Abstract

We propose a method to simultaneously measure the center frequency of a spectral feature and the frequency linewidth of the feature. The method relies on dual frequency modulation of a carrier frequency, which probes the spectral feature, and phase sensitive detection of the transmitted signal at the two modulation frequencies. The detected signals provide two servo-stabilization signals for frequency control of the carrier frequency to the resonance line center and one of the modulation frequencies to the resonance linewidth.

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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. J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, “Optical heterodyne saturation spectroscopy,” Appl. Phys. Lett. 39, 680–682 (1981).
    [CrossRef]
  3. G. C. Bjorklund, M. D. Levenson, W. Lenth, and C. Ortiz, “Frequency modulation (FM) spectroscopy. theory of lineshapes and signal-to-noise analysis,” Appl. Phys. B 32, 145–152 (1983).
    [CrossRef]
  4. D. S. Bomse, A. C. Stanton, and J. A. Silver, “Frequency modulation and wavelength modulation spectroscopies: comparison of experimental methods using a lead-salt diode laser,” Appl. Opt. 31, 718–731 (1992).
    [CrossRef] [PubMed]
  5. J. A. Silver, “Frequency-modulation spectroscopy for trace species detection: theory and comparison among experimental methods,” Appl. Opt. 31, 707–717 (1992).
    [CrossRef] [PubMed]
  6. J. M. Supplee, E. A. Whittaker, and W. Lenth, “Theoretical description of frequency modulation and wavelength modulation spectroscopy,” Appl. Opt. 33, 6294–6302 (1994).
    [CrossRef] [PubMed]
  7. G. R. Janik, C. B. Carlisle, and T. F. Gallagher, “Two-tone frequency-modulation spectroscopy,” J. Opt. Soc. Am. B 3, 1070–1074 (1986).
    [CrossRef]
  8. R. G. DeVoe and R. G. Brewer, “Laser-frequency division and stabilization,” Phys. Rev. A 30, 2827–2829 (1984).
    [CrossRef]
  9. P. Courteille, L. S. Ma, W. Neuhauser, and R. Blatt, “Frequency measurement of Te1302 resonances near 467nm,” Appl. Phys. B 59, 187–193 (1994).
    [CrossRef]
  10. R. W. P. Drever, J. L. Hall, F. W. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
    [CrossRef]
  11. K. K. Lehmann and D. Romanini, “The superposition principle and cavity ring-down spectroscopy,” J. Chem. Phys. 105, 10263–10277 (1996).
    [CrossRef]
  12. A. O’Keefe and D. A. G. Deacon, “Cavity ring-down optical spectrometer for absorption measurements using pulsed laser sources,” Rev. Sci. Instrum. 59, 2544–2551 (1988).
    [CrossRef]
  13. M. D. Levenson, B. A. Paldus, T. G. Spence, C. C. Harb, J. S. Harris Jr., and R. N. Zare, “Optical heterodyne detection in cavity ring-down spectroscopy,” Chem. Phys. Lett. 290, 335–340 (1998).
    [CrossRef]
  14. M. D. Wheeler, S. M. Newman, A. J. Orr-Ewing, and M. N. R. Ashfold, “Cavity ring-down spectroscopy,” J. Chem. Soc. Faraday Trans. 94, 337–351 (1998).
    [CrossRef]
  15. P. Zalicki and R. N. Zare, “Cavity ring-down spectroscopy for quantitative absorption measurements,” J. Chem. Phys. 102, 2708 (1995).
    [CrossRef]

1998 (2)

M. D. Levenson, B. A. Paldus, T. G. Spence, C. C. Harb, J. S. Harris Jr., and R. N. Zare, “Optical heterodyne detection in cavity ring-down spectroscopy,” Chem. Phys. Lett. 290, 335–340 (1998).
[CrossRef]

M. D. Wheeler, S. M. Newman, A. J. Orr-Ewing, and M. N. R. Ashfold, “Cavity ring-down spectroscopy,” J. Chem. Soc. Faraday Trans. 94, 337–351 (1998).
[CrossRef]

1996 (1)

K. K. Lehmann and D. Romanini, “The superposition principle and cavity ring-down spectroscopy,” J. Chem. Phys. 105, 10263–10277 (1996).
[CrossRef]

1995 (1)

P. Zalicki and R. N. Zare, “Cavity ring-down spectroscopy for quantitative absorption measurements,” J. Chem. Phys. 102, 2708 (1995).
[CrossRef]

1994 (2)

P. Courteille, L. S. Ma, W. Neuhauser, and R. Blatt, “Frequency measurement of Te1302 resonances near 467nm,” Appl. Phys. B 59, 187–193 (1994).
[CrossRef]

J. M. Supplee, E. A. Whittaker, and W. Lenth, “Theoretical description of frequency modulation and wavelength modulation spectroscopy,” Appl. Opt. 33, 6294–6302 (1994).
[CrossRef] [PubMed]

1992 (2)

1988 (1)

A. O’Keefe and D. A. G. Deacon, “Cavity ring-down optical spectrometer for absorption measurements using pulsed laser sources,” Rev. Sci. Instrum. 59, 2544–2551 (1988).
[CrossRef]

1986 (1)

1984 (1)

R. G. DeVoe and R. G. Brewer, “Laser-frequency division and stabilization,” Phys. Rev. A 30, 2827–2829 (1984).
[CrossRef]

1983 (2)

G. C. Bjorklund, M. D. Levenson, W. Lenth, and C. Ortiz, “Frequency modulation (FM) spectroscopy. theory of lineshapes and signal-to-noise analysis,” Appl. Phys. B 32, 145–152 (1983).
[CrossRef]

R. W. P. Drever, J. L. Hall, F. W. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

1981 (1)

J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, “Optical heterodyne saturation spectroscopy,” Appl. Phys. Lett. 39, 680–682 (1981).
[CrossRef]

1980 (1)

Ashfold, M. N. R.

M. D. Wheeler, S. M. Newman, A. J. Orr-Ewing, and M. N. R. Ashfold, “Cavity ring-down spectroscopy,” J. Chem. Soc. Faraday Trans. 94, 337–351 (1998).
[CrossRef]

Baer, T.

J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, “Optical heterodyne saturation spectroscopy,” Appl. Phys. Lett. 39, 680–682 (1981).
[CrossRef]

Bjorklund, G. C.

G. C. Bjorklund, M. D. Levenson, W. Lenth, and C. Ortiz, “Frequency modulation (FM) spectroscopy. theory of lineshapes and signal-to-noise analysis,” Appl. Phys. B 32, 145–152 (1983).
[CrossRef]

G. C. Bjorklund, “Frequency-modulation spectroscopy: a new method for measuring weak absorptions and dispersions,” Opt. Lett. 5, 15–17 (1980).
[CrossRef] [PubMed]

Blatt, R.

P. Courteille, L. S. Ma, W. Neuhauser, and R. Blatt, “Frequency measurement of Te1302 resonances near 467nm,” Appl. Phys. B 59, 187–193 (1994).
[CrossRef]

Bomse, D. S.

Brewer, R. G.

R. G. DeVoe and R. G. Brewer, “Laser-frequency division and stabilization,” Phys. Rev. A 30, 2827–2829 (1984).
[CrossRef]

Carlisle, C. B.

Courteille, P.

P. Courteille, L. S. Ma, W. Neuhauser, and R. Blatt, “Frequency measurement of Te1302 resonances near 467nm,” Appl. Phys. B 59, 187–193 (1994).
[CrossRef]

Deacon, D. A. G.

A. O’Keefe and D. A. G. Deacon, “Cavity ring-down optical spectrometer for absorption measurements using pulsed laser sources,” Rev. Sci. Instrum. 59, 2544–2551 (1988).
[CrossRef]

DeVoe, R. G.

R. G. DeVoe and R. G. Brewer, “Laser-frequency division and stabilization,” Phys. Rev. A 30, 2827–2829 (1984).
[CrossRef]

Drever, R. W. P.

R. W. P. Drever, J. L. Hall, F. W. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Ford, G. M.

R. W. P. Drever, J. L. Hall, F. W. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Gallagher, T. F.

Hall, J. L.

R. W. P. Drever, J. L. Hall, F. W. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, “Optical heterodyne saturation spectroscopy,” Appl. Phys. Lett. 39, 680–682 (1981).
[CrossRef]

Harb, C. C.

M. D. Levenson, B. A. Paldus, T. G. Spence, C. C. Harb, J. S. Harris Jr., and R. N. Zare, “Optical heterodyne detection in cavity ring-down spectroscopy,” Chem. Phys. Lett. 290, 335–340 (1998).
[CrossRef]

Harris, J. S.

M. D. Levenson, B. A. Paldus, T. G. Spence, C. C. Harb, J. S. Harris Jr., and R. N. Zare, “Optical heterodyne detection in cavity ring-down spectroscopy,” Chem. Phys. Lett. 290, 335–340 (1998).
[CrossRef]

Hollberg, L.

J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, “Optical heterodyne saturation spectroscopy,” Appl. Phys. Lett. 39, 680–682 (1981).
[CrossRef]

Hough, J.

R. W. P. Drever, J. L. Hall, F. W. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Janik, G. R.

Kowalski, F. W.

R. W. P. Drever, J. L. Hall, F. W. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Lehmann, K. K.

K. K. Lehmann and D. Romanini, “The superposition principle and cavity ring-down spectroscopy,” J. Chem. Phys. 105, 10263–10277 (1996).
[CrossRef]

Lenth, W.

J. M. Supplee, E. A. Whittaker, and W. Lenth, “Theoretical description of frequency modulation and wavelength modulation spectroscopy,” Appl. Opt. 33, 6294–6302 (1994).
[CrossRef] [PubMed]

G. C. Bjorklund, M. D. Levenson, W. Lenth, and C. Ortiz, “Frequency modulation (FM) spectroscopy. theory of lineshapes and signal-to-noise analysis,” Appl. Phys. B 32, 145–152 (1983).
[CrossRef]

Levenson, M. D.

M. D. Levenson, B. A. Paldus, T. G. Spence, C. C. Harb, J. S. Harris Jr., and R. N. Zare, “Optical heterodyne detection in cavity ring-down spectroscopy,” Chem. Phys. Lett. 290, 335–340 (1998).
[CrossRef]

G. C. Bjorklund, M. D. Levenson, W. Lenth, and C. Ortiz, “Frequency modulation (FM) spectroscopy. theory of lineshapes and signal-to-noise analysis,” Appl. Phys. B 32, 145–152 (1983).
[CrossRef]

Ma, L. S.

P. Courteille, L. S. Ma, W. Neuhauser, and R. Blatt, “Frequency measurement of Te1302 resonances near 467nm,” Appl. Phys. B 59, 187–193 (1994).
[CrossRef]

Munley, A. J.

R. W. P. Drever, J. L. Hall, F. W. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Neuhauser, W.

P. Courteille, L. S. Ma, W. Neuhauser, and R. Blatt, “Frequency measurement of Te1302 resonances near 467nm,” Appl. Phys. B 59, 187–193 (1994).
[CrossRef]

Newman, S. M.

M. D. Wheeler, S. M. Newman, A. J. Orr-Ewing, and M. N. R. Ashfold, “Cavity ring-down spectroscopy,” J. Chem. Soc. Faraday Trans. 94, 337–351 (1998).
[CrossRef]

O’Keefe, A.

A. O’Keefe and D. A. G. Deacon, “Cavity ring-down optical spectrometer for absorption measurements using pulsed laser sources,” Rev. Sci. Instrum. 59, 2544–2551 (1988).
[CrossRef]

Orr-Ewing, A. J.

M. D. Wheeler, S. M. Newman, A. J. Orr-Ewing, and M. N. R. Ashfold, “Cavity ring-down spectroscopy,” J. Chem. Soc. Faraday Trans. 94, 337–351 (1998).
[CrossRef]

Ortiz, C.

G. C. Bjorklund, M. D. Levenson, W. Lenth, and C. Ortiz, “Frequency modulation (FM) spectroscopy. theory of lineshapes and signal-to-noise analysis,” Appl. Phys. B 32, 145–152 (1983).
[CrossRef]

Paldus, B. A.

M. D. Levenson, B. A. Paldus, T. G. Spence, C. C. Harb, J. S. Harris Jr., and R. N. Zare, “Optical heterodyne detection in cavity ring-down spectroscopy,” Chem. Phys. Lett. 290, 335–340 (1998).
[CrossRef]

Robinson, H. G.

J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, “Optical heterodyne saturation spectroscopy,” Appl. Phys. Lett. 39, 680–682 (1981).
[CrossRef]

Romanini, D.

K. K. Lehmann and D. Romanini, “The superposition principle and cavity ring-down spectroscopy,” J. Chem. Phys. 105, 10263–10277 (1996).
[CrossRef]

Silver, J. A.

Spence, T. G.

M. D. Levenson, B. A. Paldus, T. G. Spence, C. C. Harb, J. S. Harris Jr., and R. N. Zare, “Optical heterodyne detection in cavity ring-down spectroscopy,” Chem. Phys. Lett. 290, 335–340 (1998).
[CrossRef]

Stanton, A. C.

Supplee, J. M.

Ward, H.

R. W. P. Drever, J. L. Hall, F. W. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Wheeler, M. D.

M. D. Wheeler, S. M. Newman, A. J. Orr-Ewing, and M. N. R. Ashfold, “Cavity ring-down spectroscopy,” J. Chem. Soc. Faraday Trans. 94, 337–351 (1998).
[CrossRef]

Whittaker, E. A.

Zalicki, P.

P. Zalicki and R. N. Zare, “Cavity ring-down spectroscopy for quantitative absorption measurements,” J. Chem. Phys. 102, 2708 (1995).
[CrossRef]

Zare, R. N.

M. D. Levenson, B. A. Paldus, T. G. Spence, C. C. Harb, J. S. Harris Jr., and R. N. Zare, “Optical heterodyne detection in cavity ring-down spectroscopy,” Chem. Phys. Lett. 290, 335–340 (1998).
[CrossRef]

P. Zalicki and R. N. Zare, “Cavity ring-down spectroscopy for quantitative absorption measurements,” J. Chem. Phys. 102, 2708 (1995).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. B (3)

G. C. Bjorklund, M. D. Levenson, W. Lenth, and C. Ortiz, “Frequency modulation (FM) spectroscopy. theory of lineshapes and signal-to-noise analysis,” Appl. Phys. B 32, 145–152 (1983).
[CrossRef]

P. Courteille, L. S. Ma, W. Neuhauser, and R. Blatt, “Frequency measurement of Te1302 resonances near 467nm,” Appl. Phys. B 59, 187–193 (1994).
[CrossRef]

R. W. P. Drever, J. L. Hall, F. W. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Appl. Phys. Lett. (1)

J. L. Hall, L. Hollberg, T. Baer, and H. G. Robinson, “Optical heterodyne saturation spectroscopy,” Appl. Phys. Lett. 39, 680–682 (1981).
[CrossRef]

Chem. Phys. Lett. (1)

M. D. Levenson, B. A. Paldus, T. G. Spence, C. C. Harb, J. S. Harris Jr., and R. N. Zare, “Optical heterodyne detection in cavity ring-down spectroscopy,” Chem. Phys. Lett. 290, 335–340 (1998).
[CrossRef]

J. Chem. Phys. (2)

P. Zalicki and R. N. Zare, “Cavity ring-down spectroscopy for quantitative absorption measurements,” J. Chem. Phys. 102, 2708 (1995).
[CrossRef]

K. K. Lehmann and D. Romanini, “The superposition principle and cavity ring-down spectroscopy,” J. Chem. Phys. 105, 10263–10277 (1996).
[CrossRef]

J. Chem. Soc. Faraday Trans. (1)

M. D. Wheeler, S. M. Newman, A. J. Orr-Ewing, and M. N. R. Ashfold, “Cavity ring-down spectroscopy,” J. Chem. Soc. Faraday Trans. 94, 337–351 (1998).
[CrossRef]

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

Opt. Lett. (1)

Phys. Rev. A (1)

R. G. DeVoe and R. G. Brewer, “Laser-frequency division and stabilization,” Phys. Rev. A 30, 2827–2829 (1984).
[CrossRef]

Rev. Sci. Instrum. (1)

A. O’Keefe and D. A. G. Deacon, “Cavity ring-down optical spectrometer for absorption measurements using pulsed laser sources,” Rev. Sci. Instrum. 59, 2544–2551 (1988).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Schematic of standard FM signal detected at the first harmonic of the modulation frequency. As the modulated laser spectrum is scanned over the absorption profile, in-phase detection at the modulation frequency produces the FM signals shown. The frequency scales of the different parts are not identical.

Fig. 2
Fig. 2

Schematic of FM signal detected at the second harmonic of the modulation frequency. As the modulated laser spectrum is scanned over the absorption profile, in-phase detection at the modulation frequency produces the FM signals shown. The frequency scales of the different parts are not identical.

Fig. 3
Fig. 3

Schematic of DFM spectroscopy. (a) Output spectrum of a laser that is modulated at frequency ω 1 , which is itself modulated at frequency ω 2 . (b) In-phase signal detected at the second harmonic of the second modulation frequency as the modulated laser spectrum is scanned over the absorption profile.

Fig. 4
Fig. 4

Dual FM spectra obtained by in-phase detection at the second harmonic of the second modulation frequency, as the laser is scanned over the resonance. Each spectrum corresponds to a different value of the first modulation frequency, given in terms of the Lorentzian half-width Γ. The special case ω 1 = 0.577 Γ is where the center of the spectrum exhibits a zero crossing. For each spectrum, the frequency scale is identical.

Fig. 5
Fig. 5

Signal strength of the second harmonic of the second modulation frequency, as the value of the first modulation frequency is varied, while the laser frequency is held equal to the Lorentzian resonance line center. The solid curve corresponds to in-phase detection, and the dashed curve represents quadrature-phase detection.

Fig. 6
Fig. 6

Dependence of the zero-crossing point of the in-phase signal (see Fig. 5) on the second modulation frequency ω 2 . The filled circles are the results of a numerical calculation, and the solid curve is an approximation to the results.

Fig. 7
Fig. 7

Signal strength of the in-phase second harmonic of the second modulation frequency, as the value of the first modulation frequency is varied, while the laser frequency is held equal to the Gaussian resonance line center.

Fig. 8
Fig. 8

General schematic of DFM spectroscopy. FM and DFM servo control signals stabilize ω to the resonance ω 0 and ω 1 to the linewidth.

Equations (33)

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

E in = E 0 e i ω t ,
E in = E 0 e i ( ω t + α sin ( ω 1 t ) ) ,
E in = E 0 e i ω t m = J m ( α ) e i m ω 1 t ,
E in = E 0 { α 2 e i ( ω + ω 1 ) t + e i ω t α 2 e i ( ω ω 1 ) t } .
E out = E 0 { T 1 α 2 e i ( ω + ω 1 ) t + T 0 e i ω t T 1 α 2 e i ( ω ω 1 ) t } ,
T m = exp ( δ m i ϕ m ) ,
T m 1 δ m i ϕ m .
| E out | 2 = E 0 2 exp ( 2 δ 0 ) { 1 α ( δ 1 δ 1 ) cos ω 1 t + α ( ϕ 1 2 ϕ 0 + ϕ 1 ) sin ω 1 t } ,
d δ ( ω ) d ω δ 1 δ 1 2 ω 1 ,
d 2 ϕ ( ω ) d ω 2 ϕ 1 2 ϕ 0 + ϕ 1 ω 1 2 .
| E out | 2 E 0 2 exp ( 2 δ 0 ) { 1 2 α ω 1 ( d δ d ω ) cos ω 1 t + α ω 1 2 ( d 2 ϕ d ω 2 ) sin ω 1 t } .
E in = E 0 { α 2 8 e i ( ω + 2 ω 1 ) t + α 2 e i ( ω + ω 1 ) t + e i ω t α 2 e i ( ω ω 1 ) t + α 2 8 e i ( ω 2 ω 1 ) t } .
E out = E 0 { T 2 α 2 8 e i ( ω + 2 ω 1 ) t + T 1 α 2 e i ( ω + ω 1 ) t + T 0 e i ω t T 1 α 2 e i ( ω ω 1 ) t + T 2 α 2 8 e i ( ω 2 ω 1 ) t } .
| E out | 2 ω 1 2 = E 0 2 exp ( 2 δ 0 ) α 2 4 { ( δ 2 + 2 δ 1 2 δ 0 + 2 δ 1 δ 2 ) cos 2 ω 1 t + ( ϕ 2 2 ϕ 1 + 2 ϕ 1 ϕ 2 ) sin 2 ω 1 t } .
δ 2 + 2 δ 1 2 δ 0 + 2 δ 1 δ 2 = ( δ 2 2 δ 1 + δ 0 ) ( δ 0 2 δ 1 + δ 2 ) ω 1 2 d 2 δ ( ω ) d ω 2 | ω + ω 1 ω 1 2 d 2 δ ( ω ) d ω 2 | ω ω 1 ,
δ 2 + 2 δ 1 2 δ 0 + 2 δ 1 δ 2 2 ω 1 2 d 2 δ ( ω ) d ω 2 | ω .
ϕ 2 2 ϕ 1 + 2 ϕ 1 ϕ 2 = ( ϕ 2 2 ϕ 1 + ϕ 0 ) ( ϕ 0 2 ϕ 1 + ϕ 2 ) ω 1 2 d 2 ϕ ( ω ) d ω 2 | ω + ω 1 ω 1 2 d 2 ϕ ( ω ) d ω 2 | ω ω 1 ,
ϕ 2 2 ϕ 1 + 2 ϕ 1 ϕ 2 2 ω 1 3 d 3 ϕ ( ω ) d ω 3 | ω .
| E out | 2 ω 1 2 E 0 2 exp ( 2 δ 0 ) α 2 4 { 2 ω 1 2 d 2 δ ( ω ) d ω 2 | ω cos 2 ω 1 t + 2 ω 1 3 d 3 ϕ ( ω ) d ω 3 | ω sin 2 ω 1 t } .
E in = E 0 e i ( ω t + α sin ( ω 1 t + β sin ( ω 2 t ) ) ) ,
E in = E 0 e i ω t m , n = J m ( α ) J n ( m β ) e i m ω 1 t e i n ω 2 t .
E in = E 0 { α 2 e i ( ω + ω 1 ) t [ β 2 8 e i 2 ω 2 t + β 2 e i ω 2 t + 1 β 2 e i ω 2 t + β 2 8 e i 2 ω 2 t ] + e i ω t α 2 e i ( ω ω 1 ) t [ β 2 8 e i 2 ω 2 t β 2 e i ω 2 t + 1 + β 2 e i ω 2 t + β 2 8 e i 2 ω 2 t ] } .
E out = E 0 { α 2 e i ( ω + ω 1 ) t [ T 1 2 β 2 8 e i 2 ω 2 t + T 1 1 β 2 e i ω 2 t + T 1 0 T 1 1 β 2 e i ω 2 t + T 1 2 β 2 8 e i 2 ω 2 t ] + T 0 0 e i ω t α 2 e i ( ω ω 1 ) t [ T 1 2 β 2 8 e i 2 ω 2 t T 1 1 β 2 e i ω 2 t + T 1 0 + T 1 1 β 2 e i ω 2 t + T 1 2 β 2 8 e i 2 ω 2 t ] } ,
| E out | 2 ω 2 2 E 0 2 exp ( 2 δ 0 ) α 2 4 β 2 4 { 2 ω 2 2 ( d 2 δ ( ω ) d ω 2 | ω + ω 1 + d 2 δ ( ω ) d ω 2 | ω ω 1 ) cos 2 ω 2 t + 2 ω 2 3 ( d 3 ϕ ( ω ) d ω 3 | ω + ω 1 + d 3 ϕ ( ω ) d ω 3 | ω ω 1 ) sin 2 ω 2 t .
| E out | cos 2 ω 2 t 2 ( d 2 δ ( ω ) d ω 2 | ω 0 + ω 1 + d 2 δ ( ω ) d ω 2 | ω 0 ω 1 ) .
δ ( ω ) = A 2 Γ 2 ( ω ω 0 ) 2 + Γ 2 ,
ϕ ( ω ) = A 2 Γ ( ω ω 0 ) ( ω ω 0 ) 2 + Γ 2 .
d 2 δ ( ω ) d ω 2 = A Γ 2 [ ( ω ω 0 ) 2 + Γ 2 ] 3 [ 3 ( ω ω 0 ) 2 Γ 2 ] .
d 3 ϕ ( ω ) d ω 3 = 3 A Γ [ ( ω ω 0 ) 2 + Γ 2 ] 4 [ ( ω ω 0 ) 4 6 ( ω ω 0 ) 2 Γ 2 + Γ 4 ] .
| E out | 2 ω 2 2 E 0 2 exp ( 2 δ 0 ) α 2 4 β 2 4 { 4 ω 2 2 A Γ 2 [ ω 1 2 + Γ 2 ] 3 [ Γ 2 3 ω 1 2 ] cos 2 ω 2 t + 12 ω 2 3 A Γ [ ω 1 2 + Γ 2 ] 4 [ ω 1 4 6 ω 1 2 Γ 2 + Γ 4 ] sin 2 ω 2 t } .
δ ( ω ) = A 2 e ( ω ω 0 ) 2 Γ 2 .
d 2 δ ( ω ) d ω 2 = A Γ 4 e ( ω ω 0 ) 2 Γ 2 [ 2 ( ω ω 0 ) 2 Γ 2 ] .
| E out | cos 2 ω 2 t 2 E 0 2 exp ( 2 δ 0 ) α 2 4 β 2 4 4 ω 2 2 A Γ 4 e ω 1 2 Γ 2 [ Γ 2 2 ω 1 2 ] ,

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