# Coherence scanning interferometry: linear theory of surface measurement

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.

Topics: Coherence scanning interferometry (63%), Point spread function (59%), Scattering (56%), Born approximation (55%), Inverse scattering problem (55%)

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

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Richard Leach

^{1}, Claudiu Giusca^{2}, Han Haitjema, Christopher J. Evans^{3}+1 more•Institutions (4)TL;DR: The calibration and verification infrastructure to support areal surface texture measurement and characterisation will be reviewed and the concept and current infrastructure for determining the metrological characteristics of instruments will be highlighted.

Abstract: In this paper, the calibration and verification infrastructure to support areal surface texture measurement and characterisation will be reviewed. A short historical overview of the subject will be given, along with a discussion of the most common instruments and directions of current international standards. Traceability and uncertainty will be discussed, followed by a presentation of the latest developments in software and material measurement standards. The concept and current infrastructure for determining the metrological characteristics of instruments will be highlighted and future research requirements will be presented.

77 citations

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TL;DR: It is shown that by calibrating the instrument correctly and using appropriate methods to extract phase from the resulting fringes (such as frequency domain analysis), CSI is capable of measuring the topography of surfaces with varying tilt with sub-nanometre accuracy.

Abstract: Although coherence scanning interferometry (CSI) is capable of measuring surface topography with sub-nanometre precision, it is well known that the performance of measuring instruments depends strongly on the local tilt and curvature of the sample surface. Based on 3D linear systems theory, however, a recent analysis of fringe generation in CSI provides a method to characterize the performance of surface measuring instruments and offers considerable insight into the origins of these errors. Furthermore, from the measurement of a precision sphere, a process to calibrate and partially correct instruments has been proposed. This paper presents, for the first time, a critical look at the calibration and correction process. Computational techniques are used to investigate the effects of radius error and measurement noise introduced during the calibration process for the measurement of spherical and sinusoidal profiles. Care is taken to illustrate the residual tilt and curvature dependent errors in a manner that will allow users to estimate measurement uncertainty. It is shown that by calibrating the instrument correctly and using appropriate methods to extract phase from the resulting fringes (such as frequency domain analysis), CSI is capable of measuring the topography of surfaces with varying tilt with sub-nanometre accuracy.

48 citations

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Rahul Mandal

^{1}, Jeremy M. Coupland^{1}, Richard Leach^{2}, Daniel Ian Mansfield^{3}•Institutions (3)TL;DR: A new method of calibration and adjustment using a silica micro-sphere as a calibration artifact is introduced and a straightforward method to correct for phase and amplitude imperfections in the TF is described using a modified inverse filter.

Abstract: When applied to the measurement of smooth surfaces, coherence scanning interferometry can be described by a three-dimensional linear filtering operation that is characterized either by the point-spread function in the space domain or equivalently by the transfer function (TF) in the spatial frequency domain. For an ideal, aberration-free instrument, these characteristics are defined uniquely by the numerical aperture of the objective lens and the bandwidth of the illumination source. In practice, however, physical imperfections such as those in lens aberrations, reference focus, and source alignment mean that the instrument performance is not ideal. Currently, these imperfections often go unnoticed as the instrument performance is typically only verified using rectilinear artifacts such as step heights and lateral grids. If an object of varying slope is measured, however, significant errors are often observed as the surface gradient increases. In this paper, a new method of calibration and adjustment using a silica micro-sphere as a calibration artifact is introduced. The silica microsphere was used to compute the point-spread and TF characteristics of the instrument, and the effect of these characteristics on instrument performance is discussed. Finally, a straightforward method to correct for phase and amplitude imperfections in the TF is described using a modified inverse filter.

30 citations

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Richard Leach

^{1}, Albert Weckenmann^{2}, Jeremy M. Coupland^{3}, Wito Hartmann^{2}•Institutions (3)Abstract: When using dimensional measuring instruments it is assumed that there is a property of the object, which we call surface, that is present before during and after the measurement, i.e. the surface is a fundamental property of an object that can, by appropriate means, be used to measure geometry. This paper will attempt to show that the fundamental property ‘surface’ does not exist in any simple form and that all the information we can have about a surface is the measurement data, which will include measurement uncertainty. Measurement data, or what will be referred to as the measured surface, is all that really exists. In this paper the basic physical differences between mechanically, electromagnetically and electrically measured surfaces are highlighted and discussed and accompanied by measurement results on a roughness artefact.

19 citations

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TL;DR: An uncertainty estimation method for estimating the measurement uncertainty due to the surface gradient of the workpiece is developed based on the mathematical expression of an uncertainty estimation model which is derived and verified through a series of experiments.

Abstract: Although the scanning white light interferometer can provide measurement results with subnanometer resolution, the measurement accuracy is far from perfect. The surface roughness and surface gradient have significant influence on the measurement uncertainty since the corresponding height differences within a single CCD pixel cannot be resolved. This paper presents an uncertainty estimation method for estimating the measurement uncertainty due to the surface gradient of the workpiece. The method is developed based on the mathematical expression of an uncertainty estimation model which is derived and verified through a series of experiments. The results show that there is a notable similarity between the predicted uncertainty from the uncertainty estimation model and the experimental measurement uncertainty, which demonstrates the effectiveness of the method. With the establishment of the proposed uncertainty estimation method, the uncertainty associated with the measurement result can be determined conveniently.

19 citations

##### References

More filters

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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.

4,312 citations

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01 Jan 1963Abstract: 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.

3,499 citations

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Abstract: A solution is presented 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. This solution, together with a result obtained in another recent publication, relating to the determination of the complex amplitude distribution of scattered fields from measurements of the intensity transmission functions of holograms, makes it possible to calculate the distribution of the (generally complex) refractive index throughout the object. In general many holograms are needed. corresponding to different directions of illumination of the object.

1,166 citations

### "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]....

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01 Jan 1978TL;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.

707 citations

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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.

482 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]....

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