This article presents a new robust, precise, high-frequency focal-distance-modulated confocal point sensor for probing in coordinate measuring systems (CMSs). While maintaining the known advantages of the confocal measurement principle, the sensor represents an innovative combination of a fiber-coupled confocal illumination and detection with a tuneable, acoustically driven gradient-index fluid lens for modulation of the focus distance and a novel signal processing utilizing a lock-in amplifier. The new arrangement is able to achieve an approximately linear characteristic curve for the optimized feedback control of the CMS in scanning sample mode. This article emphasizes the optical application and the signal processing of the setup.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. INTRODUCTION AND LITERATURE REVIEW
Miniaturization and micro-structuring of complex components and corresponding tightening of manufacturing tolerances create completely new challenges in dimensional metrology. Using micro or nano coordinate measuring systems (CMSs), it is already possible to realize highly accurate three-dimensional relative motions between sensor and measured surface in ranges of a few millimeters and with resolutions in the nanometer or sub-nanometer range . Conventional confocal systems such as confocal laser scanning microscopes have to make several time-consuming lateral scans of the object surface at different distances from the objective. The chromatic aberration of the objective can be used to detect the point distance using a spectrometer [2,3]. For these systems, only one surface scan is necessary, but the size of the measured point varies because of the different wavelengths used and the resulting diameters of the focused spots. However, there is still a demand for high-precision sensors that are suitable to probe objects with axial resolutions in the nanometer range in a reasonable amount of time. Therefore, the main goal of the investigations presented here is to improve the accuracy and speed of a confocal measurement while maintaining the known advantages of the confocal principle. To achieve this, an innovative sensor was developed that combines a fiber-coupled confocal illumination and detection principle with a tunable, acoustically driven gradient-index fluid (TAG) lens  for the focal-distance modulation and a novel signal processing utilizing a lock-in amplifier. This sensor is targeted to the scanning sample mode of a nano CMS in order to significantly improve accuracy and to reduce measurement uncertainty and the probing time.
The confocal principle, invented in 1957 by Minsky , and particularly its use as a point sensor in confocal laser scanning microscopy (CLSM)  have become highly significant for optical surface metrology . The literature also shows that the axial stepwise capturing of images with the controlled motion of the objective can be replaced by control of focal length with an adaptive lens, which offers several advantages . Moreover, it is known that a TAG lens can be applied for high-speed varifocal imaging , confocal microscopy involving sampling intensity curves and evaluation of the height away from the curve maximum , and also two-photon microscopy .
The novel sensor and signal processing method presented here uses a principle based on the high-frequency modulation of the focal length of the optical system. A TAG lens combined with the conventional microscope objective enables modulation in the nanometer range up to several micrometers. A TAG lens utilizes the principle of an acoustic fluctuation generated by a piezoelectric actuator. It generates sound waves in the liquid, which then cause oscillations in the density and thus its index of refraction . The high-speed focus variation associated with the deployment of the TAG lens is used to sample the intensities only in the vicinity of the intensity maximum. The signal evaluated from the higher harmonics of the modulated intensity is used to control the distance between the sample and the objective to remain in the focus of the objective lens during the lateral scan over the sample surface.
Also covered here is a new method of digital confocal signal processing and narrow-band-pass filtering by using a lock-in amplifier as well as a fiber coupling. Finally, preparations are shown to make the sensor fit for use in a nano CMS.
2. OPTICAL SETUP
Figure 1 shows the basic optical setup of the new sensor (cf. Fig. 9 with the implementation). To physically separate the heat source from the sensor and from the metrological frame of the nano CMS, the laser source (1) and photodetector (2) are coupled to a 630 nm single-mode fiber coupler (3) with a splitting ratio of 50:50. The coherent light source is a randomly polarized helium–neon laser with an output power of 0.8 mW, a wavelength of 632.8 nm, and a percentage of TEM00 more than 95%. For the detection of the reflected light, a high-speed variable-gain temperature-compensated Si avalanche photodetector with a maximum conversion gain of and an output bandwidth of up to 400 MHz is coupled to the fiber to provide a sufficient sampling rate. The fiber end of the common port of the -coupler is used as the aperture for the illumination and detection in accordance with the confocal principle.
A fiber port (4) adjustable with five degrees of freedom and with an aspheric lens and an effective focal length of 11 mm is used to collimate the laser light leaving the fiber, which has a core diameter of 3.5 μm. This results in a beam diameter of 1.8 mm with respect to the maximum entrance pupil of the microscope objective used (7). The collimated beam passes the TAG lens (5) and is modified into a convergent or divergent beam through the sinusoidal modulation of the optical refraction power of the TAG lens. To focus the beam on the measured surface, an infinity-corrected Nikon microscope objective is applied that has an extended long working distance of 4.5 mm and . The beam-splitter cube (6) between the TAG lens and the microscope objective enables the simultaneous microscopic observation of the laser spot and its surrounding area with an image sensor (12) in order to facilitate the adjustment and orientation with respect to the measured surface (8). A second beam-splitter cube (10) between the first beam splitter and the image sensor enables the bright-field illumination of the surface with a white light source (13) in order to provide sufficient light for optical observation. The tube lens (11) brings the image surface to the image sensor (12), a high-speed USB 3.0 camera from IDS (Imaging Development Systems GmbH) with a maximum frame rate of 152 fps and a CMOSIS with 2.23 megapixels. The focused spot of the confocal sensor is clearly visible in the images of the camera system and allows the orientation and selection of the region of interest on the sample surface for the measurements.
The power stability of the laser is not affected by the light returning from the sensor to the laser. This is due to the permanent high-frequency modulation of the focal length and the intensity using the TAG lens. An additional optical isolator does not provide noticeable improvement in power stability. As expected, however, power fluctuations could be detected when the modulation was switched off.
3. THEORETICAL SIGNAL AND SIGNAL PROCESSING
The basic paraxial theory of confocal microscopy was established by Sheppard and Choudhury in 1977 . The state of the art was summarized by Wilson and Sheppard in 1984 , and it was shown that the axial distribution of the normalized intensity function gives the response of an ideal confocal system (punctiform aperture) for the reflected intensity of a plane mirror  by
The resulting normalized intensity on the detector is numerically calculated using Eq. (1) and substituting with Eq. (2) and with Eq. (3), as shown in Fig. 2, where gives the position of the stage of the CMS and indicates the position where the surface of the plane mirror is equal to the focal plane of the objective.
The real characteristic was measured with an axial scan of a plane mirror at constant speed . Needless to say, the full width at half maximum (FWHM) of the measured normalized intensity function for a detection and illumination aperture given by the core diameter of the fiber is wider than the calculated normalized intensity function of an ideal confocal system with the assumption of infinitely small apertures. Using a extended long working distance microscope objective with numerical aperture leads to a theoretical FWHM of approximately 0.7 μm and a measured value of 1.5 μm. In conventional confocal microscopes, the topography of the sample is determined by searching the maximum value from such curves, knowing the assigned axial scan position. Due to the missing slope of the intensity curve at the maximum value or in its immediate vicinity, this signal is not directly suitable as a feedback signal for the control of the distance from the sample to the objective during a lateral scan.
The superposition with a modulation of the focal plane by the TAG lens allows the derivation of a better linear characteristic for height tracking by evaluating the higher harmonics of the signal with a lock-in amplifier. In addition, this characteristic can be normalized in order to reduce the dependency on the reflectivity of the sample surface. This superposition, caused by the sinusoidal modulation of the optical refraction power of the TAG lens with a frequency and an amplitude with a linear movement of the stage in the axial direction at a given scanning speed , yields a mathematical description of the distance from the sample to the modulated focal plane of the lens combination (TAG lens plus objective) relative to the focal plane of the objective:
The product corresponds to the relative position of the stage (cf. Fig. 3). The normalized intensity function with the modulation given by Eq. (4), depending on the time (Fig. 4), can be calculated using Eq. (1) and substituting with Eq. (2).
The distance was calculated using Eq. (4) for a time range of (where time indicates the moment the position of the stage and the relative distance are at the focal plane of the objective), a scanning speed of 0.04 m/s, and a focus modulation of , as shown in Fig. 3. The vertical lines indicate the moments of the intensity maximums. A more realistic scanning speed is in the range of a few micrometers per second, but this would lead to an unrecognizable plot.
Applying Eqs. (1)–(3) for the previously illustrated distance to the focal plane given by Eq. (4) leads to a normalized intensity signal , as shown in Fig. 4. Obviously, the distances of the intensity maxima as well as the signal form are characteristic of the stage positions. However, obtaining a characteristic curve requires appropriate signal processing. Therefore, an innovative method using a lock-in amplifier is presented below.
Generally, lock-in amplification involves the integration of the modulation of an input signal with a reference signal with the same frequency or a higher harmonic of the frequency of the input signal, as shown in Eq. (6). This leads to the output signal6. Applying Eqs. (5)–(7) for the first, second, and third harmonics (, 2, 3) at a scanning speed of 0.4 mm/s leads to the output signals in Fig. 5.
Due to the very close maxima, the input signal (normalized intensity function) is drawn in gray.
The phase shift of the first and third harmonics is zero, and the phase shift of the second harmonic is . Using a lock-in amplifier provides both an approximately linear characteristic curve within the range of the focus modulation and the suppression of all irrelevant frequencies by a very narrow-band-pass filter.
Usually, lock-in amplifiers scan all frequencies to “lock in” the frequency of the input signal. Since the reference signal is the modulation signal driving the TAG lens, the TAG lens controller passes a trigger signal to the lock-in amplifier to adjust the phase and frequency of the reference signal. The multiplication of the measured signal from Fig. 4 with a fundamental-frequency sinusoidal signal as shown in Fig. 6 or with a higher harmonic essentially provides a homodyne detector followed by a low-pass filter, which integrates the signal at defined periods. This allows the detection of a very noisy signal with a low or negative signal-to-noise ratio (SNR).
Looking at Eq. (5), it becomes clear that the output signal not only depends on the phase but also on the amplitude of the input signal.
Calibrating the characteristic curve using a plane mirror and then measuring an object with a different index of reflection would cause a significant amplification error. In addition, inhomogeneity of the index of reflection on the measured surface would cause linearity errors when performing a lateral scan.
Therefore, a normalization is done through the division of the third harmonic by the second harmonic, assuming that both signals change proportionally. It is also possible to divide the first by the second harmonic, but this leads to a less linear characteristic curve.
The linearity of the quotient signal is even better compared to the signal of the first or third harmonic (cf. red lines in Figs. 5 and 6). Because such a division is not a standard function in common lock-in amplifiers, it is performed on the signal processor of the CMS. In addition, the development of digital signal processing on the basis of a field programmable gate array (FPGA) with a middle-range-speed analogue-to-digital converter (ADC) with a sampling rate of 50 MHz is in progress. This will be both sufficient for data processing and quite cost efficient compared to common lock-in amplifiers available on the market. With custom signal processing, it is also possible to apply a moving average or peak detection filter to the raw signal to get a scaling factor instead of using the second harmonic.
Reducing the amplitude of the focus modulation leads to an elimination of the inflection point of the first harmonic (cf. Fig. 8). Furthermore, it seems to be beneficial not to detect or amplify the second maximum of the normalized intensity function, as shown in Fig. 2.
Another approach to achieve amplitude-independent signal processing involves using a bi-Gaussian fit on a single oscillation period or the mean of several periods and determine the distance between the centers of both Gaussian distributions. This works well in offline signal processing but requires too much computational power for a real-time application.
The resulting linear characteristic curve of the sensor can be used to control the stage and sample position for vertical motion of a nano CMS to hold the measured surface in the focal plane of the objective. A lateral scan with a nano CMS is necessary to capture a profile line on the surface. The profile line may then be evaluated either by using only the measured vertical stage position or by also bringing in the calibrated characteristic curve of the sensor to get the controller deviations as height values to add.
4. MEASUREMENTS WITH A NANO CMS
To evaluate the sensor, it was integrated and tested in a nano CMS, in this case a Nanopositioning and Nanomeasuring Machine (NMM-1) developed at the Ilmenau University of Technology and designed for measurements within a measuring volume of (cf. Fig. 9) . The interferometric length measuring systems and the drives enable the movement of the stage with a resolution of less than 0.1 nm and a positioning uncertainty of less than 10 nm in all three axes under very stable environment conditions (e.g., temperature stability). The design of the NMM-1 allows the integration of several different probe system types. However, in order to fully utilize the measurement instruction set, the probe system must be electrically connected to the NMM-1. The electrical signals are simultaneously sampled with 16-bit ADCs along with the machine’s own interferometer length and angle values. The interferometric measuring system of the NMM-1 provides the possibility to calibrate the probe system using the machine itself . Traceability of the measurements to the definition of the meter is achieved by the calibration of the frequency-stabilized helium–neon lasers. To determine the characteristic curves of the sensor, scans with a simultaneous sampling of the lock-in signals and the interferometric signals of the position were performed involving five repetitions of 25 measurements at a scanning speed of 6 μm/s. The excitation frequency of the TAG lens was 70 kHz (first resonant frequency), and the excitation amplitude varied between 10% and 45%.
The measured surface is a gold-coated plane mirror and the measurements were performed with different focus modulation amplitudes and different excitation powers of the TAG lens controller. As an example, the corresponding output signals of two measurements with 25% and 45% excitation power of the TAG lens are illustrated in Figs. 10 and 11. The maximum error of the characteristic curves for each position was determined by performing, five repetitions of 25 measurements with the driving powers 10%, 15%, 20%, 25%, 30%, 35%, 40%, and 45% (1000 measurements summarized). In addition, 500 measurements were performed within a time of 30 min and with a TAG lens excitation amplitude of 25%, and then 500 measurements were done with the same excitation power but within a time of 19 h. As an example, the characteristic curves of 25 measurements are shown in Figs. 12 and 13. The maximum range for each position of the stage of these 25 repetitions demonstrates the short time stability of the linearity of the characteristic curve, shown as an example for 45% and 25% in Figs. 14 and 15.
The maximum ranges and mean values of the five repetitions of 25 measurements for different TAG lens excitation amplitudes are presented in Table 1 and illustrated in Fig. 16 with the ranges of over these five repetitions. The focus amplitude was estimated by the apex of the first-harmonic output signal.
The mean, maximum, and range of become minimal at an excitation amplitude of 25%. As shown in Fig. 8, it is assumed that 25% excitation power yields the ideal focus modulation amplitude, preventing the second maximum and the area with a low slope of the intensity function from being reached. On the other hand, the deviation also seems to become higher if the excitation power is too low (cf. Fig. 16 at 15%) because the slope of the intensity function is also small at its peak. To validate this assumption, a better model for the simulation is required and could be given by . This model explains the discrepancy of the measured intensity function and the narrower normalized intensity function predicted by the currently used model, as seen in Fig. 2.
For 500 characteristic curves measured within 20 min with a TAG lens excitation amplitude of 25%, the maximum deviation over all positions is 55 nm. The maximum deviation of the 500 characteristic curves measured over 19 h is about 133 nm over all positions (cf. Fig. 17). The long-term stability is highly correlated with temperature (Pearson correlation coefficient ), which was simultaneously measured (cf. Fig. 18). There, the zero offset drift is 98 nm. This means that a partially mathematical temperature compensation is theoretically possible and currently under test.
Lateral closed-loop line scans in which the position is controlled using the characteristic curves shown are currently able to be performed and show promising results. After these scans have been validated on different measured surfaces, the results will be presented soon.
Moreover, FPGA-based digital signal processing with custom IPs has been developed and is currently under test.
The ambient temperature seems to influence the results significantly through thermal drifts, although the temperature of the measuring room is being controlled at . To reduce the thermal influence and achieve better temperature stability, a custom housing with a temperature control for the nano CMS has been designed and is being built.
A theoretical approach of allowing measurements independent of the reflective properties of the surface involves calculating the ratio between the first and the second harmonics or between the third and the second harmonics. This approach will soon be validated on objects that exhibit changing reflective properties. Tactile measurements may be used for comparison. Also, the task-specific measurement uncertainty will then be used to evaluate and improve the sensor.
The focal-distance-modulated confocal point sensor presented here is adequate for probing surfaces using micro or nano CMSs. The novel signal processing on the basis of a lock-in amplifier yields good results over the range of a few nanometers. A major source of errors was identified as the temperature deviations; thus, an improvement in temperature stability is necessary in order to be able to identify further influences.
Deutsche Forschungsgemeinschaft (DFG) (310357385, GZ: HA 5915/10-1).
The authors wish to thank the German Research Foundation (Deutsche Forschungsgemeinschaft) (DFG) for funding the project.
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