Fresnel coefficients (for example, Hecht E 1987, p. 94-96) relate the amplitudes of the electric field of a plane monochromatic electromagnetic wave in two semi-infinite uniform media with refractive indices, m1 and m2, respectively (relative to vacuum). The wave is incident at the plane interface between the two media from within medium 1, and is both reflected (coefficient r) into the same medium and transmitted (coefficient t) into medium 2. The simplified definitions of these coefficients given here apply to media with magnetic permeabilities approximately equal to that of vacuum (µ0, magnetic constant).
The reflection coefficients are as follows:
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r× =
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=
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m1 cosθ1 - m2 cosθ2
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m1 cosθ1 + m2 cosθ2
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(1)
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r || =
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=
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m2 cosθ1 - m1 cosθ2
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m2 cosθ1 + m1 cosθ2
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(2)
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where the subscript i denotes the incident wave, r denotes the reflected wave (in the same medium), Ê0 is the amplitude of the electric field of the wave, θ1 and θ2 are the incidence and refraction angles respectively. These angles are measured between the normal to the interface and the direction of the wave. Symbols × and || denote the components of the electric field vector that are respectively perpendicular to the incidence plane and parallel to it. The incidence plane is defined by the incident, reflection, and refraction directions and is assumed to be perpendicular to the interface between the two media.
The transmission coefficients are as follows (the subscript t denotes the transmitted wave):
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t× =
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=
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2m1 cosθ1
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m1 cosθ1 + m2 cosθ2
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(3)
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t || =
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=
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2m1 cosθ1
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m2 cosθ1 + m1 cosθ2
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(4)
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If the refractive index, m, of either medium or both media is complex, then the Fresnel coefficients are also complex quantities. Furthermore, in addition to their dependence on the incidence angle, these coefficients are also functions of the wavelength of the electromagnetic radiation, because of the wavelength-dependence of the refractive index.
