Difficulty: Introductory to Intermediate | Prerequisites: Ray model of light and reflection (Section 18.1, 18.2), basic trigonometry (sine, inverse sine).
Refraction is what happens when light crosses from one transparent material into another and changes direction. This section builds directly on the ray model: you still draw rays as straight lines, but now they bend at interfaces. Snell's law is the workhorse equation, and total internal reflection (TIR) is its most dramatic consequence. If you are comfortable with the law of reflection and measuring angles from the normal, you are ready for this material.
When light moves from one material to another, it bends (refracts) according to Snell's law: n₁ sin θ₁ = n₂ sin θ₂. Going into a higher-index material bends the ray toward the normal; going into a lower-index material bends it away. If the angle is steep enough when going from higher to lower index, the light cannot escape at all, and you get total internal reflection.
Refraction
The transmission of light from one medium to another, accompanied by a change in direction at the boundary. Think of it as light bending when it crosses the border between two materials.
Index of refraction (n)
A dimensionless number that describes how much a material slows light down relative to vacuum. Defined as n = c/v, where c is the speed of light in vacuum and v is its speed in the material. A higher n means slower light. Vacuum has n = 1 exactly; air is n = 1.00 (accepted value for problems); water is n = 1.33; typical glass is n = 1.50.
Snell's law
The equation relating the angles and indices of refraction on either side of an interface: n₁ sin θ₁ = n₂ sin θ₂. In simple terms, it tells you exactly how much the ray bends when crossing from one material into another.
Total internal reflection (TIR)
The complete reflection of light at a boundary when the angle of incidence exceeds the critical angle. The ray cannot cross into the second medium and instead bounces back entirely. Think of it as the surface acting like a perfect mirror.
Critical angle (θ_c)
The angle of incidence at which the refracted ray would travel exactly along the surface (θ₂ = 90°). For any angle greater than θ_c, the light undergoes total internal reflection. The critical angle exists only when light travels from a higher-index material into a lower-index material (n₁ > n₂).
When a ray crosses from one transparent medium into another, it changes direction. The angle of incidence θ₁ and the angle of refraction θ₂ are both measured from the normal.
Higher-index material (n₂ > n₁): the ray bends toward the normal, making a smaller angle. Light slows down and "turns in."
Lower-index material (n₂ < n₁): the ray bends away from the normal, making a larger angle. Light speeds up and "turns out."
Medium | n |
|---|---|
Vacuum | 1 (exactly) |
Air (accepted for problems) | 1.00 |
Water | 1.33 |
Oil | 1.46 |
Glass (typical) | 1.50 |
Polystyrene plastic | 1.59 |
Cubic zirconia | 2.18 |
Diamond | 2.42 |
Light in air (n = 1.00) hits glass (n = 1.50) at 30° incidence.
n₁ sin θ₁ = n₂ sin θ₂
(1.00) sin 30° = (1.50) sin θ₂
0.5 = 1.5 sin θ₂
sin θ₂ = 0.333
θ₂ = 19.5°
The ray bends toward the normal, as expected when entering a higher-index material.
TIR occurs only when light travels from a higher-index medium into a lower-index medium (n₁ > n₂). As the angle of incidence increases, the refracted ray bends further and further away from the normal. At some point the refracted angle reaches 90°, meaning the refracted ray would skim along the surface. That angle of incidence is the critical angle θ_c.
For any angle of incidence greater than θ_c, 100% of the light reflects back into the first medium. No light crosses into the second medium.
θ_c = sin⁻¹(n₂ / n₁), valid only when n₂ < n₁.
There is no critical angle and no TIR when n₂ > n₁.
Glass to air: θ_c = sin⁻¹(1.00 / 1.50) = 42°
Water to air: θ_c = sin⁻¹(1.00 / 1.33) = 48.8°
The source works through a problem with turpentine (n = 1.472) floating on water (n = 1.333), with light arriving from air at 24.8°.
Air to turpentine: (1.472) sin θ₂ = (1.00) sin 24.8°, giving θ₂ = 16.5° (light bends toward the normal).
Note: the source solves this as finding the refraction angle in air first, getting 38.1°, then uses that for the next step.
Turpentine to water: (1.472) sin 38.1° = (1.333) sin θ₂, giving θ₂ = 42.9° (light bends away from the normal because water has a lower index than turpentine).
Critical angle at turpentine-water interface: θ_c = sin⁻¹(1.333 / 1.472) = 64.9°. Any incidence angle above this in the turpentine produces TIR, and no light enters the water.
Snell's law:
n_1 \sin\theta_1 = n_2 \sin\theta_2Critical angle for total internal reflection (requires n₁ > n₂):
\theta_c = \sin^{-1}\left(\frac{n_2}{n_1}\right)Index of refraction:
n = \frac{c}{v}where c is the speed of light in vacuum and v is the speed in the material.
Fibre-optic cables rely on total internal reflection to keep light trapped inside a thin glass strand over kilometres of distance. This is how internet data travels at the speed of light through undersea cables. Diamonds sparkle because their very high index of refraction (n = 2.42) gives them a small critical angle, so most light entering the top face undergoes TIR and bounces around inside before exiting, producing the characteristic brilliance.
Students often think light always bends toward the normal. It bends toward the normal only when entering a higher-index material. When entering a lower-index material, it bends away.
A common error is trying to apply TIR when light goes from a lower-index to a higher-index material. TIR does not happen in that direction. There is no critical angle when n₂ > n₁.
Students sometimes forget that all angles are measured from the normal, not the surface. If a problem states the angle "from the surface," subtract from 90° first.
When solving multi-layer problems, students occasionally try to skip a layer. You must apply Snell's law at each interface separately.
⚠️ Snell's law problems are among the most commonly tested calculations in this unit. Expect to be given two indices and an angle, then asked to find the refracted angle.
⚠️ Know the critical angle formula cold. You will likely be asked to compute θ_c for a given pair of materials and then determine whether TIR occurs at a given angle.
⚠️ Multi-layer problems (like the turpentine-on-water example) test whether you can chain Snell's law across two boundaries. These are popular exam questions.
True or false: light bends toward the normal when entering a material with a lower index of refraction. (False: it bends away from the normal.)
Fill in the blank: total internal reflection occurs only when light travels from a ______ index material to a ______ index material. (higher, lower)
True or false: the critical angle for glass (n = 1.50) to air is about 42°. (True)
Fill in the blank: Snell's law states n₁ sin θ₁ = ______. (n₂ sin θ₂)
True or false: if n₂ > n₁, there is a critical angle for TIR. (False: no critical angle exists when n₂ > n₁.)
Q: Light travels from water (n = 1.33) into glass (n = 1.50) at an angle of incidence of 40°. What is the angle of refraction?
A: Apply Snell's law. (1.33) sin 40° = (1.50) sin θ₂. So sin θ₂ = (1.33 × 0.643) / 1.50 = 0.570. θ₂ = sin⁻¹(0.570) = 34.8°. The ray bends toward the normal because it enters a higher-index material.
Q: What is the critical angle for light travelling from diamond (n = 2.42) into air (n = 1.00)?
A: θ_c = sin⁻¹(1.00 / 2.42) = sin⁻¹(0.413) = 24.4°. Any angle of incidence above 24.4° inside the diamond produces total internal reflection.
Q: A ray in turpentine (n = 1.472) hits the turpentine-water (n = 1.333) interface at 70°. Does the ray refract into the water, or does TIR occur?
A: First find the critical angle: θ_c = sin⁻¹(1.333 / 1.472) = 64.9°. Since 70° > 64.9°, total internal reflection occurs. No light enters the water.
Q: Explain why TIR cannot occur when light goes from air into glass.
A: TIR requires the light to be in the higher-index medium (n₁ > n₂). Air (n = 1.00) has a lower index than glass (n = 1.50), so the condition is not met. There is no critical angle in this direction.
Refraction is the physical basis for how lenses work (Sections 18.5 and 18.7). A converging lens bends rays toward the optical axis by applying Snell's law at two curved surfaces. Total internal reflection connects to fibre optics and is revisited in wave optics when you study evanescent waves. The index-of-refraction table will reappear in dispersion problems, where different wavelengths of light have slightly different values of n in the same material, producing rainbows.
Refraction, Snell's law, index of refraction, refractive index, bending of light, total internal reflection, TIR, critical angle, angle of incidence, angle of refraction, fibre optics, optical density, n₁ sin θ₁ = n₂ sin θ₂, PHYS 116, University of Washington, general physics optics, Chapter 18