Fresnel Equations Calculator (Reflection and Transmission)

What the Fresnel equations describe

When light hits the boundary between two transparent materials with different refractive indices, part of it reflects back and part transmits through. The Fresnel equations, derived by Augustin-Jean Fresnel in 1821, give the exact fraction of amplitude reflected for each of the two independent polarization states:

r_s = (n₁ cos θᵢ − n₂ cos θₜ) / (n₁ cos θᵢ + n₂ cos θₜ) (s-polarization, electric field perpendicular to plane of incidence) r_p = (n₂ cos θᵢ − n₁ cos θₜ) / (n₂ cos θᵢ + n₁ cos θₜ) (p-polarization, electric field in plane of incidence)

Where n₁ is the incident medium’s refractive index, n₂ is the transmitted medium’s index, θᵢ is the angle of incidence measured from the surface normal, and θₜ is the angle of refraction (transmission) determined by Snell’s law: n₁ sin θᵢ = n₂ sin θₜ.

The reflectance (fraction of intensity reflected) is the square of the amplitude coefficient: R = r². The transmittance is T = 1 − R for non-absorbing media. The two polarizations reflect differently at every angle except normal incidence, which is why polarized sunglasses reduce glare from horizontal surfaces.

Normal incidence: the simple case

When light hits the surface head-on (θᵢ = 0), both Fresnel formulas collapse to the same expression:

R = ((n₁ − n₂) / (n₁ + n₂))²

For ordinary window glass (n = 1.5) in air (n = 1.0), this gives R = 0.04, or 4 percent reflection per surface. The other 96 percent transmits. A camera lens with 7 glass elements loses nearly half its light to these reflections without anti-reflective coatings. Modern lens coatings exploit thin-film interference to cancel the Fresnel reflection over the visible spectrum, often reducing per-surface reflection below 0.5 percent.

For a diamond (n = 2.42) in air, normal-incidence R = ((1 − 2.42)/(1 + 2.42))² = 0.172, about 17 percent reflection. Combined with the high critical angle (24°), this is what gives diamonds their characteristic brilliance: most light entering the top either reflects internally several times or comes back out the same surface.

Brewster’s angle: where p-polarization vanishes

At a specific angle of incidence, the p-polarized reflection coefficient r_p goes to exactly zero. The reflected light at this angle is completely s-polarized, which means a polarizing filter oriented to block s-polarization would block all the reflection. This angle is called Brewster’s angle (or the polarization angle):

θ_B = arctan(n₂ / n₁)

For air-to-glass (n₁ = 1, n₂ = 1.5), Brewster’s angle is arctan(1.5) ≈ 56.3°. This is why polarized sunglasses with horizontally-oriented polarizers eliminate glare from horizontal surfaces (water, hood of a car, road) so effectively: the reflected glare is mostly s-polarized at angles near Brewster’s. Photographers use Brewster’s angle to take pictures through water or windows without seeing reflections.

Laser cavities exploit this same principle. A “Brewster window” inserted at the polarization angle introduces zero reflection loss for one polarization while reflecting the other out of the cavity. This produces a linearly polarized output beam.

Total internal reflection: when the angle exceeds critical

When light goes from a denser medium to a less dense one (n₁ > n₂), Snell’s law gives sin θₜ = (n₁/n₂) sin θᵢ. Because n₁/n₂ > 1, sin θₜ can exceed 1 at sufficiently large θᵢ, which is impossible for a real angle. Physically, the light cannot transmit; it all reflects. The threshold is the critical angle:

θ_c = arcsin(n₂ / n₁)

For glass to air, θ_c = arcsin(1/1.5) ≈ 41.8°. Beyond this angle, R = 1 exactly for both polarizations. This is total internal reflection (TIR), and it is what makes fiber-optic cables work: light entering one end of the fiber undergoes TIR at every reflection off the cladding boundary and propagates with very low loss. The same effect creates the rainbow of colors in a cut diamond and the silvery look of water seen from underwater at a steep angle.

A worked example: light hitting water at 45 degrees

A laser pointer (n₁ = 1.0 air, n₂ = 1.33 water) hits the water surface at θᵢ = 45°. From Snell’s law:

sin θₜ = (1.0 / 1.33) × sin 45° = 0.752 × 0.707 = 0.532, so θₜ = 32.1°

Now the Fresnel coefficients:

cos 45° = 0.7071, cos 32.1° = 0.8472 r_s = (1.0 × 0.7071 − 1.33 × 0.8472) / (1.0 × 0.7071 + 1.33 × 0.8472) = (0.7071 − 1.127) / (0.7071 + 1.127) = −0.420 / 1.834 = −0.229 r_p = (1.33 × 0.7071 − 1.0 × 0.8472) / (1.33 × 0.7071 + 1.0 × 0.8472) = (0.9404 − 0.8472) / (0.9404 + 0.8472) = 0.0932 / 1.788 = 0.0521

R_s = 0.0524 (5.2 percent s-reflection) R_p = 0.0027 (0.27 percent p-reflection)

Total reflection for unpolarized light: R = (R_s + R_p)/2 = 0.0276, or 2.76 percent of intensity reflected. At 45° on water, most of the light gets through. The reflected component is strongly s-polarized, by a factor of about 20 to 1.

Where Fresnel equations matter in practice

Anti-reflective coatings on eyeglasses, camera lenses, and solar panels are designed using Fresnel-based interference calculations. A single-layer MgF₂ coating (n = 1.38) cuts reflection from glass from 4 percent to about 1 percent in the middle of the visible spectrum. Multi-layer coatings can achieve below 0.1 percent across the entire visible range.

Fiber-optic communications rely on TIR for low-loss propagation. Bending a fiber too sharply causes the local angle of incidence to drop below the critical angle, light leaks out, and signal weakens.

In photography and cinema, polarizing filters at Brewster’s angle eliminate reflections from non-metallic surfaces. Underwater photography is impossible without compensating for surface reflection at oblique angles.

Solar panel design tries to minimize Fresnel reflection at the air-to-glass interface, using textured coverglass and anti-reflective films. A 1 percent reduction in reflection means 1 percent more electricity generated for the panel’s entire lifetime.

Architectural glass and one-way mirrors use partial reflection at calibrated values. A two-way mirror is just heavily coated glass at near-normal incidence, with the visible reflection driven by Fresnel.

Limitations

The Fresnel equations assume both media are non-absorbing dielectrics with real-valued refractive indices. For metals (silver, gold, aluminum), the refractive index is complex, and the formulas extend to a complex-valued version that still works but with phase shifts and absorption included. The qualitative picture (high reflection, polarization differences) is similar but the math is more involved.

The equations also assume plane waves and flat interfaces. Curved surfaces, microscopic roughness, and edge effects modify reflections in real systems. For typical optical surfaces, the smooth-interface Fresnel result is accurate to a few percent.