# I Dielectric slab and angle of incidence

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1. Dec 13, 2016

### EmilyRuck

Hello!

Let's consider a plane wave represented by a ray, propagating in a 2D dielectric slab. It has a medium with refractive index $n_1$ as its core and a medium with refractive index $n_2$, $n_2 < n_1$, as its cladding. In order for this ray to represent a mode, it must satisfy two conditions:

- total internal reflection: the angle of incidence $\theta_i$ upon the dielectric interface between the core and the cladding should be such that $\sin \theta_i > n_2 / n_1$;

- self-consistency: $2 k_1 d \cos (\theta_i) + 2 \varphi_r = 2 m \pi$, where $d$ is the core thickness, $k_1$ is the wavenumber of the plane wave inside the core and $\varphi_r$ is the phase shift due to reflection at the dielectric interface.

More details can be found here, with [VIII-36].

This gives only a certain (finite) number of allowed angles $\theta_{i,m}$ for the ray, if the frequency $f$ in $k_1 = 2 \pi f \sqrt{\mu_0 \epsilon_1}$ is known.
When exciting the guide with a signal, should this signal be forced to impinge on the core/cladding interface with one of those angles? How can such a signal be usefully generated, ensuring that its $k_1$ vector will have the allowed angle of incidence?

Last edited by a moderator: Dec 13, 2016
2. Dec 18, 2016

### Greg Bernhardt

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.

3. Jan 16, 2017

### EmilyRuck

Maybe my question was simpler than it seems. In fiber optics the light source must generate a signal whose angle of incidence is not greater than the acceptance angle, in order for the signal to be guided.
As regards dielectric slab guides, instead: is the condition about the angles more restrictive than for the fibers? That is: not only there is a limit-angle, but also a limited number of single accepted angles (so that not a range of angles is admitted, but only a finite number of values for the angles). Is this so?