## Linear discrete diffraction and transverse localization of light in two-dimensional backbone lattices |

Optics Express, Vol. 18, Issue 19, pp. 20170-20182 (2010)

http://dx.doi.org/10.1364/OE.18.020170

Acrobat PDF (1573 KB)

### Abstract

We study the linear discrete diffraction characteristics of light in two-dimensional backbone lattices. It is found that, as the refractive index modulation depth of the backbone lattice increases, high-order band gaps become open and broad in sequence, and the allowed band curves of the Floquet-Bloch modes become flat gradually. As a result, the diffraction pattern at the exit face converges gradually for both the on-site and off-site excitation cases. Particularly, when the refractive index modulation depth of the backbone lattice is high enough, for example, on the order of 0.01 for a square lattice, the light wave propagating in the backbone lattice will be localized in transverse dimension for both the on-site and off-site excitation cases. This is because only the first several allowed bands with nearly flat band curves are excited in the lattice, and the transverse expansion velocities of the Floquet-Bloch modes in these flat allowed bands approach to zero. Such a linear transverse localization of light may have potential applications in navigating light propagation dynamics and optical signal processing.

© 2010 OSA

## 1. Introduction

1. D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature **424**(6950), 817–823 (2003). [CrossRef] [PubMed]

2. J. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N. Efremidis, “Spatial photonics in nonlinear waveguide arrays,” Opt. Express **13**(6), 1780–1796 (2005). [CrossRef] [PubMed]

*discrete diffraction*can be tracked back to 1965 when Jones first theoretically addressed the issue in one-dimensional (1D) optical fiber arrays [3

3. A. L. Jones, “Coupling of Optical Fibers and Scattering in Fibers,” J. Opt. Soc. Am. **55**(3), 261–269 (1965). [CrossRef]

4. S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett. **22**(1), 46–47 (1973). [CrossRef]

5. D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. **13**(9), 794–796 (1988). [CrossRef] [PubMed]

6. F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. **463**(1-3), 1–126 (2008). [CrossRef]

8. H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete Spatial Optical Solitons in Waveguide Arrays,” Phys. Rev. Lett. **81**(16), 3383–3386 (1998). [CrossRef]

15. M. J. Ablowitz and Z. H. Musslimani, “Discrete diffraction managed spatial solitons,” Phys. Rev. Lett. **87**(25), 254102 (2001). [CrossRef] [PubMed]

16. N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, “Two-dimensional optical lattice solitons,” Phys. Rev. Lett. **91**(21), 213906 (2003). [CrossRef] [PubMed]

22. O. Borovkova, V. Lobanov, A. Sukhorukova, and A. Sukhorukov, “Discrete diffraction and waveguiding of optical beams in a cascade-induced lattice,” Bull. Russ. Acad. Sci. Phys. **72**(5), 718–720 (2008). [CrossRef]

23. O. Manela, M. Segev, and D. N. Christodoulides, “Nondiffracting beams in periodic media,” Opt. Lett. **30**(19), 2611–2613 (2005). [CrossRef] [PubMed]

16. N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, “Two-dimensional optical lattice solitons,” Phys. Rev. Lett. **91**(21), 213906 (2003). [CrossRef] [PubMed]

*q*(

*k*) of the extended Floquet–Bloch (FB) eigenmodes [24] of the lattice which consists of allowed bands and forbidden band gaps and determines the key features of the wave diffraction in the lattice. The paper is organized as follows: in section 2, the band structure of the spatially extended FB eigenmodes of the square backbone lattice is studied, in comparison with that of the traditional square sinusoidal lattice. In section 3, the discrete diffraction behavior with on-site excitation (the light is coupled into a single site with an index maximum) in the square backbone lattice with increasing index modulation depth is studied, the results are also compared with those in the sinusoidal lattice. In section 4, the discrete diffraction behaviors with off-site excitation (the light is injected into a single site with an index minimum) in both the square backbone and sinusoidal lattices are studied. In section 5, transverse localization of light in linear region in strongly modulated square backbone and sinusoidal lattices will be illustrated, and the underlying mechanism for this transversely localized light will be presented based on the band structures of the excited FB modes of the lattices and the light excitation configurations. In section 6, the band structure and beam diffraction features of hexagonal lattices are investigated in comparison with those of square ones. Finally, we summarize the work in section 7.

_{x},k_{y}## 2. Band structure of the spatially extended FB eigenmodes of the square backbone lattice

*ψ*(

*x,y,z*) of the light in a 2D periodic lattice can be described by the following dimensionless equationwhere

*V*(

*x,y*) is a periodic index potential proportional to the refractive index modulation Δ

*n*(

*x,y*) of the lattice, and

*V*(

*x,y*) is given bywhere

*I*(

*x,y*)

*= A*cos

^{2}^{2}(

*πx*)cos

^{2}(

*πy*) is the normalized intensity of the lattice-forming beam, as in experiment the square backbone lattice can be optically induced by the interference of two or more pairs of the laser beams in photorefractive materials [17

17. 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**(6928), 147–150 (2003). [CrossRef] [PubMed]

25. N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **66**(4 Pt 2), 046602 (2002). [CrossRef] [PubMed]

26. C. Lou, X. S. Wang, J. J. Xu, Z. G. Chen, and J. Yang, “Nonlinear spectrum reshaping and gap-soliton-train trapping in optically induced photonic structures,” Phys. Rev. Lett. **98**(21), 213903 (2007). [CrossRef] [PubMed]

*a*is set to be 11 μm, which is also served as the unit length of the whole simulation procedure,

*A*is set to be 1.21, and

^{2}*V*is the index potential depth, which is always negative for the backbone lattices and its magnitude controls the refractive index modulation depth of the lattice synchronously. Correspondingly, for the square sinusoidal lattice as shown in Fig. 1(b), the periodic index potential is expressed as

_{0}*V*(

*x,y*) = -(

*V*/2)(sin

_{0}^{2}(

*πx*) + sin

^{2}(

*πy*)), in which

*V*is always positive.

_{0}*ψ*(

*x,y,z*)

*=*exp(

*-iqz*)exp(

*ik*)

_{x}x + ik_{y}y*U*(

_{k}*x,y*), in which

*U*(

_{k}*x,y*) possesses the same periodicity as the lattice,

*k*are the transverse components of the Bloch wave vector, and

_{x}, k_{y}*q*is the propagation constant. The band structure of the lattice can then be obtained by solving numerically the corresponding eigenvalue equation [27]:employing the plane-wave expansion (PWE) method [28

28. S. P. Guo and S. Albin, “Simple plane wave implementation for photonic crystal calculations,” Opt. Express **11**(2), 167–175 (2003). [CrossRef] [PubMed]

*u*(

_{k}*x,y*) = exp(

*ik*)

_{x}x + ik_{y}y*U*(

_{k}*x,y*) obeys Bloch’s theorem.

*V*of the square backbone lattice. The bands are depicted in the multicolor regions, and the blank regions represent the band gaps. The first gap is open when the potential depth

_{0}*V*is deeper than

_{0}16. N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, “Two-dimensional optical lattice solitons,” Phys. Rev. Lett. **91**(21), 213906 (2003). [CrossRef] [PubMed]

*V*170.0 is illustrated. The refractive index modulation depth Δ

_{0}= -*n*of the lattice is proportional to its index potential depth

_{0}*V*. For the specific experiment in Ref [17

_{0}17. 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**(6928), 147–150 (2003). [CrossRef] [PubMed]

*n*required to open the first and second band gaps equals to 1.05 × 10

_{0}^{−4}[16

**91**(21), 213906 (2003). [CrossRef] [PubMed]

^{−4}, respectively, with an operating wavelength

*λ*= 0.5 μm and a background refractive index

*n*= 2.3, which can be acquired with typical photorefractive materials in experiment. Higher order band gaps get open in sequence with a further increase in the magnitude of the potential depth

_{0}*V*.

_{0}*V*along the direction

_{0}*V*of the backbone lattice is set to be

_{0}*n*of 2.24 × 10

_{0}^{−5}, 2.24 × 10

^{−4}and 8.95 × 10

^{−4}with the aforementioned parameters of SBN:75 crystal, respectively. The potential depths are selected in such a way that no band gap is open for the

*V*= −5 case (see Fig. 3(a)); the first band gap gets open while the second band gap is still not for the

_{0}*V*= −50 case (see Fig. 3(b)); and for the

_{0}*V*= −200 case both the first and the second band gaps are already open, as shown in Fig. 3(c). In comparison, we also give the band structure of the square sinusoidal lattice with a potential depth

_{0}*V*= 90.4 in Fig. 3(d). For a Kerr-type optical lattice where

_{0}*V*= (

_{0}*2πn*)

_{0}a/λ^{2}Δ

*n*[16

_{0}**91**(21), 213906 (2003). [CrossRef] [PubMed]

17. 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**(6928), 147–150 (2003). [CrossRef] [PubMed]

^{−4}with

*λ =*0.5 μm and

*n*2.3. As is seen, with the same refractive index modulation depth, the gaps in the sinusoidal lattice are much broader than those in the backbone one. We also notice that the allowed band curves become flatter with the increase of the magnitude of the potential depth, which will have a significant effect on the light propagation dynamics in the periodic lattices, as we will discuss in detail in the following sections.

_{0}=## 3. Beam diffraction features in the square backbone lattice with on-site excitation

*λ =*0.5 μm. The beam propagates in the lattice for 6000 unit length, corresponding to a total propagation length of 6.6 cm in real space. Figure 4 shows the linear diffraction features of light in square backbone lattices with various index potential depth

*V*for the on-site excitation case, in which the index potential depth

_{0}*V*is

_{0}*y*= −1/2 slice (since for on-site excitation in the square backbone lattice, the beam is focused into the index maximum at position (−1/2,-1/2) according to the coordinates in Fig. 1(a)). For comparison, the intensity distribution pattern at the exit face and the beam propagation dynamics in the

*y*= 0 plane in the square sinusoidal lattice with a potential depth

*V*= 90.4 for the on-site excitation case are also depicted in Figs. 4(g)-4(h). For the sinusoidal lattice the incident beam is focused into the index maximum at position (0,0) according to the coordinates in Fig. 1(b), so that the light propagation slice is chosen as the

_{0}*y*= 0 plane. It is seen that, for both the square backbone and sinusoidal lattices, the beam propagation dynamics shows typical discrete diffraction characteristics in all cases with on-site excitation, which is very different from the diffraction behavior in the homogeneous media.

*q*/∂

*k*

_{⊥}, in which

## 4. Beam diffraction features in the square backbone lattice with off-site excitation

*y*= 0 slice is shown in Figs. 5(b), 5(d) and 5(f) (For off-site excitation, the light is focused into the site with an index minimum at position (0,0) according to the coordinates in Fig. 1(a)). For comparison, we also study the light propagation dynamics in the off-site excitation case in the square sinusoidal lattice with

*V*= 90.4. The intensity distribution of the diffraction pattern at the exit face and the beam propagation dynamics in the

_{0}*y =*−1/2 slice are respectively shown in Fig. 5(g) and Fig. 5(h), which hold the similar diffraction patterns as those with on-site excitation in the sinusoidal lattice (see Figs. 4(g)-4(h)).

## 5. Transverse localization of light in highly modulated square lattices

*y =*−1/2 slice in the square backbone lattice while Fig. 6(f) is the propagation dynamics in the

*y =*0 slice in the square sinusoidal lattice. The propagation length is also 6000 unit length and the refractive index modulation depth is set to be Δ

*n*= 0.0224, corresponding to an index potential depth

_{0}*V*= −5000 for the backbone lattice and

_{0}*V*= 2263.4 for the sinusoidal lattice according to the aforementioned proportional relations between the refractive index modulation depth and the index potential depth in section 2. Although a lattice with a refractive index modulation depth on the order of 0.01 may not be produced in typical photorefractive crystals by using the optical induction technique, it may be written directly through multi-photon-induced refractive index changes using a focused femtosecond laser beam [31

_{0}31. K. M. Davis, K. Miura, N. Sugimoto, and K. Hirao, “Writing waveguides in glass with a femtosecond laser,” Opt. Lett. **21**(21), 1729–1731 (1996). [CrossRef] [PubMed]

*q*below the dashed blue curves which are determined by Snell’s law and can be expressed analytically as

*V*for the backbone lattice, as can be seen from Eqs. (2) and (3). It is seen that the band curves of the excited FB modes are nearly flat, indicating that the transverse expansion velocities of these FB modes approach to zero and the transverse expansion of the beam in the lattice almost stops, therefore, the light coupled into the lattices is localized in the transverse dimension. On the other hand, in an analogy to the periodic finite potential well in quantum mechanics, with the increase in the potential depth of the periodic well, spatially localized states associated with discrete energy levels will appear in the well (see Eq. (3)). More interestingly, with a further increase in the refractive index modulation depth, it is possible to selectively excite the FB modes within the lowest allowed band.

_{0}## 6. Band structure and beam diffraction features in hexagonal lattices

32. P. R. Villeneuve and M. Piché, “Photonic band gaps in two-dimensional square and hexagonal lattices,” Phys. Rev. B Condens. Matter **46**(8), 4969–4972 (1992). [CrossRef] [PubMed]

34. T. J. Alexander, A. S. Desyatnikov, and Y. S. Kivshar, “Multivortex solitons in triangular photonic lattices,” Opt. Lett. **32**(10), 1293–1295 (2007). [CrossRef] [PubMed]

33. C. R. Rosberg, D. N. Neshev, A. A. Sukhorukov, W. Krolikowski, and Y. S. Kivshar, “Observation of nonlinear self-trapping in triangular photonic lattices,” Opt. Lett. **32**(4), 397–399 (2007). [CrossRef] [PubMed]

35. O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. **98**(10), 103901 (2007). [CrossRef] [PubMed]

36. J. C. Knight, T. A. Birks, P. S. J. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. **21**(19), 1547–1549 (1996). [CrossRef] [PubMed]

*V*(

*x,y*) of the hexagonal backbone lattice can also be expressed by Eq. (2) but with a normalized lattice-forming beam intensity in the form of

*A*1.21 and a lattice period

^{2}=*a*= 11 μm. Meanwhile, the eigenvalue bands of the propagation modes as a function of the potential depth

*V*of the hexagonal backbone lattice is also shown in Fig. 8(b), in which the inset is a magnified plot of the region where the second band gap is open. We note that the first band gap is open at

_{0}*V*= −9, and the second band gap gets open at

_{0}*V*= −90, respectively. It is evident that the index potential depths required to open the band gaps in the hexagonal backbone lattice are indeed much lower than those of the square backbone lattice with the same lattice period (see Fig. 2), just as the case for the hexagonal sinusoidal lattices [33

_{0}33. C. R. Rosberg, D. N. Neshev, A. A. Sukhorukov, W. Krolikowski, and Y. S. Kivshar, “Observation of nonlinear self-trapping in triangular photonic lattices,” Opt. Lett. **32**(4), 397–399 (2007). [CrossRef] [PubMed]

34. T. J. Alexander, A. S. Desyatnikov, and Y. S. Kivshar, “Multivortex solitons in triangular photonic lattices,” Opt. Lett. **32**(10), 1293–1295 (2007). [CrossRef] [PubMed]

*V*= −5000 for the on-site excitation case. The light propagates in the lattice with 6000 unit length. It is clearly seen that the light propagation is still of typical discrete diffraction characteristics in the hexagonal backbone lattice with

_{0}*V*= −5000, in strong contrast to the transversely localized state of light in the square backbone lattice with the same potential depth (Fig. 6(b)). The reason is that more propagation modes are excited and the transverse expansion velocities of the excited FB modes are much larger in the hexagonal backbone lattice as compared to those in the square backbone lattice. Figures 9(b) and 9(c) show the band profile of the first band along the direction

_{0}*V*= −5000. The transverse expansion velocities of the FB modes near the M symmetric point in the hexagonal backbone lattice are found to be about 12 times larger than those in the square backbone lattice. This means that larger index potential depth is required to localize the light transversely in the hexagonal backbone lattice as compared to the square backbone lattice. As an example, Fig. 10 shows the band structure of the hexagonal backbone lattice with

_{0}*V*= −60000 and the 2D output intensity distribution at a propagation length of 6000 unit length with on-site and off-site excitation, respectively. As expected, the band curves of the FB modes become much flatter in this case and the light is nearly localized in two transverse dimensions in the hexagonal backbone lattice with both on-site and off-site excitation.

_{0}32. P. R. Villeneuve and M. Piché, “Photonic band gaps in two-dimensional square and hexagonal lattices,” Phys. Rev. B Condens. Matter **46**(8), 4969–4972 (1992). [CrossRef] [PubMed]

34. T. J. Alexander, A. S. Desyatnikov, and Y. S. Kivshar, “Multivortex solitons in triangular photonic lattices,” Opt. Lett. **32**(10), 1293–1295 (2007). [CrossRef] [PubMed]

*V*= 2263.4 and with a lattice period

_{0}*a*= 11 μm, where the inset shows the index potential structural sketch of the hexagonal sinusoidal lattice. Again, the blue dashed curve represents the propagation constants determined by Snell’s law and reachable by the exciting beam. We note that only the first band of the FB modes with a flat band profile can be excited in this case, indicating that the light will be transversely localized in the lattice with both on-site and off-site excitation, as shown clearly in Figs. 11(b) and 11(c), respectively.

## 7. Conclusions

## Acknowledgements

## References and links

1. | D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature |

2. | J. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N. Efremidis, “Spatial photonics in nonlinear waveguide arrays,” Opt. Express |

3. | A. L. Jones, “Coupling of Optical Fibers and Scattering in Fibers,” J. Opt. Soc. Am. |

4. | S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett. |

5. | D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. |

6. | F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. |

7. | Y. S. Kivshar, and G. P. Agrawal, |

8. | H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete Spatial Optical Solitons in Waveguide Arrays,” Phys. Rev. Lett. |

9. | T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett. |

10. | H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. |

11. | J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. |

12. | D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, “Spatial solitons in optically induced gratings,” Opt. Lett. |

13. | F. Chen, M. Stepić, C. E. Rüter, D. Runde, D. Kip, V. Shandarov, O. Manela, and M. Segev, “Discrete diffraction and spatial gap solitons in photovoltaic LiNbO3 waveguide arrays,” Opt. Express |

14. | 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. |

15. | M. J. Ablowitz and Z. H. Musslimani, “Discrete diffraction managed spatial solitons,” Phys. Rev. Lett. |

16. | N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, “Two-dimensional optical lattice solitons,” Phys. Rev. Lett. |

17. | 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 |

18. | Z. Chen, H. Martin, E. D. Eugenieva, J. Xu, and A. Bezryadina, “Anisotropic enhancement of discrete diffraction and formation of two-dimensional discrete-soliton trains,” Phys. Rev. Lett. |

19. | X. S. Wang, A. Bezryadina, Z. G. Chen, K. G. Makris, D. N. Christodoulides, and G. I. Stegeman, “Observation of two-dimensional surface solitons,” Phys. Rev. Lett. |

20. | D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. G. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. |

21. | X. Qi, G. Zhang, N. Xu, Y. Qi, B. Han, Y. Fu, C. Duan, and J. Xu, “Linear and nonlinear discrete light propagation in weakly modulated large-area two-dimensional photonic lattice slab in LiNbO |

22. | O. Borovkova, V. Lobanov, A. Sukhorukova, and A. Sukhorukov, “Discrete diffraction and waveguiding of optical beams in a cascade-induced lattice,” Bull. Russ. Acad. Sci. Phys. |

23. | O. Manela, M. Segev, and D. N. Christodoulides, “Nondiffracting beams in periodic media,” Opt. Lett. |

24. | R. J. Elliott, and A. F. Gibson, |

25. | N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

26. | C. Lou, X. S. Wang, J. J. Xu, Z. G. Chen, and J. Yang, “Nonlinear spectrum reshaping and gap-soliton-train trapping in optically induced photonic structures,” Phys. Rev. Lett. |

27. | N. K. Efremidis, J. W. Feischer, G. Bartal, O. Cohen, H. Buljan, D. N. Christodoulides, and M. Segev, “Introduction to Solitons in Photonic Lattices,” in |

28. | S. P. Guo and S. Albin, “Simple plane wave implementation for photonic crystal calculations,” Opt. Express |

29. | K. Kawano, and T. Kitoh, |

30. | B. Lv, |

31. | K. M. Davis, K. Miura, N. Sugimoto, and K. Hirao, “Writing waveguides in glass with a femtosecond laser,” Opt. Lett. |

32. | P. R. Villeneuve and M. Piché, “Photonic band gaps in two-dimensional square and hexagonal lattices,” Phys. Rev. B Condens. Matter |

33. | C. R. Rosberg, D. N. Neshev, A. A. Sukhorukov, W. Krolikowski, and Y. S. Kivshar, “Observation of nonlinear self-trapping in triangular photonic lattices,” Opt. Lett. |

34. | T. J. Alexander, A. S. Desyatnikov, and Y. S. Kivshar, “Multivortex solitons in triangular photonic lattices,” Opt. Lett. |

35. | O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. |

36. | J. C. Knight, T. A. Birks, P. S. J. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. |

**OCIS Codes**

(050.1970) Diffraction and gratings : Diffractive optics

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: July 22, 2010

Revised Manuscript: August 27, 2010

Manuscript Accepted: August 28, 2010

Published: September 7, 2010

**Citation**

Yiling Qi and Guoquan Zhang, "Linear discrete diffraction and transverse localization of light in two-dimensional backbone lattices," Opt. Express **18**, 20170-20182 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-19-20170

Sort: Year | Journal | Reset

### References

- D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003). [CrossRef] [PubMed]
- J. Fleischer, G. Bartal, O. Cohen, T. Schwartz, O. Manela, B. Freedman, M. Segev, H. Buljan, and N. Efremidis, “Spatial photonics in nonlinear waveguide arrays,” Opt. Express 13(6), 1780–1796 (2005). [CrossRef] [PubMed]
- A. L. Jones, “Coupling of Optical Fibers and Scattering in Fibers,” J. Opt. Soc. Am. 55(3), 261–269 (1965). [CrossRef]
- S. Somekh, E. Garmire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett. 22(1), 46–47 (1973). [CrossRef]
- D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. 13(9), 794–796 (1988). [CrossRef] [PubMed]
- F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep. 463(1-3), 1–126 (2008). [CrossRef]
- Y. S. Kivshar and G. P. Agrawal, Optical solitons: from fibers to phototonic crystals (Academic Press, 2003).
- H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete Spatial Optical Solitons in Waveguide Arrays,” Phys. Rev. Lett. 81(16), 3383–3386 (1998). [CrossRef]
- T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer, and F. Lederer, “Anomalous refraction and diffraction in discrete optical systems,” Phys. Rev. Lett. 88(9), 093901 (2002). [CrossRef] [PubMed]
- H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85(9), 1863–1866 (2000). [CrossRef] [PubMed]
- J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. 90(2), 023902 (2003). [CrossRef] [PubMed]
- D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, “Spatial solitons in optically induced gratings,” Opt. Lett. 28(9), 710–712 (2003). [CrossRef] [PubMed]
- F. Chen, M. Stepić, C. E. Rüter, D. Runde, D. Kip, V. Shandarov, O. Manela, and M. Segev, “Discrete diffraction and spatial gap solitons in photovoltaic LiNbO3 waveguide arrays,” Opt. Express 13(11), 4314–4324 (2005). [CrossRef] [PubMed]
- 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(5), 053902 (2003). [CrossRef] [PubMed]
- M. J. Ablowitz and Z. H. Musslimani, “Discrete diffraction managed spatial solitons,” Phys. Rev. Lett. 87(25), 254102 (2001). [CrossRef] [PubMed]
- N. K. Efremidis, J. Hudock, D. N. Christodoulides, J. W. Fleischer, O. Cohen, and M. Segev, “Two-dimensional optical lattice solitons,” Phys. Rev. Lett. 91(21), 213906 (2003). [CrossRef] [PubMed]
- 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(6928), 147–150 (2003). [CrossRef] [PubMed]
- Z. Chen, H. Martin, E. D. Eugenieva, J. Xu, and A. Bezryadina, “Anisotropic enhancement of discrete diffraction and formation of two-dimensional discrete-soliton trains,” Phys. Rev. Lett. 92(14), 143902 (2004). [CrossRef] [PubMed]
- X. S. Wang, A. Bezryadina, Z. G. Chen, K. G. Makris, D. N. Christodoulides, and G. I. Stegeman, “Observation of two-dimensional surface solitons,” Phys. Rev. Lett. 98(12), 123903 (2007). [CrossRef] [PubMed]
- D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, Y. S. Kivshar, H. Martin, I. Makasyuk, and Z. G. Chen, “Observation of discrete vortex solitons in optically induced photonic lattices,” Phys. Rev. Lett. 92(12), 123903 (2004). [CrossRef] [PubMed]
- X. Qi, G. Zhang, N. Xu, Y. Qi, B. Han, Y. Fu, C. Duan, and J. Xu, “Linear and nonlinear discrete light propagation in weakly modulated large-area two-dimensional photonic lattice slab in LiNbO3:Fe crystal,” Opt. Express 17(25), 23078–23084 (2009). [CrossRef]
- O. Borovkova, V. Lobanov, A. Sukhorukova, and A. Sukhorukov, “Discrete diffraction and waveguiding of optical beams in a cascade-induced lattice,” Bull. Russ. Acad. Sci. Phys. 72(5), 718–720 (2008). [CrossRef]
- O. Manela, M. Segev, and D. N. Christodoulides, “Nondiffracting beams in periodic media,” Opt. Lett. 30(19), 2611–2613 (2005). [CrossRef] [PubMed]
- R. J. Elliott and A. F. Gibson, An Introduction to Solid State Physics and its Applications (The Macmillan Press, 1974).
- N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(4 Pt 2), 046602 (2002). [CrossRef] [PubMed]
- C. Lou, X. S. Wang, J. J. Xu, Z. G. Chen, and J. Yang, “Nonlinear spectrum reshaping and gap-soliton-train trapping in optically induced photonic structures,” Phys. Rev. Lett. 98(21), 213903 (2007). [CrossRef] [PubMed]
- N. K. Efremidis, J. W. Feischer, G. Bartal, O. Cohen, H. Buljan, D. N. Christodoulides, and M. Segev, “Introduction to Solitons in Photonic Lattices,” in Nonlinearities in Periodic Structures and Metamaterials, C. Denz, S. Flach, and Y. S. Kivshar, eds., (Springer, 2008), p. 295.
- S. P. Guo and S. Albin, “Simple plane wave implementation for photonic crystal calculations,” Opt. Express 11(2), 167–175 (2003). [CrossRef] [PubMed]
- K. Kawano and T. Kitoh, Introduction to optical waveguide analysis:Solving Maxwell's Equations and the Schrodinger Equation (John Wiley & Sons, Inc., 2001).
- B. Lv, Laser Optics: Beam Characterization, Propagation and Transformation, Resonator Technology and Physics (Higher Education Press, 2003).
- K. M. Davis, K. Miura, N. Sugimoto, and K. Hirao, “Writing waveguides in glass with a femtosecond laser,” Opt. Lett. 21(21), 1729–1731 (1996). [CrossRef] [PubMed]
- P. R. Villeneuve and M. Piché, “Photonic band gaps in two-dimensional square and hexagonal lattices,” Phys. Rev. B Condens. Matter 46(8), 4969–4972 (1992). [CrossRef] [PubMed]
- C. R. Rosberg, D. N. Neshev, A. A. Sukhorukov, W. Krolikowski, and Y. S. Kivshar, “Observation of nonlinear self-trapping in triangular photonic lattices,” Opt. Lett. 32(4), 397–399 (2007). [CrossRef] [PubMed]
- T. J. Alexander, A. S. Desyatnikov, and Y. S. Kivshar, “Multivortex solitons in triangular photonic lattices,” Opt. Lett. 32(10), 1293–1295 (2007). [CrossRef] [PubMed]
- O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98(10), 103901 (2007). [CrossRef] [PubMed]
- J. C. Knight, T. A. Birks, P. S. J. Russell, and D. M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21(19), 1547–1549 (1996). [CrossRef] [PubMed]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.