Rising and falling time of amplified picosecond optical
pulses by semiconductor optical amplifiers
Guang-Qiong Xia
a,*, Zheng-Mao Wu
a, Gong-Ru Lin
baDepartment of Physics, Southwest Normal University, Chongqing 400715, PR China
bInstitute of Electro-Optical Engineering, National Chiao Tung University, 1001, Ta Hsueh Rd., Hsinchu 300, Taiwan, ROC
Received 2 March 2003; received in revised form 26 June 2003; accepted 9 September 2003
Abstract
After adopting the theoretical model that includes several physical mechanisms such as the position- and time-dependent carrier lifetime, the gain saturation caused by the depletion of carrier density owing to the stimulated emission, the gain compression induced by the intraband process of carrier heating and spectral hole burning, the gain asymmetry and shift, both the rising and falling time of amplified picosecond optical pulses by the semiconductor optical amplifiers (SOAs) have been investigated numerically. The results show that with the increase of the bias current of SOAs or the length of SOAs, the rising time will decrease and the falling time increase; the input pulse with a large peak power will accelerate the rising time shortening and the falling time lengthening; the gain compression has an obvious influence on the rising and falling time for several picosecond width input pulses and gives approximately no effect for the input pulses in the tens of picosecond range; the gain asymmetry and shift affects the rising and falling time.
Ó 2003 Elsevier B.V. All rights reserved.
PACS: 42.55.Px; 42.60.Da; 42.65.Re
Keywords: Semiconductor optical amplifiers; Picosecond optical pulses; Rising and falling time
1. Introduction
With the development of semiconductor
tech-nology, the semiconductor optical amplifiers
(SOAs) have been largely improved in perfor-mances and received considerable attention due to their potential applications in optical fiber
com-munication systems as high-speed switching, all-optical wavelength conversion, 2R or 3R regener-ation, in-line amplification etc. Accordingly, the dynamic response of the SOA to ultra-short optical pulses has therefore been a key subject and has been extensively investigated both theoretically and experimentally [1–7]. Even so, we have noticed that most of the relevant reports focus on the gain dy-namics or pulse evolution, and few pay special concentrations on the asymmetry of the amplified optical pulses though some phenomena may be
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hidden in relevant researches [8,9]. The rising time (defined as the time that the amplitude of the am-plified pulse by SOA increases from 10% to 90% of the amplified pulse peak at the leading edge) and falling time (defined as the time that the amplitude of the amplified pulse by SOA decreases from 90% to 10% of the amplified pulse peak at the tailing edge) are useful to quantify the degree of the pulse asymmetry for the amplified pulse being single peak contribution. (In fact, we discover, through calculations, that this condition can usually be satisfied for the input pulse with a peak power no more than about 1 W. However, if the input pulse has a larger peak power, the amplified pulse will be distorted obviously and appear multi-peak contri-bution. Under this circumstance the rising and falling times may not be suitable for characterizing the pulse.) Based on above considerations, the ris-ing and fallris-ing times are investigated numerically for input pulses with different width and peak power or for the SOA with different bias current and length. Regardless of the complications of the amplified process of ultra-short optical pulses by SOAs, the theory modeling the dynamic process has been gradually improved in last ten years. The initial model for describing pulse amplification of SOA is presented by Agrawal and Olsson where the gain saturation of the SOA caused by the depletion of carrier density owing to the stimulated emission was included [1], and then was developed after taking into account the gain compression induced by intraband process of carrier heating and spectral hole burning [2–4], gain asymmetry and shift [4,10,11], position- and time-dependent carrier lifetime [10,11,13]. Nevertheless, few studies adopt a model that includes all above mechanisms due to the difficulty of the numerical simulations. In this paper, a relatively complete model has been used after overcoming the simulative problem and then how these mechanisms affect the rising and falling time of amplified picosecond optical pulses can be specified.
2. Theory
Assuming that the reflectivity of both facets of SOA are equal to zero, after synthesizing the
the-oretical model mentioned above, the rate equa-tions describing a optical pulse amplification by SOAs can be written as
oNðz; T Þ oT ¼ I eV F ðN Þ gmðz; T Þ hxr Pðz; T Þ; ð1Þ oPðz; T Þ oz ¼ gmðz; T ÞP ðz; T Þ ð2Þ and FðN Þ ¼ AN þ BN2þ CN3¼ N =s c; ð3Þ
where N is the carrier density, T (¼ t z=vg; vg is
the group velocity in the SOA) is measured in a reference frame moving with the pulse, I is the injection current, e is the electron charge, V is the
volume of the active layer, hx is the photon
en-ergy, r is the cross-sectional area of the active layer, P is the optical power, A; B, and C (their values usually vary with different SOAs and should be determined by comparison with experiments) characterize the nonradiative recombination, the spontaneous emission and the Auger process,
re-spectively, sC is the position- and time-dependent
carrier lifetime which sometimes is regarded as a constant in order to simplify the simulation (a comparison of the amplified pulse between the position- and time-dependent carrier lifetime and the constant carrier lifetime can be seen in [13],
where a obvious difference can be observed), gm is
the gain coefficient and can be expressed by gmðz; T Þ ¼
CgðN Þ
1þ eP ðz; T Þ; ð4Þ
where C is the confinement factor, e is the gain compression factor which is phenomenologically introduced to describe the effects of the carrier heating and spectral hole burning, and
gðN Þ ¼ aðN N0Þ a1ðk kNÞ 2 þ a2ðk kNÞ 3 : ð5Þ Here, a is the differential gain coefficient, a1and a2
are empirically determined constants and charac-terize the width and asymmetry of the gain profile, N0is the transparency carrier density, kN describes
the shift of the gain peak, which is represented by
where k0 is the gain peak wavelength at
transpar-ency, and a3 is a empirical constant showing the
shift of the gain peak.
Based on Eqs. (1)–(6), the temporal shape of amplified pulse for a given input pulse passing through a SOA can be simulated numerically (the detailed method is not involved in this Letter due to the limited space), and both the rising and falling times can be investigated thus far.
3. Results and discussion
For simplicity, the input pulse is supposed to be
a Gaussian profile with Pinexp½ðT =T0Þ
2
, where
Pin is the peak power, T0 characterizes the pulse
width. Furthermore, the input pulse is assumed to be a single-frequency light (the central wavelength is equal to 1.55 lm), which is reasonable because a picosecond pulse has a relative narrow spectral width compared with the bandwidth of the SOA.
The used data in calculations are: L¼ 0:50 103
m, r¼ 0:18 1012 m2, a¼ 2:5 1020 m2, a 1¼ 7:4 1018 m3, a 2¼ 3:155 1025 m4, a3¼ 3 1032 m4, e¼ 0:2 W1, N 0¼ 1:1 1024 m3, A¼ 1:5108, B¼ 2:5 1017 m3 s1, C¼ 9:4 1041
m6 s1, the values for A; B, and C are based on
[10], C¼ 0:3; k0¼ 1:55 lm.
Fig. 1 shows the resulting saturation charac-teristics by plotting gain versus input energy for
the bias current of the SOA I¼ 100 mA and the
input pulse widths T0¼ 20 ps (curve a) and 2 ps
(curve b), respectively, where all the physical mechanisms mentioned above have been included. Clearly, the 3-dB saturation energy (defined as the input energy for which the gain is half the unsat-urated value) is lower for 2 ps as compared to 20 ps pulse. The reason that results in the difference is mainly due to the gain compression. Calculations show that if neglecting the gain compression, the difference is negligible.
In Fig. 2, the variations of the rising and falling time with the bias current of the SOA have been
plotted for Pin¼ 10 mW after focusing on different
physical mechanisms, where in figure (a) T0¼ 20 ps
and in (b) T0¼ 2 ps, respectively. From this
dia-gram, it can be seen, as expected, that the rising time of the amplified pulse by SOA is usually shorter than that of the input pulse and the falling time is longer than that of the input pulse; with the increase of the bias current of the SOA, the rising time further shortens and the falling time further lengthens. These behaviors observed are well known and are mainly due to the gain saturation induced by the depletion of the carriers owing to the stimulated emission, as observed in [1]. The gain compression caused by the carrier heating and the spectral hole burning slows down the ris-ing time shortenris-ing and the fallris-ing time lengthen-ing. For an input pulse with several picoseconds width, the effect of the gain compression is evident, as already seen in [2]. For an input pulse in the tens of picosecond width, the effect of the gain com-pression is slight owing to a relative small char-acteristic time (50–100 fs for spectral hole burning and 700–1.3 ps for carrier heating) of these intra-band processes, which is similar to the result of [3]. From this diagram, it can also be concluded that the gain asymmetry and shift of the SOA, which has been considered in wavelength conversion or WDM systems [10–12] and however is not, to our knowledge, included in investigating the temporal change of the pulse by the SOA, has an important influence on the rising and falling time. The effect of the gain asymmetry and shift depends on the input pulse wavelength and the gain distribution of
Fig. 1. Amplifier gain versus input energy for the bias current of the SOA I¼ 100 mA with pulse widths T0¼ 20 ps (curve a)
the SOA. In this paper, the input pulse wavelength and the gain peak wavelength at transparency have been taken as 1.55 lm. As a result, with the increase of the bias current from the transparency current, since the gain profile shifts to short wavelengths, the acquired gain of the input pulse will reduce compared to the case where the gain asymmetry and shift is not considered, which in-evitably weakens the gain saturation effect and
then slows down the rising time shortening and the falling time lengthening with the increase of the bias current.
In Fig. 3, the variations of the rising and falling time with the length of SOA have been plotted
under different physical mechanisms for
N¼ 4:0 1024 m3, P
in¼ 10 mW, where in figure
(a) T0¼ 20 ps and in (b) T0¼ 2 ps, respectively.
From this diagram, it can be seen that with the increase of the length of the SOA, the rising time shortens and the falling time lengthens, as showed in [9]. Because in calculations the carrier density N maintains a constant, enlarging the SOAÕs length is equivalent to fix the length of the SOA and
Fig. 3. Variations of the rising and falling time with the length of SOA for N¼ 4:0 1024m3, P
in¼ 10 mW, where in figure
(a) T0 ¼ 20 ps and in (b) T0¼ 2 ps, respectively.
Fig. 2. Variation of the rising and falling time with the bias current of the SOA for Pin¼ 10 mW after focusing on different
physical mechanisms: (––), without gain compression, gain asymmetry and shift (i.e., e; a1; a2and a3are taken as zero); (j),
with gain compression, without gain asymmetry and shift (i.e., e has a value, a1; a2and a3are taken as zero); (–), without gain
compression, with gain asymmetry and shift (i.e., e¼ 0; a1; a2
and a3 have values); (d), with gain compression, gain
asym-metry and shift (i.e., e; a1; a2and a3have values), where in figure
increase the biased current of the SOA, then a similar tendency to Fig. 2 can be observed.
It is well known that the input pulse peak power severely affects the amplified process. In Fig. 4, for
the bias current of the SOA I¼ 100 mA, the
de-pendence of the rising and falling time on the input pulse peak power has been shown after consider-ing different physical mechanisms, where in figure
(a) T0¼ 20 ps and in (b) T0¼ 2 ps, respectively.
From this diagram, it can be found that with the increase of the input pulse peak power, the rising time of the amplified pulse shortens and the falling time lengthens, which is because the increase of the
input power peak power speeds the gain saturation effect. At the meantime, for an input pulse with several picoseconds width, the effect of the gain compression enhances with the increase of the in-put pulse peak power. If one neglects the gain compression, the gain asymmetry and shift, and compares the results with the case including the gain compression, without the gain asymmetry and shift, an approximate 0.8 ps difference of the
rising time for Pin¼ 1 W can be seen [3].
From above results and [2,3], one can predict that the input pulse width of course takes an effect on the rising and falling time. In Fig. 5, the rising and falling time (normalized to T0Þ vs. T0have been
shown under different physical mechanisms for
I ¼ 100 mA, Pin¼ 10 mW. From this diagram, it
can be seen that a wider pulse raises slightly the shortening degree of the rising time and raises largely the lengthening degree of the falling time. Also, a wider pulse weakens the effect of the gain compression, which is in agreement with general insights on the gain compression [2,3].
Acknowledgements
Guang-Qiong Xia and Zheng-Mao Wu ac-knowledge the support from the Commission of Science and Technology of Chongqing City of P.R. China and the Key Project of Chinese
Fig. 4. Dependence of the rising and falling time on the input pulse peak power after considering different physical mecha-nisms for I¼ 100 mA, where in figure (a) T0¼ 20 ps and in (b)
T0¼ 2 ps, respectively.
Fig. 5. Rising and falling time (normalized to T0) vs. T0under
Ministry of Education (Grant No. 03140), and would like to express thanks to Prof. Dr. J. Chen for helpful discussions and suggestions. Moreover, the authors would like to express thanks to the reviewers of this paper for their fruitful suggestions. References
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