Lee K.K. Lectures on Dynamical Systems, Structural Stability and Their Applications
Подождите немного. Документ загружается.
scale,
the
carrier
recombination
rate
is
about
one
nanosecond,
longer
than
the
field
couplings,
which
are
of
the
order
of
30
-
50
psec,
or
the
round
trip
time
of
the
external
cavity.
Nonetheless,
in
order
to
study
the
dynamics
of
the
two
semiconductor
lasers,
without
mixing
the
instabilities
and
chaos
induced
by
the
nonlinear
delayed
feedback
[Ikeda
1979],
the
external
cavity
has
to
be
very
short.
Indeed,
in
order
to
aviod
the
complexity
of
the
nonlinear
delayed
feedback
instability,
the
external
cavity
round
trip
time
should
not
be
greater
than
0.1
nsec.
So,
the
main
effort
of
the
study
should
be
on
the
coupling
dynamics
of
the
field
amplitudes
and
phases,
that
is,
the
study
of
the
four-dimensional
subspace
of
the
field
amplitudes
and
phases.
Theoretical
study
of
the
coupled
six
equations,
Eqs.(7.8-6),
is
straight
forward
in
the
sense
that
with
appropriate
numerical
schemes
and
large
enough
computer,
one
can
have
detailed
numerical
simulations
and
beautiful
graphics
for
the
dynamics
of
the
system.
And
indeed,
we
can
theoretically
answer
the
questions
we
have
posed
earlier.
What
we
are
interested
in,
concerned
with,
and
proposing
to
do,
instead,
is
to
find
an
experimental
approach
to
study
the
system,
which
will
be
more
satisfying.
As
we
have
pointed
out
earlier,
direct
measurements
are
hopeless
due
to
the
extremely
fast
coupling
of
the
system.
This
is
true
even
with
the
help
of
the
very
recent
silicon
photodetectors
invented
by
Seigmann
[1989],
which
have
detector
response
time
of
a
few
psec.
This
is
because
the
detectors
can
not
have
a
sampling
rate
of
100
GHz
or
higher.
Indeed,
it
is
unlikely
such
a
device
will
be
available
in
the
near
future.
It
is
then
clear
that
some
means
of
direct
detection
with
a
much
slower
sampling
rate
is
necessary.
In
the
following,
we
will
propose
a
set
of
experiments
to
be
performed,
which
can
unambiguously
map
out
the
coupling
dynamics
of
the
four-dimensional
subspace
of
the
fields
and
phases
of
two
lasers
experimentally.
The
idea
is
to
use
a
laser
medium
which
has
a
long
radiative
decay
rate,
394
such
as
those
solid-state
lasers,
e.g.,
Nd:YAG
laser,
which
has
decay
rate
of
0.23
msec,
or
gas
lasers,
such
as
co
2
laser,
which
has
decay
rate
of
0.4
msec.
If
the
external
cavity
round
trip
time
is
about
1
or
2
nsec,
then
the
coupling
time
can
only
be
longer
than
this,
which
can
be
resolved
by
the
state
of
art
detectors
with
time
resolution
of
0.1
nsec.
In
fact,
we may
not
need
such
fast
detectors.
This
is
because
even
though
the
external
cavity
round
trip
time
is
about
2
nsec,
due
to
relatively
"low"
gain
of
these
laser
media,
effective
coupling
may
take
several
round
trips,
thus
the
coupling
time
is
on
the
order
of
tens
of
nsec.
At
any
rate,
this
is
not
very
important
to
our
discussion,
we
only
want
to
point
out
that
relatively
fast
yet
inexpensive
detectors
will
be
sufficient.
Even
for
these
slow
decay
rate
laser
media,
we
still
have
a
set
of
six
coupled,
nonlinear
differential
equations
to
deal
with,
which
included
two
laser
rate
equations.
But
since
the
coupling
time
is
stretched
rather
long,
compared
with
semiconductor
lasers,
to
a few
to
tens
of
nsecs,
to
accommodate
the
slow
detector
response
time,
it
is
still
far
too
short
compared
with
the
radiative
decay
time
of
the
laser
which
is
of
the
order
of
0.2
to
0.4
msec.
Thus,
the
two
laser
rate
equations
can
be
considered
as
steady-state.
Thus,
we
can
study
the
nonlinear
coupling
of
the
fields
and
phases
of
the
two
lasers
using
existing
technology.
Here
we
shall
briefly
discuss
the
four
control
parameters
for
the
study
of
the
dynamics
of
nonlinear
coupling
of
two
(say
solid-state)
lasers.
For
the
sake
of
illustration,
let
us
here
consider
the
coupling
of
two
small
Nd:YAG
lasers.
Let
these
two
Nd:YAG
lasers
be
monolithic
ones,
such
as
the
one
for
frequency
stable
application
consatructed
by
Zhou
et
al
[1985],
but
with
somewhat
larger
gain
volume.
And
they
are
temperature
controlled
with
electric
coolers.
Let
laser
A
be
the
"reference"
one,
that
is,
except
the
output
power,
no
other
parameters
will
be
varied.
Thus,
this
laser
can
be
truely
monolithic
in
design.
The
other
laser,
laser
B,
has
PZT
mirror
as
the
back
mirror,
395
thus
the
laser
is
slightly
tunable.
The
power
reflectivity
of
the
outcoupling
mirror
is
the
same
as
the
"reference"
laser
so
that
they
all
have
the
same
amount
of
feedback.
Outside
the
outcoupler
of
laser
B,
there
is
an
electro-optic
phase
modulator,
this
can
change
the
relative
phase
of
these
two
lasers.
And
in
front
of
the
coupling
mirror,
we
can
put
a
calibrated
neutral
density
filter
to
attenuate
and
to
control
the
mutual
coupling
of
these
two
lasers.
By
controlling
the
pump
power
of
laser
B,
we
can
also
vary
the
output
power
density
of
laser
B.
This
in
turn
will
change
the
relative
power
density
of
the
two
lasers,
and
consequently,
the
relative
feedback
amount
of
these
two
lasers.
Of
course,
there
are
other
control
parameters
one
can
vary,
but
in
order
to
closely
simulate
a
diode
phased
array,
these
four
control
parameters
for
the
four-dimensional
subspace
are
the
only
relevant
ones.
For
diodes,
the
inject
current
densities
are
the
other
relevant
parameters.
In
solving
Eqs.(7.8-6c,d,e,f),
we
can
restate
these
equations
by
subtracting
these
two
set
of
equations
and
obtaining
the
set
of
two
nonlinear
differential
equations
of
the
difference
of
field
amplitude,
AE,
the
difference
of
resonance
frequencies,
A~,
and
the
difference
of
the
phases
of
these
two
lasers,
A~.
Together
with
the
other
control
parameter,
the
feedback
coupling
coefficient
x,
we
have
the
completely
deterministic
system.
We
have
also
simplified
not
only
the
system,
but
also
made
the
experiment
simpler
because
we
only
have
to
measure
all
the
variables
or
control
parameters
in
their
difference
(or
relative
values).
Nonetheless,
this
simplification
should
not
overshadow
the
crucial
importance
of
"slowing
down"
the
dynamical
coupling
of
two
lasers
by
using
lasers
of
very
slow
radiative
decay
rates.
We
believe
only
if
this
four-dimensional
subspace
is
well
understood
experimentally,
then
do
we
have
a
chance
to
understand
experimentally
the
full
system,
i.e.,
the
coupling
of
two
semiconductor
lasers.
396
Now
let
us
comment
on
the
semiconductor
laser
phased
array.
Of
course,
there
are
many
ways
to
phase
lock
semiconductor
laser
array,
but
the
most
esthetic
as
well
as
the
simplest
from
the
system
point
of
view
is
the
mutual
feedback
approach
such
as
the
self-imaging
of
the
Talbot
cavity
(Jansen
et
al
1989,
D'Amato
et
al
1989).
In
principle,
if
the
two-dimensional
array
has
a
very
large
number
of
"identical"
elements
(in
both
the
geometric
and
physical
sense),
then
with
a
very
short
time
the
array
will
self-
adjust
to
the
external
cavity
and
a
stable
phase
locked
two-
dimensional
array
can
be
formed.
Nonetheless,
the
physical
characteristics
of
each
of
the
elements
are
critical
to
the
phase
locking
process
as
well
as
the
stability
of
such
locking.
As
we
have
pointed
out
earlier,
the
locking
is
less
sensitive
to
the
feedback
amount
and
the
field
intensity
and
gain
inside
each
semiconductor
lasers,
and
the
locking
is
very
sensitive
to
the
frequency
detuning
and
the
phase.
But
these
two
most
sensitive
variables
are
very
much
determined
by
the
semiconductor
laser
material
processing.
Indeed,
in
studying
this
3N-dimensional
system,
where
N
is
the
number
of
semiconductor
lasers
and
N
is
very
large,
one
has
to
resort
to
nonlinear
stochastic
differential
equations.
The
solution,
if
it
exists,
can
give
the
statistical
properties
or
requirements
of
the
semiconductor
material
processing.
Which
in
turn
can
give
the
material
tolerance
of
the
semiconductor
laser
phased
array.
Now
we
are
getting
into
the
real
application,
namely,
the
material
engineering.
An
interesting
problem
closely
related
to
the
problem
discussed
above
is
the
random
neural
networks.
Recently,
Sompolinsky
and
Crisanti
(1988]
have
studied
a
continuous-
time
dynamical
model
of
a
network
of
N
nonlinear
elements
interacting
via
random
asymmetric
couplings.
They
have
found
that
by
using
a
self-consistent
mean-field
one
can
predict
a
transition
from
a
stationary
pahse
to
a
chaotic
phase
occurring
at
a
critical
value
of
the
gain
parameter.
In
fact,
for
most
region
of
the
gain
parameters,
the
only
397
stable
solution
is
the
chaotic
solution.
398
References
N.
B.
Abraham,
in
Far
From
Equilibrium
Phase
Transitions,
ed.
by
L.
Garrido,
Springer-Verlag,
1988.
N.
B.
Abraham,
in
1989
Yearbook
of
Encyclopedia
of
Physical
Science
and
Technology,
Academic
Press,
1989.
N.
B.
Abraham,
F.
T.
Arecchi
and
L.
A.
Lugiato,
ed.,
Instabilities
and
Chaos
in
Quantum
Optics
II,
Plenum,
1988.
N.
B.
Abraham,
T.
Chyba,
M.
Coleman,
R.
s.
Gioggia,
N.
Halas,
L.
M.
Hoffer,
s.
N.
Liu,
M.
Maeda
and
J.
c.
Wesson,
in
Laser
Physics,
ed.
by
J.
D.
Harvey
and
D.
F.
Walls,
Springer-Verlag,
1983.
N.
B.
Abraham,
D.
Dangoisse,
P.
Glorieux,
and
P.
Mandel,
Opt.
Soc.
Am.
B,
2
(1985)
23.
N.
B.
Abraham,
L.
A.
Lugiato,
P.
Mandel,
L.
M.
Narducci
and
D. K.
Bandy,
Opt.
Soc.
Am.
B,
2
(1985)
35.
N.
B.
Abraham,
L.
A.
Lugiato
and
L.
M.
Narducci,
Opt.
Soc.
Am.
B,
2
(1985)
7.
R.
Abraham
and
J.
Marsden,
Foundations
of
Mechanics,
Second
Ed.,
Benjamin,
1978.
R.
Abraham,
J.
Marsden
and
T.
Ratiu,
Manifolds,
Tensor
Analysis,
and
Applications,
Addison-Wesley,
1983.
R.
Abraham
and
s.
Smale,
in
Global
Analysis,
ed.
by
s. s.
Chern
and
s.
Smale,
1970.
J.
F.
Adams,
Ann.
Math.
75
(1962)
603.
v.
s.
Afraimovich
and
Ya.
B.
Pesin,
in
Mathematical
Physics
Reviews,
Vol.6,
ed.
by
Ya.
G.
Sinai,
Harwood
Academic
Pub.,
1987.
v.
s.
Afraimovich
and
L.
P.
Sil'nikov,
Sov.
Math.
Dokl.,
15
(1974)
206.
A.
K.
Agarwal,
K.
Banerjee
and
J.
K.
Bhattacharjee,
Phys.
Lett.
A,
119
(1986)
280.
J.
K.
Aggarwal
and
M.
Vidyasagar,
ed.,
Nonlinear
systems:
Stability
Analysis,
Academic
Press,
1977.
G.
P.
Agrawal,
Appl.
Phys.
Lett.,
49
(1986)
1013.
G.
P.
Agrawal,
Phys.
Rev.
Lett.,
59
(1987)
880.
399
A.
M.
Albano,
J.
Abounadi,
T.
H.
Chyba,
c.
E.
Searle,
and
s.
Yong,
Opt.
Soc.
Am.
B, 2
(1985)
47.
Am.
Math.
Soc.
Translation
Ser.
1,
Vol.
9
(1962).
L.
w.
Anacker
and
R.
Kopelman,
Phys.
Rev.
Lett.,
58
(1987)
289.
u.
an
der
Heiden,
in
Differential-Difference
Equations,
ed.
by
Collatz
et
al,
Birkhauser
Verlag,
1983.
u.
an
der
Heiden
and
H.-o.
Walther,
J.
Diff.
Eqs.,
47
(1983)
273.
R.
D.
Anderson,
ed.,
Symposium
on
Infinite
Dimensional
Topology,
Princeton,
1972.
R.
F.
s.
Andrade,
J.
Math.
Phys.,
23
(1982)
2271.
A. A.
Andronov
and
L.
s.
Pontryagin,
Dokl.
Akad.
Nauk.
SSSR,
14
(1937)
247.
A. A.
Andronov,
E.
A.
Vitt
and
s.
E.
Khaiken,
Theory
of
oscillators,
Pergamon,
1966.
D.
Anosov,
Sov.
Math.
Dokl.,
3
(1962)
1068.
H. A.
Antosiewicz,
in
S.
Lefschetz,
1958.
F.
T.
Arecchi,
W.
Gadomski,
and
R.
Meucci,
Phys.
Rev.
A, 34
(1986)
1617.
F.
T.
Arecchi,
w.
Gadomski,
R.
Meucci
and
J.
A.
Roversi,
Opt.
Communi.,
70
(1989)
155.
F.
T.
Arecchi,
R.
Meucci,
and
w.
Gadomski,
Phys.
Rev.
Lett.,
58
(1987)
2205.
F.
Argoul,
A.
Arneodo
and
P.
Richetti,
Phys.
Lett.
A,
120
(1987)
269.
D.
Armbruster,
z.
Phys.
B, 53
(1983)
157.
R. A.
Armstrong
and
R. McGehee,
J.
Theor.
Biol.,
56
(1976)
499.
A.
Arneodo,
P.
Coullet
and
C.
Tresser,
Phys.
Lett.
A,
79
(1980)
259.
L.
Arnold,
w.
Horsthemake
and
R.
Lefever,
z.
Physik
B,
29
(1978)
367.
V.
I.
Arnold,
Russ.
Math.
Surv.,
18
(1963)
9;
85.
v.
I.
Arnold,
Mathematical
Methods
of
Classical
Mechanics,
Springer-Verlag,
1978.
v.
Arzt
et
al,
z.
Phys.,
197
(1966)
207.
400
U.
Ascher,
P.
A.
Markowich,
C.
Schmeiser,
H.
Steinruck
and
R.
Weiss,
SIAM
J.
Appl.
Math.,
49
(1989)
165.
M.
F.
Atiyah,
Sem
Bourbaki,
Mai
1963.
M.
F.
Atiyah
and
I.
M.
Singer,
Bull.
Am.
Math.
Soc.
69
(1963)
422.
M.
F.
Atiyah
and
I.
M.
Singer,
Ann.
Math.
87
(1968)
484;
546.
D.
Auerbach,
P.
cvitanovic,
J.-P.
Eckmann,
G.
Gunaratne
and
I.
Procaccia,
Phys.
Rev.
Lett.,
58
(1987)
2387.
J.
Auslander
and
P.
Seibert,
in
Nonlinear
Differential
Equations
and
Nonlinear
Mechanics,
ed.
by
J.
P.
LaSalle
and
s.
Lefschetz,
Academic
Press,
1963.
L.
Auslander,
L.
Green
and
F.
Hahn,
ed.,
Flows
on
Homogeneous
Spaces,
Princeton,
1963.
L.
Auslander
and
R.
E.
MacKenzie,
Introduction
to
Differential
Manifolds,
McGraw-Hill,
1963.
A.
Avez,
A.
Blaquiere
and
A.
Marzollo,
Dynamical
systems
and
Microphysics,
Academic
Press,
1982.
s.
M.
Baer
and
T.
Erneux,
SIAM
J.
Appl.
Math.,
46
(1986)
721.
s.
M.
Baer,
T.
Erneux
and
J.
Rinzel,
SIAM
J.
Appl.
Math.,
49
(1989)
55.
P.
Bak,
Phys.
Today,
Dec.
1986,
38.
P.
Bak,
c.
Tang
and
K.
Wiesenfeld,
Phys.
Rev.
Lett.,
59
(1987)
381.
H.
Baltes,
Inverse
Scattering
in
Optics,
Springer-Verlag,
1980.
D. K.
Bandy,
L.
M.
Narducci,
and
L.
A.
Lugiato,
J.
Opt.
Soc.
Am.
B
2,
(1985)
148.
D. K.
Bandy,
L.
M.
Narducci,
c.
A.
Pennise,
and
L.
A.
Lugiato,
in
Coherence
and
Quantum
Optics
V,
ed.
by
L.
Mandel
and
E.
Wolf,
Plenum,
1984.
c.
Bardos
and
D.
Bessis,
ed.,
Bifurcation
Phenomena
in
Mathematical
Physics
and
Related
Topics,
D.
Reidel,
1980.
c.
Bardos,
J.
M.
Lasry,
and
M.
Schatzman,
ed.,
Bifurcation
and
Nonlinear
Eigenvalue
Problems,
Springer-Verlag,
1980.
401
v.
Bargmann,
Group
Representations
in
Mathematical
Physics,
Springer-Verlag,
1970.
J.
D.
Barrow,
Phys.
Rev.
Lett.,
46
(1981)
963.
J.
D.
Barrow,
Gen.
Rel.
&
Gravit.,
14
(1982)
523.
J.
D.
Barrow,
Phys.
Rep.,
85
(1982)
1.
K.
Baumgartel,
U.
Motschmann,
and
K.
Sauer,
in
Nonlinear
and
TUbulent
Processes
in
Physics,
ed.
by
R. z.
Sagdeev,
Hardwood
Academic
Pub.,
1984.
D.
Baums,
w.
Elsasser,
and
E.
o.
Gobel,
Phys.
Rev.
Lett.,
63
(1989)
155.
A. D.
Bazykin,
Int.
Inst.
Appl.
Syst.
Analysis,
Laxenburg,
1976.
T.
Bedford
and
J.
Swift,
ed.,
New
Directions
in
Dynamical
Systems,
Cambridge
U.
Press,
1988.
A. R.
Bednarek
and
L.
Cesari,
Ed.,
Dynamical
Systems
II,
Academic
Press,
1982.
G.
I.
Bell
and
E.
c.
Anderson,
Biophys.
J.,
7
(1967)
329.
E.
Beltrami,
Mathematics
for
Dynamic
Modeling,
Academic
Press,
1987.
J.
Bentley
and
N.
B.
Abraham,
Opt.
Commun.
41
(1982)
52.
E.
Beretta
and
F.
Solimano,
SIAM
J.
Appl.
Math.,
48
(1988)
607.
P.
Berge,
Y.
Pomeau
and
c.
Vidal,
Order
within
Chaos,
Wiley,
1987.
M.
s.
Berger,
Nonlinearity
and
Functional
Analysis,
Academic,
1977.
M.
s.
Berger
and
P.
c.
Fife,
Comm.
Pure
&
Appl.
Math.,
21
(1968)
227.
M.
v.
Berry,
J.
Phys.
A, 8
(1975)
566.
M.
V.
Berry,
J.
Phys.
A,
10
(1977)
2061.
M.
v.
Berry,
I.
c.
Percival
and
N.
o.
Weiss,
ed.,
Dynamical
Chaos,
Princeton
Univ.
Press,
1987.
M.
V.
Berry
and
c.
Upstill,
in
Progress
in
Optics,
V.18,
ed.
by
E.
Wolf,
North-Holland,
1980.
V.
I.
Bespalov
and
V.
I.
Talanov,
Sov.
Phys.
JETP
Lett.,
3
(1966)
307.
N.
P.
Bhatia
and
V.
Lakshmikantham,
Mich.
Math.
J.,
12
402
(1965)
183.
N.
P.
Bhatia
and
G.
P.
szego,
Dynamical
systems:
Stability
Theory
and
Applications,
Springer-Verlag,
1967.
N.
P.
Bhatia
and
G.
P.
Szego,
Stability
Theory
of
Dynamical
Systems,
Springer-Verlag,
1970.
c.
K.
Biebricher,
in
Evolutionary
Biology,
16,
ed.
by
M.
K.
Hecht
et
al,
Plenum
Press,
1983.
o.
Biham
and
w.
Wenzel,
Phys.
Rev.
Lett.,
63
(1989)
819.
G.
D.
Birkhoff,
Dynamical
Systems,
Am.
Math.
Soc.,
1927.
J.
s.
Birman
and
R.
F.
Williams,
Top.,
22
(1983)
47.
J.
s.
Birman
and
R.
F.
Williams,
Contemp.
Math.,
20
(1983)
1.
B.
Birnir
and
P.
J.
Morrison,
Univ.
of
Texas,
Inst.
for
Fusion
Studies,
#243
(1986).
A.
R.
Bishop,
L.
J.
Campbell
and
P.
J.
Channell,
ed.,
Fronts,
Interfaces
and
Patterns,
North-Holland,
1984.
A.
R.
Bishop,
D.
Campbell,
and
B.
Nicolaenko,
ed.,
Nonlinear
Problems:
Present
and
Future,
North-Holland,
1982.
A.
R.
Bishop,
K.
Fesser,
P.
s.
Lomdahl,
W.
c.
Kerr,
M.
B.
Williams,
and
s.
E.
Trullinger,
Phys.
Rev.
Lett.,
so
(1983)
1095.
A.
R.
Bishop,
D.
w.
McLaughlin,
M.
G.
Forest
and
E.
A.
overman
II,
Phys.
Lett.
A,
127
(1988)
335.
R.
L.
Bishop
and
R.
J.
Crittenden,
Geometry
of
Manifolds,
Academic
Press,
1964.
D.
J.
Biswas
and
R.
G.
Harrison,
Appl.
Phys.
Lett.,
47
(1985)
198.
s.
Bleher,
E.
Ott
and
c.
Grebogi,
Phys.
Rev.
Lett.,
63
(1989)
919.
K.
Bleuler
et
al
(ed),
Differential
Geometric
Methods
in
Mathematical
Physics,
Springer-Verlag,
Vol.l
1977;
Vol.2
1979.
K.
J.
Blow,
N.J.
Doran,
and
D. Wood,
Opt.
Lett.,
12
(1987)
202.
E.
J.
Bochove,
Opt.
Lett.,
11
(1986)
727.
Yu.
L.
Bolotin,
V.
Yu.
Gonchar,
V.
N.
Tarasov
and
N.
A.
Chekanov,
Phys.
Lett.
A,
135
(1989)
29.
403