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Dynamical characteristics of an optically injected semiconductor laser
1 September 2000
Optics Communications 183 Ž2000. 195–205 www.elsevier.comrlocateroptcom
Dynamical characteristics of an optically injected semiconductor laser S.K. Hwang, J.M. Liu ) Department of Electrical Engineering, UniÕersity of California, Los Angeles, CA 90095-159410, USA Received 28 March 2000; received in revised form 21 June 2000; accepted 10 July 2000
Abstract The dependence of the dynamical characteristics of an optically injected semiconductor laser on its intrinsic and operational parameters are studied. The effect of the nonlinear gain is to suppress the instability of the system. The differential gain effect makes the system be easily driven into instability by an externally injected field. Variations in the spontaneous carrier relaxation rate are found to have little effect on the dynamics of the system. The linewidth enhancement factor is observed to strongly affect the dynamical behavior of the system. A large linewidth enhancement factor leads to rich nonlinear dynamics of the system. The influence of the injection current on the dynamics of the system is indirect through the differential gain effect and the nonlinear gain effect. The balance of the competition between these two effects determines the dynamical behavior of the system when the injection current is varied. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction Nonlinear dynamics of an optically injected semiconductor laser is a research subject of great interest in the past few years due to its profound physics and potential technological applications. It was numerically predicted w1x and experimentally demonstrated w2x that an optically injected semiconductor laser follows a period-doubling route to chaos. Recently, a number of efforts have been dedicated to understanding the mechanism of bifurcation w3–6x and route to chaos w2,7,8x of this system under different operating conditions. A map was also experimentally obtained ) Corresponding author. Fax: q1 310 206 8495; e-mail: [email protected]
w9,10x to show the regions of different dynamical states of an optically injected GaAs laser. Different semiconductor lasers generally have different intrinsic properties which lead to different dynamical characteristics for the lasers. This implies that a deep understanding of the dependence of the dynamical behavior on laser parameters in an optically injected semiconductor laser will be of great value in the effort to harness the nonlinear dynamics of the system. Few attempts, however, have been made to address this subject. The few studies w1,6x on this subject mainly focused on one parameter, namely the linewidth enhancement factor, which is not complete and sufficient for us to well understand the dynamical behavior of the system as a function of different laser parameters. The main purpose of
0030-4018r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 0 0 . 0 0 8 6 5 - 8
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this study is, therefore, to comprehensively and systematically investigate the influence of different laser parameters, such as the injection current, the differential gain, and the nonlinear gain, on the dynamical characteristics of the system. In our investigation of this subject, maps are numerically obtained and compared, which show the regions of different dynamical states of the system with different values for the parameters of interest. The advantages of doing so are that the maps not only tell us the general effect of a specific parameter on the system from a global point of view, but also show us how the regions of different dynamical states change when the value of a particular laser parameter varies. The latter will be practically helpful in choosing a laser for certain desired dynamical characteristics under optical injection. As we will see later, it is possible to choose a laser with a set of parameters that minimize the chaotic regions if stable injection locking is desired. Conversely, it is also possible to choose a laser with a different set of parameters that lead to large chaotic regions in the case when laser chaos is the subject of interest. It has been suggested w11–14x that the nonlinear gain effect can be neglected in the theoretical analysis of the optical phase because the refractive index change due to this effect is found to be negligible at the peak gain wavelength. To learn how this may affect the dynamical behavior of the system, we also study this issue in our analysis. Differences in the dynamical behavior of the system are found when the contribution from the nonlinear gain effect to the optical phase is ignored. It is observed, however, that whether or not we consider the nonlinear gain effect in our analysis of the optical phase, the general effects of the laser parameters on the system are the same. Further investigation is suggested to study whether we need to take the refractive index change due to the nonlinear gain effect into consideration in the theoretical analysis. The remainder of this paper is outlined as follows. In Section 2, we describe the coupled equation model that fully characterizes the dynamics of an optically injected semiconductor laser. The laser parameters that are involved in determining the dynamics of the system are briefly introduced and discussed. In Section 3, numerical results and discussions on the effects of different laser parameters on the dynamical
behavior of the system are presented. Finally, we conclude in Section 4.
2. Coupled equation model A single-mode model of a semiconductor laser under external optical injection is proposed w2x in a form which shows explicitly the dependence of the dynamics on specific parameters. Remarkably, this model has been demonstrated to reproduce all the experimentally observed phenomena in this system w2,9,10x. The model is cast as follows: da s dt
1 gc gn 2
gs J˜
n˜ y gp Ž 2 a q a2 .
= Ž 1 q a . q jgc cos Ž V t q f . q Fa df sy dt
b gc gn
gs J˜
2 y
jgc 1qa
d n˜ dt
Ž 1.
n˜ y gp Ž 2 a q a2 .
sin Ž V t q f . q
Ff 1qa
Ž 2.
2
s ygs n˜ y gn Ž 1 q a . n˜ y gs J˜Ž 2 a q a2 . q
gsgp gc
J˜Ž 2 a q a 2 . Ž 1 q a .
2
Ž 3.
Here, a is the normalized field amplitude of the injected laser. f is the phase difference between the injection field and the injected laser. n˜ the normalized carrier density of the injected laser. gc , gs , gn and gp are the cavity decay rate, spontaneous carrier relaxation rate, differential carrier relaxation rate, and nonlinear carrier relaxation rate, respectively w15x. b is the linewidth enhancement factor. J˜ is the injection current parameter. The injection parameter j is the strength of the injection field received by the injected laser. V is the frequency detuning of the injection field from the free-running frequency of the injected laser. Fa and Ff are the normalized Langevin noise-source parameters that are characterized by a spontaneous emission rate R sp w16x. The dynamics of a semiconductor laser depends on the five intrinsic laser parameters, gc , gs , gn , gp , and b, as well as on the three operational parame˜ j , and f s Vr2p . In addition, the spontaters, J, neous emission rate R sp also has a certain effect on
S.K. Hwang, J.M. Liu r Optics Communications 183 (2000) 195–205
the dynamics of a laser w16x. The operational parameters are what one can control during the operation of a semiconductor laser under optical injection in order to control the dynamics of the laser. The intrinsic laser parameters are determined by the material and the structure of the laser. However, gn , gp , as well as R sp , vary with the operational parameter J˜ through their dependence on the laser power w15x. Therefore, these intrinsic laser parameters are not completely independent of the operating condition of the laser. To gain more insight of how the laser parameters affect the dynamical behavior of the system and the interactions among laser parameters from the model, we can further simplify Eqs. Ž1. – Ž3.. Since the dynamics of the system is independent of the time scale at which we observe, we can normalize the state variables and the laser parameters to one of the time constants involved in the dynamics of the system. Here, we choose the cavity decay rate, gc , Žequivalently, photon lifetime, tc s 1rgc . as our reference, and normalize the variables and parameters to it. The model can then be rewritten as follows: da
1 s
dt
2
gˆn nˆ y gˆp Ž 2 a q a2 . Ž 1 q a .
ˆ q f . q fa q j cos Ž Vt df
b sy
dt
2
gˆn nˆ y mgˆp Ž 2 a q a 2 . j
y 1qa d nˆ dt
Ž 4.
ˆ qf . q sin Ž Vt
ff
Ž 5.
1qa
s ygˆs nˆ y Ž 2 a q a2 . y gˆn nˆ y gˆp Ž 2 a q a2 . Ž 1 q a .
2
Ž 6.
˜ gˆs s gsrgc , gˆn s where t s trtc , nˆ s n˜gcrgs J, ˆ gnrgc , gˆp s gprgc , V s Vrgc , f a s Fargc , and ff s Ffrgc . To infer how the dynamical behavior of the system varies with or without the nonlinear gain effect in the analysis of the optical phase, a coefficient m is introduced before the second term of the phase equation. Therefore, m s 1 indicates the inclusion of this effect in our analysis and m s 0 represents the case when this effect is absent. The dynamics of a given semiconductor laser can be controlled by the three operational parameters. The dependence of the nonlinear dynamical states on
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these parameters, however, is quite complicated. Therefore, it is useful to map the states as a function of f and j at a given value of J.˜ To map the nonlinear dynamical states of the laser as a function of the operational parameters, Eqs. Ž4. – Ž6. are solved numerically for each set of parameters using second-order Runge–Kutta method. The effect of the noise sources on the nonlinear dynamics of the system has been investigated in Ref. w16x. Since the main purpose here is to study the general effect of different laser parameters on the dynamics of the system, local variations at a given operating condition due to laser noise are not the subject of interest in this paper. Therefore, the noise terms f a and ff are ignored in our simulation. In our investigation, we use the experimentally characterized parameters of an index-guided GaAsrAlGaAs quantum-well laser w15x as the starting point. The intrinsic parameters of this laser were experimentally determined to be gc s 2.6 = 10 11 sy1 , gˆs s 6.92 = 10y3 , gˆn s 4.23 J˜= 10y3 , gˆp s 9.62 J˜= 10y3 , and b s 6 w15,17x. To indicate the regions of different dynamical states of the system in the map, the following symbols are used: S for stable injection locking, P1 for limit-cycle oscillation, P2 for period-doubling, hatched regions for period-quadrupling, and black regions for deterministic chaos. There are other instabilities, such as motions of higher periodicity, in some very small parts of the chaotic regions. As we will see later, local variations of the dynamical states due to different values of a laser parameter are observed. Since our concern in this study is the general effects of laser parameters on the dynamics of the system, the investigation on the local variations is not considered. It is a topic of our future study.
3. Analysis and discussion For a given semiconductor laser under a certain operating condition, the values of the intrinsic parameters of the laser are determined. However, different lasers generally have intrinsic parameters of different values which lead to different characteristics for the lasers. Therefore, it is important for us to first understand how the dynamical behavior of the system depends on the intrinsic parameters be-
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fore we study the effect of the operational parameters on the dynamics of the system. 3.1. Effect of intrinsic parameters Figs. 1 and 2 show the maps of the dynamical states of the optically injected semiconductor laser as a function of the frequency detuning, f, and the injection parameter, j , at two different values of the normalized nonlinear carrier relaxation parameter, gˆp . If we first compare Fig. 1Ža. with Fig. 2Ža., it is observed that as the value of gˆp is reduced, the instability regions as well as the chaotic regions enlarge significantly and dominate the entire map while the stable-locking region is largely suppressed. This can be explained as follows. The normalized nonlinear carrier relaxation parameter, gˆp , represents the effect of the nonlinear gain on the system, which is generally believed to result from the spectral hole
Fig. 2. Numerically obtained maps of dynamical states when gˆp is reduced to 1.924=10y2 but the other parameters are kept the same as those used in Fig. 1, with Ža. m s1 and Žb. m s 0.
Fig. 1. Mapping of the numerically observed dynamical states as the frequency detuning versus the optical injection level is varied. The injected laser system is operated at J˜s 4 with bs6, gˆs s 6.92=10y3 , gˆn s1.692=10y2 , and gˆp s 3.843=10y2 . Ža. m s1 and Žb. m s 0. The symbols are defined in the text.
burning w11,13,18,19x and carrier heating w20,21x. This effect is found to lead to an increased damping of the relaxation oscillation. This will result in the suppression of the instability induced by the external perturbation in an optically injected semiconductor laser. Therefore, the effect of gˆp on the dynamics of the system is to stabilize the system. The same conclusion can be drawn by comparing Fig. 1Žb. with Fig. 2Žb., which are obtained with m s 0. This implies that the general effect of gˆp on the dynamics of the system is not affected by the presence or absence of the nonlinear gain effect in the analysis of the optical phase. If we next compare Fig. 1Ža. with Fig. 1Žb., or Fig. 2Ža. with Fig. 2Žb., it is observed that the system becomes more unstable when the nonlinear gain effect is ignored in the analysis of the optical phase. This can be understood based on the result observed previously for the general effect of gˆp : the effect of the nonlinear gain is to suppress the occurrence of the instability. Therefore, the neglect of the nonlinear gain effect in the analy-
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sis of the optical phase will make the system more sensitive to the external perturbation. This will then drives the system more unstable. This can also be understood from Eq. Ž5. by letting m s 0. Now we find that without the nonlinear gain term in Eq. Ž5., a change in the carrier density will result in a large fluctuation of the variation in the optical phase. This temporally varying optical phase is equivalent to a fluctuating frequency, which can de-stabilize the system. Therefore, without the competition from the nonlinear gain effect to counteract the effect of the carrier density fluctuation on the optical phase, the differential gain effect alone will easily make the system unstable. Let us next investigate the effect of the normalized differential carrier relaxation parameter, gˆn , on the dynamics of the system. The results obtained with a reduced gˆn are shown in Fig. 3. By comparing Fig. 3Ža. with Fig. 1Ža., it is observed that when the value of gˆn is reduced while the other parameters are kept unchanged, the instability regions shrink dra-
Fig. 3. Numerically obtained maps of dynamical states when gˆn is reduced to 8.46=10y3 but the other parameters are kept the same as those used in Fig. 1, with Ža. m s1 and Žb. m s 0.
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matically. Meanwhile, the stable-locking region expands upward and becomes dominant. Clearly, the nonlinear characteristics of an optically injected semiconductor laser are a strong function of gˆn : the larger the value of gˆn is, the more unstable the system is. The reason is that the normalized differential carrier relaxation parameter, gˆn , represents the effect of the differential gain on the system. A higher differential gain results not only in a higher resonance frequency but also in a reduced damping of the relaxation oscillation w22,23x. This suggests that a semiconductor laser with a higher differential gain is much easier to be driven into instability by the perturbation of an externally injected field. The same conclusion can be obtained if we compare Fig. 3Žb. with Fig. 1Žb., which are simulated with m s 0. This suggests that the general effect of gˆn is the same with or without considering the nonlinear gain effect in the analysis of the optical phase. If we compare Fig. 3Ža. with Fig. 3Žb., it is observed that a dramatic change happens when the nonlinear gain effect is neglected in the analysis of the optical phase: the dynamical behavior of the system becomes more unstable with m s 0 compared to m s 1. This change is expected as explained in the previous discussion that the system becomes more sensitive to the external perturbation when the nonlinear gain effect is ignored. Fig. 4 shows how the behavior of the system changes when the normalized spontaneous carrier relaxation parameter, gˆs , is increased while other parameters are kept the same as those used in Fig. 1. By comparing Fig. 4Ža. to Fig. 1Ža., the dynamical characteristics of the system is found to remain almost unchanged if gˆs is increased. Only minor variations are observed in the chaotic regions and in the stable-locking region as gˆs is increased: the chaotic regions seem to shrink a little whereas the stable-locking region enlarges only slightly. In fact, when we vary the value of gˆs between 1 = 10y3 and 9 = 10y3 , no significant change in the dynamical behavior of the system is observed. This can be explained as follows. The normalized spontaneous carrier relaxation parameter, gˆs , represents the decay rate of the carrier due to spontaneous emission. Since spontaneous emission behaves as a small perturbation to the coherent optical field oscillating inside the cavity, the smaller the amount of spontaneous emis-
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and 10. This linewidth enhancement factor is generally believed to be the key element in determining the nonlinear behavior of a semiconductor laser that is subject to external perturbation w1,6,10x. Fig. 5 shows the maps of the dynamical states of an optically injected semiconductor laser when the value of b is increased from 6 to 8 while keeping other parameters the same as those used in Fig. 1. By comparing Fig. 5Ža. with Fig. 1Ža., it is observed that the nonlinear dynamics of the system is indeed a strong function of the linewidth enhancement factor: a seemingly small increase in the value of b can drive the system significantly unstable. This is expected because the parameter b couples the amplitude and phase of the laser field. As can be seen from the coupled equations of the laser system, this coupling contributes significantly to the nonlinear dynamics of the system. If we compare Fig. 5Žb. with Fig. 1Žb., which are obtained with m s 0, the same dependence of the dynamical behavior on the linewidth enhancement factor is observed. Again, the
Fig. 4. Numerically obtained maps of dynamical states when gˆs is increased to 9=10y3 but the other parameters are kept the same as those used in Fig. 1, with Ža. m s1 and Žb. m s 0.
sion is, the more stable the system is. However, the effect of the spontaneous emission perturbation on the system is so insignificant compared to the other factors that only minor variations occur when the value of gˆs is changed. This can be understood from Eqs. Ž4. – Ž6. that gˆs only appears in Eq. Ž6., which describes the behavior of the carriers. The same behavior of gˆs can also be observed if we compare Fig. 4Žb. with Fig. 1Žb., which are obtained with m s 0. This, again, tells us that the neglect of the nonlinear gain effect in the analysis of the optical phase does not affect the general effect of gˆs on the dynamics of the system. By comparing Fig. 4Ža. to Fig. 4Žb., we again observe that the system becomes more unstable when the nonlinear gain effect is neglected in the analysis of the optical phase, as expected. One of the main differences between a semiconductor laser and other laser systems is that a semiconductor laser has a non-zero linewidth enhancement factor, which usually has a value between 1
Fig. 5. Numerically obtained maps of dynamical states when b is increased to 8 but the other parameters kept the same as those used in Fig. 1, with Ža. m s1 and Žb. m s 0.
S.K. Hwang, J.M. Liu r Optics Communications 183 (2000) 195–205
neglect of the nonlinear gain effect in the analysis of the optical phase makes the system more unstable if we compare Fig. 5Ža. with Fig. 5Žb.. An interesting observation is found in Fig. 5 that the two originally separated high-order instability regions ŽP2 and above. that are surrounded by the P1 region seem to begin to merge. Indeed, if b is increased to 10, one big high-order instability region is formed from the merging of these two, which dominates over half of the map. Moreover, the chaotic regions merge as well. On the contrary, if b is decreased from 6, the two high-order instability regions as well as the chaotic regions become smaller than those shown in Fig. 1Ža. and separate more far apart from each other. When b s 0, no chaotic behavior is found. Meanwhile, for b s 0, the stablelocking region becomes symmetric around the zero frequency detuning, i.e., f s 0 GHz. The latter suggests that the linewidth enhancement factor determines the asymmetric characteristic of the stablelocking region around the zero frequency detuning observed in the system w6,24x. Similar results regarding the increase of complexity and the change in the two high-order instability regions, as well as the asymmetric characteristics of the stable-locking region, with the increase in the value of linewidth enhancement factor have also been observed in Ref. w6x. Let us now discuss the effect of the refractive index change due to the nonlinear gain on the dynamics of the system. As has been pointed out, the neglect of the nonlinear gain effect in the analysis of the optical phase does not affect the general effect of each laser parameter on the dynamical behavior of the system. However, because of the absence of the competition from this nonlinear gain effect with the differential gain effect in the phase when m s 0, the system becomes more unstable. To see whether or not it is necessary to consider the nonlinear gain effect in the analysis of the phase, let us study the laser system with the set of the parameters used in Ref. w10x. In this manner, we can compare the simulation results directly with previous experimental data w10x. The numerically and experimentally obtained maps are presented in Fig. 6 and Fig. 7, respectively. If we first study Fig. 6Ža. and Žb., it is observed that there are no significant variations between them
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Fig. 6. Mapping of the dynamical states observed numerically with Ža. m s1 and Žb. m s 0. The injected laser system is operated at J˜s 0.667 with bs 4, gˆs s6.075=10y3 , gˆn s 5.56= 10y3 , and gˆp s1.0=10y2 .
except that the upper boundary of the stable-locking region is lower when m s 0. Besides, the instability regions as well as the chaotic regions enlarge slightly when m s 0. These phenomena are apparently quite different from our previous observation that the neglect of the nonlinear gain effect in the analysis of the phase drives the system dramatically unstable. This can be understood as follows. The values of gˆn and gˆp used in our previous discussion are several times larger than those used in Fig. 6. Besides, the value of b is 6 instead of 4 in the previous analysis. According to our previous discussion, the dynamical behavior of the system is found to be a strong function of these three parameters. Therefore, larger values of these parameters will make the change significant in our previous observation if we neglect the effect of the nonlinear gain in the analysis of the optical phase. This suggests that, depending on the values of these parameters, the dynamical behavior of the system may or may not be significantly differ-
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Fig. 7. Experimentally obtained mapping with the same operating conditions and laser parameters used in Fig. 6. The map is originally published in Ref. w10x and is reproduced here for the comparison with the numerical results shown in Fig. 6.
ent if the nonlinear gain effect is ignored in the analysis of the optical phase. To further address this issue, let us compare our simulation results with the experimental observation. By comparing Fig. 6 to Fig. 7, we find that the behavior of the system in Fig. 6Žb. seems to follow closer to that in Fig. 7. That is, the upper boundary of the stable-locking region in Fig. 6Žb. is closer to that in Fig. 7. However, significant differences are still observed between Fig. 6Žb. and Fig. 7: the instability regions and the chaotic regions in Fig. 6Žb. do not match exactly with those seen in Fig. 7. These observations together with the results discussed in the last paragraph suggest that, based on these data alone, we still cannot conclusively determine whether or not we need to consider the refractive index change due to the nonlinear gain effect in the theoretical analysis. In order to conclusively clarify this issue, more experimental data are needed. This suggests a further investigation on this issue.
already considered in all of the maps presented and discussed throughout this paper. The only operational parameter left to be considered is the normalized injection current level J.˜ Fig. 8 shows the map of the dynamical states of the optically injected semiconductor laser at a higher injection current level than that used in Fig. 1. By comparing Fig. 8
3.2. Effect of operational parameters Among the three operational parameters, the detuning frequency f and the injection parameter j are
Fig. 8. Numerically obtained maps of dynamical states when J˜s6 with gˆn s 2.538=10y2 , gˆp s 5.772=10y2 , and m s1. The values of b and gˆs are the same as those used in Fig. 1.
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with Fig. 1Ža., we find that the stable-locking region for J˜s 6 is larger than that for J˜s 4. Moreover, the upper stable-locking boundary for J˜s 6 shifts upward. Besides, it is also observed that as J˜ is increased from 4 to 6, the instability regions move up to the side corresponding to a positive value of the detuning frequency and shift to the right side corresponding to a larger value of the injection parameter. One intuitive explanation to the above observation is that when a laser is operated at a higher injection current level, the coherent optical power stored in the cavity is higher, thus allowing the laser to be more resistant to the perturbation of the externally injected optical field. This is why stronger externally optical perturbation is required to observe instabilities and chaos in the system at a higher injection current level as that shown in Fig. 8. However, this cannot explain why the instability regions as well as the chaotic regions expand as J˜ is increased from 4 to 6 because they are expected to shrink based on the above argument. In fact, as the injection current level J˜ is increased from 1 to 10, we observe that the instability regions, including the chaotic regions, enlarge. In addition, they move up to the side corresponding to a positive value of the detuning frequency and shift to the right side corresponding to a larger value of the injection parameter. More interestingly, it is found that the upper boundary of the stable-locking region initially keeps moving downward as the value of J˜ is increased from 1, gradually stops at J˜s 4, and starts to move upward if J˜ is further increased. These observations apparently cannot be explained by the above argument either, though this argument does predict the shift of the instability regions. Apparently, the single argument can only explain part of the observed phenomena. Some inherent elements of the system must be involved in the mechanism of determining the dynamics of the optically injected semiconductor laser when the injection current is varied. By revisiting Section 2, we find that the operational parameter J˜ actually does not appear in the normalized Eqs. Ž4. – Ž6.. The effect of this operational parameter J˜ on the dynamical characteristics of the system is implicit. It is through the dependence of the laser dynamics on the intrinsic parameters, gˆn and gˆp , that J˜ affects the behavior of the
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system. Based on the discussion on gˆn and gˆp , we know that their effects on the system are quite opposite to each other: gˆn tries to de-stabilize the dynamics of the system whereas gˆp tends to stabilize the dynamics. Therefore, varying the operational parameter J˜ for a given laser leads to the competition between gˆn and gˆp because both gˆn and gˆp vary linearly with J.˜ Depending on the relative strength of one to the other, the dynamical characteristics of the system may vary differently. If we compare a series of maps of the dynamical states when gˆn is increased, the instability regions expand toward the right side corresponding to a larger value of the injection parameter. On the other hand, if we study the maps of the dynamical states when gˆp is increased, the instability regions not only shrink but are also pushed up by the stable-locking region toward the side corresponding to a positive value of the frequency detuning as the stable-locking region expands. These observations imply that when J˜ is increased and thus the values of gˆn and gˆp are simultaneously increased with it, the instability regions would move up toward the side corresponding to a positive value of the detuning frequency and shift to the right side corresponding to a larger value of the injection parameter. Therefore, we believe that the mechanism in determining the dynamical behavior of the optically injected semiconductor laser when varying J˜ is the balance of the competition between gˆn and gˆp . This can be understood from Eqs. Ž4. – Ž6. by noting that the two effects on the behavior of the system are opposite in sign.
4. Conclusion The effects of the intrinsic and operational laser parameters on the dynamical characteristics of an optically injected semiconductor laser are comprehensively investigated. A number of maps showing the regions of different dynamical states of the system are numerically obtained and compared when the parameter of interest is varied while the other parameters are kept fixed to see the effect of each parameter on the dynamics of the system. The effect of the nonlinear gain, which is represented by gˆp , is to suppress the instability of a semiconductor laser that is subject to an external
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optical perturbation. Hence, gˆp tends to stabilize the system. On the contrary, the effect of the differential gain, which is represented by gˆn , is to make the laser system be easily driven into instability by the perturbation of an external optical injection. Therefore, the larger the value of gˆn is, the more unstable the system is. The effect of the normalized spontaneous carrier relaxation, gˆs , on the dynamical characteristics of the system is observed to be insignificant. For the linewidth enhancement factor b, it is observed that an increase in the value of b tends to de-stabilize the system when the laser is subject to external optical injection. It is also found that the linewidth enhancement factor determines the asymmetric characteristic of the stable-locking region around the zero frequency detuning. The changes of the dynamical characteristics in the system when the nonlinear gain effect is ignored in the analysis of the phase of the optical field is also studied. It is found that the general behavior of each laser parameter is not affected by such a neglect. However, when gˆp is ignored in the phase analysis, the competitive balance between gˆn and gˆp is reduced to the advantage of gˆn , resulting in increased instability for the system. A comparison between numerical and experimental results is also made. Further investigation is suggested to study this issue. ˜ it is found that For the operational parameter, J, the effect of the injection current enters the dynamics indirectly through two intrinsic parameters, the differential gain effect and the nonlinear gain effect. As J˜ is varied, both effects increase and compete with each other. The resulting balance between the competition of these two effects determines the dynamical behavior of the system. A tendency of increase in sizes of the instability regions as well as the chaotic regions is still clearly observed as J˜ is increased. These regions are also found to move to the side corresponding to a positive value of the frequency detuning and to a larger value of the injection parameter as J˜ is increased. Finally, we like to compare the simulation method used in this study to the bifurcation analysis used in Ref. w6x. The analysis and discussions throughout this paper are based on the numerical results using the simulation method introduced in Section 2. The advantage of the numerical simulation is that it reproduces all the observed phenomena in an optically
injected semiconductor laser system w2,9,10x, thus providing detailed information of the dynamics of the system. For example, the characteristics of each dynamical state, the noise effect on the dynamics w16,25x, and the transitions from one state to another can be studied in our simulation. The disadvantage of the simulation method is that systematic studies on the dynamics are not easy and are time-consuming. To avoid this problem, Wieczorek and the coworkers w6x applied the fundamental bifurcation theory to the study of the dynamics of the system, which is capable of easily producing bifurcation lines Žborders of regions of certain dynamics. which give insight into how different dynamics are related to one another to allow simple systematic studies. Their method, therefore, leads to a quick and complete picture of the dynamics of the system from a global viewpoint though detailed characteristics of the dynamics are not obtained using that method. Let us take the existence of bistability or multistability in the presence of coexisting attractors as an example. Using the bifurcation theory, it is easy to analyze whether there are coexisting attractors under the same operating condition in the system to search for the existence of bistability or multistability. However, the characteristics of the coexisting attractors cannot be obtained using the bifurcation analysis. With the simulation method, the existence of the bistability or multistability is found by numerically simulating the system with different initial conditions under the same operating condition. Compared to the bifurcation method, this is apparently not the best way to search for coexisting attractors because it is time-consuming. However, the simulation method can provide us with detailed information about the characteristics of the coexisting attractors including their basins, and their susceptibility to noise and other external perturbations. We have actually applied our simulation tools to the study w26x of the detailed characteristics of the bistability found in the locking region of the system, which cannot be obtained by a simple bifurcation analysis.
Acknowledgements The authors wish to acknowledge T.B. Simpson for helpful discussions. This work is supported by
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the US Army Research Office under the contract No. DAAG55-98-1-0269.
References w1x J. Sacher, D. Baums, P. Panknin, W. Elsasser, E.O. Gobel, Phys. Rev. A 45 Ž1992. 1893. w2x T.B. Simpson, J.M. Liu, A. Gavrielides, V. Kovanis, P.M. Alsing, Appl. Phys. Lett. 64 Ž1994. 3539; Phys. Rev. A 51 Ž1995. 4181. w3x T. Erneux, V. Kovanis, A. Gavrielides, P.M. Alsing, Phys. Rev. A 53 Ž1996. 53. w4x B. Krauskopf, W.A. van der Graaf, D. Lenstra, Quantum Semiclass. Opt. 9 Ž1997. 797. w5x P.C. De Jagher, W.A. van der Graaf, D. Lenstra, Quantum Semiclass. Opt. 8 Ž1996. 805. w6x S. Wieczorek, B. Krauskopf, D. Lenstra, Opt. Commun. 172 Ž1999. 279. w7x V. Annovazzi-Lodi, S. Donati, M. Manna, IEEE J. Quantum Electron. 30 Ž1994. 1537. w8x E.K. Lee, H.S. Pang, J.D. Park, H. Lee, Phys. Rev. A 47 Ž1993. 736. w9x V. Kovanis, A. Gavrielides, T.B. Simpson, J.M. Liu, Appl. Phys. Lett. 67 Ž1995. 2780.
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w10x T.B. Simpson, J.M. Liu, K.F. Huang, K. Tai, Quantum Semiclass. Opt. 9 Ž1997. 765. w11x R.F. Kazarinov, C.H. Henry, R.A. Logan, J. Appl. Phys. 53 Ž1982. 4631. w12x C.H. Henry, J. Lightwave Tech. 4 Ž1986. 298. w13x G.P. Agrawal, IEEE J. Quantum Electron. 26 Ž1990. 1901. w14x Y. Yao, G.H. Duan, P. Gallion, IEEE Photon. Technol. Lett. 4 Ž1992. 1103. w15x J.M. Liu, T.B. Simpson, IEEE J. Quantum Electron. 30 Ž1994. 957; IEEE Photon. Technol. Lett. 4 Ž1993. 380. w16x J.B. Gao, S.K. Hwang, J.M. Liu, Phys. Rev. A 59 Ž1999. 1582. w17x T.B. Simpson, private communication. w18x M. Yamada, Y. Suematsu, J. Appl. Phys. 52 Ž1981. 2653. w19x G.P. Agrawal, J. Appl. Phys. 63 Ž1988. 1232. w20x M. Grupen, K. Hess, Appl. Phys. Lett. 70 Ž1997. 808. w21x M.P. Kesler, E.P. Ippen, Appl. Phys. Lett. 51 Ž1987. 1765. w22x M.C. Tatham, I.F. Lealman, C.P. Seltzer, L.D. Westbrook, D.M. Cooper, IEEE J. Quantum Electron. 28 Ž1992. 408. w23x Y. Arakawa, A. Yariv, IEEE J. Quantum Electron. 21 Ž1985. 1666; IEEE J. Quantum Electron. 22 Ž1986. 1887. w24x J. Petitbon, P. Gallion, G. Debarge, C. Chabran, IEEE J. Quantum Electron. 24 Ž1988. 148. w25x S.K. Hwang, J.B. Gao, J.M. Liu, Phys. Rev. E 61 Ž2000. 5162. w26x S.K. Hwang, J.M. Liu, Opt. Commun. 169 Ž1999. 167.
Optics Communications 183 Ž2000. 195–205 www.elsevier.comrlocateroptcom
Dynamical characteristics of an optically injected semiconductor laser S.K. Hwang, J.M. Liu ) Department of Electrical Engineering, UniÕersity of California, Los Angeles, CA 90095-159410, USA Received 28 March 2000; received in revised form 21 June 2000; accepted 10 July 2000
Abstract The dependence of the dynamical characteristics of an optically injected semiconductor laser on its intrinsic and operational parameters are studied. The effect of the nonlinear gain is to suppress the instability of the system. The differential gain effect makes the system be easily driven into instability by an externally injected field. Variations in the spontaneous carrier relaxation rate are found to have little effect on the dynamics of the system. The linewidth enhancement factor is observed to strongly affect the dynamical behavior of the system. A large linewidth enhancement factor leads to rich nonlinear dynamics of the system. The influence of the injection current on the dynamics of the system is indirect through the differential gain effect and the nonlinear gain effect. The balance of the competition between these two effects determines the dynamical behavior of the system when the injection current is varied. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction Nonlinear dynamics of an optically injected semiconductor laser is a research subject of great interest in the past few years due to its profound physics and potential technological applications. It was numerically predicted w1x and experimentally demonstrated w2x that an optically injected semiconductor laser follows a period-doubling route to chaos. Recently, a number of efforts have been dedicated to understanding the mechanism of bifurcation w3–6x and route to chaos w2,7,8x of this system under different operating conditions. A map was also experimentally obtained ) Corresponding author. Fax: q1 310 206 8495; e-mail: [email protected]
w9,10x to show the regions of different dynamical states of an optically injected GaAs laser. Different semiconductor lasers generally have different intrinsic properties which lead to different dynamical characteristics for the lasers. This implies that a deep understanding of the dependence of the dynamical behavior on laser parameters in an optically injected semiconductor laser will be of great value in the effort to harness the nonlinear dynamics of the system. Few attempts, however, have been made to address this subject. The few studies w1,6x on this subject mainly focused on one parameter, namely the linewidth enhancement factor, which is not complete and sufficient for us to well understand the dynamical behavior of the system as a function of different laser parameters. The main purpose of
0030-4018r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 0 0 . 0 0 8 6 5 - 8
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this study is, therefore, to comprehensively and systematically investigate the influence of different laser parameters, such as the injection current, the differential gain, and the nonlinear gain, on the dynamical characteristics of the system. In our investigation of this subject, maps are numerically obtained and compared, which show the regions of different dynamical states of the system with different values for the parameters of interest. The advantages of doing so are that the maps not only tell us the general effect of a specific parameter on the system from a global point of view, but also show us how the regions of different dynamical states change when the value of a particular laser parameter varies. The latter will be practically helpful in choosing a laser for certain desired dynamical characteristics under optical injection. As we will see later, it is possible to choose a laser with a set of parameters that minimize the chaotic regions if stable injection locking is desired. Conversely, it is also possible to choose a laser with a different set of parameters that lead to large chaotic regions in the case when laser chaos is the subject of interest. It has been suggested w11–14x that the nonlinear gain effect can be neglected in the theoretical analysis of the optical phase because the refractive index change due to this effect is found to be negligible at the peak gain wavelength. To learn how this may affect the dynamical behavior of the system, we also study this issue in our analysis. Differences in the dynamical behavior of the system are found when the contribution from the nonlinear gain effect to the optical phase is ignored. It is observed, however, that whether or not we consider the nonlinear gain effect in our analysis of the optical phase, the general effects of the laser parameters on the system are the same. Further investigation is suggested to study whether we need to take the refractive index change due to the nonlinear gain effect into consideration in the theoretical analysis. The remainder of this paper is outlined as follows. In Section 2, we describe the coupled equation model that fully characterizes the dynamics of an optically injected semiconductor laser. The laser parameters that are involved in determining the dynamics of the system are briefly introduced and discussed. In Section 3, numerical results and discussions on the effects of different laser parameters on the dynamical
behavior of the system are presented. Finally, we conclude in Section 4.
2. Coupled equation model A single-mode model of a semiconductor laser under external optical injection is proposed w2x in a form which shows explicitly the dependence of the dynamics on specific parameters. Remarkably, this model has been demonstrated to reproduce all the experimentally observed phenomena in this system w2,9,10x. The model is cast as follows: da s dt
1 gc gn 2
gs J˜
n˜ y gp Ž 2 a q a2 .
= Ž 1 q a . q jgc cos Ž V t q f . q Fa df sy dt
b gc gn
gs J˜
2 y
jgc 1qa
d n˜ dt
Ž 1.
n˜ y gp Ž 2 a q a2 .
sin Ž V t q f . q
Ff 1qa
Ž 2.
2
s ygs n˜ y gn Ž 1 q a . n˜ y gs J˜Ž 2 a q a2 . q
gsgp gc
J˜Ž 2 a q a 2 . Ž 1 q a .
2
Ž 3.
Here, a is the normalized field amplitude of the injected laser. f is the phase difference between the injection field and the injected laser. n˜ the normalized carrier density of the injected laser. gc , gs , gn and gp are the cavity decay rate, spontaneous carrier relaxation rate, differential carrier relaxation rate, and nonlinear carrier relaxation rate, respectively w15x. b is the linewidth enhancement factor. J˜ is the injection current parameter. The injection parameter j is the strength of the injection field received by the injected laser. V is the frequency detuning of the injection field from the free-running frequency of the injected laser. Fa and Ff are the normalized Langevin noise-source parameters that are characterized by a spontaneous emission rate R sp w16x. The dynamics of a semiconductor laser depends on the five intrinsic laser parameters, gc , gs , gn , gp , and b, as well as on the three operational parame˜ j , and f s Vr2p . In addition, the spontaters, J, neous emission rate R sp also has a certain effect on
S.K. Hwang, J.M. Liu r Optics Communications 183 (2000) 195–205
the dynamics of a laser w16x. The operational parameters are what one can control during the operation of a semiconductor laser under optical injection in order to control the dynamics of the laser. The intrinsic laser parameters are determined by the material and the structure of the laser. However, gn , gp , as well as R sp , vary with the operational parameter J˜ through their dependence on the laser power w15x. Therefore, these intrinsic laser parameters are not completely independent of the operating condition of the laser. To gain more insight of how the laser parameters affect the dynamical behavior of the system and the interactions among laser parameters from the model, we can further simplify Eqs. Ž1. – Ž3.. Since the dynamics of the system is independent of the time scale at which we observe, we can normalize the state variables and the laser parameters to one of the time constants involved in the dynamics of the system. Here, we choose the cavity decay rate, gc , Žequivalently, photon lifetime, tc s 1rgc . as our reference, and normalize the variables and parameters to it. The model can then be rewritten as follows: da
1 s
dt
2
gˆn nˆ y gˆp Ž 2 a q a2 . Ž 1 q a .
ˆ q f . q fa q j cos Ž Vt df
b sy
dt
2
gˆn nˆ y mgˆp Ž 2 a q a 2 . j
y 1qa d nˆ dt
Ž 4.
ˆ qf . q sin Ž Vt
ff
Ž 5.
1qa
s ygˆs nˆ y Ž 2 a q a2 . y gˆn nˆ y gˆp Ž 2 a q a2 . Ž 1 q a .
2
Ž 6.
˜ gˆs s gsrgc , gˆn s where t s trtc , nˆ s n˜gcrgs J, ˆ gnrgc , gˆp s gprgc , V s Vrgc , f a s Fargc , and ff s Ffrgc . To infer how the dynamical behavior of the system varies with or without the nonlinear gain effect in the analysis of the optical phase, a coefficient m is introduced before the second term of the phase equation. Therefore, m s 1 indicates the inclusion of this effect in our analysis and m s 0 represents the case when this effect is absent. The dynamics of a given semiconductor laser can be controlled by the three operational parameters. The dependence of the nonlinear dynamical states on
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these parameters, however, is quite complicated. Therefore, it is useful to map the states as a function of f and j at a given value of J.˜ To map the nonlinear dynamical states of the laser as a function of the operational parameters, Eqs. Ž4. – Ž6. are solved numerically for each set of parameters using second-order Runge–Kutta method. The effect of the noise sources on the nonlinear dynamics of the system has been investigated in Ref. w16x. Since the main purpose here is to study the general effect of different laser parameters on the dynamics of the system, local variations at a given operating condition due to laser noise are not the subject of interest in this paper. Therefore, the noise terms f a and ff are ignored in our simulation. In our investigation, we use the experimentally characterized parameters of an index-guided GaAsrAlGaAs quantum-well laser w15x as the starting point. The intrinsic parameters of this laser were experimentally determined to be gc s 2.6 = 10 11 sy1 , gˆs s 6.92 = 10y3 , gˆn s 4.23 J˜= 10y3 , gˆp s 9.62 J˜= 10y3 , and b s 6 w15,17x. To indicate the regions of different dynamical states of the system in the map, the following symbols are used: S for stable injection locking, P1 for limit-cycle oscillation, P2 for period-doubling, hatched regions for period-quadrupling, and black regions for deterministic chaos. There are other instabilities, such as motions of higher periodicity, in some very small parts of the chaotic regions. As we will see later, local variations of the dynamical states due to different values of a laser parameter are observed. Since our concern in this study is the general effects of laser parameters on the dynamics of the system, the investigation on the local variations is not considered. It is a topic of our future study.
3. Analysis and discussion For a given semiconductor laser under a certain operating condition, the values of the intrinsic parameters of the laser are determined. However, different lasers generally have intrinsic parameters of different values which lead to different characteristics for the lasers. Therefore, it is important for us to first understand how the dynamical behavior of the system depends on the intrinsic parameters be-
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fore we study the effect of the operational parameters on the dynamics of the system. 3.1. Effect of intrinsic parameters Figs. 1 and 2 show the maps of the dynamical states of the optically injected semiconductor laser as a function of the frequency detuning, f, and the injection parameter, j , at two different values of the normalized nonlinear carrier relaxation parameter, gˆp . If we first compare Fig. 1Ža. with Fig. 2Ža., it is observed that as the value of gˆp is reduced, the instability regions as well as the chaotic regions enlarge significantly and dominate the entire map while the stable-locking region is largely suppressed. This can be explained as follows. The normalized nonlinear carrier relaxation parameter, gˆp , represents the effect of the nonlinear gain on the system, which is generally believed to result from the spectral hole
Fig. 2. Numerically obtained maps of dynamical states when gˆp is reduced to 1.924=10y2 but the other parameters are kept the same as those used in Fig. 1, with Ža. m s1 and Žb. m s 0.
Fig. 1. Mapping of the numerically observed dynamical states as the frequency detuning versus the optical injection level is varied. The injected laser system is operated at J˜s 4 with bs6, gˆs s 6.92=10y3 , gˆn s1.692=10y2 , and gˆp s 3.843=10y2 . Ža. m s1 and Žb. m s 0. The symbols are defined in the text.
burning w11,13,18,19x and carrier heating w20,21x. This effect is found to lead to an increased damping of the relaxation oscillation. This will result in the suppression of the instability induced by the external perturbation in an optically injected semiconductor laser. Therefore, the effect of gˆp on the dynamics of the system is to stabilize the system. The same conclusion can be drawn by comparing Fig. 1Žb. with Fig. 2Žb., which are obtained with m s 0. This implies that the general effect of gˆp on the dynamics of the system is not affected by the presence or absence of the nonlinear gain effect in the analysis of the optical phase. If we next compare Fig. 1Ža. with Fig. 1Žb., or Fig. 2Ža. with Fig. 2Žb., it is observed that the system becomes more unstable when the nonlinear gain effect is ignored in the analysis of the optical phase. This can be understood based on the result observed previously for the general effect of gˆp : the effect of the nonlinear gain is to suppress the occurrence of the instability. Therefore, the neglect of the nonlinear gain effect in the analy-
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sis of the optical phase will make the system more sensitive to the external perturbation. This will then drives the system more unstable. This can also be understood from Eq. Ž5. by letting m s 0. Now we find that without the nonlinear gain term in Eq. Ž5., a change in the carrier density will result in a large fluctuation of the variation in the optical phase. This temporally varying optical phase is equivalent to a fluctuating frequency, which can de-stabilize the system. Therefore, without the competition from the nonlinear gain effect to counteract the effect of the carrier density fluctuation on the optical phase, the differential gain effect alone will easily make the system unstable. Let us next investigate the effect of the normalized differential carrier relaxation parameter, gˆn , on the dynamics of the system. The results obtained with a reduced gˆn are shown in Fig. 3. By comparing Fig. 3Ža. with Fig. 1Ža., it is observed that when the value of gˆn is reduced while the other parameters are kept unchanged, the instability regions shrink dra-
Fig. 3. Numerically obtained maps of dynamical states when gˆn is reduced to 8.46=10y3 but the other parameters are kept the same as those used in Fig. 1, with Ža. m s1 and Žb. m s 0.
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matically. Meanwhile, the stable-locking region expands upward and becomes dominant. Clearly, the nonlinear characteristics of an optically injected semiconductor laser are a strong function of gˆn : the larger the value of gˆn is, the more unstable the system is. The reason is that the normalized differential carrier relaxation parameter, gˆn , represents the effect of the differential gain on the system. A higher differential gain results not only in a higher resonance frequency but also in a reduced damping of the relaxation oscillation w22,23x. This suggests that a semiconductor laser with a higher differential gain is much easier to be driven into instability by the perturbation of an externally injected field. The same conclusion can be obtained if we compare Fig. 3Žb. with Fig. 1Žb., which are simulated with m s 0. This suggests that the general effect of gˆn is the same with or without considering the nonlinear gain effect in the analysis of the optical phase. If we compare Fig. 3Ža. with Fig. 3Žb., it is observed that a dramatic change happens when the nonlinear gain effect is neglected in the analysis of the optical phase: the dynamical behavior of the system becomes more unstable with m s 0 compared to m s 1. This change is expected as explained in the previous discussion that the system becomes more sensitive to the external perturbation when the nonlinear gain effect is ignored. Fig. 4 shows how the behavior of the system changes when the normalized spontaneous carrier relaxation parameter, gˆs , is increased while other parameters are kept the same as those used in Fig. 1. By comparing Fig. 4Ža. to Fig. 1Ža., the dynamical characteristics of the system is found to remain almost unchanged if gˆs is increased. Only minor variations are observed in the chaotic regions and in the stable-locking region as gˆs is increased: the chaotic regions seem to shrink a little whereas the stable-locking region enlarges only slightly. In fact, when we vary the value of gˆs between 1 = 10y3 and 9 = 10y3 , no significant change in the dynamical behavior of the system is observed. This can be explained as follows. The normalized spontaneous carrier relaxation parameter, gˆs , represents the decay rate of the carrier due to spontaneous emission. Since spontaneous emission behaves as a small perturbation to the coherent optical field oscillating inside the cavity, the smaller the amount of spontaneous emis-
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and 10. This linewidth enhancement factor is generally believed to be the key element in determining the nonlinear behavior of a semiconductor laser that is subject to external perturbation w1,6,10x. Fig. 5 shows the maps of the dynamical states of an optically injected semiconductor laser when the value of b is increased from 6 to 8 while keeping other parameters the same as those used in Fig. 1. By comparing Fig. 5Ža. with Fig. 1Ža., it is observed that the nonlinear dynamics of the system is indeed a strong function of the linewidth enhancement factor: a seemingly small increase in the value of b can drive the system significantly unstable. This is expected because the parameter b couples the amplitude and phase of the laser field. As can be seen from the coupled equations of the laser system, this coupling contributes significantly to the nonlinear dynamics of the system. If we compare Fig. 5Žb. with Fig. 1Žb., which are obtained with m s 0, the same dependence of the dynamical behavior on the linewidth enhancement factor is observed. Again, the
Fig. 4. Numerically obtained maps of dynamical states when gˆs is increased to 9=10y3 but the other parameters are kept the same as those used in Fig. 1, with Ža. m s1 and Žb. m s 0.
sion is, the more stable the system is. However, the effect of the spontaneous emission perturbation on the system is so insignificant compared to the other factors that only minor variations occur when the value of gˆs is changed. This can be understood from Eqs. Ž4. – Ž6. that gˆs only appears in Eq. Ž6., which describes the behavior of the carriers. The same behavior of gˆs can also be observed if we compare Fig. 4Žb. with Fig. 1Žb., which are obtained with m s 0. This, again, tells us that the neglect of the nonlinear gain effect in the analysis of the optical phase does not affect the general effect of gˆs on the dynamics of the system. By comparing Fig. 4Ža. to Fig. 4Žb., we again observe that the system becomes more unstable when the nonlinear gain effect is neglected in the analysis of the optical phase, as expected. One of the main differences between a semiconductor laser and other laser systems is that a semiconductor laser has a non-zero linewidth enhancement factor, which usually has a value between 1
Fig. 5. Numerically obtained maps of dynamical states when b is increased to 8 but the other parameters kept the same as those used in Fig. 1, with Ža. m s1 and Žb. m s 0.
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neglect of the nonlinear gain effect in the analysis of the optical phase makes the system more unstable if we compare Fig. 5Ža. with Fig. 5Žb.. An interesting observation is found in Fig. 5 that the two originally separated high-order instability regions ŽP2 and above. that are surrounded by the P1 region seem to begin to merge. Indeed, if b is increased to 10, one big high-order instability region is formed from the merging of these two, which dominates over half of the map. Moreover, the chaotic regions merge as well. On the contrary, if b is decreased from 6, the two high-order instability regions as well as the chaotic regions become smaller than those shown in Fig. 1Ža. and separate more far apart from each other. When b s 0, no chaotic behavior is found. Meanwhile, for b s 0, the stablelocking region becomes symmetric around the zero frequency detuning, i.e., f s 0 GHz. The latter suggests that the linewidth enhancement factor determines the asymmetric characteristic of the stablelocking region around the zero frequency detuning observed in the system w6,24x. Similar results regarding the increase of complexity and the change in the two high-order instability regions, as well as the asymmetric characteristics of the stable-locking region, with the increase in the value of linewidth enhancement factor have also been observed in Ref. w6x. Let us now discuss the effect of the refractive index change due to the nonlinear gain on the dynamics of the system. As has been pointed out, the neglect of the nonlinear gain effect in the analysis of the optical phase does not affect the general effect of each laser parameter on the dynamical behavior of the system. However, because of the absence of the competition from this nonlinear gain effect with the differential gain effect in the phase when m s 0, the system becomes more unstable. To see whether or not it is necessary to consider the nonlinear gain effect in the analysis of the phase, let us study the laser system with the set of the parameters used in Ref. w10x. In this manner, we can compare the simulation results directly with previous experimental data w10x. The numerically and experimentally obtained maps are presented in Fig. 6 and Fig. 7, respectively. If we first study Fig. 6Ža. and Žb., it is observed that there are no significant variations between them
201
Fig. 6. Mapping of the dynamical states observed numerically with Ža. m s1 and Žb. m s 0. The injected laser system is operated at J˜s 0.667 with bs 4, gˆs s6.075=10y3 , gˆn s 5.56= 10y3 , and gˆp s1.0=10y2 .
except that the upper boundary of the stable-locking region is lower when m s 0. Besides, the instability regions as well as the chaotic regions enlarge slightly when m s 0. These phenomena are apparently quite different from our previous observation that the neglect of the nonlinear gain effect in the analysis of the phase drives the system dramatically unstable. This can be understood as follows. The values of gˆn and gˆp used in our previous discussion are several times larger than those used in Fig. 6. Besides, the value of b is 6 instead of 4 in the previous analysis. According to our previous discussion, the dynamical behavior of the system is found to be a strong function of these three parameters. Therefore, larger values of these parameters will make the change significant in our previous observation if we neglect the effect of the nonlinear gain in the analysis of the optical phase. This suggests that, depending on the values of these parameters, the dynamical behavior of the system may or may not be significantly differ-
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Fig. 7. Experimentally obtained mapping with the same operating conditions and laser parameters used in Fig. 6. The map is originally published in Ref. w10x and is reproduced here for the comparison with the numerical results shown in Fig. 6.
ent if the nonlinear gain effect is ignored in the analysis of the optical phase. To further address this issue, let us compare our simulation results with the experimental observation. By comparing Fig. 6 to Fig. 7, we find that the behavior of the system in Fig. 6Žb. seems to follow closer to that in Fig. 7. That is, the upper boundary of the stable-locking region in Fig. 6Žb. is closer to that in Fig. 7. However, significant differences are still observed between Fig. 6Žb. and Fig. 7: the instability regions and the chaotic regions in Fig. 6Žb. do not match exactly with those seen in Fig. 7. These observations together with the results discussed in the last paragraph suggest that, based on these data alone, we still cannot conclusively determine whether or not we need to consider the refractive index change due to the nonlinear gain effect in the theoretical analysis. In order to conclusively clarify this issue, more experimental data are needed. This suggests a further investigation on this issue.
already considered in all of the maps presented and discussed throughout this paper. The only operational parameter left to be considered is the normalized injection current level J.˜ Fig. 8 shows the map of the dynamical states of the optically injected semiconductor laser at a higher injection current level than that used in Fig. 1. By comparing Fig. 8
3.2. Effect of operational parameters Among the three operational parameters, the detuning frequency f and the injection parameter j are
Fig. 8. Numerically obtained maps of dynamical states when J˜s6 with gˆn s 2.538=10y2 , gˆp s 5.772=10y2 , and m s1. The values of b and gˆs are the same as those used in Fig. 1.
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with Fig. 1Ža., we find that the stable-locking region for J˜s 6 is larger than that for J˜s 4. Moreover, the upper stable-locking boundary for J˜s 6 shifts upward. Besides, it is also observed that as J˜ is increased from 4 to 6, the instability regions move up to the side corresponding to a positive value of the detuning frequency and shift to the right side corresponding to a larger value of the injection parameter. One intuitive explanation to the above observation is that when a laser is operated at a higher injection current level, the coherent optical power stored in the cavity is higher, thus allowing the laser to be more resistant to the perturbation of the externally injected optical field. This is why stronger externally optical perturbation is required to observe instabilities and chaos in the system at a higher injection current level as that shown in Fig. 8. However, this cannot explain why the instability regions as well as the chaotic regions expand as J˜ is increased from 4 to 6 because they are expected to shrink based on the above argument. In fact, as the injection current level J˜ is increased from 1 to 10, we observe that the instability regions, including the chaotic regions, enlarge. In addition, they move up to the side corresponding to a positive value of the detuning frequency and shift to the right side corresponding to a larger value of the injection parameter. More interestingly, it is found that the upper boundary of the stable-locking region initially keeps moving downward as the value of J˜ is increased from 1, gradually stops at J˜s 4, and starts to move upward if J˜ is further increased. These observations apparently cannot be explained by the above argument either, though this argument does predict the shift of the instability regions. Apparently, the single argument can only explain part of the observed phenomena. Some inherent elements of the system must be involved in the mechanism of determining the dynamics of the optically injected semiconductor laser when the injection current is varied. By revisiting Section 2, we find that the operational parameter J˜ actually does not appear in the normalized Eqs. Ž4. – Ž6.. The effect of this operational parameter J˜ on the dynamical characteristics of the system is implicit. It is through the dependence of the laser dynamics on the intrinsic parameters, gˆn and gˆp , that J˜ affects the behavior of the
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system. Based on the discussion on gˆn and gˆp , we know that their effects on the system are quite opposite to each other: gˆn tries to de-stabilize the dynamics of the system whereas gˆp tends to stabilize the dynamics. Therefore, varying the operational parameter J˜ for a given laser leads to the competition between gˆn and gˆp because both gˆn and gˆp vary linearly with J.˜ Depending on the relative strength of one to the other, the dynamical characteristics of the system may vary differently. If we compare a series of maps of the dynamical states when gˆn is increased, the instability regions expand toward the right side corresponding to a larger value of the injection parameter. On the other hand, if we study the maps of the dynamical states when gˆp is increased, the instability regions not only shrink but are also pushed up by the stable-locking region toward the side corresponding to a positive value of the frequency detuning as the stable-locking region expands. These observations imply that when J˜ is increased and thus the values of gˆn and gˆp are simultaneously increased with it, the instability regions would move up toward the side corresponding to a positive value of the detuning frequency and shift to the right side corresponding to a larger value of the injection parameter. Therefore, we believe that the mechanism in determining the dynamical behavior of the optically injected semiconductor laser when varying J˜ is the balance of the competition between gˆn and gˆp . This can be understood from Eqs. Ž4. – Ž6. by noting that the two effects on the behavior of the system are opposite in sign.
4. Conclusion The effects of the intrinsic and operational laser parameters on the dynamical characteristics of an optically injected semiconductor laser are comprehensively investigated. A number of maps showing the regions of different dynamical states of the system are numerically obtained and compared when the parameter of interest is varied while the other parameters are kept fixed to see the effect of each parameter on the dynamics of the system. The effect of the nonlinear gain, which is represented by gˆp , is to suppress the instability of a semiconductor laser that is subject to an external
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optical perturbation. Hence, gˆp tends to stabilize the system. On the contrary, the effect of the differential gain, which is represented by gˆn , is to make the laser system be easily driven into instability by the perturbation of an external optical injection. Therefore, the larger the value of gˆn is, the more unstable the system is. The effect of the normalized spontaneous carrier relaxation, gˆs , on the dynamical characteristics of the system is observed to be insignificant. For the linewidth enhancement factor b, it is observed that an increase in the value of b tends to de-stabilize the system when the laser is subject to external optical injection. It is also found that the linewidth enhancement factor determines the asymmetric characteristic of the stable-locking region around the zero frequency detuning. The changes of the dynamical characteristics in the system when the nonlinear gain effect is ignored in the analysis of the phase of the optical field is also studied. It is found that the general behavior of each laser parameter is not affected by such a neglect. However, when gˆp is ignored in the phase analysis, the competitive balance between gˆn and gˆp is reduced to the advantage of gˆn , resulting in increased instability for the system. A comparison between numerical and experimental results is also made. Further investigation is suggested to study this issue. ˜ it is found that For the operational parameter, J, the effect of the injection current enters the dynamics indirectly through two intrinsic parameters, the differential gain effect and the nonlinear gain effect. As J˜ is varied, both effects increase and compete with each other. The resulting balance between the competition of these two effects determines the dynamical behavior of the system. A tendency of increase in sizes of the instability regions as well as the chaotic regions is still clearly observed as J˜ is increased. These regions are also found to move to the side corresponding to a positive value of the frequency detuning and to a larger value of the injection parameter as J˜ is increased. Finally, we like to compare the simulation method used in this study to the bifurcation analysis used in Ref. w6x. The analysis and discussions throughout this paper are based on the numerical results using the simulation method introduced in Section 2. The advantage of the numerical simulation is that it reproduces all the observed phenomena in an optically
injected semiconductor laser system w2,9,10x, thus providing detailed information of the dynamics of the system. For example, the characteristics of each dynamical state, the noise effect on the dynamics w16,25x, and the transitions from one state to another can be studied in our simulation. The disadvantage of the simulation method is that systematic studies on the dynamics are not easy and are time-consuming. To avoid this problem, Wieczorek and the coworkers w6x applied the fundamental bifurcation theory to the study of the dynamics of the system, which is capable of easily producing bifurcation lines Žborders of regions of certain dynamics. which give insight into how different dynamics are related to one another to allow simple systematic studies. Their method, therefore, leads to a quick and complete picture of the dynamics of the system from a global viewpoint though detailed characteristics of the dynamics are not obtained using that method. Let us take the existence of bistability or multistability in the presence of coexisting attractors as an example. Using the bifurcation theory, it is easy to analyze whether there are coexisting attractors under the same operating condition in the system to search for the existence of bistability or multistability. However, the characteristics of the coexisting attractors cannot be obtained using the bifurcation analysis. With the simulation method, the existence of the bistability or multistability is found by numerically simulating the system with different initial conditions under the same operating condition. Compared to the bifurcation method, this is apparently not the best way to search for coexisting attractors because it is time-consuming. However, the simulation method can provide us with detailed information about the characteristics of the coexisting attractors including their basins, and their susceptibility to noise and other external perturbations. We have actually applied our simulation tools to the study w26x of the detailed characteristics of the bistability found in the locking region of the system, which cannot be obtained by a simple bifurcation analysis.
Acknowledgements The authors wish to acknowledge T.B. Simpson for helpful discussions. This work is supported by
S.K. Hwang, J.M. Liu r Optics Communications 183 (2000) 195–205
the US Army Research Office under the contract No. DAAG55-98-1-0269.
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