Abstract

To evaluate the uncertainty of concentration measurement using cavity ringdown spectroscopy, we analytically derived expressions for uncertainty for parameters, such as temperature, laser frequency, and ringdown time deviation, from the model equation. The uncertainties that are due to systematic errors in a practical cavity ringdown system were assessed through an experimental study of the PQ(35) transition in an A band of molecular oxygen. We found that, except for the line strength that is regarded as a reference value independent of the measurement, the laser frequency jitter is the largest uncertainty source in the system. Some practical requirements for minimizing the uncertainty in concentration measurements are discussed. We also demonstrated determination of the line strength of the PQ(35) transition line of oxygen to be 8.63(3) × 10-27 cm-1 with a relative uncertainty of less than 0.4%.

© 2001 Optical Society of America

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References

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  1. D. J. Santeler, “Gas dynamics in residual gas analyzer calibration,” J. Vac. Sci. Technol. A 5, 129–133 (1987).
    [CrossRef]
  2. L. Lieszkovszky, A. R. Filippelli, C. R. Tilford, “Metrological characteristics of a group of quadrupole partial pressure analyzers,” J. Vac. Sci. Technol. A 8, 3838–3854 (1990).
    [CrossRef]
  3. W. Meienburg, H. Neckel, J. Wolfrum, “In situ measurement of ammonia with a 13CO2–waveguide laser system,” Appl. Phys. B 51, 94–98 (1990).
    [CrossRef]
  4. S. Cheskis, “Quantitative measurements of absolute concentrations of intermediate species in flames,” Prog. Energy Combust. Sci. 25, 233–252 (1999).
    [CrossRef]
  5. A. O’Keefe, D. A. Deacon, “Cavity ringdown optical spectrometer for absorption measurements in an optical interferometer,” Rev. Sci. Instrum. 59, 2544–2551 (1988).
    [CrossRef]
  6. J. J. Scherer, J. B. Paul, A. O’Keefe, R. J. Saykally, “Cavity ringdown laser absorption spectroscopy: history, development, and application to pulsed molecular beams,” Chem. Rev. 97, 25–51 (1997).
    [CrossRef] [PubMed]
  7. J. J. Scherer, D. J. Rakestraw, “Cavity ringdown laser absorption spectroscopy detection of formyl (HCO) radial in a low pressure flame,” Chem. Phys. Lett. 265, 169–176 (1997).
    [CrossRef]
  8. X. Mercier, P. Jamette, J. F. Pauwels, P. Desgroux, “Absolute CH concentration measurements by cavity ringdown spectroscopy in an atmospheric diffusion flame,” Chem. Phys. Lett. 305, 334–342 (1999).
    [CrossRef]
  9. J. P. Booth, G. Cunge, L. Biennier, D. Romanini, A. Kachanov, “Ultraviolet cavity ringdown spectroscopy of free radicals in etching plasmas,” Chem. Phys. Lett. 317, 631–636 (2000).
    [CrossRef]
  10. M. Hippler, M. Quack, “Cw cavity ringdown infrared absorption spectroscopy in pulsed supersonic jets: nitrous oxide and methane,” Chem. Phys. Lett. 314, 273–281 (1999).
    [CrossRef]
  11. F. Mayinger, Optical Measurements (Springer-Verlag, Berlin, 1994), pp. 273–287.
  12. R. D. van Zee, J. T. Hodges, J. P. Looney, “Pulsed, single-mode cavity ringdown spectroscopy,” Appl. Opt. 38, 3951–3960 (1999).
    [CrossRef]
  13. D. Romanini, K. K. Lehmann, “Ringdown cavity absorption spectroscopy of the very weak HCN overtone bands with six, seven, and eight stretching quanta,” J. Chem. Phys. 99, 6287–6301 (1993).
    [CrossRef]
  14. J. Y. Lee, H.-W. Lee, J. W. Hahn, “Time domain study on cavity ringdown signals from a Fabry–Perot cavity under pulsed laser excitations,” Jpn. J. Appl. Phys. 38, 6287–6297 (1999).
    [CrossRef]
  15. K. J. Ritter, T. D. Wilkerson, “High-resolution spectroscopy of the oxygen A-band,” J. Mol. Spectrosc. 121, 1–19 (1987).
    [CrossRef]
  16. J. W. Hahn, Y. S. Yoo, J. Y. Lee, J. W. Kim, H.-W. Lee, “Cavity ringdown spectroscopy with a continuous-wave laser: calculation of coupling efficiency and a new spectrometer design,” Appl. Opt. 38, 1859–1866 (1999).
    [CrossRef]
  17. L. R. Brown, C. Plymate, “Experimental line parameters of the oxygen A band at 760 nm,” J. Mol. Spectrosc. 199, 166–179 (2000).
    [CrossRef] [PubMed]
  18. The software of HITRAN96 was developed under contract F19628-91-C-0132 for the Phillips Laboratory Geophysics Directorate under the direction of Laurence Rothman, Ontar Corporation, 9 Village Way, North Andover, Mass. 01845.

2000 (2)

J. P. Booth, G. Cunge, L. Biennier, D. Romanini, A. Kachanov, “Ultraviolet cavity ringdown spectroscopy of free radicals in etching plasmas,” Chem. Phys. Lett. 317, 631–636 (2000).
[CrossRef]

L. R. Brown, C. Plymate, “Experimental line parameters of the oxygen A band at 760 nm,” J. Mol. Spectrosc. 199, 166–179 (2000).
[CrossRef] [PubMed]

1999 (6)

R. D. van Zee, J. T. Hodges, J. P. Looney, “Pulsed, single-mode cavity ringdown spectroscopy,” Appl. Opt. 38, 3951–3960 (1999).
[CrossRef]

J. W. Hahn, Y. S. Yoo, J. Y. Lee, J. W. Kim, H.-W. Lee, “Cavity ringdown spectroscopy with a continuous-wave laser: calculation of coupling efficiency and a new spectrometer design,” Appl. Opt. 38, 1859–1866 (1999).
[CrossRef]

M. Hippler, M. Quack, “Cw cavity ringdown infrared absorption spectroscopy in pulsed supersonic jets: nitrous oxide and methane,” Chem. Phys. Lett. 314, 273–281 (1999).
[CrossRef]

J. Y. Lee, H.-W. Lee, J. W. Hahn, “Time domain study on cavity ringdown signals from a Fabry–Perot cavity under pulsed laser excitations,” Jpn. J. Appl. Phys. 38, 6287–6297 (1999).
[CrossRef]

S. Cheskis, “Quantitative measurements of absolute concentrations of intermediate species in flames,” Prog. Energy Combust. Sci. 25, 233–252 (1999).
[CrossRef]

X. Mercier, P. Jamette, J. F. Pauwels, P. Desgroux, “Absolute CH concentration measurements by cavity ringdown spectroscopy in an atmospheric diffusion flame,” Chem. Phys. Lett. 305, 334–342 (1999).
[CrossRef]

1997 (2)

J. J. Scherer, J. B. Paul, A. O’Keefe, R. J. Saykally, “Cavity ringdown laser absorption spectroscopy: history, development, and application to pulsed molecular beams,” Chem. Rev. 97, 25–51 (1997).
[CrossRef] [PubMed]

J. J. Scherer, D. J. Rakestraw, “Cavity ringdown laser absorption spectroscopy detection of formyl (HCO) radial in a low pressure flame,” Chem. Phys. Lett. 265, 169–176 (1997).
[CrossRef]

1993 (1)

D. Romanini, K. K. Lehmann, “Ringdown cavity absorption spectroscopy of the very weak HCN overtone bands with six, seven, and eight stretching quanta,” J. Chem. Phys. 99, 6287–6301 (1993).
[CrossRef]

1990 (2)

L. Lieszkovszky, A. R. Filippelli, C. R. Tilford, “Metrological characteristics of a group of quadrupole partial pressure analyzers,” J. Vac. Sci. Technol. A 8, 3838–3854 (1990).
[CrossRef]

W. Meienburg, H. Neckel, J. Wolfrum, “In situ measurement of ammonia with a 13CO2–waveguide laser system,” Appl. Phys. B 51, 94–98 (1990).
[CrossRef]

1988 (1)

A. O’Keefe, D. A. Deacon, “Cavity ringdown optical spectrometer for absorption measurements in an optical interferometer,” Rev. Sci. Instrum. 59, 2544–2551 (1988).
[CrossRef]

1987 (2)

D. J. Santeler, “Gas dynamics in residual gas analyzer calibration,” J. Vac. Sci. Technol. A 5, 129–133 (1987).
[CrossRef]

K. J. Ritter, T. D. Wilkerson, “High-resolution spectroscopy of the oxygen A-band,” J. Mol. Spectrosc. 121, 1–19 (1987).
[CrossRef]

Biennier, L.

J. P. Booth, G. Cunge, L. Biennier, D. Romanini, A. Kachanov, “Ultraviolet cavity ringdown spectroscopy of free radicals in etching plasmas,” Chem. Phys. Lett. 317, 631–636 (2000).
[CrossRef]

Booth, J. P.

J. P. Booth, G. Cunge, L. Biennier, D. Romanini, A. Kachanov, “Ultraviolet cavity ringdown spectroscopy of free radicals in etching plasmas,” Chem. Phys. Lett. 317, 631–636 (2000).
[CrossRef]

Brown, L. R.

L. R. Brown, C. Plymate, “Experimental line parameters of the oxygen A band at 760 nm,” J. Mol. Spectrosc. 199, 166–179 (2000).
[CrossRef] [PubMed]

Cheskis, S.

S. Cheskis, “Quantitative measurements of absolute concentrations of intermediate species in flames,” Prog. Energy Combust. Sci. 25, 233–252 (1999).
[CrossRef]

Cunge, G.

J. P. Booth, G. Cunge, L. Biennier, D. Romanini, A. Kachanov, “Ultraviolet cavity ringdown spectroscopy of free radicals in etching plasmas,” Chem. Phys. Lett. 317, 631–636 (2000).
[CrossRef]

Deacon, D. A.

A. O’Keefe, D. A. Deacon, “Cavity ringdown optical spectrometer for absorption measurements in an optical interferometer,” Rev. Sci. Instrum. 59, 2544–2551 (1988).
[CrossRef]

Desgroux, P.

X. Mercier, P. Jamette, J. F. Pauwels, P. Desgroux, “Absolute CH concentration measurements by cavity ringdown spectroscopy in an atmospheric diffusion flame,” Chem. Phys. Lett. 305, 334–342 (1999).
[CrossRef]

Filippelli, A. R.

L. Lieszkovszky, A. R. Filippelli, C. R. Tilford, “Metrological characteristics of a group of quadrupole partial pressure analyzers,” J. Vac. Sci. Technol. A 8, 3838–3854 (1990).
[CrossRef]

Hahn, J. W.

J. Y. Lee, H.-W. Lee, J. W. Hahn, “Time domain study on cavity ringdown signals from a Fabry–Perot cavity under pulsed laser excitations,” Jpn. J. Appl. Phys. 38, 6287–6297 (1999).
[CrossRef]

J. W. Hahn, Y. S. Yoo, J. Y. Lee, J. W. Kim, H.-W. Lee, “Cavity ringdown spectroscopy with a continuous-wave laser: calculation of coupling efficiency and a new spectrometer design,” Appl. Opt. 38, 1859–1866 (1999).
[CrossRef]

Hippler, M.

M. Hippler, M. Quack, “Cw cavity ringdown infrared absorption spectroscopy in pulsed supersonic jets: nitrous oxide and methane,” Chem. Phys. Lett. 314, 273–281 (1999).
[CrossRef]

Hodges, J. T.

Jamette, P.

X. Mercier, P. Jamette, J. F. Pauwels, P. Desgroux, “Absolute CH concentration measurements by cavity ringdown spectroscopy in an atmospheric diffusion flame,” Chem. Phys. Lett. 305, 334–342 (1999).
[CrossRef]

Kachanov, A.

J. P. Booth, G. Cunge, L. Biennier, D. Romanini, A. Kachanov, “Ultraviolet cavity ringdown spectroscopy of free radicals in etching plasmas,” Chem. Phys. Lett. 317, 631–636 (2000).
[CrossRef]

Kim, J. W.

Lee, H.-W.

J. W. Hahn, Y. S. Yoo, J. Y. Lee, J. W. Kim, H.-W. Lee, “Cavity ringdown spectroscopy with a continuous-wave laser: calculation of coupling efficiency and a new spectrometer design,” Appl. Opt. 38, 1859–1866 (1999).
[CrossRef]

J. Y. Lee, H.-W. Lee, J. W. Hahn, “Time domain study on cavity ringdown signals from a Fabry–Perot cavity under pulsed laser excitations,” Jpn. J. Appl. Phys. 38, 6287–6297 (1999).
[CrossRef]

Lee, J. Y.

J. Y. Lee, H.-W. Lee, J. W. Hahn, “Time domain study on cavity ringdown signals from a Fabry–Perot cavity under pulsed laser excitations,” Jpn. J. Appl. Phys. 38, 6287–6297 (1999).
[CrossRef]

J. W. Hahn, Y. S. Yoo, J. Y. Lee, J. W. Kim, H.-W. Lee, “Cavity ringdown spectroscopy with a continuous-wave laser: calculation of coupling efficiency and a new spectrometer design,” Appl. Opt. 38, 1859–1866 (1999).
[CrossRef]

Lehmann, K. K.

D. Romanini, K. K. Lehmann, “Ringdown cavity absorption spectroscopy of the very weak HCN overtone bands with six, seven, and eight stretching quanta,” J. Chem. Phys. 99, 6287–6301 (1993).
[CrossRef]

Lieszkovszky, L.

L. Lieszkovszky, A. R. Filippelli, C. R. Tilford, “Metrological characteristics of a group of quadrupole partial pressure analyzers,” J. Vac. Sci. Technol. A 8, 3838–3854 (1990).
[CrossRef]

Looney, J. P.

Mayinger, F.

F. Mayinger, Optical Measurements (Springer-Verlag, Berlin, 1994), pp. 273–287.

Meienburg, W.

W. Meienburg, H. Neckel, J. Wolfrum, “In situ measurement of ammonia with a 13CO2–waveguide laser system,” Appl. Phys. B 51, 94–98 (1990).
[CrossRef]

Mercier, X.

X. Mercier, P. Jamette, J. F. Pauwels, P. Desgroux, “Absolute CH concentration measurements by cavity ringdown spectroscopy in an atmospheric diffusion flame,” Chem. Phys. Lett. 305, 334–342 (1999).
[CrossRef]

Neckel, H.

W. Meienburg, H. Neckel, J. Wolfrum, “In situ measurement of ammonia with a 13CO2–waveguide laser system,” Appl. Phys. B 51, 94–98 (1990).
[CrossRef]

O’Keefe, A.

J. J. Scherer, J. B. Paul, A. O’Keefe, R. J. Saykally, “Cavity ringdown laser absorption spectroscopy: history, development, and application to pulsed molecular beams,” Chem. Rev. 97, 25–51 (1997).
[CrossRef] [PubMed]

A. O’Keefe, D. A. Deacon, “Cavity ringdown optical spectrometer for absorption measurements in an optical interferometer,” Rev. Sci. Instrum. 59, 2544–2551 (1988).
[CrossRef]

Paul, J. B.

J. J. Scherer, J. B. Paul, A. O’Keefe, R. J. Saykally, “Cavity ringdown laser absorption spectroscopy: history, development, and application to pulsed molecular beams,” Chem. Rev. 97, 25–51 (1997).
[CrossRef] [PubMed]

Pauwels, J. F.

X. Mercier, P. Jamette, J. F. Pauwels, P. Desgroux, “Absolute CH concentration measurements by cavity ringdown spectroscopy in an atmospheric diffusion flame,” Chem. Phys. Lett. 305, 334–342 (1999).
[CrossRef]

Plymate, C.

L. R. Brown, C. Plymate, “Experimental line parameters of the oxygen A band at 760 nm,” J. Mol. Spectrosc. 199, 166–179 (2000).
[CrossRef] [PubMed]

Quack, M.

M. Hippler, M. Quack, “Cw cavity ringdown infrared absorption spectroscopy in pulsed supersonic jets: nitrous oxide and methane,” Chem. Phys. Lett. 314, 273–281 (1999).
[CrossRef]

Rakestraw, D. J.

J. J. Scherer, D. J. Rakestraw, “Cavity ringdown laser absorption spectroscopy detection of formyl (HCO) radial in a low pressure flame,” Chem. Phys. Lett. 265, 169–176 (1997).
[CrossRef]

Ritter, K. J.

K. J. Ritter, T. D. Wilkerson, “High-resolution spectroscopy of the oxygen A-band,” J. Mol. Spectrosc. 121, 1–19 (1987).
[CrossRef]

Romanini, D.

J. P. Booth, G. Cunge, L. Biennier, D. Romanini, A. Kachanov, “Ultraviolet cavity ringdown spectroscopy of free radicals in etching plasmas,” Chem. Phys. Lett. 317, 631–636 (2000).
[CrossRef]

D. Romanini, K. K. Lehmann, “Ringdown cavity absorption spectroscopy of the very weak HCN overtone bands with six, seven, and eight stretching quanta,” J. Chem. Phys. 99, 6287–6301 (1993).
[CrossRef]

Santeler, D. J.

D. J. Santeler, “Gas dynamics in residual gas analyzer calibration,” J. Vac. Sci. Technol. A 5, 129–133 (1987).
[CrossRef]

Saykally, R. J.

J. J. Scherer, J. B. Paul, A. O’Keefe, R. J. Saykally, “Cavity ringdown laser absorption spectroscopy: history, development, and application to pulsed molecular beams,” Chem. Rev. 97, 25–51 (1997).
[CrossRef] [PubMed]

Scherer, J. J.

J. J. Scherer, D. J. Rakestraw, “Cavity ringdown laser absorption spectroscopy detection of formyl (HCO) radial in a low pressure flame,” Chem. Phys. Lett. 265, 169–176 (1997).
[CrossRef]

J. J. Scherer, J. B. Paul, A. O’Keefe, R. J. Saykally, “Cavity ringdown laser absorption spectroscopy: history, development, and application to pulsed molecular beams,” Chem. Rev. 97, 25–51 (1997).
[CrossRef] [PubMed]

Tilford, C. R.

L. Lieszkovszky, A. R. Filippelli, C. R. Tilford, “Metrological characteristics of a group of quadrupole partial pressure analyzers,” J. Vac. Sci. Technol. A 8, 3838–3854 (1990).
[CrossRef]

van Zee, R. D.

Wilkerson, T. D.

K. J. Ritter, T. D. Wilkerson, “High-resolution spectroscopy of the oxygen A-band,” J. Mol. Spectrosc. 121, 1–19 (1987).
[CrossRef]

Wolfrum, J.

W. Meienburg, H. Neckel, J. Wolfrum, “In situ measurement of ammonia with a 13CO2–waveguide laser system,” Appl. Phys. B 51, 94–98 (1990).
[CrossRef]

Yoo, Y. S.

Appl. Opt. (2)

Appl. Phys. B (1)

W. Meienburg, H. Neckel, J. Wolfrum, “In situ measurement of ammonia with a 13CO2–waveguide laser system,” Appl. Phys. B 51, 94–98 (1990).
[CrossRef]

Chem. Phys. Lett. (4)

J. J. Scherer, D. J. Rakestraw, “Cavity ringdown laser absorption spectroscopy detection of formyl (HCO) radial in a low pressure flame,” Chem. Phys. Lett. 265, 169–176 (1997).
[CrossRef]

X. Mercier, P. Jamette, J. F. Pauwels, P. Desgroux, “Absolute CH concentration measurements by cavity ringdown spectroscopy in an atmospheric diffusion flame,” Chem. Phys. Lett. 305, 334–342 (1999).
[CrossRef]

J. P. Booth, G. Cunge, L. Biennier, D. Romanini, A. Kachanov, “Ultraviolet cavity ringdown spectroscopy of free radicals in etching plasmas,” Chem. Phys. Lett. 317, 631–636 (2000).
[CrossRef]

M. Hippler, M. Quack, “Cw cavity ringdown infrared absorption spectroscopy in pulsed supersonic jets: nitrous oxide and methane,” Chem. Phys. Lett. 314, 273–281 (1999).
[CrossRef]

Chem. Rev. (1)

J. J. Scherer, J. B. Paul, A. O’Keefe, R. J. Saykally, “Cavity ringdown laser absorption spectroscopy: history, development, and application to pulsed molecular beams,” Chem. Rev. 97, 25–51 (1997).
[CrossRef] [PubMed]

J. Chem. Phys. (1)

D. Romanini, K. K. Lehmann, “Ringdown cavity absorption spectroscopy of the very weak HCN overtone bands with six, seven, and eight stretching quanta,” J. Chem. Phys. 99, 6287–6301 (1993).
[CrossRef]

J. Mol. Spectrosc. (2)

K. J. Ritter, T. D. Wilkerson, “High-resolution spectroscopy of the oxygen A-band,” J. Mol. Spectrosc. 121, 1–19 (1987).
[CrossRef]

L. R. Brown, C. Plymate, “Experimental line parameters of the oxygen A band at 760 nm,” J. Mol. Spectrosc. 199, 166–179 (2000).
[CrossRef] [PubMed]

J. Vac. Sci. Technol. A (2)

D. J. Santeler, “Gas dynamics in residual gas analyzer calibration,” J. Vac. Sci. Technol. A 5, 129–133 (1987).
[CrossRef]

L. Lieszkovszky, A. R. Filippelli, C. R. Tilford, “Metrological characteristics of a group of quadrupole partial pressure analyzers,” J. Vac. Sci. Technol. A 8, 3838–3854 (1990).
[CrossRef]

Jpn. J. Appl. Phys. (1)

J. Y. Lee, H.-W. Lee, J. W. Hahn, “Time domain study on cavity ringdown signals from a Fabry–Perot cavity under pulsed laser excitations,” Jpn. J. Appl. Phys. 38, 6287–6297 (1999).
[CrossRef]

Prog. Energy Combust. Sci. (1)

S. Cheskis, “Quantitative measurements of absolute concentrations of intermediate species in flames,” Prog. Energy Combust. Sci. 25, 233–252 (1999).
[CrossRef]

Rev. Sci. Instrum. (1)

A. O’Keefe, D. A. Deacon, “Cavity ringdown optical spectrometer for absorption measurements in an optical interferometer,” Rev. Sci. Instrum. 59, 2544–2551 (1988).
[CrossRef]

Other (2)

F. Mayinger, Optical Measurements (Springer-Verlag, Berlin, 1994), pp. 273–287.

The software of HITRAN96 was developed under contract F19628-91-C-0132 for the Phillips Laboratory Geophysics Directorate under the direction of Laurence Rothman, Ontar Corporation, 9 Village Way, North Andover, Mass. 01845.

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Figures (6)

Fig. 1
Fig. 1

Temperature dependence of the line intensity ρS of the PQ(35) transition in the oxygen A band. Ground energy of PQ(35) is 1803.2 cm-1: solid circles, experimental data; solid line, calculation from Eq. (6).

Fig. 2
Fig. 2

Schematic setup of a cw CRDS: D, diaphragm.

Fig. 3
Fig. 3

Spectrum of the PQ(35) transition and the fit of the Galatry profile to the data. The pressure was 3.0 × 104 Pa. The best-fit parameters suggest a center frequency of 12,966.8093 cm-1, ρS of 0.622 cm-1, a Doppler width of 0.0146 cm-1, and a collision-broadened width of 0.009 cm-1.

Fig. 4
Fig. 4

Measured line intensities at several pressure readings from the pressure gauge.

Fig. 5
Fig. 5

Center frequency and collision-broadened linewidth: □, center frequency; ●, collision-broadened linewidth of the PQ(35) transition as a function of oxygen pressure; dotted and solid lines, least-squares fits of the measured data. From the slopes of the two straight lines the pressure shift and the broadening coefficient are determined as 0.0068 and 0.029 cm-1/atm, respectively.

Fig. 6
Fig. 6

(a) Ringdown signal trace of an evacuated cavity and the fit to an exponential function. The ringdown time results in 50.78(3) µs. (b) Residuals (solid line) superimposed on the shot-noise level (dotted line) of the ringdown signal for comparison.

Tables (2)

Tables Icon

Table 1 Relative Uncertainties in Concentration Measurements from Several Sources

Tables Icon

Table 2 Experimentally Attained Relative Uncertainties of Concentration

Equations (14)

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

α=tr2l1τ-1τ0,
αi=ρST1π gvi,
ρ=MST,
M=πi αii gvi.
uρ2ρ2=uM2M2+uST2ST2,
uM2M2=π i uαi2i αi2+πi gvi2×igvivi2 uvi2.
i=0NNF0Fdv.
uααshot=2egηV01τ1/2,
uααconst=VnoiseV02fs1τ1/2,
π i uαi2i αi2=2ln 2π1/21NW131/2uαp2αp2+12-13uαV2αV2 for a Gaussian,2 1π1NW38uαp2αp2+18uαV2αV2 for a Lorentzian,
πi gvi2igvivi2 uvi2=π ln 231/2NWuv2vD2π16NWuv2vD2
ST=ST0QT0QTexp-hcEJk1T-1T0,
uST2=ST/ST02uST02+ST/T2uT2.
uST2ST2=uST02ST02+CT2uT2T2,

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