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 [
1–3]. 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
[
9–11]. 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 [
14–18]. 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
[
21–23]. 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
r4π
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
δjkj − 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
δjkj − 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
1∕λ 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
r4π
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
rk
2
0
Z
Gr − r
0
Δ
B
r
0
E
s
r
0
E
r
r
0
d
3
r
0
; (7)
where Gre
j2πkjrj
∕4πjrj; is the free-space Green’s
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
rk
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
1∕4π
2
k
2
0
n
2
− 1E
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 Green’s 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
re
2πjk
r
·r
, the boundary field and its normal
derivative can be written [
20],
E
t
r1 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
imaging—that 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 Green’s 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 Green’s 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
ArWr
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 Green’s 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
δjr
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
δjk
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
rδjk
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
j∕4πk
0
δjk
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
j∕4πk
0
δjk
0
j − k
0
step
k
0
·
ˆ
o∕k
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
IrjE
m
r − E
r
rj
2
jE
r
rj
2
jE
m
rj
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),
OrE
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
r4πjRAr4πjRWr
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
