Stabilized photorefractive running
holograms, with arbitrarily selected

phase shift, for material characterization

Ivan de Oliveira,1 Agnaldo A. Freschi,2 Igor Fier,3

and Jaime Frejlich4,∗

1Faculdade de Tecnologia/ UNICAMP, Limeira-SP, Brazil
2Universidade Federal do ABC/UFABC, Santo André-SP, Brazil

3Dept. de F́ısica, Instituto de Geociências e Cîencias Exatas/ UNESP, Rio Claro-SP, Brazil
4Instituto de F́ısica/ UNICAMP, Campinas-SP, Brazil

∗frejlich@ifi.unicamp.br

Abstract: We report on the recording of stabilized running holograms
in photorefractive materials with arbitrarily selected phase shiftϕ between
the transmitted and difracted beams propagating along the same direction
behind the photorefractive crystal. The dependence of the diffraction
efficiency and of the hologram speed onϕ, in such stabilized holograms,
can be easily measured and used for material characterization. In this
communication we applied for the first time this technique for studying
and characterizing hole-electron competition in a nominally undoped
titanosillenite crystal sample.

© 2012 Optical Society of America

OCIS codes: (160.5320) Photorefractive materials; (050.7330) Volume gratings.

References and links
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13. I. de Oliveira and J. Frejlich, “Gain and stability in photorefractive two-wave mixing,” Phys. Rev. A64, 033806
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1. Introduction

The generation of running holograms in photorefractive materials were thoroughly studied
since long, for analysing the recording process itself [1–6] and for material characteriza-
tion [6–9].

The use of self-stabilization techniques for holographic recording [10, 11] in conditions
where a nonstationary hologram arises was shown [12] to produce so called fringe-locked run-
ning holograms that were later used for analysing the recording process [13] and for material
and hole-electron competition [14] characterization in photorefractives [15, 16]. The effect of
bulk light absorption on the dielectric relaxation and its effect on running hologram record-
ing was demonstrated [17] and taken into account for improving the mathematical model for
feedback-operated running hologram recording [18]. Such feedback controlled running holo-
grams under strong light absorption were used for material characterization too [19,20]. The use
of a feedback setup enabling stabilized recording with arbitrarily selected holographic phase
shift ϕ (the phase shift between the transmitted and diffracted beams along the same direc-
tion behind the crystal) was shown [21] to produce photorefractive running holograms with
controlled speed and diffraction efficiency. In this way almost environmental perturbation-free
(stabilized) running holograms with known and fixed phase shift values could be produced [22]
and used for photorefractive materials characterization [23]. Still very important is the fact
that the latter technique allows accurately computingϕ directly from the zeroing of the error

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signal, which is a combination of the first and second temporal harmonics produced by the
piezoelectric-driven phase modulation in the setup, without requiring the knowledge of sev-
eral setup parameters like the amplification of the lock-in amplifiers and the response of the
piezoelectric [7] that may be quite nonlinear indeed.

In this paper we take advantage of such stabilized running holograms, with arbitrarily se-
lected ϕ, for characterizing nominally undoped photorefractive titanosillenite crystals. The
diffraction efficiency (η) and the running hologram speedv (or detunningΩ = Kv, K being
the hologram wave vector value) are measured as a function of the arbitrarily selectedϕ, for
characterizing some parameters of the electron as well as of the hole photoactive centers in the
samples.

2. Theory

The present theoretical development is based on the two-photoactive centers model [24–26],
one center (donors) for electrons and the other (acceptors) for holes. The strong light absorp-
tion effect on the response time is also considered. The space-charge field evolution based on
electron and on hole charge carriers are respectively described by the couple of equations [9]:

τsc1
∂Esc1(t)

∂ t
+Esc1(t)+κ12Esc2(t)+mEeff1e−ıΩt = 0 (1)

τsc2
∂Esc2(t)

∂ t
+Esc2(t)+κ21Esc1(t)+mEeff2e−ıΩt = 0 (2)

with the parameters

Eeff j ≡ E0 + ıED j

1+K2l2
sj − ıKlE j

(3)

1
τscj

=
1

τM j

1+K2l2
s j− ıKlE j

1+K2L2
D j − ıKLE j

= ωR j + ıωI j (4)

τM j = εε0/(qµ jN j) (5)

κ12 =
ξ1

1+K2l2
s1− ıKlE1

κ21 =
ξ2

1+K2l2
s2− ıKlE2

(6)

ED1 = KkBT/q ED2 = −KkBT/q (7)

KLE j = K2L2
D jE0/ED j (8)

KlE j = E0/Eq j (9)

whereı ≡
√
−1, the subindexj = 1 is always for electrons andj = 2 for holes, withE0 being

the externally applied electric field,kB the Boltzmann constant,T the absolute temperature,ε0

the permittivity of vacuum,ε the dielectric constant andq the absolute value of the electron
electric charge. The density of free charge carriers and their mobility in the extended states
(Conduction or Valence Bands) areN j andµ j , respectively, andEq j is the limit value of the
space-charge field. The phenomenological parameterξ j accounts for the effect of the Debye
length (ls j) on the electric coupling when the interacting charges are spatially separated, as
proposed elsewhere [27]. The solution of Eqs. (1) and (2) is:

Esc(t) = Esc1(t)+Esc2(t) = −mEst
sce−iΩt (10)

whereEst
sc assumes the simplified expression:

Est
sc = Eeff1

ωR1 + ıωI1

ωR1 + ıωI1− ıΩ
+Eeff2

ωR2 + ıωI2

ωR2 + ıωI2− ıΩ
(11)

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for ξ1,2 [27] being small enough to assume thatκ12 ≈ κ21 ≈ 0.
The diffraction efficiencyη and the phase shiftϕ, accounting on self-diffraction and bulk

absorption effects, can be written as [28]:

η =
2β 2

1+β 2

cosh(̄Γd/2)−cos(γ̄d/2)

β 2e−Γ̄d/2 + eΓ̄d/2
(12)

tanϕ = − sin(γ̄d/2)
1−β 2

1+β 2

[

cosh(̄Γd/2)−cos(̄γd/2)
]

+sinh(Γ̄d/2)
(13)

with β 2 being the pump-to-signal beam intensity ratio and

Γ̄d ≡
∫ d

0
(Γ)dz Γ ≡ 2πn3reff

λ
ℑ{Est

sc} (14)

γ̄d ≡
∫ d

0
(γ)dz γ ≡ 2πn3reff

λ
ℜ{Est

sc} (15)

Hereℜ{} andℑ{} represent the real and the imaginary parts respectively andz is the coordinate
along the crystal thickness. The quantitiesΓ̄ andγ̄ are here introduced in order to take account
on the effect of the bulk absorptionα on theτM j , as reported elsewhere [18]. In fact, because of
the relatively large value ofα, τM j = τM j(z)does depend on the coordinatez along the crystal
thickness:

τM j(z) =τM j(0)eαz τM j(0) =
εε0(kBT/q)hν
qL2

D jαI(0)Φ j
(16)

with Φ j being the quantum efficiency for charge carriers (electrons or holes) photogeneration
with light of photonic energyhν . ConsequentlyωR j andωI j do also depend onzas well as the
derivedEst

sc, Γ andγ quantities. It was already shown elsewhere [7, 18], that using the spatial
averages̄Γ andγ̄ instead of the plainΓ andγ, in the expressions forη and tanϕ, the effect of
the bulk absorption on these quantities are effectively taken into account.

3. Experiment and results

Holographic gratings were recorded using aλ = 514.5 nm laser light on ad = 2.35 mm thick
nominally undoped photorefractive titanosillenite crystal Bi12TiO20 (labelled BTO-J13) pro-
duced by Prof. J.F. Carvalho at the Institute of Physics of the Universidade Federal de Goiás
(UFG), Goîania-GO, Brazil. The crystal is used in a transverse configuration with symmetri-
cally incident recording beams, with the electric fieldE0 applied along the [110]-axis, parallel
to the hologram wavevector~K, and the [001]-axis perpendicular to the incidence plane. The
bulk absorption of the crystal at the operating wavelength wasα ≈ 10.50 cm−1. The recording
was carried out using self-stabilization with an arbitrarily selected phase shiftϕ as described
elsewhere [21,22].

The experimental procedure is as follows: An external electric fieldE0 is applied and a phase
shift ϕ is fixed at a chosen value by means of the stabilization feedback loop so that the latter
makes the recording pattern of fringes to run at a speedv (Ω = Kv) in order to allow recording
a photorefractive hologram with the chosen parametersE0 andϕ. The diffraction efficiency of
this running hologram is measured in these conditions and plotted (•) in Fig. 1 as a function of
ϕ, for a constantE0. The detunningΩ = Kv as a function ofϕ is also measured and plotted in
Fig. 2. From the two figures above,η is plotted as a function ofΩ in Fig. 3. The continuous
curve in Fig. 3 is the best theoretical fitting of Eq. (12) to the experimental data, with the

#160623 - $15.00 USD Received 5 Jan 2012; revised 28 Jan 2012; accepted 28 Jan 2012; published 3 Feb 2012
(C) 2012 OSA 1 March 2012 / Vol. 2,  No. 3 / OPTICAL MATERIALS EXPRESS  231



-80 -60 -40 -20 0 20 40 60
0.0

0.2

0.4

0.6

0.8

1.0

j HdegreeL

Η
H%
L

Fig. 1. Diffraction efficiencyη data (•) plotted as a function ofϕ for E0 = 5.6 kV/cm and
K = 9.9 µm−1, β 2 = 57 with total incident irradianceI0 = 25.4 mW/cm2, together with a
theoretical curve with the parameters reported in the first row in Table 1.

resulting material parameters displayed in the 3rd row of Table 1. Because of the lowK-value
involved in this experiment, the best fitted parametersLD1 andls1 (with K2L2

D1 ≪ 1 andK2l2
s1≪

1) are not much reliable. Nevertheless this set of parameters are useful as a starting point to find
out theoretical curves (continuous lines) fairly well adjusting the experimental data in Figs. 1
and 2, whereK2L2

D j andK2l2
s j are not anymore neglegibly small because a largerK value is

used. The resulting parameters from the continuous curves in Figs. 1 and 2 are displayed in the
1st and 2nd rows, respectively, in Table 1.

Table 1. Material Parameters

DATA E0 K Φ1 Φ2 LD1 LD2 ls1 ls2

(kV/cm) (µm−1) (nm) (nm) (nm) (nm)
Fig. 1 5.6 9.9 0.64 0.011 163 485 50 287
Fig. 2 5.6 9.9 0.40 0.011 170 485 50 287
Fig. 3 6.5 2.1 0.64 0.011 163 485 125 287

Parameters in 3rd column were obtained by direct data fitting.

The results in Table 1 show that the most reliable parameters found out for electrons and
holesare:Φ1 = 0.40−0.64 andΦ2 ≈ 0.011,LD1 = 163−170 nm and ŁD2 ≈ 485 nm,ls1≈ 50
nm andls2≈ 287 nm. These parameters, at least for electrons, are not far from those (Φ1 = 0.33,
LD1 = 160 nm andls1 = 38 nm) already published [20] for the same sample. The parameters
for holes, on the other hand, are roughly coherent with their expected lower concentration and
experimentally verified reduced participation in the recording process.

The hologram recorded on this undoped titanosillenite crystal was experimentally shown
(as already reported before [29]) to be easily and rapidly erased usingλ = 1200 nm light
from a LED (light emitting diode) having a photonic energy slightly above 1 eV. The latter

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-80 -60 -40 -20 0 20 40 60

-45

-30

-15

0

15

30

j HdegreeL

W
Hr

a
d
�s
L

Fig. 2. DetunningΩ = Kv data (•) plotted as a function ofϕ for E0 = 5.6 kV/cm and
K = 9.9 µm−1, β 2 = 57 with total incident irradianceI0 = 25.4 mW/cm2, together with a
theoretical curve with the parameters reported in the second row in Table 1.

is close to the energy gap between the top of the Valence Band and the Fermi level, where
the main photoactive centers are localized an where it is expected that the space-charge field-
based hologram be recorded. As optical recording is known to occur mainly by electrons, that
need to be excited to the bottom of the Conduction Band (2.2 eV above the Fermi level), we
should conclude that holes (to be excited to the top of the Valence Band 1 eV below the Fermi
level) are responsible for 1200 nm light erasure. This result indicates that holes are actually
participating (although in a much lower proportion than electrons) in optical recording (and
erasure) in undoped sillenites, in agreement with our detection and measurement of material
parameters of holes in the present running hologram experiments.

4. Conclusions

The present paper shows that running hologram experiments produced under carefully con-
trolled conditions may be useful for characterizing photorefractive materials and even detect
minoritary (more thanl2

s2/l2
s1≈ 30-fold lower concentration) charge carriers (holes in this case)

even if a relatively large number of unknown parameters are involved. In the present case, the
combined use of large and small values forK helped to optimize the parameters to be found
out from experimental data fitting: from lowK values we may overlook some of the diffusion
and Debye lengths so as to find out reliable values for the remaining parameters. Electrons
and holes being electrically uncoupled, as formulated in Eq. (11), was experimentally verified
and enabled the independent determination of parameters for both type of charge carriers here.
From data displayed in Table 1 for electrons and for holes it becomes evident that parameters
for electrons are much more relevant for curve shaping than are those for holes, so that the lat-
ter remain rather unchanged while the former vary sensibly from one experiment to the other.
This is probably due to the fact that electrons are majority carriers and therefore have far larger
influence in determining the adjustment of theoretical curves.

#160623 - $15.00 USD Received 5 Jan 2012; revised 28 Jan 2012; accepted 28 Jan 2012; published 3 Feb 2012
(C) 2012 OSA 1 March 2012 / Vol. 2,  No. 3 / OPTICAL MATERIALS EXPRESS  233



-60 -40 -20 0 20
0.0

0.2

0.4

0.6

0.8

1.0

WHrad�sL

Η
H%
L

Fig. 3. Diffraction efficiencyη data (•) plotted as a function of detunningΩ = Kv for
E0 = 6.5 kV/cm andK = 2.1 µm−1, β 2 = 29, with total incident irradianceI0 = 17.3
mW/cm2, together with the theoretical best fitting curve with the parameters as reported in
the third row in Table 1.

Acknowledgments

We acknowledge partial financial support from the Conselho de Desenvolvimento Cientı́fico e
Tecnoĺogico (CNPq), Fundaç̃ao de Amparòa Pesquisa do Estado de São Paulo (FAPESP),
FUNCAMP-Fundo de Apoio ao Ensino, Pesquisa e Extensão (FAEPEX-PAPIDC) and
Coordenaç̃ao de Aperfeiçoamento de Pessoal de Nı́vel Superior (CAPES) all from Brazil. We
are grateful to Prof. Dr. Jesiel F. Carvalho from the Institute of Physics of the Federal University
of Goiás-GO, Brazil, for the excellent photorefractive crystal sample used in this research.

#160623 - $15.00 USD Received 5 Jan 2012; revised 28 Jan 2012; accepted 28 Jan 2012; published 3 Feb 2012
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