Noise evaluation of a point autofocus surface topography measuring instrument

In this work, the measurement noise of a point autofocus surface topography measuring instrument is evaluated, as the first step towards establishing a route to traceability for this type of instrument. The evaluation is based on the determination of the metrological characteristics for noise as outlined in draft ISO specification standards by using a calibrated optical flat. The static noise and repeatability of the autofocus sensor are evaluated. The influence of environmental disturbances on the measured surface topography and the built-in software to compensate for such influences are also investigated. The instrument was found to have a measurement noise of approximately 2 nm or, when expressed with the measurement bandwidth, 0.4 nm for a single-point measurement.


Introduction
Surface texture affects the functional properties of engineered surfaces, such as surface energy (e.g. capillarity, wetting, adhesion), optical (e.g. reflectivity, absorption, diffraction) and thermal (e.g. conduction and radiation heat transfer mech anism) characteristics. Surfaces can also be engineered for biocompatibility, mechanical fatigue, hydrodynamic and tri bological performance [1,2]. Optimisation of surface texture is required in various applications to improve the performance of products. For example, automobile engine parts (cylinder liners, piston pins and oil rings) with optimised surface tex ture have been demonstrated to improve fuel efficiency and service life [3][4][5]; when coating cutting tools, surface texture is critical in affecting wettability and interactions between the coating and substrate, which in turn determines the wear behaviour and life of the coated tools [6]. Texturing the sur face of medical implants is used to improve osseointegration at the implantbone interface [7]; Fresnel lenses utilise surface texturing to achieve focusing power at significantly lower thickness and mass; and subwavelength optical structures have been used to produce antireflective surfaces, polarizers and beam splitters [8].
To optimise the performance of the abovementioned prod ucts, it is important to have confidence in the accuracy of surface topography measurements, which can be ensured by establishing traceability. This can be achieved by evaluating the metrological characteristics (MCs) of the surface topog raphy measuring instrument using calibrated artefacts [9]. The MCs for a contact stylus instrument, a coherence scan ning interferometer and an imaging confocal microscope have been investigated [10][11][12], and there has been some research on characterising noise for a focus variation instrument [13]. In this work, we evaluate the MCs for a point autofocus instru ment (PAI) for the measurement of areal surface topography. The PAI is an optical measuring instrument that automatically focuses a laser beam to a single point on the surface and raster scans an area of interest [14]. PAIs are often used to measure optics, cutting tools, and microgears. The general charac teristics and MCs of PAIs are introduced in the specification standard ISO 25178605 [15]. However, there is currently no established method for determining the MCs specifically for PAI. As the first step towards establishing traceability of this type of instrument, the measurement noise, static noise and autofocus repeatability of a commercial instrument (Mitaka Kohki MLP3SP) are evaluated. The remaining MCs will be addressed in future papers.
The rest of this paper is structured as follows: section 2 provides an introduction to the measurement mechanism of PAIs; section 3 describes the methodology used to evaluate the MCs as defined in ISO 25178605; section 4 presents the results; and section 5 concludes the findings.

Instrumentation
A PAI is a noncontact, optical areal topography measuring instrument, which consists of a laser source, a microscope objec tive, an autofocus mechanism and a precision moving stage [14,15]. The laser beam is focused onto the surface so that the focal spot defines a height of a single point on the surface. The PAI used for this work is a commercial instrument (MLP3SP) hosted in the laboratory of the Manufacturing Metrology Team of the University of Nottingham. In this instrument, autofocus is achieved using the beamoffset method [14]: the incident beam passes through one side of the objective lens and is focused onto a point on the sample surface; the reflected beam passes through the opposite side of the objective lens and is received by the autofocus sensor. The detected laser spot displacement is used as the feedback signal in the autofocus mechanism to adjust the position of the objective lens. When the objective lens is at an infocus position, surface height is computed as the sum of the position of the vertical zaxis and the autofocus (AF) axis [14]. Movement along the x, y and z axes is determined by linear scales with a nominal resolution of 10 nm, while movement along the AF axis is determined by a linear scale with a nominal resolu tion of 1 nm. The instrument features a chamber that shields the measurement from external environmental disturbances.

Methodology
The MCs of an optical measuring instrument are influenced by several factors, such as environmental, mechanical and elec trical noise, optical aberrations and mathematical algorithms. To assess the contribution of each individual factor would be time consuming and often unnecessary for the end user. Thus, an input-output model has been introduced [14] to account for the influence factors using the MCs introduced in the draft stan dard ISO/DIS 25178600 [16], and defined as characteristics of the measuring equipment, which may influence the result of measurement, may require calibration and have an immediate contribution to measurement uncertainty. Eight MCs are included in the draft standard specification: measurement noise, flatness deviation, amplification coefficient, linearity deviation, x-y perpendicularity deviation, topographic spatial resolution, topography fidelity and maximum measurable local slope [9].
Measurement noise N M is defined in ISO/DIS 25178600 as the noise added to the output signal occurring during the normal use of the instrument [16]. Measurement noise is a dynamic phenomenon, which is affected by the motion of the drive unit as well as instrument internal noise and environ mental disturbances. Determination of N M is achieved through areal surface topography measurement of a calibrated optical flat artefact. Furthermore, static noise and autofocus repeata bility are investigated in order to separate the contribution of the drive unit and that of environmental disturbances. All measure ments were performed in a temperaturecontrolled laboratory environment (20 °C ± 0.5 °C), unless otherwise stated.

Measurement noise
Two methods have been proposed in the literature [10] to eval uate N M : the subtraction method and the averaging method. Both methods require repeated measurement of a calibrated optical flat and describe the N M in terms of the root mean square height of the surface S q . The subtraction method evalu ates N M by subtracting consecutively measured surface topog raphies to try to remove the effect of the finite topography of the flat. As the subtraction combines the variances of two identical probability distributions that each characterise the noise of the instrument, N M can be estimated using the topog raphy resulting from the subtraction of the two, divided by the square root of two: The averaging method is based on the assumption that the noise contribution to S q decreases when averaging multiple measurements, i.e. noise is statistically stationary, and that the measured surface topography can be considered as made up of the 'true' topography and the noise contribution. With repeated measurements of the same surface area, measure ment noise can be estimated by: where n is the number of averaged topographies and S qn is the root mean square height of the averaged topography [17]. Furthermore, the measurement noise uncertainty contribution (following [10]) propagates with a normal distribution with null expectation and a variance equal to the measurement noise squared, and can be computed accordingly: In this work, N M is evaluated over fifteen repeated measure ments of an optical flat from a set of artefacts calibrated by the National Physical Laboratory, UK (NPLBNT 019) [18], with a nominal Sz value of 4 nm, with an expanded uncertainty of 10 nm (note that this specification is from the certificate).
The measurement settings are shown in table 1. The choice of the scanning pitch and stepping pitch results from a trade off between lateral resolution and measurement duration [19], where the highest lateral resolution is used in the scanning direction while a larger stepping pitch is selected to reduce the measurement duration to approximately one hour.
Before applying the evaluation methodologies, postpro cessing of the acquired topographies is performed using the commercial software MountainsMap ® , which includes: • levelling the surface by subtraction of the leastsquares mean plane; and • removing outliers, mostly due to contamination, by applying a threshold of 0.5% and 99.5% of the material ratio.
Measurement noise is a common performance specifica tion cited by instrument manufacturers, as it aids in quanti fying measurement repeatability and vertical resolution. In particular, the definition of vertical resolution is not consistent among various instrument manufacturers and may cause dif ficulty when comparing different instruments [20]. Although N M is an effective alternative to quantify the minimum detect able vertical distance, without the need of either defining the vertical resolution or designing a dedicated material measure, it can be affected by the temporal bandwidth of the measure ment. For example, various types of environmental distur bance can introduce noise in different bandwidths; and noise can be reduced by averaging signals over a longer duration [20]. Therefore, to create a common reference frame for describing measurement noise, it is necessary to describe N M along with the associated measurement bandwidth, expressing it in terms of noise equivalent height, in nm, divided by the square root of the data acquisition rate, in height points per second.

Static noise
Static noise evaluation complements N M when describing the noise affecting the instrument. The investigated PAI, despite being an optical instrument, is not an imaging system; on the other hand, due to its working principle, it can be treated as an optical equivalent of a contact stylus. Therefore, this work evaluates the static noise on the basis of ISO 25178701 [21]: the laser beam is focused on a calibrated optical flat (from NPLBNT 019) and fluctuations in the height of the mea sured point are recorded. Static noise is then computed as the standard deviation of the recorded surface height signal; and describes the repeatability in the vertical direction, which is affected by both the z and AF axes.
The height of a single point (focal spot) is recorded for a period of fifteen minutes. As the instrument is not in measure ment mode during this investigation, the fluctuations in sur face height can only be displayed on the instrument screen, but not saved as a file. Therefore, surface height information is retrieved from the video recording of the monitor screen at twentyfour frames per second, and then sampling the signal at 2 Hz to avoid undersampling.

Autofocus sensor repeatability
The autofocus repeatability R AF is a characteristic introduced in ISO 25178605 [15] specific to the working mechanism of PAI. R AF aims at characterising the AF axis while excluding other influence factors, such as the z axis fluctuations and environ mental disturbances. R AF is determined using an internal instrument software function, which repeatedly focuses the laser beam onto the same point and records the position of the autofocus sensor. In this work, 1500 measure ments were repeated with a sampling interval of 1.5 s. The measurement procedure was then repeated five times. R AF is determined as the standard deviation of the recorded AF axis readings.

Environmental effects
Before evaluating noise, an issue affecting the measured topography needed to be addressed. When measuring an optical flat in a temperaturecontrolled environment, notice able deviation from the ideal geometry is present on the mea sured topography, as shown in figure 1(a). The topography consists of an overall waviness superimposed on the nominal topography of the optical flat. The deviation indicates a drift in surface height over time (an approximately onehour period in this case). A builtin function is available in the instrument software to compensate for the drift, and when enabled, sig nificantly reduces the previously observed deviation, as shown in figure 1(b). The compensation function regularly corrects  the topography by actively monitoring the magnitude of drift in the height of a predefined point and subtracting this mag nitude from the measured profiles. The frequency of applying the correction can be specified, which ranges from applying it to every scanned profile, to a set number of profiles. With the increased correction frequency, drift is better compensated, however, measurement duration is increased. As this func tion is recommended by the instrument manufacturer for areal measurement, it was enabled for all evaluations of measure ment noise in this study. To minimise the influence of drift, compensation was set to be performed as often as possible (i.e. on every scanned profile). However, due to time delays in monitoring the drift and noise in the AF axis, errors are inevi tably introduced during compensation, resulting in height changes between scanned profiles as shown in figure 1(b). Even though drift can be largely compensated by the built in function, it is important to first understand the cause of the drift. In this section, the nature of the drift is analysed, and the potential cause of the drift is explored. Given the periodic behaviour of the deviation and the nature of the temperature control in the laboratory, it was suspected that the intermit tent switching on of the air conditioning system had caused the deviation. Therefore, an investigation was conducted to assess whether a correlation between the periodic deviation and temperature fluctuations can be found. The investigation also aims to determine the effectiveness of the builtin drift compensation function.
The instrument is modelled as a black box, which receives as input the nominal topography combined with the noise signal, and outputs the measured surface topography. In particular, the noise can be considered as made up of several contrib utions, where that due to the temperature fluctuations is the main focus of this section. To verify the correlation between the temperature fluctuations and the periodic deviation in the measured topography, the temperature inside the instrument measurement chamber during the measurement period is recorded by a resistance temperature detector PT1000 two wire probe with a nominal accuracy of ±0.15 °C.
Linear systems theory guarantees that the input and the output signals have at least the same harmonic content in their spectra, i.e. a harmonic at the same frequency; therefore, a common harmonic is searched between the recorded temper ature and the deviation in the resulting surface topography. The frequency spectrum of the recorded temperature is com puted using a fast Fourier transform (FFT) algorithm. The deviation in the topography is represented by the mean surface profile along the stepping direction, as shown in figure 2.
To consistently compare the frequency spectra of the temperature and topographical signals, the spatial sampling frequency of the mean surface profile (1 µm −1 ) along the step ping direction needs to be converted to a temporal frequency f T considering an overall measurement duration of fiftyfive minutes: Fifteen repeated measurements of the same area of the flat were carried out with the setup shown in table 1; temperature inside the instrument chamber was recorded at a sampling fre quency of 1 Hz. Figure 3 shows the recorded chamber temperature during the fifteen repeated areal measurements. It can be seen that the first measurement was associated with a steep increase in temperature in the measurement chamber, which is likely due to instrument warmup. Figure 4 shows the frequency spectra of the inchamber temperature and the mean surface profile for one of the meas urements. The presence of a common harmonic indicates a potential correlation between the two signals. The harmonic frequencies found in all fifteen measurements were analysed, and the results are shown in figure 5. The overlap of the expanded uncertainty intervals, with coverage factor k equal to 2, between the inchamber temperature and the topograph ical deviations suggests a degree of correlation. Therefore, temperature fluctuations in the laboratory are believed to be the cause of the waviness added to the measured topography. Figure 6 shows that after the builtin drift compensation function is applied, a principal harmonic is no longer present  in the frequency spectra of the mean surface profile and the magnitude decreases significantly, indicating that the drift induced by temperature fluctuation is effectively compensated.
To further demonstrate that temperature variation is the main cause of the drift in surface height, figure 7(a) shows an example profile along with the corresponding chamber temperature during the measurement period. A good correla tion between chamber temperature and surface height can be observed in figure 7(b).
The influence of temperature on measured surface height, as indicated in figure 7, is a common source of measure ment error, and has been reported with contact probes [22], displacement transducers (e.g. strain gauges, piezoelectric, variable resistance and variable inductance displacement transducers) [23,24], nanoscale sensors, inductive probes, capacitive probes and laser interferometer [25].  (1) and (2).
It was found that N M values stabilise at approximately 2 nm. Given that the least discernible digit of the AF sensor is 1 nm, N M and its contribution to uncertainty were determined to be 2 nm, according to equation (3).
With both the subtraction and the averaging methods, the number of repeated measurements required to reach a stable value of N M cannot be easily determined and depends on the instrument being evaluated. In the case of the PAI under evalu ation, five repeated measurements were found to be sufficient, which is less than the fifteen measurements performed in this work and less than the number suggested in literature for other instruments [10].
To further evaluate the effectiveness of the builtin drift compensation function, N M values determined both with and without applying the drift compensation function were com pared. It was found that N M can be as large as 20 nm, in the worst case obtained using the averaging method. Stabilisation of the N M value also became more difficult to achieve, as temperature fluctuations were different during each repeated measurement. In contrast, when drift compensation was applied, N M was reduced to 2 nm and stabilisation of noise values was achieved within five repeated measurements every time, indicating stable behaviour. Measurements were also performed on a shop floor without any temperature control measures; and similar noise values were found, indicating that the drift compensation function was effective in both labora tory and manufacturing environments and that strict temper ature control is not necessary for the instrument.
With the spatial sampling settings described in table 1, a total of 1001 × 101 points were measured in approximately fiftyfive minutes, resulting in a measurement bandwidth of 30.6 Hz. As a result, the bandwidth specification of measure ment noise was determined to be 0.4 nm √ Hz −1 for a single point measurement.

Static noise
Static noise was evaluated to complete the description of the noise affecting the instrument and it excludes any noise in the  drive unit involved in raster scanning. The recorded height of a single point on the optical flat is shown in figure 9. The same drift that affected the areal measurement was found in the recorded height signal. Since such drift would have been compensated for in areal measurement, it is reasonable to remove the drift when analysing the static noise of the instrument. Removal of the drift was achieved by applying a high pass filter with a cutoff frequency associated with the fundamental frequency of the inchamber temperature, which was found to be approximately 10 mHz. Static noise, determined as the standard deviation of the residual height, is computed to be 2 nm. Spikes with magnitudes of approxi mately 10 nm are observed in figure 9, which are due to the fluctuations in the z axis position, where the smallest dis cernible difference in the encoder scale is 10 nm. The spikes were not removed before applying the high pass filter in order to conform to the definition of static noise, which accounts for noise in both the vertical axis and the auto focus sensor. Figure 10 shows the repeatedly measured height of a single point on the optical flat. Similar to the observation in    Figure 8. Surface topographies used to determine measurement noise: (a) the mean of fifteen repeatedly measured topographies, and (b) the difference between two consecutively measured topographies after subtraction. section 4.3, drift in the measurement is present in the form of an oscillation caused by the periodical regulation of room temperature in the laboratory. As the definition of R AF in ISO 25178605 [15] excludes the influence of environmental dis turbance it is, therefore, necessary to remove the drift using a highpass filter with a cutoff frequency of 4 mHz, which was assessed to be the fundamental frequency of the inchamber temperature fluctuation during R AF evaluation. The resulting autofocus sensor repeatability was found to be 5 nm.

Autofocus sensor repeatability
It is worth pointing out that different cutoff frequencies were used when applying the Gaussian filters in sections 4.3 and 4.4. This is because the two measurements were obtained in separate sessions, during which temperature was found to vary in different fashions and frequencies. Therefore, the appropriate cutoff values had to be determined by the actual frequencies of temperature variation during the invest igations. Furthermore, Gaussian filtering was only applied when determining static noise and AF repeatability, as the inbuilt drift compensation function is only available during areal measurement.

Conclusion
This work is a first step towards establishing traceability of a PAI and presents methods for evaluating the measurement noise, static noise and autofocus repeatability. The influence of environmental temperature disturbances has been inves tigated, and the effectiveness of the builtin drift compensa tion function assessed. When not applying the builtin drift compensation function, a deviation with a periodic nature was observed in the measured surface topography. The devi ation was subsequently found to be caused by environmental temperature disturbances, based on analysis of the spectra of the deviation in the topographies and that of the inchamber temperature. The correlation between inchamber temper ature and surface height was also confirmed in the temporal domain. Once the builtin drift compensation function was applied, the periodic deviation was effectively compensated; and measurement noise has been determined to be 2 nm using both the subtraction method and averaging methods or, when expressed with the measurement bandwidth, 0.4 nm √ Hz −1 for a singlepoint measurement. Additionally, static noise and autofocus repeatability were determined to be 2 nm and 5 nm, respectively. The next phase of this research is to determine the other MCs from ISO/DIS 25178600 to allow uncertainty statements to be estimated with topog raphy measurements.