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<button aria-controls="toc-Typical_values-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Typical values subsection </button> <ul id="toc-Typical_values-sublist" class="vector-toc-list"> <li id="toc-Refractive_index_below_unity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Refractive_index_below_unity"> <div class="vector-toc-text"> 3.1 Refractive index below unity </div> </a> <ul id="toc-Refractive_index_below_unity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Negative_refractive_index" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Negative_refractive_index"> <div class="vector-toc-text"> 3.2 Negative refractive index </div> </a> <ul id="toc-Negative_refractive_index-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Microscopic_explanation" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Microscopic_explanation"> <div class="vector-toc-text"> 4 Microscopic explanation </div> </a> <ul id="toc-Microscopic_explanation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dispersion" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Dispersion"> <div class="vector-toc-text"> 5 Dispersion </div> </a> <button aria-controls="toc-Dispersion-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Dispersion subsection </button> <ul id="toc-Dispersion-sublist" class="vector-toc-list"> <li id="toc-Principal_refractive_index_wavelength_ambiguity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Principal_refractive_index_wavelength_ambiguity"> <div class="vector-toc-text"> 5.1 Principal refractive index wavelength ambiguity </div> </a> <ul id="toc-Principal_refractive_index_wavelength_ambiguity-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Complex_refractive_index" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Complex_refractive_index"> <div class="vector-toc-text"> 6 Complex refractive index </div> </a> <button aria-controls="toc-Complex_refractive_index-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Complex refractive index subsection </button> <ul id="toc-Complex_refractive_index-sublist" class="vector-toc-list"> <li id="toc-X-ray_and_extreme_UV" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#X-ray_and_extreme_UV"> <div class="vector-toc-text"> 6.1 X-ray and extreme UV </div> </a> <ul id="toc-X-ray_and_extreme_UV-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Relations_to_other_quantities" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Relations_to_other_quantities"> <div class="vector-toc-text"> 7 Relations to other quantities </div> </a> <button aria-controls="toc-Relations_to_other_quantities-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Relations to other quantities subsection </button> <ul id="toc-Relations_to_other_quantities-sublist" class="vector-toc-list"> <li id="toc-Optical_path_length" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Optical_path_length"> <div class="vector-toc-text"> 7.1 Optical path length </div> </a> <ul id="toc-Optical_path_length-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Refraction" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Refraction"> <div class="vector-toc-text"> 7.2 Refraction </div> </a> <ul id="toc-Refraction-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Total_internal_reflection" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Total_internal_reflection"> <div class="vector-toc-text"> 7.3 Total internal reflection </div> </a> <ul id="toc-Total_internal_reflection-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Reflectivity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Reflectivity"> <div class="vector-toc-text"> 7.4 Reflectivity </div> </a> <ul id="toc-Reflectivity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Lenses" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Lenses"> <div class="vector-toc-text"> 7.5 Lenses </div> </a> <ul id="toc-Lenses-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Microscope_resolution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Microscope_resolution"> <div class="vector-toc-text"> 7.6 Microscope resolution </div> </a> <ul id="toc-Microscope_resolution-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Relative_permittivity_and_permeability" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relative_permittivity_and_permeability"> <div class="vector-toc-text"> 7.7 Relative permittivity and permeability </div> </a> <ul id="toc-Relative_permittivity_and_permeability-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Wave_impedance" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Wave_impedance"> <div class="vector-toc-text"> 7.8 Wave impedance </div> </a> <ul id="toc-Wave_impedance-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Density" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Density"> <div class="vector-toc-text"> 7.9 Density </div> </a> <ul id="toc-Density-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bandgap" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Bandgap"> <div class="vector-toc-text"> 7.10 Bandgap </div> </a> <ul id="toc-Bandgap-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Group_index" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Group_index"> <div class="vector-toc-text"> 7.11 Group index </div> </a> <ul id="toc-Group_index-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Velocity,_momentum,_and_polarizability" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Velocity,_momentum,_and_polarizability"> <div class="vector-toc-text"> 7.12 Velocity, momentum, and polarizability </div> </a> <ul id="toc-Velocity,_momentum,_and_polarizability-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Refractivity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Refractivity"> <div class="vector-toc-text"> 7.13 Refractivity </div> </a> <ul id="toc-Refractivity-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Nonscalar,_nonlinear,_or_nonhomogeneous_refraction" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Nonscalar,_nonlinear,_or_nonhomogeneous_refraction"> <div class="vector-toc-text"> 8 Nonscalar, nonlinear, or nonhomogeneous refraction </div> </a> <button aria-controls="toc-Nonscalar,_nonlinear,_or_nonhomogeneous_refraction-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Nonscalar, nonlinear, or nonhomogeneous refraction subsection </button> <ul id="toc-Nonscalar,_nonlinear,_or_nonhomogeneous_refraction-sublist" class="vector-toc-list"> <li id="toc-Birefringence" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Birefringence"> <div class="vector-toc-text"> 8.1 Birefringence </div> </a> <ul id="toc-Birefringence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Nonlinearity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Nonlinearity"> <div class="vector-toc-text"> 8.2 Nonlinearity </div> </a> <ul id="toc-Nonlinearity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inhomogeneity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Inhomogeneity"> <div class="vector-toc-text"> 8.3 Inhomogeneity </div> </a> <ul id="toc-Inhomogeneity-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Refractive_index_measurement" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Refractive_index_measurement"> <div class="vector-toc-text"> 9 Refractive index measurement </div> </a> <button aria-controls="toc-Refractive_index_measurement-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Refractive index measurement subsection </button> <ul id="toc-Refractive_index_measurement-sublist" class="vector-toc-list"> <li id="toc-Homogeneous_media" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Homogeneous_media"> <div class="vector-toc-text"> 9.1 Homogeneous media </div> </a> <ul id="toc-Homogeneous_media-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Refractive_index_variations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Refractive_index_variations"> <div class="vector-toc-text"> 9.2 Refractive index variations </div> </a> <ul id="toc-Refractive_index_variations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> 10 Applications </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 11 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Footnotes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Footnotes"> <div class="vector-toc-text"> 12 Footnotes </div> </a> <ul id="toc-Footnotes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> 13 References </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 14 External links </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > Toggle the table of contents </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">Refractive index</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 69 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-69" aria-hidden="true" > 69 languages </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Brekingsindeks" title="Brekingsindeks – Afrikaans" lang="af" hreflang="af" data-title="Brekingsindeks" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target">Afrikaans</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B9%D8%A7%D9%85%D9%84_%D8%A7%D9%84%D8%A7%D9%86%D9%83%D8%B3%D8%A7%D8%B1" title="معامل الانكسار – Arabic" lang="ar" hreflang="ar" data-title="معامل الانكسار" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-ast mw-list-item"><a href="https://ast.wikipedia.org/wiki/%C3%8Dndiz_de_refraici%C3%B3n" title="Índiz de refraición – Asturian" lang="ast" hreflang="ast" data-title="Índiz de refraición" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/%C4%B0%C5%9F%C4%B1%C4%9F%C4%B1n_tam_daxili_qay%C4%B1tmas%C4%B1" title="İşığın tam daxili qayıtması – Azerbaijani" lang="az" hreflang="az" data-title="İşığın tam daxili qayıtması" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target">Azərbaycanca</a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AA%E0%A7%8D%E0%A6%B0%E0%A6%A4%E0%A6%BF%E0%A6%B8%E0%A6%B0%E0%A6%BE%E0%A6%99%E0%A7%8D%E0%A6%95" title="প্রতিসরাঙ্ক – Bangla" lang="bn" hreflang="bn" data-title="প্রতিসরাঙ্ক" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target">বাংলা</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9F%D0%B0%D0%BA%D0%B0%D0%B7%D1%87%D1%8B%D0%BA_%D0%BF%D1%80%D0%B0%D0%BB%D0%B0%D0%BC%D0%BB%D0%B5%D0%BD%D0%BD%D1%8F" title="Паказчык праламлення – Belarusian" lang="be" hreflang="be" data-title="Паказчык праламлення" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BA%D0%B0%D0%B7%D0%B0%D1%82%D0%B5%D0%BB_%D0%BD%D0%B0_%D0%BF%D1%80%D0%B5%D1%87%D1%83%D0%BF%D0%B2%D0%B0%D0%BD%D0%B5" title="Показател на пречупване – Bulgarian" lang="bg" hreflang="bg" data-title="Показател на пречупване" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Indeks_prelamanja" title="Indeks prelamanja – Bosnian" lang="bs" hreflang="bs" data-title="Indeks prelamanja" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target">Bosanski</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/%C3%8Dndex_de_refracci%C3%B3" title="Índex de refracció – Catalan" lang="ca" hreflang="ca" data-title="Índex de refracció" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A5%D1%83%C3%A7%C4%83%D0%BB%D1%83_%D0%BA%C4%83%D1%82%D0%B0%D1%80%D1%82%C4%83%D1%88%C4%95" title="Хуçăлу кăтартăшĕ – Chuvash" lang="cv" hreflang="cv" data-title="Хуçăлу кăтартăшĕ" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target">Чӑвашла</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Index_lomu" title="Index lomu – Czech" lang="cs" hreflang="cs" data-title="Index lomu" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Mynegrif_plygiant" title="Mynegrif plygiant – Welsh" lang="cy" hreflang="cy" data-title="Mynegrif plygiant" data-language-autonym="Cymraeg" data-language-local-name="Welsh" class="interlanguage-link-target">Cymraeg</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Brydningsindeks" title="Brydningsindeks – Danish" lang="da" hreflang="da" data-title="Brydningsindeks" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Brechungsindex" title="Brechungsindex – German" lang="de" hreflang="de" data-title="Brechungsindex" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Murdumisn%C3%A4itaja" title="Murdumisnäitaja – Estonian" lang="et" hreflang="et" data-title="Murdumisnäitaja" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%94%CE%B5%CE%AF%CE%BA%CF%84%CE%B7%CF%82_%CE%B4%CE%B9%CE%AC%CE%B8%CE%BB%CE%B1%CF%83%CE%B7%CF%82" title="Δείκτης διάθλασης – Greek" lang="el" hreflang="el" data-title="Δείκτης διάθλασης" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/%C3%8Dndice_de_refracci%C3%B3n" title="Índice de refracción – Spanish" lang="es" hreflang="es" data-title="Índice de refracción" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Refrakta_indico" title="Refrakta indico – Esperanto" lang="eo" hreflang="eo" data-title="Refrakta indico" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Errefrakzio-indize" title="Errefrakzio-indize – Basque" lang="eu" hreflang="eu" data-title="Errefrakzio-indize" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target">Euskara</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%B6%D8%B1%DB%8C%D8%A8_%D8%B4%DA%A9%D8%B3%D8%AA" title="ضریب شکست – Persian" lang="fa" hreflang="fa" data-title="ضریب شکست" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Indice_de_r%C3%A9fraction" title="Indice de réfraction – French" lang="fr" hreflang="fr" data-title="Indice de réfraction" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Comh%C3%A9ifeacht_athraonta" title="Comhéifeacht athraonta – Irish" lang="ga" hreflang="ga" data-title="Comhéifeacht athraonta" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target">Gaeilge</a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/%C3%8Dndice_de_refracci%C3%B3n" title="Índice de refracción – Galician" lang="gl" hreflang="gl" data-title="Índice de refracción" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target">Galego</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EA%B5%B4%EC%A0%88%EB%A5%A0" title="굴절률 – Korean" lang="ko" hreflang="ko" data-title="굴절률" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B2%D5%A5%D5%AF%D5%B4%D5%A1%D5%B6_%D6%81%D5%B8%D6%82%D6%81%D5%AB%D5%B9" title="Բեկման ցուցիչ – Armenian" lang="hy" hreflang="hy" data-title="Բեկման ցուցիչ" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target">Հայերեն</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%85%E0%A4%AA%E0%A4%B5%E0%A4%B0%E0%A5%8D%E0%A4%A4%E0%A4%A8%E0%A4%BE%E0%A4%82%E0%A4%95" title="अपवर्तनांक – Hindi" lang="hi" hreflang="hi" data-title="अपवर्तनांक" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Indeks_loma" title="Indeks loma – Croatian" lang="hr" hreflang="hr" data-title="Indeks loma" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Indeks_bias" title="Indeks bias – Indonesian" lang="id" hreflang="id" data-title="Indeks bias" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Indice_di_rifrazione" title="Indice di rifrazione – Italian" lang="it" hreflang="it" data-title="Indice di rifrazione" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%A7%D7%93%D7%9D_%D7%A9%D7%91%D7%99%D7%A8%D7%94" title="מקדם שבירה – Hebrew" lang="he" hreflang="he" data-title="מקדם שבירה" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%B5%E0%B2%95%E0%B3%8D%E0%B2%B0%E0%B3%80%E0%B2%AD%E0%B2%B5%E0%B2%A8_%E0%B2%B8%E0%B3%82%E0%B2%9A%E0%B3%8D%E0%B2%AF%E0%B2%82%E0%B2%95" title="ವಕ್ರೀಭವನ ಸೂಚ್ಯಂಕ – Kannada" lang="kn" hreflang="kn" data-title="ವಕ್ರೀಭವನ ಸೂಚ್ಯಂಕ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target">ಕನ್ನಡ</a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A1%D1%8B%D0%BD%D1%83_%D0%BA%D3%A9%D1%80%D1%81%D0%B5%D1%82%D0%BA%D1%96%D1%88%D1%96" title="Сыну көрсеткіші – Kazakh" lang="kk" hreflang="kk" data-title="Сыну көрсеткіші" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target">Қазақша</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Lau%C5%A1anas_koeficients" title="Laušanas koeficients – Latvian" lang="lv" hreflang="lv" data-title="Laušanas koeficients" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/L%C5%AB%C5%BEio_rodiklis" title="Lūžio rodiklis – Lithuanian" lang="lt" hreflang="lt" data-title="Lūžio rodiklis" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/T%C3%B6r%C3%A9smutat%C3%B3" title="Törésmutató – Hungarian" lang="hu" hreflang="hu" data-title="Törésmutató" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BA%D0%B0%D0%B7%D0%B0%D1%82%D0%B5%D0%BB_%D0%BD%D0%B0_%D0%BF%D1%80%D0%B5%D0%BA%D1%80%D1%88%D1%83%D0%B2%D0%B0%D1%9A%D0%B5" title="Показател на прекршување – Macedonian" lang="mk" hreflang="mk" data-title="Показател на прекршување" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target">Македонски</a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%85%E0%B4%AA%E0%B4%B5%E0%B5%BC%E0%B4%A4%E0%B5%8D%E0%B4%A4%E0%B4%A8%E0%B4%BE%E0%B4%99%E0%B5%8D%E0%B4%95%E0%B4%82" title="അപവർത്തനാങ്കം – Malayalam" lang="ml" hreflang="ml" data-title="അപവർത്തനാങ്കം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target">മലയാളം</a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%85%E0%A4%AA%E0%A4%B5%E0%A4%B0%E0%A5%8D%E0%A4%A4%E0%A4%A8%E0%A4%BE%E0%A4%82%E0%A4%95" title="अपवर्तनांक – Marathi" lang="mr" hreflang="mr" data-title="अपवर्तनांक" data-language-autonym="मराठी" data-language-local-name="Marathi" class="interlanguage-link-target">मराठी</a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Indeks_biasan" title="Indeks biasan – Malay" lang="ms" hreflang="ms" data-title="Indeks biasan" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target">Bahasa Melayu</a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%A5%D1%83%D0%B3%D0%B0%D0%BB%D0%B0%D1%85_%D0%B8%D0%BB%D1%82%D0%B3%D1%8D%D0%B3%D1%87" title="Хугалах илтгэгч – Mongolian" lang="mn" hreflang="mn" data-title="Хугалах илтгэгч" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target">Монгол</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Brekingsindex" title="Brekingsindex – Dutch" lang="nl" hreflang="nl" data-title="Brekingsindex" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%B1%88%E6%8A%98%E7%8E%87" title="屈折率 – Japanese" lang="ja" hreflang="ja" data-title="屈折率" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Breegtaal" title="Breegtaal – Northern Frisian" lang="frr" hreflang="frr" data-title="Breegtaal" data-language-autonym="Nordfriisk" data-language-local-name="Northern Frisian" class="interlanguage-link-target">Nordfriisk</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Brytningsindeks" title="Brytningsindeks – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Brytningsindeks" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Brytingsindeks" title="Brytingsindeks – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Brytingsindeks" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-oc mw-list-item"><a href="https://oc.wikipedia.org/wiki/Indici_de_refraccion" title="Indici de refraccion – Occitan" lang="oc" hreflang="oc" data-title="Indici de refraccion" data-language-autonym="Occitan" data-language-local-name="Occitan" class="interlanguage-link-target">Occitan</a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%85%E0%A8%AA%E0%A8%B5%E0%A8%B0%E0%A8%A4%E0%A8%A8%E0%A8%BE%E0%A8%82%E0%A8%95" title="ਅਪਵਰਤਨਾਂਕ – Punjabi" lang="pa" hreflang="pa" data-title="ਅਪਵਰਤਨਾਂਕ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Wsp%C3%B3%C5%82czynnik_za%C5%82amania" title="Współczynnik załamania – Polish" lang="pl" hreflang="pl" data-title="Współczynnik załamania" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/%C3%8Dndice_refrativo" title="Índice refrativo – Portuguese" lang="pt" hreflang="pt" data-title="Índice refrativo" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Indice_de_refrac%C8%9Bie" title="Indice de refracție – Romanian" lang="ro" hreflang="ro" data-title="Indice de refracție" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru badge-Q17437796 badge-featuredarticle mw-list-item" title="featured article badge"><a href="https://ru.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BA%D0%B0%D0%B7%D0%B0%D1%82%D0%B5%D0%BB%D1%8C_%D0%BF%D1%80%D0%B5%D0%BB%D0%BE%D0%BC%D0%BB%D0%B5%D0%BD%D0%B8%D1%8F" title="Показатель преломления – Russian" lang="ru" hreflang="ru" data-title="Показатель преломления" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Refractive_index" title="Refractive index – Scots" lang="sco" hreflang="sco" data-title="Refractive index" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target">Scots</a></li><li class="interlanguage-link interwiki-stq mw-list-item"><a href="https://stq.wikipedia.org/wiki/Breektaal" title="Breektaal – Saterland Frisian" lang="stq" hreflang="stq" data-title="Breektaal" data-language-autonym="Seeltersk" data-language-local-name="Saterland Frisian" class="interlanguage-link-target">Seeltersk</a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Index_lomu" title="Index lomu – Slovak" lang="sk" hreflang="sk" data-title="Index lomu" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target">Slovenčina</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Lomni_koli%C4%8Dnik" title="Lomni količnik – Slovenian" lang="sl" hreflang="sl" data-title="Lomni količnik" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%98%D0%BD%D0%B4%D0%B5%D0%BA%D1%81_%D0%BF%D1%80%D0%B5%D0%BB%D0%B0%D0%BC%D0%B0%D1%9A%D0%B0" title="Индекс преламања – Serbian" lang="sr" hreflang="sr" data-title="Индекс преламања" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Indeks_prelamanja" title="Indeks prelamanja – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Indeks prelamanja" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Taitekerroin" title="Taitekerroin – Finnish" lang="fi" hreflang="fi" data-title="Taitekerroin" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Brytningsindex" title="Brytningsindex – Swedish" lang="sv" hreflang="sv" data-title="Brytningsindex" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%92%E0%AE%B3%E0%AE%BF%E0%AE%B5%E0%AE%BF%E0%AE%B2%E0%AE%95%E0%AE%B2%E0%AF%8D_%E0%AE%95%E0%AF%81%E0%AE%B1%E0%AE%BF%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AF%86%E0%AE%A3%E0%AF%8D" title="ஒளிவிலகல் குறிப்பெண் – Tamil" lang="ta" hreflang="ta" data-title="ஒளிவிலகல் குறிப்பெண்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target">தமிழ்</a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%B5%E0%B0%95%E0%B1%8D%E0%B0%B0%E0%B1%80%E0%B0%AD%E0%B0%B5%E0%B0%A8_%E0%B0%97%E0%B1%81%E0%B0%A3%E0%B0%95%E0%B0%82" title="వక్రీభవన గుణకం – Telugu" lang="te" hreflang="te" data-title="వక్రీభవన గుణకం" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target">తెలుగు</a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%94%E0%B8%A3%E0%B8%A3%E0%B8%8A%E0%B8%99%E0%B8%B5%E0%B8%AB%E0%B8%B1%E0%B8%81%E0%B9%80%E0%B8%AB" title="ดรรชนีหักเห – Thai" lang="th" hreflang="th" data-title="ดรรชนีหักเห" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target">ไทย</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/K%C4%B1r%C4%B1lma_indisi" title="Kırılma indisi – Turkish" lang="tr" hreflang="tr" data-title="Kırılma indisi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://uk.wikipedia.org/wiki/%D0%9F%D0%BE%D0%BA%D0%B0%D0%B7%D0%BD%D0%B8%D0%BA_%D0%B7%D0%B0%D0%BB%D0%BE%D0%BC%D0%BB%D0%B5%D0%BD%D0%BD%D1%8F" title="Показник заломлення – Ukrainian" lang="uk" hreflang="uk" data-title="Показник заломлення" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%A7%D9%86%D8%B9%D8%B7%D8%A7%D9%81_%D9%86%D9%85%D8%A7" title="انعطاف نما – Urdu" lang="ur" hreflang="ur" data-title="انعطاف نما" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target">اردو</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Chi%E1%BA%BFt_su%E1%BA%A5t" title="Chiết suất – Vietnamese" lang="vi" hreflang="vi" data-title="Chiết suất" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%8A%98%E5%B0%84%E7%8E%87" title="折射率 – Wu" lang="wuu" hreflang="wuu" data-title="折射率" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target">吴语</a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%8A%98%E5%B0%84%E7%8E%87" title="折射率 – Cantonese" lang="yue" hreflang="yue" data-title="折射率" data-language-autonym="粵語" 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class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Property in optics</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Refraction_photo.png" class="mw-file-description"><img alt="refer to caption" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Refraction_photo.png/220px-Refraction_photo.png" decoding="async" width="220" height="146" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/85/Refraction_photo.png/330px-Refraction_photo.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/85/Refraction_photo.png/440px-Refraction_photo.png 2x" data-file-width="1142" data-file-height="759" /></a><figcaption>A <a href="/wiki/Ray_(optics)" title="Ray (optics)">ray</a> of light being <a href="/wiki/Refraction" title="Refraction">refracted</a> through a glass slab</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Refraction_at_interface.svg" class="mw-file-description"><img alt="Illustration of the incidence and refraction angles" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Refraction_at_interface.svg/170px-Refraction_at_interface.svg.png" decoding="async" width="170" height="152" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Refraction_at_interface.svg/255px-Refraction_at_interface.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Refraction_at_interface.svg/340px-Refraction_at_interface.svg.png 2x" data-file-width="145" data-file-height="130" /></a><figcaption>Refraction of a light ray</figcaption></figure> In <a href="/wiki/Optics" title="Optics">optics</a>, the refractive index (or refraction index) of an <a href="/wiki/Optical_medium" title="Optical medium">optical medium</a> is the <a href="/wiki/Ratio" title="Ratio">ratio</a> of the apparent speed of light in the air or vacuum to the speed in the medium. The refractive index determines how much the path of <a href="/wiki/Light" title="Light">light</a> is bent, or <a href="/wiki/Refraction" title="Refraction">refracted</a>, when entering a material. This is described by <a href="/wiki/Snell%27s_law" title="Snell's law">Snell's law</a> of refraction, n1 sin θ1 = n2 sin θ2, where θ1 and θ2 are the <a href="/wiki/Angle_of_incidence_(optics)" title="Angle of incidence (optics)">angle of incidence</a> and angle of refraction, respectively, of a ray crossing the interface between two media with refractive indices n1 and n2. The refractive indices also determine the amount of light that is <a href="/wiki/Reflectivity" class="mw-redirect" title="Reflectivity">reflected</a> when reaching the interface, as well as the critical angle for <a href="/wiki/Total_internal_reflection" title="Total internal reflection">total internal reflection</a>, their intensity (<a href="/wiki/Fresnel_equations" title="Fresnel equations">Fresnel equations</a>) and <a href="/wiki/Brewster%27s_angle" title="Brewster's angle">Brewster's angle</a>.<a href="#cite_note-Hecht-1">[1]</a> The refractive index, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">, can be seen as the factor by which the speed and the <a href="/wiki/Wavelength" title="Wavelength">wavelength</a> of the radiation are reduced with respect to their vacuum values: the speed of light in a medium is v = c/n, and similarly the wavelength in that medium is λ = λ0/n, where λ0 is the wavelength of that light in vacuum. This implies that vacuum has a refractive index of 1, and assumes that the <a href="/wiki/Frequency" title="Frequency">frequency</a> (f = v/λ) of the wave is not affected by the refractive index. The refractive index may vary with wavelength. This causes white light to split into constituent colors when refracted. This is called <a href="/wiki/Dispersion_(optics)" title="Dispersion (optics)">dispersion</a>. This effect can be observed in <a href="/wiki/Prism_(optics)" title="Prism (optics)">prisms</a> and <a href="/wiki/Rainbow" title="Rainbow">rainbows</a>, and as <a href="/wiki/Chromatic_aberration" title="Chromatic aberration">chromatic aberration</a> in lenses. Light propagation in <a href="/wiki/Absorption_(electromagnetic_radiation)" title="Absorption (electromagnetic radiation)">absorbing</a> materials can be described using a <a href="/wiki/Complex_number" title="Complex number">complex</a>-valued refractive index.<a href="#cite_note-Attwood-2">[2]</a> The <a href="/wiki/Imaginary_number" title="Imaginary number">imaginary</a> part then handles the <a href="/wiki/Attenuation" title="Attenuation">attenuation</a>, while the <a href="/wiki/Real_number" title="Real number">real</a> part accounts for refraction. For most materials the refractive index changes with wavelength by several percent across the visible spectrum. Consequently, refractive indices for materials reported using a single value for n must specify the wavelength used in the measurement. The concept of refractive index applies across the full <a href="/wiki/Electromagnetic_spectrum" title="Electromagnetic spectrum">electromagnetic spectrum</a>, from <a href="/wiki/X-ray" title="X-ray">X-rays</a> to <a href="/wiki/Radio_wave" title="Radio wave">radio waves</a>. It can also be applied to <a href="/wiki/Wave" title="Wave">wave</a> phenomena such as <a href="/wiki/Sound" title="Sound">sound</a>. In this case, the <a href="/wiki/Speed_of_sound" title="Speed of sound">speed of sound</a> is used instead of that of light, and a reference medium other than vacuum must be chosen.<a href="#cite_note-Kinsler-3">[3]</a> Refraction also occurs in oceans when light passes into the <a href="/wiki/Halocline" title="Halocline">halocline</a> where salinity has impacted the density of the water column. For <a href="/wiki/Lens" title="Lens">lenses</a> (such as <a href="/wiki/Eye_glasses" class="mw-redirect" title="Eye glasses">eye glasses</a>), a lens made from a high refractive index material will be thinner, and hence lighter, than a conventional lens with a lower refractive index. Such lenses are generally more expensive to manufacture than conventional ones. <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2>[<a href="/w/index.php?title=Refractive_index&action=edit&section=1" title="Edit section: Definition">edit</a>]</div> The relative refractive index of an optical medium 2 with respect to another reference medium 1 (n21) is given by the ratio of speed of light in medium 1 to that in medium 2. This can be expressed as follows: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{21}={\frac {v_{1}}{v_{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{21}={\frac {v_{1}}{v_{2}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24308d3def17d6a6c21fdef45592bac20abd97de" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:10.034ex; height:5.009ex;" alt="{\displaystyle n_{21}={\frac {v_{1}}{v_{2}}}.}"> If the reference medium 1 is <a href="/wiki/Vacuum" title="Vacuum">vacuum</a>, then the refractive index of medium 2 is considered with respect to vacuum. It is simply represented as n2 and is called the absolute refractive index of medium 2. The absolute refractive index n of an optical medium is defined as the ratio of the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a> in vacuum, c = 299792458 m/s, and the <a href="/wiki/Phase_velocity" title="Phase velocity">phase velocity</a> v of light in the medium, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n={\frac {\mathrm {c} }{v}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> <mi>v</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n={\frac {\mathrm {c} }{v}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e60333900f1ff2ba950ec7b934d7e20be1f708d8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.104ex; height:4.676ex;" alt="{\displaystyle n={\frac {\mathrm {c} }{v}}.}"> Since c is constant, n is inversely proportional to v: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\propto {\frac {1}{v}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>∝</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>v</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\propto {\frac {1}{v}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b36745c6a49df2d40dbba66d5f82c57686309d95" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.139ex; height:5.176ex;" alt="{\displaystyle n\propto {\frac {1}{v}}.}"> The phase velocity is the speed at which the crests or the <a href="/wiki/Phase_(waves)" title="Phase (waves)">phase</a> of the <a href="/wiki/Wave" title="Wave">wave</a> moves, which may be different from the <a href="/wiki/Group_velocity" title="Group velocity">group velocity</a>, the speed at which the pulse of light or the <a href="/wiki/Envelope_(waves)" title="Envelope (waves)">envelope</a> of the wave moves.<a href="#cite_note-Hecht-1">[1]</a> Historically <a href="/wiki/Air" class="mw-redirect" title="Air">air</a> at a standardized <a href="/wiki/Pressure" title="Pressure">pressure</a> and <a href="/wiki/Temperature" title="Temperature">temperature</a> has been common as a reference medium. <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Refractive_index&action=edit&section=2" title="Edit section: History">edit</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Thomas_Young_(scientist).jpg" class="mw-file-description"><img alt="Stipple engraving of Thomas Young" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Thomas_Young_%28scientist%29.jpg/120px-Thomas_Young_%28scientist%29.jpg" decoding="async" width="120" height="152" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Thomas_Young_%28scientist%29.jpg/180px-Thomas_Young_%28scientist%29.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Thomas_Young_%28scientist%29.jpg/240px-Thomas_Young_%28scientist%29.jpg 2x" data-file-width="898" data-file-height="1138" /></a><figcaption><a href="/wiki/Thomas_Young_(scientist)" title="Thomas Young (scientist)">Thomas Young</a> coined the term index of refraction in 1807.</figcaption></figure> <a href="/wiki/Thomas_Young_(scientist)" title="Thomas Young (scientist)">Thomas Young</a> was presumably the person who first used, and invented, the name "index of refraction", in 1807.<a href="#cite_note-4">[4]</a> At the same time he changed this value of refractive power into a single number, instead of the traditional ratio of two numbers. The ratio had the disadvantage of different appearances. <a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a>, who called it the "proportion of the sines of incidence and refraction", wrote it as a ratio of two numbers, like "529 to 396" (or "nearly 4 to 3"; for water).<a href="#cite_note-Newton-5">[5]</a> <a href="/wiki/Francis_Hauksbee" title="Francis Hauksbee">Hauksbee</a>, who called it the "ratio of refraction", wrote it as a ratio with a fixed numerator, like "10000 to 7451.9" (for urine).<a href="#cite_note-Hauksbee-6">[6]</a> <a href="/wiki/Charles_Hutton" title="Charles Hutton">Hutton</a> wrote it as a ratio with a fixed denominator, like 1.3358 to 1 (water).<a href="#cite_note-Hutton-7">[7]</a> Young did not use a symbol for the index of refraction, in 1807. In the later years, others started using different symbols: n, m, and µ.<a href="#cite_note-Fraunhofer-8">[8]</a><a href="#cite_note-Brewster-9">[9]</a><a href="#cite_note-Herschel-10">[10]</a> The symbol n gradually prevailed. <div class="mw-heading mw-heading2"><h2 id="Typical_values">Typical values</h2>[<a href="/w/index.php?title=Refractive_index&action=edit&section=3" title="Edit section: Typical values">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/List_of_refractive_indices" title="List of refractive indices">List of refractive indices</a></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Brillanten.jpg" class="mw-file-description"><img alt="Gemstone diamonds" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Brillanten.jpg/220px-Brillanten.jpg" decoding="async" width="220" height="184" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Brillanten.jpg/330px-Brillanten.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Brillanten.jpg/440px-Brillanten.jpg 2x" data-file-width="476" data-file-height="399" /></a><figcaption><a href="/wiki/Diamond" title="Diamond">Diamonds</a> have a very high refractive index of 2.417.</figcaption></figure> Refractive index also varies with wavelength of the light as given by <a href="/wiki/Cauchy%27s_equation" title="Cauchy's equation">Cauchy's equation</a>. The most general form of this equation is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n(\lambda )=A+{\frac {B}{\lambda ^{2}}}+{\frac {C}{\lambda ^{4}}}+\cdots ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>A</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>B</mi> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>C</mi> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n(\lambda )=A+{\frac {B}{\lambda ^{2}}}+{\frac {C}{\lambda ^{4}}}+\cdots ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07a1b3866af4427675cd00f3e55868a4a1638555" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:28.17ex; height:5.676ex;" alt="{\displaystyle n(\lambda )=A+{\frac {B}{\lambda ^{2}}}+{\frac {C}{\lambda ^{4}}}+\cdots ,}"> where n is the refractive index, λ is the wavelength, and A, B, C, etc., are <a href="/wiki/Coefficient" title="Coefficient">coefficients</a> that can be determined for a material by fitting the equation to measured refractive indices at known wavelengths. The coefficients are usually quoted for λ as the <a href="/wiki/Vacuum_wavelength" class="mw-redirect" title="Vacuum wavelength">vacuum wavelength</a> in <a href="/wiki/Micrometre" title="Micrometre">micrometres</a>. Usually, it is sufficient to use a two-term form of the equation: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n(\lambda )=A+{\frac {B}{\lambda ^{2}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>A</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>B</mi> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n(\lambda )=A+{\frac {B}{\lambda ^{2}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/123b0e906afd1cfb7a74051571cd1612b2f7e1da" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:16.134ex; height:5.509ex;" alt="{\displaystyle n(\lambda )=A+{\frac {B}{\lambda ^{2}}},}"> where the coefficients A and B are determined specifically for this form of the equation. <table class="wikitable plainrowheaders" style="float:right;"> <caption>Selected refractive indices at λ = 589 nm. For references, see the extended <a href="/wiki/List_of_refractive_indices" title="List of refractive indices">List of refractive indices</a>. </caption> <tbody><tr> <th scope="col">Material </th> <th scope="col">n </th></tr> <tr> <th scope="row"><a href="/wiki/Vacuum" title="Vacuum">Vacuum</a> </th> <td>1 </td></tr> <tr> <td colspan="2" style="text-align:center;"><a href="/wiki/Gas" title="Gas">Gases</a> at <a href="/wiki/Standard_temperature_and_pressure" title="Standard temperature and pressure">0 °C and 1 atm</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Air" class="mw-redirect" title="Air">Air</a> </th> <td>1.000293 </td></tr> <tr> <th scope="row"><a href="/wiki/Helium" title="Helium">Helium</a> </th> <td>1.000036 </td></tr> <tr> <th scope="row"><a href="/wiki/Hydrogen" title="Hydrogen">Hydrogen</a> </th> <td>1.000132 </td></tr> <tr> <th scope="row"><a href="/wiki/Carbon_dioxide" title="Carbon dioxide">Carbon dioxide</a> </th> <td>1.00045 </td></tr> <tr> <td colspan="2" style="text-align:center;"><a href="/wiki/Liquid" title="Liquid">Liquids</a> at 20 °C </td></tr> <tr> <th scope="row"><a href="/wiki/Water" title="Water">Water</a> </th> <td>1.333 </td></tr> <tr> <th scope="row"><a href="/wiki/Ethanol" title="Ethanol">Ethanol</a> </th> <td>1.36 </td></tr> <tr> <th scope="row"><a href="/wiki/Olive_oil" title="Olive oil">Olive oil</a> </th> <td>1.47 </td></tr> <tr> <td colspan="2" style="text-align:center;"><a href="/wiki/Solid" title="Solid">Solids</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Ice" title="Ice">Ice</a> </th> <td>1.31 </td></tr> <tr> <th scope="row"><a href="/wiki/Fused_silica" class="mw-redirect" title="Fused silica">Fused silica</a> (quartz) </th> <td>1.46<a href="#cite_note-11">[11]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Poly(methyl_methacrylate)" title="Poly(methyl methacrylate)">PMMA</a> (acrylic, plexiglas, lucite, perspex) </th> <td>1.49 </td></tr> <tr> <th scope="row"><a href="/wiki/Soda-lime_glass" class="mw-redirect" title="Soda-lime glass">Window glass</a> </th> <td>1.52<a href="#cite_note-12">[12]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Polycarbonate" title="Polycarbonate">Polycarbonate</a> (Lexan™) </th> <td>1.58<a href="#cite_note-13">[13]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Flint_glass" title="Flint glass">Flint glass</a> (typical) </th> <td>1.69 </td></tr> <tr> <th scope="row"><a href="/wiki/Sapphire" title="Sapphire">Sapphire</a> </th> <td>1.77<a href="#cite_note-14">[14]</a> </td></tr> <tr> <th scope="row"><a href="/wiki/Cubic_zirconia" title="Cubic zirconia">Cubic zirconia</a> </th> <td>2.15 </td></tr> <tr> <th scope="row"><a href="/wiki/Diamond" title="Diamond">Diamond</a> </th> <td>2.417 </td></tr> <tr> <th scope="row"><a href="/wiki/Moissanite" title="Moissanite">Moissanite</a> </th> <td>2.65 </td></tr></tbody></table> For <a href="/wiki/Visible_light" class="mw-redirect" title="Visible light">visible light</a> most <a href="/wiki/Transparency_and_translucency" title="Transparency and translucency">transparent</a> media have refractive indices between 1 and 2. A few examples are given in the adjacent table. These values are measured at the yellow doublet <a href="/wiki/D2_line" class="mw-redirect" title="D2 line">D-line</a> of <a href="/wiki/Sodium" title="Sodium">sodium</a>, with a wavelength of 589 <a href="/wiki/Nanometers" class="mw-redirect" title="Nanometers">nanometers</a>, as is conventionally done.<a href="#cite_note-FBI-15">[15]</a> Gases at atmospheric pressure have refractive indices close to 1 because of their low density. Almost all solids and liquids have refractive indices above 1.3, with <a href="/wiki/Aerogel" title="Aerogel">aerogel</a> as the clear exception. Aerogel is a very low density solid that can be produced with refractive index in the range from 1.002 to 1.265.<a href="#cite_note-16">[16]</a> <a href="/wiki/Moissanite" title="Moissanite">Moissanite</a> lies at the other end of the range with a refractive index as high as 2.65. Most plastics have refractive indices in the range from 1.3 to 1.7, but some <a href="/wiki/High-refractive-index_polymer" title="High-refractive-index polymer">high-refractive-index polymers</a> can have values as high as 1.76.<a href="#cite_note-17">[17]</a> For <a href="/wiki/Infrared" title="Infrared">infrared</a> light refractive indices can be considerably higher. <a href="/wiki/Germanium" title="Germanium">Germanium</a> is transparent in the wavelength region from 2 to 14 μm and has a refractive index of about 4.<a href="#cite_note-18">[18]</a> A type of new materials termed "<a href="/wiki/Topological_insulators" class="mw-redirect" title="Topological insulators">topological insulators</a>", was recently found which have high refractive index of up to 6 in the near to mid infrared frequency range. Moreover, topological insulators are transparent when they have nanoscale thickness. These properties are potentially important for applications in infrared optics.<a href="#cite_note-19">[19]</a> <div class="mw-heading mw-heading3"><h3 id="Refractive_index_below_unity">Refractive index below unity</h3>[<a href="/w/index.php?title=Refractive_index&action=edit&section=4" title="Edit section: Refractive index below unity">edit</a>]</div> According to the <a href="/wiki/Theory_of_relativity" title="Theory of relativity">theory of relativity</a>, no information can travel faster than the speed of light in vacuum, but this does not mean that the refractive index cannot be less than 1. The refractive index measures the <a href="/wiki/Phase_velocity" title="Phase velocity">phase velocity</a> of light, which does not carry <a href="/wiki/Information" title="Information">information</a>.<a href="#cite_note-Als-Nielsen2011-20">[20]</a><a href="#cite_note-21">[a]</a> The phase velocity is the speed at which the crests of the wave move and can be faster than the speed of light in vacuum, and thereby give a refractive index below 1. This can occur close to <a href="/wiki/Resonance_frequency" class="mw-redirect" title="Resonance frequency">resonance frequencies</a>, for absorbing media, in <a href="/wiki/Plasma_(physics)" title="Plasma (physics)">plasmas</a>, and for <a href="/wiki/X-ray" title="X-ray">X-rays</a>. In the X-ray regime the refractive indices are lower than but very close to 1 (exceptions close to some resonance frequencies).<a href="#cite_note-CXRO-22">[21]</a> As an example, water has a refractive index of 0.99999974 = 1 − 2.6×10−7 for X-ray radiation at a photon energy of 30 <a href="/wiki/Electronvolt" title="Electronvolt">keV</a> (0.04 nm wavelength).<a href="#cite_note-CXRO-22">[21]</a> An example of a plasma with an index of refraction less than unity is Earth's <a href="/wiki/Ionosphere" title="Ionosphere">ionosphere</a>. Since the refractive index of the ionosphere (a <a href="/wiki/Plasma_(physics)" title="Plasma (physics)">plasma</a>), is less than unity, electromagnetic waves propagating through the plasma are bent "away from the normal" (see <a href="/wiki/Geometric_optics" class="mw-redirect" title="Geometric optics">Geometric optics</a>) allowing the radio wave to be refracted back toward earth, thus enabling long-distance radio communications. See also <a href="/wiki/Radio_Propagation" class="mw-redirect" title="Radio Propagation">Radio Propagation</a> and <a href="/wiki/Skywave" title="Skywave">Skywave</a>.<a href="#cite_note-23">[22]</a> <div class="mw-heading mw-heading3"><h3 id="Negative_refractive_index">Negative refractive index</h3>[<a href="/w/index.php?title=Refractive_index&action=edit&section=5" title="Edit section: Negative refractive index">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Negative_index_metamaterials" class="mw-redirect" title="Negative index metamaterials">Negative index metamaterials</a></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Split-ring_resonator_array_10K_sq_nm.jpg" class="mw-file-description"><img alt="A 3D grid of open copper rings made from interlocking standing sheets of fiberglass circuit boards" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Split-ring_resonator_array_10K_sq_nm.jpg/250px-Split-ring_resonator_array_10K_sq_nm.jpg" decoding="async" width="250" height="188" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/8/82/Split-ring_resonator_array_10K_sq_nm.jpg 1.5x" data-file-width="350" data-file-height="263" /></a><figcaption>A <a href="/wiki/Split-ring_resonator" title="Split-ring resonator">split-ring resonator</a> array arranged to produce a negative index of refraction for <a href="/wiki/Microwaves" class="mw-redirect" title="Microwaves">microwaves</a></figcaption></figure> Recent research has also demonstrated the "existence" of materials with a negative refractive index, which can occur if <a href="/wiki/Permittivity" title="Permittivity">permittivity</a> and <a href="/wiki/Magnetic_permeability" class="mw-redirect" title="Magnetic permeability">permeability</a> have simultaneous negative values.<a href="#cite_note-veselago1968-24">[23]</a> This can be achieved with periodically constructed <a href="/wiki/Metamaterials" class="mw-redirect" title="Metamaterials">metamaterials</a>. The resulting <a href="/wiki/Negative_refraction" title="Negative refraction">negative refraction</a> (i.e., a reversal of <a href="/wiki/Snell%27s_law" title="Snell's law">Snell's law</a>) offers the possibility of the <a href="/wiki/Superlens" title="Superlens">superlens</a> and other new phenomena to be actively developed by means of <a href="/wiki/Negative_index_metamaterials" class="mw-redirect" title="Negative index metamaterials">metamaterials</a>.<a href="#cite_note-25">[24]</a><a href="#cite_note-shalaev2007-26">[25]</a> <div class="mw-heading mw-heading2"><h2 id="Microscopic_explanation">Microscopic explanation</h2>[<a href="/w/index.php?title=Refractive_index&action=edit&section=6" title="Edit section: Microscopic explanation">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Thin_section_scan_crossed_polarizers_Siilinj%C3%A4rvi_R636-105.90.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Thin_section_scan_crossed_polarizers_Siilinj%C3%A4rvi_R636-105.90.jpg/220px-Thin_section_scan_crossed_polarizers_Siilinj%C3%A4rvi_R636-105.90.jpg" decoding="async" width="220" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/06/Thin_section_scan_crossed_polarizers_Siilinj%C3%A4rvi_R636-105.90.jpg/330px-Thin_section_scan_crossed_polarizers_Siilinj%C3%A4rvi_R636-105.90.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/06/Thin_section_scan_crossed_polarizers_Siilinj%C3%A4rvi_R636-105.90.jpg/440px-Thin_section_scan_crossed_polarizers_Siilinj%C3%A4rvi_R636-105.90.jpg 2x" data-file-width="3660" data-file-height="2000" /></a><figcaption>In <a href="/wiki/Optical_mineralogy" title="Optical mineralogy">optical mineralogy</a>, <a href="/wiki/Thin_section" title="Thin section">thin sections</a> are used to study rocks. The method is based on the distinct refractive indices of different <a href="/wiki/Mineral" title="Mineral">minerals</a>.</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Ewald%E2%80%93Oseen_extinction_theorem" title="Ewald–Oseen extinction theorem">Ewald–Oseen extinction theorem</a></div> At the atomic scale, an electromagnetic wave's phase velocity is slowed in a material because the <a href="/wiki/Electric_field" title="Electric field">electric field</a> creates a disturbance in the charges of each atom (primarily the <a href="/wiki/Electron" title="Electron">electrons</a>) proportional to the <a href="/wiki/Electric_susceptibility" title="Electric susceptibility">electric susceptibility</a> of the medium. (Similarly, the <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</a> creates a disturbance proportional to the <a href="/wiki/Magnetic_susceptibility" title="Magnetic susceptibility">magnetic susceptibility</a>.) As the electromagnetic fields oscillate in the wave, the charges in the material will be "shaken" back and forth at the same frequency.<a href="#cite_note-Hecht-1">[1]</a>: 67  The charges thus radiate their own electromagnetic wave that is at the same frequency, but usually with a <a href="/wiki/Phase_(waves)" title="Phase (waves)">phase delay</a>, as the charges may move out of phase with the force driving them (see <a href="/wiki/Harmonic_oscillator#Sinusoidal_driving_force" title="Harmonic oscillator">sinusoidally driven harmonic oscillator</a>). The light wave traveling in the medium is the macroscopic <a href="/wiki/Superposition_principle" title="Superposition principle">superposition (sum)</a> of all such contributions in the material: the original wave plus the waves radiated by all the moving charges. This wave is typically a wave with the same frequency but shorter wavelength than the original, leading to a slowing of the wave's phase velocity. Most of the radiation from oscillating material charges will modify the incoming wave, changing its velocity. However, some net energy will be radiated in other directions or even at other frequencies (see <a href="/wiki/Scattering" title="Scattering">scattering</a>). Depending on the relative phase of the original driving wave and the waves radiated by the charge motion, there are several possibilities: <ul><li>If the electrons emit a light wave which is 90° out of phase with the light wave shaking them, it will cause the total light wave to travel slower. This is the normal refraction of transparent materials like glass or water, and corresponds to a refractive index which is real and greater than 1.<a href="#cite_note-Feynman,_Richard_P._2011-27">[26]</a>[<a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">page needed</a>]</li> <li>If the electrons emit a light wave which is 270° out of phase with the light wave shaking them, it will cause the wave to travel faster. This is called "anomalous refraction", and is observed close to absorption lines (typically in infrared spectra), with <a href="/wiki/X-ray" title="X-ray">X-rays</a> in ordinary materials, and with radio waves in Earth's <a href="/wiki/Ionosphere" title="Ionosphere">ionosphere</a>. It corresponds to a <a href="/wiki/Permittivity" title="Permittivity">permittivity</a> less than 1, which causes the refractive index to be also less than unity and the <a href="/wiki/Phase_velocity" title="Phase velocity">phase velocity</a> of light greater than the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light in vacuum</a> c (note that the <a href="/wiki/Signal_velocity" title="Signal velocity">signal velocity</a> is still less than c, as discussed above). If the response is sufficiently strong and out-of-phase, the result is a negative value of <a href="/wiki/Permittivity" title="Permittivity">permittivity</a> and imaginary index of refraction, as observed in metals or plasma.<a href="#cite_note-Feynman,_Richard_P._2011-27">[26]</a>[<a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">page needed</a>]</li> <li>If the electrons emit a light wave which is 180° out of phase with the light wave shaking them, it will destructively interfere with the original light to reduce the total light intensity. This is <a href="/wiki/Absorption_(electromagnetic_radiation)" title="Absorption (electromagnetic radiation)">light absorption in opaque materials</a> and corresponds to an <a href="/wiki/Imaginary_number" title="Imaginary number">imaginary</a> refractive index.</li> <li>If the electrons emit a light wave which is in phase with the light wave shaking them, it will amplify the light wave. This is rare, but occurs in <a href="/wiki/Laser" title="Laser">lasers</a> due to <a href="/wiki/Stimulated_emission" title="Stimulated emission">stimulated emission</a>. It corresponds to an imaginary index of refraction, with the opposite sign to that of absorption.</li></ul> For most materials at visible-light frequencies, the phase is somewhere between 90° and 180°, corresponding to a combination of both refraction and absorption. <div class="mw-heading mw-heading2"><h2 id="Dispersion">Dispersion</h2>[<a href="/w/index.php?title=Refractive_index&action=edit&section=7" title="Edit section: Dispersion">edit</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:WhereRainbowRises.jpg" class="mw-file-description"><img alt="A rainbow" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/27/WhereRainbowRises.jpg/150px-WhereRainbowRises.jpg" decoding="async" width="150" height="226" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/27/WhereRainbowRises.jpg/225px-WhereRainbowRises.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/27/WhereRainbowRises.jpg/300px-WhereRainbowRises.jpg 2x" data-file-width="1360" data-file-height="2048" /></a><figcaption>Light of different colors has slightly different refractive indices in water and therefore shows up at different positions in the <a href="/wiki/Rainbow" title="Rainbow">rainbow</a>.</figcaption></figure> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Prism-rainbow.svg" class="mw-file-description"><img alt="A white beam of light dispersed into different colors when passing through a triangular prism" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Prism-rainbow.svg/220px-Prism-rainbow.svg.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/24/Prism-rainbow.svg/330px-Prism-rainbow.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/24/Prism-rainbow.svg/440px-Prism-rainbow.svg.png 2x" data-file-width="400" data-file-height="300" /></a><figcaption>In a triangular <a href="/wiki/Prism_(optics)" title="Prism (optics)">prism</a>, <a href="/wiki/Dispersion_(optics)" title="Dispersion (optics)">dispersion</a> causes different colors to refract at different angles, splitting <a href="/wiki/White" title="White">white light</a> into a <a href="/wiki/Rainbow" title="Rainbow">rainbow</a> of colors. The blue color is more deviated (refracted) than the red color because the refractive index of blue is higher than that of red. </figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Mplwp_dispersion_curves.svg" class="mw-file-description"><img alt="A graph showing the decrease in refractive index with increasing wavelength for different types of glass" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/04/Mplwp_dispersion_curves.svg/320px-Mplwp_dispersion_curves.svg.png" decoding="async" width="320" height="213" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/04/Mplwp_dispersion_curves.svg/480px-Mplwp_dispersion_curves.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/04/Mplwp_dispersion_curves.svg/640px-Mplwp_dispersion_curves.svg.png 2x" data-file-width="600" data-file-height="400" /></a><figcaption>The variation of refractive index with wavelength for various glasses. The shaded zone indicates the range of visible light.</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Dispersion_(optics)" title="Dispersion (optics)">Dispersion (optics)</a></div> The refractive index of materials varies with the wavelength (and <a href="/wiki/Frequency" title="Frequency">frequency</a>) of light.<a href="#cite_note-dispersion_ELPT-28">[27]</a> This is called dispersion and causes <a href="/wiki/Prism_(optics)" title="Prism (optics)">prisms</a> and <a href="/wiki/Rainbow" title="Rainbow">rainbows</a> to divide white light into its constituent spectral <a href="/wiki/Color" title="Color">colors</a>.<a href="#cite_note-hyperphysics_dispersion-29">[28]</a> As the refractive index varies with wavelength, so will the refraction angle as light goes from one material to another. Dispersion also causes the <a href="/wiki/Focal_length" title="Focal length">focal length</a> of <a href="/wiki/Lens_(optics)" class="mw-redirect" title="Lens (optics)">lenses</a> to be wavelength dependent. This is a type of <a href="/wiki/Chromatic_aberration" title="Chromatic aberration">chromatic aberration</a>, which often needs to be corrected for in imaging systems. In regions of the spectrum where the material does not absorb light, the refractive index tends to decrease with increasing wavelength, and thus increase with frequency. This is called "normal dispersion", in contrast to "anomalous dispersion", where the refractive index increases with wavelength.<a href="#cite_note-dispersion_ELPT-28">[27]</a> For visible light normal dispersion means that the refractive index is higher for blue light than for red. For optics in the visual range, the amount of dispersion of a lens material is often quantified by the <a href="/wiki/Abbe_number" title="Abbe number">Abbe number</a>:<a href="#cite_note-hyperphysics_dispersion-29">[28]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V={\frac {n_{\mathrm {yellow} }-1}{n_{\mathrm {blue} }-n_{\mathrm {red} }}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">y</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">w</mi> </mrow> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">b</mi> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> <mo>−</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> <mi mathvariant="normal">e</mi> <mi mathvariant="normal">d</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\frac {n_{\mathrm {yellow} }-1}{n_{\mathrm {blue} }-n_{\mathrm {red} }}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fddde421334a75ae3b9158895a56f4dc2a417c1e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.767ex; height:5.843ex;" alt="{\displaystyle V={\frac {n_{\mathrm {yellow} }-1}{n_{\mathrm {blue} }-n_{\mathrm {red} }}}.}"> For a more accurate description of the wavelength dependence of the refractive index, the <a href="/wiki/Sellmeier_equation" title="Sellmeier equation">Sellmeier equation</a> can be used.<a href="#cite_note-30">[29]</a> It is an empirical formula that works well in describing dispersion. Sellmeier coefficients are often quoted instead of the refractive index in tables. <div class="mw-heading mw-heading3"><h3 id="Principal_refractive_index_wavelength_ambiguity">Principal refractive index wavelength ambiguity</h3>[<a href="/w/index.php?title=Refractive_index&action=edit&section=8" title="Edit section: Principal refractive index wavelength ambiguity">edit</a>]</div> Because of dispersion, it is usually important to specify the vacuum wavelength of light for which a refractive index is measured. Typically, measurements are done at various well-defined spectral <a href="/wiki/Emission_line" class="mw-redirect" title="Emission line">emission lines</a>. Manufacturers of optical glass in general define principal index of refraction at yellow spectral line of helium (587.56 nm) and alternatively at a green spectral line of mercury (546.07 nm), called d and e lines respectively. <a href="/wiki/Abbe_number" title="Abbe number">Abbe number</a> is defined for both and denoted Vd and Ve. The spectral data provided by glass manufacturers is also often more precise for these two wavelengths.<a href="#cite_note-31">[30]</a><a href="#cite_note-32">[31]</a><a href="#cite_note-33">[32]</a><a href="#cite_note-34">[33]</a> Both, d and e spectral lines are singlets and thus are suitable to perform a very precise measurements, such as spectral goniometric method.<a href="#cite_note-35">[34]</a><a href="#cite_note-36">[35]</a> In practical applications, measurements of refractive index are performed on various refractometers, such as <a href="/wiki/Abbe_refractometer" title="Abbe refractometer">Abbe refractometer</a>. Measurement accuracy of such typical commercial devices is in the order of 0.0002.<a href="#cite_note-37">[36]</a><a href="#cite_note-38">[37]</a> Refractometers usually measure refractive index nD, defined for sodium doublet D (589.29 nm), which is actually a midpoint between two adjacent yellow spectral lines of sodium. Yellow spectral lines of helium (d) and sodium (D) are 1.73 nm apart, which can be considered negligible for typical refractometers, but can cause confusion and lead to errors if accuracy is critical. All three typical principle refractive indices definitions can be found depending on application and region,<a href="#cite_note-39">[38]</a> so a proper subscript should be used to avoid ambiguity. <div class="mw-heading mw-heading2"><h2 id="Complex_refractive_index">Complex refractive index</h2>[<a href="/w/index.php?title=Refractive_index&action=edit&section=9" title="Edit section: Complex refractive index">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Mathematical_descriptions_of_opacity" title="Mathematical descriptions of opacity">Mathematical descriptions of opacity</a></div> When light passes through a medium, some part of it will always be <a href="/wiki/Attenuation" title="Attenuation">absorbed</a>. This can be conveniently taken into account by defining a complex refractive index, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {n}}=n-i\kappa .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <munder> <mi>n</mi> <mo>_</mo> </munder> </mrow> <mo>=</mo> <mi>n</mi> <mo>−</mo> <mi>i</mi> <mi>κ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\underline {n}}=n-i\kappa .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72b544a7ca9b066f83b424470ba0961625bfed71" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.537ex; margin-bottom: -0.801ex; width:11.519ex; height:3.176ex;" alt="{\displaystyle {\underline {n}}=n-i\kappa .}"> Here, the real part n is the refractive index and indicates the <a href="/wiki/Phase_velocity" title="Phase velocity">phase velocity</a>, while the imaginary part κ is called the extinction coefficient<a href="#cite_note-DresselhausMITCourse-40">[39]</a>: 36  indicates the amount of attenuation when the electromagnetic wave propagates through the material.<a href="#cite_note-Hecht-1">[1]</a>: 128  It is related to the absorption coefficient, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{\text{abs}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>abs</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{\text{abs}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fb0f5301c597a915e47d3362c1a2ea31a8210ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.104ex; height:2.009ex;" alt="{\displaystyle \alpha _{\text{abs}}}">, through:<a href="#cite_note-DresselhausMITCourse-40">[39]</a>: 41  <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha _{\text{abs}}(\omega )={\frac {2\omega \kappa (\omega )}{c}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>abs</mtext> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>ω</mi> <mi>κ</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> <mi>c</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha _{\text{abs}}(\omega )={\frac {2\omega \kappa (\omega )}{c}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/107331132e8ad844e687778fb23156b108e33fb5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:18.496ex; height:5.676ex;" alt="{\displaystyle \alpha _{\text{abs}}(\omega )={\frac {2\omega \kappa (\omega )}{c}}}"> These values depend upon the frequency of the light used in the measurement. That κ corresponds to absorption can be seen by inserting this refractive index into the expression for <a href="/wiki/Electric_field" title="Electric field">electric field</a> of a <a href="/wiki/Plane_wave" title="Plane wave">plane</a> electromagnetic wave traveling in the x-direction. This can be done by relating the complex <a href="/wiki/Wave_number" class="mw-redirect" title="Wave number">wave number</a> k to the complex refractive index n through k = 2πn/λ0, with λ0 being the vacuum wavelength; this can be inserted into the plane wave expression for a wave travelling in the x-direction as: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {E} (x,t)&=\operatorname {Re} \!\left[\mathbf {E} _{0}e^{i({\underline {k}}x-\omega t)}\right]\\&=\operatorname {Re} \!\left[\mathbf {E} _{0}e^{i(2\pi (n+i\kappa )x/\lambda _{0}-\omega t)}\right]\\&=e^{-2\pi \kappa x/\lambda _{0}}\operatorname {Re} \!\left[\mathbf {E} _{0}e^{i(kx-\omega t)}\right].\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>Re</mi> <mspace width="negativethinmathspace" /> <mrow> <mo>[</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <munder> <mi>k</mi> <mo>_</mo> </munder> </mrow> <mi>x</mi> <mo>−</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>Re</mi> <mspace width="negativethinmathspace" /> <mrow> <mo>[</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mi>i</mi> <mi>κ</mi> <mo stretchy="false">)</mo> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>2</mn> <mi>π</mi> <mi>κ</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msup> <mi>Re</mi> <mspace width="negativethinmathspace" /> <mrow> <mo>[</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mi>x</mi> <mo>−</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {E} (x,t)&=\operatorname {Re} \!\left[\mathbf {E} _{0}e^{i({\underline {k}}x-\omega t)}\right]\\&=\operatorname {Re} \!\left[\mathbf {E} _{0}e^{i(2\pi (n+i\kappa )x/\lambda _{0}-\omega t)}\right]\\&=e^{-2\pi \kappa x/\lambda _{0}}\operatorname {Re} \!\left[\mathbf {E} _{0}e^{i(kx-\omega t)}\right].\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e3936daed1b1eca2f8adf0bc77d7e57269bb7cd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:36.5ex; height:14.509ex;" alt="{\displaystyle {\begin{aligned}\mathbf {E} (x,t)&=\operatorname {Re} \!\left[\mathbf {E} _{0}e^{i({\underline {k}}x-\omega t)}\right]\\&=\operatorname {Re} \!\left[\mathbf {E} _{0}e^{i(2\pi (n+i\kappa )x/\lambda _{0}-\omega t)}\right]\\&=e^{-2\pi \kappa x/\lambda _{0}}\operatorname {Re} \!\left[\mathbf {E} _{0}e^{i(kx-\omega t)}\right].\end{aligned}}}"> Here we see that κ gives an exponential decay, as expected from the <a href="/wiki/Beer%E2%80%93Lambert_law" title="Beer–Lambert law">Beer–Lambert law</a>. Since intensity is proportional to the square of the electric field, intensity will depend on the depth into the material as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I(x)=I_{0}e^{-4\pi \kappa x/\lambda _{0}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>4</mn> <mi>π</mi> <mi>κ</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I(x)=I_{0}e^{-4\pi \kappa x/\lambda _{0}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2984d6aab558cf064f77238164cbdfca8eef309b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.99ex; height:3.343ex;" alt="{\displaystyle I(x)=I_{0}e^{-4\pi \kappa x/\lambda _{0}}.}"> and thus the <a href="/wiki/Attenuation_coefficient#Absorption_and_scattering_coefficients" title="Attenuation coefficient">absorption coefficient</a> is α = 4πκ/λ0,<a href="#cite_note-Hecht-1">[1]</a>: 128  and the <a href="/wiki/Penetration_depth" title="Penetration depth">penetration depth</a> (the distance after which the intensity is reduced by a factor of 1/e) is δp = 1/α = λ0/4πκ. Both n and κ are dependent on the frequency. In most circumstances κ > 0 (light is absorbed) or κ = 0 (light travels forever without loss). In special situations, especially in the <a href="/wiki/Gain_medium" class="mw-redirect" title="Gain medium">gain medium</a> of <a href="/wiki/Laser" title="Laser">lasers</a>, it is also possible that κ < 0, corresponding to an amplification of the light. An alternative convention uses n = n + iκ instead of n = n − iκ, but where κ > 0 still corresponds to loss. Therefore, these two conventions are inconsistent and should not be confused. The difference is related to defining sinusoidal time dependence as Re[exp(−iωt)] versus Re[exp(+iωt)]. See <a href="/wiki/Mathematical_descriptions_of_opacity" title="Mathematical descriptions of opacity">Mathematical descriptions of opacity</a>. Dielectric loss and non-zero DC conductivity in materials cause absorption. Good dielectric materials such as glass have extremely low DC conductivity, and at low frequencies the dielectric loss is also negligible, resulting in almost no absorption. However, at higher frequencies (such as visible light), dielectric loss may increase absorption significantly, reducing the material's <a href="/wiki/Transparency_(optics)" class="mw-redirect" title="Transparency (optics)">transparency</a> to these frequencies. The real n, and imaginary κ, parts of the complex refractive index are related through the <a href="/wiki/Kramers%E2%80%93Kronig_relations" title="Kramers–Kronig relations">Kramers–Kronig relations</a>. In 1986, A.R. Forouhi and I. Bloomer deduced an <a href="/wiki/Forouhi%E2%80%93Bloomer_model" title="Forouhi–Bloomer model">equation</a> describing κ as a function of photon energy, E, applicable to amorphous materials. Forouhi and Bloomer then applied the Kramers–Kronig relation to derive the corresponding equation for <a href="/wiki/Forouhi%E2%80%93Bloomer_model" title="Forouhi–Bloomer model">n as a function of E</a>. The same formalism was applied to crystalline materials by Forouhi and Bloomer in 1988. The refractive index and extinction coefficient, n and κ, are typically measured from quantities that depend on them, such as <a href="/wiki/Fresnel_equations" title="Fresnel equations">reflectance, R, or transmittance, T</a>, or ellipsometric parameters, <a href="/wiki/Ellipsometry" title="Ellipsometry">ψ and δ</a>. The determination of n and κ from such measured quantities will involve developing a theoretical expression for R or T, or ψ and δ in terms of a valid physical model for n and κ. By fitting the theoretical model to the measured R or T, or ψ and δ using regression analysis, n and κ can be deduced. <div class="mw-heading mw-heading3"><h3 id="X-ray_and_extreme_UV">X-ray and extreme UV</h3>[<a href="/w/index.php?title=Refractive_index&action=edit&section=10" title="Edit section: X-ray and extreme UV">edit</a>]</div> For <a href="/wiki/X-ray" title="X-ray">X-ray</a> and <a href="/wiki/Extreme_ultraviolet" title="Extreme ultraviolet">extreme ultraviolet</a> radiation the complex refractive index deviates only slightly from unity and usually has a real part smaller than 1. It is therefore normally written as n = 1 − δ + iβ (or n = 1 − δ − iβ with the alternative convention mentioned above).<a href="#cite_note-Attwood-2">[2]</a> Far above the atomic resonance frequency delta can be given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta ={\frac {r_{0}\lambda ^{2}n_{\mathrm {e} }}{2\pi }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">e</mi> </mrow> </mrow> </msub> </mrow> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta ={\frac {r_{0}\lambda ^{2}n_{\mathrm {e} }}{2\pi }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f25abfb0b2a082d9d81c3e77773fc0913d9acd1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.853ex; height:5.676ex;" alt="{\displaystyle \delta ={\frac {r_{0}\lambda ^{2}n_{\mathrm {e} }}{2\pi }}}"> where r0 is the <a href="/wiki/Classical_electron_radius" title="Classical electron radius">classical electron radius</a>, λ is the X-ray wavelength, and ne is the electron density. One may assume the electron density is simply the number of electrons per atom Z multiplied by the atomic density, but more accurate calculation of the refractive index requires replacing Z with the complex <a href="/wiki/Atomic_form_factor" title="Atomic form factor">atomic form factor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=Z+f'+if''}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mi>Z</mi> <mo>+</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo>+</mo> <mi>i</mi> <msup> <mi>f</mi> <mo>″</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f=Z+f'+if''}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22edb3f7b222cdc32422c1783056a25623778719" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.004ex; height:2.843ex;" alt="{\displaystyle f=Z+f'+if''}">. It follows that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\delta &={\frac {r_{0}\lambda ^{2}}{2\pi }}(Z+f')n_{\text{atom}}\\\beta &={\frac {r_{0}\lambda ^{2}}{2\pi }}f''n_{\text{atom}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>δ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>Z</mi> <mo>+</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>atom</mtext> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mi>β</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msup> <mi>f</mi> <mo>″</mo> </msup> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>atom</mtext> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\delta &={\frac {r_{0}\lambda ^{2}}{2\pi }}(Z+f')n_{\text{atom}}\\\beta &={\frac {r_{0}\lambda ^{2}}{2\pi }}f''n_{\text{atom}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d944fa7fca3b6533e55609d3708dd69c453ebb9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:24.145ex; height:11.509ex;" alt="{\displaystyle {\begin{aligned}\delta &={\frac {r_{0}\lambda ^{2}}{2\pi }}(Z+f')n_{\text{atom}}\\\beta &={\frac {r_{0}\lambda ^{2}}{2\pi }}f''n_{\text{atom}}\end{aligned}}}"> with δ and β typically of the order of 10−5 and 10−6. <div class="mw-heading mw-heading2"><h2 id="Relations_to_other_quantities">Relations to other quantities</h2>[<a href="/w/index.php?title=Refractive_index&action=edit&section=11" title="Edit section: Relations to other quantities">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Optical_path_length">Optical path length</h3>[<a href="/w/index.php?title=Refractive_index&action=edit&section=12" title="Edit section: Optical path length">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Soap_bubble_sky.jpg" class="mw-file-description"><img alt="Soap bubble" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/Soap_bubble_sky.jpg/220px-Soap_bubble_sky.jpg" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/Soap_bubble_sky.jpg/330px-Soap_bubble_sky.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/18/Soap_bubble_sky.jpg/440px-Soap_bubble_sky.jpg 2x" data-file-width="1024" data-file-height="768" /></a><figcaption>The colors of a <a href="/wiki/Soap_bubble" title="Soap bubble">soap bubble</a> are determined by the <a href="/wiki/Optical_path_length" title="Optical path length">optical path length</a> through the thin soap film in a phenomenon called <a href="/wiki/Thin-film_interference" title="Thin-film interference">thin-film interference</a>.</figcaption></figure> <a href="/wiki/Optical_path_length" title="Optical path length">Optical path length</a> (OPL) is the product of the geometric length d of the path light follows through a system, and the index of refraction of the medium through which it propagates,<a href="#cite_note-41">[40]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{OPL}}=nd."> <semantics> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>OPL</mtext> </mrow> <mo>=</mo> <mi>n</mi> <mi>d</mi> <mo>.</mo> </mrow> <annotation encoding="application/x-tex">{\text{OPL}}=nd.</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86b4599c437fe198c742f6d60830db263df998f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.2ex; height:2.176ex;" alt="{\text{OPL}}=nd."> This is an important concept in optics because it determines the <a href="/wiki/Phase_(waves)" title="Phase (waves)">phase</a> of the light and governs <a href="/wiki/Interference_(wave_propagation)" class="mw-redirect" title="Interference (wave propagation)">interference</a> and <a href="/wiki/Diffraction" title="Diffraction">diffraction</a> of light as it propagates. According to <a href="/wiki/Fermat%27s_principle" title="Fermat's principle">Fermat's principle</a>, light rays can be characterized as those curves that <a href="/wiki/Mathematical_optimization" title="Mathematical optimization">optimize</a> the optical path length.<a href="#cite_note-Hecht-1">[1]</a>: 68–69  <div class="mw-heading mw-heading3"><h3 id="Refraction">Refraction</h3>[<a href="/w/index.php?title=Refractive_index&action=edit&section=13" title="Edit section: Refraction">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Refraction" title="Refraction">Refraction</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Snells_law.svg" class="mw-file-description"><img alt="refer to caption" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Snells_law.svg/220px-Snells_law.svg.png" decoding="async" width="220" height="122" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Snells_law.svg/330px-Snells_law.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Snells_law.svg/440px-Snells_law.svg.png 2x" data-file-width="641" data-file-height="355" /></a><figcaption><a href="/wiki/Refraction" title="Refraction">Refraction</a> of light at the interface between two media of different refractive indices, with n2 > n1. Since the <a href="/wiki/Phase_velocity" title="Phase velocity">phase velocity</a> is lower in the second medium (v2 < v1), the angle of refraction θ2 is less than the angle of incidence θ1; that is, the ray in the higher-index medium is closer to the normal.</figcaption></figure> When light moves from one medium to another, it changes direction, i.e. it is <a href="/wiki/Refraction" title="Refraction">refracted</a>. If it moves from a medium with refractive index n1 to one with refractive index n2, with an <a href="/wiki/Angle_of_incidence_(optics)" title="Angle of incidence (optics)">incidence angle</a> to the <a href="/wiki/Surface_normal" class="mw-redirect" title="Surface normal">surface normal</a> of θ1, the refraction angle θ2 can be calculated from <a href="/wiki/Snell%27s_law" title="Snell's law">Snell's law</a>:<a href="#cite_note-42">[41]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{1}\sin \theta _{1}=n_{2}\sin \theta _{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{1}\sin \theta _{1}=n_{2}\sin \theta _{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4219c4208cb4c2ffdf62de226719e59d3295e60d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.192ex; height:2.509ex;" alt="{\displaystyle n_{1}\sin \theta _{1}=n_{2}\sin \theta _{2}.}"> When light enters a material with higher refractive index, the angle of refraction will be smaller than the angle of incidence and the light will be refracted towards the normal of the surface. The higher the refractive index, the closer to the normal direction the light will travel. When passing into a medium with lower refractive index, the light will instead be refracted away from the normal, towards the surface. <div class="mw-heading mw-heading3"><h3 id="Total_internal_reflection">Total internal reflection</h3>[<a href="/w/index.php?title=Refractive_index&action=edit&section=14" title="Edit section: Total internal reflection">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Total_internal_reflection" title="Total internal reflection">Total internal reflection</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Total_internal_reflection_of_Chelonia_mydas.jpg" class="mw-file-description"><img alt="A sea turtle being reflected in the water surface above" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Total_internal_reflection_of_Chelonia_mydas.jpg/220px-Total_internal_reflection_of_Chelonia_mydas.jpg" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Total_internal_reflection_of_Chelonia_mydas.jpg/330px-Total_internal_reflection_of_Chelonia_mydas.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Total_internal_reflection_of_Chelonia_mydas.jpg/440px-Total_internal_reflection_of_Chelonia_mydas.jpg 2x" data-file-width="2000" data-file-height="1500" /></a><figcaption><a href="/wiki/Total_internal_reflection" title="Total internal reflection">Total internal reflection</a> can be seen at the air-water boundary.</figcaption></figure> If there is no angle θ2 fulfilling Snell's law, i.e., <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {n_{1}}{n_{2}}}\sin \theta _{1}>1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>></mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {n_{1}}{n_{2}}}\sin \theta _{1}>1,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb553724840547bf77b3a8038d08530af6ccaab8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:13.967ex; height:5.009ex;" alt="{\displaystyle {\frac {n_{1}}{n_{2}}}\sin \theta _{1}>1,}"> the light cannot be transmitted and will instead undergo <a href="/wiki/Total_internal_reflection" title="Total internal reflection">total internal reflection</a>.<a href="#cite_note-bornwolf-43">[42]</a>: 49–50  This occurs only when going to a less optically dense material, i.e., one with lower refractive index. To get total internal reflection the angles of incidence θ1 must be larger than the critical angle<a href="#cite_note-44">[43]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{\mathrm {c} }=\arcsin \!\left({\frac {n_{2}}{n_{1}}}\right)\!.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>arcsin</mi> <mspace width="negativethinmathspace" /> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mspace width="negativethinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{\mathrm {c} }=\arcsin \!\left({\frac {n_{2}}{n_{1}}}\right)\!.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abdeb75dccb736e1c92f32939b6d122ad0ff7eea" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.466ex; height:6.176ex;" alt="{\displaystyle \theta _{\mathrm {c} }=\arcsin \!\left({\frac {n_{2}}{n_{1}}}\right)\!.}"> <div class="mw-heading mw-heading3"><h3 id="Reflectivity">Reflectivity</h3>[<a href="/w/index.php?title=Refractive_index&action=edit&section=15" title="Edit section: Reflectivity">edit</a>]</div> Apart from the transmitted light there is also a <a href="/wiki/Reflection_(physics)" title="Reflection (physics)">reflected</a> part. The reflection angle is equal to the incidence angle, and the amount of light that is reflected is determined by the <a href="/wiki/Reflectivity" class="mw-redirect" title="Reflectivity">reflectivity</a> of the surface. The reflectivity can be calculated from the refractive index and the incidence angle with the <a href="/wiki/Fresnel_equations" title="Fresnel equations">Fresnel equations</a>, which for <a href="/wiki/Normal_incidence" class="mw-redirect" title="Normal incidence">normal incidence</a> reduces to<a href="#cite_note-bornwolf-43">[42]</a>: 44  <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{0}=\left|{\frac {n_{1}-n_{2}}{n_{1}+n_{2}}}\right|^{2}\!.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msup> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="negativethinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{0}=\left|{\frac {n_{1}-n_{2}}{n_{1}+n_{2}}}\right|^{2}\!.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c4322baf5f0840e267408386dd9c2ee4ff4e480" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:17.099ex; height:6.009ex;" alt="{\displaystyle R_{0}=\left|{\frac {n_{1}-n_{2}}{n_{1}+n_{2}}}\right|^{2}\!.}"> For common glass in air, n1 = 1 and n2 = 1.5, and thus about 4% of the incident power is reflected.<a href="#cite_note-ri-min-45">[44]</a> At other incidence angles the reflectivity will also depend on the <a href="/wiki/Polarization_(waves)" title="Polarization (waves)">polarization</a> of the incoming light. At a certain angle called <a href="/wiki/Brewster%27s_angle" title="Brewster's angle">Brewster's angle</a>, p-polarized light (light with the electric field in the <a href="/wiki/Plane_of_incidence" title="Plane of incidence">plane of incidence</a>) will be totally transmitted. Brewster's angle can be calculated from the two refractive indices of the interface as <a href="#cite_note-Hecht-1">[1]</a>: 245  <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta _{\mathsf {B}}=\arctan \left({\frac {n_{2}}{n_{1}}}\right)~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>θ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">B</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta _{\mathsf {B}}=\arctan \left({\frac {n_{2}}{n_{1}}}\right)~.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3030c2bc6c22a448d591c3348c8edadd681df702" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.917ex; height:6.176ex;" alt="{\displaystyle \theta _{\mathsf {B}}=\arctan \left({\frac {n_{2}}{n_{1}}}\right)~.}"> <div class="mw-heading mw-heading3"><h3 id="Lenses">Lenses</h3>[<a href="/w/index.php?title=Refractive_index&action=edit&section=16" title="Edit section: Lenses">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Lupa.na.encyklopedii.jpg" class="mw-file-description"><img alt="A magnifying glass" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Lupa.na.encyklopedii.jpg/220px-Lupa.na.encyklopedii.jpg" decoding="async" width="220" height="120" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Lupa.na.encyklopedii.jpg/330px-Lupa.na.encyklopedii.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Lupa.na.encyklopedii.jpg/440px-Lupa.na.encyklopedii.jpg 2x" data-file-width="1971" data-file-height="1074" /></a><figcaption>The <a href="/wiki/Optical_power" title="Optical power">power</a> of a <a href="/wiki/Magnifying_glass" title="Magnifying glass">magnifying glass</a> is determined by the shape and refractive index of the lens.</figcaption></figure> The <a href="/wiki/Focal_length" title="Focal length">focal length</a> of a <a href="/wiki/Lens_(optics)" class="mw-redirect" title="Lens (optics)">lens</a> is determined by its refractive index n and the <a href="/wiki/Radius_of_curvature_(optics)" title="Radius of curvature (optics)">radii of curvature</a> R1 and R2 of its surfaces. The power of a <a href="/wiki/Thin_lens" title="Thin lens">thin lens</a> in air is given by the simplified version of the <a href="/wiki/Lensmaker%27s_formula" class="mw-redirect" title="Lensmaker's formula">Lensmaker's formula</a>:<a href="#cite_note-46">[45]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{f}}=(n-1)\left[{\frac {1}{R_{1}}}-{\frac {1}{R_{2}}}\right]\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>f</mi> </mfrac> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{f}}=(n-1)\left[{\frac {1}{R_{1}}}-{\frac {1}{R_{2}}}\right]\ ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f04247634ee3ba3bcbf081cf902a4412900e571" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.026ex; height:6.176ex;" alt="{\displaystyle {\frac {1}{f}}=(n-1)\left[{\frac {1}{R_{1}}}-{\frac {1}{R_{2}}}\right]\ ,}"> where f is the focal length of the lens. <div class="mw-heading mw-heading3"><h3 id="Microscope_resolution">Microscope resolution</h3>[<a href="/w/index.php?title=Refractive_index&action=edit&section=17" title="Edit section: Microscope resolution">edit</a>]</div> The <a href="/wiki/Optical_resolution" title="Optical resolution">resolution</a> of a good optical <a href="/wiki/Microscope" title="Microscope">microscope</a> is mainly determined by the <a href="/wiki/Numerical_aperture" title="Numerical aperture">numerical aperture</a> (ANum) of its <a href="/wiki/Objective_lens" class="mw-redirect" title="Objective lens">objective lens</a>. The numerical aperture in turn is determined by the refractive index n of the medium filling the space between the sample and the lens and the half collection angle of light θ according to Carlsson (2007):<a href="#cite_note-Carlsson-47">[46]</a>: 6  <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{\mathrm {Num} }=n\sin \theta ~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">N</mi> <mi mathvariant="normal">u</mi> <mi mathvariant="normal">m</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>n</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{\mathrm {Num} }=n\sin \theta ~.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/848b6995eb015df58768285b8315e1da88f0367c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.932ex; height:2.509ex;" alt="{\displaystyle A_{\mathrm {Num} }=n\sin \theta ~.}"> For this reason <a href="/wiki/Oil_immersion" title="Oil immersion">oil immersion</a> is commonly used to obtain high resolution in microscopy. In this technique the objective is dipped into a drop of high refractive index immersion oil on the sample under study.<a href="#cite_note-Carlsson-47">[46]</a>: 14  <div class="mw-heading mw-heading3"><h3 id="Relative_permittivity_and_permeability">Relative permittivity and permeability</h3>[<a href="/w/index.php?title=Refractive_index&action=edit&section=18" title="Edit section: Relative permittivity and permeability">edit</a>]</div> The refractive index of electromagnetic radiation equals <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n={\sqrt {\varepsilon _{\mathrm {r} }\mu _{\mathrm {r} }}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n={\sqrt {\varepsilon _{\mathrm {r} }\mu _{\mathrm {r} }}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/906056a0558a19ab45d22545fa4e8fc7a51a3f55" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.315ex; height:3.009ex;" alt="{\displaystyle n={\sqrt {\varepsilon _{\mathrm {r} }\mu _{\mathrm {r} }}},}"> where εr is the material's <a href="/wiki/Relative_permittivity" title="Relative permittivity">relative permittivity</a>, and μr is its <a href="/wiki/Permeability_(electromagnetism)" title="Permeability (electromagnetism)">relative permeability</a>.<a href="#cite_note-bleaney-48">[47]</a>: 229  The refractive index is used for optics in <a href="/wiki/Fresnel_equations" title="Fresnel equations">Fresnel equations</a> and <a href="/wiki/Snell%27s_law" title="Snell's law">Snell's law</a>; while the relative permittivity and permeability are used in <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a> and electronics. Most naturally occurring materials are non-magnetic at optical frequencies, that is μr is very close to 1, therefore n is approximately √εr.<a href="#cite_note-49">[48]</a> In this particular case, the complex relative permittivity εr, with real and imaginary parts εr and ɛ̃r, and the complex refractive index n, with real and imaginary parts n and κ (the latter called the "extinction coefficient"), follow the relation <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {\varepsilon }}_{\mathrm {r} }=\varepsilon _{\mathrm {r} }+i{\tilde {\varepsilon }}_{\mathrm {r} }={\underline {n}}^{2}=(n+i\kappa )^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <munder> <mi>ε</mi> <mo>_</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> <mo>+</mo> <mi>i</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ε</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <munder> <mi>n</mi> <mo>_</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mi>i</mi> <mi>κ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\underline {\varepsilon }}_{\mathrm {r} }=\varepsilon _{\mathrm {r} }+i{\tilde {\varepsilon }}_{\mathrm {r} }={\underline {n}}^{2}=(n+i\kappa )^{2},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8a3fce7d5fc4aea9867b7ad5a750dd821049c1a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.648ex; margin-bottom: -0.69ex; width:31.509ex; height:3.676ex;" alt="{\displaystyle {\underline {\varepsilon }}_{\mathrm {r} }=\varepsilon _{\mathrm {r} }+i{\tilde {\varepsilon }}_{\mathrm {r} }={\underline {n}}^{2}=(n+i\kappa )^{2},}"> and their components are related by:<a href="#cite_note-50">[49]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\varepsilon _{\mathrm {r} }&=n^{2}-\kappa ^{2}\,,\\{\tilde {\varepsilon }}_{\mathrm {r} }&=2n\kappa \,,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ε</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>n</mi> <mi>κ</mi> <mspace width="thinmathspace" /> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\varepsilon _{\mathrm {r} }&=n^{2}-\kappa ^{2}\,,\\{\tilde {\varepsilon }}_{\mathrm {r} }&=2n\kappa \,,\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc4ed059993a6530c039906397668cba50617e23" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.799ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\varepsilon _{\mathrm {r} }&=n^{2}-\kappa ^{2}\,,\\{\tilde {\varepsilon }}_{\mathrm {r} }&=2n\kappa \,,\end{aligned}}}"> and: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}n&={\sqrt {\frac {|{\underline {\varepsilon }}_{\mathrm {r} }|+\varepsilon _{\mathrm {r} }}{2}}},\\\kappa &={\sqrt {\frac {|{\underline {\varepsilon }}_{\mathrm {r} }|-\varepsilon _{\mathrm {r} }}{2}}}.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>n</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <munder> <mi>ε</mi> <mo>_</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </msqrt> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>κ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <munder> <mi>ε</mi> <mo>_</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>−</mo> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> </mrow> <mn>2</mn> </mfrac> </msqrt> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}n&={\sqrt {\frac {|{\underline {\varepsilon }}_{\mathrm {r} }|+\varepsilon _{\mathrm {r} }}{2}}},\\\kappa &={\sqrt {\frac {|{\underline {\varepsilon }}_{\mathrm {r} }|-\varepsilon _{\mathrm {r} }}{2}}}.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/114ff800ea6dec6b7e1d25028e6f20a062870dfd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; width:17.185ex; height:15.509ex;" alt="{\displaystyle {\begin{aligned}n&={\sqrt {\frac {|{\underline {\varepsilon }}_{\mathrm {r} }|+\varepsilon _{\mathrm {r} }}{2}}},\\\kappa &={\sqrt {\frac {|{\underline {\varepsilon }}_{\mathrm {r} }|-\varepsilon _{\mathrm {r} }}{2}}}.\end{aligned}}}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |{\underline {\varepsilon }}_{\mathrm {r} }|={\sqrt {\varepsilon _{\mathrm {r} }^{2}+{\tilde {\varepsilon }}_{\mathrm {r} }^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <munder> <mi>ε</mi> <mo>_</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <msubsup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>+</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>ε</mi> <mo stretchy="false">~</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\underline {\varepsilon }}_{\mathrm {r} }|={\sqrt {\varepsilon _{\mathrm {r} }^{2}+{\tilde {\varepsilon }}_{\mathrm {r} }^{2}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9f3390ec65b72c342503d5932a7b5a2520a17b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:16.144ex; height:4.843ex;" alt="{\displaystyle |{\underline {\varepsilon }}_{\mathrm {r} }|={\sqrt {\varepsilon _{\mathrm {r} }^{2}+{\tilde {\varepsilon }}_{\mathrm {r} }^{2}}}}"> is the <a href="/wiki/Modulus_of_complex_number" class="mw-redirect" title="Modulus of complex number">complex modulus</a>. <div class="mw-heading mw-heading3"><h3 id="Wave_impedance">Wave impedance</h3>[<a href="/w/index.php?title=Refractive_index&action=edit&section=19" title="Edit section: Wave impedance">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Wave_impedance" title="Wave impedance">Wave impedance</a></div> The wave impedance of a plane electromagnetic wave in a non-conductive medium is given by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}Z&={\sqrt {\frac {\mu }{\varepsilon }}}={\sqrt {\frac {\mu _{\mathrm {0} }\mu _{\mathrm {r} }}{\varepsilon _{\mathrm {0} }\varepsilon _{\mathrm {r} }}}}={\sqrt {\frac {\mu _{\mathrm {0} }}{\varepsilon _{\mathrm {0} }}}}{\sqrt {\frac {\mu _{\mathrm {r} }}{\varepsilon _{\mathrm {r} }}}}\\&=Z_{0}{\sqrt {\frac {\mu _{\mathrm {r} }}{\varepsilon _{\mathrm {r} }}}}\\&=Z_{0}{\frac {\mu _{\mathrm {r} }}{n}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>Z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>μ</mi> <mi>ε</mi> </mfrac> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </mrow> </msub> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> </mrow> <mrow> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </mrow> </msub> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </msqrt> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </mrow> </msub> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </mrow> </msub> </mfrac> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> </mfrac> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> <msub> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> </mfrac> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">r</mi> </mrow> </mrow> </msub> <mi>n</mi> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}Z&={\sqrt {\frac {\mu }{\varepsilon }}}={\sqrt {\frac {\mu _{\mathrm {0} }\mu _{\mathrm {r} }}{\varepsilon _{\mathrm {0} }\varepsilon _{\mathrm {r} }}}}={\sqrt {\frac {\mu _{\mathrm {0} }}{\varepsilon _{\mathrm {0} }}}}{\sqrt {\frac {\mu _{\mathrm {r} }}{\varepsilon _{\mathrm {r} }}}}\\&=Z_{0}{\sqrt {\frac {\mu _{\mathrm {r} }}{\varepsilon _{\mathrm {r} }}}}\\&=Z_{0}{\frac {\mu _{\mathrm {r} }}{n}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f7d9649cadaed54f20834ca620fee2a36fa601e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.338ex; width:35.237ex; height:17.843ex;" alt="{\displaystyle {\begin{aligned}Z&={\sqrt {\frac {\mu }{\varepsilon }}}={\sqrt {\frac {\mu _{\mathrm {0} }\mu _{\mathrm {r} }}{\varepsilon _{\mathrm {0} }\varepsilon _{\mathrm {r} }}}}={\sqrt {\frac {\mu _{\mathrm {0} }}{\varepsilon _{\mathrm {0} }}}}{\sqrt {\frac {\mu _{\mathrm {r} }}{\varepsilon _{\mathrm {r} }}}}\\&=Z_{0}{\sqrt {\frac {\mu _{\mathrm {r} }}{\varepsilon _{\mathrm {r} }}}}\\&=Z_{0}{\frac {\mu _{\mathrm {r} }}{n}}\end{aligned}}}"> where Z0 is the vacuum wave impedance, μ and ε are the absolute permeability and permittivity of the medium, εr is the material's <a href="/wiki/Relative_permittivity" title="Relative permittivity">relative permittivity</a>, and μr is its <a href="/wiki/Permeability_(electromagnetism)" title="Permeability (electromagnetism)">relative permeability</a>. In non-magnetic media (that is, in materials with μr = 1), <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z={Z_{0} \over n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>n</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z={Z_{0} \over n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f244614a9304d1fdaba7b14e585fb7fed54b0a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.257ex; height:5.343ex;" alt="{\displaystyle Z={Z_{0} \over n}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n={Z_{0} \over Z}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>Z</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n={Z_{0} \over Z}\,.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dca763b9993a2f006a10f360f4efbfc33af46937" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.005ex; height:5.343ex;" alt="{\displaystyle n={Z_{0} \over Z}\,.}"> Thus refractive index in a non-magnetic media is the ratio of the vacuum wave impedance to the wave impedance of the medium. The reflectivity R0 between two media can thus be expressed both by the wave impedances and the refractive indices as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}R_{0}&=\left|{\frac {n_{1}-n_{2}}{n_{1}+n_{2}}}\right|^{2}\\&=\left|{\frac {Z_{2}-Z_{1}}{Z_{2}+Z_{1}}}\right|^{2}\,.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}R_{0}&=\left|{\frac {n_{1}-n_{2}}{n_{1}+n_{2}}}\right|^{2}\\&=\left|{\frac {Z_{2}-Z_{1}}{Z_{2}+Z_{1}}}\right|^{2}\,.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59469111a75c97933b1ac83bf57e1011a61e99de" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:19.01ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}R_{0}&=\left|{\frac {n_{1}-n_{2}}{n_{1}+n_{2}}}\right|^{2}\\&=\left|{\frac {Z_{2}-Z_{1}}{Z_{2}+Z_{1}}}\right|^{2}\,.\end{aligned}}}"> <div class="mw-heading mw-heading3"><h3 id="Density">Density</h3>[<a href="/w/index.php?title=Refractive_index&action=edit&section=20" title="Edit section: Density">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Density-nd.GIF" class="mw-file-description"><img alt="A scatter plot showing a strong correlation between glass density and refractive index for different glasses" src="//upload.wikimedia.org/wikipedia/en/thumb/3/3b/Density-nd.GIF/370px-Density-nd.GIF" decoding="async" width="370" height="253" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/3/3b/Density-nd.GIF/555px-Density-nd.GIF 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/3/3b/Density-nd.GIF/740px-Density-nd.GIF 2x" data-file-width="911" data-file-height="623" /></a><figcaption>The relation between the refractive index and the density of <a href="/wiki/Silicate_glass" class="mw-redirect" title="Silicate glass">silicate</a> and <a href="/wiki/Borosilicate_glass" title="Borosilicate glass">borosilicate glasses</a><a href="#cite_note-51">[50]</a></figcaption></figure> In general, it is assumed that the refractive index of a glass increases with its <a href="/wiki/Density" title="Density">density</a>. However, there does not exist an overall linear relationship between the refractive index and the density for all silicate and borosilicate glasses. A relatively high refractive index and low density can be obtained with glasses containing light metal oxides such as <a href="/wiki/Lithium_oxide" title="Lithium oxide"><style data-mw-deduplicate="TemplateStyles:r1123817410">'"`UNIQ--templatestyles-000000E6-QINU`"'</style>Li2O</a> and <a href="/wiki/Magnesium_oxide" title="Magnesium oxide"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1123817410">MgO</a>, while the opposite trend is observed with glasses containing <a href="/wiki/Lead(II)_oxide" title="Lead(II) oxide"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1123817410">PbO</a> and <a href="/wiki/Barium_oxide" title="Barium oxide"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1123817410">BaO</a> as seen in the diagram at the right. Many oils (such as <a href="/wiki/Olive_oil" title="Olive oil">olive oil</a>) and <a href="/wiki/Ethanol" title="Ethanol">ethanol</a> are examples of liquids that are more refractive, but less dense, than water, contrary to the general correlation between density and refractive index. For air, n - 1 is proportional to the density of the gas as long as the chemical composition does not change.<a href="#cite_note-52">[51]</a> This means that it is also proportional to the pressure and inversely proportional to the temperature for <a href="/wiki/Ideal_gas_law" title="Ideal gas law">ideal gases</a>. For liquids the same observation can be made as for gases, for instance, the refractive index in alkanes increases nearly perfectly linear with the density. On the other hand, for carboxylic acids, the density decreases with increasing number of C-atoms within the homologeous series. The simple explanation of this finding is that it is not density, but the molar concentration of the chromophore that counts. In homologeous series, this is the excitation of the C-H-bonding. August Beer must have intuitively known that when he gave Hans H. Landolt in 1862 the tip to investigate the refractive index of compounds of homologeous series.<a href="#cite_note-53">[52]</a> While Landolt did not find this relationship, since, at this time dispersion theory was in its infancy, he had the idea of molar refractivity which can even be assigned to single atoms.<a href="#cite_note-54">[53]</a> Based on this concept, the refractive indices of organic materials can be calculated. <div class="mw-heading mw-heading3"><h3 id="Bandgap">Bandgap</h3>[<a href="/w/index.php?title=Refractive_index&action=edit&section=21" title="Edit section: Bandgap">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Annotated_Eg_vs_n.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/Annotated_Eg_vs_n.png/220px-Annotated_Eg_vs_n.png" decoding="async" width="220" height="167" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/95/Annotated_Eg_vs_n.png/330px-Annotated_Eg_vs_n.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/95/Annotated_Eg_vs_n.png/440px-Annotated_Eg_vs_n.png 2x" data-file-width="1721" data-file-height="1310" /></a><figcaption>A scatter plot of bandgap energy versus optical refractive index for many common IV, III-V, and II-VI semiconducting elements / compounds. </figcaption></figure> The optical refractive index of a semiconductor tends to increase as the <a href="/wiki/Band_gap" title="Band gap">bandgap energy</a> decreases. Many attempts<a href="#cite_note-55">[54]</a> have been made to model this relationship beginning with T. S. Moses in 1949<a href="#cite_note-56">[55]</a>. Empirical models can match experimental data over a wide range of materials and yet fail for important cases like InSb, PbS, and Ge.<a href="#cite_note-57">[56]</a> This negative correlation between refractive index and bandgap energy, along with a negative correlation between bandgap and temperature, means that many semiconductors exhibit a positive correlation between refractive index and temperature<a href="#cite_note-58">[57]</a>. This is the opposite of most materials, where the refractive index decreases with temperature as a result of a decreasing material density. <div class="mw-heading mw-heading3"><h3 id="Group_index">Group index</h3>[<a href="/w/index.php?title=Refractive_index&action=edit&section=22" title="Edit section: Group index">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">"Group index" redirects here. Not to be confused with <a href="/wiki/Index_of_a_subgroup" title="Index of a subgroup">Index of a subgroup</a>.</div> Sometimes, a "group velocity refractive index", usually called the group index is defined:[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{\mathrm {g} }={\frac {\mathrm {c} }{v_{\mathrm {g} }}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">g</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">g</mi> </mrow> </mrow> </msub> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{\mathrm {g} }={\frac {\mathrm {c} }{v_{\mathrm {g} }}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7abe045a498c0725f692b2a131d0150206bb6963" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:9.212ex; height:5.343ex;" alt="{\displaystyle n_{\mathrm {g} }={\frac {\mathrm {c} }{v_{\mathrm {g} }}},}"> where vg is the <a href="/wiki/Group_velocity" title="Group velocity">group velocity</a>. This value should not be confused with n, which is always defined with respect to the <a href="/wiki/Phase_velocity" title="Phase velocity">phase velocity</a>. When the <a href="/wiki/Dispersion_(optics)" title="Dispersion (optics)">dispersion</a> is small, the group velocity can be linked to the phase velocity by the relation<a href="#cite_note-bornwolf-43">[42]</a>: 22  <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{\mathrm {g} }=v-\lambda {\frac {\mathrm {d} v}{\mathrm {d} \lambda }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">g</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>v</mi> <mo>−</mo> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>v</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>λ</mi> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{\mathrm {g} }=v-\lambda {\frac {\mathrm {d} v}{\mathrm {d} \lambda }},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac0bd2c98e1254b6279d6267126c499302a6d1cb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.734ex; height:5.509ex;" alt="{\displaystyle v_{\mathrm {g} }=v-\lambda {\frac {\mathrm {d} v}{\mathrm {d} \lambda }},}"> where λ is the wavelength in the medium. In this case the group index can thus be written in terms of the wavelength dependence of the refractive index as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{\mathrm {g} }={\frac {n}{1+{\frac {\lambda }{n}}{\frac {\mathrm {d} n}{\mathrm {d} \lambda }}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">g</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>λ</mi> <mi>n</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>n</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>λ</mi> </mrow> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{\mathrm {g} }={\frac {n}{1+{\frac {\lambda }{n}}{\frac {\mathrm {d} n}{\mathrm {d} \lambda }}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5268a1c7c95fe1ab51c68abafadf9a1fbbb4d532" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:15.592ex; height:6.343ex;" alt="{\displaystyle n_{\mathrm {g} }={\frac {n}{1+{\frac {\lambda }{n}}{\frac {\mathrm {d} n}{\mathrm {d} \lambda }}}}.}"> When the refractive index of a medium is known as a function of the vacuum wavelength (instead of the wavelength in the medium), the corresponding expressions for the group velocity and index are (for all values of dispersion)<a href="#cite_note-59">[58]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}v_{\mathrm {g} }&=\mathrm {c} \!\left(n-\lambda _{0}{\frac {\mathrm {d} n}{\mathrm {d} \lambda _{0}}}\right)^{-1}\!,\\n_{\mathrm {g} }&=n-\lambda _{0}{\frac {\mathrm {d} n}{\mathrm {d} \lambda _{0}}},\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">g</mi> </mrow> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> <mspace width="negativethinmathspace" /> <msup> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mo>−</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>n</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mspace width="negativethinmathspace" /> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">g</mi> </mrow> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>n</mi> <mo>−</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>n</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}v_{\mathrm {g} }&=\mathrm {c} \!\left(n-\lambda _{0}{\frac {\mathrm {d} n}{\mathrm {d} \lambda _{0}}}\right)^{-1}\!,\\n_{\mathrm {g} }&=n-\lambda _{0}{\frac {\mathrm {d} n}{\mathrm {d} \lambda _{0}}},\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce9fb0b15020fe26c7c326a65c4dc795516b2ed7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:24.141ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}v_{\mathrm {g} }&=\mathrm {c} \!\left(n-\lambda _{0}{\frac {\mathrm {d} n}{\mathrm {d} \lambda _{0}}}\right)^{-1}\!,\\n_{\mathrm {g} }&=n-\lambda _{0}{\frac {\mathrm {d} n}{\mathrm {d} \lambda _{0}}},\end{aligned}}}"> where λ0 is the wavelength in vacuum. <div class="mw-heading mw-heading3"><h3 id="Velocity,_momentum,_and_polarizability">Velocity, momentum, and polarizability</h3>[<a href="/w/index.php?title=Refractive_index&action=edit&section=23" title="Edit section: Velocity, momentum, and polarizability">edit</a>]</div> As shown in the <a href="/wiki/Fizeau_experiment" title="Fizeau experiment">Fizeau experiment</a>, when light is transmitted through a moving medium, its speed relative to an observer traveling with speed v in the same direction as the light is: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}V&={\frac {\mathrm {c} }{n}}+{\frac {v\left(1-{\frac {1}{n^{2}}}\right)}{1+{\frac {v}{cn}}}}\\&\approx {\frac {\mathrm {c} }{n}}+v\left(1-{\frac {1}{n^{2}}}\right)\,.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>V</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>v</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>v</mi> <mrow> <mi>c</mi> <mi>n</mi> </mrow> </mfrac> </mrow> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> <mi>n</mi> </mfrac> </mrow> <mo>+</mo> <mi>v</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}V&={\frac {\mathrm {c} }{n}}+{\frac {v\left(1-{\frac {1}{n^{2}}}\right)}{1+{\frac {v}{cn}}}}\\&\approx {\frac {\mathrm {c} }{n}}+v\left(1-{\frac {1}{n^{2}}}\right)\,.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce53f0f8e4f6a1e3e3706f71a1bf7fa52ce70fca" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:24.353ex; height:14.843ex;" alt="{\displaystyle {\begin{aligned}V&={\frac {\mathrm {c} }{n}}+{\frac {v\left(1-{\frac {1}{n^{2}}}\right)}{1+{\frac {v}{cn}}}}\\&\approx {\frac {\mathrm {c} }{n}}+v\left(1-{\frac {1}{n^{2}}}\right)\,.\end{aligned}}}"> The momentum of photons in a medium of refractive index n is a complex and <a href="/wiki/Abraham%E2%80%93Minkowski_controversy" title="Abraham–Minkowski controversy">controversial</a> issue with two different values having different physical interpretations.<a href="#cite_note-60">[59]</a> The refractive index of a substance can be related to its <a href="/wiki/Polarizability" title="Polarizability">polarizability</a> with the <a href="/wiki/Lorentz%E2%80%93Lorenz_equation" class="mw-redirect" title="Lorentz–Lorenz equation">Lorentz–Lorenz equation</a> or to the <a href="/wiki/Molar_refractivity" title="Molar refractivity">molar refractivities</a> of its constituents by the <a href="/wiki/Gladstone%E2%80%93Dale_relation" title="Gladstone–Dale relation">Gladstone–Dale relation</a>. <div class="mw-heading mw-heading3"><h3 id="Refractivity">Refractivity</h3>[<a href="/w/index.php?title=Refractive_index&action=edit&section=24" title="Edit section: Refractivity">edit</a>]</div> In atmospheric applications, refractivity is defined as N = n – 1, often rescaled as either<a href="#cite_note-61">[60]</a> N = 106 (n – 1)<a href="#cite_note-62">[61]</a><a href="#cite_note-63">[62]</a> or N = 108 (n – 1);<a href="#cite_note-64">[63]</a> the multiplication factors are used because the refractive index for air, n deviates from unity by at most a few parts per ten thousand. <a href="/wiki/Molar_refractivity" title="Molar refractivity">Molar refractivity</a>, on the other hand, is a measure of the total <a href="/wiki/Polarizability" title="Polarizability">polarizability</a> of a <a href="/wiki/Mole_(unit)" title="Mole (unit)">mole</a> of a substance and can be calculated from the refractive index as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\frac {M}{\rho }}\cdot {\frac {n^{2}-1}{n^{2}+2}}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>M</mi> <mi>ρ</mi> </mfrac> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\frac {M}{\rho }}\cdot {\frac {n^{2}-1}{n^{2}+2}}\ ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d16492c0bbd02729e3ec9488f21aa2847084c04e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:18.314ex; height:6.176ex;" alt="{\displaystyle A={\frac {M}{\rho }}\cdot {\frac {n^{2}-1}{n^{2}+2}}\ ,}"> where ρ is the <a href="/wiki/Density" title="Density">density</a>, and M is the <a href="/wiki/Molar_mass" title="Molar mass">molar mass</a>.<a href="#cite_note-bornwolf-43">[42]</a>: 93  <div class="mw-heading mw-heading2"><h2 id="Nonscalar,_nonlinear,_or_nonhomogeneous_refraction">Nonscalar, nonlinear, or nonhomogeneous refraction</h2>[<a href="/w/index.php?title=Refractive_index&action=edit&section=25" title="Edit section: Nonscalar, nonlinear, or nonhomogeneous refraction">edit</a>]</div> So far, we have assumed that refraction is given by linear equations involving a spatially constant, scalar refractive index. These assumptions can break down in different ways, to be described in the following subsections. <div class="mw-heading mw-heading3"><h3 id="Birefringence">Birefringence</h3>[<a href="/w/index.php?title=Refractive_index&action=edit&section=26" title="Edit section: Birefringence">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Birefringence" title="Birefringence">Birefringence</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Calcite.jpg" class="mw-file-description"><img alt="A crystal giving a double image of the text behind it" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Calcite.jpg/220px-Calcite.jpg" decoding="async" width="220" height="97" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Calcite.jpg/330px-Calcite.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Calcite.jpg/440px-Calcite.jpg 2x" data-file-width="750" data-file-height="329" /></a><figcaption>A <a href="/wiki/Calcite" title="Calcite">calcite</a> crystal laid upon a paper with some letters showing <a href="/wiki/Double_refraction" class="mw-redirect" title="Double refraction">double refraction</a></figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Plastic_Protractor_Polarized_05375.jpg" class="mw-file-description"><img alt="A transparent plastic protractor with smoothly varying bright colors" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Plastic_Protractor_Polarized_05375.jpg/220px-Plastic_Protractor_Polarized_05375.jpg" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Plastic_Protractor_Polarized_05375.jpg/330px-Plastic_Protractor_Polarized_05375.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Plastic_Protractor_Polarized_05375.jpg/440px-Plastic_Protractor_Polarized_05375.jpg 2x" data-file-width="2048" data-file-height="1536" /></a><figcaption> Birefringent materials can give rise to colors when placed between crossed polarizers. This is the basis for <a href="/wiki/Photoelasticity" title="Photoelasticity">photoelasticity</a>.</figcaption></figure> In some materials, the refractive index depends on the <a href="/wiki/Polarization_(waves)" title="Polarization (waves)">polarization</a> and propagation direction of the light.<a href="#cite_note-65">[64]</a> This is called <a href="/wiki/Birefringence" title="Birefringence">birefringence</a> or optical <a href="/wiki/Anisotropy" title="Anisotropy">anisotropy</a>. In the simplest form, uniaxial birefringence, there is only one special direction in the material. This axis is known as the <a href="/wiki/Optic_axis_of_a_crystal" title="Optic axis of a crystal">optical axis</a> of the material.<a href="#cite_note-Hecht-1">[1]</a>: 230  Light with linear polarization perpendicular to this axis will experience an ordinary refractive index no while light polarized in parallel will experience an extraordinary refractive index ne.<a href="#cite_note-Hecht-1">[1]</a>: 236  The birefringence of the material is the difference between these indices of refraction, Δn = ne − no.<a href="#cite_note-Hecht-1">[1]</a>: 237  Light propagating in the direction of the optical axis will not be affected by the birefringence since the refractive index will be no independent of polarization. For other propagation directions the light will split into two linearly polarized beams. For light traveling perpendicularly to the optical axis the beams will have the same direction.<a href="#cite_note-Hecht-1">[1]</a>: 233  This can be used to change the polarization direction of linearly polarized light or to convert between linear, circular, and elliptical polarizations with <a href="/wiki/Waveplate" title="Waveplate">waveplates</a>.<a href="#cite_note-Hecht-1">[1]</a>: 237  Many <a href="/wiki/Crystal" title="Crystal">crystals</a> are naturally birefringent, but <a href="/wiki/Isotropic" class="mw-redirect" title="Isotropic">isotropic</a> materials such as <a href="/wiki/Plastic" title="Plastic">plastics</a> and <a href="/wiki/Glass" title="Glass">glass</a> can also often be made birefringent by introducing a preferred direction through, e.g., an external force or electric field. This effect is called <a href="/wiki/Photoelasticity" title="Photoelasticity">photoelasticity</a>, and can be used to reveal stresses in structures. The birefringent material is placed between crossed <a href="/wiki/Polarizers" class="mw-redirect" title="Polarizers">polarizers</a>. A change in birefringence alters the polarization and thereby the fraction of light that is transmitted through the second polarizer. In the more general case of trirefringent materials described by the field of <a href="/wiki/Crystal_optics" title="Crystal optics">crystal optics</a>, the dielectric constant is a rank-2 <a href="/wiki/Tensor" title="Tensor">tensor</a> (a 3 by 3 matrix). In this case the propagation of light cannot simply be described by refractive indices except for polarizations along principal axes. <div class="mw-heading mw-heading3"><h3 id="Nonlinearity">Nonlinearity</h3>[<a href="/w/index.php?title=Refractive_index&action=edit&section=27" title="Edit section: Nonlinearity">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Nonlinear_optics" title="Nonlinear optics">Nonlinear optics</a></div> The strong <a href="/wiki/Electric_field" title="Electric field">electric field</a> of high intensity light (such as the output of a <a href="/wiki/Laser" title="Laser">laser</a>) may cause a medium's refractive index to vary as the light passes through it, giving rise to <a href="/wiki/Nonlinear_optics" title="Nonlinear optics">nonlinear optics</a>.<a href="#cite_note-Hecht-1">[1]</a>: 502  If the index varies quadratically with the field (linearly with the intensity), it is called the <a href="/wiki/Kerr_effect" title="Kerr effect">optical Kerr effect</a> and causes phenomena such as <a href="/wiki/Self-focusing" title="Self-focusing">self-focusing</a> and <a href="/wiki/Self-phase_modulation" title="Self-phase modulation">self-phase modulation</a>.<a href="#cite_note-Hecht-1">[1]</a>: 264  If the index varies linearly with the field (a nontrivial linear coefficient is only possible in materials that do not possess <a href="/wiki/Inversion_symmetry" class="mw-redirect" title="Inversion symmetry">inversion symmetry</a>), it is known as the <a href="/wiki/Pockels_effect" title="Pockels effect">Pockels effect</a>.<a href="#cite_note-Hecht-1">[1]</a>: 265  <div class="mw-heading mw-heading3"><h3 id="Inhomogeneity">Inhomogeneity</h3>[<a href="/w/index.php?title=Refractive_index&action=edit&section=28" title="Edit section: Inhomogeneity">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Grin-lens.png" class="mw-file-description"><img alt="Illustration with gradually bending rays of light in a thick slab of glass" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Grin-lens.png/220px-Grin-lens.png" decoding="async" width="220" height="121" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2c/Grin-lens.png/330px-Grin-lens.png 1.5x, //upload.wikimedia.org/wikipedia/commons/2/2c/Grin-lens.png 2x" data-file-width="350" data-file-height="193" /></a><figcaption>A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens.</figcaption></figure> If the refractive index of a medium is not constant but varies gradually with the position, the material is known as a gradient-index (GRIN) medium and is described by <a href="/wiki/Gradient_index_optics" class="mw-redirect" title="Gradient index optics">gradient index optics</a>.<a href="#cite_note-Hecht-1">[1]</a>: 273  Light traveling through such a medium can be bent or focused, and this effect can be exploited to produce <a href="/wiki/Lens_(optics)" class="mw-redirect" title="Lens (optics)">lenses</a>, some <a href="/wiki/Optical_fiber" title="Optical fiber">optical fibers</a>, and other devices. Introducing <abbr title="gradient-index">GRIN</abbr> elements in the design of an optical system can greatly simplify the system, reducing the number of elements by as much as a third while maintaining overall performance.<a href="#cite_note-Hecht-1">[1]</a>: 276  The crystalline lens of the human eye is an example of a <abbr title="gradient-index">GRIN</abbr> lens with a refractive index varying from about 1.406 in the inner core to approximately 1.386 at the less dense cortex.<a href="#cite_note-Hecht-1">[1]</a>: 203  Some common <a href="/wiki/Mirage" title="Mirage">mirages</a> are caused by a spatially varying refractive index of <a href="/wiki/Earth%27s_atmosphere" class="mw-redirect" title="Earth's atmosphere">air</a>. <div class="mw-heading mw-heading2"><h2 id="Refractive_index_measurement">Refractive index measurement</h2>[<a href="/w/index.php?title=Refractive_index&action=edit&section=29" title="Edit section: Refractive index measurement">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Homogeneous_media">Homogeneous media</h3>[<a href="/w/index.php?title=Refractive_index&action=edit&section=30" title="Edit section: Homogeneous media">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="/wiki/Refractometry" title="Refractometry">Refractometry</a> and <a href="/wiki/Refractometer" title="Refractometer">Refractometer</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Pulfrich_refraktometer_en.png" class="mw-file-description"><img alt="Illustration of a refractometer measuring the refraction angle of light passing from a sample into a prism along the interface" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Pulfrich_refraktometer_en.png/220px-Pulfrich_refraktometer_en.png" decoding="async" width="220" height="136" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Pulfrich_refraktometer_en.png/330px-Pulfrich_refraktometer_en.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Pulfrich_refraktometer_en.png/440px-Pulfrich_refraktometer_en.png 2x" data-file-width="860" data-file-height="533" /></a><figcaption>The principle of many refractometers</figcaption></figure> The refractive index of liquids or solids can be measured with <a href="/wiki/Refractometer" title="Refractometer">refractometers</a>. They typically measure some angle of refraction or the critical angle for total internal reflection. The first <a href="/wiki/Abbe_refractometer" title="Abbe refractometer">laboratory refractometers</a> sold commercially were developed by <a href="/wiki/Ernst_Abbe" title="Ernst Abbe">Ernst Abbe</a> in the late 19th century.<a href="#cite_note-66">[65]</a> The same principles are still used today. In this instrument, a thin layer of the liquid to be measured is placed between two prisms. Light is shone through the liquid at incidence angles all the way up to 90°, i.e., light rays <a href="/wiki/Parallel_(geometry)" title="Parallel (geometry)">parallel</a> to the surface. The second prism should have an index of refraction higher than that of the liquid, so that light only enters the prism at angles smaller than the critical angle for total reflection. This angle can then be measured either by looking through a <a href="/wiki/Telescope" title="Telescope">telescope</a>,[<a href="/wiki/Wikipedia:Please_clarify" title="Wikipedia:Please clarify">clarification needed</a>] or with a digital <a href="/wiki/Photodetector" title="Photodetector">photodetector</a> placed in the focal plane of a lens. The refractive index n of the liquid can then be calculated from the maximum transmission angle θ as n = nG sin θ, where nG is the refractive index of the prism.<a href="#cite_note-67">[66]</a> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Refractometer.jpg" class="mw-file-description"><img alt="A small cylindrical refractometer with a surface for the sample at one end and an eye piece to look into at the other end" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Refractometer.jpg/220px-Refractometer.jpg" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Refractometer.jpg/330px-Refractometer.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Refractometer.jpg/440px-Refractometer.jpg 2x" data-file-width="1600" data-file-height="1200" /></a><figcaption>A handheld refractometer used to measure the sugar content of fruits</figcaption></figure> This type of device is commonly used in <a href="/wiki/Chemistry" title="Chemistry">chemical</a> laboratories for identification of <a href="/wiki/Chemical_substance" title="Chemical substance">substances</a> and for <a href="/wiki/Quality_control" title="Quality control">quality control</a>. <a href="/wiki/Digital_handheld_refractometer" title="Digital handheld refractometer">Handheld variants</a> are used in <a href="/wiki/Agriculture" title="Agriculture">agriculture</a> by, e.g., <a href="/wiki/Wine_maker" class="mw-redirect" title="Wine maker">wine makers</a> to determine <a href="/wiki/Brix" title="Brix">sugar content</a> in <a href="/wiki/Grape" title="Grape">grape</a> juice, and <a href="/wiki/Inline_process_refractometer" title="Inline process refractometer">inline process refractometers</a> are used in, e.g., <a href="/wiki/Chemical_industry" title="Chemical industry">chemical</a> and <a href="/wiki/Pharmaceutical_industry" title="Pharmaceutical industry">pharmaceutical industry</a> for <a href="/wiki/Process_control" class="mw-redirect" title="Process control">process control</a>. In <a href="/wiki/Gemology" title="Gemology">gemology</a>, a different type of refractometer is used to measure the index of refraction and birefringence of <a href="/wiki/Gemstones" class="mw-redirect" title="Gemstones">gemstones</a>. The gem is placed on a high refractive index prism and illuminated from below. A high refractive index contact liquid is used to achieve optical contact between the gem and the prism. At small incidence angles most of the light will be transmitted into the gem, but at high angles total internal reflection will occur in the prism. The critical angle is normally measured by looking through a telescope.<a href="#cite_note-68">[67]</a> <div class="mw-heading mw-heading3"><h3 id="Refractive_index_variations">Refractive index variations</h3>[<a href="/w/index.php?title=Refractive_index&action=edit&section=31" title="Edit section: Refractive index variations">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Phase-contrast_imaging" title="Phase-contrast imaging">Phase-contrast imaging</a></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:S_cerevisiae_under_DIC_microscopy.jpg" class="mw-file-description"><img alt="Budding yeast cells with dark borders to the upper left and bright borders to lower right" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/S_cerevisiae_under_DIC_microscopy.jpg/220px-S_cerevisiae_under_DIC_microscopy.jpg" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d9/S_cerevisiae_under_DIC_microscopy.jpg/330px-S_cerevisiae_under_DIC_microscopy.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d9/S_cerevisiae_under_DIC_microscopy.jpg/440px-S_cerevisiae_under_DIC_microscopy.jpg 2x" data-file-width="1560" data-file-height="1560" /></a><figcaption>A <a href="/wiki/Differential_interference_contrast_microscopy" title="Differential interference contrast microscopy">differential interference contrast microscopy</a> image of <a href="/wiki/Budding_yeast" class="mw-redirect" title="Budding yeast">budding yeast</a> cells</figcaption></figure> Unstained biological structures appear mostly transparent under <a href="/wiki/Bright-field_microscopy" title="Bright-field microscopy">bright-field microscopy</a> as most cellular structures do not attenuate appreciable quantities of light. Nevertheless, the variation in the materials that constitute these structures also corresponds to a variation in the refractive index. The following techniques convert such variation into measurable amplitude differences: To measure the spatial variation of the refractive index in a sample <a href="/wiki/Phase-contrast_imaging" title="Phase-contrast imaging">phase-contrast imaging</a> methods are used. These methods measure the variations in <a href="/wiki/Phase_(waves)" title="Phase (waves)">phase</a> of the light wave exiting the sample. The phase is proportional to the <a href="/wiki/Optical_path_length" title="Optical path length">optical path length</a> the light ray has traversed, and thus gives a measure of the <a href="/wiki/Integral" title="Integral">integral</a> of the refractive index along the ray path. The phase cannot be measured directly at optical or higher frequencies, and therefore needs to be converted into <a href="/wiki/Intensity_(physics)" title="Intensity (physics)">intensity</a> by <a href="/wiki/Interference_(optics)" class="mw-redirect" title="Interference (optics)">interference</a> with a reference beam. In the visual spectrum this is done using Zernike <a href="/wiki/Phase-contrast_microscopy" title="Phase-contrast microscopy">phase-contrast microscopy</a>, <a href="/wiki/Differential_interference_contrast_microscopy" title="Differential interference contrast microscopy">differential interference contrast microscopy</a> (DIC), or <a href="/wiki/Interferometry" title="Interferometry">interferometry</a>. Zernike phase-contrast microscopy introduces a phase shift to the low <a href="/wiki/Spatial_frequency" title="Spatial frequency">spatial frequency</a> components of the <a href="/wiki/Real_image" title="Real image">image</a> with a phase-shifting <a href="/wiki/Annulus_(geometry)" class="mw-redirect" title="Annulus (geometry)">annulus</a> in the <a href="/wiki/Fourier_optics" title="Fourier optics">Fourier plane</a> of the sample, so that high-spatial-frequency parts of the image can interfere with the low-frequency reference beam. In <abbr title="differential interference contrast microscopy">DIC</abbr> the illumination is split up into two beams that are given different polarizations, are phase shifted differently, and are shifted transversely with slightly different amounts. After the specimen, the two parts are made to interfere, giving an image of the derivative of the optical path length in the direction of the difference in the transverse shift.<a href="#cite_note-Carlsson-47">[46]</a> In interferometry the illumination is split up into two beams by a <a href="/wiki/Beam_splitter" title="Beam splitter">partially reflective mirror</a>. One of the beams is let through the sample before they are combined to interfere and give a direct image of the phase shifts. If the optical path length variations are more than a wavelength the image will contain fringes. There exist several <a href="/wiki/Phase-contrast_X-ray_imaging" title="Phase-contrast X-ray imaging">phase-contrast X-ray imaging</a> techniques to determine 2D or 3D spatial distribution of refractive index of samples in the X-ray regime.<a href="#cite_note-69">[68]</a> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2>[<a href="/w/index.php?title=Refractive_index&action=edit&section=32" title="Edit section: Applications">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Unreferenced_section plainlinks metadata ambox ambox-content ambox-Unreferenced" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></div></td><td class="mbox-text"><div class="mbox-text-span">This section does not <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">cite</a> any <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">sources</a>. Please help <a href="/wiki/Special:EditPage/Refractive_index" title="Special:EditPage/Refractive index">improve this section</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and <a href="/wiki/Wikipedia:Verifiability#Burden_of_evidence" title="Wikipedia:Verifiability">removed</a>. (September 2014) (<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a>)</div></td></tr></tbody></table> The refractive index is an important property of the components of any <a href="/wiki/Optical_instrument" title="Optical instrument">optical instrument</a>. It determines the focusing power of lenses, the dispersive power of prisms, the reflectivity of <a href="/wiki/Anti-reflective_coating" title="Anti-reflective coating">lens coatings</a>, and the light-guiding nature of <a href="/wiki/Optical_fiber" title="Optical fiber">optical fiber</a>. Since the refractive index is a fundamental physical property of a substance, it is often used to identify a particular substance, confirm its purity, or measure its concentration. The refractive index is used to measure solids, liquids, and gases. Most commonly it is used to measure the concentration of a solute in an <a href="/wiki/Aqueous_solution" title="Aqueous solution">aqueous solution</a>. It can also be used as a useful tool to differentiate between different types of gemstone, due to the unique <a href="/wiki/Chatoyancy" title="Chatoyancy">chatoyance</a> each individual stone displays. A <a href="/wiki/Refractometer" title="Refractometer">refractometer</a> is the instrument used to measure the refractive index. For a solution of sugar, the refractive index can be used to determine the sugar content (see <a href="/wiki/Brix" title="Brix">Brix</a>). <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Refractive_index&action=edit&section=33" title="Edit section: See also">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 18em;"> <ul><li><a href="/wiki/Calculation_of_glass_properties" title="Calculation of glass properties">Calculation of glass properties</a></li> <li><a href="/wiki/Clausius%E2%80%93Mossotti_relation" title="Clausius–Mossotti relation">Clausius–Mossotti relation</a></li> <li><a href="/wiki/Ellipsometry" title="Ellipsometry">Ellipsometry</a></li> <li><a href="/wiki/Fermat%27s_principle" title="Fermat's principle">Fermat's principle</a></li> <li><a href="/wiki/Index_ellipsoid" title="Index ellipsoid">Index ellipsoid</a></li> <li><a href="/wiki/Index-matching_material" title="Index-matching material">Index-matching material</a></li> <li><a href="/wiki/Laser_Schlieren_deflectometry" class="mw-redirect" title="Laser Schlieren deflectometry">Laser Schlieren deflectometry</a></li> <li><a href="/wiki/Optical_properties_of_water_and_ice" title="Optical properties of water and ice">Optical properties of water and ice</a></li> <li><a href="/wiki/Phase-contrast_X-ray_imaging" title="Phase-contrast X-ray imaging">Phase-contrast X-ray imaging</a></li> <li><a href="/wiki/Prism-coupling_refractometry" class="mw-redirect" title="Prism-coupling refractometry">Prism-coupling refractometry</a></li> <li><a href="/wiki/Velocity_factor" title="Velocity factor">Velocity factor</a></li></ul> </div> <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Footnotes">Footnotes</h2>[<a href="/w/index.php?title=Refractive_index&action=edit&section=34" title="Edit section: Footnotes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-21"><a href="#cite_ref-21">^</a> One consequence of the real part of n being less than unity is that it implies that the phase velocity inside the material, <style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style>⁠c/n⁠, is larger than the velocity of light, c. This does not, however, violate the law of relativity, which requires that only signals carrying information do not travel faster than c. Such signals move with the group velocity, not with the phase velocity, and it can be shown that the group velocity is in fact less than c.<a href="#cite_note-Als-Nielsen2011-20">[20]</a> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Refractive_index&action=edit&section=35" title="Edit section: References">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 25em;"> <ol class="references"> <li id="cite_note-Hecht-1">^ <a href="#cite_ref-Hecht_1-0">a</a> <a href="#cite_ref-Hecht_1-1">b</a> <a href="#cite_ref-Hecht_1-2">c</a> <a href="#cite_ref-Hecht_1-3">d</a> <a href="#cite_ref-Hecht_1-4">e</a> <a href="#cite_ref-Hecht_1-5">f</a> <a href="#cite_ref-Hecht_1-6">g</a> <a href="#cite_ref-Hecht_1-7">h</a> <a href="#cite_ref-Hecht_1-8">i</a> <a href="#cite_ref-Hecht_1-9">j</a> <a href="#cite_ref-Hecht_1-10">k</a> <a href="#cite_ref-Hecht_1-11">l</a> <a href="#cite_ref-Hecht_1-12">m</a> <a href="#cite_ref-Hecht_1-13">n</a> <a href="#cite_ref-Hecht_1-14">o</a> <a href="#cite_ref-Hecht_1-15">p</a> <a href="#cite_ref-Hecht_1-16">q</a> <a href="#cite_ref-Hecht_1-17">r</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFHecht,_Eugene2002" class="citation book cs1">Hecht, Eugene (2002). Optics. Addison-Wesley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-321-18878-6" title="Special:BookSources/978-0-321-18878-6"><bdi>978-0-321-18878-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Optics&rft.pub=Addison-Wesley&rft.date=2002&rft.isbn=978-0-321-18878-6&rft.au=Hecht%2C+Eugene&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARefractive+index" class="Z3988"> </li> <li id="cite_note-Attwood-2">^ <a href="#cite_ref-Attwood_2-0">a</a> <a href="#cite_ref-Attwood_2-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAttwood,_David1999" class="citation book cs1">Attwood, David (1999). Soft X-rays and extreme ultraviolet radiation: principles and applications. Cambridge University Press. p. 60. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-02997-1" title="Special:BookSources/978-0-521-02997-1"><bdi>978-0-521-02997-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Soft+X-rays+and+extreme+ultraviolet+radiation%3A+principles+and+applications&rft.pages=60&rft.pub=Cambridge+University+Press&rft.date=1999&rft.isbn=978-0-521-02997-1&rft.au=Attwood%2C+David&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARefractive+index" class="Z3988"> </li> <li id="cite_note-Kinsler-3"><a href="#cite_ref-Kinsler_3-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKinsler2000" class="citation book cs1">Kinsler, Lawrence E. (2000). <a rel="nofollow" class="external text" href="https://archive.org/details/fundamentalsacou00kins_265">Fundamentals of Acoustics</a>. John Wiley. p. <a rel="nofollow" class="external text" href="https://archive.org/details/fundamentalsacou00kins_265/page/n151">136</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-84789-2" title="Special:BookSources/978-0-471-84789-2"><bdi>978-0-471-84789-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+Acoustics&rft.pages=136&rft.pub=John+Wiley&rft.date=2000&rft.isbn=978-0-471-84789-2&rft.aulast=Kinsler&rft.aufirst=Lawrence+E.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ffundamentalsacou00kins_265&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARefractive+index" class="Z3988"> </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFYoung1807" class="citation book cs1">Young, Thomas (1807). <a rel="nofollow" class="external text" href="https://archive.org/details/lecturescourseof01younrich">A course of lectures on natural philosophy and the mechanical arts</a>. J. Johnson. p. <a rel="nofollow" class="external text" href="https://archive.org/details/lecturescourseof01younrich/page/413">413</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=A+course+of+lectures+on+natural+philosophy+and+the+mechanical+arts&rft.pages=413&rft.pub=J.+Johnson&rft.date=1807&rft.aulast=Young&rft.aufirst=Thomas&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Flecturescourseof01younrich&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARefractive+index" class="Z3988"> </li> <li id="cite_note-Newton-5"><a href="#cite_ref-Newton_5-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFNewton1730" class="citation book cs1">Newton, Isaac (1730). <a rel="nofollow" class="external text" href="https://archive.org/details/opticksoratreat00newtgoog">Opticks: Or, A Treatise of the Reflections, Refractions, Inflections and Colours of Light</a>. William Innys at the West-End of St. Paul's. p. <a rel="nofollow" class="external text" href="https://archive.org/details/opticksoratreat00newtgoog/page/n273">247</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Opticks%3A+Or%2C+A+Treatise+of+the+Reflections%2C+Refractions%2C+Inflections+and+Colours+of+Light&rft.pages=247&rft.pub=William+Innys+at+the+West-End+of+St.+Paul%27s&rft.date=1730&rft.aulast=Newton&rft.aufirst=Isaac&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fopticksoratreat00newtgoog&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARefractive+index" class="Z3988"> </li> <li id="cite_note-Hauksbee-6"><a href="#cite_ref-Hauksbee_6-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHauksbee1710" class="citation journal cs1">Hauksbee, Francis (1710). "A Description of the Apparatus for Making Experiments on the Refractions of Fluids". Philosophical Transactions of the Royal Society of London. 27 (325–336): 207. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frstl.1710.0015">10.1098/rstl.1710.0015</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:186208526">186208526</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Transactions+of+the+Royal+Society+of+London&rft.atitle=A+Description+of+the+Apparatus+for+Making+Experiments+on+the+Refractions+of+Fluids&rft.volume=27&rft.issue=%3Cspan+class%3D%22nowrap%22%3E325%E2%80%93%3C%2Fspan%3E336&rft.pages=207&rft.date=1710&rft_id=info%3Adoi%2F10.1098%2Frstl.1710.0015&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A186208526%23id-name%3DS2CID&rft.aulast=Hauksbee&rft.aufirst=Francis&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARefractive+index" class="Z3988"> </li> <li id="cite_note-Hutton-7"><a href="#cite_ref-Hutton_7-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHutton1795" class="citation book cs1">Hutton, Charles (1795). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=lsdJAAAAMAAJ&pg=PA299">Philosophical and mathematical dictionary</a>. p. 299. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170222031446/https://books.google.com/books?id=lsdJAAAAMAAJ&pg=PA299">Archived</a> from the original on 2017-02-22.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Philosophical+and+mathematical+dictionary&rft.pages=299&rft.date=1795&rft.aulast=Hutton&rft.aufirst=Charles&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DlsdJAAAAMAAJ%26pg%3DPA299&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARefractive+index" class="Z3988"> </li> <li id="cite_note-Fraunhofer-8"><a href="#cite_ref-Fraunhofer_8-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvon_Fraunhofer1817" class="citation journal cs1 cs1-prop-foreign-lang-source"><a href="/wiki/Joseph_von_Fraunhofer" title="Joseph von Fraunhofer">von Fraunhofer, Joseph</a> (1817). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=lMRSAAAAcAAJ&pg=PA208">"Bestimmung des Brechungs und Farbenzerstreuungs Vermogens verschiedener Glasarten"</a> [Determination of the Refraction and Color Scattering Power of Different Types of Glass]. Denkschriften der Königlichen Akademie der Wissenschaften zu München [Journal of the Royal Academy of Sciences in Munich] (in German). 5: 208. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170222074213/https://books.google.com/books?id=lMRSAAAAcAAJ&pg=PA208">Archived</a> from the original on 2017-02-22.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Denkschriften+der+K%C3%B6niglichen+Akademie+der+Wissenschaften+zu+M%C3%BCnchen&rft.atitle=Bestimmung+des+Brechungs+und+Farbenzerstreuungs+Vermogens+verschiedener+Glasarten&rft.volume=5&rft.pages=208&rft.date=1817&rft.aulast=von+Fraunhofer&rft.aufirst=Joseph&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DlMRSAAAAcAAJ%26pg%3DPA208&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARefractive+index" class="Z3988"> Exponent des Brechungsverhältnisses is index of refraction </li> <li id="cite_note-Brewster-9"><a href="#cite_ref-Brewster_9-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrewster1815" class="citation journal cs1"><a href="/wiki/David_Brewster" title="David Brewster">Brewster, David</a> (1815). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=GhpRAAAAYAAJ&pg=PA124">"On the structure of doubly refracting crystals"</a>. Philosophical Magazine. 45 (202): 126. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F14786441508638398">10.1080/14786441508638398</a>. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170222103726/https://books.google.com/books?id=GhpRAAAAYAAJ&pg=PA124">Archived</a> from the original on 2017-02-22.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Magazine&rft.atitle=On+the+structure+of+doubly+refracting+crystals&rft.volume=45&rft.issue=202&rft.pages=126&rft.date=1815&rft_id=info%3Adoi%2F10.1080%2F14786441508638398&rft.aulast=Brewster&rft.aufirst=David&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DGhpRAAAAYAAJ%26pg%3DPA124&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARefractive+index" class="Z3988"> </li> <li id="cite_note-Herschel-10"><a href="#cite_ref-Herschel_10-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHerschel1828" class="citation book cs1"><a href="/wiki/John_Herschel" title="John Herschel">Herschel, John F. W.</a> (1828). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Lo4_AAAAcAAJ">On the Theory of Light</a>. p. 368. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20151124000212/https://books.google.com/books?id=Lo4_AAAAcAAJ&printsec=frontcover">Archived</a> from the original on 2015-11-24.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=On+the+Theory+of+Light&rft.pages=368&rft.date=1828&rft.aulast=Herschel&rft.aufirst=John+F.+W.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DLo4_AAAAcAAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3ARefractive+index" class="Z3988"> </li> <li id="cite_note-11"><a href="#cite_ref-11">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMalitson1965" class="citation web cs1">Malitson (1965). <a rel="nofollow" class="external text" href="https://refractiveindex.info/?shelf=glass&book=fused_silica&page=Malitson">"Refractive Index Database"</a>. refractiveindex.info. 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