Plot the first-order autocorrelation functions for the following light sources: perfect monochromatic source, a thermal lamp((Lc=1 μm, a led =100 μm), lasersLc=1 m and 1 km

clc;

close all;

clear all;

% Parameters

tau = linspace(-10, 10, 1000); % Time delay (in arbitrary units)

omega_0 = 2*pi; % Angular frequency for monochromatic light

delta_omega_LED = 2*pi*1e6; % Linewidth for LED (assumed)

tau_c_thermal = 0.5; % Coherence time for thermal lamp (assumed)

% Autocorrelation functions

g_monochromatic = exp(1i*omega_0*tau);

g_thermal = exp(-abs(tau)/tau_c_thermal);

g_LED = exp(-abs(delta_omega_LED*tau/2)) .* exp(1i*omega_0*tau);

% Plotting

figure;

subplot(2,2,1);

plot(tau, real(g_monochromatic), 'b', 'LineWidth', 1.5);

title('Perfect Monochromatic Source');

xlabel('\tau');

ylabel('Re[g^{(1)}(\tau)]');

subplot(2,2,2);

plot(tau, real(g_thermal), 'r', 'LineWidth', 1.5);

title('Thermal Lamp');

xlabel('\tau');

ylabel('Re[g^{(1)}(\tau)]');

subplot(2,2,3);

plot(tau, real(g_LED), 'g', 'LineWidth', 1.5);

title('Light Emitting Diode (LED)');

xlabel('\tau');

ylabel('Re[g^{(1)}(\tau)]');

subplot(2,2,4);

plot(tau, abs(g_LED), 'm', 'LineWidth', 1.5);

title('Laser Source (Coherence Length 1 km)');

xlabel('\tau');

ylabel('|g^{(1)}(\tau)|');

sgtitle('First-Order Autocorrelation Functions for Different Light Sources');

I was using this code but not getting the correct graph.Using these formulae

Answer 1

[William Rose]

@OMEIR HUSSAIN,

[Edit: fix spelling errors.]

You can format your code as code, by highlighting your code, then click the "Code" icon above. Then click the green Run arrow to run it.

clear;
% Parameters
tau = linspace(-10, 10, 1000); % Time delay (in arbitrary units)
omega_0 = 2*pi; % Angular frequency for monochromatic light
delta_omega_LED = 2*pi*1e6; % Linewidth for LED (assumed)
tau_c_thermal = 0.5; % Coherence time for thermal lamp (assumed)
% Autocorrelation functions
g_monochromatic = exp(1i*omega_0*tau);
g_thermal = exp(-abs(tau)/tau_c_thermal);
g_LED = exp(-abs(delta_omega_LED*tau/2)) .* exp(1i*omega_0*tau);
% Plotting
figure;
subplot(2,2,1);
plot(tau, real(g_monochromatic), 'b', 'LineWidth', 1.5);
title('Perfect Monochromatic Source');
xlabel('\tau');
ylabel('Re[g^{(1)}(\tau)]');
subplot(2,2,2);
plot(tau, real(g_thermal), 'r', 'LineWidth', 1.5);
title('Thermal Lamp');
xlabel('\tau');
ylabel('Re[g^{(1)}(\tau)]');
subplot(2,2,3);
plot(tau, real(g_LED), 'g', 'LineWidth', 1.5);
title('Light Emitting Diode (LED)');
xlabel('\tau');
ylabel('Re[g^{(1)}(\tau)]');
subplot(2,2,4);
plot(tau, abs(g_LED), 'm', 'LineWidth', 1.5);
title('Laser Source (Coherence Length 1 km)');
xlabel('\tau');
ylabel('|g^{(1)}(\tau)|');
sgtitle('First-Order Autocorrelation Functions for Different Light Sources');

The two formulas in the jpeg which you attached say that

• the autocorrelation for monchromatic light is a cosine wave, and
• the autocorrelation for a source with non-zero linewidth is a cosine wave with a Gaussian envelope.

The code does not implement the formula for non-zero linewith correctly, because it does not square the term . But that is not the reason for the zero autocorrealation in the lower panels. The reason the autocorrelation is zero in the lower panels is that delta_omega_LED is very large compared to the light frequency. In your code , the light has frequency 1 cycle per unit time. The linewidth is 2*pi*10^6 radians per unit time. Therefor, by the time tau=0.02 (the tau value closest to tau=0), , and exp(-10^4)=0, so g_LED=0, for the range of tau which you plot (except tau=0, which should=1).

Let's re-run your code, correcting the squaring issue, and changing delta_omega_LED to 0.5 instead of 2*pi*1e6.

% Parameters
tau = linspace(-10, 10, 1000); % Time delay (in arbitrary units)
omega_0 = 2*pi; % Angular frequency for monochromatic light
delta_omega_LED = 0.5; % Linewidth for LED (assumed)
tau_c_thermal = 0.5; % Coherence time for thermal lamp (assumed)
% Autocorrelation functions
g_monochromatic = exp(1i*omega_0*tau);
g_thermal = exp(-abs(tau)/tau_c_thermal);
g_LED = exp(-(delta_omega_LED*tau/2).^2) .* exp(1i*omega_0*tau);
% Plotting
figure;
subplot(2,2,1);
plot(tau, real(g_monochromatic), 'b', 'LineWidth', 1.5);
title('Perfect Monochromatic Source');
xlabel('\tau');
ylabel('Re[g^{(1)}(\tau)]');
subplot(2,2,2);
plot(tau, real(g_thermal), 'r', 'LineWidth', 1.5);
title('Thermal Lamp');
xlabel('\tau');
ylabel('Re[g^{(1)}(\tau)]');
subplot(2,2,3);
plot(tau, real(g_LED), 'g', 'LineWidth', 1.5);
title('Light Emitting Diode (LED)');
xlabel('\tau');
ylabel('Re[g^{(1)}(\tau)]');
subplot(2,2,4);
plot(tau, abs(g_LED), 'm', 'LineWidth', 1.5);
title('Laser Source (Coherence Length 1 km)');
xlabel('\tau');
ylabel('|g^{(1)}(\tau)|');
sgtitle('First-Order Autocorrelation Functions for Different Light Sources');

Therefore I think you should re-evaluate the value for delta_omega_LED. In your code comments, you say the optical frequency is in arbitrary units. But the plot title says Coherence Length=1 km, which is not an arbitrary unit. I don't know if this is relevant.

Thank you for introducing me to "sgtitle()". I didn't know about that command.

[William Rose]

@OMEIR HUSSAIN,

Coherence length and linewidth are related as follows (source):

therefore , and .

Let's assume the laser wavelength = 1 . Then , and ., and the period of the light is approximately 3e-15 s. Let the range for tau be +-3x10^-14 s = +-10 periods of the light wave. This matches the range for tau in your script: +-10 periods.

tau_c_thermal in your script was 0.5 periods, so we will set tau_c_thermal=1.5e-15 s.

% Parameters
tau = linspace(-3e-14,3e-14,1000);  % Time delay [s]
omega_0 = 2e15;                     % laser frequency [rad/s]
delta_omega_LED = 6e5;              % linewidth for LED with Lcoh=1 km [rad/s]
tau_c_thermal = 1.5e-15;            % Coherence time for thermal lamp [s]
% Autocorrelation functions
g_monochromatic = exp(1i*omega_0*tau);
g_thermal = exp(-abs(tau)/tau_c_thermal);
g_LED = exp(-(delta_omega_LED*tau/2).^2) .* exp(1i*omega_0*tau);
% Plotting
figure;
subplot(2,2,1); plot(tau, real(g_monochromatic), 'b');
title('Monochromatic Laser');
xlabel('\tau'); ylabel('Re[g^{(1)}(\tau)]');
subplot(2,2,2); plot(tau, real(g_thermal), 'r');
title('Thermal Lamp');
xlabel('\tau'); ylabel('Re[g^{(1)}(\tau)]');
subplot(2,2,3); plot(tau, real(g_LED), 'g');
title('Laser, L_{coh}=1 km');
xlabel('\tau'); ylabel('Re[g^{(1)}(\tau)]');
subplot(2,2,4); plot(tau, abs(g_LED), 'm');
title('Laser, L_{coh}=1 km'); ylim([.99,1.01])
xlabel('\tau'); ylabel('|g^{(1)}(\tau)|');
sgtitle('First-Order Autocorrelation Functions for Different Light Sources');

We see above that the coherence for the laser with Lcoh=1 km is a perfect cosine wave, or unity, over the tau range shown here. We need to use a much larger range for tau, to see the roll-off in coherence. When using a much larger range for tau, we are likely to have aliasing, when sampling the real part of g^(1)(tau). See below, where the tau range for the lower two plots is larger than for the upper two plots.

% Parameters
tau1 = linspace(-3e-14,3e-14,1000); % Time delay [s]
tau2 = linspace(-1e-5,1e-5,1000);   % Time delay [s]
omega_0 = 2e15;                     % laser frequency [rad/s]
delta_omega_LED = 6e5;              % linewidth for LED with Lcoh=1 km [rad/s]
tau_c_thermal = 1.5e-15;            % Coherence time for thermal lamp [s]
% Autocorrelation functions
g_monochromatic = exp(1i*omega_0*tau1);
g_thermal = exp(-abs(tau1)/tau_c_thermal);
g_LED = exp(-(delta_omega_LED*tau2/2).^2) .* exp(1i*omega_0*tau2);
% Plotting
figure;
subplot(2,2,1); plot(tau1, real(g_monochromatic), 'b');
title('Monochromatic Laser');
xlabel('\tau'); ylabel('Re[g^{(1)}(\tau)]');
subplot(2,2,2); plot(tau1, real(g_thermal), 'r');
title('Thermal Lamp');
xlabel('\tau'); ylabel('Re[g^{(1)}(\tau)]');
subplot(2,2,3); plot(tau2, real(g_LED), 'g');
title('Laser, L_{coh}=1 km');
xlabel('\tau'); ylabel('Re[g^{(1)}(\tau)]');
subplot(2,2,4); plot(tau2, abs(g_LED), 'm');
title('Laser, L_{coh}=1 km');
xlabel('\tau'); ylabel('|g^{(1)}(\tau)|');
sgtitle('First-Order Autocorrelation Functions for Different Light Sources');

Note the different ranges for tau, in the upper and lower plots above.

[OMEIR HUSSAIN]

Thank you but they should be 5 graphs .Above you for the laser you have titled both the graphs with the same L value that is 1km.I think the graph for led is missing thank you.

[William Rose]

@OMEIR HUSSAIN, your script makes four plots. I shortened the titles since they were too long to fit. YOu plotted the real part at (2,2,3) and the magnitude at (2,2,4). I did not change that.