# Absorption of Light

## Absorption of Light

the decrease in intensity of optical radiation (light) as it passes through a material medium owing to its interaction with the medium. In the process of absorption, the energy of the light is converted to different forms of internal energy of the medium; it may be completely or partially reemitted by the medium at frequencies other than the frequency of the absorbed radiation.

The principal law describing light absorption is the Bouguer-Lambert law

I = I0l-kλI

which gives the relation between the intensity I of the light transmitted through a layer of a medium of thickness l and the intensity I0 of the initial light flux. The coefficient kλ, which is independent of I, I0, and l, is called the absorption coefficient; it generally differs for different wavelengths of light λ. The law was experimentally established in 1729 by P. Bouguer and derived theoretically from very simple assumptions in 1760 by J. Lambert. In essence, it states that when a light flux passes through a layer of a substance, the fraction of the radiation absorbed depends only on the absorption coefficient and the thickness of the layer. Thus, in differential notation, which is equivalent to the original form of the Bouguer-Lambert law, dI/I = –kλdl. The physical significance of the law is that the absorption coefficient is independent of I and l, a result verified experimentally by S. I. Vavilov for an approximately 1020-fold variation in I.

Figure 1. Schematic representation of several pairs of absorption lines in sodium vapor. The set of lines corresponds to the set of natural oscillation frequencies of “optical” electrons in the atom. Up to 50 pairs of such lines are observed in sodium, but for the sake of simplicity only three are illustrated here. Since the absorption maxima are extremely narrow, the scale of the figure is greatly distorted.

The dependence of kλ on λ is called the absorption spectrum of the substance. For isolated atoms—for example, in rarefied gases—the spectrum has the form of a set of narrow lines; that is, kλ is nonzero only in certain narrow wavelength bands with widths of tenths or hundredths of an angstrom (Å). The bands correspond to the natural oscillation frequencies of electrons within the atoms; these electrons resonate with the transmitted radiation and therefore absorb energy from it (Figure 1). The absorption spectra of individual molecules also correspond to natural frequencies—the natural frequencies of oscillations of the atoms themselves within the molecules. The atoms are much heavier than the electrons, and their oscillations are much slower. Molecular absorption spectra occupy considerably wider wavelength regions, called absorption bands, with widths ranging from a few angstroms to thousands of angstroms. Finally, light absorption by liquids and solids is usually characterized by very broad bands—thousands and tens of thousands of angstroms in width—with large values of kλ, which varies continuously (Figure 2). This effect is, qualitatively speaking, a result of the strong interaction between particles in condensed media: the energy given by the light to a single particle is rapidly imparted to a whole set of particles. In other words, not only individual particles but numerous bonds between particles resonate with the light wave. This fact is exemplified by the change in light absorption by molecular gases with increasing pressure—the higher the pressure (the stronger the interaction of the particles), the more diffuse the absorption bands, which at high pressures become similar to the absorption spectra of liquids.

Figure 2. Schematic representation of a broad absorption band

Bouguer expressed the conviction that not the thickness but the mass of matter contained in this thickness is important for light absorption. Later, in 1852, the German scientist A. Beer experimentally confirmed this hypothesis. He showed that when light is absorbed by the molecules of a gas or of a substance dissolved in an essentially nonabsorbing solvent, the absorption coefficient is proportional to the number of absorbing molecules per unit volume and consequently per unit path length of the light wave. In other words, the absorption coefficient is proportional to the concentration c: kλ = κλc, a relationship known as Beer’s law. Thus, the law of light absorption acquired the form of the Bouguer-Lambert-Beer law:

I = I0e-kλeI

where κλ is independent of the concentration and characterizes a molecule of the absorbing substance. The physical meaning of Beer’s law is that light absorption by molecules is independent of the molecules’ interaction with their environment. In real gases—even at low pressures—and in solutions, numerous deviations from the law are observed.

The above applies to media of comparatively small optical thickness, equal, if scattering of the light is disregarded, to kλl. As kλl increases, the light absorption by the medium is intensified at all frequencies—the absorption lines and bands broaden. An explanation of this effect is given by the quantum theory of light absorption, which takes into account in particular the multiple scattering of photons in an optically thick medium; the scattering is accompanied by a change in the photons’ frequency and, ultimately, by absorption of the photons by the particles of the medium. For sufficiently large kλl, the medium absorbs as an ideal black body all radiation penetrating into it.

In conducting media, such as metals and plasmas, the light energy is transferred not only to bound electrons but also, often preferentially, to free electrons. In such a medium, kλ is strongly dependent on the medium’s electrical conductivity σ. The strong light absorption in conducting media has a very marked effect on all processes of light propagation within the media. This fact is formally taken into account by the inclusion of a term containing kλ in the expression for the complex index of refraction of the medium. In the somewhat idealized case of light absorption solely by free, that is, conduction, electrons, nkλ = 4 πσ/c, where n is the real part of the refractive index and c is the speed of light. Measurements of light absorption by metals permit many of the metals’ characteristic properties to be determined; the experimental data here are described well by the modern quantum theory of metal optics. The quantity χ is often used in theoretical calculations. The relation between it and kλ is given by the equation

where λ is the wavelength of the light in a vacuum but not in the medium. If (nχ) is equal to 1, then in a layer of the medium of thickness λ the light intensity is decreased to 1/e—that is, to approximately 1/100,000—of its initial value. Very strong light absorption is characteristic of metals, at least in the visible and infrared regions of the spectrum. Accordingly, at M. Planck’s suggestion, the light absorption by media with () ≥ 1 might be called metallic absorption of light.

In terms of the quantum theory, during light absorption the electrons in absorbing atoms, ions, molecules, or solids move from lower energy levels to higher levels. A reverse transition to the ground state or to a lower excited state can be accomplished with or without emission of a photon. In the latter case, the energy of the excited particle can—as in a collision with another particle—be converted into kinetic energy of the colliding particles. The type of reverse transition determines the form of energy in the medium into which the energy of the absorbed light is converted.

In light fluxes of extremely high intensity, light absorption by many media ceases to obey the Bouguer-Lambert law—kλ begins to depend on I. The relation between I and I0 becomes nonlinear (nonlinear light absorption). This effect may arise, in particular, because a very large fraction of the absorbing particles, in moving to an excited state and remaining there for a comparatively long time, experiences a change in or total loss of the ability to absorb light; such a change, of course, substantially alters the character of the light absorption by the medium. It should be noted that Vavilov’s experiments showing the observance of the Bouguer-Lambert law even at high intensities were carried out with substances whose molecules were excited for very brief periods of approximately 10-8 sec. The fraction of excited molecules in the substances was therefore always small.

Of particular interest is the artificial creation in an absorbing medium of a population inversion of energy levels, where the number of particles excited to the higher level is greater than the number in the lower level. In this case, the probability that each photon in the incident flux will cause the emission of another photon of precisely the same type is higher than the probability that the photon will be absorbed. As a result, the intensity of the emerging flux I exceeds the intensity of the incident flux I0—that is, light amplification occurs. This phenomenon formally corresponds to a negative kλ in the Bouguer-Lambert law and therefore is called negative absorption. The operation of optical quantum amplifiers and optical quantum generators (lasers) is based on the negative absorption of light.

Extensive use is made of light absorption in various fields of science and technology. For example, many highly sensitive methods of quantitative and qualitative chemical analysis—such as absorption spectral analysis, spectrophotometry, and colorimetry—are based on light absorption. The form of the absorption spectrum can be related to the chemical structure of the substance and used for such purposes as establishing the presence of certain bonds (for example, the hydrogen bond) in molecules, investigating the nature of the flow of electrons in metals, and elucidating the band structure of semiconductors. The absorption coefficient can be determined in either transmitted or reflected light, since the intensity and polarization of reflected light depend on kλ, as expressed by the Fresnel equations.

### REFERENCES

Landsberg, G. S. Optika, 4th ed. Moscow, 1957. (Obshchii kurs fiziki, vol. 3.)
Born, M., and E. Wolf. Osnovy optiki, 2nd ed. Moscow, 1973. (Translated from English.)
El’iashevich, M. A. Atomnaia i molekuliarnaia spektroskopiia. Moscow, 1962.
Heitler, W. Kvantovaia teoriia izlucheniia. Moscow, 1956. (Translated from English.)
Sokolov, A. V. Opticheskie svoistva metallov. Moscow, 1961.
Moss, T. Opticheskie svoistva poluprovodnikov. Moscow, 1961. (Translated from English.)

A. P. GAGARIN

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