Free Brewster's Angle Calculator

Understanding Brewster's Angle and Polarized Light

When light crosses the interface between two different transparent media, the angle of incidence determines how much is reflected and how much is transmitted. At a particular incident angle known as Brewster's angle, the reflected wave becomes fully linearly polarized. This effect, described by Brewster's Law, is fundamental to many optical technologies, from anti-glare sunglasses to camera polarization filters. A dedicated Brewster Angle Calculator can quickly compute this angle based on the refractive indices of the two materials.

What Is Light Polarization?

Light is an electromagnetic wave whose electric and magnetic fields oscillate perpendicular to the direction of propagation. In natural (unpolarized) light, these vibrations occur in every possible orientation. When the fields are restricted to a specific orientation or pattern, the light is said to be polarized. The three basic forms are:

• Linear polarization — the fields oscillate in a single, fixed plane.
• Circular polarization — the field direction rotates uniformly as the wave advances.
• Elliptical polarization — the field tip traces an ellipse in the plane perpendicular to the wave vector.

Reflection from a dielectric surface can itself produce linear polarization, especially when the incident ray meets the surface at Brewster's angle.

The Condition for Perfect Polarization: Brewster's Angle

Consider a light ray traveling through a medium of refractive index $n_1$ and striking a second medium with index $n_2$. At the interface, part of the energy is reflected and part is refracted. The reflected beam becomes completely linearly polarized when the angle of incidence equals Brewster's angle $\alpha_B$. The physical reason is that at this angle the reflected and refracted rays are perpendicular to each other: $\alpha_B + \theta_t = 90^\circ$, where $\theta_t$ is the angle of refraction.

Using Snell's law ($n_1 \sin\alpha_B = n_2 \sin\theta_t$) and the perpendicular condition, one substitutes $\sin\theta_t = \cos\alpha_B$ to obtain:

$$n_1 \sin\alpha_B = n_2 \cos\alpha_B$$

which simplifies to:

$$\tan\alpha_B = \frac{n_2}{n_1}$$

Therefore,

$$\alpha_B = \arctan\left(\frac{n_2}{n_1}\right)$$

• $\alpha_B$ — Brewster's angle (the polarization angle).
• $n_1$ — refractive index of the initial medium (typically air, $n_1\approx 1$).
• $n_2$ — refractive index of the reflecting medium.

For example, for water ($n_2 = 1.33$) Brewster's angle is about $53^\circ$; for common glass ($n_2 = 1.5$) it is roughly $56^\circ$. This tool also works as a Polarization Angle Calculator or a Refractive Index Calculator — if you know either angle and one refractive index, you can rearrange the formula to find the unknown index.

Practical Applications of Brewster's Angle

Polarized Sunglasses — Sunlight reflecting from horizontal surfaces (water, roads) is strongly polarized horizontally. Sunglasses with vertically aligned polarizing lenses exploit Brewster's condition to block this glare, improving visual comfort and safety.

Photography — A polarizing filter placed in front of the camera lens can eliminate reflections from non-metallic surfaces such as water or glass. By rotating the filter, photographers control how much reflected light is cut, allowing them to see beneath the water surface or enhance the contrast of clouds against a blue sky.

Optical Design — Brewster's angle is used to design beam-splitters that produce polarized beams without absorption, to minimize reflection losses in laser cavities through Brewster windows, and to create anti-glare coatings. Understanding the underlying Brewster's Law is essential for engineers working in photonics. A Brewster's Law Calculator provides the quick numeric result needed during design.

How to Use the Calculator

Enter the refractive indices of the two media (the incident medium and the reflecting medium). The Polarized Light Calculator immediately returns Brewster's angle in degrees and, optionally, the complementary refracted angle. No manual trigonometric lookup is required; the calculator performs the arctangent automatically. It is equally suitable for students verifying homework problems and for professionals selecting optical components.