This part presents the adopted strategy for the design and the high-precision fabrication method of the sensor. This sensor consists of two fiber-optic probes associated to a 3-dimensional reflective grating of aluminium alloy. The following paragraph introduces the sensor principle.
2.1 Sensor principle
The optical fiber used in this study consists of an electronic system, of fiber-optic probes (assembly of one central emission fiber and four reception fibers placed in periphery, the overall fibers are placed in the same support), along with a mirror reflecting the light. The electronic system consists of a light transmitter, a light emitting diode (LED), which sends light in a wavelength of about 670 nm, with a numerical aperture of 0.17, this numerical aperture of the LED is defined as the sine of the half aperture angle “u” (u=10°) of the light emitting diode. The second component of the electronic system is the photodiode (Ph.D.), having an active detection zone of 1 mm diameter. These two components are integrated into a synchronous detection making it independent of the ambient luminous intensity, which could influence the measures.
The five optical fibers are multimode step index in polymethyl (PMMA) and are assembled in a steel ferrule, having a diameter of 2 mm (OMRON Company [1]).
In the fiber-optic probe, the emission fiber is placed at the center, it has a diameter of ( 486 ± 5) µm and a numerical aperture of 0.46; each of the four reception fibers has a diameter of (240 ± 5) µm, with a numerical aperture identical to that of the emission fiber. In the first study, the reflector has been a flat surface as shown in the following figure.
The functionality principle of the sensor is based on light injection into an emission fiber, which illuminates the planar mirror in motion, relatively to the fiber-optic sensor, then to the detection of the reflected light.
This injected light is guided by the emission fiber to the extremity of the probe , situated in front of the mirror. The emerging light of the probe is reflected by the mirror in translational motion, for which, the flatness of the surface is equal to λ/10 ( λ € [400 nm;700 nm]. A part of the reflected light is injected in the four reception fibers, then guided to the photodiode , which collects this light and then transforms it to an electric current. Having a perpendicular translation of the mirror axis with respect to the surface of the fiber-optic probe , the quantity of the collected light changes, and that provides a variation of the output voltage (Fig. 2 (a), (b)).
Figure (2,b) shows the response curve of the sensor, as it is seen, this curve is formed of four zones. The first zone (zone 1) is the dead zone, where the sensor detects nothing, the mirror is still very close to the probe, and along with a separation of several dozens of micrometers between the emission fiber and the reception fibers , the reflected light of the mirror does not reach the reception fibers. The second zone (zone 2), corresponds to the zone where the sensor starts to detect the reflected light, but, the sensitivity is still very low. The third zone (zone 3) is the most important zone for linear displacements, as it has a high sensitivity with the best resolution; this high sensitivity is only available on a small range, approximately 200 µm, for a linearity criterion of 1 % on the full scale voltage output [0;10 V]. The last zone is zone 4, which illustrates a voltage drop of the sensor output, and that happens when the mirror moves away from the probe , because the reflected light does not totally reach the reception fibers of the probe. So, it is always preferred to work in zone 3, as it is a linear zone along with a high sensitivity [8]. However the measurement range of that zone, for the axial displacement is limited to 200 µm, which is not suitable for applications requiring large strokes. For that reason, and in order to increase this low measurement range, the displacement direction of the flat mirror should be different from the normal vector orientation of its surface resulting in a lateral displacement direction with an angle, defined in this study as ɛ, this new configuration has provided an inclined step with this defined angle (figure 3).
In this configuration of the inclined step, the classical axial measurement range (MRaxial)will increase by a factor of (sin ɛ)-1 , giving a new lateral measurement range (MRlateral) following this equation:
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)
At the same time, the lateral displacement congiguration modifies the values of the sensor sensitivity (Saxial) and its resolution (Raxial), as a function of sin ɛ, which is explained in the following equations:
![](data:image/png;base64,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)
This inclined step, has certain geometric parameters in addition to the angle ɛ, the following figure illustrates the inclined step, in addition to its corresponding geometric parameters:
The following table illustrates the signification of each parameter.
Table 1: Geometric model parameters.
Symbol
|
Quantity
|
Φef (µm)
|
Emission fiber diameter
|
Φ (µm)
|
Probe diameter
|
β
|
Emission fiber numerical aperture
|
d (µm)
|
Distance between probe head & grating
|
dx (µm)
|
Distance : x.sinɛ
|
ds (µm)
|
Security distance
|
ɛ (°)
|
Grating angle
|
x (µm)
|
Lateral position
|
l (µm)
|
Step length
|
h (µm)
|
Step height
|
z (µm)
|
Illuminated zone diameter
|
The sensor performances are influenced by the geometric parameters of the grating. The geometric model considers the optimization of the sensor resolution Rɛ (the resolution has to be as minimum as possible).
![](data:image/png;base64,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)
Where (R) is the limit of resolution of the sensor in the common use.
It can be deduced that ɛ has to be large in order to decrease the sensir limit of resolution, we can also observe that :
![](data:image/png;base64,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)
So, the choice of the step length and height influence tha value of the sensor limit of resolution and measurement range.
This previous measurement configuration described in the previous figure cannot provide millimetric ranges with resolution in the order of tens of nanometers. To increase the range to several millimeters by preserving a good resolution, the inclined mirror configuration was duplicated; this duplication leads to a grating of successive flat mirrors. In the case of a grating of flat mirrors, two fiber-optic probes are used to avoid the transition between two consecutive steps (Figure 5) [7, 8].
The objective of this work is the linear displacement measurement for a rotating axis, such as a spindle, over an extended range (minimum 10 mm), without being disturbed by the rotational motion. For that reason, the sensor reflective grating has to have a 3D shape in order to provide a valid measurement even if the sensor rotates along its axis of symmetry. The schematic diagram of this grating associated to the fiber-optic probes is shown in the following figure.
2.2 The design of the cones-assembled grating
The reflector part of the sensor is a group of successive cones of certain properties and performances [9]. For that reason, a new design was developed to analyze the sensor response to curved reflectors. The research papers illustrated in [10, 11] present a full study of that model; for which, the response curve of the sensor at a fixed radius of curvature (Rc), was obtained.
In the two previous studies mentioned in [10,11], the influence of the curved reflector on the sensor performances was modelled. The surface of the reflector grating is convex. The property of having a convex reflector will modify the sensor performances as compared to the previous sensor with a flat mirror reflector. Different geometric parameters were considered, in order to observe the influence of the convex surface on the sensor performances. These parameters are shown in table 2 and figure 7.
Table 2: Geometric model results.
Symbol
|
definition
|
Ø (µm)
|
The probe diameter
|
Rc (mm)
|
The radius of curvature
|
M
|
The tangential point
|
nM
|
The normal vector to tangent
|
d0 (mm)
|
The initial distance between the reflector and the probe
|
α (°)
|
The incidence angle
|
δ (°)
|
The angle between nM and the symmetrical axis of the probe
|
e (µm)
|
The spacing between the emission fiber and the reception fiber
|
Δ (µm)
|
The spacing between the normal vector of the flat surface and the reflected beam from the curved surface intercepted by the sensor
|
The flat mirror configuration was used as a reference, it was found out that the higher the radius of curvature is, better will be the nomalised light intensity detected in the linear part of the curve, and in consequence, better will be the performance of the sensor. The following figure presents the results of different radii of curvature (Rc) as a function of the displacement (d0).
The geometric model previously described has been validated experimentally by fabricating two pieces of cylinders with different diameters : piece (A) with small diameters (40, 30, 20, 10 mm) and piece (B) with large diameters (55, 50, 40, 35 mm). These two pieces have been fabricated with a single crystal diamond tool on a high precision turning machine. For each cylinder in the two pieces, the calibration curve of the sensor has been obtained by moving the sensor away from the cylinder reflecting surface. The linear sensitivity has been calculated for each curve with a linearity criterion of 1%. The following figure illustrates the experimental set-up used, it consists of the two reflecting pieces together with a fiber-optic probe.
The experimental sensitivities have been compared to the theoretical sensitivities (zone 3 of figure 2), in order to test the validity of the geometric model (figure 10).
A diameter of 50 mm has been chosen to fabricate the cones’ assembled grating, in order to guarantee a good performance of the sensor. A first conical grating prototype has been fabricated and characterized geometrically.
After that, the long-range displacement operation with the conical grating was simulated with a second geometric design.
The geometric model of the cones-assembled grating provided the geometric parameters necessary for a proper functionality for the sensor.
The following figure illustrates the geometric design for one step of the cones-assembled grating.
The previously described design provided an optimal value for each parameter, as shown in table 3.
Table 3: Geometric model results.
Geometric parameter
|
Numerical value
|
l (µm)
|
1478
|
C (µm)
|
100
|
lmax (µm)
|
1578.1
|
hmax (µm)
|
119
|
hpmax (µm)
|
155.34
|
ε (°)
|
4.60
|
ɣ (°)
|
130
|
For this model, the angle γ was fixed at 130°. This angle should be higher than 90°, in contrary to the grating developed in the previous study, for which the angle was fixed at 90° (figure 4), and this is due to the tool footprint on the surface of the grating , which is suitable for the fabrication process.
The values of the parameter l and hmax were respectively 1478.1 µm and 119 µm. The angle Ԑ and the segment hpmax are deduced geometrically as shown in the following equations:
![](data:image/png;base64,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)
As explained previously, the geometric parameters influence the sensor resolution regarding the resolution, the measurement range and the sensitivity.
The cones-assembled grating has high requirements, regarding the precision of the geometric dimensions and the necessity of very low surface roughness. For those reasons, the grating has been manufactured with a high precision machine and a single crystal diamond tool [12, 13].
2.3 Machining process
The following paragraph illustrates the fabrication procedure, which had been carried out using a high-precision lathe and a single crystal diamond tool on aluminium alloy.
2.3.1 Characteristics of the high precision lathe and the machining tool
The sensor sensitivity and resolution are improved by a high reflectivity factor (R > 95%) of the grating, which depends on a low roughness (Ra < 10 nm) and also of the flatness of each step in the grating. Besides, to ensure the highest linearity of the sensor, the flatness has to be in the order of magnitude of (optical quality), where is the wavelength of the optical source. Finally, the length and the height of each step must be highly reproducible and very close to the geometric values provided by the geometrical model in order to ensure an accurate value of the angle Ԑ. For those reasons, a high precision lathe was chosen, thanks to its adaptability to 3D axis-symmetrical pieces (Figure 12).
This high precision machine is a prototype lathe, designed by SnecmaTM Motor, as shown in figure 5. The two slideways (X-and Z-axis) are guided by hydrostatic-bearings offering high damping and high stiffness; fixed on a massive granite block (1.5 tonnes), which rests on four self-leveling pneumatic isolators. The straightness of both slides is better than 0.3 µm over a displacement of 100 mm. The rotating spindle is fixed on the Z-axis, along with magnetic bearings [12].
Despite the high performances of the lathe, some defaults have been identified and measured. These residual defaults are due to a small misalignment between the spindle axis of rotation and the Z-axis, on which the spindle is fixed, as well as a small perpendicularity misalignment between the X-axis and the Z-axis. As a consequence, when a cylinder is fabricated, it will have a slight conical effect; with an increasing diameter, apart from the spindle side, the measured angle of this cone is approximately 0.014°. These defaults have been taken into account during the machining process and in the measurement of the geometrical parameters of the fabricated grating.
To get a cylinder with the smallest conical effect, this conical default should be compensated in the following manner: For 10 mm displacement of the Z-axis, 2.5 µm of the X-axis should be shifted. This compensation has been considered while manufacturing the cones-assembled grating.
To obtain a polished-mirror characteristic for the cones-assembled grating (a nanometric roughness), a single-crystal diamond tool, with a small radius of curvature (R=100 µm), was used [14], as well as an aluminum alloy 2017 to fabricate the cones-assembled grating.
2.3.2 The machining technique
To get high surface qualities for the reflector part of the sensor, the depth cut has to be less than 10 µm [12]. The manufacturing technique has two main stages: During the initial phase, the surface of the raw cylinder was dressed and turned. Lubrication with spray was added, which facilitates the cut process, and allows getting high surface roughness.
The second stage is the fabrication of the cones-assembled grating on the cylinder. This phase is done in two steps: firstly, every conical step has been fabricated with seven successive cuts, then, a finishing cut (8th cut) was done to get the corresponding shape of the cones-assembled grating. For each cone, two consecutive trajectories were programmed as previously explained in [9]. The depth of the six first cuts was fixed to 18 µm while it was 10 µm for the seventh.
Figure 13 shows the cones-assembled grating. The eighth cut allowed having a polished-mirror surface for the first five steps, whereas the last five steps which have not yet had the eighth cut are not reflective.
The high precision fabrication technique allowed getting the geometric parameters, obtained from the theoretical design. Then, it has been characterized geometrically using the NanofocusTM µscan optical profilometer (figure 14). The steps profile was measured (figure 15) and compared with the theoretical dimensions (Table 4).
Table 4: Comparison between the theoretical values and the measured values for the geometric parameters of the cones-assembled grating.
|
Total length (µm)
|
Step height (µm)
|
Ԑ (°)
|
Simulation
|
1578.1
|
119.0
|
4.60
|
Measurement
|
1578.5 ±2.8
|
115.3±1.2
|
4.5±0.1
|
It is observed from these results, that the measured values are close to the theoretical ones; the percentage error for the step length is 0.02%, concerning the step height, the failure is 3.11%, and for the angle ε the error is 2.17%.
The surface characteristics for the cones-assembled grating were identified using an interferometric microscope, which gave a roughness value of (51.4 ± 2.7) nm and an average arithmetic roughness (rms) of (63 ±3.7) nm, which is suitable for high surface reflectivity. Figure 16 illustrates the surface characteristics for the first grating step (Ra = 53 nm, arithmetic roughness (rms) = 71 nm).