What is the refractive index?

The refractive index (n) of a material is a dimensionless quantity that describes how much light is bent when it passes from one medium to another. The refractive index is defined as the ratio of the speed of light in a vacuum to the speed of light in the material:

n = c/v
where c is the speed of light in a vacuum and v is the speed of light in the material.

What is the absorption coefficient?

The absorption coefficient (α) of a material is a measure of how much light is absorbed by the material as it passes through it. The absorption coefficient is proportional to the concentration of absorbing species in the material and the cross-section of the absorbing species for absorbing light. The absorption coefficient is usually expressed in inverse length units (e.g., cm⁻¹).

The relationship between the refractive index and the absorption coefficient is given by the Kramers-Kronig relations, which state that the real and imaginary parts of a material's response (such as the refractive index and the absorption coefficient) are related through a mathematical relationship.

Kramers-Kronig relations

The Kramers-Kronig equation is a mathematical expression that describes the relationship between the real and imaginary parts of a material's response to an applied electromagnetic field. It is based on the Kramers-Kronig relations, which state that the real and imaginary parts of a material's response are mathematically related.

The Kramers-Kronig equation is usually written as:

f(ω) = 1/π * P∫∞_0 (g(ξ)/ξ - g(ω)) / (ξ - ω) dξ

where f(ω) is the real part of the response, g(ω) is the imaginary part of the response, and ω is the angular frequency of the applied field. The symbol P represents the Cauchy Principal Value of the integral, which means that the integral is taken in such a way as to avoid the singularity at ω = ξ.

The Kramers-Kronig equation can be used to predict the real part of a material's response from measurements of its imaginary part, and vice versa. This makes it a useful tool in the analysis of optical and electromagnetic phenomena, as well as in the study of quantum systems.

It should be noted that the Kramers-Kronig equation is a mathematical relationship and not a physical law. The equation assumes that the response of the material is a linear and causal function of the applied field, which is true for many materials but not for all. The validity of the equation should always be carefully checked before using it to predict the properties of a material.