Problem $2$

A symmetric step$-$index waveguide is designed by sandwiching a thin film between two glass substrates with $\backslash ($ n$\_\{1\}=1\mathrm{.}71,$ n$\_\{3\}=1\mathrm{.}48\backslash )$ and the free$-$space wavelength is $\backslash (\backslash$lambda$\_\{0\}=870\backslash$mathrm$\{$~nm$\}\backslash ).$

a$)$ To simplify the design, the waveguide was first modeled as a mirror waveguide with dielectric material $\backslash (\backslash$mathrm$\{$n$\}\_\{1\}\backslash )$ between perfect mirrors $($Fig$.2\mathrm{.}2)$:

$2\mathrm{.}2$

What is the greatest thickness, $\backslash ($ d $\backslash ),$ allowing only two TE modes to propagate in this mirror waveguide?

Find the bounce angles, $\backslash (\backslash$theta$\_\{\backslash$mathrm$\{$m$\}\}\backslash )$ and group velocities for these two modes $($use thickness $\backslash ($ d $\backslash )$ found above$)$

Find the cut$-$off wavelength for this waveguide?

b$)$ Now, a model of a dielectric step$-$index waveguide as in Fig. $2\mathrm{.}1$ is considered. What are the main differences between these two models? Would you expect the results calculated with the mirror waveguide model to be valid?

Considering the thickness $\backslash ($ d $\backslash ),$ calculated in part $($a$),$ find the number of guided modes, the acceptance angle $\backslash ($ j $\backslash ),$ and the cut$-$off wavelength for the new waveguide? For what wavelengths does this waveguide support only a single guided mode?

c$)$ Unfortunately, fabricating the waveguide in Fig. $1\mathrm{.}1$ the technician misread the design and failed to seal the film $($the slab with $\backslash ($ n$\_\{$l$\}\backslash ))$ with the top glass substrate $($leaving the film exposed to air, as shown in Fig. $2\mathrm{.}3).$ The technician hopes that the waveguide still operates as expected $($i$.$e$.$ there is the same designed number of guided modes and acceptance angle$).$ Do you agree $($explain qualitatively$)?$