Penetration depth
Encyclopedia
Penetration Depth is a measure of how deep light or any electromagnetic radiation
Electromagnetic radiation
Electromagnetic radiation is a form of energy that exhibits wave-like behavior as it travels through space...

 can penetrate into a material. 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.

When electromagnetic radiation is incident on the surface of a material, it may be (partly) reflected from that surface and there will be a field containing energy transmitted into the material. This electromagnetic field interacts with the atoms and electrons inside the material. Depending on the nature of the material, the electromagnetic field might travel very far into the material, or may die out very quickly. For a given material, penetration depth will generally be a function of wavelength
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...

.

According to Beer-Lambert law
Beer-Lambert law
In optics, the Beer–Lambert law, also known as Beer's law or the Lambert–Beer law or the Beer–Lambert–Bouguer law relates the absorption of light to the properties of the material through which the light is travelling.-Equations:The law states that there is a logarithmic dependence between the...

, 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 but one term that describes the decay of electromagnetic waves inside a material. The above definition refers to the depth at which the intensity or power of the field decays to 1/e of its surface value. In many contexts one is concentrating on the field quantities themselves: the electric and magnetic fields in the case of electromagnetic waves. 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 or about 13% of its surface value:


Note that is identical to the skin depth, the latter term usually applying to metals in reference to the decay of electrical currents
Electric current
Electric current is a flow of electric charge through a medium.This charge is typically carried by moving electrons in a conductor such as wire...

 (which follow the decay in the electric or magnetic field due to a plane wave incident on a bulk conductor). The attenuation constant is also identical to the (negative) real part of the propagation constant
Propagation constant
The propagation constant of an electromagnetic wave is a measure of the change undergone by the amplitude of the wave as it propagates in a given direction. The quantity being measured can be the voltage or current in a circuit or a field vector such as electric field strength or flux density...

, which may also be referred to as using a notation inconsistent with the above use. When referencing a source one must always be careful to note whether a number such as or refers to the decay of the field itself, or of the intensity (power) associated with that field. It can also be ambiguous as to whether a positive number describes attenuation
Attenuation
In physics, attenuation is the gradual loss in intensity of any kind of flux through a medium. For instance, sunlight is attenuated by dark glasses, X-rays are attenuated by lead, and light and sound are attenuated by water.In electrical engineering and telecommunications, attenuation affects the...

 (reduction of the field) or gain
Gain
In electronics, gain is a measure of the ability of a circuit to increase the power or amplitude of a signal from the input to the output. It is usually defined as the mean ratio of the signal output of a system to the signal input of the same system. It may also be defined on a logarithmic scale,...

; 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 (based on intensity) the following relationship holds:


where denotes the complex index of refraction, is the radian frequency of the radiation, and c is the speed of light in vacuum. 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 signficance, 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 are further detailed in the article: Mathematical descriptions of opacity
Mathematical descriptions of opacity
When an electromagnetic wave travels through a medium in which it gets absorbed , 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 absorbed...

.

It should also be noted that we are only specifying the decay of the field which may be due to absorption
Absorption (electromagnetic radiation)
In physics, absorption of electromagnetic radiation is the way by which the energy of a photon is taken up by matter, typically the electrons of an atom. Thus, the electromagnetic energy is transformed to other forms of energy for example, to heat. The absorption of light during wave propagation is...

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 detailed above.
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