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Dynamics of high gain ber laser arrays JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. , NO. , MONTH 2004
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Dynamics of high gain ber laser arrays
Jeffrey L. Rogers, Slaven Peles, and Kurt Wiesenfeld
(Invited Paper)
Abstract Recent experiments [1] have shown that a small
number of ber lasers can spontaneously form coherent states
when suitably coupled. The observed synchrony persisted for a
long time without any active control. In this paper we develop a
dynamical model for ber laser arrays that is valid in the high
gain regime. Analysis and simulations of the model nd behaviors
in qualitative agreement with those observed in the experiments.
Index Terms coupled lasers, coherence, synchrony, two way
coupling, optical bers, laser array.
I. I
NTRODUCTION
I
N differentiating the coherence displayed by optical masers
(lasers) from classical coherent behavior Max Garbuny
wrote the synchronism which is a condition of coherence
also extends to the timing of electronic interactions ... This,
in broad outline, is the nature of coherent interactions. [2].
The resulting macroscopic coherence from single lasers has
led to a host of applications. Yet there are situations where an
array of mutually coherent lasers offers distinct advantages.
One notable example is the high-power optical source. The
power output from a single laser is constrained by its physical
characteristics. Common constraints include heat dissipation,
mechanical breakdown, and power conversion efciency. In
contrast, relatively low-power ber lasers can have excellent
heat dissipation properties, relatively wide stability range, and
greater than 70% pump conversion efciency. An array of
these lasers provides an attractive alternative to the single
monolithic source.
To take full advantage of coherence array components must
maintain zero relative phases. The far eld spatial power dis-
tribution depends crucially on two aspects of the array electric
elds: (1) the degree of coherence of the elds produced by
distinct array elements and (2) the relative phases of these
elds at the array output. In this paper, we call two lasers
coherent if their respective electric elds maintain a constant
relative phase
1
. If this phase is zero the lasers are said to
be in an inphase state; if the relative phase is the lasers are
described as in an anti-phase state. Due to inevitable variations
between individuals, an array of N isolated lasers will be
incoherent and the resulting spatial intensity distribution is
simply the sum of the intensity patterns from each laser. Since
adding the incoherent elds is equivalent to a random walk the
peak intensity will increase as N . In contrast, if all N lasers are
inphase the output intensity is concentrated in a narrow pencil
at broadside (zero angle). This state is particularly attractive
since its peak intensity grows as N
2
.
1
Weaker denitions of coherence are possible, and may be relevant depend-
ing on circumstances.
For these reasons a longstanding goal of optics research
has been to produce laser arrays that are stable in the inphase
state. While these investigations have produced a number
of approaches and interesting results, the ultimate goal of
designing inphase arrays in a manner that is scalable with laser
number has remained elusive. A number of these approaches
are variations of the master oscillator multiple amplier
(MOPA) concept. As the name implies MOPAs use an array of
ampliers to boost a master signal. Due to inevitable amplier
variations these arrays require active control of the relative
phases through additional circuitry that limit both array stabil-
ity and scalability. Recent MOPA architectures have utilized
ber laser developments to report output powers of 100 W
[3]. Evolving this approach a step further, MOPA ampliers
have been replaced by lasers (oscillators) [4]. These injection
locking approaches rely on slaving each individual array laser
to the master source. A common implementation is arrays
of evanescently coupled diode lasers. While diode arrays
form coherent states, the anti-phase state is most commonly
observed [5]. Even if the inphase state can be formed in an
array of diodes, the degree of control required to maintain
this state as well as the basic physical architecture leaves the
scalability to large N of this solution an open and challenging
question [6], [7], [8].
An alternative to these approaches is to utilize mutual
synchronization to produce inphase arrays. Mutual synchro-
nization has the advantage of being passive (no active control
requirement). This approach also does not require a master
source, rather it utilizes intrinsic nonlinearity and appropriate
laser connectivity to facilitate a frequency and phase distribu-
tion that the array elements select. The phenomenon is familiar
from nonlinear science studies in several areas of physics,
biology, and engineering [9], [10], [11]: under the right
circumstances a collection of coupled nonlinear oscillators will
spontaneously form a synchronized dynamical state. At this
level of description, the laser array dynamics is governed by a
set of coupled nonlinear oscillator equations, either differential
equations [12], [13] or iterative maps. Meanwhile, the detailed
nature of the quantum mechanics of the inverted population
and the electromagnetic cavity are not directly modeled. By
focusing on the broader essentials of the problem we can
investigate array sizes spanning several orders of magnitude
and a wide range of coupling architectures in a reasonable
amount of time.
A primary motivation for this paper is the recent experimen-
tal work on coupled ber arrays at HRL Laboratories, LLC [1].
The experiments were based on very general lessons from cou-
pled oscillator and network studies. Here, we develop a more
detailed theoretical description of the coupled laser system,
keeping as general as possible the overall array architecture. JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. , NO. , MONTH 2004
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....
F(z,t)
reflector
....
E(z,t)
coupler
gain sections
face
Fig. 1.
Sketch of the laser array. Each component is described in the text
(Secs. II and III).
In Sec. II we derive a set of coupled nonlinear iterative maps
for the system. The general formulation allows for a variety
of distinct coupling schemes (Sec. III). In Sec. IV we examine
the model analytically and numerically in the limiting case of
a single laser; identifying three distinct types of behavior that
reproduce qualitatively those seen in experiments. Finally, in
Sec. V, we present some preliminary numerical results for the
array.
II. D
ESCRIPTION OF THE SYSTEM
We consider a set of N ber lasers as sketched in Fig. 1.
On one end (z = 0) each separate ber has a 100% reective
mirror. In the experiments at HRL Laboratories, LLC [1] ber
gratings with better than 98% reectivity were used. At the
other cavity end the output face mirror has reectivity r which
in practice is rather small. The lasers are independent except
over the region labeled coupler (z c ), where the elds
mix. This coupling region may take on several forms in the
laboratory. In between the mirrors are gain sections (g
z g
+
) that in practice may use various gain media. The
gain sections are connected to the cavity ends by non-gain,
no-loss ber sections.
In what follows, we calculate the change in the eld
amplitudes over one cavity round trip. In this way we are led
quite naturally to a set of coupled iterative maps that govern
the dynamics of the electric elds. We then deduce the iterative
maps that describe the evolution of the gains.
A. Evolution of the electric elds
We decompose the electric elds into counter-propagating
waves. Let E
n
(z, t) be the complex phasor amplitude for
the right-going wave in the n
th
ber and let F
n
(z, t) be the
corresponding quantity for the left-going wave. A cavity round
trip will begin with a right-going wave just prior to the output
face at z = L. The round trip is then made of the following
steps: the waves (1) bounce off of the partially reecting output
face; (2) propagate through the coupler (during this part the
amplitudes mix); (3) freely propagate to the edge of the gain
regions; (4) are amplied in passing through the gain regions;
(5) freely propagate to the reective mirrors; (6) reect back;
(7) propagate to the edge of the gain regions; (8) are amplied
again; (9) freely propagate to the coupler; and nally (10) pass
through the coupler to the output face.
We now consider each of these steps in turn. At the coupler
output face a fraction of each eld is instantaneously reected
by a factor 0 < r < 1 and with a phase shift, producing left
propagating waves
F
n
(L, t) = rE
n
(L, t).
(1)
Next, as F
n
passes through the coupler each wave is coupled
to other members of the array. Upon emerging at t