nLab photonic crystal

Idea

A photonic crystal (Benisty et a. 1999, going back to Yablonovitch 1987, John 1987) (often: a waveguide array, see Trompeter et al. 2003) is a material with periodic dielectric structure/refractive index in which light waves behave like Bloch waves of electrons in an actual crystal, such as in that they form energy bands separated by band gaps (Yablonovitch 1987). (Of course, the underlying material may itself consist of actual crystalline structures, such as graphene, see e.g. BBKKL10).

By suitably engineering photonic crystals they may emulate various phenomena seen elsewhere in nature (for instance transport properties analogous to those in semiconductors) or not seen anywhere else, whence one also speaks of “meta-materials”.

In much the same way, there are phononic crystals.

For instance, many aspects of topological phases of matter (topological insulators, semimetals, …) have analogs realizations in photonic crystals (“topological photonics”, see the references below). Notably nodal lines have been demonstrated in photonic semimetals (Park, Wong, Zhang& Oh 2021, Park, Gao, Zhang & Oh 2022) and similarly for phonons (Peng, Bouhon, Monserrat & Slager 2022).

Related concepts

• crystal

• phononic crystal

References

General

Original articles:

• Eli Yablonovitch, Inhibited Spontaneous Emission in Solid-State Physics and Electronics, Phys. Rev. Lett. 58 2059 (1987) [doi:10.1103/PhysRevLett.58.2059]

• Sajeev John, Strong localization of photons in certain disordered dielectric superlattices Phys. Rev. Lett. 58 2486 (1987) [doi:10.1103/PhysRevLett.58.2486]

Further articles:

• Thomas F. Krauss, Richard M. De La Rue, Stuart Brand, Two-dimensional photonic-bandgap structures operating at near-infrared wavelengths, Nature 383 (1996) 699–702 [doi:10.1038/383699a0]

• H. Benisty et al., Optical and confinement properties of two-dimensional photonic crystals, Journal of Lightwave Technology 17 11 (1999) 2063-2077 [doi:10.1109/50.802996]

• D. N. Chigrin & C. M. Sotomayor Torres, Periodic thin-film interference filters as one-dimensional photonic crystals, Optics and Spectroscopy 91 (2001) 484–489 [doi:10.1134/1.1405232]

Early history:

• David Lindley, Landmarks – The Birth of Photonic Crystals, Physics 6 94 (2013) [physics:v6/94]

Review of experimental realizations:

• Steven G. Johnson, Photonic Crystals: Periodic Surprises in Electromagnetism, lecture notes (2003) [webpage, pdf]

and for waveguide arrays:

• Henrike Trompeter et al., Tailoring guided modes in waveguide arrays, 11 25 (2003) 3404-3411 [doi:10.1364/OE.11.003404]

Further review:

• Ivan L.Garanovicha, Stefano Longhib, Andrey A.Sukhorukova, Yuri S. Kivshar, Light propagation and localization in modulated photonic lattices and waveguides, Physics Reports 518 1–2 (2012) 1-79 [doi:10.1016/j.physrep.2012.03.005]

On the effective appearance of the Schrödinger equation for electromagnetic waves in photonic crystals:

• Max Lein, Taking Inspiration from Quantum-Wave Analogies — Recent Results for Photonic Crystals, in Springer Proceedings in Mathematics & Statistics 270 (2018) [arXiv:1804.01234, doi:10.1007/978-3-030-01602-9_10]

Textbook account:

• John D. Joannopoulos, Steven G. Johnson, Joshua N. Winn, and Robert D. Meade, Photonic Crystals: Molding the Flow of Light [webpage, ISBN:9780691124568]

Description of photonic crystals by a non-linear Schrödinger equation?:

• A. Pankov, Periodic Nonlinear Schrödinger Equation with Application to Photonic Crystals, Milan J. Math. 73 (2005) 259–287 [doi:10.1007/s00032-005-0047-8]

• Anatoli Babin, Alexander Figotin, Nonlinear Photonic Crystals: IV. Nonlinear Schrödinger Equation Regime 15 (2005) 145-228 [arXiv:math-ph/0409079, doi:10.1080/17455030500196929]

• Waleed Alrefai, Schrödinger equation for propagation in photonic crystal fibers Eureka Physics & Engineering 1 (2016) [doi:10.21303/2461-4262.2016.00021]

Realization with graphene:

• Oleg L. Berman, Vladimir S. Boyko, Roman Ya. Kezerashvili, Anton A. Kolesnikov, Yurii E. Lozovik, Graphene-based photonic crystal, Physics Letters A 374 (2010) 4784-4786 [arXiv:1012.4143, doi:10.1016/j.physleta.2010.09.064]

Photonic analog of graphene:

• Observation of unconventional edge states in ‘photonic graphene’, Nature Materials 13 (2014) 57–62 [doi:10.1038/nmat3783]

See also:

• Wikipedia, Photonic crystal

Claim of photonic hyperbolic tesselations:

• Alicia J. Kollár, Mattias Fitzpatrick, Andrew A. Houck, Hyperbolic Lattices in Circuit Quantum Electrodynamics, Nature 571 (2019) 45–50 [arXiv:1802.09549, doi:10.1038/s41586-019-1348-3]

Topological photonics

Original idea of topological photonics:

• Duncan Haldane, S. Raghu, Possible Realization of Directional Optical Waveguides in Photonic Crystals with Broken Time-Reversal Symmetry, Phys. Rev. Lett. 100 (2008) 013904 [doi:10.1103/PhysRevLett.100.013904]

Solitons

On solitonic excitations in waveguide arrays:

• Nikolaos K. Efremidis et al., Spatial photonics in nonlinear waveguide arrays, Optics Express 13 6 (2005) 1780-1796 [doi:10.1364/OPEX.13.001780]

• Thawatchai Mayteevarunyoo, Boris A. Malomed, Solitons in one-dimensional photonic crystals, Journal of the Optical Society of America B 25 11 (2008) 1854-1863 [doi:10.1364/JOSAB.25.001854]

• Amaria Javed, Alaa Shaheen U. Al Khawaja, Amplifying optical signals with discrete solitons in waveguide arrays, Physics Letters A 384 26 (2020) 126654 [doi:10.1016/j.physleta.2020.126654]

• Alaa Shaheen, Amaria Javed, U. Al Khawaja, Adding binary numbers with discrete solitons in waveguide arrays, Phys. Scr. 95 (2020) 085107 [arXiv:2108.01406, doi:10.1088/1402-4896/aba2b2]

Photonic topological phases

On photonic analogs of the quantum Hall effect:

• Zheng Wang, Yidong Chong, J. D. Joannopoulos, Marin Soljačić, Observation of unidirectional backscattering-immune topological electromagnetic states, Nature 461 (2009) 772–775 [doi:10.1038/nature08293]

On photonic analogs of topological insulators (photonic topological insulators):

• Photonic topological insulators, Nature Materials 12 (2013) 233-239 [doi:10.1038/nmat3520]

On photonic analogs of topological semimetals (photonic topological semimetals):

• Shuqi sehn et al., A Review of Topological Semimetal Phases in Photonic Artificial Microstructures, Front. Phys., 16 [doi:10.3389/fphy.2021.771481]

• Ruey-Lin Chern and You-Zhong Yu, Photonic topological semimetals in bigyrotropic metamaterials, 30 14 (2022) 25162-25176 [doi:10.1364/OE.459097]

Photonic analog of nodal lines:

• Shuang Zhang et al., Experimental observation of photonic nodal line degeneracies in metacrystals, Nature Communications 9 950 (2018) [doi:10.1038/s41467-018-03407-5]

Movement of nodal points in photonic crystals:

• Yong-Heng Lu et al., Observing movement of Dirac cones from single-photon dynamics, Phys. Rev. B 103 064304 (2021) [doi:10.1103/PhysRevB.103.064304]

• María Blanco de Paz, Alejandro González-Tudela, Paloma Arroyo Huidobro, Manipulating Generalized Dirac Cones In Quantum Metasurfaces [arXiv:2203.11195]

On something like anyon braiding in photonic crystals:

• Xu-Lin Zhang et al., Non-Abelian braiding on photonic chips, Nature Photonics 16 (2022) 390–395 [arXiv:2112.01776, doi:10.1038/s41566-022-00976-2]

See also:

• Hui Liu et al., Topological phases and non-Hermitian topology in photonic artificial microstructures, Nanophotonic (Feb. 2023) [arXiv:10.1515/nanoph-2022-0778]

Application to holonomic quantum computation:

• Julien Pinske, Lucas Teuber, Stefan Scheel: Highly degenerate photonic waveguide structures for holonomic computation, Phys. Rev. A 101 062314 (2020) [doi:10.1103/PhysRevA.101.062314]

• Vera Neef, Julien Pinske, Friederike Klauck, Lucas Teuber, Mark Kremer et al.: Experimental Realization of a non-Abelian $U(3)$ Holonomy, in: 2021 Conference on Lasers and Electro-Optics (CLEO), IEEE (2021) [ieee:9572414]

• Julien Pinske, Stefan Scheel, Symmetry-protected non-Abelian geometric phases in optical waveguides with nonorthogonal modes, Phys. Rev. A 105 013507 (2022) [doi:10.1103/PhysRevA.105.013507, arXiv:2105.04859]

• Vera Neef, Julien Pinske, Friederike Klauck, Lucas Teuber, Mark Kremer, Max Ehrhardt, Matthias Heinrich, Stefan Scheel Alexander Szameit: Three-dimensional non-Abelian quantum holonomy, Nat. Phys. 19 (2023) 30–34 [doi:10.1038/s41567-022-01807-5]

Topological lasers

On topological lasers:

• Babak Bahari, Abdoulaye Ndao, Felipe Vallini, Abdelkrim El Amili, Yeshaiahu Fainman, Boubacar Kanté, Nonreciprocal lasing in topological cavities of arbitrary geometries, Science 358 6363 (2017) 636-640 [doi:10.1126/science.aao4551, pdf]

• Gal Harari, Miguel A. Bandres, Yaakov Lumer, Mikael C. Rechtsman, Y. D. Chong, Mercedeh Khajavikhan, Demetrios N. Christodoulides, Mordechai Segev, Topological insulator laser: Theory, Science 359 6 (2018) [doi:10.1126/science.aar4003, pdf]

• Natsuko Ishida, Yasutomo Ota, Wenbo Lin, Tim Byrnes, Yasuhiko Arakawa and Satoshi Iwamoto, A large-scale single-mode array laser based on a topological edge mode, Nanophotonics 11 9 (2022) 2169–2181 [doi:10.1515/nanoph-2021-0608]

Anyons in momentum-space

On non-trivial braiding of nodal points in the Brillouin torus of topological semi-metals (“braiding in momentum space”):

• Junyeong Ahn, Sungjoon Park, Bohm-Jung Yang, Failure of Nielsen-Ninomiya theorem and fragile topology in two-dimensional systems with space-time inversion symmetry: application to twisted bilayer graphene at magic angle, Phys. Rev. X 9 (2019) 021013 $[$doi:10.1103/PhysRevX.9.021013, arXiv:1808.05375$]$

  “here are band crossing points, henceforth called vortices”

• QuanSheng Wu, Alexey A. Soluyanov, Tomáš Bzdušek, Non-Abelian band topology in noninteracting metals, Science 365 (2019) 1273-1277 $[$arXiv:1808.07469, doi:10.1126/science.aau8740$]$

  “fundamental group of complement of nodal points/lines considered above (3)”

• Apoorv Tiwari, Tomáš Bzdušek, Non-Abelian topology of nodal-line rings in PT-symmetric systems, Phys. Rev. B 101 (2020) 195130 $[$doi:10.1103/PhysRevB.101.195130$]$

“a new type non-Abelian ‘braiding’ of nodal-line rings inside the momentum space”

• Adrien Bouhon, QuanSheng Wu, Robert-Jan Slager, Hongming Weng, Oleg V. Yazyev, Tomáš Bzdušek, Non-Abelian reciprocal braiding of Weyl points and its manifestation in ZrTe, Nature Physics 16 (2020) 1137-1143 $[$arXiv:1907.10611, doi:10.1038/s41567-020-0967-9$]$

“Here we report that Weyl points in three-dimensional (3D) systems with $\mathcal{C}_2\mathcal{T}$ symmetry carry non-Abelian topological charges. These charges are transformed via non-trivial phase factors that arise upon braiding the nodes inside the reciprocal momentum space.”

Braiding of Dirac points in twisted bilayer graphene:

• Jian Kang, Oskar Vafek, Non-Abelian Dirac node braiding and near-degeneracy of correlated phases at odd integer filling in magic angle twisted bilayer graphene, Phys. Rev. B 102 (2020) 035161 $[$arXiv:2002.10360, doi:10.1103/PhysRevB.102.035161$]$

• Bin Jiang, Adrien Bouhon, Zhi-Kang Lin, Xiaoxi Zhou, Bo Hou, Feng Li, Robert-Jan Slager, Jian-Hua Jiang Experimental observation of non-Abelian topological acoustic semimetals and their phase transitions, Nature Physics 17 (2021) 1239-1246 $[$arXiv:2104.13397, doi:10.1038/s41567-021-01340-x$]$

  (analog realization in phononic crystals)

Here, we consider an exotic type of topological phases beyond the above paradigms that, instead, depend on topological charge conversion processes when band nodes are braided with respect to each other in momentum space or recombined over the Brillouin zone. The braiding of band nodes is in some sense the reciprocal space analog of the non-Abelian braiding of particles in real space.

$$...$$

we experimentally observe non-Abelian topological semimetals and their evolutions using acoustic Bloch bands in kagome acoustic metamaterials. By tuning the geometry of the metamaterials, we experimentally confirm the creation, annihilation, moving, merging and splitting of the topological band nodes in multiple bandgaps and the associated non-Abelian topological phase transitions

• Haedong Park, Stephan Wong, Xiao Zhang, and Sang Soon Oh, Non-Abelian Charged Nodal Links in a Dielectric Photonic Crystal, ACS Photonics 8 (2021) 2746–2754 [doi:10.1021/acsphotonics.1c00876]

• Siyu Chen, Adrien Bouhon, Robert-Jan Slager, Bartomeu Monserrat, Non-Abelian braiding of Weyl nodes via symmetry-constrained phase transitions (formerly: Manipulation and braiding of Weyl nodes using symmetry-constrained phase transitions), Phys. Rev. B 105 (2022) L081117 $[$arXiv:2108.10330, doi:10.1103/PhysRevB.105.L081117$]$

  “Our work opens up routes to readily manipulate Weyl nodes using only slight external parameter changes, paving the way for the practical realization of reciprocal space braiding.”

• Bo Peng, Adrien Bouhon, Robert-Jan Slager, Bartomeu Monserrat, Multi-gap topology and non-Abelian braiding of phonons from first principles, Phys. Rev. B 105 (2022) 085115 [arXiv:2111.05872, doi:10.1103/PhysRevB.105.085115]

  (analog realization in phononic crystals)

new opportunities for exploring non-Abelian braiding of band crossing points (nodes) in reciprocal space, providing an alternative to the real space braiding exploited by other strategies.

Real space braiding is practically constrained to boundary states, which has made experimental observation and manipulation difficult; instead, reciprocal space braiding occurs in the bulk states of the band structures and we demonstrate in this work that this provides a straightforward platform for non-Abelian braiding.

• Bo Peng, Adrien Bouhon, Bartomeu Monserrat, Robert-Jan Slager, Phonons as a platform for non-Abelian braiding and its manifestation in layered silicates, Nature Communications 13 423 (2022) [doi:10.1038/s41467-022-28046-9]

  (analog realization in phononic crystals)

  it is possible to controllably braid Kagome band nodes in monolayer $\mathrm{Si}_2 \mathrm{O}_3$ using strain and/or an external electric field.

• Haedong Park, Wenlong Gao, Xiao Zhang, Sang Soon Oh, Nodal lines in momentum space: topological invariants and recent realizations in photonic and other systems, Nanophotonics 11 11 (2022) 2779–2801 [doi:10.1515/nanoph-2021-0692]

  (analog realization in photonic crystals)

• Adrien Bouhon, Robert-Jan Slager, Multi-gap topological conversion of Euler class via band-node braiding: minimal models, PT-linked nodal rings, and chiral heirs [arXiv:2203.16741]

See also:

• Robert-Jan Slager, Adrien Bouhon, Fatma Nur Ünal, Floquet multi-gap topology: Non-Abelian braiding and anomalous Dirac string phase, Nature Communications 15 1144 (2024) [arXiv:2208.12824, doi:10.1038/s41467-024-45302-2]

• Huahui Qiu et al., Minimal non-abelian nodal braiding in ideal metamaterials, Nature Communications 14 1261 (2023) [doi:10.1038/s41467-023-36952-9]

• Wojciech J. Jankowski, Mohammedreza Noormandipour, Adrien Bouhon, Robert-Jan Slager, Disorder-induced topological quantum phase transitions in Euler semimetals [arXiv:2306.13084]

• Seung Hun Lee, Yuting Qian, Bohm-Jung Yang, Euler band topology in spin-orbit coupled magnetic systems [arXiv:2404.16383]

  “Based on first-principles calculations, we report that such nodal point braiding in 2D electronic bands can be realized in a MSWI candidate, the bilayer $ZrTe_5$ with in-plane ferromagnetism under pressure. […] one can expect that the braiding of nodes can be achieved in 2D bilayer $ZrTe_5$ under the influence of an external in-plane Zeeman field.”

Incidentally, references indicating that the required toroidal (or yet higher genus) geometry for anyonic topological order in position space is dubious (as opposed to the evident toroidal geometry of the momentum-space Brillouin torus): Lan 19, p. 1, ….

Knotted nodal lines in 3d semimetals

Beware that various authors consider braids/knots formed by nodal lines in 3d semimetals, i.e. knotted nodal lines in 3 spatial dimensions, as opposed to worldlines (in 2+1 spacetime dimensions) of nodal points in effectively 2d semimetals needed for the anyon-braiding considered above.

An argument that these nodal lines in 3d space, nevertheless, may be controlled by Chern-Simons theory:

• Biao Lian, Cumrun Vafa, Farzan Vafa, Shou-Cheng Zhang, Chern-Simons theory and Wilson loops in the Brillouin zone, Phys. Rev. B 95 (2017) 094512 [doi:10.1103/PhysRevB.95.094512]