Abstract

We present a simple numerical procedure that allows the frequency-dependent absorption and dispersion line profiles to be extracted from the in-phase and in-quadrature components of a spectrum obtained by frequency-modulation techniques. The procedure is independent of the underlying line shapes and takes into account contributions by additional amplitude modulation. With the help of sum rules the parameters of this amplitude modulation can be determined.

© 1999 Optical Society of America

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References

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  1. G. C. Bjorklund, “Frequency-modulation spectroscopy: a new method for measuring weak absorptions and dispersions,” Opt. Lett. 5, 15–17 (1980).
    [CrossRef] [PubMed]
  2. G. C. Bjorklund, M. D. Levenson, W. Lenth, and C. Ortiz, “Frequency modulation (FM) spectroscopy,” Appl. Phys. B 32, 145–152 (1983).
    [CrossRef]
  3. M. Gehrtz, G. C. Bjorklund, and E. A. Whittaker, “Quantum-limited laser frequency-modulation spectroscopy,” J. Opt. Soc. Am. B 2, 1510–1525 (1985).
    [CrossRef]
  4. S. Brandt, A. Nagel, R. Wynands, and D. Meschede, “Buffer-gas-induced linewidth reduction of coherent dark resonances to below 50 Hz,” Phys. Rev. A 56, R1063–R1066 (1997).
    [CrossRef]
  5. S. W. North, X. S. Zheng, R. Fei, and G. E. Hall, “Line shape analysis of Doppler broadened frequency-modulated line spectra,” J. Chem. Phys. 104, 2129–2135 (1996).
    [CrossRef]
  6. B. A. Woody, and L. Lynds, “Frequency-modulated laser absorption spectroscopy of the HF fourth overtone,” Appl. Opt. 25, 2148–2153 (1986).
    [CrossRef] [PubMed]
  7. G. Alzetta, A. Gozzini, L. Moi, and G. Orriols, “An experimental method for the observation of r.f. transitions and laser beat resonances in oriented Na vapour,” Nuovo Cimento B 13, 5–20 (1976).
    [CrossRef]
  8. E. Arimondo, “Coherent population trapping in laser spectroscopy,” Prog. Opt. 35, 257–354 (1996).
    [CrossRef]
  9. R. Wynands and A. Nagel, “Precision spectroscopy with coherent dark states,” Appl. Phys. B 68, 1–25 (1999).
    [CrossRef]
  10. W. Lenth, “High frequency heterodyne spectroscopy with current-modulated diode lasers,” IEEE J. Quantum Electron. QE-20, 1045–1050 (1984).
    [CrossRef]
  11. C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. QE-18, 259–264 (1982).
    [CrossRef]
  12. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1989).

1999 (1)

R. Wynands and A. Nagel, “Precision spectroscopy with coherent dark states,” Appl. Phys. B 68, 1–25 (1999).
[CrossRef]

1997 (1)

S. Brandt, A. Nagel, R. Wynands, and D. Meschede, “Buffer-gas-induced linewidth reduction of coherent dark resonances to below 50 Hz,” Phys. Rev. A 56, R1063–R1066 (1997).
[CrossRef]

1996 (2)

S. W. North, X. S. Zheng, R. Fei, and G. E. Hall, “Line shape analysis of Doppler broadened frequency-modulated line spectra,” J. Chem. Phys. 104, 2129–2135 (1996).
[CrossRef]

E. Arimondo, “Coherent population trapping in laser spectroscopy,” Prog. Opt. 35, 257–354 (1996).
[CrossRef]

1986 (1)

1985 (1)

1984 (1)

W. Lenth, “High frequency heterodyne spectroscopy with current-modulated diode lasers,” IEEE J. Quantum Electron. QE-20, 1045–1050 (1984).
[CrossRef]

1983 (1)

G. C. Bjorklund, M. D. Levenson, W. Lenth, and C. Ortiz, “Frequency modulation (FM) spectroscopy,” Appl. Phys. B 32, 145–152 (1983).
[CrossRef]

1982 (1)

C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. QE-18, 259–264 (1982).
[CrossRef]

1980 (1)

1976 (1)

G. Alzetta, A. Gozzini, L. Moi, and G. Orriols, “An experimental method for the observation of r.f. transitions and laser beat resonances in oriented Na vapour,” Nuovo Cimento B 13, 5–20 (1976).
[CrossRef]

Alzetta, G.

G. Alzetta, A. Gozzini, L. Moi, and G. Orriols, “An experimental method for the observation of r.f. transitions and laser beat resonances in oriented Na vapour,” Nuovo Cimento B 13, 5–20 (1976).
[CrossRef]

Arimondo, E.

E. Arimondo, “Coherent population trapping in laser spectroscopy,” Prog. Opt. 35, 257–354 (1996).
[CrossRef]

Bjorklund, G. C.

Brandt, S.

S. Brandt, A. Nagel, R. Wynands, and D. Meschede, “Buffer-gas-induced linewidth reduction of coherent dark resonances to below 50 Hz,” Phys. Rev. A 56, R1063–R1066 (1997).
[CrossRef]

Fei, R.

S. W. North, X. S. Zheng, R. Fei, and G. E. Hall, “Line shape analysis of Doppler broadened frequency-modulated line spectra,” J. Chem. Phys. 104, 2129–2135 (1996).
[CrossRef]

Gehrtz, M.

Gozzini, A.

G. Alzetta, A. Gozzini, L. Moi, and G. Orriols, “An experimental method for the observation of r.f. transitions and laser beat resonances in oriented Na vapour,” Nuovo Cimento B 13, 5–20 (1976).
[CrossRef]

Hall, G. E.

S. W. North, X. S. Zheng, R. Fei, and G. E. Hall, “Line shape analysis of Doppler broadened frequency-modulated line spectra,” J. Chem. Phys. 104, 2129–2135 (1996).
[CrossRef]

Henry, C. H.

C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. QE-18, 259–264 (1982).
[CrossRef]

Lenth, W.

W. Lenth, “High frequency heterodyne spectroscopy with current-modulated diode lasers,” IEEE J. Quantum Electron. QE-20, 1045–1050 (1984).
[CrossRef]

G. C. Bjorklund, M. D. Levenson, W. Lenth, and C. Ortiz, “Frequency modulation (FM) spectroscopy,” Appl. Phys. B 32, 145–152 (1983).
[CrossRef]

Levenson, M. D.

G. C. Bjorklund, M. D. Levenson, W. Lenth, and C. Ortiz, “Frequency modulation (FM) spectroscopy,” Appl. Phys. B 32, 145–152 (1983).
[CrossRef]

Lynds, L.

Meschede, D.

S. Brandt, A. Nagel, R. Wynands, and D. Meschede, “Buffer-gas-induced linewidth reduction of coherent dark resonances to below 50 Hz,” Phys. Rev. A 56, R1063–R1066 (1997).
[CrossRef]

Moi, L.

G. Alzetta, A. Gozzini, L. Moi, and G. Orriols, “An experimental method for the observation of r.f. transitions and laser beat resonances in oriented Na vapour,” Nuovo Cimento B 13, 5–20 (1976).
[CrossRef]

Nagel, A.

R. Wynands and A. Nagel, “Precision spectroscopy with coherent dark states,” Appl. Phys. B 68, 1–25 (1999).
[CrossRef]

S. Brandt, A. Nagel, R. Wynands, and D. Meschede, “Buffer-gas-induced linewidth reduction of coherent dark resonances to below 50 Hz,” Phys. Rev. A 56, R1063–R1066 (1997).
[CrossRef]

North, S. W.

S. W. North, X. S. Zheng, R. Fei, and G. E. Hall, “Line shape analysis of Doppler broadened frequency-modulated line spectra,” J. Chem. Phys. 104, 2129–2135 (1996).
[CrossRef]

Orriols, G.

G. Alzetta, A. Gozzini, L. Moi, and G. Orriols, “An experimental method for the observation of r.f. transitions and laser beat resonances in oriented Na vapour,” Nuovo Cimento B 13, 5–20 (1976).
[CrossRef]

Ortiz, C.

G. C. Bjorklund, M. D. Levenson, W. Lenth, and C. Ortiz, “Frequency modulation (FM) spectroscopy,” Appl. Phys. B 32, 145–152 (1983).
[CrossRef]

Whittaker, E. A.

Woody, B. A.

Wynands, R.

R. Wynands and A. Nagel, “Precision spectroscopy with coherent dark states,” Appl. Phys. B 68, 1–25 (1999).
[CrossRef]

S. Brandt, A. Nagel, R. Wynands, and D. Meschede, “Buffer-gas-induced linewidth reduction of coherent dark resonances to below 50 Hz,” Phys. Rev. A 56, R1063–R1066 (1997).
[CrossRef]

Zheng, X. S.

S. W. North, X. S. Zheng, R. Fei, and G. E. Hall, “Line shape analysis of Doppler broadened frequency-modulated line spectra,” J. Chem. Phys. 104, 2129–2135 (1996).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B (2)

G. C. Bjorklund, M. D. Levenson, W. Lenth, and C. Ortiz, “Frequency modulation (FM) spectroscopy,” Appl. Phys. B 32, 145–152 (1983).
[CrossRef]

R. Wynands and A. Nagel, “Precision spectroscopy with coherent dark states,” Appl. Phys. B 68, 1–25 (1999).
[CrossRef]

IEEE J. Quantum Electron. (2)

W. Lenth, “High frequency heterodyne spectroscopy with current-modulated diode lasers,” IEEE J. Quantum Electron. QE-20, 1045–1050 (1984).
[CrossRef]

C. H. Henry, “Theory of the linewidth of semiconductor lasers,” IEEE J. Quantum Electron. QE-18, 259–264 (1982).
[CrossRef]

J. Chem. Phys. (1)

S. W. North, X. S. Zheng, R. Fei, and G. E. Hall, “Line shape analysis of Doppler broadened frequency-modulated line spectra,” J. Chem. Phys. 104, 2129–2135 (1996).
[CrossRef]

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

Nuovo Cimento B (1)

G. Alzetta, A. Gozzini, L. Moi, and G. Orriols, “An experimental method for the observation of r.f. transitions and laser beat resonances in oriented Na vapour,” Nuovo Cimento B 13, 5–20 (1976).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

S. Brandt, A. Nagel, R. Wynands, and D. Meschede, “Buffer-gas-induced linewidth reduction of coherent dark resonances to below 50 Hz,” Phys. Rev. A 56, R1063–R1066 (1997).
[CrossRef]

Prog. Opt. (1)

E. Arimondo, “Coherent population trapping in laser spectroscopy,” Prog. Opt. 35, 257–354 (1996).
[CrossRef]

Other (1)

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1989).

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Figures (4)

Fig. 1
Fig. 1

Some examples of FM line shapes for several ratios r of modulation frequency to full width of the Lorentzian absorption–dispersion profile: a, r=0.1; b, r=0.5; c, r=5. All spectra are shown to the same scale.

Fig. 2
Fig. 2

a, b, Simulated FM–AM spectra for a perfectly Lorentzian dispersion and absorption profile of width Δω. Parameters are k=100, ωm=(20/3)Δω, M=1, φ=0°, R=0.1, and ψ=25°. c, d, Reconstructed line profiles. e–n, Reconstructions for incorrect parameter values, showing the sensitivity to each parameter and its characteristic signature in the reconstructed spectra: e, f, φ=2°; g, h, ψ=0°; i, j, R=0.2; k, l, φ=2°, ψ=0°, R=0.2; m, n, k=99.

Fig. 3
Fig. 3

a, b, Experimental spectra of coherent population-trapping resonance in cesium vapor, c, d, -67° phase-shifted spectra; e, f, the completed reconstruction with M=1, R=0.1, ψ=-40°. The circles in f mark small spurious features that are due to the propagation of experimental noise through the numerical inversion procedure.

Fig. 4
Fig. 4

When Ci (Si) sits on a constant offset signal, the reconstructed spectrum δi (ϕi) shows characteristic steps of length k that approximate a linear (quadratic) background because δi is similar to the first integral of Ci and ϕi is similar to the second integral of Si.

Equations (18)

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E(t)=E0[1+R sin(ωmt+ψ)]cos(ωt+M sin ωmt).
ωinst=d(ωt+M sin ωmt)dt=ω+Mωm cos ωmt,
E(t)=E0[1+R sin(ωmt+ψ)]ν=- Jν(M)×exp(ω+νωm)t+c.c.,
E(t)=ν aν exp(ω+νωm)t+c.c.,
aν=Jν(M)-iR2Jν-1(M)exp(iψ)+iR2Jν+1(M)exp(-iψ).
Iphotoexp[-2δ0(ω)][S(ω)sin ωmt+C(ω)cos ωmt].
S(ω)=J0(M)J1(M)(ϕ+1+ϕ-1-2ϕ0)+J1(M)J2(M)(ϕ+2+ϕ-2-ϕ+1-ϕ-1)+(1/2)RJ02(M)[-(ϕ-1-ϕ+1)sin ψ+(2+2δ0-δ+1-δ-1)cos ψ]+(1/2)RJ12(M)[-(ϕ-2-ϕ+2)sin ψ+(4+6δ0-2δ+1-2δ-1-δ+2-δ-2)cos ψ],
C(ω)=J0(M)J1(M)(δ-1-δ+1)+J1(M)J2(M)(δ-2-δ+2+δ-1-δ+1)+(1/2)RJ02(M)[(2+2δ0-δ+1-δ-1)sin ψ+(ϕ-1-ϕ+1)cos ψ]+(1/2)RJ12(M)×[(4+6δ0-2δ+1-2δ-1-δ+2-δ-2)×sin ψ+(ϕ-2-ϕ+2)cos ψ].
X(ω)=η exp[-2δ0(ω)][S(ω)cos φ-C(ω)sin φ],
Y(ω)=η exp[-2δ0(ω)][S(ω)sin φ+C(ω)cos φ].
Iiexp(-2δi)(Si sin ωmt+Ci cos ωmt),
Si=J0(M)J1(M)(ϕi+k+ϕi-k-2ϕi)+J1(M)J2(M)(ϕi+2k+ϕi-2k-ϕi+k-ϕi-k)+(1/2)RJ02(M)[-(ϕi-k-ϕi+k)sin ψ+(2+2δi-δi+k-δi-k)cos ψ]+(1/2)RJ12(M)[-(ϕi-2k-ϕi+2k)sin ψ+(4+6δi-2δi+k-2δi-k-δi+2k-δi-2k)cos ψ],
Si=Xi cos φ+Yi sin φ,
Ci=-Xi sin φ+Yi cos φ.
IX=ηR[J02(M)+2J12(M)]cos(ψ+φ)1ni Xi,
IY=ηR[J02(M)+2J12(M)]sin(ψ+φ)1ni Yi.
ψ=arctanIYIX-φ,
R=IX2+IY2η[J02(M)+2J12(M)].

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