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Abstract. Excitation of nonlinear surface waves is studied at
the SBN-75 crystal ë air interface. The SBN-75 crystal is
characterised by the diffusion-type photorefractive nonlinear-
ity, the surface wave being excited in the surface layer and
the depth of its penetration into the crystal being determined
by the nonlinear crystal parameters. Because of a large
refraction coefécient in SBN-75, the penetration depth of the
surface wave into the air is small; therefore, the nonlinear
surface wave is localised in the surface crystal layer several
micrometers in thickness. The oscillating intensity distribu-
tion of the surface wave at the output end of the crystal is
measured. The oscillation period is determined by the angle of
incidence of the exciting beam. The nonlinear wave is excited
not only at the crystal ë air interface but also in the case
when the active surface of the crystal is covered by an
electrode, for example, an aquadag layer. This circumstance
opens up a possibility of studying new properties of the
surface waves when an external electric éeld is applied to the
crystal.
Keywords: photorefractive crystal, nonlinear surface waves.
1. Introduction
By now most papers have studied the nonlinear propaga-
tion of light in media with the Kerr nonlinearity. This
nonlinearity affects directly the light beam phase, which
allows one to compensate for the phase diffraction effect.
However, the nonlinear susceptibility w3responsible for the
Kerr effect is relatively small and that is why light
intensities of no less than 1 MW cmÿ2are required to
produce non-diverging beams. In this case, photorefractive
crystals, in which the nonlinear regimes of wave interactions
manifest themselves already at the intensities smaller than
1Wcm
ÿ2
, have a signiécant advantage.
The photorefractive effect has many stages: light energy
absorption, formation of free electrons and holes, their
transfer and trapping followed by the production of bulk
charge regions. After this charge, the electrooptical effect
forms a spatial refractive index distribution of light, which
in turn, interacts with the wave front of propagated
radiation and changes it. Photorefraction was discovered
by Ashkin and his colleagues in 1966 [1]. Two years later
Chen used this phenomenon to write phase holograms and
nowadays it is widely used for data processing [2].
In 1995 Mexican scientists performed an analysis of the
nonlinear surface waves in photorefractive crystals with the
diffusion mechanism of nonlinearity [3]. These guided and
transform-limited waves can propagate along the photo-
refractive crystal ë air (metal) interface. In this paper, we
study self-channelling of a light wave along the surface of
the photorefractive crystal without an initially prepared
waveguide layer.
2. Surface waves
Consider propagation of extraordinary TM-polarised light
along the zaxis directed along the photorefractive crystal ë
air interface (Fig. 1) [4]. Because we study the grazing
propagation of light, the anisotropy of the refractive index
will be neglected. The magnetic éeld component of the
surface wave, H(x,z), along the yaxis satisées the equation
H2Hx;zk2
Hx;z0, (1)
where kk0n2Dn(x);k02p=l0;l0is the light wave-
length in vacuum; n2is the unperturbed refractive index of
the crystal; Dn(x)is the nonlinear addition to the refractive
B.A. Usievich, D.Kh. Nurligareev, V.A. Sychugov, L.I. Lykov, N.V.
Bogodaev A.M. Prokhorov General Physics Institute, Russian Academy
of Sciences, ul. Vavilova 38, 119991 Moscow, Russia;
e-mail: borisu@kapella.gpi.ru
Received 30 September 2009; revision received 22 March 2010
Kvantovaya Elektronika 40 (5) 437 ë 440 (2010)
Translated by I.A. Ulitkin
PACSnumbe rs:42.70.Nq; 73.20.Mf; 42.65.WiSURFACE WAVES
DOI:10.1070/QE2010v040n05ABEH014223
Nonlinear surface waves on the boundary
of a photorefractive crystal
B.A. Usievich, D.Kh. Nurligareev, V.A. Sychugov, L.I. Ivleva, P.A. Lykov, N.V. Bogodaev
023/912 ë VOLO ë 29/vii-10 ë SVERKA ë 4 ÒÑÎÑÔ ÍÑÏÒ. å 1
Quantum Electronics 40 (5) 437 ë 440 (2010) ß2010 Kvantovaya Elektronika and Turpion Ltd
y
L
Iinc
Iref
x
Isw
n11
n22:35
z
Figure 1. Scheme of surface-wave excitation and propagation: Iinc ,Iref
and Isw are the incident, reêected, and surface waves, respectively.
index n2. The stationary solution for the éeld distribution in
the crystal is sought for in the form H(x,z)A(x)exp (ibz),
where bis the surface-wave propagation constant.
The nonlinear addition Dn(x)can be represented as a
result of action of the diffusion mechanism of nonlinearity
[5]:
Dnx 1
2n3
2reff
kBT
q
1
Ix Id
d
dxIx Id, (2)
where qis the electron charge; reff is the effective
electrooptical coefécient; kBis the Boltzmann constant; T
is the temperature; I(x)/ jA(x)j2is the surface-wave
intensity; Idis the dark intensity.
Note that expression (2) demonstrates the disparity of
two directions of the xaxis appearing due to the fact that
the crystal is polarised in the direction of the xaxis
(coinciding with the crystallographic axis c).
When the dark intensity Idis small compared to the
surface-wave intensity, it can be neglected and equation (1)
can be written in the form
d2Ax
dx22k2
0n4
2reff
kBT
q
dAx
dxÿk2
0n2
2ÿb2Ax
expibz 0, (3)
where exp (ibz)describes the established surface-wave
propagation in the direction of the zaxis. In this case,
equation (3) can be simpliéed:
d2Ax
dx2gdAx
dxÿk2
0n2
2ÿb2Ax 0, (4)
where g2k2
0n4
2reffkBT=q. The eigenvalues of equation (4)
have the form l1;2 ÿg=2(g2=4b2ÿk2
0n2
2)1=2.
When b<(k2
0n2
2ÿg2=4)1=2, the surface-wave amplitude
is
Ax expÿgx=2cos ÿk2
0n2
2ÿb2ÿg2=41=2xj, (5)
where 2jis the phase difference of incident and reêected
light beams producing an interference pattern. The surface-
wave amplitude decreases exponentially, experiencing oscil-
lations.
When b>(k2
0n2
2ÿg2=4)1=2, the éeld amplitude is
Ax c1exp ÿg=2ÿb2ÿk2
0n2
2g2=41=2x
c2exp ÿg=2ÿÿb2ÿk2
0n2
2g2=41=2x. (6)
This case corresponds to the nonoscillating proéle of the
surface-wave amplitude and can be observed only at small
grazing angles of the beam forming the surface wave.
In air the surface-wave amplitude is
Ux U0exp ÿb2ÿk2
0n2
11=2x, (7)
where n1is the refractive index of air. When b>kn1, the
éeld amplitude decreases exponentially while moving away
from the crystal boundary. The continuity conditions for
tangential éeld components at the photorefractive crystal ë
air interface yield the equalities
Uxjx0Axjx0,1
e1
qUx
qx
x0
1
e2
qAx
qx
x0
, (8)
where e1is the dielectric constant of air; e2is the dielectric
constant of the photorefractive crystal.
By applying continuity condition (8) to equations (5)
and (7), we obtain
jarctan e2
ÿk2
0n2
2ÿb2ÿg2=41=2
ÿb2ÿk2
0n121=2
e1
g
2e2. (9)
The surface-wave propagation constant bis related to the
angle of incidence of the exciting light onto the interface, b,
by the expression
bk0n2sin y(10)
(note that the caxis of the crystal is directed along the
normal to the interface).
By using the above expressions, we can determine, for
the given angle of incidence y, the surface-wave amplitude
distribution. This wave has a characteristic éeld oscillation
period
L2p
ÿk2
0n2
2cos2yÿg2=41=2(11)
and penetrates into the photorefractive crystal to the depth
d2=g. (12)
It is known that the light propagates in a conventional
waveguide in the form of discrete modes characterised by
their propagation constants, the éeld distribution across the
waveguide and the angle of incidence of exciting radiation
on the waveguide. The mentioned characteristics of the
radiation modes are determined by the fuléllment of the
transverse resonance condition required to realise the wave-
guide propagation of light. However, for such a waveguide
structure as the photorefractive crystal ë air interface, the
transverse resonance condition is not necessary for the wave
propagation.
Photorefractive surface waves as well as waveguide
modes are characterised by the propagation constant, the
wave éeld distribution across the direction of the wave
propagation and the angle of incidence y. However, the
surface modes of the photorefractive crystal signiécantly
differ from the modes of ordinary waveguides. Because the
attenuation of the mode éeld inside the photorefractive
crystal is determined by the photorefractive effect and is
independent of the wave phase, the necessity to satisfy the
boundary condition stands no longer and the mode spec-
trum becomes continuous, i.e. this mode can exist at any
propagation constant band, hence, at any angle of incidence
of exciting radiation.
Surface electromagnetic waves can propagate along the
interface of isotropic media whose dielectric constants have
different signs. Thus, the propagation of the éeld deep inside
the media is related to the purely imaginary value of the
wave number in a medium with a negative dielectric
constant and to the fuléllment of the condition for total
internal reêection (TIR) in a medium with a positive
dielectric constant. The requirement to change the sign
of the dielectric constant with passing through the interface
438 B.A. Usievich, D.Kh. Nurligareev, V.A. Sychugov, et al.
appears due to the necessity of matching the continuity
condition of the tangential éeld components at the medium
interface and the exponential decrease in the éeld ampli-
tudes on either side of the interface.
For the surface wave to exist at the photorefractive
crystal ë isotropic dielectric (air) interface, the change of the
sign of the dielectric constant is not required.
3. Experimental setup for studying surface waves
in photorefractive crystals
The aim of our investigations was to excite and record the
surface waves at the linear medium ë photorefractive crystal
interface and to check the mode concept for the nonlinear
surface waves. As a photorefractive medium we used
optical elements made of single crystal SBN-75 solid
solution of barium strontium niobate (SrxBa1ÿxNb2O6)
characterised by the electrooptical coefécient for extraordi-
nary polarisation reff 750 pm Vÿ1.
The problem of excitation of the surface waves at the
linear medium ë photorefractive crystal interface requires
the solution of a number of problems. The érst of them
consists in the fact that érst of all it is necessary to énd that
photorefractive crystal surface where the surface wave can
be excited most eféciently. In papers [6, 7], where the surface
wave was obtained, this surface is ambiguously; therefore, it
is needed to énd the criterion for determining this surface.
Such a criterion, in our opinion, is the direction of the
fanning effect, which was érst described in paper [8]. This
effect consists in photoinduced scattering of extraordinary
polarised light propagating in the direction perpendicular to
the caxis of the photorefractive crystal. The surface on
which the nonlinear surface wave can be excited should have
a normal directed opposite to the spontaneous polarisation
vector Ps(or the direction of the caxis).
Figure 2 shows the photograph of the fanning beam
from a 0.44-mm He ë Cd laser obtained on the screen placed
behind the SBN-75 crystal, whose caxis is perpendicular to
the direction of incidence of the He ë Cd-laser beam with
extraordinary polarisation. The fanning region has a pear-
like shape elongated in the negative direction of the caxis
from the spot of the incident laser beam.
Figure 3a shows the scheme of the experimental setup we
used to study the peculiarities of the surface-wave exciation
at the air ë SBN-75 crystal interface. The angle of incidence
of the exciting light beam onto the crystal, a, is related to the
angle of incidence onto the internal crystal surface, y, by the
expression
sin an2cos y. (13)
Nonlinear surface waves were previously excited by two
ways. The érst method [6] is excitation through the sample
end face, exciting radiation propagating along the same face
as the surface wave. At the end of this face there appears the
problem of separation of two waves for detection. The
authors of paper [6] used a near-éeld microscope to detect
the wave éeld intensity distribution at the output end face of
the crystal.
In the second excitation method [7] (see Fig. 3b), the
surface and excitation waves are separated initially because
the surface wave appears due to the TIR of the excited light
incident onto the crystal surface. In this case, the surface
wave propagates along the crystal surface, while the exciting
reêected wave propagates deep inside the photorefractive
crystal at the TIR angle. In the output plane these waves are
spatially separated and can be detected apart. Figure 4
presents the photograph of the radiation spot of the surface
wave obtained at the SBN-75 sample output with the help of
the 330optical system. This allowed us to detect clearly on
the screen the éeld distribution and to measure the period of
spatial surface-wave oscillations. However, detection of the
waves can cause some diféculties if the images of the output
waves are overlapped. Images can be overlapped due to the
fanning effect inherent in the exciting and reêected waves.
cc
Figure 2. Photograph of the fanning effect on the screen after pro-
pagation of the laser beam through the SBN-75 crystal.
SBN-75 crystal
Incident
wave
c
Reêected wave
Surface wave
a
b
Propagated wave
89
1
1
1
2 2 36
a
4
5
7
Figure 3. Optical scheme of the experiment (a) and scheme of surface-
wave excita tion in the SBN-75 crystal (b): ( 1) mirror; ( 2) polarisers; ( 3)
long-focus lens; ( 4) sample; ( 5) microobjective; ( 6) goniometer; ( 7)
screen; ( 8) polarisation rotation unit; ( 9) He ë Cd laser.
SSuurrffaacceewwaavvee
PPrrooppaaggaatteeddwwaavvee
Figure 4. Light intensity distribution at the output crystal end face in the
case of the surface-wave excitation.
Nonlinear surface waves on the boundary of a photorefractive crystal 439
First experiments on surface-wave excitation in the SBN-75
crystal showed that when the angle of incidence is y898
and the exciting beam intensity is chosen correctly, the
problem of overlapping output radiation spots of separated
waves does not appear because of the strong depletion of the
reêected wave due to the fanning effect on the way to the
output plane of the crystal after the TIR. The path length L
of the light beam after exciting the surface wave is 9 mm so
that the output beam intensity decreases by three times
compared to the surface-wave intensity.
The authors of paper [9] pointed out that the fanning
effect is most strongly pronounced in the blue region of the
spectrum. Therefore, the decrease in the exciting radiation
intensity at l0:44 mm is quite expected. Besides, to reduce
the inêuence of the fanning effect on the surface-wave
excitation process, we deliberately chose the excitation
region at the beginning of the photorefractive crystal.
Figure 5 presents the dependence of the oscillation
period of the mode éeld intensity of the surface wave in
the SBN-75 crystal on the angle of incidence of this mode, a.
According to the above concept of the surface-wave modes
[4] the oscillation period is described by relation (11).
Comparison of our experimental dependence with the
dependence calculated by expression (11) shows their
satisfactory coincidence. Thus, veriécation of the concept
of surface-wave modes can be considered as completed.
4. Results and discussion
Photorefractive SBN-75 crystals have a high refractive
index and, according to expression (8), a small éeld
amplitude of the surface wave in air. This determines the
losses when the surface wave propagates on the crystal
surface due to the light scattering. Usually, the SBN-75
crystal is monodomenized by depositing aquadag electrodes
on its surface and applying the voltage to these electrodes.
One of the surfaces is used to excite nonlinear waves in the
SBN-75 crystal. We obtained for the SBN-75 crystal a
surface wave even when aquadag electrodes were deposited
on its surface. This indicates that the scattering losses of the
surface wave are indeed low, removal of the aquadag from
the crystal surface having no effect on the wave prop-
agation. Low light scattering losses indicated, érst of all,
the small éeld amplitude in air; therefore, the nonlinear
surface waves in photorefractive crystals can be most likely
called near-surface waves whose éeld is concentrated below
the crystal surface. Ordinary linear surface waves in
waveguides can be excited due to the tunnel coupling,
for example, by using a prism for coupling light into the
waveguide. In the case of photorefractive crystals whose nis
large and the éeld in air is small, this coupling is
complicated and the experiments similar to those in
paper [10] are unlikely to be performed. However, a
prism with a circular base [11] can be obviously used to
detect the nonlinear surface waves.
5. Conclusions
Analysing the theory of nonlinear surface-wave modes and
experiments on thier excitation at the SBN-75 photo-
refractive crystal ëair interface, we have found the
possibility of the surface-wave excitation at rather low
radiation powers of the He ë Cd laser (0.5 ë 12 mW, l
0:44 mm).
We have found in the experiments that depositing an
electrode layer on the active crystal surface does not
introduce any noticeable additional losses caused by prop-
agation of surface waves in the SBN-75 crystal. Applying
the external electric éeld to these electrodes will allow
further investigation of it inêuence on the propagation of
the surface waves in this crystal.
Acknowledgements. The authors thank V.V. Osiko for
constant interest in this work. This work was supported by
the Russian Foundation for Basic Research (Grant No. 07-
02-00064).
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Lmm
2
3
4
5
6
7
8
9
10
2
1
1.5 2.0 2.5 3.0 3.5 adeg
Figure 5. Theoretical ( 1) and experimental ( 2) dependences of the
oscillation period of the surface-wave intensity on the angle of incidence
of light onto the crystal end face.
440 B.A. Usievich, D.Kh. Nurligareev, V.A. Sychugov, et al.
