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- ͒ ͑ ͒ ⌽ ͑ ͒ ͵ 2Ј ͑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 ⌿ ͑ ͒ ͵ 2Ј ͑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, . 7 2 n=0 m=0 ͑ ͒ S x ͑r͒ · ٌ͔ln ͑r͖͒⌽ ͑rٌ͓͒ − In these two equations, the off-diagonal expansion coeffi- n E,n ͒ ͑ ͒ ⌽ ͑ ͒ ٌ ͑ ͔͒ ٌ S͑x͓͒͑ ⌳ cients nm are given by + E,n r · ln n r n r =0, 13 where is the magnetic permeability of the medium. The ͒͑ ء͒ ͵ 2 ͵ 2 ͑ ͒ ͑ ͒͑ ⌳ nm = d 1 d 2 n 1, Wxy 1, 2, m 2, , third equation may be disregarded at sufficiently high fre- ⌺ ⌺ ⌿ ͑ ͒ quency. There is a similar set of equations involving E,n r ͑8͒ S͑y͒͑ ͒ and n r . where ⌺ is the source plane. As shown in the Appendix, In the half-space into which the field propagates, the when the modal expansion coefficients satisfy the equation cross-spectral density matrix can be expressed in the form 2 ͒ S͑x͒͑⌬ ء ␥ ␦ , where ␦ is the Kronecker ␦ symbol, the= ͉ ⌳͉ nm n m nm nm ͑ ͒ ͑͒⌽ ͑ ͒ ⌽ ͑ ͒ ik n r1,r2 WJ r1,r2, = ͚ n E,n r1 E,n r2 e scalar mode expansion can be recast as an electromagnetic n coherent mode expansion identical to the expansion previ- ͒ S͑y͒͑⌬ ء ͓ ͔ ␥ ͑͒⌿ ͑ ͒ ⌿ ͑ ͒ ik m r1,r2 ously introduced 8,9 . + ͚ m E,m r1 E,m r2 e m ء + ͚ ͚ ⌳ ͑͒⌽ ͑r ͒ III. GEOMETRICAL OPTICS nm E,n 1 m n It has been shown ͓4͔ that for scalar waves an eikonal ͓S͑y͒͑ ͒ S͑x͒͑ ͔͒ ⌿ ͑r ͒eik m r2 − n r1 approach to propagating the coherent modes of a field can be E,m 2 ͒ ͑ ء͒⌿͑ ء⌳ applied to certain classes of sources. In the electromagnetic + ͚ ͚ nm E,m r1 case, the propagated modes are given by the expression m n
043831-2 GEOMETRICAL OPTICS LIMIT OF STOCHASTIC … PHYSICAL REVIEW A 77, 043831 ͑2008͒
͑ ͒ Wxx r ,r , 0 WJ ͑r ,r ,͒ = ͫ 1 2 ͬ. ͑16͒ 1 2 ͑ ͒ 0 Wyy r1,r2, In the geometrical limit, the modal decomposition takes the form ء ͑ ͒ ͑͒ ͑ ͒ −ikz1 ͑ ͒ ikz2 Wxx r1,r2, = ͚ n E,n 1 e E,n 2 e , n ͑17͒
ء ͑ ͒ ␥ ͑͒ ͑ ͒ −ikz1 ͑ ͒ ikz2 Wyy r1,r2, = ͚ n E,n 1 e E,n 2 e , n ͑18͒
FIG. 1. A diagram of an external ring cavity. A ray that is only W ͑r ,r ,͒ =0. ͑19͒ transmitted through the mirrors R and RЈ ͑not reflected͒ and travels xy 1 2 a distance d=2s1 +s2 +s3 +s4. The perimeter of the cavity is Each mode can be represented by a set of rays that propagate l=2s1 +2s2. in the z direction. At the two mirrors R and RЈ, every incident ray creates two outgoing rays: One that continues on the ͓S͑x͒͑ ͒ S͑y͒͑ ͔͒ straight line path with an amplitude t ͑t ͒ times the original ⌽ ͑r ͒eik n r2 − m r1 , ͑14͒ te tm E,n 2 amplitude and one that continues in a direction governed by the law of reflection with amplitude r ͑r ͒. Hence, the field ⌬S͑i͒͑ ͒ S͑i͒͑ ͒ S͑i͒͑ ͒ te tm where m r1 ,r2 = m r2 − m r1 . The cross-spectral den- at the detector is comprised of a series of collections of rays sity matrix for the propagated field is thus the sum of matri- for each mode: A collection of rays that did not reflect at ces, each of which represents a coherent field. In the source either R or RЈ, a collection that reflected once at each of plane, this decomposition has a scalar form. However, upon these mirrors, etc. In the detection plane, the effect of the propagation, the modes must be described vectorially. Each series of reflections and transmissions can be described by coherent mode is made up of two factors: An outer product the formulas of vectors and a relative phase, both of which depend on two ͑D͒͑͒ ͑ ͒ ͑͒ ͑ ͒ points. E,n = Tte kl E,n , 20
͑D͒͑͒ ͑ ͒ ͑͒ ͑ ͒ E,n = Ttm kl E,n , 21 IV. DEGREE OF POLARIZATION ͑D͒ ͑D͒ where E,n and E,n are the propagated coherent modes at Electromagnetic beams and electromagnetic plane waves the detector and make up special subsets of electromagnetic fields in which t tЈ eikd the field has a dominant direction of propagation, taken to be T ͑kl͒ = te te , ͑22͒ te Ј ikl the z direction. In such a case, both the electric and the 1−rtertee magnetic fields oscillate perpendicular to the propagation di- Ј ikd rection. The spectral degree of polarization of the field is ttmt e T ͑kl͒ = tm . ͑23͒ given by the expression ͓13͔ tm Ј ikl 1−rtmrtme Here d is the single pass length around the cavity to the 4 det WJ ͑r,r,͒ ͑r,͒ − ͑r,͒ detector, and l is the length of the cavity. Note that the total P ͑ ͒ϵͱ 1 2 r 1− = , transmission coefficients are independent of the mode index. ͓ ͑ ͔͒2 ͑r,͒ + ͑r,͒ Tr WJ r,r, 1 2 For this reason, the cross-spectral density matrix elements ͑15͒ are proportional to the original cross-spectral density matrix. In the detection plane, the cross-spectral density takes the Ն form where 1 2 are the eigenvalues of the cross-spectral den- sity matrix when both its arguments are evaluated at r. ͑D͒͑ ͒ Wxx 1, 2, 0 To illustrate the previous analysis, consider a laser beam WJ ͑ , ,͒ = ͫ ͬ, ͑24͒ 1 2 0 ͑D͒͑ ͒ incident on a ring cavity as shown in Fig. 1. Suppose that the Wyy 1, 2, Ј Ј ͑ˆ͒ Ј ͑ˆ͒ mirror R has reflection coefficients rte k and rtm k and where transmission coefficients tЈ ͑kˆ͒ and tЈ ͑kˆ͒ for the transverse ͑D͒͑ ͒ ͑ ͉͒ ͉2 ͑ ͒ te tm Wxx 1, 2, = Wxx 1, 2, Tte , 25 electric and magnetic fields, respectively. There are likewise reflection and transmission coefficients for the mirror R. The ͑D͒ W ͑ , ,͒ = W ͑ , ,͉͒T ͉2. ͑26͒ other two mirrors are assumed to be perfectly reflecting. yy 1 2 yy 1 2 tm Consider the case when the cross-spectral density matrix Because the TE and TM coefficients for reflection and trans- of the incident light has the form mission are, in general, different, the degree of polarization
043831-3 SCHOONOVER et al. PHYSICAL REVIEW A 77, 043831 ͑2008͒
a b
P(z) P(z)
kl kl
FIG. 2. ͑Color online͒ The degree of polarization, calculated by the geometrical optics model, at the output as a function of cavity size parameter kl. In panel ͑a͒, the incident field is more TE polarized, that is, the spectral density for the TE component of the field is 3 times larger than the spectral density for the TM component of the field, and the incident field has degree of polarization P=0.5. In panel ͑b͒, the incident field is more TM polarized, that is, the spectral density for the TM component of the field is 3 times larger than the spectral density for the TE component of the field, and the incident field has degree of polarization P=0.5. The mirrors R and RЈ are identical dielectric ⑀ ⑀ mirrors with thickness t=31 , =11.34 0, and = 0. at the output can be altered by changing the path length Assume that the source is broadband with ͑͒=␥͑͒ ͑ ͒2 around the cavity, i.e., changing the eikonal of the output ⌳͑͒ ͑ − c ͒ = =exp − 2 . It is clear that at any point in the half- 2 field. In Fig. 2, the degree of polarization at the detector is P plotted against the size of the cavity. As the cavity size be- space, the spectral degree of polarization =1. For broad- comes larger, the degree of polarization changes periodically. band light, it is appropriate to define the degree of polariza- ͓͑ ͔ ͒ In panel ͑a͒, the degree of polarization falls off rapidly from tion in terms of the mutual coherence matrix 14 , pp. 174ff the initial value of P=0.5. In panel ͑b͒, the degree of polar- rather than in terms of the cross-spectral density matrix as ization increases well above the initial value. The specific choices of mirror parameters make the cavity preferentially 4 det ⌫J͑r,r,0͒ P͑r͒ = ͱ1− . ͑29͒ favor the TM polarization. Mirrors can be designed with dif- ͓Tr ⌫J͑r,r,0͔͒2 ferent parameters to favor the TE polarization or to lessen the change in the state of polarization. For any plane z=zp in the crystal, the equal-time mutual As another example, consider a planar source with a uni- coherence matrix takes the form form cross-spectral density for all pairs of points. The cross- n − n spectral density matrix takes the form ˜ ⌳˜ͩ f s ͪ ͑0͒ zp c ͑͒ ⌳͑͒ ⌫͑ ͒ ͑ ͒ ͑ ͒ ͫ ͬ ͑ ͒ J zp,zp,0 = , 30 27 . ء = , , WJ ΅ nf + ns − ء ΄ ͒ ␥͑͒͑ ⌳ 2 1 ⌳˜ ͩ z ͪ ˜␥͑0͒ c p This source generates a z-directed plane wave. Suppose the propagated field is incident upon a biaxial medium having where tilde denotes Fourier transform. It is apparent that the the fast axis in the x direction and slow axis in the y direc- degree of polarization of the field in the crystal changes as a S͑x͒ tion. The eikonal along the fast axis is =nfz and the ei- function of propagation distance, viz., konal along the slow axis is S͑y͒ =n z. From Eq. ͑14͒, the s 2 2 2 n z cross-spectral density of the field in the media takes the form P͑z͒ = expͩ− dif ͪ, ͑31͒ ͑up to a nonessential prefactor͒ 2c2 ͑ ͒ ͑ ͒ ͑͒eik0nf z2−z1 ⌳͑͒eik0 nsz2−nfz1 / ͑ ͒ ͫ ͬ where ndif=ns −nf and the approximation c 1 has , ͒ ͑ ͒ ͑ ء = ,WJ r1,r2 ⌳ ͑͒eik0 nfz2−nsz1 ␥͑͒eik0ns z2−z1 been used. In Fig. 3, the degree of polarization as a function of axial ͑28͒ distance z is shown. The values for bandwidth and difference / where k0 = c, and ns and nf are the refractive indices along in refractive index are typical of those in a polarization sen- the slow axis and fast axis, respectively. sitive optical coherence tomography imaging experiment
043831-4 GEOMETRICAL OPTICS LIMIT OF STOCHASTIC … PHYSICAL REVIEW A 77, 043831 ͑2008͒
1 ͓7,8͔ may be expressed into the same form. The electromag- netic coherent mode decomposition developed previously 0.8 ͓8,9͔ is expressed in terms of solutions to the matrix integral
0.6 equation P(z) ء ء 0.4 ͵ 2Ј ͑ Ј ͒ ͑Ј ͒ ͑͒ ͑ ͒ ͑ ͒ d WJ , , · en , = n en , . A1
0.2 ͑͒ The en and n are the right eigenvectors and eigenvalues
0 of WJ ͑,Ј,͒. When the off-diagonal elements of WJ are 0 2 4 6 8 10 ͑ ͒ Distance [mm] nonzero, Eq. A1 represents a set of coupled equations. When the off-diagonal elements are identically zero, the two FIG. 3. ͑Color online͒ The degree of polarization, P͑z͒, calcu- equations become uncoupled. The cross-spectral density ma- lated from the present theory, as a function of distance traversed trix may then be expressed in the form ء through the biaxial media with ndif=0.0019 and =8.37 13 15 WJ ͑ , ,͒ = ͚ ͑͒e ͑ ,͒ e ͑ ,͒. ͑A2͒ ϫ10 rad/s. The centerline frequency is c =1.45ϫ10 rad/s. 1 2 n n 1 n 2 n ͓15͔. After the beam travels a distance 4 mm through the The recently introduced scalar mode decomposition of a biaxial medium, its degree of polarization changes from the stochastic source ͓7͔ can be expressed in terms of solutions initial value of 1 to a value of 0.1. to two uncoupled scalar equations, Although in the preceding case the spectral degree of po- ͒ ͑ ͒͑ءЈ ͒ ͑͒͑ءlarization is invariant on propagation, it is clear that the tem- ͵ 2 ͑ Ј ͒ d Wxx , , n , = n n , , A3 poral degree of polarization changes drastically. The change is a consequence of the different phase that each spectral component accumulates on propagation. ͒ ͑ ͒͑ءЈ ͒ ␥ ͑͒͑ء2 ͑ Ј ͒ ͵ d Wyy , , n , = n n , . A4 V. CONCLUSION The diagonal elements of the cross-spectral density matrix Modal decomposition of planar electromagnetic, second- are then expressed as in the form of Eqs. ͑4͒ and ͑5͒, and the ary, partially coherent sources has been developed and the expansion of the off-diagonal elements is given by Eqs. propagation from such sources has been considered in the ͑6͒–͑8͒. When the expansion coefficients take the form short wavelength limit. The electric cross-spectral density ␣ ͑͒ ⌳ ͑͒ ͱ ͑͒ͱ␥ ͑͒ i n ␦ ͑ ͒ matrix of the propagated field has also been studied; specifi- mn = n m e nm, A5 cally, a geometrical interpretation of changes in the degree of a set of vector modes E ͑,͒ can be constructed as follows: polarization due to propagation has been considered. The n ␣ ͑͒ E ͑ ͒ ͱ ͑͒ ͑ ͒ ͱ␥ ͑͒ i n ͑ ͒ examples make it clear that a geometrical model can be use- n , = n n , xˆ + n e n , yˆ . ful for analysis in either the space-time or in the space- ͑A6͒ frequency domains. Unlike the vectors in Eq. ͑A1͒, these vectors, although or- ACKNOWLEDGMENTS thogonal, are not normalized. Their normalized form is E ͑ ͒ R.W.S. and P.S.C. acknowledge support by the National n , EЈ͑,͒ = , ͑A7͒ Science Foundation ͑NSF͒ under CAREER Grant No. n ͱ ͑͒ 0239265. E.W. acknowledges support from the U.S. Air n Force Office of Scientific Research ͑USAFOSR͒ under Grant where No. F49260-03-1-0138, and from the Air Force Research ͒ ͑ ͒ E ͑ ͒͑ءLaboratory ͑AFRC͒ under Contract No. FA 9451-04-C-0296.  ͑͒ ͵ 2E n = d n , · n , . A8 J.C.S. acknowledges support by the NSF under Grant No. DMR 0425780 and the USAFOSR under Grant No. FA It is then not difficult to show that the expression 9550-07-1-0096. ͒ ͒ EЈ͑ ͒͑͑ء ͒  ͑͒EЈ͑ WJ 1, 2, = ͚ n n 1, n 2, A9 APPENDIX n In this appendix, it is shown that under certain circum- represents the original cross-spectral density matrix and is stances, the two kinds of coherent mode decompositions also in the form of Eq. ͑A2͒.
043831-5 SCHOONOVER et al. PHYSICAL REVIEW A 77, 043831 ͑2008͒
͓1͔ A. Sommerfeld and J. Runge, Ann. Phys. 340, 277 ͑1911͒. 2205 ͑2004͒. ͓2͔ S. Rytov, Dokl. Akad. Nauk SSSR 18, 263 ͑1938͒. ͓10͔ C. Tai, Dyadic Green Functions in Electromagnetic Theory ͓3͔ M. Born and E. Wolf, Principles of Optics, 7th ed. ͑Cambridge ͑IEEE Press, Piscataway, NJ, 1994͒. ͒ University Press, Cambridge, 1999 . ͓11͔ H. Liu, G. Mu, and L. Lin, J. Opt. Soc. Am. A 23, 2208 ͓ ͔ 4 A. M. Zysk, P. S. Carney, and J. C. Schotland, Phys. Rev. Lett. ͑2006͒. 95, 043904 ͑2005͒. ͓12͔ M. Alonso, O. Korotkova, and E. Wolf, J. Mod. Opt. 53, 969 ͓5͔ A. Starikov and E. Wolf, J. Opt. Soc. Am. 72, 923 ͑1982͒. ͑2006͒. ͓6͔ L. Mandel and E. Wolf, Optical Coherence and Quantum Op- ͓13͔ E. Wolf, Phys. Lett. A 312, 263 ͑2003͒. tics ͑Cambridge University Press, Cambridge, 1995͒. ͓ ͔ ͓7͔ K. Kim and E. Wolf, Opt. Commun. 261,19͑2006͒. 14 E. Wolf, Introduction to the Theory of Coherence and Polar- ͑ ͒ ͓8͔ F. Gori, M. Santarsiero, R. Simon, G. Piquero, R. Borghi, and ization Cambridge University Press, Cambridge, 2007 . ͓ ͔ G. Guattari, J. Opt. Soc. Am. A 20,78͑2003͒. 15 J. J. Pasquesi, S. C. Schlachter, M. D. Boppart, E. Chaney, S. ͓9͔ J. Tervo, T. Setälä, and A. Friberg, J. Opt. Soc. Am. A 21, J. Kaufman, and S. A. Boppart, Opt. Express 14, 1547 ͑2006͒.
043831-6