Fresnel equations

These ratios are generally complex, describing not only the relative amplitudes but also the phase shifts at the interface.

In the diagram on the right, an incident plane wave in the direction of the ray IO strikes the interface between two media of refractive indices n1 and n2 at point O.

The angles that the incident, reflected and refracted rays make to the normal of the interface are given as θi, θr and θt, respectively.

The second form of each equation is derived from the first by eliminating θt using Snell's law and trigonometric identities.

For the case of light traversing from a less dense medium into a denser one at 45° incidence (θ = 45°), it follows algebraically from the above equations that Rp equals the square of Rs:

This relationship is only valid for the simple case of a single plane interface between two homogeneous materials, not for films on substrates, where a more complex analysis is required.

[citation needed] This is useful because measurements at normal incidence can be difficult to achieve in an experimental setup since the incoming beam and the detector will obstruct each other.

The above equations relating powers (which could be measured with a photometer for instance) are derived from the Fresnel equations which solve the physical problem in terms of electromagnetic field complex amplitudes, i.e., considering phase shifts in addition to their amplitudes.

Those underlying equations supply generally complex-valued ratios of those EM fields and may take several different forms, depending on the formalism used.

In the case of an interface into an absorbing material (where n is complex) or total internal reflection, the angle of transmission does not generally evaluate to a real number.

On the other hand, calculation of the power transmission coefficient T is less straightforward, since the light travels in different directions in the two media.

In the case of total internal reflection where the power transmission T is zero, t nevertheless describes the electric field (including its phase) just beyond the interface.

These phase shifts are different for s and p waves, which is the well-known principle by which total internal reflection is used to effect polarization transformations.

[17] Although at normal incidence these expressions reduce to 0/0, one can see that they yield the correct results in the limit as θi → 0.

An example of interference between reflections is the iridescent colours seen in a soap bubble or in thin oil films on water.

A quantitative analysis of these effects is based on the Fresnel equations, but with additional calculations to account for interference.

In 1815, the dependence of the polarizing angle on the refractive index was determined experimentally by David Brewster.

[20] But the reason for that dependence was such a deep mystery that in late 1817, Thomas Young was moved to write: [T]he great difficulty of all, which is to assign a sufficient reason for the reflection or nonreflection of a polarised ray, will probably long remain, to mortify the vanity of an ambitious philosophy, completely unresolved by any theory.

[29] Four weeks before he presented his completed theory of total internal reflection and the rhomb, Fresnel submitted a memoir [30] in which he introduced the needed terms linear polarization, circular polarization, and elliptical polarization,[31] and in which he explained optical rotation as a species of birefringence: linearly-polarized light can be resolved into two circularly-polarized components rotating in opposite directions, and if these propagate at different speeds, the phase difference between them — hence the orientation of their linearly-polarized resultant — will vary continuously with distance.

In order to compute meaningful Fresnel coefficients, we must assume that the medium is (approximately) linear and homogeneous.

where ϵ and μ are scalars, known respectively as the (electric) permittivity and the (magnetic) permeability of the medium.

Solving for k gives As usual, we drop the time-dependent factor e−iωt, which is understood to multiply every complex field quantity.

Let the angle of refraction, measured in the same sense, be θt, where the subscript t stands for transmitted (reserving r for reflected).

The corresponding dot products in the phasor form (3) are Hence: For the s polarization, the E field is parallel to the z axis and may therefore be described by its component in the z direction.

Then, since H = YE, At the interface, by the usual interface conditions for electromagnetic fields, the tangential components of the E and H fields must be continuous; that is, When we substitute from equations (8) to (10) and then from (7), the exponential factors cancel out, so that the interface conditions reduce to the simultaneous equations which are easily solved for rs and ts, yielding and At normal incidence (θi = θt = 0), indicated by an additional subscript 0, these results become and At grazing incidence (θi → 90°), we have cos θi → 0, hence rs → −1 and ts → 0.

Inside a lossless dielectric (the usual case), E and H are in phase, and at right angles to each other and to the wave vector k; so, for s polarization, using the z and xy components of E and H respectively (or for p polarization, using the xy and −z components of E and H), the irradiance in the direction of k is given simply by EH/2, which is E2/2Z in a medium of intrinsic impedance Z = 1/Y.

To compute the irradiance in the direction normal to the interface, as we shall require in the definition of the power transmission coefficient, we could use only the x component (rather than the full xy component) of H or E or, equivalently, simply multiply EH/2 by the proper geometric factor, obtaining (E2/2Z)cos θ.

Note that when comparing the powers of two such waves in the same medium and with the same cos θ, the impedance and geometric factors mentioned above are identical and cancel out.

In the case of an interface between two lossless media (for which ϵ and μ are real and positive), one can obtain these results directly using the squared magnitudes of the amplitude transmission coefficients that we found earlier in equations (14) and (22).

Substituting cos θi for sin θt in Snell's law, we readily obtain for Brewster's angle.