2006
OPTICS LETTERS / Vol. 31, No. 13 / July 1, 2006
Square-wave self-modulation in diode lasers with
polarization-rotated optical feedback
Athanasios Gavrielides
Air Force Research Laboratory, Directed Energy Directorate AFRL/DELO, 3550 Aberdeen Avenue SE, Kirtland AFB,
New Mexico 87117
Thomas Erneux
Université Libre de Bruxelles, Optique Nonlinéaire Théorique, Campus Plaine, Code Postal 231, 1050 Bruxelles,
Belgium
David W. Sukow, Guinevere Burner, Taylor McLachlan, John Miller, and Jake Amonette
Department of Physics and Engineering, Washington and Lee University, 204 West Washington Street, Lexington,
Virginia 24450
Received March 6, 2006; revised April 18, 2006; accepted April 20, 2006; posted April 24, 2006 (Doc. ID 68731)
The square-wave response of edge-emitting diode lasers subject to a delayed polarization-rotated optical
feedback is studied in detail. Specifically, the polarization state of the feedback is rotated such that the natural laser mode is coupled into the orthogonal, unsupported mode. Square-wave self-modulated polarization
intensities oscillating in antiphase are observed experimentally. We find numerically that these oscillations
naturally appear for a broad range of values of parameters, provided that the feedback is sufficiently strong
and the differential losses in the normally unsupported polarization mode are small. We then investigate the
laser equations analytically and find that the square-wave oscillations are the result of a bifurcation
phenomenon. © 2006 Optical Society of America
OCIS codes: 140.2020, 140.5960, 190.0190.
Optical feedback from an external reflector is a wellknown technique for inducing dynamic effects in
semiconductor lasers. The behavior that results depends on many factors, such as the laser architecture
and operating conditions, the external cavity length
and configuration, and the strength of the feedback.
Recently, the effect of polarization-rotated optical
feedback (PROF) has raised considerable interest.
In this setup, the polarization state of the optical
feedback is orthogonal to the laser’s natural operating mode. Such systems are of interest from a variety
of standpoints. First, their dynamic properties can be
investigated in more detail than for conventional optical feedback,1–5 and they also apply for purely optoelectronic feedback systems.5,6 Second, the effect of
PROF on vertical-cavity surface-emitting lasers has
been examined in several laboratories. The feedback
typically gives rise to switching between the dominant polarization mode and the orthogonal mode that
is nearly degenerate due to the cylindrical symmetry
of such devices.7–12 Polarization self-modulation has
a variety of useful applications stemming from the
production of optical pulses at multi-GHz repetition
rates without the need for high-speed electronics. But
edge-emitting lasers can also be made to exhibit
pulse trains generated by polarization selfmodulation, although the losses in the orthogonal
mode are typically higher than in vertical-cavity
surface-emitting lasers.13,14
If a quarter-wave plate is used to create PROF, it
allows mutual coupling between the polarization
modes as well as multiple cavity round-trips. In this
Letter, we use a Faraday rotator to avoid multiple
round-trip reflections and allow only unidirectional
0146-9592/06/132006-3/$15.00
coupling between the natural horizontal polarization
(TE) mode and the unsupported orthogonal (TM)
mode. This configuration also simplifies the analysis
since the TM mode, even when activated by feedback,
cannot influence the TE mode directly via optical injection. Recent studies of the dynamics of this
system15,16 and its synchronization properties17 demonstrate the need to consider the intensities of both
modes as independent variables. Rate equations have
been used and reproduce the experimental observations. The objective of this Letter is to show analytically that square-wave output naturally appears in
PROF systems.
The experimental apparatus is described in Fig. 1.
We use an index-guided SDL-5401 laser diode (LD)
with wavelength ␭ = 817.9 nm and current threshold
of 18.48 mA. It is stabilized in temperature and
pumped at 28.02 mA, which produces single-mode
Fig. 1. Schematic diagram of the experimental system.
ND, neutral density filter; PD1, PD2, photodetectors; AMP,
amplifiers. Other abbreviations defined in text.
© 2006 Optical Society of America
July 1, 2006 / Vol. 31, No. 13 / OPTICS LETTERS
Fig. 2. Experimental time series of the square-wave selfmodulated oscillations. Top and bottom, TE and TM mode
intensities, respectively. (a) and (b) ␶ = 2.60 ns, (c) and (d)
␶ = 6.92 ns.
operation with or without PROF. An external cavity
creates the PROF and controls its strength. After the
laser beam emerges from a collimating lens (CL), it
passes through a nonpolarizing beam splitter (BS1,
T = 70%). It continues through a Faraday rotator
(ROT), which is a Faraday isolator with its input polarizer removed and output polarizer at 45° from
horizontal. After emerging from the rotator, the beam
hits a rotatable polarizer (POL) for attenuation and
strikes a highly reflective mirror (HR). The retroreflected beam passes through the polarizer and reenters the Faraday rotator through its output polarizer, ensuring its 45° orientation. It rotates an additional 45° to vertical and is reinjected into the laser’s
TM mode. This system prevents multiple roundtrips, since any vertically polarized light reflected
from the laser facet will be extinguished by the Faraday’s output polarizer. Polarization-resolved detection is accomplished via the signal reflected from BS1
(30% of the incident ray), which strikes a second
beam splitter (BS2, R = 50%). The two resulting
beams enter detection arms identical to those used in
Sukow et al.17
With this experimental system, we observe squarewave oscillatory regimes. Two pairs of square-wave
outputs are shown in Fig. 2. The round-trip power
transmission in the cavity is 37.6%; strong feedback
is required for one to observe square waves. The oscillations of the two mode intensities are in antiphase
and exhibit a period close to twice the round-trip
time. Damped relaxation oscillations appear at every
intensity jump.
To simulate the square-wave oscillations, we consider rate equations similar to those used by Heil et
al.15 In dimensionless form,16 the following three
equations for the (complex) TE and TM fields, E1 and
E2, respectively, and the carrier density N are
dE1
共1兲
= 共1 + i␣兲NE1 ,
dt
dE2
dt
= 共1 + i␣兲k共N − ␤兲E2 + ␩E1共t − ␶兲,
dN
T
dt
= P − N − 共1 + 2N兲共兩E1兩2 + 兩E2兩2兲.
2007
describes the TM mode with k and ␤ ⬎ 0 defined as
the ratio of the TM and TE gain coefficients and the
differential losses, respectively. The last term in Eq.
(2) models the reinjection of the delayed, rotated TE
field. Finally, Eq. (3) describes the carrier density,
which depends on the intensities of both fields. All
parameters are dimensionless. In addition to k and ␤,
the laser parameters include linewidth enhancement
factor ␣, the ratio T of the carrier lifetime to the cavity lifetime, and the pump parameter above threshold, P. The feedback is controlled by its strength ␩
and its delay ␶. The range of parameters for the
square-wave regimes appears to be fairly broad and
requires only small values of ␤. Laser parameters
representative of a SDL diode laser are18 T = 150, ␣
= 2, and P = 0.5. Equations (1)–(3) have been studied
systematically16 for different values of k = 0.3– 1, ␤
= 0.03– 0.09, ␩ = 0.1– 0.4, and ␶ = 102 – 104. By gradually increasing ␩, the nonzero intensity steady state
transfers its stability to sustained relaxation oscillations that are progressively modulated by a
2␶-periodic square-wave. Above a critical value of ␩,
the square-wave oscillations dominate and exhibit
only decaying relaxation oscillations at the beginning
of each square wave, as shown in Fig. 3. In simulations, we find that square-wave oscillations with
more than two plateaus and periods longer than 2␶
are possible. However, the third plateau is unstable,
is not seen experimentally, and does not enter into
the analysis to follow. In simulations, a small amount
of noise of amplitude 10−6 is introduced in the righthand sides of Eqs. (1) and (2) to prevent the trajectory from residing near the unstable plateau.
After introducing Ej = Aj exp共i␾j兲 共j = 1 , 2兲 and the
new time s ⬅ t / ␶ into Eqs. (1)–(3), we neglect all ␶−1
small terms multiplying the time derivatives. The resulting algebraic equations are
NA1 = 0,
共4兲
k共N − ␤兲A2 + ␩A1共s − 1兲cos共⌽兲 = 0,
共5兲
␣关N共s − 1兲 − k共N − ␤兲兴A2 − ␩A1共s − 1兲sin共⌽兲 = 0,
P − N − 共1 + 2N兲共A12 + A22兲 = 0,
共6兲
共7兲
共2兲
共3兲
Equation (1) describes the TE mode, which lases
naturally since it has the lowest losses. Equation (2)
Fig. 3. Long-time numerical solution of Eqs. (1)–(3). The
values of the parameters are P = 0.5, ␣ = 2, T = 150, ␶ = 103,
␩ = 0.35, k = 1 / 3, and ␤ = 0.09. The horizontal broken lines
emphasize the plateaus of the square-wave oscillations:
兩E1 兩 ⯝ 0.7 and 兩E2 兩 ⯝ 1.1.
2008
OPTICS LETTERS / Vol. 31, No. 13 / July 1, 2006
Fig. 4. Bifurcation diagram of the square-wave selfmodulated solutions. P = 0.5, ␣ = 2, ␩ = 0.35, and k = 1 / 3. The
figure represents the difference between the values of 兩E2 兩
−兩E1兩 for each plateau of the square wave as a function of ␤.
If ␤ ⬎ ␤c, chaotic oscillations appear.
where ⌽ ⬅ ␾1共s − 1兲 − ␾2. Equations (4)–(7) relate the
values of the dependent variables at time s to their
values at time s − 1. Equation (4) is satisfied if either
N = 0 or A1 = 0. We anticipate that the square-wave solution exhibits two successive stages characterized by
(1) N = 0 and A1共s − 1兲 = 0 and (2) A1 = 0 and N共s − 1兲
= 0. For stage (1), we then find from the remaining
equations that A2 = 0 and A1 = 冑P. For stage (2) now
supplemented by A1共s − 1兲 = 冑P, we find from Eqs. (5)
and (6) that tan共⌽兲 = ␣ and
k共N − ␤兲A2 + ␩
冑
P
1 + ␣2
= 0.
共8兲
Finally, from Eq. (7) with A1 = 0, we determine A2 as
A22 =
P−N
1 + 2N
共9兲
.
Equations (8) and (9) lead to the bifurcation diagram
of the square-wave solutions. In Fig. 4, we represent
the difference between the values of the two polarization plateaus, i.e., A2 − A1, where A2 is determined
from Eq. (9) and A1 = 冑P. Using ␤ as the bifurcation
parameter, the square-wave solution exists if N 艋 0,
i.e., if
␤ 艋 ␤c =
␩
k冑共1 + ␣2兲
.
共10兲
If ␩ = 0.35, k = 1 / 3, and ␣ = 2, then ␤c ⯝ 0.47. Using
these values of ␩, k, ␣, P = 0.5, T = 150, and ␶ = 103, we
have verified numerically from Eqs. (1)–(3) that a
stable square-wave regime exists at ␤ = 0.46 but disappears at ␤ = 0.48 into chaotic oscillations as were
found previously in this system.15,17 For a given laser
with fixed differential losses, Eq. (10) provides a condition on the minimum feedback at which square
waves are possible. This agrees with numerical findings that require a sufficiently strong ␩, since higher
␩ implies a higher ␤c. In both the experiments and
the numerical simulations, the square-wave oscilla-
tions are supplemented by decaying relaxation oscillations. They can be studied analytically by investigating the stability of the plateaus with respect to
the fast time scale of the relaxation oscillation. The
results of this analysis will be presented elsewhere.
In summary, we have found experimentally and
theoretically that stable square-wave self-modulated
oscillations of a period close to twice the round-trip
time naturally appear for edge-emitting diode lasers
subject to PROF. The main condition is that the differential losses in the TM mode are small so the two
polarization modes may behave independently. In
terms of this parameter, we then showed analytically
that there exists a bifurcation point for the two main
plateaus of square-wave oscillations above which a
square-wave regime is no longer possible.
This material is based upon work supported by the
U.S. National Science Foundation under CAREER
grant 0239413. T. Erneux acknowledges the support
of the Fonds National de la Recherche Scientifique
(Belgium) and the InterUniversity Attraction Pole
program of the Belgian government. D. Sukow’s
e-mail address is sukowd@wlu.edu.
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