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Optics Communications 247 (2005) 39–48
www.elsevier.com/locate/optcom
Nearly 100% diffraction efficiency fixed holograms inoxidized iron-doped LiNbO3 crystals using
self-stabilized recording technique
Ivan de Oliveira a, Jaime Frejlich a,*, Luis Arizmendi b, Mercedes Carrascosa b
a Laboratorio de Optica, Solid state physics departement, IFGW – UNICAMP, State University of Campinas,
Caixa Postal 6165, 13083-970 Campinas-SP, Brazilb Depto. Fisica de Materiales C-IV, Universidad Autonoma de Madrid, 28049 Madrid, Spain
Received 5 August 2004; received in revised form 5 November 2004; accepted 10 November 2004
Abstract
We analyse the advantages and limitations of self-stabilized holographic recording for fixing in photorefractive iron-
doped LiNbO3 crystals. We study the effect of the degree of oxidation on the development of the fixed grating, confirm
the importance of highly oxidized crystals to obtain fixed holograms with high diffraction efficiency and put it on a
quantitative basis. We also discuss the effect of iron-concentration and the way oxidation may introduce dopant-
saturation effects in the recording. Accounting on the different parameters here considered we propose a very simple
and well performing procedure to produce nearly 100% diffraction efficiency fixed gratings.
� 2004 Elsevier B.V. All rights reserved.
PACS: 42.40.Pa; 42.65.Hw; 42.70.Nq; 42.70.Ln; 42.70.MpKeywords: Volume hologram; Photo refractive; Lithium niobate; Holography; Optical recording; Fixing; Nonlinear optics; Dynamic
holography
1. Introduction
Photorefractive materials are known to allow
recording reversible holograms which are essential
0030-4018/$ - see front matter � 2004 Elsevier B.V. All rights reserv
doi:10.1016/j.optcom.2004.11.046
* Corresponding author. Tel.: +55 19 2393136; fax: +55 19
2393137.
E-mail address: [email protected] (J. Frejlich).
for phase conjugation as well as image and data
processing and other applications. Reversibility
however is highly inconvenient for permanent data
storage and for permanent volume holographic
optical components and that is why continuous ef-
forts have been carried out since long ago in orderto find out a procedure to fix holograms in these
materials. Fixing is in general based on thermal
ed.
40 I.de Oliveira et al. / Optics Communications 247 (2005) 39–48
procedures where the originally recorded elec-
tronic grating is substituted by a p-phase shifted
non-photosensitive ionic one. Successful fixing
have been already reported for LiNbO3:Fe (LNb)
[1–5], for Bi12TiO20 (BTO) [6], Bi12SiO20 (BSO)[7] and other materials [8]. Fixing in copper-doped
LiNbO3 has also been reported [9]. Manganese-
doped LiNbO3 has shown rather poor results
[10]. A large diffraction efficiency g � 0.5 for the
fixed hologram in LiNbO3:Fe was already re-
ported [11] but requiring development with intense
light pulses. Good results were reported [12] by
simultaneous compensation during recording butit requires holographic recording at relatively high
temperature that is quite difficult and needs special
equipment to be carried out. A non thermal fixa-
tion based on doubly doped LiNbO3:Mn:Fe crys-
tals has been already reported [13] where g = 0.32
was achieved. In this case the hologram is written
on the shallower photosensitive centers and stored
in the deeper ones which are not affected by thereading wavelength so that the hologram can be
read without been erased. Higher values were re-
cently claimed to be obtained in LiNbO3:Ce:Cu
[14]. Long lifetime holograms in Bi2TeO5 not
requiring any fixing procedure were also reported
[15].
In this paper we focus on thermal fixing,
emphasize on the nature of the LiNbO3 crystaland on the features of self-stabilized holographic
recording in order to allow an adequate choice of
the simplest and reliable procedure to produce
the most efficient fixed hologram.
2. Hologram fixing
The usual fixing process essentially consists in
three steps: the recording of an electronic grating,
the compensation of this grating by non-photosen-
sitive ions (usually H+) and the development under
spatially uniform white light in order to partially
erase the photosensitive electronic grating and
get a partially uncompensated ionic non-photosen-
sitive (therefore stable) grating. The electronicgrating is usually holographically recorded, for a
time tR at room temperature on a LiNbO3:Fe crys-
tal, that is described by [16,17]
EscðtRÞ ¼ �mEeff 1� e�tR=ssc� �
; ð1Þ
where:
Eeff ¼Eph þ iED
1þ K2l2s � iKlphNþ
D
ND
� Eph
1� iKlphNþ
D
ND
; ð2Þ
Eph / NþD; ð3Þ
K2l2s ¼K2�e0kBTq2ðNDÞeff
; ðNDÞeff �Nþ
DðND � NþDÞ
ND
;
ð4Þ
1=ssc ¼ xR þ ixI; ð5Þ
xR ¼ 1
sM
1þ K2l2s1þ K2L2
D
� 1
sM/ ND � Nþ
D
NþD
; ð6Þ
xI ¼ � 1
sM
Klphð1þ K2L2
DÞ2
NþD
ND
� �KlphsM
NþD
ND
; ð7Þ
Klph ¼Eph
Eq
/ ND=ðND � NþDÞ; ð8Þ
Eq ¼qðNDÞeffK�e0
: ð9Þ
The m is the complex pattern-of-fringes modula-
tion coefficient, Esc(t) is the amplitude of the
space-charge field modulation of the recorded
grating, Eq is the latter maximum possible value(at jmj = 1), sM = �e0/r is the Maxwell relaxation
time with � being the effective dielectric constant,
e0 the electric permittivity of vacuum and r the
conductivity. Eph and ED are the photovoltaic
and the diffusion fields, respectively, NþD and ND
are the concentration of the acceptors (concentra-
tion of Fe3+) and the total (acceptors Fe3+ plus do-
nors Fe2+) photoactive centers in the sample,respectively. The parameters ls and LD are the De-
bye and the diffusion lengths, respectively. The
approximate relations in Eqs. (2), (6) and (7) de-
rive from the assumption that K2L2D � 1;
K2l2s � 1 and ED � Eph. The proportionalities in
Eqs. (3), (6) and (8) are discussed elsewhere [18].
Afterwards the crystal is heated to allow the H+
ions in the sample to diffuse so as to completelycompensate the previously recorded electronic
I.de Oliveira et al. / Optics Communications 247 (2005) 39–48 41
grating in which case the ionic grating space-
charge field is
Eh ¼ �EscðtRÞ ¼ mEeff 1� e�tR=ssc� �
: ð10Þ
The sample is then cooled down to room tem-
perature and illuminated with an intense incoher-
ent white light to develop the fixed grating. In
this way the electronic grating is (partially) erased
and phase-shifted leaving a (partially) uncompen-
sated non-photosensitive and stable ionic grating.
The development process is described by[5,19,20]:
sscdEscðtÞ=dt þ EscðtÞ þ nEh ¼ 0; ð11Þ
1=n ¼ 1þ K2l2s � iKlphNþD=ND
� 1� iKlphNþD=ND: ð12Þ
The overall field is ETscðtÞ ¼ EscðtÞ þ Eh with Eh
being the expression in Eq. (10), Esc(t) being the
solution of Eq. (11) and n being an electric cou-
pling constant. The expression of ETsc, taking into
account the complete initial grating compensation
condition ETscð0Þ ¼ 0, is
ETscðtÞ ¼ Ehð1� nÞð1� e�t=sscÞ; ð13Þ
where t = 0 is the starting of illumination in the
development process. It is worth noting the impor-
tant role of the coupling parameter n: The closerj1 � nj approaches unity, the better the perfor-
mance of the development process. For highly
doped crystals we may write
K2l2s � 1; ð14Þ
so that:
j1� nj � jxIsMjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ x2
Is2M
p ð15Þ
with
jxIsMj / NþD=ðND � Nþ
DÞ ¼ ½Fe3þ�=½Fe2þ�: ð16Þ
For strongly oxidized crystals it is jxIsMjP1 so
that j1 � nj approaches 1 in which case a favorable
development condition is achieved. On the otherhand for very reduced (and still highly doped)
crystals it is j1 � nj ! 0 and therefore jETscð1Þj
! 0.
3. Holographic recording
The holographic recording in LiNbO3 should
necessarily be somewhat stabilized because of the
possibility of environmental phase perturbationsdue to the long time scale needed for recording
in these materials. For oxidized samples, that exhi-
bit the best development performance, the record-
ing is still much more slower (sM � [Fe3+]/[Fe2+])
and stabilization in this case is therefore particu-
larly important. It is possible to stabilize the
recording pattern of fringes using the interference
of the light transmitted and reflected on an auxil-iary glassplate fixed close to the sample [21]. It is
also possible to use the interference of the trans-
mitted and reflected beams on one of the lateral
faces of the own sample to operate the stabiliza-
tion system [12]. However, both such methods
are far less efficient than self-stabilization [22–24].
The latter method uses the own recorded holo-
gram as a reference and in this way the stabiliza-tion is much more effective because it is related
to the very place where recording is actually
occurring.
3.1. Self-stabilization
Self-stabilized recording was already described
in the literature [22–24] and will be briefly dis-cussed below. Self-stabilization does necessarily
lead to running holograms which speed increases
as the holographic phase / (the phase shift be-
tween the recording pattern of light and the result-
ing hologram) shifts away from / = p. In fact it
was already shown [22–24] that self-stabilization
dynamically fixes the holographic phase shift to
/ = p. If the open loop (non stabilized) value forthe sample in the setup is already / = p (as is the
case for strongly reduced samples) the stabilization
will proceed in stationary conditions. Instead, for
samples where the open loop holographic phase
value is different from p, the stabilization produces
a moving hologram because the system tends
dynamically to keep the holographic phase fixed
to p during the process. The hologram speed willbe higher the larger the mismatching between the
feedback-imposed and the open-loop values of /.In order to qualitatively predict how fast the
Table 1
LiNbO3:Fe samples
Sample [Fe2+]/[Fe3+]a [H+]
cm�3b
d
(mm)
jxIsMjc [Fe] 1019
cm�3
LNb1 0.002 3.4 · 1017 1.5 2.0–2.2 2
LNb2 0.0037 2.2 · 1019 0.96 0.73–1 2
LNb3 0.013 3.2 · 1017 1.39 0.1 2
LNb4 0.013 6.4 · 1018 0.35 – 20
LNb5 0.03 – 0.85 – 2
d, Sample thickness; [Fe], total iron concentration.a Measured from the absorption spectrum at k = 477 nm
[25].b Measured from the absorption spectrum at k = 2870 nm
[26].c Measured from data fitting.
Fig. 2. Measurement of the running hologram speed for the
sample LNb1, b2 � 1, I0S þ I0R � 17 mW=cm2 and K = 10/lm.
The oscillating shape curve is the interference of the transmitted
plus reflected beams in a glassplate fixed close to the sample. Its
decreasing amplitude is due to scattering of light in the sample.
The filled circles represent the computed pattern-of-fringes
speed, corrected from scattering and the dashed curve is only a
42 I.de Oliveira et al. / Optics Communications 247 (2005) 39–48
hologram will move, we compute the open-loop
hologram phase shift /, from Eq. (1), for the elec-
tronic grating during recording as
tan/ ¼ IfEscðtÞg=RfEscðtÞg/ Nþ
D=ðND � NþDÞ; ð17Þ
where Ifg and Rfg represent the imaginary and
real parts, respectively. The result is plotted in
Fig. 1 for hypothetical samples of different degree
of oxidation. All three curves there start at / = pand move away from p during the process. This
means that the hologram should move slower atthe beginning of the process. Fig. 1 also shows that
the more oxidized is the sample, the faster and fur-
ther its phase shifts away from / = p. This means
that the more oxidized the sample the faster the
hologram should move in self-stabilized recording
regime. In order to experimentally confirm these
conclusions we measured the pattern-of-fringes
movement during self-stabilized recording on themost oxidized sample LNb1 (see Table 1). The
movement of fringes was measured from
the changes in the interference (IG) of the transmit-
ted and reflected beams in a glassplate fixed close
to the self-stabilized sample. The result is plotted
in Fig. 2 where the speed of the hologram (mea-
sured in terms of phase Kv) is increasing approxi-
mately from 0.2 to 0.5 rad/min in the first 3000 srecording time. This result is in agreement with
our prediction above stating that the hologram
speed should increase with time during the self-
Fig. 1. Holographic phase shift / during recording for [Fe2+]/
[Fe3+] = 0.1 (thin dashing), 0.02 (thicker dashing) and 0.003
(thickest dashing), with b2 � 1, I0R þ I0S � 16 mW=cm2 and
K = 10/lm.
guide for the eyes.
stabilized recording process. The same experiment
was carried out on the least oxidized sample
(LNb5) in similar conditions with the speed being
Kv < 0.03 rad/min. This result does also confirm
that reduced crystals move slower during self-sta-
bilized recording.
In order to achieve a highly efficient fixed grat-ing it is also necessary to start with an adequately
high value for Esc(tR) = �Eh. Unfortunately, if
self-stabilized holographic recording is used, we
are limited to an upper limit corresponding to
g = 1 because the recording automatically stops
at this point [22]. Several possibilities do exist in
order to overcome this g = 1 limit restriction.
I.de Oliveira et al. / Optics Communications 247 (2005) 39–48 43
One such possibility is simultaneous recording and
compensation at high temperature as already re-
ported in [12]. Another possibility is recording
with ordinarily polarized light. In this case the
g = 1 limit results in a roughly 3-fold higher mod-ulation, than for the case of extraordinary polari-
zation, because of the lower value of the
corresponding electro-optic coefficient. The whole
process is a very simple and reproductible one as
described below.
Fig. 3. Experimental self-stabilized setup: The piezoelectric-
supported mirror PZT that is feed from the oscillator OSC
produces the phase modulation and the necessary phase
corrections during recording. IXS and I2XS are detected in the
irradiance behind the crystal using adequately tuned (to X and
2X, respectively) lock-in amplifiers. The I2XS term is used as error
signal in the feedback loop. The spatial period is D � 0.63 lm,
the wavelength is k = 514.5 nm with I0S þ I0R ¼ 16 mW=cm2
and I0S=I0R � 1.
4. Experimental
From the discussion above we conclude that it
is very convenient to use oxidized crystals. It is
also important to use a stabilized recording tech-
nique. If self-stabilization is chosen however, as
in the present case, the simultaneous ionic com-
pensation during recording is an interesting choice.
The latter however requires holographic recordingat high temperature that is a rather difficult pro-
cess, even with the help of a feedback stabilization
system, because the whole setup should be placed
under vacuum to avoid thermal convection effects.
Our present choice here is a three-step process with
ordinarily polarized recording light: First record-
ing at room temperature, then compensating at
high temperature and finally developing at roomtemperature. The process is a very simple one
and requires no special equipment at all.
The more oxidized sample (LNb1) was pre-
pared by heating at 950 �C in dry oxygen for 2
h. Sample LNb4 was also annealed in oxygen at
900 �C. Sample LNb3 was produced by keeping
it in vacuum at 750 �C for 10 h. Sample LNb2 is
as grown. The [Fe2+] was computed from theabsorption at k = 477 nm [25] and the [H+] was
computed from the absorption at 2875 nm [26].
Some data about these samples, including their
thickness d, are reported in Table 1.
The holographic recording is carried out in the
setup schematically shown in Fig. 3 with 514.5 nm
wavelength ordinarily polarized recording beams
of similar irradiance I0S þ I0R � 16 mW=cm2;I0S=I
0R � 1 and a spatial period D = 0.63 lm. All
samples were short-circuited with conductive silver
glue. The setup and its operation is described in
details in the literature [22,24] and allows: (a)
recording during long periods without being af-
fected by environmental perturbations on the set-up and (b) recording a non-tilted Bragg-matched
hologram which allows (in the absence of dopant
saturation) always to reach g = 1. The setup re-
quires one of the recording beams to be phase
modulated with a small amplitude wd and a large
frequency X compared to the photorefractive
recording time. A first and a second harmonic
terms in X are measured in the irradiance behindthe crystal that are, respectively [21,22]:
IX ¼ 4J 1ðwdÞffiffiffiffiffiffiffiffiffiI0SI
0R
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigð1� gÞ
psinu; ð18Þ
I2X ¼ 4J 2ðwdÞffiffiffiffiffiffiffiffiffiI0SI
0R
q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigð1� gÞ
pcosu; ð19Þ
u ¼ /� p=2 ð20Þwith u being the phase shift between the transmit-
ted and diffracted beams behind the crystal. The
I2X term is used as error signal in the feedbackloop so that u is automatically fixed by the system
to u = p/2, as described in details in the literature
[22]. In this case the first harmonic is automatically
set to IX /ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigð1� gÞ
pand can be used to follow
Fig. 4. Evolution of IX (in arbitrary units) during self-stabilized
holographic recording using ordinary and extraordinary polar-
ized 514.5 nm wavelength recording beams in different exper-
imental runs with similar irradiances on the same LNb3 sample.
44 I.de Oliveira et al. / Optics Communications 247 (2005) 39–48
the recording evolution since we know that
(neglecting scattering) IXS ¼ 0 for g = 0,1 and IXSis maximum at g = 0.5. The absolute value of ghowever, was always measured from the direct dif-
fraction of the recording beams (either the ordi-narily or extraordinarily polarized one whichever
actually used for recording), as described in the
literature [27]. These beams are automatically
in-Bragg with the grating being measured. In order
to neglect bulk absorption and interfaces looses,
diffraction efficiency was always defined here as
g = Id/(Id + It), with Id and It being the diffracted
and transmitted beams, respectively.As already reported elsewhere [22,28] the
recording automatically stops as g = 1 is reached
and the hologram is kept at this value as long as
the feedback in the setup is operating. On the other
hand, in order to get a highly modulated fixed
grating at the end of the process, it is necessary
to produce as much large an electronic grating as
possible in this initial step. That is why the record-ing is carried out using ordinarily polarized light.
In fact the diffraction efficiency g, for a non-tilted
in-Bragg grating, is
g ¼ sin2jd ð21Þ
with
j ¼ 12pn3effreff jEscj=ðk cos h0Þ: ð22Þ
For ordinarily polarized light it is
j ¼ 12pn3or13jEscj=ðk cos h0Þ; ð23Þ
whereas for extraordinarily polarized light it is
j ¼ 12pn3er33jEscj=ðk cos h0Þ; ð24Þ
where k is the light wavelength, h 0 is the incidenceangle inside the crystal, with [29] r13 = 9.6 pm/V,
r33 = 30.9 pm/V and [30] no = 2.33, ne = 2.25.
Accordingly, achieving g = 1 in ordinary light
leads to a space-charge field that is roughly
ðn3er33Þ=ðn3or13Þ � 3 fold larger than if it were re-
corded with extraordinarily polarized light. This
straightforward feature is illustrated in Fig. 4
showing the IXS evolution in two different experi-ments with the same sample and similar irradi-
ances, one with ordinary and the other with
extraordinary polarized recording beams: it is
evident that the latter proceeded for a time that
is much shorter than for ordinary polarization
thus meaning a much lower recorded space-charge
field amplitude too. In the following both record-
ing with ordinary and extraordinary polarization
were carried out for comparison.After recording, the diffraction efficiency gelo;e of
the resulting electronic grating was measured. Here
‘‘el’’ stands for ‘‘electronic’’ grating whereas the
sub-indexes ‘‘o’’ and ‘‘e’’ are for ‘‘ordinary’’ and
‘‘extraordinary’’ polarization, respectively, which-
ever applies. Then the sample is placed inside an
oven for 20 min at 120–150 �C in order to promote
H+ diffusion to compensate the previously recordedelectronic grating. Afterwards the sample is allowed
to cool down to room temperature and replaced at
its original position in the setup and is illuminated
with intense incoherent white light through infrared
filters to avoid heating. In this development step the
electronic grating is partially erased and phase
shifted [31] so that a partially uncompensated ionic
grating results that is not photosensitive: an overallstable fixed grating is thus produced that is not
erased during readout. The evolution of g during
development can be formulated by substituting
Eq. (13) into Eqs. (21) and (22) and replacing
j1 � nj by its expression in Eq. (15) leading to
g ¼ sin2 pn3effreff jEhxIsMjd2k cos h0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ x2
Is2M
p j1� e�ðxRþixIÞtj" #
:
ð25Þ
I.de Oliveira et al. / Optics Communications 247 (2005) 39–48 45
Fig. 5 shows the evolution of the measured g for
the overall grating in sample LNb2 during devel-
opment and the best theoretical fit to Eq. (25)
for the ordinary polarization. From this fit we
compute xIsM [31]. The latter fit is theoreticallyconverted for the extraordinary polarization and
shown in the same figure. Fig. 6 shows the same
information but for the more oxidized sample
LNb1. Same procedure was carried out for
LNb3. Diffraction efficiencies gfixo;e (the index ‘‘fix’’
standing for ‘‘fixed’’ grating and ‘‘o’’ and ‘‘e’’
standing for ordinary and extraordinary polariza-
Fig. 5. Evolution of g (spots) measured during white light
development for the LNb2 sample and best fit (continuous
curve) to theory. The dashed curve is the theoretical conversion
of the continuous curve to extraordinary polarization.
Table 2
Fixing efficiency in LiNbO3:Fe samples
Samples gelo gfixo gele
LNb1 0.10
LNb1 0.15
LNb1 0.182
LNb1 0.37 0.22
LNb1 1.0
LNb1 0.68 0.35
LNb2 0.73 0.19
LNb2 1.0
LNb2 0.75
LNb3 1.00
LNb3 1.0 0.03–0.05
gelo , Diffraction efficiency of the electronic grating before fixing and m
for extraordinarily polarized light; gfixo , diffraction efficiency of the fixed
before but for extraordinarily polarized light; RD, development ratio w
for measurement; ‘‘symbol’’: in last column identifies the correspondi
tions, respectively) and other relevant data are re-
ported in Table 2 where the nature (ordinary or
extraordinary) of the polarization actually used
for recording is indicated. Data in Table 2 do con-
firm the good performance of the more oxidizedsamples as far as development is concerned. The
efficiency of development may be measured by
the development ratio RD that is here defined as
in [2]
RD � ½jd�fix=½jd�elh i
o;eð26Þ
Fig. 6. Evolution of g (spots) measured during white light
development for the LNb1 sample and best fit (continuous
curve) to theory. The dashed curve is the theoretical conversion
of the continuous curve to extraordinary polarization.
gfixe RD Symbol
Ord Ext
0.045 0.66 s
0.0774 0.71 s
0.096 0.71 s
0.73 50.73 0.65 4
0.65 5
0.44 50.3 0.37 40.332 0.59 s
0.02 0.09 40.11–0.14 5
easured under ordinarily polarized light; gele , same as before but
grating measured under ordinarily polarized light; gfixe , same as
ith ‘‘Ord’’ or ‘‘Ext’’ indicating the polarization of the light used
ng data in Fig. 7.
Fig. 7. Development ratio RD plotted as a function of
xIsM=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ x2
I s2M
pfor the three available samples: Electronic
grating recorded with ordinarily (5), with extraordinarily (4)
polarized light and low efficiency electronic grating recorded
with extraordinarily polarized light (s). See Table 2 for details.
The dashed line is the plot of the theoretical Eq. (27), for
comparison.
46 I.de Oliveira et al. / Optics Communications 247 (2005) 39–48
and may be computed from Eq. (25) to be
RD ¼ j1� nj � jxIsMjffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ x2
Is2M
p : ð27Þ
The experimental RD vs: xIsM=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ x2
Is2M
pdata
are plotted in Fig. 7 for three of the samples and
compared to the theoretical relation (dashed line)
in Eq. (27).
4.1. Multiple recording-compensation cycles
A question may arise about repeating the
recording-compensation cycle several times in or-der to produce larger compensated (and therefore
fixed) gratings. After recording an electronic grat-
ing until g = 1 is reached, the sample is compen-
sated in an oven at high temperature. Then the
sample is slightly white-light developed (at room
temperature) to get a slightly uncompensated elec-
tronic grating to enable operating the self-stabi-
lized recording in the next step. Then the sampleis carefully replaced in the setup, using a specially
designed holder, and recording (at room tempera-
ture) of the electronic grating is resumed until
g = 1 is reached again. The whole cycle is repeated
at will or until dopant saturation is reached. In this
way a very large compensated grating would be
obtained that would be fully developed at the
end of the process. Unfortunately this process is
not practical. In fact the differential equation rul-
ing this process is the same as in Eq. (11) except
for a new term in the right-hand side to provide
for the recording with a moving (with angularspeed X) pattern-of-fringes of visibility m
sscdEscðtÞ=dt þ EscðtÞ þ nEh ¼ �mEeff e�iXt: ð28Þ
A moving pattern of fringes is here considered
because of self-stabilization that produces, in gen-
eral, a moving pattern of recording fringes due toholographic phase mismatching as discussed be-
fore. The overall space-charge field ETscðtÞ ¼
EscðtÞ þ Eh is
ETscðtÞ ¼ �mEeff
xR þ ixI
xR þ iðxI �XÞ e�iXt
þ mEeff
xR þ ixI
xR þ iðxI �XÞ � ð1� nÞEh
� �e�t=ssc
þ ð1� nÞEh; ð29Þ
where Esc(t) is the solution of Eq. (28) with
ETscð0Þ � 0 because we assume the electronic grat-
ing is almost completely compensated at t = 0.
The first term in the right-hand side in Eq. (29)
correspond to the steady state electronic grating
being recorded and moving with phase shift Xalong with the moving recording pattern of
fringes. The second term is the one being erased
with a complex time constant 1/ssc and the third
term is a constant fixed grating. All three termsare mutually phase-shifted so that we are not able
to superimpose unshifted successive gratings to
produce a progressively larger grating to be fixed
and developed, except for the case X � 0 that is
true only for strongly reduced samples that, on
their turn, are not suitable for fixing.
5. Discussion
An important fact in this paper is the use of self-
stabilization to enable the long-term recording of
good quality untilted gratings onto the particularly
low sensitive oxidized samples that are required in
this process. It is also possible to carry out good
quality holographic recording for a long time with-
out stabilization too but it is always a very difficult
I.de Oliveira et al. / Optics Communications 247 (2005) 39–48 47
task that requires expensive equipment and very
skillful manpower. Self-stabilization, on the other
hand, is very simple to operate and very reliable
but it introduces some limitations as illustrated
by the impossibility to operate it in multiplerecording-compensation cycles as discussed in Sec-
tion 4.1. Most such limitations arise from the fact
that self-stabilization produces running holograms
which speed varies throughout the recording
process.
The recording with ordinarily polarized light is
essential in order to achieve a large g for the fixed
grating when measured with extraordinarily polar-ized light. The use of oxidized samples with high
[Fe3+]/[Fe2+]-ratio and sufficient amount of H+ is
also essential in order to achieve a large g for the
fixed grating. The [H+] in our samples was shown
to be large enough since we were always able to
completely compensate the electronic grating.
The good performance of oxidized samples, on
the other hand, is due to their characteristic cou-pling constant n that allows, at the same time, to
considerably erase and to shift the electronic grat-
ing during development with white light thus pro-
ducing a larger overall remaining grating [31].
The relation in Eq. (27) allows one to approxi-
mately select the required degree of oxidation
(xIsM � [Fe3+]/[Fe2+]) in the sample to achieve a
predetermined development performance RD asillustrated in Fig. 7. This figure and data in Table
2 also show that experimental data are always be-
low the theoretical limit for RD, except for the less
oxidized sample LNb3. Data for LNb2 are also
clearly below the theoretical prediction except for
the case of small diffraction efficiency. For the
more oxidized sample LNb1 however, data are al-
ways lower than predicted even for the case ofsmall diffraction efficiency. We believe this fact
may be due to effective dopant depletion. Such a
depletion did apparently occur with samples
LNb1 and LNb2 as evidenced by the impossibility
to record an electronic grating approaching g = 1
(see Table 2) with ordinarily polarized light. In fact
note that although all three samples LNb1, LNb2
and LNb3 have the same Fe-contents, the effectiveFe concentration (ND)eff (that determines dopant
saturation) in samples LNb1 and LNb2 is roughly
6–3-fold lower, respectively, than for LNb3. More-
over the maximum possible electronic grating dif-
fraction efficiency is determined by Eeff in Eq. (2)
jEeff j �Ephffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð1þ K2l2s Þ2 þ K2l2phðNþ
D=NDÞ2q
6Ephffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1þ K2l2phðNþD=NDÞ2
q : ð30Þ
For the hypothetical limit case of very oxidized
samples, with KlphNþD=ND � 1 (although never
achieved here), Eq. (30) simplifies to
jEeff j 6Eph
KlphNþD=ND
/ ðND � NþDÞ ¼ ½Fe2þ� ð31Þ
showing that the more oxidized the sample thelower Eeff and consequently the lower g for the
electronic grating. This overall effect of effective
dopant depletion upon sample LNb1 may perhaps
be at the origin of its larger difference between
experimental and theoretical values for RD
compared to the less oxidized samples.
In any case we want a large value for Eeff in or-
der to achieve a large electronic grating in the firststage of the fixing process. In this case it is neces-
sary to increase [Fe2+]. For doing this and still
keep the desired [Fe3+]/[Fe2+]-ratio the total
amount of Fe-concentration should be increased.
Therefore, the higher the oxidation degree of the
sample the higher the total Fe-concentration
needed in order to avoid Fe2+-shortage to achieve
a large index-of-refraction modulation. Unfortu-nately the increase in Fe-concentration leads to
electron tunneling [32,33] that has a deleterious ef-
fect on holographic recording. This was probably
the case of sample LNb4 that has 10-fold higher
Fe-concentration than the other samples and
where we were able to record only rather small
and rapidly dark-erasable holograms.
6. Conclusions
We report a very simple, reproductible and wellperforming three-step-process using self-stabiliza-
tion and ordinarily polarized recording light, at
room temperature. We showed the interest of
self-stabilized holographic recording to produce
high diffraction efficiency fixed holograms. We
48 I.de Oliveira et al. / Optics Communications 247 (2005) 39–48
confirmed previously published results showing
the importance of using oxidized samples and
put it in a quantitative basis that enables to
approximately predict the final result. A 100%
diffraction efficiency fixed grating for use with lightof fixed polarization is obtained in this way as
illustrated by Fig. 5. A good performing (�80%
for extraordinary and �40% for ordinary polariza-
tions, respectively) fixed grating may be also ob-
tained for use with variable polarization direction
light as illustrated by Fig. 6.
Acknowledgments
We acknowledge the partial financial support
from Fundacao de amparo a Pesquisa do Estado
do Sao Paulo, Brazil, Conselho Nacional de
Desenvolvimento Cientıfico e Tenologico, Brazil,
Grant TIC2001-0605, Spain. We acknowledge
Dra. Veronica Bermudez form the Crystal Grow-ing Laboratory of the UAM for the good quality
lithium niobate crystals that enabled this research.
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