Lee K.K. Lectures on Dynamical Systems, Structural Stability and Their Applications
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operators
of
photons
of
the
mode
i,
and
in
classical
treatment
they
are
merely
c-number
time-dependent
complex
amplitudes.
Of
course,
in
some
cases,
it
is
preferable
to
use
the
field
instead
of
the
mode
decomposition.
Also,
due
to
the
quantum-classical
correspondence,
one
can
replace
the
quantum
correlation
by
classical
averages.
For
simplicity,
let
us
consider
single
mode
lasers,
so
that
we
can
drop
the
index
i.
In
a
region
not
too
far
above
and
below
the
laser
threshold,
the
mode
amplitude
a•(t)
obeys
the
simple
equation
da•(t)/dt
=-
Ka•(t)
+
Ga•(t)
+
F+(t),
(7.2-37)
where
K
accounts
for
the
losses
by
the
mirrors,
refraction,
absorption
due
to
impurities,
etc.,
G
describes
the
gain
by
the
stimulated
emission,
and
F•(t)
represents
the
fluctuation
or
noise
of
the
amplitude.
This
noise
term
can
stem
from
the
spontaneous
emission
of
the
atoms
into
all
modes,
the
interaction
of
the
atoms
with
lattice
vibrations,
the
pumplight,
etc.
For
more
detail,
see
Haken
(1970]
or
Sargent,
Scully
and
Lamb
(1974].
As
before,
we
assume
that
the
statistical
average
of
the
fluctuation
vanishes,
i.e.,
<F+
(t)
> =
<F"
(t)
> = 0
(7.
2-38)
and
they
have
correlation
function
<F+(t)F"(t')>
= C
cS(t-t'),
(7.2-39)
as
in
Eq.(7.2-12).
Eq.(7.2-39)
expresses
the
fact
that
the
fluctuations
have
a
very
short
"memory"
compared
to
other
time
constants
in
the
systems.
The
constant
c
depends
on
the
cavity
width,
the
number
of
thermal
photons,
damping
constants,
occupation
numbers
of
the
individual
atoms,
etc.
The
gain
function
G
is
proportional
to
the
number
of
excited
atoms
N
2
minus
the
number
of
atoms
in
the
ground
state
N
1
•
Furthermore,
the
gain
depends
on
the
line
shape
of
the
atoms.
The
closer
the
laser
frequency
n
to
the
atomic
resonance
v,
the
larger
the
gain.
For
the
homogeneous
Lorentzian
linewidth
with
half-width
r,
the
real
part
of
the
gain
is
(Haken
1970;
v.
Arzt
et
al
1966;
Fleck
1966]:
ReG=
(N
2
-
N
1
)rlgl•![r•
+
(n-
v)•
],
(7.2-40)
334
here
constant
g
contains
the
optical
matrix
element.
It
is
important
to
notice
that
the
inversion
N
2
-
N
1
is
lowered
due
to
the
process
of
stimulated
emission.
That
is,
for
not
too
high
laser
amplitudes,
we
have
the
instantaneous
inversion
N
2
-
N
1
=
(N
2
-
N
1
)
0
-
constant·a•(t)a(t),
(N
2
-
N
1
)
0
= 0
0
•
(7.
2-41)
The
first
term
on
the
right
hand
side,
(N
2
-
N
1
)
0
,
is
the
unsaturated
inversion,
the
second
term
describes
the
cowering
of
the
inversion
due
to
laser
action.
In
Eq.(7.2-41)
we
assume
that
the
atomic
inversion
responds
immediately
to
the
field.
Substituting
Eq.(7.2-41)
into
Eq.(7.2-40)
we
obtain
the
saturated
gain:
G
8
=
Gu
- f3a•a.
(7.
2-42)
Introducing
Eq.(7.2-42)
into
Eq.(7.2-37)
we
have
the
basic
laser
equation
derived
previously
[Haken
&
Sanermann
1963;
Lamb
1964;
Haken
1964].
da•;dt
= -
Ka•
+
Gua•
- f3a•ca•a) + F•.
(7.2-43)
This
is
the
familiar
equation
of
Eq.(7.2-10).
It
should
also
be
mentioned
that
without
the
noise
term
F+,
Eq.(7.2-43)
is
similar
to
the
Cuffing
equation
which
describes
the
behavior
of
the
hardened
spring
with
cubic
stiffness
term.
As
we
have
pointed
out
in
Chapter
4,
this
system
has
a
unique
closed
orbit
which
is
the
n-set
of
all
the
orbits
except
the
fixed
point.
Furthermore,
this
system
is
auto-oscillatory
since
all
solutions
(except
one)
tend
to
become
periodic
as
time
increases.
335
Once
again,
we
may
interpret
Eq.(7.2-43)
as
the
equation
of
a
strongly
overdamped
motion
of
a
particle
in
the
potential
field
V =
(K
-
Gu)
la+l
2
/2
+
~la+l
4
/4.
In
addition,
the
particle
experiences
random
pushes
by
the
fluctuating
force
F+(t).
Let
us
first
discuss
the
situation
below
threshold.
Fig.7.2.2a
shows
the
potential
V
in
one
dimension
for
Gu
- K
S o.
After
each
push
excerted
by
the
fluctuating
force
on
the
particle,
it
falls
down
the
slope
of
potential
hill.
When
we
multiply
E
(or
a+)
by
exp(int)
to
obtain
E(t)
of
Eq.(7.2-34)
and
consider
a
sequence
of
random
pushes,
Fig.7.2.2b
results.
As
was
shown
by
Mandel
&
Wolf
[1961],
the
field
amplitude
is
Gaussian
distributed
as
is
represented
by
Fig.7.2.2c.
Fig.7.2.2
Laser
below
threshold,
G - K < o.
v
q,E
a)
Potential.
b)
Real
part
of
field
amplitude
vs.
time.
336
f
(q)
c)
Gaussian
distribution
of
field
amplitude.
For
above
laser
threshold,
Gu- K > o,
the
potential
curve
of
Fig.7.2.3a
applies.
The
state
E = 0
has
become
unstable
and
is
replaced
by
a
new
stable
state
E
0
+
0.
For
the
moment,
if
we
ignore
the
fluctuations,
a
coherent
wave
emerges
as
in
Fig.7.2.3b.
When
we
take
into
account
the
impact
of
fluctuations,
we
must
resort
to
higher
than
one-dimensional
potential,
Fig.7.2.3c.
The
random
pushes
in
the
central
direction
will
result
the
same
effect
as
in
the
one-dimensional
situation,
but
the
random
pushes
in
the
tangential
direction
will
cause
phase
diffusion.
Indeed,
this
is
the
basis
leading
to
the
prediction
of
Haken
[1964]
that
laser
light
is
amplitude
stablized
with
small
superimposed
amplitude
fluctuations
and
a
phase
diffusion.
Furthermore,
if
the
corresponding
Langevin-type
equation,
Eq.(7.2-10
or
7.2-43)
is
converted
into
a
Fokker-Planck
equation,
Eq.(7.2-25),
the
steady-state
distribution
function
can
be
easily
obtained
by
using
the
adiabatic
elimination
method
[Risken
1965].
These
and
further
properties
of
laser
light
were
further
studied
both
theoretically
and
experimentally
by
a
number
of
authors,
for
instance
the
references
cited
in
Haken
[1970],
Sargent,
Scully
and
Lamb
[1974],
and
the
recent
book
by
Milonni
and
Eberly
[1988].
337
v
(a)
(b)
E - q exp(iwt)
Fig.7.2.3
Laser
above
threshold,
G - K > o.
a)
Potential
in
one
dimension.
b)
Real
part
of
field
without
fluctuations.
f(q,, q,)
(c)
q
Fig.7.2.3
c)
Potential
in
two
dimensions.
d)
Laser
light
distribution
function.
[Haken
1986]
It
has
to
be
said
that
the
laser
was
recognized
as
the
first
example
of
a
nonequilibrium
phase
transition
and
a
perfect
analogy
to
the
Landau
theory
of
phase
transition
could
be
established
[Graham
and
Haken
1968,1970;
Haken
1983].
When
we
plot
the
stable
amplitude
E
0
vs.
(Gu-
K),
we
obtain
the
bifurcation
diagram
of
laser
light,
Fig.7.2.4.
Qo,
Eo
Fig.
7.2.4
Bifurcation
diagram
of
laser
light
338
It
should
also
be
pointed
out
that
if
one
considers
the
classical
dispersion
theory
of
the
electromagnetic
waves,
then
the
electric
field
satisfies
the
wave
equation,
- V'B +
(1/c•
)a•B;at•
+
47facaB;at;c•
=-
47fa•p;at•;c•,
(7.2-44)
where
ac
is
the
conductivity
which
describes
the
damping
of
the
field,
and
P
is
the
macroscopic
polarization.
One
can
think
of
atoms
dispersed
in
a medium
and
we may
represent
the
polarization
as
a sum
over
the
individual
atomic
contributions
at
xi
by
P(x,t)
=
:Ei
6(x-
xi)pi(t),
(7.2-45)
where
pi
is
the
dipole
moment
of
the
i-th
atom.
Then
the
field
equation
(7.2-44)
is
supplemented
by
the
equation
of
the
atom
i,
a•
p;fat•
+
2aapi/at
+
p•
pi
= e• B
(x,
t)
jm,
(7.
2-46)
where
a
is
the
damping
constant
of
the
atoms.
Once
again,
we
can
see
that
Eq.(7.2-46)
is
a
special
case
of
Eq.(7.2-9).
Indeed,
if
one
pursued
this
further
by
decomposing
the
dipole
moments
in
terms
of
raising
and
lowering
operator
for
atomic
levels,
and
using
the
interacting
Hamiltonian,
one
can
get
field
equations,
the
equation
for
the
atomic
dipole
moments,
and
the
equation
for
the
atomic
inversion.
We
shall
not
get
into
any
more
details
here.
Interested
readers
please
see,
e.g.,
Haken
(1970],
Sargent,
Scully
and
Lamb
[1974].
Statistical
concept
of
physical
processes,
not
too
far
from
equilibrium,
was
first
established
in
the
theory
of
Brownian
motion
(Langevin
1908].
The
theory
of
Fox
and
Uhlenbeck
(1970],
resulted
in
a
general
stochastic
theory
for
the
linear
dynamical
behavior
of
thermodynamical
systems
close
to
equilibrium,
which
includes
the
Langevin
theory
of
Brownian
motion,
the
Onsager
and
Machlup
theory
[1953]
for
irreversible
processes,
the
linearized
fluctuating
equations
of
Landau
and
Lifshitz
for
hydrodynamics
(1959],
and
the
linearized
fluctuating
Boltzmann
equation
as
special
cases.
For
more
detailed
discussions
of
the
general
theory
of
stochastical
processes,
see,
for
instance,
Gardiner
(1985].
339
As
has
been
pointed
out
by
Fox
[1972],
in
the
general
theory
as
well
as
in
each
of
the
above
special
cases,
the
mathematical
descriptions
involve
either
linear
partial
integra-
differential
equations
or
set
of
linear
inhomogeneous
equations.
These
inhomogeneous
terms
are
the
usual
stochastic
(or
fluctuating)
driving
forces
of
the
processes
frequently
termed
the
Langevin
fluctuations.
These
terms
are
being
referred
to
as
additive
fluctuations
for
additive
stochastic
processes.
Fox
[1972]
has
since
systematically
introduced
the
stochastic
driving
forces
for
homogeneous
equations
in
a
multiplicative
way,
and
such
processes
are
called
multiplicative
stochastic
processes.
Multiplicative
stochastic
processes
arise
naturally
in
many
disciplines
of
science.
Fox
[1972]
has
established
some
mathematical
groundwork
for
multiplicative
stochastic
processes
(MSP),
and
pointed
out
the
relevance
of
MSP
in
nonequilibrium
statistical
mechanics.
Shortly
after
lasers
were
invented,
it
became
clear
that
both
lasers
and
nonlinear
optical
processes
were
examples
of
dynamical
processes
far
from
thermodynamical
equilibrium.
Later,
Schenzle
and
Brand
[1979],
motivated
by
laser
theory
and
nonlinear
processes,
demonstrated
that:
(i)
an
ensemble
of
two-level
atoms
interacting
with
plane
electromagnetic
waves,
described
by
the
well-known
Maxwell-Block
equations;
(ii)
parametric
down
conversion
as
well
as
parametric
up
conversion;
(iii)
stimulated
Raman
scattering;
and
(iv)
autocatalytic
reactions
in
biochemistry,
are
examples
of
nonlinear
processes
whose
stochastic
fluctuations
are
multiplicative.
They
have
obtained
some
very
interesting
results.
It
is
natural
to
expect
that
many
real
systems
have
both
types
of
fluctuations,
that
is,
there
may
be
a
mixture
of
both
processes.
Indeed,
they
have
discussed
such
systems
with
mixture
of
both
fluctuations
[1979].
Some
further
discussions
on
multiplicative
fluctuations
in
nonlinear
optics
and
the
reduction
of
phase
noise
in
coherent
anti-Stokes
Raman
scattering
have
been
discussed
by
Lee
[1990].
There
are
many
unanswered
questions
on
the
340
fluctuations
of
nonlinear
phenomena
are
awaiting
for
further
study.
In
general,
relaxation
oscillations
characterized
by
two
quite
different
time
scales
can
be
described
by
xt
=
f(x,y,~,E)
and
Yt
=
Eg(x,y,~,E),
where
at
is
the
partial
derivative
of
a
with
respect
to
t,
~
<< 1
and
~
is
the
control
parameter.
Baer
and
Erneux
[1986]
have
shown
how
the
harmonic
oscillations
near
the
bifurcation
point
progressively
change
to
become
pulsed,
triangular
oscillations.
There
are
several
classics
which
deal
with
nonlinear
oscillations.
One
of
them
is
by
Minorsky
[1962].
It
has
a
great
deal
of
information
and
results
worked
out.
Indeed,
many
recent
results
for
applications
of
nonlinear
oscillations
can
be
found
in
Minorsky
[1962].
Another
book
by
Krasnosel'skii,
Burd
and
Kolesov
[1973)
is
also
a
very
useful
source.
Another
classic
is
the
one
by
Nayfeh
and
Mook
[1979].
This
one
provides
many
details
of
low
dimensional
nonlinear
oscillations
and
their
dynamical
evolutions.
In
the
1950'
and
1960's,
Lefschetz
edited
a
seriers
of
volumes
on
the
contributions
to
the
theory
of
nonlinear
oscillations
[1950,
1950, 1956,
1958,
1960].
7.3
Optical
instabilities
One
of
the
unexpected
features
of
early
development
of
laser
systems
was
the
presence
of
output
pulsations
even
under
steady
pumping
conditions.
It
did
not
take
long
to
recognize
this
result
as
an
important
aspect
of
lasers.
In
fact,
spiking
was
observed
in
some
masers
even
before
the
discovery
of
lasers
[Makhov
et
al,
1958;
Kikuchi
et
al,
1959;
Makhov
et
al,
1960].
As
early
as
1958,
Khaldre
and
Khokhlov
[1958],
Gurtovnick
[1958],
and
Oraevskii
[1959)
linked
the
output
pulsations
to
the
emergence
of
dynamical
instabilities.
Uspenskii
[1963;
1964],
Korobkin
and
Uspenskii
[1964]
already
in
the
early
1960's
studied
the
instability
phenomena
for
homogeneously
broadened
laser
341
systems.
The
current
efforts
in
optical
instabilites
seek
to
answer
the
same
overall
questions
as
the
ones
which
manifested
themselves
in
other
disciplines,
namely:
What
are
the
origins
and
the
functions
of
the
evolutionary
structures
of
the
dynamical
systems?
Are
there
universal
laws
which
demand
the
growth
of
certain
instabilities
and
structures?
What
differentiates
among
the
many
possible
routes
of
a
system
to
certain
macroscopic
behavior?
The
earliest
theoretical
models
of
laser
action
were
based
on
the
description
of
the
energy
exchanges
between
a
collection
of
inverted
two-level
atoms
and
the
cavity
field.
Later
on,
we
have
found
that
the
rate
equation
approach
is
inadequate
to
provide
a
faithful
description
of
the
observed
instabilities
[Hofelich-Abate
and
Hofelich,
1968].
A
very
significant
advance
in
the
field
of
laser
instabilities
was
the
one
by
Haken
[1975],
who
established
the
homeomorphism
between
the
single-mode
laser
model
and
the
Lorenz
equations.
Such
a
homeomorphism
between
the
single-mode
laser
and
the
Lorenz
equations
establishes
that
deterministic
chaos
is
also
a
part
of
chaotic
laser
behavior
as
long
as
the
single-mode
approximation
is
sufficiently
accurate
for
the
laser
system.
More
importantly,
such
a
homeomorphism
unifies
seemingly
different
phenomena
from
different
disciplines,
and
Synergetics
provides
the
motivation
and
the
guidance
for
an
organized
approach
to
the
problem
of
dynamical
systems
and
chaotic
behavior
in
general,
and
laser
instabilities
in
particular.
In
the
following,
we
shall
discuss
briefly
the
Maxwell-Bloch
equations
for
a
simple
ring
cavity
and
demonstrate
some
interesting
features
of
a
ring
cavity,
such
as:
(i)
if
the
detuning
of
the
incident
light
with
the
absorber
introduced,
in
a
stationary
situation
the
transmitted
field
becomes
a
multi-valued
function
of
the
incident
field;
(ii)
the
stationary
solution
is
not
always
stable
even
when
it
belongs
to
the
branch
with
positive
differential
gain,
in
fact,
in
some
cases
the
transmitted
field
exhibits
a
chaotic
behavior.
342
Let
the
simple
ring
cavity
be
the
following:
ER
EI
Er
Ml
I·
L
·I
M2
where
E
1
,
ET
and
ER
are
the
incident,
transmitted,
and
reflected
field
respectively.
L
is
the
length
of
the
sample
cell
containing
a
two
level
absorber
(for
simplicity,
homogeneously
broadened)
and
Ly
the
total
length
of
the
optical
path
in
the
ring
cavity.
Also
assume
the
reflectivity
of
M
1
and
M
2
be
R
and
1
for
M
3
and
M
4
•
Let
E(t,z)
be
the
complex
envelope
of
the
electric
field,
then
we
have
the
following
boundary
conditions
E(t,O)
= T E
1
(t)
+ R
E(t-
ljc,L)exp(ikLy),
Ey(t)
= T
E(t,L)exp(ikL),
where
T = 1 - R
and
1 =
Ly
-
L.
The
propagation
of
the
electric
field
in
the
non-linear
absorber
can
be
described
by
the
Maxwell-Block
equations
(Sargent,
Scully
and
Lamb
1974]:
aEjaz
41fin~ka,
aN;ar
- r
1
(N
+
1/z)
+
i~(a*E
-
aE*);z,
aa;ar
(iAn -
ri)a
-
i~NE,
(7.
3-1a)
(7.
3-1b)
(7.3-1c)
here
T = t -
z/c
is
the
retarded
time,
a
is
the
dimensionless
polarization
and
N =
~(N
1
-
N
2
),
~the
transition
dipole
moment,
and
An
= Q - n
(where
n
is
the
transtion
frequency
of
the
two
level
atom)
is
the
detuning
343