Compute the Brewster angle (polarization angle) for any two dielectric media. Visualize the incident, reflected, and transmitted rays at the interface, and explore Fresnel reflectance for s‑ and p‑polarized light. Ideal for optics students, engineers, and physics educators.
Brewster's angle (also called the polarization angle) is the angle of incidence at which light with a particular polarization is perfectly transmitted through a transparent dielectric surface, with no reflection. At this angle, the reflected and refracted rays are perpendicular to each other, and the reflected light is completely linearly polarized perpendicular to the plane of incidence (s‑polarization). This phenomenon was first described by the Scottish physicist Sir David Brewster in 1815.
The Brewster angle is given by the simple relation:
where n1 is the refractive index of the incident medium and n2 that of the transmitting medium. When light strikes the interface at this angle, the reflected and refracted rays are orthogonal (θB + θt = 90°), and the Fresnel reflection coefficient for p‑polarized light drops to zero.
Sir David Brewster (1781–1868) was a Scottish physicist, mathematician, and inventor who made groundbreaking contributions to optics. While experimenting with polarized light, he observed that at a specific angle of incidence, the reflected ray became completely polarized. He published his findings in 1815, establishing what we now call Brewster's Law. Brewster also invented the kaleidoscope and improved the stereoscope, but his work on polarization remains his most enduring scientific legacy.
Interestingly, Brewster's angle was independently noted by Étienne-Louis Malus a few years earlier, but Brewster was the first to publish a comprehensive quantitative treatment. The phenomenon is now a cornerstone of physical optics, with applications ranging from laser physics to telecommunications.
At Brewster's angle, the reflected and transmitted rays are perpendicular:
θB + θt = 90°
Using Snell's law: n1 sin θB = n2 sin θt
Since θt = 90° − θB, we have sin θt = cos θB, so:
n1 sin θB = n2 cos θB ⟹ tan θB = n2 / n1
The Fresnel reflection coefficient for p‑polarization vanishes at this angle, while the s‑polarization coefficient remains finite.
The Fresnel equations describe the reflection and transmission of light at a dielectric interface. At Brewster's angle, the reflectance for p‑polarized light is exactly zero, while the s‑polarized reflectance is given by:
This means that at the Brewster angle, the reflected beam is 100% s‑polarized (perpendicular to the plane of incidence). The transmitted beam contains both polarizations, but the p‑polarized component is preferentially transmitted.
The degree of polarization of the reflected light is theoretically 100% at the Brewster angle, assuming the incident light is unpolarized. In practice, surface roughness and material absorption reduce this value slightly.
| Interface (n₁ → n₂) | n₁ | n₂ | θB (°) | θt (°) | Rs at θB |
|---|---|---|---|---|---|
| Air → Glass (crown) | 1.000 | 1.520 | 56.7 | 33.3 | 0.042 |
| Air → Water | 1.000 | 1.333 | 53.1 | 36.9 | 0.020 |
| Air → Diamond | 1.000 | 2.417 | 67.5 | 22.5 | 0.172 |
| Glass → Air | 1.520 | 1.000 | 33.3 | 56.7 | 0.042 |
| Water → Glass | 1.333 | 1.520 | 48.8 | 41.2 | 0.005 |
| Glass → Diamond | 1.520 | 2.417 | 57.8 | 32.2 | 0.077 |
In a helium‑neon laser, the gain tube is often sealed with windows set at Brewster's angle to the optical axis. This configuration transmits the p‑polarized laser light with near‑zero loss while reflecting the s‑polarized component out of the cavity. The result is a highly linearly polarized output beam with minimal intracavity loss. The Brewster window also helps suppress unwanted longitudinal modes, improving the laser's spectral purity. This design is used in many commercial and research lasers, from He‑Ne to solid‑state systems.
Key takeaway: Brewster's angle is not just a theoretical curiosity — it is a practical engineering principle that enables high‑performance optical systems.
When one of the media has a complex refractive index (i.e., absorption), the Brewster condition generalizes to a complex angle. The reflectance for p‑polarization has a minimum but does not reach zero. This is important in understanding reflection from metals and semiconductors, where the imaginary part of the refractive index leads to residual reflection even at the "pseudo‑Brewster" angle.
At Brewster's angle, the reflected beam undergoes a lateral shift (the Goos‑Hänchen shift) that is particularly large for p‑polarized light. This effect, first observed in 1947, is a direct consequence of the phase change upon total internal reflection and has been studied in the context of quantum mechanics and surface plasmon resonance.
In anisotropic crystals, the refractive index depends on the polarization and propagation direction. The Brewster angle condition becomes polarization‑dependent, leading to complex optical phenomena such as conical refraction and polarization mode conversion.