Beam propagation method: Difference between revisions

Revision as of 15:59, 31 May 2014

Revision as of 08:00, 21 August 2014

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</math>

with the field written as,

:<math>E(x,y,z,t)=\psi(x,y,z)\exp(-j\omega t)</math>.

Now the spatial dependence of this field is written according to any one [[Transverse mode|TE or TM]] polarizations

Line 28:

Now the solution when replaced into the Helmholtz equation follows,

:<math>

\left[\frac{\partial^2 }{\partial x^2} + k_0^2(n^2 - \nu^2) \right]A(x,y) = \pm 2 jk_0 \nu \frac{\partial A_k(x,zy)}{\partial zy}

</math>

With the aim to calculate the field at all points of space for all times, we only need to compute the function

<math>A(x,y)</math> for all space, and then we are able to reconstruct <math>\psi(x,y,z)</math>. Since the solution

is for the time-harmonic Helmholtz equation, we only need to calculate it over one time period. We can

visualize the fields along the propagation direction, or the cross section waveguide modes.