Part II: Electromagnetic Waves | Chapter 6

Dispersion

Why different frequencies travel at different speeds

6.1 Phase and Group Velocity

Derivation 1: Two Types of Velocity

For a monochromatic wave $e^{i(kx - \omega t)}$, the phase velocityis the speed at which surfaces of constant phase move:

$$v_p = \frac{\omega}{k}$$

A wave packet (pulse) is a superposition of many frequencies. Consider two waves with slightly different frequencies:

$$u = \cos(k_1 x - \omega_1 t) + \cos(k_2 x - \omega_2 t)$$$$= 2\cos\!\left(\frac{\Delta k}{2}x - \frac{\Delta\omega}{2}t\right)\cos\!\left(\bar{k}x - \bar{\omega}t\right)$$

The envelope (modulation) moves at the group velocity:

Group Velocity

$$v_g = \frac{d\omega}{dk}$$

The group velocity is the velocity at which energy and information propagate. It equals the phase velocity only when there is no dispersion ($\omega = ck$).

The relationship between phase and group velocity:

$$v_g = v_p + k\frac{dv_p}{dk} = v_p - \lambda\frac{dv_p}{d\lambda}$$

Normal Dispersion

$dv_p/d\lambda > 0$: $v_g < v_p$. Shorter wavelengths travel slower. This is the typical case for glass in the visible range — blue light is refracted more than red.

Anomalous Dispersion

$dv_p/d\lambda < 0$: $v_g > v_p$. Occurs near absorption resonances. The group velocity can exceed $c$ or even become negative, but this does not violate relativity (no information travels superluminally).

6.2 The Lorentz Oscillator Model

Derivation 2: Frequency-Dependent Refractive Index

Model bound electrons in a dielectric as damped harmonic oscillators driven by the electric field of the wave. The equation of motion for the displacement $\vec{r}$of an electron:

$$m_e\ddot{\vec{r}} + m_e\gamma\dot{\vec{r}} + m_e\omega_0^2\vec{r} = -e\vec{E}_0 e^{-i\omega t}$$

The polarization $\vec{P} = -Ne\vec{r}$ gives the dielectric function:

Lorentz Dielectric Function

$$\epsilon(\omega) = 1 + \frac{\omega_p^2}{\omega_0^2 - \omega^2 - i\gamma\omega}$$

where $\omega_p = \sqrt{Ne^2/(m_e\epsilon_0)}$ is the plasma frequency. The complex refractive index $\tilde{n} = \sqrt{\epsilon}$ has real part (refraction) and imaginary part (absorption).

The real and imaginary parts of $\tilde{n} = n + i\kappa$ determine the refractive index and extinction coefficient:

$$n^2 - \kappa^2 = 1 + \frac{\omega_p^2(\omega_0^2 - \omega^2)}{(\omega_0^2 - \omega^2)^2 + \gamma^2\omega^2}$$$$2n\kappa = \frac{\omega_p^2\gamma\omega}{(\omega_0^2 - \omega^2)^2 + \gamma^2\omega^2}$$

6.3 Kramers-Kronig Relations

Derivation 3: Causality Constraints

The requirement of causality — that the response cannot precede the stimulus — imposes a fundamental relationship between the real and imaginary parts of any linear response function. For the susceptibility $\chi(\omega) = \chi'(\omega) + i\chi''(\omega)$:

Kramers-Kronig Relations

$$\chi'(\omega) = \frac{1}{\pi}\mathcal{P}\!\int_{-\infty}^{\infty} \frac{\chi''(\omega')}{\omega' - \omega}\,d\omega'$$$$\chi''(\omega) = -\frac{1}{\pi}\mathcal{P}\!\int_{-\infty}^{\infty} \frac{\chi'(\omega')}{\omega' - \omega}\,d\omega'$$

where $\mathcal{P}$ denotes the Cauchy principal value. These relations mean that if you know the absorption spectrum, you can compute the refractive index at all frequencies, and vice versa. They are a consequence of the analyticity of$\chi(\omega)$ in the upper half of the complex $\omega$-plane.

Physical significance: Kramers-Kronig relations are universal: they apply to any linear causal system, not just optics. They relate acoustic impedance to absorption in acoustics, real and imaginary parts of impedance in electronics, and even have analogs in scattering theory (dispersion relations for the S-matrix).

6.4 Pulse Broadening in Dispersive Media

Derivation 4: Group Velocity Dispersion

Expand the dispersion relation to second order around the carrier frequency $\omega_0$:

$$k(\omega) \approx k_0 + \frac{1}{v_g}(\omega - \omega_0) + \frac{1}{2}k''(\omega - \omega_0)^2$$

where $k'' = d^2k/d\omega^2$ is the group velocity dispersion(GVD) parameter. A Gaussian pulse with initial width $\tau_0$ broadens as it propagates:

Pulse Broadening

$$\tau(z) = \tau_0\sqrt{1 + \left(\frac{k'' z}{\tau_0^2}\right)^2}$$

The characteristic dispersion length is $L_D = \tau_0^2/|k''|$. For$z \ll L_D$, the pulse barely broadens; for $z \gg L_D$,$\tau \approx |k''|z/\tau_0$, and shorter pulses broaden faster.

In optical fibers, GVD limits the bit rate for long-distance communication. The dispersion parameter is conventionally written as:

$$D = -\frac{2\pi c}{\lambda^2}k'' \quad \text{(ps/(nm}\cdot\text{km))}$$

6.5 Sellmeier Equation and Material Dispersion

Derivation 5: Multi-Resonance Model

Real materials have multiple absorption resonances. Generalizing the Lorentz model to include several oscillator strengths $f_j$ at frequencies $\omega_j$:

Sellmeier Equation

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

where $B_j$ are oscillator strengths and $C_j = (2\pi c/\omega_j)^2$are resonance wavelengths squared. For fused silica (the fiber optic material), three terms suffice to match the refractive index to better than $10^{-5}$across the visible and near-IR.

Fused silica has a zero-dispersion wavelength near 1.27 $\mu$m, where $k'' = 0$. Telecom fibers operate near 1.55 $\mu$m (minimum loss), where dispersion can be managed by dispersion-shifted or dispersion-compensating fibers.

6.5b Plasma Dispersion and the Plasma Frequency

A plasma (ionized gas) has free electrons that respond to EM waves. Setting$\omega_0 = 0$ and $\gamma = 0$ in the Lorentz model gives the cold plasma dielectric function:

Plasma Dispersion

$$\epsilon(\omega) = 1 - \frac{\omega_p^2}{\omega^2} \quad \Longrightarrow \quad \omega^2 = \omega_p^2 + c^2k^2$$

where $\omega_p = \sqrt{n_e e^2/(m_e\epsilon_0)}$ is the plasma frequency. Waves with $\omega < \omega_p$ cannot propagate (evanescent). The plasma frequency for the ionosphere is about 10 MHz, which is why AM radio bounces off the ionosphere but FM and TV signals pass through.

Phase velocity

$v_p = c/\sqrt{1 - \omega_p^2/\omega^2} > c$

Exceeds c but carries no information

Group velocity

$v_g = c\sqrt{1 - \omega_p^2/\omega^2} < c$

Always subluminal; carries energy

The product $v_p \cdot v_g = c^2$ is a general property of this dispersion relation. Metals have $\omega_p$ in the UV range ($\sim 10^{16}$ rad/s), which is why they are reflective in the visible (below $\omega_p$) but transparent in the UV (above $\omega_p$).

6.5c Signal Velocity and Relativity

In regions of anomalous dispersion, the group velocity can exceed $c$, become zero, or even turn negative. Does this violate special relativity?

No. The group velocity is the speed of the pulse envelope, which is well-defined only for narrow-bandwidth pulses in weakly dispersive media. In strongly dispersive regions, the pulse distorts and the group velocity loses its meaning as a signal velocity.

Sommerfeld and Brillouin (1914) showed that the front velocity — the speed of the earliest detectable signal — always equals $c$:

Velocity Hierarchy

$$v_{\text{front}} = c \geq v_{\text{signal}} \geq v_{\text{energy}} \geq 0$$

Causality is never violated. Experiments by Wang et al. (2000) demonstrated superluminal group velocity in a gain medium, but careful analysis shows no information traveled faster than $c$. The pulse appears to "arrive early" only because the leading edge is reshaped by the medium's gain profile.

6.5d Worked Example: Fiber Optic Pulse Spreading

Problem: A 10 ps laser pulse at 1.55 $\mu$m propagates through 100 km of standard single-mode fiber with GVD parameter $D = 17$ ps/(nm$\cdot$km). The spectral width of the pulse is $\Delta\lambda = 0.1$ nm. Find the pulse duration after transmission.

Solution

The temporal broadening due to GVD is:

$$\Delta t_{\text{GVD}} = |D| \cdot L \cdot \Delta\lambda = 17 \times 100 \times 0.1 = 170 \text{ ps}$$

The output pulse width is approximately:

$$\tau_{\text{out}} = \sqrt{\tau_0^2 + \Delta t_{\text{GVD}}^2} = \sqrt{10^2 + 170^2} \approx 170.3 \text{ ps}$$

The pulse has broadened by a factor of 17! This severely limits the bit rate. To achieve 10 Gbit/s (100 ps bit slot), one needs either dispersion compensation, lower-dispersion fiber, or soliton propagation (where nonlinearity balances GVD).

6.6 Applications

Prisms and Rainbows

Newton's prism experiment (1666) demonstrated that white light is composed of colors. The dispersion of glass ($dn/d\lambda < 0$ in the visible) separates wavelengths by refraction angle, with violet bending more than red.

Chirped Pulse Amplification

GVD is used deliberately in ultrafast lasers. A short pulse is stretched (chirped) using dispersion, amplified at low peak power, then recompressed. Strickland and Mourou received the 2018 Nobel Prize for this technique.

Optical Fiber Communication

Dispersion limits the bandwidth of optical fibers. Managing GVD through fiber design, dispersion compensation, and soliton propagation enables terabit-per-second data rates over transoceanic distances.

Pulsar Timing

Radio pulses from pulsars are dispersed by the interstellar medium (free electron plasma). Lower frequencies arrive later, with delay proportional to the dispersion measure$\text{DM} = \int n_e\,dl$. This allows measuring the electron column density.

6.6b Solitons: When Nonlinearity Balances Dispersion

In a medium with both dispersion and nonlinearity, a remarkable phenomenon can occur: the pulse-broadening effect of GVD can be exactly balanced by the self-focusing effect of the intensity-dependent refractive index ($n = n_0 + n_2 I$), creating a soliton that propagates without changing shape.

The nonlinear Schrodinger equation governs pulse propagation in fibers:

Nonlinear Schrodinger Equation

$$i\frac{\partial A}{\partial z} = \frac{k''}{2}\frac{\partial^2 A}{\partial t^2} - \gamma|A|^2 A$$

where $\gamma = n_2\omega_0/(cA_{\text{eff}})$ is the nonlinear coefficient. The fundamental soliton solution is a hyperbolic secant pulse that propagates indefinitely without broadening. Soliton-based communication systems have been demonstrated at terabit-per-second rates over transoceanic distances.

Water wave solitons: John Scott Russell first observed a soliton in 1834 in a canal near Edinburgh: a heap of water formed by a suddenly stopped barge traveled over a mile without changing shape. He called it the "Wave of Translation" and spent years studying it, but his observations were not explained theoretically until the work of Korteweg and de Vries in 1895.

6.7 Historical Context

Isaac Newton (1666): Demonstrated that white light is a mixture of colors using a prism. Newton incorrectly interpreted dispersion as evidence for the corpuscular theory of light, but the phenomenon is elegantly explained by the wave theory.

Augustin Cauchy (1836): Proposed the first empirical dispersion formula $n = A + B/\lambda^2 + C/\lambda^4$, fitting experimental data for transparent materials. The Sellmeier equation later provided a physical basis.

Hendrik Kramers and Ralph Kronig (1926-27): Independently derived the dispersion relations connecting absorption and refraction, using causality and complex analysis. These relations have become fundamental tools in spectroscopy and scattering theory.

Lord Rayleigh (1881): Distinguished between phase velocity and group velocity, showing that energy travels at the group velocity. Hamilton had earlier (1839) introduced the concept of group velocity in the context of wave optics.

Arnold Sommerfeld and Leon Brillouin (1914): Showed that even when the group velocity exceeds $c$ (in anomalous dispersion), the signal velocity (front velocity) never exceeds $c$, resolving the apparent conflict with special relativity.

6.7b Dispersion Management in Practice

Modern optical fiber systems use several strategies to mitigate the effects of dispersion:

Dispersion-Shifted Fiber (DSF)

By modifying the waveguide geometry, the zero-dispersion wavelength can be shifted to 1.55 $\mu$m (the minimum-loss window). However, zero dispersion enhances nonlinear effects like four-wave mixing, which can cause crosstalk between WDM channels.

Dispersion Compensation

Alternating segments of fiber with opposite-sign GVD (dispersion compensation fiber, chirped fiber Bragg gratings) can cancel the total accumulated dispersion while maintaining small local dispersion to suppress nonlinear effects.

Digital Signal Processing

Modern coherent optical receivers digitize the received signal and compensate dispersion electronically using digital filters. This approach can correct for both chromatic dispersion and polarization-mode dispersion in real time, enabling 400 Gbit/s per channel over thousands of kilometers.

6.8 Python Simulation

This simulation visualizes the Lorentz dielectric function, phase vs group velocity, Gaussian pulse broadening, and the Sellmeier dispersion for fused silica.

Dispersion: Lorentz Model, Phase/Group Velocity, Pulse Broadening, and Sellmeier

Python

script.py148 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

# ============================================================
# Dispersion: Phase/Group Velocity, Kramers-Kronig, Pulse Broadening
# ============================================================

fig, axes = plt.subplots(2, 2, figsize=(12, 10))
fig.suptitle('Dispersion in Optical Media', fontsize=16, fontweight='bold', color='white')
fig.patch.set_facecolor('#0a0a0a')

for ax in axes.flat:
    ax.set_facecolor('#0a0a0a')
    ax.tick_params(colors='white')
    ax.xaxis.label.set_color('white')
    ax.yaxis.label.set_color('white')
    ax.title.set_color('white')
    for spine in ax.spines.values():
        spine.set_color('#333')

# --- Panel 1: Lorentz model: n and kappa vs frequency ---
ax1 = axes[0, 0]
omega0 = 1.0  # resonance frequency (normalized)
gamma = 0.1
omega_p = 0.5

omega = np.linspace(0.01, 2.5, 2000)
epsilon_r = 1 + omega_p**2 * (omega0**2 - omega**2) / ((omega0**2 - omega**2)**2 + gamma**2 * omega**2)
epsilon_i = omega_p**2 * gamma * omega / ((omega0**2 - omega**2)**2 + gamma**2 * omega**2)

# Complex refractive index
eps_complex = epsilon_r + 1j * epsilon_i
n_complex = np.sqrt(eps_complex)
n_real = np.real(n_complex)
kappa = np.imag(n_complex)

ax1.plot(omega / omega0, n_real, color='#22d3ee', linewidth=2.5, label='n (refractive index)')
ax1.plot(omega / omega0, kappa, color='#f97316', linewidth=2.5, label='kappa (extinction)')
ax1.axvline(x=1, color='white', linewidth=0.5, linestyle=':', alpha=0.3)
ax1.axhline(y=1, color='white', linewidth=0.5, linestyle=':', alpha=0.3)
ax1.set_xlabel('omega / omega_0')
ax1.set_ylabel('n, kappa')
ax1.set_title('Lorentz Model: n and kappa')
ax1.legend(fontsize=8, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')
ax1.set_ylim(-0.5, 3)
ax1.annotate('anomalous\ndispersion', xy=(1.0, 0.5), fontsize=8, color='#ef4444',
            ha='center')

# --- Panel 2: Phase and group velocity ---
ax2 = axes[0, 1]
# Use deep water wave dispersion as clear example: omega = sqrt(g*k)
k_arr = np.linspace(0.1, 5, 500)
g_const = 1.0
omega_water = np.sqrt(g_const * k_arr)
vp_water = omega_water / k_arr
vg_water = 0.5 * np.sqrt(g_const / k_arr)

ax2.plot(k_arr, vp_water, color='#22d3ee', linewidth=2.5, label='v_p = omega/k')
ax2.plot(k_arr, vg_water, color='#f97316', linewidth=2.5, label='v_g = d(omega)/dk')
ax2.set_xlabel('Wavenumber k')
ax2.set_ylabel('Velocity')
ax2.set_title('Phase vs Group Velocity (Deep Water Waves)')
ax2.legend(fontsize=8, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')
ax2.text(3.0, 0.55, 'v_g = v_p / 2', color='#4ade80', fontsize=10)

# --- Panel 3: Gaussian pulse broadening ---
ax3 = axes[1, 0]
x = np.linspace(-20, 40, 1000)
tau0 = 3.0  # initial pulse width
k_pp = 0.5  # GVD parameter

for z_val, color, alpha_val in [(0, '#22d3ee', 1.0), (10, '#f97316', 0.8),
                                 (30, '#a855f7', 0.6), (80, '#4ade80', 0.4)]:
    tau_z = tau0 * np.sqrt(1 + (k_pp * z_val / tau0**2)**2)
    vg_pulse = 1.0
    pulse = np.exp(-(x - vg_pulse * z_val * 0.1)**2 / (2 * tau_z**2)) / (tau_z / tau0)
    ax3.plot(x, pulse, color=color, linewidth=2, alpha=alpha_val,
             label='z = ' + str(z_val) + ', tau = ' + str(round(tau_z, 1)))

ax3.set_xlabel('Position x')
ax3.set_ylabel('Amplitude')
ax3.set_title('Gaussian Pulse Broadening (GVD)')
ax3.legend(fontsize=7, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white')

# --- Panel 4: Sellmeier dispersion for fused silica ---
ax4 = axes[1, 1]
# Sellmeier coefficients for fused silica
B = [0.6961663, 0.4079426, 0.8974794]
C = [0.0684043**2, 0.1162414**2, 9.896161**2]  # in microns^2

lam = np.linspace(0.3, 2.0, 1000)  # wavelength in microns
n_sq = 1.0
for j in range(3):
    n_sq = n_sq + B[j] * lam**2 / (lam**2 - C[j])
n_silica = np.sqrt(n_sq)

# Group index
dn_dlam = np.gradient(n_silica, lam)
ng = n_silica - lam * dn_dlam

# GVD
d2n_dlam2 = np.gradient(dn_dlam, lam)
D_param = -lam * d2n_dlam2 / (0.3)  # approximate, normalized

ax4_twin = ax4.twinx()
ax4.plot(lam, n_silica, color='#22d3ee', linewidth=2.5, label='n (phase index)')
ax4.plot(lam, ng, color='#f97316', linewidth=2.5, label='n_g (group index)')
ax4_twin.plot(lam, -lam / 0.3 * d2n_dlam2 * 1e3, color='#a855f7', linewidth=2, linestyle='--',
              label='GVD (arb. units)', alpha=0.7)
ax4_twin.axhline(y=0, color='white', linewidth=0.5, linestyle=':', alpha=0.3)
ax4_twin.set_ylabel('GVD (arb.)', color='#a855f7')
ax4_twin.tick_params(axis='y', colors='#a855f7')

# Mark zero dispersion wavelength
zd_idx = np.argmin(np.abs(d2n_dlam2))
ax4.axvline(x=lam[zd_idx], color='#4ade80', linewidth=1, linestyle='--', alpha=0.7)
ax4.text(lam[zd_idx] + 0.05, 1.47, 'ZDW', color='#4ade80', fontsize=9)

ax4.set_xlabel('Wavelength (microns)')
ax4.set_ylabel('Refractive index')
ax4.set_title('Fused Silica: Sellmeier Dispersion')
ax4.legend(fontsize=7, facecolor='#0a0a0a', edgecolor='#333', labelcolor='white', loc='upper right')

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a0a')
plt.close()
print("Plot saved to output.png")

print("\n" + "=" * 60)
print("DISPERSION: NUMERICAL RESULTS")
print("=" * 60)
print("\n--- Lorentz Model ---")
print("  Resonance: omega_0 = " + str(omega0))
print("  Damping: gamma = " + str(gamma))
print("  Plasma frequency: omega_p = " + str(omega_p))
print("\n--- Fused Silica Refractive Index ---")
for lam_val in [0.4, 0.55, 0.7, 1.0, 1.3, 1.55]:
    idx = np.argmin(np.abs(lam - lam_val))
    print("  lambda = " + str(lam_val) + " um: n = " + str(round(n_silica[idx], 5)) + ", n_g = " + str(round(ng[idx], 5)))
print("\n--- Zero Dispersion Wavelength ---")
print("  ZDW ~ " + str(round(lam[zd_idx], 3)) + " um")
print("\n--- Pulse Broadening ---")
print("  tau(z) = tau_0 * sqrt(1 + (k'' z / tau_0^2)^2)")
print("  Dispersion length: L_D = tau_0^2 / |k''|")
print("=" * 60)

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Summary

Core Equations

Phase velocity: $v_p = \omega/k$
Group velocity: $v_g = d\omega/dk$
Lorentz: $\epsilon = 1 + \omega_p^2/(\omega_0^2 - \omega^2 - i\gamma\omega)$
Pulse broadening: $\tau = \tau_0\sqrt{1 + (k''z/\tau_0^2)^2}$
Kramers-Kronig connects $n(\omega)$ and $\kappa(\omega)$

Key Insights

Energy travels at group velocity, not phase velocity
Anomalous dispersion near resonances
Causality links absorption and refraction
GVD broadens pulses, limits fiber bandwidth
Sellmeier equation models real materials accurately

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