The dielectric slab waveguide shown in Fig. 8.20 is infinite in extent in the x- and y-directions, having refractive indices n1 n2. Find the maximum angle of incidence θm for total internal reflection at the interface between the core and cladding.

Question

The dielectric slab waveguide shown in Fig. 8.20 is infinite in extent in the x- and y-directions, having refractive indices [latex] n_{1} \gt n_{2} [/latex] . Find the maximum angle of incidence [latex] heta_{m}[/latex] for total internal reflection at the interface between the core and cladding.

Accepted Answer

Applying Snell’s law to the entrance surface, we have

[latex] n_{o} \sin heta _{m} = n_{1} \sin (90^{\circ } - heta _{c}) \ \quad \quad \quad \quad = n_{1} \cos heta _{c} [/latex]

The critical angle at the interface between the core and cladding is

[latex]\sin heta _{c} = n_{2} / n_{1} [/latex]

Combining the two equations, we have

[latex] n_{o} \sin heta _{m} = n_{1} \sqrt{1 - (n_{2} / n_{1})^{2}} [/latex]

The maximum angle of incidence is therefore

[latex] heta _{m} = \sin ^{ - 1} \left[ \frac{1}{n_{o}} \sqrt{ n^{2}_{1} - n^{2}_{2}} ight] [/latex]

The wave incident at an angle less than [latex] heta_{m}[/latex] is guided through the dielectric slab by successive total internal reflections at the interface between the core and cladding.

Question 8.20: The dielectric slab waveguide shown in Fig. 8.20 is infinite......

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