Fresnel Equations
Calculate how much light is reflected and transmitted at a surface using the Fresnel equations.
Essential for lens coatings, fiber optics, and glass design.
The Formula
rp = (n2 cos θi − n1 cos θt) / (n2 cos θi + n1 cos θt)
Reflectance: R = r² | Transmittance: T = 1 − R
The Fresnel equations, developed by French physicist Augustin-Jean Fresnel in the 1820s, describe exactly how much light is reflected and how much is transmitted when a beam hits the boundary between two transparent materials. Every time you look through a window and see a faint reflection of yourself, or notice glare on a lake surface, you are observing the Fresnel equations at work. The equations give two separate reflection coefficients because light can be polarized in two orientations. The s-polarized component has its electric field perpendicular to the plane of incidence, while the p-polarized component has its electric field parallel to that plane. These two components reflect differently at every angle except normal incidence, which is why polarized sunglasses can selectively block glare.
At normal incidence (light hitting straight on), the equations simplify dramatically. The reflectance becomes R = ((n1 - n2) / (n1 + n2))². For ordinary glass with a refractive index of 1.5 in air, this gives about 4% reflection per surface. A camera lens with 7 glass elements would lose nearly half its light to reflections without anti-reflective coatings. Modern lens coatings work by creating thin layers whose Fresnel reflections cancel each other through destructive interference. The Fresnel equations are also fundamental to fiber optic communications, where engineers must minimize reflection losses at every connector and splice. In solar panel design, reducing Fresnel reflection from the glass cover directly increases energy output. The equations predict that at Brewster's angle, the p-polarized reflection drops to zero, a fact exploited in laser cavity design to produce polarized beams.
Variables
| Symbol | Meaning |
|---|---|
| rs | Reflection coefficient for s-polarized light (dimensionless) |
| rp | Reflection coefficient for p-polarized light (dimensionless) |
| n1 | Refractive index of the first medium (e.g., air = 1.0) |
| n2 | Refractive index of the second medium (e.g., glass = 1.5) |
| θi | Angle of incidence (degrees or radians) |
| θt | Angle of transmission/refraction (found using Snell's Law) |
| R | Reflectance, the fraction of light power reflected (0 to 1) |
| T | Transmittance, the fraction of light power transmitted (0 to 1) |
Example 1: Light Hitting Glass at Normal Incidence
A beam of light in air (n = 1.0) hits a glass surface (n = 1.5) straight on. What percentage is reflected?
At normal incidence (θ = 0), both equations simplify to:
r = (n1 − n2) / (n1 + n2) = (1.0 − 1.5) / (1.0 + 1.5)
r = −0.5 / 2.5 = −0.2
R = r² = 0.04
4% of the light is reflected at each glass surface
Example 2: Light Entering Water at 30 Degrees
Light hits a calm water surface (n = 1.33) at 30 degrees from normal. Find the reflectance for unpolarized light.
First, find θt using Snell's Law: sin θt = (1.0 / 1.33) × sin 30° = 0.376, so θt = 22.1°
rs = (1.0 × cos 30° − 1.33 × cos 22.1°) / (1.0 × cos 30° + 1.33 × cos 22.1°)
rs = (0.866 − 1.232) / (0.866 + 1.232) = −0.174
rp = (1.33 × cos 30° − 1.0 × cos 22.1°) / (1.33 × cos 30° + 1.0 × cos 22.1°)
rp = (1.152 − 0.927) / (1.152 + 0.927) = 0.108
R = (Rs + Rp) / 2 = (0.030 + 0.012) / 2
R ≈ 2.1% reflected for unpolarized light at 30 degrees
When to Use It
The Fresnel equations apply whenever light crosses a boundary between two transparent materials.
- Designing anti-reflective coatings for camera lenses and eyeglasses
- Calculating losses in fiber optic connectors and splices
- Optimizing solar panel glass covers for maximum light transmission
- Understanding glare on water, glass, and polished surfaces
- Laser cavity mirror design and polarization control
- Computer graphics rendering for realistic reflections