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Coherence scanning interferometry: linear theory of surface measurement

01 Jun 2013-Applied Optics (Optical Society of America)-Vol. 52, Iss: 16, pp 3662-3670
TL;DR: A difference between the filter characteristics derived in each case is found and the equivalence of the two approaches when applied to a weakly scattering object is explained.
Abstract: The characterization of imaging methods as three-dimensional (3D) linear filtering operations provides a useful way to compare the 3D performance of optical surface topography measuring instruments, such as coherence scanning interferometry, confocal and structured light microscopy. In this way, the imaging system is defined in terms of the point spread function in the space domain or equivalently by the transfer function in the spatial frequency domain. The derivation of these characteristics usually involves making the Born approximation, which is strictly only applicable to weakly scattering objects; however, for the case of surface scattering, the system is linear if multiple scattering is assumed to be negligible and the Kirchhoff approximation is assumed. A difference between the filter characteristics derived in each case is found. However this paper discusses these differences and explains the equivalence of the two approaches when applied to a weakly scattering object.

Summary (2 min read)

JE E63, E65, F36, F47 JOURNAL OF ECONOMIC LITERATURE CLASSIFICATION NUMBER

  • KW Exchange Rate Mechanism, Germany, rational expectations, transformation KEY WORDS AB ABSTRACT (Abstract follows.).
  • In particular, unification was expected to give rise to an increase in German aggregate demand that would put upward pressure on output, inflation, and the exchange rate, and downward pressure on the current account balance in Germany.
  • The model simulations also highlighted the contractionary effects of high German interest rates on other member countries of the Exchange Rate Mechanism of the European Monetary System.
  • 1Gagnon is Senior Economist in the Division of International Finance, Board of Governors of the Federal Reserve System; Masson is Assistant Director in the Research Department, International Monetary Fund; and McKibbin is Professor of Economics at Australian National University and Nonresident Senior Fellow at the Brookings Institution.
  • For the first time a socialist planned economy was being transformed into a capitalist market economy, and the transformation was occurring almost overnight.

II. An Overview of Developments Since Unification

  • This section provides a capsule summary of the evolution of the main macroeconomic variables since economic unification in July 1990.
  • Nevertheless, an overall picture emerges of a sharp fall in East German output, followed by a period of sustained growth that is faster than in the West.
  • Investment is relatively strong in the East, but large fiscal transfers continue from West to East.
  • Clearly, a first priority was to reduce the extent of state ownership and to convert the economy to private ownership.
  • Moreover, new investment has been flowing into East Germany, and there are reports that new plants operating in the eastern part of Germany are as profitable as those in the West.

III. The Models and Methods of Simulating the Impacts of Unification

  • In this section the authors summarize the models used in the early studies of German unification and give an overview of the approach and results of each of the original studies.
  • The last few months of 1989 saw large population flows from the East, and substantial migration continued early in 1990.
  • It is likely that a severe slowdown would result in East Germany (still not explicitly part of the model) and that this would lead to pressure on the Bundesbank to ease policy.
  • In MX3, the monetary authorities are assumed to move short-term interest rates in response to deviations of a target variable from its targeted level.
  • Investment increases due to the higher marginal product of capital in East Germany.

IV. Results from the Models and Key Issues Raised in the Early Studies

  • In section III the authors outlined three studies, focusing on the characteristics of the models, the approach to modeling unification, and the issues that were addressed.
  • In this section the authors present some key results from each model and draw out some overall lessons from the studies and their implications for model development and policy formulation.
  • Table 5 contains standardized results for MULTIMOD, Table 6 contains standardized results for MSG2, and Table 7 contains standardized results for MX3.
  • The scenario ordering is based on the original studies and is not necessarily the same across models.
  • In the standard trade equations used by most models, exports depend on foreign income and imports depend on domestic income.

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Coherence scanning interferometry: linear
theory of surface measurement
Jeremy Coupland,
1,
* Rahul Mandal,
1
Kanik Palodhi,
1
and Richard Leach
2
1
Mechanical and Manufacturing Engineering, Loughborough University, Loughborough, LE11 3TU, UK
2
Engineering Measurement Division, National Physical Laboratory, Teddington, TW11 0LW, UK
*Corresponding author: j.m.coupland@lboro.ac.uk
Received 21 December 2012; revised 4 April 2013; accepted 15 April 2013;
posted 23 April 2013 (Doc. ID 182295); published 22 May 2013
The characterization of imaging methods as three-dimensional (3D) linear filtering operations provides a
useful way to compare the 3D performance of optical surface topography measuring instruments, such as
coherence scanning interferometry, confocal and structured light microscopy. In this way, the imaging
system is defined in terms of the point spread function in the space domain or equivalently by the transfer
function in the spatial frequency domain. The derivation of these characteristics usually involves making
the Born approximation, which is strictly only applicable to weakly scattering objects; however, for the
case of surface scattering, the system is linear if multiple scattering is assumed to be negligible and the
Kirchhoff approximation is assumed. A difference between the filter characteristics derived in each case
is found. However this paper discusses these differences and explains the equivalence of the two
approaches when applied to a weakly scattering object. © 2013 Optical Society of America
OCIS codes: (070.0070) Fourier optics and signal processing; (090.0090) Holography; (120.0120)
Instrumentation, measurement, and metrology; (180.0180) Microscopy; (240.0240) Optics at surfaces;
(290.0290) Scattering.
http://dx.doi.org/10.1364/AO.52.003662
1. Introduction
Coherence scanning interferometry (CSI) is a three-
dimensional (3D) imaging technique that is used to
measure areal surface topography. It combines the
vertical resolution of an interferometer with the
lateral resolution of a high-power microscope and
provides a fast, noncontacting alternative to contact
stylus profilometers [
13]. CSI typically utilizes
broadband, incandescent, or LED sources and Mirau
interference objectives to record the interference
between the light scattered by the object and that
reflected from a reference surface as the objective is
scanned though focus [
4]. Since the source illumina-
tion is limited in both temporal and spatial coher-
ence, the interference fringes are observed over a
finite scan range and it is relatively straightforward
to locate the bright zero-order fringe that identifies
when path length is balanced in the interferogram.
For this reason, CSI is particularly useful for the
measurement of discontinuous surfaces, such as
those produced in the microelectronics industry.
Despite these significant advantages, CSI exhibits
certain problems that restrict its use as a traceable
measurement tool particularly when it is used to
measure sloped artifacts [
5]. Since CSI instruments
are typically calibrated using step height standards
and lateral calibration artifacts of a waffle plate
design [
6], problems with the measurement of sloped
artifacts often go unnoticed [
7,8]. In order to over-
come these deficiencies and improve measurement
quality a calibration and adjustment method using
spherical artifacts has recently been considered
[
911]. Since all slope angles are equally represented
by a spherical surface (smaller than the field of view
of the CSI), slope-related errors are immediately ap-
parent. More importantly, however, an interferogram
1559-128X/13/163662-09$15.00/0
© 2013 Optical Society of America
3662 APPLIED OPTICS / Vol. 52, No. 16 / 1 June 2013

of a spherical artifact provides the information nec-
essary to define the resolution of the instrument and,
in some cases, compensate for errors introduced by
lens aberrations [
9].
The theory that underpins this work is based on a
linear theory of 3D imaging that was first published
in the context of optical holography by Wolf [
12], and
Dandliker and Weiss [
13]. Using this approach the
performance of 3D imaging techniques including dig-
ital holography, confocal microscopy, CSI, and other
interferometers can be compared in terms of linear
system theory [
1418]. In this way the system is
characterized either in the space domain by the point
spread function (PSF) or equivalently in the (spatial)
frequency domain, by the transfer function (TF). As
the image of a point-like object, the PSF provides a
direct measure of the 3D resolution cell of the
imaging system while the TF describes how the
phase and amplitude of the spatial frequencies
present within the object are modified by the imaging
system. For the case of CSI, the phase of the TF is
of primary importance since the surface height is
deduced from this quantity.
Although the linear theory provides a good means
to compare the theoretical performance of different
imaging systems, it rests on the assumption of weak
scattering and the validity of the Born approxima-
tion [
19]. In essence, weak scattering implies that the
object causes a small perturbation to the illuminat-
ing field. The Born approximation can be assumed
when there are small changes in refractive index or
small objects, such as particles suspended in fluid,
but is not generally applicable to the comparatively
large changes in refractive index that are typical of
3D scattering objects. When light is scattered from
the interface between homogenous media, however,
it is not necessary to assume the Born appro ximation
and, providing that there is no multiple scattering
and the surface is smooth at the optical scale, the
process is also linear. A detailed analysis of surface
scattering has been presented by Beckmann and
Spizzichino [
20] and this forms the basis of inverse
scattering methods that attempt to deduce surface
topography from measurements of the scattered field
[
2123]. In this case, the surface boundary conditions
are assumed and the object can be replaced by an in-
finitely thin foil-like object, which follows the surface
topography and henceforth will be called the foil
model of the surface. As pointed out by Sheppard in
the context of confocal microscopy [
24,25], an appar-
ent consequence of the surface scattering approach is
that the effective TF (and PSF) of the measuring
instrument is modified.
In this paper, the derivation of the foil model of
the surface is presented and the associated PSF and
TF are defined. Starting from the integral form of
the Helmholtz equation, the differences between the
analyses based on the Born approximation and the
surface scattering approach are contrasted. It is
shown that there is a small but significant difference
between the PSFs and TFs; however, the two
approaches yield exactly the same result when both
the numerical aperture and refractive change tend
to zero.
2. Theory
It is shown elsewhere [17] that the output, O
B
r,of
a CSI instrument can be written as a 3D linear
filtering operation that is characterized in the space
domain by the convolution,
O
B
r
Z
H
B
r r
0
Δ
B
r
0
d
3
r
0
; (1)
where Δ
B
r4π
2
1 n
2
r defines the object in
terms of the refractive index n an d H
B
r r
0
is the
PSF given by
H
B
r
Z
G
2
NA
r;k
0
k
2
0
Sk
0
dk
0
. (2)
In this expression, Sk
0
is the spectral density
expressed as a function of the wavenumber k
0
and
G
NA
r;k
0
is the PSF of an imaging system of numeri-
cal aperture N
A
given by
G
NA
r;k
0

Z
j
4πk
0
δjkj k
0
step
×
k ·
ˆ
o
k
0

1 N
2
A
q
e
j2πk·r
d
3
k; (3)
where
ˆ
o is a unit vector in the direction of the optical
axis and δx and stepx represent a Dirac delta
function and a Heaviside step function, respectively.
Equivalently the filtering operation is defined in the
frequency domain (k-space) by the relation
~
O
B
k
~
Δ
B
k
~
H
B
k; (4)
where tilde denotes Fourier transformation such
that H
B
k
R
H
B
re
jπ2k·r
d
3
k represents the TF
and is given by
~
H
B
k
ZZ
~
G
NA
k
0
;k
0
~
G
NA
k k
0
;k
0
d
3
k
0
k
2
0
Sk
0
dk
0
;
(5)
where
~
G
NA
k;k
0

j
4πk
0
δjkj k
0
step
k ·
ˆ
o
k
0

1 N
2
A
q
.
(6)
It is noted that Eqs. (
1)(6) differ slightly from
those given in reference [
17], as some numerical
constants have been included in the object function
and the alternative definition of wavenumber k
0
1λ has been used here. The equations rest on the
1 June 2013 / Vol. 52, No. 16 / APPLIED OPTICS 3663

assumption of weak scattering, or in other words,
that the incident field is weakly perturbed by the
object. This is reasonable for objects that are charac-
terized by small variations in refractive index, such
as cellular tissue, but is rarely justified for general
3D objects. For the case of strong surface scattering
from the interface between two homogenous media
however, providing multiple scattering is negligible,
the process can also be considered linear. In order to
relate these two apparentl y disparate processes
scattering by an object characterized by the function
Δ
B
r4π
2
1 n
2
r is considered, as shown in
Fig.
1.
If the object is illuminated by the reference field
E
r
r, then scattered field denoted by E
s
r, is given
by the integral form of the Helmholtz equation such
that [
17]
E
s
rk
2
0
Z
Gr r
0
Δ
B
r
0
E
s
r
0
E
r
r
0
d
3
r
0
; (7)
where Gre
j2πkjrj
4πjrj; is the free-space Greens
function that defines a poin t source. It is noted that
the scattered field is in general a nonlinear function
of the object function Δ
B
r
0
; however, the process is
linearized by assuming that the term E
s
r
0
;inthe
integrand is negligible. This is the well-known Born
approximation, which is applicable to weak scatter-
ing events [
19]. It is clear, however, that the only
contribution to the integral is from regions where
Δ
B
r
0
4π
2
1 n
2
is nonzero (i.e., from the volume
occupied by the object itself) and the scattered field
can, therefore, be written as the volume integral,
E
s
rk
2
0
Z
V
Gr r
0
Δ
B
r
0
E
t
r
0
d
3
r
0
; (8)
where E
t
r is the transmitted field (i.e., that inside
the object boundary) and V denotes the object
volume.
Since inside the object
2
4π
2
n
2
k
2
0
E
t
r
0
0
and
2
4π
2
k
2
0
Gr r
0
0, then Gr r
0
E
t
r
0

14π
2
k
2
0
n
2
1E
t
r
0
2
Gr r
0
Gr r
0
2
E
t
r
0
.
Substitution gives
E
s
r
Z
V
Gr r
0
2
E
t
r
0
E
t
r
0
2
Gr r
0
d
3
r
0
;
(9)
and applying Greens theorem it is found
E
s
r
Z
S
Gr r
0
E
t
r
0
n
E
t
r
0
Gr r
0
n
ds;
(10)
where S denotes the object boundary. Equation (
10)
is the Kirchhoff integral [
19]. It is exact, but hardly
surprising, as it merely shows the well-known result
that the scattered field from the medium can be writ-
ten purely in terms of the field at the object boundary.
However, it is now straightforward to linearize the
scattering process by assuming appropriate boun-
dary conditions. Followi ng Beckman and Spizzichino
[
20], if the surface is illu minated by a unit amplitude
plane wave, propagating with wave vector, k
r
, such
that E
r
re
2πjk
r
·r
, the boundary field and its normal
derivative can be written [
20],
E
t
r1 Re
2πjk
r
·r
; (11)
E
t
r
n
2πjk
r
·
ˆ
n
S
1 Re
2πjk
r
·r
; (12)
where
ˆ
n
S
is the outward surface normal (as shown
in Fig.
1) and R is the Fresnel amplitude reflection
coefficient, which is assumed to be constant over the
range of scattering angles of interest. Beckmann
and Spizzichino have discussed the validity of these
boundary conditions in detail [
20] but for the
purposes of this paper it is noted that:
(i) The surface must be slowly varying on the
optical scale such that the local radius of curvature
is more than the wavelength. This is the Kirchhoff
or physical optics approximation [
19].
(ii) For a perfect conductor the reflection coefficient
is indeed constant (R 1).
(iii) More generally, the reflection coefficient de-
pends on polarization but the sum of reflection
coefficients for orthogonal polarization states is
approximately constant for angles of incidence that
are less than 45 deg.
(iv) For a dielectric, the field at the lower boundary
and its gradient may depart markedly from those
given in Eqs. (
11) and (12) due to propagation
through the object. However, this component of the
field will generally be separable from that scattered
from the top boundary using CSI due to the extra
path length traveled.
In order to explain the output of a CSI instrument
it is first necessary to consider the process of far-field
imagingthat is, measuring or reconstructing a field
Fig. 1. Scattering from a 3D object.
3664 APPLIED OPTICS / Vol. 52, No. 16 / 1 June 2013

solely from the information present at a distant boun-
dary. It is shown in Appendix
A that propagation to
and from a distant boundary is a linear filtering
operation that is characterized by a PSF that de-
pends on the numerical aperture of the instrument.
In the following a similar process will be followed to
give an expression for the measured scattered field.
First, consider the field that propagates from the
upper surface to a point r
b
; on a distant boundary Σ,
as shown in Fig.
2. Since the boundary is at a large
distance, r
b
r and the far-field Greens function
can be written
Gr
b
r
e
2πjk
0
jr
b
j
4πjr
b
j
e
2πjk
0
r·
r
b
jr
b
j
. (13)
The normal derivative of the Greens function is,
therefore,
Gr
b
r
n
2πjGr
b
rk
0
r
b
jr
b
j
·
ˆ
n
S
. (14)
Substituting Eqs. (
11)(14) into the Kirchhoff
integral of Eq. (
10), the scattered field at the distan t
boundary is given by
E
s
r
b
j
e
2πjk
0
jr
b
j
2jr
b
j
Z
S
e
2πj
k
0
r·
r
b
jr
b
j
k
r
·r
R
k
0
r
b
jr
b
j
k
r
k
0
r
b
jr
b
j
k
r

·
ˆ
n
S
ds: (15)
In accordance with comment (iv) above, a region
of interest on the upper surface of the object can be
defined by the function, Ar, given by
ArWr
x
;r
y
δr
z
sr
x
;r
y
; (16)
where Wr
x
;r
y
is a window function. Using the
sifting properties of the Dirac delta function [
26], the
scattered field can be written as an indefinite inte-
gral such that,
E
s
r
b
j
e
2πjk
0
jr
b
j
2jr
b
j
Z
e
2π j
k
0
r
b
jr
b
j
k
r
·r
R
k
0
r
b
jr
b
j
k
r
k
0
r
b
jr
b
j
k
r

·
ˆ
n
S
Ar
ˆ
n
S
· z
d
3
r: (17)
If it is assumed that this field can be measured,
for example using digital holography, an expression
for the measured field can be obtained. In a similar
manner to the derivation presented in Appendix
A,
the measured field, E
0
m
r
0
, can be written as the
Kirchhoff integral,
E
0
m
r
0

Z
Σ
G
r
0
r
b
E
s
r
b
n
E
s
r
b
G
r
0
r
b
n
ds:
(18)
Using the far-field Greens function and assuming,
without loss of generality, that the boundary surface
is spherical,
E
0
m
r
0

k
0
2
Z
Σ
1
jr
b
j
2
Z
e
2π j
k
0
r
b
jr
b
j
k
r
·r
R
k
0
r
b
jr
b
j
k
r
k
0
r
b
jr
b
j
k
r

·
ˆ
n
S
×
Ar
ˆ
n
S
· z
d
3
re
2πjk
0
r
0
r
b
jr
b
j
ds:
(19)
Using the sifting properties of the Dirac delta func-
tion once again, the measured field can be written as
the indefinite integral
E
0
m
r
0

k
0
2
Z
1
jr
b
j
2
Z
e
2πj
k
0
r
b
jr
b
j
k
r
·r
R
k
0
r
b
jr
b
j
k
r
k
0
r
b
jr
b
j
k
r

·
ˆ
n
S
Ar
ˆ
n
S
· z
d
3
re
2πjk
0
r
0
r
b
jr
b
j
δjr
b
j
r
0
d
3
r
b
. (20)
Making the substitution, k
0
k
0
r
b
r
0
, it is found,
E
0
m
r
0

1
2k
0
Z
Z
e
2π jk
0
k
r
·r
Rk
0
k
r
k
0
k
r

·
ˆ
n
S
Ar
ˆ
n
S
:z
d
3
r
δjk
0
j k
0
e
2πjk
0
·r
0
d
3
k
0
. (21)
A further simplification can be made by consider-
ing the phase within the bracketed integral in
Eq. (
21). Since the phase of the complex exponential
changes in the direction defined by k
0
k
r
, only re-
gions of the surface where the surface normal is in
this direction will contribute to the integral. This
is the principle of stationary phase and is illustrated
Fig. 2. Surface scattering to a distant boundary.
1 June 2013 / Vol. 52, No. 16 / APPLIED OPTICS 3665

in Fig. 3. Noting that case in these regions the term
k
0
k
r
·
ˆ
n
S
is negligible and
ˆ
n
S
k
0
k
r
jk
0
k
r
j,
Eq. (
21) becomes
E
0
m
r
0

R
2k
0
ZZ
e
2πjk
0
k
r
·r
jk
0
k
r
j
2
k
0
k
r
· z
Ard
3
rδjk
0
j
k
0
e
2πjk
0
·r
0
d
3
k
0
. (22)
With reference to Appendix
A, an ideal imaging sys-
tem that collects the field over the whole surface of
the boundary sphere has a TF given by
~
G
ideal
k
0

j4πk
0
δjk
0
j k
0
. Making this substitution,
E
0
m
r
0
4πjR
ZZ
e
2π jk
0
k
r
·r
jk
0
k
r
j
2
2k
0
k
r
· z
Ard
3
r
~
G
ideal
k
0
e
2πjk
0
·r
0
d
3
k
0
. (23)
For an instrument of limited numerical aperture,
however, the TF is
~
G
NA
k
0
;k
0
j4πk
0
δjk
0
j k
0
step
k
0
·
ˆ
ok
0

1 N
2
A
q
, and the measured field
E
m
r
0
is given by
E
m
r
0
4πjR
ZZ
e
2π jk
0
k
r
·r
jk
0
k
r
j
2
2k
0
k
r
· z
Ard
3
r
~
G
NA
k
0
;k
0
e
2πjk
0
·r
0
d
3
k
0
. (24)
Equation (
24) describes the field measured by an
unspecified coherent instrument operating in the
far-field with restricted numerical aperture, when
the surface of interest is illuminated by a plane
monochromatic wave propagating in the direction
of the wave vector, k
r
. In order to derive an expres-
sion for the response of a CSI, it is now necessary to
consider this type of instrument in more detail. CSI
records the interference between light scattered
from the surface of interest and that reflected from
a reference flat as the surface is scanned through
focus (i.e., scanned in the axial direction). Typically,
a Mirau objective utilizing an internal reference flat,
as shown in Fig.
4, is used for this purpose.
In this way, the intensity recorded by the camera
is proportional to that in the object plane of the
objective, which is a far-field measurement of the in-
terference between the measured scattered field
E
m
r and the reference field E
r
r, as shown in
Fig.
4. Note that the reference field is E
r
r due
to reflection at the reference surface. Accordingly,
the measured intensity, Ir, in the resulting inter-
ferogram is given by
IrjE
m
r E
r
rj
2
jE
r
rj
2
jE
m
rj
2
E
m
r
E
r
r E
m
rE
r
r
.
(25)
In a similar manner to off-axis holography, these
terms are separable in the frequency domain (see for
example [
17]) so the output of a CSI instrument is
defined as the modulated (fringe) component of the
interferogram given by the fourth term in Eq. (
25),
OrE
m
rE
r
r
. (26)
Returning to the expression obtained for surface
scattering with plane wave illumination, the output
is, O
F
r
0
E
m
r
0
e
2π jk
r
·r
0
and substituting the mea-
sured scattered field E
m
r
0
from Eq. (24), then
O
F
r
0

ZZ
e
2πjk
0
k
r
·r
jk
0
k
r
j
2
2k
0
k
r
· z
Δ
F
rd
3
r
~
G
NA
k
0
;k
0
e
2πjk
0
k
r
·r
0
d
3
k
0
; (27)
where the surface is defined by the function, Δ
F
r,
given by
Δ
F
r4πjRAr4πjRWr
x
;r
y
δr
z
sr
x
;r
y
.
(28)
Fig. 3. Principle of stationary phase.
Fig. 4. Superposition of the reference and scattered fields in a
Mirau objective.
3666 APPLIED OPTICS / Vol. 52, No. 16 / 1 June 2013

Citations
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Journal ArticleDOI
TL;DR: Optical interference microscopes are widely used in topography and roughness measurement on the micro and nanoscale as mentioned in this paper, however, despite their wide range of scientific and industrial applications, systematic d...
Abstract: Optical interference microscopes are widespread in topography and roughness measurement on the micro- and nanoscale. In spite of a wide range of scientific and industrial applications, systematic d...

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Proceedings ArticleDOI
21 Jun 2019
TL;DR: In this article, a boundary elements method is used as a rigorous scattering model to calculate the scattered field at a distant boundary, and the CSI signal is calculated by considering the image formation as back-propagation of the scattering field, combined with the reflected reference field.
Abstract: Coherence scanning interferometry (CSI) is a well-established technique for measuring surface topography based on the coherence envelope and phase of interference fringes. The most commonly used surface reconstruction methods, i.e. frequency domain analysis, the envelope detection method, and the correlogram correlation method, obtain the phase of the measured field for each pixel and, from this obtain the surface height, by assuming the two are directly proportional. For surfaces with minor deviations from a plane, it is straightforward to show that the scattered field’s phase is a linear function of surface height. An alternative approach known as the “foil model” gives more generally the scattered field as the result of a linear filtering process operating on a “foil” representation of the surface. This model assumes that the surface slowly varies on the optical scale and that there is no multiple scattering. However, for surfaces that are rough at the optical scale or have coherent features (e.g. vee-grooves), the effect of multiple scattering cannot be neglected and remains a problem for reconstruction methods. Linear reconstruction methods cannot provide accurate surface topographies for complex surfaces, since for such surfaces, the measurement process of CSI is fundamentally non-linear. To develop an advanced reconstruction method for CSI, an accurate model of the imaging process is required. In this paper, a boundary elements method is used as a rigorous scattering model to calculate the scattered field at a distant boundary. Then, the CSI signal is calculated by considering the image formation as back-propagation of the scattered field, combined with the reflected reference field. Through this approach, the optical response of a CSI system can be predicted rigorously for almost any arbitrary surface geometry. Future work will include a comprehensive experimental verification of this model, and development of the non-linear surface reconstruction algorithm.

12 citations

Proceedings ArticleDOI
03 Sep 2019
TL;DR: In this paper, the authors review the fundamentals of imaging interferometry and describe ways of defining instrument response, including the linear instrument transfer function, and propose pathways for further improving performance on difficult surface structures using advanced modeling techniques.
Abstract: Interferometers for the measurement of surface form and texture have a reputation for high performance. However, the results for many types of surface features can deviate from the expectation of one cycle of phase shift per half wavelength of surface height. Here we review the fundamentals of imaging interferometry and describe ways of defining instrument response, including the linear instrument transfer function. These considerations define practical regimes of linear behavior that are usually satisfied for traditional uses of interferometers; but that are increasingly challenged by applications involving complex textures and high surface slopes. We conclude by proposing pathways for further improving performance on difficult surface structures using advanced modeling techniques.

12 citations


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  • ...The 3D Fourier transform of the impulse response provides a 3D transfer function that fully describes the filtering process from the foil-like surface to the imaged light field [47, 48]....

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TL;DR: In this article, the wetting contact angle of a sessile drop is acquired aerially using confocal techniques to measure the radius and the height of a droplet deposited on a planar surface.
Abstract: A method is presented in which the wetting contact angle of a sessile drop is acquired aerially using confocal techniques to measure the radius and the height of a droplet deposited on a planar surface The repeatability of this method is typically less than 025°, and often less than 01°, for droplet diameters less than 1 mm To evaluate accuracy of this method, an instrument uncertainty budget is developed, which predicts a combined uncertainty of 091° for a 1 mm diameter water droplet with a contact angle of 110° For droplets having diameters less than 1 mm and contact angles between 15° and 160°, these droplets approach spherical shape and their contact angles can be computed analytically with less than 1% error For larger droplets, gravitational deformation needs to be considered

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  • ...Such a specification standard would necessarily address slope dependent measurement errors [14] and a standardised approach to evaluate, and possibly compensate for, the optical transfer function [20]....

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TL;DR: It is concluded that 3sBSM provides an accurate solution to electromagnetic scattering from a bandlimited surface and efficiently avoids the singular surface integrals and special basis functions proposed by others.
Abstract: This paper describes a novel Boundary Source Method (BSM) applied to the vector calculation of electromagnetic fields from a surface defined by the interface between homogenous, isotropic media. In this way, the reflected and transmitted fields are represented as an expansion of the electric fields generated by a basis of orthogonal electric and magnetic dipole sources that are tangential to, and evenly distributed over the surface of interest. The dipole moments required to generate these fields are then calculated according to the extinction theorem of Ewald and Oseen applied at control points situated at either side of the boundary. It is shown that the sources are essentially vector-equivalent Huygens’ wavelets applied at discrete points at the boundary and special attention is given to their placement and the corresponding placement of control points according to the Nyquist sampling criteria. The central result of this paper is that the extinction theorem should be applied at control points situated at a distance d = 3s (where s is the separation of the sources) and consequently we refer to the method as 3sBSM. The method is applied to reflection at a plane dielectric surface and a spherical dielectric sphere and good agreement is demonstrated in comparison with the Fresnel equations and Mie series expansion respectively (even at resonance). We conclude that 3sBSM provides an accurate solution to electromagnetic scattering from a bandlimited surface and efficiently avoids the singular surface integrals and special basis functions proposed by others.

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References
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TL;DR: The theory of interference and interferometers has been studied extensively in the field of geometrical optics, see as discussed by the authors for a survey of the basic properties of the electromagnetic field.
Abstract: Historical introduction 1. Basic properties of the electromagnetic field 2. Electromagnetic potentials and polarization 3. Foundations of geometrical optics 4. Geometrical theory of optical imaging 5. Geometrical theory of aberrations 6. Image-forming instruments 7. Elements of the theory of interference and interferometers 8. Elements of the theory of diffraction 9. The diffraction theory of aberrations 10. Interference and diffraction with partially coherent light 11. Rigorous diffraction theory 12. Diffraction of light by ultrasonic waves 13. Scattering from inhomogeneous media 14. Optics of metals 15. Optics of crystals 16. Appendices Author index Subject index.

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TL;DR: The scattering of electromagnetic waves from rough surfaces PDF is available at the online library of the University of Southern California as mentioned in this paper, where a complete collection of electromagnetic wave from rough surface books can be found.
Abstract: THE SCATTERING OF ELECTROMAGNETIC WAVES FROM ROUGH SURFACES PDF Are you looking for the scattering of electromagnetic waves from rough surfaces Books? Now, you will be happy that at this time the scattering of electromagnetic waves from rough surfaces PDF is available at our online library. With our complete resources, you could find the scattering of electromagnetic waves from rough surfaces PDF or just found any kind of Books for your readings everyday.

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TL;DR: In this article, a solution to an inverse scattering problem that arises in the application of holography to the determination of the three-dimensional structure of weakly scattering semi-transparent objects is presented.

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"Coherence scanning interferometry: ..." refers methods in this paper

  • ...The theory that underpins this work is based on a linear theory of 3D imaging that was first published in the context of optical holography by Wolf [12], and Dandliker and Weiss [13]....

    [...]

Book
01 Jan 1978
TL;DR: This book describes the representation of Physical Quantities by Mathematical Functions and the applications of Linear Filters and Two-Dimensional Convolution and Fourier Transformation.
Abstract: Representation of Physical Quantities by Mathematical Functions. Special Functions. Harmonic Analysis. Mathematical Operators and Physical Systems. Convolution. The Fourier Transform. Characteristics and Applications of Linear Filters. Two-Dimensional Convolution and Fourier Transformation. The Propagation and Diffraction of Optical Wave Fields. Image-Forming Systems. Appendices. Index.

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TL;DR: A correlation microscope based on the Mirau interferometer configuration using a thin silicon nitride film beam splitter is constructed, which predicts accurately both the transverse resolution at a sharp edge and the range resolution for a perfect plane reflector.
Abstract: We have constructed a correlation microscope based on the Mirau interferometer configuration using a thin silicon nitride film beam splitter. This microscope provides the amplitude and phase information for the reflected signal from a sample located on the microscope-object plane. The device is remarkably insensitive to vibrations and is self-correcting for spherical and chromatic range aberrations of the objective. An imaging theory for the correlation microscope has been derived, which predicts accurately both the transverse resolution at a sharp edge and the range resolution for a perfect plane reflector. The range resolution is slightly better than that for a scanning optical microscope using a lens with the same aperture.

491 citations


"Coherence scanning interferometry: ..." refers background in this paper

  • ...CSI typically utilizes broadband, incandescent, or LED sources and Mirau interference objectives to record the interference between the light scattered by the object and that reflected from a reference surface as the objective is scanned though focus [4]....

    [...]

Frequently Asked Questions (12)
Q1. What have the authors contributed in "Coherence scanning interferometry: linear theory of surface measurement" ?

The characterization of imagingmethods as three-dimensional ( 3D ) linear filtering operations provides a useful way to compare the 3D performance of optical surface topography measuring instruments, such as coherence scanning interferometry, confocal and structured light microscopy. However this paper discusses these differences and explains the equivalence of the two approaches when applied to a weakly scattering object. 

More generally, the reflection coefficient depends on polarization but the sum of reflection coefficients for orthogonal polarization states is approximately constant for angles of incidence that are less than 45 deg. 

For the case of strong surface scattering from the interface between two homogenous media however, providing multiple scattering is negligible, the process can also be considered linear. 

It is noted that the scattered field is in general a nonlinear function of the object function ΔB r0 ; however, the process is linearized by assuming that the term Es r0 ; in the integrand is negligible. 

When the surface of a homogenous object is measured, it can be replaced by an infinitely thin foil-like membrane, which has been called the “foil model” of the surface. 

if the spectral density of the source as a function of wavenumber is S k0 , then integrating over all illumination wave vectors, kr, within the numerical aperture and all wavenumbers, k0 that are defined by the function ~GNA −kr; k0 , the output of the CSI can be written~OF k ~ΔF k ~HF k ; (31)where ~H k is the TF and is given by~HF k jkj2 2k · zZZ ~GNA kr; k0 ~GNA k− kr; k0 d3krS k0 dk0. 

For a dielectric, the field at the lower boundary and its gradient may depart markedly from those given in Eqs. (11) and (12) due to propagation through the object. 

As pointed out by Sheppard in the context of confocal microscopy [24,25], an apparent consequence of the surface scattering approach is that the effective TF (and PSF) of the measuring instrument is modified. 

In this case the Kirchhoff or physical optics approximation is assumed, which implies that the surface is slowly varying (such that the local radius of curvature is larger than the wavelength). 

More importantly, however, an interferogram 1559-128X/13/163662-09$15.00/0 © 2013 Optical Society of America3662 APPLIED OPTICS / Vol. 52, No. 16 / 1 June 2013of a spherical artifact provides the information necessary to define the resolution of the instrument and, in some cases, compensate for errors introduced by lens aberrations [9]. 

The equations rest on the1 June 2013 / Vol. 52, No. 16 / APPLIED OPTICS 3663assumption of weak scattering, or in other words, that the incident field is weakly perturbed by the object. 

As has been shown elsewhere [17,18], these characteristics can be deduced for a range of far-field measurement instruments, including CSI, if the scattered field is a linear function of the object function, but in general, this is only true if some approximations are made.