IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 46, NO. 3, MARCH 2010 421
Analysis of an Optically Injected Semiconductor
Laser for Microwave Generation
Sze-Chun Chan, Member, IEEE
Abstract—The nonlinear dynamical period-one oscillation of an
optically injected semiconductor laser is investigated analytically.
The oscillation is commonly observed when the injection is mod-
erately strong and positively detuned from the Hopf bifurcation
boundary. The laser emits continuous-wave optical signal with pe-
riodic intensity oscillation. Since the oscillation frequency is widely
tunable beyond the relaxation oscillation frequency, the system
can be regarded as a high-speed photonic microwave source. In
this paper, analytical solution of the oscillation is presented for the
first time. By applying a two-wavelength approximation to the rate
equations, we obtain mathematical expressions that characterize
the oscillation. The analysis explains the physical origin of the
periodic intensity oscillation as the beating between two wave-
lengths, namely, the injected wavelength and the cavity resonance
wavelength. As the injection strength increases, the optical gain
reduces, the cavity is red-shifted through the antiguidance effect,
and so the beat frequency increases continuously. The theoretical
analysis is useful for designing the system for photonic microwave
applications.
Index Terms—Injection-locked oscillators, microwave genera-
tion, nonlinear dynamics, optical injection, semiconductor lasers.
I. INTRODUCTION
S
INGLE-MODE semiconductor lasers subject to constant
optical injection have been of great interest in microwave
photonics [1]–[9]. Under different operating conditions, the
laser exhibits a number of dynamical states such as stable
locking, four-wave mixing, period-one oscillation, period-two
oscillation, quasi-periodic oscillation, and chaotic oscillation.
The simplest state of stable locking has been applied for
modulation bandwidth enhancement, chirp reduction, and
noise suppression; while the most complicated state of chaotic
oscillation has been used in secure communication and chaotic
ranging [10]–[12].
In between the two extremes lies the period-one oscillation
state. The state is typically found in a large region of the param-
eter space, where the injection strength is moderately strong
and the injection frequency is positively detuned from the
so-called Hopf bifurcation boundary. In the period-one state,
the output intensity of the laser exhibits high-speed single-pe-
riod oscillation. The oscillation frequency can be continuously
Manuscript received April 26, 2009; revised June 26, 2009. Current version
published February 24, 2010. This work was supported by a Grant from City
University of Hong Kong (Project No. 7200110) and a Grant from the Research
Grant Council of Hong Kong, China (Project No. CityU 111308).
The author is with the Electronic Engineering Department, City University
of Hong Kong, Hong Kong, China (e-mail: scchan@cityu.edu.hk).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/JQE.2009.2028900
tuned far beyond the relaxation resonance frequency of the
laser. The period-one oscillation can be optically controlled,
easily locked, and deeply modulated. As a photonic microwave
source, the laser in period-one oscillation has been applied for
narrow-linewidth microwave signal generation, radio-over-fiber
(RoF) subcarrier transmission, wavelength conversion, signal
AM-to-FM conversion, and remote target detection [13]–[17].
Despite the many potential applications of period-one os-
cillation, most related studies have been conducted through
numerical simulations only. The nonlinear nature of the laser
system prohibits exact analytical investigations. There have
been a few excellent reports that qualitatively explained the
oscillation using perturbation analysis, multi-time scale anal-
ysis, or bifurcation analysis [18]–[23]. However, quantitative
characterization of the period-one oscillation for microwave
generation, to the best of our knowledge, has not been reported.
In this paper, analytical solution of the oscillation based on a
two-wavelength approximation is reported for the first time. The
laser emission is approximated to be comprised of two domi-
nating wavelengths. We obtain theoretical results that relate the
oscillation characteristics to the operating conditions, intrinsic
laser parameters, and injection parameters. In particular, math-
ematical expressions are obtained for the microwave power, op-
tical power, and injection strength, where the injection detuning
frequency and the period-one oscillation frequency are treated
as input variables.
Most importantly, the results explain the physical mechanism
behind the period-one oscillation. The oscillation can be viewed
as the beating of two dominating wavelengths. One is regener-
ated from the optical injection while the other is emitted near
the cavity resonance wavelength. When the injection strength
increases, the optical gain normally decreases due to saturation,
the cavity resonance is then red-shifted through the antiguid-
ance effect. As a result, the beat frequency increases contin-
uously with the injection strength. The role of the cavity res-
onance shifting has been speculated and simulated previously
[14], [24], [25], but it is confirmed and evaluated by our anal-
ysis for the first time. In fact, the mechanism is only approxi-
mately valid when the antiguidance factor is large and the gain
compression factor is small.
The well-established rate-equation model is used throughout
the analysis. The model has been proposed based on funda-
mental laser physics. Thus far, numerical simulations on the rate
equations have consistently yielded excellent quantitative agree-
ment with experiments, which have been conducted extensively
over many years [1], [2], [6], [7], [9]. Building on these findings,
numerical simulations are employed to verify our analytical re-
sults in the present paper. Nevertheless, all of our laser param-
eters were obtained experimentally. They were extracted from
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422 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 46, NO. 3, MARCH 2010
a 1.3- m laser using a four-wave mixing parameter extraction
technique [26], [27]. Therefore, we focus on establishing our
theoretical results using the realistic simulation results as our
reference. The results are in good agreement.
Following this introduction, the theoretical model is pre-
sented in Section II. Then, the two-wavelength approximation
is presented in Section III. Analytical results are compared to
numerical results in Section IV, which are then followed by
discussions and conclusion in Sections V and VI, respectively.
II. T
HEORETICAL
MODEL
Semiconductor lasers are dynamically class B lasers which
do not require consideration of the polarization [3], [24]. The
dynamical behavior is fully described by the temporal evolu-
tion of the complex optical field and the charge carrier density.
So a single-mode semiconductor laser under constant, coherent
optical injection is a three-dimensional system, which can be
described by state variables
. Here, is the com-
plex intracavity field amplitude in reference to the optical fre-
quency of the injection and
is the charge carrier density of
the laser. For simplicity, the state variables are normalized to
become
and , where
and are the free-running values of and , respectively.
Note that
is complex while is real.
The laser is controlled by the optical injection parameters
, which denote the injection strength and the injection
detuning frequency, respectively. The injection strength is de-
fined as the injected field strength normalized to the emitted
field strength of a free-running laser. The injection frequency
detuning is defined as the offset frequency of the injection mea-
sured from the free-running frequency of the laser.
By normalizing the established model [14], [24], [26], the rate
equations that govern
under optical injection of
are given by
(1)
(2)
where the gain
is given by
(3)
In (1)–(3),
is the cavity decay rate, is the spontaneous car-
rier relaxation rate,
is the differential carrier relaxation rate,
is the nonlinear carrier relaxation rate, is the antiguidance
factor,
is the confinement factor of the optical mode inside
the gain medium, and
is the normalized bias current above
threshold. These dynamical parameters are fixed as long as the
bias and the temperature are kept constant. They are indepen-
dent of the injection parameters and the laser dynamics. The
gain g depends on both carrier and photon densities [14], [28].
The expression in (3) is obtained by noting that the free-running
photon density equals
. Detailed discussion on the
dynamical parameters can be found in the literatures [26], [29].
The values of the parameters are summarized in Table I. The
laser is biased at
and emits about 4.5 mW of optical
TABLE I
S
UMMARY OF
SYMBOLS
power. The relaxation resonance frequency is
, which is equal to GHz. It should be men-
tioned that, without optical injection, the laser cannot be di-
rectly modulated much beyond
using conventional methods.
However, with optical injection, the laser can be driven into pe-
riod-one oscillation at frequencies several times higher than
.
Throughout this paper, both theoretical analysis and numer-
ical simulation are based on the above rate-equation model in
(1)–(3). All simulation results are obtained from the second-
order Runge-Kutta numerical integration method with time step
and time span of 238 fs and 125 ns, respectively. Besides, ac-
cording to (3),
represents the gain compression effect of the
laser. While the complete model with
is more consistent with
experiments, it is sometimes ignored in related studies by set-
ting
[30], [31]. For completeness, both cases with and
without
will be examined in the rest of the paper.
Different laser dynamics are observed over the injection
parameter space
. Through comprehensive simulations,
Figs. 1(a) and (b) show various dynamic regions for
and , respectively. The figures are mappings
of the sideband suppression ratio
, which will be discussed
in the next section. At this point, it suffices to observe the
clear boundaries between different dynamic regions. These
boundaries can also be determined by bifurcation analysis and
small-signal analysis [7], [24]. In each map, there are small re-
gions of complicated nonlinear dynamics including period-two
oscillation, high-order periodic, quasi-periodic, chaos, and
unlocked dynamics. There is also a region of stable locking that
is bounded from above by a Hopf bifurcation line. When
is increased above the Hopf bifurcation line, a large region of
period-one oscillation is found. The laser exhibits single-period
oscillation at a tunable frequency
. The corresponding inten-
sity oscillation can be converted into microwave signal using a
high-speed photodetector. Thus, it is important to analyze the
characteristics of the period-one oscillations.
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CHAN: ANALYSIS OF AN OPTICALLY INJECTED SEMICONDUCTOR LASER FOR MICROWAVE GENERATION 423
Fig. 1. Mappings of dynamic regions of the optically injected semiconductor
laser with (a)
=0
and (b)
=1
:
91
2
10 s
. The sideband suppression
ratio
R
is presented in color.
As an example of period-one oscillation, a simulated optical
spectrum is shown as the solid curve in Fig. 2. The optical
frequency axis is offset to the free-running frequency of the
laser. The injection parameters are set as
GHz . The spectrum consists of the regenerative compo-
nent at
and sidebands equally separated by , where
GHz in this case. Additionally, the dashed curve of
Fig. 2 shows the spectrum when we set
, while keeping
everything else unchanged. Comparing the two spectra reveals
that
does not have significant impact on the oscillation.
III. T
WO-WAVELENGTH APPROXIMATION
Under period-one oscillation, the state variables can
be expressed in terms of Fourier components at multiples of the
oscillation frequency
. Due to the laser nonlinearities, it is
difficult to solve for all the components. We seek an approx-
imate solution to the problem by retaining only the strongest
of these components. According to Fig. 2, the strongest optical
components for
are at offset frequencies and .It
is because the former is the direct regeneration of the injec-
tion while the latter is closest to the original cavity resonance
Fig. 2. Simulated optical spectrum of a period-one oscillation state at
=
0
:
32
and
=2
2
30
GHz. The generated period-one oscillation frequency
is
=2
2
38
GHz. Solid and dashed curves are obtained with and without
, respectively.
at the zero offset frequency. In fact, these two components con-
sistently dominate the optical spectrum even when the injection
parameters are changed.
In order to quantify such observation, we define the sideband
suppression ratio
by treating the laser as a two-wavelength
light source. The weaker of the two dominating components is
first chosen. It is then compared with the strongest component
among the rest of the sidebands. The power difference in decibel
is defined as
. For instance, in the spectrum shown as the solid
curve in Fig. 2, the component at
is slightly weaker
than that at
. It is thus compared with the strongest sideband,
which turns out to be located at
. The sideband sup-
pression ratio is thus determined as
. Mappings of
over the injection parameter space are shown in Fig. 1. Abrupt
changes occur at the boundaries between different dynamics be-
cause of sudden changes in the oscillation frequency and the
spectrum. Within the period-one oscillation region,
gradually
increases with both
and . Comparing the two mappings re-
veals that the gain compression effect of
tends to reduce .
For the cases with and without
, is always greater than 10
dB when
is above 31 and 0 GHz, respectively. There-
fore, it is reasonable to keep only the strongest components
and over a large range of injection conditions. Such
two-wavelength approximation will be used throughout the the-
oretical analysis. It should be pointed out that as
is reduced
by decreasing
, the two-wavelength approximation and the
performance of the analysis degrade. For example, the anal-
ysis does not yield good results for our laser when
for
. Extension of the analysis beyond the two-wavelength
approximation is beyond the current scope.
Applying the two-wavelength approximation, the field
is
written as
(4)
where
is regenerated from the injection and is generated
by the laser in period-one oscillation (Fig. 2). It follows that
the output intensity of the laser, after normalizing to the free-
running intensity, is given by
(5)
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424 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 46, NO. 3, MARCH 2010
where is the time-averaged intensity and
is the complex amplitude of the intensity oscillation
at
. Here, c.c. stands for complex conjugate. Similarly, since
the oscillation frequency
is typically much higher than the
spontaneous carrier relaxation rate
(6)
where
is the time-averaged carrier density and is the os-
cillation amplitude of the carrier density at the period-one fre-
quency. The harmonics of the oscillation are neglected.
The approximations in (4)–(6) will be adopted throughout the
analysis. The analysis is based on the rate equations (1) and (2).
They are rewritten as
(7)
(8)
(9)
(10)
where
(11)
and
(12)
(13)
Inspecting (12) and (1) reveals that
is a very important phys-
ical quantity. It is the shift of the cavity resonance frequency
under the influence of optical injection. Usually, optical injec-
tion reduces the charge carrier density of the cavity. The time-
averaged gain
thus reduces from its free-running value .
The reduction of gain induces, through the antiguidance factor
,
an increase of the refractive index. The corresponding effective
cavity length increases and so the cavity resonance frequency
is shifted by
. The frequency shift usually becomes more and
more negative as the injection strength increases. The frequency
plays an important role in the tunability of the period-one oscil-
lation frequency, as the next section will show.
IV. RESULTS
In this section, the period-one oscillation characteristics are
derived using the simplified rate equations (7)–(10). For any
given injection detuning frequency
, the generated period-one
oscillation frequency
can be tuned by varying the injection
Fig. 3. Regenerative optical component
j
a
j
as a function of the period-one
oscillation frequency
. The solid curves represent analytical results while the
closed symbols represent numerical results, where
(2
) =20
GHz (cir-
cles), 30 GHz (triangles), 40 GHz (squares), and 50 GHz (diamonds). (a)
=
0
. (b)
=1
:
91
2
10 s
.
strength . Thus, both and are treated as input variables in
the following derivations of the other quantities, which include
the regenerative optical component
, the cavity resonance
frequency shift
, the average output intensity , the generated
period-one optical component
, the generated microwave
power
, and the required injection strength .
The resultant analytical expressions are verified through
extensive numerical simulations. The results are presented in
Figs. 3–8. The solid curves are from analytical results while
the closed symbols are from numerical results. The analytical
results are obtained from directly evaluating the expressions to
be presented in this section. The numerical results are obtained
from the numerical integration presented in Section II. In each
figure, the results are shown for
, 30, 40, and
50 GHz, which are marked by circles, triangles, squares, and
diamonds, respectively. For completeness, both cases with
and are shown in every part (a) and
part (b), respectively.
A. Cavity Shift
By combining (8) and (10), we can eliminate
using
to yield
(14)
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CHAN: ANALYSIS OF AN OPTICALLY INJECTED SEMICONDUCTOR LASER FOR MICROWAVE GENERATION 425
Fig. 4. Cavity resonance frequency shift
as a function of the period-one os-
cillation frequency
. The solid curves represent analytical results while the
closed symbols represent numerical results, where
(2
) =20
GHz (cir-
cles), 30 GHz (triangles), 40 GHz (squares), and 50 GHz (diamonds). (a)
=
0
. (b)
=1
:
91
2
10 s
.
for . Solving the real and imaginary parts, we obtain
(15)
and
(16)
Both
and depend strongly on . Some additional
approximations are helpful in simplifying the results. For most
semiconductor lasers,
, , and
even under injection. Also, is always
the fastest rate among the parameters in the model and is much
higher than the magnitude of the cavity resonance frequency
shift
. As a result, we assume that (i)
and (ii) . These are the only extra assumptions
applied in the analysis other than the two-wavelength approx-
imation. They allow evaluation of
and as functions of
using (15) and (16), respectively.
Fig. 3 shows the regenerative optical component
as a
function of
under different . The analytical results in solid
curves agree well with the numerical results in closed symbols.
Each curve follows a similar trend. When
is equal to ,
the laser must be under simple four-wave mixing, where the
laser is just slightly perturbed by the injection. This occurs if the
injection strength is infinitesimally small and so
is zero.
Fig. 5. Average optical output intensity
s
as a function of the period-one
oscillation frequency
. The solid curves represent analytical results while
the closed symbols represent numerical results, where
(2
)
=20
GHz
(circles), 30 GHz (triangles), 40 GHz (squares), and 50 GHz (diamonds).
(a)
=0
. (b)
=1
:
91
2
10 s
.
Fig. 6. Generated optical component
j
a
j
as a function of the period-one os-
cillation frequency
. The solid curves represent analytical results while the
closed symbols represent numerical results, where
(2
) =20
GHz (cir-
cles), 30 GHz (triangles), 40 GHz (squares), and 50 GHz (diamonds). (a)
=
0
. (b)
=1
:
91
2
10 s
.
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