Abstract

Measurements of the third-order nonlinear electric susceptibility component χxyyx3 by means of self-induced changes in the state of polarization of an intense elliptically polarized laser beam are reported. The simultaneous observation of the transverse intensity profile and of the losses allows us to show the influence of field transverse gradients on the polarization state. We show that the plane-wave model cannot explain our experimental results, even at low input intensities. The influence of spatial dispersion on nonlinear susceptibility is demonstrated.

© 1997 Optical Society of America

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References

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  1. G. Rivoire, in Modern Nonlinear Optics, M. Evans, ed. (Wiley, New York, 1993), Part 1, pp. 217–247.
  2. M. Sheik-Bahae, A. A. Said, T. H. Vei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990);T. Xia, D. J. Hagan, M. Sheik-Bahae, and E. W. Van Stryland, “Eclipsing Z-scan measurement of λ/104 wave-front distortion,” Opt. Lett. 19, 317 (1994).
    [CrossRef] [PubMed]
  3. G. Boudebs, M. Chis, and J. P. Bourdin, “Third-order susceptibility measurements by nonlinear image processing,” J. Opt. Soc. Am. B 13, 1450 (1996).
    [CrossRef]
  4. P. D. Maker, R. W. Terhune, and C. M. Savage, “Intensity dependent changes in the refractive index of liquids,” Phys. Rev. Lett. 12, 507 (1964);P. D. Maker and R. W. Terhune, “Study of optical effects due to induced polarization third order in the electric field strength,” Phys. Rev. 137, A801 (1965).
    [CrossRef]
  5. C. C. Wang, “Nonlinear susceptibility constant and self-focusing of optical beams in liquids,” Phys. Rev. 152, 149 (1966).
    [CrossRef]
  6. S. Kielich, “Nonlinear change in light depolarization ratio due to optical reorientation effect,” Phys. Lett. 25A, 517 (1967).
    [CrossRef]
  7. P. X. Nguyen and G. Rivoire, “Evolution of the polarization state of an intense electromagnetic field in a nonlinear medium,” Opt. Acta 25, 233 (1978);P. X. Nguyen, J. L. Ferrier, J. Gazengel, and G. Rivoire, “Polarization of light pulses in nonlinear isotropic media,” Opt. Commun. 46, 329 (1983).
    [CrossRef]
  8. A. Barthelemy and C. Froelhy, presented at the Conference on Mathematical and Physical Problems Connected with Propagation in Quadratic Nonlinear media, Paris, February, 14, 1996.
  9. A. J. Van Wonderen, “Influence of transverse effects on self-induced polarization changes in an isotropic Kerr medium,” J. Opt. Soc. Am. B 14, 1118 (1997).
    [CrossRef]
  10. D. Wang and G. Rivoire, “Large spectral band width SRWS in CS2,” J. Chem. Phys. 98, 9279 (1993);A. Fahmi, J. P. Bourdin, R. Chevalier, X. N. Phu, and G. Rivoire, “Influence of SRWS in χ3 measurements and wave mixing experiments in CS2,” Nonlin. Opt. 12, 165 (1995).
    [CrossRef]
  11. J. P. Bourdin, N. P. Xuan, G. Rivoire, and J. M. Nunzi, “Polarization properties of the orientational response in phase conjugation,” Nonlin. Opt. 7, 1 (1994).
  12. M. Lefkin, “Mesure des susceptibilités nonlinéaire l'ordre trois par auto-modification le l'état de polarisation d'une mode lumineuse:Rôle des gradients transverses,” Ph.D. dissertation No. 232 (University of Angers, France, 1996); M. Vampouille, “Instabilités spatio-temporelles d'impulsions laser brève après propagation nonlinéaire daus CS2,” Habilitation à Diriger des Recherches diploma (University of Limoges, France, 1993); J. M. Soto-Crespo, D. R. Heatley, E. M. Wright, and N. N. Akhmediev, “Stability of the higher-bound states in a saturable self-focusing medium,” Phys. Rev. A 44, 636 (1991);J. H. Marburger, Prog. Quantum Electron. 4, 35 (1975);O. Svelto, “Self-focusing,self-trapping, and self-phase modulation of laser beams,” in Progress in Quantum Electronics, J. H. Sanders and S. Stenholm, eds. (Pergamon, New York, 1977), Vol. 5, pp. 1–68.
    [CrossRef] [PubMed]

1997 (1)

1996 (1)

1994 (1)

J. P. Bourdin, N. P. Xuan, G. Rivoire, and J. M. Nunzi, “Polarization properties of the orientational response in phase conjugation,” Nonlin. Opt. 7, 1 (1994).

1993 (1)

D. Wang and G. Rivoire, “Large spectral band width SRWS in CS2,” J. Chem. Phys. 98, 9279 (1993);A. Fahmi, J. P. Bourdin, R. Chevalier, X. N. Phu, and G. Rivoire, “Influence of SRWS in χ3 measurements and wave mixing experiments in CS2,” Nonlin. Opt. 12, 165 (1995).
[CrossRef]

1990 (1)

M. Sheik-Bahae, A. A. Said, T. H. Vei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990);T. Xia, D. J. Hagan, M. Sheik-Bahae, and E. W. Van Stryland, “Eclipsing Z-scan measurement of λ/104 wave-front distortion,” Opt. Lett. 19, 317 (1994).
[CrossRef] [PubMed]

1978 (1)

P. X. Nguyen and G. Rivoire, “Evolution of the polarization state of an intense electromagnetic field in a nonlinear medium,” Opt. Acta 25, 233 (1978);P. X. Nguyen, J. L. Ferrier, J. Gazengel, and G. Rivoire, “Polarization of light pulses in nonlinear isotropic media,” Opt. Commun. 46, 329 (1983).
[CrossRef]

1967 (1)

S. Kielich, “Nonlinear change in light depolarization ratio due to optical reorientation effect,” Phys. Lett. 25A, 517 (1967).
[CrossRef]

1966 (1)

C. C. Wang, “Nonlinear susceptibility constant and self-focusing of optical beams in liquids,” Phys. Rev. 152, 149 (1966).
[CrossRef]

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 (1964);P. D. Maker and R. W. Terhune, “Study of optical effects due to induced polarization third order in the electric field strength,” Phys. Rev. 137, A801 (1965).
[CrossRef]

Barthelemy, A.

A. Barthelemy and C. Froelhy, presented at the Conference on Mathematical and Physical Problems Connected with Propagation in Quadratic Nonlinear media, Paris, February, 14, 1996.

Boudebs, G.

Bourdin, J. P.

G. Boudebs, M. Chis, and J. P. Bourdin, “Third-order susceptibility measurements by nonlinear image processing,” J. Opt. Soc. Am. B 13, 1450 (1996).
[CrossRef]

J. P. Bourdin, N. P. Xuan, G. Rivoire, and J. M. Nunzi, “Polarization properties of the orientational response in phase conjugation,” Nonlin. Opt. 7, 1 (1994).

Chis, M.

Froelhy, C.

A. Barthelemy and C. Froelhy, presented at the Conference on Mathematical and Physical Problems Connected with Propagation in Quadratic Nonlinear media, Paris, February, 14, 1996.

Hagan, D. J.

M. Sheik-Bahae, A. A. Said, T. H. Vei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990);T. Xia, D. J. Hagan, M. Sheik-Bahae, and E. W. Van Stryland, “Eclipsing Z-scan measurement of λ/104 wave-front distortion,” Opt. Lett. 19, 317 (1994).
[CrossRef] [PubMed]

Kielich, S.

S. Kielich, “Nonlinear change in light depolarization ratio due to optical reorientation effect,” Phys. Lett. 25A, 517 (1967).
[CrossRef]

Lefkin, M.

M. Lefkin, “Mesure des susceptibilités nonlinéaire l'ordre trois par auto-modification le l'état de polarisation d'une mode lumineuse:Rôle des gradients transverses,” Ph.D. dissertation No. 232 (University of Angers, France, 1996); M. Vampouille, “Instabilités spatio-temporelles d'impulsions laser brève après propagation nonlinéaire daus CS2,” Habilitation à Diriger des Recherches diploma (University of Limoges, France, 1993); J. M. Soto-Crespo, D. R. Heatley, E. M. Wright, and N. N. Akhmediev, “Stability of the higher-bound states in a saturable self-focusing medium,” Phys. Rev. A 44, 636 (1991);J. H. Marburger, Prog. Quantum Electron. 4, 35 (1975);O. Svelto, “Self-focusing,self-trapping, and self-phase modulation of laser beams,” in Progress in Quantum Electronics, J. H. Sanders and S. Stenholm, eds. (Pergamon, New York, 1977), Vol. 5, pp. 1–68.
[CrossRef] [PubMed]

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 (1964);P. D. Maker and R. W. Terhune, “Study of optical effects due to induced polarization third order in the electric field strength,” Phys. Rev. 137, A801 (1965).
[CrossRef]

Nguyen, P. X.

P. X. Nguyen and G. Rivoire, “Evolution of the polarization state of an intense electromagnetic field in a nonlinear medium,” Opt. Acta 25, 233 (1978);P. X. Nguyen, J. L. Ferrier, J. Gazengel, and G. Rivoire, “Polarization of light pulses in nonlinear isotropic media,” Opt. Commun. 46, 329 (1983).
[CrossRef]

Nunzi, J. M.

J. P. Bourdin, N. P. Xuan, G. Rivoire, and J. M. Nunzi, “Polarization properties of the orientational response in phase conjugation,” Nonlin. Opt. 7, 1 (1994).

Rivoire, G.

J. P. Bourdin, N. P. Xuan, G. Rivoire, and J. M. Nunzi, “Polarization properties of the orientational response in phase conjugation,” Nonlin. Opt. 7, 1 (1994).

D. Wang and G. Rivoire, “Large spectral band width SRWS in CS2,” J. Chem. Phys. 98, 9279 (1993);A. Fahmi, J. P. Bourdin, R. Chevalier, X. N. Phu, and G. Rivoire, “Influence of SRWS in χ3 measurements and wave mixing experiments in CS2,” Nonlin. Opt. 12, 165 (1995).
[CrossRef]

P. X. Nguyen and G. Rivoire, “Evolution of the polarization state of an intense electromagnetic field in a nonlinear medium,” Opt. Acta 25, 233 (1978);P. X. Nguyen, J. L. Ferrier, J. Gazengel, and G. Rivoire, “Polarization of light pulses in nonlinear isotropic media,” Opt. Commun. 46, 329 (1983).
[CrossRef]

G. Rivoire, in Modern Nonlinear Optics, M. Evans, ed. (Wiley, New York, 1993), Part 1, pp. 217–247.

Said, A. A.

M. Sheik-Bahae, A. A. Said, T. H. Vei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990);T. Xia, D. J. Hagan, M. Sheik-Bahae, and E. W. Van Stryland, “Eclipsing Z-scan measurement of λ/104 wave-front distortion,” Opt. Lett. 19, 317 (1994).
[CrossRef] [PubMed]

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 (1964);P. D. Maker and R. W. Terhune, “Study of optical effects due to induced polarization third order in the electric field strength,” Phys. Rev. 137, A801 (1965).
[CrossRef]

Sheik-Bahae, M.

M. Sheik-Bahae, A. A. Said, T. H. Vei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990);T. Xia, D. J. Hagan, M. Sheik-Bahae, and E. W. Van Stryland, “Eclipsing Z-scan measurement of λ/104 wave-front distortion,” Opt. Lett. 19, 317 (1994).
[CrossRef] [PubMed]

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 (1964);P. D. Maker and R. W. Terhune, “Study of optical effects due to induced polarization third order in the electric field strength,” Phys. Rev. 137, A801 (1965).
[CrossRef]

Van Stryland, E. W.

M. Sheik-Bahae, A. A. Said, T. H. Vei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990);T. Xia, D. J. Hagan, M. Sheik-Bahae, and E. W. Van Stryland, “Eclipsing Z-scan measurement of λ/104 wave-front distortion,” Opt. Lett. 19, 317 (1994).
[CrossRef] [PubMed]

Van Wonderen, A. J.

Vei, T. H.

M. Sheik-Bahae, A. A. Said, T. H. Vei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990);T. Xia, D. J. Hagan, M. Sheik-Bahae, and E. W. Van Stryland, “Eclipsing Z-scan measurement of λ/104 wave-front distortion,” Opt. Lett. 19, 317 (1994).
[CrossRef] [PubMed]

Wang, C. C.

C. C. Wang, “Nonlinear susceptibility constant and self-focusing of optical beams in liquids,” Phys. Rev. 152, 149 (1966).
[CrossRef]

Wang, D.

D. Wang and G. Rivoire, “Large spectral band width SRWS in CS2,” J. Chem. Phys. 98, 9279 (1993);A. Fahmi, J. P. Bourdin, R. Chevalier, X. N. Phu, and G. Rivoire, “Influence of SRWS in χ3 measurements and wave mixing experiments in CS2,” Nonlin. Opt. 12, 165 (1995).
[CrossRef]

Xuan, N. P.

J. P. Bourdin, N. P. Xuan, G. Rivoire, and J. M. Nunzi, “Polarization properties of the orientational response in phase conjugation,” Nonlin. Opt. 7, 1 (1994).

IEEE J. Quantum Electron. (1)

M. Sheik-Bahae, A. A. Said, T. H. Vei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760 (1990);T. Xia, D. J. Hagan, M. Sheik-Bahae, and E. W. Van Stryland, “Eclipsing Z-scan measurement of λ/104 wave-front distortion,” Opt. Lett. 19, 317 (1994).
[CrossRef] [PubMed]

J. Chem. Phys. (1)

D. Wang and G. Rivoire, “Large spectral band width SRWS in CS2,” J. Chem. Phys. 98, 9279 (1993);A. Fahmi, J. P. Bourdin, R. Chevalier, X. N. Phu, and G. Rivoire, “Influence of SRWS in χ3 measurements and wave mixing experiments in CS2,” Nonlin. Opt. 12, 165 (1995).
[CrossRef]

J. Opt. Soc. Am. B (2)

Nonlin. Opt. (1)

J. P. Bourdin, N. P. Xuan, G. Rivoire, and J. M. Nunzi, “Polarization properties of the orientational response in phase conjugation,” Nonlin. Opt. 7, 1 (1994).

Opt. Acta (1)

P. X. Nguyen and G. Rivoire, “Evolution of the polarization state of an intense electromagnetic field in a nonlinear medium,” Opt. Acta 25, 233 (1978);P. X. Nguyen, J. L. Ferrier, J. Gazengel, and G. Rivoire, “Polarization of light pulses in nonlinear isotropic media,” Opt. Commun. 46, 329 (1983).
[CrossRef]

Phys. Lett. (1)

S. Kielich, “Nonlinear change in light depolarization ratio due to optical reorientation effect,” Phys. Lett. 25A, 517 (1967).
[CrossRef]

Phys. Rev. (1)

C. C. Wang, “Nonlinear susceptibility constant and self-focusing of optical beams in liquids,” Phys. Rev. 152, 149 (1966).
[CrossRef]

Phys. Rev. Lett. (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 (1964);P. D. Maker and R. W. Terhune, “Study of optical effects due to induced polarization third order in the electric field strength,” Phys. Rev. 137, A801 (1965).
[CrossRef]

Other (3)

G. Rivoire, in Modern Nonlinear Optics, M. Evans, ed. (Wiley, New York, 1993), Part 1, pp. 217–247.

A. Barthelemy and C. Froelhy, presented at the Conference on Mathematical and Physical Problems Connected with Propagation in Quadratic Nonlinear media, Paris, February, 14, 1996.

M. Lefkin, “Mesure des susceptibilités nonlinéaire l'ordre trois par auto-modification le l'état de polarisation d'une mode lumineuse:Rôle des gradients transverses,” Ph.D. dissertation No. 232 (University of Angers, France, 1996); M. Vampouille, “Instabilités spatio-temporelles d'impulsions laser brève après propagation nonlinéaire daus CS2,” Habilitation à Diriger des Recherches diploma (University of Limoges, France, 1993); J. M. Soto-Crespo, D. R. Heatley, E. M. Wright, and N. N. Akhmediev, “Stability of the higher-bound states in a saturable self-focusing medium,” Phys. Rev. A 44, 636 (1991);J. H. Marburger, Prog. Quantum Electron. 4, 35 (1975);O. Svelto, “Self-focusing,self-trapping, and self-phase modulation of laser beams,” in Progress in Quantum Electronics, J. H. Sanders and S. Stenholm, eds. (Pergamon, New York, 1977), Vol. 5, pp. 1–68.
[CrossRef] [PubMed]

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Figures (5)

Fig. 1
Fig. 1

Typical laser transverse intensity distribution(open circles) compared with a Gaussian shape (solid curve).

Fig. 2
Fig. 2

Experimental arrangement: A, afocal system; BS, beam splitter; λ/2: half-wave plate; λ/4, quarter-wave plate; G, Glan prism; C, nonlinear medium. (a) Measurements: W's, Wollaston prisms; F,F1,F2,Fc, calibrated neutral-density filters; Ds, Dc, D1, D2, photodiodes; L1, L2, lenses. (b) Controls: CCD camera for beam transverse shape control and spectrograph for scattering control; abbreviations are as for (a).

Fig. 3
Fig. 3

Experimental results for CS2. (a) Cell length l=1 mm, (b) cell length l=5 mm. (a.1), (b.1), Transmission T versus input intensity I0. (a.2), (b.2), ρ=ln[(1+e)/(1-e)], polarization ellipse characteristic, where e is the ellipticity. (a.3), (b.3), θ, rotation of polarization ellipse versus I0. (a.4), (b.4), Evolution of the transverse spatial profile of the output pulse with I0.

Fig. 4
Fig. 4

Ellipse rotation θ versus input intensity I0 in the absence of transverse profile changes (I0<IT): (a) CS2 (carbon disulfide) for l=1, 2, 5 mm; (b) C6H7N (α-picoline) for l =1, 2, 5, 10 mm.

Fig. 5
Fig. 5

Measured ξxyyx versus cell length l for a plane-wave model.

Tables (2)

Tables Icon

Table 1 Intensity Values IT Above Which an Ellipticity Change Is Observed

Tables Icon

Table 2 χxyyx Values Deduced from Rotation Measurements

Equations (51)

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z+n0ctA-ikΔtA=2iπωn0cPNL-α0A,
PNL(χxxyy+χxyxy)|A|2A+χxyyxAA*A.
1I±I±z=-2α0-2a(I++I-)-2bI,
φ±z=a(I++I-)+bI,
A±=12(Ax+iAy),
A±=|A±|exp(jφ±),
I±=nc2π|A±|2,
a=a+ia=12π2ωn02c2(χxxyy+χxyxy),
b=b+ib=24π2ωn02c2χxyyx;
epw(z)=e0,
θpw(z)-θ0(z)=Δθpw(z)=be01+e02I0z,
ki=sin2(θ-αi)+e2 cos2(θ-αi)cos2(θ-αi)+e2 sin2(θ-αi)=(e02+1)-(e02-1)cos 2(αi-θ)(e02+1)+(e02-1)cos 2(αi-θ).
I±z+I±kt2φ±+1k(tI±)(tφ±)=0,
φ±z-12k|A±|t2|A±|+1k(tφ±)(tφ±)
=a(I++I-)+bI-.
φ+(z=0)=φ-(z=0)=0(r),
A±(z=0)=A0±f(r),
I±z(z=0)=0,
φ±z(z=0)=a(I++I-)+2bI-+12kt2f(r)f(r)=P±+Q,
φ±=Pz+ε±(z),
zε±(z)=-14kz2I02(a+bq0±)2R(r),
1I±I±z=-1kI0(a+bq0±)zS(r),
q0±=(1±e0)22(1+e02),
R(r)=(f2/r)2,
S(r)=2fr2+1rf2r+f2r2.
Δθd(r)=I0be01+e02z×f2(r)-I06kz2(2a+b)R(r),
ρd(r)=ρ0-I0be01+e02z2kS(r),
δθ(r)=Δθ¯d-ΔθPW=I0be01+e02zI06kz2(2a+b)R(r)
Δθ¯d=I0be01+e02z12-I06kz2(2a+b)R¯
=θPW2-δθ¯d,
ρ¯d=ρ0-I0bke01+e02z2S¯,
G¯=0G(r)f2(r)2πrdr0f2(r)2πrdr.
R(r)=r2σ4exp-r2σ2,
S(r)=1σ2exp-r22σ2-2+r2σ2+r2σ2exp-r22σ2
R¯=29σ2,S¯=-518σ2.
Δθ¯d=ΔθPW21-Azλσ2,
ρ¯d-ρ0=Bzλσ2,
A=2ΔθPW2a+bb154π1+e02e0,
B=ΔθPW536π,
ΔθPW=I0be01+e02z=2Δθ,
2a+bb=χxxxxχxyyx.
χxyyx=ξxyyx1-Azλσ2.
W1/2=120Iz=0(r)1±e02-1e02+1×cos2[αi-pI(r)]2πrdr.
W1/2=W(z=0)21±πppI00e02-1e02+1cos 2(αi-u)σ2du;
W1/2=W(z=0)21±e02-1e02+1sin pI0pI0cos(2αi-pI0).
W1W2=e02+1-(e02-1)cos(2αi-pI0)Ue02+1+(e02-1)cos(2αi-pI0)U,
W1W2=e02+1-(e02-1)cos(2αi-θ)e02+1+(e02-1)cos(2αi-θ).
cos 2(αi-θ)=U cos(2αi-pI0).
θ=pI0/2.
Q=λ4πσ02r2σ02-1.
Δn0λ216π2σ02.

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