Solving PDE involving Gaussian fields.

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Discussion Overview

The discussion revolves around solving a set of coupled partial differential equations (PDEs) that describe the behavior of field amplitudes in the context of laser beams, particularly when considering Gaussian distributions in the transverse direction. Participants explore numerical methods and the implications of using Gaussian profiles versus plane wave assumptions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant presents a set of coupled differential equations governing the amplitudes E1, E2, and E3, initially assuming constant amplitudes for plane waves.
  • Another participant clarifies that the equations involve nonlinear interactions, specifically sum frequency generation in a nonlinear crystal.
  • There is a suggestion to consider Gaussian beams, which complicates the numerical solution due to the need for propagation analysis.
  • One participant proposes an intermediate approach of assuming constant q parameters for Gaussian beams to simplify the problem.
  • A later reply acknowledges the common practice of solving such problems with plane waves and agrees to initially neglect diffraction effects while considering a Gaussian profile.
  • Another participant expresses that while the proposed approach may not be "physically blameless," it is a reasonable solution given the circumstances.

Areas of Agreement / Disagreement

Participants express a mix of agreement and uncertainty regarding the best approach to solve the equations with Gaussian distributions. While some suggest simplifying assumptions, others acknowledge the complexity involved in accurately modeling the problem.

Contextual Notes

Participants note the limitations of assuming constant q parameters and the potential need to incorporate diffraction effects later in the analysis. The discussion reflects varying degrees of confidence in the proposed methods and assumptions.

vivek.iitd
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I need to solve three coupled differential equations. The equations are as follows:

dE1/dz = f(E2,E3)
dE2/dz = f(E1,E3)
dE3/dz = f(E1,E2)

Where E1,E2,E3 represents field amplitudes. In case of plane waves these amplitudes will be constant in the transverse direction therefore i can directly solve these equation given their initial values. Although, i would like to, how should i solve these equations if i have a Gaussian distribution in the transverse direction? Do i need to solve it by creating a mesh and solving these equations for each pairs of points?

I thank you for your time.
 
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sorry, equations are as follows

dE1/dz = f(E2,E3)
dE2/dz = g(E1,E3)
dE3/dz = h(E1,E2).
 
I am solving it numerically.
 
Do they represent laser beams?
 
yes they represent laser beams.
 
Actually it is a nonlinear phenomenon (sum frequency generation). To give a physical aspect to this problem it is like this, two beams (with amplitudes E1 and E2) are incident on a nonlinear crystal and they mix up with each other to give third beam (E3) this whole phenomenon is represented by these coupled equations. The direction of propagation is 'z'. In case of plane waves the amplitude will be constant in the transverse direction (plane perpendicular to 'z'), therefore i can directly solve these equations using numerical method. Although, I want to get more realistic, so i am taking Gaussian distribution. Therefore in the transverse direction, amplitude distribution is Gaussian. I am confused as how to solve these equations in that case.
 
Many times, the sum-difference problem is analyzed assuming plane waves.
If you want to consider Gaussian beams you must propagate them but the algorithm is more complex.
As an intermediate approach, I'd suggest assuming the gaussian beams have constant q paremeters. This isn't strictly true but simplifies the job. This way, at every radial coordinate r you have a gaussian amplitude and you apply the original differential equations at a given radial coordinate.
 
Thank you for your reply. You are right that usually these problems are solved assuming plane waves because they are simplest to solve.
As you suggested I will assume q parameter to be constant as of now i.e. neglecting diffraction and other effects. After that i may incorporate that also using split-step method.
Also to solve for Gaussian profile, i guess i will have to discretize along the radial direction. And after that i can apply these ODEs by taking one step at a time along radial direction.

I think this is the correct way to solve these problem for Gaussian profile right?
 
Even though it isn´t a "physically blameless" approach, I'd say it's a fair solution
 

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