Dispersion, Drude-Lorentz Theory & Optical Properties of Matter

Definition of Optical Dispersion

In a physical material medium, the phase velocity $v$ of an electromagnetic wave depends on the temporal frequency $\omega$ (or free-space wavelength $\lambda_0$) of the wave:

where $n(\omega)$ is the frequency-dependent refractive index. The variation of the refractive index with wavelength or frequency, $\frac{dn}{d\lambda}$ or $\frac{dn}{d\omega}$, is known as optical dispersion.

Because different spectral colors travel at different velocities, white light passing through a glass prism separates into its constituent spectral hues.

1. Normal Dispersion ($\frac{dn}{d\lambda} < 0$)

In regions of the electromagnetic spectrum far away from any atomic or molecular resonant absorption bands, the refractive index decreases monotonically with increasing wavelength:

Short wavelengths (blue light) experience a larger refractive index and bend more sharply than long wavelengths (red light):

In normal dispersion regimes, the empirical behavior is accurately modeled by Cauchy's Equation (Augustin-Louis Cauchy 1836):

where $A, B, C$ are empirical positive constants characteristic of the optical glass.

2. Anomalous Dispersion ($\frac{dn}{d\lambda} > 0$)

In the immediate vicinity of an absorption band (where the incident photon frequency matches an atomic or molecular transition frequency), the situation reverses abruptly:

Here, longer wavelengths experience a greater index of refraction than shorter wavelengths. Within this anomalous dispersion zone, the material exhibits intense resonant absorption of electromagnetic energy.

Microscopic Mechanical Equation of Motion

Hendrik Lorentz and Paul Drude formulated a microscopic classical model of dielectrics by picturing an atom as a nucleus surrounded by bound electrons. An electron of mass $m_e$ and charge $-e$ is subject to:

1. Electrostatic Restoring Force: $\mathbf{F}_{\text{restoring}} = -m_e \omega_0^2 \mathbf{r}$, where $\omega_0$ is the natural resonant frequency of atomic binding.

2. Dissipative Frictional Damping Force: $\mathbf{F}_{\text{damping}} = -m_e \gamma \frac{d\mathbf{r}}{dt}$, where $\gamma$ accounts for radiative damping and atomic collisions.

3. Driving Electric Force: $\mathbf{F}_{\text{drive}} = -e \mathbf{E}(t) = -e \mathbf{E}_0 e^{-i\omega t}$.

Newton's second law for the electron displacement is:

Looking for steady-state sinusoidal solutions $\mathbf{r}(t) = \mathbf{r}_0 e^{-i\omega t}$:

Induced Macroscopic Polarization ($\mathbf{P}$)

Let the medium contain $N$ atoms per unit volume, with $Z$ electrons per atom distributed among different natural resonant frequencies $\omega_j$ with oscillator strengths $f_j$ (satisfying the Thomas-Reiche-Kuhn sum rule $\sum_j f_j = Z$). The macroscopic electric dipole polarization density is:

The Complex Dielectric Function

From the macroscopic relation $\mathbf{D} = \epsilon_0 \mathbf{E} + \mathbf{P} \equiv \epsilon(\omega) \mathbf{E} = \epsilon_r(\omega) \epsilon_0 \mathbf{E}$:

This fundamental relation is the Drude-Lorentz Complex Dielectric Function.

Real and Imaginary Optical Constants

The complex index of refraction $\tilde{n}$ is defined as the square root of the complex relative permittivity (for non-magnetic media $\mu_r \approx 1$):

where:

• $n(\omega)$ is the real refractive index (governing phase velocity $v = c/n$ and refraction angles via Snell's law).
• $\kappa(\omega)$ is the extinction coefficient (governing optical absorption and wave attenuation).

Squaring both sides:

Equating real and imaginary parts:

For a single dominant resonant transition of frequency $\omega_0$ and oscillator strength $f$:

Physical Consequences Across the Spectrum

1. Well Below Resonance ($\omega \ll \omega_0$):

$\text{Im}(\tilde{\epsilon}_r) \approx 0$, meaning $\kappa \approx 0$ (transparent medium). $\text{Re}(\tilde{\epsilon}_r) > 1$, and $n$ increases with $\omega$ (normal dispersion).

2. Near Resonance ($\omega \approx \omega_0$):

$\text{Im}(\tilde{\epsilon}_r)$ reaches a sharp Lorentzian peak, causing strong resonant absorption. Simultaneously, $\text{Re}(\tilde{\epsilon}_r)$ drops steeply, producing a negative slope $\frac{dn}{d\omega} < 0$ (anomalous dispersion).

3. Well Above Resonance ($\omega \gg \omega_0$):

The medium becomes transparent again, with $n < 1$, approaching $n \to 1$ asymptotically at X-ray frequencies.

Derivation of the Sellmeier Formula

In optical materials such as crown glass, fused silica, and quartz, the damping constants $\gamma_j$ are typically much smaller than the resonant frequencies $\omega_j$. In transparent optical windows far from absorption lines ($\gamma_j \ll |\omega_j - \omega|$):

where $B_j = \frac{N e^2 f_j}{\epsilon_0 m_e \omega_j^2}$.

Converting from angular frequencies $\omega$ to free-space wavelengths $\lambda$ using $\omega = 2\pi c/\lambda$ yields the Sellmeier Dispersion Formula (Wolfgang von Sellmeier 1871):

where $C_j = \lambda_j^2$ represents the square of the absorption resonance wavelengths, and $B_j$ are dimensionless empirical coefficients.

For fused silica ($ ext{SiO}_2$), a standard three-term Sellmeier equation accurately predicts refractive index across the entire range from ultraviolet ($0.21\,\mu\text{m}$) to mid-infrared ($3.71\,\mu\text{m}$) with precision exceeding $10^{-5}$, enabling the design of precision camera lenses and fiber optic telecommunication systems.

Metals as an Unbound Electron Plasma

In electrical conductors and metals (such as silver, gold, and copper), valence electrons are not bound to individual atomic cores ($\omega_0 = 0$). They move freely through a background of positive lattice ions, experiencing collisions with an average relaxation time $\tau$ (damping rate $\gamma = 1/\tau$).

Setting $\omega_0 = 0$ and $\gamma = 1/\tau$ in the equation of motion:

Complex AC Conductivity ($\tilde{\sigma}(\omega)$)

The macroscopic conduction current density is $\mathbf{J} = -n_e e \mathbf{v} = \tilde{\sigma}(\omega) \mathbf{E}$, where:

and $\sigma_0 = \frac{n_e e^2 \tau}{m_e}$ is the standard DC electrical conductivity.

Drude Dielectric Permittivity of Metals

Substituting into Maxwell's equations:

where $\omega_p = \sqrt{\frac{n_e e^2}{\epsilon_0 m_e}}$ is the bulk plasma frequency of the metal.

High-Frequency Regime ($\omega \tau \gg 1$)

At optical frequencies (visible and UV), $\omega \tau \gg 1$, and collision damping can be neglected ($1/\tau \to 0$):

1. Below the Plasma Frequency ($\omega < \omega_p$):

$\epsilon_r(\omega) < 0$. The refractive index is purely imaginary: $\tilde{n} = i\kappa$. The wave cannot propagate and reflects totally ($R = 1.00$). This explains the brilliant silvery luster and mirror-like reflectance of metals.

2. Above the Plasma Frequency ($\omega > \omega_p$):

$\epsilon_r(\omega) > 0$. The metal becomes transparent to electromagnetic radiation! For alkali metals, $\omega_p$ lies in the near-ultraviolet, producing the famous ultraviolet transparency of metals discovered experimentally by Wood in 1933.

The Microscopic Local Field ($\mathbf{E}_{\text{local}}$)

In a dilute gas, molecules are separated by large distances, so the local electric field polarizing a molecule is simply the macroscopic applied field $\mathbf{E}$. However, in dense liquids and condensed solids, each molecule is polarized not only by external sources, but also by the intense dipolar electric fields of all neighboring polarized molecules.

By constructing a virtual spherical cavity (Lorentz sphere) around a given molecule, Hendrik Lorentz proved that the local field is:

where $\frac{\mathbf{P}}{3\epsilon_0}$ is the depolarization field contribution from the inner surface of the Lorentz cavity.

The Clausius-Mossotti Relation

The induced dipole moment of a single molecule is $\mathbf{p} = \alpha \mathbf{E}_{\text{local}}$, where $\alpha$ is the microscopic molecular polarizability. The macroscopic polarization is:

Using $\mathbf{P} = \epsilon_0 (\epsilon_r - 1) \mathbf{E}$:

Rearranging yields the celebrated Clausius-Mossotti Relation:

The Lorentz-Lorenz Formula for Optical Frequencies

At optical frequencies, Maxwell's relation gives $\epsilon_r = n^2$. Substituting into Clausius-Mossotti yields the Lorentz-Lorenz Equation:

This equation links a macroscopic optical observable—the index of refraction $n$—directly to microscopic quantum atomic parameters: number density $N$ and molecular polarizability $\alpha$.