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<title>Dispersion (optics)</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Dispersion (optics)</span></span>

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<p><b>Dispersion</b> is the phenomenon in which the <a href="Phase_velocity" title="Phase velocity">phase velocity</a> of a <a href="Wave" title="Wave">wave</a> depends on its frequency.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Sometimes the term <b>chromatic dispersion</b> is used to refer to <a href="Optics" title="Optics">optics</a> specifically, as opposed to <a href="Wave_propagation" class="mw-redirect" title="Wave propagation">wave propagation</a> in general. A medium having this common property may be termed a <b>dispersive medium</b>.

</p><p>Although the term is used in the field of optics to describe <a href="Light" title="Light">light</a> and other <a href="Electromagnetic_wave" class="mw-redirect" title="Electromagnetic wave">electromagnetic waves</a>, dispersion in the same sense can apply to any sort of wave motion such as <a href="Acoustic_dispersion" title="Acoustic dispersion">acoustic dispersion</a> in the case of sound and seismic waves, and in <a href="Gravity_wave" title="Gravity wave">gravity waves</a> (ocean waves). Within optics, dispersion is a property of telecommunication signals along <a href="Transmission_line" title="Transmission line">transmission lines</a> (such as <a href="Microwaves" class="mw-redirect" title="Microwaves">microwaves</a> in <a href="Coaxial_cable" title="Coaxial cable">coaxial cable</a>) or the <a href="Pulse_(signal_processing)" title="Pulse (signal processing)">pulses</a> of light in <a href="Optical_fiber" title="Optical fiber">optical fiber</a>.

</p><p>In optics, one important and familiar consequence of dispersion is the change in the angle of <a href="Refraction" title="Refraction">refraction</a> of different colors of light,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> as seen in the spectrum produced by a dispersive <a href="Prism_(optics)" title="Prism (optics)">prism</a> and in <a href="Chromatic_aberration" title="Chromatic aberration">chromatic aberration</a> of lenses. Design of compound <a href="Achromatic_lens" title="Achromatic lens">achromatic lenses</a>, in which chromatic aberration is largely cancelled, uses a quantification of a glass's dispersion given by its <a href="Abbe_number" title="Abbe number">Abbe number</a> <i>V</i>, where <i>lower</i> Abbe numbers correspond to <i>greater</i> dispersion over the <a href="Visible_spectrum" title="Visible spectrum">visible spectrum</a>. In some applications such as telecommunications, the absolute phase of a wave is often not important but only the propagation of <a href="Wave_packet" title="Wave packet">wave packets</a> or "pulses"; in that case one is interested only in variations of <a href="Group_velocity" title="Group velocity">group velocity</a> with frequency, so-called <a href="#Group-velocity_dispersion">group-velocity dispersion</a>.

</p><p>All common <a href="Transmission_media" class="mw-redirect" title="Transmission media">transmission media</a> also vary in <a href="Attenuation" title="Attenuation">attenuation</a> (normalized to transmission length) as a function of frequency, leading to <a href="Attenuation_distortion" title="Attenuation distortion">attenuation distortion</a>; this is not dispersion, although sometimes reflections at closely spaced <a href="Impedance_matching" title="Impedance matching">impedance boundaries</a> (e.g. crimped segments in a cable) can produce signal distortion which further aggravates inconsistent transit time as observed across signal bandwidth.

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<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<p>Dispersion causes a <a href="Rainbow" title="Rainbow">rainbow's</a> spatial separation of a white <a href="Light" title="Light">light</a> into components of different <a href="Wavelengths" class="mw-redirect" title="Wavelengths">wavelengths</a> (different <a href="Color" title="Color">colors</a>). However, dispersion also has an effect in many other circumstances: for example, <a href="Group-velocity_dispersion" title="Group-velocity dispersion">group-velocity dispersion</a> causes <a href="Pulse_(signal_processing)" title="Pulse (signal processing)">pulses</a> to spread in <a href="Optical_fiber" title="Optical fiber">optical fibers</a>, degrading signals over long distances; also, a cancellation between group-velocity dispersion and <a href="Nonlinear_system" title="Nonlinear system">nonlinear</a> effects leads to <a href="Soliton" title="Soliton">soliton</a> waves.

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<div class="mw-heading mw-heading2"><h2 id="Material_and_waveguide_dispersion">Material and waveguide dispersion</h2></div>

<p>Most often, chromatic dispersion refers to bulk material dispersion, that is, the change in <a href="Refractive_index" title="Refractive index">refractive index</a> with optical frequency. However, in a <a href="Waveguide" title="Waveguide">waveguide</a> there is also the phenomenon of <i>waveguide dispersion</i>, in which case a wave's <a href="Phase_velocity" title="Phase velocity">phase velocity</a> in a structure depends on its frequency simply due to the structure's geometry. More generally, "waveguide" dispersion can occur for waves propagating through any inhomogeneous structure (e.g., a <a href="Photonic_crystal" title="Photonic crystal">photonic crystal</a>), whether or not the waves are confined to some region. In a waveguide, <i>both</i> types of dispersion will generally be present, although they are not strictly additive. For example, in fiber optics the material and waveguide dispersion can effectively cancel each other out to produce a <a href="Zero-dispersion_wavelength" title="Zero-dispersion wavelength">zero-dispersion wavelength</a>, important for fast <a href="Fiber-optic_communication" title="Fiber-optic communication">fiber-optic communication</a>.

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<div class="mw-heading mw-heading2"><h2 id="Material_dispersion_in_optics">Material dispersion in optics</h2></div>

<p>Material dispersion can be a desirable or undesirable effect in optical applications. The dispersion of light by glass prisms is used to construct <a href="Spectrometer" title="Spectrometer">spectrometers</a> and <a href="Spectroradiometer" title="Spectroradiometer">spectroradiometers</a>. However, in lenses, dispersion causes <a href="Chromatic_aberration" title="Chromatic aberration">chromatic aberration</a>, an undesired effect that may degrade images in microscopes, telescopes, and photographic objectives.

</p><p>The <i><a href="Phase_velocity" title="Phase velocity">phase velocity</a></i> <i>v</i> of a wave in a given uniform medium is given by

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<annotation encoding="application/x-tex">{\displaystyle v={\frac {c}{n}},}</annotation>

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</math></span><img src="./c95ed0ac458c266eec2492800b4e1cab3d28738e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.104ex; height:4.676ex;" alt="{\displaystyle v={\frac {c}{n}},}" loading="lazy"></span></dd></dl>

<p>where <i>c</i> is the <a href="Speed_of_light" title="Speed of light">speed of light</a> in vacuum, and <i>n</i> is the <a href="Refractive_index" title="Refractive index">refractive index</a> of the medium.

</p><p>In general, the refractive index is some function of the frequency <i>f</i> of the light, thus <i>n</i> = <i>n</i>(<i>f</i>), or alternatively, with respect to the wave's wavelength <i>n</i> = <i>n</i>(<i>λ</i>). The wavelength dependence of a material's refractive index is usually quantified by its <a href="Abbe_number" title="Abbe number">Abbe number</a> or its coefficients in an empirical formula such as the <a href="Cauchy's_equation" title="Cauchy's equation">Cauchy</a> or <a href="Sellmeier_equation" title="Sellmeier equation">Sellmeier equations</a>.

</p><p>Because of the <a href="Kramers%E2%80%93Kronig_relations" title="Kramers–Kronig relations">Kramers–Kronig relations</a>, the wavelength dependence of the real part of the refractive index is related to the material <a href="Absorption_(electromagnetic_radiation)" title="Absorption (electromagnetic radiation)">absorption</a>, described by the imaginary part of the refractive index (also called the <a href="Refractive_index#Dispersion_and_absorption" title="Refractive index">extinction coefficient</a>). In particular, for non-magnetic materials (<a href="Permeability_(electromagnetism)" title="Permeability (electromagnetism)"><i>μ</i></a> = <a href="Magnetic_constant" class="mw-redirect" title="Magnetic constant"><i>μ</i><sub>0</sub></a>), the <a href="Linear_response_function" title="Linear response function">susceptibility</a> <i>χ</i> that appears in the Kramers–Kronig relations is the <a href="Electric_susceptibility" title="Electric susceptibility">electric susceptibility</a> <i>χ</i><sub>e</sub> = <i>n</i><sup>2</sup> − 1.

</p><p>The most commonly seen consequence of dispersion in optics is the separation of <a href="Electromagnetic_spectrum#Visible_radiation_(light)" title="Electromagnetic spectrum">white light</a> into a <a href="Optical_spectrum" class="mw-redirect" title="Optical spectrum">color spectrum</a> by a <a href="Triangular_prism_(optics)" class="mw-redirect" title="Triangular prism (optics)">prism</a>. From <a href="Snell's_law" title="Snell's law">Snell's law</a> it can be seen that the angle of <a href="Refraction" title="Refraction">refraction</a> of light in a prism depends on the refractive index of the prism material. Since that refractive index varies with wavelength, it follows that the angle that the light is refracted by will also vary with wavelength, causing an angular separation of the colors known as <i>angular dispersion</i>.

</p><p>For visible light, the refraction index <i>n</i> of most transparent materials (e.g., air, glasses) decreases with increasing wavelength <i>λ</i>:

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<annotation encoding="application/x-tex">{\displaystyle 1<n(\lambda _{\text{red}})<n(\lambda _{\text{yellow}})<n(\lambda _{\text{blue}}),}</annotation>

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</math></span><img src="./9f89af74348e87d663f9d2d29cf86392d4573c0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:35.304ex; height:3.009ex;" alt="{\displaystyle 1<n(\lambda _{\text{red}})<n(\lambda _{\text{yellow}})<n(\lambda _{\text{blue}}),}" loading="lazy"></span></dd></dl>

<p>or generally,

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<annotation encoding="application/x-tex">{\displaystyle {\frac {dn}{d\lambda }}<0.}</annotation>

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</math></span><img src="./219116c712f00de8a9eb303d0835dd69acd3f584.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.354ex; height:5.509ex;" alt="{\displaystyle {\frac {dn}{d\lambda }}<0.}" loading="lazy"></span></dd></dl>

<p>In this case, the medium is said to have <i>normal dispersion</i>. However, if the index increases with increasing wavelength (which is typically the case in the ultraviolet<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>), the medium is said to have <i>anomalous dispersion</i>.

</p><p>At the interface of such a material with air or vacuum (index of ~1), Snell's law predicts that light incident at an angle <i>θ</i> to the <a href="Surface_normal" class="mw-redirect" title="Surface normal">normal</a> will be refracted at an angle arcsin(<style data-mw-deduplicate="TemplateStyles:r1214402035">

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</style><span class="sfrac">⁠<span class="tion"><span class="num">sin <i>θ</i></span><span class="sr-only">/</span><span class="den"><i>n</i></span></span>⁠</span>). Thus, in the case of normal dispersion, blue light, with a higher refractive index, will be bent more strongly than red light, resulting in the well-known <a href="Rainbow" title="Rainbow">rainbow</a> pattern.

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<div class="mw-heading mw-heading2"><h2 id="Group-velocity_dispersion">Group-velocity dispersion</h2></div>

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</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Group-velocity_dispersion" title="Group-velocity dispersion">Group-velocity dispersion</a></div>

<p>Beyond simply describing a change in the phase velocity over wavelength, a more serious consequence of dispersion in many applications is termed <a href="Group-velocity_dispersion" title="Group-velocity dispersion">group-velocity dispersion</a> (GVD). While phase velocity <i>v</i> is defined as <i>v</i> = <i>c</i>/<i>n</i>, this describes only one frequency component. When different frequency components are combined, as when considering a signal or a pulse, one is often more interested in the <a href="Group_velocity" title="Group velocity">group velocity</a>, which describes the speed at which a pulse or information superimposed on a wave (modulation) propagates. In the accompanying animation, it can be seen that the wave itself (orange-brown) travels at a phase velocity much faster than the speed of the <i>envelope</i> (black), which corresponds to the group velocity. This pulse might be a communications signal, for instance, and its information only travels at the group velocity rate, even though it consists of wavefronts advancing at a faster rate (the phase velocity).

</p><p>It is possible to calculate the group velocity from the refractive-index curve <i>n</i>(<i>ω</i>) or more directly from the wavenumber <i>k</i> = <i>ωn</i>/<i>c</i>, where <i>ω</i> is the radian frequency <i>ω</i> = 2<i>πf</i>. Whereas one expression for the phase velocity is <i>v</i><sub>p</sub> = <i>ω</i>/<i>k</i>, the group velocity can be expressed using the <a href="Derivative" title="Derivative">derivative</a>: <i>v</i><sub>g</sub> = <i>dω</i>/<i>dk</i>. Or in terms of the phase velocity <i>v</i><sub>p</sub>,

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<annotation encoding="application/x-tex">{\displaystyle v_{\text{g}}={\frac {v_{\text{p}}}{1-{\dfrac {\omega }{v_{\text{p}}}}{\dfrac {dv_{\text{p}}}{d\omega }}}}.}</annotation>

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</math></span><img src="./4655eaaf09dda2bdcd93729a893efcb3618d0a4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:18.202ex; height:9.176ex;" alt="{\displaystyle v_{\text{g}}={\frac {v_{\text{p}}}{1-{\dfrac {\omega }{v_{\text{p}}}}{\dfrac {dv_{\text{p}}}{d\omega }}}}.}" loading="lazy"></span></dd></dl>

<p>When dispersion is present, not only the group velocity is not equal to the phase velocity, but generally it itself varies with wavelength. This is known as group-velocity dispersion and causes a short pulse of light to be broadened, as the different-frequency components within the pulse travel at different velocities. Group-velocity dispersion is quantified as the derivative of the <i>reciprocal</i> of the group velocity with respect to <a href="Angular_frequency" title="Angular frequency">angular frequency</a>, which results in <i>group-velocity dispersion</i> = <i>d</i><sup>2</sup><i>k</i>/<i>dω</i><sup>2</sup>.

</p><p>If a light pulse is propagated through a material with positive group-velocity dispersion, then the shorter-wavelength components travel slower than the longer-wavelength components. The pulse therefore becomes <i>positively <a href="Chirp" title="Chirp">chirped</a></i>, or <i>up-chirped</i>, increasing in frequency with time. On the other hand, if a pulse travels through a material with negative group-velocity dispersion, shorter-wavelength components travel faster than the longer ones, and the pulse becomes <i>negatively chirped</i>, or <i>down-chirped</i>, decreasing in frequency with time.

</p><p>An everyday example of a negatively chirped signal in the acoustic domain is that of an approaching train hitting deformities on a welded track. The sound caused by the train itself is impulsive and travels much faster in the metal tracks than in air, so that the train can be heard well before it arrives. However, from afar it is not heard as causing impulses, but leads to a distinctive descending chirp, amidst reverberation caused by the complexity of the vibrational modes of the track. Group-velocity dispersion can be heard in that the volume of the sounds stays audible for a surprisingly long time, up to several seconds.

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<div class="mw-heading mw-heading2"><h2 id="Dispersion_control">Dispersion control</h2></div>

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<p>The result of GVD, whether negative or positive, is ultimately temporal spreading of the pulse. This makes dispersion management extremely important in optical communications systems based on optical fiber, since if dispersion is too high, a group of pulses representing a bit-stream will spread in time and merge, rendering the bit-stream unintelligible. This limits the length of fiber that a signal can be sent down without regeneration. One possible answer to this problem is to send signals down the optical fibre at a wavelength where the GVD is zero (e.g., around 1.3–1.5 μm in <a href="Silica" class="mw-redirect" title="Silica">silica</a> <a href="Fibres" class="mw-redirect" title="Fibres">fibres</a>), so pulses at this wavelength suffer minimal spreading from dispersion. In practice, however, this approach causes more problems than it solves because zero GVD unacceptably amplifies other nonlinear effects (such as <a href="Four-wave_mixing" title="Four-wave mixing">four-wave mixing</a>). Another possible option is to use <a href="Soliton_(optics)" title="Soliton (optics)">soliton</a> pulses in the regime of negative dispersion, a form of optical pulse which uses a <a href="Nonlinear_optics" title="Nonlinear optics">nonlinear optical</a> effect to self-maintain its shape. Solitons have the practical problem, however, that they require a certain power level to be maintained in the pulse for the nonlinear effect to be of the correct strength. Instead, the solution that is currently used in practice is to perform dispersion compensation, typically by matching the fiber with another fiber of opposite-sign dispersion so that the dispersion effects cancel; such compensation is ultimately limited by nonlinear effects such as <a href="Self-phase_modulation" title="Self-phase modulation">self-phase modulation</a>, which interact with dispersion to make it very difficult to undo.

</p><p>Dispersion control is also important in <a href="Laser" title="Laser">lasers</a> that produce <a href="Ultrashort_pulse" title="Ultrashort pulse">short pulses</a>. The overall dispersion of the <a href="Laser_construction" title="Laser construction">optical resonator</a> is a major factor in determining the duration of the pulses emitted by the laser. A pair of <a href="Prism_(optics)" title="Prism (optics)">prisms</a> can be arranged to produce net negative dispersion, which can be used to balance the usually positive dispersion of the laser medium. <a href="Diffraction_grating" title="Diffraction grating">Diffraction gratings</a> can also be used to produce dispersive effects; these are often used in high-power laser amplifier systems. Recently, an alternative to prisms and gratings has been developed: <a href="Chirped_mirror" title="Chirped mirror">chirped mirrors</a>. These dielectric mirrors are coated so that different wavelengths have different penetration lengths, and therefore different group delays. The coating layers can be tailored to achieve a net negative dispersion.

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<div class="mw-heading mw-heading2"><h2 id="In_waveguides">In waveguides</h2></div>

<p><a href="Waveguide" title="Waveguide">Waveguides</a> are highly dispersive due to their geometry (rather than just to their material composition). <a href="Optical_fiber" title="Optical fiber">Optical fibers</a> are a sort of waveguide for optical frequencies (light) widely used in modern telecommunications systems. The rate at which data can be transported on a single fiber is limited by pulse broadening due to chromatic dispersion among other phenomena.

</p><p>In general, for a waveguide mode with an <a href="Angular_frequency" title="Angular frequency">angular frequency</a> <i>ω</i>(<i>β</i>) at a <a href="Propagation_constant" title="Propagation constant">propagation constant</a> <i>β</i> (so that the electromagnetic fields in the propagation direction <i>z</i> oscillate proportional to <i>e</i><sup><i>i</i>(<i>βz</i>−<i>ωt</i>)</sup>), the group-velocity dispersion parameter <i>D</i> is defined as<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

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<annotation encoding="application/x-tex">{\displaystyle D=-{\frac {2\pi c}{\lambda ^{2}}}{\frac {d^{2}\beta }{d\omega ^{2}}}={\frac {2\pi c}{v_{g}^{2}\lambda ^{2}}}{\frac {dv_{g}}{d\omega }},}</annotation>

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</math></span><img src="./edb07f3506801c836b0abd8bf33abc51ef67cdf6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:29.094ex; height:6.676ex;" alt="{\displaystyle D=-{\frac {2\pi c}{\lambda ^{2}}}{\frac {d^{2}\beta }{d\omega ^{2}}}={\frac {2\pi c}{v_{g}^{2}\lambda ^{2}}}{\frac {dv_{g}}{d\omega }},}" loading="lazy"></span></dd></dl>

<p>where <i>λ</i> = 2<span class="texhtml mvar" style="font-style:italic;">π</span><i>c</i>/<i>ω</i> is the vacuum wavelength, and <i>v</i><sub>g</sub> = <i>dω</i>/<i>dβ</i> is the group velocity. This formula generalizes the one in the previous section for homogeneous media and includes both waveguide dispersion and material dispersion. The reason for defining the dispersion in this way is that |<i>D</i>| is the (asymptotic) temporal pulse spreading Δ<i>t</i> per unit bandwidth

Δ<i>λ</i> per unit distance travelled, commonly reported in <a href="Picosecond" title="Picosecond">ps</a>/(<a href="Nanometre" title="Nanometre">nm</a>⋅<a href="Kilometre" title="Kilometre">km</a>) for optical fibers.

</p><p>In the case of <a href="Multi-mode_optical_fiber" title="Multi-mode optical fiber">multi-mode optical fibers</a>, so-called <a href="Modal_dispersion" title="Modal dispersion">modal dispersion</a> will also lead to pulse broadening. Even in <a href="Single-mode_fiber" class="mw-redirect" title="Single-mode fiber">single-mode fibers</a>, pulse broadening can occur as a result of <a href="Polarization_mode_dispersion" title="Polarization mode dispersion">polarization mode dispersion</a> (since there are still two polarization modes). These are <i>not</i> examples of chromatic dispersion, as they are not dependent on the wavelength or <a href="Spectral_linewidth" class="mw-redirect" title="Spectral linewidth">bandwidth</a> of the pulses propagated.

</p>

<div class="mw-heading mw-heading2"><h2 id="Higher-order_dispersion_over_broad_bandwidths">Higher-order dispersion over broad bandwidths</h2></div>

<p>When a broad range of frequencies (a broad bandwidth) is present in a single wavepacket, such as in an <a href="Ultrashort_pulse" title="Ultrashort pulse">ultrashort pulse</a> or a <a href="Chirp" title="Chirp">chirped</a> pulse or other forms of <a href="Spread_spectrum" title="Spread spectrum">spread spectrum</a> transmission, it may not be accurate to approximate the dispersion by a constant over the entire bandwidth, and more complex calculations are required to compute effects such as pulse spreading.

</p><p>In particular, the dispersion parameter <i>D</i> defined above is obtained from only one derivative of the group velocity. Higher derivatives are known as <i>higher-order dispersion</i>.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> These terms are simply a <a href="Taylor_series" title="Taylor series">Taylor series</a> expansion of the <a href="Dispersion_relation" title="Dispersion relation">dispersion relation</a> <i>β</i>(<i>ω</i>) of the medium or waveguide around some particular frequency. Their effects can be computed via numerical evaluation of <a href="Fourier_transform" title="Fourier transform">Fourier transforms</a> of the waveform, via integration of higher-order <a href="Slowly_varying_envelope_approximation" title="Slowly varying envelope approximation">slowly varying envelope approximations</a>, by a <a href="Split-step_method" title="Split-step method">split-step method</a> (which can use the exact dispersion relation rather than a Taylor series), or by direct simulation of the full <a href="Maxwell's_equations" title="Maxwell's equations">Maxwell's equations</a> rather than an approximate envelope equation.

</p>

<div class="mw-heading mw-heading2"><h2 id="Spatial_dispersion">Spatial dispersion</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Spatial_dispersion" title="Spatial dispersion">Spatial dispersion</a></div>

<p>In electromagnetics and optics, the term <i>dispersion</i> generally refers to aforementioned temporal or frequency dispersion. Spatial dispersion refers to the non-local response of the medium to the space; this can be reworded as the wavevector dependence of the permittivity. For an exemplary <a href="Anisotropy" title="Anisotropy">anisotropic</a> medium, the spatial relation between <a href="Electric_field" title="Electric field">electric</a> and <a href="Electric_displacement_field" title="Electric displacement field">electric displacement field</a> can be expressed as a <a href="Convolution" title="Convolution">convolution</a>:<sup id="cite_ref-landau_8-0" class="reference"><a href="#cite_note-landau-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>

</p>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msup>

</msup>

<msub>

</mrow>

</msub>

<msup>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D_{i}(t,r)=E_{i}(t,r)+\int _{0}^{\infty }\int f_{ik}(\tau ;r,r')E_{k}(t-\tau ,r')\,dV'\,d\tau ,}</annotation>

</semantics>

</math></span><img src="./f20ddfc753193bfeaae4c07ccb907f12ba53d315.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:58.524ex; height:5.843ex;" alt="{\displaystyle D_{i}(t,r)=E_{i}(t,r)+\int _{0}^{\infty }\int f_{ik}(\tau ;r,r')E_{k}(t-\tau ,r')\,dV'\,d\tau ,}" loading="lazy"></span></dd></dl>

<p>where the <a href="Integral_transform" title="Integral transform">kernel</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{ik}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{ik}}</annotation>

</semantics>

</math></span><img src="./20ddbf2a5f8cebb8027e1cf228e52b2c8bb124dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.795ex; height:2.509ex;" alt="{\displaystyle f_{ik}}" loading="lazy"></span> is dielectric response (susceptibility); its indices make it in general a <a href="Tensor" title="Tensor">tensor</a> to account for the anisotropy of the medium. Spatial dispersion is negligible in most macroscopic cases, where the scale of variation of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{k}(t-\tau ,r')}">

<msub>

</mrow>

</msub>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E_{k}(t-\tau ,r')}</annotation>

</semantics>

</math></span><img src="./e275aa80738846bfb515b5547b22414e265ee5ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.263ex; height:3.009ex;" alt="{\displaystyle E_{k}(t-\tau ,r')}" loading="lazy"></span> is much larger than atomic dimensions, because the dielectric kernel dies out at macroscopic distances. Nevertheless, it can result in non-negligible macroscopic effects, particularly in conducting media such as <a href="Metal" title="Metal">metals</a>, <a href="Electrolyte" title="Electrolyte">electrolytes</a> and <a href="Plasma_(physics)" title="Plasma (physics)">plasmas</a>. Spatial dispersion also plays role in <a href="Optical_rotation" title="Optical rotation">optical activity</a> and <a href="Doppler_broadening" title="Doppler broadening">Doppler broadening</a>,<sup id="cite_ref-landau_8-1" class="reference"><a href="#cite_note-landau-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> as well as in the theory of <a href="Metamaterial" title="Metamaterial">metamaterials</a>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="In_gemology">In gemology</h2></div>

<caption>Dispersion values of minerals<sup id="cite_ref-b1_10-0" class="reference"><a href="#cite_note-b1-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>

</caption>

<th>Mineral name</th>

</th></tr>

<tr>

<td><a href="Hematite" title="Hematite">Hematite</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Cinnabar" title="Cinnabar">Cinnabar</a> (HgS)</td>

<td>—

</td></tr>

<tr>

<td>synth. <a href="Rutile" title="Rutile">Rutile</a></td>

<td>0.190

</td></tr>

<tr>

<td><a href="Rutile" title="Rutile">Rutile</a> (TiO<sub>2</sub>)</td>

<td>0.120–0.180

</td></tr>

<tr>

<td><a href="Anatase" title="Anatase">Anatase</a> (TiO<sub>2</sub>)</td>

<td>—

</td></tr>

<tr>

<td><a href="Wulfenite" title="Wulfenite">Wulfenite</a></td>

<td>0.133

</td></tr>

<tr>

<td><a href="Vanadinite" title="Vanadinite">Vanadinite</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Fabulite" class="mw-redirect" title="Fabulite">Fabulite</a></td>

<td>0.109

</td></tr>

<tr>

<td><a href="Sphalerite" title="Sphalerite">Sphalerite</a> (ZnS)</td>

<td>0.088

</td></tr>

<tr>

<td><a href="Sulfur" title="Sulfur">Sulfur</a> (S)</td>

<td>—

</td></tr>

<tr>

<td><a href="Stibiotantalite" title="Stibiotantalite">Stibiotantalite</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Goethite" title="Goethite">Goethite</a> (FeO(OH))</td>

<td>—

</td></tr>

<tr>

<td><a href="Brookite" title="Brookite">Brookite</a> (TiO<sub>2</sub>)</td>

<td>0.12–1.80

</td></tr>

<tr>

<td><a href="Linobate" class="mw-redirect" title="Linobate">Linobate</a></td>

<td>0.075

</td></tr>

<tr>

<td><a href="Zincite" title="Zincite">Zincite</a> (ZnO)</td>

<td>—

</td></tr>

<tr>

<td>synth. <a href="Moissanite" title="Moissanite">Moissanite</a> (SiC)</td>

<td>—

</td></tr>

<tr>

<td><a href="Cassiterite" title="Cassiterite">Cassiterite</a> (SnO<sub>2</sub>)</td>

<td>0.035

</td></tr>

<tr>

<td><a href="Zirconia" class="mw-redirect" title="Zirconia">Zirconia</a> (ZrO<sub>2</sub>)</td>

<td>0.035

</td></tr>

<tr>

<td><a href="Powellite" title="Powellite">Powellite</a> (CaMoO<sub>4</sub>)</td>

<td>—

</td></tr>

<tr>

<td><a href="Andradite" title="Andradite">Andradite</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Demantoid" title="Demantoid">Demantoid</a></td>

<td>0.034

</td></tr>

<tr>

<td><a href="Cerussite" title="Cerussite">Cerussite</a></td>

<td>0.033–0.050

</td></tr>

<tr>

<td><a href="Titanite" title="Titanite">Titanite</a></td>

<td>0.019–0.038

</td></tr>

<tr>

<td><a href="Benitoite" title="Benitoite">Benitoite</a></td>

<td>0.026

</td></tr>

<tr>

<td><a href="Anglesite" title="Anglesite">Anglesite</a></td>

<td>0.025

</td></tr>

<tr>

<td><a href="Diamond" title="Diamond">Diamond</a> (C)</td>

<td>0.025

</td></tr>

<tr>

<td>synth. <a href="Cassiterite" title="Cassiterite">Cassiterite</a> (SnO<sub>2</sub>)</td>

<td>—

</td></tr>

<tr>

<td><a href="Flint_glass" title="Flint glass">Flint glass</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Hyacinth_(mineral)" class="mw-redirect" title="Hyacinth (mineral)">Hyacinth</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Jargoon" title="Jargoon">Jargoon</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Starlite" title="Starlite">Starlite</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Scheelite" title="Scheelite">Scheelite</a></td>

<td>0.026

</td></tr>

<tr>

<td><a href="Zircon" title="Zircon">Zircon</a> (ZrSiO<sub>4</sub>)</td>

<td>0.022

</td></tr>

<tr>

<td>0.022

</td></tr>

<tr>

<td><a href="Dioptase" title="Dioptase">Dioptase</a></td>

<td>0.021

</td></tr>

<tr>

<td>Whe Vinay wellite</td>

<td>—

</td></tr>

<tr>

<td><a href="Gypsum" title="Gypsum">Gypsum</a></td>

<td>0.008

</td></tr>

<tr>

<td><a href="Alabaster" title="Alabaster">Alabaster</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Epidote" title="Epidote">Epidote</a></td>

<td>0.012–0.027

</td></tr>

<tr>

<td><a href="Tanzanite" title="Tanzanite">Tanzanite</a></td>

<td>0.011

</td></tr>

<tr>

<td><a href="Thulite" title="Thulite">Thulite</a></td>

<td>0.011

</td></tr>

<tr>

<td><a href="Zoisite" title="Zoisite">Zoisite</a></td>

<td>—

</td></tr>

<tr>

<td>0.015

</td></tr>

<tr>

<td><a href="Spessartine" title="Spessartine">Spessartine</a></td>

<td>0.015

</td></tr>

<tr>

<td><a href="Uvarovite" title="Uvarovite">Uvarovite</a></td>

<td>0.014–0.021

</td></tr>

<tr>

<td><a href="Almandine" title="Almandine">Almandine</a></td>

<td>0.013–0.016

</td></tr>

<tr>

<td><a href="Hessonite" class="mw-redirect" title="Hessonite">Hessonite</a></td>

<td>0.013–0.015

</td></tr>

<tr>

<td><a href="Willemite" title="Willemite">Willemite</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Pleonaste" class="mw-redirect" title="Pleonaste">Pleonaste</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Rhodolite" title="Rhodolite">Rhodolite</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Boracite" title="Boracite">Boracite</a></td>

<td>0.012

</td></tr>

<tr>

<td><a href="Cryolite" title="Cryolite">Cryolite</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Staurolite" title="Staurolite">Staurolite</a></td>

<td>0.012–0.013

</td></tr>

<tr>

<td><a href="Pyrope" title="Pyrope">Pyrope</a></td>

<td>0.013–0.016

</td></tr>

<tr>

<td><a href="Diaspore" title="Diaspore">Diaspore</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Grossular" title="Grossular">Grossular</a></td>

<td>0.012

</td></tr>

<tr>

<td><a href="Hemimorphite" title="Hemimorphite">Hemimorphite</a></td>

<td>0.013

</td></tr>

<tr>

<td><a href="Kyanite" title="Kyanite">Kyanite</a></td>

<td>0.011

</td></tr>

<tr>

<td><a href="Peridot" title="Peridot">Peridot</a></td>

<td>0.012–0.013

</td></tr>

<tr>

<td><a href="Spinel" title="Spinel">Spinel</a></td>

<td>0.011

</td></tr>

<tr>

<td><a href="Vesuvianite" title="Vesuvianite">Vesuvianite</a></td>

<td>0.014

</td></tr>

<tr>

<td><a href="Gahnite" title="Gahnite">Gahnite</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Clinozoisite" title="Clinozoisite">Clinozoisite</a></td>

<td>0.011–0.014

</td></tr>

<tr>

<td><a href="Labradorite" title="Labradorite">Labradorite</a></td>

<td>0.010

</td></tr>

<tr>

<td><a href="Axinite" title="Axinite">Axinite</a></td>

<td>0.011

</td></tr>

<tr>

<td><a href="Diopside" title="Diopside">Diopside</a></td>

<td>0.01

</td></tr>

<tr>

<td><a href="Ekanite" title="Ekanite">Ekanite</a></td>

<td>0.012

</td></tr>

<tr>

<td><a href="Corundum" title="Corundum">Corundum</a> (Al<sub>2</sub>O<sub>3</sub>)</td>

<td>0.011

</td></tr>

<tr>

<td>synth. <a href="Corundum" title="Corundum">Corundum</a></td>

<td>0.011

</td></tr>

<tr>

<td>0.011

</td></tr>

<tr>

<td><a href="Sapphire" title="Sapphire">Sapphire</a> (Al<sub>2</sub>O<sub>3</sub>)</td>

<td>0.011

</td></tr>

<tr>

<td><a href="Kornerupine" title="Kornerupine">Kornerupine</a></td>

<td>0.010

</td></tr>

<tr>

<td><a href="Sinhalite" title="Sinhalite">Sinhalite</a></td>

<td>0.010

</td></tr>

<tr>

<td><a href="Sodalite" title="Sodalite">Sodalite</a></td>

<td>0.009

</td></tr>

<tr>

<td><a href="Rhodizite" class="mw-redirect" title="Rhodizite">Rhodizite</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Hiddenite" title="Hiddenite">Hiddenite</a></td>

<td>0.010

</td></tr>

<tr>

<td><a href="Kunzite" class="mw-redirect" title="Kunzite">Kunzite</a></td>

<td>0.010

</td></tr>

<tr>

<td><a href="Spodumene" title="Spodumene">Spodumene</a></td>

<td>0.010

</td></tr>

<tr>

<td><a href="Tourmaline" title="Tourmaline">Tourmaline</a></td>

<td>0.009–0.011

</td></tr>

<tr>

<td><a href="Cordierite" title="Cordierite">Cordierite</a></td>

<td>0.009

</td></tr>

<tr>

<td><a href="Danburite" title="Danburite">Danburite</a></td>

<td>0.009

</td></tr>

<tr>

<td><a href="Herderite" title="Herderite">Herderite</a></td>

<td>0.008–0.009

</td></tr>

<tr>

<td><a href="Rubellite" title="Rubellite">Rubellite</a></td>

<td>0.008–0.009

</td></tr>

<tr>

<td>Achroite</td>

<td>—

</td></tr>

<tr>

<td><a href="Dravite" class="mw-redirect" title="Dravite">Dravite</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Elbaite" title="Elbaite">Elbaite</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Indicolite" class="mw-redirect" title="Indicolite">Indicolite</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Liddicoatite" class="mw-redirect" title="Liddicoatite">Liddicoatite</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Scapolite" title="Scapolite">Scapolite</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Schorl" class="mw-redirect" title="Schorl">Schorl</a></td>

<td>—

</td></tr>

<tr>

<td>Verdelite</td>

<td>—

</td></tr>

<tr>

<td><a href="Andalusite" title="Andalusite">Andalusite</a></td>

<td>0.009

</td></tr>

<tr>

<td><a href="Baryte" title="Baryte">Baryte</a> (BaSO<sub>4</sub>)</td>

<td>0.009

</td></tr>

<tr>

<td><a href="Euclase" title="Euclase">Euclase</a></td>

<td>0.009

</td></tr>

<tr>

<td><a href="Datolite" title="Datolite">Datolite</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Alexandrite" class="mw-redirect" title="Alexandrite">Alexandrite</a></td>

<td>0.011

</td></tr>

<tr>

<td><a href="Chrysoberyl" title="Chrysoberyl">Chrysoberyl</a></td>

<td>0.011

</td></tr>

<tr>

<td><a href="Rhodochrosite" title="Rhodochrosite">Rhodochrosite</a></td>

<td>0.010–0.020

</td></tr>

<tr>

<td><a href="Sillimanite" title="Sillimanite">Sillimanite</a></td>

<td>0.009–0.012

</td></tr>

<tr>

<td><a href="Hambergite" title="Hambergite">Hambergite</a></td>

<td>0.009–0.010

</td></tr>

<tr>

<td><a href="Pyroxmangite" title="Pyroxmangite">Pyroxmangite</a></td>

<td>—

</td></tr>

<tr>

<td>synth. <a href="Scheelite" title="Scheelite">Scheelite</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Smithsonite" title="Smithsonite">Smithsonite</a></td>

<td>0.008–0.017

</td></tr>

<tr>

<td><a href="Amblygonite" title="Amblygonite">Amblygonite</a></td>

<td>0.008

</td></tr>

<tr>

<td><a href="Aquamarine_(gemstone)" class="mw-redirect" title="Aquamarine (gemstone)">Aquamarine</a></td>

<td>0.009–0.013

</td></tr>

<tr>

<td><a href="Beryl" title="Beryl">Beryl</a></td>

<td>0.009–0.013

</td></tr>

<tr>

<td><a href="Emerald" title="Emerald">Emerald</a></td>

<td>0.009–0.013

</td></tr>

<tr>

<td><a href="Heliodor" class="mw-redirect" title="Heliodor">Heliodor</a></td>

<td>0.009–0.013

</td></tr>

<tr>

<td><a href="Morganite" class="mw-redirect" title="Morganite">Morganite</a></td>

<td>0.009–0.013

</td></tr>

<tr>

<td><a href="Brazilianite" title="Brazilianite">Brazilianite</a></td>

<td>0.008

</td></tr>

<tr>

<td><a href="Celestine_(mineral)" title="Celestine (mineral)">Celestine</a></td>

<td>0.008

</td></tr>

<tr>

<td><a href="Topaz" title="Topaz">Topaz</a></td>

<td>0.008

</td></tr>

<tr>

<td><a href="Goshenite" class="mw-redirect" title="Goshenite">Goshenite</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Apatite" title="Apatite">Apatite</a></td>

<td>0.008–0.010

</td></tr>

<tr>

<td><a href="Aventurine" title="Aventurine">Aventurine</a></td>

<td>0.008

</td></tr>

<tr>

<td><a href="Amethyst" title="Amethyst">Amethyst</a> (SiO<sub>2</sub>)</td>

<td>0.008

</td></tr>

<tr>

<td><a href="Citrine_quartz" class="mw-redirect" title="Citrine quartz">Citrine quartz</a></td>

<td>0.008

</td></tr>

<tr>

<td><a href="Prasiolite" title="Prasiolite">Prasiolite</a></td>

<td>0.008

</td></tr>

<tr>

<td><a href="Quartz" title="Quartz">Quartz</a> (SiO<sub>2</sub>)</td>

<td>0.008

</td></tr>

<tr>

<td>Rose quartz (SiO<sub>2</sub>)</td>

<td>0.008

</td></tr>

<tr>

<td>Smoky quartz (SiO<sub>2</sub>)</td>

<td>0.008

</td></tr>

<tr>

<td><a href="Anhydrite" title="Anhydrite">Anhydrite</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Dolomite_(mineral)" title="Dolomite (mineral)">Dolomite</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Morion_(mineral)" class="mw-redirect" title="Morion (mineral)">Morion</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Feldspar" title="Feldspar">Feldspar</a></td>

<td>0.008

</td></tr>

<tr>

<td><a href="Moonstone_(gemstone)" title="Moonstone (gemstone)">Moonstone</a></td>

<td>0.008

</td></tr>

<tr>

<td><a href="Orthoclase" title="Orthoclase">Orthoclase</a></td>

<td>0.008

</td></tr>

<tr>

<td><a href="Pollucite" title="Pollucite">Pollucite</a></td>

<td>0.007

</td></tr>

<tr>

<td><a href="Albite" title="Albite">Albite</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Bytownite" title="Bytownite">Bytownite</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Emerald#Synthetic_emerald" title="Emerald">synth. Emerald</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Magnesite" title="Magnesite">Magnesite</a> (MgCO<sub>3</sub>)</td>

<td>—

</td></tr>

<tr>

<td><a href="Sanidine" title="Sanidine">Sanidine</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Sunstone" title="Sunstone">Sunstone</a></td>

<td>—

</td></tr>

<tr>

<td>synth. <a href="Alexandrite" class="mw-redirect" title="Alexandrite">Alexandrite</a></td>

<td>—

</td></tr>

<tr>

<td>synth. <a href="Sapphire" title="Sapphire">Sapphire</a> (Al<sub>2</sub>O<sub>3</sub>)</td>

<td>—

</td></tr>

<tr>

<td><a href="Phosphophyllite" title="Phosphophyllite">Phosphophyllite</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Phenakite" title="Phenakite">Phenakite</a></td>

<td>0.009

</td></tr>

<tr>

<td><a href="Cancrinite" title="Cancrinite">Cancrinite</a></td>

<td>0.008–0.009

</td></tr>

<tr>

<td><a href="Leucite" title="Leucite">Leucite</a></td>

<td>0.008

</td></tr>

<tr>

<td><a href="Enstatite" title="Enstatite">Enstatite</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Obsidian" title="Obsidian">Obsidian</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Anorthite" title="Anorthite">Anorthite</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Actinolite" title="Actinolite">Actinolite</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Jeremejevite" title="Jeremejevite">Jeremejevite</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Nepheline" title="Nepheline">Nepheline</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Apophyllite" title="Apophyllite">Apophyllite</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Hauyne" title="Hauyne">Hauyne</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Natrolite" title="Natrolite">Natrolite</a></td>

<td>—

</td></tr>

<tr>

<td>synth. <a href="Quartz" title="Quartz">Quartz</a> (SiO<sub>2</sub>)</td>

<td>—

</td></tr>

<tr>

<td><a href="Aragonite" title="Aragonite">Aragonite</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Augelite" title="Augelite">Augelite</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Beryllonite" title="Beryllonite">Beryllonite</a></td>

<td>0.007

</td></tr>

<tr>

<td><a href="Strontianite" title="Strontianite">Strontianite</a></td>

<td>—

</td></tr>

<tr>

<td><a href="Calcite" title="Calcite">Calcite</a> (CaCO<sub>3</sub>)</td>

<td>0.013–0.014

</td></tr>

<tr>

<td><a href="Fluorite" title="Fluorite">Fluorite</a> (CaF<sub>2</sub>)</td>

<td>0.004

</td></tr>

<tr>

<td><a href="Tremolite" title="Tremolite">Tremolite</a></td>

<td>—

</td></tr></tbody></table>

<p>In the <a href="Technical_terminology" class="mw-redirect" title="Technical terminology">technical terminology</a> of <a href="Gemology" title="Gemology">gemology</a>, <i>dispersion</i> is the difference in the refractive index of a material at the B and G (686.7 <a href="Nanometre" title="Nanometre">nm</a> and 430.8 nm) or C and F (656.3 nm and 486.1 nm) <a href="Fraunhofer_lines" title="Fraunhofer lines">Fraunhofer wavelengths</a>, and is meant to express the degree to which a prism cut from the <a href="Gemstone" title="Gemstone">gemstone</a> demonstrates "fire". Fire is a colloquial term used by gemologists to describe a gemstone's dispersive nature or lack thereof. Dispersion is a material property. The amount of fire demonstrated by a given gemstone is a function of the gemstone's facet angles, the polish quality, the lighting environment, the material's refractive index, the saturation of color, and the orientation of the viewer relative to the gemstone.<sup id="cite_ref-b1_10-1" class="reference"><a href="#cite_note-b1-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading2"><h2 id="In_imaging">In imaging</h2></div>

<p>In photographic and microscopic lenses, dispersion causes <a href="Chromatic_aberration" title="Chromatic aberration">chromatic aberration</a>, which causes the different colors in the image not to overlap properly. Various techniques have been developed to counteract this, such as the use of <a href="Achromat" class="mw-redirect" title="Achromat">achromats</a>, multielement lenses with glasses of different dispersion. They are constructed in such a way that the chromatic aberrations of the different parts cancel out.

</p>

<div class="mw-heading mw-heading2"><h2 id="Pulsar_emissions">Pulsar emissions</h2></div>

<p><a href="Pulsar" title="Pulsar">Pulsars</a> are spinning neutron stars that emit <a href="Pulse" title="Pulse">pulses</a> at very regular intervals ranging from milliseconds to seconds. Astronomers believe that the pulses are emitted simultaneously over a wide range of frequencies. However, as observed on Earth, the components of each pulse emitted at higher radio frequencies arrive before those emitted at lower frequencies. This dispersion occurs because of the ionized component of the <a href="Interstellar_medium" title="Interstellar medium">interstellar medium</a>, mainly the free electrons, which make the group velocity frequency-dependent. The extra delay added at a frequency <span class="texhtml mvar" style="font-style:italic;">ν</span> is

</p>

<msub>

</mrow>

</msub>

<mrow>

<mfrac>

<msup>

</mrow>

</msup>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle t=k_{\text{DM}}\cdot \left({\frac {\text{DM}}{\nu ^{2}}}\right),}</annotation>

</semantics>

</math></span><img src="./cb646404459711889fac8390be43f194036177bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.021ex; height:6.176ex;" alt="{\displaystyle t=k_{\text{DM}}\cdot \left({\frac {\text{DM}}{\nu ^{2}}}\right),}" loading="lazy"></span></dd></dl>

<p>where the dispersion constant <i>k</i><sub>DM</sub> is given by<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup>

</p>

<msub>

</mrow>

</msub>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k_{\text{DM}}={\frac {e^{2}}{2\pi m_{\text{e}}c}}\approx 4.149~{\text{GHz}}^{2}\,{\text{pc}}^{-1}\,{\text{cm}}^{3}\,{\text{ms}},}</annotation>

</semantics>

</math></span><img src="./ba570a851d3137ecb1c538a05fe47c6be431784e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:42.615ex; height:6.009ex;" alt="{\displaystyle k_{\text{DM}}={\frac {e^{2}}{2\pi m_{\text{e}}c}}\approx 4.149~{\text{GHz}}^{2}\,{\text{pc}}^{-1}\,{\text{cm}}^{3}\,{\text{ms}},}" loading="lazy"></span></dd></dl>

<p>and the <b>dispersion measure</b> (DM) is the column density of free electrons (<a href="Total_electron_content" title="Total electron content">total electron content</a>) – i.e. the <a href="Number_density" title="Number density">number density</a> of electrons <i>n</i><sub>e</sub> integrated along the path traveled by the photon from the pulsar to the Earth – and is given by

</p>

</mrow>

</mrow>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{DM}}=\int _{0}^{d}n_{e}\,dl}</annotation>

</semantics>

</math></span><img src="./20b47933c1ffa865965e56051e003f61044b34e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.638ex; height:6.343ex;" alt="{\displaystyle {\text{DM}}=\int _{0}^{d}n_{e}\,dl}" loading="lazy"></span></dd></dl>

<p>with units of <a href="Parsec" title="Parsec">parsecs</a> per cubic centimetre (1 pc/cm<sup>3</sup> = 30.857<span style="margin:0 .15em 0 .25em">×</span>10<sup><span class="nowrap">21</span></sup> m<sup>−2</sup>).<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup>

</p><p>Typically for astronomical observations, this delay cannot be measured directly, since the emission time is unknown. What <i>can</i> be measured is the difference in arrival times at two different frequencies. The delay Δ<i>t</i> between a high-frequency <span class="texhtml mvar" style="font-style:italic;">ν</span><sub>hi</sub> and a low-frequency <span class="texhtml mvar" style="font-style:italic;">ν</span><sub>lo</sub> component of a pulse will be

</p>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

<mrow>

<mfrac>

</mrow>

</mrow>

</msubsup>

</mfrac>

</mrow>

<mfrac>

</mrow>

</mrow>

</msubsup>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta t=k_{\text{DM}}\cdot {\text{DM}}\cdot \left({\frac {1}{\nu _{\text{lo}}^{2}}}-{\frac {1}{\nu _{\text{hi}}^{2}}}\right).}</annotation>

</semantics>

</math></span><img src="./8ef033fbd3038928b580f61cfeaf71e0677fc046.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.985ex; height:7.509ex;" alt="{\displaystyle \Delta t=k_{\text{DM}}\cdot {\text{DM}}\cdot \left({\frac {1}{\nu _{\text{lo}}^{2}}}-{\frac {1}{\nu _{\text{hi}}^{2}}}\right).}" loading="lazy"></span></dd></dl>

<p>Rewriting the above equation in terms of Δ<i>t</i> allows one to determine the DM by measuring pulse arrival times at multiple frequencies. This in turn can be used to study the interstellar medium, as well as allow observations of pulsars at different frequencies to be combined.

</p>

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<ul><li><a href="Calculation_of_glass_properties" title="Calculation of glass properties">Calculation of glass properties</a> incl. dispersion</li>

<li><a href="Cauchy's_equation" title="Cauchy's equation">Cauchy's equation</a></li>

<li><a href="Dispersion_relation" title="Dispersion relation">Dispersion relation</a></li>

<li><a href="Fast_radio_burst" title="Fast radio burst">Fast radio burst</a> (astronomy)</li>

<li><a href="Fluctuation_theorem" title="Fluctuation theorem">Fluctuation theorem</a></li>

<li><a href="Green%E2%80%93Kubo_relations" title="Green–Kubo relations">Green–Kubo relations</a></li>

<li><a href="Group_delay" class="mw-redirect" title="Group delay">Group delay</a></li>

<li><a href="Intramodal_dispersion" title="Intramodal dispersion">Intramodal dispersion</a></li>

<li><a href="Kramers%E2%80%93Kronig_relations" title="Kramers–Kronig relations">Kramers–Kronig relations</a></li>

<li><a href="Linear_response_function" title="Linear response function">Linear response function</a></li>

<li><a href="Multiple-prism_dispersion_theory" title="Multiple-prism dispersion theory">Multiple-prism dispersion theory</a></li>

<li><a href="Sellmeier_equation" title="Sellmeier equation">Sellmeier equation</a></li>

<li><a href="Ultrashort_pulse" title="Ultrashort pulse">Ultrashort pulse</a></li>

<li><a href="Virtually_imaged_phased_array" title="Virtually imaged phased array">Virtually imaged phased array</a></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFBornWolf1999" class="citation book cs1"><a href="Max_Born" title="Max Born">Born, Max</a>; Wolf, Emil (October 1999). <i><a href="Principles_of_Optics" title="Principles of Optics">Principles of Optics</a></i>. Cambridge: <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/principlesofopti0006born/page/n43">14</a>–24. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-64222-1</bdi>.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.proximion.com/about-dispersion">Dispersion Compensation</a>. Retrieved 25-08-2015.</span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://glassproperties.com/dispersion/">Calculation of the Mean Dispersion of Glasses</a>.</span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Born, M. and Wolf, E. (1980) "<a href="Principles_of_Optics" title="Principles of Optics">Principles of Optics</a>", 6th ed., p. 93. Pergamon Press.</span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Ramaswami, Rajiv and Sivarajan, Kumar N. (1998) <i>Optical Networks: A Practical Perspective</i>. Academic Press: London.</span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.rp-photonics.com/chromatic_dispersion.html">Chromatic Dispersion</a>, <i>Encyclopedia of Laser Physics and Technology</i> (Wiley, 2008).</span>

</li>

<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><cite id="CITEREFMaiCampbellWhitingKang2020" class="citation journal cs1">Mai, Wending; Campbell, Sawyer D.; Whiting, Eric B.; Kang, Lei; Werner, Pingjuan L.; Chen, Yifan; <a href="Douglas_Werner" title="Douglas Werner">Werner, Douglas H.</a> (2020-10-01). <a rel="nofollow" class="external text" href="https://doi.org/10.1364%2FOME.399414">"Prismatic discontinuous Galerkin time domain method with an integrated generalized dispersion model for efficient optical metasurface analysis"</a>. <i><a href="Optical_Materials_Express" title="Optical Materials Express">Optical Materials Express</a></i>. <b>10</b> (10): <span class="nowrap">2542–</span>2559. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2020OMExp..10.2542M">2020OMExp..10.2542M</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1364%2FOME.399414">10.1364/OME.399414</a></span>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/2159-3930">2159-3930</a>.</cite></span>

</li>

<li id="cite_note-landau-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-landau_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-landau_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFLandauLifshitzPitaevskii1984" class="citation book cs1"><a href="Lev_Landau" title="Lev Landau">Landau, L. D.</a>; <a href="Evgeny_Lifshitz" title="Evgeny Lifshitz">Lifshitz, E. M.</a>; <a href="Lev_Pitaevskii" title="Lev Pitaevskii">Pitaevskii, L. P.</a> (1984). <a href="Course_of_Theoretical_Physics" title="Course of Theoretical Physics"><i>Electrodynamics of Continuous Media</i></a>. Vol. 8 (2nd ed.). <a href="Butterworth-Heinemann" title="Butterworth-Heinemann">Butterworth-Heinemann</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-7506-2634-7</bdi>.</cite></span>

</li>

<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><cite id="CITEREFDemetriadouPendry2008" class="citation journal cs1">Demetriadou, A.; <a href="John_Pendry" title="John Pendry">Pendry, J. B.</a> (1 July 2008). "Taming spatial dispersion in wire metamaterial". <i><a href="Journal_of_Physics%3A_Condensed_Matter" title="Journal of Physics: Condensed Matter">Journal of Physics: Condensed Matter</a></i>. <b>20</b> (29): 295222. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2008JPCM...20C5222D">2008JPCM...20C5222D</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1088%2F0953-8984%2F20%2F29%2F295222">10.1088/0953-8984/20/29/295222</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120249447">120249447</a>.</cite></span>

</li>

<li id="cite_note-b1-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-b1_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-b1_10-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFSchumann2009" class="citation book cs1">Schumann, Walter (2009). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=V9PqVxpxeiEC&pg=PA42"><i>Gemstones of the World</i></a> (4th newly revised & expanded ed.). Sterling Publishing Company. pp. <span class="nowrap">41–</span>42. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4027-6829-3</bdi><span class="reference-accessdate">. Retrieved <span class="nowrap">31 December</span> 2011</span>.</cite></span>

</li>

<li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.gemsociety.org/article/gemstone-dispersion/">"What is gemstone dispersion?"</a>. <i>International Gem Society (GemSociety.org)</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2015-03-09</span></span>.</cite></span>

</li>

<li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text">"Single-Dish Radio Astronomy: Techniques and Applications", ASP Conference Proceedings, vol. 278. Edited by Snezana Stanimirovic, <a href="Daniel_R._Altschuler" title="Daniel R. Altschuler">Daniel Altschuler</a>, Paul Goldsmith, and Chris Salter. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>1-58381-120-6</bdi>. San Francisco: Astronomical Society of the Pacific, 2002, p. 251–269.</span>

</li>

<li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text">Lorimer, D. R., and Kramer, M., <i>Handbook of Pulsar Astronomy</i>, vol. 4 of Cambridge Observing Handbooks for Research Astronomers (<a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, Cambridge, U.K.; New York, U.S.A, 2005), 1st edition.</span>

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

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<div class="side-box-text plainlist">Wikimedia Commons has media related to <a href="https://commons.wikimedia.org/wiki/Dispersion" class="extiw external" title="commons:Dispersion"><span style="font-style:italic; font-weight:bold;">Dispersion</span></a>.</div></div>

</div>

<ul><li><a rel="nofollow" class="external text" href="http://wiki.math.toronto.edu/DispersiveWiki/index.php/Main_Page">Dispersive Wiki</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170723091053/http://wiki.math.toronto.edu/DispersiveWiki/index.php/Main_Page">Archived</a> 2017-07-23 at the <a href="Wayback_Machine" title="Wayback Machine">Wayback Machine</a> – discussing the mathematical aspects of dispersion.</li>

<li><a rel="nofollow" class="external text" href="http://www.rp-photonics.com/dispersion.html">Dispersion</a> – Encyclopedia of Laser Physics and Technology</li>

<li><a rel="nofollow" class="external text" href="http://qed.wikina.org/dispersion/">Animations demonstrating optical dispersion</a> by QED</li>

<li><a rel="nofollow" class="external text" href="http://webdemo.inue.uni-stuttgart.de/webdemos/02_lectures/uebertragungstechnik_2/chromatic_dispersion/">Interactive webdemo for chromatic dispersion</a> Institute of Telecommunications, University of Stuttgart</li></ul>

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</style><div id="Glass_science_topics124" style="font-size:114%;margin:0 4em"><a href="Glass" title="Glass">Glass</a> science topics</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basics</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Glass" title="Glass">Glass</a></li>

<li><a href="Glass_transition" title="Glass transition">Glass transition</a></li>

<li><a href="Supercooling" title="Supercooling">Supercooling</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Formulation</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="AgInSbTe" title="AgInSbTe">AgInSbTe</a></li>

<li><a href="Bioglass" class="mw-redirect" title="Bioglass">Bioglass</a></li>

<li><a href="Borophosphosilicate_glass" title="Borophosphosilicate glass">Borophosphosilicate glass</a></li>

<li><a href="Borosilicate_glass" title="Borosilicate glass">Borosilicate glass</a></li>

<li><a href="Ceramic_glaze" title="Ceramic glaze">Ceramic glaze</a></li>

<li><a href="Chalcogenide_glass" title="Chalcogenide glass">Chalcogenide glass</a></li>

<li><a href="Cobalt_glass" title="Cobalt glass">Cobalt glass</a></li>

<li><a href="Cranberry_glass" title="Cranberry glass">Cranberry glass</a></li>

<li><a href="Crown_glass_(optics)" title="Crown glass (optics)">Crown glass</a></li>

<li><a href="Flint_glass" title="Flint glass">Flint glass</a></li>

<li><a href="Fluorosilicate_glass" title="Fluorosilicate glass">Fluorosilicate glass</a></li>

<li><a href="Fused_quartz" title="Fused quartz">Fused quartz</a></li>

<li><a href="GeSbTe" title="GeSbTe">GeSbTe</a></li>

<li><a href="Cranberry_glass" title="Cranberry glass">Gold ruby glass</a></li>

<li><a href="Lead_glass" title="Lead glass">Lead glass</a></li>

<li><a href="Milk_glass" title="Milk glass">Milk glass</a></li>

<li><a href="Phosphosilicate_glass" title="Phosphosilicate glass">Phosphosilicate glass</a></li>

<li><a href="Photochromic_lens" title="Photochromic lens">Photochromic lens glass</a></li>

<li><a href="Glass#Silicate_glass" title="Glass">Silicate glass</a></li>

<li><a href="Soda%E2%80%93lime_glass" title="Soda–lime glass">Soda–lime glass</a></li>

<li><a href="Sodium_hexametaphosphate" title="Sodium hexametaphosphate">Sodium hexametaphosphate</a></li>

<li><a href="Sodium_silicate" title="Sodium silicate">Soluble glass</a></li>

<li><a href="Tellurite_glass" title="Tellurite glass">Tellurite glass</a></li>

<li><a href="Thoriated_glass" title="Thoriated glass">Thoriated glass</a></li>

<li><a href="Ultra_low_expansion_glass" title="Ultra low expansion glass">Ultra low expansion glass</a></li>

<li><a href="Uranium_glass" title="Uranium glass">Uranium glass</a></li>

<li><a href="Vitreous_enamel" title="Vitreous enamel">Vitreous enamel</a></li>

<li><a href="Wood's_glass" title="Wood's glass">Wood's glass</a></li>

<li><a href="ZBLAN" title="ZBLAN">ZBLAN</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Glass-ceramic" title="Glass-ceramic">Glass-ceramics</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bioactive_glass" title="Bioactive glass">Bioactive glass</a></li>

<li><a href="CorningWare" title="CorningWare">CorningWare</a></li>

<li><a href="Glass-ceramic-to-metal_seals" title="Glass-ceramic-to-metal seals">Glass-ceramic-to-metal seals</a></li>

<li><a href="Macor" title="Macor">Macor</a></li>

<li><a href="Zerodur" title="Zerodur">Zerodur</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Preparation</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Annealing_(glass)" title="Annealing (glass)">Annealing</a></li>

<li><a href="Chemical_vapor_deposition" title="Chemical vapor deposition">Chemical vapor deposition</a></li>

<li><a href="Glass_batch_calculation" title="Glass batch calculation">Glass batch calculation</a></li>

<li><a href="Glass_production" title="Glass production">Glass forming</a></li>

<li><a href="Glass_production#Hot_end" title="Glass production">Glass melting</a></li>

<li><a href="Calculation_of_glass_properties" title="Calculation of glass properties">Glass modeling</a></li>

<li><a href="Ion_implantation" title="Ion implantation">Ion implantation</a></li>

<li><a href="Liquidus" class="mw-redirect" title="Liquidus">Liquidus temperature</a></li>

<li><a href="Sol%E2%80%93gel_process" title="Sol–gel process">sol–gel technique</a></li>

<li><a href="Viscosity#Viscosity_of_amorphous_materials" title="Viscosity">Viscosity</a></li>

<li><a href="Vitrification" title="Vitrification">Vitrification</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Optics" title="Optics">Optics</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Achromatic_lens" title="Achromatic lens">Achromat</a></li>

<li><a href="Gradient-index_optics" title="Gradient-index optics">Gradient-index optics</a></li>

<li><a href="Hydrogen_darkening" title="Hydrogen darkening">Hydrogen darkening</a></li>

<li><a href="Optical_amplifier" title="Optical amplifier">Optical amplifier</a></li>

<li><a href="Optical_fiber" title="Optical fiber">Optical fiber</a></li>

<li><a href="Optical_lens_design" title="Optical lens design">Optical lens design</a></li>

<li><a href="Photochromic_lens" title="Photochromic lens">Photochromic lens</a></li>

<li><a href="Photosensitive_glass" title="Photosensitive glass">Photosensitive glass</a></li>

<li><a href="Refraction" title="Refraction">Refraction</a></li>

<li><a href="Transparency_and_translucency" title="Transparency and translucency">Transparent materials</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Surface<br>modification</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Anti-reflective_coating" title="Anti-reflective coating">Anti-reflective coating</a></li>

<li><a href="Chemically_strengthened_glass" title="Chemically strengthened glass">Chemically strengthened glass</a></li>

<li><a href="Corrosion#Corrosion_of_glasses" title="Corrosion">Corrosion</a></li>

<li><a href="Dealkalization" title="Dealkalization">Dealkalization</a></li>

<li><a href="DNA_microarray" title="DNA microarray">DNA microarray</a></li>

<li><a href="Hydrogen_darkening" title="Hydrogen darkening">Hydrogen darkening</a></li>

<li><a href="Insulated_glazing" title="Insulated glazing">Insulated glazing</a></li>

<li><a href="Porous_glass" title="Porous glass">Porous glass</a></li>

<li><a href="Self-cleaning_glass" title="Self-cleaning glass">Self-cleaning glass</a></li>

<li><a href="Sol%E2%80%93gel_process" title="Sol–gel process">sol–gel technique</a></li>

<li><a href="Tempered_glass" title="Tempered glass">Tempered glass</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Diverse<br>topics</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Conservation_and_restoration_of_glass_objects" title="Conservation and restoration of glass objects">Conservation and restoration of glass objects</a></li>

<li><a href="Glass-coated_wire" title="Glass-coated wire">Glass-coated wire</a></li>

<li><a href="Safety_glass" title="Safety glass">Safety glass</a></li>

<li><a href="Glass_databases" title="Glass databases">Glass databases</a></li>

<li><a href="Glass_electrode" title="Glass electrode">Glass electrode</a></li>

<li><a href="Glass_fiber_reinforced_concrete" title="Glass fiber reinforced concrete">Glass fiber reinforced concrete</a></li>

<li><a href="Glass_ionomer_cement" title="Glass ionomer cement">Glass ionomer cement</a></li>

<li><a href="Glass_microsphere" title="Glass microsphere">Glass microspheres</a></li>

<li><a href="Fiberglass" title="Fiberglass">Glass-reinforced plastic</a></li>

<li><a href="Glass_cloth" title="Glass cloth">Glass cloth</a></li>

<li><a href="Glass-to-metal_seal" title="Glass-to-metal seal">Glass-to-metal seal</a></li>

<li><a href="Porous_glass" title="Porous glass">Porous glass</a></li>

<li><a href="Prince_Rupert's_drop" title="Prince Rupert's drop">Prince Rupert's drops</a></li>

<li><a href="Radioactive_waste#Vitrification" title="Radioactive waste">Radioactive waste vitrification</a></li>

<li><a href="Windshield" title="Windshield">Windshield</a></li>

<li><a href="Glass_fiber" title="Glass fiber">Glass fiber</a></li></ul>

</div></td></tr></tbody></table></div>

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