On the optical characterization of media
(introductory notes)

Among the fundamental characteristics of a material/medium such as chemical composition, density, conductivity, viscosity, one should probably give the first place to the optical constants - the refractive index $n$ and the extinction coefficient $k$. These quantities describe the interaction of the electromagnetic radiation with a medium and well reacts to variations of its composition and structure. Therefore, the optical methods of measurements of $n$ and $k$, having a high accuracy, speed, and possibility of non-destructing and distant control, became wide spread in practice of the physical/chemical analysis.

The optical characteristics are known to be in particular sensitive to changes of the state of materials in the regions of absorption bands. The position of spectral bands is characterized by one of the following quantities: the frequency $f$, the wavelength $\lambda = c/f$ or the wavenumber $\nu = 1/\lambda$. Instead of the frequency $f$ one can use also the circular frequency $\omega = 2 \pi f$.

In the physical optics one usually considers the electromagnetic radiation in the wavelength interval from 1 nm to 1 mm. Generally, there are the following range notation: 10-200 nm is called the vacuum ultraviolet (vacUV) spectral region, that of 200-400 nm the ultraviolet (UV) one, that of 400-800 nm the visual one, that of 800 nm and further the infrared (IR) one. In some regions radiation is often characterized by the photon energy $\sigma$ in units of electron-Volt (eV) equal to the energy of an electron passed the potential difference of one volt (1 eV = 8.0659 $10^3 cm^{-1}$).

The bands in the UV, visual and near IR regions are usually connected with electron absorption, and those in the middle and far IR regions with molecular and lattice absorptions. Between the bands there are transparent regions where the values of $k$ are usually close to 0.

The electromagnetic radiation propagating in a medium is characterazed by the amplitudes of the electric $\vec{E}$ and magnetic $\vec{H}$ vector oscillations, frequency, polarization state and direction of propagation determined by the wavevector $\vec{k}$. The expression for a monochromatic plane wave in an isotropic medium is

\begin{displaymath} \vec{E} = \vec{E}_0 \exp \left\{- 2 \pi {\rm i} f \left[t +... ...t(n+k{\rm i}\right) \frac{\vec{k}\vec{r}}{c} \right]\right\}, \end{displaymath} (1)

where $t$ is time, and $\vec{r}$ the radius-vector.

The parameter $m = n+k$i in Eq.(1) describes the optical properties of the medium where radiation propagates. The quantity $m$ is called the complex refractive index, its real part $n$ is equal to the ratio of the velocity of electromagnetic radiation in vacuum to the phase velocity of radiation in the medium, and its imaginary part $k$ characterizes the decrease of the radiation intensity in the medium because of absorption.

The refractive index $n$ and the extinction coefficient $k$ of a medium are functions of the frequency, and in the case of anisotropic media $n$ and $k$ also depend on the direction of radiation propagation $\vec{k}$. In the last case the values of $n$ and $k$ are tensors of the second order.

The intensity of radiation in a medium is proportional to $\vert E_0\vert^2$ and its absorption is described by the Bouguer-Lambert-Beer law. According to this law, the intensity of light $I$ after its propagation through a material slab of the thickness $d$ is connected with the initial intensity $I_0$ as follows:

\begin{displaymath} I = I_0 \exp\left( - \alpha d\right). \end{displaymath} (2)

The value $A = (I_0 - I) / I_0$ being the ratio of the intensity absorbed by the slab to the incident one is called the absorption coefficient.

From Eqs.(1)-(2) one obtains the expression for the absorption coefficient

\begin{displaymath} \alpha = 4 \pi k / \lambda = 4 \pi \nu k. \end{displaymath} (3)

To characterize absorption by a medium, one also uses the quantity $T = I/I_0$ 100 (in percent) being the ratio of the intensityof the transmitted radiation to that of the incident one and called the transmission coefficient (or transmittance).

For values of $k < 10^{-4}$, to describe the optical properties one often uses the quantity $D = \log T^{-1}$ called the optical density.

The optical constants $n$ and $k$ allow one to estimate not only transmittance and absorption of radiation, but also its reflection at the border of two media. To calculate the reflection coefficient (or reflectance) from the values of $n$ and $k$ for two bordering media and the incidence angle $\theta$ one can usitilize the Fresnel formula. Note that the values of the transmission ($T$), absorption ($A$) and reflection ($R$) coefficients, which characterize the properties of a sample, are connected

\begin{displaymath} T + A + R = 1 . \end{displaymath} (4)

In the electromagnetic theory one often uses not $n$ and $k$, but the dielectric functions (or permittivity) defined as $\varepsilon = m^2$. As it follows from its definition,

\begin{displaymath} \varepsilon = \varepsilon _1 + \varepsilon _1 {\rm i}, \end{displaymath} (5)

where $\varepsilon _1 = n^2 - k^2, \varepsilon _2 = 2 n k$.

In an anisotropic medium, the permittivity is a symmetrical tensor and can be transformed to the main axes

\begin{displaymath} \langle \varepsilon \rangle = \begin{tabular}[c]{\vert ccc... ...on _b$\ & 0 \\ 0 & 0 & $\varepsilon _c$\ \\ \end{tabular} \end{displaymath} (6)

where $\varepsilon _a, \varepsilon _b, \varepsilon _c$ are the principle values of the permittivity tensor, and consequently, $n_a = \sqrt{\varepsilon _a}, n_b = \sqrt{\varepsilon _b}, n_c = \sqrt{\varepsilon _c}$ are the principal refractive indices.

According to the optical properties, crystals are divided in 3 groups:

  1. cubic crystals when $n_a = n_b = n_c$ (these crystals are optically isotropic);
  2. single-axis crystals when $n_a = n_b \ne n_c$ (these crystals are trigonal, tetragonal and hexagonal. The direction for which $n_a = n_b$ is called the optical axis. For a wave propagating in this direction birefringence does not occur);
  3. biaxial crystals when $n_a \ne n_b \ne n_c$ (these crystals are monoclinic, triclinic and rhombic).

For single-axis crystals, the coinciding values of the refractive index are called the refractive index of the ordinary ray, the third principal refractive index is the principal refractive index of the extraordinary ray.

Like isotropic media, crystals are characterized not only by the refractive indices, but also by the absorption indices that can be combined in a tensor complex refractive index

\begin{displaymath} \langle m_{ik} \rangle = \langle (n + k {\rm i})_{ik} \rangle, \end{displaymath} (7)

where the principal axes of the $\langle n \rangle$ and $\langle k \rangle$ can be different.

The theory of interaction of light with a material that accounts for the frequency dependence of the optical functions $n, k, \varepsilon _1, \varepsilon _2$ was developed by Lorentz and Drude. Despite the difference with the contemporary quantum-mechanical approach, the results of this classical theory remain formally valid and correctly describe the main features of the process of interaction of radiation with a medium.

A material medium in the classical theory is described by a set of harmonic oscillators characterized by the eigenfrequency $f_0$, the oscillator strength $\rho$ and the damping $\gamma$. Using the Maxwell equations for electromagnetic radiation and the equations of the classical mechanics for description of oscillator charge motion, one can obtain the expression for the frequency dependence of the dielectric permittivity

\begin{displaymath} \varepsilon _1 = n^2 - k^2 = \varepsilon _\infty + \sum_j... ...j}^2 - f^2} {(f_{0j}^2 - f^2)^2 + f_{0j}^2 f^2 \gamma_j^2}, \end{displaymath} (8)


\begin{displaymath} \varepsilon _2 = 2 n k = \sum_j 4 \pi \rho_j f_{0j}^2 \fr... ... \gamma_j } {(f_{0j}^2 - f^2)^2 + f_{0j}^2 f^2 \gamma_j^2}, \end{displaymath} (9)

where $j$ is the number of the oscillator.

Denoting the high-frequency dielectric permittivity by $\varepsilon _\infty$, the expression for $\varepsilon $ can written via $f_{\rm LO}$ and $f_{\rm TO}$ being the frequencies of the longitudal and transversal oscillations of crystal lattice, respectively. Taking into account the damping of the oscillations ( $\gamma_{\rm LO}$ and $\gamma_{\rm TO}$), allowing one to describe the anharmonicity, the dielectric permittivity is

\begin{displaymath} \varepsilon = \varepsilon _\infty \prod_j \frac{f_{{\rm L... ...j}} {f_{{\rm TO}j}^2 - f^2 - {\rm i} f \gamma_{{\rm TO}j}}. \end{displaymath} (10)

Using these equations one can describe the values of $n$ and $k$ in a wide spectral range both for isotropic and anisotropic media.

For more information on the optical constant theory we refer the reader to the chapter of the classical C.F.Bohren & D.R.Huffman book [1] and a recent good textbook by M.Fox [2]. Some technical aspects of obtaing the optical constants are described briefly in another chapter of the book [1] and in more detail, e.g., in the books edited by E.D.Palik [3,4,5].

The optical constants for atmospheric aerosols are well presented in the HITRAN database (see http://www.hitran.com). Those for materials of astronomical interest are collected by us in the JPDOC database (http://www.astro.spbu.ru/JPDOC). This WWW database includes references to the papers where the optical constants were measured or calculated and original data obtained in the laboratory of the Friedrich Schiller University, Jena. A description of the JPDOC is given in [6,7].

(After V.M.Zolotarev)

 References:

   1.
Bohren C.F., Huffman D.R. (1983) Absorption and Scattering of Light by Small Particles. J.Wiley & Sons, New York.

   2.
Fox M. (2001) Optical Properties of Solids. Oxford Univ. Press.

   3.
Palik E.D. (ed.) (1985) Handbook of Optical Constants of Solids. Academic Press, New York.

   4.
Palik E.D. (ed.) (1991) Handbook of Optical Constants of Solids, II. Academic Press, New York.

   5.
Palik E.D. (ed.) (1998) Handbook of Optical Constants of Solids, III. Academic Press, New York.

   6.
Henning Th., Il'in V.B., Krivova N.A., Michel B., and Voshchinnikov N.V. (1999) WWW database of optical constants for astronomy. Astron. Astrophys. Suppl. 136, 405.

   7.
Jaeger C., Il'in V.B., Henning Th., Mutschke H., Fabian D., Semenov D.A., and Voshchinnikov N.V. (2003) A database of optical constants of cosmic dust analogs. J. Quant. Spectrosc. Rad. Transf., in press.
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Created by V.I.
Last modified: 20/02/03, V.I.