Brewster’s Angle and Polarization of Light
When light travels from one medium to another, such as from air to water or air to glass, it undergoes both reflection and refraction. Brewster’s angle is a special angle of incidence at which light with a specific polarization is perfectly transmitted through the interface with no reflection.
What is Brewster’s Angle?
Brewster’s angle (also known as the polarization angle) is the angle of incidence at which light with p-polarization (polarized in the plane of incidence) is perfectly transmitted through a dielectric surface, with no reflection. It was discovered by Scottish physicist Sir David Brewster in 1812.
Where:
- θᵦ is Brewster’s angle (measured from the normal to the surface)
- n₁ is the refractive index of the first medium (where the light is coming from)
- n₂ is the refractive index of the second medium (where the light is being transmitted to)
The Physics Behind Brewster’s Angle
At Brewster’s angle, the reflected and refracted rays are perpendicular to each other. This geometric condition leads to some fascinating physical consequences:
- Perfect Polarization: Light reflected at Brewster’s angle is completely polarized perpendicular to the plane of incidence (s-polarization).
- No p-polarized Reflection: Light with polarization parallel to the plane of incidence (p-polarization) experiences zero reflection at Brewster’s angle.
- 90° Between Rays: The reflected and refracted rays form a 90° angle at Brewster’s angle.
Example Calculation:
For light traveling from air (n₁ = 1.0) to glass (n₂ = 1.5):
θᵦ = arctan(1.5/1.0) = arctan(1.5) ≈ 56.3°
This means that when light hits glass at an angle of 56.3° from the normal, any light polarized parallel to the plane of incidence will be completely transmitted with no reflection.
Reflection Coefficients and Fresnel Equations
The behavior of light at an interface is described by the Fresnel equations, which give the reflection and transmission coefficients for different polarizations of light:
For p-polarization (parallel):
rp = [n₂cosθᵢ – n₁cosθₜ] / [n₂cosθᵢ + n₁cosθₜ]
For s-polarization (perpendicular):
rs = [n₁cosθᵢ – n₂cosθₜ] / [n₁cosθᵢ + n₂cosθₜ]
Where rp and rs are the reflection coefficients, θᵢ is the angle of incidence, and θₜ is the angle of transmission (calculated using Snell’s law).
At Brewster’s angle, rp becomes zero, meaning no reflection for p-polarized light.
Critical Angle and Total Internal Reflection
While related to Brewster’s angle, the critical angle is another important concept in optics:
The critical angle only exists when light travels from a medium with higher refractive index to one with lower refractive index (e.g., from glass to air). When the angle of incidence exceeds the critical angle, all light is reflected back into the first medium – a phenomenon called total internal reflection.
Important Note: Brewster’s angle exists for any interface between different materials, but the critical angle only exists when light travels from a higher refractive index medium to a lower one (n₁ > n₂).
Applications of Brewster’s Angle
- Polarizing Filters: Brewster’s angle is used in the design of polarizing filters and optical instruments.
- Photography: Photographers use polarizing filters to reduce glare from non-metallic surfaces by utilizing Brewster’s angle principles.
- LCD Displays: Liquid crystal displays (LCDs) rely on polarization effects related to Brewster’s angle.
- Thin-Film Coatings: Brewster’s angle is considered in the design of anti-reflective coatings for lenses and optical elements.
- Laser Systems: Brewster windows are used in laser systems to minimize reflection losses while maintaining polarization.
Common Values of Brewster’s Angle
Here are some common interfaces and their corresponding Brewster’s angles:
- Air (n=1.0) to Water (n=1.33): θᵦ ≈ 53.1°
- Air (n=1.0) to Crown Glass (n=1.52): θᵦ ≈ 56.7°
- Air (n=1.0) to Diamond (n=2.42): θᵦ ≈ 67.5°
- Water (n=1.33) to Crown Glass (n=1.52): θᵦ ≈ 48.8°
Experimental Verification
You can observe Brewster’s angle in everyday life. Look at a non-metallic surface like a lake or a window at different angles while wearing polarized sunglasses. Rotate the glasses and you’ll notice that at a certain angle (Brewster’s angle), the glare is minimized when the glasses are oriented to block the s-polarized reflected light.
Real-Life Example:
When taking photographs of water surfaces, photographers often use polarizing filters rotated to the correct orientation to eliminate reflections. This works because light reflected from water at Brewster’s angle (approximately 53°) is strongly polarized.
Historical Context
Sir David Brewster (1781-1868) discovered this phenomenon while studying light polarization. His work was part of a broader scientific exploration of light properties in the early 19th century that helped establish the wave theory of light. The angle bearing his name has become a fundamental concept in optical physics and has practical applications that continue to be relevant today.
