56.3099
deg
0.982794
rad
33.6901
deg
14.7929
%
1.5
56.3099
deg
0.982794
rad
33.6901
deg
14.7929
%
1.5
The Brewster’s Angle Calculator computes the polarizing angle at which light reflected from a surface is completely p-polarized—meaning the reflected beam contains only the s-polarization component while the p-component is entirely transmitted. This remarkable phenomenon, discovered by Sir David Brewster in 1815, is a cornerstone of polarization optics.
When unpolarized light strikes a dielectric interface (such as air–glass), both reflected and transmitted beams are partially polarized. At one specific angle of incidence, known as Brewster’s angle θB, the reflected and refracted rays become perpendicular to each other. Under this geometric condition, the oscillating dipoles in the second medium cannot radiate in the direction of the reflected beam for the p-polarization, so the reflected light is purely s-polarized.
Brewster’s angle is given by the elegant formula $$\theta_B = \arctan\!\left(\frac{n_2}{n_1}\right)$$ where n₁ is the refractive index of the incident medium and n₂ is that of the transmitting medium. For air–glass (n₂ = 1.5), θB ≈ 56.3°.
Brewster’s angle has numerous practical applications. Polarizing filters for cameras exploit Brewster reflection to reduce glare from water, glass, and roads. Laser cavities use Brewster windows—optical flats tilted at θB—to ensure that the intracavity beam is linearly polarized with zero reflection loss for the p-polarization. In telecommunications, Brewster-angle fiber couplers minimize back-reflections.
Photographers rely on Brewster’s angle when using circular polarizing filters to cut reflections from shop windows or enhance sky contrast. The effect is strongest when the camera axis is roughly perpendicular to the sun direction, because the reflected skylight is partially polarized at Brewster’s angle off atmospheric molecules.
This calculator provides the Brewster angle in radians and degrees, the corresponding refraction angle, and the s-polarization reflectance at that angle (which is the only nonzero reflectance component at θB).
Brewster’s angle is derived from Snell’s law combined with the condition that reflected and refracted rays are perpendicular:
$$n_1 \sin\theta_B = n_2 \sin\theta_r \quad\text{and}\quad \theta_B + \theta_r = 90^\circ$$
Substituting θr = 90° − θB:
$$n_1 \sin\theta_B = n_2 \cos\theta_B \implies \tan\theta_B = \frac{n_2}{n_1}$$
$$\theta_B = \arctan\!\left(\frac{n_2}{n_1}\right)$$
At this angle the Fresnel reflection coefficient for p-polarization is exactly zero. The s-polarization reflectance is computed using the Fresnel equation:
$$R_s = \left(\frac{n_1\cos\theta_B - n_2\cos\theta_r}{n_1\cos\theta_B + n_2\cos\theta_r}\right)^{\!2}$$
The Brewster angle tells you the exact incidence angle at which reflected light is 100% s-polarized. The refraction angle at Brewster’s angle is always the complement (90° − θB). The s-polarization reflectance shows how much of the s-component is reflected; for air–glass this is about 15%, meaning 85% of the s-polarized light is transmitted along with 100% of the p-polarized light.
Inputs
Results
Light traveling in air (n₁ = 1.0) strikes crown glass (n₂ = 1.5). Brewster’s angle is 56.3°. The reflected beam is fully s-polarized; about 14.9% of the s-component is reflected.
Inputs
Results
Sunlight reflecting off a lake surface. At θ<sub>B</sub> ≈ 53.1° the glare is completely p-polarization-free. A polarizing filter aligned to block s-polarization eliminates this glare entirely.
Brewster’s angle (also called the polarizing angle) is the angle of incidence at which light reflected from a dielectric surface is completely linearly polarized in the s-direction (perpendicular to the plane of incidence). At this angle, the p-polarization reflection coefficient drops to zero.
At Brewster’s angle the reflected and refracted rays are exactly 90° apart. The reflected p-component would require the induced dipoles in the refracting medium to radiate along their oscillation axis, which is physically impossible. Therefore, no p-polarized light can be reflected.
Not in the classical sense. Metals have complex refractive indices (with an imaginary absorption component), so the p-polarization reflectance never truly reaches zero. However, there is a pseudo-Brewster angle where the p-reflectance reaches a minimum. True Brewster’s angle applies strictly to transparent dielectric materials.
When light reflects off surfaces like water or glass near Brewster’s angle, the reflected light is predominantly s-polarized. A polarizing filter oriented to block s-polarization can therefore eliminate most of this reflected glare, which is why photographers use circular polarizers to cut reflections from windows and water surfaces.
Both angles are derived from Snell’s law but describe different phenomena. The critical angle applies only when going from a denser to a less dense medium (n₁ > n₂) and marks the onset of total internal reflection. Brewster’s angle exists for either direction and marks complete polarization of reflected light. For a glass–air interface, the Brewster angle from the glass side and the critical angle are related but not equal.
Yes. When n₂ > n₁ (light entering a denser medium), θB = arctan(n₂/n₁) > 45°. For air–glass, θB ≈ 56.3°. When n₂ < n₁ (light leaving the denser medium), θB < 45°. In both cases, θB + θr = 90°.
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