Brewster's Angle Calculator
📚 Understanding Brewster's Angle and Light Polarization
What is Brewster's Angle?
Brewster's angle, named after physicist David Brewster who discovered it in 1815, is the specific incidence angle at which light reflecting from a boundary between two transparent media becomes completely linearly polarized. At this unique angle, the reflected light contains only s-polarized light (perpendicular to the plane of incidence), while all p-polarized light (parallel to the plane of incidence) is transmitted into the second medium with zero reflection. This phenomenon occurs because at Brewster's angle, the reflected and refracted rays are perpendicular to each other.
The Brewster's Angle Formula
| Concept | Formula | Application |
|---|---|---|
| Brewster's Angle (Basic) | tan(θ_B) = n₂/n₁ | Fundamental relationship; direct calculation from indices |
| Brewster's Angle (Direct) | θ_B = arctan(n₂/n₁) | Angle in radians or degrees; standard form |
| Perpendicularity Condition | θ_B + θ₂ = 90° | Reflected and refracted rays perpendicular |
| Snell's Law at Brewster's | n₁ sin(θ_B) = n₂ cos(θ_B) | Derivation relationship; physical basis |
| Refraction Angle | θ₂ = 90° - θ_B | Transmitted ray angle at Brewster condition |
Derivation from Snell's Law and Physical Principles
The derivation of Brewster's angle begins with Snell's law: n₁sin(θ₁) = n₂sin(θ₂), which describes light refraction at a boundary. The key insight is that at Brewster's angle, the reflected and refracted rays are perpendicular to each other, meaning θ₁ + θ₂ = 90°. This perpendicularity condition is crucial because when the electric field oscillations in the plane of incidence (p-polarized) would need to radiate back along the reflected direction, they cannot do so when the ray is perpendicular. Therefore, sin(θ₂) = sin(90° - θ₁) = cos(θ₁). Substituting into Snell's law gives n₁sin(θ_B) = n₂cos(θ_B). Dividing both sides by cos(θ_B) yields n₁tan(θ_B) = n₂, so tan(θ_B) = n₂/n₁, and therefore θ_B = arctan(n₂/n₁).
Brewster's Angles for Common Material Interfaces
| Medium 1 | Medium 2 | n₁ | n₂ | Brewster's Angle |
|---|---|---|---|---|
| Air | Water | 1.00 | 1.33 | 53.1° |
| Air | Glass | 1.00 | 1.50 | 56.3° |
| Air | Diamond | 1.00 | 2.42 | 67.5° |
| Air | Ice | 1.00 | 1.31 | 52.7° |
| Water | Glass | 1.33 | 1.50 | 41.8° |
| Glass | Diamond | 1.50 | 2.42 | 58.6° |
Polarization at Brewster's Angle
At Brewster's angle, the reflected light is completely s-polarized (perpendicular to the plane of incidence), while the p-polarized component is fully transmitted. This creates perfect linear polarization of the reflected light. When unpolarized light (containing equal s and p components) reflects at θ_B, only the s component is reflected, producing 100% linearly polarized reflected light. This property is fundamental to many optical applications, from photography to laser technology. The transmitted light at Brewster's angle contains both s and p components, but at reduced intensity due to partial transmission.
Applications in Optical Systems
Photography and Glare Reduction: Polarizing filters exploit Brewster's angle to eliminate reflections from water surfaces, glass, and roads. When these surfaces are viewed at angles near Brewster's angle, reflections become strongly s-polarized. A polarizing filter oriented perpendicular to this polarization blocks glare while allowing scene light through, dramatically improving contrast and color saturation.
Laser Technology: In laser cavities, Brewster windows (optical elements positioned at Brewster's angle) minimize reflection losses for p-polarized light while reflecting s-polarized light. This preferentially forces the laser to oscillate only in p-polarization, producing linearly polarized output with high efficiency and stability.
Anti-Reflection Coatings: Optical designers use Brewster angle principles in broadband anti-reflection coating specifications. By understanding how light behaves at Brewster's angle, engineers optimize multi-layer coatings to minimize Fresnel losses across spectral ranges.
Polarimetry and Optical Testing: Brewster's angle is fundamental to polarimetric measurements and optical characterization. Precise angle measurements enable determination of material properties and optical quality.
Wavelength Dependence of Brewster's Angle
Because refractive indices vary with wavelength (dispersion), Brewster's angle is also wavelength-dependent. For example, typical optical glass has n ≈ 1.52 in the infrared but n ≈ 1.53 in the ultraviolet. As wavelength changes, the refractive index ratio n₂/n₁ changes, shifting θ_B = arctan(n₂/n₁). Optical systems operating over broad spectral ranges must account for this wavelength dependence through careful design, wavelength-specific calculations, or broadband anti-reflection coatings optimized for the intended wavelength range.
Why RevisionTown's Brewster's Angle Calculator?
Calculating Brewster's angle requires precise mathematical computations, unit conversions between radians and degrees, and understanding physical relationships. Our advanced calculator eliminates errors by automatically computing θ_B = arctan(n₂/n₁) from refractive indices, supporting common materials with preset values, displaying results in both degrees and radians, and providing comprehensive optical analysis. Whether designing optical systems, setting up laboratory experiments, or teaching optics principles, this calculator ensures accuracy and provides educational insights into light polarization and interface behavior.
❓ Frequently Asked Questions
Brewster's angle (θ_B) is the incident angle at which reflected light from a boundary between two media is completely polarized. The formula is tan(θ_B) = n₂/n₁, where n₁ is the refractive index of the incident medium and n₂ is the refractive index of the transmitting medium. Solving for angle gives θ_B = arctan(n₂/n₁). At this special angle, the reflected and refracted rays are perpendicular (90°) to each other. The reflected light contains only s-polarized (perpendicular) light, while p-polarized (parallel) light is fully transmitted with zero reflection.
Brewster's angle represents the unique incidence angle where reflected light becomes perfectly linearly polarized. When unpolarized light hits an interface at θ_B, the reflected light contains only the s-polarized component (perpendicular to the plane of incidence), while p-polarized light (parallel to the plane) is completely transmitted. This occurs because at this specific angle, the reflected and refracted rays form a 90° angle, preventing p-polarized electric field oscillations from radiating back along the reflection direction.
Starting from Snell's law n₁sin(θ₁) = n₂sin(θ₂) and Brewster's condition θ₁ + θ₂ = 90°, we get θ₂ = 90° - θ₁. Substituting sin(θ₂) = cos(θ₁) into Snell's law gives n₁sin(θ_B) = n₂cos(θ_B). Dividing both sides by cos(θ_B) yields n₁tan(θ_B) = n₂, so tan(θ_B) = n₂/n₁. Taking the arctangent gives θ_B = arctan(n₂/n₁).
For air-to-glass (n₁ = 1.00, n₂ = 1.50), θ_B ≈ 56.3°. For air-to-water (n₂ = 1.33), θ_B ≈ 53.1°. For air-to-ice (n₂ = 1.31), θ_B ≈ 52.7°. For air-to-diamond (n₂ = 2.42), θ_B ≈ 67.5°. For water-to-glass (n₁ = 1.33, n₂ = 1.50), θ_B ≈ 41.8°. As the refractive index ratio increases, Brewster's angle increases toward 90°.
Polarizing filters for cameras exploit Brewster's angle by suppressing reflections from surfaces like water, glass, and roads. When light reflects near Brewster's angle, the reflected component becomes strongly s-polarized. A polarizing filter oriented to block this polarization dramatically reduces glare and reflections, enhancing image contrast and color saturation. Photographers rotate the polarizer while viewing through the lens to maximize the suppression of unwanted reflections.
At Brewster's angle, reflected light is completely s-polarized (perpendicular to the plane of incidence). The p-polarized component is fully transmitted into the second medium with zero reflection. At angles different from Brewster's angle, both s and p components are reflected, resulting in partial polarization. This angular dependence is fundamental to polarimetric techniques and essential for designing anti-reflection optical systems.
Yes, Brewster's angle depends on wavelength because refractive indices are wavelength-dependent (dispersion). As wavelength changes, both n₁ and n₂ change, so the ratio n₂/n₁ shifts, making θ_B = arctan(n₂/n₁) change. Optical systems operating over broad wavelengths require detailed calculations for specific wavelengths or use broadband designs to account for this wavelength dependence.
In laser cavities, Brewster windows (optical elements positioned at Brewster's angle) minimize reflection losses for p-polarized light while reflecting s-polarized light. This preferentially encourages the laser to oscillate only in p-polarization, producing linearly polarized output. Brewster windows reduce cavity losses, improve laser efficiency, and control output polarization in high-precision optical systems.