FRINGES Generates synthetic fringe patterns

by Chuck DiMarzio Northeastern University May 2009

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!! CRC Press, Boca Raton, FL, 2011.
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Contents

Equation

$$E = E_{ref} + E_{sig} $$
$$ I = E E^* $$

Coordinates

xx=(-1:0.01:1);
[x1,y1]=meshgrid(xx,xx); % x,y rectangular coordinates
rsq=x1.^2+y1.^2;         % r = radial coordinate

Parameters

lambda=632.8; % wavelength in nm
bumpheight=2*lambda; % Represents a bump on a mirror
smallbumpheight=lambda/5; %
bumpwidth=0.3;  % width of the bump in x1,y1
bumpcenter=[0.15,0.4];
%
% Delta is twice the bumpheight because the light reflects from the mirror
% and thus the path length change is twice the change in the surface
% position.  The bump has a Gaussian shape.
%
relrad=((x1-bumpcenter(1)).^2+(y1-bumpcenter(2)).^2)/(bumpwidth/2).^2;

Optical Path Length for a bump and a small bump

Delta_bump=2*bumpheight*exp(-relrad);
Delta_smallbump=2*smallbumpheight*exp(-relrad);

Reference Wave at z=0, t = 0

Eref=zeros(size(x1));
Eref(rsq<1)=1;   % Plane wave in circular aperture

Tilted plane-wave signal, with tilt coefficient a1

$$ \Delta_{tilt} = a1 x1 $$
$$ \Delta_{tilt} = 8 \lambda $$ at $$ x1==2 $$

a1=8*lambda/2;
Esigflat=exp(1i*2*pi/lambda*a1*x1);
%
%
E = Eref+Esigflat;
I = (abs(E)).^2;
fig1=figure;imagesc(xx,xx,I);axis image;colormap(flipud(gray));axis off;

Tilted plane--wave signal with bumps

Esig=Esigflat.*exp(1i*2*pi/lambda*Delta_bump);
E = Eref+Esig;
I = (abs(E)).^2;
fig2=figure;imagesc(xx,xx,I);axis image;colormap(flipud(gray));axis off;

Esig=Esigflat.*exp(1i*2*pi/lambda*Delta_smallbump);
E = Eref+Esig;
I = (abs(E)).^2;
fig3=figure;imagesc(xx,xx,I);axis image;colormap(flipud(gray));axis off;

Curved, spherical signal wavefront

$$ Delta_f = b1 \rho^2$$
$$ Delta_f = 4 \lambda $$ at $$ x_1=1$$

b1=4*lambda;
Esigsph=exp(1i*2*pi/lambda*b1*rsq);

E = Eref+Esigsph;
I = (abs(E)).^2;
fig4=figure;imagesc(xx,xx,I);axis image;colormap(flipud(gray));axis off;

Curved signal wave with bumps

Esig=Esigsph.*exp(1i*2*pi/lambda*Delta_bump);
E = Eref+Esig;
I = (abs(E)).^2;
fig5=figure;imagesc(xx,xx,I);axis image;colormap(flipud(gray));axis off;

Esig=Esigsph.*exp(1i*2*pi/lambda*Delta_smallbump);
E = Eref+Esig;
I = (abs(E)).^2;
fig6=figure;imagesc(xx,xx,I);axis image;colormap(flipud(gray));axis off;