Brewster's angle

When light encounters a boundary between two media with different refractive indices, some of it is usually reflected as shown in the figure above. The fraction that is reflected is described by the Fresnel equations, and depends on the incoming light's polarization and angle of incidence.

The Fresnel equations predict that light with the p polarization (electric field polarized in the same plane as the incident ray and the surface normal at the point of incidence) will not be reflected if the angle of incidence is

${\displaystyle \theta _{\mathrm {B} }=\arctan \!\left({\frac {n_{2}}{n_{1}}}\right)\!,}$

where n₁ is the refractive index of the initial medium through which the light propagates (the "incident medium"), and n₂ is the index of the other medium. This equation is known as Brewster's law, and the angle defined by it is Brewster's angle.

The physical mechanism for this can be qualitatively understood from the manner in which electric dipoles in the media respond to p-polarized light. One can imagine that light incident on the surface is absorbed, and then re-radiated by oscillating electric dipoles at the interface between the two media. The polarization of freely propagating light is always perpendicular to the direction in which the light is travelling. The dipoles that produce the transmitted (refracted) light oscillate in the polarization direction of that light. These same oscillating dipoles also generate the reflected light. However, dipoles do not radiate any energy in the direction of the dipole moment. If the refracted light is p-polarized and propagates exactly perpendicular to the direction in which the light is predicted to be specularly reflected, the dipoles point along the specular reflection direction and therefore no light can be reflected. (See diagram, above)

With simple geometry this condition can be expressed as

${\displaystyle \theta _{1}+\theta _{2}=90^{\circ },}$

where θ₁ is the angle of reflection (or incidence) and θ₂ is the angle of refraction.

Using Snell's law,

${\displaystyle n_{1}\sin \theta _{1}=n_{2}\sin \theta _{2},}$

one can calculate the incident angle θ₁ = θ_B at which no light is reflected:

${\displaystyle n_{1}\sin \theta _{\mathrm {B} }=n_{2}\sin(90^{\circ }-\theta _{\mathrm {B} })=n_{2}\cos \theta _{\mathrm {B} }.}$

Solving for θ_B gives

${\displaystyle \theta _{\mathrm {B} }=\arctan \!\left({\frac {n_{2}}{n_{1}}}\right)\!.}$

For a glass medium (n₂ ≈ 1.5) in air (n₁ ≈ 1), Brewster's angle for visible light is approximately 56°, while for an air-water interface (n₂ ≈ 1.33), it is approximately 53°. Since the refractive index for a given medium changes depending on the wavelength of light, Brewster's angle will also vary with wavelength.

The phenomenon of light being polarized by reflection from a surface at a particular angle was first observed by Étienne-Louis Malus in 1808.[3] He attempted to relate the polarizing angle to the refractive index of the material, but was frustrated by the inconsistent quality of glasses available at that time. In 1815, Brewster experimented with higher-quality materials and showed that this angle was a function of the refractive index, defining Brewster's law.

Brewster's angle is often referred to as the "polarizing angle", because light that reflects from a surface at this angle is entirely polarized perpendicular to the plane of incidence ("s-polarized"). A glass plate or a stack of plates placed at Brewster's angle in a light beam can, thus, be used as a polarizer. The concept of a polarizing angle can be extended to the concept of a Brewster wavenumber to cover planar interfaces between two linear bianisotropic materials. In the case of reflection at Brewster's angle, the reflected and refracted rays are mutually perpendicular.

For magnetic materials, Brewster's angle can exist for only one of the incident wave polarizations, as determined by the relative strengths of the dielectric permittivity and magnetic permeability.[4] This has implications for the existence of generalized Brewster angles for dielectric metasurfaces.[5]

Polarized sunglasses use the principle of Brewster's angle to reduce glare from the sun reflecting off horizontal surfaces such as water or road. In a large range of angles around Brewster's angle, the reflection of p-polarized light is lower than s-polarized light. Thus, if the sun is low in the sky, reflected light is strongly s-polarized. Polarizing sunglasses use a polarizing material such as Polaroid sheets to block horizontally-polarized light, preferentially blocking reflections from horizontal surfaces. The effect is strongest with smooth surfaces such as water, but reflections from roads and the ground are also reduced.

Photographers use the same principle to remove reflections from water so that they can photograph objects beneath the surface. In this case, the polarizing filter camera attachment can be rotated to be at the correct angle (see figure).

Photographs taken of a window with a camera polarizer filter rotated to two different angles. In the picture at left, the polarizer is aligned with the polarization angle of the window reflection. In the picture at right, the polarizer has been rotated 90°, eliminating the heavily polarized reflected sunlight.

When recording a hologram, light is typically incident at Brewster's angle. Because the incident light is p-polarized, it is not back reflected from the transparent back-plane of the holographic film. This avoids unwanted interference effects in the hologram.

Brewster angle prisms are used in laser physics. The polarized laser light enters the prism at Brewster's angle without any reflective losses.

In surface science, Brewster angle microscopes are used in imaging layers of particles or molecules at air-liquid interfaces. By using a laser aimed at Brewster's angle to the interface, the pure liquid appears black in the image whereas molecule layers give out a reflection that can be detected and presented with a camera.

Brewster windows

A Brewster window

Gas lasers typically use a window tilted at Brewster's angle to allow the beam to leave the laser tube. Since the window reflects some s-polarized light but no p-polarized light, the round trip loss for the s polarization is higher than that of the p polarization. This causes the laser's output to be p polarized due to competition between the two modes.[6]

When the reflecting surface is absorbing, reflectivity at parallel polarization (p) goes through a non-zero minimum at the so-called pseudo-Brewster's angle.[7][8]

• Brewster angle microscope
• Total internal reflection

• Lakhtakia, A. (1992). "General schema for the Brewster conditions" (PDF). Optik. 90 (4): 184–186.

• Brewster's Angle Extraction from Wolfram Research
• Brewster window at RP-photonics.com
• TE, TM Reflection Coefficients – interactive phase and magnitude plots showing Brewster's angle