“Mathematical descriptions of opacity”的意思、由来-开放百科全书

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.

Background: unattenuated wave

Description

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

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

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:

Complex conjugate ambiguity

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.^[1] The two definitions are complex conjugates of each other.

Attenuation coefficient

One way to incorporate attenuation into the mathematical description of the wave is via an attenuation coefficient:^[2]

The attenuation coefficient, in turn, is simply related to several other quantities:

Penetration depth and skin depth

Penetration depth

Skin depth

Physically, the penetration depth is the distance which the wave can travel before its intensity reduces by a factor of 1/e

0.37. The skin depth is the distance which the wave can travel before its amplitude reduces by that same factor.

The absorption coefficient is related to the penetration depth and skin depth by

Complex angular wavenumber and propagation constant

Complex angular wavenumber

Another way to incorporate attenuation is to use the complex angular wavenumber:^[4]^[6]

In accordance with the ambiguity noted above, some authors use the complex conjugate definition:^[7]

Propagation constant

A closely related approach, especially common in the theory of transmission lines, uses the propagation constant:^[8]^[9]

Comparing the two equations, the propagation constant and the complex angular wavenumber are related by:

This quantity is also called the attenuation constant,^[7]^[13] sometimes denoted α.

Unfortunately, the notation is not always consistent. For example,

is sometimes called "propagation constant" instead of γ, which swaps the real and imaginary parts.^[11]

Complex refractive index

Recall that in nonattenuating media, the refractive index and angular wavenumber are related by:

A complex refractive index can therefore be defined in terms of the complex angular wavenumber defined above:

In accordance with the ambiguity noted above, some authors use the complex conjugate definition, where the (still positive) extinction coefficient is minus the imaginary part of

.^[1]^[12]

Complex electric permittivity

In nonattenuating media, the electric permittivity and refractive index are related by:

In attenuating media, the same relation is used, but the permittivity is allowed to be a complex number, called complex electric permittivity:^[2]

AC conductivity

Another way to incorporate attenuation is through the electric conductivity, as follows.^[13]

One of the equations governing electromagnetic wave propagation is the Maxwell-Ampere law:

where σ is the (real, but frequency-dependent) electrical conductivity, called AC conductivity.

If the current J was not included explicitly (through Ohm's law), but only implicitly (through a complex permittivity), the quantity in parentheses would be simply the complex electric permittivity. Therefore,

Notes

1. ^¹For the definition of complex refractive index with a positive imaginary part, see [https://books.google.com/books?id=K9YJ950kBDsC&pg=PA6 Optical Properties of Solids, by Mark Fox, p. 6]. For the definition of complex refractive index with a negative imaginary part, see [https://books.google.com/books?id=qFl1mSZTtIcC&pg=PA588 Handbook of infrared optical materials, by Paul Klocek, p. 588].
2. ^¹²Griffiths, section 9.4.3.
3. ^IUPAC Compendium of Chemical Terminology
4. ^¹Griffiths, section 9.4.1.
5. ^Jackson, Section 5.18A
6. ^¹Jackson, Section 7.5.B
7. ^¹{{cite book|url=https://books.google.com/books?id=Uq924mcshMkC&pg=PA35|page=35|title=Integrated Photonics|isbn=978-0-470-84868-5|author1=Lifante|first1=Ginés|year=2003}}
8. ^"Propagation constant", in ATIS Telecom Glossary 2007
9. ^{{cite book|url=https://books.google.com/books?id=AzLYk1qaaz8C&pg=PA93 |page=93|title=Adv Imaging and Electron Physics|isbn=978-0-08-057758-6|date=1995-03-27|volume=92|author1=P. W. Hawkes |author2= B. Kazan}}
10. ^¹{{cite book|url=https://books.google.com/books?id=KpY1hpKKwdQC&pg=PA132 |page=132|title=Electric Power Transmission and Distribution|isbn=9788131707913|date=2008-09-01|author=S. Sivanagaraju}}
11. ^See, for example, Encyclopedia of laser physics and technology
12. ^Pankove, pp. 87-89
13. ^Jackson, section 7.5C