Download Length Scales Analysis of Wave Scattering from Rough Surfaces

Indian Journal of Science and Technology, Vol 8(16), DOI: 10.17485/ijst/2015/v8i16/64930, July 2015
ISSN (Print) : 0974-6846
ISSN (Online) : 0974-5645
Length Scales Analysis of Wave Scattering from
Rough Surfaces
M. Salami∗
Department of Physics, Shahrood Branch, Islamic Azad University, Shahrood, Iran; [email protected]
Abstract
The aim is to study surface characteristics through scattered wave intensity in frame work of the Kirchhoff wave theory.
There are three length scales which scaling behavior between them play role in rough surface scattering, wave-length (λ)
of the incident wave, the roughness (σ) and the correlation length (ξ) of the surface. In this work we show the effective
role of the correlation length in surface scattering. Up to now, some of the reports for wave scattering from rough surfaces
are based on the product of the wave number and surface roughness kσ without consideration to correlation length. For
λ  x, correlation length has no effects on the properties of the scattered wave and kσ is the appropriate parameter to
study the rough surfaces scattering problem. But, if ξ and λ are in comparable range, scattered wave depends on how kσ is
­chosen by k or σ. In this case, kσ is not suitable to present wave scattering from rough surface. In other word, for a constant
ka, ­scattered wave could depend on any of the parameters k or σ. To justify our statements we compare our theoretical
results with experimental data for the intensity profile obtained from a self-affine rough surface. We show that by ­changing
the parameters ξ, λ, σ, and the Hurst exponent, the intensity profiles obtained by theory and numerical estimation ­overlap
when λ is much longer than ξ. But interestingly, the profiles start to diverge when ξ tends to λ. This provides a good
­understanding of the role of the characteristics of the surface on the profile of scattered intensity.
Keywords: Correlation Length, Kirchhoff Theory, Roughness, Wave Scattering
1. Introduction
Roughness effects on scattering intensity have been
­investigated by many researchers for the past few decades.
Kirchhoff theory is one of the most used methods to study
wave scattering problem1-5. It is suitable for rough surface
with roughness is same order and smaller than wavelength, which is based on electromagnetic principles.
Also, the surfaces with slight roughness are approximated
by Rayleigh-Rice theory, which is a perturbed boundary
condition method6. The scattered intensity depends on
the properties of the rough surface and the incident field.
Evaluating the intensity from a rough surface with known
properties is called the direct problem7-9. The opposite
question of the direct problem is called the inverse problem which is determining the properties of a surface by
using the information embedded in measured scattered
intensity10-15.
*Author for correspondence
Some of the scattering studies are based on ­variations
in dimensionless kσ parameter; where k and σ are two
length scale in the system3,16-18. These studies they do not
consider the effects of the third length scale in the system which is correlation length ξ. Although, some reports
showed how correlation length plays effective role in
wave scattering in Rayleigh-Rice framework6,19-22, it could
be useful to consider this effect in Kirchhoff theory, too.
Schiffer indicated when ξ comparable with λ, the effect are
losing in reflection, shifting in Brewster angle to smaller
ones and slight reddening of scattering light6 in rayleighRice theory. Also, Yanguas-Gil et al.22 showed when the
correlation length is smaller
than the wavelength (ξ < λ),
2
the single parameter wa contains the information of wave
x
scattering phenomena in Rayleigh-Rice framework.
These three length scales (λ, σ and ξ) depend on
dimensions will affect the scattered wave. When λ  x,
the correlation length has no effect on the scattered wave
d a monochromatic wave illuminates the rough surface with finite dimensions and R 0
ity. According to Dirichlet boundary
Length Scales Analysis of Wave Scattering from Rough Surfaces
size of the runner shoes can be interpreted as observation scale or wavelength of incident beam. If the runner
wants to run fast, the size of the rocks and the height mean
squared of the rocks are important. Really small rocks
with small height root mean squared have no effect in the
speed of the runner. This situation in a scattering system
is equivalent to the wavelength larger than the surface
roughness (σ) which means the diffuse scattering intensity is small. On the other hand if the size of his shoes is
comparable to the size of the rocks, he has to slow down
to be able to go through the troughs made by rocks. If
the height mean squared are big, the runner shoes will be
trapped between the rocks that makes him to slow down.
Figure
1. The
geometry
usedused
for for
wave
scattering
from
a
Figure
1. The
geometry
wave
scattering
from
This means that by decreasing wavelength, the fluctuation
rough
surface.
HereHere
0\ isθ1the
angle
between
a rough
surface.
is the
angle
betweenthe
theincident
incident
of the surface is more intuitive and the diffuse scatterbeam
andand
thethe
normal
the
scattered
beam
normalofofthe
thesurface.
surface. θ02 2isisthe
scattered
angle
ing intensity increases. By decreasing the wavelength,
the angle
between
the incident
and planes.
angle
3 isangle
andand
θ3 is0the
between
the incident
and scattered
the diffuse scattering intensity has some reduction. This
andnumbers
k sc are the
numbers
for
scattered
k incwave
kinc andplanes.
ksc are the
for wave
incident
and scattered
resembles the situation where the sizes of the rocks or the
incident
scattered beam respectively.
beam and
respectively.
height mean squared are larger than the size of the runλ=1000ξ
λ=ξ
ner’s shoes. In this situation the parameters of the rocks
σ
σ
λ
λ
has less effect in the speed of the runner. This means if the
ns, R 0 = -1 for such surfaces3.
wavelength is smaller than the fluctuations of the rough
surface, the scattering intensity decrease. In addition to
h these assumptions we can obtain that the coherent part of scattered light wave is:
these parameters, the complexity of the surface roughness
σ
σ
is another key determining point in the diffuse scatter(a)
(b)
Figure 2: Dependence of diffuse scattered intensity per k ξ on kσ for angles θ = θ = 0 , θ = 5 and ing intensity. This means a surface that is covered by the
H = 1, (a) λ = 1000ξ, (b) λ = ξ.
Figure 2. Dependence of diffuse scattered intensity per
periodical peaks and troughs will have different diffuse
2 2
on kσ for angles
θ1 = θ33 = 0°, θ2 = 5° and H = 1, (a) λ =
4.2k ξExperimental
approach
scattering intensity from a surface with height fluctua1000ξ, (b) λ = ξ.
In this section we study the diffuse scattering intensity with the presence of three length scale for rough tions. This parameter of a rough surface is described as
andWe
kσsuppose
is a suitable
present
wave
scattering
surfaces.
that ξ and λ parameter
are in comparableto
range.
As mentioned
above
to change kσ we can Hurst exponent, H.
from
the
rough
surface
(see
Figure
2(a)).
But
when
ξ and
either change k or σ. We discuss about the effects of changing each of these parameters
individually.
λ are inwe the
same
range
dimensionless
Furthermore,
will show
our results
withanother
different values
of correlation length,parameter
ξ, and Hurst exponent, 2. Kirchhoff Theory
kξ will appear as well. We need to know how kσ is changed
H, on self-affine surfaces.
Any incident field can be written as yinc (r ) = exp ( −ik inc .r ) ,
(see Figure 2(b)). To make any changes in kσ, we can
4.2.1 Effect of variation in kσ
where kinc is the wave number of the incident beam and r
either change k or σ. Different values of k or σ may result
Fig. 3 and Fig. 4 show the diffuse scattering intensity as a function of scattering angle for different kσ is the position vector. The incident beam is scattered from
in the same value for kσ. In this paper, we focus on scatat two
different
values of Hurst
exponent. In the former
one the wavelength
is constant, λ = 633nm a rectangular rough surface area with following conditering
problem
in framework
of Kirchhoff
approximation
andwhen
the σ is variable
and
in
the
latter
one
the
height
root
mean
squared
is
constant,
σ = 50nm,
there are three length scales in the system, while
ξ when tions: –X < x0 ≤ X and –Y < y0 ≤ Y. The scattered field
sc
the and
wavelength
canin
modify
a variable.
In both figures we found the diffuse scattering intensity at from the mentioned rough surface is shown by y . By
λ are
theassame
range.
different values
kσ betweenthe
0.2 andconcept
2.0 with the intervals
of 0.2.relation
These values are
in agreement with using Kirchhoff theory one can determine the distribuTo ofclarify
of the
between
tion of coherent and diffuse parts of scattered wave from
­roughness,
and wavelength in a scatexperimental
values incorrelation
previous works [13,length
14].
3
tering
system
we
can
represent
the
problem
with
a
runner
In both figures the diffuse scattering intensity is calculated for two different values of Hurst exponent. a rough surfaces .
Kirchhoff theory is calculated based on three
a surface
up the
ofones
rocks
with (Hdifferent
By running
comparing the on
two graphs
on the topmade
(H = 1) with
on the bottom
= 0.5), we can see
3
h
­
ypotheses
. (1) The roughness of each point of the sursizes.
The
longitudinal
size
of
a
rock
and
the
height
of
that
that the diffuses scattering intensity has smaller standard deviation at H = 0.5. Also, the peak of the
face
is
assumed
to have the same optical behavior as its
indicate
correlation
length,
ξ,
and
the
height
root
mean
diffuse scattering intensity is higher for smaller value of the Hurst exponent.
tangent plane. Fresnel laws can thus be locally applied. (2)
squared, σ, respectively. The speed of the runner depends
8 height root mean squared
The surface reflectivity R0 is independent of the position
on the longitudinal size and the
on the rough surface and the local angle of incidence. (3)
of the rocks in addition to the size of his shoes. Here the
(b)
0.25
Diffuse scattering intensity per k 2ξ 2
Diffuse scattering intensity per k 2ξ 2
(a)
0.3
0.25
=constant
=constant
0.2
0.15
0.05
0.1
0.05
0.5
1
k
1.5
2
0
2 2
2
=constant
=constant
0.2
0.15
0.1
0
0.3
Vol 8 (16) | July 2015 | www.indjst.org
0.5
1
1.5
k
1
3
2
◦
2
◦
Indian Journal of Science and Technology
M. Salami
Calculations are performed in the far-field, which works
when the surface area S0 is small. For all the calculations
in this study, we assumed a monochromatic wave illuminates the rough surface with finite dimensions and R0
reflectivity. According to Dirichlet boundary
conditions, R 0 = –1 for such surfaces3.
Through these assumptions we can obtain that the
coherent part of scattered light wave is:
where AM = 4XY is the area of the mean ­scattering ­surface
and Cor(R) is the height-height correlation ­function and
it is defined as Cor ( R ) =
h (x + R) h ( x )
s2
.
3. Scattering from Self-affine
Surface
One of the main groups of the rough surface is ­represented
by self-affine fractal scaling, defined by Mandelbrot in
SM
terms of fractional Brownian motion23. Lets consider a
1  Aa

surface
with+ aBb
single-valued
height function, z(r) of
+ c
Where the phase function is f ( x0 + y0 ) = Ax0 + By0 + Ch ( xrough
0 , y0 ) , F = 
 r = (x, y). All rough surfaces
C
C vector,
2 positional
the in-plane
1  Aa Bb 
are characterized by two parameters, the mean-square
Ax0 + By0 + Ch ( x0 , y0 ) , F = 
+
+ c and

C
2 C
2 1
roughness s =< z (r ) > 2 and z (r ) = h (r ) − < h (r ) > . As
y sc (r ) =
−ikeikr
2F (q1 , q2 , q3 )
4pr
∫ exp ikf (x , y ) dx dy , (1)
0
0
A = sin q1 − sin q2 cos q3
B = − sin q2 sin q3
C = − ( cos q1 + cos q2 )
a = sin q1 (1 − R0 ) + sin q2 cos q3 (1 + R0 )
b = sin q2 sin q3 (1 + R0 )
c = cos q2 (1 + R0 ) − cos q1 (1 − R0 )
0
0
mentioned earlier, h(r) is the height function and < … >
is the spacial average over a planar reference surface. We
assume that z(r) – z(r’) is a stochastic Gaussian variable
whose distribution depends on the difference of the two
position vectors r and r’, (x’ – x, y’ – y). Using the height
function we can define the structure function as S(R)
=<[z(r’) – z(r)]2 >, where R =| r’ – r |. The average is taken
over all pairs of points on the surface that are separated
horizontally by the distance R. S(R) could be defined in
terms of the height-height correlation function that was
< z ( R ) z (0 ) >
defined earlier, Cor ( R ) =
, as,
s2
The coherent field could be derived from the ­average
amplitude of the scattered field and the intensity of
the coherent field for a surface with a Gaussian height
­distribution,
I coh = y sc
y sc∗ = e − g I 0 , (2)
where g = k 2C 2a 2 and Io is the coherent scattered intensity
from a smooth surface with the same size as the rough
surface.
The average diffuse field intensity can also be ­calculated
using ysc,
I d = y sc y sc∗ − y sc
=
k2 F 2
∞
(
)
y sc∗
gCor R
AM e − g J 0 kR A2 + B 2 e ( ) − 1 RdR, (3)


2pr 2
0
∫
Vol 8 (16) | July 2015 | www.indjst.org
g ( R ) = 2 < z (r ) > −2 < z (r ) z (r ′ ) >= 2s 2 −
2
2 < z ( R ) z (0) >= 2s 2 − 2s 2Cor ( R ) .
(4)
If the surface exhibits self-affine roughness, S(R) will
scale as S(R) ∝α R2H, 24, where 0 < H < 1 is referred to
as Hurst exponent 25. The Hurst exponent represents the
degree of the surface irregularity.
Large values of H correspond to smooth height-height
fluctuations, while the small values shows the jagged and
irregular surfaces at the short length scale. The meanSquare Roughness, S(R), of any physical self-affine surface
will saturate at sufficiently large horizontal lengths. Thus
S(R) is characterized by a correlation length,
S ( R ) aR2 H , For R  x,
S ( R ) = 2s 2 , For R  x (5)
Indian Journal of Science and Technology
3
Length Scales Analysis of Wave Scattering from Rough Surfaces
By using the diffuse reflectivity data26,27 the correlation
function for a self-affine surfaces can be written as28,
− Rx
Cor ( R ) = e ( ) 2H
(6)
When R  x, there will be no correlation, Cor(R) =
0 and for R  x correlation function can be written as
2H
 R
Cor ( R )  1 −  
This indicates for short length scale
 x
r  x, the correlation function shows power-law behavior29.
At the largest possible value for Hurst exponent,
smooth height-hight fluctuations, the correlation function
is Gaussian and Eq. 3 for the diffuse scattered intensity
can be written as3:
< I d >=
k 2 F 2 x2e − g
4pr
2
(
)
 k 2 A2 + B 2 x 2 
gn
 (7)
exp 
AM ∑
4n
n =1 n ! n


∞
For a Gaussian noise surface with H = 0.5, we can
obtain the following expression for the diffuse scattered
intensity from Eq. 318:
< I d >=
k2 F 2
2pr 2
gn
n =1 n !
x2
∞
AM e − g ∑
(
)
3
 k 2 A2 + B 2 x 2  2

n2  1 +
n2


.
(8)
4. Result and Discussion
It was stated earlier, when the system is characterized
by the three length scales k, σ, and ξ, the dimensionless
parameter kσ is unable to give a complete description of the
scattering problem from the rough surface. Hence, a new
dimensionless parameter kξ is introduced which enables a
comprehensive study on the scattering phenomena. Hence,
the combined effects of the wavelength, roughness and
correlation length on the scattered waves need to be studied. The formalism presented here would provide basis for
both theoretical studies and experimental observations,
in a sense that two approaches are proposed based on the
physical applications. To comply with the theoretical modeling needs, the chosen variable would be kξ which is more
convenient due to its dimensionless nature, and to comply
with the observers needs in order to calculate the scattered
intensity, values of the wave-length and roughness have
4
Vol 8 (16) | July 2015 | www.indjst.org
been chosen close to expected measurements. These two
approaches are p
­ resented in the following sections.
4.1 Theoretical Approach
The scattered intensity depends on the altitude correlation
function of the rough surface, so the area of the correlated section (the part of the surface that has correlated
altitudes) also plays an important role. To calculate the
intensity of the scattered wave, we use r = 0 and r as the
lower and upper limits of the integral respectively, where
r and ξ have the same order of magnitude. For the case
r  x, the correlated function and the integrand will be
zero. On the other hand we know that a correlated surface
is proportional to ξ2.
Also we should keep in mind that ξ is a characteristic
length of the surface which possesses a scaling behavior.
This implies consideration of the scale of observation; this
indicates the importance of the incident wave length to
the surface. Consequently the parameter kξ would show
its importance. Since the scattered field intensity depends
on ξ2, in order to gain its independence from the parameters ξ and λ, it is divided by k2ξ2. In such a condition, we
obtained an expression independent of the parameters ξ,
σ, λ. When the correlated regime is much smaller than
the wave length, the intensity only depends on the kσ (see
Figure 2(a)). However when ξ is as of the order of λ its
effects on the scattered wave intensity becomes prominent
and the curve indicating the variations of the diffused
intensity in terms of kσ becomes obviously dependent on
k or σ (see Figure 2(b)).
4.2 Experimental Approach
In this section we study the diffuse scattering intensity
with the presence of three length scale for rough surfaces.
We suppose that ξ and λ are in comparable range. As
mentioned above to change k σ we can either change k or
σ. We discuss about the effects of changing each of these
parameters individually. Furthermore, we will show our
results with different values of correlation length, ξ, and
Hurst exponent, H, on self-affine surfaces.
4.2.1 Effect of Variation in kσ
Figure 3 and Figure 4 show the diffuse scattering intensity as a function of scattering angle for different kσ at two
different values of Hurst exponent. In the former one the
wavelength is constant, λ = 633nm and the σ is variable and
in the latter one the height root mean squared is constant,
Indian Journal of Science and Technology
M. Salami
(a) kσ <0.6, H=1, λ =633nm
(a) kσ <0.6, H=1, λ =633nm
(b) kσ >0.6, H=1, λ =633nm
(b) kσ >0.6, H=1, λ =633nm
σ = 50 nm, when the wavelength can modify as a variable.
In both Figures we found the diffuse scattering intensity at
different values of kσ between 0.2 and 2.0 with the intervals
of 0.2. These values are in agreement with experimental
values in previous works13,14.
In both figures the diffuse scattering intensity is
­
calculated for two different values of Hurst exponent. By
Scattering Angel (Degree)
Scattering Angel (Degree)
comparing the two graphs on the top (H = 1) with the
Scattering Angel (Degree)
Scattering Angel (Degree)
(c) kσ <0.6,
(d) kσ >0.6,
(a) kH=0.5,
<0.6,λ =633nm
H=1, λ =633nm
(b) kH=0.5,
>0.6,λ =633nm
H=1, λ =633nm
σ
σ
(c) kσ <0.6, H=0.5, λ =633nm
(d) kσ >0.6, H=0.5, λ =633nm
ones on the bottom (H = 0.5), we can see that the diffuses
(a) kσ <0.6, H=1, λ =633nm
(b) kσ >0.6, H=1, λ =633nm
k σ =0.2
k σ =0.6
k σ =0.6 scattering intensity has smaller standard deviation at H
kσ =0.2
k σ =0.4
=0.2
k σ =0.8
=0.6
k σ =0.8
kσ =0.4
k σ =0.6
=0.4
k σ =1
=0.8
kkσσ=0.6
kkσσ=0.2
=1
=0.6
k σ =0.6
k σ =1.2
=1
kkσσ=0.8
kσ =0.4
=1.2 = 0.5. Also, the peak of the diffuse scattering intensity is
k σ =1.4
=1.2
kkσσ=1
kσ =0.6
=1.4
k σ =1.6
=1.4
kkσσ=1.2
=1.6 higher for smaller value of the Hurst exponent.
k σ =1.8
=1.6
kkσσ=1.4
=1.8
k σ =2
=1.8
kkσσ=1.6
=2
In all different cases, we found a threshold value for
k σ =2
k σ =1.8
k σ =2
kσ where the diffuse scattering intensity has its own maximum value. This threshold value is happening where the
Scattering Angel (Degree)
Scattering Angel
(Degree)Angel (Degree)
correlation length, height mean squared and the waveScattering
Angel
(Degree)
Scattering
Scattering Angel (Degree)
Scattering Angel (Degree)
Scattering
(Degree)
Scattering
(Degree)
(c) kσ <0.6, H=0.5,
=633nm
(d) kσ >0.6, H=0.5,
=633nm
λ Angel
λ Angel
length are in the same order of magnitude. In both Figure
Figure 3: The
scattering
of scattering
angleH=0.5,
with constant
= 633nm.
(c)diffuse
kσ <0.6,
H=0.5, λintensity
=633nm as aa function
(d) kσ >0.6,
=633nmλ
λconstant
Figure
3: The
function
of scattering
angleangle
with and
λ = between
633nm.
The graphs
are diffuse
plottedscattering
for valuesintensity
of kσ andasHurst
exponent.
The incident
the angle
3 and Figure 4, the graphs on the left and right show the
k
=0.2
σ
σ
k
=0.6
◦
The
graphs are
for planes
values are
of kσ
and θHurst
exponent.
The incident angle and the angle between
the incident
andplotted
scattered
equal,
1 = θk3σ =
=0.40◦ and the correlation length is ξ = 1000nm.
k σ =0.8
the Figure
incident and
are equal,
θ1 = kθkσ3σ=0.2
=
and the correlation
length
is ξ = 1000nm.
kkσσ=0.6
3. scattered
Theplanes
diffuse
scattering
as
a
function
of
=0.60 intensity
=1
diffuse scattering intensity for the kσ values smaller and
k σ =0.4
kkσσ=0.8
=1.2
k σ =0.6
kkσσ=1
=1.4
In all different cases,
we found
a threshold
value
for=kσ633nm.
where the diffuse
scattering
intensity
has larger than the threshold value respectively. The rate of
scattering
angle
with
constant
λ
The
graphs
are
kkσσ=1.2
=1.6
In all different cases, we found a threshold value for kσ where the diffuse scattering intensity
has
kkσσ=1.4
=1.8
forvalue.
values
of kσ value
andis Hurst
The length,
incident
its plotted
own maximum
This threshold
happeningexponent.
where the correlation
heightkkσmean
changing the diffuse scattering intensity is larger for the
=2
σ=1.6
its own maximum value. This threshold value is happening where the correlation length, heightk σmean
=1.8
k σ =2
angle
and
the angle
incident
and3 and
scattered
squared
and the
wavelength
are in thebetween
same order of the
magnitude.
In both Fig.
Fig. 4, the graphs
kσ values smaller than the threshold value.
squared and the wavelength are in the same order of magnitude. In both Fig. 3 and Fig. 4, the graphs
planes are equal, θ1 = scattering
θ3 = 0°intensity
and for
thethecorrelation
length is
In Figure 3, where the wavelength is constant, the
on the left and right show the diffuse
kσ values smaller and larger than the
on the left and right show the diffuse scattering intensity for the kσ values smaller and larger than the
ξ = 1000nm.
standard
deviation of diffuse scattering intensity is getting
(b) kdiffuse
H=1, σ =50nm
Angel of
(Degree)
σ =50nm Scattering
threshold
valueH=1,
respectively.
The rate
changing the
intensity
is larger for the kσ
(a) kσ <1,
σ >1, scattering
Scattering Angel (Degree)
threshold value respectively. The rate of changing the diffuse scattering intensity
is larger for the kσ
Scattering Angel (Degree)
larger by increasing kσ. However when the height mean
(b) kσ >1, H=1, σ =50nmScattering Angel (Degree)
(a) smaller
kσ <1, than
H=1, σ
=50nm
values
the
threshold value.
kσ =1
values smaller
than
the threshold
value. kσ =0.2
Figure
3: The
diffuse scattering
intensity as a function of scattering angle withkσconstant
λ = 633nm.
kσ =0.4
=1.2
squared is constant (σ in Figure 4) the standard deviation
The
graphs
arediffuse
plottedscattering
forisvalues
of kσthe
and
Hurst
exponent.
The
incident
angle
the angle
between
In Fig.
3, where
the
wavelength
constant,
standard
deviation
of diffuse
scattering
intensity
kσ intensity
=0.6
=1.4
σ
Figure
3: The
as
a function
of scattering
angle
withkσkand
constant
λ is= 633nm.
kσ =0.2
=1
In Fig.the
3, where theand
wavelength
isplanes
constant,
the standard
of diffuse
scattering
intensity
is
kσ =0.8
kσ =1.6
scattered
arekσequal,
θ1 = θexponent.
= 0◦ andThe
the
correlation
length
ξ=
1000nm.
3 deviation
kσ =0.4
kσ
=1.2is
of diffuse scattering intensity decreases as kσ increases.
The incident
graphs are plotted
for values
of
and
Hurst
incident
angle
and
the
angle
between
kkσ =1
kσ=1.4
=1.8
kσ
σ =0.6
the incident and scattered planes
are equal, θ1 = θ3 = 0◦ and the correlation length
kσ=1.6
=2is ξ = 1000nm.
kσ =0.8
kσ
Furthermore, by decreasing Hurst exponent due to the
kσ =1 a threshold value for kσ where the diffuse kscattering
σ =1.8
In all different cases, we found
intensity has
kσ =2
9
In all different cases, we found a threshold
value for kσ where the diffuse scattering intensity
has
increased
irregularity of the surface roughness, the diffuse
9
its own maximum value. This threshold value is happening where the correlation length, height mean
increases. The curve of the diffuse intensity is getits own maximum value. This threshold value is happening where the correlation length, heightintensity
mean
squared and the wavelength are in the same order of magnitude. In both Fig. 3 and Fig. 4, the graphs
squared and the wavelength are in the same order of magnitude. In both Fig. 3 and Fig. 4, the ting
graphs sharper and the standard deviation of that decreases.
on the left and right show the diffuse scattering intensity for the kσ values smaller and larger than the
Scattering Angel (Degree)
Scattering Angel (Degree)
on the left and right show the diffuse scattering intensity for the kσ values smaller and larger than theIn some of the studies the scattering intensity has been
(c) kσ <0.8,
H=0.5,
(d) kσ >0.8,
H=0.5,
=50nm
σscattering
threshold
value
respectively.
the(b)
diffuse
intensity is larger for the kσ
kScattering
H=1,
σ =50nm
σ =50nmThe rate of changing
(a) kScattering
<1,=50nm
H=1,
σσ
σ >1,
Angel
(Degree)
Angel
(Degree)
studied as a function of dimensionless kσ. As discussed
value
respectively. The rate of changing
the H=0.5,
diffuseσscattering
intensity is larger for the kσ
(c) kσthreshold
<0.8, H=0.5,
=50nm
(d) kσ >0.8,
=50nm
σ
(b) kσ >1, H=1, σ =50nm kσ =0.8
σ =50nm
(a)smaller
kσ <1, than
H=1,the
values
threshold
value.
kσ =0.2
kσ =0.2
kσ =1 before the changes in kσ could be the result of variation
kσ =1
kσ =0.4
kσ =1.2
values smaller than the threshold
value. kσ =0.4
kσ=0.8
=1.2
kσ
kσ =0.6
=0.2 is constant,
In Fig. 3, where the wavelength
scattering
intensity
is
kσ =0.6 the standard deviation of diffuse
kσ
=1.4
kσ =0.2
kσ =1 in the wavelength or the height mean squared paramekσ=1=1.4
kσ
kσ =0.8
=0.4
kσ =0.8
kσ =1.6
kσ =0.4 the standard deviation of diffuse
kσ intensity
=1.2
In Fig. 3, where the wavelength
is
kσscattering
=1.6
kσ
=1.2
kσ =0.6is constant,
kσ =1
kσ =1.8
kσ =0.6
k
=1.4
σ
kσ
kσ=1.4
=1.8
kσ =0.8
kσ =2 ters. Our result show that not only the value of kσ is one
kσ =0.8
kσ =1.6
kσ
kσ=1.6
=2
kσ =1
kσ =1.8
kσ =1.8
k
=2 of the key parameters, but also how this parameter has
σ
9
kσ =2
9
been changed is important too. Our results show that the
scattered intensity varies by changing the wavelength and
the height mean squared individually. In another word,
Scattering
Angel
(Degree)
Scattering
Angel
(Degree)
Scattering Angel (Degree)
Scattering Angel (Degree)
(c) kScattering
(d) kScattering
>0.8,AngelH=0.5,
σ <0.8,AngelH=0.5,
σ
(Degree)σ =50nm
(Degree)σ =50nm
studying the diffuse scattering intensity as a function of
Scattering Angel (Degree)
Scattering Angel (Degree)
Figure 4: The diffuse scattering intensity as a function of scattering angle with constant σ = 50nm.
(c)
kσ <0.8,
H=0.5,
(d) kσ >0.8,
H=0.5,
σ =50nm
σ =50nm
The
graphs
are
plotted
for
different
values
of
kσ
and
Hurst
exponent.
The
incident
angle
and
the
angle
σ
k
=0.8
kσ =0.2of scattering angle with constant σ = 50nm. kσ does not depict all the features of the scattered intenFigure 4: The diffuse scattering intensity as a function
kσ is
=1
=0.4 θ1 = θ3 = 0◦ and the correlation length
between
the are
incident
and
planesofare
and
The graphs
plotted
for scattered
different values
kσ equal
andkkσσHurst
exponent. The incident angle and the angle
kσ =1.2sity and we need to consider the effects of the wavelength
=0.6
kσ =0.8
◦ as a function
ξbetween
= Figure
1000nm.
kσ =0.2
4. and
Thescattered
diffuse
the incident
planesscattering
are equal
the correlation length
is=1.4
kσ
1 = θ3 = 0 and
kσ and
=0.8 θintensity
kσ =1
kσ =0.4
kσ =1.6
ξ = 1000nm.
kσ =0.6
σ =1.8and the height mean squared as well.
of scattering angle with constant
σ = 50nm. The graphs kkσσk=1.2
=1.4
kσ =0.8
σ =2
kthe
getting larger by increasing kσ. However when the
height mean squared is constant (σ in Fig. 4)
kσ =1.6
Furthermore, by adding more parameters to our ­system
arelarger
plotted
for different
of kσ
and
Hurst
exponent.
=1.8
getting
by increasing
kσ. However values
when the height
mean
squared
is constant
(σ in Fig. 4)kσthe
σ
k
=2
standard deviation of diffuse scattering intensity decreases as kσ increases. Furthermore, by decreasing
such as correlation length, ξ, other dimensionless parameThe deviation
incident
angle
and intensity
the angle
between
the Furthermore,
incident byand
standard
of diffuse
scattering
decreases
as kσ increases.
decreasing
Hurst exponent due to the increased irregularity of the surface roughness, the diffuse intensity increases. ter such as kξ effects the scattering intensity. We will discuss
scattered
planes
are
equal
and
θ
=
θ
=
0°
and
the
correlation
1 surface
3 roughness, the diffuse intensity increases.
Hurst exponent due to the increased irregularity of the
The length
curve of the
intensity is getting sharper and the standard deviation of that decreases.
the effect of this parameter in the next section. To increase
is diffuse
ξ = 1000nm
Scattering Angel (Degree)
Scattering Angel (Degree)
26
26
20
18
Diffuse
Diffuse
Scattering
Scattering
Intensity
Intensity
(ar.un.)
(ar.un.)
22
20
18
16
16
14
14
12
12
10
108
68
46
24
02
0
-50
0
50
-50
0
50
k σ =0.6
k σ =0.8
=0.6
k σ =1
=0.8
k σ =1.2
=1
k σ =1.4
=1.2
k σ =1.6
=1.4
k σ =1.8
=1.6
k σ =2
=1.8
k σ =2
24
22
22
20
20
18
18
16
16
14
14
12
12
10
108
68
46
24
02
0
-50
0
50
-50
0
50
26
40
26
40
35
24
26
22
24
20
22
18
20
16
18
14
16
12
14
10
12
8
10
6
8
4
6
2
4
-50
0
2
-50
40
35
24
26
22
24
20
22
18
20
16
18
14
16
12
14
10
12
8
10
6
8
4
6
2
4
0
-50
2
-50
0
35
30
30
25
25
20
20
15
15
10
10
5
5
0
0
0
-50
0
0
50
050
-50
50
0
35
30
30
25
25
20
20
15
15
10
10
5
5
0
0
50
Diffuse
Scattering
Intensity
(ar.un.)
Diffuse
Scattering
Intensity
(ar.un.)
40
Diffuse
Scattering
Intensity
(ar.un.)
Diffuse
Scattering
Intensity
(ar.un.)
Diffuse
Diffuse
Scattering
Scattering
Intensity
Intensity
(ar.un.)
(ar.un.)
26
24
kσ =0.2
kσ =0.4
=0.2
kσ =0.6
=0.4
kσ =0.6
24
22
Diffuse
Diffuse
Scattering
Scattering
Intensity
Intensity
(ar.un.)
(ar.un.)
Diffuse
Diffuse
Scattering
Scattering
Intensity
Intensity
(ar.un.)
(ar.un.)
26
24
3025
2520
2015
1510
105
50
-50
0
-50
50
0
50
70
DiffuseDiffuse
Scattering
Intensity
(ar.un.) (ar.un.)
Scattering
Intensity
5060
4050
3040
2030
1020
-20
-20
80
60
20
0
20
40
20
0
2015
1510
105
50
-50
0
-20 0
-20
0
0
-20
20
0
20
30
40
20
30
10
20
100
60
60
40
40
20
20
0
0
20
0
20
-20
0
-20
20
0
20
60 50
50 40
40 30
30 20
20 10
10 0
-20
-200
80
0 -20
0
-20
20 0
20
0
20
20
80
80
Diffuse Scattering Intensity (ar.un.)
Diffuse Scattering Intensity (ar.un.)
Diffuse Scattering Intensity (ar.un.)
Diffuse Scattering Intensity (ar.un.)
50
40
50
70
20 10
50
0
50
60
70 60
40 30
0
-50
70 60
0
-20 10
0
2520
80
30 20
20
3025
80
50 40
40
3530
60
70
0
60 50
60
50
50
4035
70
Diffuse Scattering Intensity (ar.un.)
Diffuse Scattering Intensity (ar.un.)
80
0
Diffuse
Scattering
Intensity
(ar.un.)(ar.un.)
Diffuse
Scattering
Intensity
Scattering
Intensity
DiffuseDiffuse
Scattering
Intensity
(ar.un.) (ar.un.)
0
6070
0
Diffuse
Scattering
Intensity
Diffuse
Scattering
Intensity
(ar.un.)(ar.un.)
Diffuse
Scattering
Intensity
(ar.un.)
Diffuse
Scattering
Intensity
(ar.un.)
3530
70
010
50
0
50
0
40
4035
Diffuse Scattering Intensity (ar.un.)
Diffuse Scattering Intensity (ar.un.)
Diffuse
Scattering
Intensity
(ar.un.)
Diffuse
Scattering
Intensity
(ar.un.)
40
0
-50
0
-50
80
60
60
60
60
40
40
40
40
20
20
20
20
0
-20
0
20
0
-20
0
20
The curve of the diffuse intensity is getting sharper and the standard deviation of that decreases.
0
0
-20 scattering
0
20 has been studied as a function
In some of the studies
the
intensity
of dimensionless
-20
0
20 kσ. As
Scattering Angel (Degree)
Scattering Angel (Degree)
In some of
the studies
scattering
intensity
has been
as aoffunction
of dimensionless
kσ. As σ = 50nm.
Figure
4: Thethediffuse
scattering
intensity
as astudied
function
scattering
angle with constant
discussed before
the
changes
in
kσ
could
be
the
result
of
variation
in
the
wavelength
or
the
height
mean
The graphs are plotted for different values of kσ and Hurst exponent. The incident angle
and the angle
discussed before
the
in kσscattering
could
be intensity
the result
of are
the
height
meanσ = length
Figure
4: changes
The
diffuse
as
avariation
function
ofthe
scattering
withthe
constant
50nm. is
between
the incident
and
scattered
planes
equal in
and
θ1wavelength
= θ3 =angle
0◦orand
correlation
Vol parameters.
8 The
(16)
|1000nm.
July
2015
| www.indjst.org
squared
Our
result
show
not only
theofvalue
of kσ
is one
of the key
but and
alsothe angle
are
plotted
for that
different
values
kσ and
Hurst
exponent.
Theparameters,
incident angle
ξ =graphs
◦
squared parameters.
Our
result show
that not only
the are
value
of kσand
is one
of the
but also length is
between the
incident
and scattered
planes
equal
θ1 =
θ3 =key
0 parameters,
and the correlation
ξ = 1000nm.
getting larger by increasing kσ. However when the height mean squared is constant (σ in Fig. 4) the
10
10
Indian Journal of Science and Technology
5
Length Scales Analysis of Wave Scattering from Rough Surfaces
kσ while the wavelength is constant, both the height mean
squared, σ, and the correlation length, ξ, should change to
keep the Hurst exponent ­constant. If the variation in kσ
is due to the changes in the wavelength, both the height
mean squared, σ, and the correlation length, ξ, remain
constant for a constant value of the Hurst exponent.
Even though the parameters that we used to ­calculate
the diffuse scattering intensity is different in the top and
the bottom graphs, they show the similar behavior when
kσ increases. This is due to the relative magnitude of the
wavelength and the roughness of the surface. In the top
graphs λ is constant for the top graphs, but σ has to increase
to enlarge kσ. This means the wavelength is decreasing
compare to σ. On the other hand, σ is constant in the
bottom graphs, so to enlarge kσ, λ has to decrease. This
also results in small wavelength compare to the roughness of the surface. When the wavelength is smaller than
the roughness of a surface, the diffuse scattering intensity
decreases. This is similar to the runner example that we
mentioned earlier. If we assume that the Hurst exponent
of a surface is constant, the changes in correlation length,
ξ, is coupled with that of the height mean squared, σ. Our
results show that the scattering intensity for a surface
with constant Hurst exponent not only depends on the
wavelength of the incident beam, but also it changes by
changing the σ and ξ.
4.2.2 Effect of correlation Length
Figure 5 shows the diffuse scattering intensity as a ­function
of kσ for two different values of the Hurst exponent H =
0.5, 1.0. For the top graphs λ is constant and for the ones
at the bottom σ is constant. In all four graphs of Figure
5, the diffuse scattering intensity is calculated for three
values of the correlation length, ξ = 500, 1000, 1500 nm.
Besides, for all the graphs the angle between the incident
beam and the normal of the plane and the angle between
the incident and scattered planes are kept constant, θ1 = θ3
= 0°. The diffuse scattered intensity is calculated for one
specific scattered angle, θ2 = 5°.
In Figure 5 (a and b) the diffuse scattering intensity
increases as kσ increases due to the increment of σ. Both
of these graphs show a maximum value for intensity at a
specific value of kσ that is the same as the threshold value
mentioned in the previous section. Similar behavior can
be seen in Figure. 5 (d) where σ is kept constant and kσ
increases be decreasing λ. However, in Figure 5 (c), the
intensity ends up to a plateau after reaching to its maximum value. This result is confirmed by Figure 4 (b), where
shows that the changes of the diffuse scattering intensity is
small at θ2 = 5°.
5. Conclusion
The scattered wave from a rough surfaces depends on the
roughness parameters such as σ, ξ and λ. In the limiting
case of l x, only two characteristic lengths; λ which
is the observation scale, and σ which is the scale of surface, or their ratio comes in to effect on the scattered wave
intensity. This means that in this case kσ is important.
This limit has been the case of consideration by other
­researchers.
λ
λ
But if ξ and λ are in comparable range, the system posλ
λ
sesses three characteristic lengths; λ the observation scale,
σ and ξ which are the scales of the rough surface. In this
(b) H=0.5, λ =633nm
(a) H=1, λ=633nm
(b) H=0.5, λ =633nm
(a) H=1, λ=633nm
30
45
30
45
case, kσ does not prove to be a suitable parameter in order
ξ =500nm
ξ =500nm
=500nm
ξ =500nm
40
40
to present wave
scattering from ξξthe
rough surface, unless
=1000nm
=1000nm
ξ =1000nm
ξ =1000nm
ξ
25
25
=1500nm
=1500nm
=1500nm
=1500nm
ξ
ξ
ξ
ξ
35
35
the
consideration
of
a
varying
k
or
σ
is studied for a conσ
σ
20
30
λ =633nm
20
30 λ=633nm
λ =633nm (b) H=0.5,
(a) H=1,
(b) H=0.5,
λ=633nm
(a) H=1,
stant kσ. In other words is must be understood that which
30
45
σ
σ
30
45
25
25
15 ξ =500nm
ξ =500nm
of the parameters
and which is constant.
15
=500nm
ξ =500nm k or σ is varying
ξ
40
40
20
ξ =1000nm
ξ =1000nm
20
ξ =1000nm
ξ =1000nm
25
25
=1500nm
=1500nm
=1500nm
=1500nm
ξ
ξ
ξ
ξ
This has been emphasized in this work. Besides, we study
35
35
10
15
10
15
20the scattering phenomena including all three length
30
20
30
10
10
5
5
25
25
scales,
k, σ and ξ, both theoretically and e­ xperimentally.
5
5
15
15
20
20
0
0
To study
the relation
of the
scattered
wave intensity
0
0
0.5
1
1.50.5
2 1
1
1.50.5
2 1
1.5
2 0.5
1.5
2
kσ
kσ
kσ
kσ
10
15
10
15
Figure 5: Dependence of diffuse scattered intensity on kσ for different correlation length (ξ = 500, 1000, and kσ when ξ and λ are comparable, we used a self-af10 θ = θ = 0 , θ = 5 .
1500nm) and for
10
(c) H=1,
σ =50nm
(d) H=0.5, σ =50nm
(c)angles
H=1,
σ =50nm
(d) H=0.5, σ =50nm
5
Figure 5: Dependence of diffuse scattered intensity on kσ for different correlation
length (ξ = 500, 1000, 5fine rough surface. We compare two regimes for the kσ
40
100
40
100
5
5
1500nm) and for angles θ = θ = 0 , θ = 5 .
wavelength is changed
relative
magnitude
the wavelength and ξ
the
roughness
the surface. In intensity
the
top graphson
λ is constant
=500nmis σ = cont. and
Figure
5. of Dependence
of=500nm
diffuseof scattered
kσ for 0parameter, ξone
ξ =500nm
ξ =500nm
0
0
0
=1000nm
=1000nm
0.5 ξ
1.5
2 0.5
0.5 ξ =1000nm 2 1
1.5
ξ =1000nm 2
ξ
0.5
1
1.5
21
1
=1500nm
=1500nm
k=
ξenlarge
ξ =1500nm
σ the
the 1.5
other
is λ k=σ const. andξ σ=1500nm
is changed. The result
80
k
ξ1500nm)
σ wavelength
σ
80
thefor
top magnitude
graphs,
butofσthe
has
to increaseand
to length
kσ.(ξThis
means
the
wavelength
is decreasing
compare
relative
the roughness
of
surface.
In the
top graphs
λ is
constant
for kand
different
correlation
500,
1000,
and
30
30
shows
the diffuse scattering intensity in both of them
angles
θσ1hand,
= σθto3σ=50nm
=is 0°,
θto2 enlarge
=in 5°.
σ =50nm
to for
σ.topOn
the(c)
other
constant
the kσ.
bottom
so H=0.5,
to
enlarge
λ has to decrease.
(d) H=0.5,
the
graphs,
butH=1,
has
increase
Thisgraphs,
means(d)
the
wavelength
is decreasing
compare
(c) H=1, σ =50nm
σkσ,
=50nm
(b) H=0.5, =633nm
(a) H=1, =633nm
30
ξ =500nm
ξ =1000nm
ξ =1500nm
25
15
20
10
15
5
1
kσ
(c) H=1,
=50nm
0
100
1.5
0.5
1
1.5
kσ
40
60
20
40
0
20
0
60
100
40
80
20
60
0
1
1.5
kσ
0.5
1
2
1.5
kσ
1
3
1
3
◦
◦
2
2
2
10
15
5
10
0
5
0.5
1
1.5
(d) H=0.5,
=50nm
0
0.5
1
1.5
40
kσ
kσ
◦
40
30
30
20
20
10
10
0
0
2
2
ξ =500nm
ξ =1000nm
ξ =1500nm
(d) H=0.5, =50nm
ξ =500nm
ξ =1000nm
ξ =1500nm
0.5
tering Intensity
(ar.un.) Intensity (ar.un.)
Diffuse Scattering
Diffuse Scattering
Intensity (ar.un.)
Diffuse Scattering
Intensity (ar.un.)
60
80
2
ξ =500nm
ξ =1000nm
ξ =1500nm
(c) H=1, =50nm
80
100
2
15
20
ξ =500nm
ξ =1000nm
ξ =1500nm
0.5
1
1.5
2
0.5
1
1.5
2
kσ
kσ
◦
60
100
40
tering Intensity
(ar.un.) Intensity (ar.un.)
Diffuse Scattering
5
0.5
ξ =500nm
ξ =1000nm
ξ =1500nm
20
25
Diffuse Scattering Intensity
Diffuse (ar.un.)
Scattering Intensity (ar.un.)
30
20
10
0
ξ =500nm
ξ =1000nm
ξ =1500nm
30
ering Intensity
Diffuse (ar.un.)
Scattering Intensity (ar.un.)
35
25
Diffuse Scattering
Intensity (ar.un.)
Diffuse Scattering
Intensity (ar.un.)
ξ =500nm
ξ =1000nm
ξ =1500nm
40
30
Diffuse Scattering Intensity
(ar.un.) Intensity (ar.un.)
Diffuse Scattering
Diffuse Scattering
Intensity (ar.un.)
Diffuse Scattering
Intensity (ar.un.)
Diffuse Scattering Intensity
Diffuse (ar.un.)
Scattering Intensity (ar.un.)
ering Intensity
Diffuse (ar.un.)
Scattering Intensity (ar.un.)
45
35
(b) H=0.5,
=633nm
25
Diffuse Scattering
Intensity (ar.un.)
Diffuse Scattering
Intensity (ar.un.)
40
Diffuse Scattering Intensity
(ar.un.) Intensity (ar.un.)
Diffuse Scattering
45
(a) H=1, =633nm
40
This
in small
compare
the roughness
surface. kσ,
When
thetowavelength
to
σ. also
On results
the other
hand,wavelength
σ is constant
in thetobottom
graphs, of
so the
to enlarge
λ has
decrease.
6
20 ξ =500nm
20
ξ =500nm
ξ =1000nm
ξ =1000nm
=1500nm
is smaller
than the
of
a
surface,
the
diffuse
scattering
intensity
This
similar to
This
also results
in roughness
small
wavelength
compare
to
the
roughness
of
the
surface.
When
theiswavelength
=1500nm
40
ξ decreases.
80
ξ
Vol 8 (16) | July 2015 | www.indjst.org
30
30
thesmaller
runnerthan
example
that we mentioned
earlier.
If we scattering
assume that
the10Hurst
exponent
of isa similar
surface to
is
is
the roughness
of a surface,
the diffuse
intensity
decreases.
This
20
60
10
constant,
changesthat
in correlation
length,
ξ, is coupled
with that
theHurst
height
mean squared,
σ. Our
the
runnerthe
example
we mentioned
earlier.
If we assume
that ofthe
exponent
of a surface
is
20
20
results show
the scattering
intensity
forξ,aissurface
with
constant
exponent
only depends
constant,
thethat
changes
in
length,
coupled
with
that of Hurst
the
meannot
squared,
σ. Our
0 height
0 correlation
40
0
ξ =500nm
ξ =1000nm
ξ =1500nm
ξ =500nm
ξ =1000nm
ξ =1500nm
Indian Journal of Science and Technology
M. Salami
have a threshold value for kσ which happening when the
correlation length, height mean squared and wave length
have comparable value. It is noted that by existence of
correlation length ξ, kσ is not the only dimensionless
parameter. Therefore we must consider by variation of ξ
which parameters (k or σ) are being changed.
The role of the Hurst exponent has been investigated
for self-affine rough surfaces. Such an examina­tion was
performed over a wide range of surface topographies,
from logarithmic (H = 0) to a power-law self-affine rough
surface, 0 < H < 1. The roughness exponent H has a strong
impact on the diffused part of wave scattering mainly for
relatively large correlation lengths. Therefore, Hurst exponent must be taken carefully into account before deducing
the roughness correlation lengths from wave scattering
measurements.
6. Acknowledgement
The authors would like to thank the research council
of Islamic Azad Uni­versity of Shahrood for financial
­support.
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