Fresnel Reflection and Transmission Coefficients

When an electromagnetic wave encounters an interface between two dielectric media with different ε (permittivity) and μ (permeability), the wave undergoes reflection and transmission.

The following Fresnel Coefficients mathematically describe the reflection and transmission at oblique incidence, assuming the wave is propagating from region 1 to region 2. As the boundary conditions at the interface must be taken into consideration, parallel polarization E_|| (electric field parallel to the plane of incidence) and perpendicular polarization E_⊥ (electric field parallel to the plane of incidence) will be considered separately.

For both polarizations, Snell's Law of reflection and refraction is:

Θ_i = Θ_r angle of incidence equals angle of reflection, yo
β₁ sin Θ_i = β₂ sin Θ_t

where Θ_i and Θ_r and Θ_t are the angles of incidence, reflection, and transmission, and β₁ and β₂ are the wavenumbers in dielectric 1 and 2, respectively.

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Perpendicular (E_⊥) Polarization

The Reflection coefficients R_⊥ and T_⊥ are:

R_⊥ = (η₂ cos Θ_i - η₁ cos Θ_t) / (η₂ cos Θ_i + η₁ cos Θ_t)

T_⊥ = 2 η₂ cos Θ_i / (η₂ cos Θ_i + η₁ cos Θ_t)

where η is the characteristic impedance of the dielectric, η = sqrt(μ/ε).

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Parallel (E_||) Polarization

The Reflection coefficients R_|| and T_|| are:

R_|| = (-η₁ cos Θ_i + η₂ cos Θ_t) / (η₁ cos Θ_i + η₂ cos Θ_t)

T_|| = 2 η₂ cos Θ_i / (η₁ cos Θ_i + η₂ cos Θ_t)