EXERCISE 8.2-2 The Asymmetric Planar Waveguide. Examine the TE field in an asymmetric planar waveg- uide consisting of a dielectric slab of width d and refractive index ni placed on a substrate of lower refractive index n2 and covered with a medium of refractive index ng < n2 < n,, as illustrated in Fig. 8.2-10. (a) Determine an expression for the maximum inclination angle 0 of plane waves undergoing total internal reflection, and the corresponding numerical aperture NA of the waveguide. (b) Write an expression for the self-consistency condition, similar to (8.2-4). (c) Determine an approximate expression for the number of modes M (valid when M is very large). d n Figure 8.2-10 Asymmetric planar waveguide.

EXERCISE 8.2-2 The Asymmetric Planar Waveguide. Examine the TE field in an asymmetric planar waveg- uide consisting of a dielectric slab of width d and refractive index ni placed on a substrate of lower refractive index n2 and covered with a medium of refractive index ng < n2 < n,, as illustrated in Fig. 8.2-10. (a) Determine an expression for the maximum inclination angle 0 of plane waves undergoing total internal reflection, and the corresponding numerical aperture NA of the waveguide. (b) Write an expression for the self-consistency condition, similar to (8.2-4). (c) Determine an approximate expression for the number of modes M (valid when M is very large). d n Figure 8.2-10 Asymmetric planar waveguide.