Complex refractive index

A single real refractive index describes a transparent material. To include absorption, the index is made complex,

so that a plane wave $\exp [i(\tilde{n}{k}_{0}z-\omega t)]$ with vacuum wavenumber $k_0=2\pi \mathrm{/}\lambda$ decays as it travels. The real part $n$ fixes the phase velocity and refraction; the imaginary part $\kappa$, called the extinction coefficient, fixes the loss. (Some texts, especially in engineering, write $n-i\kappa$ with the opposite time convention; the physics is the same.)

Absorption coefficient

The field decays as $\exp (-\kappa k_{0}z)$, so the intensity decays as $\exp (-\alpha z)$ with

the absorption coefficient of the Beer-Lambert law. Small values of $\kappa$ matter: at 1550 nm, $\kappa ={10}^{-4}$ gives $\alpha$ = 8.1 cm⁻¹, a loss of 35 dB/cm, far more than any useful waveguide can tolerate, and a low-loss waveguide at 1 dB/cm corresponds to $\kappa$ of about 3 × 10⁻⁶. Strongly absorbing materials, such as metals, have $\kappa$ of order 1 or more, and light penetrates only a fraction of a wavelength: $\kappa$ = 3.4 at 633 nm gives an intensity penetration depth $1\mathrm{/}\alpha$ of 15 nm. Semiconductors just above their band gap absorb far less: $\alpha$ of about ${10}^4$ cm⁻¹ is $\kappa$ of about 0.07 at 870 nm and a penetration depth of 1 µm, and $\kappa$ approaches 1 only well above the gap.

Relation to permittivity and reflection

The complex index is the square root of the complex relative permittivity, $\tilde{\epsilon }={\epsilon }_1+i{\epsilon }_2={\tilde{n}}^2$, so ${\epsilon }_1=n^2-{\kappa }^2$ and ${\epsilon }_2=2n\kappa$. The Fresnel equations hold with $\tilde{n}$ in place of $n$; at normal incidence from air the reflectance is

4% for glass ($n$ = 1.5, $\kappa$ = 0) and 94% for a metal-like index of $0.2+3.4i$; the Fresnel equations calculator extends this to any angle and polarization. Metals are reflective mainly because $\kappa$ is large. The real and imaginary parts are not independent: the Kramers-Kronig relations connect the whole absorption spectrum to the dispersion of $n$, which is how absorption changes such as the free-carrier effect in silicon come with index changes.

Measurement

For thin films, $n$ and $\kappa$ are measured by spectroscopic ellipsometry, which fits the change in polarization on reflection to a layer model, or by fitting reflection and transmission spectra. For bulk samples with weak absorption, $\kappa$ follows from transmission through two thicknesses (removing the surface reflections), and $n$ from a prism goniometer or refractometer. For strongly absorbing materials, reflection measurements over a wide spectrum are converted to $n$ and $\kappa$ through the Kramers-Kronig relations.

References: M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), Ch. 14; E. D. Palik (ed.), Handbook of Optical Constants of Solids (Academic Press, 1985); H. Fujiwara, Spectroscopic Ellipsometry (Wiley, 2007).