Abstract

The emission and synchronization of mutually-coupled semiconductor lasers with short cavities has been already recorded, with transversely unstable solutions existing within the chaotic synchronization manifold. Noise and laser-mismatch induced instabilities cause short de-synchronization events within the overall generalized synchronization, influencing the pragmatism of using these signals in secure data exchange applications. However, such operation can be functional for user authentication and sensing applications by assessing a time-averaged performance of synchrony. Until now, this has not been examined either in large-scale laser network configurations or in large transmission coupling paths, as real network implementations oblige. Here we present the first implementation of a fully-coupled fiber network with up to 16 semiconductor lasers, independently controlled and coupled through long interacting cavities. High level of consistent global or cluster synchrony via chaotic signals is demonstrated among all devices of the same origin and under appropriate operation. Devices that are not identical fail to synchronize at any condition, when coupled to the network. Under multiplexed operation, groups of lasers that emit at spectral distances as low as 50pm are shown to preserve intra-cluster synchronization when transmitted in the same fiber-optic channel, despite their large bandwidth of emitted signals.

© 2016 Optical Society of America

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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]

2015 (1)

2014 (3)

J. P. Toomey and D. M. Kane, “Mapping the dynamic complexity of a semiconductor laser with optical feedback using permutation entropy,” Opt. Express 22(2), 1713–1725 (2014).
[Crossref] [PubMed]

J. Lehnert, P. Hövel, A. Selivanov, A. Fradkov, and E. Schöll, “Controlling cluster synchronization by adapting the topology,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 90(4), 042914 (2014).
[Crossref] [PubMed]

L. M. Pecora, F. Sorrentino, A. M. Hagerstrom, T. E. Murphy, and R. Roy, “Cluster synchronization and isolated desynchronization in complex networks with symmetries,” Nat. Commun. 5, 4079 (2014).
[Crossref] [PubMed]

2013 (1)

C. R. S. Williams, T. E. Murphy, R. Roy, F. Sorrentino, T. Dahms, and E. Schöll, “Experimental observations of group synchrony in a system of chaotic optoelectronic oscillators,” Phys. Rev. Lett. 110(6), 064104 (2013).
[Crossref] [PubMed]

2012 (7)

V. Flunkert and E. Schöll, “Chaos synchronization in networks of delay-coupled lasers: role of the coupling phases,” New J. Phys. 14(3), 033039 (2012).
[Crossref]

M. Bourmpos, A. Argyris, and D. Syvridis, “Sensitivity analysis of a star optical network based on mutually coupled semiconductor lasers,” J. Lightwave Technol. 30(16), 2618–2624 (2012).
[Crossref]

A. M. Hagerstrom, T. E. Murphy, R. Roy, P. Hövel, I. Omelchenko, and E. Schöll, “Experimental observation of chimeras in coupled-map lattices,” Nat. Phys. 8(9), 658–661 (2012).
[Crossref]

M. Nixon, M. Fridman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Controlling synchronization in large laser networks,” Phys. Rev. Lett. 108(21), 214101 (2012).
[Crossref] [PubMed]

T. Dahms, J. Lehnert, and E. Schöll, “Cluster and group synchronization in delay-coupled networks,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 86(1), 016202 (2012).
[Crossref] [PubMed]

Y. Aviad, I. Reidler, M. Zigzag, M. Rosenbluh, and I. Kanter, “Synchronization in small networks of time-delay coupled chaotic diode lasers,” Opt. Express 20(4), 4352–4359 (2012).
[Crossref] [PubMed]

J. Tiana-Alsina, K. Hicke, X. Porte, M. C. Soriano, M. C. Torrent, J. Garcia-Ojalvo, and I. Fischer, “Zero-lag synchronization and bubbling in delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 85(2), 026209 (2012).
[Crossref] [PubMed]

2011 (3)

M. Nixon, M. Friedman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Synchronized cluster formation in coupled laser networks,” Phys. Rev. Lett. 106(22), 223901 (2011).
[Crossref] [PubMed]

K. Hicke, O. D’Huys, V. Flunkert, E. Schöll, J. Danckaert, and I. Fischer, “Mismatch and synchronization: influence of asymmetries in systems of two delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(5), 056211 (2011).
[Crossref] [PubMed]

I. Kanter, E. Kopelowitz, R. Vardi, M. Zigzag, W. Kinzel, M. Abeles, and D. Cohen, “Nonlocal mechanism for cluster synchronization in neural circuits,” Eur. Phys. Lett. 93(6), 66001 (2011).
[Crossref]

2010 (3)

A. E. Motter, “Nonlinear dynamics: Spontaneous synchrony breaking,” Nat. Phys. 6(3), 164–165 (2010).
[Crossref]

J. Zamora-Munt, C. Masoller, J. Garcia-Ojalvo, and R. Roy, “Crowd synchrony and quorum sensing in delay-coupled lasers,” Phys. Rev. Lett. 105(26), 264101 (2010).
[Crossref] [PubMed]

M. Fridman, M. Nixon, N. Davidson, and A. A. Friesem, “Passive phase locking of 25 fiber lasers,” Opt. Lett. 35(9), 1434–1436 (2010).
[Crossref] [PubMed]

2009 (2)

M. Nixon, M. Fridman, E. Ronen, A. A. Friesem, and N. Davidson, “Phase locking of two fiber lasers with time-delayed coupling,” Opt. Lett. 34(12), 1864–1866 (2009).
[Crossref] [PubMed]

V. Flunkert, O. D’Huys, J. Danckaert, I. Fischer, and E. Schöll, “Bubbling in delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(6), 065201 (2009).
[Crossref] [PubMed]

2008 (3)

O. D’Huys, R. Vicente, T. Erneux, J. Danckaert, and I. Fischer, “Synchronization properties of network motifs: influence of coupling delay and symmetry,” Chaos 18(3), 037116 (2008).
[Crossref] [PubMed]

S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, “Critical phenomena in complex networks,” Rev. Mod. Phys. 80(4), 1275–1335 (2008).
[Crossref]

A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, and C. Zhou, “Synchronization in complex networks,” Phys. Rep. 469(3), 93–153 (2008).
[Crossref]

2007 (2)

B. B. Zhou and R. Roy, “Isochronal synchrony and bidirectional communication with delay-coupled nonlinear oscillators,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(2), 026205 (2007).
[Crossref] [PubMed]

F. Sorrentino and E. Ott, “Network synchronization of groups,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(5), 056114 (2007).
[Crossref] [PubMed]

2005 (1)

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

2004 (3)

J. Mulet, C. R. Mirasso, T. Heil, and I. Fischer, “Synchronization scenario of two distant mutually coupled semiconductor lasers,” J. Opt. B. Quantum Semiclass Opt. 6(1), 97–105 (2004).
[Crossref]

D. M. Abrams and S. H. Strogatz, “Chimera states for coupled oscillators,” Phys. Rev. Lett. 93(17), 174102 (2004).
[Crossref] [PubMed]

F. Rogister, K. S. Thornburg, L. Fabiny, M. Möller, and R. Roy, “Power-law spatial correlations in arrays of locally coupled lasers,” Phys. Rev. Lett. 92(9), 093905 (2004).
[Crossref] [PubMed]

2002 (1)

C. Bandt and B. Pompe, “Permutation entropy: A natural complexity measure for time series,” Phys. Rev. Lett. 88(17), 174102 (2002).
[Crossref] [PubMed]

2001 (1)

S. H. Strogatz, “Exploring complex networks,” Nature 410(6825), 268–276 (2001).
[Crossref] [PubMed]

1998 (2)

L. M. Pecora and T. L. Carroll, “Master stability functions for synchronized coupled systems,” Phys. Rev. Lett. 80(10), 2109–2112 (1998).
[Crossref]

D. J. Watts and S. H. Strogatz, “Collective dynamics of ‘small-world’ networks,” Nature 393(6684), 440–442 (1998).
[Crossref] [PubMed]

1997 (1)

A. Hohl, A. Gavrielides, T. Erneux, and V. Kovanis, “Localized synchronization in two coupled nonidentical semiconductor lasers,” Phys. Rev. Lett. 78(25), 4745–4748 (1997).
[Crossref]

1996 (2)

M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, “Phase synchronization of chaotic oscillators,” Phys. Rev. Lett. 76(11), 1804–1807 (1996).
[Crossref] [PubMed]

L. Kocarev and U. Parlitz, “Generalized synchronization, predictability, and equivalence of unidirectionally coupled dynamical systems,” Phys. Rev. Lett. 76(11), 1816–1819 (1996).
[Crossref] [PubMed]

1995 (2)

E. Diehl and J. D. Sterman, “Effects of feedback complexity on dynamic decision making,” Organ. Behav. Hum. Decis. Process. 62(2), 198–215 (1995).
[Crossref]

Y. Kuramoto, “Scaling behavior of turbulent oscillators with non-local interaction,” Prog. Theor. Phys. 94(3), 321–330 (1995).
[Crossref]

1994 (2)

R. Roy and K. S. Thornburg., “Experimental synchronization of chaotic lasers,” Phys. Rev. Lett. 72(13), 2009–2012 (1994).
[Crossref] [PubMed]

P. Ashwin, J. Buescu, and I. Stewart, “Bubbling of attractors and synchronisation of chaotic oscillators,” Phys. Lett. A 193(2), 126–139 (1994).
[Crossref]

1993 (1)

H. D. I. Abarbanel, R. Brown, J. J. Sidorowich, and L. S. Tsimring, “The analysis of observed chaotic data in physical systems,” Rev. Mod. Phys. 65(4), 1331–1392 (1993).
[Crossref]

1990 (2)

E. Ott, C. Grebogi, and J. A. Yorke, “Controlling chaos,” Phys. Rev. Lett. 64(11), 1196–1199 (1990).
[Crossref] [PubMed]

K. Aihara, T. Takabe, and M. Toyoda, “Chaotic neural networks,” Phys. Lett. A 144(6-7), 333–340 (1990).
[Crossref]

1989 (1)

J. D. Sterman, “Deterministic chaos in an experimental economic system,” J. Econ. Behav. Organ. 12(1), 1–28 (1989).
[Crossref]

1980 (1)

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16(3), 347–355 (1980).
[Crossref]

1961 (1)

R. Fitzhugh, “Impulses and physiological states in theoretical models of nerve membrane,” Biophys. J. 1(6), 445–466 (1961).
[Crossref] [PubMed]

Abarbanel, H. D. I.

H. D. I. Abarbanel, R. Brown, J. J. Sidorowich, and L. S. Tsimring, “The analysis of observed chaotic data in physical systems,” Rev. Mod. Phys. 65(4), 1331–1392 (1993).
[Crossref]

Abeles, M.

I. Kanter, E. Kopelowitz, R. Vardi, M. Zigzag, W. Kinzel, M. Abeles, and D. Cohen, “Nonlocal mechanism for cluster synchronization in neural circuits,” Eur. Phys. Lett. 93(6), 66001 (2011).
[Crossref]

Abrams, D. M.

D. M. Abrams and S. H. Strogatz, “Chimera states for coupled oscillators,” Phys. Rev. Lett. 93(17), 174102 (2004).
[Crossref] [PubMed]

Aihara, K.

K. Aihara, T. Takabe, and M. Toyoda, “Chaotic neural networks,” Phys. Lett. A 144(6-7), 333–340 (1990).
[Crossref]

Annovazzi-Lodi, V.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

Arenas, A.

A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, and C. Zhou, “Synchronization in complex networks,” Phys. Rep. 469(3), 93–153 (2008).
[Crossref]

Argyris, A.

M. Bourmpos, A. Argyris, and D. Syvridis, “Sensitivity analysis of a star optical network based on mutually coupled semiconductor lasers,” J. Lightwave Technol. 30(16), 2618–2624 (2012).
[Crossref]

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

Ashwin, P.

P. Ashwin, J. Buescu, and I. Stewart, “Bubbling of attractors and synchronisation of chaotic oscillators,” Phys. Lett. A 193(2), 126–139 (1994).
[Crossref]

Aviad, Y.

Bandt, C.

C. Bandt and B. Pompe, “Permutation entropy: A natural complexity measure for time series,” Phys. Rev. Lett. 88(17), 174102 (2002).
[Crossref] [PubMed]

Bourmpos, M.

Brown, R.

H. D. I. Abarbanel, R. Brown, J. J. Sidorowich, and L. S. Tsimring, “The analysis of observed chaotic data in physical systems,” Rev. Mod. Phys. 65(4), 1331–1392 (1993).
[Crossref]

Brunner, D.

Buescu, J.

P. Ashwin, J. Buescu, and I. Stewart, “Bubbling of attractors and synchronisation of chaotic oscillators,” Phys. Lett. A 193(2), 126–139 (1994).
[Crossref]

Carroll, T. L.

L. M. Pecora and T. L. Carroll, “Master stability functions for synchronized coupled systems,” Phys. Rev. Lett. 80(10), 2109–2112 (1998).
[Crossref]

Cohen, D.

I. Kanter, E. Kopelowitz, R. Vardi, M. Zigzag, W. Kinzel, M. Abeles, and D. Cohen, “Nonlocal mechanism for cluster synchronization in neural circuits,” Eur. Phys. Lett. 93(6), 66001 (2011).
[Crossref]

Colet, P.

A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. García-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438(7066), 343–346 (2005).
[Crossref] [PubMed]

D’Huys, O.

K. Hicke, O. D’Huys, V. Flunkert, E. Schöll, J. Danckaert, and I. Fischer, “Mismatch and synchronization: influence of asymmetries in systems of two delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(5), 056211 (2011).
[Crossref] [PubMed]

V. Flunkert, O. D’Huys, J. Danckaert, I. Fischer, and E. Schöll, “Bubbling in delay-coupled lasers,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(6), 065201 (2009).
[Crossref] [PubMed]

O. D’Huys, R. Vicente, T. Erneux, J. Danckaert, and I. Fischer, “Synchronization properties of network motifs: influence of coupling delay and symmetry,” Chaos 18(3), 037116 (2008).
[Crossref] [PubMed]

Dahms, T.

C. R. S. Williams, T. E. Murphy, R. Roy, F. Sorrentino, T. Dahms, and E. Schöll, “Experimental observations of group synchrony in a system of chaotic optoelectronic oscillators,” Phys. Rev. Lett. 110(6), 064104 (2013).
[Crossref] [PubMed]

T. Dahms, J. Lehnert, and E. Schöll, “Cluster and group synchronization in delay-coupled networks,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 86(1), 016202 (2012).
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J. Lehnert, P. Hövel, A. Selivanov, A. Fradkov, and E. Schöll, “Controlling cluster synchronization by adapting the topology,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 90(4), 042914 (2014).
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J. Zamora-Munt, C. Masoller, J. Garcia-Ojalvo, and R. Roy, “Crowd synchrony and quorum sensing in delay-coupled lasers,” Phys. Rev. Lett. 105(26), 264101 (2010).
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F. Rogister, K. S. Thornburg, L. Fabiny, M. Möller, and R. Roy, “Power-law spatial correlations in arrays of locally coupled lasers,” Phys. Rev. Lett. 92(9), 093905 (2004).
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M. Nixon, M. Fridman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Controlling synchronization in large laser networks,” Phys. Rev. Lett. 108(21), 214101 (2012).
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M. Nixon, M. Fridman, E. Ronen, A. A. Friesem, and N. Davidson, “Phase locking of two fiber lasers with time-delayed coupling,” Opt. Lett. 34(12), 1864–1866 (2009).
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A. M. Hagerstrom, T. E. Murphy, R. Roy, P. Hövel, I. Omelchenko, and E. Schöll, “Experimental observation of chimeras in coupled-map lattices,” Nat. Phys. 8(9), 658–661 (2012).
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M. Nixon, M. Fridman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Controlling synchronization in large laser networks,” Phys. Rev. Lett. 108(21), 214101 (2012).
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M. Nixon, M. Friedman, E. Ronen, A. A. Friesem, N. Davidson, and I. Kanter, “Synchronized cluster formation in coupled laser networks,” Phys. Rev. Lett. 106(22), 223901 (2011).
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M. Nixon, M. Fridman, E. Ronen, A. A. Friesem, and N. Davidson, “Phase locking of two fiber lasers with time-delayed coupling,” Opt. Lett. 34(12), 1864–1866 (2009).
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[Crossref]

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J. Lehnert, P. Hövel, A. Selivanov, A. Fradkov, and E. Schöll, “Controlling cluster synchronization by adapting the topology,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 90(4), 042914 (2014).
[Crossref] [PubMed]

C. R. S. Williams, T. E. Murphy, R. Roy, F. Sorrentino, T. Dahms, and E. Schöll, “Experimental observations of group synchrony in a system of chaotic optoelectronic oscillators,” Phys. Rev. Lett. 110(6), 064104 (2013).
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[Crossref]

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[Crossref]

T. Dahms, J. Lehnert, and E. Schöll, “Cluster and group synchronization in delay-coupled networks,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 86(1), 016202 (2012).
[Crossref] [PubMed]

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[Crossref] [PubMed]

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[Crossref] [PubMed]

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J. Lehnert, P. Hövel, A. Selivanov, A. Fradkov, and E. Schöll, “Controlling cluster synchronization by adapting the topology,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 90(4), 042914 (2014).
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Supplementary Material (1)

NameDescription
» Visualization 1: MP4 (2579 KB)      Mapping of the pairwise max-CC of the 16-SLs' network

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Figures (7)

Fig. 1
Fig. 1 Full-mesh-type network with optically-coupled 16 SLs. (a) Laser network topology: PC: Polarization controller, 1x2 and 1x8: optical couplers, EDFA: 25dB-gain Erbium-doped fiber amplifier, OF: Optical filter, PM: Inline optical power monitor, ATT: Optical attenuator. (b) Monitoring stage: It includes optical signal detection from laser's output port in an optical spectrum analyzer (OSA), as well as electrical signal detection in a four-channel real-time oscilloscope (OSC) - by employing equal number of digital photoreceivers - and in a radio-frequency analyzer (RFA). (c) Biasing/temperature conditions of the uncoupled solitary 16 SLs (#: laser identification number) for identical wavelength emission (λ = 1549.600) and for different levels of near-threshold optical power emission (black rectangles: PL#em = −15dBm, red circles: PL#em = −20dBm, blue triangles: PL#em = −25dBm).

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Fig. 2
Fig. 2 Effect of coupling strength on the correlated emission of a 16-laser coupled network. Averaged-CC values between two pairs of SLs (L#1-L#2 and L#6-L#7) vs. the applied injection ratio, when the emitted power from the lasers is set to −15dBm. Timetraces in insets show the dynamics of the emitted signals for different coupling strengths. Gray region indicates synchronized network coupling conditions through chaotic signals.

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Fig. 3
Fig. 3 Temporal evolution of L#1 and L#2 emission, as well as their difference in a coupled 16-laser network, when R = 0.2dB. De-synchronziation events appear when power dropouts occur and are minimized for optimal operating and coupling conditions. In the right column, a detail in the temporal region where a de-synchronization event takes place is provided.

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Fig. 4
Fig. 4 Cross-correlation mapping of the experimentally built 16-laser coupled network under synchronized conditions. Max-CC among pairs of lasers, considering (a) the detection-limited bandwidth emitted signals (8GHz), and (b) the filtered emitted signals with 2GHz bandwidth. The power of lasers' emissions is −15dBm, frequency detuning is minimized and R = 0.2dB.

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Fig. 5
Fig. 5 Effects of a frequency detuned laser (L#16) in correlated emission and NPE of emitted signals, in a 16-laser coupled network. (a) Pairwise average-CC evaluation of L#14, L#15 and L#16, for (a1) low-level (R = −12dB) and (a2) high-level (R = 0.2dB) injection ratios. Low injection conditions establish a highly sensitive system to wavelength emission mismatches, while high injection conditions establish a consistently synchronized system. (b) Normalized permutation entropy mapping of a laser with fixed conditions in the network (L#14) vs. ordinal pattern and applied delay, for: (b1) R = −12dB and frequency matched L#16, (b2) R = −12dB and L#16 with frequency detuning of 3.5GHz, (b3) R = 0.2dB and frequency matched L#16, (b4) R = 0.2dB and L#16 with frequency detuning of 17.6GHz. Contour plots are used for better transient screening.

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Fig. 6
Fig. 6 Cluster synchronization in an 8-SL coupled network configuration. Cross-correlation mapping with (a) zero-detuned wavelength laser emission, and (b) with cluster synchronization among two quartets of lasers (L#1-L#4 and L#5-L#8) that are 50pm spaced in wavelength, when R = 0.8dB. (c) Timeseries of laser emission (L#1: yellow, L#2: green, L#6: purple, L#7: pink) for the cases of (a) (top) and (b) (bottom). Y-axis shift on timeseries is applied only for screening purposes. (d) Averaged-CC between lasers vs. the injection ratio level (two lasers from each cluster are monitored), for the cluster synchronization conditions of case (b). The optical power of all lasers' emission is −15dBm.

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Fig. 7
Fig. 7 Synchrony comparison in an 8-node coupled network that includes 7 identical SLs and 1 SL from a different manufacturer. The comparison is made between the worst performance among the 7 identical lasers (black rectangles) and the best performance of synchronization between the different SL and the 7 identical lasers (red circles), versus the different SL’s emitted optical power and for (a) R = −9dBm and (b) R = −0.5dBm. All uncoupled identical lasers emit optical power −15dBm, while all uncoupled lasers operate at λ = 1549.600nm.

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Equations (6)

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R L# = C 2 ⋅ P inj tot P em L#
SNR= σ chaoti c signal 2 σ syste m noise 2
d E i (t) dt =jΔ ω i ⋅ E i (t)+ 1 2 (1+ja)[ G i (t)− t ph −1 )⋅ E i (t)+κ⋅ ∑ m E m (t− τ im ) e −i ω 0 τ im + D ξ(t)
d N i (t) dt = I e − N i (t) t s − G i (t)⋅ | E i (t) | 2
G i (t)= g n ⋅[ N i (t)− N 0 ]⋅ (1+s⋅ | E i (t) | 2 ) −1
H=H(P)= S(P) S max = − ∑ i=1 D! p( π i )⋅lnp( π i ) lnD!

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