Abstract

A relationship for the complex amplitude of generalized paraxial Hermite–Gaussian (HG) beams is deduced. We show that under certain parameters, these beams transform into the familiar HG modes and elegant HG beams. The orbital angular momentum (OAM) of a linear combination of two generalized HG beams with a phase shift of π/2, with their double indices composed of adjacent integer numbers taken in direct and inverse order, is calculated. The modulus of the OAM is shown to be an integer number for the combination of two HG modes, always equal to unity for the superposition of two elegant HG beams, and a fractional number for two hybrid HG beams. Interestingly, a linear combination of two such HG modes also presents a mode that is characterized by a nonzero OAM and the lack of radial symmetry but does not rotate during propagation.

© 2014 Optical Society of America

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  1. L. Allen, M. W. Beijersergen, 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, 8185–8189 (1992).
    [CrossRef]
  2. M. Berry, J. Nye, and F. Wright, “The elliptic umbilic diffraction catastrophe,” Phil. Trans. R. Soc. Lond. 291, 453–484 (1979).
    [CrossRef]
  3. J. M. Vaughan and D. Willetts, “Interference properties of a light-beam having a helical wave surface,” Opt. Commun. 30, 263–267 (1979).
    [CrossRef]
  4. P. Coullet, G. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
    [CrossRef]
  5. V. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser-beam with screw dislocations in the wavefront,” JETP Lett. 52, 429–431 (1990).
  6. S. N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, “The rotor phase filter,” J. Mod. Opt. 39, 1147–1154 (1992).
    [CrossRef]
  7. E. G. Abramochkin and V. G. Volostnikov, “Beam transformation and nontransformed beams,” Opt. Commun. 83, 123–135 (1991).
    [CrossRef]
  8. M. W. Beijersbergen, L. Allen, H. E. Van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
    [CrossRef]
  9. A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photon. 3, 161–204 (2011).
    [CrossRef]
  10. V. V. Kotlyar and A. A. Kovalev, Vortex Laser Beams (Novaya Tekhnika, 2012) [in Russian].
  11. S. Vyas and P. Senthilkumaran, “Interferometric optical vortex array generator,” Appl. Opt. 46, 2893–2898 (2007).
    [CrossRef]
  12. E. Fraczek and G. Budzyn, “An analysis of an optical vortices interferometer with focused beam,” Opt. Appl. 39, 91–99 (2009).
  13. B. K. Singh, G. Singh, P. Senthilkumaran, and D. S. Metha, “Generation of optical vortex array using single-element reversed-wavefront folding interferometer,” Int. J. Opt. 2012, 689612 (2012).
    [CrossRef]
  14. P. Vaity, A. Aadhi, and R. Singh, “Formation of optical vortices through superposition of two Gaussian beams,” Appl. Opt. 52, 6652–6656 (2013).
    [CrossRef]
  15. Y. Shen, G. T. Campbell, B. Hage, H. Zou, B. C. Buchler, and P. K. Lam, “Generation and interferometric analysis of high charge optical vortices,” J. Opt. 15, 044005 (2013).
    [CrossRef]
  16. J. B. Gotte, K. O’Holleran, D. Precce, F. Flossman, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express 16, 993–1006 (2008).
    [CrossRef]
  17. E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A 6, S157–S161 (2004).
    [CrossRef]
  18. D. P. O’Dwyer, C. F. Phelan, Y. P. Rakovich, P. R. Eastham, J. C. Lunney, and J. F. Donegan, “Generation of continuously tunable fractional orbital angular momentum using internal conical diffraction,” Opt. Express 18, 16480–16485 (2010).
    [CrossRef]
  19. H. Kogelnik and T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
    [CrossRef]
  20. A. E. Siegman, “Hermite-Gaussian functions of complex argument as optical beam eigenfunction,” J. Opt. Soc. Am. 63, 1093–1094 (1973).
    [CrossRef]
  21. F. Gori, “Why is the Fresnel transform so little known?” in Current Trends in Optics, J. C. Dainty, ed. (Academic, 1994), pp. 140–148.
  22. S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Paakkonen, J. Simonen, and J. Turunen, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48, 1543–1557 (2001).
  23. J. Humblet, “Sur le moment d’impulsion d’une onde electromagnetique,” Physica 10, 585–603 (1943).
  24. A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series. Special Functions (Gordon and Breach Science, 1986).
  25. V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8–14 (2007).
    [CrossRef]
  26. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Elsevier, 2007).

2013 (2)

Y. Shen, G. T. Campbell, B. Hage, H. Zou, B. C. Buchler, and P. K. Lam, “Generation and interferometric analysis of high charge optical vortices,” J. Opt. 15, 044005 (2013).
[CrossRef]

P. Vaity, A. Aadhi, and R. Singh, “Formation of optical vortices through superposition of two Gaussian beams,” Appl. Opt. 52, 6652–6656 (2013).
[CrossRef]

2012 (1)

B. K. Singh, G. Singh, P. Senthilkumaran, and D. S. Metha, “Generation of optical vortex array using single-element reversed-wavefront folding interferometer,” Int. J. Opt. 2012, 689612 (2012).
[CrossRef]

2011 (1)

2010 (1)

2009 (1)

E. Fraczek and G. Budzyn, “An analysis of an optical vortices interferometer with focused beam,” Opt. Appl. 39, 91–99 (2009).

2008 (1)

2007 (2)

S. Vyas and P. Senthilkumaran, “Interferometric optical vortex array generator,” Appl. Opt. 46, 2893–2898 (2007).
[CrossRef]

V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8–14 (2007).
[CrossRef]

2004 (1)

E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A 6, S157–S161 (2004).
[CrossRef]

2001 (1)

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Paakkonen, J. Simonen, and J. Turunen, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48, 1543–1557 (2001).

1993 (1)

M. W. Beijersbergen, L. Allen, H. E. Van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

1992 (2)

S. N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, “The rotor phase filter,” J. Mod. Opt. 39, 1147–1154 (1992).
[CrossRef]

L. Allen, M. W. Beijersergen, 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, 8185–8189 (1992).
[CrossRef]

1991 (1)

E. G. Abramochkin and V. G. Volostnikov, “Beam transformation and nontransformed beams,” Opt. Commun. 83, 123–135 (1991).
[CrossRef]

1990 (1)

V. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser-beam with screw dislocations in the wavefront,” JETP Lett. 52, 429–431 (1990).

1989 (1)

P. Coullet, G. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[CrossRef]

1979 (2)

M. Berry, J. Nye, and F. Wright, “The elliptic umbilic diffraction catastrophe,” Phil. Trans. R. Soc. Lond. 291, 453–484 (1979).
[CrossRef]

J. M. Vaughan and D. Willetts, “Interference properties of a light-beam having a helical wave surface,” Opt. Commun. 30, 263–267 (1979).
[CrossRef]

1973 (1)

1966 (1)

H. Kogelnik and T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

1943 (1)

J. Humblet, “Sur le moment d’impulsion d’une onde electromagnetique,” Physica 10, 585–603 (1943).

Aadhi, A.

Abramochkin, E. G.

E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A 6, S157–S161 (2004).
[CrossRef]

E. G. Abramochkin and V. G. Volostnikov, “Beam transformation and nontransformed beams,” Opt. Commun. 83, 123–135 (1991).
[CrossRef]

Allen, L.

M. W. Beijersbergen, L. Allen, H. E. Van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

L. Allen, M. W. Beijersergen, 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, 8185–8189 (1992).
[CrossRef]

Barnett, S. M.

Bazhenov, V.

V. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser-beam with screw dislocations in the wavefront,” JETP Lett. 52, 429–431 (1990).

Beijersbergen, M. W.

M. W. Beijersbergen, L. Allen, H. E. Van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

Beijersergen, M. W.

L. Allen, M. W. Beijersergen, 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, 8185–8189 (1992).
[CrossRef]

Berry, M.

M. Berry, J. Nye, and F. Wright, “The elliptic umbilic diffraction catastrophe,” Phil. Trans. R. Soc. Lond. 291, 453–484 (1979).
[CrossRef]

Brychkov, Yu. A.

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series. Special Functions (Gordon and Breach Science, 1986).

Buchler, B. C.

Y. Shen, G. T. Campbell, B. Hage, H. Zou, B. C. Buchler, and P. K. Lam, “Generation and interferometric analysis of high charge optical vortices,” J. Opt. 15, 044005 (2013).
[CrossRef]

Budzyn, G.

E. Fraczek and G. Budzyn, “An analysis of an optical vortices interferometer with focused beam,” Opt. Appl. 39, 91–99 (2009).

Campbell, G. T.

Y. Shen, G. T. Campbell, B. Hage, H. Zou, B. C. Buchler, and P. K. Lam, “Generation and interferometric analysis of high charge optical vortices,” J. Opt. 15, 044005 (2013).
[CrossRef]

Coullet, P.

P. Coullet, G. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[CrossRef]

Donegan, J. F.

Eastham, P. R.

Flossman, F.

Fraczek, E.

E. Fraczek and G. Budzyn, “An analysis of an optical vortices interferometer with focused beam,” Opt. Appl. 39, 91–99 (2009).

Franke-Arnold, S.

Gil, G.

P. Coullet, G. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[CrossRef]

Gori, F.

F. Gori, “Why is the Fresnel transform so little known?” in Current Trends in Optics, J. C. Dainty, ed. (Academic, 1994), pp. 140–148.

Gotte, J. B.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Elsevier, 2007).

Hage, B.

Y. Shen, G. T. Campbell, B. Hage, H. Zou, B. C. Buchler, and P. K. Lam, “Generation and interferometric analysis of high charge optical vortices,” J. Opt. 15, 044005 (2013).
[CrossRef]

Humblet, J.

J. Humblet, “Sur le moment d’impulsion d’une onde electromagnetique,” Physica 10, 585–603 (1943).

Khonina, S. N.

V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8–14 (2007).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Paakkonen, J. Simonen, and J. Turunen, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48, 1543–1557 (2001).

S. N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, “The rotor phase filter,” J. Mod. Opt. 39, 1147–1154 (1992).
[CrossRef]

Kogelnik, H.

H. Kogelnik and T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Kotlyar, V. V.

V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8–14 (2007).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Paakkonen, J. Simonen, and J. Turunen, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48, 1543–1557 (2001).

S. N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, “The rotor phase filter,” J. Mod. Opt. 39, 1147–1154 (1992).
[CrossRef]

V. V. Kotlyar and A. A. Kovalev, Vortex Laser Beams (Novaya Tekhnika, 2012) [in Russian].

Kovalev, A. A.

V. V. Kotlyar and A. A. Kovalev, Vortex Laser Beams (Novaya Tekhnika, 2012) [in Russian].

Lam, P. K.

Y. Shen, G. T. Campbell, B. Hage, H. Zou, B. C. Buchler, and P. K. Lam, “Generation and interferometric analysis of high charge optical vortices,” J. Opt. 15, 044005 (2013).
[CrossRef]

Li, T.

H. Kogelnik and T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Lunney, J. C.

Marichev, O. I.

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series. Special Functions (Gordon and Breach Science, 1986).

Metha, D. S.

B. K. Singh, G. Singh, P. Senthilkumaran, and D. S. Metha, “Generation of optical vortex array using single-element reversed-wavefront folding interferometer,” Int. J. Opt. 2012, 689612 (2012).
[CrossRef]

Nye, J.

M. Berry, J. Nye, and F. Wright, “The elliptic umbilic diffraction catastrophe,” Phil. Trans. R. Soc. Lond. 291, 453–484 (1979).
[CrossRef]

O’Dwyer, D. P.

O’Holleran, K.

Paakkonen, P.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Paakkonen, J. Simonen, and J. Turunen, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48, 1543–1557 (2001).

Padgett, M. J.

Phelan, C. F.

Precce, D.

Prudnikov, A. P.

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series. Special Functions (Gordon and Breach Science, 1986).

Rakovich, Y. P.

Rocca, F.

P. Coullet, G. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Elsevier, 2007).

Senthilkumaran, P.

B. K. Singh, G. Singh, P. Senthilkumaran, and D. S. Metha, “Generation of optical vortex array using single-element reversed-wavefront folding interferometer,” Int. J. Opt. 2012, 689612 (2012).
[CrossRef]

S. Vyas and P. Senthilkumaran, “Interferometric optical vortex array generator,” Appl. Opt. 46, 2893–2898 (2007).
[CrossRef]

Shen, Y.

Y. Shen, G. T. Campbell, B. Hage, H. Zou, B. C. Buchler, and P. K. Lam, “Generation and interferometric analysis of high charge optical vortices,” J. Opt. 15, 044005 (2013).
[CrossRef]

Shinkarev, M. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, “The rotor phase filter,” J. Mod. Opt. 39, 1147–1154 (1992).
[CrossRef]

Siegman, A. E.

Simonen, J.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Paakkonen, J. Simonen, and J. Turunen, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48, 1543–1557 (2001).

Singh, B. K.

B. K. Singh, G. Singh, P. Senthilkumaran, and D. S. Metha, “Generation of optical vortex array using single-element reversed-wavefront folding interferometer,” Int. J. Opt. 2012, 689612 (2012).
[CrossRef]

Singh, G.

B. K. Singh, G. Singh, P. Senthilkumaran, and D. S. Metha, “Generation of optical vortex array using single-element reversed-wavefront folding interferometer,” Int. J. Opt. 2012, 689612 (2012).
[CrossRef]

Singh, R.

Skidanov, R. V.

V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8–14 (2007).
[CrossRef]

Soifer, V. A.

V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8–14 (2007).
[CrossRef]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Paakkonen, J. Simonen, and J. Turunen, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48, 1543–1557 (2001).

S. N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, “The rotor phase filter,” J. Mod. Opt. 39, 1147–1154 (1992).
[CrossRef]

Soskin, M. S.

V. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser-beam with screw dislocations in the wavefront,” JETP Lett. 52, 429–431 (1990).

Spreeuw, R. J. C.

L. Allen, M. W. Beijersergen, 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, 8185–8189 (1992).
[CrossRef]

Turunen, J.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Paakkonen, J. Simonen, and J. Turunen, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48, 1543–1557 (2001).

Uspleniev, G. V.

S. N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, “The rotor phase filter,” J. Mod. Opt. 39, 1147–1154 (1992).
[CrossRef]

Vaity, P.

Van der Veen, H. E.

M. W. Beijersbergen, L. Allen, H. E. Van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

Vasnetsov, M. V.

V. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser-beam with screw dislocations in the wavefront,” JETP Lett. 52, 429–431 (1990).

Vaughan, J. M.

J. M. Vaughan and D. Willetts, “Interference properties of a light-beam having a helical wave surface,” Opt. Commun. 30, 263–267 (1979).
[CrossRef]

Volostnikov, V. G.

E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A 6, S157–S161 (2004).
[CrossRef]

E. G. Abramochkin and V. G. Volostnikov, “Beam transformation and nontransformed beams,” Opt. Commun. 83, 123–135 (1991).
[CrossRef]

Vyas, S.

Willetts, D.

J. M. Vaughan and D. Willetts, “Interference properties of a light-beam having a helical wave surface,” Opt. Commun. 30, 263–267 (1979).
[CrossRef]

Woerdman, J. P.

M. W. Beijersbergen, L. Allen, H. E. Van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

L. Allen, M. W. Beijersergen, 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, 8185–8189 (1992).
[CrossRef]

Wright, F.

M. Berry, J. Nye, and F. Wright, “The elliptic umbilic diffraction catastrophe,” Phil. Trans. R. Soc. Lond. 291, 453–484 (1979).
[CrossRef]

Yao, A. M.

Zou, H.

Y. Shen, G. T. Campbell, B. Hage, H. Zou, B. C. Buchler, and P. K. Lam, “Generation and interferometric analysis of high charge optical vortices,” J. Opt. 15, 044005 (2013).
[CrossRef]

Adv. Opt. Photon. (1)

Appl. Opt. (2)

Int. J. Opt. (1)

B. K. Singh, G. Singh, P. Senthilkumaran, and D. S. Metha, “Generation of optical vortex array using single-element reversed-wavefront folding interferometer,” Int. J. Opt. 2012, 689612 (2012).
[CrossRef]

J. Mod. Opt. (2)

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, P. Paakkonen, J. Simonen, and J. Turunen, “An analysis of the angular momentum of a light field in terms of angular harmonics,” J. Mod. Opt. 48, 1543–1557 (2001).

S. N. Khonina, V. V. Kotlyar, M. V. Shinkarev, V. A. Soifer, and G. V. Uspleniev, “The rotor phase filter,” J. Mod. Opt. 39, 1147–1154 (1992).
[CrossRef]

J. Opt. (1)

Y. Shen, G. T. Campbell, B. Hage, H. Zou, B. C. Buchler, and P. K. Lam, “Generation and interferometric analysis of high charge optical vortices,” J. Opt. 15, 044005 (2013).
[CrossRef]

J. Opt. A (1)

E. G. Abramochkin and V. G. Volostnikov, “Generalized Gaussian beams,” J. Opt. A 6, S157–S161 (2004).
[CrossRef]

J. Opt. Soc. Am. (1)

JETP Lett. (1)

V. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Laser-beam with screw dislocations in the wavefront,” JETP Lett. 52, 429–431 (1990).

Opt. Appl. (1)

E. Fraczek and G. Budzyn, “An analysis of an optical vortices interferometer with focused beam,” Opt. Appl. 39, 91–99 (2009).

Opt. Commun. (5)

E. G. Abramochkin and V. G. Volostnikov, “Beam transformation and nontransformed beams,” Opt. Commun. 83, 123–135 (1991).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. Van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

J. M. Vaughan and D. Willetts, “Interference properties of a light-beam having a helical wave surface,” Opt. Commun. 30, 263–267 (1979).
[CrossRef]

P. Coullet, G. Gil, and F. Rocca, “Optical vortices,” Opt. Commun. 73, 403–408 (1989).
[CrossRef]

V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274, 8–14 (2007).
[CrossRef]

Opt. Express (2)

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

M. Berry, J. Nye, and F. Wright, “The elliptic umbilic diffraction catastrophe,” Phil. Trans. R. Soc. Lond. 291, 453–484 (1979).
[CrossRef]

Phys. Rev. A (1)

L. Allen, M. W. Beijersergen, 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, 8185–8189 (1992).
[CrossRef]

Physica (1)

J. Humblet, “Sur le moment d’impulsion d’une onde electromagnetique,” Physica 10, 585–603 (1943).

Proc. IEEE (1)

H. Kogelnik and T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Other (4)

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Elsevier, 2007).

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series. Special Functions (Gordon and Breach Science, 1986).

F. Gori, “Why is the Fresnel transform so little known?” in Current Trends in Optics, J. C. Dainty, ed. (Academic, 1994), pp. 140–148.

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Figures (5)

Fig. 1.
Fig. 1.

Intensity of the beam [Eq. (28)] (a) without and (b) with the carrier frequency at w=2λ, b=c=2/w (two HG modes).

Fig. 2.
Fig. 2.

Intensity of the beam [Eq. (28)] (a) without and (b) with the carrier frequency at w=2λ, b=c=1/w (two elegant HG beams).

Fig. 3.
Fig. 3.

Intensity of the beam [Eq. (28)] (a) without and (b) with the carrier frequency at w=2λ, b=1/(7λ), c=1/(3λ) (two generalized HG beams with different widths on the axes).

Fig. 4.
Fig. 4.

Intensity of the beam [Eq. (28)] (a) without and (b) with the carrier frequency at w=2λ, b=1/(5λ), c=1/(5λ) (two generalized HG beams with equal width on the axes).

Fig. 5.
Fig. 5.

Patterns of (a) intensity and (b) phase of the beam [Eq. (28)] with a carrier frequency for the parameters w=2λ, b=c=2/w (mode).

Equations (47)

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E1(x,z)=(iz0z)1/2in×(1iz0z)(n+1)/2[(ab)21+iz0z]n/2×exp[(xa(z))2+ikx22R(z)]Hn(xb(z)),
z0=ka22,a(z)=a[1+(zz0)2]1/2,R(z)=z[1+(z0z)2],b(z)=b(zz0)(1iz0z)1/2[(ab)21+iz0z]1/2.
En(x,z)=in[aa(z)]Hn[2xa(z)]×exp[x2a2(z)+ikx22R(z)i(n+1/2)arctg(zz0)].
Ee(x,z)=(q(z))(n+1)/2×exp[(xaq(z))2]Hn(xaq(z)),
Eh(x,y,z=0)=exp[(xa)2(yc)2]×Hm(2xa)Hn(yc).
E(x,y,0)=exp[w22(x2+y2)]×[H2p(cx)H2s+1(cy)+iγH2s+1(cx)H2p(cy)],
Jz=Im{R2E*(xEyyEx)dxdy}.
Jz=4γc+xexp(w2x2)H2p(cx)H2s+1(cx)dx×[2p+exp(w2y2)H2s+1(cy)H2p1(cy)dy(2s+1)+exp(w2y2)H2p(cy)H2s(cy)dy].
Jz=4πγ[(2p)!(2s+1)!]2w4(p+s+1)×k=0min(p,s+1)[(s+1)c2kw2](c2w2)p+s2k(2c2)2k(pk)!(s+1k)!(2k)!×[k=0min(s,p1)(2c2)2k+1(c2w2)p+s2k1(p1k)!(sk)!(2k+1)!k=0min(p,s)(2c2)2k(c2w2)p+s2k(pk)!(sk)!(2k)!].
exp(px2){H2m+1(bx)H2n+1(cx)H2m(bx)H2n(cx)}dx=(2m+δ)!(2n+δ)!πp×k=0min(m,n)(b2pp)mk(c2pp)nk(2bcp)2k+δ(mk)!(nk)!(2k+δ)!,
xexp(px2)H2m(bx)H2n+1(cx)dx=π(2m)!(2n+1)!cpm+n+3/2×k=0min(m,n+1)(n+1)c2kp(mk)!(n+1k)!(2k)!×(b2p)mk(c2p)nk(2bc)2k.
Jz=24p+2πγ[(2p+1)!]2w2.
I=R2E*Edxdy=π(1+γ2)w221+4p(2p)!(2p+1)!.
JzI=(2γ1+γ2)(2p+1).
Em(x,y)=exp[w22(x2+y2)]×[H2p(wx)H2p+1(wy)+iH2p+1(wx)H2p(wy)]
JzI=(2p+1).
Em(x,y,0)=exp[w22(x2+y2)]×[Hn(wx)Hn+1(wy)+iγHn+1(wx)Hn(wy)],
JzI=2γ(n+1)1+γ2.
Ee(x,y)=exp[w22(x2+y2)]×[H2p(wx2)H2p+1(wy2)+iγH2p+1(wx2)H2p(wy2)].
JzI=2γ1+γ2.
Ee(x,y)=exp[w22(x2+y2)]×[H2p(wx2)H2p+k(wy2)+iγH2p+k(wx2)H2p(wy2)],
JzI=(γ1+γ2)k(4p+k)Γ2(2p+k/2)Γ(2p+1/2)Γ(2p+k+1/2),
JzI=(2γ1+γ2)3(4p+1)(4p+5).
Eh1(x,y)=exp[w22(x2+y2)]×[H2p(wx2)H2p+1(wy)+iγH2p+1(wx)H2p(wy2)].
JzI=(2γ1+γ2)(2p+1)!(4p1)!!.
Eh2(x,y)=exp[w22(x2+y2)]×[H2p(wx)H2p+1(wy2)+iγH2p+1(wx2)H2p(wy)],
JzI=(2γ1+γ2)(2p+1)!(2p+1)(p21)(4p+1)!!.
E(x,y,0)=exp(x2+y2w2)×[H2p(bx)H2s+1(cy)+iγH2s+1(cx)H2p(by)].
I(x,y,z=0)=|E(x,y,0)+Cexp(iαx)|2,
E(x,y)=exp(x2+y22σ2)[(xr0)+iy]=exp(r22σ2)[rexp(iϕ)r0],
JzI=Im{02πdϕ0rdrE*Eϕ}[02πdϕ0rdr|E|2]1.
JzI=11+(r0σ)2.
E(x,y)=exp(x2+y22σ2)[(xr0)+iy]×[(x+r0)+iy]=exp(r22σ2)×[r2exp(2iϕ)r02].
JzI=42+(r0σ)4.
{tanθ=zz0,ρ=r01+(zz0)2,
{tan(2θ)=(2zz0)[1(zz0)2]1/2,ρ=r01+(zz0)2.
E(x,y)=exp(x2+y22σ2)[(xr0)+iy]×[(x+r1)iy]=exp(r22σ2)×[r2r0r1+rr1exp(iϕ)rr0exp(iϕ)].
JzI=(r12r02)(r1r0)2+2σ2+(r0r1σ)2.
E˜(x,y,0)=exp[w22(x2+y2)]×[H2p+1(bx)H2p(cy)+iγH2p(cx)H2p+1(by)],
JzI=(2γ1+γ2)(2p+1),
JzI=2γ1+γ2,
JzI=(2γ1+γ2)(2p+1)!(2p+1)(p21)(4p+1)!!,
JzI=(2γ1+γ2)(2p+1)!(4p1)!!.
E(x,z)=ik2πz+E(u,0)exp[ik2z(ux)2]du,
E(u,0)=Hn(ub)exp[(ua)2],
E(x,z)=ik2πz+Hn(ub)exp[u2a2+ik2z(ux)2]du.
+exp[(xy)2]Hn(αx)dx=π(1α2)n/2Hn(αy1α2)

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