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svmodes.m
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svmodes.m
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function [phi,neff] = svmodes (lambda, guess, nmodes, dx, dy, eps, boundary, field);
% This function calculates the modes of a dielectric waveguide
% using the semivectorial finite difference method.
%
% USAGE:
%
% [phi,neff] = svmodes (lambda, guess, nmodes, dx, dy, eps, ...
% boundary, field);
%
% INPUT:
%
% lambda - optical wavelength
% guess - scalar shift to apply when calculating the eigenvalues.
% This routine will return the eigenpairs which have an
% effective index closest to this guess
% nmodes - the number of modes to calculate
% dx - horizontal grid spacing
% dy - vertical grid spacing
% eps - index mesh (= n^2(x,y))
% boundary - 4 letter string specifying boundary conditions to be
% applied at the edges of the computation window.
% boundary(1) = North boundary condition
% boundary(2) = South boundary condition
% boundary(3) = East boundary condition
% boundary(4) = West boundary condition
% The following boundary conditions are supported:
% 'A' - field is antisymmetric
% 'S' - field is symmetric
% '0' - field is zero immediately outside of the
% boundary.
% field - must be 'EX', 'EY', or 'scalar'
%
% OUTPUT:
%
% phi - three-dimensional vector containing the requested
% field component for each computed mode
% neff - vector of modal effective indices
%
% AUTHOR: Thomas E. Murphy (tem@umd.edu)
boundary = upper(boundary);
[nx,ny] = size(eps);
% now we pad eps on all sides by one grid point
eps = [eps(:,1),eps,eps(:,ny)];
eps = [eps(1,:); eps ; eps(nx,:)];
% compute free-space wavevector
k = 2*pi/lambda;
if isscalar(dx)
dx = dx*ones(nx+2,1); % uniform grid
else
dx = dx(:); % convert to column vector
dx = [dx(1);dx;dx(length(dx))]; % pad dx on top and bottom
end
if isscalar(dy)
dy = dy*ones(1,ny+2); % uniform grid
else
dy = dy(:); % convert to column vector
dy = [dy(1);dy;dy(length(dy))].'; % pad dy on top and bottom
end
n = ones(1,nx*ny); n(:) = ones(nx,1)*(dy(3:ny+2)+dy(2:ny+1))/2;
s = ones(1,nx*ny); s(:) = ones(nx,1)*(dy(1:ny)+dy(2:ny+1))/2;
e = ones(1,nx*ny); e(:) = (dx(3:nx+2)+dx(2:nx+1))/2*ones(1,ny);
w = ones(1,nx*ny); w(:) = (dx(1:nx)+dx(2:nx+1))/2*ones(1,ny);
p = ones(1,nx*ny); p(:) = dx(2:nx+1)*ones(1,ny);
q = ones(1,nx*ny); q(:) = ones(nx,1)*dy(2:ny+1);
en = ones(1,nx*ny); en(:) = eps(2:nx+1 ,3:ny+2);
es = ones(1,nx*ny); es(:) = eps(2:nx+1 ,1:ny);
ee = ones(1,nx*ny); ee(:) = eps(3:nx+2 ,2:ny+1);
ew = ones(1,nx*ny); ew(:) = eps(1:nx ,2:ny+1);
ep = ones(1,nx*ny); ep(:) = eps(2:nx+1 ,2:ny+1);
switch lower(field)
case 'ex'
an = 2./n./(n+s);
as = 2./s./(n+s);
ae = 8*(p.*(ep-ew)+2.*w.*ew).*ee./...
((p.*(ep-ee)+2.*e.*ee).*(p.^2.*(ep-ew)+4.*w.^2.*ew) + ...
(p.*(ep-ew)+2.*w.*ew).*(p.^2.*(ep-ee)+4.*e.^2.*ee));
aw = 8*(p.*(ep-ee)+2.*e.*ee).*ew./...
((p.*(ep-ee)+2.*e.*ee).*(p.^2.*(ep-ew)+4.*w.^2.*ew) + ...
(p.*(ep-ew)+2.*w.*ew).*(p.^2.*(ep-ee)+4.*e.^2.*ee));
ap = ep.*k^2 - an - as - ae.*ep./ee - aw.*ep./ew;
case 'ey'
an = 8*(q.*(ep-es)+2.*s.*es).*en./...
((q.*(ep-en)+2.*n.*en).*(q.^2.*(ep-es)+4.*s.^2.*es) + ...
(q.*(ep-es)+2.*s.*es).*(q.^2.*(ep-en)+4.*n.^2.*en));
as = 8*(q.*(ep-en)+2.*n.*en).*es./...
((q.*(ep-en)+2.*n.*en).*(q.^2.*(ep-es)+4.*s.^2.*es) + ...
(q.*(ep-es)+2.*s.*es).*(q.^2.*(ep-en)+4.*n.^2.*en));
ae = 2./e./(e+w);
aw = 2./w./(e+w);
ap = ep.*k^2 - an.*ep./en - as.*ep./es - ae - aw;
case 'scalar'
an = 2./n./(n+s);
as = 2./s./(n+s);
ae = 2./e./(e+w);
aw = 2./w./(e+w);
ap = ep.*k^2 - an - as - ae - aw;
end
ii = zeros(nx,ny);
ii(:) = (1:nx*ny);
% Modify matrix elements to account for boundary conditions
% north boundary
ib = zeros(1,nx);
ib(:) = ii(1:nx,ny);
if (boundary(1) == 'S')
ap(ib) = ap(ib) + an(ib);
elseif (boundary(1) == 'A')
ap(ib) = ap(ib) - an(ib);
end
% south boundary
ib = zeros(1,nx);
ib(:) = ii(1:nx,1);
if (boundary(2) == 'S')
ap(ib) = ap(ib) + as(ib);
elseif (boundary(2) == 'A')
ap(ib) = ap(ib) - as(ib);
end
% east boundary
ib = zeros(1,ny);
ib(:) = ii(nx,1:ny);
if (boundary(3) == 'S')
ap(ib) = ap(ib) + ae(ib);
elseif (boundary(3) == 'A')
ap(ib) = ap(ib) - ae(ib);
end
% west boundary
ib = zeros(1,ny);
ib(:) = ii(1,1:ny);
if (boundary(4) == 'S')
ap(ib) = ap(ib) + aw(ib);
elseif (boundary(4) == 'A')
ap(ib) = ap(ib) - aw(ib);
end
iall = zeros(1,nx*ny); iall(:) = ii;
in = zeros(1,nx*(ny-1)); in(:) = ii(1:nx,2:ny);
is = zeros(1,nx*(ny-1)); is(:) = ii(1:nx,1:(ny-1));
ie = zeros(1,(nx-1)*ny); ie(:) = ii(2:nx,1:ny);
iw = zeros(1,(nx-1)*ny); iw(:) = ii(1:(nx-1),1:ny);
A = sparse ([iall,iw,ie,is,in], ...
[iall,ie,iw,in,is], ...
[ap(iall),ae(iw),aw(ie),an(is),as(in)]);
shift = (2*pi*guess/lambda)^2;
options.tol = 1e-12;
options.disp = 0; % suppress output
options.isreal = isreal(A);
clear an as ae aw ap in is ie iw iall ii en es ee ew ep ...
n s e w p q;
[v,d] = eigs(A,speye(size(A)),nmodes,shift,options);
neff = lambda*sqrt(diag(d))/(2*pi);
phi = zeros(nx,ny,nmodes);
temp = zeros(nx,ny);
for k = 1:nmodes;
temp(:) = v(:,k)/max(abs(v(:,k)));
phi(:,:,k) = temp;
end;