Abstract

A theory to treat the propagation of an intense pump pulse and a weak probe pulse in nonlinear media is presented. We give a general formalism applicable to the pump–probe propagation in a variety of nonlinear optical media. Based on this formalism, we obtain first an analytic, closed-form solution of pump–probe propagation in a third-order nonlinear medium, including linear absorption, nonlinear refraction, and two-photon absorption. The effects of velocity mismatch are also considered in our results. The general formalism leads further to analytic results for pump–probe propagation in a medium with both third- and fifth-order nonlinearity.

© 2004 Optical Society of America

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References

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    [CrossRef]
  7. L. Luo, L. Chen, Z. R. Qiu, X. Y. Yu, D. C. Dai, J. Y. Zhou, and J. Kuhl, “Measurement of femtosecond, resonant, nonlinear refraction in Nd:YVO4 by degenerate pump–probe spectroscopy,” J. Appl. Phys. 89, 8342–8344 (2001).
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    [CrossRef]
  10. G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989).
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  13. L. Lepetit, G. Cheriaux, and M. Joffre, “Linear techniques of phase measurement by femtosecond spectral interferometry for applications in spectroscopy,” J. Opt. Soc. Am. B 12, 2467–2474 (1995).
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  14. D. N. Fittinghoff, J. L. Bowie, J. N. Sweetser, R. T. Jennings, M. A. Krumbugel, K. W. DeLong, R. Trebino, and I. A. Walmsley, “Measurement of the intensity and phase of ultraweak, ultrashort laser pulses,” Opt. Lett. 21, 884–886 (1996).
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2002

J. M. Harbold, F. O. Ilday, F. W. Wise, and B. G. Aitken, “Highly nonlinear Ge–As–Se and Ge–As–S–Se glasses for all-optical switching,” IEEE Photon. Technol. Lett. 14, 822–824 (2002).
[CrossRef]

2001

L. Luo, L. Chen, Z. R. Qiu, X. Y. Yu, D. C. Dai, J. Y. Zhou, and J. Kuhl, “Measurement of femtosecond, resonant, nonlinear refraction in Nd:YVO4 by degenerate pump–probe spectroscopy,” J. Appl. Phys. 89, 8342–8344 (2001).
[CrossRef]

K. Ekvall, C. Lundevall, and P. Van der Medlen, “Studies of the fifth-order nonlinear susceptibility of ultraviolet-grade fused silica,” Opt. Lett. 26, 896–898 (2001).
[CrossRef]

2000

1999

1997

1996

1995

Aitken, B. G.

J. M. Harbold, F. O. Ilday, F. W. Wise, and B. G. Aitken, “Highly nonlinear Ge–As–Se and Ge–As–S–Se glasses for all-optical switching,” IEEE Photon. Technol. Lett. 14, 822–824 (2002).
[CrossRef]

Borrelli, N. F.

Bowie, J. L.

Chen, L.

L. Luo, L. Chen, Z. R. Qiu, X. Y. Yu, D. C. Dai, J. Y. Zhou, and J. Kuhl, “Measurement of femtosecond, resonant, nonlinear refraction in Nd:YVO4 by degenerate pump–probe spectroscopy,” J. Appl. Phys. 89, 8342–8344 (2001).
[CrossRef]

Cheriaux, G.

Chi, C. H.

Chiu, T. L.

Dai, D. C.

L. Luo, L. Chen, Z. R. Qiu, X. Y. Yu, D. C. Dai, J. Y. Zhou, and J. Kuhl, “Measurement of femtosecond, resonant, nonlinear refraction in Nd:YVO4 by degenerate pump–probe spectroscopy,” J. Appl. Phys. 89, 8342–8344 (2001).
[CrossRef]

DeLong, K. W.

Ekvall, K.

Fittinghoff, D. N.

Harbold, J. M.

J. M. Harbold, F. O. Ilday, F. W. Wise, and B. G. Aitken, “Highly nonlinear Ge–As–Se and Ge–As–S–Se glasses for all-optical switching,” IEEE Photon. Technol. Lett. 14, 822–824 (2002).
[CrossRef]

Ilday, F. O.

J. M. Harbold, F. O. Ilday, F. W. Wise, and B. G. Aitken, “Highly nonlinear Ge–As–Se and Ge–As–S–Se glasses for all-optical switching,” IEEE Photon. Technol. Lett. 14, 822–824 (2002).
[CrossRef]

Jennings, R. T.

Joffre, M.

Kang, I.

Krauss, T. D.

Krumbugel, M. A.

Kuhl, J.

L. Luo, L. Chen, Z. R. Qiu, X. Y. Yu, D. C. Dai, J. Y. Zhou, and J. Kuhl, “Measurement of femtosecond, resonant, nonlinear refraction in Nd:YVO4 by degenerate pump–probe spectroscopy,” J. Appl. Phys. 89, 8342–8344 (2001).
[CrossRef]

Lepetit, L.

Lundevall, C.

Luo, L.

L. Luo, L. Chen, Z. R. Qiu, X. Y. Yu, D. C. Dai, J. Y. Zhou, and J. Kuhl, “Measurement of femtosecond, resonant, nonlinear refraction in Nd:YVO4 by degenerate pump–probe spectroscopy,” J. Appl. Phys. 89, 8342–8344 (2001).
[CrossRef]

Qiu, Z. R.

L. Luo, L. Chen, Z. R. Qiu, X. Y. Yu, D. C. Dai, J. Y. Zhou, and J. Kuhl, “Measurement of femtosecond, resonant, nonlinear refraction in Nd:YVO4 by degenerate pump–probe spectroscopy,” J. Appl. Phys. 89, 8342–8344 (2001).
[CrossRef]

Shore, K. A.

Smolorz, S.

Spencer, P. S.

Sun, C. K.

Sweetser, J. N.

Trebino, R.

Van der Medlen, P.

Walmsley, I. A.

Wang, J. K.

Wise, F. W.

Yu, X. Y.

L. Luo, L. Chen, Z. R. Qiu, X. Y. Yu, D. C. Dai, J. Y. Zhou, and J. Kuhl, “Measurement of femtosecond, resonant, nonlinear refraction in Nd:YVO4 by degenerate pump–probe spectroscopy,” J. Appl. Phys. 89, 8342–8344 (2001).
[CrossRef]

Zhou, J. Y.

L. Luo, L. Chen, Z. R. Qiu, X. Y. Yu, D. C. Dai, J. Y. Zhou, and J. Kuhl, “Measurement of femtosecond, resonant, nonlinear refraction in Nd:YVO4 by degenerate pump–probe spectroscopy,” J. Appl. Phys. 89, 8342–8344 (2001).
[CrossRef]

IEEE Photon. Technol. Lett.

J. M. Harbold, F. O. Ilday, F. W. Wise, and B. G. Aitken, “Highly nonlinear Ge–As–Se and Ge–As–S–Se glasses for all-optical switching,” IEEE Photon. Technol. Lett. 14, 822–824 (2002).
[CrossRef]

J. Appl. Phys.

L. Luo, L. Chen, Z. R. Qiu, X. Y. Yu, D. C. Dai, J. Y. Zhou, and J. Kuhl, “Measurement of femtosecond, resonant, nonlinear refraction in Nd:YVO4 by degenerate pump–probe spectroscopy,” J. Appl. Phys. 89, 8342–8344 (2001).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Lett.

Other

G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989).

L. Cohen, Time–Frequency Analysis (Prentice-Hall, Englewood Cliffs, N.J., 1995).

R. Trebino, Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses (Kluwer Academic, Boston, Mass., 2002).

S. Mukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University, New York, 1995).

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Figures (8)

Fig. 1
Fig. 1

Probe spectral profile for various two-photon absorption coefficients. Time delay is td=0.5τ, velocity mismatch is z=τ, nonlinear refraction is αzA2=π. Two-photon absorption coefficients are βzA2=0, solid curve; βzA2=0.1π, dotted curve; βzA2=0.2π, dashed curve.

Fig. 2
Fig. 2

Temporal profiles of probe amplitude (upper) and phase (lower) for various two-photon absorption coefficients. Parameters are the same as for Fig. 1.

Fig. 3
Fig. 3

Probe spectral profile for various two-photon absorption coefficients. Time delay is td=-0.5τ, velocity mismatch is z=τ, nonlinear refraction is αzA2=π. Two-photon absorption coefficients are βzA2=0, solid curve; βzA2=0.1π, dotted curve; βzA2=0.2π, dashed curve.

Fig. 4
Fig. 4

Probe spectral profile for various two-photon absorption coefficients. Time delay is td=0, velocity mismatch is z=0, nonlinear refraction is αzA2=π. Two-photon absorption coefficients are βzA2=0, solid curve; βzA2=0.1π, dotted curve; βzA2=0.2π, dashed curve.

Fig. 5
Fig. 5

Calculated pump–probe signals dS(ω, td). The incident probe is chirp-free. Parameters are z=0, αzA2=π, βzA2=0.2π.

Fig. 6
Fig. 6

Calculated pump–probe signals dS(ω, td). The incident probe is a chirped Gaussian pulse. Parameters are the same as for Fig. 5.

Fig. 7
Fig. 7

Calculated pump–probe signals dS(ω, td). The pump and probe have the same velocity z=0. The nonlinear refraction is αzA2=0.4, and the fifth-order susceptibility is γzA4=-0.16, δzA4=0.08.

Fig. 8
Fig. 8

Calculated pump–probe signals dS(ω, td). The pump and probe have the same velocity z=0. However, the intensity of the pump pulse is four times larger compared with that of Fig. 7. The nonlinear refraction is αzA2=1.6, and the fifth-order susceptibility is γzA4=-2.56, δzA4=1.28.

Equations (34)

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zz-1c2ttΦ(z, t)=4πc2ttP(z, t),
P(z, t)=χ(1)Φ(z, t)+Pnl(z, t),
Pnl(z, t)=n=1χ(2n+1)|Φ(z, t)|2nΦ(z, t)
Φ(z, t)=Eu(z, t)exp(ikuz-iωut)+Er(z, t)exp(ikrz-iωrt),
izEu+1vutEu+Fu(|Eu|)Eu=0,
izEr+1vrtEr+Fr(|Eu|)Er=0.
Fu(x)=iηu+(αu+iβu)x2+,
Fr(x)=iηr+2(αr+iβr)x2+ .
Eu(z, T)U(z, T)exp[iϕ(z, T)],
(z+T)U=-Fu(i)(U)U,
(z+T)ϕ=Fu(r)(U),
U(z, T)=Γ[h0(T-z)+z],
ddxΓ(x)=-Fu(i)[Γ(x)]Γ(x),
Γ[h0(T)]=U0(T).
ϕ(z, T)=ϕ0(T-z)+0zdzFu(r)[U(z, T-z+z)].
Er(z, T)=Er(0, T-σz)×expi0zdzFr[U(z, T-σz+σz)],
Γ(x)=ηuexp(2ηux)-βu1/2,
Er(z, T)=Er(0, T-σz)exp-ηrz+i0zdz2ηu(αr+iβr)/βu1+ηu/βuU02[T-σz+(σ-1)z]exp(2ηuz)-1,
σ=1Er(0, T-z)exp-ηrz+iαr+iβrβulog1+βuU02(T-z)[1-exp(-2ηuz)]ηu.
Er(z, T)=Er(0, T-σz)×exp-ηrz+i2αr0zdzU02[T-σz+(σ-1)z]exp(-2ηuz).
Eu(0, T)=A exp(-T2/τ2),
Er(0, T)=B exp[-(T-td)2/τ2],
Er(z, T)=Er(0, T-σz)×expi(αr+iβr)0zdz× U02[T-σz+(σ-1)z]1/2+βuzU02[T-σz+(σ-1)z],
σ=1Er(0, T-z)expiαr+iβrβu×log[1+2βuzU02(T-z)].
S(ω, td)=dTf(T-td)g(T)exp[i(ω-ωr)T]2,
g(T)=expiαr+iβrβu log[1+2βuLIu(T)].
dS(ω, td)=S(ω, td)-S0(ω),
Er(0, T)=B exp[-(1-i)(T-td)2/τ2],
Fu(x)=αux2+(γu+iδu)x4,
Fr(x)=2αrx2+3(γr+iδr)x4,
ddxΓ(x)+δuΓ(x)5=0,
U(z, T)=U04(T-z)1+4δuzU04(T-z)-1/4,
Er(z, T)=Er(0, T-σz)expi0zdz×2αrU02[T-σz+(σ-1)z]1+4δuzU04[T-σz+(σ-1)z]+3(γr+iδr)U04[T-σz+(σ-1)z]1+4δuzU04[T-σz+(σ-1)z]
σ=1Er(0, T-z)×expi4αrzU02(T-z)1+4δuzU04(T-z)+1+i3γr+i3δr4δu log[1+4δuzU04(T-z)].

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