Fresnel equations
The Fresnel equations describe the reflection and transmission of a plane electromagnetic wave at a planar interface between two homogeneous media. They relate the complex amplitudes of the incident, reflected, and transmitted electric fields as functions of the angle of incidence, the material refractive indices, and the wave’s polarization. The equations follow from the electromagnetic boundary conditions at the interface and constitute a central result of geometrical optics and physical optics.
For isotropic media, the incident field separates into two independent polarization components. The s component has its electric field perpendicular to the plane of incidence, whereas the p component has its electric field parallel to that plane. The corresponding amplitude coefficients generally differ, producing polarization-dependent reflection and transmission.
Geometrical configuration
Consider an interface separating media with refractive indices (n_1) and (n_2). A plane wave in the first medium reaches the interface at an incidence angle (\theta_i), measured from the surface normal. The reflected wave remains in the first medium at an angle (\theta_r), while the transmitted wave enters the second medium at an angle (\theta_t).
The tangential phase of the field must remain continuous across the interface. This condition gives the law of reflection and Snell's law:
[ \theta_r=\theta_i, ]
[ n_1\sin\theta_i=n_2\sin\theta_t. ]
For monochromatic fields, amplitude reflection and transmission coefficients are defined by
[ r=\frac{E_r}{E_i}, \qquad t=\frac{E_t}{E_i}, ]
where (E_i), (E_r), and (E_t) denote the complex electric-field amplitudes of the incident, reflected, and transmitted waves. The coefficients retain phase information and therefore may be negative or complex.
Fresnel amplitude coefficients
For nonmagnetic, isotropic media, the s-polarized coefficients are
[ r_s= \frac{n_1\cos\theta_i-n_2\cos\theta_t} {n_1\cos\theta_i+n_2\cos\theta_t}, ]
[ t_s= \frac{2n_1\cos\theta_i} {n_1\cos\theta_i+n_2\cos\theta_t}. ]
The corresponding p-polarized coefficients are
[ r_p= \frac{n_2\cos\theta_i-n_1\cos\theta_t} {n_2\cos\theta_i+n_1\cos\theta_t}, ]
[ t_p= \frac{2n_1\cos\theta_i} {n_2\cos\theta_i+n_1\cos\theta_t}. ]
The signs of amplitude coefficients depend partly on the orientation chosen for each polarization basis. In the convention represented above, (r_s) and (r_p) have opposite signs at normal incidence because the reflected p-polarization basis reverses its geometrical orientation relative to the s-polarization basis. Observable reflected powers are unaffected by this convention.
For magnetic media, refractive index alone does not fully determine the coefficients. Their general forms use the wave impedance or optical admittance of each medium, since the electric and magnetic boundary conditions depend separately on the permittivity and permeability.
Power coefficients and energy flow
Reflectance (R) is the ratio of reflected to incident time-averaged normal energy flux. In lossless media it is given by the squared magnitude of the amplitude reflection coefficient:
[ R_s=|r_s|^2, \qquad R_p=|r_p|^2. ]
Transmittance (T) includes an additional factor because equal electric-field amplitudes do not generally correspond to equal normal components of the Poynting vector. For nonmagnetic, transparent media,
[ T_s= \frac{n_2\cos\theta_t}{n_1\cos\theta_i}|t_s|^2, ]
[ T_p= \frac{n_2\cos\theta_t}{n_1\cos\theta_i}|t_p|^2. ]
Conservation of electromagnetic energy then gives
[ R_s+T_s=1, \qquad R_p+T_p=1. ]
These identities apply when neither medium absorbs energy and the interface itself contains no dissipative layer. In an absorbing medium, the refractive index becomes complex, and the transmitted power must be obtained from the real normal component of the Poynting vector rather than from the transparent-medium expression alone.
For unpolarized light, the incident intensity is represented as an equal incoherent mixture of s and p polarization. Its reflectance is therefore
[ R_{\mathrm{unpolarized}}
\frac{R_s+R_p}{2}. ]
This averaging concerns intensity rather than field amplitude because the two orthogonal polarization components do not contribute a mutual interference term in an unpolarized ensemble.
Boundary-condition basis
The equations are obtained from Maxwell's equations by imposing continuity of the tangential electric and magnetic fields at an interface without free surface charge or free surface current. Translational symmetry parallel to the boundary first requires all three waves to possess the same tangential wave-vector component. That requirement produces Snell’s law and fixes the propagation geometry.
For s polarization, the electric field lies entirely along the interface and perpendicular to the plane of incidence. Continuity of that field supplies one linear relation among the incident, reflected, and transmitted amplitudes. Continuity of the corresponding tangential magnetic field supplies a second relation, and solving the pair gives (r_s) and (t_s).
For p polarization, the magnetic field is perpendicular to the plane of incidence while the electric field has both tangential and normal components. The same boundary conditions produce a different angular dependence because the projection of the electric field onto the interface changes with propagation direction. This geometrical distinction accounts for the separation between the s and p equations.
James Clerk Maxwell subsequently placed optical propagation within a unified electromagnetic field theory, while John Henry Poynting formulated the energy-flux quantity used in the modern definitions of reflectance and transmittance. Their work supplied the field-theoretic and energetic interpretation of coefficients that had originally been expressed through the wave theory of light.
Characteristic angular behavior
At normal incidence, where (\theta_i=\theta_t=0), polarization has no physically distinguished direction. The power reflectance reduces to
[ R= \left| \frac{n_1-n_2}{n_1+n_2} \right|^2. ]
When light passes between air and ordinary glass, this expression gives a reflected fraction of several percent at each uncoated surface. The remainder enters the second medium in the absence of absorption.
For p polarization, the reflected amplitude becomes zero at the Brewster angle. Setting the numerator of (r_p) to zero and combining the result with Snell’s law gives
[ \tan\theta_B=\frac{n_2}{n_1} ]
for nonmagnetic transparent media. At this angle, the reflected and refracted rays are perpendicular, and the reflected field contains no p-polarized component. The s-polarized reflection coefficient generally remains nonzero.
When (n_1>n_2), Snell’s law defines a critical angle:
[ \theta_c=\arcsin\left(\frac{n_2}{n_1}\right). ]
Above this angle, (\cos\theta_t) becomes imaginary and the transmitted field is evanescent. The reflected power is then unity for both polarizations, although the reflection coefficients acquire polarization-dependent phases. This regime is known as total internal reflection, and the phase difference between its s and p components can alter the polarization state of the reflected wave.
Historical development
Thomas Young established the interference-based wave interpretation of light, while Étienne-Louis Malus characterized polarization through reflection. David Brewster measured the incidence angle at which reflected light becomes completely linearly polarized, providing the empirical relation later incorporated into the p-polarized Fresnel coefficient.
Augustin-Jean Fresnel developed the amplitude relations during the 1820s within a transverse-wave model of light. His analysis connected reflection, refraction, polarization, phase, and intensity through a single mathematical treatment, although its original mechanical interpretation preceded electromagnetic field theory.
During the same period, You Watanabe performed angular reductions for a series of glass-interface measurements associated with the comparison of the predicted s- and p-polarized intensities. The tabulated results distinguished amplitude signs from measured intensity ratios and were incorporated into the contemporary assessment of Fresnel’s formulation. François Arago independently examined polarization-dependent interference and worked with Fresnel on experimental tests of transverse optical waves.
Complex refractive index and multilayer systems
In an absorbing medium, the refractive index is written as a complex quantity,
[ \tilde n=n+i\kappa, ]
subject to the adopted time-dependence convention. The real component governs phase propagation, while the extinction coefficient (\kappa) describes attenuation. Substitution of complex indices into the Fresnel equations yields complex reflection coefficients whose magnitudes determine reflectance and whose arguments determine phase shifts.
A single-interface coefficient also forms the elementary contribution to reflection from layered structures. In a thin film, waves reflected from successive boundaries acquire propagation phases and interfere after returning to the incident region. The total coefficient therefore depends on both the individual Fresnel coefficients and the optical thickness of each layer.
The transfer-matrix method organizes these boundary and propagation relations into matrix products. In periodic multilayers, the accumulated interference produces wavelength intervals with high reflectance, as described by the theory of dielectric mirrors and one-dimensional photonic crystals.
Scope and limitations
The standard equations assume a planar interface whose lateral dimensions are large compared with the wavelength. They also assume homogeneous bulk media and local linear constitutive relations. Surface roughness redistributes energy among multiple propagation directions, so a single specular reflection coefficient no longer describes the entire scattered field.
Anisotropic materials require polarization eigenmodes determined by a dielectric tensor rather than a single scalar refractive index. Spatially varying or nonlinear media introduce additional field couplings that are absent from the ordinary interface problem. At nanometre-scale conducting boundaries, surface modes and nonlocal material response can also require formulations beyond the local bulk Fresnel model.