The continuing trend towards smaller dimensions in manufacturing of technical parts sees an increasing use of techniques such as electro-chemical machining (ECM) or laser-chemical machining (LCM) that employ an electrolytic fluid in combination with a localized energy input to achieve a contactless material removal for the production of micro-structures [1] with high aspect ratios and sharp surface gradients. The advantages over conventional techniques such as micro-milling or laser ablation, e.g., are lower cost, lack of tool wear, and reduced thermal stress induction [2]. Since the workpiece is submerged in a fluid, the in situ measurement that is needed for quality assessment or closed-loop control is not trivial. The challenging manufacturing environment hinders the use of conventional measurement techniques for several reasons: The general lack of accessibility to the workpiece prevents the use of tactile geometry acquisition, demonstrating the need for contactless optical measurement methods. Optical micro-topography measurement techniques can be divided into interferometric methods (such as displacement interferometry [3,4] and digital holography), as well as non-interferometric techniques (such as laser-scanning confocal microscopy [4], light sheet microscopy, or HiLo microscopy [5]). However, these techniques are prone to the in situ conditions of the LCM process such as process-induced currents, thermal gradients, and refractive index fluctuations for the case of interferometric methods [3,6], or high surface angles or curvatures for the case of confocal microscopy [7]. For the application in environments where the surface of the measured object is completely covered by a fluid, the in situ topography measurement of highly curved micro-surfaces remains a challenging task for optical metrology. In contrast to other optical measurement systems, confocal fluorescence microscopy shows promise as an in situ capable measurement technique, because it does not detect the light reflected by the specimen such as conventional confocal techniques, but rather the light emitted by the fluorescent fluid covering it. Thus, light is detected even under angles $>75°$ from the surface normal [4], allowing the measurement of a specimen with sharp edges and high surface angles [7,8], with a lateral resolution comparable to conventional confocal microscopy. The positions of the specimen surface and the fluid surface are determined through the change in the fluorescence signal, while the confocal volume is scanned vertically (perpendicular to the fluid surface) through the fluid. Consequently, this method was already successfully applied for the measurement of metallic micro-spheres with high curvatures by coating the specimen surface with a thin fluorescent film $<100\mathrm{nm}$ [7] to increase the angular range of the scattered light. As a result, the otherwise occurring “bat wing” artifacts, where the surface position uncertainty is higher at steep edges due to the reflected light not being captured by the objective, were not present [7]. Another application utilized the method for an in situ measurement of the tool wear of a cutting tool edge thinly covered (< 113 μm) by a fluorescent cutting fluid [8,9]. For micro-manufacturing applications such as LCM and ECM, however, the specimens are neither submerged in a thin fluid layer nor a thinly deposited fluorophore film of 0.1 μm–100 μm [7,10], but thicker fluid layers $>1\mathrm{mm}$ [6]. The fluorescence signal in thick fluid layers exhibits dependencies on the fluorophore concentration, fluid depth, and index of refraction that could not be observed in thin fluid layers. Thus, the development of a signal processing method, which is based on a physical model of the fluorescence signal considering these effects, is necessary for the accurate measurement of geometric parameters of specimens submerged in thick fluid layers. The aim of this Letter is to determine the potential of the proposed measurement approach, based on confocal fluorescence microscopy, for the in situ measurement in micro-manufacturing applications where micro-structures are submerged in fluid layers $>1\mathrm{mm}$. For this purpose, the uncertainty of a step height measurement is compared to the theoretically achievable uncertainty of an ideal, shot-noise-limited signal, and the influence of the measurement parameters fluid depth and fluorophore concentration on the achievable uncertainty is evaluated.

The confocal fluorescence microscopy setup (see Fig. 1) is based on the well-established confocal microscopy technique: Light from a green diode laser ($\lambda =532\mathrm{nm}$) is expanded by a Keplerian beam expander and redirected by a beam splitter to the objective lens ($\mathrm{NA}=0.42$, $\mathrm{WD}=20\mathrm{mm}$), exciting the fluorophore molecules dissolved in the fluid (dilute aqueous solution of Rhodamine B) surrounding the specimen. The specimen container is positioned using a three-axis linear stage to enable a scanning of the position of the focus through the fluid. Both the fluorescence light emitted by the fluid ($\lambda =565\mathrm{nm}$), as well as the scattered excitation light ($\lambda =532\mathrm{nm}$), are collected by the objective lens, passed back through the beam splitter and are focused on a pinhole (diameter: 5 μm) that attenuates light originating far from the focal plane, thus producing the confocal effect. Behind the pinhole, a notch filter with a center wavelength of 532 nm and a full width at half-maximum of 17 nm separates the excitation light from the fluorescence light, which is ultimately detected by a charge-coupled device.

 

Fig. 1. Diagram of the confocal fluorescence microscopy setup. The fluorescence signal (bottom right) is detected during the vertical movement of the confocal volume through the fluid.

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The measurement principle is based on the detection of the fluorescence intensity emitted from the fluid covering the specimen using the confocal principle, which prevents fluorescence light not originating from a volume within the focus of the objective (confocal volume) from contributing to the fluorescence signal. Scanning the confocal volume of the excitation laser vertically (in $z$-direction) through the fluid produces a characteristic fluorescence intensity signal (see Fig. 1). For values of $z$ far outside the boundaries of the fluid $z_0<z<z_1$, no signal is detected, since no fluorescent fluid is present within the confines of the confocal volume. Using a suitable signal processing of the measured fluorescence signal, the desired surface position $z_0$ of the specimen is obtained.

For the purpose of modeling the confocally acquired fluorescence intensity, the confocal volume needs to be considered first. The confocal volume function $I(\mathit{r},z)$ describes the spatially distributed contribution of each infinitesimal volume element to the detected fluorescence light power. As such, the total detected fluorescence signal corresponds to the integral of the confocal volume function over each dimension $(x,y,z)$. In a first approximation, the confocal volume function $I(\mathit{r},z)$ can be described by a three-dimensional Gaussian function [11,12]:

The fluorescence intensity signal resulting from a measurement of a single lateral point on a submerged specimen with a referenced step geometry is shown in Fig. 2.

 

Fig. 2. Measured fluorescence intensity signal $\tilde{S}(z_i)$ with the fitted model function $S(z)$ [see Eq. (3)] of a single $(x,y)$-point. The parameter $z_0$ (surface position) resulting from the fit is circled.

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The different media are shaded according to the model parameters resulting from the fit, revealing a fluid depth of 2.6 mm. The surface position profile of the step ($z_0$ at different $y$-positions) was determined twice for two different $x$-positions; see Fig. 3(a) (29 measurements) and Fig. 3(b) (23 measurements). After correcting the total tilt of the step by subtracting a linear regression of the lower surface from all points, the step height $h$ was determined by subtracting the mean lower from the mean upper surface position, meaning that both propagate into the resulting step height uncertainty:

 

Fig. 3. Results of the surface position ($z_0$) measurements of a step with a nominal height of 253.5(2) μm. Both results (a) and (b) were measured at different lateral positions $x$.

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The step height results were verified by a reference measurement using the existing method of tactile profilometry, resulting in $h_{\mathrm{ref}}=(253.5\pm 0.2)\mathrm{\mu m}$. A comparison with the results obtained using confocal fluorescence microscopy shows no systematic deviations. However, the measured surface positions show a relatively large stochastic scattering (up to 20% with respect to the step height), most likely caused by the model uncertainties (such as the assumptions mentioned on page 2) and detector noise.

The inhomogeneous surface quality of the submerged step object and a cross-sensitivity of the model fit regarding starting and boundary parameters are also likely contributors.

The first measurement results demonstrate, in principle, the suitability of confocal fluorescence microscopy to determine geometry parameters such as the step height in thick fluid layers using a model-based signal processing. However, the current uncertainties of 8.8 μm and 11.4 μm exceed typical requirements of $\approx 1\mathrm{\mu m}$ for micro-manufacturing metrology. In order to assess the potential of the measurement approach, the lower uncertainty limit for the surface position and the step height measurement is derived and compared to the experimental uncertainties.

To obtain the uncertainty, ${\sigma }_{z_0}=\sqrt{\mathrm{Var}(z_0)}$ of the surface position $z_0$ from the measurement with the nonlinear least squares approximation approach, the covariance matrix of the estimator ${\underset{\_}{\theta }}_S={[{\widehat{S}}_0,\widehat{C},\widehat{ϵ},\widehat{\mathrm{\Xi }},{\widehat{z}}_0,{\widehat{z}}_1]}^T$ based on the fit function $S(z)$ from Eq. (3) is calculated. Applying an uncertainty propagation calculation to the least squares estimator results in the estimator’s covariance matrix [14]:

To estimate the lowest achievable surface position uncertainty ${\sigma }_{z_0}$, an intensity signal solely influenced by Poisson distributed shot noise is considered at first. Based on the full-well depth ($12000{\mathrm{e}}^{-}$) and the quantum efficiency (0.40) of the used CCD sensor chip (SONY ICX205AL), the mean number of detected photons $N(z_i)$ is calculated with the signal model function $S(z)$ from Eq. (3). Since the detected signal is the sum over $20\times 20$ pixels, the maximum amounts to $1.15·{10}^6$photons. According to the Poisson distribution, the variance of the shot noise consequently corresponds to the mean number of photons: ${\sigma }_{\tilde{S}(z_i)}^2=N(z_i)$. With this variance entered in Eq. (7), the achievable uncertainty with the estimator of Eq. (5) in case of shot noise amounts to ${\sigma }_{z_0}=0.16\mathrm{\mu m}$ for a single surface position measurement. Note that the estimated uncertainty was verified by Monte Carlo simulations.

The uncertainty propagation of Eq. (4) results in a step height uncertainty of ${\sigma }_h=0.07\mathrm{\mu m}$ for the 23 position measurements of the second experiment; see Fig. 3(b). As a result, the shot noise limit is about a factor of 125 smaller than the lowest standard deviation of the experimental results. This illustrates the potential of the measurement approach when decreasing the uncertainty of the fluorescence signal to the physical limit.

The uncertainty calculation is repeated, but with the average variance of the measured fluorescence signal $\tilde{S}(z_i)$ from Fig. 2 around the model curve $S(z)$, i.e., ${\sigma }_{\tilde{S}(z_i)}^2=1.26·{10}^9$, in units of photons. As a result, the calculated surface position uncertainty is ${\sigma }_{z_0}=8.56\mathrm{\mu m}$, and the propagated uncertainty for the step height measurement based again on 23 surface positions is ${\sigma }_h=3.75\mathrm{\mu m}$. The estimated uncertainty is a factor of 54 larger than the shot noise limit, indicating the presence of additional noise sources and model uncertainties. However, the lowest standard deviation of the experimental results [see Fig. 3(b)] is still a factor of about 2.3 larger, which suggest varying measurement conditions such as the shape of the fluid surface or the surface micro-topography of the specimen.

In order to analyze the effect of the fluid depth $d_{\mathrm{f}}=(z_1-z_0)$ and the concentration-dependent attenuation coefficient $ϵ$ on the uncertainty ${\sigma }_{z_0}$, calculated with the average variance of the signal around the model, the relation between the three quantities is illustrated in Figs. 4 and 5. The calculations reveal that pairs of $(ϵ,d_{\mathrm{f}})$ exist for which the uncertainty ${\sigma }_{z_0}$ of the surface position is at a minimum (shown in black circles/squares). The minimum decreases with higher $ϵ$ and lower fluid depths $d_{\mathrm{f}}$, which can be approximated by a power function shown as green and blue curves in Figs. 4 and 5, respectively. The resulting trade-off between $d_{\mathrm{f}}$ and $ϵ$ enables the selection of a suitable fluorophore concentration for any particular application-dependent fluid depth to achieve a minimal measurement uncertainty.

 

Fig. 4. Calculated uncertainty of the surface position $z_0$ as a function of fluid depth $d_{\mathrm{f}}=z_1-z_0$ for different attenuation coefficients $ϵ$. The uncertainty minima (black squares) are fitted with a power function (green) over $d_{\mathrm{f}}$ with respect to $ϵ$.

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Fig. 5. Calculated uncertainty of the surface position $z_0$ as a function of attenuation coefficients $ϵ$ for different fluid depths $d_{\mathrm{f}}=z_1-z_0$. The uncertainty minima (black circles) are fitted with a power function (blue) over $ϵ$ with respect to $d_{\mathrm{f}}$.

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The comparison between the lowest measured step height uncertainty (8.8 μm) and the calculated uncertainties using the variances of an ideal, shot-noise-limited signal (0.07 μm) and of the fit-residuum (3.75 μm) demonstrate the untapped potential of the measurement technique. The results indicate that the step height uncertainty is currently mainly limited by model uncertainties or model assumptions, respectively, detector noise and natural surface height variations of the specimen.

In conclusion, a sub-micrometer measurement uncertainty seems feasible with the proposed model-based measurement approach, since the minimal achievable surface position uncertainty for a measurement with a fluid thickness of 2.6 mm amounts to 0.07 μm. The theoretical uncertainty of the measurement approach was shown to be lower than the experiment, suggesting that the measurement system can be optimized by identifying and eliminating other sources of uncertainty. Hence, the main sources of model uncertainties such as the assumptions of the confocal volume shape, influence of the specimen reflectivity, or negligible fluorescence absorption need to be studied in more detail in future studies.