Fresnel Reflection Calculator
Calculate the percentage of light reflected at an optical interface for s- or p-polarization using the Fresnel equations. Indispensable for anti-reflection coating design, laser optics, and photography filter analysis.
About this calculator
When light crosses the boundary between two transparent media, some fraction is reflected and the rest is transmitted. The Fresnel equations quantify this split based on the angle of incidence and the refractive indices on both sides. For s-polarization (electric field perpendicular to the plane of incidence): r_s = (n₁ cos θ₁ − n₂ cos θ₂) / (n₁ cos θ₁ + n₂ cos θ₂), and reflectance R_s = r_s². For p-polarization (electric field parallel to the plane of incidence): r_p = (n₂ cos θ₁ − n₁ cos θ₂) / (n₂ cos θ₁ + n₁ cos θ₂), and R_p = r_p². The refracted angle θ₂ is found from Snell's law: n₁ sin θ₁ = n₂ sin θ₂. When θ₁ reaches Brewster's angle, R_p drops to zero — the basis for polarizing filters. At the critical angle for total internal reflection, R = 100%.
How to use
Suppose light travels from air (n₁ = 1.0) into glass (n₂ = 1.5) at an incident angle of 30° with s-polarization. Step 1: find θ₂ via Snell's law: sin θ₂ = (1.0/1.5) × sin 30° = 0.333, so θ₂ ≈ 19.47°. Step 2: cos θ₁ = cos 30° ≈ 0.866; cos θ₂ ≈ 0.943. Step 3: r_s = (1.0 × 0.866 − 1.5 × 0.943) / (1.0 × 0.866 + 1.5 × 0.943) = (0.866 − 1.414) / (0.866 + 1.414) = −0.548 / 2.280 ≈ −0.240. Step 4: R_s = (−0.240)² ≈ 0.058 = 5.8%. Enter these values in the calculator to verify instantly.
Frequently asked questions
What is Brewster's angle and why does p-polarization reflectance drop to zero there?
Brewster's angle θ_B is the incidence angle at which the reflected and refracted rays are exactly perpendicular, satisfying tan θ_B = n₂ / n₁. At this angle the oscillating dipoles in the medium that would radiate the reflected p-polarized wave are aligned along the reflection direction, so they cannot emit in that direction and reflectance falls to zero. For an air-glass interface (n = 1.5), θ_B ≈ 56.3°. This principle is exploited in Brewster windows used in laser cavities to produce a linearly polarized output beam with zero reflection loss.
How do Fresnel reflections affect multi-element camera lenses?
Each uncoated glass-air surface reflects about 4–5% of incident light. A lens with ten elements has twenty surfaces, potentially losing more than half the light to reflections while also generating ghost images and flare. Modern anti-reflection coatings use thin-film interference to destructively cancel the Fresnel reflection, reducing each surface loss to less than 0.1%. Understanding per-surface Fresnel reflectance helps lens designers assess how many coating layers are needed and whether a simpler design with fewer elements might be optically superior.
What happens to Fresnel reflection when the incident angle exceeds the critical angle?
When light travels from a denser medium to a less dense medium (n₁ > n₂) and the incidence angle exceeds the critical angle θ_c = arcsin(n₂/n₁), Snell's law has no real solution for θ₂. This is total internal reflection — 100% of the light is reflected back into the first medium. The critical angle for glass-to-air (n₁ = 1.5, n₂ = 1.0) is about 41.8°. Total internal reflection is the physical mechanism that confines light inside optical fiber cores, making fiber-optic communications possible.