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The temperature dependence of the bending loss light energy in multimode optical fibers is reported and analyzed. The work described in this paper aims to extend an initial previous analysis concerning planar optical waveguides, light energy loss, to circular optical waveguides. The paper also presents à novel intrinsic fiber optic sensing device base on this study allowing to measure temperatures parameters. The simulation results are validated theoretically in the case of silica/silicone optical fiber. A comparison is done between results obtained with an optical fiber and the results obtained from the previous curved optical planar waveguide study. It is showed that the bending losses and the temperature measurement range depend on the curvature radius of an optical fiber or waveguide and the kind of the optical waveguide on which the sensing process is implemented.

A curvature effect is easily reached with optical fibers; therefore many laboratories have investigated the effects of curvatures on optical fiber measuring responses. Gratings implemented in optical fibers provide measuring performances related to bending that were also investigated [1-12]. Therefore, the power attenuation coefficient of bent fibers is one of the parameters that must be determined for using the fiber as a transducer.

In previous paper [

In this a previous work [

The purpose of this new work is to extend the analysis of the planar optical waveguide response to the temperature [

An optical fiber is a good example of cylindrical optical waveguide. The optical fiber bending loss phenomena is used as a transduction effect in some types of intrinsic optical fiber sensors (temperature, displacement, strain…) [1,2,13-20].

The step-index optical fiber that we use is curved during its manufacturing at a high temperature. A geometrical modeling is used to describe the light propagation in the optical fiber and to determine the light power attenuation at the output of the fiber due to the fiber bending. In this case the geometrical approach is similar to the one presented for the planar waveguide sensing modeling.

The guidance of the core rays in a straigth step-index optical fiber is achieved by ensuring that the propagation angle q, satisfies the condition:, where the critical angle, q_{c}, and the critical angle, a_{c}, are given respectively, at room-temperature (T_{0} = 20˚C) by:

and. The numerical aperture (NA) is given at T_{0} by:

. n_{1} and n_{2} are the core and the cladding refractive index respectively [

When the optical fiber is bent with a curvature radius R (figure 1), the local numerical aperture at a given location of the optical fiber curved part will be changed. In a meridional-plane of the optical fiber, when the position angle at the beginning of the bend is = 0˚ or 180˚, the optical fiber behavior becomes identical to the one an optical planar waveguide [_{0}, by:

where r is the fiber core radius and is the abscissa on the input optical fiber aperture where the origin is “O”. “r_{0}” will satisfy the relation:. However, in all others optical fiber planes, when and 180˚, the local numerical aperture, , can be calculated by using Equation (1), where the quantity replaces and “” to satisfy the relation:. Finally, the local numerical aperture of a curved step-index optical fiber is given by:

where is required to satisfy the relation:.

Equation (2) becomes identical to equation: when R is infinite (straight optical fiber case) and becomes identical to equation (1) when f = 0˚ or 180˚ (bent step-index optical planar waveguide case). We used the silica/silicone step-index optical fiber with the following characteristics at T_{0}, for a wavelength l = 633 nm: n_{1} = 1.4570, n_{2} = 1401, 2r = 200 µm and NA = 0.4000.

In _{0 }for various values of R, as a function of the position “r_{0}” with a value of f. It is clearly seen that the local numerical aperture increase with the increase of the values of R

and r_{0}. The maximum of the local numerical aperture is obtained when the angle f is equal to 180˚.

The ray paths in the core of a step index fiber are straight lines, but the geometrical description is more complicated than in a planar waveguide, that we analyzed previously [

In the

() relative to ox'y'z'. The

vector OP is given by:

where the distance of the point “P” from the fiber center is r_{0}, if we let r be a point along the ray distant “L” from “P” then:

If the ray meets the torus at “” then we can write, where “” is the position of the first reflection. At “”, we have:

And this gives the angular distance x around the axis of the bent optical fiber curvature:

x_{1} corresponds to angle. From Equation (6), we deduce an equation which will provide, as a function of the position “L”, the intersection of the optical ray with the bent optical fiber core-cladding interface. The smallest real positive solution of the equation represents the distance “L_{1}”.

From the point of the first reflection “” on the optical fiber torus, the optical ray is reflected to another point on the surface of the torus (core-cladding interface) around the bend.

After the first reflection all solutions for the distance “L” to the next reflection are given as solutions of the cubic problem. For obtaining the cubic equation, we must replace r_{0} by r in the quadratic equation.

Some simple coordinates rotations and translations simplify the calculation of the incident and the reflected ray angles at each reflection point [

With: (see Equation (8)).

A rotation around “” will bring the local x-axis tangential to the surface of the torus at. This rotation is z. All the others parameters were defined previously.

Having determined the geometry of the ray path, we can then calculate the fractional power loss at each reflection point along a given path by using the following equation:

where g is the attenuation coefficient of each ray, which varies from one reflection to the next one along the bent optical fiber, it is given by:

where is the angular separation between two successive reflections and N is the total number of reflections. We can then use the Generalized Fresnel’s Law to calculate the transmission coefficient at each reflection point along a given path.

An algebraic expression for the transmission coefficient of the refracted rays (), is given by [3,11,12]:

where and are given by: and

and:.

For tunneling rays when, the transmission coefficient at a reflection point of the radius is given by:

where the parameter “V” is given by:

The parameter is the radius of curvature of the core-cladding interface in the incidence plane at “0”. It is defined by the normal to the interface and the incident ray direction. It is given by [3,11,12]:

where and.

When, the transmission coefficient is givenby:

But there are some optical rays that reach the interface with incidence angles close to.

Finally, the total intensity at the end of the angular length “” of the bent part of the optical waveguide is found by a quadruple summation of Equation (9) across the cross-sectional area (r,f) of the optical fiber at X'X and the distribution of the ray angle (q,y) [3,11,12]:

We plot in

In

skews rays can pass through the regions of high attenuation, although they may initially have a low attenuation.

The optical fiber sensor that we propose is an intrinsic optical fiber sensor; the sensitive element is the curved part of the fiber (

For a given radius (R = R_{0}) of the optical fiber curvature and from equation (2), the local numerical aperture inside this bent portion of the fiber according to the temperature is written in the following way:

where the core refractive index and the cladding refractive index are written according to the temperature (T) in the following way [^{.}

The coefficient and are respectively the thermo-optic coefficient of core the refractive index and the thermo-optic coefficient of cladding the refractive index. K_{1} = –3.78 × 10^{−4}/˚C and K_{2} = 1.7744 × 10^{−5}/˚C.

Silicone-based polymers possess a unique set of properties that makes them highly suitable for optical applications. The excellent thermal stability (−115˚C to 260˚C) allows this material to be useful for high temperature sensing applications [22-26].

In this optical fiber the small positive thermo-optic coefficient effect in inorganic glass waveguides used as the core is canceled out by using the negative thermooptic coefficient of polymers used to constitute the cladding.

For an applied temperature T, we plot in figure 7 the local numerical apertures in an optical fiber according to the value of “”. We observe that the local numerical aperture increases when the applied temperature increases. Consequently, an optical ray unguided at room temperature becomes guided at temperature greater than T_{0}.

In this section, we analyze the effect of the temperature variations on the light propagation in a curved optical fiber. We present the effects of the curvature radius on the optical fiber temperature response. For this analysis, the curvature radius R and the length of the bent part of the optical fiber (x) are given. The refractive index of the core and the cladding of the optical fiber depend on the temperature. The geometrical model is used to evaluate the light output power according to the temperature. The output power at the end of the transducer bent part of the fiber is given by:

For a given curvature length (x = 2pR_{0}), we plot in _{0}.

The response curve of the optical fiber operating as a

temperature sensor that we propose in _{0}) increases when the local numerical aperture is increasing up to a saturation level. Each curve is corresponding to the recovery of the losses induced by the refracted optical rays and the recovery of the losses induced by the tunnel effect, up to saturation. We can say that the temperature sensing range of the temperature optical fiber that we propose depends on the curvature radius of the optical fiber.

We deduce from

• The part of the response curve between Tc1 and T_{0} is corresponding to intensity losses caused only by the temperature effect.

• The part of the response curve between T_{0} and Tc2 is corresponding to intensity losses caused only by the optical fiber curvature.

• The linear region of the optical fiber temperature sensor response is increasing when the curvature radius of the bent fiber is decreasing.

The sensor sensitivity is given by:

where P(R_{0},T) is the output intensity. We deduce from

_{0} = 2 mm. In the liner zone between 20˚C and 180˚C, this sensor has a sensitivity of 0.004˚C^{−1}.

The response to temperature variations on the bending

light power loss of a multimode optical fiber with different bent fiber curvature radii have been analyzed. It has been found that the bending losses due to the internal optical fiber numerical aperture variations increase when the fiber bending angle increase. The more important losses are caused by refraction and tunnel effects. We have shown that a bent optical fiber can be used as a temperature transducer. The use of an optical fiber curved during its manufacturing at high temperature allows to minimize some residual mechanical effects, and allows to use rigorously the geometrical approach to describe the light propagation, to evaluate the losses in light power output values and to calibrate the temperature sensor. We have shown that if we use, for example a silica/ silicone fiber as a transducer, we can obtain good performances with an excellent sensitivity and an excellent linearity associated to a large temperature measurement range, mainly because the thermo-optic effect value on the silicone is negative and important.