We describe a method for increasing the diffraction efficiency of multiply exposed photorefractive holograms by periodic copying. The method is experimentally demonstrated with photorefractive and thermoplastic recording media

Brady, David

Hsu, Ken

Psaltis, Demetri

English

Caltech Authors - Main

July 15,1990 / Vol. 15, No. 14 / OPTICS LETTERS 817
Periodically refreshed multiply exposed photorefractive
holograms
David Brady, Ken Hsu, and Demetri Psaltis
Department of Electrical Engineering, California Institute of Technology, Pasadena, California 91125
Received June 12, 1989; accepted April 19, 1990
We describe a method for increasing the diffraction efficiency of multiply exposed photorefractive holograms by
periodic copying. The method is experimentally demonstrated with photorefractive and thermoplastic recording
media.
Photorefractive holograms are electro-optic index per-
turbations formed by the redistribution of photogen-
erated charge among local trapping sites. The charge
pattern associated with a hologram is driven by its own
space-charge field to decay exponentially with a time
constant characterized by the dielectric relaxation
time. In the dark, this time may be days or months.
For applications in adaptive holography and holo-
graphic information storage, multiple exposures are
used to form photorefractive holograms. Since the
relaxation time decreases substantially during expo-
sure, recording an uncorrelated hologram causes the
partial erasure of previously recorded holograms. An
exposure schedule by which an arbitrary number of
holograms of equal diffraction efficiency may be re-
corded is described in Ref. 1. Slightly different sched-
ules directed at the same goal are described in Refs. 2
and 3. Unfortunately, the recording of multiple holo-
grams according to this exposure schedule results in a
diffraction efficiency for each hologram proportional
to M-2 , where M is the number of holograms recorded.
This decrease in the diffraction efficiency limits the
number of exposures to at most several hundred.
We describe a system in which part of the decrease
in the diffraction efficiency of a multiply exposed ho-
logram is recovered by periodic copying between two
holographic media. Palais and Wise4 demonstrated
that copying a weak hologram onto a second medium,
using large modulation depth, dramatically increases
the diffraction efficiency of the second hologram.
Johnson et al. 5 applied this technique to improve the
diffraction efficiency of multiply exposed silver halide
films. Here we demonstrate that periodic copying
between two dynamic media improves the diffraction
efficiency by a factor of M. A related result that we
derive in this Letter is that periodic copying between
two holograms results in a stable diffraction efficiency
when an indefinitely long sequence of exposures is
performed. Long sequences of exposures are used in
adaptive holographic systems, such as optical percep-
trons.6 Cross-modulation gratings and Bragg degen-
eracy problems have been considered7 for holographic
associative memories, and we will not discuss these
issues specifically for the periodic copying system con-
sidered here.
The architecture of the system is shown in Fig. 1. A
series of holograms between a reference plane wave
and a set of signal beams is recorded in a strontium
barium niobate (SBN):Ce crystal. Shutters S4 and S5
are closed during this operation. We use plane waves
as signal beams in our experiments. Different beams
are generated by rotation of the mirror RM. Let Ir
and I, represent the intensities of the reference and
signal beams, respectively. The diffraction efficiency
of therecorded holograms is monitored by illuminat-
ing the crystal with the phase-conjugated reference
beam, R1*. The path of the diffracted readout beam
to the output charge-coupled device (CCD) is shown as
a dashed line in the figure. A self-pumped BaTiO3
phase-conjugate mirror is used to generate the conju-
gate wave. In addition to providing automatic align-
ment of the conjugate beam, the phase-conjugate mir-
ror compensates for phase distortions due to imperfec-
tions in the SBN. When the diffraction efficiency of
the holograms becomes unacceptably low, the record-
ed holograms are copied from the SBN into a second
holographic medium, which in our experiments is a
thermoplastic plate. The thermoplastic hologram is
formed by using the light diffracted by the SBN holo-
gram and reference wave R2. Shutters S2 and S4 are
closed. The hologram written on the thermoplastic
plate is copied back to the SBN by reference beams Ri
and R2* (see Fig. 1). The intensity of R2* is selected
to make the intensity of the signal beam diffracted
from the thermoplastic equal to the original signal
intensity, I. Shutters S1 and S5 are closed during
this step. The result is a rejuvenated hologram of
_1__{} _ RM Camera
-I- C2RM
BaTiO3
Fig. 1. Experimental system.
0146-9592/90/140817-03$2.00/0 © 1990 Optical Society of America
818 OPTICS LETTERS / Vol. 15, No. 14 / July 15, 1990
each of the signal beams in the SBN. At this point, we
begin adding a new series of holograms to those al-
ready stored in the SBN. When the diffraction effi-
ciency of the holograms again falls unacceptably low,
we repeat the copying process.
The temporal behavior of photorefractive holo-
grams may be described by growth in the amplitude of
the index perturbation proportional to (1 - e-It) dur-
ing recording and decay proportional to e-It during
the recording of successive holograms.8' 9 I is the spa-
tially averaged recording intensity, and a is a constant
determined by materials parameters and the record-
ing geometry. The amplitude of the perturbation cor-
responding to the mth hologram when M holograms
are recorded is IAm = Ao[i - exp(-aItmn)]exp(- M acitm,, (1)
where AO is the saturation amplitude and tm is the
recording time of the mth hologram. The perturba-
tion created in the photorefractive hologram is
-Nr) = E Aj.(r), (2)
m
where fm(r) describes the normalized spatial structure
of the mth hologram. Requiring that Am be a constant
for all m results in the following exposure schedule:
tm = (al) ln[ + (m -)X] (3)
where x = 1 - exp(-tlaI). The schedule described in
Ref. 1 corresponds to Eq. (3) for the case x = 1 and (t1
ca). We have recorded and reconstructed as many
as Iii holograms in SBN by using this schedule with-
out copying. The uniformity of the holograms was
good, but, as expected, the diffraction efficiency per
hologram was less than 10- 4 times the saturation dif-
fraction efficiency for a single hologram.
We begin the experiment by recording a series of
holograms in the system of Fig. 1 with Ml exposures on
the SBN following the schedule of Eq. (3) for x = 1.
When tm, is substituted from Eq. (3) into Eq. (1), the
amplitude of the space charge for each hologram is
found to be Am = Ao/M, for all m. If Ml is large
enough, then the diffraction efficiency per hologram is
small, and the diffraction efficiency of the mth holo-
gram, 77m, is proportional to IAmi2. This implies that
7m = no/Mi 2, where no is proportional to IAO12. After
Ml exposures, we copy the holograms summed in the
SBN onto the thermoplastic. We then copy back to
the SBN as described above. The amplitude of the
saturation space charge in a photorefractive hologram
is proportional to the modulation depth with which it
is recorded. Since the intensity of the signal beam
reconstructed from the thermoplastic hologram is
equal to Is, the intensity of each of the M1 equally
recorded components is I/Ml. The modulation
depth for each component is the ratio of (IrIs/Ml)l"2 to
the total recording intensity (Ir + Is). Since this mod-
ulation depth is ~/Ml times less than the modulation
depth for one hologram, the amplitude of the space-
charge grating for each component is Am = Ao/IVMl.
eqm is thus no/Ml. The total diffraction efficiency, n7T =
Minm, is restored to the saturation value of no.
We now begin recording a new series of holograms in
the SBN. The first hologram of each new series is
written until its amplitude is equal to the partially
erased amplitude of the copied holograms. Let Am(i)
and tjm be the amplitude and the exposure time of the,
mth hologram recorded in the SBN in the jth record-
ing cycle, respectively. We would like tjm to be such
that Am(Z) is constant for all j and m. Assuming that
this is the case, we use an argument similar to that
described in the previous paragraph to conclude that
Am(X) = Ao/(E§l-I MI)1/2 at the start of the jth recording
cycle, i.e., just after the hologram is copied for the (j -
1)th time. Mj is the number of exposures that we
make in the jth cycle. The perturbation in the crystal
at the start of the jth recording cycle is
j-1 Ml
(r')= E E Am(')fm(l)(r')
1=1 m=l
AO j-1 Ml
= j-1 1/2 E Efm(l) (r'). (4)
aMl) 1=1 m=l
The amplitude of the first hologram is equal to the
partially erased amplitude of the copied holograms if
AO[i - exp(-tjlal)] = AO exp(-t 1 laD.
(E Ml
1
)
(5)
Let x(]) = 1 - exp(-tj 1 aI); Eq. (5) yields xUi) = 1 -[(=y:l M1 )112 /i + (E~l--2 M1)l/2 ]. Letting x = x(Z) in the
exposure schedule of Eq. (3), we make Mj exposures in
the jth cycle to record F,1=1 Ml holograms with equal
amplitudes. In order to maintain a constant diffrac-
tion efficiency from the thermoplastic, we select Mj
such that the total diffraction efficiency of the
summed hologram on the SBN falls back to its value
after the first M1 exposures, i.e., 70/M1. Each time M
= E Mi holograms are copied back and forth, the
diffraction efficiency for each hologram is restored to
i/M.
Figure 2 is a log-log plot of the mean diffraction-
diffraction efficiency per hologram relative to the dif-
fraction efficiency for a single exposure versus the
number of exposures. The figure shows experimental
results for recording as many as 25 holograms in a
SBN:Ce crystal. The diffracted phase-conjugate ref-
erence for each hologram was monitored by the CCD
shown in Fig. 1. The solid curve in Fig. 2 corresponds
to the theoretical M- 2 decay in the diffraction efficien-
cy per hologram when Eq. (2) is followed with x = 1.
The *'s are experimental data points for the mean
diffracted power of holograms stored according to this
schedule. The dashed curve shows the theoretical
path followed by the diffraction efficiency per holo-
gram with periodic copying when Mi = 5. The #'s
show experimental data points found when this ap-
proach was followed. The dotted curve corresponds
to a decrease in diffraction efficiency proportional to
July 15, 1990 / Vol. 15, No. 14 / OPTICS LETTERS 819
Ed 0.05_0
raz.
0 0.01 '
E 0.0t5 \s
NUMBER OF EXPOSURES
Fig. 2. Relative diffraction efficiency of each hologram ver-
sus number of exposures (log-log scale).
M-'. Thermoplastic holograms were made after 5, 10,
and 15 exposures.
One might like to implement periodically refreshed
storage by using two photorefractive media. Unfortu-
nately, copying between two photorefractive crystals
is complicated by the decay of the hologram being
copied while the second hologram is recorded. Sup-
pose that we wish to copy a hologram between two
identical photorefractive crystals. As we read out the
first hologram, the light intensity diffracted from it
decays exponentially at a rate proportional to the
readout intensity. The rate at which the second holo-
gram builds up is determined by the overall intensity
incident upon it. If the reference beam for the second
hologram is too bright, then the modulation depth is
too weak; if it is too low, then the second hologram
does not build up sufficiently before the first hologram
decays. If we set the intensity of the reference beam
such that the modulation depth is constant at the
second hologram, then the overall diffraction efficien-
cy of the copied hologram can exceed the efficiency of
the original only if the strength of the perturbation
produced in the crystal by a single exposure exceeds
the strength required to achieve (in principle) 100%
diffraction efficiency. This can be accomplished by
selecting a crystal with t t appropriate combination of
electro-optic coefficient, dielectric constant, index,
and thickness. For example, the critical thickness for
holograms using the r5t electro-optic coefficient in
barium titanate is approximately 2 mm. Since crys-
tals of such thickness are readily available, it should be
possible to extend the copying technique to an all-
photorefractive system.
We have described periodic copying between two
holograms, using two key simplifications: The holo-
graphic media are planar, which implies that they can
be copied in one exposure, and there is a fixed number
of exposures M that need to be done. The implemen-
tation of learning algorithms in optical neural net-
worksi is ta nt class of problems that requires
an arbitrary number of exposures. From Eq. (4) we
see that the perturbation that accumulates in the crys-
tal is simply the exoue of all the exposures, indepen-
dently of their number. If the individual exposures
are statistically independent, then the sum in Eq. (4)
will grow in proportion to iM. This will precisely
counterbalance the M1/ 2 factor in Eq. (4), yielding a
steady-state diffraction efficiency that is independent
of M. If the individual exposures are correlated (a
case that we have not analyzed in this Letter), then the
sum can grow faster than GAl. In this case the reduc-
tion in diffraction efficiency due to an additional expo-
sure would have been less that what we calculated.
The copying process recovers this smaller loss in dif-
fraction efficiency, and the steady-state overall effi-
ciency remains unchanged and independent of M.
The extension of periodically refreshed recording to
holograms utilizing volume degrees of freedom is com-
plicated by the fact that a volume hologram cannot in
general be evaluated or copied in a single exposure. 10' 11
We define the rank R of a hologram to be the minimum
number of exposures needed to form the hologram. If
the complete hologram is not known a priori, i.e., if we
wish to record holograms adaptively, it is not possible
to ensure that the minimal number of exposures will
be used in recording. The copying technique de-
scribed here allows us to recover the loss in diffraction
efficiency that is due to exposing a hologram more
than R times. In our experiments R was limited to 1
by our use of a single reference beam. For holograms
of higher rank, R references would be required. Using
appropriate reference signals to copy a stored holo-
gram results in an increase in the diffraction efficiency
of each component of the hologram by the factor MA
R2, compared with the efficiency obtained without
copying, where M is the number of exposures used to
form the original hologram. In the simplest case the
steady-state efficiency is again independent of M.
This research is supported by the Defense Ad-
vanced Research Projects Agency. The authors thank
Rockwell International and Ratnakar Neurgaonkar
for the SBN used in these experiments. David Brady
acknowledges the support of the Aerojet Corporation
through the Caltech Program for Advanced Technol-
ogy.
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6. D. Psaltis, D. Brady, X.-G. Gu, and K. Hsu, in Optical
Processing and Computing, H. Arsenault, ed. (Academ-
ic, Orlando, Fla., 1989).
7. H. Lee, X. G. Gu, and D. Psaltis, J. Appl. Phys. 65, 2191
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Periodically refreshed multiply exposed photorefractive holograms

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Periodically refreshed multiply exposed photorefractive holograms

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