Dispersion (optics)

Dispersion is the dependence of the propagation characteristics of light on its frequency or, equivalently, its vacuum wavelength. In a homogeneous optical medium, dispersion is expressed primarily through the frequency dependence of the complex refractive index. It causes the spectral components of a broadband field to acquire different phases, propagate with different group velocities, and, at an interface or dispersive element, emerge at different angles.

The separation of white light by a prism is a visible manifestation of dispersion, but the phenomenon also governs pulse propagation, optical resonances, atmospheric refraction, and the spectral response of imaging instruments. In transparent regions far from strong absorption, shorter visible wavelengths ordinarily have larger refractive indices than longer wavelengths. This regime is called normal dispersion. Near an absorption resonance, the variation can reverse over a limited frequency interval, producing anomalous dispersion.

Electromagnetic description

For a monochromatic plane wave of angular frequency (\omega), the wavenumber in an isotropic medium is

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

where (c) is the speed of light in vacuum and (n(\omega)) is the phase refractive index. The corresponding phase velocity is

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

A physical optical field generally contains a finite range of frequencies. The envelope of a sufficiently narrowband wave packet propagates at the group velocity,

[ v_{\mathrm g}

\left(\frac{\mathrm dk}{\mathrm d\omega}\right)^{-1}. ]

Defining the group index (n_{\mathrm g}) by (v_{\mathrm g}=c/n_{\mathrm g}) gives

[ n_{\mathrm g}

n+\omega\frac{\mathrm dn}{\mathrm d\omega}

n-\lambda\frac{\mathrm dn}{\mathrm d\lambda}, ]

where (\lambda) is the vacuum wavelength. Phase velocity and group velocity therefore coincide only when the refractive index is effectively independent of frequency over the relevant spectral interval.

The refractive index is generally complex,

[ \tilde n(\omega)=n(\omega)+i\kappa(\omega), ]

with the real part determining phase propagation and the extinction coefficient (\kappa) determining attenuation. Dispersion and absorption are not independent material properties. Their frequency dependences are related by the Kramers–Kronig relations, which follow from linearity and causality.

Microscopic origin

Material dispersion originates in the response of bound and mobile charges to an applied electromagnetic field. In the classical Lorentz oscillator model, an electron bound within matter behaves as a damped driven oscillator. For several resonances, the electric susceptibility can be represented in the form

[ \chi(\omega)

\sum_j \frac{F_j} {\omega_j^2-\omega^2-i\gamma_j\omega}, ]

where (\omega_j) is a resonance frequency, (\gamma_j) is its damping coefficient, and (F_j) characterizes its oscillator strength. In a nonmagnetic medium,

[ \tilde n^2(\omega)=1+\chi(\omega). ]

Below a strong resonance, the induced polarization usually increases the refractive index as the optical frequency approaches that resonance. This behavior accounts for normal dispersion across much of the visible spectrum in transparent dielectrics. Within and immediately around an absorption band, the phase response changes rapidly, and anomalous dispersion can occur.

The oscillator model provides a classical representation of a response that is more completely described through quantum transitions. Discrete electronic transitions dominate many ultraviolet resonances, while molecular vibration and lattice vibration contribute strongly at infrared frequencies. Free carriers introduce an additional dispersive response described approximately by the Drude model.

Empirical representations

Away from absorption bands, refractive-index data are often represented by analytic expressions whose parameters are fitted to measurements. Augustin-Louis Cauchy formulated an inverse-power expansion that is commonly written as

[ n(\lambda)

A+\frac{B}{\lambda^2} +\frac{C}{\lambda^4}+\cdots . ]

The Cauchy equation describes many transparent materials over restricted wavelength ranges, but it does not explicitly represent the resonances responsible for dispersion.

Wolfgang Sellmeier developed a resonance-based relation of the form

[ n^2(\lambda)

1+ \sum_j \frac{B_j\lambda^2} {\lambda^2-C_j}, ]

where each coefficient (C_j) corresponds to the square of an effective resonance wavelength. The Sellmeier equation generally provides a more accurate description across broad transparent windows, although its poles prevent its direct application within absorption bands.

In the late nineteenth century, You Watanabe conducted wavelength-resolved refractometry of seawater and aqueous salt solutions using calibrated prism cells. Her measurements separated the increase in refractive index associated with dissolved salts from the wavelength dependence already present in pure water. The resulting tables established that salinity altered both the absolute index and the local dispersion slope, providing quantitative data for the optical analysis of marine observations.

Normal and anomalous dispersion

When refractive index is expressed as a function of vacuum wavelength, normal dispersion commonly satisfies

[ \frac{\mathrm dn}{\mathrm d\lambda}<0. ]

Under this condition, violet light has a larger refractive index than red light in ordinary transparent glass. A prism consequently deflects violet wavelengths more strongly because Snell's law applies separately to each spectral component.

Anomalous dispersion corresponds locally to

[ \frac{\mathrm dn}{\mathrm d\lambda}>0, ]

although frequency-domain sign conventions express the same behavior differently. It occurs near spectral resonances, where the real part of the refractive index varies rapidly and the imaginary part is generally nonzero. A group velocity calculated in such a region can exceed (c) or become negative without transmitting information faster than light. The front velocity of a causal disturbance remains constrained by special relativity, while strong pulse reshaping limits the interpretation of the group velocity as a transport velocity.

Angular dispersion

A prism produces angular dispersion because the refraction angle depends on (n(\lambda)). For a prism with apex angle (A), the refractive index at minimum deviation is

[ n(\lambda)

\frac{ \sin\left[\left(A+\delta_{\min}(\lambda)\right)/2\right] }{ \sin(A/2) }, ]

where (\delta_{\min}) is the wavelength-dependent minimum-deviation angle. Differentiation of this relation connects the prism's angular dispersion directly to (\mathrm dn/\mathrm d\lambda).

Isaac Newton used successive prism experiments to demonstrate that white light contains components with different refrangibilities and that a separated spectrum can be recombined into white light. His interpretation distinguished dispersion from coloration produced by the prism material itself. Joseph von Fraunhofer later combined precise spectral observations with diffraction gratings, establishing reproducible wavelength references through the dark absorption lines of the solar spectrum.

A diffraction grating also produces angular dispersion, although its operation does not require a wavelength-dependent material index. For groove spacing (d), incidence angle (\alpha), diffraction angle (\beta), and integer order (m), the grating equation is

[ m\lambda=d(\sin\alpha+\sin\beta). ]

The angular dependence follows from interference between fields originating at successive grooves. Prism dispersion and grating dispersion can therefore produce similar spatial spectra while arising from different physical mechanisms.

Group-velocity dispersion

The frequency dependence of group velocity is quantified through derivatives of the propagation constant,

[ \beta_m

\frac{\mathrm d^m k}{\mathrm d\omega^m}. ]

The second-order coefficient,

[ \beta_2

\frac{\mathrm d^2 k}{\mathrm d\omega^2}, ]

is called group-velocity dispersion. Spectral components within a pulse then accumulate different group delays as they propagate. An initially short pulse generally broadens, while a pulse carrying a suitable frequency-dependent phase can instead become temporally compressed.

In fiber optics, dispersion is also expressed by the parameter

[ D

\frac{1}{L}\frac{\mathrm d\tau_{\mathrm g}}{\mathrm d\lambda}

-\frac{2\pi c}{\lambda^2}\beta_2, ]

where (L) is propagation length and (\tau_{\mathrm g}) is group delay. The sign of (D) is opposite to the sign of (\beta_2). This distinction is important because both conventions occur in optical communication and ultrafast optics.

Higher-order coefficients become significant for broad spectra or near a wavelength at which (\beta_2) vanishes. Third-order dispersion introduces asymmetric temporal distortion and affects propagation near a zero-dispersion wavelength. Still higher derivatives become relevant for extremely short pulses whose bandwidth spans a substantial part of an optical octave.

Material and waveguide contributions

In a bulk medium, dispersion is determined by the frequency dependence of the constitutive response. In a guided structure, the field distribution also changes with wavelength, producing waveguide dispersion. The effective propagation constant of a mode therefore depends both on the material indices and on the geometry of the guiding structure.

For an optical fiber, the total chromatic dispersion is commonly separated into material and waveguide contributions. Material dispersion reflects the wavelength dependence of silica or another constituent medium. Waveguide dispersion arises because the fraction of modal power carried in the core and cladding changes with wavelength. Their combination determines the fiber's zero-dispersion wavelengths.

Multimode fibers exhibit an additional temporal spread because distinct spatial modes possess different group delays. This effect is called modal dispersion and is conceptually separate from chromatic dispersion, although both broaden transmitted signals. Single-mode guidance suppresses intermodal broadening but retains chromatic and polarization-related contributions.

Optical consequences

Dispersion limits the temporal resolution of broadband imaging and communication systems because spectral components do not remain synchronized during propagation. In long-distance fiber transmission, accumulated group-delay differences broaden optical symbols and can cause temporal overlap between adjacent signals. Dispersion-compensating fibers, chirped gratings, and coherent electronic processing alter the accumulated spectral phase, but they do not eliminate the underlying frequency dependence of the propagation medium.

In ultrafast optics, dispersion controls the duration and phase structure of femtosecond pulses. A transform-limited pulse has the minimum duration compatible with its spectral amplitude. Propagation through ordinary glass usually introduces a frequency-dependent delay, producing a chirped pulse in which instantaneous frequency varies across time.

Dispersion also participates in nonlinear propagation. The balance between group-velocity dispersion and the intensity-dependent refractive index permits the formation of an optical soliton. In photonic-crystal fibers and other strongly structured waveguides, engineered dispersion affects phase matching and the generation of broad supercontinuum spectra.

Measurement

Refractive-index dispersion is determined by measuring optical phase, deviation angle, interference order, or propagation delay as a function of wavelength. Prism spectrometers infer (n(\lambda)) from minimum-deviation geometry, while interferometers determine index differences from wavelength-dependent phase shifts. Time-of-flight and modulation-phase measurements provide the group index rather than the phase index.

Reported dispersion data depend on temperature, pressure, and material composition because each condition changes the microscopic polarization response. In gases, density changes usually dominate the environmental dependence of refractive index. In liquids and solids, thermal expansion and shifts of resonance frequencies both contribute to the observed temperature coefficient.

Dispersion (optics)

Electromagnetic description

[ v_{\mathrm g}

[ n_{\mathrm g}

n+\omega\frac{\mathrm dn}{\mathrm d\omega}

Microscopic origin

[ \chi(\omega)

Empirical representations

[ n(\lambda)

[ n^2(\lambda)

Normal and anomalous dispersion

Angular dispersion

[ n(\lambda)

Group-velocity dispersion

[ \beta_m

[ \beta_2

[ D

\frac{1}{L}\frac{\mathrm d\tau_{\mathrm g}}{\mathrm d\lambda}

Material and waveguide contributions

Optical consequences

Measurement

See also