Abstract

The interaction of two closely spaced resonances in degenerate four-wave-mixing spectra is investigated theoretically. Degenerate four-wave-mixing spectra for the phase-conjugate geometry (counterpropagating pump beams) are calculated by integrating the time-dependent density-matrix equations at numerous grid points along the phase-matching axis and summing the polarization contribution from each of these grid points. Both purely homogeneously broadened resonances and resonances that are both collision and Doppler broadened are considered. The homogeneously broadened results are found to agree with the results of the widely used Abrams–Lind expression for the case of stationary absorbers exposed to a steady-state excitation. We examine the effects of power broadening, pressure broadening, transition dipole moment, and population on the spectra of two neighboring homogeneously broadened resonances. As the laser power is increased, the real parts of the resonance susceptibilities destructively interfere with each other in the region between the lines, resulting in a significant and persistent dip in reflectivity. When Doppler broadening is included in the calculations, similar interference effects are observed, although at first the interference dip begins to disappear until the homogeneous part of the line shapes is broadened significantly by saturation. Using this numerical approach, we obtain good agreement with experimental results of primarily Doppler-broadened neighboring resonances.

© 1997 Optical Society of America

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References

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  1. R. L. Farrow and D. J. Rakestraw, “Detection of trace molecular species using degenerate four-wave mixing,” Science 257, 1894–1900 (1992).
    [Crossref] [PubMed]
  2. K. Kohse-Höinghaus, “Laser techniques for the quantitative detection of reactive intermediates in combustion systems,” Prog. Energy Combust. Sci. 20, 203–279 (1994).
    [Crossref]
  3. A. C. Eckbreth, Laser Diagnostics for Combustion Temperature and Species, 2nd ed. (Gordon and Breach, Amsterdam, The Netherlands, 1996).
  4. T. Dreier and D. J. Rakestraw, “Measurement of OH rotational temperatures in a flame using degenerate four-wave mixing,” Opt. Lett. 15, 72–74 (1990).
    [Crossref] [PubMed]
  5. D. A. Feikema, E. Domingues, and M.-J. Cottereau, “OH rotational temperature and number density measurements in high-pressure laminar flames using double phase-conjugate four-wave mixing,” Appl. Phys. B 55, 424–429 (1992).
    [Crossref]
  6. A. P. Smith and A. G. Astill, “Temperature measurement using degenerate four-wave mixing with non-saturating laser powers,” Appl. Phys. B 58, 459–466 (1994).
    [Crossref]
  7. B. Yip, P. M. Danehy, and R. K. Hanson, “Degenerate four-wave mixing temperature measurements in a flame,” Opt. Lett. 17, 751–753 (1992).
    [Crossref] [PubMed]
  8. R. L. Abrams and R. C. Lind, “Degenerate four-wave mixing in absorbing media,” Opt. Lett. 2, 94–96 (1978); erratum,  3, 205 (1978).
    [Crossref] [PubMed]
  9. R. W. Boyd, Nonlinear Optics (Academic, Boston, 1992), p. 191.
  10. R. P. Lucht, R. L. Farrow, and D. J. Rakestraw, “Saturation effects in gas-phase degenerate four-wave mixing spectroscopy:nonperturbative calculations,” J. Opt. Soc. Am. B 10, 1508–1520 (1993).
    [Crossref]
  11. T. A. Reichardt and R. P. Lucht, “Effect of Doppler broadening on quantitative concentration measurements with degenerate four-wave mixing spectroscopy,” J. Opt. Soc. Am. B 13, 1107–1119 (1996).
    [Crossref]
  12. M. S. Brown, L. A. Rahn, and R. P. Lucht, “Degenerate four-wave mixing line shapes of hydroxyl at high pump intensities,” Appl. Opt. 34, 3274–3280 (1995).
    [Crossref] [PubMed]
  13. R. L. Farrow, D. J. Rakestraw, and T. Dreier, “Investigation of the dependence of degenerate four-wave mixing line intensities on transition dipole moment,” J. Opt. Soc. Am. B 9, 1770–1777 (1992).
    [Crossref]
  14. R. L. Farrow and D. J. Rakestraw, Combustion Research Facility, Sandia National Laboratories, Livermore, California 94550, and R. L. Vander Wal, Nyma, Inc., NASA Lewis Research Center, Cleveland, Ohio 44135 (personal communication).

1996 (1)

1995 (1)

1994 (2)

A. P. Smith and A. G. Astill, “Temperature measurement using degenerate four-wave mixing with non-saturating laser powers,” Appl. Phys. B 58, 459–466 (1994).
[Crossref]

K. Kohse-Höinghaus, “Laser techniques for the quantitative detection of reactive intermediates in combustion systems,” Prog. Energy Combust. Sci. 20, 203–279 (1994).
[Crossref]

1993 (1)

1992 (4)

R. L. Farrow and D. J. Rakestraw, “Detection of trace molecular species using degenerate four-wave mixing,” Science 257, 1894–1900 (1992).
[Crossref] [PubMed]

D. A. Feikema, E. Domingues, and M.-J. Cottereau, “OH rotational temperature and number density measurements in high-pressure laminar flames using double phase-conjugate four-wave mixing,” Appl. Phys. B 55, 424–429 (1992).
[Crossref]

B. Yip, P. M. Danehy, and R. K. Hanson, “Degenerate four-wave mixing temperature measurements in a flame,” Opt. Lett. 17, 751–753 (1992).
[Crossref] [PubMed]

R. L. Farrow, D. J. Rakestraw, and T. Dreier, “Investigation of the dependence of degenerate four-wave mixing line intensities on transition dipole moment,” J. Opt. Soc. Am. B 9, 1770–1777 (1992).
[Crossref]

1990 (1)

1978 (1)

Abrams, R. L.

Astill, A. G.

A. P. Smith and A. G. Astill, “Temperature measurement using degenerate four-wave mixing with non-saturating laser powers,” Appl. Phys. B 58, 459–466 (1994).
[Crossref]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, Boston, 1992), p. 191.

Brown, M. S.

Cottereau, M.-J.

D. A. Feikema, E. Domingues, and M.-J. Cottereau, “OH rotational temperature and number density measurements in high-pressure laminar flames using double phase-conjugate four-wave mixing,” Appl. Phys. B 55, 424–429 (1992).
[Crossref]

Danehy, P. M.

Domingues, E.

D. A. Feikema, E. Domingues, and M.-J. Cottereau, “OH rotational temperature and number density measurements in high-pressure laminar flames using double phase-conjugate four-wave mixing,” Appl. Phys. B 55, 424–429 (1992).
[Crossref]

Dreier, T.

Eckbreth, A. C.

A. C. Eckbreth, Laser Diagnostics for Combustion Temperature and Species, 2nd ed. (Gordon and Breach, Amsterdam, The Netherlands, 1996).

Farrow, R. L.

R. P. Lucht, R. L. Farrow, and D. J. Rakestraw, “Saturation effects in gas-phase degenerate four-wave mixing spectroscopy:nonperturbative calculations,” J. Opt. Soc. Am. B 10, 1508–1520 (1993).
[Crossref]

R. L. Farrow, D. J. Rakestraw, and T. Dreier, “Investigation of the dependence of degenerate four-wave mixing line intensities on transition dipole moment,” J. Opt. Soc. Am. B 9, 1770–1777 (1992).
[Crossref]

R. L. Farrow and D. J. Rakestraw, “Detection of trace molecular species using degenerate four-wave mixing,” Science 257, 1894–1900 (1992).
[Crossref] [PubMed]

R. L. Farrow and D. J. Rakestraw, Combustion Research Facility, Sandia National Laboratories, Livermore, California 94550, and R. L. Vander Wal, Nyma, Inc., NASA Lewis Research Center, Cleveland, Ohio 44135 (personal communication).

Feikema, D. A.

D. A. Feikema, E. Domingues, and M.-J. Cottereau, “OH rotational temperature and number density measurements in high-pressure laminar flames using double phase-conjugate four-wave mixing,” Appl. Phys. B 55, 424–429 (1992).
[Crossref]

Hanson, R. K.

Kohse-Höinghaus, K.

K. Kohse-Höinghaus, “Laser techniques for the quantitative detection of reactive intermediates in combustion systems,” Prog. Energy Combust. Sci. 20, 203–279 (1994).
[Crossref]

Lind, R. C.

Lucht, R. P.

Rahn, L. A.

Rakestraw, D. J.

Reichardt, T. A.

Smith, A. P.

A. P. Smith and A. G. Astill, “Temperature measurement using degenerate four-wave mixing with non-saturating laser powers,” Appl. Phys. B 58, 459–466 (1994).
[Crossref]

Yip, B.

Appl. Opt. (1)

Appl. Phys. B (2)

D. A. Feikema, E. Domingues, and M.-J. Cottereau, “OH rotational temperature and number density measurements in high-pressure laminar flames using double phase-conjugate four-wave mixing,” Appl. Phys. B 55, 424–429 (1992).
[Crossref]

A. P. Smith and A. G. Astill, “Temperature measurement using degenerate four-wave mixing with non-saturating laser powers,” Appl. Phys. B 58, 459–466 (1994).
[Crossref]

J. Opt. Soc. Am. B (3)

Opt. Lett. (3)

Prog. Energy Combust. Sci. (1)

K. Kohse-Höinghaus, “Laser techniques for the quantitative detection of reactive intermediates in combustion systems,” Prog. Energy Combust. Sci. 20, 203–279 (1994).
[Crossref]

Science (1)

R. L. Farrow and D. J. Rakestraw, “Detection of trace molecular species using degenerate four-wave mixing,” Science 257, 1894–1900 (1992).
[Crossref] [PubMed]

Other (3)

A. C. Eckbreth, Laser Diagnostics for Combustion Temperature and Species, 2nd ed. (Gordon and Breach, Amsterdam, The Netherlands, 1996).

R. W. Boyd, Nonlinear Optics (Academic, Boston, 1992), p. 191.

R. L. Farrow and D. J. Rakestraw, Combustion Research Facility, Sandia National Laboratories, Livermore, California 94550, and R. L. Vander Wal, Nyma, Inc., NASA Lewis Research Center, Cleveland, Ohio 44135 (personal communication).

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Figures (8)

Fig. 1
Fig. 1

DFWM reflectivity as a function of laser detuning ω˜L -ω˜a=(ωL-ωa)/2πc for two neighboring homogeneously broadened resonances (γ=5×109 s-1, ω˜b-ω˜a=0.1 cm-1) for (a) Ipump/Isat=0.1 and (b) Ipump/Isat=10. The results of the Abrams–Lind model and the DNI calculations are shown for each laser intensity.

Fig. 2
Fig. 2

DNI calculations for two neighboring homogeneously broadened resonances (ω˜b-ω˜a=0.1 cm-1) at different laser intensities (left column) and for different collision rates (right column).

Fig. 3
Fig. 3

Values of βreal (left column) and βimag (right column) for three cases (ω˜b-ω˜a=0.1 cm-1) to demonstrate laser saturation and pressure broadening. The top two figures represent the base case, the middle two figures show the effect of laser saturation, and the bottom two figures show the effect of pressure broadening.

Fig. 4
Fig. 4

DNI results for μ12,b=0.707μ12,a, ρ11,a=ρ11,b, γ=2.5×109 s-1, and a line spacing of ω˜b-ω˜a=0.1 cm-1 (left column) and ω˜b-ω˜a=0.05 cm-1 (right column) for three different laser intensities.

Fig. 5
Fig. 5

DNI results for μ12,b=μ12,a, ρ11,b=0.707ρ11,a, γ=2.5×109 s-1, and a line spacing of ω˜b-ω˜a=0.1 cm-1 (left column) and ω˜b-ω˜a=0.05 cm-1 (right column) for three different laser intensities.

Fig. 6
Fig. 6

DNI calculations for two neighboring resonances that are both collision and Doppler broadened [γ=2.5×109 s-1, Δω˜D=(ΔωD/2πc)=0.1 cm-1, ω˜b-ω˜a=0.1 cm-1] at different laser intensities.

Fig. 7
Fig. 7

Experimental results showing the effects of saturation broadening (left column) and pressure broadening (right column). For the spectra in the left column the buffer gas pressure was held constant at 0 mTorr, and the laser power was increased. For the right column, the laser power was held constant at 0.02 MW/cm2, and the buffer gas pressure was increased.

Fig. 8
Fig. 8

Comparison of data obtained for the P1(6) and P1(11) lines of NO with the results of DNI calculations for Δω˜D=0.1 cm-1. (a) For the experimental data the pump laser intensity was 0.20 MW/cm2 and the buffer gas pressure was 0 mTorr, while for the DNI calculations the peak pump laser intensity was 0.03 MW/cm2 and γ=2.5×108 s-1. (b) The experimental pump laser intensity was 0.02 MW/cm2 and the buffer gas pressure was 20 Torr, while for the DNI calculations the peak pump intensity was 0.003 MW/cm2 and γ=1.75 ×109 s-1. For both (a) and (b) the probe beam intensity was set equal to one fourth the intensity of the pump beams.

Equations (14)

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ρ11,a(z, t)t=-i(V12,aρ21,a-ρ12,aV21,a)+Γaρ22,a,
ρ21,a(z, t)t=-ρ21,a(iωa+γa)-iV21,a(ρ11,a-ρ22,a),
ρ11,b(z, t)t=-i(V12,bρ21,b-ρ12,bV21,b)+Γbρ22,b,
ρ21,b(z, t)t=-ρ21,b(iωb+γb)-iV21,b(ρ11,b-ρ22,b),
β=α0 i+Δx1+(Δx)22Ipump/Isat(1+4Ipump/Isat)3/2.
Isat=Isat0[1+(Δx)2],
Isat0=0c2γΓ2μ212=2π2cγΔωCA21λ3.
R=|βL|2=L2[(βreal,a+βreal,b)2+(βimag,a+βimag,b)2],
βreal,a=α0,a Δxa1+(Δxa)22Ipump/Isat,a(1+4Ipump/Isat,a)3/2,
βreal,b=α0,b Δxb1+(Δxb)22Ipump/Isat,b(1+4Ipump/Isat,b)3/2,
βimag,a=α0,a 11+(Δxa)22Ipump/Isat,a(1+4Ipump/Isat,a)3/2,
βimag,b=α0,b 11+(Δxb)22Ipump/Isat,b(1+4Ipump/Isat,b)3/2.
Ωpump=μ21Apump2.
Ωsat=(γΓ)1/22,

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