Light as a Wave — Question 7

Question 7

Huygens wavelets recover the refraction angle

A plane wave crosses a plane boundary from index $n_1=1.00$ into $n_2=1.50$. During a common time interval $\Delta t$, its wavefront advances distances $v_1\Delta t$ and $v_2\Delta t$. With angles measured from the normal, the construction gives $\sin\theta_1=v_1\Delta t/AB$ and $\sin\theta_2=v_2\Delta t/AB$. Let $\theta_1=30^\circ$.

1. Eliminate the construction length AB and time interval to derive the refraction law.

2. Find the refracted angle and the wavelength ratio across the boundary.

Question 7 - Solution

Huygens wavelets recover the refraction angle

(a) Method and calculation

$$\frac{\sin\theta_1}{\sin\theta_2}=\frac{v_1}{v_2}=\frac{n_2}{n_1}.$$$$\boxed{n_1\sin\theta_1=n_2\sin\theta_2}.$$

(b) Interpretation and further calculation

$$\boxed{\theta_2=\sin^{-1}(1/3)=19.471^\circ}.$$$$f_2=f_1,\quad\boxed{\lambda_2/\lambda_1=v_2/v_1=n_1/n_2=2/3}.$$

Physical check

The same time interval is essential: the wavefront construction compares simultaneous phase surfaces, while frequency remains continuous at a stationary boundary.