- Email: [email protected]
Response characteristics of direct current modulation on a bandwidth-enhanced semiconductor laser under strong injection locking
1 January 2000
Optics Communications 173 Ž2000. 349–355 www.elsevier.comrlocateroptcom
Response characteristics of direct current modulation on a bandwidth-enhanced semiconductor laser under strong injection locking H.F. Chen a , J.M. Liu a
a,)
, T.B. Simpson
b
Department of Electrical Engineering, UniÕersity of California, Los Angeles, Los Angeles, CA 90095-159410, USA b Jaycor, Inc., P.O. Box 85154, San Diego, CA 92186-5154, USA Received 29 January 1999; received in revised form 17 September 1999; accepted 26 October 1999
Abstract The response characteristics of direct current modulation on an injection-locked semiconductor laser are investigated. When a semiconductor laser is strongly injection locked in a stable locking state, the benefits of a significantly enhanced modulation bandwidth, a broadband noise reduction, and a large modulation dynamic range can be realized. A reduction on the frequency chirping can also be realized at a sufficiently large modulation index, though not at small modulation indices due to the effect of the laser noise. In a locking-unlocking bistable state, a large modulation current can unlock the laser. In a state near or beyond the Hopf bifurcation boundary, the dynamic instability of the laser can lead to high broadband noise and large frequency chirping. The effects of the distortion and the noise compression on the response of the current modulation with digital signals are investigated with eye patterns. q 2000 Elsevier Science B.V. All rights reserved. Keywords: Injection-locked oscillators; Current modulation; Eye pattern; Semiconductor lasers
High-speed direct current modulation on a semiconductor laser has important applications in optical communication systems. Recent studies w1–4x have demonstrated that it is possible to significantly enhance the modulation bandwidth, and simultaneously reduce the broadband noise of a semiconductor laser by strong optical injection locking. The requirements for simultaneous fulfillment of modulation bandwidth enhancement, broadband noise reduction, and stable injection-locking can be satisfied over a wide
) Corresponding author. Fax: q1-310-206-8495; e-mail: [email protected]
range of operating conditions w4x. Under certain operating conditions, a more than three-fold enhancement on the modulation bandwidth relative to the modulation bandwidth of the same laser in the free-running condition can be accomplished w3,4x. In addition, it has long been realized that injection locking can reduce frequency chirping in a semiconductor laser w5,6x, though the effect of the laser noise was not considered in the previous analyses regarding the chirp reduction. The results of these studies clearly point to the practical usefulness of properly injection-locked semiconductor lasers for high-speed current modulation applications. In this paper, we compare the characteristics of the current modulation response of a semiconductor
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 9 9 . 0 0 6 4 6 - X
350
H.F. Chen et al.r Optics Communications 173 (2000) 349–355
laser undergoing strong optical injection with that of the response of an identical laser under free-running operation, including the effect of the laser intrinsic noise. We compare the dynamic range of modulation, the distortion of the large-signal modulation, the frequency chirping and the stability of the laser under modulation. The characteristics of the modulation response for the laser operated in locking regions with different dynamic characteristics are analyzed and compared to quantify the benefit of injection-locking in a stable locking region. The limitations imposed by the intrinsic spontaneous laser noise on these characteristics of the modulation response are also studied. We consider an injection-locked single-mode semiconductor laser subject to direct current modulation. The laser is biased above its threshold for oscillation with a constant injection current density, J. In the free-running condition without current modulation, the laser oscillates at an optical frequency v 0 with a steady-state field amplitude A 0 and a steadystate carrier density N0 . The optical injection field, which has a field amplitude A i and an optical frequency v i , is characterized by the frequency detuning f s Vr2p s Ž v i y v 0 .r2p and the normalized dimensionless injection parameter j s h < A i
simulated using the following coupled equations w4,8x: da 1 gc gn s n˜ y gp Ž 2 a q a2 . Ž 1 q a . dt 2 gs J˜ q jgc cos Ž V t q f Ž t . . q Fa , df b gc gn sy n˜ y gp Ž 2 a q a 2 . dt 2 gs J˜ y
jgc 1qa
d n˜ dt
sin Ž V t q f Ž t . . q
Ž 1.
Ff 1qa
,
Ž 2.
2
s ygs n˜ y gn Ž 1 q a . n˜ y gs J˜Ž 2 a q a 2 . q
gs gp gc
2 Ž 2 a q a2 . Ž 1 q a . J˜
q mgs Ž J˜q 1 . cos Ž V m t . ,
Ž 3.
where gs is the spontaneous carrier relaxation rate, gn is the differential carrier relaxation rate, gp is the nonlinear carrier relaxation rate, and b is the linewidth enhancement factor. The parameter J˜s Ž Jred y gs N0 .rgs N0 , where e is the electronic charge and d is the active layer thickness of the laser, is the differential injection current density above threshold, normalized to the threshold current density. Fa and Ff are normalized noise-source parameters related to the spontaneous emission rate R sp of the laser w4x. The characteristics of a given semiconductor laser subject to optical injection are determined by the ˜ the three operational parameters: the bias point J, injection parameter j , and the frequency detuning f. In this study, we consider an SDL model 5301-G1 index-guided GaAsrAlGaAs quantum-well laser biased at J˜s 2r3, the characteristics of which have been mapped and thoroughly studied under different injection-locking conditions w4x. Throughout this letter, the injection parameter is fixed at a moderate level of j s 0.2 while a few different values of frequency detuning representing different locking conditions are chosen. At j s 0.2, the stable locking region is bounded by f s 1 GHz and f s y13 GHz, where f s 1 GHz is the Hopf bifurcation boundary. Between f s y13 GHz and f s y22 GHz is a region of locking-unlocking bistability, where the laser can be either locked or unlocked depending on the initial condition. The laser cannot be locked
H.F. Chen et al.r Optics Communications 173 (2000) 349–355
when the frequency detuning is more negative than y22 GHz. Relative to the free-running laser, a broadband noise reduction occurs in the locked region when the injection field is negatively detuned beyond f s y3 GHz. The three representative values of frequency detuning chosen in this study are f s 1 GHz on the Hopf bifurcation boundary Ždata represented by dash-dotted curves in the figures., f s y10 GHz in the stable locking region Ždata represented by solid curves., and f s y18 GHz in the locking-unlocking bistability region Ždata represented by dashed curves.. In addition, the modulation characteristics of the laser in its free-running condition are also presented Žin dotted curves. as a reference for gauging the improvement of performance due to optical injection. The upper plot in Fig. 1 shows the current modulation response, calculated with a modulation index m s 1%, and normalized to the low-frequency response of the laser in its free-running condition, for the four different operating conditions. The improvement over the free-running situation in terms of enhanced bandwidth due to stable injection locking can be easily seen. Except around the resonance peak of the case corresponding to f s 1 GHz on the Hopf
Fig. 1. Current modulation response normalized to the modulation index and to the low-frequency response, when ms1%, of the free-running laser. Dash-dotted curve: Injection locking at f s1 GHz. Solid curve: Injection locking at f sy10 GHz. Dashed curve: Injection locking at f sy18 GHz. Dotted curve: Free-running. The curves in the upper plot are the response curves for ms1%, and those in the lower plot are the response curves for ms100%. The 0 dB in the lower plot corresponds to the 0 dB in the upper plot in order to make all the response curves comparable.
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bifurcation boundary, the modulation response in each case remains virtually undistorted as the modulation index is increased up to m s 10%. Each modulation response curve begins to show a slight distortion as the modulation index is increased above m s 10%. The response around half the resonance frequency of each operating condition has the least distortion and that near the resonance frequency shows the most significant distortion. The lower plot in Fig. 1 shows the distorted current modulation response when the modulation index reaches m s 100%. At a given modulation strength, negatively shifting the frequency detuning of the injected optical field generally reduces the distortion in the current modulation response if the laser remains stably locked. However, when the laser is injection-locked in the bistability region, a high modulation index, corresponding to m G 50% for the bistable case studied here, can cause instability by unlocking the laser. As a result, the modulation response in such an operating condition becomes very irregular, as can be seen from the dashed curve in the lower plot of Fig. 1. For weak current modulation with small values of modulation index, the modulation response will be obscured by the intrinsic laser noise. To study this effect, we carried out our simulation with and without the noise source for different values of m. In our simulations, the laser noise induced insignificant differences between the overall response due to the combined modulation current and intrinsic noise and the modulation response alone when the modulation index m is larger than 1%. Below m s 1%, the relative importance of the laser noise gradually increases. Below m s 0.1%, the laser noise induces fluctuations in the response that obscure the modulation response. The distortion in the response curve at large values of m and the fluctuations caused by the laser noise at small values of m respectively can be used as the upper and lower limits to compare the dynamic range of the modulation index for the different operating conditions. This is shown in Fig. 2. The upper part of the figure represents the maximum modulation index in each operating condition, as a function of the modulation frequency, for a response distortion of less than 1 dB. The lower part shows the minimum modulation index in each operating
H.F. Chen et al.r Optics Communications 173 (2000) 349–355
352
Fig. 2. Dynamic range of the current modulation response. The upper part of the figure represents the maximum modulation index, as a function of the modulation frequency, in each operating condition for a response distortion less than 1 dB. The lower part shows the minimum modulation index in each operating condition for the modulation response to be easily distinguishable from the fluctuations caused by the laser noise. Each curve corresponds directly to the curves in Fig. 1 that have the same style.
condition for the modulation response to be 3 dB above the fluctuations caused by the laser noise. We can see from these results that stable injection locking significantly increases the modulation dynamic range of a laser besides enhancing its modulation bandwidth. This effect is particularly pronounced at low modulation indices because stable injection locking also reduces the broadband noise of the laser. The effects of the response distortion and the noise compression on the current modulation with digital signals can be observed through the eye patterns. The eye patterns are generated by modulating the current of the semiconductor laser under the various operating conditions with a train of random raised-cosine functions: n
Jm s
CK h Ž t y KT . ,
Ý
Ž 4.
Ks0
sin hŽ t . s
pt
pb t
ž / ž / ž /
pt T
T
cos
1y
T 2bt 2
,
Ž 5.
T
where CK is a series of the random numbers with the value 0 or 1, which represents digitized information, and T s 1rfm , where the modulation frequency f m
presents the bit rate. The value of b is chosen to be 0.3 in this simulation. The eye patterns shown in Fig. 3 are generated by choosing the bit rate equal to the corresponding resonance frequency of each operating condition. Three different modulation indexes m s 10%, 50%, and 100% are chosen to present the advantages of the laser injection-locked in the stablelocking region. We can see that the eye patterns obtained in the condition when the laser is injectionlocked at f s y10 GHz have clearer eye opening with less distortion or less noise than those obtained in other operating conditions studied here. This observation can be quantified through the measurement of the relative eye opening. The relative eye opening as a function of the modulation frequency in various operating conditions is shown in Fig. 4. The relative eye opening shows the same tendency as the noise compression. Since injection-locking the semiconductor laser at f s y18 GHz gains better noise compression than injection-locking it at f s y10 GHz, the relative eye opening obtained at f s y18 GHz is larger than that obtained at f s y10 GHz, which in turn is larger than that obtained in the free- running condition. However, when the laser is injection-locked with a frequency detuning f s y18 GHz, the modulation signal with a modulation index m G 50% unlocks the laser, resulting in zero eye opening in this situation. When the frequency detuning is positively shifted beyond the noise compression region, the eye opening also rapidly decreases to zero. Therefore, the eye opening obtained in the operating condition with f s 1 GHz is zero for a modulation index of any value. Current modulation on a semiconductor laser modulates not only the laser intensity, but also its frequency because a change in the carrier density causes a change in the index of refraction of the laser medium. This change in the index of refraction generates frequency chirping, which can place a considerable limitation on the modulation bit rate and has to be considered. In this paper, the frequency chirping is measured by the normalized chirp to power ratio ŽCPR. w5x, which is defined as follows: CPR s
1
df
2ps
dt
,
Ž 6.
H.F. Chen et al.r Optics Communications 173 (2000) 349–355
353
Fig. 3. Eye pattern. The bit rate is chosen as the corresponding resonance frequency for each operating condition: the bit rate f m s 2.9 GHz is chosen for the free-running operation, f m s 4.7 GHz for f s y18 GHz, and f m s 6.8 GHz for f s y10 GHz. The eye opening obtained from the operating condition f s 1 GHz is zero for the range of the bit rate we concern, so the eye patterns are not shown. The intensity, normalized to < A 0 < 2 , of the eye patterns is the differential intensity above or below the corresponding field intensity of the injection-locked laser without current modulation for each operating condition.
where s is the modulation response. Because the linewidth enhancement factor b, also defined as the
X Fig. 4. Relative eye opening Ž Dr D ., where the measurements of X the quantities D and D are marked in Fig. 3. The modulation indexes for each operating condition are ms10%, 50%, and 100%. Each curve corresponds directly to the curves in Fig. 1 that have the same style. The relative eye opening for the operating condition f sy18 GHz with mG 50% is zero; that for the operating condition f s1 GHz is zero with the modulation index of any value.
chirping parameter, is directly responsible for the frequency chirping of a semiconductor laser under current modulation, there exists a relationship between the parameter b and the CPR. In the case of small-signal modulation with the effect of the laser noise ignored, the relationship between the chirping parameter b and the CPR can be obtained by linearizing the rate equation around the locking operating point w8x. This relationship between b and the CPR can be expressed as follows: 1
df
2ps
dt
(
where U s
Vs
jgc aL
Vm 2p
jgc aL
sin f L ,
)
b
V m2 q Ž U y Vrb . V m2 q Ž U q Vb .
cos f L ,
2
2
,
Ž 7. Ž 8. Ž 9.
where f L is the phase of the intracavity laser field relative to the injection field, and a L is the field
H.F. Chen et al.r Optics Communications 173 (2000) 349–355
354
amplitude, normalized to A 0 , of the injection-locked laser w8x. An effective linewidth enhancement factor beff , which is the modified chirping parameter under injection locking, can be defined as follows:
)
beff s b
V m2 q Ž U y Vrb . V m2 q Ž U q Vb .
2
2
.
Ž 10 .
This effective chirping parameter beff is equal to b when the laser is free running. The effective chirping parameter beff as a function of the modulation frequency f m is shown in Fig. 5. We can see that injection-locking the laser at f s y10 GHz reduces the effective chirping parameter more than injection-locking the laser at f s y18 GHz does. Further positively shifting the frequency detuning can reduce the effective chirping parameter more until the boundary of the Hopf bifurcation is reached. When the modulation frequency increases, the value of beff will eventually converge to that of b. When the effect of the intrinsic noise or that of the nonlinearity of the laser on the frequency chirping are significant, however, the simple relationship in Eq. Ž7. between the CPR and b is no longer valid. In this situation, it is not possible to simply represent the frequency chirping with a effective chirping parameter. Then the measurement of the frequency chirping including the effects of the intrinsic noise and the nonlinearity of the laser dynamics under a large modulation current is better quantified directly with the CPR. The effect of the intrinsic laser noise on the frequency chirping is realized by the fact that the
Fig. 5. Effective chirping parameter. Each curve corresponds directly to the curves in Fig. 1 that have the same style.
Fig. 6. Frequency chirping measured by the chirp to power ratio ŽCPR., in which the amount of the chirping has been normalized to Gigahertz. Each curve corresponds directly to the curves in Fig. 1 that have the same style. Parts Ža. and Žb. correspond to ms1%. The curves in part Ža. were obtained with full consideration of the laser noise while those in part Žb. were obtained by ignoring the laser noise. Part Žc. corresponds to ms 30%, with two curves for each operating condition: The lower one is obtained by ignoring the noise to serve as a reference; the upper one is obtained with full consideration of the laser noise.
intrinsic noise can cause the phase of the laser field to fluctuate. The fluctuations induced by the intrinsic noise can obscure the modulation response in the frequency domain. Previous studies on the frequency chirping of a semiconductor laser under direct current modulation have ignored the effect of the laser noise w5,6x. However, our investigation reveals a dramatic effect of the noise on the frequency chirping. The results of the frequency chirping are presented in Fig. 6. Both parts Ža. and Žb. correspond to the modulation index m s 1%. The curves in part Žb. were obtained by ignoring the laser noise. They serve as a reference for comparison with the curves in part Ža. to clearly demonstrate the effect of the noise on the frequency chirping. We can see from the curves in part Žb. that if the laser noise were not present, significant reduction of the frequency chirping could be achieved by optical injection, and positively shifting the frequency detuning could further reduce the frequency chirping. In reality, however, when the modulation index is small, the chirp is dominated by the laser noise, as can be seen by comparing the curves in parts Ža. and Žb.. As a result, the chirp follows the same tendency as the
H.F. Chen et al.r Optics Communications 173 (2000) 349–355
power noise. Therefore, reduction of the frequency chirping in a semiconductor laser is not always guaranteed by injection locking. If we increase the modulation index within the range where the chirp is still dominated by the noise, the chirp does not change, but the modulation response increases. Thus all the CPR curves will shift downward until they match their respective CPR curves obtained by ignoring the laser noise. At this point, the modulation intensity dominates the chirp, and the effect of the noise is barely seen except at relatively low modulation frequencies. A significant reduction on the frequency chirping by injection locking compared to the free running condition can then be expected. This result is shown in part Žc. of Fig. 6, corresponding to m s 30%. In part Žc., there are two curves for each operating condition: The lower one is obtained by ignoring the noise to serve as a reference; and the upper one is obtained with full consideration of the laser noise. At or above this modulation index, m s 30%, the chirp is reduced by increasing the injection parameter or by positively shifting the frequency detuning. When the frequency detuning is positively shifted beyond the Hopf bifurcation boundary, the laser enters a state of instability and the frequency chirping suddenly becomes larger than that in the free-running condition. If we increase the modulation index up to m s 100%, the CPR curve for the stable locking state remains smooth and increases slightly but those for the other three operating conditions become irregular and increase dramatically. From the analyses presented above, we can conclude that when a semiconductor laser is strongly injection locked in a stable locking state, the benefits of a significantly enhanced modulation bandwidth, a broadband noise reduction, a large modulation dynamic range, and a better eye opening can be real-
355
ized. A reduction on the frequency chirping can also be realized at a sufficiently large modulation index, though not at small modulation indices due to the effect of the laser noise. A semiconductor laser injection-locked in a locking-unlocking bistable state cannot fully enjoy such benefits because a large modulation current can unlock the laser. Nor can one operated in a state near or beyond the Hopf bifurcation boundary because of the high broadband noise and the large frequency chirping associated with the instability of the laser. Finally, a semiconductor laser operated in a stable injection-locking state generally has better current modulation characteristics than in its free-running state.
Acknowledgements This work was supported by the Army Research Office under contract DAAG-55-98-C-0038 and contract DAAG-55-98-1-0269.
References w1x T.B. Simpson, J.M. Liu, A. Gavrielides, IEEE Photon. Technol. Lett. 7 Ž1995. 709. w2x J. Wang, M.K. Halder, L. Li, F.V.C. Mendis, IEEE Photon. Technol. Lett. 8 Ž1996. 34. w3x T.B. Simpson, J.M. Liu, IEEE Photon. Technol. Lett. 9 Ž1997. 1322. w4x J.M. Liu, H.F. Chen, X.J. Meng, T.B. Simpson, IEEE Photon. Technol. Lett. 9 Ž1997. 1325. w5x S. Piazzolla, P. Spano, M. Tamburrini, IEEE J. Quantum Electron. QE-22 Ž1986. 2219. w6x G. Yabre, J. Lightwave Technol. 14 Ž1996. 2367. w7x J.M. Liu, T.B. Simpson, IEEE J. Quantum Electron. 30 Ž1994. 957. w8x T.B. Simpson, J.M. Liu, A. Gavrielides, IEEE J. Quantum Electron. 32 Ž1996. 1456.
Optics Communications 173 Ž2000. 349–355 www.elsevier.comrlocateroptcom
Response characteristics of direct current modulation on a bandwidth-enhanced semiconductor laser under strong injection locking H.F. Chen a , J.M. Liu a
a,)
, T.B. Simpson
b
Department of Electrical Engineering, UniÕersity of California, Los Angeles, Los Angeles, CA 90095-159410, USA b Jaycor, Inc., P.O. Box 85154, San Diego, CA 92186-5154, USA Received 29 January 1999; received in revised form 17 September 1999; accepted 26 October 1999
Abstract The response characteristics of direct current modulation on an injection-locked semiconductor laser are investigated. When a semiconductor laser is strongly injection locked in a stable locking state, the benefits of a significantly enhanced modulation bandwidth, a broadband noise reduction, and a large modulation dynamic range can be realized. A reduction on the frequency chirping can also be realized at a sufficiently large modulation index, though not at small modulation indices due to the effect of the laser noise. In a locking-unlocking bistable state, a large modulation current can unlock the laser. In a state near or beyond the Hopf bifurcation boundary, the dynamic instability of the laser can lead to high broadband noise and large frequency chirping. The effects of the distortion and the noise compression on the response of the current modulation with digital signals are investigated with eye patterns. q 2000 Elsevier Science B.V. All rights reserved. Keywords: Injection-locked oscillators; Current modulation; Eye pattern; Semiconductor lasers
High-speed direct current modulation on a semiconductor laser has important applications in optical communication systems. Recent studies w1–4x have demonstrated that it is possible to significantly enhance the modulation bandwidth, and simultaneously reduce the broadband noise of a semiconductor laser by strong optical injection locking. The requirements for simultaneous fulfillment of modulation bandwidth enhancement, broadband noise reduction, and stable injection-locking can be satisfied over a wide
) Corresponding author. Fax: q1-310-206-8495; e-mail: [email protected]
range of operating conditions w4x. Under certain operating conditions, a more than three-fold enhancement on the modulation bandwidth relative to the modulation bandwidth of the same laser in the free-running condition can be accomplished w3,4x. In addition, it has long been realized that injection locking can reduce frequency chirping in a semiconductor laser w5,6x, though the effect of the laser noise was not considered in the previous analyses regarding the chirp reduction. The results of these studies clearly point to the practical usefulness of properly injection-locked semiconductor lasers for high-speed current modulation applications. In this paper, we compare the characteristics of the current modulation response of a semiconductor
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 9 9 . 0 0 6 4 6 - X
350
H.F. Chen et al.r Optics Communications 173 (2000) 349–355
laser undergoing strong optical injection with that of the response of an identical laser under free-running operation, including the effect of the laser intrinsic noise. We compare the dynamic range of modulation, the distortion of the large-signal modulation, the frequency chirping and the stability of the laser under modulation. The characteristics of the modulation response for the laser operated in locking regions with different dynamic characteristics are analyzed and compared to quantify the benefit of injection-locking in a stable locking region. The limitations imposed by the intrinsic spontaneous laser noise on these characteristics of the modulation response are also studied. We consider an injection-locked single-mode semiconductor laser subject to direct current modulation. The laser is biased above its threshold for oscillation with a constant injection current density, J. In the free-running condition without current modulation, the laser oscillates at an optical frequency v 0 with a steady-state field amplitude A 0 and a steadystate carrier density N0 . The optical injection field, which has a field amplitude A i and an optical frequency v i , is characterized by the frequency detuning f s Vr2p s Ž v i y v 0 .r2p and the normalized dimensionless injection parameter j s h < A i
simulated using the following coupled equations w4,8x: da 1 gc gn s n˜ y gp Ž 2 a q a2 . Ž 1 q a . dt 2 gs J˜ q jgc cos Ž V t q f Ž t . . q Fa , df b gc gn sy n˜ y gp Ž 2 a q a 2 . dt 2 gs J˜ y
jgc 1qa
d n˜ dt
sin Ž V t q f Ž t . . q
Ž 1.
Ff 1qa
,
Ž 2.
2
s ygs n˜ y gn Ž 1 q a . n˜ y gs J˜Ž 2 a q a 2 . q
gs gp gc
2 Ž 2 a q a2 . Ž 1 q a . J˜
q mgs Ž J˜q 1 . cos Ž V m t . ,
Ž 3.
where gs is the spontaneous carrier relaxation rate, gn is the differential carrier relaxation rate, gp is the nonlinear carrier relaxation rate, and b is the linewidth enhancement factor. The parameter J˜s Ž Jred y gs N0 .rgs N0 , where e is the electronic charge and d is the active layer thickness of the laser, is the differential injection current density above threshold, normalized to the threshold current density. Fa and Ff are normalized noise-source parameters related to the spontaneous emission rate R sp of the laser w4x. The characteristics of a given semiconductor laser subject to optical injection are determined by the ˜ the three operational parameters: the bias point J, injection parameter j , and the frequency detuning f. In this study, we consider an SDL model 5301-G1 index-guided GaAsrAlGaAs quantum-well laser biased at J˜s 2r3, the characteristics of which have been mapped and thoroughly studied under different injection-locking conditions w4x. Throughout this letter, the injection parameter is fixed at a moderate level of j s 0.2 while a few different values of frequency detuning representing different locking conditions are chosen. At j s 0.2, the stable locking region is bounded by f s 1 GHz and f s y13 GHz, where f s 1 GHz is the Hopf bifurcation boundary. Between f s y13 GHz and f s y22 GHz is a region of locking-unlocking bistability, where the laser can be either locked or unlocked depending on the initial condition. The laser cannot be locked
H.F. Chen et al.r Optics Communications 173 (2000) 349–355
when the frequency detuning is more negative than y22 GHz. Relative to the free-running laser, a broadband noise reduction occurs in the locked region when the injection field is negatively detuned beyond f s y3 GHz. The three representative values of frequency detuning chosen in this study are f s 1 GHz on the Hopf bifurcation boundary Ždata represented by dash-dotted curves in the figures., f s y10 GHz in the stable locking region Ždata represented by solid curves., and f s y18 GHz in the locking-unlocking bistability region Ždata represented by dashed curves.. In addition, the modulation characteristics of the laser in its free-running condition are also presented Žin dotted curves. as a reference for gauging the improvement of performance due to optical injection. The upper plot in Fig. 1 shows the current modulation response, calculated with a modulation index m s 1%, and normalized to the low-frequency response of the laser in its free-running condition, for the four different operating conditions. The improvement over the free-running situation in terms of enhanced bandwidth due to stable injection locking can be easily seen. Except around the resonance peak of the case corresponding to f s 1 GHz on the Hopf
Fig. 1. Current modulation response normalized to the modulation index and to the low-frequency response, when ms1%, of the free-running laser. Dash-dotted curve: Injection locking at f s1 GHz. Solid curve: Injection locking at f sy10 GHz. Dashed curve: Injection locking at f sy18 GHz. Dotted curve: Free-running. The curves in the upper plot are the response curves for ms1%, and those in the lower plot are the response curves for ms100%. The 0 dB in the lower plot corresponds to the 0 dB in the upper plot in order to make all the response curves comparable.
351
bifurcation boundary, the modulation response in each case remains virtually undistorted as the modulation index is increased up to m s 10%. Each modulation response curve begins to show a slight distortion as the modulation index is increased above m s 10%. The response around half the resonance frequency of each operating condition has the least distortion and that near the resonance frequency shows the most significant distortion. The lower plot in Fig. 1 shows the distorted current modulation response when the modulation index reaches m s 100%. At a given modulation strength, negatively shifting the frequency detuning of the injected optical field generally reduces the distortion in the current modulation response if the laser remains stably locked. However, when the laser is injection-locked in the bistability region, a high modulation index, corresponding to m G 50% for the bistable case studied here, can cause instability by unlocking the laser. As a result, the modulation response in such an operating condition becomes very irregular, as can be seen from the dashed curve in the lower plot of Fig. 1. For weak current modulation with small values of modulation index, the modulation response will be obscured by the intrinsic laser noise. To study this effect, we carried out our simulation with and without the noise source for different values of m. In our simulations, the laser noise induced insignificant differences between the overall response due to the combined modulation current and intrinsic noise and the modulation response alone when the modulation index m is larger than 1%. Below m s 1%, the relative importance of the laser noise gradually increases. Below m s 0.1%, the laser noise induces fluctuations in the response that obscure the modulation response. The distortion in the response curve at large values of m and the fluctuations caused by the laser noise at small values of m respectively can be used as the upper and lower limits to compare the dynamic range of the modulation index for the different operating conditions. This is shown in Fig. 2. The upper part of the figure represents the maximum modulation index in each operating condition, as a function of the modulation frequency, for a response distortion of less than 1 dB. The lower part shows the minimum modulation index in each operating
H.F. Chen et al.r Optics Communications 173 (2000) 349–355
352
Fig. 2. Dynamic range of the current modulation response. The upper part of the figure represents the maximum modulation index, as a function of the modulation frequency, in each operating condition for a response distortion less than 1 dB. The lower part shows the minimum modulation index in each operating condition for the modulation response to be easily distinguishable from the fluctuations caused by the laser noise. Each curve corresponds directly to the curves in Fig. 1 that have the same style.
condition for the modulation response to be 3 dB above the fluctuations caused by the laser noise. We can see from these results that stable injection locking significantly increases the modulation dynamic range of a laser besides enhancing its modulation bandwidth. This effect is particularly pronounced at low modulation indices because stable injection locking also reduces the broadband noise of the laser. The effects of the response distortion and the noise compression on the current modulation with digital signals can be observed through the eye patterns. The eye patterns are generated by modulating the current of the semiconductor laser under the various operating conditions with a train of random raised-cosine functions: n
Jm s
CK h Ž t y KT . ,
Ý
Ž 4.
Ks0
sin hŽ t . s
pt
pb t
ž / ž / ž /
pt T
T
cos
1y
T 2bt 2
,
Ž 5.
T
where CK is a series of the random numbers with the value 0 or 1, which represents digitized information, and T s 1rfm , where the modulation frequency f m
presents the bit rate. The value of b is chosen to be 0.3 in this simulation. The eye patterns shown in Fig. 3 are generated by choosing the bit rate equal to the corresponding resonance frequency of each operating condition. Three different modulation indexes m s 10%, 50%, and 100% are chosen to present the advantages of the laser injection-locked in the stablelocking region. We can see that the eye patterns obtained in the condition when the laser is injectionlocked at f s y10 GHz have clearer eye opening with less distortion or less noise than those obtained in other operating conditions studied here. This observation can be quantified through the measurement of the relative eye opening. The relative eye opening as a function of the modulation frequency in various operating conditions is shown in Fig. 4. The relative eye opening shows the same tendency as the noise compression. Since injection-locking the semiconductor laser at f s y18 GHz gains better noise compression than injection-locking it at f s y10 GHz, the relative eye opening obtained at f s y18 GHz is larger than that obtained at f s y10 GHz, which in turn is larger than that obtained in the free- running condition. However, when the laser is injection-locked with a frequency detuning f s y18 GHz, the modulation signal with a modulation index m G 50% unlocks the laser, resulting in zero eye opening in this situation. When the frequency detuning is positively shifted beyond the noise compression region, the eye opening also rapidly decreases to zero. Therefore, the eye opening obtained in the operating condition with f s 1 GHz is zero for a modulation index of any value. Current modulation on a semiconductor laser modulates not only the laser intensity, but also its frequency because a change in the carrier density causes a change in the index of refraction of the laser medium. This change in the index of refraction generates frequency chirping, which can place a considerable limitation on the modulation bit rate and has to be considered. In this paper, the frequency chirping is measured by the normalized chirp to power ratio ŽCPR. w5x, which is defined as follows: CPR s
1
df
2ps
dt
,
Ž 6.
H.F. Chen et al.r Optics Communications 173 (2000) 349–355
353
Fig. 3. Eye pattern. The bit rate is chosen as the corresponding resonance frequency for each operating condition: the bit rate f m s 2.9 GHz is chosen for the free-running operation, f m s 4.7 GHz for f s y18 GHz, and f m s 6.8 GHz for f s y10 GHz. The eye opening obtained from the operating condition f s 1 GHz is zero for the range of the bit rate we concern, so the eye patterns are not shown. The intensity, normalized to < A 0 < 2 , of the eye patterns is the differential intensity above or below the corresponding field intensity of the injection-locked laser without current modulation for each operating condition.
where s is the modulation response. Because the linewidth enhancement factor b, also defined as the
X Fig. 4. Relative eye opening Ž Dr D ., where the measurements of X the quantities D and D are marked in Fig. 3. The modulation indexes for each operating condition are ms10%, 50%, and 100%. Each curve corresponds directly to the curves in Fig. 1 that have the same style. The relative eye opening for the operating condition f sy18 GHz with mG 50% is zero; that for the operating condition f s1 GHz is zero with the modulation index of any value.
chirping parameter, is directly responsible for the frequency chirping of a semiconductor laser under current modulation, there exists a relationship between the parameter b and the CPR. In the case of small-signal modulation with the effect of the laser noise ignored, the relationship between the chirping parameter b and the CPR can be obtained by linearizing the rate equation around the locking operating point w8x. This relationship between b and the CPR can be expressed as follows: 1
df
2ps
dt
(
where U s
Vs
jgc aL
Vm 2p
jgc aL
sin f L ,
)
b
V m2 q Ž U y Vrb . V m2 q Ž U q Vb .
cos f L ,
2
2
,
Ž 7. Ž 8. Ž 9.
where f L is the phase of the intracavity laser field relative to the injection field, and a L is the field
H.F. Chen et al.r Optics Communications 173 (2000) 349–355
354
amplitude, normalized to A 0 , of the injection-locked laser w8x. An effective linewidth enhancement factor beff , which is the modified chirping parameter under injection locking, can be defined as follows:
)
beff s b
V m2 q Ž U y Vrb . V m2 q Ž U q Vb .
2
2
.
Ž 10 .
This effective chirping parameter beff is equal to b when the laser is free running. The effective chirping parameter beff as a function of the modulation frequency f m is shown in Fig. 5. We can see that injection-locking the laser at f s y10 GHz reduces the effective chirping parameter more than injection-locking the laser at f s y18 GHz does. Further positively shifting the frequency detuning can reduce the effective chirping parameter more until the boundary of the Hopf bifurcation is reached. When the modulation frequency increases, the value of beff will eventually converge to that of b. When the effect of the intrinsic noise or that of the nonlinearity of the laser on the frequency chirping are significant, however, the simple relationship in Eq. Ž7. between the CPR and b is no longer valid. In this situation, it is not possible to simply represent the frequency chirping with a effective chirping parameter. Then the measurement of the frequency chirping including the effects of the intrinsic noise and the nonlinearity of the laser dynamics under a large modulation current is better quantified directly with the CPR. The effect of the intrinsic laser noise on the frequency chirping is realized by the fact that the
Fig. 5. Effective chirping parameter. Each curve corresponds directly to the curves in Fig. 1 that have the same style.
Fig. 6. Frequency chirping measured by the chirp to power ratio ŽCPR., in which the amount of the chirping has been normalized to Gigahertz. Each curve corresponds directly to the curves in Fig. 1 that have the same style. Parts Ža. and Žb. correspond to ms1%. The curves in part Ža. were obtained with full consideration of the laser noise while those in part Žb. were obtained by ignoring the laser noise. Part Žc. corresponds to ms 30%, with two curves for each operating condition: The lower one is obtained by ignoring the noise to serve as a reference; the upper one is obtained with full consideration of the laser noise.
intrinsic noise can cause the phase of the laser field to fluctuate. The fluctuations induced by the intrinsic noise can obscure the modulation response in the frequency domain. Previous studies on the frequency chirping of a semiconductor laser under direct current modulation have ignored the effect of the laser noise w5,6x. However, our investigation reveals a dramatic effect of the noise on the frequency chirping. The results of the frequency chirping are presented in Fig. 6. Both parts Ža. and Žb. correspond to the modulation index m s 1%. The curves in part Žb. were obtained by ignoring the laser noise. They serve as a reference for comparison with the curves in part Ža. to clearly demonstrate the effect of the noise on the frequency chirping. We can see from the curves in part Žb. that if the laser noise were not present, significant reduction of the frequency chirping could be achieved by optical injection, and positively shifting the frequency detuning could further reduce the frequency chirping. In reality, however, when the modulation index is small, the chirp is dominated by the laser noise, as can be seen by comparing the curves in parts Ža. and Žb.. As a result, the chirp follows the same tendency as the
H.F. Chen et al.r Optics Communications 173 (2000) 349–355
power noise. Therefore, reduction of the frequency chirping in a semiconductor laser is not always guaranteed by injection locking. If we increase the modulation index within the range where the chirp is still dominated by the noise, the chirp does not change, but the modulation response increases. Thus all the CPR curves will shift downward until they match their respective CPR curves obtained by ignoring the laser noise. At this point, the modulation intensity dominates the chirp, and the effect of the noise is barely seen except at relatively low modulation frequencies. A significant reduction on the frequency chirping by injection locking compared to the free running condition can then be expected. This result is shown in part Žc. of Fig. 6, corresponding to m s 30%. In part Žc., there are two curves for each operating condition: The lower one is obtained by ignoring the noise to serve as a reference; and the upper one is obtained with full consideration of the laser noise. At or above this modulation index, m s 30%, the chirp is reduced by increasing the injection parameter or by positively shifting the frequency detuning. When the frequency detuning is positively shifted beyond the Hopf bifurcation boundary, the laser enters a state of instability and the frequency chirping suddenly becomes larger than that in the free-running condition. If we increase the modulation index up to m s 100%, the CPR curve for the stable locking state remains smooth and increases slightly but those for the other three operating conditions become irregular and increase dramatically. From the analyses presented above, we can conclude that when a semiconductor laser is strongly injection locked in a stable locking state, the benefits of a significantly enhanced modulation bandwidth, a broadband noise reduction, a large modulation dynamic range, and a better eye opening can be real-
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ized. A reduction on the frequency chirping can also be realized at a sufficiently large modulation index, though not at small modulation indices due to the effect of the laser noise. A semiconductor laser injection-locked in a locking-unlocking bistable state cannot fully enjoy such benefits because a large modulation current can unlock the laser. Nor can one operated in a state near or beyond the Hopf bifurcation boundary because of the high broadband noise and the large frequency chirping associated with the instability of the laser. Finally, a semiconductor laser operated in a stable injection-locking state generally has better current modulation characteristics than in its free-running state.
Acknowledgements This work was supported by the Army Research Office under contract DAAG-55-98-C-0038 and contract DAAG-55-98-1-0269.
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