Download Enhanced Nonlinear Optical Response from a Stee and Gently

264
Brazilian Journal of Physics, vol. 27/A, no. 4, december, 1997
Enhanced Nonlinear Optical Response from a Stee
and Gently Sloping Low Relief Surface:
What is the Dierence?
R.Z. Vitlina, G.I. Surdutovich, V. Baranauskas
DSIF/FEEC, Unicamp, Cx.P.6101, Campinas - SP
Received February 2, 1997
The second-order eective nonlinear susceptibility of a low relief interface between two media
with nonlinear susceptibilities is calculated analytically for the cases of steep and gently sloping elongated periodical irregularities. An enhancement of the nonlinearity occurs for steep
relief irregularities only under a suciently large ratio of the linear dielectric permittivities
of these media.
I. Introduction
Heterogeneous composite materials such as optical
bers, amorphous and polycrystalline semiconductors,
two-dimensional structures on a surface and etc. are
widely used nowadays in optoelectronics and dierent
devices for propagation of light. Interest in composites
arises since their optical properties can dier dramatically from those of the constituent components. One of
the rst eective - medium theories to deal with optical
properties of a composite material comprised of spherical mesoscopic impurities embedded in a host medium
was given by Maxwell Garnett [1], who considered the
linear response of metallic inclusion particles and was
able to explain the colors of metallic colloids. In several
articles [2-5] the analysis was extended to the case of
nonlinear optical media. It was shown that due to the
local eld eects a composite can possess a greater (enhanced) eective nonlinear susceptibility than any of
its components. In the case of a third-order susceptibility the predicted enhancement was recently observed
experimentally for a composite structure of alternating
sub-wavelength-thick layers[6].
When two components are intermixed over a distance much smaller than the wavelength of light but
still large compared with the interatomic distance then
the linear and nonlinear susceptibilities of each constituent material are essentially the same as those of
a bulk sample of the material. At the same time, the
propagation of light through the composite can be described by spatially averaged values of these susceptibilities, i.e. a concept of the eective equivalent homogeneous medium may be introduced. Here we calculate the eective second - order nonlinear susceptibility, , of a new structure: a low relief interface between
two materials possessing second - order nonlinearities.
We obtained analytical solutions for the cases of steep
and gently sloping periodic two-dimensional (2D) low
reliefs. Both lowness and periodicity are essential for
obtaining such solutions. Since the electric eld in
composites is nonuniformly distributed between the two
constituents, then the eective nonlinearity may be enhanced due to the concentration of the electric eld in
a component of greater nonlinearity. We demonstrate
that such an enhancement takes place if the more nonlinear component has the smaller linear refractive index.
Low - relief - surface
Let us examine the reection of light from a periodic elongated low-relief interface between two nonlinear media with linear permittivities 1 and 2 and
second-order susceptibilities ^1 and ^1 . The term low
here means that the period ` and height h of the irregularities are much smaller than the wavelength of the
R. Z. Vitlina et al.
incident radiation, whereas the relation between ` and
h may be arbitrary. A particular case of such a relief
(steep triangular irregularities with h `) is given in
Fig. 1. For linear media this problem was considered
265
earlier in [7], where it was shown, that a low - relief surface, in terms of electromagnetic wave reection, to be
equivalent to a homogeneous but anisotropic thin lm.
Figure 1. A low -relief layer with periodic irregularities of a triangular shape on the interface between ambient medium with
(2)
permittivity 1 and susceptibility (2)
1 , and a medium with permittivity 2 and susceptibility 2 . The sublayer of a thickness
`2
d has f2 (z ) = ` .
The developed approach is analogous to going over
from the microscopic to the macroscopic Maxwell equation in a medium. The ne details of the relief do not
aect the reected radiation. In fact, the reection is
only inuenced by a certain limited amount of information about the relief. Really, spatially nonuniform
elds, which depend on the shape of the relief, vary over
distances comparable to the parameters `; h << .
Therefore, the quasistatic approximation is sucient
for calculating the \microelds". These elds satisfy
the Laplace equation, and their spatially varing part
is known to fall o exponentially with distance from
the interface. Since the minimum rate of this decay is
of the order of the reciprocal dimensions of the relief,
only spatially uniform elds survive outside a narrow
surface layer [7,8]. As far as wave zone begins at distances which are comparable to the wavelength, where
the microelds have decayed, the relief and its shape
aect the scattering only through the \relief-averaged"
elds.
Thus, the problem is reduced to calculation of the
eective permittivity tensor ^ef of a lm of thickness
h, including all irregularities. This tecnique allows the
numerical calculation of all the components of the effective tensor for arbitrary ratio of the parameters `; h.
For a relief with plane faces in the limiting cases of \gen-
266
Brazilian Journal of Physics, vol. 27/A, no. 4, december, 1997
tly sloping;; (` h) and steep (` h) irregularities
this may be done analytically. The particular case of
reection of a normally incident wave from a low relief
interface between two linear media has been considered
in [9] in a somewhat dierent context.
Here, we interested in a process of surface second
harmonic generation (SSHG) from a low relief surface
in these two limiting cases and so should calculate the
eective second-order susceptibility ^(2)ef of the interface layer. Similar to the approach used in Ref.[6], we
introduce the eective susceptibility ^(2)ef in the following way
P~ (2!) = ^(1)ef E (2!) + ^(2)ef E~ (!)E~ (!); (1)
where P~ (2!); E~ (!), E~ (2!) are Fourier transforms of
the polarization and the electric eld at the frequencies
2! and !; ^(1)ef = ^ef ; 1=4 and and (2)ef are the
eective susceptibilities of the equivalent homogeneous
surface layer.
a) Gently sloping irregularities: h `
According to the theoretical analysis of [7] for linear media in this case we may apply the perturbation theory in the parameter h=`. Let the 2D surface
relief be described by a periodic symmetric function
'(x) = '(x + `), '(x) = '(;x). To rst order in the
parameter h=`, only diagonal components of a tensor
^ef are nonzero [7]:
1 + 2 1 ; A h ;
ef
=
xx
2
`
2
h
1 2
ef
zz = + 1 ; A ` ;
1
2
ef
yy = 1(1 ; c) + 2c ;
(2)
where constant
A depends on the form of the relief,
R`
1
c = h` 0 '(x)dx is the concentration of material 2
in the interface layer. The zeroth order terms, under
h=` ! 0; correspond to a smooth boundary, i.e. to the
substitution of the inhomogeneous two - layered (with
permittivities jep1; 2 and thickness h=2 each) lm by
the homogeneous lm with the permittivity tensor ^ef .
The presence of this eective lm results only in an arbitrary change in the common phase of the amplitude
reection coecients, which is irrelevant. Similarly, for
the nonlinear media the presence of the eective lm in
the limiting case h=` ! 0 leads to an eective nonlinear
susceptibility tensor with components:
(2)
(2)
1xij + 2xij
;
2
!
ef
(2)
(2)
(2)ef
1zij
2zij
zz
zij = 2 + ; i; j = x; y; z: (3)
1
2
Such renormalization also results in a change only
in the common phase of the amplitude of SSHG. Therefore, gently sloping irregularities on the interface cannot
induce any enhancement eect and, on the whole, their
inuence on SSHG is negligibly small.
ef
(2)
=
xij
b) Steep symmetric irragularities:h `
We divide the irregularity layer of height h (Fig.1)
into sublayers of thickness d in such a way that the condition ` d h holds. The inequality d h allows
us to ignore the slope of the faces with respect to the z
axis in evaluating of the component xx and zz of one
of the sublayers, while the inequality ` h allows us to
treat the set of irregularities in a sublayer of thickness
d as a ne - layered medium with a normal directed
along x-axis. As a result, the components of the sublayers eective permittivity tensor, which is diagonal
for symmetric irregularities, may be written as
xx (z ) = f (z )1+2 f (z ) ;
2 1
1 2
yy (z ) = xx(z ) = 1 f1 (z ) + 2f2 (z );
(4)
where 1 and 2 are the permittivities of the ambient
and substrate media, and f1 (z ) and f2 (z ) are volume
fractions of these constituents in the sublayer.
Now, when the eective permittivity tensor of the
sublayer is known, we may write out the all components
of the tensor ^(2) (z ) of this sublayer:
c
2
(2)
xxx (2!; z ) = xx (2!; z )[xx (!; z )] Pxxx (!; 2!; z )
R. Z. Vitlina et al.
267
xxi(2!z )
ixx (2!; z )
ijx(2!; z )
xij (2!; z )
ijk (2!; z )
=
=
=
=
=
xx(2!; z )xx (!; z )Pxxi(!; 2!; z )
[xx(!; z )]2Bixx (!; 2!; z )
xx(!; z )Cijx(!; 2!; z )
xx(!; z )Bxij (!; 2!; z )
Aijk (!; 2!; z )
(5)
where
fn (z ) nijk
n (2! )
n=1;2
n=1;2
X fn (z )
X
fn (z ) Bijk =
D
nijk
ijk =
2
(
!
)
(
!
)n (2!) nijk
n=1;2 n
n=1;2 n
X fn
X
fn(z )
Tijk =
P
(6)
nijk
ijk =
2
n (! )
n (2! )n (! )
n=1;2
n=1;2
As a result, we have a homogeneous sublayer with an eective linear permittivity ^(z ) and nonlinear susceptibility
^2(z ) at the height z . In this way, we reduce the problem of calculation of the properties of the interface layer
of a thickness h to the calculation of the inhomogeneous ne-layered structure consisting of sublayers with known
parameters. The calculation of the dielectric permittivity tensor ^ef of the h - layer may be performed in a
straightforward manner [7]:
Aijk =
X
1Z
ef
=
xx
h
fn (z )nijk
h=2
;h=2
Cijk =
X
;1 1
xx (z )dz ; (ef
zz )
h
Z
h=2
;h=2
;zz1 (z )dz;
(7)
whereas the component ef
yy is given by Eq. (2) since for 2D relief it depends only on concentrations of the materials
in a layer and does not depend on the shape of the irregularities.
The calculation of the eective tensor (2)ef may be performed as a generalization of a known procedure for two
component ne-layered alternating structure in [6] for the case of multiple (actually innite) number of components.
The nal result is expressed through the sublayer quantities, Eqs. (5). The most important tensor components are
presented below in an explicit form:
zz (2!)[zz (!)]2 Z h=2 Azzz (!; 2!; z )dz
ef
(2
!
)
=
zzz
h
;h=2 zz (2!; z )[ ; zz (!; z )]2
Z h=2
1
2
ef
zzz (2! ) =
h ;h=2 xx(2!; z )[xx (!; z )] Pxxx(!; 2!; z )dz
Z h=2
ef
2
xx (2!; z ) C (!; 2!; z )dz
[
zz (! )]
ef
xzz (2!) =
xzz
h
;h=2 2zz (!; z )
Z h=2
ef
xx(2!; z )xx (!; z ) D (!; 2!; z )dz
zz (! )
ef
(2
!
)
=
xxz
xxz
h ;h=2
zz (!; z )
Z h=2
ef
xx(!; z ) T (!; 2!; z )dz
ef
zz (! )zz (2! )
(8)
ef
(2
!
)
=
zxz
zxz
h
;h=2 zz (!; z )zz (2!; z )
To obtain concrete results it is necessary to assume a certain model of the shape of the irregularities. As an
example, we present results of the calculations for a triangular relief. In this case the function '(x) has the form:
2h ; l
x ;l=2 x 0
'(x) = 2lh ; 4l +
; x 0 x l=2
l
4
268
Brazilian Journal of Physics, vol. 27/A, no. 4, december, 1997
From Eqs. (7) we obtain [7]
1 2 ln 2 ; ef = 1 2 ; ef = 22
(9)
ef
xx =
zz
2 ; 1 1 yy 2
ln ;21
If one makes the simplifying assumptions that the linear dielectric constants are not frequency dependent, i.e.
1(2!) = 1 (!) and 2 (2!) = 2(!), the nal formulas become more visible. We present explicit expressions for the
most enhanced components of the eective tensor ^(2)ef :
xxx
(2)ef
1 (2) ;
= 21 (2)
1xxx +
22 2xxx
1
zzz
(2)ef
= 21 1ln;12
1 2
2
!3
(
(2)
(2)
1zzz
2zzz
+
1
2
)
(10)
For = 21 = 10 the estimation for enhanced components gives
(2)
(2)
(2)
(2)
xxx 52xxx ;
zzz 32zzz :
ef
The reason for the greatest enhancement of the (2)
xxx
component is evident, since a steep relief presents maximal screening for the electric eld in the x-direction.
As far as there is no screening at all of the eld
along the y-axis, values of the components with indices
iyy; yiy; yyi(i = x; z ) are the smallest.
Conclusion
We have demonstrated that gently sloping low relief
irregularities do not inuence the nonlinear properties
of the interface between two media. In contrast, quite
a sizeable enhancement of the second-order nonlinear
properties of the interface layer with steep low relief
periodic irregularities may be achieved. The most enef
(2)ef
hanced are the (2)
components of the efxxx and zzz
fective tensor. The degree of the enhancement increases
with an increase in the ratio = 1 =2 of the linear permittivities of the ambient and substrate media, while
the electric eld amplitude becomes concentrated in the
more nonlinear component. The nal characteristics of
SSHG strongly depend on the specic form of the substrate medium tensor ^(2)
2 . Of future interest is the case
of the asymmetric steep irregularities. If due to symmetric properties of the tensor ^(2)
SHG for a at sur2
face is absent, then all SSHG eects will be connected
with a presence of low-relief asymmetric irregularities.
d
Acnowledgments
We are grateful to Prof. R.W. Boyd for useful discussions and Dr. Steven Durrant for reading of the
text. This work is supported by the Brazilian Research
Council (CNPq).
References
1. Maxwell Garnett, Phil.Trans.R.Soc.,London, 203,
385 (1904).
2. J.W. Haus, R. Ingura and C.W. Bowden, Phys.
Rev. A. 40, 5729 (1989).
3. D.S. Chelma and D.A.B. Miller, Opt. Lett. 11,
522 (1986).
4. J.P. Sipe and R.W. Boyd, Phys.Rev.A 46,1614
(1992).
5. R.W. Boyd and J.E. Sipe, JOSA B11, 297 (1994).
6. G.L. Fisher, R.W. Boyd, R.J. Gehr, S.A. Jenekle,
J.A. Osaheni, J.E. Sipe and L.A. Weller- Brophy,
Phys. Rev. Lett. 74, 1871 (1995).
7. R.Z. Vitlina and A.M. Dykhne, Sov.Phys. JETP,
72(6),983 (1991).
8. R.Z. Vitlina, Sov. Phys. Poverhnost 3, 50 (1991).
9. D.E. Aspnes, Phys. Rev. B41, 10334 (1990).