Dispersion (Optics) · aiwiki.page

Dispersion in optics is the dependence of light’s propagation properties on frequency or wavelength. In a transparent, homogeneous material, it usually means that the refractive index, and therefore the phase velocity of light, varies with frequency. More generally, dispersion describes the frequency dependence of a wave’s propagation constant, including effects caused by optical structures rather than by the material alone. Familiar manifestations include the separation of white light into colors by a prism and the changing shape of a light pulse as it propagates. (ocw.mit.edu)

Physical origin

An electromagnetic wave drives the charged constituents of matter, particularly electrons, into oscillatory motion. Their response contributes to the electromagnetic field propagating through the material. Because this response depends on how the driving frequency compares with the material’s resonant frequencies, the resulting refractive index is frequency-dependent. Refraction is therefore not generally described by one constant index valid throughout the electromagnetic spectrum. (feynmanlectures.caltech.edu)

A classical description treats bound charges as driven, damped oscillators. Far from a resonance, their response can yield a transparent region with smoothly varying refractive index. Near a resonance, both the refractive index and absorption can change strongly. In absorbing media, the index is represented by a complex number: its real part describes phase propagation, while its imaginary part describes attenuation under the chosen wave convention. Dispersion and absorption are thus related aspects of the material’s electromagnetic response. (feynmanlectures.caltech.edu)

Color separation and refraction

At a stationary interface, refraction obeys Snell’s law,

n_1(\lambda)\sin\theta_1=n_2(\lambda)\sin\theta_2,

where $\lambda$ denotes vacuum wavelength and the angles are measured from the interface normal. If the refractive indices vary with wavelength, different spectral components generally refract through different angles. A prism uses successive refractions at nonparallel surfaces to produce an angularly separated spectrum. In ordinary visible-light glasses, violet light has a higher refractive index than red light and is deviated more strongly. (ocw.mit.edu)

The same wavelength dependence contributes to a rainbow. Sunlight undergoes refraction on entering and leaving water droplets, together with internal reflection. Different wavelengths emerge in different preferred directions. Dispersion supplies the color separation, while droplet geometry and reflection determine the overall pattern. (science.nasa.gov)

Color separation is not unique to material dispersion. A diffraction grating separates wavelengths through diffraction and interference. This is also called angular dispersion, but its physical origin differs from the frequency-dependent refractive index of a prism. (science.nasa.gov)

Phase velocity, group velocity, and pulse dispersion

For a plane wave in a transparent homogeneous medium, the propagation constant is

k(\omega)=\frac{n(\omega)\omega}{c},

where $\omega$ is angular frequency and $c$ is the speed of light in vacuum. The phase velocity, describing the motion of a constant-phase surface, is

v_{\mathrm p}=\frac{\omega}{k}=\frac{c}{n(\omega)}.

For a sufficiently narrowband wave packet in a weakly absorbing medium, the envelope travels approximately at the group velocity,

v_{\mathrm g} =\left(\frac{dk}{d\omega}\right)^{-1} =\frac{c}{n+\omega\,dn/d\omega}.

The corresponding group index is

n_{\mathrm g}=n+\omega\frac{dn}{d\omega} =n-\lambda\frac{dn}{d\lambda}.

Phase and group velocities therefore need not be equal. (ocw.mit.edu)

A pulse contains a range of frequencies. Differences in their group delays can alter its duration and shape. Group-velocity dispersion (GVD) is quantified by

\beta_2=\frac{d^2k}{d\omega^2}.

For propagation through a uniform length $L$ , the group-delay dispersion is $L\beta_2$ , commonly expressed in square femtoseconds. Higher derivatives describe higher-order dispersion. (ocw.mit.edu)

Second-order dispersion broadens an initially transform-limited Gaussian pulse and produces a frequency variation across its envelope, called chirp. It can instead compress an already chirped pulse when the imposed dispersion compensates its existing spectral phase. Thus dispersion does not invariably lengthen every pulse. (ocw.mit.edu)

Normal and anomalous dispersion

In the traditional refractive-index terminology, normal dispersion means that refractive index increases with frequency, or decreases with vacuum wavelength:

\frac{dn}{d\omega}>0, \qquad \frac{dn}{d\lambda}<0.

This behavior occurs across much of the visible transparency range of common optical glasses. Anomalous dispersion denotes the opposite slope and is often associated with spectral regions near absorption resonances. (ocw.mit.edu)

In pulse and fiber optics, the same words commonly classify GVD: normal GVD means $\beta_2>0$ , and anomalous GVD means $\beta_2<0$ . These definitions are not identical to those based on the first derivative of refractive index. For a homogeneous medium,

\beta_2 =\frac{\lambda^3}{2\pi c^2} \frac{d^2n}{d\lambda^2}.

GVD therefore depends on the curvature of the refractive-index curve, not merely its slope. A description of “normal dispersion” must be interpreted in its stated context. (ocw.mit.edu)

Material, waveguide, and modal dispersion

Several mechanisms are distinguished in optical transmission:

Material dispersion arises from the frequency-dependent refractive indices of the constituent materials.
Waveguide dispersion arises because confinement and the spatial distribution of a guided mode vary with frequency. It can occur even when the constituent material indices are treated as constant.
Intermodal dispersion occurs when different guided modes have different group delays. A pulse distributed among several modes can consequently spread in time. (ocw.mit.edu)

In an optical fiber, a mode is described by its propagation constant $\beta(\omega)$ , rather than simply by the bulk-material wavenumber. Its effective refractive index is $n_{\mathrm{eff}}=c\beta/\omega$ . Material properties and waveguide geometry together determine its chromatic dispersion. Single-mode operation removes intermodal spreading among spatial modes, but does not remove chromatic dispersion. Fiber design can modify the frequency dependence of propagation by changing the refractive-index profile and operating wavelength. (feynmanlectures.caltech.edu)

Fiber optics often uses the dispersion parameter

D=\frac{1}{L}\frac{d\tau_{\mathrm g}}{d\lambda} =-\frac{2\pi c}{\lambda^2}\beta_2,

where $\tau_{\mathrm g}$ is group delay. It is commonly expressed in picoseconds per nanometer per kilometer. The sign of $D$ is opposite that of $\beta_2$ . (ocw.mit.edu)

Quantifying material dispersion

Optical-glass specifications often characterize visible dispersion through the Abbe number,

V_d=\frac{n_d-1}{n_F-n_C},

using refractive indices at designated spectral lines: approximately 587.6 nm for $d$ , 486.1 nm for $F$ , and 656.3 nm for $C$ . A larger Abbe number indicates weaker relative visible dispersion. It is useful for comparing glasses but does not describe their complete wavelength dependence. Partial-dispersion data are needed for more precise chromatic correction. (media.schott.com)

A widely used wavelength-dependent representation is the Sellmeier equation, commonly written

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

The coefficients are fitted for a particular material over a specified wavelength range. Wavelength units must match those used in the coefficient definition. Such fits allow calculation of refractive index and its derivatives, but should not be extrapolated indiscriminately into absorption bands or beyond their validated ranges. (media.schott.com)

Applications and compensation

Dispersion is useful in spectroscopy, where prisms spatially separate spectral components for analysis. In imaging systems, however, it produces chromatic aberration: different wavelengths may focus at different axial positions or produce different image magnifications. An achromatic lens combines elements with different dispersions to reduce these errors. Correction at selected wavelengths does not necessarily eliminate residual color errors throughout an entire spectral band. (ocw.mit.edu)

In telecommunications, chromatic dispersion can distort optical signals accumulated over long fiber paths. Its effects depend on spectral bandwidth, transmission length, material dispersion, and waveguide design. Managing these factors is part of designing a transmission system. (web.mit.edu)

For ultrashort laser pulses, even relatively short paths through glass can contribute appreciable dispersion. Compensation optics introduce an opposing spectral phase. Chirped mirrors use wavelength-dependent reflection delays, often together with adjustable glass wedges, to control pulse duration. Experiments have used this combination both to compress pulses and to characterize their spectral phase. (thorlabs.us)

Historical development and interpretive limits

Prism experiments were central to Isaac Newton’s investigation of light and color. In correspondence published in 1672, he described differently colored rays as having different degrees of refrangibility and discussed recombining separated colors to produce white light. These experiments supported the view that a prism separates components already present in white light rather than simply creating colors by modifying it. (newtonproject.ox.ac.uk)

Dispersion should not be equated with every form of optical spreading. For ordinary plane waves in vacuum, $k=\omega/c$ , so phase and group velocities are $c$ and GVD is zero. Nevertheless, a finite beam can spread through diffraction. A guided structure can also be dispersive even when its material is not. (feynmanlectures.caltech.edu)

Finally, phase velocity is not generally the velocity of information transmission. A refractive index below unity can give a phase velocity above $c$ without permitting faster-than-light signals. Near strong resonances, absorption and pulse reshaping also limit a simple interpretation of envelope motion: signal propagation depends on the medium’s response over a range of frequencies, not on its refractive index at one frequency alone. (feynmanlectures.caltech.edu)

References

Lecture 13: Dispersive Medium, Phase Velocity, Group Velocityocw.mit.edu
Chapter 2: Classical Electromagnetism and Opticsocw.mit.edu
Lecture 4: Ray Optics, EM Optics, Guided Wave Opticsocw.mit.edu
014 Lecture 18 Notesweb.mit.edu
Optics: Lecture Notesocw.mit.edu
TIE-29: Refractive Index and Dispersionmedia.schott.com
Wave Behaviorsscience.nasa.gov
Earthnasa.gov