Penetration depth

It is defined as the depth at which the intensity of the radiation inside the material falls to 1/e (about 37%) of its original value at (or more properly, just beneath) the surface.

This electromagnetic field interacts with the atoms and electrons inside the material.

For a given material, penetration depth will generally be a function of wavelength.

According to Beer–Lambert law, the intensity of an electromagnetic wave inside a material falls off exponentially from the surface as If

denotes the penetration depth, we have Penetration depth is one term that describes the decay of electromagnetic waves inside of a material.

Since the power of a wave in a particular medium is proportional to the square of a field quantity, one may speak of a penetration depth at which the magnitude of the electric (or magnetic) field has decayed to 1/e of its surface value, and at which point the power of the wave has thereby decreased to

is identical to the skin depth, the latter term usually applying to metals in reference to the decay of electrical currents (which follow the decay in the electric or magnetic field due to a plane wave incident on a bulk conductor).

is also identical to the (negative) real part of the propagation constant, which may also be referred to as

It can also be ambiguous as to whether a positive number describes attenuation (reduction of the field) or gain; this is usually obvious from the context.

The attenuation constant for an electromagnetic wave at normal incidence on a material is also proportional to the imaginary part of the material's refractive index n. Using the above definition of

is the radian frequency of the radiation, c is the speed of light in vacuum and

is very much a function of frequency, as is its imaginary part which is often not mentioned (it is essentially zero for transparent dielectrics).

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

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.