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Relative permittivity and permeability

The refractive index of electromagnetic radiation equals n = ε r μ r, {\displaystyle n={\sqrt {\varepsilon _{\mathrm {r} }\mu _{\mathrm {r} }}},} where ε is the material's relative permittivity, and μ is its relative permeability. The refractive index is used for optics in Fresnel equations and Snell's law; while the relative permittivity and permeability are used in Maxwell's equations and electronics. Most naturally occurring materials are non-magnetic at optical frequencies, that is μ is very close to 1, therefore n is approximately √ ε. In this particular case, the complex relative permittivity ε, with real and imaginary parts ε and ɛ̃, and the complex refractive index n, with real and imaginary parts n and κ (the latter called the "extinction coefficient"), follow the relation ε _ r = ε r + i ε ~ r = n _ 2 = (n + i κ) 2, {\displaystyle {\underline {\varepsilon }}_{\mathrm {r} }=\varepsilon _{\mathrm {r} }+i{\tilde {\varepsilon }}_{\mathrm {r} }={\underline {n}}^{2}=(n+i\kappa)^{2},}

and their components are related by: ε r = n 2 − κ 2, ε ~ r = 2 n κ, {\displaystyle {\begin{aligned}\varepsilon _{\mathrm {r} }&=n^{2}-\kappa ^{2}\,,\\{\tilde {\varepsilon }}_{\mathrm {r} }&=2n\kappa \,,\end{aligned}}}

and:

n = | ε _ r | + ε r 2, κ = | ε _ r | − ε r 2. {\displaystyle {\begin{aligned}n&={\sqrt {\frac {|{\underline {\varepsilon }}_{\mathrm {r} }|+\varepsilon _{\mathrm {r} }}{2}}},\\\kappa &={\sqrt {\frac {|{\underline {\varepsilon }}_{\mathrm {r} }|-\varepsilon _{\mathrm {r} }}{2}}}.\end{aligned}}}

where | ε _ r | = ε r 2 + ε ~ r 2 {\displaystyle |{\underline {\varepsilon }}_{\mathrm {r} }|={\sqrt {\varepsilon _{\mathrm {r} }^{2}+{\tilde {\varepsilon }}_{\mathrm {r} }^{2}}}} is the complex modulus.