Energy Transfer and Message Filtering in Chaos Communications Using Injection locked Laser Diodes

Transcription

1 181 Energy Transfer and Message Filtering in Chaos Communications Using Injection locked Laser Diodes Atsushi Murakami* and K. Alan Shore School of Informatics, University of Wales, Bangor, Dean Street, Bangor, Gwynedd, LL57 1UT, Wales-United Kingdom, Tel: +44 (0) ; Fax: + 44 (0) ; murakami@informatics.bangor.ac.uk Abstract- Message-signal filtering in optical chaos communications using injection-locked laser diodes (LDs) are theoretically studied In the driven response of injection-locked LDs, the driving force is supplied to drive field and carrier with energy transfer. The driving-energy transfer occurs for sinusoidal signals masked by chaos, resulting in message signal filtering in the chaos communications. Index Terms- Maximum Laser diodes, chaos, secure communications. I. INTRODUCTION Injection locking is a fundamental technique to control or synchronize local oscillators by injection of a master oscillator s signal. In particular, injection locking in laser diodes (LD) has been extensively studied for applications in optical communications [1,2]. Recently, LD injection locking has been found to have an advantage in the fascinating issue of chaos secure communications in laser systems due to its ability of generating synchronous response to injected chaotic signals, i.e., a kind of chaos synchronization in laser systems [3-7]. It has been recognized that injection-locking induced chaos synchronization is a phenomenon which is difficult to be treated within conventional complete chaos synchronization theory [8,9] in the following respects : this kind of synchronization can be observed 1) in a system condition distinct from the synchronization solutions of the systems equations and 2) for a wide range of parameter variations in the strong injection locking regime (with the injection ratio exceeding several tens of percents of the injected laser outputs) [6]. These observations are contrary to the prediction of conventional theory that requires strictly matched pair of driving and driven systems. Furthermore, the ability to separate message signals from chaotic carrier has been experimentally observed in this synchronization state. This so-called chaos pass filtering (CPF) [4,10] has been of great interest in the subject of laser chaos synchronization, and message signal filtering based on CPF including synchronization of chaos has been vigorously investigated. Recently, we have theoretically revealed several aspects of the synchronization and chaos pass filtering related to laser injection locking. Synchronization of chaos is a result of a kind of resonant response of the injection-locked LD with broadened modulation-bandwidth [11]. Phase locking plays an important role in the synchronization the synchronization of the chaotic optical-phase oscillations is highly accurate and dominates over that of the intensity oscillation [12]. Moreover, the driven laser in the CPF scheme fully differentiates sinusoidal signals and the chaotic carrier. The laser shows a frequency-dependent response to the sinusoidal part which follows the fundamental driven response of the injection-locked LDs, while synchronizing with the chaos in a frequency range below the resonant frequency [13]. We have discussed this kind of synchronization and the related phenomenon from the viewpoint of enhanced linearity due to strong injection locking. Very recently, we have analytically shown that driving signal energy is transferred between the field and carrier in the laser cavity of injection-locked LDs [14]. It can be considered that the energy transfer would play an important role for the message-signal transmission in the chaos secure

2 182 communications scheme. In this paper, we further develop the energy transfer theory to show a novel aspect of chaos-pass filtering in injection-locked LDs. II. THEORETICAL BACKGROUND As a theoretical background, we briefly review the fundamental driven response of stronglyinjection-locked slave LDs (i.e., their response to sinusoidal optical signals) [14]. The response of the laser diodes driven with optical signals from a master LD through optical injection can be analyzed by the following set of rate equations; de(t) = 1 dt 2 (1 + iα)g[n(t) N ]E(t) th +ξe inj (t)exp(iδωt) (1) dn(t) = J γ N N(t) {γ p + g[n(t) N th ]} E(t) 2 dt (2) where E is the slowly varying complex fieldamplitude, and N is the carrier number. α = 3.5 is the linewidth enhancement factor, g = ns -1 the linear gain coefficient, N th = is the threshold carrier number for the solitary operation, γ p =500ns -1 and γ N =0.5ns -1 represent the photon and carrier decay rates, and J is the pump current normalized by electron charge e. The injection effect is contained in the second term on the right-hand sides of Eq. (1). E inj (t) is the injected timevarying complex field-amplitude from the master, and Δω = ω M ω S is the frequency detuning denoting the angular-frequency difference between the injecting (master) and injected (slave) lasers. ξ =1/τ in (1 r 0 2)r inj / r 0 denotes the injection strength, in which τ in = 7 ps is the round-trip time in the laser cavity, r 0 =0.548 represents the amplitude reflectivity of the laser exit facet, and r inj represents the rate of the injected electric field amplitude. In actual analysis, the complex field is further divided its real amplitude and optical phase, E(t) = A(t)exp[iφ(t)]. All the other parameter values used in our study are listed in Ref. [12]. To analytically obtain the driven response, use is made of a small signal approximation around steady states of the variables: x(t) = x s +δx(t), ( x = A, φ, N, A inj, and φ inj ), substituting these into Eqs. (1) and (2), and linearizing them under small signal approximation (details are given in Ref. [14]). From the derived set of linearized equations for δa, δφ, and δn, the driven response for each variable can be expressed via δa /δa inj, δφ /δφ inj, or δn /δa inj, respectively. Furthermore, these are complex and so can be divided into the magnitude and phase in the form of R k = G k exp(iθ k ), (k= A, φ, and N ), where G and θ are referred to as response gain and phase shift. Figure 1 presents the fundamental driven response of the injection-locked LDs. Figs. 1(a) and (b) present the gain and phase shift, respectively. The parameter values are: pump current J =1.3J th (J th being the threshold current); strong injection ratio of r inj = 50% ; frequency detuning Δω /2π =-0.1GHz (see the injection-locking diagram in Ref. [15]). This figure explicitly shows first that a phase locking occurs for oscillating phase signal, in the range Fig. 1. Fundamental driven response of the injectionlocked LDs, response gain in (a) and phase shift in (b), analytically derived from the rate equations, Eqs. (1) and (2). The locking parameters are the injection ratio of r inj = 50% and the frequency detuning of Δω /2π = 0.1 GHz at the pump current J = 1.3J th.

3 183 below the relaxation oscillation frequency ( f R 2.4GHz for the parameters). This is simply presented by unity of the phase gain in (a) and zero phase-shift in (b), i.e., G φ =1 and θ φ = 0, respectively. This means that optical phase of the driven LD linearly responds to the driving phase signal, namely, the relation φ(t) φ inj (t) = φ s φ inj s = φ L, where φ L is the steady-state locked-phase, is established. Second, one can see clearly that transfer of the drivingoscillation energy occurs between A and N. In the lower frequency, G A is very small, and it increases in the higher frequency range. While G N has a large value in the lower frequency, comparable for that at the resonant peak. Then it decreases for higher frequency. Clearly, in the lower frequency range, the oscillation energy of the driving signal is supplied mostly to drive the carriers, resulting in suppression of field oscillation. From the result, one scenario of the LD s driven response appears, namely, the driving phase signal is used to establish phase locking state on which the amplitude signal drives both the field amplitude and carrier sharing its driving oscillation energy between them. In Ref. [13], it has been shown that, in the CPF case, the filtering property of sinusoidal signals masked by chaos corresponds to the G A curve being independent of the signal magnitude. Then, we consider here that the above scenario, i.e., the transfer of energy based on the phase locking, can be also applied to the case of CPF response. In the following, we present some numerical results on CPF response of the injection-locked LD and examined in terms of the fundamental driven response. III. NUMERICAL RESULTS As a CPF scheme, a chaotic transmitter (master) LD with optical delayed feedback from an external reflector and stand-alone receiver (slave) LD injected by the transmitter laser output are considered. Rate equations of the transmitter then are written in similar forms to Eqs. (1) and (2), just replacing the injection term of Eq. (1) with the feedback term κ fb E(t τ )exp( iω T τ ) where 2 κ fb = 1/τ in (1 r 0 )r fb / r 0 is the feedback strength with its feedback ratio r fb, ω T is the angular emission frequency, and τ is the feedback delay time. For the receiver equations, now E inj E T. A sinusoidal signal corresponding to a message in a chaos communication scheme is imposed by application of sinusoidal modulation to the transmitter s pump current. For ease in comparison with the previous result of CPF (Ref. [13]), we apply the same modulation scheme with sum of three different sinusoidal modulations to the pump current, in the form of 3 J T = J T b + m i (J T b J th ) sin(2πf i t). (3) i=1 Parameters for the chaotic transmitter are the bias pump current J T b =1.3J th, the delay time τ =1ns, and feedback ratio r fb =1.5 %. For the current modulations, different modulation frequencies are chosen, f 1,2,3 =0.2, 1.0, and 5.0GHz with the corresponding modulation indices m 1,2,3 =5, 5, and 10%, respectively. All the other parameters are the same as those in Fig. 1. First, we confirm phase locking in this case. The result is shown in Fig. 2. These are temporal waveforms of optical phase oscillations, for which φ T (t), φ R (t), and Δφ = φ R φ T represent optical-phase oscillations of the transmitter, the receiver, and their difference, respectively. φ T must include the three different sinusoidal oscillations in the chaotic fluctuation through the current modulations. Then, we find that the receiver generates highly accurate synchronization in φ R. The difference betweenφ T and φ R is 4.94rad that is found to be identical to the locked-phase from the steady state analysis, Δφ = φ L [12]. This verifies that accurate synchronization in the receiver s phase results from phase locking, and the whole oscillations of the transmitter s phase with the chaos and sinusoids are re-generated in the phase response of the receiver. This is the same as the case of no modulation, i.e., just the case of synchronized chaotic

4 184 Fig. 2. Receiver s phase locking for the external injection of chaotic oscillation and three different modulations of the transmitter LD with optical delayed feedback. oscillations reported in Ref. [12].Under phase locking, the CPF response is shown in the other variables, the field amplitude and carrier number. Figure 3 represents power spectra of the transmitter s output intensity S T ( f ), the receiver S R ( f ), and the receiver s carrier S NR ( f ), respectively. The receiver s spectra are vertically shifted for ease of comparison. Note that all the spectra are the result from taking average of 50 shots of the chaotic data in order to clarify the spectral structure (for details see Ref. [13]). The transmitter s driving signal has clear peaks corresponding to the three modulations in the broadband chaos.by comparison of S T ( f ) and S R ( f ), the receiver generates a very similar chaotic spectrum except for the higher frequency components around 10 GHz that is amplified compared to the transmitter. It is noted that the amplification corresponds to the resonance peak in the driven response of G A in Fig. 1. As discussed in ref. [11,13], this kind of synchronization related to the laser injection locking is not the so-called complete chaossynchronization, but a kind of resonant response. While for the sinusoidal signals the response differs in each modulation peak, namely, the left two peaks are attenuated and the right peak is excited. As shown in Ref. [13], these responses mostly follow the fundamental laser driven-response, G A and θ A, in Fig. 1. In a communications scheme, the attenuation essentially enables message signals to be filtered out of chaotic carriers. Fig. 3. Power spectra of the transmitter s driving signal ( S T ) and the receiver s response ( S R R and S N ) vertically shifted. It has been shown in Ref. [13] that the bandwidth for the message filtering is in a few GHz range up to the relaxation oscillation frequency, f R, while signals exceeding f R are excited. We now consider the carrier s spectrum. It is explicitly shown that S NR ( f ) has a higher value in the lower frequency range, similar to the G N curve in Fig. 1. Then, energy transfer scenario is considered for the CPF case, in the same manner as the fundamental driven response. Coupling of the field (output intensity) and carrier is evaluated from the spectra by taking S NR / S R. The result is shown in Fig. 4. Solid line is the analytical curve obtained from the small signal approximation of Eq. (2) in the following form δn δp = γ p + g(n s N th) (4) γ R + iω where δp is the small-signal output intensity converted form the field amplitude by δp = 2A s δa. It is very interesting that the coupling obtained from the spectra finely corresponds to the analytical curve. Since the curve completely reflects the fundamental response characteristics in the field and carrier as shown in Fig. 1 (Ref. [14]), the correspondence of the two results in Fig. 4 explicitly verifies that transfer of oscillation energy of the driving signal between the field amplitude (power) and carrier also occurs in the CPF response. For the modulation signals, most part of the driving signal is supplied to drive the carrier oscillation in the lower frequency range, and subsequently, the signal

5 185 Fig. 4. Coupling between the field (intensity) and the carrier oscillations. Solid and gray lines represent the analytical curve of Eq. (4) and the numerical one obtained by S R N /S R, in Fig. 3. attenuation causes in the response. While, in the higher frequency range, the driving energy is transferred from the carrier to the field, inducing signal excitation. Then, we have to consider why the response to chaotic part is maintained to be synchronous, because the energy transfer must be also applied to the whole chaotic oscillations as the same manner as the sinusoidal parts. First, it is noted that the chaotic oscillation used here is dense around the relaxation oscillation frequency f R. When one pays attention to the fact that the relaxation oscillation is inherently caused by coupling between the field and carrier, the energy transfer between them must be balanced around at f R. Consequently, the oscillation energy of the driving chaotic part is uniformly supplied to field and carrier, which may be the cause to provide the homogenous gain and the resultant synchronous response to chaos to the receiver response. While in the other frequency range except for the main oscillation around f R, the spectrum S T has a very small intensity and no clear peaks. It is considered that such trivial components do not supply adequate force to drive the field or the carrier. While, if chaotic oscillation has a moderate amount of spectral intensity even in the lower or higher frequency range, the transfer must cause. This can be confirmed in the experiment of Ref. [4], in which the driving chaos having strong peaks (corresponding to external cavity modes) in the lower frequency range is used (see Fig. 5 of Ref. [4]). There, one can see that the attenuation effect is observed even for the chaotic parts in the lower frequency range. Fig. 5. CPF response for strong chaos of the transmitter with the high feedback ratio r fb = 10 %. Power spectra and response gain are shown in (a) and (b), respectively. In (b), the response gain for the weak chaos (Fig. 3) is presented by gray line for ease in comparison. To demonstrate that, a CPF response is calculated for a strong driving-chaos having homogeneously strong oscillation components for whole frequency range. The result is presented in Fig. 5. The driving chaos is induced by higher feedback strength, in which the feedback ratio is chosen r fb =10 % with the same delay as that in Fig. 3. The power spectrum of the driving chaos and the receiver response is shown in Fig. 5(a). The response gain is shown in Fig. 5(b). Another response gain curve corresponding to Fig. 3 is also shown by the gray line for ease of comparison. In this case, the gain curve for the strong chaos is not homogeneous in contrast to the gray curve. It is found that attenuation or excitation of the chaotic parts is slightly caused in the lower or higher frequency range, reflecting the fundamental driven response in Fig. 1. The causal relationship for the energy transfer is now clarified, and this agrees well with the experimental result of Ref. [4]. IV. DISCUSSIONS We have developed a simple treatment of the synchronization of chaos and the chaos-pass filtering from the viewpoint of energy transfer in the fundamental response of the injection-locked

6 186 LD. One can say that such energy transfer can be caused even in the case of the receiver exhibiting complete synchronization because the phenomenon does not depend on the external parameters which determine the condition for the complete synchronization. However, this does not happen because, in general, complete synchronization completely re-produces the driving signal over the whole frequency range of chaos. In respect of that, it should be first noted that phase locking does not occurs in the complete case, because the phase of the receiver must be identical to that of the driving signal, i.e., φ R φ T = 2nπ( φ L ) from the above reason. Moreover, since complete synchronization is very sensitive to perturbation to the synchronization state, large modulation (single or multiple) as used here must completely destroy such synchronization. As also discussed in Ref. [13], we consider that there exists a kind of linearity enhancement behind the seemingly complicated chaos synchronization and CPF of the injection-locked LD. That clearly appears in the superposition principle of laser response to chaos and multiple modulations, in the firm phase-locking that must suppress degree of freedom of the receiver laser, or in the fact that the simple interpretation of these phenomena based on the employed small-signal-analysis as shown in Fig. 1. In Ref. [14], we have already discussed this enhanced linearity from the viewpoint of how the major nonlinearity of the injection locked LD can be suppressed. It should be also noted that the extremely strong optical injection forces the locked receiver LD to be apart from lasing operation. This point has been discussed in Ref. [15] that the strong injection can cause significant carrier-reduction, which shifts cavity resonance frequency up to a few tens of GHz. Consequently, the injection-locked LD in this strong locking regime emits the locked field with an injected frequency that is detuned from its cavity resonant frequency. The frequency detuning between them appears as the resonant peak in the driven response in Fig. 1 (about 14 GHz in this case) [14]. This indicates a possibility that the locked LDs behaves, not as a laser, but as an amplifier that just supplies optical gain to the injected light. We consider these factors probably provide the driven LD a relatively simple rule, which governs synchronization of chaos and chaos-pass filtering occurring in the strong injection locking of LDs. However, we should also mention that the injection-locked LD is not a linear system in spite of having the major nonlinearity suppressed. This is clearly understood from the fact that, if the receiver is a linear system, its responses to all the modulation signals and chaos must be entirely on the fundamental response curve, i.e., transfer function for linear systems in this case. However, in fact, an explicit difference exists in the responses to them. It is considered that the difference in the receiver response results from the residual nonlinearity that cannot be suppressed due to the strong optical injection. Moreover, we consider that much deeper insight would be obtained by taking a viewpoint of coherence of signals. This means that sinusoidal signals can be referred to as coherent, while chaotic signals which have many frequency components, i.e., broadband, and do not have distinct oscillations are incoherent. This viewpoint of signal coherence provides a new insight of driving-response efficiency. Namely, this implies that coherent signals such as sinusoidal ones can transmit their driving force effectively to the driven systems. While, it can be considered that incoherent chaotic signals having random amplitude and phase may cause energy loss in driving systems. This signal coherence is also important for the viewpoint of energy transfer that have been discussed in the present paper, and theory based on the above insight should be developed in future study. V. CONCLUSIONS In conclusion, we have presented analytical and numerical results on the driven response of the strongly-injection-locked laser diodes to reveal the physical mechanism of synchronization of chaos and chaos-pass filtering related to the injection locking. From the fundamental laser response, we propose a simple rule that phase locking occurs for the optical-phase signals and transfer of energy between the field and carrier is caused. This scenario is applied to the case of a

7 187 CPF response of the receiver LD with injection locking. Then, it is numerically demonstrated that phase locking occurs even for the chaotic oscillation superimposed by the three different modulations. The power spectra of the field (power) and carrier in the receiver response explicitly show clear evidence verifying the energy transfer followed by the fundamental laser response. It is pointed out that the energy transfer between the field and carrier is balanced at f R. Subsequently, it is remarkable in the lower and higher frequency range, causing attenuation and excitation of modulation signals superimposed on the chaotic oscillation. Synchronization of chaos itself is also discussed from the viewpoint of the balance point at f R that provides a homogeneous response gain to a weak chaotic oscillation whose main oscillation is dense at the same f R. It is shown that this synchronous state with homogeneous gain is altered for the case of strong chaotic oscillation. In this case, the response to the broad chaotic signal also exhibits slight attenuation and excitation in the lower and higher frequency ranges as similar to the case of sinusoidal driving signals, which reflects the fundamental driven response of the injection locked LD. In summary, chaos-pass filtering of the injectionlocked LD that is driven by both chaos and imposed sinusoidal signals can be understood from the viewpoint of transfer of energy of the driving amplitude (intensity) signal between the field and carrier based on the firm phase locking by the driving phase signal. ACKNOWLEDGMENT This work is funded by JSPS (Japan Society for the Promotion of Science) Postdoctoral Fellowships For Research Abroad. REFERENCES [1] R. Lang, Injection locking properties of a semiconductor laser, IEEE J. Quantum Electron., vol. 18, pp , [2] F. Mogensen, H. Olesen, and G. Jacobsen, Locking conditions and stability properties for a semiconductor lasers with external light injection, IEEE J. Quantum Electron., vol. 21, pp , [3] Y. Takiguchi, H. Fujino, and J. Ohtsubo, Experimental synchronization of chaotic oscillations in externally injected semiconductor lasers in a low-frequency fluctuation regime, Opt. Lett., vol. 24, pp , [4] I. Fischer, Y. Liu, and P. Davis, Synchronization of chaotic semiconductor laser dynamics on subnanosecond time scales and its potential for chaos communication, Phys. Rev. A, vol. 62, p (R), [5] H. Fujino and J. Ohtsubo, Experimental synchronization on chaotic oscillations in external-cavity semiconductor lasers, Opt. Lett., vol. 25, pp , [6] A. Murakami and J. Ohtsubo, Synchronization of feedback-induced chaos in semiconductor lasers by optical injection, Phys. Rev. A, vol. 65, p , [7] Y. Liu, P. Davis, Y. Takiguchi, T. Aida, S. Saito, and J. M. Liu, Injection locking and synchronization of periodic and chaotic signals in semiconductor lasers, IEEE J. Quantum Electron. vol. 39, pp , [8] L. M. Pecora and T. L. Carroll, Driving systems with chaotic signals, Phys. Rev. A, vol. 44, pp , [9] H. U. Voss, Anticipating chaotic synchronization, Phys. Rev. E, vol. 61, pp , [10] A, Uchida, Y. Liu, and P. Davis, Characteristics of chaotic masking in synchronized semiconductor lasers, IEEE J. Quantum Electron. vol. 39, pp , [11] A. Murakami, "Synchronization of chaos due to linear response in optically driven semiconductor lasers," Phys. Rev. E, vol. 65, p , [12] A. Murakami, Phase locking and chaos synchronization in injection-locked semiconductor lasers, IEEE J. Quantum Electron., vol. 39, pp , [13] A. Murakami and K. A. Shore, Chaos-pass filtering in Injection-Locked Semiconductor Lasers, Phys. Rev. A, vol.72, p , [14] A. Murakami and K. A. Shore, Analogy between optically-driven Injection-locked laser diodes and driven damped linear oscillators, Phys. Rev. A, vol. 73, p , [15] A. Murakami, K. Kawashima, and K. Atsuki, Cavity resonance shift and bandwidth enhancement in semiconductor lasers with strong light injection, IEEE J. Quantum Electron. vol. 39, pp , 2003.