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Geometrical Optics Limit of Stochastic Electromagnetic Fields University of Pennsylvania ScholarlyCommons Departmental Papers (BE) Department of Bioengineering April 2008 Geometrical optics limit of stochastic electromagnetic fields Robert W. Schoonover University of Illinois Adam M. Zysk University of Illinois P. Scott Carney University of Illinois John C. Schotland University of Pennsylvania, [email protected] Emil Wolfe University of Rochester Follow this and additional works at: https://repository.upenn.edu/be_papers Recommended Citation Schoonover, R. W., Zysk, A. M., Carney, P. S., Schotland, J. C., & Wolfe, E. (2008). Geometrical optics limit of stochastic electromagnetic fields. Retrieved from https://repository.upenn.edu/be_papers/115 Copyright American Physical Society. Reprinted from American Physical Society, Volume 77, Article 043831, April 2008, 6 pages. Publisher URL: http://link.aps.org/abstract/PRA/v77/e043831 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/be_papers/115 For more information, please contact [email protected]. Geometrical optics limit of stochastic electromagnetic fields Abstract A method is described which elucidates propagation of an electromagnetic field generated by a stochastic, electromagnetic source within the short wavelength limit. The results can be used to determine statistical properties of fields using ar y tracing methods. Keywords light propagation, ray tracing Comments Copyright American Physical Society. Reprinted from American Physical Society, Volume 77, Article 043831, April 2008, 6 pages. Publisher URL: http://link.aps.org/abstract/PRA/v77/e043831 This journal article is available at ScholarlyCommons: https://repository.upenn.edu/be_papers/115 PHYSICAL REVIEW A 77, 043831 ͑2008͒ Geometrical optics limit of stochastic electromagnetic fields Robert W. Schoonover, Adam M. Zysk,* and P. Scott Carney Department of Electrical and Computer Engineering and The Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA John C. Schotland Department of Bioengineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA Emil Wolf Department of Physics and Astronomy and the Institute of Optics, University of Rochester, Rochester, New York 14627, USA ͑Received 10 September 2007; published 24 April 2008͒ A method is described which elucidates propagation of an electromagnetic field generated by a stochastic, electromagnetic source within the short wavelength limit. The results can be used to determine statistical properties of fields using ray tracing methods. DOI: 10.1103/PhysRevA.77.043831 PACS number͑s͒: 42.15.Ϫi, 42.25.Kb, 42.25.Ja I. INTRODUCTION though not necessarily mutually orthogonal, are also uncor- related and thus the propagated cross-spectral density matrix The behavior of monochromatic sources and fields at can be reconstituted by adding the matrices formed by taking short wavelengths can be described, in many cases, by geo- the outer product of each propagated mode with itself and metrical optics. The foundations of geometrical optics of sca- multiplying by the appropriate weight. Propagation of elec- lar wave fields have been discussed by Sommerfeld and tromagnetic fields can be described using dyadic Green func- ͓ ͔ Runge in a classic paper 1 , and their analysis has been tion methods. However, the implementation of such methods ͓ ͔͑ ͓ ͔ ͒ generalized by Rytov 2 see also 3 , Chap. 3 to mono- may be prohibitively numerically intensive. Thus it is desir- chromatic electromagnetic fields. able to describe the propagation by approximate asymptotic A generalization of the results of Sommerfeld, Runge, and methods. It will be shown that for certain systems, the propa- Rytov to broader classes of fields presents some difficulties, gation can be adequately carried out within the framework of especially when these fields are stochastic. However, a solu- geometrical optics, thus greatly reducing the computational ͓ ͔ tion for stochastic scalar fields was obtained 4 via the so- complexity of the analysis and making available a wealth of called coherent mode representation of scalar wave fields of computational tools for ray-tracing. The main part of this ͑ ͓ ͔ ͓ ͔ ͒ any state of coherence see 5 or 6 , Sec. 4.1 .Inthe paper is organized as follows: In Sec. II, the scalar decom- present paper, that analysis is extended to stochastic, planar, position of planar, stochastic, electromagnetic sources is re- secondary electromagnetic sources and to the fields that they viewed; in Sec. III, the propagation of the modes is consid- generate. ered within the high-frequency limit; finally, in Sec. IV, two In the high-frequency limit of electrodynamics, the vector examples are given to illustrate this geometrical method for fields are often approximated by scalar waves or by rays, computing the degree of polarization. depending on the application. The second-order correlations of fields in the scalar approximation have been extensively studied ͓6͔. The average second-order correlation properties II. MODAL DECOMPOSITION of electromagnetic fields can be described by correlation ma- Consider a random, statistically stationary, secondary trices in either the temporal or in the spectral domain. The electromagnetic planar source represented by the mutual co- time domain representation is often employed because detec- herence matrix tors necessarily time integrate any signal received. The spec- tral domain representation, however, offers certain advan- ⌫J͑␳ ,␳ ,␶͒ϵ͓⌫ ͑␳ ,␳ ,␶͔͒ ͑1͒ tages, especially in connection with propagation in 1 2 ij 1 2 dispersive and absorptive media. ͒ ͑ ␳ ͒ ͑␳ ␶͔͒͑͘ء ͓͗ The electromagnetic cross-spectral density matrix of a = Ei 1,0 Ej 2, , 2 planar source, which describes the spatial correlations in the ␳ ͑ ͒ spectral domain, may be expressed in terms of so-called co- where the k k=1,2 , are position vectors of a point in the ␶ herent modes ͓7͔. These modes are orthogonal and also mu- transverse plane, is a time delay, i and j label the x or y tually uncorrelated. The modes may be propagated, and components, the asterisk denotes complex conjugation and the brackets denote an ensemble average over the fluctuating electric field. The Fourier transform of the mutual coherence matrix is the cross-spectral density matrix, *Present address: Medical Imaging Research Center, Electrical and Computer Engineering Department, Illinois Institute of Tech- ͑␳ ␳ ␻͒ϵ͵ ␶⌫͑␳ ␳ ␶͒ i␻␶ ͑ ͒ nology, Chicago, IL 60616. WJ 1, 2, d J 1, 2, e . 3 1050-2947/2008/77͑4͒/043831͑6͒ 043831-1 ©2008 The American Physical Society SCHOONOVER et al. PHYSICAL REVIEW A 77, 043831 ͑2008͒ A random, planar, stochastic electromagnetic source con- ͒ ͑ ͒ ⌽ ͑ ␻͒ ͵ 2␳Ј␾ ͑␳Ј ␻͒ ٌЈ ϫ ͑␳Ј sidered in the space-frequency domain can be described by n r, =− d n , yˆ · GJ ,r , 9 two sets of scalar modes ͓7͔. The cross-spectral density at ⌺ every point in the source plane can be written in terms of ͒ ͑ ͒ these modes. Propagation of the modes may be expressed by ⌿ ͑ ␻͒ ͵ 2␳Ј␺ ͑␳Ј ␻͒ ٌЈ ϫ ͑␳Ј n r, = d n , xˆ · GJ ,r , 10 the method of dyadic Green functions. However, the imple- ⌺ mentation of this approach may be quite complicated in prac- tice. In many cases of interest, methods of geometrical optics where GJ͑r,rЈ͒ is the dyadic Green function for the wave may be applied to each of the coherent modes which sim- equation in the half-space ͓10͔. This method of propagating plify the calculation. It is the purpose of this paper to de- the modes has been previously discussed ͓11͔ and the propa- velop such an approach. gation of the correlation matrices using an approximation of Modal expansions have previously been expressed in the dyadic Green function has been treated elsewhere ͓12͔. terms of the solutions to two- or three-dimensional coupled For sources that give rise to beams, the z component of the Fredholm integral equations ͓8,9͔. However, solving these field can be neglected and the cross-spectral density matrix integral equations is not often tractable. Instead, a different remains a 2ϫ2 matrix. In general, though, the cross-spectral modal representation has been introduced in which the diag- density matrix cannot be reduced to a 2ϫ2 representation. onal elements of the cross-spectral density are expressed in It is assumed that the propagated modes may be expanded ͕␾ ͖ terms of two sets of independent coherent modes n and in a series by asymptotic evaluation of the integrals in Eqs. ͕␺ ͖ n which are solutions of uncoupled integral equations: ͑9͒ and ͑10͒ for sufficiently large values of the wave number ϱ k=␻/c. The leading order term may be well approximated ␳ ␻͒␾ ͑␳ ␻͒ ͑ ͒ by the expression͑ء␳ ␳ ␻͒ ␭ ͑␻͒␾͑ Wxx 1, 2, = ͚ n n 1, n 2, , 4 n=0 S͑x͒͑ ͒ ⌽ ͑ ␻͒ ⌽ ͑ ͒ ik n r ͑ ͒ n r, = E,n r e , 11 ϱ ⌽ ͑ ͒ S͑x͒͑ ͒ where E,n r and n r are the frequency-independent am- ␳ ␻͒␺ ͑␳ ␻͒ ͑ ͒ ⌽ ͑ ␻͒͑ء␳ ␳ ␻͒ ␥ ͑␻͒␺͑ Wyy 1, 2, = ͚ n n 1, n 2, . 5 plitude and the eikonal of n r, , respectively, and the n=0 superscript x refers to a Cartesian component of the electric ␾ ␭ field. Upon substituting this expression into Maxwell’s equa- Here n and n are the eigenfunctions and the eigenvalues, respectively, of W , and ␺ and ␥ are the eigenfunctions tions, one obtains three coupled first-order equations of dif- xx n n ferent power in k ͑see ͓3͔, Chap. 3͒. The first is the eikonal and the eigenvalues of Wyy. The off-diagonal elements of the cross-spectral density matrix may be expressed in the form equation, which takes the form ͒ ͑ ͒ S͑x͒͑ ͉͒2 2ٌ͉͑ ϱ ϱ n r = n r , 12 ͒ ͑ ␳ ␻͒␺ ͑␳ ␻͒͑ء␳ ␳ ␻͒ ⌳ ͑␻͒␾͑ Wxy 1, 2, = ͚ ͚ nm n 1, m 2, , 6 where n͑r͒ is the refractive index. The second equation is the m n=0 =0 so-called transport equation for the field amplitudes, ϱ ϱ ͑ ͒ 1 ͑ ͒ ͒ ͑ S x ͑ ͒ ٌ͔⌽ ͑ ͒ ٌ͕2S xٌ͓ ء ء ͑␳ ␳ ␻͒ ⌳ ͑␻͒␺ ͑␳ ␻͒␾ ͑␳ ␻͒ ͑ ͒ n r · E,n r + n r Wyx 1, 2, = ͚ ͚ nm m 1, n 2, .
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