Abstract

A new approach to measure the second order correlation function g(2) and the coherence time was investigated. The g(2) was calculated from the photon pair time interval distribution by direct numerical self-convolution with the high order correction. The accuracy of this method was examined using an optical fiber based Hanbury-Brown-Twiss interferometer with a pseudo-thermal light source. We found that the significance of the high order correction is related to the factor Īτc, which is the overlapping of the photon wave packets. A novel technique was also demonstrated to measure the coherence time τc of a light source using the random phase modulation. This method is more suitable for a weak light source with a long coherence time using a simple experimental setup.

© 2016 Optical Society of America

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References

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    [Crossref]
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    [Crossref] [PubMed]
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    [PubMed]
  4. K. Nakayama, Y. Yoshikawa, H. Matsumoto, Y. Torii, and T. Kuga, “Precise intensity correlation measurement for atomic resonance fluorescence from optical molasses,” Opt. Express 18, 6604–6612 (2010).
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  5. M. Das, A. Shirasaki, K. P. Nayak, M. Morinaga, F. Le Kien, and K. Hakuta, “Measurement of fluorescence emission spectrum of few strongly driven atoms using an optical nanofiber,” Opt. Express 18, 17154–17164 (2010).
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  7. L. Fleury, J.-M. Segura, G. Zumofen, B. Hecht, and U. P. Wild, “Nonclassical photon statistics in single-molecule fluorescence at room temperature,” Phys. Rev. Lett. 84, 1148–1151 (2000).
    [Crossref] [PubMed]
  8. S. Du, P. Kolchin, C. Belthangady, G. Y. Yin, and S. E. Harris, “Subnatural Linewidth Biphotons with Controllable Temporal Length,” Phys. Rev. Lett. 100, 183603 (2008).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]

2010 (2)

2009 (1)

L. Yuan, Z. Yu-Chi, Z. Peng-Fei, G. Yan-Qiang, L. Gang, W. Jun-Min, and Z. Tian-Cai, “Experimental Study on Coherence Time of a Light Field with Single Photon Counting,” Chinese Phys. Lett. 26074205 (2009).
[Crossref]

2008 (1)

S. Du, P. Kolchin, C. Belthangady, G. Y. Yin, and S. E. Harris, “Subnatural Linewidth Biphotons with Controllable Temporal Length,” Phys. Rev. Lett. 100, 183603 (2008).
[Crossref] [PubMed]

2006 (1)

J. S. Neergaard-Nielsen, B. M. Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, “Generation of a Superposition of Odd Photon Number States for Quantum Information Networks,” Phys. Rev. Lett. 97, 083604 (2006).
[Crossref] [PubMed]

2004 (1)

G. Scarcelli, A. Valencia, and Y. Shih, “Experimental study of the momentum correlation of a pseudothermal field in the photon-counting regime,” Phys. Rev. A 70, 051802 (2004).
[Crossref]

2001 (1)

L. Fleury, J.-M. Segura, G. Zumofen, B. Hecht, and U. P. Wild, “Photon statistics in single-molecule fluorescence at room temperature,” J. Luminescence 94, 805–809 (2001).
[Crossref]

2000 (2)

L. Fleury, J.-M. Segura, G. Zumofen, B. Hecht, and U. P. Wild, “Nonclassical photon statistics in single-molecule fluorescence at room temperature,” Phys. Rev. Lett. 84, 1148–1151 (2000).
[Crossref] [PubMed]

P. Michler, A. Imamoğlu, M. D. Mason, and P. J. Carson, “Quantum correlation among photons from a single quantum dot at room temperature,” Nature 406968 (2000).
[PubMed]

1998 (1)

M. B. Plenio and P. L. Knight, “The quantum-jump approach to dissipative dynamics in quantum optics,” Rev. Mod. Phys. 70101 (1998).
[Crossref]

1996 (1)

F. De Martini, G. Di Giuseppe, and M. Marrocco, “Single-mode generation of quantum photon states by excited single molecules in a microcavity trap,” Phys. Rev. Lett. 76900 (1996).
[Crossref] [PubMed]

1986 (1)

L. E. Richter, H. I. Mandelberg, and M. S. Kruger, “Linewidth determination from self-heterodyne measurements with subcoherence delay times,” IEEE J. Quantum Electron. 22, 2070 (1986).
[Crossref]

1983 (1)

S. Reynaud, “La fluorescence de résonance: etude par la méthode de l’atome habillé,” Annales de Physique 8315 (1983).

1980 (1)

T. Okoshi, K. Kikuchi, and A. Nakayama, “Novel Method for High-Resolution Measurement of Laser Output Spectrum,” Electron. Lett. 16, 630–631 (1980).
[Crossref]

1971 (1)

1964 (1)

W. Martienssen, “Coherence and Fluctuations in Light Beams,” Am. J. Phys. 32, 919 (1964).
[Crossref]

Belthangady, C.

S. Du, P. Kolchin, C. Belthangady, G. Y. Yin, and S. E. Harris, “Subnatural Linewidth Biphotons with Controllable Temporal Length,” Phys. Rev. Lett. 100, 183603 (2008).
[Crossref] [PubMed]

Carson, P. J.

P. Michler, A. Imamoğlu, M. D. Mason, and P. J. Carson, “Quantum correlation among photons from a single quantum dot at room temperature,” Nature 406968 (2000).
[PubMed]

Das, M.

De Martini, F.

F. De Martini, G. Di Giuseppe, and M. Marrocco, “Single-mode generation of quantum photon states by excited single molecules in a microcavity trap,” Phys. Rev. Lett. 76900 (1996).
[Crossref] [PubMed]

Di Giuseppe, G.

F. De Martini, G. Di Giuseppe, and M. Marrocco, “Single-mode generation of quantum photon states by excited single molecules in a microcavity trap,” Phys. Rev. Lett. 76900 (1996).
[Crossref] [PubMed]

Du, S.

S. Du, P. Kolchin, C. Belthangady, G. Y. Yin, and S. E. Harris, “Subnatural Linewidth Biphotons with Controllable Temporal Length,” Phys. Rev. Lett. 100, 183603 (2008).
[Crossref] [PubMed]

Estes, L. E.

Fleury, L.

L. Fleury, J.-M. Segura, G. Zumofen, B. Hecht, and U. P. Wild, “Photon statistics in single-molecule fluorescence at room temperature,” J. Luminescence 94, 805–809 (2001).
[Crossref]

L. Fleury, J.-M. Segura, G. Zumofen, B. Hecht, and U. P. Wild, “Nonclassical photon statistics in single-molecule fluorescence at room temperature,” Phys. Rev. Lett. 84, 1148–1151 (2000).
[Crossref] [PubMed]

Gang, L.

L. Yuan, Z. Yu-Chi, Z. Peng-Fei, G. Yan-Qiang, L. Gang, W. Jun-Min, and Z. Tian-Cai, “Experimental Study on Coherence Time of a Light Field with Single Photon Counting,” Chinese Phys. Lett. 26074205 (2009).
[Crossref]

Hakuta, K.

Harris, S. E.

S. Du, P. Kolchin, C. Belthangady, G. Y. Yin, and S. E. Harris, “Subnatural Linewidth Biphotons with Controllable Temporal Length,” Phys. Rev. Lett. 100, 183603 (2008).
[Crossref] [PubMed]

Hecht, B.

L. Fleury, J.-M. Segura, G. Zumofen, B. Hecht, and U. P. Wild, “Photon statistics in single-molecule fluorescence at room temperature,” J. Luminescence 94, 805–809 (2001).
[Crossref]

L. Fleury, J.-M. Segura, G. Zumofen, B. Hecht, and U. P. Wild, “Nonclassical photon statistics in single-molecule fluorescence at room temperature,” Phys. Rev. Lett. 84, 1148–1151 (2000).
[Crossref] [PubMed]

Hettich, C.

J. S. Neergaard-Nielsen, B. M. Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, “Generation of a Superposition of Odd Photon Number States for Quantum Information Networks,” Phys. Rev. Lett. 97, 083604 (2006).
[Crossref] [PubMed]

Imamoglu, A.

P. Michler, A. Imamoğlu, M. D. Mason, and P. J. Carson, “Quantum correlation among photons from a single quantum dot at room temperature,” Nature 406968 (2000).
[PubMed]

Jun-Min, W.

L. Yuan, Z. Yu-Chi, Z. Peng-Fei, G. Yan-Qiang, L. Gang, W. Jun-Min, and Z. Tian-Cai, “Experimental Study on Coherence Time of a Light Field with Single Photon Counting,” Chinese Phys. Lett. 26074205 (2009).
[Crossref]

Kikuchi, K.

T. Okoshi, K. Kikuchi, and A. Nakayama, “Novel Method for High-Resolution Measurement of Laser Output Spectrum,” Electron. Lett. 16, 630–631 (1980).
[Crossref]

Knight, P. L.

M. B. Plenio and P. L. Knight, “The quantum-jump approach to dissipative dynamics in quantum optics,” Rev. Mod. Phys. 70101 (1998).
[Crossref]

Kolchin, P.

S. Du, P. Kolchin, C. Belthangady, G. Y. Yin, and S. E. Harris, “Subnatural Linewidth Biphotons with Controllable Temporal Length,” Phys. Rev. Lett. 100, 183603 (2008).
[Crossref] [PubMed]

Kruger, M. S.

L. E. Richter, H. I. Mandelberg, and M. S. Kruger, “Linewidth determination from self-heterodyne measurements with subcoherence delay times,” IEEE J. Quantum Electron. 22, 2070 (1986).
[Crossref]

Kuga, T.

Le Kien, F.

Lorenzo, u.

Loudon, R.

R. Loudon, “The Quantum Theory of Light,” (Oxford University Press2000).

Mandelberg, H. I.

L. E. Richter, H. I. Mandelberg, and M. S. Kruger, “Linewidth determination from self-heterodyne measurements with subcoherence delay times,” IEEE J. Quantum Electron. 22, 2070 (1986).
[Crossref]

Marrocco, M.

F. De Martini, G. Di Giuseppe, and M. Marrocco, “Single-mode generation of quantum photon states by excited single molecules in a microcavity trap,” Phys. Rev. Lett. 76900 (1996).
[Crossref] [PubMed]

Martienssen, W.

W. Martienssen, “Coherence and Fluctuations in Light Beams,” Am. J. Phys. 32, 919 (1964).
[Crossref]

Mason, M. D.

P. Michler, A. Imamoğlu, M. D. Mason, and P. J. Carson, “Quantum correlation among photons from a single quantum dot at room temperature,” Nature 406968 (2000).
[PubMed]

Matsumoto, H.

Michler, P.

P. Michler, A. Imamoğlu, M. D. Mason, and P. J. Carson, “Quantum correlation among photons from a single quantum dot at room temperature,” Nature 406968 (2000).
[PubMed]

Mølmer, K.

J. S. Neergaard-Nielsen, B. M. Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, “Generation of a Superposition of Odd Photon Number States for Quantum Information Networks,” Phys. Rev. Lett. 97, 083604 (2006).
[Crossref] [PubMed]

Morinaga, M.

Nakayama, A.

T. Okoshi, K. Kikuchi, and A. Nakayama, “Novel Method for High-Resolution Measurement of Laser Output Spectrum,” Electron. Lett. 16, 630–631 (1980).
[Crossref]

Nakayama, K.

Narducci,

Nayak, K. P.

Neergaard-Nielsen, J. S.

J. S. Neergaard-Nielsen, B. M. Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, “Generation of a Superposition of Odd Photon Number States for Quantum Information Networks,” Phys. Rev. Lett. 97, 083604 (2006).
[Crossref] [PubMed]

Nielsen, B. M.

J. S. Neergaard-Nielsen, B. M. Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, “Generation of a Superposition of Odd Photon Number States for Quantum Information Networks,” Phys. Rev. Lett. 97, 083604 (2006).
[Crossref] [PubMed]

Okoshi, T.

T. Okoshi, K. Kikuchi, and A. Nakayama, “Novel Method for High-Resolution Measurement of Laser Output Spectrum,” Electron. Lett. 16, 630–631 (1980).
[Crossref]

Peng-Fei, Z.

L. Yuan, Z. Yu-Chi, Z. Peng-Fei, G. Yan-Qiang, L. Gang, W. Jun-Min, and Z. Tian-Cai, “Experimental Study on Coherence Time of a Light Field with Single Photon Counting,” Chinese Phys. Lett. 26074205 (2009).
[Crossref]

Plenio, M. B.

M. B. Plenio and P. L. Knight, “The quantum-jump approach to dissipative dynamics in quantum optics,” Rev. Mod. Phys. 70101 (1998).
[Crossref]

Polzik, E. S.

J. S. Neergaard-Nielsen, B. M. Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, “Generation of a Superposition of Odd Photon Number States for Quantum Information Networks,” Phys. Rev. Lett. 97, 083604 (2006).
[Crossref] [PubMed]

Reynaud, S.

S. Reynaud, “La fluorescence de résonance: etude par la méthode de l’atome habillé,” Annales de Physique 8315 (1983).

Richard, A. T.

Richter, L. E.

L. E. Richter, H. I. Mandelberg, and M. S. Kruger, “Linewidth determination from self-heterodyne measurements with subcoherence delay times,” IEEE J. Quantum Electron. 22, 2070 (1986).
[Crossref]

Scarcelli, G.

G. Scarcelli, A. Valencia, and Y. Shih, “Experimental study of the momentum correlation of a pseudothermal field in the photon-counting regime,” Phys. Rev. A 70, 051802 (2004).
[Crossref]

Segura, J.-M.

L. Fleury, J.-M. Segura, G. Zumofen, B. Hecht, and U. P. Wild, “Photon statistics in single-molecule fluorescence at room temperature,” J. Luminescence 94, 805–809 (2001).
[Crossref]

L. Fleury, J.-M. Segura, G. Zumofen, B. Hecht, and U. P. Wild, “Nonclassical photon statistics in single-molecule fluorescence at room temperature,” Phys. Rev. Lett. 84, 1148–1151 (2000).
[Crossref] [PubMed]

Shih, Y.

G. Scarcelli, A. Valencia, and Y. Shih, “Experimental study of the momentum correlation of a pseudothermal field in the photon-counting regime,” Phys. Rev. A 70, 051802 (2004).
[Crossref]

Shirasaki, A.

Tian-Cai, Z.

L. Yuan, Z. Yu-Chi, Z. Peng-Fei, G. Yan-Qiang, L. Gang, W. Jun-Min, and Z. Tian-Cai, “Experimental Study on Coherence Time of a Light Field with Single Photon Counting,” Chinese Phys. Lett. 26074205 (2009).
[Crossref]

Torii, Y.

Valencia, A.

G. Scarcelli, A. Valencia, and Y. Shih, “Experimental study of the momentum correlation of a pseudothermal field in the photon-counting regime,” Phys. Rev. A 70, 051802 (2004).
[Crossref]

Wild, U. P.

L. Fleury, J.-M. Segura, G. Zumofen, B. Hecht, and U. P. Wild, “Photon statistics in single-molecule fluorescence at room temperature,” J. Luminescence 94, 805–809 (2001).
[Crossref]

L. Fleury, J.-M. Segura, G. Zumofen, B. Hecht, and U. P. Wild, “Nonclassical photon statistics in single-molecule fluorescence at room temperature,” Phys. Rev. Lett. 84, 1148–1151 (2000).
[Crossref] [PubMed]

Yan-Qiang, G.

L. Yuan, Z. Yu-Chi, Z. Peng-Fei, G. Yan-Qiang, L. Gang, W. Jun-Min, and Z. Tian-Cai, “Experimental Study on Coherence Time of a Light Field with Single Photon Counting,” Chinese Phys. Lett. 26074205 (2009).
[Crossref]

Yin, G. Y.

S. Du, P. Kolchin, C. Belthangady, G. Y. Yin, and S. E. Harris, “Subnatural Linewidth Biphotons with Controllable Temporal Length,” Phys. Rev. Lett. 100, 183603 (2008).
[Crossref] [PubMed]

Yoshikawa, Y.

Yuan, L.

L. Yuan, Z. Yu-Chi, Z. Peng-Fei, G. Yan-Qiang, L. Gang, W. Jun-Min, and Z. Tian-Cai, “Experimental Study on Coherence Time of a Light Field with Single Photon Counting,” Chinese Phys. Lett. 26074205 (2009).
[Crossref]

Yu-Chi, Z.

L. Yuan, Z. Yu-Chi, Z. Peng-Fei, G. Yan-Qiang, L. Gang, W. Jun-Min, and Z. Tian-Cai, “Experimental Study on Coherence Time of a Light Field with Single Photon Counting,” Chinese Phys. Lett. 26074205 (2009).
[Crossref]

Zumofen, G.

L. Fleury, J.-M. Segura, G. Zumofen, B. Hecht, and U. P. Wild, “Photon statistics in single-molecule fluorescence at room temperature,” J. Luminescence 94, 805–809 (2001).
[Crossref]

L. Fleury, J.-M. Segura, G. Zumofen, B. Hecht, and U. P. Wild, “Nonclassical photon statistics in single-molecule fluorescence at room temperature,” Phys. Rev. Lett. 84, 1148–1151 (2000).
[Crossref] [PubMed]

Am. J. Phys. (1)

W. Martienssen, “Coherence and Fluctuations in Light Beams,” Am. J. Phys. 32, 919 (1964).
[Crossref]

Annales de Physique (1)

S. Reynaud, “La fluorescence de résonance: etude par la méthode de l’atome habillé,” Annales de Physique 8315 (1983).

Chinese Phys. Lett. (1)

L. Yuan, Z. Yu-Chi, Z. Peng-Fei, G. Yan-Qiang, L. Gang, W. Jun-Min, and Z. Tian-Cai, “Experimental Study on Coherence Time of a Light Field with Single Photon Counting,” Chinese Phys. Lett. 26074205 (2009).
[Crossref]

Electron. Lett. (1)

T. Okoshi, K. Kikuchi, and A. Nakayama, “Novel Method for High-Resolution Measurement of Laser Output Spectrum,” Electron. Lett. 16, 630–631 (1980).
[Crossref]

IEEE J. Quantum Electron. (1)

L. E. Richter, H. I. Mandelberg, and M. S. Kruger, “Linewidth determination from self-heterodyne measurements with subcoherence delay times,” IEEE J. Quantum Electron. 22, 2070 (1986).
[Crossref]

J. Luminescence (1)

L. Fleury, J.-M. Segura, G. Zumofen, B. Hecht, and U. P. Wild, “Photon statistics in single-molecule fluorescence at room temperature,” J. Luminescence 94, 805–809 (2001).
[Crossref]

J. Opt. Soc. Am. (1)

Nature (1)

P. Michler, A. Imamoğlu, M. D. Mason, and P. J. Carson, “Quantum correlation among photons from a single quantum dot at room temperature,” Nature 406968 (2000).
[PubMed]

Opt. Express (2)

Phys. Rev. A (1)

G. Scarcelli, A. Valencia, and Y. Shih, “Experimental study of the momentum correlation of a pseudothermal field in the photon-counting regime,” Phys. Rev. A 70, 051802 (2004).
[Crossref]

Phys. Rev. Lett. (4)

F. De Martini, G. Di Giuseppe, and M. Marrocco, “Single-mode generation of quantum photon states by excited single molecules in a microcavity trap,” Phys. Rev. Lett. 76900 (1996).
[Crossref] [PubMed]

L. Fleury, J.-M. Segura, G. Zumofen, B. Hecht, and U. P. Wild, “Nonclassical photon statistics in single-molecule fluorescence at room temperature,” Phys. Rev. Lett. 84, 1148–1151 (2000).
[Crossref] [PubMed]

S. Du, P. Kolchin, C. Belthangady, G. Y. Yin, and S. E. Harris, “Subnatural Linewidth Biphotons with Controllable Temporal Length,” Phys. Rev. Lett. 100, 183603 (2008).
[Crossref] [PubMed]

J. S. Neergaard-Nielsen, B. M. Nielsen, C. Hettich, K. Mølmer, and E. S. Polzik, “Generation of a Superposition of Odd Photon Number States for Quantum Information Networks,” Phys. Rev. Lett. 97, 083604 (2006).
[Crossref] [PubMed]

Rev. Mod. Phys. (1)

M. B. Plenio and P. L. Knight, “The quantum-jump approach to dissipative dynamics in quantum optics,” Rev. Mod. Phys. 70101 (1998).
[Crossref]

Other (1)

R. Loudon, “The Quantum Theory of Light,” (Oxford University Press2000).

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Figures (5)

Fig. 1
Fig. 1 The simplified HBT experimental scheme. A clock (counter) is triggered by the photon received from the ”START” detector, then is stopped by the subsequent photon received from the ”STOP” detector. The time intervals were measured using the clock, and then recorded as a histogram.
Fig. 2
Fig. 2 The experimental set-up. To generate a pseudo-thermal light source, a single longitudinal mode 632 nm He-Ne laser, passing through two polarizers for controlling the incident power, is focused on a rotating wheel with a surface of sandpaper. The back scattering of light was collected into a fiber splitter without any collimator. One of SPCM (PMT1) was as the START to trigger the universal counter for time interval measurement. The other SPCM (PMT2) was the ”STOP”. The time intervals were recorded by a computer for subsequent off-line analysis. The second order correlation function of the pseudo-thermal light was then calculated from the histogram of the recoded time intervals.
Fig. 3
Fig. 3 Typical J(τ) with various orders of Dn(τ) at a 20 Hz rotating frequency. The coherence time of the pseudo-thermal light source τc is ~ 10µsec and Ī ~ 4 × 104 photons/sec. The results of 4th and 6th order are in a very good agreement within the region τ < 100 µsec, where is important for deriving the coherence time. The bin size = 100 nsec was used.
Fig. 4
Fig. 4 The second order correlation functions g(2)(τ) with rotating frequencies 0 Hz, 20 Hz, 100 Hz, 300 Hz, 500 Hz and 900 Hz. The black lines are the fitting functions A + B e 2 τ / τ c . Except the 0 Hz, the resulted coherence times τc are 28.10(80) µs, 7.40(11) µs, 3.00(5) µs, 1.75(2) µs and 0.96(3) µs, respectively. The inset shows a Gaussian fit (thin red line) of the g(2)(τ) with a rotating frequency 700 Hz, in comparison with the exponential decay fit (thin blue line). The fitting residual and χ2 show that the exponential decay function is slightly better than the Gaussian.
Fig. 5
Fig. 5 τcωr v.s. ωr. At the low frequency regime, the uncorrected τc (red dot) strongly deviates from the high-order corrected τc (black square) and the theoretical model (blue line). The corrected τc are in very good agreement with the theory, which gives τcωr = ((ωrτ0) 1 + k) 1. The coherence time τ0 of the incident light (He-Ne laser) was derived as 74(15) µs from the fitting parameter of the theoretical model. The inset shows a typical beat-note signal of two HeNe lasers with a RBW=3 kHz. The measured (-3db) linewidth is 6.5(1.3) kHz. Assuming equal linewidth of the two lasers, the coherence time is 97(20) µs.

Equations (22)

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

J ( τ ) = I ¯ g ( 2 ) ( τ )
J ( τ ) = K ( τ ) + K ( τ ) K ( τ ) + ...... = n = 1 K n ( τ )
J ˜ ( s ) = K ˜ ( s ) 1 K ˜ ( s )
I ¯ g ^ ( 2 ) ( ω ) = J ^ ( ω ) = n = 1 K ^ n ( ω ) = K ^ ( ω ) 1 K ^ ( ω )
K ^ ( ω ) = I ¯ g ^ ( 2 ) ( ω ) 1 + I ¯ g ^ ( 2 ) ( ω )
g ( 2 ) ( τ ) = 1 + | g ( 1 ) ( τ ) | 2
K ^ ( ω ) = I ¯ g ^ ( 1 ) ( ω ) 2 1 + I ¯ g ^ ( 1 ) ( ω ) 2
g ( 1 ) ( τ ) = e τ τ c
g ^ ( 1 ) ( ω ) 2 = τ c 1 + ( ω τ c / 2 ) 2
K ^ ( ω ) = I ¯ τ c 1 + ( ω τ c / 2 ) 2 + I ¯ τ c
I ¯ τ c 5 4 + I ¯ τ c < 1
Δ J ^ m ( ω ) = n = m + 1 K ^ n ( ω ) 1 K ^ n ( ω ) = K ^ m ( ω )
D ( τ ) = n = 1 1 2 n K n ( τ )
D ( τ ) = 1 2 K ( τ ) + 1 4 K 2 ( τ ) + 1 8 K 3 ( τ ) + 1 16 K 4 ( τ ) D 2 ( τ ) = 1 4 K 2 ( τ ) + 2 8 K 3 ( τ ) + 3 16 K 4 ( τ ) D 3 ( τ ) = 1 8 K 3 ( τ ) + 3 16 K 4 ( τ ) D 4 ( τ ) = 1 16 K 4 ( τ )
2 n = 1 D n ( τ ) = 2 ( 1 2 ) n = 1 K n ( τ ) = J ( τ )
D ( τ , ε ) = n = 1 ( 1 2 + ε ) n 1 ( 1 2 + ε ) K n ( τ )
g ( 2 ) ( τ ) = 1 + e 2 τ / τ c
1 τ c = k ω r
τ c ω r = 1 k = const .
δ ω m = δ ω 0 + k ω r
1 τ c = 1 τ 0 + k ω r
τ c ω r = 1 1 ω r τ 0 + k

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