Propagating a Gaussian-Profile EM Wave in a Medium

[andrew300591]

Homework Statement

Hi everyone,

As part of my final year project I am modelling the propagation of an electromagnetic wave through a medium of refractive index, n. I begin at the boundary between vacuum and the medium, x = 0 and propagate forward to some distance x.

I have initially tried modelling a vacuum throughout i.e. n = 1. In this case I expected that the initial Gaussian wave should propagate forward unchanged, however I have not been able to achieve this result - could you please help me find out why?

I am attempting this issue in the Python programming language.

Homework Equations

(1) E(x=0,t) = e^(-0.5 * (t / τ)**2)
(2) E(ω,t) = F[E(x=0,t)]
(3) E(ω,t) * e^(ikz) = E(ω,t) * e^(iωnz/c)
(4) E(x,t) = F^-1[E(ω,t) * e^(iωnz/c)]
* F[] represents a forward Fourier transform, F^-1[] represents an inverse Fourier transform
** τ represents the width of the initial Gaussian profile, I have set τ = 1

The Attempt at a Solution

So far to do this I have tried:

(i) I begin with a wave profile, E(x=0,t) that is Gaussian in time (equation 1 below)
(ii) Perform a forward Fourier transform (equation 2) to calculate the angular spectrum of the initial Gaussian profile
(iii) Multiply the angular spectrum by an exponential factor (equation 3) to propagate forward in space
(iv) Inverse Fourier transform the product of the exponential and the forward Fourier transform to obtain the wave profile at distance x, E(x,t)

 

[Nate810]

(equation 4)However my results do not match the expected result of the wave profile being unchanged. I would be grateful if someone could help me identify where I'm going wrong.