Abstract

Allan variance has been used to characterize the slow drift of a near-IR distributed feedback laser-based continuous wave cavity ringdown spectroscopy (CW-CRDS) system. Long-term drift in the cavity loss rate, highly correlated with changes in ambient pressure but not temperature, is observed. With differential measurement of on- and off-peak decay rates, the drift between them largely cancels out, but some residual drift remains if the lasers are detuned more than a few hundred megahertz from each other. A sensitivity to bulk cavity loss (1σ) of 4.4×1012cm1 has been obtained during an optimum integration time of 30  min with our CW-CRDS setup, which corresponds to the methane detection limit (3σ) in N2 of 0.24 parts in 109 by volume (ppbv) at 20Torr or 29 parts in 1012 by volume (pptv) at 760Torr pressure. The stability of our system is demonstrated by measuring sub-ppbv methane in N2 at 760Torr through recording the spectrum of methane lines with our setup.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  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]
  2. K.Busch and M.Busch, eds., Cavity Ringdown Spectroscopy--An Ultratrace-Absorption Measurement Technique, ACS Symposium Series (American Chemical Society, 1999), Vol. 720.
    [CrossRef]
  3. P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for Physical Sciences, 2nd ed. (McGraw-Hill, 1992).
  4. K. K. Lehmann and H. Huang, “Optimal signal processing in cavity ring-down spectroscopy,” in Frontiers of Molecular Spectroscopy, J. Laane, ed. (Elsevier, 2008). Chap. 18, pp. 623-657.
  5. D. W. Allan, “Statistics of atomic frequency standards,” Proc. IEEE 54, 221-230 (1966).
    [CrossRef]
  6. H. Huang and K. K. Lehmann, “Noise in cavity ring-down spectroscopy caused by transverse mode coupling,” Opt. Express 15, 8745-8759 (2007).
    [CrossRef] [PubMed]
  7. P. Werle, R. Mucke, and F. Slemr, “The limits of signal averaging in atmospheric trace-gas monitoring by tunable diode-laser absorption spectroscopy (TDLAS),” Appl. Phys. B 57, 131-139 (1993).
    [CrossRef]
  8. D. D. Nelson, B. McManusa, S. Urbanski, S. Herndon, and M. S. Zahniser, “High precision measurements of atmospheric nitrous oxide and methane using thermoelectrically cooled mid-infrared quantum cascade lasers and detectors,” Spectrochim. Acta A 60, 3325-3335 (2004).
    [CrossRef]
  9. D. V. Land, A. P. Levick, and J. W. Hand, “The use of Allan deviation for the measurement of the noise and drift performance of microwave radiometers,” Meas. Sci. Technol. 18, 1917-1928 (2007).
    [CrossRef]
  10. J. M. Langridge, S. M. Ball, A. J. L. Shillings, and R. L. Jones, “A broadband absorption spectrometer using light emitting diodes for ultrasensitive, in situ trace gas detection,” Rev. Sci. Instrum. 79, 123110 (2008).
    [CrossRef]
  11. T. Wu, W. Zhao, W. Chen, W. Zhang, and X. Gao, “Incoherent broadband cavity enhanced absorption spectroscopy for in situ measurements of NO2 with a blue light emitting diode,” Appl. Phys. B 94, 85-94 (2009).
    [CrossRef]
  12. J. B. Dudek, P. B. Tarsa, A. Velasquez, M. Wladyslawski, P. Rabinowitz, and K. K. Lehmann, “Trace moisture detection using continuous-wave cavity ring-down spectroscopy,” Anal. Chem. 75, 4599-4605 (2003).
    [CrossRef] [PubMed]
  13. H. Huang and K. K. Lehmann, “Effects of birefringence and polarization dependent loss of supermirrors in cavity ring-down spectroscopy,” Appl. Opt. 47, 3817-3827 (2008).
    [CrossRef] [PubMed]
  14. H. Huang and K. K. Lehmann, “Noise caused by finite extinction ratio of light modulator in CW cavity ring-down spectroscopy,” Appl. Phys. B 94, 355-366 (2009).
    [CrossRef]
  15. We obtained this information from Hamamatsu Photonics.
  16. L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, B. M., B. L. R., M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, “The HITRAN 2004 molecular spectroscopic database,” J. Quant. Spectr. Rad. Trans. 96, 139-204 (2005).
    [CrossRef]

2009 (2)

T. Wu, W. Zhao, W. Chen, W. Zhang, and X. Gao, “Incoherent broadband cavity enhanced absorption spectroscopy for in situ measurements of NO2 with a blue light emitting diode,” Appl. Phys. B 94, 85-94 (2009).
[CrossRef]

H. Huang and K. K. Lehmann, “Noise caused by finite extinction ratio of light modulator in CW cavity ring-down spectroscopy,” Appl. Phys. B 94, 355-366 (2009).
[CrossRef]

2008 (2)

J. M. Langridge, S. M. Ball, A. J. L. Shillings, and R. L. Jones, “A broadband absorption spectrometer using light emitting diodes for ultrasensitive, in situ trace gas detection,” Rev. Sci. Instrum. 79, 123110 (2008).
[CrossRef]

H. Huang and K. K. Lehmann, “Effects of birefringence and polarization dependent loss of supermirrors in cavity ring-down spectroscopy,” Appl. Opt. 47, 3817-3827 (2008).
[CrossRef] [PubMed]

2007 (2)

D. V. Land, A. P. Levick, and J. W. Hand, “The use of Allan deviation for the measurement of the noise and drift performance of microwave radiometers,” Meas. Sci. Technol. 18, 1917-1928 (2007).
[CrossRef]

H. Huang and K. K. Lehmann, “Noise in cavity ring-down spectroscopy caused by transverse mode coupling,” Opt. Express 15, 8745-8759 (2007).
[CrossRef] [PubMed]

2005 (1)

L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, B. M., B. L. R., M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, “The HITRAN 2004 molecular spectroscopic database,” J. Quant. Spectr. Rad. Trans. 96, 139-204 (2005).
[CrossRef]

2004 (1)

D. D. Nelson, B. McManusa, S. Urbanski, S. Herndon, and M. S. Zahniser, “High precision measurements of atmospheric nitrous oxide and methane using thermoelectrically cooled mid-infrared quantum cascade lasers and detectors,” Spectrochim. Acta A 60, 3325-3335 (2004).
[CrossRef]

2003 (1)

J. B. Dudek, P. B. Tarsa, A. Velasquez, M. Wladyslawski, P. Rabinowitz, and K. K. Lehmann, “Trace moisture detection using continuous-wave cavity ring-down spectroscopy,” Anal. Chem. 75, 4599-4605 (2003).
[CrossRef] [PubMed]

1993 (1)

P. Werle, R. Mucke, and F. Slemr, “The limits of signal averaging in atmospheric trace-gas monitoring by tunable diode-laser absorption spectroscopy (TDLAS),” Appl. Phys. B 57, 131-139 (1993).
[CrossRef]

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]

1966 (1)

D. W. Allan, “Statistics of atomic frequency standards,” Proc. IEEE 54, 221-230 (1966).
[CrossRef]

Allan, D. W.

D. W. Allan, “Statistics of atomic frequency standards,” Proc. IEEE 54, 221-230 (1966).
[CrossRef]

Ball, S. M.

J. M. Langridge, S. M. Ball, A. J. L. Shillings, and R. L. Jones, “A broadband absorption spectrometer using light emitting diodes for ultrasensitive, in situ trace gas detection,” Rev. Sci. Instrum. 79, 123110 (2008).
[CrossRef]

Barbe, A.

L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, B. M., B. L. R., M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, “The HITRAN 2004 molecular spectroscopic database,” J. Quant. Spectr. Rad. Trans. 96, 139-204 (2005).
[CrossRef]

Benner, D. C.

L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, B. M., B. L. R., M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, “The HITRAN 2004 molecular spectroscopic database,” J. Quant. Spectr. Rad. Trans. 96, 139-204 (2005).
[CrossRef]

Bevington, P. R.

P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for Physical Sciences, 2nd ed. (McGraw-Hill, 1992).

Carleer, M. R.

L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, B. M., B. L. R., M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, “The HITRAN 2004 molecular spectroscopic database,” J. Quant. Spectr. Rad. Trans. 96, 139-204 (2005).
[CrossRef]

Chackerian, C.

L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, B. M., B. L. R., M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, “The HITRAN 2004 molecular spectroscopic database,” J. Quant. Spectr. Rad. Trans. 96, 139-204 (2005).
[CrossRef]

Chance, K.

L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, B. M., B. L. R., M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, “The HITRAN 2004 molecular spectroscopic database,” J. Quant. Spectr. Rad. Trans. 96, 139-204 (2005).
[CrossRef]

Chen, W.

T. Wu, W. Zhao, W. Chen, W. Zhang, and X. Gao, “Incoherent broadband cavity enhanced absorption spectroscopy for in situ measurements of NO2 with a blue light emitting diode,” Appl. Phys. B 94, 85-94 (2009).
[CrossRef]

Coudert, L. H.

L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, B. M., B. L. R., M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, “The HITRAN 2004 molecular spectroscopic database,” J. Quant. Spectr. Rad. Trans. 96, 139-204 (2005).
[CrossRef]

Dana, V.

L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, B. M., B. L. R., M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, “The HITRAN 2004 molecular spectroscopic database,” J. Quant. Spectr. Rad. Trans. 96, 139-204 (2005).
[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]

Devi, V. M.

L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, B. M., B. L. R., M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, “The HITRAN 2004 molecular spectroscopic database,” J. Quant. Spectr. Rad. Trans. 96, 139-204 (2005).
[CrossRef]

Dudek, J. B.

J. B. Dudek, P. B. Tarsa, A. Velasquez, M. Wladyslawski, P. Rabinowitz, and K. K. Lehmann, “Trace moisture detection using continuous-wave cavity ring-down spectroscopy,” Anal. Chem. 75, 4599-4605 (2003).
[CrossRef] [PubMed]

Flaud, J. M.

L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, B. M., B. L. R., M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, “The HITRAN 2004 molecular spectroscopic database,” J. Quant. Spectr. Rad. Trans. 96, 139-204 (2005).
[CrossRef]

Gamache, R. R.

L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, B. M., B. L. R., M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, “The HITRAN 2004 molecular spectroscopic database,” J. Quant. Spectr. Rad. Trans. 96, 139-204 (2005).
[CrossRef]

Gao, X.

T. Wu, W. Zhao, W. Chen, W. Zhang, and X. Gao, “Incoherent broadband cavity enhanced absorption spectroscopy for in situ measurements of NO2 with a blue light emitting diode,” Appl. Phys. B 94, 85-94 (2009).
[CrossRef]

Goldman, A.

L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, B. M., B. L. R., M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, “The HITRAN 2004 molecular spectroscopic database,” J. Quant. Spectr. Rad. Trans. 96, 139-204 (2005).
[CrossRef]

Hand, J. W.

D. V. Land, A. P. Levick, and J. W. Hand, “The use of Allan deviation for the measurement of the noise and drift performance of microwave radiometers,” Meas. Sci. Technol. 18, 1917-1928 (2007).
[CrossRef]

Hartmann, J. M.

L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, B. M., B. L. R., M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, “The HITRAN 2004 molecular spectroscopic database,” J. Quant. Spectr. Rad. Trans. 96, 139-204 (2005).
[CrossRef]

Herndon, S.

D. D. Nelson, B. McManusa, S. Urbanski, S. Herndon, and M. S. Zahniser, “High precision measurements of atmospheric nitrous oxide and methane using thermoelectrically cooled mid-infrared quantum cascade lasers and detectors,” Spectrochim. Acta A 60, 3325-3335 (2004).
[CrossRef]

Huang, H.

H. Huang and K. K. Lehmann, “Noise caused by finite extinction ratio of light modulator in CW cavity ring-down spectroscopy,” Appl. Phys. B 94, 355-366 (2009).
[CrossRef]

H. Huang and K. K. Lehmann, “Effects of birefringence and polarization dependent loss of supermirrors in cavity ring-down spectroscopy,” Appl. Opt. 47, 3817-3827 (2008).
[CrossRef] [PubMed]

H. Huang and K. K. Lehmann, “Noise in cavity ring-down spectroscopy caused by transverse mode coupling,” Opt. Express 15, 8745-8759 (2007).
[CrossRef] [PubMed]

K. K. Lehmann and H. Huang, “Optimal signal processing in cavity ring-down spectroscopy,” in Frontiers of Molecular Spectroscopy, J. Laane, ed. (Elsevier, 2008). Chap. 18, pp. 623-657.

Jacquemart, D.

L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, B. M., B. L. R., M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, “The HITRAN 2004 molecular spectroscopic database,” J. Quant. Spectr. Rad. Trans. 96, 139-204 (2005).
[CrossRef]

Jones, R. L.

J. M. Langridge, S. M. Ball, A. J. L. Shillings, and R. L. Jones, “A broadband absorption spectrometer using light emitting diodes for ultrasensitive, in situ trace gas detection,” Rev. Sci. Instrum. 79, 123110 (2008).
[CrossRef]

Jucks, K. W.

L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, B. M., B. L. R., M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, “The HITRAN 2004 molecular spectroscopic database,” J. Quant. Spectr. Rad. Trans. 96, 139-204 (2005).
[CrossRef]

Land, D. V.

D. V. Land, A. P. Levick, and J. W. Hand, “The use of Allan deviation for the measurement of the noise and drift performance of microwave radiometers,” Meas. Sci. Technol. 18, 1917-1928 (2007).
[CrossRef]

Langridge, J. M.

J. M. Langridge, S. M. Ball, A. J. L. Shillings, and R. L. Jones, “A broadband absorption spectrometer using light emitting diodes for ultrasensitive, in situ trace gas detection,” Rev. Sci. Instrum. 79, 123110 (2008).
[CrossRef]

Lehmann, K. K.

H. Huang and K. K. Lehmann, “Noise caused by finite extinction ratio of light modulator in CW cavity ring-down spectroscopy,” Appl. Phys. B 94, 355-366 (2009).
[CrossRef]

H. Huang and K. K. Lehmann, “Effects of birefringence and polarization dependent loss of supermirrors in cavity ring-down spectroscopy,” Appl. Opt. 47, 3817-3827 (2008).
[CrossRef] [PubMed]

H. Huang and K. K. Lehmann, “Noise in cavity ring-down spectroscopy caused by transverse mode coupling,” Opt. Express 15, 8745-8759 (2007).
[CrossRef] [PubMed]

J. B. Dudek, P. B. Tarsa, A. Velasquez, M. Wladyslawski, P. Rabinowitz, and K. K. Lehmann, “Trace moisture detection using continuous-wave cavity ring-down spectroscopy,” Anal. Chem. 75, 4599-4605 (2003).
[CrossRef] [PubMed]

K. K. Lehmann and H. Huang, “Optimal signal processing in cavity ring-down spectroscopy,” in Frontiers of Molecular Spectroscopy, J. Laane, ed. (Elsevier, 2008). Chap. 18, pp. 623-657.

Levick, A. P.

D. V. Land, A. P. Levick, and J. W. Hand, “The use of Allan deviation for the measurement of the noise and drift performance of microwave radiometers,” Meas. Sci. Technol. 18, 1917-1928 (2007).
[CrossRef]

Maki, A. G.

L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, B. M., B. L. R., M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, “The HITRAN 2004 molecular spectroscopic database,” J. Quant. Spectr. Rad. Trans. 96, 139-204 (2005).
[CrossRef]

Mandin, J. Y.

L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, B. M., B. L. R., M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, “The HITRAN 2004 molecular spectroscopic database,” J. Quant. Spectr. Rad. Trans. 96, 139-204 (2005).
[CrossRef]

Massie, S. T.

L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, B. M., B. L. R., M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, “The HITRAN 2004 molecular spectroscopic database,” J. Quant. Spectr. Rad. Trans. 96, 139-204 (2005).
[CrossRef]

McManusa, B.

D. D. Nelson, B. McManusa, S. Urbanski, S. Herndon, and M. S. Zahniser, “High precision measurements of atmospheric nitrous oxide and methane using thermoelectrically cooled mid-infrared quantum cascade lasers and detectors,” Spectrochim. Acta A 60, 3325-3335 (2004).
[CrossRef]

Mucke, R.

P. Werle, R. Mucke, and F. Slemr, “The limits of signal averaging in atmospheric trace-gas monitoring by tunable diode-laser absorption spectroscopy (TDLAS),” Appl. Phys. B 57, 131-139 (1993).
[CrossRef]

Nelson, D. D.

D. D. Nelson, B. McManusa, S. Urbanski, S. Herndon, and M. S. Zahniser, “High precision measurements of atmospheric nitrous oxide and methane using thermoelectrically cooled mid-infrared quantum cascade lasers and detectors,” Spectrochim. Acta A 60, 3325-3335 (2004).
[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]

Orphal, J.

L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, B. M., B. L. R., M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, “The HITRAN 2004 molecular spectroscopic database,” J. Quant. Spectr. Rad. Trans. 96, 139-204 (2005).
[CrossRef]

Perrin, A.

L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, B. M., B. L. R., M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, “The HITRAN 2004 molecular spectroscopic database,” J. Quant. Spectr. Rad. Trans. 96, 139-204 (2005).
[CrossRef]

Rabinowitz, P.

J. B. Dudek, P. B. Tarsa, A. Velasquez, M. Wladyslawski, P. Rabinowitz, and K. K. Lehmann, “Trace moisture detection using continuous-wave cavity ring-down spectroscopy,” Anal. Chem. 75, 4599-4605 (2003).
[CrossRef] [PubMed]

Rinsland, C. P.

L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, B. M., B. L. R., M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, “The HITRAN 2004 molecular spectroscopic database,” J. Quant. Spectr. Rad. Trans. 96, 139-204 (2005).
[CrossRef]

Robinson, D. K.

P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for Physical Sciences, 2nd ed. (McGraw-Hill, 1992).

Rothman, L. S.

L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, B. M., B. L. R., M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, “The HITRAN 2004 molecular spectroscopic database,” J. Quant. Spectr. Rad. Trans. 96, 139-204 (2005).
[CrossRef]

Shillings, A. J. L.

J. M. Langridge, S. M. Ball, A. J. L. Shillings, and R. L. Jones, “A broadband absorption spectrometer using light emitting diodes for ultrasensitive, in situ trace gas detection,” Rev. Sci. Instrum. 79, 123110 (2008).
[CrossRef]

Slemr, F.

P. Werle, R. Mucke, and F. Slemr, “The limits of signal averaging in atmospheric trace-gas monitoring by tunable diode-laser absorption spectroscopy (TDLAS),” Appl. Phys. B 57, 131-139 (1993).
[CrossRef]

Smith, M. A. H.

L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, B. M., B. L. R., M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, “The HITRAN 2004 molecular spectroscopic database,” J. Quant. Spectr. Rad. Trans. 96, 139-204 (2005).
[CrossRef]

Tarsa, P. B.

J. B. Dudek, P. B. Tarsa, A. Velasquez, M. Wladyslawski, P. Rabinowitz, and K. K. Lehmann, “Trace moisture detection using continuous-wave cavity ring-down spectroscopy,” Anal. Chem. 75, 4599-4605 (2003).
[CrossRef] [PubMed]

Tennyson, J.

L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, B. M., B. L. R., M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, “The HITRAN 2004 molecular spectroscopic database,” J. Quant. Spectr. Rad. Trans. 96, 139-204 (2005).
[CrossRef]

Tolchenov, R. N.

L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, B. M., B. L. R., M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, “The HITRAN 2004 molecular spectroscopic database,” J. Quant. Spectr. Rad. Trans. 96, 139-204 (2005).
[CrossRef]

Toth, R. A.

L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, B. M., B. L. R., M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, “The HITRAN 2004 molecular spectroscopic database,” J. Quant. Spectr. Rad. Trans. 96, 139-204 (2005).
[CrossRef]

Urbanski, S.

D. D. Nelson, B. McManusa, S. Urbanski, S. Herndon, and M. S. Zahniser, “High precision measurements of atmospheric nitrous oxide and methane using thermoelectrically cooled mid-infrared quantum cascade lasers and detectors,” Spectrochim. Acta A 60, 3325-3335 (2004).
[CrossRef]

Vander Auwera, J.

L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, B. M., B. L. R., M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, “The HITRAN 2004 molecular spectroscopic database,” J. Quant. Spectr. Rad. Trans. 96, 139-204 (2005).
[CrossRef]

Varanasi, P.

L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, B. M., B. L. R., M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, “The HITRAN 2004 molecular spectroscopic database,” J. Quant. Spectr. Rad. Trans. 96, 139-204 (2005).
[CrossRef]

Velasquez, A.

J. B. Dudek, P. B. Tarsa, A. Velasquez, M. Wladyslawski, P. Rabinowitz, and K. K. Lehmann, “Trace moisture detection using continuous-wave cavity ring-down spectroscopy,” Anal. Chem. 75, 4599-4605 (2003).
[CrossRef] [PubMed]

Wagner, G.

L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, B. M., B. L. R., M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, “The HITRAN 2004 molecular spectroscopic database,” J. Quant. Spectr. Rad. Trans. 96, 139-204 (2005).
[CrossRef]

Werle, P.

P. Werle, R. Mucke, and F. Slemr, “The limits of signal averaging in atmospheric trace-gas monitoring by tunable diode-laser absorption spectroscopy (TDLAS),” Appl. Phys. B 57, 131-139 (1993).
[CrossRef]

Wladyslawski, M.

J. B. Dudek, P. B. Tarsa, A. Velasquez, M. Wladyslawski, P. Rabinowitz, and K. K. Lehmann, “Trace moisture detection using continuous-wave cavity ring-down spectroscopy,” Anal. Chem. 75, 4599-4605 (2003).
[CrossRef] [PubMed]

Wu, T.

T. Wu, W. Zhao, W. Chen, W. Zhang, and X. Gao, “Incoherent broadband cavity enhanced absorption spectroscopy for in situ measurements of NO2 with a blue light emitting diode,” Appl. Phys. B 94, 85-94 (2009).
[CrossRef]

Zahniser, M. S.

D. D. Nelson, B. McManusa, S. Urbanski, S. Herndon, and M. S. Zahniser, “High precision measurements of atmospheric nitrous oxide and methane using thermoelectrically cooled mid-infrared quantum cascade lasers and detectors,” Spectrochim. Acta A 60, 3325-3335 (2004).
[CrossRef]

Zhang, W.

T. Wu, W. Zhao, W. Chen, W. Zhang, and X. Gao, “Incoherent broadband cavity enhanced absorption spectroscopy for in situ measurements of NO2 with a blue light emitting diode,” Appl. Phys. B 94, 85-94 (2009).
[CrossRef]

Zhao, W.

T. Wu, W. Zhao, W. Chen, W. Zhang, and X. Gao, “Incoherent broadband cavity enhanced absorption spectroscopy for in situ measurements of NO2 with a blue light emitting diode,” Appl. Phys. B 94, 85-94 (2009).
[CrossRef]

Anal. Chem. (1)

J. B. Dudek, P. B. Tarsa, A. Velasquez, M. Wladyslawski, P. Rabinowitz, and K. K. Lehmann, “Trace moisture detection using continuous-wave cavity ring-down spectroscopy,” Anal. Chem. 75, 4599-4605 (2003).
[CrossRef] [PubMed]

Appl. Opt. (1)

Appl. Phys. B (3)

H. Huang and K. K. Lehmann, “Noise caused by finite extinction ratio of light modulator in CW cavity ring-down spectroscopy,” Appl. Phys. B 94, 355-366 (2009).
[CrossRef]

T. Wu, W. Zhao, W. Chen, W. Zhang, and X. Gao, “Incoherent broadband cavity enhanced absorption spectroscopy for in situ measurements of NO2 with a blue light emitting diode,” Appl. Phys. B 94, 85-94 (2009).
[CrossRef]

P. Werle, R. Mucke, and F. Slemr, “The limits of signal averaging in atmospheric trace-gas monitoring by tunable diode-laser absorption spectroscopy (TDLAS),” Appl. Phys. B 57, 131-139 (1993).
[CrossRef]

J. Quant. Spectr. Rad. Trans. (1)

L. S. Rothman, D. Jacquemart, A. Barbe, D. C. Benner, B. M., B. L. R., M. R. Carleer, C. Chackerian, K. Chance, L. H. Coudert, V. Dana, V. M. Devi, J. M. Flaud, R. R. Gamache, A. Goldman, J. M. Hartmann, K. W. Jucks, A. G. Maki, J. Y. Mandin, S. T. Massie, J. Orphal, A. Perrin, C. P. Rinsland, M. A. H. Smith, J. Tennyson, R. N. Tolchenov, R. A. Toth, J. Vander Auwera, P. Varanasi, and G. Wagner, “The HITRAN 2004 molecular spectroscopic database,” J. Quant. Spectr. Rad. Trans. 96, 139-204 (2005).
[CrossRef]

Meas. Sci. Technol. (1)

D. V. Land, A. P. Levick, and J. W. Hand, “The use of Allan deviation for the measurement of the noise and drift performance of microwave radiometers,” Meas. Sci. Technol. 18, 1917-1928 (2007).
[CrossRef]

Opt. Express (1)

Proc. IEEE (1)

D. W. Allan, “Statistics of atomic frequency standards,” Proc. IEEE 54, 221-230 (1966).
[CrossRef]

Rev. Sci. Instrum. (2)

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]

J. M. Langridge, S. M. Ball, A. J. L. Shillings, and R. L. Jones, “A broadband absorption spectrometer using light emitting diodes for ultrasensitive, in situ trace gas detection,” Rev. Sci. Instrum. 79, 123110 (2008).
[CrossRef]

Spectrochim. Acta A (1)

D. D. Nelson, B. McManusa, S. Urbanski, S. Herndon, and M. S. Zahniser, “High precision measurements of atmospheric nitrous oxide and methane using thermoelectrically cooled mid-infrared quantum cascade lasers and detectors,” Spectrochim. Acta A 60, 3325-3335 (2004).
[CrossRef]

Other (4)

K.Busch and M.Busch, eds., Cavity Ringdown Spectroscopy--An Ultratrace-Absorption Measurement Technique, ACS Symposium Series (American Chemical Society, 1999), Vol. 720.
[CrossRef]

P. R. Bevington and D. K. Robinson, Data Reduction and Error Analysis for Physical Sciences, 2nd ed. (McGraw-Hill, 1992).

K. K. Lehmann and H. Huang, “Optimal signal processing in cavity ring-down spectroscopy,” in Frontiers of Molecular Spectroscopy, J. Laane, ed. (Elsevier, 2008). Chap. 18, pp. 623-657.

We obtained this information from Hamamatsu Photonics.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Measured decay time constants τ 1 and τ 2 drift with time. The horizontal axis is data point indices, which can be converted into time if divided by the ringdown rate 4.9 Hz . τ 1 and τ 2 measure cavity loss with two diode lasers that are 23 GHz separated in the frequency.

Fig. 2
Fig. 2

Allan plots of two algorithms with the data in Fig. 1. The modified algorithm (thinner curves) generates smoother Allan plots. For clarity, the Allan plots have been separated for (A)  k 1 , (B)  k 2 , and (C)  k 1 k 2 .

Fig. 3
Fig. 3

Setup diagram: thick lines, laser beams; thin lines, signal transition; curves, optical fibers; OS, optical switch; C, collimator; FI, Faraday isolator; M, mirror; MML, mode matching lenses; AD, AOM driver; CCB, control circuit board; RDC, ringdown cavity; OPM, off-axis parabolic mirror; PZT, three piezo transducers; SM1, front supermirror; SM2, back supermirror; TS, trigger signal; D, detector; and PC, computer.

Fig. 4
Fig. 4

Spectrum of R4 lines of 2 ν 3 band (antisymmetric C H stretch) of methane in nitrogen, recorded at a total pressure of 15.71 Torr and room temperature. The curve is a simulated spectrum based on the HITRAN database [16]. Four line intensities are also shown in the figure. The measured methane concentration is 33 ppbv . The Doppler width of each line is about 563 MHz at 300 K .

Fig. 5
Fig. 5

Allan plots of four typical results. The first minimum of k 1 k 2 Allan plot is labeled by an arrow. (A) Allan plots of Δ ν = 20 MHz . The drift between k 1 and k 2 canceled out perfectly. (B) Allan plots of the data with optical feedback from the detector to the cavity mode. (C) Allan plots of the data in Fig. 1. The drift between k 1 and k 2 canceled out partially. (D) Drift experiment with 20   SCCM nitrogen gas flowing at 760 Torr in the cell. The single-channel ringdown rate is 6.9 Hz , 8.6 Hz , 4.9 Hz , and 4.7 Hz for (A), (B), (C), and (D), respectively.

Fig. 6
Fig. 6

Summary of drift experiments. Both (A) the final sensitivity δ α and (B) the optimum integration time depend on the frequency difference Δ ν .

Fig. 7
Fig. 7

(A) Lab pressure change in Baratron voltage readings ( 3 V correspond to 760 Torr ) during the experiment of Fig. 1. (B)  τ 1 drifts with time. (C)  τ 2 drifts with time. In (B) and (C), residuals after subtraction of linearly scaled pressure drift in (A) are also displayed below the two drift traces with right vertical axes. The data in all three panels have been averaged by every successive 20 decays.

Fig. 8
Fig. 8

Trace methane detection in nitrogen with flow rates of (A)  40   SCCM , (B)  60   SCCM , (C)  80   SCCM , (D)  100   SCCM , and (E)  120   SCCM , respectively. Because of pressure broadening, the four lines shown in Fig. 4 are not resolved at 760 Torr pressure. The fitted residual is shown below each spectrum. The measured methane concentrations are (A)  0.69 ppbv , (B)  0.50 ppbv , (C)  0.34 ppbv , (D)  0.27 ppbv , and (E)  0.21 ppbv , respectively, with an uncertainty ( 3 σ ) of 0.19 ppbv .

Equations (14)

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

α = 1 c [ k k 0 ] ,
A n ( p ) = 1 p i = 1 + p ( n 1 ) p n x i , n = 1 m .
σ A , n 2 ( p ) = 1 2 [ A n ( p ) A n + 1 ( p ) ] 2 , n = 1 m 1.
σ A 2 ( p ) t = 1 2 ( m 1 ) i = 1 m 1 [ A i ( p ) A i + 1 ( p ) ] 2 .
Var ( σ A 2 ( p ) t ) = 3 m 4 ( m 1 ) 2 σ 4 p 2 N p 3 σ 4 N p .
σ A 2 ( p ) e = 1 2 s 2 p 2 + σ 2 p ,
p min = ( σ s ) 2 / 3
σ A 2 ( p min ) e = 3 2 ( s σ 2 ) 2 / 3 .
σ A 2 ( p ) e = σ 2 p + D 3 p ( 2 p 2 + 1 ) ,
p min = D + 3 σ 2 2 D
σ A 2 ( p min ) e = 8 3 D ( D + 3 σ 2 ) .
Var ( σ A 2 ( p ) t ) = 3 m 4 ( m 1 ) 2 σ 4 p 2 + 2 ( m 1 ) ( D 3 p ( 2 p 2 + 1 ) 2 ) 2 + 4 ( σ 2 p ( m 1 ) ) ( D 3 p ( 2 p 2 + 1 ) ) .
A n ( p ) = 1 p i = n p + n 1 x i ,
σ A 2 ( p ) t = 1 2 ( N + 1 2 p ) n = 1 N 2 p [ A n ( p ) A n + p ( p ) ] 2 .

Metrics