Criterion method of GCCS for three-node VCSEL networks with delay coupling

Abstract

A criterion method of GCCS (Globally Complete Chaos Synchronization) for three-node VCSEL (Vertical Cavity Surface Emitting Laser) networks with delay coupling is provided, including steps of: providing a delay-coupled VCSEL network consisting of three identical units and dynamic equations of the VCSEL network; providing assumptions of an outer-coupling matrix and a unitary matrix under the dynamic equations of the VCSEL network; in the three-node VCSEL network, determining rate equations of i-VCSEL, determining dynamic equations of a synchronization manifold, and determining a master-stability equation; calculating three maximum Lyapunov exponents; determining a stability of a synchronization state of the three-node VCSEL network, and determining whether the synchronization manifold of the VCSEL network is a chaotic waveform. Through a master-stability function, the method for determining whether the GCCS is achieved among all node lasers is provided, which solves a difficult problem of GCCS criterion for the VCSEL networks.

Claims

1. A criterion method of GCCS (Globally Complete Chaos Synchronization) for three-node VCSEL (Vertical Cavity Surface Emitting Laser) networks with delay coupling, comprising steps of: (1), providing a delay-coupled VCSEL network consisting of three identical units, which comprises specific steps of: using multiple optical isolators in optical paths for performing the GCCS, so as to guarantee unidirectional propagation of optical wave; processing the three identical units with mutual coupling through multiple beam splitters; providing optical feedbacks to a first VCSEL node and a second VCSEL node through a first mirror and a second minor, respectively; wherein when a bias current, an optical feedback strength and an injection strength are fixed to certain values, all of the three identical units emit two chaotic polarization components; because polarization directions thereof are along directions of an x-axis and a y-axis, the chaotic polarization components are defined as x linear polarization and y linear polarization; dynamic equations of the VCSEL network are that: ${\stackrel{g}{X}}_i\left(t\right)=F\left(X_i\left(t\right)\right)+\underset{j=1}{\overset{3}{.Math.}}{K}_{\mathrm{ij}}H\left(X_i\left(t\right),X_j\left(t-\tau \right)\right),i=1,2,3;$ wherein: in the equations, X.sub.i (t)=[X.sub.i1(t), X.sub.i2 (t), X.sub.i3 (t), X.sub.i4 (t), X.sub.i5(t), X.sub.i6 (t)].sup.T ∈R.sup.6 represents a state variable of an i.sup.th node; F: R.sup.6.fwdarw.R.sup.6 represents a nonlinear vector-valued function of an isolated node; H: R.sup.6×R.sup.6.fwdarw.R.sup.6 represents an inner-coupling function of the i.sup.th node and a j.sup.th node; τ is a coupling time-delay, which is assumed to be the same for all links; K=(K.sub.ij).sub.3×3 ∈R.sup.3×3 is an outer-coupling matrix which describes a coupling topology and a strength of each link in the network; K.sub.ii is a self-feedback strength of the i.sup.th node; and, K.sub.ij is an injection strength from the j.sup.th node to the i.sup.th node; (2), providing two assumptions, wherein: for a first assumption, the sum of each row in the outer-coupling matrix K=(K.sub.ij).sub.3×3 ∈R.sup.3×3 is assumed to be a same constant (constant row sum for short) that: $\sigma =\underset{j=1}{\overset{3}{.Math.}}{K}_{\mathrm{ij}},i=1,2,3;$ for a second assumption, it is assumed that a unitary matrix U exists, which makes K.sup.T=UΛU.sup.−1; the first assumption ensures an existence of an invariant synchronization manifold; and, for a given constant row sum σ, a dynamic in the synchronization manifold of the VCSEL network is that:
(t)=F(S(t))+σH(S(t),S(t−τ)); a maximum Lyapunov exponent λ.sub.max is a function of σ and λ.sub.k, and is calculated through a master-stability equation of:
(t)=Aξ(t)+λ.sub.kDH(S(t−τ))ξ(t−τ); in the equation, A=DF(S(t))+σDH (S(t)); λ.sub.k is an eigenvalue of K, and k=1, 2, 3; DF and DH are Jacobian matrices calculated on the synchronization manifold; an MSF (Master-Stability Function) is able to be obtained through calculating the maximum Lyapunov exponent; the constant row sum σ is an eigenvalue of the outer-coupling matrix K; when λ.sub.k=σ, the equation of (t)=Aξ(t)+λ.sub.kDH(S(t−τ))ξ(t−τ) is the master-stability equation for the synchronization manifold, and σ is related to disturbances within the synchronization manifold; and, transversal eigenvalues refer to all eigenvalues except for the eigenvalue σ; (3), based on a spin-flip model and the dynamic equations of the VCSEL network, obtaining rate equations for an i-VCSEL in the three-node VCSEL networks that: $\frac{{\mathrm{dE}}_{x,y}^i\left(t\right)}{\mathrm{dt}}=\left\{\kappa \left[N^i\left(t\right)-1\right]m{\gamma }_a\right\}{E}_{x.y}^i\left(t\right)-\kappa n^i\left(t\right)E_{y,x}^i\left(t\right)\left\{\sin \left[{\phi }_y^i\left(t\right)-{\phi }_x^i\left(t\right)\right]\pm \alpha \cos \left[{\phi }_y^i\left(t\right)-{\phi }_x^i\left(t\right)\right]\right\}+K_{i1}{E}_{x,y}^1\left(t-\tau \right)\cos \left[\omega \tau +{\phi }_{x,y}^i\left(t\right)-{\phi }_{x,y}^1\left(t-\tau \right)\right]+K_{i2}{E}_{x,y}^2\left(t-\tau \right)\cos \left[\omega \tau +{\phi }_{x,y}^i\left(t\right)-{\phi }_{x,y}^2\left(t-\tau \right)\right]+K_{i3}{E}_{x,y}^3\left(t-\tau \right)\cos \left[\omega \tau +{\phi }_{x,y}^i\left(t\right)-{\phi }_{x,y}^3\left(t-\tau \right)\right],\frac{d{\phi }_{x,y}^i\left(t\right)}{\mathrm{dt}}=\kappa \alpha \left[N^i\left(t\right)-1\right]m{\gamma }_p\pm \kappa n^i\left(t\right)\frac{E_{y,x}^i\left(t\right)}{E_{x,y}^i\left(t\right)}\left\{\cos \left[{\phi }_y^i\left(t\right)-{\phi }_x^i\left(t\right)\right]m\alpha \sin \left[{\phi }_y^i\left(t\right)-{\phi }_x^i\left(t\right)\right]\right\}-K_{i1}\frac{E_{x,y}^1\left(t-\tau \right)}{E_{x,y}^i\left(t\right)}\sin \left[\omega \tau +{\phi }_{x,y}^i\left(t\right)-{\phi }_{x,y}^1\left(t-\tau \right)\right]-K_{i2}\frac{E_{x,y}^2\left(t-\tau \right)}{E_{x,y}^i\left(t\right)}\sin \left[\omega \tau +{\phi }_{x,y}^i\left(t\right)-{\phi }_{x,y}^2\left(t-\tau \right)\right]-K_{i3}\frac{E_{x,y}^3\left(t-\tau \right)}{E_{x,y}^i\left(t\right)}\sin \left[\omega \tau +{\phi }_{x,y}^i\left(t\right)-{\phi }_{x,y}^3\left(t-\tau \right)\right],\frac{{\mathrm{dN}}^i\left(t\right)}{\mathrm{dt}}={\gamma }_e\left\{\mu -N^i\left(t\right)\left[1+{\left(E_x^i\left(t\right)\right)}^2+{\left(E_y^i\left(t\right)\right)}^2\right]+2n^i\left(t\right)E_x^i\left(t\right)E_y^i\left(t\right)\sin \left({\phi }_y^i\left(t\right)-{\phi }_x^i\left(t\right)\right)\right\},\frac{{\mathrm{dn}}^i\left(t\right)}{\mathrm{dt}}=-{\gamma }_s{n}^i\left(t\right)-{\gamma }_e\left\{n^i\left(t\right)\left[{\left(E_x^i\left(t\right)\right)}^2+{\left(E_y^i\left(t\right)\right)}^2\right]-2N^i\left(t\right)E_x^i\left(t\right)E_y^i\left(t\right)\sin \left[{\phi }_y^i\left(t\right)-{\phi }_x^i\left(t\right)\right]\right\},$ wherein: in the equations, a superscript i represents the i-VCSEL; subscripts x and y respectively represent an x linear polarization mode and a y linear polarization mode; E is a slowly varying real amplitude of field; φ is a real phase of the field; N is a total carrier inversion between a conduction band and a valence band; n is a difference between carrier inversions for a spin-up radiation channel and a spin-down radiation channel; κ is a decay rate of the field; α is a linewidth enhancement factor; γ.sub.e is a decay rate of total carrier population; γ.sub.s is a spin-flip rate; γ.sub.a and γ.sub.p are linear anisotropies, respectively representing dichroism and birefringence; μ is a normalized injection current; and, central angular frequencies ω of all node VCSELs are assumed to be the same; (4), according to the equation of (t)=F(S(t))+σH(S(t), S(t−τ)), obtaining dynamic equations of the synchronization manifold of the three-node VCSEL network that: $\frac{{\mathrm{dE}}_{\mathrm{sx},y}\left(t\right)}{\mathrm{dt}}=\left\{\kappa \left[N_s\left(t\right)-1\right]m{\gamma }_a\right\}{E}_{\mathrm{sx},y}\left(t\right)-\kappa n_s\left(t\right)E_{\mathrm{sy},x}\left(t\right)\left\{\sin \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\pm \alpha \cos \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\right\}+\sigma E_{\mathrm{sx},y}\left(t-\tau \right)\cos \left[{\phi }_{\mathrm{sx},y}\left(t\right)-{\phi }_{\mathrm{sx},y}\left(t-\tau \right)+\omega \tau \right],\frac{d{\phi }_{\mathrm{sx},y}\left(t\right)}{\mathrm{dt}}=\left\{\kappa \alpha \left[N_s\left(t\right)-1\right]m{\gamma }_p\right\}\pm \frac{\kappa n_s\left(t\right)E_{\mathrm{sy},x}\left(t\right)}{E_{\mathrm{sx},y}\left(t\right)}\left\{\cos \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]m\alpha \sin \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\right\}-\frac{\sigma E_{\mathrm{sx},y}\left(t-\tau \right)}{E_{\mathrm{sx},y}\left(t\right)}\sin \left[{\sigma }_{\mathrm{sx},y}\left(t\right)-{\phi }_{\mathrm{sx},y}\left(t-\tau \right)+\omega \tau \right],\frac{{\mathrm{dN}}_s\left(t\right)}{\mathrm{dt}}={\gamma }_e\left\{\mu -N_s\left(t\right)\left[1+E_{\mathrm{sx}}^2\left(t\right)+E_{\mathrm{sy}}^2\left(t\right)\right]+2n_s\left(t\right)E_{\mathrm{sx}}\left(t\right)E_{\mathrm{sy}}\left(t\right)\sin \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\right\},\frac{{\mathrm{dn}}_s\left(t\right)}{\mathrm{dt}}=-{\gamma }_s{n}_s\left(t\right)-{\gamma }_e\left\{n_s\left(t\right)\left[E_{\mathrm{sx}}^2\left(t\right)+E_{\mathrm{sy}}^2\left(t\right)\right]-2N_s\left(t\right)E_{\mathrm{sx}}\left(t\right)E_{\mathrm{sy}}\left(t\right)\sin \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\right\},$ (5), according to the equation of (t)=Aξ(t)+λ.sub.kDH(S(t−τ))ξ(t−τ), obtaining master-stability equations for the three-node VCSEL network that: $\frac{d\Delta E_{x,y}\left(t\right)}{\mathrm{dt}}=\left[\kappa \left(N_s\left(t\right)-1\right)m{\gamma }_a\right]\Delta E_{x,y}\left(t\right)-\kappa n_s\left(t\right)\left\{\sin \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\pm \alpha \cos \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\right\}\Delta E_{y,x}\left(t\right)-\kappa n_s\left(t\right)E_{\mathrm{sy},x}\left(t\right)\left\{-\cos \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\pm \alpha \sin \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\right\}\Delta {\phi }_x\left(t\right)-\sigma E_{\mathrm{sx},y}\left(t-\tau \right)\sin \left[{\phi }_{\mathrm{sx},y}\left(t\right)-{\phi }_{\mathrm{sx},y}\left(t-\tau \right)+\omega \tau \right]\Delta {\phi }_{x,y}\left(t\right)-\kappa n_s\left(t\right)E_{\mathrm{sy},x}\left(t\right)\left\{\cos \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]m\alpha \sin \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\right\}\Delta {\phi }_y\left(t\right)+\kappa E_{\mathrm{sx},y}\left(t\right)\Delta N\left(t\right)-\kappa E_{\mathrm{sy},x}\left(t\right)\left\{\sin \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\pm \alpha \cos \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\right\}\Delta n\left(t\right)+{\lambda }_k\left\{\begin{array}{c}\cos \left[{\phi }_{\mathrm{sx},y}\left(t\right)-{\phi }_{\mathrm{sx},y}\left(t-\tau \right)+\omega \tau \right]\Delta E_{x,y}\left(t-\tau \right)+\\ E_{\mathrm{sx},y}\left(t-\tau \right)\sin \left[{\phi }_{\mathrm{sx},y}\left(t\right)-{\phi }_{\mathrm{sx},y}\left(t-\tau \right)+\omega \tau \right]\Delta {\phi }_{x,y}\left(t-\tau \right)\end{array}\right\},\frac{d\Delta {\phi }_{x,y}\left(t\right)}{\mathrm{dt}}=\frac{m\kappa n_s\left(t\right)E_{\mathrm{sy},x}\left(t\right)}{E_{\mathrm{sx},y}^2\left(t\right)}\left\{\cos \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]m\alpha \sin \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\right\}\Delta E_{x,y}\left(t\right)+\frac{\sigma E_{\mathrm{sx},y}\left(t-\tau \right)}{E_{\mathrm{sx},y}^2\left(t\right)}\sin \left[{\phi }_{\mathrm{sx},y}\left(t\right)-{\phi }_{\mathrm{sx},y}\left(t-\tau \right)+\omega \tau \right]\Delta E_{x,y}\left(t\right)\pm \frac{\kappa n_s\left(t\right)}{E_{\mathrm{sx},y}\left(t\right)}\left\{\cos \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]m\alpha \sin \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\right\}\Delta E_{y,x}\left(t\right)\pm \frac{\kappa n_s\left(t\right)E_{\mathrm{sy},x}\left(t\right)}{E_{\mathrm{sx},y}\left(t\right)}\left\{\sin \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\pm \alpha \cos \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\right\}\Delta {\phi }_x\left(t\right)-\frac{\sigma E_{\mathrm{sx},y}\left(t-\tau \right)}{E_{\mathrm{sx},y}\left(t\right)}\cos \left[{\phi }_{\mathrm{sx},y}\left(t\right)-{\phi }_{\mathrm{sx},y}\left(t-\tau \right)+\omega \tau \right)\Delta {\phi }_{x,y}\left(t\right)\pm \frac{\kappa n_s\left(t\right)E_{\mathrm{sy},x}\left(t\right)}{E_{\mathrm{sx},y}\left(t\right)}\left\{-\sin \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]m\alpha \cos \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\right\}\Delta {\phi }_y\left(t\right)+\kappa \alpha \Delta N\left(t\right)\pm \frac{\kappa E_{\mathrm{sy},x}\left(t\right)}{E_{\mathrm{sx},y}\left(t\right)}\left\{\cos \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]m\alpha \sin \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\right\}\Delta n\left(t\right)+{\lambda }_k\left\{\frac{-\sin \left[{\phi }_{\mathrm{sx},y}\left(t\right)-{\phi }_{\mathrm{sx},y}\left(t-\tau \right)+\omega \tau \right]}{E_{\mathrm{sx},y}\left(t\right)}\Delta E_{x,y}\left(t-\tau \right)+\frac{E_{\mathrm{sx},y}\left(t-\tau \right)}{E_{\mathrm{sx},y}\left(t\right)}\cos \left[{\phi }_{\mathrm{sx},y}\left(t\right)-{\phi }_{\mathrm{sx},y}\left(t-\tau \right)+\omega \tau \right]\Delta {\phi }_{x,y}\left(t-\tau \right)\right\},\frac{d\Delta N\left(t\right)}{\mathrm{dt}}=\left\{-2{\gamma }_e{N}_s\left(t\right)E_{\mathrm{sx}}\left(t\right)+2{\gamma }_e{n}_s\left(t\right)E_{\mathrm{sy}}\left(t\right)\sin \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\right\}\Delta E_x\left(t\right)+{\gamma }_e\left\{-2N_s\left(t\right)E_{\mathrm{sy}}\left(t\right)+2n_s\left(t\right)E_{\mathrm{sx}}\left(t\right)\sin \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\right\}\Delta E_y\left(t\right)-2{\gamma }_e{n}_s\left(t\right)E_{\mathrm{sx}}\left(t\right)E_{\mathrm{sy}}\left(t\right)\cos \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\Delta {\phi }_x\left(t\right)+2{\gamma }_e{n}_s\left(t\right)E_{\mathrm{sx}}\left(t\right)E_{\mathrm{sy}}\left(t\right)\cos \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\Delta {\phi }_y\left(t\right)-{\gamma }_e\left[1+E_{\mathrm{sx}}^2\left(t\right)+E_{\mathrm{sy}}^2\left(t\right)\right]\Delta N\left(t\right)+2{\gamma }_e{E}_{\mathrm{sx}}\left(t\right)E_{\mathrm{sy}}\left(t\right)\sin \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\Delta n\left(t\right),\frac{d\Delta n\left(t\right)}{\mathrm{dt}}=\left\{-2{\gamma }_e{N}_s\left(t\right)E_{\mathrm{sx}}\left(t\right)+2{\gamma }_e{n}_s\left(t\right)E_{\mathrm{sy}}\left(t\right)\sin \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\right\}\Delta E_x\left(t\right)+{\gamma }_e\left\{-2N_s\left(t\right)E_{\mathrm{sy}}\left(t\right)+2n_s\left(t\right)E_{\mathrm{sx}}\left(t\right)\sin \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\right\}\Delta E_y\left(t\right)-2{\gamma }_e{N}_s\left(t\right)E_{\mathrm{sx}}\left(t\right)E_{\mathrm{sy}}\left(t\right)\cos \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\Delta {\phi }_x\left(t\right)+2{\gamma }_e{N}_s\left(t\right)E_{\mathrm{sx}}\left(t\right)E_{\mathrm{sy}}\left(t\right)\cos \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\Delta {\phi }_y\left(t\right)+2{\gamma }_e{E}_{\mathrm{sx}}\left(t\right)E_{\mathrm{sy}}\left(t\right)\sin \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\Delta N\left(t\right)-\left\{{\gamma }_s+{\gamma }_e\left[E_{\mathrm{sx}}^2\left(t\right)+E_{\mathrm{sy}}^2\left(t\right)\right]\right\}\Delta n\left(t\right).$ (6), according to the equations in the steps (4) and (5), calculating three maximum Lyapunov exponents, determining a stability of a synchronization state of the three-node VCSEL network, and determining whether the synchronization manifold of the three-node VCSEL network is a chaotic waveform.

2. The criterion method of the GCCS for the three-node VCSEL networks with the delay coupling, as recited in claim 1, wherein: in the step (1), when t.fwdarw.∞, if x.sub.1(t)=x.sub.2(t)=x.sub.3(t)=S(t), the VCSEL network realizes globally complete synchronization, wherein S(t) represents the synchronization manifold of the VCSEL network; when S(t) is a chaotic waveform, the VCSEL network realizes the GCCS.

3. The criterion method of the GCCS for the three-node VCSEL networks with the delay coupling, as recited in claim 1, wherein: in the step (2), for a given coupling topology with the constant row sum σ, if λ.sub.1 max and λ.sub.2 max are respectively negative at the transversal eigenvalues λ.sub.1 and λ.sub.2 the synchronization state is stable; if λ.sub.3 max is positive at λ.sub.3=σ, temporal traces of the synchronization manifold is chaotic.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 is a flow chart of a criterion method of GCCS (Globally Complete Chaos Synchronization) for three-node VCSEL (Vertical Cavity Surface Emitting Laser) networks with delay coupling according to the present invention.

(2) FIG. 2 is a sketch view of an MSF (Master-Stability Function) for the three-node VCSEL network in a parameter space of σ and λ.sub.k according to the present invention.

(3) FIG. 3 is a sketch view of a ring topology based on three mutually coupled VCSELs according to the present invention.

(4) FIG. 4 is a sketch view of detailed optical paths of the ring topology based on the three mutually coupled VCSELs according to the present invention.

(5) FIG. 5 is a sketch view of temporal evolutions of synchronization errors for the three-node VCSEL network according to the present invention.

(6) FIG. 6 is a sketch view of temporal traces of a synchronization manifold for the three-node VCSEL network according to the present invention.

(7) FIG. 7 is a sketch view of a topology of a bus-shaped three-node VCSEL network according to the present invention.

(8) FIG. 8 is a sketch view of detailed optical paths of the bus-shaped three-node VCSEL network according to the present invention.

(9) FIG. 9 is a sketch view of temporal evolutions of synchronization errors for the bus-shaped three-node VCSEL network according to the present invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

(10) Technical solutions of the present invention are further described in detail with accompanying drawings as follows, but the protection scope of the present invention is not limited thereto.

(11) As shown in FIG. 1, a criterion method of GCCS (Globally Complete Chaos Synchronization) for three-node VCSEL (Vertical Cavity Surface Emitting Laser) networks with delay coupling comprises steps of:

(12) (1), providing a delay-coupled VCSEL network consisting of three identical units, wherein dynamic equations of the VCSEL network are that:

(13) ${\stackrel{g}{X}}_i\left(t\right)=F\left(X_i\left(t\right)\right)+\underset{j=1}{\overset{3}{.Math.}}{K}_{\mathrm{ij}}H\left(X_i\left(t\right),X_j\left(t-\tau \right)\right),i=1,2,3;$

(14) wherein: in the equations, X.sub.i(t)=[X.sub.i1(t), X.sub.i2(t), X.sub.i3(t), X.sub.i4(t), X.sub.i5(t), X.sub.i6(t)].sup.T∈R.sup.6 represents a state variable of an i.sup.th node; F:R.sup.6.fwdarw.R.sup.6 represents a nonlinear vector-valued function of an isolated node; H:R.sup.6×R.sup.6.fwdarw.R.sup.6 represents an inner-coupling function of the i.sup.th node and a j.sup.th node; r is a coupling time-delay, which is assumed to be the same for all links; K=(K.sub.ij).sub.3×3∈R.sup.3×3 is an outer-coupling matrix which describes a coupling topology and a strength of each link in the network; K.sub.ii is a self-feedback strength of the i.sup.th node; and, K.sub.ij is an injection strength from the j.sup.th node to the i.sup.th node;

(15) (2), providing two assumptions, wherein:

(16) for a first assumption, the sum of each row in the outer-coupling matrix K=(K.sub.ij).sub.3×3∈R.sup.3×3 is assumed to be a same constant (constant row sum for short) that:

(17) $\sigma =\underset{j=1}{\overset{3}{.Math.}}{K}_{\mathrm{ij}},i=1,2,3;$

(18) for a second assumption, it is assumed that a unitary matrix U exists, which makes K.sup.T=UΛU.sup.−1;

(19) the first assumption ensures an existence of an invariant synchronization manifold; and, for a given constant row sum σ, a dynamic in the synchronization manifold of the VCSEL network is that:
(t)=F(S(t))+σH(S(t),S(t−τ)),

(20) a maximum Lyapunov exponent λ.sub.max is a function of σ and λ.sub.k, and is calculated through a master-stability equation of:
(t)=Aξ(t)+λ.sub.kDH(S(t−τ))ξt(t−τ);

(21) in the equation, A=DF(S(t))+σDH(S(t); λ.sub.k is an eigenvalue of K, and k=1, 2, 3; DF and DH are Jacobian matrices calculated on the synchronization manifold; an MSF is able to be obtained through calculating the maximum Lyapunov exponent; the constant row sum σ is an eigenvalue of the outer-coupling matrix K; when λ.sub.k=σ, the equation of (t)=Aξ(t)+λ.sub.kDH(S(t−τ))ξ(t−τ) is the master-stability equation for the synchronization manifold, and σ is related to disturbances within the synchronization manifold; and, transversal eigenvalues refer to all eigenvalues except for the eigenvalue σ;

(22) (3), based on a spin-flip model and the dynamic equations of the VCSEL network, obtaining rate equations for an i-VCSEL in the three-node VCSEL networks that:

(23) $\frac{{\mathrm{dE}}_{x,y}^i\left(t\right)}{\mathrm{dt}}=\left\{\kappa \left[N^i\left(t\right)-1\right]m{\gamma }_a\right\}{E}_{x,y}^i\left(t\right)-\kappa n^i\left(t\right)E_{y,x}^i\left(t\right)\left\{\sin \left[{\phi }_y^i\left(t\right)-{\phi }_x^i\left(t\right)\right]\pm \alpha \cos \left[{\phi }_y^i\left(t\right)-{\phi }_x^i\left(t\right)\right]\right\}+K_{i1}{E}_{x,y}^1\left(t-\tau \right)\cos \left[\omega \tau +{\phi }_{x,y}^i\left(t\right)-{\phi }_{x,y}^1\left(t-\tau \right)\right]+K_{i2}{E}_{x,y}^2\left(t-\tau \right)\cos \left[\omega \tau +{\phi }_{x,y}^i\left(t\right)-{\phi }_{x,y}^2\left(t-\tau \right)\right]+K_{i3}{E}_{x,y}^3\left(t-\tau \right)\cos \left[\omega \tau +{\phi }_{x,y}^i\left(t\right)-{\phi }_{x,y}^3\left(t-\tau \right)\right],\frac{d{\phi }_{x,y}^i\left(t\right)}{\mathrm{dt}}=\kappa \alpha \left[N^i\left(t\right)-1\right]m{\gamma }_p\pm \kappa n^i\left(t\right)\frac{E_{y,x}^i\left(t\right)}{E_{x,y}^i\left(t\right)}\left\{\cos \left[{\phi }_y^i\left(t\right)-{\phi }_x^i\left(t\right)\right]m\alpha \sin \left[{\phi }_y^i\left(t\right)-{\phi }_x^i\left(t\right)\right]\right\}-K_{i1}\frac{E_{x,y}^1\left(t-\tau \right)}{E_{x,y}^i\left(t\right)}\sin \left[\omega \tau +{\phi }_{x,y}^i\left(t\right)-{\phi }_{x,y}^1\left(t-\tau \right)\right]-K_{i2}\frac{E_{x,y}^2\left(t-\tau \right)}{E_{x,y}^i\left(t\right)}\sin \left[\omega \tau +{\phi }_{x,y}^i\left(t\right)-{\phi }_{x,y}^2\left(t-\tau \right)\right]-K_{i3}\frac{E_{x,y}^3\left(t-\tau \right)}{E_{x,y}^i\left(t\right)}\sin \left[\omega \tau +{\phi }_{x,y}^i\left(t\right)-{\phi }_{x,y}^3\left(t-\tau \right)\right],\frac{{\mathrm{dN}}^i\left(t\right)}{\mathrm{dt}}={\gamma }_e\left\{\mu -N^i\left(t\right)\left[1+{\left(E_x^i\left(t\right)\right)}^2+{\left(E_y^i\left(t\right)\right)}^2\right]+2n^i\left(t\right)E_x^i\left(t\right)E_y^i\left(t\right)\sin \left({\phi }_y^i\left(t\right)-{\phi }_x^i\left(t\right)\right)\right\},\frac{{\mathrm{dn}}^i\left(t\right)}{\mathrm{dt}}=-{\gamma }_s{n}^i\left(t\right)-{\gamma }_e\left\{n^i\left(t\right)\left[{\left(E_x^i\left(t\right)\right)}^2+{\left(E_y^i\left(t\right)\right)}^2\right]-2N^i\left(t\right)E_x^i\left(t\right)E_y^i\left(t\right)\sin \left[{\phi }_y^i\left(t\right)-{\phi }_x^i\left(t\right)\right]\right\},$

(24) wherein: in the equations, a superscript i represents the i-VCSEL; subscripts x and y respectively represent an x linear polarization mode and a y linear polarization mode; E is a slowly varying real amplitude of field; φ is a real phase of the field; N is a total carrier inversion between a conduction band and a valence band; n is a difference between carrier inversions for a spin-up radiation channel and a spin-down radiation channel; κ is a decay rate of the field; α is a linewidth enhancement factor; γ.sub.e is a decay rate of total carrier population; γ.sub.s is a spin-flip rate; γ.sub.a and γ.sub.p are linear anisotropies, respectively representing dichroism and birefringence; μ is a normalized injection current; and, central angular frequencies ω of all node VCSELs are assumed to be the same;

(25) (4), according to the equation of (t)=F(S(t))+σH(S(t),S(t−τ)), obtaining dynamic equations of the synchronization manifold of the three-node VCSEL network that:

(26) $\frac{{\mathrm{dE}}_{\mathrm{sx},y}\left(t\right)}{\mathrm{dt}}=\left\{\kappa \left[N_s\left(t\right)-1\right]m{\gamma }_a\right\}{E}_{\mathrm{sx},y}\left(t\right)-\kappa n_s\left(t\right)E_{\mathrm{sy},x}\left(t\right)\left\{\sin \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\pm \alpha \cos \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\right\}+\sigma E_{\mathrm{sx},y}\left(t-\tau \right)\cos \left[{\phi }_{\mathrm{sx},y}\left(t\right)-{\phi }_{\mathrm{sx},y}\left(t-\tau \right)+\omega \tau \right],\frac{d{\phi }_{\mathrm{sx},y}\left(t\right)}{\mathrm{dt}}=\left\{\kappa \alpha \left[N_s\left(t\right)-1\right]m{\gamma }_p\right\}\pm \frac{\kappa n_s\left(t\right)E_{\mathrm{sy},x}\left(t\right)}{E_{\mathrm{sx},y}\left(t\right)}\left\{\cos \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]m\alpha \sin \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\right\}-\frac{\sigma E_{\mathrm{sx},y}\left(t-\tau \right)}{E_{\mathrm{sx},y}\left(t\right)}\sin \left[{\sigma }_{\mathrm{sx},y}\left(t\right)-{\phi }_{\mathrm{sx},y}\left(t-\tau \right)+\omega \tau \right],\frac{{\mathrm{dN}}_s\left(t\right)}{\mathrm{dt}}={\gamma }_e\left\{\mu -N_s\left(t\right)\left[1+E_{\mathrm{sx}}^2\left(t\right)+E_{\mathrm{sy}}^2\left(t\right)\right]+2n_s\left(t\right)E_{\mathrm{sx}}\left(t\right)E_{\mathrm{sy}}\left(t\right)\sin \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\right\},\frac{{\mathrm{dn}}_s\left(t\right)}{\mathrm{dt}}=-{\gamma }_s{n}_s\left(t\right)-{\gamma }_e\left\{n_s\left(t\right)\left[E_{\mathrm{sx}}^2\left(t\right)+E_{\mathrm{sx}}^2\left(t\right)\right]-2N_s\left(t\right)E_{\mathrm{sx}}\left(t\right)E_{\mathrm{sy}}\left(t\right)\sin \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\right\},$

(27) (5), according to the equation of (t)=Aξ(t)−λ.sub.kDH(S(t−τ))ξ(t−τ), obtaining master-stability equations for the three-node VCSEL network that:

(28) 0$\frac{d\Delta E_{x,y}\left(t\right)}{\mathrm{dt}}=\left[\kappa \left(N_s\left(t\right)-1\right)m{\gamma }_a\right]\Delta E_{x,y}\left(t\right)-\kappa n_s\left(t\right)\left\{\sin \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\pm \alpha \cos \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\right\}\Delta E_{y,x}\left(t\right)-\kappa n_s\left(t\right)E_{\mathrm{sy},x}\left(t\right)\left\{-\cos \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\pm \alpha \sin \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\right\}\Delta {\phi }_x\left(t\right)-\sigma E_{\mathrm{sx},y}\left(t-\tau \right)\sin \left[{\phi }_{\mathrm{sx},y}\left(t\right)-{\phi }_{\mathrm{sx},y}\left(t-\tau \right)+\omega \tau \right]\Delta {\phi }_{x,y}\left(t\right)-\kappa n_s\left(t\right)E_{\mathrm{sy},x}\left(t\right)\left\{\cos \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]m\alpha \sin \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\right\}\Delta {\phi }_y\left(t\right)+\kappa E_{\mathrm{sx},y}\left(t\right)\Delta N\left(t\right)-\kappa E_{\mathrm{sy},x}\left(t\right)\left\{\sin \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\pm \alpha \cos \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\right\}\Delta n\left(t\right)+{\lambda }_k\left\{\begin{array}{c}\cos \left[{\phi }_{\mathrm{sx},y}\left(t\right)-{\phi }_{\mathrm{sx},y}\left(t-\tau \right)+\omega \tau \right]\Delta E_{x,y}\left(t-\tau \right)+\\ E_{\mathrm{sx},y}\left(t-\tau \right)\sin \left[{\phi }_{\mathrm{sx},y}\left(t\right)-{\phi }_{\mathrm{sx},y}\left(t-\tau \right)+\omega \tau \right]\Delta {\phi }_{x,y}\left(t-\tau \right)\end{array}\right\},\frac{d\Delta {\phi }_{x,y}\left(t\right)}{\mathrm{dt}}=\frac{m\kappa n_s\left(t\right)E_{\mathrm{sy},x}\left(t\right)}{E_{\mathrm{sx},y}^2\left(t\right)}\left\{\cos \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]m\alpha \sin \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\right\}\Delta E_{x,y}\left(t\right)+\frac{\sigma E_{\mathrm{sx},y}\left(t-\tau \right)}{E_{\mathrm{sx},y}^2\left(t\right)}\sin \left[{\phi }_{\mathrm{sx},y}\left(t\right)-{\phi }_{\mathrm{sx},y}\left(t-\tau \right)+\omega \tau \right]\Delta E_{x,y}\left(t\right)\pm \frac{\kappa n_s\left(t\right)}{E_{\mathrm{sx},y}\left(t\right)}\left\{\cos \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]m\alpha \sin \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\right\}\Delta E_{y,x}\left(t\right)\pm \frac{\kappa n_s\left(t\right)E_{\mathrm{sy},x}\left(t\right)}{E_{\mathrm{sx},y}\left(t\right)}\left\{\sin \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\pm \alpha \cos \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\right\}\Delta {\phi }_x\left(t\right)-\frac{\sigma E_{\mathrm{sx},y}\left(t-\tau \right)}{E_{\mathrm{sx},y}\left(t\right)}\cos \left[{\phi }_{\mathrm{sx},y}\left(t\right)-{\phi }_{\mathrm{sx},y}\left(t-\tau \right)+\omega \tau \right)\Delta {\phi }_{x,y}\left(t\right)\pm \frac{\kappa n_s\left(t\right)E_{\mathrm{sy},x}\left(t\right)}{E_{\mathrm{sx},y}\left(t\right)}\left\{-\sin \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]m\alpha \cos \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\right\}\Delta {\phi }_y\left(t\right)+\kappa \alpha \Delta N\left(t\right)\pm \frac{\kappa E_{\mathrm{sy},x}\left(t\right)}{E_{\mathrm{sx},y}\left(t\right)}\left\{\cos \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]m\alpha \sin \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\right\}\Delta n\left(t\right)+{\lambda }_k\left\{\frac{-\sin \left[{\phi }_{\mathrm{sx},y}\left(t\right)-{\phi }_{\mathrm{sx},y}\left(t-\tau \right)+\omega \tau \right]}{E_{\mathrm{sx},y}\left(t\right)}\Delta E_{x,y}\left(t-\tau \right)+\frac{E_{\mathrm{sx},y}\left(t-\tau \right)}{E_{\mathrm{sx},y}\left(t\right)}\cos \left[{\phi }_{\mathrm{sx},y}\left(t\right)-{\phi }_{\mathrm{sx},y}\left(t-\tau \right)+\omega \tau \right]\Delta {\phi }_{x,y}\left(t-\tau \right)\right\},\frac{d\Delta N\left(t\right)}{\mathrm{dt}}=\left\{-2{\gamma }_e{N}_s\left(t\right)E_{\mathrm{sx}}\left(t\right)+2{\gamma }_e{n}_s\left(t\right)E_{\mathrm{sy}}\left(t\right)\sin \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\right\}\Delta E_x\left(t\right)+{\gamma }_e\left\{-2N_s\left(t\right)E_{\mathrm{sy}}\left(t\right)+2n_s\left(t\right)E_{\mathrm{sx}}\left(t\right)\sin \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\right\}\Delta E_y\left(t\right)-2{\gamma }_e{n}_s\left(t\right)E_{\mathrm{sx}}\left(t\right)E_{\mathrm{sy}}\left(t\right)\cos \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\Delta {\phi }_x\left(t\right)+2{\gamma }_e{n}_s\left(t\right)E_{\mathrm{sx}}\left(t\right)E_{\mathrm{sy}}\left(t\right)\cos \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\Delta {\phi }_y\left(t\right)-{\gamma }_e\left[1+E_{\mathrm{sx}}^2\left(t\right)+E_{\mathrm{sy}}^2\left(t\right)\right]\Delta N\left(t\right)+2{\gamma }_e{E}_{\mathrm{sx}}\left(t\right)E_{\mathrm{sy}}\left(t\right)\sin \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\Delta n\left(t\right),\frac{d\Delta n\left(t\right)}{\mathrm{dt}}=\left\{-2{\gamma }_e{N}_s\left(t\right)E_{\mathrm{sx}}\left(t\right)+2{\gamma }_e{n}_s\left(t\right)E_{\mathrm{sy}}\left(t\right)\sin \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\right\}\Delta E_x\left(t\right)+{\gamma }_e\left\{-2N_s\left(t\right)E_{\mathrm{sy}}\left(t\right)+2n_s\left(t\right)E_{\mathrm{sx}}\left(t\right)\sin \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\right\}\Delta E_y\left(t\right)-2{\gamma }_e{N}_s\left(t\right)E_{\mathrm{sx}}\left(t\right)E_{\mathrm{sy}}\left(t\right)\cos \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\Delta {\phi }_x\left(t\right)+2{\gamma }_e{N}_s\left(t\right)E_{\mathrm{sx}}\left(t\right)E_{\mathrm{sy}}\left(t\right)\cos \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\Delta {\phi }_y\left(t\right)+2{\gamma }_e{E}_{\mathrm{sx}}\left(t\right)E_{\mathrm{sy}}\left(t\right)\sin \left[{\phi }_{\mathrm{sy}}\left(t\right)-{\phi }_{\mathrm{sx}}\left(t\right)\right]\Delta N\left(t\right)-\left\{{\gamma }_s+{\gamma }_e\left[E_{\mathrm{sx}}^2\left(t\right)+E_{\mathrm{sy}}^2\left(t\right)\right]\right\}\Delta n\left(t\right).$