (95) 

k

2

K x( )ρ′2 ~ k

2

K L( )ρ′2 = k

2

1 α1

L2 ρ′2 = k 2W0

2

4L2 ρ′2 ~ W0

2

λL

⎛

⎝

⎜

⎞

⎠

⎟ ρ′2

λL

⎛

⎝

⎜ ⎞

⎠

⎟ (99) 
NASA/TM—2011-216974 22 

But by assumption, W0

2 << λL . However, in most propagation scenarios, ρ′ > λL in the significant 

range of integration over ρ′; there is thus an indeterminate size for the last term of Equation (99). At this 

point, without further analysis, it will be taken to be ~1. Therefore, once again the condition 

k

2

K x( )ρ′2 ~ k

2

K L( )ρ′2 ~ 1 (100) 

obtains and Equation (95) gives for a convergent integral condition 

A = −1+

k

2

K L( )ρ2 < 0 (101) 

which is identical to the result in Equation (98) of Case (i) (Sec. 6.1). 

Using the definitions given above, Equations (98) and (101) reduce to 

πα1

1+ α1

2L2 <1 (102) 

which can be further reduced to 

ρ2 W0

2

1+ λ2L2 π2W0

4 <1 (103) 

This finally can be simplified once more to yield the constraint 

ρ2 < W0

2 +

λ2L2

π2W0

2 (104) 

existing between the beam wave and propagation quantities. 

This condition holds for all types of collimated beam waves and can also be a good indicator for 

convergent and divergent beam waves. It gives the possible range of values of the transverse coordinate 

for which the intensity of a beam wave remains stable within Rytov theory. For plane (W0 → ∞) and 

spherical waves (W0 → 0), the condition holds trivially. The condition shows itself for nominal values of

W0. For example, as shown in Figure 3, a case in which the intensity divergence was noted to occur is 

given by W0 = 0.005 m, λ =0.63 μm and L = 2500 m. This gives through Equation (104) ρ < 0.1 m, which 

is just before the intensity divergence sets in. However, the case for which W0 = 0.05 m shown in 

Figure 4, i.e., right at the Fresnel zone length, gives ρ < 0.05 m; the intensity divergence occurred well 

before this value. The discrepancy is be due to the fact that an asymptotic analysis on either side of the 

quantity λL was used to derive Equation (104). One can expect a disagreement in situations in which 

W0 ~ λL . A more complete analysis than that given above is needed to cover such cases. Suffice it to 

say that the condition W0 ~ λL precludes the use of Rytov theory to describe the behavior of beam 

wave intensity. 

Thus the source of intensity divergences for beam waves within the confines of Rytov theory has 

been identified and an expression is given for the stable range of intensity predictions. The fact that the 

MCF remains stable over these transverse ranges is connected with the fact that the difference coordinate 

over which the MCF is evaluated is symmetric about the beam wave axis; the centroid coordinate is taken 

to be zero in these cases. Thus, the divergences that arise off axis are more or less canceled by performing 

the field comparison across the difference coordinates as is done to obtain the MCF. 
NASA/TM—2011-216974 23 

7.0 Conclusion 

A closed form solution for the MCF based on the first and second Rytov approximations has been 

derived and compared with that from strong fluctuation theory. The agreement between these two 

approaches is quite good in all beam wave cases that satisfy the weak fluctuation case save for the 

intensity distribution for a collimated beam wave. In the region where the initial beam waist size is on the 

order of the first Fresnel zone length, the solution in the Rytov case diverges. The source of this 

divergence is found to be in the way a beam wave is modeled within the Rytov approach; it arises from 

the ratio of the complex amplitudes that occurs within the theory.

NASA/TM—2011-216974 25 

Appendix A 

One has for a beam wave of waist radius W0 and a phase front radius of curvature R0 both measured 

at the output aperture of the transmitter, an initial field distribution of unit amplitude given by (Refs. 2 

and 3) 

 

U0



( ) r = U0 x,

 ( ) ρ = 1

1+ iαx

exp − kα

2

⎛

⎝

⎜ ⎞

⎠

⎟ ρ2

1+ iαx

⎡

⎣

⎢ ⎤

⎦

⎥ (A1) 

where 

α≡α1 + iα2 , α1 ≡ 2

kW0

2 , α2 ≡ 1

R0

 (A2) 

using Equations (9) and (11) with Equation (A1), one obtains the correlation expressions 

 

ψ1 L,

 ( ) ρ1 ψ1 L, ( ) ρ2 = − k 2

4

0

L

∫ exp i



κ⋅

 ( ) Q 

−∞

∞

∫ H2 ( ) L − x,κ Fε



( ) κ d 2κdx (A3) 

and 

 

ψ1 L,

 ( ) ρ1 ψ1

* L,

 ( ) ρ2 = k 2

4

0

L

∫ exp i



κ⋅

 ( ) P 

−∞

∞

∫ H L( ) − x,κ

2

Fε



( ) κ d 2κdx (A4) 

where the complex coordinates are 



Q ≡ γ ρ1 −  ( ) ρ2 and 



P ≡ γρ1 − γ*ρ2 with 

γ = γ( ) L, x ≡ 1+ iαx

1+ iαL = γ R − iγ I , γ I > 0

γ R ≡ (1− α2 x)(1− α2L) + α1

2Lx

(1− α2L)2 + α1

2L2 , γ I ≡ α1(L − x)

(1− α2L)2 + α1

2L2

 (A5) 

and 

H L( ) − x,κ ≡ exp − iκ2

2k ( ) L − x γ

⎡

⎣

⎢ ⎤

⎦

⎥. (A6) 

Finally, the two dimensional spectral amplitude of the permittivity fluctuations is given by 

 

Fε



( ) κ = 2πΦε



( ) κ where 

Φε



( ) κ is the associated three-dimensional spectral density. 

The solution to the second Rytov approximation yields (Ref. 4) 

 

ψ2 L,

 ( ) ρ = ψ2 ( ) L = −i

k

8

0

L

∫

0

x

∫ −∞

∞

∫ γ 2 ( ) x, x ′ H2 ( ) x − ′ x ,κ κ2Fε



( ) κ d 2κdx dx ′ (A7) 

a result that is independent of the transverse position. 

From Equations (21) to (23), one obtains in the case of an isotropic spatial spectrum of fluctuations 

(Ref. 3) 
NASA/TM—2011-216974 26 

 

Bχ

S

L,



ρ1,

( ) ρ2 = ( ) 2π

2 k 2

8

⎛

⎝

⎜ ⎞

⎠

⎟

0

L

∫

0

∞

∫ Re J0 ( ) κP exp − κ2

k ( ) L − x γ I

⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎩

∓J0 ( ) κQ exp − iκ2

k ( ) L − x γ

⎡

⎣

⎢ ⎤

⎦

⎥

⎫

⎬

⎭

Φε ( ) κ κdκdx

 (A8) 

and 

 

BχS L,



ρ1,

 ( ) ρ2 = −( ) 2π

2 k 2

8

⎛

⎝

⎜ ⎞

⎠

⎟

0

L

∫

0

∞

∫ Im J0 ( ) κP exp − κ2

k ( ) L − x γ I

⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎩

+J0 ( ) κQ exp − iκ2

k ( ) L − x γ

⎡

⎣

⎢ ⎤

⎦

⎥

⎫

⎬

⎭

Φε ( ) κ κdκdx

 (A9) 

In particular, 

 

σχ

S

2 L,

( ) ρ = ( ) 2π

2 k 2

8

⎛

⎝

⎜ ⎞

⎠

⎟

0

L

∫

0

∞

∫ Re J0 ( ) 2iγ Iκρ exp − κ2

k ( ) L − x γ I

⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎩

∓exp − iκ2

k ( ) L − x γ

⎡

⎣

⎢ ⎤

⎦

⎥

⎫

⎬

⎭

Φε ( ) κ κdκdx

 (A10) 

and 

 

σχS L,

 ( ) ρ = −( ) 2π

2 k 2

8

⎛

⎝

⎜ ⎞

⎠

⎟

0

L

∫

0

∞

∫ Im J0 ( ) 2iγ Iρκ exp − κ2

k ( ) L − x γ I

⎡

⎣

⎢ ⎤

⎦

⎥

⎧

⎨

⎩

+exp − iκ2

k ( ) L − x γ

⎡

⎣

⎢ ⎤

⎦

⎥

⎫

⎬

⎭

Φε ( ) κ κdκdx

 (A11) 

In addition, one has for the log-amplitude and phase structure functions 

 

Dχ

S

L,



ρ1,

( ) ρ2 = ( ) 2π

2 k 2

8

⎛

⎝

⎜ ⎞

⎠

⎟

0

L

∫

0

∞

∫ Re⎧

⎨

⎩ ( ) J0 ( ) 2iγ Iκρ1 + J0 ( ) 2iγ Iκρ2 − 2J0 ( ) κP

⋅exp − κ2

k ( ) L − x γ I

⎡

⎣

⎢ ⎤

⎦

⎥ ∓ 2 1( ) − J0 ( ) κQ exp − iκ2

k ( ) L − x γ

⎡

⎣

⎢ ⎤

⎦

⎥

⎫

⎬

⎭

Φε ( ) κ κdκdx

 (A12) 

Finally, for the mean log-amplitude and phase fluctuations (Ref. 4) 

χ( ) L

S L( )

⎫

⎬

⎪

⎭

⎪

= −( ) 2π

2 Re

Im

⎧

⎨

⎩

⎫

⎬

⎭

ik

8

⎛

⎝

⎜ ⎞

⎠

⎟

0

L

∫

0

x

∫

0

∞

∫ γ 2 ( ) x, x ′ exp − iκ2

k ( ) x − ′ x γ( ) x, x ′ ⎡

⎣

⎢ ⎤

⎦

⎥κ2 ⋅Φε ( ) κ κdκdx dx ′ (A13) 

In these expressions, γ I = − Im{ }γ > 0. Before an evaluation of these various equations is given for the 

Kolmogorov spectrum of turbulence, an expression for the MCF will be developed. 
NASA/TM—2011-216974 27 

Appendix B 

Using the Kolmogorov spectrum for permittivity fluctuations Φε ( ) κ = 0.033Cε

2κ−11/3 where, in terms 

of the refractive index structure constant Cε

2 = 4Cn

2, and evaluating the requisite functions needed in 

Equation (38), one obtains the following results from Equations (39), (A10), (A12), and (A13) 

 

DW ( ) 1,2 ≡ DW L,



ρ1,

 ( ) ρ2 = Dχ L,

ρ1,

 ( ) ρ2 + DS L,

ρ1,

 ( ) ρ2

= −0.816Cn

2k 7/6L11/6 α1L

( ) 1− α2L 2

+ α1

2L2

⎛

⎝

⎜

⎞

⎠

⎟

5/6

⋅ 1F1 − 5

6

,1;

2ρ1

2

W 2

⎛

⎝

⎜

⎞

⎠

⎟+1F1 − 5

6

,1;

2ρ2

2

W 2

⎛

⎝

⎜

⎞

⎠

⎟

⎛

⎝

⎜

⎞

⎠

⎟ + 4.352k 2Cn

2

0

L

∫ L − x

k

⎛

⎝

⎜ ⎞

⎠

⎟ γ I

⎛

⎝

⎜ ⎞

⎠

⎟

5/6

⋅Re 1F1 − 5

6

,1;− kP12

2

4( ) L − x γ I

⎛

⎝

⎜

⎞

⎠

⎟

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

⎞

⎠

⎟ dx

 (B1) 

σχ

2 ( ) L,ρ = 2.176k 7/6L11/6Cn

2 Re 6

11

⎛

⎝

⎜ ⎞

⎠

⎟i 5/6 2F1 − 5

6

,

11

6 ,

17

6 ;

iαL

1+ iαL

⎛

⎝

⎜ ⎞

⎠

⎟ ⎧

⎨

⎩

⎫

⎬

⎭

⎡

⎣

⎢

− 3

8

⎛

⎝

⎜ ⎞

⎠

⎟ α1L

( ) 1− α2L 2

+ α1

2L2

⎛

⎝

⎜

⎞

⎠

⎟

5/6

1F1 − 5

6

,1; 2ρ2

W 2

⎛

⎝

⎜ ⎞

⎠

⎟

⎤

⎦

⎥

⎥

⎥

 (B2) 

χ = −2.176k 7/6Cn

2L11/6 Re 6

11

⎛

⎝

⎜ ⎞

⎠

⎟i 5/6 2F1 − 5

6

,

11

6 ,

17

6 ;

iαL

1+ iαL

⎛

⎝

⎜ ⎞

⎠

⎟ ⎧

⎨

⎩

⎫

⎬

⎭

 (B3) 

ΔχS = 2.176k 2Cn

2

0

L

∫ ( ) L − x γ I

k

⎛

⎝

⎜ ⎞

⎠

⎟

5/6

Im 1F1 − 5

6

,1;− kP21

2

4( ) L − x γ I

⎛

⎝

⎜

⎞

⎠

⎟−1F1 − 5

6

,1;− kP12

2

4( ) L − x γ I

⎛

⎝

⎜

⎞

⎠

⎟

⎧

⎨

⎪

⎩

⎪

⎫

⎬

⎪

⎭

⎪

dx (B4) 
NASA/TM—2011-216974 28 

References 

1. V.I. Tatarskii, Wave Propagation in a Turbulent Medium, McGraw-Hill, New York, 1961. 

2. V.I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation, U.S. Dept. of 

Commerce, TT–68–50464, Springfield, VA, 1971. 

3. A. Ishimaru, “Fluctuations of a Beam Wave Propagating Through a Locally Homogeneous Medium,” 

Radio Sci. 4 (4), pp. 295–305 (1969). 

4. R.M. Manning, “Beam Wave Propagation within the Second Rytov Perturbation Approximation,” 

Radiophysics and Quantum Electron. 39 (4), pp. 287–295 (1996). 

5. R.M. Manning, “Theoretical Investigation of Millimeter Wave Propagation Through a Clear 

Atmosphere,” Proc. SPIE 410, Laser Beam Propagation in the Atmosphere, J.C. Leader, ed., 

pp. 119–136 (1983). 

6. A.D. Wheelon, “Skewed Distribution of Irradiance Predicted by the Second-Order Rytov 

Approximation,” JOSA A 18 (11), pp. 2789–2798 (2001). 

7. A. Ishimaru, Wave Propagation and Scattering in Random Media, Academic Press, New York, NY, 

Vol. 2, 1978. 
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The Atmospheric Mutual Coherence Function From the First and Second Rytov 

Approximations and Its Comparison to That of Strong Fluctuation Theory 

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6. AUTHOR(S)

Manning, Robert, M. 

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13. SUPPLEMENTARY NOTES

14. ABSTRACT

An expression for the mutual coherence function (MCF) of an electromagnetic beam wave propagating through atmospheric turbulence is 

derived within the confines of the Rytov approximation. It is shown that both the first and second Rytov approximations are required. The 

Rytov MCF is then compared to that which issues from the parabolic equation method of strong fluctuation theory. The agreement is found 

to be quite good in the weak fluctuation case. However, an instability is observed for the special case of beam wave intensities. The source 

of the instabilities is identified to be the characteristic way beam wave amplitudes are treated within the Rytov method. 

15. SUBJECT TERMS

Electromagnetic wave transmission; Wave propagation; Coherent scattering; Coherent radiation 

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