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Reduction of light-induced refractive-index
changes by decreased modulation
of light patterns in photorefractive crystals
U. van Stevendaal, K. Buse, H. Malz, H. Veenhuis, and E. Kra
¨tzig
Fachbereich Physik, Universita
¨t Osnabru
¨ck, D-49069 Osnabru
¨ck, Germany
Received February 5, 1998; revised manuscript received September 8, 1998
In holographic experiments many photorefractive crystals show refractive-index changes much smaller than
one would expect from the known electro-optic coefficients and space-charge fields. We show that degrada-
tion of the interference pattern is the origin of this effect. The degree of modulation of a light grating in
photorefractive crystals is measured by three methods and compared with the modulation of the grating of
the incident light. All methods, measurement of the amplitudes of fundamental and second-order gratings,
of the grating amplitudes in a crystal with and without another crystal in front of it, and of the drift currents
through an inhomogeneously illuminated sample, yield consistent results: In lithium niobate there is almost
no degradation of the light pattern. However, in our barium titanate and potassium tantalate–niobate
samples the degree of modulation is smaller, and the reduction factor is 0.55 and 0.60, respectively. Inhomo-
geneities of refractive-index changes are shown to be the origin of the effect. © 1998 Optical Society of
America [S0740-3224(98)01512-4]
OCIS codes: 100.2650, 160.2260, 160.5320, 210.4810.
1. INTRODUCTION
Photorefractive crystals are of special interest for many
advanced and promising applications. Lithium niobate
(LiNbO3) is suitable for volume holographic storage be-
cause of its large storage times and excellent optical
homogeneity.1–5Barium titanate (BaTiO3) and potas-
sium tantalate–niobate @KTa12xNbxO3(KTN)#are
rather interesting for, e.g., image amplification, phase
conjugation, and real-time optical information
processing.1,6
The photorefractive effect in electro-optic crystals has
been intensively investigated1,7,8; however, often the in-
dex changes are much smaller than expected. We con-
sider two coherent plane waves intersecting inside a pho-
torefractive crystal. The light intensity Iis given by
I5I0@11minput cos~Kz!#, (1)
where Kis the absolute value of grating vector Kand zis
the spatial coordinate along K. The degree of modulation
minput 52(I1
wI2
w)1/2/I0and the total intensity I05I1
w
1I2
wdepend on the intensities I1
wand I2
wof the two writ-
ing beams. The light excites charge carriers, which mi-
grate because of drift, diffusion, and the bulk photovoltaic
effect. A space-charge field Esc builds up, which produces
variations in the refractive indices by means of the
electro-optic effect.
Space-charge fields Esc along the zdirection (crystallo-
graphic caxes in, e.g., LiNbO3, BaTiO3, and KTN) yield
the refractive-index changes
Dno~e!52
~
1/2!no~e!
3r13~33!Esc , (2)
where no(e)are the refractive indices and r13(33)are the
linear electro-optic coefficients (contracted indices) for or-
dinary (o) and extraordinary (e) light polarization, respec-
tively.
The electro-optic coefficients of, e.g., BaTiO3and KTN
are known from interferometric measurements. Bulk
photovoltaic effects in BaTiO3and KTN make no signifi-
cant contribution to the space-charge field. Thus for a
fully modulated light pattern the space-charge field Esc
should be equal to (ED
21E0
2)1/2 if the transport lengths
are small and if space-charge limiting effects are
negligible.9Here ED5(kBT/e)Kis the diffusion field,
kBis the Boltzmann constant, Tis the absolute tempera-
ture, and eis the elementary charge. An externally ap-
plied electric field is denoted E0. Thus Esc can be calcu-
lated and, together with the electro-optic coefficients and
Eq. (2), yields a prediction for the refractive-index
changes. However, the Dnvalues obtained in holo-
graphic experiments are usually much smaller.10–14
Even if we consider that for holographic gratings the crys-
tal is partially clamped, a remarkable difference between
calculated and measured values remains. For example,
Loheide et al.10–12 measured refractive-index changes
that are approximately 2–5 times smaller than expected.
The main focus of the present investigation is to ex-
plore the origin of the reduced refractive-index changes.
It is reasonable to suppose that a reduction of the degree
of modulation of the light-interference pattern plays an
important role in these changes. Thus we develop meth-
ods that are useful in obtaining information about the de-
gree of modulation of interference patterns inside photo-
refractive crystals. We apply these methods to the
2868 J. Opt. Soc. Am. B/Vol. 15, No. 12/December 1998 van Stevendaal et al.
0740-3224/98/122868-09$15.00 © 1998 Optical Society of America
investigation of several materials. The results are pre-
sented and discussed with the aim to find out whether de-
grees of reduced modulation play a significant role in
these changes.
2. EXPERIMENTAL METHODS
A. Crystals
LiNbO3, BaTiO3, and KTN crystals were supplied by the
Crystal Growing Group of the Department of Physics,
University of Osnabru
¨ck. LiNbO3crystals of the congru-
ently melting composition were grown by the Czochralski
technique; BaTiO3and KTa0.52Nb0.48O3, by the top-seeded
solution growth method. Table 1 lists the dimensions of
the samples used and the kinds and concentrations of
dopants. In the experiments the light propagates
through the crystal along the aaxis. The dimension of
the sample in the c-axis direction is denoted c. In Table
1 the photoconductivity nonlinearity parameters xare
also shown for the samples investigated. We obtain
these parameters from intensity-dependent measure-
ments of the photoconductivity
s
and fits of the phenom-
enological relation
s
}Ixto the experimental data,15 with
an exponent xin the range 0.5 <x<1.0.
Special attention is paid to the preparation of BaTiO3.
After the samples are cut, the surfaces are polished to op-
tical quality. Heating the crystals close to the
paraelectric–ferroelectric phase transition and cooling
them to room temperature with an applied electric field
yield single-domain, completely poled crystals. To en-
sure that the samples are single domain, electro-optic co-
efficients are measured interferometrically, and the re-
sults correspond to literature values.16 Furthermore, we
also proved with piezoelectric measurements of some
samples that the crystals are completely poled. BaTiO3
samples 2–4 are cut from a single crystal. After an ex-
perimental characterization of BaTiO3sample 5, the sur-
faces perpendicular to the aaxis are antireflection coated
with SiO2for light of 515-nm wavelength. The coated
sample is denoted 5a.
B. Experimental Setup
Holographic gratings are recorded with a conventional
two-beam interference setup. The beam of an Ar-ion la-
ser of 515-nm wavelength is spatially filtered, expanded
by approximately a factor of 5 in diameter, and divided
into two beams that we use to write the hologram. To
prevent holographic scattering, the writing beams are or-
dinarily polarized. With the help of neutral-density fil-
ters, one beam can be attenuated to vary the degree of
modulation minput of the light-intensity pattern from 0 to
1. The total writing intensity I0is kept constant at ;1.6
kW m22, and grating vector Kwith K51.73
m
m21is
aligned parallel to the caxis of the crystal (symmetrical
incidence). To read the hologram we use the 633-nm line
ofaHe–Ne laser. The beam is also spatially filtered and
expanded, and the light polarization is chosen to be either
ordinary or extraordinary. To avoid erasure effects
caused by the read-out light we keep the intensity of the
reading beam small (I548Wm
22
). The crystal sur-
faces perpendicular to the caxis are contacted by silver-
paste electrodes. These electrodes are contacted by thin
wires by drops of silver paste. Therefore the crystal can
be short-circuited, or an external electric field can be ap-
plied. Details of direct conductivity measurements of ox-
idic crystals are described in Ref. 17. By measuring the
intensities of diffracted and transmitted beams, Idiff and
Itrans , we get the diffraction efficiency
h
5Idiff /(Idiff
1Itrans). Diffraction efficiency values are monitored
continuously, and the light-induced refractive-index
changes Dnare calculated from Kogelnik’s formula.18
C. Amplitudes of Kand 2KGratings
The light pattern is of sinusoidal shape, and a refractive-
index grating of the same spatial frequency Kis obtained.
However, gratings of higher harmonics can also be gener-
ated. The Kand 2 Kgratings are read subsequently by
the He–Ne laser light, which enters the crystal at Bragg
angles QKand Q2K, respectively. These angles are de-
termined by the relations K54
p
sin QK/land 2K
54
p
sin Q2K/l, where lis the wavelength of the He–Ne
Table 1. Notation, Dimensions, Doping Concentrations in the Melt,
Absorption Coefficients
a
o,515 nm for Ordinarily Polarized Light of Wavelength 515 nm,
and Photoconductivity Nonlinearity Parameters xof the Samples Investigateda
Sample Number Notation a3b3c(mm 3mm 3mm) Dopingb
a
o,515 nm (m21)x
LiNbO31 63515 1.9 34.0 32.5 930-ppm Cu 1317 1.0
LiNbO32 T02-7 0.8 36.7 38.9 370-ppm Fe 306 1.0
LiNbO33 T02-69 0.4 34.1 34.7 370-ppm Fe 391 1.0
LiNbO34 T02-76 0.4 34.1 34.8 370-ppm Fe 385 1.0
BaTiO31 Gr87 2.8 34.8 35.1 50-ppm Fe 12 0.74
BaTiO32 Gr94.2.1 2.1 32.4 34.3 100-ppm Fe 24 0.62
BaTiO33 Gr94.2.2 0.8 32.4 34.3 100-ppm Fe 24 0.62
BaTiO34 Gr94.2.3 0.4 32.4 34.3 100-ppm Fe 24 0.62
BaTiO35 Gr94.3 2.4 33.9 33.1 100-ppm Fe 23 0.62
BaTiO36 Gr97.1 3.5 34.5 34.5 150-ppm Fe 26 0.72
KTN 1 G69.2.2 2.1 34.1 33.7 5000-ppm Fe 71 1.0
KTN 2 G70.1.1.1.1 0.9 32.9 32.9 10,000-ppm Fe 202 1.0
aThe first dimension, a, is the thickness of the crystals along the direction of light propagation, and the third dimension, c, denotes the length along the
crystallographic caxis.
bppm, parts in 106.
van Stevendaal et al. Vol. 15, No. 12/December 1998/J. Opt. Soc. Am. B 2869
laser light. The samples are always short-circuited, the
degree of modulation of the interference pattern is varied
in the range 0.3 <minput <1, and the steady-state dif-
fraction efficiencies are measured. Figure 1 is a sche-
matic drawing of the arrangement, and Fig. 2 shows a
typical Dn(t) curve for a hologram-recording process.
To compare measured and expected values we have to
derive expressions for the space-charge field. To deter-
mine whether holes or electrons are the dominant charge
carriers we measure refractive-index changes Dn, beam-
coupling coefficients G, and photoconductivity
s
ph as func-
tions of external electric fields, spatial frequencies, and
light intensities. Taking into account a one-center
charge-transport model with electron–hole competition
and fitting the corresponding equations to all experimen-
tal data, we get expressions for electron and hole contri-
butions to the photoconductivity. Loheide et al.10–12
showed with this technique that hole conductivity makes
the major contribution (more than 90%) to the overall con-
ductivity in KTN crystals. We carefully perform a simi-
lar investigation with BaTiO3samples used in this re-
search. The results show that the electron conductivity
is also negligible in these samples. Thus the following
theoretical considerations are made in terms of a model
with monopolar hole conductivity. Analogous results are
obtained for monopolar electron conduction.
The current density jis given by
j5e
m
hE 2eD
]
h/
]
z, (3)
where eis the elementary charge,
m
and hare the mobil-
ity and the concentration of holes in the valence band, re-
spectively, Eis the electric field, and Dis the diffusion
constant for holes in the valence band. Depending on the
material used, it might be necessary also to consider a
bulk photovoltaic current. All spatially dependent quan-
tities (j, h, E) are Fourier developed in Kup to second or-
der. For j, e.g., we write
j5j~0!1~1/2!@j~1!exp~iKz!1j~1!
*exp~2iKz!#
1~1/2!@j~2!exp~i2Kz!1j~2!
*exp~2i2Kz!#. (4)
By the subscripts (0), (1), and (2) we denote zeroth, first,
and second Fourier components, respectively, and by a su-
perscript asterisk the complex-conjugated term.
The conductivity is given by
s
5e
m
h, and we allow for
sublinear photoconductivity according to the phenomeno-
logical relation
s
}Ixbetween conductivity
s
and light
intensity I, with an exponent xin the range 0.5 <x
<1.0.15 The degree of modulation minput of the incom-
ing interference pattern can be reduced within the crystal
to mcrystal because of reflection, scattering, etc. From a
Fourier development of
$
I0@11mcrystal cos(Kz)
#
%
xwe ob-
tain the coefficients
s
(0),
s
(1), and
s
(2). These coeffi-
cients are real because the light intensity is an even func-
tion of z. The continuity equation
]
j/
]
z1
r
˙50, with
r
˙50 in the steady state, leads to j(0)5constant and
j(1)5j(1)
*5j(2)5j(2)
*50. Together with Eq. (3) these
equations yield the first Fourier component of the space-
charge field:
E~1!5i
a
ED
F
11
1
2
b
22
b
~
a
/2!2212
b
G
3
F
12
S
a
2
D
2
1
1
4
b
2
~
a
/2!221
G
21
, (5)
with
a
5
s
(1)/
s
(0)and
b
5
s
(2)/
s
(0). Here EDis the
diffusion field. For the second order of the space-charge
field we obtain
Fig. 1. Schematic drawing of the arrangement to measure
refractive-index changes of first- and second-order gratings Dno
K
and Dno
2K. Intensities and wavelength of the writing beams are
I1
wand I2
wand l5515 nm. Gratings with spatial frequencies K
and 2Kare read under the angles QKand Q2K, respectively,
with laser light of IK
r5I2K
r548Wm
22and l5633 nm. The
subscripts trans and diff denote the intensities of transmitted
and diffracted beams, respectively.
Fig. 2. Refractive-index change Dne
Kof the Kgrating as a func-
tion of recording time tfor BaTiO3sample 1. After 100 s the
writing beams are switched off and the hologram is erased. (Re-
cording intensity, 1.6 kW m22; wavelength, 515 nm; degree of
modulation of the light pattern, minput 50.92; reading intensity,
48Wm
22
; wavelength, l5633 nm.)
2870 J. Opt. Soc. Am. B/Vol. 15, No. 12/December 1998 van Stevendaal et al.
E~2!52
a
2
E
~
1
!
H
124
b
a
2
F
12
S
a
2
D
2
1
1
4
b
2
~
a
/2!221
G
3
F
11
1
2
b
22
b
~
a
/2!2212
b
G
21
J
. (6)
For x51, i.e., a linear photoconductivity, the second
Fourier order
s
(2)is zero, which leads to
b
50 and
a
5
s
(1)/
s
(0)5mcrystal . In this case Eqs. (5) and (6) be-
come
E~1!5imcrystal ED
F
12
S
mcrystal
2
D
2
G
21
,
E~2!52
m
crystal
2E~1!. (7)
The refractive-index amplitudes of the Kgrating, DnK,
and of the 2Kgrating, Dn2K, are then given by
u
Dno~e!
K
u
5
U
1
2no~e!
3r13~33!E~1!
U
, (8)
u
Dno~e!
2K
u
5
U
1
2no~e!
3r13~33!E~2!
U
. (9)
From Eqs. (7)–(9) it is obvious that the ratio between the
amplitudes
u
Dno(e)
2K/Dno(e)
K
u
yields mcrystal/2. This is a use-
ful way to measure mcrystal and to permit comparison with
minput .
In the case of LiNbO3the same calculations can be
done, e.g., by assuming dominant electron conduction and
negligible diffusion processes in comparison with photo-
voltaic effects. Then the first Fourier component of the
space-charge field [Eq. (7)] is real and proportional to the
photovoltaic field Ephv .
D. Grating Amplitudes in a Crystal without and with
Another Crystal in Front of It
Another method to obtain information on whether a crys-
tal decreases the degree of modulation of the pattern is
based on the use of a LiNbO3crystal as a reference
sample. Again we short-circuit the crystal and write a
hologram with extraordinarily polarized beams and a
grating vector Kparallel to the crystallographic caxis un-
til the steady-state situation is reached. The degree of
modulation of the light-intensity pattern is minput
'0.63. By reading the hologram with ordinarily polar-
ized light of the He–Ne laser we obtain the maximum
value of the refractive-index change Dno
ref . After erasing
the hologram we place another crystal in front of the ref-
erence crystal, with the caxis perpendicular to the grat-
ing vector (see Fig. 3). Writing a hologram again yields
the value Dno
cry . This configuration avoids, e.g., beam-
coupling and diffraction effects from the first crystal.
Only diffraction from the reference crystal has to be taken
into account because the relevant elements of the linear
electro-optic coefficients of the sample in front of the ref-
erence crystal are zero in this writing configuration.
However, the additional sample may degrade the degree
of modulation of the light pattern in the reference crystal,
and this degradation will reduce Dno. The refractive-
index change Dnois proportional to the space-charge field
Esc and to mcrystal for small modulation degrees. There-
fore the ratio Dno
cry/Dno
ref corresponds directly to the re-
duction of the degree of modulation of the light-intensity
pattern that is due to the sample that has been placed in
front of the reference crystal.
E. Drift Currents through Inhomogeneously
Illuminated Samples
We use another novel approach to determine the effective
degree of modulation of the light pattern in the crystals.
We illuminate the crystal with the intensity pattern given
Fig. 3. Schematic drawing of the arrangement to measure
refractive-index changes in reference crystal LiNbO31 without
and with another sample placed in front of it. The writing
beams, with intensities I1
wand I2
w, are extraordinarily polarized
with respect to the reference crystal, and the wavelength is l
5515 nm. The reading beam has an intensity Ir548Wm
22
at l5633 nm and is ordinarily polarized. Grating vector Kof
the hologram is parallel to the crystallographic caxis for the ref-
erence crystal and perpendicular to cfor any other test crystal
placed in front of LiNbO3sample 1.
Fig. 4. Schematic drawing of the arrangement to measure the
external current density jext . The wavelength of the illuminat-
ing beams with intensities I1
wand I2
wis l5515 nm. With the
help of a voltage supply (V) , an external electric field is applied
to the crystal and the current is measured by an electrometer (A).
van Stevendaal et al. Vol. 15, No. 12/December 1998/J. Opt. Soc. Am. B 2871
by Eq. (1) and vary the degree of modulation minput from 0
to 1. We apply an external voltage Uext from 0.5 to 1 kV
to the crystal surfaces that are perpendicular to the caxis
and measure directly the resultant external current iext
with the help of a Keithley 617 electrometer (see Fig. 4).
Bulk photovoltaic currents, which are measured in the
short-circuited situation, are subtracted.
The idea behind use of this technique is that the cur-
rent should approach zero if the crystal is illuminated
with a sinusoidal light pattern that is fully modulated in
the crystal; i.e., mcrystal 51. Any degradation of the de-
gree of modulation yields light in the nominally dark (in-
sulating) regions and causes an increase of the current.
To treat this result quantitatively, we consider the cur-
rent density given by jext(z)5
s
(z)E(z). Because of the
continuity equation, the current density is spatially con-
stant in the steady state @jext(z)5jext#. The generated
field E(z) is modulated, because the conductivity is modu-
lated according to the light pattern. The relation be-
tween applied voltage and modulated field E(z) is given
by U5
*
0
cE(z)dz, where cdenotes the crystal length in
the direction of the applied electric field. Taking into ac-
count the relation between conductivity and light inten-
sity,
s
(z)}
$
I0@11mcrystal cos( Kz)#
%
x, we obtain an ex-
pression for the external current density that depends on
the degree of modulation mcrystal :
jext }
H
E
0
c
@11mcrystal cos~Kz!#2xdz
J
21
. (10)
In general, we have to solve the integral equation numeri-
cally. However, for a linear relation between conductiv-
ity and light intensity (x51 ) , expression (10) simplifies
to jext }(1 2mcrystal
2)1/2.
F. Imaging of the Interference Pattern
The use of a CCD detector is appropriate for getting fur-
ther information about the light-interference pattern.
We illuminate the crystal with the intensity pattern given
by Eq. (1). The writing beams are again ordinarily polar-
ized, and the degree of modulation is minput '1. The
spatial frequency is K50.9
m
m21. We observe the
light-interference pattern in the steady state at the rear
crystal surface with the help of the CCD detector.
3. EXPERIMENTAL RESULTS
A. Amplitudes of Kand 2KGratings
Figure 5(a) shows the measured amplitudes of the Kand
2Kgratings in LiNbO3sample 1. The values of Dno
Kand
Dno
2Kdepend on the degree of modulation minput . For
small degrees of modulation (minput ,0.7) the refractive-
index changes of the Kgrating increase approximately
linearly with minput , but the situation for minput .0.7 is
quite different: In this case the increase is much greater.
In the whole range of the minput values the refractive-
index changes of the 2Kgrating are always smaller than
those of the Kgrating, as expected from Eq. (7). For
small degrees of modulation, Dno
2Kincreases quadrati-
cally with minput , whereas for minput .0.7 the increase is
much more pronounced. The dependences of Dno
Kand
Dno
2Kon the degree of modulation minput are well de-
scribed by Eqs. (7)–(9) with mcrystal 5minput if the (phase-
shifted) diffusion field iEDis replaced by Ephv , where we
consider the photovoltaic field as the main charge-driving
force in LiNbO3. For the calculation of the photovoltaic
field Ephv 5
b
31 /(
s
ph /I) the photovoltaic coefficient
b
31
51.7 31029V21and the specific photoconductivity
s
ph /I51.9 310215 (Vm)21are used. These param-
eters are measured directly by photoelectrical techniques.
Furthermore, we need the electro-optic coefficient r13
512 pm V21and the refractive index no(633 nm)
52.29.19
Similar measurements are performed with BaTiO3
sample 1, and the results are shown in Fig. 5(b). Again,
refractive-index changes for the 2Kgrating are smaller
than for the Kgrating. But, in the whole range investi-
gated, the dependence of Dne
Kon minput is nearly linear
and the dependence of Dne
2Kon minput is quadratic. If we
plot the refractive-index changes calculated from Eqs. (5),
(6), (8), and (9), a remarkable difference between these
values (dashed curves) and the measured data becomes
obvious. For this calculation we use the photoconductiv-
ity nonlinearity coefficient x50.74 presented in Table 1,
the effective electro-optic coefficient r33
eff 580 pm V21for
BaTiO3in holographic experiments,20 the refractive index
ne(633 nm) 52.36,21 and the diffusion field ED
5KkBT/e(K51.73
m
m21,T5295 K). The Fourier
components
s
(0),
s
(1), and
s
(2)of the photoconductivity
Fig. 5. Refractive-index changes Dno
Kand Dno
2Kof the Kand 2K
gratings (ordinary light polarization) versus degree of modula-
tion minput of the incident light pattern (a) for LiNbO3sample 1
and (b) for BaTiO3sample 1 (Dne
Kand Dne
2Kextraordinary light
polarization). The symbols are measured results, and the
curves are fits as described in Subsections 2.C and 3.A. (Dashed
curves, reduction factor R151; solid curves, reduction factor
R150.51.)
2872 J. Opt. Soc. Am. B/Vol. 15, No. 12/December 1998 van Stevendaal et al.
s
ph are deduced from the Fourier development of
s
ph
}
$
I0@11minput cos(Kz)#
%
0.74 for each value of minput .
A better description of the experimental data is obtained
if we introduce a reduction factor R1and if we use instead
of minput the scaled term R1minput 5mcrystal. Then a fit
of Eqs. (5) and (6) to the experimental data with only one
parameter R1can describe both curves simultaneously.
For BaTiO3sample 1 the reduction factor is R150.51
60.05.
In Table 2 the obtained reduction factors are given
for all samples investigated. The LiNbO3samples show
values of R1close to 1, but the coefficient R1is signifi-
cantly smaller for the other crystals. Within the measur-
ing accuracy the reduction factors seem to be equal for all
uncoated BaTiO3samples used. It should be emphasized
that R1for the coated sample is appreciably higher (R1
'0.63) than for the uncoated one ( R1'0.55) . The
KTN samples show values in the range 0.5–0.6.
B. Grating Amplitudes in a Crystal without and with
Another Crystal in Front of It
Placing another crystal in front of the reference crystal
reduces the saturation values of the refractive-index grat-
ing amplitudes from Dno
ref to Dno
cry . The reference crys-
tal is LiNbO3sample 3, and in Table 3 the reduction fac-
tors R25Dno
cry/Dno
ref are listed that were obtained for
different crystals through which the recording beams had
to pass.
The quotient R25Dno
cry/Dno
ref is 1.0 if the light passes
through a LiNbO3crystal, but the BaTiO3samples reduce
the saturation value of refractive-index changes in the
reference crystal; the reduction factor is 0.6 60.1.
There is no significant dependence on crystal thickness;
R2is equal within the measuring accuracy for BaTiO3
samples 2, 3, and 4, which were cut from the same crystal.
C. Drift Currents through Inhomogeneously
Illuminated Samples
In Fig. 6 a typical dependence of the external current
density jext on the degree of modulation minput is shown
for LiNbO3sample 1. Because of the linearity between
conductivity and light intensity in LiNbO3the depen-
dence of jext on mcrystal is well described by the relation
jext }(1 2mcrystal
2)1/2. The current density has reached
its maximum value for mcrystal 50 and goes to zero for
mcrystal 51. We again take into consideration a reduc-
tion factor R35mcrystal /minput . Then a fit of relation
(10) to the experimental results yields a value for the re-
duction of the degree of modulation: mcrystal ,minput
<1 yields R3,1, and thus the current does not reach 0
even for minput 51. In Table 4 the reduction factors R3
obtained by this method for several samples are listed.
Whereas R3is almost 1 for the LiNbO3sample, only up-
per limits can be given for the other crystals. The reason
for this phenomenon is that in these samples the current
density shows no pronounced dependence on the degree of
Fig. 6. Current density jext as a function of degree of modulation
minput for LiNbO3sample 1. The symbols are measured results,
and the solid curve is a fit as described in Subsections 2.E and
3.C.
Table 2. Scaling Factor R1of the Degree of Modu-
lation of the Light Pattern Deduced from the Ra-
tio between the Amplitudes of Kand 2KGratingsa
Sample Number R1
LiNbO31 1.1 60.1
LiNbO32 0.85 60.1
BaTiO31 0.51 60.05
BaTiO35 0.55 60.06
BaTiO35a 0.63 60.06
BaTiO36 0.42 60.04
KTN 1 0.61 60.06
KTN 2 0.47 60.05
aFor details, see Subsections 2.C and 3.A of the text. The error ranges
are given by the reproducibility. Sample BaTiO35a is the same as
sample 5 but with antireflection-coated surfaces perpendicular to the a
axis.
Table 3. Scaling Factor R2of the Degree of Modu-
lation of the Light Pattern Deduced from Reduc-
tion of the Grating Amplitude in LiNbO3Sample 3
by Placement of Another Sample in Front of Ita
Sample Number R2
LiNbO33 1.0 60.2
LiNbO311LiNbO33 1.0 60.2
LiNbO321LiNbO33 0.9 60.2
LiNbO341LiNbO33 1.0 60.2
BaTiO311LiNbO33 0.55 60.1
BaTiO321LiNbO33 0.65 60.1
BaTiO331LiNbO33 0.6 60.1
BaTiO341LiNbO33 0.6 60.1
aFor details see Subsections 2.D and 3.B of the text. The error ranges
are given by the reproducibility.
Table 4. Scaling Factor R3of the Modulation De-
gree of the Light Pattern Deduced from Measure-
ments of the Drift Current through Crystalsa
Sample Number R3
LiNbO31 0.95
BaTiO31,0.7
BaTiO35,0.7
BaTiO36,0.6
aFor details see Subsections 2.E and 3.C of the text.
van Stevendaal et al. Vol. 15, No. 12/December 1998/J. Opt. Soc. Am. B 2873
light modulation. Nevertheless, we observed a tendency
for the values of mcrystal to be reduced in BaTiO3samples.
D. Imaging of the Interference Pattern
The image taken from the rear side of LiNbO3crystal 3 is
shown in Fig. 7(a). It can be seen that the light-intensity
pattern is fully and nearly homogeneously modulated.
In the image both the dark and the bright fringes pass
unbroken from the top to the bottom of the picture.
In Fig. 7(b) a typical pattern for BaTiO3is shown for
sample 4. Unlike for the LiNbO3sample the light-
intensity pattern is not homogeneously modulated. The
dark fringes are interrupted by bright regions and vice
versa. There also exist bright regions that are much
brighter than others. This effect can be observed in all
BaTiO3and KTN samples.
4. DISCUSSION
A. Amplitudes of Kand 2KGratings
In many theoretical analyses of photorefractive effects,
Fourier development only up to first order is used. How-
ever, our experiments show impressively that a pro-
nounced 2Kgrating is present even for minput 50.5 [Fig.
5(a)]. Thus development up to the first Fourier order is
appropriate only for minput ,0.5. In crystals for which
the degree of modulation of the light pattern in the crys-
tal mcrystal is significantly reduced compared with minput ,
the linear approximation might also be valid for minput
.0.5. In any case, it is important to be aware of 2K
gratings and higher harmonics, which can, e.g., cause un-
desired cross-talk effects in holographic storage devices in
which many holograms of different Kvalues are multi-
plexed.
The dependence of Dno
Kon minput is well described by
Eqs. (7) and (8) with mcrystal 5minput for LiNbO3sample 1
[Fig. 5(a)]. Both the linear increase of the Kgrating for
minput ,0.7 and the more pronounced increase for
minput .0.7 agree well with the theoretically predicted
dependence. The nonlinearity for minput close to 1 is in-
deed strong: For minput 50.95, e.g., the refractive-index
changes are approximately a factor of 1.3 larger than one
would expect from a linear extrapolation of the values for
small minput . The quadratic increase of the amplitude
Dno
2Kof the 2Kgrating for minput ,0.7 and the stronger
increase for minput .0.7 are excellently explained by our
Eqs. (7) and (9). For minput 50.95 there is also a factor
of 1.3 between the quadratically extrapolated and the cor-
rectly calculated and measured values.
From the experiments it turns out that measurement
of the amplitudes of the Kand the 2Krefractive-index
gratings is useful in yielding information about the de-
gree of modulation of the light pattern inside the crystal.
With Eqs. (7)–(9) and the measured Dno
Kand Dno
2Kwe
calculate mcrystal . As we have already mentioned, there
is good agreement between mcrystal and minput for LiNbO3
sample 1. For LiNbO3crystal 2 the mcrystal value is
slightly reduced: R150.85 60.09.
However, in other materials the measured R1values
are much smaller (Table 2), ;0.55 for BaTiO3and ;0.60
for KTN. Note that the influence of the sublinear photo-
conductivity has already been considered. In BaTiO3
sample 1, e.g., we have
s
}I0.74, which yields a reduction
of E(1)by a factor of 0.77 for minput 50.93. Before dis-
cussing the origin of the small R1values observed in
BaTiO3and KTN, we treat the results of the other experi-
mental methods.
B. Grating Amplitudes in a Crystal with and without
Another Crystal in Front of It
As we have already emphasized, in the configuration used
(Fig. 3) there is no diffraction from a grating in the first
crystal. Thus a decrease of diffraction efficiency because
a sample has been placed in front of the reference crystal
can result only from a distortion of the interference pat-
tern. From Table 3 it becomes obvious that LiNbO3does
not distort the light pattern appreciably; only for sample
2 a degradation (R250.9) is measured.
However, BaTiO3reduces the refractive-index changes,
the space-charge field, and the degree of effective light-
pattern modulation in the crystal behind the BaTiO3
sample; the factor is R2'0.6. Although the thicknesses
of BaTiO3samples 2–4 differ considerably, the values R2
are equal within the measuring accuracy. The degrada-
tion of the light pattern is independent of the thickness of
the material.
C. Drift Currents through Inhomogeneously
Illuminated Samples
In LiNbO3we get a strong decrease of the drift currents if
the crystal is illuminated with a fully modulated light
pattern. The dark regions are insulating, which explains
the high resistivity. The relation jext }(1 2mcrystal
2)1/2
Fig. 7. Images of light-interference patterns recorded with a
CCD detector from the rear sides (a) of LiNbO3sample 3 and (b)
of BaTiO3sample 4. The wavelength of the illuminating beams
with the total writing intensity I051.6 kW m22is l
5515 nm, and the degree of modulation of the light pattern is
minput 51. The spatial frequency is K50.9
m
m21.
2874 J. Opt. Soc. Am. B/Vol. 15, No. 12/December 1998 van Stevendaal et al.
makes this method precise for measurements with
mcrystal .0.8. In LiNbO3sample 1 we get R3'0.95,
which shows that there is almost no degradation of the
light pattern in LiNbO3.
The dependences of jext on minput for the BaTiO3crys-
tals are remarkably different. The degrees of modulation
mcrystal in these materials are so small that it is hard to
detect a decrease of jext at all, even for minput 51. Thus,
in these crystals, only the upper limits of mcrystal and R3
5mcrystal /minput are obtained.
D. Imaging of the Interference Pattern
As we can see from Fig. 7, there is a significant difference
between the light-interference patterns in the LiNbO3
and the BaTiO3crystals. The pattern at the rear side of
the BaTiO3sample is distorted. But there is no remark-
able increase in the homogeneous background light.
Thus light scattering is not reducing the degree of modu-
lation of the light pattern. Instead, it is highly probable
that inside the sample refractive-index inhomogeneities
are present that cause the interrupted dark and bright
fringes. We consider two light beams, which interfere
constructively in a determined region inside the crystal.
If one beam passes through refractive-index inhomogene-
ities, the optical path length changes. By an increase of
half of the wavelength, e.g., the two beams interfere de-
structively. This might be the reason that we observe
dark regions in areas that are supposed to be bright. In
the same way bright regions in dark areas can be ex-
plained.
Another consequence of refractive-index inhomogene-
ities could be that a beam is refracted in dark regions and
does not interfere destructively with the other beams. It
is also possible that beams are refracted in bright regions,
which makes these areas much brighter than others.
This might be an explanation for the bright spots ob-
served in Fig. 7(b).
E. Comparison of the Results
A comparison of the results shown in Tables 2–4 reveals
that the reduction factors of the degree of modulation of
the interference pattern that are determined by three dif-
ferent and independent methods agree quite well, which
underlines our conclusion that the explanation of the ex-
perimental results is correct. Furthermore, it is suffi-
cient to use just one reduction factor, R'R1'R2
'R3.
The measurements indicate that the LiNbO3crystals
show all the expected refractive-index changes (R'1).
At first glance sample 2 is an exception. However, this
crystal has a much smaller absorption than LiNbO3crys-
tal 1 (Table 1). Most probably the degree of modulation
is reduced in sample 2 by light reflected from the crystal
surfaces. Considering just this effect, we expect mcrystal
52(I1
wI2
w)1/2/(I01Iinc), where Iinc is the intensity of in-
coherent background illumination. When we take into
account multiple internal reflections, Iinc is given by Iinc
5I0Qexp(2
a
od)/@12Qexp(2
a
od)#, with Q5(no
21)2/(no11)2. Here
a
ois the absorption coefficient
and nois the refractive index for ordinarily polarized light
of 515-nm wavelength. With no52.33 (Ref. 22) and
a
o
5306 m21we get for equal I1
wand I2
wthe result that
mcrystal /minput 50.9, which agrees well with the mea-
sured Rfor sample 2. Thus antireflection coating of this
sample could provide R'1, too. A crucial advantage of
LiNbO3emerges: The crystal quality of LiNbO3is so
high that there is no degradation of the interference pat-
tern and that, when light reflections inside the crystal are
taken into account, all the refractive-index changes are
obtained.
The Rvalues measured for BaTiO3and KTN are so
small that they cannot be explained by light reflection
only. Space-charge limiting effects are carefully ruled
out by measurement of the saturation values of the
refractive-index gratings for different spatial frequencies
and externally applied electric fields. Even anti-
reflection-coated BaTiO3still shows R'0.6. Because R
depends only slightly on the thickness of the samples, ho-
lographic scattering cannot be the only reason for this ef-
fect. Instead, a distortion of the light-interference pat-
tern inside the crystal because of refractive-index
inhomogeneities (Subsection 4.D) can cause a degradation
of the degree of modulation. Large refractive-index inho-
mogeneities can arise from crystal imperfections such as
microcracks, microcrystals of different structure, and
changes in composition. Refractive-index changes of Dn
50.1 are sufficient to move a fringe of the interference
pattern by half a period, even if the size of the defect is
less than 1
m
m. The causes of the refractive-index inho-
mogeneities are still unknown.
5. SUMMARY AND CONCLUSIONS
We have developed and used the following three different
methods to measure the degree of modulation of a light
pattern in crystals: (1) The amplitudes of fundamental
and second-order gratings are measured. The ratio
yields information about the degree of modulation of the
pattern in the crystal. These experiments also reveal
that strong second-order gratings are present if the modu-
lation degree of the light pattern exceeds 0.5. (2) Grating
amplitudes in a crystal with and without another crystal
in front of it are determined. The light passes through
the first crystal. However, just this passing can cause a
significant reduction in the degree of modulation of the in-
terference pattern behind the first sample. (3) Drift cur-
rents through an inhomogeneously illuminated sample
are measured. Background illumination caused by, e.g.,
scattered light makes it impossible to establish dark and
insulating regions, even if the input light pattern is fully
modulated.
All these methods yield consistent results: In LiNbO3
there is almost no degradation of the light pattern. How-
ever, in our BaTiO3and KTN samples the modulation de-
gree is smaller and the reduction factors are 0.55 and
0.60, respectively.
ACKNOWLEDGMENTS
We thank S. Odoulov and J. Neumann for valuable dis-
cussions. The financial support of the Deutsche For-
schungsgemeinschaft (SFB 225, C5) is gratefully ac-
knowledged.
van Stevendaal et al. Vol. 15, No. 12/December 1998/J. Opt. Soc. Am. B 2875
U. van Stevendaal’s e-mail address is uvst@physik.uni-
osnabrueck.de.
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