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Abstract
A novel radius of the curvature measurement method for optical spherical surfaces using absolute interferometry is proposed. A measurement setup is designed and built around a common-path Fizeau interferometer. The cavity length (volume of air between reference and tested surfaces) can be measured by the absolute wavelength tuning interferometry. An interconnection of data from three different tunable laser diodes (central wavelengths 780, 785 and 852 nm) allows us to measure the cavity length with uncertainty from tens to hundreds of nanometres. Once the reference radius of curvature is known/measured/calibrated, the radius of surface under test can be computed applying the value of the cavity length. The radius of curvature is measured directly in confocal position of the interferometer with relative precision of about 10 ppm. Moreover, unlike standard radius measurement by interferometry, the uncertainty of the introduced method can be optimized by selecting a suitable transmission sphere. In the paper, the method is described, tested, and verified by measuring several specimens featuring different radii of curvature. The results are analysed and furthermore compared to other measurement device.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Spherical optics, although obsolete in the eyes of early adopters of aspheres, still plays a crucial role in optical system construction. Where fast optics, superpolished surfaces, low optical noise systems or affordable systems are crucial, spherical elements remain inevitable or, at least, the first choice. This is due to the fact that production of one element featuring aspherical surface with certain requirements, such as extremely low micro-roughness, low subsurface damage, or lack of mid spatial frequencies, is more demanding and more expensive than production of (albeit several) spherical elements with stringent specification. This is also usually true for adjustment of aspheres within optical systems. Thus, optical designers tend to use only the necessary number of aspherical surfaces in the system. At any rate, specifications of spherical surfaces for power laser optics, cine maker’s optics, analytical optical systems, optical systems for space, etc., are continuously tightening. One of the most critical values of the spherical element specification set are surfaces radii of curvature [1]. Two decades ago, the specification of the radius of an optical surface was around one percent from nominal, today one tenth of a percent is not out of our world, and half of a per mille has started to appear. It is very difficult to measure radius with such precision due to the low repeatability of the standard process, which is poor thanks to operator influence and other (systematic) errors. Moreover, the geometry of the surface could cause difficulties in reaching sufficiently low uncertainty. Leading groups specialized in precision optics measurement are working on the development of a repeatable measurement method with the aim to reach resolution under 100 nm for radius in hundreds of millimetres [2,3]. However, the measurement procedure is rather cumbersome and special on purpose developed hardware and well managed process is crucial.
Radius measurement is an elementary skill in optics work. There are essentially three standard methods for radius measurement used in optical workshops: optical rings, calibrating glass [4], and measuring of displacement between the interferometer confocal and cat’s eye position [5]. All of them have some inherent shortcomings. Optical rings are not very precise and, for some geometries, can even be difficult to apply; relative measurement error dramatically increases for longer radii, and the risk of surface scratching is enormous, especially when soft optical material is considered. Calibrating glass measurement is less prone to scratching and could be sufficient even for longer radii, however, the original calibrating glass has to be made and characterized for every measured radius. For cat’s eye-confocal radius measurements, the interferometer must be equipped with an additional precision length measuring system. For the most precision techniques, some laser interferometers are used to measure distance [6–11]. Distance measurement introduces its own inherent drawbacks, such as alignment and straightness of rails, Abbe error, thermal expansion, etc. In addition to that, there are many surfaces for which this technique cannot be used. For long radii or concave surfaces, it is difficult to measure in a given configuration due to the accessible travel in the optical axis. Sometimes, there are issues when measuring in the cat’s eye position due to reflections or imperfections in the surface vertex or due to a hole in the surface centre, e.g., for telescope mirrors. Other alternative techniques based on optical coherence tomography [12], Fabry-Perrot [13], short coherence approach [14], using controllable source spectrum [15], or partially compensation interferometry for aspheric surfaces [16,17] require specific hardware and their applicability may be limited.
In recent years, the TOPTEC Centre team dealing with precision optical metrology research developed a method based on absolute interferometry [18] which is a candidate for a fast, traceable, and precise method with a potential for full automation. The method is built around a classical common-path Fizeau interferometer and the radius of curvature is measured within one shot, i.e., without the need for any mechanical movement. However, in order to reach the absolute interferometric measurement, several wavelengths and a certain degree of laser tunability are employed. This helps us to virtually interconnect the different wavelengths and achieve low measurement uncertainty.
2. Principle of the method
Digital interferometry is a precise, accurate and full-field measurement method, which is sensitive to the phase change of an optical wave. The principle of the Fizeau interferometer for spherical surface measurement is illustrated in Fig. 1. The last surface of a transmission sphere facing a measured surface is called the reference surface and acts as a beam splitter. On the reference surface, part of the light is reflected back and propagates to a digital camera while the rest of the beam propagates through a cavity (air volume between the reference and the measured surfaces) to the measured surface.
Fig. 1. a) Principle of radius measurement by Fizeau interferometer; b) image of the absolute wavelength tuning Fizeau interferometer (AWA).
After a reflection of the light beam on the measured surface, a part of the light propagates back through the interferometer following the same path and impinges the camera sensor. Due to this common-path configuration, many aberrations occurring inside the interferometer are cancelled out. This is the significant advantage of Fizeau interferometry over, e.g., Twyman-Green arrangements [19], which are also used in optical workshops. Superposition of the two waves (reflected from the measured surface and the reference surface) generates an interference pattern, which is captured by a digital camera and processed in order to retrieve the wavefront difference between the two surfaces.
When measuring the radius of a spherical surface R by absolute interferometry, the measured surface is placed in the confocal position, i.e., the centre of the measured surface radius R coincides with the centre of the reference surface, which has the radius ${R_{TS}}$, see Fig. 1. Then the surface radius can be determined using the cavity length L and the radius of the reference surface ${R_{TS}}$
In (1), the coordinates $x,\; \; y$ represent the right handed Cartesian coordinates system, where the $\; z$ axis is coincident with the optical axis of the measured surface. The cavity length L also comprises (subsumes) irregularities of the measured W and reference ${W_{TS}}$ surfaces. However, the irregularities W and ${W_{TS}}$ can be measured by standard interferometry and subtracted and, for the scope of the paper, we can, therefore, assume $W = {W_{TS}} = 0$. The reference radius ${R_{TS}}$ is supposed to be known (previously measured), so the cavity length L fully determines the measured surface radius R. However, the cavity length cannot be measured by standard interferometric techniques due to their limited measurement range given by 2π ambiguity. Therefore, an absolute measurement approach must be applied.
The interference pattern I,
is captured by the digital camera. In the interference Eq. (2), an additional component A represents the mean intensity of interfering waves, the term B stands for a multiplicative component representing the interference modulation depth, and $\phi $ is an interference phase. As a rule, both the reference surface and the measured surface, which are made of glass, are uncoated, so approximately 4% of light is reflected from each surface and thus a high contrast interference pattern can be observed. Introducing the optical wavenumber in the air k, which is inversely proportional to the wavelength in the air of the laser source$\; \lambda $, the interference phase is related to the cavity lengths L asThe inherent drawback of the interferometric technique is its $2\pi $ ambiguity. When measuring the interference phase $\phi $ (using, e.g., the phase shifting technique [20]), due to the harmonic nature of the light wave, the interference phase ${\phi _W}\; $is wrapped in an interval [$- \pi ,\pi $] so that
The relative (wrapped) phase ${\phi _W}$ can be very precisely measured while the integer multiple N remains unknown. The rewritten Eq. (5) using (4),
has, therefore, two unknowns ($N$, $L$) and cannot be solved. For a smooth and continuous surface, the integer N can be determined relatively within the surface area using spatial unwrapping [21]; however, that only applies for irregularity measurement, not for cavity length measurement. This drawback can be addressed by studying the rate at which the phase ${\phi _W}$ in Eq. (6) changes with the wavenumber $k$:As a result, the constant term $N\; 2\pi $ in (6) disappears and the cavity length L can be computed as
For clarity reasons, the phase change $d{\phi _W}$ will be denoted as $\mathrm{\Delta }{\phi _W}$, the wavenumber change $dk\sim \Delta k$, and the coordinates $x,y$ will be omitted further in the text. It is also more convenient to use wavelength change $\Delta \lambda $ around a central wavelength $\lambda $ instead of the wavenumber change $\Delta k$, where $\mathrm{\Delta }k \approx{-} 2\pi \Delta \lambda /{\lambda ^2}$.
Naturally, measuring the cavity length L suffers from uncertainties. The cavity length measurement uncertainty ${u_L}$ can be defined as
In (9), there is an uncertainty in the wavelength change $\Delta \lambda $ denoted as ${u_{\Delta \lambda }}$ as well as in the phase term $\mathrm{\Delta }{\phi _W}$ denoted as ${u_{\Delta {\phi _W}}}$.
It is obvious from Eq. (9) that the longer wavelength tuning span $\Delta \lambda $, the less sensitive the measurement method is to error sources and, thus, the better the precision which can be achieved. To explore the required wavelength tuning span $\mathrm{\Delta }\lambda $ in order to achieve the required precision of the cavity length measurement ${u_L}$, both terms in the equations (9) can be treated separately. Starting with sensitivity to ${u_{\Delta \lambda }}$, we can assume ${u_{\Delta {\phi _W}}} = 0rad$ and substitute (8) into (9) resulting in
In (10), $\Delta \lambda $ is the wavelength shift and ${u_{\Delta \lambda }}$ is its uncertainty. Assuming the wavelength shift uncertainty ${u_{\Delta \lambda }}$ to be $0.02\; pm$ and the required relative precision $\frac{{{u_L}}}{L} = 1 \times {10^{ - 6}}$, the minimum wavelength scan is $\Delta \lambda \approx 20\; nm$.
The other uncertainty source, the phase error ${u_{\Delta {\phi _W}}}$ in Eq. (9), is a sum of different distortion sources, e.g., electronic noise, background intensity variation, intensity ratio of reference and object wave, diffraction patterns from dust particles, and environmental distortions. Moreover, in case of a transmission sphere settling or a slight movement of any part, the phase error is integrated over the wavelength scanning time. Assuming ${u_{\Delta \lambda }} = 0\; nm$, the required tuning range $\Delta \lambda $ can be estimated from (9) to be
Assuming the required cavity length measurement uncertainty ${u_L} = 1\; \mu m$ and the phase error ${u_{\Delta {\phi _W}}} = 0.2\pi $, the required tuning range yields $\Delta \lambda \approx 30nm$. The phase error ${u_{\Delta {\phi _W}}}\; $depends on the environmental conditions and the cavity length, so the required wavelength range varies accordingly. In any case, to achieve precision meeting current standards, the wavelength tuning span must be in in the order of tens on nanometres. Coherent sources exhibiting such mode-hop-free range are rare or very expensive. We propose to replace the long wavelength tuning range by a combination of more sources (three in this particular case) with shorter tuning ranges. The combination of the measured data from the different sources comes from our earlier work [22] and enables us to effectively reach the wavelength span of $\Delta \lambda \approx 80\; nm$ (corresponding to the synthetic wavelength $\Lambda = \frac{{{\lambda _{min}}{\lambda _{max}}}}{{{\lambda _{max}} - {\lambda _{min}}}} \approx 9\; \mu m$) and thus to achieve both absolute and very precise measurements at the same time.
3. Experiments
3.1 Experimental arrangements
The developed absolute wavelength tuning interferometer (AWA) is shown in Fig. 1(b). It consists of a Fizeau-type interferometer and a 6-axis stage working in a vertical position. An achromatic transmission sphere (TS), designed for the wavelength range of 760–860 nm, is used to transform the collimated output beam of the interferometer into a spherical wavefront. The last surface of the TS (the reference surface) is used as the beamsplitter to split the reference and the object wave. For the measurement, we used a TS with an optical diameter of 4” with a F-number of 10.58. The form error of the TS reference surface is certified to be λ/40 peak-to-valley and therefore it is not further considered. The reference radius value was measured as cat’s eye-confocal distance using Zygo DynaFiz interferometer to be 1074.9 ± 0.01 mm.
The interferometer is designed for a wavelength range of 760–860 nm. The laser beam is generated by a tunable laser set consisting of three distributed feedback (DFB) laser diodes (LD) controlled by a temperature and a current source. The change of the wavelength is achieved using the control of the LD temperature. The range of generated wavelengths for individual LDs is about 1.5 nm while their central wavelengths are 780, 785, and 852 nm, respectively. Interferograms are captured at a frame rate of 25 frames per second (FPS) by a u-eye IDS camera UI-3370CP-M with 2048 × 2048 pixels. The wavelength for each frame is measured by the HighFinesse WS6 wavelength meter with a wavelength shift uncertainty of ${u_{\Delta \lambda }} = 0.02\; pm$. The measured values are for wavelength in the air under current environmental conditions. The measurement time is below one minute and approximately the same time is needed to process the data.
The nominal radii of the specimens reported in this paper and their corresponding cavity lengths are summarized in Table 1.
Table 1. Nominal parameters of the measured surfaces.
3.2 Measurement procedure
The measurement procedure is illustrated in Fig. 2. When the measurement starts, wavelength is continuously swept over the tuning range of the first LD (${\lambda _1} = 780nm$). The longer a cavity, the finer the required sampling of interferograms in the time domain. For proper evaluation (by phase-shifting as described below), the phase step between two consecutive interferograms should not exceed $\pi /2$. Substituting the phase step ${u_{\Delta {\phi _W}}} = \pi /2\; $in Eq. (11), the wavelength step yields ${\lambda ^2}/4L$. Hence, the cavity length must be taken into account when initializing the measurement. The directly controlled variable is not the wavelength $\lambda \; $but the LD temperature $\textrm{T}$, related as $\mathrm{\Delta }\lambda /{\Delta \textrm{T}} \approx 0.05\; nm/K$ in our case.
Fig. 2. a) The beginning of a sequence of interferograms captured during wavelength tuning; b) Measured intensity values (blue) and computed phase values (black) as a function of the wavenumber change within the pixel denoted by the blue square in a).
The result of the measurements is a sequence of interferograms captured at different wavelengths. In order to retrieve the phase ${\phi _W}$, the phase shifting (PS) method is applied. The applied PS technique requires several interferograms with the known phase shift α between them resulting in a set of m equations
where $i = 1..m$ is an integer. For the solution of the nonlinear system of equations, the Gaussian least squares approach is used [23]. Since the wavelength is measured and the nominal cavity length ${L_0}\; $is known (see Table 1), the phase shift can be computed asThe PS uses eight frames and is sequentially applied along the full time (wavelength) range. As a result, the phase value at a given point (x, y) on the surface increases with an increase in the wavenumber, see Eq. (8), and, thus, it can be easily unwrapped using conventional 1D phase unwrapping algorithms. The absolute value of the cavity length is then computed independently for every single pixel as the phase slope with respect to the optical wavenumber. Finally, the surface radius can be computed using Eq. (1).
In order to increase the measurement precision, the results from the three LDs denoted as LD1, LD2, and LD3 are combined [22] so that the final synthetic (shortest) wavelength is determined by the wavelengths which are the most apart. The middle laser diode LD2 is used to assure proper interconnection between LD1 and LD3.
4. Results and discussion
4.1 Testing of the limits of the method
The first testing configuration, introduced in Fig. 3, was used to verify the principle of the method and to determine repeatability as well as measurement accuracy with minimal influence of environmental conditions. No transmission sphere was mounted on the interferometer and a plane-parallel plate was placed in the tested cavity and aligned so that both its surfaces reflected the laser beam back to the interferometer. The observed interference pattern (see Fig. 4(a)) is a result of superposition of beams reflected from the front and the back surface of the plane-parallel plate. Thickness T, refractive index of the glass$\; n$, and difference between the irregularities of both the surfaces $\Delta W$ determine the measured optical path difference ($OPD$):
Fig. 3. a) Experimental arrangement for testing and verification of the method; b) interferogram; c) phase map retrieved by phase-shifting; d) wavefront error computed by spatial unwrapping of the phase map in c) – common single wavelength interferometry; e) results obtained by absolute interferometry involving thickness of the specimen (note the colorbar); f) comparison between single wavelength interferometry (red) and absolute interferometry (blue).
The aim of this testing was to simulate ‘ideal ‘conditions defining the limits of the measurement technique. This configuration, where both the interfering surfaces are fixed to each other by a glass bulk, differs from radii measurement by having much more stable conditions. Effects like TS settling, slight movement of the specimens, null position error, and environmental conditions are mostly supressed. Moreover, the cavity length is shorter when compared to cavity lengths used for most spherical surface measurements. The phase error ${u_{\Delta {\phi _W}}}$ can, therefore, be assumed to be $0.02\pi $. The expected measurement uncertainty can be estimated using Eq. (9) as ${u_L} \approx 39\; nm.$
The measurements were repeated ten times with an approximately five-minute time gap between them. One pixel (in Fig. 3(c); denoted by the blue dot) with coordinates $x ={-} 20\; mm,\; y = 0\; mm$ is chosen for the statistical analysis. The average OPD from the set of ten measurements is 39.476 mm while the measurement repeatability defined as the standard deviation is 11 nm.
In further analysis, the results of absolute interferometric measurement (Fig. 3(e)) are compared to single-wavelength interferometry (Fig. 3(d)). Single wavelength interferometry suffers from the 2π ambiguity and thus only the surface irregularity difference $2n\Delta W({x,y} )$ can be measured while the cavity length $2nT$ remains unmeasurable. Therefore, only the irregularity difference $2n\Delta W({x,y} )$ can be the subject of the comparison. Moreover, in the single wavelength approach, a spatial unwrapping algorithm must be applied to the measured phase map ${\phi _W}$ (Fig. 3(c)). On the other hand, the single wavelength is shorter than the synthetic wavelength used in absolute interferometry by a factor of $\mathrm{\Lambda }/{\lambda _{LD1}} \approx 12$ and thus the precision of the single wavelength measurement of $2n\Delta W({x,y} )$ is supposed to be proportionally higher. The comparison of the two methods is demonstrated in Fig. 3(f). The standard deviation of the high spatial frequency data (considered as noise) for the single wavelength (red line) is approximately four times lower than the absolute interferometric measurement (blue line). Considering the three-sigma rule, this result is in agreement with the expected results. Apart from the expected higher noise, the measurement of the surface irregularity difference $2n\Delta W$ by absolute interferometry is in perfect agreement with the single wavelength approach. Moreover, both the approaches can be advantageously combined. It is also worth noting that if we knew (measured) the glass refractive index, this method would enable us to measure the glass thickness T with an unprecedented accuracy and repeatability.
4.2 Measurements of surface radii
The radii of three different specimens (see Table 1) were measured using the absolute interferometric method in the configuration illustrated in Fig. 1. The measurements were repeated ten times with a time gap of several minutes between them. The room temperature during the measurements was 22.8°C. Repeatability of the absolute interferometric method in workshop conditions is defined as the standard deviation STD. The results of the radii measurements are summarized in Table 2.
Table 2. Results of ten repeated measurements (units are mm)
The results were also compared to a conventional measurement instrument LuphoScan 260 HD [24] which claims a low uncertainty. The results of the radius measurements are summarized in Table 3 while the measured form errors (irregularities) after a best fit sphere removal are summarized in Fig. 4.
Table 3. Radii of curvature (in mm) measured by absolute interferometry (AWA) and LuphoScan instrument
Fig. 4. Surface irregularities of three specimens with various nominal radii measured by absolute interferometry (AWA) and LuphoScan.
There are three main uncertainty sources in the measurement: (i) cavity length measurement uncertainty ${u_L}$, (ii) uncertainty of reference surface radius of curvature ${u_{\; {R_{TS}}}}$, and (iii) uncertainty related to the null cavity error ${u_{\; {R_C}}}$. The influence of the dispersion of the transmission sphere can be neglected due to its achromatic design. While ${u_L}$ is the inherent measurement uncertainty by absolute interferometry, ${u_{\; {R_{TS}}}}$, and ${u_{\; {R_C}}}$ can be optimized by a proper characterization and adjustment of the system.
The uncertainty of the cavity length measurement ${u_L}$ depends on the phase uncertainty ${u_{\Delta {\phi _W}}}$ and an uncertainty in the wavelength change ${u_{\Delta \lambda }}$, see Eq. (9). Due to the measurement time about 1 minute, the phase error ${u_{\Delta {\phi _W}}}$ depends on the environmental conditions like vibrations, air turbulences or temperature variation. As mentioned in section 3.2, using PS sequentially in the wavelength range, we obtain a set of phase maps related to different wavelengths. The phase error ${u_{\Delta {\phi _W}}}$ can be estimated as the standard deviation of all the phase maps (independently in each pixel) that can be visualized as the standard deviation map, see Fig. 5. Substituting the maximum value of the STD map ${u_{\Delta {\phi _W}}} = 0.09\pi $ in Eq. (9) results in cavity length uncertainty ${u_L} \approx 171nm$.
The uncertainty of the radius measurement ${u_R}$ is furthermore affected by the uncertainty of the radius of the transmission sphere (in our case ${u_{\; {R_{TS}}}}\sim 10\mu m$) and the displacement of the spherical surface centre from the null position at the centre of curvature denoted as ‘cat’s eye’ in Fig. 1(a) (the surface is not sitting perfectly in the confocal position). The displacement will appear as a defocus in the interferometrically measured phase. The null cavity error ${u_{\; {R_C}}}$ can be compensated based on the sag equation for a sphere as
where ${W_{DEF}}$ is the peak-to-valley of the defocus measured by a Zernike polynomial decomposition and F stands for the F-number of the spherical element part measured by the interferometer. Although ${u_{\; {R_C}}}$ can be used to compensate the null cavity error, an error in the F and/or ${W_{DEF}}$ is naturally transferred in the null correction term ${u_{\; {R_C}}}$. This error is relative, thus ${u_{\; {R_C}}}$ errors are smaller for fast, well-nulled cavities. Assuming an F-number $F = 10.58 \pm 1\%$ and the measured residual defocus about ${W_{DEF}} = 200\; nm\; \pm 1\%$, the resulting defocus compensation is ${u_{\; {R_C}}} = 109.2 \pm 3\; \mu m$, i.e. the uncertainty after the compensation is ${u_{\; {R_{CE}}}} ={\pm} 3\; \mu m$. Finally, the uncertainty values for AWA measurements in Table 3 are calculated as ${u_R} = \sqrt {ST{D^2} + {u_L}^2 + {u_{\; {R_{CE}}}}^2 + {u_{\; {R_{TS}}}}^2} $.It is worth noting that the uncertainties of the defocus compensator and the radius of the transmission sphere are much higher than the cavity length measurement uncertainty in the particular case. However, the uncertainty ${u_{\; {R_{TS}}}}$ can be lowered by a more accurate measurement of ${R_{TS}}$ or a calibration using a calibrated spherical surface with a low radius uncertainty. Furthermore, the null cavity error can be lowered by taking a number of cavity length measurements at several small steps through the confocal null position [11]. A least-squares linear best fit to the recorded defocus coefficient/cavity length pairs can then be used to estimate the ‘null’ cavity length. Such an approach has an uncertainty level comparable to ${u_L}$. This possibility of improving the method requires further study which is beyond the scope of this paper and, therefore, is not dealt with in detail here.
It is important to emphasize that this approach does not suffer from any errors related to the adjustment of the optical rail (bench) which occurs during standard measuring of the radius of curvature by cat’s eye-confocal distance measurement, such as Abbe offset, cosine error, or axial runout. Moreover, the standard radius measurement approach inherently exhibits higher uncertainty for long spherical surfaces with large radii due to the need for longer displacement, i.e., the measurement uncertainty is proportional to the radius value. On the other hand, the measurement uncertainty of the proposed absolute interferometry technique is related to the cavity length. As a result, the cavity length can be adjusted using a suitable transmission sphere so that, even for large radii, a very low uncertainty ${u_L}$ is achieved.
5. Conclusion
The paper presents a newly proposed and developed absolute interferometric method for precision radius of curvature measurement. The introduced method has several properties enabling its use in new generation production lines of smart factories and has a high potential for full automation. The radius of curvature is measured directly in the confocal position, hence there is no need for the specimen movement and any additional length measurement devices. Inherent sources of errors, such as Abbe error, are, therefore, mitigated by the working principle of the method.
In the paper, the measurement principle is described and analysed in order to reveal error sources and their contributions to the total uncertainty budget. Assuming uncertainty sources inherited by the system (omitting environmental conditions), the cavity length measurement uncertainty ranges in tens of nanometres. This prediction has been verified and confirmed by a simplified arrangement simulating almost ideal measurement conditions. Further, radii of curvature of several specimens were measured under standard optical workshop conditions. Repeatability of the measurements was computed to be in units of microns, which represents a relative precision of about 10 ppm. Some ways to further improve the precision of the measurement are proposed and discussed. The radii of curvature of the specimens were also measured with another state-of-the-art commercial measurement device and a comparison showed very good agreement.
Funding
Ministerstvo Školství, Mládeže a Tělovýchovy (CZ.02.1.01/0.0/0.0/16_026/0008390).
Acknowledgement
This work has been carried out within the Framework of the Partnership for Excellence in Superprecise Optics project (Reg. No. CZ.02.1.01/0.0/0.0/16_026/0008390) and co-funded from European Structural and Investment Funds.
Disclosures
The authors declare no conflicts of interest.
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