Fresnel Equations - Definitions and Power Equations

Definitions and Power Equations

In the diagram on the right, an incident light ray IO strikes the interface between two media of refractive indices n₁ and n₂ at point O. Part of the ray is reflected as ray OR and part refracted as ray OT. The angles that the incident, reflected and refracted rays make to the normal of the interface are given as θ_i, θ_r and θ_t, respectively.

The relationship between these angles is given by the law of reflection:

and Snell's law:

The fraction of the incident power that is reflected from the interface is given by the reflectance R and the fraction that is refracted is given by the transmittance T. The media are assumed to be non-magnetic.

The calculations of R and T depend on polarisation of the incident ray. Two cases are analyzed:

The incident light is s-polarized. That means its electric field is in the plane of the interface (perpendicular to the plane of the diagram above).
The incident light is p-polarized. That means its electric field is in a perpendicular direction to s-polarized above (in the plane of the diagram above).

For the s-polarized light, the reflection coefficient is given by

$R_\mathrm{s} = \left|\frac{n_1\cos\theta_{\mathrm{i}}-n_2\cos\theta_{\mathrm{t}}}{n_1\cos\theta_{\mathrm{i}}+n_2\cos\theta_{\mathrm{t}}}\right|^2 =\left|\frac{n_1\cos\theta_{\mathrm{i}}-n_2\sqrt{1-\left(\frac{n_1}{n_2} \sin\theta_{\mathrm{i}}\right)^2}}{n_1\cos\theta_{\mathrm{i}}+n_2\sqrt{1-\left(\frac{n_1}{n_2} \sin\theta_{\mathrm{i}}\right)^2}}\right|^2$ ,

where the second form is derived from the first by eliminating θ_t using Snell's law and trigonometric identities.

For the p-polarized light, the R is given by

$R_\mathrm{p} = \left|\frac{n_1\cos\theta_{\mathrm{t}}-n_2\cos\theta_{\mathrm{i}}}{n_1\cos\theta_{\mathrm{t}}+n_2\cos\theta_{\mathrm{i}}}\right|^2 =\left|\frac{n_1\sqrt{1-\left(\frac{n_1}{n_2} \sin\theta_{\mathrm{i}}\right)^2}-n_2\cos\theta_{\mathrm{i}}}{n_1\sqrt{1-\left(\frac{n_1}{n_2} \sin\theta_{\mathrm{i}}\right)^2}+n_2\cos\theta_{\mathrm{i}}}\right|^2$ .

As a consequence of the conservation of energy, the transmission coefficients are given by

and

These relationships hold only for power coefficients, not for amplitude coefficients as defined below.

If the incident light is unpolarised (containing an equal mix of s- and p-polarisations), the reflection coefficient is

For common glass, the reflection coefficient is about 4%. Note that reflection by a window is from the front side as well as the back side, and that some of the light bounces back and forth a number of times between the two sides. The combined reflection coefficient for this case is 2R/(1 + R), when interference can be neglected (see below).

The discussion given here assumes that the permeability μ is equal to the vacuum permeability μ₀ in both media. This is approximately true for most dielectric materials, but not for some other types of material. The completely general Fresnel equations are more complicated.

Read more about this topic: Fresnel Equations

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