Abstract

Photonic crystals (PhCs) act on light in two different ways: confinement and modification of propagation. Both phenomena rely on the complex interplay between multiply scattered waves that can form what is known as a Bloch mode. Here, we present a technique that allows direct imaging of Bloch modes, both in real space and in k-space. The technique gives access to the location of the field maxima inside the PhC, the dispersion relation, equifrequency surfaces, as well as reflection and transmission coefficients. Our key advance is that we retrieve the desired information comprehensively, without postprocessing or cumbersome near-field scanning techniques, even for modes that are nominally lossless, i.e., below the light line. To highlight the potential of the technique, we extract the dispersion curve of a coupled cavity waveguide consisting of as many as 100 cavities, as well as the equifrequency surfaces and polarization properties of a PhC beam splitter.

© 2007 Optical Society of America

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  1. T. F. Krauss, R. M. De La Rue, and S. Brand, "Two-dimensional photonic-bandgap structures operating at near-infrared wavelengths," Nature 383, 699-702 (1996).
    [CrossRef]
  2. X. Letartre, C. Seassal, C. Grillet, P. Rojo-Romeo, P. Viktorovitch, M. Le Vassor d'Yerville, D. Cassagne, and C. Jouanin, "Group velocity and propagation losses measurement in a single-line photonic-crystal waveguide on InP membranes," Appl. Phys. Lett. 79, 2312-2314 (2001).
    [CrossRef]
  3. M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, "Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs," Phys. Rev. Lett. 87, 253902 (2001).
    [CrossRef] [PubMed]
  4. T. Baba and T. Matsumoto, "Resolution of photonic crystal superprism," Appl. Phys. Lett. 81, 2325-2327 (2002).
    [CrossRef]
  5. M. J. Steel, R. Zoli, C. Grillet, R. C. McPhedran, C. Martijn de Sterke, A. Norton, P. Bassi, and B. J. Eggleton, "Analytic properties of photonic crystal superprism parameters," Phys. Rev. E 71, 056608 (2005).
    [CrossRef]
  6. H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, "Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides," Phys. Rev. Lett. 94, 123901 (2005).
    [CrossRef] [PubMed]
  7. H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, "Real-space observation of ultraslow light in photonic crystal waveguides," Phys. Rev. Lett. 94, 073903 (2005).
    [CrossRef] [PubMed]
  8. R. Wüest, D. Erni, P. Strasser, F. Robin, H. Jäckel, B. C. Buchler, A. F. Koenderink, V. Sandoghdar, and R. Harbers, "A standing-wave meter to measure dispersion and loss of photonic-crystal waveguides," Appl. Phys. Lett. 87, 261110 (2005).
    [CrossRef]
  9. P. Kramper, M. Kafesaki, C. M. Soukoulis, A. Birner, F. Mller, U. Gsele, R. B. Wehrspohn, J. Mlynek, and V. Sandoghdar, "Near-field visualization of light confinement in a photonic crystal microresonator," Opt. Lett. 29, 174-176 (2004).
    [CrossRef] [PubMed]
  10. N. Louvion, D. Gérard, J. Mouette, F. de Fornel, C. Seassal, X. Letartre, A. Rahmani, and S. Callard, "Local observation and spectroscopy of optical modes in an active photonic-crystal microcavity," Phys. Rev. Lett. 94, 113907 (2005).
    [CrossRef] [PubMed]
  11. R. Zengerle, "Light propagation in singly and doubly periodic planar waveguides," J. Mod. Opt. 34, 1589-1617 (1987).
    [CrossRef]
  12. M. Galli, D. Bajoni, M. Patrini, G. Guizzetti, D. Gerace, L. C. Andreani, M. Belotti, and Y. Chen, Phys. Rev. B 72, 125322 (2005).
    [CrossRef]
  13. M. Loncar, D. Nedeljkovic, T. P. Pearsall, J. Vuckovic, A. Scherer, S. Kuchinsky, and D. C. Allan, "Experimental and theoretical confirmation of Bloch-mode light propagation in planar photonic crystal waveguides," Appl. Phys. Lett. 80, 1689-1691 (2002).
    [CrossRef]
  14. G. Bartal, O. Cohen, H. Buljan, J. W. Fleischer, O. Manela, and M. Segev, "Brillouin zone spectroscopy of nonlinear photonic lattices," Phys. Rev. Lett. 94, 163902 (2005).
    [CrossRef] [PubMed]
  15. P. St. J. Russell, "Optics of Floquet-Bloch waves in dielectric gratings," Appl. Phys. B 39, 231-246 (1986).
    [CrossRef]
  16. E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopuolou, and C. M. Soukoulis, "Negative refraction by photonic crystals," Nature 423, 604-605 (2003).
    [CrossRef] [PubMed]
  17. C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, "All-angle negative refraction without negative effective index," Phys. Rev. B 65, 201104 (2002).
    [CrossRef]
  18. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58, 10096-10099 (1998).
    [CrossRef]
  19. S. Yang, J. H. Page, Z. Liu, M. L. Cowan, C. T. Chan, and P. Sheng, "Focusing of sound in a 3D phononic crystal," Phys. Rev. Lett. 93, 024301 (2004).
    [CrossRef] [PubMed]
  20. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Self-collimating phenomena in photonic crystals," Appl. Phys. Lett. 74, 1212-1214 (1999).
    [CrossRef]
  21. P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljacic, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, "Achieving centimeter-scale supercollimation in a large-area two-dimensional photonic crystal," Nat. Mater. 5, 93-96 (2006).
    [CrossRef] [PubMed]
  22. R. Ferrini, A. Berrier, L. A. Dunbar, R. Houdré, M. Mulot, S. Anand, S. de Rossi, and A. Talneau, "Minimization of out-of-plane losses in planar photonic crystals by optimizing the vertical waveguide," Appl. Phys. Lett. 85, 3998-4000 (2004).
    [CrossRef]
  23. Note that here the PhC structure is excited in the second band of the dispersion diagram. As a result the group velocity in the first Brillouin zone is positive for "negative" wave vectors value, hence the image of the FW propagating wave in the Fourier space is located at −kx.
  24. N. Le Thomas, R. Houdré, L. H. Frandsen, J. Fage-Pedersen, A. V. Lavrinenko, and P. I. Borel, "Grating-assisted superresolution of slow waves in Fourier space," Phys. Rev. B 76, 035103 (2007).
    [CrossRef]
  25. M. I. Kobolov and C. Fabre, "Quantum limits on optical resolution," Phys. Rev. Lett. 85, 3789-3792 (2000).
    [CrossRef]
  26. C. K. Rushforth and R. W. Harris, "Restoration, resolution, and noise," J. Opt. Soc. Am. 58, 539-545 (1968).
    [CrossRef]
  27. N. Stefanou and A. Modinos, "Impurity bands in photonic insulator," Phys. Rev. B 57, 12127-12133 (1998).
    [CrossRef]
  28. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, "Coupled-resonator optical waveguide: a proposal and analysis," Opt. Lett. 24, 711-713 (1999).
    [CrossRef]
  29. M. F. Yanik, W. Suh, Z. Wang, and S. Fan, "Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency," Phys. Rev. Lett. 93, 233903 (2004).
    [CrossRef] [PubMed]
  30. M. F. Yanik and S. Fan, "Stopping light all-optically," Phys. Rev. Lett. 92, 083901 (2004).
    [CrossRef] [PubMed]
  31. B.-S. Song, S. Noda, T. Asano, and Y. Akahane, "Ultra-high-Q photonic double-heterostructure nanocavity," Astrophys. J., Suppl. Ser. 4, 93-96 (2005).
  32. M. L. Povinelli and S. Fan, "Radiation loss of coupled-resonator waveguides in photonic-crystal slabs," Appl. Phys. Lett. 89, 191114 (2006).
    [CrossRef]
  33. R. Mitra, Computer Techniques for Electromagnetics (Pergamon, 1973), Chap 2.
  34. E. E. Orlova, J. N. Hovenier, T. O. Klaassen, I. Kasalynas, A. J. L. Adam, J. R. Gao, T. M. Klapwijk, B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, "Antenna model for wire lasers," Phys. Rev. Lett. 96, 173904 (2006).
    [CrossRef] [PubMed]
  35. B. Lombardet, L. A. Dunbar, R. Ferrini, and R. Houdré, "Fourier analysis of Bloch wave propagation in photonic crystals," J. Opt. Soc. Am. B 22, 1179-1190 (2005).
    [CrossRef]

2007

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

2006

P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljacic, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, "Achieving centimeter-scale supercollimation in a large-area two-dimensional photonic crystal," Nat. Mater. 5, 93-96 (2006).
[CrossRef] [PubMed]

M. L. Povinelli and S. Fan, "Radiation loss of coupled-resonator waveguides in photonic-crystal slabs," Appl. Phys. Lett. 89, 191114 (2006).
[CrossRef]

E. E. Orlova, J. N. Hovenier, T. O. Klaassen, I. Kasalynas, A. J. L. Adam, J. R. Gao, T. M. Klapwijk, B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, "Antenna model for wire lasers," Phys. Rev. Lett. 96, 173904 (2006).
[CrossRef] [PubMed]

2005

B.-S. Song, S. Noda, T. Asano, and Y. Akahane, "Ultra-high-Q photonic double-heterostructure nanocavity," Astrophys. J., Suppl. Ser. 4, 93-96 (2005).

M. J. Steel, R. Zoli, C. Grillet, R. C. McPhedran, C. Martijn de Sterke, A. Norton, P. Bassi, and B. J. Eggleton, "Analytic properties of photonic crystal superprism parameters," Phys. Rev. E 71, 056608 (2005).
[CrossRef]

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, "Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides," Phys. Rev. Lett. 94, 123901 (2005).
[CrossRef] [PubMed]

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, "Real-space observation of ultraslow light in photonic crystal waveguides," Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

R. Wüest, D. Erni, P. Strasser, F. Robin, H. Jäckel, B. C. Buchler, A. F. Koenderink, V. Sandoghdar, and R. Harbers, "A standing-wave meter to measure dispersion and loss of photonic-crystal waveguides," Appl. Phys. Lett. 87, 261110 (2005).
[CrossRef]

B. Lombardet, L. A. Dunbar, R. Ferrini, and R. Houdré, "Fourier analysis of Bloch wave propagation in photonic crystals," J. Opt. Soc. Am. B 22, 1179-1190 (2005).
[CrossRef]

N. Louvion, D. Gérard, J. Mouette, F. de Fornel, C. Seassal, X. Letartre, A. Rahmani, and S. Callard, "Local observation and spectroscopy of optical modes in an active photonic-crystal microcavity," Phys. Rev. Lett. 94, 113907 (2005).
[CrossRef] [PubMed]

M. Galli, D. Bajoni, M. Patrini, G. Guizzetti, D. Gerace, L. C. Andreani, M. Belotti, and Y. Chen, Phys. Rev. B 72, 125322 (2005).
[CrossRef]

G. Bartal, O. Cohen, H. Buljan, J. W. Fleischer, O. Manela, and M. Segev, "Brillouin zone spectroscopy of nonlinear photonic lattices," Phys. Rev. Lett. 94, 163902 (2005).
[CrossRef] [PubMed]

2004

R. Ferrini, A. Berrier, L. A. Dunbar, R. Houdré, M. Mulot, S. Anand, S. de Rossi, and A. Talneau, "Minimization of out-of-plane losses in planar photonic crystals by optimizing the vertical waveguide," Appl. Phys. Lett. 85, 3998-4000 (2004).
[CrossRef]

S. Yang, J. H. Page, Z. Liu, M. L. Cowan, C. T. Chan, and P. Sheng, "Focusing of sound in a 3D phononic crystal," Phys. Rev. Lett. 93, 024301 (2004).
[CrossRef] [PubMed]

P. Kramper, M. Kafesaki, C. M. Soukoulis, A. Birner, F. Mller, U. Gsele, R. B. Wehrspohn, J. Mlynek, and V. Sandoghdar, "Near-field visualization of light confinement in a photonic crystal microresonator," Opt. Lett. 29, 174-176 (2004).
[CrossRef] [PubMed]

M. F. Yanik, W. Suh, Z. Wang, and S. Fan, "Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency," Phys. Rev. Lett. 93, 233903 (2004).
[CrossRef] [PubMed]

M. F. Yanik and S. Fan, "Stopping light all-optically," Phys. Rev. Lett. 92, 083901 (2004).
[CrossRef] [PubMed]

2003

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopuolou, and C. M. Soukoulis, "Negative refraction by photonic crystals," Nature 423, 604-605 (2003).
[CrossRef] [PubMed]

2002

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

M. Loncar, D. Nedeljkovic, T. P. Pearsall, J. Vuckovic, A. Scherer, S. Kuchinsky, and D. C. Allan, "Experimental and theoretical confirmation of Bloch-mode light propagation in planar photonic crystal waveguides," Appl. Phys. Lett. 80, 1689-1691 (2002).
[CrossRef]

T. Baba and T. Matsumoto, "Resolution of photonic crystal superprism," Appl. Phys. Lett. 81, 2325-2327 (2002).
[CrossRef]

2001

X. Letartre, C. Seassal, C. Grillet, P. Rojo-Romeo, P. Viktorovitch, M. Le Vassor d'Yerville, D. Cassagne, and C. Jouanin, "Group velocity and propagation losses measurement in a single-line photonic-crystal waveguide on InP membranes," Appl. Phys. Lett. 79, 2312-2314 (2001).
[CrossRef]

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, "Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs," Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

2000

M. I. Kobolov and C. Fabre, "Quantum limits on optical resolution," Phys. Rev. Lett. 85, 3789-3792 (2000).
[CrossRef]

1999

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Self-collimating phenomena in photonic crystals," Appl. Phys. Lett. 74, 1212-1214 (1999).
[CrossRef]

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, "Coupled-resonator optical waveguide: a proposal and analysis," Opt. Lett. 24, 711-713 (1999).
[CrossRef]

1998

N. Stefanou and A. Modinos, "Impurity bands in photonic insulator," Phys. Rev. B 57, 12127-12133 (1998).
[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, 10096-10099 (1998).
[CrossRef]

1996

T. F. Krauss, R. M. De La Rue, and S. Brand, "Two-dimensional photonic-bandgap structures operating at near-infrared wavelengths," Nature 383, 699-702 (1996).
[CrossRef]

1987

R. Zengerle, "Light propagation in singly and doubly periodic planar waveguides," J. Mod. Opt. 34, 1589-1617 (1987).
[CrossRef]

1986

P. St. J. Russell, "Optics of Floquet-Bloch waves in dielectric gratings," Appl. Phys. B 39, 231-246 (1986).
[CrossRef]

1968

Appl. Phys. B

P. St. J. Russell, "Optics of Floquet-Bloch waves in dielectric gratings," Appl. Phys. B 39, 231-246 (1986).
[CrossRef]

Appl. Phys. Lett.

M. Loncar, D. Nedeljkovic, T. P. Pearsall, J. Vuckovic, A. Scherer, S. Kuchinsky, and D. C. Allan, "Experimental and theoretical confirmation of Bloch-mode light propagation in planar photonic crystal waveguides," Appl. Phys. Lett. 80, 1689-1691 (2002).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Self-collimating phenomena in photonic crystals," Appl. Phys. Lett. 74, 1212-1214 (1999).
[CrossRef]

R. Ferrini, A. Berrier, L. A. Dunbar, R. Houdré, M. Mulot, S. Anand, S. de Rossi, and A. Talneau, "Minimization of out-of-plane losses in planar photonic crystals by optimizing the vertical waveguide," Appl. Phys. Lett. 85, 3998-4000 (2004).
[CrossRef]

T. Baba and T. Matsumoto, "Resolution of photonic crystal superprism," Appl. Phys. Lett. 81, 2325-2327 (2002).
[CrossRef]

X. Letartre, C. Seassal, C. Grillet, P. Rojo-Romeo, P. Viktorovitch, M. Le Vassor d'Yerville, D. Cassagne, and C. Jouanin, "Group velocity and propagation losses measurement in a single-line photonic-crystal waveguide on InP membranes," Appl. Phys. Lett. 79, 2312-2314 (2001).
[CrossRef]

R. Wüest, D. Erni, P. Strasser, F. Robin, H. Jäckel, B. C. Buchler, A. F. Koenderink, V. Sandoghdar, and R. Harbers, "A standing-wave meter to measure dispersion and loss of photonic-crystal waveguides," Appl. Phys. Lett. 87, 261110 (2005).
[CrossRef]

M. L. Povinelli and S. Fan, "Radiation loss of coupled-resonator waveguides in photonic-crystal slabs," Appl. Phys. Lett. 89, 191114 (2006).
[CrossRef]

Astrophys. J., Suppl. Ser.

B.-S. Song, S. Noda, T. Asano, and Y. Akahane, "Ultra-high-Q photonic double-heterostructure nanocavity," Astrophys. J., Suppl. Ser. 4, 93-96 (2005).

J. Mod. Opt.

R. Zengerle, "Light propagation in singly and doubly periodic planar waveguides," J. Mod. Opt. 34, 1589-1617 (1987).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

Nat. Mater.

P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljacic, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, "Achieving centimeter-scale supercollimation in a large-area two-dimensional photonic crystal," Nat. Mater. 5, 93-96 (2006).
[CrossRef] [PubMed]

Nature

T. F. Krauss, R. M. De La Rue, and S. Brand, "Two-dimensional photonic-bandgap structures operating at near-infrared wavelengths," Nature 383, 699-702 (1996).
[CrossRef]

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopuolou, and C. M. Soukoulis, "Negative refraction by photonic crystals," Nature 423, 604-605 (2003).
[CrossRef] [PubMed]

Opt. Lett.

Phys. Rev. B

N. Stefanou and A. Modinos, "Impurity bands in photonic insulator," Phys. Rev. B 57, 12127-12133 (1998).
[CrossRef]

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, "All-angle negative refraction without negative effective index," Phys. Rev. B 65, 201104 (2002).
[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, 10096-10099 (1998).
[CrossRef]

M. Galli, D. Bajoni, M. Patrini, G. Guizzetti, D. Gerace, L. C. Andreani, M. Belotti, and Y. Chen, Phys. Rev. B 72, 125322 (2005).
[CrossRef]

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

Phys. Rev. E

M. J. Steel, R. Zoli, C. Grillet, R. C. McPhedran, C. Martijn de Sterke, A. Norton, P. Bassi, and B. J. Eggleton, "Analytic properties of photonic crystal superprism parameters," Phys. Rev. E 71, 056608 (2005).
[CrossRef]

Phys. Rev. Lett.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, "Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides," Phys. Rev. Lett. 94, 123901 (2005).
[CrossRef] [PubMed]

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, "Real-space observation of ultraslow light in photonic crystal waveguides," Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

M. F. Yanik, W. Suh, Z. Wang, and S. Fan, "Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency," Phys. Rev. Lett. 93, 233903 (2004).
[CrossRef] [PubMed]

M. F. Yanik and S. Fan, "Stopping light all-optically," Phys. Rev. Lett. 92, 083901 (2004).
[CrossRef] [PubMed]

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, "Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs," Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

E. E. Orlova, J. N. Hovenier, T. O. Klaassen, I. Kasalynas, A. J. L. Adam, J. R. Gao, T. M. Klapwijk, B. S. Williams, S. Kumar, Q. Hu, and J. L. Reno, "Antenna model for wire lasers," Phys. Rev. Lett. 96, 173904 (2006).
[CrossRef] [PubMed]

M. I. Kobolov and C. Fabre, "Quantum limits on optical resolution," Phys. Rev. Lett. 85, 3789-3792 (2000).
[CrossRef]

N. Louvion, D. Gérard, J. Mouette, F. de Fornel, C. Seassal, X. Letartre, A. Rahmani, and S. Callard, "Local observation and spectroscopy of optical modes in an active photonic-crystal microcavity," Phys. Rev. Lett. 94, 113907 (2005).
[CrossRef] [PubMed]

G. Bartal, O. Cohen, H. Buljan, J. W. Fleischer, O. Manela, and M. Segev, "Brillouin zone spectroscopy of nonlinear photonic lattices," Phys. Rev. Lett. 94, 163902 (2005).
[CrossRef] [PubMed]

S. Yang, J. H. Page, Z. Liu, M. L. Cowan, C. T. Chan, and P. Sheng, "Focusing of sound in a 3D phononic crystal," Phys. Rev. Lett. 93, 024301 (2004).
[CrossRef] [PubMed]

Other

Note that here the PhC structure is excited in the second band of the dispersion diagram. As a result the group velocity in the first Brillouin zone is positive for "negative" wave vectors value, hence the image of the FW propagating wave in the Fourier space is located at −kx.

R. Mitra, Computer Techniques for Electromagnetics (Pergamon, 1973), Chap 2.

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Figures (7)

Fig. 1
Fig. 1

Experimental setup; O microscope objective (NA 0.9), L 1 , L 2 , and L 3 achromatic lenses and on the left electron microscopy image of a PhC tile with the access ridge waveguides. The green path corresponds to the near-field imaging and the blue corresponds to the far-field imaging with L 2 removed. ξ and x are the Fourier plane and intermediate image plane, respectively.

Fig. 2
Fig. 2

(a)–(d) Near-field image of the Bloch wave mode in a square lattice PhC tile of 80 periods for different reduced energies u. Below each real space image is shown the corresponding far-field image from (e) to (h). The thin black curves in the lower half space of the far-field frame are the 2D PWE theoretical EFSs starting from u = 0.315 in the center and drawn with a step of 0.005 . The far-field image is limited by the maximum propagating wave vector k max = 0.2636 × 2 π a (the red circle in Figs. 2e, 2f, 2g, 2h delimits the output pupil of the objective that defines k max ). The corresponding diffraction limited spatial resolution is 0.85 μ m .

Fig. 3
Fig. 3

Dispersion curves for a square lattice PhC for TE (blue) and TM (green) polarization, plotted versus the modulus of the wave vector. The points represent the experimental data whereas the lines are for the theoretical PWE calculation ( f = 25 % , n e f f T E = 3.28 , and n e f f T M = 3.22 ). The dotted line is the light line of the microscope objective. Two experimental data points below this light line are highlighted with circle-cross points. The horizontal error bars are given by the size of the points while the vertical ones are far below the size of the points.

Fig. 4
Fig. 4

Experimental Fourier space spectra of square lattice PhC tiles operating in the self-collimating regime ( u = 0.283 ) for TE polarization, with length: 9 (1), 18, (2), and 36 μ m (3). Thin dark curve: sinc fits. The spatial frequency range outside the bandwidth of the setup is highlighted in gray. Inset: log-scale plot of spectrum 2 corresponding to the 18 μ m long tile.

Fig. 5
Fig. 5

(a) Experimental Fourier space spectrum (log-scale) of a 9.7 μ m long W1 waveguide (dark curve), analytical continuation of the experimental spectrum (blue curve), and theoretical spectrum (red curve) at λ = 1565 nm ( u = 0.281 ) . FWi: Forward propagating Bloch wave in the i m e Brillouin zone, BWi: Backward propagating Bloch wave in the i m e Brillouin zone. Dotted and dashed lines: light lines of the objective and of air, respectively. (b) Experimental dispersion curve determined from the analytical continuation of the Fourier space spectrum (circles), and theoretical dispersion curve (blue curve). (c) End-fire optical transmission through the W1 waveguide. (d) Electron microscopy image of the W1 waveguide ( a = 440 nm ) structure.

Fig. 6
Fig. 6

(a) Electron microscopy image of the coupled cavity waveguide, (b) and (c) real space and Fourier space images, respectively, of the CCW excited at λ = 1535 nm ( u = 0.287 ) . The light radiated from the 100 cavities is observed as bright spots along the entire waveguide in real space and appears as vertical straight lines in the far field separated by the reciprocal vector G s c of the CCW supercell. (d) Dispersion diagram of the CCW in its first, second, third, and fourth Brillouin zones, plotted versus the modulus of the wave vector, and with a = 440 nm chosen for the normalization. Purple, blue, dark, and green experimental dots: position in k-space of the d 1 , d 2 , d 3 , and d 4 lines, respectively. Square dots: experimental dispersion curve of the 9.7 μ m long W1 waveguide (see Fig. 3). Gray curves: theoretical 2D plane wave expansion calculations, with 30% and 2.65 as fitting parameters for the filling factor and the effective index, respectively. The deviation in the slope of the dispersion curve can be explained by the discrepancy between the idealized 2D model and the real 3D experiment, and not from the number of plane waves (3411) used in the model. Dotted and dashed lines: light lines of the objective and of air, respectively.

Fig. 7
Fig. 7

(a) and (b) near-field image of the TM and TE waves, respectively, propagating in the double slab polarizer beam splitter (color coded: increasing intensity from blue to red) superimposed with an optical microscope image of the structure (black and white), (c) emission diagram of the TE wave imaged in the Fourier space at λ = 1550 nm , (d) same as (c) with the emission of the second splitter filtered in the intermediate image plane, (e) reflection coefficient of the first slab deduced from the far-field pattern measured for different wavelengths as in (c).

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