Abstract

We introduce a new class of paraxial optical beams exhibiting discrete-like diffraction patterns reminiscent to those observed in periodic evanescently coupled waveguide lattices. It is demonstrated that such paraxial beams are analytically described in terms of generalized Bessel functions. Such effects are elucidated via pertinent examples.

©2013 Optical Society of America

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  1. D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424(6950), 817–823 (2003).
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  2. H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Bräuer, and U. Peschel, “Visual Observation of Zener Tunneling,” Phys. Rev. Lett. 96(2), 023901 (2006).
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  3. S. Longhi, “Jaynes-Cummings photonic superlattices,” Opt. Lett. 36(17), 3407–3409 (2011).
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  4. R. Keil, A. Perez-Leija, P. Aleahmad, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Observation of Bloch-like revivals in semi-infinite Glauber-Fock photonic lattices,” Opt. Lett. 37(18), 3801–3803 (2012).
    [Crossref] [PubMed]
  5. R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical Analogue of Displaced Fock States and Quantum Correlations in Glauber-Fock Photonic Lattices,” Phys. Rev. Lett. 107(10), 103601 (2011).
    [Crossref] [PubMed]
  6. A. Perez-Leija, R. Keil, A. Szameit, A. F. Abouraddy, H. Moya-Cessa, and D. N. Christodoulides, “Tailoring the Correlation and Anticorrelation behavior of Path-entangled Photons in Glauber-Fock Oscillator Lattices,” Phys. Rev. A 85(1), 013848 (2012).
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  9. Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson Localization and Nonlinearity in One-Dimensional Disordered Photonic Lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
    [Crossref] [PubMed]
  10. A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science 329(5998), 1500–1503 (2010).
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  11. Y. Bromberg, Y. Lahini, and Y. Silberberg, “Bloch Oscillations of Path-Entangled Photons,” Phys. Rev. Lett. 105(26), 263604 (2010).
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    [Crossref] [PubMed]
  16. A. Szameit, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, “Long-range interaction in waveguide lattices,” Phys. Rev. A 77(4), 043804 (2008).
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    [Crossref]
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    [Crossref] [PubMed]
  22. N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. 35(23), 4045–4047 (2010).
    [Crossref] [PubMed]
  23. N. K. Efremdis, P. Zhang, Z. Chen, D. N. Christodoulides, C. E. Ruter, and D. Kip, “Wave propagation in waveguide arrays with alternating positive and negative couplings,” Phys. Rev. A 81(5), 053817 (2010).
    [Crossref]
  24. N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. 35(23), 4045–4047 (2010).
    [Crossref] [PubMed]
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2013 (2)

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L. C. Kwek, B. M. Rodríguez-Lara, A. Szameit, and D. N. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87(1), 012309 (2013).
[Crossref]

A. Perez-Leija, R. Keil, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Perfect transfer of path-entangled photons in Jx photonic lattices,” Phys. Rev. A 87(2), 022303 (2013).
[Crossref]

2012 (4)

R. Keil, A. Perez-Leija, P. Aleahmad, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Observation of Bloch-like revivals in semi-infinite Glauber-Fock photonic lattices,” Opt. Lett. 37(18), 3801–3803 (2012).
[Crossref] [PubMed]

A. Perez-Leija, R. Keil, A. Szameit, A. F. Abouraddy, H. Moya-Cessa, and D. N. Christodoulides, “Tailoring the Correlation and Anticorrelation behavior of Path-entangled Photons in Glauber-Fock Oscillator Lattices,” Phys. Rev. A 85(1), 013848 (2012).
[Crossref]

M. Bellec, G. M. Nikolopoulos, and S. Tzortzakis, “Faithful communication Hamiltonian in photonic lattices,” Opt. Lett. 37(21), 4504–4506 (2012).
[Crossref] [PubMed]

J. M. Zeuner, N. K. Efremidis, R. Keil, F. Dreisow, D. N. Christodoulides, A. Tünnermann, S. Nolte, and A. Szameit, “Optical Analogues for massless Dirac particles and conical diffraction in one dimension,” Phys. Rev. Lett. 109(2), 023602 (2012).
[Crossref] [PubMed]

2011 (2)

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical Analogue of Displaced Fock States and Quantum Correlations in Glauber-Fock Photonic Lattices,” Phys. Rev. Lett. 107(10), 103601 (2011).
[Crossref] [PubMed]

S. Longhi, “Jaynes-Cummings photonic superlattices,” Opt. Lett. 36(17), 3407–3409 (2011).
[Crossref] [PubMed]

2010 (5)

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science 329(5998), 1500–1503 (2010).
[Crossref] [PubMed]

Y. Bromberg, Y. Lahini, and Y. Silberberg, “Bloch Oscillations of Path-Entangled Photons,” Phys. Rev. Lett. 105(26), 263604 (2010).
[Crossref] [PubMed]

N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. 35(23), 4045–4047 (2010).
[Crossref] [PubMed]

N. K. Efremdis, P. Zhang, Z. Chen, D. N. Christodoulides, C. E. Ruter, and D. Kip, “Wave propagation in waveguide arrays with alternating positive and negative couplings,” Phys. Rev. A 81(5), 053817 (2010).
[Crossref]

N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. 35(23), 4045–4047 (2010).
[Crossref] [PubMed]

2009 (1)

A. Rai and G. S. Agarwal, “Possibility of coherent phenomena such as Bloch oscillations with single photons via W states,” Phys. Rev. A 79(5), 053849 (2009).
[Crossref]

2008 (3)

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson Localization and Nonlinearity in One-Dimensional Disordered Photonic Lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[Crossref] [PubMed]

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, and A. Tünnermann, “Second-order coupling in femtosecond-laser-written waveguide arrays,” Opt. Lett. 33(22), 2689–2691 (2008).
[Crossref] [PubMed]

A. Szameit, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, “Long-range interaction in waveguide lattices,” Phys. Rev. A 77(4), 043804 (2008).
[Crossref]

2006 (1)

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Bräuer, and U. Peschel, “Visual Observation of Zener Tunneling,” Phys. Rev. Lett. 96(2), 023901 (2006).
[Crossref] [PubMed]

2005 (1)

2003 (1)

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

1991 (1)

G. Dattoli, A. Torre, S. Lorenzzutta, and G. Maino, “Theory of generalized Bessel functions II,” Nuovo Cim. 106(1), 21–51 (1991).
[Crossref]

1990 (1)

G. Dattoli, L. Giannessi, L. Mezi, and A. Torre, “Theory of generalized Bessel functions,” Nuovo Cim. 105(3), 327–348 (1990).
[Crossref]

1965 (1)

Abouraddy, A. F.

A. Perez-Leija, R. Keil, A. Szameit, A. F. Abouraddy, H. Moya-Cessa, and D. N. Christodoulides, “Tailoring the Correlation and Anticorrelation behavior of Path-entangled Photons in Glauber-Fock Oscillator Lattices,” Phys. Rev. A 85(1), 013848 (2012).
[Crossref]

Agarwal, G. S.

A. Rai and G. S. Agarwal, “Possibility of coherent phenomena such as Bloch oscillations with single photons via W states,” Phys. Rev. A 79(5), 053849 (2009).
[Crossref]

Aleahmad, P.

Avidan, A.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson Localization and Nonlinearity in One-Dimensional Disordered Photonic Lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[Crossref] [PubMed]

Bandres, M. A.

Bellec, M.

Bräuer, A.

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Bräuer, and U. Peschel, “Visual Observation of Zener Tunneling,” Phys. Rev. Lett. 96(2), 023901 (2006).
[Crossref] [PubMed]

Bromberg, Y.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science 329(5998), 1500–1503 (2010).
[Crossref] [PubMed]

Y. Bromberg, Y. Lahini, and Y. Silberberg, “Bloch Oscillations of Path-Entangled Photons,” Phys. Rev. Lett. 105(26), 263604 (2010).
[Crossref] [PubMed]

Chen, Z.

N. K. Efremdis, P. Zhang, Z. Chen, D. N. Christodoulides, C. E. Ruter, and D. Kip, “Wave propagation in waveguide arrays with alternating positive and negative couplings,” Phys. Rev. A 81(5), 053817 (2010).
[Crossref]

Christodoulides, D. N.

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L. C. Kwek, B. M. Rodríguez-Lara, A. Szameit, and D. N. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87(1), 012309 (2013).
[Crossref]

A. Perez-Leija, R. Keil, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Perfect transfer of path-entangled photons in Jx photonic lattices,” Phys. Rev. A 87(2), 022303 (2013).
[Crossref]

A. Perez-Leija, R. Keil, A. Szameit, A. F. Abouraddy, H. Moya-Cessa, and D. N. Christodoulides, “Tailoring the Correlation and Anticorrelation behavior of Path-entangled Photons in Glauber-Fock Oscillator Lattices,” Phys. Rev. A 85(1), 013848 (2012).
[Crossref]

J. M. Zeuner, N. K. Efremidis, R. Keil, F. Dreisow, D. N. Christodoulides, A. Tünnermann, S. Nolte, and A. Szameit, “Optical Analogues for massless Dirac particles and conical diffraction in one dimension,” Phys. Rev. Lett. 109(2), 023602 (2012).
[Crossref] [PubMed]

R. Keil, A. Perez-Leija, P. Aleahmad, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Observation of Bloch-like revivals in semi-infinite Glauber-Fock photonic lattices,” Opt. Lett. 37(18), 3801–3803 (2012).
[Crossref] [PubMed]

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical Analogue of Displaced Fock States and Quantum Correlations in Glauber-Fock Photonic Lattices,” Phys. Rev. Lett. 107(10), 103601 (2011).
[Crossref] [PubMed]

N. K. Efremdis, P. Zhang, Z. Chen, D. N. Christodoulides, C. E. Ruter, and D. Kip, “Wave propagation in waveguide arrays with alternating positive and negative couplings,” Phys. Rev. A 81(5), 053817 (2010).
[Crossref]

N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. 35(23), 4045–4047 (2010).
[Crossref] [PubMed]

N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. 35(23), 4045–4047 (2010).
[Crossref] [PubMed]

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson Localization and Nonlinearity in One-Dimensional Disordered Photonic Lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[Crossref] [PubMed]

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

Dattoli, G.

G. Dattoli, A. Torre, S. Lorenzzutta, and G. Maino, “Theory of generalized Bessel functions II,” Nuovo Cim. 106(1), 21–51 (1991).
[Crossref]

G. Dattoli, L. Giannessi, L. Mezi, and A. Torre, “Theory of generalized Bessel functions,” Nuovo Cim. 105(3), 327–348 (1990).
[Crossref]

Dreisow, F.

J. M. Zeuner, N. K. Efremidis, R. Keil, F. Dreisow, D. N. Christodoulides, A. Tünnermann, S. Nolte, and A. Szameit, “Optical Analogues for massless Dirac particles and conical diffraction in one dimension,” Phys. Rev. Lett. 109(2), 023602 (2012).
[Crossref] [PubMed]

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical Analogue of Displaced Fock States and Quantum Correlations in Glauber-Fock Photonic Lattices,” Phys. Rev. Lett. 107(10), 103601 (2011).
[Crossref] [PubMed]

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, and A. Tünnermann, “Second-order coupling in femtosecond-laser-written waveguide arrays,” Opt. Lett. 33(22), 2689–2691 (2008).
[Crossref] [PubMed]

Efremdis, N. K.

N. K. Efremdis, P. Zhang, Z. Chen, D. N. Christodoulides, C. E. Ruter, and D. Kip, “Wave propagation in waveguide arrays with alternating positive and negative couplings,” Phys. Rev. A 81(5), 053817 (2010).
[Crossref]

Efremidis, N. K.

J. M. Zeuner, N. K. Efremidis, R. Keil, F. Dreisow, D. N. Christodoulides, A. Tünnermann, S. Nolte, and A. Szameit, “Optical Analogues for massless Dirac particles and conical diffraction in one dimension,” Phys. Rev. Lett. 109(2), 023602 (2012).
[Crossref] [PubMed]

N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. 35(23), 4045–4047 (2010).
[Crossref] [PubMed]

N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. 35(23), 4045–4047 (2010).
[Crossref] [PubMed]

Giannessi, L.

G. Dattoli, L. Giannessi, L. Mezi, and A. Torre, “Theory of generalized Bessel functions,” Nuovo Cim. 105(3), 327–348 (1990).
[Crossref]

Gutiérrez-Vega, J. C.

Heinrich, M.

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical Analogue of Displaced Fock States and Quantum Correlations in Glauber-Fock Photonic Lattices,” Phys. Rev. Lett. 107(10), 103601 (2011).
[Crossref] [PubMed]

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, and A. Tünnermann, “Second-order coupling in femtosecond-laser-written waveguide arrays,” Opt. Lett. 33(22), 2689–2691 (2008).
[Crossref] [PubMed]

Ismail, N.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science 329(5998), 1500–1503 (2010).
[Crossref] [PubMed]

Jones, A. L.

Kay, A.

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L. C. Kwek, B. M. Rodríguez-Lara, A. Szameit, and D. N. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87(1), 012309 (2013).
[Crossref]

Keil, R.

A. Perez-Leija, R. Keil, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Perfect transfer of path-entangled photons in Jx photonic lattices,” Phys. Rev. A 87(2), 022303 (2013).
[Crossref]

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L. C. Kwek, B. M. Rodríguez-Lara, A. Szameit, and D. N. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87(1), 012309 (2013).
[Crossref]

A. Perez-Leija, R. Keil, A. Szameit, A. F. Abouraddy, H. Moya-Cessa, and D. N. Christodoulides, “Tailoring the Correlation and Anticorrelation behavior of Path-entangled Photons in Glauber-Fock Oscillator Lattices,” Phys. Rev. A 85(1), 013848 (2012).
[Crossref]

R. Keil, A. Perez-Leija, P. Aleahmad, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Observation of Bloch-like revivals in semi-infinite Glauber-Fock photonic lattices,” Opt. Lett. 37(18), 3801–3803 (2012).
[Crossref] [PubMed]

J. M. Zeuner, N. K. Efremidis, R. Keil, F. Dreisow, D. N. Christodoulides, A. Tünnermann, S. Nolte, and A. Szameit, “Optical Analogues for massless Dirac particles and conical diffraction in one dimension,” Phys. Rev. Lett. 109(2), 023602 (2012).
[Crossref] [PubMed]

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical Analogue of Displaced Fock States and Quantum Correlations in Glauber-Fock Photonic Lattices,” Phys. Rev. Lett. 107(10), 103601 (2011).
[Crossref] [PubMed]

Kip, D.

N. K. Efremdis, P. Zhang, Z. Chen, D. N. Christodoulides, C. E. Ruter, and D. Kip, “Wave propagation in waveguide arrays with alternating positive and negative couplings,” Phys. Rev. A 81(5), 053817 (2010).
[Crossref]

Kwek, L. C.

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L. C. Kwek, B. M. Rodríguez-Lara, A. Szameit, and D. N. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87(1), 012309 (2013).
[Crossref]

Lahini, Y.

Y. Bromberg, Y. Lahini, and Y. Silberberg, “Bloch Oscillations of Path-Entangled Photons,” Phys. Rev. Lett. 105(26), 263604 (2010).
[Crossref] [PubMed]

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science 329(5998), 1500–1503 (2010).
[Crossref] [PubMed]

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson Localization and Nonlinearity in One-Dimensional Disordered Photonic Lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[Crossref] [PubMed]

Lederer, F.

A. Szameit, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, “Long-range interaction in waveguide lattices,” Phys. Rev. A 77(4), 043804 (2008).
[Crossref]

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Bräuer, and U. Peschel, “Visual Observation of Zener Tunneling,” Phys. Rev. Lett. 96(2), 023901 (2006).
[Crossref] [PubMed]

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

Lobino, M.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science 329(5998), 1500–1503 (2010).
[Crossref] [PubMed]

Longhi, S.

Lorenzzutta, S.

G. Dattoli, A. Torre, S. Lorenzzutta, and G. Maino, “Theory of generalized Bessel functions II,” Nuovo Cim. 106(1), 21–51 (1991).
[Crossref]

Maino, G.

G. Dattoli, A. Torre, S. Lorenzzutta, and G. Maino, “Theory of generalized Bessel functions II,” Nuovo Cim. 106(1), 21–51 (1991).
[Crossref]

Matsuda, N.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science 329(5998), 1500–1503 (2010).
[Crossref] [PubMed]

Matthews, J. C. F.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science 329(5998), 1500–1503 (2010).
[Crossref] [PubMed]

Mezi, L.

G. Dattoli, L. Giannessi, L. Mezi, and A. Torre, “Theory of generalized Bessel functions,” Nuovo Cim. 105(3), 327–348 (1990).
[Crossref]

Michaelis, D.

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Bräuer, and U. Peschel, “Visual Observation of Zener Tunneling,” Phys. Rev. Lett. 96(2), 023901 (2006).
[Crossref] [PubMed]

Morandotti, R.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson Localization and Nonlinearity in One-Dimensional Disordered Photonic Lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[Crossref] [PubMed]

Moya-Cessa, H.

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L. C. Kwek, B. M. Rodríguez-Lara, A. Szameit, and D. N. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87(1), 012309 (2013).
[Crossref]

A. Perez-Leija, R. Keil, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Perfect transfer of path-entangled photons in Jx photonic lattices,” Phys. Rev. A 87(2), 022303 (2013).
[Crossref]

A. Perez-Leija, R. Keil, A. Szameit, A. F. Abouraddy, H. Moya-Cessa, and D. N. Christodoulides, “Tailoring the Correlation and Anticorrelation behavior of Path-entangled Photons in Glauber-Fock Oscillator Lattices,” Phys. Rev. A 85(1), 013848 (2012).
[Crossref]

R. Keil, A. Perez-Leija, P. Aleahmad, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Observation of Bloch-like revivals in semi-infinite Glauber-Fock photonic lattices,” Opt. Lett. 37(18), 3801–3803 (2012).
[Crossref] [PubMed]

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical Analogue of Displaced Fock States and Quantum Correlations in Glauber-Fock Photonic Lattices,” Phys. Rev. Lett. 107(10), 103601 (2011).
[Crossref] [PubMed]

Nikolopoulos, G. M.

Nolte, S.

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L. C. Kwek, B. M. Rodríguez-Lara, A. Szameit, and D. N. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87(1), 012309 (2013).
[Crossref]

R. Keil, A. Perez-Leija, P. Aleahmad, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Observation of Bloch-like revivals in semi-infinite Glauber-Fock photonic lattices,” Opt. Lett. 37(18), 3801–3803 (2012).
[Crossref] [PubMed]

J. M. Zeuner, N. K. Efremidis, R. Keil, F. Dreisow, D. N. Christodoulides, A. Tünnermann, S. Nolte, and A. Szameit, “Optical Analogues for massless Dirac particles and conical diffraction in one dimension,” Phys. Rev. Lett. 109(2), 023602 (2012).
[Crossref] [PubMed]

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical Analogue of Displaced Fock States and Quantum Correlations in Glauber-Fock Photonic Lattices,” Phys. Rev. Lett. 107(10), 103601 (2011).
[Crossref] [PubMed]

A. Szameit, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, “Long-range interaction in waveguide lattices,” Phys. Rev. A 77(4), 043804 (2008).
[Crossref]

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, and A. Tünnermann, “Second-order coupling in femtosecond-laser-written waveguide arrays,” Opt. Lett. 33(22), 2689–2691 (2008).
[Crossref] [PubMed]

OBrien, J. L.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science 329(5998), 1500–1503 (2010).
[Crossref] [PubMed]

Perez-Leija, A.

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L. C. Kwek, B. M. Rodríguez-Lara, A. Szameit, and D. N. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87(1), 012309 (2013).
[Crossref]

A. Perez-Leija, R. Keil, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Perfect transfer of path-entangled photons in Jx photonic lattices,” Phys. Rev. A 87(2), 022303 (2013).
[Crossref]

A. Perez-Leija, R. Keil, A. Szameit, A. F. Abouraddy, H. Moya-Cessa, and D. N. Christodoulides, “Tailoring the Correlation and Anticorrelation behavior of Path-entangled Photons in Glauber-Fock Oscillator Lattices,” Phys. Rev. A 85(1), 013848 (2012).
[Crossref]

R. Keil, A. Perez-Leija, P. Aleahmad, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Observation of Bloch-like revivals in semi-infinite Glauber-Fock photonic lattices,” Opt. Lett. 37(18), 3801–3803 (2012).
[Crossref] [PubMed]

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical Analogue of Displaced Fock States and Quantum Correlations in Glauber-Fock Photonic Lattices,” Phys. Rev. Lett. 107(10), 103601 (2011).
[Crossref] [PubMed]

Pertsch, T.

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, and A. Tünnermann, “Second-order coupling in femtosecond-laser-written waveguide arrays,” Opt. Lett. 33(22), 2689–2691 (2008).
[Crossref] [PubMed]

A. Szameit, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, “Long-range interaction in waveguide lattices,” Phys. Rev. A 77(4), 043804 (2008).
[Crossref]

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Bräuer, and U. Peschel, “Visual Observation of Zener Tunneling,” Phys. Rev. Lett. 96(2), 023901 (2006).
[Crossref] [PubMed]

Peruzzo, A.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science 329(5998), 1500–1503 (2010).
[Crossref] [PubMed]

Peschel, U.

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Bräuer, and U. Peschel, “Visual Observation of Zener Tunneling,” Phys. Rev. Lett. 96(2), 023901 (2006).
[Crossref] [PubMed]

Politi, A.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science 329(5998), 1500–1503 (2010).
[Crossref] [PubMed]

Poulios, K.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science 329(5998), 1500–1503 (2010).
[Crossref] [PubMed]

Pozzi, F.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson Localization and Nonlinearity in One-Dimensional Disordered Photonic Lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[Crossref] [PubMed]

Rai, A.

A. Rai and G. S. Agarwal, “Possibility of coherent phenomena such as Bloch oscillations with single photons via W states,” Phys. Rev. A 79(5), 053849 (2009).
[Crossref]

Rodríguez-Lara, B. M.

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L. C. Kwek, B. M. Rodríguez-Lara, A. Szameit, and D. N. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87(1), 012309 (2013).
[Crossref]

Ruter, C. E.

N. K. Efremdis, P. Zhang, Z. Chen, D. N. Christodoulides, C. E. Ruter, and D. Kip, “Wave propagation in waveguide arrays with alternating positive and negative couplings,” Phys. Rev. A 81(5), 053817 (2010).
[Crossref]

Silberberg, Y.

Y. Bromberg, Y. Lahini, and Y. Silberberg, “Bloch Oscillations of Path-Entangled Photons,” Phys. Rev. Lett. 105(26), 263604 (2010).
[Crossref] [PubMed]

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science 329(5998), 1500–1503 (2010).
[Crossref] [PubMed]

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson Localization and Nonlinearity in One-Dimensional Disordered Photonic Lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[Crossref] [PubMed]

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

Sorel, M.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson Localization and Nonlinearity in One-Dimensional Disordered Photonic Lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[Crossref] [PubMed]

Streppel, U.

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Bräuer, and U. Peschel, “Visual Observation of Zener Tunneling,” Phys. Rev. Lett. 96(2), 023901 (2006).
[Crossref] [PubMed]

Szameit, A.

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L. C. Kwek, B. M. Rodríguez-Lara, A. Szameit, and D. N. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87(1), 012309 (2013).
[Crossref]

A. Perez-Leija, R. Keil, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Perfect transfer of path-entangled photons in Jx photonic lattices,” Phys. Rev. A 87(2), 022303 (2013).
[Crossref]

A. Perez-Leija, R. Keil, A. Szameit, A. F. Abouraddy, H. Moya-Cessa, and D. N. Christodoulides, “Tailoring the Correlation and Anticorrelation behavior of Path-entangled Photons in Glauber-Fock Oscillator Lattices,” Phys. Rev. A 85(1), 013848 (2012).
[Crossref]

J. M. Zeuner, N. K. Efremidis, R. Keil, F. Dreisow, D. N. Christodoulides, A. Tünnermann, S. Nolte, and A. Szameit, “Optical Analogues for massless Dirac particles and conical diffraction in one dimension,” Phys. Rev. Lett. 109(2), 023602 (2012).
[Crossref] [PubMed]

R. Keil, A. Perez-Leija, P. Aleahmad, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Observation of Bloch-like revivals in semi-infinite Glauber-Fock photonic lattices,” Opt. Lett. 37(18), 3801–3803 (2012).
[Crossref] [PubMed]

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical Analogue of Displaced Fock States and Quantum Correlations in Glauber-Fock Photonic Lattices,” Phys. Rev. Lett. 107(10), 103601 (2011).
[Crossref] [PubMed]

A. Szameit, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, “Long-range interaction in waveguide lattices,” Phys. Rev. A 77(4), 043804 (2008).
[Crossref]

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, and A. Tünnermann, “Second-order coupling in femtosecond-laser-written waveguide arrays,” Opt. Lett. 33(22), 2689–2691 (2008).
[Crossref] [PubMed]

Thompson, M. G.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science 329(5998), 1500–1503 (2010).
[Crossref] [PubMed]

Torre, A.

G. Dattoli, A. Torre, S. Lorenzzutta, and G. Maino, “Theory of generalized Bessel functions II,” Nuovo Cim. 106(1), 21–51 (1991).
[Crossref]

G. Dattoli, L. Giannessi, L. Mezi, and A. Torre, “Theory of generalized Bessel functions,” Nuovo Cim. 105(3), 327–348 (1990).
[Crossref]

Trompeter, H.

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Bräuer, and U. Peschel, “Visual Observation of Zener Tunneling,” Phys. Rev. Lett. 96(2), 023901 (2006).
[Crossref] [PubMed]

Tünnermann, A.

J. M. Zeuner, N. K. Efremidis, R. Keil, F. Dreisow, D. N. Christodoulides, A. Tünnermann, S. Nolte, and A. Szameit, “Optical Analogues for massless Dirac particles and conical diffraction in one dimension,” Phys. Rev. Lett. 109(2), 023602 (2012).
[Crossref] [PubMed]

A. Szameit, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, “Long-range interaction in waveguide lattices,” Phys. Rev. A 77(4), 043804 (2008).
[Crossref]

F. Dreisow, A. Szameit, M. Heinrich, T. Pertsch, S. Nolte, and A. Tünnermann, “Second-order coupling in femtosecond-laser-written waveguide arrays,” Opt. Lett. 33(22), 2689–2691 (2008).
[Crossref] [PubMed]

Tzortzakis, S.

Wörhoff, K.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science 329(5998), 1500–1503 (2010).
[Crossref] [PubMed]

Zeuner, J. M.

J. M. Zeuner, N. K. Efremidis, R. Keil, F. Dreisow, D. N. Christodoulides, A. Tünnermann, S. Nolte, and A. Szameit, “Optical Analogues for massless Dirac particles and conical diffraction in one dimension,” Phys. Rev. Lett. 109(2), 023602 (2012).
[Crossref] [PubMed]

Zhang, P.

N. K. Efremdis, P. Zhang, Z. Chen, D. N. Christodoulides, C. E. Ruter, and D. Kip, “Wave propagation in waveguide arrays with alternating positive and negative couplings,” Phys. Rev. A 81(5), 053817 (2010).
[Crossref]

Zhou, X. Q.

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science 329(5998), 1500–1503 (2010).
[Crossref] [PubMed]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Nature (1)

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

Nuovo Cim. (2)

G. Dattoli, L. Giannessi, L. Mezi, and A. Torre, “Theory of generalized Bessel functions,” Nuovo Cim. 105(3), 327–348 (1990).
[Crossref]

G. Dattoli, A. Torre, S. Lorenzzutta, and G. Maino, “Theory of generalized Bessel functions II,” Nuovo Cim. 106(1), 21–51 (1991).
[Crossref]

Opt. Lett. (6)

Phys. Rev. A (6)

N. K. Efremdis, P. Zhang, Z. Chen, D. N. Christodoulides, C. E. Ruter, and D. Kip, “Wave propagation in waveguide arrays with alternating positive and negative couplings,” Phys. Rev. A 81(5), 053817 (2010).
[Crossref]

A. Perez-Leija, R. Keil, A. Szameit, A. F. Abouraddy, H. Moya-Cessa, and D. N. Christodoulides, “Tailoring the Correlation and Anticorrelation behavior of Path-entangled Photons in Glauber-Fock Oscillator Lattices,” Phys. Rev. A 85(1), 013848 (2012).
[Crossref]

A. Rai and G. S. Agarwal, “Possibility of coherent phenomena such as Bloch oscillations with single photons via W states,” Phys. Rev. A 79(5), 053849 (2009).
[Crossref]

A. Szameit, T. Pertsch, S. Nolte, A. Tünnermann, and F. Lederer, “Long-range interaction in waveguide lattices,” Phys. Rev. A 77(4), 043804 (2008).
[Crossref]

A. Perez-Leija, R. Keil, A. Kay, H. Moya-Cessa, S. Nolte, L. C. Kwek, B. M. Rodríguez-Lara, A. Szameit, and D. N. Christodoulides, “Coherent quantum transport in photonic lattices,” Phys. Rev. A 87(1), 012309 (2013).
[Crossref]

A. Perez-Leija, R. Keil, H. Moya-Cessa, A. Szameit, and D. N. Christodoulides, “Perfect transfer of path-entangled photons in Jx photonic lattices,” Phys. Rev. A 87(2), 022303 (2013).
[Crossref]

Phys. Rev. Lett. (5)

Y. Bromberg, Y. Lahini, and Y. Silberberg, “Bloch Oscillations of Path-Entangled Photons,” Phys. Rev. Lett. 105(26), 263604 (2010).
[Crossref] [PubMed]

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson Localization and Nonlinearity in One-Dimensional Disordered Photonic Lattices,” Phys. Rev. Lett. 100(1), 013906 (2008).
[Crossref] [PubMed]

R. Keil, A. Perez-Leija, F. Dreisow, M. Heinrich, H. Moya-Cessa, S. Nolte, D. N. Christodoulides, and A. Szameit, “Classical Analogue of Displaced Fock States and Quantum Correlations in Glauber-Fock Photonic Lattices,” Phys. Rev. Lett. 107(10), 103601 (2011).
[Crossref] [PubMed]

H. Trompeter, T. Pertsch, F. Lederer, D. Michaelis, U. Streppel, A. Bräuer, and U. Peschel, “Visual Observation of Zener Tunneling,” Phys. Rev. Lett. 96(2), 023901 (2006).
[Crossref] [PubMed]

J. M. Zeuner, N. K. Efremidis, R. Keil, F. Dreisow, D. N. Christodoulides, A. Tünnermann, S. Nolte, and A. Szameit, “Optical Analogues for massless Dirac particles and conical diffraction in one dimension,” Phys. Rev. Lett. 109(2), 023602 (2012).
[Crossref] [PubMed]

Science (1)

A. Peruzzo, M. Lobino, J. C. F. Matthews, N. Matsuda, A. Politi, K. Poulios, X. Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, and J. L. OBrien, “Quantum Walks of Correlated Photons,” Science 329(5998), 1500–1503 (2010).
[Crossref] [PubMed]

Other (5)

J. C. F. Matthews, K. Poulios, J. D. A. Meinecke, A. Politi, A. Peruzzo, N. Ismail, K. Wörhoff, M. G. Thompson, and J. L. O'Brien, “Simulating quantum statistics with entangled photons: a continuous transition from bosons to fermions,” Preprint: arXiv:1106.1166.

J. W. Goodman, Introduction to Fourier Optics, McGraw-Hill Physical and Quantum Electronics Series (Roberts, 2005).

H. Moya-Cessa and F. Soto-Eguibar, Differential Equations: An Operational Approach (Rinton Press, 2011).

C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge, 2005).

W. H. Louisell, Quantum and Statistical Properties of Radiation (Wiley, 1973).

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Figures (4)

Fig. 1
Fig. 1 Transverse view of an array of evanescently coupled single mode waveguides. κ 1 is the coupling strength between nearest neighbors whereas κ 2 represents the couplings between next-nearest neighbors.

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Fig. 2
Fig. 2 (a) and (c) depict the intensity evolution of the initial fields ψ( x,0 )=exp( − x 2 / 2 σ 2 ) J 3 ( αx ) and ψ( x,0 )=exp( − x 2 / 2 σ 2 ) J 6 ( αx ) , respectively. The parameters used are α=1 and σ= 10 −3 . (b) and (d) show the corresponding initial field profile.

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Fig. 3
Fig. 3 Theoretical comparison of light propagation through free space for the initial field profile ψ( x,0 )=exp( − x 2 / 2 σ 2 ) J 0 ( αx ) (a-c), and in a periodic photonic lattice of 30 elements, under a single site excitation (d-f). Figs. (a, d) depict the initial field distributions, (b, e) the normalized intensity evolution, and (c, f) the intensity profiles at the output. The parameters used for the paraxial beams are α=1 and σ= 10 −3 . For the waveguide array we assumed coupling coefficients κ 1 =1 c m −1 and κ 2 =0.1 c m −1 .

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Fig. 4
Fig. 4 (a) Intensity propagation of the field ψ( x,0 )=exp( − x 2 / 2 σ 2 ) J 0 ( αx ) and (b) the absolute square of the Gaussian envelope exp( − x 2 / 2 σ 2 ) , the parameters used are the same as in Fig. (3). In this case, we found that FWHM=3mm , whereas l=3.3cm .

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Equations (23)

Equations on this page are rendered with MathJax. Learn more.

i d E m dZ + κ 1 ( E m+1 + E m−1 )+ κ 2 ( E m+2 + E m−2 )=0,
E m ( Z )= ∑ l=−∞ ∞ i m−l J m−2l ( 2 κ 1 Z ) J l ( 2 κ 2 Z ) ,
E m ( Z )= i m J m ( 2 κ 1 Z,2 κ 2 Z;−i ).
J m ( x,y ;1 )= 1 2π ∫ −π π exp[ i( xsin( ϕ )+ysin( 2ϕ )−mϕ ) ] dϕ,
2m J m ( x,y;r )=x[ J m−1 ( x,y;r )+ J m+1 ( x,y;r )]+2y( rJ m−2 ( x,y;r )+ 1 r J m−2 ( x,y;r ) ),
T(x,y;r,t) = ∑ m=−∞ ∞ t m J m ( x,y; r ) =exp( x 2 ( t− 1 t )+ y 2 ( r t 2 − 1 r t 2 ) ).
i ∂ψ ∂ξ =− 1 2 ∂ 2 ψ ∂ s 2 ,
ψ( s,ξ )= ∫ −∞ ∞ ∫ −∞ ∞ F( w−u )G( u )exp( i ξ 2 w 2 )exp( isw )  dudw.
ψ( s,ξ )=exp( g 1 s 2 −i g 2 ) ∫ −∞ ∞ F( v )exp( ( i ξ 2 + g 3 ) v 2 )exp( isv exp( −i2 g 2 ) )dv,
ψ( s,ξ )= 1 2π exp( g 1 ( ξ ) s 2 −i g 2 ( ξ ) ) ∫ −π π exp( −inϕ+iAsin( ϕ )+iB sin 2 ( ϕ ) ) dϕ,
ψ( s,ξ )=exp( g 1 ( ξ ) s 2 −i( g 2 ( ξ )− B 2 ) ) ∑ m=−∞ ∞ ( −i ) m J n+2m ( A ) J m ( B/2 ) .
ψ( s,ξ )=exp( i ξ 2 p ^ 2 ) ∫ −∞ ∞ G( u )exp( isu )du ∫ −∞ ∞ F( v )exp( isv )dv .
ψ( s,ξ )= ∫ −∞ ∞ G( u )exp( iu[ s+ξ p ^ ] )exp( i ξ 2 p ^ 2 )du ∫ −∞ ∞ F( v )exp( isv )dv .
ψ( s,ξ )= ∫ −∞ ∞ G( u )( ∫ −∞ ∞ F( v )exp( iξ u 2 2 )exp( ius )exp( iξu p ^ ) exp( iξ p ^ 2 2 )exp( isv )dv )du .
exp( iξ p ^ 2 2 )exp( isv )= ∑ k=0 ∞ ( iξ /2 ) k k! ( −i ∂ ∂s ) 2k exp( isv )=exp( iξ v 2 2 )exp( isv )
ψ( s,ξ )= ∫ −∞ ∞ G( u )( ∫ −∞ ∞ F( v )exp( iξ ( u+v ) 2 /2 ) exp( is( u+v ) )dv )du
ψ( s,ξ )= ∫ −∞ ∞ G( u ) ( ∫ −∞ ∞ F( w−u )exp( iξ w 2 /2 )exp( isw )dw )du.
ψ( s,ξ )=exp( −[ s 2 +ξ( s p ^ + p ^ s )+ ξ 2 p ^ 2 ] / 2 σ 2 )exp( iξ p ^ 2 /2 )f( s ).
exp( − [ s 2 +ξ( s p ^ + p ^ s )+ ξ 2 p ^ 2 ] 2 σ 2 )=exp( g 1 ( ξ ) s 2 )exp( g 2 ( ξ )( s p ^ + p ^ s ) )exp( g 3 ( ξ ) p ^ 2 ).
ψ( s,ξ )=exp( g 1 s 2 )exp( g 2 ( s p ^ + p ^ s ) ) ∫ −∞ ∞ F( v )exp( ( i ξ 2 + g 3 ) v 2 )exp( isv )dv .
ψ( s,ξ )=exp( g 1 s 2 −i g 2 ) ∫ −∞ ∞ F( v )exp( ( i ξ 2 + g 3 ) v 2 )exp( isv exp( −i2 g 2 ) )dv.
ψ( s,ξ )= 1 2π exp( g 1 s 2 −i g 2 ) ∫ −π π exp( −inϕ+iAsinϕ+iB sin 2 ϕ ) dϕ,
ψ( s,ξ )= 1 2π exp( g 1 s 2 −i( g 2 −B/2 ) ) ∑ j,m=0 ∞ 1 j!m! ( − iB 4 ) j+m ∫ −π π exp( −i( n+2j−2m )ϕ+iAsinϕ )dϕ

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