eugenio r. m´endez departamento de optica´gea.df.uba.ar/giambiagi/material/mendez_c01.pdf ·...

Light scattering by randomly rough surfacesEugenio R. Mendez

Departamento de Optica

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

X2

X1

X3

k

θsθ0

φs

q

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

x1

x3

3 1ζx = (x )

Surface scattering vs. volume scattering

Introduction: Mirror

Scattering Angle [deg]

-60 -30 0 30 60

∂Rs(

θ s)/

∂θs

specular

Specular reflection by a mirror

Introduction: Weakly rough surface


-60 -30 0 30 60

<∂R

s(θ s

)/∂θ

s>

-60 -30 0 30 60

∂Rs(

θ s)/

∂θs

0


specular+ diffuse

Scattering by a weakly rough surface (δ << λ)

Introduction: Rough surface


-90 -60 -30 0 30 60 90

<∂R

s(θ s

)/∂θ

s>

-90 -60 -30 0 30 60 90

∂Rs(

θ s)/

∂θs


diffuse

"incoherent"

Scattering by a rough surface (δ ∼ λ)

Introduction: Speckle

(a) (b)

(a) Speckle from a weakly rough surface. (b) Speckle from a rough surface.

Intensity fluctuations:

PI(x) =1

〈I〉exp{−x/〈I〉}

Universal law!

Introduction: Rougher surface (larger slopes)


-90 -60 -30 0 30 60 90

<∂R

s(θ s

)/∂θ

s>

"coherent effect"enhancedbackscattering

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. They can be

classified as:

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.


X3

X2

X1

ζ(x1)

X3

X2

X1

ζ(x1,x2)

(a) (b)

Examples of randomly rough surfaces. (a) Surface with two-dimensional roughness

and (b) Surface with one-dimensional roughness.


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.


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.


A basic quantity for the description of the random process is the two-point height

correlation function

〈ζ(x1)ζ(x′1)〉 = δ2W (|x1 − x′1|),

where the angled brackets represent an average over an ensemble of realizations of

the surface, and

δ2 = 〈ζ2(x1)〉.

The parameter δ represents the standard deviation or the rms height of the surface.

The fact that the autocorrelation function W (|x1 − x′1|) depends on the coordinates

x1 and x′1 only through their difference is a reflection of the assumed stationarity of

ζ(x1).


For perturbative calculations it is often necessary, to introduce the Fourier integral

representation of the surface profile function,

ζ(x1) =

∫dk

2πeikx1ζ(k),

where k is a wave vector. For stationary surfaces, the two-point correlation of the

Fourier coefficient ζ(k) is given by

〈ζ(k)ζ(k′)〉 = 2πδ(k+ k′)δ2g(|k|).

The function g(|k|) is called the power spectrum of the surface roughness, and is

defined by

g(|k|) =

∫dx1e

−ikx1W (|x1|).

It is a non-negative function of |k|, and is normalized according to∫dk

2πg(|k|) = 1.


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.

PZ(ζ1, ζ2) =1

2πδ2√

1− ρ212

exp

{−ζ21 + ζ22 − 2W12 ζ1ζ2

2(1−W12)δ2

},

with

W12 =〈ζ1ζ2〉δ2

,

and 〈ζ〉 = 0.


One also has that

MZ(ω1, ω2) =

∫ ∞

−∞

∫ ∞

−∞ei(ω1ζ1+ω2ζ2)PZ(ζ1, ζ2)dζ1dζ2 =

⟨ei(ω1ζ1+ω2ζ2)

⟩= exp

{−(δ2/2)[W12 ω

21 + 2ω1ω2 +W12 ω

22]

},

PZ(ζ) =1√2πδ

exp{−ζ2/(2δ2)

},

MZ(ω) =

∫ ∞

−∞eiωζPZ(ζ)dζ =

⟨eiωζ

⟩= exp

{−δ2ω2/2

}


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., Mendez, 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.


Another common assumption is that the correlation function W (|x1|) is also Gaussian:

W (|x1|) = exp(−x21/a

2),

with the corresponding power spectrum

g(|k|) =√πa exp(−a2k2/4).

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.)


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 − x′1|) =⟨[ζ(x1)− ζ(x′1)]

2⟩

= 2δ2[1−W (|x1 − x′1|)

],

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,

g(|k|) =

∫dx1e

−ikx1

(1−

D(|x1|)2δ2

).


A gaussian-correlated photoresist surface vs. a ground glass 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 + ξ2k2,

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 + ξ2k2)ν+1/2,

where Aν is a constant, and we have found it convenient to define the parameter

ν = (n− 1)/2.


From the normalization condition, we find that

Aν = 2ξ√πΓ(2ν + 1)

Γ(ν),

where Γ(2ν + 1) is the gamma function. The corresponding correlation function can

then be written as

Wν(|x1|) =(|x1|/ξ)ν

2ν−1Γ(ν)Kν(|x1|/ξ) .

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.


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) ,

W5/2(|x1|) =

(1 + |x1|/ξ+

1

3(|x1|/ξ)2

)exp(−|x1|/ξ) .

The family of K-correlation functions provide a versatile model for the characteri-

zation 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 →∞.

.

-5

0

5

0 200 400 600 800 1000

Hei

ght [

µm]

-5

0

5

0 200 400 600 800 1000

Hei

ght [

µm]

-5

0

5

0 200 400 600 800 1000

Hei

ght [

µm]

0

5

0 200 400 600 800 1000

Hei

ght [

µm]

Position [µm]

(a)

(b)

(c)

(d)

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

ψ =N∑j=1

aj =1√N

N∑j=1

αjeiφj.

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.


Let

ψr = <e{ψ} =1√N

N∑j=1

αj cosφj,

ψi = =m{ψ} =1√N

N∑j=1

αj sinφj.

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.


〈ψr〉 =1√N

N∑j=1

〈αj cosφj〉 =1√N

N∑j=1

〈αj〉〈cosφj〉 = 0,

〈ψi〉 =1√N

N∑j=1

〈αj sinφj〉 =1√N

N∑j=1

〈αj〉〈sinφj〉 = 0.

This result follows from the fact that 〈cosφj〉 = 〈sinφj〉 = 0, because the phase is

uniformly distributed on (−π, π).


Similarly,

〈ψ2r 〉 =

1

N

N∑j,k=1

〈αjαk〉〈cosφj cosφk〉 =< α2 >

2= σ2,

〈ψ2i 〉 =

1

N

N∑j,k=1

〈αjαk〉〈sinφj sinφk〉 =< α2 >

2= σ2,

〈ψrψi〉 =1

N

N∑j,k=1

〈αjαk〉〈cosφj sinφk〉 = 0,

where we have used the results

〈cosφj cosφk〉 = 〈sinφj sinφk〉 = 1/2; j = k,

and zero for j 6= k, and

〈cosφj sinφk〉 = 0,

for all j and k.


Thus, the real and imaginary parts of ψ have zero means, equal variances and are

uncorrelated

PΨ(ψr, ψi) =1

2πσ2exp

{ψ2r + ψ2

i

2σ2

}.

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

a =√ψ2r + ψ2

i ; θ = tan−1ψi/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

σ2exp{−a2/(2σ2)},

for a ≥ 0.

The amplitude follows a Rayleigh distribution.


The intensity is given by the transformation

I = a2; a =√

(I).

Then

PI(I) = PA

(a =

√I) ∣∣∣∣dadI

∣∣∣∣=

1

2σ2exp{−I/(2σ2)},

for I ≥ 0. But since 〈I〉 = 2σ2, we can write

PI(I) =1

〈I〉exp{−I/〈I〉}.

Universal law!

The standard deviation σI = 〈I〉, and the speckle contrast σI/〈I〉 = 1.


(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

R(q|k) = A0

∫dx1P (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

CR(q, q′) = 〈R(q)R∗(q′)〉 = I0

∫ ∫dx1dx

′1P (x1)P

∗(x′1)〈e−i[φ(x1)−φ(x′1)]〉e−i(q−k)x1ei(q′−k)x′1

If we assume that the diffuser is δ−correlated we can write

CR(q − q′) = I0

∫dx1|P (x1)|2e−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!

eugenio r. m´endez departamento de optica´gea.df.uba.ar/giambiagi/material/mendez_c01.pdf ·...

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