Refractive Index - Complex Index of Refraction and Absorption

Complex Index of Refraction and Absorption

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 index of refraction,

Here, the real part of the refractive index indicates the phase speed, while the imaginary part indicates the amount of absorption loss when the electromagnetic wave propagates through the material.

That 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 -direction. We can do this by relating the wave number to the refractive index through, with being the vacuum wavelength. With complex wave number and refractive index this can be inserted into the plane wave expression as

Here we see that gives an exponential decay, as expected from the Beer–Lambert law. Since intensity is proportional to the square of the electric field, the absorption coefficient becomes .

κ is often called the extinction coefficient in physics although this has a different definition within chemistry. Both n and κ are dependent on the frequency. In most circumstances (light is absorbed) or (light travels forever without loss). In special situations, especially in the gain medium of lasers, it is also possible that, corresponding to an amplification of the light.

An alternative convention uses instead of, but where 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 versus . 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 (κ ≈ 0). However, at higher frequencies (such as visible light), dielectric loss may increase absorption significantly, reducing the material's transparency to these frequencies.

The real and imaginary parts of the complex refractive index are related through the Kramers–Kronig relations. For example, one can determine a material's full complex refractive index as a function of wavelength from an absorption spectrum of the material.

For X-ray and extreme ultraviolet radiation the complex refractive index deviates only slightly from unity and usually has a real part smaller than 1. It is therefore normally written as (or ).