Light as a Wave — Question 8

Question 8

Carrier phase and pulse-envelope delay probe different indices

A narrowband pulse traverses $L=10\,\mathrm{mm}$ of a transparent material whose local dispersion is $n(\omega)=1.50+0.020(\omega-\omega_0)/\omega_0$. Use $k(\omega)=n(\omega)\omega/c$ and $c=3.00\times10^8\,\mathrm{m/s}$. Compare delays with the same length in vacuum.

1. Find the phase index, group index $n_g=c\,dk/d\omega$, and excess group delay at the carrier.

2. For carrier vacuum wavelength $600\,\mathrm{nm}$, calculate the excess carrier phase without reducing it modulo $2\pi$.

Question 8 - Solution

Carrier phase and pulse-envelope delay probe different indices

(a) Method and calculation

$$n(\omega_0)=1.50,\quad n_g=n+\omega\frac{dn}{d\omega}=\boxed{1.520}.$$$$\boxed{\Delta\tau_g=\frac{(n_g-1)L}{c}=17.333\,\mathrm{ps}}.$$

(b) Interpretation and further calculation

$$\Delta\phi_0=\frac{2\pi}{\lambda_0}[n(\omega_0)-1]L.$$$$\boxed{\Delta\phi_0=\frac{50000\pi}{3}\ \mathrm{rad}\simeq5.2360\times10^4\,\mathrm{rad}}.$$

Physical check

The phase index governs the carrier; the derivative of phase with frequency governs envelope delay in the narrowband, weak-distortion limit.