## Analysis of the use of tapered graded-index polymer optical fibers for refractive-index sensors

Optics Express, Vol. 16, Issue 21, pp. 16616-16631 (2008)

http://dx.doi.org/10.1364/OE.16.016616

Acrobat PDF (547 KB)

### Abstract

The behavior of tapered graded-index polymer optical fibers is analyzed computationally for different refractive indices of the surrounding medium. This serves to clarify the main parameters affecting their possible performance as refractive-index sensors and extends an existing study of similar structures in glass fibers. The ray-tracing method is employed, its specific implementation is explained, and its results are compared with experimental ones, both from our laboratory and from the literature. The results show that the current commercial graded-index polymer optical fibers can be used to measure a large range of refractive indices with several advantages over glass fibers.

© 2008 Optical Society of America

## 1. Introduction

1. J. Villatoro, D. Monzón-Hernández, and D. Luna-Moreno, “In-line optical fiber sensors based on cladded multimode tapered fibers,” Appl. Opt. **43**, 5933–5938 (2004). [CrossRef] [PubMed]

3. S. Xue, M.A. van Eijkelenborg, G. W. Barton, and P. Hambley, “Theoretical, Numerical, and Experimental Analysis of Optical Fiber Tapering,” J. Lightwave Technol. **25**, 1169–1176 (2007). [CrossRef]

5. J. Zubia and J. Arrue, “Plastic Optical Fibers: An Introduction to Their Technological Processes and Applications,” Opt. Fiber Technol. **7**, 101–140 (2001). [CrossRef]

7. M. Lomer, J. Arrue, C. Jáuregui, P. Aiestaran, J. Zubia, and J.M. López Higuera, “Lateral polishing of bends in plastic optical fibres applied to a multipoint liquid-level measurement sensor,” Sens. Actuators A **137**, 68–73 (2007). [CrossRef]

8. J. Arrue, G. Aldabaldetreku, G. Durana, J. Zubia, and F. Jiménez, “Computational research on the behaviour of bent plastic optical fibres in communications links and sensing applications,” in *Recent Research Developments in Optics* , S.G. Pandalai, ed. (Research Signpost, Kerala, India, 2005), Chap. 5.

1. J. Villatoro, D. Monzón-Hernández, and D. Luna-Moreno, “In-line optical fiber sensors based on cladded multimode tapered fibers,” Appl. Opt. **43**, 5933–5938 (2004). [CrossRef] [PubMed]

2. G. Shangping and S. Albin, “Transmission property and evanescent wave absorption of cladded multimode fiber tapers,” Opt. Express **11**, 215–223 (2003). [CrossRef]

*n*), since all rays escaping from the core into the cladding would also escape from the cladding if the outer medium’s refractive index is greater than or equal to

_{2}*n*. Since

_{2}*n*depends on the material used to manufacture the fiber, which cannot be arbitrarily changed, each fiber will have its own upper bound of measurable refractive indices. In Table 1 we summarize the main characteristics of the commercial fibers studied, where

_{2}*n*is the refractive index of the outer medium,

*ρ*

_{0}is the core radius,

*ρ*is the radius of the cladding,

_{0, clad}*n*

_{1}and

*n*

_{2}are the core and the cladding refractive indices respectively, and

*g*is the refractive index profile exponent [9]. Note that the OM-Giga and the GigaPOF fibers have the same characteristics, despite being from different manufacturers [10

10. Chromis FiberOptics Co., “Chromis Fiberoptics,” (Head office in 6 Powder Horn Dr., Warren, NJ 07059, USA). http://www.chromisfiber.com.

11. FiberFin Inc., “FiberFin,” (Head office in 201 Beaver Street, Yorkville, Illinois, USA.). http://www.fiberfin.com.

12. Asahi Glass Co. Ltd., “Asahi Glass,” (Lucina Division; Head office in 1-12-1, Yurakucho, Chiyoda-ku, Tokyo 100–8405, Japan). http://www.lucina.jp/eg_lucina/productsengf2.htm.

### 2. Implementation of the ray-tracing method

8. J. Arrue, G. Aldabaldetreku, G. Durana, J. Zubia, and F. Jiménez, “Computational research on the behaviour of bent plastic optical fibres in communications links and sensing applications,” in *Recent Research Developments in Optics* , S.G. Pandalai, ed. (Research Signpost, Kerala, India, 2005), Chap. 5.

*α*is only slightly greater than the critical angle

_{i}*α*. On the other hand, when

_{c}*α*<

_{i}*α*, refraction occurs, the Fresnel transmission coefficient

_{c}*T*being approximately 1 if

*α*/

_{i}*α*< 0.9 [8

_{c}8. J. Arrue, G. Aldabaldetreku, G. Durana, J. Zubia, and F. Jiménez, “Computational research on the behaviour of bent plastic optical fibres in communications links and sensing applications,” in *Recent Research Developments in Optics* , S.G. Pandalai, ed. (Research Signpost, Kerala, India, 2005), Chap. 5.

*α*is closer to

_{i}*α*, a precise value of

_{c}*T*is given by a generalized Fresnel transmission coefficient that takes into account the local curvature at the interface,

*T*being less than 1. Therefore, by using

*T*=1 whenever

*α*<

_{i}*α*and

_{c}*T*=0 otherwise, we are obtaining a value of attenuation slightly greater than the real one. Thus, our results provide an upper limit for the attenuation of the tapers considered, serving this value as a conservative reference, in the sense that in practice slightly smaller attenuations can be expected.

### 2.1 Discretization of the eikonal equation

**r**is the position vector of a generic point in the curve,

*s*is its natural parameter (distance from any given point

**r**

_{0}along the ray path),

*n*(

**r**) is the refractive index as a function of the point in space, and ∇

*n*(

**r**) is its gradient. Note also that

*d*

**r**/

*ds*is the unit tangent

**t**to the ray path at

_{unit}**r**. Figure 1 illustrates this.

*n*(

**r**) is sufficiently smooth. At points where

*n*is discontinuous, reflection or refraction occurs as governed by Snell’s law. When an analytic or exact solution for the ray path does not exist, as usually happens, the eikonal equation must be solved numerically in order to obtain the trajectory of the light ray approximately.

*Recent Research Developments in Optics* , S.G. Pandalai, ed. (Research Signpost, Kerala, India, 2005), Chap. 5.

13. A. Sharma, D. Vizia, and A. K. Ghatak, “Tracing rays through graded-index media: a new method,” Appl. Opt. **21**, 984–987 (1982). [CrossRef] [PubMed]

**t**to the ray path as

**t**=

*d*=

**r**/dt*n*

**t**, we finally write:

_{unit}**r**(position vector of the ray at the beginning of its trajectory) and

_{0}**t**

_{0}, which is

*n*(

**r**

_{0}) times the unit tangent

**t**to the ray at

_{unit}**r**. Note that, unlike the eikonal equation, Eq. (3) is very similar in structure to a set of ordinary differential equations

_{0}**u**″=

**f**(

*t*,

**u**) with initial conditions (

*t*

_{0},

**u**

_{0}), but without

*t*appearing on the right-hand side of the equations, and with three components in each equation (

*x*,

*y*and

*z*), which can be handled using matrix notation. Therefore, starting with (

*t*

_{0},

**r**

_{0}), a succession of points (

*t*

_{1},

**r**

_{1}), (

*t*

_{2},

**r**

_{2})… can be generated by means of any of the standard algorithms for solving sets of ordinary differential equations with initial conditions. For our calculations, we have used Runge-Kutta’’s third-order algorithm [13

13. A. Sharma, D. Vizia, and A. K. Ghatak, “Tracing rays through graded-index media: a new method,” Appl. Opt. **21**, 984–987 (1982). [CrossRef] [PubMed]

*h*is the step size chosen to run the algorithm. The smaller

*h*is, the higher the accuracy is, but also the computational cost. Since

*h*cannot be arbitrarily low, it must be selected carefully in order for the global (accumulated) error at the end of ray paths to be acceptable. The selection was made by comparing the numerical solutions obtained with different step sizes for a given ray path. By using a very exact ray path as reference, i.e. one obtained numerically with a very small step size

*h*(close to 10

^{-6}m, with which rounding errors are still negligible using double-precision arithmetic), we found that a value of

*h*on the order of 10

^{-5}m was sufficient for the global error to be negligible in the typical lengths of a few centimeters needed in our studies.

### 2.2 Particular case of a tapered GI fiber with core and cladding

**r**of the ray path. The position along the path is specified by means of a parameter

*p*that determines the fiber cross section where the point of the ray path lies. For the type of geometries studied here,

*p*was chosen to be the coordinate along the fiber symmetry axis (i.e.

*p*=

*z*in Fig. 2). The geometry module also returns some other geometrical parameters, such as the radii of the cross sections of the core and the cladding as functions of

*p*.

15. T.A. Birks and Y. W. Li, “The shape of fiber tapers,” J. Lightwave Technol. **10**, 432–438 (1992). [CrossRef]

*n*(

**r**) and its gradient ∇

*n*(

**r**) (

**grad**

*n*) as functions of the

*x*,

*y*and

*z*coordinates (see also Fig. 1). The surfaces of constant refractive index are represented in Fig. 2 by dotted lines, which are their intersections with the meridional plane. There are three parts that should be considered for the calculation of

**grad**

*n*: the narrowing section of length

*L*, the central part of constant diameter (waist) of length

*L*

_{0}, and the expanding section following it, of length

*L*.

*z*axis at the point where the fiber begins to be tapered, the core radius (interface with the cladding) follows a decreasing exponential given by

*ρ*and

_{0}*ρ*are the core radii of the undeformed fiber and of the waist, respectively, and

_{min}*L*is the length of the section [1

1. J. Villatoro, D. Monzón-Hernández, and D. Luna-Moreno, “In-line optical fiber sensors based on cladded multimode tapered fibers,” Appl. Opt. **43**, 5933–5938 (2004). [CrossRef] [PubMed]

*L*(if the width of the heating zone were varied, a variety of taper shapes could be obtained). The density of the core can be assumed to remain constant after deformation, and the refractive index

_{0}*n*to depend only on the dopant concentration. Under these assumptions, the surfaces of constant refractive index

*n*are, in the 2-D view, exponentials proportional to that of the interface with the cladding. Therefore, if

*d*is the variable distance from a point on the dotted line to the fiber symmetry axis, and

*r*is its initial value at the entrance to the taper, we have:

*n*(

*d,z*) is readily obtainable, since

*n*(

*r*) is a known function (a quasiparabolic profile with exponent

*g*) given by

*n*

_{1}is the maximum refractive index, which takes place at the fiber symmetry axis, and

*n*

_{2}is the minimum one, i.e. that of the cladding [9]. The result is:

*d*=(

*x*

^{2}+

*y*

^{2})

^{1/2}, the three-dimensional gradient of

*n*is obtained as:

*d*for (

*x*

^{2}+

*y*

^{2})

^{1/2}, yields

**grad**

*n*as an explicit function of

*x*,

*y*and

*z*, as needed to solve the eikonal equation numerically.

*n*(

*d*)=

*n*

_{1}(1-2Δ

*ρ*-

_{min}^{g}

*d*

^{g})

^{1/2}, and the procedure is the same as for the narrowing section.

*z*should be substituted by

*z*-(

*L*+

*L*

_{0}).

**n**to the surface at the point where it reaches the interface with the external medium, so as to be able to determine the incidence angle and to compare it with the critical angle of refraction [14].

**n**to the interface can be obtained analytically by taking advantage of the existing axial symmetry. For example, for the narrowing section the result can be obtained in two steps, by first considering the unit normal in two dimensions

**n**

_{2D}, i.e. in the meridional plane, which is given by (see Fig. 3):

**n**

_{2D}is necessary to take the absolute value of the derivative

*dρ*(

*z*)/

*dz*.

**n**can be obtained by rotating

**n**

_{2D}around the fiber axis, since its axial and radial components are equal to those of

**n**

_{2D}, resulting:

**n**

_{2D}also appears in this section, this time in order to take the correct orientation backwards.

### 2.3 Tapers with sinusoidal shape

16. J. Arrue, F. Jimémez, M. Lomer, G. Aldabaldetreku, G. Durana, and J. Zubia “Characterization of tapered, polished or uncladded SI and GI POF geometries for use in tapers and multipoint sensors,” in *Proceedings of 15 ^{th} International Conference on Plastic Optical Fibers and Applications POF’2006* , (Korea, 2006), pp. 187–192.

## 3. Validation of the numerical algorithms and experimental results

### 3.1 Validation of the numerical algorithms by comparison with our own experiments

*narrowing ratio*(

*NR*) the quotient

*ρ*/

_{min}*ρ*(which will be an important parameter in the rest of the paper), we have

_{max}*NR*∈[0.55,0.65] for the 0.6-mm taper and

*NR*∈[0.35,0.55] for the 0.4-mm one. The measurements of their attenuations have been carried out using a low-numerical-aperture light source (

*NA*≈0.16). Table 2 summarizes the experimental and the computational results obtained for both tapers:

### 3.2 Comparison of our numerical results with experimental ones reported in the literature

**43**, 5933–5938 (2004). [CrossRef] [PubMed]

*L*and

*L*, namely

_{0}*L*=

*L*

_{0}ln(1/

*NR*) [1

**43**, 5933–5938 (2004). [CrossRef] [PubMed]

15. T.A. Birks and Y. W. Li, “The shape of fiber tapers,” J. Lightwave Technol. **10**, 432–438 (1992). [CrossRef]

*L*

_{0}, the length

*L*is also conveniently adjusted.

*L*

_{0}, whatever its speed, inasmuch it is much larger than the elongation speed. The length

*L*becomes larger when the narrowing ratio

*NR*=

*ρ*/

_{min,clad}*ρ*decreases [1

_{0,clad}**43**, 5933–5938 (2004). [CrossRef] [PubMed]

*ρ*being the radius of the cladding at the waist (the narrowing ratio applies both to the core and to the cladding).

_{min,clad}*n*on the fraction of power reaching the output end of the sensor, which has been either calculated by using two different methods based on ray-tracing, or measured in the laboratory [1

**43**, 5933–5938 (2004). [CrossRef] [PubMed]

**43**, 5933–5938 (2004). [CrossRef] [PubMed]

*NR*=40/62.5=0.64, which is the highest value used in the measurements, together with our own computational results for the same value of

*NR*, obtained as described in the previous section. Although our aim is to analyze GI-POF tapers instead of glass ones, in this way we have had an experimental reference with which to be able to compare some of our results. Although there were other experimental results available for lower values of

*NR*, such as

*NR*=0.48 or lower, we have not used them in order to ensure a sufficiently high number of modes propagating along the fiber for a reliable application of the ray-tracing method [5,9]. Since the number of modes

*M*inside the core can be approximated by

*M*≈160 at the waist of the taper in the case of the experimental results in [1

**43**, 5933–5938 (2004). [CrossRef] [PubMed]

*NR*=0.64 and

*λ*=850 nm. This number is on the verge of validity of the ray-tracing method; however, the real number of modes propagating inside the fiber at the waist of the taper increases considerably if

*n*<

*n*=1.459, since total internal reflections at the external interface of the cladding serve to guide modes along the fiber. In the case of a POF we will deal with thousands or millions of modes, so the applicability of the method will be out of doubt.

_{2}**43**, 5933–5938 (2004). [CrossRef] [PubMed]

*NR*=0.7 and

*NR*=0.6). These results were calculated by using a numerical integration technique proposed in [2

2. G. Shangping and S. Albin, “Transmission property and evanescent wave absorption of cladded multimode fiber tapers,” Opt. Express **11**, 215–223 (2003). [CrossRef]

2. G. Shangping and S. Albin, “Transmission property and evanescent wave absorption of cladded multimode fiber tapers,” Opt. Express **11**, 215–223 (2003). [CrossRef]

*NR*=0.7 and

*NR*=0.6 with the squared blue line). When

*n*decreases and approaches 1.43, our results agree better with the experimental ones. One of the possible reasons for the agreement not to be complete is that it is impossible in practice to reproduce the exact illumination conditions at the entrance of the device, since these conditions change from one LED to another and depend on the distance from the LED to the fiber, etc. Our results have been obtained with a light source of 10 000 rays initially parallel to the fiber axis and uniformly illuminating its whole input cross section. The length of the waist is

*L*

_{0}=5 mm.

*L*. In Fig. 7(b)

*L*is larger than in Fig. 7(a) (5 mm against 2 mm), but the waist radius is the same in both cases (

*ρ*=20 µm), as well as the optical fiber employed (the glass fiber of Table 1). We can notice that the ray path from the entrance of the taper to the point where it enters the cladding has more oscillations for

_{min,clad}*L*=5 mm (Fig. 7(b)) than for

*L*=2 mm (in both cases it is approximately sinusoidal with a decreasing amplitude). Moreover, in a hypothetical case in which the narrowing section of the taper were much shorter than 2 mm, the ray would tend to reach the cladding without oscillations, because, in such a case, the surfaces of constant refractive index would be almost perpendicular to the ray, and the radial component of the gradient of the refractive index would decrease considerably. As a consequence, the light power attenuation would be much higher. In contrast, for moderate lengths the total refraction loss becomes greater when

*L*increases. This indicates that the amplitude of the oscillations of the ray path does not decrease as rapidly as the diameter of the fiber. As for the expanding section following the waist of the taper, the attenuation tends to be small, because the diameter of the fiber increases with

*z*. Anyway, there can be refracting rays in this section as well. For example, this could be the case of the ray plotted in Fig. 3 for a sufficiently high outer refractive index

*n*.

*L*close to 0) tend to yield high losses, because in such small distances many rays cannot bend sharply enough to avoid the core-cladding interface standing almost perpendicularly to their trajectories. As a consequence, losses decrease as

*L*increases for small values of

*L*(and a given narrowing ratio). However, this tendency does not continue forever, and losses in the tapered section start to increase again when

*L*becomes large enough. Figure 8 helps to explain this effect. It shows two tapered sections with the same narrowing ratio (

*NR*=0.3) but with different lengths

*L*, namely 2 mm and 10 mm. In both cases, the core of the tapered fiber has been depicted together with the trajectories of some light rays. These rays are identical at the entrances of both geometries (the 2-mm taper and the 10-mm one). Therefore, we can compare the number of rays that reach the core-cladding interface in both cases. Let us consider, to facilitate the explanation, that any ray entering the cladding will escape from the fiber (i.e. that the refractive index of the outer medium is high enough). The ray paths have been obtained numerically (considering that the fiber is the Lucina of Table 1). In this way, we can notice that 5 of the rays considered escape when

*L*=2 mm, whereas this number is greater (7) when

*L*has been increased to 10 mm. Although the depicted rays have been chosen arbitrarily, the fact that losses can increase when

*L*increases is not a coincidence. In fact, we can notice that the rays that escape are those whose initial distance to the fiber symmetry axis is greater than a certain threshold value for each taper, which establishes the limit between escaping or not. This distance is smaller in the 10-mm taper considered than in the 2-mm one, and it would decrease slowly if we chose larger values of

*L*(although the rate of change would tend to decrease). The dashed lines in Fig. 8 also help to understand what is happening.

*L*

_{0}is not very important if it is large enough. This can be easily explained: because of the translational symmetry in the straight section, the trajectory of each ray becomes periodical in shape [9], so the ray will be lost by refraction within one period, or never.

## 4. Sensor design with a commercial GI POF

10. Chromis FiberOptics Co., “Chromis Fiberoptics,” (Head office in 6 Powder Horn Dr., Warren, NJ 07059, USA). http://www.chromisfiber.com.

11. FiberFin Inc., “FiberFin,” (Head office in 201 Beaver Street, Yorkville, Illinois, USA.). http://www.fiberfin.com.

12. Asahi Glass Co. Ltd., “Asahi Glass,” (Lucina Division; Head office in 1-12-1, Yurakucho, Chiyoda-ku, Tokyo 100–8405, Japan). http://www.lucina.jp/eg_lucina/productsengf2.htm.

*L*

_{0}=5 mm for the length of the waist, which is a value that has already been reported in the manufacturing process [1

**43**, 5933–5938 (2004). [CrossRef] [PubMed]

*NR*and the outer refractive index

*n*accompanying its corresponding critical angle of refraction

*α*. This will facilitate the comparison with the Lucina fiber, since its value of

_{c}*n*

_{2}is different, which means that the same value of

*α*corresponds to a different value of

_{c}*n*, namely

*n*=

*n*

^{2}sin(

*α*). Therefore, the corresponding refractive indices in Fig. 9 range from 1.292 to 1.492. On the other hand, by linearly spacing the critical angle instead of the corresponding refractive index, the differences between the slopes of the curves of constant

_{c}*NR*are seen more clearly, since these curves would be nearly vertical for

*n*=

*n*

_{2}if the refractive indices were linearly spaced. The slope of the curves of constant

*NR*is smaller when

*NR*is very close to 1, as will be shown more clearly in Fig. 10. In all cases, the light source employed in the computational simulations is the one explained in the previous section (parallel rays uniformly illuminating the whole cross section).

*α*=90°, there are losses for any value of the narrowing ratio, except for

_{c}*NR*=1, since the light rays whose turning points [9] are initially very close to the core-cladding interface will eventually refract into the cladding and escape for sufficiently large values of

*L*. This is so because the radius of the taper decreases more rapidly than the distance to the fiber axis of the successive turning points.

*NR*. This fact is more clearly seen in Fig. 10, which also serves to compare the behavior of the OM-Giga/GigaPOF fibers with that of the Lucina.

*P*/

_{out}*P*). For example, for

_{in}*NR*=0.4 and

*α*=88°,

_{c}*P*

_{out}*/P*=0.32 for the OM-Giga/GigaPOF fibers, whereas

_{in}*P*/

_{out}*P*=0.20 for the Lucina fiber.

_{in}*P*/

_{out}*P*=0.31, which almost coincides with the result corresponding to the OM-Giga/GigaPOF.

_{in}*P*

_{1}on the first contour line) until it reaches the outer surface of a thin cladding (point

*P*

_{2}on the second contour line) or, alternatively, of a thick one (point

*P*

_{3}on the third contour line). Since a ray within the cladding propagates following a straight line, its angle of incidence onto the interface between the cladding and the outer medium depends only on the normal to it (

**n**

_{2}or

**n**

_{3}). Figure 11 shows an angle of incidence

*α*that is typically smaller than

_{3}*α*(for the geometry represented,

_{2}*α*=71.3° and

_{2}*α*=67.4°). Therefore, a ray that would reflect at

_{3}*P*

_{2}if the cladding were narrow, could refract and get lost at

*P*

_{3}if the cladding were thicker. This turns out to be the dominant effect explaining the higher losses for thicker claddings.

*NR*influences the range of measurable refractive indices with each type of fiber. For example, if

*NR*=0.4 we can measure values from

*n*=1.34 (

*α*=90°) to

_{c}*n*=1.10 (

*α*=55°), whereas, in the case of the OM-Giga/GigaPOF fibers, the range is smaller and extends from

_{c}*n*=1.492 (

*α*=90°) to

_{c}*n*=1.352 (

*α*=65°). If the narrowing ratio were 0.7 instead of 0.4, we could only measure refractive indices down to 1.469 (

_{c}*α*=80°) in the case of the OM-Giga/GigaPOF fibers, and down to 1.259 (

_{c}*α*=70°) in the case of the Lucina fiber. The maximum measurable refractive index is always

_{c}*n*=

*n*

_{2}. Therefore, the range of measurable refractive indices can be adjusted at will up to n2, although the narrowing ratio should not be too close to 1 for losses to occur. Note that, if the range of attenuations does not change much, any increase in the range of the measurable refractive indices must be achieved at the expense of the sensitivity of the sensor to variations of

*n*, and vice-versa.

18. J. Zubia, J. Arrue, G. Fuster, and D. Kalymnios, “Light power behavior when bending plastic optical fibers,” IEE P-Optoelectron. **145**, 313–318 (1998). [CrossRef]

**43**, 5933–5938 (2004). [CrossRef] [PubMed]

*NR*=0.7 and a Lucina fiber, the range of refractive indices extends from 1.34 (

*α*=90°) to 1.259 (

_{c}*α*=70°), so the difference is Δ

_{c}*n*(

*Lucina,NR*=0.7)=0.081, whereas, in the case of a glass fiber with

*NR*=0.64 only, the difference is Δ

*n*

_{(glass,NR=0.64)}=0.029 (see the computational curve in Fig. 6). As for the OM-Giga/GigaPOF fibers, the range is not so large because the cladding is much thinner, as explained before. In any case, the OM-Giga/GigaPOF could be preferred over the Lucina fiber when the advantage of having a large fiber diameter is an important issue, since their diameter is larger than that of the Lucina and, to a greater extent, larger than that of the conventional glass GI fibers. Notice that, even in the hypothetical case of having two fibers with the same geometry, profile exponent and numerical aperture, the maximum measurable refractive index would be different if the cladding is of a different material (

*n*

_{2}is not a parameter that can be changed for a given material).

3. S. Xue, M.A. van Eijkelenborg, G. W. Barton, and P. Hambley, “Theoretical, Numerical, and Experimental Analysis of Optical Fiber Tapering,” J. Lightwave Technol. **25**, 1169–1176 (2007). [CrossRef]

20. M. Kezmah and D. Donlagic, “Multimode all-fiber quasi-distributed refractometer sensor array and crosstalk mitigation,” Appl. Opt. **46**, 4081–4091 (2007). [CrossRef] [PubMed]

*NR*. For example, for

*NR*=40/62.5=0.64 the difference between the maximum and the minimum measurable values of

*n*in an etched glass taper is Δ

*n*

_{(glass,etched)}=0.035, whereas this difference is smaller with a stretched glass taper, namely, Δ

*n*(

*glass,stretched*)=0.029. All in all, in both cases, the range is much smaller than that obtained by us for the Lucina fiber with the same narrowing ratio, which is Δ

*n*

_{(Lucina,NR=0.64)}=0.13.

## 5. Conclusions

## Acknowledgments

^{th}Research Framework Programme, under projects TEC2006-13273-C03-01, GIU05/03 and EJIE07/12, HEGATEK-05 and SHMSENS, S-PE07CA05, and AISHAII, respectively.

## References and links

1. | J. Villatoro, D. Monzón-Hernández, and D. Luna-Moreno, “In-line optical fiber sensors based on cladded multimode tapered fibers,” Appl. Opt. |

2. | G. Shangping and S. Albin, “Transmission property and evanescent wave absorption of cladded multimode fiber tapers,” Opt. Express |

3. | S. Xue, M.A. van Eijkelenborg, G. W. Barton, and P. Hambley, “Theoretical, Numerical, and Experimental Analysis of Optical Fiber Tapering,” J. Lightwave Technol. |

4. | O. ZiemannH. PoiselA. BachmannJ. VinogradovSpecial problems measuring POFin |

5. | J. Zubia and J. Arrue, “Plastic Optical Fibers: An Introduction to Their Technological Processes and Applications,” Opt. Fiber Technol. |

6. | J. Munisami and D. Kalymnios, “High NA POF performance versus the requirements of the recent standard ISO/IEC JTC 1 FDIS 24702,” in |

7. | M. Lomer, J. Arrue, C. Jáuregui, P. Aiestaran, J. Zubia, and J.M. López Higuera, “Lateral polishing of bends in plastic optical fibres applied to a multipoint liquid-level measurement sensor,” Sens. Actuators A |

8. | J. Arrue, G. Aldabaldetreku, G. Durana, J. Zubia, and F. Jiménez, “Computational research on the behaviour of bent plastic optical fibres in communications links and sensing applications,” in |

9. | A. W. Snyder and J. D. Love, |

10. | Chromis FiberOptics Co., “Chromis Fiberoptics,” (Head office in 6 Powder Horn Dr., Warren, NJ 07059, USA). http://www.chromisfiber.com. |

11. | FiberFin Inc., “FiberFin,” (Head office in 201 Beaver Street, Yorkville, Illinois, USA.). http://www.fiberfin.com. |

12. | Asahi Glass Co. Ltd., “Asahi Glass,” (Lucina Division; Head office in 1-12-1, Yurakucho, Chiyoda-ku, Tokyo 100–8405, Japan). http://www.lucina.jp/eg_lucina/productsengf2.htm. |

13. | A. Sharma, D. Vizia, and A. K. Ghatak, “Tracing rays through graded-index media: a new method,” Appl. Opt. |

14. | F. Jiménez, J. Arrue, G. Aldabaldetreku, and J. Zubia, “Numerical Simulation of Light Propagation in Plastic Optical Fibres of Arbitrary 3D Geometry,” WSEAS Trans. Math. |

15. | T.A. Birks and Y. W. Li, “The shape of fiber tapers,” J. Lightwave Technol. |

16. | J. Arrue, F. Jimémez, M. Lomer, G. Aldabaldetreku, G. Durana, and J. Zubia “Characterization of tapered, polished or uncladded SI and GI POF geometries for use in tapers and multipoint sensors,” in |

17. | C. McAtamney, A. Cronin, R. Sherlock, G.M. O’Connor, and T. J. Glynn, “Reproducible method for fabricating fused biconical tapered couplers using a CO |

18. | J. Zubia, J. Arrue, G. Fuster, and D. Kalymnios, “Light power behavior when bending plastic optical fibers,” IEE P-Optoelectron. |

19. | J. Mateo, I. Garces, and A. Losada, “A novel technique to fabricate low loss POF tapers,” in |

20. | M. Kezmah and D. Donlagic, “Multimode all-fiber quasi-distributed refractometer sensor array and crosstalk mitigation,” Appl. Opt. |

**OCIS Codes**

(060.2370) Fiber optics and optical communications : Fiber optics sensors

(080.2720) Geometric optics : Mathematical methods (general)

(250.5460) Optoelectronics : Polymer waveguides

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: August 11, 2008

Revised Manuscript: September 21, 2008

Manuscript Accepted: September 22, 2008

Published: October 2, 2008

**Citation**

J. Arrue, F. Jiménez, G. Aldabaldetreku, G. Durana, J. Zubia, M. Lomer, and J. Mateo, "Analysis of the use of tapered graded-index polymer optical fibers for refractive-index Sensors," Opt. Express **16**, 16616-16631 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-21-16616

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### References

- J. Villatoro, D. Monzón-Hernández, and D. Luna-Moreno, "In-line optical fiber sensors based on cladded multimode tapered fibers," Appl. Opt. 43, 5933-5938 (2004). [CrossRef] [PubMed]
- G. Shangping and S. Albin, "Transmission property and evanescent wave absorption of cladded multimode fiber tapers," Opt. Express 11, 215-223 (2003). [CrossRef]
- S. Xue, M. A. van Eijkelenborg, G.W. Barton, and P. Hambley, "Theoretical, Numerical, and Experimental Analysis of Optical Fiber Tapering," J. Lightwave Technol. 25, 1169-1176 (2007). [CrossRef]
- O. Ziemann, H. Poisel, A. Bachmann, and J. Vinogradov, "Special problems measuring POF," in POF Modelling: Theory, Measurement and Application, C.-A. Bunge and H. Poisel, eds., (Books on Demand GmbH, Norderstedt, Germany, 2008).
- J. Zubia and J. Arrue, "Plastic Optical Fibers: An Introduction to Their Technological Processes and Applications," Opt. Fiber Technol. 7, 101-140 (2001). [CrossRef]
- J. Munisami and D. Kalymnios, "High NA POF performance versus the requirements of the recent standard ISO/IEC JTC 1 FDIS 24702," in Proceedings of 15th International Conference on Plastic Optical Fibers and Applications POF�??2006, (Korea, 2006), pp. 102-109.
- M. Lomer, J. Arrue, C. Jáuregui, P. Aiestaran, J. Zubia, and J.M. López Higuera, "Lateral polishing of bends in plastic optical fibres applied to a multipoint liquid-level measurement sensor," Sens. Actuators, A 137, 68-73 (2007). [CrossRef]
- J. Arrue, G. Aldabaldetreku, G. Durana, J. Zubia, and F. Jiménez, "Computational research on the behaviour of bent plastic optical fibres in communications links and sensing applications," in Recent Research Developments in Optics, S. G. Pandalai, ed., (Research Signpost, Kerala, India, 2005), Chap. 5.
- A. W. Snyder and J. D. Love, Optical waveguide theory (Chapman & Hall, New York, 1983).
- Chromis FiberOptics Co., "Chromis Fiberoptics," (Head office in 6 Powder Horn Dr., Warren, NJ 07059, USA). http://www.chromisfiber.com.
- FiberFin Inc., "FiberFin," (Head office in 201 Beaver Street, Yorkville, Illinois, USA.).http://www.fiberfin.com.
- Asahi Glass Co. Ltd., "Asahi Glass," (Lucina Division; Head office in 1-12-1, Yurakucho, Chiyoda-ku, Tokyo 100-8405, Japan). http://www.lucina.jp/eg_lucina/productsengf2.htm.
- A. Sharma, D. Vizia, and A. K. Ghatak, "Tracing rays through graded-index media: a new method," Appl. Opt. 21, 984-987 (1982). [CrossRef] [PubMed]
- F. Jiménez, J. Arrue, G. Aldabaldetreku, and J. Zubia, "Numerical Simulation of Light Propagation in Plastic Optical Fibres of Arbitrary 3D Geometry," WSEAS Trans. Math. 3, 824-829 (2004).
- T. A. Birks and Y. W. Li, "The shape of fiber tapers," J. Lightwave Technol. 10, 432-438 (1992). [CrossRef]
- J. Arrue, F. Jimémez, M. Lomer, G. Aldabaldetreku, G. Durana, and J. Zubia "Characterization of tapered, polished or uncladded SI and GI POF geometries for use in tapers and multipoint sensors," in Proceedings of 15th International Conference on Plastic Optical Fibers and Applications POF�??2006, (Korea, 2006), pp. 187-192.
- C. McAtamney, A. Cronin, R. Sherlock, G. M. O�??Connor, and T. J. Glynn, "Reproducible method for fabricating fused biconical tapered couplers using a CO2 laser based process," in Proceedings of 3rd International WLT Conference on Lasers in Manufacturing, (Munich, 2005), pp. 673-678.
- J. Zubia, J. Arrue, G. Fuster, and D. Kalymnios, "Light power behavior when bending plastic optical fibers," IEE Proc.Optoelectron. 145, 313-318 (1998). [CrossRef]
- J. Mateo, I. Garces, and A. Losada, "A novel technique to fabricate low loss POF tapers," in Proceedings of 9th International Conference on Plastic Optical Fibers and Applications POF�??2000, (Boston, 2000), pp. 72-76.
- M. Kezmah and D. Donlagic, "Multimode all-fiber quasi-distributed refractometer sensor array and cross-talk mitigation," Appl. Opt. 46, 4081-4091 (2007). [CrossRef] [PubMed]

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