Phase Noise in Semiconductor Lasers

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298
JOURNAL
OF
LIGHTWAVE TECHNOLOGY, VOL. LT-4, NO.
3,
MARCH
1986
Phase Noise in Semiconductor Lasers
CHARLES
H.
HENRY
(Invited Paper)
Abstract-The subject
of
phase noise in semiconductor lasers is re-
viewed. The description
of
noise in lasers and those aspects of phase
noise that are relevant to optical communications are emphasized. The
topics covered include: Langevin forces; laser linewidth above thresh-
old and helow threshold; line structure due to relaxation oscillations;
phase fluctuations; line narrowing by a passive cavity section and by
external feedback; coherence collapse due
to
optical feedback; the shot
noise limits of several schemes
of
coherent optical communication, and
the linewidth required to approach these ideal limits.
I.
INTRODUCTION
A
FASCINATING ASPECT of any single-mode laser
is the high degree of spectral purity of the laser ra-
diation. A related property is the long coherence time of
the optical field: the time interval during which two com-
ponents of the field emitted at different times from the
laser can stably interfere. The behavior of the laser is sim-
ilar to that of other oscillators. Above threshold, the am-
plitude of the optical field is nearly fixed, but the phase
may take any value. The wandering of phase can be de-
scribed as Brownian motion or phase diffusion
[
l].
This
relatively slow wandering of phase determines both the
laser linewidth and the coherence time. Rapid phase fluc-
tuations are also of great importance; they introduce er-
rors in coherent optical communications and contribute to
the shape of the wings of the laser line.
Interest in the linewidth was present
from
the very be-
ginning of laser physics. In their first paper proposing the
laser, Schawlow and Townes [2] derived a formula for the
linewidth, predicted that the lineshape would be Lorent-
zian and that the linewidth would narrow inversely with
laser power,
so
that the linewidth-power product
AvPo
is
constant. This formula is only valid below threshold. Lax
[3] pointed out that above threshold, the amplitude fluc-
tuations of the laser are stabilized and this
is
accompanied
by a
X 2
reduction in
AvPo.
We will refer to this correction
as the modified Schawlow-Townes formula. The detailed
change in the linewidth through the threshold region was
calculated by Hempstead and Lax [4]. The extreme nar-
rowness of the linewidth of gas lasers, made measurement
of
the intrinsic linewidth a formidable problem. However,
Gerhardt
et
al.
[5] finally succeeded by using a 500-m
folded interferometer and by operating their He-Ne laser
Manuscript received
July
1,
1985;
revised October
16, 1985.
The
author
is
with
AT&T
Laboratories, Murray Hill,
NJ
07974.
IEEE
Log
Number
8406714.
at microwatt power levels. Their measurement confirmed
quantitatively the theoretical predictions for linewidth and
coherence length, including the power dependence of the
AvPo
reduction calculated by Hempstead and Lax
[4].
The interest in linewidth and phase noise was renewed
in the last few years by a burst of activity stimulated by
the availability of single-mode semiconductor lasers con-
tinuously operating at room temperature and the expec-
tation that these devices would find many applications re-
quiring a high degree of coherence, such as interferometry
and coherent lightwave communications. The first careful
linewidth studies of AlGaAs lasers were made by Fleming
and Mooradian
[6].
They observed the Lorentzian shape
and the line narrowing inversely with power as expected,
but surprisingly, they found that the linewidth was about
50
times greater than that predicted
by
the modified
Schawlow-Townes formula. This linewidth enhancement
was explained by the author as primarily due to the change
in the cavity resonance frequency with gain
[7].
This re-
sults in a correction of
1
+
a2
to the modified Schawlow-
Townes formula, where the linewidth parameter
01
=
An
'/
An"
is the ratio of the changes in the real and imaginary
parts of the refractive index with change in carrier num-
ber.
In
AlGaAs and InGaAsP lasers alpha is about
4-7
[7],
[8]-[I21 (one smaller value has been reported [13]).
A
similar correction is expected to occur in gas lasers [3],
[
141,
however the correction in this case is expected to be
of
order unity or less [14] and zero when the cavity mode
is tuned to the center of the transition line.
In
a semicon-
ductor laser, the laser line occurs at the foot of a steep
absorption edge and this causes
a
to be large [8]. An ad-
ditional factor of about
X 2
in linewidth results because,
in semiconductor lasers, the population associated with
laser transition is not fully inverted [6],
[7].
In addition to the increased linewidth, semiconductor
lasers depart from the expectations of the classical laser
theory in another respect. It was found
by
Daino
et
al.
[15] that the line shape is not a perfect Lorentzian, but
has satellite peaks far out in the wings that are separated
from the main peak by multiples of the relaxation oscil-
lation frequency of the laser. This structure is a conse-
quence of the large value of
Q!
that causes a coupling of
phase and amplitude fluctuations
[
161,
[
171.
Vahala
et
al.
[I81 showed that opposite satellite peaks differ in height
due to the correlation of amplitude and phase fluctuations.
The broad linewidth
of
the semiconductor laser
(AvPo
0733-8724/86/0300-0298$01
.OO
O
1986
IEEE
HENRY: PHASE
NOISE IN
SEMICONDUCTOR LASERS
299
=
50
-
100 MHz
*
mW) makes conventional semicon-
ductor lasers unacceptable for many applications requir-
ing a high degree of coherence, e.g., high-resolution
spectroscopy, interferometric sensors, and coherent opti-
cal communications. The broad linewidth can be over-
come in a number of ways. Perhaps the most successful
way is to add a passive section to the laser cavity by an-
tireflection
(AR)
coating the laser facet and adding an ex-
ternal reflector several tens of centimeters away from the
facet. The laser linewidth narrows approximately as the
square of the external reflector separation. Using
a
dif-
fraction grating as the external reflector aids single-mode
operation, stability, and allows the laser to be tuned over
hundreds of angstroms [19]. Wyatt and Devlin [20] have
reported stable single-mode operation with a linewidth of
only
10
kHz with such devices.
The aim of this paper is to review phase noise in semi-
conductor lasers.
No
attempt will be made to broadly sur-
vey all work in this field, instead we will concentrate in
moredetail on what seems to be the essentials of the subject,
especially those aspects that are relevant to optical com-
munications. In Section
11,
we will develop the equations
governing intensity and phase fluctuations in semiconduc-
tor lasers. We will take up
the
linewidth, lineshape, and
time dependence of phase fluctuations in Section 111. We
will deal with line narrowing by means of a passive sec-
tion and by optical feedback in Section IV. We will also
discuss the instability of coherence collapse brought about
by optical feedback in this section. In Section V, we re-
view the detectability limits for several keying schemes
used in coherent optical communications and the line-
width requirements to reach these limits.
A
brief summary
is given in Section VI.
11. FLUCTUATIONS
IN
SEMI CONDUCTOR
LASERS
A.
Classical Description of Quantum
Noise
Before describing noise in semiconductor lasers, we
should comment on how to picture the radiation field of a
laser. This is confusing because
of
the complimentary
particle and wave descriptions of light. Phase noise, the
subject of this paper, is a wave phenomenon, while shot
noise, which limits the detectability
of
coherent waves, is
a particle phenomenon. We shall think of laser radiation
as a classical wave field described by a complex ampli-
tude
p
(t ).
The shot noise aspects of laser radiation will be
thought of as noise associated with the carriers which re-
sults from generation and detection of light, because en-
ergy can only be added or removed from the radiation
field in quanta of
Aw.
This viewpoint may not be the only possible one, but it
is both simple and consistent with a number
of
fundamen-
tal studies of radiation and lasers:
1)
Hanbury-Brown and Twiss [21] showed that when
radiation is divided between two detectors, each detected
signal consists of a correlated part proportional to the fluc-
tuating light intensity
Z(t)
=
I
/3
( t )
plus uncorrelated shot
noise, which can be thought of as occurring in detection.
2) Using the quantum theory of radiation, Glauber [22]
showed that each mode of
the
free-radiation field pos-
sesses a continuous set of coherent states. The optical field
associated with these states oscillates sinusoidally at the
mode angular frequency
w
and approaches the field of a
classical monochromatic wave when the expectation value
for the energy of the coherent state is many
Aw
and un-
certainties demanded by Heisenberg’s principle are neg-
ligible.
A
coherent state is a pure quantum mechanical
state, not a mixture, but a state that does not have a def-
inite energy. When the energy of a coherent state is mea-
sured, destroying the state, discrete energies
nfiw
are
found that have a Poisson distribution. The optical field
of a single-mode laser can be thought
of
as a mixture of
coherent states, that becomes nearly a pure state far above
threshold.
3)
By using a basis of coherent states, Lax and Louise11
[23], [24] showed that, even in the case of nonfree fields,
the averages of essentially all measurable laser properties
can be expressed in terms of averages of a classical
(c-
number) wave field
P(t )
associated in a precise manner
with the quantum field amplitudes. Starting with a
fully
quantum mechanical model of a laser, they transformed
the quantum problem into a classical problem
of
calculat-
ing the statistical properties of a fluctuating wave field.
They derived the Langevin rate equations and the Fok-
ker-Planck equations that
P(t )
and the probability distri-
bution
P( p,
t)
satisfy. We will discuss the Langevin rate
equations in the next section.
B.
Langevin Rate Equations
quantity, as
We will describe fhe optical field of the laser, a real
E(x, t )
=
B[P(t)
@(x>
+
P(t)*
+(x>*]
(1)
where
/3
(t )
gives the time dependence,
+(x)
gives the spa-
tial dependence of the optical mode, and
B
is a constant.
It is convenient to choose
B
so
that the average intensity
,
(
I )
equals the average number of photons in the mode
(
P
)
(electromagnetic energy/Awo). The complex ampli-
tude
p
( t )
can be expressed in terms of two real quantities:
intensity
Z(t)
and phase
4(t )
P(t)
=
~ ( t )
exp
(
-
iwot
-
i4
(t)).
(2)
The behavior
of
a semiconductor laser is described by the
field amplitude
/3
and carrier number
N
that controls the
gain of the mode. The laser can thus be described as a
classical system having 3 real variables:
p’
,
p”
,
and
N
or
I,
4,
and
N,
where
p
=
/3’
+
ip”.
The laser, like many classical systems having several
variables fluctuating in time, satisfies a set of first-order
ordinary differential equations that include random Lan-
gevin forces [25]. Consider a system with variables
a
=
(al,
a2,
*
*
).
The Langevin rate equations are
u i
=
Ai @,
t )
+
F&),
i
=
1, 2,
- -
*
.
(3)
The
Ai
are chosen
so
that
(
Fi (t ))
=
0. Without the Lan-
gevin force,
( 3)
is the set
of
rate equations governing the
average values (with a small correction when products of
small fluctuations are included).
The role of the Langevin force is to account for how
the statistical distribution of the variables
a( t )
changes in
time. For example, suppose that at
t
=
0,
the variables
are described by
ao.
After a time
t,
the fluctuations in the
system change
a.
into a distribution of values
a
described
by probability distribution P
(a).
This distribution could
be found by using a computer to integrate ( 3) many times
from
0
to
t
with a suitably chosen random force.
In choosing
Fi
(t ),
we will make the common assump-
tion that the system is Markoffian
[25],
i.e., that the ran-
dom forces have no memory and the correlation of their
products function is a delta function:
( Fi ( t ) )
F,( u) )
=
2Dg6(t
-
U)
(4)
where
D,
is called the diffusion coefficient or diffusion
matrix
[25].
This name is appropriate because the random
forces cause
P( a)
to spread in a way analogous to diffu-
sion. The spread is limited the drift term by
Ai,
which tries
to restore
a
to the steady-state value.
The Markoffian assumption is justified in the case of the
semiconductor laser for which the major source of fluc-
tuations is spontaneous emission. This process is only
correlated for a carrier scattering time that is of order
s,
a negligibly short time. Only the diffusion coef-
ficients are needed to calculate mean square fluctuations.
However, for calculations of the line shape and of error
rates, additional assumptions about the Langevin forces
must be made. These amount to assuming that the Lan-
gevin forces have approximately Gaussian amplitude dis-
tributions.
The Langevin rate equation for
P( t )
can be derived
either by using the method of Lax and Louise11
[23], [24]
or semiclassically by adding a Langevin force to the wave
equation
[26], [27].
The resulting equation is
where
f l
and
Fp
are complex and
wo
is the cavity resonance
frequency at threshold when the net gain
AG
is zero. The
linewidth parameter
a
determines the change in the cavity
angular resonance frequency with net gain. The optical
field is coupled to the carriers by the dependence of
AG
on the carrier number
N.
In the absence of Fp and fluc-
tuations in
N,
P
increases exponentially with
AGt/2
and
has an angular frequency of
wo
+
aAG/2
and
I
increases
exponentially with
AGt.
For a Fabry-Perot cavity of uniform semiconductor
material, it is easily established that
CY
=
An ‘/An

[7].
When the carrier number changes, the modes shifts in a
way to keep the real part
of
the propagation constant
k‘
=
wn
‘/c
constant. This changes the angular frequency by
Aw
=
-wug An
‘ l c
and changes the gain
G
by
AG
=
-2wAn
“ug/c.
The ratio of these two changes is
0112.
The diffusion coefficients relating Fp and Fp* are given
t
F,grt)dt
.
REAL
0
Fig. 1.
The
change of complex field amplitude
p(t)
during a
short
time
T.
The exp
( - w,t )
time dependence has been removed.
2Dpp* R
(7)
where
R
is the spontaneous rate. The physical content of
(6)
and
(7)
can be seen in Fig.
1,
which shows that during
a short time T, the Langevin force changes the complex
value
/3
by Fp
dt.
Equation
(6)
follows from stationarity
(that
(
Fp
( t
+
7)
Fp
( t ) )
is independent
of t )
and insures
that the change in
0
in Fig.
1
has a random angle
in
the
complex
P
plane
[28,
section
1181.
Equations
(6)
and
(7)
show that
Fp
has two independent components and that
the diffusion coefficient for a product of a component of
Fp in any direction with itself is
Rf 2.
Applying the law of
cosines to the triangle in Fig.
1,
it is easily established
that the average change in
I,
during a short time
T,
is
RT
[7].
Therefore it is reasonable to refer to
R
as the spon-
taneous emission rate.
For a closed cavity, such as an index guided mode with
high reflecting ends, it can be shown
[27]
that
and
where
el/
is the separation of quasi-Fermi levels of the
conduction and valence bands of the semiconductor.
Evaluations of
n,,,
for AlGaAs and InGaAsP lasers give
values of about
2.6
and
1.6,
respectively
[29]-[31].
This
parameter
n,,,
characterizes the incomplete inversion of the
levels associated with the lasing transition and goes to
1
when inversion is complete. Equations
(8)
and (9) are es-
sentially a statement of the fluctuation-dissipation theo-
rem
[25], [28,
sect.
1231.
They follow from the re-
quirement that in equilibrium, with no losses from the
cavity, the spontaneous emission rate
R
due to fluctua-
tions
Fp
must be equal to the rate of dissipation
-
G
(
I
)
,
HENRY: PHASE
NOISE IN
SEMICONDUCTOR LASERS
30
1
i
I
/
SLOPE
=
36.7
1
0
0.5
1
.@
1.5
2.0 2.5
I NVERSE
POWER
(mW-')
Fig. 2. Linewidth of an AlGaAs laser versus inverse power at three tem-
peratures. From Welford and Mooradian
[30].
where
( I )
is the equilibrium photon number given by
-nsp
[27]. (In this case,
G,
fiwo
-
el/,
and
nSp
are nega-
tive.) In the case of open resonators, such as encountered
in gain-guided structures or in lasers with low reflection
facets, relation
(8)
no longer holds and
R
can be much
greater than Gn,. This was shown by Petermann [32] for
gain-guided structures, and by Ujihara [33] and the author
[27] for the case of low reflecting facets. In the case of a
conventional semiconductor laser with uncoated cleaved
facets,, this enhancement is only about 13 percent [27].
Above laser threshold, the coupling of
I
and
N
must be
taken into account. In this case, it is more convenient to
express the equations for the field
6
in terms, of
I
and phase
4.
The transformation of (5)-(7) is easily done using
(2)
and applying the general procedure for transforming Lan-
gevin equations given by Lax [34]. This results
in
Z
=
AGI
+
R
+'F,(t) (10)
.a
4
=
-
AG
+
F+(t)
2
(1
1)
with diffusion coefficients
R
2011
=
2RI,
2D++
=
-
2
I'
2D,,
=
0.
(12)
Equations (12) can easily be understood with the aid of
Fig. 1. The fluctuations in
I
are due to
F,
given by the
product
of
2.$1
and the component of
Fp
parallel to and
the fluctuation in
4
is due to
F6
given by
the
component
of
Fp
perpendicular to
/3
divided by
$I.
The lack of cor-
relation of these two components of
Fp
leads to
D,+
=
0.
The transformation
p,
p*
to I,
r$
leading to (10)-(12) is
valid for Markoffian random forces; however, Fp and
Fp*
are not Markoffian on the time scale of optical fre-
quencies. A careful discussion of this transformation,
jus-
tifying
(lo)-( 12)
has been given by Lax
[
1
,
sect. VI.
The rate equation for the carrier number is
fi
=
C
-
S
-
GI
+
FN( t )
03)
where C is the current (carrieds),
S
is
the
spontaneous
emission rate into nonlasing modes (radiative and nonra-
diative), and GI is the
net
rate of stimulated emission. The
gain G is actually the difference between the rates
of
emis-
sion and absorption
G
=
R
-
A.
The diffusion coefficients
relating
N
with
I
and
q5
are
2DNN
=
C
+
S
+
( R
+
A)
I,
2 0 ~ 1
=
-2RI,
2016
=
0.
(14)
These diffusion coefficients may be obtained directly by
the methods of Lax and Louise11 [23], [24], but are more
difficult to obtain by other methods. The diffusion coeffi-
cients for carrier number
N
and photon number
P
may be
simply derived, because both quantities undergo shot
noise fluctuations (during emission and absorption events,
N
and
P
change oppositely by 1). For various
ai,
fluc-
tuating purely from shot noise, the diffusion coefficients
can be determined by inspection
of
the rate equations 1251.
The diffusion coefficient 2Dii is just the sum
of
rates in
and rates out and 20, is the negative of the sum the rates
for which
i
and
j
both change. For example, 2DNN
=
C
+
S
+
AP
+
R( P
+
1) and
2DNp
=
-(AP
+
R( P
+
1)).
However, the physically useful quantity is intensity
I,
not photon number
P.
While
(
I )
=
(
P ),
Iand
P
have
different distributions and different diffusion coefficients.
We can think of the instantaneous distribution of
P
as a
Poisson distribution determined the average value
I(t).
This leads to
the
relation
(f?
=
( I 2 >
+
( 1 )
(15)
which can be rigorously derived. This relation was used
by McCumber [35], Lax [36], and Shimpe [37] to relate
the diffusion coefficients of
P
and
I.
It can be used to de-
rive (14).
111. POWER
SPECTRA
OF
SEMICONDUCTOR LASERS
A. Below
Threshold Operation
Laser operation naturally divides into two regimes that
are relatively easy to describe: below threshold operation,
in which
-
AG is large and fluctuations in
N
that alter AG,
can be neglected; and above threshold operation, where
fluctuations
of
I
and
N
are stabilized and may be regarded
as small deviations from the steady-state values. The tran-
sition region near threshold has been calculated for gas
lasers, but not for semiconductor lasers [4], [14]. In this
section, we will consider operation below threshold.
The power spectrum is usually measured by passing the
laser radiation through a scanning Fabry-Perot interfer-
ometer and measuring the power or by adding a nearly
monochromatic (local oscillator) field, having a fre-
quency substantially different from the laser mode, to the
laser field and then measuring the power in the beat spec-
trum. In both methods, the measured spectrum is propor-
tional to
Wp
( w),
the spectral density of the Fourier trans-
form of
p
( t )
P(w> =
-
p(t)
exp
(iot)
dt.
(16)
&
--
302
JOURNAL
OF
LIGHTWAVE
TECHNOLOGY,
VOL.
LT-4. NO.
3,
MARCH
1986
It is easily shown by applying the principle of stationarity,
that is steady-state operation, there is no correlation be-
tween different Fourier components of any of the system
variables or Langevin forces
[28,
sect.
1181
( P ( W') *
P( 0 ) )
=
Wp ( W)
6( W
-
W')
(17)
and that
m
Wp(w) =
s
(
P(t)*
P(O)>
exp
( i w0
dt.
(18)
Below threshold, with AG regarded as constant, the
Fourier component of the field amplitude
(a)
can be cal-
culated immediately by writing
( 5 )
in terms of its Fourier
components
- m
Fp
( W)
P( w)
=
(
AG
2
1
"2"
(19)
i
W ~ + C Y - - W
--
where
Fp( w)
is the Fourier transform of
Fp,
defined ex-
actly like
(0)
in
(16).
The autocorrelation functions of
the Fourier components have the same diffusion coeffi-
cients as the time components. For example,
(
F~( w') * F~( w) )
=
R~ ( w'
-
w).
(20)
The calculation of
(
P( w') * P( w) )
using this relation re-
sults in
showing that the laser below threshold has a Lorentz
lineshape and a linewidth
of
-AG
R
Av =- - -
-
(22)
27r
27rI
where
AG
=
- R/Z
used in the last equality can be ob-
tained by taking the average of
(10).
When
I
is related to
optical power, (22)
is
the Schawlow-Townes linewidth
formula. It is only valid below threshold. One application
of this formula is to give the noise bandwidth of a laser
amplifier.
B.
Lineshape
and
Phase Noise in
a
Semiconductor
Laser Operating Above Threshold
Above threshold, intensity fluctuations become stabi-
lized. This stabilization process greatly increases the
linewidth-power product. Assuming that amplitude fluc-
tuations in
P
(t)
are negligible, we can use
(2)
to express
the correlation function
(
6
(t)"
/3
(0))
as
(
P@>*
P(O>>
=
I
exp (exp
(i&(t)))
(23)
where
I
is the average intensity and where
A4
(t )
=
4
(t)
-
c$
(0). The neglect of fluctuations in the amplitude of
P
in (23) is not entirely correct. Vahala
et
al.
[
181
showed
that correlations of phase and intensity fluctuations at the
relaxation frequency could explain about a 20-percent
asymmetry in the structure occurring in the tails of the
lineshape. For simplicity, we will neglect this effect.
In general (exp
( i
A4
( t ) ) )
is
difficult to evaluate, but in
the case where A4( t ) is
a
Gaussian variable (one with a
Gaussian probability distribution), it is easily shown by
integration that
[
11
(exp ( i A4W) )
=
exp
1-i
(A4(t)*)I.
(24)
Thus, if
Aq5(t)
is a Gaussian variable, evaluation
of
the
field correlation reduces to evaluating the mean square of
A4
( t ).
The justification that
A4
(t>
is a Gaussian variable rests
on the fact that above threshold, the deviations of
I
and
N
from the steady-state values are small. In this case, the
equations obeyed by these deviations and
4
(lo), (1
l ),
(13)
are nearly linear and driven by nearly Gaussian Lan-
gevin forces. The solution of a set of linear Langevin
equations driven by Gaussian forces will also be Gaussian
distributed
[25].
The force Fo is Gaussian. It arises from
the additive contributions of many independent sources of
spontaneous emission throughout the cavity
[27]
and by
the central limit theorem [38], this is sufficient
to
ensure
that each component of
Fp
will be
a
Gaussian variable.
The forces
FI
and F+ are formed by products of compo-
nents of
Fp
and
I'
To the extent that fluctuations in
I
can be neglected, these forces are Gaussian variables. The
force
FN
arises from shot noise and should have a Poisson
distribution, but the event rate is
so
high (about 107/ns),
that it is also a Gaussian variable.
The understanding of the Lorentzian linewidth of a laser
above threshold, does not require a complete solution of
(lo),
(1
I),
(13).
The line broadening is primarily due to
low frequency fluctuations. At low frequencies, the
suppression of intensity fluctuations is extremely good.
We can make use of this by neglecting
Z
and solve for AG
in
(10)
Substituting this expression into
(1
I ),
and dropping the
constant
RII
term, which only causes
a
small frequency
shift, we arrive at an equation for
6
6
=
F,(t)
- -
FI(t).
01
21
(26)
The phase change during a time
t,
A4
=
+( t )
-
4(0)
is
given by integrating this equation
A+
=
1'
F,(t)
dt
-
-
F,(t)
dl.
(27)
Squaring and averaging this equation with the aid of
(4)
and the diffusion coefficients (12) we find
R
(A$(t )')
=
-
(1
+
a2)
t.
21
Substitution
of
(28) into (24) and (23) shows that the
21
a
st
0
0
(28)
HENRY:
PHASE NOISE
IN SEMICONDUCTOR LASERS
303
1
I
1
I
10,000
11,000
12,000
13,000 14,000
ENERGY
(Cm-’)
Fig.
3.
Spectrum
of
the real and imaginary changes in refractive index of
the
GaAs
active layer of
a
buried heterostructure laser when
i t
is excited
from low currents up to threshold;
hvI.
is
the laser energy. From Henry
et
al.
[8].
field autocorrelation function decays exponentially with
correlation time
T,,,,
=
414
1
+
a2)
R.
Substitution of
(28)
into (23) and using (1
8)
results in a Lorentz lineshape,
with a full width of
I
R
4lrz
Av=- - - ( l
+ C Y )
2
which differs from the below threshold case by (1
+
a’)/
2.
The factor of
$
is due
to
the suppression of intensity
fluctuations that contribute to the linewidth below thresh-
old. The factor
1
+
a2
is due to the increased phase
changes brought on by this suppression.
To compare with the experiment, it is necessary to
reexpress
R
and
I
in (29). We can use
(8)
to write
R
in
terms of
G
and write
G
=
gu,
=
-In
(R,) ug/(qL)
where
R,
is the facet power reflectivity,
TJ
is
the quantum
effi-
ciency, and
L
is the cavity length. The facet power and
intensity are related by 2P0
=
TJGZ~V.
With these changes,
the linewidth is given by
Equation (30) was confirmed by Welford and Mooradian
[30], who measured both the linewidth and the other pa-
rameters as a function of temperature (Fig. 2). However,
they found an unexpected effect. In addition to the line-
width varying as
P;‘,
they found a power independent
component that is nearly negligible
at
room temperature,
but
is
substantial at low temperature. The power indepen-
dent component has not been satisfactorily explained as
yet 1391.
C.
Line
Structure
High-frequency fluctuations
the wings of the lineshape and
short-time durations. It is just
contribute to structure in
to the change in
q5
during
these short-time duration
changes in phase that can lead to errors in optical com-
munications. To compute
( A4
( Q2)
with no restriction on
time, we have to solve the full equations
(lo),
(1
l),
(13),
but linearized to describe small oscillations about the
steady state, The equations governing the deviations in
4,
I,
and
N
are obtained by expanding
I
and
N
as
Z(t)
=
z
+
p
(t)
N( t )
=
N
+
n(t)
(31)
and expanding
S,
G,
and
aAG
as
s ( t )
s
f
SNfl(t)
(32)
G(t)
=
G
+
G,n(t)
-
G,p(t)
(33)
aAG
=
aGNn(t)
(34)
which results in the linear equations
4
=
aGNn
+
F,(t)
(35)
p
=
GNZ~Z
-
+
F/(t )
(36)
=
-rNn
-
Gp
+
Fv(t)
(37)
where the damping coefficients
=
G,Z
+
R/I
and
rN
=
GNZ
+
SN.
Equations (32)-(34) account for the change
in spontaneous emission rate with carrier number, the
change in gain with carrier number and light intensity,
and the change in mode frequency with carrier number.
The change in gain with light intensity (gain saturation)
is necessary to account for the large damping of relaxation
oscillations that is observed in index guided lasers and the
increase in damping with light intensity
[
171, [40]. It is
very likely that gain saturation results from spectral hole
burning [41]. Spectral hole burning causes a gain change
that is nearly symmetric about the laser line 1411. The
dispersive change in refractive index associated with this
change should be zero at the laser line and for that reason
makes no contribution to (34). On the other hand, the
change in gain associated with a change in carrier number
occurs primarily at higher photon energies where
it
results
in a decrease in absorption. This is illustrated in Fig.
3,
where the changes in refractive index with carrier density
An

and
An

for AlGaAs are plotted versus energy
[B].
The fact that the laser line occurs in the tail of a steep
absorption edge and in the tail of the gain change explains
why
a
typically has values of 4-7 in index guided lasers
operated at room temperature.
The means square of
Aq5(t)
is easily found by solving
(35)-(37)
[
171. This
is
done by writing these equations in
terms of their Fourier components, solving for
( 4
( w) ’ )
and then converting back to
( A4
( t ) 2)
by contour integra-
tion. We will only give the result
R
(
A4(t )2)
=
-
(1
+
a2A)
t
21
+
a2A
[COS
(36)
-
exp
(-rt)
cos
(Qt
-
36)]
2 r cos 6
(3
8)
where
Q
GyGG,vI)”2
is the relaxation frequency,
r
=
-1
..
~.~
-.
304
JOURNAL
OF
LIGHTWAVE TECHNOLOGY, VOL. LT-4,
NO.
3,
MARCH
1986
n
t
(nsec)
Fig.
4.
Mean square phase change
( A+( t )')
versus time
for
weak damp-
ing, moderate damping, and the linear approximation. From Henry
[17].
(I?/
f
rN)/2
is the damping rate,
A
=
[(l
-+
r,/Q2
+
I':X/(G2RZ)]/(l
+
rhrI'l/GGNI)2
is a constant near that is slightly less than unity and
6
=
tan-'
(I'iQ)
is a small angle [17].
The average
( A+
(t )2)
is the sum
of
a linear term and a
damped oscillatory term. This is illustrated in Fig. 4 [17].
The linear term gives rise to the linewidth and the corre-
lation time, while the oscillatory term causes the satellites
to occur, separated from the main by multiples of the re-
laxation oscillation frequency. The spectral shapes arising
from the values of
(Ad,
( t ) 2)
plotted in Fig. 4 are shown
in Fig.
5.
The three curves correspond to weak damping
in which gain saturation is neglected, gain saturation ap-
propriate for spectral hole burning in GaAs, and the linear
approximation (28), in which relaxation oscillations are
neglected and the lineshape is exactly Lorentzian. They
were plotted by evaluating (23j,
(24),
(38) with the aid of
a fast Fourier transform program
[
171. The side peaks tend
to be of order
1
percent of the main peak, because of the
strong damping of relaxation oscillations in index guided
lasers. The average
(Ad,
( t ) 2)
has been directly measured
by Diano
1151,
and by Eichen and Melman [42], who
found an excellent fit of the data
on 1.3-pm InGaAsP las-
ers using (38) with
A
=
1.
Their data is shown in Fig.
6.
They also found good agreement between the Fourier
transform of exp
(-4
(Ar$(t)'))
and the spectrum rnea-
sured with a scanning Fabry-Perot interferometer.
IV.
LINE
NARROWING
A.
Ideal Fabry-Perot Cavity with
a
Passive Section
The linewidth power product of conventional semicon-
ductor lasers can be dramatically reduced by adding a pas-
sive section to the laser cavity. This is done by
4R
coat-
ing one end and adding an external reflector separated by
about 10-20 cm from the semiconductor chip. The spec-
trum of such a laser fabricated by Olsson [43] is shown
in Fig. 7. The data was taken by beating two similar ex-
Po=lmW
I - 3.l xl o~
$=I
52ghz
cY.5
3
V-Uo
(ghz)
Fig.
5.
Power spectrum of the laser line calculated for the three (A$(r)*)
functions of Fig. 4-weak damping, intermediate damping, and the lin-
ear approximation which gives a Lorentz line shape.
From
Henry
[17].
2.0
c
1
I
/
0.0
I 1
I
l o l l!
0.0
0.4 0.8
1.2 1.6
2.0
dns)
Fig.
6.
Direct measurement
of
(A+(r)')
and comparison with a theoretical
fit
using (38). After Eichen and Melman [42].
ternal cavity lasers together. The lineshape is Lorentzian
in shape and corresponds to a
AvP,
of about
10
kHz
*
mW
for each laser. This is a reduction of about lo4 from
the laser
AvPo
=
100
MHz
.
mW €or each isolated laser
prior to AR coating.
This enormous linewidth reduction of a laser with a long
passive section Lp can
be
readily understood. The de-
crease of linewidth in proportion to Li 2 can be understood
by consideration of
(29),
which applies to a uniform laser
cavity. A long passive section will decrease the average
spontaneous emission rate
-
Li' and increase the average
intensity Z
-
Lp, for a fixed Po, resulting in a linewidth
reduction proportional to Li 2.
In [27], the linewidth of a Fabry-Perot cavity with a
HENRY: PHASE NOISE IN SEMICONDUCTOR LASERS
305
0
2
I ’
I
m
n
I I
i
0
W
v)
LT
W
3
0
n
a
-
60
-80
t
-1
-100
I I
-
50
50
AV
( MHz)
Fig.
7.
Beat spectrum
of
two
1.5Sym
external cavity lasers having grating
reflectors
15
cm from the laser diode. The center of the spectrum is as
1.328
MHz,
From
the data
of
Olsson
[43].
passive section was calculated under the ideal assump-
tions of a perfect AR coating and loss from the passive
end being the same as from the cleaved uncoated end. It
was found that the linewidth forinulas
(29)-(30)
are re-
duced by a simple factor given by the square of the ratio
of the time spent in the active section to the time spent in
both sections, where the time in each section equals
Lh,.
This factor is
(39)
where
a
and
p
refer to the active and passive sections. The
right-hand expression in
(39)
applies in the case of a long
air-filled passive section, where
nRa
is the group index of
the semiconductor. For
Lp
=
15
cm,
L,
=
300
pm,
and
n;,
=
4,
this reduction
is
about
1
.5104,
so
that a linewidth
of 150 Mhz would be reduced
to
10 kHz. This is only
slightly larger than the reduction observed by Olsson
[43].
Linewidth reduction favors a long external cavity, how-
ever, the longitudinal mode spacing decreases as
L-’
and
therefore single-mode operation becomes more difficult to
obtain for large
L.
For a 15-cm external cavity, the lon-
gitudinal mode spacing is about
1
GHz or
0.08
A
at
1.55
pm. All but one of these modes can be suppressed by add-
ing frequency selective filters to the external cavity. A
bandwidth of a few angstroms can be achieved by using
a diffraction grating as the external reflecting element
[20],
[43].
Additional mode selection can be achieved by plac-
ing an etalon in the external cavity
[43],
by coupled-cav-
ity effects due to an imperfect AR coating and due to op-
tical nonlinearities
[41].
Piezoelectric tuning of the length
of
the
external cavity can be used to select out one mode
WI.
B.
Line Narrowing
by
External Feedback
The analysis in Section IV-A is restricted to the case of
an ideal passive section with a perfect AR coating of the
laser facet. Here we review an analysis developed by Kik-
uchi and Okoschi
[44]
and Agrawal
[45]
that is applicable
for arbitrary laser facet reflectivities and weak or moder-
ate levels of feedback.
It
is useful to.think
of
external
feedback as a form of self-locking. The laser field
(t )
is
injection-locked to a field that was emitted earlier from
the laser and
is
reinjected after making a round trip in the
external cavity. Feedback is described by adding a term
KP
(t
-
T ~ )
to the equation for the laser field (5) that rep-
resents the effect of the field coupled back into the laser
cavity after a delay
T~
i
AG
2
p
=
-io0
+
-
(1
-
io!)
*
P( t )
+
K P ( t
-
70)
f
Fp(t).
(40)
An equation
of
this type was first used by Lang and
Ko-
bayashi
[46].
It is readily shown by considering steady-
state field propagation that
where
Ro
is the power reflectivity of the external reflector
including coupling losses,
R,
is the facet power reflectiv-
ity, and
T,
is the round-trip time in the semiconductor cav-
ity. Equation
(40)
is not valid for strong levels of feed-
back
KT,
>>
1,
since multiple round trips in the external
cavity have been neglected.
Equation
(40)
is more conveniently discussed by chang-
ing variables to
p
(t )
=
p
( t )

exp ( - i wt ). The steady state
is found by keeping
6’
constant and fluctuations are found
by solving the equation for
0’
under various approxima-
tions. The transformed equation is
-iAw
+
-
(1
-
ior)
2
P( t ) ’,+ P( t
-
T~)’
exp
(iCP)
+
Fpt (t ).
(42)
where
CP
=
w 0
and
Aw
=
w
-
wo.
The steady-state op-
erating point is found by setting
6’
and
Fp,
to
zero. Also
A G
=
-2K
COS
CP
(43)
Aw
=
-K(Q
cos
CP
+
sin
a).
(44)
The different values of
CP
form an ellipse of solutions
AG
versus
Aw,
where
CP
is the steady-state phase angle be-
tween the cavity field and the feedback field; see Fig. 8.
The additional relation
CP
=
W T ~,
results in a discrete set
of
values for
CP,
corresponding to the composite cavity
modes. The ellipse forms the locking range over which
self-locking can take place.
In
practice the locking range
is restricted to negative values of
AG,
the portion of the
range for which gain reduction takes place. If the locking
range covers many external cavity modes, the laser will
operate on one or several modes having
the
greatest gain
reduction. If a grating is used as an externai reflector, one
can tune across the locking range. But in the range of pos-
itive AG the laser will choose to operate on
a
different
longitudinal mode, for which feedback from the grating
is negligible. For strong feedback, the laser can be below
threshold except for part of the range with the most neg-
306
JOURNAL OF LIGHTW.4VE TECHNOLOGY,
VOL.
LT-4,
NO.
3, MARCH 1986
L I I l I I I I'I I I ~ l 1'-
@
=
180"
2 -
-
l -
-
Y -
I
4 -
C J O
-I
-
~
Aw:w-w0
a =
6 5
-2
-
-
Au=-aK
I I I I I I
l l l ~,l l
-5
0
5
AW( K.]
Fig.
8.
Gain change versus angular frequency change for the steady states
of
operation of a laser with reflective feedback. The laser modes are
actually points along the ellipse that become closely spaced when the
locking range contains many
modes.
The
angle is the phase angle be-
tween the injected field
p ( f
-
T(,)
and the cavity field
@(I ).
From Henry
and Kazarjnov 1.541.
ative AG. Decreased values of AG are observed by in-
creased light intensity [47]. The change in intensity is
proportional to the feedback parameter
K.
Equation (42) is a nonlinear differential-difference
equation containing fields at two different times. This
complication has prevented exact solution. The injected
field suppresses high-frequency phase fluctuations that al-
ter the phase of the laser field relative to the injected field.
Only phase changes taking place in a time large compared
to the round-trip time
T~
are unsuppressed. The low-fre-
quency behavior can be found by expanding
P( t
-
70)
'
1441,
1451
/3(t
-
To)/
=
P( t )/
-
T"P'(t)'.
(45)
This approximation changes the field equation to
0'
=
(1
'+
KT0
exp
(i+)
(46)
where AG represents the deviation from the steady state
and terms set to zero by the steady-state condition have
been dropped. This equation is essentially the same as
( 5 )
(with
w,
removed by transforming to
P')
except for the
complex constant dividing the right-hand side. This alters
the effective value for
01
and the Langevin force. With
these changes, the linewidth can be calculated in the same
manner as in the derivation of (29). The result found by
Agrawal [45] is
Av
=
A
vo
[ I
+
K T ~
(cos
CP
-
01
sin
+)12
(47)
where
Av,
is
the linewidth in the absence of feedback
given by (29) and
(30).
For
a
=
0, (47) reduces to the
result derived by Kikuchi and Okoschi [44]. The denom-
inator is positive and results in line narrowing all along
the lower portion of the ellipse for negative
d G( 2 ~ cos
(CP)
>
0). Line broadening is possible, but only at very
low feedback levels, when there is a single mode on the
ellipse. Then adjusting the cavity length will move the
mode onto different regions of the ellipse for negative AG
and narrowing and broadening can be observed. At higher
levels of feedback, there will be several modes or more
on the ellipse and lasing will occur for the mode of lowest
gain. The factor which controls the line narrowing is
K T ~.
For
K T ~
>>
1,
narrowing is proportional to
(ngaLU/Lp)*,
the same factor as we found in the analysis in Section
IV-
A. The effect of feedback on the linewidth becomes neg-
ligible when
~7~
becomes much less than unity. For ex-
ample, with L,
=
300 hm and
Lp
=
30
cm,
K T ~
is unity
for
Ro/R,
=
lop5.
This is the reason why for accurate
linewidth measurements, external reflection feedback
must be reduced to about
60
dB.
Suris and Tager [48] have managed to solve the nonlin-
ear differential-difference equation (42), without making
expansion (45), but only for the case
01
=
0.
They show
that the above analysis (47), predicting a single laser line
narrowed by feedback, is only valid if the linewidth of the
isolated laser
Avo
is small compared to the external cavity
mode spacing. For example, if
Avo
=
100
MHz,
this cor-
responds to Lp
<
150
cm. Otherwise, the semiconductor
laser emission associated with external cavity modes con-
tains a number of narrow lines associated with external
cavity modes with an envelope width equal to Avo.
C. Coherence Collapse
According to (46), line narrowing should increase with
increasing feedback. However Lenstra
et ai.
[49] reported
that at relatively high-feedback levels, the laser line be-
comes enormously broadened. They called this effect co-
herence collapse. This phenomenon was encountered in a
different way by Temkin
et
ai.
[50].
They studied the be-
havior of lasers under moderately high levels of feedback
from distant reflectors and found that they were unstable.
Some of their data on this instability is shown in Fig. 9.
Coherent feedback reduces the threshold and increases the
light intensity of the laser. It takes about
10
round trips
in the external cavity for the laser to
go
from initially on
to the state of coherent feedback. This corresponds to
moving from the origin to the bottom of the ellipse
in
Fig.
8. Just as the steady state is reached, the laser becomes
unstable. It suddenly returns to the initial operation at
higher threshold and lower intensity in which it is not ben-
efiting from coherent feedback. Then the build-up process
starts all over again. This instability had been observed
previously, but had not been explained
[51]-[53].
The line
broadening observed by Lenstra
et
al.
[49] appears to be
merely the frequency chirp associated with this cycle.
Kazarinov and the author have recently explained this
instability [54]. It corresponds to a large and rapid phase
fluctuation in which the laser jumps out of its self-locked
state. This instability is puzzling because prior to its on-
set, the operating point of the laser reaches a steady state
at the bottom of the ellipse with
CP
=
0 near the long
wavelength end of the locking range. Analyses of stability
of injection locking show that the short wavelength end
is
HENRY: PHASE
NOISE IN
SEMICONDUCTOR LASERS
L=
30
Fig.
9.
Time dependence of the intensity of a laser exhibiting instability
due to reflective feedback. Three levels of feedback are shown ranging
from power field reflectivities of
-5
dB in the upper figure
to
-15 dB
i n
the lower figure. The arrows indicate zero intensity. The steps in
i n-
tensity build
up
during successive
to
round trips in the external cavity.
Coherence collapse occurs after the steady state is reached. The jitter is
due to the stochastic nature
of
the instability. From Temkin
et
al.
[ S O].
unstable and exhibits high-frequency self-pulsations, but
the long wavelength end of the locking range, near the
bottom of the ellipse, is stable and has strongly damped
relaxation oscillations
[ 55],
[ 56],
[12]. However, this
conclusion is the result of a conventional linear stability
analysis and coherence collapse results from nonlinear ef-
fects that are brought about by a rare but large fluctuation
in spontaneous emission intensity. Our analysis consisted
of approximately solving
(42),
with
P( t
-
7,)’
regarded
as constant, without linearizing this equation.
Consider a large fluctuation in spontaneous emission
which decreases the mode intensity. This reduces stimu-
lated emission and increases the carrier number. The car-
rier number change causes a momentary change in mode
frequency that reduces the alignment of the laser field
P(t)
and the injected field
j3(t
-
7,)
from
@
=
0 to
A+.
This
in turn decreases the stimulated emission in the cavity in
proportion to cos
(A+).
The decrease in stimulated emis-
sion increases the carrier number, which causes
A+
to in-
crease further, driving the laser out of its locked
+
=
0
state. There, other forces acting in the laser which try to
restore the steady state. These are just the forces that give
rise
to
relaxation oscillations. For small fluctuations, the
restoring forces win out, but for a large fluctuation in
1
1 1 I,,,,/
0.02
0.01
0.1
0.2
10.002
AWC
Fig.
10.
Average instability time
t,
versus the fractional decrease in the
threshold current
AC/C,
a parameter proportional to feedback parameter
K.
The set
of
curves
t,
are
for
different intensities. Parameter
I,
is shown
in the inset. The inset
shows
the kink in the light versus current relation
resulting from the change in
t,
in going from a
to
c.
From Henry and
Kazarinov
[54].
spontaneous emission, the laser will jump out
of
lock.
The phenomenon has a threshold and can be described by
fluctuations that take the system over a barrier
[54].
An interesting aspect of this phenomenon is that for
strong enough feedback, self-locking becomes very
strong, the barrier becomes very high, and the probability
of jumping over the barrier becomes negligible; stability
is restored. This is illustrated in Fig.
10,
where the av-
erage time for instability to occur
is
plotted versus the
degree of feedback. For strong feedback, the average in-
stability time becomes exponentially large. This renewed
onset of stability for strong feedback was observed by
Temkin
et
al.
[50]
using
a
laser with an AR-coated facet.
Our calculation also showed that the stability is greater
near threshold, where the light intensity is small, than well
above threshold. This change in stability as the light in-
tensity increases give rise to a “kinked” light versus in-
tensity relation that is illustrated in Fig.
10
and was ob-
served by Temkin et
al.
[50].
v.
LINEWIDTH REQUIREMENTS
IN
COHERENT
OPTICAL
COMMUNICATIONS
A.
Shot
Noise Limit
Coherent optical communications is currently being
rapidly developed in many laboratories. This interest is
stimulated by several well-known advantages of this
method: by mixing the weak received field with a local
oscillator field, the detected signal can be sufficiently in-
creased and filtered
so
that noise in subsequent amplifi-
cation becomes negligible and the physical limit of de-
tectability, due to shot noise, can be approached; the
communication channels can be closeIy spaced, enabling
a single optical amplifier to simultaneously amplify many
channels.
The schemes for transmitting digital information are the
same
in
optical communications as earlier developed for
microwave communications
[57].
However, in optical
308
JOURNAL
OF
LIGHTWAVE TECHNOLOGY.
VOL.
LT-4, NO.
3,
MARCH
1986
communications, shot noise rather than thermal noise lim-
its detectability and phase noise, which is negligible in
the microwave case, hinders attainment of the fundamen-
tal limits. The physical limit of shot noise is not easily
reached. It seems that the closer
a
modulation method is
to the shot noise limit in the absence of phase noise, the
more sensitive it is to phase noise. In this section, we will
first consider the shot noise limit, which can in principle
be reached using homodyne phase shift keying (PSK). In
the next section, we will consider the more practical het-
erodyne schemes of differential phase shift keying
(DPSK), frequency shift keying
(FSK)
and amplitude shift
keying
(ASK),
and the linewidth requirements that lasers
used in these methods must satisfy.
In coherent lightwave detection, the incident optical
signal field
/3.A(t)
is mixed with
a
local oscillator field
Pe(t)
and then detected by
a
quantum detector, such
as a
p-i-n
diode. We can model the detector by the simple equation
for the number of carriers
n
generated during detection
h = - - +
n
T d
V d I P A
$-
fie(’
+
F,!(t)
(48)
where
T[,
is the inverse detector response time,
F,
is
a
Langevin force describing the shot noise fluctuations, and
qdI/3,.,
+
is the average rate of optical generation of
carriers
at
the detector. For frequencies less than the de-
tector response time, we can neglect
h
in
(48).
The de-
tected signals
s(t )
=
n/T,
is given by
s(t)
=
2Td( l Af B) l i 2
cos
( WAR
+
dAB)
+
F,~(t)
(49)
where
wAB
=
wA
-
W B
and
dAB
=
+A
-
+B.
The Langevin
force is due
to
shot noise of generation of carriers. The
diffusion coefficient
of
this force will be given by the av-
erage rate of generation
2 Dr m
=
qdr B
(50)
where generation due to
I,.,
has been neglected.
In principle, the shot noise limit can be reached
i n
ho-
modyne (PSK). In this case,
wA
=
wR
and the bits of

1

and “0” are generated by altering
dA
by
T.
It is assumed
that aside from this modulation, and
+*
are kept equal.
(A method for doing this is by means of
a
phase-locked
loop [-58].) The signal integrated over one bit time T is
given by an average value
( s)
=
2 q c l ( ~ A ~ B ) ” 2 ~
plus
a
fluctuation part
As
=
F,,(t)
dt.
The mean square of
As
is
(As’)
=
2D,, T
=
rd
I,
T. The number of photocarriers
generated in each bit will be Poisson distributed. Since
the average number
of
detected carriers per bit
is
large
compared to unity, this distribution is closely approxi-
mated by a Gaussian and
F,,(t)
can be regarded
as
a
Gauss-
ian variable. The signal associated with
“ I ”
and “0”
will be two Gaussian distributions centered at
the
two val-
ues of
(s).
The error probability PE is found by integrat-
ing the tails of the distribution that extend beyond zero.
This is an error function, but aside from
a
prefactor
of
order unity, it is
DPSK
P C
+
Ps
I
I NT COMP
FSK
ASK
P C
+
Ps
r
I NT
COMP
Fig.
11.
Schematic diagram of the methods
of
detection for: differential
phase shift keying
(DPSK).
frequency phase-shift keying (FSK), and
amplitude shift keying (ASK). The components are optical detector
(DET), handpass amplifiers
w,,~. w,,
w2,
time delay
T,
integrator (INT)
for time
T,
and decision circuit or comparator
(COMP).
7 -
DP S K
Ab’*
=
Au,
-
m
6 -
a
I
4 -
9
I
.,
100
200
300 400
500
600
Rg
A
VA
Fig. 12. Average bit energy
P,
(photons) versus the ratio of hit rate to
laser linewidth, necessary
to
achieve an error rate of
lo-’.
From
J.
Salz
1581.
=
exp
(
-2Ps)
(5
1)
where
PB
=
qZA
T the average number of detected signal
photons in each bit reaching the detector. Setting P,
=
IOp9,
we find
PR
=
10.
This is the shot noise limit, the
minimum detectable power limited by shot noise
[-58].
B.
Heterodyne Detection
Ideally, the homodyne PSK scheme is more sensitive
than other methods. However, it requires phase locking
the local oscillator to the received field. This is
a
complex
procedure and one that
is
very sensitive to phase noise
[58).
The heterodyne methods DPSK, FSK, and ASK are
more resistant to phase noise and nearly
as
sensitive
as
PSK. The detection methods for the 3 schemes we will
discuss are sketched in Fig.
11.
The minimum value of
Ps for these schemes are not easily calculated owing to
the nonlinear nature of the detection process that involves
squaring or taking
a
product of the detected signals.
How-
ever, these limits have been worked out using the same
method
as
for the microwave case
1-57],
[ 58],
but with
HENRY: PHASE
NOISE
IN SEMICONDUCTOR
LASERS
309
shot noise replacing thermal noise. Here, we will limit
ourselves to a qualitative discussion.
In the PSK case, we saw that
PE
depends exponentially
on the squared average signal divided by the averaged
mean square noise. This is generally true for all the de-
tection schemes. All heterodyne detection schemes mix
PA
and
PB
that differ in angular frequency by
@AB.
In this
case, the noise is unchanged, but the signal is sinusoidal
(49)
and the average squared signal is half that of the
homodyne case. This increases
PB
by
3
dB.
The most sensitive
of
the three heterodyne schemes is
DSPK. In this scheme, phase is modulated as in PSK.
After detection, the detected signal is divided, one half is
delayed by the bit time
T,
and the two channels are mul-
tiplied together. The resulting signal is proportional to cos
[ $AB( t )
-
$AB( t
-
T) ],
which changes sign when the two
successive bits differ. Ideally, this detection method suf-
fers only a 3-dB loss in sensitivity due to heterodyne de-
tection.
The signal does not depend on absolute phase
so
that it
will not be affected by slow phase drifts. There will be a
power penalty (increase in
PB),
however, if the angular
argument of the cosine, which
is
normally
0
or
T,
changes
due to phase diffusion during the bit time
T.
If the prob-
ability of a phase change
$A
-
qi B
of greater than
n/2
exceeds
l ow9,
unacceptable errors will occur regardless
of
the
signal power. We can readily estimate the linewidth
at which this will occur. The phase change
A$,.,
-
A$B
is
a
Gaussian variable. If we assume that both of the lasers
generating fields
A
and
B
have strongly damped relaxation
oscillations, then the mean square change of
$A
-
(bB
dur-
ing time
T
can be related to the linewidths
AvA
and
AvB
by
( 28)
and
( 29): ( A$:)
+
( A$;)
=
2nT(AvA
+
AYE).
The error probability is found by integrating a Gaussian
distribution with this mean square value to find the area
of
the
tail beyond
n/2.
The error probability is given by
the Gaussian distribution evaluated at
a12,
aside from a
prefactor
of
order unity. This results in
where
RB
=
liT
is the bit rate
[59].
If we set
AvA
=
AvA
and equate
PE
to we find
RB/A\vA
=
210.
This is the
minimum allowable bit-rate linewidth ratio for DPSK re-
gardless of power. Salz
[58]
finds that to nearly reach the
ideal limit of
PB,
occurring in the absence of phase noise
(3 dB above the shot noise limit,
PE
=
20) RB/AvA
=
500-600
is needed. His results are shown in Fig.
12.
The linewidth requirements for
FSK
and ASK are less
stringent than for DPSK, but this is purchased at the price
of
reduced sensitivity in the absence of phase noise and
increased bandwidth
of
the electronics. In FSK,
uA
is
modulated and
wAB
switches between two values,
w1
and
w2.
The detected signal is divided and each half
is
passed
through a narrow-band amplifier. Each signal is then “en-
velope detected” by squaring and integrating. The two
signals are then compared to determine whether the bit is
“1”
or
“0”.
Passing the signal through two bandpass
filters doubles the mean square noise compared to DPSK,
introducing an additional 3-dB increase in PB
[57], [58]
In ASK, the signal field is modulated on and
off.
After
detection, the signal is passed through a bandpass ampli-
fier and then envelope-detected by squaring. In this case,
the noise
is
the same as in DSPK, but the signal change
in going from
“1”
to
“0”
is reduced by a factor of
2.
This change increases
PB
by
6
dB
(Ps
=
80).
An approximate analysis of the effect of phase noise on
FSK and ASK has been made by Garrett and Jacobsen
[60],
[61].
They assume that the effect of phase noise is
to cause average frequency shifts
Aj.
When averaged over
a bit time
(PB
=
40).
Af
=
A$A
-
A 6 B
23rT
(53)
Af
has a Gaussian distribution for
Aj.
They calculate the
effect of phase noise on error rates by calculating
PE
as
a
function of Afand convoluting this with the Gaussian dis-
tribution.
Their analysis shows that
FSK
and ASK are much less
sensitive to phase noise than DSPK. For example, FSK
has a minimum linewidth requirement similar to DPSK.
If
the two frequencies in FSK are separated by
mRB,
where
rn
=
1-3,
a value of
Af
exceeding
mRB/2
will cause an
error. This will occur if
A$A
-
A+B
=
a m.
In the case
of
DPSK, an error occurs when the corresponding quan-
tity is
~ 1 2,
Therefore, for
m
>
1/2,
the linewidth require-
ments are greater for DSPK than for
FSK.
The error rate
for this frequency change is derived in the same way as
( 52)
and
is
given by
Comparing these two equations, we see that minimum
linewidth for FSK is
4m2
greater than for DSPK.
VI.
SUMMARY
Noise in lasers can be rigorously described in terms of
fluctuations of a complex classical wave field P(t) that has
both intensity and phase fluctuations. Below threshold,
both fluctuations contribute equally to the broadening of
the laser line, which is Lorentzian in shape and has a width
given by the Schawlow-Townes formula. Above thresh-
old, the suppression of low-frequency intensity fluctua-
tions by changes in carrier number changes causes an ad-
ditional broadening
of
( 1
+
a2)/2,
where
CY
=
4-7 for
AlGaAs and InGaAsP lasers at room temperature. This
results in
AuP,
=
60-120
MHz
-
mW. This lineshape
above threshold is approximately Lorentzian, but has
small side peaks separated from the line center by the re-
laxation oscillation frequency.
Enormous line narrowing can be achieved by extending
the laser cavity with a passive section. Linewidths of only
10
kHz have been achieved while still maintaining single-
3 10
JOURNAL OF
LIGHTWAVE
TECHNOLOGY, VOL.
LT-4,
NO.
3,
MARCH
1986
mode operation by
AR
coating one facet and adding
a
passive section about 15 cm long. Line narrowing by op-
tical feedback is a consequence of self-locking of the op-
tical field by the delayed field returning after external re-
flection. The instability of coherence collapse can occur
in which a large and rapid phase fluctuation causes the
laser to jump out
of
the self-locked state. This instability
is suppressed by very strong optical feedback.
The detection sensitivity in digital coherent optical
communications is limited by shot noise occurring when
light is detected and by phase noise. While the linewidth
is due to low-frequency phase fluctuations, errors in
op-
tical communications are caused by phase changes occur-
ring during one bit time. The minimum number of pho-
tons per bit
PB
depends on the method of digital
modulation. The shot noise limit
of
PB
=
10
for an error
probability of lop9 can be achieved by homodyne PSK,
but this scheme
is
extremely sensitive to phase noise. Het-
erodyne
DPSK
has a
P,
3 dB more than the shot noise
limit, which can be approached for
RBI Av A
=
500.
The
linewidth requirements for heterodyne FSK and ASK are
an order
of
magnitude less restrictive on linewidth than
DPSK and have
PB
that are
6
and
9
dB greater than the
shot noise limit, respectively.
ACKNOWLEDGMENT
The author wishes to thank
N.
A. Olsson,
J.
Salz,
M.
Lax,
R.
Schimpe, and G.
P.
Agrawal for stimulating dis-
cussions.
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*
Charles
H.
Henry
was born in Chicago,
IL,
in
1937. He received the M.S. degree in physics from
the University of Chicago, Chicago,
IL,
in 1959,
and the Ph.D. degree in physics from the Univer-
sity of Illinois, Urbana, in 1965.
Since 1965, he has been a member of the
staff
in the Semiconductor Electronics Research De-
partment, AT&T Bell Laboratories, Murray Hill,
NJ. From 1971 to 1976, he served as head of this
department. His research is primarily on the phys-
ics associated with light emittine device technol-
I
v
1985.
and Techniques.
New York: McGraw-Hill, 1966, Ch. 7. American Association for the Advancement of Science.
.~
ogy. He is the author of over 80 published papers.
[57]
M.
Schwartz, W.
R.
Bennett, and
S.
Stein,
Communication Systems
Dr. Henry is a Fellow of the American Physical Society and of the