## Self-accelerating beams in photonic crystals |

Optics Express, Vol. 21, Issue 7, pp. 8886-8896 (2013)

http://dx.doi.org/10.1364/OE.21.008886

Acrobat PDF (2886 KB)

### Abstract

We find accelerating beams in a general periodic optical system, such as photonic crystal slabs, honeycomb lattices, and various metamaterials. These beams retain a shape-preserving profile while bending to highly non-paraxial angles along a circular-like trajectory. The properties of such beams depend on the crystal lattice structure: on a small-scale, the fine features of the beams profile are uniquely derived from the exact structure of the crystalline cells, while on a large-scale the beam only depends on the periodicity of the lattice, asymptotically reaching the free-space analytic solutions when the wavelength is much larger than the cell size. We demonstrate such beams in a 2D Kronig-Penney separable model, but our methodology of finding such solutions is general, predicting accelerating beams in any periodic structure. This highlights how light can be guided through a general system by only tailoring the incoming field, without altering the structure itself.

© 2013 OSA

## Introduction

1. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. **32**(8), 979–981 (2007). [CrossRef] [PubMed]

2. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. **99**(21), 213901 (2007). [CrossRef] [PubMed]

3. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. **33**(3), 207–209 (2008). [CrossRef] [PubMed]

5. J. A. Davis, M. J. Mintry, M. A. Bandres, and D. M. Cottrell, “Observation of accelerating parabolic beams,” Opt. Express **16**(17), 12866–12871 (2008). [CrossRef] [PubMed]

6. F. Courvoisier, A. Mathis, L. Froehly, R. Giust, L. Furfaro, P. A. Lacourt, M. Jacquot, and J. M. Dudley, “Sending femtosecond pulses in circles: highly nonparaxial accelerating beams,” Opt. Lett. **37**(10), 1736–1738 (2012). [CrossRef] [PubMed]

8. E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. **106**(21), 213903 (2011). [CrossRef] [PubMed]

9. P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science **324**(5924), 229–232 (2009). [CrossRef] [PubMed]

10. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics **2**(11), 675–678 (2008). [CrossRef]

11. A. Salandrino and D. N. Christodoulides, “Airy plasmon: a nondiffracting surface wave,” Opt. Lett. **35**(12), 2082–2084 (2010). [CrossRef] [PubMed]

12. 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]

13. I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett. **106**(21), 213903 (2011). [CrossRef] [PubMed]

17. Y. Hu, Z. Sun, D. Bongiovanni, D. Song, C. Lou, J. Xu, Z. Chen, and R. Morandotti, “Reshaping the trajectory and spectrum of nonlinear Airy beams,” Opt. Lett. **37**(15), 3201–3203 (2012). [CrossRef] [PubMed]

**is not homogeneous**– hence the bending beam experiences a refractive index that varies during propagation, and has to compensate for that. To overcome this difficulty, Ref [18] has applied the tight binding model and predicted discrete accelerating beams in photonic waveguide arrays [18]. This naturally gives rise to an intriguing question: is it possible to design a self-accelerating beam in the actual continuous system with a periodically-varying refractive index, without using the tight binding approximation? In two recent papers [19,20

20. I. D. Chremmos and N. K. Efremidis, “Band-specific phase engineering for curving and focusing light in waveguide arrays,” Phys. Rev. A **85**(6), 063830 (2012). [CrossRef]

20. I. D. Chremmos and N. K. Efremidis, “Band-specific phase engineering for curving and focusing light in waveguide arrays,” Phys. Rev. A **85**(6), 063830 (2012). [CrossRef]

22. K. G. Makris, D. N. Christodoulides, O. Peleg, M. Segev, and D. Kip, “Optical transitions and Rabi oscillations in waveguide arrays,” Opt. Express **16**(14), 10309–10314 (2008). [CrossRef] [PubMed]

23. K. Shandarova, C. E. Rüter, D. Kip, K. G. Makris, D. N. Christodoulides, O. Peleg, and M. Segev, “Experimental observation of Rabi oscillations in photonic lattices,” Phys. Rev. Lett. **102**(12), 123905 (2009). [CrossRef] [PubMed]

24. B. Alfassi, O. Peleg, N. Moiseyev, and M. Segev, “Diverging rabi oscillations in subwavelength photonic lattices,” Phys. Rev. Lett. **106**(7), 073901 (2011). [CrossRef] [PubMed]

26. 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]

27. I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting accelerating wave packets of Maxwell’s equations,” Phys. Rev. Lett. **108**(16), 163901 (2012). [CrossRef] [PubMed]

**both**the propagation direction and in the other transverse direction, namely, a photonic crystal slab [28

28. S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B **60**(8), 5751–5758 (1999). [CrossRef]

29. P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljacić, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, “Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal,” Nat. Mater. **5**(2), 93–96 (2006). [CrossRef] [PubMed]

27. I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting accelerating wave packets of Maxwell’s equations,” Phys. Rev. Lett. **108**(16), 163901 (2012). [CrossRef] [PubMed]

**higher-order self-accelerating beams**. These include small-scale features and large-scale trajectories that do not exist in the free-space solutions. Finally, we show that certain solutions, which are highly affected by the crystal lattice structure, can reach deep subwavelength scales. For example, for a 500 nanometers wavelength in silicon, we find solutions displaying lobes of width ranging from about 1/15 of the wavelength to less than 1/50 of the wavelength. In that particular example, the whole beam is bending in a circle with a radius of one micron. To illustrate all of the above, we use a 2D periodic structure based on the superposition of two 1D Kronig-Penney lattices, which yields analytic solutions. This said, our method applies to any periodic structure in any number of dimensions.

## Methodology

*ε(x+nL*[for all integers

_{x},z+mL_{z})=ε(x,z)*n,m*]. Here, Maxwell's equations can be reduced to the scalar case, by assuming a transverse electric field

**E**(

*x,z*)

*=ŷE*(

*x,z*) which is polarized in the

*y*direction. In reality, instead of infinite thickness in the

*y*axis, we further assume that the slab (see Fig. 1(a) for a typical example) is uniform in the

*y*direction, and much thicker than

*L*and

_{x}*L*. Then, Maxwell's equations reduces to a Helmholtz-type equation for the electric field, with

_{z}*k*, and

_{0}=2π/λ*λ*being the free-space wavelength.Each solution can always be described by a superposition of Bloch modes

**k**

*=(k*(inside the Brillouin zone) and a periodic functionThese modes compose the band structure of the system (Fig. 1(b)). Restricting to a monochromatic light, we cut the band structure and get one (or more) closed curves (solid and dashed green-on-black curve in Fig. 1(b)). Any monochromatic solution is a superposition of modes along such a curve:with any arbitrary weight function

_{x,}k_{z})*w(*

**k**

*)*.

**How do we construct beams that accelerate inside lattices?**This problem translates to how to find the proper weight function

*w(*

**k**

*)*. To do that, we add a 3rd virtual dimension and find the family of beams which are non-diffracting along this new axis [30

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

27. I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting accelerating wave packets of Maxwell’s equations,” Phys. Rev. Lett. **108**(16), 163901 (2012). [CrossRef] [PubMed]

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

**108**(16), 163901 (2012). [CrossRef] [PubMed]

**108**(16), 163901 (2012). [CrossRef] [PubMed]

*η*We suggest two natural ways to choose this parameterization: the exact method is to measure the curve length and divide it evenly by infinitesimal curve lengths, inside the [0,2

*π*] interval; alternatively, the simple way to do that is to use the angle associated with the wave-vector

**k**(

*k*=

_{θ}*η*) as the parameter. This way, the actual integration parameter is simply the angle, but it relates to the actual point through an indirect relation – from the angle

*θ*to

*k*and then to

_{θ}**k**. The second method is possible only for “simple curves”, where the relation is well defined, meaning that there is only one point on the curve for each angle

*k*. Both methods are exactly identical in the free-space situation. In practice, we find that in standard periodic structures both methods yields almost the same results, hence we henceforth work with the simple parameterization.

_{θ}*k*is [0,

_{θ}*π*], covering half of the curve, which corresponds to

**k**values of positive group velocity. Moreover, we assign each mode with a phase

*αk*. The parameter

_{θ}*α*is proportional to the angular momentum of the beam, just like the

*α*parameter familiar from the corresponding problem in free-space [27

**108**(16), 163901 (2012). [CrossRef] [PubMed]

*α*was also the order of the “half-Bessel” function that represented the analytic solution. Here, as in free-space, the parameter

*α*plays the important role of controlling the radius of the trajectory, which again is approximately equal to

*α/k*. In the lattice, the effective wavelength depends on how much of the wavefunction is in air and how much is in glass, hence we calculate this

_{effective}*k*by averaging the local radius over the whole monochromatic curve. Therefore, the radius of the beam's trajectory scales linearly with

_{effective}*α*, but its dependence on the wavelength is not strictly linear, but rather depends on the band, which indirectly depends on the cell structure and size. Note that in order to make this solution shape-preserving, a proper normalization of the modes is needed. The modes are not square integrable, but they are periodic, hence we can normalize each mode by the square integral over the Brillouin zone. This yields a self-accelerating solution which is shape-preserving to within a good approximation (the variations in the shape are in response to the lattice structure),as shown in Fig. 2(b) and the following Fig. 3. The Poynting vector is breathing as well, while keeping its average front parallel to the circular trajectories of the beam lobes. All the simulations in this work are carried out in the seperable Kronig-Penney model:

*ε(x,z)=½(ε*.

_{Kronig-Penney}(x)+ε_{Kronig-Penney}(z))**k**satisfying the Kronig-Penney relation, which is a transcendent equation that describes the band structure. To find these wave-vectors, we take a grid covering the Brillouin zone, and filter our “illegitimate” Block modes (Bloch modes that do not have a real propagation constant). The grid is chosen such that the desired curve is described by a dense enough set of wave-vectors

**k**. Still, summing over all these Bloch modes with appropriate phases (Eq. (5)) is not good enough, since the density of Bloch modes is changing along the curve. Here comes the normalization stage, where we can either normalize the weights such that the density is azimuthally uniform (corresponding to the “simple” parameterization mentioned above), or normalize the weights according to the local curve length, by calculating the distance between adjacent points in the Brillouin zone (corresponding to the “exact” parameterization mentioned above). All figures in this paper are made using the latter (“simple”) option. Last of all, we remove the non-causal wave-vectors

**k**, by filtering out those with negative

*k*.

_{z}*w*in Eq. (4) (different than the case of Eq. (5)) can intentionally cause the beam to “breath”, creating a periodically shape-preserving dynamics, as is seen in Fig. 2(c). These are the first members of a very wide family of periodically self-accelerating beams, which are superpositions of the basic shape-preserving beams of Eq. (5), in exact analogy to what we know from free-space non-paraxial acceleration [27

**108**(16), 163901 (2012). [CrossRef] [PubMed]

## High-order self-accelerating beams

**108**(16), 163901 (2012). [CrossRef] [PubMed]

**must**consider higher bands. Below, we choose the ninth band, as a typical example of high-order accelerating beams.

*α*. But, the lattice cell is now stretched or squeezed to different scales, shifting the curve up or down on the band structure. Note that this is completely analogous to changing the wavelength while keeping the cell scale constant, since these are the only two length scales in the problem. Figure 3(a) is simulated near the bottom of the first band, Fig. 3(b) is simulated near the bottom of the ninth band, Fig. 3(c) is simulated near the top of the ninth band, and Fig. 3(d) is simulated near the top of the first band (see them marked on Fig. 2(d)). We use these as examples in order to emphasize three important conclusions: The first conclusion is that the overall profile of the accelerating beam in the photonic crystal is only affected by the shape and radius of the curve, and is independent of the band index. It is easy to see an example by comparing Fig. 3(a) and Fig. 3(b), where the overall profiles of the beams are clearly the same, although the fine-features are totally different. The Poynting vectors support this conclusion since from an averaged perspective they are approximately the same. The second conclusion is that the fine features of the beams only depend on the band index, and not on the specific curve. We enlarge several cells to highlight the similarity between the single-cell-modes of Fig. 3(b) and Fig. 3(c), and their clear difference from the single-cell-mode of Fig. 3(a) and Fig. 3(d). Note that the single-cell-mode of the first band has a single peak (Fig. 3(d)) while that of the ninth band has three peaks in each dimensions (total of nine), as is intuitively expected. The number of peaks in each cell is n

^{2}, where n is the index of the band. Interestingly, in all examples the flow of the Poynting vector is slightly drawn from its main circular course, concentrating around intensity peaks. The third conclusion we would like to emphasize is that the width of the lobes of the accelerating beams is inversely proportional to the radius of the monochromatic curve. When the curve is near the bottom of the (ninth) band, as in Fig. 3(b), it has a small radius, therefore having very wide lobes (the main lobe is ~ 25 cells). On the other side, when the curve is near the top of the same (ninth) band, as in Figs. 3(c)-3(d), the radius is as large as it can be, leading to a beam with lobes' thickness of only a single cell. This extreme case is also where the trajectory is the farthest from a pure circular trajectory. The intermediate case is seen in Fig. 2, where the curve is chosen at the middle of the (first) band, leading to lobes with a width of several (~2-3) cells.

*k*, which may be insufficient to describe more complicated curves. When the curve is considerably different from a circle, one needs to assign proper weights around the curve, which is done numerically by calculating the local curve length of each section. Our current method still restricts us to the (relatively simple) bands which have “cone-like” shape. This allows us to solve for self-accelerating beams in all bands in their first appearance, before folding. This is because each band is “cone-like” in its first Brillouin zone. This raises the question what parameterization is needed to find additional self-accelerating beams in the farther Brillouin zones of each mode. Most probably, the exact angle should be recalculated to account for the folding.

_{θ}## Discussion and conclusion

**non-isotropic**material. This model would have “half-Mathieu” self-accelerating beams [32

32. P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. **109**(19), 193901 (2012). [CrossRef] [PubMed]

33. P. Aleahmad, M.-A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. **109**(20), 203902 (2012). [CrossRef] [PubMed]

**108**(16), 163901 (2012). [CrossRef] [PubMed]

*x*and

*z*, giving a stretched band structure, where the almost-circular curves becomes almost-elliptical after the stretch. An appropriate elliptical parameterization (generalizing the simple azimuthal angle parameter) yields self-accelerating beams in such systems. Furthermore, when one axis is extremely stretched (compared to the other axes), we get one dimensional periodic system. Therefore, we get the discrete accelerating beams [18], and all their generalizations [19–21], by extreme-scales of our model. This result is of course expected, since paraxial models must be found as limit cases of the Helmholtz equation, as also happened in the generalization of paraxial accelerating beams [1

1. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. **32**(8), 979–981 (2007). [CrossRef] [PubMed]

2. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. **99**(21), 213901 (2007). [CrossRef] [PubMed]

**108**(16), 163901 (2012). [CrossRef] [PubMed]

34. L. Levi, M. Rechtsman, B. Freedman, T. Schwartz, O. Manela, and M. Segev, “Disorder-enhanced transport in photonic quasicrystals,” Science **332**(6037), 1541–1544 (2011). [CrossRef] [PubMed]

37. 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**(7088), 1166–1169 (2006). [CrossRef] [PubMed]

38. M. Rechtsman, A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, and M. Segev, “Amorphous photonic lattices: band gaps, effective mass, and suppressed transport,” Phys. Rev. Lett. **106**(19), 193904 (2011). [CrossRef] [PubMed]

39. M. Florescu, S. Torquato, and P. J. Steinhardt, “Designer disordered materials with large, complete photonic band gaps,” Proc. Natl. Acad. Sci. U.S.A. **106**(49), 20658–20663 (2009). [CrossRef] [PubMed]

40. T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature **446**(7131), 52–55 (2007). [CrossRef] [PubMed]

41. L. Levi, Y. Krivolapov, S. Fishman, and M. Segev, “Hyper-transport of light and stochastic acceleration by evolving disorder,” Nat. Phys. **8**(12), 912–917 (2012). [CrossRef]

42. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. **47**(3), 264–267 (1979). [CrossRef]

## Acknowledgments

## References and links

1. | G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. |

2. | G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. |

3. | G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. |

4. | M. A. Bandres, “Accelerating parabolic beams,” Opt. Lett. |

5. | J. A. Davis, M. J. Mintry, M. A. Bandres, and D. M. Cottrell, “Observation of accelerating parabolic beams,” Opt. Express |

6. | F. Courvoisier, A. Mathis, L. Froehly, R. Giust, L. Furfaro, P. A. Lacourt, M. Jacquot, and J. M. Dudley, “Sending femtosecond pulses in circles: highly nonparaxial accelerating beams,” Opt. Lett. |

7. | I. Chremmos, Z. Chen, D. N. Christodoulides, and N. K. Efremidis, “Abruptly autofocusing and autodefocusing optical beams with arbitrary caustics,” Phys. Rev. A |

8. | E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. |

9. | P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science |

10. | J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics |

11. | A. Salandrino and D. N. Christodoulides, “Airy plasmon: a nondiffracting surface wave,” Opt. Lett. |

12. | 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. |

13. | I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett. |

14. | A. Lotti, D. Faccio, A. Couairon, D. G. Papazoglou, P. Panagiotopoulos, D. Abdollahpour, and S. Tzortzakis, “Stationary nonlinear Airy beams,” Phys. Rev. A |

15. | R. Bekenstein and M. Segev, “Self-accelerating optical beams in highly nonlocal nonlinear media,” Opt. Express |

16. | I. Dolev, I. Kaminer, A. Shapira, M. Segev, and A. Arie, “Experimental observation of self-accelerating beams in quadratic nonlinear media,” Phys. Rev. Lett. |

17. | Y. Hu, Z. Sun, D. Bongiovanni, D. Song, C. Lou, J. Xu, Z. Chen, and R. Morandotti, “Reshaping the trajectory and spectrum of nonlinear Airy beams,” Opt. Lett. |

18. | R. E-, “Ganainy, K. G. Makris, M. A. Miri, D. N. Christodoulides, and Z. Chen, “Discrete beam acceleration in uniform waveguide arrays,” Phys. Rev. A |

19. | K. G. Makris, R. El-Ganainy, X. Qi, Z. Chen, and D. N. Christodoulides, “Accelerating and diffractionless beams in optical lattices,” CLEO Technical Digest, Paper JTu3K.6 (2012). |

20. | I. D. Chremmos and N. K. Efremidis, “Band-specific phase engineering for curving and focusing light in waveguide arrays,” Phys. Rev. A |

21. | X. Qi, R. El-Ganainy, P. Zang, K. G. Makris, D. N. Christodoulides, and Z. Chen, “Observation of accelerating Wannier-Stark beams in optically induced photonic lattices,” CLEO Technical Digest, Paper QM3E.2 (2012). |

22. | K. G. Makris, D. N. Christodoulides, O. Peleg, M. Segev, and D. Kip, “Optical transitions and Rabi oscillations in waveguide arrays,” Opt. Express |

23. | K. Shandarova, C. E. Rüter, D. Kip, K. G. Makris, D. N. Christodoulides, O. Peleg, and M. Segev, “Experimental observation of Rabi oscillations in photonic lattices,” Phys. Rev. Lett. |

24. | B. Alfassi, O. Peleg, N. Moiseyev, and M. Segev, “Diverging rabi oscillations in subwavelength photonic lattices,” Phys. Rev. Lett. |

25. | R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. |

26. | T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer, and F. Lederer, “Optical Bloch oscillations in temperature tuned waveguide arrays,” Phys. Rev. Lett. |

27. | I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting accelerating wave packets of Maxwell’s equations,” Phys. Rev. Lett. |

28. | S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B |

29. | P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljacić, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, “Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal,” Nat. Mater. |

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

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

32. | P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. |

33. | P. Aleahmad, M.-A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. |

34. | L. Levi, M. Rechtsman, B. Freedman, T. Schwartz, O. Manela, and M. Segev, “Disorder-enhanced transport in photonic quasicrystals,” Science |

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

36. | M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg, and M. C. Netti, “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature |

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

38. | M. Rechtsman, A. Szameit, F. Dreisow, M. Heinrich, R. Keil, S. Nolte, and M. Segev, “Amorphous photonic lattices: band gaps, effective mass, and suppressed transport,” Phys. Rev. Lett. |

39. | M. Florescu, S. Torquato, and P. J. Steinhardt, “Designer disordered materials with large, complete photonic band gaps,” Proc. Natl. Acad. Sci. U.S.A. |

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

41. | L. Levi, Y. Krivolapov, S. Fishman, and M. Segev, “Hyper-transport of light and stochastic acceleration by evolving disorder,” Nat. Phys. |

42. | M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. |

**OCIS Codes**

(260.1960) Physical optics : Diffraction theory

(260.2110) Physical optics : Electromagnetic optics

(260.2065) Physical optics : Effective medium theory

(160.5298) Materials : Photonic crystals

(070.7345) Fourier optics and signal processing : Wave propagation

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: February 19, 2013

Revised Manuscript: March 24, 2013

Manuscript Accepted: March 25, 2013

Published: April 3, 2013

**Citation**

Ido Kaminer, Jonathan Nemirovsky, Konstantinos G. Makris, and Mordechai Segev, "Self-accelerating beams in photonic crystals," Opt. Express **21**, 8886-8896 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-7-8886

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