Have a look at these articles.

a) https://www.lpl.arizona.edu/hamilton/.../Electromagnetic_Waves_in_Free_Space.pdf

b) https://www.utdallas.edu/~dlm/.../FFTandMatLab-wanjun%20huang.pdf

c) https://ocw.mit.edu/.../electrical...and...electromagnetics-and.../MIT6_013S09_chap02.

From what I understand your question, you were talking about two coupled fields. So actually, they're not independent. The problem is that you didn't know how to solve the coupled equations. Do I understand your problem correctly?

I have gone through few methods for solving coupled wave equations , but for my case one of the equation is somewhat similar to NLSWE , so can u please help me with some methodology , or algorithm for solving these equations which are as,

attached in image below

Dear Asim Asrar ,

There're 2 ways happened to me that can be employed:

1/ You transform these 2 equations into frequency domain, then the time derivation will become a multiplication factor (iwt) .

Then you can use suitable ODE solvers of Matlab to solve (ode45 for example).

2/ You may solve these 2 equation which is a system of 1st order partial differential equations by using system of PDE solvers. You may find here : https://www.mathworks.com/help/matlab/math/partial-differential-equations.html .

I hope this helps.

Dear Hoa Nguyen Phuoc Trung ,

Thenks alot for the suggestion, sir i have one more question that-

i am representing electric field E1 as a gaussian pulse

so do i need to take E2 as the same gaussain pulse .

or there needs to be some change in between them.

Code for my fundamental input gaussian pulse is as-

Pi=30*10^13; %input('Enter the value of input power in mW ')

t=101.95*10^-15; %input('Enter the value of input pulse width in seconds ')

tau=-300*10^-15:0.01*10^-15:300*10^-15; %input('Enter time period with upper(U), lower(L) and interval between upper and lower interval(I) in this format L:I:U')

L=800*10^-9;

dt=1/t;

% dt=10^-4.9/t;

c=3*1e8;

z=30*10^-3;

deff=2.064*10^-12;

pi=3.1415926535;

esn=8.854*10^-12

I0=30*10^13;

E0=sqrt(Pi); %for ii=0.1:0.1:(s/10) %1*10^-7:1*10^-7:s %different fiber lengths

E1=E0*exp(-(tau/t).^2); % generation of an gaussian input pulse

subplot(2,1,1);

plot(tau,abs(E1).^2)

Awaiting your response.

Dear Asim,

In the paper that you attached, E1, and E2 are fundamental and harmonic fields. So if you represent E1 as gaussian at frequency w, E2 is also gaussian at frequency 2w.

In your code above, there is no information about the frequency in the field.

It should be something like:

E1= sqrt(Pi) e^(i*w*t-i*beta*z) e^[ -(t/T0)^2]

where beta is the propagation constant.

Dear Hoa Nguyen Phuoc Trung ,

thenks alot for the assistance, if we have to incorporate Group Velocity dispersion (GVD) to our simulations , where should we need to impede this parameter, as from the coupled equations we do not have any factor for Group Velocity dispersion.

Awaiting your valuable response.

Dear Asim,

The group velocity dispersion is included in the presentation of E1 and E2. When you look at E1= sqrt(Pi) e^(i*w*t-i*beta*z) e^[ -(t/T0)^2] , beta is the propagation constant, which is beta = n * w/c ; n is the refractive index at frequency w (in waveguide, n will become effective refractive index, but let's talk about it at another time). n (and so beta) will tell you how group velocity disperses by relating to the 1st and 2nd derivative to frequency of n, and so of beta.

GVD = 1/c (2 dn/dw + w d2n/dw2).

So to sum up, by representing the E1 = function(w,beta,t) you already include the dispersion property of the medium.

PS: you may read some books of waveguide and nonlinear optics for more details (one good book is Nonlinear fiber optics of Gowind Agrawal).

Dear Hoa Nguyen Phuoc Trung ,

in the equation for fundamental electric filed " E1= sqrt(Pi) e^(i*w*t-i*beta*z) e^[ -(t/T0)^2] " , I think the factor beta is wave vector as beta being = n*w/c.

so how come E1=function (GVD).

can u please provide me reference for the equation of electric field " E1= sqrt(Pi) e^(i*w*t-i*beta*z) e^[ -(t/T0)^2] "

as i am dealing with bulk BBO and other non linear crystals . can u suggest me some lierature(viz - papers or books ) for getting more clarity over the GVM and other factors.

as the reference book " Nonlinear fibre optics of Gowind Agrawal " is for optical fibres , so is it feasible to concepts and formula of fibre optics to bulk materials.

Also please suggest me some books to have more clarity over the concepts and physics for bulk crystals.