Calculation of Reflection and Transmission Coefficients in scuff-transmission

Transcription

1 Calculation of Reflection and Transmission Coefficients in scuff-transmission Homer Reid May 9, 2015 Contents 1 The Setup 2 2 Scattering coefficients from surface currents Computation of b(q)

2 Homer Reid: scuff-transmission 2 1 The Setup scuff-transmission considers geometries with 2D periodicity, i.e. the structure consists of a unit-cell geometry of finite extent in the z direction that is infinitely periodically replicated in both the x and y directions. The structure is illuminated either from below (the default) or from above by a plane wave with propagation vector k confined to the xz plane. Working at angular frequency ω, let the free-space wavelength be k 0 = ω c, and let the relative permittivity and permeability of the lowermost and uppermost regions in the geometry be ɛ L,U and µ L,U. The wavenumber, refractive index, and relative wave impedance in the uppermost and lowermost regions are k L = n L k 0, n L ɛ L µ L, Z L = k U = n U k 0, n U ɛ U µ U, Z U = µl ɛ L µu ɛ U. I will refer to region from which the planewave originates (either the uppermost or lowermost homogeneous region in the scuff-em geometry) as the incident region. The region into which the planewave eventually emanates is the transmitted region. I will use the sub/superscripts I, T to denote these quantities; thus the wavenumber and relative wave impedance in the incident and transmitted regions are { } } kl, Z L, k U, Z U, wave incident from below {k I, Z I, k T, Z T = { } ku, Z U, k L, Z L, wave incident from above In what follows, I will use the symbols k I, k R, k T respectively to denote the propagation vectors of the incident, reflected, and transmitted waves. I will take these vectors always to live in the xz plane (i.e. k has no y component, k y = 0). I will let θ I and θ T be the angles of incidence and transmission (the angles between the incident and transmitted wavevector and the z axis). wave incident from below: k I = k L sin θ I ˆx + cos θ I ẑ wave incident from above: k I = k U sin θ I ˆx cos θ I ẑ The reflected wavevector is wave incident from below: k R = k L sin θ I ˆx cos θ I ẑ wave incident from above: k R = k L sin θ I ˆx + cos θ I ẑ The transmitted wavevector is wave incident from below: k T = k U sin θ T ˆx + cos θ T ẑ wave incident from above: k T = k L sin θ T ˆx cos θ T ẑ

3 Homer Reid: scuff-transmission 3 The incident and transmitted angles are related by n I sin θ I = n T sin θ T. For a general vector v, I will define v = v/ v to be a unit vector in the direction of v. Definition of scattering coefficients The incident, reflected, and transmitted fields may be written in the form E I (x) = E 0 ɛ I e iki x E R (x) = re 0 ɛ R e ikr x E T (x) = te 0 ɛ T e ikt x H I (x) = H 0 ɛ I e iki x l H R (x) = rh 0 ɛ R e ikr x H T (x) = th 0 ɛ T, e ikt x (1) where E 0 is the incident field magnitude, ɛ I,R,T are E-field polarization vectors, ɛ I,R,T are H-field polarization vectors, and we have H 0 i k E 0 Z I Z 0, H 0 i k E 0 Z T Z 0, ɛ k ɛ, ɛ = k ɛ. The polarization vectors are given by for the TE case: ɛ I TE = ɛ R TE = ɛ T TE = ŷ, ɛ I,R,T TE for the TM case: ɛ I TM = ɛ R TM = ɛ T TM = ŷ ɛ I,R,T TM = k I,R,T ŷ = k I,R,T ŷ Equations (1) define the reflection and transmission coefficients r and t computed by scuff-transmission.

4 Homer Reid: scuff-transmission 4 2 Scattering coefficients from surface currents Next we consider an extended structure described by Bloch-periodic boundary conditions, i.e. all fields and currents satisfy Q(x + L) = e ikb L Q(x) (2) where Q is a field (E or H) or a surface current (K or N) and the Bloch wavevector is 1 k B = k I sin θ I ˆx = k T sin θ T ˆx. For plane waves like (1), equation (2) actually holds for any arbitrary vector L; for our purposes we will only need to use it for certain special vectors L determined by the structure of the lattice in our PBC geometry. We will derive expressions for the plane-wave reflection and transmission coefficients in terms of the surface-current distribution in the unit cell of the structure. Fields from surface currents On the other hand, the scattered E fields in the incident and transmitted regions may be obtained in the usual way from the surface-current distributions on the surfaces bounding those regions. For example, at points in the transmitted medium, the scattered (that is, transmitted) E field is given by { } E T (x) = Γ EE;T (x, x ) K(x ) + Γ EM;T (x, x ) N(x ) dx S T { } = ik T Z 0 Z T G(k T ; x, x ) K(x ) + C(k T ; x, x ) N(x ) dx (3) S where S T is the surface bounding the transmitted region and G, C are the homogeneous dyadic GFs for that region. Using the Bloch periodicity of the surface currents, i.e. { } { } K(x + L) K(x) = e ikb L N(x + L) N(x) we can restrict the surface integral in (3) to just the lattice unit cell: { } E T (x) = ik T Z 0 Z T G(k T ; x, x ) K(x ) + C(k T ; x, x ) N(x ) dx (4) UC where the periodic Green s functions are { } G(x, x ) { } G(x, x + L) e ikb L C(x, x ) C(x, x + L) L (5) 1 Recall our convention that the propagation vector lives in the xy plane.

5 Homer Reid: scuff-transmission 5 I now invoke the following representation of the dyadic Green s functions (Chew, 1995): for z > z, G(k; ρ, z; ρ, z dq ) = (2π) G ± 2 (k; q)e iq (ρ ρ ) e ±iqz(z z ) (6a) C(k; ρ, z; ρ, z dq ) = (2π) C ± 2 (k; q)e iq (ρ ρ ) e ±iqz(z z ) (6b) where q = (q x, q y ) is a two-dimensional Fourier wavevector, dq = dq x dq y, q z = k2 q 2, ± = sign(z z ), and G ± (k; q) = i 2q z C ± (k; q) = i 2q z k k 2 0 ±q z q y q z 0 q x q y q x 0. q 2 x q x q y ±q x q z q y q x q 2 y ±q y q z ±q z q x ±q z q y q 2 z Inserting (6) into (5), I obtain, for the periodic version of e.g. the G kernel, G(k; ρ, z; ρ dq, z) = (2π) G ± 2 (k; q)e iq (ρ ρ ) e ±iqz(z z ) e i(kb q) L = V 1 UC q=k B+Γ G ± (k; q)e iq (ρ ρ ) e ±iqz(z z ) L }{{} V BZ Γ δ(q k Γ) where the sum over Γ runs over reciprocal lattice vectors; the prefactor V BZ, the volume of the Brillouin zone, is related to the unit-cell volume by V BZ = 4π 2 /V UC for a 2D square lattice. Similarly, we find C(k; ρ, z; ρ, z) = V 1 UC q=k B+Γ C ± (k; q)e iq (ρ ρ ) e ±iqz(z z ). Keeping only the Γ = 0 term in these sums, the scattered E-fields in the uppermost and lowermost regions are thus E upper (x) = e i(kuxx+kuzz)[ ik U Z 0 Z G+ U (k U ; k B ) K ] U (k B ) + ik C+ U (k U ; k )Ñ(k B B) (7) E lower (x) = e i(klxx klzz)[ ik L Z 0 Z G U (k L ; k B ) K(k ] B ) + ik C L (k L ; k )Ñ(k B B) where e.g. KU is something like the two-dimensional Fourier transform of the surface currents on the boundary of the uppermost region R U : K U (k B ) 1 K(ρ, z )e ikb ρ iq z z dx, qz 2 = k 2 U k B 2. V UC R U (8)

6 Homer Reid: scuff-transmission 6 with K L and ÑU,L defined similarly. Comparing this to (1c), we see that the transmission and reflection coefficients for the polarization defined by polarization vector ɛ are given by { } t = ik r U Z 0 Z U ɛ G+ (k U, k B ) K U (k B ) + ik U ɛ C+ (k U, k )Ñ(k B B) (9) { } r = ik t L Z 0 Z L ɛ G (k L, k B ) K L (k B ) + ik L ɛ C (k L, k )Ñ(k B B) (10) The expressions on the RHS compute the upper (lower) quantities on the LHS for the case in which the plane wave is incident from below (above). The K and Ñ quantities are given by sums of contributions from individual basis functions; for example, K U (q) = 1 V UC b α R U k α bα (q), Ñ U (q) = Z 0 V UC b α R U n α bα (q) where the sums are over all RWG basis functions that live on surfaces bounding the uppermost medium and {k α, n α } are the surface-current coefficients obtained as the solution to the scuff-em scattering problem. 2.1 Computation of b(q) For RWG functions the quantity b(q) may be evaluated in closed form: b α (q) b α (x)e iq x dx sup b α 1 u { ( ) = l α du dv e iq [Q+ +ua + +vb] ua + + vb 0 0 ( ) } e iq [Q +ua +vb] ua + vb where = l α {e iq Q+[ ( ) f 1 q A +, q B A + ( ) ] + f 2 q A +, q B B e iq Q [ ( ) f 1 q A, q B A ( ) ] } + f 2 q A, q B B f 1 (x, y) = f 2 (x, y) = 1 u u 0 0 ue i(ux+vy) dv du ve i(ux+vy) dv du.