Polarization Effects Explained — Fresnel Equations for Reflection and TransmissionLight is an electromagnetic wave; its electric and magnetic fields oscillate perpendicular to the direction of propagation. When light encounters an interface between two different media (for example, air and glass), part of the wave is reflected and part is transmitted (refracted). The Fresnel equations describe how much of the incident light is reflected or transmitted and how the polarization state of the light is affected. This article explains polarization basics, derives the Fresnel formulas for reflection and transmission coefficients, explores special cases (Brewster’s angle and total internal reflection), and outlines practical implications and applications.
1. Fundamentals of Polarization
Polarization describes the orientation of the electric field vector of an electromagnetic wave. Common polarization states:
- Linear polarization: the electric field oscillates in a fixed direction (e.g., vertical or horizontal).
- Circular polarization: the electric field rotates at a constant magnitude, tracing a circle in the transverse plane.
- Elliptical polarization: a general case where the tip of the electric field traces an ellipse.
At an interface, it’s customary to decompose an incident wave into two orthogonal linear polarization components relative to the plane of incidence (the plane defined by the incident ray and the surface normal):
- s-polarization (senkrecht, German for perpendicular): electric field perpendicular to the plane of incidence (also called TE — transverse electric).
- p-polarization (parallel): electric field parallel to the plane of incidence (also called TM — transverse magnetic).
These two polarizations interact differently with the boundary because boundary conditions apply to the tangential components of the electric and magnetic fields.
2. Boundary Conditions and Physical Setup
Consider a plane wave incident from medium 1 (refractive index n1) onto a planar interface with medium 2 (refractive index n2) at angle θi relative to the normal. The reflected wave leaves medium 1 at angle θr, and the transmitted (refracted) wave enters medium 2 at angle θt. By geometry and Snell’s law:
- θr = θi
- n1 sin θi = n2 sin θt
Electromagnetic boundary conditions at the interface require continuity of tangential components of the electric field E and the magnetic field H (or equivalently the magnetic induction B and electric displacement D depending on materials). For non-magnetic, isotropic, linear media (μ1 = μ2 ≈ μ0), the relevant conditions reduce to matching tangential E and H across the boundary.
Applying these conditions to s- and p-polarizations gives different sets of equations because the orientations of E and H relative to the plane change.
3. Fresnel Reflection and Transmission Coefficients (Amplitude)
Define amplitude reflection coefficients rs (s-polarized) and rp (p-polarized), and amplitude transmission coefficients ts and tp. These relate reflected and transmitted electric field amplitudes to the incident amplitude.
For non-magnetic media (μ1 = μ2), the standard Fresnel amplitude coefficients are:
-
s-polarization: rs = (n1 cos θi – n2 cos θt) / (n1 cos θi + n2 cos θt) ts = (2 n1 cos θi) / (n1 cos θi + n2 cos θt)
-
p-polarization: rp = (n2 cos θi – n1 cos θt) / (n2 cos θi + n1 cos θt) tp = (2 n1 cos θi) / (n2 cos θi + n1 cos θt)
Signs depend on chosen field conventions; these forms assume E-field amplitudes measured parallel to the chosen polarization directions and consistent phase conventions.
Notes:
- ts and tp as written give the ratio of transmitted to incident field amplitudes, taking into account field-component geometry but not directly power. Multiplying by appropriate ratios of cosines and refractive indices converts to transmitted power fractions.
4. Reflectance and Transmittance (Power Coefficients)
Most practical interest lies in power fractions: reflectance R and transmittance T (the fraction of incident intensity reflected or transmitted). For light in non-absorbing media:
- Rs = |rs|^2
- Rp = |rp|^2
For transmittance, because intensity depends on the medium’s refractive index and propagation angle, use:
- Ts = (n2 cos θt / n1 cos θi) |ts|^2
- Tp = (n2 cos θt / n1 cos θi) |tp|^2
Energy conservation requires Rs + Ts = 1 for each polarization when there are no absorptive losses.
5. Special Cases
Brewster’s angle
- For p-polarized light there exists an angle θB where the reflected amplitude rp = 0, so Rp = 0. This Brewster angle satisfies: tan θB = n2 / n1
- At θB, reflected light is purely s-polarized. This principle is used in glare-reducing polarizers and in determining refractive indices experimentally.
Normal incidence (θi = 0)
- cos θi = cos θt = 1, so rs = (n1 – n2)/(n1 + n2) and rp = (n2 – n1)/(n2 + n1) = -rs.
- Both polarizations behave identically at normal incidence; reflectance R = ((n1 – n2)/(n1 + n2))^2.
Total internal reflection (TIR)
- Occurs when n1 > n2 and θi > θc where sin θc = n2 / n1. Beyond θc, θt becomes complex and transmitted waves are evanescent (do not propagate into medium 2). In this regime:
- |rs| = |rp| = 1 (total reflected power), but the reflection imparts a polarization-dependent phase shift between s and p components.
- That phase difference is exploited in devices like phase retarders and internal-reflection polarizers.
Phase shifts on reflection
- Even when reflectance is 1 (TIR), the reflected wave can experience a phase shift φs or φp. The differential phase Δφ = φp – φs can convert linear polarization into elliptical or circular polarization upon reflection, an effect used in Fresnel rhombs.
6. Complex Refractive Indices and Absorbing Media
If medium 2 is absorbing, its refractive index n2 is complex: n2 = n’ + iκ (where κ is the extinction coefficient). Fresnel coefficients become complex-valued with magnitudes less than unity; reflectance and transmittance formulas must be adapted. For metals, reflectance typically remains high and strongly polarization- and angle-dependent; p-polarization can show pronounced features near plasma resonances.
7. Practical Applications
- Anti-reflection coatings: layers designed so reflected amplitudes from different surfaces cancel for targeted wavelengths and polarizations, reducing Rs and Rp.
- Polarizing beamsplitters: use different reflection/transmission for s and p polarizations to separate components.
- Optical sensing and ellipsometry: measurement of polarization changes on reflection reveals thin-film thicknesses, refractive indices, and surface properties.
- Photography and vision: linear polarizing filters reduce glare (preferentially removing p- or s-polarized reflection depending on geometry).
- Fiber optics and total internal reflection devices: exploit TIR to confine light with minimal loss.
8. Numerical Example
Consider light from air (n1 = 1.0) hitting glass (n2 = 1.5) at θi = 45°. Use Snell’s law: sin θt = (n1/n2) sin θi = (⁄1.5)·sin 45° ≈ 0.4714 so θt ≈ 28.1°.
Compute rs: rs = (1.0·cos45° – 1.5·cos28.1°) / (1.0·cos45° + 1.5·cos28.1°) Cosines: cos45°≈0.7071, cos28.1°≈0.8820 Numerator ≈ (0.7071 – 1.3230) = -0.6159 Denominator ≈ (0.7071 + 1.3230) = 2.0301 rs ≈ -0.3034 → Rs ≈ 0.0921 (9.2% reflected for s)
Compute rp: rp = (1.5·cos45° – 1.0·cos28.1°) / (1.5·cos45° + 1.0·cos28.1°) Numerator ≈ (1.0607 – 0.8820) = 0.1787 Denominator ≈ (1.0607 + 0.8820) = 1.9427 rp ≈ 0.0920 → Rp ≈ 0.00846 (0.85% reflected for p)
This example shows how reflection can be strongly polarization-dependent at oblique incidence.
9. Measurement and Ellipsometry
Ellipsometry measures the amplitude ratio and phase difference between p and s reflected components. It reports these as Ψ and Δ, where:
- tan Ψ = |rp/rs|
- Δ = arg(rp) – arg(rs)
From measured Ψ and Δ, one can infer complex refractive indices and film thicknesses with high precision.
10. Summary
- The Fresnel equations quantify how s- and p-polarized components reflect and transmit at interfaces.
- Reflectance and transmittance depend on angle, refractive indices, and polarization.
- Brewster’s angle and total internal reflection are key phenomena arising from Fresnel behavior.
- Polarization-dependent reflection is widely used in optics for filtering, sensing, and controlling light.
Mathematically and experimentally, Fresnel’s laws remain fundamental to classical optics — essential for designing coatings, polarizers, sensors, and modern photonic devices.
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