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Appl. Phys. B 68, 1007–1012 (1999) / DOI 10.1007/s003409900017 Applied Physics B
Lasers
and Optics
Springer-Verlag 1999
Nonlinearity of a response in photorefractive crystals with
a square-wave applied field
S.M. Shandarov1, N.I. Nazhestkina2, O.V. Kobozev1, A.A. Kamshilin2
1Department of Electronic Devices, State University of Control Systems and Radioelectronics, 40 Lenin Avenue, Tomsk 634050, Russia
(Fax: +7-3822/414-321, E-mail: shand@stack.ru)
2Department of Physics, University of Joensuu, P.O. Box 111, FIN-80101, Joensuu, Finland
Received: 18 November 1998/Revised version: 26 January 1999/Published online: 12 April 1999
Abstract. The space-charge-field formationin a photorefrac-
tive crystal illuminated by a one-dimensional light pattern is
studied when an alternating square-wave electric field is ap-
plied to the crystal. We derive the general nonlinear equation
for the time-averaged distribution of the space-charge field
and employ the simplified versions of this equation to analyze
the space-charge-field formation in the case of a Gaussian
beam and interference light pattern. The borders of the ap-
plicability of simplified equations and analytic expressions to
describe a self-action of light beams and the large modula-
tion effects under two-beam coupling are estimated from the
numerical calculations.
PACS: 42.65.Hw; 42.70.Nq
The ac-field technique can significantly increase the nonlin-
ear response of photorefractive crystals to the light illumina-
tion [1]. This increasing is especially pronounced in wide-
gap-semiconductors and sillenite crystals possessing a long
drift length of the charge carriers. Applying this technique,
two-wave-mixinggain is significantly increased [1–6], beam
fanning and phase conjugation efficiency is improved [7,
8], spatial subharmonics [9–11] and photorefractive surface
waves [12,13] are observed. A standard linear model of
the material response, which is valid for the low modula-
tion depth of the light-interference pattern, is not univer-
sally true for analysis of the listed effects. Swinburne et al.
have predicted the nonlinear dependence of the two-beam-
coupling gain on the modulation index of an interference
pattern [14]. They derived an approximate analytical expres-
sion for the space-charge field in a crystal under an alternating
square-wave field neglecting charge diffusion and trap satura-
tion. Comparing numerical solutions of the nonlinear material
equations [15,16] to the analytic model [14], one can con-
clude that the latter is useful only when the grating fringe
spacing is much larger than the drift length.
Dedicated to Prof. Dr. Eckard Krätzig on the occasion of his 60th birthday.
It is currently believed that the material nonlinearity is
the reason for the subharmonic instabilities in sillenite crys-
tals. To calculate numerically the threshold conditions for
Bi12SiO20 crystals under a square-wave electric field, the
space-charge field was assumed as a superposition of the fun-
damental and subharmonic gratings [10, 17]. Photorefractive
grating’s instability in respect of the parametric excitation of
space-charge waves was used by Sturman et al. to estimate the
conditions of the spatial subharmonics generation [18].
In this paper we derive a nonlinear differential equation
for the time-averaged space-charge field that is formed in
the steady-state regime in a crystal under application of the
square-wave electric field when the crystal is exposed by the
one-dimensional light pattern. No restrictions on the shape
and modulation depth of the light distribution are suggested.
A simplified version of this equation neglecting diffusion
of the charge carriers is then applied to analyze formation
of the space-charge field caused by a Gaussian light beam.
It is shown that the linear approximation is valid when the
drift length is much smaller than the beam size. Necessity
of considering the trap saturation for moderate drift lengths
is proved. We have also obtained an analytical expression
for the fundamental harmonic of the space-charge field tak-
ing into account diffusion of the charge carriers and trap
saturation.
1 Model and nonlinear equation
The response of a photorefractive crystal to illumination by
the one-dimensional light pattern with an intensity I(x)is
described by a set of nonlinear material equations [19]. We
restrict our consideration to the band-transportmodel consid-
ering only one type of charge carriers, electron, and one pho-
toactive impurity level. The relevant equations can be written
as
∂N+
D
∂t=(β +sI)(ND−N+
D)−γRneN+
D,(1)
1008
∂ESC
∂x=e
εN+
D−NA−ne,(2)
∂
∂xε∂ESC
∂t+eµne(ESC +E0)+kBTµ∂ne
∂x=0,(3)
where NDis the total density of donors, N+
Dis the density
of ionized donors, NAis the density of compensating ac-
ceptors centers, neis the density of free electrons, sis the
light-excitation cross section, γRis the two-body recombi-
nation coefficient, βis the thermal-excitation rate, εis the
static dielectric constant, µis the electron mobility, kBis the
Boltzmann’s constant, Tis the absolute temperature, E0is the
external homogeneous electric field (which may be a function
of time), ESC is the space-charge field, and eis the elementary
charge. We assume that all functions in (1)–(3) are predom-
inantly varied along the xdirection. In addition, an external
electric field E0of square-wave form is applied to the crystal
along the xaxis (Fig. 1). In the case of the ac-field applica-
tion, the illumination of the whole crystal is not necessary.
Let us consider the typical configuration of the experiment
in which the middle layer of the crystal is illuminated and
the dark (light-free) regions are adjacent to the electrodes
(Fig. 1). The transverse size of the crystal (distance between
the electrodes) is d.
Using the adiabatic approximation under the continuous-
wave low-intensity illumination, the conditions neNA
and ∂ne/∂t=0 are obeyed. Assuming additionally that
∂ESC/∂xe(ND−NA)/ε, (1) and (2) can be reduced:
ne=
(β +sI)(ND−NA)τR−ε
eNAγR
∂2ESC
∂t∂x
1+ε
eNA
∂ESC
∂x
.(4)
Applying an approach similar to the analysis of the bright
screening soliton in 1+1 dimensions [20], following ap-
proximations are considered while integrating the continuity
equation (3). First, we assume that the light field is well
localized in the middle layer of the crystal. Therefore, spa-
tial derivatives of the space-charge field, ESC(x), and elec-
tron density, ne(x), vanish at the crystal faces x=±d/2,
i.e. dESC/dx|±d/2=dne/dx|±d/2=0. Second, we consider
Fig.1. Schematic diagram illustrating the applied square-wave field and
one-dimensional light field distribution in photorefractive crystal. The light
field exists within the middle layer and the dark areas are adjacent to the
electrodes. The crystal size along the xaxis is equal to d
the light intensity at the crystal faces, I(±d/2), being much
smaller than the dark irradiance, Id=β/s. Finally, we be-
lieve that the boundary values of the space-charge field tend
to zero, i.e., ˜
ESC(±d/2)=0, since the light field is assumed
to be localized in a layer with the transverse size of lSd.
Then the integrating of (3) immediately yields
ε∂ESC
∂t+eµne[ESC +E0(t)]+kBTµ∂ne
∂x=δ(t), (5)
where δ(t)=σdE0(t)=eµβ(ND−NA)τRE0(t)is the current
density in the dark area of the crystal near the electrodes, σdis
the dark conductivity, and τR=(γRNA)−1is the recombina-
tion time.
Next, we exclude the density of free electrons, ne, from
(4) and (5) and derive two equations for space-charge fields,
E+
SC(x,t)and E−
SC(x,t), for positive and negative half-periods
of the applied square-wave field, respectively. In the diffu-
sion current, kBTµ∂ne/∂x, only the terms that are linear and
quadratic in E±
SC are taken into account. Thereafter, we con-
sider a space-charge field at the steady state under an applied
square-wave field of high frequency, f0aeαI/(εE0hω),
where ais the characteristic size of a spatial inhomogene-
ity in the light distribution, αis the absorption coefficient,
and hωis the photon energy. It is known that weak oscilla-
tions of the space-charge field around its mean value occur
at the high frequency if the crystal is illuminated by an in-
terference pattern (ais equal to the fringe spacing Λ) with
small modulation index [4,18]. Nevertheless, we assume
that this modulation is negligible. Then time averaging of
the equations for E+
SC(x,t)and E−
SC(x,t)over their respec-
tive half-periods and employment of evident relationships,
hE+
SC(x,t)i=hE
−
SC(x,t)i= ˜
E
SC(x)and h∂E+
SC(x,t)/∂ti=
−h∂E−
SC(x,t)/∂ti= ˜
D
SC(x), yield two nonlinear equations
for unknown functions ˜
ESC(x)and ˜
DSC(x). After some sim-
ple manipulations one obtains
˜
ESC −τdIdLE
Id+I
d˜
DSC
dx+kBT
e"1
Id+I
dI
dx−LA
E0
d2˜
ESC
dx2#
× 1−LA
E0
d˜
ESC
dx!=0,(6)
˜
DSC 1+LA
E0
d˜
ESC
dx!−LE˜
ESC
E0
d˜
DSC
dx−LA
τd
d˜
ESC
dx
−L2
Dd2˜
DSC
dx2+L2
DLA
E0
d˜
DSC
dx
d2˜
ESC
dx2=− I
I
dτ
dE
0,(7)
where τd=ε/σdis the dielectric relaxation time in the
dark area of the crystal, LE=µτRE0is the drift length,
LA=εE0/(eNA)is the length of electron tightening by the
electric field E0,andL
D=(µτRkBT/e)1/2is the diffusion
length. To obtain the equation for ˜
ESC, it is convenient to ex-
clude the function ˜
DSC from (6) and (7). In the case of interest
when the drift length LEfar exceeds the lengths LD,LA,and
the Debye screening length, LS=[k
BTε/(e2NA)]1/2, the non-
linear equation for time-averaged space-charge-field can be
presented in the form
1009
2L2
SLE˜
ESC
E0
d3˜
ESC
dx3
−"LELA 1−˜
E2
SC
E2
0!+L2
S+L2
D−2L2
SLE
E0 d˜
ESC
dx+˜
ESC
Id+I
dI
dx!# d2˜
ESC
dx2
−(2(LE−LA)˜
ESC
E0+LA(2LE−LA)˜
ESC
E2
0
d˜
ESC
dx
−"LE−LA 1−˜
E2
SC
E2
0!+2L2
S−3L2
D#1
Id+I
dI
dx)d˜
ESC
dx
+"1−1
Id+I LE˜
ESC
E0
dI
dx+2L2
Dd2I
dx2!# ˜
ESC +LEE0+kBT
e1
Id+I
dI
dx=0.
(8)
Here we also assume that the following inequalities are satis-
fied
LA
a1,LD
a1,LS
a1,(9)
and restrict our consideration to the terms that are linear
and quadratic on the small parameters LA/a,LD/aand
LS/a. Equation (8) can be applied to an analysis of var-
ied photorefractive phenomena not resorting to strong re-
strictions on the modulation depth of the light intensity
distribution.
2 Space-charge field induced by Gaussian beam
Let us consider the distribution of space-charge field in
a crystal of the sillenite family, such as Bi12SiO20 and
Bi12TiO20 , caused by a Gaussian beam with the light intensity
distribution,
I(x)=I0exp −x2
a2.(10)
Note that the half-width aof the beam is assumed to be
smaller than the transverse dimension dof the crystal,
ad(see Fig. 1). It should be underlined that the areas
adjacent the electrodes are always suggested to be non-
illuminated. The typical material parameters of these crys-
tals, which are NA≈1022 m−3,µτR=10−10 –10−11 m2/V,
and ε=4.5×10−10 F/m[18], lead to the following es-
timations of the characteristic lengths: LE=10–100 µm,
LA=0.28 µm(at E0=10 kV/cm), LD=0.5–1.6µm,and
L
S≈0.08 µm. If the Gaussian beam’s size a>10 µm,the
terms involving L2
S,L2
D,andL
2
Ain (8) can be neglected. Ad-
ditionally neglecting the diffusion term (kBT/eLEE0), we
can simplify (8)
LELA 1−˜
E2
SC
E2
0!d2˜
ESC
dx2+"2(LE−LA)˜
ESC
E0
+2LELA˜
ESC
E2
0
d˜
ESC
dx−LELA 1−˜
E2
SC
E2
0!1
Id+I
dI
dx#d˜
ESC
dx
−LEE0 1−˜
E2
SC
E2
0!1
Id+I
dI
dx−˜
ESC =0.(11)
In the linear approximation and in absence of the trap satura-
tion (LA=0), the space-charge field can be expressed in the
explicit form
˜
ESC =−LEE
0
I
d+I
dI
dx.(12)
The magnitude of the space-charge field in the linear approx-
imation can be estimated as ˜
ESC ≈E0LE/a. Increasing either
the applied field, E0, or the parameter µτRof a crystal, we
must take into account nonlinearterms of the second order in
respect to ˜
ESC(x)in (11):
˜
ESC −LE
E0(Id+I)
d
dx(Id+I)˜
E2
SC=−LEE
0
I
d+I
dI
dx.(13)
To demonstrate the borders of applicability of (11)–(13),
we calculate the distribution of the space-charge field for
a=20 µm,Id/I0=0.1andE
0=10 kV/cm varying the ratio
LE/a. It was found that in the limit of the short drift length,
LE/a1, the linear approximation of (12) gives the same
result as the solution of both (11) and (13). The solid curve
in Fig. 2 relates to the solution of (13) and it was calculated
for µτR=2×10−12 m2/V(LE=2µm)andL
A=0.25 µm.
The linear approximation (dotted line) differs from the exact
solution no more than 4% near the extremes of the space-
charge-field distribution. It is seen in Fig. 2 that increasing
the drift length up to the value of 4µmleads to apprecia-
ble difference between the linear approximation (dotted line)
and solution of (13) (solid line). Note that (13) possesses
the asymptotic solution in the form of the linear function
[˜
ESC(x)=bx]at xaonly when LE<0.25a√1+Id/I0.
For these drift lengths, the solution of (13) coincides with
that of (11). The space-charge-field distribution at the large
drift length, LE>0.25a√1+Id/I0, can be correctly de-
scribed by (11), which takes into account the trap satura-
tion. These results are shown in Fig. 3 by solid lines calcu-
lated for LE=8µmand LE=20 µm. Dotted lines in Fig. 3
correspond to the solutions of (13) for the same values of
LE. One can see that these solutions have a discontinu-
ity at x=0 and these are different from the correct solu-
tion of (11) in the neighborhood of this point. As follows
from Figs. 2, 3, the approximate solutions give an overes-
timated space-charge field. Naturally, the space-charge field
Fig.2. The distribution of space-charge field created by the Gaussian beam
with size a=20 µmfor drift lengths of LE=2µmand LE=4µmat E0=
10 kV/cm,LA=0.25 µmand Id/I0=0.1. The dotted curves are the linear
approximation given by (12), and solid curves are numerical solutions of
(11) and (13), which are coincided
1010
Fig.3. The distribution of space-charge field created by the Gaussian beam
with size a=20 µmfor drift lengths of LE=8µmand LE=20 µmat
E0=10 kV/cm,LA=0.25 µmand Id/I0=0.1. The solid and dotted
curves are the numerical solutions of (11) and (13), respectively
given by the correct solution is limited by the applied field
E0.Note that peculiarities of the space-charge-field distri-
bution discussed above may cause beam fanning during its
propagation in the crystal under an applied field. The nonlin-
ear effects of this kind have been considered for a crystal with
the strong diffusion nonlinearity[21].
3 Photorefractive gratings at the large modulation
Analyzing the large modulation effects in the case of a pho-
torefractive grating, we restrict our consideration to the non-
linear terms of the second order in respect to the space-charge
field in (8):
2L2
SLE˜
ESC
E0
d3˜
ESC
dx3−LELA+L2
S+L2
D−2L2
SLE
E0
× d˜
ESC
dx+˜
ESC
Id+I
dI
dx!# d2˜
ESC
dx2−"2(LE−LA)˜
ESC
E0
−(LELA+2L2
S−3L2
D)1
Id+I
dI
dxd˜
ESC
dx
+"1−1
Id+I LE˜
ESC
E0
dI
dx+2L2
Dd2I
dx2!# ˜
ESC
+LEE0+kBT
e1
Id+I
dI
dx=0.(14)
The light intensity distribution inside the crystal is taken in
the form
I(x)=I0(1+mcos(Kx))P(x), (15)
where I0is the mean light intensity, mis the modulation in-
dex, K=2π/Λ is the grating wave number, and P(x)is the
slowly varying envelope function. This function is concerned
with an intensity shape of the light beams that create an in-
terference pattern. When the interference pattern is created by
Gaussian beams, their half-width, a0,isassumedtosatisfy
the inequalities LE,Λa0d. Therefore, influence of the
slowly varying envelope functionon the space-charge field of
the grating can be ignored. Moreover, the envelope function
has to provide the localization of the illuminated layer in the
middle of the crystal and existence of the light-free areas near
the electrodes. Note that nonlinear terms ˜
E2
SC(d2˜
ESC/dx2),
˜
ESC(d˜
ESC/dx)2,and ˜
E2
SC(d˜
ESC/dx), which were omitted in
(8), are of the third order in respect to spatial harmonic ampli-
tudes of the grating. To get a simple analytic expression, we
represent the space-charge field without consideration of high
spatial harmonics in the form
˜
ESC(x)=E1
2exp(iKx)+c.c. ,(16)
where c.c. means complex conjugate. Substituting (15) and
(16) into (14) and ignoring the dark irradiance Id, we obtain
the amplitude of the fundamental harmonic:
E1=iE01+K2(LELA+L2
S+L2
D)
mK(LE+2LA)+2K3L2
SLE/2(17)
−s1+K2(LELA+L2
S+L2
D)2
+m2(E0/Eµ+ED/E0)K(LE+2LA)+2K3L2
SLE
mK(LE+2LA)+2K3L2
SLE/2,
where Eµ=(KµτR)−1is the drift field and ED=KkBT/e
is the diffusion field. The nonlinear analytic model given by
(18) takes into accounts the trap saturation and diffusion con-
tribution of the charge redistribution. Therefore, it is also
valid for gratings with fringe spacing smaller than the drift
length. However, the limitations on the amplitudes of the fun-
damental space-charge field, E1<E0, and its second spatial
harmonic, E2E1, must be obeyed. Taking into account the
second harmonic in the expansion for space-charge field al-
lows us to determine its amplitude E2and to define more
exactly the amplitude of the fundamental grating. Here the
equations for E1and E2can be written as
m
4E0K(LE+2LA)+2K3LEL2
SE2
1
−i1+K2(LELA+L2
S+L2
D)E1
+1
E0K(LE−LA)+5K3LEL2
SE1E2
+m
2E0KLE+8K3LEL2
SE2
2
−im
21+6K2LELA+8K2L2
S−K2L2
DE2
+mE2
0
Eµ+ED=0,(18)
E2=−m(1+6K2L2
D−K2L2
S)E
1−i2[K(LE−LA)+2K3LEL2
S]E
2
1/E
0
21+4K2(LELA+L2
S+L2
D)+im(KLA+2K3LEL2
S)E1/E0.
(19)
Figure 4 shows the amplitude of a fundamental harmonic
normalized to the modulation index and calculated by (17)
as a function of the fringe spacing for various mat µτR=
1011
10−11 m2/V,NA=1022 m−3,ε=5×10−10 F/mand E0=
10 kV/cm. Note that E1has only the imaginary part. The
space-charge field calculated using the standard linear ap-
proximation [1,3] is also shown in Fig. 4. It can be seen
that the nonlinearity of a photorefractive response is most
pronounced at the maximum of this curve. The noticeable
nonlinearity takes place for m>0.1 if the listed above crystal
parameters and applied electric field are used in the numerical
simulation.
The magnitude of the second harmonic can be estimated
substituting the solution for E1from (17) into (19). Figure 5
shows the amplitude of the second harmonic normalized to
E1versus the fringe spacing in this case. It is evident that this
approach provides a reasonable accuracy for m<0.1 only.
Similar dependencies for E1and E2calculated from (18) and
(19) are shown in Figs. 6 and 7, respectively. The curve cal-
culated for m=0.01 in Fig. 6 exactly repeats behavior of the
space-charge field in the standard linear approximation shown
in Fig. 4. However, if the second harmonic is considered, the
nonlinearity of the fundamental harmonic relative to the mod-
ulation index is stronger than (17) suggests. Note that the
fundamental component of the grating is weakly dependent
on the fringe spacing at Λ>5µmfor the modulation index
m=0.4.
Comparing Figs. 5 and 7, we conclude that the second
harmonic calculated from the analytic expressions (17) and
(19) is overestimated compare to the joint solution of (18)
and (19). One can see from Fig. 7 that the second spatial har-
monic peaks at the fringe spacing Λ=15 µmand becomes
comparable with the fundamental component of the grating
for m>0.2. It is evident that the borders of applicability of
(18) and (19) to the analysis of large modulation effects can
be determined by the numerical integration of (14).
4 Conclusion
In contrast to the approach that uses presentation of the space-
charge field as a superposition of different spatial harmonics,
we have derived the nonlinear differential equation describing
the space-charge-field formation for illumination by a one-
dimensional light pattern of a general view. This equation is
valid at the steady state when a square-wave field of the high
Fig.4. Normalized amplitude of the fundamental harmonic of the
space-charge field versus the grating fringe spacing for various mod-
ulation index at E0=10 kV/cm,µτR=10−11 m2/V,NA=1022 m−3,
ε=5×10−10 F/m.Thethin solid curve is the standard linear approach and
the curves at m=0.1, 0.2, and 0.4 are calculated by (14)
Fig.5. The amplitude of second harmonic normalized to the fundamental
component versus the grating fringe spacing for various modulation index
calculated by (17) and (19) at the same conditions as in Fig. 4
Fig.6. Normalized amplitude of the fundamental harmonic of the space-
charge field versus the grating fringe spacing for various modulation index
calculated by the solution of the set (18) and (19) at E0=10 kV/cm,
µτR=10−11 m2/V,NA=1022 m−3,ε=5×10−10 F/m
frequency is applied to the crystal over a wide range of the
spatial inhomogeneity’s characteristic size of the light distri-
bution. For the case of the Gaussian beam, we have compared
the analytic (in the linear approximation) and two numeri-
Fig.7. The amplitude of the second harmonic normalized to the funda-
mental component versus the grating fringe spacing for various modulation
index calculated by the solution of the set (18) and (19) at the same condi-
tions as in Fig. 6
1012
cal solutions of three simplified equations. It is shown that
the trap saturation should be taken into account for large
drift lengths: LE>0.25a√1+Id/I0. The linear approxima-
tion holds at short drift lengths: LEa.
We have also considered the nonlinear dependence of the
amplitude of the photorefractive grating’s fundamental har-
monic on the intensity-modulation index. The derived ana-
lytic expression of the fundamental harmonic can be appli-
cable over a wide range of the fringe spacing but it provides
a reasonable accuracy for m<0.1 only. The amplitude of the
grating’s fundamental component was also calculated taking
into account the second spatial harmonic.
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