Dispersion

The refractive index varies with frequency:

$n(\omega) = 1 + \frac{Ne^2}{2m_e\varepsilon_0} \sum_j \frac{f_j}{\omega_{0j}^2 - \omega^2 - i\gamma_j\omega}$

Where $N$ is the electron density, $f_j$ are oscillator strengths, $\omega_{0j}$ are resonance frequencies, and $\gamma_j$ are damping constants.

• Normal dispersion ($dn/d\lambda \lt 0$): away from resonances, $n$ decreases with increasing $\lambda$.
• Anomalous dispersion ($dn/d\lambda \gt 0$): near resonances, $n$ increases with $\lambda$.

• Phase velocity: $v_p = \omega/k = c/n$.
• Group velocity: $v_g = d\omega/dk = c/(n + \omega\, dn/d\omega)$.

For normal dispersion, $v_g \lt v_p$. In regions of anomalous dispersion, $v_g$ can exceed $c$ or even become negative, but this does not violate causality (signal velocity remains $\leq c$).

Starting from the Lorentz oscillator model for a single resonance:

$n^2(\omega) = 1 + \frac{Ne^2}{m_e\varepsilon_0}\frac{1}{\omega_0^2 - \omega^2 - i\gamma\omega}$

The real part $n(\omega) = \mathrm{Re}\sqrt{\epsilon(\omega)}$ gives the refractive index. The imaginary part gives absorption:

$\alpha_{\mathrm{abs}} = \frac{2\omega}{c}\,\mathrm{Im}\, n(\omega)$

Worked example. For X-rays ($\omega \gg \omega_0$):

$n \approx 1 - \frac{Ne^2}{2m_e\varepsilon_0\omega^2} = 1 - \frac{\omega_p^2}{2\omega^2}$

Where $\omega_p = \sqrt{Ne^2/(m_e\varepsilon_0)}$ is the plasma frequency. Since $n \lt 1$, X-rays undergo total external reflection at grazing incidence.

Since $n$ depends on $\lambda$, a lens has different focal lengths for different wavelengths. The longitudinal chromatic aberration is:

$\Delta f = f(\lambda_1) - f(\lambda_2)$

Achromatic doublet. Two lenses of different materials (e.g., crown and flint glass) with different dispersive powers are combined to cancel chromatic aberration at two wavelengths. The condition is:

$\frac{\omega_1}{f_1} + \frac{\omega_2}{f_2} = 0$

Where $\omega_i = (n_{i,F} - n_{i,C})/(n_{i,d} - 1)$ is the Abbe number for glass $i$.

The Cauchy equation provides an empirical fit for normal dispersion:

$n(\lambda) = A + \frac{B}{\lambda^2} + \frac{C}{\lambda^4}$

where $A, B, C$ are material constants determined experimentally.

The Sellmeier equation is more accurate, especially near resonances:

$n^2(\lambda) = 1 + \sum_j \frac{B_j \lambda^2}{\lambda^2 - \lambda_j^2}$

where $\lambda_j$ are the resonance wavelengths and $B_j$ are oscillator strengths.

In optical fiber communication, dispersion broadens pulses and limits the bit rate. The dispersion parameter $D$ is defined as:

$D = \frac{d}{d\lambda}\left(\frac{1}{v_g}\right) = -\frac{2\pi c}{\lambda^2}\frac{d^2\beta}{d\omega^2}$

where $\beta(\omega)$ is the propagation constant. $D$ has units of ps/(nm$\cdot$km).

For standard single-mode fiber, $D = 0$ near $\lambda = 1.31\ \mu$m (zero-dispersion wavelength). Dispersion-shifted fibers move this zero to $1.55\ \mu$m where attenuation is minimum.

Pulse broadening. A pulse with spectral width $\Delta\lambda$ broadens by:

$\Delta\tau = D\, L\, \Delta\lambda$

where $L$ is the fiber length. For a 10 km fiber with $D = 17$ ps/(nm$\cdot$km) and $\Delta\lambda = 1$ nm, the broadening is $\Delta\tau = 170$ ps.

A prism disperses white light into its constituent colors. The deviation angle $\delta$ for a prism with apex angle $A$ is:

$\delta = \theta_1 + \arcsin\left(n\sin\left(A - \arcsin\frac{\sin\theta_1}{n}\right)\right) - A$

At minimum deviation $\delta_m$:

$n = \frac{\sin\frac{A + \delta_m}{2}}{\sin\frac{A}{2}}$

Worked example. A glass prism with $A = 60^\circ$ gives $\delta_m = 53.7^\circ$ for sodium light. The refractive index is:

$n = \frac{\sin((60 + 53.7)/2)}{\sin(30^\circ)} = \frac{\sin 56.85^\circ}{0.5} = \frac{0.837}{0.5} = 1.674$

Practice problem. A flint glass prism has $n_F = 1.734$ (blue), $n_D = 1.720$ (yellow), $n_C = 1.713$ (red). For $A = 60^\circ$, find the angular dispersion $\delta_F - \delta_C$ at minimum deviation.

In addition to material dispersion, waveguides exhibit waveguide dispersion because the effective index depends on the confinement geometry. The total dispersion in a fiber is:

$D_{\mathrm{total}} = D_{\mathrm{material}} + D_{\mathrm{waveguide}}$

Waveguide dispersion can be engineered by varying the core-cladding index difference and core radius, enabling dispersion-flattened and dispersion-shifted fibers.

• Normal dispersion: $dn/d\lambda < 0$, group velocity $< c$.
• Anomalous dispersion: $dn/d\lambda > 0$ near resonances.
• The Lorentz oscillator model connects microscopic resonances to macroscopic dispersion.
• Chromatic aberration is corrected using achromatic doublets with different Abbe numbers.
• Material and waveguide dispersion together determine pulse broadening in optical fibers.
• Prisms disperse light via wavelength-dependent deviation angles.

Problem 1. A crown glass prism has $n = 1.52$ at $\lambda = 589$ nm. Compute the minimum deviation angle for $A = 60^\circ$.

Problem 2. An optical fiber has $D = 17$ ps/(nm$\cdot$km) at 1550 nm. A 1 nm bandwidth signal propagates 50 km. What is the pulse broadening in ps?

Problem 3. Show that the group velocity can be written as $v_g = c/(n + \omega dn/d\omega)$ by differentiating the dispersion relation.