Abstract

We demonstrate an approach to generate a class of pseudonondiffracting optical beams with the transverse shapes related to the superlattice structures. For constructing the superlattice waves, we consider a coherent superposition of two identical lattice waves with a specific relative angle in the azimuthal direction. We theoretically derive the general conditions of the relative angles for superlattice waves. In the experiment, a mask with multiple apertures which fulfill the conditions for superlattice structures is utilized to generate the pseudonondiffracting superlattice beams. With the analytical wave functions and experimental patterns, the pseudonondiffracting optical beams with a variety of structures can be generated systematically.

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  1. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A4(4), 651–654 (1987).
    [CrossRef]
  2. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58(15), 1499–1501 (1987).
    [CrossRef] [PubMed]
  3. V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature419(6903), 145–147 (2002).
    [CrossRef] [PubMed]
  4. D. McGloin, V. Garcés-Chávez, and K. Dholakia, “Interfering Bessel beams for optical micromanipulation,” Opt. Lett.28(8), 657–659 (2003).
    [CrossRef] [PubMed]
  5. J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun.197(4-6), 239–245 (2001).
    [CrossRef]
  6. Z. Ding, H. Ren, Y. Zhao, J. S. Nelson, and Z. Chen, “High-resolution optical coherence tomography over a large depth range with an axicon lens,” Opt. Lett.27(4), 243–245 (2002).
    [CrossRef] [PubMed]
  7. C. Yu, M. R. Wang, A. J. Varela, and B. Chen, “High-density non-diffracting beam array for optical interconnection,” Opt. Commun.177(1-6), 369–376 (2000).
    [CrossRef]
  8. Z. Bouchal, “Nondiffracting optical beams-physical properties, experiments, and applications,” Czech. J. Phys.53(7), 537–578 (2003).
    [CrossRef]
  9. M. Boguslawski, P. Rose, and C. Denz, “Increasing the structural variety of discrete nondiffracting wave fields,” Phys. Rev. A84(1), 013832 (2011).
    [CrossRef]
  10. M. Boguslawski, P. Rose, and C. Denz, “Nondiffracting kagome lattice,” Appl. Phys. Lett.98(6), 061111 (2011).
    [CrossRef]
  11. P. Rose, M. Boguslawski, and C. Denz, “Nonlinear lattice structures based on families of complex nondiffracting beams,” New J. Phys.14(3), 033018 (2012).
    [CrossRef]
  12. Y. F. Chen, H. C. Liang, Y. C. Lin, Y. S. Tzeng, K. W. Su, and K. F. Huang, “Generation of optical crystals and quasicrystal beams: Kaleidoscopic patterns and phase singularity,” Phys. Rev. A83(5), 053813 (2011).
    [CrossRef]
  13. A. Kudrolli, B. Pier, and J. P. Gollub, Physica, “Superlattice patterns in surface waves,” Physica D123(1-4), 99–111 (1998).
  14. M. Silber and M. R. E. Proctor, “Nonlinear Competition between Small and Large Hexagonal Patterns,” Phys. Rev. Lett.81(12), 2450–2453 (1998).
    [CrossRef]
  15. H. Arbell and J. Fineberg, “Spatial and Temporal Dynamics of Two Interacting Modesin Parametrically Driven Surface Waves,” Phys. Rev. Lett.81(20), 4384–4387 (1998).
    [CrossRef]
  16. H. J. Pi, S. Park, J. Lee, and K. J. Lee, “Superlattice, Rhombus, Square, And Hexagonal Standing Waves In Magnetically Driven Ferrofluid Surface,” Phys. Rev. Lett.84(23), 5316–5319 (2000).
    [CrossRef] [PubMed]
  17. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci.336(1605), 165–190 (1974).
    [CrossRef]
  18. M. S. Soskin and M. V. Vasnetsov, “Singular Optics,” Prog. Opt.42, 219–276 (2001).
    [CrossRef]

2012 (1)

P. Rose, M. Boguslawski, and C. Denz, “Nonlinear lattice structures based on families of complex nondiffracting beams,” New J. Phys.14(3), 033018 (2012).
[CrossRef]

2011 (3)

Y. F. Chen, H. C. Liang, Y. C. Lin, Y. S. Tzeng, K. W. Su, and K. F. Huang, “Generation of optical crystals and quasicrystal beams: Kaleidoscopic patterns and phase singularity,” Phys. Rev. A83(5), 053813 (2011).
[CrossRef]

M. Boguslawski, P. Rose, and C. Denz, “Increasing the structural variety of discrete nondiffracting wave fields,” Phys. Rev. A84(1), 013832 (2011).
[CrossRef]

M. Boguslawski, P. Rose, and C. Denz, “Nondiffracting kagome lattice,” Appl. Phys. Lett.98(6), 061111 (2011).
[CrossRef]

2003 (2)

D. McGloin, V. Garcés-Chávez, and K. Dholakia, “Interfering Bessel beams for optical micromanipulation,” Opt. Lett.28(8), 657–659 (2003).
[CrossRef] [PubMed]

Z. Bouchal, “Nondiffracting optical beams-physical properties, experiments, and applications,” Czech. J. Phys.53(7), 537–578 (2003).
[CrossRef]

2002 (2)

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature419(6903), 145–147 (2002).
[CrossRef] [PubMed]

Z. Ding, H. Ren, Y. Zhao, J. S. Nelson, and Z. Chen, “High-resolution optical coherence tomography over a large depth range with an axicon lens,” Opt. Lett.27(4), 243–245 (2002).
[CrossRef] [PubMed]

2001 (2)

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun.197(4-6), 239–245 (2001).
[CrossRef]

M. S. Soskin and M. V. Vasnetsov, “Singular Optics,” Prog. Opt.42, 219–276 (2001).
[CrossRef]

2000 (2)

C. Yu, M. R. Wang, A. J. Varela, and B. Chen, “High-density non-diffracting beam array for optical interconnection,” Opt. Commun.177(1-6), 369–376 (2000).
[CrossRef]

H. J. Pi, S. Park, J. Lee, and K. J. Lee, “Superlattice, Rhombus, Square, And Hexagonal Standing Waves In Magnetically Driven Ferrofluid Surface,” Phys. Rev. Lett.84(23), 5316–5319 (2000).
[CrossRef] [PubMed]

1998 (3)

A. Kudrolli, B. Pier, and J. P. Gollub, Physica, “Superlattice patterns in surface waves,” Physica D123(1-4), 99–111 (1998).

M. Silber and M. R. E. Proctor, “Nonlinear Competition between Small and Large Hexagonal Patterns,” Phys. Rev. Lett.81(12), 2450–2453 (1998).
[CrossRef]

H. Arbell and J. Fineberg, “Spatial and Temporal Dynamics of Two Interacting Modesin Parametrically Driven Surface Waves,” Phys. Rev. Lett.81(20), 4384–4387 (1998).
[CrossRef]

1987 (2)

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A4(4), 651–654 (1987).
[CrossRef]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

1974 (1)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci.336(1605), 165–190 (1974).
[CrossRef]

Arbell, H.

H. Arbell and J. Fineberg, “Spatial and Temporal Dynamics of Two Interacting Modesin Parametrically Driven Surface Waves,” Phys. Rev. Lett.81(20), 4384–4387 (1998).
[CrossRef]

Arlt, J.

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun.197(4-6), 239–245 (2001).
[CrossRef]

Berry, M. V.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci.336(1605), 165–190 (1974).
[CrossRef]

Boguslawski, M.

P. Rose, M. Boguslawski, and C. Denz, “Nonlinear lattice structures based on families of complex nondiffracting beams,” New J. Phys.14(3), 033018 (2012).
[CrossRef]

M. Boguslawski, P. Rose, and C. Denz, “Increasing the structural variety of discrete nondiffracting wave fields,” Phys. Rev. A84(1), 013832 (2011).
[CrossRef]

M. Boguslawski, P. Rose, and C. Denz, “Nondiffracting kagome lattice,” Appl. Phys. Lett.98(6), 061111 (2011).
[CrossRef]

Bouchal, Z.

Z. Bouchal, “Nondiffracting optical beams-physical properties, experiments, and applications,” Czech. J. Phys.53(7), 537–578 (2003).
[CrossRef]

Chen, B.

C. Yu, M. R. Wang, A. J. Varela, and B. Chen, “High-density non-diffracting beam array for optical interconnection,” Opt. Commun.177(1-6), 369–376 (2000).
[CrossRef]

Chen, Y. F.

Y. F. Chen, H. C. Liang, Y. C. Lin, Y. S. Tzeng, K. W. Su, and K. F. Huang, “Generation of optical crystals and quasicrystal beams: Kaleidoscopic patterns and phase singularity,” Phys. Rev. A83(5), 053813 (2011).
[CrossRef]

Chen, Z.

Denz, C.

P. Rose, M. Boguslawski, and C. Denz, “Nonlinear lattice structures based on families of complex nondiffracting beams,” New J. Phys.14(3), 033018 (2012).
[CrossRef]

M. Boguslawski, P. Rose, and C. Denz, “Nondiffracting kagome lattice,” Appl. Phys. Lett.98(6), 061111 (2011).
[CrossRef]

M. Boguslawski, P. Rose, and C. Denz, “Increasing the structural variety of discrete nondiffracting wave fields,” Phys. Rev. A84(1), 013832 (2011).
[CrossRef]

Dholakia, K.

D. McGloin, V. Garcés-Chávez, and K. Dholakia, “Interfering Bessel beams for optical micromanipulation,” Opt. Lett.28(8), 657–659 (2003).
[CrossRef] [PubMed]

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature419(6903), 145–147 (2002).
[CrossRef] [PubMed]

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun.197(4-6), 239–245 (2001).
[CrossRef]

Ding, Z.

Durnin, J.

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A4(4), 651–654 (1987).
[CrossRef]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

Fineberg, J.

H. Arbell and J. Fineberg, “Spatial and Temporal Dynamics of Two Interacting Modesin Parametrically Driven Surface Waves,” Phys. Rev. Lett.81(20), 4384–4387 (1998).
[CrossRef]

Garces-Chavez, V.

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun.197(4-6), 239–245 (2001).
[CrossRef]

Garcés-Chávez, V.

D. McGloin, V. Garcés-Chávez, and K. Dholakia, “Interfering Bessel beams for optical micromanipulation,” Opt. Lett.28(8), 657–659 (2003).
[CrossRef] [PubMed]

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature419(6903), 145–147 (2002).
[CrossRef] [PubMed]

Gollub, J. P.

A. Kudrolli, B. Pier, and J. P. Gollub, Physica, “Superlattice patterns in surface waves,” Physica D123(1-4), 99–111 (1998).

Huang, K. F.

Y. F. Chen, H. C. Liang, Y. C. Lin, Y. S. Tzeng, K. W. Su, and K. F. Huang, “Generation of optical crystals and quasicrystal beams: Kaleidoscopic patterns and phase singularity,” Phys. Rev. A83(5), 053813 (2011).
[CrossRef]

Kudrolli, A.

A. Kudrolli, B. Pier, and J. P. Gollub, Physica, “Superlattice patterns in surface waves,” Physica D123(1-4), 99–111 (1998).

Lee, J.

H. J. Pi, S. Park, J. Lee, and K. J. Lee, “Superlattice, Rhombus, Square, And Hexagonal Standing Waves In Magnetically Driven Ferrofluid Surface,” Phys. Rev. Lett.84(23), 5316–5319 (2000).
[CrossRef] [PubMed]

Lee, K. J.

H. J. Pi, S. Park, J. Lee, and K. J. Lee, “Superlattice, Rhombus, Square, And Hexagonal Standing Waves In Magnetically Driven Ferrofluid Surface,” Phys. Rev. Lett.84(23), 5316–5319 (2000).
[CrossRef] [PubMed]

Liang, H. C.

Y. F. Chen, H. C. Liang, Y. C. Lin, Y. S. Tzeng, K. W. Su, and K. F. Huang, “Generation of optical crystals and quasicrystal beams: Kaleidoscopic patterns and phase singularity,” Phys. Rev. A83(5), 053813 (2011).
[CrossRef]

Lin, Y. C.

Y. F. Chen, H. C. Liang, Y. C. Lin, Y. S. Tzeng, K. W. Su, and K. F. Huang, “Generation of optical crystals and quasicrystal beams: Kaleidoscopic patterns and phase singularity,” Phys. Rev. A83(5), 053813 (2011).
[CrossRef]

McGloin, D.

D. McGloin, V. Garcés-Chávez, and K. Dholakia, “Interfering Bessel beams for optical micromanipulation,” Opt. Lett.28(8), 657–659 (2003).
[CrossRef] [PubMed]

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature419(6903), 145–147 (2002).
[CrossRef] [PubMed]

Melville, H.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature419(6903), 145–147 (2002).
[CrossRef] [PubMed]

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

Nelson, J. S.

Nye, J. F.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci.336(1605), 165–190 (1974).
[CrossRef]

Park, S.

H. J. Pi, S. Park, J. Lee, and K. J. Lee, “Superlattice, Rhombus, Square, And Hexagonal Standing Waves In Magnetically Driven Ferrofluid Surface,” Phys. Rev. Lett.84(23), 5316–5319 (2000).
[CrossRef] [PubMed]

Pi, H. J.

H. J. Pi, S. Park, J. Lee, and K. J. Lee, “Superlattice, Rhombus, Square, And Hexagonal Standing Waves In Magnetically Driven Ferrofluid Surface,” Phys. Rev. Lett.84(23), 5316–5319 (2000).
[CrossRef] [PubMed]

Pier, B.

A. Kudrolli, B. Pier, and J. P. Gollub, Physica, “Superlattice patterns in surface waves,” Physica D123(1-4), 99–111 (1998).

Proctor, M. R. E.

M. Silber and M. R. E. Proctor, “Nonlinear Competition between Small and Large Hexagonal Patterns,” Phys. Rev. Lett.81(12), 2450–2453 (1998).
[CrossRef]

Ren, H.

Rose, P.

P. Rose, M. Boguslawski, and C. Denz, “Nonlinear lattice structures based on families of complex nondiffracting beams,” New J. Phys.14(3), 033018 (2012).
[CrossRef]

M. Boguslawski, P. Rose, and C. Denz, “Nondiffracting kagome lattice,” Appl. Phys. Lett.98(6), 061111 (2011).
[CrossRef]

M. Boguslawski, P. Rose, and C. Denz, “Increasing the structural variety of discrete nondiffracting wave fields,” Phys. Rev. A84(1), 013832 (2011).
[CrossRef]

Sibbett, W.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature419(6903), 145–147 (2002).
[CrossRef] [PubMed]

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun.197(4-6), 239–245 (2001).
[CrossRef]

Silber, M.

M. Silber and M. R. E. Proctor, “Nonlinear Competition between Small and Large Hexagonal Patterns,” Phys. Rev. Lett.81(12), 2450–2453 (1998).
[CrossRef]

Soskin, M. S.

M. S. Soskin and M. V. Vasnetsov, “Singular Optics,” Prog. Opt.42, 219–276 (2001).
[CrossRef]

Su, K. W.

Y. F. Chen, H. C. Liang, Y. C. Lin, Y. S. Tzeng, K. W. Su, and K. F. Huang, “Generation of optical crystals and quasicrystal beams: Kaleidoscopic patterns and phase singularity,” Phys. Rev. A83(5), 053813 (2011).
[CrossRef]

Tzeng, Y. S.

Y. F. Chen, H. C. Liang, Y. C. Lin, Y. S. Tzeng, K. W. Su, and K. F. Huang, “Generation of optical crystals and quasicrystal beams: Kaleidoscopic patterns and phase singularity,” Phys. Rev. A83(5), 053813 (2011).
[CrossRef]

Varela, A. J.

C. Yu, M. R. Wang, A. J. Varela, and B. Chen, “High-density non-diffracting beam array for optical interconnection,” Opt. Commun.177(1-6), 369–376 (2000).
[CrossRef]

Vasnetsov, M. V.

M. S. Soskin and M. V. Vasnetsov, “Singular Optics,” Prog. Opt.42, 219–276 (2001).
[CrossRef]

Wang, M. R.

C. Yu, M. R. Wang, A. J. Varela, and B. Chen, “High-density non-diffracting beam array for optical interconnection,” Opt. Commun.177(1-6), 369–376 (2000).
[CrossRef]

Yu, C.

C. Yu, M. R. Wang, A. J. Varela, and B. Chen, “High-density non-diffracting beam array for optical interconnection,” Opt. Commun.177(1-6), 369–376 (2000).
[CrossRef]

Zhao, Y.

Appl. Phys. Lett. (1)

M. Boguslawski, P. Rose, and C. Denz, “Nondiffracting kagome lattice,” Appl. Phys. Lett.98(6), 061111 (2011).
[CrossRef]

Czech. J. Phys. (1)

Z. Bouchal, “Nondiffracting optical beams-physical properties, experiments, and applications,” Czech. J. Phys.53(7), 537–578 (2003).
[CrossRef]

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

Nature (1)

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature419(6903), 145–147 (2002).
[CrossRef] [PubMed]

New J. Phys. (1)

P. Rose, M. Boguslawski, and C. Denz, “Nonlinear lattice structures based on families of complex nondiffracting beams,” New J. Phys.14(3), 033018 (2012).
[CrossRef]

Opt. Commun. (2)

J. Arlt, V. Garces-Chavez, W. Sibbett, and K. Dholakia, “Optical micromanipulation using a Bessel light beam,” Opt. Commun.197(4-6), 239–245 (2001).
[CrossRef]

C. Yu, M. R. Wang, A. J. Varela, and B. Chen, “High-density non-diffracting beam array for optical interconnection,” Opt. Commun.177(1-6), 369–376 (2000).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (2)

M. Boguslawski, P. Rose, and C. Denz, “Increasing the structural variety of discrete nondiffracting wave fields,” Phys. Rev. A84(1), 013832 (2011).
[CrossRef]

Y. F. Chen, H. C. Liang, Y. C. Lin, Y. S. Tzeng, K. W. Su, and K. F. Huang, “Generation of optical crystals and quasicrystal beams: Kaleidoscopic patterns and phase singularity,” Phys. Rev. A83(5), 053813 (2011).
[CrossRef]

Phys. Rev. Lett. (4)

M. Silber and M. R. E. Proctor, “Nonlinear Competition between Small and Large Hexagonal Patterns,” Phys. Rev. Lett.81(12), 2450–2453 (1998).
[CrossRef]

H. Arbell and J. Fineberg, “Spatial and Temporal Dynamics of Two Interacting Modesin Parametrically Driven Surface Waves,” Phys. Rev. Lett.81(20), 4384–4387 (1998).
[CrossRef]

H. J. Pi, S. Park, J. Lee, and K. J. Lee, “Superlattice, Rhombus, Square, And Hexagonal Standing Waves In Magnetically Driven Ferrofluid Surface,” Phys. Rev. Lett.84(23), 5316–5319 (2000).
[CrossRef] [PubMed]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

Physica D (1)

A. Kudrolli, B. Pier, and J. P. Gollub, Physica, “Superlattice patterns in surface waves,” Physica D123(1-4), 99–111 (1998).

Proc. R. Soc. Lond. A Math. Phys. Sci. (1)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci.336(1605), 165–190 (1974).
[CrossRef]

Prog. Opt. (1)

M. S. Soskin and M. V. Vasnetsov, “Singular Optics,” Prog. Opt.42, 219–276 (2001).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

The schematic diagrams of the wave vectors of the superposed waves Ψ q ( ρ,ϕ; K s , Δ q ) with (a) q = 3, (b) q = 4, and (c) q = 6.

Fig. 2
Fig. 2

Numerical patterns for the intensity of the superlattice waves | Ψ q ( ρ,ϕ; K s , Δ q ) | 2 with q = 4.

Fig. 3
Fig. 3

Numerical patterns for the intensity of the superlattice waves | Ψ q ( ρ,ϕ; K s , Δ q ) | 2 with (a)-(c) q = 3, and (d)-(f) q = 6.

Fig. 4
Fig. 4

(a)-(c) Numerically patterns for the output intensity profiles of pseudonondiffracting optical superlattice patterns with different radii of apertures.

Fig. 5
Fig. 5

Experimental setup for generating pseudonondiffracting optical beams with superlattice structures.

Fig. 6
Fig. 6

Experimental patterns observed for pseudonondiffracting optical superlattice beams with q = 4 under the optimal alignment.

Fig. 7
Fig. 7

Experimental patterns observed for pseudonondiffracting optical superlattice beams with (a)-(c) q = 3, and (d)-(f) q = 6.

Fig. 8
Fig. 8

(a)-(c) Contour plots of phase fields Θ q ( ρ,ϕ; K s , Δ q ) for the boxed regions shown in Fig. 3(a)-3(c), respectively.

Equations (18)

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

ψ q ( ρ,ϕ; K s )= 1 q s=0 q1 e i K s ρ ,
Ψ q ( ρ,ϕ; K s , Δ q )= ψ q ( ρ,ϕ; K s )+ ψ q ( ρ,ϕ; K s ),
{ K s = n s b 1 + m s b 2 K s = n s b 1 + m s b 2 | K s |=b n s 2 + m s 2 =b n s 2 + m s 2 =| K s | ,
{ n s = m s m s = n s .
K s = m s b 1 + n s b 2 .
Δ q = cos 1 ( K 0 K 0 | K 0 || K 0 | )= cos 1 ( 2 n 0 m 0 n 0 2 + m 0 2 ), for q=4.
b=| b 1 |=| b 2 |= K n 0 2 + m 0 2 , for q=4.
Δ q = cos 1 ( n 0 2 m 0 2 +4 n 0 m 0 2( n 0 2 + m 0 2 n 0 m 0 ) ), for q=3 or 6,
b= K n 0 2 + m 0 2 n 0 m 0 , for q=3 or 6.
E o ( ρ,ϕ,z )= i e i 2π λ ( f+z ) λf E i ( ρ , ϕ ) e i 2π λ ρ 2 2f ( 1 z f ) e i 2πρ ρ λf cos( ϕ ϕ ) ρ d ρ d ϕ ,
E i ( ρ , ϕ )=δ( ρ R ),
E i ( ρ , ϕ )= 1 q δ( ρ R ) s=0 q1 δ( ϕ 2πs q ) ,
E i ( ρ , ϕ )= δ( ρ R ) q s=0 q1 [ δ( ϕ 2πs q )+δ( ϕ 2πs q Δ q ) ] ,
E i ( ρ , ϕ )= ( 2 π a 2 ) 1 2 s=0 q1 { exp[ ρ 2 + R 2 2 ρ Rcos( ϕ ( 2πs /q ) ) a 2 ] +exp[ ρ 2 + R 2 2 ρ Rcos( ϕ ( 2πs /q ) Δ q ) a 2 ] },
E o ( ρ,ϕ;f )= ( 2 π a 2 ) 1 2 e i 2π λ ( 2f ) e R 2 a 2 iλf × e ρ 2 a 2 e 2R ρ a 2 cos( ϕ ϕ ) e i 2πρ ρ λf cos( ϕ ϕ ) ρ d ρ d ϕ .
e α x 2 2βx dx= π α e β 2 /α .
E o ( ρ,ϕ;f )= π a 2 e i 2π λ ( 2f ) e π a 2 λf ρ 2 iλf s=0 q1 [ e i 2πR λf ρcos( ϕ 2πs q ) + e i 2πR λf ρcos( ϕ 2πs q Δ q ) ] .
Θ q ( ρ,ϕ; K s , Δ q )= tan 1 { Im[ Ψ q ( ρ,ϕ; K s , Δ q ) ] / Re[ Ψ q ( ρ,ϕ; K s , Δ q ) ] },

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