The amplitude coefficients for the case of internal reflection r_{┴} , t_{┴}, r_{‖} and T_{‖} are plotted in Fig (a) for flint glass to air. There is a critical angle of incidence θ_{c} such that if θ_{i} > θ_{c} Snell’s law cannot be satisfied as sin θ_{t} would need to exceed 1. For θ_{i} > θ_{c} there will be no trsmcmicksion, t = 0, but there will be total internal reflection, ӏrl = 1. From Snell’s law with θ_{t} = 90° the critical angle is

**θ _{c} = sin^{-1} (n_{t}/n_{i})**

For internal reflection n_{t} < n_{i}, and so **√ [(n _{t} / n_{i})^{2} – sin^{2} θ_{i}]** is either real if < Ot or imaginary if θ

_{i}> θ

_{c}. Hence, for θ

_{i}> θ

_{c}the amplitude reflection coefficients are real, and so the phase shift is ϕ = 0° if r > 0 and ϕ = 180° if r < 0, as was the case for external reflection. However, for θ

_{i}> θ

_{c}the amplitude reflection coefficients are complex, and what is plotted in Fig (b) for θ

_{i}> θ

_{c}is actually ӏr

_{┴}ӏ and ӏT

_{‖}ӏ.

Adopting the convention used in optics texts for the sign of ϕ the amplitude reflection coefficient is defined r = ӏ r ӏ e^{-iϕ} The phase shift ϕ may then be derived by noting that for θ_{i} > θ_{c} equation may be written in the form;

and so the phase shift is: **ϕ = 2 tan ^{-1} (b/a)**

where **b = √ [sin ^{2} θ_{i} — (n_{t}/n_{i})^{2}]** , and for perpendicular polarisation a = cos θ

_{i}and for parallel polarisation

**a = (n**. The resulting phase shifts are plotted in Fig (b).

_{t}/n_{i})^{2 }cos θ_{i}**Figure: (a) Amplitude reflection and transmission coefficients, and (b) phase shift, for internal reflection within flint glass. (For θ _{i} > θ_{c} in part (a) r_{┴} and r_{‖} is plotted.)**

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