Fresnel Equations
Interactive visualization of reflection and refraction at an optical interface. Explore how light behaves at boundaries with different refractive indices, including s-polarization, p-polarization, and the Brewster angle.
Reflection & Transmission Coefficients
- θâ (Angle of Incidence): Adjust the angle at which light hits the interface (0° to 89°)
- nâ (Refractive Index): Set the refractive index of the second medium (glass ~1.5, water ~1.33, diamond ~2.4)
- s-polarization: Electric field perpendicular to the plane of incidence
- p-polarization: Electric field parallel to the plane of incidence
- Brewster Angle: Angle where p-polarized light has zero reflection (highlighted in yellow)
- Critical Angle: Beyond this angle, total internal reflection occurs (nâ > nâ)
- Use preset buttons for common configurations
đ Physics Background
đ Reflection & Refraction of Light
When light encounters an interface between two media with different refractive indices, part of the light is reflected and part is refracted (transmitted). The behavior depends on the angle of incidence and the polarization state of the light.
The Fresnel equations describe the reflection and transmission coefficients for the electric field amplitudes, which depend on the polarization direction relative to the plane of incidence.
đ Snell's Law
The direction of the refracted ray is governed by Snell's Law:
nâ sin(θâ) = nâ sin(θâ)
Where:
- nâ = refractive index of incident medium (typically 1.0 for air)
- nâ = refractive index of transmission medium
- θâ = angle of incidence (measured from normal)
- θâ = angle of refraction (measured from normal)
đ˘ Fresnel Equations
The amplitude reflection coefficients for s- and p-polarization are:
rs = (nâ cos θâ â nâ cos θâ) / (nâ cos θâ + nâ cos θâ)
rp = (nâ cos θâ â nâ cos θâ) / (nâ cos θâ + nâ cos θâ)
The intensity reflectances are:
Rs = |rs|²
Rp = |rp|²
For energy conservation, the transmittances are: Ts = 1 â Rs and Tp = 1 â Rp (corrected for the ratio of cosines and refractive indices).
đŻ Special Angles
â Brewster's Angle
Condition: θâ + θâ = 90°
θB = arctan(nâ / nâ)
At the Brewster angle, p-polarized light has zero reflection. All p-polarized light is transmitted, and reflected light is 100% s-polarized. This principle is used in polarizing beam splitters and laser windows.
đ Critical Angle (Total Internal Reflection)
Condition: Occurs when nâ > nâ (light going from denser to less dense medium)
θc = arcsin(nâ / nâ)
When θâ > θc, all light is reflected and none is transmitted. This is called Total Internal Reflection (TIR) and is the principle behind fiber optics and prism reflectors.
đŹ Polarization States
đ¸ s-Polarization (TE)
The electric field oscillates perpendicular to the plane of incidence (the plane containing the incident ray and the surface normal). "s" comes from the German word "senkrecht" (perpendicular).
đš p-Polarization (TM)
The electric field oscillates parallel to the plane of incidence. "p" comes from "parallel". At Brewster's angle, p-polarized light experiences zero reflection.
đ Applications
- Polarizing Sunglasses: Block horizontally polarized glare from reflections
- Anti-reflective Coatings: Designed to minimize reflection using interference
- Fiber Optics: Use total internal reflection to guide light over long distances
- Laser Cavities: Brewster windows allow lossless transmission for one polarization
- Photography: Polarizing filters reduce glare and enhance contrast
- Optical Instruments: Beam splitters and combiners in microscopes and telescopes