Total Internal Reflection — Question 9

Question 9

Evanescent penetration grows near the critical angle

For n1 greater than n2, define $\theta_c=\sin^{-1}(n_2/n_1)$ and let $\theta=\theta_c+\delta$ with small positive delta in radians. The evanescent decay constant is $\kappa=k_0\sqrt{n_1^2\sin^2\theta-n_2^2}$, where k0 is fixed.

1. Expand the expression inside the square root to first order in delta.

2. Derive the leading penetration-depth dependence and predict the change when delta is reduced by a factor of four.

Question 9 - Solution

Evanescent penetration grows near the critical angle

(a) Method and calculation

$$\sin^2(\theta_c+\delta)=\sin^2\theta_c+2\sin\theta_c\cos\theta_c\,\delta+O(\delta^2).$$$$n_1^2\sin^2\theta-n_2^2\simeq2n_2\sqrt{n_1^2-n_2^2}\,\delta.$$

(b) Interpretation and further calculation

$$\boxed{\ell_E=\kappa^{-1}\simeq\frac{1}{k_0\sqrt{2n_2\sqrt{n_1^2-n_2^2}\,\delta}}}.$$$$\boxed{\ell_E\propto\delta^{-1/2},\quad\ell_E(\delta/4)\simeq2\ell_E(\delta)}.$$

Physical check

This ideal plane-wave result diverges as the threshold is approached from above; finite beam size limits the usefulness of that limiting model.