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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    [Crossref]
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    [Crossref]
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    [Crossref]
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  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]
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  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. 501537–1549 (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]
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1994 (2)

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 194, 191–196 (1994).
[Crossref] [PubMed]

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]

1991 (1)

1990 (1)

1989 (2)

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]

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]

1988 (1)

1980 (1)

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 64, 579–588 (1986)].

1978 (2)

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. 71, 327–331 (1989).
[Crossref]

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]

1977 (2)

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

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]

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 (3)

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

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]

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. 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 10, 1443–1446 (1993).
[Crossref]

1973 (1)

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

1972 (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]

1971 (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)].

1966 (2)

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. 501537–1549 (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]

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. 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. 46, 329–333 (1983); N. Pfeffer, F. Charra, and J. M. Nunzi, “Phase and frequency resolution of picosecond optical Kerr nonlinearities,” Opt. Lett. 16, 1987–1989 (1991).
[Crossref] [PubMed]

1964 (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]

Akhmanov, S. A.

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. 501537–1549 (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]

Akhmediev, N.

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]

Arakelyan, S. M.

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 64, 579–588 (1986)].

Arons, J.

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]

Born, M.

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

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]

Buryak, A.

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]

Chilingaryan, Yu. S.

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 64, 579–588 (1986)].

Feit, M. D.

Fleck, J. A.

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]

Galstyan, S. R.

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 64, 579–588 (1986)].

Garibyan, O. V.

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 64, 579–588 (1986)].

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]

Gradshteyn, I. S.

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

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]

Herrmann, J.

Hewlett, S. J.

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 194, 191–196 (1994).
[Crossref] [PubMed]

Karayan, A. S.

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 64, 579–588 (1986)].

Khokhlov, R. V.

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. 501537–1549 (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]

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]

Lefkir, M.

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.

Longhurst, R. S.

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

Loree, T. R.

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. 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 10, 1443–1446 (1993).
[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]

Maker, P. D.

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]

Max, C. E.

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]

McCall, G. H.

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. 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 10, 1443–1446 (1993).
[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.

Mitchell, D. J.

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 194, 191–196 (1994).
[Crossref] [PubMed]

Namba, K.

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]

Nguyen Phu, X.

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. 71, 327–331 (1989).
[Crossref]

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]

Rivoire, G.

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. 71, 327–331 (1989).
[Crossref]

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.

Ryzhik, I. M.

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

Saikan, S.

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]

Savage, C. M.

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]

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)].

Shen, Y. R.

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

Snyder, A. W.

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 194, 191–196 (1994).
[Crossref] [PubMed]

Soto-Crespo, J. M.

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]

Sukhorukov, A. P.

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. 501537–1549 (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]

Svelto, O.

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

Terhune, R. W.

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]

Thorne, J. M.

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. 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 10, 1443–1446 (1993).
[Crossref]

van Wonderen, A. J.

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.

Wang, C. C.

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. 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. 46, 329–333 (1983); N. Pfeffer, F. Charra, and J. M. Nunzi, “Phase and frequency resolution of picosecond optical Kerr nonlinearities,” Opt. Lett. 16, 1987–1989 (1991).
[Crossref] [PubMed]

Wolf, E.

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

Xuan, N. P.

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.

Yariv, A.

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]

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

Yeh, P.

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]

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)].

Zheludev, N. I.

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]

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. Appl. Phys. (1)

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. 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 10, 1443–1446 (1993).
[Crossref]

J. Opt. Soc. Am. B (3)

Opt. Acta (1)

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. 71, 327–331 (1989).
[Crossref]

Opt. Commun. (3)

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]

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]

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]

Phys. Fluids (1)

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]

Phys. Rev. (1)

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. 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. 46, 329–333 (1983); N. Pfeffer, F. Charra, and J. M. Nunzi, “Phase and frequency resolution of picosecond optical Kerr nonlinearities,” Opt. Lett. 16, 1987–1989 (1991).
[Crossref] [PubMed]

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]

Phys. Rev. Lett. (2)

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]

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 194, 191–196 (1994).
[Crossref] [PubMed]

Pis’ma Zh. Eksp. Teor. Fiz. (1)

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 64, 579–588 (1986)].

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]

Usp. Fiz. Nauk (1)

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]

Zh. Eksp. Teor. Fiz. (2)

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)].

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. 501537–1549 (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]

Other (7)

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.

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

See p. 552 of Ref. 16.

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

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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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