Abstract

We discuss the characteristics of photon-number squeezing generated in a spectrally filtered Raman Stokes pulse obtained from a nonlinear fiber. The Raman Stokes pulse generated from a microstructure fiber with a femtosecond input laser pulse exhibits far higher photon-number squeezing. We studied the quantum correlation established among the frequency modes in the Raman Stokes pulse by using numerical calculations and experiments. To design an intra-Stokes pulse quantum correlation through the optical nonlinearities of a fiber to obtain higher photon-number squeezing, we shaped the input laser pulse by using a self-learning closed-loop control approach.

© 2007 Optical Society of America

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  1. S. Spälter, M. Burk, U. Strössner, M. Böhm, A. Sizmann, and G. Leuchs, "Photon number squeezing of spectrally filtered sub-picosecond optical solitons," Europhys. Lett. 38, 335-340 (1997).
    [CrossRef]
  2. S. Spälter, M. Burk, U. Strössner, A. Sizmann, and G. Leuchs, "Propagation of quantum properties of sub-picosecond solitons in a fiber," Opt. Express 2, 77-83 (1998).
    [CrossRef]
  3. K. Hirosawa, H. Furumochi, A. Tada, F. Kannari, M. Takeoka, and M. Sasaki, "Photon number squeezing of ultrabroadband laser pulses generated by microstructure fibers," Phys. Rev. Lett. 94, 203601 (2005).
    [CrossRef] [PubMed]
  4. M. Takeoka, D. Fujishima, and F. Kannari, "Optimization of ultrashort-pulse squeezing by spectral filtering with the Fourier pulse-shaping technique," Opt. Lett. 26, 1592-1594 (2001).
    [CrossRef]
  5. K. Hirosawa, H. Furumochi, A. Tada, F. Kannari, M. Takeoka, and M. Sasaki, "Generation of quantum correlation between co-propagating pulses in optical fibers," presented at the European Quantum Electronics Conference, Munich, Germany, June 12-17, 2005.
  6. A. M. Weiner, S. Oudin, D. D. Leaird, and D. H. Reitze, "Shaping of femtosecond pulses using phase-only filters designed by simulated annealing," J. Opt. Soc. Am. A 10, 1112-1120 (1993).
    [CrossRef]
  7. D. Hollenbeck and C. D. Cantrell, "Multiple-vibrational-mode model for fiber-optic Raman gain spectrum and response function," J. Opt. Soc. Am. B 19, 2886-2891 (2002).
    [CrossRef]
  8. M. Shirasaki and H. A. Haus, "Squeezing of pulses in a nonlinear interferometer," J. Opt. Soc. Am. B 7, 30-34 (1990).
    [CrossRef]
  9. C. R. Doerr, M. Shirasaki, and F. I. Khatri, "Simulation of pulsed squeezing in optical fiber with chromatic dispersion," J. Opt. Soc. Am. B 11, 143-149 (1994).
    [CrossRef]
  10. N. Nishizawa, T. Horio, M. Mori, T. Goto, and K. Yamane, "Effect of group-velocity dispersion on photon-number squeezing of optical pulses using optical fibers and spectral filter," Jpn. J. Appl. Phys. Part 1 38, 1961-1965 (1999).
    [CrossRef]
  11. Y. Lai and S.-S. Yu, "General quantum theory of nonlinear optical-pulse propagation," Phys. Rev. A 51, 817-829 (1995).
    [CrossRef] [PubMed]
  12. S. J. Carter, P. D. Drumond, M. D. Reid, and R. M. Shelby, "Squeezing of quantum solitons," Phys. Rev. Lett. 58, 1841-1844 (1987).
    [CrossRef] [PubMed]
  13. P. D. Drumond and S. J. Carter, "Quantum-field theory of squeezing in solitons," J. Opt. Soc. Am. B 4, 1565-1573 (1987).
    [CrossRef]
  14. P. D. Drumond and J. F. Corney, "Quantum noise in optical fibers. I. Stochastic equations," J. Opt. Soc. Am. B 18, 139-152 (2001).
    [CrossRef]
  15. J. F. Corney, P. D. Drummond, J. Heersink, V. Josse, G. Leuchs, and U. L. Andersen, "Many-body quantum dynamics of polarization squeezing in optical fibers," Phys. Rev. Lett. 97, 23606 (2006).
    [CrossRef]
  16. S. Spälter, N. Korolkova, F. König, A. Sizmann, and G. Leuchs, "Observation of multimode quantum correlations in fiber optical solitons," Phys. Rev. Lett. 81, 786-789 (1998).
    [CrossRef]

2006

J. F. Corney, P. D. Drummond, J. Heersink, V. Josse, G. Leuchs, and U. L. Andersen, "Many-body quantum dynamics of polarization squeezing in optical fibers," Phys. Rev. Lett. 97, 23606 (2006).
[CrossRef]

2005

K. Hirosawa, H. Furumochi, A. Tada, F. Kannari, M. Takeoka, and M. Sasaki, "Photon number squeezing of ultrabroadband laser pulses generated by microstructure fibers," Phys. Rev. Lett. 94, 203601 (2005).
[CrossRef] [PubMed]

2002

2001

1999

N. Nishizawa, T. Horio, M. Mori, T. Goto, and K. Yamane, "Effect of group-velocity dispersion on photon-number squeezing of optical pulses using optical fibers and spectral filter," Jpn. J. Appl. Phys. Part 1 38, 1961-1965 (1999).
[CrossRef]

1998

S. Spälter, M. Burk, U. Strössner, A. Sizmann, and G. Leuchs, "Propagation of quantum properties of sub-picosecond solitons in a fiber," Opt. Express 2, 77-83 (1998).
[CrossRef]

S. Spälter, N. Korolkova, F. König, A. Sizmann, and G. Leuchs, "Observation of multimode quantum correlations in fiber optical solitons," Phys. Rev. Lett. 81, 786-789 (1998).
[CrossRef]

1997

S. Spälter, M. Burk, U. Strössner, M. Böhm, A. Sizmann, and G. Leuchs, "Photon number squeezing of spectrally filtered sub-picosecond optical solitons," Europhys. Lett. 38, 335-340 (1997).
[CrossRef]

1995

Y. Lai and S.-S. Yu, "General quantum theory of nonlinear optical-pulse propagation," Phys. Rev. A 51, 817-829 (1995).
[CrossRef] [PubMed]

1994

1993

1990

1987

S. J. Carter, P. D. Drumond, M. D. Reid, and R. M. Shelby, "Squeezing of quantum solitons," Phys. Rev. Lett. 58, 1841-1844 (1987).
[CrossRef] [PubMed]

P. D. Drumond and S. J. Carter, "Quantum-field theory of squeezing in solitons," J. Opt. Soc. Am. B 4, 1565-1573 (1987).
[CrossRef]

Andersen, U. L.

J. F. Corney, P. D. Drummond, J. Heersink, V. Josse, G. Leuchs, and U. L. Andersen, "Many-body quantum dynamics of polarization squeezing in optical fibers," Phys. Rev. Lett. 97, 23606 (2006).
[CrossRef]

Böhm, M.

S. Spälter, M. Burk, U. Strössner, M. Böhm, A. Sizmann, and G. Leuchs, "Photon number squeezing of spectrally filtered sub-picosecond optical solitons," Europhys. Lett. 38, 335-340 (1997).
[CrossRef]

Burk, M.

S. Spälter, M. Burk, U. Strössner, A. Sizmann, and G. Leuchs, "Propagation of quantum properties of sub-picosecond solitons in a fiber," Opt. Express 2, 77-83 (1998).
[CrossRef]

S. Spälter, M. Burk, U. Strössner, M. Böhm, A. Sizmann, and G. Leuchs, "Photon number squeezing of spectrally filtered sub-picosecond optical solitons," Europhys. Lett. 38, 335-340 (1997).
[CrossRef]

Cantrell, C. D.

Carter, S. J.

P. D. Drumond and S. J. Carter, "Quantum-field theory of squeezing in solitons," J. Opt. Soc. Am. B 4, 1565-1573 (1987).
[CrossRef]

S. J. Carter, P. D. Drumond, M. D. Reid, and R. M. Shelby, "Squeezing of quantum solitons," Phys. Rev. Lett. 58, 1841-1844 (1987).
[CrossRef] [PubMed]

Corney, J. F.

J. F. Corney, P. D. Drummond, J. Heersink, V. Josse, G. Leuchs, and U. L. Andersen, "Many-body quantum dynamics of polarization squeezing in optical fibers," Phys. Rev. Lett. 97, 23606 (2006).
[CrossRef]

P. D. Drumond and J. F. Corney, "Quantum noise in optical fibers. I. Stochastic equations," J. Opt. Soc. Am. B 18, 139-152 (2001).
[CrossRef]

Doerr, C. R.

Drummond, P. D.

J. F. Corney, P. D. Drummond, J. Heersink, V. Josse, G. Leuchs, and U. L. Andersen, "Many-body quantum dynamics of polarization squeezing in optical fibers," Phys. Rev. Lett. 97, 23606 (2006).
[CrossRef]

Drumond, P. D.

Fujishima, D.

Furumochi, H.

K. Hirosawa, H. Furumochi, A. Tada, F. Kannari, M. Takeoka, and M. Sasaki, "Photon number squeezing of ultrabroadband laser pulses generated by microstructure fibers," Phys. Rev. Lett. 94, 203601 (2005).
[CrossRef] [PubMed]

K. Hirosawa, H. Furumochi, A. Tada, F. Kannari, M. Takeoka, and M. Sasaki, "Generation of quantum correlation between co-propagating pulses in optical fibers," presented at the European Quantum Electronics Conference, Munich, Germany, June 12-17, 2005.

Goto, T.

N. Nishizawa, T. Horio, M. Mori, T. Goto, and K. Yamane, "Effect of group-velocity dispersion on photon-number squeezing of optical pulses using optical fibers and spectral filter," Jpn. J. Appl. Phys. Part 1 38, 1961-1965 (1999).
[CrossRef]

Haus, H. A.

Heersink, J.

J. F. Corney, P. D. Drummond, J. Heersink, V. Josse, G. Leuchs, and U. L. Andersen, "Many-body quantum dynamics of polarization squeezing in optical fibers," Phys. Rev. Lett. 97, 23606 (2006).
[CrossRef]

Hirosawa, K.

K. Hirosawa, H. Furumochi, A. Tada, F. Kannari, M. Takeoka, and M. Sasaki, "Photon number squeezing of ultrabroadband laser pulses generated by microstructure fibers," Phys. Rev. Lett. 94, 203601 (2005).
[CrossRef] [PubMed]

K. Hirosawa, H. Furumochi, A. Tada, F. Kannari, M. Takeoka, and M. Sasaki, "Generation of quantum correlation between co-propagating pulses in optical fibers," presented at the European Quantum Electronics Conference, Munich, Germany, June 12-17, 2005.

Hollenbeck, D.

Horio, T.

N. Nishizawa, T. Horio, M. Mori, T. Goto, and K. Yamane, "Effect of group-velocity dispersion on photon-number squeezing of optical pulses using optical fibers and spectral filter," Jpn. J. Appl. Phys. Part 1 38, 1961-1965 (1999).
[CrossRef]

Josse, V.

J. F. Corney, P. D. Drummond, J. Heersink, V. Josse, G. Leuchs, and U. L. Andersen, "Many-body quantum dynamics of polarization squeezing in optical fibers," Phys. Rev. Lett. 97, 23606 (2006).
[CrossRef]

Kannari, F.

K. Hirosawa, H. Furumochi, A. Tada, F. Kannari, M. Takeoka, and M. Sasaki, "Photon number squeezing of ultrabroadband laser pulses generated by microstructure fibers," Phys. Rev. Lett. 94, 203601 (2005).
[CrossRef] [PubMed]

M. Takeoka, D. Fujishima, and F. Kannari, "Optimization of ultrashort-pulse squeezing by spectral filtering with the Fourier pulse-shaping technique," Opt. Lett. 26, 1592-1594 (2001).
[CrossRef]

K. Hirosawa, H. Furumochi, A. Tada, F. Kannari, M. Takeoka, and M. Sasaki, "Generation of quantum correlation between co-propagating pulses in optical fibers," presented at the European Quantum Electronics Conference, Munich, Germany, June 12-17, 2005.

Khatri, F. I.

König, F.

S. Spälter, N. Korolkova, F. König, A. Sizmann, and G. Leuchs, "Observation of multimode quantum correlations in fiber optical solitons," Phys. Rev. Lett. 81, 786-789 (1998).
[CrossRef]

Korolkova, N.

S. Spälter, N. Korolkova, F. König, A. Sizmann, and G. Leuchs, "Observation of multimode quantum correlations in fiber optical solitons," Phys. Rev. Lett. 81, 786-789 (1998).
[CrossRef]

Lai, Y.

Y. Lai and S.-S. Yu, "General quantum theory of nonlinear optical-pulse propagation," Phys. Rev. A 51, 817-829 (1995).
[CrossRef] [PubMed]

Leaird, D. D.

Leuchs, G.

J. F. Corney, P. D. Drummond, J. Heersink, V. Josse, G. Leuchs, and U. L. Andersen, "Many-body quantum dynamics of polarization squeezing in optical fibers," Phys. Rev. Lett. 97, 23606 (2006).
[CrossRef]

S. Spälter, N. Korolkova, F. König, A. Sizmann, and G. Leuchs, "Observation of multimode quantum correlations in fiber optical solitons," Phys. Rev. Lett. 81, 786-789 (1998).
[CrossRef]

S. Spälter, M. Burk, U. Strössner, A. Sizmann, and G. Leuchs, "Propagation of quantum properties of sub-picosecond solitons in a fiber," Opt. Express 2, 77-83 (1998).
[CrossRef]

S. Spälter, M. Burk, U. Strössner, M. Böhm, A. Sizmann, and G. Leuchs, "Photon number squeezing of spectrally filtered sub-picosecond optical solitons," Europhys. Lett. 38, 335-340 (1997).
[CrossRef]

Mori, M.

N. Nishizawa, T. Horio, M. Mori, T. Goto, and K. Yamane, "Effect of group-velocity dispersion on photon-number squeezing of optical pulses using optical fibers and spectral filter," Jpn. J. Appl. Phys. Part 1 38, 1961-1965 (1999).
[CrossRef]

Nishizawa, N.

N. Nishizawa, T. Horio, M. Mori, T. Goto, and K. Yamane, "Effect of group-velocity dispersion on photon-number squeezing of optical pulses using optical fibers and spectral filter," Jpn. J. Appl. Phys. Part 1 38, 1961-1965 (1999).
[CrossRef]

Oudin, S.

Reid, M. D.

S. J. Carter, P. D. Drumond, M. D. Reid, and R. M. Shelby, "Squeezing of quantum solitons," Phys. Rev. Lett. 58, 1841-1844 (1987).
[CrossRef] [PubMed]

Reitze, D. H.

Sasaki, M.

K. Hirosawa, H. Furumochi, A. Tada, F. Kannari, M. Takeoka, and M. Sasaki, "Photon number squeezing of ultrabroadband laser pulses generated by microstructure fibers," Phys. Rev. Lett. 94, 203601 (2005).
[CrossRef] [PubMed]

K. Hirosawa, H. Furumochi, A. Tada, F. Kannari, M. Takeoka, and M. Sasaki, "Generation of quantum correlation between co-propagating pulses in optical fibers," presented at the European Quantum Electronics Conference, Munich, Germany, June 12-17, 2005.

Shelby, R. M.

S. J. Carter, P. D. Drumond, M. D. Reid, and R. M. Shelby, "Squeezing of quantum solitons," Phys. Rev. Lett. 58, 1841-1844 (1987).
[CrossRef] [PubMed]

Shirasaki, M.

Sizmann, A.

S. Spälter, M. Burk, U. Strössner, A. Sizmann, and G. Leuchs, "Propagation of quantum properties of sub-picosecond solitons in a fiber," Opt. Express 2, 77-83 (1998).
[CrossRef]

S. Spälter, N. Korolkova, F. König, A. Sizmann, and G. Leuchs, "Observation of multimode quantum correlations in fiber optical solitons," Phys. Rev. Lett. 81, 786-789 (1998).
[CrossRef]

S. Spälter, M. Burk, U. Strössner, M. Böhm, A. Sizmann, and G. Leuchs, "Photon number squeezing of spectrally filtered sub-picosecond optical solitons," Europhys. Lett. 38, 335-340 (1997).
[CrossRef]

Spälter, S.

S. Spälter, M. Burk, U. Strössner, A. Sizmann, and G. Leuchs, "Propagation of quantum properties of sub-picosecond solitons in a fiber," Opt. Express 2, 77-83 (1998).
[CrossRef]

S. Spälter, N. Korolkova, F. König, A. Sizmann, and G. Leuchs, "Observation of multimode quantum correlations in fiber optical solitons," Phys. Rev. Lett. 81, 786-789 (1998).
[CrossRef]

S. Spälter, M. Burk, U. Strössner, M. Böhm, A. Sizmann, and G. Leuchs, "Photon number squeezing of spectrally filtered sub-picosecond optical solitons," Europhys. Lett. 38, 335-340 (1997).
[CrossRef]

Strössner, U.

S. Spälter, M. Burk, U. Strössner, A. Sizmann, and G. Leuchs, "Propagation of quantum properties of sub-picosecond solitons in a fiber," Opt. Express 2, 77-83 (1998).
[CrossRef]

S. Spälter, M. Burk, U. Strössner, M. Böhm, A. Sizmann, and G. Leuchs, "Photon number squeezing of spectrally filtered sub-picosecond optical solitons," Europhys. Lett. 38, 335-340 (1997).
[CrossRef]

Tada, A.

K. Hirosawa, H. Furumochi, A. Tada, F. Kannari, M. Takeoka, and M. Sasaki, "Photon number squeezing of ultrabroadband laser pulses generated by microstructure fibers," Phys. Rev. Lett. 94, 203601 (2005).
[CrossRef] [PubMed]

K. Hirosawa, H. Furumochi, A. Tada, F. Kannari, M. Takeoka, and M. Sasaki, "Generation of quantum correlation between co-propagating pulses in optical fibers," presented at the European Quantum Electronics Conference, Munich, Germany, June 12-17, 2005.

Takeoka, M.

K. Hirosawa, H. Furumochi, A. Tada, F. Kannari, M. Takeoka, and M. Sasaki, "Photon number squeezing of ultrabroadband laser pulses generated by microstructure fibers," Phys. Rev. Lett. 94, 203601 (2005).
[CrossRef] [PubMed]

M. Takeoka, D. Fujishima, and F. Kannari, "Optimization of ultrashort-pulse squeezing by spectral filtering with the Fourier pulse-shaping technique," Opt. Lett. 26, 1592-1594 (2001).
[CrossRef]

K. Hirosawa, H. Furumochi, A. Tada, F. Kannari, M. Takeoka, and M. Sasaki, "Generation of quantum correlation between co-propagating pulses in optical fibers," presented at the European Quantum Electronics Conference, Munich, Germany, June 12-17, 2005.

Weiner, A. M.

Yamane, K.

N. Nishizawa, T. Horio, M. Mori, T. Goto, and K. Yamane, "Effect of group-velocity dispersion on photon-number squeezing of optical pulses using optical fibers and spectral filter," Jpn. J. Appl. Phys. Part 1 38, 1961-1965 (1999).
[CrossRef]

Yu, S.-S.

Y. Lai and S.-S. Yu, "General quantum theory of nonlinear optical-pulse propagation," Phys. Rev. A 51, 817-829 (1995).
[CrossRef] [PubMed]

Europhys. Lett.

S. Spälter, M. Burk, U. Strössner, M. Böhm, A. Sizmann, and G. Leuchs, "Photon number squeezing of spectrally filtered sub-picosecond optical solitons," Europhys. Lett. 38, 335-340 (1997).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Jpn. J. Appl. Phys. Part 1

N. Nishizawa, T. Horio, M. Mori, T. Goto, and K. Yamane, "Effect of group-velocity dispersion on photon-number squeezing of optical pulses using optical fibers and spectral filter," Jpn. J. Appl. Phys. Part 1 38, 1961-1965 (1999).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

Y. Lai and S.-S. Yu, "General quantum theory of nonlinear optical-pulse propagation," Phys. Rev. A 51, 817-829 (1995).
[CrossRef] [PubMed]

Phys. Rev. Lett.

S. J. Carter, P. D. Drumond, M. D. Reid, and R. M. Shelby, "Squeezing of quantum solitons," Phys. Rev. Lett. 58, 1841-1844 (1987).
[CrossRef] [PubMed]

K. Hirosawa, H. Furumochi, A. Tada, F. Kannari, M. Takeoka, and M. Sasaki, "Photon number squeezing of ultrabroadband laser pulses generated by microstructure fibers," Phys. Rev. Lett. 94, 203601 (2005).
[CrossRef] [PubMed]

J. F. Corney, P. D. Drummond, J. Heersink, V. Josse, G. Leuchs, and U. L. Andersen, "Many-body quantum dynamics of polarization squeezing in optical fibers," Phys. Rev. Lett. 97, 23606 (2006).
[CrossRef]

S. Spälter, N. Korolkova, F. König, A. Sizmann, and G. Leuchs, "Observation of multimode quantum correlations in fiber optical solitons," Phys. Rev. Lett. 81, 786-789 (1998).
[CrossRef]

Other

K. Hirosawa, H. Furumochi, A. Tada, F. Kannari, M. Takeoka, and M. Sasaki, "Generation of quantum correlation between co-propagating pulses in optical fibers," presented at the European Quantum Electronics Conference, Munich, Germany, June 12-17, 2005.

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Figures (11)

Fig. 1
Fig. 1

Schematic of the experimental setup. SLM, spatial light modulator; PD, photodiode; MF, microstructure fiber; S.A. algorithm, simulated annealing algorithm.

Fig. 2
Fig. 2

Noise levels relative to the shot noise. The spectrum was selected and preserved with a LPF for different cutoff wavelengths at a coupled laser energy of 110 pJ into a 40 cm long MF.

Fig. 3
Fig. 3

Dependence of noise levels relative to the shot noise on the launched laser energy into a 40 cm long MF. The cutoff wavelength of the LPF was adjusted so that the highest squeezing was obtainable.

Fig. 4
Fig. 4

Numerical model calculations of dependence of noise levels relative to the shot noise on the fiber length. The cutoff wavelength of the LPF was adjusted so that the highest squeezing was obtainable. The launched laser peak power was 750 W .

Fig. 5
Fig. 5

(a) Numerical simulation results of the output spectrum and the noise levels relative to shot noise for various LPFs and HPFs. (b) Map of intrapulse quantum correlations. The fiber length was 30 cm , and the input laser peak power was 500 W .

Fig. 6
Fig. 6

Numerical simulation results of the output spectrum and the noise levels relative to shot noise for various LPFs. The fiber length was 10 cm , and the input laser peak power was 750 W . The contribution of SRS was adjusted: (a) f R = 0.18 , (b) f R = 0.0 , and (c) f R = 1.0 .

Fig. 7
Fig. 7

Noise levels relative to shot noise obtained with the LPF for various linear chirp rates. The fiber length was 40 cm , and the input laser energy was 105 pJ .

Fig. 8
Fig. 8

Noise levels relative to the shot noise obtained with the LPF for various linear chirp rates. The fiber length was 40 cm , and the input laser energy was 145 pJ .

Fig. 9
Fig. 9

Results of adaptive pulse-shape control. The input laser energy was 80 pJ , and the fiber length was 40 cm . The LPF cutoff wavelength was set at 880 nm . (a) Evolution of squeezing level (Fano factor) during iterative closed-loop control. (b) Output spectra both prior to and after optimization. (c) The spectrum of the input pulse and the optimized phase.

Fig. 10
Fig. 10

Results of adaptive pulse-shape control. The input laser energy was 140 pJ , and the fiber length was 40 cm . The LPF cutoff wavelength was set at 910 nm . (a) Evolution of squeezing level (Fano factor) during iterative closed-loop control. (b) Output spectra both prior to and after optimization. (c) The spectrum of the input pulse and the optimized phase.

Fig. 11
Fig. 11

Results of adaptive pulse-shape control. The input laser energy was 97 pJ , and the fiber length was 40 cm . The BPF cutoff wavelengths were set at 890 and 960 nm . (a) Evolution of squeezing level (Fano factor) during iterative closed-loop control. (b) Output spectra both prior to and after optimization. (c) The spectrum of the input pulse and the optimized phase.

Equations (2)

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A ( z , t ) z j n 2 j n β n n ! n t n A ( z , t ) = j γ ( 1 f R ) A ( z , t ) 2 A ( z , t ) + j f R γ A ( z , t ) t h R ( t τ ) A ( z , τ ) 2 d τ .
C ( i , j ) = cov ( n i , n j ) Δ n i 2 Δ n j 2 δ i j n i Δ n i 2 .

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