1076
OPTICS
LETTERS /
Vol. 17, No. 15
/ August 1, 1992
Limitation
on holographic
storage in
photorefractive
waveguides
Mordechai Segev
and Amnon Yariv
California
Institute of Technology,
Pasadena, California 91125
Joseph Shamir
Department
of Electrical Engineering, Technion-Israel
Institute of Technology, Haifa 32000,
Israel
Received March 30, 1992
We show that photorefractive
waveguide devices are subject
to a limitation on their holographic
storage capabil-
ity owing to transversely
nonuniform nonlinear losses to
evanescent and radiation modes.
Recent developments
in the growth of photorefrac-
tive (PR) thin
films,' as well as in optical fibers,
2
'
3
offer practical uses of
the PR waveguides for ap-
plications such as multimode-to-single-mode
con-
version,4`7 low-power all-optical switching,
8
and
high-capacity
holographic storage.
9
In this
Letter we draw attention to the fact
that
there
exists a basic limitation
on the performance
of
PR waveguide devices,
which is due to the irrevers-
ible coupling of power from
the guided modes to
evanescent and radiation modes. The resulting
non-
linear loss reduces the storage capability
and the
resolution
of the stored images in PR slab wave-
guides and fibers,
especially in the proposed PR fiber
bundles.
9
The
limitation arises from transversely
nonuniform losses, which cannot
be compensated
for, even by an ideal phase-conjugate
reconstruc-
tion
10
of the stored hologram. This translates
to re-
ductions in the
resolution of the stored images, in
the angular selectivity,
and in the multiple-image
storage capability. We analyze
the sources for this
loss and its dynamics and demonstrate
its influence
in two
applications: as a nonlinear mode coupler
(a
funneling device
4
) and
as a holographic memory with
an ideal phase-conjugate
reconstruction.'
0
As an
example
we demonstrate the loss of pictorial infor-
mation in a crude
image that consists of two spatial
guided modes stored
in a two-dimensional (slab)
waveguide.
Consider a simplified model of a slab
dielectric
waveguide,
in which the guided modes are repre-
sented by their
propagation constant f3i, where i is
the guided-mode serial number
(i = 1 is the lowest-
order
mode), and form a discrete set. The propaga-
tion constant is i3i =
k cos Oi, where Oi is the
angle of
propagation with respect to the
waveguide axis z
and
k = con/c (n is the refractive index of the slab).
The spectrum
of guided modes is restricted to the
range 0oil 7< j0,, where 0, is the
critical angle for prop-
agation
in the waveguide. For angles larger than
0,
(and smaller than iT/2),
the light propagates in a
continuum of radiation modes.
Both guided and
radiation
modes are characterized by a real propa-
gation constant
/3. The
complete spectrum
of
waveguide modes also
includes a continuum of evan-
escent modes." These
propagate in the waveguide
at angles occupying the same range as
the guided
modes
but that consists of the complementary plane-
wave basis, i.e.,
at all angles that do not correspond
to guided modes. The
evanescent modes have com-
plex propagation constants, since
they do not satisfy
the waveguide boundary condition for real
83. They
represent
plane waves with equal-amplitude planes
perpendicular
to their
phase fronts
(equal-phase
planes), and
hence they are sometimes called inho-
mogeneous waves." As
a result of their complex /3,
the intensity of the
evanescent modes decays expo-
nentially with z, and they all but
disappear within a
distance of a few wavelengths. Their
influence is
significant
only as
spatial transients
in the vicinity
of inhomogeneities
and discontinuities in the wave-
guide. We represent
these modes by their complex
propagation constant Gq + iaq,
where 6q = k cos 0q
and q
= k sin Oq, with
0
q their angle of propagation.
The
holographic recording in a PR waveguide
is
made through the interaction
between the pairs of
guided modes. The role of a reference
wave can be
played
by each of the guided modes or any subgroup
of them. For simplicity,
we restrict ourselves to
transmission gratings only. This recording
process
results
in a dynamic volume hologram, which is,
in
principle, identical
to the one that caused the non-
linear mode-coupling
effects.
4 8
,
0
Note that our
analysis remains valid
for reflection gratings, as
demonstrated in the storage scheme
of Ref. 9. Con-
sidering the interaction
between pairs
of guided
modes only, and assuming
that the nonlinear inter-
action (the dynamic volume hologram)
is solely a
power exchange
between pairs
of modes,
4
1
8
one can
write a dynamic
equation for the intensity Ii of each
individual guided mode i:
di
1
N
dz I =
(1)
where
Io = Y
2
NIi and is constant with z (we neglect
here the normalization
to Ii given in Refs. 1 and 2).
0146-9592/92/151076-03$5.00/0
©
1992 Optical Society of America
August
1, 1992 / Vol. 17,
No. 15 / OPTICS LETTERS
1077
The PR
intensity coupling
coefficient,
rij
= rij(qi, qj),
between
each
pair
of plane
waves qi
and qj is
calcu-
lated
given the
material
parameters
and the
polar-
ization of
the waves,
4
and its values
are real. In
Eq. (1) we neglect all
the uniform losses to
the guided
modes,
including material
absorption and light
scattering
owing
to inhomogeneities.
The es-
sential
reason
for this is that
in an ideal
hologram
reconstruction,
transversely uniform
losses can be
tolerated,
since they affect
only the efficiency,
whereas a nonuniform
loss
degrades
the reconstruc-
tion quality.
Inclusion of the interaction with
the continua of
both
the ordinary
radiation
and
the evanescent
modes
[represented
by their
intensities
Iq(z)]
yields
dIi
Ii (4
dI_
= (z)
dz
Io(z)
LIi(Z)rii(qi
qj) -
| Iq(z)I`(qikq)dq
for each
individual guided mode i.
Note that the
absolute
intensity
guided
in the waveguide
[Io(z)]
may vary with
z since light power is
constantly es-
caping
from the waveguide
core owing
to coupling to
the radiative
modes,
which do not
confine their
power to the
vicinity of
the core. Equation
(2)
can
be simplified by recalling that in any waveguide
the
intensity of
both the radiation
and the evanescent
modes decays much faster than can be
replenished
by
the PR gain.
Nevertheless,
we assume
that there
is always a small amount of guided
energy in these
modes, owing
to scattering from
the guided modes.
Assuming homogeneous distribution
of scattering
centers,
we may take
the ratio of the
intensity
within
these modes
Iq(z) to
the total intensity
Io(z)
to be a constant
for a given
system: 71
= Iq(z)/Io(z).
This constant
can be calculated if the scatterers are
assumed
to
be known'12
3
(this
scattered
noise was
used to
explain and
analyze the Fanning
effect12
and the PR backscattering'
3
).
Under this assump-
tion Eq. (2) simplifies to
d1i = (z)>
Ij(z) r F(q, q) - qG ,
dz j=,
LIOWz
(3)
where Gi = fJk r(qi, q)dq.
In principle, a radiative
mode can either "milk"
energy
from a guided mode or transfer power
to it
(the direction of the energy transfer is
determined
by the sign of
Fi). However, while energy scattered
out of a coherent
guided mode is almost entirely
lost,
electromagnetic energy
scattered randomly
into
that mode
is not phase matched to the
propagating
mode (the phase of the
light in the radiative mode is
random), and
thus its contribution is negligible. As
a result, this
nonlinear process is not reciprocal, i.e.,
energy
that escaped from the guided modes cannot
be recovered
[for the same reason that we neglected
the nonlinear interaction between pairs
of non-
guided modes in Eq. (2)]. The
effective nonlinear
loss is calculated
by accounting for li = ri(qi, q) > 0
only,
and Gi is defined as an integral over the
loss
region only,
Gi = r(qi, q)dq,
(4)
where
Vqi is the modal region
for which Fi > 0,
for an
individual guided
mode i. The total nonlinear
loss
as a result
of the coupling to the nonguided
modes is
given by the
change in the abolute light intensity,
dIo(z) N dI(z)_
d=E d =
dz i=1
dz
N
-,q2
Ii(z)Gi .
i=l
(5)
Since in general Gi depends
on the mode number i,
the nonlinear loss is transversely
nonuniform. Each
individual guided mode
experiences its own loss,
which is not necessarily
identical to the losses of the
other
guided modes, and this loss does not depend
on z only but depends on the transverse
coordinate
as well.
Up to this point we have formulated
the mode-
coupling process in a
PR waveguide. For the given
boundary conditions
at z = 0, one can refer to it as a
recording
process of the PR volume hologram in
the
waveguide. The ideal reconstruction
process con-
sists of a propagation in the opposite
direction, start-
ing with a mode distribution that
is the output of the
recording process,
i.e., a phase-conjugate reconstruc-
tion.'
0
When
the modes propagate in the negative
z direction,
the mode-coupling dynamics in Eq. (3)
is
changed.
The coupling between pairs of guided
modes simply reverses the sign (Fij -
ri), but the
nonlinear effective loss term, -'rGi, remains
nega-
tive (Gi > 0), yet it may differ from
the one for
propagation in the positive z direction.
In any case,
the nonlinear loss remains nonuniform
and does not
turn into effective gain for the reason
described
above. In general, a nonuniform
loss deteriorates
the quality of the reconstructed
hologram in any
reconstruction scheme,
including an ideal phase-
conjugate reconstruction.'
4
As an
example, we demonstrate the degradation
of
a simple
stored image, as a result of the nonuniform
nonlinear loss, for an ideal phase-conjugate
recon-
struction. Consider a BaTiO
3
PR slab waveguide of
.,
3 -
>I
2
1.5
1
0.5
0
... . . .. . .. ...
......... . . .. .............................
.. . .. . . .. . . .. . .
0
0.1 0.2 0.3 0.4
0.5
Z [cm]
Fig. 1. Dynamics of the guided modes' intensity
along
the PR slab waveguide. The solid
curves show the evolu-
tion of the funneling and
antifunneling effects for the
noise-free case,
where the original intensities are per-
fectly reconstructed at each point along the
propagation
direction z; the dashed curves show
the reconstruction for
the case of a background noise
ratio of 71 = 10-4.
1078 OPTICS LETTERS / Vol. 17, No. 15 / August 1, 1992
the material parameters of Ref. 4 and of a thickness
that permits only two
guided modes. We launch a
crude image that consists of two guided modes of
identical intensity from the facet z = 0 and observe
its dynamics in a 5-cm-long waveguide, where the
z axis coincides with the +c crystalline axis. The
two guided modes, I, and I2, satisfy Eq. (3), with
the boundary conditions I1(0) = I2(0) = 1. The re-
construction of this volume hologram is made
by feeding back the mode outputs, by a
phase-
conjugate mirror with a reflectivity of unity, and ex-
amining the reconstruction at the front facet z = 0.
The solid curves in Fig. 1 show the calculated mode-
coupling dynamics I, and I2 versus z (only the range
o c z - 0.5 cm is shown) for the case of 'q = 0 (no
energy in the nonguided modes). The geometry
of this proposed experiment is of a funneling de-
vice,
4
'
8
and all the energy is transfered from the
higher-order mode I2 to the lower-order one I,. The
reconstruction, as expected, restores the original
inputs, and the evolution of the mode intensities re-
traces itself until full reconstruction Ii*(o) =
I2*(0) = 1 (where the asterisk represents the phase-
conjugate nature of the reconstruction) is obtained.
A real-life experiment is demonstrated by the
dashed curves in Fig. 1, where we assumed that the
background scattered noise results in a value of
71 = 10-' (see Refs. 12 and 13). This time the mode
evolution did not retrace itself, and the recon-
structed values were I1*(0) = 1.0757 and I2*(0) =
0.9057 (in the forward positive z propagation the
dashed and the solid curves almost coincide and are
practically indistinguishable). The overall power
loss in this process is 0.0188, which corresponds to
less than 1% of the input power, but the modes show
intensity deviations of + 7.57% and -9.43% from the
optimal reconstructed values. The nonuniform loss
affects the mode-coupling process between the
guided modes and thus gives rise to a nonreciprocity
in this process.
Examination of the above results for the two ap-
plications of interest (nonlinear mode couplers and
holographic memories) finds the former to be less af-
fected by coupling to evanescent modes. The mode
conversion efficiency, from the higher mode (2) to
the lower one (1), is almost unaffected by the nonlin-
ear loss, but the distortion of the reconstructed im-
age is large. The interaction with the radiative
modes can be somewhat reduced by a reduction in
the density of the scattering centers, which thus re-
duces the seeding of these modes. Nevertheless a
finite amount of light will always exist in these
evanescent modes, and it will always interact with,
and lead to a milking of, the guided modes. The
lowest limit on the density of the scattering centers
in the PR crystals is the dopants' density (which also
determines the storage capacity),
but in practice the
actual scatterer density is much higher. The non-
linear loss affects thin waveguides more than thick
ones, and it disappears completely in bulk media.
Therefore the proposed storage scheme,
9
which uses
bundles of thin waveguides, is affected by it more
strongly
than is the multimode fiber scheme. In
both cases, one should reconsider the figure of merit
obtained by using PR waveguides instead of bulk
crystals as storage media. It is not obvious that the
benefits of uniformity
in the material,
9
which are
improved in waveguides, are worth the limitations
on the resolution and the storage capacity that are
introduced by the nonlinear loss.
In conclusion, we described a basic limitation on
the performance of photorefractive waveguide
devices and its influence on their storage capability
and their operation as nonlinear mode couplers. We
suggest that the storage capacity and the resolution
of stored images in photorefractive waveguide
devices are limited by effective nonlinear loss, in-
duced by coupling to radiative and evanescent
modes, rather than strictly by the material volume.
This research was supported by the U.S. Army Of-
fice of Scientific Research, Durham, North Carolina.
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