2.5 Dispersion

Dispersion in electromagnetic waves

Dispersion is the phenomenon where a wave's propagation velocity depends on its frequency. It governs how electromagnetic waves interact with matter across the spectrum, and it places real, quantifiable limits on optical system performance and data transmission rates. The mathematical framework here builds directly on the wave equation in media from earlier in the unit.

Frequency dependence of wave velocity

In vacuum, all frequencies travel at $c$. In a material medium, that symmetry breaks: the phase velocity $v_p = \frac{c}{n(\omega)}$ becomes frequency-dependent because the refractive index $n$ itself depends on $\omega$.

• Higher-frequency components typically travel slower than lower-frequency ones in most transparent materials (like glass in the visible range).
• This frequency-dependent velocity causes different spectral components to separate spatially as a wave propagates. Prisms splitting white light into colors and rainbows are both direct consequences.

Refractive index vs. frequency

The refractive index $n(\omega)$ quantifies how much slower light travels in a medium compared to vacuum: $v_p = \frac{c}{n(\omega)}$.

In dispersive media, $n$ is not a constant but a function of frequency. Two regimes matter:

• Normal dispersion: $\frac{dn}{d\omega} > 0$ (refractive index increases with frequency, so higher frequencies are slower). This is the typical behavior in transparent regions away from resonances.
• Anomalous dispersion: $\frac{dn}{d\omega} < 0$ (refractive index decreases with increasing frequency). This occurs near absorption resonances.

Normal vs. anomalous dispersion

Normal dispersion dominates in spectral regions far from any resonance of the medium. Most transparent glasses and crystals show normal dispersion across the visible spectrum, which is why a prism bends violet light more than red.

Anomalous dispersion appears in narrow frequency bands near absorption resonances. Here the refractive index can decrease sharply with increasing frequency, and absorption becomes significant. The term "anomalous" is historical; the behavior is fully predicted by the Lorentz model and is not unusual once you account for resonance effects.

Dispersion in optical media

Optical media (glasses, crystals, gases) exhibit dispersion because of how their atomic and molecular structure interacts with the oscillating electric field of light. The key physics is the coupling between the field and bound charges at characteristic resonance frequencies.

Absorption and emission of light

Atoms and molecules have discrete energy levels, which define a set of resonance frequencies $\omega_0$. When incident light has a frequency near $\omega_0$:

1. The oscillating electric field drives bound electrons at close to their natural frequency.
2. Energy transfers efficiently from the field to the medium (absorption). Electrons are excited to higher energy states.
3. Excited electrons eventually return to lower states, emitting photons at the resonance frequency.

Away from resonance, the medium is largely transparent but still refracts light because the bound electrons oscillate slightly out of phase with the driving field.

Resonance frequencies of media

Each material has its own set of resonance frequencies determined by its electronic, vibrational, and rotational energy level structure. Near these resonances:

• The refractive index changes rapidly with frequency.
• The absorption coefficient peaks sharply.
• The transition from normal to anomalous dispersion occurs.

For example, ordinary glass has strong UV absorption resonances, which is why $n(\omega)$ increases steadily through the visible (normal dispersion) as you approach those UV resonances from below.

Kramers-Kronig relations

The Kramers-Kronig relations connect the real and imaginary parts of the complex refractive index $\tilde{n}(\omega) = n(\omega) + i\kappa(\omega)$, where $n$ governs refraction and $\kappa$ governs absorption.

These relations state that if you know the absorption spectrum $\kappa(\omega)$ over all frequencies, you can compute $n(\omega)$, and vice versa. Formally:

$n(\omega) - 1 = \frac{2}{\pi} \mathcal{P} \int_0^{\infty} \frac{\omega' \kappa(\omega')}{\omega'^2 - \omega^2} d\omega'$

where $\mathcal{P}$ denotes the Cauchy principal value.

The physical content is profound: dispersion and absorption are not independent properties. They are two manifestations of the same underlying causal response of the medium to the electromagnetic field. Any material that absorbs at some frequency must also show dispersion, and the Kramers-Kronig relations tell you exactly how.

Dispersion and wave propagation

Dispersion has its most dramatic effects on pulses and wavepackets, which are superpositions of many frequency components. The distinction between phase velocity and group velocity becomes essential here.

Group vs. phase velocity

Phase velocity $v_p = \frac{\omega}{k}$ is the speed at which a single-frequency plane wave's phase fronts advance. It describes a monochromatic wave but not the motion of information or energy.

Group velocity $v_g = \frac{d\omega}{dk}$ is the speed at which the envelope of a wavepacket propagates. For a pulse in a dispersive medium:

$v_g = \frac{c}{n(\omega) + \omega \frac{dn}{d\omega}}$

In a non-dispersive medium, $\frac{dn}{d\omega} = 0$, so $v_g = v_p$. In a dispersive medium, they differ. For normal dispersion ($\frac{dn}{d\omega} > 0$), the group velocity is less than the phase velocity.

Pulse broadening and distortion

A short pulse contains a broad spread of frequencies (this follows from Fourier analysis). When that pulse enters a dispersive medium:

1. Each frequency component travels at a slightly different phase velocity.
2. The components that were initially in phase begin to dephase.
3. The pulse envelope stretches in time (broadens) and its peak intensity drops.
4. If dispersion is strong enough, the pulse shape distorts as well.

The amount of broadening depends on the group velocity dispersion (GVD), characterized by $\frac{d^2 k}{d\omega^2}$, and on the bandwidth of the pulse. Broader-bandwidth (shorter) pulses are more susceptible.

Dispersion-induced limitations in communication

In fiber-optic communication, dispersion directly limits performance:

• A transmitted pulse broadens as it propagates, eventually overlapping with neighboring pulses. This is called inter-symbol interference.
• The maximum bit rate $B$ and transmission length $L$ are roughly constrained by $B^2 \cdot L \cdot D \lesssim 1$, where $D$ is the dispersion parameter (typically in units of ps/(nm·km)).
• Standard single-mode fiber has $D \approx 17 \text{ ps/(nm·km)}$ at 1550 nm, which becomes a serious constraint at data rates above 10 Gb/s over long distances.

Compensation strategies include dispersion-compensating fibers (with large negative $D$), chirped fiber Bragg gratings, and digital signal processing at the receiver.

Mathematical treatment of dispersion

Several models describe the frequency-dependent response of materials, ranging from first-principles classical models to empirical fitting formulas.

Lorentz oscillator model

The Lorentz model treats each bound electron as a damped harmonic oscillator driven by the electric field of the wave. For a single resonance at frequency $\omega_0$ with damping constant $\gamma$:

$\tilde{\epsilon}(\omega) = 1 + \frac{N e^2 / (m \epsilon_0)}{\omega_0^2 - \omega^2 - i\gamma\omega}$

where $N$ is the number density of oscillators, $e$ and $m$ are the electron charge and mass.

This single expression captures all the essential dispersion physics:

• Far below resonance ($\omega \ll \omega_0$): $n$ is nearly constant and slightly greater than 1.
• Approaching resonance from below: $n$ increases (normal dispersion).
• Near resonance: $n$ changes rapidly, absorption peaks, and anomalous dispersion appears.
• Far above resonance ($\omega \gg \omega_0$): $n$ approaches 1 from below.

For real materials with multiple resonances, you sum contributions from each oscillator.

Sellmeier and Cauchy equations

The Sellmeier equation is the standard empirical formula for $n(\lambda)$ in transparent regions:

$n^2(\lambda) = 1 + \sum_i \frac{B_i \lambda^2}{\lambda^2 - C_i}$

Each term corresponds to a resonance at wavelength $\sqrt{C_i}$, with strength $B_i$. The coefficients are tabulated for common optical materials (BK7 glass, fused silica, etc.) and are accurate to several decimal places across wide wavelength ranges.

The Cauchy equation is a simpler polynomial approximation valid far from resonances:

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

It's less accurate than Sellmeier but useful for quick estimates and for fitting data over limited wavelength ranges.

Dispersion curves and diagrams

Dispersion curves plot $n(\omega)$ or $n(\lambda)$ across a spectral range. On these plots:

• Normal dispersion regions show $n$ increasing smoothly with $\omega$.
• Anomalous dispersion regions appear as dips in $n$ near absorption bands.
• Absorption bands themselves show up as peaks in $\kappa(\omega)$.

An alternative representation is the $\omega$-$k$ diagram, where the slope gives the phase velocity and the local slope of the tangent gives the group velocity. Straight lines through the origin indicate non-dispersive propagation; curvature indicates dispersion.

Applications and examples

Dispersion in optical fibers

Optical fibers used in telecommunications exhibit two main sources of dispersion:

• Material dispersion: arises from the wavelength dependence of the silica refractive index. It dominates at shorter wavelengths and passes through zero near 1.3 µm.
• Waveguide dispersion: arises because the mode's confinement in the core depends on wavelength. It can be engineered by changing the fiber's index profile.

The total chromatic dispersion is the sum of both contributions. In standard single-mode fiber, the zero-dispersion wavelength falls near 1310 nm. Dispersion-shifted fibers move this zero to 1550 nm (the minimum-loss window). Dispersion-flattened fibers maintain low dispersion across a broad band for wavelength-division multiplexed (WDM) systems.

Prism and grating dispersion

Prisms and diffraction gratings both separate light by wavelength, but through different mechanisms:

• Prisms rely on the material's dispersion: Snell's law $n_1 \sin\theta_1 = n_2 \sin\theta_2$ gives a wavelength-dependent refraction angle because $n_2 = n_2(\lambda)$. The angular dispersion of a prism is $\frac{d\theta}{d\lambda} = \frac{t}{d} \frac{dn}{d\lambda}$, where $t$ is the base length and $d$ the beam diameter.
• Diffraction gratings disperse light via interference: the grating equation $d(\sin\theta_i + \sin\theta_m) = m\lambda$ gives angular dispersion $\frac{d\theta_m}{d\lambda} = \frac{m}{d\cos\theta_m}$, which depends on the grating spacing $d$ and diffraction order $m$, not on material dispersion.

Both are used in spectrometers, monochromators, and pulse compressors. Gratings generally offer higher dispersion and resolving power than prisms.

Atmospheric dispersion and rainbows

Atmospheric dispersion occurs because the refractive index of air depends on wavelength (roughly following a Cauchy-type relation). Effects include:

• Rainbows: Sunlight entering a water droplet undergoes refraction, internal reflection, and refraction again. Because $n_{\text{water}}$ varies with wavelength (about 1.344 for red to 1.343 for violet at visible wavelengths... actually the variation is from about 1.331 for red to 1.344 for violet), different colors emerge at different angles. The primary rainbow appears at roughly 42° from the anti-solar point, with red on the outside and violet on the inside.
• Atmospheric chromatic aberration: Stars near the horizon appear slightly elongated into short vertical spectra because refraction through the atmosphere is wavelength-dependent. Adaptive optics systems on telescopes include atmospheric dispersion correctors to compensate.

Dispersion compensation techniques

Dispersion compensation is critical in both telecommunications and ultrafast optics. The core idea is to introduce dispersion of the opposite sign to cancel accumulated pulse broadening.

• Dispersion-compensating fiber (DCF): Specialty fiber with large negative dispersion (e.g., $D \approx -80 \text{ ps/(nm·km)}$) inserted periodically along a transmission link.
• Chirped fiber Bragg gratings: Reflect different wavelengths from different positions along the grating, introducing a controlled delay versus wavelength.
• Prism and grating pairs: Used in ultrafast laser systems. A pair of prisms or gratings introduces negative group velocity dispersion, compensating for the positive GVD of laser gain media and other optics. This is essential for generating and maintaining femtosecond pulses.
• Chirped mirrors: Multilayer dielectric mirrors designed so that different wavelengths penetrate to different depths before reflecting, providing precise dispersion control over broad bandwidths.