Mathematical descriptions of opacity

When an electromagnetic wave travels through a medium in which it gets attenuated (this is called an "opaque" or "attenuating" medium), it undergoes exponential decay as described by the Beer–Lambert law. However, there are many possible ways to characterize the wave and how quickly it is attenuated. This article describes the mathematical relationships among:

Note that in many of these cases there are multiple, conflicting definitions and conventions in common use. This article is not necessarily comprehensive or universal.

An electromagnetic wave propagating in the +z-direction is conventionally described by the equation:

where

The wavelength is, by definition,

For a given frequency, the wavelength of an electromagnetic wave is affected by the material in which it is propagating. The vacuum wavelength (the wavelength that a wave of this frequency would have if it were propagating in vacuum) is

where c is the speed of light in vacuum.

In the absence of attenuation, the index of refraction (also called refractive index) is the ratio of these two wavelengths, i.e.,

The intensity of the wave is proportional to the square of the amplitude, time-averaged over many oscillations of the wave, which amounts to:

Note that this intensity is independent of the location z, a sign that this wave is not attenuating with distance. We define I₀ to equal this constant intensity:

Because

either expression can be used interchangeably. Generally, physicists and chemists use the convention on the left (with e^−iωt), while electrical engineers use the convention on the right (with e^+iωt, for example see electrical impedance). The distinction is irrelevant for an unattenuated wave, but becomes relevant in some cases below. For example, there are two definitions of complex refractive index, one with a positive imaginary part and one with a negative imaginary part, derived from the two different conventions. The two definitions are complex conjugates of each other.

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