Brewster Angle and Polarization Calculator

Find the angle at which reflected light is fully polarized, plus how much light reflects at any angle.

What it does

Light that bounces off a surface comes back partly polarized, and at one particular angle it comes back completely polarized. That angle is the Brewster angle, and it is the reason a polarizing film in sunglasses removes the glare from a wet road while leaving the rest of the scene alone.

OpenStax University Physics sets out the law. The reflected light is completely polarized at an angle of reflection given by the tangent of that angle equal to the ratio of the two refractive indices, and that angle is the Brewster angle, named after the nineteenth-century Scottish physicist who found it. The same section gives the recipe for the everyday case: light reflected at the Brewster angle could be completely blocked by a good polarizing filter held with its axis vertical.

The geometry behind it is simple. At the Brewster angle the reflected ray and the refracted ray are at ninety degrees to each other, which is why the page reports the angle of refraction next to the angle of incidence. The physics is a little subtler, and it is the reason the reflected light comes back polarized in one direction only: the vibration of the light in the plane of incidence does not match the vibration of the light in the second medium, so that component is not reflected at all.

This page computes the Fresnel reflectances for both components of the light at any angle of incidence, the angle of refraction, the total reflected fraction for unpolarized light and how strongly the reflected beam is polarized. It also reports the Brewster angle for the pair of media you choose, so you can see at a glance how far your incidence angle sits from it.

The model is for a smooth, flat interface between two smooth, non-absorbing media, which is why it does not describe a reflection off a scratched or dusty surface. It also assumes the light is incident from the denser medium onto the less dense one, or the other way round, in either case, since the law is written for the direction of travel and the page follows the ray.

How to use

1. Pick the medium the light starts in and the medium it enters.
2. Enter the angle of incidence, measured from the normal to the surface.
3. Read the Brewster angle for that pair of media, which is where the reflected light is fully polarized.
4. Read the angle of refraction, which follows from Snell's law at the angle you entered.
5. Compare the two reflected fractions. The parallel component always reflects less than the perpendicular one, and it drops to zero at the Brewster angle.
6. Read the polarization of the reflected beam. The closer your angle is to the Brewster angle, the more completely polarized the reflection is.

Formula

theta B

Brewster angle of incidence, in degrees.

n1 and n2

Refractive indices of the first and second medium.

theta i

Angle of incidence, measured from the normal.

theta t

Angle of refraction, measured from the normal.

Rs and Rp

Reflected fractions of the perpendicular and the parallel component.

The subscript s stands for the component perpendicular to the plane of incidence and p for the component parallel to it. At the Brewster angle the reflected and refracted rays are at right angles, so the two angles add to 90 degrees, and Rp is exactly zero there. The degree of polarization of the reflected beam assumes the incident light is unpolarized, which is what ordinary light is.

Worked example

Reference table

Limits

• The calculation is for a smooth flat interface between two smooth, non-absorbing media, so it does not describe rough or dusty surfaces.
• It assumes the incident light is unpolarized, which is the usual case but not the case for laser light or light that has already been through a polarizer.
• There is no absorption in either medium, so the transmitance is simply one minus the reflectance.
• Only a single interface is considered, so a coated lens or a stack of surfaces is out of scope.

Common mistakes

• Measuring the angle from the surface instead of from the normal. The Brewster angle for air to glass is 56.66 degrees from the normal, which is 33.34 degrees from the surface.
• Using the refraction angle as the Brewster angle. The Brewster angle is the angle of incidence.
• Expecting the reflected light to be polarized in the plane of incidence. The parallel component is the one that disappears, so what is left is polarized across it.
• Applying the law to light going from a dense medium to a less dense one and expecting the angle to disappear. The law still holds, but the angle is smaller and total internal reflection can intervene first.
• Forgetting that the polarization is a property of the reflected beam only. The refracted light is only partly polarized.

Try it yourself

Quick checks. Answers sit tucked under each question.

Questions

Sources

• OpenStax University Physics Volume 3, 1.7: polarization, Malus's law and Brewster's law
• Physics LibreTexts: polarization, Brewster's angle and the polarising glossary
• OpenStax College Physics 2e, 27.8: polarization by reflection

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