1992 OPTICS LETTERS / Vol. 27, No. 22 / November 15, 2002
Alternate oscillations in semiconductor ring lasers
M. Sorel and P. J. R. Laybourn
Department of Electronics and Electrical Engineering, University of Glasgow, Oakfield Avenue, Glasgow G12 8LT, Scotland, UK
A. Scirè and S. Balle
Instituto Mediterráneo de Estudios Avanzados, Consejo Superior de Investigaciones Cientificas– Universidad de las Islas Baleares,
Campus Universidad de las Islas Balears, E-07071 Palma de Mallorca, Spain
G. Giuliani, R. Miglierina, and S. Donati
Dipartimento di Elettronica, Università di Pavia, Via Ferrata 1, I-27100 Pavia, Italy
Received June 3, 2002
We report on fabrication and characterization of single-longitudinal- and transverse-mode semiconductor ring
lasers. A bifurcation from bidirectional stable operation to a regime with alternate oscillations of the coun-
terpropagating modes was observed experimentally and is theoretically explained through a two-mode model.
Analytical expressions for the onset and the frequency of the oscillations are derived, and
L
I curves numeri-
cally evaluated. Good quantitative agreement between theory and measurements made over a large number
of tested devices is obtained. © 2002 Optical Society of America
OCIS codes: 140.3560, 230.3120, 190.3100.
The two-mode dynamics in ring laser systems has been
the subject of a large number of experimental and theo-
retical investigations.
1
From the fundamental point of
view, analysis of the peculiar symmetry properties of
the ring laser provides insight for nonlinear dynamics
studies,
2
enhancing the knowledge of bifurcation the-
ory applied to symmetry groups.
3
Besides, the devel-
opment of the ring laser gyroscope has raised great
practical interest, enhancing efforts toward a full un-
derstanding of the ring laser physics.
4
More recently,
semiconductor ring lasers
5
(SRLs) have been demon-
strated and studied because of their interesting fea-
tures: They do not require cleaved facets or gratings
for optical feedback, and thus monolithic integration
is easily achievable. They are promising candidates
for wavelength filtering, multiplexing–demultiplexing
applications, electrical and all-optical switching, and
bistable devices for optical memories.
6
In this Letter
we report, for the first time to our knowledge, the ex-
perimental observation of an oscillatory bidirectional
regime in SRLs. Analytical expressions for oscillation
threshold and frequency are calculated, and the
L
I
curves are numerically reproduced with a two-mode
model. A quantitative agreement between theory and
measurements made over a large number of devices is
obtained.
SRLs with 2-mm-wide single-transverse-mode ridge
waveguides were fabricated in a double-quantum-well
GaAs兾AlGaAs structure with a 1-mm ring radius; the
latter value was chosen to minimize bending losses.
A sketch and a photographic detail of the device are
shown in Fig. 1. An output straight waveguide with
the same structure is directionally coupled to the ring,
providing cross-power transmission of ⬃10%.The
output waveguide has a 5
±
tilt angle with respect to
cleaved chip facets to minimize backref lections, and
it terminates at each end in a reverse-biased contact
that acts as an integrated photodetector for mode 1
(counterclockwise) and for mode 2 (clockwise). The
reverse-biased contacts further reduce the optical
feedback from the end facets. Figure 2(a) shows
experimental L
I curves measured at 25
±
C from the
two integrated photodiodes, PD1 and PD2, as a func-
tion of dc current injected into the ring contact. The
photocurrents from PD1 and PD2 are analyzed in their
dc components, and they are also fed to a 50-V digital
oscilloscope and to a rf spectrum analyzer with which
we investigate the dynamics. Three distinct operat-
ing regimes are identified, labeled A–C in Fig. 2(a).
In regime A, which occurs from threshold (195 mA) to
250 mA, the ring laser operates bidirectionally; the
intensities of the two counterpropagating modes are
cw. Regime B, which occurs from 250 to 285 mA, is
again bidirectional, but in this case oscillatory behav-
ior in the modal power is observed. The intensities
of modes 1 and 2 are harmonically modulated at a
frequency in the tens-of-megahertz range: When the
intensity of mode 1 reaches a maximum, the intensity
of mode 2 reaches a minimum, and vice versa. We call
this regime alternate oscillations (AO). In regime C
Fig. 1. (a) Geometry of the SRL device and layout of the
contacts. PD1, PD2, photodiodes. (b) Detail of the output
waveguide.
0146-9592/02/221992-03$15.00/0 © 2002 Optical Society of America
November 15, 2002 / Vol. 27, No. 22 / OPTICS LETTERS 1993
Fig. 2. (a) Experimental (dc) L I curves of a sample
SRL device. (b) Theoretical time-averaged L
I curve
obtained with a 苷 3.5, s 苷 0.005, c 苷 0.01, k
c
苷 0.0044,
k
d
苷 0.000327, g 苷 0.002, and t
p
苷 10 ps. The Hopf
bifurcation boundary is m
H
苷 1.4.
(above 285 mA) the laser operates quasi-unidirection-
ally, and no oscillation occurs. When the current
is further increased in regime C, switching between
the two modes is observed, as previously reported for
SRLs.
7
Optical spectrum measurements reveal that
the SRL operates in a single longitudinal mode and
that in bidirectional operation the wavelengths of the
two counterpropagating modes are locked to the same
value. Figure 3(a) reproduces oscilloscope traces of
the intensities of modes 1 and 2 in regime B, showing
AO. As the ring laser current is increased, the
oscillation frequency decreases almost linearly, while
the waveforms of the intensity oscillations become
distorted into triangular shapes. Several SRLs from
different batches were measured, and all devices
exhibited the same behavior reported here. Regimes
with bidirectional oscillatory instability were previ-
ously reported in gas
8
and dye
9
ring lasers, but they
were never observed in SRLs.
The theoretical analysis is based on a set of di-
mensionless semiclassical Lamb equations for the two
(slowly varying) complex amplitudes of counterpropa-
gating fields E
1
and E
2
. The applicability of this
theory to the devices that we fabricated is guaran-
teed by the single-transverse- and longitudinal-mode
operation reported experimentally, by the absence
of relevant feedback from outside the cavity and the
reduced scattering loss achieved through the choice
of a shallow-etched ridge structure and of a tilted
and weakly coupled output waveguide. The Lamb
equations read as
ᠨ
E
1, 2
苷 共1 1 ia兲关N共1 2 sjE
1, 2
j
2
2 cjE
2, 1
j
2
兲 2 1兴
3 E
1, 2
2 共k
d
1 ik
c
兲E
2, 1
, (1)
where a accounts for phase –amplitude coupling; the
self- and cross-saturation coefficients are given by s
and c, and the parameters k
d
and k
c
represent the
dissipative and the conservative components of the
backscattering, respectively.
8
Carrier density N
obeys the usual rate equation:
ᠨ
N 苷 g关m2N 2 N共1 2 sjE
1
j
2
2 cjE
2
j
2
兲 jE
1
j
2
2 N 共1 2 sjE
2
j
2
2 cjE
1
j
2
兲 jE
2
j
2
兴 , (2)
where m is the dimensionless pump (m 苷 1 at laser
threshold). In Eqs. (1) and (2) the dimensionless time
is rescaled by photon lifetime t
p
. Parameter g is
the ratio of photon lifetime t
p
to carrier lifetime t
s
.
The set of Eqs. (1) and (2) written for the intensities
S
1, 2
苷 jE
1, 2
j
2
reverts to a previous model,
10
except for
inclusion of backscattering terms. In a real system
the ideal symmetry along the ring is never met, for
many reasons: imperfections in waveguide, output
coupler, and scattering centers. Any break in the
invariance symmetry along the ring translates into a
source of coupling between the two counterpropagat-
ing fields.
11
Thus backscattering terms have to be
considered. In our approach, k
d
and k
c
are regarded
as fitting parameters because their actual values are,
in principle, technology dependent. According to a
previous analysis,
12
saturation parameters c and s in
a SRL fulfill the condition that c兾s . 1.
By substituting the general solution E
1, 2
苷
Q
1, 2
共t兲exp关iv
1, 2
t 1 if
1,2
共t兲兴 into Eqs. (1) and (2)
we find the steady-state solutions by setting all
the derivatives to zero. We find a symmetric
共Q
1
苷 Q
2
苷 Q兲 steady state (SS). When we introduce
c 苷 f
2
2f
1
and I 苷 2Q
2
,SS苷 共v
1, 2
, c, I
0
, N
0
兲 is
given by v
1
苷 v
2
苷 2ak
d
1 k
c
, c
0
苷 p:
I
0
苷
N
0
2 1 1 k
d
hN
0
,
(3)
N
0
苷
m
1 1 I
0
2hI
0
2
,
(4)
where h 苷 共c 1 s兲兾2. After linearization of
the perturbations defined by E
1, 2
苷 共
p
I
0
兾2 1
a
1, 2
兲exp共iv
1,2
t 1 if
1, 2
兲 and N 苷 N
0
1D, a linear
stability analysis of solutions [Eqs. (3) and (4)] is
performed
13
; a
1, 2
are complex perturbations of the
field amplitudes, and D is a real perturbation of the
carrier variables. To first order in the perturbations,
Fig. 3. Time traces of the intensities of modes 1 and
2 in the AO regime. (a) Experimental single-shot
acquisition measured from PD1 and PD2 on a 50-V
oscilloscope: vertical scale, 50 mV兾division; horizontal
scale, 10 ns兾division; ring current, 275 mA. (b) Result
obtained through numerical simulations (solid curves)
and from application of Eq. (8) (dashed curves): vertical
scale, arbitrary units; horizontal scale, 10 ns兾division.
The parameters are the same as for Fig. 2, and m 苷 1.58.
1994 OPTICS LETTERS / Vol. 27, No. 22 / November 15, 2002
the system of Eqs. (1) and (2) decouples into two sub-
sets. The first subset accounts for the total intensity
stability; it contains the variables
S 苷 a
2
1 a
1
and D,
and it is always stable. The second subset contains
the complex variable R 苷 a
2
2 a
1
and describes the
stability of one field with respect to the other:
ᠨ
R 苷
1
/
2
共1 1 ia兲N
0
I
0
共c 2 s兲共R 1 R
ⴱ
兲
2 2共k
d
1 ik
c
兲R , (5)
ᠨ
R
ⴱ
苷
1
/
2
共1 2 ia兲N
0
I
0
共c 2 s兲共R 1 R
ⴱ
兲
2 2共k
d
2 ik
c
兲R
ⴱ
. (6)
The calculation of the eigenvalues associated with
Eqs. (5) and (6) shows that near threshold the SS
is stable. As the pump is increased, the SS loses
stability through a Hopf bifurcation when
4k
d
苷 N
0
I
0
共c 2 s兲 , (7)
exhibiting pulsating behavior at the frequency
V 苷 2
∑
k
d
2
1 k
c
2
2
N
0
I
0
2
共k
d
1ak
c
兲
∏
1兾2
. (8)
We remark that an in-phase solution 共c 苷 0兲 also
exists, but it is always unstable and we do not
consider it here. When I
0
and N
0
are expressed
as functions of m, a particular value of m 苷 m
H
satisfies Eq. (7). Thus, for 1 ,m,m
H
, bidirec-
tional stable operation is predicted; when m.m
H
,
oscillatory behavior takes place. This oscillation
represents a limit cycle on the variables R and R
ⴱ
and corresponds to AO of the two mode intensities
jE
1, 2
j
2
. The AO angular frequency at onset is ob-
tained by substitution of Eq. (7) into Eq. (8), yielding
V
H
苷 共k
c
2
2 k
d
2
1 2ak
d
k
c
兲
1兾2
. When the pump is
increased further, the oscillation frequency decreases,
until oscillations disappear when two new quasi-
unidirectional solutions become stable. In that case
the output power is concentrated primarily in one
direction, and no pulsation occurs. The analytical
description of this transition is rather involved and
will be the subject of further investigations.
Figures 2(b) and 3(b) illustrate numerical results
obtained by Eqs. (1) and (2) through a standard
Runge–Kutta algorithm. We found that the pump
interval for the AO regime widens when k
c
is increased
and shrinks when k
d
is increased. The observed SRL
dynamics can be interpreted as follows: Because of
the strong cross-gain saturation, the semiconductor
medium tends to select unidirectional operation.
However, because of backscattering, the pure unidi-
rectional state is not a solution for the ring cavity, and
bidirectional regimes are favored. The tendency to
unidirectional behavior is recovered at a higher pump
level, at which nonlinear gain imposes a stronger
mode selection. The AO regime reported here has
some analogies to the AO induced by conservative
backscattering in He– Ne ring lasers.
8
However, an
important difference lies in the fact that the He–Ne
ring laser supports bidirectional operation (clockwise
and counterclockwise waves carry the same amount of
energy during the pulsation) because c兾s , 1.
In conclusion, we have fabricated and tested
single-longitudinal- and transverse-mode semiconduc-
tor ring lasers. A bifurcation from bidirectional stable
operation to oscillatory behavior has been measured.
and it was described through a two-mode approach,
yielding analytical expressions for the oscillations’
onset and frequency near the bifurcation; the whole
experimental L
I curve is well reproduced by theory.
The authors acknowledge technical support from
the Nanoelectronics Research Center and the dry
etching group of the Department of Electronics and
Electrical Engineering, Glasgow University, and from
A. Fanzio. A. Scirè thanks the European Union for
the financial support of a Marie Curie fellowship
under contract HPMF-CT-2000-00617. A. Scirè’s
e-mail address is scire@imedea.uib.es.
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