Abstract

We study the dynamics of a paired optical vortex (OV) generated by second-harmonic generation (SHG) using sub-picosecond pulses. By changing the position of a thin nonlinear crystal along the propagation direction, we observe a rotation of two vortices with changing separation distance. The dynamics is well explained by SHG with a beam walk-off, which introduces a contamination of zero-order Laguerre-Gaussian beam (LG0) together with topological charge doubling. The quantitative analysis indicates that the rotation angle of the OVs manifests the Gouy phase while the splitting provides the walk-off angle of the crystal. We also show that the subtraction of LG0 is realized by the superposition of LG0 with an anti-balanced phase in the pump.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. K. T. Gahagan and G. A. Swartzlander, Jr., “Optical vortex trapping of particles,” Opt. Lett. 21, 827–829 (1996).
    [CrossRef] [PubMed]
  2. D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
    [CrossRef] [PubMed]
  3. Y. Tokizane, K. Shimatake, Y. Toda, K. Oka, M. Tsubota, S. Tanda, and R. Morita, “Global evaluation of closed-loop electron dynamics in quasi-one-dimensional conductors using polarization vortices,” Opt. Express 17, 24198–24207 (2009).
    [CrossRef]
  4. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004).
    [CrossRef] [PubMed]
  5. 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, 8185–8189 (1992).
    [CrossRef] [PubMed]
  6. M. Berry, “Making waves in physics,” Nature 403, 21–21 (2000).
    [CrossRef] [PubMed]
  7. M. J. Paz-Alonso and H. Michinel, “Superfluidlike Motion of Vortices in Light Condensates,” Phys. Rev. Lett. 94, 093901-1–4 (2005).
    [CrossRef] [PubMed]
  8. I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light-beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
    [CrossRef]
  9. K. Staliunas, “Dynamics of optical vortices in a laser beam,” Opt. Commun. 90, 123–127 (1992).
    [CrossRef]
  10. G. Indebetow, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
    [CrossRef]
  11. M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
    [CrossRef]
  12. D. Rozas, C. T. Law, G. A. Swartzlander, “Propagation dynamics of optical vortices,” J. Opt. Soc. Am. B 14, 3054–3065 (1997).
    [CrossRef]
  13. . I. Freund, “Optical vortex trajectories,” Opt. Commun. 181, 19–33 (2000). I. D. Maleev, and G. A. Swartzlander, “Composite optical vortices,” J. Opt. Soc. Am. B 20, 1169–1176 (2003).
    [CrossRef]
  14. K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, “Second-harmonic generation and the orbital angular momentum of light,” Phys. Rev. A 54, R3742–3745 (1996).
    [CrossRef] [PubMed]
  15. A. Berzanskis, A. Matijosius, A. Piskarskas, V. Smilgevicius, and A. Stabinis, “Sum-frequency mixing of optical vortices in nonlinear crystals,” Opt. Commun. 150, 372–280 (1998).
    [CrossRef]
  16. F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901-1–4 (2005).
    [CrossRef] [PubMed]
  17. A. Dreischuh, D. N. Neshev, V. Z. Kolev, S. Saltiel, M. Samoc, W. Krolikowski, and Y. S. Kivshar, “Nonlinear dynamics of two-color optical vortices in lithium niobate crystals,” Opt. Express 16, 5406–5420 (2008).
    [CrossRef] [PubMed]
  18. 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, 123–132 (1993).
    [CrossRef]
  19. Y. Yoshikawa and H. Sasada, “Versatile generation of optical vortices based on paraxial mode expansion,” J. Opt. Soc. Am. A 19, 2127–2133 (2002).
    [CrossRef]
  20. K. Kato, “Second-harmonic generation to 2048 Å in β -BaB2O4,” IEEE J. Quantum Electron.,  QE-22, 1013–1014 (1986).
    [CrossRef]
  21. S. M. Baumann, D. M. Kalb, L. H. MacMillan, E. J. Galvez, “Propagation dynamics of optical vortices due to Gouy phase,” Opt. Express 17, 9818–9827 (2009).
    [CrossRef] [PubMed]
  22. J. H. Chow, G. de Vine, M. B. Gray, and D. E. McClelland, “Measurement of Gouy phase evolution by use of spatial mode interference,” Opt. Lett. 29, 2339–2341 (2004).
    [CrossRef] [PubMed]
  23. J. Hamazaki, Y. Mineta, K. Oka, and R. Morita, "Direct observation of Gouy phase shift in a propagating optical vortex,” Opt. Express 14, 8382–8392 (2006).
    [CrossRef] [PubMed]
  24. A. Bahabad and A. Arie, “Generation of optical vortex beams by nonlinear wave mixing,” Opt. Express 15, 17619–17624 (2007).
    [CrossRef] [PubMed]
  25. Y. Ueno, Y. Toda, S. Adachi, R. Morita and T. Tawara, “Coherent transfer of orbital angular momentum to excitons by optical four-wave mixing,” Opt. Express 17, 20567–20574 (2009).
    [CrossRef] [PubMed]

2009

2008

2007

2006

2004

2003

2002

2000

M. Berry, “Making waves in physics,” Nature 403, 21–21 (2000).
[CrossRef] [PubMed]

1998

A. Berzanskis, A. Matijosius, A. Piskarskas, V. Smilgevicius, and A. Stabinis, “Sum-frequency mixing of optical vortices in nonlinear crystals,” Opt. Commun. 150, 372–280 (1998).
[CrossRef]

1997

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[CrossRef]

D. Rozas, C. T. Law, G. A. Swartzlander, “Propagation dynamics of optical vortices,” J. Opt. Soc. Am. B 14, 3054–3065 (1997).
[CrossRef]

1996

K. T. Gahagan and G. A. Swartzlander, Jr., “Optical vortex trapping of particles,” Opt. Lett. 21, 827–829 (1996).
[CrossRef] [PubMed]

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, “Second-harmonic generation and the orbital angular momentum of light,” Phys. Rev. A 54, R3742–3745 (1996).
[CrossRef] [PubMed]

1993

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, 123–132 (1993).
[CrossRef]

I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light-beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

G. Indebetow, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

1992

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, 8185–8189 (1992).
[CrossRef] [PubMed]

K. Staliunas, “Dynamics of optical vortices in a laser beam,” Opt. Commun. 90, 123–127 (1992).
[CrossRef]

1986

K. Kato, “Second-harmonic generation to 2048 Å in β -BaB2O4,” IEEE J. Quantum Electron.,  QE-22, 1013–1014 (1986).
[CrossRef]

Adachi, S.

Allen, L.

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, “Second-harmonic generation and the orbital angular momentum of light,” Phys. Rev. A 54, R3742–3745 (1996).
[CrossRef] [PubMed]

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, 123–132 (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, 8185–8189 (1992).
[CrossRef] [PubMed]

Arie, A.

Bahabad, A.

Barnett, S. M.

Basistiy, I. V.

I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light-beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

Baumann, S. M.

Bazhenov, V. Y.

I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light-beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[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, 123–132 (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, 8185–8189 (1992).
[CrossRef] [PubMed]

Ber?zanskis, A.

A. Berzanskis, A. Matijosius, A. Piskarskas, V. Smilgevicius, and A. Stabinis, “Sum-frequency mixing of optical vortices in nonlinear crystals,” Opt. Commun. 150, 372–280 (1998).
[CrossRef]

Berry, M.

M. Berry, “Making waves in physics,” Nature 403, 21–21 (2000).
[CrossRef] [PubMed]

Chow, J. H.

Courtial, J.

de Vine, G.

Dholakia, K.

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, “Second-harmonic generation and the orbital angular momentum of light,” Phys. Rev. A 54, R3742–3745 (1996).
[CrossRef] [PubMed]

Dreischuh, A.

Franke-Arnold, S.

Freund, I.

Gahagan, K. T.

Galvez, E. J.

Gibson, G.

Gorshkov, V. N.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[CrossRef]

Gray, M. B.

Grier, D. G.

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[CrossRef] [PubMed]

Hamazaki, J.

Heckenberg, N. R.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[CrossRef]

Indebetow, G.

G. Indebetow, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

Kalb, D. M.

Kato, K.

K. Kato, “Second-harmonic generation to 2048 Å in β -BaB2O4,” IEEE J. Quantum Electron.,  QE-22, 1013–1014 (1986).
[CrossRef]

Kivshar, Y. S.

Kolev, V. Z.

Krolikowski, W.

Law, C. T.

MacMillan, L. H.

Malos, J. T.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[CrossRef]

Matijo?sius, A.

A. Berzanskis, A. Matijosius, A. Piskarskas, V. Smilgevicius, and A. Stabinis, “Sum-frequency mixing of optical vortices in nonlinear crystals,” Opt. Commun. 150, 372–280 (1998).
[CrossRef]

McClelland, D. E.

Mineta, Y.

Morita, R.

Neshev, D. N.

Oka, K.

Padgett, M. J.

Pas’ko, V.

Piskarskas, A.

A. Berzanskis, A. Matijosius, A. Piskarskas, V. Smilgevicius, and A. Stabinis, “Sum-frequency mixing of optical vortices in nonlinear crystals,” Opt. Commun. 150, 372–280 (1998).
[CrossRef]

Rozas, D.

Saltiel, S.

Samoc, M.

Sasada, H.

Shimatake, K.

Simpson, N. B.

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, “Second-harmonic generation and the orbital angular momentum of light,” Phys. Rev. A 54, R3742–3745 (1996).
[CrossRef] [PubMed]

Smilgevi?cius, V.

A. Berzanskis, A. Matijosius, A. Piskarskas, V. Smilgevicius, and A. Stabinis, “Sum-frequency mixing of optical vortices in nonlinear crystals,” Opt. Commun. 150, 372–280 (1998).
[CrossRef]

Soskin, M. S.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[CrossRef]

I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light-beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

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, 8185–8189 (1992).
[CrossRef] [PubMed]

Stabinis, A.

A. Berzanskis, A. Matijosius, A. Piskarskas, V. Smilgevicius, and A. Stabinis, “Sum-frequency mixing of optical vortices in nonlinear crystals,” Opt. Commun. 150, 372–280 (1998).
[CrossRef]

Staliunas, K.

K. Staliunas, “Dynamics of optical vortices in a laser beam,” Opt. Commun. 90, 123–127 (1992).
[CrossRef]

Swartzlander, G. A.

Tanda, S.

Tawara, T.

Toda, Y.

Tokizane, Y.

Tsubota, M.

Ueno, Y.

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, 123–132 (1993).
[CrossRef]

Vasnetsov, M.

Vasnetsov, M. V.

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[CrossRef]

I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light-beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[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, 123–132 (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, 8185–8189 (1992).
[CrossRef] [PubMed]

Yoshikawa, Y.

IEEE J. Quantum Electron.

K. Kato, “Second-harmonic generation to 2048 Å in β -BaB2O4,” IEEE J. Quantum Electron.,  QE-22, 1013–1014 (1986).
[CrossRef]

J. Mod. Opt.

G. Indebetow, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Nature

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[CrossRef] [PubMed]

M. Berry, “Making waves in physics,” Nature 403, 21–21 (2000).
[CrossRef] [PubMed]

Opt. Commun.

I. V. Basistiy, V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Optics of light-beams with screw dislocations,” Opt. Commun. 103, 422–428 (1993).
[CrossRef]

K. Staliunas, “Dynamics of optical vortices in a laser beam,” Opt. Commun. 90, 123–127 (1992).
[CrossRef]

A. Berzanskis, A. Matijosius, A. Piskarskas, V. Smilgevicius, and A. Stabinis, “Sum-frequency mixing of optical vortices in nonlinear crystals,” Opt. Commun. 150, 372–280 (1998).
[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, 123–132 (1993).
[CrossRef]

Opt. Express

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004).
[CrossRef] [PubMed]

J. Hamazaki, Y. Mineta, K. Oka, and R. Morita, "Direct observation of Gouy phase shift in a propagating optical vortex,” Opt. Express 14, 8382–8392 (2006).
[CrossRef] [PubMed]

A. Bahabad and A. Arie, “Generation of optical vortex beams by nonlinear wave mixing,” Opt. Express 15, 17619–17624 (2007).
[CrossRef] [PubMed]

A. Dreischuh, D. N. Neshev, V. Z. Kolev, S. Saltiel, M. Samoc, W. Krolikowski, and Y. S. Kivshar, “Nonlinear dynamics of two-color optical vortices in lithium niobate crystals,” Opt. Express 16, 5406–5420 (2008).
[CrossRef] [PubMed]

S. M. Baumann, D. M. Kalb, L. H. MacMillan, E. J. Galvez, “Propagation dynamics of optical vortices due to Gouy phase,” Opt. Express 17, 9818–9827 (2009).
[CrossRef] [PubMed]

Y. Ueno, Y. Toda, S. Adachi, R. Morita and T. Tawara, “Coherent transfer of orbital angular momentum to excitons by optical four-wave mixing,” Opt. Express 17, 20567–20574 (2009).
[CrossRef] [PubMed]

Y. Tokizane, K. Shimatake, Y. Toda, K. Oka, M. Tsubota, S. Tanda, and R. Morita, “Global evaluation of closed-loop electron dynamics in quasi-one-dimensional conductors using polarization vortices,” Opt. Express 17, 24198–24207 (2009).
[CrossRef]

Opt. Lett.

Phys. Rev. A

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, 8185–8189 (1992).
[CrossRef] [PubMed]

K. Dholakia, N. B. Simpson, M. J. Padgett, and L. Allen, “Second-harmonic generation and the orbital angular momentum of light,” Phys. Rev. A 54, R3742–3745 (1996).
[CrossRef] [PubMed]

M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos, and N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1997).
[CrossRef]

Other

M. J. Paz-Alonso and H. Michinel, “Superfluidlike Motion of Vortices in Light Condensates,” Phys. Rev. Lett. 94, 093901-1–4 (2005).
[CrossRef] [PubMed]

F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Polarization singularities from unfolding an optical vortex through a birefringent crystal,” Phys. Rev. Lett. 95, 253901-1–4 (2005).
[CrossRef] [PubMed]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1.
Fig. 1.

(a) Schematic illustration of SHG setup. The HG00 (TEM00) beam from 100 femtosecond (fs) mode-locked Ti:sapphire (ML: TiS) is converted to a doughnut-like LG1 by using a spatial phase shifter (SPS) and an astigmatic mode converter (AMC). The second-harmonic OVs are generated by focusing onto a thin BBO (ø 5 mm × 0.1 mm) crystal on a motorized linear stage using a conventional lens (f = 100 mm), and detected by a charge-coupled device (CCD) after passing through a UV-pass filter. The development of the second-harmonic vortices is observed by changing the crystal position along the propagation direction. (b) Observed intensity distribution of fundamental LG1 (left) and its interferogram with HG00 (right). (c) Schematic illustration of the second-harmonic OVs with a charge splitting. The distance (rv ) and rotation angle (ϕv ) between two vortices are denoted.

Fig. 2.
Fig. 2.

Observed intensity distributions of a paired second-harmonic vortex at various positions of the BBO crystal (z′).

Fig. 3.
Fig. 3.

(a) Trajectories of the second-harmonic vortices at various z′. Plots of different colors show typical positions. Evolutions of (b) ϕv and (c) rv as a function of z′/zR .

Fig. 4.
Fig. 4.

Calculated intensity (upper) and phase (lower) distributions of the paired second-harmonic vortex at various z′.

Fig. 5.
Fig. 5.

(a) Calculated trajectories of the second-harmonic vortices at various z′. Evolutions of (b) ϕv and (c) rv as a function of z′/zR . Experimental plots of Fig. 3 are also shown by gray color.

Fig. 6.
Fig. 6.

Intensity distributions of a paired second-harmonic vortex obtained by introducing LG 1 + e i φ p LG 0 . The arrows indicate the vortices for clarity. The crystal is fixed at z′ = 0. The splitting is reduced significantly by using LG0 pump pulses with an anti-balanced phase φp ~ π. Intensity distributions of SHG obtained by the individual pump pulses (left:LG1, right:LG0) are also shown in the lower part.

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

E ( r , φ , z , t ) = E 0 u ( r , φ , z ) exp [ i ( kz ω t ) ] ,
u ( r , φ , z ) = a ( r , φ , z ) e i Φ G ( z ) = w 0 w ( z ) ( r w ( z ) ) e i φ e r 2 w ( z ) 2 e i Φ G ( z ) .
Φ G ( z ) = ( + 1 ) tan 1 ( z z R ) , w ( z ) = w 0 1 + ( z z R ) 2 ,
a ( x , y , z ) = w 0 w ( z ) ( x ± iy w ( z ) ) e ( x 2 + y 2 ) w ( z ) 2 ,
E S z = α E S x + g E F E F , E F z = 0 ,
E S z = α E S x + g E F ( x , y , z ) E F ( x , y , z ) .
E S ( x 1 , y , z ) = g 0 L E F ( x 1 , y , z ) E F ( x 1 , y , z ) dz .
F ( ξ , η ) = e 2 η 2 0 1 ( ξ γ t + i η ) 2 e 2 ( ξ γ t ) 2 dt .
F ( ξ , η ) = e 2 η 2 0 1 ( ξ γ t + i η ) 2 e 2 ( ξ γ t ) 2 dt
[ ξ γ 2 + i ( η γ 2 3 ) ] [ ξ γ 2 + i ( η + γ 2 3 ) ] .
u 2 S ( x , y , z ) = ga 1 ( x s 1 , y s 2 , z ) a 1 ( x s 1 , y + s 2 , z ) e i 2 Φ G 1 ( z ) ,
u 2 S ( x , y , z ) = ge 2 s 2 2 w ( z ) 2 ( w 0 w ( z ) ) 2 ( x s 1 + iy ) 2 + s 2 2 w ( z ) 2 e { 2 ( x s 1 ) 2 + 2 y 2 } w ( z ) 2 e i 2 Φ G 1 ( z )
= ge 2 s 2 2 w ( z ) 2 e i ( Φ G 0 ( z ) + Φ G 2 ( z ) ) [ u 2 ( x s 1 , y , z ) e i Φ G 2 ( z ) + ( s 2 w ( z ) ) 2 u 0 ( x s 1 , y , z ) e i Φ G 0 ( z ) ] ,

Metrics