The attenuation constant for an electromagnetic wave at normal incidence on a material is also proportional to the imaginary part of the materials refractive index n'. Using the above definition of   (based on intensity) the following relationship holds:
where  denotes the complex index of refraction,  is the radian frequency of the radiation, c is the speed of light in vacuum and  is the wavelength. Note that   is very much a function of frequency, as is its imaginary part which is often not mentioned (it is essentially zero for transparent dielectrics). The complex refractive index of metals is also infrequently mentioned but has the same significance, leading to a penetration depth (or skin depth ) accurately given by a formula which is valid up to microwave frequencies.
Relationships between these and other ways of specifying the decay of an electromagnetic field can be expressed by mathematical descriptions of opacity.
This is only specifying the decay of the field which may be due to absorption of the electromagnetic energy in a lossy medium or may simply describe the penetration of the field in a medium where no loss occurs (or a combination of the two). For instance, a hypothetical substance may have a complex index of refraction  . A wave will enter that medium without significant reflection and will be totally absorbed in the medium with a penetration depth (in field strength) of , where   is the vacuum wavelength. A different hypothetical material with a complex index of refraction   will also have a penetration depth of 16 wavelengths, however in this case the wave will be perfectly reflected from the material! No actual absorption of the radiation takes place, however the electric and magnetic fields extend well into the substance. In either case the penetration depth is found directly from the imaginary part of the material's refractive index as is detailed above.