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Two-wave mixing in „111…-cut Bi12SiO20 and Bi12TiO20 crystals: Characterization
and comparison with the general orientation
V. P. Kamenov, Yi Hu, E. Shamonina, and K. H. Ringhofer
Department of Physics, University of Osnabru
¨ck, D-49069 Osnabru
¨ck, Germany
V. Ya. Gayvoronsky
Institute of Physics, National Academy of Sciences, 252650 Kiev, Ukraine
共Received 20 December 1999兲
We study the process of two-wave mixing 共TWM兲in optically active, electro-optic, and elasto-optic BSO
and BTO crystals. We calculate the TWM gain for arbitrary crystal cut and optimize the energy exchange. For
the (111) cut, by choosing an appropriate coordinate system, we obtain a simple analytical solution for the
components of the coupling tensor which allows us to optimize analytically the TWM gain with respect to the
grating orientation and the initial light polarization.
PACS number共s兲: 42.70.Nq, 42.65.Hw
I. INTRODUCTION
The energy exchange of TWM in sillenite crystals is the
subject of many studies. In these crystals, in addition to the
primary electro-optic 共Pockels兲effect, there exists also the
elasto-optic effect 共or secondary electro-optic effect兲as well
as optical activity. It is known that optical activity affects
TWM strongly 关1,2兴. After the pioneerring work of Izvanov
et al. 关3兴, the role of the elasto-optic effect for sillenite crys-
tals was investigated intensively 关4–22兴. Recently, it was
shown that the elasto-optic effect leads to surface relief grat-
ings 关19兴and to surface waves 关20兴. In most of the TWM
studies in sillenites the intermediate role of all three effects
mentioned above must be considered.
For real-time holographic interferometry in sillenites un-
der diffusion recording 关23兴, it is important to optimize the
energy exchange. Many papers were devoted to such an op-
timization 关5,6,8–10,13,17,18,21,22兴. They show that the in-
fluence of optical activity and of the elasto-optic effect is
significant. Nevertheless, for a long time there was a wide-
spread belief that the optimum orientation is obtained for the
grating orientation K
储
关111兴共and for polarization 关111兴of
the recording beams兲since both the electro-optic and elasto-
optic effect are most pronounced along the space diagonal of
the unit cell 关4兴. In Refs. 关18兴and 关21兴the simultaneous
influence of optical activity and of the elasto-optic and
electro-optic effect is investigated for (110)-cut crystals. It is
shown that owing to the natural optical activity in sillenites
the optimum grating orientation, as well as the optimum light
polarization, moves away from the 关111兴direction and de-
pends strongly on the crystal thickness. Corresponding re-
sults were obtained later for the (111) cut 关22兴. These recent
results question the most popular cut (110) as the optimum
one so that a detailed study of two-wave mixing with arbi-
trary crystal orientation involving all known effects is neces-
sary. Reference 关17兴makes an attempt to treat this problem
analytically. However, the considerations there are very gen-
eral and do not give a clear answer.
In order to give an overall picture we derive the general
orientational dependence of the gain for crystals with point
group symmetry 共23兲共in particular BSO and BTO兲and op-
timize it, taking into account optical activity as well as the
elasto-optic and electro-optic effect.
In the last few years there was a growing interest in
(111)-cut crystals as photorefractive medium in TWM and
four-wave mixing experiments 关22,24–27兴. The authors of
Ref. 关24兴show that the diffraction efficiency for s-polarized
waves in (111)-cut GaAs crystals does not depend on the
grating orientation. In this work the elasto-optic effect is ne-
glected. The dependence of the TWM gain for s- and
p-polarized waves on the grating orientation in the (111)
plane without taking into account optical activity and the
elasto-optic effect is given in Ref. 关27兴. It is shown in Ref.
关25兴that reflexion gratings in (111)-cut BSO crystals can be
used for dynamic interferometry, whereby it is possible to
write a grating by orthogonally polarized waves. The optimi-
zation of the TWM gain for (111)-cut BSO crystals is given
in Ref. 关22兴. However, the expressions given there are quite
complicated and it is difficult to understand their physical
meaning. It is a common disturbing feature of the elasto-
optic effect that the dependence of its contribution to the
photorefractivity on the grating orientation can become very
complicated.
In this paper we give a simple analytical expression for
the coupling tensor of a (111)-cut crystal which simplifies
the propagation equations significantly. This allows us to de-
scribe TWM under diffusion recording in optically active,
electro-optic and elasto-optic BSO and BTO crystals analyti-
cally and to present the optimal gain with respect to the
grating orientation and the initial light polarization in a
simple form.
The structure of the paper is as follows. In Sec. II we
present the coupled wave equations for TWM in sillenites
under diffusion recording. Sec. III optimizes the gain for
general crystal orientation. In Sec. IV we stress the special
cut (111) and characterize its coupling tensor and its gain
analytically.
II. COUPLED WAVE EQUATIONS
In our considerations a pump wave Epexp关i(kp•r⫺
t)兴
and a signal wave Esexp关i(ks•r⫺
t)兴interact in an arbi-
PHYSICAL REVIEW E AUGUST 2000VOLUME 62, NUMBER 2
PRE 62
1063-651X/2000/62共2兲/2863共8兲/$15.00 2863 ©2000 The American Physical Society
trarily oriented sillenite crystal. We assume that the inci-
dence angles of both waves are equal and small so that the
amplitudes Epand Eshave only xand ycomponents and the
grating vector K⫽kp⫺kslies in the xy-plane. In this
paraxial approximation the waves themselves propagate
through the medium along an arbitrary zdirection.
The coupled wave equations for the slowly varying am-
plitudes Epand Esin the steady state have the form
Es
⬘⫽i%
2•Es⫹ig*H•Ep,
共1兲
Ep
⬘⫽i%
2•Ep⫹igH•Es,
where %is the rotatory power,
2is the 2nd component of
the vector of Pauli matrices, His the coupling tensor 共with
dimensionless components兲, and gis the coupling constant,
proportional to the space-charge field Esc . For diffusion re-
cording
g⫽i⌫m
2⫽⫺
n3r41
S
Esc ,⌫⫽
n3r41
S
KkBT/e
1⫹共KrD兲2,
共2兲
where m⫽2Es
*•Ep/(
兩
Es
兩
2⫹
兩
Ep
兩
2) is the complex light
modulation depth, nthe refractive index, r41
Sthe clamped
electro-optic coefficient, the wavelength, rD
⫽(
⑀
¯
⑀
0kBT/Nte2)1/2 the Debye radius, kBBoltzmann’s con-
stant, Tthe temperature, ethe electron charge,
⑀
¯
⑀
0the di-
electric permittivity, Ntthe effective trap density, and K
⫽
兩
K
兩
.
The coupling tensor His proportional to the induced
change of the inverse dielectric permittivity tensor, H
⫽⫺⌬
⑀
⫺1/(r41
SEsc). The changes of
⑀
⫺1are via the linear
electro-optic effect and the elasto-optic effect,
⌬
⑀
ij
⫺1⫽Esc共rijq
S⫹pijkl
␥
kneqnmnlnm兲nq,共3兲
where the tensor
␥
kn is the inverse of the Christoffel tensor
⌫kn⫽ckinj
Eninj;byrijq
S,pijkl ,ckinj
E, and eqnm we denote the
components of the linear electro-optic, elasto-optic, elastic-
ity, and piezoelectric tensors, respectively; (n1,n2,n3) are
the components of the unit vector K/Kpointing in the direc-
tion of the grating vector in the crystallographic coordinate
system. The second term in Eq. 共3兲corresponds to the con-
tribution of the elasto-optic effect: 共i兲the space-charge field
creates stress in the crystal through the inverse piezoelectric
effect (eqnm); 共ii兲consequently, the crystal is deformed by
Hooke’s law with elastic moduli (ckinj
E); 共iii兲finally, the de-
formations change the inverse dielectric permittivity tensor
(pijkl).
Performing the multiplications in Eq. 共3兲we obtain for the
components of ⌬
⑀
⫺1in the crystallographic coordinate sys-
tem,
⌬
⑀
ij
⫺1⫽Escr41
Sns⫹EscGp4
兵
Ani
2nj
2⫹B共ni
2⫹nj
2兲
⫹C共ni
4⫹nj
4兲ns
2⫹Dn1
2n2
2n3
2
其
ns,共4兲
⌬
⑀
ii
⫺1⫽EscG
兵
B共p1⫹p2⫹p3兲⫹E共p1ni
2⫹p2nj
2⫹p3ns
2兲
⫹F共p1ni
4⫹p2nj
4⫹p3ns
4兲⫹C共p1nj
2ns
2⫹p2ns
2ni
2
⫹p3ni
2nj
2兲
其
n1n2n3,
with
A⫽⫺共c2⫹c4兲共c1⫺c2⫹2c4兲⫺2c4共c1⫺c4兲,
B⫽c1c4,
C⫽共c1⫹c2兲共c1⫺c2⫺2c4兲,
D⫽⫺共c2⫹c4兲共c1⫺c2⫺2c4兲,
E⫽⫺共c2⫹c4兲共c1⫺c2兲⫺c4共c1⫺c4兲,
F⫽共c2⫹c4兲共c1⫺c2⫺2c4兲,
G⫽2e41
兵
c1c4
2⫹共c1⫹2c2⫹c4兲共c1⫺c2⫺2c4兲2n1
2n2
2n3
2
⫹c4共c1⫹c2兲共c1⫺c2⫺2c4兲共n1
2n2
2⫹n1
2n3
2⫹n2
2n3
2兲
其
⫺1.
Here the indices
兵
ijs
其
are cyclic permutations of the se-
quence
兵
123
其
; the parameters (p1,p2,p3,p4)
⫽(p11
E,p12
E,p13
E,p44
E), (c1,c2,c4)⫽(c11
E,c12
E,c44
E), and e41
are the nonzero components of the elasto-optic, elastic, and
piezoelectric tensors, respectively. Note that the electro-optic
effect contributes only to the off-diagonal elements of ⌬
⑀
⫺1
关see the first term on the right-hand side of ⌬
⑀
ij
⫺1in Eq. 共4兲兴.
To calculate the coupling tensor H, we need to transform
⌬
⑀
⫺1to the laboratory coordinate system with the z-axis
coinciding with the propagation direction of the light,
Hij⫽⫺ 1
r41
SEsc
ei•⌬
⑀
⫺1•ej,i⫽x,y,j⫽x,y,共5兲
where ex,yare the unit vectors along the xand yaxes of the
laboratory coordinate system. In contrast to the z-direction,
which is determined from the crystal cut, the choice of ex,yis
free.
III. OPTIMIZATION FOR ARBITRARY CRYSTAL CUT
In this section we optimize the energy exchange between
the pump and the signal beam at a given crystal thickness by
varying the direction of light propagation 共i.e., the crystal
cut兲, the grating orientation, and the light polarization. We
consider sillenite crystals which are optically active and pi-
ezoelectric. The optical absorption is neglected because it
has no influence on the gain, G,
G⫽
兩
Es
兩
2⫺
兩
Es
0
兩
2
兩
Es
0
兩
2,共6兲
where Es
0is the signal beam amplitude in the absence and Es
the corresponding amplitude in the presence of the pump
beam, both values being measured behind the crystal.
2864 PRE 62V. P. KAMENOV et al.
In our calculations the pump and the signal waves are
linearly polarized with the same input polarization, which is
one of the conditions for the strongest coupling of the waves
关17兴. Choosing K⫽rD
⫺1we ensure that the space-charge field
in the crystal has the largest possible amplitude with respect
to the angle of incidence 关1兴.
The crystal parameters used in the calculations below are
关18兴:⫽514.5 nm, n⫽2.615,
⑀
¯
⫽56, %⫽38.6°/mm,
r41
S⫽5.0⫻10⫺12 m/V, e41⫽1.12 C/m2,c
兵
1,2,4
其
⫽
兵
12.96,
2.99,2.45
其
⫻1010 N/m2,p
兵
1,2,3,4
其
⫽
兵
0.16,0.13,0.12,0.015
其
,
and Nt⫽1022 m⫺3for BSO; ⫽633 nm, n⫽2.58,
⑀
¯
⫽47, %⫽6.3°/mm, r41
S⫽4.75⫻10⫺12 m/V, e41⫽1.1
C/m2,c
兵
1,2,4
其
⫽
兵
13.7,2.8,2.6
其
⫻1010 N/m2,p
兵
1,2,3,4
其
⫽
兵
0.173,⫺0.0015,⫺0.0015,⫺0.005
其
, and Nt⫽1022 m⫺3
for BTO.
A. TWM without optical activity
Let us now neglect optical activity, i.e., we consider a
very thin crystal. For TWM in an optical nonactive medium,
under diffusion recoding the interacting waves keep their ini-
tial linear polarization. The additional rotation of the light
polarization caused from the nonlinear interaction of the
waves can also be neglected because the crystal is thin.
Examining Eqs. 共1兲for %⫽0 one can see that the only
parameter responsible for the optimization of the energy ex-
change is the so called effective coupling 关27兴,
Ceff⫽es•H•ep,共7兲
where epand esare the polarization vectors corresponding to
the wave amplitudes Epand Es共in our treatment es⬅ep).
From Eqs. 共4兲,共5兲, and 共7兲we calculate the effective cou-
pling for arbitrary crystal orientation. In order to get the
strongest energy exchange for every cut we maximize Ceff
with respect to all grating orientations and all light polariza-
tion vectors in the xy plane 共remember that in our paraxial
approximation the z-axis coincides always with the propaga-
tion direction兲.
In Fig. 1 the maximal coupling coefficient for each crystal
cut is drawn in the direction of the surface normal of this cut
for a BSO crystal 共for BTO the dependence is qualitatively
the same兲. The symmetry of the figure corresponds to the
symmetry of the point group 共23兲. In cubic crystals the space
diagonal 共e.g., 关111兴) has threefold symmetry, which can be
clearly seen from the figure. The twofold symmetry of the
diagonal of one of the cube faces 共e.g., 关110兴) is also well
emphasized. The basic symmetry for the crystallographic
axes 关100兴,关010兴, and 关001兴is twofold. Higher symmetry
is not forbidden, and these axes show fourfold symmetry.
The absolute minimum 共namely zero兲occurs when the light
propagates along one of the crystallographic axes
关100兴,关010兴, and 关001兴. In this case the waves are not
coupled at all. The absolute maximum is achieved for the cut
with ez
储
关110兴. In order to explain this, we recall that for this
crystal cut the grating vector Kand the 关11
¯
1兴-direction lie
in the same plane. Consequently, the grating can be oriented
along the space diagonal, which is the condition for maximal
energy exchange if there is no optical activity. Actually all
propagation directions which lie in the plane perpendicular
to one of the four space diagonals of the cube fulfill this
condition and give maximal coupling 共see the bold curve in
Fig. 1兲. An additional feature of Fig. 1 is that there is a local
minimum which appears when ezis parallel to 关111兴共or to
any other space diagonal兲.
B. TWM with optical activity
For diffusion recording, optical activity in sillenite crys-
tals can usually not be neglected. Without applied external
field the initial linear polarizations of the light waves remains
linear within the crystal but their angles of polarization
change strongly with the crystal thickness owing to natural
optical activity. Since it is impossible to keep the optimum
polarization everywhere inside the crystal, the effective cou-
pling alters periodically with the thickness. The resulting en-
ergy exchange will not reach the maximum value possible
for an optically nonactive crystal because the maximal effec-
tive coupling can be achieved only locally. Correspondingly,
as shown for the (110) cut, the optimum grating orientation
is generally not in the 关111兴direction 关18兴. Here we will
additionally optimize the TWM gain with respect to the crys-
tal orientation. For any given propagation direction we look
for grating orientation and initial polarization angles for
which the energy exchange between the beams becomes
maximum.
Including optical activity in Eqs. 共1兲makes the optimiza-
tion more mathematically complicated. In our analysis we
use the approximation of constant light modulation along the
crystal thickness and the approximation of strong optical ac-
tivity. The argument for the first approximation is that the
diffraction efficiency in sillenites is small enough so that the
dynamic change of the light modulation is negligible. The
validity of the second approximation is based on the value of
the ratio g
˜
/%, with g
˜
⫽⌫

/(

⫹1), where

is the beam
intensity ratio. For typical BSO parameters we find g
˜
/%
⫽0.06 (

⫽1) and g
˜
/%⫽0.12 (

→⬁) so that optical ac-
tivity dominates. For BTO, g
˜
/%is in the range from
0.23 (

⫽1) to 0.57 (

→⬁) so that we may only expect
qualitative agreement with experiment.
The practical consequences of the above approximations
are that 共i兲gis constant along the crystal width; 共ii兲the
FIG. 1. Orientational dependence of the normalized maximal
effective coupling, Ceff , for BSO. The maximal effective coupling
for a given cut is drawn in the direction of the normal to the crystal
surface.
PRE 62 2865
TWO-WAVE MIXING IN 共111兲-CUT Bi12SiO20 AND . . .
eigenmodes of the system are 共left or right兲circularly polar-
ized waves. Using 共i兲and 共ii兲we obtain,
G⫽⌫d


⫹1关共H11⫹H22兲⫹共H11⫺H22兲
cos共2
⫺%d兲
⫹2H12
sin共2
⫺%d兲兴,共8兲
with
⫽sin共%d兲/共%d兲.
Here dis the crystal thickness, and
is the polarization
angle measured from the xaxis of the laboratory coordinate
system. For the first time an analytical expression of this
form for the gain was given in Ref. 关18兴but it was not
realized there that it is even valid for an arbitrary crystal cut
if only the coupling tensor components and the polarization
angle are measured in the appropriate coordinate system.
The strongest influence of optical activity on the TWM
process is to be expected when the polarization vector rotates
by 180° during the propagation in the crystal and doing so
scans all possible angles of polarization. In this case the re-
sulting gain is independent of the initial light polarization.
We have seen that (110) is the optimum cut without optical
activity. It can be expected that the strongest shift of the
optimum crystal orientation from (110) is at %d⫽180°
关18,21兴. The optimization of Eq. 共8兲for BSO for %d⫽180°
is shown in Fig. 2 共for BTO we obtain qualitatively the same
dependence兲. The symmetry of Fig. 1 is here present again.
The absolute and local minima 共e.g., ez
储
关001兴and ez
储
关111兴,
respectively兲are preserved too. Contrary to the case %⫽0,
the propagation directions perpendicular to 关111兴are no
longer equivalent. Before all the planes perpendicular to the
space diagonals gave the propagation directions which cor-
respond to the absolute maximum of the gain. Now there are
only six equivalent maxima:
关110兴,关11
¯
0兴,关011
¯
兴,关101
¯
兴,关101兴, and 关011兴.Itisin-
teresting to note that the optimization gives the same result
for each %d⬎0. In general, the closer %dis to 180° the
bigger the difference is between the maximum gain for 关110兴
and the other propagation directions perpendicular to one of
the space diagonals of the cell. That is, (110) remains the
optimum cut for every crystal thickness. At the same time
the corresponding optimum orientations of the grating vector
and of the light polarization vector 关both of them lie in the
(110) plane兴depend strongly on the crystal thickness
关18,21兴.
IV. SPECIAL CUT „111…
In the previous section we saw that (110) and (111) are
the most interesting crystal cuts. The (110) cut is character-
ized by the strongest energy exchange while the (111) cut
possesses the highest symmetry 共for cubic crystals兲.Aswe
will show, owing to its symmetry properties, the (111) cut
allows an easy analytical treatment.
The coupling tensor elements are important factors for
wave-mixing. In this section we characterize the coupling
tensor for the (111) cut and present it in a convenient form.
In Sec. II we have mentioned that there is no restriction on
the choice of the x- and y-axis of the laboratory coordinate
system. In many cases it is useful to choose the x-axis par-
allel to the Kvector, so that the light polarization angle is
measured from the plane of incidence; this is convenient for
experiments. Another possibility, often used for analytical
calculations, is to have fixed x- and y-axes. We shall consider
a coordinate system which rotates with the grating vector. In
such a system the symmetry of the (111) cut is most clearly
expressed and a significant simplification of the coupling
tensor for BSO and BTO crystals is achieved.
A. Another form of the coupling tensor
Let us choose a laboratory coordinate system with axes
ez
储
关111兴and exalways parallel to K. The angle ⌿between
the 关11
¯
0兴axis and the Kvector 共see Fig. 3兲indicates the
rotation of the grating vector or the coordinate system around
the 关111兴axis.
From Eqs. 共4兲and 共5兲we calculate the coupling tensor
components, Hij
(111) , for the (111) cut,
Hij
(111)⫽
兺
k关Ak
ijsin k⌿⫹Bk
ijcos k⌿兴
1⫹B6cos 6⌿⫹Hij
0.共9兲
Here the summation index ktakes the values 3 and 9. The
coefficients Ak
ij ,Bk
ij , and B6depend only on the material
constants of the crystals and characterize the elasto-optic ef-
FIG. 2. Orientational dependence of the normalized optimum
gain for BSO at
d⫽180° 共see the caption of Fig. 1兲.
FIG. 3. Geometrical scheme for the optical configuration of
(111)-cut crystals.
2866 PRE 62
V. P. KAMENOV et al.
fect. They are given in the Appendix. The parameter Hij
0in
Eq. 共9兲denotes the components of the coupling tensor with-
out elasto-optic contribution,
H11
0⫽⫺H22
0⫽
冑
6
3sin 3⌿,H12
0⫽
冑
6
3cos 3⌿.共10兲
The following symmetry can be checked directly:
Hij
(111)(⌿)⫽Hij
(111)(⌿⫾120°)⫽⫺Hij
(111)(⌿⫾180°).
B. New form of the coupling tensor
for BSO and BTO crystals
The set of Eqs. 共9兲and 共10兲is valid for each crystal with
cubic symmetry. A useful simplification is possible if we
restrict our attention to BSO and BTO. The evaluation of the
coefficient B6关see Eqs. 共A1兲in the Appendix兴gives B6⯝
⫺0.04 for both BSO and BTO. Consequently we can neglect
the denominator in the first term of Eq. 共9兲,1⫹B6cos 6⌿
⯝1. Similarly, we obtain for the coefficients Ak
ij and Bk
ij that
A3
11⫽0.35Ⰷ
兵
A9
11,B3
11,B9
11
其
,A3
22⫽0.29Ⰷ
兵
A9
22,B3
22,B9
22
其
, and
B3
12⫽0.11Ⰷ
兵
A3
12,A9
12,B9
12
其
for BSO; A3
22⫽0.24
Ⰷ
兵
A9
22,B3
22,B9
22
其
and 1Ⰷ
兵
Ak
11,Bk
11 ,Ak
12 ,Bk
12
其
k⫽3,9 for BTO.
Finally, we rewrite the coupling tensor in the form
H11
(111)⯝a11sin 3⌿,H22
(111)⯝a22sin 3⌿,
H12
(111)⯝a12cos 3⌿,共11兲
where a11⫽
冑
6/3⫹A3
11 ,a12⫽
冑
6/3⫹B3
12 , and a22⫽⫺
冑
6/3
⫹A3
22 for BSO; a11⫽a12⫽
冑
6/3 and a22⫽⫺
冑
6/3⫹A3
22 for
BTO. The accuracy of this approximation is 2% for BSO
and 4% for BTO. The tensor H(111) in Eq. 共11兲has the same
angular dependence as H0, except that every element is nor-
malized differently owing to the elasto-optic effect.
A graphical motivation of the above simplification is
shown in Fig. 4. We compare the angular dependence of the
nonsimplified coupling tensor from Eq. 共9兲共solid line兲with
the pure electro-optic tensor Hij
(0) 共dashed line兲. As one can
see, the role of the elasto-optic effect is different for both
crystals. Its contribution to H11
(111) and H22
(111) for BTO is
negligibly small in contrast to BSO where all three coeffi-
cients are influenced. However, for both crystals the Hij
(111)
dependences show practically the same behavior as the Hij
0
dependences but with different amplitudes, which is the base
of our approximation. The components of the simplified ten-
sor of Eq. 共11兲are not plotted in Fig. 4, because they are
graphically indistinguishable from the nonsimplified one. Fi-
nally, we would like to emphasize that our simplification is
valid only in the appropriate coordinate system 共Fig. 3兲.
C. TWM gain in the new representation
The new simplified coupling tensor includes all known
effects and can be used with a high degree of accuracy for
the analytical analysis of (111)-cut BSO and BTO crystals.
We can, for instance, calculate explicitly the dependence of
the gain on the grating vector orientation, ⌿. From Eqs. 共8兲
and 共11兲we obtain,
G⫽⌫d


⫹1关h1sin 3⌿⫹h2
sin 3⌿cos共2
⫺%d兲
⫹h3
cos 3 ⌿sin共2
⫺%d兲兴,共12兲
with
h1⫽a11⫹a22 ,h2⫽a11⫺a22 ,h3⫽2a12 .
Optimization of Eq. 共12兲with respect to the polarization
angle gives for the maximum gain, G
兩
max ,
G
兩
max⫽⌫d


⫹1共h1sin 3⌿⫹
兩
兩
冑
h3
2⫺共h3
2⫺h2
2兲sin23⌿兲.
共13兲
The corresponding initial light polarization,
opt ,is
FIG. 4. Dependences of the coupling tensor components Hij(⌿)
for (111)-cut BSO 共a兲, and BTO 共b兲. The dashed lines correspond
to the dependences without elasto-optic effect.
PRE 62 2867
TWO-WAVE MIXING IN 共111兲-CUT Bi12SiO20 AND . . .
opt⫺%d
2⫽
再
1
2arctan
冉
h3
h2cot 3⌿
冊
⫹90°
共⫺sin共%d兲sin共3⌿兲兲 for
⫽0
arbitrary for
⫽0,
共14兲
where
is Heaviside’s function. It is responsible for the typical 90°-jump of the polarization which characterizes the point
where the minimum and the maximum gain are equal 关18兴.
After optimization of Eq. 共13兲with respect to the angle ⌿we obtain the maximum gain Gmax(d) and the optimum grating
orientation ⌿opt for a given crystal thickness d,
Gmax共d兲⫽
冦
⌫d


⫹1
h3共h1
2⫹
2
兩
h3
2⫺h2
2
兩
兲
冑
2共h3
2⫺h2
2兲2⫹h1
2共h3
2⫺h2
2兲
for
兩
兩
⭓
th
⌫d


⫹1共h1⫹h2
兩
兩
兲for
兩
兩
⬍
th ,
共15兲
⌿opt⫽
再
1
3arcsin h1h3
冑
2共h3
2⫺h2
2兲2⫹h1
2共h3
2⫺h2
2兲
for
兩
兩
⭓
th
30° for
兩
兩
⬍
th ,
共16兲
where
th⫽h1h2/
兩
h3
2⫺h2
2
兩
. We assumed in the above equa-
tions that %⬎0 and took into account that h1,2,3⬎0.
In Figs. 5共a兲and 5共b兲the dependence of ⌿opt and
opt on
the crystal thickness is plotted. For BSO the threshold pa-
rameter
th is greater than one. Since
兩
兩
⫽
兩
sin(%d)/(%d)
兩
is
always smaller than one, the optimum grating orientation as
well as the initial light polarization for BSO are independent
of the crystal thickness. On the other hand, for BTO ⌿opt and
opt depend on the crystal thickness. Up to %d⯝90° 共e.g.,
兩
兩
⫽
th) there are two branches which correspond to the
maximum gain. At
兩
兩
⭐
th they stick together and the opti-
mum grating orientation and initial light polarization are the
same as for BSO, ⌿opt⫽30° and
opt⫽%d/2.
Let us now discuss the origin of such behavior. After
excluding the elasto-optic contribution in Eqs. 共12兲–共16兲
共i.e., putting h1⫽0 and h2⫽h3⫽2
冑
6/3), it can be directly
seen that for a given crystal thickness the maximal gain is
achieved for arbitrary grating orientation, i.e., G
兩
max
⫽Gmax(d). The presence of the elasto-optic effect gives a
preference direction for the energy exchange. Since there is
also optical activity, the optimal grating orientation depends
in principle on the crystal thickness. The form of this depen-
dence is determined from the ratio between the strengths of
the elasto-optic effect and the optical activity.
It is worth mentioning that in some papers 关4兴, the electro-
optic and the elasto-optic coefficients for BSO are negative.
In this case the energy flow will be in the opposite direction.
Consequently, the optimum grating orientation will be ro-
tated by 180° from our result.
Lastly, we will compare our analysis with the already
published experimental results from Ref. 关22兴, where TWM
measurements are performed in a 2.1-mm-thick (111)-cut
BSO crystal with a He-Ne laser 共the rotatory power is
⫽21.4°/mm at ⫽633 nm). The maximum TWM gain is
measured when the grating vector is parallel to the
关1
¯
1
¯
2兴-axis 共with periodicity 120°) and the light polarization
is inclined at 23° to the plane of incidence. In our notation
these conditions correspond to ⌿opt⫽30 and
opt⫽23°. The
theoretical prediction from Eqs. 共14兲and 共16兲共see also Fig.
5兲gives ⌿opt⫽30 and
opt⫽22.5°. This result is in very
good agreement with the experiment.
V. CONCLUSIONS
We investigated TWM in cubic crystals. In this study all
known effects were taken into account: optical activity,
FIG. 5. Dependences ⌿opt(
d)共a兲and
opt(
d)共b兲for BSO
共dashed line兲and BTO 共solid line兲.
2868 PRE 62
V. P. KAMENOV et al.
electro-optic effect, and elasto-optic effect. We showed that
the (110) cut corresponds to the largest possible TWM gain
even if optical activity and the elasto-optic effect are taken
into account. We derived a simple analytical expression for
the components of the coupling tensor of (111)-cut BTO and
BSO crystals which allowed us to optimize the TWM gain
analytically with respect to the grating orientation and the
initial light polarization.
ACKNOWLEDGMENT
We acknowledge financial support by the Deutsche Fors-
chungsgemeinschaft 共Sonderforschungsbereich 225, Gra-
duiertenkolleg ‘‘Mikrostruktur Oxidischer Kristalle,’’
Emmy-Noether-Programm兲.
APPENDIX
For (111)-cut cubic crystals the coupling tensor Hij in the
coordinate system shown in Fig. 3 is given by Eq. 共9兲. The
corresponding coefficients are
A3
11⫽
冑
6
648
e41
r41
冋
p1a1⫹
冉
p2⫹p3
2⫹2p4
冊
a2
册
,共A1兲
A3
22⫽
冑
6
648
e41
r41
冋
共p1⫺2p4兲a4⫹p2⫹p3
2a5
册
,
A9
11⫽⫺A9
22⫽B9
12⫽
冑
6
648
e41
r41
冉
p1⫺p2⫹p3
2⫺2p4
冊
a3,
A3
12⫽
冑
2
432
e41
r41 共p2⫺p3兲a6,
B3
11⫽⫺B9
11⫽⫺B3
22⫽B9
22⫽A9
12⫽
冑
2
432
e41
r41 共p2⫺p3兲a3,
B3
12⫽
冑
6
648
e41
r41
冋
⫺
冉
p1⫺p2⫹p3
2
冊
a3⫹2p4a7
册
,
B6⫽⫺ 1
108b共c1⫹2c2⫹c4兲共c1⫺c2⫺2c4兲2,
where aiand bare combinations of the elasticity moduli ci,
a1⫽3共c1
2⫹4c2
2⫺2c4
2⫺5c1c2⫹5c1c4⫺11c2c4兲/b,
a2⫽3共5c1
2⫹2c2
2⫹2c4
2⫺7c1c2⫹19c1c4⫺13c2c4兲/b,
a3⫽⫺共c1⫹2c2⫹c4兲共c1⫺c2⫺2c4兲/b,
a4⫽3共3c1
2⫹2c4
2⫺3c1c2⫹11c1c4⫺5c2c4兲/b,
a5⫽3共3c1
2⫹6c2
2⫺2c4
2⫺9c1c2⫹13c1c4⫺19c2c4兲/b,
a6⫽⫺3共c1⫹2c2⫹c4兲共c1⫺c2⫹2c4兲/b,
a7⫽共26c1
2⫺25c2
2⫹2c4
2⫺c1c2⫹55c1c4⫺49c2c4兲/b,
b⫽c1c4
2⫹1
108共c1⫺c2⫺2c4兲关27c4共c1⫹c2兲
⫹共c1⫹2c2⫹c4兲共c1⫺c2⫺2c4兲兴.
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