1382
IEEE
JOURNAL
OF
QUANTUM
ELECTRONICS,
VOL.
QE-15,
NO.
12,
DECEMBER
1979
Coherence
of
a Room-Temperature
CW
GaAs/GaAIAs Injection Laser
AXEL R. REISINGER,
C.
D. DAVID, JR.,
K.
L. LAWLEY,
AND
AMNON YARIV,
FELLOW,
IEEE
Absrmct-The temporal coherence
of
a stripe-geometry double-
heterojunction GaAs/GaAlAs laser operating
CW
at room temperature
was
determined. A heterodyne detection scheme was used involving
the
mixing of the laser field with a frequency-shifted and time-delayed
image of itself in
an
interferometer. Because the laser device oscillated
in several longitudinal modes, the autocorrelation function of its output
exhibited resonances for specific time delays. The rate at which the
amplitude of these resonances decreased with increasing time delays
provided a measure of
an
apparent coherence length associated with
individual longitudinal modes. The coherence length,
so
defied, was
found to increase linearly with drive current in excess of threshold.
This
observation is interpreted
as
evidence that the intrinsic linewidth
of a longitudinal mode is inversely proportional
to
the coherent optical
power in that mode. Apparent coherence lengths were a few centi-
meters for a few milliwatts of total optical power emitted per facet.
For a perfectly balanced interferometer, a
sharp
heterodyne beat
signal
was
also
observed when the laser device was operated considerably
below threshold, i.e., in the
LED
mode.
T
I.
INTRODUCTION
HE COHERENCE ,properties of conventional GaAs in-
jection .lasers have been investigated by a number of
authors
[
11
-
[8]
.
All of these experiments were concerned
with homojunction devices operated either CW
[l]
,
[3], [4]
,
[8]
or pulsed [2], [5]
-[7]
at liquid nitrogen temperature
[2]
,
[4]
-
181
or below
[
13
,
[3]
.
The intrinsic laser linewidth
was determined by several methods, including the use of a
Fabry-Perot interferometer
[
11 ,heterodyne mixing of adjacent
longitudinal modes in large laser cavities [2], [3]
,
and homo-
dyne detection of correlated noise in an unbalanced Michelson
interferometer [4]. Spectral information can also be obtained
indirectly from measurements of the temporal coherence of
the laser by holographic techniques [5]
-
[8],
which amount to
a determination of the visibility of interference fringes in an
unbalanced interferometer. The best reported result on a
GaAs homojunction laser was a linewidth substantially smaller
than 150
kHz
(the detection system’s resolution) for a device
operated CW at
77
K
with
250
mW of optical power emitted
in
a single mode [4].
This
paper describes measurements of the coherence length
of
a
GaAs/GaAlAs double-heterojunction stripe-geometry
multimode laser operated CW at room temperature with an
Manuscript received April 4, 1979; revised July 5,1979.
A. Reisinger was with Texas Instruments, Inc. Dallas
TX.
He is now
with Optical Information Systems, Exxon Enterprises, Inc., Elmsford,
NY
10523.
Dallas, TX 75222.
C.
D. David, Jr. and
K.
L.
Lawley are with Texas Instruments, Inc.,
A.
Yariv is with the Department of Electrical Engineering, California
Institute of Technology, Pasadena,
CA
91 109.
100,
MHz
SPECTRUM
BOXCAR
ANALYZER
INTEGRATOR
x-v
RECORDER
Fig.
1.
Schematic
of
the experimental apparatus:
1
=
laser,
2
=
1OX
microscope objective,
3,
6,
and lO=apertures,
4
and
8
=beam
splitters,
5
=
acoustooptic Bragg cell, 7
=
stationary mirrors, 9
=
beam
stop,
11
=
photomultiplier,
12
=
movable mirror,
13
=
translation
stage, and 14
=
electric motor drive.
optical power output of a few milliwatts per facet. The ex-
perimental techniques used involved heterodyne mixing of
the laser field with a frequency shifted image of itself
in
an
unbalanced interferometer.
11.
EXPERIMENTAL APPARATUS
A schematic of the experimental apparatus is shown in Fig.
1.
The output of the laser source is collimated by means of a
1OX
microscope objective, reduced in diameter by an aperture,
and divided into two arms by a first beam splitter. One arm
goes through an acoustooptic Bragg cell operated at a frequency
of
100 MHz. An aperture is used to pass only the Bragg-
diffracted light beam, which has been frequency shifted by the
acoustic frequency. This beam is then redirected by means of
two stationary mirrors toward a detector. The second arm
of the interferometer is reflected back by a movable mirror,
traverses the first beam splitter, and is recombined with the
first arm by means of a second beam splitter. The detector
is a photomultiplier with
an
S-1
response (RCA-7102). It
sees the superposition of the laser field with a frequency shifted
image of itself.
In
this sense, the experiment amounts to a
heterodyne detection scheme, with the laser field playing the
role of local oscillator, and the frequency shifted field acting
as the signal [9]. The purpose of the movable mirror mounted
on a translation stage, which is driven by an electric motor, is
0018-9197/79/1200-1382$00.75
0
1979 IEEE
REISINGER
et
al.:
COHERENCE
OF
A ROOM-TEMPERATURE CW LASER
200
400
600
DRIVE CURRENT
(nl)
Fig.
2.
Laser power output (one side) versus drive current
for
the laser
device used
in
these experimerlts.
to introduce an adjustable path length difference between the
two
arms
of
the interferometer. The photocurrent generated
in the photomultiplier is displayed
on
a
spectrum analyzer
(Tektronix, Type
491).
The injection laser used was.a double-heterostructure stripe-
geometry room-temperature
CW
GaAs/GaAlAs laser fabricated
at Texas Instruments. The optical power output versus drive
current of this particular device is showri in Fig.
2.
In spite of
the relatively high-threshold current of
575
mA, no appreciable
degradation was observed during the course of the experiment
(estimated operation
-250
h). As evidenced by the spectrum
(taken
25
mA above threshold) shown in Fig.
3,
the laser oscil-
lated in a large number of longitudinal modes. Laser emission
peaked around
8900
A,
and the mode spacing was about
2.7
A,
consistent with
a
Fabry-Perot cavity length of
300
pm.
111.
RESULTS
It was found that the amplitude of the heterodyne beat signal
at
100
MHz
depended very critically
on
the position of the
movalile mirror. The effect can be dramatically demonstrated
by slowly scanning the mirror position with the electric motor,
monitoring the amplitude of the beat sigh1 with a boxcar
integrator, and displayhi it directly
on
a
X-Y
recorder. The
result is shown in Fig.
4.
The existence of the resonances for
discrete mirror positions separated by about
1.5
mm
is directly
traceable to the multimode nature of the
laser
[6]
,
[
101
.
Each
longitudinal .mode of Fig.
3
generates its, own beat signal at
100
MHz.
However, there is a phase difference
A@
between
the contributions
to
the beat signal from two adjacent longitu-
dinal modes, since they oscillate at different frequencies. This
phase difference can be written as
A@
=
Ak
*
(2x)
(1)
where
2x
is the path length difference between the two arms
1383
I-
-
1tlLO,lI,2,,
8bOO
8850
8900
8S60
WAVELENGTH
Fig.
3.
Output spectrum
of
the laser
25
mA
above threshold.
I""I'
BOO
mA
MIRROR
POSITION
Inn)
(ARBITRARV
OR161N)
Fig.
4.
Amplitude
of
heterodyne beat signal
(100
MHz)
versus position
of
movable mirror.
of the interferometer and
Ak
is the wave-vector difference
between two adjacent longitudinal modes. The factor of
2
comes about because when the mirror
is
displaced by an
amount
x,
the path length imbalance changes by twice that
amount. The contributions to the beat signal arising from the
individual laser modes will add up cooperatively only when
they are all in phase, i.e., when
A@
of
(1)
obeys the relation
A@
=
2Mn
(2)
where
N
is an integer. The quantity
Ak
and the longitudinal
mode spacing
Ah
are related by
Ak
=
2n
Ah/h2.
(3)
1384
IEEE JOURNAL
OF
QUANTUM ELECTRONICS,
VOL.
QE-15, NO. 12, DECEMBER 1979
Equations
(1)-(3)
imply that
a
strong beat signal will be
observed for mirror positions Separated by
Ax
=
h2/(2AA).
(4)
With
h
=
8900
a
and
Ah
=
2.7
a
in
(4),
the spatial periodicity
Ax.
is 1.47 mm, in good agreement with the experheritally
observed value. One expects the sharpness of the resonances
as a function of mirror positidn to increase with the number of
longitudinal modes.
It
is
shown in Appendix
I
that the magnitude of the com-
ponent
ia
of the photocurrent at angular frequency
i2
is
proportional to the amplitude of the autocorrelation fun&ion
gg)
(7)
of the laser field
in
0:
I~$)(T)I
.
(5)
It is further shown that for!a multimode laser, whose individual
modes have a Lorentzian profile, the autocorrelation function
can be written
ii~
the form
2
gg'(i)=CAkexp-jwk7exp-yk171
(6)
k
where each subscript
k
refers to a particular longitudinal mode
oscillating at angular frequency
wk,
with a half-width at half-
power of
yk.
The coefficients
Ak,
which are a measure of the
amount of coherent optical power in mode
k,
are normalized
so
that
CAk
=
1.
(7)
k
Equation
(6)
is obtained by taking the Fourier trans%rm of
the power spectral density of a multimode laser source (Wiener-
Khintchine theorem
[
111).
If for simplicity we assume that
all the modes have the same spectral width
yo,
(6)
reduces to
gg)(7)
=
exp
lyb
171
Ak
exp
-iwkr
.
(8)
The summ&@n over
k
is, of course, responsible for the reso-
nant behavior exemplified by the result of Fig.
4.
The amp&
tude of
gg)
(7)
is miximum when all the exponential terms in
the summation are
in
phase. For regularly spaced modes, the
condition is
k
.,
Aw
.
T=
2n
(9)
where
Aw
is the angular frequency difference between adjacent
modes. Since
7
=
2x/c
(e,=
speed of light), it is trivial to show
that. the above condition is equivalent to
(4).
When this con-
dition is met, the beat signal
in
as expressed in
(S),
is maximum
and given by
1
(in
Imax
a
exp
-yo
I
7
1
.
(10)
Referring to Fig.
4,
the envelope of the peaks does appear to
go through a maximum (corresponding to a balanced inter-
ferometer) and to decay on either side of this maximum, in
qualitative agreement with (10). While the continuous scan
experiment
of
Fig.
4
demonstrates the sharpness of the spatial
resonances, accurate measurements of the maximum ampli-
I
I I
I
I
I I
I
t
I
I
1
I I
I
I
1
-4
-3
-2
-1
e:
0
~
1::
2
3
4
MIRROR
POSITION
(cm)
Fig.
5.
Envelope of the peak sigIlal versus pasition of the movable
mirror for drive currents of
580,
600,
and
620
mA,
respectively.
Position
0
corresponds to balanced interferometer.
4
i
f
:/
/
f
i
III1_1IIIII
600
650
DRIVE
CURRENT
(nl)
Fig.
6.
Coherence length versus drive current. The circles and triangles
correspond
to
two
sets
of measurements.
tude of each peak require that the position of the movable
mirror be tuned manually and its orientation optimized for
each peak. Such careful measurements were made at several
drive currents. Peak amplitudes versus mirror position are
shown in Fig.
5
for drive currents of
580,
600,
and
620
mA,
respectively. The width of the envelope increases with increas-
ing drive current.
A
coherence length can be defined,as the
distance at which the envelope drops by a factor of
I/e
from
its maximum value (with balanced interferometer). Fig.
6
is a
plot of the coherence length, defined in this manner, versus
drive current. The plot appears linear and intersects the
horizontal axis at or near the threshold current of
575
mA.
Since coherence length and frequency bandwidth are inverse of
each other, and since laser output is-to a first approximation-
proportional to drive current in excess of threshold, Fig.
6
is interpreted as evidence that the intrinsic linewidth of the
longitudinal modes is inversely proportional to the CW optical
laser output
[
13
,
[12].
A
reduction of the linewidth, with
REISINGER
et
al.:
COHERENCE
OF
A
ROOM-TEMPERATURE
cw
LASER
1385
increasing drive current, has been demonstrated previously
in PbSnTe
CW
lasers operating at
4.2
K
[
121
.
IV.
DISCUSSION
Possible sources of error that would cause erroneous measure-
ments of the coherence length are
1)
misalignment of the two
interfering beams,
,2)
nonuniform linewidth for the various
longitudinal modes, and
3)
unequal frequency spacing beiween
adjacent longitudinal modes over the emission spectrum.
With a He-Ne laser we could observe a beat signal whose
amplitude was constant ‘within
20
percent for path length
differences up to
50
cm. Our alignment procedure thus appears
sufficiently accurate to eliminate this potential problem.
Rather than assigning the same spectral width
yo
to all
longitudinal modes, it is more reasonable to assume that the
width of each longitudinal mode is inversely proportional to
the amount of coherent optical power
id
that mode
[13].
Thus, we write
Yk
=
Yo
@oIAk)
(11)
where the subscript
o
refers to the dominant longitudinal mode.
Each coefficient Ak can be evaluated from the spectral data
of Fig.
3
as being proportional to the area under the peak
corresponding to mode
k,
subject to the normalization con-
dition of
(7).
We can anticipate that this correction will tend
to shorten the “apparent” coherence length, although not by
a very large amount, since the broadest modes are also the
weakest.
Perhaps more important is the effect of unequal spacing in
the frequency domain between adjacent pairs of longitudinal
modes. This second-order dispersion phenomenon has been
used previously to determine the group index of water
[14]
.
The average frequency spacing between adjacent longitudinal
modes in the present laser was
6v
100
GHz. The magni-
tude of the second-order dispersion
ti(’)
(v)
is usually
50-
100
MHz
[
15
J
.
Since
6(’)
(v)
can be defined by
6(’)(v)=(Vk+l
-
Vk)-
(vk-
%-I)
(12)
it is easy to show recursively that
wk
-
wg
Ek2*6~tk2~6‘2)(v) (13)
where the subscripts
o
and
k
have the same meaning
as
in
(1
1).
The influence of both nonuniform linewidth and second-
order dispersion can be ascertained by substituting
(1
1)
and
(13)
into
(6).
The result is
Ig$)(T)I
=
exp
-y07
exp/(wk
-
oo)7
Ik
.
exp
-yo
[@oIAk)
-
11
1
*
(14)
Fig.
7
is a plot of the magnitude of the autocorrelation
function versus mirror position, calculated from
(14)
with the
following parameter values:
yo
=
1.5
X
10”
rad/s (Lcoh
=
c/y,
=
2
cm) and
6(’)
(v)
=
50
MHz. The relative mode ampli-
tudes Ak were determined on the basis of Fig.
3.
If the con-
tribution
of
the dominant peak alone, represented by the
dotted curve, is used to define the “intrinsic” coherence
length Lcoh, the “abparent” coherence length L&h is reduced
GAW
-
1.5E+18 RADISEC
f
€4.6
2
8.5
3
8.4
8
8.3
8.2
a.
I
a.a
0
2
4
6
8
18
12
I4
16
18
NTRROR
POSITION
Coni)
Fig.
7.
Computer plot of the magnitude of the laser field autocorrela-
tion function versus mirror position calculated from the spectrum
of
Fig.
3.
The dotted curve is the contribution of the dominant
peak alone.
I
I
I
I
I
Ill1
I
1
I
IIlli-j
1
10
100
INTRINSIC COHERENCE LENGTH
Icm)
..
Fig.
8.
“Apparent” coherent length versus “intrinsic” coherent length
illustrating the effect of nonuniform linewidth and increasing second-
order dispersion
6
(’)
(v).
in this particular case by a factor of about
1.6.
Fig.
8
is a log-
log graph ofLLoh plotted ag&st Lcoh for karious conditions.
The straight dotted line with slope
1
corresponkls~ to
a
uni-
form linewidth
-yo
for
all
modes and no second-order dispersion.
In this case, Lcoh and L&h are equal in the seqse that the
envelope of the correlation peaks as a function of time delay
7
coincides with the correlation function associated with the
dominant longitudinal mode alone. The straight solid line
includes the effect on nonuniform linewidth only (but still
no
second-order dispersion). The “apparent” coherense length is
1386
IEEE
JOURNAL
OF
QUANTUM
ELECTRONICS, VOL.
QE-15,
NO.
12,
DECEMBER
1979
reduced by a factor of about 1.6. The solid curves in Fig.
8
show the additional contribution of second-order dispersion,
which causes a further shortening of the apparent coherence
length. The effect is most noticeable for long intrinsic coher-
ence lengths and large values of the parameter
6(’)
(v).
Of
interest is the fact that for large intrinsic linewidths
Avo
(or
short
Lcoh,
since
Lcoh
=
c/rrAv,),
second-order dispersion is
relatively unimportant. The implication of Fig.
8
is that if
indeed
L&h
increases in proportion to coherent optical power
output, as the theory predicts,
Lcoh
at first should increase
linearly with drive current and eventually should tend to
saturate.
No
evidence of saturation was observed in the present
experiment. It should be emphasized, however, that Fig.
8
was obtained specifically with the spectrum of Fig.
3
and
assumes, rather restrictively, that this spectrum remains
independent of lasing conditions. A more thorough quantita-
tive analysis would require a detailed knowledge of the evolu-
tion of the coherent emission spectrum as a function of drive
current. With these restrictions in mind, one infers from
Figs.
7
and
8
an “intrinsic” coherence length of order of
magnitude
10
cm at a power output level of
%
mW/facet.
This corresponds to an “intrinsic” linewidth of ‘1 GHz.
On the basis of Fig.
3
it is estimated that about 7 percent of
the total optical power is contained in one of the dominant
longitudinal modes. One concludes that if the total power
were emitted in
a
single longitudinal mode, the linewidth at
that power level would be expected to be around 70 MHz.
It
is
interesting to note in passing that, when the mirror
position is adjusted for a perfectly balanced interferometer,
the beat signal continues to be observed with a drive current as
low as
350
mA. This current is considerably below threshold
(see Fig. 2), and the radiation emitted by the device is then
essentially incoherent and very broad-band, covering about
200
A.
Accordingly, the coherence length is extremely small,
and the beat signal disappears as soon as the interferometer is
detuned. Furthermore, with incoherent light, the width of the
beat signal as seen on the spectrum analyzer continues to be as
narrow as that observed when the device is operated in the
laser mode (the bandwidth appears limited by the electronics
driving the acoustooptic modulator). This initially surprising
result can in fact be understood if one recognizes that the
response of a photomultiplier (a square law detector) to
a
broad-band incoherent-optical field (one which obeys Gaussian
statistics)
[ll]
gives rise to a photocurrent whose spectrum
consists of a delta function at zero frequency (a strong dc
component) superposed on a broad background [16]
,
[17]
.
In the present heterodyne detection scheme, the optical
field is mixed not only with itself but
also
with an image of
itself, which has been frequency shifted by 100 MHz. One
would then expect the photocurrent spectrum to display an
additional delta function at
100
MHz. A proof of this intuitive
argument is outlined in Appendix 11. It is this additional delta
function (in practice a narrow peak) which was observed.
Heterodyne mixing of incoherent light beams has been demon-
strated previously
[
181
.
V. SUMMARY
The apparent coherence length
of
a stripe-geometry double-
heterojunction GaAs/GaAlAs laser operated CW at room
temperature was measured by a heterodyne detection scheme
which involves mixing the laser field with a frequency-shifted
age
of itself. Because of the multimode nature of the laser
output, the autocorrelation function of the optical field, as
determined in an unbalanced interferometer, displayed a
resonant behavior for specific time delays. The apparent
coherence length was shown to increase linearly with drive
current in excess of threshold. This observation is interpreted
as evidence that the intrinsic linewidth is inversely proportional
to the coherent optical power output.
APPENDIX
I
PHOTOCURRENT
AND
AUTOCORRELATION FUNCTION
OF
OPTICAL FIELD
The total optical field
ET
incident on the photocathode of
the detector can be written as
ET(t)
=
E(t)
t
exp
jat
E(t
t
T)
(15)
where the first term represents the time dependence of the
laser field, and the second term describes its image shifted in
frequency by the acoustic angular frequency
52
and delayed in
time by the amount
T
corresponding to the imbalance between
the two arms of the interferometer. For simplicity, both fields
have been assumed to have equal amplitudes. The instanta-
neous photocurrent
i(t)
is proportional to the total field in-
tensity
ET(^)
E$(t)
where the star
*
denotes the complex
‘j
conjugate
i(t)
a
ET(t)
E$(t).
(16)
Substitution of (15) into (16) gives rise to four terms, as is
common in all interference problems. In particular, one of the
two cross terms represents the instantaneous photocurrent
component
in
(t)
at angular frequency
t
52
ia
(t)
a
E”(t)
E(t
t
T)
exp
jilt
.
(17)
An interference experiment of the type discussed here can
only provide a measure of the statistical average
(in)
of the
amplitude of the photocurrent at frequency
52:
(ia)a
(E*(t)E(t
t
7)).
(18)
The right-hand side of
(18)
is related to the first-order (cam-
plex) autocorrelation function
gg)
(7)
of the laser field alone,
which is defied as
g2)
(7)
=
(E(t)E”(t
t
T))/(E(t)E*(t))
.
(19)
It follows that the average magnitude of the photocurrent
component at angular frequency
52
is given by
pa)
I
a
lgg)(T)
I
*
(20)
The power spectral density of a single-laser mode with a
Lorentzian profile
is
Application of the Wiener-Khintchine theorem gives
g$)(T)
=
exp
-jooT
exp
-
yo
ITI
.
(22)
For
K
distinct laser modes, each with Lorentzian profile, the
power spectral density becomes