August 1966
1045
JOURNAL OF THE OPTICAL SOCIETY OF AMERICA
VOLUME 56, NUMBER 8
AUGUST 1966
Validity of the Rytov Approximation in Optical Propagation Calculations*
W. P. BROWN, JR.
Hughes Research Laboratories, Malibu, California 90265
(Received 13 January 1966)
The applicability of the Rytov approximation to the calculation of the characteristics of optical propagation in a weakly inhomogeneous random medium is investigated. The condition that the mean square value
of the second term in the associated perturbation expansion be smaller than that of the first is adopted as a
criterion for the validity of the Rytov approximation. It is shown that there is a very severe range limitation
on the validity of the Rytov approximation for optical propagation in the lower portions of the earth's
atmosphere. Furthermore, comparison of the limiting form for large xzkl of the validity condition derived
in this paper with the condition on the validity of the Born approximation obtained independently by
Mintzer, and by Kay and Silverman reveals that the Rytov and Born approximations have the same domain
of validity in this limiting case. The equivalence of the validity conditions for the Rytov and Born approximations contradicts the statements of Tatarski and Chernov who contend that the Rytov approximation
is superior to the Born approximation. It is conjectured that the Rytov and Born approximations have the
same domain of validity for all x/klP.
INDEx HEADINGS: Atmospheric optics.
I. INTRODUCTION
THE work reported in this paper was prompted by
Tthe
remarks of Hufnagel and Stanley' regarding
the validity of the Rytov approximation employed in
the work of Obukhov, 2 Chernov, 3 and Tatarski.A On the
basis of a qualitative argument which appears to be substantiated by experimental data, they conclude that the
validity of the Rytov approximation is limited in the
case of optical propagation in the atmosphere. At present there appears to be no general agreement among
those interested in optical propagation, concerning this
* Work supported by the U. S. Air Force Office of Scientific
Research, under Contract AF 49(638)-1439.
' R. E. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 52
(1964).
2 A. M. Obukhov, Izvestiya Akad. Nauk., Geophys. Ser. 2, 155
(1953) [translated by WV.C. Hoffman in Project RAND Report
T-47 (1955)].
3 L. A. Chernov, Wave Propagation in a Randomt Medium
(McGraw-Hill Book Co., Inc., New York, 1960).
4V.
I. Tatarski, Wave Propagation in a Turbulent Medium
(McGraw-Hill Book Co., Inc., New York, 1961).
question. The purpose of this paper is to corroborate
mathematically the qualitative conclusions of Hufnagel
and Stanley. The applicability of the Rytov approximation to the calculation of the characteristics of optical
propagation in a weakly inhomogeneous atmosphere is
investigated.
The point of departure for the calculations is the observation that the wave-fluctuation 'approximations
derived by the method of Rytov are simply the first
terms in perturbation expansions valid for sufficiently
weak atmospheric inhomogeneities. The condition that
the mean square value of the second term in the perturbation expansion be smaller than that of the first is
adopted as a criterion for the validity of these results.
Violation of this condition implies that the perturbation
expansion is not uniformly valid and thus that the approximation cannot be improved by including additional terms in the expansion. Experience with similar
problems suggests that the nonuniformity of the perturbation expansion occurs at large propagation distances. This prediction is verified by the calculations.
W. P. BROWN,
1046
JR.
Vol. 56
Since this discussion is concerned primarily with the
validity of the Rytov approximation as applied by
Chernov and Tatarski, we have attempted to follow
their analysis closely to facilitate a comparison. In
order to obtain a quantitative estimate of the mean
square value of the second term in the perturbation expansion, however, it is necessary to assume that refrac-
the imaginary part to the perturbation in the wavefront
(phase). Substitution of (2) in (1) yields the following
nonlinear equation for it'(r):
tive index of the atmosphere is a homogeneous isotropic
Rytov obtains an approximate solution of (3) by assuming that the terms V~q(r)* Vi,1(r) and k262n512(r) can be
neglected. According to Rytov (and those who have
adopted his approach), the legitimacy of this assumption
is guaranteed if the following conditions are satisfied:
gaussian random process. This assumption is more restrictive than that of Tatarski, who assumes only that
the refractive index is locally homogeneous and isotropic. There is good reason to believe, however, that
the limitation on the applicability of the Rytov approximation derived in this paper also applies for more general
random media. The nonuniformity of the perturbation
expansion is primarily a defect associated with the
method of approximation and, as such, should not depend critically on the details of the statistical characteristics of the medium.
V2Pj(r)+ V~p(r) *E2k 2,6o(r)+n7,Pj(r)0
+
2k 260(r)nj1(r) + *232n12(r)= 0.
5<<1i,
(3) \
(4)
I Vb I1[<<k.
(5)
The calculations given in this paper reveal that (4) and
(5) are not sufficient conditions to ensure the validity
of the approximation obtained when the abovementioned terms are neglected.
II. METHOD OF RYTOV-PERTURBATION
EXPANSION
B. Perturbation Expansion
A. Formulation of the Problem
Suppose that the change of the refractive index in the
time required for propagation is negligibly small. Then,
under the additional hypothesis that the change of the
index over a distance of one wavelength
Implicit in the approach of Chernov and Tatarski is
the assumption that Al1(r)possesses an asymptotic expansion in powers of the small parameter
is small, the
00
characteristics of electromagnetic waves propagating in
an inhomogeneous medium can be derived from the
solution of the reduced wave equation,
3:
#z(r)= B E 6j 'j(r).
(6)
5
[V 2 +k 2 n 2 (r)]U(r) = 0,
When (6) is substituted in (3) and the coefficients of the
(1)
where k is the wavenumber =27r/N, n(r) is the refractive index, and the harmonic time dependence eriwt
has been assumed. In the case of wave propagation
various powers of
tion of (1) is written in the form
U(r)= exp[#o(r)+#i(r),
2k24n(r)n 1 (r),
Xo(r) =
-
Xl(r) =
- k 2 nl12(r)
- Vtlo* VV'io,
(8)
(9)
In-1
X.(r)
=-
Z
v~Pj,(r)- vq/i,0 0 psj(r),
On, 2.
(10)
p==O
(2) To complete the specification of the slj(r), conditions
with Vo(r) chosen such that expE[o(r)] is the solution
(1) in the limit of zero S (i.e., no fluctuations).
(7)
where
sents the mean value of the refractive index, 3 is a di-
mensionless parameter much less than unity, and nl(r)
is a random function of position that accounts for the
fluctuations in refractive index.
In accordance with the method of Rytov, the solu-
are equated to zero, a recursive sys-
(V2 +2Vipo-V)tfrim(r)= X.(r),
in a
turbulent atmosphere, the refractive index is composed of the sum of two terms, n(r) =no(r)+bni(r). The
term no(r) is a sure function (nonrandom) which repre-
a
tem of partial differential equations is obtained for the
&ij(r).The rnth term in (6) satisfies the equation
It is
apparent that the real part of 4+(r)is related to the perturbation in the amplitude of U(r) for nonzero a and
I W. C. Hoffman, IRE Trans. Antennas Propagation Special
Suppl., S301 (1959).
must be assigned at the boundaries of the domain of inhomogeneity. Plane-wave propagation in the x direction
is considered and, in conformity with the assumptions
of Chernov and Tatarski, the tli(r) are required to remain bounded when x t-> Wc.
Equation (7) is solved for the case of propagation in a
medium characterized by a refractive index that has a
mean value of unity, no(r)= 1, and a normalized random
VALIDITY
August 1966
OF THE RYTOV APPROXIMATION
component given by
It is assumed thati/jr)
xl(r) = g(x)vi(r).
(11)
If
Vlj(r=
The quantity g(x) is a sure function of the coordinate
along the initial direction of propagation. The functional
form of g(x) is assumed to be such that the integrals
which appear in the solution of (7) remain bounded.
This factor is included in anticipation of the nonuniformity of (6) for propagation in an unbounded inhomogeneous medium. The function P1(r) is taken to be
a homogeneous isotropic random field; as such, it
has the two-dimensional Fourier-Stieltjes spectral
representation.6
vi(r) =
ff
exp[i. *rtd1dN(xx),
(16)
where dkj(vi,x) is a random process determined by the
equation that results when (16) is substituted in (7).
Under this hypothesis, the source function Xj(r) in (7)
can also be represented by a two-dimensional FourierStieltjes integral
Xj(r)=
f
exp[ii. rt]dXj(ux).
(17)
The quantity dXj(ti,x) is a random process determined,
in general, by dN(;,x) and a convolution of the
e^,,+
d*'rmcx), with m< j.
e zuz,
(13)
(14)
The quantity dN(c,x) is a random process that satisfies
the relation
(dN(Icx)dN*(,cx/))av = (ic-)/ic,
X-X' I )dndic', (15)
Substitution of (16) and (17) in (7) yields the
following differential equation for d4't(x,x) (note that
to(r)=ikx for the case of plane wave propagation in a
medium where no(r) = 1):
(allaxl+i2k8ldx-KI)d~j(scx)=dX
1(E,x)
(18)
with Ki2=x . The general solution of (18) is easily
obtained by variation of parameters. When the boundedness condition for x oo is invoked, the solution
becomes
where the ( ),vdenotes a statistical average, 3(c- vC')is
a two-dimensional delta function, and F,(jrx-x'j)
is
the two-dimensional spectral density of Pi(r).
dx' exp{i[k- (k2 -K2 ) ]x'}dXj(ic,x')
(exp-(i(k-(k2-K2)]x}f
+exp- {ilkj+ (k2-K2)2]X}f
The quantity dXo(cx)
exp[i.-rjdtj(t,x),
(12)
rt= eyy+ezz.
dtj(cx)=-
has the spectral representation
-00
with (ei= unit vector in ith direction)
'K =
1047
dx' exp{i[k+ (k2- K2)2]x'}dXj(1C,x')). (19)
is easily obtained from (8), (11), (12), and (17):
dXo(x,x) =
-
2k2 g(x)dN(x,x).
(20)
Substitution of (20) in (19) then yields an expressionfor the two-dimensional Fourier-Stieltjes transform of the first
term in the perturbation expansion (6).
d4ro(xx)=
ik2
(k2 -
/
exp-(i[k-(k
2
X
-
2
K)
]x}j
dx' exp{i~k-(k 2 -K2)
-K
x'}g(x')dX(cx')
K~2)
+exp- (i~k+(k 2-
K2)
]X}f
2
dx' exp{i[k+ (k2-Ki2)]x')g(x')dN (i,x')).
(21)
Equation (21) reduces to the result of Tatarski 7 if it is assumed that g(x') is unity in the interval 0<x'<x and
goes smoothly to zero outside this interval (approximately zero within a few correlation lengths from the end points
x'=0, x). In this case, it can be shown that the second integral in (21) is negligible and the first can be approxi-
6A. M. Yaglom, An Introduction to threTheory of Stationary Random Functions, R. A. Silverman, Ed. (Prentice Hall, Englewood
Cliffs, N. J., 1962).
V. I. Tatarski, Ref. 4, pp. 128-130.
4
A
X n
WV. P.
1(148
BROWN,
mated by replacing (k2-K2)' with k- ( 2/2k), and the limits
dio(Ex)-ik exp- [-x]
f
JR.
Vol. 56
to x by 0 to X. Thus
-C
dx' exp+[i-x ]dX(x,').
(22)
Similarly, a relatively straightforward calculation reveals that under the same assumptions the two-dimensional
Fourier-Stieltjes transform of the second term in the perturbation expansion is approximately
ik
z
K2
K
+-exp-pFt]
2k
L2kJ
7dxyexp-X
[•x']2
ff
(
'dPo(K-itr)dto(4r')
(23)
with d'Fo(cx) given by (22).
Tatarski8 has calculated the mean square value of the real and imaginary parts of the approximation to 4u10(r)
obtained from (22). Addition of his results yields an expression for the mean square value of #1 0(r).
(4lo(r)VPro*(r))vz..2k2xff duf
duE4(x,u).
(24)
An estimate of the mean square value of #1 1(r) is derived in the following section. It is shown later that
82(#ll(r)&n*(r))av is comparable in magnitude
to (iio(rAQio*(r))Xv when x is sufficiently large.
III. ESTIMATION OF THE MEAN SQUARE VALUE OF THE SECOND TERM IN
THE PERTURBATION EXPANSION
In this section the mean square value of 4'1(r) is estimated for the case in which (1) Pi(r) is a gaussian random
process, and (2) the second-order moments of vP(r) are gaussian. The first assumption permits expression of the
fourth-order moments of the refractive-index fluctuations in terms of a permutation of the products of the secondorder moments. When the relation between the second-order moments and the two-dimensional spectral density
of vi(r) is taken into account, it is found that the fourth-order moments have the following representation
4
(TI dN(ici,Xi))av= { (Xl+
i=1
x2)8(K
3 +K 4)F,(l1,Xl2)Fv(t
3 ,Xa34)+8(xi+ a3)6(c2+K4)F,(c1,Xl3)Fv(x
2 ,X7l
24 )
+ 6(x±1+ C4) 6c(1
2 + x 3 )F,(61,X1 4)Fd(i 2 ,x 2 3)} dYc1dX2 dc3dX4 ,
(25)
where xij =
x-jx . The second assumption is made to simplify the task of evaluating the integrals that appear in
(#fr)1l*(r)Xav. With this assumption, the two-dimensional spectral density of the ijth moment of vi(r) canrbe
written
2
Fp(xi,xi;) = (12 /4,r) exp[-Kc 2 12/4-Xij21 /]
1
(26)
where I is the correlation distance of the fluctuations.
A formal representation of the mean square value of i/l'(r) in terms of the two-dimensional Fourier-Stieltjes
transform dTI(i,x) is obtained when (16) (with j= 1) is multiplied by its complex conjugate and a statistical average taken.
ff11
x00
exp[i(it-
) r,](dtl(;cx)d *(ca,,x))av.
(27)
In the case of interest here, i.e., long distance propagation, the contribution of the first term in (23) to
(#611(r)46l1*(r)).v
is negligible in comparison with that of the second term.
8 See Ref. 4, pp. 130-137.
VALIDITY
August 1966
OF THE RYTOV APPROXIMATION
1 049
Thus, the statistical average (dt(xcx)dt*(c 0 x)) 5 7 is approximated by
[ (K2-Ka2)X]
j 5 dx'f dx" exp~fXK
1
(dtl(cx)d4l*(Vcax))av~ -e
x
ff ff (xC
Ka,2Xf/
"
Y.'()a
I">rc"(d4o(c-Yc',xO)dto(cXf)dta*(,a
K')
-xc
K2XI-
-c",
x")dro(i",x"))>av.
(28)
-cc
The dependence of (tn(r)ta*(r))av on the spectral characteristics of the refractive-index fluctuations is made explicit when the results expressed in (22), (25), (27), and (28) are combined. Upon rearrangement of the resultant
expression, we find that
(
k- f2 dKii /
(r)
di4
j
4 ii d
pX
dx'f
j
indx"I
px'
if"dA/
dvii dv2I
Jo
Jo
Jo
-00
-0:
(C'
ma"
dv4(MiA+M2),
(29)
Jo
where M 1 and M 2 denote the functions
2
M1 = K12K2
exp(i/2k(-K2
(Vl+v2)]+K22[2x"-
[2x'-
(V3+ V4)]})F6(itVl2)F,(ic2,v3
2
M2I=2 (iKi 1cK2)
exp{i/2k[2xi: 1c2(x -X")+Kl'(Vl-V3)+K2
2
4)
(V2-v 4)]}F96(c,vn3)Fv(62,v2
4)
with vpj= jvi-vjI.
A. Estimation of the Integrals Involving Ml
The rather formidable task of evaluating the integrals in (29) is greatly simplified by the gaussian correlation
assumption (2) and because the present paper is concerned with the value of (29) when x>>d,k>>1. Consider first
the term involving Ml. Note that Ml is the product of a function of Kci,x', VI, V2and the complex conjugate of the
same function with il, x', VI, V2replaced by x2, x", V3, V4.Thus, the contribution of Ml to (4ln(r)itc*(r))av is given by
(k2/4)LIL 1 * where L 1 is the integral
L1=,
J
d.cij
drif
dx'j
dV2 21iexp{ - (iKc12 2k)[2x'- (V1+V
2 )]}F,(c1i,v 12).
(30)
When the variables v,, V2in (30) are replaced by new variables u= v 1 -v 2 , W=Vl+V2,the resultant integral with respect to w is easily evaluated. This yields
1=-i2k l
d.
dx'
duF,(l A)j[.exp-
Jo o
[K1 (u- 2x')-i
i itiit] xpi
2
2k
.
(31)
From (26) it is apparent that F,(Ei,u) is very nearly zero for all values of Kiand u outside the domain ci(, I-1,
u l. Consequently, the exponentials exp[r4iKl2u/2k] in (31) can be replaced by unity since K1 2u/k( 1/ki<<1 in
the domain where F,(xi,u) is appreciably different from zero.
i - i2kJf dcij dx'[t- exp- (i )] j duFci,u).
(32)
Finally, note that the bracketed quantity in (32) is very small unless x't hi2 . Consequently, in view of the fact
that F,(Ri,u) rapidly approaches zero when u> l,it is permissible to replace the upper limit in the integral over it by
infinity. When this is done and F/(ci,u) is expressed in the form (26),9 the evaluation of the integrals in (32) is
relatively straightforward. The result is
L,--i(r)kxl[1+i(k1
2
/4x) ln(1+i4x/kh 2)].
(33)
Actually, the assumption of gaussian correlation implicit in the use of (26) is not required in the evaluation of (32). It is possible to
proceed in the manner of Tatarski, i.e., let
Io F,(xl,u)du= r4(KO),
where k is the three-dimensional spectral density of the refractive index fluctuations. In this case q(K,O)need not correspond to a gaussian
correlation function. Equation (26) is employed in the calculation of L, however, to conform with the approach required in the evaluation of the integrals involving M2 .
W. P. BROWN,
1050
JR.
Vol. 56
Thus, the contribution of Ml to (qt1(r)qt,,*(r))av is
(k2l4)LL,*-
(.7r/4)(kl)1y2 T(y),
(34a)
2
where -yis the dimensionless variable x/k1 and
T(,y) = [1- (1/4,y) tan7'4y] 2 + (1/64-y2)[ln(1+
16y 2 )] 2 .
(34b)
B. Estimation of the Integrals Involving M2
Integration over the variables vim,V2, v3, and V4 is facilitated by the introduction of new variables Ui=vi-vi+2,
is written in the form (26) and it is
2 (i= 1, 2). When the two-dimensional spectral density F,(xifii)
again noted that exp[iKti2i/2kJ- 1 in the region where the integral F, is appreciably different from zero, the integrals over ui, wi are easily evaluated. The result can be written in the form
Wi= Vi+Vi+
dvlJ
dV2J
dvJ
dV4M2-(X1 lc2)2W(X1',X',1)exp(-E(K,2+K22)12
14]+iE'cl-21k](xl'-x")),
(35)
where
18
W(x',x",l)=-'1+exp327r2
-(X I'-Xf/)2_
exp--
1
X12\
-exp-
X12\
21
X-
- 2-
fx'/e
l
eU2 dite"/-
I Jo1J
X"
2-
e-2du±X'/J
-Udu+
I
I
1
X'-_X'[/
e--'du.
JoI
le2
(36)
Next substitute (35) in (29), interchange the order of integration with respect to the dus and x', x", and express
the dxi in the polar form dxi= KidKid6i.
After evaluating the integrals with respect to the angles si,
it is found that
the contribution of M2 to (ftl(r)+tl*(r))av can be expressed in the form
2k2
k2
-4LLl*=4
where W(x',x",l)
2
x
dx'J dx"V(x',",1)Q(xx
(37)
,1),
is the function defined in (36) and Q(x',x",l) is
Q8xx",l)= - 47r2
dxj
f
dK2{KiK2Jo[EKK2(x'-x")/k]
The integral over K2 in (38) is the zero-order Hankel
transform of the function K2' exp(-K 2 2 12/4). Substitution of the tabulated value of this transforms in (38),
integration of the result over Ki, and differentiation
with respect to x'-x" as indicated in (38) yields the
following explicit representation
of Q.
Q(x',x",l)= (272/18)[1- 121 2 )/(1n2 +l);'1
(39)
where X denotes the dimensionless variable (x'- x")/k12 .
Finally, note that in the major portion of the domain
of integration in (37), W(x',x",l) can be approximated
by the function obtained when the upper limits of the
integrals in (36) are replaced by infinity. When written
in terms of the variable i defined above and a variable
v= (x'+x")/k1 2 , the approximation assumes the form
W(x',x",l)'-(18/128r)(kl)2 (u-
2v)2.
(40)
The portions of the domain of integration in (37) where
(40) is not valid are confined to the immediate vicinity
10W. Magnus and F. Oberhettinger, Formulas and Tlheoremsfor
the Functions of Mathlematical Physics (Chelsea Publishing Co.,
New York, 1954), p. 137.
2 2
expE-(K
)1
_2/ 4]}
(38)
of the lines x'= 0, x"= 0, x'= x". Since these regions are
no more than a few correlation lengths in width, the
error incurred in approximating W(x',x",l) everywhere
by (40) is negligible. Substitution
of (39) and (40) Jin
(37) and evaluation of the resultant integral yields~the
following expression for the contribution of M2 to
(tiu(r)tlu*(r))av,
(tlu(r)ipln*(r))av,-k2/4LLi*= [Er(kl)6 /256]P(y),
(41)
where Oyagain denotes the dimensionless variable x/k12
and
P(-y) = 1 9 2-y 2 - 144-ytan'2-y+
28 ln(4-y
2
+ 1)-
140/3
49/3
128/3 y4 + 6,y 2- 7/6
± -- +(,Y,+
4')
(42)
(0Y2 4
IV. A CRITERION FOR THE VALIDITY OF
THE RYTOV APPROXIMATION
As mentioned in Sec. II, the conditions 5K<1,
IV/,1o(r)I<<1 are not sufficient to guarantee the validity
VALIDITY OF THE RYTOV APPROXIMATION
August 1966
of
P40(r)as an approximation to i64(r). A more basic re-
quirement is that the perturbation expansion (6) asymptotically represent #l(r). An asymptotic expansion is
characterized by the fact that each successive term in
the expansion is smaller than the preceding term for
sufficiently small values of the perturbation parameter."
In view of this, and since the various terms in (6) are
random functions which are describable only in statistical terms, it is assumed that the validity of the expansion is related to the relative magnitude of the mean
square value of successive terms. In particular, the
condition that 62 times the mean square value of i',i(r)
is less than the mean square value of i',o(r) is adopted
as a criterion for the validity of the Rytov approximation technique:
31(,P11r+li
r>a
(43)
(r)).v<v
s(r)4,11*
In effect, this condition imposes a limitation on the
acceptable values of the parameters x/l and ki for a
given value of 6. When (43) is not satisfied, an entirely
new approach to the approximation problem is required.
An integral representation of the mean square value
of 4'10(r) in terms of the two-dimensional spectral density
of the refractive
index fluctuations
is given in (24).
Substitution of the gaussian spectral-density function
employed in the calculations of Sec. III in (24) and
evaluation of the resultant integrals yields
assume that the nonuniformity of the approximation
techniques (Rytov or Born) is independent of the details of the physical configuration considered and of the
type of statistics which describe the medium.
The equivalence of the validity conditions for the
Rytov and Born approximations contradicts the statements of Tatarski,' 4 Chernov,"1 and, more recently,
de Wolf,'6 who contend that the Rytov approximation
has a larger domain of validity than the Born approximation. The results derived in this paper prove definitely
that this is not the case when y>>4. Furthermore, the
fact that both approximations suffer from the same type
of nonuniformity leads to the conjecture that the Rytov
and Born approximations have the same domain of
validity for all y.
The Rytov and Born approximations are employed
in situations where the geometrical-optics approximation is invalid. Thus, to place the condition on the Rytov
approximation in proper perspective, it is worthwhile
to derive a condition on the validity of geometrical
optics consistent with the criterion for validity adopted
for the Rytov approximation. This criterion is stated in
(43). The mean square value of the first term in the
geometrical-optics approximation is related to the phase
fluctuations and that of the second to the amplitude
fluctuations. Tatarski shows that these quantities are
given by the expressions'
(44)
(,r)1(k1)1,y.
for the mean square value of #,d(r) in (43) gives the fol(w)'5%1(kl)Vy[T(y)+(1/256)PQ'6/1
< 1-
(45)
The significance of (45) is discussed in the following
section.
V. DISCUSSION AND SUMMARY
Note first that when y>>1,the bracketed quantity in
(45) approaches unity. In this case, the validity criterion is identical, to within a constant, with the condition on the Born approximation derived independently
by Mintzer'2 and by Kay and Silverman.'3
(46)
x< [(7r) 2162]-l.
Mintzer
considered the propagation
of waves excited
by a point source in a three-dimensional random medium
whereas Kay and Silverman considered the propagation
of a plane wave
through
a random
stack of one-
dimensional dielectric slabs. In view of the equality of
the validity conditions for these diverse problems and
the problem considered in this paper it is reasonable to
11A. ErdElyi, Asymptotic Expansions (Dover Publications,
Inc., New York, 1956), pp. 8-13.
12D. Mintzer, J. Acoust. Soc. Am. 25, 1107 (1953).
"1I. Kay and R. A. Silverman, Nuovo Cimento Suppl. 9 (Ser.
10) 626 (1958).
(7r)tl 2 k2 xl,
(47)
(X )av= (8/3 ) (,or)132X3/13.
(48)
(S'2).=
Substitution of this result and those derived in Sec. III
lowing condition on the parameters 6, hi, and -y(= x/kl 2 ).
1051
Substitution of (47) and (48) in the validity criterion
(X2)av< (Si2 1)., and simplification
of the result
yields
the condition
x<0.61k1',
(i.e., y<0.61).
(49)
This condition agrees with the result obtained by
Tatarskill on the basis of a physical argument. In a
given situation, comparison of the limitations imposed
on x by the conditions (45) and (49) indicates the utility
of employing the more complicated Rytov approximation in place of the relatively simple geomnetrical-optics
approximation.
The nature of the limitation placed on the applicability of the Rytov (or Born) approximation by the
inequality (45) is illustrated by Fig. 1. The three curves
shown in this figure are obtained by solving equation
(45) with an equality sign. The domain of validity at a
given wavelength includes all points below the correspondingly labeled curve. These curves were computed for a correlation distance of 1.0 cm. Such a dis-
tance is typical of the inner scale for sea level atmos14V. I. Tatarski, Ref. 4, pp. 122-126.
1"A. M. Obukhov, Ref. 2, pp. 58-66.
16D. A. de Wolf, J. Opt. Soc. Am. 55, 812 (1965).
'7 V. I. Tatarski, Ref. 4, p. 148.
18V. I. Tatarski, Ref. 4, pp. 120-121.
W. P. BROWN,
1052
JR.
Vol. 56
classified as very weak turbulence, when SX10'-1 to
relatively strong turbulence,2 0 when 3X10--10. It is
apparent that the limitation on the Rytov (or Born)
approximation
can be rather severe in the case of propa-
gation over horizontal paths at sea level. In such cases,
very little is gained in going to the Rytov approximation when the light is in the red or shorter wavelength
102
portions of the spectrum. As seen from Fig. 1, the do-
main of validity of the Rytov and geometrical-optics
approximations
6328
lo0
A
4880
For example, when
OPTICS
DOMAIN OF VALtDITYFOR GEOMETRICAL
lo-'
0
I
1
2
3
coalesce
at 6X 10i-5.2
for 6328-A
radiation and at smaller values of a for the shorterwavelength radiation. Furthermore, in this portion of
the spectrum neither the Rytov nor the geometricaloptics approximation is valid at the distances currently
envisaged for coherent optical propagation experiments.
lo,
4
5
6I
7
8
9
1I
10
a x lo,,
FIG. 1. Limiting curves for the domains of vadiditv.
l= 1 cm, N=4880 1, 6328 i, and 3.39 M.
pheric turbulence.'2 Generally, the domains of validity
shrink as I is increased (when y>>A, the maximum range
is proportional to l-l). Also, it is noted from (45) that
for a given y, the maximum allowable 6 is proportional
to Xd.Thus, the limiting curve for the domain of validity
at a wavelength not depicted in Fig. 1 can be obtained
by multiplying the values of 6 on one of the given curves
by the three-halves power of the ratio of wavelengths.
Finally, the maximum propagation distance over which
the Rytov approximation is valid for a given degree of
turbulence (i.e., a given 6) can be obtained from Fig. 1
2
.
by multiplying the maximum allowable -y by Wd
Available data on refractive-index fluctuations indicate that 6 is typically in the range 1<6X108<10 for
sea-level atmospheric turbulence. The turbulence that
causes these fluctuations ranges from what may be
'"R. E. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 57
(1964;.
AX108 is
2, the maximum allowable
Since corange at 6328 A is 2.2 km (xmax=ky12,nax).
herent optical propagation experiments over horizontal
16-km paths have already been conducted,2' it is apparent that there is a need for a new theory of shortwavelength propagation valid at these distances.
The situation is somewhat different in the infrared portion of the spectrum, where it appears that existing
theory is adequate for most practical situations (at
ranges less than a few kilometers).
In summary, it has been shown that when 7y>>Z,the
Rytov and Born approximations have the same domain
of validity and it has been conjectured that the same is
true for all 'y. In addition, it has been shown that there
is a very severe range limitation on the validity of the
existing theories of propagation when X< 0.6 y. The
limitation in the infrared portion of the spectrum is not
nearly so severe, but even here it is necessary to exclude
situations
in which the range is greater than a few
kilometers.
20 Generally, the strength of the turbulence decreases as the
altitude above the earth is increased. The data tabulated by
Hufnagel and Stanley' indicate that the magnitude of the fluctuation is decreased by a factor of 10 at 3 km.
21F. E. Goodwin, paper presented at the conference on Atmospheric Limitations on Optical Propagation, Boulder, Colorado
(1965).
Martin Shenker (left) and L. F. Barcus, both of Farrand
Optical Co., at Rochester Section's 50th Anniversary Symposium
on the Use of Very Large Computers in Lens Design, 16 November
1965.