By using Time dependent group velocity dispersion parameter plot the graph between intensity and T/To

I wants to write a matlab code in which firstly by using fourier transformation of group velocity dispersion parameter beta2= (omega/c)* (double differentiation of wrt omega) (1-(mew*(omega- omega0)/(omega-omega0)^2 + L^2)^(1/2)then by using gaussian pulse intensity equation of dispersion U=To/(To^2/-i*beta2*z)^(1/2)*exp(-T^2/2*(To^2/-i*beta2*z)) in which T0 has a fix value but T is in range of -10To to 10To then plot the graph between intensity and T/T0

Answer 1

[Hitesh]

Hi Aayushi,

Kindly follow the below steps to achieve time dependent group velocity dispersion parameter and plot the graph between intensity and T/To. Set physical constants and parameters such as the speed of light 'c', central angular frequency 'omega0', and pulse width 'To'.

Define the Range for Time 'T':

• Create a range for 'T' from (-10 \times T_0) to (10 \times T_0).
• Compute 'T_div_To' as T divided by To for plotting.

Calculate the Group Velocity Dispersion Parameter 'beta2':

• Define a frequency range 'omega' around 'omega0'.
• Calculate the term involving 'omega' and its second derivative with respect to 'omega' using numerical differentiation.
• Compute 'beta2' using the expression derived from the second derivative.

Interpolate 'beta2' at 'omega0':

• Use interpolation to find the value of 'beta2' at 'omega0' to ensure it is compatible with the size of the time array T.

Calculate the Intensity of the Gaussian Pulse:

• Use the Gaussian pulse equation to calculate the field 'U' as a function of 'T'.
• Compute the intensity as the squared magnitude of 'U'.

Kindly refer to the below code for better understanding of the steps metioned above.

% Define omega for beta2 calculation
omega = linspace(omega0 - 5e13, omega0 + 5e13, 1000); % Frequency range

% Calculate the second derivative manually
term = 1 - mu * (omega - omega0) ./ sqrt((omega - omega0).^2 + L^2);
d_term_domega = gradient(term, omega); % First derivative
d2_term_domega2 = gradient(d_term_domega, omega); % Second derivative

% Calculate beta2
beta2 = (omega/c) .* d2_term_domega2;

% Interpolate beta2 at omega0
beta2_interp = interp1(omega, beta2, omega0, 'linear', 'extrap');

% Calculate intensity using the Gaussian pulse equation
z = 1; % Propagation distance (example value)
U = To ./ sqrt(To^2 - 1i * beta2_interp * z) .* exp(-T.^2 ./ (2 * (To^2 - 1i * beta2_interp * z)));

% Calculate intensity
intensity = abs(U).^2;

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For more information regarding the "gradient" function, kindly refer to the following MATLAB documentation:

https://www.mathworks.com/help/matlab/ref/gradient.html