Phase-locked sustainment of photorefractive holograms using phase
conjugation
Yong Qiao Demetri Psaltis, Claire Gu,~) John Hong,‘)
Pochi Yeh ‘) and R R Neurgaonkae
Department
Lf
Elect&l &gineering, California Institute
of
Technology, Pasadena, California 91125
(Received i9 April 1991; accepted for publication 22 July 1991)
A method for sustaining multiply exposed photorefractive holograms, in a phase-locked
fashion, by using a pair of phase-conjugating mirrors is described. It is shown that a steady
state exists where the overall diffraction efficiency is independent of the number of
holographic exposures and the tlnal holograms are exactly in phase.with the initial ones.
Both analytical and experimental results are presented.
In this communication, we describe a system in which
a multiply exposed photorefractive hologram can reach a
steady state with overall efficiency independent of the num-
ber of holographic exposures M. Previously reported expo-
sure schedules’,’ for such multiply.exposed holograms re-
sult in a diffraction efficiency of the individual holograms
proportional to M- ‘, while diffraction efficiency of the
composite hologram scales as M- ’ if the individual expo-
sures are statistically independent. Furthermore, the
phases of the recorded holograms in this system remain
locked, which is not true for a previously reported copying
method.3 The system diagram is shown in Fig. 1, .where the
primary hologram is complemented by two phase-conju-
gating mirrors (PCMs). In this system, the PCMs are pho-
torefractive crystals in the four-wave mixing configuration.
They must share the same pair of pump beams so that the
phase-conjugate beams retain the same relative phase. The
basic idea of this system is to record a primary hologram
with external beams, read out this primary hologram with
the reference beam Oj, and finally copy the hologram that is
read out back to the same crystal using the two PCMs. For
photorefractive holograms produced only by diffusion,
there is a phase shift of ?r/2 between the interference pat-
tern and the corresponding hologram. When the reference
beam oj is on and if the crystal axis is oriented properly, the
interference pattern formed by the reference beam or and
the diffracted beam ti will create a hologram that is exactly
in phase with the original hologram.4 When these two
beams are phase conjugated (to produce the beams 0; and
ti), the hologram that the phase conjugate beams create is
exactly in phase with the original hologram, and therefore
the latter gets enhanced and sustained..
We assume that the hologram is recorded with a plane-
wave reference beam whose angle is selected from one of
several possible positions. An arbitrary signal beam can
also be decomposed into a set of plane-wave components.
The hologram can then be described as a superposition of
gratings, each being the result of the interference between
the jth reference beam and the ith plane-wave component
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of the signal beam. Let Eij,i denote the amplitude of the
space-charge field recorded in the photorefractive crystal
that corresponds to the (ij)th grating. Using the fact that
the hologram is phase locked in this system, the first-order
dynamic equation describing the formation and decay of
hologram can be written as5
7tdl&gl/df= - IQ,,ll + mijl-q.
(1)
In Pq. ( 1 ), mu is the modulation depth of the interference
pattern, given by
mij=2t;o;/I0,
(2)
where IO is the total illuminating intensity; rt is the char-
acteristic time constant, which can be written as
r*= 7-;[I().
(3)
7; and {ES1 are real parameters that depend upon the crys-
tal properties and the recording geometry. The amplitude
diffraction efficiency of the (ij)th grating is denoted by
wii and it is related to the space-charge field by
W~‘S~(BIqy,II),
(4)
where p depends on the effective electro-optic coefficient of
the crystal, the hologram thickness, and the recording
wavelength.
If we defiue
Yij’PIGyJ I
(5)
and
c=WI&I,
then a set of simplified equations is obtained:
(6)
dy; 1
x=-$ ( -4)Yij-t ct;oj>,
(7)
wii= sin (y$ ,
(8)
where we have used Eqs. (2) and (3).
We first consider the case of single reference beam with
IV gratings recorded in the crystal. With the reference
beam on (see Fig. 1 for illustration), the dynamics of the
PCM system are described by E!qs. (7) and (S), with
t; =ArlWij,
(9)
4646
J. Appt. Phys. 70 (8), 15 October 1991
0021-8979/91/084846-03$03.00
@ 1991 American Institute of Physics
4646
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PCM 1
Amplitude
PCM 2
Amplitude
Reflectivity
Reflectivity
FIG. 1. Schematic diagram for the hologram sustaining system with a
single reference beam.
,
(10)
I,-,=A2 +A2r T 2 W~j-+ A”
k=l
A is the real amplitude of the reference beam, and r1 and
r2 are the amplitude reflectivities of the two PCMs. Sub-
stituting Eqs. (9)-( 11) into Eq. (7)) we obtain
a+(p2-11)
k=I
,
(12)
where a = 1 +
l/r
z and p = q/r2 In deriving E?q. ( 12),
we have assumed that c, T;, and rl are all independent of
the grating index i. This assumption is valid if the spatial
bandwidth of the signal beam is small.
The steady state of the system is obtained by setting
dyii/dt = 0 in Eq. ( 12):
CPWlli II-j
Yii'a+ (p2-1)Z~==1W~j'
(13)
The steady-state diffraction efficiency wgj2 can be solved
from the above equation and Eq. (8). Assuming low dif-
fraction efficiencies, a sufficient but not necessary condition
for nonzero steady state is
p > max{a/c, 1).
( 14)
It can be shown, using straightforward perturbation anal-
ysis, that the steady state is stable under this condition. For
typical photorefractive crystals, c = 0.2-10. In the case of
small c, the steady-state overall diffraction efficiency satis-
fies the condition BF=1w$)2 g 1, which implies that
Wp’
z y(Z) and dxi@ =: 1. The latter is actually
the undlpleted reference approximation. With these ap-
proximations, Eq. (13) can be solved explicitly, and it
yields
(s)2-cp - =-
k=l
wU
-pq=W
(15)
With the approximation wy
z yii, Eq. (12) also shows that
all the gratings rise or decay with the same time constant,
0.06
0.04 L----J-.
0 10 20 360 700
t
FTG. 2. Numerical simulation of Eq. (12) in which the primary hologram
consists of two gratings with different initial amplitudes. The simulation
was performed with the following parameters: p = 8, a = 2, c = 0.37,
v,,(O) = 0.08, yzi(O) = 0.05, and (r:A’/T;) = 1.
(s) Id _
which implies that wil :wy
l<i,k<N. Here w;’
- wF):w$? for any i,k,
represents the initial value of wii So
Eq. (15) can be rewritten as
w!?j2
w!~~2=)71 IJ
rl
8~~,w~yi *
(16)
This property of grating strength normalization is very use-
ful in many applications including neural network
implementationG9 since it effectively prevents interconnec-
tion weights from either decaying or saturating. If the pri-
mary hologram is formed through a sequence of M expo-
sures using the exposure schedule of Ref. 2, then w$@’
- M- 2 for all jj pairs. Therefore we can see from Eq. ( 16)
that the steady-state diffraction efficiency wg)’ is indepen-
dent of M. For large values of c, the above approximations
do not hold and we must solve Eqs. (8) and ( 13) for the
exact steady states.
The approximation wU
z yii used to derive the steady
state takes into account only the first term in the expansion
of the sine function in Eq. (8). This approximation, how-
ever, is insufficient when the overall diffraction efficiency
starts approaching its steady-state value ~1. When that
happens, dyv/dt z 0 and the higher-order terms of the sine
expansion cannot be ignored in the dynamic equation ( 12).
These higher-order terms, according to our model, have an
equalizing effect that will lead the system to a final steady
state where all the holographic gratings reach the same
diffraction efficiency. This same steady-state diffraction ef-
ficiency can be found by solving Eqs. (8) and ( 13). For
large p and low diffraction efficiencies, this equalizing pro-
cess occurs much slower than the grating normalization
process we discussed above, so that in practice we usually
observe the latter case as a quasisteady state. Shown in Fig.
2 is a numerical simulation of Eq. (12) in which the pri-
mary hologram consists of two gratings with different ini-
tial amplitudes. Initially the ratio of the strengths of the
two gratiflgs remains constant until a quasisteady state is
4647 J. Appl. Phys., Vol. 70, No. 8, 15 October 1991 Qiao et
a/.
4847
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‘-..--L----A
2.00 300 400 500
time (seconds)
FIG. 3. Experimental results for the hologram sustaining system.
r2
= 1 in all the experiments. For p = 6.63, the same steady-state diffraction
efficiency is reached when we start with either low (0) or high (A)
diffraction efficiency. For p = 2.35, the diffraction efficiency decays to
zero (0).
reached. Afterwards, the strengths of the two gratings
slowly converge to a common final steady-state value.
The experimental system consists of an SBN crystal as
the primary hologram and a BaTiOs crystal for the PC&.
The first experiment examines the dynamics of a single
grating recorded in this system. The relevant parameters
are rz = 1 and c = 0.37. Figure 3 shows three experimental
curves measuring the changes in diffraction efficiency with
time. Whenp = rt = 6.63, condition ( 14) was satisfied and
the system reached an overall steady-state diffraction effi-
ciency of about 0.845%; independent of the initial condi-
tion. For comparison, the theoretical value for the steady-
state diffraction efficiency is 71 = 1.06% from Eq. (15).
The discrepancy between the experimental and theoretical
results may be due to the wave-mixing effect in the SBN
and the dependence of phase conjugate reflectivity on the
probe intensity. When p was reduced to 2.35, however, the
system did not have a nonzero steady state and thus the
grating decayed to zero as predicted.
The second experiment investigates the steady-state be-
havior of multiple gratings recorded in the system. This
was done by recording the Fourier transform hologram of
an image, which consisted of multiple gratings resulting
from different spatial frequency components of the Fourier
transform. Figure 4(a) shows the reconstruction of the
image from
the
SBN when it was first recorded and Fig.
4(b) shows the steady-state hologram. Although there is
some distortion in the steady-state hologram, it can be seen
that the grating normalization effect is dominant since all
the spatial frequency components are roughly proportional
to their initial conditions.
In order to store information in a volume hologram,
we need multiple reference beams. For the multiple refer-
ence beam case, assuming that there are N plane-wave
components in the signal beam and R reference beams,
there are two possible ways of sustaining them. One way is
to bring in the reference beams cyclically, and the other is
to use mutually incoherent reference beams and have them
on simultaneously. Following derivations similar to the
(b)
FIG. 4. (a) The reconstruction of the Fourier transform hologram of an
image initially recorded in the SBN crystal. (b) The steady-state response
of the hologram stored in the SBN with the initial condition being a
hologram of the image shown in (a).
single reference case, it can
schemes lead to the same
state:
be shown that .both of these
grating-normalization steady
WY2 =
UP2
II
fl
g&h@
2. (17)
From Eq. ( 17), the number of reference beams that can be
supported is bounded by
R <cp/a=crlr2/( 1 + ri).
(18)
The authors gratefully acknowledge helpful discus-
sions with Hsin-Yu Li; ‘At Caltech, this work was sup-
ported by the Defense Advanced Research Projects
Agency and the Air ‘Force Office of Scientific Research.
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J. Appl. Phys., Vol. 70, No. 8, 15 October 1001
Qiao 8f
al.
4648
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