The conventional signal model of confocal sensors is well established and has proven to be exceptionally robust especially when measuring rough surfaces. Its physical derivation however is explicitly based on plane surfaces or point like objects, respectively. Here we show experimental results of a confocal point sensor measurement of a surface standard. The results illustrate the rise of severe artifacts when measuring curved surfaces. On this basis, we present a systematic extension of the conventional signal model that is proven to be capable of qualitatively explaining these artifacts.
©2012 Optical Society of America
Scanning confocal microscopy is a well established imaging tool, that has gained a lot of attraction due to its superior lateral resolution and depth discrimination compared to conventional microscopy . It has found a vast number of applications in diverse fields ranging from engineering to life sciences . In fact, the principle has been so successful over the years, that its concept and realization has been put on Natures list of milestones in light microscopy . The need to minimize the influence of mechanical vibrations and to minimize inspection times in industrial surface metrology soon led to a class of single shot confocal point sensors [4, 5]. These sensors exploit chromatic aberrations in the optical system and thereby get rid of axial, mechanical scanning. A second class of confocal systems for surface inspection minimizes the need for transverse scanning by measuring multiple points in the image field simultaneously [6,7]. Even though the fundamental ideas of these systems have been developed considerable time ago, clever implementations and combinations with other measurement principles are still subject to ongoing research [8–12]. However, the fundamental signal model of all confocal systems is based on the classical image formation theory for optical microscopes [1, 13]. It is explicitly derived for the two special cases of plane surfaces and point like measurement objects only [1, 13]. Nevertheless, it forms the basis for the classical signal processing approaches to calculate surface height data from confocal signals . Existing approaches to simulate the behavior of confocal sensors based either on ray tracing  or on rigorous methods [16–19] suffer from their substantial computational cost. Additionally, they generally offer little intuitive understanding of the sensor behavior.
Therefore, in section 1 we review the conventional signal model of confocal systems as well as the common signal processing in confocal surface metrology. In section 2 we present an exemplary confocal measurement of a curved surface, that shows significant deviations from the true surface profile. In section 3, an improved signal model is introduced, that is capable of qualitatively reproducing the artifacts observed in section 2 in simulations and provides an intuitive picture of the problem. Finally, section 5 presents numerical evaluation of the new signal model, showing a good agreement with the experimental data and section 6 concludes this contribution.
2. Review of the conventional signal model
The signal of a true single mode detector , e.g. a single mode fibre or an ideal point detector, is given by the inner product of the scattered field impinching on the detector and the complex conjugate of the mode of the detector . In an ideal confocal setup, this detector mode is equal to the illumination field and the detected intensity may be written as:Eq. (1), inserting a two dimensional delta function for the illumination field. The propagation of the illumination field through the optical system and the interaction with the measurement object can be expressed by operators and respectively. Where is the displacement of the object plane from the focal plane of the system and coordinate systems in the detector plane and object plane are labeled according to Fig. 1 .
The scattered field reaching the detector is calculated by successively propagating the illumination field to the measurement object, interacting the field with the object and propagating it back to the detector. Therefore, using the respective operators, Eq. (1) becomesFig. 1 to be ideal and accumulate the aberrations of the optical setup into the aberration function of lens 2. Neglecting constant phase terms and factors of proportionality, propagation up to lens 2 is given by the Fourier Transform. Propagation through a lens and into the vicinity of its focus is a standard problem in Fourier Optics  and the propagator, transforming the illuminating detector mode into the focal region of the system, can be written asFig. 2 , propagation of the scattered field in the focal region of the system back to the detector is represented as :Fig. 2, therefore is the identity operator. Consequently, the second and third Fourier Transforms in Eq. (5) can be simplified using the property of duality of the Fourier Transform, i.e. , to get
An ideal confocal point sensor employs a point source and a point detector whose mode is described by a two dimensional delta function . Using the following definition of the Fourier TransformEq. (6) further simplifies and we get
The integral of Eq. (8) has maximum value if the aberration function of lens 2 is symmetric and if at the same time the phase term vanishes, i.e. . If we expand the square root in the exponential in a Taylor Series retaining only the factor of second order, i.e. applying the paraxial approximation and neglecting constant phase factors again, we can switch to cylindrical coordinates and get 24] , that is normalized to the numerical aperture of the system such that , we get the well known formula for the confocal signal [e.g 1,13.]14]1], the resulting error is presumably object independent. Therefore it is assumed that a suitable calibration measurement takes care of this error along with nonlinearities of the axial scanning mechanism .
In the following section it will be shown that the signal model of Eq. (10) and the corresponding signal processing according to Eq. (11), despite being very robust when measuring locally plane objects, can produce severe artifacts when measuring locally curved surfaces.
3. Measurement artifacts
The Physikalisch-Technische Bundesanstalt recently released a so called chirp calibration standard as a natural equivalent to 2D-resolution standards for characterization of 3D-sensors . Figure 3 shows a reference measurement of the surface profile of this chirp calibration standard. The measurement was conducted by the PTB with a tactile stylus apparatus with 2 µm stylus tip. The surface of the chirp calibration standard consists of discrete cosine intercepts of amplitude 0.5 µm, that vary in wavelength in steps of 10% from 91 µm to 10 µm and back to 91 µm. The surface profile is assembled in such a way that the individual cosine intercepts meet in their height maxima, thereby producing a height profile continuous in height and slope but discontinuous in curvature at the maxima. The chirp calibration standard is manufactured by high precision diamond turning in nickel on a copper substrate and the reference measurement shows, that the surface profile can be assumed to be of perfectly sinusoidal shape on the scale of our experiments.
A measurement of this chirp calibration normal with a custom build chromatic confocal point sensor with 50x magnification, 0.5 NA and 830nm centre wavelength was recently published . The processing of the recorded confocal intensity signals was done by calculation of the centre of gravity according to Eq. (11) and the resulting measurement of the surface profile showed considerable and systematic deviation from the reference measurement. It can be clearly seen from Fig. 4 that the artifacts in the measurement are strongly depending on the measurement object. Particularly, the local surface curvature seems to have a strong influence on the measurement result in this case.
This comes to no surprise as the underlying signal model from Eq. (10) was explicitly derived for plane surfaces only. As the spot size of our confocal sensor can be approximated to about 2 µm, this assumption of locally plane surfaces becomes more and more invalid at the investigated cosine intercepts of wavelength 91 µm to 24 µm.
4. The improved signal model
In the derivation of the standard signal model in section 2, it was argued that the object operator , describing the interaction of the illumination field and a perfectly reflecting plane mirror is the identity operator. This simple model for the interaction of illumination light and measurement object proved to be inadequate for the experiment presented in section 1. Therefore, the natural starting point for an improvement of the signal model is a more accurate modeling of this operator . We start by identifiying the result of the object operator in Eq. (5) as the scattered field in the object plane . Expressing the remaining Fourier Transform operators in integral form we getEq. (12) can be rewritten to yieldEq. (3) it can be seen that second factor in the overlap integral is the illumination field in the object plane. Note that instead of calculating the overlap integral of the illuminating mode and the backscattered field in the plane of the detector, Eq. (13) calculates the overlap integral in the plane of the measurement object. This way the computational effort of numerically propagating the scattered field back through the optical system is eliminated. Depending on the measurement object, the backscattered field may be of complicated shape and the effort of propagating such a field through the optical system would be substantial. Rewriting the backscattered field in terms of the illuminating field and the object operator, Eq. (13) takes the following concise form:
In general, reflection and refraction of scalar wavefields at curved interfaces is still a complicated problem . However, the so called Thin Element Approximation (TEA) is a widely applied method for approximating the propagation of scalar light fields through thin optical elements . Comparisons with more rigorous methods show that TEA performs well at surfaces with height variations, that are small compared to its lateral features . The surface of the PTB chirp standard at the measured positions shows a sinusoidal height variation of 1 µm over periods of about 30 µm. Therefore, following the ideas of TEA, the object operator acting on an incident field is written as:Fig. 5 . Using this operator in Eq. (14) results in:Eq. (17) pictures the signal forming process in confocal sensors as calculating the overlap, i.e. the resemblance, of the local surface profile with a series of defocused wavefronts. Figure 6 illustrates how this behaviour leads to measurement artifacts at locally curved surfaces. If the measurement surface locally has the shape of the illuminating wavefront, the incoming field will be imaged onto itself. In this case, the confocal signal following Eq. (17) will be maximum. This result illustratively explains the sensitivity of confocal measurements to local surface curvature, that other authors predicted from rigorous simulations .
5. Simulation results
Comparing the results to the experimental data of Fig. 4, a qualitatively good agreement between simulation and experiment around the minimum of the surface profile is achieved. The simulation produces symmetrical artifacts around the maxima of the surface profile. This is the expected behavior of an ideal system, producing symmetric wavefronts around focus. The measurement data however does not show this symmetry. Introducing residual aberrations into confocal system of the simulation certainly solves this problem. Figure 8 shows the simulation result for the cosine intercept with 39 µm wavelength where a spherical aberration of has been incorporated into the pupil function of the system. Comparing Fig. 7 and Fig. 8 shows that aberrations in the optical system break the symmetry suggested by the ideal signal model producing an excellent overall agreement between simulation and experiment for a wavelength of the surface of 91 µm and 39 µm. The fact that the agreement becomes worse for a wavelength of 24 µm most probably marks the limit of validity of the Thin Element Approximation of the object interaction as the aspect ratio of the surface becomes higher. Also second order reflections, that are neglected in our object operator, might already become relevant at this wavelength.
A detailed review of the derivation of the standard signal model recovered that it is only valid for locally plane measurement objects. We presented experimental proof that this fact can lead to severe artifacts when measuring locally curved surfaces. Furthermore, we have reformulated the conventional signal model in terms of an overlap integral in the object plane. This model involves an operator describing the interaction of the illuminating light field and the measurement object in spatial coordinates. Even though any operator, including operators involving rigorous calculations could be applied here, we have exemplarily shown that an object operator based on scalar Thin Element Approximation is capable of qualitatively reproducing the measurement artifacts observed at locally curved surfaces. This methodology additionally provided an illustrative picture of the signal forming process that explains the observed artifacts in an intuitive way.
The financial support of the Bundesministerium für Bildung und Foschung (BMBF) under the grant 13N10386 is explicitly acknowledged. Additionally the publication of this work was supported by the German Research Foundation (DFG) within the funding program “Open Access Publishing”.
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