Electronic excitation of space charge waves in photorefractive materials
V. A. Kalinin and L. Solymar
Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, United Kingdom
͑Received 11 June 1996; accepted for publication 30 October 1996͒
The excitation of a space charge wave in a uniformly illuminated photorefractive crystal by an
applied ac voltage is studied theoretically. A spatial differential equation for the ac component of the
space charge field is derived and boundary conditions applicable both to Ohmic and rectifying
junctions are obtained leading to an analytical solution for the space charge wave amplitude. It is
shown that the low frequency space charge wave associated with the recharge of impurities can be
efficiently excited with an amplitude approximately equal to the applied ac field. For the parameters
of a BSO crystal, the space charge wave region declines within a few grating spacings near the
cathode. The possibility of experimental observation of the effect is discussed. © 1996 American
Institute of Physics. ͓S0003-6951͑96͒04653-0͔
Optical signal processing has been in the forefront of
interest for quite some time. One of the techniques widely
used is to deflect an optical wave by a grating which may be
produced by a variety of means, e.g., mechanical, optical, or
acoustic.
The aim of this letter is to propose a novel way of gen-
erating a grating in the form of an electronically excited
space charge wave ͑SCW͒. It is a wave based on the viola-
tion of space charge neutrality on a drifting electron beam.
The periodic variation of charge density is accompanied by a
space charge electric field. If the wave propagates in a pho-
torefractive material ͑which is electro-optic by definition͒
then the space charge electric field will give rise to a periodic
variation of the index of refraction, therefore we have a grat-
ing.
How can we excite a SCW? It can be done by illuminat-
ing a photoconductor by two optical beams incident at an
angle relative to each other. The resonant interactions be-
tween two such beams found by Huignard and Marrackchi
1
and Refregier et al.
2
were indeed interpreted ͑Stepanov
et al.,
3
Furman
4
͒ as three-wave interactions where the third
wave was a SCW. There is thus no doubt that a SCW, cre-
ated by the optical waves themselves, can affect the optical
waves. The efficiency of excitation can be very high ͑as high
as 50%͒ as has been shown recently.
5
However the problem
of electronic excitation has not so far been addressed al-
though some indirect evidence was provided by Zhdanova
et al.
6
who measured resonance peaks in the impedance of a
piece of bulk, compensated germanium.
How would we go about exciting a SCW? The basic
configuration is shown in Fig. 1͑a͒ where both a dc voltage
of magnitude V
0
and an ac voltage of amplitude V
1
are ap-
plied to a photorefractive material. The presence of a dc field
assures that a SCW with a well-defined dispersion relation-
ship can exist in the material. The frequency of the ac field,
⍀, will determine the frequency of the SCW and the corre-
sponding wave number, K, can then be obtained from the
dispersion relationship.
We shall start the mathematical treatment with the ma-
terials equations of the Kiev group
7,8
which describe the in-
terrelations between ionized donor density ͑from where elec-
trons are liberated by the incident light͒, electron density,
current, and electric field in a photorefractive material in the
absence of a photovoltaic effect. Using the method and ap-
proximations described in detail by Solymar et al.
9
and as-
suming that the space charge electric field is small in com-
parison with the applied dc field so that nonlinear terms can
be neglected, it is straightforward to derive a single partial
differential equation in terms of the space charge electric
field
ץ
2
E
ץ
x
ץ
t
ϩ
ͩ
1
r
ϩ
1
d
ͪ
ץ
E
ץ
t
ϩ
E
r
d
ϪD
ץ
3
E
ץ
x
2
ץ
t
ϩ
ץ
2
E
ץ
t
2
ϭ
1
⑀⑀
0
ͩ
ץ
j
t
ץ
t
ϩ
j
t
r
ͪ
, ͑1͒
where
ϭ
E
0
is the electron drift velocity,
is the mobil-
ity, E
0
is the applied dc field, D is the diffusion constant,
d
ϭ
⑀⑀
0
/
is the dielectric relaxation time,
r
is the electron
recombination time,
ϭe
n
0
is the conductivity, e is the
electron charge, n
0
is the average density of electrons,
⑀
and
⑀
0
are relative and free space permittivity, respectively, and
j
t
is the total current.
FIG. 1. ͑a͒ Geometry of the problem, ͑b͒ a typical spatial distribution of the
ac electric field amplitude in the crystal.
4265Appl. Phys. Lett. 69 (27), 30 December 1996 0003-6951/96/69(27)/4265/3/$10.00 © 1996 American Institute of Physics