MATLAB: HOW TO FIND THE FULL WIDTH AT HALF MAXIMUM

fourier opticsfwhmImage Processing Toolboxpoint spread function

Hi,

I have a problem in the following code:

clear all;
clc;
close all;
lamda=531e-9;
k=2*pi/lamda;
[x,y]=meshgrid(-1*lamda:(2*lamda/127):1*lamda);
r=sqrt(x.^2+y.^2);
w=10^-3;
z=10^-3;
z1=sqrt(w.^2+z.^2);
N.A=w/z1;
v=((k*r));
I=(2*bessel(1,v)./(v)).^2;
I0=max(max(I));
I2=I/I0;
I3=0;
figure(1)
plot(I2(64,:))
%figure(2)
%imagesc(I2),colormap gray
%For full width of airy pattern 
% a=19;
% v1=k*w/z;
% d1=a/v1 % d1=1.605*10^-6m;
 % For full width at half maximum
 b=46; 
 v2=k*w/z;
   d2=b/v2 %d2=6.760*10^-7m;

%%%%%%%%%%%

when you run the above code, you will find figure(1) and I am in need to find the half width full maximum(FWHM) of the said fig. using matlab. It will be a great help for me if anyone help me out in this regard…

Thanking You!

Best Answer

MOHD: First change bessel to besselj like Eric advised. Then add this onto your code.

% Find the max
plot(I2(64,:), 'bo-')
grid on;
oneProfile = I2(64,:);
maxValue = max(oneProfile)
% Find where it's more than half the max.
aboveHalfMax = oneProfile > maxValue/2
% Get the first and last index where it's more than the half max.
firstIndex = find(aboveHalfMax, 1, 'first')
lastIndex = find(aboveHalfMax, 1, 'last')
% Draw lines there
hold on;
line([firstIndex firstIndex], [0 maxValue/2], 'Color', 'r', 'LineWidth', 2);
line([lastIndex lastIndex], [0 maxValue/2], 'Color', 'r', 'LineWidth', 2);
line([firstIndex lastIndex], [maxValue/2 maxValue/2], 'Color', 'r', 'LineWidth', 2);

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The following code lets you simulate the PSF associated with spherical aberration (as well as diffraction). You need to specify the PSF sampling pitch, the wavelength, the aperture diameter, the system focal length, the amount of spherical aberration, and the PSF array size.

The idea is to calculate a wavefront with spherical aberration and then use the Fourier transform to calculate the PSF. You need to be careful with sampling, hence the assert() statement.

You can easily include other aberrations as well, if desired. Simply modify how the wavefront is calculated.

You can check that this code is working by setting RMS_SA to zero (i.e., simulate a diffraction-limited system, in which case the PSF is an Airy disk). Then the first null of the PSF should have a radius of 1.22*lambda*focal_length/aperture_diameter.

You'll have to do some work to simulate a spectrally broad system. In that case you could simulate a number of monochromatic PSFs and then sum them together. You would need to know how the spherical aberration varies as a function of wavelength (variation as the reciprocal of wavelength would be a good estimate - i.e., longer wavelengths have less aberration) as well as the spectral output of the source, the spectral transmittance of the optical system, and the spectral responsivity of the detector. The product of these spectral terms would be used as the weights in a weighted average to calculate the broadband PSF from the monochromatic PSFs.

Once you get the PSF you can convolve this with your scene to simulate imaging performance. imfilter() may work, but I like convnfft ( http://www.mathworks.com/matlabcentral/fileexchange/24504-fft-based-convolution) from the File Exchange. You could use conv2() as well.

Good luck,

Eric

%%Define parameters
clear all;close all;
psf_sampling = 0.5e-6;%focal plane sampling in meters
lambda = 0.6328e-6;%wavelength in meters
N = 256;
aperture_diameter = 0.0254;%meters; 1 inch
focal_length = 5*aperture_diameter;%meters
RMS_SA = 0.25;%RMS spherical aberration content in waves
%%Calculate pupil plane sampling
delta_fx = 1/(psf_sampling*N);
x_pupil = (-fix(N/2):fix((N-1)/2)) * delta_fx * lambda * focal_length;
[X_pupil,Y_pupil] = meshgrid(x_pupil);
R_pupil = sqrt(X_pupil.^2 + Y_pupil.^2);
R_norm = R_pupil/(aperture_diameter/2);%Pupil normalized to aperture diameter
assert(max(R_norm(:))>=sqrt(2),'Sampling is not sufficient to reconstruct the entire wavefront.');
%%Create wavefront
W = RMS_SA * sqrt(5) * (6*R_norm.^4 - 6*R_norm.^2 + 1);%Spherical Aberration wavefront
W(R_norm>1) = 0;
E = exp(1i*2*pi*W);%Complex amplitude
E(R_norm>1) = 0;%Impose aperture size
figure;imagesc(angle(E)/(2*pi));colorbar;title('Wavefront Phase (waves)');
%%Create point-spread function
psf = abs(fftshift(fft2(ifftshift(E)))).^2;
psf = psf/sum(psf(:));%Normalize to unity energy
x_psf = (-fix(N/2):fix((N-1)/2)) * psf_sampling;
figure;imagesc(x_psf*1e6,x_psf*1e6,psf);
title(sprintf('PSF with %.4f waves RMS of Spherical Aberration', RMS_SA));
xlabel(sprintf('Position (\\mum)'));
ylabel(sprintf('Position (\\mum)'));

MATLAB: Eigenvalue matrix error subscript

in the last line your are trying to index K with

K((lamda^2)+(n^2))

this turns out to be a negative number.

Best Answer

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