Small-signal-equivalent circuits for a semiconductor laser
Osman Kibar, Daniel Van Blerkom, Chi Fan, Philippe J. Marchand, and Sadik C. Esener
Passive electrical circuits whose voltage and current equations are exactly equivalent to the small-signal
rate equations of a semiconductor laser are derived to model an electrically modulated laser ~verified to
be the same as that given in the literature!, an optically modulated laser ~i.e., a laser used as an optical
amplifier!, and a multimode laser. These circuits offer a fast and efficient simulation tool with little
computational complexity in which the small-signal assumption ~i.e., small modulation range! is neither
violated nor insufficient for the simulation. © 1998 Optical Society of America
OCIS code: 140.5900.
1. Introduction
Semiconductor lasers have been extensively modeled
in terms of equivalent electrical circuits1–5 to simulate their large-signal behavior. However, because
the laser behavior includes nonlinear effects, accurate models tend to be computationally demanding.
For simulations for which speed is favored over accuracy or the operation of the laser is limited to a
small modulation range, it is more advantageous to
model the small-signal behavior of the laser. In this
case the laser equations are reduced to linear differential equations, and the electrical circuits that
model the laser are significantly simplified. Previously, based on the impedance characteristics of a
laser,6 an equivalent electrical circuit was proposed
to model the small-signal behavior of lasers in the
presence of small perturbations.7 This circuit has
been further modified to include the effects of lateral
carrier diffusion8 and of package parasitics9 under
electrical modulation.
In this paper we exploit the similarities between
semiconductor lasers and electrical RLC circuits to
repeat the derivations of the above-described electrical circuits, which exactly model the small-signal behavior of semiconductor lasers under electrical
modulation. We then extend our derivations to include optical modulation ~i.e., when a laser is used as
an optical amplifier! and multimode lasers in which
The authors are with the Department of Electrical and Computer Engineering, University of California, San Diego, La Jolla,
California 92093-0407.
Received 10 November 1997; revised manuscript received 24
April 1998.
0003-6935y98y266136-04$15.00y0
© 1998 Optical Society of America
6136
APPLIED OPTICS y Vol. 37, No. 26 y 10 September 1998
the secondary modes can be longitudinal, spatial, or
polarization modes. In Section 2 we state the smallsignal rate equations and outline the procedure to
convert these equations into the current and the voltage equations of RLC circuits. This procedure is
then applied to electrical modulation, and the resulting circuit is compared with the previously reported
circuits for verification of our model. In Sections 3
and 4 we modify the equations and our electrical
circuit to include optical modulation as well as multimode operation of a laser. Section 5 reviews the
results of the modeling and concludes the paper.
2. Modeling Small-Signal Laser Behavior
Rate equations can be used for a simple treatment of
the frequency-domain or time-domain behaviors of
the electron and the photon numbers in a laser cavity.
The reader is referred to any textbook on lasers for a
discussion of the rate equations and their validity for
various laser phenomena ~see, e.g., Ref. 10!. The
general rate equations for a single-mode laser are
Ṗ~t! 5 GP~t! 2 gP~t! 1 Rsp,
Ṅ~t! 5
Ibias
2 ge N~t! 2 GP~t!,
q
(1a)
(1b)
where P and N are the total number of photons in the
lasing mode and electrons in the excited state inside
the cavity, respectively. The dots over P and N represent time derivatives. The three terms on the
right-hand side ~RHS! of Eq. ~1a! are, in order, the
stimulated emission rate, the total photon-loss rate
including the loss through the mirrors, and the spontaneous emission rate into the lasing mode. The
three terms on the RHS in Eq. ~1b! are, in order, the
injection rate of electrons into the gain medium, the
electron-loss rate ~including nonradiative recombination, spontaneous emission, and Auger recombination!, and the loss rate of electrons that results from
stimulated emission.
The linearized small-signal rate equations in the
presence of a small perturbation ~i.e., if we assume
dP ,, P and dN ,, N! are10
dṖ~t! 5 2GPdP~t! 1 sN3PdN~t!,
dṄ~t! 5 2GNdN~t! 2 sP3NdP~t!,
Rsp
2 GP P,
P
(2b)
(3a)
GN 5 ge 1 geN N 1 GN P,
(3c)
sP3N 5 G 1 GP P,
(3d)
where the subscripts P or N on the RHS of the equations mean the derivatives with respect to P or N,
respectively.
At this point we note that the small-signal rate
equations are two linearly coupled differential equations similar to the voltage and the current equations
of a RLC circuit. In a laser cavity the electrons and
the photons exchange energy through absorption and
emission with the various loss mechanisms dissipating energy in the cavity. Similarly, in a RLC circuit
the capacitor and the inductor exchange energy, and
the resistor dissipates energy out of the circuit. In
addition, the continuity conditions of the electrical
circuit ~i.e., voltage across the capacitor and current
across the inductor are continuous! are equivalent to
those of the laser cavity ~i.e., the changes in the electron and the photon numbers inside the cavity are
continuous if rate equations are being utilized!. Motivated by these analogies, we make the assumption
that the excited electrons in a laser cavity can be
represented with the charge across a capacitor, and
the photons in a given mode of the laser with the
magnetic flux linkage of an inductor:
C~t!
,
q~Hys!
dN~t! 5
Q~t!
,
q
iC~t! 5 2GN CvC~t! 2
(4)
where division by the electronic charge q and by the
fixed units of @Henry per second ~Hys!# ensure proper
units for the components in the electrical circuit.
Assigning electrons to an inductor and photons to a
capacitor still would have given us the same results,
sP3NLiL~t!
.
~Hys!
(5a)
(5b)
Equations ~5a! and ~5b! are the voltage and the current equations, respectively, of the circuit given in
Fig. 1 ~with a proper choice of component values!, and
this circuit is exactly the same as the one given in Ref.
7 to model the small-signal behavior of a single-mode
laser in the presence of small perturbation. The output light power of the laser @dPout~t!# is the product of
the deviation of the photon number from the equilibrium value @dP~t!#, the unit photon energy ~hn!, and
the photon-loss rate through the output mirror ~am!,
so from Eqs. ~4! and Fig. 1 we have
dPout~t! 5 am
(3b)
sN3P 5 GN P 1 RspN ,
dP~t! 5
vL~t! 5 2GP LiL~t! 1 sN3PCvC~t!~Hys!,
(2a)
where GP and GN are the decay rates of fluctuations of
the photon and the electron numbers, respectively,
and sN3P and sP3N are the coupling coefficients between the photon- and the electron-number fluctuations, respectively. Note that the electron and the
photon numbers in Eqs. ~2! ~dN and dP! refer to the
deviations from the equilibrium values at a given bias
of the laser. At a given bias point, which fixes P and
N, the four coefficients in Eqs. ~2! are given as
GP 5
but the component values in the electrical circuits
would turn out to be negative.
We use Eqs. ~4! and the standard equations for a
capacitor and an inductor ~i.e., Q 5 CVC, iC 5 dQydt
and C 5 LiL, VL 5 dCydt! to convert Eqs. ~2! into
voltage and current equations:
LiL~t!
hn.
q~Hys!
Similarly, for arbitrary external electrical modulation @im~t!#, Eq. ~2b! is modified as
dṄ~t! 5 2GN dN~t! 2 sP3N dP~t! 1
im~t!
,
q
(6)
such that the current equation ~5b! becomes
iC~t! 5 2GN CvC~t! 2
sP3NLiL~t!
1 im~t!.
~Hys!
(7)
The circuit is then modified as shown in Fig. 2, and
including the electrical parasitics associated with an
external current source gives us the circuit published
in Ref. 9.
Fig. 1. Small-signal circuit model of a single-mode laser in the
presence of small perturbation:
dP~t! 5
C5
LiL~t!
CvC~t!
~Hys!
,
, dN~t! 5
,L5
q ~Hys!
q
sP3N
1
~Hys! sN3P
~Hys! GP
, Rn 5
, Rp 5
.
~Hys! sN3P
GN
sP3N
10 September 1998 y Vol. 37, No. 26 y APPLIED OPTICS
6137
Ṅ~t! 5
Ibias
2 ge N~t! 2
q
( G P ~t!,
k
k
(12b)
k
where k is the mode number @i.e., Eq. ~12a! is actually
a set of k equations for k modes#. Following the
same procedure as for the single-mode laser, we derive the small-signal rate equations @analogous to
Eqs. ~2!#:
Fig. 2. Small-signal circuit model of a single-mode laser with
electrical modulation.
3. Optical Modulation
By choosing an appropriate bias point, one can operate a laser as an optical amplifier.11–14 In this case
the photon number for a given laser mode is externally modulated, so Eq. ~1a! is changed to
Ṗ~t! 5 GP~t! 2 gP~t! 1 Rsp 1 hc
Pin~t!
,
hn
(8)
where Pin~t! is the input light power. As a result,
Eq. ~2a! is modified as
dṖ~t! 5 2GPdP~t! 1 sN3PdN~t! 1 hc
Pin~t!
,
hn
(9)
and in return the voltage equation ~5a! becomes
vL~t! 5 2GP LiL~t! 1 sN3PCvC~t!~Hys!
1
q~Hys!hc Pin~t!
,
hn
(10)
T1~1 2 R2!
.
1 1 R1 R2 2 2~R1 R2!1y2cos d
(13a1)
dṖk~t! 5 2GPkdPk~t! 1 sN3PkdN~t!,
(13ak)
dṄ~t! 5 2GN dN~t! 2
(11)
In Eq. ~11! the subscript 1 refers to the input mirror
and T and R are the transmittance and the reflectance, respectively, of each mirror ~with T 1 R 1 A 5
1, and A the mirror absorption!. The term d is the
mismatch between the wavelength of the input signal
and the optical round-trip path length of the cavity.
Equation ~11! is obtained by slight modification of the
procedure given in Ref. 15 in which the transmittance
of a passive Fabry–Perot interferometer is calculated.
The additional term in Eq. ~10! adds to the voltage
across the inductor, so it is equivalent to placing a
voltage source in series with the inductor–resistor
pair in the equivalent electrical circuit ~Fig. 3!.
(s
Pk3N
dPk~t!,
(13b)
k
where the expressions for the decay rates and the
coupling coefficients @i.e., Eqs. ~3!# are now given separately for each mode simply by addition of a subscript, from 1 to k, to each of the P terms.
Using Eqs. ~4! in Eqs. ~13! along with the standard
current–voltage equations for a capacitor and an inductor and rearranging the q and the Henry per second terms, we obtain one current equation and k
voltage equations ~one for each of the k modes!:
vL1~t! 5 2GP1L1 iL1~t! 1 sN3P1CvC~t!~Hys!,
·
·
·
vLk~t! 5 2GPkLk iLk~t! 1 sN3PkCvC~t!~Hys!,
iC~t! 5 2GN CvC~t! 2
(
k
where hc is the coupling efficiency of the optical input
signal into the Fabry–Perot cavity of the laser:
hc 5
dṖ1~t! 5 2GP1dP1~t! 1 sN3P1dN~t!,
·
·
·
sPk3NLk iLk~t!
~Hys!
,
(14a1)
(14ak)
(14b)
The circuit that satisfies all the above equations is
not trivial, and the resulting circuit is shown in Fig.
4 along with the component values. Note that each
inductor Lj , which represents an optical mode, is now
accompanied by a second inductor Lxj to account for
the reduction in the coupling coefficient of that mode
with the electrons. The output power of each mode
is proportional to only Lj and not Lxj. The dominant
mode is given by the left-most inductor–resistor pair
4. Multimode Lasers
The model for a multimode laser is more involved.
When more than one mode exists in the laser, the
rate equations ~1! become
Ṗk~t! 5 Gk Pk~t! 2 gk Pk~t! 1 Rspk,
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(12a)
APPLIED OPTICS y Vol. 37, No. 26 y 10 September 1998
Fig. 3. Small-signal circuit model of a single-mode laser with
q ~Hys! h c P in~t!
.
optical modulation: V in~t! 5
hn
~i.e., mode 1 in Fig. 4!. This mode has the highest
coupling with the electrons, so it has the smallest Rpj,
Lj , and Lxj. The addition of external modulation to
the circuit is exactly the same as was explained in
Sections 1–3, that is, a current source parallel to Rn
is added to account for electrical modulation and a
voltage source is added in series with an inductor–
resistor pair, whose mode matches that of the optical
input signal ~shown in Fig. 4!.
~i.e., the modulation range is small compared with
the equilibrium values at the given bias condition!.
Therefore these circuits provide a fast and accurate
simulation tool with little computational complexity
for small-signal laser behavior. These circuits can
further be used in combination with more advanced
optoelectronic computer-aided-design programs that
use mathematical techniques to obtain the largesignal behavior of a laser through piecing together
small-signal behaviors at various regimes.
5. Conclusion
The authors thank Daniel Hartmann for useful
discussions. This study is supported by the U.S. Defense Advanced Research Projects Agency under
grant F49620-97-1-0285.
A simple procedure, which was used to model the
small-signal behavior of a semiconductor laser as an
equivalent electrical circuit, has first been applied to
an electrically modulated laser, and the resulting
equivalent circuit has been verified to be exactly the
same as that given in the literature. The procedure
has then been used to model the laser behavior under
optical modulation ~i.e., when the laser is used as an
optical amplifier!. Finally, the model has been extended to include multimode operation of the laser.
In all cases it has been shown that the voltage and the
current equations of the electrical circuits are exactly
equivalent to the small-signal rate equations of the
semiconductor laser.
The electrical circuits presented in this paper are
exact models for a multimode semiconductor laser
under external modulation ~electrical or optical! as
long as the small-signal assumption is not violated
Fig. 4. Small-signal circuit model of a multimode laser:
Rn 5
~Hys! sN3P1
Lxj 5 Lj
RP j 5
Vinj~t! 5
S
GN
sN3P1
sN3Pj
,C5
D
21 ,
~Hys! GPj sN3P1
sPj3N
sN3Pj
,
1
~Hys!
, Lj 5
~ j 5 1, . . . , k!,
~Hys! sN3P1
sPj3N
~ j 5 1, . . . , k!,
~ j 5 1, . . . , k!,
q ~Hys! hcPinj~t! sN3P1
,
hn
sN3Pj
~ j 5 1, . . . , k!.
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