2.1. Derivation of the 3D Transfer Function for Monochromatic Light
With respect to the following derivation, we refer to our paper [
19] using the Ewald or McCutchen sphere construction according to
Figure 1c [
7,
13] considering that the angles of incidence and the scattering angles are limited by the NA of the microscope objective. The side view of the resulting Ewald limiting sphere (
Figure 1) represents the cut-off in the spatial frequency domain due to the NA of the objective lens [
12,
21,
24].
For the calculation of the 2D MTF, two uniformly-filled circular apertures are correlated [
20]. The generalized physical situation is explained by
Figure 2, where the incident and the scattered wave vectors form spherical caps of radius
(see
Figure 2a. The NA limits the lateral extension of these caps, which are inverted with respect to each other as shown in
Figure 1c because of the different propagation direction of the incident and scattered waves with respect to the optical axis of the objective lens. The lateral and vertical shift of the caps represents the lateral and the axial spatial frequency, respectively. Under the assumption that the caps must intersect, the maximum axial shift is
, whereas the minimum axial shift equals
(see
Figure 1c and
Figure 2a). Contributions of the lateral spatial frequency components to the imaging process can be expected as long as
. Integration along the
-axis results in the sum of the areas
and area
, which equals the intersection area of two laterally shifted circular apertures (see
Figure 2c).
results via integration from
to
. The corresponding lines of intersection of the two caps are full circles tilted by the angle
(see
Figure 2b). Hence, area
resulting from integration is an ellipse (see
Figure 2c). Area
is attributed to the integration range from
to
and represents the difference between the intersecting area of the two circles and
. The areas
and
used to obtain the 3D TF are related to
and
by
As shown in
Figure 2b, area
is given by a circle of radius
. In the
-plane, the points of intersection of circles of radius
centered around the origin and point
P are the end points of the two vectors
and
and the starting points of the vectors
and
. These vectors exhibit that each point
P of the TF can be reached by two different ray paths in the
-plane as described by Quartel and Sheppard [
16]. Due to the symmetry of the configuration,
where
The angle
is given by
In order to calculate the 3D TF related to a point scatterer, we used the derivations
and
[
19]. According to the projection slice theorem [
25], this leads to the familiar MTF formula for a diffraction limited system by
-integration. Here, we are interested in the transfer function that holds for specularly reflective surfaces. In the ideal case, according to energy conservation, the light reflected or diffracted at a specular surface hitting the open aperture of the objective lens will contribute to the measured signal without loss of energy. Under this assumption for a certain
value, the 3D TF should no longer depend on the
coordinate. This is achieved if the multiplication by
is renounced and the derivations
and
are used to define the 3D TF. As a consequence, the intensity distribution of the reflected and diffracted light in the pupil plane is no longer uniform. According to
Figure 2b,
is given by
and, therefore,
where the negative sign agrees with the fact that the circle of the largest radius is assigned to the smallest
-value.
The contribution to the TF coming from the area
originates from skew rays [
17], the wave vectors of which leave the plane of incidence (
-plane). Consistent with [
19], the derivation of
with respect to
results in
Hence, the desired formula for the 3D transfer function is:
where
and
are given by (
14) and (
15). This result is in agreement with earlier calculations of interference signals arising from perfectly adjusted perfect mirrors, i.e., for
[
26,
27]. In the following,
is normalized to a maximum value of
. The dependency on
in the argument indicates that
is the 3D TF for the monochromatic case, where
(
16) is an analytical formula that calculates the 3D TF for specularly reflective and diffractive surfaces, comprising often used calibration standards in CSI. The differences between the 3D TFs obtained in [
19] for scattering surfaces and the 3D TFs according to (
16) are shown in
Figure 3 and
Figure 4 for a wavelength
of 500 nm and an NA of 0.55 (
Figure 3) as well as 0.9 (
Figure 4). Note that, due to the rotational symmetry with respect to the
-axis, the results presented in this paper are cross sections of
in the
-plane. In the paraxial case, i.e., for small NA values, the differences between the TFs for specular reflection and scattering will be negligible. However, for an NA of 0.9, differences of up to 0.25 appear. The deviations between the results can be explained by the different scattering behaviour in the two situations, which is well known from confocal microscopy [
21] (Chapter 1), [
22] (Chapter 3). For the point scatterer, a constant intensity distribution of the scattered light is assumed to appear in the pupil plane of the objective lens. This leads to a scattering characteristic, which is similar to a Lambertian emitter, i.e., for a constant
, the scattered light intensity is maximum in the direction of reflection (
) and falls down continuously until the backscattering direction (
) is reached. This can be obtained from the factor
according to (
12), which represents the major difference in the calculation of the TF for a point scatterer and a specular surface. Equation (
12) can be rewritten as:
On the other hand, the irradiance of reflected light incident on a perfect mirror under an angle
must be the same whether the mirror is perfectly aligned in the
-plane or if it is tilted by an angle
such that the plane wave is incident normal to the surface. This is achieved by (
14), which no longer depends on the
coordinate. The 3D TF discussed so far holds for conventional microscopes. Therefore, it should be mentioned that, due to the symmetry of the arrangement according to
Figure 2a,b and the uniform pupil illumination, it equals the 3D TF of an interference microscope [
6,
19]. As a consequence of the different 3D TFs shown in
Figure 3 and
Figure 4, the MTFs also resulting from
-integration will be different as
Figure 5 confirms. For the point scatterer, the MTF exactly agrees with the MTF of a diffraction limited system [
19], whereas in the specular case at high NA values, the resulting MTF comes closer to a linear function (see
Figure 5b). Additionally, note that the plateau of
for
enables the perfect reconstruction of the weak phase grating as long as
which is the Abbe criterion of lateral resolution.
2.3. 3D Transfer Functions and Surface Topography Reconstruction
This section is intended to investigate the impact of the shape of the 3D TF of an interference microscope on the related surface topography reconstruction capabilities. Recent publications by different researchers elucidating this subject [
5,
7,
10,
28] come to the conclusion that, for the case of a uniformly-filled illumination pupil and sufficiently small surface height deviations, the transfer function of a CSI instrument takes the form of the familiar modulation transfer function (MTF) given by the autocorrelation function of a circular pupil [
20]. In our nomenclature, this means that, in order to obtain the CSI measurement result in the spatial frequency domain, for a certain axial spatial frequency value, the scattered field
would be multiplied by the 2D MTF, which depends solely on the coordinate
defined in (
9) due to the rotational symmetry. However, according to the above sections, our results differ with respect to the following two points:
Sheppard and Larkin [
26] introduce the
-factor
for an aplanatic system with uniformly filled pupils. This NA-factor describes the increasing fringe spacing depending on the NA of the optical system and, therefore, it can be used to define the equivalent wavelength by
or the equivalent axial spatial frequency
Equation (
20) can be derived as the center of gravity of the transfer function
i.e.,
In practical applications of CSI, the evaluation wavelength
is mostly chosen such that
. A slightly different alternative is to choose an evaluation wavelength
which corresponds to a maximum bandwidth of the corresponding partial transfer function
(see the blue line in
Figure 1d). This results in
If maximum lateral resolution is required, the evaluation wavelength should be chosen such that
as marked by the dashed black line in
Figure 1d. The three functions
,
, and
are plotted in
Figure 8.
Once the evaluation wavelength is chosen, the corresponding partial transfer function
is calculated and multiplied by the Fourier transformed phase object
resulting in the filtered Fourier transformed phase object
Fourier transform with respect to the transverse
coordinates leads to the filtered phase object
from which the reconstructed surface profile surface can be obtained by
where
represents the imaginary part and
the real part.
Figure 9 depicts cross sectional views of the 3D TFs according to
Figure 6 and
Figure 7 for NA = 0.55 and
µ
(
Figure 9a) as well as for NA = 0.9 and
µ
(
Figure 9b). Note that the values of
are chosen slightly higher than
. Thus,
shows a plateau even for spectral distributions of slightly broader FWHM. For the lower NA, the cross sectional views in
Figure 9a exhibit a strong dependence on the spectral FWHM, whereas, in
Figure 9b, the changes are less significant.
Figure 10a shows the surface profile
and the reconstructed surface profile
of a cosine-shaped surface according to (
2) of
nm peak-to-valley (PV) amplitude and 0.5 µm period length. For the corresponding phase object
(
1) represents a satisfactory approximation. The absolute value of the Fourier transform of
with respect to the
x coordinate (see (
5)) is displayed in
Figure 10b together with the partial transfer function
for
µ
and an FWHM of 2.5 nm (see
Figure 9a) as well as the product
Due to the rectangular shape of
the reconstructed profile shows only slight amplitude deviations caused by higher diffraction orders which are not considered in the reconstruction. For
Figure 10c,d, the same profile and evaluation wavelength but an FWHM of 100 nm (see
Figure 9a) is presumed. Since the value of the partial transfer function
is below 0.5 at the
-position of the spatial frequency of the surface, the amplitude of the reconstructed surface profile is less than half of the original amplitude too.
Figure 11 shows surface profiles
and reconstructed surface profiles
of a cosine-shaped surface of 550 nm peak-to-valley (PV) amplitude and 5 µm period length. In this case, the corresponding Fourier transformed phase object
comprises numerous diffraction orders as it can be seen in
Figure 11b,d. In
Figure 11b, for an FWHM of 2.5 nm, higher order spatial frequency contributions are cut by the corresponding partial transfer function
for
µ
. For the broader spectral bandwidth according to
Figure 11d, corresponding to an FWHM of 100 nm, the higher order diffraction maxima are damped down significantly. However, in both cases, the reconstructed profiles agree quite well with the original input profiles (see
Figure 11a,c).
In
Figure 12, an input profile with 50 nm PV-amplitude and 0.3 µm period length is presumed. This period is close to the resolution limit of 0.28 µm for NA = 0.9 and
nm. According to
Figure 12a for
µ
, the first order diffraction component is cut by the corresponding partial TF in
Figure 12b, such that the cosinusoidal structure does not appear in the reconstructed profile. However, if
is reduced to 13.0 µ
(
Figure 12c,d), the bandwidth of the partial TF increases and the cosinusoidal structure is resolved, although the PV-amplitude is reduced approximately by a factor of 3. This is in accordance with our experimental results published earlier [
6,
29].
All results of surface profile reconstruction shown so far are based on the analysis of the phase of the corresponding interference signals at the certain wavelength
. However, in practical applications, CSI signal processing is usually a combination of coherence peak detection and phase analysis [
30,
31,
32]. Since the 3D TF is defined in the 3D spatial frequency domain, it is obvious to determine the maximum position of the signal envelope in the spatial frequency domain, too. As pointed out by de Groot and Deck [
30], the envelope’s position equals the derivative of the phase with respect to the axial spatial frequency
Considering (
1) and (
2), the phase of
results in
Thus, the surface topography can be reconstructed by:
where
is an infinitesimal increment with respect to the
coordinate.
In order to obtain the envelope position in the spatial frequency domain, we first define based on (
5)
and
Then, the derivative of the phase function is calculated numerically and the reconstructed surface profile called coherence profile results in:
Note that and are normalized such that as long as and as long as . Under these assumptions, .
Since (
34) contains the difference of the electric fields depending on
at
and
, we can describe the influence of the instrument by the use of the envelope transfer function defined as the normalized difference quotient
where
As long as
, the reconstructed coherence profile equals the original profile, i.e.,
. Unfortunately, the function
no longer shows a simple low-pass filter characteristic and thus the results of envelope and phase evaluation may be different. The occurring problem is explained by
Figure 13, which shows
and cross sections of
for constant values of
µ
and
µ
belonging to the real and the imaginary part of
. The horizontal dashed red line in
Figure 13a and the vertical dashed red line in (b) correspond to
µ
. Equation (
35) results in the normalized derivative of
with respect to
at
.
Figure 13b shows that
is positive for
µ
, whereas it is negative with a much higher gradient for
µ
. As a consequence, the transfer function
according to
Figure 14b,d results, which leads to the reconstructed coherence profiles in
Figure 14a,c depending on the period of the input surface. Due to the large absolute value and the negative sign of
for
µ
, the reconstructed profile in
Figure 14c is inverted and the amplitude is approximately seven times the amplitude of the original profile. This effect has already been obtained from experimental investigations [
33]. If the period of the surface is doubled (i.e.,
µm), the evaluation of the coherence peak position results in
Figure 14a,b. In this case, the reconstructed profile perfectly agrees with the original profile.
Note that the enhancement-effect of high-frequency contributions depends on the coherence length of the contributing light. Furthermore, the NA of the system affects its extension along the
-axis. It should be mentioned that the same phenomenon appears at higher surface slopes, where the reflected light is attributed to higher transverse spatial frequency contributions [
6,
9,
34,
35,
36]. Furthermore, the effect hardly depends on whether the 3D TF of a specular surface or a point scatterer is presumed to be valid.
As a final example,
Figure 15 shows the measured topography of a diamond milled aluminum mirror as a real-world object. The comparison of the results of envelope analysis in
Figure 15a and phase analysis in (b) exhibits that the envelope result is characterized by high-frequency components, whereas the result of phase evaluation appears to be low-pass filtered. We suppose that this is a consequence of the high spatial frequency enhancement introduced in
Figure 13 and
Figure 14. However, since the NA is 0.55 in
Figure 15, the bandwidth for high-frequency enhancement is even broader in this case. In both cases, the sawtooth-like grooves of approximately 50 nm depth originating from the manufacturing process can be recognized. Although the theoretical derivation of the 3D TF reported in
Section 2.1 neglects the central obscuration due to the reference mirror in a Mirau interferometer, the results according to
Figure 15 confirm the basic effect of different spatial frequency transfer characteristics depending on whether the coherence or the phase profile is analyzed.