Understanding Fresnel Laws: Polarization Effects at Dielectric Interfaces### Introduction
Light interacting with a boundary between two dielectric media undergoes reflection and transmission. The Fresnel laws (or Fresnel equations) describe how much of an incident electromagnetic wave is reflected and transmitted at such an interface, and how those amounts depend on the wave’s polarization, angle of incidence, and the optical properties (refractive indices) of the two media. Understanding these laws is essential in optics, photonics, remote sensing, and many engineering applications like anti-reflection coatings, polarizers, and optical sensors.
Electromagnetic basis and boundary conditions
Maxwell’s equations govern light as an electromagnetic wave. At a planar boundary between two linear, isotropic, non-magnetic dielectrics (with refractive indices n1 and n2), the tangential components of the electric field E and magnetic field H must be continuous across the interface. Applying these boundary conditions to incident, reflected, and transmitted plane waves yields the Fresnel coefficients — complex amplitude ratios for reflection ® and transmission (t) for the two fundamental polarizations.
The two polarization states considered are:
- s-polarization (perpendicular, sometimes called TE): electric field perpendicular to the plane of incidence.
- p-polarization (parallel, sometimes called TM): electric field parallel to the plane of incidence.
Fresnel amplitude coefficients
Let θ1 be the angle of incidence in medium 1, θ2 the angle of transmission in medium 2, related by Snell’s law: n1 sin θ1 = n2 sin θ2.
For s-polarization: r_s = (n1 cos θ1 – n2 cos θ2) / (n1 cos θ1 + n2 cos θ2) t_s = (2 n1 cos θ1) / (n1 cos θ1 + n2 cos θ2)
For p-polarization: r_p = (n2 cos θ1 – n1 cos θ2) / (n2 cos θ1 + n1 cos θ2) t_p = (2 n1 cos θ1) / (n2 cos θ1 + n1 cos θ2)
These r and t are amplitude reflection and transmission coefficients. They can be complex if either medium has absorption (complex refractive index).
Reflectance and transmittance (power coefficients)
Reflectance R and transmittance T are the fractions of incident power reflected and transmitted. For non-absorbing media:
R_s = |r_s|^2 R_p = |r_p|^2
T takes into account the ratio of wave impedances or the cosine factors due to energy flux normal components:
T_s = (n2 cos θ2 / n1 cos θ1) |t_s|^2 T_p = (n2 cos θ2 / n1 cos θ1) |t_p|^2
Energy conservation ensures R + T = 1 for lossless media.
Angle and polarization dependence
- Normal incidence (θ1 = 0): r_s = r_p = (n1 – n2)/(n1 + n2). Polarization has no effect at normal incidence.
- Oblique incidence: reflectance differs between s and p. Generally, R_s > R_p for dielectric-to-dielectric transitions at oblique angles.
- Brewster’s angle θ_B: for p-polarized light, r_p = 0 when tan θ_B = n2 / n1. At this angle there is zero reflection for p-polarized light and the reflected light is purely s-polarized.
- Total internal reflection (TIR): occurs when n1 > n2 and θ1 exceeds the critical angle θ_c = arcsin(n2/n1). Beyond θ_c, θ2 becomes complex, reflection is total (R = 1) and transmitted wave becomes evanescent (decays exponentially).
Phase shifts
Reflected waves acquire polarization-dependent phase shifts. For dielectric interfaces without absorption:
- r_s is negative (phase shift of π) when light reflects from a medium with higher refractive index (n2 > n1) at normal incidence; otherwise phase behavior follows the signs of the numerator/denominator.
- r_p crosses through zero at Brewster’s angle; phase changes rapidly in that region. In TIR, r becomes complex with unit magnitude; the phase shift varies with angle and differs for s and p, causing effects like Goos–Hänchen and Imbert–Fedorov shifts (lateral and transverse beam shifts).
Effects in practical optics
- Anti-reflection coatings: use destructive interference to minimize R at target wavelengths/angles by engineering thin-film stacks; design uses Fresnel equations plus phase accumulation in layers.
- Polarizers and beamsplitters: exploit difference between R_s and R_p (e.g., Brewster windows transmit p while reflecting s).
- Optical sensors and ellipsometry: measure amplitude ratios and phase differences between s and p to extract thin-film thicknesses and refractive indices.
- Fiber optics and waveguides: Fresnel reflections at connectors or facet interfaces can cause back-reflection; index matching and angled cleaves reduce reflections.
Example calculation
Consider light going from air (n1 = 1.0) to glass (n2 = 1.5) at θ1 = 45°. First compute θ2 via Snell’s law: sin θ2 = n1/n2 * sin θ1 = (⁄1.5)*sin45° ≈ 0.4714 ⇒ θ2 ≈ 28.1°. Compute r_s: r_s = (1·cos45° – 1.5·cos28.1°) / (1·cos45° + 1.5·cos28.1°) Numerically, cos45°≈0.7071, cos28.1°≈0.8829, r_s ≈ (0.7071 – 1.3243)/(0.7071 + 1.3243) ≈ (-0.6172)/(2.0314) ≈ -0.3038 ⇒ R_s ≈ 0.0923 (9.23% reflected). Compute r_p similarly yields R_p ≈ 0.004 (≈0.4% reflected), illustrating strong polarization dependence at this angle.
Extensions: absorbing media and anisotropic interfaces
- Complex refractive indices: for metals or absorbing dielectrics, n = n’ + iκ. Fresnel coefficients become complex; reflectance can be high even at normal incidence.
- Anisotropic or birefringent media: polarization states couple; simple s/p separation may not hold. Fresnel-like boundary conditions exist but require tensorial dielectric permittivity and solving for eigenpolarizations.
- Multilayer stacks and coated surfaces: use transfer-matrix methods (characteristic matrices) built from Fresnel coefficients and phase factors to compute overall reflection/transmission across wavelengths and angles.
Experimental measurement techniques
- Reflectometry: measures reflectance vs. angle/wavelength for s and p.
- Ellipsometry: measures ratio of complex reflection coefficients (ρ = r_p / r_s) to deduce film thickness and complex refractive index.
- Polarimetric imaging: uses polarization-sensitive detectors to map changes in R_s and R_p across samples or scenes.
Summary
Fresnel laws quantitatively connect the electromagnetic boundary conditions to observable reflectance, transmittance, and phase behavior at dielectric interfaces, with strong dependence on polarization and angle. They provide the foundation for designing coatings, polarizing optics, sensors, and for understanding phenomena such as Brewster’s angle, total internal reflection, and polarization-dependent phase shifts.
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