Chaos synchronization in two unidirectionally coupled vertical-cavity surface-emitting lasers (VCSELs) with open-loop configuration is studied numerically. We consider two low-order transverse modes for weakly index-guided VCSELs and investigate the generalized type of chaos synchronization for both global coupling and mode-selective coupling configurations. It is found that the synchronization can be obtained between each mode of the transmitter and the receiver lasers when all modes of the transmitter are injected into the receiver equally and coupled with the corresponding modes of the receiver laser. For mode-selective cases, the modes with strong optical injection can achieve the synchronization; while the rest modes without coupling can not obtain the synchronization any more even though the injection is very strong. The results afford an opportunity to multichannel optical secure communications.
© 2006 Optical Society of America
The application of chaotic systems to private communications has been a field of special interest in recent years. Although the dynamics of chaotic systems are continually unstable, they are also bounded. This leads the variables of the system to oscillate in a noisy but deterministic way. Due to these features, it is difficult or impossible to predict the future states of the system. Previous methods for generating chaotic signals simply employ electrical circuitry to create the background noise, but it is very difficult to push the circuitry to the high frequencies needed for high-speed communications . While in semiconductor laser systems the chaotic operation can be obtained easily by introducing external perturbation, such as external optical feedback, current modulation, optoelectronic feedback, optical injection, and so on [2–9]. In addition, the dimension of the chaotic signal generated in electric system is relatively low, which greatly degrades the unpredictability of the chaotic signal and the security of the communication system. Contrarily, the dimension of the chaotic optical signal, especially that generated in delayed-feedback system, is often much higher than that of the electrically generated one. Moreover, semiconductor lasers have many other attractive advantages, such as easy to compact, high modulation, fast response, etc; therefore, the chaotic optical communications have attracted more and more considerations and already become a very hot research topic [10–29].
A common chaotic optical communication system is composed of two subsystems, i.e. the transmitter and the receiver systems. In the transmission part, one or more external perturbations are often introduced to drive the transmitter laser to operate chaotically. Among these various methods that drive a semiconductor laser to show chaotic output, the external optical feedback is most widely used for communication purpose [11–22]. By using certain chaotic encoding method, such as CM (chaos modulation), CMS (chaos masking), or CSK (chaos shift keying), the message is hidden in the chaotic carrier generated by the transmitter. Compared with the chaotic carrier, the message is often too weak to be distinguished from the chaotic signal. The chaotic carrier together with the message is injected directly into the receiver laser cavity. These two systems can synchronize when the operation conditions are appropriate. After achieving the synchronization, the message can be extracted in the receiver in virtue of the chaotic filter effects . It is obvious that the synchronization of chaotic systems plays a crucial role in chaotic communications, since almost all of the chaotic communication schemes available are based on the synchronization between the receiver laser and the transmitter laser. Recent studies show that two kinds of chaos synchronization (complete synchronization and generalized synchronization) originating from different physical backgrounds can be found with different conditions [11–13]. The synchronization conditions, the synchronization robust, and the comparison of different system configurations have been widely studied by many researchers [11–24, 26, 28, 29]. Relevant experiments have also been done to verify the theoretical predictions [21, 22, 25–27]. However, most of these previous works were for the conventional edge-emitting lasers. In contrast with the significant achievements of the edge-emitting lasers, relatively little attention has been paid for the vertical-cavity surface-emitting lasers (VCSELs) [25, 27–29]. In recent twenty years, VCSELs have achieved considerable improvements and already become the most attractive laser source in optoelectronic fields. Compared with the conventional strip lasers, VCSELs have many desirable features, including an ultra-low threshold current, single-longitudinal-mode operation, circular output beam, wafer-scale integrability, high-speed modulation capacity, etc. Due to the important role played by VCSELs in the future optical communications, it is necessary to investigate the VCSELs relating issues thoroughly, including the optical secure communications based on VCSELs. Spencer et al. have investigated the optical synchronization of chaotic external-cavity VCSELs by using a traveling wave model [28, 29]. Experimentally, Fujiwara et al have found the chaos synchronization of two mutually coupled VCSELs , and the synchronization of two unidirectionally coupled VCSELs has also been reported by Hong et al. . However, in past papers the multi-transverse-mode characteristics of VCSELs have seldom been taken into account. Due to the considerable width of the laser cavity, VCSELs often operate with multiple transverse modes. Actually, the multi-mode behaviors have already attracted much consideration in these years [30–34]. Hence, it is essential to include the transverse mode characteristics when we investigate the application of VCSELs to chaotic optical communications.
In this paper, considering the spatial dependences of the rate equations, the chaos synchronization of two unidirectionally coupled VCSELs with open-loop configuration is studied by establishing and solving the master-slave-type multi-transverse-mode rate equations. In this study, both the transmitter and the receiver VCSELs are assumed to have a weakly index-guided structure and can operate with two low-order transverse modes simultaneously, and only the generalized type of synchronization is taken into account since the other type of synchronization (complete synchronization or anticipating synchronization) is very difficult to be realized in practical system. For the generalized synchronization, the synchronization performance, the system correlation degree, and the influences of the injection strength on chaos synchronization degree are investigated for two different cases: global injection (all-to-all coupling) and mode-selective injection.
2. Theoretical model
In this study, an open-loop unidirectionally coupled VCSEL system is examined. The detailed setup has been given by previous papers [31–33]; so we do not give it here. The only difference of our configuration to the past one is the laser sources, namely, both the transmitter and the receiver are chosen to be VCSELs (M-VCSEL and S-VCSEL) operating with two low-order transverse modes. Both VCSELs are assumed to have a weakly index-guided structure with cylindrically symmetric geometry. The schematic diagram of the considered VCSELs can be found in Refs.  and . The laser active region is consisted of three quantum wells with thickness dw for each. Along the longitudinal direction (z direction), the distance between two distributed Bragg reflectors defines the length of laser cavity. The radiuses of the active region and the cladding layer are taken to be Rcore and Rclad , respectively. The refractive index is n(r) = ncore for r<Rcore and n(r) = nclad for Rcore <r<Rclad . The injection current I(r) is taken to be of a disc form, i.e. I(r) =I 0 for r<Rd and I(r)=0 otherwise, where Rd is the radius of the current injection region.
Normally, for weakly step-index-guided VCSELs, the potential optical mode profiles are determined by the build-in index guiding introduced by the transverse refractive step in the surrounding region. According to the optical waveguide theory, the linear polarized modes are proper to represent the corresponding transverse modes, where the superscripts m and n denote the corresponding azimuthal and radial orders; s and c denote the sine and cosine azimuthal modes, respectively. In this paper, to simplify the calculations, only two low-order cosine azimuthal modes (LP01 and LP11) are taken into account. However, the more transverse mode cases can still be studied in a straight forward manner.
By using the method of separating variables, the spatially dependent electric field Ei (r, φ, t) of the ith transverse mode can be written as
where i= 1 and i = 2 correspond to the LP01 and LP11 modes, respectively. Ei (t) and ψi (r, ψ) are the spatially independent average electric field and its corresponding transverse distribution function. t, r and φ are the temporal, radial, and azimuthal variables, respectively. The electric distribution is normalized in both the longitudinal and the transverse directions so that |Ei (t)| is proportional to the photon density Pi (t) of the ith mode. For the special weakly index-guided structure, the transverse mode profiles can be easily obtained by using the optical waveguide theory
where ci is the normalization coefficient satisfying following relationship
and J i-1 and K i-1 are the i-1th order Bessel functions of the first and second kinds, respectively. and -(ncladk 0)2 are the eigenvalues, which can be obtained by solving following eigenvalue equations 
here βi is the propagation constant and k 0 is the vacuum wave number.
Introducing the spatial dependences of the transverse modes into the rate equations, we obtain a set of equations including the transverse mode characteristics of VCSELs. Normally, the mathematical model of external-cavity or optical-injection semiconductor lasers is taken to be of the well-known Lang-Kobayashi form . Besides, as shown in many past papers, two sets of rate equations are required to describe respectively the dynamics of the transmitter and the receiver lasers. In a word, the master-slave rate equations of VCSELs including the spatial effects and carrier diffusion are given by
here j = √-1 , βc is the linewidth broadening factor, Γz is the longitudinal field confinement, τp is the photon lifetime, Dn is the carrier diffusion coefficient in the active region, q is the electric charge, V = 3dw is the volume of the active layer, τe is the carrier lifetime, vg is the group velocity, g0 is the gain coefficient, and N 0 is the transparent carrier density. We can see that the distribution of carriers shows strongly spatial dependences. Different transverse modes have different spatial distributions; therefore, the optical gains of them are actually provided by different local carriers and these modes can interact through the competition for the carriers [31–33]. During the lasing process, the total energy in the model is conserved, i.e. the number of the photons created is balanced with the number of the carriers lost.
The last terms on the right-hand-side of equations (6) and (7) stand for the feedback and injection effects, respectively, where k and kc are the corresponding feedback and injection parameters, τ is the feedback delays time, and τc is the propagation time of the carrier signal from the transmitter to the receiver. and denote the angular frequencies of the ith mode of transmitter and receiver lasers, respectively. ∆υ = ( - )/(2π) is the frequency detuning between the two lasers. Besides, the azimuthal distribution is also averaged by azimuthal integrating to avoid the complex three-dimensional integration. We assume that both modes have the same value of ∆υ . The optical gain of the ith transverse mode is obtained by completing following two-dimensional integration
Equations (1)–(9) give the detailed description of the unidirectionally coupled multi-transverse-mode VCSELs with open-loop configuration. In fact, the model obtained here is composed of two sets of differential equations with infinite dimension. In stead of the conventional rate equations which solely consider the longitudinal modes, our model includes the temporal, radial, and azimuthal variables. Hence, common numerical methods such as the Runge-Kutta method can not solve this model. From the mathematical viewpoint, the divisions in the considered domains, including the time and the space, are needed. In this paper, the finite difference method is used to obtain the solutions. The solutions can be very close to the accurate ones as long as the integration steps are chosen to be small enough. Since we have averaged the azimuthal distribution though azimuthal integration, hence the division in the azimuthal direction is not needed. This greatly simplifies our calculations. The time and the space integration steps used in this study are 0. 1ps and 0.04μm, respectively. To better characterize the laser behaviors, denser divisions of the considered domains are required; however, this will inevitably increase the computational efforts greatly. Therefore, there is a typical tradeoff between the model accuracy and the calculation speed. In this study, the bias current I 0 and the current injection radius Rd are taken to be 1mA and 2.1μm, respectively, so that both modes (LP01 and LP11 modes) can be excited simultaneously. The other parameters can be found in Ref. .
It is obvious that the spatial distributions of the considered two transverse modes must be obtained firstly. In fact, as shown in Eqs. (2) and (3), the transverse mode distributions for the given parameters are determined by the eigenvalues u and w , which can be obtained by solving the eigenvalue equations; therefore, the propagation constant ultimately decides the spatial distribution of a special transverse mode. However, the eigenvalue equations are some transcendent equations which can not be solved directly. In this paper, by using computer simulation, we numerically calculate the eigenvalue equations and obtain the propagation constants corresponding to these two transverse modes. The values of the obtained propagation constants and eigenvalues are listed in Table 1. In section 3, using these parameters together with the typical device parameters, we will solve the rate equations numerically and investigate the chaos synchronization relating issues under different coupling conditions detailedly.
3. Results and discussions
First, we briefly review the chaotic behaviors of M-VCSEL which is subject to strong external optical feedback. Solving the corresponding time-delay differential equations composed of two transverse mode equations and one carrier density equation, one can obtain the numerical solutions. It has been shown both experimentally and numerically that external cavity length and external feedback strength play important roles in determining the output dynamics of an external-cavity semiconductor laser. When the external cavity length is long enough as well as the feedback strength is strong enough, the laser can exhibit a rich variety of nonlinear behaviors, including bifurcation, multi-periodic oscillation, and chaotic fluctuation. Figure 1 shows the chaotic behaviors shown by the M-VCSEL, where the feedback conditions are chosen as k =8ns -1 and Lext =2cm. In this figure, (a), (b), and (c) correspond to the dynamics of the output power, the optical phase, and the carrier density, respectively. With the parameters under consideration, the free-running powers of the two modes are different, i.e. the strength of LP01 mode is relatively larger than that of LP11 mode due to its stronger coupling with the carrier profile. With strong feedback, as shown in Fig. 1(a), the power of LP01 mode is always larger than that of LP11 mode, and both modes show noise-like oscillations in the time domain. The fluctuation amplitudes are very large, which are beneficial to mask the message efficiently. Similarly, the optical phases of both modes also show chaotic oscillations, that is to say the frequency chirps as well as the laser operating frequencies will vary with the same rule. Due to the noise-like variations of the modal strengths, the consumptions of both modes to the carriers also show randomicity, which leads the carrier density to show similar noise-like fluctuation. The temporal variation of carrier density is plotted in Fig. 1(c), where the chaotic behaviors can be seen clearly. As explained previously, the chaotic output is very complex and ultra-sensitive to the system parameters and initial conditions. It is very difficult to decode the message without a proper receiver due to the high frequencies involved and the large number of dynamic degrees of freedom of the chaotic carriers. In other words, the chaos synchronization can not be achieved without proper system parameters, including the device parameters and operating parameters.
It has been indicated that two different types of chaos synchronization can be obtained in the unidirectionally coupled semiconductor laser system. One is complete synchronization, which is the result of perfect symmetry between the transmitter and the receiver. The synchronization degree of complete type is very high; however, it is very difficult to be realized in practical environment. The other type of chaos synchronization is the generalized type, which originates from the strong optical injection locking and amplification. Although the output strength of the receiver is relatively stronger than that of the transmitter, the variation rules are almost the same. Compared with complete synchronization, the synchronization degree of the second type is relatively low; however, it dose not require a severe symmetry between the two parts of the communication system and has a relatively larger tolerance to the parameter mismatch. Therefore, the generalized chaos synchronization is easy to be realized and most of the chaos synchronizations found experimentally correspond to this type. In following parts we will solely focus on this type of synchronization.
Figure 2 gives the time series of the modal powers of the transmitter and the receiver VCSELs for zero-detuning case (the frequency detuning effects are always neglected in this paper for the sake of simplicity). Moreover, the correlation curves of them are also plotted in the third column. Here, all modal chaotic outputs of the transmitter are injected into the cavity of the receiver laser, namely the so-called global injection/coupling. The three subfigures shown in the first column of Fig. 2 (from top to bottom) represent the chaotic series of the total output power, the power of LP01 mode, and the power of LP11 mode, respectively. It is obvious that all they show chaotic oscillations introduced by the strong optical feedback (k =8ns -1 and Lext =2cm). All these chaotic outputs of M-VCSEL are injected into the resonant cavity of S-VCSEL. The injection strength is chosen to be so strong (kc = 533ns -1) that the receiver laser can easily be locked by the transmitter laser. Hence, as shown in the second column of Fig. 2, the total output power and the powers of both modes show the same variation trends with those of the transmitter laser. Due to the amplification effects of the strong optical injection, the output power of S-VCSEL is relatively stronger than M-VCSEL’s. In addition, the injection delay is taken to be zero ( Lc = 0cm) in order to estimate the synchronization degree and calculate the correlation curve without shifting any of these two chaotic signals. The correlation curves of the total output, LP01 mode output, and LP11 mode output are given in the third column of Fig. 2, respectively. From these curves, we can get that the synchronization quality is very high.
Figure 3 plots the time series and power spectra for the considered transverse modes of the two VCSELs. Here, the feedback and injection delays have been changed slightly (τ=1ns and τc =3ns) to investigate the time delay between the chaotic signals of the two lasers. In Fig. 3, the black line denotes the LP01 mode and the gray line stands for the LP11 mode. Since the synchronization considered currently is the generalized type; therefore, according to the previous papers [12, 13, 20], the time delay of the two lasers’ outputs should be τc , i.e. 3ns for the parameters here, as shown in Fig. 3(a) and (c). Figures 3(b) and (d) give the power spectra corresponding to Fig. 3(a) and (c), respectively, by applying fast Fourier transformation (FFT) to the time series of modal powers. As shown in Fig. 3(b), the power spectra of both modes are broadened greatly compared with the free-running case (the free-running power spectra are not given here). There are so many frequency components in the spectrum of the chaotic signal; therefore, as introduced previously the chaotic carrier used in the communication system is very complex and it is difficult to be predicted or duplicated. Due to the strong optical injection of the chaotic signal from the transmitter, the power spectra of the modes of the slave laser can not keep its original simple forms (free-running power spectrum) any more, but show the similar broadened chaotic spectra with those of the transmitter laser. Moreover, the spectrum intensities are relatively larger than those of the injecting chaotic signals, including LP01 and LP11 modes. From these figures, we can see that with these parameters and working conditions, the output intensities and the corresponding power spectra of the two modes achieve the synchronization with high performance.
We now focus on the cross-correlation functions for the generalized synchronization. As discussed in previous papers, we can determine which type of synchronization is achieved by analyzing the correlation function. Here, for the two-transverse-mode VCSELs, we reexamine this matter. The cross-correlation function of the ith-mode is often defined as
where Pm,i and Ps,i are the power of the ith-mode of the transmitter and receiver, respectively. ∆t is the time delay between the two sets of the chaotic signals of these two lasers. The brackets denote the temporal averaging. For the calculation of the correlation function of the total modal power, the powers of the transmitter and the receiver should be the addition of the powers of the LP01 and LP11 modes. The simulation results are plotted in Fig. 4, where (a), (b), and (c) correspond to the total power, the LP01 mode, and the LP11 mode, respectively. We consider that the system is synchronized when the global maximum of the correlation function is over 0.9. At the same time, the time value corresponding to that maximum value indicates the kind of synchronization. From these three small figures in Fig. 4, we can easily find that the maximum of the correlation coefficient happens at the value of ∆t=3ns, which equals to the transmission delay of the chaotic signal emitted from the transmitter by the square. This indicates that the synchronization achieved here is the generalized type, where the slave laser responds the received chaotic signal immediately. In fact, the delay is the result of the propagation of the optical signal in the channel. It is worthy pointing out that the time value corresponding to the maximum of the correlation coefficient should be 2ns (τc -τ) if the complete type of synchronization is considered. The detailed discussions can be found in some other papers [12, 13, 16].
We next examine the synchronization performances of the mode-selective injection configurations. Figure 5 shows the simulation results for the LP01-injection case. It is more convenient here to compare these results with those of the global coupling case. As shown in Fig. 5, we can see that the synchronization performance gets worse compared with Fig. 2, since the equivalent injection strength without the coupling of LP11 mode is actually decreased for a fixed value of kc . Simultaneously, it can also be seen that the LP11 mode of the slave laser is completely suppressed by the LP01-mode-selective injection. In fact, the mode-selective injection can be used to improve the single-mode characteristics, which has attracted much attention already. Figure 6 displays the similar results to those of Fig. 5. In Fig. 6 only the LP11 mode is injected into the slave laser. Unlike the case of Fig. 5, the LP01 mode has not been suppressed thoroughly by the LP01 mode injection due to its relatively larger free-running strength. In addition, the synchronization performance is degraded further compared with LP01-injection case. From the physical viewpoint, it is ultimately the result of the further decrease of the equivalent injection strength. We have pointed out that the free-running strength of LP11 mode is weaker than that of LP01 mode; therefore, with the same injection parameter the coupling between the transmitter and the receiver is weaker for the LP11-injection case. From Fig. 5 and Fig. 6 we can also conclude that when the mode-selective injection is considered the other modes without optical injection can not achieve synchronization any more, being different from the modes with optical injection.
Finally, we investigate the effects of injection parameter kc on the system synchronization degree. Figure 7 displays the variations of cross-correlation coefficient C(∆t = τc ) with respect to kc for different coupling configurations, where (a), (b), and (c) correspond to all-to-all injection (global injection), LP01-injection, and LP11-injection, respectively. These figures clearly display how the two VCSELs of the secure communication system evolve from free-running state to the synchronization with the increase of the coupling degree. Here, to simplify the calculation of the correlation coefficient, the transmission delay is taken to be 0ns. As shown in Fig. 7(a), for weak injection cases, the correlation coefficients for both modes as well as the total laser power are very small, that is to say the synchronization degree of the system is very low. With the increase of kc , however, the correlation coefficients begin to increase. Especially, when kc is larger than 300ns -1 all these coefficients are near 1, which shows that the system achieves an accurate synchronization under strong optical injection. However, for LP01 mode injection [Fig. 7(b)], the results are different since the two modes have different coupling designs. We can see that, for strong optical injections, only LP01 modes of these two lasers can obtain synchronization, while the rest modes without optical coupling can not achieve synchronization. With the synchronization of LP01 mode, the total power is also synchronized since the LP11 mode is actually suppressed by the mode-selective injection and the total laser power is solely provided by the synchronized LP01 mode at this time. Due to the lack of optical coupling, the correlation degree of LP11 modes keeps very low. Besides, we can also find from Fig. 7(b) that values of the system correlation coefficient for strong optical injection cases are somewhat smaller than those of the global injection case (compared with the maximum shown in Fig. 4). This indicates that the synchronization degree is actually degraded by the single mode coupling. This coincides well with the observation obtained by analyzing the correlation curves shown in Fig. 2 and 5. The case of LP11 mode injection is also examined and the simulation results are given in Fig. 7(c), where similar behaviors are also observed. The differences Fig. 7(b) and Fig. 7(c) are: (1) the LP01 mode is not completely suppressed by the mode-selective injection the correlation coefficient dose not become as small as LP01-injection case; (2) the system synchronization performance is degraded further due to the further decrease of the equivalent injection level. However, this degradation can easily be compensated by increasing the injection strength. From the utilizing point of view, the multi-mode VCSELs and mode-selective injection configuration provide an opportunity for multichannel optical secure communications [35, 36].
In this paper, we have investigated the chaos synchronization of two unidirectionally coupled VCSELs with open-loop setup, where the chaotic master laser has been established by optical feedback from an external mirror. The lasers under consideration are assumed to have the weakly index-guided structure and can operate simultaneously with two low-order transverse modes. For both global and mode-selective injections, the generalized type of chaos synchronization are studied. It is found through computer simulation that the modal powers, the carrier density, and the optical phases of the master VCSEL subject to strong external optical feedback can exhibit chaotic behaviors. Injecting the chaotic signal emitted from the transmitter into the receiver and properly selecting the injection conditions, the system can be well synchronized. For global injection case where the outputs of the M-VCSEL’s two modes are injected into the cavity of S-VCSEL equally, the powers of both modes as well as the laser total output show synchronous behaviors with high quality. The correlation curves, the temporal series, and the corresponding power spectra are also examined, and the similar behaviors to those obtained in conventional edge-emitting lasers are found. For mode-selective coupling cases where only one of these two modes is injected into the slaver, the modes with injection can still achieve synchronization; while the other two modes without coupling can not obtain synchronization any more since there is no sufficient coupling between them. In addition, for the case of mode-selective injection, the synchronization performance is degraded somewhat due to the decrease of the equivalent injection strength.
Finally, for different coupling cases, we study the influence of the injection parameter on the synchronization performance. It is found that, with the increase of the injection strength, the correlation level of the modes with optical coupling can be very high; however, the correlation coefficient of the other two modes without optical coupling keeps a relatively small value even though the optical injection is very strong. It is worth pointing out that the multichannel communication basing on multi-mode semiconductor lasers (multi-longitudinal EELs or multi-transverse-mode VCSELs) has already been investigated by several authors numerically [16, 35, 36]. From experiment point of view, we can use the frequency-selective filter in the transmission line to select a certain mode for realizing the mode-selective optical injection, since different transverse modes always have different frequencies; besides, utilizing special mode-selection techniques to change the mode distribution of the transmitter laser, the mode-selective injection can also be implemented in practical systems. In the receiver part, as indicated in Ref. , some band-pass filters and low-pass filters are needed to decode the signals hidden in the corresponding chaotic outputs. In virtue of the multi-mode synchronization, the chaotic optical communication system can achieve multiplexed transmission with higher communication speed. Therefore, it is expected that the relevant results will greatly stimulate the experimental investigations.
This work was supported by the National Natural Science Foundation of China under Grant 10174057 and 90201011, the Key Project of Chinese Ministry of Education under Grant 2005-105148, and the Doctor Innovation Fund of Southwest Jiaotong University.
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