Abstract

For optical systems, periodic dielectric materials could mold the flow of light, producing abundant attractive properties. Analogous with the electrical bandstructure of solid materials, the dispersion of the optical states of photonic structures are essential to characterize the optical properties, while those states below the light cone are not detectable by far-field measurement experimentally. A method for far-field detection has been developed for observing states below the light cone with compound lattices. The basic mechanism involved is that periodic weak scattering leads to band folding, making the states out of the light cone to occur inside. By using polarization-resolved momentum-space imaging spectroscopy, the band structures and iso-frequency contours of plasmonic lattices with different dimensions and symmetries are experimentally mapped out in good agreement with the simulation.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2011).
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    [Crossref]
  3. A. Martínez and J. Marti, “Negative refraction in two-dimensional photonic crystals: Role of lattice orientation and interface termination,” Phys. Rev. B 71(23), 235115 (2005).
    [Crossref]
  4. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58(16), R10096 (1998).
    [Crossref]
  5. J. Amet, G. Ulliac, F. Baida, and M.-P. Bernal, “Experimental evidence of enhanced electro-optic control on a lithium niobate photonic crystal superprism,” Appl. Phys. Lett. 96(10), 103111 (2010).
    [Crossref]
  6. X. Yu and S. Fan, “Bends and splitters for self-collimated beams in photonic crystals,” Appl. Phys. Lett. 83(16), 3251–3253 (2003).
    [Crossref]
  7. D. Tang, L. Chen, and W. Ding, “Efficient beaming from photonic crystal waveguides via self-collimation effect,” Appl. Phys. Lett. 89(13), 131120 (2006).
    [Crossref]
  8. L. Shi, H. Yin, X. Zhu, X. Liu, and J. Zi, “Direct observation of iso-frequency contour of surface modes in defective photonic crystals in real space,” Appl. Phys. Lett. 97(25), 251111 (2010).
    [Crossref]
  9. J. A. Gerber, S. Berweger, B. T. O’Callahan, and M. B. Raschke, “Phase-resolved surface plasmon interferometry of graphene,” Phys. Rev. Lett. 113(5), 055502 (2014).
    [Crossref]
  10. N. Le Thomas, R. Houdré, L. H. Frandsen, J. Fage-Pedersen, A. Lavrinenko, and P. I. Borel, “Grating-assisted superresolution of slow waves in fourier space,” Phys. Rev. B 76(3), 035103 (2007).
    [Crossref]
  11. J. Jágerská, N. Le Thomas, R. Houdré, J. Bolten, C. Moormann, T. Wahlbrink, J. Ctyrokỳ, M. Waldow, and M. Först, “Dispersion properties of silicon nanophotonic waveguides investigated with fourier optics,” Opt. Lett. 32(18), 2723–2725 (2007).
    [Crossref]
  12. N. Le Thomas, R. Houdré, D. Beggs, and T. Krauss, “Fourier space imaging of light localization at a photonic band-edge located below the light cone,” Phys. Rev. B 79(3), 033305 (2009).
    [Crossref]
  13. E. C. Regan, Y. Igarashi, B. Zhen, I. Kaminer, C. W. Hsu, Y. Shen, J. D. Joannopoulos, and M. Soljačić, “Direct imaging of isofrequency contours in photonic structures,” Sci. Adv. 2(11), e1601591 (2016).
    [Crossref]
  14. 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(18), 186103 (2018).
    [Crossref]
  15. H. M. Doeleman, F. Monticone, W. Hollander, A. Alù, and A. F. Koenderink, “Experimental observation of a polarization vortex at an optical bound state in the continuum,” Nat. Photonics 12(7), 397–401 (2018).
    [Crossref]
  16. A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329(5994), 930–933 (2010).
    [Crossref]
  17. S. Fan and J. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65(23), 235112 (2002).
    [Crossref]
  18. M. Povinelli, S. G. Johnson, and J. Joannopoulos, “Slow-light, band-edge waveguides for tunable time delays,” Opt. Express 13(18), 7145–7159 (2005).
    [Crossref]
  19. X. Wu, Y. Meng, J. Tian, Y. Huang, H. Xiang, D. Han, and W. Wen, “Direct observation of valley-polarized topological edge states in designer surface plasmon crystals,” Nat. Commun. 8(1), 1304 (2017).
    [Crossref]
  20. W.-J. Chen, S.-J. Jiang, X.-D. Chen, B. Zhu, L. Zhou, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5(1), 5782 (2014).
    [Crossref]
  21. X. Huang, M. Xiao, Z.-Q. Zhang, and C. T. Chan, “Sufficient condition for the existence of interface states in some two-dimensional photonic crystals,” Phys. Rev. B 90(7), 075423 (2014).
    [Crossref]
  22. D. Han, F. Wu, X. Li, C. Xu, X. Liu, and J. Zi, “Transmission and absorption of metallic films coated with corrugated dielectric layers,” Appl. Phys. Lett. 89(9), 091104 (2006).
    [Crossref]

2018 (2)

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(18), 186103 (2018).
[Crossref]

H. M. Doeleman, F. Monticone, W. Hollander, A. Alù, and A. F. Koenderink, “Experimental observation of a polarization vortex at an optical bound state in the continuum,” Nat. Photonics 12(7), 397–401 (2018).
[Crossref]

2017 (1)

X. Wu, Y. Meng, J. Tian, Y. Huang, H. Xiang, D. Han, and W. Wen, “Direct observation of valley-polarized topological edge states in designer surface plasmon crystals,” Nat. Commun. 8(1), 1304 (2017).
[Crossref]

2016 (1)

E. C. Regan, Y. Igarashi, B. Zhen, I. Kaminer, C. W. Hsu, Y. Shen, J. D. Joannopoulos, and M. Soljačić, “Direct imaging of isofrequency contours in photonic structures,” Sci. Adv. 2(11), e1601591 (2016).
[Crossref]

2014 (3)

J. A. Gerber, S. Berweger, B. T. O’Callahan, and M. B. Raschke, “Phase-resolved surface plasmon interferometry of graphene,” Phys. Rev. Lett. 113(5), 055502 (2014).
[Crossref]

W.-J. Chen, S.-J. Jiang, X.-D. Chen, B. Zhu, L. Zhou, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5(1), 5782 (2014).
[Crossref]

X. Huang, M. Xiao, Z.-Q. Zhang, and C. T. Chan, “Sufficient condition for the existence of interface states in some two-dimensional photonic crystals,” Phys. Rev. B 90(7), 075423 (2014).
[Crossref]

2010 (3)

J. Amet, G. Ulliac, F. Baida, and M.-P. Bernal, “Experimental evidence of enhanced electro-optic control on a lithium niobate photonic crystal superprism,” Appl. Phys. Lett. 96(10), 103111 (2010).
[Crossref]

L. Shi, H. Yin, X. Zhu, X. Liu, and J. Zi, “Direct observation of iso-frequency contour of surface modes in defective photonic crystals in real space,” Appl. Phys. Lett. 97(25), 251111 (2010).
[Crossref]

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329(5994), 930–933 (2010).
[Crossref]

2009 (1)

N. Le Thomas, R. Houdré, D. Beggs, and T. Krauss, “Fourier space imaging of light localization at a photonic band-edge located below the light cone,” Phys. Rev. B 79(3), 033305 (2009).
[Crossref]

2007 (2)

N. Le Thomas, R. Houdré, L. H. Frandsen, J. Fage-Pedersen, A. Lavrinenko, and P. I. Borel, “Grating-assisted superresolution of slow waves in fourier space,” Phys. Rev. B 76(3), 035103 (2007).
[Crossref]

J. Jágerská, N. Le Thomas, R. Houdré, J. Bolten, C. Moormann, T. Wahlbrink, J. Ctyrokỳ, M. Waldow, and M. Först, “Dispersion properties of silicon nanophotonic waveguides investigated with fourier optics,” Opt. Lett. 32(18), 2723–2725 (2007).
[Crossref]

2006 (2)

D. Tang, L. Chen, and W. Ding, “Efficient beaming from photonic crystal waveguides via self-collimation effect,” Appl. Phys. Lett. 89(13), 131120 (2006).
[Crossref]

D. Han, F. Wu, X. Li, C. Xu, X. Liu, and J. Zi, “Transmission and absorption of metallic films coated with corrugated dielectric layers,” Appl. Phys. Lett. 89(9), 091104 (2006).
[Crossref]

2005 (2)

A. Martínez and J. Marti, “Negative refraction in two-dimensional photonic crystals: Role of lattice orientation and interface termination,” Phys. Rev. B 71(23), 235115 (2005).
[Crossref]

M. Povinelli, S. G. Johnson, and J. Joannopoulos, “Slow-light, band-edge waveguides for tunable time delays,” Opt. Express 13(18), 7145–7159 (2005).
[Crossref]

2003 (1)

X. Yu and S. Fan, “Bends and splitters for self-collimated beams in photonic crystals,” Appl. Phys. Lett. 83(16), 3251–3253 (2003).
[Crossref]

2002 (2)

C. Luo, S. G. Johnson, J. Joannopoulos, and J. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65(20), 201104 (2002).
[Crossref]

S. Fan and J. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65(23), 235112 (2002).
[Crossref]

1998 (1)

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58(16), R10096 (1998).
[Crossref]

Alù, A.

H. M. Doeleman, F. Monticone, W. Hollander, A. Alù, and A. F. Koenderink, “Experimental observation of a polarization vortex at an optical bound state in the continuum,” Nat. Photonics 12(7), 397–401 (2018).
[Crossref]

Amet, J.

J. Amet, G. Ulliac, F. Baida, and M.-P. Bernal, “Experimental evidence of enhanced electro-optic control on a lithium niobate photonic crystal superprism,” Appl. Phys. Lett. 96(10), 103111 (2010).
[Crossref]

Baida, F.

J. Amet, G. Ulliac, F. Baida, and M.-P. Bernal, “Experimental evidence of enhanced electro-optic control on a lithium niobate photonic crystal superprism,” Appl. Phys. Lett. 96(10), 103111 (2010).
[Crossref]

Beggs, D.

N. Le Thomas, R. Houdré, D. Beggs, and T. Krauss, “Fourier space imaging of light localization at a photonic band-edge located below the light cone,” Phys. Rev. B 79(3), 033305 (2009).
[Crossref]

Bernal, M.-P.

J. Amet, G. Ulliac, F. Baida, and M.-P. Bernal, “Experimental evidence of enhanced electro-optic control on a lithium niobate photonic crystal superprism,” Appl. Phys. Lett. 96(10), 103111 (2010).
[Crossref]

Berweger, S.

J. A. Gerber, S. Berweger, B. T. O’Callahan, and M. B. Raschke, “Phase-resolved surface plasmon interferometry of graphene,” Phys. Rev. Lett. 113(5), 055502 (2014).
[Crossref]

Bolten, J.

Borel, P. I.

N. Le Thomas, R. Houdré, L. H. Frandsen, J. Fage-Pedersen, A. Lavrinenko, and P. I. Borel, “Grating-assisted superresolution of slow waves in fourier space,” Phys. Rev. B 76(3), 035103 (2007).
[Crossref]

Chan, C. T.

W.-J. Chen, S.-J. Jiang, X.-D. Chen, B. Zhu, L. Zhou, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5(1), 5782 (2014).
[Crossref]

X. Huang, M. Xiao, Z.-Q. Zhang, and C. T. Chan, “Sufficient condition for the existence of interface states in some two-dimensional photonic crystals,” Phys. Rev. B 90(7), 075423 (2014).
[Crossref]

Chen, A.

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(18), 186103 (2018).
[Crossref]

Chen, L.

D. Tang, L. Chen, and W. Ding, “Efficient beaming from photonic crystal waveguides via self-collimation effect,” Appl. Phys. Lett. 89(13), 131120 (2006).
[Crossref]

Chen, W.-J.

W.-J. Chen, S.-J. Jiang, X.-D. Chen, B. Zhu, L. Zhou, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5(1), 5782 (2014).
[Crossref]

Chen, X.-D.

W.-J. Chen, S.-J. Jiang, X.-D. Chen, B. Zhu, L. Zhou, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5(1), 5782 (2014).
[Crossref]

Ctyrok?, J.

Curto, A. G.

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329(5994), 930–933 (2010).
[Crossref]

Ding, W.

D. Tang, L. Chen, and W. Ding, “Efficient beaming from photonic crystal waveguides via self-collimation effect,” Appl. Phys. Lett. 89(13), 131120 (2006).
[Crossref]

Doeleman, H. M.

H. M. Doeleman, F. Monticone, W. Hollander, A. Alù, and A. F. Koenderink, “Experimental observation of a polarization vortex at an optical bound state in the continuum,” Nat. Photonics 12(7), 397–401 (2018).
[Crossref]

Dong, J.-W.

W.-J. Chen, S.-J. Jiang, X.-D. Chen, B. Zhu, L. Zhou, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5(1), 5782 (2014).
[Crossref]

Fage-Pedersen, J.

N. Le Thomas, R. Houdré, L. H. Frandsen, J. Fage-Pedersen, A. Lavrinenko, and P. I. Borel, “Grating-assisted superresolution of slow waves in fourier space,” Phys. Rev. B 76(3), 035103 (2007).
[Crossref]

Fan, S.

X. Yu and S. Fan, “Bends and splitters for self-collimated beams in photonic crystals,” Appl. Phys. Lett. 83(16), 3251–3253 (2003).
[Crossref]

S. Fan and J. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65(23), 235112 (2002).
[Crossref]

Först, M.

Frandsen, L. H.

N. Le Thomas, R. Houdré, L. H. Frandsen, J. Fage-Pedersen, A. Lavrinenko, and P. I. Borel, “Grating-assisted superresolution of slow waves in fourier space,” Phys. Rev. B 76(3), 035103 (2007).
[Crossref]

Gerber, J. A.

J. A. Gerber, S. Berweger, B. T. O’Callahan, and M. B. Raschke, “Phase-resolved surface plasmon interferometry of graphene,” Phys. Rev. Lett. 113(5), 055502 (2014).
[Crossref]

Guan, F.

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(18), 186103 (2018).
[Crossref]

Han, D.

X. Wu, Y. Meng, J. Tian, Y. Huang, H. Xiang, D. Han, and W. Wen, “Direct observation of valley-polarized topological edge states in designer surface plasmon crystals,” Nat. Commun. 8(1), 1304 (2017).
[Crossref]

D. Han, F. Wu, X. Li, C. Xu, X. Liu, and J. Zi, “Transmission and absorption of metallic films coated with corrugated dielectric layers,” Appl. Phys. Lett. 89(9), 091104 (2006).
[Crossref]

Hollander, W.

H. M. Doeleman, F. Monticone, W. Hollander, A. Alù, and A. F. Koenderink, “Experimental observation of a polarization vortex at an optical bound state in the continuum,” Nat. Photonics 12(7), 397–401 (2018).
[Crossref]

Houdré, R.

N. Le Thomas, R. Houdré, D. Beggs, and T. Krauss, “Fourier space imaging of light localization at a photonic band-edge located below the light cone,” Phys. Rev. B 79(3), 033305 (2009).
[Crossref]

J. Jágerská, N. Le Thomas, R. Houdré, J. Bolten, C. Moormann, T. Wahlbrink, J. Ctyrokỳ, M. Waldow, and M. Först, “Dispersion properties of silicon nanophotonic waveguides investigated with fourier optics,” Opt. Lett. 32(18), 2723–2725 (2007).
[Crossref]

N. Le Thomas, R. Houdré, L. H. Frandsen, J. Fage-Pedersen, A. Lavrinenko, and P. I. Borel, “Grating-assisted superresolution of slow waves in fourier space,” Phys. Rev. B 76(3), 035103 (2007).
[Crossref]

Hsu, C. W.

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(18), 186103 (2018).
[Crossref]

E. C. Regan, Y. Igarashi, B. Zhen, I. Kaminer, C. W. Hsu, Y. Shen, J. D. Joannopoulos, and M. Soljačić, “Direct imaging of isofrequency contours in photonic structures,” Sci. Adv. 2(11), e1601591 (2016).
[Crossref]

Huang, X.

X. Huang, M. Xiao, Z.-Q. Zhang, and C. T. Chan, “Sufficient condition for the existence of interface states in some two-dimensional photonic crystals,” Phys. Rev. B 90(7), 075423 (2014).
[Crossref]

Huang, Y.

X. Wu, Y. Meng, J. Tian, Y. Huang, H. Xiang, D. Han, and W. Wen, “Direct observation of valley-polarized topological edge states in designer surface plasmon crystals,” Nat. Commun. 8(1), 1304 (2017).
[Crossref]

Igarashi, Y.

E. C. Regan, Y. Igarashi, B. Zhen, I. Kaminer, C. W. Hsu, Y. Shen, J. D. Joannopoulos, and M. Soljačić, “Direct imaging of isofrequency contours in photonic structures,” Sci. Adv. 2(11), e1601591 (2016).
[Crossref]

Jágerská, J.

Jiang, S.-J.

W.-J. Chen, S.-J. Jiang, X.-D. Chen, B. Zhu, L. Zhou, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5(1), 5782 (2014).
[Crossref]

Joannopoulos, J.

M. Povinelli, S. G. Johnson, and J. Joannopoulos, “Slow-light, band-edge waveguides for tunable time delays,” Opt. Express 13(18), 7145–7159 (2005).
[Crossref]

S. Fan and J. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65(23), 235112 (2002).
[Crossref]

C. Luo, S. G. Johnson, J. Joannopoulos, and J. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65(20), 201104 (2002).
[Crossref]

Joannopoulos, J. D.

E. C. Regan, Y. Igarashi, B. Zhen, I. Kaminer, C. W. Hsu, Y. Shen, J. D. Joannopoulos, and M. Soljačić, “Direct imaging of isofrequency contours in photonic structures,” Sci. Adv. 2(11), e1601591 (2016).
[Crossref]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2011).

Johnson, S. G.

M. Povinelli, S. G. Johnson, and J. Joannopoulos, “Slow-light, band-edge waveguides for tunable time delays,” Opt. Express 13(18), 7145–7159 (2005).
[Crossref]

C. Luo, S. G. Johnson, J. Joannopoulos, and J. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65(20), 201104 (2002).
[Crossref]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2011).

Kaminer, I.

E. C. Regan, Y. Igarashi, B. Zhen, I. Kaminer, C. W. Hsu, Y. Shen, J. D. Joannopoulos, and M. Soljačić, “Direct imaging of isofrequency contours in photonic structures,” Sci. Adv. 2(11), e1601591 (2016).
[Crossref]

Kawakami, S.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58(16), R10096 (1998).
[Crossref]

Kawashima, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58(16), R10096 (1998).
[Crossref]

Koenderink, A. F.

H. M. Doeleman, F. Monticone, W. Hollander, A. Alù, and A. F. Koenderink, “Experimental observation of a polarization vortex at an optical bound state in the continuum,” Nat. Photonics 12(7), 397–401 (2018).
[Crossref]

Kosaka, H.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58(16), R10096 (1998).
[Crossref]

Krauss, T.

N. Le Thomas, R. Houdré, D. Beggs, and T. Krauss, “Fourier space imaging of light localization at a photonic band-edge located below the light cone,” Phys. Rev. B 79(3), 033305 (2009).
[Crossref]

Kreuzer, M. P.

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329(5994), 930–933 (2010).
[Crossref]

Lavrinenko, A.

N. Le Thomas, R. Houdré, L. H. Frandsen, J. Fage-Pedersen, A. Lavrinenko, and P. I. Borel, “Grating-assisted superresolution of slow waves in fourier space,” Phys. Rev. B 76(3), 035103 (2007).
[Crossref]

Le Thomas, N.

N. Le Thomas, R. Houdré, D. Beggs, and T. Krauss, “Fourier space imaging of light localization at a photonic band-edge located below the light cone,” Phys. Rev. B 79(3), 033305 (2009).
[Crossref]

J. Jágerská, N. Le Thomas, R. Houdré, J. Bolten, C. Moormann, T. Wahlbrink, J. Ctyrokỳ, M. Waldow, and M. Först, “Dispersion properties of silicon nanophotonic waveguides investigated with fourier optics,” Opt. Lett. 32(18), 2723–2725 (2007).
[Crossref]

N. Le Thomas, R. Houdré, L. H. Frandsen, J. Fage-Pedersen, A. Lavrinenko, and P. I. Borel, “Grating-assisted superresolution of slow waves in fourier space,” Phys. Rev. B 76(3), 035103 (2007).
[Crossref]

Li, X.

D. Han, F. Wu, X. Li, C. Xu, X. Liu, and J. Zi, “Transmission and absorption of metallic films coated with corrugated dielectric layers,” Appl. Phys. Lett. 89(9), 091104 (2006).
[Crossref]

Liu, W.

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(18), 186103 (2018).
[Crossref]

Liu, X.

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(18), 186103 (2018).
[Crossref]

L. Shi, H. Yin, X. Zhu, X. Liu, and J. Zi, “Direct observation of iso-frequency contour of surface modes in defective photonic crystals in real space,” Appl. Phys. Lett. 97(25), 251111 (2010).
[Crossref]

D. Han, F. Wu, X. Li, C. Xu, X. Liu, and J. Zi, “Transmission and absorption of metallic films coated with corrugated dielectric layers,” Appl. Phys. Lett. 89(9), 091104 (2006).
[Crossref]

Lu, L.

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(18), 186103 (2018).
[Crossref]

Luo, C.

C. Luo, S. G. Johnson, J. Joannopoulos, and J. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65(20), 201104 (2002).
[Crossref]

Marti, J.

A. Martínez and J. Marti, “Negative refraction in two-dimensional photonic crystals: Role of lattice orientation and interface termination,” Phys. Rev. B 71(23), 235115 (2005).
[Crossref]

Martínez, A.

A. Martínez and J. Marti, “Negative refraction in two-dimensional photonic crystals: Role of lattice orientation and interface termination,” Phys. Rev. B 71(23), 235115 (2005).
[Crossref]

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2011).

Meng, Y.

X. Wu, Y. Meng, J. Tian, Y. Huang, H. Xiang, D. Han, and W. Wen, “Direct observation of valley-polarized topological edge states in designer surface plasmon crystals,” Nat. Commun. 8(1), 1304 (2017).
[Crossref]

Monticone, F.

H. M. Doeleman, F. Monticone, W. Hollander, A. Alù, and A. F. Koenderink, “Experimental observation of a polarization vortex at an optical bound state in the continuum,” Nat. Photonics 12(7), 397–401 (2018).
[Crossref]

Moormann, C.

Notomi, M.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58(16), R10096 (1998).
[Crossref]

O’Callahan, B. T.

J. A. Gerber, S. Berweger, B. T. O’Callahan, and M. B. Raschke, “Phase-resolved surface plasmon interferometry of graphene,” Phys. Rev. Lett. 113(5), 055502 (2014).
[Crossref]

Pendry, J.

C. Luo, S. G. Johnson, J. Joannopoulos, and J. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65(20), 201104 (2002).
[Crossref]

Povinelli, M.

Quidant, R.

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329(5994), 930–933 (2010).
[Crossref]

Raschke, M. B.

J. A. Gerber, S. Berweger, B. T. O’Callahan, and M. B. Raschke, “Phase-resolved surface plasmon interferometry of graphene,” Phys. Rev. Lett. 113(5), 055502 (2014).
[Crossref]

Regan, E. C.

E. C. Regan, Y. Igarashi, B. Zhen, I. Kaminer, C. W. Hsu, Y. Shen, J. D. Joannopoulos, and M. Soljačić, “Direct imaging of isofrequency contours in photonic structures,” Sci. Adv. 2(11), e1601591 (2016).
[Crossref]

Sato, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58(16), R10096 (1998).
[Crossref]

Shen, Y.

E. C. Regan, Y. Igarashi, B. Zhen, I. Kaminer, C. W. Hsu, Y. Shen, J. D. Joannopoulos, and M. Soljačić, “Direct imaging of isofrequency contours in photonic structures,” Sci. Adv. 2(11), e1601591 (2016).
[Crossref]

Shi, L.

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(18), 186103 (2018).
[Crossref]

L. Shi, H. Yin, X. Zhu, X. Liu, and J. Zi, “Direct observation of iso-frequency contour of surface modes in defective photonic crystals in real space,” Appl. Phys. Lett. 97(25), 251111 (2010).
[Crossref]

Soljacic, M.

E. C. Regan, Y. Igarashi, B. Zhen, I. Kaminer, C. W. Hsu, Y. Shen, J. D. Joannopoulos, and M. Soljačić, “Direct imaging of isofrequency contours in photonic structures,” Sci. Adv. 2(11), e1601591 (2016).
[Crossref]

Tamamura, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58(16), R10096 (1998).
[Crossref]

Taminiau, T. H.

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329(5994), 930–933 (2010).
[Crossref]

Tang, D.

D. Tang, L. Chen, and W. Ding, “Efficient beaming from photonic crystal waveguides via self-collimation effect,” Appl. Phys. Lett. 89(13), 131120 (2006).
[Crossref]

Tian, J.

X. Wu, Y. Meng, J. Tian, Y. Huang, H. Xiang, D. Han, and W. Wen, “Direct observation of valley-polarized topological edge states in designer surface plasmon crystals,” Nat. Commun. 8(1), 1304 (2017).
[Crossref]

Tomita, A.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58(16), R10096 (1998).
[Crossref]

Ulliac, G.

J. Amet, G. Ulliac, F. Baida, and M.-P. Bernal, “Experimental evidence of enhanced electro-optic control on a lithium niobate photonic crystal superprism,” Appl. Phys. Lett. 96(10), 103111 (2010).
[Crossref]

van Hulst, N. F.

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329(5994), 930–933 (2010).
[Crossref]

Volpe, G.

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329(5994), 930–933 (2010).
[Crossref]

Wahlbrink, T.

Waldow, M.

Wang, B.

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(18), 186103 (2018).
[Crossref]

Wen, W.

X. Wu, Y. Meng, J. Tian, Y. Huang, H. Xiang, D. Han, and W. Wen, “Direct observation of valley-polarized topological edge states in designer surface plasmon crystals,” Nat. Commun. 8(1), 1304 (2017).
[Crossref]

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2011).

Wu, F.

D. Han, F. Wu, X. Li, C. Xu, X. Liu, and J. Zi, “Transmission and absorption of metallic films coated with corrugated dielectric layers,” Appl. Phys. Lett. 89(9), 091104 (2006).
[Crossref]

Wu, X.

X. Wu, Y. Meng, J. Tian, Y. Huang, H. Xiang, D. Han, and W. Wen, “Direct observation of valley-polarized topological edge states in designer surface plasmon crystals,” Nat. Commun. 8(1), 1304 (2017).
[Crossref]

Xiang, H.

X. Wu, Y. Meng, J. Tian, Y. Huang, H. Xiang, D. Han, and W. Wen, “Direct observation of valley-polarized topological edge states in designer surface plasmon crystals,” Nat. Commun. 8(1), 1304 (2017).
[Crossref]

Xiao, M.

X. Huang, M. Xiao, Z.-Q. Zhang, and C. T. Chan, “Sufficient condition for the existence of interface states in some two-dimensional photonic crystals,” Phys. Rev. B 90(7), 075423 (2014).
[Crossref]

Xu, C.

D. Han, F. Wu, X. Li, C. Xu, X. Liu, and J. Zi, “Transmission and absorption of metallic films coated with corrugated dielectric layers,” Appl. Phys. Lett. 89(9), 091104 (2006).
[Crossref]

Yin, H.

L. Shi, H. Yin, X. Zhu, X. Liu, and J. Zi, “Direct observation of iso-frequency contour of surface modes in defective photonic crystals in real space,” Appl. Phys. Lett. 97(25), 251111 (2010).
[Crossref]

Yu, X.

X. Yu and S. Fan, “Bends and splitters for self-collimated beams in photonic crystals,” Appl. Phys. Lett. 83(16), 3251–3253 (2003).
[Crossref]

Zhang, Y.

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(18), 186103 (2018).
[Crossref]

Zhang, Z.-Q.

X. Huang, M. Xiao, Z.-Q. Zhang, and C. T. Chan, “Sufficient condition for the existence of interface states in some two-dimensional photonic crystals,” Phys. Rev. B 90(7), 075423 (2014).
[Crossref]

Zhen, B.

E. C. Regan, Y. Igarashi, B. Zhen, I. Kaminer, C. W. Hsu, Y. Shen, J. D. Joannopoulos, and M. Soljačić, “Direct imaging of isofrequency contours in photonic structures,” Sci. Adv. 2(11), e1601591 (2016).
[Crossref]

Zhou, L.

W.-J. Chen, S.-J. Jiang, X.-D. Chen, B. Zhu, L. Zhou, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5(1), 5782 (2014).
[Crossref]

Zhu, B.

W.-J. Chen, S.-J. Jiang, X.-D. Chen, B. Zhu, L. Zhou, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5(1), 5782 (2014).
[Crossref]

Zhu, X.

L. Shi, H. Yin, X. Zhu, X. Liu, and J. Zi, “Direct observation of iso-frequency contour of surface modes in defective photonic crystals in real space,” Appl. Phys. Lett. 97(25), 251111 (2010).
[Crossref]

Zi, J.

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(18), 186103 (2018).
[Crossref]

L. Shi, H. Yin, X. Zhu, X. Liu, and J. Zi, “Direct observation of iso-frequency contour of surface modes in defective photonic crystals in real space,” Appl. Phys. Lett. 97(25), 251111 (2010).
[Crossref]

D. Han, F. Wu, X. Li, C. Xu, X. Liu, and J. Zi, “Transmission and absorption of metallic films coated with corrugated dielectric layers,” Appl. Phys. Lett. 89(9), 091104 (2006).
[Crossref]

Appl. Phys. Lett. (5)

J. Amet, G. Ulliac, F. Baida, and M.-P. Bernal, “Experimental evidence of enhanced electro-optic control on a lithium niobate photonic crystal superprism,” Appl. Phys. Lett. 96(10), 103111 (2010).
[Crossref]

X. Yu and S. Fan, “Bends and splitters for self-collimated beams in photonic crystals,” Appl. Phys. Lett. 83(16), 3251–3253 (2003).
[Crossref]

D. Tang, L. Chen, and W. Ding, “Efficient beaming from photonic crystal waveguides via self-collimation effect,” Appl. Phys. Lett. 89(13), 131120 (2006).
[Crossref]

L. Shi, H. Yin, X. Zhu, X. Liu, and J. Zi, “Direct observation of iso-frequency contour of surface modes in defective photonic crystals in real space,” Appl. Phys. Lett. 97(25), 251111 (2010).
[Crossref]

D. Han, F. Wu, X. Li, C. Xu, X. Liu, and J. Zi, “Transmission and absorption of metallic films coated with corrugated dielectric layers,” Appl. Phys. Lett. 89(9), 091104 (2006).
[Crossref]

Nat. Commun. (2)

X. Wu, Y. Meng, J. Tian, Y. Huang, H. Xiang, D. Han, and W. Wen, “Direct observation of valley-polarized topological edge states in designer surface plasmon crystals,” Nat. Commun. 8(1), 1304 (2017).
[Crossref]

W.-J. Chen, S.-J. Jiang, X.-D. Chen, B. Zhu, L. Zhou, J.-W. Dong, and C. T. Chan, “Experimental realization of photonic topological insulator in a uniaxial metacrystal waveguide,” Nat. Commun. 5(1), 5782 (2014).
[Crossref]

Nat. Photonics (1)

H. M. Doeleman, F. Monticone, W. Hollander, A. Alù, and A. F. Koenderink, “Experimental observation of a polarization vortex at an optical bound state in the continuum,” Nat. Photonics 12(7), 397–401 (2018).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. B (7)

N. Le Thomas, R. Houdré, D. Beggs, and T. Krauss, “Fourier space imaging of light localization at a photonic band-edge located below the light cone,” Phys. Rev. B 79(3), 033305 (2009).
[Crossref]

X. Huang, M. Xiao, Z.-Q. Zhang, and C. T. Chan, “Sufficient condition for the existence of interface states in some two-dimensional photonic crystals,” Phys. Rev. B 90(7), 075423 (2014).
[Crossref]

S. Fan and J. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65(23), 235112 (2002).
[Crossref]

C. Luo, S. G. Johnson, J. Joannopoulos, and J. Pendry, “All-angle negative refraction without negative effective index,” Phys. Rev. B 65(20), 201104 (2002).
[Crossref]

A. Martínez and J. Marti, “Negative refraction in two-dimensional photonic crystals: Role of lattice orientation and interface termination,” Phys. Rev. B 71(23), 235115 (2005).
[Crossref]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58(16), R10096 (1998).
[Crossref]

N. Le Thomas, R. Houdré, L. H. Frandsen, J. Fage-Pedersen, A. Lavrinenko, and P. I. Borel, “Grating-assisted superresolution of slow waves in fourier space,” Phys. Rev. B 76(3), 035103 (2007).
[Crossref]

Phys. Rev. Lett. (2)

J. A. Gerber, S. Berweger, B. T. O’Callahan, and M. B. Raschke, “Phase-resolved surface plasmon interferometry of graphene,” Phys. Rev. Lett. 113(5), 055502 (2014).
[Crossref]

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(18), 186103 (2018).
[Crossref]

Sci. Adv. (1)

E. C. Regan, Y. Igarashi, B. Zhen, I. Kaminer, C. W. Hsu, Y. Shen, J. D. Joannopoulos, and M. Soljačić, “Direct imaging of isofrequency contours in photonic structures,” Sci. Adv. 2(11), e1601591 (2016).
[Crossref]

Science (1)

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329(5994), 930–933 (2010).
[Crossref]

Other (1)

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 2011).

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Figures (4)

Fig. 1.
Fig. 1. Schematic views and corresponding band structures simulated along $x$ -direction for (a) simple and (b) compound lattices of the one-dimensional PMMA grating on flat Ag substrate. The periods are $a$ and $2a$ for these two lattices respectively, with $a = 260$ nm. The width of air for (a) is $l = 70$ nm, while two widths of air for (b) are $l = 70$ nm and $l^{\prime } = 100$ nm. The red line is light line. (c) Iso-frequency contours of the one-dimensional compound lattice: area inside the red circle corresponds to the cross section of light cone; solid black circles are the iso-frequency contours of the one-dimensional simple lattice, which are spaced by $2\pi /a$ , the base vector of the reciprocal lattice. Dashed circles are iso-frequency contours folded by periodic weak scattering of the compound lattice. (d) Field distributions of the states represented by the four colored points(red,blue,yellow,purple) in (a) and (b). Nearly same fields for red (blue) and yellow(purple) points show the physical essence of the periodic weak scattering. (e) Positions of the two bands at the boundary of first Brillouin zone(FBZ). $\Delta l$ is the variation of the width of air changed. The corresponding relative change of band gap $\Delta d/ d =(d^{\prime } - d)/ d$ is shown in (f).
Fig. 2.
Fig. 2. (a) Schematic view of the experimental setup. L, Lens. (b-c) SEM images and corresponding band structures (extinction spectra) of simple lattice (b) and its compound counterpart (c), obtained experimentally. The scale bar is 250 nm. White regions correspond to the states out of the numerical aperture of the microscope. To compare with the simulation results, the light line is also shown as red line. (d) Three typical iso-frequency contours with 450, 463 and 512 THz for the compound lattice show the evolution of iso-frequency contours when the frequency increases. The green one corresponds to the case shown in Fig. 1(c).
Fig. 3.
Fig. 3. (a) Zoomed band structures of compound lattice shown in Fig. 2(c). Band structures with negative wave vectors are also shown. A reciprocal lattice vector along $k_{x}$ direction of the compound lattice is $2 \pi /2a = \pi /a$ . (b) The obtained band structure of the 1D simple lattice, with the simulation results denoted by orange circles for comparisons. The red line is light line.
Fig. 4.
Fig. 4. Experimental band structures along $\Gamma$ -X (a) and $\Gamma$ -M (b) of 2D square lattice. The lattice constant is 360 nm. The radius of air circle is 90 nm. (c) and (d) show the measured band structures for two designed compound lattices for the two symmetry directions. The radius of the bigger air holes is 120 nm, while that of the smaller ones is fixed to 90 nm. The scale bar is 500 nm. (e) and (f) are the obtained bands of simple 2D square lattice, with the simulated results denoted by orange circles. The red line is light line. (g) and (h) give the corresponding iso-frequency contours at the frequencies 353 and 435 THz, respectively.

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