# Propagating higher order Hermite Gaussian modes. What are complex amplitude coefficients?

I've been tasked with writing a code (in MatLab, but I'm currently using Mathematica because I don't know MatLab %\ ...) to simulate the propagation of a Gaussian beam. I don't really know anything about optics, so I'm learning on the fly using this manual kindly provided to me by my adviser. I'm stuck trying to figure out what the general closed form of a complex amplitude coefficient a[j,n,m] (p.74 and also below) would look like :( Could someone explain it please?

i have looked through various articles and presentations such as http://www.colorado.edu/physics/phys4510/phys4510_fa05/Chapter5.pdf (and others i can't post links too because of my low reputation) and haven't seen mention of said complex amplitude parameter anywhere else. Lots of complex amplitudes, but not parameters. I don't think they are the same thing, are they?

• We don't debug code here, that can be done at either stackoverflow or Computational Science. Also, it's useful to ask a question when posting on a question & answer site. – Kyle Kanos May 15 '15 at 2:46
• I'm not asking to debug code ... i posted it to show that I'm working on something ... and I did ask a question ... I asked what the general closed form of the complex amplitude coefficient a mentioned in the manual is ... did you even read what I wrote?.. – Raksha May 15 '15 at 2:59
• I did read it, and the text parser that is my brain sees a single question mark that is a request to "dumb down" some mathematics. The thing that you call a question seems to be a statement saying that you're stuck on something and not a question in any meaningful manner. – Kyle Kanos May 15 '15 at 3:02
• It is better, there's an actual question now. Note though that your PDF link is broken (leads to the image). – Kyle Kanos May 15 '15 at 3:11
• I think Hermite orthagonality can be used to Fourier transform as described here en.wikipedia.org/wiki/Hermite_polynomials#Orthogonality but I'm not sure. Could someone confirm? It looks like it'll be horrible this way thought because of the argument of H and the extra k/2R term in the exponential... – Raksha May 15 '15 at 19:25

Turns out that due to orthogonality relations of Hermite-Gauss poly's, Hermite-Gauss modes are orthonormal, so $$\int \int u_{n,m} \left(u_{n',m'}\right){}^*dxdy=\delta _{m,m'} \delta _{n,n'}$$