Regularities of Signal and Sensitivity Variation of a Reflection Fiber Optopair Sensor Dependent on the Angle between Axes of Fiber Tips

Abstract. Regularities of variation of output signal U of one reflection fiber optopair dependent on the distance h between active fiber tips and light reflecting body-mirror (U -h characteristics) and on the angle 2θ between the FAT axes, the distance between FAT being minimal ( b = bmin), have been explored by modelling and experimentally. The parameters of U -h characteristics have been established: maximal sensitivi ty Smax(h), localization and values of maxima and inflection points (+,−) of a function U(h), length of interval∆h in which the output signal U(h) is linear (98 % of sensitivity maxima), as well as dependences of these parameters on the an gleθ d distanceh. It has been shown that the experimental results are well described by the formulas improved by the authors previously. It has been demonstrated that reflec tion fiber optopair sensitivity Smax to displacement considerably increases with an increase of the angleθ. It has been defined that, with an increase of the angle θ up to20◦, sensitivity increases up to 30 times when active fiber tips axes are almost parallel and the diamet er of he fiber core is100 μm, and125 μm with cladding. Apart from that, Smax increases almost exponentially up to θ = 20◦. A drawback of such an RFP is that with an increase of the angle θ, the size of the sensor head also increases. However, due to their consid erably increased sensitivity, they can be and are wide used.


Introduction
Reflection fiber optic sensors are widely used in measurements.They are usually comprised of a parallel fiber tip optopair [1][2][3][4].Fiber optopairs are frequently used the active fiber tips (FAT) axes of which form an angle 2θ [4][5][6][7][8][9][10].Unfortunately, no analysis has been made so far in order to decide when it is reasonable to use one or another reflection fiber optopair (RFP), in which FAT axes form an angle θ (0 ≤ θ < 90 • ), b = b min , and to determine the regularities of U -h characteristics (UHC) and sensitivity variation depending on the distance h to the light reflecting body and on the angle θ.

Experimental set-up
RFP U -h characteristics (UHC) were measured by using a circuit shown in Fig. 1b.Fibers A, L (WF 100/125 P 0.22) were installed in SMA 905 connectors.The angle between FAT axes is 2θ.The maximal output signal and minimal distance b min were controlled by a special mounting desk (Fig. 2).SMA emitter (H22E4020IR), of 15 dBm power, λ max = 850 nm and a stabilized current supply 80 mA were used.Output power of the sensor was measured by a precise fiber emission gauge LP-5025-8.Fibers A, L and the mirror (Au) were fastened on a precise xyz positioning device under a microscope.The positioning step was controlled by an electronic device and, in addition, by a micrometer (±0.5 µm).The angle 2θ between FAT axes was defined by the microscope (Fig. 2) scale indices.Fig. 1. a -Arrangement of fiber tips in a measuring head.b -Measuring scheme of an optopair output signal.LP-5025-8 is a fiber emitting light power meter.SMA-5 is a connector.Emitter -H22E4020IR.L is a light emitting fiber.A is a light receiving fiber.h is the distance to the mirror.h0 denotes the peak position of U -h characteristics.h ′ 0 is the distance when fiber tips touch the mirror.

Modelling and experimental results
In order to determine UHC in theory, mathematical model was applied [6,11].If the distance between the centers of fiber tips A and L is minimal, i.e., b min (θ) = 2a cos θ (Fig. 1b), then the UHC of a sensor (a signal emerging in fiber A) is expressed by the function U (hθ): here where a is the radius of fiber cladding, a 0 is the radius of fiber core, and C 0 , k, m are constants defined in the experiment.In a particular case, where θ = 0 • , formula (1), becomes (because x(h) = −2a and z(h) = 2h) Experimental UHC are presented in Fig. 3. Measurements were taken at the angles θ between FAT axes 0 • , 25 • , 34.4 • and 44 • .
UHC curves have a single maximum each the value of which is increasing and its position h max is decreasing with an increase of the angle θ.The part of higher sensitivity of the curve is before reaching the maximum and of lower sensitivity after it.As shown by the experimental results, the part of higher sensitivity of the curve is decreasing with an increase of the angle between the FAT axes and it vanishes at the angle θ larger than 40 • , therefore we have the experimental points of decreasing values that depend on h.
As seen from Fig. 3, the modeling curves are well congruent with the experimental results.Therefore we can establish the regularities of dependence of the UHC on the angle θ by means of simulation.
In the modeling curves the part of higher sensitivity of a curve vanishes at shorter distances h.This is conditioned by the fact that in modelling the light out of emitting fiber is assumed to be propagated as out of a point source, the position of which is coincident with the intersection point of the fiber axis and the tip plane.This point (Fig. 1b) is as far from the mirror plane as the distance h − 2a sin θ.Only the first point of experimental curves can be measured via such a distance.Therefore the first point of experimental curves is a distant as h ′ 0 = a sin θ that is a distance when the mirror touches the fiber tips (Fig. 1b).This fact is defined by observing via microscope.The main parameters of UHC are presented in Figs. 4, 6, 7 and 8.
The principal parameter of RFP U -h characteristics that determines metrological abilities of displacement sensors is sensitivity to displacement S max (Fig. 6).In addition, (Fig. 8) illustrates variation of the dependence of the inflection point (−) position h − (curve 1), peak position h p (curve 2), inflection point (+) position h + (curve 3), peak signal value U p (curve 4), values U − of the inflection point (−) (curve 5) and that of inflection point (+) (curve 6) on θ.The length of interval ∆h in which the output signals U (h) are linear (98 % of sensitivity maxima), and their dependence on the angle θ are demonstrated in Fig. 7. RFP sensitivity S max dependence on the distance h is shown in Fig. 4. As illustrated by calculation results, RFP U -h characteristics have higher positive sensitivity peak S max+ and a lower negative sensitivity peak S max− .The positions of sensitivity peaks are congruent with the inflection points h + and h − of UHC.At these points there is the highest measurement sensitivity and the percentage interval of linearity.) and that calculated by the authors, and by [9] in receiving fiber A.
Fig. 7 illustrates that the linearity interval is increasing with a decrease of the angle θ between the FAT axes, while the sensitivity S max diminishing.It has been established that RFP sensitivity is increasing almost exponentially from 0 • to 40 • with an increase of the angle θ (Fig. 6).The negative RFP sensitivity is equal to S = 0.149 µW/µm as θ = 5   Experimental and modeling results are well congruent.It has been shown that the experimental research results are well expressed by the above formulas improved by us [3].
We have proved that sensitivity of a fiber pair to displacement S max increases with increase of the angle θ.The sensitivity S max is lowest when fiber active tips axes are parallel (θ = 0 • ).The sensitivity S max of a fiber optopair, the diameter of the fiber core of which is 100 µm (with cladding 125 µm), increases about 30 times after an increase of the angle θ up to 30 • , moreover, is increasing almost exponentially.The position h p of the U -h characteristics peak and that of inflection points h + , h − , as well as the linearity interval are exponentially decreasing with an increase of the angle θ.
The drawback of such a fiber optopair is the fact that with an increase of the angle θ, the size of the sensor head also increases.However, due to their higher sensitivity, they can be and are wide used.

Fig. 6 .
Fig.7illustrates that the linearity interval is increasing with a decrease of the angle θ between the FAT axes, while the sensitivity S max diminishing.It has been established that RFP sensitivity is increasing almost exponentially from 0 • to 40 • with an increase of the angle θ (Fig.6).The negative RFP sensitivity is equal to S = 0.149 µW/µm as θ = 5 • , and to S = 0.750 µW/µm as θ = 25 • , i.e., the sensitivity increases 5 times.The positive RFP sensitivity increases from 0.046 µW/µm to 1.200 µW/µm, i.e., almost

Fig. 7 .
Fig. 7. Length of the interval in which the output signal U (h) is linear (98 % of sensitivity maxima).