Abstract

We generate higher-order azimuthally modulated Bessel optical lattices in photorefractive crystals by employing a phase-imprinting technique. We report on the experimental observation of self-trapping and nonlinear localization of light in such segmented lattices in the form of ring-shaped and single-site states. The experimental results agree well with numerical simulations accounting for an anisotropic and spatially nonlocal nonlinear response of photorefractive crystals.

© 2006 Optical Society of America

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    [CrossRef] [PubMed]
  5. D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, "Spatial solitons in optically induced gratings," Opt. Lett. 28,710-712 (2003).
    [CrossRef] [PubMed]
  6. H. Martin, E. D. Eugenieva, Z. G. Chen, and D. N. Christodoulides, "Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices," Phys. Rev. Lett. 92,123902-4 (2004).
    [CrossRef] [PubMed]
  7. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Rotary solitons in Bessel optical lattices," Phys. Rev. Lett. 93,093904-4 (2004).Q2
    [CrossRef] [PubMed]
  8. X. Wang, Z. Chen, and P. G. Kevrekidis, "Observation of discrete solitons and soliton rotation in optically induced periodic ring lattices", Phys. Rev. Lett. 96,083904-4 (2005).Q3
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  20. A. S. Desyatnikov, D. N. Neshev, Yu. S. Kivshar, N. Sagemerten, D. Trager, J. Jagers, C. Denz, and Y. V. Kartashov, "Nonlinear photonic lattices in anisotropic nonlocal self-focusing media," Opt. Lett. 30,869-871 (2005).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  24. J. Rogel-Salazar, G. H. C. New, and S. Chavez-Cerda, "Bessel-Gauss beam optical resonator," Opt. Commun. 190,117-122 (2001).
    [CrossRef]
  25. R. Fischer, D. Träger, D. N. Neshev, A. A. Sukhorukov,W. Krolikowski, C. Denz, and Yu. S. Kivshar, "Reduce dsymmetry two-dimensional solitons in square photonic lattices," Phys. Rev. Lett. 96, 023905 (2006).
    [CrossRef] [PubMed]

2006

R. Fischer, D. Träger, D. N. Neshev, A. A. Sukhorukov,W. Krolikowski, C. Denz, and Yu. S. Kivshar, "Reduce dsymmetry two-dimensional solitons in square photonic lattices," Phys. Rev. Lett. 96, 023905 (2006).
[CrossRef] [PubMed]

2005

A. S. Desyatnikov, D. N. Neshev, Yu. S. Kivshar, N. Sagemerten, D. Trager, J. Jagers, C. Denz, and Y. V. Kartashov, "Nonlinear photonic lattices in anisotropic nonlocal self-focusing media," Opt. Lett. 30,869-871 (2005).
[CrossRef] [PubMed]

X. Wang, Z. Chen, and P. G. Kevrekidis, "Observation of discrete solitons and soliton rotation in optically induced periodic ring lattices", Phys. Rev. Lett. 96,083904-4 (2005).Q3
[CrossRef]

D. McGloin and K. Dholakia, "Bessel beams: diffraction in a new light," Contemp. Phys. 46,15-28 (2005).
[CrossRef]

2004

Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, "Stable soliton complexes and azimuthal switching in modulated Bessel optical lattices," Phys. Rev. E 70,065602-4 (2004).Q6
[CrossRef]

C. Lopez-Mariscal, J. C. Gutierrez-Vega, and S. Chavez-Cerda, "Production of high-order Bessel beams with a Mach-Zehnder interferometer," Appl. Optics 43,5060-5063 (2004).
[CrossRef]

H. Martin, E. D. Eugenieva, Z. G. Chen, and D. N. Christodoulides, "Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices," Phys. Rev. Lett. 92,123902-4 (2004).
[CrossRef] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Rotary solitons in Bessel optical lattices," Phys. Rev. Lett. 93,093904-4 (2004).Q2
[CrossRef] [PubMed]

2003

2002

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 66,046602-5 (2002).Q1
[CrossRef]

2001

J. Rogel-Salazar, G. H. C. New, and S. Chavez-Cerda, "Bessel-Gauss beam optical resonator," Opt. Commun. 190,117-122 (2001).
[CrossRef]

1998

W. X. Cong, N. X. Chen, and B. Y. Gu, "Generation of nondiffracting beams by diffractive phase elements," J. Opt. Soc. Am. A 15,2362-2364 (1998).
[CrossRef]

A. A. Zozulya, D. Z. Anderson, A. V. Mamaev, and M. Saffman, "Solitary attractors and low-order filamentation in anisotropic self-focusing media," Phys. Rev. A 57,522-534 (1998).
[CrossRef]

1993

12. T. Wulle and S. Herminghaus, "Nonlinear optics of bessel beams," Phys. Rev. Lett. 70,1401-1404 (1993);Erratum: Phys. Rev. Lett. 71,209 (1993). Q4Q5
[CrossRef] [PubMed]

1992

1989

1987

J. Durnin, "Exact solutions for nondiffracting beams. I. The scalar theory," J. Opt. Soc. Am. A 4,651-654 (1987).
[CrossRef]

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58,1499-1501 (1987).
[CrossRef] [PubMed]

Anderson, D. Z.

A. A. Zozulya, D. Z. Anderson, A. V. Mamaev, and M. Saffman, "Solitary attractors and low-order filamentation in anisotropic self-focusing media," Phys. Rev. A 57,522-534 (1998).
[CrossRef]

Brown, D. L.

Chattrapiban, N.

Chavez-Cerda, S.

C. Lopez-Mariscal, J. C. Gutierrez-Vega, and S. Chavez-Cerda, "Production of high-order Bessel beams with a Mach-Zehnder interferometer," Appl. Optics 43,5060-5063 (2004).
[CrossRef]

J. Rogel-Salazar, G. H. C. New, and S. Chavez-Cerda, "Bessel-Gauss beam optical resonator," Opt. Commun. 190,117-122 (2001).
[CrossRef]

Chen, N. X.

Chen, Z.

X. Wang, Z. Chen, and P. G. Kevrekidis, "Observation of discrete solitons and soliton rotation in optically induced periodic ring lattices", Phys. Rev. Lett. 96,083904-4 (2005).Q3
[CrossRef]

Chen, Z. G.

H. Martin, E. D. Eugenieva, Z. G. Chen, and D. N. Christodoulides, "Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices," Phys. Rev. Lett. 92,123902-4 (2004).
[CrossRef] [PubMed]

Christodoulides, D. N.

H. Martin, E. D. Eugenieva, Z. G. Chen, and D. N. Christodoulides, "Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices," Phys. Rev. Lett. 92,123902-4 (2004).
[CrossRef] [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 422,147-150 (2003).
[CrossRef] [PubMed]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, "Discretizing light behaviour in linear and nonlinear waveguide lattices," Nature 424,817-823 (2003).
[CrossRef] [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 66,046602-5 (2002).Q1
[CrossRef]

Cofield, D.

Cong, W. X.

Denz, C.

R. Fischer, D. Träger, D. N. Neshev, A. A. Sukhorukov,W. Krolikowski, C. Denz, and Yu. S. Kivshar, "Reduce dsymmetry two-dimensional solitons in square photonic lattices," Phys. Rev. Lett. 96, 023905 (2006).
[CrossRef] [PubMed]

A. S. Desyatnikov, D. N. Neshev, Yu. S. Kivshar, N. Sagemerten, D. Trager, J. Jagers, C. Denz, and Y. V. Kartashov, "Nonlinear photonic lattices in anisotropic nonlocal self-focusing media," Opt. Lett. 30,869-871 (2005).
[CrossRef] [PubMed]

Desyatnikov, A. S.

Dholakia, K.

Durnin, J.

J. Durnin, "Exact solutions for nondiffracting beams. I. The scalar theory," J. Opt. Soc. Am. A 4,651-654 (1987).
[CrossRef]

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58,1499-1501 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

Efremidis, N. K.

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 422,147-150 (2003).
[CrossRef] [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 66,046602-5 (2002).Q1
[CrossRef]

Egorov, A. A.

Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, "Stable soliton complexes and azimuthal switching in modulated Bessel optical lattices," Phys. Rev. E 70,065602-4 (2004).Q6
[CrossRef]

Eugenieva, E. D.

H. Martin, E. D. Eugenieva, Z. G. Chen, and D. N. Christodoulides, "Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices," Phys. Rev. Lett. 92,123902-4 (2004).
[CrossRef] [PubMed]

Fischer, R.

R. Fischer, D. Träger, D. N. Neshev, A. A. Sukhorukov,W. Krolikowski, C. Denz, and Yu. S. Kivshar, "Reduce dsymmetry two-dimensional solitons in square photonic lattices," Phys. Rev. Lett. 96, 023905 (2006).
[CrossRef] [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 422,147-150 (2003).
[CrossRef] [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 66,046602-5 (2002).Q1
[CrossRef]

Friberg, A. T.

Gu, B. Y.

Gutierrez-Vega, J. C.

C. Lopez-Mariscal, J. C. Gutierrez-Vega, and S. Chavez-Cerda, "Production of high-order Bessel beams with a Mach-Zehnder interferometer," Appl. Optics 43,5060-5063 (2004).
[CrossRef]

Herminghaus, S.

12. T. Wulle and S. Herminghaus, "Nonlinear optics of bessel beams," Phys. Rev. Lett. 70,1401-1404 (1993);Erratum: Phys. Rev. Lett. 71,209 (1993). Q4Q5
[CrossRef] [PubMed]

Hill, W. T.

Huang, H.

Indebetouw, G.

Jagers, J.

Kartashov, Y. V.

A. S. Desyatnikov, D. N. Neshev, Yu. S. Kivshar, N. Sagemerten, D. Trager, J. Jagers, C. Denz, and Y. V. Kartashov, "Nonlinear photonic lattices in anisotropic nonlocal self-focusing media," Opt. Lett. 30,869-871 (2005).
[CrossRef] [PubMed]

Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, "Stable soliton complexes and azimuthal switching in modulated Bessel optical lattices," Phys. Rev. E 70,065602-4 (2004).Q6
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Rotary solitons in Bessel optical lattices," Phys. Rev. Lett. 93,093904-4 (2004).Q2
[CrossRef] [PubMed]

Kevrekidis, P. G.

X. Wang, Z. Chen, and P. G. Kevrekidis, "Observation of discrete solitons and soliton rotation in optically induced periodic ring lattices", Phys. Rev. Lett. 96,083904-4 (2005).Q3
[CrossRef]

Kivshar, Y.

Kivshar, Yu. S.

R. Fischer, D. Träger, D. N. Neshev, A. A. Sukhorukov,W. Krolikowski, C. Denz, and Yu. S. Kivshar, "Reduce dsymmetry two-dimensional solitons in square photonic lattices," Phys. Rev. Lett. 96, 023905 (2006).
[CrossRef] [PubMed]

A. S. Desyatnikov, D. N. Neshev, Yu. S. Kivshar, N. Sagemerten, D. Trager, J. Jagers, C. Denz, and Y. V. Kartashov, "Nonlinear photonic lattices in anisotropic nonlocal self-focusing media," Opt. Lett. 30,869-871 (2005).
[CrossRef] [PubMed]

Krolikowski, W.

R. Fischer, D. Träger, D. N. Neshev, A. A. Sukhorukov,W. Krolikowski, C. Denz, and Yu. S. Kivshar, "Reduce dsymmetry two-dimensional solitons in square photonic lattices," Phys. Rev. Lett. 96, 023905 (2006).
[CrossRef] [PubMed]

D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, "Spatial solitons in optically induced gratings," Opt. Lett. 28,710-712 (2003).
[CrossRef] [PubMed]

Lederer, F.

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

Lee, W. M.

Lin, Y.

Lopez-Mariscal, C.

C. Lopez-Mariscal, J. C. Gutierrez-Vega, and S. Chavez-Cerda, "Production of high-order Bessel beams with a Mach-Zehnder interferometer," Appl. Optics 43,5060-5063 (2004).
[CrossRef]

Mamaev, A. V.

A. A. Zozulya, D. Z. Anderson, A. V. Mamaev, and M. Saffman, "Solitary attractors and low-order filamentation in anisotropic self-focusing media," Phys. Rev. A 57,522-534 (1998).
[CrossRef]

Martin, H.

H. Martin, E. D. Eugenieva, Z. G. Chen, and D. N. Christodoulides, "Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices," Phys. Rev. Lett. 92,123902-4 (2004).
[CrossRef] [PubMed]

McGloin, D.

Melville, H.

Miceli, J. J.

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58,1499-1501 (1987).
[CrossRef] [PubMed]

Neshev, D.

Neshev, D. N.

R. Fischer, D. Träger, D. N. Neshev, A. A. Sukhorukov,W. Krolikowski, C. Denz, and Yu. S. Kivshar, "Reduce dsymmetry two-dimensional solitons in square photonic lattices," Phys. Rev. Lett. 96, 023905 (2006).
[CrossRef] [PubMed]

A. S. Desyatnikov, D. N. Neshev, Yu. S. Kivshar, N. Sagemerten, D. Trager, J. Jagers, C. Denz, and Y. V. Kartashov, "Nonlinear photonic lattices in anisotropic nonlocal self-focusing media," Opt. Lett. 30,869-871 (2005).
[CrossRef] [PubMed]

New, G. H. C.

J. Rogel-Salazar, G. H. C. New, and S. Chavez-Cerda, "Bessel-Gauss beam optical resonator," Opt. Commun. 190,117-122 (2001).
[CrossRef]

Ostrovskaya, E.

Rogel-Salazar, J.

J. Rogel-Salazar, G. H. C. New, and S. Chavez-Cerda, "Bessel-Gauss beam optical resonator," Opt. Commun. 190,117-122 (2001).
[CrossRef]

Rogers, E. A.

Roy, R.

Saffman, M.

A. A. Zozulya, D. Z. Anderson, A. V. Mamaev, and M. Saffman, "Solitary attractors and low-order filamentation in anisotropic self-focusing media," Phys. Rev. A 57,522-534 (1998).
[CrossRef]

Sagemerten, N.

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 66,046602-5 (2002).Q1
[CrossRef]

Segev, M.

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 422,147-150 (2003).
[CrossRef] [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 66,046602-5 (2002).Q1
[CrossRef]

Seka, W.

Sibbett, W.

Silberberg, Y.

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

Spalding, G. C.

Sukhorukov, A. A.

R. Fischer, D. Träger, D. N. Neshev, A. A. Sukhorukov,W. Krolikowski, C. Denz, and Yu. S. Kivshar, "Reduce dsymmetry two-dimensional solitons in square photonic lattices," Phys. Rev. Lett. 96, 023905 (2006).
[CrossRef] [PubMed]

Tao, S. H.

Torner, L.

Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, "Stable soliton complexes and azimuthal switching in modulated Bessel optical lattices," Phys. Rev. E 70,065602-4 (2004).Q6
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Rotary solitons in Bessel optical lattices," Phys. Rev. Lett. 93,093904-4 (2004).Q2
[CrossRef] [PubMed]

Trager, D.

Träger, D.

R. Fischer, D. Träger, D. N. Neshev, A. A. Sukhorukov,W. Krolikowski, C. Denz, and Yu. S. Kivshar, "Reduce dsymmetry two-dimensional solitons in square photonic lattices," Phys. Rev. Lett. 96, 023905 (2006).
[CrossRef] [PubMed]

Turunen, J.

Vasara, A.

Vysloukh, V. A.

Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, "Stable soliton complexes and azimuthal switching in modulated Bessel optical lattices," Phys. Rev. E 70,065602-4 (2004).Q6
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, "Rotary solitons in Bessel optical lattices," Phys. Rev. Lett. 93,093904-4 (2004).Q2
[CrossRef] [PubMed]

Wang, X.

X. Wang, Z. Chen, and P. G. Kevrekidis, "Observation of discrete solitons and soliton rotation in optically induced periodic ring lattices", Phys. Rev. Lett. 96,083904-4 (2005).Q3
[CrossRef]

Wulle, T.

12. T. Wulle and S. Herminghaus, "Nonlinear optics of bessel beams," Phys. Rev. Lett. 70,1401-1404 (1993);Erratum: Phys. Rev. Lett. 71,209 (1993). Q4Q5
[CrossRef] [PubMed]

Yuan, X. C.

Zozulya, A. A.

A. A. Zozulya, D. Z. Anderson, A. V. Mamaev, and M. Saffman, "Solitary attractors and low-order filamentation in anisotropic self-focusing media," Phys. Rev. A 57,522-534 (1998).
[CrossRef]

Appl. Opt.

Appl. Optics

C. Lopez-Mariscal, J. C. Gutierrez-Vega, and S. Chavez-Cerda, "Production of high-order Bessel beams with a Mach-Zehnder interferometer," Appl. Optics 43,5060-5063 (2004).
[CrossRef]

Contemp. Phys.

D. McGloin and K. Dholakia, "Bessel beams: diffraction in a new light," Contemp. Phys. 46,15-28 (2005).
[CrossRef]

J. Opt. Soc. Am. A

Nature

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

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Figures (3)

Fig. 1.
Fig. 1.

Schematic structure of the experimental setup for generating the modulated Bessel-like beams with a phase imprinting: PPM - programmable phase modulator; PBS - polarizing beam splitter; λ/2 - half-wave plate; L - lenses; M - mirrors; BS - beam splitter; FF - Fourier filter; SBN -photorefractive crystal, Strontium Barium Niobate; CCD - camera. Inset on the left: Images of the Bessel lattice, localized state, and its interferogram.

Fig. 2.
Fig. 2.

Experimental images: (a) Intensity profile of the generated third-order Bessel lattice; (b) linear diffraction of the probe beam without the ilattice, the outer radius of the black circle shows the size of the input beam at the crystal front facet; (c-h) linear diffraction (top row) and nonlinear self-focusing (bottom row) of the input beam positioned (c,d) at the lattice center, (e,f) between two cites, and (g,h) at one site of the first lattice ring.

Fig. 3.
Fig. 3.

Results of numerical simulations with the anisotropic nonlocal model Eqs. (3)–(5): (a) Intensity of the azimuthally modulated Bessel beam and (b) corresponding induced refractive index profile ~ ∂ x φ; parameters are A = 2.5 and γ = 2. (c,d) Examples of the ring-like and single-site solitons corresponding to the experimental data shown in Figs. 2(d) and (h), respectively. The soliton with β = 0.13 and the peak intensity max|E|2 = 0.05 in (c) is close to the bifurcation point from the corresponding linear guided wave, while the soliton in (d) with β = 0.3 and max |E|2 = 0.97 is essentially in the nonlinear regime.

Equations (5)

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E ( r , φ ) = E 0 J n ( ρ w ) cos ( n φ ) ,
i z E + 2 E + γ x φ E = 0 ,
2 φ + φ In ( 1 + I ) = x In ( 1 + I ) .
I = E 2 + A 2 J 3 2 ( r ) cos 2 ( 3 φ )
β U + 2 U + γ x φ U = 0 .

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