Download Light scattering by randomly rough surfaces

Light scattering by randomly rough surfaces
Eugenio R. Méndez
Departamento de Óptica
Outline
• Phenomenological description of the problem.
• Statistical characterization of randomly rough surfaces
• Statistical properties of the scattered field.
• Theoretical approaches:
- The Kirchhoff approximation.
- Perturbation theory and the Rayleigh method.
- Numerical techniques.
• Examples.
• Inverse scattering problems.
- Determination of statistical parameters.
- Profilometry.
Introduction
X3
θ0
θs
X2
q
k
φs
X1
ζ(x1,x2)
Light scattering by a randomly rough surface.
• Ocean surface, ground glass, grown layer of a semiconductor, etc.
• Probed by EM waves. Scales from nano to meters and more.
• Applications in: remote sensing, NDT, semiconductor industry, etc.
Introduction
x3
x 3 = ζ (x 1)
x1
Surface scattering
vs.
volume scattering
Introduction: Mirror
∂Rs(θs)/∂θs
specular
-60
-30
0
30
Scattering Angle [deg]
Specular reflection by a mirror
60
Introduction: Weakly rough surface
∂Rs(θs)/∂θs
<∂Rs(θs)/∂θs>
specular
+ diffuse
0
-60
-30
0
30
Scattering Angle [deg]
60
-60
-30
0
30
Scattering Angle [deg]
Scattering by a weakly rough surface (δ << λ)
60
Introduction: Rough surface
∂Rs(θs)/∂θs
<∂Rs(θs)/∂θs>
"incoherent"
-90
-60
-30
0
30
60
90
diffuse
-90
Scattering Angle [deg]
Scattering by a rough surface (δ ∼ λ)
-60
-30
0
30
Scattering Angle [deg]
60
90
Introduction: Speckle
(a)
(b)
(a) Speckle from a weakly rough surface. (b) Speckle from a rough surface.
Intensity fluctuations:
PI (x) =
1
exp{−x/hIi}
hIi
Universal law!
Introduction: Rougher surface (larger slopes)
<∂Rs(θs)/∂θs>
"coherent effect"
enhanced
backscattering
-90
-60
-30
0
30
60
90
Scattering Angle [deg]
Scattering by a surface with large slopes (δ/a ∼ 1)
Introduction: Coherent effects
Enhanced backscattering
Reciprocal paths interfere constructively
In other directions, the coherence is lost.
Introduction: Multiple scattering
i) Excitation of surface plasmon-polaritons
ii) Large slopes (geometrical optics)
Introduction
Scattering problem:
Given a surface, specified in statistical terms, calculate the scattered field
(also specified in statistical terms, average field, mean intensity, correlations, etc.).
Tough problem
Combination of an EM problem with a problem of statistical modeling.
EM problem - Gaussian groove. No analytical solution for the general problem.
Statistical problem - Not many models for n−dimensional distributions.
Statistical characterization of the surface
Familiar scene in optical laboratories: The bright spot caused by the scattering of a
laser beam on an optical surface, such as a mirror.
=> On the scale of the wavelength of visible light, most surfaces are rough.
The roughness cannot be easily described in detail and is better suited for a
statistical description
The surfaces are assumed planar in the absence of the roughness.
classified as:
They can be
2D - departure from the plane depends on x1 and x2 .
1D - departure from the plane depends on only x1 .
To simplify the presentation we concentrate on 1D case. The 2D case constitutes a
simple extension.
Statistical characterization of the surface
X3
X3
X2
X2
X1
ζ(x1,x2)
(a)
X1
ζ(x1)
(b)
Examples of randomly rough surfaces. (a) Surface with two-dimensional roughness
and (b) Surface with one-dimensional roughness.
Statistical characterization of the surface
We assume that the surface profile can be represented by a continuos single-valued
function of x1 (no reentrant surfaces).
The surface profile function ζ(x1 ) - realization of a stationary random process. Without
loss of generality, also assumed to be zero-mean.
The n-order joint Probability Density Function (PDF) of surface heights is
PZ (ζ1 , ζ2 , ..., ζj , ..., ζn),
where the ζj = ζ(x(j)
1 ) are the surface heights at specified points in space.
The characteristic function MZ (ω1 , ..., ωn) is given by the n-order Fourier transform of
PZ (ζ1 , ..., ζn).
A complete description of a random process would involve knowledge of the nth-order
joint probability density function for all n.
Statistical characterization of the surface
Normally, only a partial description of the process is available.
In some cases, however, this partial description is sufficient to solve a given scattering
problem.
For instance, with the theories based on the Kirchhof approximation, knowledge of
the second order probability density function is enough to calculate the mean field and
the mean intensity in the far field.
Moreover, for the case in which many irregularities of the surface are illuminated,
the speckle has Gaussian statistics and these two quantities completely specify the
scattered field in statistical terms.
Statistical characterization of the surface
A basic quantity for the description of the random process is the two-point height
correlation function
hζ(x1 )ζ(x01 )i = δ 2 W (|x1 − x01 |),
where the angled brackets represent an average over an ensemble of realizations of
the surface, and
δ 2 = hζ 2 (x1 )i.
The parameter δ represents the standard deviation or the rms height of the surface.
The fact that the autocorrelation function W (|x1 − x01 |) depends on the coordinates
x1 and x01 only through their difference is a reflection of the assumed stationarity of
ζ(x1 ).
Statistical characterization of the surface
For perturbative calculations it is often necessary, to introduce the Fourier integral
representation of the surface profile function,
Z
dk ikx1
ζ(x1 ) =
e ζ̂(k),
2π
where k is a wave vector. For stationary surfaces, the two-point correlation of the
Fourier coefficient ζ̂(k) is given by
hζ̂(k)ζ̂(k0 )i = 2πδ(k + k0 )δ 2 g(|k|).
The function g(|k|) is called the power spectrum of the surface roughness, and is
defined by
Z
g(|k|) =
dx1 e−ikx1 W (|x1 |).
It is a non-negative function of |k|, and is normalized according to
Z
dk
g(|k|) = 1.
2π
Statistical characterization of the surface
A common assumption made in scattering theory is that the surface profile function
constitutes a realization of a Gaussian random process.
For such processes, the joint probability density functions are known to all orders and,
for zero-mean processes, are completely determined by the two-point height correlation
function.
2
1
ζ1 + ζ22 − 2W12 ζ1 ζ2
p
,
PZ (ζ1 , ζ2 ) =
exp −
2
2
2(1
−
W
)δ
2πδ
1 − ρ12
12 2
with
W12 =
and hζi = 0.
hζ1 ζ2 i
,
δ2
Statistical characterization of the surface
One also has that
Z
MZ (ω1 , ω2 ) =
∞
Z
∞
D
i(ω1 ζ1 +ω2 ζ2 )
e
i(ω1 ζ1 +ω2 ζ2 )
PZ (ζ1 , ζ2 )dζ1 dζ2 = e
−∞ −∞
2
2
2
= exp −(δ /2)[W12 ω1 + 2ω1 ω2 + W12 ω2 ] ,
PZ (ζ) = √
1
2πδ
Z
MZ (ω) =
exp −ζ 2 /(2δ 2 ) ,
∞
iωζ
iωζ
e PZ (ζ)dζ = e
−∞
2 2 = exp −δ ω /2
E
Statistical characterization of the surface
Many surfaces of practical interest, however, have statistics that are non-Gaussian
The problem has not received the attention it deserves. Some approaches can be
found in:
– Beckmann, P. 1973. IEEE TAP AP-21, 169-175).
– Kim, M. J., Méndez, E. R., and O’Donnell, K. A. (1987). J. Modern Optics 34, 1107-1119.
– Tatarskii, V. I. (1995). Waves in Random Media 5, 243-252.
– Tatarskii, V. V. and Tatarskii, V. I. (1996). Waves in Random Media 6, 419-435.
Main problem - there are not many kinds of random processes ζ(x1 ) for which the
n-order joint PDF is known.
Often, one has only a histogram of heights and an estimate of a correlation function.
The problem is then to find a joint second order PDF of heights that allows us to
obtain a solution to the scattering problem.
For simplicity, in what follows we shall assume that the statistics of the surface are
Gaussian.
Statistical characterization of the surface
Another common assumption is that the correlation function W (|x1 |) is also Gaussian:
W (|x1 |) = exp(−x21 /a2 ),
with the corresponding power spectrum
√
g(|k|) = πa exp(−a2 k2 /4).
δ - standard deviation of heights or rms height; measure of the variations in height.
a - correlation length; measure of the lateral scale of the irregularities.
For a zero-mean Gaussian random process with a Gaussian correlation function, the
process is completely specified by the parameters δ and a.
This kind of surfaces are often called single scale surfaces.
(Note: the assumptions of Gaussian statistics and Gaussian correlation should not be confused.)
Statistical characterization of the surface
Surfaces found in practice have many lateral scales and are better modeled as random
processes with power law spectra.
At least over some range of spatial frequencies they behave as fractal surfaces.
Fractal surfaces cannot be described in terms of the standard deviation of heights and
the height correlation function.
In such case, one normally works with the structure function
D(|x1 − x01 |) = [ζ(x1 ) − ζ(x01 )]2
2
0
= 2δ 1 − W (|x1 − x1 |) ,
where the last equality only holds for cases in which δ and W (|x1 |) exist.
So, in terms of the structure function, the power spectrum is given by,
Z
D(|x
|)
1
.
g(|k|) =
dx1 e−ikx1 1 −
2δ 2
Statistical characterization of the surface
A gaussian-correlated photoresist surface vs. a ground glass surface.
Statistical characterization of the surface
One-dimensional fractal surfaces have a power spectrum of the form
g(|k|) =
Kn
,
|k|n
with 1 < n < 3.
The fractal dimension D is related to the exponent n of the power law through the
expression
D = (5 − n)/2.
– n = 1 corresponds to an extreme fractal (D = 2),
– n = 2 corresponds to a Brownian fractal (D = 1.5),
– n = 3 to a marginal fractal (D = 1).
For the case of two-dimensional surfaces, the exponent must be increased by one.
In practice, power law spectra must be truncated at some stage (inner and outer
scale). In this respect, surfaces whose correlation function is a negative exponential
function provide an interesting model. The correlation function
W (|x1 |) = exp(−|x1 |/ξ),
is associated with the Lorentzian power spectrum
g(|k|) =
2ξ
,
1 + ξ 2k2
where ξ is a parameter.
We see that for large k the behavior is that of a Brownian fractal. For small k, on the
other hand, the power spectrum is well behaved.
This permits the use of the standard formalism in terms of the standard deviation and
correlation function, and suggests an extension to power spectra of the kind
g(|k|) =
Aν
,
(1 + ξ 2 k2 )ν+1/2
where Aν is a constant, and we have found it convenient to define the parameter
ν = (n − 1)/2.
Statistical characterization of the surface
From the normalization condition, we find that
√ Γ(2ν + 1)
Aν = 2ξ π
,
Γ(ν)
where Γ(2ν + 1) is the gamma function. The corresponding correlation function can
then be written as
(|x1 |/ξ)ν
Wν (|x1 |) = ν−1
Kν (|x1 |/ξ) .
2
Γ(ν)
In this expression, Kν (z) represents a modified Bessel function of order ν.
To model fractal surfaces 0 < ν < 1, but it is clear that the expressions are also valid
for ν > 1.
Statistical characterization of the surface
We note that for semi-integer values of ν the expression has simple representations.
We have, for instance, the following three correlation functions:
W1/2 (|x1 |) = exp(−|x1 |/ξ) ,
W3/2 (|x1 |) = (1 + |x1 |/ξ) exp(−|x1 |/ξ) ,
1
W5/2 (|x1 |) = 1 + |x1 |/ξ + (|x1 |/ξ)2 exp(−|x1 |/ξ) .
3
The family of K-correlation functions provide a versatile model for the characterization of rough surfaces, encompassing behaviors that can range from the negative
exponential correlation for ν = 1/2 to the case of the Gaussian correlation in the limit
ν → ∞.
Height [µm]
.
5
(a)
0
Height [µm]
-5
5
0
200
400
600
800
1000
200
400
600
800
1000
200
400
600
800
1000
200
400
600
Position [µm]
800
1000
(b)
0
Height [µm]
-5
5
0
(c)
0
-5
Height [µm]
0
(d)
5
0
0
Realizations of surface profiles corresponding to different kinds of random processes. (a) Gaussian
random process with a Gaussian correlation. (b) Gaussian random process with a negative
exponential correlation. (c) Gaussian random process with a power law spectrum. (d) Surface with
negative exponential statistics and Gaussian correlation.
Statistics of the field
At a point P in the far field
N
X
N
1 X
ψ=
aj = √
αj eiφj .
N j=1
j=1
Random walk in the complex plane.
Assumptions:
1. αj and φj are statistically independent.
2. αj are identically distributed with mean < α > and second moment < α2 >.
3. The phases are uniformly distributed on (−π, π).
The first assumption is the most important.
Statistics of the field
Let
N
1 X
ψr = <e{ψ} = √
αj cos φj ,
N j=1
N
1 X
ψi = =m{ψ} = √
αj sin φj .
N j=1
Both, ψr and ψr arise from many independent random contributions.
The Central Limit Theorem indicates that when N >> 1, both should approach
Gaussian distributions.
Thus, ψr and ψr are joint Gaussian variables and the field ψ is a Complex Gaussian
random variable.
Statistics of the field
N
N
1 X
1 X
hαj cos φj i = √
hαj ihcos φj i = 0,
hψr i = √
N j=1
N j=1
N
N
1 X
1 X
hψii = √
hαj sin φj i = √
hαj ihsin φj i = 0.
N j=1
N j=1
This result follows from the fact that hcos φj i = hsin φj i = 0, because the phase is
uniformly distributed on (−π, π).
Statistics of the field
Similarly,
N
X
1
< α2 >
2
hψr i =
= σ2,
hαj αk ihcos φj cos φk i =
N
2
j,k=1
hψi2 i
N
1 X
< α2 >
=
= σ2,
hαj αk ihsin φj sin φk i =
N
2
j,k=1
N
1 X
hψr ψii =
hαj αk ihcos φj sin φk i = 0,
N
j,k=1
where we have used the results
hcos φj cos φk i = hsin φj sin φk i = 1/2;
and zero for j 6= k, and
hcos φj sin φk i = 0,
for all j and k.
j = k,
Statistics of the field
Thus, the real and imaginary parts of ψ have zero means, equal variances and are
uncorrelated
1
PΨ (ψr , ψi) =
exp
2πσ 2
ψr2 + ψi2
2σ 2
.
The field is a circular complex Gaussian random variable.
The contours of equal probability are circles in the complex plane.
Statistics of the field: Amplitude and phase
The amplitude and phase are given by
q
a = ψr2 + ψi2 ;
θ = tan−1 ψi/ψr .
The inverse transform is
ψr = a cos θ;
ψi = a sin θ.
Since the Jacobian of the transform is a, we can write
PA,Θ (a, θ) = PΨ (ψr = a cos θ, ψi = a sin θ) · a .
Integrating over θ one gets
PA(a) =
a
2
2
exp{−a
/(2σ
)},
σ2
for a ≥ 0.
The amplitude follows a Rayleigh distribution.
Statistics of the field
The intensity is given by the transformation
2
I=a ;
a=
p
(I).
Then
PI (I) = PA
√ da a = I dI
1
2
exp{−I/(2σ
)},
2σ 2
for I ≥ 0. But since hIi = 2σ 2 , we can write
=
PI (I) =
1
exp{−I/hIi}.
hIi
Universal law!
The standard deviation σI = hIi, and the speckle contrast σI /hIi = 1.
Statistics of the field
(a)
(b)
(a) Speckle from a weakly rough surface. (b) Speckle from a rough surface.
Speckle conveys (practically) no information about the properties of the scattering
surface.
To determine the lateral scale of the pattern, we consider a simple expression for the
scattering amplitude
Z
R(q|k) = A0 dx1 P (x1 )e−iφ(x1 ) e−i(q−k)x1 ,
where q = (ω/c) sin θs and k = (ω/c) sin θ0 , P (x1 ) is the amplitude of the illuminating
field, and φ(x1 ) is the random phase introduced by the irregularities.
The correlation can be written as
Z Z
0
0
0
CR (q, q 0 ) = hR(q)R∗ (q 0 )i = I0
dx1 dx01 P (x1 )P ∗ (x01 )he−i[φ(x1 )−φ(x1 )] ie−i(q−k)x1 ei(q −k)x1
If we assume that the diffuser is δ−correlated we can write
Z
0
CR (q − q 0 ) = I0 dx1 |P (x1 )|2 e−i(q−q )x1 .
Fourier transform of the illumination on the surface.
Analogue of the van Cittert-Zernike theorem of coherence theory.
Independent of the roughness properties!