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Complex refractive index

When light passes through a medium, some part of it will always be absorbed. This can be conveniently taken into account by defining a complex refractive index, n _ = n + i κ. {\displaystyle {\underline {n}}=n+i\kappa.}

Here, the real part n is the refractive index and indicates the phase velocity, while the imaginary part κ is called the extinction coefficient indicates the amount of attenuation when the electromagnetic wave propagates through the material. It is related to the absorption coefficient, α abs {\displaystyle \alpha _{\text{abs}}}, through: α abs (ω) = 2 ω κ (ω) c {\displaystyle \alpha _{\text{abs}}(\omega)={\frac {2\omega \kappa (\omega)}{c}}} These values depend upon the frequency of the light used in the measurement.

That κ corresponds to absorption can be seen by inserting this refractive index into the expression for electric field of a plane electromagnetic wave traveling in the x -direction. This can be done by relating the complex wave number k to the complex refractive index n through k = 2π n / λ, with λ being the vacuum wavelength; this can be inserted into the plane wave expression for a wave travelling in the x -direction as: E (x, t) = Re = Re = e − 2 π κ x / λ 0 Re. {\displaystyle {\begin{aligned}\mathbf {E} (x,t)&=\operatorname {Re} \!\left\\&=\operatorname {Re} \!\left\\&=e^{-2\pi \kappa x/\lambda _{0}}\operatorname {Re} \!\left.\end{aligned}}}

Here we see that κ gives an exponential decay, as expected from the Beer–Lambert law. Since intensity is proportional to the square of the electric field, intensity will depend on the depth into the material as

I (x) = I 0 e − 4 π κ x / λ 0. {\displaystyle I(x)=I_{0}e^{-4\pi \kappa x/\lambda _{0}}.}

and thus the absorption coefficient is α = 4π κ / λ, and the penetration depth (the distance after which the intensity is reduced by a factor of 1/ e) is δ = 1/ α = λ /4π κ.

Both n and κ are dependent on the frequency. In most circumstances κ > 0 (light is absorbed) or κ = 0 (light travels forever without loss). In special situations, especially in the gain medium of lasers, it is also possible that κ < 0, corresponding to an amplification of the light.

An alternative convention uses n = n + iκ instead of n = n − iκ, but where κ > 0 still corresponds to loss. Therefore, these two conventions are inconsistent and should not be confused. The difference is related to defining sinusoidal time dependence as Re versus Re. See Mathematical descriptions of opacity.

Dielectric loss and non-zero DC conductivity in materials cause absorption. Good dielectric materials such as glass have extremely low DC conductivity, and at low frequencies the dielectric loss is also negligible, resulting in almost no absorption. However, at higher frequencies (such as visible light), dielectric loss may increase absorption significantly, reducing the material's transparency to these frequencies.

The real n, and imaginary κ, parts of the complex refractive index are related through the Kramers–Kronig relations. In 1986, A.R. Forouhi and I. Bloomer deduced an equation describing κ as a function of photon energy, E, applicable to amorphous materials. Forouhi and Bloomer then applied the Kramers–Kronig relation to derive the corresponding equation for n as a function of E. The same formalism was applied to crystalline materials by Forouhi and Bloomer in 1988.

The refractive index and extinction coefficient, n and κ, are typically measured from quantities that depend on them, such as reflectance, R, or transmittance, T, or ellipsometric parameters, ψ and δ. The determination of n and κ from such measured quantities will involve developing a theoretical expression for R or T, or ψ and δ in terms of a valid physical model for n and κ. By fitting the theoretical model to the measured R or T, or ψ and δ using regression analysis, n and κ can be deduced.