Vol.
5,
No.
8/August
1988/J.
Opt.
Soc.
Am.
B
1783
Theory
of
intracavity-pumped
photorefractive
phase-conjugate
mirror
R.
Yahalom
and
A.
Yariv
Thomas
!.
Watson,
Jr.
Laboratories
of
Applied
Physics,
California
Institute
of Technology,
Pasadena,
California
91125
Received
November
30,
1987;
accepted
April
7,
1988
We
present
a new
type
of
phase-conjugate
mirror
that
is
based
on
an
externally
driven
Fabry-Perot
interferometer
with
intracavity-pumped
photorefractive
material,
which
is
probed
by
the
signal
beam.
It
is
shown
theoretically
that
such
a
configuration
leads
to
multivalued
solutions
and
possibly
to bistability.
This
configuration
also
permits
optical
control
of
the
resonator
output
and
electrical
control
of the
phase-conjugate
reflectivity.
INTRODUCTION
During
the
past
years
there
has
been
considerable
research
on
optical
phase
conjugation
based
on
photorefractive
mate-
rials,
and
many
novel
devices
have
been
demonstrated.'
Photorefractive
crystals
were
shown
to
have
exceptionally
high
nonlinear
gain
and
were
proved
to
be
effective
for
low-
energy
cw
applications.
2
In
this
paper
we
propose
to
com-
bine
these
properties
with
the
inherent
nonlinear
behavior
of
a Fabry-Perot
interferometer.
The
use
of
an
externally
driven
nonlinear
cavity
with
a
Kerr-like
medium
as
an
optically
bistable
device
has
been
well
documented.
3
The
probing
of
such
a
device
by
a
weak
signal
beam,
thereby
combining
degenerate
four-wave
mix-
ing
(FWM)
with
the
cavity
operation,
was
also
investigated.
4
However,
to
our
knowledge
no
attention
has
been
given
to
the
behavior
of an
intracavity
photorefractive
material
when
it
is
probed
by
an
external
(not
necessarily
weak)
signal
beam
that
is
not
part
of
the
cavity.
A
typical
arrangement
for
such
a (cavity-coupled)
phase-
conjugate
mirror
(CC-PCM)
is
shown
in Fig.
1.
The
pump
field
E
is
coupled
into
the
cavity
by
a
mirror
(Ml)
with
amplitude
reflectivity
r.
The
internal
fields
are
E
(left
propagating)
and
E
2
(right
propagating).
The
crystal
is
probed
by
a
signal
beam,
E
4
,
and
the
phase-conjugate
field
is
E
3
.
In
this
paper
we
discuss
the
specific
configuration
of
Fig.
1.
Other
directions
of
the
c
axis
and
the
signal
beam
are
also
possible
(Fig.
2).
Whereas
the
detailed
structure
of
the
results
depends
on
the
chosen
configuration,
the
main
attri-
butes
of
a
CC-PCM
are
common
to
all
and
will
be
demon-
strated
here
with
the
configuration
of
Fig.
1
only.
In
this
configuration
a
strong
internal
field
E
2
is
diffracted
prefer-
entially
into
E
3
.
E
may
thus
be
of
the
same
order
of
magni-
tude
as E
4
,
giving
rise
to
a
relatively
large
modulation
depth
of
EE
4
* and
to
high
phase-conjugate
mirror
(PCM)
reflec-
tivity.
if
we
assume
the
usual
7r/2
phase
shift
between
the
interfer-
ence
pattern
and
the
induced
space-charge
field,
i.e.,
a real
effective
coupling
constant
-y.
5
The
notation
that
we
use
is
similar
to
that
of
Ref.
1.
For
simplicity,
we
assume
that
the
transmission
grating
is
dominant,
that
the
nonlinear
medi-
um
is
lossless,
and
that
all
interacting
beams
are
plane
waves.
The
basic
equations
are'
dAl
-__=
PA
4
,
dz
dA
4
*
-
dz
dA
2
*
d_
=
PA
*'
dz
dA
3
dz
-
A
2
,
r =
(AlA
4
*
+
A
2
*A3)/IT,
(1)
IT
=1
I,
I
=
lAil',
*=1
where
y is
the
(in
general)
complex
coupling
constant
and
Ai
are
the
field
amplitudes.
The
most
general
solution
of
Eq.
(1)
is given
by
RPc
= 41cl
2
/(Q/T
+
A)
2
,
(2)
where
Q =
(A
2
+
41cl
2
)/
2
,
T
= -tanh(lQ/2IT),
A
=
I2(1)
- I,(O)
-
I4(0),
THEORETICAL
ANALYSIS
AND
RESULTS
[n
the
discussion
that
follows
we
assume
that
all
fields
are
of
the
same
frequency.
In
such
a
case
the
fields
E
and
E
2
do
not
suffer
any
phase
shifts
due
to the
nonlinear
interaction,
(3)
RPC
is
the
PCM
intensity
reflectivity;
and
Ic12
is
given
by
[I
c2
-
I,(0)I
2
(1)](Q/T
+
A)
2
+
41c12[I2(l)
+
Re(T)Q/ITI
2
]
=
0.
(4)
For
real
y
this
solution
can
be
manipulated
further.
We
0740-3224/88/081783-05$02.00
©
1988
Optical
Society
of America
R. Yahalom
and
A. Yariv
1784
J. Opt. Soc.
Am. B/Vol.
5, No.
8/August 1988
r1
Two other important outputs are the resonator transmit-
tance (To) and reflectance Ro), which are found to be
Ro = 1 - [I2(l) - 1c1
2
/I(l)]/q,
To = (1 - r2
2
)[1
2
(l) - RPcJq.
(11)
(12)
Fig. 1. Intracavity
pumped PCM. This specific configuration
is
the one analyzed
in the paper.
Fig. 2. Examples of
three other possible
configurations of a CC-
PCM (in terms of the c axis and probe direction).
normalize all intensities by the probe intensity, so that
14(0) = 1. It can be shown that Eqs. (2)-(4) lead to the
solution
c= [11(0)12(1) -lcl']/[Q/T + I2(1)],
(5)
where 1c12 can now be solved to be
Icl,
2
2
= (1(0)[I2(l)
-RJ11/2 (R. )1/
2
)
2
.
(6)
Equations (5) and (6) still must
be solved numerically in a
self-consistent way. I2(1) and I(0) are the known inputs,
and they are given in terms of the boundary conditions
1E
0
1
2
-q = [ 12(C) + r,
2
1c
2
/I2(l) - 2r11clcos(0 + )]/(1 -r2).
(7)
Here 0 is the round-trip (geometrical) phase delay, and is
the additional phase delay caused by the nonlinear interac-
tion during the round trip through the crystal. Note that for
real y, 6 is equal to zero. The second boundary condition is
I,(0) = r
2
2
1
2
(0) r
2
2
[1
2
(l) - RPC (8)
We can therefore
find that
A =
r
2
2
Rp, + I
2
(l)(1-r
2
2
)-1,
(9)
IT = 1 + I2(l)(1 + r
2
2
) _ r
2
2
RPC. (10)
When Eqs. (7)-(10) are used, the PCM reflectivity [Eqs. (5)
and (6)] is determined by the control parameters q, 0, r
1
, r
2
j
only.
The solution described above shows a highly nonlinear
dependence of RPC not only on the input pump-probe ratio q
(as expected for the FWM process) but also on the cavity
parameters that can be electrically controlled. (0, as an
example, can be piezoelectrically changed.) This depen-
dence also leads, as will be demonstrated, to multivalued
solutions even for relatively small coupling y.
We see
that both Ro and To depend
on the PCM reflectivity.
It is
possible to control the cavity
outputs optically by
changing q, and multivalued solutions are also possible.
In
Fig. 3 we show the
results (Rpc,
Ro) for r, = 0 and
r2 = 1,
i.e., for a normal
FWM process when one of the pump beams
is supplied by the reflection
on mirror M2 of the second
pump. In Figs. 3(a) and 3(b) we see that for yl = 3 both
log(Rpc) and log(Ro) are almost linear in log(q) when I <
I4(0) (q < 1) and saturate at near unity when q > 1. This
solution is single valued for all values of q. In Figs. 3(c) and
3(d) the case of yl = 6 is shown. Here a second solution
appears for q > 1, which is due to dominance of the E2*E3
grating.'
These results
are to be compared with Fig.
4, in which
similar calculations were
done for r, = 0.9, r
2
=1. In Figs.
4(a) and 4(b) (l = 3) we see the appearance of three solu-
tions for q S 2, for which we have a jump of approximately 2
orders of magnitude in Rpc, which brings about the possibili-
ty of bistable behavior as a function of q. The cavity reflec-
tivity Ro is a constant ('1) for the lower branch
of RPC but is
increasing as q decreases for the upper branches
of Rpc It is,
therefore, possible to change the resonator output by con-
trolling the optical input q. This behavior can be under-
stood by observing the internal fields. The lower values of
Rpc are the results of destructive interference between E
1
(l)
and E
0
. [El(l) in this case originates mainly from the dif-
fraction of E
4
but can be relatively large.] As a consequence,
E
2
(l), E
2
(0), and E
3
(0) (-Rpc)
are small. In the case
of the
upper branches of Rpc the opposite is correct. E(l) is large
because of the constructive interference of E
0
and E1(l). As
a result, the term ElE
4
* leads to a strong grating upon which
E
2
(l) is diffracted
to give higher values of the PCM reflectiv-
ity.
In Figs. 4(c) and 4(d) we see similar results for yl = 6. The
main differences are a bigger jump (of factor 103) in Rp. at
the bistable point q _ 3 and the appearances of multivalued
solutions for q > 1, as occurred also in the case r, = 0.
As y is increased, the solutions become richer and more
complicated. As an example, we show in Fig. 5 the behavior
of Ro for yl = 10.
In all the above examples the cavity was tuned to reso-
nance conditions ( = 0). When the cavity is slightly de-
tuned, the bistable loop is even clearer. In Fig. 6 we show
Rp. for yl = 3 and 0 = 5°. The corresponding expected
hysteresis loop, assuming initial conditions q = 0, is also
displayed.
As we mentioned above, it is also possible to control the
PCM reflectivity by changing the cavity round-trip phase.
In Fig. 7 log(Rpc) is shown as a function of 0 for yl = 3,
loglo(q) = 0.2,0.4,0.8. We observe that for logl
0
, (q) = 0.2 if
initially the system is on the upper
branches of RPC [see also
Fig. 4(a)], then, as 0 is increased, self-oscillations cannot
contribute substantially to the PCM reflectivity; at 0 40
the reflectivity drops approximately 3 orders of magnitude.
For higher values of q a smooth control of RPC as a function of
0 is possible.
R. Yahalom and A. Yariv
R.
Yahalom
and
A.
Yariv
Vol.
5,
No.
8/August
1988/J.
Opt.
Soc.
Am.
B
1785
3
-2
-1
0
LOG,0
(a)
(a)
1
2
:
0
-1
U-2
c
-3
0
..
-5
-6
-7
-IQ
3
-2
-1
0
1
LOG
10
(q)
(C)
2
-3
-2
-1
0
1
2
3
LW
10 tqJ
LOG
10
(q)
(b)
(d)
Fig.
3.
Log-log
plots
of
the
phase-conjugate
reflectivity
Rpc
and
the
cavity
reflectivityRo
as
a function
of
the
pump-probe
ratio
q for
the
case
of
an
open
cavity
(r
1
=
0).
(a)
0
0-
(C)
2
Y2 =6
rI
=
0.9
a1
0=0
-1
-2-
.3
I
1
I
-3
-2
-1
0
1
2
3
L
OG
1
0
(q)
(d)
(b)
Fig.
4.
Rpc
and
Ro
as
a
function
of
q
for
the
case
r
=
0.9.
0
-1
-2
-3
-4
-5
-
r1=0
-6
-7
-811e
0
0
-j
0
0
0
-J
72=6
r1
=
0
l
l l~~~~~~~r
=
~~~
\
~~~~~r1=0
2
-
-3
-2
-1
0
1
2
3
0
0
0
Cr
8
i
Z.
I
v
-
1
1786
J.
Opt.
Soc. Am.
B/Vol.
5,
No. 8/August
1988
4
3
2
-6
0
0
0~~~~~~~~~~~~~
-4
-2
0
2
4
LOG
1
0
(q)
Fig. 5.
A plot of
the cavity
reflectivity
Ro
as a function
of the
input
parameter q
for a high-gain
(yl = 10)
crystal.
2
0
t -3 it /
rue= 3
d-)
-4 -
r1
=0.9
o
0 = 5
-j -5
-4
-6 _
-7
-8
-3
-2
-1
0
1
2
LOGQ (q)
Fig.
6. Rp. and
Rb as a
function of
q for a detuned
resonator:
yl =
3, 0 = 5O.
PCM
5
). It is
impossible
to predict
from
the steady-state
analysis
how
much of
the incoming
probe
field
E
4
partici-
pates
in the
usual degenerate
FWM
process
(thus
creating
a
fixed
grating
with
the
pump
field)
and what
part
of E
4
is
channeled
into detuned
self-oscillations
(creating a
moving
grating).
A detailed
dynamic
investigation
is needed.
We
note, however,
that
for a high-finesse
resonator,
detuned
oscillations
are limited
only
to a small
cavity
detuning an-
gle,
5
beyond
which
they cannot
be
sustained.
The main
features
of
the dependence
of RP,
on 0 for
small values
of
q (q
< 1)
can therefore
also
be found from
our previous
analysis.
Another
means for
controlling
the phase-conjugate
reflec-
tivity
and the
cavity behavior
is
detuning
of the probe
fre-
quency
relative
to the
pump frequency.
Because
of the
frequency
offset between
the
pump and
the signals
beams,
the light
interference
grating
moves
in space.
Because
of
the
finite response
time
(r) of the
grating in
the photorefrac-
tive material,
a phase
lag is induced
between
the
index grat-
ing and
the interference
grating;
the
coupling
constant
be-
comes
complex
and
is given
by
y =
Yo/(1 +
ir),
(13)
where
'y is the
coupling
constant
with
zero
detuning
and
6 is
the detuning
frequency.
To see
the effect
of complex
y, we follow
the procedure
of
Ref. 5 and
find that
for our
boundary
conditions,
with I4(0)
= 1, the round-trip
phase is given
by
[JIm
{ YI4(
1
+
f +
34
+
34
/f)}
J1= O
[fffl-
A
dz,
(14)
where
-2[21c2
+ (Q/T
+ A)I
2
(l)]
(Q/P
+ A)[Q/T
+ 212() - A]
2(l)(1
-
12) -
A
14 =
1
- 12134
(15)
(16)
0
-0.5k
-1.0
0-
1-0
-
0o
1.5k
2.0[
2.5[
-3.0
0
20
40 60
80
100 120
140 160
180
CAVITY
DETUNING
ANGLE,
Fig. 7. The
dependence
of Rp, on
the cavity
detuning angle
0, for -yl
= 3 and three
values
of the pump-probe
ratio q.
In
the above
calculations
we did
not include
the possible
appearance
of detuned
oscillations
inside
the cavity
when
0
- 27rm,
which
may be
important
for q <
1 even when
the
external
pump and
probe fields
are of
equal frequencies.
Such
oscillations
are known
to happen
in the
absence
of
external pumping
(q
= 0, i.e.,
a detuned
linear self-pumped
I[Q/IT
+
2I2() -
A] 1
2
1C1
2
12 =
1[21c12 +
(QIT +
A)I
2
(1)]1
2
= 41c
2
/(Q/T
+ A)1
2
.
Here, T
41c12)1/2.
ly.
(17)
(18)
= tanhfyQ(1
-
z)/2[2I2(l)
- A]}
and Q
= (A
2
+
Ic and A are
given by
Eqs. (4)
and (9), respective-
From Eq.
(14) we
see that
the dependence
of
I2(1) (and,
therefore,
of Rpc)
on the
boundary
condition
(i.e., on
the
pump-probe
ratio q) becomes
even
more
involved,
as it is
now
a function
of 4p[I
2
(l)] [note
that I2(1)
also depends
on 6]:
q = (I2(l)
+
r 12C12/I2(l)
- 2r
1
1lccosVt/[I
2
(l)])/(1
-
ri2).
(19)
When
Eq. (19)
is combined
with Eqs.
(2)-(4),
(9), (10),
and
(13)-(18),
the explicit
solutions
for Rp,
can be
found.
CONCLUSION
We
have suggested
and
explored
the theory
of a CC-PCM
when
the crystal
is intracavity
pumped.
We
have shown
that
this system
can
have bistability
and
multivalued
solu-
tions as
a function
of the pump-probe
ratio q and
detuning
6,
even
for smaller
nonlinear
gain y.
When
q is changed,
the
- - -
-
r*. .
_
_
*-*-~~-_LOG1
(q)= 08
0.8
LOG
1 0
(q)
= 0.4
__
___ LOGIO(q) = 0.2
I I
t I
I I
I I
R. Yahalom and A. Yariv
Vol.
5,
No.
8/August
1988/J.
Opt.
Soc.
Am.
B
1787
cavity
output
can
be
optically
controlled
and
the
PCM
re-
flectivity
can
be
controlled
externally
by
varying
the
cavity
parameters
(0)
electrically.
ACKNOWLEDGMENTS
This
research
was
supported
by
the
U.S.
Air
Force
Office
of
Scientific
Research
and
by
the
U.S.
Army
Research
Office.
REFERENCES
1.
M.
Cronin-Golomb,
B.
Fischer,
J.
0.
White,
and
A.
Yariv,
IEEE
J.
Quantum
Electron.
QE-20,
12
(1984).
2.
For
a
review,
see
T.
J.
Hall,
R.
Jaura,
L.
M.
Connors,
and
P.
D.
Foote,
Prog.
Quantum
Electron.
10,
77
(1985).
3.
H.
M.
Gibbs,
Optical
Bistability
(Academic,
New
York,
1985).
4.
G.
P.
Agrawal,
J.
Opt.
Soc.
Am.
73,
654
(1983).
5.
S.
K.
Kwong,
M.
Cronin-Golomb,
and
A.
Yariv,
IEEE
J. Quan-
tum
Electron.
QE-22,
1508
(1986).
R.
Yahalom
and
A.
Yariv