Content uploaded by Alexei Kamshilin
Author content
All content in this area was uploaded by Alexei Kamshilin on Apr 17, 2015
Content may be subject to copyright.
Asymmetry of two-wave coupling in cubic
photorefractive crystals
Hemmo Tuovinen, Alexei A. Kamshilin, and Timo Jaaskelainen
Va
¨isa
¨la
¨Laboratory, Department of Physics, University of Joensuu, P.O. Box 111, FIN-80101, Joensuu, Finland
Received May 19, 1997; revised manuscript received August 19, 1997
Analysis of two-wave coupling in cubic crystal classes 23 and 4
¯
3mis presented for arbitrary orientation of the
grating vector and the applied electric field. We show that the most efficient two-wave coupling is achieved
when both the grating vector and the external electric field are parallel to the ^1
¯
11&axis in the (110) crystal cut.
However, beam coupling is asymmetric in this geometry: The enhancement coefficient may differ by orders of
magnitude whether interacting beams propagate in the ^110&direction or backwards. Because of this asym-
metry, the far-field fanning patterns of photorefractive Bi12TiO20 crystals strongly depend on the direction of
the pump-beam propagation. A theoretical model is able to explain main features of the observed fanning
patterns if the piezoelectric and photoelastic effects are incorporated into the system of vectorial coupled-wave
equations. © 1997 Optical Society of America [S0740-3224(97)01612-3]
1. INTRODUCTION
Photorefractive two-wave coupling in electro-optic crys-
tals has been studied extensively for its potentials in
many applications. Much attention has been focused on
materials of cubic symmetry such as the optically active
sillenite crystals Bi12SiO20 and Bi12TiO20 (23 symmetry
group) and the isotropic crystals GaAs, InP, and CdTe
(4
¯
3msymmetry group) because of their high carrier mo-
bility, which permits achievement of fast response time.
The symmetry properties of the photorefractive two-beam
coupling in such crystals have been studied both theoreti-
cally and experimentally by several research groups.1–5
These properties provide possibilities to realize effective
energy exchange between two coherent beams in a two-
wave coupling geometry, which has been demonstrated to
be very useful in signal and image processing6and in in-
terferometric measuring systems.7Moreover, two-beam
coupling and a coherent amplification of weak beams un-
derlie the photorefractive beam-fanning effect8that in
turn is the base of photorefractive optical oscillators9and
of both self-pumped and mutually pumped phase
conjugators.10,11 An optimal design of these devices
should take into account an orientational dependence of
light-induced scattering, which is mainly defined by sym-
metry properties of two-wave coupling. For example, the
asymmetrical scattering-light distribution in a Bi12SiO20
crystal under external electric field can be matched by
mapping the two-wave-coupling gain of a variable grating
orientation.12 In the cited work,12 the external electric
field was parallel to the ^001&axis while the interacting
beams were propagating under small angles to the ^110&
axis. Since the early paper of Marrakchi et al.13 this ge-
ometry was incorrectly considered as a most effective one
for two-beam coupling in cubic crystals. Only six years
later it was pointed out by Stepanov14 (and confirmed by
other authors15,16) that the optimal geometry of two-wave
coupling is achieved in these crystals when both the ex-
ternal electric field and the grating vector are parallel to
the ^1
¯
11&axis.
Two-wave coupling in photorefractive crystals was
theoretically investigated in a number of papers.17–20
The most advanced theory was recently presented by Ped-
ersen and Johansen,21 which includes almost all effects of
photorefractive beam coupling under a general angle of
incidence: optical activity, externally induced linear bi-
refringence, self diffraction, and depleted pump beam.
However, this theory does not consider piezoelectric and
photoelastic effects despite the importance of these effects
for the explanation of the experimental results, which
was pointed out by several research groups.12,22,23 Un-
fortunately, each of these works neglects one or more ef-
fects listed above. Moreover, to the best of our knowl-
edge, no attempt has been made before to compare the
two-wave-coupling gain for the beams propagating along
the ^110&axis with the counterdirectionally propagating
beams. The only exception is the early paper of Fabre
et al.,24 where the particular case of the grating vector’s
being parallel to the ^001&axis was considered, but, as we
show in this work, there is no asymmetry of the gain fac-
tor in this geometry if the counterpropagating beams are
considered.
The present paper is devoted to detailed theoretical
and experimental study of the most effective geometry of
two-wave coupling in the (110) cut of cubic crystals, when
both the grating vector and the electric field are parallel
to the ^1
¯
11&axis. Particularly, we demonstrate that the
gain for beams propagating along the ^110&axis signifi-
cantly differs from the gain for the same beams but
propagating backwards (along the
^
110
&
axis). If in one
direction maximal beam coupling is observed for beams
linearly polarized along the external electric field, then
for counterpropagating beams the optimal polarization is
the orthogonal one, and the net amplification is much
weaker. We also show that the important peculiarities of
Tuovinen et al. Vol. 14, No. 12/December 1997/J. Opt. Soc. Am. B 3383
0740-3224/97/123383-10$10.00 © 1997 Optical Society of America
light-induced scattering (angular distribution of the fan-
out light) in sillenite crystals can be explained by map-
ping the two-wave-coupling gain with the obligatory con-
sideration of the piezoelectric and photoelastic effects.
2. THEORETICAL FORMULATION
Referring to Fig. 1, we consider the intersection of two co-
herent polarized beams, Rand S, inside a cubic photore-
fractive crystal with the angle 2
u
between them. The
bulk refractive index of the crystal is n0at the wave-
length l. These beams produce an interference pattern,
which is recorded in the crystal as a space-charge-field
grating with the vector KG, given by KG5kR2kS,
where kRand kSare the wave vectors of the beam Rand
S, respectively. This grating modulates the permittivity
tensor of the crystal in the form
D
e
ˆ52
1
2n
0
4
r
41@EGexp~iKG•r!
1EG
*exp~2iKG•r!#H
ˆG, (1)
where EGis the complex amplitude of the fundamental
component of the space-charge field and H
ˆGis the cou-
pling tensor, which describes the relative changes of the
permittivity-tensor components. It is convenient to sepa-
rate this tensor into two parts, H
ˆG5H
ˆG11r41
21•H
ˆG2,
the first of which describes the influence of the pure
electro-optic effect, and the second corresponds to the pi-
ezoelectric and photoelastic effects. The electro-optic
part includes only the direction cosines eGx ,eGy , and eGz
of the grating field in the direction of principal crystallo-
graphic axes:
H
ˆG15
F
0eGz eGy
eGz 0eGx
eGy eGx 0
G
. (2)
The influence of the piezoelectric and photoelastic ef-
fects can be introduced by the way similar to that pro-
posed by Shandarov et al.25 First, we calculate the ten-
sor of elastic moduli, G
ˆG, for the grating field EGwith the
direction cosines eGx ,eGy , and eGz . This tensor is given
by
where c11 5c22 5c33 ,c12 5c13 5c23 , and c44 5c55
5c66 are nonzero coefficients of the rigidity tensor for
both symmetry groups of cubic electro-optic crystals.
Second, we introduce piezoelectric coefficients in the form
of a pseudovector PG:
PG52e14
S
eGyeGz
eGxeGz
eGxeGy
D
G
ˆG
21, (4)
where e14 5e15 5e36 is a nonvanishing coefficient of the
piezoelectric tensor for both the 4
¯
3mand 23 point-group
symmetries. Finally, the elements of the H
ˆG2coupling
tensor, which describes the change of permittivity tensor
owing to the piezoelectric and photoelastic effects, can be
expressed by25
HG2~x,x!5p11eGxPGx 1p12eGyPGy 1p13eGzPGz ,
HG2~y,y!5p11eGyPGy 1p12eGzPGz 1p13eGxPGx ,
HG2~z,z!5p11eGzPGz 1p12eGxPGx 1p13eGyPGy ,
HG2~x,y!5HG2~y,x!5p44~eGxPGy 1eGyPGx!,
HG2~x,z!5HG2~z,x!5p44~eGxPGz 1eGzPGx!,
HG2~y,z!5HG2~z,y!5p44~eGyPGz 1eGzPGy!,
(5)
where PGx ,PGy , and PGz are projections of the piezoelec-
tric pseudovector on the principal crystallographic axes
and p11 5p22 5p33 ,p12 5p23 5p31 ,p13 5p21 5p32 ,
and p44 5p55 5p66 are nonvanishing coefficients of the
elasto-optic tensor for sillenite crystals (23 point-group
symmetry). There is additional diminishing of the num-
Fig. 1. Two-beam-coupling geometry in the cubic photorefrac-
tive crystal. The grating vector KGlies in the plane of incidence
of the interacting beams and makes an angle
c
with the electric
field vector E0. Vectors qRM ,qRE ,qSM , and qSE are unit vec-
tors in the directions of the TM and TE modes of the Rand S
beams, respectively. kRand kSare the wave vectors of the in-
teracting beams.
G
ˆG5
F
c11eGx
21c44~eGy
21eGz
2!~c12 1c44!eGxeGy ~c12 1c44!eGxeGz
~c12 1c44!eGxeGy c11eGy
21c44~eGx
21eGz
2!~c12 1c44!eGyeGz
~c12 1c44!eGxeGz ~c12 1c44!eGyeGz c11eGz
21c44~eGx
21eGy
2!
G
, (3)
3384 J. Opt. Soc. Am. B/Vol. 14, No. 12/December 1997 Tuovinen et al.
ber of independent elasto-optic coefficients for crystals of
4
¯
3mgroup: p12 5p23 5p31 5p13 5p21 5p32 .
We consider two-beam coupling under an alternating
external electric field of square-wave form. Such a field
is known to be very efficient for hologram recording in
crystals with a long drift length.26 The external electric
field of the amplitude E0is applied so that it may not co-
incide with the grating vector KG. In this case, the ex-
ternally induced birefringence of the crystal is described
in a similar way as for the grating field [Eq. (1)] but the
corresponding coupling matrix H
ˆEshould be calculated
with the direction cosines of the external electric field.
The grating space-charge field, EG, which is generated by
the redistribution of photoinduced carriers inside the pho-
torefractive crystal, depends on the external electric field,
E0. The linearized model of the hologram formation in
BSO-type crystals under an external ac field was devel-
oped by Stepanov and Petrov,26 who demonstrated that
the space-charge field of the grating is phase shifted 90°
with respect to the interference pattern. This fact is fa-
vorable for beam coupling. In the steady-state satura-
tion limit, the space-charge field amplitude EGis given
by26
EG5m
ED~11KG
2LD
2!1E0KGLEcos2
c
~11KG
2LD
2!~11KG
2LS
2!1KG
2LALEcos2
c
,
(6)
where mis the modulation of the interference pattern
created by interacting waves (one should notice that it
varies with the coordinate because of self diffraction),
c
is
the angle between the external electric field and the grat-
ing vector, EDis the diffusion field, LDand LEare the
photoelectron diffusion and drift lengths, respectively, LS
is the Debye screening length, and LAis the length of
electron tightening by the electric field E0:
ED5KG
kBT
e,LD5
A
D
t
,LE5
mt
E0,
LS5
A
ee
0kBT
e2NA
,LA5
ee
0E0
eNA
. (7)
Here kBis the Boltzmann constant, Tis the absolute tem-
perature, eis the electron charge, NAis the trap density,
t
is the photoelectron lifetime,
m
is the charge mobility,
e
is the low-frequency dielectric constant,
e
0is the free-
space permittivity, and Dis the diffusion coefficient.
The optical fields Rand Sare assumed to be of trans-
verse nature and these are characterized by transverse-
electric and transverse-magnetic components, denoted by
the subscripts Eand M, respectively (see Fig. 1). RM,
RE,SM, and SEare of a complex nature (to describe a
general state of polarization) and are dependent on the
coordinate z.21 The modulation of the interference pat-
tern created by these beams is given by
m52R•S*
u
R
u
21
u
S
u
2. (8)
We consider the simultaneous diffraction of two light
waves on the basis of the system of coupled-wave equa-
tions presented by Pedersen and Johansen,21 additionally
including the piezoelectric and photoelastic effects in the
description of the coupling tensors H
ˆEand H
ˆGas shown
above. The coupled-wave equation may be written in the
form
dRM
dz52
a
R
M2
r
R
E2i
k
0
~
H
MM
ERM1HME
ERE!
1
k
G
IT
~HMM
GSM1HME
GSE!~SM
*RM1SE
*RE!,
(9)
dRE
dz52
a
R
E1
r
R
M2i
k
0
~
H
EM
ERM1HEE
ERE!
1
k
G
IT
~HEM
GSM1HEE
GSE!~SM
*RM1SE
*RE!,
(10)
dSM
dz52
a
S
M2
r
S
E2i
k
0
~
H
MM
ESM1HME
ESE!
2
k
G
IT
~HMM
GRM1HME
GRE!~RM
*SM1RE
*SE!,
(11)
dSE
dz52
a
S
E1
r
S
M2i
k
0
~
H
EM
ESM1HEE
ESE!
2
k
G
IT
~HEM
GRM1HEE
GRE!~RM
*SM1RE
*SE!,
(12)
where
a
is the absorption coefficient,
r
is the rotatory
power, IT5
u
R
u
21
u
S
u
2is the total intensity of light,
k
0
and
k
Gare the coupling coefficients for the external elec-
tric field and the grating field, respectively:
k
05
p
n0
3r41E0
l,
k
G5
p
n0
3r41EG
2lm, (13)
where lis the wavelength, n0is the average refractive in-
dex, and r41 is the electro-optic coefficient. Coupling-
matrix elements Hi,j
Eand Hi,j
Ggovern the orientational
property of two-wave coupling3and are readily calculated
as a tensor–vector product:
HMM
G5~qRM •H
ˆG•qRM!,
HME
G5~qRM •H
ˆG•qRE!,
HEM
G5~qRE •H
ˆG•qRM!,
HEE
G5~qRE •H
ˆG•qRE!,(14)
where the vectors qRM ,qRE ,qSM , and qSE are unit vec-
tors in the directions of the TM and TE modes of two
beams. Coupling-matrix elements for the external elec-
tric field, Hi,j
E, are expressed in a similar way but with
the tensor H
ˆE, which differs from H
ˆGwhen the external
field does not coincide with the grating vector.
The system of the coupled-wave equations (9)–(12) de-
scribes two-wave coupling of the polarized beams, which
arbitrarily propagate in a cubic photorefractive crystal,
with optical activity, externally induced birefringence via
Tuovinen et al. Vol. 14, No. 12/December 1997/J. Opt. Soc. Am. B 3385
electro-optic, piezoelectric, and photoelastic effects, self
diffraction, and depleted pump beam taken into account.
In this work we use the paraxial approximation just be-
cause our experiments were carried out for grating peri-
ods larger then 1
m
m. For such gratings the rigorous
theory of two-wave coupling gives the same result as the
paraxial approximation.21 Here we also assume that the
unit polarization vectors of the Rbeam are equal to that
of Sbeam, i.e., qRM 5qSM and qRE 5qSE , that is correct
in the paraxial approximation. The system of Eqs. (9) –
(12) is a three-dimensional description of the two-wave-
mixing process in photorefractive cubic crystals. We are
not able to find a general solution of this system. There-
fore we will analyze the system of Eqs. (9)–(12) for the
most commonly used cases and solve it numerically for
the optimal geometry of two-wave coupling in cubic crys-
tals.
3. TWO-WAVE COUPLING IN THE
OPTIMAL GEOMETRY
Let the interacting beams propagate along the ^110&axis.
If the grating vector is parallel to the ^001&axis, we have
so-called two-wave-mixing geometry.13 In this case, the
coupling matrix coefficients Hi,j
Gcalculated from Eq. (14)
are reduced to
HMM
G50, HME
G50,
HEM
G50, HEE
G521, (15)
if the piezoelectric and photoelastic effects are neglected.
Therefore we can conclude directly from Eqs. (11) and (12)
that only the signal beam, which is orthogonally polarized
to the grating vector SE, will get the enhancement be-
cause the coupling coefficient for dSMis equal to zero.
This is a well-known fact for this geometry, pointed out by
many authors (e.g., Ref. 13). The grating vector is paral-
lel to the ^110&axis in another popular geometry of wave
coupling in cubic crystals called diffraction geometry. In
this geometry the coupling coefficients Hi,j
Gare given by
HMM
G50, HME
G521,
HEM
G521, HEE
G50. (16)
It means the cross-polarization coupling of interacting
beams and can be regarded as a physical reason for an-
isotropic diffraction in this geometry, the phenomenon,
which is also very well known.13,27 For grating vectors
parallel to the ^1
¯
11&axis, the coupling coefficients have
the following values
HMM
G521.1547, HME
G50,
HEM
G50, HEE
G50.5774. (17)
One can see here that the coefficient HMM
G, responsible
for coupling of the components parallel to the grating vec-
tor, has a 15% higher magnitude than the maximal cou-
pling coefficient in the two-wave-mixing geometry.
Therefore a weak beam experiences the highest enhance-
ment in this geometry, which can be called an optimal
one.14,15 Note the opposite sign of coupling coefficients
for orthogonal TE and TM modes in Eq. (17). This means
that, while the TM-polarized component of the Sbeam
gets an enhancement, the TE-polarized component will be
suppressed during the coupling. The situation is
changed if the interacting beams propagate in the oppo-
site direction along the
^
110
&
axis. Now the coupling co-
efficients are
HMM
G520.3849, HME
G50.5443,
HEM
G50.5443, HEE
G50.9623, (18)
resulting in stronger suppression of the TM polarization
and in relatively weaker enhancement of the TE polariza-
tion. The reason of the change of sign for the coupling
coefficients is illustrated by Fig. 2, where we show the
unit vectors qRE and qRM in respect to the crystallo-
graphic axes when the beams propagate in ^110&axis [Fig.
2(a)] and backwards [Fig. 2(b)]. One can see that the di-
rection cosines for TM components change their signs to
the opposite, while those for TE polarization remain the
same. This change of sign for TM direction cosines also
results in the opposite form for H
ˆG[Eq. (2)]. Conse-
quently, we can see from Eq. (14) that the coupling coef-
ficients responsible for the coupling of components of the
same polarization (HMM
Gand HEE
G) change their value for
backward-propagating beams and the cross-polarization
coupling coefficients (HGand HEM
G) appear. Therefore, in
the diffraction geometry (the grating vector parallel to the
^110&axis), there is no change in beam coupling for
backward-propagating beams, while in the two-wave ge-
ometry (the grating vector parallel to the ^001&axis), the
direction of the energy flow has been observed to change
to the opposite (the change of the coupling-coefficient
sign).24 In the optimal geometry (the grating vector par-
allel to the ^1
¯
11&axis), behavior of two-wave coupling is
more complicated: In one direction, the TM-polarized
beam gets maximal amplification, whereas in the opposite
direction, maximal enhancement is achieved for orthogo-
Fig. 2. Disposition of the electric field E0and unit vectors qRM
and qRE of the polarization modes in respect to the ^001&and
^11
¯
0&axes for beams propagating in the ^110&direction (a) and
backward (b).
c
15arcsin @1/(3)1/2#if the electric field is paral-
lel to the ^1
¯
11&axis.
3386 J. Opt. Soc. Am. B/Vol. 14, No. 12/December 1997 Tuovinen et al.
nal TE-polarized beam. We confirm this conclusion by
experiments with photorefractive BTO crystals.
4. BEAM FANNING IN BTO CRYSTALS
Experiments have been carried out with three different
BTO samples measuring 1.85 33.84 38.1 mm3(BTO-1),
0.92 34.4 39.8 mm3(BTO-2), and 0.92 34.5
312.67 mm3(BTO-3). The samples were cut from a
single crystal so that the longest side is parallel to the
^110&axis, as is shown in Fig. 3. Silver-evaporated elec-
trodes enabled the external electric field to be parallel to
the ^1
¯
11&axis. An alternating bipolar voltage of square-
wave form with a peak amplitude up to 5 kV and a rep-
etition rate of about 40 Hz was applied to the sample to
enhance the beam coupling.26 We used a nonexpanded
beam from a He–Ne laser at the wavelength of 632.8 nm
with output power 20 mW to pump our samples. The
single pump beam was directed either along the ^110&axis
or counterdirectionally along the
^
110
&
axis. A self-
induced scattering pattern was observed on a screen lo-
cated approximately 0.3 m behind the crystal.
In sillenite crystals, the photorefractive response is
small if no electric field is applied to the crystal. There-
fore only tiny scattering caused by input-face nonunifor-
mity and–or by bulk imperfections was observed in the
far field when the pump beam was transmitted through
the sample without bias electric field. Otherwise, strong-
fanning scattering appears after applying the ac electric
field to the crystal. It is now well established that the
strength of the scattering pattern reflects the angular dis-
tribution of the two-wave gain.8,28 As it follows from
coupled-wave equations (9)–(12), the coupling coefficient
depends on the polarization state of the interacting beam
for any geometry of the experiments. Therefore the po-
larization state of the pump beam should affect the distri-
bution of fanning light. Experimental photographs of the
scattering patterns shown in Fig. 4 clearly demonstrate
this difference. When the pump beam is linearly polar-
ized in the horizontal plane (parallel to both the electric
field and the ^1
¯
11&axis, Fig. 3), the strongest self-induced
scattering is observed just in the direction of the electric
field, as is shown in Fig. 4(a). However, if the pump
beam has TE polarization, the fanning is a three-lobed
structure consisting of a small lobe directed as in the pre-
vious case along the ^1
¯
11&axis and two side lobes or pet-
als [Fig. 4(b)]. The most intensive petal makes an angle
of about 56° with the ^1
¯
11&axis. Therefore the strongest
enhancement of scattering light occurs here for gratings
significantly tilted in respect to the external electric field.
Another fanning distribution is observed if the same
sample is illuminated by the same pump beam but from
the opposite side. In the experimental setup we simply
rotated the sample by 180° around the ^11
¯
2&axis (Fig. 3).
In this case, the TE-polarized pump produces strong fan-
ning along the externally applied electric field vector [Fig.
4(d)] while the TM-polarized beam gives rise to a scatter-
ing pattern [Fig. 4(c)] similar to that obtained with the
vertically polarized pump in the previous case [Fig. 4(b)].
Total intensity of scattering patterns on Figs. 4(c) and
4(d) is weaker compared with that in similar pictures of
Figs. 4(a) and 4(b) respectively. Examining the grating
Fig. 3. Schematic of the experimental setup for the observation of beam fanning in a BTO crystal. Crystal is pumped by a He–Ne laser
at 632.8 nm. The linear polarizer controls the polarization state of the pump beam. The inset shows one of the scattering patterns
displayed on the screen.
Tuovinen et al. Vol. 14, No. 12/December 1997/J. Opt. Soc. Am. B 3387
direction parallel to the electric field only, we may con-
clude that the theoretical prediction obtained in the pre-
vious section is proven to be true by the experiment. Re-
ally, the strongest enhancement of weak beams is
observed for the TM-polarized pump beam propagating in
^110&direction, whereas for the counterpropagating beam
it is the TE-polarized beam that gets the maximal en-
hancement just in accordance with the sign of coupling co-
efficients in Eqs. (17) and (18), respectively.
The position of the scattering-pattern petals can be cor-
rectly predicted by mapping the two-wave gain in photo-
refractive crystals in a way similar to that which has been
recently done for BSO (Ref. 12) and BaTiO3(Ref. 28) crys-
tals. We assume that the fanning intensity is large in a
direction for which two-beam-coupling gain between the
primary beam and a signal beam traveling in that direc-
tion is large. As a convenient approximation, we calcu-
lated the coupling coefficient between the pump plane
wave and small-intensity seed wave propagating in an ar-
bitrary direction. The system of coupled equations (9) –
(12) has been numerically solved using the parameters of
the BTO crystal shown in Table 1. Below we present re-
Fig. 4. Photographs of scattering patterns obtained by a CCD camera for the BTO-2 sample. The directly transmitted beam was
blocked. White arrows show the polarization state of the input beam. Pictures (a) and (b) were taken for the pump beam propagating
along the ^110&axis while the beam propagation is counterdirectional for pictures (c) and (d). Applied ac voltage is 63.5 kV.
Table 1. Parameters Assumed in the Calculations for Bi12TiO20 at l5632.8 nm
Parameter Notation Value Unit Reference
Refractive index n02.58 29
Absorption
a
0.5 cm21
Optical rotary power
r
6.5 deg/mm 29
Electro-optic coefficient r41 4.74 pm/V 30
Trap density NA231016 cm2331
Dielectric constant
e
47 29
Lifetime-mobility product
mt
1.7 31027cm2/V 31
Elastic tensor components c11 13.7 31010 N/m232
c12 2.8 31010 N/m232
c44 2.6 31010 N/m232
Photoelastic constants p11 20.055 30
p44 0.0035 30
p12 1p13 0.295 30
Piezoelectric coefficient e14 1.1 C/m232
3388 J. Opt. Soc. Am. B/Vol. 14, No. 12/December 1997 Tuovinen et al.
sults of calculations for the fanning patterns in the geom-
etry, with initial conditions corresponding to the photo-
graphs of Fig. 4. Figure 5 shows a contour map of the
intensity-enhancement coefficient as a function of the
propagation direction of a weak signal beam with the in-
put intensity 1026times smaller than the pump. The
pump beam is either TM [Figs. 5(a) and 5(c)] or TE [Figs.
5(b) and 5(d)] polarized and is assumed to propagate in ei-
ther the ^110&[Figs. 5(a) and 5(b)] or
^
110
&
[Figs. 5(c) and
5(d)] direction, which corresponds to the center of the
plot. The two coordinate axes have been chosen in such a
way that each point on the plot corresponds to the point of
incidence of a signal ray on a flat screen placed as far be-
hind the crystal as the one used for the photographs of
Fig. 4. The horizontal axis is parallel to the ^1
¯
11&axis
and the vertical axis is parallel to the ^11
¯
2&axis. As one
can see from Fig. 5, the numerical simulation predicts the
main feature that the strongest coupling along the
electric-field vector occurs for orthogonal polarizations
when the pump beam propagates in the opposite direc-
tion. Moreover, the theory correctly predicts the angle of
strongest coupling in the case of TE-polarized pump
propagating along ^110&direction [Fig. 5(b)] and for the
counterpropagating TM-polarized pump [Fig. 5(c)]. The
tilt angle
u
Pof the side petal calculated from the theoret-
ical dependencies of Figs. 5(b) and 5(c) is equal to 57°,
which is in good agreement with the value of 56° mea-
sured from the experimental photographs of Figs. 4(b)
and 4(c).
Nevertheless, far from all peculiarities of the scattering
patterns can be explained by the developed theory. For
example, there is always small scattering in the direction
of the electric field in Figs. 4(b) and 4(c), which is not re-
vealed by the theoretical calculation. Another disagree-
ment between the theory and experiment is which of the
polarization states of the pump beam is needed to get the
maximal fanning effect. The theory predicts that the
maximal scattering should be exactly either the TM or TE
in the geometry corresponding to Figs. 5(a) and 5(c) and
5(b) and 5(d), respectively. However, the maximal inten-
sity of the scattering pattern is achieved for different po-
larizations in different samples, as shown in Table 2,
where the angle of input polarization is measured in re-
spect to the ^1
¯
11&axis. We should note that the optimal
polarization does not depend on the applied electric field.
It can be seen from Table 2 that the samples BTO-1 and
Fig. 5. Theoretical prediction of the relative intensity (normalized to 100) calculated as a contour map of the two-wave-mixing enhance-
ment for different interaction geometries. The interaction length is 10 mm and ac electric field is 35 kV/cm. Arrows in each graph
show the polarization state of the input pump beam. Graphs (a) and (b) are calculated for the pump beam propagating in the ^110&
direction, while the pump beam propagates in the
^
110
&
direction for graphs (c) and (d). The pump propagation corresponds to the
center of each graph. Coordinates are shown in relative units: The angle between the interacting beams of 15° corresponds to 1.11.
Tuovinen et al. Vol. 14, No. 12/December 1997/J. Opt. Soc. Am. B 3389
BTO-2 have practically the same optimum input-
polarization angles in spite of the difference in their
thicknesses. Nevertheless, the sample BTO-3, which is
longer by almost 3 mm, has quite different optimum po-
larization angles. From one hand, the observed discrep-
ancies may be explained by the fact that the theory was
developed under the approximation of coupling between
two plane waves only, but in reality the intensity of fan-
ning light is comparable with the pump-beam intensity.
Therefore interaction among adjacent fanned beams can-
not be neglected. This additional interaction in optically
active and birefringent media should result in changes of
both the intensity and polarization states of the interact-
ing beams. In the longer sample, the larger part of the
pump beam is transferred to the fanning light, yielding
the different optimal polarization. On the other hand,
the crystallographic orientations of our samples are de-
fined with an accuracy of 15° only. This uncertainty in
the sample orientation also results in the measured value
of the optimal polarization. Another important factor,
which we cannot control experimentally, is the optical
quality of the sample faces and volume. This factor de-
fines the initial distribution of the scattered light and
naturally affects the optimal polarization for enhanced
scattering. For a more quantitative analysis one would
need to account also for the effect of the multiwave inter-
action between various components of fanning light, finite
beam size, reflections at the crystal surfaces, and initial
distribution of scattered light.
Similar beam-coupling asymmetry in respect to the
counterpropagating beams should also be observable in
semiconductor crystals of the 4
¯
3mgroup (such as GaAs).
The system of coupled-wave equations for these crystals
is readily derived from Eqs. (9)–(12) by excluding the
term responsible for the optical activity (
r
50, for these
crystals). Absence of the optical activity does not change
the values of coupling coefficients Hi,j
Gin Eqs. (17) and
(18), which are responsible for the asymmetry in the op-
timal geometry of beam coupling. Crystals of both fami-
lies belong to symmetry groups without the inversion cen-
ter, therefore the axis ^1
¯
11&, which is a cube diagonal, is a
polar axis. This is the main physical reason of the ob-
served asymmetry of beam coupling and, consequently, of
fanning scattering on the direction of the pump beam
propagation.
5. INFLUENCE OF PIEZOELECTRIC AND
PHOTOELASTIC EFFECTS IN
DIFFERENT GEOMETRIES
As has been demonstrated by many authors,2,12,22,23,28 the
correct description of the self-diffraction process in photo-
refractive crystals should account for the contribution of
the piezoelectric and photoelastic effects. This conclu-
sion is also confirmed by comparison of our experiment
and numerical simulation for fanning light distribution in
BTO crystals. To demonstrate the influence of these ef-
fects, we plot the contour maps in Fig. 6, which are calcu-
lated using the same parameters as for graphs in Fig. 5(a)
and 5(b), but considering the clamped electro-optic tensor
described only by Eq. 2. One can see from Fig. 6(b) that
the angle of strongest coupling for the TE-polarized pump
[
u
Ein Fig. 6(b)] is about 38°, that is, one and a half times
smaller than in the experimentally observed pattern.
However, there is no difference in the orientation of the
scattering pattern between calculations with and without
considering piezoelectric and photoelastic effects for the
TM-polarized beam [compare Figs. 5(a) and 6(a)].
Our numerical simulation of two-beam coupling con-
firms the experimental fact that the maximal two-wave
coupling in cubic crystals occurs when both the external
electric field and the grating vector are parallel to the axis
^1
¯
11&. It should be noted that the influences of the piezo-
electric and photoelastic effects are also maximal in this
geometry. Change of coupling coefficients caused by
these effects is different for different geometries, as we il-
Fig. 6. Relative intensity of the enhanced signal beam calcu-
lated with the same parameters as for Figs. 5(a) and 5(b) but con-
sidering the electro-optic effect without piezoelectric and photo-
elastic contributions. The graph (a) is calculated for TM-
polarized pump beam and the graph (b) is calculated for TE-
polarized beam propagating in the ^110&direction.
Table 2. Optimal Polarization of the Pump Beam
to get Maximal Fanning
Geometry of Figure Theory
Experiment
BTO-1 BTO-2 BTO-3
4(a) 0° 30° 31° 0°
4(b) 90° 120° 126° 90°
4(c) 0° 64° 68° 225°
4(d) 90° 154° 160° 65°
3390 J. Opt. Soc. Am. B/Vol. 14, No. 12/December 1997 Tuovinen et al.
lustrate in Table 3, where we put together the coupling
coefficients in the form of matrices similar to those of Eqs.
(15)–(17). One can see that mechanical stresses and
strains caused by the spatially modulated electric field of
holographic grating has almost no influence on diffraction
in the geometries of the grating vector being parallel ei-
ther to the ^001&or ^11
¯
0&axis. It is worthy to note that
the relative 38% increase of the coupling coefficient HMM
G
in the optimal geometry may result in a gain-factor im-
provement by orders of magnitude for the long interaction
length. A surplus of the gain coefficient given by piezo-
electricity depends on the material parameters. For ex-
ample, in the case of GaAs, HMM
G521.389 and HEE
G
50.481 with the parameters reported in Ref. 22.
6. CONCLUSION
Analysis of coupled-wave equations for the cubic photore-
fractive crystal classes 23 and 4
¯
3mshows that the most
efficient geometry for two-wave coupling is achieved when
both the grating vector and external electric field are par-
allel to the ^1
¯
11&axis in the (110) crystal cut. However,
strong asymmetry of beam coupling occurs in this geom-
etry. Differently polarized beams get maximal enhance-
ment whether the beams travel in the ^110&direction or
backward. The direction ^110&cannot be distinguished
from the opposite
^
110
&
direction by the X-ray orientation
method. Therefore sometimes simple rotation of the
sample by 180° may result in higher coefficients of the
weak beam enhancement. Because of this asymmetry,
the far-field fanning patterns of photorefractive Bi12TiO20
crystals strongly depend on the direction of the pump-
beam propagation. A theoretical model explains the
main features of the observed fanning patterns if the pi-
ezoelectric and photoelastic effects are incorporated into
the system of vectorial coupled-wave equations. Two-
beam-coupling asymmetry should be taken into account
for optimal design of mutually pumped phase conjugators.
Note that previously reported phase conjugators in the
optimal geometry of sillenite crystals, with the electric
field being parallel to the ^1
¯
11&axis, were pumped from
the opposite sample’s sides by the horizontally polarized
beams,33 which is not the optimal polarization.
ACKNOWLEDGMENTS
The authors are grateful to Victor V. Prokofiev and Ervin
Nippolainen for the growth and preparation of the BTO
samples. The financial support of this research provided
by the Academy of Finland is greatly appreciated.
REFERENCES
1. J. Strait, J. D. Reed, and N. V. Kukhtarev, ‘‘Orientational
dependence of photorefractive two-beam coupling in
InP:Fe,’’ Opt. Lett. 15, 209–211 (1990).
2. V. I. Volkov, Y. F. Kargin, N. V. Kukhtarev, A. V. Privalko,
T. I. Siemenec, S. M. Shandarov, and V. V. Shepelevitch,
‘‘Influence of the photoelasticity on self-diffraction of light
in electrooptic crystals,’’ Sov. J. Quantum Electron. 21,
1122–1125 (1991).
3. Y. Ding and H. J. Eichler, ‘‘Crystal orientation dependence
of the photorefractive four-wave mixing in compound semi-
conductors of symmetry group 4
¯
3m,’’ Opt. Commun. 110,
456–464 (1994).
4. A. Roy and K. Singh, ‘‘Cross and parallel polarization cou-
pling in transmission-type degenerate four-wave mixing in
compound semiconductor photorefractive crystals: orien-
tational dependence,’’ J. Mod. Opt. 41, 987–1000 (1994).
5. H. Tuovinen, A. A. Kamshilin, R. Ravattinen, V. V.
Prokofiev, and T. Jaaskelainen, ‘‘Two-wave mixing and fan-
ning effect in Bi12TiO20 under an alternating electric field,’’
Opt. Eng. 34, 2641–2647 (1995).
6. S. Mallick and H. Rajbenbach, ‘‘Photorefractive diffraction
in sillenite crystals: application in image processing,’’ Opt.
Comput. Process. 3,3–18 (1995).
7. A. Blouin and J. Monchalin, ‘‘Detection of ultrasonic motion
of a scattering surface by two-wave mixing in a photorefrac-
tive GaAs crystal,’’ Appl. Phys. Lett. 65, 932–934 (1994).
8. V. V. Voronov, I. R. Dorosh, Y. S. Kuzminov, and N. V.
Tkachenko, ‘‘Photoinduced light scattering in cerium-doped
barium strontium niobate crystals,’’ Sov. J. Quantum Elec-
tron. 10, 1346–1349 (1980).
9. J. O. White, M. Cronin-Golomb, B. Fischer, and A. Yariv,
‘‘Coherent oscillation by self-induced gratings in the photo-
refractive crystal BaTiO3,’’ Appl. Phys. Lett. 40, 450 –452
(1982).
10. M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv,
‘‘Passive phase-conjugate mirror based on self-induced os-
cillation in an optical ring cavity,’’ Appl. Phys. Lett. 42,
919–921 (1983).
11. S. Weiss, S. Sternklar, and B. Fischer, ‘‘Double phase-
conjugated mirror: analysis, demonstration, and applica-
tions,’’ Opt. Lett. 12, 114–116 (1987).
12. H. C. Ellin and L. Solymar, ‘‘Light scattering in bismuth
silicate: matching of experimental results,’’ Opt. Commun.
130,85–88 (1996).
13. A. Marrakchi, J. P. Huignard, and P. Gunter, ‘‘Diffraction
efficiency and energy transfer in two-wave mixing experi-
ments with Bi12SiO20 crystals,’’ Appl. Phys. 24, 131–138
(1981).
14. S. I. Stepanov, ‘‘Dynamic holography,’’ in Optics and Infor-
mation Age, H. H. Arsenault, ed., Proc. SPIE 813, 497–500
(1987).
15. N. V. Kukhtarev, T. I. Siemenec, and P. Hribek, ‘‘The influ-
ence of photoelasticity on the self-diffraction of light in cu-
bic photorefractive crystals,’’ Ferroelectr. Lett. Sect. 13,
29–35 (1991).
16. B. Sugg, F. Kahmann, R. A. Rupp, P. Delaye, and G.
Roosen, ‘‘Diffraction and two-beam-coupling in GaAs along
the ^111&direction,’’ Opt. Commun. 102,6–12 (1993).
17. D. L. Staebler and J. J. Amodey, ‘‘Coupled-wave analysis of
holographic storage in LiNbO3,’’ J. Appl. Phys. 43, 1042 –
1049 (1972).
18. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin,
and V. L. Vinetskii, ‘‘Holographic storage in electrooptic
crystals. II. Beam coupling-light amplification,’’ Ferro-
electrics 22, 961–964 (1979).
Table 3. Coupling Coefficients for Various
Orientations of the Grating Vector
in the (110) Cut for BTO Crystals
Grating
Orientation
Coupling Coefficients
Considering
Electro-Optics Only
Including
Piezoelectricity
^001&
S
00
021
DS
00
021
D
^11
¯
0&
S
021
210
DS
021.031
21.031 0
D
^1
¯
11&
S
21.155 0
0 0.577
DS
21.384 0
0 0.367
D
Tuovinen et al. Vol. 14, No. 12/December 1997/J. Opt. Soc. Am. B 3391
19. A. Marrakchi, R. V. Johnson, and A. R. Tanguay, Jr., ‘‘Po-
larization properties of enhanced self-diffraction in sillenite
crystals,’’ IEEE J. Quantum Electron. 23, 2142–2151
(1987).
20. P. Yeh, ‘‘Photorefractive two-beam coupling in cubic crys-
tals,’’ J. Opt. Soc. Am. B 4, 1382–1386 (1987).
21. H. C. Pedersen and P. M. Johansen, ‘‘Analysis of wave cou-
pling in photorefractive cubic media far from the paraxial
limit,’’ J. Opt. Soc. Am. B 12, 592–599 (1996).
22. G. Pauliat, P. Mathey, and G. Roosen, ‘‘Influence of piezo-
electricity on the photorefractive effect,’’ J. Opt. Soc. Am. B
8, 1942–1946 (1991).
23. V. V. Shepelevitch, N. N. Egorov, and V. Shepelevich, ‘‘Ori-
entation and polarization effects of two-beam coupling in a
cubic optically active photorefractive piezoelectric BSO
crystals,’’ J. Opt. Soc. Am. B 11, 1394–1402 (1994).
24. J. C. Fabre, J. M. C. Jonathan, and G. Roosen, ‘‘Photore-
fractive beam coupling in GaAs and InP generated by nano-
second light pulses,’’ J. Opt. Soc. Am. B 5, 1730–1736
(1988).
25. S. M. Shandarov, V. V. Shepelevitch, and N. D. Khat’kov,
‘‘Variation of the permittivity tensor in cubic photorefrac-
tive crystals under the influence of the electric field of a ho-
lographic grating,’’ Opt. Spectrosc. 70, 627–630 (1991).
26. S. I. Stepanov and M. P. Petrov, ‘‘Efficient unstationary ho-
lographic recording in photorefractive crystals under an ex-
ternal alternating electric field,’’ Opt. Commun. 53, 292–
295 (1985).
27. M. P. Petrov, S. V. Miridonov, S. I. Stepanov, and V. V. Ku-
likov, ‘‘Light diffraction and nonlinear image processing in
electrooptic Bi12SiO20 crystals,’’ Opt. Commun. 31, 301–305
(1979).
28. G. Montemezzani, A. A. Zozulya, L. Czaia, D. Z. Anderson,
M. Zgonik, and P. Gunter, ‘‘Origin of the lobe structure in
photorefractive beam fanning,’’ Phys. Rev. A 52, 1791–1794
(1995).
29. S. L. Sochava, E. V. Mokrushina, V. V. Prokofiev, and S. I.
Stepanov, ‘‘Experimental comparison of the ac field and the
moving-grating holographic-recording techniques for
Bi12SiO20 and Bi12TiO20 photorefractive crystals,’’ J. Opt.
Soc. Am. B 10, 1600–1604 (1993).
30. S. M. Shandarov, A. Emelyanov, O. V. Kobozev, and A. V.
Reshet’ko, ‘‘Photorefractive properties of doped sillenite
crystals,’’ in Nonlinear Optics of Low-dimensional Struc-
tures and New Materials, V. I. Emel’yanov and V. Y.
Panchenko, eds., Proc. SPIE 2801, 221–230 (1996).
31. A. A. Kamshilin, E. Raita, V. V. Prokofiev, and T. Jaaskel-
ainen, ‘‘Nonlinear self-channeling of a laser beam at the
surface of a photorefractive fiber,’’ Appl. Phys. Lett. 67,
3242–3244 (1995).
32. K. S. Aleksandrov, V. S. Bondarenko, M. P. Zaitseva, B. P.
Sorokin, Y. I. Kokorin, V. M. Zrazhevskii, A. M. Sysoev, and
B. V. Sobolev, ‘‘Complex investigation of nonlinear electro-
mechanical properties of crystal with structure of sillenite,’’
Sov. Phys. Solid State 26, 2167–2174 (1984).
33. A. A. Kamshilin, V. V. Prokofiev, and T. Jaaskelainen,
‘‘Beam fanning and double phase conjugation in a fiber-like
photorefractive sample,’’ IEEE J. Quantum Electron. 31,
1642–1647 (1995).
3392 J. Opt. Soc. Am. B/Vol. 14, No. 12/December 1997 Tuovinen et al.
