Theory of partial coherence for weakly periodic media

Transcription

1 2682 J. Opt. Soc.Am. A/Vol. 22, No. 12/December 2005 B. J. Hoenders and M. Bertolotti Theory of partial coherence for weakly periodic media Bernhard J. Hoenders University of Groningen, Institute for Theoretical Physics and Materials Science Centre, Nijenborgh 4, NL-9747 AG Groningen, The Netherlands, and Dipartimento di Energetica, Università "La Sapienza" di Ramo, Via Scarpa 16, 00161, Rome, Italy Mario Bertolotti Dipartimento di Energetica, Università "La Sapienza" di Roma, Via Scarpa 16, 00161, Rome, Italy Received January 21, 2005; accepted February 24, 2005 The second-order theory of partial coherence for scalar and TE or TM fields is developed for weakly periodic media, and the van Cittert-Zernike theorem of classical coherence theory is generalized for such media. The coherence properties of a wave field, generated by a quasi-homogeneous source distribution at the entrance plane of a finite weakly periodic medium, are calculated both inside such a structure and in the far field. The second-order theory of partial coherence for pulse propagation through weakly periodic media is also developed Optical Society of America GelS codes: , , , , , INTRODUCTION The theory of partial coherence, rooted already in the 19th century, has been extensively analyzed for classical1 as well as for quantized fields.2 The theory, however, has not been extended to the analysis of the propagation of coherence functions through inhomogeneous structures such as, e.g., photonic bandgap structures. Although early work on the theory of partial coherence of optical cavities that were considered as a periodic succession of reflections can be found in Refs. 3 and 4, the theoretical treatment of the problem still lacks a general formulation. It is therefore the aim of this paper to construct the theory of partial coherence for wave propagation in weak media that takes into account the spatial and temporal statistical properties of the incoming field. We will restrict ourselves to weakly scattering media, i.e., it is assumed that the propagation of the pertinent field quantities can be described in first-order Born approximation. This restriction allows us to explicitly calculate pertinent quantities such as the second-order coherence function. Moreover, from the results for a similar problem arising in solidstate physics (the theory of electrons weakly bounded to a periodic potentiai5), it is to be expected that this special case already reveals the intrinsic physical properties of the problem such as the occurrence of bandgaps and the description of the field in terms of the Bloch wave functions. The results obtained in Section 2 justify this expectation. We first develop a theory of deterministic TE/TM wavefield propagation in a weakly periodic medium. Then the second-order theory of partial coherence is developed. Both cases of semi-infinite and finite photonic structures are considered and a generalization of the van Cittert- Zernike theorem is obtained. Coherence theory for pulses propagating through both semi-infinite and finite photonic structures is considered next. 2. PROPAGATION IN WEAKLY PHOTONIC MEDIA We consider propagation of electromagnetic TE or TM waves through a weak photonic material with constant magnetic permeability and a periodically varying dielectric function ε(r), r=(x,y). We restrict ourselves to twodimensional problems, which enables us to decompose the vectorial electromagnetic problem into two scalar problems, i.e., TE and TM fields. Then the governing equation for time-harmonic TE waves with frequency ω reads as if ω=ck0 and n2=ε(r). The magnetic field follows from the Maxwell equation xe=-ab/ t. For the TM case the pertinent equations reads as (1) and the electric field follows from the Maxwell equation xh=ad/ t (the geometry is depicted in Fig. 1). The calculation of the propagation of the partial coherence function for weakly periodic media will proceed in three steps: (a) the determination of the basic set of functions (plane waves for free space), (b) the solution of the deterministic boundary-value problem, and (c) the introduction of the statistics. (2) /05/ /$ Optical Society of America

2 B. J. Hoendersand M. Bertolotti Vol.22, No. 12/December2005/J. Opt. Soc.Am.A 2683 first-order Born approximation. They are the basic set of functions for the description of the field, generalizing the set of plane waves for a homogeneous space as the complete basis for the solution space of the propagator of Eq. (3). The denominator of these wave functions could become zero: k02 - (k + n)2 = O. (8) Fig. 1. Geometryofthe system. The equation to be solved in first-order Born approximation reads as where ψ denotes Ez or Hz, respectively, if for simplicity the term Vη(r) V is neglected in Eq. (2). The function (3) This condition is easily recognized as the von Laue condition, the basic condition for the occurrence of Bragg reflection in crystals. If this case occurs, the approximation of Eq (7) breaks down and another expression for Eq. (7) has to be used. 5 We now turn our attention to the second problem to be solved, namely, the solution of the Dirichlet boundaryvalue problem This problem will also be solved in a perturbative way To this end, we recall6 that the field ψ(r) can be written in terms of the first-order Bloch functions [Eq (7)]because they are the complete set of modes in the R2 space for the solutions of Eq. (7): n2(r) = Σ an exp(in r), a0 = 1 n (4) denotes the periodically varying refractive index of the medium k0=ω/c, and r=(x,y). The vector (9) nx = 0, ± 1, ± 2,..., n x = 0, ± 1, ± 2,.., (5) not to be confused with the scalar refractive index n2(r), denotes the successive orders in the Fourier expansion of the refractive index n 2 (r), which is a periodical function with periodicity a in the x direction and periodicity b in the y direction. For free space the basic set of modes is the set of plane waves A(k)exp(ik r). The first-order approximation to the free-space solutions is then obtained by the ansatz: k 2 = k02, (6) (10) where r'σ=(x,o) and kσ=(ky,o). Taking the Fourier transform of both sides of Eq. (10) we end up with an equation for A(kσ): (11) where ψ~(kσ)denotes the Fourier transform of ψ(rσ) and nσ=(ny,o). The solution ofeq. (11)for A(k σ ) is obtained in the first-order Born approximation iterating Eq. (11) once: with unknown coefficients dn. The reason we try the solution of Eq (6) is that, effectively in the first-order Born approximation, the term k02n2(r)ψ(r) is replaced by k02n2(r) A(k)exp(ik r) and each term occurring in Eq. (6) is an eigenvector of 2 with eigenvalue (k+n)2 Inserting Eq. (6) into Eq. (3) with the replacement mentioned above of the second term occurring on the left-hand side of Eq. (3) and equating the coefficients of the terms exp(in r) lead to Inserting Eq. (12) into Eq. (9) yields (12) This approximate solution of the wave equation is known in solid-state physics as the nearly free binding case 5 The functions of Eq. (7) are the Bloch wave functions in the (7) This integral describes the propagation of a field ψ(r) from its boundary values at the plane z = 0 and is the generalization of the plane-wave spectrum representation for a field propagating in free space. To get some further insight of the properties of the field we will calculate its asymptotic behavior, viz, the far-field approximation,

3 2684 J. Opt. Soc. Am. A/Vol. 22, No. 12/December 2005 B. J. Hoenders and M. Bertolotti with respect to large values of the distance between an observer and the plane z=0 (the y axis). The detailed calculations are to be found in Refs The result reads as sources 9 through such media will be analyzed. The theory will be restricted to second-order statistics and therefore the second-order correlation function Γ(r1, r2 ;tl, t2), known as the mutual coherence function,9 is introduced: (15) where the brackets indicate that the average of the quantity ψ(rl,t1)ψ*(r2,t2) has to be taken over the appropriate space-time ensemble. It is convenient to introduce the normalized mutual coherence function by setting9 ki2=k02, kx=k0cos(φ), ky=k0sin(φ). (14) The vector s denotes the unit vector pointing from the origin of the plane x =0 to the place r of the observer: s =[cos(φ),sin(φ)] if r= r s, and φ denotes the angle subtended with the x axis. R is the distance from the origin of the plane x=0 (the y axis) to the location of the observer. We have omitted the double summation term occurring on the right-hand side of Eq. (13) because it corresponds to a second-order approximation. This far-field expansion of Eq.(14), as well as the general wave function of Eq.(13) together with the relation of Eq.(12) leads to the following observations: (a) A given field ψ(rσ) at the boundary plane x=0 manifests itself in the propagated field not only through its Fourier transform ψ~(kσ)but also by the shifted components ψ~(kσ+nσ)[see Eqs. (12) and (13)]. This property is of great importance for the behavior of the coherence in the far field. whose modulus lies between zero and one: (16) (17) The analysis of the behavior of the coherence functions of Eqs. (15) and (16) will be the central problem considered in this section. We assume that the electromagnetic field considered is quasi-monochromatic with mean frequency ν~. Inserting Eq.(14) into Eq.(15) yields (b) The perturbed wave field contains a term showing that the unperturbed wave is modulated by a periodic function whose period is equal to that of the medium. This wave is the Born approximation of the Bloch modes of the medium. We have derived the deterministic Eq. (13) describing the propagation of the field generated by a known surface distribution. No physical theory is complete without the introduction of the appropriate statistics. Therefore we will now introduce the statistical properties of the field and calculate the second-order correlation coefficient (mutual coherence function) for the field. 3. GENERALIZED VAN CITTERT-ZERNIKE THEOREM AND COHERENCE PROPERTIES OF FIELDS GENERATED BY QUASI-HOMOGENEOUS SOURCES The classical basic theory of coherence culminates in the derivation of the van Cittert-Zernike theorem.9 We will discuss the generalization of this theorem if the field propagates through a weak photonic (periodic) medium. Furthermore, the propagation of a class of interesting source distributions known as quasi-homogeneous (18) if Γ'(rl,r2;ν~)=R Γ(rl,r2;ν~). From now on we will omit the prime and therefore tacitly assume that the factor R 2 is absorbed in the mutual coherence function Γ(rl,r2;ν~). The simplest possible assumption for the state of coherence at the entrance plane is that of total incoherence, as originally assumed by van Cittert 10 and Zernike.11The coherence function at the plane z =0 then reads as Inserting Eq.(19) into Eq.(18) yields (19)

4 B. J. Hoenders and M. Bertolotti Vol. 22, No. 12/December 2005/J. Opt. Soc. Am. A 2685 if rσ= rσ1= rσ2 Another source distribution that is a generalization of the one introduced by van Cittert-Zernike is the so-called quasi-homogeneous source distribution, first introduced by9 (21) (20) Inserting Eq. (21) into Eq. (18) and omitting second-order terms, containing products of the expansion coefficients an, am leads to (22) and sσ,mσ denote again the projection of the vectors s,m on the entrance plane. In conclusion, there are two basic effects: (a) Equations (20) and (22) show that, in the far-field, functions with argument S2σ-slσ±mσ occur. This implies a shift of the coherence functions involved over distances ±mσ. The coherence between two points of the wave field, which for homogeneous media always tends to zero if the distance between Slσ and S2σ becomes larger, for photonic materials is periodically repeating itself, be it with a weight factor am(k02-(k+mσ+m)2)-1. (b) Equations (20) and (22) show that to the unperturbed coherence function a perturbation is added that equals the unperturbed coherence function, We will now turn attention to the influence of the finite size of the structure on the coherence of the field ensuing after the crystal. We consider a crystal of finite width a that is incident on a field with a given state of coherence [see Eqs. (19) and (21)] at the entrance plane. Our aim is to solve the deterministic Dirichlet problem for this structure first, i.e., to calculate the field inside and outside the crystal generated by a given distribution ψ(r) at the entrance plane x=o, and then introduce the statistics of the problem (see Fig. 2 for the geometry and the various wave amplitudes of the plane waves inside and outside the medium). To this end we recall that the functions of Eq. (7), i.e., (23) are the complete basis for the field satisfying Eq. (3) in the whole space R3. So they certainly can be used for the representation of the field inside the finite crystal. Expression (23) represents waves traveling to the right, because inside the slab waves traveling to the left also exist, multiplied by a periodic function whose period equals that of the photonic medium. 4. FINITE-SIZED PHOTONIC CRYSTALS Fig. 2. Geometry and various wave amplitudes of the plane waves inside and outside the medium.

5 2686 J. Opt. Soc. Am. A/Vol. 22, No. 12/December 2005 B. J. Hoenders and M. Bertolotti so another set of modes is introduced: (32) (24) if a tilde above a function means its Fourier transform with respect to r σ. At the boundary x=a the field is continuous: For later use we introduce the functions (25) (33) and the first-order derivatives of the field are continuous: and a similar equation B0(kσ, to the left: rσ) with propagating waves (26) (27) (34) Equations (32)-(34) are a set of linear equations to be solved for the unknown spectral distributions A(kσ), B(k σ ), and C(k σ ). We are interested only in the first-order approximate solution of these equations, namely, solutions up to first order in the periodic perturbation. The solutions of the unperturbed system read as and a similar equation for B 1 (k σ, r σ) with A replaced by B, respectively. We further introduce (35a) and a similar equation for B2(kσ,rσ): (28) (35b) (29) As usual for the solution of a Dirichlet problem the completeness of the set of Bloch modes makes sure that we can make the following ansatz for the field inside the crystal: With these solutions we obtain the first-order approximate solutions of the system of Eqs. (32)-(34), iterating the set of equations once.as we are interested only in the spectral distribution at the end surface of the crystal, we will present only the result for C(kσ(2)): (30) The field ψout(r) outside the end plane x=a admits the representation in free space (31) with a spectral density C(kσ(2)) to be determined by the continuity conditions. Now we impose the conditions for the field at the boundaries: At x=o the field is equal to the given distribution ψ(rσ),which taking the Fourier transform with respect to rσ of the functions involved leads to (36) It can be shown9 that in the far zone the second-order coherence function, generated by a state of coherence

6 B. J. Hoenders and M. Bertolotti Vol. 22, No. 12/December 2005/J. Opt. Soc. Am. A 2687 at the plane x=a (37): (37) is equal to the Fourier transform of Eq. (38) where the unit vectors s((21,)2) denote the direction from the origin of the plane x=a to the point of observation. We are therefore interested in the Fourier transform of <ψ(r1(2)σ)ψ*(r2(2)σ» in the plane x=a. Using Eq. (31) leads to (39) where the values of C(k((12)σ,2σ)) are given by Eqs. (36) and (35a). These equations show that Γ(r1(2)σ,r2(2)σ;x=a) is linearly related to the coherence function <ψ~(k1(2)σ) ψ~* (k2(2)σ» that follows from the two choices for coherence distributions in the entrance plane that we took earlier: The van Cittert-Zernike coherence function [Eq. (19)] and the quasi-homogeneous coherence function, namely, Eq. (21). Then the Fourier transforms of these two distributions read, respectively, as 5. COHERENCE PROPERTIES OF PULSES For pulses a similar set of equations is obtained. The response theory developed for a single plane wave leads straightforwardly to the description of the scattering and transmission of a propagating pulse solving the pertinent initial problem for wave functions ψ satisfying the wave equation [Eq. (3)]. To derive the relevant equations we therefore formulate the initial value problem and give its solution in terms of a double Fourier transform over time frequency wand space frequency k. We assume that at t = 0 the pulse is located outside the photonic crystal medium and that we therefore have to solve only the initial value problem for the medium outside the photonic crystal structure. Theorem 1. Let ψ be a solution of the wave equation [Eq. (3)]. Suppose that at time t =0 the initial distribution ψ(r,t=0)=f(r), r=(x,z) (the initial pulse or pulse train), and the first-order time derivative ψt(r,t=0) =g(r) of ψ are given. Then the function ψ is uniquely determined and reads as (42) where ω=c k, and a(k) and b(k) are the solution to the set of equations D(k) + E(k) = f~(k + nσ) (43) (40) and for the general quasi-homogeneous source distribution as given by Eq. (21) we have and - iωd(k) + iωe(k) = g~(k), (44) (45) (41) Combining Eqs. (35a), (36), (38), (40), and (41) then shows that the second-order coherence function in the far field has the same structure as Eq. (22), which leads to the following conclusions concerning the influence of the periodicity of the medium on the spatial coherence properties of a quasi-monochromatic field after propagation through a photonic medium: The partial coherence of a wave field in the far field after propagation through a periodic medium is a quasiperiodic function of the partial coherence function generated by the homogeneous medium. The periodicity of the partial coherence of the wave field is transverse to the propagation direction, i.e., the cause of the change of the coherence properties of the wave field is due entirely to the periodicity of the medium in the direction transverse to the propagation direction and not to the periodicity in the longitudinal direction. The proof of this theorem is obtained immediately from the representation of Eq. (42) and imposing the initial value conditions f(r) and g(r). We have not yet required that, at t=0, the wave packet move to the right, which leads to the condition (46) Corollary 1. If the wave packet moves to the right, the solution of the one-dimensional wave equation reads as (47) The proof of Eq. (47) follows from Eqs. (43) and (44) and the observation that Eq. (46) in Fourier space reads as Then it follows that E(k)=0 g~(k) = iωf~(k). (48) and D(k)=f~(k).

7 2688 J. Opt. Soc. Am. A/Vol. 22, No. 12/December 2005 B. J. Hoenders and M. Bertolotti After this preliminary survey of results for the propagation of a pulse in a two-dimensional homogeneous space we are now in the position to formulate analytically the propagation of a pulse of arbitrary shape and initially fully located outside the photonic crystal medium and moving to the right through a photonic crystal medium in terms of a Fourier decomposition in space and time. All we have to do is to replace a(k)exp((ik r)) by the full set of pertinent plane-wave modes for the problem at hand comprising the reflected mode, the modes inside the medium, and the mode outside the end plane of the crystal. These modes will now be constructed (see Fig. 2). The field ψ(-)(r) in free space, with labell, outside the entrance plane of the crystal is represented by the following ansatz: ω = c k(1) (49) The continuity conditions at the boundary x=a yield (54) and the continuity conditions for the first-order derivatives of the field at the boundary x=a lead to (56) Again this set of equations is to be solved by iterating once, viz, inserting the unperturbed solutions into the perturbation occurring in the summation The relations between the coefficients of the unperturbed solutions follow from the classical theory of scattering of an electromagnetic wave by a slab of width a (Refs. 12 and 13) and read as if k(1)=(kσ(1),kx(l)) and the coefficient D(k(1)) denotes the spectral component of the incoming pulse The field inside the crystal with label 2 is represented by with (50) E(unp)(k(1))= RD(unp)(k(1)), C(unp)(k(3))= TD(unp)(k(3)), (57) where R denotes the reflection coefficient, (51a) (58) (51b) if k(2)= (k(2),k(2)). The field outside the end plane x=a of σ x the crystal with label 3 admits the representation and T the transmission coefficient, (59) ω = c k(3)1 (52) if ifk(3) = (k(3),k(3)). The continuity conditions for the field at σ x the boundary x = 0 yield (60) and the continuity conditions for the first-order derivatives of the field at the boundary x = 0 lead to (53) We will need only the first-order approximation for the spectral density C(k(3)). This quantity is obtained by inserting the unperturbed values of A and B, Eq. (57), into Eq. (55), yielding

8 B. J. Hoenders and M. Bertolotti Vol. 22, No. 12/December 2005/J. Opt. Soc. Am. A 2689 (61) Wehave determined the appropriate modes for our system consisting of a photonic crystal embedded in two different homogeneous spaces. We are now in the position to solve the problem of scattering and transmission of a pulse by a finite photonic crystal with length a. To this end we recall the corollary 1 and observe that Eqs. (61) and (57) give the relation between C(k(3) and the known value D(unp)(k(l) (see corollary 1). Equation (52) then leads to the basic result for the representation of the pulse outside the end plane of the crystal: Theorem 2. Suppose that at time t=o a pulse with shape ψ(r,t=o)= ::n(k)exp(ik r)dk is moving to the right and is scattered and transmitted by a photonic crystal medium. The representation of the resulting field then reads as if ω = c k(3). (62) The mutual coherence function of Eq. (15) then becomes The rather complicated expression ofeq. (63), however, admits a simple physical interpretation if we observe that Eqs. (57) and (61) show that C(k(3) is a linear combination of spectral densities D, laterally shifted with respect to the x axis, where D denotes the Fourier transform of the pulse at time t=o, (see corollary 1). These values of D, weighted by k02-(k(2)+nσ+n)2, have to be integrated over the whole k(3)space. But for values of k(3)near the Bragg resonances the perturbation analysis breaks down and waves are generated with amplitudes like exp(in r). These waves are interpreted as the waves connected with multiple reflection. Hence they lead to a series of shifted pulses along distances n outside the structure of finite widths equal to the width of the incoming pulse. Summarizing, the analysis given above and Eqs. (61) and (63) then lead to the following conclusions for the temporal coherence properties of the field after passage through the photonic structure: A pulse of finite width w, scattered and transmitted by a weakly scattering photonic crystal, will generate a set of pulses laterally shifted with respect to the x axis, with distances equal to multiples of the periodicity of the crystal. A pulse in exit from the structure is made by the sum of a number of pulses. All these pulses are coherent with each other, and therefore, considering the envelope of the distribution, the coherence time is increased. The total length of the system of pulses increases in time because of the multiple reflections, and this translates into a narrowing of its spectrum. This is exactly the filtering action of the structure. 6. DISCUSSION The second-order theory of partial coherence is developed for the case of TE or TM waves propagating through a weakly period medium. These particular choices for the electromagnetic field, which means that we restrict ourselves to two-dimensional problems, reduce the originally vectorial problem to two scalar problems that considerably simplify the analysis of the problem. However, the dependence on the polarization of the partial coherence properties of the field can therefore not be completely analyzed, so that the full vectorial theory has to be developed as a generalization of the theory in this paper. Although we restricted ourselves to weak periodic media, the essential properties of the problem pertinent to periodic media show up: The basic wave functions involved [Eq. (7)] are the Bloch wave functions of the problem and the bandgap that already develops as explained below Eq. (7). So the first-order Born perturbative analysis chosen by us suffices for a qualitative analysis of the exact problem. One of the core results of the analysis is the observation, drawn from Eq. (12), that a given field ψ(rσ) at the boundary plane x=o manifests itself in the propagated field not only through its Fourier transform ψ~(kσ)but also by the shifted components ψ~(kσ+nσ)[see Eqs. (12) and (13)]. This result expresses mathematically the periodic reflection of the waves in the lateral direction. These lateral waves explain the periodic shifts of the coherence showing up in the generalization of the van Cittert-Zernike theorem, observed at the end of Section 3. The finite size of the medium is fully taken into account and its effects on the wave field are exactly calculated. To this end we used for the representation of the wave field inside the finite medium the wave functions of Eq. (7), which represent the field in the infinite space occupied by the periodic medium. We would like to stress the fact that there is no contradiction in use of wave functions that are valid in infinite space for problems pertaining to finite space, as is sometimes objected: The infinite space functions [Eq. (7)] are ultimately the set of basis functions for all the solutions of the basic equation [Eq. (1)] in infinite space, which therefore certainly includes the solutions of Eq. (3) in a finite part of R3. In Section 4 we developed the theory of the propagation of the partial coherence function in a weak periodic medium given a distribution at a plane. Using the tech-

9 2690 J. Opt. Soc. Am. A/Vol. 22, No. 12/December 2005 B. J. Hoenders and M. Bertolotti niques developed in this section, a similar theory is developed in Section 5 for the more general case of an incoming pulse scattered by a weak periodic medium. This culminated in Eq. (63), which admitted, however, a simple physical interpretation: A pulse of finite width w, scattered and transmitted by a weakly scattering photonic crystal, will generate a set of laterally shifted pulses with distances equal to multiples of the periodicity of the crystal. A pulse in exit from the structure is made by the sum of a number of pulses. All these pulses are coherent with each other, and therefore, considering the envelope of the distribution, the coherence time is increased. The total length of the system of pulses increases in time because of the multiple reflections, and this translates into a narrowing of its spectrum. This is exactly the filtering action of the structure. B. J. Hoenders' addressisb.j.hoenders@rug.nl. REFERENCES 1. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975). 2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995). 3. E. Wolf, "Spatial coherence of resonant modes in a maser interferometer," Phys. Lett. 3, (1963). 4. W. Streifer, "Spatial coherence in periodic systems," J. Opt. Soc. Am. 56, (1966). 5. R. Becker, Theorie der Elektriztät (B.G. Teubner, 1969), Vol. 3, Chap. A2, p E. C. Titchmarsh, Eigenfunction Expansions (Oxford U. Press, Oxford, 1958), Vol B. J. Hoenders and D. N. Pattanayak, "Interaction of a moving charged particle with a spatially dispersive medium. I. Structure of the electromagnetic field," Phys. Rev. D 13, (1976). 8. B. J. Hoenders and D. N. Pattanayak, "Interaction of a moving charged particle with a spatially dispersive medium. II. Cerenkov and transition radiation," Phys. Rev. D 13, (1976). 9. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995), Chaps. 1-9, pp P. H. van Cittert, "Die wahrscheinliche Schwingungsverteilung in einer von einer Lichtquelle direkt oder mittels einer Linse beleuchteten Ebene," Physica (Amsterdam) 1, (1934). 11. F. Zernike, "The concept of degree of coherence and its applications to optical problems," Physica (Amsterdam) 5, (1938). 12. P. Yeh, A. Yariv, and C.-S. Hong, "Electromagnetic propagation in periodic stratified media. 1. General theory," J. Opt. Soc. Am. 67, (1977). 13. P. Yeh, Optical Waves in Layered Media, Wiley Series in Pure and Applied Optics (Wiley, 1988).