Reflection and Refraction — Question 1

Question 1

A wavelength change at a boundary leaves optical frequency unchanged

A monochromatic beam travels from index n1=1.20 into index n2=1.60 at 40 degrees to the normal. Its wavelength in the first medium is 600 nm. The media are stationary, transparent, and nondispersive at this frequency. Use c=3.00 times 10 to the eighth m/s.

1. Find the refracted and reflected angles.

2. Find the frequency, wavelength in medium 2, and phase-speed ratio.

Question 1 - Solution

A wavelength change at a boundary leaves optical frequency unchanged

(a) Method and calculation

$$n_1\sin\theta_1=n_2\sin\theta_2\Rightarrow\boxed{\theta_2=\sin^{-1}(0.75\sin40^\circ)=28.822^\circ}.$$$$\boxed{\theta_r=40^\circ}.$$

(b) Interpretation and further calculation

$$f=\frac{c}{n_1\lambda_1}=\boxed{4.16667\times10^{14}\,\mathrm{Hz}}.$$$$\boxed{\lambda_2=\lambda_1n_1/n_2=450\,\mathrm{nm},\quad v_2/v_1=n_1/n_2=0.750}.$$

Physical check

A stationary interface preserves temporal frequency; the smaller phase speed produces a shorter wavelength.