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Electromagnetic Waves & Antennas Toolbox

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Electromagnetic Waves & Antennas Toolbox

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06 Feb 2004 (Updated )

Companion Software

multidiel(n,L,lambda,theta,pol)
% multidiel.m - reflection response of isotropic or birefringent multilayer structure
%
%          na | n1 | n2 | ... | nM | nb
% left medium | L1 | L2 | ... | LM | right medium 
%   interface 1    2    3     M   M+1
%
% Usage: [Gamma,Z] = multidiel(n,L,lambda,theta,pol)
%        [Gamma,Z] = multidiel(n,L,lambda,theta)       (equivalent to pol='te')
%        [Gamma,Z] = multidiel(n,L,lambda)             (equivalent to theta=0)
%
% n      = isotropic 1x(M+2), uniaxial 2x(M+2), or biaxial 3x(M+2), matrix of refractive indices
% L      = vector of optical lengths of layers, in units of lambda_0
% lambda = vector of free-space wavelengths at which to evaluate the reflection response
% theta  = incidence angle from left medium (in degrees)
% pol    = for 'tm' or 'te', parallel or perpendicular, p or s, polarizations
%
% Gamma = reflection response at interface-1 into left medium evaluated at lambda 
% Z     = transverse wave impedance at interface-1 in units of eta_a (left medium)
%
% notes: M is the number of layers (M >= 0)
%
%        n = [na, n1, n2, ..., nM, nb]        = 1x(M+2) row vector of isotropic indices
%
%            [ na1  n11  n12  ...  n1M  nb1 ]   3x(M+2) matrix of birefringent indices, 
%        n = [ na2  n21  n22  ...  n2M  nb2 ] = if 2x(M+2), it is extended to 3x(M+2)
%            [ na3  n31  n32  ...  n3M  nb3 ]   by repeating the top row
%
%        optical lengths are in units of a reference free-space wavelength lambda_0:
%        for i=1,2,...,M,  L(i) = n(1,i) * l(i), for TM, 
%                          L(i) = n(2,i) * l(i), for TE,
%        TM and TE L(i) are the same in isotropic case. If M=0, use L=[].
%
%        lambda is also in units of lambda_0, that is, lambda/lambda_0 = f_0/f
%
%        reflectance = |Gamma|^2, input impedance = Z = (1+Gamma)./(1-Gamma)
%
%        delta(i) = 2*pi*[n(1,i) * l(i) * sqrt(1 - (Na*sin(theta))^2 ./ n(3,i).^2))]/lambda, for TM
%        delta(i) = 2*pi*[n(2,i) * l(i) * sqrt(1 - (Na*sin(theta))^2 ./ n(2,i).^2))]/lambda, for TE
%
%        if n(3,i)=n(3,i+1)=Na, then will get NaN's at theta=90 because of 0/0, (see also FRESNEL)

% S. J. Orfanidis - 2000 - www.ece.rutgers.edu/~orfanidi/ewa

function [Gamma,Z] = multidiel(n,L,lambda,theta,pol)

if nargin==0, help multidiel; return; end
if nargin<=4, pol='te'; end
if nargin==3, theta=0; end

if size(n,2)==1, n = n'; end                            % in case n is entered as column 

K = size(n,1);                                          % birefringence dimension
M = size(n,2)-2;                                        % number of layers

if K==1, n = [n; n; n]; end                             % isotropic case
if K==2, n = [n(1,:); n]; end                           % uniaxial case

if M==0, L = []; end                                    % single interface, no slabs

theta = theta * pi/180;

if pol=='te',
    Nsin2 = (n(2,1)*sin(theta))^2;                      % (Na*sin(tha))^2              
    c = sqrt(1 - Nsin2 ./ n(2,:).^2);                   % coefficient ci, or cos(th(i)) in isotropic case
    nT = n(2,:) .* c;                                   % transverse refractive indices
    r = n2r(nT);                                        % r(i) = (nT(i-1)-nT(i)) / (nT(i-1)+nT(i))
else
    Nsin2 = (n(1,1)*n(3,1)*sin(theta))^2 / (n(3,1)^2*cos(theta)^2 + n(1,1)^2*sin(theta)^2);
    c = sqrt(1 - Nsin2 ./ n(3,:).^2);
    nTinv = c ./ n(1,:);                                % nTinv(i) = 1/nT(i) to avoid NaNs
    r = -n2r(nTinv);                                    % minus sign because n2r(n) = -n2r(1./n)
end

if M>0,
    L = L .* c(2:M+1);                                  % polarization-dependent optical lengths
end

Gamma = r(M+1) * ones(1,length(lambda));                % initialize Gamma at right-most interface

for i = M:-1:1,                                         % forward layer recursion 
    delta = 2*pi*L(i)./lambda;                          % phase thickness in i-th layer
    z = exp(-2*j*delta);                          
    Gamma = (r(i) + Gamma.*z) ./ (1 + r(i)*Gamma.*z);
end

Z = (1 + Gamma) ./ (1 - Gamma);






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