Abstract

We present a new solution of the paraxial equation based on the Pearcey function, which is related to the Airy function and describes diffraction about a cusp caustic. The Pearcey beam displays properties similar not only to Airy beams but also Gaussian and Bessel beams. These properties include an inherent auto-focusing effect, as well as form-invariance on propagation and self-healing. We describe the theory of propagating Pearcey beams and present experimental verification of their auto-focusing and self-healing behaviour.

© 2012 OSA

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  1. A. E. Siegman, Lasers (University Science Books, 1986).
  2. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A4, 651–654 (1987).
    [CrossRef]
  3. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  4. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys.47, 264–267 (1979).
    [CrossRef]
  5. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett.32, 979–981 (2007).
    [CrossRef] [PubMed]
  6. M. A. Bandres and J. C. Gutiérres-Vega, “Ince-Gaussian beams,” Opt. Lett.29, 144–146 (2004).
    [CrossRef] [PubMed]
  7. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett.25, 1493–1495 (2000).
    [CrossRef]
  8. M. A. Bandres, “Accelerating parabolic beams,” Opt. Lett.33, 1678–1680 (2008).
    [CrossRef] [PubMed]
  9. M. V. Berry and C. J. Howls, “Integrals with coalescing saddles,” http://dlmf.nist.gov/36.2 (Digital Library of Mathematical Functions, National Institute of Standards and Technology, 2012).
  10. T. Poston and I. Stewart, Catastrophe Theory and its Applications (Dover Publications Inc., 1997).
  11. M. V. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt.18, 257–346 (1980).
    [CrossRef]
  12. G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett.7, 684–685 (1971).
    [CrossRef]
  13. M. Mazilu, D. J. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light beats the spread: non-diffracting beams,” Laser Photon. Rev.4, 529–547 (2010).
    [CrossRef]
  14. F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photon.4, 780–785 (2010).
    [CrossRef]
  15. Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun.151, 207–211 (1998).
    [CrossRef]
  16. 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. A5, 837–843 (2011).
    [CrossRef]
  17. V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature419, 145–147 (2002).
    [CrossRef] [PubMed]
  18. P. Vaity and R. P. Singh, “Self-healing property of optical ring lattice,” Opt. Lett.36, 2994–2996 (2011).
    [CrossRef] [PubMed]
  19. F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun.64, 491–495 (1987).
    [CrossRef]
  20. J. E. Morris, M. Mazilu, J. Baumgartl, T. Cizmar, and K. Dholakia, “Propagation characteristics of Airy beams: dependence upon spatial coherence and wavelength,” Opt. Express17, 13236–13245 (2009).
    [CrossRef] [PubMed]
  21. J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express16, 12880–12891 (2008).
    [CrossRef] [PubMed]
  22. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photon.2, 675–678 (2008).
    [CrossRef]
  23. M. V. Berry and C. J. Howls, “Infinity interpreted,” Phys. World6, 35–39 (1993).
  24. J. F. Nye, Natural Focusing and Fine Structure of Light (IoP Publishing, 1999).
  25. E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett.106, 213902 (2011).
    [CrossRef] [PubMed]
  26. T. Pearcey, “The structure of an electromagnetic field in the neighbourhood of a cusp of a caustic,” Phil. Mag. S. 7 37, 311–317 (1946).
  27. M. V. Berry, J. F. Nye, and F. J. Wright, “The elliptic umbilic diffraction catastrophe,” Phil. Trans. R. Soc. A291, 453–484 (1979).
    [CrossRef]
  28. J. J. Stamnes, Waves in Focal Regions (Taylor & Francis, 1986).
  29. M. A. Bandres and M. Guizar-Sicairos, “Paraxial group,” Opt. Lett.34, 13–15 (2009).
    [CrossRef]
  30. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publications Inc., 1965).
  31. M. V. Berry, “Faster than Fourier,” in Quantum Coherence and Reality, J. S. Anandan and J. L. Safko, eds. (World Scientific, 1992), 55–65.
  32. M. R. Dennis and J. Lindberg “Natural superoscillation of random functions in one and more dimensions,” Proc. SPIE7394, article 73940A (2009).
    [CrossRef]
  33. E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M R Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nature Mat.11, 432–435 (2012).
    [CrossRef]
  34. J. Baumgartl, S. Kosmeier, M. Mazilu, E. T. F. Rogers, N. I. Zheludev, and K. Dholakia, “Far field subwavelength focusing using optical eigenmodes,” Appl. Phy. Lett.98, 181109 (2011).
    [CrossRef]
  35. M. Anguiano-Morales, A. Martínez, M. D. Iturbe-Castillo, S. Chávez-Cerda, and N. Alcalá-Ochoa, “Self-healing property of a caustic optical beam,” Appl. Opt.46, 8284–8290 (2007).
    [CrossRef] [PubMed]
  36. N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett.35, 4045–4047 (2010).
    [CrossRef] [PubMed]
  37. D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, “Observation of abruptly autofocusing waves,” Opt. Lett.36, 1842–1844 (2011).
    [CrossRef] [PubMed]

2012 (1)

E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M R Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nature Mat.11, 432–435 (2012).
[CrossRef]

2011 (5)

J. Baumgartl, S. Kosmeier, M. Mazilu, E. T. F. Rogers, N. I. Zheludev, and K. Dholakia, “Far field subwavelength focusing using optical eigenmodes,” Appl. Phy. Lett.98, 181109 (2011).
[CrossRef]

E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett.106, 213902 (2011).
[CrossRef] [PubMed]

D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, “Observation of abruptly autofocusing waves,” Opt. Lett.36, 1842–1844 (2011).
[CrossRef] [PubMed]

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. A5, 837–843 (2011).
[CrossRef]

P. Vaity and R. P. Singh, “Self-healing property of optical ring lattice,” Opt. Lett.36, 2994–2996 (2011).
[CrossRef] [PubMed]

2010 (3)

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

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photon.4, 780–785 (2010).
[CrossRef]

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

2009 (3)

2008 (3)

2007 (2)

2004 (1)

2002 (1)

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

2000 (1)

1998 (1)

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

1993 (1)

M. V. Berry and C. J. Howls, “Infinity interpreted,” Phys. World6, 35–39 (1993).

1987 (3)

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun.64, 491–495 (1987).
[CrossRef]

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

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

1980 (1)

M. V. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt.18, 257–346 (1980).
[CrossRef]

1979 (2)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys.47, 264–267 (1979).
[CrossRef]

M. V. Berry, J. F. Nye, and F. J. Wright, “The elliptic umbilic diffraction catastrophe,” Phil. Trans. R. Soc. A291, 453–484 (1979).
[CrossRef]

1971 (1)

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett.7, 684–685 (1971).
[CrossRef]

1946 (1)

T. Pearcey, “The structure of an electromagnetic field in the neighbourhood of a cusp of a caustic,” Phil. Mag. S. 7 37, 311–317 (1946).

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publications Inc., 1965).

Alcalá-Ochoa, N.

Anguiano-Morales, M.

Balazs, N. L.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys.47, 264–267 (1979).
[CrossRef]

Bandres, M. A.

Baumgartl, J.

J. Baumgartl, S. Kosmeier, M. Mazilu, E. T. F. Rogers, N. I. Zheludev, and K. Dholakia, “Far field subwavelength focusing using optical eigenmodes,” Appl. Phy. Lett.98, 181109 (2011).
[CrossRef]

J. E. Morris, M. Mazilu, J. Baumgartl, T. Cizmar, and K. Dholakia, “Propagation characteristics of Airy beams: dependence upon spatial coherence and wavelength,” Opt. Express17, 13236–13245 (2009).
[CrossRef] [PubMed]

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photon.2, 675–678 (2008).
[CrossRef]

Berry, M. V.

M. V. Berry and C. J. Howls, “Infinity interpreted,” Phys. World6, 35–39 (1993).

M. V. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt.18, 257–346 (1980).
[CrossRef]

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys.47, 264–267 (1979).
[CrossRef]

M. V. Berry, J. F. Nye, and F. J. Wright, “The elliptic umbilic diffraction catastrophe,” Phil. Trans. R. Soc. A291, 453–484 (1979).
[CrossRef]

M. V. Berry, “Faster than Fourier,” in Quantum Coherence and Reality, J. S. Anandan and J. L. Safko, eds. (World Scientific, 1992), 55–65.

Bouchal, Z.

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

Broky, J.

Chad, J. E.

E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M R Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nature Mat.11, 432–435 (2012).
[CrossRef]

Chávez-Cerda, S.

Chlup, M.

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

Christodoulides, D. N.

Cizmar, T.

Dennis, M R

E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M R Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nature Mat.11, 432–435 (2012).
[CrossRef]

Dennis, M. R.

M. R. Dennis and J. Lindberg “Natural superoscillation of random functions in one and more dimensions,” Proc. SPIE7394, article 73940A (2009).
[CrossRef]

Deschamps, G. A.

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett.7, 684–685 (1971).
[CrossRef]

Dholakia, K.

J. Baumgartl, S. Kosmeier, M. Mazilu, E. T. F. Rogers, N. I. Zheludev, and K. Dholakia, “Far field subwavelength focusing using optical eigenmodes,” Appl. Phy. Lett.98, 181109 (2011).
[CrossRef]

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

J. E. Morris, M. Mazilu, J. Baumgartl, T. Cizmar, and K. Dholakia, “Propagation characteristics of Airy beams: dependence upon spatial coherence and wavelength,” Opt. Express17, 13236–13245 (2009).
[CrossRef] [PubMed]

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photon.2, 675–678 (2008).
[CrossRef]

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

Dogariu, A.

Durnin, J.

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

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

Eberly, J. H.

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

Efremidis, N. K.

Fahrbach, F. O.

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photon.4, 780–785 (2010).
[CrossRef]

Garces-Chavez, V.

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

Gori, F.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun.64, 491–495 (1987).
[CrossRef]

Greenfield, E.

E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett.106, 213902 (2011).
[CrossRef] [PubMed]

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun.64, 491–495 (1987).
[CrossRef]

Guizar-Sicairos, M.

Gunn-Moore, F.

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

Gutiérres-Vega, J. C.

Gutiérrez-Vega, J. C.

Howls, C. J.

M. V. Berry and C. J. Howls, “Infinity interpreted,” Phys. World6, 35–39 (1993).

Iturbe-Castillo, M. D.

Kosmeier, S.

J. Baumgartl, S. Kosmeier, M. Mazilu, E. T. F. Rogers, N. I. Zheludev, and K. Dholakia, “Far field subwavelength focusing using optical eigenmodes,” Appl. Phy. Lett.98, 181109 (2011).
[CrossRef]

Kozawa, Y.

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. A5, 837–843 (2011).
[CrossRef]

Lindberg, J.

E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M R Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nature Mat.11, 432–435 (2012).
[CrossRef]

M. R. Dennis and J. Lindberg “Natural superoscillation of random functions in one and more dimensions,” Proc. SPIE7394, article 73940A (2009).
[CrossRef]

Martínez, A.

Mazilu, M.

J. Baumgartl, S. Kosmeier, M. Mazilu, E. T. F. Rogers, N. I. Zheludev, and K. Dholakia, “Far field subwavelength focusing using optical eigenmodes,” Appl. Phy. Lett.98, 181109 (2011).
[CrossRef]

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

J. E. Morris, M. Mazilu, J. Baumgartl, T. Cizmar, and K. Dholakia, “Propagation characteristics of Airy beams: dependence upon spatial coherence and wavelength,” Opt. Express17, 13236–13245 (2009).
[CrossRef] [PubMed]

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photon.2, 675–678 (2008).
[CrossRef]

McGloin, D.

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

Melville, H.

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

Miceli, J. J.

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

Morris, J. E.

Nye, J. F.

M. V. Berry, J. F. Nye, and F. J. Wright, “The elliptic umbilic diffraction catastrophe,” Phil. Trans. R. Soc. A291, 453–484 (1979).
[CrossRef]

J. F. Nye, Natural Focusing and Fine Structure of Light (IoP Publishing, 1999).

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun.64, 491–495 (1987).
[CrossRef]

Papazoglou, D. G.

Pearcey, T.

T. Pearcey, “The structure of an electromagnetic field in the neighbourhood of a cusp of a caustic,” Phil. Mag. S. 7 37, 311–317 (1946).

Poston, T.

T. Poston and I. Stewart, Catastrophe Theory and its Applications (Dover Publications Inc., 1997).

Raz, O.

E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett.106, 213902 (2011).
[CrossRef] [PubMed]

Rogers, E. T. F.

E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M R Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nature Mat.11, 432–435 (2012).
[CrossRef]

J. Baumgartl, S. Kosmeier, M. Mazilu, E. T. F. Rogers, N. I. Zheludev, and K. Dholakia, “Far field subwavelength focusing using optical eigenmodes,” Appl. Phy. Lett.98, 181109 (2011).
[CrossRef]

Rohrbach, A.

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photon.4, 780–785 (2010).
[CrossRef]

Roy, T.

E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M R Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nature Mat.11, 432–435 (2012).
[CrossRef]

Sato, S.

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. A5, 837–843 (2011).
[CrossRef]

Savo, S.

E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M R Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nature Mat.11, 432–435 (2012).
[CrossRef]

Segev, M.

E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett.106, 213902 (2011).
[CrossRef] [PubMed]

Sibbett, W.

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

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986).

Simon, P.

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photon.4, 780–785 (2010).
[CrossRef]

Singh, R. P.

Siviloglou, G. A.

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions (Taylor & Francis, 1986).

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publications Inc., 1965).

Stevenson, D. J.

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

Stewart, I.

T. Poston and I. Stewart, Catastrophe Theory and its Applications (Dover Publications Inc., 1997).

Tzortzakis, S.

Upstill, C.

M. V. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt.18, 257–346 (1980).
[CrossRef]

Vaity, P.

Vyas, S.

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. A5, 837–843 (2011).
[CrossRef]

Wagner, J.

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

Walasik, W.

E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett.106, 213902 (2011).
[CrossRef] [PubMed]

Wright, F. J.

M. V. Berry, J. F. Nye, and F. J. Wright, “The elliptic umbilic diffraction catastrophe,” Phil. Trans. R. Soc. A291, 453–484 (1979).
[CrossRef]

Zheludev, N. I.

E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M R Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nature Mat.11, 432–435 (2012).
[CrossRef]

J. Baumgartl, S. Kosmeier, M. Mazilu, E. T. F. Rogers, N. I. Zheludev, and K. Dholakia, “Far field subwavelength focusing using optical eigenmodes,” Appl. Phy. Lett.98, 181109 (2011).
[CrossRef]

Am. J. Phys. (1)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys.47, 264–267 (1979).
[CrossRef]

Appl. Opt. (1)

Appl. Phy. Lett. (1)

J. Baumgartl, S. Kosmeier, M. Mazilu, E. T. F. Rogers, N. I. Zheludev, and K. Dholakia, “Far field subwavelength focusing using optical eigenmodes,” Appl. Phy. Lett.98, 181109 (2011).
[CrossRef]

Electron. Lett. (1)

G. A. Deschamps, “Gaussian beam as a bundle of complex rays,” Electron. Lett.7, 684–685 (1971).
[CrossRef]

J. Opt. Soc. Am. A (2)

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. A5, 837–843 (2011).
[CrossRef]

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

Laser Photon. Rev. (1)

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

Nat. Photon. (2)

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photon.4, 780–785 (2010).
[CrossRef]

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photon.2, 675–678 (2008).
[CrossRef]

Nature (1)

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

Nature Mat. (1)

E. T. F. Rogers, J. Lindberg, T. Roy, S. Savo, J. E. Chad, M R Dennis, and N. I. Zheludev, “A super-oscillatory lens optical microscope for subwavelength imaging,” Nature Mat.11, 432–435 (2012).
[CrossRef]

Opt. Commun. (2)

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun.64, 491–495 (1987).
[CrossRef]

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

Opt. Express (2)

Opt. Lett. (8)

Phil. Mag. S. (1)

T. Pearcey, “The structure of an electromagnetic field in the neighbourhood of a cusp of a caustic,” Phil. Mag. S. 7 37, 311–317 (1946).

Phil. Trans. R. Soc. A (1)

M. V. Berry, J. F. Nye, and F. J. Wright, “The elliptic umbilic diffraction catastrophe,” Phil. Trans. R. Soc. A291, 453–484 (1979).
[CrossRef]

Phys. Rev. Lett. (2)

E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett.106, 213902 (2011).
[CrossRef] [PubMed]

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

Phys. World (1)

M. V. Berry and C. J. Howls, “Infinity interpreted,” Phys. World6, 35–39 (1993).

Proc. SPIE (1)

M. R. Dennis and J. Lindberg “Natural superoscillation of random functions in one and more dimensions,” Proc. SPIE7394, article 73940A (2009).
[CrossRef]

Prog. Opt. (1)

M. V. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt.18, 257–346 (1980).
[CrossRef]

Other (7)

A. E. Siegman, Lasers (University Science Books, 1986).

M. V. Berry and C. J. Howls, “Integrals with coalescing saddles,” http://dlmf.nist.gov/36.2 (Digital Library of Mathematical Functions, National Institute of Standards and Technology, 2012).

T. Poston and I. Stewart, Catastrophe Theory and its Applications (Dover Publications Inc., 1997).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover Publications Inc., 1965).

M. V. Berry, “Faster than Fourier,” in Quantum Coherence and Reality, J. S. Anandan and J. L. Safko, eds. (World Scientific, 1992), 55–65.

J. F. Nye, Natural Focusing and Fine Structure of Light (IoP Publishing, 1999).

J. J. Stamnes, Waves in Focal Regions (Taylor & Francis, 1986).

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Figures (6)

Fig. 1
Fig. 1

Transverse profile intensities of the Pearcey beam, with parameters x0 = y0 = 10−4 m and where z e 2 k y 0 2 0.251 m and k is the wavenumber for wavelength λ = 500 nm; (a) intensity of the Pearcey function for z = 0 m; (b) the Pearcey beam at z = 0.8ze m; (c) at z = 0.975ze m; (d) at z = 1.025ze m; (e) z = 1.2ze m; (f) z = 2ze m. The cusp underlying the Pearcey pattern is shown as a white dashed line in (a). Upon propagation, the cusp - and therefore the shape of the Pearcey pattern - flattens out to a line, then inverts after a singular plane at z = ze.

Fig. 2
Fig. 2

Transverse intensity of a Pearcey-Gauss beam as z increases for x0 = y0 = 0.1 mm, w0 = 2.0 mm and λ = 500 nm. The scaling and inversion of the pattern is still evident, however, there now exists a small hourglass-shaped focal point that was absent in case of the unmodulated Pearcey beam. The intensities of each image are not on the same scale.

Fig. 3
Fig. 3

Intensity of the Pearcey-Gauss beam (for x0 = y0 = 0.1 mm, w0 = 2.0 mm and λ = 500 nm) at the essential focusing plane; (a) magnification of the essential focus of Fig. 2; the dashed lines correspond to the intensity cross-sections of (c) and (d); (b) intensity of the Fourier distribution of the Pearcey-Gauss beam according to Eq. (9), which mimics the δ-line parabola of Eq. (3); (c) the short-dashed line shows the intensity cross-section of the essential focus in the x-direction; (d) intensity cross-section along the long-dashed line of (a) in the y-direction.

Fig. 4
Fig. 4

Experimental setup; (a) schematic of the experiment, where Li are lenses, SLM is the spatial light modulator, CCD is the charge coupled device camera, PBS is a polarizing beam splitter; the focal widths of lenses are f1 = 25 mm, f2 = 100 cm, f3 = 680 mm, f4 = 400 mm and f5 = 800 mm; (b) image encoded on the SLM. The Fourier transform of the Pearcey-Gauss beam describes a parabola with phase given by Eq. (9). The hue indicates the phase while brightness describes the corresponding intensity. The SLM was used in the standard first-order diffraction configuration.

Fig. 5
Fig. 5

Experimental observation of the Pearcey-Gauss beam for consecutive propagation distances. The collapse of the beam to a point is clearly visible, as well as the predicted inversion. These results agree with the theoretical and numerical predictions. The propagation distances are given, inset in the images.

Fig. 6
Fig. 6

Experimental images of the self-healing of the Pearcey beam from an arbitrary perturbation. The obstacle was cylindrical, with a rectangular projection, and the area blocked is indicated by the white line in (a). It is clear that the Pearcey beam recovers from the initial perturbation and still collapses and inverts after its essential focus.

Equations (12)

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Pe ( X , Y ) d s exp [ i ( s 4 + s 2 Y + s X ) ] ,
( 2 y 3 y 0 ) 3 + ( x x 0 ) 2 = 0 .
Pe ˜ ( k x x 0 , k y y 0 ) = 1 ( 2 π ) 2 d x d y Pe ( x x 0 , y y 0 ) e i k x x i k y y = x 0 y 0 e i k x 4 x 0 4 δ ( k x 2 x 0 2 k y y 0 ) ,
Pe beam ( x , y , z ) = i k 2 π z d x d y Pe ( x x 0 , y y 0 ) exp ( i k 2 z [ ( x x ) 2 + ( y y ) 2 ] ) = 1 ( 1 z / z e ) 1 4 Pe ( x x 0 ( 1 z / z e ) 1 4 , y z y 0 / 2 k x 0 2 y 0 ( 1 z / z e ) 1 2 ) ,
( 2 3 y z y 0 / 2 k x 0 2 y 0 ( 1 z / z e ) 1 2 ) 3 + ( x x 0 ( 1 z / z e ) 1 4 ) 2 = 0 .
Pe beam ( x , y , z e ) = e i π / 4 π y / y 0 y 0 2 / x 0 2 exp ( i x 2 y 0 4 ( y x 0 2 y 0 3 ) ) .
G ( x , y , z ) = 1 ( 1 + i z / z R ) exp ( x 2 + y 2 w 0 2 ( 1 + i z / z R ) ) ,
PeG ( x , y , z ) = G ( x , y , z ) [ 1 z / ζ ( z ) z e ] 1 4 Pe ( x x 0 ζ ( z ) [ 1 z / ζ ( z ) z e ] 1 4 , y z y 0 / 2 k x 0 2 y 0 ζ ( z ) [ 1 z / ζ ( z ) z e ] 1 2 ) ,
PeG ˜ ( k x , k y ) = w 2 exp [ w 2 ( k x 2 + k y 2 ) / 4 ] 4 π ( 1 + i w 2 / 4 y 0 2 ) 1 / 4 Pe ( w 2 k x 2 i x 0 ( 1 + i w 2 / 4 y 0 2 ) 1 / 4 , w 2 ( k y y 0 / 2 x 0 2 ) 2 i y 0 ( 1 + i w 2 / 4 y 0 2 ) 1 / 2 ) ,
PeG ( x , y 0 3 / x 0 2 , z e ) = exp ( x 2 y 0 6 / x 0 4 w 0 2 ζ ( z e ) ) w 0 2 y 0 ζ ( z e ) 3 / 2 [ 2 Γ ( 5 4 ) F 0 2 ( ; 1 2 , 3 4 ; w 0 2 x 4 2 10 ζ ( z e ) 3 x 0 4 y 0 2 ) + w 0 x 2 2 5 ζ ( z e ) 3 / 2 y 0 x 0 2 Γ ( 1 4 ) F 0 2 ( ; 5 4 , 3 2 ; w 0 2 x 4 2 10 ζ ( z e ) 3 x 0 4 y 0 2 ) ] ,
PeG ( 0 , y , z e ) = exp ( y 2 w 0 2 ζ ( z e ) ) e i π / 4 w 0 4 ζ ( z e ) y 0 y y 0 y 0 2 x 0 2 exp [ w 0 2 32 ζ ( z e ) ( y y 0 2 y 0 x 0 2 ) 2 ] × K 1 4 [ w 0 2 32 ζ ( z e ) ] ( y y 0 2 y 0 x 0 2 ) 2 ,
U ( x , y ) = d s exp [ i ( s 2 n + y s n + x s m ) ] ,

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