20.4.1. Light dispersion

20.4.1.1. First experiences

20.4.1.2. Classical dispersion

20.4.1.2.1. Drude-Lorentz model

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\[n(\omega)-1 = \frac{q^2 N}{2 \varepsilon_0 m} \frac{1}{\omega_0^2 - \omega^2 + i \gamma \omega}\]

Remark. The refractive index \(n\) is a complex number, with

\[\begin{split}\begin{aligned} \text{re} \{ n \}(\omega) & = 1 + \Delta n_0 \omega^2_0 \frac{\omega_0^2 - \omega^2}{(\omega_0^2 - \omega)^2 + \gamma^2 \omega^2} \\ \text{im} \{ n \}(\omega) & = - \Delta n_0 \omega^2_0 \frac{\gamma \omega}{(\omega_0^2 - \omega)^2 + \gamma^2 \omega^2} \ \le 0 \ , \quad \forall \omega \ge 0 \\ \end{aligned}\end{split}\]

Here the condition \(\omega \ge 0\) is required by the condition that the incoming wave travels from left to right.

Thus, a travelling wave

\[\varphi(x,t) = e^{i (\omega t - k x)} = e^{i \omega \left( t - \frac{x}{c} \right)} = e^{i \omega \left( t - n \frac{x}{c_0} \right)} \ ,\]

thus contains both a oscillating and a dampening contribution by \(n = \text{re}\{ n \} + i \, \text{im}\{ n \}\),

\[\varphi(x,t) = e^{\frac{\omega}{c_0} \text{im}\{ n(\omega) \} x} e^{i \omega \left( t - \text{re}\{ n(\omega) \} \frac{x}{c_0} \right)}\]

As the imaginary part of the refractive index is negative for all the positive frequencies, than the real exponential represents a damping term. The minimum value of \(\text{im}\{ n \}(\omega)\) for a slightly damped second-order oscillator occurs approximately at the natural frequency, \(\omega_{max damp} \simeq \omega_0\). This represents an absorption of radiation by the matter.

Remark. Drude-Lorentz model in the form \(n - 1 = \delta\) is compatible with empirical formulas in the form \(n^2 - 1 = \widetilde{\delta}\), as the correction is “small enough” for a linear approximation

\[n^2 = ( 1 + \delta )^2 \simeq 1 + 2 \delta \ .\]

20.4.1.2.2. Einstein

20.4.1.2.3. Ladenburg

Ladenburg wrote the energy balance at thermodynamical equilibrium, from transitions between each pair of states \(i\), \(k > i\), using Einstein’s coefficients from his quantum theory of radiation interacting with matter. The total amount of energy emitted by \(N_k\) molecules in the state \(k\) transitioning to a lower state \(i\) is

\[J = \underbrace{h \nu_{ik}}_{\Delta E_{ik}} N_k \left[ A_{ki} + B_{ki} u(\nu_{ik}) \right] \ .\]

At thermal equilibrium, this energy is equal to the energy absorbed by \(N_i\) molecules in the state \(i\),

\[A = h \nu_{ik} N_i B_{ik} u(\nu_{ik}) \ .\]

Exploiting the relations between Einstein coefficients (20.3), Ladenburg wrote the emitted and absorbed energy as a function of the spontaneous emission foefficient \(A_{ki}\) (allowing to relate dispersion with other phenomena involving emission),

\[J = A = h \nu_{ik} u(\nu_{ik}) N_i B_{ik} \ . \]