Nonlinear Bloch modes in two-dimensional photonic lattices
Optics Express, Vol. 14, Issue 5, pp. 1913-1923 (2006)
http://dx.doi.org/10.1364/OE.14.001913
Acrobat PDF (275 KB)
Abstract
We generate experimentally different types of two-dimensional Bloch waves of a square photonic lattice by employing the phase imprinting technique. We probe the local dispersion of the Bloch modes in the photonic lattice by analyzing the linear diffraction of beams associated with the high-symmetry points of the Brillouin zone, and also distinguish the regimes of normal, anomalous, and anisotropic diffraction through observations of nonlinear self-action effects.
© 2006 Optical Society of America
1. Introduction
2. P. S. Russell, “Bloch wave analysis of dispersion and pulse-propagation in pure distributed feedback structures,” J. Mod. Opt. 38, 1599–1619 (1991). [CrossRef]
3. P. St. J. Russell, T. A. Birks, and F. D. Lloyd Lucas, “Photonic Bloch waves and photonic band gaps,” in Confined Electrons and Photons, E. Burstein and C. Weisbuch, eds., (1995), pp. 585–633. [CrossRef]
3. P. St. J. Russell, T. A. Birks, and F. D. Lloyd Lucas, “Photonic Bloch waves and photonic band gaps,” in Confined Electrons and Photons, E. Burstein and C. Weisbuch, eds., (1995), pp. 585–633. [CrossRef]
4. H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863–1866 (2000). [CrossRef] [PubMed]
5. T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett. 88, 093901–4 (2002). [CrossRef] [PubMed]
6. M. Lončar, D. NedeljkoviĆ, T. P. Pearsall, J. VučkoviĆ, 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]
7. P. E. Barclay, K. Srinivasan, M. Borselli, and O. Painter, “Probing the dispersive and spatial properties of photonic crystal waveguides via highly efficient coupling from fiber tapers,” Appl. Phys. Lett. 85, 4–6 (2004). [CrossRef]
8. S. I. Bozhevolnyi, V. S. Volkov, T. Sondergaard, A. Boltasseva, P. I. Borel, and M. Kristensen, “Near-field imaging of light propagation in photonic crystal waveguides: explicit role of Bloch harmonics,” Phys. Rev. B 66, 235204–9 (2002). [CrossRef]
9. H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. Hulst van, T. F. Krauss, and L. Kuipers, “Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides,” Phys. Rev. Lett. 94, 123901–4 (2005). [CrossRef] [PubMed]
10. R. J. P. Engelen, T. J. Karle, H. Gersen, J. P. Korterik, T. F. Krauss, L. Kuipers, and N. F. Hulst van, “Local probing of Bloch mode dispersion in a photonic crystal waveguide,” Opt. Express 13, 4457–4464 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-12-4457. [CrossRef] [PubMed]
12. W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical-response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987). [CrossRef] [PubMed]
13. D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled wave-guides,” Opt. Lett. 13, 794–796 (1988). [CrossRef] [PubMed]
15. J. Feng, “Alternative scheme for studying gap solitons in an infinite periodic Kerr medium,” Opt. Lett. 18, 1302–1304 (1993). [CrossRef] [PubMed]
16. R. F. Nabiev, P. Yeh, and D. Botez, “Spatial gap solitons in periodic nonlinear structures,” Opt. Lett. 18, 1612–1614 (1993). [CrossRef] [PubMed]
17. S. John and N. Akozbek, “Nonlinear-optical solitary waves in a photonic band-gap,” Phys. Rev. Lett. 71, 1168–1171 (1993). [CrossRef] [PubMed]
18. N. Akozbek and S. John, “Optical solitary waves in two- and three-dimensional nonlinear photonic band-gap structures,” Phys. Rev. E 57, 2287–2319 (1998). [CrossRef]
19. S. F. Mingaleev and Yu. S. Kivshar, “Self-trapping and stable localized modes in nonlinear photonic crystals,” Phys. Rev. Lett. 86, 5474–5477 (2001). [CrossRef] [PubMed]
20. B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2993 (1997). [CrossRef]
21. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton trains in photonic lattices,” Opt. Express 12, 2831 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-13-2831. [CrossRef] [PubMed]
22. J. Hudock, N. K. Efremidis, and D. N. Christodoulides, “Anisotropic diffraction and elliptic discrete solitons in two-dimensional waveguide arrays,” Opt. Lett. 29, 268–270 (2004). [CrossRef] [PubMed]
23. D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 90, 053902–4 (2003). [CrossRef] [PubMed]
24. A. A. Sukhorukov, D. Neshev, W. Królikowski, and Yu. S. Kivshar, “Nonlinear Bloch-wave interaction and Bragg scattering in optically induced lattices,” Phys. Rev. Lett. 92, 093901–4 (2004). [CrossRef] [PubMed]
25. C. M. de Sterke, “Theory of modulational instability in fiber Bragg gratings,” J. Opt. Soc. Am. B 15, 2660–2667 (1998). [CrossRef]
26. J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. 92, 163902–4 (2004). [CrossRef] [PubMed]
27. R. Iwanow, G. I. Stegeman, R. Schiek, Y. Min, and W. Sohler, “Discrete modulational instability in periodically poled lithium niobate waveguide arrays,” Opt. Express 13, 7794–7799 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-20-7794. [CrossRef] [PubMed]
28. M. StepiĆ, C. Wirth, C. Rüter, and D. Kip, “Experimental observation of modulational instability in self-defocusing nonlinear waveguide arrays,” Opt. Lett. 31, 247–249 (2006). [CrossRef] [PubMed]
30. B. J. Eggleton, C. M. de Sterke, A. B. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and tunable multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998). [CrossRef]
31. 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, 163902–4 (2005). [CrossRef] [PubMed]
2. Two-dimensional Bloch waves: theoretical background
32. N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photore-fractive optically induced photonic lattices,” Phys. Rev. E 66, 046602–5 (2002). [CrossRef]
3. Experimental arrangements
32. N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photore-fractive optically induced photonic lattices,” Phys. Rev. E 66, 046602–5 (2002). [CrossRef]
4. Excitation of the Bloch modes of the first band
4.1. Γ_{1}-point
34. J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147–150 (2003). [CrossRef] [PubMed]
35. H. Martin, E. D. Eugenieva, Z. G. Chen, and D. N. Christodoulides, “Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices,” Phys. Rev. Lett. 92, 123902–4 (2004). [CrossRef] [PubMed]
34. J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147–150 (2003). [CrossRef] [PubMed]
35. H. Martin, E. D. Eugenieva, Z. G. Chen, and D. N. Christodoulides, “Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices,” Phys. Rev. Lett. 92, 123902–4 (2004). [CrossRef] [PubMed]
34. J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147–150 (2003). [CrossRef] [PubMed]
35. H. Martin, E. D. Eugenieva, Z. G. Chen, and D. N. Christodoulides, “Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices,” Phys. Rev. Lett. 92, 123902–4 (2004). [CrossRef] [PubMed]
4.2. X_{1}-point
36. O. Cohen, B. Freedman, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Grating-mediated waveguiding,” Phys. Rev. Lett. 93, 103902–4 (2004). [CrossRef] [PubMed]
37. B. Freedman, O. Cohen, O. Manela, M. Segev, J. W. Fleischer, and D. N. Christodoulides, “Grating-mediated wave guiding and holographic solitons,” J. Opt. Soc. Am. B 22, 1349–1355 (2005). [CrossRef]
4.3. M_{1}-point
5. Excitation of the Bloch modes of the second band
5.1. X_{2}-point
17. S. John and N. Akozbek, “Nonlinear-optical solitary waves in a photonic band-gap,” Phys. Rev. Lett. 71, 1168–1171 (1993). [CrossRef] [PubMed]
18. N. Akozbek and S. John, “Optical solitary waves in two- and three-dimensional nonlinear photonic band-gap structures,” Phys. Rev. E 57, 2287–2319 (1998). [CrossRef]
38. R. Fischer, D. Träger, D. N. Neshev, A. A. Sukhorukov, W. Królikowski, C. Denz, and Yu. S. Kivshar, “Reduced-symmetry two-dimensional solitons in photonic lattices,” Phys. Rev. Lett. 96, 023905–4 (2006). [CrossRef] [PubMed]
17. S. John and N. Akozbek, “Nonlinear-optical solitary waves in a photonic band-gap,” Phys. Rev. Lett. 71, 1168–1171 (1993). [CrossRef] [PubMed]
18. N. Akozbek and S. John, “Optical solitary waves in two- and three-dimensional nonlinear photonic band-gap structures,” Phys. Rev. E 57, 2287–2319 (1998). [CrossRef]
39. O. Manela, O. Cohen, G. Bartal, J. W. Fleischer, and M. Segev, “Two-dimensional higher-band vortex lattice solitons,” Opt. Lett. 29, 2049–2051 (2004). [CrossRef] [PubMed]
40. G. Bartal, O. Manela, O. Cohen, J. W. Fleischer, and M. Segev, “Observation of second-band vortex solitons in 2D photonic lattices,” Phys. Rev. Lett. 95, 053904–4 (2005). [CrossRef] [PubMed]
5.2. M_{2}-point
6. Conclusions
Acknowledgments
References and links
1. | J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995). |
2. | P. S. Russell, “Bloch wave analysis of dispersion and pulse-propagation in pure distributed feedback structures,” J. Mod. Opt. 38, 1599–1619 (1991). [CrossRef] |
3. | P. St. J. Russell, T. A. Birks, and F. D. Lloyd Lucas, “Photonic Bloch waves and photonic band gaps,” in Confined Electrons and Photons, E. Burstein and C. Weisbuch, eds., (1995), pp. 585–633. [CrossRef] |
4. | H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85, 1863–1866 (2000). [CrossRef] [PubMed] |
5. | T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett. 88, 093901–4 (2002). [CrossRef] [PubMed] |
6. | M. Lončar, D. NedeljkoviĆ, T. P. Pearsall, J. VučkoviĆ, 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] |
7. | P. E. Barclay, K. Srinivasan, M. Borselli, and O. Painter, “Probing the dispersive and spatial properties of photonic crystal waveguides via highly efficient coupling from fiber tapers,” Appl. Phys. Lett. 85, 4–6 (2004). [CrossRef] |
8. | S. I. Bozhevolnyi, V. S. Volkov, T. Sondergaard, A. Boltasseva, P. I. Borel, and M. Kristensen, “Near-field imaging of light propagation in photonic crystal waveguides: explicit role of Bloch harmonics,” Phys. Rev. B 66, 235204–9 (2002). [CrossRef] |
9. | H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. Hulst van, T. F. Krauss, and L. Kuipers, “Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides,” Phys. Rev. Lett. 94, 123901–4 (2005). [CrossRef] [PubMed] |
10. | R. J. P. Engelen, T. J. Karle, H. Gersen, J. P. Korterik, T. F. Krauss, L. Kuipers, and N. F. Hulst van, “Local probing of Bloch mode dispersion in a photonic crystal waveguide,” Opt. Express 13, 4457–4464 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-12-4457. [CrossRef] [PubMed] |
11. | Yu. I. Voloshchenko, Yu. N. Ryzhov, and V. E. Sotin, “Stationary waves in non-linear, periodically modulated media with higher group retardation,” Zh. Tekh. Fiz. 51, 902–907 (1981) (in Russian) [English translation: Tech. Phys. 26, 541-544 (1981)]. |
12. | W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical-response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987). [CrossRef] [PubMed] |
13. | D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled wave-guides,” Opt. Lett. 13, 794–796 (1988). [CrossRef] [PubMed] |
14. | C. M. de Sterke and J. E. Sipe, “Gap solitons,” in Progress in Optics, E. Wolf, ed., (North-Holland, Amsterdam, 1994), Vol. XXXIII, pp. 203–260. |
15. | J. Feng, “Alternative scheme for studying gap solitons in an infinite periodic Kerr medium,” Opt. Lett. 18, 1302–1304 (1993). [CrossRef] [PubMed] |
16. | R. F. Nabiev, P. Yeh, and D. Botez, “Spatial gap solitons in periodic nonlinear structures,” Opt. Lett. 18, 1612–1614 (1993). [CrossRef] [PubMed] |
17. | S. John and N. Akozbek, “Nonlinear-optical solitary waves in a photonic band-gap,” Phys. Rev. Lett. 71, 1168–1171 (1993). [CrossRef] [PubMed] |
18. | N. Akozbek and S. John, “Optical solitary waves in two- and three-dimensional nonlinear photonic band-gap structures,” Phys. Rev. E 57, 2287–2319 (1998). [CrossRef] |
19. | S. F. Mingaleev and Yu. S. Kivshar, “Self-trapping and stable localized modes in nonlinear photonic crystals,” Phys. Rev. Lett. 86, 5474–5477 (2001). [CrossRef] [PubMed] |
20. | B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Nonlinear pulse propagation in Bragg gratings,” J. Opt. Soc. Am. B 14, 2980–2993 (1997). [CrossRef] |
21. | Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton trains in photonic lattices,” Opt. Express 12, 2831 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-13-2831. [CrossRef] [PubMed] |
22. | J. Hudock, N. K. Efremidis, and D. N. Christodoulides, “Anisotropic diffraction and elliptic discrete solitons in two-dimensional waveguide arrays,” Opt. Lett. 29, 268–270 (2004). [CrossRef] [PubMed] |
23. | D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 90, 053902–4 (2003). [CrossRef] [PubMed] |
24. | A. A. Sukhorukov, D. Neshev, W. Królikowski, and Yu. S. Kivshar, “Nonlinear Bloch-wave interaction and Bragg scattering in optically induced lattices,” Phys. Rev. Lett. 92, 093901–4 (2004). [CrossRef] [PubMed] |
25. | C. M. de Sterke, “Theory of modulational instability in fiber Bragg gratings,” J. Opt. Soc. Am. B 15, 2660–2667 (1998). [CrossRef] |
26. | J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo, M. Sorel, and J. S. Aitchison, “Experimental observation of discrete modulational instability,” Phys. Rev. Lett. 92, 163902–4 (2004). [CrossRef] [PubMed] |
27. | R. Iwanow, G. I. Stegeman, R. Schiek, Y. Min, and W. Sohler, “Discrete modulational instability in periodically poled lithium niobate waveguide arrays,” Opt. Express 13, 7794–7799 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-20-7794. [CrossRef] [PubMed] |
28. | M. StepiĆ, C. Wirth, C. Rüter, and D. Kip, “Experimental observation of modulational instability in self-defocusing nonlinear waveguide arrays,” Opt. Lett. 31, 247–249 (2006). [CrossRef] [PubMed] |
29. | M. Chauvet, G. Fu, G. Salamo, J. W. Fleischer, and M. Segev, “Experimental Observation of Discrete Modulation Instability in 1-D Nonlinear Waveguide Arrays,” In Nonlinear Guided Waves and Their Applications, Postconference ed. OSA p. WD39 (Optical Society of America, Washington DC, 2005). |
30. | B. J. Eggleton, C. M. de Sterke, A. B. Aceves, J. E. Sipe, T. A. Strasser, and R. E. Slusher, “Modulational instability and tunable multiple soliton generation in apodized fiber gratings,” Opt. Commun. 149, 267–271 (1998). [CrossRef] |
31. | 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, 163902–4 (2005). [CrossRef] [PubMed] |
32. | N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photore-fractive optically induced photonic lattices,” Phys. Rev. E 66, 046602–5 (2002). [CrossRef] |
33. | Nonlinear Photonic Crystals, Vol. 10 of Springer Series in Photonics, R. E. Slusher and B. J. Eggleton, eds., (Springer-Verlag, Berlin, 2003). |
34. | J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147–150 (2003). [CrossRef] [PubMed] |
35. | H. Martin, E. D. Eugenieva, Z. G. Chen, and D. N. Christodoulides, “Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices,” Phys. Rev. Lett. 92, 123902–4 (2004). [CrossRef] [PubMed] |
36. | O. Cohen, B. Freedman, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Grating-mediated waveguiding,” Phys. Rev. Lett. 93, 103902–4 (2004). [CrossRef] [PubMed] |
37. | B. Freedman, O. Cohen, O. Manela, M. Segev, J. W. Fleischer, and D. N. Christodoulides, “Grating-mediated wave guiding and holographic solitons,” J. Opt. Soc. Am. B 22, 1349–1355 (2005). [CrossRef] |
38. | R. Fischer, D. Träger, D. N. Neshev, A. A. Sukhorukov, W. Królikowski, C. Denz, and Yu. S. Kivshar, “Reduced-symmetry two-dimensional solitons in photonic lattices,” Phys. Rev. Lett. 96, 023905–4 (2006). [CrossRef] [PubMed] |
39. | O. Manela, O. Cohen, G. Bartal, J. W. Fleischer, and M. Segev, “Two-dimensional higher-band vortex lattice solitons,” Opt. Lett. 29, 2049–2051 (2004). [CrossRef] [PubMed] |
40. | G. Bartal, O. Manela, O. Cohen, J. W. Fleischer, and M. Segev, “Observation of second-band vortex solitons in 2D photonic lattices,” Phys. Rev. Lett. 95, 053904–4 (2005). [CrossRef] [PubMed] |
OCIS Codes
(050.1950) Diffraction and gratings : Diffraction gratings
(190.4420) Nonlinear optics : Nonlinear optics, transverse effects in
(190.5940) Nonlinear optics : Self-action effects
ToC Category:
Nonlinear Optics
History
Original Manuscript: January 9, 2006
Revised Manuscript: February 15, 2006
Manuscript Accepted: February 17, 2006
Published: March 6, 2006
Citation
Denis Träger, Robert Fischer, Dragomir N. Neshev, Andrey A. Sukhorukov, Cornelia Denz, Wieslaw Królikowski, and Yuri S. Kivshar, "Nonlinear Bloch modes in two-dimensional photonic lattices," Opt. Express 14, 1913-1923 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-5-1913
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References
- J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).
- P. S. Russell, "Bloch wave analysis of dispersion and pulse-propagation in pure distributed feedback structures," J. Mod. Opt. 38,1599-1619 (1991). [CrossRef]
- P. St. J. Russell, T. A. Birks, and F. D. Lloyd Lucas, "Photonic Bloch waves and photonic band gaps," in Confined Electrons and Photons, E. Burstein and C. Weisbuch, eds., (1995), pp. 585-633. [CrossRef]
- H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Diffraction management," Phys. Rev. Lett. 85,1863-1866 (2000). [CrossRef] [PubMed]
- T. Pertsch, T. Zentgraf, U. Peschel, A. Brauer, and F. Lederer, "Anomalous refraction and diffraction in discrete optical systems," Phys. Rev. Lett. 88,093901-4 (2002). [CrossRef] [PubMed]
- M. Lon¡car, D. Nedeljković, T. P. Pearsall, J. Vu¡cković, 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]
- P. E. Barclay, K. Srinivasan, M. Borselli, and O. Painter, "Probing the dispersive and spatial properties of photonic crystal waveguides via highly efficient coupling from fiber tapers," Appl. Phys. Lett. 85,4-6 (2004). [CrossRef]
- S. I. Bozhevolnyi, V. S. Volkov, T. Sondergaard, A. Boltasseva, P. I. Borel, and M. Kristensen, "Near-field imaging of light propagation in photonic crystal waveguides: explicit role of Bloch harmonics," Phys. Rev. B 66,2352049 (2002). [CrossRef]
- H. Gersen, T. J. Karle, R. J. P. Engelen,W. Bogaerts, J. P. Korterik, N. F. Hulst, van, T. F. Krauss, and L. Kuipers, "Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides," Phys. Rev. Lett. 94,1239014 (2005). [CrossRef] [PubMed]
- R. J. P. Engelen, T. J. Karle, H. Gersen, J. P. Korterik, T. F. Krauss, L. Kuipers, and N. F. Hulst, van, "Local probing of Bloch mode dispersion in a photonic crystal waveguide," Opt. Express 13,4457-4464 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-12-4457. [CrossRef] [PubMed]
- Yu. I. Voloshchenko, Yu. N. Ryzhov, and V. E. Sotin, "Stationary waves in non-linear, periodically modulated media with higher group retardation," Zh. Tekh. Fiz. 51,902-907 (1981) (in Russian) [English translation: Tech. Phys. 26, 541-544 (1981)].
- W. Chen and D. L. Mills, "Gap solitons and the nonlinear optical-response of superlattices," Phys. Rev. Lett. 58,160-163 (1987). [CrossRef] [PubMed]
- D. N. Christodoulides and R. I. Joseph, "Discrete self-focusing in nonlinear arrays of coupled wave-guides," Opt. Lett. 13,794-796 (1988). [CrossRef] [PubMed]
- C. M. de Sterke and J. E. Sipe, "Gap solitons," in Progress in Optics, E. Wolf, ed., (North-Holland, Amsterdam, 1994), Vol. XXXIII, pp. 203-260.
- J. Feng, "Alternative scheme for studying gap solitons in an infinite periodic Kerr medium," Opt. Lett. 18,1302-1304 (1993). [CrossRef] [PubMed]
- R. F. Nabiev, P. Yeh, and D. Botez, "Spatial gap solitons in periodic nonlinear structures," Opt. Lett. 18,1612-1614 (1993). [CrossRef] [PubMed]
- S. John and N. Akozbek, "Nonlinear-optical solitary waves in a photonic band-gap," Phys. Rev. Lett. 71,1168-1171 (1993). [CrossRef] [PubMed]
- N. Akozbek and S. John, "Optical solitary waves in two- and three-dimensional nonlinear photonic band-gap structures," Phys. Rev. E 57,2287-2319 (1998). [CrossRef]
- S. F. Mingaleev and Yu. S. Kivshar, "Self-trapping and stable localized modes in nonlinear photonic crystals," Phys. Rev. Lett. 86,5474-5477 (2001). [CrossRef] [PubMed]
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