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Abstract
A technique for measurement of the thickness of optical elements using absolute wavelength scanning interferometry is presented in this paper. To achieve high-grade optical components and systems, the thickness of both planar and non-planar optical components must be measured with an accuracy of a few micrometers. The proposed technique is based on the Fizeau interferometer and interconnects data from three different tunable laser diodes yielding a long effective wavelength range and thus low measurement uncertainty. The uncertainty of the central thickness measurement ranges from hundreds of nanometers to a few microns. The method allows to measure the thickness of both flat optical elements as well as lenses with curved surfaces. Moreover, the areal information provided by the interferometry and its high angle sensitivity help to quickly and precisely align the measured component and reduce misalignment errors. The results of thickness measurements have been validated and cross-tested with other techniques. In addition to the thickness, the technique provides some additional information (wedge, surface form error) in the case of flat samples and can be easily and quickly modified (mounting of a Fizeau transmission sphere) to measure other essential parameters of optical elements. Thus, this one approach can replace many single-purpose measuring devices while maintaining high accuracy.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The thickness of an optical component is the material thickness of the component measured usually at the center, specifically called the central thickness. The effect of inaccurate element thickness depends strongly on the shape of the element and the properties of the entire optical system. Typically, however, it can manifest itself by shifting the image plane, affecting the effective focal length of the system, or the beam displacement. It is therefore very important to control the thickness for both planar and non-planar optical components, i.e., lenses and is a part of ISO 10110 standard [1]. Manufacturing tolerances for center thickness are +/-0.20 mm for typical quality, +/-0.050 mm for precision quality, and +/-0.010 mm for high quality. For some specific applications, e.g., in the semiconductor industry, the tolerances can be even tighter and therefore very accurate and precise thickness testing must be applied in the manufacturing chain.
Mostly single point methods are used for the thickness measurement of both plane parallel plates and lenses. Apart from widely used tactile measurements [2,3], there are non-contact optical approaches based on confocal microscopy and white-light interferometry [4–6], and two-arm or single-arm interferometry [7–15] with low-coherence light source providing thickness measurement with low uncertainty.
Area methods involving thickness measurements are mainly limited to advanced interferometric techniques. However, interferometric methods have only been described for testing plane parallel plates and not elements with curved surfaces (lenses). Fourier-transform phase shifting interferometry or wavelength scanning interferometry [16–20] have been developed for testing the testing parallel plates. Due to a relatively short tuning range (typically less than 20 GHz) and thus high uncertainty these techniques are limited to thickness profiling (non-homogeneity measurement) rather than measurement of the material thickness of components. A longer tuning range interferometer with a widely tunable (infra-red) telecom IR laser diode (tuning range 5THz) has been reported [21]. However, for the IR spectral region, an expensive low-resolution InGaAs camera must be used.
In this paper, we introduce an absolute wavelength tuning interferometric technique that can measure the thickness of both parallel plates and lenses with curved surfaces. Several wavelengths with a certain degree of laser tunability are employed during measurement. This helps us to virtually interconnect the different wavelengths and achieve a very long effective tuning range (80nm∼30THz) and thus very low measurement uncertainty [22]. The interferometer works in (near-IR) NIR with a high-resolution CMOS camera. The magnitude of the uncertainty is close to single wavelength measurement uncertainty but with a measurement range of up to approximately 1 meter. Unlike single point (and often single purpose) methods, it provides areal information helping to quickly and precisely align the measured component in the measurement device. As a result, misalignment uncertainties are rapidly decreased. In addition, form error, wedge, and inhomogeneity can be standardly measured [23] for flat samples surface without modification. After a quick mounting of a transmission sphere, the same technique can also be used for the measurement of the radius of curvature and/or surface form errors of lenses [24].
2. Methods
2.1 Absolute wavelength scanning interferometry for thickness measurement
Fizeau interferometry is used to measure phase difference $\phi $ between waves reflected from surfaces forming a cavity as illustrated in Fig. 1. Surfaces within the cavity reflect some amount of light and act as beam splitters. Assuming first-order interference and two surfaces cavity, the phase difference yields in the superposition of the two waves that is captured by a digital camera as modulated intensity fringe pattern:
where A stands for the additional term and the multiplicative term B represents the modulation of the interference signal. The phase difference depends on optical wavenumber k, which is inversely proportional to the wavelength of the laser source λ: $k = 2\pi /\lambda $. The phase difference in (2) includes information about the distribution of the refractive index of the medium ${n_M}$, the physical gap (thickness) between surfaces L including surface irregularities. In optical workshops, the refractive index value ${n_M}$ is usually known (air or very well-defined/measured optical glass) with low uncertainty and physical thickness: can be retrieved from the phase difference and known wavelength of light. However, only the wrapped phase difference ${\phi _W} = \phi mod({2\pi } )$ naturally constrained by $- \pi < {\phi _W} \le \pi $ can be measured using, e.g., phase shifting technique [25]. The wrapped phase difference can be expressed by where ${\phi _W}$ can be very accurately measured while the integer multiple N remains unknown. The wrapped phase difference can still be very useful to measure, e.g., surface form irregularity of smooth optical surfaces [3], however, it cannot be used for the thickness L measurement unless $L < \lambda /({2{n_M}} )$ for which $N = 0$. The unambiguity issue can be addressed by studying the rate at which the phase ${\phi _W}\textrm{}$ in (4) changes with the wavenumber (function arguments are omitted for clarity):The wavenumber change $dk$ is realized by scanning the wavelength of the laser source:
The term $2\pi N$ is no longer presented in (5) and the physical gap L can be ambiguously calculated as:
Note the term describing chromatic dispersion of the refractive index plays an important role, particularly for solid media such as glasses. Let us define effective wavenumber change
including the dispersion for a specific medium which simplifies (7) to:The refractive index dispersion relation for optical glasses is well measured [26] so the wavenumber change $dk$ can be straightforwardly transformed into the effective value $d{k_{eff}}$ and the physical gap L can be computed as a slope of the linear function in (9).
From (9) follows that the longer wavelength (wavenumber) scanning interval, the better accuracy can be achieved. Typically, the mode-hop-free wavelength range is about 1 nm. Some external cavity lasers achieving the tuning range up to 20 nm are rare or very expensive. We propose to replace the long wavelength tuning range with the combination of three sources with shorter tuning ranges. The way to combine measured data from different laser sources is based on the extrapolation of the waveform between the most distant wavelengths. A medium laser source is used to ensure proper coupling, see [22]. As a result, an effective wavelength range of 80 nm can be achieved, providing both absolute and very accurate measurements.
2.2 Experimental arrangement
The principle scheme of the interferometer is shown in Fig. 1. This particular scheme is for flat samples measurement, but the testing space (TS) can be easily adjusted for any measurement configuration, see Fig. 2. The interferometer is designed for the wavelengths 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 wavelength is tuned by the LD temperature. The wavelength tuning interval for individual LDs is about 1.5 nm while their central wavelengths are 780, 785, and 852 nm, respectively. The LDs’ central wavelengths were chosen to successfully extrapolate the waveform between two distant wavelengths L1 and L3. The short wavelength gap between LD1 and LD2 provides an intermediate step to assure proper extrapolation [22]. Light from lasers is delivered to the interferometer through a fiber switch (FS) and the diameter of the collimated output beam is 100 mm. Interferograms are captured by a u-eye camera (CAM) UI-5370CP with 2048 × 2048 pixels with a frame rate of 25 frames per second (FPS). The camera can focus on the measured sample by moving along the optical z-axis. The wavelength for each frame is measured by the HighFinesse WS6 wavelength meter with the uncertainty $0.01pm$. The measurement time is within one minute and approximately the same time is needed to process the data.
Fig. 2. Configurations for the thickness measurement: a) measurement of flat elements with transmission plane - air cavity; b) measurement of free flat elements - glass cavity; c) lens thickness measurement.
Firstly, the wavelength is continuously tuned over the tuning range of the first LD (central wavelength ${\lambda _1} = 780nm$) resulting in a cosine signal for each pixel, see (1). The wavelength step is set to obtain i samples within one period: ${\dot{\phi }_W} = 2\pi /i$, where i is an integer. From (5) and (6) follows that wavelength step can be estimated as
and depends on the cavity length. Note ${L_0}$ represents nominal (or estimated) cavity length and the chromatic dispersion term is omitted here. The sampling parameter i should be set with some margin to assure sufficient sampling. Typically $i\sim 8$ is a good value to keep sufficient sampling and a reasonable amount of data. The directly controlled parameter of the arrangement is not the wavelength $\lambda \textrm{}$ but temperature T driven by the LD temperature controller. The relation between LD temperature change $\mathrm{\Delta }T$ and the wavelength change $\mathrm{\Delta }\lambda $ was experimentally determined as $\mathrm{\Delta }\lambda /\mathrm{\Delta }T \approx 0.05\textrm{}nm/K$ for the specific arrangement. Hence, the vector of LD temperature values can be loaded onto the LD driver. The result of measurements is therefore a sequence of interferograms captured at different wavelengths. The phase shifting (PS) method [25] is used to retrieve the phase ${\phi _W}$. PS uses 8 frames and is applied sequentially over the full wavelength range. The retrieved phase values increase linearly with the effective wavenumber and can be unwrapped using conventional 1D phase unwrapping algorithms. The absolute value of the cavity length is then computed independently for each pixel using (9). This procedure is performed for each laser diode LD1, LD2, and LD3, and measurement accuracy is increased by combining their results [22].2.3 Configurations for thickness measurement
There are several configurations where the proposed technique can be advantageously used for thickness measurement. Namely, it is the measurement of (i) the thickness of multiple flat elements optically bonded to a glass pad during the polishing process, (ii) the thickness of free flat elements (e.g., after debonding), and (iii) the thickness of non-flat elements (lenses).
Flat optical surfaces are widely used for example in the optical or semiconductor industry. The thickness of flat elements is very often an essential parameter and must be controlled during the polishing procedure to reach a value within the required tolerances. Multiple flat surfaces are usually polished (e.g., by chemical mechanical polishing with planetary kinematics) simultaneously, where several flat surfaces are optically bonded to a glass pad. Multiple elements polishing saves time, but more importantly, better results can be achieved. Both thickness and flatness are controlled separately by a single-point thickness gauge a flatness interferometer. This cumbersome procedure can be replaced by a single measurement providing more complex and more accurate results. The measurement configuration is shown in Fig. 2(a). There is a Fizeau plane reference surface mounted to the interferometer head. On the reference surface, part of the light is reflected and propagates to a digital camera while the rest of the beam propagates through the air cavity to the measured surface and reflects back to the interferometer. The air cavity length can be accurately measured and hence all required parameters such as flatness, thickness, and a wedge of all the elements can be controlled within one measurement.
Individual flat elements (e.g., after debonding) also need to be controlled. The configuration of the testing section for such measurement is in Fig. 2(b). The measured cavity is the glass from which the element is made. Thickness is measured for each pixel so a wedge can also be obtained. Several elements can be measured simultaneously.
Apart from flat elements, there are elements with non-flat (e.g., spherical or aspherical) surfaces – lenses (see Fig. 2(c)) for which thickness measurement is also necessary as it influences the back-focal plane and other parameters of optical systems. If the cavity consists of two flat surfaces reflecting similar wavefront curvatures, a low-density fringe pattern is captured by an interferometer. This is a conventional interferometric arrangement. However, surfaces with different surface curvatures significantly modulate the reflected wavefronts, which create a pattern with high fringe density and cause vignetting of the light. Assuming an element to be aligned with the optical axis of the interferometer, there is no curvature in the vertices of both surfaces (normal to the interferometer’s optical axis). Wavefronts reflected from vertex areas of front and back surfaces are naturally biased (delayed) due to the thickness of the lens and have similar wavefront curvature that increases with distance from the vertex. In other words, it is an interference of two spherical waves virtually emitted from different points on the interferometer optical axis.
It is important to evaluate the wavefront difference between the two spherical waves to validate the applicability of the method for the lens thickness measurement. For the quantitative analysis, let us assume a geometry in Fig. 3 and the paraxial approximation. A biconvex lens with radii of curvatures ${R_1}$, ${R_2}$, refractive index of material $n,$ and the thickness t is placed in the measurement cavity at the distance $\xi $ from the interferometer. Both surfaces reflect spherical waves virtually emitted from different points with the distance $\mathrm{\Delta }R$ between them. Radii of curvature of wavefronts entering the interferometer reflected from the front surface (FS)
and from the back surface (BS)Fig. 3. Geometry for lens thickness measurements. Green rays are reflected from the front surface while dark blue rays are reflected from the back surface. Note, the illumination beam impinging the lens is collimated and is not shown in the picture for clarity.
The assumption must not always be met, however, it can provide a rule of thumb for estimating the limits further in the text. The wavefront difference (13) along with the physical gap between surfaces (the center thickness) creates the measured cavity $L({x,y} )= \mathrm{\Delta }Z({x,y} )+ t$ and can be measured by the absolute interferometry, see (9). The lens thickness t is then determined as the length of the cavity at the vertex position which holds $\mathrm{\Delta }Z({0,0} )= 0$, $L({0,0} )= t$, respectively.
This approach has naturally some limits as for “fast” surfaces (short radius of curvature) the measurement becomes more challenging. There are two major limiting conditions for a successful center thickness measurement: (i) sufficient amount of light captured by the camera, and (ii) acceptable slope of the wavefront difference $\mathrm{\Delta }Z$. To estimate the minimal measurable radius of curvature, let us again approximate the surface sag Z around the vertex position with parabolic approximation $Z(r )\approx \frac{{{r^2}}}{{2{R_1}}}$, where $r = \sqrt {{x^2} + {y^2}} $ denotes the radial distance from the vertex. Ray impinging in the z-direction will be reflected with an angle
The stop aperture of the interferometer (see S in Fig. 1) allows only the rays impinging the interferometer with an angle below ${\alpha _{MAX}} = 0.015rad$ to pass the optical system. Therefore only rays fulfilling $\alpha < {\alpha _{MAX}}$ will reach the camera sensor. This condition is valid for the front surface reflection while rays reflected from the back surface are influenced by refraction on the front surface, reflection on the back surface, and the second refraction on the front surface. However, the assumption of reflection from the front surface can provide an estimate of the minimum measurable radius of curvature given the amount of light passing through the optical system. For proper evaluation and sufficient light hitting the camera (with a reasonable exposure time), the safe side condition is collecting light from the circular vertex area of the lens with a diameter of at least $2r\sim 0.45mm$. The value was estimated as the spot size diameter of 9 pixels on the camera sensor when the lens surface is focused. Due to diverging/converging wavefronts impinging the camera, the interferogram spot size can be adjusted by moving the camera along the optical axis. However, the bigger the spot size is, the longer exposure is required to keep a reasonable signal-to-noise ratio. Collecting light from a smaller area of the lens leads to either a weak intensity signal (long exposure time) or a small reflection spot. Recalling ${\alpha _{MAX}}$, the minimal measurable radius of curvature can therefore be estimated as ${R_1} > 30mm$. This value applies to an interferometer with a beam output diameter of 100 mm and other hardware parameters and may differ for others.
However, using long exposure times can reduce the limiting condition by a factor of two or more at the cost of more complicated alignment.
It is also important to consider the curvatures of both superposing wavefronts. As the wavefront difference (13) and (14) increases with r, the Nyquist sampling criterion can be quickly violated. Although the absolute interferometer measures the cavity length independently in each pixel and beyond the sampling criterion, the visibility of fringes decreases when the Nyquist condition is violated with a factor of 2. Let us, therefore, assume the limiting wavefront difference slope for the maximal angle ${\alpha _{MAX}}$ in (14) to be $\frac{{dZ}}{{dr}} < \lambda $. Assuming magnification of the interferometer M, and pixel pitch $\varepsilon $, the maximal $\mathrm{\Delta }R$ can be estimated as:
Substituting typical parameters used for measurements $\varepsilon = 5\mu m$, $M = 10$, and $\xi = 150mm$, the maximal virtual source distance meeting the slope criterion is $\mathrm{\Delta }R = 78mm$. Assuming N-BK7 bi-convex lens with ${R_1} = {R_2} = 30mm$ (chosen concerning the first condition) and thickness $t = 10mm$ and substituting the parameters in (11) and (12), the source distance is $\mathrm{\Delta }{R_{30}} = 16mm$. The condition $\mathrm{\Delta }R > \mathrm{\Delta }{R_{30}}$ is easily met and hence the slope for a lens with 30 mm radii of curvature is measurable. Some examples of simulated interferograms for the measurement of lenses with different radii of curvature are shown in Fig. 4.
Fig. 4. Examples of lens thickness measurements of bi-convex lenses with different radii of curvatures: a) R1 = R2 = 100 mm; b) R1 = R2 = 60 mm; c) R1 = R2 = 30 mm. Note the red circle denotes interferograms created by wavefronts passed through the optical system of the interferometer.
3. Results
3.1 Thickness measurement of plan parallel elements – bonded to a glass pad
During the iterative polishing process, the plan parallel plates are often optically bonded to a glass pad and must therefore be measured in the configuration Fig. 2(a) with the transmission plane. The ability of the absolute interferometry to measure under such conditions was demonstrated by measuring two plan parallel elements with a nominal thickness of 9.06 mm and diameter of 30 mm optically bonded to a 100 mm diameter glass pad, see Fig. 5(d). The front side of the glass pad to which the elements were bonded was polished with flatness better than 50 nm Peak-to-Valley to allow for the optical bonding. The back surface of the glass pad was ground. The elements as well as the glass pad were made of fused silica. The measured sample was inserted in the measured beam of the absolute interferometer equipped with a standard transmission flat. The cavity length between the reference surface of the transmission sphere and the glass pad was approximately ${L_0} = 200mm$. Assuming the cavity length ${L_0}$ and ${n_M} = 1$ (air), values of temperature were generated using (10) and applied to the LDs temperature controller. A sequence of interferograms and wavelengths was captured and stored in computer memory. The data were further processed using (9). For illustration, the interferogram captured at the wavelength $\lambda = 780nm$ is displayed in Fig. 5(a).
Fig. 5. Configuration 1: a) an interferogram captured by the camera; b) 3D topography within the full field of view; c) 2D maps of measured values; d) the photograph of the measured sample - two plan parallel elements optically bonded to the glass pad.
Result of the data processing is the distance between the reference and measured surface in each point (x,y). To remove the overall tilt of the measured sample within the interferogram cavity, the first-order polynomial function (tip/tilt) was fitted to values linked to the glass pad and subtracted from the measured data. Figure 5(b) shows a 3D map of measured distances after tip/tilt removal and subtraction of the mean distance between the reference surface and glass pad (offset). Results are therefore biased to have zero value at the glass pad distance and thickness values at each point are related to the glass pad, see Fig. 5(c). The main objective of the measurement task is to obtain the center thickness value (see Table 1), however, wedge, as well as surface form error, can also be obtained from the measured data as follows from Fig. 5(c). Within one measurement (approximately 1 minute), all elements fitting into the interferometer’s field of view can be measured simultaneously. Both the repeatability and the reproducibility of the measurement defined as the standard deviation of ten measurements are around 1 micrometer. To validate the measured values, the coordinate measuring machine (CMM) Mitutoyo LEGEX 774 was used to measure the central thicknesses of the elements. Firstly, the glass pad was horizontally aligned with the coordinate system of the CMM and the central thickness was then measured with reference to a point on the glass pad by a tactile probe. Results in Table 1 show a good agreement between AWA and CMM measurements.
Table 1. Measured values of flat elements in configuration 1
3.2 Thickness measurement of plan parallel elements
The same plan parallel elements were measured after debonding from the pad by configuration in Fig. 2(b). The elements were freely placed on a soft black pad to prevent them from scratching and to avoid strong specular reflections from the pad. The transmission flat needed for configuration 1 was unmounted from the interferometer. In configuration 2, interference between the front and back surfaces was observed, see Fig. 6(a). Multiple samples can be measured simultaneously unless they fit within the interferometer’s field of view. The measurement procedure was practically identical to the previous one. The temperatures for the LDs controller were set using (10), where the nominal cavity length ${L_0} = 9.06$ and ${n_M} = 1.4537$ (fused silica for $\lambda = 780$) were assumed. Absolute interferometric measurement provides thickness in each point with reference to the front surface see Fig. 6(b). It is worth noting that unlike the air cavity in configuration 1, chromatic dispersion in (9) must be considered here. The refractive index and its change were computed for each wavelength using the Sellmeier equation [27] with parameters obtained from the glass manufacturer. Central thickness as well as wedge can be easily extracted from the measured data. Furthermore, the surface from error summed up for both surfaces can be computed. Configuration with more cavities would be necessary to evaluate surface form error separately for both surfaces [18,23].
Single point thickness measurement instrument OptiSurf LTM by trioptics was used to validate the results, see Table 2. Both flat samples were consecutively measured. The best measurement accuracy provided by the manufacturer is ±0.5µm, however, measurements are also sensitive to aligning/centering due to the single-point nature of measurement. Transmission measurement also introduces refractive index error into the total uncertainty. Reproducibility (measurement after reinserting) of measurement by OptiSurf LTM was in the range of a few micrometers and repeatability below one micrometer.
Table 2. Measured values of flat elements in configuration 2
3.3 Thickness measurement of curved surfaces (lenses)
The absolute interferometry can also be applied for the central thickness measurement of lenses, see Fig. 2(c). Due to vignetting of light reflected from surfaces with high angles with respect to the interferometer optical axis, only a small portion of light reflected from both surfaces close to their vertices passes through the interferometer and reach the camera sensor. Such reflections might be called “ghosts” in optical workshop terminology and their presence is usually undesirable. However, they can be advantageously used for central thickness measurement. The reflections from both surfaces are aligned to be overlapping with slight tip/tilt adjustments of the measured lens and the final alignment is done to observe concentric interference fringes. The alignment can be very accurate due to the high angular sensitivity of interferometers. The misalignment error with a significant margin can be estimated as ${\alpha _{MAX}}/3$ and its contribution to the uncertainty budget $t\left( {\frac{1}{{cos({{\alpha_{MAX}}/3n\textrm{}} )}} - 1} \right) \approx 50nm\textrm{}$ is therefore negligible.
The measurement procedure does not differ from the previous configuration; however, several features must be optimized as the curvature of wavefronts varies with distance. Apart from the tip/tilt alignment to assure both reflections from the front and the back surface are overlapping and the interferometer and lens axes are aligned, the sample and the camera are moved along the interferometer’s optical to scale the interferogram size and adjust its intensity. This can be advantageously used to optimize fringe spacing regarding the camera pitch.
Two lenses denoted LA1384 and LD1464 made of N-BK7 glass with nominal central thicknesses of 8.2 mm and 3 mm were aligned and measured. In Fig. 7(a), the full interferometer’s field of view is shown. Apart from the halo caused by the edges and barrel of the lens, the interference structure due to the overlapping reflections from both surfaces is visible. The zoomed interference pattern is in Fig. 7(b), while the evaluated cavity length is in Fig. 7(c). As aforementioned, the size of the interference pattern can be adjusted by refocusing so the lateral dimensions in Fig. 7(c) do not have to represent correct physical dimensions related to the lens, however, the areal information can be used to precisely align the lens and accurately find the extreme value representing the central thickness. It is worth noting that in the middle of the interferometer field of view, there might be reflections from lenses from which the interferometer is built, therefore it is not preferable to place the measured lens in the center of the interferometer cavity.
Fig. 7. Configuration 3: a) image captured by the camera– lens LD1464; b) zoomed interferogram – lens LD1464; c) 2D map of measured values – lens LD1464; d) image captured by the camera– lens LD1384; e) zoomed interferogram – lens LD384; f) 2D map of measured values – lens LD1384.
The measured values were again compared to measurements by OptiSurf LTM and the results are summarized in Table 3.
Table 3. Measured thicknesses of lenses in configuration 3
4. Discussion
Some limits of central thickness measurements by absolute interferometry (for example, the maximum curvature of surfaces) have been mentioned in previous chapters. In terms of uncertainty, different configurations (see Fig. 2) have various sensitivities to uncertainty sources. The cavity in the configuration illustrated in Fig. 2(a) is formed by air and two independent surfaces. Moreover, the cavity length is usually longer than in configurations b), c) where the cavity is created by a bulk of glass. Therefore, the first configuration is more sensitive to ambient disturbances like vibrations, temperature variation, and air turbulences. Placing the elements close to the interferometer helps to reduce these effects. In configurations b) and c), on the other hand, the refractive index of the glass is always known with some uncertainty. Furthermore, the third configuration suffers more significantly from electronic noise as the signal-to-noise ratio is lower due to vignetting of light.
To quantitatively estimate the uncertainty of measurement, let us recall (7)
Individual uncertainty sources are summarized in Table 4.
Table 4. Table of uncertainty sources
Substituting measured values along with the uncertainty values in the table into (18), the computed thickness measurement uncertainties range from 0.6µm (configuration 3 – LD1464) to values slightly above 1 µm (configuration 1 – the air cavity). The repeatability of measurement is in the same order (∼1µm) so we can estimate the uncertainty of the central thickness measurement to range from hundreds of nanometers to a few microns. Note that the uncertainty due to alignment error is typically lower and therefore was omitted here.
5. Conclusions
The paper introduces a technique for measurement of the thickness of optical elements using absolute wavelength scanning interferometry. The cavity length (between reference and tested surfaces) can be measured with low uncertainty by the absolute interferometry due to the interconnection of data from three different tunable laser diodes providing a long effective wavelength range. The approach is applicable for various geometries of the elements (flats, spheres, aspheres, and freeforms) and due to areal information and the high angular sensitivity of interferometers, the alignment error is minimal. Furthermore, additional information (wedge, surface form error) can be retrieved from the measured data in the case of plane-parallel plate samples. It is also worth mentioning, the same arrangement (interferometer) equipped with an appropriate transmission sphere can be used to measure radii of curvature or surface form error for spherical as well as aspherical surfaces. It follows that the presented approach makes it possible to characterize many essential parameters of optical elements with only one arrangement and thus replaces many single-purpose devices commonly used in the production chain while maintaining the precision required for the production of high-end optical components.
The absolute wavelength scanning interferometry was verified by measuring the thicknesses of plane parallel plates as well as curved surfaces (lenses) in three configurations. Firstly, the thicknesses of plane parallel plates optically bonded to a glass pad were measured with reference to the glass pad. The thickness of the elements (nominal thickness 9.06 mm) was measured with 1 micrometer uncertainty. The results were validated using a coordinate measuring machine. The same plane parallel plates were further debonded from the glass pad and their thicknesses were re-measured in the second configuration, where the cavity is formed by the front and back surfaces of the element. The results were cross-tested with the first configuration and further validated using a single-point low-coherence interferometer. Last but not least, the thickness of bi-convex and bi-concave lenses was measured using the absolute wavelength scanning interferometry with an uncertainty below one micrometer. The results were again compared with the single-point low-coherence interferometer.
Funding
Technická Univerzita v Liberci (PURE-2020-3010); Ministerstvo Školství, Mládeže a Tělovýchovy (CZ.02.1.01/0.0/0.0/16_026/0008390).
Acknowledgments
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. This research was also supported by the Grant Program of the Technical University of Liberec (project number PURE-2020-3010).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. ISO 10110-1:2019, Optics and photonics — Preparation of drawings for optical elements and systems — Part 1: General. Retrieved from https://www.iso.org/cms/render/live/en/sites/isoorg/contents/data/standard/05/75/57574.html.
2. L. C. Zhang, X. M. Wu, H. Z. Li, and H. X. Sun, “Research on measurement of center thickness of the lens with a hole,” in S. Elect, ed., Proceedings of SPIE (2016), 9684.
3. S. Malacara Inc, Optical Shop Testing (Wiley, 1992).
4. M. Kunkel and J. Schulze, “Noncontact measurement of central lens thickness,” Glass Science and Technology 78(5), 245–247 (2005).
5. S. Kim, J. Na, M. J. Kim, and B. H. Lee, “Simultaneous measurement of refractive index and thickness by combining low-coherence interferometry and confocal optics,” Opt. Express 16(8), 5516–5526 (2008). [CrossRef]
6. J. Na, H. Y. Choi, E. S. Choi, C. Lee, and B. H. Lee, “Self-referenced spectral interferometry for simultaneous measurements of thickness and refractive index,” Appl. Opt. 48(13), 2461–2467 (2009). [CrossRef]
7. G. Coppola, P. Ferraro, M. Iodice, and S. De Nicola, “Method for measuring the refractive index and the thickness of transparent plates with a lateral-shear, wavelength-scanning interferometer,” Appl. Opt. 42(19), 3882–3887 (2003). [CrossRef]
8. K. N. Zhang, O. Sasaki, S. J. Luo, T. Suzuki, Y. X. Liu, and J. X. Pu, “A dual-scanning white-light interferometer for exact thickness measurement of a large-thickness glass plate,” Meas. Sci. Technol. 31(4), 045009 (2020). [CrossRef]
9. Z. C. Jian, C. C. Hsu, and D. C. Su, “Improved technique for measuring refractive index and thickness of a transparent plate,” Opt. Commun. 226(1-6), 135–140 (2003). [CrossRef]
10. S. A. Alexandrov and I. V. Chernyh, “Interference method for determination of the refractive index and thickness,” Opt. Eng 39(9), 2480–2486 (2000). [CrossRef]
11. G. D. Gillen and S. Guha, “Use of Michelson and Fabry-Perot interferometry for independent determination of the refractive index and physical thickness of wafers,” Appl. Opt. 44(3), 344–347 (2005). [CrossRef]
12. J. Y. Liu and F. Lei, “Measurement of Lens-Center Thickness Based on Low-Coherence Interference with Transmitted Illumination,” Laser & Optoelectronics Progress 56(12), 3 (2019).
13. W. V. Sorin and D. F. Gray, “Simultaneous thickness and group index measurement using optical low-coherence reflectometry,” IEEE Photonics Technol. Lett. 4(1), 105–107 (1992). [CrossRef]
14. T. GmbH, “https://trioptics.com/products/optisurf-ltm-lens-center-thickness-measurement/”.
15. S. C. Zilio, “Simultaneous thickness and group index measurement with a single arm low-coherence interferometer,” Opt. Express 22(22), 27392–27397 (2014). [CrossRef]
16. Y. Kim, K. Hibino, N. Sugita, and M. Mitsuishi, “Absolute optical thickness measurement of transparent plate using excess fraction method and wavelength-tuning Fizeau interferometer,” Opt. Express 23(4), 4065–4073 (2015). [CrossRef]
17. T. Fukano and I. Yamaguchi, “Separation of measurement of the refractive index and the geometrical thickness by use of a wavelength-scanning interferometer with a confocal microscope,” Appl. Opt. 38(19), 4065–4073 (1999). [CrossRef]
18. L. L. Deck, “Fourier-transform phase-shifting interferometry,” Appl. Opt. 42(13), 2354–2365 (2003). [CrossRef]
19. O. Matousek, F. Prochaska, and J. Kredba, “Laser rods characterization by Fourier transform phase-shifting interferometry,” in T. Optoelectron Syst, ed., Proceedings of SPIE (2019), 11385.
20. P. de Groot, “Measurement of transparent plates with wavelength-tuned phase-shifting interferometry,” Appl. Opt. 39(16), 2658–2663 (2000). [CrossRef]
21. L. L. Deck, “Absolute distance measurements using FTPSI with a widely tunable IR laser,” in W. T. C. U. W. C. E. Architecture ed., Proceedings of SPIE (2002), 4778, pp. 218–226.
22. M. Peca, P. Psota, P. Vojtíšek, and V. Lédl, “Absolute and relative surface profile interferometry using multiple frequency-scanned lasers” (International Society for Optics and Photonics, 2016), 10151, p. 101510 H.
23. O. Matoušek, V. Lédl, P. Psota, and P. Vojtíšek, “Methods for refractive-index homogeneity calculation using Fourier-transform phase-shifting interferometry,” (International Society for Optics and Photonics, 2016), 10151, p. 101510Y.
24. J. Kredba, P. Psota, M. Stašík, V. Lédl, L. Veselý, and J. Nečásek, “Absolute interferometry for fast and precise radius measurement,” Opt. Express 29(8), 12531–12542 (2021). [CrossRef]
25. L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. 29(2), 183–185 (2004). [CrossRef]
26. “Optical Glass | SCHOTT,” https://www.schott.com/en-gb/products/optical-glass-p1000267.
27. W. Sellmeier, “Ueber die durch die Aetherschwingungen erregten Mitschwingungen der Körpertheilchen und deren Rückwirkung auf die ersteren, besonders zur Erklärung der Dispersion und ihrer Anomalien,” Ann. Phys. 223(11), 386–403 (1872). [CrossRef]
