Dual-channel chaos synchronization and communication based on unidirectionally coupled VCSELs with polarization-rotated optical feedback and polarization-rotated optical injection

Jiao Liu; Zheng-Mao Wu; Guang-Qiong Xia

doi:10.1364/OE.17.012619

1. Introduction

Since the first demonstration of chaos synchronization by Pecora and Carroll [1], different chaotic synchronization schemes and their applications have been reported. Chaotic synchronization based on semiconductor lasers (SLs) and their applications in secure communication have attracted considerable interests during the past decades [2–24]. As one of the microchip lasers, vertical-cavity surface-emitting lasers (VCSELs) exhibit many advantages over the conventional edge-emitting semiconductor lasers (EELs), such as single longitudinal-mode operation, low threshold current, circular output beam with narrow divergence, low cost and easy large-scale integration into two-dimensional arrays etc. Generally, the output of the VCSEL includes two orthogonal linear polarization (LP) modes (i. e., x LP mode and y LP mode) due to weak material and cavity anisotropies, which in our opinions may afford a possibility to realize dual-channel communication by separately using the two LP modes. There were many experimental and theoretical reports on the synchronization and communication characteristics of two unidirectionally coupled VCSELs [7–16], where different systematical constructions, such as polarization-preserved coupled system with polarization-preserved optical feedback at the transmitter VCSEL (T-VCSEL) port and polarization-preserved optical injection at the receiver VCSEL (R-VCSEL) port [8–10], the system with polarization-preserved optical feedback and orthogonal optical injection [11–14] or system with orthogonal optical feedback and orthogonal optical injection [15], are taken into consideration respectively. However, we have noticed that most of relevant works usually focused on the synchronization properties of total output of VCSELs. In this paper, a novel dual-channel chaos synchronization system based on two unidirectionally coupled VCSELs subject to polarization-rotated optical feedback at the T-VCSEL port and polarization-rotated optical injection at the R-VCSEL port is presented, and the synchronization performances of such configuration and the influence of mismatched parameters on synchronization for each LP mode and the total output are investigated numerically. Also, communication performances between each pair of LP modes of two VCSELs have been preliminarily examined.

2. System model and theory

Figure 1 is the schematic diagram of the polarization-rotated coupled system. The output of T-VCSEL is divided into x and y LP modes by a polarization beam splitter (PBS). Then, different messages can be independently encoded on each LP mode. These two orthogonal LP modes are recombined by PBS2 before passing through a half-wave plate (HWP). Here, the angles between the x direction and the fast axis of the HWP are 45°. As a result, the x LP mode is rotated into the y direction polarization, while the y LP mode is rotated into the x direction polarization before being fed back into the T-VCSEL and being injected into the R-VCSEL. Optical isolator (ISO) is used to ensure the light unidirectional transmission.

Fig. 1 Schematic diagram for the dual-channel chaos communication system based on the unidirectionally VCSELs subject to polarization-rotated optical feedback and polarization-rotated optical injection. T-VCSEL: transmitter-VCSEL; R-VCSEL: receiver-VCSEL; PBS: polarization beam splitter; HWP: half-wave plate; BS: beam splitter; ISO: optical isolator; M: mirror; PD: photodetector.

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Based on the spin-flip model (SFM) [25], for such a proposed system, the rate equations for the T-VCSEL with polarization-rotated optical feedback and the R-VCSEL with polarization-rotated optical injection can be described by:

\frac{d E_{x, y}^{T}}{d t} = k (1 + i α) [(N^{T} - 1) E_{x, y}^{T} \pm i n^{T} E_{y, x}^{T}] \pm (- γ_{a} - i γ_{p}) E_{x, y}^{T} + f E_{y, x}^{T} (t - τ) e^{- i ω^{T} τ} + F_{x, y}^{T}

\frac{d E_{x, y}^{R}}{d t} =k (1 + i α) [(N^{R} - 1) E_{x, y}^{R} \pm i n^{R} E_{y, x}^{R}] \pm (- γ_{a} - i γ_{p}) E_{x, y}^{R} + η E_{y, x}^{T} (t - τ_{c}) e^{-i ω^{T} τ_{c} + i Δ ω t} + F_{x, y}^{R}

\frac{d N^{T, R}}{d t} = - γ [N^{T, R} - μ + N^{T, R} (| E_{x}^{T, R} |^{2} + | E_{y}^{T, R} |^{2})] - i γ n^{T, R} (E_{y}^{T, R} E_{x}^{T, R}^{*} - E_{x}^{T, R} E_{y}^{T, R}^{*})

\frac{d n^{T, R}}{d t} = - γ_{s} n^{T, R} - γ n^{T, R} (| E_{x}^{T, R} |^{2} + | E_{y}^{T, R} |^{2}) - i γ N^{T, R} (E_{y}^{T, R} E_{x}^{T, R}^{*} - E_{x}^{T, R} E_{y}^{T, R}^{*})

where superscripts T and R stand for T-VCSEL and R-VCSEL, respectively, and subscripts x and y represents x and y LP modes, respectively. E is the slowly varied complex amplitude of the field, N is the total carrier inversion between the conduction and valence bands, n accounts for the difference between carrier inversions for the spin-up and spin-down radiation channels, k is the decay rate of field, α is the line-width enhancement factor, γ is the decay rate of total carrier population, γ_s is the spin-flip rate, γ_a and γ_p are the linear anisotropies representing dichroism and birefringence, respectively, μ is the normalized injection current (μ takes the value 1 at threshold), τ is the feedback delay time and τ_c is the propagation delay time between the T-VCSEL and R-VCSEL, f is the feedback rate and η is the injection rate. ω^T and ω ^R are respectively the optical frequencies of T-VCSEL and R-VCSEL at the solitary laser threshold in the absence of linear anisotropies, Δω = ω ^T -ω ^R is the frequency detuning, and spontaneous emission noises are modeled by following Langevin sources:

F_{x}^{T, R} = \sqrt{β_{s p} / 2} (\sqrt{N^{T, R} + n^{T, R}} ξ_{1}^{T, R} + \sqrt{N^{T, R} - n^{T, R}} ξ_{2}^{T, R})

F_{y}^{T, R} = - i \sqrt{β_{s p} / 2} (\sqrt{N^{T, R} + n^{T, R}} ξ_{1}^{T, R} - \sqrt{N^{T, R} - n^{T, R}} ξ_{2}^{T, R})

where ξ₁ and ξ₂ indicate independent Gaussian white noise with zero mean and unitary variance, and β_sp is spontaneous emission rate [8].

3. Results and discussion

The rate Eqs. (1)-(4) can be numerically solved with the fourth-order Runge-Kutta method. During the calculations, the parameters are used as follows [8,25]: α = 3, k = 300GHz, γ = 1GHz, γ_a = 0.1GHz, γ_p = 10GHz, γ_s = 50GHz, ω^T = 2.2176 × 10¹⁵rad/s (the corresponding central wavelength is 850 nm), Δω = 0, β_sp = 10⁻⁶GHz, τ = 3ns, τ_c = 5ns, μ = 1.3, f = η = 15GHz.

3.1 LP mode intensity

Figure 2(a) gives P-I curve for each LP mode and the total output of a VCSEL subject to external polarization-rotated optical feedback. For comparison, the corresponding result for polarization-preserved optical feedback system is given in Fig. 2(b). In the numerical simulation, the intensities are averaged over the time window of 1.5μs. As shown in this diagram, the averaged intensity of x LP mode is always equivalent to that of y LP mode for the case of polarization-rotated optical feedback. However, for the case of polarization-preserved optical feedback, the average intensities of these two LP modes are very different.

Fig. 2 P-I curves for (a) polarization-rotated optical feedback and (b) polarization-preserved optical feedback.

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3.2 Complete chaotic synchronization

Chaos synchronization can be divided into complete synchronization and injection-locking synchronization. The time delay between the two chaotic waveforms is a key to distinguish from these two types of synchronization. The delay time is Δτ = τ_c-τ for complete synchronization, while it is τ_c for injection-locking synchronization. In this paper, we only focus on the complete synchronization. The time series of x LP, y LP and total output intensities of the two lasers are shown in Fig. 3(a) . As shown in these figures, the temporal waveforms for each pair of corresponding LP modes and the total output between the two lasers are almost the same but for a time lag Δτ.

Fig. 3 (a) Time series of x LP, y LP and total output, and (b) corresponding cross-correlation coefficients.

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To specifically describe the synchronization quality between the two lasers, the quality of chaos synchronization and its time shift can be quantified by calculating the following shifted correlation coefficient C(Δt):

C_{x, y} (Δ t) = \frac{〈 [I_{x, y}^{T} (t - Δ t) - 〈 I_{x, y}^{T} (t - Δ t) 〉] [I_{x, y}^{R} (t) - 〈 I_{x, y}^{R} (t) 〉] 〉}{{〈 {[I_{x, y}^{T} (t - Δ t) - 〈 I_{x, y}^{T} (t - Δ t) 〉]}^{2} 〉}^{1 / 2} {〈 {[I_{x, y}^{R} (t) - 〈 I_{x, y}^{R} (t) 〉]}^{2} 〉}^{1 / 2}}

where Δt is the time shift between these two lasers output, the bracket 〈〉 denote temporal average, I = |E|² is the output intensity of the laser. A large value of C indicates that good synchronization has been achieved. For perfect chaotic synchronization, C equals to 1. Figure 3(b) gives the cross correlation coefficients corresponding to Fig. 3(a) as a function of time shift. It can be seen that the quality of chaotic synchronization is very relevant to time shift. The peaks are very close to 1 for Δt = 2ns, which means that complete synchronization can be achieved with 2ns time lag for each pair of corresponding LP modes and the total output, respectively.

3.3 Effects of internal mismatched parameters on synchronization quality

In order to achieve complete synchronization, it is important to match corresponding parameters of the T-VCSEL and the R-VCSEL. Though external parameters such as current can be controlled easily, internal parameters are difficult to be accurately controlled. Therefore, it is important to investigate the influences of the intrinsic mismatched parameters on the quality of chaos synchronization. For convenience, we fix intrinsic parameters of T-VCSEL, and only change the intrinsic parameters of R-VCSEL. The relative mismatched parameters are defined as:

Δ α = \frac{α^{R} - α^{T}}{α^{T}}, Δ k = \frac{k^{R} - k^{T}}{k^{T}}, Δ γ = \frac{γ^{R} - γ^{T}}{γ^{T}}, Δ γ_{p} = \frac{γ_{p}^{R} - γ_{p}^{T}}{γ_{p}^{T}}, Δ γ_{s} = \frac{γ_{s}^{R} - γ_{s}^{T}}{γ_{s}^{T}}, Δ γ_{a} = \frac{γ_{a}^{R} - γ_{a}^{T}}{γ_{a}^{T}}

Figure 4(a) shows the variation of the maximum of cross correlation coefficient of each LP mode and total output with different internal mismatched parameters. For comparison, we also calculate the influence of mismatched parameters on synchronization for polarization-preserved coupled system, and the corresponding results are shown in Fig. 4(b). As seen from these figures, the effects of mismatched α, γ, κ, and γ_p on synchronization quality are relatively larger than that of mismatched γ_s, γ_a. It can also be observed that the polarization-rotated coupled system keeps a certain sensitive to mismatched parameters but possesses better robustness to mismatched parameters than that for polarization-preserved coupled system. It should be pointed out that above results are obtained for only one mismatched parameter. And in practice, there always exist multiple mismatched parameters. Under this condition, the system synchronization may further worsen. Therefore, one should select such T-VCSEL and R-VCSEL with good parameters matching in order to improve system synchronization quality. Additionally, for the polarization-preserved coupled system, almost same variation trends are obtained for x LP and the total output due to a very weak y LP mode output under above given conditions.

Fig. 4 Maximum of cross correlation coefficient between x LP, y LP and total output of T-VCSEL and R-VCSEL versus different mismatched parameters, where (a) and (b) is for polarization-rotated coupled system and polarization-preserved coupled system, respectively, dotted, solid and dashed lines correspond to x LP, y LP and total output, respectively.

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3.4 Message encoding and decoding

Finally, we will briefly examine the communication performances of this proposed system. Different encoding and decoding techniques such as chaos shift keying (CSK), chaos masking (CMS) and chaos modulation (CM) have been proposed and investigated for secure message transmission based on chaotic synchronization. In this paper, message is assumed to be encoded by means of additive chaos modulation (ACM). For the ACM encryption scheme as shown in Fig. 1, the message is mixed with the chaotic carrier in the nonlinear system, and then a new chaotic state different from the original one will be conformed. Because the symmetry of system can be maintained under the ACM scheme, the complete chaos synchronization can be achieved in the system and an excellent synchronous signal can be obtained at the receiver port in principle. The message encoding is carried out by modulating the chaotic carrier of the T-VCSEL so that $E_{x, y}^{T}^{'} (t) = E_{x, y}^{T} (t) (1 + m_{x, y} (t))$ [3,17], where the modulation index is 5% and the modulation frequency is 500MHz. By using a Fourth-Butterworth low-pass filter, signals can be demodulated by filtering the intensity difference between T-VSEL output and R-VCSEL output for each LP mode. Figure 5 displays the encoding m_x,y(t) and decoding message m_x,y^’(t) for each LP mode, respectively. From this figure, one can clearly observe that sent message carried by these two LP modes can be extracted satisfactorily.

Fig. 5 Original message m_x,y(t) and recovered message m_x,y^’(t), where (a) and (b) correspond to x LP mode and y LP mode, respectively.

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4. Conclusions

In summary, a novel dual-channel chaos communication system by using the two LP modes of VCSELs is presented. In such a system, a VCSEL with polarization-rotated optical feedback is used as a transmitter and a VCSEL subject to polarization-rotated optical injection is used as a receiver. The synchronization characteristics and the influence of mismatched parameters on synchronization performances have been numerically investigated. Similar to polarization-preserved synchronization system, complete synchronization can also be achieved between each pair of the LP modes and the total output of T-VCSEL and R-VCSEL for this polarization-rotated coupled configuration. Internal mismatched parameters between T-VCSEL and R-VCSEL have influence on synchronization quality, but this system possesses better robustness to mismatched parameters compared with the polarization-preserved synchronization system. Additionally, the communication performances are preliminarily examined by using different message carried separately by two LP modes.

Acknowledgments

This work was supported by the Open Fund of the State Key Lab of Millimeter Waves of China under Grant K200805 and the Natural Science Foundation Project of Chongqing City of China under Grant 2007BB2333.

References and links

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Dual-channel chaos synchronization and communication based on unidirectionally coupled VCSELs with polarization-rotated optical feedback and polarization-rotated optical injection

Abstract

1. Introduction

2. System model and theory

3. Results and discussion

3.1 LP mode intensity

3.2 Complete chaotic synchronization

3.3 Effects of internal mismatched parameters on synchronization quality

3.4 Message encoding and decoding

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (5)

Equations (8)

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