Light as a Wave — Question 10

Question 10

A tilted plane wave has different apparent spatial periods on two axes

In a uniform dielectric of index n, a plane wave has phase $\phi=k_0n(x\cos\theta+y\sin\theta)-\omega t$. Let $\lambda_0=600\,\mathrm{nm}$, $n=1.50$, and $\theta=30^\circ$. Spatial periods along x and y are measured while the other coordinate and time remain fixed.

1. Find both apparent periods and the actual wavelength along the propagation direction.

2. Write the equation of an equal-phase line and determine its slope.

Question 10 - Solution

A tilted plane wave has different apparent spatial periods on two axes

(a) Method and calculation

$$\boxed{\lambda_{\parallel}=\lambda_0/n=400\,\mathrm{nm}};\quad\Lambda_x=\frac{\lambda_0}{n\cos\theta},\quad\Lambda_y=\frac{\lambda_0}{n\sin\theta}.$$$$\boxed{\Lambda_x=461.880\,\mathrm{nm},\quad\Lambda_y=800\,\mathrm{nm}}.$$

(b) Interpretation and further calculation

$$\phi=\phi_*\Longrightarrow x\cos\theta+y\sin\theta=\frac{\phi_*+\omega t}{k_0n}.$$$$\boxed{\frac{dy}{dx}=-\cot\theta=-\sqrt3};\quad\nabla\phi=k_0n(\cos\theta,\sin\theta).$$

Physical check

The wave vector is perpendicular to each equal-phase line. A period measured along an oblique coordinate axis exceeds the true wavelength.