Abstract

We present a study of two-beam coupling energy transfer arising from electronic and nuclear (i.e., Raman) contributions to the third-order refractive nonlinearity. Calculations of transient two-beam coupling in fused silica with no free parameters agree reasonably well with experiments. Numerical calculations of two-beam coupling in a medium with a purely electronic nonlinearity modeled by a Debye-type response are presented. The calculations indicate that it should be possible, although challenging, to measure electronic two-beam coupling for response times as short as ∼0.1 fs. Experiments designed to isolate the electronic response fail to produce two-beam coupling signals but do indicate that the time scale of the electronic response is well below 1 fs.

© 2000 Optical Society of America

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  1. J. A. Yeazell, M. Mallalieu, and C. R. Stroud, Jr., “Observation of the collapse and revival of a Rydberg electronic wave packet,” Phys. Rev. Lett. 64, 2007–2010 (1990).
    [CrossRef] [PubMed]
  2. A. Baltuska, Z. Wei, M. S. Pshenichnikov, D. A. Wiersma, and R. Szipöz, “All-solid-state cavity-dumped sub-5-fs laser,” Appl. Phys. B 65, 175–188 (1997).
    [CrossRef]
  3. A. B. Grudinin, E. M. Dianov, D. V. Korobkin, A. M. Prokhorov, V. N. Serkin, and D. V. Haidarov, “Decay of femtosecond pulses in single-mode optical fibers,” JETP Lett. 46, 221–225 (1987).
  4. I. P. Christov, H. C. Kapteyn, M. M. Murnane, Ch.-P. Huang, and J. Zhou, “Space-time focusing of femtosecond pulses in a Ti:sapphire laser,” Opt. Lett. 20, 309–311 (1995).
    [CrossRef] [PubMed]
  5. A. Dogariu, T. Xia, A. A. Said, E. W. Van Stryland, and N. Bloembergen, “Purely refractive transient energy transfer by stimulated Rayleigh-wing scattering,” J. Opt. Soc. Am. B 14, 796–803 (1997).
    [CrossRef]
  6. N. Tang and R. L. Sutherland, “Time-domain theory for pump–probe experiments with chirped pulses,” J. Opt. Soc. Am. B 14, 3412–3423 (1997).
    [CrossRef]
  7. A. Dogariu and D. J. Hagan, “Low frequency Raman gain measurements using chirped pulses,” Opt. Express 1, 73–76 (1997).
    [CrossRef] [PubMed]
  8. R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif., 1992), Chap. 6.4.
  9. H. J. Eichler, M. Glotz, A. Kummrow, K. Richter, and X. Yang, “Picosecond pulse amplification by coherent wave mixing in silicon,” Phys. Rev. A 35, 4673–4679 (1987).
    [CrossRef] [PubMed]
  10. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995), p. 67.
  11. A. E. Siegman, Lasers (University Science, Sausolito, Calif., 1986), pp. 333–334.
  12. J. Ranka and A. Gaeta, “Breakdown of the slowly varying envelope approximation in the self-focusing of ultrashort pulses,” Opt. Lett. 23, 534–536 (1998).
    [CrossRef]
  13. S. Smolorz and F. Wise, “Measurement of the nonlinear optical response of optical fiber materials by use of spectrally resolved two-beam coupling,” Opt. Lett. 24, 1103–1105 (1999).
    [CrossRef]
  14. B. Proctor and F. Wise, “Quartz prism sequence for reduction of cubic phase in a mode-locked Ti:sapphire laser,” Opt. Lett. 17, 1295–1297 (1992).
    [CrossRef] [PubMed]
  15. I. Kang, T. Krauss, and F. Wise, “Sensitive measurement of nonlinear refraction and two-photon absorption by spectrally resolved two-beam coupling,” Opt. Lett. 22, 1077–1079 (1997).
    [CrossRef] [PubMed]
  16. G. Turrell, Infrared and Raman Spectra of Crystals (Academic, London, 1972), Chap. 4.
  17. U. Müller, Anorganische Strukturchemie (Teubner, Stuttgart, 1992), p. 50.
  18. R. Adair, L. L. Chase, and S. A. Payne, “Nonlinear refractive index measurements of optical crystals,” Phys. Rev. B 39, 3337–3350 (1998).
    [CrossRef]
  19. R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif. 1992), Chap. 8.
  20. R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif. 1992), Chap. 1.
  21. A. E. Siegman, Lasers (University Science, Sausolito, Calif., 1986), pp. 944–947.
  22. R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif., 1992), pp. 279–280.

1999 (1)

1998 (2)

J. Ranka and A. Gaeta, “Breakdown of the slowly varying envelope approximation in the self-focusing of ultrashort pulses,” Opt. Lett. 23, 534–536 (1998).
[CrossRef]

R. Adair, L. L. Chase, and S. A. Payne, “Nonlinear refractive index measurements of optical crystals,” Phys. Rev. B 39, 3337–3350 (1998).
[CrossRef]

1997 (5)

1995 (1)

1992 (1)

1990 (1)

J. A. Yeazell, M. Mallalieu, and C. R. Stroud, Jr., “Observation of the collapse and revival of a Rydberg electronic wave packet,” Phys. Rev. Lett. 64, 2007–2010 (1990).
[CrossRef] [PubMed]

1987 (2)

A. B. Grudinin, E. M. Dianov, D. V. Korobkin, A. M. Prokhorov, V. N. Serkin, and D. V. Haidarov, “Decay of femtosecond pulses in single-mode optical fibers,” JETP Lett. 46, 221–225 (1987).

H. J. Eichler, M. Glotz, A. Kummrow, K. Richter, and X. Yang, “Picosecond pulse amplification by coherent wave mixing in silicon,” Phys. Rev. A 35, 4673–4679 (1987).
[CrossRef] [PubMed]

Adair, R.

R. Adair, L. L. Chase, and S. A. Payne, “Nonlinear refractive index measurements of optical crystals,” Phys. Rev. B 39, 3337–3350 (1998).
[CrossRef]

Baltuska, A.

A. Baltuska, Z. Wei, M. S. Pshenichnikov, D. A. Wiersma, and R. Szipöz, “All-solid-state cavity-dumped sub-5-fs laser,” Appl. Phys. B 65, 175–188 (1997).
[CrossRef]

Bloembergen, N.

Chase, L. L.

R. Adair, L. L. Chase, and S. A. Payne, “Nonlinear refractive index measurements of optical crystals,” Phys. Rev. B 39, 3337–3350 (1998).
[CrossRef]

Christov, I. P.

Dianov, E. M.

A. B. Grudinin, E. M. Dianov, D. V. Korobkin, A. M. Prokhorov, V. N. Serkin, and D. V. Haidarov, “Decay of femtosecond pulses in single-mode optical fibers,” JETP Lett. 46, 221–225 (1987).

Dogariu, A.

Eichler, H. J.

H. J. Eichler, M. Glotz, A. Kummrow, K. Richter, and X. Yang, “Picosecond pulse amplification by coherent wave mixing in silicon,” Phys. Rev. A 35, 4673–4679 (1987).
[CrossRef] [PubMed]

Gaeta, A.

Glotz, M.

H. J. Eichler, M. Glotz, A. Kummrow, K. Richter, and X. Yang, “Picosecond pulse amplification by coherent wave mixing in silicon,” Phys. Rev. A 35, 4673–4679 (1987).
[CrossRef] [PubMed]

Grudinin, A. B.

A. B. Grudinin, E. M. Dianov, D. V. Korobkin, A. M. Prokhorov, V. N. Serkin, and D. V. Haidarov, “Decay of femtosecond pulses in single-mode optical fibers,” JETP Lett. 46, 221–225 (1987).

Hagan, D. J.

Haidarov, D. V.

A. B. Grudinin, E. M. Dianov, D. V. Korobkin, A. M. Prokhorov, V. N. Serkin, and D. V. Haidarov, “Decay of femtosecond pulses in single-mode optical fibers,” JETP Lett. 46, 221–225 (1987).

Huang, Ch.-P.

Kang, I.

Kapteyn, H. C.

Korobkin, D. V.

A. B. Grudinin, E. M. Dianov, D. V. Korobkin, A. M. Prokhorov, V. N. Serkin, and D. V. Haidarov, “Decay of femtosecond pulses in single-mode optical fibers,” JETP Lett. 46, 221–225 (1987).

Krauss, T.

Kummrow, A.

H. J. Eichler, M. Glotz, A. Kummrow, K. Richter, and X. Yang, “Picosecond pulse amplification by coherent wave mixing in silicon,” Phys. Rev. A 35, 4673–4679 (1987).
[CrossRef] [PubMed]

Mallalieu, M.

J. A. Yeazell, M. Mallalieu, and C. R. Stroud, Jr., “Observation of the collapse and revival of a Rydberg electronic wave packet,” Phys. Rev. Lett. 64, 2007–2010 (1990).
[CrossRef] [PubMed]

Murnane, M. M.

Payne, S. A.

R. Adair, L. L. Chase, and S. A. Payne, “Nonlinear refractive index measurements of optical crystals,” Phys. Rev. B 39, 3337–3350 (1998).
[CrossRef]

Proctor, B.

Prokhorov, A. M.

A. B. Grudinin, E. M. Dianov, D. V. Korobkin, A. M. Prokhorov, V. N. Serkin, and D. V. Haidarov, “Decay of femtosecond pulses in single-mode optical fibers,” JETP Lett. 46, 221–225 (1987).

Pshenichnikov, M. S.

A. Baltuska, Z. Wei, M. S. Pshenichnikov, D. A. Wiersma, and R. Szipöz, “All-solid-state cavity-dumped sub-5-fs laser,” Appl. Phys. B 65, 175–188 (1997).
[CrossRef]

Ranka, J.

Richter, K.

H. J. Eichler, M. Glotz, A. Kummrow, K. Richter, and X. Yang, “Picosecond pulse amplification by coherent wave mixing in silicon,” Phys. Rev. A 35, 4673–4679 (1987).
[CrossRef] [PubMed]

Said, A. A.

Serkin, V. N.

A. B. Grudinin, E. M. Dianov, D. V. Korobkin, A. M. Prokhorov, V. N. Serkin, and D. V. Haidarov, “Decay of femtosecond pulses in single-mode optical fibers,” JETP Lett. 46, 221–225 (1987).

Smolorz, S.

Stroud Jr., C. R.

J. A. Yeazell, M. Mallalieu, and C. R. Stroud, Jr., “Observation of the collapse and revival of a Rydberg electronic wave packet,” Phys. Rev. Lett. 64, 2007–2010 (1990).
[CrossRef] [PubMed]

Sutherland, R. L.

Szipöz, R.

A. Baltuska, Z. Wei, M. S. Pshenichnikov, D. A. Wiersma, and R. Szipöz, “All-solid-state cavity-dumped sub-5-fs laser,” Appl. Phys. B 65, 175–188 (1997).
[CrossRef]

Tang, N.

Van Stryland, E. W.

Wei, Z.

A. Baltuska, Z. Wei, M. S. Pshenichnikov, D. A. Wiersma, and R. Szipöz, “All-solid-state cavity-dumped sub-5-fs laser,” Appl. Phys. B 65, 175–188 (1997).
[CrossRef]

Wiersma, D. A.

A. Baltuska, Z. Wei, M. S. Pshenichnikov, D. A. Wiersma, and R. Szipöz, “All-solid-state cavity-dumped sub-5-fs laser,” Appl. Phys. B 65, 175–188 (1997).
[CrossRef]

Wise, F.

Xia, T.

Yang, X.

H. J. Eichler, M. Glotz, A. Kummrow, K. Richter, and X. Yang, “Picosecond pulse amplification by coherent wave mixing in silicon,” Phys. Rev. A 35, 4673–4679 (1987).
[CrossRef] [PubMed]

Yeazell, J. A.

J. A. Yeazell, M. Mallalieu, and C. R. Stroud, Jr., “Observation of the collapse and revival of a Rydberg electronic wave packet,” Phys. Rev. Lett. 64, 2007–2010 (1990).
[CrossRef] [PubMed]

Zhou, J.

Appl. Phys. B (1)

A. Baltuska, Z. Wei, M. S. Pshenichnikov, D. A. Wiersma, and R. Szipöz, “All-solid-state cavity-dumped sub-5-fs laser,” Appl. Phys. B 65, 175–188 (1997).
[CrossRef]

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

JETP Lett. (1)

A. B. Grudinin, E. M. Dianov, D. V. Korobkin, A. M. Prokhorov, V. N. Serkin, and D. V. Haidarov, “Decay of femtosecond pulses in single-mode optical fibers,” JETP Lett. 46, 221–225 (1987).

Opt. Express (1)

Opt. Lett. (5)

Phys. Rev. A (1)

H. J. Eichler, M. Glotz, A. Kummrow, K. Richter, and X. Yang, “Picosecond pulse amplification by coherent wave mixing in silicon,” Phys. Rev. A 35, 4673–4679 (1987).
[CrossRef] [PubMed]

Phys. Rev. B (1)

R. Adair, L. L. Chase, and S. A. Payne, “Nonlinear refractive index measurements of optical crystals,” Phys. Rev. B 39, 3337–3350 (1998).
[CrossRef]

Phys. Rev. Lett. (1)

J. A. Yeazell, M. Mallalieu, and C. R. Stroud, Jr., “Observation of the collapse and revival of a Rydberg electronic wave packet,” Phys. Rev. Lett. 64, 2007–2010 (1990).
[CrossRef] [PubMed]

Other (9)

R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif. 1992), Chap. 8.

R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif. 1992), Chap. 1.

A. E. Siegman, Lasers (University Science, Sausolito, Calif., 1986), pp. 944–947.

R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif., 1992), pp. 279–280.

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, San Diego, Calif., 1995), p. 67.

A. E. Siegman, Lasers (University Science, Sausolito, Calif., 1986), pp. 333–334.

G. Turrell, Infrared and Raman Spectra of Crystals (Academic, London, 1972), Chap. 4.

U. Müller, Anorganische Strukturchemie (Teubner, Stuttgart, 1992), p. 50.

R. W. Boyd, Nonlinear Optics (Academic, San Diego, Calif., 1992), Chap. 6.4.

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Figures (7)

Fig. 1
Fig. 1

Calculated TBC signals for the full and nuclear response of fused silica, with and without chirp. The calculations are made with parameters nominally identical to those in the experiments: 30-fs pulses with nanojoule energies are focused to intensities of 20 GW/cm2.

Fig. 2
Fig. 2

Illustration of the origin of chirp-free TBC signals: For negative delay τ, the probe pulse experiences a redshift in frequency owing to the positive slope of Δn, for positive delay τ, a blueshift. This leads to a gain in energy from the pump for negative delay, and a loss of energy to the pump for positive delay.

Fig. 3
Fig. 3

Calculated TBC signals with zero and small chirp for two exponential responses.

Fig. 4
Fig. 4

Magnitude of purely electronic TBC signals as a function of the nonlinear index of refraction n2, for |C|>0 (left) and C=0 (right). The symbols are the results of simulations, the lines linear and quadratic fits, respectively.

Fig. 5
Fig. 5

Magnitude of the calculated TBC signal in KBr, with varying chirp parameter C and response time TR. The symbols are the results of calculations, the curves are provided merely to guide the eye. These calculations assume Gaussian pulses with t0=10 fs and intensity 30 GW/cm2.

Fig. 6
Fig. 6

Schematic of the TBC experiment. In experiments performed with pulses shorter than 20 fs, the lens is replaced by a curved mirror to minimize dispersion. Mirror M1 is passed by the pulses coming from the laser, but it picks off the pulses that have traversed the prism sequence and directs them into the experiment. For that purpose, mirror M2 imparts a small incline to the pulses that is canceled by the alignment of mirror M1. BS, beam splitter; D, signal detector; R, reference detector.

Fig. 7
Fig. 7

TBC signals in fused silica. Solid curves, experimental signal; dashed curves, calculated signal adjusted to match the experimental ΔT/T.

Tables (2)

Tables Icon

Table 1 Experimental ΔT/T is Larger Than the Calculated Value by the Indicated Factor

Tables Icon

Table 2 Nonlinear Index of Refraction of Alkali Halides, Determined Using Spectrally Resolved TBC with 17-fs Pulses Centered at 775 nm a

Equations (27)

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dA1dz=i4πk1A1|A2|2-tRNL(t-t)dt+exp(iΔωt)-tRNL(t-t)exp(-iΔωt)dt,
dA1*dz=-i4πk1A1*|A2|2-tRNL(t-t)dt+exp(-iΔωt)-tRNL(t-t)exp(iΔωt)dt.
ddz |A1|2=A1dA1*dz+A1*dA1dz=-8πkRNL0ΔωTR1+Δω2TR2 |A1|2|A2|2,
ddz |A2|2=8πkRNL0ΔωTR1+Δω2TR2 |A1|2|A2|2.
A(t)=A exp-t2(1+iC)2t02(1+C2)(Ref.10),
δω=-tCt02(1+C2)(Ref.11).
2z2-n02c22t2E(z, t)=4πc22t2 PNL(z, t),
PNL,j(r, t)=i,k,lEi(r, t)-tdtRNL,jikl(t-t)×Ek(r, t)El(r, t).
E(z, t)=A1(t-τ)e1exp{-i[ω1(t-τ)-k1z]}+A1*(t-τ)e1*exp{i[ω1(t-τ)-k1z]}+A2(t)e2exp[-i(ω2t-k2z)]+A2*(t)e2*exp[i(ω2t-k2z)],
dA1dz=i2πk1[A1(t-τ)H1+A2(t)H4],
H1=-tRNL(t-t)[2|A1(t-τ)|2+2|A2(t)|2]dt,
H4=exp(iΔωt)-tRNL(t-t)2A1(t-τ)×A2*(t)exp(-iΔωt)dt.
2x2+2y2+2z2E(r, t)-1c22t2E(r, t)=4πc22t2P(r, t).
PL,j(r, t)=l(n02-1)4π El(r, t),
PNL,j(r, t)=i,k,lEi(r, t)-tdtRNL,jikl(t-t)×Ek(r, t)El(r, t),
E(z, t)=e(A1(t-τ)exp{-i[ω1(t-τ)-k1z]}+A1*(t-τ)exp{i[ω1(t-τ)-k1z]}+A2(t)exp[-i(ω2t-k2z)]+A2*(t)exp[i(ω2t-k2z)]).
2z2-n02c22t2E(z, t)=4πc22t2 PNL(z, t).
l.h.s.=2ik1A1(t-τ)z-k12A1(t-τ)+2A1(t-τ)z2exp{-i[ω1(t-τ)-k1z]}+-2ik1A1*(t-τ)z-k12A1*(t-τ)+2A1*(t-τ)z2exp{i[ω1(t-τ)-k1z]}+n02c22iω1A1(t-τ)t+ω12A1(t-τ)-2A1(t-τ)t2exp{-i[ω1(t-τ)-k1z]}+n02c2-2iω1A1*(t-τ)t+ω12A1*(t-τ)-2A1*(t-τ)t2exp{i[ω1(t-τ)-k1z]}+2ik2A2(t)z-k22A2(t)+2A2(t)z2exp[-i(ω2t-k2z)]+-2ik2A2*(t)z-k22A2*(t)+2A2*(t)z2exp[i(ω2t-k2z)]+n02c22iω2A2(t)t+ω22A2(t)-2A2(t)t2exp[-i(ω2t-k2z)]+n02c2-2iω2A2*(t)t+ω22A2*(t)-2A2*(t)t2exp[i(ω2t-k2z)].
4πc22t2 PNL(z, t)-ω124πc2 PNL(z, t)=-ω124πc2(A1(t-τ)exp{-i[ω1(t-τ)-k1z]}+A2(t)exp[-i(ω2t-k2z)]+c.c.)×-tRNL(t-t)(2|A1(t-τ)|2+2|A2(t)|2+A1(t-τ)2exp{-2i[ω1(t-τ)-k1z]}+A1*(t-τ)2exp{2i[ω1(t-τ)-k1z]}+2A1*(t-τ)A2exp[i(Δωt-ω1τ+Δkz)]+A2(t)exp[-2i(ω2t-k2z)]+2A1(t-τ)A2*(t)exp[-i(Δωt-ω1τ+Δkz)]+2A1*(t-τ)A2*(t)exp[i(Ωt-ω1τ-Kz)]+A2*(t)2exp[2i(ω2t-k2z)]+2A1(t-τ)A2(t)exp[-i(Ωt-ω1τ-Kz)])dt,
2Az2k Az,2At2ω At,
dA1dz=i2πk1[A1(t-τ)H1+A1*(t-τ)H2+A2(t)H4+A2(t)*H3],
H1=-tRNL(t-t)[2|A1(t-τ)|2+2|A2(t)|2]dt,
H2=exp(2iω1t)-tRNL(t-t)A1(t-τ)2×exp(-2iω1t)dt,
H3=exp(iΩt)-tRNL(t-t)2A1(t-τ)A2(t)×exp(-iΩt)dt,
H4=exp(iΔωt)-tRNL(t-t)2A1(t-τ)A2*(t)×exp(-iΔωt)dt.
dA1dz=i2πk1[A1(t-τ)H1+A2(t)H4],
dA2dz=i2πk2[A2(t)H1+A1(t-τ)H4*],

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