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Diffstat (limited to 'transverse/HermiteGaussianE.m')
| -rw-r--r-- | transverse/HermiteGaussianE.m | 134 |
1 files changed, 134 insertions, 0 deletions
diff --git a/transverse/HermiteGaussianE.m b/transverse/HermiteGaussianE.m new file mode 100644 index 0000000..ced8f6f --- /dev/null +++ b/transverse/HermiteGaussianE.m @@ -0,0 +1,134 @@ +function z=HermiteGaussianE(params,x,varargin)
+% Returns the complex field amplitude of a single polarization Hermite-Gaussian mode.
+%----------------------------------------------------------------------------------------------
+% PROGRAM: HermiteGaussianE
+% AUTHOR: Andri M. Gretarsson
+% DATE: 6/26/03
+%
+% SYNTAX: z=HermiteGaussianE([l,m,q <,lambda,a>],x <,y>);
+% <...> indicates optional arguments
+%
+% Returns the complex field amplitude of a single polarization Hermite-Gaussian mode.
+% The domain of the function is specified in cartesian coordinates x,y.
+% If the input argument a is not specified or is equal to 1, the area under the
+% surface z(x,y) is equal to 1. If z=z(x), i.e. the one dimensional form is being used,
+% then the area under the surface formed by rotating z(x) around the z axis equals 1.
+% In short the functions are normalized and form an orthonormal basis wrt. the inner
+% product formed by multiplying two HermiteE's together and integrating over area.
+% The formula is taken from "Principles of Lasers" by Orazio Svelto, 4th ed. Section 5.5.1.3.
+%
+% Note that it is possible to specify the parameters l,m,q,a,lambda as column vectors. In that case,
+% the matrix returned z, will be a stack of 2D matrixes (i.e. a 3D matrix), where successive 2D matrixes
+% in the stack (i.e. successive layers of the 3D matrix) correpsond to successive rows of the
+% l,m,q,a,lambda columns
+%
+% INPUT ARGUMENTS:
+% ----------------
+% l,m = Hermite Gaussian mode numbers (integers, positive or zero). Can be a scalar or a column vector.
+% q = complex radius of curvature of the beam (1/q = 1/R + i*lambda/pi/w^2).
+% Can be a scalar or a column vector.
+% a = complex amplitude (includes phase, e.g. for a beam that
+% has been propageted with an ABCD matrix a = 1/(A + [B/q1])^(1+l+m) ).
+% See for example, O. Svelto, Principles of Lasers 4th ed. eqn. 4.7.30.
+% Can be a scalar or a column vector.
+% lambda = Wavelength of the light. Can be a scalar or a column vector. Default is 1064e-9;
+% x = x position vector or a 2D matrix generated by meshgrid.
+% y = y position vector or a 2D matrix generated by meshgrid.
+% If y is not specified then the default y is used:
+% if x is 1D, y=zeros(size(x)). If x is 2D, y=x.
+%
+% OUTPUT ARGUMENTS:
+% -----------------
+% z(i,j) = Complex field of the Hermite Gaussian mode with parameters l,m,... at
+% ( x(i,j),y(i,j) ). May be a vector or a matrix depending on whether
+% x and y are vectors or matrixes. If in addition, l,m,q,... are columns,
+% then z(i,j,k) will correspond to the complex field amplitude of the
+% Hermite Gaussian mode corresponding to the parameters l(k),m(k),...
+%
+% NOTES:
+% ------
+% If x and y are not vectors but matrixes generated by
+% the matlab function meshgrid, then the output variable z
+% is a matrix rather than a vector. The matrix form allows
+% the function HermiteGaussianE to have a plane as it's domain
+% rather than a curve.
+%
+% If the parameters l,m,q,p,lambda are equal length column vectors rather
+% than scalars, z is a size(x)*length(lambda) matrix. E.g. if size(x) is n*n
+% then each level z(:,:,k) is a 2D field of a Hermite Gaussian with
+% the parameters given by [l(k),m(k),q(k),lambda(k),a(k)].
+%
+% EXAMPLE 1 (2D : two TEM modes):
+% w=[0.001; 0.001];
+% xseed=[-3*max(w):max(w)/30:3*max(w)]; yseed=xseed';
+% [x,y]=meshgrid(xseed,yseed);
+% lambda = [1.064e-6 ; 1.064e-6];
+% R = [-30 ; -30];
+% q = (1./R - i* lambda./pi./w.^2).^(-1); a=[1;1];
+% l=[1;2]; m=[0;2];
+% E=HermiteGaussianE([l,m,q,lambda,a],x,y);
+% subplot(2,1,1); h1=pcolor(x,y,abs(E(:,:,1))); set(h1,'EdgeColor','none'); axis square;
+% subplot(2,1,2); h2=pcolor(x,y,abs(E(:,:,2))); set(h2,'EdgeColor','none'); axis square; shg;
+%
+% EXAMPLE 2 (1D : four TEM modes):
+% w=[1,2,3,4].'; x=[-10:0.001:10]; lambda=[1,1,1,1].'*656e-9; R=[1,1,1,1].'*1000;
+% q = (1./R - i* lambda./pi./w.^2).^(-1); a=[1,1,1,1].'; l = [0,1,2,3].'; m=[0,0,0,0].';
+% E=HermiteGaussianE([l,m,q,lambda,a],x); I=E.*conj(E); phi=angle(E);
+% plot(x,I(:,1),x,I(:,2),x,I(:,3),x,I(:,4));
+%
+% Last updated: July 18, 2004 by AMG
+%----------------------------------------------------------------------------------------------
+%% SYNTAX: z=HermiteGaussianE([l,m,q <,lambda,a>],x <,y>);
+%----------------------------------------------------------------------------------------------
+
+defaultcoord2=0;
+if nargin>=3, y=varargin{1}; else defaultcoord2=1; end
+
+if min(size(x))==1
+ if size(x,1)<size(x,2), x=x'; end %make x and y columnar
+ if defaultcoord2
+ y=zeros(size(x));
+ else
+ if size(y,1)<size(y,2), y=y'; end
+ end
+
+end
+
+if min(size(x)) > 1
+ z=zeros(size(x,1),size(x,2),size(params,1)); % need this since zeros(size(y),10) gives a 2D matrix even if y is 2D! (Matlab feature.)
+ if defaultcoord2, y=transpose(x); end
+else
+ z=zeros(length(x),size(params,1));
+end
+
+l=params(:,1);
+m=params(:,2);
+q=params(:,3);
+if size(params,2)>=4
+ lambda=params(:,4);
+else
+ lambda=1064e-9;
+end
+if size(params,2)>=5
+ a=params(:,5);
+else
+ a=ones(size(q));
+end
+
+w=w_(q,lambda);
+
+if min(size(x))>=2 % x is a matrix (2D array)
+ for u=1:size(params,1)
+ z(:,:,u) = a(u)...
+ .* sqrt((2/pi))/2^((l(u)+m(u))/2)/sqrt(factorial(l(u))*factorial(m(u)))/w(u)...
+ .* hermitepoly(l(u),sqrt(2).*x/w(u)).*hermitepoly(m(u),sqrt(2).*y/w(u))...
+ .* exp( -i*2*pi/lambda(u)*(x.^2+y.^2)/2/q(u) ) ;
+ end
+else
+ for u=1:size(params,1) % x is a vector (1D array)
+ z(:,u) = a(u)...
+ .* sqrt((2/pi))/2^((l(u)+m(u))/2)/sqrt(factorial(l(u))*factorial(m(u)))/w(u)...
+ .* hermitepoly(l(u),sqrt(2).*x/w(u)).*hermitepoly(m(u),sqrt(2).*y/w(u))...
+ .* exp( -i*2*pi/lambda(u)*(x.^2+y.^2)/2/q(u) );
+ end
+end
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