Content deleted Content added
redundan categories |
→Principles: Confusion between a 2D and a 3D model leading to errors in the equations. |
||
Line 11:
</math>
with the field written as,
:<math>E(x,y,z,t)=\psi(x,y)\exp(-j\omega t)</math>.<!-- Where is the variable z in the right hand side of equation? It should be in the complex amplitude: Phi(x,y,z). -->
Now the spatial dependence of this field is written according to any one [[Transverse mode|TE or TM]] polarizations
:<math>\psi(x,y) = A(x,y)\exp(+jk_o\nu y)
</math>,<!-- Confusion between a transverse direction called y and the propagation direction called y too. The propagation direction should be called z and replace y in the exponential term. -->
with the envelope
:<math>A(x,y)
Line 21 ⟶ 22:
:<math>
\frac{\partial^2( A(x,y) )}{\partial y^2} = 0
</math><!-- The amplitude should vary slowly along the propagation, hence the derivative should be done with respect to z. -->
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,y)}{\partial y}
</math><!-- Same problem. The left operator should contain a double derivative with respect to y and the derivative in the right hand side of the equation should be realized with respect to z. -->
With the aim to calculate the field at all points of space for all times, we only need to compute the function
| |||