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Nthorder Multifrequency Coherence Functions: a Functional Path Integral Approach Conrad M Nthorder multifrequency coherence functions: A functional path integral approach Conrad M. Rose and Ioannis M. Besieris Citation: Journal of Mathematical Physics 20, 1530 (1979); doi: 10.1063/1.524213 View online: http://dx.doi.org/10.1063/1.524213 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/20/7?ver=pdfcov Published by the AIP Publishing This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.173.125.76 On: Mon, 24 Mar 2014 17:26:14 Nth-order multifrequency coherence functions: A functional path integral approach~ Conrad M. Rose Naval Surface Weapons Center, Dahlgren, Virginia 22448 loannis M. Besieris Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061 (Received 21 August 1978) A functional (or path) integral applicable to a broad class of randomly perturbed media is constructed for the nth-order multi frequency coherence function (a quantity intimately linked to nth-order pulse statistics). This path integral is subsequently carried out explicitly in the case of a nondispersive, deterministically homogeneous medium, with a simplified (quadratic) Kolmogorov spectrum, and a series of new results are derived. Special cases dealing with the two-frequency mutual coherence function for plane and beam pulsed waves are considered, and comparisons are made with previously reported findings. 1. INTRODUCTION gation in the atmosphere and in lightguides, and underwater sound wave propagation) where the restriction to spatially There has been recently a mounting interest in the sub­ ject of propagation of pulsed signals through randomly per­ planar sources must be lifted. The problem of propagation of turbed media. Proposed high data rate communication sys­ pulsed beam waves in randomly perturbed environment has been investigated by the method of "temporal moments. "16 tems at millimeter and optical frequencies, remote sensing Since this technique is based on the two-frequency mutual schemes, low-frequency underwater sound signaling and de­ coherence function, it yields only information at the level of tection, interpretation of signals emitted by extraterrestrial second-order pulse statistics (e.g., mean arrival time, mean radio sources such as pulsars, all require a quantitative as­ square pulse width, etc.). Pulse shapes cannot be obtained sessment of stochastic pulse broadening. The latter leads to directly by this method; however, they can be synthesized by an irreversible degradation of a signal, in contradistinction superimposing individual temporals moments. An alterna­ to dispersive pulse spreading, which is a reversible phenom­ tive technique which, at least in principle, enables one to enon. (The receiver is usually equipped with "built-in disper­ compute approximately multifrequency coherence func­ sion" in order to make optimal use of the additional signal tions for arbitrary beam waves propagating in a random me­ bandwidth). dium was proposed recently by Fante. 17 It is based on the Earlier contributions in this area (cf., e.g., Refs. 1-3) phase-screen approximation along with the extended Huy­ were confined to weakly turbulent media and/or short prop­ gens-Fresnel principle. So far, only results pertaining to the agation paths, for which methods such as the Born and Ry­ two-frequency mutual coherence function have been report­ tov approximations were adequate. More recent theoretical ed in the literature. IS analyses which account for multiple scattering, large-scale inhomogeneities and long propagation distances are based One of the reasons why the aforementioned approaches for the most part on the parabolic equation for the complex (with the exception, perhaps, of the one suggested by l7 field amplitude and the Markov random process approxima­ Fante ) have not yielded sufficient information in connec­ tion" Within the framework of this formalism, it has been tion with multifrequency coherence functions for beams in a recognized that complete information about transient signal random medium is that the study of the asymptotic (or even propagation in random media requires the solutions of mo­ exact in special cases) behavior of these functions based on ment equations for the wave field at different frequencies and their governing local equations is a nontrivial problem. In different positions. Although a complete set of such equa­ contradistinction, recently formulated methods based on tions can be derived (cf. e.g., Ref. 5), solutions are available functional path integration (cf. Refs. 19-21) have the dis­ only for the two-frequency mutual coherence function, and tinct advantage that they work on a global rather than a local these results (both analytical and numerical) are generally level, thus making the algorithmic derivation of asymptotic 6 15 restricted to plane wave calculations. - solutions to higher-order moments easier. There exist physical situations (e.g., laser beam propa- It is our specific intent in this exposition to use the func­ tional path integration approach in order to evaluate nth­ order coherence functions for arbitrary source distributions. a)Research supported in part by the Office of Naval Research under con­ tract No. NOOOI4-76-C-0056 and the Graduate and Internal Research These evaluations will be performed within the confines of a Programs of the Naval Surface Weapons Center. simplified (quadratic) Kolmogorov spectrum. (Similar re- 1530 J. Math. Phys. 20(7). July 1979 0022-2488/79/071530-09$01.00 © 1979 American Institute of Physics 1530 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.173.125.76 On: Mon, 24 Mar 2014 17:26:14 suIts, for a single frequency and a specific source distribu­ \j2E (r,w;a) + k 2(w)[1 + f2(r,w) tion, have been determined by Furutsu22 using a different + E1(r,w;a)]E (r,w;a) = 0. (2.5) technique. ) It is clear that f2(r,w) accounts for deterministic inhomoge­ The structure of the paper can be outlined as follows: neities in the medium and, in light of a statement made earli­ Several preliminary concepts pertaining to pulse propaga­ er, it vanishes in the absence of such background profiles. On tion within the domain of validity of the quasioptics (or para­ the other hand, f1(r,w;a), a zero-mean random function, is bolic) approximation are developed in Sec. 2. A functional directly associated with the superimposed random effects. path integral applicable to a broad class of random media is constructed at the beginning of Sec. 3 for the nth-order mul­ For plane and beam propagation in the z-direction, it is tifrequency coherence function (a quantity intimately linked convenient to resort to the transformation to nth-order pulse statistics). This path integral is subse­ E (r,w;a) = 1ft (x,z,w;a)exp(ikz);r = (x,z),k = k (w). (2.6) quently carried out explicitly in the case of a nondispersive, deterministically homogeneous medium, with a simplified In the quasioptical description, the slowly varying complex (quadratic) Kolmogorov spectrum, and a series of new re­ random amplitude function 1ft (x,z,w;a) is described exceed­ suIts are derived. Special cases dealing with the two-frequen­ ingly well by the stochastic complex parabolic equation cy mutual coherence function for plane and beam pulsed i --Ift(x,z,w;a)a waves are considered in Sec. 4, where comparisons are also k az made with previously reported findings. Finally, the possi­ 1 2 bility ofasymptotic expansions of our general exact results in = - --'\7xtf!(x,z,m;a) 2k 2 the partially and fully saturated regimes are briefly discussed - !klx,z,w) El(x,z,w;a)] 1ft (x,z,w;a), z> 0. in Sec. 5. + (2.7a) In the presence of a deterministic profile (E +O), the para­ 2. PRELIMINARY CONCEPTS 2 bolic equation (2.7a) constitutes a valid approximation to The preliminary analysis, as well as the notation, in this (2.5) if the normals to the wavefronts in the unperturbed section will be specialized to the case of electromagnetic problem, where fl = 0, remain close to the z axis. wave propagation in a random channel. It should be empha­ Corresponding to the boundary condition E (x, sized, however, that the resulting stochastic complex para­ O,w;a)-Eo(x,w;a) for (2.5), one has the "initial" condition bolic equation is equally applicable in other physical areas (e.g., random underwater sound propagation). (2.7b) which incorporates all the information concerning the tem­ A. The quasioptics approximation poral frequency spectrum and the spatial distribution of the source at the initial plane z 0. Ignoring depolarization effects, the transverse, com­ = plex, electric field of radiation is governed by the stochastic In our subsequent formulation based on the functional Helmholtz equation path integral technique we shall require the fundamental so­ lution (referred to alternatively as the propagator or Green's \j2E (r,w;a) + (W/c)2Er (r,w;a)E (r,w;a) = 0, rEI? 3. (2. I) function) of (2.7). This quantity, denoted here by Here, w is the angular frequency, c is the speed of light in G (x,x',z,w;a), provides a link between the wavefunction vacuo, and Er (r,w;a )-the relative permittivity provided the 1ft (x,z,w;a),z > 0, and the boundary condition Ifto(x,w;a), medium is nonmagnetic-is a dimensionless, scalar, random viz., function depending on a parameter aEA, (A ,F,P) being an underlying probability measure space. If, in addition to dis­ Ift(x,z,w;a) = ( dx'G(x,x',z,w;a)lfto(x',w;a), (2.8) persion, the medium is characterized by either gain or loss, JR' the relative permittivity Eir,w;a) is complex.
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