Laser Travelling wave rate equations solved numerically

In summary, the author used method of lines to discritizze the spatial variable in order to solve for the forward and backward propagating photon density.
  • #1
chitambira
4
0
have a wave equation:

(∂^2 p)/〖∂z〗^2 -1/c^2 (∂^2 p)/〖∂t〗^2

In my case, (fields propagation within a semiconductor laser)
Which can be factored into forward propagating photon density:

(∂p^+)/∂t+C_g (∂p^+)/∂z=C_g Gp^++〖1/2 βR〗_sp

And backward propagation wave (photon density):

〖∂p〗^-/∂t-C_g (∂p^-)/∂t=C_g Gp^-+〖1/2 βR〗_sp

The photons are due to direct modulation of a laser, with the carrier density given by:

∂n/∂t=J/qd-n/τ_nr -〖R_sp-C〗_g G(p^++p^-)

I applied method of lines to discritizze the spatial variable, using backward difference to Forward photon and forward difference to backward photon density to obtain a system of first order ODEs, I then applied first order Finite differences for the time variable to obtain the following equations:

Equation 1
(p^+ (t+1,z)-p^+ (t,z))/Δt+C_g (p^+ (t,z)-p^+ (t-1,z))/Δz=C_g.G〖(n(t,z) ).p〗^++〖1/2 β.R〗_sp (n(t,z))

Equation 2
(p^- (t+1,z)-p^- (t,z))/Δt-C_g (p^- (t,z)-p^- (t,z+1))/Δz=C_g.G〖(n(t,z) ).(p〗^++〖1/2 β.R〗_sp (n(t,z))

Equation 3
(n(t,z)-n(t-1,z))/Δt=J(t,z)/qd-n(t,z)/τ_nr - R_sp (n(t,z) )G〖(n(t,z) )-〖1/2 C〗_g G〖(n(t,z) ) 〖.(p〗〗^+ (t,z)+p〗^- (t,z)+p^+ (t,z-1)+p^- (t,z-1) )

I now want to solve these using MATLAB
Can anybody help with any recipe, particularly the best algorithm or code to solve such a scenario varying in both time and space?

I have attached a word document with better readable equations
 

Attachments

  • I have a wave equation.doc
    115.5 KB · Views: 368
Physics news on Phys.org
  • #2
The equations that you wrote down (the ones you get after discretization) are almost the algorithm. They can be translated directly into code. Have you ever programmed a finite difference method before? If not, do you have an example that you can use to get you started?

You don't talk about boundary conditions; you need to think about this before you can implement the method. Also, I notice that you use backward differences for the time variable in the third equation. Is that on purpose or a typo?
 
  • #3
backward difference on third equation is done on purpose, I am trying to use two coupled meshes. I have never programmed a FD before especially in this case where discritization has been done to both the time and spatial variables.
Boundary conditions are that p(0, z) = n(0,z) = 0 (causality)
p^+(t,0) = kp^-(t,0)
p^+(t,L) = kp^-(t,L)

where L, is the total length to the right boundary
 

1. What are laser travelling wave rate equations?

Laser travelling wave rate equations are mathematical equations that describe the behavior of a laser beam as it travels through a medium. They take into account factors such as the gain and loss of energy, as well as the speed and direction of the beam.

2. How are laser travelling wave rate equations solved?

Laser travelling wave rate equations are typically solved numerically using computational methods. This involves breaking the equations down into smaller, more manageable parts and using algorithms to solve for the desired variables.

3. What is the importance of solving laser travelling wave rate equations numerically?

Numerical solutions of laser travelling wave rate equations allow for a deeper understanding of the behavior of laser beams in different conditions. This can aid in the design and optimization of laser systems, as well as predicting their performance.

4. What are some challenges in solving laser travelling wave rate equations numerically?

One of the main challenges in solving laser travelling wave rate equations numerically is the complexity of the equations themselves. They often involve multiple variables and can be difficult to solve accurately and efficiently. Additionally, the choice of numerical method and parameters can greatly affect the results.

5. How are the results of numerical solutions of laser travelling wave rate equations validated?

The results of numerical solutions of laser travelling wave rate equations can be validated by comparing them to experimental data. If the numerical solutions closely match the observed behavior in the laboratory, it can be considered a successful validation of the model. Additionally, sensitivity analysis can be performed to understand the impact of different parameters on the results.

Similar threads

  • Differential Equations
Replies
1
Views
707
  • Differential Equations
Replies
1
Views
2K
  • Calculus and Beyond Homework Help
Replies
7
Views
510
Replies
5
Views
1K
  • Differential Equations
Replies
3
Views
3K
  • Differential Equations
Replies
1
Views
1K
  • Differential Equations
Replies
5
Views
2K
  • Differential Equations
Replies
1
Views
1K
  • Differential Equations
Replies
3
Views
1K
Replies
7
Views
1K
Back
Top