Brewster's Angle Calculator
This calculator works out Brewster's angle, the specific angle of incidence at which light reflecting off a boundary between two transparent media becomes completely polarised, useful for anyone working with optics, photography filters, laser design or polarised sunglasses. Brewster's angle matters because it identifies exactly where a polarising filter has the strongest effect, and explains why glare from water, glass and wet roads fades when you view those surfaces from a particular angle. You enter the refractive index of the incident medium (n1) and the transmitted medium (n2), either by typing your own values or choosing a preset such as air to glass, air to water, air to diamond or water to glass. The calculator instantly returns Brewster's angle in degrees and radians, plus the refracted angle at that same point worked out from Snell's Law, along with a breakdown showing the ratio n2/n1, the arctan step, and confirmation that incidence and refraction sum to exactly 90 degrees. A verdict line below the results explains whether light is passing into a denser or less dense medium and confirms the reflected ray is fully s-polarised at that angle. Try the presets first to see familiar examples, then switch to custom values for your own materials. Refractive indices vary with wavelength, so for precise work use the index matching your actual light source rather than a generic figure.
1. Media Preset
2. About the Inputs
The refractive index (n) of a medium is the ratio of the speed of light in a vacuum to the speed of light in that medium. Air is approximately 1.000, water is 1.333, and common glass ranges from 1.45 to 1.9 depending on type.
Brewster's angle is measured from the normal (perpendicular) to the surface, not from the surface itself.
At Brewster's angle, the refracted and reflected rays are at exactly 90 degrees to each other.
Calculation Breakdown
Geometry at Brewster's Angle
What is Brewster's Angle?
Brewster's angle (symbol theta_B, also called the polarisation angle) is named after Scottish physicist Sir David Brewster, who discovered it experimentally in 1815. It is the angle of incidence at which light travelling between two dielectric (non-conducting) media produces a reflected ray that is completely polarised in the s-direction (perpendicular to the plane of incidence). At this angle, no p-polarised (parallel) component appears in the reflected ray; it is entirely transmitted into the second medium.
The physical reason is that p-polarised oscillating dipoles in the second medium, when aligned in the direction of the reflected beam, cannot radiate energy in that direction. This only occurs when the reflected and refracted rays are exactly 90 degrees apart, which is the geometric condition that Brewster's Law captures.
The Formula
Brewster's Law states:
tan(θB) = n2 / n1
where n1 is the refractive index of the incident medium (the medium the light is coming from) and n2 is the refractive index of the transmitted medium (the medium the light is entering). Rearranging gives:
θB = arctan(n2 / n1)
At Brewster's angle, the refracted angle satisfies Snell's Law (n1 sin(θB) = n2 sin(θr)), and the two angles sum to exactly 90 degrees: theta_B + theta_r = 90 degrees.
Worked Example
For light passing from air (n1 = 1.000) into ordinary crown (window) glass (n2 = 1.500):
- Ratio: n2 / n1 = 1.500 / 1.000 = 1.500
- Brewster's angle: arctan(1.500) = 56.31 degrees
- Refracted angle (Snell's Law): arcsin(sin(56.31) / 1.500) = 33.69 degrees
- Check: 56.31 + 33.69 = 90.00 degrees (confirmed)
This matches the calculator's default output for the air-to-glass preset.
Common Brewster Angles
| Media (n1 to n2) | n1 | n2 | Brewster's Angle |
|---|---|---|---|
| Air to water | 1.000 | 1.333 | 53.12 deg |
| Air to glass (n=1.5) | 1.000 | 1.500 | 56.31 deg |
| Air to fused silica | 1.000 | 1.460 | 55.60 deg |
| Air to diamond | 1.000 | 2.417 | 67.52 deg |
| Water to glass | 1.333 | 1.500 | 48.37 deg |
| Glass to air (internal) | 1.500 | 1.000 | 33.69 deg |
Note that the Brewster angle for glass-to-air is exactly 90 minus the air-to-glass Brewster angle (33.69 + 56.31 = 90). This is because n2/n1 is the reciprocal in each direction, and arctan(x) + arctan(1/x) = 90 degrees for any positive x.
Practical Applications
- Polarising filters (photography): A circular polariser on a camera lens removes glare from non-metallic surfaces (water, glass, wet roads) most effectively when the camera is aimed at about Brewster's angle to the surface.
- Polarised sunglasses: Horizontal glare reflected from roads and water is s-polarised. Polarised lenses block s-polarised light, cutting glare without reducing overall brightness as much as tinted lenses.
- Brewster windows in lasers: Gas laser tubes often use windows set at Brewster's angle so that p-polarised light passes through with zero reflection loss, improving efficiency and producing a polarised beam.
- Ellipsometry: Thin film thickness and optical properties are measured by analysing reflected polarised light near Brewster's angle.
- Anti-reflection coatings: Thin film coatings exploit Brewster's angle effects to minimise reflections in camera lenses and glasses.
Related Calculators
- Maths and Stats Calculators
- Snell's Law Calculator: calculate the angle of refraction at any interface.
- Angle of Refraction Calculator: find refracted angle from incidence angle and refractive indices.
- Refractive Index Calculator: calculate refractive index from speed or angle data.
- Angle of Incidence Calculator: work with incident rays and surface normals.
Method and sources: Formula: theta_B = arctan(n2/n1), from Brewster's Law (1815). Refractive index values from the NIST Handbook of Optical Constants and CRC Handbook of Chemistry and Physics. The relationship theta_B + theta_r = 90 deg follows from Snell's Law and is the geometric basis of complete polarisation at this angle.
This calculator assumes monochromatic light and non-absorbing (dielectric) media. Refractive indices are wavelength-dependent; for precise work use n values at your specific wavelength. Metals and other absorbing media require a complex refractive index treatment and do not produce a simple Brewster angle.