The Fresnel Equations

Definition

When a light wave strikes a smooth interface between two transparent media, part of it reflects and part refracts. The Fresnel equations quantify exactly how much of the electric-field amplitude is reflected and transmitted, and the answer depends on two things: the angle of incidence and the polarization of the wave relative to the plane of incidence (the plane containing the incident ray and the surface normal).

We split any polarization into two independent components:

• s-polarization (from German senkrecht, "perpendicular"; also called TE, transverse-electric): the electric field is perpendicular to the plane of incidence.
• p-polarization ("parallel"; also called TM, transverse-magnetic): the electric field lies in the plane of incidence.

For non-magnetic dielectrics (relative permeability μ ≈ 1), the amplitude coefficients are:

r_s = (n₁·cosθᵢ − n₂·cosθₜ) / (n₁·cosθᵢ + n₂·cosθₜ)
t_s = (2·n₁·cosθᵢ)          / (n₁·cosθᵢ + n₂·cosθₜ)

r_p = (n₂·cosθᵢ − n₁·cosθₜ) / (n₂·cosθᵢ + n₁·cosθₜ)
t_p = (2·n₁·cosθᵢ)          / (n₂·cosθᵢ + n₁·cosθₜ)

where the symbols are:

• n₁ — refractive index of the incident medium (dimensionless).
• n₂ — refractive index of the transmitting medium (dimensionless).
• θᵢ — angle of incidence, measured from the surface normal (radians or degrees).
• θₜ — angle of refraction, fixed by Snell's law n₁·sinθᵢ = n₂·sinθₜ.
• r, t — amplitude reflection and transmission coefficients (ratios of reflected/transmitted E-field amplitude to incident E-field amplitude; dimensionless, can be negative or complex).

Note the sign convention matters: the formulas above use the "Fresnel" convention in which a positive r_p corresponds to a specific choice of the reference direction for the field. Different textbooks flip the sign of r_p, which changes whether the phase jump appears in r_s or r_p — but the physics (the reflectance and the location of the zero) is identical.

From amplitudes to power: reflectance and transmittance

What a detector or your eye actually measures is power, not field amplitude. The reflectance R is the fraction of incident power that reflects, and the transmittance T is the fraction transmitted:

R = |r|²
T = (n₂·cosθₜ) / (n₁·cosθᵢ) · |t|²

The reflected beam stays in the same medium and keeps the same cross-sectional area, so R is simply |r|². The transmitted beam is a different story: it bends, so its cross-section changes by a factor cosθₜ/cosθᵢ, and it travels in a medium of different wave impedance, contributing the factor n₂/n₁. Multiplying these gives the geometric-and-impedance correction that turns |t|² into a true power fraction. A very common beginner error is to write T = |t|² — that is dimensionally a field ratio squared, not a power ratio, and it does not conserve energy.

For lossless media, the Fresnel equations guarantee, for each polarization independently:

R + T = 1

This is a direct consequence of the continuity of the tangential E and H fields (Maxwell's boundary conditions) and the conservation of the normal component of the time-averaged Poynting vector.

Brewster's angle: the disappearing reflection

Look at the p-polarization numerator, n₂cosθᵢ − n₁cosθₜ. There exists a special angle where this vanishes and r_p = 0 exactly — no p-polarized light reflects at all. Using Snell's law, that condition simplifies beautifully to:

θ_B = arctan(n₂ / n₁)

This is Brewster's angle, named for David Brewster (1815). At θ_B the reflected and refracted rays are exactly 90° apart (θ_B + θₜ = 90°). The physical picture: the refracted wave drives oscillating dipoles in the second medium, and a dipole radiates nothing along its own axis. When the reflected ray would point along that dipole axis, the p-polarized reflection is forbidden — leaving reflected light that is 100% s-polarized. This is exactly how polarized sunglasses cut glare off wet roads and water: the glare, reflected near Brewster's angle, is predominantly s-polarized (horizontal), so a vertically-oriented polarizer blocks it. For an air→glass interface (n₁ = 1, n₂ = 1.52), θ_B = arctan(1.52) ≈ 56.7°.

Phase changes on reflection

Because r can be negative or complex, the reflected wave can pick up a phase shift relative to the incident wave.

• External reflection (n₁ < n₂), below Brewster: r_s is negative for all angles, a 180° (π) phase flip — the famous "half-wave loss" responsible for the dark central fringe in Newton's rings and for how anti-reflection coatings and thin-film interference are designed.
• p-polarization sign flip: r_p starts positive at normal incidence (in the convention above), passes through zero at Brewster's angle, and becomes negative beyond it — a π phase change concentrated right at θ_B.
• Internal reflection (n₁ > n₂), below critical angle: both r_s and r_p are real; there is no π jump of the s-wave at low angles.
• Total internal reflection: r becomes complex with |r| = 1, so the phase shift varies continuously with angle and differs between s and p. Fresnel exploited this in his rhomb, a glass prism that uses two internal reflections to convert linear to circular polarization — an achromatic quarter-wave device.

Total internal reflection and the evanescent wave

When light goes from a denser to a rarer medium (n₁ > n₂) and θᵢ exceeds the critical angle

θ_c = arcsin(n₂ / n₁)

Snell's law demands sinθₜ = (n₁/n₂)·sinθᵢ > 1, which is impossible for a real angle. Formally cosθₜ = √(1 − sin²θₜ) becomes purely imaginary. Substitute that into the Fresnel equations and r turns into a complex number whose magnitude is exactly 1, so R = |r|² = 1: every photon is reflected. This is the principle behind optical fibers, diamond's brilliance, and the total-internal-reflection prisms in binoculars. For a glass→air interface (n₁ = 1.52, n₂ = 1), θ_c = arcsin(1/1.52) ≈ 41.1°.

The transmitted field does not simply vanish: it becomes an evanescent wave that clings to the surface and decays exponentially into the rarer medium over a fraction of a wavelength. It carries zero time-averaged power across the boundary (consistent with R = 1), but a second surface brought within a wavelength can "frustrate" the total reflection and tap that energy — the optical analog of quantum tunneling.

Worked example: air to glass at normal incidence

Set θᵢ = 0, so θₜ = 0 and all cosines equal 1. Both polarizations degenerate to the same formula:

r = (n₁ − n₂) / (n₁ + n₂)
R = ((n₁ − n₂) / (n₁ + n₂))²

For air (n₁ = 1) and crown glass (n₂ = 1.52):

r = (1 − 1.52) / (1 + 1.52) = −0.52 / 2.52 = −0.206
R = (−0.206)² = 0.0426  →  about 4.3%
T = 1 − R = 0.957       →  about 95.7%

Each air-glass surface reflects roughly 4%. A single windowpane has two surfaces, so about 8% is lost to reflection (ignoring the tiny interference between them) and about 92% transmits. This is exactly why quarter-wave anti-reflection coatings — which use destructive interference between the two reflections off a thin film — are standard on camera lenses, eyeglasses, and solar cells.

Key values at a glance

Interface (n₁ → n₂)	Normal-incidence R	Brewster angle θ_B	Critical angle θ_c
Air → water (1.00 → 1.33)	2.0%	53.1°	— (n₂ > n₁)
Air → crown glass (1.00 → 1.52)	4.3%	56.7°	—
Air → diamond (1.00 → 2.42)	17.2%	67.5°	—
Air → silicon (1.00 → 3.88)	34.8%	75.5°	—
Water → air (1.33 → 1.00)	2.0%	36.9°	48.8°
Glass → air (1.52 → 1.00)	4.3%	33.3°	41.1°
Diamond → air (2.42 → 1.00)	17.2%	22.5°	24.4°

Notice R at normal incidence is the same going either way across an interface — reflectance is symmetric — but Brewster and critical angles are not.

JavaScript — computing the Fresnel coefficients

// Fresnel amplitude + power coefficients for a lossless interface.
// Angles in radians. Handles total internal reflection via complex cosθₜ.

function fresnel(n1, n2, thetaI) {
  const cosI = Math.cos(thetaI);
  const sinT = (n1 / n2) * Math.sin(thetaI);      // Snell's law
  const sin2T = sinT * sinT;

  if (sin2T <= 1) {
    // Ordinary refraction — everything real
    const cosT = Math.sqrt(1 - sin2T);
    const rs = (n1 * cosI - n2 * cosT) / (n1 * cosI + n2 * cosT);
    const rp = (n2 * cosI - n1 * cosT) / (n2 * cosI + n1 * cosT);
    const ts = (2 * n1 * cosI) / (n1 * cosI + n2 * cosT);
    const tp = (2 * n1 * cosI) / (n2 * cosI + n1 * cosT);
    const Rs = rs * rs, Rp = rp * rp;
    const geom = (n2 * cosT) / (n1 * cosI);
    const Ts = geom * ts * ts, Tp = geom * tp * tp;
    return { Rs, Rp, Ts, Tp, R: (Rs + Rp) / 2, T: (Ts + Tp) / 2, tir: false };
  } else {
    // Total internal reflection — cosθₜ is imaginary, |r| = 1, R = 1
    return { Rs: 1, Rp: 1, Ts: 0, Tp: 0, R: 1, T: 0, tir: true };
  }
}

const deg = d => d * Math.PI / 180;

// Air -> glass at normal incidence: expect R ~ 4.3%
console.log(fresnel(1.0, 1.52, 0).R.toFixed(4));       // 0.0426

// Brewster angle for air -> glass: r_p (hence Rp) should be ~0
const thetaB = Math.atan(1.52 / 1.0);                  // 0.9889 rad = 56.7°
console.log((thetaB * 180 / Math.PI).toFixed(1));      // 56.7
console.log(fresnel(1.0, 1.52, thetaB).Rp.toExponential(2)); // ~0.00e+0

// Glass -> air past the critical angle (41.1°): total internal reflection
console.log(fresnel(1.52, 1.0, deg(45)).R);            // 1  (tir = true)

// Energy conservation check at an arbitrary angle
const f = fresnel(1.0, 1.33, deg(30));
console.log((f.R + f.T).toFixed(6));                    // 1.000000

History and why it mattered

Fresnel published these equations in 1823, decades before James Clerk Maxwell's electromagnetic theory of light (1865). Fresnel worked from a mechanical, elastic-solid model of the luminiferous aether, treating light as transverse vibrations. Remarkably, his results were exactly right, and when Maxwell's equations arrived they reproduced the Fresnel coefficients from the boundary conditions on E and H — one of the great confirmations that light is an electromagnetic wave. Fresnel's insistence on transverse (not longitudinal) vibrations, needed to explain polarization-dependent reflection, was a decisive blow to the older corpuscular picture and a triumph of the wave theory.

Where the Fresnel equations show up

• Anti-reflection coatings. Quarter-wave films on lenses and solar panels use the Fresnel amplitudes and thin-film interference to cancel the ~4% surface reflection.
• Polarizing beam splitters & Brewster windows. Laser cavities use Brewster-angle windows to give zero loss for p-polarization, forcing the laser to lase in a single, pure polarization.
• Optical fibers. Guiding relies on total internal reflection; R = 1 at the core-cladding boundary keeps light trapped over kilometers.
• Ellipsometry. Measuring the complex ratio r_p/r_s versus angle determines film thickness and refractive index to sub-nanometer precision in semiconductor manufacturing.
• Computer graphics. Physically-based renderers use the Schlick approximation to the Fresnel reflectance for realistic glints on water, glass, and skin at grazing angles.
• Radar & radio. The same equations govern reflection of radio waves off the ground and sea, including the Brewster-angle "pseudo-Brewster" dip for lossy surfaces.

Common misconceptions

• Thinking T = |t|². The transmittance needs the factor (n₂cosθₜ)/(n₁cosθᵢ) to account for the refracted beam's changed area and impedance. Only then does R + T = 1.
• Assuming r + t = 1. That is never required; it is R + T that conserves energy. In fact t can exceed 1 (the transmitted amplitude can be larger than the incident amplitude) without violating anything, because power depends on impedance and area too.
• Believing total internal reflection means zero field in medium 2. An evanescent wave penetrates a fraction of a wavelength; it just carries no net power. Bring a second surface close and you can frustrate the reflection.
• Confusing Brewster's angle with the critical angle. Brewster (r_p → 0, both directions of propagation) exists for any interface; the critical angle (onset of TIR) only exists going from dense to rare (n₁ > n₂).
• Forgetting the phase flip. External reflection carries a π phase shift; ignoring it gives wrong thin-film and interference predictions (e.g., mislabelling Newton's-rings fringes).
• Using the equations for absorbing metals as-is. For metals the index is complex (n → n + iκ); the Fresnel equations still hold but with complex refractive indices, and R + T = 1 becomes R + T + A = 1 once absorption A is included.