Total Internal Reflection — Question 10

Question 10

Guided zigzag rays take longer than axial rays

A planar guide has core index n1, lower-index cladding n2, length L, and separation D between its parallel side walls. A guided ray makes angle beta with the guide axis. Ignore reflection phase delays and material dispersion.

1. Find the transit time and axial distance between successive wall reflections.

2. Find the limiting ratio of the slowest meridional-ray transit time to the axial transit time allowed by total internal reflection.

Question 10 - Solution

Guided zigzag rays take longer than axial rays

(a) Method and calculation

$$\ell=L/\cos\beta\Rightarrow\boxed{t(\beta)=n_1L/(c\cos\beta)}.$$$$\tan\beta=D/\Delta x\Rightarrow\boxed{\Delta x=D/\tan\beta}.$$

(b) Interpretation and further calculation

$$\theta_{\mathrm{wall}}=90^\circ-\beta>\theta_c\Rightarrow\cos\beta>n_2/n_1.$$$$t_0=n_1L/c\Rightarrow\boxed{t/t_0< n_1/n_2}\quad\text{with limiting supremum}\ n_1/n_2.$$

Physical check

All rays have the same speed c/n1 in this model, but oblique rays cover longer paths; the critical ray itself is the limiting case.