## Multiplexing complex two-dimensional photonic superlattices |

Optics Express, Vol. 20, Issue 24, pp. 27331-27343 (2012)

http://dx.doi.org/10.1364/OE.20.027331

Acrobat PDF (4323 KB)

### Abstract

We introduce a universal method to optically induce multiperiodic photonic complex superstructures bearing two-dimensional (2D) refractive index modulations over several centimeters of elongation. These superstructures result from the accomplished superposition of 2D fundamental periodic structures. To find the specific sets of fundamentals, we combine particular spatial frequencies of the respective Fourier series expansions, which enables us to use nondiffracting beams in the experiment showing periodic 2D intensity modulation in order to successively develop the desired multiperiodic structures. We present the generation of 2D photonic staircase, hexagonal wire mesh and ratchet structures, whose succeeded generation is confirmed by phase resolving methods using digital-holographic techniques to detect the induced refractive index pattern.

© 2012 OSA

## 1. Introduction

1. E. Yablonovitch, T. J. Gmitter, and K. M. Leung, “Photonic band structure: The face-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. **67**, 2295–2298 (1991). [CrossRef] [PubMed]

2. M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Bragg diffraction of finite beams by thick gratings,” J. Opt. Soc. Am. **70**, 300–304 (1980). [CrossRef]

3. R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, “Optical analogue of electronic Bloch oscillations,” Phys. Rev. Lett. **91**, 263902 (2003). [CrossRef]

5. T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature **446**, 52–55 (2007). [CrossRef] [PubMed]

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

12. B. Terhalle, T. Richter, A. S. Desyatnikov, D. N. Neshev, W. Krolikowski, F. Kaiser, C. Denz, and Y. S. Kivshar, “Observation of multivortex solitons in photonic lattices,” Phys. Rev. Lett. **101**, 013903 (2008). [CrossRef] [PubMed]

13. Y. S. Chan, C. T. Chan, and Z. Y. Liu, “Photonic band gaps in two dimensional photonic quasicrystals,” Phys. Rev. Lett. **80**, 956–959 (1998). [CrossRef]

14. B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides, and J. W. Fleischer, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature **440**, 1166–1169 (2006). [CrossRef] [PubMed]

15. Y. V. Kartashov, A. A Egorov, V. A. Vysloukh, and L. Torner, “Stable soliton complexes and azimuthal switching in modulated Bessel optical lattices,” Phys. Rev. E **70**, 065602(R) (2004). [CrossRef]

19. F. Ye, D. Mihalache, and B. Hu, “Elliptic vortices in composite Mathieu lattices,” Phys. Rev. A **79**, 053852 (2009). [CrossRef]

20. D. M. Jović, M. R. Belić, and C. Denz, “Anderson localization of light at the interface between linear and nonlinear dielectric media with an optically induced photonic lattice,” Phys. Rev. A **85**, 031801(R) (2012). [CrossRef]

21. S. Longhi, “Klein tunneling in binary photonic superlattices,” Phys. Rev. B **81**, 075012 (2010). [CrossRef]

23. F. Dreisow, M. Heinrich, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, “Classical simulation of relativistic Zitterbewegung in photonic lattices,” Phys. Rev. Lett. **105**, 143902 (2010). [CrossRef]

24. M. Heinrich, Y. V. Kartashov, L. P. R. Ramirez, A. Szameit, F. Dreisow, R. Keil, S. Nolte, A. Tünnermann, V. A. Vysloukh, and L. Torner, “Observation of two-dimensional superlattice solitons,” Opt. Lett. **34**, 3701–3703 (2009). [CrossRef] [PubMed]

25. M. Heinrich, R. Keil, F. Dreisow, A. Tünnermann, A. Szameit, and S. Nolte, “Nonlinear discrete optics in femtosecond laser-written photonic lattices,” Appl. Phys. B **104**, 469–480 (2011). [CrossRef]

26. M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature **404**, 53–56 (2000). [CrossRef] [PubMed]

7. D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, “Spatial solitons in optically induced gratings,” Opt. Lett. **28**, 710–712 (2003). [CrossRef] [PubMed]

8. 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]

27. B. Terhalle, A. S. Desyatnikov, D. N. Neshev, W. Krolikowski, C. Denz, and Y. S. Kivshar, “Dynamic diffraction and interband transitions in two-dimensional photonic lattices,” Phys. Rev. Lett. **106**, 083902 (2011). [CrossRef] [PubMed]

28. A. S. Desyatnikov, N. Sagemerten, R. Fischer, B. Terhalle, D. Träger, D. N. Neshev, A. Dreischuh, C. Denz, W. Krolikowski, and Y. S. Kivshar, “Two-dimensional self-trapped nonlinear photonic lattices,” Opt. Express **14**, 2851–2863 (2006). [CrossRef] [PubMed]

29. P. Rose, M. Boguslawski, and C. Denz, “Nonlinear lattice structures based on families of complex nondiffracting beams,” New J. Phys. **14**, 033018 (2012). [CrossRef]

30. J. Xavier, M. Boguslawski, P. Rose, J. Joseph, and C. Denz, “Reconfigurable optically induced quasicrystallographic three-dimensional complex nonlinear photonic lattice structures,” Adv. Mater. **22**, 356–360 (2010). [CrossRef] [PubMed]

31. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A **4**, 651–654 (1987). [CrossRef]

33. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. **29**, 44–46 (2004). [CrossRef] [PubMed]

34. C. Denz, G. Pauliat, G. Roosen, and T. Tschudi, “Volume hologram multiplexing using a deterministic phase encoding method,” Opt. Commun. **85**, 171–176 (1991). [CrossRef]

35. Y. Taketomi, J. E. Ford, H. Sasaki, J. Ma, Y. Fainman, and S. H. Lee “Incremental recording for photorefractive hologram multiplexing,” Opt. Lett. **16**, 1774–1776 (1991). [CrossRef] [PubMed]

36. P. Rose, B. Terhalle, J. Imbrock, and C. Denz, “Optically induced photonic superlattices by holographic multiplexing,” J. Phys. D: Appl. Phys. **41**, 224004 (2008). [CrossRef]

37. M. Boguslawski, A. Kelberer, P. Rose, and C. Denz, “Photonic ratchet superlattices by optical multiplexing,” Opt. Lett. **37**, 797–799 (2012). [CrossRef] [PubMed]

## 2. Multiperiodic structures and set of their fundamentals

38. J. Becker, P. Rose, M. Boguslawski, and C. Denz, “Systematic approach to complex periodic vortex and helix lattices,” Opt. Express **19**, 9848–9862 (2011). [CrossRef] [PubMed]

39. M. Boguslawski, P. Rose, and C. Denz, “Increasing the structural variety of discrete nondiffracting wave fields,” Phys. Rev. A **84**, 013832 (2011). [CrossRef]

*M*is designed by combining 1D lattice structures of equal modulation frequency resulting in a set of 2D fundaments of varying periodicity, which accounts for a series expansion of the desired structure. This procedure will for all of our examples end up in different sets of ten 2D basis lattice beams, where the series expansion is symbolically written as Here,

*a*is the weighting coefficient of the

_{m}*m*th order term and

*I*describes a 2D basis intensity. Further, the vector

**r**= (

*x*,

*y*)

*characterizes a point in 2D space and*

^{T}*j*= 1, 2 (or depending on the rotational symmetry of the structure

*j*= 1, 2, 3, respectively) represents a modulation vector in the corresponding 2D Fourier space. In general, all following field distributions and their intensities are specified as 2D functions exclusively in a transverse plane. The thereto orthogonal direction

*z*can be identified with the direction of propagation. The field evolution in this direction as well as the temporal development are consequently neglected in the following considerations, as these behaviors can be examined by a multiplication of the field distribution of interest with the term exp(

*i*(

*k*

_{||}

*z*−

*ωt*)), where

*k*

_{3D}in the direction of propagation and

*k*describes the modulation in transverse direction.

29. P. Rose, M. Boguslawski, and C. Denz, “Nonlinear lattice structures based on families of complex nondiffracting beams,” New J. Phys. **14**, 033018 (2012). [CrossRef]

39. M. Boguslawski, P. Rose, and C. Denz, “Increasing the structural variety of discrete nondiffracting wave fields,” Phys. Rev. A **84**, 013832 (2011). [CrossRef]

### 2.1. Staircase superlattice

*m*lead to the derived series expansions with 2D basis sets consisting of nondiffracting beam intensities.

_{2D}with respect to 1D fundamentals. To keep the intensity non-negative, we adjoin an offset term: where

*m*determines the order of the expansion term. Thus, for every expansion term

*m*two

*sin*-modulations with orthogonal modulation vectors

_{2D}. The length of the modulation vectors are given by the arbitrarily chosen lattice period of the first order

*g*

_{1}and the order

*m*, hence: In the next step we will combine the 1D intensity contributions of a particular value of

*m*. These terms reveal modulation vectors

*m*in Eq. (2) can be rewritten as

*ϕ*for a particular

_{j}*m*with

*j*= 1, 2 using

*m*. Now it becomes obvious that the wave field whose absolute square value appears in Eq. (4) equals the intensity of a field distribution consisting of four interfering plane waves. This field belongs to the four-fold discrete NDBs, whose transverse intensity distribution is presented in Fig. 1(c) and which is referred as Ψ

_{4,1}(

**r**) in Ref. [39

39. M. Boguslawski, P. Rose, and C. Denz, “Increasing the structural variety of discrete nondiffracting wave fields,” Phys. Rev. A **84**, 013832 (2011). [CrossRef]

*π*/4, where plane waves of opposite spatial frequency have opposite sign, resulting in a real space translation of the structure in transverse direction (cf. Ref. [39

**84**, 013832 (2011). [CrossRef]

### 2.2. Hexagonal wire mesh superlattice

40. A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. **6**, 183–191 (2007). [CrossRef] [PubMed]

*ϕ*with in this case

_{j}*j*= 1, 2, 3 valid for all orders

*m*. The term in square brackets appearing in Eq. (6) and representing the unweighted intensity of every expansion order

*m*can be rewritten to Now we have to find a 2D basis for the expansion, for instance given by the intensity of a hexagonal nondiffracting beam, which consists of three plane waves conforming the spatial frequency conditions of a nondiffracting beam (cf. Ref. [39

**84**, 013832 (2011). [CrossRef]

*ϕ*

_{1}=

*φ*

_{1}−

*φ*

_{2},

*ϕ*

_{2}=

*φ*

_{1}−

*φ*

_{3},

*ϕ*

_{3}=

*φ*

_{2}−

*φ*

_{3}. Now, the arguments of the three plane waves in Eq. (9) are summations of the formerly introduced phase functions

*ϕ*.

_{j}### 2.3. Ratchet superlattice

41. T. Salger, S. Kling, T. Hecking, C. Geckeler, L. Morales-Molina, and M. Weitz, “Directed transport of atoms in a Hamiltonian quantum ratchet,” Science **326**, 1241–1243 (2009). [CrossRef] [PubMed]

42. P. Hänggi and F. Marchesoni, “Artificial Brownian motors: Controlling transport on the nanoscale,” Rev. Mod. Phys. **81**, 387–442 (2009). [CrossRef]

*m*from two mutual orthogonal vectors and are chosen to be With the previously introduced phase function

*ϕ*=

_{j}**k**

_{j}**r**/2, where

*j*= 1, 2, and under consideration of Eq. (4), every expansion term of Λ

_{2}

*can be expressed through the earlier introduced wave field Ψ*

_{D}_{4,1}(

**r**,

*k*), which again is depicted in Fig. 3(c). Hence, the series expansion for a multiperiodic 2D ratchet intensity distribution Λ

_{m}_{2}

*composed of a 2D set of fundamental nondiffracting beam intensities is where exclusively the weighting factor 1/(2*

_{D}*m*) as well as the relation between the modulation vectors of every order

*m*determines the difference between Eq. (5) and Eq. (13).

## 3. Experimental implementation

*λ*= 532 nm and an output power of approximately 80 mW is phase modulated by a computer controlled spatial light modulator (PSLM, Holoeye ‘Pluto’, 1920 × 1080 pixels). The phase pattern given to the PSLM and the displayed pattern of the following amplitude modulator for spatial frequency filtering (ASLM, Holoeye ‘LC-R 2500’, 1024 × 768 pixels) are chosen to implement particular nondiffracting beams according to certain expansion terms. The phase distribution includes both amplitude as well as phase information about the desired wave field which is numerically calculated beforehand. Besides phase modulation, amplitude modulation can be achieved by an intensity weighted blazed grating to diffract the relevant light into a certain diffraction order [43

43. J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno “Encoding amplitude information onto phase-only filters,” Appl. Opt. **38**, 5004–5013 (1999). [CrossRef]

*t*is easily adaptable via a short switching time of the PSLM.

_{m}^{2}at the imaging plane of the PSLM, the laser power over this area is about 300 μW. Into this volume, whose longitudinal length extends several centimeters [29

29. P. Rose, M. Boguslawski, and C. Denz, “Nonlinear lattice structures based on families of complex nondiffracting beams,” New J. Phys. **14**, 033018 (2012). [CrossRef]

*c*axis) in order to minimize the influence of the induced refractive index change on the writing light due to a small electro-optical coefficient [36

36. P. Rose, B. Terhalle, J. Imbrock, and C. Denz, “Optically induced photonic superlattices by holographic multiplexing,” J. Phys. D: Appl. Phys. **41**, 224004 (2008). [CrossRef]

35. Y. Taketomi, J. E. Ford, H. Sasaki, J. Ma, Y. Fainman, and S. H. Lee “Incremental recording for photorefractive hologram multiplexing,” Opt. Lett. **16**, 1774–1776 (1991). [CrossRef] [PubMed]

*τ*, and also gets exponentially erased (

_{w}*τ*) due to illumination with non-identical intensity distributions. This can be summarized by with Δ

_{e}*n*

_{0}= Δ

*n*(

*t*= 0) and Δ

*n*as the saturation value of the refractive index change. In general, a written refractive index modulation is persistent in the SBN crystal until further illuminations.

_{sat}*m*is weighted by a factor depending on

*m*. This weighting factor is realized in the experiment by adapting the illumination time

*t*for each lattice-inducing beam. Thereby, rather than controlling the external voltage or intensity of each fundamental structure, we regulate

_{m}*t*to weight a specific induction term. Hence, we determine the net illumination duration for one sequence of illumination to

_{m}## 4. Analysis of photonic structures

44. B. Kemper and G. von Bally, “Digital holographic microscopy for live cell applications and technical inspection,” Appl. Opt. **47**, A52–A61 (2008). [CrossRef] [PubMed]

*c*axis to operate with a high electro-optical coefficient, which is approximately 6 times larger than the one for the writing procedure.

*n*caused by the optical induction process can be extracted by carrying out Δ

*n*= Δ

*φλ*/(2

*πd*), where

*d*is the optical path length of the probe beam through the medium.

^{2}. In addition to these illustrations, the induced structures are presented as animated surface plots supporting an enhanced spatial impression of all refractive index modulations (cf. Media 1, Media 2, Media 3).

*sinc*function. We note that this specific rotational symmetry particularly results from the diamond orientation of the fundamental structures and is not the ordinary case for photorefractive media with drift dominated charge carrier transport processes causing an orientation anisotropy [46

46. B. Terhalle, A. S. Desyatnikov, C. Bersch, D. Träger, L. Tang, J. Imbrock, Y. S. Kivshar, and C. Denz, “Anisotropic photonic lattices and discrete solitons in photorefractive media,” Appl. Phys. B **86**, 399–405 (2007). [CrossRef]

*c*axis compared to any other direction. Thus, vertical lines perpendicular to the

*c*axis are more pronounced than the ones that are inclined by 30° to the

*c*axis. The far field picture presented in the inset of Fig. 5(d) and the surface plot in Fig. 5(f) emphasize the influence of the crystal’s orientation anisotropy, additionally.

37. M. Boguslawski, A. Kelberer, P. Rose, and C. Denz, “Photonic ratchet superlattices by optical multiplexing,” Opt. Lett. **37**, 797–799 (2012). [CrossRef] [PubMed]

*c*axis, respectively. Altogether, especially the ratchet structure is preferred to be applied as a light director and marks a throughout relevant system for propagation experiments in the linear and in the nonlinear regime, as well.

36. P. Rose, B. Terhalle, J. Imbrock, and C. Denz, “Optically induced photonic superlattices by holographic multiplexing,” J. Phys. D: Appl. Phys. **41**, 224004 (2008). [CrossRef]

*t*

_{1}= 20 s,

*t*

_{2}= 80 s,

*t*

_{3}= 140 s, and

*t*

_{4}= 200 s. The formation of the staircase photonic structure becomes obvious already after

*t*

_{3}= 140 s, and also the far field intensity reveals characteristic details of the desired structure after several seconds of illumination.

## 5. Conclusion

## Acknowledgments

## References and links

1. | E. Yablonovitch, T. J. Gmitter, and K. M. Leung, “Photonic band structure: The face-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. |

2. | M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Bragg diffraction of finite beams by thick gratings,” J. Opt. Soc. Am. |

3. | R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, “Optical analogue of electronic Bloch oscillations,” Phys. Rev. Lett. |

4. | H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett. |

5. | T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature |

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

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

8. | 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 |

9. | O. Manela, O. Cohen, G. Bartal, J. W. Fleischer, and M. Segev, “Two-dimensional higher-band vortex lattice solitons,” Opt. Lett. |

10. | 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. |

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

12. | B. Terhalle, T. Richter, A. S. Desyatnikov, D. N. Neshev, W. Krolikowski, F. Kaiser, C. Denz, and Y. S. Kivshar, “Observation of multivortex solitons in photonic lattices,” Phys. Rev. Lett. |

13. | Y. S. Chan, C. T. Chan, and Z. Y. Liu, “Photonic band gaps in two dimensional photonic quasicrystals,” Phys. Rev. Lett. |

14. | B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides, and J. W. Fleischer, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature |

15. | Y. V. Kartashov, A. A Egorov, V. A. Vysloukh, and L. Torner, “Stable soliton complexes and azimuthal switching in modulated Bessel optical lattices,” Phys. Rev. E |

16. | Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Rotary solitons in bessel optical lattices,” Phys. Rev. Lett. |

17. | Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton shape and mobility control in optical lattices,” Prog. Optics |

18. | S. Huang, P. Zhang, X. Wang, and Z. Chen, “Observation of soliton interaction and planetlike orbiting in Bessel-like photonic lattices,” Opt. Lett. |

19. | F. Ye, D. Mihalache, and B. Hu, “Elliptic vortices in composite Mathieu lattices,” Phys. Rev. A |

20. | D. M. Jović, M. R. Belić, and C. Denz, “Anderson localization of light at the interface between linear and nonlinear dielectric media with an optically induced photonic lattice,” Phys. Rev. A |

21. | S. Longhi, “Klein tunneling in binary photonic superlattices,” Phys. Rev. B |

22. | S. Longhi, “Photonic analog of Zitterbewegung in binary waveguide arrays,” Opt. Lett. |

23. | F. Dreisow, M. Heinrich, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, “Classical simulation of relativistic Zitterbewegung in photonic lattices,” Phys. Rev. Lett. |

24. | M. Heinrich, Y. V. Kartashov, L. P. R. Ramirez, A. Szameit, F. Dreisow, R. Keil, S. Nolte, A. Tünnermann, V. A. Vysloukh, and L. Torner, “Observation of two-dimensional superlattice solitons,” Opt. Lett. |

25. | M. Heinrich, R. Keil, F. Dreisow, A. Tünnermann, A. Szameit, and S. Nolte, “Nonlinear discrete optics in femtosecond laser-written photonic lattices,” Appl. Phys. B |

26. | M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature |

27. | B. Terhalle, A. S. Desyatnikov, D. N. Neshev, W. Krolikowski, C. Denz, and Y. S. Kivshar, “Dynamic diffraction and interband transitions in two-dimensional photonic lattices,” Phys. Rev. Lett. |

28. | A. S. Desyatnikov, N. Sagemerten, R. Fischer, B. Terhalle, D. Träger, D. N. Neshev, A. Dreischuh, C. Denz, W. Krolikowski, and Y. S. Kivshar, “Two-dimensional self-trapped nonlinear photonic lattices,” Opt. Express |

29. | P. Rose, M. Boguslawski, and C. Denz, “Nonlinear lattice structures based on families of complex nondiffracting beams,” New J. Phys. |

30. | J. Xavier, M. Boguslawski, P. Rose, J. Joseph, and C. Denz, “Reconfigurable optically induced quasicrystallographic three-dimensional complex nonlinear photonic lattice structures,” Adv. Mater. |

31. | J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A |

32. | Z. Bouchal, “Nondiffracting optical beams: Physical properties, experiments, and applications,” Czech. J. Phys. |

33. | M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. |

34. | C. Denz, G. Pauliat, G. Roosen, and T. Tschudi, “Volume hologram multiplexing using a deterministic phase encoding method,” Opt. Commun. |

35. | Y. Taketomi, J. E. Ford, H. Sasaki, J. Ma, Y. Fainman, and S. H. Lee “Incremental recording for photorefractive hologram multiplexing,” Opt. Lett. |

36. | P. Rose, B. Terhalle, J. Imbrock, and C. Denz, “Optically induced photonic superlattices by holographic multiplexing,” J. Phys. D: Appl. Phys. |

37. | M. Boguslawski, A. Kelberer, P. Rose, and C. Denz, “Photonic ratchet superlattices by optical multiplexing,” Opt. Lett. |

38. | J. Becker, P. Rose, M. Boguslawski, and C. Denz, “Systematic approach to complex periodic vortex and helix lattices,” Opt. Express |

39. | M. Boguslawski, P. Rose, and C. Denz, “Increasing the structural variety of discrete nondiffracting wave fields,” Phys. Rev. A |

40. | A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. |

41. | T. Salger, S. Kling, T. Hecking, C. Geckeler, L. Morales-Molina, and M. Weitz, “Directed transport of atoms in a Hamiltonian quantum ratchet,” Science |

42. | P. Hänggi and F. Marchesoni, “Artificial Brownian motors: Controlling transport on the nanoscale,” Rev. Mod. Phys. |

43. | J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno “Encoding amplitude information onto phase-only filters,” Appl. Opt. |

44. | B. Kemper and G. von Bally, “Digital holographic microscopy for live cell applications and technical inspection,” Appl. Opt. |

45. | U. Schnars and W. Jueptner, |

46. | B. Terhalle, A. S. Desyatnikov, C. Bersch, D. Träger, L. Tang, J. Imbrock, Y. S. Kivshar, and C. Denz, “Anisotropic photonic lattices and discrete solitons in photorefractive media,” Appl. Phys. B |

**OCIS Codes**

(090.4220) Holography : Multiplex holography

(100.5070) Image processing : Phase retrieval

(220.4000) Optical design and fabrication : Microstructure fabrication

(070.6120) Fourier optics and signal processing : Spatial light modulators

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: August 23, 2012

Revised Manuscript: October 12, 2012

Manuscript Accepted: October 19, 2012

Published: November 19, 2012

**Virtual Issues**

Nonlinear Photonics (2012) *Optics Express*

**Citation**

Martin Boguslawski, Andreas Kelberer, Patrick Rose, and Cornelia Denz, "Multiplexing complex two-dimensional photonic superlattices," Opt. Express **20**, 27331-27343 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-24-27331

Sort: Year | Journal | Reset

### References

- E. Yablonovitch, T. J. Gmitter, and K. M. Leung, “Photonic band structure: The face-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett.67, 2295–2298 (1991). [CrossRef] [PubMed]
- M. G. Moharam, T. K. Gaylord, and R. Magnusson, “Bragg diffraction of finite beams by thick gratings,” J. Opt. Soc. Am.70, 300–304 (1980). [CrossRef]
- R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, “Optical analogue of electronic Bloch oscillations,” Phys. Rev. Lett.91, 263902 (2003). [CrossRef]
- H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” Phys. Rev. Lett.96, 053903 (2006). [CrossRef] [PubMed]
- T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature446, 52–55 (2007). [CrossRef] [PubMed]
- D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett.13, 794–796 (1988). [CrossRef] [PubMed]
- D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, “Spatial solitons in optically induced gratings,” Opt. Lett.28, 710–712 (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,” Nature422, 147–150 (2003). [CrossRef] [PubMed]
- 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]
- 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 (2005). [CrossRef] [PubMed]
- T. J. Alexander, A. S. Desyatnikov, and Y. S. Kivshar, “Multivortex solitons in triangular photonic lattices,” Opt. Lett.32, 1293–1295 (2007). [CrossRef] [PubMed]
- B. Terhalle, T. Richter, A. S. Desyatnikov, D. N. Neshev, W. Krolikowski, F. Kaiser, C. Denz, and Y. S. Kivshar, “Observation of multivortex solitons in photonic lattices,” Phys. Rev. Lett.101, 013903 (2008). [CrossRef] [PubMed]
- Y. S. Chan, C. T. Chan, and Z. Y. Liu, “Photonic band gaps in two dimensional photonic quasicrystals,” Phys. Rev. Lett.80, 956–959 (1998). [CrossRef]
- B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides, and J. W. Fleischer, “Wave and defect dynamics in nonlinear photonic quasicrystals,” Nature440, 1166–1169 (2006). [CrossRef] [PubMed]
- Y. V. Kartashov, A. A Egorov, V. A. Vysloukh, and L. Torner, “Stable soliton complexes and azimuthal switching in modulated Bessel optical lattices,” Phys. Rev. E70, 065602(R) (2004). [CrossRef]
- Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Rotary solitons in bessel optical lattices,” Phys. Rev. Lett.93, 093904 (2004). [CrossRef] [PubMed]
- Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton shape and mobility control in optical lattices,” Prog. Optics52, 63–148 (2009). [CrossRef]
- S. Huang, P. Zhang, X. Wang, and Z. Chen, “Observation of soliton interaction and planetlike orbiting in Bessel-like photonic lattices,” Opt. Lett.35, 2284–2286 (2010). [CrossRef] [PubMed]
- F. Ye, D. Mihalache, and B. Hu, “Elliptic vortices in composite Mathieu lattices,” Phys. Rev. A79, 053852 (2009). [CrossRef]
- D. M. Jović, M. R. Belić, and C. Denz, “Anderson localization of light at the interface between linear and nonlinear dielectric media with an optically induced photonic lattice,” Phys. Rev. A85, 031801(R) (2012). [CrossRef]
- S. Longhi, “Klein tunneling in binary photonic superlattices,” Phys. Rev. B81, 075012 (2010). [CrossRef]
- S. Longhi, “Photonic analog of Zitterbewegung in binary waveguide arrays,” Opt. Lett.35, 235–237 (2010). [CrossRef] [PubMed]
- F. Dreisow, M. Heinrich, R. Keil, A. Tünnermann, S. Nolte, S. Longhi, and A. Szameit, “Classical simulation of relativistic Zitterbewegung in photonic lattices,” Phys. Rev. Lett.105, 143902 (2010). [CrossRef]
- M. Heinrich, Y. V. Kartashov, L. P. R. Ramirez, A. Szameit, F. Dreisow, R. Keil, S. Nolte, A. Tünnermann, V. A. Vysloukh, and L. Torner, “Observation of two-dimensional superlattice solitons,” Opt. Lett.34, 3701–3703 (2009). [CrossRef] [PubMed]
- M. Heinrich, R. Keil, F. Dreisow, A. Tünnermann, A. Szameit, and S. Nolte, “Nonlinear discrete optics in femtosecond laser-written photonic lattices,” Appl. Phys. B104, 469–480 (2011). [CrossRef]
- M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, “Fabrication of photonic crystals for the visible spectrum by holographic lithography,” Nature404, 53–56 (2000). [CrossRef] [PubMed]
- B. Terhalle, A. S. Desyatnikov, D. N. Neshev, W. Krolikowski, C. Denz, and Y. S. Kivshar, “Dynamic diffraction and interband transitions in two-dimensional photonic lattices,” Phys. Rev. Lett.106, 083902 (2011). [CrossRef] [PubMed]
- A. S. Desyatnikov, N. Sagemerten, R. Fischer, B. Terhalle, D. Träger, D. N. Neshev, A. Dreischuh, C. Denz, W. Krolikowski, and Y. S. Kivshar, “Two-dimensional self-trapped nonlinear photonic lattices,” Opt. Express14, 2851–2863 (2006). [CrossRef] [PubMed]
- P. Rose, M. Boguslawski, and C. Denz, “Nonlinear lattice structures based on families of complex nondiffracting beams,” New J. Phys.14, 033018 (2012). [CrossRef]
- J. Xavier, M. Boguslawski, P. Rose, J. Joseph, and C. Denz, “Reconfigurable optically induced quasicrystallographic three-dimensional complex nonlinear photonic lattice structures,” Adv. Mater.22, 356–360 (2010). [CrossRef] [PubMed]
- J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A4, 651–654 (1987). [CrossRef]
- Z. Bouchal, “Nondiffracting optical beams: Physical properties, experiments, and applications,” Czech. J. Phys.53, 537–578 (2003). [CrossRef]
- M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett.29, 44–46 (2004). [CrossRef] [PubMed]
- C. Denz, G. Pauliat, G. Roosen, and T. Tschudi, “Volume hologram multiplexing using a deterministic phase encoding method,” Opt. Commun.85, 171–176 (1991). [CrossRef]
- Y. Taketomi, J. E. Ford, H. Sasaki, J. Ma, Y. Fainman, and S. H. Lee “Incremental recording for photorefractive hologram multiplexing,” Opt. Lett.16, 1774–1776 (1991). [CrossRef] [PubMed]
- P. Rose, B. Terhalle, J. Imbrock, and C. Denz, “Optically induced photonic superlattices by holographic multiplexing,” J. Phys. D: Appl. Phys.41, 224004 (2008). [CrossRef]
- M. Boguslawski, A. Kelberer, P. Rose, and C. Denz, “Photonic ratchet superlattices by optical multiplexing,” Opt. Lett.37, 797–799 (2012). [CrossRef] [PubMed]
- J. Becker, P. Rose, M. Boguslawski, and C. Denz, “Systematic approach to complex periodic vortex and helix lattices,” Opt. Express19, 9848–9862 (2011). [CrossRef] [PubMed]
- M. Boguslawski, P. Rose, and C. Denz, “Increasing the structural variety of discrete nondiffracting wave fields,” Phys. Rev. A84, 013832 (2011). [CrossRef]
- A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater.6, 183–191 (2007). [CrossRef] [PubMed]
- T. Salger, S. Kling, T. Hecking, C. Geckeler, L. Morales-Molina, and M. Weitz, “Directed transport of atoms in a Hamiltonian quantum ratchet,” Science326, 1241–1243 (2009). [CrossRef] [PubMed]
- P. Hänggi and F. Marchesoni, “Artificial Brownian motors: Controlling transport on the nanoscale,” Rev. Mod. Phys.81, 387–442 (2009). [CrossRef]
- J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno “Encoding amplitude information onto phase-only filters,” Appl. Opt.38, 5004–5013 (1999). [CrossRef]
- B. Kemper and G. von Bally, “Digital holographic microscopy for live cell applications and technical inspection,” Appl. Opt.47, A52–A61 (2008). [CrossRef] [PubMed]
- U. Schnars and W. Jueptner, Digital Holography (Springer, 2005).
- B. Terhalle, A. S. Desyatnikov, C. Bersch, D. Träger, L. Tang, J. Imbrock, Y. S. Kivshar, and C. Denz, “Anisotropic photonic lattices and discrete solitons in photorefractive media,” Appl. Phys. B86, 399–405 (2007). [CrossRef]

## 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.

### Supplementary Material

» Media 1: AVI (6020 KB)

» Media 2: AVI (5837 KB)

» Media 3: AVI (6200 KB)

» Media 4: AVI (1131 KB)

« Previous Article | Next Article »

OSA is a member of CrossRef.