Abstract

We report an extension of the spectrally resolved two-beam coupling technique to measure the nonlinear intensity index of refraction (n2I) and the two-photon absorption coefficient (β) by use of chirped laser pulses. The linear chirp parameter b is incorporated into the derivation of a more general model than the previous one [Opt. Lett. 22, 1077 (1997)]. We have also analyzed the validity of this linear chirp model through a comparison of the experimental results for fused silica with the numerically accurate calculation that considers higher-order chirps obtained by second-harmonic generation frequency-resolved optical gating. The results show that this method potentially can be used to extract the chirp. Finally, we applied this transient spectrally resolved nonlinear transmittance spectroscopy to semiconductor-doped glasses to extract their n2I and β.

© 1999 Optical Society of America

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References

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  1. R. L. Sutherland, Handbook of Nonlinear Optics (Marcel Dekker, New York 1996), and references therein.
  2. M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. QE-6, 760 (1990), and references therein.
    [CrossRef]
  3. X. Kang, T. Krauss, and F. Wise, Opt. Lett. 22, 1077 (1997).
    [CrossRef] [PubMed]
  4. J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic, San Diego, Calif. 1995).
  5. G. Taft, A. Rundquist, M. Murnane, I. P. Christov, H. C. Kapteyn, K. W. DeLong, D. N. Fittinghoff, M. A. Krumbügel, J. N. Sweetser, and R. Trebino, IEEE J. Sel. Top. Quantum Electron. 2, 575 (1996).
    [CrossRef]
  6. D. Milam, Appl. Opt. 37, 546 (1998).
    [CrossRef]
  7. In the calculation of the simulated nonlinear transmittance signals we used the complete phase derived from the FROG measurement. The best Gaussian fit to the intensity data was adopted in the calculation. A calculation with the actual intensity data was also performed, which resulted in a worse match with the experimental nonlinear transmittance data, perhaps because of an error in the intensity retrieved from the FROG measurement.
  8. We also performed the calculation by using different orders of polynomials to fit the extracted phase from the FROG measurement. We found that the amplitude of the simulated transient when only the linear chirp was used did not agree with the experimental data. When the higher-order chirps were included in the calculation, the whole body of experimental data could be reproduced. A close comparison of the experimental and the simulation results has shown that there are still slight differences between them. Furthermore, the sample length is one hundredth of the dispersion length LD. The effect of GVD is normally regarded to be negligible [G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif. 1989), Chapter 4]. It is thus conceivable that this discrepancy is due to errors in intensity and phase of the laser pulses extracted from the FROG measurement. This fact has further demonstrated that the transient nonlinear transmittance technique is more sensitive to the chirp than is FROG.
  9. A. J. Taylor, G. Rodriguez, and T. S. Clement, Opt. Lett. 21, 1812 (1996).
    [CrossRef] [PubMed]
  10. G. Rodriguez and A. J. Taylor, Opt. Lett. 23, 858 (1998).
    [CrossRef]
  11. G. P. Banfi, V. Degiorgio, and D. Ricard, Adv. Phys. 47, 447 (1998).
    [CrossRef]
  12. M. G. Bawendi, M. L. Steigerwald, and L. E. Brus, Annu. Rev. Phys. Chem. 41, 477 (1990); Y. Wang and N. Herron, J. Phys. Chem. 95, 525 (1991); A. P. Alivisatos, J. Phys. Chem. JPCHAX 100, 13, 226 (1996); A. Tomasulo and M. C. Ramakrishna, J. Chem. Phys. JCPSA6 105, 3612 (1996).
    [CrossRef]
  13. G. P. Banfi, V. Degiorgio, D. Fortusini, and H. M. Tan, Appl. Phys. Lett. 67, 13 (1995).
    [CrossRef]

1998

1997

1996

G. Taft, A. Rundquist, M. Murnane, I. P. Christov, H. C. Kapteyn, K. W. DeLong, D. N. Fittinghoff, M. A. Krumbügel, J. N. Sweetser, and R. Trebino, IEEE J. Sel. Top. Quantum Electron. 2, 575 (1996).
[CrossRef]

A. J. Taylor, G. Rodriguez, and T. S. Clement, Opt. Lett. 21, 1812 (1996).
[CrossRef] [PubMed]

1995

G. P. Banfi, V. Degiorgio, D. Fortusini, and H. M. Tan, Appl. Phys. Lett. 67, 13 (1995).
[CrossRef]

1990

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. QE-6, 760 (1990), and references therein.
[CrossRef]

Banfi, G. P.

G. P. Banfi, V. Degiorgio, and D. Ricard, Adv. Phys. 47, 447 (1998).
[CrossRef]

G. P. Banfi, V. Degiorgio, D. Fortusini, and H. M. Tan, Appl. Phys. Lett. 67, 13 (1995).
[CrossRef]

Christov, I. P.

G. Taft, A. Rundquist, M. Murnane, I. P. Christov, H. C. Kapteyn, K. W. DeLong, D. N. Fittinghoff, M. A. Krumbügel, J. N. Sweetser, and R. Trebino, IEEE J. Sel. Top. Quantum Electron. 2, 575 (1996).
[CrossRef]

Clement, T. S.

Degiorgio, V.

G. P. Banfi, V. Degiorgio, and D. Ricard, Adv. Phys. 47, 447 (1998).
[CrossRef]

G. P. Banfi, V. Degiorgio, D. Fortusini, and H. M. Tan, Appl. Phys. Lett. 67, 13 (1995).
[CrossRef]

DeLong, K. W.

G. Taft, A. Rundquist, M. Murnane, I. P. Christov, H. C. Kapteyn, K. W. DeLong, D. N. Fittinghoff, M. A. Krumbügel, J. N. Sweetser, and R. Trebino, IEEE J. Sel. Top. Quantum Electron. 2, 575 (1996).
[CrossRef]

Fittinghoff, D. N.

G. Taft, A. Rundquist, M. Murnane, I. P. Christov, H. C. Kapteyn, K. W. DeLong, D. N. Fittinghoff, M. A. Krumbügel, J. N. Sweetser, and R. Trebino, IEEE J. Sel. Top. Quantum Electron. 2, 575 (1996).
[CrossRef]

Fortusini, D.

G. P. Banfi, V. Degiorgio, D. Fortusini, and H. M. Tan, Appl. Phys. Lett. 67, 13 (1995).
[CrossRef]

Hagan, D. J.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. QE-6, 760 (1990), and references therein.
[CrossRef]

Kang, X.

Kapteyn, H. C.

G. Taft, A. Rundquist, M. Murnane, I. P. Christov, H. C. Kapteyn, K. W. DeLong, D. N. Fittinghoff, M. A. Krumbügel, J. N. Sweetser, and R. Trebino, IEEE J. Sel. Top. Quantum Electron. 2, 575 (1996).
[CrossRef]

Krauss, T.

Krumbügel, M. A.

G. Taft, A. Rundquist, M. Murnane, I. P. Christov, H. C. Kapteyn, K. W. DeLong, D. N. Fittinghoff, M. A. Krumbügel, J. N. Sweetser, and R. Trebino, IEEE J. Sel. Top. Quantum Electron. 2, 575 (1996).
[CrossRef]

Milam, D.

Murnane, M.

G. Taft, A. Rundquist, M. Murnane, I. P. Christov, H. C. Kapteyn, K. W. DeLong, D. N. Fittinghoff, M. A. Krumbügel, J. N. Sweetser, and R. Trebino, IEEE J. Sel. Top. Quantum Electron. 2, 575 (1996).
[CrossRef]

Ricard, D.

G. P. Banfi, V. Degiorgio, and D. Ricard, Adv. Phys. 47, 447 (1998).
[CrossRef]

Rodriguez, G.

Rundquist, A.

G. Taft, A. Rundquist, M. Murnane, I. P. Christov, H. C. Kapteyn, K. W. DeLong, D. N. Fittinghoff, M. A. Krumbügel, J. N. Sweetser, and R. Trebino, IEEE J. Sel. Top. Quantum Electron. 2, 575 (1996).
[CrossRef]

Said, A. A.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. QE-6, 760 (1990), and references therein.
[CrossRef]

Sheik-Bahae, M.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. QE-6, 760 (1990), and references therein.
[CrossRef]

Sweetser, J. N.

G. Taft, A. Rundquist, M. Murnane, I. P. Christov, H. C. Kapteyn, K. W. DeLong, D. N. Fittinghoff, M. A. Krumbügel, J. N. Sweetser, and R. Trebino, IEEE J. Sel. Top. Quantum Electron. 2, 575 (1996).
[CrossRef]

Taft, G.

G. Taft, A. Rundquist, M. Murnane, I. P. Christov, H. C. Kapteyn, K. W. DeLong, D. N. Fittinghoff, M. A. Krumbügel, J. N. Sweetser, and R. Trebino, IEEE J. Sel. Top. Quantum Electron. 2, 575 (1996).
[CrossRef]

Tan, H. M.

G. P. Banfi, V. Degiorgio, D. Fortusini, and H. M. Tan, Appl. Phys. Lett. 67, 13 (1995).
[CrossRef]

Taylor, A. J.

Trebino, R.

G. Taft, A. Rundquist, M. Murnane, I. P. Christov, H. C. Kapteyn, K. W. DeLong, D. N. Fittinghoff, M. A. Krumbügel, J. N. Sweetser, and R. Trebino, IEEE J. Sel. Top. Quantum Electron. 2, 575 (1996).
[CrossRef]

Van Stryland, E. W.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. QE-6, 760 (1990), and references therein.
[CrossRef]

Wei, T.-H.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. QE-6, 760 (1990), and references therein.
[CrossRef]

Wise, F.

Adv. Phys.

G. P. Banfi, V. Degiorgio, and D. Ricard, Adv. Phys. 47, 447 (1998).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

G. P. Banfi, V. Degiorgio, D. Fortusini, and H. M. Tan, Appl. Phys. Lett. 67, 13 (1995).
[CrossRef]

IEEE J. Quantum Electron.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. QE-6, 760 (1990), and references therein.
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

G. Taft, A. Rundquist, M. Murnane, I. P. Christov, H. C. Kapteyn, K. W. DeLong, D. N. Fittinghoff, M. A. Krumbügel, J. N. Sweetser, and R. Trebino, IEEE J. Sel. Top. Quantum Electron. 2, 575 (1996).
[CrossRef]

Opt. Lett.

Other

In the calculation of the simulated nonlinear transmittance signals we used the complete phase derived from the FROG measurement. The best Gaussian fit to the intensity data was adopted in the calculation. A calculation with the actual intensity data was also performed, which resulted in a worse match with the experimental nonlinear transmittance data, perhaps because of an error in the intensity retrieved from the FROG measurement.

We also performed the calculation by using different orders of polynomials to fit the extracted phase from the FROG measurement. We found that the amplitude of the simulated transient when only the linear chirp was used did not agree with the experimental data. When the higher-order chirps were included in the calculation, the whole body of experimental data could be reproduced. A close comparison of the experimental and the simulation results has shown that there are still slight differences between them. Furthermore, the sample length is one hundredth of the dispersion length LD. The effect of GVD is normally regarded to be negligible [G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif. 1989), Chapter 4]. It is thus conceivable that this discrepancy is due to errors in intensity and phase of the laser pulses extracted from the FROG measurement. This fact has further demonstrated that the transient nonlinear transmittance technique is more sensitive to the chirp than is FROG.

J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic, San Diego, Calif. 1995).

R. L. Sutherland, Handbook of Nonlinear Optics (Marcel Dekker, New York 1996), and references therein.

M. G. Bawendi, M. L. Steigerwald, and L. E. Brus, Annu. Rev. Phys. Chem. 41, 477 (1990); Y. Wang and N. Herron, J. Phys. Chem. 95, 525 (1991); A. P. Alivisatos, J. Phys. Chem. JPCHAX 100, 13, 226 (1996); A. Tomasulo and M. C. Ramakrishna, J. Chem. Phys. JCPSA6 105, 3612 (1996).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Simulated nonlinear transmittance transients with the chirp-free model at three frequency detunings δ. (a) Envelope function f(Δt), (b) total transient signal. The induced phase shift ΔΦ is 4×10-4 rad.

Fig. 2
Fig. 2

Simulated nonlinear transmittance transients with the linear-chirp model at three frequency detunings δ. (a) Envelope function f(Δt), (b) total transient signal. Arrows denote the peak positions of f(Δt). b=9×1025 s-2, ΔΦ=3.8×10-4 rad.

Fig. 3
Fig. 3

Simulated nonlinear transmittance transients with the linear-chirp model for three linear chirp parameters b. (a) Envelope function f(Δt), (b) total transient signal. Arrows denote the peak positions of f(Δt). δ=-5×1013 s-1; ΔΦ=3.8×10-4 rad.

Fig. 4
Fig. 4

Experimental layout for the transient nonlinear transmittance measurements. Mirrors M1–M7 are Ag coated. BS, beam splitter (reflection/transmission, 90/10); CH, chopper; L, plano–convex lens (f=11.4 cm); S, sample; PD, photodiode detector; PMT, photomultiplier tube.

Fig. 5
Fig. 5

Autocorrelation trace and spectrum of the femtosecond laser pulses at the output of the mode-locked Ti:sapphire oscillator. Solid curves, Gaussian fitting curves. The fitted pulse-width parameter is τ, and the full width at half-maximum ΔfFWHM of the laser spectrum in frequency is shown (ΔλFWHM=11.5 nm).

Fig. 6
Fig. 6

Normalized nonlinear transmittance signals obtained for 2-mm-thick fused silica at two detection wavelengths. Open circles, experimental data; solid curves, best-fitting curves with the linear-chirp model. Their fitted results are listed in Table 1. Dashed curves, fitting curves for the chirp-free model.

Fig. 7
Fig. 7

Intensity and phase data of the laser pulses extracted from the SHG FROG spectrograms. (a) FROG spectrogram, (b) corresponding intensity I(t) and phase ϕ(t). Filled circles, intensity data; open squares, phase data. Solid curve, the best Gaussian fit to the intensity. The fitted pulse-width parameter is 84.2 fs.

Fig. 8
Fig. 8

Simulated results for the normalized nonlinear transmittance measurements at two detection wavelengths. The chirp was obtained by the FROG trace (described in text). Open circles, simulation data; solid curves, the fitting curves with the linear-chirp model. Fitting results are listed in Table 2.

Fig. 9
Fig. 9

Normalized nonlinear transmittance signal obtained for 3-mm RG610 semiconductor-doped glass at 781.5 nm. Open circles, experimental data; solid curve, fitting curve with the linear-chirp model. Fitting results are listed in Table 3.

Fig. 10
Fig. 10

Normalized nonlinear transmittance signal obtained for 3-mm OG550 semiconductor-doped glass at 781.5 nm. Open circles, experimental data; solid curve, fitting curves with the linear-chirp model. Fitting results are listed in Table 3.

Tables (3)

Tables Icon

Table 1 Results Obtained by Fitting the Experimental Data on 2-mm Fused Silica with the Linear-Chirp Modela

Tables Icon

Table 2 Results Obtained by Fitting the Simulated Transients at Two Frequency Detuningsa

Tables Icon

Table 3 Results Obtained by Fitting the Experimental data at 781.5 nm on 3-mm Semiconductor-Doped Glasses with the Linear-Chirp Modela

Equations (34)

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

Ipuz+1vgIput=-(α+βIpu)Ipu,
ϕpuz+1vgϕput=-kn2IIpu,
Iprz+1vgIprt=-(α+2βIpu)Ipr,
ϕprz+1vgϕprt=-2kn2IIpu.
ΔTT(Δt, δ)=23exp(δ2τ2/6)exp[-2(Δt)2/3τ2]×[2ΔΦ sin(2δΔt/3)-q cos(2δΔt/3)],
E(t)=E0 exp(-t2/τ2+ibt2+iω0t),
Epr(z=L, ω)=FT[Epr(z=L, t-Δt)]=FTEpr(z=0, t-Δt)×exp-i2ω0cn0n2ILIpu(t)-βLIpu(t)
FTEpr(z=0, t-Δt)1-i2ΔΦ×exp-2t2τ2-q exp-2t2τ2=FT[Epr(z=0, t-Δτ)]+FT[ΔEpr(z=L, t-Δt)],
ΔTT(ω)=|Epr(z=L, ω)|2-|Epr(z=0, ω)|2|Epr(z=0, ω)|2=|Epr(z=0, ω)+ΔEpr|2-|Epr(z=0, ω)|2|Epr(z=0, ω)|22 ReΔEprEpr(z=0, ω)=2 ReFT[ΔEpr(z=L, t-Δt)]FT[Epr(z=0, t)],
Epr(z=0, ω)=Epr0πτ21-ibτ21/2 exp-δ2τ24(1-ibτ2).
FT[ΔEpr(z=L, t-Δt)]=Epr0(i2ΔΦ+q)πτ23-ibτ21/2×exp(4Δt+iδτ2)24τ2(3-ibτ2)-2(Δt)2τ2.
ΔTT(Δt, δ)=2(3+b2τ4)2+4b2τ4(9+b2τ4)21/4×exp[g(b, δ, τ)]×exp-[(Δt)-Ts(b, δ, τ)]2[Γ(b, τ)]2×(2ΔΦ sin Θ-q cos Θ),
Ts=-δbτ42(3+b2τ4),
Γ=τ9+b2τ42(3+b2τ4)1/2,
g=δ2τ22(1+b2τ4)(3+b2τ4),
Θ=4b9+b2τ4Δt+3δ4b2-δ24b(1+b2τ4)-12tan-12bτ23+b2τ4.
E(t)=Epr0 exp-t2τ2exp(iω0t)exp[iϕ(t)],
ΔTT(ω, Δt)=|Epr(z=L, ω, Δt)|2-|Epr(z=0, ω)|2|Epr(z=0, ω)|2,
ΔTT(Δt, δ)=f(δ, τ, Δt)ζ(δ, Δt, ΔΦ, q),
f(δ, τ, Δt)=23expδ2τ26exp-2Δt23τ2,
ζ(δ, Δt, ΔΦ, q)=[(2ΔΦ)2+q2]1/2 sin2δΔt3-θ,
tan θ=q2ΔΦ.
Epr(z=0, t)=Epr0 exp(-t2/τ2)exp(iω0t),
ΔEpr(z=L, t)=Epr0(i2ΔΦ+q)
×exp-2(Δt)23τ2exp-3τ2t+23Δt2
×exp(iω0t)
=Epr0(i2ΔΦ+q)×exp-2(Δt)23τ2exp-3τ2t2×exp(iω0t)exp-i2ω0Δt3,
f(Δt)=h(b, τ)exp[g(b, δ, τ)]×exp{-[Δt-Ts(b, δ, τ)]2/[Γ(b, τ)]2},
2ΔΦ sin Θ-q cos Θ.
ΔEpr(z=L, t)=Epr0(i2ΔΦ+q)exp-2(Δt)23τ2×exp-3τ2t2exp(iω0t)exp(ibt2)×exp-i2ω0Δt3×exp-ib4Δt3t+49(Δt)2,
expi2δΔt3exp-ib4Δt3t+49(Δt)2.
I(x, y)=I0 exp(-x2/σx2)exp(-y2/σy2),
Δϕ(z)=-ω0cn2IA(z)A0Ipu0δz,
Δϕtotal=-ω0cA0n2IIpu0z0-L/2z0+L/2A(z)dz,

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