Abstract

Self-accelerating Airy beams, which are nondiffracting waves in the form of an Airy function that propagate in free space with constant acceleration, have received considerable attention in recent years. They are typically generated by manipulation of the phase front of the wave by means of specially designed optical elements. Here we show that autofocusing, radially symmetric Airy waves can form spontaneously as a laser beam propagates in a defocusing, nonlocal thermal nonlinear medium, inside a cylindrical channel with a reflective boundary. The beam forms a ring-shaped optical caustic, which, following reflection from the boundary, converges to a focal point. We demonstrate this new method experimentally and numerically, and present a semi-classical analytical model for the wave dynamics that shows that the self-generated, radially symmetric wave is indeed a caustic with an Airy-function profile. In the hydrodynamic representation of the nonlinear wave equation, the ring-shaped caustic that we describe can be interpreted as a shock wave that forms as the “photonic fluid” bounces off the reflective boundary. These results suggest a very simple and accessible, yet mathematically accurate, way to obtain autofocusing radially symmetric Airy waves for various applications.

© 2015 Optical Society of America

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References

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  1. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264 (1979).
    [Crossref]
  2. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007).
    [Crossref]
  3. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
    [Crossref]
  4. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008).
    [Crossref]
  5. S. Jia, J. C. Vaughan, and X. Zhuang, “Isotropic three-dimensional superresolution imaging with a self-bending point spread function,” Nat. Photonics 8, 302–306 (2014).
    [Crossref]
  6. A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101, 071110 (2012).
    [Crossref]
  7. P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009).
    [Crossref]
  8. A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4, 103–106 (2010).
    [Crossref]
  9. T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3, 395–398 (2009).
    [Crossref]
  10. I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett. 106, 213903 (2011).
    [Crossref]
  11. R. Bekenstein and M. Segev, “Self-accelerating optical beams in highly nonlocal nonlinear media,” Opt. Express 19, 23706 (2011).
    [Crossref]
  12. E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. 106, 213902 (2011).
    [Crossref]
  13. N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, “Generation of electron Airy beams,” Nature 494, 331–335 (2013).
    [Crossref]
  14. A. Libster-Hershko, I. Epstein, and A. Arie, “Rapidly accelerating Mathieu and Weber surface plasmon beams,” Phys. Rev. Lett. 113, 123902 (2014).
    [Crossref]
  15. N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. 35, 4045–4047 (2010).
    [Crossref]
  16. D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, “Long-transient effects in lasers with inserted liquid samples,” Opt. Lett. 36, 1842–1844 (2011).
    [Crossref]
  17. S. A. Akhmanov, D. P. Krindach, A. P. Sukhorukov, and R. V. Khokhlov, “Nonlinear defocusing of laser beams,” Pis’ma Zh. Eksp. Teor. Fiz. 6, 509 (1967), JETP Lett. 6, 38 (1967).
  18. J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
    [Crossref]
  19. A. Minovich, D. N. Neshev, A. Dreischuh, W. Krolikowski, and Y. S. Kivshar, “Experimental reconstruction of nonlocal response of thermal nonlinear optical media,” Opt. Lett. 32, 1599–1601 (2007).
    [Crossref]
  20. A. W. Snyder and D. J. Mitchell, “Accessible solitons,” Science 276, 1538–1541 (1997).
    [Crossref]
  21. D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Laguerre and Hermite soliton clusters in nonlocal nonlinear media,” Phys. Rev. Lett. 98, 053901 (2007).
    [Crossref]
  22. M. V. Berry, “Uniform approximation for potential scattering involving a rainbow,” Proc. Phys. Soc. 89, 479–490 (1966).
    [Crossref]
  23. C. Chester, B. Friedman, and F. Ursell, “An extension of the method of steepest descents,” Proc. Cambridge Philos. Soc. 53, 599–611 (1957).
    [Crossref]
  24. E. Madelung, “Quantum theory in hydrodynamical form,” Z. Phys. 40, 322 (1927).
    [Crossref]
  25. W. Wan, S. Jia, and J. W. Fleischer, “Dispersive superfluid-like shock waves in nonlinear optics,” Nat. Phys. 3, 46–51 (2006).
    [Crossref]

2014 (2)

S. Jia, J. C. Vaughan, and X. Zhuang, “Isotropic three-dimensional superresolution imaging with a self-bending point spread function,” Nat. Photonics 8, 302–306 (2014).
[Crossref]

A. Libster-Hershko, I. Epstein, and A. Arie, “Rapidly accelerating Mathieu and Weber surface plasmon beams,” Phys. Rev. Lett. 113, 123902 (2014).
[Crossref]

2013 (1)

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, “Generation of electron Airy beams,” Nature 494, 331–335 (2013).
[Crossref]

2012 (1)

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101, 071110 (2012).
[Crossref]

2011 (4)

I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett. 106, 213903 (2011).
[Crossref]

R. Bekenstein and M. Segev, “Self-accelerating optical beams in highly nonlocal nonlinear media,” Opt. Express 19, 23706 (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]

D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, “Long-transient effects in lasers with inserted liquid samples,” Opt. Lett. 36, 1842–1844 (2011).
[Crossref]

2010 (2)

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

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4, 103–106 (2010).
[Crossref]

2009 (2)

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3, 395–398 (2009).
[Crossref]

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009).
[Crossref]

2008 (1)

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

2007 (4)

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007).
[Crossref]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

A. Minovich, D. N. Neshev, A. Dreischuh, W. Krolikowski, and Y. S. Kivshar, “Experimental reconstruction of nonlocal response of thermal nonlinear optical media,” Opt. Lett. 32, 1599–1601 (2007).
[Crossref]

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Laguerre and Hermite soliton clusters in nonlocal nonlinear media,” Phys. Rev. Lett. 98, 053901 (2007).
[Crossref]

2006 (1)

W. Wan, S. Jia, and J. W. Fleischer, “Dispersive superfluid-like shock waves in nonlinear optics,” Nat. Phys. 3, 46–51 (2006).
[Crossref]

1997 (1)

A. W. Snyder and D. J. Mitchell, “Accessible solitons,” Science 276, 1538–1541 (1997).
[Crossref]

1979 (1)

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

1967 (1)

S. A. Akhmanov, D. P. Krindach, A. P. Sukhorukov, and R. V. Khokhlov, “Nonlinear defocusing of laser beams,” Pis’ma Zh. Eksp. Teor. Fiz. 6, 509 (1967), JETP Lett. 6, 38 (1967).

1966 (1)

M. V. Berry, “Uniform approximation for potential scattering involving a rainbow,” Proc. Phys. Soc. 89, 479–490 (1966).
[Crossref]

1965 (1)

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[Crossref]

1957 (1)

C. Chester, B. Friedman, and F. Ursell, “An extension of the method of steepest descents,” Proc. Cambridge Philos. Soc. 53, 599–611 (1957).
[Crossref]

1927 (1)

E. Madelung, “Quantum theory in hydrodynamical form,” Z. Phys. 40, 322 (1927).
[Crossref]

Akhmanov, S. A.

S. A. Akhmanov, D. P. Krindach, A. P. Sukhorukov, and R. V. Khokhlov, “Nonlinear defocusing of laser beams,” Pis’ma Zh. Eksp. Teor. Fiz. 6, 509 (1967), JETP Lett. 6, 38 (1967).

Arie, A.

A. Libster-Hershko, I. Epstein, and A. Arie, “Rapidly accelerating Mathieu and Weber surface plasmon beams,” Phys. Rev. Lett. 113, 123902 (2014).
[Crossref]

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, “Generation of electron Airy beams,” Nature 494, 331–335 (2013).
[Crossref]

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3, 395–398 (2009).
[Crossref]

Balazs, N. L.

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

Baumgartl, J.

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

Bekenstein, R.

Berry, M. V.

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

M. V. Berry, “Uniform approximation for potential scattering involving a rainbow,” Proc. Phys. Soc. 89, 479–490 (1966).
[Crossref]

Broky, J.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

Buccoliero, D.

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Laguerre and Hermite soliton clusters in nonlocal nonlinear media,” Phys. Rev. Lett. 98, 053901 (2007).
[Crossref]

Chester, C.

C. Chester, B. Friedman, and F. Ursell, “An extension of the method of steepest descents,” Proc. Cambridge Philos. Soc. 53, 599–611 (1957).
[Crossref]

Chong, A.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4, 103–106 (2010).
[Crossref]

Christodoulides, D. N.

I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett. 106, 213903 (2011).
[Crossref]

D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, “Long-transient effects in lasers with inserted liquid samples,” Opt. Lett. 36, 1842–1844 (2011).
[Crossref]

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

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4, 103–106 (2010).
[Crossref]

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009).
[Crossref]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007).
[Crossref]

Courvoisier, F.

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101, 071110 (2012).
[Crossref]

Desyatnikov, A. S.

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Laguerre and Hermite soliton clusters in nonlocal nonlinear media,” Phys. Rev. Lett. 98, 053901 (2007).
[Crossref]

Dholakia, K.

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

Dogariu, A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

Dreischuh, A.

Dudley, J. M.

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101, 071110 (2012).
[Crossref]

Efremidis, N. K.

Ellenbogen, T.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3, 395–398 (2009).
[Crossref]

Epstein, I.

A. Libster-Hershko, I. Epstein, and A. Arie, “Rapidly accelerating Mathieu and Weber surface plasmon beams,” Phys. Rev. Lett. 113, 123902 (2014).
[Crossref]

Fleischer, J. W.

W. Wan, S. Jia, and J. W. Fleischer, “Dispersive superfluid-like shock waves in nonlinear optics,” Nat. Phys. 3, 46–51 (2006).
[Crossref]

Friedman, B.

C. Chester, B. Friedman, and F. Ursell, “An extension of the method of steepest descents,” Proc. Cambridge Philos. Soc. 53, 599–611 (1957).
[Crossref]

Froehly, L.

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101, 071110 (2012).
[Crossref]

Furfaro, L.

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101, 071110 (2012).
[Crossref]

Ganany-Padowicz, A.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3, 395–398 (2009).
[Crossref]

Gordon, J. P.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[Crossref]

Gover, A.

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, “Generation of electron Airy beams,” Nature 494, 331–335 (2013).
[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]

Jacquot, M.

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101, 071110 (2012).
[Crossref]

Jia, S.

S. Jia, J. C. Vaughan, and X. Zhuang, “Isotropic three-dimensional superresolution imaging with a self-bending point spread function,” Nat. Photonics 8, 302–306 (2014).
[Crossref]

W. Wan, S. Jia, and J. W. Fleischer, “Dispersive superfluid-like shock waves in nonlinear optics,” Nat. Phys. 3, 46–51 (2006).
[Crossref]

Kaminer, I.

I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett. 106, 213903 (2011).
[Crossref]

Khokhlov, R. V.

S. A. Akhmanov, D. P. Krindach, A. P. Sukhorukov, and R. V. Khokhlov, “Nonlinear defocusing of laser beams,” Pis’ma Zh. Eksp. Teor. Fiz. 6, 509 (1967), JETP Lett. 6, 38 (1967).

Kivshar, Y. S.

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Laguerre and Hermite soliton clusters in nonlocal nonlinear media,” Phys. Rev. Lett. 98, 053901 (2007).
[Crossref]

A. Minovich, D. N. Neshev, A. Dreischuh, W. Krolikowski, and Y. S. Kivshar, “Experimental reconstruction of nonlocal response of thermal nonlinear optical media,” Opt. Lett. 32, 1599–1601 (2007).
[Crossref]

Kolesik, M.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009).
[Crossref]

Krindach, D. P.

S. A. Akhmanov, D. P. Krindach, A. P. Sukhorukov, and R. V. Khokhlov, “Nonlinear defocusing of laser beams,” Pis’ma Zh. Eksp. Teor. Fiz. 6, 509 (1967), JETP Lett. 6, 38 (1967).

Krolikowski, W.

A. Minovich, D. N. Neshev, A. Dreischuh, W. Krolikowski, and Y. S. Kivshar, “Experimental reconstruction of nonlocal response of thermal nonlinear optical media,” Opt. Lett. 32, 1599–1601 (2007).
[Crossref]

D. Buccoliero, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Laguerre and Hermite soliton clusters in nonlocal nonlinear media,” Phys. Rev. Lett. 98, 053901 (2007).
[Crossref]

Lacourt, P. A.

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101, 071110 (2012).
[Crossref]

Leite, R. C. C.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[Crossref]

Lereah, Y.

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, “Generation of electron Airy beams,” Nature 494, 331–335 (2013).
[Crossref]

Libster-Hershko, A.

A. Libster-Hershko, I. Epstein, and A. Arie, “Rapidly accelerating Mathieu and Weber surface plasmon beams,” Phys. Rev. Lett. 113, 123902 (2014).
[Crossref]

Lilach, Y.

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, “Generation of electron Airy beams,” Nature 494, 331–335 (2013).
[Crossref]

Madelung, E.

E. Madelung, “Quantum theory in hydrodynamical form,” Z. Phys. 40, 322 (1927).
[Crossref]

Mathis, A.

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101, 071110 (2012).
[Crossref]

Mazilu, M.

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

Minovich, A.

Mitchell, D. J.

A. W. Snyder and D. J. Mitchell, “Accessible solitons,” Science 276, 1538–1541 (1997).
[Crossref]

Moloney, J. V.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009).
[Crossref]

Moore, R. S.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[Crossref]

Neshev, D. N.

Papazoglou, D. G.

Polynkin, P.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009).
[Crossref]

Porto, S. P. S.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965).
[Crossref]

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]

Renninger, W. H.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4, 103–106 (2010).
[Crossref]

Segev, M.

I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett. 106, 213903 (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]

R. Bekenstein and M. Segev, “Self-accelerating optical beams in highly nonlocal nonlinear media,” Opt. Express 19, 23706 (2011).
[Crossref]

Siviloglou, G. A.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009).
[Crossref]

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Supplementary Material (1)

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» Supplement 1: PDF (1327 KB)      Supplemental Document

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Figures (7)

Fig. 1.
Fig. 1. (a) Setup described in the text. (b) Sketch explaining the formation of a caustic from partial waves reflected off the boundary.
Fig. 2.
Fig. 2. (a) Numerical simulation of Eq. (1) with a 3-mm-diameter, 8-cm-long channel, a centered, 1-mm-diameter Gaussian input beam, the parabolic potential described in the text, U(0)=9000m1, β0=1.181×107m1, and g|A(0)|2=100m1; (a) is a contour plot of the beam intensity as a function of the radial coordinate and the propagation distance inside the channel; (b) is the intensity profile at the output of the channel; (c) self-acceleration and focusing to a point of the output beam as it propagates in free space behind the channel. See also Supplement 1.
Fig. 3.
Fig. 3. Experimental: (a) image of the output plane of the 3 mm channel for a 1 W input beam (note: the cylindrical channel has an opening, i.e., a groove on one side, which results in a cutoff piece that is seen ejected to the left); (b) image of the fluorescence at the output plane of the channel for a 3.5 W input beam and a 27% duty cycle; (c) the ring in (b) focuses to a point after propagating 6cm in free space.
Fig. 4.
Fig. 4. (a) Classical evolution of the Lagrangian manifold in phase space for short times. (b) Corresponding density profiles—the red dashed line corresponds to the initial density, which is arbitrarily taken to be A0(r)=1r2; the purple solid line is the semi-classical result Eq. (6); and the black dots are brute-force numerical solutions of Eq. (2); without loss of generality we take f(r)=(1+cosπr)/2.
Fig. 5.
Fig. 5. (a) Classical evolution of the Lagrangian manifold in phase space for t=400, corresponding to a propagation distance of 8 cm in the experiment—a caustic appears at r=Rc(t). (b) The three classical trajectories q(i)(t) correspond to the three momenta pi in (a); q1(t) and q2(t) merge exactly for r=Rc(t).
Fig. 6.
Fig. 6. (a) Numerical A(r,t) (red dashed line) and crude semi-classical result Eq. (7) (blue solid line) for t=400 (z=8cm); the semi-classical result diverges at the caustic. (b) Same as (a), for the semi-classical result Eq. (8), supplemented with the contribution ψ(3) from Eq. (7).
Fig. 7.
Fig. 7. (a) Numerical A(r,t) (red dashed line) and semi-classical result Eq. (10) (blue solid line) for t=400 (z=8cm). (b) Sketch of semi-classical trajectories (black lines) and the (time-dependent) caustic (red lines) obtained from Eq. (10); the curves show the evolution of A(r,t) See also Supplement 1.

Equations (11)

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iψz=12β02ψ+U(x,y)ψ+g|ψ|2ψ.
itψ=[12mr2+f(r)]ψ(r,t).
ψ(r,t)=A(r,t)eiS(r,t),
St=12m[1A2Ar2(Sr)2]f,
A2t+r(A21mSr)=0.
tS+H(r,rS)=0,
ψ(r,t)=|q0q|1/2A0(q0(r,t))eiS(q0(r,t),t),
ψ(r,t)=b{1,2,3}ψ(b)(r,t)=b{1,2,3}eiμbπ/2A0(q(b)(r,t))|q0(b)q|1/2eiS(q(b)(r,t)),
ψ(1)(r,t)+ψ(2)(r,t)=2πlA0(q˜)|q0(1,2)p|Rc1/2Ai(Rcrl)ei(Sc+kc(rRc))+3π4i,
l3=122qp2|Rc;Sc=S(q˜,t).
ψ(1)(r,t)+ψ(2)(r,t)=πeiSc+iπ[(A0(q0(1)(r,t))|q0(1)q|12+A0(q0(2)(r,t))|q0(2)q|12)ς14Ai(ς)+i(A0(q0(2)(r,t))|q0(2)q|12A0(q0(1)(r,t))|q0(1)q|12)ς14Ai(ς)];ς=ς(r,t)=[34(S(q0(2)(r,t),t)S(q0(1)(r,t),t))]2/3;Sc=Sc(r,t)=S(q0(1)(r,t),t)+S(q0(2)(r,t),t)2.

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