Abstract

Allowing for an isotropic optical nonlinearity of third order, we derive approximate Gaussian solutions of the paraxial wave equation. They describe a monochromatic elliptically polarized light signal of finite extension both in space and in time. In contrast to plane-wave results, the state of polarization is found to be strongly nonuniform in the transverse direction. Both the orientation and the shape of the polarization ellipse are not conserved. For low intensities, parameters measuring the induced optical activity depend on the intensity quadratically instead of linearly. For high intensities, saturation sets in. It is argued that by means of nonlinear ellipsometry one can evaluate two tensor components, namely, Re χxyyx(3) and Re χxxxx(3). Upon taking into account transverse as well as temporal effects, the magnitude of these components may decrease by a factor of 10.

© 1997 Optical Society of America

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References

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  1. P. D. Maker, R. W. Terhune, and C. M. Savage, “Intensity-dependent changes in the refractive index of liquids,” Phys. Rev. Lett. 12, 507–509 (1964); P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. 137, A801–A818 (1965).
    [CrossRef]
  2. A. Owyoung, “Ellipse rotation studies in laser host materials,” IEEE J. Quantum Electron. QE-9, 1064–1069 (1973).
    [CrossRef]
  3. S. M. Arakelyan, S. R. Galstyan, O. V. Garibyan, A. S. Karayan, and Yu. S. Chilingaryan, “Strong, nonlinear, optical activity in the nematic phase of a liquid crystal,” Pis’ma Zh. Eksp. Teor. Fiz. 32, 561–565 (1980) [JETP Lett. 32, 543–547 (1980)]; S. A. Boiko, M. I. Dykman, M. P. Lisitsa, V. I. Sidorenko, and G. G. Tarasov, “Variation of resonance-radiation polarization due to self-induced dichroism in a KCl:Li crystal with FA centers,” Opt. Spektrosk. 58, 1055–1058 (1985) [Opt. Spectrosc. 58, 645–647 (1985)]; S. A. Akhmanov, N. I. Zheludev, and R. S. Zadoyan, “Picosecond spectroscopy of nonlinear optical activity and nonlinear absorption in gallium arsenide,” Zh. Eksp. Teor. Fiz. 91, 984–1000 (1986) [Sov. Phys. JETP SPHJAR 64, 579–588 (1986)].
  4. A. Owyoung, R. W. Hellwarth, and N. George, “Intensity-induced changes in optical polarizations in glasses,” Phys. Rev. B 5, 628–633 (1972).
    [CrossRef]
  5. C. C. Wang, “Nonlinear susceptibility constants and self-focusing of optical beams in liquids,” Phys. Rev. 152, 149–156 (1966); R. W. Hellwarth, A. Owyoung, and N. George, “Origin of the nonlinear refractive index of liquid CCl4,” Phys. Rev. A 4, 2342–2347 (1971); J. M. Cherlow, T. T. Yang, and R. W. Hellwarth, “Nonlinear optical susceptibilities of solvents,” IEEE J. Quantum Electron. IEJQA7 QE-12, 644–646 (1976); X. Nguyen Phu, J. L. Ferrier, J. Gazengel, and G. Rivoire, “Polarization of picosecond light pulses in nonlinear isotropic media,” Opt. Commun. OPCOB8 46, 329–333 (1983); N. Pfeffer, F. Charra, and J. M. Nunzi, “Phase and frequency resolution of picosecond optical Kerr nonlinearities,” Opt. Lett. OPLEDP 16, 1987–1989 (1991).
    [CrossRef] [PubMed]
  6. S. Saikan and K. Namba, “Intensity dependent polarization change in the D1 and D2 resonance lines of sodium,” Opt. Commun. 23, 73–76 (1977); D. V. Vlasov, R. A. Garaev, V. V. Korobkin, and R. V. Serov, “Measurement of nonlinear polarizability of air,” Zh. Eksp. Teor. Fiz. 76, 2039–2045 (1979) [Sov. Phys. JETP 49, 1033–1036 (1979)].
    [CrossRef]
  7. D. M. Pennington, M. A. Henesian, and R. W. Hellwarth, “Nonlinear index of air at 1.053 μm,” Phys. Rev. A 39, 3003–3009 (1989).
    [CrossRef] [PubMed]
  8. J. M. Thorne, T. R. Loree, and G. H. McCall, “Intensity filtration of laser light,” J. Appl. Phys. 45, 3072–3078 (1974); K. Sala and M. C. Richardson, “A passive nonresonant technique for pulse contrast enhancement and gain isolation,” J. Appl. Phys. 49, 2268–2276 (1978); D. V. Murphy and R. K. Chang, “Pulse stretching of Q-switched laser emission by intracavity nonlinear ellipse rotation,” Opt. Commun. OPCOB8 23, 268–272 (1977); V. L. Kalashnikov, V. P. Kalosha, V. P. Mikhailov, I. G. Poloyko, and M. I. Demchuk, “Self-mode locking of continuous-wave solid-state lasers by a nonlinear Kerr polarization modulator,” J. Opt. Soc. Am. B JOBPDE 10, 1443–1446 (1993).
    [CrossRef]
  9. For recent reviews, see N. I. Zheludev, “Polarization instability and multistability in nonlinear optics,” Usp. Fiz. Nauk 157, 683–717 (1989) [Sov. Phys. Usp. 32, 357–375 (1989)]; D. David, D. D. Holm, and M. V. Tratnik, “Hamiltonian chaos in nonlinear optical polarization dynamics,” Phys. Rep. 187, 281–367 (1990).
    [CrossRef]
  10. K. Hayata, A. Misawa, and M. Koshiba, “Spatial polarization instabilities due to transverse effects in nonlinear guided-wave systems,” J. Opt. Soc. Am. B 7, 1268–1280 (1990).
    [CrossRef]
  11. A. L. Gaeta and R. W. Boyd, “Transverse instabilities in the polarizations and intensities of counterpropagating light waves,” Phys. Rev. A 48, 1610–1624 (1993).
    [CrossRef] [PubMed]
  12. N. Akhmediev, A. Buryak, and J. M. Soto-Crespo, “Elliptically polarized solitons in birefringent optical fibers,” Opt. Commun. 112, 278–282 (1994); Y. Silberberg and Y. Barad, “Rotating vector solitary waves in isotropic fibers,” Opt. Lett. 20, 246–248 (1995).
    [CrossRef] [PubMed]
  13. A. W. Snyder, S. J. Hewlett, and D. J. Mitchell, “Dynamic spatial solitons,” Phys. Rev. Lett. 72, 1012–1015 (1994); A. W. Snyder, D. J. Mitchell, and Y. Chen, “Spatial solitons of Maxwell’s equations,” Opt. Lett. 19, 524–526 (1994); M. Haelterman and A. P. Sheppard, “The elliptically polarized fundamental vector soliton of isotropic Kerr media,” Phys. Lett. A PYLAAG 194, 191–196 (1994).
    [CrossRef] [PubMed]
  14. J. Arons and C. E. Max, “Self-precession and frequency shift for electromagnetic waves in homogeneous plasmas,” Phys. Fluids 17, 1983–1994 (1974); B. Chakraborty, S. N. Paul, M. Khan, and B. Bhattacharyya, “Wave-precession and related nonlinear effects in plasmas,” Phys. Rep. 114, 181–317 (1984).
    [CrossRef]
  15. M. D. Feit and J. A. Fleck, Jr., “Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B 5, 633–640 (1988); S. Chi and Q. Guo, “Vector theory of self-focusing of an optical beam in Kerr media,” Opt. Lett. 20, 1598–1600 (1995); Y. Chen and J. Atai, “Solitary waves of Maxwell’s equations in nonlinear anisotropic media,” J. Mod. Opt. JMOPEW 42, 1649–1658 (1995).
    [CrossRef] [PubMed]
  16. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Secs. 1.3 and 1.4.
  17. X. Nguyen Phu and G. Rivoire, “Evolution of the polarization state of an intense electromagnetic field in a nonlinear medium,” Opt. Acta 25, 233–246 (1978); D. V. Vlasov, V. V. Korobkin, and R. V. Serov, “Nonlinear precession of elliptically polarized Gaussian beams,” Kvantovaya Elektron. (Moscow) 6, 1542–1546 (1979) [Sov. J. Quantum Electron. 9, 904–907 (1979)]; V. P. Nayyar, A. Kumar, and A. Garg, “Elliptically polarized Gaussian wave fields in nonlinear optics,” Opt. Commun. OPCOB8 71, 327–331 (1989).
    [CrossRef]
  18. See p. 552 of Ref. 16.
  19. M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [CrossRef]
  20. A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1989), Chaps. 6 and 18.
  21. O. Svelto, “Self-focusing, self-trapping, and self-phase modulation of laser beams,” Prog. Opt. 12, 1–51 (1974).
    [CrossRef]
  22. R. W. Hellwarth, “Third-order optical susceptibilities of liquids and solids,” Prog. Quantum Electron. 5, 1–68 (1977).
    [CrossRef]
  23. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, London, 1980), entry 2.202.
  24. V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz. 61, 118–134 (1971) [Sov. Phys. JETP 34, 62–69 (1972)].
  25. A. Yariv and P. Yeh, “The application of Gaussian beam formalism to optical propagation in nonlinear media,” Opt. Commun. 27, 295–298 (1978); P. P. Banerjee, R. M. Misra, and M. Maghraoui, “Theoretical and experimental studies of propagation of beams through a finite sample of a cubically nonlinear material,” J. Opt. Soc. Am. B 8, 1072–1080 (1991).
    [CrossRef]
  26. J. Herrmann, “Propagation of ultrashort light pulses in fibers with saturable nonlinearity in the normal-dispersion region,” J. Opt. Soc. Am. B 8, 1507–1511 (1991); Y. Chen, “Variational principle for vector spatial solitons and nonlinear modes,” Opt. Commun. 84, 355–358 (1991); M. Karlsson, “Optical beams in saturable self-focusing media,” Phys. Rev. A PLRAAN 46, 2726–2734 (1992).
    [CrossRef] [PubMed]
  27. S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Self-focusing and self-trapping of intense light beams in a nonlinear medium,” Zh. Eksp. Teor. Fiz. 50 1537–1549 (1966) [Sov. Phys. JETP 23, 1025–1033 (1966)]; C. S. Wang, “Propagation of an intense light beam in a nonlinear medium,” Phys. Rev. 173, 908–917 (1968).
    [CrossRef]
  28. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap. 17.
  29. R. S. Longhurst, Geometrical and Physical Optics, 3rd ed. (Longman, London, 1973), p. 542.
  30. M. Lefkir, N. P. Xuan, A. J. van Wonderen, and G. Rivoire, “Stabilité de l’état de polarisation en régime picoseconde,” contribution to Quatrième Colloque sur les Lasers et l’Optique Quantique, Palaiseau, France, November 6–8, 1995.

1993 (1)

A. L. Gaeta and R. W. Boyd, “Transverse instabilities in the polarizations and intensities of counterpropagating light waves,” Phys. Rev. A 48, 1610–1624 (1993).
[CrossRef] [PubMed]

1990 (1)

1989 (1)

D. M. Pennington, M. A. Henesian, and R. W. Hellwarth, “Nonlinear index of air at 1.053 μm,” Phys. Rev. A 39, 3003–3009 (1989).
[CrossRef] [PubMed]

1977 (1)

R. W. Hellwarth, “Third-order optical susceptibilities of liquids and solids,” Prog. Quantum Electron. 5, 1–68 (1977).
[CrossRef]

1975 (1)

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

1974 (1)

O. Svelto, “Self-focusing, self-trapping, and self-phase modulation of laser beams,” Prog. Opt. 12, 1–51 (1974).
[CrossRef]

1973 (1)

A. Owyoung, “Ellipse rotation studies in laser host materials,” IEEE J. Quantum Electron. QE-9, 1064–1069 (1973).
[CrossRef]

1972 (2)

A. Owyoung, R. W. Hellwarth, and N. George, “Intensity-induced changes in optical polarizations in glasses,” Phys. Rev. B 5, 628–633 (1972).
[CrossRef]

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz. 61, 118–134 (1971) [Sov. Phys. JETP 34, 62–69 (1972)].

Boyd, R. W.

A. L. Gaeta and R. W. Boyd, “Transverse instabilities in the polarizations and intensities of counterpropagating light waves,” Phys. Rev. A 48, 1610–1624 (1993).
[CrossRef] [PubMed]

Gaeta, A. L.

A. L. Gaeta and R. W. Boyd, “Transverse instabilities in the polarizations and intensities of counterpropagating light waves,” Phys. Rev. A 48, 1610–1624 (1993).
[CrossRef] [PubMed]

George, N.

A. Owyoung, R. W. Hellwarth, and N. George, “Intensity-induced changes in optical polarizations in glasses,” Phys. Rev. B 5, 628–633 (1972).
[CrossRef]

Hayata, K.

Hellwarth, R. W.

D. M. Pennington, M. A. Henesian, and R. W. Hellwarth, “Nonlinear index of air at 1.053 μm,” Phys. Rev. A 39, 3003–3009 (1989).
[CrossRef] [PubMed]

R. W. Hellwarth, “Third-order optical susceptibilities of liquids and solids,” Prog. Quantum Electron. 5, 1–68 (1977).
[CrossRef]

A. Owyoung, R. W. Hellwarth, and N. George, “Intensity-induced changes in optical polarizations in glasses,” Phys. Rev. B 5, 628–633 (1972).
[CrossRef]

Henesian, M. A.

D. M. Pennington, M. A. Henesian, and R. W. Hellwarth, “Nonlinear index of air at 1.053 μm,” Phys. Rev. A 39, 3003–3009 (1989).
[CrossRef] [PubMed]

Koshiba, M.

Lax, M.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Louisell, W. H.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

McKnight, W. B.

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Misawa, A.

Owyoung, A.

A. Owyoung, “Ellipse rotation studies in laser host materials,” IEEE J. Quantum Electron. QE-9, 1064–1069 (1973).
[CrossRef]

A. Owyoung, R. W. Hellwarth, and N. George, “Intensity-induced changes in optical polarizations in glasses,” Phys. Rev. B 5, 628–633 (1972).
[CrossRef]

Pennington, D. M.

D. M. Pennington, M. A. Henesian, and R. W. Hellwarth, “Nonlinear index of air at 1.053 μm,” Phys. Rev. A 39, 3003–3009 (1989).
[CrossRef] [PubMed]

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz. 61, 118–134 (1971) [Sov. Phys. JETP 34, 62–69 (1972)].

Svelto, O.

O. Svelto, “Self-focusing, self-trapping, and self-phase modulation of laser beams,” Prog. Opt. 12, 1–51 (1974).
[CrossRef]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz. 61, 118–134 (1971) [Sov. Phys. JETP 34, 62–69 (1972)].

IEEE J. Quantum Electron. (1)

A. Owyoung, “Ellipse rotation studies in laser host materials,” IEEE J. Quantum Electron. QE-9, 1064–1069 (1973).
[CrossRef]

J. Opt. Soc. Am. B (1)

Phys. Rev. A (3)

A. L. Gaeta and R. W. Boyd, “Transverse instabilities in the polarizations and intensities of counterpropagating light waves,” Phys. Rev. A 48, 1610–1624 (1993).
[CrossRef] [PubMed]

D. M. Pennington, M. A. Henesian, and R. W. Hellwarth, “Nonlinear index of air at 1.053 μm,” Phys. Rev. A 39, 3003–3009 (1989).
[CrossRef] [PubMed]

M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Phys. Rev. B (1)

A. Owyoung, R. W. Hellwarth, and N. George, “Intensity-induced changes in optical polarizations in glasses,” Phys. Rev. B 5, 628–633 (1972).
[CrossRef]

Prog. Opt. (1)

O. Svelto, “Self-focusing, self-trapping, and self-phase modulation of laser beams,” Prog. Opt. 12, 1–51 (1974).
[CrossRef]

Prog. Quantum Electron. (1)

R. W. Hellwarth, “Third-order optical susceptibilities of liquids and solids,” Prog. Quantum Electron. 5, 1–68 (1977).
[CrossRef]

Sov. Phys. JETP (1)

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz. 61, 118–134 (1971) [Sov. Phys. JETP 34, 62–69 (1972)].

Other (21)

A. Yariv and P. Yeh, “The application of Gaussian beam formalism to optical propagation in nonlinear media,” Opt. Commun. 27, 295–298 (1978); P. P. Banerjee, R. M. Misra, and M. Maghraoui, “Theoretical and experimental studies of propagation of beams through a finite sample of a cubically nonlinear material,” J. Opt. Soc. Am. B 8, 1072–1080 (1991).
[CrossRef]

J. Herrmann, “Propagation of ultrashort light pulses in fibers with saturable nonlinearity in the normal-dispersion region,” J. Opt. Soc. Am. B 8, 1507–1511 (1991); Y. Chen, “Variational principle for vector spatial solitons and nonlinear modes,” Opt. Commun. 84, 355–358 (1991); M. Karlsson, “Optical beams in saturable self-focusing media,” Phys. Rev. A PLRAAN 46, 2726–2734 (1992).
[CrossRef] [PubMed]

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Self-focusing and self-trapping of intense light beams in a nonlinear medium,” Zh. Eksp. Teor. Fiz. 50 1537–1549 (1966) [Sov. Phys. JETP 23, 1025–1033 (1966)]; C. S. Wang, “Propagation of an intense light beam in a nonlinear medium,” Phys. Rev. 173, 908–917 (1968).
[CrossRef]

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap. 17.

R. S. Longhurst, Geometrical and Physical Optics, 3rd ed. (Longman, London, 1973), p. 542.

M. Lefkir, N. P. Xuan, A. J. van Wonderen, and G. Rivoire, “Stabilité de l’état de polarisation en régime picoseconde,” contribution to Quatrième Colloque sur les Lasers et l’Optique Quantique, Palaiseau, France, November 6–8, 1995.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, London, 1980), entry 2.202.

P. D. Maker, R. W. Terhune, and C. M. Savage, “Intensity-dependent changes in the refractive index of liquids,” Phys. Rev. Lett. 12, 507–509 (1964); P. D. Maker and R. W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. 137, A801–A818 (1965).
[CrossRef]

C. C. Wang, “Nonlinear susceptibility constants and self-focusing of optical beams in liquids,” Phys. Rev. 152, 149–156 (1966); R. W. Hellwarth, A. Owyoung, and N. George, “Origin of the nonlinear refractive index of liquid CCl4,” Phys. Rev. A 4, 2342–2347 (1971); J. M. Cherlow, T. T. Yang, and R. W. Hellwarth, “Nonlinear optical susceptibilities of solvents,” IEEE J. Quantum Electron. IEJQA7 QE-12, 644–646 (1976); X. Nguyen Phu, J. L. Ferrier, J. Gazengel, and G. Rivoire, “Polarization of picosecond light pulses in nonlinear isotropic media,” Opt. Commun. OPCOB8 46, 329–333 (1983); N. Pfeffer, F. Charra, and J. M. Nunzi, “Phase and frequency resolution of picosecond optical Kerr nonlinearities,” Opt. Lett. OPLEDP 16, 1987–1989 (1991).
[CrossRef] [PubMed]

S. Saikan and K. Namba, “Intensity dependent polarization change in the D1 and D2 resonance lines of sodium,” Opt. Commun. 23, 73–76 (1977); D. V. Vlasov, R. A. Garaev, V. V. Korobkin, and R. V. Serov, “Measurement of nonlinear polarizability of air,” Zh. Eksp. Teor. Fiz. 76, 2039–2045 (1979) [Sov. Phys. JETP 49, 1033–1036 (1979)].
[CrossRef]

S. M. Arakelyan, S. R. Galstyan, O. V. Garibyan, A. S. Karayan, and Yu. S. Chilingaryan, “Strong, nonlinear, optical activity in the nematic phase of a liquid crystal,” Pis’ma Zh. Eksp. Teor. Fiz. 32, 561–565 (1980) [JETP Lett. 32, 543–547 (1980)]; S. A. Boiko, M. I. Dykman, M. P. Lisitsa, V. I. Sidorenko, and G. G. Tarasov, “Variation of resonance-radiation polarization due to self-induced dichroism in a KCl:Li crystal with FA centers,” Opt. Spektrosk. 58, 1055–1058 (1985) [Opt. Spectrosc. 58, 645–647 (1985)]; S. A. Akhmanov, N. I. Zheludev, and R. S. Zadoyan, “Picosecond spectroscopy of nonlinear optical activity and nonlinear absorption in gallium arsenide,” Zh. Eksp. Teor. Fiz. 91, 984–1000 (1986) [Sov. Phys. JETP SPHJAR 64, 579–588 (1986)].

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, New York, 1989), Chaps. 6 and 18.

J. M. Thorne, T. R. Loree, and G. H. McCall, “Intensity filtration of laser light,” J. Appl. Phys. 45, 3072–3078 (1974); K. Sala and M. C. Richardson, “A passive nonresonant technique for pulse contrast enhancement and gain isolation,” J. Appl. Phys. 49, 2268–2276 (1978); D. V. Murphy and R. K. Chang, “Pulse stretching of Q-switched laser emission by intracavity nonlinear ellipse rotation,” Opt. Commun. OPCOB8 23, 268–272 (1977); V. L. Kalashnikov, V. P. Kalosha, V. P. Mikhailov, I. G. Poloyko, and M. I. Demchuk, “Self-mode locking of continuous-wave solid-state lasers by a nonlinear Kerr polarization modulator,” J. Opt. Soc. Am. B JOBPDE 10, 1443–1446 (1993).
[CrossRef]

For recent reviews, see N. I. Zheludev, “Polarization instability and multistability in nonlinear optics,” Usp. Fiz. Nauk 157, 683–717 (1989) [Sov. Phys. Usp. 32, 357–375 (1989)]; D. David, D. D. Holm, and M. V. Tratnik, “Hamiltonian chaos in nonlinear optical polarization dynamics,” Phys. Rep. 187, 281–367 (1990).
[CrossRef]

N. Akhmediev, A. Buryak, and J. M. Soto-Crespo, “Elliptically polarized solitons in birefringent optical fibers,” Opt. Commun. 112, 278–282 (1994); Y. Silberberg and Y. Barad, “Rotating vector solitary waves in isotropic fibers,” Opt. Lett. 20, 246–248 (1995).
[CrossRef] [PubMed]

A. W. Snyder, S. J. Hewlett, and D. J. Mitchell, “Dynamic spatial solitons,” Phys. Rev. Lett. 72, 1012–1015 (1994); A. W. Snyder, D. J. Mitchell, and Y. Chen, “Spatial solitons of Maxwell’s equations,” Opt. Lett. 19, 524–526 (1994); M. Haelterman and A. P. Sheppard, “The elliptically polarized fundamental vector soliton of isotropic Kerr media,” Phys. Lett. A PYLAAG 194, 191–196 (1994).
[CrossRef] [PubMed]

J. Arons and C. E. Max, “Self-precession and frequency shift for electromagnetic waves in homogeneous plasmas,” Phys. Fluids 17, 1983–1994 (1974); B. Chakraborty, S. N. Paul, M. Khan, and B. Bhattacharyya, “Wave-precession and related nonlinear effects in plasmas,” Phys. Rep. 114, 181–317 (1984).
[CrossRef]

M. D. Feit and J. A. Fleck, Jr., “Beam nonparaxiality, filament formation, and beam breakup in the self-focusing of optical beams,” J. Opt. Soc. Am. B 5, 633–640 (1988); S. Chi and Q. Guo, “Vector theory of self-focusing of an optical beam in Kerr media,” Opt. Lett. 20, 1598–1600 (1995); Y. Chen and J. Atai, “Solitary waves of Maxwell’s equations in nonlinear anisotropic media,” J. Mod. Opt. JMOPEW 42, 1649–1658 (1995).
[CrossRef] [PubMed]

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Secs. 1.3 and 1.4.

X. Nguyen Phu and G. Rivoire, “Evolution of the polarization state of an intense electromagnetic field in a nonlinear medium,” Opt. Acta 25, 233–246 (1978); D. V. Vlasov, V. V. Korobkin, and R. V. Serov, “Nonlinear precession of elliptically polarized Gaussian beams,” Kvantovaya Elektron. (Moscow) 6, 1542–1546 (1979) [Sov. J. Quantum Electron. 9, 904–907 (1979)]; V. P. Nayyar, A. Kumar, and A. Garg, “Elliptically polarized Gaussian wave fields in nonlinear optics,” Opt. Commun. OPCOB8 71, 327–331 (1989).
[CrossRef]

See p. 552 of Ref. 16.

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Figures (3)

Fig. 1
Fig. 1

Plots of parameters ρT and ΔT as a function of the dimensionless pulse energy ξ for e0=0.3, η=1, κ=3, Λ =0.01 (curve a), Λ=0.05 (curve b), and Λ=0.25 (curve c). Parameters (a) ρT and (b) ΔT fully characterize the self-induced changes in the state of polarization of the optical pulse. The relations that determine ρT and ΔT are collected in the appendix.

Fig. 2
Fig. 2

Plot of parameters ρT and ΔT as a function of the dimensionless pulse energy ξ for Λ=0.5. The other parameters are the same as in Fig. 1. The straight line in (b) corresponds to the PCW result for parameter ΔT.

Fig. 3
Fig. 3

Plot of parameters ρ and Δ as a function of ξ in the cw case, with Λ=5. The other parameters are the same as in Fig. 1.

Equations (116)

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

Ecl=E exp[i(kz-ωt)]+c.c.,
Hcl=H exp[i(kz-ωt)]+c.c.,
Hz=iλ2πwHx(x/w)+Hy(y/w)+iλ2πlHz(z/l),
(1+4πχ)·Ecl+O(f2)=0,
×H+ikeˆz×H+iωc-1(E+4πP)=4πσc-1E,
×E+ikeˆz×E-iωc-1H=0,
Ez+12ikt2E=2πiωcnPNL-αE,
PNL,j=3k,l,mχjklm(3)EkElEm*.
χjklm(3)=jklmRjjRkkRllRmmχjklm(3).
χjklm(3)=μ1δjkδlm+μ2δjlδkm+μ3δjmδkl.
f1,maxjklm|χjklm(3)||E|2/f2O(10),
A+A-=121i1-iExEy
A±=|A±|exp(ia±).
|A+|2z+1k|A+|2t2a++1k(t|A+|2)·(ta+)
=-2α|A+|2-2a(|A+|2+|A-|2)|A+|2-2b|A+|2|A-|2,
a+z-12k|A+|t2|A+|+12k(ta+)·(ta+)
=a(|A+|2+|A-|2)+b|A-|2.
Ecl=2(|A+|+|A-|)R(θ)S(e)R(-ηΦ)eˆx,
R(θ)=cos θ-sin θsin θcos θ,S(e)=100e.
2θ=a+-a-,e=||A+|-|A-|||A+|+|A-|,
η=sgn(|A-|-|A+|),Φ=kz-ωt+1/2(a++a-).
ddz|A+|2=-2α|A+|2-2a(|A+|2+|A-|2)|A+|2-2b|A+|2|A-|2,
ddz(|A+|2-|A-|2)2+γ|A+|2|A-|2=0,γ=ba.
e(z)=e(z=0)e0,
|A±|2=1/2I (1ηe)21+e2.
I(z)/I(0)=[1+2aI(0)z]-1,
θ(z)-θ0=ηbe02a(1+e02)log[I(0)/I(z)],
θ(z)-θ0=12πηωe0zcn(1+e02)I(0)χxyyx(3).
ζ(2a+b)I(0)z=(1+K)1/21ψ(z)dx(1+Kxμ)-1/2,
K=4e02(1-e02)2,μ=2γ2+γ=2χxyyx(3)/χxxxx(3).
1+ζψ1+2-μ2ζ2/(2-μ),
I(z)/I(0)=[1+Kψ(z)μ]1/2/[(1+K)1/2ψ(z)],
e(z)=K1/2ψ(z)μ/2/{1+[1+Kψ(z)μ]1/2}.
[|A+(z)|2-|A-(z)|2]2=I(z)2-I(0)2ψ(z)-2/(1+K).
θ(z)-θ0=ηb2blog1+e(z)1-e(z)1-e01+e0.
Aˆ±(s)=A±(z)exp(αz),2αs=1-exp(-2αz).
A±(z, α)=exp(-αz)A±1-exp(-2αz)2α, α=0.
zlog|A+|2|A-|2+1k|A+|2t·(|A+|2ta+)-1k|A-|2t
·(|A-|2ta-)=0.
e(z=0, r)=e0,θ(z=0, r)=θ0,
|A+||A-|=1-ηe01+ηe0.
rr|A+|2 θr=0,
θz=b(|A-|2-|A+|2).
e0(1+ηe0)2|A+|2r=0.
a±(z, r)=a±,0(z)+1/2a±,1(z)kr2.
a+,1-1 |A+|2z+2|A+|2+r |A±|2r=-2αa+,1-1|A+|2.
J+(z, r)=|A+(z, r)|2p+(z)-1 exp(2αz),
p+(z)=exp-20zdsa+,1(s).
|A+(z, r)|2=p+(z)exp(-2αz)|A+[z=0, rp+(z)1/2]|2.
|A±(z=0, r)|2=|A±,0|2 exp(-r2/w02).
da+,0dz+l-1p+=a|A+,0|2p++(a+b)|A-,0|2p-,
da+,1dz-l-2p+2+a+,12=-2l-1a|A+,0|2p+2-2l-1(a+b)|A-,0|2p-2,
a+,1(z=0)=a-,1(z=0)=a0/l2,
a+,0(z)=a+,0(0)-arctan(u++v+z)+arctan(u+),
a+,1(z)=v+ u++v+z1+(u++v+z)2,
p+(z)=1+u+21+(u++v+z)2.
u+(a=b=0)=a0l-1,
v+(a=b=0)=l-1+a02l-3.
l-1l-1-a|A+,0|2-(a+b)|A-,0|2.
A+(z, r)=A+,01+u+21+(u++v+z)21/2×exp-r22w021+u+21+(u++v+z)2+1/2ikr2v+ u++v+z1+(u++v+z)2-i arctan(u++v+z)+i arctan(u+),
w+2=w02 1+(u++v+z)21+u+2.
|(2a+b)|I0l1,r/w01,
e(z, r)=e0+O(l-2),
2θ(z, r)=2θ(0)+zb(|A-,0|2-|A+,0|2)+O(l-3).
(12π)-1n2|(2a+b)|I0lO(10).
1-e(z, r)1+e(z, r)=1-e01+e01+v+2z21+v-2z21/2×expr2z2(v-2-v+2)2w02(1+v+2z2)(1+v-2z2),
2θ(z, r)-2θ0=arctan(v-z)-arctan(v+z)+kzr2(v+2-v-2)2(1+v+2z2)(1+v-2z2),
v±=l-1-a|A±,0|2-(a+b)|A,0|2.
limze(z, r)=(1+e0)|v-|-(1-e0)|v+|(1+e0)|v-|+(1-e0)|v+|.
e(z, r)=e0+1/4z2(1-e02)(v-2-v+2)(1-r2/w02)+O(z4).
θ(z, r)-θ0=z(v--v+)(1-r2/w02)+O[|(2a+b)I0|2]+O(z3).
Epol(x, y, z=L, t)=eˆeˆ·Ecl(x, y, z=L, t),
Epol=12eˆ[A˜+ exp(-iα)+A˜- exp(iα)]×exp[i(kz-ωt)]+c.c.
A˜±(z, r)=A˜±,0(z+q±)-1 exp[1/2ikr2(z+q±)-1],
A˜+,0=-ilA+,0 1-iv+L1-iv+2lL,q+=-L+il1-iv+2lL.
I=dxdy|S|,S=c4πEpol×Hpol.
A˜+x(kA˜+)r|q+|10w0l=O(f),
S=cn4π|A˜+ exp(-iα)+A˜- exp(iα)|2eˆz.
I=1/2I0+1/2cn ReA˜+,0A˜-,0*(z+q+)(z+q-*)×exp(-2iα)0dy×exp1/2iky1z+q+-1z+q-*,
I0=1/2cnw02(|A+,0|2+|A-,0|2).
I=1/2I0+1/2I0 1-e021+e02ρ cos(2α-2θ0-Δ),
ρ=1+(1+l2/L2)(v+2L2-v-2L2)24(1+v+2L2)(1+v-2L2)-1/2,
Δ=arctan(v-L)-arctan(v+L)+arctanl(v+2L-v-2L)2+v+2L2+v-2L2.
ρ=1,Δ=(v--v+)L.
ξ=2e01+e0224πω2I0nc3χxyyx(3),Λ=Ll.
v±L=Λ[1-1/2(β±η)ξ],
β=1+e022e0κ,κ=χxxxx(3)/χxyyx(3).
ρ=1-1/2(1+Λ-2)-1ξ2+O(ξ3),
Δ=ηβΛ2(1+Λ2)ξ2+O(ξ3).
ρ=limξρ=1+4β2(1+Λ-2)(1-β2)2-1/2,
Δ=limξΔ=η arctan2βΛ(1+β2).
A±(z=0, r, t)=A±,0h(t)exp(-1/2r2/w02).
A±,0A±,0h(t-vg-1z),
E-dtI(t)=1/2E0+1/2E0 1-e021+e02h¯-1-dth(t)2ρ(t)×cos[2α-2θ0-Δ(t)],
1+e021-e02[E(E0, α)-E(E0, α+π/2)]/E0
=cos(2α-2θ0)G+sin(2α-2θ0)H.
G=h¯-1-dth(t)2ρ(t)cos[Δ(t)],
H=h¯-1-dth(t)2ρ(t)sin[Δ(t)].
ρT=(G2+H2)1/2,ΔT=arctan(H/G).
(ρT, ΔT)(ξ=0)=(ρ, Δ)(ξ=0),
limξ(ρT, ΔT)=limξ(ρ, Δ).
limxh¯-1-dth(t)2f[xh(t)2]=limxf(x),
h(t)=exp(-1/2t2/T2).
ξGauss=2e01+e0248(π log 2)1/2ω2E0nτc3χxyyx(3)=log 16π1/2ξcw.
(12π)-1n2βξO(10).
(χxyyx(3))PCW=Λ-1 dΔTdξ(χxyyx(3))TT10(χxyyx(3))TT.
ρT=(G2+H2)1/2,ΔT=arctan(H/G),
G=2π0dt exp(-t2)ρ(t)cos[Δ(t)],
H=2π0dt exp(-t2)ρ(t)sin[Δ(t)],
ρ(t)=1+Λ2(1+Λ2)(δ+2-δ-2)24(1+Λ2δ+2)(1+Λ2δ-2)-1/2,
Δ(t)=arctan(Λδ-)-arctan(Λδ+)+arctanΛ(δ+2-δ-2)2+Λ2δ+2+Λ2δ-2,
δ±(t)=1-1/2(β±η)ξ exp(-t2),β=1+e022e0κ,
ξ=2e01+e0248(π log 2)1/2ω2E0nτc3χxyyx(3),
Λ=Ll,
l=nωc-1w02.
χxyyx(3)c2(4π107)-1χxyyx(3)

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