Playground
Interactive refraction simulator: drag the incidence angle slider to see the refracted ray bend in real time across an air-glass interface.
Variables
| Symbol | Name | SI | Dimension | Range |
|---|---|---|---|---|
| Refractive index (medium 1) Refractive index of the incident medium | dimensionless | 1 | 1 – 2.5 | |
| Angle of incidence Angle between incident ray and the normal | rad | 1 | 0 – 1.5708 | |
| Refractive index (medium 2) Refractive index of the refracting medium | dimensionless | 1 | 1 – 2.5 | |
| Angle of refractionoutput Angle between refracted ray and the normal | rad | 1 | 0 – 1.5708 |
Deep dive
Derivation
From Fermat's principle of least time: light takes the path that minimizes travel time between two points. At an interface, minimizing t = d1/(c/n1) + d2/(c/n2) with respect to the entry point yields n1*sin(theta1) = n2*sin(theta2). Alternatively, from Huygens' principle: wavelets in the second medium travel at v2 = c/n2, causing the wavefront to tilt, producing the same relation.
Experimental verification
Verified by Snellius (1621) using protractors and glass prisms. Modern verification uses laser beams at precise angles through calibrated optical media with angular resolution < 0.01 degrees.
Common misconceptions
- Light 'decides' to bend — it follows the path of least time naturally, not by choice.
- Refraction only applies to light — it applies to all waves crossing an interface (sound, seismic, etc.).
- The angle is measured from the surface — it is always measured from the normal to the surface.
Real-world applications
- Optical fiber design: total internal reflection keeps light trapped inside the core.
- Corrective lenses: curved surfaces use refraction to focus light on the retina.
- Mirage formation: continuous refraction in heated air layers bends light upward.
- Diamond cutting: high refractive index (n=2.42) creates total internal reflection sparkle.
Worked examples
Light entering water from air
Given:
- n_1:
- 1
- n_2:
- 1.33
- theta_1:
- 0.7854
Find: theta_2
Solution
theta_2 = arcsin(n1*sin(theta1)/n2) = arcsin(1.00 * sin(45°) / 1.33) = arcsin(0.5317) = 32.1° = 0.5605 rad
Critical angle for glass-to-air
Given:
- n_1:
- 1.5
- n_2:
- 1
Find: theta_critical
Solution
theta_c = arcsin(n2/n1) = arcsin(1.0/1.5) = arcsin(0.6667) = 41.8° = 0.7297 rad
Scenarios
What if…
- scenario:
- What if n2 < n1 (denser to rarer)?
- answer:
- Light bends away from the normal. Beyond the critical angle, total internal reflection occurs — no light crosses the boundary.
- scenario:
- What if the medium has a negative refractive index?
- answer:
- In metamaterials with n < 0, the refracted ray bends to the same side of the normal — 'negative refraction' enables superlenses.
- scenario:
- What if wavelength changes?
- answer:
- Refractive index is wavelength-dependent (dispersion). Different colors refract at different angles — this is how prisms split white light into a rainbow.
Limiting cases
- condition:
- theta_1 → 0
- result:
- theta_2 → 0
- explanation:
- Normal incidence means no bending.
- condition:
- n_1 = n_2
- result:
- theta_2 = theta_1
- explanation:
- Same medium on both sides means no refraction.
- condition:
- theta_1 → arcsin(n_2/n_1) (n_1 > n_2)
- result:
- theta_2 → 90°
- explanation:
- Critical angle: total internal reflection begins.
Context
Willebrord Snellius · 1621
Discovered empirically by Snellius; independently derived by Descartes in 1637 using corpuscular theory. Ibn Sahl described an equivalent law in 984 AD.
Hook
Why does a swimming pool look shallower than it really is?
A light ray passes from air (n=1.00) into water (n=1.33) at 45 degrees. Find the refraction angle using Snell's law.
Dimensions:
- lhs:
- n₁·sin(θ₁) → [1]·[1] = [1]
- rhs:
- n₂·sin(θ₂) → [1]·[1] = [1]
- check:
- Both sides are dimensionless. ✓
Validity: Valid for isotropic, homogeneous media. Breaks down for metamaterials with negative refractive index and at scales comparable to wavelength (diffraction regime).