Abstract

The second harmonic generation in the bulk of an isotropic chiral liquid with spatial or temporal nonlocality of its quadratic response is studied analytically. The fundamental pulsed beam has Gaussian time envelope and carries a polarization singularity at its axis. The influence of the topological charge and the handedness of the polarization singularity on the polarization state of the signal beam is revealed.

© 2017 Optical Society of America

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References

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  1. J. E. Sipe, D. J. Moss, and H. M. van Driel, “Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals,” Phys. Rev. B 35, 1129–1141 (1987).
    [Crossref]
  2. Y. Shen, “Surface properties probed by second-harmonic and sum-frequency generation,” Nature 337, 519–525 (1989).
    [Crossref]
  3. M. Beresna, P. G. Kazansky, Y. Svirko, M. Barkauskas, and R. Danielius, “High average power second harmonic generation in air,” App. Phys. Lett. 95, 121502 (2009).
    [Crossref]
  4. M. M. Coles, M. D. Williams, and D. L. Andrews, “Second harmonic generation in isotropic media: six-wave mixing of optical vortices,” Opt. Express 21, 12783–12789 (2013).
    [Crossref] [PubMed]
  5. A. Bahabad and A. Arie, “Generation of optical vortex beams by nonlinear wave mixing,” Opt. Express 15, 17619–17624 (2007).
    [Crossref] [PubMed]
  6. P. Guyot-Sionnest and Y. R. Shen, “Local and nonlocal surface nonlinearities for surface optical second-harmonic generation,” Phys. Rev. B 35, 4420–4426 (1987).
    [Crossref]
  7. S. N. Volkov, N. I. Koroteev, and V. A. Makarov, “Second-harmonic generation in the interior of an isotropic medium with quadratic nonlinearity by a focused inhomogeneously polarized pump beam,” J. Exp. Theor. Phys. 86, 687–695 (1998).
    [Crossref]
  8. S. Slussarenko, A. Murauski, T. Du, V. Chigrinov, L. Marrucci, and E. Santamato, “Tunable liquid crystal q-plates with arbitrary topological charge,” Opt. Express 19, 4085–4090 (2011).
    [Crossref] [PubMed]
  9. C. F. Phelan, K. E. Ballantine, P. R. Eastham, J. F. Donegan, and J. G. Lunney, “Conical diffraction of a gaussian beam with a two crystal cascade,” Opt. Express 20, 13201–13207 (2012).
    [Crossref] [PubMed]
  10. B. Yang and E. Brasselet, “Arbitrary vortex arrays realized from optical winding of frustrated chiral liquid crystals,” J. Opt 15, 044021 (2013).
    [Crossref]
  11. M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Chapter 5 singular optics: Optical vortices and polarization singularities,” (Elsevier, 2009), pp. 293–363.
  12. K. S. Grigoriev, V. A. Makarov, and I. A. Perezhogin, “Polarization singularities in a sum-frequency light beam generated by a bichromatic singular beam in the bulk of an isotropic nonlinear chiral medium,” Phys. Rev. A 92, 023814 (2015).
    [Crossref]
  13. K. S. Grigoriev, V. A. Makarov, and I. A. Perezhogin, “Formation of the lines of circular polarization in a second harmonic beam generated from the surface of an isotropic medium with nonlocal nonlinear response in the case of normal incidence,” J. Opt. 18, 014004 (2016).
    [Crossref]
  14. Y. R. Shen, The Principles of Nonlinear Optics, Wiley Series in Pure and Applied Optics (J. Wiley, 1984).
  15. E. J. Galvez, B. L. Rojec, V. Kumar, and N. K. Viswanathan, “Generation of isolated asymmetric umbilics in light’s polarization,” Phys. Rev. A 89, 031801 (2014).
    [Crossref]
  16. P. M. Rentzepis, J. A. Giordmaine, and K. W. Wecht, “Coherent optical mixing in optically active liquids,” Phys. Rev. Lett. 16, 792–794 (1966).
    [Crossref]
  17. A. P. Shkurinov, A. V. Dubrovskii, and N. I. Koroteev, “Second harmonic generation in an optically active liquid: Experimental observation of a fourth-order optical nonlinearity due to molecular chirality,” Phys. Rev. Lett. 70, 1085–1088 (1993).
    [Crossref] [PubMed]
  18. V. Agranovič and V. Ginzburg, Spatial Dispersion in Crystal Optics and the Theory of Excitons, no. 18 in Interscience monographs and texts in physics and astronomy, v. 18 (Interscience Publishers, 1966).
  19. J. A. Giordmaine, “Nonlinear optical properties of liquids,” Phys. Rev. 138, A1599–A1606 (1965).
    [Crossref]
  20. E. G. Hanson, Y. R. Shen, and G. K. L. Wong, “Experimental study of self-focusing in a liquid crystalline medium,” Appl. Phys. 14, 65–77 (1977).
    [Crossref]

2016 (1)

K. S. Grigoriev, V. A. Makarov, and I. A. Perezhogin, “Formation of the lines of circular polarization in a second harmonic beam generated from the surface of an isotropic medium with nonlocal nonlinear response in the case of normal incidence,” J. Opt. 18, 014004 (2016).
[Crossref]

2015 (1)

K. S. Grigoriev, V. A. Makarov, and I. A. Perezhogin, “Polarization singularities in a sum-frequency light beam generated by a bichromatic singular beam in the bulk of an isotropic nonlinear chiral medium,” Phys. Rev. A 92, 023814 (2015).
[Crossref]

2014 (1)

E. J. Galvez, B. L. Rojec, V. Kumar, and N. K. Viswanathan, “Generation of isolated asymmetric umbilics in light’s polarization,” Phys. Rev. A 89, 031801 (2014).
[Crossref]

2013 (2)

B. Yang and E. Brasselet, “Arbitrary vortex arrays realized from optical winding of frustrated chiral liquid crystals,” J. Opt 15, 044021 (2013).
[Crossref]

M. M. Coles, M. D. Williams, and D. L. Andrews, “Second harmonic generation in isotropic media: six-wave mixing of optical vortices,” Opt. Express 21, 12783–12789 (2013).
[Crossref] [PubMed]

2012 (1)

2011 (1)

2009 (1)

M. Beresna, P. G. Kazansky, Y. Svirko, M. Barkauskas, and R. Danielius, “High average power second harmonic generation in air,” App. Phys. Lett. 95, 121502 (2009).
[Crossref]

2007 (1)

1998 (1)

S. N. Volkov, N. I. Koroteev, and V. A. Makarov, “Second-harmonic generation in the interior of an isotropic medium with quadratic nonlinearity by a focused inhomogeneously polarized pump beam,” J. Exp. Theor. Phys. 86, 687–695 (1998).
[Crossref]

1993 (1)

A. P. Shkurinov, A. V. Dubrovskii, and N. I. Koroteev, “Second harmonic generation in an optically active liquid: Experimental observation of a fourth-order optical nonlinearity due to molecular chirality,” Phys. Rev. Lett. 70, 1085–1088 (1993).
[Crossref] [PubMed]

1989 (1)

Y. Shen, “Surface properties probed by second-harmonic and sum-frequency generation,” Nature 337, 519–525 (1989).
[Crossref]

1987 (2)

J. E. Sipe, D. J. Moss, and H. M. van Driel, “Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals,” Phys. Rev. B 35, 1129–1141 (1987).
[Crossref]

P. Guyot-Sionnest and Y. R. Shen, “Local and nonlocal surface nonlinearities for surface optical second-harmonic generation,” Phys. Rev. B 35, 4420–4426 (1987).
[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. 14, 65–77 (1977).
[Crossref]

1966 (1)

P. M. Rentzepis, J. A. Giordmaine, and K. W. Wecht, “Coherent optical mixing in optically active liquids,” Phys. Rev. Lett. 16, 792–794 (1966).
[Crossref]

1965 (1)

J. A. Giordmaine, “Nonlinear optical properties of liquids,” Phys. Rev. 138, A1599–A1606 (1965).
[Crossref]

Agranovic, V.

V. Agranovič and V. Ginzburg, Spatial Dispersion in Crystal Optics and the Theory of Excitons, no. 18 in Interscience monographs and texts in physics and astronomy, v. 18 (Interscience Publishers, 1966).

Andrews, D. L.

Arie, A.

Bahabad, A.

Ballantine, K. E.

Barkauskas, M.

M. Beresna, P. G. Kazansky, Y. Svirko, M. Barkauskas, and R. Danielius, “High average power second harmonic generation in air,” App. Phys. Lett. 95, 121502 (2009).
[Crossref]

Beresna, M.

M. Beresna, P. G. Kazansky, Y. Svirko, M. Barkauskas, and R. Danielius, “High average power second harmonic generation in air,” App. Phys. Lett. 95, 121502 (2009).
[Crossref]

Brasselet, E.

B. Yang and E. Brasselet, “Arbitrary vortex arrays realized from optical winding of frustrated chiral liquid crystals,” J. Opt 15, 044021 (2013).
[Crossref]

Chigrinov, V.

Coles, M. M.

Danielius, R.

M. Beresna, P. G. Kazansky, Y. Svirko, M. Barkauskas, and R. Danielius, “High average power second harmonic generation in air,” App. Phys. Lett. 95, 121502 (2009).
[Crossref]

Dennis, M. R.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Chapter 5 singular optics: Optical vortices and polarization singularities,” (Elsevier, 2009), pp. 293–363.

Donegan, J. F.

Du, T.

Dubrovskii, A. V.

A. P. Shkurinov, A. V. Dubrovskii, and N. I. Koroteev, “Second harmonic generation in an optically active liquid: Experimental observation of a fourth-order optical nonlinearity due to molecular chirality,” Phys. Rev. Lett. 70, 1085–1088 (1993).
[Crossref] [PubMed]

Eastham, P. R.

Galvez, E. J.

E. J. Galvez, B. L. Rojec, V. Kumar, and N. K. Viswanathan, “Generation of isolated asymmetric umbilics in light’s polarization,” Phys. Rev. A 89, 031801 (2014).
[Crossref]

Ginzburg, V.

V. Agranovič and V. Ginzburg, Spatial Dispersion in Crystal Optics and the Theory of Excitons, no. 18 in Interscience monographs and texts in physics and astronomy, v. 18 (Interscience Publishers, 1966).

Giordmaine, J. A.

P. M. Rentzepis, J. A. Giordmaine, and K. W. Wecht, “Coherent optical mixing in optically active liquids,” Phys. Rev. Lett. 16, 792–794 (1966).
[Crossref]

J. A. Giordmaine, “Nonlinear optical properties of liquids,” Phys. Rev. 138, A1599–A1606 (1965).
[Crossref]

Grigoriev, K. S.

K. S. Grigoriev, V. A. Makarov, and I. A. Perezhogin, “Formation of the lines of circular polarization in a second harmonic beam generated from the surface of an isotropic medium with nonlocal nonlinear response in the case of normal incidence,” J. Opt. 18, 014004 (2016).
[Crossref]

K. S. Grigoriev, V. A. Makarov, and I. A. Perezhogin, “Polarization singularities in a sum-frequency light beam generated by a bichromatic singular beam in the bulk of an isotropic nonlinear chiral medium,” Phys. Rev. A 92, 023814 (2015).
[Crossref]

Guyot-Sionnest, P.

P. Guyot-Sionnest and Y. R. Shen, “Local and nonlocal surface nonlinearities for surface optical second-harmonic generation,” Phys. Rev. B 35, 4420–4426 (1987).
[Crossref]

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. 14, 65–77 (1977).
[Crossref]

Kazansky, P. G.

M. Beresna, P. G. Kazansky, Y. Svirko, M. Barkauskas, and R. Danielius, “High average power second harmonic generation in air,” App. Phys. Lett. 95, 121502 (2009).
[Crossref]

Koroteev, N. I.

S. N. Volkov, N. I. Koroteev, and V. A. Makarov, “Second-harmonic generation in the interior of an isotropic medium with quadratic nonlinearity by a focused inhomogeneously polarized pump beam,” J. Exp. Theor. Phys. 86, 687–695 (1998).
[Crossref]

A. P. Shkurinov, A. V. Dubrovskii, and N. I. Koroteev, “Second harmonic generation in an optically active liquid: Experimental observation of a fourth-order optical nonlinearity due to molecular chirality,” Phys. Rev. Lett. 70, 1085–1088 (1993).
[Crossref] [PubMed]

Kumar, V.

E. J. Galvez, B. L. Rojec, V. Kumar, and N. K. Viswanathan, “Generation of isolated asymmetric umbilics in light’s polarization,” Phys. Rev. A 89, 031801 (2014).
[Crossref]

Lunney, J. G.

Makarov, V. A.

K. S. Grigoriev, V. A. Makarov, and I. A. Perezhogin, “Formation of the lines of circular polarization in a second harmonic beam generated from the surface of an isotropic medium with nonlocal nonlinear response in the case of normal incidence,” J. Opt. 18, 014004 (2016).
[Crossref]

K. S. Grigoriev, V. A. Makarov, and I. A. Perezhogin, “Polarization singularities in a sum-frequency light beam generated by a bichromatic singular beam in the bulk of an isotropic nonlinear chiral medium,” Phys. Rev. A 92, 023814 (2015).
[Crossref]

S. N. Volkov, N. I. Koroteev, and V. A. Makarov, “Second-harmonic generation in the interior of an isotropic medium with quadratic nonlinearity by a focused inhomogeneously polarized pump beam,” J. Exp. Theor. Phys. 86, 687–695 (1998).
[Crossref]

Marrucci, L.

Moss, D. J.

J. E. Sipe, D. J. Moss, and H. M. van Driel, “Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals,” Phys. Rev. B 35, 1129–1141 (1987).
[Crossref]

Murauski, A.

O’Holleran, K.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Chapter 5 singular optics: Optical vortices and polarization singularities,” (Elsevier, 2009), pp. 293–363.

Padgett, M. J.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Chapter 5 singular optics: Optical vortices and polarization singularities,” (Elsevier, 2009), pp. 293–363.

Perezhogin, I. A.

K. S. Grigoriev, V. A. Makarov, and I. A. Perezhogin, “Formation of the lines of circular polarization in a second harmonic beam generated from the surface of an isotropic medium with nonlocal nonlinear response in the case of normal incidence,” J. Opt. 18, 014004 (2016).
[Crossref]

K. S. Grigoriev, V. A. Makarov, and I. A. Perezhogin, “Polarization singularities in a sum-frequency light beam generated by a bichromatic singular beam in the bulk of an isotropic nonlinear chiral medium,” Phys. Rev. A 92, 023814 (2015).
[Crossref]

Phelan, C. F.

Rentzepis, P. M.

P. M. Rentzepis, J. A. Giordmaine, and K. W. Wecht, “Coherent optical mixing in optically active liquids,” Phys. Rev. Lett. 16, 792–794 (1966).
[Crossref]

Rojec, B. L.

E. J. Galvez, B. L. Rojec, V. Kumar, and N. K. Viswanathan, “Generation of isolated asymmetric umbilics in light’s polarization,” Phys. Rev. A 89, 031801 (2014).
[Crossref]

Santamato, E.

Shen, Y.

Y. Shen, “Surface properties probed by second-harmonic and sum-frequency generation,” Nature 337, 519–525 (1989).
[Crossref]

Shen, Y. R.

P. Guyot-Sionnest and Y. R. Shen, “Local and nonlocal surface nonlinearities for surface optical second-harmonic generation,” Phys. Rev. B 35, 4420–4426 (1987).
[Crossref]

E. G. Hanson, Y. R. Shen, and G. K. L. Wong, “Experimental study of self-focusing in a liquid crystalline medium,” Appl. Phys. 14, 65–77 (1977).
[Crossref]

Y. R. Shen, The Principles of Nonlinear Optics, Wiley Series in Pure and Applied Optics (J. Wiley, 1984).

Shkurinov, A. P.

A. P. Shkurinov, A. V. Dubrovskii, and N. I. Koroteev, “Second harmonic generation in an optically active liquid: Experimental observation of a fourth-order optical nonlinearity due to molecular chirality,” Phys. Rev. Lett. 70, 1085–1088 (1993).
[Crossref] [PubMed]

Sipe, J. E.

J. E. Sipe, D. J. Moss, and H. M. van Driel, “Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals,” Phys. Rev. B 35, 1129–1141 (1987).
[Crossref]

Slussarenko, S.

Svirko, Y.

M. Beresna, P. G. Kazansky, Y. Svirko, M. Barkauskas, and R. Danielius, “High average power second harmonic generation in air,” App. Phys. Lett. 95, 121502 (2009).
[Crossref]

van Driel, H. M.

J. E. Sipe, D. J. Moss, and H. M. van Driel, “Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals,” Phys. Rev. B 35, 1129–1141 (1987).
[Crossref]

Viswanathan, N. K.

E. J. Galvez, B. L. Rojec, V. Kumar, and N. K. Viswanathan, “Generation of isolated asymmetric umbilics in light’s polarization,” Phys. Rev. A 89, 031801 (2014).
[Crossref]

Volkov, S. N.

S. N. Volkov, N. I. Koroteev, and V. A. Makarov, “Second-harmonic generation in the interior of an isotropic medium with quadratic nonlinearity by a focused inhomogeneously polarized pump beam,” J. Exp. Theor. Phys. 86, 687–695 (1998).
[Crossref]

Wecht, K. W.

P. M. Rentzepis, J. A. Giordmaine, and K. W. Wecht, “Coherent optical mixing in optically active liquids,” Phys. Rev. Lett. 16, 792–794 (1966).
[Crossref]

Williams, M. D.

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. 14, 65–77 (1977).
[Crossref]

Yang, B.

B. Yang and E. Brasselet, “Arbitrary vortex arrays realized from optical winding of frustrated chiral liquid crystals,” J. Opt 15, 044021 (2013).
[Crossref]

App. Phys. Lett. (1)

M. Beresna, P. G. Kazansky, Y. Svirko, M. Barkauskas, and R. Danielius, “High average power second harmonic generation in air,” App. Phys. Lett. 95, 121502 (2009).
[Crossref]

Appl. Phys. (1)

E. G. Hanson, Y. R. Shen, and G. K. L. Wong, “Experimental study of self-focusing in a liquid crystalline medium,” Appl. Phys. 14, 65–77 (1977).
[Crossref]

J. Exp. Theor. Phys. (1)

S. N. Volkov, N. I. Koroteev, and V. A. Makarov, “Second-harmonic generation in the interior of an isotropic medium with quadratic nonlinearity by a focused inhomogeneously polarized pump beam,” J. Exp. Theor. Phys. 86, 687–695 (1998).
[Crossref]

J. Opt (1)

B. Yang and E. Brasselet, “Arbitrary vortex arrays realized from optical winding of frustrated chiral liquid crystals,” J. Opt 15, 044021 (2013).
[Crossref]

J. Opt. (1)

K. S. Grigoriev, V. A. Makarov, and I. A. Perezhogin, “Formation of the lines of circular polarization in a second harmonic beam generated from the surface of an isotropic medium with nonlocal nonlinear response in the case of normal incidence,” J. Opt. 18, 014004 (2016).
[Crossref]

Nature (1)

Y. Shen, “Surface properties probed by second-harmonic and sum-frequency generation,” Nature 337, 519–525 (1989).
[Crossref]

Opt. Express (4)

Phys. Rev. (1)

J. A. Giordmaine, “Nonlinear optical properties of liquids,” Phys. Rev. 138, A1599–A1606 (1965).
[Crossref]

Phys. Rev. A (2)

K. S. Grigoriev, V. A. Makarov, and I. A. Perezhogin, “Polarization singularities in a sum-frequency light beam generated by a bichromatic singular beam in the bulk of an isotropic nonlinear chiral medium,” Phys. Rev. A 92, 023814 (2015).
[Crossref]

E. J. Galvez, B. L. Rojec, V. Kumar, and N. K. Viswanathan, “Generation of isolated asymmetric umbilics in light’s polarization,” Phys. Rev. A 89, 031801 (2014).
[Crossref]

Phys. Rev. B (2)

P. Guyot-Sionnest and Y. R. Shen, “Local and nonlocal surface nonlinearities for surface optical second-harmonic generation,” Phys. Rev. B 35, 4420–4426 (1987).
[Crossref]

J. E. Sipe, D. J. Moss, and H. M. van Driel, “Phenomenological theory of optical second- and third-harmonic generation from cubic centrosymmetric crystals,” Phys. Rev. B 35, 1129–1141 (1987).
[Crossref]

Phys. Rev. Lett. (2)

P. M. Rentzepis, J. A. Giordmaine, and K. W. Wecht, “Coherent optical mixing in optically active liquids,” Phys. Rev. Lett. 16, 792–794 (1966).
[Crossref]

A. P. Shkurinov, A. V. Dubrovskii, and N. I. Koroteev, “Second harmonic generation in an optically active liquid: Experimental observation of a fourth-order optical nonlinearity due to molecular chirality,” Phys. Rev. Lett. 70, 1085–1088 (1993).
[Crossref] [PubMed]

Other (3)

V. Agranovič and V. Ginzburg, Spatial Dispersion in Crystal Optics and the Theory of Excitons, no. 18 in Interscience monographs and texts in physics and astronomy, v. 18 (Interscience Publishers, 1966).

Y. R. Shen, The Principles of Nonlinear Optics, Wiley Series in Pure and Applied Optics (J. Wiley, 1984).

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Chapter 5 singular optics: Optical vortices and polarization singularities,” (Elsevier, 2009), pp. 293–363.

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Figures (3)

Fig. 1
Fig. 1 The dependences of the (a) peak and (b) integral power of the SH pulsed beam on z-coodrinate. The curves are calculated at Δk = 0.5/ld (thin solid line), Δk = 0 (thick line) and Δk = −0.5/ld (dashed line). The dependences are normalized on the maximum values of power for the positive phase mismatch. The other parameters are ŭ−1 = 0, lds/ld = 5.
Fig. 2
Fig. 2 Polarization distribution of fundamental pulse at z = 0 (a,c) and the corrseponding signal pulse (b,d), generated in the bulk of the medium with spatially nonlocal nonlinear response, at z = ld, integrated over time. Left-hand polarization ellipses are filled and right-hand ones are empty. The C-point in the fundamendal beam is marked by filled (charge 1/2) and opened (charge −1/2) circle. The parameters of the fundamental beam are p = 3 2, q = 0.5 exp(iπ/3) (a,b) and p = 0.5, q = 3 2 exp ( 2 i π / 3 ) (c,d). The other parameters are Δk = −0.5/ld, ŭ−1 = 0, lds/ld = 10.
Fig. 3
Fig. 3 Polarization distribution of fundamental pulse at z = 0 (a,b,c and e,f,g) in three different time layers (−τ0 (a,e), 0 (b,f) and τ0 (c,g)) and the corrseponding signal pulse (d,h), generated in the bulk of the medium with temporally nonlocal nonlinear response, at z = ld, integrated over time. Left-hand polarization ellipses are filled and right-hand ones are empty. The C-point in the fundamendal beam is marked by filled (charge 1/2) and opened (charge −1/2) circle. The parameters of the fundamental beam are p = 3 2, q = 0.5 exp(iπ/3) (a,b,c) and p = 0.5, q = 3 2 exp ( 2 i π / 3 ) (e,f,g). The other parameters are Δk = −0.5/ld, ŭ−1 = 0, lds/ld = 10, δ/τ0 = 0.45.

Equations (38)

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[ ( z + 1 u ω t ) i 2 k ω Δ + i k ω   2 2 t 2 ] ± ( ω ) = 0 ,
± ( ω ) ( r , z , t ) = f ± ( r , z ) G ( r , z , t ) ,
G ( r , z , t ) = 1 β z ( z ) β t ( z ) exp [ r 2 w 2 β z ( z ) [ t ( z l 0 ) / u ω ] 2 τ 0 2 β t 2 ( z ) ]
β z ( z ) = 1 + i ( z l 0 ) / l d ,
β t ( z ) = 1 i ( z l 0 ) / l d s ,
E = ( + ( ω ) e + + ( ω ) e ) exp ( i ω t + i k ω ( z l 0 ) ) .
( ω ) ( r , z , t ) i k ω [ 1 + g ( z , t ) ] [ ( e + L + ( r ) ) + ( e L ( r ) ) ] G ( r , z , t ) .
L ± ( r , z ) = f ± 2 r f ± w 2 β z ,
g ( z , t ) = 2 i [ t ( z l 0 ) / u ω ] β t 2 k ω u ω τ 0 2 ,
P i ( 2 ) ( r , t ) = t d t 1 t d t 2 d r 1 d r 2 χ i j k ( r 1 , r 2 , t 1 , t 2 ) E j ( r r 1 , t t 1 ) E k ( r r 2 , t t 2 ) ,
χ i j k ( r 1 , r 2 , t 1 , t 2 ) = χ i j k ( r 1 , r 2 ) δ ( t 1 ) δ ( t 2 ) ,
P i ( 2 ) = γ i j k l E j E k r l
γ i j i j = γ 1 , γ i j j i = γ 2 , γ i i j j = γ 3 , γ i i i i = γ 1 + γ 2 + γ 3 .
P ( 2 ) = γ 1 E ( E ) + 1 2 γ 2 ( E E ) + γ 3 ( E ) E .
P s ( 2 ) = i γ 3 [ e z , f + f f f + ] G 2 exp ( 2 i ω t + 2 i k ω ( z l 0 ) ) .
f ± ( r , z ) = E L p ( x + i y ) + q ( x i y ) 2 w β z ( z ) , f ( r , z ) = E G
P s ( 2 ) = ± γ 3 E G E L p e q e + w β z G 2 exp ( 2 i ω t + 2 i k ω ( z l 0 ) ) .
[ ( z + 1 u 2 ω t ) i 2 k 2 ω Δ + i k 2 ω   2 2 t 2 ] ± ( 2 ω ) = 2 π i k 2 ω ε 2 ω P s ± ( 2 ω ) exp ( i Δ k ( z l 0 ) ) .
P s ( 2 ) = ( P s + ( 2 ω ) e + + P s ( 2 ω ) e ) exp ( 2 i ω t + 2 i k ω ( z l 0 ) )
( 2 ω ) ( r , t ) = ± γ 3 E G E L 2 π i k 2 ω l d w ε 2 ω ( p e q e + ) J nloc ( r , t )
J nlco ( r , z , t ) = l 0 / l d ζ β ˜ z 1 ( ζ ) d ζ B z ( ζ , ζ ) B t ( ζ , ζ ) × × exp ( i Δ k l d ζ 2 r 2 β ˜ z ( ζ ) w 2 B z ( ζ , ζ ) 2 [ t ( z l 0 ) / u 2 ω + ζ l d / u ] 2 β ˜ t 2 ( ζ ) τ 0 2 B t 2 ( ζ , ζ ) ) ,
B z ( ζ , ζ ) = β ˜ z 2 ( ζ ) + i ζ ζ k 2 ω / ( 2 k ω ) β ˜ z ( ζ ) ,
B t 2 ( ζ , ζ ) = β ˜ t 4 ( ζ ) i ζ ζ k ω   / ( 2 k 2 ω   ) l d l d s β ˜ t 2 ( ζ ) ,
β ˜ z ( ζ ) = 1 + i ζ , β ˜ t ( ζ ) = 1 i ζ l d / l d s .
S 3 ( 2 ω ) = | p | 2 | q | 2 .
| P s ( 2 ) | γ 3 E G E L 1 w = 1 k ω w ( k ω γ 3 ) E G E L
χ i j k ( r 1 , r 2 , t 1 , t 2 ) = χ i j k ( t 1 , t 2 ) δ ( r 1 ) δ ( r 2 ) .
P i ( 2 ω ) { χ i j k ( ω , ω ) j ( ω ) k ( ω ) + i χ i j k ( ω 1 , ω 2 ) ω 2 | ω , ω j ( ω ) k ( ω ) t + i χ i j k ( ω 1 , ω 2 ) ω 1 | ω , ω j ( ω ) t k ( ω ) } .
χ i j k ( ω 1 , ω 2 ) = 0 d t 1 0 d t 2 χ i j k ( t 1 , t 2 ) exp ( i ω 1 t 1 + i ω 2 t 2 ) .
χ i j k ( ω 1 , ω 2 ) ω 1 | ω , ω = χ i k j ( ω 1 , ω 2 ) ω 2 | ω , ω .
P ( 2 ω ) i γ t [ ω t , ω ] ,
Σ ± ( ω ) ( r , z , t ) = 1 ± ( ω ) ( r , z , t δ ) + 2 ± ( ω ) ( r , z , t + δ ) .
P s ( 2 ω ) = 4 δ β t 2 k ω τ 0 2 γ t [ e z , ( f 2 + + f 1 + f 1 + + f 2 + ) e + + ( f 2 f 1 f 1 f 2 ) e + 1 2 ( f 2 f 1 + f 1 f 2 + + f 2 + f 1 f 1 + f 2 ) ] G 2 exp ( 2 δ 2 τ 0 2 β t 2 ) .
P s ( 2 ω ) = ± γ t E G E L 4 i δ w β t 2 β z k ω τ 0 2 ( p e + + q e ) G 2 exp ( 2 δ 2 τ 0 2 β t 2 )
( 2 ω ) = γ t E G E L 8 π k 2 ω l d δ w ε 2 ω k ω τ 0 2 ( p e + + q e ) J loc ( r , t ) ,
J loc ( r , z , t ) = l 0 / l d ζ β ˜ z 1 ( ζ ) β ˜ t 2 ( ζ ) d ζ B z ( ζ , ζ ) B t ( ζ , ζ ) × exp ( i Δ k l d ζ 2 r 2 β ˜ z ( ζ ) w 2 B z ( ζ , ζ ) 2 [ t ( z l 0 ) / u 2 ω + ζ l d / u ] 2 β ˜ t 2 ( ζ ) τ 0 2 B t 2 ( ζ , ζ ) 2 δ 2 τ 0 2 β ˜ t 2 ( ζ ) ) ,
S 3 ( 2 ω ) = | q | 2 | p | 2
| P s ( 2 ω ) | γ t E G E L δ k ω w τ 0 2 = 1 k ω w ( δ / τ 0 ) ( γ t / τ 0 ) E G E L

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