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Coupling of orthogonally polarized waves and
vectorial coherent oscillation
in periodically poled LiNbO3:Y:Fe
D. Barilov and A. Shumelyuk
Institute of Physics, National Academy of Sciences, 03650, Kiev-39, Ukraine
L. Hesselink
Stanford University, Room 353 Durand Building, Stanford, California 94305-4035
B. Sturman
International Institute for Nonlinear Studies, Koptyug Avenue 1, 630090, Novosibirsk, Russia
S. Odoulov
Institute of Physics, National Academy of Sciences, 03650, Kiev-39, Ukraine
Received November 9, 2002; revised manuscript received March 31, 2003
Photorefractive grating recording with two orthogonally polarized eigenwaves (ordinary and extraordinary)
and vectorial coherent oscillation in a backward four-wave mixing scheme in iron-doped periodically poled
lithium niobate are reported. © 2003 Optical Society of America
OCIS codes: 190.4720, 210.4810, 090.2900, 090.7330.
1. INTRODUCTION
Periodically poled ferroelectric crystals are known as
promising materials for quasi-phase-matched frequency
conversion owing to their
(2)nonlinearity.1–3At low in-
tensities, these artificially tailored materials also exhibit
strong photorefractive nonlinearity.4It has been shown
that iron-doped periodically poled lithium niobate (PPLN)
has the same sensitivity to holographic grating recording
as homogeneously poled material but exhibits an inhib-
ited response at low spatial frequencies; i.e., this material
is optical-damage free.5,6
Efficient recording of Bragg-matched three-dimension-
al index gratings is possible in PPLN:Fe because of the
photovoltaic charge transport.7The nonlinear change of
optical permittivity ⌬
⑀
is proportional in this case to the
product of an effective photovoltaic constant

eff and an
effective electro-optic constant reff . Both

eff and reff are
sensitive to inversion of the crystal polar axis; they
change sign in each new domain of the opposite sponta-
neous polarization. The sign of the product

effreff re-
mains, however, the same throughout the whole layered
structure; this ensures continuity of the recorded phase
grating and facilitates attaining high diffraction effi-
ciency.
Until now, efficient recording of index gratings was
achieved in PPLN by use of longitudinal photovoltaic
currents,5i.e., currents that propagate along the optical z
axis and are proportional to the diagonal components of
the photovoltaic tensor. Charge separation occurs in this
case owing to the spatial intensity modulation produced
by either ordinary or extraordinary light waves.
It is known, however, that the photovoltaic tensor pos-
sesses nondiagonal components in lithium niobate
(LiNbO3) ; the corresponding photovoltaic currents can be
excited only when ordinary and extraordinary light waves
are present simultaneously in the sample.7This fact has
allowed for pure polarization recording (in the absence of
spatial modulation of the light intensity) of the index
grating in single-domain LiNbO3:Fe crystals.8
In this paper we prove the possibility of such polariza-
tion grating recording in PPLN:Y:Fe. The feasibility of
polarization recording in the single-domain case does not
guarantee its feasibility in PPLN because a large amount
of yttrium is deliberately introduced into the melt to sta-
bilize the final domain structure.9
We report (i) efficient polarization recording of index
gratings with high spatial frequencies and inhibited opti-
cal damage, (ii) strong intensity redistribution for the cor-
responding orthogonally polarized recording waves (vec-
torial coupling), (iii) efficient generation of phase-
conjugate waves in a backward four-wave mixing
geometry, and (iv) coherent high-quality oscillation in a
cavity formed by conventional and phase-conjugate mir-
rors. Finally, using the experimental data, we evaluate
the effective photovoltaic field that characterizes the po-
larization recording.
Barilov et al. Vol. 20, No. 8/ August 2003/J. Opt. Soc. Am. B 1649
0740-3224/2003/081649-07$15.00 © 2003 Optical Society of America
2. PHYSICAL BACKGROUND
Photorefractive nonlinearity is caused by a light-induced
charge separation leading to spatial variations of the op-
tical permittivity tensor
␦
⑀
ˆowing to the linear electro-
optic effect. For the Cartesian components of the permit-
tivity tensor one can write
␦
⑀
mn共r兲⯝⫺n4rmnlEl共r兲, (1)
where n⫽(no⫹ne)/2 ⯝no,eis the average refractive
index, noand neare refractive indices for ordinary and
extraordinary waves, respectively, rmnl is an electro-optic
tensor, and Elis the lth component of the space-charge
field. The largest components of the electro-optic tensor
are r333 ⬅r33 ,r113 ⬅r13 , and r131 ⬅r51 . Components
r33 and r13 are responsible for coupling between waves of
the same polarization, extraordinary (e) and ordinary (o),
respectively, whereas component r51 is responsible for the
mutual coupling of oand ewaves.
In PPLN:Fe, the light-induced charge transport is pre-
dominantly due to the bulk photovoltaic effect.7The mth
component of photovoltaic current density jis
jm⫽

mnlAnAl*, (2)
where

mnl is the photovoltaic tensor and Ais the com-
plex slowly varying (in time) amplitude of the electric
light-field vector such that I⫽
兩
A
兩
2is the light intensity.
The photovoltaic tensor is generally complex; it can be
represented in the form7

mnl ⫽

mnl
L⫹i
␦
mnk

kl
C, (3)
where

ˆLand

ˆCare real tensors that describe the so-
called linear and circular photovoltaic currents and
␦
mnk
is the unit antisymmetric tensor. Tensor

mnl
Lis symmet-
ric to permutation of subscripts nand l. Its independent
components are

33
L⬅

333
L,

31
L⬅

311
L, and

15
L⬅

131
L.
Components

33
Land

31
Lcorrespond to the longitudinal
currents induced by light polarized parallel and perpen-
dicular, respectively, to the optical axis. Component

15
L
is responsible for the current flowing perpendicular to the
polar axis; to excite this transverse current, the light field
has to possess simultaneously components that are paral-
lel and perpendicular to the zaxis. In other words, oand
ewaves have to be present simultaneously in the crystal.
Component

12
Cis also responsible for the transverse pho-
tovoltaic current induced by oand ewaves. Because the
wave vectors of these waves are different, the transverse
currents are spatially oscillating. The main difference
between the transverse spatially oscillating currents re-
lated to components

15
Land

12
Cis that their distributions
are shifted with respect to each other by a quarter of a pe-
riod. It is known that for iron-doped lithium niobate the
inequality

12
CⰇ

15
Lholds true.10
The photovoltaic currents result in formation of space-
charge fields. To describe them it is convenient to intro-
duce the photovoltaic field
Em
pv ⫽

mnlAnAl*/
, (4)
where
⫽
Iis the photoconductivity and
is the spe-
cific photoconductivity.
In accordance with the notation of Eq. (4), we label the
photovoltaic fields that are due to the linear photovoltaic
currents propagating along the zaxis and excited by (e)
and (o) waves E33
Land E31
L, respectively. Similarly, fields
E15
Land E12
Crefer to photovoltaic tensor components

15
L
and

12
C. They characterize the result of charge separa-
tion caused by the spatially oscillating transverse photo-
voltaic currents. In LiNbO3doped with iron or copper
longitudinal fields E33
Land E31
Lrange from a few tens to
hundreds of kilovolts per centimeter. Transverse field
E12
Cin LiNbO3:Fe is roughly 1 order of magnitude smaller
than the longitudinal fields and considerably larger than
E15
L.
The space-charge fields for the gratings recorded in
PPLN in different geometries and with different orienta-
tions and polarizations of two light beams were calculated
in Refs. 4 and 11. For the polarization recording of a
grating with grating vector Kperpendicular to the crystal
caxis (see Fig. 1) the amplitude of the fundamental Fou-
rier component of the space-charge field is
EK
共0兲⬇⫺iE12
C共s
ˆe
ˆo兲2
冋
1⫺
2
Kx0
tanh
冉
Kx0
2
冊
册
AoAe*
I0
.
(5)
Here x0is a half-period of the domain structure, Ao,eare
the scalar amplitudes of ordinary and extraordinary re-
cording waves, e
ˆois the unit polarization vector for the o
wave, and s
ˆ⫽K/Kis the unit grating vector. This ex-
pression includes two K-dependent factors; one (in brack-
ets) is due to the periodic poling,11 and the other,
共s
ˆe
ˆo兲2⫽
4
2
4
2⫹关no⫺ne兲/no]2, (6)
is caused by the crystal birefringence.12 Here
is the
half-angle between the recording beams inside the
sample, which is, because of refraction, small enough for
sin
⯝
to be considered. Spatial frequency Kof the re-
corded grating is expressed by
as follows:
K⫽k
兵
4
2⫹关共no⫺ne兲/no兴2
其
1/2, (7)
Fig. 1. Schematic representation of photorefractive grating re-
cording with two orthogonally polarized light waves.
1650 J. Opt. Soc. Am. B/ Vol. 20, No. 8/ August 2003 Barilov et al.
where k⫽2
no/is the light wave number. Note
that expression (5) also describes the space-charge field
in the single-domain case that corresponds to the limit
Kx0→⬁, i.e., to Eq. (7).
Each of the two K-dependent factors in expression (5)
reduces the space-charge field; the smaller the spatial fre-
quency, the stronger the reduction. The cutoff spatial
frequency for polarization grating recording (it can be in-
troduced as the frequency for which the amplitude of the
space-charge field becomes two times smaller than its
maximum value) depends on domain size x0. This is il-
lustrated in Fig. 2, which shows the spatial frequency de-
pendences for several representative values of the domain
sizes, namely, x0⫽10 , 20, 50
m, and also for the single-
domain case, x0⫽⬁.
To emphasize the role of birefringence for the Kdepen-
dence of the space-charge field, we have set no⫽nein
Fig. 2A. This case is relevant also to periodically poled
LiTaO3, which possesses only a small birefringence at
room temperature. For LiNbO3the cutoff spatial fre-
quency depends weakly on domain size; see Fig. 2B.
There are several main options for experimental evalu-
ation of the photovoltaic fields. One can measure, e.g.,
gain factor ⌫for two-beam coupling in the geometry of
Fig. 1, diffraction efficiency
of the recorded photorefrac-
tive grating (Fig. 3A), phase-conjugate reflectivity Rpc in
the backward four-wave mixing geometry of Fig. 3B, or
threshold coupling strength (⌫l)th for a coherent oscilla-
tion (Fig. 3C). The relations that specify the quantities
introduced are listed below.
The gain factor for coupling of two orthogonally polar-
ized light waves is12
⌫⫽
2
no2ne2r51

12
C
cos
共s
ˆe
ˆo兲2
冋
1⫺
2
Kx0
tanh
冉
Kx0
2
冊
册
. (8)
The only photovoltaic field entering this expression is
E12
C⫽

12
C/
.
In the actual case
兩
E15
L
兩
Ⰶ
兩
E12
C
兩
, which corresponds to a
nonlocal phototefractive response, the expression for the
diffraction efficiency is similar to that known for the dif-
fusion nonlinearity12,13:
⫽
m
1⫹m
关exp共⌫l/2兲⫺1兴2
关mexp共⌫l兲⫹1兴, (9)
where m⫽Ie(0)/Io(0 ) is the input beam ratio and lis the
crystal thickness.
Fig. 2. Calculated space-charge field versus spatial frequency
for polarization grating recording. Domain size x0is ⬁for the
solid curves, 50
m for dashed curves, 20
m for dashed–dotted
curves and 10
m for dashed–double-dotted curves. The crystal
birefringence no⫺neis zero for A and 0.1 for B.
Fig. 3. Schematic representation of measurements of A polar-
ization anisotropic diffraction efficiency, B backward-polarization
four-wave mixing, and C, the semilinear coherent oscillator with
two counterpropagating ordinary pump beams.
Barilov et al. Vol. 20, No. 8/ August 2003/J. Opt. Soc. Am. B 1651
When
兩
E15
L
兩
Ⰶ
兩
E12
C
兩
, the phase-conjugate reflectivity for
four-wave mixing with two counterpropagating ordinary
pump beams and an e-polarized signal beam (Fig. 3B) can
be expressed as14
Rpc ⫽
冉
⌫l
4⫺⌫l
冊
2
. (10)
Taking finally into account that the threshold condition
for the coherent oscillation in a semilinear cavity (Fig. 3C)
with reflectivity Rof a conventional mirror reads as
RRpc ⫽1, one can obtain for the threshold coupling
strength
共⌫l兲th ⫽
4
1⫹
冑
R
.(11)
Equations (8)–(11) allow photovoltaic field E12
Cto be cal-
culated from experimental values of
,Rpc ,or(⌫l)th or
directly from ⌫[see Eq. (8)]. To evaluate the component
of the photovoltaic tensor

12
C, one must measure specific
photoconductivity
independently.
3. RECORDING OF THREE-DIMENSIONAL
PHASE GRATINGS WITH ORTHOGONALLY
POLARIZED LIGHT WAVES
In our experiment a light beam from a frequency-doubled
diode-pumped Nd3⫹:YAG laser (single mode, single fre-
quency, ⫽0.53
m, 100-mW output power) was used for
anisotropic grating recording. A 1-mm-thick LiNbO3
sample, K243, synthesized at the Department of Physics,
Moscow State University, contained 0.74 wt. % of yttrium
and 0.06 wt. % of iron. The input–output optically fin-
ished sample faces and the zaxis were parallel to the do-
main walls, whereas the xaxis was normal to the domain
walls. The domain structure period was approximately 8
m.
Owing to the special features of the growth technique
used,9the sample is inhomogeneous: It contains two pe-
ripheral PPLN areas separated by a nearly 3-mm-wide
single-domain area. This structure allows for compari-
son of the results obtained with PPLN and single-domain
crystals of identical compositions and dimensions.
Two recording beams (obtained from an unexpanded
1-mm-waist laser beam) impinge symmetrically at an
angle ⫾
0upon the sample in the XY plane (Fig. 1). The
diffraction efficiency is measured at saturation, with only
one of two recording waves incident upon the sample and
the other one blocked. It is calculated as the ratio of dif-
fracted intensity component Idto total intensity It⫹Id
transmitted through the sample:
⫽
Id
Id⫹It
. (12)
Intensity gain factor ⌫
˜
is calculated from the standard
relation
⌫
˜
⫽
1
lln
冋
Ie共l兲
Ie共0兲
Io共0兲
Io共l兲
册
. (13)
For sufficiently small and large input beam ratios we in-
deed have ⌫
˜
⫽⌫.
Figure 4 shows the dependences
(m) (Fig. 4A) and
⌫
˜
(m) (Fig. 4B) on beam ratio for a grating recorded in the
PPLN area of the sample. Note the rather high values of
the diffraction efficiency (⬃60%) for the polarization re-
cording and also the difference of the optimum (for
)
pump ratio mfrom unity; the latter feature is typical of
media with nonlocal nonlinear response. The small-
Fig. 4. Dependence on pump ratio of A, diffraction efficiency and
B, the two-beam coupling gain factor.
Fig. 5. Measured diffraction efficiency versus spatial frequency
for polarization grating recording. Open squares and filled
circles, single-domain and PPLN areas, respectively. Solid and
dashed curves, theoretical dependences (see text).
1652 J. Opt. Soc. Am. B/ Vol. 20, No. 8/ August 2003 Barilov et al.
signal gain factor (⌫
˜
⬇⌫) for e-waves approaches 40
cm⫺1. It should be mentioned also that for beam ratio
m⭐20 we observe rather strong light-induced scatter-
ing, both parametric6and wide-angular. This competing
nonlinear effect strongly depletes the e-waves and re-
duces the measured values of ⌫.
The spatial-frequency dependences of
for the single-
domain and PPLN areas of the sample are illustrated in
Fig. 5. Equal pump intensities ( m⫽1) are used in this
experiment. The solid and dashed curves show the cor-
responding dependences calculated from Eq. (8) and ex-
pression (5) for 2x0⫽⬁and 2 x0⫽8
m, respectively.
The maximum attainable value of the space-charge field
is ⬃7.5 kV/cm for this fit.
4. COHERENT OPTICAL OSCILLATION
A photorefractive crystal pumped by two counterpropa-
gating waves can serve as a phase-conjugate mirror (see,
e.g., Ref. 15). With reflectivity Rpc larger than 100% (the
amplified phase-conjugate reflectivity), this mirror can
form (together with a conventional mirror) a cavity (see
Fig. 3C) in which coherent oscillation occurs.16
The experimental arrangement used for the study of
self-oscillation in this semilinear cavity15 is depicted in
Fig. 6. Note that this type of oscillator (with a phase-
conjugate mirror) is important because of its ability to
compensate for intracavity phase distortions.15,17
Coherent oscillation with bulk homogeneously poled
LiNbO3:Fe has been known for a long time.18 –20 Unfor-
tunately, the angular divergence of the oscillation wave
here is a few orders of magnitude larger than the diffrac-
tion limit (of the order of a few degrees19) because of
strong optical damage.
We have repeated these experiments, using PPLN:Y:Fe
instead of bulk iron-doped lithium niobate. The same
Nd3⫹:YAG laser is used to form two o-polarized pump
waves. The optical (and also polar) axis of the sample is
perpendicular to the pump plane; see Fig. 6. The spon-
taneously arising oscillation waves are epolarized. The
distance between the sample and mirror M’s is 5 cm.
With the diameter of the pumped area inside the sample
⬃1 mm, the Fresnel number of the cavity is ⬇20. The
angle between the cavity axis and the pump direction is
⬇20°; i.e., the fundamental grating spacing is ⌳⫽2
/K
⯝1.6
m.
Under the conditions described, our 1-mm-thick PPLN
sample exhibits well-developed coherent oscillation. It is
not possible, however, to achieve self-oscillation with
e-polarized pump beams because of the unfavorable
(o→e) direction of the nonlinear energy exchange be-
tween the crystal eigenmodes; a similar situation takes
place in the single-domain case.12
Figure 7 shows the far-field pattern of output radiation
for our coherent oscillator recorded with a CCD camera
(inset) and the intensity distribution in its cross section.
The angular divergence (FWHM) here is ⬃10⫺3rad, i.e.,
at least 1 order of magnitude smaller than that reported
earlier for single-domain crystals.19
As distinct from the coherent oscillators based on diffu-
sion photorefractive nonlinearity, the largest output in-
tensity in our case is obtained at equal pump intensities
(Fig. 8). The oscillation disappears abruptly for m
⭐0.1 and m⭓10; i.e., the oscillation threshold is highly
pronounced.
Fig. 6. Experimental arrangement for the study of coherent os-
cillation in PPLN: M’s, flat mirrors; /2’s, phase retarders; BS,
beam splitter.
Fig. 7. Far-field light pattern recorded with a CCD-camera (in-
set) and the corresponding intensity distribution for the oscilla-
tion wave. The length of the semilinear cavity is 5 cm; the cav-
ity axis makes an angle of 20° with the pump direction.
Fig. 8. Oscillation intensity versus beam ratio.
Barilov et al. Vol. 20, No. 8/ August 2003/J. Opt. Soc. Am. B 1653
5. DISCUSSION AND EVALUATION OF
PHOTOVOLTAIC PARAMETERS
Our experimental results show that PPLN:Y:Fe is a ma-
terial that is appropriate for photorefractive recording
with two orthogonally polarized crystal eigenwaves, an
ordinary wave and an extraordinary wave. Even with a
relatively thin sample, l⫽0.1 cm, diffraction efficiency
⬇60 % and coupling strength ⌫l⬇4 are achieved.
The direct estimate ⌫l⬇3.7 cm⫺1from the data of Fig.
4B fits well the values extracted from our measurements
of the diffraction efficiency. By setting
⫽0.57 and
m⫽12 in Eq. (9), we obtain easily ⌫l⫽4. One more in-
dependent estimate of the product ⌫lcan be obtained
from the position of the maximum of the dependence
(m). As follows from Eq. (9), the largest diffraction ef-
ficiency has to be reached at the beam ratio
mmax ⫽exp
冉
⫺
⌫l
2
冊
. (14)
Using the experimental value mmax ⫽12 (Fig. 4B), we
have ⌫l⫽5, which is not far from the direct estimate
made above.
With the gain factor known, we can evaluate the pho-
tovoltaic field. Using Eq. (8) and the values n⫽2.3,
r51 ⫽28 ⫻10⫺10 cm/V, and ⫽0.53
m, we have E12
C
⫽

12
C/
⬇7.5 kV/cm. This value is compatible with the
data known for the single-domain LiNbO3:Fe crystals.
We can conclude therefore that the presence of yttrium
does not much (if at all) influence the photovoltaic prop-
erties of iron-doped PPLN samples.
The photovoltaic parameters can also be evaluated
from the characteristics of the coherent oscillation. The
presence of this threshold phenomenon proves unambigu-
ously that our PPLN sample ensures the amplified phase-
conjugate reflectivity, Rpc ⬎100%. This result might be
surprising at first glance because the diffraction efficiency
of a grating recorded with the optimized beam ratio does
not exceed 60%. However, there is no contradiction here
because we are dealing with a rather specific (in-phase)
superposition of two photorefractive gratings, each re-
corded by two copropagating waves. The initial grating
recorded by the signal wave together with the copropagat-
ing pump wave is coherently enhanced by the grating re-
corded with the other pump wave and the generated
phase-conjugate wave. This results in the highest pos-
sible efficiency of phase conjugation by means of back-
ward four-wave mixing. Note that in classic
(3)media
these two gratings are mutually shifted by
/2 and in
crystals with gradient photorefractive nonlinearity they
are exactly out of phase (mutually shifted by
). Conse-
quently the net phase-conjugate reflectivity is smaller (at
the same coupling strength) for these nonlinearities than
for the nonlinearity caused by the circular photovoltaic ef-
fect. To attain values Rpc ⬎1 the product ⌫lshould be
at least larger than 2, as follows from Eq. (10), which in-
deed does not contradict our previous estimates.
The value of ⌫lcan also be extracted from the experi-
mental data on the threshold pump ratio. The relevant
threshold condition is known from the theory of polariza-
tion four-wave coupling.14 For the case under study
(
兩
E12
C
兩
Ⰷ
兩
E15
L
兩
) we have
Rpc ⫽
1⫺⌬2
关1⫺⌬tanh共⌫l⌬/4兲兴2, (15)
where ⌬⫽(m⫺1)/(m⫹1). With mth ⫽0.1 we arrive
at coupling strength ⌫l⬇3. This value is somewhat
smaller than that extracted from two-beam coupling ex-
periments, most probably because of cavity losses such as
from Fresnel reflections from the crystal face and crystal
absorption. This estimate in fact gives only the lowest
limit for the threshold coupling strength.
It should be noted that with ⌫lapproaching 4 the os-
cillator considered is still below the threshold of coherent
mirrorless oscillation predicted in Ref. 12 and experimen-
tally observed in bulk homogeneously poled crystals.21
6. CONCLUSIONS
Periodically poled lithium niobate codoped with iron and
yttrium possesses almost the same photorefractive sensi-
tivity to polarization grating recording as that of bulk ho-
mogeneously poled crystals. This means that even a
large amount of yttrium in the samples does not strongly
reduce the photorefractive nonlinearity related to the iron
centers.
Highly efficient frequency-degenerate wave mixing is
possible with PPLN:Y:Fe in classic backward-wave geom-
etry (phase conjugation, various photorefractive oscilla-
tors) as well as in forward-wave geometry (parametric
amplification of copropagating coherent seed waves).
By varying domain size x0it is possible to control the
cutoff spatial frequency4,5,22,23 below which the photore-
fractive sensitivity of periodically poled material de-
creases. This control can be accomplished with periodi-
cally poled lithium niobate, but the technique can be
especially effective for crystals with low birefringence,
such as periodically poled lithium tantalite.
ACKNOWLEDGMENTS
We are grateful to I. Naumova for furnishing PPLN
samples. Partial financial support from the U.S. Civilian
Research and Development Foundation in Ukraine (grant
UP2-2122) is gratefully acknowledged.
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