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Dispersion of backward-propagating waves in a surface defect on a three-dimensional photonic band-gap crystal

Timon J. Vreman1, Melissa J. Goodwin1, Lars J. Corbijn van Willenswaard1,2, William L. Barnes3,1, Ad Lagendijk1, and Willem L. Vos1,*

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Phys. Rev. B 112, 155305 – Published 20 October, 2025

DOI: https://doi.org/10.1103/jckd-3l5d

Abstract

We experimentally study the dispersion relation of waves in a thin quasi-two-dimensional (2D) defect layer with periodic nanopores that sits on a 3D photonic band-gap crystal made from silicon by CMOS-compatible methods. The nanostructures are probed by momentum-resolved broadband near-infrared imaging of p-polarized reflected light as a function of off-axis wave vectors. We identify surface defect modes at frequencies inside the 3D photonic band gap with a narrow relative linewidth (Δω/ω=0.028), which are absent in defect-free 3D photonic band-gap crystals. We calculate the dispersion of the states with relevant mode symmetries using a plane-wave-expansion supercell method, with structural parameters directly extracted from scanning electron microscope images. The calculated bands match very well with the measured data. The slope of the dispersion curves of the surface defect states is negative in one off-axis direction, corresponding to backward-propagating waves in that direction where the phase velocity and the group velocity point in opposite directions, as confirmed by finite-difference time-domain simulations. We also present a didactic and analytic model of a 2D grating sandwiched between vacuum and a negative real effective ɛ′<0 that mimics the 3D photonic band gap. The model's dispersion agrees with the experiments and with the full theory and shows that the backward propagation is caused by the surface grating. We discuss possible applications, including a device that senses the output direction of photons emitted by embedded quantum emitters in response to their emission frequency.

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References (85)

  1. K. Sakoda, Optical Properties of Photonic Crystals, Springer Series in Optical Sciences Vol. 80 (Springer, Berlin, 2005).
  2. J. Joannopoulos, S. G. Johnson, J. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University Press, Princeton, NJ, 2008).
  3. J.-M. Lourtioz, H. Benisty, V. Berger, J.-M. Gérard, D. Maystre, A. Tchelnokov, and D. Pagnoux, Photonic Crystals (Springer, Berlin, 2008).
  4. Edited by M. A. Noginov, G. Dewar, M. W. McCall, and N. I. Zheludev, Tutorials in Complex Photonic Media (SPIE, Bellingham, WA, 2009).
  5. L. Novotny and B. Hecht, Principles of Nano-optics, 2nd ed. (Cambridge University, Cambridge, UK, 2012).
  6. Edited by M. Ghulinyan and L. Pavesi, Light Localisation and Lasing: Random and Quasi-random Photonic Structures (Cambridge University Press, Cambridge, UK, 2014).
  7. The dielectric constant ɛ=ɛ′+iɛ′′ is written as a sum of a real part ɛ′=Re{ɛ} and an imaginary part ɛ′′=Im{ɛ}.
  8. M. van Exter and A. Lagendijk, Ultrashort surface-plasmon and phonon dynamics, Phys. Rev. Lett. 60, 49 (1988).
  9. W. L. Barnes, A. Dereux, and T. W. Ebbesen, Surface plasmon subwavelength optics, Nature (London) 424, 824 (2003).
  10. S. A. Maier et al., Plasmonics: Fundamentals and Applications (Springer, New York, 2007), Vol. 1.
  11. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (John Wiley & Sons, Hoboken, NJ, 2019).
  12. F. H. L. Koppens, D. E. Chang, and J. García de Abajo, Graphene plasmonics: A platform for strong light-matter interactions, Nano Lett. 11, 3370 (2011).
  13. Photonic crystals are nanostructures with spatially periodic variations of the dielectric constant, with length scales a of the order of the wavelength of light: a≃λ.
  14. O. Takayama, A. A. Bogdanov, and A. V. Lavrinenko, Photonic surface waves on metamaterial interfaces, J. Phys.: Condens. Matter 29, 463001 (2017).
  15. K. Ishizaki and S. Noda, Manipulation of photons at the surface of three-dimensional photonic crystals, Nature (London) 460, 367 (2009).
  16. K. Ishizaki, K. Gondaira, Y. Ota, K. Suzuki, and S. Noda, Nanocavities at the surface of three-dimensional photonic crystals, Opt. Express 21, 10590 (2013).
  17. T. F. Krauss, R. M. De La Rue, and S. Brand, Two-dimensional photonic-bandgap structures operating at near-infrared wavelengths, Nature (London) 383, 699 (1996).
  18. T. Baba, N. Fukaya, and J. Yonekura, Integrated Photonics Research (Optica, Washington, DC, 1999), p. RTuE3.
  19. S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, Guided modes in photonic crystal slabs, Phys. Rev. B 60, 5751 (1999).
  20. D. Sharma, S. B. Hasan, R. Saive, J. J. W. van der Vegt, and W. L. Vos, Enhanced absorption in thin and ultrathin silicon films by 3D photonic band gap back reflectors, Opt. Express 29, 41023 (2021).
  21. S. Fan and J. D. Joannopoulos, Analysis of guided resonances in photonic crystal slabs, Phys. Rev. B 65, 235112 (2002).
  22. A. Rosenberg, M. W. Carter, J. A. Casey, M. Kim, R. T. Holm, R. L. Henry, C. R. Eddy, V. Shamamian, K. Bussmann, S. Shi et al., Guided resonances in asymmetrical GaN photonic crystal slabs observed in the visible spectrum, Opt. Express 13, 6564 (2005).
  23. Y. Zhang, A. Chen, W. Liu, C. W. Hsu, B. Wang, F. Guan, X. Liu, L. Shi, L. Lu, and J. Zi, Observation of polarization vortices in momentum space, Phys. Rev. Lett. 120, 186103 (2018).
  24. L. Chen, K. A. Morgan, G. A. Alzaidy, C.-C. Huang, Y.-L. D. Ho, M. P. Taverne, X. Zheng, Z. Ren, Z. Feng, I. Zeimpekis et al., Observation of complete photonic bandgap in low refractive index contrast inverse rod-connected diamond structured chalcogenides, ACS Photon. 6, 1248 (2019).
  25. S. Cueff, L. Berguiga, and H. S. Nguyen, Fourier imaging for nanophotonics, Nanophotonics 13, 841 (2024).
  26. S. Foteinopoulou, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, Backward surface waves at photonic crystals, Phys. Rev. B 75, 245116 (2007).
  27. S. Foteinopoulou, G. Kenanakis, N. Katsarakis, I. Tsiapa, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, Experimental verification of backward wave propagation at photonic crystal surfaces, Appl. Phys. Lett. 91, 214102 (2007).
  28. A. P. Vinogradov, A. V. Dorofeenko, A. M. Merzlikin, and A. A. Lisyansky, Surface states in photonic crystals, Phys. Usp. 53, 243 (2010).
  29. J. Hu, S. Tian, Y. Yang, S. Zhuang, and H. Guo, Dispersion engineering in unidirectional excitation of the surface wave of photonic crystal, Opt. Lett. 43, 5319 (2018).
  30. D. Yilmaz, A. Yeltik, and H. Kurt, Highly controlled Bloch wave propagation in surfaces with broken symmetry, Opt. Lett. 43, 2660 (2018).
  31. E. Gonzalez-Valencia, I. Del Villar, and P. Torres, Bloch waves at the surface of a single-layer coating D-shaped photonic crystal fiber, Opt. Lett. 45, 2547 (2020).
  32. K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, Photonic band gaps in three dimensions: New layer-by-layer periodic structures, Solid State Commun. 89, 413 (1994).
  33. M. Maldovan and E. L. Thomas, Diamond-structured photonic crystals, Nat. Mater. 3, 593 (2004).
  34. L. A. Woldering, A. P. Mosk, R. W. Tjerkstra, and W. L. Vos, The influence of fabrication deviations on the photonic band gap of three-dimensional inverse woodpile nanostructures, J. Appl. Phys. 105, 093108 (2009).
  35. J. M. van den Broek, L. A. Woldering, R. W. Tjerkstra, F. B. Segerink, I. D. Setija, and W. L. Vos, Inverse-woodpile photonic band gap crystals with a cubic diamondlike structure made from single-crystalline silicon, Adv. Funct. Mater. 22, 25 (2012).
  36. D. Devashish, S. B. Hasan, J. J. W. van der Vegt, and W. L. Vos, Reflectivity calculated for a three-dimensional silicon photonic band gap crystal with finite support, Phys. Rev. B 95, 155141 (2017).
  37. S. B. Hasan, A. P. Mosk, W. L. Vos, and A. Lagendijk, Finite-size scaling of the density of states in photonic band-gap crystals, Phys. Rev. Lett. 120, 237402 (2018).
  38. M. Adhikary, R. Uppu, C. A. M. Harteveld, D. A. Grishina, and W. L. Vos, Experimental probe of a complete 3D photonic band gap, Opt. Express 28, 2683 (2020).
  39. S. R. Huisman, R. V. Nair, L. A. Woldering, M. D. Leistikow, A. P. Mosk, and W. L. Vos, Signature of a three-dimensional photonic band gap observed on silicon inverse woodpile photonic crystals, Phys. Rev. B 83, 205313 (2011).
  40. To match with the symmetry tables, this notation differs from different work of our group where the pores were taken to be parallel to the x and z directions.
  41. T. Hahn, U. Shmueli, and J. C. W. Arthur, International Tables for Crystallography (Reidel, Dordrecht, 1983), Vol. 1.
  42. M. I. Aroyo, A. Kirov, C. Capillas, J. M. Perez-Mato, and H. Wondratschek, Bilbao crystallographic server. II. Representations of crystallographic point groups and space groups, Acta Crystallogr. A 62, 115 (2006).
  43. K. Sakoda, Transmittance and Bragg reflectivity of two-dimensional photonic lattices, Phys. Rev. B 52, 8992 (1995).
  44. S. Foteinopoulou and C. M. Soukoulis, Electromagnetic wave propagation in two-dimensional photonic crystals: A study of anomalous refractive effects, Phys. Rev. B 72, 165112 (2005).
  45. D. A. Grishina, C. A. M. Harteveld, L. A. Woldering, and W. L. Vos, Method for making a single-step etch mask for 3D monolithic nanostructures, Nanotechnology 26, 505302 (2015).
  46. M. J. Goodwin, C. A. M. Harteveld, M. J. de Boer, and W. L. Vos, Deep reactive ion etching of cylindrical nanopores in silicon for photonic crystals, Nanotechnology 34, 225301 (2023).
  47. C. P. Mavidis, A. C. Tasolamprou, S. B. Hasan, T. Koschny, E. N. Economou, M. Kafesaki, C. M. Soukoulis, and W. L. Vos, The local density of optical states in the 3d band gap of a finite photonic crystal, Phys. Rev. B 101, 235309 (2020).
  48. W. L. Vos and H. M. van Driel, Higher order Bragg diffraction by strongly photonic fcc crystals: Onset of a photonic bandgap, Phys. Lett. A 272, 101 (2000).
  49. S. Noda, K. Tomoda, N. Yamamoto, and A. Chutinan, Full three-dimensional photonic bandgap crystals at near-infrared wavelengths, Science 289, 604 (2000).
  50. M. J. A. de Dood, B. Gralak, A. Polman, and J. G. Fleming, Superstructure and finite-size effects in a Si photonic woodpile crystal, Phys. Rev. B 67, 035322 (2003).
  51. M. Qi, E. Lidorikis, P. T. Rakich, S. G. Johnson, J. D. Joannopoulos, E. P. Ippen, and H. I. Smith, A three-dimensional optical photonic crystal with designed point defects, Nature (London) 429, 538 (2004).
  52. W. Man, M. Megens, P. J. Steinhardt, and P. M. Chaikin, Experimental measurement of the photonic properties of icosahedral quasicrystals, Nature (London) 436, 993 (2005).
  53. S. Takahashi, K. Suzuki, M. Okano, M. Imada, T. Nakamori, Y. Ota, K. Ishizaki, and S. Noda, Direct creation of three-dimensional photonic crystals by a top-down approach, Nat. Mater. 8, 721 (2009).
  54. K. Suzuki, K. Ishizaki, Y. Ota, and S. Noda, Surface modes of three-dimensional photonic crystals constructed using a top-down approach, Opt. Express 19, 25651 (2011).
  55. N. Le Thomas, R. Houdré, M. V. Kotlyar, D. O'Brien, and T. F. Krauss, Exploring light propagating in photonic crystals with Fourier optics, J. Opt. Soc. Am. B 24, 2964 (2007).
  56. Y. Zhang, M. Zhao, J. Wang, W. Liu, B. Wang, S. Hu, G. Lu, A. Chen, J. Cui, W. Zhang, C. W. Hsu, X. Liu, L. Shi, H. Yin, and J. Zi, Momentum-space imaging spectroscopy for the study of nanophotonic materials, Sci. Bull. 66, 824 (2021).
  57. Y. Zhang, Z. Che, W. Liu, J. Wang, M. Zhao, F. Guan, X. Liu, L. Shi, and J. Zi, Unfolded band structures of photonic quasicrystals and moiré superlattices, Phys. Rev. B 105, 165304 (2022).
  58. R. Uppu, M. Adhikary, C. A. M. Harteveld, and W. L. Vos, Spatially shaping waves to penetrate deep inside a forbidden gap, Phys. Rev. Lett. 126, 177402 (2021).
  59. See Supplemental Material at http://link.aps.org/supplemental/10.1103/jckd-3l5d for a video of the 3D dataset of momentum-resolved reflectivity on bar 15F crystal 2 without surface defect.
  60. See Supplemental Material at http://link.aps.org/supplemental/10.1103/jckd-3l5d for a video of the 3D dataset of momentum-resolved reflectivity on bar 15F crystal 2 with a surface defect.
  61. See Supplemental Material at http://link.aps.org/supplemental/10.1103/jckd-3l5d for a video of the 3D dataset of momentum-resolved reflectivity on bar 15F crystal 3 without surface defect.
  62. See Supplemental Material at http://link.aps.org/supplemental/10.1103/jckd-3l5d for a video of the 3D dataset of momentum-resolved reflectivity on bar 15F crystal 3 with a surface defect.
  63. A complete study of all refraction effects requires equifrequency surfaces, which also enable further analysis of the excitation of Bloch modes by a particular kin from the direction of the group velocity Refs. [2, 44].
  64. S. G. Johnson and J. D. Joannopoulos, Block-iterative frequency-domain methods for maxwell's equations in a planewave basis, Opt. Express 8, 173 (2001).
  65. L. Elcoro, B. Bradlyn, Z. Wang, M. G. Vergniory, J. Cano, C. Felser, B. A. Bernevig, D. Orobengoa, G. de la Flor, and M. I. Aroyo, Double crystallographic groups and their representations on the Bilbao crystallographic server, J. Appl. Crystallogr. 50, 1457 (2017).
  66. The wave number ν̃=ω/2πc0 is in spectroscopic tradition gauged in cm−1 that is equivalent to the older unit kayser Ref. [85].
  67. M. Notomi, Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap, Phys. Rev. B 62, 10696 (2000).
  68. Bloch modes inside the infinite photonic crystal adhere to the Laue condition: A Bloch mode with wave vector k′ is equivalent to one with wave vector k′+G.
  69. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, Meep: A flexible free-software package for electromagnetic simulations by the FDTD method, Comput. Phys. Commun. 181, 687 (2010).
  70. R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Electromagnetic Bloch waves at the surface of a photonic crystal, Phys. Rev. B 44, 10961 (1991).
  71. R. Hillebrand, S. Senz, W. Hergert, and U. Gösele, Macroporous-silicon-based three-dimensional photonic crystal with a large complete band gap, J. Appl. Phys. 94, 2758 (2003).
  72. J. Schilling, J. White, A. Scherer, G. Stupian, R. Hillebrand, and U. Gösele, Three-dimensional macroporous silicon photonic crystal with large photonic band gap, Appl. Phys. Lett. 86, 011101 (2005).
  73. F. García-Santamaría, M. Xu, V. Lousse, S. Fan, P. V. Braun, and J. A. Lewis, A germanium inverse woodpile structure with a large photonic band gap, Adv. Mater. 19, 1567 (2007).
  74. Resonant modes and transmission in a waveguide cavity, http://meep.readthedocs.io/en/latest/Python_Tutorials/Resonant_Modes_and_Transmission_in_a_Waveguide_Cavity/#band-diagram, accessed 22-November-2024).
  75. See Supplemental Material at http://link.aps.org/supplemental/10.1103/jckd-3l5d for a video of the simulations of Fig. 8.
  76. L. J. Corbijn van Willenswaard, S. Smeets, N. Renaud, M. Schlottbom, J. J. W. van der Vegt, and W. L. Vos, Non-utopian optical properties computed of a tomographically reconstructed real photonic nanostructure, Opt. Express 32, 32028 (2024).
  77. ɛ3′ was estimated from our FDTD simulations from the gradient of the electric field strength into the bulk of the band gap crystal (cf. Fig. 8). It turns out that the precise value of ɛ3′ is not critical, as long as ɛ3′≪ɛ2′.
  78. A. F. Koenderink, Emission and transport of light in photonic crystals, Ph.D. thesis, Universiteit van Amsterdam, 2003.
  79. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge University Press, Cambridge, UK, 2013).
  80. S. A. Hack, J. J. W. van der Vegt, and W. L. Vos, Cartesian light: Unconventional propagation of light in a three-dimensional superlattice of coupled cavities within a three-dimensional photonic band gap, Phys. Rev. B 99, 115308 (2019).
  81. T. J. Vreman, M. J. Goodwin, L. J. Corbijn van Willenswaard, W. J. Barnes, A. Lagendijk, and W. L. Vos, [Dataset], Zenodo (2025), https://doi.org/10.5281/zenodo.14845289.
  82. The symmetry properties of Bloch modes follow from the irreducible representations of the group of the wave vector. The given wave vectors are on the Σ high symmetry line for which there are four irreducible representations: Σ1 to Σ4.
  83. Just like Bloch modes of the bulk structure, Bloch modes in the surface defect wave have an energy density that is mostly present in a structure with space symmetry group 141. Therefore, we expect that the symmetry of the bulk PC bands and the surface defect mode are not much different, and we use the same symmetry matrices for the modes of the bulk PC as for the surface defect modes.
  84. To provide an additional example, the effect of the disturbance of air modes with a surface mode is clearer for the modes excitable by s-polarized light.
  85. Wikipedia contributors, Wavenumber, Wikipedia https://en.wikipedia.org/w/index.php?title=Wavenumber&oldid=1235416429.

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Phys. Rev. B 112, 155305– Published 20 October, 2025
  • Received 10 February 2025
  • Revised 28 August 2025
  • Accepted 1 October 2025
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