Compliant Glass Mechanism Instrumented with a Bragg Grating to Measure Indentation Force

Abstract

This paper presents a force sensor made of a compliant glass mechanism instrumented with a waveguide and a Bragg grating, measuring the reflected wavelength shift produced by the strain in the compliant element generated by the applied force. The compliant element geometry and material have been chosen for the sensor to be spliced or manufactured at the extremity of an optical fiber, enabling possible insertion of the instrument in the bronchial tree after embedding in a proper catheter. The context of this research is the mechanical discrimination between healthy and cancerous lung tissues based on their mechanical signature. The paper proposes a comprehensive study including the mechanical design of the structure and the optimization of the production parameters, thanks to an experimental parametric study. After experimental characterization of the mechanism stiffness, the optical response to a mechanical force is reproduced with two different samples on two different days (more than 25 repetitions). The conclusion is that a fair linear and repeatable response is observed (±26 mN) for forces ranging from 0 to 250 mN.

1. Introduction

This paper presents a new force sensor in the form of an indentation tip that paves the way for mechanical discrimination of healthy vs. cancerous lung tissues by indentation.

Lung cancer represented 2.5 million new cases worldwide in 2022 [1]. It is the deadliest cancer, representing 18% of all cancer deaths. Early diagnosis is crucial to increase the survival rate at 5 years, from 65% for patients diagnosed at stage I to only 5% at stage IV. Nowadays, a biopsy is required to confirm the diagnosis, while most nodules are difficult to access, as they are located in the so-called lung periphery [2]. Therefore, thinner sensors or those mounted on bronchoscopes could help reduce the number of biopsies or even provide an alternative when the targeted tissues cannot be reached by biopsy tools. In vivo sensors are excellent alternatives, but biochemical sensors cannot give accurate spatial information, since the binding of proteins and antibodies happens all along the anatomical pathway inside the patient. Contrarily, mechanical indentation sensors could circumvent this difficulty by offering a localized, fast indentation force measurement. Beyond lungs, such a tool would help characterize many other tissues, such as the bladder, endometrial tissue in gynecology, the heart, and digestive tissues, or lymph nodes. Even for organs such as the prostate or tongue, which are relatively easier to access, a mechanical diagnostic could help to better target the biopsies. Ref. [3] indeed reports the modulus ratio between normal and cancerous cells using an atomic force microscope for various tissues. More generally, mechanical properties are known to be a key marker for cell growth [4].

The goal of this research, as exposed in this paper, is to design and develop a new sensor for insertion in the operating channel of a bronchoscope (inner diameter down to $1.2 \mathrm{m}\mathrm{m}$) to mechanically characterize lung tissue in vivo. This paper reports only preliminary results on the force-sensing element, equipped with a waveguide and a Bragg grating in order to measure the force applied to the fiber tip during its insertion into tissues.

The assumption is that this force can be converted into tissue stiffness, further used to discriminate between healthy and tumoral tissues.

Scientifically, the mechanical characterization of lung tissues is important for optimizing ventilation treatment [5], monitoring of arterial hypertension and pulmonary fibrosis [6], or optimizing the culture media with adequate stiffness to develop new drugs against lung cancer mutations [7]. Adapted from [8], Figure 1 displays healthy tissues’ stiffness (Young modulus) ranging between 1 and 5 kPa, while cancerous tissues are up to five times stiffer according to [9].

Several comments arise from these data. First, while cancerous tissues are stiffer than healthy tissues, cancer cells are softer than healthy ones, contributing to measurement dispersion. Secondly and consequently, the scale and therefore the measurement technique impact the results. At the macro-scale, mechanical properties of the lungs are measured with volume-pressure measurements, punch indentation, or tensile testing of isolated biopsied tissues [15]. At smaller scales, punch indentation is replaced by AFM indentation with micrometer-size tips [6]. In addition to indentation, magnetic resonance elastography is promising, but it is not yet an established method in bronchoscopy [6].

Thirdly, lung tissue stiffness also depends on the respiration cycle through the transpulmonary pressure [9]. Finally, the post-processing of the indentation data also impacts the results. In the literature, Young modulus E is typically deduced from contact force measurement (Hertz contact mechanics).

This paper relies on the use of Bragg gratings for embedded force sensing in medical instruments. Ref. [21] summarizes the benefits of such systems in medical applications (sensitivity, biocompatibility, chemical inertness, and insensitivity to magnetic field) and reports many application fields (endovascular intervention in the cardiac area, retinal microsurgery, prostate intervention, laparoscopic and robotic surgery, and flexible endoscopy). Most architectures combine three to five fibers (typically one Bragg grating for temperature auto-calibration and three of them for the three degrees-of-freedom: one compression and two bendings), disposed around a central or external compliant structure made of aluminum flexure, hollow stainless-steel tube, titanium wire shaft, or hollow nitinol tube with cuts. These designs are limited by the materials of the compliant structure and by the high level of assembly complexity. The material limitation is given by the maximal elastic strain $\sigma /E$ ratio: based on this criterion, glass is better than titanium alloys, steel, aluminum, or nitinol. Figure 2 summarizes the main features of these Bragg grating-based solutions found in the literature.

The second element is the current trend to use monolithic mechanisms, i.e., produced in a single-step production without assembly, as reported by, for instance, ref. [32] for a vitrectomy mechanism in eye surgery.

Finally, glass shaping at a microscale is considerably improved thanks to femtosecond laser pulse processing, allowing the fine production of 3D structures in glass parts. The illumination of glass first leads to an increase in the refractive index, and at a higher exposure dose, to a local densification surrounded by a zone with tensile stress, promoting a higher etching rate in KOH [33]. At a larger tensile stress, the appearance of cracks limits the etching rate. This allows the precise shaping of glass. In the medical domain, it has enabled the emergence of bistable mechanisms limiting the stroke, such as puncturing needles in vein canulation [34].

More specifically, in terms of femtosecond laser pulses engineering of glass, it has been demonstrated that three types of refractive index modulations can be generated depending on the femtosecond laser pulses properties [35,36,37,38,39], i.e., pulse energy and duration, repetition rate, scanning speed and direction, and polarization. With increasing energy density levels, the silica glass undergoes different structural changes: (1) continuous densification enabling an increase in the refractive index to produce optical waveguides; (2) micro/nano gratings or shaping (ablation) upon subsequent exposure to an etchant chemical; and (3) formation of voids. All these changes have already been successfully implemented, as reported in [40,41].

The main idea followed in this paper is shown in Figure 3. Under the application of an external force, the compliant structure, further described in Section 2.1 deforms according to the model presented in Section 2.2. Thanks to the fabrication and characterization processes explained in Section 2.3 and Section 2.4, the following results could be obtained (Section 3): mechanical stiffness of the mechanism and its optical response to a mechanical force, thanks to the analysis of wavelength shift in a light beam reflected by a Bragg grating engraved in the compliant part of the tip. The observed limitations are discussed in Section 4 and conclusions are drawn in Section 5.

2. Materials and Methods

2.1. Mechanical Principle

Figure 4 provides two successive designs. Design $\#1$ converts the force F applied upwards to the tip into a bending deformation of the vertical thin beam of thickness h and length L. The force lever arm has a length s. The out-of-plane dimension b (not represented) depends on the glass slide thickness used to produce the samples (see later on). The right side of this beam undergoes traction ($ϵ>0$ in red) while the left side undergoes compression ($ϵ<0$ in blue). Consequently, since the sample is in glass, it can be instrumented with a waveguide and a Bragg grating patterned in the non-zero $ϵ$ area. Plugging an optical fiber in this instrument (aligned with the waveguide) enables its connection to a Bragg spectrum meter for reading the strain level. Thanks to the mechanical model or calibration, this strain can further be converted into the force F applied to the tip. As further explained, this design $\#1$ is subjected to a transverse force T when the tip is applied to or inserted in a substrate.

A second design (design $\#2$) has thus been designed symmetrically. The geometry of these designs is explained in Figure 4 and

Table 1. Geometrical properties as designed and measured for designs $\#1$ and $\#2$.

(All Sizes in $\mathsf{\mu }$m)	$\mathit{b}$	$\mathit{h}$	$\mathit{L}$	$\mathit{s}$	$\mathbf{\Delta }\mathit{x}$	$\mathbf{\Delta }\mathit{y}$
Design $\#1$ (as designed)	500	40	600	375	100	n.a.
Design $\#1$ (sample)	498	35	606	378	106	n.a.
Design $\#2$ (as designed)	500	60	1900	500	150	70
Design $\#2$ (as measured, sample $\#4$)	495	59	1905	501	144	68
Design $\#2$ (as measured, sample $\#6$)	499	59	1902	500	143	67

n.a. is not applicable.

.

2.2. Mechanical Design

The mechanical design is based on the well-known Euler–Bernoulli theory, limited to the linear regime. The general sketch and a more precise forces representation are given in Figure A1 for design $\#1$ and Figure A3 for design $\#2$.

In both cases, the stiffness is defined as the ratio between the force F applied to the sharp tip and the resulting displacement ${\delta }_y$ (details are provided in Appendix A and Appendix B)

$$k_1={\displaystyle \frac{F}{{\delta }_y}}\hspace{1em}=\hspace{1em}\frac{EI}{s^{2}L}$$

(1)

$$k_2={\displaystyle \frac{F}{{\delta }_y}}\hspace{1em}=\hspace{1em}\frac{4EI}{s^{2}L}$$

(2)

2.3. Fabrication

2.3.1. Mechanical Fabrication

The raw material is made of fused silica glass slides (Siegert Wafer GmbH, Aachen, Germany $74\times 26\mathrm{m}{\mathrm{m}}^2 \pm 0.1\mathrm{m}\mathrm{m}$ with a thickness $b=500 \mathrm{\mu }\mathrm{m} \pm 20$, Ra < 1 nm). Preliminary characterization reported in [40] indicates, in similar conditions, a Young modulus equal to $E=72\mathrm{G}\mathrm{Pa}$ and an ultimate stress about $\sigma =1\mathrm{G}\mathrm{Pa}$, therefore exhibiting a remarkable elastic coefficient $\sigma /E$ and making glass a unique candidate for compliant mechanisms. Due to its biocompatibility, fused silica can also be used in the medical field.

To shape the compliant mechanism out of the silica glass slide, a femtosecond laser-assisted (wet) etching of fused silica has been used, combining a first femtosecond laser irradiation followed by an etching step in a 12 M KOH solution at 85 °C [40]. The process parameters are those reported in [41] with the × 20 objective: an energy pulse of 230 nJ, a repetition rate equal to 1000 kHz, and a writing speed equal to 950 mm/min (with perpendicular polarization). For these values, ref. [41] reports a differential etching rate equal to $130 \mathrm{\mu }\mathrm{m}/\mathrm{h}$ for the illuminated area against $0.7 \mathrm{\mu }\mathrm{m}/\mathrm{h}$ for the non-illuminated regions. These values are summarized in

Table 2. Summary of the process parameters.

	Energy (nJ)	Speed (mm/min)	Repetition Rate (kHz)	Pitch ($\mathsf{\mu }$m)	Source
Fabrication	230	950	1000	3 (//voxel waist)	[40]
				7 (//voxel height)	[40]
Waveguide	150	50	1000	0.5	Appendix C
Bragg gratings	150	15	1000	0.5	[41]

.

Examples of microscope images for designs $\#1$ and $\#2$ are shown in Figure 5.

2.3.2. Waveguides

The production of waveguides in fused silica with a femtosecond laser has been described in [41]. The exact values of the machine parameters (energy pulse, writing speed, and pitch) have been optimized (with the method of design of experiments) as given in Appendix C. As a result, the combination of parameters that maximizes the difference in refractive index is an energy of 150 nJ, a writing speed of 50 mm/min, and a pitch of $0.5\mathrm{\mu }\mathrm{m}$.

2.3.3. Bragg Gratings

The process parameters towards Bragg gratings inscription are also based on [41]. The energy pulse has been adapted to the same 150 nJ as for the waveguides. The Bragg period was chosen $\Lambda =1.1\mathrm{\mu }\mathrm{m}$, corresponding (for an order $m=2$) to a reflection spectrum centered on

$${\lambda }_{\mathrm{Bragg}}={\displaystyle \frac{2n_{\mathrm{eff}}\Lambda }{m}}=1589.5\mathrm{n}\mathrm{m}$$

(3)

and consequently a relationship between the shift $\Delta {\lambda }_{\mathrm{Bragg}}$ and the strain $ϵ$ given by

$$\Delta {\lambda }_{\mathrm{Bragg}}={\lambda }_{\mathrm{Bragg}}ϵ(1-p_e)=1.24ϵ$$

(4)

where $p_e=0.22$ is the photo-elastic coefficient of fused silica and the value 1.24 obtained for $ϵ$ expressed in $\mathrm{p}\mathrm{m}/\mu ϵ$.

2.4. Characterization

2.4.1. Mechanical Analysis Towards the Stiffness

The stiffness was measured on a home-made setup shown in Figure 6 and Figure 7. The glass structure (on the left) is mounted on a manual stage whose position is tracked with a non-contact displacement sensor (Keyence LC-2440, Osaka, Japan). An external force is applied to the mechanism tip by contacting it with a high-carbon spring steel cantilevered beam (Precision Brand 09740-20 Piece Metric Steel Feeler Gage Poc-kit Asst 12.7 mm × 127 mm Blades, Downers Grove, IL, USA) with calibrated stiffness $k_{\mathrm{cal}}$. Its displacement (hence the applied force) is measured with a second non-contact displacement laser. The applied force is given by

$$F=k_{\mathrm{cal}}{\delta }_{\mathrm{blade}}$$

(5)

and the corresponding displacement ${\delta }_y$ of the glass tip is given by

$${\delta }_y={\delta }_{\mathrm{sensor}}-{\delta }_{\mathrm{blade}}$$

(6)

Consequently, the stiffness of the compliant designs is given by

$$k_{1,2}={\displaystyle \frac{k_{\mathrm{cal}}{\delta }_{\mathrm{blade}}}{{\delta }_{\mathrm{sensor}}-{\delta }_{\mathrm{blade}}}}$$

(7)

2.4.2. Optical Analysis Towards the Bragg Shift

First, the coupling and propagation losses were estimated as follows: a dedicated glass slide was prepared including an input and output made of alignment cavities and elastic clamps, connected by a 10.8 mm long waveguide which is about 2 times larger than the travel length between the beginning of the waveguide and the position of the Bragg grating in the final sensor (Figure 8). The input was connected to an optical source EXFO model FLS2200, Eastleigh, UK (1 mW or 0 dBm) and the output to a power meter (AUA-9 Power Meter 850–1625 nm FC-PC). With the selected process parameters (speed = 50 mm/min, energy = 150 nJ, and pitch = 0.5$\mathrm{\mu }\mathrm{m}$), the transmitted power was equal to 0.070 mW (−11.54 dBm). This measure confirms the inscription of the waveguide (in the absence of a waveguide in the glass slide, the transmitted power is 30 times smaller) and the choice of the writing speed at 50 mm/min (by comparison, a waveguide printed with 30 mm/min only transmits 0.061 mW).

Two different samples of design $\#2$ (labeled 4 and 6) were then connected to a Bragg interrogator (brand fibersensing featuring a tunable laser source, a photodetector, and a circulator to acquire the reflected amplitude spectrum from the Bragg grating) with a Thorlabs GF1 optical fiber (1500–1600 nm with a cladding diameter equal to $125\mathrm{\mu }\mathrm{m} \pm 1.5$). A standard single-mode optical fiber pigtail makes the connection between the interrogator and the instrumented compliant structure. Index matching gel G608N3 (Thorlabs, Newton, NJ, USA) has been used at the interface between the optical fiber and the waveguide.

To assess the robustness of the coupling, the linear power peak of the reflected Bragg spectra was compared for zero load, reporting 12.9 and 13.2 $\mathrm{\mu }\mathrm{W}$ for the sample 4 ($P=13.05\mathrm{\mu }\mathrm{W} \pm 0.21$) and 16.5, 21.4, 21.9, and 21.5 µW for the sample 6 ($P=20.33\mathrm{\mu }\mathrm{W} \pm 0.25$).

The samples were then submitted to a force ramp ranging from 0 to about $350\mathrm{m}\mathrm{N}$ while acquiring the full reflected spectrum for each sample and for each applied force (27 spectra acquired in total). The first step of the analysis is to locate the wavelength ${\lambda }_{\max }$ corresponding to the power peak (linear units in mW), and to determine a threshold power level for step 2, corresponding to the fraction $\eta$ of the maximal power (Figure 9). This cut-off enables us to get rid of the noise far away from the peak. The chosen value is $\eta =0.48$, minimizing the noise level for the final force resolution (see Appendix D).

Because a second emerging peak is observed in many cases (likely to be the signature of bi-refringence), the position of the peak does not reflect the Bragg shift correctly. The second step, therefore, consists of calculating the wavelength ${\lambda }_{\mathrm{COG}}$ of the gravity center of the spectrum portion exhibiting powers above $\eta P_{\max }$ (Figure 10). When the spectrum is perfectly symmetric, ${\lambda }_{\mathrm{COG}}={\lambda }_{\max }$ (Figure 11).

Applying an external force to the mechanical structure varies the strain and, therefore, the reflected wavelength. Hence, the difference between identified peaks’ wavelengths leads to the peak shift (in nm or pm).

3. Results

3.1. Production of an Instrumented Compliant Mechanism and Its Optical Connection

The manufacturing of parts, waveguides, and Bragg gratings was achieved according to the process parameters reported in Table 2. Figure 12 (left) gives an example of symmetric deformation of design $\#2$ under a typical axial load. Figure 12 (right) shows the compliant mechanism and an optical fiber positioned in front of the waveguide, thanks to alignment pockets (for rough positioning) and the combination of a mechanical reference and an elastic clamp for the fine positioning. An index-matching gel is used at the tip of the optical fiber in front of the waveguide (the latter cannot be seen in this figure).

3.2. Mechanical Stiffness of Both Designs $\#1$ and $\#2$

Figure 13 shows the force-displacement characteristics of design $\#1$. Using Equation (1) and the geometrical data of Table 1, the stiffness model provides a stiffness shown by the solid red line, equal to $k_{1,\mathrm{th}}=1.48\mathrm{k}\mathrm{N}{\mathrm{m}}^{-1} \pm 0.43$ (considering $3\mathrm{\mu }\mathrm{m}$ error on the lengths and $1\%$ on E). The black points are the measures, whose linear fit leads to an experimental stiffness equal to $1.24\mathrm{k}\mathrm{N}{\mathrm{m}}^{-1} \pm 0.03$. The gray strip represents the prediction interval. Both theoretical and experimental estimations fairly overlap one another.

Similarly, for design $\#2$, results are shown in Figure 14 using Equation (2). The stiffness is measured to be $4.86\mathrm{k}\mathrm{N}{\mathrm{m}}^{-1} \pm 0.16$ for sample # 4 and $4.33\mathrm{k}\mathrm{N}{\mathrm{m}}^{-1} \pm 0.23$ for sample # 6. All these values are in line with the theoretical predictions.

3.3. Optomechanical Results for Design $\#2$

While design $\#1$ is necessarily subject to a transverse force T generated by the indented medium due to the lateral displacement of its indentation tip (which may modify the expression of the stiffness $k_1$), design $\#2$ is free of transverse force solicitation for perfectly aligned force F (shown in Figure 15a). However, even with design $\#2$, a small angular misalignment between the sensor axis and the loading direction may occur, and this may affect the response of such a compliant structure, deforming the sensor along an additional translational mode very close to the primary one in terms of stiffness (shown in Figure 15b). A countermeasure for the future could therefore foresee a waveguide (and its Bragg grating) in both the left and right arms of design $\#2$, in such a way that the sum of left and right strain signals could lead to F and the difference in signals could lead to T (see complements in Appendix B). This approach is not possible with design $\#1$, and, therefore, only design $\#2$ has been considered in the following.

In addition to the limits of this work, transverse bending (Figure 15c) and torsional (Figure 15d) deformation should also be considered. These modes are clearly stiffer as estimated from numerical mode analysis (the transversal mode exhibits a resonance frequency 2.5 times higher than the targeted working mode, and the torsional mode 3.8 times larger). The torsional mode is unlikely to be activated since it requires a lever arm, which would vanish when manufacturing the tip as a sphere with (almost) punctual contact. Still, a more detailed study should be carried out, especially concerning the transverse bending mode.

For design $\#2$, two different replicates of the sensor have been used (samples 4 and 6) in two different days. The first replicate was tested twice, while the second replicate was tested four times.

During a given test, the structure was gradually deformed by imposing an increasing tip displacement ${\delta }_y$, further converted into the corresponding strain $ϵ$ located on the Bragg grating, close to the cantilevered side (point O) at a distance from the neutral axis designed to be x_Bragg,th $=12.5\mathrm{\mu }\mathrm{m}$ (see Section 4 and the discussion on sources of errors for further explanations on x_Bragg,th):

$$ϵ={\displaystyle \frac{x_{\mathrm{Bragg},\mathrm{th}}{\delta }_y}{sL}}$$

(8)

where s and L are the values measured for samples 4 and 6 as indicated in Table 1. For each applied load, the optical spectrum was processed as indicated in Section 2.4.2, and the wavelength shift could be evaluated. The linear relationship between the shift (in pm) and the strain (expressed in microstrain, the standard scale in the field expressing $0.1\%$ deformation as 1000 microstrain) is a property of Bragg reflection, and the slope is a material property, i.e., $1.24\mathrm{p}\mathrm{m}/\mu ϵ$ for glass. Initially considering x_Bragg,th $=12.5\mathrm{\mu }\mathrm{m}$ as designed, we found a slope of $0.82\mathrm{p}\mathrm{m}/\mu ϵ$. Assuming an effective value x_Bragg,eff different from x_Bragg,th $=12.5\mathrm{\mu }\mathrm{m}$, x_Bragg,eff has therefore been fitted to match the theoretical slope for the glass, equal to $1.24\mathrm{p}\mathrm{m}/\mu ϵ$.

A different fit is led for each trial (see Figure 16 and details in Appendix E), leading to the value x_Bragg,eff $=8.3\mathrm{\mu }\mathrm{m} \pm 0.4$, which was further used in Figure 17, since we have no other experimental mean to assess the true value of $x_{\mathrm{Bragg}}$.

Next, the displacement ${\delta }_y$ was converted into the corresponding applied force using the stiffness $k_2$:

$$F=k_2{\delta }_y$$

(9)

The wavelength shift could then be plotted against the force as shown in Figure 18. The six different series of measures fairly collapse along the linear regression line, and the corresponding residuals are shown in Figure 19. On this latter figure, it can be shown that these residuals are not distributed randomly as expected when all relevant information has been extracted from the fit. The residuals instead grow quadratically with the applied force. We therefore limited our analysis to applied forces below $250\mathrm{m}\mathrm{N}$ since this threshold seems to be a turning point in the evolution of these residuals.

Let us, however, observe that up to now, the analysis has been based on the pooling of results obtained with two different sensors across six repetitions. If this provides a conservative and reliable estimation of the sensor’s performances, it can, however, be noted that in the best series of measures (${19}_7\#4$), the force error is only $5\mathrm{m}\mathrm{N}$, which is lower than 3% of the $180\mathrm{m}\mathrm{N}$ full range (Figure 20).

4. Discussion

The first source of error identified in this paper is the geometrical error measured with the microscope. On top of a typical uncertainty of $1\mathrm{\mu }\mathrm{m}/\mathrm{pixel}$, the parasitic etching illustrated in Figure 21 induces an additional error. Illuminated area typically undergoes an etching rate of about $130\mathrm{\mu }\mathrm{m}/\mathrm{h}$ while non-illuminated areas are etched at a slower pace (200× slower, $0.700 \mathrm{\mu }\mathrm{m}/\mathrm{h}$). As a consequence, the etched walls cannot be perfectly vertical (i.e., perpendicular to the glass slide) and rather exhibit a 1/200 slope (exact value is 0.0054). For a slide thickness $b=500\mathrm{\mu }\mathrm{m}$, this means a blurred area of about $1.4 \mathrm{\mu }\mathrm{m}$. We therefore consider $2.5\mathrm{\mu }\mathrm{m}$ uncertainty in the different lengths measured with the microscope.

A second source of error is related to the misalignment of the optical fiber with respect to the waveguide. Indeed, it should be noted that when the mechanism is loaded, the deflection ${\delta }_y$ is measured while the strain $ϵ$ is derived. This calculation depends on the distance between the neutral axis and the actual location of the interaction between the light beam and the Bragg grating. As shown in Figure 22 (A-A’ front view), the alignment groove section exhibits a trapezoidal section instead of a rectangular one (see Figure 23 and details in Appendix F), with a widening of the slit and a bit of undercut. The B-B’ front view in Figure gives a representation of this misalignment with respect to the waveguide (orange rectangle) and the neutral axis (green line). This misalignment is estimated at $3.85\mathrm{\mu }\mathrm{m}$ (see details in Appendix F). Since this misalignment comes from systematic over-etching, as explained, compensation strategies could be implemented in the future, such as geometric pre-corrections based on our understanding of the etching process.

Let us mention that, besides the misalignment, the shape of the waveguide might also play a role. Since the Bragg grating is not made of infinitely small points and since the light beam is Gaussian (i.e., exhibits a spatial distribution of the power), the effective distance $x_{\mathrm{Bragg},\mathrm{eff}}$ between this interaction and the neutral axis can be different from the designed one $x_{\mathrm{Bragg},\mathrm{th}}$.

Two other sources of error should be accounted for, but have not been estimated in this work. The first one arises from Equation (6), since the error on ${\delta }_y$ is the sum of the errors on ${\delta }_{\mathrm{sensor}}$ and ${\delta }_{\mathrm{blade}}$, each of which is a few micrometers. Additionally, not only ${\delta }_y$ or $ϵ$ are inaccurate: the shift estimation depends on the calculation of the Bragg wavelength, which might be impacted by birefringence as introduced in Figure 10. Such a birefringence effect is reported in the literature on fused silica specimens exposed to low-energy femtosecond pulses [42] and on fiber Bragg gratings [43,44]. This could arise from the anisotropy of the laser process: while denoting x the femto-second laser direction and y the light propagation axis, this birefringence could arise from anisotropy of optical properties along x and z. This assumption should be checked using polarized light in the future.

In this study, as well as in the targeted application, the number of cycles imposed on each sensor is supposed to be very low (<10 cycles). However, there is no stress threshold for which a glass mechanism would be 100% safe, similar to fatigue analysis with metals. The failure probability P is rather described with a so-called Weibull distribution. In reference [40] by Amez-Droz in 2024, for the same glass slides and the same manufacturing process, this failure probability $P\left(\sigma \right)$ is given as a function of the mechanical stress $\sigma$ by:

$$P=1-e^{-{(\sigma /{\sigma }_N)}^m}$$

(10)

with ${\sigma }_N=1.95\mathrm{G}\mathrm{Pa}$ and $m=4.86$. With a strain level at about $400\times {10}^{-6}$, this work presents stress levels up to $\sigma =28.8\mathrm{M}\mathrm{Pa}$, leading to a failure probability close to zero (${10}^{-9}$).

Finally, the reasons unraveling the information remaining in the residuals of Figure 19 should be further investigated.

5. Conclusions

This paper reports the integrated fabrication of a monolithic force sensor made of a compliant glass mechanism, an interfacing technique to connect it to an optical fiber aligned with an internal waveguide, and a Bragg grating used to monitor the structure’s strain during mechanical loading. An extensive characterization has been achieved (geometry, stiffness, waveguide parameters, shift-force response, shift-strain response). The result is a force sensor with a range of about $250\mathrm{m}\mathrm{N}$ with a resolution of ±26 $\mathrm{m}\mathrm{N}$. A less conservative point of view on the same data can also be given while considering the ${19}_7\#4$ series, which exhibits a force error of $5\mathrm{m}\mathrm{N}$, which is lower that 3% of the $180\mathrm{m}\mathrm{N}$ force range.

The perspectives are twofold. First, the resolution should be further increased by improving the reflected spectrum quality. Second, the shape of the indentation tip should be produced into classical geometries (cones, pyramids), allowing the application of post-processing methods relevant for viscoelastic soft biomaterials [45,46,47]. Obviously, the work reported in this paper is only a proof-of-concept on this long journey, and a much larger set of measures should be achieved to lift the statistical limits of the current dataset.