Complex refractive index and the Fresnel equations

The concept of refractive index and its variation with wavelength is well known when applied to transparent materials. So is the idea of absorbance and the way it varies with wavelength for coloured materials and metals. It is less widely understood that dispersion and absorption of light are properly considered together and can be represented as the real and imaginary parts of a single quantity, the complex refractive index, N. 

For a transparent material, the imaginary part (the extinction coefficient k) is zero it is simply related to the absorption coefficient a, measured by transmission, by 

The reflection and transmission (refraction) of light obliquely incident on the interface between two isotropic media is entirely controlled by the angle of incidence and the complex refractive indices of the media, being described by the Fresnel reflection and refraction equations (see Appendix). Originally worked out for transparent materials, these equations apply with complete generality when the refractive indices are complex rather than simple numbers. If the refractive indices are complex numbers, the angles of refraction must also be complex. For a description of the meaning of such quantities, see ref. 3. 

The form of the Fresnel equations is deceptively simple. They encapsulate the different reflectivity in the p and s planes referred to, but also describe such phenomena as internal reflection, where the refractive index of the incident medium is higher than that of the substrate, and total and frustrated total internal reflection. 

While it is comparatively easy to measure the modulus of the reflection coefficient (or rather its square), it is difficult to measure the phase change of the separate p and s components. It is, however, feasible to measure both 

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