Lecture 2: Refraction

Lecture aims to explain:

1. Refractive index

2. Snell’s Law of refraction

3. Total internal reflection

4. Dispersion of the refractive index Refractive index Propagation of light in a medium

Speed of light in vacuum: c= 299,792,458 m/s ~3x108m/s

In a medium, waves are scattered by atoms. As the scattered waves interact, the net result is the change of the phase velocity

sin(kx − ωt) vp = ω / k = ωλ / 2π

Refractive index is the ratio of speed of light in vacuum to the speed of light in matter n ≡ c / vp

Frequency is not affected, but the wavelength is λn ≡ λ0 / n Refractive indices of various materials

Data for wavelength 589 nm

Vacuum: 1 Air: 1.00029 Ice: 1.31 Water: 1.333 Glass: ~1.5 (may vary for different types of glass)

Zircon (ZrSiO4): 1.923 Diamond: 2.4 Gallium Phosphide (GaP): 3.5 Snell’s Law of Refraction Snell’s Law of Refraction

Transmitted wave moves slower: wavefront has a kink

sinθ sinθ From i = t BD AE follows Snell’s Law:

ni sinθi = nt sinθt

Dated 1621, named after Willebrord Snellius (1580-1626), a Dutch astronomer from Leiden. Now known that this law was also discovered by an Arab scientist from Baghdad in the X century, and in England ~1600. EXAMPLE 2.1: measurement of the refractive index

Find the ratio xi/xt on the figure using Snell’s law (the circle radius is R). How can this arrangement be used for the measurement of the refractive index of an unknown material. What will this ratio be for light propagating from air into water? (nw=1.33) EXAMPLE 2.2: propagation of light through multi-layer structures

Light propagating from water into air is incident on the water surface o (nw=1.33) at the angle of αint=30 . A plane parallel glass plate (ng=1.6) is brought into contact with the surface of water. Calculate the angle α between the top surface of the glass plate and the direction at which light will emerge from glass into air. Calculate α for the case when the glass plate is removed. Total internal reflection EXAMPLE 2.3: total internal reflection

Light propagating from water into air is incident on the water 0 surface (nw=1.33) at the angle of θw=60 . Calculate the angle of propagation of light in air.

θ For n i > n t the critical angle of incidence c is deduced from ni sinθc = nt

Compare glass (ng=1.5) and diamond (nd=2.4): diamond will appear more luminous because light will bounce many times inside and leave in all directions, also reflection from surfaces depends on the contrast of refractive indeces Dispersion of the refractive index Dispersion

Refractive index depends on wavelength

dn - dispersion λ d Dispersion depends on the material, and is usually negative in the visible

As an example, diamond has a very high dispersion, which explains its ability to show “fire”, or colour SUMMARY

Refractive index: ratio of speed of light in vacuum to the speed of light in the medium n ≡ c / vp

Snell’s law of refraction: θ = θ describes refraction of light ni sin i nt sin t

Total internal reflection occurs for light incident at the interface between two materials at an angle above the critical

θ For n i > n t the critical angle of incidence c is deduced from ni sinθc = nt

Dispersion of the refractive index dn is present in most materials

dλ