Beam wave two-frequency mutual-coherence function andpulse propagation in random media: an analytic solution

1. Sreenivasiah and Akira Ishimaru

Pulse propagation in a random medium is determined by the two-frequency mutual coherence functionwhich satisfies a parabolic equation. In the past, numerical solutions of this equation have been reportedfor the plane wave case. An exact analytical solution for the plane wave case has also been reported for aGaussian spectrum of refractive-index fluctuations. Using the same approximation, an exact analytic solu-tion for the more general case of an incident beam wave is presented. The solution so obtained is used tostudy the propagation characteristics of the beam wave mutual coherence function at a single frequency aswell as at two frequencies. Simple expressions are obtained which qualitatively describe the decollimatingand defocusing effects of turbulence on a propagating beam wave. The time variation of the received pulseshape, on and away from the beam axis, is studied when the medium is excited with a delta function input.The results are presented for both collimated and focused beams.

1. Introduction

Pulse propagation characteristics and temporalfluctuation of a wave in a turbulent medium are deter-mined by the two-frequency mutual coherence function,which satisfies a parabolic equation under the "Markovrandom process approximation." 1-3 For the planewave case the two-frequency mutual coherence functionhas been extensively studied using numerical solutionsto the parabolic equation.3-10 An exact analytic solu-tion has been obtained by Sreenivasiah et al. 10 underthe approximation A(O) - A(p) p 2 , which correspondsto a Gaussian spectrum of refractive-index fluctuations.However, for the case of an incident beam wave, theparabolic equation governing the two-frequency mutualcoherence function involves five independent variables,and the solution is difficult even by numerical methods.In this paper we present an exact analytic solutionunder the approximation A (0) - A (p) p

2, which has

been found to be valid, under strong fluctuation con-ditions, for a medium characterized by a Gaussianspectrum of refractive-index fluctuations. Using thesolution so obtained, we study the propagation char-acteristics of the beam wave mutual coherence function

at a single frequency as well as at two frequencies. Weobtain simple expressions which qualitatively describethe decollimating and defocusing effects of turbulenceon a propagating beam wave. In particular we will notethat the turbulence places an upper limit on the dis-tance over which a beam-remains collimated and a lowerlimit on the spot size obtainable using a focused beam.We obtain conditions under which the maximum dis-tance of collimation and the minimum spot size arelimited by the turbulence. We study the variation ofthe beam wave two-frequency mutual coherence func-tion, with the normalized difference frequency variable.The time variation of the received pulse shape with adelta function input will also be studied. We willpresent the results for both collimated and focusedbeams.

1. Parabolic Equation

Consider a wave E(, w,t) propagating along the z axisin a random medium characterized by the relative di-electric constant

fr(r,W,t) = (,(TW,t))[1 + Ej(r,x,t)]. (1)

As the wave propagates in the z direction, its phaseprogresses substantially as i(kz - wt). Therefore, wewrite

I. Sreenivasiah is with University of Colorado, Department ofElectrical Engineering, Boulder Colorado 80303, and Akira Ishimaruis with University of Washington, Department of Electrical Engi-neering, Seattle Washington 98195.

Received 29 October 1978.0003-6935/79/101613-06$00.50/0.( 1979 Optical Society of America.

E(rwt) = u(z,,w,t) exp[i(kz - t)], (2)

where k = ko( Er)1/2, and ko is the free space wave-number.

It has been shown that under the Markov approxi-mation the two-frequency mutual coherence functionF satisfies the following equation3' 6-811

15 May 1979 / Vol. 18, No. 10 / APPLIED OPTICS 1613

I 2i -+ -v2 - V2 + - fh2Ak(0) + k2A2(0)cz kl1 k 2 24 2

- lk 2 [Al(Td - Vtd) + A2(d - Vtd)]l) r = 0, (3)

where r = (u(z,p,,co,tl)u*(z,p 2 ,Co2,t 2)), and k, and k2are the wavenumbers at cw, and w 2 , respectively, V2 and

2 are the 2-D Laplacians with respect to p, and P2, re-spectively, d = P1 - 2, td = t - t2 , V is the transversewind velocity, and A1 and A2 are given by the spectraldensities cInl and 4n2 of the refractive-index fluctua-tions at co, and )2.

Ai(T = 167r2 f Jo(Kp)I)(K)KdK. (4)

In Eq. (3), we have used Taylor's "frozen turbulence"hypothesis.'

Let us assume that the refractive index is fre-quency-independent within the frequency band underconsideration, and therefore Aj(p) = A2(p) = A(p).

Extensive numerical results have been obtained inthe past for the plane wave case, V2 = V2. The purposeof this paper is to present an analytic solution for thebeam wave case under the approximation

D(p) = A(O) - A(p) p2. (5)

The conditions under which Eq. (5) may be used havebeen discussed by Ishimaru and Sreenivasiah.9 For amedium which is characterized by a Gaussian spectrumof refractive-index fluctuations, J%(K) and D(p) aregiven by

4 (K) = (n2)13 1 212)D r) /7r 4 /

D(P) = 4x/r(ni) Ii1 - exp( - I4 C_2

(6a)

(6b)

where (n2) is the variance, and is the characteristicscale size of the refractive-index fluctuations. Theconstant of proportionality C in Eq. (6b) is given by

and we made use of Eq. (6b) and also assumed that thewind velocity V = 0.

In Eq. (8b), Vd and are 2-D Laplacian operators,and Vd and Vc are the gradient operators with respectto Pd and jC, respectively.

The field intensity E of a beam wave in free space isgiven by

E(z,p-,k,t) - u(z,p,k) explik[z - (t/vo)JI,

where

u(z,p,k) =- expl - ) -1 +iaz ~_2 +i.zI

(Q1X 1 Wv,

a =- + i-= a, + iai.7r W20 Ro

In Eq. (9), Wo is the beam size at z = 0, Ro is the focallength, and X and v are the wavelength and velocity oflight in free space, respectively. At z = 0 the two-fre-quency mutual coherence function is given by

where

r(z = O) = r(0,P,Pdkckd)

= Go exp(-aopd - bopc - icopc Pd), (10)

kcar .kd4 8

bo = 4ao,

.kdco = kai -t 2 a,.

and Go is a normalizing constant, chosen such that thetotal power in the incident beam is unity, and is givenby

(11)

Ill. Analytic Solution

To obtain the solution to Eq. (8b), we define the fol-lowing Fourier transform relationship:

C = 4x/,r(nl)h' = [A(O)]/12 .

We define r = rlr 2, where

r 2 = exp-8 kdA(O)z],

and transform Eq. (3) into the following form:

+ flV + V + YVd Vc + bPl)ri (zi,,cdkc,kd) = 0,

where

1 = (ikd)/(2klk 2 ),

-y= [kc/(klk2)]

= (klk2 )/4]1[A(0)1/12 1,

Pd = P1 P2,

Pc = /2 ( + 2),

kd = k +-k2,

k = '2 (l + 2)

(7)Ij(zWdTdkckd) = 1 ri exp(-iKd -c)d.c,

(2zr)2(12)

r(zPCPdkCkd) = I, exp(iK-d * Tc)d. d

(8a) Taking the Fourier transform of Eq. (8b) we obtainthe following differential equations for I:

(8b)

+ V -Kd - TKd Vd + bp I = O

with the boundary condition

(13a)

Ii(z = 0) = Io = G' exp(-aOPd - bOKd -coKdPd ), (13b)

where

G = Go/(4-7bo)

ao = [(c2)/(4bo)] + ao (b = 1/(4bo) (c = co/(2bo) )

(13c)

We seek the solution to I in the following form:

I = Gf exp(-glpd-g 2 Kd-g3d * d), (14)

1614 APPLIED OPTICS / Vol. 18, No. 10 / 15 May 1979

J

k C2 4Go = 1 1 + Wj72WO4 2 -7)'( Ro

where f, g1, g2 , and g 3 are functions of z only, and G isa constant.

By substituting Eq. (14) into Eq. (13a) and equatingthe like terms to zero, we obtain

[(bf)1(bz) - 40g1f = 0,

[(gj)g/(oz)] - 403g2 - 6 = 0,

[(g2)/(bZ)]- 13- _Yg3 + (/4) = 0,

[(693)/(bZ) - 40g 1g3 - 2ygy = 0,

with

G = [(Gob')/(rfo)]

fo = f(z = 0)

gio = gl(z = 0) = a0

g 20 = 2 (Z = 0) = b0

(15a)

and a0, b', and cS, which are given by Eqs. (13c), mustbe evaluated at kd = 0 in this case. We also note thatG, is a function of z.

IV. Numerical Results

A. Single-Frequency Mutual Coherence Function

(15b) In Eq. (17) G, gives the intensity of the beam on the

(15c) axis. The coefficient a determines thQ transverse(15d) correlation distance of the mutual coherence function,

and b, determines the beam spot size.Let us study how the random medium affects the spot

size of the beam. We define the radius W of the beamspot such that

(15e)

g30 = 3(Z = 0) = C

We solve Eqs. (15a)-(15d) with the boundary condi-tions (15e) and, using the transform relationship (12),obtain the final solution to F, as

ri = Go Il20) I exp(-apd - bp2 - icT,* P.d) (16)

where

a l~[(g3/2)]a =1 - 3A42]

b = /092),

c = 93/(292),

/ = sec[(46)1/ 2 z + 0ol,

g1 = (i.) tan[(43()1/2Z + o],

92 - +0 + 0 + C2,(40 4) 6

3 = -[-y/(213)] + c3f,

00 = tan-l [a'4 (a)1/21,

C2= bo- 3

= +IC3 ( 2,) Ifo

and a0, b' and c'o are given by Eq. (13c).When k1 = k2 , we must take the limiting form of Eq.

(16) or solve Eq. (8b) with / = 0. In either event weobtain the following solution for P8 = r (kd = 0):

rs(z,;5cTd) = G. exp[-aspd - bsp- icsc Pd],

[r(zp,. = Wpd = o)]/[r(zpc = Pd = 0)] = exp(-2) (18)

and obtain the following equation for the beam size W,of a collimated beam:

(, 2 112 r X2Z2 2 W0\121/2

Wc ()=Wo[1 + 7r2w 3 2 )] (19)

where rL[= (k 2/4)A(O)z] is the optical distance. Inobtaining Eq. (19) we made use of Eqs. (17) and(13c).

We notice that in free space the beam size remainsclose to Wo as long as z << (irW2)/X. The beam beginsto diverge rapidly in the far-field region z > (Wo)/X.The presence of the random medium causes the beamto diverge further and this effect becomes noticeablewhen the parameters of the medium are such that

T TL (-W) = 75/2(n2)1z >> 1. (20)

The beam may start diverging even in the near field ifthe parameters are such that

2 ( Xz 12L = V (n2)1-1 Z2>> 1.3 V (21)

Thus the turbulence places an upper limit on thedistance over which the beam remains collimated. Thislimiting distance Zmax may be found by forcing equalityin Eq. (21):

Zmax= ( W d )1/3 (22)

In Fig. 1 we have plotted this limiting distance against

14

12-

(17) 10-

whereE

E

G.,= (GobW)(g2s),

as, = g- R32M/49201,

bG = 1/092s),

C.,= (g30)/(2g 2 ),

gj = 5z + a0,

92 = 1/3 '26Z + y2a~Z2 + CozZ + b,

93 = yez 2 + 2yaoz + c,

8

6

4

2-10-10

0 5 10 15 20 25 30 35 40W cm

Fig. 1. Maximum distance of collimation in the presence of turbu-lence. Zmax VS Wo.

15 May 1979 / Vol. 18, No. 10 / APPLIED OPTICS 1615

I

the aperture size Wo, with the quantity (n )/1, the ratioof the variance to the scale size of the refractive-indexfluctuations, as a parameter.

The beam spot size of a focused beam, in the presenceof turbulence, is given by

Wf ( A )[ + 2 L ( )2]1/2. (23)

and does not depend on Pc or -d, we need only to studyF1. We may further restrict our study to the rangewhere Wcohl <W coh2. Using Eqs. (26) and (27) we maywrite this inequality as

(28)

We notice from Eq. (23) that, when the inequality (20)is satisfied, the beam spot size becomes larger comparedwith the free space spot size (= Xz/rWO). We also no-tice that there is a lower limit on the minimum spot sizethat is obtainable in the presence of turbulence. Thisminimum spot size Wfmin is obtained by taking the limitof Eq. (23) as Wo - :

Wfmin = (- TL) () = ( n7 2)()1z3) . (24)

We note that the limiting spot size is proportional toz 312.

B. Two-Frequency Mutual Coherence Function

To study the propagation characteristics of the beamwave two-frequency mutual coherence function we ex-amine rF and r 2 in Eq. (8b). First, let us consider thecoherence bandwidth. This is obtained by letting Pd= Pc = 0 and examining F as a function of the differencefrequency. From Eq. (8b) we write

r = rlr2 = r, exp[-d A(0)zj = rl expE 1/2 -Ža•) (25)

where Wcoh2 is the coherence bandwidth due to the fre-quency variation of the total cross section at =[(k1k2)/4]/A(0)1 of the medium per unit volume and isgiven by

Wcoh2 = vo(V7r(n2)lz)- 1 /2 (26)

where vo is the velocity of light in free space. It is moreconvenient to study the behavior of PI if the differencefrequency cOd is normalized through the relation

(CUd 1/2 2v01I = _ ; Wcohl 2 2 (27)M-coh /-Vr(n2)z 2

and the variation of r1 is examined with respect to .In Eqs. (27) COcohl is the coherence bandwidth, for aplane wave, due to the diffusion effect. For the planewave case,5 1 0 the parameter X arises in a natural wayand leads to a universal curve rF vs i7. For the beamwave case, the variation of with -j depends on theother parameters of the problem as well, namely, thewavelength X at the center frequency, the distance ofpropagation z, aperture size Wo, and the ratio (ni)/l ofthe variance to characteristic scale size of the refrac-tive-index fluctuations. However, the qualitative be-havior of r vs X is similar to that of a plane wave case,and over a wide range of these parameters, our calcu-lations showed that the coherence bandwidth of a beamwave, due to the diffusion effect, remains between Wcohland 3 cohl-

We must note that whichever is the smaller of CWcohlor coh2 is dominant in determining the total coherencebandwidth. Since the expression for r2 is very simple

Also, in practice Cvd cannot exceed the center frequencyWC. So as long as C0,oh, and X

0coh2 are both greater than

wc, the random medium does not have any significanteffect on the two-frequency mutual coherence function.The condition cohl < c may be written

73/2(l) I > 1.

In Figs. 2 and 3 we have shown the variation of rwith respect to i7 for collimated and focused beams usingWo and PC as parameters. For our study we chose X =0.6943 m, z = 10 km, and (n2)/ = 1014 m-1. The co-herence bandwidth Wcohl corresponding to these pa-rameters is 3.4 X 1014 rad/sec, while Wc = 27.1 X 1014,and the inequality (29) is satisfied. Figure 2 shows theresults on the beam axis (Pc = Pd = 0) for a collimatedbeam with the aperture size Wo as a parameter. As theaperture size is increased, the results should approachthose of a plane wave. We found that the rF vs 7 curvefor the beam wave practically coincides with the planewave curve when Wo > 1 m. Note also that the band-width becomes maximum when Wo is close to theFresnel size (7rW = z; Wo = 4.37 cm).

In Fig. 3(a) we have shown the results for a collimatedbeam, with Pc as a parameter. The curves with solidlines are for Wo = 10 cm, while those with dashed linesare for W = 25 cm. We notice that as we move awayfrom the beam axis, the F1 vs curves get narrower, andhence the coherence bandwidth decreases. While thisnarrowing effect is significant for W = 10 cm, the F1vs X7 curves for Wo 25 cm are very close together. Thisis to be expected since F1 does not depend on PC for aplane wave, and collimated beam results approach thoseof a plane wave for large aperture sizes.

The results for a focused beam are shown in Fig. 3(b).We notice that the amplitude and phase of r1 changemore rapidly with the difference frequency as we moveaway from the beam axis. When Wo is changed from10 cm to 25 cm, there is no change in the variation of 1with 7 on the beam axis. While the amplitude curvesfor W = 25 cm are narrower compared with those of Wo= 10 cm, the phase curves are coincident for the twocases. However, this may not be true for all the valuesof Wo.

C. Pulse PropagationThe average intensity P(pc,t) at a point PC on the

receiving aperture, due to an input pulse Pi(t), is givenby

(30)

where G (pc,t) is the output power due to a delta func-tion input Pi(t) = 6(t) and is given by

1616 APPLIED OPTICS / Vol. 18, No. 10 / 15 May 1979

(29)

Vr 2) Z 3(ni - > .

4 N

Po �p�,t - -Z - _Z - t dt'," = fPi(t')G �p,,t VO

D t l o | S i i i i

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.71

Fig. 2. r(pc = Pd = 0,77) vs 7 for a collimated beam. = 0.6943 Am;z = 10 km and (n)/ = 10 4 m. Wo = mm,1 cm, 3.5 cm, 10 cm,

25 cm, and 2 1 m.

280

240

( z ~1G p,,t -- = -Sodr(zpcPd = 0,Wd)vol 27r

X exp[-iwOjt - Z/VO)], (31)

200 where

160 , r(Z,P0 ,Pd = 0,Wd) = Fl(Z,PCPd = 0,Wd)r2

= = ex i~~~!i CWd 121.(2-120 1P~~~~1(Z,Pc,Pd =0,CWd) p(2

o D [ ~~~~~~~~~~2 Uh)]

80 For the purpose of our study, we are assuming that Pi (t)

.40 is independent of Pc; however, any dependence of Pi (t)on Pc could be easily taken care of. We may write

o G [pct - (z/vo)] in the following convolution form:so

(33)

where

( -vo 2w

X exp[-iWd(t - z/vo)]dd (34)

Since 12 is independent of Pc, I2 is also independent ofPc and can be given simply by12 [ x 1/ vol (Vi/27r)1/2 T 2 T exp/2 (t ]

where T2 = 1/Wcoh2, and Wch2 is given by Eq. (26).may write I1 in the following form:

(35)

We

0vw1

a:I 1(Pctn) = 2 Sfl(zpc,Pd = 0,77) exp(-it.77 2 )d(?72 ), (36)

2rTj

where T1 = l/Wcohl and t = (t - z/vo)/T1 . The totalreceiver power is given by

Pr (t - = 5TPO - _pct-Ar(c)dc,(a(a)

(37)

0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0'7

(b)

Fig. 3. r(PcPd = ,n)/rP(P,,Pd = O0 = 0) vs v. X = 0.6943 pm; z= 10km and (n)/1 = 0-14 m- 1 . (a) Collimated beam: Wo = 10 cm;

Pc= 0,5cm,10cm,and15cm. Wo=25cm;p,=0and15cm. (b)Focused beam: Wo = 10 cm and 25 cm; Pc = 0, 5 cm, 10 cm, and 15

cm.

where A, (Pc) represents the receiver characteristics.For the purpose of this paper we will not be concerned

with the form of the input pulse shape and the receivercharacteristics since these are not affected by the me-dium. Since the expression for I2 is simple enough, wewill restrict ourselves to the study of Il(pc,tn). We usethe solution in Eqs. (16) and (36) to compute Il(pctn).The results are shown in Figs. 4(a) and 4(b) for colli-mated and focused beams, respectively. The parame-ters chosen are the same as those used in Figs. 3(a) and3(b). As seen from the figures, we note that as we moveaway from the beam axis the pulse amplitude decreasesand the width increases rapidly. This widening effectis more severe for the focused beam, especially for largeraperture sizes. For a collimated beam, the pulse am-plitude decreases more slowly as we move away from theaxis and as the aperture size Wo is increased. This isto be expected since for a plane wave the pulse ampli-tude and the shape are to be invariant with respect toPc Finally, I,(pc > O,t,) has a nonzero value for t,, <0, indicating that the incoming wavefront is concave asviewed from the receiver.

15 May 1979 / Vol. 18, No. 10 / APPLIED OPTICS 1617

1.0

1o

o 0.8-

° 0.6

6 0.4

I- 0.2

0

J U0 IG �p�,t - z = fIi�p,,t - z - t' I2(t )dt',

U'_

Pc.0 Collimated Beamz-10 km, /I . I - m- W-lOcm- W0 -25 cm

5cm

2.0 2.5 3.0 3.5tn=(t-z/v.)/ T

(a)

Focused Beamz-10km, <nl-2>/ 14

-W. 10cm---We-25cm

Ic-0; W. - 10,25 cm

5cm

5cm

0 0.5 1.0 1.5 2.0 2.5

tn=(t-Z/Vb)T,(b)

Fig. 4. I(pctj) = I(pct.)-Ti vs t = t - [z/(vo]/T 1. X = 0.6943pm; z = 10 km and (n2)/l = 10-14 m-1 . (a) Collimated beam: Wo= 10 cm and 25 cm; Pc = O. 5 cm, 10 cm, and 15 cm. (b) Focused beam:

Wo = 10 cm and 25 cm; Pc = 0, 5 cm, and 10 cm.

V. Conclusion

In this paper we obtained an analytic solution to thebeam wave two-frequency mutual coherence functionand studied its propagation characteristics. We notedthat the random medium places an upper limit on thedistance over which a beam remains collimated and alower limit on the spot size obtainable using a focusedbeam. We gave quantitative expressions showing theselimits [Eqs. (22) and (24)] and the conditions underwhich the defocusing [Eq. (20)] and the decollimatingeffect [Eq. (21)] sets in. We have graphically presentedthe variation of the mutual coherence function with thedifference frequency and the pulse shape on the re-ceiving aperture on and away from the beam axis.Though we have presented the results for a few cases,the solution as given by Eq. (16) may be used to obtainthe results for any practical situation. Finally, Eqs. (28)and (29) would enable one to determine if the turbu-lence has any frequency effects in a particular situationand if so, which of the effects, dispersive or diffuse, isdominant.

This work was supported by Deputy for ElectronicTechnology (RADC) under AF19628-77-C-0045 andNational Science Foundation under ENG77-12544.

References

1. V. I. Tatarski, The Effects of the Turbulent Atmosphere on WavePropagation (U.S. Department of Commerce, Springfield, Va.,1971).

2. Y. N. Barabanenkov, Y. A. Kravtsov, S. M. Rytov, and V. I. Ta-tarski, Sov. Phys. Usp. 13, 551 (1971).

3. A. Ishimaru, Wave Propagation and Scattering in RandomMedia, Vols. 1 and 2 (Academic, New York, 1978).

4. C. H. Liu, A. W. Wernik, K. C. Yeh, and M. Y. Youakim, RadioSci., 9, 599 (1974).

5. L. C. Lee and J. R. Jokipii, Astrophys. J. 201, 532 (1975).6. L. C. Lee, J. Math. Phys. 15,1431 (1974).7. S. T. Hong, I. Sreonivasiah, and A. Ishimaru, IEEE Trans. An-

tennas Propag. AP-25, 822 (1976).

8. I. Sreenivasiah and A. Ishimaru, "Plane wave pulse propagationthrough atmospheric turbulence at millimeter and opticalwavelengths," Department of Electrical Engineering, Universityof Washington, Research Report AFCRL-TR-74-0205 (1974).

9. I. Sreenivasiah, "Two-frequency mutual coherence function andpulse propagation in continuous random media: Forward andbackscattering solutions," Ph.D. Dissertation, University ofWashington, Seattle (1976).

10. I. Sreenivasiah, A. Ishimaru, and S. T. Hong, Radio Sci. 11, 775(1976).

11. L. Erukhimov, I. Zarnitsyna, and P. Kirsch, Izv. VUZ. Radiofizika16, 573 (1973).

1618 APPLIED OPTICS / Vol. 18, No. 10 / 15 May 1979

1.4

1.2-

1.0-

Z' 0.6-

I. -

0.2-

0

3. T

2.54

2.04

1.5 +

1.0

0.5+

0 !5 -1.0 -0.5

_Z ------ �__ -, _