farfield faurhofer diffraction power ratio mismatch

i calculated power ratio of diffraction but there seems to be some problem as power fraction is almost zero and power fraction farfield is almost one.

clc

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clear all

lambda = 332.8e-9; % wavelength of light in vacuum

k = 2*pi/lambda;

a = 3e-7; % radius of diffracting circular aperture

Io = 1.0; % relative intensity

R = 1; % distance of screen from aperture

half_angle = 13.3; % half angle of light cone

center_angle = 0; % center angle of light cone

Y = (-0.55e-0:1e-3:0.9e-0);

Z = Y; % coordinates of the screen

I(1:length(Y),1:length(Z))=0;

for i=1:length(Y)

for j=1:length(Z)

q = (Y(i).^2+Z(j).^2).^0.5;

beta = k*a*q/R;

I(i,j) = Io.*((besselj(1,(beta))/(beta+1e-12)).^2);

end

end

% Calculate fraction of power in cone

theta = atan2(Y,Z);

cone_indices = (theta > center_angle-half_angle) & (theta < center_angle+half_angle);

power_fraction = sum(sum(I(cone_indices)))/sum(sum(I));

% Calculate far field using Fraunhofer approximation

N = 1000; % number of points in far field

theta = linspace(-pi/2,pi/2,N); % angular coordinates in far field

E_theta = zeros(1,N);

for i=1:length(theta)

q = k*a*sin(theta(i)); % distance from center of aperture in far field

E_theta(i) = (Io/(2*pi*R*q))*((2*besselj(1,beta)/(beta+1e-12))*exp(-1i*k*q))/(q);

end

% Calculate power in cone in far field

theta = abs(theta)*180/pi;

cone_indices = (theta > center_angle-half_angle) & (theta < center_angle+half_angle);

power_fraction_farfield = sum(abs(E_theta(cone_indices)).^2)/sum(abs(E_theta).^2);

disp(['Power fraction within cone: ' num2str(power_fraction_farfield)]);

Answer 1

[Shaik]

Hi Sean,

The issue in your code lies in the calculation of the variable beta inside the second loop. Currently, you calculate beta outside of the loop, and its value remains constant throughout the loop iterations. To fix this, you need to calculate beta inside the loop so that it corresponds to the current value of q. Here's the modified code:

clc
close all
clear all

lambda = 332.8e-9; % wavelength of light in vacuum
k = 2*pi/lambda;

a = 3e-7; % radius of diffracting circular aperture

Io = 1.0; % relative intensity

R = 1; % distance of screen from aperture

half_angle = 13.3; % half angle of light cone
center_angle = 0; % center angle of light cone

Y = (-0.55e-0:1e-3:0.9e-0);
Z = Y; % coordinates of the screen

I(1:length(Y),1:length(Z))=0;

for i=1:length(Y)
    for j=1:length(Z)
        q = (Y(i).^2+Z(j).^2).^0.5;
        beta = k*a*q/R; % Calculate beta inside the loop
        I(i,j) = Io.*((besselj(1,(beta))/(beta+1e-12)).^2);
    end
end

% Calculate fraction of power in cone
theta = atan2(Y,Z);
cone_indices = (theta > center_angle-half_angle) & (theta < center_angle+half_angle);
power_fraction = sum(sum(I(cone_indices)))/sum(sum(I));

% Calculate far field using Fraunhofer approximation
N = 1000; % number of points in far field
theta = linspace(-pi/2,pi/2,N); % angular coordinates in far field
E_theta = zeros(1,N);

for i=1:length(theta)
    q = k*a*sin(theta(i)); % distance from center of aperture in far field
    beta = k*a*q/R; % Calculate beta inside the loop
    E_theta(i) = (Io/(2*pi*R*q))*((2*besselj(1,beta)/(beta+1e-12))*exp(-1i*k*q))/(q);
end

% Calculate power in cone in far field
theta = abs(theta)*180/pi;
cone_indices = (theta > center_angle-half_angle) & (theta < center_angle+half_angle);
power_fraction_farfield = sum(abs(E_theta(cone_indices)).^2)/sum(abs(E_theta).^2);
disp(['Power fraction within cone: ' num2str(power_fraction)]);

Power fraction within cone: 5.9128e-05

By calculating beta inside the loop, you ensure that it corresponds to the current value of q for each iteration. This should provide the correct power fraction within the cone and far field.

[Sean]

Thank you, i tried with this code but power fraction is still very low for near field and 1 for farfield