1. Introduction

In recent years, optical chaotic cryptosystems have attracted much attention for their potential applications in secure communications [1–6]. The underlying concept in such systems is that message is mixed with the noise-like output of the chaotic transmitter laser and is recovered by the receiver laser synchronized with the transmitter laser. Experimental researches have demonstrated that several GHz chaotic message can be transmitted and recovered based on diode laser cryptosystems [4, 5]. However, it will be difficult to further improve the communication rate due to the limitation of the bandwidth.

To improve the chaotic communication rate, chaotic multiplexed methods can be adopted [7, 8], and some configurations of multiple-channel chaotic communication have been presented [6–9]. Recently, a novel dual-channel optical chaotic communication system has been proposed and preliminary experimental results have been reported [6]. This scheme has some advantages such as the low cross talk between the two channels, encoding and decoding easily, expandability of the number of channels.

In this paper, a theoretical model of the dual-channel optical chaotic communication system [6] has been presented. Based on this model, the transmission function of semiconductor laser has been deduced by small-signal analysis, and the cross talk between two channels has been analyzed quantitatively. For a 250MHz modulation message, the results obtained by numerical simulation are in agreement with the experimental results of Ref. [6].

2. Theoretical model

 

Fig. 1. Schematic diagram of the dual-channel optical chaotic communication system: TL1, transmitter laser 1; TL2, transmitter laser 2; RL, receiver laser; DEC, decoder laser; BS1-BS6, beam splitters; M1-M4, mirrors; OI, optical isolators; S1-S2, message.

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The schematic diagram of the dual-channel optical chaotic communication system is shown in Fig. 1. Four single-mode diode lasers with similar internal parameters are used as transmitter laser 1 (TL1), transmitter laser 2 (TL2), receiver laser (RL), and decoder laser (DEC), respectively, where TL1 and TL2 emit at 1550nm and 1550.8nm, respectively, and have the frequency spacing of 100GHz. Both TL1 and TL2 can be driven into chaos under the appropriate feedback from the external cavity mirrors M1 and M2 [10]. The output of both TL1 and TL2 transmits simultaneously to RL through BS3 and BS4, and the output of the RL is injected into the DEC through BS6. The optical isolator (IS) ensures the light transmitting un-directionally. Practically, the lasing wavelengths of the RL and the DEC can operate simultaneously at 1550nm or 1550.8nm through adjusting their temperature and/or bias current, so the RL and DEC are able to operate at the same channel with TL1 (or TL2). The message can be introduced into transmitter lasers by directly modulating the driven currents and be recovered by comparing the input and the output of the DEC.

In such a scheme, each communication channel includes three lasers, i.e., transmitter laser (TL1 or TL2), RL, and DEC. For the transmitter laser and the DEC, the Lang-Kobayashi equations can be used to describe their dynamics subject to external feedback, which can be written as

where the subscripts T (its subscripts 1, 2 represent two different channels, respectively), D and R correspond to the transmitter laser, decoder laser and receiver laser, respectively, E is the slowly varying field amplitude, Φ is the slowly varying phase, N is the carrier number, N ₀ is the carrier number at transparency, G=g(N-N₀ )/(1+E ²/$E_s^2$) (g is the differential gain coefficient, E_s is the saturation field amplitude), α is the line-width enhancement factor, γ_P is the photon loss rate, γ_e is the total carrier loss rate, τ _L is the roundtrip time in laser cavity, τ _T is the roundtrip time in external cavity, τ_D is the transmission time between RL and DEC, k_T is the feedback coefficient of the transmitter lasers, k_D is the injection coefficient from RL to DEC, β is the spontaneous emission rate, ω is the angular frequency of the lasers, J represents the injection carrier rate, ζ is the unity intensity Langevin noise.

The dynamical behaviors of the RL are very complex because two different signals from TL1 and TL2 are simultaneously injected into RL. Troger et al. have carried out the theoretical and experimental investigations about a single-mode laser subject to external light injection form several lasers [11], where because the single-mode fiber optics setup has been used, all the injection coefficients between the transmitter lasers and the receiver laser are treated as to be equal. However, as mentioned in Ref. [11], in free-space systems, the spatial overlap between the optical mode in free space and in the receiver laser differs for each transmitter laser, so the injection coefficients between the transmitter lasers and the receiver laser are all unequal and have to be determined separately. Based on this consideration, the matrix $\left(\begin{array}{cc}{k}_{11}& k_{12}\\ k_{21}& k_{22}\end{array}\right)$ is used to express the injection coefficient, where k_ij (i,j=1,2) indicates the injection coefficient of TL_j to RL_i, and then the rate equations of RL can be expressed as

where E _R1,R2(t), Φ _R1,R2(t) and N _R1,R2(t) are the electrical amplitude, the phase and the carrier density, respectively, the subscripts R1 and R2 indicate the received system synchronized with TL1 or TL2, respectively, (Δω)_R1,R2 is the angular frequency deviation between TL1 and TL2, (Δω)R₁ =ω_R -ω₁ , (Δω)R₂ =ω_R -ω₂ , τ_inj is the transmission time from TL to RL. In this paper, in order to make the theory be suitable for the case of small channel interval, the effect of the cross injection has been taken into account though it can be ignored for the case of large channel interval. If the system is assumed to be symmetric for the two channels, matrix $\left(\begin{array}{cc}{k}_{11}& k_{12}\\ k_{21}& k_{22}\end{array}\right)$ can be written as $\left(\begin{array}{cc}{k}_{\mathit{inj}}& k_{\mathit{cro}}\\ k_{\mathit{cro}}& k_{\mathit{inj}}\end{array}\right)$, where k_inj is the injection coefficient of the same channel, and k_cro is the cross injection coefficient between two different channels, then Eq. (2) can be divided into two identical equations. When the received system synchronizes with TL1, Eq. (2) can be expressed as:

3. Transmission function

Normally, in the chaotic cryptosystem, the message recovery relies on the intensity difference between the input and output of the received system [12, 13]. For effectively decoding message, the signal attenuation should be as small as possible during transmission and be appropriately large in the received system. In this scheme, the demodulation of message is achieved by comparing the intensity difference between the input and output of DEC, so the attenuation of message in the DEC is necessary. The magnitude of attenuation can be described by the transmission function, which is similar to the small-signal response of a semiconductor laser subject to optical injection [14]. When RL is synchronized with TL1, the injection-locking state of RL is ($\overline{E_R}$, $\overline{{\Phi }_R}$, $\overline{N_R}$). Considering the intensity of modulation signal is always far smaller than that of the chaotic output, the small-signal analysis can be used. A small perturbation of TL1 output is written as δE ₁(t), then $E_1\left(t\right)=\overline{E_1}+\delta E_1\left(t\right)$, ${\theta }_{R2}\left(t\right)=\overline{{\Phi }_{R2}}+\Delta {\omega }_{R2}t+\delta {\Phi }_{R1}\left(t\right)$, $E_{R1}\left(t\right)=\overline{E_{R1}}+\delta E_{R1}\left(t\right)$, $N_{R1}\left(t\right)=\overline{N_{R1}}+\delta N_{R1}\left(t\right)$, ${\theta }_{R1}\left(t\right)=\overline{{\Phi }_{R1}}+\delta {\Phi }_{R1}\left(t\right)$. Substituting them into Eq. (3), after neglecting the gain saturation (it is reasonable because the biased current is assumed to be near the threshold current in this paper) and the langevin noise, one can obtain the following small-signal linear differential equations:

When TL1 is well synchronized with RL, the frequency detuning between the TL1 and RL should be small (usually smaller than 10 GHz) [15], and is assumed to be 0 GHz for simplicity in this paper. Because the frequency detuning between the TL2 and RL is about 100 GHz, the effect of these items including Φ_R2 can be ignored. Taking the Laplace transform to Eq. (4), the complex transmission function is

where

with

and

4. Results and discussion

4.1Chaos synchronization of lasers in the same channel

 

Fig. 2. (a). The chaotic temporal waveforms of TL1, RL, DEC.

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Fig. 2. (b). Chaotic attractors of TL1.

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Fig. 2. (c). Synchronization errors of TL1 to RL.

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Fig. 2. (d). Synchronization errors of TL1 to DEC.

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The rate equations (1)–(3) can be solved numerically with the four-order Runge-Kutta method. In the calculation, RL, DEC and TL1 are assumed to operate at the same channel, and the used data are: g=3.2×10³s^-1, α=3.0, N₀ =1.25×10⁸, E_s =2.0352×10³, γ_P =2.38×10¹¹s^-1, γ_e =6.21×10⁸s^-1, τ_L =8.5ps, τ=4ns, τ_inj =4ns, τ_D =4ns, β=5×10²s^-1, f ₁=ω ₁/2π=1.9355×10⁵GHz, f ₂=ω ₂/2π=1.9345×10⁵GHz, k₁ =k_inj =k_D =0.187, J/J_th =1.059 (where J_th is the threshold injection carrier rate, J_th =1.99×10⁸ s^-1), (Δf) _D =(Δω) _D /2π=0GHz, Δf _R1=(Δω)_R1/2π=0GHz, Δf _R2=(Δω)_R2/2π=100GHz. Figures 2(a)-2(d) give the chaotic temporal waveforms, the chaotic attractors of TL1, synchronization errors of TL1 to RL, and synchronization errors of TL1 to DEC, respectively. From these diagrams, it can be seen that both RL and DEC can be synchronized with TL1 after experiencing a transient process (about 10-20ns), which is coincided with the experimental results in Ref. [16].

4.2 Analysis of signal transmission function

 

Fig. 3. (a). Transmission functions at different injection coefficient k.

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Fig. 3. (b). Transmission functions at different frequency detuning Δf _R1.

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The transmission function strongly depends on the condition of optical injection, i.e., the injection coefficient k_inj , the frequency detuning Δf _R1 and Δf _R2. Figure 3 shows the transmission functions for different k_inj , Δf _R1 and Δf _R2. The data used in calculation are: Δf _R1=Δf _D =0GHz, Δf _R2=Δf _R1+100GHz, J/J _th =1.059, k_cro =0.1×k_inj , k ₁=k_inj =k_D =k, and the other data are the same as Fig. 2. From this diagram, it can be seen that, for different k, Δf _R1 and Δf _R2, the transmittance curves have different distributions and frequency resonant humps, so the quantity of the attenuation in the RL can be adjusted through changing k and Δf _R1. With the same method, the transmission function of the DEC can also be obtained. Because the DEC has the similar parameters as the RL, the transmission function of the DEC has the similar behaviors as Fig. 3. As mentioned above, the demodulation of message is achieved by comparing the intensity difference between the input and output of DEC, and then the attenuation of message in the DEC is necessary (i.e., the transmittance should be smaller than 0dB). Therefore, for given k and Δf _R1, the suitable range of the message frequency can be roughly estimated from this diagram.

4.3 Numerical analysis of cross talk between two channels

From Eq. (2), it can be concluded that there exists cross talk between two channels, which will decrease the quality of synchronization. The quality of synchronization can be described by the following cross-correlation function:

The bigger the C is, the higher quality of synchronization will be. If C is equal to 1, the system is synchronized completely. As shown in Fig. 4, the quality of synchronization decreases with the increase of the k_cro /k_inj value. When k_cro /k_inj varies within the regime of 0~0.3, the system is well synchronized (C≥0.99). With the further increase of k_cro /k_inj , the quality of synchronization decreases gradually.

 

Fig. 4. The cross-correlation function for different values of k_cro /k_inj .

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4.4 Message transmission and recovery

During the calculation, the biased current of the TL1 is modulated by the form of J=1.059J_th [1+m_d sin (2πf_mt)/2], where m_d and f_m are the modulation depth and the modulation frequency, respectively. Previous work [17] has shown that the external-cavity length affects the efficiency of message encoding and decoding, here f_m is taken as 250MHz, which is the first harmonic of the free spectral range of the external cavities. The other parameters are: k_inj =k₁ =k_D =0.187, (Δf) _R1 =(Δf) _D =0GHz, (Δf) _R2 =100GHz, β=5×10²s^-1, k_cro /k_inj =0.1, m_d =0.6%. The linear correlation plot of the output of TL1 and DEC is shown in Fig. 5(a), where good synchronization between TL1 and DEC can be found. As shown in Fig. 5(b), the outputs of TL2 and DEC are almost no correlation, so cross talk between two channels is very low. Figure 6(a) is the RF power spectrums of TL1, RL, DEC and the rough message. From this diagram, it can be seen that the three lasers are well synchronized, and message is well hidden in the chaotic background of the spectrum of the lasers but is clearly visible in the spectrum of rough message. These results are in agreement with Ref. [6]. After passing through a fourth-order Butterworth low-pass filter with the cutoff frequency of 1.1 f_m , the message can be well recovered, as shown in Fig. 6(b).

 

Fig. 5. (a). Correlation plot of DEC and TL1.

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Fig. 5. (b). Correlation plot of DEC and TL2.

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Fig. 6. (a). RF spectra of TL1, RL, DEC and the roughly recovered message.

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Fig. 6. (b). Recovered message after passing through a fourth-order Butterworth low-pass filter.

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5. Conclusions

In this paper, a theoretical model of dual-channel optical chaotic communication is proposed, and is confirmed to be reasonable by comparing the numerical simulation with the experimental results. Based on small-signal analysis, the transmission function of system is deduced, and the influence of different parameters on the transmission of signal is also given; The cross talk between two channels is analyzed quantitatively, and the result shows that the system is well synchronized when k_cro /k_inj is located within 0~0.3. For 250MHz modulation message, the numerical simulation shows it can be efficiently hidden during the transmission and decoded easily in the receiver. By the way, it should be pointed out that, through adding transmitter lasers and adjusting the emitting wavelength of lasers, more than two channels multiple can be established though it may be a challenge in practice.