Linear Optical Properties

Abstract

This chapter provides an extended overview of the optical properties of glasses. In Sect. 5.1 the underlying physical background of light–matter interaction is presented, where the phenomena of refraction, reflection, absorption, emission and scattering are introduced.

Most oxide glasses are transparent in the visible spectral range. This obvious fact, confirmed by every look through a window, is based on two highly nontrivial principles: (i) the existence of an electronic bandgap and (ii) the (nearly) complete absence of light scattering. Although transparent solid materials like single crystals and glasses have been known for thousands of years, understanding the existence of an electronic bandgap, an energy range where practically no absorption of electromagnetic radiation occurs, requires quantum mechanics, which just became 100 years old. If such a forbidden zone is larger than the photon energy of blue photons, the photons with the largest energy quantum in the visible spectral range, the material is visibly transparent.

$$E_{\text{band\_gap}}> h\nu_{\text{blue}}\;,$$

with $h\nu_{\text{blue}}=hc/\lambda_{\text{blue}}={\mathrm{3.18}}\,{\mathrm{eV}}$, where $h$ is Planck's constant, $c$ is the speed of light in a vacuum (or air), $\lambda$ is the wavelenght and $\nu$ is the frequency. The energy is given here in units of eV (${\mathrm{1}}\,{\mathrm{eV}}={\mathrm{1.602\times 10^{-19}}}\,{\mathrm{J}}$).

The (nearly) complete absence of light scattering in glasses has its origin in the fact that as opposed, e. g., to most ceramic materials, glasses are isotropic and extremely homogeneous on all length scales relevant for the interaction with visible light. Also, while glasses have well-defined structure on the atomic scale, a few ångstroms ($\mathrm{\AA}$), they are completely disordered and therefore homogeneous and isotropic on the larger length scales relevant for the interaction with visible light.

1 Interaction of Light with Optical Materials

In this section the general physics of the interaction of light with matter is briefly presented. A detailed insight into theoretical electrodynamics cannot be given here. The interested reader might refer to standard textbooks on electrodynamics, e. g., [5.1, 5.2].

1.1 Dielectric Function

The starting point for an analysis of any interaction between electromagnetic waves with matter are Maxwell's equations. The static interaction for the dielectric displacement and the magnetic induction is described by

$$\begin{aligned}\displaystyle\boldsymbol{\nabla}\cdot\boldsymbol{D}&\displaystyle=\rho\;,\\ \displaystyle\boldsymbol{\nabla}\cdot\boldsymbol{B}&\displaystyle=0\;,\end{aligned} $$

(5.1)

whereas the dynamic interaction of the electric and magnetic fields is given by

$$\begin{aligned}\displaystyle\boldsymbol{\nabla}\times\boldsymbol{E}&\displaystyle=-\boldsymbol{\dot{B}}\;,\\ \displaystyle\boldsymbol{\nabla}\times\boldsymbol{H}&\displaystyle=\boldsymbol{j}+\boldsymbol{\dot{D}}\;.\end{aligned}$$

(5.2)

$\boldsymbol{E}$ and $\boldsymbol{B}$ are the electric and magnetic fields; $\boldsymbol{D}$ and $\boldsymbol{H}$ are the electric displacement and the auxiliary magnetic fields; $\rho$ and $\boldsymbol{j}$ are the charge and the current density.

Material equations are needed to complete Maxwell's equations and turn them into a closed set of equations

$$\begin{aligned}\displaystyle\boldsymbol{D}&\displaystyle={\varepsilon}_{0}\boldsymbol{E}+\boldsymbol{P}\;,\\ \displaystyle\boldsymbol{B}&\displaystyle={\mu}_{0}\boldsymbol{H}+\boldsymbol{M}\;,\end{aligned} $$

(5.3)

where $\boldsymbol{P}$ and $\boldsymbol{M}$ are the polarization and magnetization densities. The vacuum permittivity (in SI units) is ${\varepsilon}_{0}={\mathrm{8.854\times 10^{-12}}}\,{\mathrm{A{\,}s/(V{\,}m)}}$ and the vacuum permeability is ${\mu}_{0}=4\uppi\times{\mathrm{10^{-7}}}\,{\mathrm{V{\,}s/(A{\,}m)}}$.

The complete optical properties for any spatial combination of matter are included in the solution of (5.1), (5.2), which are closed by using the material equations (5.3) and by using appropriate boundary conditions. For only a few special cases such a solution can be written down directly. In the following we give a few examples.

1.1.1 Wave Equation in a Vacuum

If we want to solve (5.1) and (5.2) in an infinite vacuum we have the following boundary conditions and material equations

$$\boldsymbol{P}(\boldsymbol{r})=0\;,\quad\boldsymbol{M}(\boldsymbol{r})=0\;,\quad\rho(\boldsymbol{r})=0\;,\quad\boldsymbol{j}(\boldsymbol{r})=0\;,$$

(5.4)

where $\boldsymbol{r}=(x,y,z)$ are the three spatial coordinates. With these simplest possible boundary conditions the material equations (5.3) read

$$\begin{aligned}\displaystyle\boldsymbol{D}&\displaystyle={\varepsilon}_{0}\boldsymbol{E}\;,\\ \displaystyle\boldsymbol{B}&\displaystyle={\mu}_{0}\boldsymbol{H}\;.\end{aligned}$$

(5.5)

After applying a few vector operations, one gets the wave equation for the electromagnetic field $\boldsymbol{E}$ in a vacuum

$$\boldsymbol{\Updelta}\boldsymbol{E}-{\mu}_{0}{\varepsilon}_{0}\boldsymbol{\ddot{E}}=0\;.$$

(5.6)

An identical wave equation can be derived for the magnetic field $\boldsymbol{B}$. (5.6) immediately defines the speed of light $c$ (in a vacuum), then

$$c=\sqrt{\frac{1}{{\mu}_{0}{\varepsilon}_{0}}}\;.$$

(5.7)

Equation (5.6) is generally solved by all fields that fulfill $\boldsymbol{E}(\boldsymbol{r},t)=\boldsymbol{E}_{0}\cdot f(\boldsymbol{kr}\pm\omega t)$ involving any arbitrary scalar function $f$. The most common systems of function $f$ are plane waves

$$\boldsymbol{E}_{\text{s}}(\boldsymbol{r},t)=\boldsymbol{E}_{0}\Re(\mathrm{e}^{-\mathrm{i}(\boldsymbol{k}\boldsymbol{r}-\omega t)}).$$

(5.8)

These plane waves, with a time and spatial dependent phase $\theta=\boldsymbol{k}\boldsymbol{r}-\omega t$, are described by a wave vector $\boldsymbol{k}$, an angular frequency $\omega$, and a corresponding wavelength $\lambda=2\uppi/k=2\uppi c/\omega$, where $k=|\boldsymbol{k}|$ is the absolute value of the wave vector. Describing an arbitrary field $\boldsymbol{E}$ in terms of plane waves is identical to decomposing this electrical field into its Fourier components.

1.1.2 Wave Propagationin an Ideal Transparent Medium

We can describe an ideal material by simply replacing the speed of light in a vacuum by that of the medium,

$$c_{\text{med}}\to\frac{c}{n}\;,$$

(5.9)

where $n$ is the (in this case only real) refractive index of the material. Wave propagation in a dispersing or weakly absorbing medium is considered at the end of the present section. In fact, most parts of an optical design can be done by treating optical glasses as such ideal transparent materials (Sect. 5.1.2). Even though such an ideal material cannot exist in reality, optical glasses come very close to it (for electromagnetic radiation in the visible range). For such an ideal material the wave equation (5.6) reads

$$\boldsymbol{\Delta}\boldsymbol{E}-\frac{n^{2}}{c^{2}}\,\boldsymbol{\ddot{E}}=0\;.$$

(5.10)

It is solved again by plane transverse waves, where the speed of light is now reduced to the speed of light in the transparent medium $c_{{\text{med}}}=c/n$ and the wavelength of the light wave is reduced to ${\lambda}_{{\text{med}}}=\lambda/n$.

1.1.3 Refraction and Reflection

We now derive the laws of refraction and reflection for the ideal transparent medium just described. They are obtained by solving Maxwell's equations at the (infinite) boundary between two materials of different refractive indices $n_{1}$ and $n_{2}$ (Fig. 5.1). As boundary conditions one obtains that the normal component of the electric displacement (and magnetic induction) and the tangential component of the electric (and magnetic) field have to be continuous at the interface

$$D^{\text{n}}_{1}=D^{\text{n}}_{2}\;,\quad E^{\text{t}}_{1}=E^{\text{t}}_{2}\;.$$

(5.11)

Further, a phase shift of an incoming wave occurs upon reflection

$${\theta}_{\text{r}}=\uppi-{\theta}_{\text{i}}\;,$$

(5.12)

where ${\theta}_{\text{r,i}}$ are the phases of the reflected and incident wave respectively. If we solve Maxwell's equations for an incoming plane wave (applying the boundary conditions stated above), Snell's law of refraction is obtained

$$n_{1}\sin{\alpha}_{1}=n_{2}\sin{\alpha}_{2}\;,$$

(5.13)

together with that of reflection

$${\alpha}_{\text{r}}={\alpha}_{1}\;.$$

(5.14)

Now the electric field $\boldsymbol{E}$ is decomposed into its components, which are defined relative to the plane outlined by the three beams of incoming, transmitted and reflected light. This decomposition is shown in Fig. 5.1. The coefficients for reflection and transmission are defined as

$$\begin{aligned}\displaystyle r_{\parallel}&\displaystyle=\frac{E_{0{\text{r}}}^{\parallel}}{E_{0{\text{i}}}^{\parallel}}\;;\quad t_{\parallel}=\frac{E_{0{\text{t}}}^{\parallel}}{E_{0{\text{i}}}^{\parallel}}\;,\\ \displaystyle r_{\perp}&\displaystyle=\frac{E_{0{\text{r}}}^{\perp}}{E^{\perp}_{0{\text{i}}}}\;;\quad t_{\perp}=\frac{E_{0{\text{t}}}^{\perp}}{E^{\perp}_{0{\text{i}}}}\;.\end{aligned}$$

(5.15)

The Fresnel formula for these coefficients can be derived as

$$\begin{aligned}\displaystyle r_{\perp}&\displaystyle=\frac{n_{1}\cos({\alpha}_{1})-n_{2}\cos{({\alpha}_{2})}}{n_{1}\cos(\alpha_{1})+n_{2}\cos(\alpha_{2})}=+\frac{\sin(\alpha_{1}-\alpha_{2})}{\sin(\alpha_{1}+\alpha_{2})}\;,\\ \displaystyle r_{\parallel}&\displaystyle=\frac{n_{2}\cos(\alpha_{1})-n_{1}\cos{(\alpha_{2})}}{n_{1}\cos(\alpha_{2})+n_{2}\cos(\alpha_{1})}=-\frac{\tan(\alpha_{1}-\alpha_{2})}{\tan(\alpha_{1}+\alpha_{2})}\;,\\ \displaystyle t_{\perp}&\displaystyle=\frac{2n_{1}\cos(\alpha_{1})}{n_{1}\cos(\alpha_{1})+n_{2}\cos(\alpha_{2})}\\ \displaystyle&\displaystyle=+\frac{2\sin(\alpha_{2})\cos(\alpha_{1})}{\sin(\alpha_{1}+\alpha_{2})}\;,\\ \displaystyle t_{\parallel}&\displaystyle=\frac{2n_{1}\cos(\alpha_{1})}{n_{1}\cos(\alpha_{2})+n_{2}\cos(\alpha_{1})}\\ \displaystyle&\displaystyle=+\frac{2\sin(\alpha_{2})\cos(\alpha_{1})}{\sin(\alpha_{1}+\alpha_{2})\cos(\alpha_{1}-\alpha_{2})}\;.\end{aligned}$$

(5.16)

Here the usual convention has been used that the coefficients of reflectivity obtain an additional minus sign in order to indicate back-traveling of light. The quantities that are measured in an experiment are intensities. The relationship between the intensities defines the reflectivity ($R$) and transmissivity ($T$) of a material

$$\begin{aligned}\displaystyle R_{\perp}&\displaystyle:=|r_{\perp}|^{2}\;;\quad R_{\parallel}:=|r_{\parallel}|^{2}\;,\\ \displaystyle T_{\perp}&\displaystyle:=|t_{\perp}|^{2}\;;\quad T_{\parallel}:=|t_{\parallel}|^{2}\;.\end{aligned}$$

(5.17)

The angular-dependent coefficients of reflection from (5.16) are displayed in Fig. 5.2a,b. In Fig. 5.2a,ba the case of light propagating from an optically thin medium with refractive index $n_{1}$ to an optically thicker medium with refractive index $n_{2}> n_{1}$ is plotted. At the so-called Brewster angle $\alpha_{\text{B}}$ the reflected light is completely polarized; $\alpha_{\text{B}}$ is given by the condition $\alpha_{1}+\alpha_{2}=\uppi/2$. Therefore, the Brewster angle $\alpha_{\text{B}}$ results as a solution of

$$\alpha_{1}=\frac{\uppi}{2}-\arccos\left(\frac{n_{2}}{n_{1}}\cos\alpha_{1}\right),$$

(5.18)

which gives $\alpha_{\mathrm{B}}=\arctan n_{2}/n_{1}$. In Fig. 5.2a,bb the case of light propagating from an optically thick to an optically thin medium is plotted. Here an additional special angle occurs—the angle of total reflection $\alpha_{\text{T}}$. All light approaching the surface at an angle larger than $\alpha_{\text{T}}$ is totally reflected. At $\alpha_{1}=\alpha_{\text{T}}$ the angle for refraction in the medium with refractive index $n_{2}$ is $\alpha_{2}=\uppi/2$. For $\alpha_{\text{T}}$ it follows that

$$\alpha_{\text{T}}=\arcsin\left(\frac{n_{2}}{n_{1}}\right).$$

(5.19)

Evaluating (5.16) for the special case of normal incident light as $\lim_{\alpha\to 0}$ allows one to calculate the reflectivity for normal incidence

$$R_{\text{norm}}=\left|\frac{n_{1}-n_{2}}{n_{1}+n_{2}}\right|^{2}.$$

(5.20)

Note that polarization with respect to a plane of refraction loses its meaning at normal incidence. Therefore (5.20) is valid for all polarization directions. For practical applications it is important to note that the Fresnel equations remain valid in the case of weakly absorbing media discussed at the end of the present section. In this case, only the refractive indices have to be replaced by the complex quantities of (5.36). It is further helpful to define a transmittivity and reflectivity for unpolarized light

$$R^{\text{unpol}}=\frac{E^{\parallel 2}_{0{\text{r}}}+E^{\perp 2}_{0{\text{r}}}}{E^{\parallel 2}_{0{\text{i}}}+E^{\perp 2}_{0{\text{i}}}}\;;\quad T^{\text{unpol}}=\frac{E^{\parallel 2}_{0{\text{t}}}+E^{\perp 2}_{0{\text{t}}}}{E^{\parallel 2}_{0{\text{i}}}+E^{\perp 2}_{0{\text{i}}}}\;.$$

(5.21)

Inserting the expressions for the reflection coefficients we obtain, for example, the total reflectivity as a function of the incident and refracted angular

$$R^{\text{unpol}}_{\text{all}}=\frac{E^{\parallel 2}_{0{\text{i}}}\frac{\tan^{2}(\alpha_{1}-\alpha_{2})}{\tan^{2}(\alpha_{1}+\alpha_{2})}+E^{\perp 2}_{0{\text{i}}}\frac{\sin^{2}(\alpha_{1}-\alpha_{2})}{\sin^{2}(\alpha_{1}+\alpha_{2})}}{E^{\parallel 2}_{0{\text{i}}}+E^{\perp 2}_{0{\text{i}}}}\;.$$

With the definitions from (5.21) the following sum rule must be fulfilled

$$R^{\text{unpol}}+\frac{n_{2}}{n_{1}}\frac{\cos\alpha_{2}}{\cos\alpha_{1}}T^{\text{unpol}}=1\;.$$

(5.22)

The rule provides an easy check for transmitted and reflected total intensities, especially for normal incidence.

Fig. 5.1

The polarization directions of the $\boldsymbol{E}$ and $\boldsymbol{B}$ fields for reflection and refraction at an interface between two optical materials of different refractive indices are shown. A circle indicates that the vector is perpendicular to the plane shown. The $\boldsymbol{B}$ fields (not shown) are always perpendicular to $\boldsymbol{k}$ and $\boldsymbol{E}$

Fig. 5.2a,b

The reflection coefficients are plotted as a function of incident scattering angle for light propagating from a medium of (a) smaller refractive index into a medium of larger index and (b) larger refractive index into a medium of smaller index. Here, total reflection occurs at an angle $\alpha_{\text{T}}$ and $\alpha_{\text{B}}$ is the Brewster angle

1.1.4 Wave Propagation in an Isotropic, Homogeneous Medium

We now consider wave propagation in an ideal optical material. This is a nonmagnetic, homogeneous, isotropic material, which is further a perfectly linear optical material. Considering time dependence, including retardation in the materials, (5.2) leads to

$$\boldsymbol{D}(\boldsymbol{r},t)=\varepsilon_{0}\boldsymbol{E}(\boldsymbol{r},t)+\boldsymbol{P}(\boldsymbol{r},t)\;.$$

(5.23)

The polarizability is related to the electric field via the susceptibility $\chi$ (5.24). In the case of a homogeneous isotropic material $\chi$ is a scalar function. In optically anisotropic media $\chi$ becomes a second-rank tensor

$$\boldsymbol{P}(\boldsymbol{r},t)=\int\mathrm{d}\boldsymbol{r}^{\prime}\int^{t}_{-\infty}\mathrm{d}t^{\prime}\chi(\boldsymbol{r}-\boldsymbol{r}^{\prime},t-t^{\prime})\boldsymbol{E}(\boldsymbol{r},t^{\prime})\;.$$

(5.24)

Fourier transformation in time and space deconvolutes the integral and leads to

$$\boldsymbol{P}(\boldsymbol{k},\omega)=\chi(\boldsymbol{k},\omega)\boldsymbol{E}(\boldsymbol{k},\omega)\;,$$

(5.25)

where $\chi(\boldsymbol{k},\omega)$ is, in general, a complex analytic function of the angular frequency $\omega$. The complex function $\chi(\boldsymbol{k},\omega)$ unifies the two concepts of (i) a low-frequency polarizability $\chi^{\prime}$ and (ii) a low-frequency conductivity $\sigma$ of mobile charges to a single complex quantity

$$\lim_{\omega\to 0}\chi(\omega)=\chi^{\prime}(\omega)+4\uppi\mathrm{i}\frac{\sigma(\omega)}{\omega}\;.$$

(5.26)

At larger frequencies the separation of the two concepts breaks down. A bound charge can have a dielectric response that does not differ from the response of a mobile charge when it is probed at large enough frequencies. For example, above the frequencies of optical phonon modes in the infrared () the bound charges are unable to follow the electric field, whereas below the phonon modes the charges can follow this motion (Sect. 5.1.3). The usual form in which the susceptibility enters the equations for optical purposes is via the dielectric function

$$\varepsilon(\boldsymbol{k},\omega)=1+\chi(\boldsymbol{k},\omega)\;.$$

(5.27)

The dielectric function can now be identified with the square of the refractive index by comparing (5.10) and (5.29). This is done in (5.37). Inserting the dielectric function into the material equation (5.23) after a Fourier transformation in time and space gives

$$\boldsymbol{D}(\boldsymbol{k},\omega)=\varepsilon_{0}\varepsilon(\boldsymbol{k},\omega)\boldsymbol{E}(\boldsymbol{k},\omega)\;.$$

(5.28)

Here, we restrict ourselves to ideal optically isotropic materials by neglecting the nature of the dielectric function as a second-rank tensor. Glasses are isotropic but nearly all lose their isotropy when, e. g., uniaxial stress is applied. With the same steps as in (5.5) and (5.6) a wave equation can be derived, which has the following form in Fourier space

$$\left[k^{2}-\varepsilon(\boldsymbol{k},\omega)\frac{\omega^{2}}{c^{2}}\right]\boldsymbol{E}_{0}=0\;,$$

(5.29)

where the relations $\partial^{2}/\partial t^{2}\boldsymbol{E}_{\text{s}}(\boldsymbol{r},t)\to-\omega^{2}\boldsymbol{E}_{\text{s}}(\boldsymbol{r},t)$ and $\Updelta\boldsymbol{E}_{\text{s}}(\boldsymbol{r},t)\to-k^{2}\boldsymbol{E}_{\text{s}}(\boldsymbol{r},t)$ have been used. The index s refers, as in (5.8), to the system of plane waves. The expression in brackets in (5.29) defines the dispersion relation for an optically linear, homogeneous, isotropic material.

1.1.5 Poynting Vector and Energy Transport

The energy flux density of the electric field is obtained via the Poynting vector, given by

$$\boldsymbol{S}=\boldsymbol{E}\times\boldsymbol{H}\;.$$

(5.30)

It gives the rate at which electromagnetic energy crosses a unit area and has the unit $\mathrm{W{\,}m^{-2}}$. It points in the direction of energy propagation. The time average of the absolute value of the Poynting vector $\overline{|\boldsymbol{S}|}$ is called the intensity $I$ of the electromagnetic wave

$$I=\langle|\boldsymbol{S}|\rangle=\tfrac{1}{2}|\boldsymbol{E}\times\boldsymbol{H}|$$

(5.31)

and is the energy flux density of the electromagnetic radiation. In the special case of propagation of transverse plane waves (as given in (5.8)), it simplifies to

$$I=\frac{1}{2}\frac{n}{c\mu_{0}}|\boldsymbol{E}_{0}|^{2}\;,$$

(5.32)

where in a vacuum $n=1$ is valid.

1.1.6 General Form of the Dielectric Function

For most optical materials the dielectric function has a form in which a transparent frequency (or wavelength) window is bounded at the high energy side by electron–hole excitations (dominating the ultraviolet () edge) and at the low energy site by IR absorptions given by optical phonon modes (lattice vibrations). The general form of the dielectric function is given by the Kramers–Heisenberg equation [5.3]

$$\varepsilon(\boldsymbol{k},\omega)=1+\sum_{j}\frac{\alpha_{\boldsymbol{k},j}}{\omega^{2}-\omega^{2}_{\boldsymbol{k},j}-\mathrm{i}\omega\eta_{\boldsymbol{k},j}}\;.$$

(5.33)

Here $\alpha_{\boldsymbol{k},j}$ is the amplitude, $\omega_{\boldsymbol{k},j}$ is the frequency and $\eta_{\boldsymbol{k},j}$ is the damping of the particular excitation $j$. A schematic view of the dielectric function is plotted in Fig. 5.3. Here we use a model for a transparent, homogeneous, isotropic solid (such as glass) with one generic absorption at low energies ($\omega_{\text{IR}}$ in the infrared, IR) and another one at large photon energies ($\omega_{\text{UV}}$ in the ultraviolet spectral range, UV). In the following this model solid is used to discuss optical material properties.

Fig. 5.3

Dielectric function $\varepsilon(\omega)$ for the model optical solid with one generic absorption in the infrared $\omega_{\text{IR}}$ and a second one in the ultraviolet $\omega_{\text{UV}}$. The dielectric function is plotted on a logarithmic energy scale. The solid line is the real part and the dashed line is the imaginary part of $\varepsilon(\omega)$

1.1.7 Dispersion Relation

Solving (5.29) gives two frequency-dependent solutions for the wave vector $\boldsymbol{k}$ as a function of $\omega$ since the left-hand side of (5.29) is quadratic in $\boldsymbol{k}$. Far away from absorptions the dielectric function is real and positive. Here only one solution exists, which describes the wave propagating with the speed of light in the medium. Close to an absorption two solutions exist, which are even more complicated. This means that near a resonance the dispersion of the light cannot be considered independently from the dispersion of the excitations in the material. They both form a composite new entity propagating in the medium. This is called the polariton [5.4]. For our model solid with two generic absorptions ($\omega_{\text{IR}}$ and $\omega_{\text{UV}})$, the dispersion is plotted in Fig. 5.4. Figure 5.4 shows the solution of (5.29) for the dielectric function $\varepsilon({\boldsymbol{k},\omega})$ plotted in Fig. 5.3. The frequency $\omega$ is plotted as a function of the real and imaginary part of the the wave vector $\boldsymbol{k}$. The dashed line is the dispersion in a vacuum with the speed of light as a derivative $\omega=ck$. In the regions where the imaginary part of $k$ is small there are propagating modes with a speed of light that is smaller than the vacuum speed of light. In the vicinity of absorptions anomalous propagation occurs together with strong damping.

Fig. 5.4

Dispersion relation $\omega(k)$ for the model optical solid on a double logarithmic scale. The solid line is the real part of $\boldsymbol{k}$ and the dashed line is the imaginary part of $\boldsymbol{k}$. For comparison the simply linear dispersion relation for light propagating in a vacuum $\omega=ck$ is shown with a long-dashed line

1.1.8 Wave Propagation, Phase and Group Velocity

When an electromagnetic wave propagates through a medium one can define two velocities. The phase velocity is the speed with which a certain phase propagates. It is, for example, the velocity of the wavefront maxima moving through the medium. The phase velocity is given as

$$v_{\text{ph}}=\frac{\omega}{k}\;.$$

(5.34)

In Fig. 5.5 the phase velocity of the model solid is plotted on a logarithmic frequency scale. Close to the absorption edges of the material it loses its meaning because attenuation due to the absorption processes will dominate most processes. Far away from absorptions it reaches a nearly constant value. The second velocity is the group velocity

$$v_{\text{gr}}(k)=\frac{\partial\omega}{\partial k}\;.$$

(5.35)

This is the velocity at which a complete wave packet travels through the medium and is, hence, the speed with which information can travel through the system. The phase and group velocities are plotted in Fig. 5.5 for the model solid. Reasonably far away from material absorptions the group and phase velocities approach each other. However, the group velocity is always smaller than the phase velocity.

Fig. 5.5

The group velocity $v_{\text{gr}}(k)=\partial\omega/\partial k$ (solid line) and the phase velocity $v_{\text{ph}}(k)=\omega/k$ (dashed line) are plotted on a logarithmic frequency scale. Far away from absorptions both velocities approach each other while the group velocity is always smaller than the phase velocity

1.1.9 Refractive Index

The refractive index $n$ is the most widely used physical quantity in optical design. It is the square root of the dielectric function. The dynamic refractive index is generally a complex quantity

$$\tilde{n}(\omega)=n(\omega)+\mathrm{i}\kappa(\omega)$$

(5.36)

and must fulfill the Kramers–Kronig relations [5.3]. The refractive index for our generic model solid is plotted as a function of logarithmic frequency in Fig. 5.6. In practical use, the wavelength dependence is often exploited

$$\tilde{n}(\boldsymbol{k},\lambda)=\sqrt{\varepsilon\left(\boldsymbol{k},\frac{2\uppi c}{\lambda}\right)}\;.$$

(5.37)

With a few basic steps, (5.33) can be rewritten as a function of wavelength alone. If one further restricts consideration to wavelengths that are far away from absorptions, $(\omega^{2}-\omega^{2}_{k,j})\ll\omega\eta_{\boldsymbol{k},j}$, the Sellmeier formula (Sect. 5.1.2), which is widely used for characterizing optical materials, is obtained

$$n(\lambda)^{2}\approx 1+\sum_{j}\frac{B_{j}\lambda^{2}}{\lambda^{2}-\lambda_{j}^{2}}\;,$$

(5.38)

where $\lambda_{i}=2\uppi c/\omega_{i}$ with $i\in\{(\boldsymbol{k},j)\}$ is used and $B_{j}=a_{k,j}\lambda_{k,j}/(2\uppi c)^{2}$. It describes the dispersion of a material by infinitely sharp absorption lines ($\delta$ functions), which are far away from the visible range at wavelength $\lambda_{j}$ and have the absorption strength $B_{j}$. Normally $B_{j}$ and $\lambda_{j}$ are just fitting constants to describe the dispersion of the refractive index over a certain wavelength range. The Sellmeier formula (5.38) uses three infinitely sharp absorption lines. One in the infrared wavelength range and two in the ultraviolet range of of the spectrum to fit the dispersion of material in the range of optical wavelengths. Even if only used as fitting parameters, the absorptions are connected to the microscopic fundamental absorptions of the material. In some cases $n(\lambda)$ and not $n(\lambda)^{2}$ is approximated with a Sellmeier formula. Since $n(\lambda)$ as well as $n(\lambda)^{2}$ are complex differential (analytic) functions both formulae give refractive indices and dispersions with the same accuracy. However, care has to be taken over which quantity is expressed when using a Sellmeier formula.

Fig. 5.6

The complex refractive index $n(\omega)=n^{*}(\omega)+\mathrm{i}\kappa(\omega)$ is plotted on a logarithmic frequency scale. The real part is plotted with a solid line and the imaginary part with a dashed line

1.1.10 Wave Propagationin a Weakly Absorbing Medium

In this subsection the link between the attenuation of a wave and the imaginary part of the refractive index is given. A weakly absorbing medium is defined by the imaginary part of the refractive index (5.36) as being much smaller than the real part

$$\kappa\ll n$$

(5.39)

(the coefficient $\kappa/n$ is also called the attenuation index). In this case light propagates as transverse waves through the medium. We consider two points in our medium: P${}_{1}$ and P${}_{2}$. Between these points the light travels the distance $l$. In the absence of absorption, the electric and magnetic fields at point P${}_{2}$ are given by

$$\boldsymbol{E}_{2}=\boldsymbol{E}_{1}\mathrm{e}^{\mathrm{i}\frac{\omega}{c}\tilde{n}l}\;,\quad\boldsymbol{H}_{2}=\boldsymbol{H}_{1}\mathrm{e}^{\mathrm{i}\frac{\omega}{c}\tilde{n}l}\;.$$

(5.40)

Using (5.31) we obtain for weak absorption the radiation intensity at point P${}_{1}$. The radiation intensity at P${}_{2}$ is

$$\begin{aligned}\displaystyle I_{2}&\displaystyle=\tfrac{1}{2}|\boldsymbol{E}_{2}\times\boldsymbol{H}_{2}|\;,\\ \displaystyle&\displaystyle=\tfrac{1}{2}|\boldsymbol{E}_{1}\times\boldsymbol{H}_{1}|\mathrm{e}^{-2\frac{\omega}{c}\kappa l}\;,\\ \displaystyle&\displaystyle=I_{1}\mathrm{e}^{-2\frac{\omega}{c}\kappa l}=I_{1}\mathrm{e}^{-\alpha l}\;.\end{aligned}$$

(5.41)

The absorption coefficient $\alpha$ is connected to the complex refractive index by

$$\alpha=2\frac{\omega}{c}\kappa\;;$$

(5.42)

$\alpha$ can easily be measured and its importance for optical properties is discussed in Sect. 5.1.3. Equation (5.41) is also called Lambert–Beer law.

It is also important to note that the Fresnel equations (5.16) remain valid in the case of a weakly absorbing medium if the complex refractive indices are used. As an example, we plot the reflectivity resulting from (5.20) (at the interface air–model solid) near an absorption for a complex refractive index $n_{2}\to\tilde{n}_{2}$. In Fig. 5.7 the reflectivity is plotted on a logarithmic frequency scale. Note that the absorption seems to be shifted compared to the plots of the complex dielectric function or the complex refractive index. This is due to the fact that the poles of the function $R_{\text{norm}}$ in (5.20) are shifted from the poles of $n$. Measurement of the reflectivity is of importance for reflection spectroscopy or ellipsometry.

Fig. 5.7

Reflectivity $R$ for normal incidence plotted on a logarithmic frequency scale

1.2 Linear Refraction

As already introduced in Sect. 5.1.1, two phenomena occur when light impinges upon the surface of any optical material: reflection and refraction [5.5]. The reflected light bounces off the glass surface, while the refracted light travels through the material. The amount of light that is reflected depends on the refractive index of the sample, which also affects the refractive behavior of the sample [5.6]. The refractive index of optical materials turns out to be one of the most important factors that must be considered when designing systems to transmit and modulate light [5.7]. The refractive index is a complex material property that depends on thermodynamic parameters, e. g., temperature and on wavelength [5.8]. The wavelength dependence of the refractive index is the dispersion [5.5].

1.2.1 Law of Refraction

When a light ray impinges upon a smooth glass surface, a portion is reflected and the rest is either transmitted or absorbed. The material modulates the light upon transmission. The light travels at a different velocity as it is transmitted through the glass as compared to a vacuum. As introduced in Sect. 5.1.1, the index of refraction ($n$) is defined as the ratio of the speed of light in a vacuum ($c$) to that in the material, ($c_{\text{m}}$) [5.9]

$$n=\frac{c}{c_{\text{m}}}\;.$$

(5.43)

Most commonly, the reported refractive indices are relative to the speed of light in air, rather than in a vacuum, no matter which technique is used to measure the refractive index [5.10]. The index of refraction for a vacuum is defined as 1. The index of refraction of air is $\mathrm{1.00029}$ at standard temperature (${\mathrm{25}}\,{\mathrm{{}^{\circ}\mathrm{C}}}$) and pressure (${\mathrm{1}}\,{\mathrm{atm}}$) (). Therefore, the index of refraction of optical matter ($n_{\text{rel}}$) relative to air ($n_{\text{air}}$), rather than a vacuum is [5.9]

$$n_{\text{rel}}=\frac{n_{\text{m}}}{n_{\text{air}}}\;.$$

(5.44)

The STP indices of some common compounds, and classes of compounds, are shown in Table 5.1 [5.11].

Table 5.1 Indices of common materials at standard temperature and pressure at ${\mathrm{587.56}}\,{\mathrm{nm}}$ (helium d line) (after [5.11])

As discussed in Sect. 5.1.1.1, when light hits a glass surface at an angle $\theta_{\text{i}}$, it is Fresnel-reflected back at an angle $\theta_{\text{t}}$. The angle of incidence is equal to the angle of reflection ($\theta_{\text{i}}=\theta_{\text{r}}$), as shown in Fig. 5.8 [5.12]. The percentage of light reflected for $\theta_{\text{i}}=0$ at each interface ($R$) relative to the incident intensity (Sect. 5.1.1) is dependent on the index of refraction of the two media the light is passing through, typically air ($n_{2}$) and glass ($n_{1}$) (Fig. 5.8 and (5.20)) [5.12]

$$R_{\text{norm}}=100\left({\frac{n_{1}-n_{2}}{n_{1}+n_{2}}}\right)^{2}\;.$$

(5.45)

Fresnel's formula (5.45) assumes smooth surfaces that produce only specular reflection. Diffuse reflection occurs when the surface is rough, so the incident light is reflected through a range of angles, thereby reducing the intensity of the specular reflected light at any given angle [5.12]. The specular reflection that is taken into consideration by Fresnel's relationship can be monitored and used to estimate the refractive index of samples in situ [5.13].

Fig. 5.8

Ray tracings of incident, reflected, and transmitted light from one medium to another representing the angles and indices necessary to apply Snell's law (after [5.12])

The angle of the light transmitted within the material ($\alpha_{\text{t}}$), relative to the incident light transmitted through air, is dependent on the refractive indices of the air ($n_{\text{air}}$) and solid ($n_{\text{m}}$) and the incident light angle ($\alpha_{\text{i}}$) [5.12]

$$n_{\text{air}}\sin\alpha_{\text{i}}=n_{\text{m}}\sin\alpha_{\text{t}}\;.$$

(5.46)

This is the general form of Snell's law of refraction to predict transmitted angles in media [5.14].

1.2.2 Dispersion Relationships in Glass

The refractive index of a medium is dependent upon the wavelength of the light being transmitted (Sect. 5.1.1). This wavelength dependence is dispersion, which means that different wavelengths of light will be modulated differently by the same piece of matter [5.12]. Each wavelength of light will be subject to a different index of refraction. Our model solid in Sect. 5.1.1.7 shows dispersion due to the fact that the transparent wavelength range of a solid is bounded by absorptions. One ramification of dispersion is that white light can be separated into its principal visible components through a glass prism, or a simple raindrop. It is the dispersion of white light through raindrops that causes rainbows. The dispersion of light through optical materials results in the light being refracted at various angles because of Snell's law (5.46). The various components of white light experience different indices of refraction, which leads to different angles of exiting light. The difference in refractive index with wavelength is illustrated in Fig. 5.9 for BK7 optical glass, which is a high-dispersion material.

Fig. 5.9

Dispersion present in BK7 optical glass. Common index of refraction measurement wavelengths are indicated

In normal dispersion, the index increases for shorter wavelengths of light [5.15]. Normal dispersion is valid only far away from absorption bands in the infrared wavelength range (Fig. 5.6). Water has a normal dispersion response in the visible, so the red light is refracted by a lower index, and thus a greater angle, which is why red is on top in a rainbow. Discussion of rainbow formation is presented elsewhere in depth [5.12].

Anomalous dispersion is an increase in refractive index with an increase in wavelength. Anomalous dispersion typically occurs when one approaches a long wavelength absorption like an IR vibration mode. This is also seen in our model solid in Fig. 5.6 and is further discussed in [5.9].

Due to dispersion, the index of refraction must be reported with the wavelength of measurement. The most common wavelengths at which $n$ is measured are reported in Table 5.2. These wavelengths most often correspond to common sharp emission lines of gas discharge lamps. The index can be determined most accurately $(\pm{\mathrm{1\times 10^{-6}}})$ by measuring the angle of minimum deviation of light in a prism [5.15]. However, a Pulfrich refractometer $(\pm{\mathrm{1\times 10^{-5}}})$ is most commonly used in industry. Details of measurement techniques are given in [5.16] and in Chap. 29.

Table 5.2 Wavelength of spectral lines used for measuring index of refraction, with the common designation and spectral line source (after [5.9])

The index at various wavelengths is commonly referred to by the designations in Table 5.2: i. e., $n_{\text{d}}$ is the refractive index measured at the yellow helium d line of ${\mathrm{582.5618}}\,{\mathrm{nm}}$. The dispersion is often given as a difference in $n$ at two wavelengths. For instance, the primary dispersion is given by $n_{\text{F}}-n_{\text{C}}$ (hydrogen lines) and $n_{\text{F}^{\prime}}-n_{\text{C}^{\prime}}$ (cadmium lines). The most commonly reported measure of dispersion is the Abbe number ($\nu$), which is commonly given for two sets of conditions

$$\nu_{\text{d}}=\frac{n_{\text{d}}-1}{n_{\text{F}}-n_{\text{C}}}\;,\quad\nu_{\text{e}}=\frac{n_{\text{e}}-1}{n_{{\text{F}}^{\prime}}-n_{{\text{C}}^{\prime}}}\;.$$

(5.47)

The Abbe number is a measure of the ratio of the refractive power to the dispersion. In most optical materials catalogs, a six-digit number is assigned to the solid that is dependent upon the index and the Abbe number: the first three digits are ${\mathrm{1000}}(n_{\text{d}}-1)$ and the next three digits are $10\nu_{\text{d}}$. Using this property, optical glasses, for example, are divided into two general categories: crowns and flints (Fig. 5.10). Crown glasses typically have a low index of refraction and a high Abbe number ($n_{\text{d}}<{\mathrm{1.60}}$ and $\nu_{\text{d}}> 50$), whereas flint glasses have high indices and low Abbe numbers ($n_{\text{d}}> {\mathrm{1.60}}$ and $\nu_{\text{d}}<50$) [5.15]. The terms crown and flint have historical significance in that flint glasses typically had lead oxide added to them to increase the refractive index (Fig. 5.10) and crown glasses were typically blown and had curvature – a crown. Typically, to make an achromatic optical system crown and flint lenses are combined in series (Fig. 5.10, Table 5.3).

Fig. 5.10

Abbe diagram showing index of refraction versus the Abbe number for optical glasses

Table 5.3 Commercially available optical glasses by code, glass type and manufacture (H $=$ Hoya, O $=$ Ohara, and S $=$ SCHOTT) (after [5.17, 5.18, 5.19])

Table 5.3 (continued)

Table 5.3 (continued)

Often it is desirable to have a mathematical representation of the index as a function of wavelength. A considerable number of models exist for just this purpose. Perhaps the best known, and most widely used, is the Sellmeier formula [5.15]

$$n^{2}(\lambda)-1=\frac{B_{1}\lambda^{2}}{\lambda^{2}-C_{1}}+\frac{B_{2}\lambda^{2}}{\lambda^{2}-C_{2}}+\frac{B_{3}\lambda^{2}}{\lambda^{2}-C_{3}}\;.$$

(5.48)

Most major optical material manufacturers supply the Sellmeier coefficients for their glasses on the product data sheets. Note that the coefficients $C_{1}$, $C_{2}$ and $C_{3}$ have the physical unit of a squared length and are usually given in units of $\mathrm{{\upmu}m^{2}}$. With the six Sellmeier coefficients it is possible to estimate the index of refraction at any wavelength around the visible spectral range, given that it is not near a strong absorption. Numerous other dispersion models have been developed and are presented elsewhere [5.12, 5.15].

The index of refraction is also dependent on temperature, and similar formulas and tables of constants have been developed for the differential change in $n$ with temperature [5.15]

$$\begin{aligned}\displaystyle&\displaystyle\frac{\mathrm{d}n_{\text{abs}}({\lambda,T})}{\mathrm{d}T}=\frac{\left[{n^{2}({\lambda,T_{0}})-1}\right]}{2n({\lambda,T_{0}})}\\ \displaystyle&\displaystyle\quad\times\left({D_{0}+2D_{1}\Updelta T+3D_{2}\Updelta T^{2}+\frac{E_{0}+2E_{1}\Updelta T}{\lambda^{2}-\lambda_{\text{TK}}^{2}}}\right),\end{aligned}$$

(5.49)

where $T_{0}$ is ${\mathrm{20}}\,{\mathrm{{}^{\circ}\mathrm{C}}}$, $T$ is the temperature in $\mathrm{{}^{\circ}\mathrm{C}}$, $\Updelta T$ is $T-T_{0}$, $\lambda$ is the wavelength in $\mathrm{\upmu{}m}$, and $\lambda_{\text{TK}}$ is the average effective resonance wavelength for the temperature coefficients in $\mathrm{\upmu{}m}$. The constants $E_{0}$, $E_{1}$, $D_{0}$, $D_{1}$, and $D_{2}$ are provided on the manufacturer's product data sheets for each composition. The index of refraction must be measured at a temperature of ${\mathrm{20}}\,{\mathrm{{}^{\circ}\mathrm{C}}}$.

1.3 Absorption

Absorption in a material can be viewed from two sides. Inside the glass the absorption follows the Lambert–Beer law

$$I(d)=I_{0}\mathrm{e}^{-\alpha d}\;,$$

(5.50)

which means that the light intensity, $I$ is reduced by a factor of $\mathrm{e}^{-\alpha d}$ after traveling a distance $d$ through a homogeneous absorbing medium. The quantity $\alpha$ is called the absorption coefficient and was already introduced in Sect. 5.1.1.10. The absorption coefficient has the physical unit inverse meter, m${}^{-1}$. Sometimes the absorbance, $A$, is also used, which is based on the decadic logarithm, $\log_{10}$

$$A=\frac{\alpha d}{\ln(10)}\;.$$

(5.51)

The microscopic origin of the absorption coefficient is discussed in Sect. 5.1.4. The second view on absorption comes from the experimental side. We place a sample, e. g., a parallel plate, between a light source and a detector. The attenuation of the light is now related to surface reflection, surface scattering, bulk absorption in the material and bulk scattering in the material. (In addition complicated light paths occur, which include multiple internal reflections as sketched in Fig. 5.11. Figure 5.11 shows multiple reflections followed by multiple partial transmissions and reflections in a plane parallel slab of material, e. g., glass.) The reflectivity on a surface between two media is related to the complex refractive index as

$$\begin{aligned}\displaystyle&\displaystyle R=\left|\frac{\tilde{n}-\tilde{n}_{0}}{\tilde{n}+\tilde{n}_{0}}\right|^{2}\approx\left(\frac{n-1}{n+1}\right)^{2},\\ \displaystyle&\displaystyle\text{for}\quad k\ll n\quad\text{and}\quad n_{0}=1\;.\end{aligned}$$

(5.52)

If we can neglect surface and volume scattering together with interference effects, the total reflectivity $R_{\text{t}}$ and transmission $T_{\text{t}}$ can be expressed as an infinite series

$$R_{\text{t}} =R+R(1-R)^{2}\frac{\mathrm{e}^{-2\alpha d}}{1-R^{2}\mathrm{e}^{-2\alpha d}}$$

(5.53)

$$T_{\text{t}} =(1-R)^{2}\frac{\mathrm{e}^{-\alpha d}}{1-R^{2}\mathrm{e}^{-2\alpha d}}\stackrel{[2.\ 0]}{\approx}(1-R)^{2}\mathrm{e}^{-\alpha d}\;.$$

(5.54)

The transmission can be approximated by its second-order term, which is a good approximation for most materials where $r^{2}\ll 1$. As $n$ and $\alpha$ are usually unknown, two independent measurements have to be performed, either a transmission and a reflectivity measurement ($T_{\text{t}},R_{\text{t}}$) or a combination of two transmission measurements of different thickness ($T_{1},T_{2}$). The absorption coefficient can then be determined by

$$ \begin{aligned}\displaystyle&\displaystyle\alpha(T_{\text{t}},R_{\text{t}})=\\ \displaystyle&\displaystyle\frac{1}{d}\ln\left\{\frac{(1-R_{\text{t}})^{2}-T_{\text{t}}^{2}+\sqrt{4T_{\text{t}}^{2}+[T_{\text{t}}^{2}-(1-R_{\text{t}})^{2}]^{2}}}{2T_{\text{t}}}\right\}\end{aligned}$$

(5.55)

$$ \alpha(T_{1},T_{2})_{[2.O]}=\frac{1}{d_{2}-d_{1}}\ln\left(\frac{T_{1}}{T_{2}}\right).$$

(5.56)

For the combination of two transmission methods no analytical solution exists, but the second-order approximation (neglecting all multiple reflections $> 2$) is accurate enough for many purposes ($n\approx 2\Leftrightarrow r^{2}=0.012$). Often a combination of two transmission methods turns out to be more suitable, as reflectivity measurements always have a worse signal-to-noise ratio and therefore a comparatively higher error and lower resolution. An important criterion for the quality of a transmission measurement is the sample homogeneity (which means no striae inside the two samples).

Fig. 5.11

Schematic view of light transmission through a transparent plate, where multiple reflection on both sides becomes important

An important question concerning absorption is: What happens to the absorbed radiation energy? In most cases the energy is released into lattice vibrations via a cascade of different microscopic processes like electronic transitions and multiphonon absorption processes. This means the electromagnetic radiation that is absorbed is finally heating the glass. There are a number of important exceptions where the absorbed radiation is either instantaneously released as electromagnetic radiation or is released after a certain time and with a shift in energy. This is the case when fluorescence or luminescence occurs, which is discussed in Sect. 5.1.5. Usually the fluorescence or luminescence occurs at lower photon energies (longer wavelengths) than the absorbed light. There are rare cases when many photon processes lead to upconversion, where radiation of shorter wavelength (higher photon energy) occurs. An example for this is the infrared to green upconversion in Er${}^{3+}$-doped glasses.

1.4 Microscopic Origin of Absorption

Light absorption in glass is mainly caused by the excitation of electron–hole pairs, which can form bound states as excitons that are either delocalized or bound to local impurities. Absorption can be caused by electronic transitions within the absorbing specimen, such as transition metal ions, rare earth ions or even nanosized semiconducting particles. If electrons and holes originate from different ions, the term charge transfer is also used for a particular absorption line. This is due to the fact that the excitation of an electron–hole pair transfers a charge from one ion to a second one. The 4f electron systems like the rare earth ions have very well-defined absorption lines, since the electronic transitions within the 4f shell couple only weakly to phonon modes. A quantity that is microscopically related to the absorption of a single well-defined specimen is the absorption cross-section $\sigma_{\text{abs}}(\lambda,T,\ldots)$. It is connected to the absorption coefficient $\alpha(\lambda,T,\ldots)$ as

$$\alpha(\lambda,T,\ldots)=\sum_{i}\rho_{i}\sigma_{\text{abs}}(\lambda,T,\ldots)\;,$$

(5.57)

where the sum over $i$ is the summation over all absorbing transitions, $\rho_{i}$ is the density in units of $\mathrm{m^{-3}}$ of the specimen to which the absorbing transition belongs, and $\sigma_{\text{abs}}(\lambda,T,\ldots)$ is the individual absorption cross-section of a given electronic transition. The absorption cross-section is a microscopic property of an absorbing specimen (in a given crystal field), which is wavelength dependent and has the physical unit of an area, $\mathrm{m^{2}}$. A frequency $\nu=c/\lambda$ (or wavelength) integration of the absorption cross-section of a particular well-defined absorption peak gives the oscillator strength, $P$, of the particular transition when divided by the integrated cross-section of a classic excitation

$$P=\left(\frac{1}{\sigma_{\text{int-classic}}}\right)\int_{\text{peak}}\sigma(\nu)\mathrm{d}\nu\;,$$

(5.58)

with

$$\sigma_{\text{int-classic}}=\frac{\mathrm{e}^{2}}{4\epsilon_{0}m_{0}c}\approx{\mathrm{2.65\times 10^{-6}}}\,{\mathrm{m/s}}\;.$$

The oscillator strength is therefore the relation between the measured absorption and the absorption of a given transition expected by a classical theory. The value $P$ takes into account the quantum-mechanic matrix elements of ground state and excited state as well as the multiplicity of particular spin and momentum configurations in an atom. The value $P$ therefore takes into account all spin rules and selection, rules which have a deep quantum-mechanical origin. For optically allowed transitions for the total spin $S$ and total momentum $L$ of a many-electron system the quantum-mechanic selection rules of spin conservation and parity change apply:

• Russel–Sounder selection rule: $\Updelta S=0$

• Laporte selection rule: $\Updelta L=\pm 1$.

Violations of these rules are called forbidden transitions and are of comparatively weaker intensity. The Russel–Sounder rule can be partially circumvented by the influence of spin-orbit coupling (e. g., enables 4f–4f transitions) and Laporte forbidden transitions due to vibronic coupling (e. g., enables 3d–3d transitions in octahedral environments) or ligand field interaction causing a lower symmetric environment (e. g., enables 3d–3d transitions in tetragonal environments) [5.20] (Table 5.4).

Table 5.4 Oscillator strengths for various types of transitions [5.21]

Numerous mechanisms exist for the creation of colors in glasses. Color is usually created by absorption of a particular part of the visible spectrum. In the following we discuss different types of absorption.

1.4.1 Absorption From Single Ions

Most commercial glasses are colored by intraionic transitions between the 3d or 4f energy levels of dissolved transition-metal or lanthanide ions [5.22, 5.23, 5.24].

Among the sharpest and most well-defined electronic transitions are the 4f electron systems. The electronic transitions within the 4f electron systems are both spin-forbidden and Laporte-forbidden. Therefore a relatively large concentration of these ions is needed to color a glass. Most of these transitions merely decay into phonons and therefore lead to strong fluorescence, which finds numerous applications in laser active materials, optical amplifiers and lighting technology.

One example are Nd${}^{3+}$ glasses for laser applications. A second example is the transition within the Er${}^{3+}$ spin system at the IR wavelength of optical telecommunication. Er-doped glass fibers are widely used as optic amplifiers for telecommunication applications.

d-Electron systems are often used to color glasses. The electronic transitions are symmetry-allowed but strongly couple to phonons. Therefore they only give a small contribution to fluorescence but are efficient coloring agents in glasses. For example the blue color of glasses can be created by adding the oxides of Co or Cu to a glass compositions, and green colors can be obtained by doping glasses with Fe or $\mathrm{Cr^{3+}}$.

Very efficient optic transitions are obtained by 4f–5d transitions. An example is $\mathrm{Ce^{3+}}$, which creates red colors in glasses. Here the one electron of the $\mathrm{Ce^{3+}}$ that is in the 4f shell is excited into the 5d shell.

1.4.2 Absorption by a Combination of Ions

Efficient coloration can also be provided by a combination of two coloring ions. An example is the coloration of brown glasses using Fe and V. Here the excitation of a $\mathrm{Fe^{3+}}$,$\mathrm{V^{3+}}$ pair to a $\mathrm{Fe^{2+}}$,$\mathrm{V^{4+}}$ pair causes light absorption. Such mechanisms are often called charge transfer mechanisms ().

1.4.3 Absorption by Conducting Particles

The coloration of glasses due to small conducting particles in the glass is a combination of both absorption and scattering effects. For small metallic particles, the scattering is governed specifically by Mie's theory but can also be treated in general with the Rayleigh scattering theory [5.12]. Mie scattering and Rayleigh scattering are covered in Sect. 5.1.6. For understanding the Mie scattering of metallic, conducting particles the absorptions need to be taken into account. In particular, the resonance surface plasmon is important for coloring glasses, since its absorption line depends on the size of the metal particle. For several hundred years glass makers have known how to create beautiful colored glasses with nanosized particles of gold or silver inside. For gold and silver the surface plasmon mode extends well into the visible range. The gold ruby glasses in church windows, which are colored using nanometer-sized gold colloids in the glass, are well-known examples of this.

1.4.4 Absorption by Semiconductor Particles

Semiconductor particles in glass can be made so small ($1{-}10\,{\mathrm{nm}}$) that the scattering of visible light plays a minor role, however they absorb light over a continuum of wavelengths corresponding to energies greater than the bandgap ($E_{\text{g}}$) of the semiconductor particles [5.12]. The bandgap of the semiconductor particles is controlled by the size of the particle and the chemical composition. Typically, semiconductor glasses are melted with Zn, Cd, S, Se and Te raw materials in the batch and upon casting the glasses cool colorless [5.15]. Secondary heat treatment (striking) results in the crystallization of various semiconductor crystal phases in the glass, or a mixture thereof: ZnS, ZnSe, ZnTe, CdS, CdSe, and CdTe. The size and distribution of the semiconductor particles can be controlled by the heat treatment and thus so can the optical properties of the resulting glasses. With the proper heat treatment, one glass can be struck into multiple glasses with absorptions leading to red, yellow and orange coloration [5.15]. When the bandgap of the particles is large enough, the absorption edge is shifted into the UV and the glass appears colorless. The opposite can also occur when the bandgap energy is so low that it absorbs all visible light, and the sample appears black [5.15].

Semiconductor-doped glasses are often used as low-pass filter glasses because of their sharp absorption cutoff.

1.5 Emission

For numerous applications of active optical materials the interrelation between specific absorption and emission processes [5.25] is of crucial importance. One example is a laser or an optical amplifier, where a specific absorption on a well-defined level in the optically active material is needed for pumping. This leads to the buildup of an inversion in a lower laying state with a lower energy level compared to the pump level. The inversion can either be emptied by spontaneous emission, which causes noise, or by stimulated emission, which amplifies a laser (or signal) mode.

The microscopic processes between these absorption and emission levels can have different origins, for example several levels of 4f electron systems of a rare earth. These levels only interact weakly with lattice vibrations and therefore barely show thermal broadening (zero phonon lines). Examples for laser materials are $\mathrm{Nd^{3+}}$-doped yttrium aluminate garnet () single crystals and $\mathrm{Nd^{3+}}$-doped phosphate glasses. Examples of optical amplifier materials are $\mathrm{Er^{3+}}$-doped fused silica and $\mathrm{Er^{3+}{/}Yb^{3+}}$-doped multicomponent glasses (Fig. 5.12). Here the optical emission of $\mathrm{Er^{3+}}$ in the wavelength range of optical telecommunication is used. Further laser systems are, e. g., ruby lasers ($\mathrm{Cr^{3+}}$ $+$ doped $\mathrm{Al_{2}O_{3}}$); semiconductor lasers, where the pumping is done electronically; gas lasers such as the helium–neon laser, where the energy levels of He are used for pumping and the inversion is built up in the energy levels of Ne; or dye lasers where broad $\pi$-electron systems of organic molecules are used to provide the necessary tunable energy levels. For more details on laser physics see also [5.25]. The absorption and the absorption cross-sections can be directly measured and are described in Sect. 5.1.3. As we will see, the approach to treating spectral emission of an excited material is more sophisticated.

Fig. 5.12

Plot of the emission and absorption cross-section for an Er${}^{3+}$-doped glass

1.5.1 Emission Cross-Section

Although absorption has a well-defined meaning with respect to attenuation of light and can be measured quantitatively via the absorption coefficient $\alpha$ (Sect. 5.1.1.10) it has a quantum-mechanical origin. In particular, when making absorption (and emission) quantitative the quantum-mechanical concept of transition probabilities is needed. It is also usually expressed via oscillator strengths of certain transitions. The fact that light is absorbed at certain frequencies $\omega$ (or energies $\hbar\omega$) means that light is portioned in packages (photons) of energy $E_{\text{photon}}$ given by

$$E_{\text{photon}}=\hbar\omega=\frac{hc}{\lambda}\;,$$

(5.59)

where $h$ is Planck's quantum and $\hbar=h/(2\uppi)$. The absorption cross-section (Sect. 5.1.4) is connected to the transition probability $R_{ij}$, which is the probability for absorbing a photon to induce a transition in the medium from state $i$ to state $j$. In general the transition probability depends on the detailed wavelength dependence of the transition and on the wavelength dependence of the incoming light. Therefore the transition probability is related to the cross-section as follows

$$R_{ij}=\int\sigma(\omega)\Phi(\omega)\mathrm{d}\omega=\int\frac{-2\uppi c}{\lambda^{2}}\sigma(\lambda)\Phi(\lambda)\mathrm{d}\lambda\;,$$

(5.60)

where $\Phi$ is the photon flux measured in units of $\mathrm{m^{-2}}$, which is related to the light intensity $I=\hbar\omega\Phi$.

1.5.2 Quantum Emissions

Furthermore, emission and emission cross-sections have a quantum-mechanical nature. Historically the work of Einstein not only brought its author a Nobel prize but also laid the foundation of laser physics. Einstein introduced the fundamental concepts of stimulated and spontaneous emission. The stimulated emission process produces copies of photons with identical energy and – more importantly – phase as the incident photon. This is the process behind laser activity and amplification. While stimulated emission is just the opposite process of absorption, spontaneous emission is caused by the finite lifetime of an excited state. This lifetime is responsible for noise in a laser or optical amplifier. The lifetime $\tau_{2}$ of the upper, excited level with only one transition $2\to 1$ is the inverse of the transition probability $A_{21}$

$$\tau_{2}=\frac{1}{A_{21}}\;.$$

(5.61)

In the case of different decay channels the lifetime is the inverse of a sum over all transition probabilities

$$\tau_{2}=\frac{1}{\sum_{m}A_{2m}}\;.$$

(5.62)

We now want to define an emission cross-section. If we consider an ideal system with a single nondegenerate transition and no nonradiative decay channels, the emission and absorption cross-sections are identical $\sigma^{\text{abs}}=\sigma^{\text{emis}}$.

Real systems deviate from this in the following ways:

• The upper as well as the lower laser level are composed of an ensemble of sublevels. Especially in disordered systems such as glasses an inhomogeneous broadening of the energy levels can become important. Therefore an effective density of states for each level has to be taken into account.

• The spacing of the sublevels is such that thermal occupation has to be considered, i. e., where $\Updelta E=h\Updelta\omega<k_{\text{B}}T$, where $k_{\text{B}}$ is the Boltzmann constant.

• Different degeneracies of the energy levels are involved. If the upper level has a degeneracy of $g_{2}$ and the lower level a degeneracy of $g_{1}$, the emission and absorption probabilities have to be multiplied by these degeneracy factors. If an energy state is connected with a magnetic moment, a canceling of degeneracies due to crystal field splitting can occur, which leads to an ensemble of sublevels (see above).

• There exist nonradiative processes that empty the upper state and decrease the radiative emission cross-section. At high enough energies lattice vibrations (phonons) can empty an excited state in a nonradiative way.

For these reasons the effective emission cross-section for a real system deviates from the absorption cross-section in a nontrivial way

$$\sigma^{\text{abs}}(\lambda)\neq\sigma^{\text{emis}}(\lambda)\;.$$

(5.63)

An ideal two-level system and a real system are sketched schematically in Fig. 5.13a,ba,b. The emission cross-section is related to the transition probability by

$$\sigma^{\text{emis}}(\lambda)=\frac{\lambda^{2}A_{21}}{8\uppi}S\left(\frac{c}{\lambda}\right),$$

(5.64)

where $S(c/\lambda)=S(\nu)$ is the lineshape function. The lineshape function is normalized to unity $\int S(\nu)\mathrm{d}\nu=1$. The question remains if and how it is possible to measure the emission cross-section. While for the absorption cross-section such a measurement is straightforward, it is far more complicated to complete this process for an emission cross-section. The lineshape can be determined by saturating a transition with a light source then switching off the light and measuring the resulting fluorescence, which is decaying exponentially in time, with a spectrometer. The wavelength-dependent signal of the spectrometer is proportional to the emission cross-section. If a sample of volume $V$ is irradiated with light until a stationary state is reached, the inversion level $0<N_{2}(\boldsymbol{x})<1$ is a measure of the density of the excited optical centers. The probability for an emission process to occur in the wavelength range between $\lambda$ and $\lambda+\Updelta\lambda$ is

$$W(\lambda)=\frac{\sigma^{\text{emis}}(\lambda)\Updelta\lambda}{\int\sigma^{\text{emis}}(\lambda)\mathrm{d}\lambda}\;.$$

(5.65)

Therefore the light intensity emitted by an infinitesimal volume element $\mathrm{d}V(\boldsymbol{x})$ is given by

$$\mathrm{d}I(\lambda,\boldsymbol{x})=NN_{2}\tau^{-1}\mathrm{d}V(\boldsymbol{x})h\nu W(\lambda)\;.$$

(5.66)

Here again $N$ is the concentration of optical centers, $h\nu=hc/\lambda$ is the energy of each emitted phonon with Planck's constant $h$ and $\tau$ is the lifetime of the excited state. For a measurement of $W(\lambda)$, this intensity, which is emitted with equal probability in all spatial directions, has to be collected in a spectrometer that normally covers only a small part of the spatial angle in which the emission takes place. On the way to the spectrometer the light may also travel through other volume elements causing absorption or stimulated emission. Therefore the absolute signal measured in the spectrometer is related in a nontrivial way to the emission cross-section. For measuring cross-sections the following special conditions are thus used:

• Excitation and measurement are done close to the surface of a sample to avoid the emitted light traveling through large parts of the sample. This is done either in a backscattering geometry or at a corner of the material.

• The excitation is so strong (e. g., with a laser) that saturation is reached $N_{2}\approx 1$. In this way absorption of emitted light traveling through the sample is suppressed.

• Standards are used for a comparison. These are, e. g., rare-earth ions of well-defined concentration in single crystals, glasses with well-controlled compositions or fluid organic dyes.

Even if it is a nontrivial task to measure the emission cross-sections, it is the most important microscopic quantity for the laser activity of a transition.

Fig. 5.13a,b

In (a) the schematic terms for an ideal, only lifetime-broadened emission process from an excited state to the ground state is plotted. Many real systems sketched in (b) show a splitting into different sublevels, and different degeneracies and nonradiative transitions

1.5.3 Example for Er Ions

As an example we show here the emission and absorption cross-sections of $\mathrm{Er^{3+}}$ for the transition around ${\mathrm{1550}}\,{\mathrm{nm}}$. This transition is important for telecommunication applications since it takes place in the wavelength range where silica fibers exhibit minimum light attenuation. The states involved in this transition have the spectroscopic symbols $E_{1}\to{}^{4}\mathrm{I}_{15/2}$ and $E_{2}\to{}^{4}\mathrm{I}_{13/2}$, which stand for spin, angular and total moment of the correlated eleven $\smash{{4}{\mathrm{f}}}$ electron systems that forms the $\mathrm{Er^{3+}}$ system. In Fig. 5.12 we plot an example of this transition of $\mathrm{Er^{3+}}$, which is of high technological importance. The value of the emission cross-section is influenced by the fact that the ground state is eightfold degenerate whereas the degeneracy of the excited state is sevenfold. It is further influenced by nonradiative decay channels due to lattice vibrations. The spectral shape of the cross-sections is formed by the level splitting in the local crystal field formed by the glass environment and the thermal occupation of the different sublevels. In addition there is inhomogeneous broadening due to the disordered structure of the glass.

1.6 Volume Scattering

This section deals with the interaction of light with particles inside a certain volume leading to light scattering. Before going into details the term scattering must be defined and explained. Afterwards scattering is subdivided into single scattering, multiple scattering, and coherent scattering.

1.6.1 Definition and Basics

Scattering is defined here as energy absorption of incident light followed by reemission of part of this light at the same frequency. Thus inelastic effects such as Raman scattering or Brillouin scattering are not discussed in this section.

The origins of light scattering are spatial fluctuations of the complex refractive index. From this point, volume scattering is diffraction of an electromagnetic wave at particles in a certain volume.

Obviously scattering is strongly related to diffraction, as seen from the definition of scattering. Indeed, diffraction is scattering by a flat particle [5.26].

In the following, volume scattering is subdivided into (Fig. 5.14):

• Single scattering

• Multiple scattering

• Coherent scattering.

Volume scattering can be regarded as the sum of single scattering events as long as the density of scattering particles is not too high. Mathematically this is expressed with the packing fraction $\eta$ defined by

$$\eta:=\frac{N_{\text{scat}}V_{\text{scat}}}{V_{\text{vol}}}\;,$$

(5.67)

where $N$ is the number (integer) of the single scatterer, $V_{\text{scat}}$ is the volume of a single scatterer, and $V_{\text{vol}}$ is the whole volume in which the $N$ identical scatterers are located.

Fig. 5.14

Illustration of the subdivision of volume scattering: single scattering (top), multiple scattering (middle), and coherent scattering (bottom)

Before discussing these three different subdivisions of volume scattering a few basic items must be described.

Scattering attenuates light, as does absorption by the medium itself. Both effects together are referred to as extinction, where extinction $=$ absorption $+$ scattering [5.26].

The total (over all directions) scattered power $P_{\text{scat}}$ (in units of $\mathrm{W}$) can be calculated from the scattering cross-section $\sigma_{\text{scat}}$ (in units of ${\mathrm{m^{2}}}$) by

$$P_{\text{scat}}=\sigma_{\text{scat}}I_{\text{in}}\;,$$

(5.68)

where $I_{\text{in}}$ is the incident light intensity (in units of ${\mathrm{W{\,}m^{-2}}}$). Typically one wants to know the power scattered in a certain direction (Fig. 5.15) and this is described by the differential scattering cross-section $\mathrm{d}\sigma_{\text{scat}}/\mathrm{d}\Omega$ ($\Omega$: solid angle)

$$\frac{\mathrm{d}\sigma_{\text{scat}}}{\mathrm{d}\Omega}=R^{2}\frac{I_{\text{scat}}}{I_{\text{in}}}\;,$$

(5.69)

where $R$ is the distance from scatterer to observer and $I_{\text{scat}}$ is the scattering intensity. The scattering cross-section can be obtained from the differential scattering cross-section by

$$\sigma_{\text{scat}}=\int_{4\uppi}{\frac{\mathrm{d}\sigma_{\text{scat}}}{\mathrm{d}\Omega}}\mathrm{d}\Omega\;.$$

(5.70)

Fig. 5.15

Scattering in a certain direction described by the scattering cross-section

Unfortunately the scattering cross-section and the differential scattering cross-section can only be calculated for a few geometries, such as spheres (Mie scattering). The attenuation of light due to scattering is related to the scattering cross-section and the packing fraction of scatterers as follows

$$\tau_{\text{scat}}=N_{\text{scat}}\sigma_{\text{scat}}=\eta\frac{\sigma_{\text{scat}}}{V_{\text{scat}}}\;,$$

(5.71)

which has the unit of inverse meter ($\mathrm{m^{-1}}$) and occurs in the exponent when light with starting intensity $I_{0}$ travels a distance $d$ through a scattering medium and is attenuated as

$$I=I_{0}\mathrm{e}^{-\tau_{\text{scat}}d}\;.$$

(5.72)

In the case when $\tau_{\text{scat}}d\ll 1$ optic imaging is possible and only weak scattering is present. In the case when $\tau_{\text{scat}}d\gg 1$ a sample is turbid and cannot be used for optical imaging. In this case multiple scattering is present.

1.6.2 Single Scattering (Mie Scattering)

A very important geometry of single scatterers is the sphere. In 1908 Mie [5.27] derived an analytic theory that completely describes the scattering from a (conducting or nonconducting) sphere embedded in a nonconducting medium. A compact survey of the formulas can be found in [5.28]. Many basic characteristics of scattered power can be obtained by studying scattering by a sphere. The calculated scattering cross-section normalized to the geometric cross-section of a glass sphere ($=\uppi a^{2}$ where $a$ is the sphere radius) can be seen in Fig. 5.16.

Fig. 5.16

Scattering cross-section per sphere area (cross-section) versus wavelength for a glass sphere ($n_{\text{sphere}}=1.50$) of radius $a={\mathrm{400}}\,{\mathrm{nm}}$ in air ($n_{\text{medium}}=1.0$)

Figure 5.16 demonstrates the different wavelength dependence of the scattering cross-section and thus of the scattered power according to (5.68). Depending on the geometric sphere size (characterized by the diameter $2a$) relative to the wavelength, three important regions can be identified:

• $\lambda\ll 2a/n_{\text{medium}}$, region of geometric optics. Surprisingly the term $\sigma_{\text{scat}}/(\uppi a^{2})=2$. This effect is called the extinction paradox and the factor 2 is due to diffraction effects [5.29, 5.30]. In Fig. 5.17 the factor of 2 is approached only approximately since at small wavelengths (or large scattering particles) the Mie solution becomes numerically instable.

• $\lambda\approx 2a/n_{\text{medium}}$, resonance effect, the term $\sigma_{\text{scat}}/(\uppi a^{2})$ has its maximum. Therefore the highest scattered power can be measured if the geometric size of an object is of the order of the wavelength.

• $\lambda\gg 2a/n_{\text{medium}}$, Rayleigh scattering. The scattered power is proportional to $1/\lambda^{4}$. This effect is responsible for the blue sky and red sunsets, where small particles inside the atmosphere scatter blue light ($\approx{\mathrm{400}}\,{\mathrm{nm}}$) more than red light ($\approx{\mathrm{750}}\,{\mathrm{nm}}$).

These basic characteristics of a single scatterer – here shown for the special case of a sphere – are typical for all single scatterers of arbitrary geometry. As discussed and shown in Fig. 5.16 different behaviors of the scattered power can be observed based on the size of the scatterer relative to the wavelength.

Fig. 5.17

Schematic plot of a three-level laser system via a pump level with a very short lifetime $\tau_{1}\ll\tau_{2}$ where the upper laser level (u) is populated. Depopulation can either occur via stimulated emission, where copies of photons are made or via spontaneous emission, where the probability is determined by the lifetime $\tau_{2}$

1.6.3 Multiple Scattering

In the preceding section scattering at a single isolated particle was described. Now, multiple scattering is discussed, which is the weighted superposition of many single scatterers without interference effects. Multiple scattering with interference effects is called coherent scattering and is discussed below.

Multiple scattering is present whenever the product of turbidity and sample thickness is much larger than 1 ($\tau_{\text{scat}}d\gg 1$) and can be described by diffusive light transport [5.28]. An approximation that treats scattering analytically is the Kubelka–Munck theory [5.31], which approximates light transport in a strongly scattering medium as a diffusive phenomenon in one direction. Based on the knowledge of the scattering function (described by the differential scattering cross-section) of a single scatterer, the overall scattering function can be calculated by tracing rays throughout the volume, if the scattering centers are large enough to allow for use of classical optics.

1.6.4 Coherent Scattering

Coherent scattering is scattering where the phase relation of the electromagnetic wave on different scattering centers is important and interference plays an important role for the scattered light. When dealing with materials containing scatterers separated by distances greater than the coherence length (the distance necessary for propagating waves to lose their coherence), the scattering events can be treated as independent occurrences even under multiple scattering conditions. Such scattering is called incoherent, and the resulting intensity of such radiation is simply a summation of the intensity contributions of all the independent scattering centers.

However, when the distances between scatterers are on the order of or less than the coherence length, coherent scattering effects must be considered [5.32]. In this case, wave packet interference takes place as the packet is capable of interacting with more than one scattering center at a time; several scatterers distort the photon packet simultaneously. Thus, a relationship exists between the phases of the light signals arising from the different scatterers. The events are no longer separate; they are correlated, and the resulting intensity of transmitted light is no longer a simple sum. Thus, in coherent scattering, wave packets experience a combined interaction with several scattering centers that affects their transport through the medium. In photon group interference (a quantum effect), the wave packet comes apart upon scattering but the different sinusoidal components meet again and interfere. This interference affects the intensity of light transmitted through the sample; thus, coherent scattering studies can yield valuable structural information.

As discussed, scattering is coherent when the phases of the light signals arising from different scattering centers are correlated and incoherent when the phases are not. Hence, the propagation of coherently scattered light depends strongly on the direction of the scattering vector $\boldsymbol{q}$ – the difference between the incident and outgoing wave vectors, $\boldsymbol{k}_{\text{in}}-\boldsymbol{k}_{\text{out}}$ (Fig. 5.18, right) – whereas incoherently scattered light can propagate in any direction regardless of the phase relation between waves from different scattering centers [5.33]. When the average distance between scattering centers ($d$) is on the order of the coherence length or less, the interference effects are significant and quantitatively describable by the static structure factor $S(q)$, which gives correlations in the positions of particle centers (Fig. 5.18, left). $S(q)$ is the link between the theoretical description of structural inhomogeneities and the actual experimental scattering measurements [5.34]. An extreme case of coherent scattering is a strictly periodic configuration of scattering centers. The scattering intensity shows maxima at Bragg peaks and the structure factor consists of sharp $\delta$ peaks. Note that the relation between incoherent and coherent scattering terms is usually expressed via the Debye–Waller factor.

Fig. 5.18

Left: an example of the static structure factor, for the case of hard spheres. Right: the scattering vector $\boldsymbol{q}$

The plot of the structure factor in real space is the pair correlation function, $g(r)$, which gives the probability of finding a pair of centers at a specific distance $r$ apart in a sample. The relationship between $g(r)$ and $S(q)$ is

$$S(q)=1+4\uppi\rho\int_{0}^{\infty}{r^{2}\left[{g(r)-1}\right]\frac{\sin(kr)}{kr}}\mathrm{d}r\;,$$

(5.73)

where $\rho$ is the density of scattering centers. There is a mathematical connection between the cross-section of the scatterers and the structure factor. For the general case of scattering from a correlated group of particles the scattering cross-section is

$$\sigma_{\text{scat}}=\frac{1}{\boldsymbol{k}_{0}^{2}}\int_{0}^{2\boldsymbol{k}_{0}a}{F(y,\boldsymbol{k}_{0}a)S(y)y\mathrm{d}y}\;,$$

(5.74)

where $y=qa$, $F(y,\boldsymbol{k}_{0}a)$ is the form factor, $a$ is the radius of the scatterer, and $\boldsymbol{k}_{0}$ is the incident wave vector [5.35].

Coherently scattered photons have a phase relationship and consequently exhibit more wavelike behavior. In experiments in which coherent scattering is a prominent effect, the interference between scattering paths must be considered – and can be exploited. For example, light from a monochromatic coherent source scatters from a sample and exhibits a characteristic speckle pattern – an array of bright, nonoverlapping spots due to interference effects – based on the composition and structure of the sample, as long as single scattering is the dominant effect [5.33]. Thus, such patterns contain structural information. An additional field is the analysis of backscattering cones. Another example of coherent scattering effects is evidenced by the unexpectedly high transparency of a glass ceramic – a phenomenon that cannot be explained by Rayleigh or Mie scattering theories.