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<span>Lyddane–Sachs–Teller relation</span> </div> </a> <ul id="toc-Lyddane–Sachs–Teller_relation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Surface_phonon_polariton" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Surface_phonon_polariton"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Surface phonon polariton</span> </div> </a> <ul id="toc-Surface_phonon_polariton-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Experimental_observation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Experimental_observation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Experimental observation</span> </div> </a> <button aria-controls="toc-Experimental_observation-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span 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/></a></span></td></tr><tr><td class="sidebar-content"> <div class="hlist"><ul><li><a href="/wiki/Phase_(matter)" title="Phase (matter)">Phases</a></li><li><a href="/wiki/Phase_transition" title="Phase transition">Phase transition</a></li><li><a href="/wiki/Quantum_critical_point" title="Quantum critical point">QCP</a></li></ul></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/State_of_matter" title="State of matter">States of matter</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"><ul><li><a href="/wiki/Solid" title="Solid">Solid</a></li><li><a href="/wiki/Liquid" title="Liquid">Liquid</a></li><li><a href="/wiki/Gas" title="Gas">Gas</a></li><li><a href="/wiki/Plasma_(physics)" title="Plasma (physics)">Plasma</a></li><li><a href="/wiki/Bose%E2%80%93Einstein_condensate" title="Bose–Einstein condensate">Bose–Einstein condensate</a></li><li><a href="/wiki/Bose_gas" title="Bose gas">Bose gas</a></li><li><a href="/wiki/Fermionic_condensate" title="Fermionic condensate">Fermionic condensate</a></li><li><a href="/wiki/Fermi_gas" title="Fermi gas">Fermi gas</a></li><li><a href="/wiki/Fermi_liquid_theory" title="Fermi liquid theory">Fermi liquid</a></li><li><a href="/wiki/Supersolid" title="Supersolid">Supersolid</a></li><li><a href="/wiki/Superfluidity" title="Superfluidity">Superfluidity</a></li><li><a href="/wiki/Luttinger_liquid" title="Luttinger liquid">Luttinger liquid</a></li><li><a href="/wiki/Time_crystal" title="Time crystal">Time crystal</a></li></ul></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Phase phenomena</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"><ul><li><a href="/wiki/Order_parameter" class="mw-redirect" title="Order parameter">Order parameter</a></li><li><a href="/wiki/Phase_transition" title="Phase transition">Phase transition</a></li><li><a href="/wiki/Quantum_critical_point" title="Quantum critical point">QCP</a></li></ul></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Electronic phases</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"><ul><li><a href="/wiki/Electronic_band_structure" title="Electronic band structure">Electronic band structure</a></li><li><a href="/wiki/Plasma_(physics)" title="Plasma (physics)">Plasma</a></li><li><a href="/wiki/Insulator_(electricity)" title="Insulator (electricity)">Insulator</a></li><li><a href="/wiki/Mott_insulator" title="Mott insulator">Mott insulator</a></li><li><a href="/wiki/Semiconductor" title="Semiconductor">Semiconductor</a></li><li><a href="/wiki/Semimetal" title="Semimetal">Semimetal</a></li><li><a href="/wiki/Electrical_conductor" title="Electrical conductor">Conductor</a></li><li><a href="/wiki/Superconductivity" title="Superconductivity">Superconductor</a></li><li><a href="/wiki/Thermoelectric_effect" title="Thermoelectric effect">Thermoelectric</a></li><li><a href="/wiki/Piezoelectricity" title="Piezoelectricity">Piezoelectric</a></li><li><a href="/wiki/Ferroelectricity" title="Ferroelectricity">Ferroelectric</a></li><li><a href="/wiki/Topological_insulator" title="Topological insulator">Topological insulator</a></li><li><a href="/wiki/Spin_gapless_semiconductor" title="Spin gapless semiconductor">Spin gapless semiconductor</a></li></ul></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Electronic phenomena</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"><ul><li><a href="/wiki/Quantum_Hall_effect" title="Quantum Hall effect">Quantum Hall effect</a></li><li><a href="/wiki/Spin_Hall_effect" title="Spin Hall effect">Spin Hall effect</a></li><li><a href="/wiki/Kondo_effect" title="Kondo effect">Kondo effect</a></li></ul></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Magnetic phases</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"><ul><li><a href="/wiki/Diamagnetism" title="Diamagnetism">Diamagnet</a></li><li><a href="/wiki/Superdiamagnetism" title="Superdiamagnetism">Superdiamagnet</a><br /> <a href="/wiki/Paramagnetism" title="Paramagnetism">Paramagnet</a></li><li><a href="/wiki/Superparamagnetism" title="Superparamagnetism">Superparamagnet</a><br /><a href="/wiki/Ferromagnetism" title="Ferromagnetism">Ferromagnet</a></li><li><a href="/wiki/Antiferromagnetism" title="Antiferromagnetism">Antiferromagnet</a><br /><a href="/wiki/Metamagnetism" title="Metamagnetism">Metamagnet</a></li><li><a href="/wiki/Spin_glass" title="Spin glass">Spin glass</a></li></ul></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Quasiparticle" title="Quasiparticle">Quasiparticles</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"><ul><li><a href="/wiki/Phonon" title="Phonon">Phonon</a></li><li><a href="/wiki/Exciton" title="Exciton">Exciton</a></li><li><a href="/wiki/Plasmon" title="Plasmon">Plasmon</a><br /><a href="/wiki/Polariton" title="Polariton">Polariton</a></li><li><a href="/wiki/Polaron" title="Polaron">Polaron</a></li><li><a href="/wiki/Magnon" title="Magnon">Magnon</a></li><li><a href="/wiki/Roton" title="Roton">Roton</a></li></ul></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Soft_matter" title="Soft matter">Soft matter</a></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"><ul><li><a href="/wiki/Amorphous_solid" title="Amorphous solid">Amorphous solid</a></li><li><a href="/wiki/Colloid" title="Colloid">Colloid</a></li><li><a href="/wiki/Granular_material" title="Granular material">Granular material</a></li><li><a href="/wiki/Liquid_crystal" title="Liquid crystal">Liquid crystal</a></li><li><a href="/wiki/Polymer" title="Polymer">Polymer</a></li></ul></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"><ul><li><a href="/wiki/Johannes_Diderik_van_der_Waals" title="Johannes Diderik van der Waals">Van der Waals</a></li><li><a href="/wiki/Heike_Kamerlingh_Onnes" title="Heike Kamerlingh Onnes">Onnes</a></li><li><a href="/wiki/Max_von_Laue" title="Max von Laue">von Laue</a></li><li><a href="/wiki/William_Henry_Bragg" title="William Henry Bragg">Bragg</a></li><li><a href="/wiki/Peter_Debye" title="Peter Debye">Debye</a></li><li><a href="/wiki/Felix_Bloch" title="Felix Bloch">Bloch</a></li><li><a href="/wiki/Lars_Onsager" title="Lars Onsager">Onsager</a></li><li><a href="/wiki/Neville_Mott" class="mw-redirect" title="Neville Mott">Mott</a></li><li><a href="/wiki/Rudolf_Peierls" title="Rudolf Peierls">Peierls</a></li><li><a href="/wiki/Lev_Landau" title="Lev Landau">Landau</a></li><li><a href="/wiki/Luttinger" class="mw-redirect" title="Luttinger">Luttinger</a></li><li><a href="/wiki/Philip_Warren_Anderson" class="mw-redirect" title="Philip Warren Anderson">Anderson</a></li><li><a href="/wiki/John_Hasbrouck_Van_Vleck" title="John Hasbrouck Van Vleck">Van Vleck</a></li><li><a href="/wiki/John_Hubbard_(physicist)" title="John Hubbard (physicist)">Hubbard</a></li><li><a href="/wiki/William_Bradford_Shockley" class="mw-redirect" title="William Bradford Shockley">Shockley</a></li><li><a href="/wiki/John_Bardeen" title="John Bardeen">Bardeen</a></li><li><a href="/wiki/Leon_Cooper" title="Leon Cooper">Cooper</a></li><li><a href="/wiki/John_Robert_Schrieffer" title="John Robert Schrieffer">Schrieffer</a></li><li><a href="/wiki/Brian_Josephson" title="Brian Josephson">Josephson</a></li><li><a href="/wiki/Louis_N%C3%A9el" title="Louis Néel">Louis Néel</a></li><li><a href="/wiki/Leo_Esaki" title="Leo Esaki">Esaki</a></li><li><a href="/wiki/Ivar_Giaever" title="Ivar Giaever">Giaever</a></li><li><a href="/wiki/Walter_Kohn" title="Walter Kohn">Kohn</a></li><li><a href="/wiki/Leo_Kadanoff" title="Leo Kadanoff">Kadanoff</a></li><li><a href="/wiki/Michael_Fisher" title="Michael Fisher">Fisher</a></li><li><a href="/wiki/Kenneth_G._Wilson" title="Kenneth G. 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ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Condensed_matter_physics" title="Template:Condensed matter physics"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Condensed_matter_physics" title="Template talk:Condensed matter physics"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Condensed_matter_physics" title="Special:EditPage/Template:Condensed matter physics"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p>In <a href="/wiki/Condensed_matter_physics" title="Condensed matter physics">condensed matter physics</a>, a <b>phonon polariton</b> is a type of <a href="/wiki/Quasiparticle" title="Quasiparticle">quasiparticle</a> that can form in a diatomic ionic <a href="/wiki/Crystal" title="Crystal">crystal</a> due to coupling of <a href="/wiki/Phonon" title="Phonon">transverse optical phonons</a> and <a href="/wiki/Photon" title="Photon">photons</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> They are particular type of <a href="/wiki/Polariton" title="Polariton">polariton</a>, which behave like <a href="/wiki/Boson" title="Boson">bosons</a>. Phonon polaritons occur in the region where the wavelength and energy of phonons and photons are similar, as to adhere to the <a href="/wiki/Avoided_crossing" title="Avoided crossing">avoided crossing principle</a>. </p><p>Phonon polariton spectra have traditionally been studied using <a href="/wiki/Raman_spectroscopy" title="Raman spectroscopy">Raman spectroscopy</a>.<sup id="cite_ref-:0_2-0" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The recent advances in (scattering-type) <a href="/wiki/Near-field_scanning_optical_microscope" title="Near-field scanning optical microscope">scanning near-field optical microscopy</a>((s-)SNOM) and <a href="/wiki/Atomic_force_microscopy" title="Atomic force microscopy">atomic force microscopy</a>(AFM) have made it possible to observe the polaritons in a more direct way.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Theory">Theory</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phonon_polariton&action=edit&section=1" title="Edit section: Theory"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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"><span typeof="mw:File"><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></span></div></td><td class="mbox-text"><div class="mbox-text-span">This section <b>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></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Phonon_polariton" title="Special:EditPage/Phonon polariton">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>.</span> <span class="date-container"><i>(<span class="date">April 2022</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>A phonon polariton is a type of quasiparticle that can form in some crystals due to the coupling of photons and lattice vibrations. They have properties of both light and sound waves, and can travel at very slow speeds in the material. They are useful for manipulating electromagnetic fields at nanoscale and enhancing optical phenomena.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Phonon polaritons only result from coupling of transverse optical phonons, this is due to the particular form of the <a href="/wiki/Dispersion_relation" title="Dispersion relation">dispersion relation</a> of the phonon and photon and their interaction. Photons consist of electromagnetic waves, which are always transverse. Therefore, they can only couple with transverse phonons in crystals. </p><p>Near <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {k} =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {k} =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcf61590d049476b690b9b29d4a0017aa61cdad1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.672ex; height:2.176ex;" alt="{\displaystyle \mathbf {k} =0}"></span> the dispersion relation of an acoustic phonon can be approximated as being linear, with a particular gradient giving a dispersion relation of the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{\rm {ac}}=v_{\rm {ac}}k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{\rm {ac}}=v_{\rm {ac}}k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/906233c0e63be0ad1dd8f49779324a78b34d27d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.452ex; height:2.509ex;" alt="{\displaystyle \omega _{\rm {ac}}=v_{\rm {ac}}k}"></span>, with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{\rm {ac}}}"> <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">a</mi> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{\rm {ac}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86b89c2087265259c0e2affd51a241202286e1bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.912ex; height:2.009ex;" alt="{\displaystyle v_{\rm {ac}}}"></span> the speed of the wave, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{\rm {ac}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">a</mi> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{\rm {ac}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/beece08c3dc345b7f4dbadc5cc3a4d17187e6d9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.23ex; height:2.009ex;" alt="{\displaystyle \omega _{\rm {ac}}}"></span> the <a href="/wiki/Angular_frequency" title="Angular frequency">angular frequency</a> and <i>k</i> the absolute value of the wave vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {k} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {k} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ea699cbc1f843f2e855577d57529430ec33a1ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:2.176ex;" alt="{\displaystyle \mathbf {k} }"></span>. The dispersion relation of photons is also linear, being also of the form <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{\rm {p}}=ck}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">p</mi> </mrow> </mrow> </msub> <mo>=</mo> <mi>c</mi> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{\rm {p}}=ck}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b67db057a5787196824a84b3edfc595c4940ac24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.909ex; height:2.843ex;" alt="{\displaystyle \omega _{\rm {p}}=ck}"></span>, with <i>c</i> being the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light in vacuum</a>. The difference lies in the magnitudes of their speeds, the speed of photons is many times larger than the speed for the acoustic phonons. The dispersion relations will therefore never cross each other, resulting in a lack of coupling. The dispersion relations touch at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {k} =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {k} =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcf61590d049476b690b9b29d4a0017aa61cdad1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.672ex; height:2.176ex;" alt="{\displaystyle \mathbf {k} =0}"></span>, but since the waves have no energy, no coupling will occur. </p><p>Optical phonons, by contrast, have a non-zero angular frequency at <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {k} =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {k} =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcf61590d049476b690b9b29d4a0017aa61cdad1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.672ex; height:2.176ex;" alt="{\displaystyle \mathbf {k} =0}"></span> and have a negative slope, which is also much smaller in magnitude to that of photons. This will result in the crossing of the optical phonon branch and the photon dispersion, leading to their coupling and the forming of a phonon polariton. </p> <div class="mw-heading mw-heading3"><h3 id="Dispersion_relation">Dispersion relation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phonon_polariton&action=edit&section=2" title="Edit section: Dispersion relation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The behavior of the phonon polaritons can be described by the dispersion relation. This dispersion relation is most easily derived for diatomic ion crystals with optical isotropy, for example <a href="/wiki/Sodium_chloride" title="Sodium chloride">sodium chloride</a> and <a href="/wiki/Zinc_sulfide" title="Zinc sulfide">zinc sulfide</a>. Since the atoms in the crystal are charged, any lattice vibration which changes the relative distance between the two atoms in the unit cell will change the <a href="/wiki/Dielectric_polarization" class="mw-redirect" title="Dielectric polarization">dielectric polarization</a> of the material. To describe these vibrations, it is useful to introduce the parameter w, which is given by: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {w} =\mathbf {q} {\sqrt {\frac {\mu }{V}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>μ</mi> <mi>V</mi> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {w} =\mathbf {q} {\sqrt {\frac {\mu }{V}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6cb36b343c23add5415054373d2df721ab68458" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.392ex; height:6.176ex;" alt="{\displaystyle \mathbf {w} =\mathbf {q} {\sqrt {\frac {\mu }{V}}}}"></span></dd></dl> <p>Where </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {q} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">q</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {q} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7be005a326b7ac3fe4c24bca391369f44c4c2876" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.416ex; height:2.009ex;" alt="{\displaystyle \mathbf {q} }"></span> is the displacement of the positive atom relative to the negative atom;</li></ul> <ul><li><i>μ</i> is the <a href="/wiki/Reduced_mass" title="Reduced mass">reduced mass</a> of the two atoms;</li> <li><i>V</i> is the volume of the unit cell.</li></ul> <p>Using this parameter, the behavior of the lattice vibrations for long waves can be described by the following equations:<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ddot {\mathbf {w} }}=-{\omega _{0}}^{2}\mathbf {w} +({\frac {\epsilon _{0}-\epsilon _{\infty }}{4\pi }})^{1/2}\omega _{0}\mathbf {E} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo>¨</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mrow> <mrow> <mn>4</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ddot {\mathbf {w} }}=-{\omega _{0}}^{2}\mathbf {w} +({\frac {\epsilon _{0}-\epsilon _{\infty }}{4\pi }})^{1/2}\omega _{0}\mathbf {E} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71a628956b75c12fb42fa474d3ec4c23c3fe15e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:32.423ex; height:5.009ex;" alt="{\displaystyle {\ddot {\mathbf {w} }}=-{\omega _{0}}^{2}\mathbf {w} +({\frac {\epsilon _{0}-\epsilon _{\infty }}{4\pi }})^{1/2}\omega _{0}\mathbf {E} }"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} =({\frac {\epsilon _{0}-\epsilon _{\infty }}{4\pi }})^{1/2}\omega _{0}\mathbf {w} +({\frac {\epsilon _{\infty }-1}{4\pi }})\mathbf {E} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mrow> <mrow> <mn>4</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mn>4</mn> <mi>π</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {P} =({\frac {\epsilon _{0}-\epsilon _{\infty }}{4\pi }})^{1/2}\omega _{0}\mathbf {w} +({\frac {\epsilon _{\infty }-1}{4\pi }})\mathbf {E} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f83b86ba69f650a657151ab54a3bd1379441afd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:36.424ex; height:5.176ex;" alt="{\displaystyle \mathbf {P} =({\frac {\epsilon _{0}-\epsilon _{\infty }}{4\pi }})^{1/2}\omega _{0}\mathbf {w} +({\frac {\epsilon _{\infty }-1}{4\pi }})\mathbf {E} }"></span></dd></dl> <p>Where </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\ddot {\mathbf {w} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo>¨</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\ddot {\mathbf {w} }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b48c7b73fdd8574642708cea1c61d62d211ce9bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:2.176ex;" alt="{\displaystyle {\ddot {\mathbf {w} }}}"></span> denotes the double time derivative of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {w} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {w} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20795664b5b048744a2fd88977851104cc5816f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:1.676ex;" alt="{\displaystyle \mathbf {w} }"></span></li></ul> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2cae6289b0fe626d1f9472a3416ac73e87bc5a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.998ex; height:2.009ex;" alt="{\displaystyle \epsilon _{0}}"></span> is the static <a href="/wiki/Dielectric_constant" class="mw-redirect" title="Dielectric constant">dielectric constant</a></li></ul> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon _{\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon _{\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b091179f7a4f9e553fe95f7de39c9a9be0c0200" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.82ex; height:2.009ex;" alt="{\displaystyle \epsilon _{\infty }}"></span> is the high-frequency dielectric constant</li></ul> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a713d16c489051d4f515e12b1f86061c6be799b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \omega _{0}}"></span> is the infrared dispersion frequency</li></ul> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d7f22b39d51f780fc02859059c1757c606b9de2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.757ex; height:2.176ex;" alt="{\displaystyle \mathbf {E} }"></span> is the <a href="/wiki/Electric_field" title="Electric field">electric field</a></li></ul> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {P} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0c250ef2a112c86b93c637dfa288c6d7f34ac3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle \mathbf {P} }"></span> is the dielectric polarization.</li></ul> <p>For the full coupling between the phonon and the photon, we need the four <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a> in matter. Since, macroscopically, the crystal is uncharged and there is no current, the equations can be simplified. A phonon polariton must abide all of these six equations. To find solutions to this set of equations, we write the following trial <a href="/wiki/Plane_wave" title="Plane wave">plane wave</a> solutions for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {w} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {w} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20795664b5b048744a2fd88977851104cc5816f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.931ex; height:1.676ex;" alt="{\displaystyle \mathbf {w} }"></span> , <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d7f22b39d51f780fc02859059c1757c606b9de2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.757ex; height:2.176ex;" alt="{\displaystyle \mathbf {E} }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {P} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0c250ef2a112c86b93c637dfa288c6d7f34ac3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle \mathbf {P} }"></span>: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {w} =\mathbf {w_{0}} e^{i(\mathbf {k} \cdot \mathbf {x} -\omega t)}+{\text{c.c.}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </msub> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>−</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>c.c.</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {w} =\mathbf {w_{0}} e^{i(\mathbf {k} \cdot \mathbf {x} -\omega t)}+{\text{c.c.}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55000c2428daa17defbb4e04b2e92ff73f79e47f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.847ex; height:3.176ex;" alt="{\displaystyle \mathbf {w} =\mathbf {w_{0}} e^{i(\mathbf {k} \cdot \mathbf {x} -\omega t)}+{\text{c.c.}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {P} =\mathbf {P_{0}} e^{i(\mathbf {k} \cdot \mathbf {x} -\omega t)}+{\text{c.c.}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">P</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">P</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </msub> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>−</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>c.c.</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {P} =\mathbf {P_{0}} e^{i(\mathbf {k} \cdot \mathbf {x} -\omega t)}+{\text{c.c.}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a17ce844be8d5feef11d0335e09d32000e73a661" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.638ex; height:3.176ex;" alt="{\displaystyle \mathbf {P} =\mathbf {P_{0}} e^{i(\mathbf {k} \cdot \mathbf {x} -\omega t)}+{\text{c.c.}}}"></span></dd> <dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {E} =\mathbf {E_{0}} e^{i(\mathbf {k} \cdot \mathbf {x} -\omega t)}+{\text{c.c.}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">E</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi mathvariant="bold">E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </msub> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>−</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>c.c.</mtext> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {E} =\mathbf {E_{0}} e^{i(\mathbf {k} \cdot \mathbf {x} -\omega t)}+{\text{c.c.}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb3bf9f9ff300829ba98e43b394a5c536f8dd52d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.499ex; height:3.176ex;" alt="{\displaystyle \mathbf {E} =\mathbf {E_{0}} e^{i(\mathbf {k} \cdot \mathbf {x} -\omega t)}+{\text{c.c.}}}"></span></dd></dl> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Phonon_polaritons.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Phonon_polaritons.svg/239px-Phonon_polaritons.svg.png" decoding="async" width="239" height="247" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Phonon_polaritons.svg/359px-Phonon_polaritons.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e7/Phonon_polaritons.svg/479px-Phonon_polaritons.svg.png 2x" data-file-width="436" data-file-height="450" /></a><figcaption>Dispersion relation of phonon polaritons in <a href="/wiki/Gallium_phosphide" title="Gallium phosphide">GaP</a>. Red curves are the uncoupled phonon and photon dispersion relations, black curves are the result of coupling (from top to bottom: upper polariton, LO phonon, lower polariton).</figcaption></figure> <p>Where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {k} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">k</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {k} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ea699cbc1f843f2e855577d57529430ec33a1ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:2.176ex;" alt="{\displaystyle \mathbf {k} }"></span> denotes the wave vector of the plane wave, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }"></span> the position, <i>t</i> the time, and <i>ω</i> the angular frequency. Notice that wave vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ccd4b98d198d6538010ae815ee1199baabd3493" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.843ex;" alt="{\displaystyle {\vec {k}}}"></span> should be perpendicular to the electric field and the <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</a>. Solving the resulting equations for ω and k, the magnitude of the wave vector, yields the following dispersion relation, and furthermore an expression for the optical dielectric constant:<sup id="cite_ref-:1_6-0" class="reference"><a href="#cite_note-:1-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {k^{2}c^{2}}{\omega ^{2}}}=\epsilon _{\infty }+{\frac {\epsilon _{0}-\epsilon _{\infty }}{{\omega _{0}}^{2}-\omega ^{2}}}{\omega _{0}}^{2}=\epsilon (\omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mrow> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>ϵ</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {k^{2}c^{2}}{\omega ^{2}}}=\epsilon _{\infty }+{\frac {\epsilon _{0}-\epsilon _{\infty }}{{\omega _{0}}^{2}-\omega ^{2}}}{\omega _{0}}^{2}=\epsilon (\omega )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/603edda8eaf1ff1d263d547e378ad9f3a8323b8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:34.504ex; height:6.343ex;" alt="{\displaystyle {\frac {k^{2}c^{2}}{\omega ^{2}}}=\epsilon _{\infty }+{\frac {\epsilon _{0}-\epsilon _{\infty }}{{\omega _{0}}^{2}-\omega ^{2}}}{\omega _{0}}^{2}=\epsilon (\omega )}"></span></dd></dl> <p>With <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon (\omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon (\omega )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/625118be3f4f705564abea06ef1a4b5e3d6958ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.199ex; height:2.843ex;" alt="{\displaystyle \epsilon (\omega )}"></span> the optical dielectric constant. </p><p>The solution of this dispersion relation has two branches, an upper branch and a lower branch (see also the figure). If the slope of the curve is low, the particle is said to behave "phononlike", and if the slope is high the particle behaves "photonlike", owing these names to the slopes of the regular dispersion curves for phonons and photons.<sup id="cite_ref-:2_7-0" class="reference"><a href="#cite_note-:2-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> The phonon polariton behaves phononlike for low <i>k</i> in the upper branch, and for high <i>k</i> in the lower branch. Conversely, the polariton behaves photonlike for high <i>k</i> in the upper branch, low <i>k</i> in the lower branch. </p> <div class="mw-heading mw-heading3"><h3 id="Limit_behaviour_of_the_dispersion_relation">Limit behaviour of the dispersion relation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phonon_polariton&action=edit&section=3" title="Edit section: Limit behaviour of the dispersion relation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The dispersion relation describes the behaviour of the coupling. The coupling of the phonon and the photon is the most promininent in the region where the original transverse disperion relations would have crossed. In the limit of large <i>k</i>, the solid lines of both branches approach the dotted lines, meaning, the coupling does not have a large impact on the behaviour of the vibrations. </p><p>Towards the right of the crossing point, the upper branch behaves like a photon. The physical interpretation of this effect is that the frequency becomes too high for the ions to partake in the vibration, causing them to be essentially static. This results in a dispersion relation resembling one of a regular photon in a crystal. The lower branch in this region behaves, because of their low <a href="/wiki/Phase_velocity" title="Phase velocity">phase velocity</a> compared to the photons, as regular transverse lattice vibrations.<sup id="cite_ref-:1_6-1" class="reference"><a href="#cite_note-:1-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Lyddane–Sachs–Teller_relation"><span id="Lyddane.E2.80.93Sachs.E2.80.93Teller_relation"></span>Lyddane–Sachs–Teller relation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phonon_polariton&action=edit&section=4" title="Edit section: Lyddane–Sachs–Teller relation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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">Main article: <a href="/wiki/Lyddane%E2%80%93Sachs%E2%80%93Teller_relation" title="Lyddane–Sachs–Teller relation">Lyddane–Sachs–Teller relation</a></div> <p>The longitudonal optical phonon frequency <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{L}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{L}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7284e7df80758a0946c74413f95c81c1750c755c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.009ex;" alt="{\displaystyle \omega _{L}}"></span> is defined by the zero of the equation for the dielectric constant.<sup id="cite_ref-:2_7-1" class="reference"><a href="#cite_note-:2-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> Writing the equation for the dielectric constant in a different way yields: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon (\omega )={\frac {{\omega _{0}}^{2}\epsilon _{0}-\omega ^{2}\epsilon _{\infty }}{{\omega _{0}}^{2}-\omega ^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mrow> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon (\omega )={\frac {{\omega _{0}}^{2}\epsilon _{0}-\omega ^{2}\epsilon _{\infty }}{{\omega _{0}}^{2}-\omega ^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44ed37ff17557ac7eecc7f187cff0ca67f8dab92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:21.846ex; height:6.343ex;" alt="{\displaystyle \epsilon (\omega )={\frac {{\omega _{0}}^{2}\epsilon _{0}-\omega ^{2}\epsilon _{\infty }}{{\omega _{0}}^{2}-\omega ^{2}}}}"></span></dd></dl> <p>Solving the equation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon (\omega _{L})=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϵ</mi> <mo stretchy="false">(</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon (\omega _{L})=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc4bd7111352fa03e97921d0b6bf9654436843a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.812ex; height:2.843ex;" alt="{\displaystyle \epsilon (\omega _{L})=0}"></span> yields: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {{\omega _{L}}^{2}}{{\omega _{0}}^{2}}}={\frac {\epsilon _{0}}{\epsilon _{\infty }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {{\omega _{L}}^{2}}{{\omega _{0}}^{2}}}={\frac {\epsilon _{0}}{\epsilon _{\infty }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2d1523675477714f3700d3e6fb632660b9c8060" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:11.442ex; height:6.343ex;" alt="{\displaystyle {\frac {{\omega _{L}}^{2}}{{\omega _{0}}^{2}}}={\frac {\epsilon _{0}}{\epsilon _{\infty }}}}"></span></dd></dl> <p>This equation gives the ratio of the frequency of the longitudonal optical phonon (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{L}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{L}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7284e7df80758a0946c74413f95c81c1750c755c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.797ex; height:2.009ex;" alt="{\displaystyle \omega _{L}}"></span>), to the frequency of the transverse optical phonon (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a713d16c489051d4f515e12b1f86061c6be799b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.5ex; height:2.009ex;" alt="{\displaystyle \omega _{0}}"></span>) in diatomic cubic ionic crystals, and is known as the <a href="/wiki/Lyddane%E2%80%93Sachs%E2%80%93Teller_relation" title="Lyddane–Sachs–Teller relation">Lyddane-Sachs-Teller relation</a>. The ratio <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{L}/\omega _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{L}/\omega _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71d415b87a0a9e3ed8e200fa5c3cc8e0b08bb386" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.46ex; height:2.843ex;" alt="{\displaystyle \omega _{L}/\omega _{0}}"></span> can be found using inelastic neutron scattering experiments. </p> <div class="mw-heading mw-heading2"><h2 id="Surface_phonon_polariton">Surface phonon polariton</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phonon_polariton&action=edit&section=5" title="Edit section: Surface phonon polariton"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Surface phonon polariton(SPhPs) are a specific kind of phonon polariton.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> They are formed by the coupling of optical surface phonon, instead of normal phonons, and light, resulting in an electromagnetic surface wave. They are similar to <a href="/wiki/Surface_plasmon_polariton" title="Surface plasmon polariton">surface plasmon polaritons</a>, although studied to a far lesser extent.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> The applications are far ranging from materials with negative index of refraction to high-density IR data storage.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>One other application is in the cooling of <a href="/wiki/Microelectronics" title="Microelectronics">microelectronics</a>. Phonons are the main source of heat conductivity in materials, where optical phonons contribute far less than acoustic phonons. This is because of the relatively low group velocity of optical phonons. When the thickness of the material decreases, the conductivity of via acoustic also decreases, since surface scattering increases.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> This microelectronics are getting smaller and smaller, reductions is getting more problematic. Although optical phonons themselves do not have a high thermal conductivity, SPhPs do seem to have this. So they may be an alternative means of cooling these electronic devices.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Experimental_observation">Experimental observation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phonon_polariton&action=edit&section=6" title="Edit section: Experimental observation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Most observations of phonon polaritons are made of surface phonon polaritons, since these can be easily probed by Raman spectroscopy or AFM. </p> <div class="mw-heading mw-heading3"><h3 id="Raman_spectroscopy">Raman spectroscopy  </h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phonon_polariton&action=edit&section=7" title="Edit section: Raman spectroscopy "><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As with any Raman experiment, a <a href="/wiki/Laser" title="Laser">laser</a> is pointed at the material being studied. If the correct <a href="/wiki/Wavelength" title="Wavelength">wavelength</a> is chosen, this laser can induce the formation of a polariton on the sample. Looking at the <a href="/wiki/Stokes_shift" title="Stokes shift">Stokes shifted</a> emitted radiation and by using the <a href="/wiki/Conservation_of_energy" title="Conservation of energy">conservation of energy</a> and the known laser energy, one can calculate the polariton energy, with which one can construct the dispersion relation.<sup id="cite_ref-:0_2-1" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="SNOM_and_AFM">SNOM and AFM</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phonon_polariton&action=edit&section=8" title="Edit section: SNOM and AFM"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The induction of polaritons is very similar to that in Raman experiments, with a few differences. With the extremely high spatial resolution of SNOM, one can induce polaritons very locally in the sample. This can be done continuously, producing a <a href="/wiki/Continuous_wave" title="Continuous wave">continuous wave</a>(CW) of polariton, or with an ultrafast pulse, producing a polariton with a very high temporal footprint. In both cases the polaritons are detected by the tip of the AFM, this signal is then used to calculate the energy of the polariton. One can also perform these experiments near the edge of the sample. This will result in the polaritons being <a href="/wiki/Reflection_(physics)" title="Reflection (physics)">reflected</a>. In the case of CW polaritons, <a href="/wiki/Standing_wave" title="Standing wave">standing waves</a> will be created, which will again be detected by the AFM tip. In the case of the polaritons created by the ultrafast laser, no standing wave will be created. The wave can still <a href="/wiki/Wave_interference" title="Wave interference">interfere</a> with itself the moment it is reflected of the edge. Whether one is observing on the bulk surface or close to an edge, the signal is in temporal form. One can <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a> this signal, converting the signal into frequency domain, which can used to obtain the dispersion relation.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Polaritonics_and_real-space_imaging">Polaritonics and real-space imaging</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phonon_polariton&action=edit&section=9" title="Edit section: Polaritonics and real-space imaging"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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/Polaritonics" title="Polaritonics">Polaritonics</a></div><p>Phonon polaritons also find use in the field of <a href="/wiki/Polaritonics" title="Polaritonics">polaritonics</a>, a field between <a href="/wiki/Photonics" title="Photonics">photonics</a> and <a href="/wiki/Electronics" title="Electronics">electronics</a>. In this field phonon polaritons are used for high speed signal processing and terahertz spectroscopy.<sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> The real-space imaging of phonon polaritons was made possible by projecting them onto a CCD camera.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phonon_polariton&action=edit&section=10" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Polariton" title="Polariton">Polariton</a></li> <li><a href="/wiki/Phonon" title="Phonon">Phonon</a></li> <li><a href="/wiki/Surface_plasmon_polariton" title="Surface plasmon polariton">Surface plasmon polariton</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Phonon_polariton&action=edit&section=11" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></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"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><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="CITEREFHopfield1958" class="citation journal cs1">Hopfield, J. J. 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