Abstract

The limit of sensitivity was investigated in the detection of sodium atoms in a flame with the resonant Voigt effect. An elliptically polarized beam of light from a cw dye laser was used to probe magnetically induced optical anisotropy of sodium atoms introduced into a propane–air flame. The optical path length in the flame was 8 mm, and the effective volume of the probed region was 1.1 × 10−9 m3. Sodium atoms of the number density of 9.5 × 1013 m−3 in the flame produced a signal-to-noise ratio of 120 when the cross-correlation technique was incorporated to estimate the signal energy relative to the background-noise energy.

© 1995 Optical Society of America

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References

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  1. M. Yamamoto and S. Murayama, “Analysis of resonant Voigt effect,” J. Opt. Soc. Am. 69, 781–786 (1979).
    [CrossRef]
  2. M. Yamamoto, Y. Takubo, and S. Murayama, “Detection limit of resonant magnetooptic spectroscopy,” Jpn. J. Appl. Phys. 23, 783 (1984).
    [CrossRef]
  3. M. Yamamoto, J. Hanari, and Y. Takubo, “Analytical detection sensitivity of the laser-probed resonant Voigt effect,” J. Opt. Soc. Am. B 3, 1245–1250 (1986).
    [CrossRef]
  4. M. Yamamoto, Y. Furuya, K. Amemiya, Y. Miyata, and Y. Takubo, “Laser-probed resonant Voigt effect of OH molecules,” J. Opt. Soc. Am. B 4, 1212–1216 (1987).
    [CrossRef]
  5. Y. Takubo, Y. Miyata, K. Amemiya, and M. Yamamoto, “Magneto-optic spectra of the A2Σ+ − X2Πi transition of OH,” J. Chem. Phys. 93, 176–182 (1990).
    [CrossRef]
  6. K. Muroo, Y. Takubo, and M. Yamamoto, “Resonant Voigt-effect spectra of the Rb D2transition,” in Annual Meeting, Vol. 23 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 11.
  7. K. Muroo, T. Matsunobe, Y. Shishido, Y. Takubo, and M. Yamamoto, “Resonant Voigt-effect spectrum of the rubidium D2transition,” J. Opt. Soc. Am. B 11, 409–414 (1994).
    [CrossRef]
  8. A. Corney, B. P. Kibble, and G. W. Series, “The forward scattering of resonance radiation, with special reference to double resonance and level-crossing experiments,” Proc. R. Soc. A 293, 70–93 (1966).
    [CrossRef]
  9. G. Horlick, “Detection of spectral information utilizing cross-correlation techniques,” Anal. Chem. 45, 319–324 (1973).
    [CrossRef]
  10. C. Th. J. Alkemade, “Detection of small numbers of atoms and molecules,” in Analytical Applications of Lasers, E. H. Piepmeier, ed. (Wiley, New York, 1986).
  11. L. A. Davis, R. J. Krupa, and J. D. Winefordner, “Evaluation of a laser excited Voigt effect coherent forward scattering atomic spectrometer,” Spectrochim. Acta 41B, 1167–1174 (1986).
  12. G. M. Hermann, “Coherent forward scattering atomic spectrometry,” Anal. Chem. 64, 571A–579A (1992).
    [CrossRef]
  13. J. D. Winefordner and C. Stevenson, “Linking principles with absolute detection power in atomic spectrometry: how far can we go?” Spectrochim. Acta 48B, 757–767 (1993).
  14. J. J. Kankare and R. Stephens, “A unified theory of magneto-optic phenomena in analytical atomic spectroscopy,” Spectrochim. Acta 35B, 849–864 (1980).
  15. H. Debus, W. Hanle, A. Scharmann, and P. Wirz, “Trace element detection by forward scattering using a continuum light source,” Spectrochim. Acta 36B, 1015–1021 (1981).
  16. P. Wirz, H. Debus, W. Hanle, and A. Scharmann, “Synchronous multielement detection by forward scattering in a transverse magnetic field (SYNFO),” Spectrochim. Acta 37B, 1013–1020 (1982).
  17. R. Stephens, “Linearization of the magneto-optic signals produced by the Voigt effect in atomic vapors,” Spectrochim. Acta 38B, 1077–1086 (1983).
  18. A. D. Kersey, J. B. Dawson, and D. J. Ellis, “The contributions of magnetically induced dichroism and the resonant Voigt effect to the detection of silver by atomic spectroscopy,” Spectrochim. Acta 35B, 865–873 (1980).
  19. S. L. Adler, “Photon splitting and photon dispersion in a strong magnetic field,” Ann. Phys. 67, 599–647 (1971).
    [CrossRef]
  20. E. Iacopini and E. Zavattini, “Experimental method to detect the vacuum birefringence induced by a magnetic field,” Phys. Lett. 85B, 151–154 (1979).
  21. R. Cameron, G. Cantatore, A. C. Melissinos, J. Rogers, Y. Semertzidis, H. Halama, A. Prodell, F. A. Nezrick, C. Rizzo, and E. Zavattini, “Measurement of the magnetic birefringence of neon gas,” J. Opt. Soc. Am. B 8, 520–524 (1991).
    [CrossRef]
  22. P. B. Lukins and G. L. D. Ritchie, “Magneto-optical Cotton–Mouton effect of molecular oxygen, a comparison,” Chem. Phys. Lett. 180, 551–555 (1991).
    [CrossRef]

1994 (1)

1993 (1)

J. D. Winefordner and C. Stevenson, “Linking principles with absolute detection power in atomic spectrometry: how far can we go?” Spectrochim. Acta 48B, 757–767 (1993).

1992 (1)

G. M. Hermann, “Coherent forward scattering atomic spectrometry,” Anal. Chem. 64, 571A–579A (1992).
[CrossRef]

1991 (2)

1990 (1)

Y. Takubo, Y. Miyata, K. Amemiya, and M. Yamamoto, “Magneto-optic spectra of the A2Σ+ − X2Πi transition of OH,” J. Chem. Phys. 93, 176–182 (1990).
[CrossRef]

1987 (1)

1986 (2)

M. Yamamoto, J. Hanari, and Y. Takubo, “Analytical detection sensitivity of the laser-probed resonant Voigt effect,” J. Opt. Soc. Am. B 3, 1245–1250 (1986).
[CrossRef]

L. A. Davis, R. J. Krupa, and J. D. Winefordner, “Evaluation of a laser excited Voigt effect coherent forward scattering atomic spectrometer,” Spectrochim. Acta 41B, 1167–1174 (1986).

1984 (1)

M. Yamamoto, Y. Takubo, and S. Murayama, “Detection limit of resonant magnetooptic spectroscopy,” Jpn. J. Appl. Phys. 23, 783 (1984).
[CrossRef]

1983 (1)

R. Stephens, “Linearization of the magneto-optic signals produced by the Voigt effect in atomic vapors,” Spectrochim. Acta 38B, 1077–1086 (1983).

1982 (1)

P. Wirz, H. Debus, W. Hanle, and A. Scharmann, “Synchronous multielement detection by forward scattering in a transverse magnetic field (SYNFO),” Spectrochim. Acta 37B, 1013–1020 (1982).

1981 (1)

H. Debus, W. Hanle, A. Scharmann, and P. Wirz, “Trace element detection by forward scattering using a continuum light source,” Spectrochim. Acta 36B, 1015–1021 (1981).

1980 (2)

A. D. Kersey, J. B. Dawson, and D. J. Ellis, “The contributions of magnetically induced dichroism and the resonant Voigt effect to the detection of silver by atomic spectroscopy,” Spectrochim. Acta 35B, 865–873 (1980).

J. J. Kankare and R. Stephens, “A unified theory of magneto-optic phenomena in analytical atomic spectroscopy,” Spectrochim. Acta 35B, 849–864 (1980).

1979 (2)

M. Yamamoto and S. Murayama, “Analysis of resonant Voigt effect,” J. Opt. Soc. Am. 69, 781–786 (1979).
[CrossRef]

E. Iacopini and E. Zavattini, “Experimental method to detect the vacuum birefringence induced by a magnetic field,” Phys. Lett. 85B, 151–154 (1979).

1973 (1)

G. Horlick, “Detection of spectral information utilizing cross-correlation techniques,” Anal. Chem. 45, 319–324 (1973).
[CrossRef]

1971 (1)

S. L. Adler, “Photon splitting and photon dispersion in a strong magnetic field,” Ann. Phys. 67, 599–647 (1971).
[CrossRef]

1966 (1)

A. Corney, B. P. Kibble, and G. W. Series, “The forward scattering of resonance radiation, with special reference to double resonance and level-crossing experiments,” Proc. R. Soc. A 293, 70–93 (1966).
[CrossRef]

Adler, S. L.

S. L. Adler, “Photon splitting and photon dispersion in a strong magnetic field,” Ann. Phys. 67, 599–647 (1971).
[CrossRef]

Alkemade, C. Th. J.

C. Th. J. Alkemade, “Detection of small numbers of atoms and molecules,” in Analytical Applications of Lasers, E. H. Piepmeier, ed. (Wiley, New York, 1986).

Amemiya, K.

Y. Takubo, Y. Miyata, K. Amemiya, and M. Yamamoto, “Magneto-optic spectra of the A2Σ+ − X2Πi transition of OH,” J. Chem. Phys. 93, 176–182 (1990).
[CrossRef]

M. Yamamoto, Y. Furuya, K. Amemiya, Y. Miyata, and Y. Takubo, “Laser-probed resonant Voigt effect of OH molecules,” J. Opt. Soc. Am. B 4, 1212–1216 (1987).
[CrossRef]

Cameron, R.

Cantatore, G.

Corney, A.

A. Corney, B. P. Kibble, and G. W. Series, “The forward scattering of resonance radiation, with special reference to double resonance and level-crossing experiments,” Proc. R. Soc. A 293, 70–93 (1966).
[CrossRef]

Davis, L. A.

L. A. Davis, R. J. Krupa, and J. D. Winefordner, “Evaluation of a laser excited Voigt effect coherent forward scattering atomic spectrometer,” Spectrochim. Acta 41B, 1167–1174 (1986).

Dawson, J. B.

A. D. Kersey, J. B. Dawson, and D. J. Ellis, “The contributions of magnetically induced dichroism and the resonant Voigt effect to the detection of silver by atomic spectroscopy,” Spectrochim. Acta 35B, 865–873 (1980).

Debus, H.

P. Wirz, H. Debus, W. Hanle, and A. Scharmann, “Synchronous multielement detection by forward scattering in a transverse magnetic field (SYNFO),” Spectrochim. Acta 37B, 1013–1020 (1982).

H. Debus, W. Hanle, A. Scharmann, and P. Wirz, “Trace element detection by forward scattering using a continuum light source,” Spectrochim. Acta 36B, 1015–1021 (1981).

Ellis, D. J.

A. D. Kersey, J. B. Dawson, and D. J. Ellis, “The contributions of magnetically induced dichroism and the resonant Voigt effect to the detection of silver by atomic spectroscopy,” Spectrochim. Acta 35B, 865–873 (1980).

Furuya, Y.

Halama, H.

Hanari, J.

Hanle, W.

P. Wirz, H. Debus, W. Hanle, and A. Scharmann, “Synchronous multielement detection by forward scattering in a transverse magnetic field (SYNFO),” Spectrochim. Acta 37B, 1013–1020 (1982).

H. Debus, W. Hanle, A. Scharmann, and P. Wirz, “Trace element detection by forward scattering using a continuum light source,” Spectrochim. Acta 36B, 1015–1021 (1981).

Hermann, G. M.

G. M. Hermann, “Coherent forward scattering atomic spectrometry,” Anal. Chem. 64, 571A–579A (1992).
[CrossRef]

Horlick, G.

G. Horlick, “Detection of spectral information utilizing cross-correlation techniques,” Anal. Chem. 45, 319–324 (1973).
[CrossRef]

Iacopini, E.

E. Iacopini and E. Zavattini, “Experimental method to detect the vacuum birefringence induced by a magnetic field,” Phys. Lett. 85B, 151–154 (1979).

Kankare, J. J.

J. J. Kankare and R. Stephens, “A unified theory of magneto-optic phenomena in analytical atomic spectroscopy,” Spectrochim. Acta 35B, 849–864 (1980).

Kersey, A. D.

A. D. Kersey, J. B. Dawson, and D. J. Ellis, “The contributions of magnetically induced dichroism and the resonant Voigt effect to the detection of silver by atomic spectroscopy,” Spectrochim. Acta 35B, 865–873 (1980).

Kibble, B. P.

A. Corney, B. P. Kibble, and G. W. Series, “The forward scattering of resonance radiation, with special reference to double resonance and level-crossing experiments,” Proc. R. Soc. A 293, 70–93 (1966).
[CrossRef]

Krupa, R. J.

L. A. Davis, R. J. Krupa, and J. D. Winefordner, “Evaluation of a laser excited Voigt effect coherent forward scattering atomic spectrometer,” Spectrochim. Acta 41B, 1167–1174 (1986).

Lukins, P. B.

P. B. Lukins and G. L. D. Ritchie, “Magneto-optical Cotton–Mouton effect of molecular oxygen, a comparison,” Chem. Phys. Lett. 180, 551–555 (1991).
[CrossRef]

Matsunobe, T.

Melissinos, A. C.

Miyata, Y.

Y. Takubo, Y. Miyata, K. Amemiya, and M. Yamamoto, “Magneto-optic spectra of the A2Σ+ − X2Πi transition of OH,” J. Chem. Phys. 93, 176–182 (1990).
[CrossRef]

M. Yamamoto, Y. Furuya, K. Amemiya, Y. Miyata, and Y. Takubo, “Laser-probed resonant Voigt effect of OH molecules,” J. Opt. Soc. Am. B 4, 1212–1216 (1987).
[CrossRef]

Murayama, S.

M. Yamamoto, Y. Takubo, and S. Murayama, “Detection limit of resonant magnetooptic spectroscopy,” Jpn. J. Appl. Phys. 23, 783 (1984).
[CrossRef]

M. Yamamoto and S. Murayama, “Analysis of resonant Voigt effect,” J. Opt. Soc. Am. 69, 781–786 (1979).
[CrossRef]

Muroo, K.

K. Muroo, T. Matsunobe, Y. Shishido, Y. Takubo, and M. Yamamoto, “Resonant Voigt-effect spectrum of the rubidium D2transition,” J. Opt. Soc. Am. B 11, 409–414 (1994).
[CrossRef]

K. Muroo, Y. Takubo, and M. Yamamoto, “Resonant Voigt-effect spectra of the Rb D2transition,” in Annual Meeting, Vol. 23 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 11.

Nezrick, F. A.

Prodell, A.

Ritchie, G. L. D.

P. B. Lukins and G. L. D. Ritchie, “Magneto-optical Cotton–Mouton effect of molecular oxygen, a comparison,” Chem. Phys. Lett. 180, 551–555 (1991).
[CrossRef]

Rizzo, C.

Rogers, J.

Scharmann, A.

P. Wirz, H. Debus, W. Hanle, and A. Scharmann, “Synchronous multielement detection by forward scattering in a transverse magnetic field (SYNFO),” Spectrochim. Acta 37B, 1013–1020 (1982).

H. Debus, W. Hanle, A. Scharmann, and P. Wirz, “Trace element detection by forward scattering using a continuum light source,” Spectrochim. Acta 36B, 1015–1021 (1981).

Semertzidis, Y.

Series, G. W.

A. Corney, B. P. Kibble, and G. W. Series, “The forward scattering of resonance radiation, with special reference to double resonance and level-crossing experiments,” Proc. R. Soc. A 293, 70–93 (1966).
[CrossRef]

Shishido, Y.

Stephens, R.

R. Stephens, “Linearization of the magneto-optic signals produced by the Voigt effect in atomic vapors,” Spectrochim. Acta 38B, 1077–1086 (1983).

J. J. Kankare and R. Stephens, “A unified theory of magneto-optic phenomena in analytical atomic spectroscopy,” Spectrochim. Acta 35B, 849–864 (1980).

Stevenson, C.

J. D. Winefordner and C. Stevenson, “Linking principles with absolute detection power in atomic spectrometry: how far can we go?” Spectrochim. Acta 48B, 757–767 (1993).

Takubo, Y.

K. Muroo, T. Matsunobe, Y. Shishido, Y. Takubo, and M. Yamamoto, “Resonant Voigt-effect spectrum of the rubidium D2transition,” J. Opt. Soc. Am. B 11, 409–414 (1994).
[CrossRef]

Y. Takubo, Y. Miyata, K. Amemiya, and M. Yamamoto, “Magneto-optic spectra of the A2Σ+ − X2Πi transition of OH,” J. Chem. Phys. 93, 176–182 (1990).
[CrossRef]

M. Yamamoto, Y. Furuya, K. Amemiya, Y. Miyata, and Y. Takubo, “Laser-probed resonant Voigt effect of OH molecules,” J. Opt. Soc. Am. B 4, 1212–1216 (1987).
[CrossRef]

M. Yamamoto, J. Hanari, and Y. Takubo, “Analytical detection sensitivity of the laser-probed resonant Voigt effect,” J. Opt. Soc. Am. B 3, 1245–1250 (1986).
[CrossRef]

M. Yamamoto, Y. Takubo, and S. Murayama, “Detection limit of resonant magnetooptic spectroscopy,” Jpn. J. Appl. Phys. 23, 783 (1984).
[CrossRef]

K. Muroo, Y. Takubo, and M. Yamamoto, “Resonant Voigt-effect spectra of the Rb D2transition,” in Annual Meeting, Vol. 23 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 11.

Winefordner, J. D.

J. D. Winefordner and C. Stevenson, “Linking principles with absolute detection power in atomic spectrometry: how far can we go?” Spectrochim. Acta 48B, 757–767 (1993).

L. A. Davis, R. J. Krupa, and J. D. Winefordner, “Evaluation of a laser excited Voigt effect coherent forward scattering atomic spectrometer,” Spectrochim. Acta 41B, 1167–1174 (1986).

Wirz, P.

P. Wirz, H. Debus, W. Hanle, and A. Scharmann, “Synchronous multielement detection by forward scattering in a transverse magnetic field (SYNFO),” Spectrochim. Acta 37B, 1013–1020 (1982).

H. Debus, W. Hanle, A. Scharmann, and P. Wirz, “Trace element detection by forward scattering using a continuum light source,” Spectrochim. Acta 36B, 1015–1021 (1981).

Yamamoto, M.

K. Muroo, T. Matsunobe, Y. Shishido, Y. Takubo, and M. Yamamoto, “Resonant Voigt-effect spectrum of the rubidium D2transition,” J. Opt. Soc. Am. B 11, 409–414 (1994).
[CrossRef]

Y. Takubo, Y. Miyata, K. Amemiya, and M. Yamamoto, “Magneto-optic spectra of the A2Σ+ − X2Πi transition of OH,” J. Chem. Phys. 93, 176–182 (1990).
[CrossRef]

M. Yamamoto, Y. Furuya, K. Amemiya, Y. Miyata, and Y. Takubo, “Laser-probed resonant Voigt effect of OH molecules,” J. Opt. Soc. Am. B 4, 1212–1216 (1987).
[CrossRef]

M. Yamamoto, J. Hanari, and Y. Takubo, “Analytical detection sensitivity of the laser-probed resonant Voigt effect,” J. Opt. Soc. Am. B 3, 1245–1250 (1986).
[CrossRef]

M. Yamamoto, Y. Takubo, and S. Murayama, “Detection limit of resonant magnetooptic spectroscopy,” Jpn. J. Appl. Phys. 23, 783 (1984).
[CrossRef]

M. Yamamoto and S. Murayama, “Analysis of resonant Voigt effect,” J. Opt. Soc. Am. 69, 781–786 (1979).
[CrossRef]

K. Muroo, Y. Takubo, and M. Yamamoto, “Resonant Voigt-effect spectra of the Rb D2transition,” in Annual Meeting, Vol. 23 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 11.

Zavattini, E.

Anal. Chem. (2)

G. M. Hermann, “Coherent forward scattering atomic spectrometry,” Anal. Chem. 64, 571A–579A (1992).
[CrossRef]

G. Horlick, “Detection of spectral information utilizing cross-correlation techniques,” Anal. Chem. 45, 319–324 (1973).
[CrossRef]

Ann. Phys. (1)

S. L. Adler, “Photon splitting and photon dispersion in a strong magnetic field,” Ann. Phys. 67, 599–647 (1971).
[CrossRef]

Chem. Phys. Lett. (1)

P. B. Lukins and G. L. D. Ritchie, “Magneto-optical Cotton–Mouton effect of molecular oxygen, a comparison,” Chem. Phys. Lett. 180, 551–555 (1991).
[CrossRef]

J. Chem. Phys. (1)

Y. Takubo, Y. Miyata, K. Amemiya, and M. Yamamoto, “Magneto-optic spectra of the A2Σ+ − X2Πi transition of OH,” J. Chem. Phys. 93, 176–182 (1990).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (4)

Jpn. J. Appl. Phys. (1)

M. Yamamoto, Y. Takubo, and S. Murayama, “Detection limit of resonant magnetooptic spectroscopy,” Jpn. J. Appl. Phys. 23, 783 (1984).
[CrossRef]

Phys. Lett. (1)

E. Iacopini and E. Zavattini, “Experimental method to detect the vacuum birefringence induced by a magnetic field,” Phys. Lett. 85B, 151–154 (1979).

Proc. R. Soc. A (1)

A. Corney, B. P. Kibble, and G. W. Series, “The forward scattering of resonance radiation, with special reference to double resonance and level-crossing experiments,” Proc. R. Soc. A 293, 70–93 (1966).
[CrossRef]

Spectrochim. Acta (7)

J. D. Winefordner and C. Stevenson, “Linking principles with absolute detection power in atomic spectrometry: how far can we go?” Spectrochim. Acta 48B, 757–767 (1993).

J. J. Kankare and R. Stephens, “A unified theory of magneto-optic phenomena in analytical atomic spectroscopy,” Spectrochim. Acta 35B, 849–864 (1980).

H. Debus, W. Hanle, A. Scharmann, and P. Wirz, “Trace element detection by forward scattering using a continuum light source,” Spectrochim. Acta 36B, 1015–1021 (1981).

P. Wirz, H. Debus, W. Hanle, and A. Scharmann, “Synchronous multielement detection by forward scattering in a transverse magnetic field (SYNFO),” Spectrochim. Acta 37B, 1013–1020 (1982).

R. Stephens, “Linearization of the magneto-optic signals produced by the Voigt effect in atomic vapors,” Spectrochim. Acta 38B, 1077–1086 (1983).

A. D. Kersey, J. B. Dawson, and D. J. Ellis, “The contributions of magnetically induced dichroism and the resonant Voigt effect to the detection of silver by atomic spectroscopy,” Spectrochim. Acta 35B, 865–873 (1980).

L. A. Davis, R. J. Krupa, and J. D. Winefordner, “Evaluation of a laser excited Voigt effect coherent forward scattering atomic spectrometer,” Spectrochim. Acta 41B, 1167–1174 (1986).

Other (2)

C. Th. J. Alkemade, “Detection of small numbers of atoms and molecules,” in Analytical Applications of Lasers, E. H. Piepmeier, ed. (Wiley, New York, 1986).

K. Muroo, Y. Takubo, and M. Yamamoto, “Resonant Voigt-effect spectra of the Rb D2transition,” in Annual Meeting, Vol. 23 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 11.

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Figures (2)

Fig. 1
Fig. 1

Experimental apparatus. P.M.T., photomultiplier tube. Ref., reference; Sig., signal.

Fig. 2
Fig. 2

Voigt-effect profile of the Na D2 line measured with elliptically polarized light and a perfectly crossed analyzer: Δα = 1.7 × 10−4 and Δθ = 0. Dots, experimental points; solid curve, theoretical profile; light-extinction ratio of the analyzer: a = 2.9 × 10−8; dashed curve, the mutual correlation function for the measured data (experimental points) and the theoretical profile.

Equations (5)

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I ( ω ) / I 0 = S ( ω ) + R ,
S ( ω ) = [ k L | Δ n ( ω ) | ] 2 + 2 k L Δ n ( r ) ( ω ) Δ α + 2 k L Δ n ( i ) ( ω ) Δ θ ,
R = Δ α 2 + Δ θ 2 + a ,
( SNR ) max C k L N | B ( ω ) | γ a .
SNR = 2 C k L N | B ( r ) Δ α + B ( i ) Δ θ | ( I 0 η τ ) [ ( Δ α 2 + Δ θ 2 + a ) ( I 0 η τ ) ] 1 / 2 ,

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