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A linear model of fringe generation and analysis in coherence scanning interferometry

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A LINEAR MODEL OF FRINGE GENERATION AND ANALYSIS IN COHERENCE SCANNING INTERFEROMETRY
Kanik Polodhi
1
, Jeremy Coupland
1
and Richard Leach
2
1
Department of Mechanical and Manufacturing Engineering Loughborough University, UK
2
Engineering Measurement Division National Physical Laboratory, UK
INTRODUCTION
Coherence scanning interferometry (CSI) or scanning white light interferometry (SWLI) as it is often called is an increasingly popular method to characterize engineering surfaces. Combining the lateral resolution of a high power optical microscope with the axial resolution of an interferometer, CSI often provides a more rapid and more convenient means to measure surface topography than the contacting methods that form the basis of current international standards for surface measurement. Routine use of CSI for engineering metrology has, however, prompted comparison with traditional stylus profilometry. In 1990 Hillman was the first to report that CSI measurements of a roughness standard could differ significantly from those obtained from stylus profilometry [1]. Other observations include so-
called „batwings‟ that are found at the
edge of step discontinuities [2]
and „fringe order‟
errors that occur at half-multiples of the wavelength [3]. For example, figure 1 shows a CSI measurement of a metalized sinusoidal grating. Although different models of CSI instruments differ significantly in the manner in which they estimate surface profile from the measured interferogram, they share similar optical geometries and errors of this kind are observed in all commercial instruments [4]. It should be emphasized that spurious data is most likely to occur when the surface gradient is large and consequently less of the scattered field is collected by the aperture of the objective. Nevertheless, large surface gradients are encountered when measuring rough engineering surfaces and for this reason the srcins of the error sources deserve further consideration. In a recent paper we have compared three-dimensional (3D) imaging systems including holography, confocal microscopy and CSI as linear processes [5]. This analysis is strictly correct for the case of weakly scattering objects where the object causes a small perturbation in the illuminating field. In this way each system is characterized in the space domain by its point spread function (PSF) (its response to a point object) or equivalently in the frequency domain by its transfer function, and the imaging process can be considered to be a 3D filtering operation. In most practical cases of imaging through 3D transparent media the assumption of weak scattering is rarely satisfied due to relatively large refractive index changes. For CSI measurements of surface profile, however, the assumption of weak scattering can be justified provided the field is returned to the instrument by a single scattering. In this case the linear imaging theory provides considerable insight into the behaviour of CSI measurements. In this paper the srcin of gradient induced fringe order
FIGURE 1. CSI measurement of sinusoidal grating 8
m pitch, 466 nm amplitude (pk-pk). The ideal form is shown by the dashed line. The figure clearly shows fringe order errors of 300 nm.
errors is examined using linear systems theory.
A new „foil model‟ of CSI is introduced that
illustrates the formation of white light interference fringes and the model is used to show how the PSF influences fringe order errors. The paper describes how to calculate the PSF for ideal instruments and those that exhibit axial aberration such as defocus and axial chromatic aberration.
LINEAR IMAGING THEORY
We have recently published an analysis of 3D optical imaging methods, including CSI, in terms of linear systems theory [5]. It is shown that, if the object is described by its refractive index contrast,
rr
2
n1
(1) where, n(
r
) is the refractive index, then the fringe modulation, I(
r
)
,
of the interference pattern generated by a CSI instrument can be described by the linear, shift-invariant filtering operation,
'rd''HI
3
rrrr
(2) where d
3
r‟ represents the scalar dr
x
‟dr
y
‟dr
z
‟, and
H(
r
) is the 3D PSF that defines the system response. In essence, the PSF of a CSI instrument is a function of the 3D coherent responses of the optical systems used to both illuminate and observe the light that is scattered from the object. The 3D coherent response,
0
k ,G
r
,
of an ideal imaging system with a numerical aperture N
A
, operating at wavenumber k
0
can be written in the form [5], where,
o
ˆ
is a unit vector in the direction of the optical axis and
and
step
represent a 3D Dirac delta function and a Heaviside step function, respectively. Most CSI instruments use a Mirau objective for high resolution measurements and the numerical aperture of this lens usually limits the response of both the illumination and observation functions. In this case the PSF of an ideal aberration free instrument system has been shown to be [5],
002020ideal
dk k ,Gk Sk ReH
rr
(4) where, Re{ } denotes the real part and S(k
0
) is the spectral density of the illumination. For the purposes of this paper, however, it is useful to differentiate between the responses
0i
k ,G
r
and
0o
k ,G
r
of the illumination and observation optics respectively, such that, the more general PSF can be written
00o0i0
20
dk k ,Gk ,Gk Sk ReH
rrr
(5) In essence the PSF provides the interferogram that would be observed if a point-like object were measured by a CSI instrument. Figure 2 shows a slice through the PSF of an ideal instrument of N
A
= 0.55 with a Gaussian distributed spectral density such that the central wavelength is 600 nm and the bandwidth approximately 135 nm (at 1/e
2
).
FIGURE 2. Ideal PSF (N
A
= 0.55; mean wavelength 0.6 µm; bandwidth 135 nm).
It can be seen that the PSF has a fairly compact form with fringes with a frequency of approximately half the mean wavelength extending in the axial direction. The lateral extent of the fringes (full width) is around 0.6 µm and is approximately the mean wavelength for this objective. It is noted, however, that the PSF extends to a certain extent beyond this limit and
k deN1k
ˆ
.stepk )k ()k ,(G
3r.k j2
A02000
okkr
(3)
0.6 µm
o
ˆ
this implies that the fringe pattern generated by a CSI instrument at a particular position in space depends in part on the profile of the surrounding area.
THE FOIL MODEL
Aside from the assumption of scalar diffraction, the linear theory of CSI rests, fundamentally, on the assumption of weak scattering. Weak scattering from an inhomogeneous 3D object occurs when the illuminating field is largely unperturbed by the object itself [6]. This is the case, for example, if a homogeneous medium is sparsely seeded with small particles that scatter only a small fraction of the illuminating field. In this case the fringe pattern observed at each particle location would take a form similar to figure 2. For more general inhomogeneous 3D objects, weak scattering can rarely be justified. This is because the wavelength of the field within an inhomogeneous bulk material generally varies significantly and after a propagation of just a few wavelengths the phase of the internal field will depart significantly from the illuminating field. In optical profilometry, however, the situation is significantly different since we are generally concerned with scattering from the surface of a homogeneous media. In this case, the assumption of weak scattering implies that the illuminating field at the surface is that which would be present in the absence of the object. In effect, this means that the scattered field propagates from the surface via a single scattering event. Multiple scattering will occur predominantly when attempting to measure rough or high aspect ratio surfaces [7], however, for the smooth surfaces that are the subject of this paper, it can be neglected. Because we are concerned with surface scattering it is intuitive to consider only a thin
„
foil-
like‟ layer on the object surface. Indeed, for
the case of a metal object, the illuminating field is rapidly attenuated and is only of significance within the skin depth [6] and the remainder of the object has no measurable effect. If the metal is a perfect conductor then the skin depth is infinitesimally small while the (complex) refractive index contrast is infinitely large and the scattered field will be exactly that which would be scattered from a thin foil with the same profile. In this case we can describe the object by an equivalent foil surface and can write,
'rdr,rf rrr'HI
3'y'x'z'y'x
rrr
(6) where,
yx
,rrf
describes the profile. Here, we have assumed, without loss of generality, the observation direction is aligned to z-axis. We will refer to this equation as the foil model of CSI. Although we have introduced the foil model with reference to a perfect conductor, it can be used with some validity to model surface scattering from most homogeneous engineering materials. In a real metal, for example, the inhomogeneous field within the skin depth affects the phase and amplitude of the scattered field and because of this the reflection coefficient varies as a function of the angle of incidence, and is polarization dependent [6]. For the case of CSI, however, the variation is small for the range of angles accepted by the numerical aperture of a typical instrument and the phase change results in an apparent movement of the object surface in an axial direction. For the case of a dielectric surface, the situation is similar, however, there is no phase change and although the magnitude of the reflection coefficient is relatively small it is reasonably constant over the range of angles accepted by the N
A
of the instrument. Consequently, provided that there is no multiple scattering from internal surfaces (such as thick films), scattering from dielectric and metal surfaces will closely resemble that from a perfect conductor and the foil model will remain valid. To illustrate the fringe generation process and the effect of the PSF we now consider the fringes obtained when a sinusoidal surface form is investigated using an ideal CSI instrument that is free from aberration. The sinusoidal profile is modelled as the foil defined by the profile,
gxyx
/ r2sinA,rrf
(7) where, A and
g
are the amplitude and wavelength of the grating respectively. In this work the 3D convolution of equation (2) was evaluated numerically using an FFT algorithm implemented in MATLAB software. To avoid errors introduced by the discrete numerical formulation it is necessary to ensure that both the foil model and the PSF are adequately sampled. For the case of the foil model this was
accomplished by introducing a Gaussian thickness profile such that,
2yxzyxz
T,rrf r
2exp,rrf r
(8) where T is the thickness (at 1/e
2
) point and was chosen to be approximately quarter of the mean wavelength. Figure 3 shows the interferogram generated by the foil model of CSI for the case of the ideal system specified previously when the instrument is used to measure a grating with the same pitch and amplitude as that illustrated in figure 1.
FIGURE 3. Interferogram generated by the foil model for sinusoidal grating 8
m pitch, 466 nm amplitude (pk-pk).
The dotted line, shown in figure 3 shows the surface profile that is obtained by finding the peak in the fringe envelope. It can be seen that ideal optics should be capable of identifying the correct fringe and consequently should provide a high quality measurement of the surface profile shown in figure 1. The effect of the PSF becomes more noticeable as the surface gradient increases. For example, if the pitch of the sinusoidal profile is decreased to 3.3 µm the foil model with ideal PSF generates the fringe pattern shown in figure 4. Fringe order errors similar to those shown in figure 1 are now apparent at points where the gradient is steep.
THE EFFECT OF AXIAL ABERRATIONS
It is clear that aberrations will affect the PSF and it is quite straightforward to include axial aberration into the foil model of CSI. In general terms, aberration can affect the responses,
0i
k ,G
r
and
0o
k ,G
r
of both the illumination and observation optics. When common optics are used in a Mirau objective we would expect the responses to be balanced such that
0i
k ,G
r
=
0o
k ,G
r
since the optics to perform both functions are largely the same. Important
FIGURE 4. Interferogram generated by the foil model for sinusoidal grating 3.3
m pitch, 466 nm amplitude (pk-pk).
to note, however, that although they are balanced (and appear in both arms of the interferometer), the aberrations will have a residual effect on the PSF and consequently limit the ability to measure steep artefacts. It is also possible to introduce unbalanced aberrations that affect the illumination and observation response in a different way and these can have a greater effect. For example, Mirau objectives of large numerical aperture frequently include a ring that moves the position of the beam splitter within the lens. The effect of this is to change the path length and relative focus between the reference and imaging arms of the interferometer. In general unbalanced aberration has a greater effect on the PSF than balanced aberration and the focus ring is a control that should be adjusted with great caution. In the following we illustrate the effects of unbalanced and balanced aberration.
Unbalanced Aberration: Defocus
Adjusting the position of the beamsplitter in a high N
A
Mirau objective can be modelled by introducing defocus d, into either the illumination or the observation response. From the Fourier shift theorem this is equivalent to a linear phase ramp in k-space and the observation response can be written,
k deN1k
ˆ
.stepk )k (e)k ,(G
3. j2
A0200
ˆ
. jd0o
rkok
okkr
(9) The PSF defined by equation (5) can be calculated in this case using the observation and illumination responses defined by equation (9)
and equation (4), respectively. For the typical CSI instrument defined previously the PSF corresponding to a defocus of approximately 2.7
m is shown in figure 5.
FIGURE 5. PSF with a defocus of 2.7
m
With reference to the ideal PSF shown in figure 2, it is noted that the PSF with defocus is larger in both lateral and axial extent. The effect of the defocus on the measurement of a sinusoidal profile with 4.2
m pitch is shown in figure 6.
FIGURE 6. Interferogram generated by the foil model for sinusoidal grating 4.2
m pitch, 466 nm amplitude (pk-pk).
Fringe order errors similar to those in figure 4 are now apparent although it is noted that the pitch and hence the maximum gradient of this grating is less.
Balanced Aberration: Axial Chromatic Aberration
Axial chromatic aberration can be thought of as a defocus that depends on wavelength and consequently can be modelled in a similar manner to the unbalanced defocus that was discussed previously. Writing
0
k / d
where,
is a dimensionless constant and noting that the aberration is balanced we can write,
k deN1k
ˆ
stepk )k (e)k ,(G)k ,(G
3r.k j2
A0200k o
ˆ
.k j0i0o
0
ok.krr
(10) For the typical CSI instrument defined previously, evaluation of the impulse response with axial chromatic aberration is shown in figure 7. In this case the value of
is 2
, which corresponds to a relative defocus of approximately 0.54
m between the wavelengths at the extreme of the spectrum (1/e
2
point).
FIGURE 7. PSF with axial chromatic aberration.
The effect of axial chromatic aberration on the measurement of a sinusoidal profile is shown in figure 8. In this case the errors are apparent with a grating of 8
m pitch and are quite smoothly varying.
DISCUSSION AND CONCLUSIONS
In this paper we have considered a linear theory of CSI instrumentation in which the system response can be characterized by its 3D point spread function (PSF) and have argued that this is appropriate to surface profilometry providing that multiple scattering effects are negligible. In this case the scattering from the object surface is equivalent to that from a foil-like object with the same surface profile. We call the application
0.6 µm 0.6 µm

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