Dynamic saturation in Semiconductor Optical Amplifiers: accurate model, role of carrier density, and slow light.
ABSTRACT We developed an improved model in order to predict the RF behavior and the slow light properties of the SOA valid for any experimental conditions. It takes into account the dynamic saturation of the SOA, which can be fully characterized by a simple measurement, and only relies on material fitting parameters, independent of the optical intensity and the injected current. The present model is validated by showing a good agreement with experiments for small and large modulation indices.
- [show abstract] [hide abstract]
ABSTRACT: The experimental demonstration and the far-field pattern characterization of an optically controlled phased-array antenna are described. It operates between 2.5 and 3.5 GHz and is made of 16 radiating elements. The optical control uses a two-dimensional architecture based on free-space propagation and on polarization switching by N spatial light modulators of p × p pixels. It provides 2(N-1) time-delay values and an analog control of the 0 to 2π phase for each of the p × p signals feeding the antenna (N = 5, p = 4).Applied Optics 09/1996; 35(26):5293-300. · 1.69 Impact Factor
- [show abstract] [hide abstract]
ABSTRACT: Microwave photonic filters are photonic subsystems designed with the aim of carrying equivalent tasks to those of an ordinary microwave filter within a radio frequency (RF) system or link, bringing supplementary advantages inherent to photonics such as low loss, high bandwidth, immunity to electromagnetic interference (EMI), tunability, and reconfigurability. There is an increasing interest in this subject since, on one hand, emerging broadband wireless access networks and standards spanning from universal mobile telecommunications system (UMTS) to fixed access picocellular networks and including wireless local area network (WLAN), World Interoperability for Microwave Access, Inc. (WIMAX), local multipoint distribution service (LMDS), etc., require an increase in capacity by reducing the coverage area. An enabling technology to obtain this objective is based on radio-over-fiber (RoF) systems where signal processing is carried at a central office to where signals are carried from inexpensive remote antenna units (RAUs). On the other hand, microwave photonic filters can find applications in specialized fields such as radar and photonic beamsteering of phased-arrayed antennas, where dynamical reconfiguration is an added value. This paper provides a tutorial introduction of this subject to the reader not working directly in the field but interested in getting an overall introduction of the subject and also to the researcher wishing to get a comprehensive background before working on the subject.Journal of Lightwave Technology 02/2006; 24(1):201- 229. · 2.56 Impact Factor
- [show abstract] [hide abstract]
ABSTRACT: The ability to manipulate the speed of light has recently become one of the most exciting emergent topics in optics. There are several experimental demonstrations showing the capability to slow down light more than six orders of magnitude in a variety of media, ranging from atomic vapor, solid state crystal, to semiconductors. These results have led to intensive research into new materials, devices, and system studies that examine their impact to new applications. It is believed that we are on the verge of a dramatic change in the way we envision and construct communication, processing and control systems. One direct application of slow and fast light devices is in the area of communications. One grand challenge remaining in information technology today is to store and buffer optical signals directly in optical format. As such, optical signals must be converted to electronic signals to route, switch, or be processed. This resulted in significant latencies and traffic congestions in current networks. In addition, keeping the data in optical domain during the routing process can greatly reduce the power, complexity and size of the routers. To this end, a controllable optical delay line can effectively function as an optical buffer, and the storage is proportional to the variability of the group velocity. In addition to optical buffers, slow and fast light devices can be used as tunable true-time delay elements in microwave photonics, which are important for remotely controlling phased array antenna. Other novel applications include nonlinear optics, optical signal processing, and quantum information processing. There are various approaches that can be used to vary the optical group velocity. Ultraslow or fast group velocity may result from a large material dispersion, waveguide dispersion, or both. In this paper, the authors provide a review of recent progress of slow and fast light using semiconductor devices. Specifically, they will discuss results obtained using se- miconductor quantum-well/quantum-dot absorber and optical amplifiers. Slow and fast light are controllable electrically by changing the bias current or voltage as well as optically by changing the pump laser intensity and wavelength. Delay-bandwidth tradeoff and other figures of merits are analyzedJournal of Lightwave Technology 01/2007; 24(12):4642-4654. · 2.56 Impact Factor
arXiv:0910.4064v2 [physics.optics] 19 Nov 2009
Dynamic saturation in semiconductor
optical amplifiers: accurate model, role
of carrier density, and slow light
Perrine Berger1,2, Mehdi Alouini1,3, J´ erˆ ome Bourderionnet1,
Fabien Bretenaker2, and Daniel Dolfi1
1Thales Research & Technology, 1 av. Augustin Fresnel, 91767 Palaiseau Cedex, France
2Laboratoire Aim´ e Cotton, CNRS-Universit´ e Paris Sud 11, Campus d’Orsay, 91405 Orsay
3Institut de Physique de Rennes, UMR CNRS 6251, Campus de Beaulieu, 35042 Rennes
behaviorand the slow light propertiesof the SOA valid for any experimental
conditions. It takes into account the dynamic saturation of the SOA, which
can be fully characterized by a simple measurement, and only relies
on material fitting parameters, independent of the optical intensity and
the injected current. The present model is validated by showing a good
agreement with experiments for small and large modulation indices.
We developed an improved model in order to predict the RF
© 2009 Optical Society of America
OCIS codes: (250.5980) Semiconductor optical amplifiers; (070.6020) Continuous optical sig-
References and links
1. J. Yao, “Microwave Photonics,” J. Lightwave Technol. 27, 314-335 (2009).
2. D. Dolfi, P. Joffre, J. Antoine, J-P. Huignard, D. Philippet, and P. Granger, “Experimental demonstration of a
phased-array antenna optically controlled with phase and time delays,” Appl. Opt. 35, 5293-5300 (1996).
3. J. Capmany, B. Ortega, and D. Pastor, “A Tutorial on Microwave Photonic Filters,” J. Lightwave Technol. 24,
4. C. J. Chang-Hasnain and S. L. Chuang, “Slow and Fast Light in Semiconductor Quantum-Well and Quantum-Dot
Devices,” J. Lightwave Technol. 24, 4642-4654 (2006).
5. H. Su, and S. L. Chuang, “Room temperature slow and fast light in quantum-dot semiconductor optical ampli-
fiers,” App. Phys. Lett. 88, 061102 (2006).
6. A. V. Uskov, F. G. Sedgwick, and C. J. Chang-Hasnain, “Delay Limit of Slow Light in Semiconductor Optical
Amplifiers,” IEEE Photon. Technol. Lett. 18, 731-733 (2006).
7. B. Pesala, F. Sedgwick, A. Uskov, and C. Chang-Hasnain, “Ultrahigh-bandwidth electrically tunable fast and
slow light in semiconductor optical amplifiers,” J. Opt. Soc. Am. B 25, C46-C54 (2008).
8. L. Th´ evenaz, “Slow and fast light in optical fibres,” Nature Photonics 2, 474-481 (2008).
9. P.-C. Ku, F. Sedgwick, C.J. Chang-Hasnain, P. Palinginis, T. Li, H. Wang, S.-W. Chang, and S-L. Chuang, “Slow
light in semiconductor quantum wells,” Opt. Lett. 29, 2291-2293 (2004).
10. R. Boula-Picard, M. Alouini, J. Lopez, N. Vodjdani, and J.-C. Simon, “Impact of the Gain Saturation Dynamics
in Semiconductor Optical Amplifiers on the Characteristics of an Analog Optical Link,” J. Lightwave Technol.
23, 2420-2426 (2005).
11. J. Mørk, R. Kjr, M. van der Poel, and K. Yvind, “Slow light in a semiconductor waveguide at gigahertz frequen-
cies,” Opt. Express 13, 8136-8145 (2005).
12. S. S. Maicas, F.¨Ohman, J. Capmany, and J. Mørk, “Controlling Microwave Signals by Means of Slow and Fast
Light Effects in SOA-EA Structures,” IEEE Photon. Technol. Lett. 19, 1589-1591 (2007).
13. Y. Chen, and J. Mørk, “Broadband Microwave Phase Shifter based on High Speed Cross Gain Modulation in
Quantum Dot Semiconductor Optical Amplifiers,” in International Topical Meeting on Slow and Fast Light,
2009 OSA Technical Digest (Optical Society of America, 2009).
14. G. P. Agrawal, “Population pulsations and nondegenerate four-wave mixing in semiconductor lasers and ampli-
fiers,” J. Opt. Soc. Am. B 5, 147-159 (1988).
15. E. Zhou, X. Zhang, and D. Huang, “Evaluating characteristics of semiconductor optical amplifiers using optical
pumping near the transparency,” J. Opt. Soc. Am. B 24, 2647-2657 (2007).
16. A. Capua, V. Mikhelashvili, G. Eisenstein, J. P. Reithmaier, A. Somers, A. Forchel, M. Calligaro, O. Parillaud,
and M. Krakowski, “Direct observation of the coherent spectral hole in the noise spectrum of asaturated InAs/InP
quantum dash amplifier operating near 1550 nm,” Opt. Express 16, 2141-2146 (2008).
17. J. Kim, M. Laemmlin, C. Meuer, D. Bimberg, and G. Eisenstein, “Static Gain Saturation Model of Quantum-Dot
Semiconductor Optical Amplifiers,” IEEE J. Quantum Electron. 44, 658-666 (2008).
18. S.-W. Chang, P. K. Kondratko, H. Su, and S. L. Chuang, “Slow Light Based on Coherent Population Oscillation
in Quantum Dots at Room Temperature,” IEEE J. Quantum Electron. 43, 196-205 (2007).
19. M. J. Connelly, “Wideband Semiconductor Optical Amplifier Steady-State Numerical Model,” IEEE J. Quantum
Electron. 37, 439-447 (2001).
20. Y. Chen, W. Xue, F. Ohman, and J. Mork, “Theory of optical-filtering enhanced slow and fast light effects in
semiconductor optical waveguides,” Lightwave Technology, Journal of 23, 3734-3743 (2008).
21. T. Mukai and T. Saitoh, “Detuning characteristics and conversion efficiency of nearly degenerate four-wave
mixing in a 1.5-µm traveling-wave semiconductor laser amplifier Quantum Electronics,” Quantum Electronics,
IEEE Journal of 16, 865-875 (1990)
22. A. Haug, “Evidence of the importance of auger recombination for InGaAsP lasers,” IEE Electron. Lett. 20, 85-86
23. E. Rosencher and B. Vinter, Optoelectronics (Cambridge, 2002).
24. M. Shtaif, B. Tromborg, and G. Eisenstein, “Noise spectra of semiconductor optical amplifiers: relation between
semiclassical and quantum descriptions,” Quantum Electronics, IEEE Journal of 34, 869-878 (1998).
25. A. Ouacha, Q. Chen, M. Willander, R. A. Logan, and T. Tanbun-Ek, “Recombination process and its effect on the
dc performance of inp/ingaas single-heterojunction bipolar transistors,” Journal of Applied Physics 73, 444-4447
26. L. Y. Leu, J. T. Gardner, and S. R. Forrest, “A high-gain, high-bandwidth in0.53ga0.47as/inp heterojunction pho-
totransistor for optical communications,” Journal of Applied Physics, 69, 1052-1062 (1991).
27. E. A. J. M. Bente, Y. Barbarin, M. J. R. Heck, and M. K. Smit, “Modeling of integrated extended cavity
inp/ingaasp semiconductor modelocked ring lasers,” Optical and Quantum Electronics, 40, 131-148 (2008).
28. M. Petrauskas, S. Juodkazis, V. Netikis, M. Willander, A. Ouacha, and B. Hammarlund, “Picosecond carrier
dynamics in highly excited ingaas/inp/ingaasp/inp structures,” Semiconductor science and technology, 7, 1355-
29. E. Shumakher, S. Dill, and G. Eisenstein, “Optoelectronic Oscillator Tunable by an SOA Based Slow Light
Element,” Lightwave Technology, Journal of 27, 4063-4068 (2009).
30. S. O Duill, R. F. O’Dowd, and G. Eisenstein, “On the role of high-order coherent population oscillations in slow
and fast light propagation using semiconductor optical amplifiers,” Selected Topics in Quantum Electronics,
IEEE Journal of 15, 578-584 (2009).
31. P. Berger, J. Bourderionnet, M. Alouini,F. Bretenaker and D. Dolfi, “Theoretical Study of the Spurious-Free
Dynamic Range of a Tunable Delay Line based on Slow Light in SOA,”, Optics Express 27, 20584-20597 (2009).
The generation of continuously tunable optical delays is a key element in microwave photon-
ics. Among the targeted applications, one can quote the filtering of microwave signals, the
synchronization of optoelectronics oscillators, and the control of optically fed phased array an-
tennas [1, 2, 3]. With these applications in view, large efforts are currently done in order to
develop delay lines based on slow and fast light effects [4, 5, 6, 7, 8]. To date, one of the most
mature approaches for integration in real field systems is that based on Coherent Population
Oscillations (CPO) in semiconductor structures [9, 10, 11]. This approach offers compactness,
continuous tunability of the delay throughinjected current control, and possible high-level par-
allelism [12, 13]. Obviously, the implementation of CPO effects in microwave photonics delay
lines reliesonaccuratetheoreticaldescriptionofthe underlyingmechanismsin ordertodevelop
reliable predictive models. Numerous theoretical models have been developed in the past few
years to describe CPO effects in Semiconductor Optical Amplifiers (SOAs) [14, 15, 16, 17].
They are usually based on a semi-classical description of the interaction between the carriers
and the input optical fields. These models offer a comprehensiveunderstanding of the gain sat-
uration dynamics and associated group index changes. However, on the one hand, a complete
model would require a detailed knowledge of the geometrical and material parameters of the
semiconductor structure [18, 19]. Unfortunately, most of them are unknown especially when
the SOA under consideration is a commercially available device. On the other hand, others,
simpler, assumed that both the saturation power and the carrier recombinationlifetime are con-
stant [5, 14, 20, 21]. This assumption applies when the SOA is operated at a fixed injection
current [22, 23]. However, the injection current and the input optical power have to be tuned
overa wide rangein orderto controlthe speed of light into the SOA: consequentlythis assump-
tion restricts the predictive capability of a model describing microwave-photonics delay lines
using slow light in SOA.
In this paper we derive an improved model that enables to predict the RF gain compression,
the RF phase delay, and the optical group delay and which is valid for all experimental condi-
tions for a given component. Furthermore, we show that the detailed knowledge of the inner
geometrical and material characteristics of the SOA is not requiredprovidedthat some prelimi-
nary and easy characterizationmeasurementsare conducted.This model is then experimentally
We consideran opticalcarrier modulatedby an RF signal and injected in a travelingwave SOA.
The total field is then composed of the optical carrier of complex amplitude E0and two side-
bands of complex amplitudes E1and E2. The total optical field E is normalized to include the
factor√ε0n0c0, i.e., the optical intensity is given by Iopt(z,t) =1
under small RF signal approximation. U is the DC component of the intensity, M =
The local equations for the propagation of the optical field Etotaland the evolution of carrier
density N inside the SOA are :
0) is the beat-note term at the RF frequency Ω.
where γ holds for the internal losses of the SOA, Γg(z,t) is the material modal gain, τsis
the carrier lifetime, I is the injected current, V is the volume of the active region, and ω is
the pulsation of the optical carrier E0. We introduce N(z,t) =¯N(z)+∆N(z)e−iΩt+c.c. and
g(z,t) = g(N(z,t)) = ¯ g(¯N(z)) +a(¯N(z))∆N(z,t)e−iΩt+c.c. where a is the differential gain
and Eq. 2 lead to:
∂N|¯N. The wavelength of the optical carrier is fixed. Consequently, the equations Eq. 1
U [−γ +Γg(¯N)],
whereUs(¯N) is the saturation intensity defined as: Us(¯N) =
In most of the simple models, the common approach to solve equations (3) and (4) is to
consider a and τsconstant with respect to the carrier density and thus over the whole length
of the device [5, 14, 20, 21]. This approximation does however not give account of strong
saturation conditions, with high gain and carrier density variations, which typically occur in
quantum wells structures with strong carrier confinement. In this paper, we propose to consider
the carrierdensityvariationalong the propagationaxis and its influenceon a and τs. Our central
hypothesis is that a and τscan be determined as functions of the DC component of the optical
intensityU solely, allowing these dependencies to be determined from gain measurements.
Let us first suppose that we fulfill the small signal condition. In this case, the stimulated
emission is negligible comparedto the spontaneous emission, leading to the unsaturated steady
state solution of the rate equation (Eq. 1):
q L Sact
where L is the length of the SOA, Sactis the area of the active section of the SOA. Moreover,
we also suppose in this case that the carrier density¯N is constant along the SOA. These hy-
pothesis are equivalent to consider that the amplified spontaneous emission does not saturate
the gain. A verification of this assumption will be shown in section 3. Under these conditions, a
measurement of the small signal modal gain Γg0versus I will be equivalent, owing to Eq. 5, to
a determination of the modal gain Γg versus¯N/τs. Here, Γ is the ratio Sact/Sguideof the active
to modal gain areas in the SOA.
A last relationship between
function of U. It is obtained by substituting Γg(¯N
carriers rate equation (Eq. 1):
τsand U is then required to determine the modal gain Γg as a
τs) in the saturated steady state solution of the
q L Sact
where the injected current I is now fixed by the operating conditions.
Added to the previous relationship between Γg and
Γg as a function of
and the injected current I.
To solve Eq. 4, we need to express¯N as a function ofU(z)
¯N with respect to
SOA using the well-known equation :
τs, the Eq. 6 gives another expression of
τscan be known with respect to the local
Γand I. Consequently,Γg and
and I. This is equivalentto express
and I. Consequently, we model our
τsis known as a function ofU(z)
= A¯N +B¯N2+C¯N3,
where A, B, and C, which are respectively the non-radiative, spontaneous and Auger recom-
bination coefficients, are the only parameters that will have to be fitted from the experimental
Using Eq. 7 and the fact that we have provedthat¯N/τsand Γg can be considered as function
and I. This permits to replace Eqs. (3) and (4) by the following system:
and I only, we see that¯N, Γa = Γ∂g
Γaτscan also be considered as functions
Γ g0 (m−1)
0 50 100
Γ g (m−1)
Fig. 1. (a) Experimental fiber-to-fiber gain G with respect to the output optical power Pout
at a strong current (500 mA). The double arrow indicates the range of the measured ASE
output power. (b) Experimental small signal gain Γg0as a function of the injected current I
at 1535 nm, and fitted by: Γg0=C1−C2
(c) Deduced material modal gain Γg(U) as a function of the local intensityU at 500 mA.
I, withC1= 5588.7m−1andC2= 306.1A−1.m−1.
Eqs. (8) and (9) are then numerically solved: Eq. 8 gives
Eq. 9, and the microwave transfer function of the SOA S21= γiM(L)
sertion losses, is then computed. If the output power of the RF microwave signal is wanted,
index, and R and ηphare respectively the photodiode resistive load and efficiency.
The microwave complex transfer function S21fully characterizes the slow light properties of
the SOA. Indeed the optical group delay ∆τgcan be expressed as ∆τg(Ω) =arg(S21(Ω))
group index ∆ng(Ω) =c arg(S21(Ω))
It is important to note that the recombination coefficients A, B and C are the only fitting
parameters of our model. Once obtained from experimental data, they are fixed for any other
experimental conditions. Moreover, the only geometrical required parameters are the length
L of the SOA and the active area cross section Sact. The derivation of a predictive model,
independent of the experimental conditions (current and input optical power) is then possible,
provided that the simple measurements of the total losses and the small signal gain versus
the current are conducted. The above model lies in the fact that first, the spatial variations of
the saturation parameters are taken into account, and second, their values with respect to the
local optical power are deduced from a simple measurement. These keys ideas lead to a very
convenientmodel of the microwavecomplex transfer functionof the SOA, and then of the slow
light properties of the component.It can be easily used to characterize commercialcomponents
whose design details are usually unknown, as we will experimentally show in the next section.
Γ, with the initial condi-
can be then introduced into
Sact, where Pinis the optical input power.
M(0), where γi are the in-
|2, with the initial condition M(0) = 1, where m the input modulation
, and the
In order to validate our model, we studied a commercially available SOA (InP/InGaAsP Quan-
tum Well Booster Amplifier from COVEGA). The length L of this SOA is 1.50 mm and the
active area cross-section is set at 0.06 µm2. We proposed to compare the experimental and
simulated complex transfer function S21for a large set of operating conditions (Pin,I). As ex-
plained in section 2, the study of the phase arg(S21) is equivalentto the optical groupdelay. We
can then restrain our comparison to S21. In order to fully characterize the response of the SOA
through Γg(U) as described in section 2, the preliminary step consists in measuring the total
losses and the unsaturated gain Γg0(I) for different injected currents.
The total losses are measured by the following experiment: at low current, the output optical
power is measured while a strong input optical power is sent into the SOA. When the current
Fig. 2. Experimental set-up. For small modulation index m, a laser is externally modulated
by a Mach-Zehnder modulator (MZ) (a); for large modulation index, a directly modulated
laser is used (b). In both cases, the input optical power Pinis controlled through a variable
optical attenuator; two optical isolators are used before and after the SOA. The photode-
tector (PD) restitute the RF signal. The Vector Network Analyser (VNA) is calibrated with
the whole link without the SOA, in order to measure the RF transfer function of the SOA.
is low enough, the SOA is in the absorption regime: the resulting output power is an increasing
function of the input power (absorptionsaturation). When the current is above the transparency
current, the resulting output power becomes a decreasing function of the input power (gain
saturation). Between these two regimes, i.e. at transparency,the ratio between the output power
and the input power is exactly equal to the total losses. The total losses of the SOA γiexp(−γL)
are measured to be equal to −16.4dB in our case.
To measure only the unsaturated gain Γg0despite the amplified spontaneous emission, we
measured the SOA unsaturated RF gain GRF(=|S21|2) at a RF frequency Ω well above 1/τs
(typically 20 GHz). The derivationof the modal gain Γg with respect to
gain Γg0(I) is relying on the hypothesis that the amplified stimulated emission (ASE) does not
saturate the gain. In Fig. 1a, we represent the experimental fiber-to-fiber gain with respect to
the output optical power at a strong current (500 mA) and the range of the experimental output
power of the ASE. The maximum power of the ASE is equal to 1.54 dBm. Moreover,when the
small signal measurement is performed, a maximum input optical power of 80 µW was used,
corresponding to an output optical power of 8.1 dBm for the maximum current. Consequently,
both signal and ASE output power level are well below the output power required to saturate
the gain (14.2 dBm for a 3dB gain reduction). Therefore, the experimental conditions match
our preliminary assumptions. Under these conditions, Eq. 8 can be simplified and integrated,
in Fig. 2a, we used a Vector Network Analyzer (VNA) to measure the RF gain GRFfor a small
input power which does not saturate the SOA (typically 10−80µW). The unsaturated gain of
ourSOA is displayedinFig. 1b.It is empiricallyfittedbyΓg0=C1−C2
From this simple measurement and using Eq. 6, the material modal gain Γg is then known as a
function of the local intensityU/Γ inside the SOA (Fig. 1c).
The complex RF transfer function of the SOA is measured thanks to a VNA for small and
large modulation indices (set-ups in Fig. 2). In Fig. 3 and Fig. 4, we report the corresponding
RF gains, 20log|S21|, and the measured evolution of the RF phase shift, arg(S21), as a function
of the modulationfrequencyΩ. In each of these figures, the plots labeled(a) and (b) correspond
to the evolutions of the RF gains and phase shifts versus RF frequency, for different injected
currents, while the plots labeled (c) and (d) are obtained by managing the input optical power.
τsfrom the unsaturated
20 log (|S21|) (dB)
20 log (|S21|) (dB)
I=75mAI=100mA I=250mA I=450mA
Fig. 3. Low modulation index (m = 0.06) : gain and phase shift simulations (dashed line)
and experimental data (solid line) for (a) and (b): different injected currents at Pin= 0dBm,
and for (c) and (d): different optical input powers Pinat I = 500mA. The operating wave-
length was 1535nm.
In Fig. 3 and Fig. 4, the simulation results are reported in dashed line. The best
fit values for the recombination coefficients are: A = 2×109s−1, B = 1.2×10−10cm3s−1,
C = 1.8×10−31cm6s−1. These values are in the range of what can be found in the literature
for semiconductor materials [25, 26, 27, 28]. The computed complex transfer function shows
a very good agreement with the experimental data, both at small and large modulation index,
for any experimental conditions (injected current, input optical power), and with a single set
of the fitting parameters (A, B, C): our convenient model is predictive for any experimental
In order to highlight the weight of the spatial variations of the carrier density and the satura-
tion parameters, we plotted in Fig. 5a,5b the variations of the carrier density¯N along the SOA
for the different experimental situations of Fig. 3. The subsequent variations of the modal gain
Γg and the saturation parameters Ps, τsand a, with respect to¯N, are displayed in Fig. 5c,5d.
We find at least one order of magnitude of variation for almost all these parameters, which are
20 log (|S21|) (dB)
Fig. 4. Large modulation index (m > 0.6) : gain and phase shift simulations (dashed line)
and experimental data (solid line) for (a) and (b): different injected currents at Pin= 0dBm.
The operating wavelength was 1548.5nm.
nevertheless often taken constant in literature for practical models [5, 14, 20, 21]. According
to Eq. 7, this approximation can be justified when the variations of¯N along the SOA are rela-
tively not too strong, that is for moderate bias current (< 150 mA in our case) or a high bias
current, but low optical power. However, for any other condition, and especially in the case of
quantum well or quantum dots structures, it is necessary to take into account the saturation dy-
namics along the propagation to ensure good performances of the model and robustness versus
changes in experimentalconditions.Indeed,Fig. 5 shows that consideringPs, τsand a constant,
and then Γg linear with¯N, drastically limits the range of experimental conditions (Pin,I) where
such models are valid, which forces the saturation parameters to be adjusted with the current
and/or the optical input power.
Our improved model is still easy to use, even for commercial components, but despite the
hypothesis we were compelled to make, it remains valid for a large range of experimental
conditions, with a reduced set of unknown- and thus fitted- parameters. These advantages have
been achieved by taking into account the spatial variation of the saturation parameters and by
showing that their values as a functionof the local optical power can be retrieved from a simple
measurement. It ensures that the model only relies on material fitting parameters, independent
of the optical intensity and injected current.
The slow light properties are then also modeled for a large range of the input optical powers
Pinandinjected currentsI, which is essential fromthe operationalpoint of view, since the speed
of light in SOA is controlled by these two key parameters. While the applications of slow light
in SOA aretakingshape,aconvenientandaccuratemodelwith theparameterstuningthedelays
is a necessary tool to fully characterize the effect of slow light in SOA on a microwave link, or
to develop new architectures improving the slow light properties. This model could be easily
used when an optical filtering is performed after the SOA to enhance the slow light effect, as
described in [20, 29]. In this case, Eq. 9 just has to be replaced by the corresponding coupled
equations in E1and E2. Moreover, to take into account higher order coherent population os-
cillations , the present model can be generalized using equations similar to Eq. 9 for each
harmonic of the optical intensity. The determination of Γg as a function of U is slightly more
subtle in this case: it is presented in another paper, in order to study the harmonic generation
and the intermodulation products .
x 10x 10
Γ g (m−1)
x 10 x 10
Fig. 5. (a) and (b): Simulated carrier density¯N along the SOA: (a) at a fixed input optical
power (0 dBm), for various currents; (b) at a fixed current (500 mA), for various input
optical power. (c) and (d): Simulated variations with respect to the carrier density¯N of
(c) the modal gain Γg (solid line), and the modal differential gain a (dashed line); (d) the
carrier lifetime τs(solid line), and the local saturation power Ps(dashed line).
We developed an improved but still convenient model in order to predict the RF behavior and
slow light properties of the SOA, valid for any experimental conditions (input optical power,
injected current). It takes into account the spatial variations of the saturation parameters along
the SOA, whichare fullycharacterizedbythe simple measurementofthe small signal gain.The
resulting model only relies on material fitting parameters, independent of the optical intensity
and injected current. We showed a remarkably good agreement between the model and the
experimental data, at small and large modulation indices. The ease of use and the accurate
prediction obtained for any experimental conditions will be useful to characterize the effect
of slow light in SOA on a microwave link, and to develop new designs improving the slow
light properties. The key ideas of this improvedmodel can easily be used when optical filtering
is performed after the SOA. A generalization of our approach will be carried out in a next
step, in order to determine the harmonic generation, intermodulation products and spurious
free dynamic range, for a full characterization of a SOA based opto-electronic link.
The authors acknowledgethe partial support from the ”D´ el´ egationG´ en´ erale pour l’Armement”
DGA/MRIS and from the GOSPEL EC/FET project.