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frequency-scaling property. This behavior has been illustrated w1 = ckl into a random medium. Then, according to the theoretically and experimentally for a centrally loaded dipole HuygenSFresnel principle [ 7 ] , the field at a transverse dis- undergoing square-wave modulation by a reed switch. tance p in the plane x is

REFERENCES [ 11 M. K. Hu, “On measurements of microwave E and a field distribu-

tions by using modulated scattering methods,” IRE Trans. Micro- wave Theory Tech., vol. M’IT-8, pp. 295-300, May 1960.

[2] A. J. Bahr, V. R. Frank, J. P. Petro, and L. E. Sweeney, Jr., “Radar scattering from intermittently contacting targets,” IEEE Trans. AntennasFropagat., vol. AP-25, pp. 512-518, July 1977.

[3] V. R. Frank, J. P. Petro, and A. J. Bahr, “Backscattering from a cylindrical dipole centrally loaded by a time-varying impedance,” IEEE Trans. Antennas Fropagat., vol. AP-25, pp. 356-358, May 1977.

[4] Y. Hu, “Back-scattering cross section of a center-loaded cylindrical antenna,” IRE Trans. AntennasPropagat., vol. AP-6, pp. 140-148, January 1958.

[5] J. A. Stratton, Electromapetic Theory. New York: McGraw- Hill, 1941, pp. 488-490.

Multiple-Frequency Mutual Coherence Functions for a Beam in a Random Medium

RONALD L. FANTE, SENIOR MEMBER, IEEE

where

is the additional complex phase (due to turbulence) of a spher- ical wave propagating from (0, p1) to ( x , p). If we now ignore the log amplitude x but retain the phase S in accordance with our previous discussion, it is easy to see that

Absfruct-By using the phase-screen approximation dong with the extended Huygens-Fresnel principle, we have developed a method for computing the multiple-frequency mutual coherence functions for an ( P ’ - P2I2 arbitrary beam propagating in a random medium.

2 x

I. INTRODUCTION

It has been &Own [11-[31 that the study Of where ( - ) denotes an ensemble average. Because we have ignored the log amplitude, (2) is only a fair approximation in extended weak turbulence but becomes more accurate in mod- erately strong turbulence (because (x2) saturates but ( S 2 ) does not).

We now assume that S is a Gaussian random variable, and also that ( S 2 ) 3 (S)2 Experimental evidence indicates that S is Gaussian in weak and moderately strong turbulence, but there is no evidence as to whether this is also the case in very strong turbulence. When S is Gaussian, ( 2 ) becomes

transient signal propagation in a random medium requires a knowledge of the two-frequency and other higher order mul- tiple frequency coherence functions. At present, however, results are available only for the two-frequency coherence function, and these results are generally limited to plane wave calculations. In this paper we will demonstrate how the mul- tiple-frequency coherence functions can be approximately computed for arbitrary beam waves. The technique is based on the fact that calculations which approximate the random med- ium by phase screens and then use the Huygens-Fresnel prin- ciple are in excellent agreement with experimental data for the single-frequency coherence functions [4] , [ 51 . In fact it has been demonstrated [ 61 that the local phase fluctuations intro- duced on the beam as it propagates through the random med- ium are the dominant contributions to the field and intensity fluctuations, provided the turbulence is fairly strong and the beam diameter greatly exceeds the phase coherence length p o = ( k 2 C n 2 x r 3 I 5 where k is the signal wavenumber, x is the path length and Cn2 is the index of refraction structure con- stant. We will now use these ideas to calculate the multiple fre- quency coherence functions.

11. FORMAL RESULTS Suppose we have s source in the plane x = 0 with a tram-

verse distribution u0@1) and radiating a signal at frequency

Manuscript received July 7,1977; revised November 29,1977. The author is with the Rome Air Development Center/EEA,

Hanscom Air Force Base, MA 01731.

1 Note that ( S ) is not zero, as is sometimes assumed. In fact, when x and S obey Gaussian statistics, we have shown that

where the last step holds only for relatively weak turbulence.

U.S. Government work not protected by U.S. copyright.

622 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. AP-26, NO. 4, JULY 1978

For a collimated beam we may assume that

Equation (1) can also be used to calculate the fourth moment of the field, but this result will not be presented here. If we use this distribution for UO, substitute (6) into (3), then

Calculations of D 1 2 ( p - p', 0) have been made, employing make the coordinate transformations g = p1 - p 2 , Q = (pl + the Rytov approximation, by Ishimaru [21. The generalization p 2 ) / 2 , and fhally integrate on d2q, we find for the axial two- of his result to the two-source case is frequency coherence function

where @ ( K ) is the wavenumber spectrum of the index of refraction fluctuations and /3 = I t@ - p ' ) + (1 - t)bl - p 2 ) I. Note that when k l = k2 , (5) reduces to the usual result for the phase structure function [ S i . Although (5) was computed using the Rytov method, it can be shown [ 91 that it is approx- imately valid for strong as well as weak turbulence.

111. AXIAL COHERENCE FUNCTIONS

For mathematical simplicity we now assume that p = p r = 0, so that all measurements are made along the beam axis. In this case we need to know D12(0, p1 - p 2 ) . We have shown that if A12 E I kl - k2 I / h o < 1 where ko E ( k l -k k 2 ) / 2 and

@ ( K ) = 0.033 C , 2 ( ~ 2 + L o b 2 ) - l 1 I 6 exp ( - K ~ / K ~ ~ )

where LO is the outer scale size of the turbulent eddies, that the following approximation is valid:

p 1 2 ( o , P l - p 2 ) 2 g ( A 1 2 ) --hlP1 - P 2 lV ( 6 )

where

where w E a,-,-2 - i(k1 - k2) /2x . Unfortunately, the inte- gration in (10) cannot be done in closed form2 when v = 5/3. Therefore we shall approximate hgV by a quadratic function of

= I p1 - p 2 I for all ranges of the turbulence strength param- eter o12. We shall define ho so that exp (-hot2) = e-l at the same value at which exp (-hE5I3) = e-l. We then find that ho = (0 .545k02xCn2)6 /5 when u12 < a-5/6 and ho isgiven by (9) for o12 > ( Y - ~ ] ~ . If we use this approximation in (10) and then perform the integration on d 2 t we get

F 2

where F = koao2/x is the Fresnel number of the source, a n d

F(1 + 2a02ho)

1 + 4a02ho + F 2 Q =

Equation (1 1) is the general result for the axial two-frequency ' coherence function of a collimated beam in the limit when

For relatively small values of x, the coherence bandwidth f2 and the delay time Td are controlled by the behavior of the numerator of (1 1). The coherence bandwidth is determined from g(A12) 1 as

I k1 - k2 I Q k o .

with (Y = (ko/Km2x), T = ko-7/5Cn-12/5x-11/5, a = 0.39koZC,2L05~3x, b = 0.07k07/6C,2~11/6, and p = a f 1 0.39 Cn2L5l3x o.oo13Km7~3cn2X3. Also v = 5/3 for U12 < cu-5/6, and f22 - v = 2 for crI2 > ( Y - ~ / ~ where o12 is the strength of the turbu- lence and is defined by (112 1 . 2 3 k 0 ~ / ~ C , ~ ~ ~ ~ ' ~ . Finally, and the delay time is vrovided al < c ~ - ~ / ~ we have

-= + smaller terms C 2

..

h = 0 .545kO2Cn2x ,

and for u12 S a-5/6 we get

I 0 . 3 0 K m 1 / 3 C , 2 k 0 2 ~ , S A l 2

h = 0.11 k o 1 3 ~ 6 C n 2 x 5 / 6 [ $ + (A12)-1/6],

where c is the speed of light. Note that for relatively small x (weak turbulence) the coherence bandwidth is independent of the beam size, but this is not true for the delay time. In the

( 9 )

( i f a < A 1 2 < 7 . We have developed a computer program to evaluate (10) for v =

5/3, and have shown that (1 1) is a good approximation to (10).

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plane limit (ao + a) we find that (13) reduces to Td = 2hox/ cko2 = 0 .97k02 /5Cn2~615 for uI2 < a-5/6: and Td = 2hDx/ cko2 = o.6K,113cn2X2/C for o12 % cr-5/6. These results for a and Td agree with results obtained previously [3] , [ 101 in the plane wave limit.

If x is very large and fl > 4a02ho, the coherence band- width is determined by the behavior of the denominator of (1 1) and we find

i In the plane wave limit, (14) reduces to = (2hox/ cko2)2 = o.36K,2/3cn4x4c-2 which is equivalent to the results obtained in this limit by Sreenivasiah e t al. [ 1 1 1.

r31

[41

, 15 1

[71

REFERENCES H. Su and M. Plonus, “Optical-pulse propagation in a turbulent medium,”J. Opt. SOC. Amer., vol. 61, pp. 256-260, Mar. 1971. A. Ishimaru, “Temporal frequency spectra of multifrequency waves in turbulent atmosphere,” IEEE Trans. Antennas Propagat., vol. AP-20, pp. 10-19, Jan. 1972. C. Liu, A. Wernik, and K. Yeh, “Propagation of pulse trains through a random medium,” ZEEE Trans. Antennas Propagat.,

V. Banakh, G. Krekov, V. Mironov, S. Khmelevtsov, and S. Tsvik, “Focused-laser-beam scintillations in the turbulent atmos- phere,”J. Opt. SOC. Amer., vol. 64, pp. 516-518, Apr. 1974. R. Fante, “Comparison of theories for intensity fluctuations in strong turbulence,” Radio Sci. (New Series), vol. 11, pp. 215- 219, Mar. 1976. -, “Some results on the effect of turbulence in phase- compensated systems,” J. Opt. SOC. Amer., vol. 66, pp. 730- 735, July 1976. Z. Feizulin and Y. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Radiophys. Quantum Electron., vol. 10,

V. Tatarski, “The effect of the turbulent atmosphere on wave propagation,” U.S. Dept. Commerce, Springfield, VA, 1971. R. Fante, “Some new results on propagation of EM waves in strongly turbulent media,” IEEE Trans Antenms Propagat., vol. AP-23, pp. 382-385, May 1975. S. Hong and A. Ishiiaru, “Multiple scattering effects on coherent bandwidth and pulse distortion of a wave propagating in a random distniution of particles,” Radio Sci., vol. 10, pp. 637-644, June 1975. I . Sreenivasiah, A. Ishimaru, and S. Hong, “Two-frequency mutual coherence function and pulse propagation in a random medium: an analytic solution to the plane wave case,’’ Radio

V O ~ . AP-22, pp. 624-627, July 1974.

pp. 33-35, Jan. 1967.

SCL, V O ~ . 11 , pp. 775-778, Oct. 1976.

Guided Electromagnetic Waves in a Periodically Nonuniform Tunnel

JAMES R. WAIT, FELLOW, IEEE

Abstract-A boundary value analysis is outlined for the transnission in a circular tunnel that contains a thin axial conductor. The series impedance of the latter is allowed to vary in a periodic fashion through- out its length. Simplifications of the formal mode equation are achieved by invoking quasi-static conditions used in previous studies. It is shown

Manuscript received December 15,1977. The author is with the Cooperative Institute for Research in

Environmental Sciences, University of Colorado and National Oceano- graphic and Atmospheric Administration, Boulder, CO 80309.

that the axial nonunifonnity can be represented approximately as a modification of the series impedance of the equivalent transmission line of the composite tunnel structure.

Electromagnetic wave transmission in tunnel-like structures has a number of important applications in mining technology and transportation. Analytical studies of such problems [ 11, [ 21 have usually assumed that the guiding structure is laterally uniform, although the influence and exploitation of mode con- verters have been considered [3 ] . It is our purpose to outline the analysis for an idealized periodic modulation of the lateral properties. Such results can provide insight to the physics of the wave propagation along laterally varying tunnel structures.

Insofar as possible, the notation will be the same as used previously [ 1 1 , [ 21 and necessary derivations niu be brief.

The model we chose is depicted in Fig. 1. The air-filled circular tunnel of radius a contains a thin axial conductor of radius c offset from the center of the tunnel by a distance po. The homogeneous medium external to the tunnel has conduc- tivity u,, permittivity e@, and permeability fie. For conven- ience, a cylindrical coordinate system (p, 6, z ) is chosen to be coaxial with the tunnel while an associated or displaced cylin- drical coordinate system ( p f , G‘, z) is chosen to be coaxial with the axial conductor. To be specific, pf = 0 corresponds to p = po and @ = Go. Within the tunnel, the air (in spite of the presence of noxious fumes) is assumed to have zero conduc- tivity, permittivity eo, and permeability p0. To allow for a lateral variation of the structure, we stipulate that the effective series impedance ZJz) of the axial conductor varies period- ically throughout its length. This condition can be succinctly stated as

where E,(z) is the average value of the axial electric field at the surface of the conductor and I ( z ) is the total current carried by the conductor. Actually, because of the assumed thinness of the axial conductor, this impedance condition can be applied at any convenient value of @ f (i.e., E&) E, (p’ = c, @ f = 0, z)).

The periodic property is specified by

Z,(z) = Z,(z + nd) (2 )

for all values of z where n is an integer and d is the basic period. Floquet’s theorem [4] now tells us that the axial propagation constants of the discrete modes have the form r, = r + i(2nn/d) where r is the propagation constant of a basic mode. For a time factor exp (iot), the axial current on the conductor can thus be written

for a given basic mode, for example, in the limiting case of a uniform conductor I ( z ) = Io exp (-Fz). This case was anal- yzed earlier [ 11. Thus we use the same type of formulation to obtain the following expression for the z component U of the

U.S. Government work not protected by U.S. copyright.