Discrete-like diffraction dynamics in free space

Armando Perez-Leija; Francisco Soto-Eguibar; Sabino Chavez-Cerda; Alexander Szameit; Hector Moya-Cessa; Demetrios N. Christodoulides

doi:10.1364/OE.21.017951

Discrete-like diffraction dynamics in free space

Armando Perez-Leija, Francisco Soto-Eguibar, Sabino Chavez-Cerda, Alexander Szameit, Hector Moya-Cessa, and Demetrios N. Christodoulides

Optics Express
Vol. 21,
Issue 15,
pp. 17951-17960
(2013)
•https://doi.org/10.1364/OE.21.017951

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Armando Perez-Leija, Francisco Soto-Eguibar, Sabino Chavez-Cerda, Alexander Szameit, Hector Moya-Cessa, and Demetrios N. Christodoulides, "Discrete-like diffraction dynamics in free space," Opt. Express 21, 17951-17960 (2013)

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Related Topics
Table of Contents Category
- Physical Optics
Optics & Photonics Topics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Diffraction (050.1940)
- Propagation (350.5500)
- Dispersion (260.2030)

About this Article
History
- Original Manuscript: March 22, 2013
- Revised Manuscript: June 10, 2013
- Manuscript Accepted: June 14, 2013
- Published: July 19, 2013

Accessible

Open Access

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.

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References

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Publication

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]
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]
S. Longhi, “Jaynes-Cummings photonic superlattices,” Opt. Lett. 36(17), 3407–3409 (2011).
[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. 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]
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]
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]
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]
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.
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]
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]
A. L. Jones, “Coupling of optical fibers and scattering fibers,” J. Opt. Soc. Am. 55(3), 261 (1965).
[Crossref]
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]
J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. A 22(2), 289–298 (2005).
[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]
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]
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).

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)

J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. A 22(2), 289–298 (2005).
[Crossref] [PubMed]

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)

A. L. Jones, “Coupling of optical fibers and scattering fibers,” J. Opt. Soc. Am. 55(3), 261 (1965).
[Crossref]

Abouraddy, A. F.

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.

Bandres, M. A.

J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. A 22(2), 289–298 (2005).
[Crossref] [PubMed]

Bellec, M.

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

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.

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

Chen, Z.

Christodoulides, D. N.

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]

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

Efremdis, N. K.

Efremidis, N. K.

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.

J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. A 22(2), 289–298 (2005).
[Crossref] [PubMed]

Heinrich, M.

Ismail, N.

Jones, A. L.

A. L. Jones, “Coupling of optical fibers and scattering fibers,” J. Opt. Soc. Am. 55(3), 261 (1965).
[Crossref]

Kay, A.

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]

Kip, D.

Kwek, L. C.

Lahini, Y.

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

Longhi, S.

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

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.

Matthews, J. C. F.

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.

Moya-Cessa, H.

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]

Nikolopoulos, G. M.

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

Nolte, S.

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]

OBrien, J. L.

Perez-Leija, A.

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]

Pertsch, T.

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.

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.

Poulios, K.

Pozzi, F.

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.

Ruter, C. E.

Silberberg, Y.

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

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

Thompson, M. G.

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.

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]

Tzortzakis, S.

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

Wörhoff, K.

Zeuner, J. M.

Zhang, P.

Zhou, X. Q.

J. Opt. Soc. Am. (1)

A. L. Jones, “Coupling of optical fibers and scattering fibers,” J. Opt. Soc. Am. 55(3), 261 (1965).
[Crossref]

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

J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. A 22(2), 289–298 (2005).
[Crossref] [PubMed]

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)

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

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

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

Phys. Rev. A (6)

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).
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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

(a) and (c) depict the intensity evolution of the initial fields $ψ (x, 0) = \exp (- x^{2} / 2 σ^{2}) J_{3} (α x)$ and $ψ (x, 0) = \exp (- x^{2} / 2 σ^{2}) J_{6} (α x)$ , 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

Theoretical comparison of light propagation through free space for the initial field profile $ψ (x, 0) = \exp (- x^{2} / 2 σ^{2}) J_{0} (α x)$ (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

(a) Intensity propagation of the field $ψ (x, 0) = \exp (- x^{2} / 2 σ^{2}) J_{0} (α x)$ 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 $F W H M = 3 m m$ , whereas $l = 3.3 c m$ .

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Equations (23)

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i \frac{d E_{m}}{d Z} + κ_{1} (E_{m + 1} + E_{m - 1}) + κ_{2} (E_{m + 2} + E_{m - 2}) = 0,

E_{m} (Z) = \sum_{l = - \infty}^{\infty} i^{m - l} J_{m - 2 l} (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) = \frac{1}{2 π} \int_{- π}^{π} \exp [i (x \sin (ϕ) + y \sin (2 ϕ) - m ϕ)] d ϕ,

2 m J_{m} (x, y; r) = x [J_{m - 1} (x, y; r) + J_{m + 1} (x, y; r)] + 2 y ({rJ}_{m - 2} (x, y; r) + \frac{1}{r} J_{m - 2} (x, y; r)),

T (x, y; r, t) = \sum_{m = - \infty}^{\infty} t^{m} J_{m} (x, y; r) = \exp (\frac{x}{2} (t - \frac{1}{t}) + \frac{y}{2} (r t^{2} - \frac{1}{r t^{2}})) .

i \frac{\partial ψ}{\partial ξ} = - \frac{1}{2} \frac{\partial^{2} ψ}{\partial s^{2}},

ψ (s, ξ) = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} F (w - u) G (u) \exp (i \frac{ξ}{2} w^{2}) \exp (i s w) d u d w .

\begin{array}{l} ψ (s, ξ) = \exp (g_{1} s^{2} - i g_{2}) \int_{- \infty}^{\infty} F (v) \exp ((i \frac{ξ}{2} + g_{3}) v^{2}) \exp (i s v \exp (- i 2 g_{2})) d v, \end{array}

ψ (s, ξ) = \frac{1}{2 π} \exp (g_{1} (ξ) s^{2} - i g_{2} (ξ)) \int_{- π}^{π} \exp (- i n ϕ + i A \sin (ϕ) + i B \sin^{2} (ϕ)) d ϕ,

ψ (s, ξ) = \exp (g_{1} (ξ) s^{2} - i (g_{2} (ξ) - \frac{B}{2})) \sum_{m = - \infty}^{\infty} {(- i)}^{m} J_{n + 2 m} (A) J_{m} (B / 2) .

ψ (s, ξ) = \exp (i \frac{ξ}{2} {\hat{p}}^{2}) \int_{- \infty}^{\infty} G (u) \exp (i s u) d u \int_{- \infty}^{\infty} F (v) \exp (i s v) d v .

ψ (s, ξ) = \int_{- \infty}^{\infty} G (u) \exp (i u [s + ξ \hat{p}]) \exp (i \frac{ξ}{2} {\hat{p}}^{2}) d u \int_{- \infty}^{\infty} F (v) \exp (i s v) d v .

ψ (s, ξ) = \int_{- \infty}^{\infty} G (u) (\int_{- \infty}^{\infty} F (v) \exp (\frac{i ξ u^{2}}{2}) \exp (i u s) \exp (i ξ u \hat{p}) \exp (\frac{i ξ {\hat{p}}^{2}}{2}) \exp (i s v) d v) d u .

\exp (\frac{i ξ {\hat{p}}^{2}}{2}) \exp (i s v) = \sum_{k = 0}^{\infty} \frac{{(i ξ / 2)}^{k}}{k!} {(- i \frac{\partial}{\partial s})}^{2 k} \exp (i s v) = \exp (\frac{i ξ v^{2}}{2}) \exp (i s v)

ψ (s, ξ) = \int_{- \infty}^{\infty} G (u) (\int_{- \infty}^{\infty} F (v) \exp (i ξ {(u + v)}^{2} / 2) \exp (i s (u + v)) d v) d u

ψ (s, ξ) = \int_{- \infty}^{\infty} G (u) (\int_{- \infty}^{\infty} F (w - u) \exp (i ξ w^{2} / 2) \exp (i s w) d w) d u .

ψ (s, ξ) = \exp (- [s^{2} + ξ (s \hat{p} + \hat{p} s) + ξ^{2} {\hat{p}}^{2}] / 2 σ^{2}) \exp (i ξ {\hat{p}}^{2} / 2) f (s) .

\exp (- \frac{[s^{2} + ξ (s \hat{p} + \hat{p} s) + ξ^{2} {\hat{p}}^{2}]}{2 σ^{2}}) = \exp (g_{1} (ξ) s^{2}) \exp (g_{2} (ξ) (s \hat{p} + \hat{p} s)) \exp (g_{3} (ξ) {\hat{p}}^{2}) .

ψ (s, ξ) = \exp (g_{1} s^{2}) \exp (g_{2} (s \hat{p} + \hat{p} s)) \int_{- \infty}^{\infty} F (v) \exp ((i \frac{ξ}{2} + g_{3}) v^{2}) \exp (i s v) d v .

ψ (s, ξ) = \exp (g_{1} s^{2} - i g_{2}) \int_{- \infty}^{\infty} F (v) \exp ((i \frac{ξ}{2} + g_{3}) v^{2}) \exp (i s v \exp (- i 2 g_{2})) d v .

ψ (s, ξ) = \frac{1}{2 π} \exp (g_{1} s^{2} - i g_{2}) \int_{- π}^{π} \exp (- i n ϕ + i A \sin ϕ + i B \sin^{2} ϕ) d ϕ,

ψ (s, ξ) = \frac{1}{2 π} \exp (g_{1} s^{2} - i (g_{2} - B / 2)) \sum_{j, m = 0}^{\infty} \frac{1}{j! m!} {(- \frac{i B}{4})}^{j + m} \int_{- π}^{π} \exp (- i (n + 2 j - 2 m) ϕ + i A \sin ϕ) d ϕ