💡

Fresnel Interface Coefficients

Reflection and refraction of a plane wave at a flat boundary between two media. See the complex amplitudes r and t for s and p light, the power split R + T, the Brewster and critical angles, the evanescent field of total internal reflection, and how an absorbing medium (n + iκ) such as a metal reflects. (For diffraction in the Fresnel near-field regime, see single-slit and aperture diffraction.)

Type exact values in the number boxes • Drag across the R/T or phase plot to set θ₁ • Settings are kept in the page URL

θ₁ 45° Incidence

θ₂ 28.1° Refraction

R 0.050 Reflectance (unpol.)

θB 56.3° Brewster angle

θc none Critical angle

E Unpolarized Polarization

⚡ Experiments

🎚️ Parameters

Incident medium n₁

Medium 2 index n₂ (real part)

Medium 2 extinction κ₂ (ñ₂ = n₂ + iκ₂)

Angle of incidence θ₁

Vacuum wavelength λ₀

Indices are held fixed when λ₀ changes (no dispersion). λ₀ sets lengths only: medium wavelengths, wavefront spacing and decay depths.

🔆 Incident polarization

Unpolarized s (TE) p (TM)

👁️ Diagram layers

Electric-field samples

Wavefronts (crests of Re E)

Interface geometry and fields lossless

Incident (P = 1)
Reflected (width ∝ R)
Transmitted (width ∝ T)
s field (E along y): ⊙ out of / ⊗ into screen
p field: arrow in the x–z plane

Ray widths and labels give the power fraction for the selected polarization (unpolarized = average of s and p). Wavefronts are crests of Re{E exp[i(k·r − ωt)]}, spaced λ₀/n in each medium; field samples come from the same boundary solution. The scale bar is one vacuum wavelength; the depth axis of medium 2 is magnified (and labelled) when the decay length is much shorter than λ₀.

R and T versus angle

R_s
T_s (dashed)
R_p
T_p (dashed)
R unpolarized (dotted)

Power fractions (normal energy flux), 0–1. Drag across the plot to set θ₁. Markers: θ_B (Brewster or minimum-R_p angle) and θ_c (critical angle).

Reflection phase versus angle

arg r_s
arg r_p (dashed)

Phase of the reflected field relative to the incident field at z = 0, in degrees, for exp(−iωt) and the s/p bases defined below (r_p = −r_s at normal incidence). Gaps mark ±180° wraps.

Transmitted field versus depth propagating

Time-averaged |E_t(z)|² / |E₀|² in medium 2 at x = 0 (not a power fraction). It is flat for a lossless propagating wave and falls as exp(−2 Im k_z2 z) for an evanescent or absorbed wave; the marker shows the 1/e field depth.

📈 Coefficients at θ₁

R_s (s reflection)

R_p (p reflection)

T_s (s transmission)

T_p (p transmission)

R + T: s = 1.000, p = 1.000

Complex field amplitudes (|·| ∠ phase)

r_s—

r_p—

t_s—

t_p—

Frequency f = c/λ₀ (same in both media)—

λ₁ = λ₀/n₁—

λ₂ = λ₀/n₂ (phase-front spacing in medium 2)—

k_z2/k₀ = ñ₂ cos θ₂—

1/e field depth in medium 2— (propagating)

At the Brewster angle: R_p = 0, reflected light is purely s-polarized

Total internal reflection: R = 1, T = 0, but an evanescent field (nonzero E) decays into medium 2

💡 How to use

Experiments load a configuration and say what you should observe; the note under the buttons states the expected result.
n₁, n₂, κ₂: medium 1 is lossless; medium 2 may absorb (κ₂ > 0). Glass ≈ 1.5, water ≈ 1.33, diamond ≈ 2.42; gold at 633 nm ≈ 0.20 + 3.09i.
θ₁: angle from the surface normal. Every slider has a number box for exact values; you can also drag across either angle plot.
λ₀: vacuum wavelength. It changes lengths (λ₀/n, decay depths), not R or T, because the indices are held constant.
Polarization: s or p makes the ray widths and the depth plot show that polarization; unpolarized uses the average of s and p powers.
Animate ωt advances the optical phase (slowed by about 10¹⁵); Step advances it by π/8 while paused.
Copy link / JSON / CSV / PNG below the readouts reproduce or export the current state, the angle sweep, and the figures.

Learn with this tool

Learning objectives

Derive r and t for s and p light from the boundary conditions on tangential E and H, and explain why T ≠ |t|².
Predict and measure the Brewster angle, the critical angle, and the angle-dependent phase shift of total internal reflection.
Relate an evanescent or absorbed field's decay depth to Im k_z2, and estimate a metal's reflectance from ñ = n + iκ.

Prerequisites

Plane waves, complex phasors and the exp[i(k·r − ωt)] convention (electromagnetic waves and energy flow).
Snell's law and refractive index; linear polarization (polarization).
Maxwell boundary conditions at a charge-free, current-free interface.

Model

A monochromatic plane wave in a lossless medium 1 (z < 0) meets a flat interface z = 0 with medium 2 (z > 0). Both media are linear, isotropic, homogeneous and nonmagnetic (μ = μ₀). Medium 2 may absorb: ñ₂ = n₂ + iκ₂. The tangential wavevector k_x = k₀ n₁ sin θ₁ is the same for all three waves (Snell's law), and k_zj = k₀ √(ñ_j² − (k_x/k₀)²) with Im k_z ≥ 0. Continuity of tangential E and H gives

s-polarized amplitudes

r_s=\frac{k_{z1}-k_{z2}}{k_{z1}+k_{z2}},\qquad t_s=\frac{2k_{z1}}{k_{z1}+k_{z2}}

p-polarized amplitudes

r_p=\frac{\tilde n_2^2 k_{z1}-n_1^2 k_{z2}}{\tilde n_2^2 k_{z1}+n_1^2 k_{z2}}

t_p=\frac{2n_1\tilde n_2 k_{z1}}{\tilde n_2^2 k_{z1}+n_1^2 k_{z2}}

Power coefficients

R=|r|^2,\qquad T_s=\frac{\operatorname{Re}k_{z2}}{k_{z1}}|t_s|^2

T_p=\frac{\operatorname{Re}\!\left(k_{z2}\tilde n_2^*/\tilde n_2\right)}{k_{z1}}|t_p|^2

λ₀, k₀: vacuum wavelength, k₀ = 2π/λ₀; the frequency f = c/λ₀ is the same in both media
n₁, ñ₂: refractive indices; ñ₂ = n₂ + iκ₂ with κ₂ ≥ 0 the extinction coefficient
θ₁, θ₂: angle of incidence; angle of the transmitted phase fronts, tan θ₂ = k_x/Re k_z2
s (TE): E along ŷ, normal to the plane of incidence (x–z)
p (TM): H along ŷ; each wave's E is along ê_p = ŷ × k/(ñk₀), so r_p = −r_s at θ₁ = 0
r, t: complex ratios of reflected or transmitted to incident E amplitude at z = 0
R, T: ratios of time-averaged normal energy flux (Poynting vector along z)
d: 1/e depth of |E| in medium 2, d = 1/Im k_z2; intensity falls by 1/e at d/2

Assumptions and validity: infinite plane waves (a real beam's width must be much larger than λ₀), a perfectly flat and abrupt interface, a semi-infinite medium 2 (no back surface), local linear response, and indices that do not change with λ₀ in this tool. R + T = 1 holds exactly here because medium 1 is lossless: T is the power that enters medium 2, and in an absorber all of it is dissipated.

Derivation from the boundary conditions

s polarization. Write E_y = e^ik_xx[e^ik_z1z + r e^−ik_z1z] for z < 0 and t e^{ik_xx + ik_z2z} for z > 0 (time factor e^−iωt dropped). Faraday's law, ∇ × E = iωμ₀H, gives H_x = −(k_z/ωμ₀)E_y for each plane wave (k_z → −k_z1 for the reflected wave). Continuity of E_y and H_x at z = 0 gives 1 + r = t and k_z1(1 − r) = k_z2 t, hence r_s and t_s above.

p polarization. Now H_y is the same for each wave's basis, H = ñE/η₀ ŷ, and the tangential electric field is E_x = (k_z/ñk₀)E. Continuity of H_y: n₁(1 + r) = ñ₂ t. Continuity of E_x: (k_z1/n₁)(1 − r) = (k_z2/ñ₂) t. Solving gives r_p and t_p.

Power. The normal flux of one plane wave is S_z = ½ Re(E × H*)·ẑ. For s this is |E|² Re(k_z)/(2ωμ₀); for p it is |E|² Re(k_z ñ*/ñ)/(2ωμ₀). Dividing by the incident flux gives T. The ratio of the field amplitudes squared, |t|², ignores the change of impedance and of beam cross-section, which is why T ≠ |t|² (air→glass at normal incidence: |t|² = 0.64, T = 0.96).

Special cases. r_p = 0 when ñ₂²k_z1 = n₁²k_z2; for real indices this is tan θ_B = n₂/n₁. When n₁ sin θ₁ > n₂ (real), k_z2 = iκ is imaginary, |r| = 1 and Re k_z2 = 0, so T = 0 although t ≠ 0: the field is evanescent. Then r_s = e^iφ_s with φ_s = −2 arctan[√(n₁² sin²θ₁ − n₂²)/(n₁ cos θ₁)].

Exercise 1: field ratio or power ratio?

Air (n₁ = 1) to glass (n₂ = 1.5) at θ₁ = 0. Predict |t_s|, |t_s|² and T_s.
Click Normal incidence.
Read |t_s|, R_s and T_s from the readouts.
Why is T_s larger than |t_s|², and does R + T add to 1?

Show answer

|t| = 2/(1 + 1.5) = 0.800, so |t|² = 0.640, but T = (n₂/n₁)|t|² = 1.5 × 0.64 = 0.960 and R = 0.2² = 0.040. The flux factor n₂ cos θ₂/(n₁ cos θ₁) accounts for the higher impedance ratio (intensity ∝ n|E|²) and, at oblique incidence, the change in beam cross-section. R + T = 1.000.

Exercise 2: find Brewster's angle by measurement

For air → water (n₂ = 1.33), predict θ_B and R_s there. Is θ₁ + θ₂ special?
Set n₂ = 1.33, κ₂ = 0, select p, and drag across the R/T plot until R_p is smallest.
Record θ₁, θ₂, R_p and R_s.
Why does a polarizer with a vertical transmission axis cut glare from a lake?

Show answer

θ_B = arctan 1.33 = 53.06°, θ₂ = 36.94°, so θ₁ + θ₂ = 90°: the reflected direction is along the dipole axis of the p-polarized currents in the water, so they cannot radiate into it. R_p = 0, R_s ≈ 0.077. Reflected glare near θ_B is mostly s (horizontal) polarized, which a vertical-axis polarizer blocks.

Exercise 3 (limiting cases): grazing incidence and the critical angle

Air → glass: what do R_s and R_p tend to as θ₁ → 90°? Glass → air: what happens to the 1/e depth d as θ₁ → θ_c from above?
Type θ₁ = 89.9 in the number box; then load Critical angle and raise θ₁ in 0.1° steps.
Record R at 89.9°; record d at θ_c + 0.1°, + 1° and + 10°.
Explain both limits from k_z1 and k_z2.

Show answer

As θ₁ → 90°, k_z1 → 0, so r_s → −1 and r_p → −1 (in this basis): every interface becomes a perfect mirror at grazing incidence (R ≈ 0.99 at 89.9°). At θ_c = 41.81° the normal wavevector k_z2 passes through zero, so d = 1/Im k_z2 diverges as θ₁ → θ_c⁺: d ≈ 2.55 λ₀ at θ_c + 0.1°, 0.80 λ₀ at + 1° and 0.25 λ₀ at + 10°, falling to about 0.2 λ₀ at 60°. Just below θ_c (the preset's 41.81° is 0.0003° short of it) the transmitted wave still propagates, grazing the surface at θ₂ ≈ 89.8°.

Exercise 4: why metals shine

Estimate R at normal incidence for ñ₂ = 0.20 + 3.09i, and how deep the light gets.
Click Metal mirror, then sweep θ₁.
Read R at 0°, the depth d, and the angle and value of the R_p minimum.
Why does R_p not reach zero?

Show answer

R = [(n − 1)² + κ²]/[(n + 1)² + κ²] = 10.19/10.98 = 0.927. The 7.3 % that enters is absorbed within d = λ₀/(2πκ) ≈ 33 nm (intensity depth ≈ 16 nm) at 633 nm. With complex ñ₂ the condition ñ₂²k_z1 = n₁²k_z2 has no real-angle solution, so R_p only dips to about 0.87 near 70.6° (the pseudo-Brewster angle), which ellipsometry uses to measure n and κ.

Worked example: evanescent field and phase in a glass prism

A 633 nm HeNe beam inside glass (n₁ = 1.50) meets a glass–air face at θ₁ = 60°. Find θ_c, the reflection phases, and the evanescent depth.

θ_c = arcsin(1.00/1.50) = 41.81°, so 60° gives TIR and R_s = R_p = 1.
k_x/k₀ = 1.5 sin 60° = 1.299; Im k_z2/k₀ = √(1.299² − 1) = √0.6875 = 0.829.
d = λ₀/(2π × 0.829) = 633 nm/5.21 = 121.5 nm for |E|; intensity falls by 1/e over 60.8 nm.
φ_s = −2 arctan(0.829/(1.5 cos 60°)) = −2 arctan(1.106) = −95.7°; φ_p = −2 arctan(1.5² × 0.829/(1.5 cos 60°)) = −136.2°.
|t_s| = 2k_z1/|k_z1 + k_z2| = 1.5/√(0.75² + 0.829²) = 1.34: the field just outside the glass is stronger than the incident field, yet it carries no normal power.

Load Glass→air TIR to check each number in the readouts and on the phase and depth plots. Bringing a second prism within about d of the face frustrates the TIR and lets power tunnel across (a beam-splitter cube uses this).

When the model fails

Finite beams: near θ_c and in TIR a real beam is displaced along the surface (Goos–Hänchen shift, of order d) and the plane-wave R(θ) is smeared over the beam's angular spread.
Thin layers and coatings: a film thinner than the decay length, or any layer with two surfaces, needs multiple-reflection (transfer-matrix) analysis: see thin films. A metal film of ~30 nm transmits noticeably.
Dispersion: real indices depend on λ₀; this tool holds them fixed, so the λ₀ slider does not show colour-dependent reflectance.
Rough, graded or anisotropic interfaces: scattering, index gradients and birefringent crystals change both amplitudes and polarization eigenmodes.
Nonlocal, magnetic or nonlinear media: surface plasmons at structured metals, μ ≠ μ₀ metamaterials, or intense fields are outside this linear, local model.

References

E. Hecht, Optics, 5th ed., Pearson (2017), §4.6–4.7.
M. Born and E. Wolf, Principles of Optics, 7th ed., Cambridge (1999), §1.5 and §14.2 (metals).
J. D. Jackson, Classical Electrodynamics, 3rd ed., Wiley (1999), §7.3–7.4.
B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed., Wiley (2019), §6.2.
P. B. Johnson and R. W. Christy, "Optical constants of the noble metals", Phys. Rev. B 6, 4370 (1972) (gold index used in the metal preset).

📚 Physics background

📐 Snell's law is phase matching

All three waves must oscillate together along the interface, so they share k_x. With |k| = n k₀ this gives n₁ sin θ₁ = n₂ sin θ₂. The frequency is unchanged; only the wavelength changes, λ = λ₀/n.

🎯 Brewster and critical angles

Brewster angle

\theta_B=\arctan\!\left(\frac{n_2}{n_1}\right)

At this angle Rₚ = 0 and θ₁ + θ₂ = 90°.

Critical angle

\theta_c=\arcsin\!\left(\frac{n_2}{n_1}\right)

Total internal reflection requires n₁ > n₂.

Brewster windows in laser tubes pass p light without loss, so the laser tends to oscillate in p. Beyond θ_c the s and p reflection phases differ; a Fresnel rhomb uses two TIR bounces to make a 90° retarder.

🪞 Absorbing media and metals

With ñ = n + iκ, k_z2 is complex even below θ_c. The transmitted wave is inhomogeneous: its phase fronts tilt at θ₂ while its amplitude decays along z. Large κ makes |r| close to 1, which is why polished metals are good mirrors, and R_p shows only a shallow minimum at the principal (pseudo-Brewster) angle.

🔬 What this tool does and does not simulate

Every number and every drawn field comes from the exact plane-wave solution above. Ray widths are power fractions, but ray length and beam width are schematic. The animation shows the real phase evolution slowed by about 10¹⁵; it is not a time-domain simulation of a pulse. Multiple interfaces, beams, scattering and dispersion are not modelled.