Open Access
Add to BibSonomyAbstract
We theoretically report the existence of optical Bloch oscillations (BO) of an Airy beam in a one-dimensional optically induced photonic lattice with a linear transverse index gradient. The Airy beam experiencing optical BO shows a more robust non-diffracting feature than its counterparts in free space or in a uniform photonic lattice. Interestingly, a periodical recurrence of Airy shape accompanied with constant alternation of its acceleration direction is also found during the BO. Furthermore, we demonstrate that the period and amplitude of BO of an Airy beam can be readily controlled over a wide range by varying the index gradient and/or the lattice period. Exploiting these features, we propose a scheme to rout an Airy beam to a predefined output channel without losing its characteristics by longitudinally modulating the transverse index gradient.
© 2014 Optical Society of America
1. Introduction
Airy beams, a kind of diffraction-free wave packets, have attracted enormous attention since 2007 when they were first introduced to the field of optics [1,2]. One of the most intriguing properties of an Airy beam is its ability for self-accelerating, i.e., the trajectory of light follows a parabolic curve during propagation. This unique characteristic enables potential applications such as optical micromanipulation [3,4], plasmonics routing [5–7], high-resolution microscopy [8,9], and all-optical routing [10]. The implementations of these applications require precisely controlling the propagation dynamics of Airy beams. Researchers have demonstrated that the ballistic trajectory of Airy beam can be tuned in linear region by spatial modulations of the refractive index [11–13], phase [14,15] or spatial momentum [7,16,17]. Additionally, it has been shown that the propagation of Airy beam can be controlled nonlinearly through the phase-matching technique [18], diffusion [19], and self-focusing or self-defocusing effects [20,21]. Fundamentally different to the above two approaches, yet an effective way of dynamical controlling an Airy beam is to use periodic photonic structures, such as photonic lattices [22,23] and photonic crystals [24]. Recent researches also show that Airy beams propagating in photonic lattices can give rise to interesting phenomena, such as accelerating lattice solitons [22] and accelerating Wannier-Stark (WS) states [23,25].
In fact, light propagating in a periodic photonic structure is analogous to electron travelling in a crystalline solid. Therefore, one can manipulate the propagation of a light beam in a photonic structure very much like electrons are controlled by a static electric field applied to a periodic potential. This can be exemplified by the optical Bloch oscillations (BO) that a light beam experiences a cosine-like trajectory with a non-diffracting feature when travelling through photonic structures with a linear index gradient [26–31], resembling the performance of an electron in a semiconductor supperlattice under a dc-bias voltage. Although BOs have been realized in various physical settings, such as photonic crystal [32], plasmonics [33] and dielectric waveguide arrays [28], the realization of such settings generally requires time-consuming and costly fabrications. Moreover, it is extremely difficult to achieve a dynamically tunable BO in these platforms. On the contrary, optically induced photonic lattices can be easily written, erased, and reconfigured in the photorefractive crystals [34], providing a flexible and low-cost platform to control the propagation dynamics of optical BO. In this work, we theoretically study the optical BO of an Airy beam in a one-dimensional (1D) optically induced photonic lattice with a linear transverse index gradient. It is found that during BO, an Airy beam preserves its non-spreading feature for a significantly longer propagation distance in comparison with its counterparts in free space as well as in uniform photonic lattices. The Airy beam undergoing BO also exhibits a perfect revival of the beam profile and an alternatively switched acceleration direction. Furthermore, the BO of an Airy beam shows high tunabilities in its period and amplitude by changing the index gradient and lattice period. This offers opportunities to steer the Airy beam to a specific output channel, opening new prospects for all-optical routing.
2. Theory model
The 1D photonic lattice can be optically induced in a biased photorefractive crystal, for example strontium barium niobate (SBN), by modulation the lattice writing beam with a amplitude mask [30,34]. Figure 1(a) shows a 1D optically induced photonic lattice, which is created by a lattice writing beam I0 = cos(2πx/Λ), where Λ is the lattice period. A linear transverse index gradient as depicted in Fig. 1(b) can be imprinted on a photonic lattice when the lattice induced beam is modulated transversely [30]. Then, the induced transverse refractive index pattern is given by [35]
Here, ne is the extraordinary refractive index, γ33 is the electro-optic coefficient, E0 is the biased electric field, and ρ is the linear index gradient.
Fig. 1 (a) Schematic view of one-dimensional (1D) photonic lattice (left) with its index distribution (right). (b) Schematic view of 1D photonic lattice with an index gradient (left) and its index distribution (right). (c) The first two Bloch bands of the uniform 1D photonic lattice as shown in (a).
The light propagation in a periodical structure is dominated by the band-gap structure (or diffraction relation). Figure 1(c) shows the first two Bloch bands of the uniform 1D photonics lattice shown in Fig. 1(a), which is calculated by using the Floquet-Bloch approach [36]. Within the framework of the couple mode theory [37], the first Bloch band can be analytically described as
Here, kx and kz are respectively for the x and z components of the wave vector k, β is the propagation constant of waveguide, and η denotes the coupling coefficient between the adjacent waveguides. Then, the transverse velocity of light is given byMeanwhile, under paraxial approximation, the trajectory of light in a medium with index gradient obeys [38]:where k0 is the wave vector in vacuum. Combing Eq. (3) and Eq. (4), we arrive atwhich describes the oscillating trajectory of the light, i.e. BO, in a 1D photonic lattice with a linear transverse index gradient.To numerically analyze the propagation dynamics of Airy beam in a 1D photonic lattice with a transverse index gradient, we employ the paraxial equation of diffraction [13]
Here, ψ = Ai(s/5)exp(0.04s) is the Airy wavepacket, s = (kl)1/2x is the dimensionless transverse coordinate with l = 0.5kne2γ33E0, ξ = lz is the normalized propagation distance, and α = ρ/l represents the normalized index gradient. The evolution of Airy beam can be numerically derived by solving Eq. (6) with the split-step Fourier propagation method. In the following simulations, we use realistic parameters: λ = 532 nm, ne = 2.33, γ33 = 280 pm/V, and E0 = 300 V/mm, and the corresponding normalized physical units are x0 = 1/(kl)1/2 = 2.4 μm, z0 = 1/l = 0.16 mm and ρ0 = l = 6.2/mm.3. Results and discussion
It is found that by properly engineering the parameters of the photonic lattice, the optical BO of an Airy beam can occur [see Fig. 2(a) as an example with Λ = 2, α = 0.01, where the beam exhibits a cosine-like oscillation that can be analytically described by Eq. (5)]. Though the optical BO stems from the beating of the optical WS states, a more intuitive picture of BO can be found by considering the competition between the Bragg reflections and the total internal reflections. The initially launched Airy beam has a propagation direction along the lattice, corresponding to the wave number located at the centre of the Brillouin zone. Due to the index gradient, the Airy beam propagates toward the higher index region leading to a gradually increased wave number. When the wave number reaches the boundary of the Brillouin zone, where Bragg reflection takes place, it is reversed and tilts towards the opposite boundary of the Brillouin zone. As a result, the Airy beam travels back to the lower index region. This beam experiences the total internal reflection when its wave number approaches to the centre of the Brillouin zone. The constant alternation between the Bragg reflections and the total internal reflections eventually contributes to a periodical oscillation of the Airy beam, i.e. optical BO. Note that during BO the Airy beam exhibits characteristics of non-spreading and periodically revived input excitation even at a quite long propagation distance. These interesting features have not been found during the propagation of an Airy beam either in free space [Fig. 2(b)] or in a uniform photonic lattice [Fig. 2(c)]. It can be also clearly seen in Figs. 2(d)–2(g), which represent the input excitation intensity and intensity profiles at the dashed lines of Figs. 2(a)–2(c), respectively. The Airy beam experiencing with BO maintains an almost undisturbed profile in term of intensity and width, as shown in Fig. 2(e). Those travelling in the free space and photonic lattice however loss their non-diffraction feature and decay to barely flatten patterns, as shown in Figs. 2(f) and 2(g), respectively. It is also worth noting that the convex and concave trajectories alternatively appear when the Airy beam propagating during the BO, leading to a periodically switching of the acceleration direction of the light beam. This unusual feature provides a new approach to control the acceleration direction of an Airy beam, which may be beneficial for Airy beam based optical manipulation.
Fig. 2 Propagation dynamics of Airy beam in (a) photonic lattice with a liner transverse index gradient, (b) free space, and (c) uniform photonic lattice. (d) Input intensity profile of Airy beam. (e)-(g) Cross-sections of light intensity at the dashed lines of (a)-(c), respectively.
As can be seen from Eq. (5), both the period and amplitude of BO of an Airy beam are inversely proportional to the index gradient. This provides an opportunity to modulate the propagation dynamics of an Airy beam by fine tuning the parameters of the photonic lattice. Figures 3(a)–3(c) display the BO of Airy beam in photonic lattices (Λ = 2) with three typical index gradients α = 0.006, 0.01, and 0.015, respectively. The detailed dependences of the oscillation period (ΓBO) and the amplitude (ABO) on the inverse of index gradient are respectively plotted in Figs. 3(d) and 3(e). Here, the red dots are obtained by fitting the trajectory of the beam center of Airy beam (defined as ∫s|ψ|2dr/∫|ψ|2dr) with Eq. (5). It can be seen that both the period and amplitude of BO scale with the inverse of index gradient. This trend can be well represented by the reciprocal functions ΓBO = 3.04/α and ΑBO = 0.3169/α, which are respectively depicted as solid lines in Figs. 3(d) and 3(e). Notably, the BO of Airy beam can be sustained within the index gradient region from 0.003 to 0.015, accompanying with a 5 times modulation in oscillation period and a 6 times modulation in oscillation amplitude. The modulation depth can be further increased within an extended range of index gradient at the larger lattice period, e.g. Λ = 4.
Fig. 3 Optical Bloch oscillations (BO) of Airy beam in 1D photonic lattices with index gradients (a) α = 0.006, (b) α = 0.01, and (c) α = 0.015 at Λ = 2. (d) BO period and (e) amplitude versus 1/α, where the red dots and solid lines are the simulation and fitting results, respectively.
It is also demonstrated that a tunable BO of an Airy beam can be achieved through altering the period of the photonic lattice. Figures 4(a)–4(c) show the BO of an Airy beam in photonic lattices with three typical lattice periods Λ = 2, 3, and 3.6 at α = 0.01. The oscillation period and the amplitude versus the inverse of photonic lattice period are depicted in Figs. 4(d) and 4(e), where the red dots are derived by fitting the trajectory of BO of Airy beam with Eq. (5). Similarly, we find the oscillation period is inversed related to the lattice period, and can be expressed as ΓBO = 616.2/Λ as shown by the solid line in Fig. 4(d). It is also revealed that the oscillation amplitude linearly scales with the inverse of lattices period, which is not simply seen from Eq. (5). To obtain more physical insight, we extract the coupling coefficient of the photonic lattice by fitting its first Bloch band with Eq. (2) at different lattice periods [see the blue dots in Fig. 4(e)]. It is found that the coupling coefficient linearly increases with the inverse of lattice period, following the same trend with the oscillation amplitude. Therefore, the change of the oscillation amplitude can be attributed to the lattice period varied coupling coefficient, which also confirms the linear dependence of the oscillation amplitude on lattice period indicated by Eq. (5).
Fig. 4 BO of Airy beam in 1D photonic lattices with lattice periods (a) Λ = 2, (b) Λ = 3, and (c) Λ = 3.6 at α = 0.01. (d) BO period versus 1/Λ, where the red dots and solid line are the simulation and fitting results, respectively. (e) Dependence of oscillation amplitude (red dots) and coupling coefficient 2η (blue dots) on 1/Λ.
The tunable characteristics of BO displayed in Figs. 3 and 4 can be utilized to arbitrarily control the trajectory of an Airy beam. In Fig. 5, we present the examples to steer the path of the Airy beam undergoing BO by varying the parameters of photonic lattice. The upper (lower) panel of Fig. 5(a) shows the BO of Airy beam can be gradually attenuated (amplified) by increasing the index gradient α = 0.01 + 9.5 × 10−7ξ (decreasing α = 0.01-9.5 × 10−7ξ) along the propagation direction. Interestingly, it is observed that the Airy beam still preserves the capability of recurrence even subjected to a varying longitudinal index gradient. Similarly, the beam trajectory controllability is realized by longitudinally modulating the period of the photonic lattice as shown in Fig. 5(b), where the upper and lower panels correspond to a lattice period modulation of Λ = 2.9 ± 1.8 × 10−4ξ. Because the photonic lattice can be easily reconfigured by the writing beam in a photorefractive crystal [34], it is possible to introduce a proper index gradient increment or a variation of lattice period to precisely navigate the Airy beam to a pre-selected channel without losing the beam features. The results are exemplified by Fig. 5(c), where the lattice period is Λ = 2, and the beam profiles from top to bottom respectively correspond to the incident beam, the output beams at ξ = 800 after index gradient modulations of α = 0.01-9.7 × 10−7ξ, α = 0.015-1.5 × 10−6ξ, α = 0.007 + 1.6 × 10−6ξ and α = 0.005 + 2.4 × 10−6ξ. In this way, the Airy beam is able to be distributed to a desirable output channel, suggesting a potential strategy for all-optical routing.
Fig. 5 Controlling the beam path of Airy beam by longitudinally changing (a) the index gradient and (b) period of photonic lattice. The upper and lower panels of (a) are respectively for the index gradient modulations of α = 0.01 + 9.5 × 10−7ξ and α = 0.01-9.5 × 10−7ξ, while those of (b) are for the lattice period modulations of Λ = 2.9 + 1.8 × 10−4ξ and Λ = 2.9-1.8 × 10−4ξ, respectively. (c) Assigning the Airy beam to the desirable output channel by altering the index gradient variation along the propagation direction. The intensity profiles in (c) from top to bottom are respectively for the input beam, and the output beams at ξ = 800 after index gradient modulations of α = 0.01-9.7 × 10−7ξ, α = 0.015-1.5 × 10−6ξ, α = 0.007 + 1.6 × 10−6ξ and α = 0.005 + 2.4 × 10−6ξ.
As a concluding remark, we would like to discuss the experimental feasibility of observing the Airy beam BO. In the case of Fig. 2(a), we need a photonic lattice of over 50 mm in length, which is longer than the typical length of the photonic lattice for BO experiment [30], to observe a full cycle of BO. However, with the advance of crystal growth techniques [39], the SBN crystal with length over 100 mm can be grown, making it possible to fabricate photonic lattice for observing the long-distance evolution of Airy beam BO. On the other hand, to induce the photonic lattice with large size, a spatial band pass-filter can be utilized to effectively eliminate the Talbot self-imaging effect of the lattice writing beam, ensuring an undisturbed transvers index distribution along the propagation direction [40,41]. Alternatively, the biased electric field on the SBN crystal can be increased from 300 V/mm to a moderately higher value like 500 V/mm [30]. Then, according to the normalization rule in the theory section, it just requires a distance of 30 mm to observe a cycle of BO in Fig. 2(a). Therefore, the demanded size of the photonic lattice for Airy BO can be largely decreased, enabling a more readily photonic lattice fabrication procedure.
4. Conclusion
In summary, we have theoretically demonstrated that the optical BO of an Airy beam can exist in a 1D optically induced photonic lattice having a linear transverse index gradient. Unlike its counterparts in free space or in a uniform photonic lattice, an Airy beam undergoing BO holds a robust non-spreading beam profile even after a comparably long propagation distance and exhibits a capacity of perfect recurrence of Airy shape along with an alternating acceleration direction. Moreover, it has been demonstrated that both the period and the amplitude of the BO of an Airy beam are inversely related to the index gradient and lattice period, in agreement with our theoretical prediction. Such BO enables the navigation of Airy beams to a specific output channel without losing its characteristics. Our results may find potential applications in all-optical routing.
Acknowledgments
The work of F. Xiao, B. Li, M. Wang, P. Zhang and J. Zhao was supported by the National Basic Research Program (973 Program) of China (2012CB921900), NSFC (61308031, 61377035, 11304250) and the Natural Science Basic Research Plan in Shaanxi Province (2014JQ1033). The work of W. Zhu and M. Premaratne was supported by the Australian Research Council through its Discovery Grant scheme under Grant DP140100883.
References and links
1. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007). [CrossRef] [PubMed]
2. G. Siviloglou, J. Broky, A. Dogariu, and D. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007). [CrossRef] [PubMed]
3. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photon. 2(11), 675–678 (2008). [CrossRef]
4. P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Trapping and guiding microparticles with morphing autofocusing Airy beams,” Opt. Lett. 36(15), 2883–2885 (2011). [CrossRef] [PubMed]
5. A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, “Generation and near-field imaging of Airy surface plasmons,” Phys. Rev. Lett. 107(11), 116802 (2011). [CrossRef] [PubMed]
6. L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, “Plasmonic Airy beam generated by in-plane diffraction,” Phys. Rev. Lett. 107(12), 126804 (2011). [CrossRef] [PubMed]
7. P. Zhang, S. Wang, Y. Liu, X. Yin, C. Lu, Z. Chen, and X. Zhang, “Plasmonic Airy beams with dynamically controlled trajectories,” Opt. Lett. 36(16), 3191–3193 (2011). [CrossRef] [PubMed]
8. T. Vettenburg, H. I. Dalgarno, J. Nylk, C. Coll-Lladó, D. E. Ferrier, T. Čižmár, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014). [CrossRef] [PubMed]
9. S. Jia, J. C. Vaughan, and X. Zhuang, “Isotropic three-dimensional super-resolution imaging with a self-bending point spread function,” Nat. Photon. 8(4), 302–306 (2014). [CrossRef]
10. P. Rose, F. Diebel, M. Boguslawski, and C. Denz, “Airy beam induced optical routing,” Appl. Phys. Lett. 102(10), 101101 (2013). [CrossRef]
11. W. Liu, D. N. Neshev, I. V. Shadrivov, A. E. Miroshnichenko, and Y. S. Kivshar, “Plasmonic Airy beam manipulation in linear optical potentials,” Opt. Lett. 36(7), 1164–1166 (2011). [CrossRef] [PubMed]
12. Z. Ye, S. Liu, C. Lou, P. Zhang, Y. Hu, D. Song, J. Zhao, and Z. Chen, “Acceleration control of Airy beams with optically induced refractive-index gradient,” Opt. Lett. 36(16), 3230–3232 (2011). [CrossRef] [PubMed]
13. N. K. Efremidis, “Airy trajectory engineering in dynamic linear index potentials,” Opt. Lett. 36(15), 3006–3008 (2011). [CrossRef] [PubMed]
14. E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. 106(21), 213902 (2011). [CrossRef] [PubMed]
15. M. A. Bandres and J. C. Gutiérrez-Vega, “Airy-Gauss beams and their transformation by paraxial optical systems,” Opt. Express 15(25), 16719–16728 (2007). [CrossRef] [PubMed]
16. Y. Hu, P. Zhang, C. Lou, S. Huang, J. Xu, and Z. Chen, “Optimal control of the ballistic motion of Airy beams,” Opt. Lett. 35(13), 2260–2262 (2010). [CrossRef] [PubMed]
17. Y. Hu, D. Bongiovanni, Z. Chen, and R. Morandotti, “Periodic self-accelerating beams by combined phase and amplitude modulation in the Fourier space,” Opt. Lett. 38(17), 3387–3389 (2013). [CrossRef] [PubMed]
18. T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photon. 3(7), 395–398 (2009). [CrossRef]
19. S. Jia, J. Lee, J. W. Fleischer, G. A. Siviloglou, and D. N. Christodoulides, “Diffusion-trapped Airy beams in photorefractive media,” Phys. Rev. Lett. 104(25), 253904 (2010). [CrossRef] [PubMed]
20. R. Bekenstein and M. Segev, “Self-accelerating optical beams in highly nonlocal nonlinear media,” Opt. Express 19(24), 23706–23715 (2011). [CrossRef] [PubMed]
21. I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett. 106(21), 213903 (2011). [CrossRef] [PubMed]
22. K. Makris, I. Kaminer, R. El-Ganainy, N. Efremidis, Z. Chen, M. Segev, and D. Christodoulides, “Accelerating diffraction-free beams in photonic lattices,” Opt. Lett. 39(7), 2129–2132 (2014). [CrossRef] [PubMed]
23. R. El-Ganainy, K. G. Makris, M. A. Miri, D. N. Christodoulides, and Z. Chen, “Discrete beam acceleration in uniform waveguide arrays,” Phys. Rev. A 84(2), 023842 (2011). [CrossRef]
24. K. Makris, I. Kaminer, R. El-Ganainy, N. Efremidis, Z. Chen, M. Segev, and D. Christodoulides, “Accelerating diffraction-free beams in photonic lattices,” Opt. Express 21, 8886–8896 (2013). [CrossRef] [PubMed]
25. X. Qi, K. G. Makris, R. El-Ganainy, P. Zhang, J. Bai, D. N. Christodoulides, and Z. Chen, “Observation of accelerating Wannier-Stark beams in optically induced photonic lattices,” Opt. Lett. 39, 1065–1067 (2014).
26. U. Peschel, T. Pertsch, and F. Lederer, “Optical Bloch oscillations in waveguide arrays,” Opt. Lett. 23(21), 1701–1703 (1998). [CrossRef] [PubMed]
27. T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. 83(23), 4752–4755 (1999). [CrossRef]
28. R. Morandotti, U. Peschel, J. Aitchison, H. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 83(23), 4756–4759 (1999). [CrossRef]
29. I. Chremmos and N. K. Efremidis, “Surface optical Bloch oscillations in semi-infinite waveguide arrays,” Opt. Lett. 37(11), 1892–1894 (2012). [CrossRef] [PubMed]
30. 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(5), 053903 (2006). [CrossRef] [PubMed]
31. D. Witthaut, F. Keck, H. Korsch, and S. Mossmann, “Bloch oscillations in two-dimensional lattices,” New J. Phys. 6, 41 (2004). [CrossRef]
32. A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, “Geometric potential and transport in photonic topological crystals,” Phys. Rev. Lett. 104(15), 150403 (2010). [CrossRef] [PubMed]
33. A. Block, C. Etrich, T. Limboeck, F. Bleckmann, E. Soergel, C. Rockstuhl, and S. Linden, “Bloch oscillations in plasmonic waveguide arrays,” Nat. Commun. 5, 3843 (2014). [CrossRef] [PubMed]
34. P. Zhang, S. Liu, C. Lou, F. Xiao, X. Wang, J. Zhao, J. Xu, and Z. Chen, “Incomplete Brillouin-zone spectra and controlled Bragg reflection with ionic-type photonic lattices,” Phys. Rev. A 81(4), 041801 (2010). [CrossRef]
35. 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), 046602 (2002). [CrossRef] [PubMed]
36. P. S. J. Russell, “Optics of Floquet-Bloch waves in dielectric gratings,” Appl. Phys. B 39(4), 231–246 (1986). [CrossRef]
37. H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. 85(9), 1863–1866 (2000). [CrossRef] [PubMed]
38. M. Born and E. Wolf, Principles of Optics (CUP Archive, 1999), Chap. 3.
39. L. I. Ivleva, N. V. Bogodaev, N. M. Polozkov, and V. V. Osiko, “Growth of SBN single crystals by Stepanov technique for photorefractive applications,” Opt. Mater. 4(2–3), 168–173 (1995). [CrossRef]
40. P. Zhang, R. Egger, and Z. Chen, “Optical induction of three-dimensional photonic lattices and enhancement of discrete diffraction,” Opt. Express 17(15), 13151–13156 (2009). [CrossRef] [PubMed]
41. 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(3), 356–360 (2010). [CrossRef] [PubMed]
