1. Introduction

Precise position measurement is a key requirement in modern production lines. Several measurement technologies have been investigated: laser interferometry, grating interferometry, laser triangulation, capacitive sensors, eddy current sensors, and chromatic confocal technology [1]. Laser interferometry can provide high performance with linear incremental measurements and allows simple configuration for multi-axis usage. However, laser interferometry is also vulnerable to environmental variables, such as humidity, air pressure, and temperature. In addition, because a moving mirror needs to be installed, it is difficult for the interferometer to determine the displacement (or step height) within a relatively small space. Similarly, a related method, grating interferometry, requires the installation of a scale grating and reading head on a static base and traveling platform. However, modern grating encoders enable simultaneous six-degree-of-freedom measurements with a single reading-head. Other measurement techniques like laser triangulation, capacitive/eddy current sensors, and chromatic confocal technology measure the movement of the sample directly, which simplifies measurements significantly. However, laser triangulation is limited to measurements of highly reflective or tilted surfaces, and the capacitive/eddy current technique requires the measured target to be conductive, and working distances are rather small. Chromatic confocal measurement technology has been attracting increasing attention during the past decades because of its high resolution and suitability for almost all surfaces [2].

Chromatic confocal technology is a variation of confocal technology [3,4] that uses the optical conjugate principle. Before confocal technology was used to measure displacement, it had been widely employed as scanning confocal microscopy (SCM), with a laser source, for biomedical applications [5,6]. In SCM, the measuring head is always mounted on an axial linear piezo-stage (LPS) to travel a fixed distance. Only when the sample-surface meets the focal plane of the measuring head, the maximum detection-sensitivity can be achieved during a scan.

Chromatic confocal measurements on the other hand, rely on a novel encoding-strategy for the axial displacement, using a continuous wavelength with a white-light source. When white light passes through the lens, each monochromatic light component has a different focal point on the optical axis. This is because conventional lenses have different refractive indices for different wavelengths. Assuming a suitable lens has been selected, every point on the optical axis corresponds to one specific wavelength. If the sample surface is located within the dispersion range, it must cross the axis with a focal plane that corresponds to a single wavelength. The correct wavelength can be obtained using a high-resolution spectrometric system to detect the reflected light coming from the sample surface. This effectively avoids the generation of “light-noise”. Thus, a position change on the axis corresponds to a displacement of the sample, which can be obtained by measuring the wavelengths at the start and end points. Because a certain sample-surface only uses one focal point, the reflected and detected light through the conjugated pinhole would, in theory, be monochromatic. Although scanning confocal microscopy with a linear piezo-stage makes the axial displacement resolution approximate to the LPS, using several peak-extraction algorithms [7,8], the measurement speed is severely limited by the scan speed of the LPS, for every measured position. Because there is no scanning mechanism, chromatic confocal measurements can be faster using only a single detection with the spectrometer. Hence, the technology is commercially used for in-line real-time measurement [9–11], non-contact surface profile evaluation [12–15], biomedical testing [16] and others.

When using chromatic confocal technology, the imaging quality of the focal spots is important for the accuracy of the measured wavelength. On-axis optical aberration includes both axial chromatic aberration and spherical aberration. Axial chromatic aberration is necessary to encode displacement, while spherical aberration, the most important factor to be optimized, increases the radius of the focal spot - which leads to more uncertainty about the reflective zone of the sample surface. Because the lateral resolution is strongly affected, and the reflected spectrum broadens, the peak wavelength is harder to measure.

Several possibilities have been tried to minimize spherical aberration. These include the use of independent optical elements to generate chromatic dispersion and reduce spherical aberration separately. SL, Dobson et al. [17] used a diffractive optical element (DOE) and a 40X microscope objective lens to improve measurement performance. Its dispersion range reached 55 μm for a 100 nm bandwidth, and the full width of maximum height (FWMH) of the longitudinal point-spread functions was only 2.5 μm. Similarly, Reyes, Johnson Garzon et al. [18] combined a diffractive Fresnel lens with a 50X microscope objective lens. They obtained a 300 μm dispersion range for a wavelength range of 542 nm to 938 nm, and the axial resolution was about 1 μm. Furthermore, Fleischle, David et al. [19] used gradient index lenses and DOE to obtain a 300 μm dispersion range with a 60 nm bandwidth and 2 μm spot-diameter. In addition to the difficulty in manufacturing and the high cost of the DOE or Fresnel lens, the diffraction loss is very high when visible lights pass through [20]. This can block several components of the reflected light and lead to a relatively low detection-efficiency. While the use of commercial microscope-objective lenses can reduce spherical aberration significantly, the obtained effective focal lengths are very small, which greatly limits the measurable displacement-range. The displacement range of a chromatic confocal system mainly depends on the axial chromatic dispersion. U. Minoni [21] used a highly nonlinear, micro-structured optical fiber to generate supercontinuum light between 500 nm and 900 nm and a dispersion range of about 280 μm. Johnson Garzon Reyes et al. [15] also used broadband light (400 nm to 1100 nm) and obtained a dispersion range of 200 μm. However, a supercontinuum light-source is not readily available, and the measurements are severely affected by the wide range of the material’s reflectance for a broad spectrum.

In this paper, we use common spherical lenses and a super-resolution aperture for the optical design to balance a large dispersion range with good imaging-quality. Common spherical lenses have several options to adjust parameters and meet the design targets: a large dispersion range and small spherical aberration. In addition, a super-resolution aperture is designed and manufactured in the optical path to modulate the distribution of the light beam according to previous studies [22–24]. This can, theoretically, help optimize imaging quality. Although Fujiwara, Naoki et al. [25] applied for a patent of a spatial bandpass filter (SBF), the exact parameters and its imaging quality have not been described with much detail in their patent or other reports. In this paper, we determine the most suitable parameters based on the axial spherical-aberration curves using ZEMAX software, for which the imaging results can also be obtained. In addition, this paper also introduces an optical system with an X-shaped fiber coupler by combining four components: illumination arm (IA), detection arm (RA), reference arm (RA), and measurement arm (MA). The system does not need a beam splitter, and its fiber core diameter is small enough to replace the pinholes. Both measurement arm and reference arm can be repurposed quickly to obtain the crosstalk spectrum, reference spectrum, and reflected spectrum. These spectra help determine the normalized dispersion coefficient, whose peak represents the confocal wavelength [21]. Because the normalized dispersion coefficient is not symmetric with respect to the confocal wavelength, the peak-extraction algorithms of conventional confocal microscopy [7] are not suitable for chromatic confocal systems. Here, we introduce an algorithm suitable to improve both accuracy and stability for chromatic confocal peak extraction. The preliminary peak is obtained first using a centroid algorithm. Next, the effective fitting zone is fixed before applying the Gauss fitting algorithm.

Overall, we determined the distribution around the focal point, in theory, and we carried out calibration experiments needed to obtain the wavelength-displacement relationship for the chromatic confocal system. Both stability and resolution were measured and evaluated using a series of experiments.

2. Principle

A schematic of the chromatic confocal measurement setup with an SBF is shown in Fig. 1. The white light emitted from the light source after the pinhole PH₁ can be regarded as point source. The SBF blocks part of the light, and the passing light is focused on the optical axis by a (refracting) lens. When a sample is placed in the focus plane of wavelength λ_i, the focused light is reflected to the detection pinhole PH₂ by the beam splitter and then introduced into the spectrometer for wavelength measurement. The focal points, PH₁ and PH₂, are conjugated to each other. Here, s is the object distance and l is the distance to the focal plane. For the SBF, R is the outer diameter, a is the radius of the center block zone, and b is the maximum diameter of the passing zone.

 

Fig. 1 Schematic of chromatic confocal technology.

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As shown in Fig. 1, both axial chromatic dispersion and the confocal technique are important in this system. The axial chromatic dispersion, which is caused by the different refractive indices within the used wavelength range, can be calculated. For a single lens, the focal length l(λ) can be expressed as [26]:

where n(λ) is the refractive index for wavelength λ, R₁ and R₂ are the radii of curvature of the lens. The focal length l(λ) corresponds to the refractive index n(λ). For common lenses, the larger this wavelength is, the smaller is the refractive index, and the larger is its focal length. However, n(λ) cannot be described with a simple formula with λ, hence, the relationship between l(λ) and λ is always obtained through calibration experiments. In the calibration experiments, l(λ) is provided by the linear scanning stage, and λ is from the peak extraction of the normalized dispersion coefficient between reflected intensity and reference intensity.

However, because the ideal focal point does not exist, we took the diffraction effect into account and derived the intensity distribution. According to a study by Born et al. [27,28], the normalized complex amplitude U at the image plane can be obtained using:

where P(ρ) is the transmission function of the pupil, ρ is the normalized radius of the pupil, J₀ is the zero-order Bessel function, NA is the numerical aperture of the lens, v and u correspond to the radial coordinate r and the axial distance z between the image plane and focal plane (in Fig. 1).

In this paper, P(ρ) for the SBF can be expressed as:

where R_a is the ratio of a/b.

If u equals 0, U can be calculated with Eq. (3):

Hence, we can obtain the intensity distribution for the image plane:

Similarly, for v = 0, the complex amplitude on the optical axis is:

Then, the axial intensity distribution can be acquired:

According to Eq. (5) and Eq. (7), the theoretical intensity distribution can be reached when b, s, and l are fixed. Also, with a certain optical system and monochromatic light, the NA and λ are constant, too. Under such a condition, we derived the distribution of I(v, 0) and I(0, u) under different R_a, and plotted the curves in Fig. 2. The Airy-disk radius is the value of v when I(v, 0) decreases to 0 at the first time – see Fig. 2(a). It can be seen that, as R_a increases, I(0, 0) decreases but the Airy disk radius becomes smaller. This leads to a high lateral measurement-resolution. However, the focal depth, which is derived from u when I(0, u) = 0 at the first time in Eq. (8), increases with increasing R_a – see Fig. 2(b). The increased focal depth and reduced axial intensity directly affect the axial measurement resolution. This is helpful to determine R_a.

 

Fig. 2 Intensity distribution around the focus spot for different R_a values.

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3. Optical design and simulation

Based on the above principles, a chromatic confocal displacement measurement system is designed using a prototype SBF. As shown in Fig. 3, the optical system is equipped with an X-shaped fiber coupler (FC) to connect four components: illumination arm (IA), detection arm (DA), reference arm (RA), and measurement arm (MA). The white light from the light source (LS) is converted into a light beam by an achromatic lens (AL₁). The colored glass filter (CGF) removes not needed colors, while the aperture stop (AS) blocks divergent light. Through AL₂ light is coupled into one end of the fiber coupler. Light is then split into a reference- and measurement-arm. For the measurement arm, the fiber can be regarded as point light-source. After the modulation of SBF and convergence by the chromatic dispersion lens (CDL), the light is reflected by a measurement mirror and returned to the fiber. As the fiber couple can transmit light from both sides, the fiber end of measurement arm serves as PH₁ and PH₂ simultaneously – see Fig. 1. Similarly, the reference light is reflected in the reference arm. The detection arm collects the reflected light from the measurement- and reference-arms into the spectrometer to extract the peak wavelength. In the reference- and measurement-arms, adjustable irises (AIs) are used to block or unblock the light. First, light from the illumination arm does not just spread to the reference- and measurement-arms, the detection arm also receives some of the light, namely the crosstalk light I₀. This can be achieved when both AI₁ and AI₂ are set to blocking. Then, the reference spectrum I_R and the measured spectrum I_M are detected separately, when either AI₁ or AI₂ are not blocking.

 

Fig. 3 Schematic of a chromatic confocal measurement system: light source (LS); achromatic lens (AL); colored glass filter (CGF); aperture stop (AS); adjustable iris (AI); chromatic dispersion lens (CDL); measuring mirror (MM); reference mirror (RM); fiber coupler (FC).

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I_N(λ_k) is defined as the light intensity after the SBF₁ or SBF₂ from the fiber exit in RA or MA. Because of the dispersion caused by CDL, lights of different wavelengths are reflected with different ratios to the incident intensities. To describe the ratios clearly, we set η(λ_k) as the normalized dispersion coefficient to represent the ratio at wavelength λ_k. And the reflectance of MM and RM at wavelength λ_k is described as ψ(λ_k). Then the detected light intensity in the spectrometer can be expressed as bellow:

Next, we can calculate the normalized dispersion coefficient η(λ_k) in Eq. (10), whose peak wavelength λ_m represents the focus wavelength. We can see that I_N(λ_k) and ψ(λ_k) are both eliminated, with only the detected intensities left in the expression. So the effect of the light-source-variation and reflectance-variation against wavelength can be greatly decreased, by the normalization strategy of η(λ_k).

In this system, the chromatic dispersion lens, CDL, generate most dispersion and focuses light on the sample surface, while spherical aberration determines the quality of the focus spot. Generally, convex spherical surfaces generate negative spherical aberration, while concave surfaces produce positive spherical aberration. Therefore, we chose a double convex lens and a plano-concave lens to compensate for the spherical aberration. The design is also relatively simple, the dispersion range is relatively broad, and the spherical aberration is acceptable. In addition, we use the SBF to correct the spherical aberration further. As a typical feature of super-resolution pupil technology, SBF blocks not only the stray light at the border but also the central light for the full spectrum, most of which can be reflected into the fiber. Because SBF reduces also the light of other wavelengths not just the focal wavelength, the FWMH of the reflected spectrum decreases to obtain the peak wavelength more easily. This also improves the resolution of the axial measurement.

Using ZEMAX, we simulated the optical system and obtained the axial spherical aberration curves for three representative wavelengths, 500 nm, 550 nm, and 600 nm. The relationship between axial spherical aberration and the normalized radius of aperture is shown in Fig. 4. For monochromatic light, the optical lines from the boundary of the aperture intersect outside the theoretical focal point. When SBF is not used, the maximum aberration ranges for the three wavelengths are about 1.5 mm, 1.8 mm, and 2.0 mm. These numbers are not good enough to enable accurate measurement. The values decrease greatly when the SBF is used. Its main parameters are S_a and S_b and represent the ratios a/R and b/R – see Fig. 1. Using the curves, we can find two extrema. To decrease the aberration and allow more light to pass through, both S_a and S_b can be chosen within the extreme values. According to the auxiliary lines, the ranges for the deviations (S_b-S_a) and aberrations are calculated: (0.59, 0.16), (0.50, 0.12), (0.47, 0.12). We chose 550 nm to be the central wavelength to design the SBF. Then, the aberration is small and the deviation range suitable. S_a and S_b are 0.385 and 0.885, respectively. Under this a condition, the SBF decreases the aberration from 1.8 mm to 0.12 mm, which greatly improves image quality. In addition, the real SBF is manufactured with the two parameters using three cantilever beams to support the central block - see the insert in Fig. 4.

 

Fig. 4 Axial spherical aberrations and design of SBF.

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Furthermore, for better clarity to show the performance using an SBF, the image of the monochromatic spots for different axial positions are plotted using ZEMAX - see Fig. 5. Figure 5(a) was obtained without SBF, and Fig. 5(b) was obtained with a SBF with the parameters a = 0.385, b = 0.885, R_a = 0.435. The spot sizes are smallest at the focal plane of each wavelength. With the SBF the spot sizes are smaller and stray light is reduced. The RMS diameter of the spot for 550 nm is only 10 μm in Fig. 5 (b). In addition, the other two spots in the focal plane for 550 nm are hollow in the center, which makes it easy to reflect the ring-like light outside the fiber. Hence, the SBF can enhance the advantages of the confocal principle and decrease the FWMH of the reflected spectrum.

 

Fig. 5 Spot diagrams for three wavelengths: (a) without SBF; (b) with SBF.

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4. Experiments and results

Several experiment are carried out. Figure 6 shows the experimental setup. An LED light source (Sinco Ltd.) was used to generate white light, whose effective wavelength range is from 400 nm to 700 nm. The CGF (Unionoptics, Inc. GG400) can transmit light with a wavelength larger than 400 nm. The four ALs are identical and have a focal length of 50 mm (Thorlabs, Inc.). The FC’s splitting ratio is 50:50, and its fiber core diameter is 105 μm with a NA of 0.22. The wavelength resolution of the spectrometer (Ideaoptics, inc. FX2000) is about 0.57 nm. The MM is fixed on a precise linear stage (PI Ltd. M-112.1DG1), whose repeatability precision is 0.25 μm.

 

Fig. 6 Experimental setup of the chromatic confocal system.

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A calibration experiment was carried out to acquire the relation between the displacement of the PI stage and the peak wavelength of the normalized dispersion coefficient. The PI stage was moved with a step size of 20 μm. For each position, the integration time for the spectrometer was set to 25 ms, and the average time was 10 ms to reduce noise.

Figure 7 shows the normalized dispersion coefficient curves of these positions derived using Eq. (10). As the PI stage moves in one direction, the reflected wavelength also seems to move along the wavelength axis. It can be seen that the normalized dispersion coefficient distribution is similar to a Gaussian Function, which coincides with the typical fitting algorithm in SCM. However, the normalized dispersion coefficient curves are not symmetric with respect to the confocal wavelength, and their headpieces are too rugged to directly extract the peaks using the maximum values. Thus, the peak wavelength was determined through integration. First, an approximate peak wavelength was picked up using a centroid algorithm. Then, the effective fitting zone was fixed prior to applying the Gauss fitting algorithm. The fit zone is a window with the approximate wavelength center and a window width of 160 nm. The fine peak wavelength of the fitted Gauss curve was derived for the maximum.

 

Fig. 7 The normalized dispersion coefficient and calibration curve.

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This way, the calculated focus wavelengths can be plotted showing the displacement of the PI stage - see Fig. 7(b). The displacement measurement range was 1.05 mm for a wavelength range of 490 nm - 630 nm. As mentioned above, the curve cannot be fitted using simple formula, therefore, we used a first Fourier function to fit the relationship. Its fitting error is as small as ± 2 μm, with root mean squared error (RMSE) of about 0.00084, i.e., ± 0.2% for the full measurement range.

Secondly, stability experiments were carried out when the stage was fixed at five different positions: 0.005 mm (P1), 0.255 mm (P2), 0.505 mm (P3), 0.755 mm (P4) and 1.005 mm (P5). At every position, the spectrometer detected the reflected light fifty times, and the interval time was 10 s. Using Eq. (10) and Eq. (11), the fluctuation of the calculated peak wavelengths is illustrated in Fig. 8.

 

Fig. 8 Results of the stability experiment for five fixed positions.

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It can be directly seen that the amplitude of variation in Fig. 8 (a) is larger than that of Fig. 8(b) at the same position, almost two times, which proved the effectiveness of the usage of the SBF. On the other hand, as shown in Fig. 8(b) with the SBF, the smallest fluctuation range is around ± 0.05 nm at the middle of the measuring range (P3). The main cause is that the LED generates light through fluorescence excitation, reducing the intensity stability. In addition, the peak extraction method for the asymmetric curve also contains a fitting error. But at the start and the end of the measuring range, the fluctuation range increases to about ± 0.16 nm. This is mainly caused by the focus quality reduction as the wavelength changes away from the central wavelength, 550 nm in this research. It should be noted that, however, the instability of the peak wavelength is even below the resolution of the spectrometer. According to the calibration curve in Fig. 7(b), the instability for the displacement measurement is smaller than ± 0.36 μm.

Finally, the resolution of the chromatic confocal system was determined - see Fig. 9. In the experiments, we moved the stage by a constant displacement increments from three start positions (0.065 mm, 0.505 mm, 0.98 mm). At every position, the peak wavelength was measured ten times.

 

Fig. 9 Results of the resolution experiments.

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It is obvious that displacement increments of 2.0 μm can be distinguished clearly. But when the increment is as small as 1.0 μm, some data points of a nearby step are too close to be discerned. Hence, we think a displacement increment of 1.5 μm represents a suitable resolution for our system. In addition, the resolution near the displacement increment 0.505 mm is below 1.0 μm.

5. Discussion and conclusions

In this paper, a chromatic dispersion system was designed, constructed, and evaluated. The system uses a super-resolving pupil filtering element, referred to as spatial-bandpass-filter (SBF), an X-shaped fiber coupler, and a modified peak-extraction algorithm.

Based on the theoretical analysis, the effect of the spatial bandpass filter on the focus spots was determined. The SBF parameters affect the lateral focus-spot diameter in a favorable way, while the focal depth becomes worse. This requires balancing of the SBF parameters. A combination of common spherical lenses and the SBF improves the optical dispersion range substantially. Common spherical lenses have many options to adjust parameters and meet the desired objective: a large dispersion range. The SBF was used mainly to reduce the spherical aberration and transmission loss without changing the dispersion range. This combination also greatly improved the design options for this system. We also found that a “hollow” beam spots appears, when the spot is outside the focus position with the SBF. This increases the confocal effect further and eventually improve the resolution too. The optical design with a reference arm solves the challenge caused by the large fiber-core diameter. It enables high photo-sensitivity but makes it difficult to extract the peak wavelength. The X-shaped fiber coupler comprises four components: illumination arm, detection arm, reference arm, and measuring arm. The system does not need a beam splitter, and its fiber-core diameter is small enough to replace the pinholes. Both measurement- and reference-arms can be quickly converted to determine the crosstalk spectrum, reference spectrum, and reflected spectrum. These spectra enable the derivation of the normalized dispersion coefficient, whose peak represents the confocal wavelength. Furthermore, an improved peak-extraction algorithm that uses an optimized window, solves the problem of asymmetric and rugged headpieces of the normalized dispersion coefficient curves.

The prototype described in this paper reaches a position stability of ± 0.36 μm using a commercial cost-effective white-light source, axial displacement-resolution (up to 1 μm), and a fitting error smaller than ± 0.2% for a measurement range of 1.05 mm. Although these preliminary parameters are not yet competitive with the latest commercial products, the method investigated in this study may open new doors for improved chromatic displacement-measurement systems in the future.