Abstract

We study the self-focusing and defocusing of a light beam carrying angular momentum (called twisted light) propagating in a nonlinear medium. We derive a differential equation for the beam width parameter f as a function of the propagation distance, angular frequency, beam waist and intensity of the beam. The method is based on the Wentzel-Kramers-Brillouin and the paraxial approximations. Analytical expressions for f are obtained, analyzed and illustrated for typical experimental situations.

© 2010 Optical Society of America

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  1. Y. R. Shen, The Principles of Nonlinear Optics, (Wiley, 1984).
  2. W. S. Williams, J. T. Hunt, and W. E. Warren, "Light propagation through large laser system," IEEE J. Quantum Electron. 17, 1727 (1981).
    [CrossRef]
  3. A. M. Rubenchik, M. P. Fedoruk, and S. K. Turitsyn, "Laser beam self-focusing in the atmosphere," Phys. Rev. Lett. 102, 233902 (2009).
    [CrossRef] [PubMed]
  4. G. G. Luther, J. V. Moloney, A. C. Newell, and E. M. Wright, "Self-focusing threshold in normally dispersive media," Opt. Lett. 19, 862 (1994).
    [CrossRef] [PubMed]
  5. E. G. Hanson, Y. R. Shen, and G. K. L. Wong, "Experimental study of self-focusing in a liquid crystalline medium," Appl. Phys. (Berl.) 14, 65 (1977).
    [CrossRef]
  6. S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, "Self focusing and diffraction of light in a nonlinear medium," Sov. Phys. Usp. 10, 609 (1968).
    [CrossRef]
  7. S. D. Patil, M. V. Takale, and M. B. Dongare, "Propagation of Hermite-cosh-Gaussian laser beams in n-InSb," Opt. Commun. 281, 4776 (2008).
    [CrossRef]
  8. S. D. Patil, S. T. Navare, M. V. Takale, and M. B. Dongare, "Self-focusing of cosh-Gaussian laser beams in a parabolic medium with linear absorption," Opt. Lasers Eng. 47, 604 (2009).
    [CrossRef]
  9. M. W. Beijersbergen, L. Allen, H. van Der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123 (1993).
    [CrossRef]
  10. G. Molina-Terriza, J. P. Torres, and L. Torner, "Twisted photons," Nat. Phys. 3, 305 (2007).
    [CrossRef]
  11. M. Padgett, J. Courtial, and L. Allen, "Light’s Orbital Angular Momentum," Phys. Today 57, 35 (2004).
    [CrossRef]
  12. L. Allen, M. J. Padgett, and M. Babiker, "The orbital angular momentum of light," Prog. Opt. XXIX, 291 (1999).
    [CrossRef]
  13. M. S. Sodha, A. K. Ghatak, and V. K. Tripathi, Self Focusing of lasers in plasmas and semiconductors, (New Delhi: Tata McGraw-Hill; 1974).
  14. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185 (1992).
    [CrossRef] [PubMed]
  15. S. Furhapter, A. Esacher, S. Bernet, and M. Ritsch-Marte, "Spiral phase contrast imaging in microscopy," Opt. Express 13, 689 (2005).
    [CrossRef] [PubMed]
  16. A. T. O’Neil, I. Mac Vicar, L. Allen, and M. J. Padgett, "Intrinsic and extrinsic nature of the orbital angular momentum of a light beam," Phys. Rev. Lett. 88, 053601 (2002).
    [CrossRef]
  17. N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, "The mechanical equivalence of the spin and orbital angular momentum of light: an optical spanner," Opt. Lett. 22, 52 (1997).
    [CrossRef] [PubMed]
  18. K. T. Gahagan, and G. A. Schwartzlander, "Optical vortex trapping of particles," Opt. Lett. 21, 827 (1996).
    [CrossRef] [PubMed]
  19. M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Optical alignment and spinning of laser-trapped microscopic particles," Nature 394, 348 (1998).
    [CrossRef]
  20. A. T. O’Neil, and M. J. Padgett, "Three-dimensional optical confinement of micron-sized metal particles and the de-coupling of the spin and orbital angular momentum within an optical spanner," Opt. Commun. 185, 139 (2000).
    [CrossRef]
  21. M. P. MacDonald, L. Paterson, W. Sibbett, K. Dholakia, and P. E. Bryant, "Trapping and manipulation of low-index particles in a two-dimensional interferometric optical trap," Opt. Lett. 26, 863 (2001).
    [CrossRef]
  22. A. E. Siegmann, Lasers, (University Science Book, Mill- Valley, CA, 1986), Sec. 17.5.
  23. J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, "Second-harmonic generation and the conservation of orbital angular momentum with high-order Laguerre-Gaussian modes," Phys. Rev. A 56, 4193 (1997).
    [CrossRef]
  24. L. C. Dvila-Romero, D. L. Andrews, and M. Babiker, "A quantum electrodynamics framework for the nonlinear optics of twisted beams," J. Opt. B Quantum Semiclassical Opt. 4, S66 (2002).
    [CrossRef]

2009 (2)

A. M. Rubenchik, M. P. Fedoruk, and S. K. Turitsyn, "Laser beam self-focusing in the atmosphere," Phys. Rev. Lett. 102, 233902 (2009).
[CrossRef] [PubMed]

S. D. Patil, S. T. Navare, M. V. Takale, and M. B. Dongare, "Self-focusing of cosh-Gaussian laser beams in a parabolic medium with linear absorption," Opt. Lasers Eng. 47, 604 (2009).
[CrossRef]

2008 (1)

S. D. Patil, M. V. Takale, and M. B. Dongare, "Propagation of Hermite-cosh-Gaussian laser beams in n-InSb," Opt. Commun. 281, 4776 (2008).
[CrossRef]

2007 (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, "Twisted photons," Nat. Phys. 3, 305 (2007).
[CrossRef]

2005 (1)

2004 (1)

M. Padgett, J. Courtial, and L. Allen, "Light’s Orbital Angular Momentum," Phys. Today 57, 35 (2004).
[CrossRef]

2002 (2)

A. T. O’Neil, I. Mac Vicar, L. Allen, and M. J. Padgett, "Intrinsic and extrinsic nature of the orbital angular momentum of a light beam," Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef]

L. C. Dvila-Romero, D. L. Andrews, and M. Babiker, "A quantum electrodynamics framework for the nonlinear optics of twisted beams," J. Opt. B Quantum Semiclassical Opt. 4, S66 (2002).
[CrossRef]

2001 (1)

2000 (1)

A. T. O’Neil, and M. J. Padgett, "Three-dimensional optical confinement of micron-sized metal particles and the de-coupling of the spin and orbital angular momentum within an optical spanner," Opt. Commun. 185, 139 (2000).
[CrossRef]

1999 (1)

L. Allen, M. J. Padgett, and M. Babiker, "The orbital angular momentum of light," Prog. Opt. XXIX, 291 (1999).
[CrossRef]

1998 (1)

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Optical alignment and spinning of laser-trapped microscopic particles," Nature 394, 348 (1998).
[CrossRef]

1997 (2)

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, "Second-harmonic generation and the conservation of orbital angular momentum with high-order Laguerre-Gaussian modes," Phys. Rev. A 56, 4193 (1997).
[CrossRef]

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, "The mechanical equivalence of the spin and orbital angular momentum of light: an optical spanner," Opt. Lett. 22, 52 (1997).
[CrossRef] [PubMed]

1996 (1)

1994 (1)

1993 (1)

M. W. Beijersbergen, L. Allen, H. van Der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123 (1993).
[CrossRef]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

1981 (1)

W. S. Williams, J. T. Hunt, and W. E. Warren, "Light propagation through large laser system," IEEE J. Quantum Electron. 17, 1727 (1981).
[CrossRef]

1977 (1)

E. G. Hanson, Y. R. Shen, and G. K. L. Wong, "Experimental study of self-focusing in a liquid crystalline medium," Appl. Phys. (Berl.) 14, 65 (1977).
[CrossRef]

1968 (1)

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, "Self focusing and diffraction of light in a nonlinear medium," Sov. Phys. Usp. 10, 609 (1968).
[CrossRef]

Akhmanov, S. A.

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, "Self focusing and diffraction of light in a nonlinear medium," Sov. Phys. Usp. 10, 609 (1968).
[CrossRef]

Allen, L.

M. Padgett, J. Courtial, and L. Allen, "Light’s Orbital Angular Momentum," Phys. Today 57, 35 (2004).
[CrossRef]

A. T. O’Neil, I. Mac Vicar, L. Allen, and M. J. Padgett, "Intrinsic and extrinsic nature of the orbital angular momentum of a light beam," Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef]

L. Allen, M. J. Padgett, and M. Babiker, "The orbital angular momentum of light," Prog. Opt. XXIX, 291 (1999).
[CrossRef]

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, "The mechanical equivalence of the spin and orbital angular momentum of light: an optical spanner," Opt. Lett. 22, 52 (1997).
[CrossRef] [PubMed]

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, "Second-harmonic generation and the conservation of orbital angular momentum with high-order Laguerre-Gaussian modes," Phys. Rev. A 56, 4193 (1997).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. van Der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123 (1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Andrews, D. L.

L. C. Dvila-Romero, D. L. Andrews, and M. Babiker, "A quantum electrodynamics framework for the nonlinear optics of twisted beams," J. Opt. B Quantum Semiclassical Opt. 4, S66 (2002).
[CrossRef]

Babiker, M.

L. C. Dvila-Romero, D. L. Andrews, and M. Babiker, "A quantum electrodynamics framework for the nonlinear optics of twisted beams," J. Opt. B Quantum Semiclassical Opt. 4, S66 (2002).
[CrossRef]

L. Allen, M. J. Padgett, and M. Babiker, "The orbital angular momentum of light," Prog. Opt. XXIX, 291 (1999).
[CrossRef]

Beijersbergen, M. W.

M. W. Beijersbergen, L. Allen, H. van Der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123 (1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Bernet, S.

Bryant, P. E.

Courtial, J.

M. Padgett, J. Courtial, and L. Allen, "Light’s Orbital Angular Momentum," Phys. Today 57, 35 (2004).
[CrossRef]

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, "Second-harmonic generation and the conservation of orbital angular momentum with high-order Laguerre-Gaussian modes," Phys. Rev. A 56, 4193 (1997).
[CrossRef]

Dholakia, K.

Dongare, M. B.

S. D. Patil, S. T. Navare, M. V. Takale, and M. B. Dongare, "Self-focusing of cosh-Gaussian laser beams in a parabolic medium with linear absorption," Opt. Lasers Eng. 47, 604 (2009).
[CrossRef]

S. D. Patil, M. V. Takale, and M. B. Dongare, "Propagation of Hermite-cosh-Gaussian laser beams in n-InSb," Opt. Commun. 281, 4776 (2008).
[CrossRef]

Dvila-Romero, L. C.

L. C. Dvila-Romero, D. L. Andrews, and M. Babiker, "A quantum electrodynamics framework for the nonlinear optics of twisted beams," J. Opt. B Quantum Semiclassical Opt. 4, S66 (2002).
[CrossRef]

Esacher, A.

Fedoruk, M. P.

A. M. Rubenchik, M. P. Fedoruk, and S. K. Turitsyn, "Laser beam self-focusing in the atmosphere," Phys. Rev. Lett. 102, 233902 (2009).
[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," Nature 394, 348 (1998).
[CrossRef]

Furhapter, S.

Gahagan, K. T.

Hanson, E. G.

E. G. Hanson, Y. R. Shen, and G. K. L. Wong, "Experimental study of self-focusing in a liquid crystalline medium," Appl. Phys. (Berl.) 14, 65 (1977).
[CrossRef]

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," Nature 394, 348 (1998).
[CrossRef]

Hunt, J. T.

W. S. Williams, J. T. Hunt, and W. E. Warren, "Light propagation through large laser system," IEEE J. Quantum Electron. 17, 1727 (1981).
[CrossRef]

Khokhlov, R. V.

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, "Self focusing and diffraction of light in a nonlinear medium," Sov. Phys. Usp. 10, 609 (1968).
[CrossRef]

Luther, G. G.

Mac Vicar, I.

A. T. O’Neil, I. Mac Vicar, L. Allen, and M. J. Padgett, "Intrinsic and extrinsic nature of the orbital angular momentum of a light beam," Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef]

MacDonald, M. P.

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres, and L. Torner, "Twisted photons," Nat. Phys. 3, 305 (2007).
[CrossRef]

Moloney, J. V.

Navare, S. T.

S. D. Patil, S. T. Navare, M. V. Takale, and M. B. Dongare, "Self-focusing of cosh-Gaussian laser beams in a parabolic medium with linear absorption," Opt. Lasers Eng. 47, 604 (2009).
[CrossRef]

Newell, A. C.

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," Nature 394, 348 (1998).
[CrossRef]

O’Neil, A. T.

A. T. O’Neil, I. Mac Vicar, L. Allen, and M. J. Padgett, "Intrinsic and extrinsic nature of the orbital angular momentum of a light beam," Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef]

A. T. O’Neil, and M. J. Padgett, "Three-dimensional optical confinement of micron-sized metal particles and the de-coupling of the spin and orbital angular momentum within an optical spanner," Opt. Commun. 185, 139 (2000).
[CrossRef]

Padgett, M.

M. Padgett, J. Courtial, and L. Allen, "Light’s Orbital Angular Momentum," Phys. Today 57, 35 (2004).
[CrossRef]

Padgett, M. J.

A. T. O’Neil, I. Mac Vicar, L. Allen, and M. J. Padgett, "Intrinsic and extrinsic nature of the orbital angular momentum of a light beam," Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef]

A. T. O’Neil, and M. J. Padgett, "Three-dimensional optical confinement of micron-sized metal particles and the de-coupling of the spin and orbital angular momentum within an optical spanner," Opt. Commun. 185, 139 (2000).
[CrossRef]

L. Allen, M. J. Padgett, and M. Babiker, "The orbital angular momentum of light," Prog. Opt. XXIX, 291 (1999).
[CrossRef]

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, "Second-harmonic generation and the conservation of orbital angular momentum with high-order Laguerre-Gaussian modes," Phys. Rev. A 56, 4193 (1997).
[CrossRef]

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, "The mechanical equivalence of the spin and orbital angular momentum of light: an optical spanner," Opt. Lett. 22, 52 (1997).
[CrossRef] [PubMed]

Paterson, L.

Patil, S. D.

S. D. Patil, S. T. Navare, M. V. Takale, and M. B. Dongare, "Self-focusing of cosh-Gaussian laser beams in a parabolic medium with linear absorption," Opt. Lasers Eng. 47, 604 (2009).
[CrossRef]

S. D. Patil, M. V. Takale, and M. B. Dongare, "Propagation of Hermite-cosh-Gaussian laser beams in n-InSb," Opt. Commun. 281, 4776 (2008).
[CrossRef]

Ritsch-Marte, M.

Rubenchik, A. M.

A. M. Rubenchik, M. P. Fedoruk, and S. K. Turitsyn, "Laser beam self-focusing in the atmosphere," Phys. Rev. Lett. 102, 233902 (2009).
[CrossRef] [PubMed]

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," Nature 394, 348 (1998).
[CrossRef]

Schwartzlander, G. A.

Shen, Y. R.

E. G. Hanson, Y. R. Shen, and G. K. L. Wong, "Experimental study of self-focusing in a liquid crystalline medium," Appl. Phys. (Berl.) 14, 65 (1977).
[CrossRef]

Sibbett, W.

Simpson, N. B.

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Sukhorukov, A. P.

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, "Self focusing and diffraction of light in a nonlinear medium," Sov. Phys. Usp. 10, 609 (1968).
[CrossRef]

Takale, M. V.

S. D. Patil, S. T. Navare, M. V. Takale, and M. B. Dongare, "Self-focusing of cosh-Gaussian laser beams in a parabolic medium with linear absorption," Opt. Lasers Eng. 47, 604 (2009).
[CrossRef]

S. D. Patil, M. V. Takale, and M. B. Dongare, "Propagation of Hermite-cosh-Gaussian laser beams in n-InSb," Opt. Commun. 281, 4776 (2008).
[CrossRef]

Torner, L.

G. Molina-Terriza, J. P. Torres, and L. Torner, "Twisted photons," Nat. Phys. 3, 305 (2007).
[CrossRef]

Torres, J. P.

G. Molina-Terriza, J. P. Torres, and L. Torner, "Twisted photons," Nat. Phys. 3, 305 (2007).
[CrossRef]

Turitsyn, S. K.

A. M. Rubenchik, M. P. Fedoruk, and S. K. Turitsyn, "Laser beam self-focusing in the atmosphere," Phys. Rev. Lett. 102, 233902 (2009).
[CrossRef] [PubMed]

van Der Veen, H.

M. W. Beijersbergen, L. Allen, H. van Der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123 (1993).
[CrossRef]

Warren, W. E.

W. S. Williams, J. T. Hunt, and W. E. Warren, "Light propagation through large laser system," IEEE J. Quantum Electron. 17, 1727 (1981).
[CrossRef]

Williams, W. S.

W. S. Williams, J. T. Hunt, and W. E. Warren, "Light propagation through large laser system," IEEE J. Quantum Electron. 17, 1727 (1981).
[CrossRef]

Woerdman, J. P.

M. W. Beijersbergen, L. Allen, H. van Der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123 (1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Wong, G. K. L.

E. G. Hanson, Y. R. Shen, and G. K. L. Wong, "Experimental study of self-focusing in a liquid crystalline medium," Appl. Phys. (Berl.) 14, 65 (1977).
[CrossRef]

Wright, E. M.

Appl. Phys. (Berl.) (1)

E. G. Hanson, Y. R. Shen, and G. K. L. Wong, "Experimental study of self-focusing in a liquid crystalline medium," Appl. Phys. (Berl.) 14, 65 (1977).
[CrossRef]

IEEE J. Quantum Electron. (1)

W. S. Williams, J. T. Hunt, and W. E. Warren, "Light propagation through large laser system," IEEE J. Quantum Electron. 17, 1727 (1981).
[CrossRef]

J. Opt. B Quantum Semiclassical Opt. (1)

L. C. Dvila-Romero, D. L. Andrews, and M. Babiker, "A quantum electrodynamics framework for the nonlinear optics of twisted beams," J. Opt. B Quantum Semiclassical Opt. 4, S66 (2002).
[CrossRef]

Nat. Phys. (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, "Twisted photons," Nat. Phys. 3, 305 (2007).
[CrossRef]

Nature (1)

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Optical alignment and spinning of laser-trapped microscopic particles," Nature 394, 348 (1998).
[CrossRef]

Opt. Commun. (3)

A. T. O’Neil, and M. J. Padgett, "Three-dimensional optical confinement of micron-sized metal particles and the de-coupling of the spin and orbital angular momentum within an optical spanner," Opt. Commun. 185, 139 (2000).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. van Der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123 (1993).
[CrossRef]

S. D. Patil, M. V. Takale, and M. B. Dongare, "Propagation of Hermite-cosh-Gaussian laser beams in n-InSb," Opt. Commun. 281, 4776 (2008).
[CrossRef]

Opt. Express (1)

Opt. Lasers Eng. (1)

S. D. Patil, S. T. Navare, M. V. Takale, and M. B. Dongare, "Self-focusing of cosh-Gaussian laser beams in a parabolic medium with linear absorption," Opt. Lasers Eng. 47, 604 (2009).
[CrossRef]

Opt. Lett. (4)

Phys. Rev. A (2)

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, "Second-harmonic generation and the conservation of orbital angular momentum with high-order Laguerre-Gaussian modes," Phys. Rev. A 56, 4193 (1997).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

A. T. O’Neil, I. Mac Vicar, L. Allen, and M. J. Padgett, "Intrinsic and extrinsic nature of the orbital angular momentum of a light beam," Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef]

A. M. Rubenchik, M. P. Fedoruk, and S. K. Turitsyn, "Laser beam self-focusing in the atmosphere," Phys. Rev. Lett. 102, 233902 (2009).
[CrossRef] [PubMed]

Phys. Today (1)

M. Padgett, J. Courtial, and L. Allen, "Light’s Orbital Angular Momentum," Phys. Today 57, 35 (2004).
[CrossRef]

Prog. Opt. (1)

L. Allen, M. J. Padgett, and M. Babiker, "The orbital angular momentum of light," Prog. Opt. XXIX, 291 (1999).
[CrossRef]

Sov. Phys. Usp. (1)

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, "Self focusing and diffraction of light in a nonlinear medium," Sov. Phys. Usp. 10, 609 (1968).
[CrossRef]

Other (3)

Y. R. Shen, The Principles of Nonlinear Optics, (Wiley, 1984).

M. S. Sodha, A. K. Ghatak, and V. K. Tripathi, Self Focusing of lasers in plasmas and semiconductors, (New Delhi: Tata McGraw-Hill; 1974).

A. E. Siegmann, Lasers, (University Science Book, Mill- Valley, CA, 1986), Sec. 17.5.

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Figures (1)

Fig. 1
Fig. 1

(color online) (a) Intensity in CGS units of LG beam verses the radial distance from the propagation direction (in cm) for the l = 1, and p = 0. The angular frequency is ω = 2 × 1014 rad/sec, w0 = 1 [cm], ɛ0 = 1, α = 1, E0 = 0.3 [StatV /cm]. ρ = 0.66 × 104. (b) Initial intensity profile (dotted curve) compared to the propagated intensity at ξ = 4 × 10–4.

Equations (22)

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u l p L G ( r , ϕ , z ) = C p l w ( z ) [ 2 r w ( z ) ] l exp [ r 2 w 2 ( z ) ] L p l ( 2 r 2 w 2 ( z ) ) × exp [ i k r 2 z 2 ( z 2 + z R 2 ) i l ϕ + i ( 2 p + l + 1 ) arctan ( z z R ) ] ,
u l p L G ( r , ϕ , z = 0 ) = C p l [ 2 r w 0 ] l exp [ r 2 w 0 2 ] L p l ( 2 r 2 w 0 2 ) exp ( i l ϕ ) .
ɛ ( r , z ) ɛ 0 ( z ) r 2 ɛ 2 ( z ) .
2 E + ω 2 c 2 ɛ E + ( E ɛ ɛ ) = 0
2 E + ω 2 c 2 ɛ E = 0
E ( r , ϕ , z ) = A ( r , ϕ , z ) exp [ i ( ω t k z ) ]
2 i k A z = 2 A r 2 + 1 r A r + 1 r 2 2 A ϕ 2 ω 2 c 2 ɛ 2 r 2 A .
A ( r , ϕ , z ) = A 0 ( r , z ) exp { i [ k S ( r , z ) l ϕ ] }
S = r 2 2 β ( z ) + Θ ( z ) .
β ( z ) = 1 f d f d z .
2 S z + ( S r ) 2 = 1 k 2 A 0 [ 2 A 0 r 2 + 1 r A 0 r l 2 r 2 A 0 ] ɛ 2 ɛ 0 r 2
A 0 2 z + A 0 2 r S r + A 0 2 ( 2 S r 2 + 1 r S r ) = 0
A 0 ( r , z ) = E 0 f ( 2 r w 0 f ) l exp ( r 2 w 0 2 f 2 ) L P l ( 2 r 2 w 0 2 ) .
1 f d 2 f d z 2 = 4 c 2 ω 2 ɛ 0 ( z ) w 0 4 f 4 ɛ 2 ( z ) ɛ 0 ( z ) .
f = 1 and d f d z = 0 at z = 0 .
ξ = z c w 0 2 ω
ρ = w 0 ω c
ɛ 0 ( z ) f d 2 f d ξ 2 = 4 f 4 ρ 2 w 0 2 ω 2 ( z ) .
ɛ 2 ( f ) r 2 = α E 0 2 f 2 r 2 w 0 2 f 2
1 f d 2 f d ξ 2 = 1 ɛ 0 f 4 ( 4 ρ 2 α E 0 2 ) .
f = ɛ 0 + 4 ξ 2 E 0 2 α ξ 2 ρ 2 ɛ 0 .
d 2 f d z 2 = c 2 ɛ 0 w 0 4 ω 2 ( 4 α w 0 2 ω 2 c 2 E 0 2 ) f 3 .

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