Download Quantum Phase Noise and Field Correlation in Single Frequency

JOURNAL
IEEE
OF QUANTUM ELECTRONICS, VOL.343
QE-20,
1984
APRIL
NO. 4,
[ 121 N. Holonyak, Jr., R. M. Kolbas, W. D. Laidig, B. A. Vojak, K.
Hess, R. D. Dupuis, and P. D. Dapkus, “Phonon-assisted recombination and stimulated
emission in quantum-well A1,Gal-,AsGaAs heterostructures,” J. Appl. Phys., vol. 51, pp. 1328-1337,
Mar. 1980.
[ 131 A. Sugimura, “Phonon assisted gain coefficient in AlGaAs quantum well lasers,’’ Appl. Phys. Lett., vol. 43, Oct. 1983.
[ 141 J. S. Blakemore, “Semiconducting and other major properties of
gallium arsenide,” J. Appl. Phys., vol. 53, pp. R123-R181, Oct.
1982.
[ 151 K. Itoh, M. Inoue, and I. Teramoto, “New heteroisolation stripegeometry visible-light-emitting lasers,” IEEE J. Quantum Electron., vol. QE-11, pp. 421-426, July 1975.
[ 161 W. T. Tsang, “Device characteristics of (A1Ga)As multiquantumwell heterostructure lasers grown by molecular beam epitaxy,”
Appl. Phys. Lett., vol. 38, pp. 204-207, Feb. 1981.
[ 171 W. T. Tsang, “Extremely low threshold (A1Ga)As modified multiquantum well heterostructure lasers grown by molecular beam
epitaxy,”Appl. Phys. Lett., vol. 39, pp. 786-788, Nov. 1981.
[ 181 W. Streifer, D. R. Schifres, and R. D. Burnham, “Optical analysis
of multiple-quantum-well lasers,” Appl. Opt., vol. 18, pp. 35473548, Nov. 1979.
[ 191 H. C.. Casey, Jr. and M. B. Panish, Heterostructure Lasers. New
York: Academic, 1978.
Akira Sugimura was born in Osaka, Japan, on
May 10, 1947. He received the B.S. degree in
physics from Kyoto University, Kyoto, Japan,
in1970,andthe
M.S. and Ph.D. degrees in
physics from the University of Tokyo, Tokyo,
Japan, in 1972 and 1975,respectively.
In1975hejoinedthe
Musashino Electrical
Communication Laboratory, Nippon Telegraph
and
Telephone
Public Corporation,
Tokyo,
Japan. He hasbeen engaged in research on
optical fibers and semiconductor lasers.
Dr. Sugimura is a member of the Institute of Electronics and Communication Engineers of Japan, the Physical Society of Japan, and the
Japan Society of Applied Physics.
Quantum Phase Noise and Field Correlation in Single
Frequency Semiconductor Laser Systems
PHILIPPE B. GALLION, MEMBER, IEEE, AND GUY DEBARGE
[19]. The phasenoise of a single frequencysemiconductor
laser is aparameter ofmajorimportanceinapplicationsin
which the temporal coherence properties of optical signals are
involved. Suchapplicationsaretransmissionssystemsand
sensorsinwhichphaseorfrequencymodulated
signalsare
detected by heterodyne or homodyne techniques [27]-[31]
or discriminated by direct interferometric detection[32], [ 3 3 ] .
The laser phase is also a determining parameter for incoherent
systems in which a spurious coherent mixing produces a phase
to intensity noise conversion: this is for example a case for the
square law detection of a beam which is first split by a polarINTRODUCTION
ization dispersive medium,andtheonepartis,latermixed
HE temporal coherence properties of multimode semicon- through polarization sensitive elements, with the another underductor lasers have been discussed by a number of authors going a time delay [21] .
The repercussion of spectral spread on such systems can be
[ 1] -[SI . Experimental investigations of these properties were
completely
discussed in terms of the autocorrelation function
commonly achieved by using interferometric or beating techand
spectral
density of the photocurrent; the purpose of this
niques [9] - [ 11] . The higher temporal coherence of single lonpaper
is
to
give
a simple unified treatment which can be applied
gitudinalmodesemiconductor lasers hasalsobeendiscussed
in
the
various
above
mentioned particular cases. The results are
[ 3 ], [ 121- [ 141 and experimentally investigated by the above
discussed
as
a
function
of the laser linewidth, phase matching,
mentioned methods [ I 51 -[26]. For a single-frequency conthe
balance
and
the
phase
correlation between the two mixed
stant-amplitude laser source operating for above threshold, the
beams.
In
addition
the
phase
noise sensitivity of various detecrandom quantum phase fluctuations are usually considered to
tion
schemes
is
compared.
be the major source of optical spectral spread [3], [ 121 , [ 171-
Abstract-The influence of quantum phase fluctuations which affect
single frequencysemiconductor lasers in various coherentdetection
systems is discussed in terms of photocurrent autocorrelation and spectral density functions. The general treatment given in this paper can be
applied in diverse practical cases and points out the problems of phase
correlation and phase matching between the two mixed optical beams.
In the more general case the Photocurrent spectrum is found to be composed of discrete and quasi-Lorentzian parts whose energies and spectral
spreads are discussed as a function of the laser line width, the phase
matching and the phase correlation between the two coherently combined fields.
T
QUANTUMPHASE NOISE MODEL FORSINGLE
Manuscript received July 18, 1983;revised November 14,1983.
LONGITUDINAL
MODE SEMICONDUCTQR LASERS
The authors are with Groupe Opto6lectronique
Microondes
et
D6parteThe
optical
field
emitted
far above threshold by a single
biased
mentElectroniqueet
Physique Ecole NationaleSuperieure des T616frequency laser is commonly modeledas a quasimonochromatic
communications 46, rue Barrault 75634 Paris, Cedex 13 France.
0018-9197/84/0400-0343$01 .OO 0 1984 IEEE
JOURNAL
IEEE
344
OF QUANTUM ELECTRONICS, VOL. QE-20, NO. 4, APRIL 1984
amplitude-stabilized field undergoing a phase fluctuation [ 121
(1 1
a t ) = E , exp i[w,t + @(Ol
where w, is the average optical frequency and
@ ( tis) a stochastic
process representing the random phase fluctuation leading t o
the broadening of the spectral line.
The time dependence of E , on time t is an amplitude noise.
Its contribution to the field spectrum can be neglected in spite
of its large spectrum width because its integration power
is
much smaller than the phase noise one [12] .
It is useful tointroducetheoptical
field autocorrelation
function defined as
G$)(T) = ( E * ( t ) E ( t+ T ) )
= (exp [iA@(t,
T)] ) exp ( j w , ~ )
(2)
where A@(t,T ) is the phase jitter, i.e., the randomphase change
between times t and t + T
A@(t,T ) = @(t+ T ) - @(t).
(3)
Underthe above mentionedassumptions,the
phase jitter
A@(t,T ) is usually assumed to be a zero-mean stationary random Gaussian process with the associated probability density
function being [ 151 -[34]
(4)
(A@*(T))is the mean-square phase jitter which is related to
the instanteous angular frequency-fluctuation spectrum S$ (a)
by [341
[TI
explain the experimental
values reported by Fleming and
Mooradian [13] :
where ug is the group velocity of the light in theactive medium,
hu the lasing photon energy, nSp the spontaneous emission factor [3], am the loss of the mirror, Po the output power per
facet and a the linewidth enhancement factor introduced by
Henry [ 141 .
Forchanneled-substrate-planar(CSP)-structure
Ga A1 As
devices FlemingandMooradian
[ 131 have reportedexperimental values for the product 2 y Po of 2 7 X 110 X IO6 rad .
s-l . mW.
Recently
authors
[35]-[38]
have
discussed
modified
a
model taking into account the relaxation resonance effect. A
Lorentzianlineshapewithasecondordercorrection
is then
carried out. This correction is not taken into account in this
study which refers only to the first order model (assuming a
flat frequency-fluctuationspectrum leading toa rigorously
Lorentzian line shape).
TOTALDETECTEDFIELDMODELIZATION
The purpose of this paper is to express the spectral density
of the photocurrent produced by homodyne and heterodyne
detection of two fields as a function of their linewidth, their
relative weight and their remaining phase correlation. For this
purpose we assume thatthedetectedtotal
field E T ( t ) is a
superposition of a laser field E ( t ) as expressed by equation (1)
with a time-delayed and frequency-shifted
image of itself:
E T ( t )= E(t) + aE(t + 7 , ) exp j n t
(1 1)
where a is a real factor which accounts for the amplitude ratio
between two mixed fields. For coherent optical communica(*@2(T))=
27
S@(W)dU.
(5) tion systems inwhich a weak signal is detected with a powerful
local oscillator we have a << 1. For a phase-modulated optical signal detection by the retardation method 1321, [33] a is
Using the well known relation 1341
closely one. a can take any value in the case of noncoherent
single-mode transmissions systems and unbalanced sensors and
T ) ] ) = exp [ - + (AQ2(7))]
(6 1
(exp [+jA@(t,
when the modes coupling in a single-mode fiber with polarizatherefore the laser field correlation function is expressed as
tion dispersion is involved [21] .
o T , is the time-delay which also refers to the eventual reG$)(T) = exp [-+ ( A @ * ( T ) exp
)] ( j w , ~ ) .
( 7 ) maining phase correlation between the two combined beams.
The case considered here concerns quantum phase-fluctuation For coherent communication systems, 7, is the time delay in
the optical local field generator [23], [27], [30]. For differaffectingafar
above threshold biased semiconductorlaser:
then the instantaneous angular frequency-fluctuation spectrum ential phase shiftkeyingsystemsandforphase-modulated
optical detection by the retardation method [32], [33]T , is a
S,(w) can be assumed to be flat [12]-[16] leading tothe
mean-square phase-jitter (AG2(7)) increasing linearly with the bit duration. Finally T , is the optical time delay for unbalanced
interferometric systems and the dispersion time for the polaritime delay
zation-dispersed modes in single mode fiber mixing [21]. The
= 2y 17-1
(81 optical mixing of two
fields ofequal linewidth and uncorrelated
where 2yis the angular full linewidth athalf maximum (FWHM) phases is obtained by T , = 00.
n is themeanfrequencydifferencebetweentwomixed
of the Lorentzian laser field spectrum S E ( w ) obtained by the
fields. a is theintermediaryfrequency in heterodynecomFourier transform of (7)
munication systems and the acousto-optic frequency in hetero717
dyne sensors. 52 = 0 for homodyne communications systems,
s, (w) = E; y 2 + (w
- w,)2 .
for fiber interferometers without
Bragg-cell, for the polarization dispersed modes in single mode fiber mixing and for the
2y is given by the Schawlow-Townes formula including the
phase modulated optical detection by retardation method.
With
linebroadening excess factorintroducedbyHenry[14]to
?'
1;
wr
sin -
GALLION AND DEBARGE:
FREQUENCY
SINGLE
SEMICONDUCTOR LASER SYSTEMS
345
appropriate values of CY,7, and C2 (1) for the total detected
field one can easily consider the above mentioned four cases.
Because of the square law of optical detectors, these interferometric systems are able to convert the quantumphase noise of
the laser field E(t) into intensity noise and then give a spectral
spread of photocurrent I(t). For stationary fields, the autocorrelation function RZ(7)of the photocurrent depends only on
the intensity correlation function
of the detected total field
G,g(r)
PI
~ ~ (=7eo)G$;(o) s ( T )
+02
G$$)
(I2)
(1 6 )
For laser phase noise A@2( T ) is given by (8), using the reduced
timedelay,defined
a s 7 = 277 andnotingby
0 the meanphase
G ( 2 ) ( ~ex)
difference W,T, betweenthetwomixedfields
where
e is the electron charge,
u is the detector sensitivity,
& ( T ) is the Dirac function,
Gz'(7) is the firstorderopticalintensitycorrelation
function,
[(I t c u 2 ) t 2a cos e exp
G g (7)
-
___
-
E:
(-3
1
ET
2
2
for any7
4a2 exp (-7,)
{sin h ( 1 ~ 1 -7,)
cos2 e [ I - exp ((1
i 771
1 -7,)J}
i.e., the second order optical field correlation function defined
for 0 <T<T,.
hv
(1 7 )
-.'
By virtue of the Wiener-Khintchine theorem, the spectrum
(13) ofthephotocurrent
is given by theFouriertransform
of its
autocorrelationfunction.Omittingtheshot
noise termand
where the brackets denote a timeaverage.
'using the reduced angular frequency, defined as0 = (3/2y,the
Thefirsttermin
(1 2 ) is the usualshot-noiseassociatedwith
'two-sided spectrum
is written as
the dc componentof the photocurrent and the second account
2
forthisdccomponentandfluctuations
arising from phase into
SZ((J)
=
t a2 t 2cu cos e exp
intensity noise conversion.
u2 E:
G~;(T) =
( ~ ~ ( t ) ~ $ t( Tt ))E ~
* I~
C
t (7(~) t)
(3
[1
1
PHOTOCURRENT
SPECTRUM
IN THE HOMODYNE
(S2 = 0)
t 4cu2 exp (-7,)
1tw2
DETECTIONCASE
The autocorrelation functionof the photocurrent is obtained
sin 0 7,
- by substituting (1 1) into(13) with s2 = 0. This substitutiongives
rise to 16 terms. Using (1) for the laser field after a straight-for(1 8)
(2)
only as a function
ward calculation allow us to express GET(^)
where 6(G) is the delta function.
of the phase jitter of linear combinationof @ ( t )and @(tt 7):
Let us consider one of particular cases in which a two beams
interferometer with equal optical field intensities
is tuned up
G$:(T)
=E," ((1 t cu212 t 2a(1 t a 2 )expjw,T,
for a maximum optical power output. Therefore
we have CY =
. (exp jA@(tt T , 7,)) t CC t a2 exp 2jw,7,
1 and 0 = 2 k l with k integer. Taking moreover the Gaussian
probability density function assumption for the phase jitter
(4),
* (expjA\k(t, io)> t CC t a2
one can find the following expression
derivedby Armstrong
(exp jA(a(t,7 , ) ) + CC)
(14)
1151
w7,
(J
1
and CC denotes the complex conjugate of the preceding term.
The approximately Lorentzian part of this spectrum
is plotted
$
' :()
'
can be expressed Only as a function Of
by in Fig. l(a) and l(b) for variousvaluesof the parameters%, i.e.,
using (6)
and
coherence
ratio,
time
todelay
timethe
0 = (J,T,,the i.e.,
mean phase difference between the two mixed beams.
(1 t a 2 ) 2t 4a(l t a2)cos (o,T,)
PHOTOCURRENT
SPECTRUM
IN THE HETERODYNE
(C2 # 0)
DETECTIONCASE
In this paragraph we perform a quite
similaranalysis as in
(1 1) into (13) give
the previous chapter. The substitution of
IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. QE-20, NO. 4, APRIL 1984
346
T i n dolq e0
.0
tir ratio
T i n dolay tu
t i n rut10
1
2.
.e
1. cc.S-3
2 ca2=.5
2
.7
ca2=.s
9. cns2-l
.e
m
Y
r
€LETECTICN
KnmYE mocTIW
.5
-4
.9
.2
.I
n
WuTmH
1
ff--7
.0
t i n &lo
T i n d o l q tu cohrrcnoP
t i n rotlo 0 8 0
4
.e
1. codd
2 cLaz-.S
1. cc.S-3
2 cLaz-.5
3. -1
s.
.7
-1
~
1
.8
!
K W D Y I E DETEclIW
I
1
.5
I
I
I
.4
1
-T
----!-
--
.3
I
.2
.1
-
0
0
2
9
4
s
RrmxEDRIolLDEy
o
1
2
9
EL.lm3 FFf!x€N3
4
5
(b)
Fig. 1. Lorentzian part of the homodyne spectrum as a function (a) and (b) of the time delay to coherence time ratio To
and of the phase matching parameter cos2 e.
also rise to 16 termsbut because anexplicittimedependencz
G$;(T) =E," { ( l + 01~)'+ o2 exp (~RT)
appearing in the form of exp + j R t , 10 of them average out to
. (exp jA@((t,7 , ) ) + CC)
0 ; using the previous notations, the 6 surviving terms give the
where @ ( t )have the same meaning as above (1 5).
following result [22] - [ 2 3 ] :
AND
GALLION
DEBARGE:
FREQUENCY
SINGLE
SEMICONDUCTOR LASER SYSTEMS
347
et ( T o e )= 2a2 [ 1 - exp (-T~)]
[1 t sin’
Using again the (6) G$;(T) is written
c
€,(Toe) = 1 t a2 t 2a cos e exp
(21)
In the same manner as the previous chapter., using the same
reduced variables one obtains the first order optical intensity
correlation function G$;(T):
2a2 * cos E T exp (-T),
+
Omitting again the usual shot noise term, the two-sided heterodyne photocurrent spectrum is then obtained by Fourier
transform
e exp (-7,)]
(24)
(-3 1 ’.
(251
For an in phase (e = 0) and an out of phase (8 = 7 ) optical
mixing, the eL (To, e ) arrives at a minimum value. On the other
hand in the case of a quadrature optical mixing, according to
the well known fact that the phase noise conversion efficiency
e ),arrives at a maximum value.
is optimum, the eL ( T ~
The heterodyne case is quite different. Here the photocurrent spectrum consists of a dc component, a monochromatlc
and an approximately Lorentzian component centered at the
intermediary frequency G. The first of them refers to the dc
current associated with the noninteractive superposition of the
two optical powers. The monochromatic one at the frequency
2
! expresses the remaining phase correlation between the two
mixedbeams.Thethirdandlastterm
is quasi-Lorentzian
spectrum which cancels out for To = 0 and become rigorouslyLorentzian with linewidth twice than the laser spectrum when
-
7 , = 00.
t a’ exp (-7,)
1
1ITT
+(a-2 ) 2
1
sin ( T S - i 7 ) ~ ~
- cos (W- 2 ) T o
w - 2
.
The approximately Lorentzian part
of this spectrumis plotted
i.e., the time
in Fig. 2 for various values of the parameter 7,)
delay to coherence time ratio.
Because of the mean frequency difference between the twomixed beams the photocurrent spectrumis no longer dependent
on the phase matching even in the correlated case. It is to be
noticed that because of time averaging of phase matching in
the heterodyne case the photocurrent spectrum is founded to
be the same as in the homodyne case, with cos2 0 =
The energiesin thequasi-Lorentzianpart E ~ , ~ ( T and
, ) in
, ) the spectrum are given
the monochromatic part E ~ , ~ ( Tof
by
eL, n(~,) = 201’ [ 1 - exp (-7,)]
(26)
i.
eg,
= 2a2 exp (-7,).
(27)
These energies expressions stand for an uniform transfer of
energy
from the monochromatic to the Lorentzian component
DISCUSSION
when increasing the decorrelation time7,.
The homodyne spectrum consists of two terms whose behavAs show in Fig. 3 there is no significant difference in the corior relates closely to T , and 0 values. Let us examine the phys- related spectral spread in the homodyne and heterodynecases.
ical meaning of eachterminseveraltypicalcases.Thefirst
Some comparison of these results with experimental measureterm which is a dc component refers to the simple adding of ments has been made in various particular
cases: Heterodyne
the two optical powers and to the amount of remaining phase detection of beams with uncorrelated phases [20], heterodyne
correlation between the two mixed beams. For large values of detectionofbeamswithcorrelatedphases
[23]-[24] and
the normalized time-delay T,,i.e., for completely decorrelated homodyne detection of beams with correlated phase and phase
0 vanishes. On quadrature [21]. Theresultsindicateforeach
fields: its dependence on the phase matching
case that the
the other hand for a value of T , close to zero it becomes very used theoretical description of the laser-phase random process
sensitive on the 0 value and it depends no longer on the spectral
is consistentwithexperiments.Experimentalmeasurements
spread. The second term takes the form
of an approximately in the moregeneral case would be of a great interest.
Lorentzian lineshape which vanishes out for a close to zero7,
the optical
value. For large values of T, this term stands for
CONCLUSION
mixingof two independent fields and it becomes rigorously
Lorentzian with a FWHM which is twice the original laser lineTherepercussionofquantum
laser phasenoise in various
as anopticalproductwidth. Id thiscase thedetectoracts
coherent systems has been discussed in terms of autocorreladetector whose output is the autocorrelation product of the
tion function and spectral density of the photocurrent. The
laser field spectrum. The spectrum is then no longer dependent given treatment includes the phase matching, the balance and
on the phase matching8 .
the phase correlation between two coherently combined beams.
The etlergies ofthequasi-Lorentziancomponent
eL (7,B) For stronglycorrelatedbeamsthephotocurrentspectrum
is
and of the dc component eg (‘roe) of the spectrum are given found to be a Dirac function both in the homodyne and the
heterodyne cases, and extremely dependent on phase matching
by
348
IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. QE-20, NO. 4, APRIL 1984
NORMALIZED LGRENZIAN PARTOF PHOTOCURRENT SPECTRUM
1
I
au= 8
-
I
Time delay to’coherence
is t a u
tlme ratio
For any t h e t a
.6
HETERODYNE OETECTION
.5
.4
.3
.2
.l
0
0
1
2
3
4
5
NORHALIZED DEPARTURE FROM INTERMEDIATE F E O U E N C Y
Fig. 2. Lorentzian part of the heterodyne spectrumas a function of the time delay to coherence time ratio F ~ .
NORMALIZED HALF LINEWIOTH OF THE PHOTOCURRENT SPECTRUM
8
~~
~
~~
~
~
~~~~
Heterodyne case
7
~
2
-
,
Homodyne caee
I
6
1 Cos2=0
5
4
3
2
1
TIHE DELAY TO CGMRENCE
TIMERATIO
Pig. 3. Normalized half linewidth at half maximum of the Lorentzian partof the photocurrent spectrum.
in the liomodyne case. For weakly correlated beams the spec- photocurrent spectrum is a complex mixture of the two noises
trum is Lorentzian with a FWHM which is twice the original and its derivation seems to be quite tedious.
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DEBARGE:
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SEMICONDUCTOR LASER SYSTEMS
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