Total Internal Reflection — Question 7

Question 7

Changing a narrow gap can modulate evanescent coupling exponentially

Two high-index prisms are separated by a low-index gap of width d. For sufficiently weak coupling, use the supplied model $T(d)=C e^{-2\kappa d}$, with positive constants C and kappa independent of d. This is an approximate power-transmission model, not the full boundary solution.

1. Find the gap increase needed to reduce transmission by a factor of ten.

2. Find the fractional transmission change for a small displacement delta d, and evaluate the tenfold gap increase for kappa inverse equal to 120 nm.

Question 7 - Solution

Changing a narrow gap can modulate evanescent coupling exponentially

(a) Method and calculation

$$\frac{T(d+\Delta d)}{T(d)}=e^{-2\kappa\Delta d}=1/10.$$$$\boxed{\Delta d=\ln10/(2\kappa)}.$$

(b) Interpretation and further calculation

$$\boxed{\delta T/T\simeq-2\kappa\,\delta d}\quad(|\kappa\delta d|\ll1).$$$$\boxed{\Delta d=(120\,\mathrm{nm})\ln10/2=138.155\,\mathrm{nm}}.$$

Physical check

Ratios remove the unknown prefactor C. Extrapolating this weak-coupling model to a vanishing gap is not justified.