Fresnel equations

The Fresnel formulas (after Augustin Jean Fresnel ) describe quantitatively the reflection and transmission of a plane electromagnetic wave at a plane interface. The first calculated reflection and transmission factor is the ratio of the reflected or transmitted amplitude to that of the incident wave. By squaring we obtain the reflection and the transmission factor, representing as energy intensity ratios of sizes.

• 4.1 Vertical incidence
• 4.2 Discussion of the amplitude ratios

Previews

The Fresnel formulas can be derived from the Maxwell's equations, this one uses special cases of the boundary conditions of electromagnetic waves in a charge- and current-free boundary layer:

Here, the normal to the interface, and the other sizes to describe the magnetic field and electric field in the two media. The tangential component of the electric field strength E and the magnetic field strength H are continuous at the boundary as well as the normal component of the electric flux density D and the magnetic flux density B ( tangentially and normally refers to the interface ).

Depending on the polarization of the incident wave, different boundary conditions for the incidence of an electromagnetic wave in an optical interface. Each arbitrarily polarized electromagnetic wave can be represented as a superposition of two linearly polarized waves that oscillate perpendicular to each other. As a reference plane, the plane of incidence, which is defined by the wave vector of the incident wave and the surface normal is used. One incident, randomly polarized wave can be written so as a superposition of a parallel (p ) and perpendicular ( s ) polarized to the incident plane wave:

Here, the field vector of the electric field are the unit vectors for s- and p- polarization, and the parameters corresponding to any phase shifts.

Due to the superposition principle, it is sufficient to calculate the amplitude ratios for parallel and perpendicular to the plane of incidence, linearly polarized waves.

The polarization direction (vertical or parallel to the plane of incidence ) will remain unchanged after the reflection.

General case

In the general case, both media have a different permittivity and permeability as well as a complex refractive index

Preview for equations with eliminiertem angle of refraction

In general necessary for the calculation of the degrees of reflection or transmission with the Fresnel formulas for both the refractive index of the media involved, as well as the angle of incidence and refraction.

To specify addition to these general equations also independent of the angle of refraction shape, the angle of refraction of the general form must be eliminated. Since both angles (and ) linked by the Snell's law, this can be achieved ( by means of a limiting case ) as follows:

Squaring supplies (using a trigonometric conversion ) the following relation:

Changed in these scenarios:

The case is used with the positive sign, so that later is the reflection factor r ≤ 1 as a solution.

Vertical polarization

Looking at the first component, the linear vertical (index s ) is polarized to the incident plane. It is referred to in the literature as transverse electric (TE) component.

The transmittance and reflectance. Here, the coefficient related to the electric field.

Parallel polarization

In the other case, the amplitude of the incident plane linearly parallel (index p) is considered polarized wave. It is referred to in the literature as transverse magnetic (TM) component. Here, the coefficients related to the magnetic field.

The directions of the electric field vectors corresponding respectively to the directions of the vectors and, wherein the normal vector of the plane of incidence.

Special case: the same magnetic permeability

For frequent in practice special case that the involved materials have approximately the same magnetic permeability (), e.g., for non-magnetic materials, the Fresnel formulas simplify as follows:

Special case: dielectric materials

Another special case is obtained for an ideal dielectric, in which the absorption coefficient of the complex refractive index is equal to zero. That is, the material on both sides of the interface absorbs the appropriate electromagnetic radiation is not (). The following applies:

By eliminating the complex parts of the Fresnel formulas simplify as follows:

Note: The second equal sign is obtained by applying the law of refraction and addition theorems. The assumptions made ​​here are not valid for angles of incidence of 0 ° and 90 ° and the formulas can not therefore be used. This purpose must be used from pure cosine the original form

Normal incidence

A further simplification is obtained for the case that the incident angle α is equal to 0 (normal incidence ):

If, for example visible light perpendicular to the interface air / quartz glass, the proportion is

The incident intensity independent of the polarization reflected ( see section related to the reflection and transmission coefficients).

Discussion of the amplitude ratios

There, where the amplitude coefficients are real and negative, a phase jump of on ( no phase change in real and positive ) occurs:

The amplitude ratio has a zero crossing at the Brewster angle:

Examples: Brewster angle for the air - glass and glass - air.

For the amplitude conditions are complex from a certain angle. From this critical angle or critical angle total reflection occurs. The critical angle corresponds to the angle of refraction so, that is, the wave travels at the interface along.

Example: critical angle for glass air.

Related to the reflection and transmission coefficients

Consider a beam of radiation which irradiates the surface of the interface. Thus, the beam cross sections of the incident, reflected and transmitted beam are the same, respectively. The energy per unit time and unit area by a surface whose normal ( for isotropic media the same Ausbreitungsrichutung ) is parallel to the direction of energy flow, flows, is given by the Poynting vector:

The mean energy flux density is obtained by averaging over time:

The average power, which is transported by the bundle of rays per unit time ( average power, impinging on area) corresponds to the average energy flux density times the cross-sectional area, so

Or

General ( unpolarized light) (often referred to with ρ ) defines the reflectance R as follows:

( often referred to with τ ) and transmittance T as:

The two coefficients can now be calculated using the Fresnel formulas, they are the product of the corresponding reflection and transmission factor with its complex conjugate value.

For ideal dielectrics which do not have the absorption coefficient and the reflection coefficients are real, the equations simplify to:

With the s- and p- polarized component.

Without absorption following energy flow balance applies to the individual polarization directions:

With the total amplitude (i for the incident, reflected or transmitted wave) applies without absorbing the energy flow balance: