Brewster's Angle Calculator with Fresnel Reflectance
Enter the refractive indices of two optical media (or choose from the material presets) to find the Brewster angle - the angle of incidence at which reflected light is completely p-polarized. You also get the refracted (transmitted) ray angle via Snell's law, the Fresnel reflectance coefficients Rs and Rp at the Brewster angle, and the degree of polarization. A chart shows how Rs and Rp vary across all angles of incidence from 0 to 90 degrees.
What is Brewster's angle?
Brewster's angle (also called the polarizing angle) is the specific angle of incidence at which light striking an interface between two transparent media produces a reflected beam that is completely linearly polarized in the s-direction (perpendicular to the plane of incidence). It was discovered experimentally by Sir David Brewster in 1815 and can be derived from the Fresnel equations. At this angle, the p-component of the reflected electric field vanishes completely because the reflected and refracted rays are perpendicular - their angle sum equals exactly 90 degrees. Named after the Scottish physicist, the formula is elegant: theta_B = arctan(n2 / n1), where n1 and n2 are the refractive indices of the two media.
How to use this calculator
Select a material preset for each medium or choose Custom and type in any refractive index from 1.000 upward. The calculator instantly shows Brewster's angle, the complementary refraction angle, and the Fresnel s-reflectance Rs at that angle (Rp is always exactly 0 at the Brewster angle by definition). Toggle 'Also evaluate at a specific angle' to enter any angle of incidence and see Rs, Rp, and the refraction angle at that specific angle - useful for comparing reflectance well away from the Brewster angle. The Fresnel chart plots Rs and Rp across all angles from 0 to 90 degrees so you can see exactly where the Brewster dip occurs.
The Fresnel equations and s- vs p-polarization
When light hits a surface, it splits into two polarization components. The p-component (parallel, TM mode) oscillates in the plane that contains the incident ray and the surface normal. The s-component (senkrecht, German for perpendicular, TE mode) oscillates perpendicular to that plane. The Fresnel reflectance for each component is: Rs = ((n1*cos(theta_i) - n2*cos(theta_t)) / (n1*cos(theta_i) + n2*cos(theta_t)))^2 and Rp = ((n2*cos(theta_i) - n1*cos(theta_t)) / (n2*cos(theta_i) + n1*cos(theta_t)))^2, where theta_t is the refracted angle from Snell's law: n1*sin(theta_i) = n2*sin(theta_t). At normal incidence (0 degrees), Rs = Rp, both equal ((n1 - n2) / (n1 + n2))^2. As the incidence angle increases, Rp decreases, reaches zero at Brewster's angle, then rises again toward 100% at grazing incidence.
Real-world applications of Brewster's angle
Polarizing sunglasses and camera filters exploit Brewster's angle to reduce glare. Light reflected from horizontal surfaces (water, wet roads, snow) is largely s-polarized because those surfaces lie roughly at or near the Brewster angle for sunlight. A polarizer oriented to block s-polarized light removes most of that glare. In laser optics, Brewster windows are flat glass plates tilted at Brewster's angle in a laser cavity; the p-polarized beam passes through with zero reflection loss, which forces the laser to emit purely p-polarized light and minimizes cavity losses. Optical coatings and anti-reflection layers are designed knowing the reflectance behavior from the Fresnel equations across the full angular range.
Common optical materials and their Brewster angles (air interface)
| Material | Refractive index n | Brewster's angle (deg) |
|---|---|---|
| Ice | 1.309 | 52.6 |
| Water | 1.333 | 53.1 |
| Fused quartz | 1.544 | 57.1 |
| Crown glass | 1.500 | 56.3 |
| Flint glass | 1.620 | 58.3 |
| Sapphire | 1.770 | 60.6 |
| Diamond | 2.417 | 67.5 |
Brewster angle from air (n1 = 1.000) into each material using the standard formula theta_B = arctan(n2).
Frequently asked questions
What is Brewster's angle formula?
Brewster's angle is theta_B = arctan(n2 / n1), where n1 is the refractive index of the incident medium and n2 is the refractive index of the reflecting medium. For light going from air (n1 = 1.000) into crown glass (n2 = 1.500), theta_B = arctan(1.500 / 1.000) = arctan(1.5) which is approximately 56.3 degrees.
Why is Rp exactly zero at Brewster's angle?
At Brewster's angle, the reflected and refracted rays are perpendicular to each other (theta_B + theta_t = 90 degrees). The p-component of the electric field in the reflected beam would have to oscillate along the direction of propagation, which is impossible for an electromagnetic transverse wave. The numerator of the Rp Fresnel coefficient (n2*cos(theta_i) - n1*cos(theta_t)) equals zero at this geometry, so the p-reflectance vanishes completely.
Does Brewster's angle apply when going from a denser medium to a less dense one?
Yes. You can have a Brewster angle when light travels from glass into air or from water into air, not just from air into glass. The formula is symmetric in structure, so theta_B = arctan(n2 / n1) works in both directions. However, when light travels from a denser medium into a less dense one, you also need to check for total internal reflection: if the angle of incidence exceeds the critical angle theta_c = arcsin(n2 / n1), all light is reflected and Fresnel partial reflection no longer applies.
What is the degree of polarization at Brewster's angle?
For unpolarized incident light, the degree of polarization (DOP) of the reflected beam at Brewster's angle is 100%, because all the reflected light is s-polarized (Rp = 0). For the transmitted beam the situation is different: both polarizations are transmitted, but the beam is partially polarized because the s-component is depleted more by the reflection. Stacking many glass plates at Brewster's angle can produce a highly polarized transmitted beam, which is called a pile-of-plates polarizer.
Why is Brewster's angle always greater than 45 degrees when going from air to glass?
Any glass or optical crystal has a refractive index greater than 1.0 (the index of air), so n2 / n1 > 1, which means arctan(n2 / n1) > arctan(1) = 45 degrees. The more optically dense the second medium, the larger the ratio n2 / n1, and the closer Brewster's angle approaches 90 degrees (grazing incidence). For light travelling the other way, from glass to air, n2 / n1 < 1 so Brewster's angle is less than 45 degrees.
How do polarizing filters work in photography?
A circular polarizing (CPL) filter mounted on a camera lens blocks light oscillating in one direction. When the camera is pointed so that specular reflections from glass, water or foliage arrive near Brewster's angle, those reflections are mostly s-polarized. Rotating the filter until its polarizing axis is perpendicular to the s-direction attenuates the glare while passing the scene light and sky light, which is partially polarized at 90 degrees from the sun. The result is richer colors, deeper blue skies and clearer water surfaces.