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Optical aberrations of microscope lenses are known as a source of systematic errors in confocal surface metrology, which has become one of the most popular methods to measure the surface topography of microstructures. We demonstrate that these errors are not constant over the entire field of view but also depend on the local slope angle of the microstructure and lead to significant deviations between the measured and the actual surface. It is shown by means of a full vectorial high NA numerical model that a change in the slope angle alters the shape of the intensity depth response of the microscope and leads to a shift of the intensity peak of up to several hundred nanometers. Comparative experimental data are presented which support the theoretical results. Our studies allow for correction of optical aberrations and, thus, increase the accuracy in profilometric measurements.
© 2015 Optical Society of America
1. Introduction
Confocal microscopy is a well established method in surface metrology to obtain three-dimensional topography data of microstructures. Typical applications range from micro-optics to microsystem technology. The advantage of confocal microscopy compared to other optical methods, such as white light interference microscopy, is an enhanced lateral resolution below the Rayleigh limit [1]. The axial resolution depends on the width and sampling rate of the measured intensity profile which is obtained by scanning the sample axially through the focal point of the microscope objective and, hence, on the sensitivity of the actuator which moves the sample [2, 3]. The shape of the vertical intensity profile is denoted as depth response and exhibits a sinc2 shape, if the optical elements are free of aberrations [1].
However, as any optical measuring device, confocal microscopes suffer from systematic errors, which are either a result of sample-light interactions or an erroneous optical beam path inside the microscope. The first case, for example, has been studied on grooves in the micrometer range with concave or rectangular profiles, which generate caustics and overshooting at edges in the measured topography [4, 5]. In the latter case, optical aberrations such as spherical aberrations lead to an asymmetrical depth response which deviates from a sinc2-like shape where the peak is shifted from the geometrical focal point of the microscope objective [6, 7]. The topography of a surface is usually calculated from measured intensity data by either determining the position of the maximum or center of mass of the depth response. In both cases, the asymmetrical shape affects the measured axial position of the sample and introduces an offset between the measured axial position of the sample and the geometrical focal point of the objective. If this offset stays constant over the entire field of view, the offset can be ignored since absolute position of the sample is usually not of interest.
In this paper, we show both theoretically and experimentally that the shape of the depth response depends on the surface slope and aberrations of the microscope objective and, thus, may vary significantly over the whole field of view. A similar effect was presented by Lehmann for white light interferometry where chromatic aberration of the objective leads to an erroneous depth response when imaging curved microstructures depending on the local slope of the sample [8]. To demonstrate that lens aberrations may also effect confocal measurements, we consider a reflection-type confocal microscope imaging a surface with an objective which suffers from aberrations, such as spherical aberrations, as sketched in Fig. 1. When a plane wave passes an objective, it is converted into a converging spherical wave and small amounts of phase aberrations may be added depending on the optical quality of the objective. If the illuminating beam gets reflected by the sample, the position where the rays reenter the objective and get aberrated again depends on the local surface slope. Therefore, the phase distribution inside the exit pupil of the objective depends on the aberration as well as the surface slope angle. Since the aberrations in the exit pupil strongly influence the shape of the depth response of the microscope, the response changes its shape depending on the surface’s slope angle.
Fig. 1 Beam path through a microscope objective with third order spherical aberration and phase distribution inside the entrance/exit pupil of the objective when illuminating and imaging a curved microstructure.
2. Numerical model
To verify this hypothesis, we consider a numerical model which is based on the setup sketched in Fig. 2. In the following, we give an outline of our numerical model. However, for comprehensive descriptions of similar models we refer to [9, 11, 12]. Our model is decomposed into an illuminating, scattering and imaging (collection and refocusing) part according to [9]. Also, similar simulation approaches can be found in [10] for confocal imaging of spherical objects or in [11] to simulate imaging of rectangular grooves by a microscope. The illuminating component is shown in Fig. 2(a). A microscope objective is illuminated by a quasi-monochromatic, randomly polarized light source, which generates an electrical field Ein(kx,ky) inside the entrance pupil of the objective. The Cartesian coordinates inside the entrance pupil are defined by the traverse components kx and ky of the wave vector k. The aperture of the entrance pupil is given by the numerical aperture NA of the objective such that . In the framework of our model, each light ray, which strikes the entrance pupil gets refracted by the objective and the electrical field vector Ein is rotated accordingly (see Fig. 3). After refraction, each point (kx,ky) inside the aperture of the entrance pupil defines a plane wave in spatial frequency space, which converges towards the geometrical focal point of the objective with its wavevector (kx,ky,kz). The plane waves are readily described by
where . The term W(kx,ky) takes phase aberrations into account. Integrating Eq. (1) over kx and ky yields the well known Debye integral, which describes the electrical field in the vicinity of the focal point [13]. The factor in Eq. (1) correspondes to the obliquity factor and takes energy conservation into account since the electrical field inside the plane entrance pupil is projected onto a spherical reference sphere to obtain a converging spherical wave. Axial scanning of the sample through the focus is simulated by the factor −ikz z in Eq. (1). The matrix Rθ describes the rotation of the electrical field vector Ef due to refraction by the objective and is described by where the cosine and sine terms are given by and respectively. The axis of rotation is given byFig. 2 Numerical model of a reflection-type confocal microscope consisting of an illuminating (a) and an imaging optical path (b).
To quantify different types of aberrations, the aberration phase is often decomposed into Zernike polynomials [13]. The influence of aberrations on the axial confocal response is extensively studied in literature [7, 14]. It was shown that the largest influence on the axial response is due to aberrations with a phase error, which is radially symmetric with respect to the optical axis of the objective such as spherical aberrations [14]. Therefore, we restrict our model to radial symmetric phase errors and the phase error function reads
where rk is the normalized radial coordinate inside the aperture of the objective and are the radial symmetric parts of the Zernike polynomials, which can be found in [13]. Note, that excluding asymmetrical aberrations is strictly valid only, if one considers a flat sample surface, which is not tilted with respect to the optical axis. In this case, phase errors in the detector plane cancel out due to the asymmetry of the phase errors. If the sample is tilted, the latter argument does not hold anymore and, thus, limits our model in the general case, which is discussed in detail in Sec 4. After passing the objective, light interacts with the sample. We assume that the curvature of the microstructure on the sample is locally flat in the vicinity of the focal point and, i.e., can be described as a plane mirror, which is inclined by an angle α (see Fig. 2). The interaction of each plane wave described by Eq. (1) and the sample can be calculated by means of the law of reflection. The reflected waves are collected by the objective subsequently, as shown in Fig. 2(b). The field in the exit pupil of the objective defined by the coordinates was, again, calculated by considering the refraction of plane wave by the objective as was described previously for illumination. The field in the exit pupil is subsequently focused onto the detector by the tube lens. We assume that the microscope is corrected for an infinite tube length and the field is not altered significantly when propagating through the tube. Furthermore we assume, that the microscope is strictly confocal, which implies that the detector can be assumed to be infinitesimally small. It is located in the focal point of the aberration free tube lens. The field on the detector can also be described by Debye’s integral, i.e., integrating Eq. (1) over the exit pupil of the tube lens.For the numerical simulations, the pupils of the objective and the tube lens were discretized with a patch size of 1/50×1/50(k NA)2. We chose a NA = 0.8 and a wavelength of λ = 590nm, which corresponds to the experimental parameters utilized (see below). Simulation results of the axial response curves when imaging a tilted mirror, which is inclined by slope angles α = 0°,15°,30°, are shown in Fig. 4. The actual surface is always located at the axial position z = 0. Fig. 4(a) shows the axial response in the absence of aberrations. As expected, the intensity peak of the response curve drops with an increasing slope angle as less light is reflected back into the objective. However, the shape of the curve stays symmetric and the peak is located at z = 0 for all slope angles α. In contrast, the response curves, which are simulated with spherical aberration with C40 = 0.1λ are asymmetric and the axial location of the peak is shifted away from the position of the actual surface due to the spherical aberration as indicated by the red dashed line in Fig. 4(b).
Fig. 4 Simulated confocal intensity response for various tilt angels α: a) without aberrations; b) with spherical aberration (C40 = 0.1/λ).
3. Signal evaluation
There are different methods to determine the vertical position of the surface from the axial response [3]: finding the vertical position of the intensity response, which may be determined from raw intensity data or applying a fit or low-pass filter first, or calculating the center-of-mass (COM) of the response. Especially, the first method is highly sensitive to changes in the shape of the intensity response. As shown in Fig. 4(b), when the surface location is determined using a peak search, a significant systematic error may occur, which accounts in our simulations for up to several hundred nanometer. Hence, this is a significant error, especially when measuring topographies in the lower micrometer range.
4. Experimental method and results
To verify our numerical results experimentally, we performed topography measurements on high precision ruby microspheres. A sphere is a suitable geometry for our measurement task, since it provides all inclination angles in one measurement. Its diameter was chosen such that the curvature of the surface did not generate significant phase errors in the circle of confusion of the microscope objectives. We chose an objective by performing measurements on an optical flat using various objectives and picked the particular one for our experiments which exhibits the largest amount of aberrations in the axial response (Olympus UM Flex FL 50x, NA=0.8). Since we also carried out measurements using objectives which showed almost no asymmetric confocal response due to aberrations, we can deduce that an asymmetric response is mainly caused by the objective itself and not by an misalignment of the confocal setup. The measurements where carried out using a confocal microscope with a Nipkow Disk (Nanofocus μSurf). In order to compare numerical with experimental results, the aberration coefficients Cn0 of the objective need to be known. The coefficients may be determined directly by measuring the wavefront error of the objective utilizing a Shack-Hartmann sensor [15]. Instead, we used a method similar to the one presented in [16] to determine the coefficient from the axial intensity response measured on the optical flat (SiMetrics) at a tilt angle α = 0. The confocal response was simulated according to our model and fitted to the measured response by means of least squares, where the aberration coefficients were free parameters. Additionally, the NA was chosen as free parameter since the nominal NA of the objective might differ from the actual NA in the setup, which also influences the shape of the confocal response. Both, model and fit were implemented in MatLab™ utilizing a build-in downhill simplex optimizer. The measured and fitted intensity responses are shown in Fig. 5(a) and the resulting aberration coefficients are given in table 1.
Fig. 5 a) Measured and fitted theoretical intensity response; b) Measured and simulated systematic error Δz as a function of the inclination angle α (experimental results were obtained using a peak find based algorithm).
Table 1. Effective NA and aberration coefficients obtained by least-square-fit.
Please note, that the defocus term C20 does not change the shape of the confocal response but shifts the response along the z-axis and, hence, only determines the z-position of the confocal response at an inclination angle α = 0. As discussed in Sec. 2, determining the aberration coefficients directly from the confocal response using a non-tilted and flat sample restricts our approach to radially symmetric phase aberrations. A more comprehensive way to determine the corresponding Zernike coefficients is based on interferometric techniques or utilizing a Shack-Hartmann sensor. Both methods are currently not available at our laboratory but will be included in future work to fully characterize the optical setup.
Prior to the experiments, we corrected the remaining curvature of field of the microscope by carrying out 15 topography measurements on an optical flat (SiMetrics). Each measurement was performed at a different region on the flat to minimize the influence of surface imperfections. Averaging over all topographies yields the curvature of field, which is subtracted from all further measurements. Subsequently, we performed topography measurements on five ruby spheres, which have a diameter of 300 μm and a form deviation of the 0.08 μm (data was provided by the manufacturer, Komeg GmbH [17]). The confocal intensity response was evaluated by a build-in peak find algorithm provided by the manufacturer of the microscope. To enhance the reflectivity of the spheres, they were coated with a silver layer of a few ten nanometer thickness. Since the diameter of the spheres is known with high precision, we can easily calculate the difference in height between the diameter of the spheres and the measured topography, which determines the systematical error Δz. The results averaged over all five measurements are shown in Fig. 5(b), where the error bars indicate the standard deviations. The theoretically determined error Δz for peak find and center of mass algorithms are also shown for comparison. For calculating the center of mass, we only used intensity values of the intensity response, which were larger than 0.5 times of its maximum value [3]. This approach is often used to avoid systematic errors due to irregular side slope in the intensity response. Comparing both results, it is apparent that peak find as well as center of mass algorithms are affected by a change in shape of the response, which leads to systematic errors Δz greater than 300 nm. The experimental results obtained by a peak find algorithm agree well with the experimental findings. Both exhibit a systematic error, which accounts for more than 300 nm. However, there is a discrepancy of up to 50 nm between experiment and numerical model. This deviation is caused by two factors: First, the systematic error Δz strongly depends on the aberration coefficients and, therefore, on the measurement of these parameters. Second, we had no calibrated standards including suitable spherical features available; these will be included in further measurements. Also, when the angle α increases, the total intensity, which contribute to the measurement drops, which leads to a higher standard deviation in Δz.
To account for the asymmetric case, we carried out additional simulations, where we also considered asymmetrical aberrations. To quantify how strong each type of aberration contributes to the systematic error Δz, we simulated Δz including radially symmetric aberrations such as defocus, spherical aberration of 3rd and 5th order as well as asymmetrical aberrations such as distortion, astigmatism, coma and trefoil. The corresponding coefficients for each aberration were set to 0.05λ and the peak find algorithm was applied to determine the height values from the simulated confocal responses. The results are shown in Fig. 6 and indicate that symmetrical, indicated by solid lines, as well as asymmetrical aberrations, indicated by dashed lines, lead to a significant error Δz. However, as expected, the influence of distortion and defocus is almost negligible. The strongest influence of aberrations on Δz is due to 3rd order spherical aberration, astigmatism, coma and trefoil. Especially, spherical aberration of 3rd order and trefoil lead to an error Δz ranging between 100 nm and 150 nm at an angle α = 30°. Hence, we also have to take asymmetrical aberrations into account, which can readily explain the remaining discrepancy between measurements and theoretical results presented in Fig. 5. Even, if to our own experience, the aberration coefficients decrease with higher orders and 3rd order spherical aberration is often the most significant term, for a comprehensive examination one has to consider that asymmetrical and also higher order aberrations contribute significantly to the error Δz.
Fig. 6 Simulated systematic error Δz as a function of the inclination angle α caused by various aberrations: distortion (C11), defocus (C20), astigmatism (C22), coma (C31), trefoil (C33), 3rd order spherical aberration (C40) and 5th order spherical aberration (C60); each aberration coefficient accounts for 0.05λ.
To quantify how strong the aberration coefficients Cnm contribute to the error Δz, we carried out simulations for values of the coefficients C04, C06, C11, C22, C31, C33 ranging from 0 to 0.1λ, where the tilt angle was chosen such that α = 25◦. The Zernike coefficients correspond to 3rd and 5th order spherical aberration, tilt, astigmatism, coma and trefoil, respectively. We did not consider the defocus term because it only shifts the location of the focal point along the optical axis, which does not contribute to the error Δz. The simulation results are shown in Fig. 7 and indicate a linear dependence between the coefficients Cnm and the error Δz for each type of aberration. Again, we found a strong dependence of Δz on the symmetrical as well as asymmetrical Zernike aberration terms with a maximum error larger than 200 nm. Hence, a large uncertainty when determining the aberration coefficients experimentally may also explain the discrepancy between measured and simulated error Δz in Fig. 5.
Fig. 7 Simulated systematic error Δz as a function of various Zernike coefficients Cnm: distortion (C11), astigmatism (C22), coma (C31), trefoil (C33), 3rd order spherical aberration (C40) and 5th order spherical aberration (C60); the tilt angle is fixed to α = 25°.
5. Conclusion
We discussed that an aberrated microscope objective not only leads to a asymmetric axial intensity response of a confocal profiler but also changes the shape of the response depending an the local inclination angle of the sample. This effect may occur on curved surfaces, such as sinusoidal gratings, and spherical objects as well as on inclined flanks. Due to this effect, the measured height information of the specimen suffers from systematic errors, which is dependent on the topography of the specimen. Since we observed a deviation between the measured and actual surface of more than 300 nm due to this effect, it should be noted that these errors should be taken into account as a significant source of error, especially, when performing topography measurements at structures in the lower micrometer range. By means of our numerical model, we showed that both, radially symmetric but also asymmetric aberrations contribute to this error. However, when calibrating the system carefully using calibration standards with reference microstructures including, e.g., spherical objects the error can be significantly reduced. These corrections could be done by determining the local slope of the surface and correcting the height information according to the calibration measurements.
Acknowledgment
We acknowledge support by Deutsche Forschungsgemeinschaft and Open Access Publishing Fund of Leibniz Universität Hannover.
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