About

When unpolarized light strikes a dielectric interface at Brewster's angle θ_B, the reflected beam becomes perfectly linearly polarized. The p-polarized component (electric field in the plane of incidence) has zero reflectance at this angle, leaving only s-polarized light in the reflection. Miscalculating this angle in optical system design leads to stray reflections, degraded extinction ratios in polarizers, and wasted energy in laser cavities. This calculator computes θ_B = arctan(n₂ ÷ n₁) for any pair of media, then derives the refracted angle via Snell's law and the full Fresnel reflectance curves for both polarization states.

The tool assumes non-magnetic, lossless dielectrics at optical frequencies. For absorbing media (metals, doped semiconductors), the pseudo-Brewster angle differs and requires the complex refractive index. Refractive indices listed here are for the sodium D-line (λ = 589.3 nm); chromatic dispersion shifts θ_B by up to 0.5° across the visible spectrum. Pro tip: at Brewster's angle, the reflected and refracted rays are exactly 90° apart - a useful geometric check.

Formulas

Brewster's angle is the incidence angle at which the reflected light is completely s-polarized. It occurs when the reflected and refracted rays are perpendicular (θ_r + θ_t = 90°).

θ_B = arctan(n₂n₁)

The refracted (transmitted) angle follows from Snell's law:

n₁ sin θ_i = n₂ sin θ_t

Fresnel reflectance coefficients for s- and p-polarization:

R_s = |n₁ cos θ_i − n₂ cos θ_tn₁ cos θ_i + n₂ cos θ_t|²

R_p = |n₂ cos θ_i − n₁ cos θ_tn₂ cos θ_i + n₁ cos θ_t|²

Where n₁ = refractive index of incident medium, n₂ = refractive index of transmitting medium, θ_i = angle of incidence, θ_t = angle of refraction, R_s = reflectance of s-polarized component, R_p = reflectance of p-polarized component. At θ_B, R_p = 0 exactly.

Reference Data

Material	Refractive Index (n)	Brewster's Angle from Air (°)	Wavelength	Common Use
Vacuum	1.0000	-	All	Reference standard
Air (STP)	1.0003	-	589 nm	Ambient medium
Water	1.3330	53.12	589 nm	Aquatic optics, photography
Ice	1.3090	52.61	589 nm	Atmospheric optics
Fused Silica (SiO₂)	1.4585	55.56	589 nm	Laser windows, fiber optics
Crown Glass (BK7)	1.5168	56.60	589 nm	Lenses, prisms
Flint Glass (SF11)	1.7847	60.71	589 nm	Dispersive optics
Polycarbonate	1.5860	57.76	589 nm	Eyewear, optical discs
PMMA (Acrylic)	1.4914	56.16	589 nm	Display panels, lenses
Sapphire (Al₂O₃)	1.7680	60.50	589 nm	Watch crystals, substrates
Diamond	2.4170	67.52	589 nm	Gemstones, ATR spectroscopy
Cubic Zirconia	2.1700	65.27	589 nm	Gemstone simulant
Silicon (Si)	3.4800	73.97	1550 nm	IR optics, semiconductors
Germanium (Ge)	4.0030	75.97	10600 nm	IR windows, thermal imaging
Zinc Selenide (ZnSe)	2.4030	67.40	10600 nm	CO₂ laser optics
Calcium Fluoride (CaF₂)	1.4340	55.10	589 nm	UV & IR windows
Magnesium Fluoride (MgF₂)	1.3780	54.04	589 nm	Anti-reflection coatings
Barium Titanate (BaTiO₃)	2.4100	67.45	589 nm	Electro-optic modulators
Lithium Niobate (LiNbO₃)	2.2860	66.38	589 nm	Nonlinear optics, modulators
Glycerol	1.4730	55.84	589 nm	Microscopy immersion
Ethanol	1.3610	53.69	589 nm	Solvent, spectroscopy

Frequently Asked Questions

At Brewster's angle, the refracted and reflected rays are perpendicular. The p-polarized component oscillates in the plane of incidence, and a dipole cannot radiate along its own axis. Since the reflected ray direction aligns with the p-polarization oscillation direction of the refracted wave, no p-polarized light can be reflected. The s-polarized component oscillates perpendicular to this plane and has no such geometric constraint, so it reflects normally.

No. Brewster's angle requires light passing from a lower to higher refractive index medium (external reflection, n₂ > n₁) for the standard formula. When n₁ > n₂ (internal reflection), a Brewster angle still exists at arctan(n₂ ÷ n₁), and it is always less than the critical angle for total internal reflection. Both phenomena coexist but at different angles.

Refractive index varies with wavelength (chromatic dispersion). For BK7 glass, n ranges from 1.5308 at 400 nm to 1.5098 at 700 nm, shifting Brewster's angle by approximately 0.35°. For precision applications (laser polarizers, Brewster windows), use the Sellmeier equation coefficients for your exact wavelength rather than the sodium D-line value.

Not directly. Metals have a complex refractive index n = n_r + ik, where k is the extinction coefficient. The p-polarized reflectance reaches a minimum but never zero. This angle is called the pseudo-Brewster angle. For gold at 633 nm, the pseudo-Brewster angle is approximately 75° with residual reflectance around 15%.

At Brewster's angle, θ_r + θ_t = 90°. This is used as a quick geometric verification in lab setups. If you measure the refracted ray and find it perpendicular to the reflected ray, you have confirmed Brewster's condition. This property is also the physical basis: the reflected ray direction coincides with the oscillation axis of the p-polarized dipoles in the refracted medium, which cannot radiate in that direction.

Brewster windows are placed at Brewster's angle at both ends of a gas laser tube. The p-polarized component passes through with zero reflection loss per surface, while the s-polarized component suffers ~15% loss per surface (for glass). After many round trips, the s-component is suppressed entirely, and the laser output is linearly polarized. This avoids anti-reflection coatings that could be damaged by the intracavity beam. The technique is standard in He-Ne, Ar-ion, and CO₂ lasers.