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Electromagnetic Wave Propagation in Periodic Porous Structures I. David Abrahams1 and Gregory A. Kriegsmann2 1 School of Mathematics University of Manchester Oxford Road, Manchester M13 9PL, UK 2 Department of Mathematical Sciences Center for Applied Mathematics and Statistics New Jersey Institute of Technology University Heights, Newark, NJ 07102 Abstract We employ a homogenization procedure to describe the propagation of electromagnetic waves in a dielectric structure which is doubly-periodic in the X-Y plane and of arbitrary variation in the direction of propagation, Z. The fundamental cell is composed of an arbitrarily shaped pore filled with a dielectric and the host by another. Our analysis yields the structure of the electromagnetic fields at the micro level and gives an effective medium equation at the macro level. The latter contains a simple arithmetic average of the dielectric constants and a correction term which involves a line integral around the pore. The integrand of this integral depends upon the polarization of the wave and the solution to a canonical potential problem. We approximately solve this problem for small pore volumes and for large contrasts, the pore dielectric constant being much larger than the host. We also provide an equivalent variational formulation for the potential problem and use a simple Raleigh-Ritz procedure to determine an approximate solution. For all these approximations, we provide a simple macroscopic description of electromagnetic wave propagation in our structure. 1. Introduction In this paper we model and analyze the transmission, reflection, and propagation of a plane electromagnetic wave through a composite slab of finite thickness. The composite contains a host material with an index of refraction Nh2 and a periodic arrangement of parallel dielectric cylinders with index Np2 . These cylindrical structures need not be circular and their cross sections can vary in in the direction, Z, of propagation. The slab thickness and the wavelength of the incident microwave 2 are of the same order which is much larger than the period of the structure. The problem considered here was originally motivated by a microwave assisted chemical vapour process. In this setting the slab is a porous ceramic host where Nh2 is complex and the dielectric cylinders are air filled pores with Np2 = 1. The slab is put into an environment where a gas infuses into the pores and microwaves are used to heat the host. The gas reacts with the heated pores, fills them, and creates a composite material. In more realistic and complicated models of this process [1] the slab is treated as a porous medium and linear mixing theory is used to approximate the effective dielectric constant, or index of refraction. The periodic porous structure studied in this paper allows a more refined approximation to this constant and consequently will provide a basis for a more accurate model of the heating process. The structure we consider also has motivation in optical devices, such as quantumdot arrays [2]. Here, the host material is air with Nh2 = 1 and the cylinders are semiconductors layered in the Z direction. In these devices the wavelength of the incident light and the thickness of the structure are again of the same order, but they are only a factor of three times larger than the period. Although the theory developed in this paper requires these scales to be disparate, our results may offer some qualitative information about these devices. Mathematically and physically there are two approaches to formulate our problem. In the first, see e.g. [3], equations are derived for the electric and magnetic fields in the direction of propagation, i.e. the Z direction. If the pores are circular and not changing with Z and the fields are all proportional to eiΓZ , where Γ is the propagation constant, then applying the continuity of all tangential fields across the pore boundary produces a doubly infinite system of algebraic equations. The solution of these equations give the Fourier coefficients of the Z components of the electric and magnetic fields. However, non-trivial solutions are obtained only for those values of Γ that render a zero determinant. The complexity of this problem is exacerbated by the fact that Γ appears in this determinant in a very nonlinear manner. This approach was recently taken in [4] and several ad-hoc approximations were made to determine Γ. Since this scheme required a truncation of the infinite system it produced a set of N values for the propagation constant Γi , i = 1, 2, 3...N where the truncated system is N × N . Numerical calculations showed that the {Γi } clustered about two distinct values of Γ, namely Γ1 and Γ2 . This suggested that there are only two propagation constants for this problem. The other and more fruitful approach is to compute the transverse electric and magnetic fields and then deduce the fields in the Z direction. This is the approach we take in this paper. We find that two potentials determine the transverse electric and magnetic fields. These potentials satisfy homogeneous boundary conditions across the pore contour C. In fact the potential that yields the magnetic fields has 3 two independent, elementary solutions, each giving rise to a particular polarization. (This gives credibility to the suggestion just mentioned above.) Using this solution along with the unknown potential for the electric fields we deduce the electric and magnetic fields in the Z direction. From these fields and their periodicity we are able to derive a simple macroscopic equation governing the propagation of electromagnetic waves through this structure. This equation is a second order ordinary differential equation. It contains an effective index of refraction composed of two parts. The first is just a mixture theory result and the second contains a contour integral around C whose integrand depends upon a related potential function. This function is harmonic everywhere in the cell, periodic on its boundaries, continuous across C, and has a nonhomogeneous jump in its normal derivative across this curve. The forcing on this term is related to the polarization of the propagating wave. We shall now outline the remainder of this paper. Section 2 contains the description and formulation of our problem. Section 3 presents our homogenization analysis of the Maxwell’s equations exploiting the smallness of the structure’s period to the incident wavelength. The result is the ordinary differential equation described in the preceding paragraph. Two limiting cases are considered in Section 4; the dilute limit and the strong dielectric contrast limit. Approximate formulae are given for these two scenarios. Section 5 contains a variational statement of the microstructure potential problem. A simple one term Raleigh-Ritz approximation is made to yield a simple effective index of refraction. Using the theory developed in the preceding sections we study the scattering and transmission of a plane electromagnetic wave through a finite slab. These results are contained in Section 6. Finally, Section 7 contains our conclusions. 2. Formulation A plane electromagnetic wave propagates in the positive Z direction, where X, Y , Z are Cartesian coordinates, and is normally incident upon a porous slab which occupies the region 0 < Z < L. The wave is partially transmitted through the structure and part is reflected. The pores are cylindrical in shape, with axes lying parallel to the Z, have identical shape, and are distributed in a doubly-periodic arrangement in the X-Y plane; the cross-section of the pores is arbitrary and can vary with Z. The geometry of the problem is sketched in Figure 1 for the case of circular pores. In the analysis that follows we assume that the wavelength of the incident field λ is much longer than LX and LY , the periodicity of the pore structure in the X and Y directions respectively. However, λ is of the same order as the slab thickness L. This naturally introduces the small parameter δ = LX /L which we shall exploit on 4 2 N h R 0 L Y N 2 p LX Figure 1. The doubly periodic arrangement of pores indicating the fundamental cell. developing our theory. We also introduce the cell aspect ratio l = LY /LX , which we take to be an O(1) quantity, and without loss of generality we can henceforth choose l ≥ 1. The (steady-state) electromagnetic fields are governed by Maxwell’s equations, which are given in dimensionless form by ∂E3 ∂E2 − = ikH1 , ∂y ∂z (1a) ∂E1 ∂E3 + = ikH2 , ∂x ∂z (1b) ∂E2 ∂E1 − = ikδ 2 H3 , ∂x ∂y (1c) ∂H3 ∂H2 − = −ikN 2 E1 , ∂y ∂z (2a) − ∂H1 ∂H3 + = −ikN 2 E2 , ∂x ∂z (2b) ∂H2 ∂H1 − = −ikδ 2 N 2 E3 , ∂x ∂y (2c) − 5 where the wavenumber k = 2πL/λ and N 2 is the index of refraction which takes the value Np2 in the pore regions and the value Nh2 in the host medium. In equations (1-2) the dimensionless independent spatial variables are defined by x = X/LX , y = Y /LX , and z = Z/L, and the fields by Ej = Ej′ /E0 , Hj = Z0 Hj′ /E0 , E3 = δ −1 E3′ /E0 , j = 1, 2, (3a) H3 = δ −1 Z0 H3′ /E0 , (3b) p where Z0 = µ0 /ǫ0 , µ0 is the magnetic permeability of free space, ǫ0 is the permittivity of free space, E0 is the strength of the incident wave, and the primes denote the dimensional electromagnetic fields. For simplicity of analysis, the pores and host have both been taken to have the permeability of free space. We note here that the fields in the direction of propagation have been scaled by δ, that is, they are assumed small compared to the transverse fields in the x-y plane; we shall show below that this scaling yields a consistent asymptotic result. Due to the periodicity of the structure and the normal incidence of the impinging wave it is enough to find the periodic solution to (1-2) in the fundamental cell shown in Figure 2. The region of the cell occupied by the host is denoted Rh , occupied by the pore is RP , and the whole cell is R ≡ Rh ∪ Rp . 3. Analyses In the following analyses we will only be concerned with the leading order (homogenized) electromagnetic fields in the limit as δ → 0. Accordingly, we may set δ = 0 into (1-2). From (1c) we then find that ∂E1 ∂E2 − = 0. ∂x ∂y (4a) Next, we take the partial derivative of (2a) with respect to x, the partial derivative of (2b) with respect to y, and add the resulting expressions. Using (2c) with δ = 0, this resulting expression becomes ∂ N 2 E1 ∂ N 2 E2 + = 0. (4b) ∂x ∂y Introducing the potential function Φ and setting E1 = ∂Φ/∂x and E2 = ∂Φ/∂y we find that (4a) is satisfied and (4b) yields ∇ · N 2 ∇Φ = 0, (x, y) ∈ R, (5) 6 FUNDAMENTAL CELL 2 N h R l r0 P N 2 p R H 1 Figure 2. The non-dimensionalized fundamental cell. where R denotes the region occupied by the fundamental cell. The gradient and divergence operators are with respect to the transverse variables x and y only; this notation will be followed for the remainder of this paper. We now follow a similar path using (2c) and (1a-1c) to find that ∂H2 ∂H1 − = 0, ∂x ∂y (6a) ∂H2 ∂H1 + = 0. ∂x ∂y (6b) Introducing a second potential function Ψ and setting H1 = −∂Ψ/∂y and H2 = ∂Ψ/∂x, then (6b) is satisfied and (6a) yields ∇2 Ψ = 0, (x, y) ∈ R. (7) We note that, for conciseness, here and henceforth the functional dependence of Φ and Ψ on z is implied but not written explicitly. 7 If the potentials Φ and Ψ can be found then the electric and magnetic fields in the z direction can be deduced. This follows by substituting the potentials for the transverse fields in (1a-b) and (2a-b); the first yields E3 = and the second gives ∂Φ − ikΨ + Q(z) ∂z (8a) ∂ 2Ψ ∂Φ ∂H3 = − ikN 2 , ∂y ∂z ∂x ∂x (8b) ∂2Ψ ∂Φ ∂H3 =− + ikN 2 , ∂x ∂z ∂y ∂y (8c) where Q(z) is an unknown function at this stage. Next, we deduce the boundary conditions that Φ and Ψ must satisfy in the unit cell, R. First, the electric and magnetic fields must be periodic in both x and y directions. Since the transverse fields are given by the x and y derivatives of the potentials, it follows that these are periodic too. In addition, the required periodicity of E3 implies that the right hand side of (8a) is also periodic. Secondly, we demand that the tangential electric and magnetic fields in the x-y plane are continuous across the boundary C of the pore. Expressing these transverse tangential fields in terms of their potentials we deduce (see Appendix 1), respectively, that ∂Ψ = 0, (x, y) ∈ C, (9a) [Φ]C = ∂n C where the notation [ ]C denotes the jump in the bracketed quantity across the pore boundary C, and n is a coordinate locally normal to the curve C and measured positive on the host material side. Thirdly, the tangential components of the electric and magnetic fields in the longitudinal direction are also continuous across the pore boundary. In Appendix 1 we show that this implies 2 ∂Φ N = 0, [Ψ]C = 0, (x, y) ∈ C. (9b) ∂n C Now the solution of (7) which satisfies [Ψ]C = [∂Ψ/∂n]C = 0, with periodic ∂Ψ/∂x and ∂Ψ/∂y, is Ψ = a(z)x + b(z)y, (10a) in which a(z) and b(z) are functions to be determined. This potential then gives H1 = −b(z), H2 = a(z), (x, y) ∈ R, (10b) 8 that is, the magnetic fields are constant throughout R in any plane with z constant. For the moment we take b = 0 and consequently, H1 = 0 and H2 = a(z). Then the periodicity of E3 given by (8a) implies ∂Φ ∂Φ (x, 0) = (x, l), ∂z ∂z 0 < x < 1, ∂Φ ∂Φ (0, y) = (1, y) − ika(z), ∂z ∂z These can be integrated to give Φ(x, 0) = Φ(x, l) + h(x), (11a) 0 < y < l. 0 < x < 1, Φ(0, y) = Φ(1, y) − ikA(z) + g(y), (11b) (11c) 0 < y < l, (11d) where dA/dz = a. The functions h(x) and g(y) are, in fact, constants. To see this we differentiate (11c) with respect to x and obtain h′ = Φx (x, 0) − Φx (x, l). Since Φx = E1 and this field is periodic in y, we find h′ = 0. A similar calculation holds for g(y). We may therefore take both these constants to be zero as they have no effect on the fields E1 and E2 . For piecewise constant refractive index, N 2 , we can define a new potential P by Φ = ikA{x − [N 2 ]C P }. (12) Inserting this into (5) and using [Φ]C = [N 2 ∂Φ/∂n]C = 0 we find that P satisfies ∇2 P = 0, [P ]C = 0, N 2 ∂P ∂n (x, y) ∈ R, = C ∂x , ∂n (13a) (x, y) ∈ C, (13b) and employing (11c) and (11d): P (x, 0) = P (x, l), 0 < x < 1, P (0, y) = P (1, y), 0 < y < l, (13c) that is, P is doubly periodic. If this problem can be solved then from (12) we have ∂Φ ∂P 2 E1 = , = ikA 1 − [N ]C ∂x ∂x E2 = −ikA[N 2 ]C ∂P . ∂y (14a) (14b) 9 We note here that the periodicity of E1 and E2 imply the same for ∂P/∂x and ∂P/∂y, respectively. To determine the amplitude function A we return to (8b) which becomes, on account of (10b) and a = dA/dz, d2 A ∂Φ ∂H3 = − ikN 2 . 2 ∂y dz ∂x (15a) Next, we integrate (15a) along the straight line x = x0 , 0 < y < l, where x0 is chosen to ensure that the line is in the region Rh , i.e. it does not intersect the pore region Rp . Since H3 must be periodic in y, the result is d2 A l 2 − ikNh2 dz Z 0 l ∂Φ(x0 , y) dy = 0. ∂x Inserting (12) into this relationship and dividing by l we find that d2 A + k 2 Nh 2 2 dz ( 1 1 − [N 2 ]C l Z l 0 ) ∂P (x0 , y) dy A = 0. ∂x (15b) We shall now recast the differential equation (15b) into a more useful form. First, we note that the integral in (15b) is independent of x0 . To see this we define I(x) = Rl ∂P/∂x(x, y) dy. Since P is harmonic in Rh , it follows that dI/dx = −∂P/∂y|l0 . 0 The periodicity of ∂P/∂y yields our result. Thus, the integrand in (15b) can be evaluated, without loss of generality, at x = x0 = 1. Next, since P and x are harmonic functions in R we deduce ∇ · {N 2 P ∇x − N 2 x∇P } = 0. Integrating this expression in the region Rh , and using the periodicity of P , Px , and Py , we obtain Nh2 I ∂P + x ds − Nh2 ∂n I P + ∂x ∂n ds = Nh2 Z 0 l ∂P (1, y) dy ∂x (15c) H where denotes counter clockwise integration on C, the direction derivative ∂/∂n is defined to point normally into the host region, and the superscript + denotes evaluation on the Rh side of the pore. We integrate the same expression in the pore region Rp and find −Np2 I ∂P − ds + Np2 x ∂n I P− ∂x ds = 0 ∂n (15d) 10 where the superscript − denotes evaluation on the Rp side of the pore. Adding equations (15c) and (15d), employing (13b), and noting (via the divergence theorem) H that x∂x/∂n ds = Ap , where Ap is the pore area, we arrive at Nh2 Z 0 l ∂P (1, y) dy = Ap − [N 2 ]C ∂x I P ∂x ds. ∂n Finally, combining this result with (15b) we find that A satisfies I 1 22 ∂x d2 A 2 2 + k < N > + [N ]C P ds A = 0 dz 2 l ∂n (15e) (16a) where Ap 2 Ap 2 Ah 2 [N ]C = N + N , (16b) l l p l h Ah is the area of Rh and l is the area of the fundamental cell. Equation (16a) is an effective medium equation for our periodic, porous material. If the pore shape is independent of z, then the bracketed term in (16a) is the effective (constant) index of refraction. It is the sum of a simple mixture relationship (16b) and a contour integral, which involves the microstructure of the medium. We note here several observations about our results to this point. First, if Ap = 0, then there is no pore region and < N 2 >= Nh2 . Also, the line integral in (16a) is gone so that (16c) reduces to d2 A/dz 2 + k 2 Nh2 A = 0. Furthermore, [N 2 ]C = 0, since there is no contour, and (14) reduces to E1 = ikA and E2 = 0. Recalling that H2 = dA/dz and H1 = 0, our results must reduce to transverse electromagnetic (TEM) propagation through a homogeneous slab with the electric field polarized in the x direction. This requires that Q = 0 in (8a). Our next observation is, if [N 2 ]C = 0, then Np2 = Nh2 and the results are the same as those described in the preceding paragraph. Out third observation is, if Ah = 0, then < N 2 >= Np2 and the line integral in (16a) is now around the perimeter of the fundamental cell. This integral vanishes by the periodicity of P and (16a) reduces to d2 A/dz 2 +k 2 Np2 A = 0. The periodic solution to (13) is now P = 0, so that (14) again gives E1 = ikA and E2 = 0. The magnetic fields are still given by H1 = 0 and H2 = dA/dz. Thus, we again have a TEM electromagnetic wave with the electric field polarized in the x direction. We close this section by considering the case where Ψ = b(z)y. All of our analysis carries over and the results are the same with a few important exceptions. First, equation (14) is replaced by < N 2 >= Nh2 − E1 = ∂P ∂Φ = −ikB[N 2 ]C ∂x ∂x (17a) 11 ∂Φ ∂P 2 E2 = = ikB 1 − [N ]C ∂y ∂y (17b) where B = db/dz and P satisfies all of (13) with the exception that the second equation in (13b) is replaced by N 2 ∂P ∂n = C ∂y , ∂n (x, y) ∈ C. (17c) Secondly, the function ∂x/∂n in the line integral in (16a) is replaced by ∂y/∂n. All of the observations made above for the limiting cases of Ap = 0, [N 2 ]C = 0, and Ah = 0 still hold yielding a TEM electromagnetic wave polarized with the electric field in the y direction, i.e. E1 = 0, E2 = ikB, H1 = dB/dz, and H2 = 0. 4. Further Limiting Cases In addition to the limiting cases briefly described in the previous section, there are two other physical scenarios where the integral in (16a) can be approximated or easily computed. In the first, the pore is circular and its radius is r0 ≪ 1. This is called the small pore, or low concentration, limit. If we were to rescale (13a-b) so that the pore had unit radius, then the cell boundaries would, to leading order, be scaled off to infinity. The periodic boundary conditions (13c) would be replaced by the requirement that P is bounded away from the pore. The small pore limit is often called the dilute limit, or limit of low area fraction, where the area fraction, φ, is the ratio of the pore area to the cell area, i.e. φ = Ap /l = πr02 /l. The solution to this limiting case, which satisfies the requisite conditions (13a-b), is r, cos θ 2 P =− 2 Np + Nh2 r0 , r r < r0 , r0 < r. (18a) To extend this result to incorporate the periodic boundary conditions, i.e. for higher area fractions, would require an application of a technique such as the method of matched asymptotic expansions or a complex variable approach (see e.g. Parnell & Abrahams [5]). For ease of exposition we restrict attention to just the dilute pore limit in this article. We may now compute the integral expression in (16a) using (18a), which after simplification yields [N 2 ]C d2 A 2 2 A = 0. + k Nh 1 − 2φ 2 dz 2 Np + Nh2 (18b) 12 If the pore radius is independent of z, then the bracketed term in (18b) is the effective index of refraction. It is a small perturbation in the area fraction φ. In the second physical scenario we take Np2 ≫ Nh2 , that is, there is a large dielectric contrast between the pore and the host materials. Then the second boundary condition in (13b) becomes approximately 1 ∂x ∂P − =− 2 , ∂n Np ∂n (x, y) ∈ C, (19a) where the superscript − denotes evaluation of the normal derivative just inside the curve C. In the pore region Rp we again have ∇ · {P ∇x − x∇P } = 0. Integrating this expression in Rp , using the divergence theorem, we find that I ∂x P ds = ∂n I x ∂P − ds. ∂n (19b) H Inserting (19a) into the right hand side of (19b), recalling that x∂x/∂n ds = Ap = φl, and using this result in (16a) we obtain after some simplification d2 A + k 2 Nh2 {1 + φ}A = 0. dz 2 (19c) If the pore fraction φ is independent of z, then the bracketed term is again the effective index of refraction; note that it only depends on Nh . Thus, the wave speed is essentially determined by the material with the higher propagation speed. We observe that (19c) cannot be derived from (18b) by taking the limit Np2 /Nh2 → ∞, that is, the limits of small area fraction and large contrast can not be interchanged. 5. A Variational Approach The propagation of electromagnetic waves through our periodic porous medium is governed by (16) which depends intimately on P , the solution to the boundary value problem (13). We believe that there are no exact solutions to this problem. Although evaluation of P can be obtained to a high degree of accuracy by a number of numerical methods, such calculations are likely to become intensive if the pore shape changes in z. In this section we present a variational approach, which leads to a simple approximation. First, we introduce the functional I(Q) defined by ZZ I ∂x 2 2 ds (20) I(Q) = N |∇Q| dx dy + 2 Q ∂n R 13 y 1 r 2 Np 0 r= 1/2 N2 h x 1 Figure 3. The cell geometry for the trial function P0 . where Q is taken from the set D of functions which are periodic on the cell R, continuous there, and possess piecewise smooth partial derivatives. Note that the line integral runs over the boundary of the pore. We may use standard calculus of variations arguments to deduce that the function which minimizes I is the solution to (13), and conversely, the solution of (13) minimizes I(Q). This allows us to use a Raleigh-Ritz approach to reduce (13) to a finite dimensional linear algebra problem. The procedure is straightforward and we do not present the general case here. Rather, we shall choose a single function from D, and minimize I with respect to it, i.e. employ a special one-dimensional Raleigh-Ritz approximation. For simplicity, we take the cell to be square, i.e. l = 1 and the pore to be circular with radius r0 and center (1/2, 1/2). We then introduce a larger circle with radius 1/2 and the same center as the pore; see Figure 3. As a trial function we take 1 (1 − 2 )r, r < r0 , 4r0 2 4r0 cos θ P0 = 2 (21a) 1 r0 < r < 1/2, (r − ), Np + Nh2 + 4r02 [N 2 ]C 4r 0, elsewhere. This function is harmonic within the larger circle, satisfies the boundary conditions 14 (13b), vanishes on the circumference of the larger circle and in the corners of the remaining region. We now take Q = αP0 and choose α to minimize I(Q). Inserting this value of Q into (20) and setting its derivative with respect to α to zero yields ∂x ds ∂n α = − RR , N 2 |∇P0 |2 dx dy R H P0 (21b) where the contour integral is taken around the circular pore. A straightforward calculation of the integrals in (21b) yields α = 1. Then, when P0 is inserted into the integral in (16a) the approximate amplitude equation becomes d2 A + k 2 Ω2 A = 0, 2 dz I ∂x 2 2 2 2 Ω =< N > +[N ]C P0 ds. ∂n (21c) (21d) Finally, we carry out the integration required in (21d) and find Ω2 = Nh2 ( 2φ[N 2 ]C 1− Np2 + Nh2 + 4 πφ [N 2 ]C ) (21e) where now φ = Ap = πr02 , since l = 1, and r0 may depend upon z. In Figure 4 we have plotted Ω2 /Nh2 as a function of Np2 /Nh2 for three different values of r0 . Each curve passes through the point (1, 1); that is when Np = Nh , Ω2 = Nh2 as expected. As r0 increases, the effect of the pore becomes more pronounced. The opposite is true as r0 decreases. In each case, Ω2 → γ 2 (r0 ) as Np /Nh → ∞, where γ 2 = 1 + 2Ap /(1 − 4r02 ). This behavior is qualitatively the same as given by (19c), but the asymptote differs by the factor 2/(1 − 4r02 ). This error is a result of employing the simple test function P0 . Finally we note that Ω2 given by (21e) reduces to the dilute limit (18b) as φ → 0. 6. Propagation and Reflection Up until now we have been concerned with developing a theory for electromagnetic wave propagation through a periodic, porous medium. In this section the problem of propagation through, and reflection from, a finite slab of this material is analyzed and simple formulae for the reflection and transmission coefficients will be derived. 15 3 r0 = 2/5 r0 = 1/3 r0 = 1/6 2 2 Ω /N h 2 1 0 0 1 2 2 N p/N 3 2 h Figure 4. The effective wavenumber against contrast of refractive indices. We begin by considering an incident plane electromagnetic wave whose electric field is polarized in the x direction (i.e. E2 = 0, H1 = 0). In the fundamental cell, but external to the porous slab, the electric field E1 and the magnetic field H2 may be written, respectively, as the eigenfunction expansions X E1 = eikz + ρe−ikz + e1nm ψnm eknm z , (22a) n,m H2 = eikz − ρe−ikz + X h2nm ψnm eknm z , (22b) n,m where ρ is the reflection coefficient, p the infinite sums omit the n = m = 0 term, 2πi(nx+my/l) ψnm = e , and knm = 4π 2 (n2 + m2 /l2 )/δ 2 − k 2 assuming the waves propagate in a vacuum (i.e. N = 1). We note that the first two terms, i.e. the m = n = 0 terms, in (22) represent the incident and reflected plane waves respectively. The form of these fields (22) follows from the fact all the components of the electromagnetic fields satisfy the same Helmholtz equation, namely Exx + Eyy + δ 2 (Ezz + k 2 N 2 E) = 0, as may be derived from equations (1) and (2). We note here that the infinite sums contain only evanescent modes, which decay rapidly away from the interface between 16 p n2 + m2 /l2 ≫ 1. The other the slab and free-space. This is because knm ∼ 2π δ components of the electric and magnetic fields, i.e. E2 , E3 , H1 , and H3 , contain only evanescent terms. Now in the porous slab, we take Ψ = xdA/dz and accordingly H2 = dA/dz and E1 is given by (14a). Since E1 and H2 are tangential fields at the interface z = 0, they are continuous across it. Thus, we equate them to (22a) and (22b), respectively, at z=0 and find X ∂P 2 , (23a) 1+ρ+ e1nm ψnm = ikA(0) 1 − [N ]C ∂x n,m 1−ρ+ X h2nm ψnm = n,m dA (0). dz (23b) Next, these equations are integrated over the region R at z = 0 to give, respectively ZZ 2 l(1 + ρ) = ikA(0) l − [N ]C ∂P/∂x dx dy , (24a) R l(1 − ρ) = l dA (0). dz (24b) RR Equation (24a) simplifies to (1 + ρ) = ikA(0), since R ∂P/∂x dx dy = 0. This fact is readily deduced from integrating the quantity ∇ · (P ∇x) over the region R, using the divergence theorem and the continuity of P across the pore interface. Finally, combining the simplified version of (24a): ρ = ikA(0) − 1 (25a) with (24b) we deduce dA (0) + ikA(0) = 2. (25b) dz We can repeat the same analysis at z = 1, the other interface between the porous medium and free space. In free-space beyond the slab, z > 1, the outgoing/evanescent transverse fields may be written as E1 = τ eikz + X ē1nm ψnm eknm z , (26a) X h̄2nm ψnm eknm z , (26b) n,m H2 = τ eikz + n,m 17 where τ is the transmission coefficient; as before, the electric field is polarized in the x direction. Again integrating these over R, and equating on z = 1 with the field inside the slab we deduce dA (1) − ikA(1) = 0, dz (27a) τ = ike−ik A(1). (27b) Now, equations (25b) and (27a) are the boundary conditions required by the amplitude equation (16a). This is a standard two-point boundary value problem. If the pore shape is independent of z, and the refractive indices are constant throughout the slab, then it can be solved exactly; omitting details we find in this uniform case (K + k)eiK(z−1) + (K − k)eiK(1−z) A(z) = (K 2 + k 2 ) sin K + 2iKk cos K in which I ∂x 1 22 2 ds , K = k < N > + [N ]C P l ∂n 2 2 with < N 2 > given in (16b) and the reflection and transmission coefficients are ρ= (k 2 − K 2 ) sin K , (K 2 + k 2 ) sin K + 2iKk cos K τ= 2iKke−iK . (K 2 + k 2 ) sin K + 2iKk cos K Note that, if K = k, then the slab has the same effective wavenumber as that in free-space, and we find ρ = 0, τ = 1 as expected. If the pore shape depends on z then, in general, numerical methods must be employed to obtain an approximate solution to A(z). In either case, the reflection and transmission coefficients are determined uniquely from the boundary conditions at 0 and 1. 7. Conclusions We have presented a leading order homogenization analysis of electromagnetic wave propagation through a composite slab of finite thickness. Our analysis gives an explicit description of the electromagnetic fields within the slab. Specifically, the field components are proportional to either the amplitude A(z) or its derivative dA . This amplitude satisfies a second order differential equation with an effective dz 18 index of refraction, composed of two parts. The first is a linear combination of the host and pore indices of refraction, weighted by the relative areas of the host and pore, respectively. This is just a simple mixture term. The second part contains a line integral whose integrand depends upon the wave polarization and a harmonic function P . This potential function is doubly periodic in the plane z = constant, as are its derivatives, and satisfies an inhomogeneous jump condition across the pore interface. This term describes the microstructure effect on the effective index of refraction. We have considered three approximations of this potential function, each yielding an approximate, effective index. In the first case, the pore area was assumed to be small compared to that of the host, i.e. the dilute concentration limit. The effective index in this case is Nh2 with a small correction due to the pores. In the second case, the index of the pore was assumed to be much larger than that of the host. The effective index in this case is Nh2 [1 +φ] where φ is the relative area of the pore to the unit cell. In the third case we made use of an equivalent variational statement of our potential problem. A simple one-term Raleigh-Ritz approximation was employed to produce an approximate effective index of refraction. We are presently investigating various numerical methods to better approximate our potential function, and hence the effective index of refraction. Finally, we note here that our analysis can easily be extended to handle a smooth 2 N . In this case Φ still satisfies (5), but now P is defined by Φ = ikA{x − P } for the x polarization, where, as in Section 3, A is a function of z only. It follows that P now satisfies ∂ 2 N (28) ∇ · (N 2 ∇P ) = ∂x and the same periodic boundary conditions. The derivation of the ordinary differential equation for A follows along a parallel path. The exception is that the ∂ identity ∇ · (P N 2 ∇x − xN 2 ∇P ) = (P − x) N 2 is now integrated around the ∂x entire fundamental cell. Omitting the details it is easy to demonstrate that A satisfies ZZ ∂ 2 1 d2 A 2 2 P +k < N >+ N dx dy A = 0, (29a) dz 2 l ∂x R where 1 < N >= l 2 ZZ N 2 dx dy, (29b) R where R denotes the fundamental cell. Again the effective index of refraction contains an average, or mixture, term and a part that depends upon the microstructure of the electric field. 19 Acknowledgments The work of G. A. Kriegsmann was sponsored by the Department of Energy under grant number DE-FG02-04ER25654. Appendix 1. In this Appendix we derive the boundary conditions given in equations (9a) and (9b) in the text. To begin we describe our pore surface by the dimensional equation XP = LX r0 (θ, Z/L)n̂ + Z k̂, (A1) where n̂ = (cos θ, sin θ, 0) and r0 is a smooth function of its arguments and is 2π periodic in θ. This is a very general pore surface, which reduces to a circular structure when r0 is independent of θ. We observe that the intersection of the surface with the plane z = Z/L = constant is the curve C shown in Figure 2 (in the case when r0 is independent of θ). The unit tangents to our pore surface are given by r0θ n̂ + r0 t̂ , T̂1 = p 2 2 r0 + r0θ δr0z n̂ + k̂ , T̂2 = p 2 1 + δ 2 r0z (A2) (A3) where t̂ = (− sin θ, cos θ, 0) and r0θ , r0z indicate partial derivatives of r0 with respect to the specified variable. We note that the vector T̂1 lies in the plane z = constant and is tangent to the curve C. The small parameter, δ = Lx /L, in T̂2 implies that the pore shape is slowly changing in z. Taking the curl of these two tangent vectors gives the unit normal to the pore surface p r 2 + r 2 n̂1 − δr0 r0z k̂ , (A4) N̂ = p0 2 0θ2 2 δ2 r0 + r0θ + r02 r0z p 2 ) is the unit normal to the curve C, i.e. n̂ · T̂ = where n̂1 = (−r0θ t̂+r0 n̂)/ r02 + r0θ 1 1 0. We now have the prerequisite geometry to derive the boundary conditions (9). Recall from (3) that the dimensional electric field is given by E′ = E0 {E1 î + E2 ĵ + δE3 k̂}. Expressing the Ei in terms of the potentials Φ and Ψ we have E′ = E0 { ∂Φ ∂Φ î + ĵ + δ(Φz − ikΨ)k̂}. ∂x ∂y (A5) 20 From Maxwell’s equations one can deduce [3] that the tangential component of the electric field along T̂1 must be continuous across the surface. Taking the dot product of E′ with this unit vector gives [∇Φ · T̂1 ]C = 0, where ∇ is the two-dimensional gradient operator given in terms of x and y only. But this is just the directional derivative of Φ along the curve C. From this observation we deduce that [Φ]C = a constant which, without loss of generality, we take to be zero. This is the first boundary condition in (9a). Similarly from (3) the dimensionless magnetic field is given in terms of the potential Ψ and H3 by H′ = H0 {−Ψy î + Ψx ĵ + δH3 k̂}. (A6) Its component along T̂1 must also be continuous across the pore surface, see [4]. Taking the dot product of H′ and T̂1, noting the definition of n̂1 , and using the ∂Ψ = 0. This is the second boundary same reasoning as above we deduce ∂n C condition in (9a) where the subscript denotes the normal derivative of Ψ across the curve C. To obtain the two remaining boundary conditions in (9b) we can proceed along two paths. The first would be to demand the continuity of the electric and magnetic field components along T̂2 across the surface. A second and equivalent approach, see [4], is to require that N 2 (E′ · N̂ ) and H′ · N̂ are continuous across the pore surface. We choose the latter for ease of presentation and calculation. We find by taking the dot product of (A5) with (A4) and multiplying the result by N 2 that ∂Φ + O(δ 2 ), (A7) N 2 (E′ · N̂ ) = N 2 ∂n where the O(δ 2 ) involves the component of E′ along k̂. Since we are only concerned with the leading order approximations of all the fields, we neglect the O(δ 2 ) term in (A7). Then the continuity of the resulting expression across C gives the first boundary condition in (9b). Performing the same analysis on H′ · N̂ gives the second. References (1) Deepak and J. W. Evans, “Mathematical Model for Chemical Vapor Infiltration in a Microwave-Heated Preform”, Journal of the American Ceramic Society, 76 (1993), pp. 1924-29. (2) X. Mei, M. Blumin, M. Sun, D. Kim, Z.H. Wu, and H.E. Ruda, “Highly Ordered GaAs/AlGaAs Quantum Dot Arrays on GaAs(01) Substrates”, Applied Physics Letters 82 (2003), pp. 967-969. 21 (3) D. S. Jones, Acoustic and Electromagnetic Scattering, Oxford Science Publications, Clarendon Press, Oxford (1989). (4) G. A. Kriegsmann, “Electromagnetic Propagation in Periodic Porous Structures”, Wave Motion, 36 (2002) pp. 457-472. (5) W. J. Parnell and I. D. Abrahams, “Dynamic Homogenization in Periodic Fibre Reinforced Media: Quasi-Static Limit for SH Waves”, Wave Motion, 43 (2005), pp. 474-498.