Dielectric slab and angle of incidence

[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 http://people.seas.harvard.edu/~jones/ap216/lectures/ls_1/ls1_u8/ls1_unit_8.html, 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?

 

[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?