Abstract

The discovery of a new type of soliton occurring in periodic systems is reported. This type of nonlinear excitation exists at a Dirac point of a photonic band structure, and features an oscillating tail that damps algebraically. Solitons in periodic systems are localized states traditionally supported by photonic bandgaps. Here, it is found that besides photonic bandgaps, a Dirac point in the band structure of triangular optical lattices can also sustain solitons. Apart from their theoretical impact within the soliton theory, they have many potential uses because such solitons are possible in both Kerr material and photorefractive crystals that possess self-focusing and self-defocusing nonlinearities. The findings enrich the soliton family and provide information for studies of nonlinear waves in many branches of physics.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2017 (1)

2016 (1)

2015 (2)

J. Buencuerpo, J. M. Llorens, P. Zilio, W. Raja, J. Cunha, A. Alabastri, R. P. Zaccaria, A. Martí, and T. Versloot, “Light-trapping in photon enhanced thermionic emitters,” Opt. Express 23(19), A1220–A1235 (2015).
[PubMed]

K. Xie, W. Zhang, A. D. Boardman, H. Jiang, Z. Hu, Y. Liu, M. Xie, Q. Mao, L. Hu, Q. Li, T. Yang, F. Wen, and E. Wang, “Fiber guiding at the Dirac frequency beyond photonic bandgaps,” Light Sci. Appl. 4, e304 (2015).

2014 (2)

K. Xie, H. M. Jiang, A. D. Boardman, Y. Liu, Z. H. Wu, M. Xie, P. Jiang, Q. Xu, M. Yu, and L. E. Davis, “Trapped photons at a Dirac point: a new horizon for photonic crystals,” Laser Photonics Rev. 8, 583–589 (2014).

X. Tran, S. Longhi, and F. Biancalana, “Optical analogue of relativistic Dirac solitons in binary waveguide arrays,” Ann. Phys. 340, 179–187 (2014).

2013 (1)

2012 (1)

M. J. Ablowitz and Y. Zhu, “Nonlinear Waves in Shallow Honeycomb Lattices,” SIAM J. Appl. Math. 72, 240–260 (2012).

2010 (2)

S. R. Zandbergen and M. J. A. de Dood, “Experimental observation of strong edge effects on the pseudodiffusive transport of light in photonic graphene,” Phys. Rev. Lett. 104(4), 043903 (2010).
[PubMed]

J. M. Christian, G. S. McDonald, and P. Chamorro-Posada, “Helmholtz algebraic solitons,” J. Phys. A Math. Theor. 43, 085212 (2010).

2009 (3)

O. Peleg, Y. Plotnik, N. Moiseyev, O. Cohen, and M. Segev, “Self-trapped leaky waves and their interactions,” Phys. Rev. A 80, 041801 (2009).

S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photonics Rev. 3, 243–261 (2009).

A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109–162 (2009).

2008 (4)

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78, 033834 (2008).

X. Zhang, “Observing Zitterbewegung for photons near the Dirac point of a two-dimensional photonic crystal,” Phys. Rev. Lett. 100(11), 113903 (2008).
[PubMed]

O. Bahat-Treidel, O. Peleg, and M. Segev, “Symmetry breaking in honeycomb photonic lattices,” Opt. Lett. 33(19), 2251–2253 (2008).
[PubMed]

J. Yang, “Iteration methods for stability spectra of solitary waves,” J. Comput. Phys. 227, 6862–6876 (2008).

2007 (4)

J. Yang and T. I. Lakoba, “Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations,” Stud. Appl. Math. 118, 153–197 (2007).

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

C. Brunhuber, F. G. Mertens, and Y. Gaididei, “Long-range effects on superdiffusive algebraic solitons in anharmonic chains,” Eur. Phys. J. B 57, 57–65 (2007).

2006 (1)

H. Alatas, A. A. Iskandar, M. O. Tjia, and T. P. Valkering, “Rational solitons in deep nonlinear optical Bragg grating,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(6 Pt 2), 066606 (2006).
[PubMed]

2004 (1)

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “Routing of anisotropic spatial solitons and modulational instability in liquid crystals,” Nature 432(7018), 733–737 (2004).
[PubMed]

2003 (3)

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

2002 (1)

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).
[PubMed]

1998 (1)

J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282(5393), 1476–1478 (1998).
[PubMed]

1995 (1)

Y. Nogami, F. M. Toyama, and Z. Zhao, “Nonlinear Dirac soliton in an external field,” J. Phys. Math. Gen. 28, 1413–1424 (1995).

1991 (1)

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B Condens. Matter 44(16), 8565–8571 (1991).
[PubMed]

1973 (1)

N. G. Vakhitov and A. A. Kolokolov, “Stationary solutions of the wave equation in a medium with nonlinearity saturation,” Radiophys. Quantum Electron. 16, 783–789 (1973).

Ablowitz, M. J.

M. J. Ablowitz and Y. Zhu, “Nonlinear Waves in Shallow Honeycomb Lattices,” SIAM J. Appl. Math. 72, 240–260 (2012).

Alabastri, A.

Alatas, H.

H. Alatas, A. A. Iskandar, M. O. Tjia, and T. P. Valkering, “Rational solitons in deep nonlinear optical Bragg grating,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(6 Pt 2), 066606 (2006).
[PubMed]

Alexander, T. J.

Assanto, G.

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “Routing of anisotropic spatial solitons and modulational instability in liquid crystals,” Nature 432(7018), 733–737 (2004).
[PubMed]

Bahat-Treidel, O.

Bartal, G.

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).
[PubMed]

Biancalana, F.

X. Tran, S. Longhi, and F. Biancalana, “Optical analogue of relativistic Dirac solitons in binary waveguide arrays,” Ann. Phys. 340, 179–187 (2014).

Birks, T. A.

J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282(5393), 1476–1478 (1998).
[PubMed]

Boardman, A. D.

K. Xie, W. Zhang, A. D. Boardman, H. Jiang, Z. Hu, Y. Liu, M. Xie, Q. Mao, L. Hu, Q. Li, T. Yang, F. Wen, and E. Wang, “Fiber guiding at the Dirac frequency beyond photonic bandgaps,” Light Sci. Appl. 4, e304 (2015).

K. Xie, H. M. Jiang, A. D. Boardman, Y. Liu, Z. H. Wu, M. Xie, P. Jiang, Q. Xu, M. Yu, and L. E. Davis, “Trapped photons at a Dirac point: a new horizon for photonic crystals,” Laser Photonics Rev. 8, 583–589 (2014).

Broeng, J.

J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282(5393), 1476–1478 (1998).
[PubMed]

Brunhuber, C.

C. Brunhuber, F. G. Mertens, and Y. Gaididei, “Long-range effects on superdiffusive algebraic solitons in anharmonic chains,” Eur. Phys. J. B 57, 57–65 (2007).

Buencuerpo, J.

Chamorro-Posada, P.

J. M. Christian, G. S. McDonald, and P. Chamorro-Posada, “Helmholtz algebraic solitons,” J. Phys. A Math. Theor. 43, 085212 (2010).

Christian, J. M.

J. M. Christian, G. S. McDonald, and P. Chamorro-Posada, “Helmholtz algebraic solitons,” J. Phys. A Math. Theor. 43, 085212 (2010).

Christodoulides, D. N.

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).
[PubMed]

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

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).
[PubMed]

Cohen, O.

M. Kozlov, O. Kfir, and O. Cohen, “Self-trapped leaky waves in lattices: discrete and Bragg soleakons,” Opt. Express 21(17), 19690–19700 (2013).
[PubMed]

O. Peleg, Y. Plotnik, N. Moiseyev, O. Cohen, and M. Segev, “Self-trapped leaky waves and their interactions,” Phys. Rev. A 80, 041801 (2009).

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).
[PubMed]

Conti, C.

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “Routing of anisotropic spatial solitons and modulational instability in liquid crystals,” Nature 432(7018), 733–737 (2004).
[PubMed]

Cunha, J.

Davis, L. E.

K. Xie, H. M. Jiang, A. D. Boardman, Y. Liu, Z. H. Wu, M. Xie, P. Jiang, Q. Xu, M. Yu, and L. E. Davis, “Trapped photons at a Dirac point: a new horizon for photonic crystals,” Laser Photonics Rev. 8, 583–589 (2014).

de Dood, M. J. A.

S. R. Zandbergen and M. J. A. de Dood, “Experimental observation of strong edge effects on the pseudodiffusive transport of light in photonic graphene,” Phys. Rev. Lett. 104(4), 043903 (2010).
[PubMed]

De Luca, A.

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “Routing of anisotropic spatial solitons and modulational instability in liquid crystals,” Nature 432(7018), 733–737 (2004).
[PubMed]

Desyatnikov, A. S.

Efremidis, N. K.

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).
[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).
[PubMed]

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).
[PubMed]

Fleischer, J. W.

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).
[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).
[PubMed]

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).
[PubMed]

Freedman, B.

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).
[PubMed]

Gaididei, Y.

C. Brunhuber, F. G. Mertens, and Y. Gaididei, “Long-range effects on superdiffusive algebraic solitons in anharmonic chains,” Eur. Phys. J. B 57, 57–65 (2007).

Geim, A. K.

A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109–162 (2009).

Guinea, F.

A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109–162 (2009).

Haldane, F. D. M.

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78, 033834 (2008).

Hsieh, C. I.

Hu, L.

K. Xie, W. Zhang, A. D. Boardman, H. Jiang, Z. Hu, Y. Liu, M. Xie, Q. Mao, L. Hu, Q. Li, T. Yang, F. Wen, and E. Wang, “Fiber guiding at the Dirac frequency beyond photonic bandgaps,” Light Sci. Appl. 4, e304 (2015).

Hu, Z.

K. Xie, W. Zhang, A. D. Boardman, H. Jiang, Z. Hu, Y. Liu, M. Xie, Q. Mao, L. Hu, Q. Li, T. Yang, F. Wen, and E. Wang, “Fiber guiding at the Dirac frequency beyond photonic bandgaps,” Light Sci. Appl. 4, e304 (2015).

Hudock, J.

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).
[PubMed]

Iskandar, A. A.

H. Alatas, A. A. Iskandar, M. O. Tjia, and T. P. Valkering, “Rational solitons in deep nonlinear optical Bragg grating,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(6 Pt 2), 066606 (2006).
[PubMed]

Jiang, H.

K. Xie, W. Zhang, A. D. Boardman, H. Jiang, Z. Hu, Y. Liu, M. Xie, Q. Mao, L. Hu, Q. Li, T. Yang, F. Wen, and E. Wang, “Fiber guiding at the Dirac frequency beyond photonic bandgaps,” Light Sci. Appl. 4, e304 (2015).

Jiang, H. M.

K. Xie, H. M. Jiang, A. D. Boardman, Y. Liu, Z. H. Wu, M. Xie, P. Jiang, Q. Xu, M. Yu, and L. E. Davis, “Trapped photons at a Dirac point: a new horizon for photonic crystals,” Laser Photonics Rev. 8, 583–589 (2014).

Jiang, P.

K. Xie, H. M. Jiang, A. D. Boardman, Y. Liu, Z. H. Wu, M. Xie, P. Jiang, Q. Xu, M. Yu, and L. E. Davis, “Trapped photons at a Dirac point: a new horizon for photonic crystals,” Laser Photonics Rev. 8, 583–589 (2014).

Kfir, O.

Kivshar, Y. S.

Knight, J. C.

J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282(5393), 1476–1478 (1998).
[PubMed]

Kolokolov, A. A.

N. G. Vakhitov and A. A. Kolokolov, “Stationary solutions of the wave equation in a medium with nonlinearity saturation,” Radiophys. Quantum Electron. 16, 783–789 (1973).

Kozlov, M.

Lai, C. H.

Lakoba, T. I.

J. Yang and T. I. Lakoba, “Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations,” Stud. Appl. Math. 118, 153–197 (2007).

Lederer, F.

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

Li, G.

Li, Q.

K. Xie, W. Zhang, A. D. Boardman, H. Jiang, Z. Hu, Y. Liu, M. Xie, Q. Mao, L. Hu, Q. Li, T. Yang, F. Wen, and E. Wang, “Fiber guiding at the Dirac frequency beyond photonic bandgaps,” Light Sci. Appl. 4, e304 (2015).

Liu, Y.

K. Xie, W. Zhang, A. D. Boardman, H. Jiang, Z. Hu, Y. Liu, M. Xie, Q. Mao, L. Hu, Q. Li, T. Yang, F. Wen, and E. Wang, “Fiber guiding at the Dirac frequency beyond photonic bandgaps,” Light Sci. Appl. 4, e304 (2015).

K. Xie, H. M. Jiang, A. D. Boardman, Y. Liu, Z. H. Wu, M. Xie, P. Jiang, Q. Xu, M. Yu, and L. E. Davis, “Trapped photons at a Dirac point: a new horizon for photonic crystals,” Laser Photonics Rev. 8, 583–589 (2014).

Llorens, J. M.

Longhi, S.

X. Tran, S. Longhi, and F. Biancalana, “Optical analogue of relativistic Dirac solitons in binary waveguide arrays,” Ann. Phys. 340, 179–187 (2014).

S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photonics Rev. 3, 243–261 (2009).

Manela, O.

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).
[PubMed]

Mao, Q.

K. Xie, W. Zhang, A. D. Boardman, H. Jiang, Z. Hu, Y. Liu, M. Xie, Q. Mao, L. Hu, Q. Li, T. Yang, F. Wen, and E. Wang, “Fiber guiding at the Dirac frequency beyond photonic bandgaps,” Light Sci. Appl. 4, e304 (2015).

Maradudin, A. A.

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B Condens. Matter 44(16), 8565–8571 (1991).
[PubMed]

Martí, A.

McDonald, G. S.

J. M. Christian, G. S. McDonald, and P. Chamorro-Posada, “Helmholtz algebraic solitons,” J. Phys. A Math. Theor. 43, 085212 (2010).

Mertens, F. G.

C. Brunhuber, F. G. Mertens, and Y. Gaididei, “Long-range effects on superdiffusive algebraic solitons in anharmonic chains,” Eur. Phys. J. B 57, 57–65 (2007).

Moiseyev, N.

O. Peleg, Y. Plotnik, N. Moiseyev, O. Cohen, and M. Segev, “Self-trapped leaky waves and their interactions,” Phys. Rev. A 80, 041801 (2009).

Neto, A. H. C.

A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109–162 (2009).

Nogami, Y.

Y. Nogami, F. M. Toyama, and Z. Zhao, “Nonlinear Dirac soliton in an external field,” J. Phys. Math. Gen. 28, 1413–1424 (1995).

Novoselov, K. S.

A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109–162 (2009).

Peccianti, M.

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “Routing of anisotropic spatial solitons and modulational instability in liquid crystals,” Nature 432(7018), 733–737 (2004).
[PubMed]

Peleg, O.

O. Peleg, Y. Plotnik, N. Moiseyev, O. Cohen, and M. Segev, “Self-trapped leaky waves and their interactions,” Phys. Rev. A 80, 041801 (2009).

O. Bahat-Treidel, O. Peleg, and M. Segev, “Symmetry breaking in honeycomb photonic lattices,” Opt. Lett. 33(19), 2251–2253 (2008).
[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).
[PubMed]

Peres, N. M. R.

A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109–162 (2009).

Plihal, M.

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B Condens. Matter 44(16), 8565–8571 (1991).
[PubMed]

Plotnik, Y.

O. Peleg, Y. Plotnik, N. Moiseyev, O. Cohen, and M. Segev, “Self-trapped leaky waves and their interactions,” Phys. Rev. A 80, 041801 (2009).

Raghu, S.

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78, 033834 (2008).

Raja, W.

Russell, P. S. J.

J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282(5393), 1476–1478 (1998).
[PubMed]

Schmidt, M. A.

Sears, S.

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).
[PubMed]

Segev, M.

O. Peleg, Y. Plotnik, N. Moiseyev, O. Cohen, and M. Segev, “Self-trapped leaky waves and their interactions,” Phys. Rev. A 80, 041801 (2009).

O. Bahat-Treidel, O. Peleg, and M. Segev, “Symmetry breaking in honeycomb photonic lattices,” Opt. Lett. 33(19), 2251–2253 (2008).
[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).
[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).
[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).
[PubMed]

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).
[PubMed]

Silberberg, Y.

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

Tjia, M. O.

H. Alatas, A. A. Iskandar, M. O. Tjia, and T. P. Valkering, “Rational solitons in deep nonlinear optical Bragg grating,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(6 Pt 2), 066606 (2006).
[PubMed]

Toyama, F. M.

Y. Nogami, F. M. Toyama, and Z. Zhao, “Nonlinear Dirac soliton in an external field,” J. Phys. Math. Gen. 28, 1413–1424 (1995).

Tran, X.

X. Tran, S. Longhi, and F. Biancalana, “Optical analogue of relativistic Dirac solitons in binary waveguide arrays,” Ann. Phys. 340, 179–187 (2014).

Umeton, C.

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “Routing of anisotropic spatial solitons and modulational instability in liquid crystals,” Nature 432(7018), 733–737 (2004).
[PubMed]

Vakhitov, N. G.

N. G. Vakhitov and A. A. Kolokolov, “Stationary solutions of the wave equation in a medium with nonlinearity saturation,” Radiophys. Quantum Electron. 16, 783–789 (1973).

Valkering, T. P.

H. Alatas, A. A. Iskandar, M. O. Tjia, and T. P. Valkering, “Rational solitons in deep nonlinear optical Bragg grating,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(6 Pt 2), 066606 (2006).
[PubMed]

Versloot, T.

Wang, E.

K. Xie, W. Zhang, A. D. Boardman, H. Jiang, Z. Hu, Y. Liu, M. Xie, Q. Mao, L. Hu, Q. Li, T. Yang, F. Wen, and E. Wang, “Fiber guiding at the Dirac frequency beyond photonic bandgaps,” Light Sci. Appl. 4, e304 (2015).

Wen, F.

K. Xie, W. Zhang, A. D. Boardman, H. Jiang, Z. Hu, Y. Liu, M. Xie, Q. Mao, L. Hu, Q. Li, T. Yang, F. Wen, and E. Wang, “Fiber guiding at the Dirac frequency beyond photonic bandgaps,” Light Sci. Appl. 4, e304 (2015).

Wu, Z. H.

K. Xie, H. M. Jiang, A. D. Boardman, Y. Liu, Z. H. Wu, M. Xie, P. Jiang, Q. Xu, M. Yu, and L. E. Davis, “Trapped photons at a Dirac point: a new horizon for photonic crystals,” Laser Photonics Rev. 8, 583–589 (2014).

Xie, K.

K. Xie, W. Zhang, A. D. Boardman, H. Jiang, Z. Hu, Y. Liu, M. Xie, Q. Mao, L. Hu, Q. Li, T. Yang, F. Wen, and E. Wang, “Fiber guiding at the Dirac frequency beyond photonic bandgaps,” Light Sci. Appl. 4, e304 (2015).

K. Xie, H. M. Jiang, A. D. Boardman, Y. Liu, Z. H. Wu, M. Xie, P. Jiang, Q. Xu, M. Yu, and L. E. Davis, “Trapped photons at a Dirac point: a new horizon for photonic crystals,” Laser Photonics Rev. 8, 583–589 (2014).

Xie, M.

K. Xie, W. Zhang, A. D. Boardman, H. Jiang, Z. Hu, Y. Liu, M. Xie, Q. Mao, L. Hu, Q. Li, T. Yang, F. Wen, and E. Wang, “Fiber guiding at the Dirac frequency beyond photonic bandgaps,” Light Sci. Appl. 4, e304 (2015).

K. Xie, H. M. Jiang, A. D. Boardman, Y. Liu, Z. H. Wu, M. Xie, P. Jiang, Q. Xu, M. Yu, and L. E. Davis, “Trapped photons at a Dirac point: a new horizon for photonic crystals,” Laser Photonics Rev. 8, 583–589 (2014).

Xu, Q.

K. Xie, H. M. Jiang, A. D. Boardman, Y. Liu, Z. H. Wu, M. Xie, P. Jiang, Q. Xu, M. Yu, and L. E. Davis, “Trapped photons at a Dirac point: a new horizon for photonic crystals,” Laser Photonics Rev. 8, 583–589 (2014).

Yang, J.

J. Yang, “Iteration methods for stability spectra of solitary waves,” J. Comput. Phys. 227, 6862–6876 (2008).

J. Yang and T. I. Lakoba, “Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations,” Stud. Appl. Math. 118, 153–197 (2007).

Yang, T.

K. Xie, W. Zhang, A. D. Boardman, H. Jiang, Z. Hu, Y. Liu, M. Xie, Q. Mao, L. Hu, Q. Li, T. Yang, F. Wen, and E. Wang, “Fiber guiding at the Dirac frequency beyond photonic bandgaps,” Light Sci. Appl. 4, e304 (2015).

Yu, M.

K. Xie, H. M. Jiang, A. D. Boardman, Y. Liu, Z. H. Wu, M. Xie, P. Jiang, Q. Xu, M. Yu, and L. E. Davis, “Trapped photons at a Dirac point: a new horizon for photonic crystals,” Laser Photonics Rev. 8, 583–589 (2014).

Zaccaria, R. P.

Zandbergen, S. R.

S. R. Zandbergen and M. J. A. de Dood, “Experimental observation of strong edge effects on the pseudodiffusive transport of light in photonic graphene,” Phys. Rev. Lett. 104(4), 043903 (2010).
[PubMed]

Zeisberger, M.

Zhang, W.

K. Xie, W. Zhang, A. D. Boardman, H. Jiang, Z. Hu, Y. Liu, M. Xie, Q. Mao, L. Hu, Q. Li, T. Yang, F. Wen, and E. Wang, “Fiber guiding at the Dirac frequency beyond photonic bandgaps,” Light Sci. Appl. 4, e304 (2015).

Zhang, X.

X. Zhang, “Observing Zitterbewegung for photons near the Dirac point of a two-dimensional photonic crystal,” Phys. Rev. Lett. 100(11), 113903 (2008).
[PubMed]

Zhao, Z.

Y. Nogami, F. M. Toyama, and Z. Zhao, “Nonlinear Dirac soliton in an external field,” J. Phys. Math. Gen. 28, 1413–1424 (1995).

Zhu, Y.

M. J. Ablowitz and Y. Zhu, “Nonlinear Waves in Shallow Honeycomb Lattices,” SIAM J. Appl. Math. 72, 240–260 (2012).

Zilio, P.

Ann. Phys. (1)

X. Tran, S. Longhi, and F. Biancalana, “Optical analogue of relativistic Dirac solitons in binary waveguide arrays,” Ann. Phys. 340, 179–187 (2014).

Eur. Phys. J. B (1)

C. Brunhuber, F. G. Mertens, and Y. Gaididei, “Long-range effects on superdiffusive algebraic solitons in anharmonic chains,” Eur. Phys. J. B 57, 57–65 (2007).

J. Comput. Phys. (1)

J. Yang, “Iteration methods for stability spectra of solitary waves,” J. Comput. Phys. 227, 6862–6876 (2008).

J. Phys. A Math. Theor. (1)

J. M. Christian, G. S. McDonald, and P. Chamorro-Posada, “Helmholtz algebraic solitons,” J. Phys. A Math. Theor. 43, 085212 (2010).

J. Phys. Math. Gen. (1)

Y. Nogami, F. M. Toyama, and Z. Zhao, “Nonlinear Dirac soliton in an external field,” J. Phys. Math. Gen. 28, 1413–1424 (1995).

Laser Photonics Rev. (2)

S. Longhi, “Quantum-optical analogies using photonic structures,” Laser Photonics Rev. 3, 243–261 (2009).

K. Xie, H. M. Jiang, A. D. Boardman, Y. Liu, Z. H. Wu, M. Xie, P. Jiang, Q. Xu, M. Yu, and L. E. Davis, “Trapped photons at a Dirac point: a new horizon for photonic crystals,” Laser Photonics Rev. 8, 583–589 (2014).

Light Sci. Appl. (1)

K. Xie, W. Zhang, A. D. Boardman, H. Jiang, Z. Hu, Y. Liu, M. Xie, Q. Mao, L. Hu, Q. Li, T. Yang, F. Wen, and E. Wang, “Fiber guiding at the Dirac frequency beyond photonic bandgaps,” Light Sci. Appl. 4, e304 (2015).

Nature (3)

M. Peccianti, C. Conti, G. Assanto, A. De Luca, and C. Umeton, “Routing of anisotropic spatial solitons and modulational instability in liquid crystals,” Nature 432(7018), 733–737 (2004).
[PubMed]

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

Opt. Express (4)

Opt. Lett. (2)

Phys. Rev. A (2)

O. Peleg, Y. Plotnik, N. Moiseyev, O. Cohen, and M. Segev, “Self-trapped leaky waves and their interactions,” Phys. Rev. A 80, 041801 (2009).

S. Raghu and F. D. M. Haldane, “Analogs of quantum-Hall-effect edge states in photonic crystals,” Phys. Rev. A 78, 033834 (2008).

Phys. Rev. B Condens. Matter (1)

M. Plihal and A. A. Maradudin, “Photonic band structure of two-dimensional systems: The triangular lattice,” Phys. Rev. B Condens. Matter 44(16), 8565–8571 (1991).
[PubMed]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (2)

H. Alatas, A. A. Iskandar, M. O. Tjia, and T. P. Valkering, “Rational solitons in deep nonlinear optical Bragg grating,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(6 Pt 2), 066606 (2006).
[PubMed]

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).
[PubMed]

Phys. Rev. Lett. (4)

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).
[PubMed]

S. R. Zandbergen and M. J. A. de Dood, “Experimental observation of strong edge effects on the pseudodiffusive transport of light in photonic graphene,” Phys. Rev. Lett. 104(4), 043903 (2010).
[PubMed]

X. Zhang, “Observing Zitterbewegung for photons near the Dirac point of a two-dimensional photonic crystal,” Phys. Rev. Lett. 100(11), 113903 (2008).
[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).
[PubMed]

Radiophys. Quantum Electron. (1)

N. G. Vakhitov and A. A. Kolokolov, “Stationary solutions of the wave equation in a medium with nonlinearity saturation,” Radiophys. Quantum Electron. 16, 783–789 (1973).

Rev. Mod. Phys. (1)

A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Rev. Mod. Phys. 81, 109–162 (2009).

Science (1)

J. C. Knight, J. Broeng, T. A. Birks, and P. S. J. Russell, “Photonic band gap guidance in optical fibers,” Science 282(5393), 1476–1478 (1998).
[PubMed]

SIAM J. Appl. Math. (1)

M. J. Ablowitz and Y. Zhu, “Nonlinear Waves in Shallow Honeycomb Lattices,” SIAM J. Appl. Math. 72, 240–260 (2012).

Stud. Appl. Math. (1)

J. Yang and T. I. Lakoba, “Universally-convergent squared-operator iteration methods for solitary waves in general nonlinear wave equations,” Stud. Appl. Math. 118, 153–197 (2007).

Other (3)

M. Skorobogatiy and J. Yang, Fundamentals of Photonic Crystal Guiding (Cambridge University, 2009).

K. C. Kao and G. A. Hockham, “Dielectric–fibre surface waveguides for optical frequencies,” IEE Proceedings 113, 1151–1158 (1966).

Y. S. Kivshar and G. P. Agrawal, Optical solitons: from fibers to photonic crystals, (Academic, 2003).

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Figures (13)

Fig. 1
Fig. 1 Four different lattice potentials. (a, b) Index potential of the photorefractive material for I0 = 2, χ = 0, V0 = 250 (a, self-focusing) and I0 = 1, χ = 3, V0 = −150 (b, self-defocusing). (c, d) Index potential of the Kerr nonlinear medium for V0 = 10, χ = 3/2 (c) and V0 = −35, χ = −1/3 (d). In the cases of (b) and (c) the potentials exhibit absolute index maxima on lattice sites, while in the cases of (a) and (d) the potentials exhibit absolute index minima on lattice sites.
Fig. 2
Fig. 2 Band structures (up) and 3D views of Dirac cones around the six Brillouin zone corners (down) of four different optical lattices. (a-d) correspond to the lattices shown in Fig. 1(a-d).
Fig. 3
Fig. 3 Mode formed at a defect in the potential of Fig. 1(d). Parameters of the defect are R = 2 and Vd = −43.3. Waveguide runs from Z = 0 to 75. Between Z = 0 and 20 a source with a propagation constant qD = −41.81 is imposed. In the second stage (Z≥20) the source beam is turned off and the eigenmode evolves freely on its own. (a) Amplitude evolution of the excited eigenmode. (b) The excited field pattern |U| of the eigenmode at Z = 75. (c) The product r3/2|U| on the X-axis at Z = 75. (d) The instantaneous decay rate α of the excited beam as it propagates down the waveguide. (e) The propagation constant spectrum of the excited mode. The green vertical line indicates the position of the Dirac point.
Fig. 4
Fig. 4 (a-d) Defect parameter Vd (up) and the corresponding power loss rate α (down) versus eigen propagation constant q of the localized mode for respectively the lattices shown in Fig. 1(a-d). The green vertical lines indicate the positions of the Dirac points. The red horizontal lines at α = 0.021 mark level of the critical decay rate αc, below which waveguiding is possible.
Fig. 5
Fig. 5 Dirac-point solitons in a saturable self-focusing lattice. The lattice potential is shown in Fig. 1(a). (a, b) The field profiles of the fundamental soliton (a) and the first vortex soliton (m = 1) (b) at the Dirac point qD = 115.644. (c, d) Product r3/2Φ of the fundamental soliton (c) and r3/2|Φ| of the first vortex soliton (d) on the X axis. (e, f) Power P (e) and the real part of the perturbation growth rate Re(λ) (f) versus the propagation constant q of the soliton, where “o” represents the fundamental soliton and “+” represents the vortex soliton. The grey scale in the background indicates level of linear losses of the wave in the lattice potential. The dotted vertical line indicates the position of the Dirac point.
Fig. 6
Fig. 6 Dirac-point solitons in a saturable self-defocusing lattice. The lattice potential is shown in Fig. 1(b). (a, b) The field profiles of the fundamental soliton (a) and the first vortex soliton (m = 1) (b) at the Dirac point qD = −8.67. (c) Product r3/2Φ of the fundamental soliton on the Y axis. (d) Product r3/2|Φ| of the first vortex soliton on the X axis. (e, f) Power P (e) and the real part of the perturbation growth rate Re(λ) (f) versus the propagation constant q of the soliton, where “o” represents the fundamental soliton and “+” represents the vortex soliton. The grey scale in the background indicates level of linear losses of the wave in the lattice potential. The dotted vertical line indicates the position of the Dirac point.
Fig. 7
Fig. 7 Dirac-point solitons in Kerr nonlinear media. The lattice potential is shown in Fig. 1(c). (a, b) The field profiles of the fundamental solitons in a Kerr self-focusing (σ = 1) lattice (a) and a Kerr self-defocusing (σ = −1) lattice (b) at the Dirac point qD = 29.445. (c) Product r3/2Φ of the fundamental soliton of a Kerr self-focusing lattice on the X axis. (d) Product r3/2Φ of the fundamental soliton of a Kerr self-defocusing lattice on the Y axis. (e, f) Power P (e) and real part of the perturbation growth rate Re(λ) (f) versus the propagation constant q of the soliton, where “o” corresponds to the self-focusing case and “+” corresponds to the self-defocusing case. The grey scale in the background indicates level of linear losses of the wave in the lattice potential. The dotted vertical line indicates the position of the Dirac point.
Fig. 8
Fig. 8 Dirac-point solitons in Kerr nonlinear media. The lattice potential is shown in Fig. 1(d). (a, b) The field profiles of the fundamental solitons in a Kerr self-focusing (σ = 1) lattice (a) and a Kerr self-defocusing (σ = −1) lattice (b) at the Dirac point qD = −41.81. (c, d) Product r3/2Φ of the fundamental soliton of a Kerr self-focusing lattice (c) and of a Kerr self-defocusing lattice (d) on the X axis. (e, f) Power P (e) and real part of the perturbation growth rate Re(λ) (f) versus the propagation constant q of the soliton, where “o” corresponds to the self-focusing case and “+” corresponds to the self-defocusing case. The grey scale in the background indicates level of linear losses of the wave in the lattice potential. The dotted vertical line indicates the position of the Dirac point.
Fig. 9
Fig. 9 Evolution of the first vortex Dirac-point soliton in a saturable self-focusing lattice. The initial soliton is the one shown in Fig. 5(b), with 2% of relative random noise superposed to the profile. The lattice potential is shown in Fig. 1(a). The Dirac-point soliton is shown to degrade into a gap soliton in propagation. (a) Evolution of amplitude |Φ | on the ring of the soliton in propagation. (b) Propagation constant spectrum of the soliton. The green vertical line indicates the position of the Dirac point. The second peak centered at q = 128.96 (within a bandgap) corresponds to a gap soliton. (c-e) The |Φ | field of the soliton at respectively Z = 0, 5, 10.
Fig. 10
Fig. 10 Evolution of the first vortex Dirac-point soliton in a saturable self-defocusing lattice. The initial soliton is the one shown in Fig. 6(b), with 2% of relative random noise superposed to the profile. The lattice potential is shown in Fig. 1(b). The Dirac-point soliton is shown to break up into radiation waves in propagation. (a) Evolution of amplitude |Φ| on the ring of the soliton in propagation. (b) Propagation constant spectrum of the soliton. The green vertical line indicates the position of the Dirac point. (c-e) The |Φ | field of the soliton at respectively Z = 0, 15, 30.
Fig. 11
Fig. 11 Evolution of the fundamental Dirac-point soliton in a Kerr self-focusing lattice. The initial soliton is the one shown in Fig. 7(a), with 2% of relative random noise superposed to the profile. The lattice potential is shown in Fig. 1(c). The Dirac-point soliton is shown to break up into radiation waves in propagation. (a) Evolution of amplitude |Φ | of the soliton in propagation. (b) Propagation constant spectrum of the soliton. The green vertical line indicates the position of the Dirac point. (c-e) The |Φ | field of the soliton at respectively Z = 0.0, 0.03, 0.06.
Fig. 12
Fig. 12 Evolution of the fundamental Dirac-point soliton in a Kerr self-defocusing lattice. The initial soliton is the one shown in Fig. 8(b), with 2% of relative random noise superposed to the profile. The lattice potential is shown in Fig. 1(d). The Dirac-point soliton is shown to degrade into a gap soliton in propagation. (a) Evolution of amplitude |Φ | of the soliton in propagation. (b) Propagation constant spectrum of the soliton. The green vertical line indicates the position of the Dirac point. The second peak centered at q = −47.35 (within a bandgap) corresponds to a gap soliton. (c-e) The |Φ | field of the soliton at respectively Z = 0, 32.5, 65.
Fig. 13
Fig. 13 The linear and nonlinear propagation lengths versus the initial eigenvalue q in a Kerr self-defocusing lattice potential as that shown in Fig. 1(d). Red + : Propagation distance Lsol of the fundamental Dirac-point soliton. Blue x: Damping length Llin of the corresponding linear eigenmode. The green vertical line indicates the position of the Dirac point.

Equations (7)

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( 2 x 2 + 2 y 2 + 2 z 2 )E+ k 0 2 [ n e 2 Δn ]E=0
i U Z +( 2 X 2 + 2 Y 2 )U V NL U=0
( 2 X 2 + 2 Y 2 )ΦVΦ=qΦ
G' [ δ(GG')|k+G | 2 +f(GG') ]h(G') =qh(G)
( 2 X 2 + 2 Y 2 )Φ V NL Φ=qΦ
L K =i( 1 2 σ( Φ *2 Φ 2 ) L 0 +σ|Φ | 2 1 2 σ( Φ 2 + Φ *2 ) L 0 +σ|Φ | 2 + 1 2 σ( Φ 2 + Φ *2 ) 1 2 σ( Φ 2 Φ *2 ) )
L s =i( 1 2 V 0 ( V 0 /V +|Φ | 2 ) 2 ( Φ *2 Φ 2 ) L 0 + 1 2 V 0 ( V 0 /V +|Φ | 2 ) 2 ( 2|Φ | 2 Φ 2 Φ *2 ) L 0 + 1 2 V 0 ( V 0 /V +|Φ | 2 ) 2 ( 2|Φ | 2 + Φ 2 + Φ *2 ) 1 2 V 0 ( V 0 /V +|Φ | 2 ) 2 ( Φ 2 Φ *2 ) )

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