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IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 5, NO. 3, MAY/JUNE 1999
851
The Modulation Response of a
Semiconductor Laser Amplifier
Jesper Mørk, Antonio Mecozzi, Member, IEEE, and Gadi Eisenstein, Fellow, IEEE
Abstract—We present a theoretical analysis of the modulation
response of a semiconductor laser amplifier. We find a resonance
behavior similar to the well-known relaxation oscillation resonance found in semiconductor lasers, but of a different physical
origin. The role of the waveguide (scattering) loss is investigated
in detail and is shown to influence the qualitative behavior of
the response. In particular, it is found that a certain amount
of waveguide loss may be beneficial in some cases. Finally, the
role of the microwave propagation of the modulation signals
is investigated and different feeding schemes are analyzed. The
nonlinear transparent waveguide, i.e., an amplifier saturated to
the point where the stimulated emission balances the internal
losses, is shown to be analytically solvable and is a convenient
vehicle for gaining qualitative understanding of the dynamics of
modulated semiconductor optical amplifiers.
Index Terms—Laser amplifiers, modeling, optical modulation,
semiconductor lasers.
I. INTRODUCTION
S
EMICONDUCTOR optical amplifiers (SOA’s) have been
the subject of intense research over the past decade or so
[1]. The main characteristics of an SOA compared to other
optical amplifiers are perhaps its small size, made possible by
the huge gain available in direct-bandgap semiconductors, and
the fact that it is electrically pumped. The small size and compatibility with semiconductor laser sources and semiconductor
detectors offer the possibility of photonic integration and the
electrical injection offers the possibility of simple modulation
and control schemes.
The main application of SOA’s may be within optical signal
processing. It has been demonstrated that cross-gain and crossphase saturation in an SOA (the modulation of the amplitude
or the phase of an optical beam passing through the waveguide
by injection of a control beam) can be used for various types
of signal-processing [2]–[4]. Also, coherent techniques such
as four-wave mixing (FWM) have been shown to have a
remarkable potential [5]–[7]. The success of these various
schemes can be traced to the large value of the differential
in semiconductor media; this leads to a large
gain
gain for modest injection currents, but also implies that the
semiconductor gain medium can be readily saturated. The
carrier lifetime of the semiconductor is on the order of several
Manuscript received December 2, 1998; revised May 19, 1999.
J. Mørk is with the Center for Communications, Optics and Materials,
Technical University of Denmark, DK-2800 Lyngby, Denmark.
A. Mecozzi is with Fondazione Ugo Bordoni, 00142 Rome, Italy.
G. Eisenstein is with the Department of Electrical Engineering, Technion,
Haifa 32000, Israel.
Publisher Item Identifier S 1077-260X(99)06945-2.
hundreds of picoseconds. This would seem to imply that the
characteristic frequencies of schemes relying on the saturation
of the carrier density should be of the order of a few gigahertz
at most. It has been recently demonstrated, however, that long
waveguides offer responses with speeds in the range of tens
of gigahertz [3], [8]. The large gain of long amplifiers reduces
the stimulated carrier lifetimes to values of the order of tens of
picoseconds and propagation effects also increase the amplifier
bandwidth [9], [10].
In this paper, we shall analyze the modulation response of a
semiconductor laser amplifier, i.e., the frequency dependence
of the amplitude modulation imposed on an injected CW
optical beam when the bias current is modulated. This mode
of operation has not been dealt with extensively, although it
seems important to understand whether SOA’s can be used as
efficient high-speed modulators. Also, switching fabrics based
on SOA gates rely on current modulation [1]. These may only
need moderate switching speeds but the requirements to the
turn-on and turn-off times can be severe.
We show here that the modulation response of an SOA
displays a number of interesting features. A resonance behavior, resembling the well-known relaxation resonance in
semiconductor lasers, is predicted. Since the cavity feedback
is absent, the underlying physical mechanisms are different.
We analyze this behavior in detail and find that the waveguide
(scattering loss) plays a very important role. This is similar
to the case of cross-gain modulation (XGM) in SOA’s [11],
[12], and the present analysis yields further insight into this
behavior. We show that the case of a transparent waveguide,
where stimulated emission exactly balances the scattering
losses, is particularly simple to analyze, but it still displays
many of the general features of the SOA propagation dynamics
and is convenient for understanding the underlying physics. It
has been demonstrated that at microwave frequencies above 10
GHz, the typical electrode geometry of semiconductor lasers
and amplifiers yields a large propagation loss [13], which
means that the details of the way the modulation signals are
fed to the device becomes important. We analyze different
modulation schemes that give qualitatively different results.
The paper is organized as follows: In Section II, we give
the basic model describing the large-signal modulation behavior of SOA’s and investigate the small-signal response
by numerical integration. Section III presents an analytical
investigation of the so-called transparent waveguide, where
stimulated emission exactly balances the waveguide loss. On
the basis of this simplified model, Section IV investigates
the effects of the microwave propagation characteristics of
1077–260X/99$10.00  1999 IEEE
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IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 5, NO. 3, MAY/JUNE 1999
the electrode. Section V demonstrates the application of an
alternative technique for studying the modulation response,
where the amplifier is considered as a cascade of discrete
elements [14], each of which can be simply analyzed. This
yields further insight into the dynamics of SOA’s. Finally, a
brief summary and conclusion is given in Section VI.
II. BASIC EQUATIONS AND SMALL-SIGNAL ANALYSIS
Considering small-signal modulation,
form
and
take the
(6)
(7)
, we get the
and neglecting terms, but of first order in
following equation for the average photon density along the
amplifier
The equations for the propagation of the photon density
and the rate equation for the carrier density
are
(8)
where
is the saturated gain
(1)
(9)
(2)
is the saturation photon density
Here, is the spatial coordinate along the amplifier and is
the local time, measured in a coordinate system moving with
the group velocity . The relationship with the coordinates
and
measured in a static reference coordinate system
, where
is
are:
one of the space and time dependent variables, i.e., photon
density , carrier density , or current . Furthermore, is the
is the carrier
differential gain, is the confinement factor,
is the active region volume,
is
density at transparency,
is the internal loss, which is mainly
the carrier lifetime and
due to waveguide scattering and free-carrier absorption. The
photon density is related to the instantaneous power via
(10)
and the small-signal material gain is
(11)
For the small-signal amplitude we find
(3)
(12)
is the modal area of the waveguide,
where
related to the geometric area, , by the confinement factor.
The modal gain in the equations above is
Since a cw signal is applied at the input of the amplifier,
the initial conditions are
(4)
(13)
(14)
These equations neglect ultrafast gain nonlinearities like carrier heating and spectral holeburning, which is a good approximation for moderate saturation and low to moderate
modulation frequencies, say below 20 GHz. While the inclusion of these effects lead to obvious quantitative changes,
they do not modify the qualitative conclusions of the theory.
We have also neglected the influence of spontaneous emission
on the saturation of the gain. For long amplifiers with large
gain, the level of amplified spontaneous emission can be high
enough to saturate the amplifier [11], [15] but this case is
not treated here. The implicit assumption is therefore that
the injected power is large enough to dominate the saturation
behavior.
We shall consider sinusoidal current modulation of the form
(5)
being the constant (dc) current, which is assumed
with
to be uniform along the amplifier. The form of the current
depends on the way the current
modulation amplitude
is applied, i.e., uniformly, in the form of a traveling wave,
several discrete contacts (multielectrode amplifier), etc.
Equation (8) can easily be solved numerically for the longitudinal variation of the average photon density, and once this
dependency is known, (12) can be solved for the small-signal
amplitude.
To illustrate the intrinsic propagation effects we shall first
consider the case of traveling-wave modulation, i.e., the current modulation amplitude follows the propagating optical
becomes infield, in which case the current amplitude
dependent of . Fig. 1 shows examples of the calculated
modulation response for different amplifier lengths and different values of the internal loss. Notice, that since we are
performing a small-signal analysis, the curves in Fig. 1 just
scale linearly with the current modulation amplitude. Fig. 1(a)
shows the relative response, i.e., normalized with respect to
the transmitted dc-power, and Fig. 1(b) shows the absolute
amplitude. The parameters used for these calculations are:
mW,
ps,
m ,
m ,
, and
m .
The most prominent feature seen in Fig. 1(a) is the appearance of a resonance for the 5-mm-long amplifier when
the internal loss is increased. The resonance appears for
MØRK et al.: THE MODULATION RESPONSE OF A SEMICONDUCTOR LASER AMPLIFIER
853
(a)
(a)
(b)
(b)
Fig. 1. Modulation amplitude at amplifier output versus modulation frequency for different amplifier lengths L. (a) Relative modulation response.
(b) Absolute modulation amplitude. The modulation was assumed to be a
traveling wave matched to the optical propagation speed. Line signatures
correspond to different values of the internal loss: i
20 cm01 (solid
line), i = 10 cm01 (dashed–dotted line is only shown for L = 5 mm for
the normalized response), and i = 0 (dashed). The small-signal gain was
fixed at g0 = 138 cm01 .
=
modulation frequencies on the order of 10 GHz and resembles the well-known relaxation oscillation resonance seen in
(semiconductor) lasers. Due to the absence of a laser cavity,
however, the physical origin must be quite different. While, in
the present example, the resonance is seen only for the long
amplifier, it appears also for the shorter amplifiers, i.e., when
the internal loss or the small-signal gain are increased. This is
illustrated in Fig. 2, which is for a 2-mm-long amplifier.
Fig. 1 shows that the relative response actually increases
when the internal loss is increased. The increase is larger
for smaller modulation frequencies and in the vicinity of
the resonance (if present) and approaches zero for large
modulation frequencies. It can be seen from the figure that
except for frequencies in the vicinity of the resonance, the
improvement comes from the increase in the response at zero
frequency. Since Fig. 1(b) shows that the absolute amplitude
of the modulation always decreases when the loss is increased,
it can be concluded that the improvement of the relative
response for increasing loss is due to the lowering of the
transmitted dc or average power component in combination
with the appearance of the resonance phenomenon. From
Fig. 1(b), it can be seen that the absolute amplitude of the
modulation at the output never exceeds the zero-loss case.
(c)
Fig.. 2. Modulation response for a 2-mm-long amplifier in dependence of:
(a) internal loss i , (b) small-signal gain factor g0 , and (c) injected dc power
P (0). Unless otherwise mentioned, the injected dc power was 1 mW, the loss
was 20 cm01 , and the unsaturated small-signal gain was g0 = 276 cm01 .
The modulation was assumed to be a traveling wave matched to the optical
propagation speed.
The mechanisms responsible for this behavior can be analyzed by considering the much simpler case of a so-called
transparent waveguide.
III. THE TRANSPARENT WAVEGUIDE
In the case where
(15)
we have
,
and therefore
(16)
independently of .
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IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 5, NO. 3, MAY/JUNE 1999
(18)
In the last line we assumed
to be independent of , an
assumption that will be relaxed later when we analyze the
influence of the microwave propagation characteristics. In the
.
following, we shall assume
(corresponding to the “steady-state” limit of a
For
) we get
long waveguide,
(19)
(29)
In this case, the equation for the small-signal amplitude
becomes
(17)
with
defined by
(20)
therefore becomes independent of the frequency in
and
this limit.
Let us define the relative modulation as
(21)
Equation (19) shows that the parameter governs both the
amplitude of the source term, i.e., the last term in (19), as
well as the propagation of the generated modulation compohas a low-pass
nent. The frequency dependent amplitude
.
characteristic with a cutoff frequency of
evolves according to
The absolute value of
Re
where
(22)
is the phase of the small-signal amplitude,
, and
(30)
The high- and low-frequency limits of
are
(31)
and
for
(32)
has a simple physical meaning, since for
The parameter
we have
, at which the response
If we assume that the frequency
is down by 3 dB from the low-frequency limit, is attained
in the high-frequency limit (which seems to be a reasonable
approximation from numerical examples), we get
(24)
(33)
is therefore the effective absorption coefficient
and
of the waveguide for transmitting a modulation component at
displays a
the angular frequency . It can be seen that
high-pass filter characteristic with the properties
while
The 3-dB bandwidth is a decreasing function of
the low-frequency modulation efficiency increases with .
also displays a resonance
The expression (28) for
behavior as a function of frequency similar to the effect seen
are attained,
in the numerical results. If large values of
several resonances are seen. The behavior can be understood
versus
geometrically by plotting the surface
and
and considering the locus curve that
follows for
increasing .
Equations (28) and (23) show that for vanishing imaginary
, the
part of the effective propagation constant , Im
modulation amplitude at the output decreases monotonically
with the modulation frequency , and the resonance is, therefore, associated with the phase of . The sluggish response of
the carriers to the modulated current thus acts to increase the
modulation amplitude at the output, since the attenuation of
the waveguide is effectively decreased.
From the expression (27) it can be seen that the total
modulation at coordinate is the sum of local contributions
along the waveguide with a phase factor
from coordinates
which may add in-phase or out of phase.
It can be seen that if the current amplitude is allowed dependent and chosen as
Re
(23)
(25)
for
(26)
The high-pass transmission characteristic of the waveguide
reflects that components with frequency higher than the stimulated carrier lifetime cannot excite oscillations of the carrier
density and are therefore transmitted without loss
for
, whereas lower frequency components are
attenuated due to the transfer of energy to carrier density
oscillations.
The exact solution of (19) for a nonzero current modulation
amplitude reads
(27)
(28)
(34)
MØRK et al.: THE MODULATION RESPONSE OF A SEMICONDUCTOR LASER AMPLIFIER
where is constant, the phase factor of different contributions
is cancelled (they always add in-phase) and we get
(35)
Even in this “phase-matched” case (which is different from
considered above) it is found that the
the case
modulation response exhibits a resonance. As already dishas a low-pass filter characteristic, while the
cussed,
, which reflects the transmission
factor
characteristic of the waveguide, has a high-pass characteristic.
When multiplied, the two factors act to give a maximum for
certain values of the parameters. This shows that in this case
the resonant modulation behavior occurs because for certain
parameters the drop of the carrier density modulation with
frequency is not strong enough to suppress the increase of the
waveguide transmission with frequency.
It is found that in this phasematched case the maximum of
the modulation response occurs for
the average (dc) power will depend on the distance traversed
up to the position of the modulation.
The modulation also shows a maximum for a certain modulation frequency in this case. We find that maximum modulation at the output,
is attained for
(39)
Also in this case, there is a requirement on the propagation
distance to see the maximum, i.e.,
(40)
B. Traveling-Wave Modulation
The case of a traveling-wave type of modulation corresponds to (in static reference-frame coordinates , , neglectbelow),
ing the constant phase of
c.c.
(36)
meaning that for given magnitudes of the loss and the smallsignal gain there is a critical length below which no resonance
will be observed. The numerical value of 0.796 above arises
from the numerical solution of a simple nonlinear equation.
855
(41)
(42)
, where
is the phase
with propagation constant
velocity. The corresponding current in local coordinates becomes
c.c.
(43)
where
IV. THE DISTRIBUTED NATURE
CURRENT MODULATION
OF THE
(44)
In the above, we considered the case of a current modulation amplitude which was constant in the moving coordinate
system, i.e., corresponding to a traveling microwave which is
matched exactly with the propagating optical signal. We now
consider more realistic situations where the current modulation
is either very localized or in the form of a traveling wave that
propagates with a speed different from the optical mode.
and
are the phase index for the microwave
and
propagating on the electrode and the group index for light
propagating in the waveguide.
This case can, thus, be analyzed by making in (27) the
substitution
(45)
obtaining
A. Localized Current Modulation
The results of Tauber et al. [13] show that the propagation
losses for modulated currents applied to the contacts can be
very large at microwave frequencies, i.e., in excess of 300
dB/cm for frequencies larger than 20 GHz. If this is the case,
the modulation is to be considered rather as a “point source”
located at the position of the bonding wire or transmission line.
This case can be analyzed by approximating the current as
a delta function located at the coordinate
(37)
which upon insertion in (27) yields
(38)
The result obviously depends only on the distance
from where the modulation is applied, the first part of the
amplifier is completely inactive. Notice that this only holds for
the transparent case considered here, since in the general case
(46)
we get the result previously derived for a
For
current modulation considered to be constant in the moving
corresponds to the phase
coordinate system. Since
velocity of the propagating microwave coinciding with the
,
group velocity of light propagating in the waveguide,
i.e., the constant translation speed of the local coordinate
system, this is as expected.
The case of a uniform current amplitude along the amplifier
, which means that the phase velocity
corresponds to
or the wavelength of the microwave are infinite. One expects
here the modulation amplitude of light at the output to be
very small when the light propagation delay through the
waveguide corresponds to one period of the modulation, i.e.,
.
According to the results of Tauber et al. [13] the phase
velocity of the microwave is in the range of 7%–12% of the
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IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 5, NO. 3, MAY/JUNE 1999
A. Single SOA with Zero Internal Loss: Large-Signal Analysis
Neglecting the internal loss,
the equations for the
photon density and carrier density can be formulated as
(47)
(48)
where
(49)
Fig. 3. Effect of the propagation speed of a traveling-wave type current
modulation. The group index of the optical field is fixed at ng
3:5 while
the group index n for the microwave field is varied. A negative value for n
indicates counter-propagation. The case of n = 0 corresponds to an infinite
wavelength of the microwave field and the current modulation amplitude is
thus uniform along the waveguide.
=
velocity of light in vacuum for frequencies in the range of
in the copropagating case
5–40 GHz. This means that
.
since
Fig. 3 shows calculations, using (46), of the effect of
different propagation speeds of the propagating microwave.
The figure shows that the frequency at which mismatch
between the microwave and the light leads to a dip in the
modulation response is smaller for the case of a copropagating
) than for the
traveling-wave (in the realistic case of
). This might
case of a uniform current modulation (
at first sight be somewhat surprising. It can be understood,
however, by considering the other extreme case where light
propagates much faster than the microwave. Then, a dip in
the modulation amplitude at the output is expected when the
wavelength of the microwave corresponds to the cavity length,
. Since
this gives a lower
i.e.,
frequency limit than the propagation delay of light.
At intermediate frequencies, below the “dip,” travelingwave effects may, however, increase the response over that
of the phase-matched case.
In an actual case of a single bonding wire attached to
the center part of an electrode, the result will probably be
a damped microwave propagating in both directions and one
gets a combination of the effects discussed above. Since
the propagation losses increase with frequency (from about
100–150 dB/cm at a few GHz to about 500 dB/cm at 40 GHz
according to [13]), the type of the field may even change with
frequency.
Defining a new variable
(50)
we get
(51)
(52)
Here, we have explicitly shown the transformation to the
local coordinate system, by using , the current in the static
coordinate system, as the variable. Notice that (52) is a large
signal equation, only the absence of internal loss was assumed.
Thus, even though it is an ordinary differential equation, as
opposed to the set of partial differential equations we started
with, it takes full account of the propagation effects.
We see that for zero internal loss the effect of the walkoff
is to change the modulation current according to the transformation
(53)
B. Single SOA with Zero Internal Loss: Small-Signal Analysis
Let us expand
(54)
(55)
(56)
(57)
with unknown ,
lowest order
,
, and
. Equation (52) gives, to
(58)
V. CASCADED AMPLIFIERS
We develop here an alternative technique for analyzing the
dynamics of long SOA’s by studying a cascaded chain of
several amplifiers. The basics of this technique were developed
in [14]. The technique utilizes that, for zero internal loss, a
very simple analytical solution for the dynamics of an SOA
including propagation effects can be found. The effects of loss
can be analyzed by cascading amplifiers and including a finite
loss in-between.
as the integrated gain
where
(59)
and, to first-order
(60)
MØRK et al.: THE MODULATION RESPONSE OF A SEMICONDUCTOR LASER AMPLIFIER
where
(61)
From (51), we get
(62)
Inserting (60) in the above, we obtain
(63)
where
(64)
(65)
857
All amplifiers are equal, and the loss between the amplifiers is
constant
. The input power of the line is such that the
saturated gain of each amplifier exactly compensates for the
. The second, is the case of zero loss between
loss,
the amplifiers. We will show that, as obvious for physical
reasons, this case is equivalent to a single amplifier with gain
equal to the product of the saturated gains of all the amplifiers.
1) Transparent Line: If the input power of the line is such
that the saturated gain of each of the amplifiers exactly
, we have
and
compensates for the loss
. Assume also that the applied modulation has the same
amplitude at each amplifier, but include the possibility that the
going from an
phase of the modulation increases by
amplifier to the next. Assume also that the single amplifier is
so short that the effect of the walkoff of the modulation is
negligible within the amplifier. We, therefore, have
(66)
From (58),
is the solution of
(73)
where
(67)
where
(74)
with
(68)
C. Amplifier Chain
(75)
independent of frequency. Solution of (69) is thereby
blocks, each one being the sequence
Consider a chain of
of an SOA and a linear element of loss . The output of the
th linear loss, is fed into the input of the
th amplifier,
with a possible frequency-dependent phase shift that we set
. We have
(76)
(69)
The saturated gain of the amplifiers
are the solution of
(70)
where
is the input of the th amplifier
th
Note that the modulation at the input of the fictitious
is the modulation at the th amplifier
amplifier
output times . If the phase of the modulation is such that
, (or if
and the amplifier chain
for lying in the region where
is such that
the response of the chain is large) we get
(77)
(71)
and
are given by (65) and (66) with ,
In (69),
and
replaced by
, , and
.
. The solution of (69) can be obtained
Assume
iteratively. The result is
Here, we have defined the frequency independent normalized
as
modulation amplitude
(78)
Assume now that
(79)
Expanding (65) to first order in
, we obtain
(80)
(72)
It is convenient at this point to separate two limit cases. The
first, the easiest to analyze, is the case of a transparent line.
Inserting this expansion in (77) we obtain
(81)
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IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 5, NO. 3, MAY/JUNE 1999
(84) into (83) and using in the product the cancellation of the
numerator of the
th term with the denominator of the
th, we obtain
(85)
where
(86)
N
Fig. 4. Pictorial representation of the
dependence of the high-frequency
limit of the term [(1 + )N 1], that appears in the modulation response,
(77), of a transparent line of modulated amplifiers.
R 0
The amplitude of the modulation is
is
Let us now again assume, for simplicity, that
independent of except for a phase factor that exactly cancels
the phase factor in the sum of (85). Inserting (86) into (85),
we finally obtain
(82)
The modulation shows maxima, equal to twice the normalized
modulation amplitude, for
, and zeros for
, with
and
. The number of maxima
and minima is obviously finite. They are the points that are
given by the above equations and meet condition (79) for the
validity of (82).
A pictorial representation of the high-frequency modulation
response of a chain of amplifiers is shown in Fig. 4. The
arrows represent the result of the successive addition to a
vector of unit length of a small vector in quadrature. The
occurrence of maxima and minima of the modulation response
is the result of the constructive and destructive interference
at the amplifier output of the modulation response of each
amplifier in the chain, rotated by the effect of the delayed
response of all the amplifiers downstream, which at highfrequency act like pure phase shifters.
) should be studied with
2) Zero Loss: This case (
the help of the general solution (72), which becomes
(83)
and using
(84)
We have used in (84) that the input of the
is the input of the th times its saturated gain
th amplifier
. Inserting
(87)
As expected from physical reasons, the modulation response in
this loss-free case is equal to the response of a single amplifier
with a saturated gain equal to the product of the saturated
gains of all the amplifiers in the cascade, and subject to the
. Unlike the case of a transparent
total modulation
line where maxima and minima may appear, the modulation
response shows a pure low-pass characteristic. The reason
for the qualitatively different behavior of the two cases is
related to the way each amplifier in the chain contributes
to the total modulation at the chain output. Each amplifier
contributes with a term of the product in the first line of (72).
,
In the transparent line case they are
and these terms are
whereas in the zero loss case,
. In the transparent line case and in the high
frequency limit, each amplifier contributes with a pure rotation
and this leads, as we discussed above, to interference effects
in the output modulation. In the zero loss case, on the other
is never a pure rotation unless
.
hand,
,
In this case, however, also the in-quadrature contribution
, is zero and
altogether.
proportional to
The interface effects of the transparent case cannot therefore
take place.
D. Limit
: Saturated Amplifier with Internal Loss
Assume that the amplifiers have length
. If we set
and
, and let
, we
MØRK et al.: THE MODULATION RESPONSE OF A SEMICONDUCTOR LASER AMPLIFIER
F. Inclusion of Ultrafast Dynamics
obtain by expanding to first order (67)
(88)
If the loss between the amplifiers is
implies
condition
859
, the
The inclusion of the ultrafast dynamics may be done by
using the integral approach developed in [16]. Equation (56)
of that paper can be easily extended to a modulated current,
by arguments similar to those used above. We get
(89)
From (76), substituting into
, we get
given by (65)
(98)
where (see [16, eqs. (27), (45)])
(99)
(90)
Expanding to first order, using (54)–(57) together with
where
(100)
(91)
we get
Assuming phase matching of the modulation, we get for the
efficiency
(101)
where
unchanged, with
[see (61)]. Equation (63) remains formally
(102)
(103)
(92)
(104)
E. Two Cascaded Amplifiers
It is easy, using (63), to find the efficiency of the modulation
in between, in the
for two amplifiers in a cascade with loss
general case (and for matched modulation, for simplicity). The
is given by
output of the second amplifier
is the solution of
(105)
where
(106)
(93)
where
and
The results obtained for a chain of amplifiers are therefore
easily extended to the case in which the fast response is
included.
VI. CONCLUSION
are given by
(94)
(95)
and
are the unsaturated gain of the two
and
amplifiers. In two limit cases, we obtain
(96)
(97)
we have
which is an indication
For
that the loss may increase the modulation efficiency (at zero
detuning).
We have shown, through numerical simulations and approximate analytical treatments, that the modulation response of a
semiconductor optical amplifier is heavily influenced by propagation effects. In particular, it was found that a finite amount
of waveguide internal loss, due to waveguide scattering or freecarrier absorption, can lead to the appearance of a resonance
in the modulation response. Comparison to the case of zero
internal loss shows that the loss decreases the response at all
frequencies, but not uniformly so. Two different techniques
were used for this theoretical study, namely a standard smallsignal analysis of the well-known propagation equations as
well as description in terms of cascaded amplifiers.
Experimental results for a modulated SOA were presented
in [17] and clearly show the appearance of a resonance in the
modulation response. It is to be noted that the mount used
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IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 5, NO. 3, MAY/JUNE 1999
in that case was carefully designed for high-frequency laser
applications [18]. This is a very important requirement if one
attempts to measure the intrinsic modulation characteristics
without the masking effect due to parasitic effects of the
package and microwave propagation effects of the electrodes,
and may explain why resonance effects usually are not seen
in measured amplifier modulation responses.
We notice, however, that the electrically modulated SOA
bear some common features with the optically modulated SOA
(XGM) where an intensity modulated control beam is used to
modulate the gain of the amplifier. In the latter case, resonance
behavior has been observed both experimentally [8], [9] and
theoretically [9], [12], but was not explained. Our results show
that the resonance characteristic is not related to the special
case of optical modulation but rather is a general consequence
of the propagation characteristics of long and lossy amplifiers.
[13] D. A. Tauber, R. Spickermann, R. Nagarajan, T. Reynolds, A. L.
Holmes, and J. E. Bowers, “Inherent bandwidth limits in semiconductor
lasers due to distributed microwave effects,” Appl. Phys. Lett., vol. 64,
pp. 1610–1612, Mar. 1994.
[14] A. Mecozzi and D. Marcenac, “Theory of optical amplifier chains,” J.
Lightwave Technol., vol. 16, pp. 745–756, May 1998.
[15] K. Obermann, I. Koltchanov, K. Petermann, S. Diez, R. Ludwig,
and H. G. Weber, “Noise analysis of frequency converters utilizing
semiconductor laser amplifiers,” IEEE J. Quantum Electron., vol. 33,
pp. 81–88, Jan. 1997.
[16] A. Mecozzi and J. Mørk, “Saturation effects in nondegenerate four-wave
mixing between short optical pulses in semiconductor laser amplifiers,”
IEEE J. Select. Topics Quantum Electron., vol. 3, pp. 1190–1207, Oct.
1997.
[17] H. Dong and A. Gopinath, “AlGaAs/GaAs active optical ridge waveguide switch/modulator on a semi-insulating substrate,” in Proc. Integrated Photonics Research, 1994.
[18] H. Dong, F. Williamson, and A. Gopinath, “Airbridged high-speed
AlGaAs–GaAs ridge waveguide lasers,” IEEE Photon. Technol. Lett.,
vol. 8, pp. 46–48, Jan. 1996.
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[5] A. D.’Ottavi, F. Girardin, L. Graziani, F. Martelli, P. Spano, A. Mecozzi,
S. Scotti, R. Dall’Ara, J. Eckner, and G. Guekos, “Four-wave mixing
in semiconductor optical amplifiers: A practical tool for wavelength
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[6] S. Diez, C. Schmidt, R. Ludwig, H. G. Weber, K. Obermann, S. Kindt,
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[7] D. F. Geraghty, R. B. Lee, M. Verdiell, M. Ziari, A. Mathur, and K.
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[8] D. D. Marcenac, A. E. Kelly, D. Nesset, and D. G. Moodie, “Speed and
travelling wave effects in cross-gain modulation in semiconductor optical amplifiers,” in COST239/240 Workshop on Semiconductor Optical
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[9] T. Durhuus, B. Mikkelsen, C. Joergensen, S. L. Danielsen, and K. E.
Stubkjaer, “All-optical wavelength conversion by semiconductor optical
amplifiers,” J. Lightwave Technol., vol. 14, pp. 942–954, June 1996.
[10] D. Marcenac and A. Mecozzi, “Switches and frequency converters based
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dissertation, Dept. Electromagnetic Systems, Tech. Univ. Denmark, Rep.
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Jesper Mørk was born in Denmark in 1962. He received the M.Sc. and Ph.D.
degrees in electrical engineering from the Technical University of Denmark,
Lyngby, in 1986 and 1988, respectively.
From 1988 to the shutdown in 1996, he was with Tele Danmark Research;
since 1994, as a group leader. After half a year with Ericsson in Denmark
as a Project Manager, he moved to his present position as an Associate
Research Professor at Technical University of Denmark in November 1996.
His main research interests are ultrafast carrier dynamics, photonic switching,
and nonlinear dynamics in semiconductor lasers.
Antonio Mecozzi (M’97) was born in Rome, Italy, in 1959. He received
the laurea degree in chemical engineering from the University of Rome, “La
Sapienza,” Italy, in 1983.
He joined the Optics Division of the Fondazione Ugo Bordoni, Rome,
Italy, in December 1985. From February 1991 to June 1992, he was with
the Optics Group of the Research Laboratory of Electronics, Massachusetts
Institute of Technology, Cambridge, MA, working with Prof. H. Haus. His
main research interests are quantum optics, nonlinear optics, long-distance
optical transmission, and laser theory. He is a topical editor of the OSA
publication, Optics Letters for Nonlinear Optics.
Dr. Mecozzi is a fellow of the Optical Society of America.
Gadi Eisenstein (S’80–M’80–SM’90–F’99) was born in Haifa, Israel, on June
29, 1949. He received the B. Sc. degree in electrical engineering from the
University of Santa Cara, Santa Clara, CA, and in 1975, he started his graduate
studies at the University of Minnesota under Prof. K. Champlin and the late
Prof. A. Van der Ziel. His Ph.D. work dealt with specialized Schottky barrier
detectors for far-infrared radiation.
In 1980, he joined Bell Laboratories, Holmdel, NJ, where he worked for
nine years in the field of devices and concepts for optical fiber communication
making significant contributions in the areas of optical amplifiers, short
optical pulse generators, and very fast systems. In 1989, he joined the
Technion—Israel Institute of Technology, Haifa, Israel, where he is a Professor
of Electrical Engineering. He is one of the founders of the Barbara and
Norman Seiden Center for Advanced Optoelectronics Research. He spent one
sabbatical year at the University of Minnesota from August 1997 to June 1998.