Vol. 2, No. 12/December 1985/J. Opt. Soc. Am A 2161
Numerical solutions for the fourth moment of a finite beampropagating in a random medium
Moshe Tur
School of Engineering, Tel-Aviv University, Ramat Aviv, Tel-Aviv, Israel 69978
Received February 22, 1985; accepted July 3, 1985
Numerical solutions of the.fourth-moment differential equation are obtained for a finite, initially Gaussian beampropagating in a. two-dimensional homogeneous and isotropic random medium with a Gaussian correlation func-tion. In addition to the covariance of the intensity fluctuations, the full spatial dependence of the fourth moment ofthe propagating field is described for several beam diameters and propagation lengths.
1. INTRODUCTION
The propagation of laser beams in a clear, nonattenuatingatmosphere (i.e., neglecting atomic and molecular absorp-tions as well as Rayleigh and Mie scattering) may be severelylimited by the atmospheric turbulence. Beam spread, lossof optical resolution in .imaging systems, beam jitter, anddeep intensity fluctuations in line-of-sight communicationsystems are but some of the degrading effects that are due tothe random, inhomogeneous refractive index of the atmo-sphere."' 2
Current theories of laser beam propagation in the turbu-lent atmosphere are based on moment equations that de-scribe the propagation of the various moments of the radia-tion field.3 The second-moment equation (whose solutionsdetermine the average intensity distribution across thebeam profile as well as the resolution limit of the inhomogen-eous medium) is easily solved for both plane and sphericalwaves,3' 4 and the results are equivalent to those obtainedfrom the extended Huygens-Fresnel principle. However,the equations for the higher moments have not been solvedto date by exact analytical methods. Of particular impor-tance is the fourth moment, since it contains informationabout the intensity fluctuations and their correlations.This moment has been studied by perturbation,"l2 asymptot-ic,6-8 and numerical techniques.9-13 The perturbation re-sults, all based on the Rytov approximation,' predict amonotonic increase of the normalized variance of the fluctu-ating intensity I,
with the propagation distance and/or the strength of therefractive-index fluctuations. Experimentally, it has beenshown that xoi2 saturates at a value of the order of unity,' 4
and therefore perturbation techniques are useful only whenac2 < 1. One of the first theoretical treatments that showedthat the fourth-moment equation is capable of predictingsaturation was a numerical study 9 that was followed by othernumerical solutionsl0-'2 and various asymptotic results.6 -8
Recently, a two-scale approximation algorithm was devel-oped' 5 -'7 that has produced results that are in close agree-ment with available numerical solutions for plane-wave
propagation in a two-dimensional random medium charac-terized by either a Gaussian 12"15' 1 6 or a Kolmogorov13"17 cor-relation function. During the development of the two-scalemethod, these numerical solutions proved to be extremelyuseful as a bench mark in spite of the two-dimensional na-ture of the numerical model and the particular choice of thecorrelation function.
Finite-beam propagation is more complicated than thepropagation of either plane or spherical waves. The latterwaves have the obvious property that their fields exist at allpoints in space, and therefore they cannot exhibit degrada-tions related to the finite dimensions of the beam, such asturbulence-induced fluctuations of the center of mass of thebeam as well as beam spread. This paper will present thenumerical solutions of the fourth-moment equation for aninitially Gaussian beam propagating in a two-dimensionalrandom medium with a Gaussian correlation function. It isa followup on a preliminary publication." Section 2 pre-sents the governing differential equations for both the sec-ond- and fourth-order moments, and in Section 3 we de-scribe the finite-beam solution for the second moment,which is required for the evaluation of oI2 [Eq. (1)]. Thefourth-moment equation and its physical significance arediscussed in Section 4 along with an algorithm for its solu-tion. A Born-type perturbation solution of the fourth-mo-ment equation is derived in Section 5, and in Section 6 wepresent the results of both the numerical solution and theperturbation technique.
2. THE MOMENTS EQUATIONS IN TWODIMENSIONS
Our two-dimensional model is generated from the three-dimensional physical world (with z the propagation direc-tion and x and y the transverse coordinates) by assumingtranslational invariance in they direction of all initial condi-tions as well as for the random index-of-refraction fieldn[=n(x, z)]. The form of the differential equations thatgovern the propagation of the second and the fourth mo-ments can be obtained, therefore, from their well-estab-lished3 three-dimensional counterparts simply by omittingthe y dependence. Thus, if we define the second and the
2162 J. Opt. Soc. Am. A/Vol. 2, No. 12/December 1985
fourth moments of a propagating quasi-monochromaticfield U(x, z) with frequency v by
r2 = (U(X,,z)U*(X 2 ,Z))
(* denotes complex conjugation) (2)
and
r 4 = (U(x,, z) U*(x2 , Z)U*(x3 Z)U(x4, z)) , (3)
then, in terms of scaled coordinates, the governing differen-tial equations are
ar2(G, PI S) r. 2 --- 1_____ _ = {aa - k2 [a(0) - a(s)]} r 2(G, p, s), (4)
r = z/kl"2 (5)
(ln is the characteristic length for the refractive-index fluctu-ations, and kl 2 is the distance required by the diffractionprocess to transform phase fluctuations into intensity fluc-tuations'),
The correlation function a(x, s,) can assume various func-tional forms depending on the nature of the random medi-um. In this work, attention is focused on a Gaussian corre-lation function with the following two-dimensional form:
o(x, s) = ao exp[-(X 2 + sz2)/1ln21, (15)
which yields
at) = a(0)exp(-2 ), a(0) = |;lzao- (16)
Although the Gaussian model is a poor descriptor of theindex-of-refraction fluctuations in the atmosphere, it wasbelieved that this smooth and well-behaved function wouldreduce numerical errors as well as facilitate easier compari-son with newly developed analytical solutions of the fourth-moment equation (see, e.g., Ref. 16).
3. THE SECOND MOMENT FOR A FINITEBEAM
A Fourier transformation of Eq. (4) with respect to p pro-duces a first-order partial differential equation that can besolved by the method of characteristics' 8 to give
F2(G, p, S) = J daa 2(a, S - ca)
(6) X exp{-(y/a) Lat [1 - U(lnU)/f(0)]du} exp(iajp).
(7)
(8)
(9)
(10)
(17)
P2 (a, s) is the Fourier transform of r 2(r, p, S)I t=o, namely,
We shall evaluate Eq. (18) for the case of an initiallyGaussian beam propagating in a random medium in whichthe index-of-refraction fluctuations have a Gaussian correla-tion function [Eq. (15)]. Thus, if the initial field distribu-tion is
U(x, z = 0) = U0 exp(- 2 I2)' (19)
where I U0 12 measures the initial on-axis intensity, the finalresult is
r I- p or r da' exp!4[ (S-- / - a' + )21 1
X exp 1-zy[ 2 - erf() - erf(s - 't/j3)
X exp(ia'p/O)- (20)
The mean irradiance profile of the beam at range ¢ and atransverse distance p is given by
7(D, p) = r 2(, P, S = 0)
= I U0 7' da' ex [-{ (1 + f) (a)2]exp(ia//10)
X exp k-- I -1- X [erf(a'N)) (21)
For large enough values of yA, the last exponential factorin Eq. (21) is negligibly small, unless a' is small. In theregion of small a', the error function may be expanded in aTaylor series. Retaining only the first nonvanishing term ofthe expansion, we obtain a Gaussian profile for the trans-verse irradiance distribution:
Moshe Tur
Vol. 2, No. 12/December 1985/J. Opt. Soc. Am. A 2163
(b)Fig. 1. The average intensity of an initially Gaussian beam that propagates in a medium with a Gaussian correlation function. (a) y = 0.5; (b)-y = 5 for two beam radii Do/ll = 4 on the left and Do/ll = 1 on the right. In both (a) and (b) the top row shows the range dependence of the on-axis intensity I(D, p = 0) and the two lower rows depict the transverse structure of 7(t, p = x/Il) for r = 5 and r = 10, respectively.
l() D= v)I[ expi- (22)
where
Jo = I U0 12, D() = /3(1 + .2 + 4,Y3)1/2 (23)
Actually,18 the above series-expansion process is nonuni-formly convergent in p, and Eq. (22) may not be correct forIPI - O. However, for near-axis points, we may use Eqs.(23) to define the region of multiple scatter by the twoconditions
y >> 1 3>> . (24)
-y
The second condition ensures that D(r) is dominated by the3 scattering term.Figure 1 describes the range dependence of the on-axis
irradiance I(D, 0) for , = 0.5,5 and two values of A: Do/ll = 1,4. The transverse irradiance distributions of the beam I(t,x/ll) at r = 5 and v = 10 also appear in the figure along withthe free-propagation profiles. The dashed lines are the re-sults of the approximate analysis [Eqs. (22) and (23)]. Thefit for y = 5 is excellent. But even for Py = 0.5 and r = 10, theapproximate analysis gives a fairly good estimate.
Detailed calculations of the propagation of the secondmoment of a finite beam in three-dimensional random me-dia appear in Refs. 2, 6, 18, and 19.
4. FINITE-BEAM SOLUTIONS FOR THEFOURTH MOMENT
It has already been shown" that the displaced Fourier trans-form of r 4,
M(G, i, , X, q) = -1 J r 4 ( , X, l, p, q + W#iB)exp(-iXp)dp,2 or _ ,(
(25)
satisfies a much simpler partial differential equation
aM(t, i, 77, X, q)
=[WQq + X 84X) + i 02X1= [Y163 0~077J~'~'9' (26)
which has only three independent variables I, 7, and r andtwo parameters q and X. It is also similar in form to theplane-wave equation.12
M(D, i, 77, X, q) has several useful properties. Like r 4(I, 2,77, p, q), M(¢, i, 77, X, q) = M(t, l, i, X, q) = M*(A, -i, 7, X, q).Also, by integrating Eq. (26) with respect to i7 between infi-nite limits, a conservation law is obtained (assuming that0M/0- - 0 as I q -> , a condition that is certainly valid for afinite beam):
Moshe Tur
l3.0
0
0~11.8
' 0.6
X- 0 .4
.- 3,
- 0.2
0.0
3.0
sC0.8
X 0 .6
IC;
0~ 0.2
0 .0
0.30
CO.08
X 0.06
° 0.04
613
_~: 0.02
0.0032. 40
E - I .! . . .
2164 J. Opt. Soc. Am. A/Vol. 2, No. 12/December 1985
d I M(t, 0, 7, 0, 0)dq7 = 0,dD
or
d I dp I d&7I4(t, 0,1 7, p, 0) = 0. (27)
To find r4(G, I, 7, p, q) at an arbitrary range r = A, giventhat r4(O, (, 77, p, q) (to < cf), we first calculate M(to, a7, X, q
- Xtf/fl) from the given r 4 (t = 0, I, n, p, q) and Eq. (25); thenwe march in the v direction, using Eq. (26), until we findM(f , 7, X, A - Xt//3). This is repeatedly done for as manyvalues of X as are required for the inversion of Eq. (25).Finally,
r 4(tf, r , p, q) = J M(t, ¢ , n, X, A - Xtf/#)exp[iXp]dX.
(28)
Several measurable quantities can now be derived from F4
and M(-). If I(xj, z) and I(x2, z) are the intensities at points(xi, z) and (x2, z), then the normalized covariance of theintensity fluctuations at the above two points is
where 7(t, p) is the average intensity at (1np, kln2t) as com-puted from Eq. (21). Evidently, the scintillation index aj
2
[Eq. (1)] is given by C(p, 0, D).Following Ref. 20, we note that F4 is also related to the
fluctuations of the center-of-mass coordinate of the propa-gating beam, which is defined by
Pcm = I xI(x, z)dx.-ax_
(32)
Thus the transverse behavior of M(.) also determines thevariance of the fluctuations of the beam center of mass.Until now, detailed solutions of the partial differential equa-tions for either F4 or M(-) (in the strong-scattering regime)did not exist, and (pcm2 ) was calculated using various ap-proximations that are discussed in Ref. 21.
All the above-mentioned observables are extracted fromr 4 (G, a, 7, P, q) with q = 0 and -q = 0. The importance of F4with q = 0 has been discussed in Refs. 12, 22, and 23. Nu-merical results for r 4(G, I, a, p, q = 0) with both t 5d 0 and wei0 will be presented in Section 6 along with their physicalsignificance.
5. A BORN-TYPE SOLUTION FOR r4
When -y is small, a perturbation solution can be developedfor M(r, a, 77, X, q). We expand M(-) in powers of y:
M(-) = OM(-) + 'M(-) + 2M(-) + 3M(-) + .. . (34)
0M(G, a, rn, X, q) is the solution of Eq. (26) withy = 0, i.e., itrepresents free propagation. jM(.), j _ 1 are then deter-mined from Eq. (26) by iteration:
Since OM(t, a, 77, X, q) is known, the last term in Eq. (35), tobe denoted by T(D, I, n, X, q), is known, too:
T(¢, A, 77, X, q) = yf(t, 77, q + Xt//3) X 0 M(¢, a, 77, X, q). (36)
The resulting equation
02 M(, s, 77, X, ) 2 M(r , X, + T(I , , ,X, )
(37)
can be easily solved by using Fourier techniques:
1M(w, A, tj, X q) = J dke J dk,1 exp[i(k, + k,7n - ktkj)]
X J dU' exp[iktkr] , kt, k,, A, q(),
(38)
I(.) is the intensity, and P = S-. I(x, z)dx is the consetotal power of the beam. Using our definitions and ntion, it can easily be shown that the normalized mean sqiof Pcm is directly related to M(.):
Do))27r
S J dpI(, p)]I1=-
T(r, kt, kI, X, q) = 1 2 d I d7 7 exp[-i(kt + k,,7)]rved
ota-luare
(39)
T(-) is analytically determined from the initial conditions[Eq. (19)], the scattering function f(-) [Eq. (14)], and Eq.(39). The integration over ¢' in Eq. (38) was carried outusing Simpson's rule, and a two-dimensional fast-Fourier-transform algorithm was employed to evaluate the integrals
2 over kt and k,.- 77 Although the procedure can be easily extended to j > 1, we
4 used only OM(W) + 1M(W) to estimate M(.) [Eq. (25)]. r 4 isthen evaluated using the methods of Section 4. Perturba-
(33) tion solutions based on the Rytov method appear in Refs.24-31.
Moshe Tur
X T(r, �, n, X, q).
Xf d 77 2 a2 2� a2 - - 2 a2 4dX1 0 XDq fl a-'
a +(') -q
X MG, 0, 77, X, - WM-q=OX=o
Vol. 2, No. 12/December 1985/J. Opt. Soc. Am. A 2165
6. NUMERICAL SOLUTIONS OF THEFOURTH-MOMENT EQUATION
Equation (26) was solved numerically by using methods pre-viously discussed in Refs.11 and 23. Grid spacings were A1 =A77 = 0.125-0.25 and Ai = 0.005-0.025, with boundaries at(max = m.ax = 40.5. All the results of this section wereobtained for the Gaussian correlation function [Eq. (15)]and for a moderate level of scattering given by -y = 0.5.Since program execution required a formidably long runtime and close to 90% of the core memory of a CDC 6600computer, we had to choose a particular value for -y. Thevalue Sy = 0.5 was selected, since it is large enough to givesaturation at a reasonably short range but not so large as tocause numerical problems (e.g., too small characteristic scat-tering lengths).32 The beam profile at ¢ = 0 was assumedGaussian, as in Eq. (19), and the X integration in Eq. (28) wasperformed using Simpson's rule with AX/d2(A) = 0.1-0.2,where
d2(f)= [+ ( /,32)2]; (40)
Aj is the value of ¢ at the range under consideration. Clearly,to ensure accurate Fourier transformation, one requires that
AX X p << 2r. (41)
A. Intensity Fluctuations and Their CorrelationsFigure 2 shows the fluctuation index aj [Eq. (1)] as a functionof range tfor two values of3 las well as for a plane wave. Alsoshown are perturbation results derived from the first Bornapproximation of Section 5. It is seen that, for Doll/ = 1, o-Irises quite slowly but eventually crosses the plane-wavecurve. As Do/ll increases, the range dependence of o-I ap-proaches that of a plane wave (Do/ll = oc). However, forlarge enough A, a, for the finite beam again exceeds theplane-wave saturation level. Since -y = 0.5 represents onlymoderate scattering, these results, although interesting,cannot resolve the controversy between those who claim6'33
that the variance of the finite beam saturates and those whoassert3 2 that for very strong scattering the variance of a finitebeam eventually diverges. It should be kept in mind,though, that our numerical results were obtained for a two-dimensional Gaussian-type random medium.
0.4
BORN SERIESPLANE WAVE
D,/
BORN SERIESFINITE BEAM
I I
DO/tn =4
- PLANE WAVE
I (D,/tn = sO )
I I I~~~~~~~~~~~~~~~~~
2 4 6 8 10
Fig. 2. alversus PforDo/l, =1,4, (plane wave) and y = 0.5. Alsoshown are the Born approximations for a plane wave12 and for DO/lI= 1; see Section 5.
2.0 - -D /en= 1.0 0.3
1.5 - \- .2~
w
1.0 / I-
0.10.5
2 4 6 8 10p =R/D,
Fig. 3. Off-axis dependence of aI on p = R/Do for r = 2 and Do/ll =1. Also shown is the beam intensity profile I(~, lip).
The dependence of as and beam intensity on p is depictedinFig.3for = 2andDo/ln = 1 [usingAX = O.ld 2 (# = 2)]. acincreases toward the edge of the beam, in agreement withpreviously published perturbation-based results.2 6-28 Asthe beam propagates into the medium, it spreads, and a-I willchange significantly only for values p = R/Do, which are ofthe order of the effective beam radius re [re is determinedfrom I(D, re) = I(P, 0)exp(-2); see Eq. (21)].32 In view ofrestriction (41), oI for large values of p cannot be derivedaccurately from our data. For small to moderate values of p,we have found that a, increases with p for all ranges and forboth Do/ll = 1 and Do/ll = 4. However, the incrementalchange in o-1 becomes smaller as either v or Do/ll increases.
Figure 4(a) describes the normalized covariance C1(x/2,-x/2, z = kl 2 v) = Cj(p = 0, n = X/i1, A) [Eq. (31)] of our finitebeam at the two points xl = 177/2 and x2 =-Inn/2 locatedsymmetrically around p = 0 for v = 2, Doll, = 1, and Do/ll =4. The plane-wave covariance function (for an initially unitamplitude plane wave; see Ref. 12) also appears in Fig. 4.Again, the covariance functions of the finite beam approachthe plane-wave function as Do/ll increases. C1(x/2, -x/2, z= kl, 2 r) for v = 6 and v = 10 are plotted in Figs. 4(b) and 4(c).The finite-beam characteristic transverse length for the co-variance of intensity fluctuations is of order In regardless ofDo ll, and range. This behavior is typical' of our correlationfunction [Eq. (16)], which is characterized by a single corre-lation length. Plane-wave results for two grids22 : A/ = A77
= 0.125, (max = flmax = 5.25 and At = A77 = 0.125, (max = 77max =
10.25 also appear in Fig. 4(b). At this range, the boundariesof the numerical grid are too close, the conservation law [Eq.(40) of Ref. 12] is not satisfied, and we have a spread thatincreases with x/ll (for x/ll = 0, the expected accuracy is 5%).Nevertheless, it is seen that the correlation length of theintensity fluctuations of the plane wave is shorter than thatof the finite beam.3 4' 35
We conclude this section by noting that we also foundthat, for small to moderate values of p, there is only a slightdependence of C1(p, 7), A) on p.
B. Numerical Solutions for the Spatial Behavior of r4(¢, A, p, q = 0) and M(t ~, 7, X, q = -Xk/,)Since we hope that, in the future, analytical solutions of thefourth-moment equation for the finite beam will be devel-oped, we present here more details about the spatial behav-
I .2H
0.8f
Moshe Tur
2166 J. Opt. Soc. Am. A/Vol. 2, No. 12/December 1985
XDo/dn = I
- \\\ Do/2n 4
Plane \ \-Wave(Da/n=caN\.\
I I I I,\.
y =0.5
~ 2
M(W, i, , X, q = -L/) = 1 X expl-[X/d 2(r)]2 )
X M1(G, i, 77, X, q =-At1)(42)
I I I I I I I I I I L .
2.0 X/ 3.0 _--- 4.0
(a)
y =0.5
C = 6
d2(G) is defined in Eq. (40). The particular value q =-At//was chosen for the following reason: Since we were princi-pally interested in intensity fluctuations and their correla-tions, it was necessary to evaluate r 4(&, A, 77, p, q = 0), which,according to the algorithm of Section 4, is calculated fromM(D, i, 77, X, q = - XA/3); see Eq. (28).
Figures 5 and 6 show the dependence of M1(&, A, , X, q =-XA/13) [Eq. (42)] on its transverse coordinates for = 2, 6, 10and Do/ll = 1, 4. When r = 2 and Doll, = 1 (Fig. 5) there isnot much difference (at least in shape) between the scat-tered M1(D, t = 0, w7, X, q =-XA/13) and the free-propagationM&(D, t = 0, X, A, q =-XL//3), as given by [see Eq. (19)]
Fig. 4. The normalized covariance of intensity fluctuations at twopoints x1 = +x/2 and X2 = -x/2 located symmetrically around p = 0,-y = 0.5. Also shown are plane-wave results. 12 (a) r = 2; (b) r = 6;(c) = 10.
ior of M(D, A, 77, X, q = -XA/O) and r 4 ( n, a, r, p, = 0).These results may serve to assess the accuracy of futuresolutions.
For the initial Gaussian condition of Eq. (19) and q =
-X/3, M(W, A, 77, X, q) Eq. (25) can be expressed as
This is no longer true for ¢ = 2 and Doll, = 4 (Fig. 6). Here,scattering and diffraction with their associated differentscales act together to produce the odd shapes of M1(D, t = 0,77, X, q = -AA/,8). However, as the beam propagates into themedium, the scattering process becomes dominant, and alltransverse scales for both Do/ll = 1 and Do/ll = 4 are of orderunity, that is, smaller than their free-propagation counter-parts.
One interesting feature of Figs. 5 and 6 is the similaritybetween the shapes of Mj(-) for X = 0 and X = d 2(G) in spite ofthe appreciable magnitude difference of X.
M(W, A, 0, 0) has another interesting physical significancein that it is proportional to the average of T4 (F, A, 71, p, 0) overp [see Eq. (25)].36,37 It also satisfies (with t = 0) the conser-vation law [Eqs. (27)]. Hence the areas under the solid andthe dotted curves for Mi(D, t = 0, w1, X, q = -XA/O) in Figs. 5and 6 are the same [the grid extends to 6max = 77max = 40.5,and only a part of M 1(D, A, a, 0, 0) is shown in the figures].We may conclude, therefore, that M1(P, t = 0, 7, 0, 0) [as wellas M(¢, t = 0, n, 0, 0)] first decreases with 77, with a character-istic scale of order unity, but then has a tail long enough tosatisfy the conservation law [Eqs. (27)]. This type of behav-ior has also been described by Brown 10 for the propagation ofa two-dimensional plane wave in a two-dimensional Kol-mogorov-type random medium.
The physical significance of r 4 when t X i7 0 0 has beendiscussed in Refs.12,22, and 38. When tXi X 7 0, Re r 4(&, A,77, p, q = 0) and Im F4(, A, 77, p, q = 0) measure theamplitude-phase fluctuations in the propagating field asgiven by Re r 4 = (aia 2a3a4 cos(01 - 02 - 03 + 04)) and Im F4= (aja 2 a3 a4 sin(¢, - 02 - 073 + 04)), where ai and 0i are,respectively, the amplitude and the phase of the field com-plex wave function U(xi, z).
1.0
0.5
0.0
ToNC
N
NC
1
N
i
U
Moshe Tur
-0.5
1.0
(o
0l-
C-,
NC4 0.0
U
1.0
NC
(0
N
C'.'N_.22
01! 0.0
2!
-)
NC
0
c'J 0.0
(43)
Vol. 2, No. 12/December 1985/J. Opt. Soc. Am. A 2167
Perspective views, similar to those of Refs. 12 and 38, ofRe F4 and Im r4 appear in Figs. 7 and 8 for r = 2 and r = 6.Although Re r 4 (G = 2, Do/ln = 1) and Re r 4 (t = 2, Do/la = 4)differ substantially from each other, Re r4 ( = 6, Do/ll, 4) isalmost exactly 20 X Re r 4 (G = 2, Do/ll = 1). The reason for
this high degree of similarity is probably the dominance ofscattering over diffraction for , = 6.
Although data pertaining to the spatial dependence of r4can be extracted directly from Figs. 7 and 8 (see Ref. 12), wealso present in Fig. 9 a few cuts through Figs. 7 and 8 along
0.5 -
0.0
-0I
PI -1':
C - 'I'
0.0 -0.1 , I, 1'. I. 0' . I, .
517
(a)
0.125r
_ . (X.0.5)
I - F
I.~~~~~ E
0.00
-0.025 , I . I . 1I.' i1 Io 02 577
1.0
(-
P0
0.0
1.01
K X .,
- (X 4')',
QZt
PD
Ir0.0
-0.2 I , I,!.!I.!0 7 10
0.05
.,2_.. Qp
X4)
I
. F0I f.
(X 4\ a- o.oo
-0.2 L ! I . I,0 71 4
tb)
1! l X l 0.2)
-N( 4)\
' I.I 'i;"I ', '
-0.01 , !, I I I!0 77 4
0.1 .
I ' ..
0
0 II
(x 10)7--.
0.00 ,IIIIi~IIJ
0.1
..
0
CZ
77
p .
C2S
:9-
3! oo
2 0
.1) .<
I.
0
:2
2
(c)
Fig. 5. The dependence of M1 (t, (, n, X, q = -XA/3) on X for t = 0and , = i7, = 1. - , X = O; --- , X = d2() [Eq. (40)]; .free propagation [Eq. (43)]. Two values of X are shown as well asfree-propagation solutions (y = 0). (a) fl = 1, r = 2; (b) 3 = 1, = 6;(c) / = 1, r = 10 [(X 0.5) means that the displayed quantity is twiceits actual value].
10 0
0.0
p.C.
-12
0
17
(X 0.1)
772
(c)
Fig. 6. The dependence of M1(D, (, 7, A, q = - XA/I) on 7 for t = 0and =,qB=4. ,X=0;-----,XA= d2() [Eq. (40)];.free propagation [Eq. (43)]. Two values of X are shown, as well asfree-propagation solutions (-y = 0). (a) f3 = 4, v = 2; (b) 1 = 4, r = 6;(c) 1 = 4, t = 10 [(X 0.5) means that the displayed quantity is half ofits actual value].
Moshe Tur
.,
.1
:f0.0k
77
F
2
(X 0.2)
1.6
N
A:2
'z
<~
F~
tl
f
Er
10
02.
p.
E
(0)77
10
0
I
i-0
775
77
(b)
77 4
1.0 2.0 3.0 4.0377
1.0 2.0 73. 0 4.0
(0)Fig.7. Thedependenceofr 4(, ,p=O.q =0O) on and7Xfor/ l= 1and = 2,6.information from this perspective views.
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Fig. 8. The dependence of F4 (, A, ,7, p = O, q = O) on i and 7 for1 = 4 and t = 2, 6. See Ref.information from this perspective views.
(b)
See Ref. 12 for instructions on how to extract quantitative
, t = 6 .
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12 for instructions on how to extract quantitative
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Vol. 2, No. 12/December 1985/J. Opt. Soc. Am. A 2169
Fig.9. The dependence of r 4 (, , ,p = 0, q = O) on 11 for =and = n.means that the displayed quantity is twice its actual value].
with free propagation results. Since for Do/l, = 1 the dif-fraction scale Do is of the same order as the correlation scaleof the medium l,,, the scattered r 4 (&, I, ij, p, q = 0) is notsignificantly different. from its free-propagation version.However, when Do/ll - 4 two scales can be observed in Fig. 9for P4 (G, (, 7, p, q = 0): a short scattering scale and a longdiffraction scale. Also, in Fig, 9 Im r4(, 77, p, P.q = 0)I1 ispositive for Doll, = 1 but negative for Do/l, = 4. Thisbehavior is also shared by the plane-wave fourth mnoment.1 2
7. CONCLUSION
In this paper we have presented numerical solutions of thefourth-moment equation for an initially finite, Gaussianbeam propagating in a two-dimensional random mediumcharacterized by a Gaussian correlation function. The re-sults reveal interesting characteristics of the propagationprocess and clearly show the distinction between the propa-gation of plane waves and finite beams. It is our hope thatthese solutions will prove useful in checking new anAlyticmethods for the yet unsolved fourth-moment equation<.- It isalso important that these computations be extended to high-er values of -y and t6 other correlation functions and, inparticular, to power-law, Kolmogorov-type functions.
9 1
ACKNOWLEDGMENT
The author would like to thank M. J. Beran for many helpfuldiscussions.
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