Abstract

Due to their unique ability to maintain an intensity distribution upon propagation, non-diffracting light fields are used extensively in various areas of science, including optical tweezers, nonlinear optics and quantum optics, in applications where complex transverse field distributions are required. However, the number and type of rigorously non-diffracting beams is severely limited because their symmetry is dictated by one of the coordinate system where the Helmholtz equation governing beam propagation is separable. Here, we demonstrate a powerful technique that allows the generation of a rich variety of quasi-non-diffracting optical beams featuring nearly arbitrary intensity distributions in the transverse plane. These can be readily engineered via modifications of the angular spectrum of the beam in order to meet the requirements of particular applications. Such beams are not rigorously non-diffracting but they maintain their shape over large distances, which may be tuned by varying the width of the angular spectrum. We report the generation of unique spiral patterns and patterns involving arbitrary combinations of truncated harmonic, Bessel, Mathieu, or parabolic beams occupying different spatial domains. Optical trapping experiments illustrate the opto-mechanical properties of such beams.

© 2013 OSA

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  1. M. Mazilu, J. D. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light beats the spread: “non-diffracting” beams,” Laser Photonics Rev.4(4), 529–547 (2010).
    [CrossRef]
  2. J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “Efficiency of second-harmonic generation with Bessel beams,” Phys. Rev. A60(3), 2438–2441 (1999).
    [CrossRef]
  3. J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett.90(2), 023902 (2003).
    [CrossRef] [PubMed]
  4. J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature422(6928), 147–150 (2003).
    [CrossRef] [PubMed]
  5. D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, “Spatial solitons in optically induced gratings,” Opt. Lett.28(9), 710–712 (2003).
    [CrossRef] [PubMed]
  6. H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, “Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices,” Phys. Rev. Lett.92(12), 123902 (2004).
    [CrossRef] [PubMed]
  7. F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep.463(1-3), 1–126 (2008).
    [CrossRef]
  8. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton shape and mobility control in optical lattices,” Prog. Opt.52, 63–148 (2009).
    [CrossRef]
  9. O. Morsch and M. Oberthaler, “Dynamics of Bose-Einstein condensates in optical lattices,” Rev. Mod. Phys.78(1), 179–215 (2006).
    [CrossRef]
  10. L. Pitaevskii and S. Stringari, Bose-Einstein Condensation (Oxford University, 2003).
  11. R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Phys. Rev. Lett.86(22), 5188–5191 (2001).
    [CrossRef] [PubMed]
  12. D. Schrader, I. Dotsenko, M. Khudaverdyan, Y. Miroshnychenko, A. Rauschenbeutel, and D. Meschede, “Neutral atom quantum register,” Phys. Rev. Lett.93(15), 150501 (2004).
    [CrossRef] [PubMed]
  13. I. Bloch, “Ultracold quantum gases in optical lattices,” Nat. Phys.1(1), 23–30 (2005).
    [CrossRef]
  14. S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of ultracold atomic Fermi gases,” Rev. Mod. Phys.80(4), 1215–1274 (2008).
    [CrossRef]
  15. K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics5(6), 335–342 (2011).
    [CrossRef]
  16. 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]
  17. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics2(11), 675–678 (2008).
    [CrossRef]
  18. J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip9(10), 1334–1336 (2009).
    [CrossRef] [PubMed]
  19. J. Baumgartl, T. Cizmár, M. Mazilu, V. C. Chan, A. E. Carruthers, B. A. Capron, W. McNeely, E. M. Wright, and K. Dholakia, “Optical path clearing and enhanced transmission through colloidal suspensions,” Opt. Express18(16), 17130–17140 (2010).
    [CrossRef] [PubMed]
  20. M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature394(6691), 348–350 (1998).
    [CrossRef]
  21. J. Leach, G. Sinclair, P. Jordan, J. Courtial, M. Padgett, J. Cooper, and Z. Laczik, “3D manipulation of particles into crystal structures using holographic optical tweezers,” Opt. Express12(1), 220–226 (2004).
    [CrossRef] [PubMed]
  22. P. T. Korda and D. G. Grier, “Annealing thin colloidal crystals with optical gradient forces,” J. Chem. Phys.114(17), 7570–7573 (2001).
    [CrossRef]
  23. A. Ehrlicher, T. Betz, B. Stuhrmann, D. Koch, V. Milner, M. G. Raizen, and J. Kas, “Guiding neuronal growth with light,” Proc. Natl. Acad. Sci. U.S.A.99(25), 16024–16028 (2002).
    [CrossRef] [PubMed]
  24. D. J. Carnegie, D. J. Stevenson, M. Mazilu, F. Gunn-Moore, and K. Dholakia, “Guided neuronal growth using optical line traps,” Opt. Express16(14), 10507–10517 (2008).
    [CrossRef] [PubMed]
  25. D. J. Carnegie, T. Cizmár, J. Baumgartl, F. J. Gunn-Moore, and K. Dholakia, “Automated laser guidance of neuronal growth cones using a spatial light modulator,” J. Biophotonics2(11), 682–692 (2009).
    [CrossRef] [PubMed]
  26. C. Bustamante, J. C. Macosko, and G. J. L. Wuite, “Grabbing the cat by the tail: manipulating molecules one by one,” Nat. Rev. Mol. Cell Biol.1(2), 130–136 (2000).
    [CrossRef] [PubMed]
  27. T. Nishizaka, H. Miyata, H. Yoshikawa, S. Ishiwata, and K. Kinosita., “Unbinding force of a single motor molecule of muscle measured using optical tweezers,” Nature377(6546), 251–254 (1995).
    [CrossRef] [PubMed]
  28. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-Free Beams,” Phys. Rev. Lett.58(15), 1499–1501 (1987).
    [CrossRef] [PubMed]
  29. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A4(4), 651 (1987).
    [CrossRef]
  30. E. G. Kalnins and W. Miller, “W. Lie theory and separation of variables. 9. Orthogonal R separable coordinate systems for the wave equation ψtt−Δ2ψ=0,” J. Math. Phys.17, 330 (1976).
  31. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett.25(20), 1493–1495 (2000).
    [CrossRef] [PubMed]
  32. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramirez, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun.195(1-4), 35–40 (2001).
    [CrossRef]
  33. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett.29(1), 44–46 (2004).
    [CrossRef] [PubMed]
  34. C. López-Mariscal, M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Observation of parabolic nondiffracting optical fields,” Opt. Express13(7), 2364–2369 (2005).
    [CrossRef] [PubMed]
  35. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys.47(3), 264 (1979).
    [CrossRef]
  36. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett.99(21), 213901 (2007).
    [CrossRef] [PubMed]
  37. S. López-Aguayo, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Method to generate complex quasinondiffracting optical lattices,” Phys. Rev. Lett.105(1), 013902 (2010).
    [CrossRef] [PubMed]
  38. Y. V. Kartashov, S. López-Aguayo, V. A. Vysloukh, and L. Torner, “Stripe-like quasi-nondiffracting optical lattices,” Opt. Express19(10), 9505–9511 (2011).
    [CrossRef] [PubMed]
  39. C. López-Mariscal and K. Helmerson, “Shaped nondiffracting beams,” Opt. Lett.35(8), 1215–1217 (2010).
    [CrossRef] [PubMed]
  40. J. C. Gutiérrez-Vega and C. Lopez-Mariscal, “Nondiffracting vortex beams with continuous orbital angular momentum order dependence,” J. Opt. A, Pure Appl. Opt.10(1), 015009 (2008).
    [CrossRef]
  41. D. M. Cottrell, J. M. Craven, and J. A. Davis, “Nondiffracting random intensity patterns,” Opt. Lett.32(3), 298–300 (2007).
    [CrossRef] [PubMed]
  42. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt.21(15), 2758–2769 (1982).
    [CrossRef] [PubMed]
  43. Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, “Gerchberg-Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett.21(12), 842–844 (1996).
    [CrossRef] [PubMed]
  44. M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express16(10), 7264–7278 (2008).
    [CrossRef] [PubMed]
  45. J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. A22(2), 289–298 (2005).
    [CrossRef] [PubMed]
  46. C. López-Mariscal, M. A. Bandres, and J. C. Gutierrez-Vega, “Observation of the experimental propagation properties of Helmholtz-Gauss beams,” Opt. Eng.45(6), 068001 (2006).
    [CrossRef]
  47. R. Simon and N. Mukunda, “Iwasawa decomposition in first-order optics: universal treatment of shape-invariant propagation for coherent and partially coherent beams,” J. Opt. Soc. Am. A15(8), 2146–2155 (1998).
    [CrossRef]
  48. R. Piestun, Y. Y. Schechner, and J. Shamir, “Propagation-invariant wave fields with finite energy,” J. Opt. Soc. Am. A17(2), 294–303 (2000).
    [CrossRef] [PubMed]
  49. Z. K. Bouchal, “Controlled spatial shaping of nondiffracting patterns and arrays,” Opt. Lett.27(16), 1376–1378 (2002).
    [CrossRef] [PubMed]
  50. J. D. Ring, C. J. Howls, and M. R. Dennis, “Incomplete Airy beams: finite energy from a sharp spectral cutoff,” Opt. Lett.38(10), 1639–1641 (2013).
    [CrossRef] [PubMed]
  51. J. Durnin, J. J. Miceli, and J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett.13(2), 79–80 (1988).
    [CrossRef] [PubMed]
  52. R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syrett, “Interboard optical data distribution by Bessel beam shadowing,” Opt. Commun.122(4-6), 169–177 (1996).
    [CrossRef]
  53. Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun.151(4-6), 207–211 (1998).
    [CrossRef]
  54. S. Vyas, Y. Kozawa, and S. Sato, “Self-healing of tightly focused scalar and vector Bessel-Gauss beams at the focal plane,” J. Opt. Soc. Am. A28(5), 837–843 (2011).
    [CrossRef] [PubMed]
  55. X. D. He, P. Xu, J. Wang, and M. S. Zhan, “Rotating single atoms in a ring lattice generated by a spatial light modulator,” Opt. Express17(23), 21007–21014 (2009).
    [CrossRef] [PubMed]
  56. S. B. Purnapatra, S. Bera, and P. P. Mondal, “Spatial filter based Bessel-like beam for improved penetration depth imaging in fluorescence microscopy,” Sci. Rep.2, 692 (2012).
    [CrossRef] [PubMed]
  57. J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature467(7313), 301–304 (2010).
    [CrossRef] [PubMed]
  58. M. Uchida and A. Tonomura, “Generation of electron beams carrying orbital angular momentum,” Nature464(7289), 737–739 (2010).
    [CrossRef] [PubMed]
  59. N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, “Generation of electron Airy beams,” Nature494(7437), 331–335 (2013).
    [CrossRef] [PubMed]

2013 (2)

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, “Generation of electron Airy beams,” Nature494(7437), 331–335 (2013).
[CrossRef] [PubMed]

J. D. Ring, C. J. Howls, and M. R. Dennis, “Incomplete Airy beams: finite energy from a sharp spectral cutoff,” Opt. Lett.38(10), 1639–1641 (2013).
[CrossRef] [PubMed]

2012 (1)

S. B. Purnapatra, S. Bera, and P. P. Mondal, “Spatial filter based Bessel-like beam for improved penetration depth imaging in fluorescence microscopy,” Sci. Rep.2, 692 (2012).
[CrossRef] [PubMed]

2011 (3)

2010 (6)

M. Mazilu, J. D. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light beats the spread: “non-diffracting” beams,” Laser Photonics Rev.4(4), 529–547 (2010).
[CrossRef]

S. López-Aguayo, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Method to generate complex quasinondiffracting optical lattices,” Phys. Rev. Lett.105(1), 013902 (2010).
[CrossRef] [PubMed]

C. López-Mariscal and K. Helmerson, “Shaped nondiffracting beams,” Opt. Lett.35(8), 1215–1217 (2010).
[CrossRef] [PubMed]

J. Baumgartl, T. Cizmár, M. Mazilu, V. C. Chan, A. E. Carruthers, B. A. Capron, W. McNeely, E. M. Wright, and K. Dholakia, “Optical path clearing and enhanced transmission through colloidal suspensions,” Opt. Express18(16), 17130–17140 (2010).
[CrossRef] [PubMed]

J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature467(7313), 301–304 (2010).
[CrossRef] [PubMed]

M. Uchida and A. Tonomura, “Generation of electron beams carrying orbital angular momentum,” Nature464(7289), 737–739 (2010).
[CrossRef] [PubMed]

2009 (4)

X. D. He, P. Xu, J. Wang, and M. S. Zhan, “Rotating single atoms in a ring lattice generated by a spatial light modulator,” Opt. Express17(23), 21007–21014 (2009).
[CrossRef] [PubMed]

D. J. Carnegie, T. Cizmár, J. Baumgartl, F. J. Gunn-Moore, and K. Dholakia, “Automated laser guidance of neuronal growth cones using a spatial light modulator,” J. Biophotonics2(11), 682–692 (2009).
[CrossRef] [PubMed]

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip9(10), 1334–1336 (2009).
[CrossRef] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton shape and mobility control in optical lattices,” Prog. Opt.52, 63–148 (2009).
[CrossRef]

2008 (6)

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep.463(1-3), 1–126 (2008).
[CrossRef]

S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of ultracold atomic Fermi gases,” Rev. Mod. Phys.80(4), 1215–1274 (2008).
[CrossRef]

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics2(11), 675–678 (2008).
[CrossRef]

J. C. Gutiérrez-Vega and C. Lopez-Mariscal, “Nondiffracting vortex beams with continuous orbital angular momentum order dependence,” J. Opt. A, Pure Appl. Opt.10(1), 015009 (2008).
[CrossRef]

M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express16(10), 7264–7278 (2008).
[CrossRef] [PubMed]

D. J. Carnegie, D. J. Stevenson, M. Mazilu, F. Gunn-Moore, and K. Dholakia, “Guided neuronal growth using optical line traps,” Opt. Express16(14), 10507–10517 (2008).
[CrossRef] [PubMed]

2007 (2)

D. M. Cottrell, J. M. Craven, and J. A. Davis, “Nondiffracting random intensity patterns,” Opt. Lett.32(3), 298–300 (2007).
[CrossRef] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett.99(21), 213901 (2007).
[CrossRef] [PubMed]

2006 (2)

C. López-Mariscal, M. A. Bandres, and J. C. Gutierrez-Vega, “Observation of the experimental propagation properties of Helmholtz-Gauss beams,” Opt. Eng.45(6), 068001 (2006).
[CrossRef]

O. Morsch and M. Oberthaler, “Dynamics of Bose-Einstein condensates in optical lattices,” Rev. Mod. Phys.78(1), 179–215 (2006).
[CrossRef]

2005 (3)

2004 (4)

M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett.29(1), 44–46 (2004).
[CrossRef] [PubMed]

J. Leach, G. Sinclair, P. Jordan, J. Courtial, M. Padgett, J. Cooper, and Z. Laczik, “3D manipulation of particles into crystal structures using holographic optical tweezers,” Opt. Express12(1), 220–226 (2004).
[CrossRef] [PubMed]

D. Schrader, I. Dotsenko, M. Khudaverdyan, Y. Miroshnychenko, A. Rauschenbeutel, and D. Meschede, “Neutral atom quantum register,” Phys. Rev. Lett.93(15), 150501 (2004).
[CrossRef] [PubMed]

H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, “Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices,” Phys. Rev. Lett.92(12), 123902 (2004).
[CrossRef] [PubMed]

2003 (3)

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett.90(2), 023902 (2003).
[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,” Nature422(6928), 147–150 (2003).
[CrossRef] [PubMed]

D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, “Spatial solitons in optically induced gratings,” Opt. Lett.28(9), 710–712 (2003).
[CrossRef] [PubMed]

2002 (3)

Z. K. Bouchal, “Controlled spatial shaping of nondiffracting patterns and arrays,” Opt. Lett.27(16), 1376–1378 (2002).
[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]

A. Ehrlicher, T. Betz, B. Stuhrmann, D. Koch, V. Milner, M. G. Raizen, and J. Kas, “Guiding neuronal growth with light,” Proc. Natl. Acad. Sci. U.S.A.99(25), 16024–16028 (2002).
[CrossRef] [PubMed]

2001 (3)

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramirez, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun.195(1-4), 35–40 (2001).
[CrossRef]

R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Phys. Rev. Lett.86(22), 5188–5191 (2001).
[CrossRef] [PubMed]

P. T. Korda and D. G. Grier, “Annealing thin colloidal crystals with optical gradient forces,” J. Chem. Phys.114(17), 7570–7573 (2001).
[CrossRef]

2000 (3)

1999 (1)

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “Efficiency of second-harmonic generation with Bessel beams,” Phys. Rev. A60(3), 2438–2441 (1999).
[CrossRef]

1998 (3)

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature394(6691), 348–350 (1998).
[CrossRef]

R. Simon and N. Mukunda, “Iwasawa decomposition in first-order optics: universal treatment of shape-invariant propagation for coherent and partially coherent beams,” J. Opt. Soc. Am. A15(8), 2146–2155 (1998).
[CrossRef]

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun.151(4-6), 207–211 (1998).
[CrossRef]

1996 (2)

Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, “Gerchberg-Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett.21(12), 842–844 (1996).
[CrossRef] [PubMed]

R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syrett, “Interboard optical data distribution by Bessel beam shadowing,” Opt. Commun.122(4-6), 169–177 (1996).
[CrossRef]

1995 (1)

T. Nishizaka, H. Miyata, H. Yoshikawa, S. Ishiwata, and K. Kinosita., “Unbinding force of a single motor molecule of muscle measured using optical tweezers,” Nature377(6546), 251–254 (1995).
[CrossRef] [PubMed]

1988 (1)

1987 (2)

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

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

1982 (1)

1979 (1)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys.47(3), 264 (1979).
[CrossRef]

1976 (1)

E. G. Kalnins and W. Miller, “W. Lie theory and separation of variables. 9. Orthogonal R separable coordinate systems for the wave equation ψtt−Δ2ψ=0,” J. Math. Phys.17, 330 (1976).

Allen, L.

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “Efficiency of second-harmonic generation with Bessel beams,” Phys. Rev. A60(3), 2438–2441 (1999).
[CrossRef]

Arie, A.

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, “Generation of electron Airy beams,” Nature494(7437), 331–335 (2013).
[CrossRef] [PubMed]

Arlt, J.

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “Efficiency of second-harmonic generation with Bessel beams,” Phys. Rev. A60(3), 2438–2441 (1999).
[CrossRef]

Assanto, G.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep.463(1-3), 1–126 (2008).
[CrossRef]

Balazs, N. L.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys.47(3), 264 (1979).
[CrossRef]

Bandres, M. A.

Baumgartl, J.

J. Baumgartl, T. Cizmár, M. Mazilu, V. C. Chan, A. E. Carruthers, B. A. Capron, W. McNeely, E. M. Wright, and K. Dholakia, “Optical path clearing and enhanced transmission through colloidal suspensions,” Opt. Express18(16), 17130–17140 (2010).
[CrossRef] [PubMed]

D. J. Carnegie, T. Cizmár, J. Baumgartl, F. J. Gunn-Moore, and K. Dholakia, “Automated laser guidance of neuronal growth cones using a spatial light modulator,” J. Biophotonics2(11), 682–692 (2009).
[CrossRef] [PubMed]

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip9(10), 1334–1336 (2009).
[CrossRef] [PubMed]

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics2(11), 675–678 (2008).
[CrossRef]

Bera, S.

S. B. Purnapatra, S. Bera, and P. P. Mondal, “Spatial filter based Bessel-like beam for improved penetration depth imaging in fluorescence microscopy,” Sci. Rep.2, 692 (2012).
[CrossRef] [PubMed]

Berry, M. V.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys.47(3), 264 (1979).
[CrossRef]

Betz, T.

A. Ehrlicher, T. Betz, B. Stuhrmann, D. Koch, V. Milner, M. G. Raizen, and J. Kas, “Guiding neuronal growth with light,” Proc. Natl. Acad. Sci. U.S.A.99(25), 16024–16028 (2002).
[CrossRef] [PubMed]

Bloch, I.

I. Bloch, “Ultracold quantum gases in optical lattices,” Nat. Phys.1(1), 23–30 (2005).
[CrossRef]

Boothroyd, S. A.

R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syrett, “Interboard optical data distribution by Bessel beam shadowing,” Opt. Commun.122(4-6), 169–177 (1996).
[CrossRef]

Bouchal, Z.

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun.151(4-6), 207–211 (1998).
[CrossRef]

Bouchal, Z. K.

Briegel, H. J.

R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Phys. Rev. Lett.86(22), 5188–5191 (2001).
[CrossRef] [PubMed]

Broky, J.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett.99(21), 213901 (2007).
[CrossRef] [PubMed]

Bustamante, C.

C. Bustamante, J. C. Macosko, and G. J. L. Wuite, “Grabbing the cat by the tail: manipulating molecules one by one,” Nat. Rev. Mol. Cell Biol.1(2), 130–136 (2000).
[CrossRef] [PubMed]

Capron, B. A.

Carmon, T.

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett.90(2), 023902 (2003).
[CrossRef] [PubMed]

Carnegie, D. J.

D. J. Carnegie, T. Cizmár, J. Baumgartl, F. J. Gunn-Moore, and K. Dholakia, “Automated laser guidance of neuronal growth cones using a spatial light modulator,” J. Biophotonics2(11), 682–692 (2009).
[CrossRef] [PubMed]

D. J. Carnegie, D. J. Stevenson, M. Mazilu, F. Gunn-Moore, and K. Dholakia, “Guided neuronal growth using optical line traps,” Opt. Express16(14), 10507–10517 (2008).
[CrossRef] [PubMed]

Carruthers, A. E.

Chan, V. C.

Chávez-Cerda, S.

Chen, Z.

H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, “Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices,” Phys. Rev. Lett.92(12), 123902 (2004).
[CrossRef] [PubMed]

Chlup, M.

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun.151(4-6), 207–211 (1998).
[CrossRef]

Christodoulides, D. N.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep.463(1-3), 1–126 (2008).
[CrossRef]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett.99(21), 213901 (2007).
[CrossRef] [PubMed]

H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, “Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices,” Phys. Rev. Lett.92(12), 123902 (2004).
[CrossRef] [PubMed]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett.90(2), 023902 (2003).
[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,” Nature422(6928), 147–150 (2003).
[CrossRef] [PubMed]

Chrostowski, J.

R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syrett, “Interboard optical data distribution by Bessel beam shadowing,” Opt. Commun.122(4-6), 169–177 (1996).
[CrossRef]

Cizmár, T.

J. Baumgartl, T. Cizmár, M. Mazilu, V. C. Chan, A. E. Carruthers, B. A. Capron, W. McNeely, E. M. Wright, and K. Dholakia, “Optical path clearing and enhanced transmission through colloidal suspensions,” Opt. Express18(16), 17130–17140 (2010).
[CrossRef] [PubMed]

D. J. Carnegie, T. Cizmár, J. Baumgartl, F. J. Gunn-Moore, and K. Dholakia, “Automated laser guidance of neuronal growth cones using a spatial light modulator,” J. Biophotonics2(11), 682–692 (2009).
[CrossRef] [PubMed]

Cižmár, T.

K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics5(6), 335–342 (2011).
[CrossRef]

Cooper, J.

Cottrell, D. M.

Courtial, J.

Craven, J. M.

Davis, J. A.

Day, D.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip9(10), 1334–1336 (2009).
[CrossRef] [PubMed]

Dennis, M. R.

Dholakia, K.

K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics5(6), 335–342 (2011).
[CrossRef]

J. Baumgartl, T. Cizmár, M. Mazilu, V. C. Chan, A. E. Carruthers, B. A. Capron, W. McNeely, E. M. Wright, and K. Dholakia, “Optical path clearing and enhanced transmission through colloidal suspensions,” Opt. Express18(16), 17130–17140 (2010).
[CrossRef] [PubMed]

M. Mazilu, J. D. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light beats the spread: “non-diffracting” beams,” Laser Photonics Rev.4(4), 529–547 (2010).
[CrossRef]

D. J. Carnegie, T. Cizmár, J. Baumgartl, F. J. Gunn-Moore, and K. Dholakia, “Automated laser guidance of neuronal growth cones using a spatial light modulator,” J. Biophotonics2(11), 682–692 (2009).
[CrossRef] [PubMed]

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip9(10), 1334–1336 (2009).
[CrossRef] [PubMed]

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics2(11), 675–678 (2008).
[CrossRef]

D. J. Carnegie, D. J. Stevenson, M. Mazilu, F. Gunn-Moore, and K. Dholakia, “Guided neuronal growth using optical line traps,” Opt. Express16(14), 10507–10517 (2008).
[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, K. Dholakia, L. Allen, and M. J. Padgett, “Efficiency of second-harmonic generation with Bessel beams,” Phys. Rev. A60(3), 2438–2441 (1999).
[CrossRef]

Dogariu, A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett.99(21), 213901 (2007).
[CrossRef] [PubMed]

Dorsch, R. G.

Dotsenko, I.

D. Schrader, I. Dotsenko, M. Khudaverdyan, Y. Miroshnychenko, A. Rauschenbeutel, and D. Meschede, “Neutral atom quantum register,” Phys. Rev. Lett.93(15), 150501 (2004).
[CrossRef] [PubMed]

Durnin, J.

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett.13(2), 79–80 (1988).
[CrossRef] [PubMed]

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

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,” Nature422(6928), 147–150 (2003).
[CrossRef] [PubMed]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett.90(2), 023902 (2003).
[CrossRef] [PubMed]

Ehrlicher, A.

A. Ehrlicher, T. Betz, B. Stuhrmann, D. Koch, V. Milner, M. G. Raizen, and J. Kas, “Guiding neuronal growth with light,” Proc. Natl. Acad. Sci. U.S.A.99(25), 16024–16028 (2002).
[CrossRef] [PubMed]

Eugenieva, E. D.

H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, “Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices,” Phys. Rev. Lett.92(12), 123902 (2004).
[CrossRef] [PubMed]

Fienup, J. R.

Fleischer, J. W.

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett.90(2), 023902 (2003).
[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,” Nature422(6928), 147–150 (2003).
[CrossRef] [PubMed]

Friese, M. E. J.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature394(6691), 348–350 (1998).
[CrossRef]

Garcés-Chávez, V.

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]

Giorgini, S.

S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of ultracold atomic Fermi gases,” Rev. Mod. Phys.80(4), 1215–1274 (2008).
[CrossRef]

Gover, A.

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, “Generation of electron Airy beams,” Nature494(7437), 331–335 (2013).
[CrossRef] [PubMed]

Grier, D. G.

P. T. Korda and D. G. Grier, “Annealing thin colloidal crystals with optical gradient forces,” J. Chem. Phys.114(17), 7570–7573 (2001).
[CrossRef]

Gu, M.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip9(10), 1334–1336 (2009).
[CrossRef] [PubMed]

Guizar-Sicairos, M.

Gunn-Moore, F.

M. Mazilu, J. D. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light beats the spread: “non-diffracting” beams,” Laser Photonics Rev.4(4), 529–547 (2010).
[CrossRef]

D. J. Carnegie, D. J. Stevenson, M. Mazilu, F. Gunn-Moore, and K. Dholakia, “Guided neuronal growth using optical line traps,” Opt. Express16(14), 10507–10517 (2008).
[CrossRef] [PubMed]

Gunn-Moore, F. J.

D. J. Carnegie, T. Cizmár, J. Baumgartl, F. J. Gunn-Moore, and K. Dholakia, “Automated laser guidance of neuronal growth cones using a spatial light modulator,” J. Biophotonics2(11), 682–692 (2009).
[CrossRef] [PubMed]

Gutierrez-Vega, J. C.

C. López-Mariscal, M. A. Bandres, and J. C. Gutierrez-Vega, “Observation of the experimental propagation properties of Helmholtz-Gauss beams,” Opt. Eng.45(6), 068001 (2006).
[CrossRef]

Gutiérrez-Vega, J. C.

Hannappel, G. M.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip9(10), 1334–1336 (2009).
[CrossRef] [PubMed]

He, X. D.

Heckenberg, N. R.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature394(6691), 348–350 (1998).
[CrossRef]

Helmerson, K.

Howls, C. J.

Ishiwata, S.

T. Nishizaka, H. Miyata, H. Yoshikawa, S. Ishiwata, and K. Kinosita., “Unbinding force of a single motor molecule of muscle measured using optical tweezers,” Nature377(6546), 251–254 (1995).
[CrossRef] [PubMed]

Iturbe-Castillo, M. D.

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramirez, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun.195(1-4), 35–40 (2001).
[CrossRef]

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett.25(20), 1493–1495 (2000).
[CrossRef] [PubMed]

Jordan, P.

Kalnins, E. G.

E. G. Kalnins and W. Miller, “W. Lie theory and separation of variables. 9. Orthogonal R separable coordinate systems for the wave equation ψtt−Δ2ψ=0,” J. Math. Phys.17, 330 (1976).

Kartashov, Y. V.

Y. V. Kartashov, S. López-Aguayo, V. A. Vysloukh, and L. Torner, “Stripe-like quasi-nondiffracting optical lattices,” Opt. Express19(10), 9505–9511 (2011).
[CrossRef] [PubMed]

S. López-Aguayo, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Method to generate complex quasinondiffracting optical lattices,” Phys. Rev. Lett.105(1), 013902 (2010).
[CrossRef] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton shape and mobility control in optical lattices,” Prog. Opt.52, 63–148 (2009).
[CrossRef]

Kas, J.

A. Ehrlicher, T. Betz, B. Stuhrmann, D. Koch, V. Milner, M. G. Raizen, and J. Kas, “Guiding neuronal growth with light,” Proc. Natl. Acad. Sci. U.S.A.99(25), 16024–16028 (2002).
[CrossRef] [PubMed]

Khudaverdyan, M.

D. Schrader, I. Dotsenko, M. Khudaverdyan, Y. Miroshnychenko, A. Rauschenbeutel, and D. Meschede, “Neutral atom quantum register,” Phys. Rev. Lett.93(15), 150501 (2004).
[CrossRef] [PubMed]

Kinosita, K.

T. Nishizaka, H. Miyata, H. Yoshikawa, S. Ishiwata, and K. Kinosita., “Unbinding force of a single motor molecule of muscle measured using optical tweezers,” Nature377(6546), 251–254 (1995).
[CrossRef] [PubMed]

Kivshar, Y.

Koch, D.

A. Ehrlicher, T. Betz, B. Stuhrmann, D. Koch, V. Milner, M. G. Raizen, and J. Kas, “Guiding neuronal growth with light,” Proc. Natl. Acad. Sci. U.S.A.99(25), 16024–16028 (2002).
[CrossRef] [PubMed]

Korda, P. T.

P. T. Korda and D. G. Grier, “Annealing thin colloidal crystals with optical gradient forces,” J. Chem. Phys.114(17), 7570–7573 (2001).
[CrossRef]

Kozawa, Y.

Krolikowski, W.

Laczik, Z.

Leach, J.

Lederer, F.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep.463(1-3), 1–126 (2008).
[CrossRef]

Lereah, Y.

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, “Generation of electron Airy beams,” Nature494(7437), 331–335 (2013).
[CrossRef] [PubMed]

Lilach, Y.

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, “Generation of electron Airy beams,” Nature494(7437), 331–335 (2013).
[CrossRef] [PubMed]

López-Aguayo, S.

Y. V. Kartashov, S. López-Aguayo, V. A. Vysloukh, and L. Torner, “Stripe-like quasi-nondiffracting optical lattices,” Opt. Express19(10), 9505–9511 (2011).
[CrossRef] [PubMed]

S. López-Aguayo, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Method to generate complex quasinondiffracting optical lattices,” Phys. Rev. Lett.105(1), 013902 (2010).
[CrossRef] [PubMed]

Lopez-Mariscal, C.

J. C. Gutiérrez-Vega and C. Lopez-Mariscal, “Nondiffracting vortex beams with continuous orbital angular momentum order dependence,” J. Opt. A, Pure Appl. Opt.10(1), 015009 (2008).
[CrossRef]

López-Mariscal, C.

MacDonald, R. P.

R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syrett, “Interboard optical data distribution by Bessel beam shadowing,” Opt. Commun.122(4-6), 169–177 (1996).
[CrossRef]

Macosko, J. C.

C. Bustamante, J. C. Macosko, and G. J. L. Wuite, “Grabbing the cat by the tail: manipulating molecules one by one,” Nat. Rev. Mol. Cell Biol.1(2), 130–136 (2000).
[CrossRef] [PubMed]

Martin, H.

H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, “Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices,” Phys. Rev. Lett.92(12), 123902 (2004).
[CrossRef] [PubMed]

Mazilu, M.

McGloin, D.

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]

McNeely, W.

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]

Mendlovic, D.

Meschede, D.

D. Schrader, I. Dotsenko, M. Khudaverdyan, Y. Miroshnychenko, A. Rauschenbeutel, and D. Meschede, “Neutral atom quantum register,” Phys. Rev. Lett.93(15), 150501 (2004).
[CrossRef] [PubMed]

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett.13(2), 79–80 (1988).
[CrossRef] [PubMed]

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

Miller, W.

E. G. Kalnins and W. Miller, “W. Lie theory and separation of variables. 9. Orthogonal R separable coordinate systems for the wave equation ψtt−Δ2ψ=0,” J. Math. Phys.17, 330 (1976).

Milner, V.

A. Ehrlicher, T. Betz, B. Stuhrmann, D. Koch, V. Milner, M. G. Raizen, and J. Kas, “Guiding neuronal growth with light,” Proc. Natl. Acad. Sci. U.S.A.99(25), 16024–16028 (2002).
[CrossRef] [PubMed]

Miroshnychenko, Y.

D. Schrader, I. Dotsenko, M. Khudaverdyan, Y. Miroshnychenko, A. Rauschenbeutel, and D. Meschede, “Neutral atom quantum register,” Phys. Rev. Lett.93(15), 150501 (2004).
[CrossRef] [PubMed]

Miyata, H.

T. Nishizaka, H. Miyata, H. Yoshikawa, S. Ishiwata, and K. Kinosita., “Unbinding force of a single motor molecule of muscle measured using optical tweezers,” Nature377(6546), 251–254 (1995).
[CrossRef] [PubMed]

Mondal, P. P.

S. B. Purnapatra, S. Bera, and P. P. Mondal, “Spatial filter based Bessel-like beam for improved penetration depth imaging in fluorescence microscopy,” Sci. Rep.2, 692 (2012).
[CrossRef] [PubMed]

Morsch, O.

O. Morsch and M. Oberthaler, “Dynamics of Bose-Einstein condensates in optical lattices,” Rev. Mod. Phys.78(1), 179–215 (2006).
[CrossRef]

Mukunda, N.

Neshev, D.

New, G. H. C.

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramirez, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun.195(1-4), 35–40 (2001).
[CrossRef]

Nieminen, T. A.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature394(6691), 348–350 (1998).
[CrossRef]

Nishizaka, T.

T. Nishizaka, H. Miyata, H. Yoshikawa, S. Ishiwata, and K. Kinosita., “Unbinding force of a single motor molecule of muscle measured using optical tweezers,” Nature377(6546), 251–254 (1995).
[CrossRef] [PubMed]

Oberthaler, M.

O. Morsch and M. Oberthaler, “Dynamics of Bose-Einstein condensates in optical lattices,” Rev. Mod. Phys.78(1), 179–215 (2006).
[CrossRef]

Okamoto, T.

R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syrett, “Interboard optical data distribution by Bessel beam shadowing,” Opt. Commun.122(4-6), 169–177 (1996).
[CrossRef]

Ostrovskaya, E.

Padgett, M.

Padgett, M. J.

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “Efficiency of second-harmonic generation with Bessel beams,” Phys. Rev. A60(3), 2438–2441 (1999).
[CrossRef]

Piestun, R.

Pitaevskii, L. P.

S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of ultracold atomic Fermi gases,” Rev. Mod. Phys.80(4), 1215–1274 (2008).
[CrossRef]

Purnapatra, S. B.

S. B. Purnapatra, S. Bera, and P. P. Mondal, “Spatial filter based Bessel-like beam for improved penetration depth imaging in fluorescence microscopy,” Sci. Rep.2, 692 (2012).
[CrossRef] [PubMed]

Raizen, M. G.

A. Ehrlicher, T. Betz, B. Stuhrmann, D. Koch, V. Milner, M. G. Raizen, and J. Kas, “Guiding neuronal growth with light,” Proc. Natl. Acad. Sci. U.S.A.99(25), 16024–16028 (2002).
[CrossRef] [PubMed]

Ramirez, G. A.

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramirez, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun.195(1-4), 35–40 (2001).
[CrossRef]

Rauschenbeutel, A.

D. Schrader, I. Dotsenko, M. Khudaverdyan, Y. Miroshnychenko, A. Rauschenbeutel, and D. Meschede, “Neutral atom quantum register,” Phys. Rev. Lett.93(15), 150501 (2004).
[CrossRef] [PubMed]

Raussendorf, R.

R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Phys. Rev. Lett.86(22), 5188–5191 (2001).
[CrossRef] [PubMed]

Ring, J. D.

Rodríguez-Dagnino, R. M.

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramirez, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun.195(1-4), 35–40 (2001).
[CrossRef]

Rubinsztein-Dunlop, H.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature394(6691), 348–350 (1998).
[CrossRef]

Sato, S.

Schattschneider, P.

J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature467(7313), 301–304 (2010).
[CrossRef] [PubMed]

Schechner, Y. Y.

Schrader, D.

D. Schrader, I. Dotsenko, M. Khudaverdyan, Y. Miroshnychenko, A. Rauschenbeutel, and D. Meschede, “Neutral atom quantum register,” Phys. Rev. Lett.93(15), 150501 (2004).
[CrossRef] [PubMed]

Segev, M.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep.463(1-3), 1–126 (2008).
[CrossRef]

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature422(6928), 147–150 (2003).
[CrossRef] [PubMed]

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett.90(2), 023902 (2003).
[CrossRef] [PubMed]

Shamir, J.

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]

Silberberg, Y.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep.463(1-3), 1–126 (2008).
[CrossRef]

Simon, R.

Sinclair, G.

Siviloglou, G. A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett.99(21), 213901 (2007).
[CrossRef] [PubMed]

Stegeman, G. I.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep.463(1-3), 1–126 (2008).
[CrossRef]

Stevenson, D. J.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip9(10), 1334–1336 (2009).
[CrossRef] [PubMed]

D. J. Carnegie, D. J. Stevenson, M. Mazilu, F. Gunn-Moore, and K. Dholakia, “Guided neuronal growth using optical line traps,” Opt. Express16(14), 10507–10517 (2008).
[CrossRef] [PubMed]

Stevenson, J. D.

M. Mazilu, J. D. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light beats the spread: “non-diffracting” beams,” Laser Photonics Rev.4(4), 529–547 (2010).
[CrossRef]

Stringari, S.

S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of ultracold atomic Fermi gases,” Rev. Mod. Phys.80(4), 1215–1274 (2008).
[CrossRef]

Stuhrmann, B.

A. Ehrlicher, T. Betz, B. Stuhrmann, D. Koch, V. Milner, M. G. Raizen, and J. Kas, “Guiding neuronal growth with light,” Proc. Natl. Acad. Sci. U.S.A.99(25), 16024–16028 (2002).
[CrossRef] [PubMed]

Syrett, B. A.

R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syrett, “Interboard optical data distribution by Bessel beam shadowing,” Opt. Commun.122(4-6), 169–177 (1996).
[CrossRef]

Tepichín, E.

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramirez, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun.195(1-4), 35–40 (2001).
[CrossRef]

Tian, H.

J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature467(7313), 301–304 (2010).
[CrossRef] [PubMed]

Tonomura, A.

M. Uchida and A. Tonomura, “Generation of electron beams carrying orbital angular momentum,” Nature464(7289), 737–739 (2010).
[CrossRef] [PubMed]

Torner, L.

Y. V. Kartashov, S. López-Aguayo, V. A. Vysloukh, and L. Torner, “Stripe-like quasi-nondiffracting optical lattices,” Opt. Express19(10), 9505–9511 (2011).
[CrossRef] [PubMed]

S. López-Aguayo, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Method to generate complex quasinondiffracting optical lattices,” Phys. Rev. Lett.105(1), 013902 (2010).
[CrossRef] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton shape and mobility control in optical lattices,” Prog. Opt.52, 63–148 (2009).
[CrossRef]

Uchida, M.

M. Uchida and A. Tonomura, “Generation of electron beams carrying orbital angular momentum,” Nature464(7289), 737–739 (2010).
[CrossRef] [PubMed]

Verbeeck, J.

J. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature467(7313), 301–304 (2010).
[CrossRef] [PubMed]

Voloch-Bloch, N.

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, “Generation of electron Airy beams,” Nature494(7437), 331–335 (2013).
[CrossRef] [PubMed]

Vyas, S.

Vysloukh, V. A.

Y. V. Kartashov, S. López-Aguayo, V. A. Vysloukh, and L. Torner, “Stripe-like quasi-nondiffracting optical lattices,” Opt. Express19(10), 9505–9511 (2011).
[CrossRef] [PubMed]

S. López-Aguayo, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Method to generate complex quasinondiffracting optical lattices,” Phys. Rev. Lett.105(1), 013902 (2010).
[CrossRef] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton shape and mobility control in optical lattices,” Prog. Opt.52, 63–148 (2009).
[CrossRef]

Wagner, J.

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun.151(4-6), 207–211 (1998).
[CrossRef]

Wang, J.

Wright, E. M.

Wuite, G. J. L.

C. Bustamante, J. C. Macosko, and G. J. L. Wuite, “Grabbing the cat by the tail: manipulating molecules one by one,” Nat. Rev. Mol. Cell Biol.1(2), 130–136 (2000).
[CrossRef] [PubMed]

Xu, P.

Yoshikawa, H.

T. Nishizaka, H. Miyata, H. Yoshikawa, S. Ishiwata, and K. Kinosita., “Unbinding force of a single motor molecule of muscle measured using optical tweezers,” Nature377(6546), 251–254 (1995).
[CrossRef] [PubMed]

Zalevsky, Z.

Zhan, M. S.

Am. J. Phys. (1)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys.47(3), 264 (1979).
[CrossRef]

Appl. Opt. (1)

J. Biophotonics (1)

D. J. Carnegie, T. Cizmár, J. Baumgartl, F. J. Gunn-Moore, and K. Dholakia, “Automated laser guidance of neuronal growth cones using a spatial light modulator,” J. Biophotonics2(11), 682–692 (2009).
[CrossRef] [PubMed]

J. Chem. Phys. (1)

P. T. Korda and D. G. Grier, “Annealing thin colloidal crystals with optical gradient forces,” J. Chem. Phys.114(17), 7570–7573 (2001).
[CrossRef]

J. Math. Phys. (1)

E. G. Kalnins and W. Miller, “W. Lie theory and separation of variables. 9. Orthogonal R separable coordinate systems for the wave equation ψtt−Δ2ψ=0,” J. Math. Phys.17, 330 (1976).

J. Opt. A, Pure Appl. Opt. (1)

J. C. Gutiérrez-Vega and C. Lopez-Mariscal, “Nondiffracting vortex beams with continuous orbital angular momentum order dependence,” J. Opt. A, Pure Appl. Opt.10(1), 015009 (2008).
[CrossRef]

J. Opt. Soc. Am. A (5)

Lab Chip (1)

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip9(10), 1334–1336 (2009).
[CrossRef] [PubMed]

Laser Photonics Rev. (1)

M. Mazilu, J. D. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light beats the spread: “non-diffracting” beams,” Laser Photonics Rev.4(4), 529–547 (2010).
[CrossRef]

Nat. Photonics (2)

K. Dholakia and T. Čižmár, “Shaping the future of manipulation,” Nat. Photonics5(6), 335–342 (2011).
[CrossRef]

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics2(11), 675–678 (2008).
[CrossRef]

Nat. Phys. (1)

I. Bloch, “Ultracold quantum gases in optical lattices,” Nat. Phys.1(1), 23–30 (2005).
[CrossRef]

Nat. Rev. Mol. Cell Biol. (1)

C. Bustamante, J. C. Macosko, and G. J. L. Wuite, “Grabbing the cat by the tail: manipulating molecules one by one,” Nat. Rev. Mol. Cell Biol.1(2), 130–136 (2000).
[CrossRef] [PubMed]

Nature (7)

T. Nishizaka, H. Miyata, H. Yoshikawa, S. Ishiwata, and K. Kinosita., “Unbinding force of a single motor molecule of muscle measured using optical tweezers,” Nature377(6546), 251–254 (1995).
[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,” Nature422(6928), 147–150 (2003).
[CrossRef] [PubMed]

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature394(6691), 348–350 (1998).
[CrossRef]

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. Verbeeck, H. Tian, and P. Schattschneider, “Production and application of electron vortex beams,” Nature467(7313), 301–304 (2010).
[CrossRef] [PubMed]

M. Uchida and A. Tonomura, “Generation of electron beams carrying orbital angular momentum,” Nature464(7289), 737–739 (2010).
[CrossRef] [PubMed]

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, “Generation of electron Airy beams,” Nature494(7437), 331–335 (2013).
[CrossRef] [PubMed]

Opt. Commun. (3)

R. P. MacDonald, S. A. Boothroyd, T. Okamoto, J. Chrostowski, and B. A. Syrett, “Interboard optical data distribution by Bessel beam shadowing,” Opt. Commun.122(4-6), 169–177 (1996).
[CrossRef]

Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun.151(4-6), 207–211 (1998).
[CrossRef]

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramirez, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun.195(1-4), 35–40 (2001).
[CrossRef]

Opt. Eng. (1)

C. López-Mariscal, M. A. Bandres, and J. C. Gutierrez-Vega, “Observation of the experimental propagation properties of Helmholtz-Gauss beams,” Opt. Eng.45(6), 068001 (2006).
[CrossRef]

Opt. Express (7)

Opt. Lett. (9)

Phys. Rep. (1)

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, “Discrete solitons in optics,” Phys. Rep.463(1-3), 1–126 (2008).
[CrossRef]

Phys. Rev. A (1)

J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “Efficiency of second-harmonic generation with Bessel beams,” Phys. Rev. A60(3), 2438–2441 (1999).
[CrossRef]

Phys. Rev. Lett. (7)

J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett.90(2), 023902 (2003).
[CrossRef] [PubMed]

H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, “Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices,” Phys. Rev. Lett.92(12), 123902 (2004).
[CrossRef] [PubMed]

R. Raussendorf and H. J. Briegel, “A one-way quantum computer,” Phys. Rev. Lett.86(22), 5188–5191 (2001).
[CrossRef] [PubMed]

D. Schrader, I. Dotsenko, M. Khudaverdyan, Y. Miroshnychenko, A. Rauschenbeutel, and D. Meschede, “Neutral atom quantum register,” Phys. Rev. Lett.93(15), 150501 (2004).
[CrossRef] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of Accelerating Airy Beams,” Phys. Rev. Lett.99(21), 213901 (2007).
[CrossRef] [PubMed]

S. López-Aguayo, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Method to generate complex quasinondiffracting optical lattices,” Phys. Rev. Lett.105(1), 013902 (2010).
[CrossRef] [PubMed]

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

Proc. Natl. Acad. Sci. U.S.A. (1)

A. Ehrlicher, T. Betz, B. Stuhrmann, D. Koch, V. Milner, M. G. Raizen, and J. Kas, “Guiding neuronal growth with light,” Proc. Natl. Acad. Sci. U.S.A.99(25), 16024–16028 (2002).
[CrossRef] [PubMed]

Prog. Opt. (1)

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton shape and mobility control in optical lattices,” Prog. Opt.52, 63–148 (2009).
[CrossRef]

Rev. Mod. Phys. (2)

O. Morsch and M. Oberthaler, “Dynamics of Bose-Einstein condensates in optical lattices,” Rev. Mod. Phys.78(1), 179–215 (2006).
[CrossRef]

S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of ultracold atomic Fermi gases,” Rev. Mod. Phys.80(4), 1215–1274 (2008).
[CrossRef]

Sci. Rep. (1)

S. B. Purnapatra, S. Bera, and P. P. Mondal, “Spatial filter based Bessel-like beam for improved penetration depth imaging in fluorescence microscopy,” Sci. Rep.2, 692 (2012).
[CrossRef] [PubMed]

Other (1)

L. Pitaevskii and S. Stringari, Bose-Einstein Condensation (Oxford University, 2003).

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Figures (7)

Fig. 1
Fig. 1

Experimental setup used to generate quasi-arbitrary non-diffracting optical beams using a Spatial Light Modulator (SLM) . The setup includes a collimating lens ( L 1 ) , two linear polarizers ( LP 1 and LP 2 ) surrounding the SLM to block any residual phase modulation in the SLM, the input lens of the filtering stage ( L 2 with focal distance f 2 ), the iris diaphragm used to filter the zero-order diffraction on the Fourier plane of L 2 , and the output lens ( L 3 , focal length f 3 ). The propagation distance of the quasi-non-diffracting beam is measured from a focal distance f 3 away from the output lens L 3 .

Fig. 2
Fig. 2

First line shows quasi-non-diffracting spiraling beams generated numerically for different widths of angular spectrum δ=0.1 (a), δ=0.2 (b) and δ=0.3 (c). Second to fifth lines illustrate propagation of corresponding experimentally generated patterns.

Fig. 3
Fig. 3

Experimental angular spectra (obtained from first diffraction orders in focal plane of a lens) corresponding to spiral patterns in Fig. 2.

Fig. 4
Fig. 4

Experimentally observed evolution of Gaussian (a) and Laguerre-Gaussian (b) beams with characteristic width r 0 0.1mm .

Fig. 5
Fig. 5

First line shows quasi-non-diffracting Bessel (a) and Mathieu (b) beams with removed sectors, as well as spiraling pattern (c) generated numerically for angular spectrum width δ=0.1 . Second to fourth lines illustrate propagation of corresponding experimentally generated patterns.

Fig. 6
Fig. 6

(a) Parabolic-Bessel and (b) parabolic-cosine quasi-non-diffracting beams with different symmetries in right and left half-planes. First line shows numerically constructed beams for angular spectrum width δ=0.1 . Second to fourth lines illustrate propagation of corresponding experimentally generated patterns. (c) Phase distribution in parabolic-cosine beam from (b) and numerical simulations of its propagation.

Fig. 7
Fig. 7

Polystyrene spheres distributed in the focal plane of the imaging microscope objective when (a) the whole parabolic-cosine beam passes through the optical trapping system and when only the parts of the parabolic-cosine beam that contain straight lines (b) or curves (c) pass through the pinhole.

Equations (2)

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q(η,ζ,ξ)=exp(i k ξ ξ) 0 2π G(φ)exp[i k t (ηcosφ+ζsinφ)]dφ .
2 q ξ 2 + 2 q η 2 + 2 q ζ 2 + k 2 q=0,

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