Fresnel Reflectance

Fresnel Reflectance

When electromagnetic radiation strikes a planar boundary between two nonabsorbing, nonmagnetic dielectric media with refractive indices n₁ and n₂, Snell's law n₁ sin(θ_i) = n₂ sin(θ_t) determines the refraction angle θ_t. Below the critical angle, the Fresnel equations give the power reflectances R_s = ((n₁ cos θ_i − n₂ cos θ_t)/(n₁ cos θ_i + n₂ cos θ_t))² for perpendicular (s) polarization and R_p = ((n₂ cos θ_i − n₁ cos θ_t)/(n₂ cos θ_i + n₁ cos θ_t))² for parallel (p) polarization.

Worked Example: Brewster Angle

Calculate the Brewster angle and polarized power reflectances for light traveling from air (n₁ = 1) into glass (n₂ = 3/2 = 1.5):

1. Brewster Incidence Angle: Use θ_B = arctan(n₂ / n₁) = arctan(1.5) ≈ 56.30993247402°.
2. Transmitted Angle: At Brewster's angle, θ_i + θ_t = 90°, so θ_t = 90° − 56.30993247402° ≈ 33.69006752598°.
3. Polarized Reflectances: The p-polarized reflectance vanishes identically (R_p = 0), while R_s = (25/169) ≈ 0.147928994083.
4. Unpolarized Averages: The unpolarized reflectance is R_unpol = (R_s + R_p)/2 ≈ 0.073964497041 and unpolarized transmittance is T_unpol = 1 − R_unpol ≈ 0.926035502959.

Experiment: Exploring Polarized Regimes

Use the Fresnel Reflectance interactive module to compare normal incidence, Brewster polarization, critical boundary, and total internal reflection:

1. Set Interface to "Air to glass" and Angle setting to "Normal". Both s and p reflectances equal 0.04 (4%) and transmittance equals 0.96.
2. Switch Angle setting to "Brewster" to verify that p reflectance drops to 0 while s reflectance rises to 0.147928994083.
3. Change Interface to "Glass to air" and compare "Critical" (θ_i ≈ 41.81031489578°, θ_t = 90°) with "60 degrees" (total internal reflection where transmitted angle is unavailable and R_s = R_p = 1).

Limitations

This model evaluates power reflectance and transmittance for lossless, isotropic, nonmagnetic dielectrics at discrete refractive index presets (1 and 3/2). It does not compute complex reflection phase shifts during total internal reflection, Goos-Hänchen spatial shifts, or complex refractive indices in absorbing conductors and semiconductors.

References and Further Reading

For full derivations of the electromagnetic boundary conditions and Fresnel reflection and transmission coefficients at planar dielectric interfaces, consult:

• Richard Fitzpatrick, Oscillations and Waves: Fresnel Relations