Abstract

We report a simple method to measure the third-order nonlinear refraction and absorption of bulk materials and waveguides. This method relies on the nonlinear evolution of the spectral intensity and width of femtosecond pulses, owing to self-phase modulation, according to a variable chirp introduced before propagation through the nonlinear sample. We describe the experimental details, and we present a comprehensive numerical analysis of the dispersive and the nonlinear effects that can influence the measurement. We also extend the method to characterize the time response of the fast cubic nonlinearity.

© 2001 Optical Society of America

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References

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  1. F. Louradour, E. Lopez-Lago, V. Couderc, V. Messager, and A. Barthelemy, “Dispersive-scan measurement of the fast component of the third-order nonlinearity of bulk materials and waveguides,” Opt. Lett. 24, 1361–1363 (1999).
    [CrossRef]
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    [CrossRef] [PubMed]
  3. M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. QE-26, 760–769 (1989).
  4. M. C. Gabriel, N. A. Whitaker, Jr., C. W. Dirk, M. G. Kuzyck, and M. Thakur, “Measurement of ultrafast optical nonlinearities using a modified Sagnac interferometer,” Opt. Lett. 16, 1334–1336 (1991).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  7. E. T. J. Nibbering, M. A. Franco, B. S. Prade, G. Grillon, C. Le Blanc, and A. Mysyrowicz, “Measurement of the nonlinear refractive index of transparent materials by spectral analysis after nonlinear propagation,” Opt. Commun. 119, 479–484 (1995).
    [CrossRef]
  8. A. J. Taylor, G. Rodriguez, and T. S. Clement, “Determination of n2 by direct measurement of the optical phase,” Opt. Lett. 21, 1812–1814 (1996).
    [CrossRef] [PubMed]
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    [CrossRef]
  10. G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989).
  11. R. H. Stolen, J. P. Gordon, W. J. Tomlinson, and H. A. Haus, “Raman response function of silica-core fibers,” J. Opt. Soc. Am. B 6, 1159–1166 (1989).
    [CrossRef]
  12. E. T. J. Nibbering, G. Grillon, M. A. Franco, B. S. Prade, and A. Mysyrowicz, “Determination of the inertial contribution to the nonlinear refractive index of air, N2, and O2 by use of unfocused high-intensity femtosecond laser pulses,” J. Opt. Soc. Am. B 14, 650–656 (1997).
    [CrossRef]
  13. J.-G. Tian, W. P. Zang, and G. Zhang, “Two modified z-scan methods for determination of nonlinear optical index with enhanced sensitivity,” Opt. Commun. 107, 415–419 (1994).
    [CrossRef]
  14. T. Xia, D. J. Hagan, M. Sheik-Bahae, and E. W. van Stryland, “Eclipsing z-scan measurement of λ/104 wavefront distortion,” Opt. Lett. 19, 317–319 (1994).
    [CrossRef] [PubMed]

1999 (1)

1997 (1)

1996 (1)

1995 (1)

E. T. J. Nibbering, M. A. Franco, B. S. Prade, G. Grillon, C. Le Blanc, and A. Mysyrowicz, “Measurement of the nonlinear refractive index of transparent materials by spectral analysis after nonlinear propagation,” Opt. Commun. 119, 479–484 (1995).
[CrossRef]

1994 (2)

J.-G. Tian, W. P. Zang, and G. Zhang, “Two modified z-scan methods for determination of nonlinear optical index with enhanced sensitivity,” Opt. Commun. 107, 415–419 (1994).
[CrossRef]

T. Xia, D. J. Hagan, M. Sheik-Bahae, and E. W. van Stryland, “Eclipsing z-scan measurement of λ/104 wavefront distortion,” Opt. Lett. 19, 317–319 (1994).
[CrossRef] [PubMed]

1991 (2)

1989 (3)

1984 (1)

1973 (1)

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

Barthelemy, A.

Clement, T. S.

Couderc, V.

Dirk, C. W.

Fork, R. L.

Franco, M. A.

E. T. J. Nibbering, G. Grillon, M. A. Franco, B. S. Prade, and A. Mysyrowicz, “Determination of the inertial contribution to the nonlinear refractive index of air, N2, and O2 by use of unfocused high-intensity femtosecond laser pulses,” J. Opt. Soc. Am. B 14, 650–656 (1997).
[CrossRef]

E. T. J. Nibbering, M. A. Franco, B. S. Prade, G. Grillon, C. Le Blanc, and A. Mysyrowicz, “Measurement of the nonlinear refractive index of transparent materials by spectral analysis after nonlinear propagation,” Opt. Commun. 119, 479–484 (1995).
[CrossRef]

Gabriel, M. C.

Gordon, J. P.

Grillon, G.

E. T. J. Nibbering, G. Grillon, M. A. Franco, B. S. Prade, and A. Mysyrowicz, “Determination of the inertial contribution to the nonlinear refractive index of air, N2, and O2 by use of unfocused high-intensity femtosecond laser pulses,” J. Opt. Soc. Am. B 14, 650–656 (1997).
[CrossRef]

E. T. J. Nibbering, M. A. Franco, B. S. Prade, G. Grillon, C. Le Blanc, and A. Mysyrowicz, “Measurement of the nonlinear refractive index of transparent materials by spectral analysis after nonlinear propagation,” Opt. Commun. 119, 479–484 (1995).
[CrossRef]

Hagan, D. J.

T. Xia, D. J. Hagan, M. Sheik-Bahae, and E. W. van Stryland, “Eclipsing z-scan measurement of λ/104 wavefront distortion,” Opt. Lett. 19, 317–319 (1994).
[CrossRef] [PubMed]

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. QE-26, 760–769 (1989).

Haus, H. A.

Kobayashi, T.

Kuzyck, M. G.

Le Blanc, C.

E. T. J. Nibbering, M. A. Franco, B. S. Prade, G. Grillon, C. Le Blanc, and A. Mysyrowicz, “Measurement of the nonlinear refractive index of transparent materials by spectral analysis after nonlinear propagation,” Opt. Commun. 119, 479–484 (1995).
[CrossRef]

Lopez-Lago, E.

Louradour, F.

Martinez, O. E.

Messager, V.

Minoshima, K.

Mysyrowicz, A.

E. T. J. Nibbering, G. Grillon, M. A. Franco, B. S. Prade, and A. Mysyrowicz, “Determination of the inertial contribution to the nonlinear refractive index of air, N2, and O2 by use of unfocused high-intensity femtosecond laser pulses,” J. Opt. Soc. Am. B 14, 650–656 (1997).
[CrossRef]

E. T. J. Nibbering, M. A. Franco, B. S. Prade, G. Grillon, C. Le Blanc, and A. Mysyrowicz, “Measurement of the nonlinear refractive index of transparent materials by spectral analysis after nonlinear propagation,” Opt. Commun. 119, 479–484 (1995).
[CrossRef]

Nibbering, E. T. J.

E. T. J. Nibbering, G. Grillon, M. A. Franco, B. S. Prade, and A. Mysyrowicz, “Determination of the inertial contribution to the nonlinear refractive index of air, N2, and O2 by use of unfocused high-intensity femtosecond laser pulses,” J. Opt. Soc. Am. B 14, 650–656 (1997).
[CrossRef]

E. T. J. Nibbering, M. A. Franco, B. S. Prade, G. Grillon, C. Le Blanc, and A. Mysyrowicz, “Measurement of the nonlinear refractive index of transparent materials by spectral analysis after nonlinear propagation,” Opt. Commun. 119, 479–484 (1995).
[CrossRef]

Owyoung, A.

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

Prade, B. S.

E. T. J. Nibbering, G. Grillon, M. A. Franco, B. S. Prade, and A. Mysyrowicz, “Determination of the inertial contribution to the nonlinear refractive index of air, N2, and O2 by use of unfocused high-intensity femtosecond laser pulses,” J. Opt. Soc. Am. B 14, 650–656 (1997).
[CrossRef]

E. T. J. Nibbering, M. A. Franco, B. S. Prade, G. Grillon, C. Le Blanc, and A. Mysyrowicz, “Measurement of the nonlinear refractive index of transparent materials by spectral analysis after nonlinear propagation,” Opt. Commun. 119, 479–484 (1995).
[CrossRef]

Rodriguez, G.

Said, A. A.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. QE-26, 760–769 (1989).

M. Sheik-Bahae, A. A. Said, and E. W. Stryland, “High sensitivity single-beam n2 measurements,” Opt. Lett. 14, 955–957 (1989).
[CrossRef] [PubMed]

Sheik-Bahae, M.

Stolen, R. H.

Stryland, E. W.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. QE-26, 760–769 (1989).

M. Sheik-Bahae, A. A. Said, and E. W. Stryland, “High sensitivity single-beam n2 measurements,” Opt. Lett. 14, 955–957 (1989).
[CrossRef] [PubMed]

Taiji, M.

Taylor, A. J.

Thakur, M.

Tian, J.-G.

J.-G. Tian, W. P. Zang, and G. Zhang, “Two modified z-scan methods for determination of nonlinear optical index with enhanced sensitivity,” Opt. Commun. 107, 415–419 (1994).
[CrossRef]

Tomlinson, W. J.

van Stryland, E. W.

Wei, T. H.

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. QE-26, 760–769 (1989).

Whitaker Jr., N. A.

Xia, T.

Zang, W. P.

J.-G. Tian, W. P. Zang, and G. Zhang, “Two modified z-scan methods for determination of nonlinear optical index with enhanced sensitivity,” Opt. Commun. 107, 415–419 (1994).
[CrossRef]

Zhang, G.

J.-G. Tian, W. P. Zang, and G. Zhang, “Two modified z-scan methods for determination of nonlinear optical index with enhanced sensitivity,” Opt. Commun. 107, 415–419 (1994).
[CrossRef]

IEEE J. Quantum Electron. (2)

M. Sheik-Bahae, A. A. Said, T. H. Wei, D. J. Hagan, and E. W. Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. QE-26, 760–769 (1989).

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

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

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

Opt. Commun. (2)

J.-G. Tian, W. P. Zang, and G. Zhang, “Two modified z-scan methods for determination of nonlinear optical index with enhanced sensitivity,” Opt. Commun. 107, 415–419 (1994).
[CrossRef]

E. T. J. Nibbering, M. A. Franco, B. S. Prade, G. Grillon, C. Le Blanc, and A. Mysyrowicz, “Measurement of the nonlinear refractive index of transparent materials by spectral analysis after nonlinear propagation,” Opt. Commun. 119, 479–484 (1995).
[CrossRef]

Opt. Lett. (6)

Other (1)

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

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Figures (13)

Fig. 1
Fig. 1

Experimental setup for nonlinear refraction measurement by the dispersive-scan method (D-scan). The second-order dispersion is varied from -60 000 rad fs2 to +36 000 rad fs2 by translation of the prisms.

Fig. 2
Fig. 2

Peak spectral intensity at the output of the nonlinear sample versus the dispersion introduced at the input for increasing peak nonlinear phase shifts; ΔϕNL0=0.2, 0.44, 0.68, 0.91, and 1.15 rads.

Fig. 3
Fig. 3

Output-pulse spectral width versus dispersion for the same peak nonlinear phase shifts as in Fig. 2.

Fig. 4
Fig. 4

(a) As a check, we show that without third-order dispersion there is no wavelength shift. (b)–(d) Spectrum peak wavelength shift for ΔϕNL0=0.22, 0.72, and 1.22 rad resulting from the negative third-order dispersion of the DL. ϕ varies from -100 000 rad fs3 to -54 000 rad fs3 when ϕ varies from -60 000 rad fs2 to +36 000 rad fs2.

Fig. 5
Fig. 5

Evolution of the spectral peak intensity together with the transmitted average power versus dispersion. A two-photon absorption coefficient was chosen equal to β=7×10-6 m/GW, and the nonlinear phase shift was ΔϕNL0=1.22 rad.

Fig. 6
Fig. 6

Spectral peak intensity at the output of the nonlinear sample versus the dispersion introduced at the input for increasing values of the two-photon absorption coefficient, β=0, 10-6, 10-5, 5×10-5, and 10-4 m/GW if ΔΦNL0=1.22 rad.

Fig. 7
Fig. 7

Spectral size at the output of the nonlinear sample versus the dispersion introduced at the input for increasing values of the two-photon absorption coefficient, β=0, 10-6, 10-5, 5×10-5, and 10-4 m/GW, for ΔΦNL0=1.22 rad.

Fig. 8
Fig. 8

Central-wavelength shift versus dispersion for a nonlinear time response τ=30 fs, a pulse duration T0=73 fs, and a nonlinear phase shift ΔϕNL0=1.22 rad.

Fig. 9
Fig. 9

Maximum shift of the central wavelength versus the nonlinearity time response for a constant value of ΔΦSPM equal to 1.22 rad and a pulse duration of T0=73 fs.

Fig. 10
Fig. 10

D-scan measurement of the Kerr nonlinearity of a single-mode silica-fiber sample. Data on the spectral intensity versus dispersion (open circles) were obtained with pulses of 73-fs duration (FWHMI) at λ0=830 nm. The fit to a theoretical curve (continuous curve) provides a nonlinear refraction coefficient n2=(3.1±0.19)10-11 m2/GW.

Fig. 11
Fig. 11

D-scan measurement of the Kerr nonlinearity of a single-mode silica-fiber sample. Measured spectral width (FWHMI) versus dispersion (open circles) obtained in the same situation as for Fig. 10 together with a theoretical fit carried with n2=(3.1±0.19)10-11 m2/GW.

Fig. 12
Fig. 12

D-scan measurement of the time response of the cubic nonlinearity of a single-mode silica-fiber sample. The measured spectral shift (Δλ0) versus dispersion (open circles) is obtained in the same conditions as the data in Figs. 11 and 12. The fit to a theoretical curve (continuous curve), including the influence of the third-order dispersion of the dispersive line, indicates a time response τ=4±2 fs.

Fig. 13
Fig. 13

D-scan measurement of the Kerr nonlienarity of the commercial glass SF57. The experimental data on the spectral size versus dispersion (open circles) are obtained with pulses of 85 fs of duration (FWHMI) at λ0=820 nm. The fit to a theoretical curve (continuous curve) gives a nonlinear refraction coefficient n2=(4.1±0.25)10-10 m2/GW.

Tables (1)

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Table 1 Simulation Parameters

Equations (13)

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ϕ(rads2)=2ϕ(ω)ω2ω=ω0
Ed(t)=E0|1-i2a2ϕ| exp-a2-i2a4ϕ1+4a4ϕ2t2.
Ez=iγ|E|2E,
ϕ(radfs3)=3ϕω3ω=ω0.
Ez+i2β2 2Et2-16β3 3Et3=iγ|E|2E,
Ez+i2β2 2Et2=iγ|E|2E-β|E|2E,
β=-1IoL LnΔPP0-1,
H=-t 1τ exp-t-tτ|E(t)|2dt,
H(t)=12τI0πA exp14Aτ2-tτErfcAt-12τA,
Ez+i2β2 2Et2=iγ(1-αR)|E|2E+iαRγΔnRE.
ΔnR(t)=-tf(t-t)|E(t)|2dt.
f(t)=CR exp-tτR1 sin2πtτR2+π80,
CR=1τR1+4π2τR221τR1 sin π80+2πτR2 cos π80.

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