Coefficients of Fresnel

See also: Fresnel

The coefficients of Fresnel , introduced by Augustin Jean Fresnel (1788-1827), intervene in the description of the phenomenon of reflection - Réfraction of the electromagnetic waves with the interface between two mediums, from which the index of refraction is different. They express the bonds between the amplitudes of the waves considered and transmitted compared to the amplitude of the incidental wave.

For that one introduces the coefficient of reflection in amplitude R and the coefficient of transmission in amplitude T of the electric field such as:

$r\; =\; \backslash \; frac\; \{E\_\; \{R\}\}\; \{E\_\; \{I\}\}\; \backslash \; \backslash \; \backslash \; and\; \backslash \; \backslash \; \backslash \; T\; =\; \backslash \; frac\; \{E\_\; \{T\}\}\; \{E\_\; \{I\}\}$

where E_i, E_r and E_t are the amplitudes associated respectively with the incidental, reflected and transmitted electric field (refracted).

In general, these coefficients depend:

• of the constant dielectric of the mediums of entry and exit, respectively ε₁ and ε₂
• of the frequency F of the wave incidence
• of the angles of incidence θ_i=θ₁ and refraction-transmission θ_t=θ₂,
• of the polarization of the waves. What brings to a possible polarization of a wave initially nonpolarized.

They are obtained by considering the relations of continuity to the interface of the tangential components of the magnetic electric fields and associated with the wave.

Calculations of the coefficients in a usual case

Let us consider 2 mediums, different indexes of refraction, separated by a plane interface. The incidental wave is a plane wave, of vector of wave $\backslash \; vec\; \{K\}$, and of pulsation ω.

The coefficients of Fresnel depend on the polarization of the electromagnetic field, one considers 2 cases in general:

• Transverse electric (TE): the incidental electric field is polarized perpendicular to the plan of incidence, the magnetic field is contained in the plan of incidence.

• Transverse magnetic (TM): the incidental magnetic field is polarized perpendicular to the plan of incidence, the electric field is contained in the plan of incidence.

Working hypotheses

The coefficients of Fresnel calculated here are valid only under the following assumptions on the mediums

• the mediums considered are nonmagnetic

• the mediums considered are linear, homogeneous and isotropic

One adds also a assumption design to knowing the harmonic assumption which consists with to consider electromagnetic fields at a particular frequency, and to note them like the real parts of complex sizes. This simplifies calculations and allows also to deduce from the equations in an esthetic way of the electromagnetic phenomena like absorption, the dephasing of the wave, the waves évanescentes…

Case of the electric transverse waves

Let us consider an electromagnetic plane wave:

$\backslash \; vec\; \{E\}\; \backslash \; =\; \backslash \; E\; \backslash \; exp\; \{I\; (\backslash \; vec\; \{K\}.\; \backslash \; vec\; \{R\}\; -\; \backslash \; omega.t)\}\; \backslash \; \backslash \; vec\; \{E\}\; \_\; \{\}$ where E represents the amplitude complexes there

$\backslash \; vec\; \{H\}\; \backslash \; =\; \backslash \; \backslash \; frac\; \{1\}\; \{I\; \backslash \; Omega\; \backslash \; driven\}\; (\backslash \; vec\; \{K\}\; \backslash \; times\; \backslash \; vec\; \{E\})$

If the incidental electric field is polarized perpendicular to the plan of incidence , the tangential of electric field and magnetic component are continuous:

$E\_i\; +\; E\_r\; \backslash \; =\; \backslash \; E\_t\; \backslash \; Rightarrow\; 1\; +\; r\_\; \{TE\}\; \backslash \; =\; \backslash \; t\_\; \{TE\}$

$k\_\; \{iz\}\; \backslash \; (1\; -\; r\_\; \{TE\})\; \backslash \; =\; \backslash \; k\_\; \{tz\}\; \backslash \; t\_\; \{TE\}$

The coefficients of transmission and reflection are written then:

By introducing, for each medium, the relation of dispersion $K\; =\; \backslash \; frac\; \{\backslash \; Omega\}\; \{C\}\; \backslash ,\; N$, one obtains the coefficients of Fresnel according to the characteristic of the incidence (n₁, θ₁) and of the refraction (n₂, θ₂):

Discussion: the indexes of refraction being complex, the polarization of the transmitted and reflected wave can be modified compared to the incidental wave. Even if these indices would be real, in the case $n\_\; \{2\}\; \backslash \; >\; \backslash \; n\_\; \{1\}$, it may be that the coefficient of reflection becomes negative, the considered wave is then out of phase of 180° compared to the incidental wave (see figure).
The only way of cancelling the coefficient of reflection is, by taking account of the Lois of Snell-Descartes, to have $n\_1\; =\; n\_2$. Consequently, an electric wave polarized transverse undergoes a reflection as soon as it passes in a medium of different index optical, which is not the case of a magnetic transverse wave (existence of a Angle of Brewster).

Case of the magnetic transverse waves

By introducing, for each medium, the relation of dispersion $K\; =\; \backslash \; frac\; \{\backslash \; Omega\}\; \{C\}\; \backslash ,\; N$, one obtains the coefficients of Fresnel according to the characteristic of the incidence (n₁, θ₁) and of the refraction (n₂, θ₂):

Note: it should be noted that according to the works, the signs of the coefficients of Fresnel differ. This comes from the arbitrary orientations made at the beginning. For example, to direct towards the H_r figure forwards, amounts replacing, for the calculation of R , E_r by - E_r what will change the sign of the coefficient.

Discussion: the case TM is remarkable for two reasons:

• the coefficient of reflection can become null for an angle of incidence, known as Angle of Brewster;

• in certain situations (interface metal-air), the denominator of the coefficient of reflection TM can become null (the coefficient becomes infinite!). One then obtains a considered wave and a wave transmitted without incidental wave: the study of the denominator specifies the conditions of realization then, the components of the vectors of waves are then imaginary. The process thus employs the waves évanescentes, and causes the appearance of the plasmons of surface.

Difference between the dielectric and metal mediums

The coefficients of Fresnel should be different for the dielectric ones and the metal ones, since the presence or not of currents and loads free in the mediums does not imply the same relations of passage, therefore the same coefficients of Fresnel. However in the case of much of metals known as " ohmiques" (described by one conductivity σ), it is possible to replace a homogeneous ohmic metal (ε, μ₀, σ) by dielectric homogeneous of permittivity $\backslash \; epsilon\_\; \{EFF\}\; =\; \backslash \; epsilon\; -\; \backslash \; frac\; \{I\; \backslash \; sigma\}\; \{\backslash \; Omega\}$. By this metal-dielectric equivalent description in harmonic mode, one can consider the same expressions for the coefficients of Fresnel, whether it is dielectric or an ohmic metal. In this case it is the expression of the permittivity which changes.

See too

Related articles

• Electrodynamic of the continuous mediums

• Interferometer of Fabry-Pérot
• Surfaces with high impedance

External bonds

An interesting site on the coefficients of Fresnel

Reference books

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