Abstract

A quantitative approach for calculating the noise due to the stochastic nature of multimode laser radiation in nonlinear-optical processes is presented. The model is applicable when it is appropriate to describe the nonlinear interaction with a perturbation expansion in the incoming field, and it is derived under the assumption of independent individual laser-mode intensities and phases. It is possible to separate noise contributions from mode-amplitude and -phase fluctuations, respectively, and also to identify the noise contribution from each laser source. For coherent anti-Stokes Raman scattering (CARS) thermometry, the model shows that with a single-mode pump laser the stochastic phases in the dye laser do not generate noise in the conventional approach and that amplitude fluctuations in the dye laser(s) do not (significantly) generate noise in the dual-broadband approaches. Thus, in the dual-broadband approaches, the spectral noise in the Stokes beam is not a lower limit for the noise in the CARS beam. The model seems to overestimate the noise due to phase fluctuations and to underestimate the noise due to amplitude fluctuations.

© 1988 Optical Society of America

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References

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  1. L. A. Rahn, R. L. Farrow, R. P. Lucht, “Effects of laser field statistics on coherent anti-Stokes Raman spectroscopy intensities,” Opt. Lett. 9, 223 (1984).
    [CrossRef] [PubMed]
  2. D. A. Greenhalgh, S. T. Whittley, “Mode noise in broadband CARS spectroscopy,” Appl. Opt. 24, 907 (1985).
    [CrossRef] [PubMed]
  3. R. J. Hall, “The statistical behaviour of nonresonant CARS intensities,” Opt. Commun. 56, 127 (1985).
    [CrossRef]
  4. D. R. Snelling, R. A. Sawchuk, R. E. Mueller, “Single pulse CARS noise: a comparison between single-mode and multimode pump lasers,” Appl. Opt. 24, 2771 (1985).
    [CrossRef] [PubMed]
  5. R. L. Farrow, L. A. Rahn, “Interpreting coherent anti-Stokes Raman spectra measured with multimode Nd:YAG pump lasers,” J. Opt. Soc. Am. B 2, 903 (1985).
    [CrossRef]
  6. R. J. Hall, “Theoretical analysis of non-thermal pump effects in broadband CARS spectroscopy,” Opt. Quantum Electron. 18, 319 (1986).
    [CrossRef]
  7. G. S. Agarwal, R. L. Farrow, “Theoretical modeling of two-color CARS spectra measured with a frequency-doubled, multimode pump laser,” J. Opt. Soc. Am. B 3, 1596 (1986).
    [CrossRef]
  8. R. J. Hall, D. A. Greenhalgh, “Noise properties of single-pulse CARS spectroscopy with multimode pump sources,” J. Opt. Soc. Am. B 3, 1637 (1986).
    [CrossRef]
  9. D. R. Snelling, G. J. Smallwood, R. A. Sawchuk, T. Parameswaran, “Precision of multiplex CARS temperatures using both single-mode and multimode pump lasers,” Appl. Opt. 26, 99 (1987).
    [CrossRef] [PubMed]
  10. S. Kröll, M. Aldén, T. Berglind, R. J. Hall, “Noise characteristics of single shot broadband Raman resonant CARS with single- and multimode lasers,” Appl. Opt. 26, 1068 (1987).
    [CrossRef]
  11. R. J. Hall, A. C. Eckbreth, “Coherent anti-Stokes Raman spectroscopy (CARS) application to combustion diagnostics,” Laser Appl. 5, 213 (1984).
  12. M. Pealat, P. Bouchardy, M. Lefebvre, J.-P. Taran, “Precision of multiplex CARS temperature measurements,” Appl. Opt. 24, 1012 (1985).
    [CrossRef] [PubMed]
  13. A. C. Eckbreth, G. M. Dobbs, J. H. Stufflebeam, P. A. Tellex, “CARS temperature and species measurements in augmented jet engine exhausts,” Appl. Opt. 23, 1328 (1984).
    [CrossRef] [PubMed]
  14. D. R. Snelling, T. Parameswaran, G. J. Smallwood, “Noise characteristics of single-shot broadband CARS signals,” Appl. Opt. 26, 4298 (1987).
    [CrossRef] [PubMed]
  15. A. C. Eckbreth, T. J. Anderson, “Dual broadband CARS for simultaneous, multiple species measurements,” Appl. Opt. 24, 2731 (1985).
    [CrossRef] [PubMed]
  16. A. C. Eckbreth, T. J. Anderson, “Simultaneous rotational coherent anti-Stokes Raman spectroscopy and coherent Stokes Raman spectroscopy with arbitrary pump-Stokes spectral separation,” Opt. Lett. 11, 496 (1986).
    [CrossRef] [PubMed]
  17. M. Aldén, P.-E. Bengtsson, H. Edner, “Rotational CARS generation through a multiple four-color interaction,” Appl. Opt. 25, 4493 (1986).
    [CrossRef] [PubMed]
  18. W. B. Roh, P. W. Schreiber, J. P. E. Taran, “Single-pulse coherent anti-Stokes Raman scattering,” Appl. Phys. Lett. 29, 174 (1976).
    [CrossRef]
  19. M. A. Yuratich, “Effects of laser linewidth on coherent anti-Stokes Raman spectroscopy,” Mol. Phys. 38, 625 (1979).
    [CrossRef]
  20. H. Kataoka, S. Maeda, C. Hirose, “Effects of laser linewidth on the coherent anti-Stokes Raman spectroscopy spectral profile,” Appl. Spectrosc. 36, 565 (1982).
    [CrossRef]
  21. R. E. Teets, “Accurate convolutions of coherent anti-Stokes Raman spectra,” Opt. Lett. 9, 226 (1984).
    [CrossRef] [PubMed]
  22. D. A. Greenhalgh, R. J. Hall, “A closed form solution for the CARS intensity convolution,” Opt. Commun. 57, 125 (1986).
    [CrossRef]
  23. F. Y. Yueh, E. J. Beiting, “Analytical expressions for coherent anti-Stokes Raman spectral (CARS) profiles,” Comp. Phys. Commun. 42, 65 (1986).
    [CrossRef]
  24. M. Aldén, Department of Atomic Physics, Combustion Centre, Lund Institute of Technology, Box 118, S-221 Lund, Sweden (personal communication).
  25. G. L. Eesley, Coherent Raman Spectroscopy (Pergamon, New York, 1981), Chap. 3B.
  26. M. D. Levenson, Introduction to Nonlinear Laser Spectroscopy (Academic, New York, 1982).
  27. See, e.g., in several of the contributions in the special issue on stimulated Raman and Brillouin scattering for laser beam control, J. Opt. Soc. Am. B 3 (10)(1986), and references therein.
  28. L. A. Westling, M. G. Raymer, “Intensity autocorrelation measurements and spontaneous FM phase locking in a multimode pulsed dye laser,” J. Opt. Soc. Am. B 3, 911 (1986).
    [CrossRef]
  29. D. S. King, R. R. Cavanagh, “Streak-camera analysis of XeCl- and N2-pumped dye-laser outputs,” Opt. Lett. 8, 18 (1983).
    [CrossRef] [PubMed]
  30. R. E. Teets, “CARS signals: phase matching, transverse modes, and optical effects,” Appl. Opt. 25, 855 (1986).
    [CrossRef]
  31. J. Ducuing, N. Bloembergen, “Statistical fluctuations in nonlinear optical processes,” Phys. Rev. 133, A1493 (1964).
    [CrossRef]
  32. A. M. Mood, Introduction to the Theory of Statistics (McGraw-Hill, New York, 1974), Chap. 5.
  33. M. Aldén, P. E. Bengtsson, H. Edner, S. Kröll, D. Nilsson, “Rotational CARS:comparison of different techniques with emphasis on temperature accuracies,” to be submitted to Appl. Opt.
  34. L. A. Westling, M. G. Raymer, J. J. Snyder, “Single-shot spectral measurements and mode correlations in a multimode pulsed dye laser,” J. Opt. Soc. Am. B 1, 150 (1984).
    [CrossRef]
  35. Z. W. Li, C. Radzewicz, M. G. Raymer, “Temporal smoothing of multimode dye laser pulses,” Opt. Lett. 12, 416 (1987).
    [CrossRef] [PubMed]
  36. S. Kröll, D. Sandell, “A model for calculating the noise due to the stochastic nature of multimode laser radiation in nonlinear optical processes,” in Proceedings of the International Conference on Nonlinear Optics, V. J. Corcoran, ed. (Society for Optics and Quantum Electronics, McLean, Va., to be published).

1987 (4)

1986 (10)

D. A. Greenhalgh, R. J. Hall, “A closed form solution for the CARS intensity convolution,” Opt. Commun. 57, 125 (1986).
[CrossRef]

F. Y. Yueh, E. J. Beiting, “Analytical expressions for coherent anti-Stokes Raman spectral (CARS) profiles,” Comp. Phys. Commun. 42, 65 (1986).
[CrossRef]

See, e.g., in several of the contributions in the special issue on stimulated Raman and Brillouin scattering for laser beam control, J. Opt. Soc. Am. B 3 (10)(1986), and references therein.

L. A. Westling, M. G. Raymer, “Intensity autocorrelation measurements and spontaneous FM phase locking in a multimode pulsed dye laser,” J. Opt. Soc. Am. B 3, 911 (1986).
[CrossRef]

R. E. Teets, “CARS signals: phase matching, transverse modes, and optical effects,” Appl. Opt. 25, 855 (1986).
[CrossRef]

A. C. Eckbreth, T. J. Anderson, “Simultaneous rotational coherent anti-Stokes Raman spectroscopy and coherent Stokes Raman spectroscopy with arbitrary pump-Stokes spectral separation,” Opt. Lett. 11, 496 (1986).
[CrossRef] [PubMed]

M. Aldén, P.-E. Bengtsson, H. Edner, “Rotational CARS generation through a multiple four-color interaction,” Appl. Opt. 25, 4493 (1986).
[CrossRef] [PubMed]

R. J. Hall, “Theoretical analysis of non-thermal pump effects in broadband CARS spectroscopy,” Opt. Quantum Electron. 18, 319 (1986).
[CrossRef]

G. S. Agarwal, R. L. Farrow, “Theoretical modeling of two-color CARS spectra measured with a frequency-doubled, multimode pump laser,” J. Opt. Soc. Am. B 3, 1596 (1986).
[CrossRef]

R. J. Hall, D. A. Greenhalgh, “Noise properties of single-pulse CARS spectroscopy with multimode pump sources,” J. Opt. Soc. Am. B 3, 1637 (1986).
[CrossRef]

1985 (6)

1984 (5)

1983 (1)

1982 (1)

1979 (1)

M. A. Yuratich, “Effects of laser linewidth on coherent anti-Stokes Raman spectroscopy,” Mol. Phys. 38, 625 (1979).
[CrossRef]

1976 (1)

W. B. Roh, P. W. Schreiber, J. P. E. Taran, “Single-pulse coherent anti-Stokes Raman scattering,” Appl. Phys. Lett. 29, 174 (1976).
[CrossRef]

1964 (1)

J. Ducuing, N. Bloembergen, “Statistical fluctuations in nonlinear optical processes,” Phys. Rev. 133, A1493 (1964).
[CrossRef]

Agarwal, G. S.

Aldén, M.

S. Kröll, M. Aldén, T. Berglind, R. J. Hall, “Noise characteristics of single shot broadband Raman resonant CARS with single- and multimode lasers,” Appl. Opt. 26, 1068 (1987).
[CrossRef]

M. Aldén, P.-E. Bengtsson, H. Edner, “Rotational CARS generation through a multiple four-color interaction,” Appl. Opt. 25, 4493 (1986).
[CrossRef] [PubMed]

M. Aldén, Department of Atomic Physics, Combustion Centre, Lund Institute of Technology, Box 118, S-221 Lund, Sweden (personal communication).

M. Aldén, P. E. Bengtsson, H. Edner, S. Kröll, D. Nilsson, “Rotational CARS:comparison of different techniques with emphasis on temperature accuracies,” to be submitted to Appl. Opt.

Anderson, T. J.

Beiting, E. J.

F. Y. Yueh, E. J. Beiting, “Analytical expressions for coherent anti-Stokes Raman spectral (CARS) profiles,” Comp. Phys. Commun. 42, 65 (1986).
[CrossRef]

Bengtsson, P. E.

M. Aldén, P. E. Bengtsson, H. Edner, S. Kröll, D. Nilsson, “Rotational CARS:comparison of different techniques with emphasis on temperature accuracies,” to be submitted to Appl. Opt.

Bengtsson, P.-E.

Berglind, T.

Bloembergen, N.

J. Ducuing, N. Bloembergen, “Statistical fluctuations in nonlinear optical processes,” Phys. Rev. 133, A1493 (1964).
[CrossRef]

Bouchardy, P.

Cavanagh, R. R.

Dobbs, G. M.

Ducuing, J.

J. Ducuing, N. Bloembergen, “Statistical fluctuations in nonlinear optical processes,” Phys. Rev. 133, A1493 (1964).
[CrossRef]

Eckbreth, A. C.

Edner, H.

M. Aldén, P.-E. Bengtsson, H. Edner, “Rotational CARS generation through a multiple four-color interaction,” Appl. Opt. 25, 4493 (1986).
[CrossRef] [PubMed]

M. Aldén, P. E. Bengtsson, H. Edner, S. Kröll, D. Nilsson, “Rotational CARS:comparison of different techniques with emphasis on temperature accuracies,” to be submitted to Appl. Opt.

Eesley, G. L.

G. L. Eesley, Coherent Raman Spectroscopy (Pergamon, New York, 1981), Chap. 3B.

Farrow, R. L.

Greenhalgh, D. A.

Hall, R. J.

S. Kröll, M. Aldén, T. Berglind, R. J. Hall, “Noise characteristics of single shot broadband Raman resonant CARS with single- and multimode lasers,” Appl. Opt. 26, 1068 (1987).
[CrossRef]

R. J. Hall, D. A. Greenhalgh, “Noise properties of single-pulse CARS spectroscopy with multimode pump sources,” J. Opt. Soc. Am. B 3, 1637 (1986).
[CrossRef]

R. J. Hall, “Theoretical analysis of non-thermal pump effects in broadband CARS spectroscopy,” Opt. Quantum Electron. 18, 319 (1986).
[CrossRef]

D. A. Greenhalgh, R. J. Hall, “A closed form solution for the CARS intensity convolution,” Opt. Commun. 57, 125 (1986).
[CrossRef]

R. J. Hall, “The statistical behaviour of nonresonant CARS intensities,” Opt. Commun. 56, 127 (1985).
[CrossRef]

R. J. Hall, A. C. Eckbreth, “Coherent anti-Stokes Raman spectroscopy (CARS) application to combustion diagnostics,” Laser Appl. 5, 213 (1984).

Hirose, C.

Kataoka, H.

King, D. S.

Kröll, S.

S. Kröll, M. Aldén, T. Berglind, R. J. Hall, “Noise characteristics of single shot broadband Raman resonant CARS with single- and multimode lasers,” Appl. Opt. 26, 1068 (1987).
[CrossRef]

M. Aldén, P. E. Bengtsson, H. Edner, S. Kröll, D. Nilsson, “Rotational CARS:comparison of different techniques with emphasis on temperature accuracies,” to be submitted to Appl. Opt.

S. Kröll, D. Sandell, “A model for calculating the noise due to the stochastic nature of multimode laser radiation in nonlinear optical processes,” in Proceedings of the International Conference on Nonlinear Optics, V. J. Corcoran, ed. (Society for Optics and Quantum Electronics, McLean, Va., to be published).

Lefebvre, M.

Levenson, M. D.

M. D. Levenson, Introduction to Nonlinear Laser Spectroscopy (Academic, New York, 1982).

Li, Z. W.

Lucht, R. P.

Maeda, S.

Mood, A. M.

A. M. Mood, Introduction to the Theory of Statistics (McGraw-Hill, New York, 1974), Chap. 5.

Mueller, R. E.

Nilsson, D.

M. Aldén, P. E. Bengtsson, H. Edner, S. Kröll, D. Nilsson, “Rotational CARS:comparison of different techniques with emphasis on temperature accuracies,” to be submitted to Appl. Opt.

Parameswaran, T.

Pealat, M.

Radzewicz, C.

Rahn, L. A.

Raymer, M. G.

Roh, W. B.

W. B. Roh, P. W. Schreiber, J. P. E. Taran, “Single-pulse coherent anti-Stokes Raman scattering,” Appl. Phys. Lett. 29, 174 (1976).
[CrossRef]

Sandell, D.

S. Kröll, D. Sandell, “A model for calculating the noise due to the stochastic nature of multimode laser radiation in nonlinear optical processes,” in Proceedings of the International Conference on Nonlinear Optics, V. J. Corcoran, ed. (Society for Optics and Quantum Electronics, McLean, Va., to be published).

Sawchuk, R. A.

Schreiber, P. W.

W. B. Roh, P. W. Schreiber, J. P. E. Taran, “Single-pulse coherent anti-Stokes Raman scattering,” Appl. Phys. Lett. 29, 174 (1976).
[CrossRef]

Smallwood, G. J.

Snelling, D. R.

Snyder, J. J.

Stufflebeam, J. H.

Taran, J. P. E.

W. B. Roh, P. W. Schreiber, J. P. E. Taran, “Single-pulse coherent anti-Stokes Raman scattering,” Appl. Phys. Lett. 29, 174 (1976).
[CrossRef]

Taran, J.-P.

Teets, R. E.

Tellex, P. A.

Westling, L. A.

Whittley, S. T.

Yueh, F. Y.

F. Y. Yueh, E. J. Beiting, “Analytical expressions for coherent anti-Stokes Raman spectral (CARS) profiles,” Comp. Phys. Commun. 42, 65 (1986).
[CrossRef]

Yuratich, M. A.

M. A. Yuratich, “Effects of laser linewidth on coherent anti-Stokes Raman spectroscopy,” Mol. Phys. 38, 625 (1979).
[CrossRef]

Appl. Opt. (10)

D. A. Greenhalgh, S. T. Whittley, “Mode noise in broadband CARS spectroscopy,” Appl. Opt. 24, 907 (1985).
[CrossRef] [PubMed]

D. R. Snelling, R. A. Sawchuk, R. E. Mueller, “Single pulse CARS noise: a comparison between single-mode and multimode pump lasers,” Appl. Opt. 24, 2771 (1985).
[CrossRef] [PubMed]

D. R. Snelling, G. J. Smallwood, R. A. Sawchuk, T. Parameswaran, “Precision of multiplex CARS temperatures using both single-mode and multimode pump lasers,” Appl. Opt. 26, 99 (1987).
[CrossRef] [PubMed]

S. Kröll, M. Aldén, T. Berglind, R. J. Hall, “Noise characteristics of single shot broadband Raman resonant CARS with single- and multimode lasers,” Appl. Opt. 26, 1068 (1987).
[CrossRef]

M. Pealat, P. Bouchardy, M. Lefebvre, J.-P. Taran, “Precision of multiplex CARS temperature measurements,” Appl. Opt. 24, 1012 (1985).
[CrossRef] [PubMed]

A. C. Eckbreth, G. M. Dobbs, J. H. Stufflebeam, P. A. Tellex, “CARS temperature and species measurements in augmented jet engine exhausts,” Appl. Opt. 23, 1328 (1984).
[CrossRef] [PubMed]

D. R. Snelling, T. Parameswaran, G. J. Smallwood, “Noise characteristics of single-shot broadband CARS signals,” Appl. Opt. 26, 4298 (1987).
[CrossRef] [PubMed]

A. C. Eckbreth, T. J. Anderson, “Dual broadband CARS for simultaneous, multiple species measurements,” Appl. Opt. 24, 2731 (1985).
[CrossRef] [PubMed]

M. Aldén, P.-E. Bengtsson, H. Edner, “Rotational CARS generation through a multiple four-color interaction,” Appl. Opt. 25, 4493 (1986).
[CrossRef] [PubMed]

R. E. Teets, “CARS signals: phase matching, transverse modes, and optical effects,” Appl. Opt. 25, 855 (1986).
[CrossRef]

Appl. Phys. Lett. (1)

W. B. Roh, P. W. Schreiber, J. P. E. Taran, “Single-pulse coherent anti-Stokes Raman scattering,” Appl. Phys. Lett. 29, 174 (1976).
[CrossRef]

Appl. Spectrosc. (1)

Comp. Phys. Commun. (1)

F. Y. Yueh, E. J. Beiting, “Analytical expressions for coherent anti-Stokes Raman spectral (CARS) profiles,” Comp. Phys. Commun. 42, 65 (1986).
[CrossRef]

J. Opt. Soc. Am. B (6)

Laser Appl. (1)

R. J. Hall, A. C. Eckbreth, “Coherent anti-Stokes Raman spectroscopy (CARS) application to combustion diagnostics,” Laser Appl. 5, 213 (1984).

Mol. Phys. (1)

M. A. Yuratich, “Effects of laser linewidth on coherent anti-Stokes Raman spectroscopy,” Mol. Phys. 38, 625 (1979).
[CrossRef]

Opt. Commun. (2)

R. J. Hall, “The statistical behaviour of nonresonant CARS intensities,” Opt. Commun. 56, 127 (1985).
[CrossRef]

D. A. Greenhalgh, R. J. Hall, “A closed form solution for the CARS intensity convolution,” Opt. Commun. 57, 125 (1986).
[CrossRef]

Opt. Lett. (5)

Opt. Quantum Electron. (1)

R. J. Hall, “Theoretical analysis of non-thermal pump effects in broadband CARS spectroscopy,” Opt. Quantum Electron. 18, 319 (1986).
[CrossRef]

Phys. Rev. (1)

J. Ducuing, N. Bloembergen, “Statistical fluctuations in nonlinear optical processes,” Phys. Rev. 133, A1493 (1964).
[CrossRef]

Other (6)

A. M. Mood, Introduction to the Theory of Statistics (McGraw-Hill, New York, 1974), Chap. 5.

M. Aldén, P. E. Bengtsson, H. Edner, S. Kröll, D. Nilsson, “Rotational CARS:comparison of different techniques with emphasis on temperature accuracies,” to be submitted to Appl. Opt.

S. Kröll, D. Sandell, “A model for calculating the noise due to the stochastic nature of multimode laser radiation in nonlinear optical processes,” in Proceedings of the International Conference on Nonlinear Optics, V. J. Corcoran, ed. (Society for Optics and Quantum Electronics, McLean, Va., to be published).

M. Aldén, Department of Atomic Physics, Combustion Centre, Lund Institute of Technology, Box 118, S-221 Lund, Sweden (personal communication).

G. L. Eesley, Coherent Raman Spectroscopy (Pergamon, New York, 1981), Chap. 3B.

M. D. Levenson, Introduction to Nonlinear Laser Spectroscopy (Academic, New York, 1982).

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Figures (2)

Fig. 1
Fig. 1

Conceptual picture of the CARS process. ωa, ωb, and ωc represent the frequencies of the incoming photons, and ωas represents the frequency of the CARS photon. A Raman resonance is indicated.

Fig. 2
Fig. 2

A conceptual picture of an instrument function with the spectral width W and two anti-Stokes modes with a frequency separation Δω transmitted through the detection window. The two modes will give rise to a beat signal at frequency Δω in the detector. The horizontal scale is frequency, and the vertical scale is intensity (arbitrary units).

Tables (5)

Tables Icon

Table 1 Laser Types Used for Each of the Steps in the CARS Process Depicted in Fig. 1

Tables Icon

Table 2 Definition of Symbols

Tables Icon

Table 3 Values Used When Calculating the Noise in Tables 4 and 5a

Tables Icon

Table 4 Calculated Noise for Nonresonant CARS

Tables Icon

Table 5 Calculated Noise in a Spectrum Where a Single Raman Resonance Contributes to the Signal in Each Diode

Equations (114)

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E a = a exp [ i ( ω a t + θ a ) ] , E b = b exp [ i ( ω a t + θ a ) ] ,
I 0 = | E a + E b | 2 = a 2 + b 2 + 2 a b cos [ ( ω a ω b ) t + θ a θ b ] .
2 a b cos [ ( ω a ω b ) t + θ a θ b ]
E a = k a k exp { 1 2 ( k Ω a Γ a ) 2 + i [ ( ω a 0 + k Ω a ) t + θ k ] } + c . c . , E b = r b r exp { 1 2 ( r Ω b Γ b ) 2 + i [ ( ω b 0 + r Ω b ) t + θ r ] } + c . c . , E c = m c m exp { 1 2 ( m Ω c Γ c ) 2 + i [ ( ω c 0 + m Ω c ) t + ψ m ] } + c . c .
P ( 3 ) χ ( 3 ) E a E b E c ,
ω as 0 = ω a 0 ω b 0 + ω c 0
E as krm E as ( ω k r m ) = k r m χ k r m ( 3 ) a k b r c m × exp { 1 2 [ ( k Ω a Γ a ) 2 + ( r Ω b Γ b ) 2 + ( m Ω c Γ c ) 2 ] } × exp { i [ ω a 0 ω b 0 + ω c 0 + k Ω a r Ω b + m Ω c ) t + θ k θ r + ψ m ] } + c . c . ,
I = k l r s m n a k a l b r b s c m c n exp [ i ( θ k θ l ϕ r + ϕ s + ψ m ψ n ) ] × H ( k , , l , r , s , m , n ) .
E ( I ) = i a i b i c k r m H ( k , k , r , r , m , m ) ,
V ( I ) = ( i a i b i c ) 2 k l r s m n [ H ( klrrmm ) H ( klssnn ) + H ( kkrsmm ) H ( llrsnn ) + H ( kkrrmn ) H ( llssmn ) + H ( kkrsmn ) H ( llrsmn ) + H ( klrrmn ) H ( klssmn ) + H ( klrsmm ) H ( klrsnn ) + H ( klrsmn ) H ( klrsmn ) ] .
k = 1 N ( 2 f k ) ! k = 1 N ( f k ! ) 2
k = 1 N f k !
k r m E as ( ω k r m ) S α ( ω k r m ) ,
k r m l s n 1 T T / 2 T / 2 E as ( ω k r m ) S α ( ω k r m ) E as ¯ ( ω l s n ) S α ¯ ( ω l s n ) d t = I ( Δ α ) ,
H Δ α ( k , l , r , s , m , n ) = χ k r m ( 2 ) χ l s n ( 37 ) ¯ exp { 1 2 [ ( k 2 + l 2 ) Ω a 2 Γ a 2 + ( r 2 + s 2 ) Ω b 2 Γ b 2 + ( m 2 + n 2 ) Ω c 2 Γ c 2 ] } × exp { 1 2 [ ( k Ω a r Ω b + m Ω c Δ α ) 2 + ( l Ω a s Ω b + n Ω c Δ α ) 2 W 2 ] } sinc ( Δ ω T 2 ) ,
sinc ( x ) = sin ( x ) x Δ ω = ( k l ) Ω a ( r s ) Ω b + ( m n ) Ω c ,
Δ α = ω α ω a 0 + ω b 0 ω c 0 .
I N ( Δ α ) = I ( Δ α ) β I ( Δ β ) ,
Ω x min ( W , Γ x ) ,
T Ω x π / 2 .
H ( k , l , r , s , m , n ) = exp { 1 2 [ ( k 2 + l 2 ) Ω a 2 Γ a 2 + ( r 2 + s 2 ) Ω b 2 Γ b 2 + ( m 2 + n 2 ) Ω c 2 Γ c 2 ] } × exp { 1 2 [ ( k Ω a r Ω b + m Ω c Δ α ) 2 + ( l Ω a s Ω b + n Ω c Δ α ) 2 W 2 ] } × sinc { T 2 [ ( k l ) Ω a ( r s ) Ω b + ( m n ) Ω c ] } ( k Ω a r Ω b Δ α + i Γ r ) ( l Ω a s Ω b Δ α i Γ r ) ,
Γ a , Γ b , Γ c , W Γ r 1 / T ,
Δ ω = ( k l + m n ) Ω a ( r s ) Ω b = P Ω a + S Ω b ,
σ ( I ) I = 1 ( 2 π ) 1 / 4 { x = a , b , c Ω x Γ x Γ 1 [ ( Γ 2 Γ x 2 ) ] 1 / 2 + 2 π T x = a , b , c [ 1 W 2 + Γ x 2 1 Γ 2 ] 1 / 2 + 2 π W T } 1 / 2 .
H ( k , l , r , s , m , n ) = exp { 1 2 [ ( k 2 l 2 ) Ω a 2 Γ a 2 + ( r 2 s 2 ) Ω b 2 Γ b 2 + + ( m 2 n 2 ) Ω c 2 Γ c 2 + + ( k Ω a r Ω b + m Ω c Δ α ) 2 + ( 1 Ω a s Ω b n Ω c Δ α ) 2 W 2 ] } × sinc { T 2 [ ( k l ) Ω a ( r s ) Ω b + ( m n ) Ω c ] } ,
V ( F ) = k V ( x i ) [ ( F x i ) 2 ] E ,
k V ( a k ) [ ( I N a k ) 2 ] E
Γ x ( Γ a 2 + Γ b 2 + Γ c 2 + W 2 ) 1 / 2
( Ω b / 2 π Γ r ) 1 / 2
1 / ( Γ r T ) 1 / 2
I j = klrsmn a k a l b r b s c m c n exp { i [ ( θ k θ l ) ( ϕ r ϕ s ) + ( ψ m ψ n ) ] } H j ( k , l , r , s , m , n ) , j = 1 , 2 ,
C ( Z , W ) = C { [ Z E ( Z ) ] [ W E ( W ) ] ¯ } = E Z ( W ¯ ) E ( Z ) E ( W ) ¯ .
E ( X 2 ) = m ,
E ( X 4 ) = 2 m 2 .
E ( e i k Y ) = 1 if k = 0 , and 0 otherwise .
E ( I 1 ) = k l r s m n E ( a k a l ) E ( b r b s ) E ( c m c n ) E { exp [ i ( θ k θ l ) ] } × E { exp [ i ( ϕ s ϕ r ) ] } E exp [ i ( ψ m ψ n ) ] } × H 1 ( k , l , r , s , m , n ) .
E { exp [ i ( θ k θ l ) ] } = E [ exp ( i θ k ) ] E [ e ( i θ l ) ] = 0 .
E ( I 1 ) = k r m E ( a k 2 ) E ( b r 2 ) E ( c m 2 ) H 1 ( k , k , r , r , m , m ) .
E ( I 1 ) = ( i a i b i c ) k r m H 1 ( k , k , r , r , m , m ) .
I j = n = 1 8 S j ( n ) , j = 1 , 2 ,
S j ( 1 ) = klrsmn k l , r s , m n a k a l b r b s c m c n exp { i [ θ k θ l ) + ( ϕ s ϕ r ) + ( ψ m ψ n ) ] } H j ( k , l , r , s , m , n ) , S j ( 2 ) = klrsm k l , r s a k a l b r b s c m 2 exp { i [ θ k θ l ) + ( ϕ s ϕ r ) ] } H j ( k , l , r , s , m , n ) , S j ( 3 ) = klrmn k l , m n a k a l b r 2 c m c n exp { i [ θ k θ l ) + ( ψ m ψ n ) ] } H j ( k , l , r , r , m , n ) , S j ( 4 ) = krsmn r s , m n a k 2 b r b s c m c n exp { i [ θ s θ r ) + ( ψ m ψ n ) ] } H j ( k , k , r , s , m , n ) , S j ( 5 ) = klrm k l a k a l b r 2 c m 2 exp i [ θ k θ l ) ] H j ( k , l , r , r , m , m ) , S j ( 6 ) = krsm r s a k 2 b r b s c m 2 exp [ i ( ϕ s ϕ r ) ] H j ( k , k , r , s , m , m ) , S j ( 7 ) = krmn m n a k 2 b r 2 c m c n exp [ i ( ψ m ψ n ) ] H j ( k , k , r , r , m , n ) , S j ( 8 ) = k r m a k 2 b r 2 c m 2 H j ( k , k , r , r , m , m ) , j = 1 , 2 .
C ( I 1 , I 2 ) = u = 1 8 C [ S 1 ( u ) , S 2 ( u ) ] .
C [ S 1 ( u ) , S 2 ( u ) ] , u = 1 , 8 : C [ S 1 ( 1 ) , S 2 ( 1 ) ] = klrsnopqtu υ k l , r s , m n o p , q t , u υ H 1 ( k , l , r , s , m , n ) H 2 ( o , p , q , t , u , υ ) ¯ C 1 ,
C 1 = C ( a k a l b r b s c m c n exp { i [ ( θ k θ l ) + ( ϕ s ϕ r ) + ( ψ m ψ n ) ] } , a o a p b q b t c u c υ exp { i [ ( θ o θ p ) + ( ϕ t ϕ q ) + ( ψ u ψ υ ) ] } ) .
E ( a k a l b r b s c m c n exp { i [ ( θ k θ l ) + ( ϕ s ϕ r ) + ( ψ m ψ n ) ] } ) = 0
C 1 = E ( a k a l a o a p ) E ( b r b s b q b t ) E ( c m c n c u c υ ) × E { exp [ i ( θ k θ l + θ p θ o ) ] } E { exp [ i ( ϕ s ϕ r + ϕ q ϕ t ) ] } × E { exp [ i ( ψ m ψ n + ψ u ψ υ ] } .
C 1 = ( i a i b i c ) 2
C [ S 1 ( 1 ) , S 2 ( 1 ) ] = ( i a i b i c ) 2 klrsmn k l , r s , m n H 1 ( k , l , r , s , m , n ) H 2 ( k , l , r , s , m , n ) ¯ , C [ S 1 ( 2 ) , S 2 ( 2 ) ] = klrsmopqtn k l , r s , o p , q t H 1 ( k , l , r , s , m , m ) H 2 ( o , p , q , t , n , n ) ¯ C 2 ,
C 2 = C ( a k a l b r b s c m 2 ) exp { i [ ( θ k θ l ) + ( ϕ s ϕ r ) ] } , a o a p b q b t c n 2 exp { i [ ( θ o θ p ) + ϕ q ϕ ] } ) .
C 2 = E ( a k a l a o a p ) E ( b r b s b q b t ) E ( c m 2 c n 2 ) × E { exp [ i ( θ k θ l + θ p θ o ) ] } × E { exp [ i ( ϕ s ϕ r + ϕ q ϕ t ) ] } = E ( a k 2 a l 2 ) E ( b r 2 b s 2 ) E ( c m 2 c n 2 ) = ( i a i b ) 2 E ( c m 2 c n 2 ) ,
C 2 = ( i a i b ) 2 E ( c m 4 ) = 2 ( i a i b i c ) 2 .
C 2 = ( i a i b ) 2 E ( c m 2 ) E ( c n 2 ) = ( i a i b i c ) 2 .
C [ S 1 ( 2 ) , S 2 ( 2 ) ] = ( i a i b i c ) 2 [ klrsmn k l , r s , m n H 1 ( k , l , r , s , m , m ) × H 2 ( k , l , r , s , n , n ) ¯ + 2 klrsm k l , r s H 1 ( k , l , r , s , m , m ) × H 2 ( k , l , r , s , m , m ) ¯ ] .
C [ S 1 ( 3 ) , S 2 ( 3 ) ] = ( i a i b i c ) 2 [ klrsmn k l , r s , m n H 1 ( k , l , r , r , m , n ) H 2 ( k , l , s , s , m , n ) ¯ , + 2 klrmn k l , m n H 1 ( k , l , r , r , m , n ) H 2 ( k , l , r , r , m , n ) ¯ ] .
C [ S 1 ( 4 ) , S 2 ( 4 ) ] = ( i a i b i c ) 2 [ klrsmn k l , r s , m n H 1 ( k , k , r , s , m , n ) H 2 ( l , l , r , s , m , n ) ¯ + 2 klrmn r s , m n H 1 ( k , k , r , s , m , n ) H 2 ( k , k , r , s , m , n ) ¯ ] .
C [ S 1 ( 5 ) , S 2 ( 5 ) ] = klrmopsn k l , o p H 1 ( k , l , r , r , m , m ) H 2 ( o , p , s , s , n , n ) ¯ C 3 ,
C 3 = C { a k a l b r 2 c m 2 exp [ i ( θ k θ l ) ] , a o a p b s 2 c n 2 exp [ i ( θ o θ p ) ] } .
C 3 = i a 2 E ( b r 2 b s 2 ) E ( c m 2 c n 2 ) if k = o and l = p , and 0 otherwise .
r = s , m n C 3 = 4 ( i a i b i c ) 2 , r = s , m n C 3 = 2 ( i a i b i c ) 2 , r s , m = n C 3 = 2 ( i a i b i c ) 2 , r s , m n C 3 = ( i a i b i c ) 2
C [ S 1 ( 5 ) , S 2 ( 5 ) ] = ( i a i b i c ) 2 [ klrsmn k l , r s , m n H 1 ( k , l , r , r , m , m ) H 2 ( k , l , s , s , n , n ) ¯ + 2 klrsm k l , r s H 1 ( k , l , r , r , m , m ) H 2 ( k , l , s , s , m , m ) ¯ + 2 klrmn k l , m n H 1 ( k , l , r , r , m , m ) H 2 ( k , l , r , r , n , n ) ¯ + 4 k l r m k l H 1 ( k , l , r , r , m , m ) H 2 ( k , l , r , r , m , m ) ¯ ] .
C [ S 1 ( 6 ) , S 2 ( 6 ) ] = ( i a i b i c ) 2 [ klrsmn k l , r s , m n H 1 ( k , k , r , s , m , m ) H 2 ( l , l , r , s , n , n ) ¯ + 2 klrsm k l , r s H 1 ( k , k , r , s , m , m ) H 2 ( l , l , r , s , m , m ) ¯ + 2 krsmn r s , m n H 1 ( k , k , r , s , m , m ) H 2 ( k , k , r , s , n , n ) ¯ + 4 krsm r s , H 1 ( k , k , r , s , m , m ) H 2 ( k , k , r , s , m , m ) ¯ ] ,
C [ S 1 ( 7 ) , S 2 ( 7 ) ] = ( i a i b i c ) 2 [ klrsmn k l , r s , m n H 1 ( k , k , r , r , m , n ) H 2 ( l , l , s , s , m , n ) ¯ + 2 klrmn k l , m n H 1 ( k , k , r , r , m , n ) H 2 ( l , l , r , r , m , n ) ¯ + 2 krsmn r s , m n H 1 ( k , k , r , s , m , n ) H 2 ( k , k , s , s , m , n ) ¯ + 4 krmn m n H 1 ( k , k , r , r , m , n ) H 2 ( k , k , r , r , m , n ) ¯ ] .
C [ S 1 ( 8 ) , S 2 ( 8 ) ] = klrsmn H 1 ( k , k , r , r , m , m ) H 2 ( l , l , s , s , n , n ) ¯ C 4 ,
C 4 = C ( a k 2 b r 2 c m 2 , a l 2 b s 2 c n 2 ) = E ( a k 2 a l 2 ) E ( b r 2 b s 2 ) E ( c m 2 c n 2 ) ( i a i b i c ) 2 .
C [ S 1 ( 8 ) , S 2 ( 8 ) ] = ( i a i b i c ) 2 [ 7 k r m H 1 ( k , k , r , r , m , n ) H 2 ( k , k , r , r , m , m ) ¯ + 3 krmn m n H 1 ( k , k , r , r , m , m ) H 2 ( k , k , r , r , n , n ) ¯ + 3 krsm r s H 1 ( k , k , r , r , m , m ) H 2 ( k , k , s , s , m , m ) ¯ + 3 klrm k l H 1 ( k , k , r , r , m , m ) H 2 ( l , l , r , r , m , m ) ¯ + krsmn r s , m n H 1 ( k , k , r , r , m , m ) H 2 ( k , k , s , s , n , n ) ¯ + klrmn k l , m n H 1 ( k , k , r , r , m , m ) H 2 ( l , l , r , r , n , n ) ¯ + klrsm k l , r s H 1 ( k , k , r , r , m , m ) H 2 ( l , l , s , s , m , m ) ¯ ] .
C ( I 1 , I 2 ) = ( i a i b i c ) 2 [ H 1 ( k , l , r , s , m , n ) H 2 ( k , l , r , s , m , n ) ¯ + H 1 ( k , l , r , s , m , m ) H 2 ( k , l , r , s , n , n ) ¯ + H 1 ( k , l , r , r , m , n ) H 2 ( k , l , s , s , m , n ) ¯ + H 1 ( k , k , r , s , m , n ) H 2 ( l , l , r , s , m , n ) ¯ + H 1 ( k , l , r , r , m , m ) H 2 ( k , l , s , s , n , n ) ¯ + H 1 ( k , k , r , s , m , m ) H 2 ( l , l , r , s , n , n ) ¯ + H 1 ( k , k , r , r , m , n ) H 2 ( l , l , s , s , m , n ) ¯ ] ,
E ( I ) = ( i a i b i c ) k r m H ( k , k , r , r , m , m ) , V ( I ) = ( i a i b i c ) 2 klrsmn [ | H ( k , l , r , s , m , n ) | 2 + H ( k , l , r , s , m , m ) H ( k , l , r , s , n , n ) ¯ + H ( k , l , r , r , m , n ) H ( k , l , s , s , m , n ) ¯ + H ( k , k , r , s , m , n ) H ( l , l , r , s , m , n ) ¯ + H ( k , l , r , r , m , m ) H ( k , l , s , s , n , n ) ¯ + H ( k , k , r , s , m , m ) H ( l , l , r , s , n , n ) ¯ + H ( k , k , r , r , m , n ) H ( l , l , s , s , m , n ) ¯ ] .
I ( Δ ) = klrsmn a k a l b r b s c m c n exp { i [ ( θ k θ l ) ( ϕ r ϕ s ) + ( ψ m ψ n ) ] } H Δ ( k , l , r , s , m , n ) ,
H Δ ( k , l , r , s , m , n ) = exp { 1 2 [ Ω a 2 Γ a 2 ( k 2 + l 2 ) + Ω b 2 Γ b 2 ( r 2 + s 2 ) + Ω c 2 Γ c 2 ( m 2 + m 2 ) ] } × exp { 1 2 W 2 [ ( Ω a k Ω b r + Ω c m Δ ) 2 + ( Ω a l Ω b s + Ω c n Δ ) 2 } × sinc { T 2 [ Ω a ( k l ) Ω b ( r s ) + Ω c ( m n ) ] } .
Ω x min ( W , Γ x ) , x = a , b , c .
k r m H Δ ( k , k , , r , r , m , m ) R { exp [ ( Ω a 2 Γ a 2 k 2 + Ω b 2 Γ b 2 r 2 + Ω c 2 Γ c 2 m 2 ) ] exp [ 1 W 2 ( k Ω a r Ω b + m Ω c Δ ) 2 ] } d m d r d k .
R exp ( p x 2 ± q x ) d x = exp { q 2 / ( 4 p ) } ( π / p ) 1 / 2 ,
E [ I ( Δ ) ] ( i a i b i c ) π ( π ) 1 / 2 ( Γ a Γ b Γ c Ω a Ω b Ω c ) W Γ exp ( Δ 2 / Γ 2 ) ,
C Δ = π ( π ) 1 / 2 ( Γ a Γ b Γ c Ω a Ω b Ω c ) W Γ exp ( Δ 2 / Γ 2 ) .
f ( a ) = R exp ( a x 2 ) sinc 2 ( x ) d x .
C [ I ( Δ α ) , I ( Δ β ) ] ( i a i b i c ) 2 C Δ α C Δ β 1 ( 2 π ) 1 / 2 × { 2 T W ( 1 W 2 Γ 2 ) 1 / 2 f [ 2 Γ 2 T 2 W 2 ( Γ 2 W 2 ) ] × exp [ 1 2 ( Δ α Δ β ) 2 Γ 2 W 2 Γ 2 W 2 ] + x = a , b , c ( Ω x Γ x ( 1 Γ x 2 Γ 2 ) 1 / 2 × exp [ 1 2 ( Δ α Δ β ) 2 Γ x 2 Γ 2 ( Γ 2 Γ x 2 ) ] + 2 T { [ ( W 2 + Γ x 2 ) ( 1 W 2 + Γ x 2 Γ 2 ) ] 1 / 2 × f [ 2 ( Γ 2 Γ x 2 ) T 2 W 2 ( Γ 2 W 2 Γ x 2 ) ] × exp [ 1 2 ( Δ α Δ β ) 2 Γ 2 W 2 Γ x 2 Γ 2 ( W 2 + Γ x 2 ) ] } ) } ,
σ ( I ) I = [ V ( I ) E ( I ) 2 ] 1 / 2 ( 2 π ) 1 / 4 [ x = a , b , c [ Ω x Γ x ( 1 Γ x 2 Γ 2 ) 1 / 2 ] + 2 T x = a , b , c ( f { 2 W 2 T 2 [ 1 + W 2 Γ 2 ( Γ x 2 + W 2 ) ] } × { ( W 2 + Γ x 2 ) [ 1 ( W 2 + Γ x 2 ) Γ 2 ] } 1 / 2 ) + 2 T W ( 1 W 2 Γ 2 ) 1 / 2 f [ 2 Γ 2 T 2 W 2 ( Γ 2 W 2 ) ] ] 1 / 2 .
I N ( Δ α ) = I ( Δ α ) / β I ( Δ β ) ,
E ( X Y ) E ( X ) E ( Y ) V ( X Y ) V ( X ) E ( Y ) 2 + V ( Y ) E ( Y ) 2 E ( Y ) 4 2 Re [ C ( X , Y ) ] E ( Y ) 2 E ( Y ) 3 ,
V ( X / Y ) E ( X / Y ) 2 V ( X ) E ( X ) 2 + V ( Y ) E ( Y ) 2 2 Re [ C ( X , Y ) ] E ( X ) E ( Y ) .
E ( X ) = E [ I ( Δ α ) ] , E ( Y ) = E [ β I ( Δ β ) ] = β E [ I ( Δ β ) ] , V ( X ) = V [ I ( Δ α ) = C [ I ( Δ α ) , I ( Δ α ) ] , C ( X , Y ) = C [ I ( Δ α ) , β I ( Δ β ) ] = β C [ I ( Δ α ) , I ( Δ β ) ] , V ( Y ) = V [ β I ( Δ β ) ] = C [ α I ( Δ α ) ] = α β C [ I ( Δ α ) , I ( Δ β ) ] .
F Δ α = 1 2 β [ Δ β 2 Γ 2 1 Γ 2 ( Δ β Δ α ) 2 x 2 ( 1 x ) ] β exp ( Δ β 2 / Γ 2 ) + β γ [ 1 Γ 2 ( Δ β 2 + Δ γ 2 ) 1 Γ 2 ( Δ β Δ γ ) 2 x 2 ( 1 x ) ] [ β exp ( Δ β 2 / Γ 2 ) ] 2 ,
V [ I N ( Δ α ) ] E [ I N ( Δ α ) ] 2 1 ( 2 π ) 1 / 2 ( 2 T W ( 1 W 2 Γ 2 ) 1 / 2 f [ 2 Γ 2 T 2 W 2 ( Γ 2 W 2 ) ] × F Δ α ( W 2 Γ 2 ) + 2 T a , b , c { [ ( W 2 + Γ x 2 ) ( 1 ( W 2 + Γ x 2 ) Γ 2 ) ] 1 / 2 × f [ 2 ( Γ 2 Γ x 2 ) T 2 W 2 ( Γ 2 W 2 Γ x 2 ) ] F Δ α ( 1 ( W 2 + Γ x 2 ) Γ 2 ) } + x = a , b , c [ Ω x Γ x ( 1 Γ x 2 Γ 2 ) 1 / 2 F Δ α ( Γ x 2 Γ 2 ) ] ) .
Φ ( x ) = 1 ( 2 π ) 1 / 2 x exp ( y 2 / 2 ) d y .
f ( a ) f ( 0 ) = o a f ( y ) d y , f ( 0 ) = R sinc 2 ( x ) d x = π , f ( y ) = R sinc 2 ( x ) exp ( y x 2 ) d x = 1 2 ( π / y ) 1 / 2 [ 1 exp ( 1 / y ) ] .
o a f ( y ) d y = ( π ) 1 / 2 2 [ o a y 1 / 2 exp ( 1 / y ) d y o a y 1 / 2 d y ] = ( π ) 1 / 2 2 ( 2 ( a ) 1 / 2 exp ( 1 / a ) 4 ( π ) 1 / 2 × { 1 Φ [ ( 2 / a ) 1 / 2 ] } 2 ( a ) 1 / 2 ) .
f ( a ) = π 2 π { 1 Φ [ ( 2 / a ) 1 / 2 ] } ( π a ) 1 / 2 [ 1 exp ( 1 / a ) ] .
( 1 x 1 x 3 ) γ ( x ) < 1 Φ ( x ) < 1 x γ ( x ) ,
1 Φ ( x ) exp ( x 2 / 2 ) / [ x ( 2 π ) 1 / 2 ]
1 Φ [ ( 2 / a ) 1 / 2 ] ( a / 2 ) 1 / 2 exp ( 1 / a ) / ( 2 π ) 1 / 2 = ½ ( a / π ) 1 / 2 exp ( 1 / a ) .
f ( a ) π ( π a ) 1 / 2 exp ( 1 / a ) ( π a ) 1 / 2 + ( π a ) 1 / 2 exp ( 1 / a ) = π ( π a ) 1 / 2 .
F Δ ( 1 ) 1 , F Δ ( ) < ,
F k = υ = 1 k f υ , k 1 ,
I j = u { k = 1 N [ x = 2 F k 1 + 1 2 F k a k [ u ( x ) ] × exp ( i y = F k 1 + 1 F k { θ k [ u ( 2 y 1 ) ] θ k [ u ( 2 y ) ] } ) × H j [ u ( 1 ) , u ( 2 ) , , u ( 2 F N ) ] ] } .
P n ( 2 m ) [ H ( k 1 , , k n , k n + 1 , , k n + 2 m , k n + 2 m + 1 , , k 2 N ) ] = ( j = 1 m Σ i j A n ( m ) i 1 i j ) H ( k 1 , , k n , k n + 1 , k i 1 , k n + 3 , k i 2 , , k n + 2 m 1 , k i m , k n + 2 m + 1 , , k 2 N ) ,
j = 1 m i j A n ( m ) i 1 i j = i m A n ( m ) i 1 i m i 3 A n ( m ) i 1 i 2 i 3 i 2 A n ( m ) i 1 i 2 Σ i 1 A n ( m ) .
H 1 ( s 1 , , s n , k n + 1 , , k n + 2 m , t n + 2 m + 1 , , t 2 N ) × H 2 ( u 1 , , u n , k n + 1 , , k n + 2 m , υ n + 2 m + 1 , , υ 2 N ) ,
P n ( 2 m ) [ H 1 ( s 1 , , s n , k n + 1 , , k n + 2 m , t n + 2 m + 1 , , t 2 N ) × H 2 ( u 1 , , u n , k n + 1 , , k n + 2 m , υ n + 2 m + 1 , , υ 2 N ) ] = ( j = 1 2 m Σ i j A n ( m ) i 1 i j ) [ H 1 ( s 1 , , s n , k n + 1 , k i 1 , k n + 3 , k i 2 , , k n + 2 m 1 , k i m , t n + 2 m + 1 , t 2 N ) × H 2 ( u 1 , , u n , k i m + 1 , k n + 2 , k i m + 2 , k n + 4 , , k i 2 m , k n + 2 m , υ n + 2 m + 1 , , υ 2 N ) ] ,
E ( I 1 ) = ( υ = 1 N m υ f υ ) { [ n = 1 N P 2 F n 1 ( 2 f n ) ] × H 1 [ u ( 1 , , u ( 2 N ) ] } ,
E ( I 1 I ¯ 2 ) = ( υ = 1 N m υ f υ ) { [ n = 1 N P 2 F n 1 ( 2 f n ) ] × H 1 [ u ( 1 , , u ( 2 N ) ] H 2 [ u ( 1 ) , u ( 2 N ) ] ¯ } ,
C ( I 1 , I 2 ) = E ( I 1 I ¯ 2 ) E ( I 1 ) E ( I 2 ) ¯
V ( I 1 ) = E ( | I 1 | 2 ) | E ( I 1 ) | 2 .
σ ( I ) I = 1 ( 2 π ) 1 / 4 [ ( 4 + 2 ) Ω a Γ a Γ 2 + Ω b ( W 2 + 2 Γ a 2 ) 1 / 2 + 8 T ( W 2 + 2 Γ a 2 ) 1 / 2 + 2 π W T ] 1 / 2 .
σ ( I ) I = 1 ( 2 π ) 1 / 4 ( Ω b W ) 1 / 2 .
σ ( I ) I = 1 ( 2 π ) 1 / 4 [ x = a , b , c Ω x Γ x Γ 1 ( Γ 2 Γ x 2 ) 1 / 2 + 2 π T x = a , b , c ( 1 W 2 + Γ x 2 1 Γ 2 ) 1 / 2 + 2 π W T ] 1 / 2 .
σ ( I ) I = 1 ( 2 π ) 1 / 4 [ x = a , b Ω x Γ x Γ 1 ( Γ 2 Γ x 2 ) 1 / 2 + 2 π W T ] 1 / 2 .
σ ( I ) I = 1 ( 2 π ) 1 / 4 [ 1 2 Γ Γ a Ω a ( W 2 + Γ a 2 ) 1 / 2 + Ω b Γ b Γ 2 + 2 π W T ] 1 / 2 .
σ ( I ) I = 1 ( 2 π ) 1 / 4 ( Ω b W ) 1 / 2 .
σ ( I ) I = { 1 ( 2 π ) 1 / 2 Ω a Γ a { ( 1 + Γ a 2 W 2 ) 1 / 2 + ( 1 + Γ a 2 Γ b 2 ) 1 / 2 × [ 1 + 2 ( 1 + Γ a 2 2 W 2 + Γ a 2 ) 1 / 2 ] } × F δ + 1 ( 2 π ) 1 / 2 Ω b Γ a ( 1 + Γ a 2 Γ b 2 ) + ( 2 π ) 1 / 2 1 T Γ a [ ( 1 + Γ a 2 W 2 ) 1 / 2 + 4 ( 1 + Γ a 2 Γ b 2 ) 1 / 2 × ( 1 + Γ a 2 2 W 2 + Γ a 2 ) ] + 1 T Γ r } 1 / 2 .
σ ( I ) I = 1 ( 2 π ) 1 / 2 ( Ω b Γ r ) 1 / 2 .
σ ( I ) I = [ 1 ( 2 π ) 1 / 2 Ω c Γ c ( 1 + Γ c 2 W 2 ) 1 / 2 F δ ( 2 π ) 1 / 2 T Γ c ( 1 + Γ c 2 W 2 ) + 1 T Γ r ] 1 / 2 .
σ ( I ) I = 1 ( T Γ r ) 1 / 2 .
σ ( I ) I = [ 1 ( 2 π ) 1 / 2 Ω c Γ c ( 1 + Γ c 2 W 2 ) F δ ( 2 π ) 1 / 2 T Γ c ( 1 + Γ c 2 W 2 ) + 1 2 π Ω a Γ r ] 1 / 2 .
σ ( I ) I = 1 ( 2 π ) 1 / 2 ( Ω a Γ r ) 1 / 2 .

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