Brewster's Angle Calculator – Polarization Angle
Find the Brewster's angle at which reflected light becomes perfectly polarized for any pair of media by entering their refractive indices.
Enter the refractive indices of the two media to instantly calculate the Brewster's (polarization) angle in degrees.
Brewster's Angle Calculator – Polarization Angle
Find the Brewster's angle at which reflected light becomes perfectly polarized for any pair of media by entering their refractive indices.
About Brewster's Angle
Brewster's angle, also called the polarization angle, is the specific angle of incidence at which light traveling from one medium into another is reflected with perfect linear polarization. When unpolarized light strikes a surface at Brewster's angle, the reflected beam contains only the component of the electric field that oscillates parallel to the surface (s-polarization), while the transmitted (refracted) beam is partially polarized with the complementary component.
The phenomenon was discovered by the Scottish physicist Sir David Brewster in 1815. He found empirically that the angle of polarization depended on the refractive indices of the two media involved, and he stated what is now known as Brewster's Law: the tangent of Brewster's angle θ_B equals the ratio of the second medium's refractive index n₂ to the first medium's refractive index n₁. In formula form: tan(θ_B) = n₂ / n₁, which gives θ_B = arctan(n₂ / n₁).
An important geometrical consequence of Brewster's Law is that at the polarization angle, the reflected and refracted rays are exactly perpendicular to each other — the angle between them is always 90°. This occurs because the oscillating dipoles in the refracting medium that would re-radiate in the direction of the reflected beam are oriented along that direction, so they cannot emit radiation, causing the p-polarized component of the reflected beam to vanish entirely.
Brewster's angle has numerous practical applications across optics and photonics. In laser technology, Brewster windows are flat optical components mounted at Brewster's angle in a laser cavity to allow the intracavity beam to pass through with zero reflection loss while simultaneously generating a linearly polarized output. Polarizing sunglasses exploit the same principle: because glare from horizontal surfaces such as water or roads reflects at or near Brewster's angle for visible light, a vertically oriented polarizing filter blocks most of the reflected glare while transmitting the direct scene light.
In photography, a circular polarizing filter rotates to the orientation that cancels reflections off glass, water, or paint, improving color saturation and reducing haze. In optical fiber communications, connectors polished at an angle close to Brewster's angle for the fiber-air interface reduce back-reflections that could disturb laser sources. Remote sensing and ellipsometry use precise measurements at Brewster's angle to characterize thin film thicknesses and optical properties of surfaces with sub-nanometer accuracy.
For common optical materials, Brewster's angle from air (n₁ ≈ 1.00) is approximately 56° for crown glass (n = 1.52), 53° for water (n = 1.33), and 67° for diamond (n = 2.42). The angle is larger when the second medium has a higher refractive index, because a larger index ratio requires a steeper angle of incidence for the reflected and refracted rays to remain perpendicular.
Brewster's Angle Examples
Common material pairs and their Brewster's angles at visible-light wavelengths.
| Media Pair | Brewster's Angle | Application |
|---|---|---|
| Air (n₁ = 1.00) → Glass (n₂ = 1.50) | 56.31° | Classic optics example. At this angle reflected light from glass is completely polarized. Brewster windows in lasers use this geometry. |
| Air (n₁ = 1.00) → Water (n₂ = 1.33) | 53.06° | Glare on water surfaces is maximally polarized near this angle. Polarized sunglasses block this reflected component. |
| Water (n₁ = 1.33) → Glass (n₂ = 1.50) | 48.44° | Relevant for underwater optics. Polarization angle is lower than air-to-glass because the index contrast is smaller. |
| Air (n₁ = 1.00) → Diamond (n₂ = 2.42) | 67.51° | Diamond's high refractive index produces a steep Brewster angle. Relevant in gemology and high-index optical coatings. |
How to Use the Brewster's Angle Calculator
- Enter the refractive index of the first medium (n₁), the medium in which the incident light travels. For air or vacuum, use 1.00.
- Enter the refractive index of the second medium (n₂), the medium the light is entering. Look up the value in optical data tables for your material.
- Click Calculate. The Brewster's angle is shown in degrees, computed from θ_B = arctan(n₂ / n₁).
- Use the result to orient a Brewster window, select a polarizing filter angle, or set up a reflection-based polarimetry experiment.
- Click Reset to clear both fields and start a new calculation for a different material pair.
Frequently Asked Questions
What is Brewster's Law?
Brewster's Law states that the tangent of the polarization angle equals the ratio of the second medium's refractive index to the first: tan(θ_B) = n₂ / n₁. At this angle of incidence, the reflected beam is completely linearly polarized and the reflected and refracted rays are perpendicular to each other.
Why is reflected light polarized at Brewster's angle?
When light hits the interface at Brewster's angle, the refracted beam travels at exactly 90° to the direction the reflected beam would take. The oscillating dipoles in the second medium that emit the reflected light are aligned along the p-polarization direction and cannot radiate in that direction (dipole radiation vanishes along the dipole axis), so the p-component of the reflected beam is zero. Only the s-component (perpendicular to the plane of incidence) is reflected.
Does Brewster's angle depend on wavelength?
Yes, slightly. Because refractive indices vary with wavelength (a phenomenon called dispersion), Brewster's angle changes with the color of light. For most common optical materials the variation across visible wavelengths is small — typically less than 1°. For high-precision polarimetry or broadband applications, wavelength-specific index values should be used.
What happens if light hits the surface at an angle other than Brewster's angle?
Outside Brewster's angle, the reflected light is partially polarized — both polarization components are present, but the s-component dominates in reflection. At normal incidence (0°) both components reflect equally and the light remains unpolarized after reflection. Only at exactly Brewster's angle is the reflected beam completely s-polarized.
How are Brewster windows used in lasers?
A Brewster window is a flat glass plate inserted into a laser cavity at Brewster's angle. The intracavity beam passes through with essentially zero reflection loss for the p-polarized component, while also experiencing zero Fresnel reflection. This eliminates spurious reflections that would affect cavity stability, and the resulting output is inherently linearly polarized, making Brewster windows indispensable in gas lasers such as HeNe and Ar-ion designs.
Can I use the calculator for total internal reflection?
Brewster's angle exists for light traveling in either direction through an interface and does not require n₁ < n₂. However, if n₁ > n₂ and the angle of incidence exceeds the critical angle, total internal reflection occurs and there is no transmitted ray. In that regime Brewster's angle still has a mathematical value from arctan(n₂/n₁), but it may be less than the critical angle, meaning the surface behaves differently. Always check whether total internal reflection applies before relying on Brewster's Law.