The amplitude-phase coupling factor α (linewidth enhancement factor) is compared for typical semiconductor quantum-well and bulk double heterostructure lasers. As a direct consequence of the reduction of the differential gain, there is no reduction of α in single-quantum-well lasers compared to bulk lasers. The number of quantum wells strongly affects the amplitude-phase coupling in quantum-well lasers. It is shown that the interband transition induced amplitude-phase coupling dominates that induced by the plasma effect of carriers in typical quantum-well lasers. By considering the spontaneous emission factor in the spectral linewidth, the authors show that there is an optimal number of quantum wells for achieving the narrowest spectral linewidth

Chen, T. R.

Yariv, A.

Zhao, B.

English

Caltech Authors - Main

1027 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 29. NO. 4. APRIL 1993 
A Comparison of Amplitude-Phase Coupling and Linewidth Enhancement in 
Semiconductor Quantum- Well and Bulk Lasers 
B. Zhao, T .  R. Chen, and A.  Yariv 
Abstract-The amplitude-phase coupling factor a (linewidth 
enhancement factor) is compared in typical semiconductor 
quantum-well and bulk double heterostructure lasers. As a di- 
rect consequence of the reduction of the differential gain, there 
is no reduction of a in the single quantum-well lasers compared 
to the bulk lasers. The number of quantum wells strongly af- 
fects the amplitude-phase coupling in quantum-well lasers. It 
is shown that the interband transition induced amplitude-phase 
coupling still dominates over that induced by the plasma effect 
of carriers in the typical quantum-well lasers. By considering 
the spontaneous emission factor in the spectral linewidth we 
show that there is an optimal quantum-well number for achiev- 
ing the narrowest spectral linewidth. 
INTRODUCTION 
N semiconductor lasers there is a fundamental linewidth I enhancement resulting from coupling between sponta- 
neous emission induced amplitude and phase perturba- 
tion. This leads to the modified Schawlow-Townes for- 
mula for the spectral linewidth of semiconductor lasers 
V I ,  P I ;  
v; hvga,n,(l + a*) 
AV = (1) 81rP 
where vg is the group velocity of the light, hv is the lasing 
energy, g is the modal gain, am is the mirror loss, P is 
the optical output power, nSp is the spontaneous emission 
factor, and a is the linewidth enhancement factor which 
reflects the amplitude-phase coupling in the lasers. The 
amplitude-phase coupling factor a is of fundamental im- 
portance in semiconductor lasers. It plays a crucial role 
in determining the field linewidth, the chirping character- 
istics, and in any attempts to control these characteristics. 
The reduction of the spectral linewidth of semiconduc- 
tor lasers is important in optical communication, since a 
narrower spectral linewidth enables a larger number of 
communication channels by reducing the pulse spreading 
in digital systems and in analog systems, which are all 
influenced by signal distortion due to fiber dispersion. The 
Manuscript received June 5 ,  1992. This work was supported by DARPA, 
The authors are with T. J .  Watson Sr. Laboratories of Applied Physics, 
IEEE Log Number 9207492. 
the Office of Naval Research, and the NSF. 
California Institute of  Technology, Pasadena, CA 91 125. 
amplitude to phase coupling is also the major factor caus- 
ing undesired frequency modulation (FM) (chirping) as- 
sociated with the amplitude modulation (AM) [3] in 
semiconductor lasers. And in some circumstances the 
amplitude noise (or the phase noise) can be reduced 
through the decorrelation of phase-amplitude coupling [4] 
in semiconductor lasers. 
It has been believed that quantum-well (QW) lasers 
would be a good candidate to achieve a much narrow 
spectral linewidth due to the reduction of linewidth en- 
hancement factor a compared to that of bulk lasers [ 5 ] ,  
[6]. The single quantum-well (SQW) lasers were expected 
to be superior to their multiple quantum-well (MQW) 
counterparts in the spectral linewidth due to the smaller 
spontaneous emission factor nSp in SQW lasers [7]. The 
reduction of the a factor in the QW lasers has been attrib- 
uted to the differential gain enhancement compared to bulk 
lasers [5], [6]. However, in our recent study, we found 
that there is a significant modification to the expected dif- 
ferential gain enhancement in QW lasers if the unavoid- 
able thermal population of injected carriers in the optical 
confining region (state filling effect) is included [8], [9]. 
In this paper, we present an analysis which shows how 
the amplitude-phase coupling factor a and the spectral 
linewidth in QW lasers are influenced by the modification 
of the differential gain enhancement due to state filling. 
The linewidth enhancement factor a is defined as 
(3 1 x ( N )  = X r  + ixi 
where N is the injected carrier density, x r  and xi are real 
and imaginary part of the effective optical susceptibility 
0018-9197/93$03.00 O 1993 IEEE 
1028 
where 
IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 29, NO. 4,  APRIL 1993 
(6) 
V T 2  C;(€ - E )  = 
(A/T2)2 + (€ - E ) 2  
and 
€ - E  
(FZ/T*)~  + (E  - E)2  C,(E - E )  = (7) 
E is L e  photon energy, T2 is the collisional dephasing 
time, f, and fh are the quasi-Fermi distribution functions 
for electrons and holes, respectively, p i , ,  is the density of 
states function which represents either the step-like two- 
dimensional (2-D) reduced density of states for the QW 
structures or the parabolic three-dimensional (3-D) den- 
sity of states for the bulk double heterostructure (DH), i 
designates either light holes ( j  = I )  or heavy holes ( j  = 
h ) ,  A , ( € )  includes the transition dipole moment and the 
polarization modification factor for the dipole moment in 
the QW structure. r is the conventional confinement fac- 
tor for the DH structures and is equal to 1 / t  for the QW 
structures, where t is the effective lateral width of the op- 
tical mode. The modal gain of the laser devices is related 
to Xi by 
where n, is the modal effective refractive index. 
We calculated the a parameter at the modal gain peak 
for typical GaAs-AlGaAs QW and DH lasers. For QW 
structure we assume a symmetric 4000 A AIo sGao 5 A ~ -  
Alo 25Gao 75 As graded index separ!te confinement hetero- 
structure (GRINSCH) with 100 A GaAs quantum wells 
at its center. For DH lasers we assume an Al, 35Gao 6 5 A ~ -  
GaAs DH with GaAs active layer thickness 0.1 pm. The 
calculation is made for the dominant TE mode in the DH 
lasers and in the GRINSCH SQW, three quantum wells 
(3QW), and five quantum wells (5QW) lasers, respec- 
tively. We have assumed parabolic and uncoupled sub- 
bands. Typical values of effective masses for electrons 
and holes in GaAs material are adopted and T2 = 0.1 ps 
is used. 
In Fig. 1 we show the a parameter at the gain peak as 
a function of the modal gain. It is important to note that 
there is no reduction of a parameter in the SQW structure. 
On the contrary, the a parameter in SQW structure is 
much larger than that in DH structure. The explanation to 
this is the following. A decrease of carrier density (due to 
spontaneous emission) results in a decrease of quasi-Fermi 
energies. The change of Fermi energies is relatively small 
due to the unavoidable thermal population of carriers 
among the large density of states in the SCH confining 
region. The carrier density change in the SCH confining 
region contributes relatively little to the change of x, due 
to the nonresonance nature of C,(€ - E )  [see (6)] and 
the flat feature of the 2-D step-like density of states in the 
'  
0 
Modal Gain ( cm-' ) 
Fig. 1 .  The linewidth enhancement factor LY (interband transition compo- 
nent) at optical gain peak as a function of the modal gain in the bulk DH 
lasers and the QW lasers with different number of quantum wells. 
QW. The resulting change in the denominator of (2) is 
thus small. On the other hand, the carrier density change 
in SCH confining region cantributes a relatively large 
amount to the change of X ,  due to the long tail of C,(E 
- E )  at high energy [see (7)]. Apparently, a large carrier 
population in the SCH region will enhance this effect. This 
leads to an increase of the a parameter in the SQW lasers. 
The abrupt drop of the a curve for the SQW structure in 
Fig. 1 is due to the onset of the second quantized state 
lasing which leads to a differential gain [or the denomi- 
nator in (2)] enhancement. Fig. 1 also shows that the use 
of MQW leads to a reduction CY. This is attributed to the 
reduction of the state filling in the SCH optical confining 
region [9], which results in an additional differential gain 
enhancement in MQW lasers. However, the reduction of 
Q in the MQW structures is not significant in comparison 
to the a in bulk DH structures. 
Generally the amplitude and phase coupling in semi- 
conductor lasers consists of an interband transition com- 
ponent (as we have discussed above and shown in Fig. 1) 
and a free-carrier component. The free-carrier component 
of a stems from the plasma effect of injected carriers and 
has been shown to be much smaller than the interband 
transition component in bulk lasers [lo]. We plotted in 
Fig. 2 the calculated free-carrier component of a at the 
optical gain peak in the assumed QW and bulk lasers as a 
function of the modal gain. It was suggested that the car- 
rier fluctuation in the optical confining layers affect the 
amplitude-phase coupling through a significantly large 
free-carrier component of a in SCH QW lasers [ 113. We 
show in Fig. 2, comparing with Fig. 1, that the interband 
transition component of a is still dominant in the QW las- 
ers. Fig. 2 shows that the free-carrier component of a is 
larger in SQW lasers and is smaller in MQW lasers com- 
pared to the bulk lasers. The explanation is referred to the 
differential gain behavior in these lasers. 
IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 29, NO. 4,  APRIL 1993 1029 
B c 
ij 
8 
I
LL 
c 0
U) 
._ 
._ 
E w 
U) 
3 
e, 
([I 
c 
0. 
0 
" 
0.0 - - = I  
0 20 4 0  60 80 100 
Modal Gain ( cm-l ) 
Fig. 2. The linewidth enhancement factor ci (free-camer component) at 
optical gain peak as a function of the modal gain in the bulk DH lasers and 
the QW lasers with different number of quantum wells. 
Equation (1) shows that, in addition to the a parameter, 
the spontaneous emission factor nsp also plays an impor- 
tant role in determining the spectral linewidth of semi- 
conductor lasers. nsp is the ratio of the spontaneous emis- 
sion rate into the lasing mode to the stimulated emission 
rate and is given by 
- E , = l , h s  A f ( ~ ) p f , r ( ~ ) f Y h ~ f ( ~  - E )  dG 
c , = , h S A f ( G ) p f , r ( G ) ( f ,  + h  - 1)%!(€ - E )  d F '  
(9) 
The calculated spontaneous emission factor at the gain 
peak as a function of the modal gain for the assumed QW 
and bulk DH lasers is plotted in Fig. 3 .  It shows that QW 
lasers possess lower nsp than that of DH lasers. The SQW 
lasers have the smallest nJp. This is due to the 2-D flat 
density of states distribution in the QW structures, which 
makes it necessary to have a larger inversion in QW struc- 
tures compared to bulk DH structures in order to achieve 
the same modal gain. 
Equation (1) shows that n,(l + a2)  is the figure of 
merit for the spectral linewidth in semiconductor lasers of 
different structures. Notice that a decreases and nsp in- 
creases in the QW lasers as the number of quantum wells 
increases, there might be an optimal quantum-well num- 
ber for achieving the narrowest spectral linewidth. To 
compare the linewidth enhancement in different laser 
structures, we plot nsp ( 1  + a2 ) at the gain peak as a func- 
tion of the modal gain in Fig. 4, where both the interband 
transition component and the free-carrier component of a 
are included. It can be seen that MQW lasers possess nar- 
rower linewidth compared to bulk lasers and this is attrib- 
uted mainly to the smaller nsp in the MQW lasers. The 
linewidth of the SQW lasers can be either smaller or larger 
than that of the bulk lasers depending on the modal gain. 
Fig. 3 .  The spontaneous emission factor nrp at optical gain peak as a func- 
tion of the modal gain in the bulk DH lasers and the QW lasers with dif- 
ferent number of quantum wells. 
80 
a 
c* 40  
20 
1 
0 20 40 60 E O  100 
Modal Gain ( cm-' ) 
Fig. 4. The figure of merit for the spectral linewidth enhancement in semi- 
conductor lasers n,( l  + a') at optical gain peak as a function of the modal 
gain in the bulk DH lasers and the QW lasers with different number of 
quantum wells. 
CONCLUSION 
In conclusion, we find that there is an enhancement in 
the amplitude-phase coupling in SQW lasers compared 
with that of bulk DH lasers. This is due mainly to the 
reduction in differential gain caused by the state filling 
effect in SQW lasers. The amplitude-phase coupling is re- 
duced in MQW lasers due to the extra differential gain 
enhancement. It is shown that the interband transition in- 
duced amplitude-phase coupling is dominant compared to 
that induced by the plasma effect of injected carriers in 
typical QW lasers. Compared to bulk lasers, the reduction 
of amplitude-phase coupling is not significant in MQW 
lasers. Due to the smaller spontaneous emission factor nsp 
I030 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 29, NO. 4,  APRIL 1993 
in QW lasers, the QW lasers may have narrower linewidth 
than that of the bulk lasers and there is an optimal QW 
well number for achieving the narrowest spectral line- 
width in the QW lasers. 
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[4] K.  Vahala and M. Newkirk, “Intensity noise reduction in semicon- 
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A comparison of amplitude-phase coupling and linewidth enhancement in semiconductor quantum-well and bulk lasers

Effect of state-filling on the modulation response and the threshold current of quantum well lasers,”

Intensity noise reduction in semiconductor lasers by amplitude-phase decorrelation,”

Linewidth broadening of SCH quantum well lasers enhanced by camer fluctuation in optical guiding Iayers,”

Linewidth enhancement factor for quantum-well lasers,”

Measurement of the linewidth enhancement factor CY of semiconductor lasers,”

On the linewidth enhancement factor CY in semiconductor injection lasers,”

Quantum noise and dynamics in quantum well and quantum wire lasers,”

Spectral dependence of the change in refractive index due to camer injection in GaAs lasers,”

The extra differential gain enhancement in multiple quantum well lasers,” IEEEPhoton.

Theory of gain, modulation response, and spectral linewidth in AlGaAs quantum well lasers,’’

Theory of the linewidth of semiconductor lasers,”

http://authors.library.caltech.edu/6841/1/ZHAieeejqe93.pdf

A comparison of amplitude-phase coupling and linewidth enhancement in semiconductor quantum-well and bulk lasers

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