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DOI: 10.1007/s00340-002-0933-5
Appl. Phys. B 75, 67–73 (2002)
Lasers and Optics
Applied Physics B
n.c. deliolanis1,✉
a.g. apostolidis1
e.d. vanidhis1
d.g. papazoglou2
Photorefractive properties of (110)and (111)-
cut sillenite crystals when external electric field
is applied along the direction of the optimum
diffraction efficiency
1Aristotle University of Thessaloniki, Department of Physics, Solid State Section 313-1,
541 24 Thessaloniki, Greece
2Foundation for Research and Technology-Hellas (FORTH), Institute of Electronic Structure & Laser,
P.O. Box 1527, Vassilika Vouton, Heraklion 71110, Crete, Greece
Received: 5 November 2001/Revised version: 18 April 2002
Published online: 8 August 2002 • © Springer-Verlag 2002
ABSTRACT We study the influence of the application of an ex-
ternal electric field on the grating formed in (110)and (111)-cut
photorefractive sillenite crystals. Optimum conditions for ap-
plication of the bias field are examined, taking into account
the shift of the grating orientation for maximum diffraction
efficiency along the crystal depth. The contribution of the piezo-
electric effect is considered both in the electro-optic tensor and
space charge field calculations. An analysis on the diffractive
properties under these optimum conditions shows an increase
of 30% in maximum diffraction efficiency in Bi12GeO20 ,and
experimental verification for this is provided.
PACS 42.65.Hw; 42.70.Mp; 42.40.Pa
1 Introduction
The optical and diffractive properties of sillenite
crystals have been studied for almost 20 years [1, 2]. Opti-
mization of the diffraction efficiency (DE) and gain is very
important in applications in the field of image and signal
processing. It has been established that the diffractive prop-
erties depend upon various parameters, such as the direc-
tion of polarization, grating orientation, the externally ap-
plied electric field direction, crystal depth and crystal cut
[3– 7]. Those parameters influence the DE and gain, and var-
ious studies for their optimization in photorefractive crys-
tals have appeared in the literature [8–11]. Optical activity,
piezoelectric and photoelastic effects are usually included
in the theoretical calculations involved. Optimization of dif-
fraction efficiency is important for applications involving
holographic storage, such as optical memories and holo-
graphic interferometers, while optimization of gain is im-
portant for optical signal processing applications. In most
of the literature, the optimization of gain is studied with re-
spect to the polarization and grating orientation for gratings
recorded under the diffusion regime. According to Shamon-
ina et al. [12] and Shepelevich et al. [13], optimum grating
vector orientation for maximum gain in (110)-cut sillenite
crystals is a function of the product of optical activity and
✉Fax: +30-31/998019, E-mail: optlab@auth.gr
crystal depth. However, the above publications do not con-
sider the influence of the piezoelectric effect on the space
charge field through the disturbance of the static dielectric
constant.
The recording of long wavelength volume gratings (Λ>
10 µm) can be enhanced by applying a DC electric field to the
crystal. In this case, drift is the main charge transfer mechan-
ism. DE versus grating vector orientation of (110)and (111)-
cut sillenite crystals under the application of a stationary elec-
tric field along [110]and [110], respectively is considered in
our previous work [14–16]. Analytical formulae for the space
charge field, refractive index perturbations, and diffraction
efficiency versus grating vector orientation and input polar-
ization are given, taking into account the piezoelectric con-
tribution and optical activity. In (110)-cut Bi12GeO20 (BGO)
the DE is zero when the grating vector is parallel to the ap-
plied field, and two maxima appear at ±23.3◦[14]. Similarly,
on the (111)-cut two maxima appear at ±20◦with respect to
[110][16]. In both cases, those maxima are not close to the
direction expected, according to Shamonina et al. [5, 12] and
Shepelevich et al. [13], because the externally applied field si-
nusoidally modifies the distribution of the space charge field
and, consequently, enhances the DE in the direction of its ap-
plication. It is obvious that the application of external bias in
the optimum direction would result in an increase in the max-
imum DE.
This paper initially answers the question of which is the
optimum grating and input polarization orientation for obtain-
ing maximum diffraction efficiency for (110)and (111)-cut
sillenite crystals. It then answers the question of what the pho-
torefractive properties of these gratings are when the electric
field is applied in the optimum direction. First, the basic the-
ory and an analysis to determine the optimum bias direction
are presented, including the piezoelectric influence on both
the space charge field and refractive index modulation calcu-
lations. Finally, the diffraction efficiency is calculated when
an external electric field is applied in the previously calculated
optimized directions for both cuts. Experimental verification
on the (110)-cut is provided.
2 Basic theory
The (110)-cut (also called the Huigniard configura-
tion) and the more recent (111)-cut which are examined in this
68 Applied Physics B – Lasers and Optics
a
b
FIGURE 1 Geometry of both crystal configurations. a(110)-cut, and
b(111)-cut. Gis the grating vector oriented at an angle γGto the x-axis. The
angle θstands for the polarization orientation of the readout beam A.Here
Oxand Ozare the principal axes of the modulated indicatrix, situated at an
angle ψwith respect to the x-axis. The box inside stands for the crystal cut,
for which external bias can be applied along the optimum direction δ
paper are both illustrated in Fig. 1. The (111)-configuration
possesses a three-fold rotation axis, while the (110)-cut ex-
hibits no symmetry element. Two collimated light beams
propagate at small angles along the ydirection ([110]and
[1 1 1], respectively), having the same polarization direction,
and form a volume grating inside the crystal. The diffrac-
tion efficiency is calculated [14] by assuming that light is
diffracted from elementary grating slices and then by integrat-
ing the intensity of the diffracted beams throughout the crystal
depth, yielding:
η=R+Qcos(2θ+φ) . (1)
Diffraction efficiency is modulated sinusoidally by the input
polarization angle θ,Qis the amplitude, Ris the dc term, and
φis the phase given by the formulae:
R=π2L2
4λ2(∆no
x+∆no
z)2+(∆no
x−∆no
z)2sinc2(oL),
Q=π2L2
2λ2(∆no
x)2−(∆no
z)2sinc(oL), (2)
φ=oL−2ψ,
where ois the optical rotatory power and Lis the crystal
depth. Here ∆no
xand ∆no
zare the amplitudes of the pertur-
bations of the refractive index ellipsoid along the Oxand
Ozprincipal axes, and ψis the orientation angle of the
Oxprinciple axis with respect to the Ox axis. ∆no
xand
∆no
zare related to the diffraction efficiency of the elemen-
tary gratings parallel to the principal axes. They generally
depend upon the crystal configuration and the grating vec-
tor orientation γG, and they are directly proportional to the
space charge field amplitude Eo
sc. Analytical formulae for
∆no
x,∆no
z,andψfor propagation at small angles along
[110]and [111], in which the piezoelectric effect contribu-
tion is taken into account, are given in References [14, 17]
and [16], respectively. It is obvious from (1) that the max-
imum diffraction efficiency ηmax and optimum polarization
angle θmax are:
ηmax =R+|Q|,(3a)
θmax =−φ
2=ψ−1
2oL,Q>0,
−φ
2+90◦=ψ−1
2oL+90◦,Q<0.(3b)
2.1 Evaluation of the space charge electric field under
external bias
To enhance the space charge field build up process,
a bias electric field is externally applied to the crystal. For
a fairly long grating period (Λ>10 µm), drift is the princi-
pal mechanism for charge transfer as compared to diffusion.
The main difference between the drift and diffusion mech-
anisms is that when the former is dominant, space charge is
transferred along the applied field direction, while when the
latter dominates the carriers’ move along the grating gradient.
However, the electric bias in a particular direction, forming an
angle δwith the Ox-axis, will not favor those gratings which
deviate strongly from the direction of application of the ex-
ternal electric field. This means that the grating vector forms
an angle γG−δwith the displacement of the carrier distribu-
tion. The space charge amplitude o
sc will not be proportional
to the displacement itself, but for small carrier displacements
compared to the grating period o
sc will be proportional to
the projection of the displacement to the grating vector, or
[1,6,14]:
o
sc ∝cos(γG−δ). (4)
For grating vectors lying perpendicular to the external electric
field (γG−δ=90◦), the space charge drifts along the interfer-
ence fringes, and no space charge field builds up.
Apart from the electro-optic coefficients, the piezoelec-
tric effect modifies the static dielectric constant as well. This
means that the effective static dielectric constant (which in-
cludes the piezoelectric effect contribution) should be used in
the Poisson equation to derive the space charge field ampli-
tude. Using the Poisson equation (after Günter et al. [18]) and
following the analysis in References [14, 16], we obtain the
formula for the electric field amplitude:
Eo
sc =Λ
o
sc
2πεεo
(1+p.c.)−1,(5)
where the piezoelectric contribution (p.c.) is:
p.c.(110)=e2
εεo
cos2(γG)
C[4sin
2(γG)A+cos2(γG)B](6a)
and
p.c.(111)=e2
εεo
1
9D8√3(A1−A2)cos(γG)sin3(γG)
−(A1+A2−2A3)cos(4γG)+2cos(2γG)
+3(A1+A2+A3).(6b)
Equations (6a) and (6b) apply to the (110)and (111)configu-
rations, respectively. The p.c. term modifies the space charge
DELIOLANIS et al. Photorefractive properties of sillenites when bias is optimally oriented 69
FIGURE 2 Influence of the piezoelectric effect on space charge field ampli-
tude (1+p.c.)−1vs grating vector orientation (γG). (110)-cut (solid line)and
(111)-cut (dashed line)BGOcrystal
field distribution due to the piezoelectric influence. A,B,C,
D,A1,A2,and A3are functions of the grating vector, which
depend upon the piezoelectric and elastic tensor coefficients,
and which are analytically calculated in References [14] and
[16]. In Fig. 2 (see (5)), the factor (1+p.c.)−1is plotted vs γG
for BGO and for both crystal cuts. Parameters used for the cal-
culation in the case of BGO are: the piezoelectric coefficient
[19] e=e123 =0.98 C/m2, and the elastic coefficients c1=
C1111 =12.84 ×1010 N/m2,c2=C1122 =2.94 ×1010 N/m2,
c3=C2323 =2.55 ×1010 N/m2. In both cases, the p.c. re-
duces the space charge field amplitude by up to 8%.Using(4)
and assuming that the space charge field is proportional to the
externally applied field Eo, the fraction Λ
o
sc/2πεεocan be
substituted by cos(γG−δ)αEo,andEo
sc becomes:
Eo
sc =cos(γG−δ)αEo(1+p.c.)−1.(7)
Here αis a scaling parameter between the applied electric
field Eoand induced space charge field Eo
sc. It is representing
a quantitative measure of the efficiency of the photorefractive
recording, and its value can be calculated by fitting the experi-
mental data of diffraction efficiency measurements with (1)
(see Sect. 4.1).
3 Determination of maximum diffraction efficiency
grating orientation
The externally applied static electric field can dra-
matically affect the space charge field build up, as shown by
(7). Proper selection of the field orientation can maximize
the diffraction efficiency. However, the choice is not quite
straightforward. The maximum DE expression given in (3a)
can be analyzed into a product of two factors.
The first factor depends upon the direction of application
of the external electric field δ, and the second (ηind ) upon the
crystal configuration and on the grating vector orientation, be-
ing independent of δ. This separation is possible because δis
appearing only on the cosine term cos(γG−δ) of the Eo
sc in
(7), and also because ∆no
xand ∆no
zare proportional to Eo
sc.
Consequently, the cosine factor squared can be pulled out of
(3a), and it can become ηmax =cos2(γG−δ) ηind . The obvious
procedure to maximize ηmax is to find the optimum γG(called
γopt), for which ηind is maximized, and then to apply the elec-
tric bias along γopt so that cos(γG−δ) =1.
In Fig. 3 ηind is plotted versus γGand Lfor BGO. The pa-
rameters are (for λ=645 nm): electro-optic coefficient [20]
r=r41 =3.14 pm/V,o=+20.5◦/mm,ε=51.5and the
photoelastic coefficients [21] p1=p1111 =0.12,p2+p3=
p1122 +p1133 =0.19,p4=p2323 =0.01. Figure 3a refers
to the (110)-cut, where two symmetric humps with respect
to the [110](γG=0) direction appear. Their maxima are
apparently deviating from the [111]and [111]directions
(γG=±arctan(1/√2)≈±35.3◦). The maxima range be-
tween γG=±35.3◦for a very thin crystal to ±49.5◦for
a13 mm Bi12 GeO20 crystal, and their trace is shown in the
contour plot at the bottom of Fig. 3a. The maximum di-
vergence is observed when L=8.8mm (or oL=π). At
the (111)-cut (Fig. 3b), multiple humps appear having a 60◦
period, and the maxima are located at γG=±π/6and ±π/2
throughout the entire crystal thickness range without exhibit-
ing any deviation.
Although the effect of the cosine modulation of ηmax due
to the electric field application is not considered above, the
piezoelectric influence on the dielectric constant and, conse-
quently, on the space charge field amplitude is still taken into
account. This consideration results in a more accurate calcu-
lation of the optimum grating vector orientation γopt, contrary
to the case where the piezoelectric effect is not considered
in the space charge field build up. In Fig. 4, γopt is plotted
versus the crystal thickness Lfor the (110)-cut of BGO and
a
b
FIGURE 3 δ-independent diffraction efficiency (ηind) versus grating vector
orientation (γG) and crystal depth (L)fora(110)-cut BGO and b(111)-cut
BGO (arbitrary units)
70 Applied Physics B – Lasers and Optics
FIGURE 4 Grating optimum direction γopt versus crystal depth Lfor
(110)-cut BGO and BTO crystals. Four cases: BGO with p.c. (1) and without
p.c. term (2) (bottom and left axes). BTO with p.c. (3) and without p.c. term
(4) (top and right axes). The straight lines (5) are drawn to visualize the [111]
direction (for both γopt axes) and the [001]direction (for the right axis only)
Bi12TiO20 (BTO) crystals; there are two cases calculated, in-
cluding and omitting the p.c. term in (5). The parameters used
for BTO are: [17, 22] (for λ=633 nm) the electro-optic coef-
ficient r=r41 =5.3pm/V,o=+6.3◦/mm,ε=47, the pho-
toelastic coefficients p1=p1111 =0.173,p2+p3=p1122 +
p1133 =−0.003,p4=p2323 =−0.005, the piezoelectric coef-
ficient e=e123 =1.1C/m2, and the elastic coefficients c1=
C1111 =13.7×1010 N/m2,c2=C1122 =2.8×1010 N/m2,
ab
cd
FIGURE 5 Diffraction efficiency amplitude
|Q|,dclevel R, polarization angle θmax,and
normalized maximum DE ηmax versus grat-
ing vector orientation γGand crystal depth
Lfor a (110)-cut BGO crystal with electric
field (Eo=6kV/cm) applied along the opti-
mum direction δ=γopt (L)(see Fig. 4, line 1).
Here, ηmax is normalized for each crystal depth
by dividing it with the local maximum at this
depth
c3=C2323 =2.6×1010 N/m2. Optimum orientation is di-
verging about 1◦towards the [110]direction for BGO crystals,
which is due to the negative gradient of the (1+p.c.)−1when
γGranges between 35◦and 55◦(see Fig. 2). For BTO crys-
tals the infuence of the p.c. term is more obvious. By omitting
the p.c. γopt exhibits a peak when L=π/o=28.5mm,butif
p.c. is included, a plateau appears at γopt =π/2, and the dis-
placement of ηmax is up to 30◦. Optimum orientation when
allowing for p.c. is closer to the [001]direction, because the
gradient of (1+p.c.)−1is positive for γG>55◦. On the con-
trary, γopt is not drifting at the (111)cut, since it is fixed at
30◦and (1+p.c.)−1at this angle is maximum (the gradient
is zero). Thus, a more simplified approach could be used for
optimization of the [111]configuration.
4 Results and discussion
In this section, diffractive properties of gratings
recorded under optimum recording conditions are examined.
Optimization of the (110)configuration involves a continu-
ous selection of the orientation of the bias electric field, while
optimization of the (111)configuration is achieved by keep-
ing a constant orientation of δ=30◦. Figure 5 illustrates the
dependence of diffractive quantities |Q|,R,θmax and nor-
malized ηmax when an electric bias field (Eo=6kV/cm)is
applied along γopt for the (110)-cut case. In this case the di-
rection of the applied bias field δis a function of thickness,
and ranges from 35.3–49.5◦for L=0to13mm. It is obvious
that the sinusoidal Eo
sc distribution shapes the amplitude |Q|
and the dc term R, and consequently, ηmax. By qualitatively
comparing Fig. 3a and Fig. 5d, we can see that the two sym-
DELIOLANIS et al. Photorefractive properties of sillenites when bias is optimally oriented 71
metric humps, present in Fig. 3a, become disproportionate –
the thicker the crystal is, the larger the disproportion between
the two humps. The ratio of the two peaks is 1:0.35 for a very
thin crystal, and it turns out to be 1:0.1for a thick 13 mm
crystal. In Fig. 5c, π/2jumps of the polarization orientation
θmax corresponding to the maximum are observed on two oc-
casions: when γG=0([110]direction) and when oL=π,
which is for L=8.8mm (phase jumps are expected when
oL=kπ,wherekis an integer). In addition, |Q|is zero at
these locations.
In Fig. 6 the diffractive quantities for the (111)-cut under
optimum bias conditions are presented. In this case the op-
timum orientation for the (DE) optimization is constant (i.e.
δ=30◦) over the entire crystal thickness. By qualitatively
comparing Fig. 3b and Fig. 6d, we can see that one of three
humps present in Fig. 3b is enhanced (the γG=30◦hump),
and the ratio of the peaks is 1:0.47 for a very thin crystal and
1:0.36 for a thick 13 mm crystal. In the (111)case, |Q|=0
and π/2phase jumps of the θmax appear when oL=πand
when γG=0,±60◦([110],[101]and [011]directions). The
longitudinal phase jumps in both crystal cuts take place when
the grating is recorded in a direction in which the condition
∆no
x=∆no
zis fulfilled. This means that diffraction effi-
ciency is the same along either of the principal axes of the
modulated indicatrix (i.e. Oxand Oz, see Fig. 1). There-
fore, linearly polarized light in an arbitrary direction splits
into the two principal axes Oxand Ozand the two compo-
nents are equally diffracted. The propagation process does not
introduce an extra amplitude or phase difference between two
diffracted components of the beam, so they combine and the
amplitude is independent of light polarization (|Q|=0).
ab
cd
FIGURE 6 Diffraction efficiency amplitude
|Q|,dclevel R, polarization angle θmax,and
normalized maximum DE ηmax versus grat-
ing vector orientation γGand crystal depth L
for a (111)-cut BGO crystal with electric field
(Eo=6kV/cm) applied along δ=30◦. Here,
ηmax is normalized like Fig. 5
4.1 Experimental results
From an experimental point of view, the crystal
should be cut having its side faces normal to one of the op-
timum directions in order to apply the electric field. In this
study the case of 3mm (110)-cut Bi12 GeO20 qcrystalisex-
amined experimentally. The deviation of γopt from the [111]
direction is 2.2◦for this thickness, and the side faces of the
crystal are cut so that external bias can be applied along
δ=+37.5◦. The grating is recorded by projecting the Fourier
filtered image of a Ronchi grating onto the crystal illumi-
nated by a quasi-monochromatic 546 nm unpolarized light
obtained from a Hg-Xe arc lamp. The image is filtered at the
Fourier plane by letting through only the symmetrical first
order diffraction beams to project only the basic harmonic si-
nusoidal image onto the crystal. The crystal dimensions are
12 ×10 ×3mm
3, the beam intensity modulation is m=1,the
grating frequency is 50 lines/mm, the applied electric field is
Eo=6kV/cm, and the exposure is 0.4mJ/cm2. During the
readout, 645 nm polarized light is used while the applied high
voltage is switched off. The DE was calculated from the ratio
of the intensity of the diffracted beam to that of the transmit-
ted beam. This effectively takes out any effects imposed by
absorption or Fresnel reflection loses.
In Fig. 7 both experimental and theoretical results of the
amplitude |Q|and the dc level Rof diffraction efficiency as
a function of the grating vector for the (110)-cut BGO are pre-
sented. The value of the scaling parameter calculated by fitting
is α=0.75, corresponding to a space charge field amplitude of
Eo
sc =4.37 kV/cm at γG=+37.5◦. In addition, the polariza-
tion angle θmax at which the maximum diffraction efficiency is
72 Applied Physics B – Lasers and Optics
FIGURE 7 Diffraction efficiency versus grating vector orientation (γG)for
a3mm(110)-cut Bi12GeO20 when external electric field E0=6kV/cm is
applied along δ=+37.5◦(in line 7 E0is applied along [110]). 1 – theoretical
R, 2 – theoretical |Q|, 3 – theoretical ηmax , 4 – experimental R,5–experi-
mental |Q|, 6 – experimental ηmax, 7 – theoretical ηmax for E0=6kV/cm
applied along [110]
achieved is depicted in Fig. 8. The theoretical and experimen-
tal results generally match with each other, and some small
mismatch in diffraction efficiency measurements may be at-
tributed to a slightly misoriented cut. Maximum diffraction
efficiency ηmax =+6.3% occurs when the grating angle is
γG=+37.5◦and the input polarization angle is θ=3.6◦.The
symmetric lobe is still observable, but its peak is reduced to
a value of ηmax =1.8% and shifted to γG=−10.2◦, because
for this grating orientation, Eo
sc is less than 50% with regard to
its peak value at the enhanced lobe area.
The advantage of applying the electric field along the op-
timum direction in comparison with the common case, where
it is applied along [110], is the increase of about 30% in the
maximum diffraction efficiency (see Fig. 7, lines 3 and 7). Re-
garding γGand θ, a selective unidirectional behavior arises
due to the space charge field angular distribution. It can be
FIGURE 8 Theoretical results and experimental data for θmax versus grat-
ing vector orientation (γG)fora3mm(110)-cut Bi12GeO20 crystal. Eo=
6kV/cm is applied along δ=+37.5◦
seen from Fig. 7 that the range of γGforwhichDE(line3)is
enhanced above the peak of the regular bias along [110] (line
7, ηmax =4.8%) is limited between 19–57◦.Furthermore,at
γG=37.5◦, DE sensitivity on the polarization angle is also
increased. As a result, the maximum DE is achieved when
θ=θmax =+3.6◦, but is zeroed when θ=θmax ±90◦.This
means that applying an electric field along the optimum di-
rection in a (110)-cut B12 GeO20 crystal results in selective
behavior of the diffraction efficiency in tgerms of both the
orientations of the grating vector and the input polarization.
5 Conclusions
In this paper, diffraction properties of gratings
recorded in (110)and (111)sillenite crystals are studied
under optimum conditions. The application of the electric
field strongly influences the build up of the space charge field
and the phase grating, and enhances the diffraction efficiency
when the grating vector is oriented parallel to the direction of
application. The optimum application direction of the elec-
tric field to enhance DE coincides with the grating direction
for which maximum DE occurs. An analysis is made to deter-
mine this optimum direction, in which it is necessary for the
influences on the DE related to the crystal configuration and
bias orientation are separated. While no particular difficul-
ties arise in the selection of optimum orientation for the (111)
case, finding the optimum grating orientation for the (110)
case involves calculation of the maximum diffraction effi-
ciency shift along the crystal depth. The deviation is generally
small for thin crystals, but it is considerable for larger crystal
thicknesses. Allowing for the piezoelectric influence in both
refractive index and space charge field calculations leads to
more accurate results in terms of the optimization parameters
of the DE. This effect is not very profound for the thin 3mm
BGO crystal, which is examined experimentally. However,
this would not be the case for BTO crystals, since in some
cases allowing for the piezoelectric contribution in the static
dielectric constant leads to a 30◦displacement of the diffrac-
tion efficiency maxima. For the diffusion recording theme,
which is isotropic, the diffraction efficiency is exhibiting vari-
ous minima and maxima, but when an electric field is applied,
one of the maxima is enhanced while the others are dimin-
ished. A 30% increase of the DE with respect to the usual
direction of application was achieved for (110)-cut BGO. In
this case, a quite sensitive behavior of DE arises regarding the
grating and polarization direction. Finally, examination of the
strong influence of the external electric field orientation on the
build up of the space charge field for various crystal configu-
rations could lead to very interesting diffractive properties.
ACKNOWLEDGEMENTS N.C. Deliolanis is supported by the
Greek State Scholarship Foundation.
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