Modes in a cylindrical dielectric waveguide

[krakatoa]

Homework Statement

Find the modes of a cylindrical dielectric waveguide of permitivity $$\epsilon$$ and radious $$a$$ with $$\epsilon > \epsilon_0$$ surrounded by vaccum

Relevant Equations

Planar waves in the maxwell ecuations result into:

$$( \nabla_t^2 + (\mu\epsilon\omega^2 - k^2))E_z = 0$$
for waveguides
where:
$$\nabla_t F = F_x + F_y$$

Note: imagine z axis along the cylinder

I pretend to use the ecuation twice, once for the interior and another for the vaccum, so if I use the cilindrical coordinates for $$\nabla_t^2$$ it results in two Bessel equations, one for the interior and another fot the vaccum.
In the vaccum, the fields should experiment a exponential decay, in my book says that for this restriccion I should put (in the vaccum) $$k^2 - mu\epsilon\omega^2$$ instead the original constants, but I don't understean why... also I don't understeand what are the boundary conditions to proceed to resolve my bessel's equations.
any help or any similar solved problem?

 

[Delta2]

I am afraid I can't be much of a help here but I think in the vacuum the equation would have to be $$(\nabla^2_t+\mu_0\epsilon_0\omega^2)E_z=0$$, that is no term $k^2E_z$ in the vacuum cause that is a dissipative term and vacuum doesn't do dissipation.

 

[baby_1]

Hello,

I pretend to use the equation twice, once for the interior and another for the vacuum, so if I use the cylindrical coordinates for $$\nabla_t^2$$ it results in two Bessel equations, one for the interior and another for the vacuum.

You didn't mention about the material of your cylindrical wave guide. If it is PEC, you need to calculate the wave inside of the wave guide by boundary condition where $E_t=H_t=0$.

For solving your problem, follow these rules

1. Write the $\nabla^2 \vec{E}_{z}=(j\omega \mu \sigma-\omega \mu\varepsilon) \vec{E}_{z}=\gamma ^2\vec{E}_{z}$ in cylindrical form
2. Use sepration of variables to solve the above equation( convert it to Bessel function form just based on r variable)

 

[Delta2]

You didn't mention about the material of your cylindrical wave guide.

The way I understood it is that there is no material, it is just the dielectric in cylindrical shape, surrounded by vacuum.

 

[baby_1]

Dear Delta2,
if cylindrical dielectric wave-guide without the PEC body is assumed the user should calculate the above equation which has been obtained by separation of variables for outside and inside of wave-guide and then connect them via boundary conditions. But I thought that the user is newbie to solve a simple cylindrical dielectric wave-guide which has been surrounded by PEC.