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%----------------------------------------------------------------------------------------------
% PROGRAM: decompose
% AUTHOR: Andri M. Gretarsson
% DATE: 7/10/04
%
% SYNTAX: [coeffs,tmat]=decompose(z1,domain,type,terms,[q <,lambda,accuracy>]);
% <...> indicates optional arguments
%
% Calculates the coefficients of the decomposition of the function z1 into
% Hermite Gaussian modes or Laguerre Gaussian modes.
%
% INPUT ARGUMENTS:
% ----------------
% z1 = The values of the function to be decomposed. 2D (nxn) matrix
% domain = the domain values at which the values in z1 are specified.
% The domain is a nxmx2 array where domain(:,:,1) and
% domain(:,:,2) are the x and y meshes corresponding to the
% values in z1. (These meshes are often generated using meshgrid.m.)
% type = 'hg' for a Hermite Gaussian mode decomposition
% 'lg' for a Laguerre Gaussian mode decomposition
% terms = This argument can be specified either as a scalar or as a matrix.
% If it is specified as a scalar, it indicates the number of
% terms out to which to calculate the decomposition. Since the
% Hermite and Laguerre bases are two dimensional and are specified
% by a 2D argument (l,m) or (p,m), default rules are applied to
% decide exactly which terms to calculate. The output
% matrix tmat indicates what terms were calculated.
%
% When terms is specified as a matrix, it indicates exactly which
% terms to calculate. For Hermite Gaussian modes (type='hg')
% terms is a LxM matrix where a 1 in the (i,j)th entry indicates
% that the term TEM(l,m)=(i-1,j-1) will be calculated. A zero
% indicates it will not be calculated. For example:
%
% 1 0 0
% terms = 0 1 0
% 0 1 0
%
% indicates that the coefficients of the Hermite Gaussian modes
% (0,0), (1,1) and (2,1) will be calculated but no others.
%
% For Laguerre Gaussian modes (type='lg') the terms matrix is
% interpreted similarly. However, since the (p,m) mode can
% have either positive or negative m terms the terms matrix is now
% a PxMx2 matrix, where terms(:,:,1) indicates which positive-m
% terms to calculate and terms(:,:,2) indicates which negative-m
% terms to calculate. However, note that since m=0 must not
% be duplicated, the first column in terms(:,:,2) is always zero.
% Thus for example:
%
% 1 0 0
% terms(:,:,1) = 0 1 0
% 0 1 0
%
% 0 0 0
% terms(:,:,2) = 0 1 0
% 0 1 1
%
% indicates that the coefficients of the Laguerre Gaussian modes
% (0,0), (1,1), (2,1), (1,-1), (2,-1) and (2,-2) will be calculated.
%
% q = the complex radius of curvature "q" of the Gaussian basis.
% lambda = wavelength of the light in the Gaussian basis. Default is 1.064 microns
% accuracy = only calculate the coefficients of the decomposition to the
% specified accuracy. Actually rounds each result to the nearest
% increment of accuracy. For example, if accuracy=0.3, then a coefficient
% of 1.54 would be rounded to 1.5 while a coefficient of 1.56 would be
% rounded to 1.8. Accuracy of 0 applies no rounding. Specifying
% accuracy other than "0" does not speed up the calculation.
%
% OUTPUT ARGUMENTS:
% -----------------
% coeffs = the coefficients of the terms in the expansion. coeffs is a matrix
% the same size a tmat.
% tmat = coefficient request matrix. Often this would be specified by the user
% by the input argument "terms", in other words tmat=terms. However, as explained
% under the definition of the input argument terms, this function
% allows the user to specify only how many terms he wants, in other words terms can be
% a scalar and not a matrix. In this case tmat must be calculated according to
% default rules. In this case it is convenient to have the terms request matrix
% as an output argument. Each entry in the coefficient request matrix tmat correspond
% to a particular term of the expansion (as explained above). If the tmat(i,j) is 1,
% the coefficient of that term is calculated and returned in coeffs(i,j). If tmat(i,j)
% is 0, the corresponding coefficient was not calculated and coeffs(i,j) is set to 0
% regardless of whether the contribution of that term to z1 is really zero.
%
% EXAMPLE 1 (Hermite Gaussian, only number of terms specified):
% [x,y]=meshgrid([-pi/2:0.1:pi/2],[-pi/2:0.1:pi/2]); z1=cos(sqrt(x.^2+y.^2));
% clear domain; domain(:,:,1)=x; domain(:,:,2)=y;
% w=pi/4; R=1e3; lambda=1e-6; q=(1./R - i* lambda./pi./w.^2).^(-1);
% [coeffs,tmat]=decompose(z1,domain,'hg',140,[q,lambda,1e-6])
% z1recomposed=recompose(domain,'hg',coeffs,[q,lambda,1e-6]);
% subplot(3,1,1); h=pcolor(x,y,abs(z1).^2); set(h,'EdgeColor','none'); axis square; colorbar
% subplot(3,1,2); h=pcolor(x,y,abs(z1recomposed).^2); set(h,'EdgeColor','none'); axis square; colorbar
% subplot(3,1,3); h=pcolor(x,y,abs(z1).^2-abs(z1recomposed).^2); set(h,'EdgeColor','none'); axis square; colorbar; shg;
%
% Last updated: July 18, 2004 by AMG
%----------------------------------------------------------------------------------------------
% SYNTAX: [coeffs,tmat]=decompose(z1,domain,type,terms,[q <,lambda,accuracy>]);
%----------------------------------------------------------------------------------------------
function [coeffs,tmat]=decompose(z1,domain,type,terms,params)
% HERMITE GAUSSIAN EXPANSION --------------------------------------------------------------------------
if strcmpi(type,'hg')
q=params(1);
if length(params)>=2, lambda=params(2); else lambda=1.064e-6; end
if length(params)>=3, accuracy=params(3); else accuracy=1e-4; end
if size(terms,1)==1 && size(terms,2)==1 % Make terms request matrix
n=ceil(sqrt(terms));
tmat=ones(n,n);
if n^2>terms
tmat(end,end-ceil((n^2-terms)/2)+1:end)=0;
tmat(end-floor((n^2-terms)/2):end-1,end)=0;
end
else
tmat=terms;
end
coeffs=zeros(size(tmat));
for s=1:size(tmat,1) % Calculate the coefficients
l=s-1;
for t=1:size(tmat,2)
m=t-1;
if tmat(s,t)==1
z2=HermiteGaussianE([l,m,q,lambda],domain(:,:,1),domain(:,:,2));
coeffs(s,t)=overlap(z1,conj(z2),domain,1,accuracy);
else
coeffs(s,t)=0;
end
end
end
% LAGUERRE GAUSSIAN EXPANSION --------------------------------------------------------------------------
elseif strcmpi(type,'lg')
q=params(1);
if length(params)>=2, lambda=params(2); else lambda=1.064e-6; end
if length(params)>=3, accuracy=params(3); else accuracy=1e-4; end
if size(terms,1)==1 && size(terms,2)==1 % Make terms request matrix
n=ceil( (1+sqrt(1+8*terms))/4 );
tmat=ones(n,n,2); tmat(:,1,2)=0;
ndiff=2*n^2-n-terms;
if ndiff>0
rdiff=ceil(ndiff/2); ldiff=floor(ndiff/2);
tmat(end,end-ceil(rdiff/2)+1:end,1)=0;
tmat(end-floor(rdiff/2):end-1,end,1)=0;
tmat(end,end-ceil(ldiff/2)+1:end,2)=0;
tmat(end-floor(ldiff/2):end-1,end,2)=0;
end
%disp(' '); dispmat(tmat(:,:,1)); disp(' '); dispmat(tmat(:,:,2)); disp(' '); disp(num2str(sum(sum(sum(tmat)))));
else
tmat=terms;
end
clear coeffs;
coeffs(:,:,1)=zeros(size(tmat,1),size(tmat,2));
coeffs(:,:,2)=zeros(size(tmat,1),size(tmat,2));
for s=1:size(tmat,1) % Calculate the coefficients
p=s-1;
for t=1:size(tmat,2)
m=t-1;
if tmat(s,t,1)==1 % coeff requested
z2=LaguerreGaussianE([p,m,q,lambda],domain(:,:,1),domain(:,:,2),'pol');
coeffs(s,t,1)=overlap(z1,conj(z2),domain,domain(:,:,1),accuracy);
else
coeffs(s,t,1)=0;
end
if tmat(s,t,2)==1 % coeff requested
z2=LaguerreGaussianE([p,-m,q,lambda],domain(:,:,1),domain(:,:,2),'pol');
coeffs(s,t,2)=overlap(z1,conj(z2),domain,domain(:,:,1),accuracy);
else
coeffs(s,t,2)=0;
end
end
end
end
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