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Photonic topological insulator
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<h2 id="history">History</h2> <p>Topological order in solid state systems has been studied in condensed matter physics since the discovery of <a href="/facts/Quantum_Hall_effect/yK3zcCgu">integer quantum Hall effect</a>. But topological matter attracted considerable interest from the physics community after the proposals for possible observation of symmetry-protected topological phases (or the so-called <i><a href="/facts/Topological_insulator/pbrbxG9X">topological insulators</a></i>) in <a href="/facts/Graphene/QDGUBaPt">graphene</a>,<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> and experimental observation of a 2D topological insulator in CdTe/HgTe/CdTe quantum wells in 2007.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a><a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p><p>In 2008, <a href="/facts/Duncan_Haldane/Hj4rQvdA">Haldane</a> and Raghu proposed that unidirectional electromagnetic states analogous to (integer) quantum Hall states can be realized in nonreciprocal magnetic <a href="/facts/Photonic_crystal/R763CtuW">photonic crystals</a>.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> This prediction was first realized in 2009 in the microwave frequency regime.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> This was followed by the proposals for analogous <a href="/facts/Quantum_spin_Hall_effect/26hdSS9L">quantum spin Hall states</a> of electromagnetic waves that are now known as photonic topological insulators.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a><a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> It was later found that topological electromagnetic states can exist in continuous media as well--theoretical and numerical study has confirmed the existence of topological Langmuir-cyclotron waves in continuous magnetized plasmas.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a><a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> </p> <h2 id="platforms">Platforms</h2> <p>Photonic topological insulators are designed using various photonic platforms including optical waveguide arrays,<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> coupled ring resonators,<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> bi-anisotropic <a href="/facts/Metamaterial/3DeLU6ME">meta-materials</a>, and photonic crystals.<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> More recently, they have been realized in 2D dielectric<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> and plasmonic<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> meta-surfaces. Despite the theoretical prediction,<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a><a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a> no experimental demonstration of photonic topological insulator in continuous media has been reported. </p> <h2 id="chern-number">Chern number</h2> <p>As an important figure of merit for characterizing the quantized collective behaviors of the wavefunction, Chern number is the topological invariant of quantum Hall insulators. Chern number also identifies the topological properties of the photonic topological insulators (PTIs), thus it is of crucial importance in PTI design. The full-wave finite-difference frequency-domain (FDFD) method based MATLAB program for computing the Chern number has been written.<a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a> Recently, the finite-difference method has been extended to analyze the topological invariant of non-Hermitian topological dielectric photonic crystals by first-principle Wilson loop calculation. <a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a> All MATLAB codes can be found at GitHub website. <a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Symmetry-protected_topological_order/h8CVbg1M">Symmetry-protected topological order</a></li> <li><a href="/facts/Metamaterial/3DeLU6ME">Metamaterial</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Lu, Ling; Joannopoulos, John D.; Soljačić, Marin (November 2014). "Topological photonics". 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PMID 37000869. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10065437" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10065437</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><p>Fu, Yichen; Qin, Hong (2021-06-24). "Topological phases and bulk-edge correspondence of magnetized cold plasmas". Nature Communications. 12 (1): 3924. doi:10.1038/s41467-021-24189-3. ISSN 2041-1723. PMC 8225675. PMID 34168159. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8225675" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8225675</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li> <li id="fn:19"><p>Zhao, Ran; Xie, Guo-Da; Chen, Menglin L. N.; Lan, Zhihao; Huang, Zhixiang; Sha, Wei E. I. (2020-02-17). "First-principle calculation of Chern number in gyrotropic photonic crystals". Optics Express. 28 (4): 4638–4649. arXiv:2001.08913. Bibcode:2020OExpr..28.4638Z. doi:10.1364/OE.380077. ISSN 1094-4087. PMID 32121697. 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