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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 24 — Nov. 22, 2010
  • pp: 24536–24555
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A comprehensive analysis of scattering in polymer optical fibers

Gotzon Aldabaldetreku, Iñaki Bikandi, María Asunción Illarramendi, Gaizka Durana, and Joseba Zubia  »View Author Affiliations


Optics Express, Vol. 18, Issue 24, pp. 24536-24555 (2010)
http://dx.doi.org/10.1364/OE.18.024536


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Abstract

The aim of this paper is to investigate the properties of the inhomogeneities that give rise to light scattering in polymer optical fibers (POFs). We perform several measurements in two commercial POFs of identical characteristics: these measurements, based on the side-illumination technique, consist in the detection of the total amount of scattered light guided along a POF sample under different launching conditions and in the acquisition of the corresponding near- and far-field patterns. We carry out complementary computer simulations considering inhomogeneities of different sizes at different positions inside the POF. The comparison of these simulated results with the experimental measurements will provide us with valuable information about the size and placement of the most influential inhomogeneities.

© 2010 Optical Society of America

1. Introduction

In the last few years, polymer optical fibers (POFs) have been widely used both for short-haul communications links, where distances to cover are generally less than one kilometre [1

1. O. Ziemann, J. Krauser, P. E. Zamzow, and W. Daum, POF Handbook: Optical Short Range Transmission Systems (Springer, Berlin, 2008), 2nd ed.

], and for a whole range of different sensing applications [2

2. D. Kalymnios, P. Scully, J. Zubia, and H. Poisel, “POF sensors overview,” in 13th international plastic optical fibres conference 2004: Proceedings, (Nuremberg (Germany), 2004), pp. 237–244.

]. This is mainly due to their robustness, large core diameters, and high numerical apertures, which facilitate handling and light coupling [3

3. T. Kaino, “Polymer optical fibers,” in Polymers for lightwave and integrated optics, L. A. Hornak, ed. (Marcel Dekker, Inc., New York, 1992), chap. 1.

, 4

4. J. Zubia and J. Arrue, “Plastic optical fibers: An introduction to their technological processes and applications,” Opt. Fiber Technol.7, 101–140 (2001), http://dx.doi.org/10.1006/ofte.2000.0355. [CrossRef]

].

If compared to earlier works, one of the main novelties in this paper is that we measure not only the total amount of scattered light intensity that propagates along an step-index (SI) POF sample, but also the experimental near- and far-field patterns obtained for this scattered light. Additionally, we carry out simulations using an elaborate computational model based on the ray-tracing method. The results obtained with such a model will provide us with much more information about the inhomogeneities than if we used other theoretical scattering models available in the literature [13

13. C.-A. Bunge, R. Kruglov, and H. Poisel, “Rayleigh and Mie scattering in polymer optical fibers,” J. Lightwave Technol.24, 3137–46 (2006), http://dx.doi.org/10.1109/JLT.2006.878077. [CrossRef]

]. The main reason is that, whereas the theoretical models only calculate the total amount of scattered intensity carried by rays whose propagation direction lies within the acceptance cone of the fiber, our model is capable of providing a complete description of the scattered power distribution of rays at the output end of the fiber, being such a distribution a key element in the calculation of the numerical near- and far-field patterns. Thus, the results obtained with this computational model will be compared to the experimental results with the purpose of obtaining information about the mean size and the placement of the most influential inhomogeneities that cause light scattering in POFs.

The structure of the paper is as follows. First, the main characteristics of the investigated SI POFs are briefly introduced. Then, the experimental set-up used to measure the total scattered light intensity and the corresponding near- and far-field patterns is described. After that, the experimental results and their most relevant features are presented and discussed. Next, the computer simulations are explicated, to follow with the discussion of the numerical results. Afterwards, the experimental and simulation results are compared. Finally, we summarize the main conclusions.

2. Characteristics of the analyzed POFs

We have measured two commercial poly-methylmethacrylate (PMMA) based SI POFs from different manufacturers: the PGU-CD1001-22-E POF (abbreviated to PGU POF from now on) from Toray [16

16. Toray Industries Inc., “Raytela Plastic Optical Fiber,” http://www.toray.co.jp/english/raytela/index.html.

] and the Super ESKA SK-40 POF (named as SK) from Mitsubishi [17

17. Mitsubishi Rayon Co., Ltd., “Super ESKA Plastic Optical Fiber,” http://www.pofeska.com/.

]. Both fibers share the same physical properties, which are summarized in Table 1.

Table 1. Specifications of the investigated POFs

table-icon
View This Table

There is, however, one difference: the SK POF is a bare fiber, which only consists of the core and the cladding layers, whereas the PGU POF has an additional surrounding cover. Therefore, and for the sake of comparison, we have carefully stripped the cover from the PGU POF, exposing its cladding to the air.

3. Experiment

3.1. Experimental set-up

Figure 1 shows the experimental set-up employed to measure the scattered intensity in the investigated POFs using the side-illumination technique [18

18. M. G. Kuzyk, Polymer Fiber Optics: Materials, Physics, and Applications (Taylor and Francis, Boca Raton, 2007).

].

Fig. 1 Experimental set-up used to measure the scattered intensity in POFs. Legend: POL: linear polarizer; L1: plano-concave lens (f′ = −40 mm); L2: plano-convex lens (f′ = +200 mm); L3: symmetric-convex lens (f′ = +150 mm); L4: symmetric-convex lens (f′ = +50 mm); BS: beam splitter; NDF: absorptive neutral densitive filter; CAM: frame grabber; OBJ: 0.1–NA objective; xy POS: xy–micropositioner; RS: rotation stage; RPD: reference photodetector (with attenuator); RMM: reference multimeter; SPD: signal photodetector (without attenuator); SMM: signal multimeter.

A 633-nm-wavelength and 2 mW He-Ne laser is used as the light source. The polarization of the emitted beam can be selected between horizontal (parallel to the xz–plane) and vertical by means of a Glan-Thompson linear polarizer. With the purpose of reducing as much as possible the final spot size incident on the POF sample, the laser beam is collimated and expanded using lenses L1 and L2. The laser beam crosses a beam splitter in order to obtain a reference signal and cancel the laser light intensity fluctuations. As to the other light beam, it crosses a 0.1–NA objective and impinges on the lateral surface of the POF sample. We have chosen this objective as a trade-off, since it is necessary to produce at the focal plane a spot size as small as possible and, at the same time, the divergence of the beam must be kept as low as possible. This way, we ensure that the volume of fiber illuminated by the laser beam is the lowest one. In this case, the obtained spot size at the focal plane is 6.7 ±0.5 μm (measured at 1/e2 of the maximum value).

The POF sample is held by two xy–micropositioners standing on a rotation stage. On the one hand, we use the xy–micropositioners to change the lateral y–position of the point of incidence on the POF sample and to adjust accordingly the distance of this point of incidence from the objective so as to make it always coincide with the focal plane; this way, we can scan the fiber surface laterally along the xy–plane, which constitutes the first kind of measurement, namely, the lateral scan. On the other hand, the rotation stage, driven by a motion controller, allows us to change the angle of incidence α (also named as launching angle) along the xz–plane (in this case, the lateral y–position of the point of incidence is always fixed to y = 0 μm); therefore, in this second kind of measurement we are able to perform an angular scan of the fiber. Both kinds of measurements are represented in Fig. 2.

Fig. 2 Geometrical arrangement of the POF relative to the incident beam for each measurement method.

In all these measurements, the length of the POF sample is 17 cm. Such a short length is absolutely necessary to be able to detect the faint scattered signal that reaches the output end of the POF sample. Furthermore, using a short fiber has an additional advantage, since the effects of mode coupling are still not significant [8

8. D. Gloge, “Optical power flow in multimode fibers,” Bell Syst. Tech. J. 51, 1767–1783 (1972).

,19

19. G. Jiang, R. F. Shi, and A. F. Garito, “Mode coupling and equilibrium mode distribution conditions in plastic optical fibers,” IEEE Photon. Technol. Lett.9, 1128–1131 (1997), http://dx.doi.org/10.1109/68.605524. [CrossRef]

22

22. M. A. Losada, J. Mateo, I. Garcés, J. Zubia, J. A. Casao, and P. Pérez-Vela, “Analysis of strained plastic optical fibers,” IEEE Photon. Technol. Lett.16, 1513–1515 (2004), http://dx.doi.org/10.1109/LPT.2004.826780. [CrossRef]

] and, therefore, they can be safely neglected.

Going back to Fig. 1, two digital multimeters, attached to the reference and signal photodetectors, send their measurements to a computer, which stores all the data in a file for further processing. The process described here is semi-automated: most of the parts of the system (more specifically, the motion controller, the rotation stage and both multimeters) are controlled by the computer via GPIB by means of custom-built LabVIEW programs: however, the Glan-Thompson linear polarizer and the xy–micropositioners can only be adjusted manually. For this reason, whereas the step size for the angular scan can be as good as 0.5 degrees, we are much more limited for the lateral scan, with step sizes of 25 and 50 μm.

With regard to the measurement of the near- and far-field patterns of the scattered light that is guided along the POF sample to the output end, a slight variation in the experimental set-up is necessary. This modification consists simply in replacing the signal photodetector and the multimeter by the Hamamatsu LEPAS optical beam measurement system [23

23. Hamamatsu Photonics K. K., “LEPAS–12 optical beam measurement system,” http://sales.hamamatsu.com/en/products/system-division/laser-fiber-optic-measurement/beam-analysis.php.

].

An image of the incident spot on the lateral surface of the fiber is recorded by a frame grabber installed at the bottom arm of the beam splitter: we use an absorptive neutral density filter so as not to saturate the camera with the intense reflected laser beam and two lenses L3 and L4 that magnify the images to be recorded. These images are essential in the process of adjusting the x–position of the POF sample so as to make the point of incidence coincide with the focal plane of the objective. Figure 3 shows an example of the images obtained by the frame grabber.

Fig. 3 Still taken from the frame grabber which shows at its center the incident spot on the lateral surface of the SK POF (at y = 0 μm). The dashed lines superimposed on the photograph delimit the upper and lower boundaries of the fiber.

3.2. Experimental results

We show first the experimental results obtained for the lateral and angular scans in Figs. 4 and 5, respectively. In both cases, each measurement has been repeated three times and the mean value has been taken in order to obtain more accurate values (the vertical bars denote the uncertainty associated with each measurement).

Fig. 4 Experimental results for lateral scan. Step size is of 50 μm between −400 μm and + 400 μm, and it is further reduced to 25 μm within intervals [−600 μm, −400 μm] and [+ 400 μm, +600 μm] in order to obtain a more detailed measurement of the abrupt change in intensity around y = ±ρcore (ρcore = ϕcore/2). The measured intensity I is divided by the intensity I0 obtained by the RPD reference photodiode (see also Fig. 1).
Fig. 5 Experimental results for angular scan. Step size is of 0.5° between 45° and +45°. The measured intensity I is divided by the intensity I0 obtained by the RPD reference photodiode (see also Fig. 1).

With regard to the lateral scan (Fig. 4), two intense peaks stand out when the point of incidence is placed at the core/cladding interface (y = ±ρcore, being ρcore = ϕcore/2). Whereas the asymmetries observed in these peaks are straightforwardly attributable to tolerances in the diameters of the fiber core and cladding, it is more difficult to find out the reasons that would explain the exaggerated nature of these peaks (the measured values around y = ±ρcore are, in some instances, about eight times larger than in the case where the point of incidence is placed at y = 0 μm, in the same way as in Refs. [12

12. H. Poisel, A. Hager, V. Levin, and K.-F. Klein, “Lateral coupling to polymer optical fibres,” in 7th international plastic optical fibres conference 1998: Proceedings, (Berlin (Germany), 1998), pp. 114–116.

] and [13

13. C.-A. Bunge, R. Kruglov, and H. Poisel, “Rayleigh and Mie scattering in polymer optical fibers,” J. Lightwave Technol.24, 3137–46 (2006), http://dx.doi.org/10.1109/JLT.2006.878077. [CrossRef]

]). We believe that these unexpectedly high intensity values arise partly as a consequence of the finite spot size of the incident laser beam. This is better illustrated with the help of Fig. 6. Since in the vicinity of y = ±ρcore the laser beam impinging on the lateral surface of the POF sample illuminates a larger area than in the case of y = 0 μm, it seems reasonable to think that, in such conditions, the laser beam will collide with a larger number of inhomogeneities, giving rise to higher scattered intensities [15

15. I. Bikandi, M. A. Illarramendi, J. Zubia, G. Aldabaldetreku, G. Durana, and L. Bazzana, “Dependence of fluorescence in POFs doped with conjugated polymers on launching conditions,” in POF 2009 Conference Proceedings (CD-ROM), (Sydney (Australia), 2009). Paper no. 32, http://igigroup.net/osc3/index.php?cPath=23.

]. Obviously, and assuming that the total amount of incident energy flux is the same in both cases (both at y = ±ρcore and at y = 0 μm), the power density reduces in accordance with the area projected by this incident laser beam; however, we believe that the larger number of inhomogeneities exposed to this incident laser beam at y = ±ρcore far compensates for this power density reduction.

Fig. 6 Effect of the finite spot size in the lateral scan. The area illuminated by the incident laser beam on the lateral surface of the POF depends on the y–position of the point of incidence. At y = +ρcore the projected area is larger than at y = 0 μm.

As to the angular scan (Fig. 5), we can observe that the scattered intensity collected at the output end of each POF sample increases as α increases (in the positive sense); in other words, the scattered intensity increases as the laser beam is directed to the forward direction of propagation along the POF sample (towards the positive z–axis). This result suggests that the average radiation pattern of the scattered intensity is quite directive, a feature whose implications will be further discussed in Sections 4 and 5.

All in all, from the above results it is clear that the scattered intensity shows a very weak dependence on the polarization. Additionally, if we compare the amounts of scattered intensities collected on each POF sample, we can observe that these intensities are slightly higher in the case of the SK POF, though these differences are not significant at all, as we would expect for POFs of identical characteristics.

Fig. 7 Experimental near- and far-field patterns of the PGU POF for different lateral y–positions. Measurements carried out at α = 0° and using horizontal polarization. The color scale denotes intensity counts (notice that the maximum value of the color scale in (c) has been divided by four, i.e. (7 × 104)/4 = 1.75 × 104).
Fig. 8 Experimental near- and far-field patterns of the PGU POF for different launching angles α. Measurements carried out at y = 0 μm and using horizontal polarization. The color scale denotes intensity counts (notice the variation of the maximum value of the color scale on each near-field pattern; it has been reduced to (7 × 104)/4 = 1.75 × 104 in (a) and to (7 × 104)/2 = 3.5 × 104 in (c)).
Fig. 9 Experimental near- and far-field patterns of the SK POF for different launching angles α. Measurements carried out at y = 0 μm and using vertical polarization. The color scale denotes intensity counts (notice the variation of the maximum value of the color scale on each near-field pattern; it has been reduced to (7 × 104)/4 = 1.75 × 104 in (a) and to (7 × 104)/2 = 3.5 × 104 in (c)).

First of all, we analyze in Fig. 7 the results obtained for the PGU POF and for three different lateral y–positions when using horizontal polarization. Here we have omitted the results corresponding to vertical polarization since they are practically the same (in agreement with the insensitivity to polarization exhibited by the results of the lateral and angular scans). As to the results obtained for the SK POF, we have also omitted them because they are practically identical to those obtained for the PGU POF, both qualitatively and quantitatively; in this sense, any conclusion drawn from the PGU POF will also be applicable to the SK POF.

If we now focus our attention on the near-field patterns shown on the left-hand side, we can see that, irrespective of the lateral offset of the laser beam, the maximum intensity is confined in a perimeter delimited by the interface separating the core from the cladding (at r = ρcore).

Turning our attention to the far-field patterns shown on the right-hand side, it can be observed that the maximum intensity takes always place at a far-field angle θ (relative to the z–axis) of approximately 30°. This value corresponds, precisely, to the theoretical acceptance angle θmax calculated from arcsinNA (in our case, θmax = arcsin 0.5 = 30°). This angle delimits two regions of very different behaviors: thus, the scattered intensity collected at angles below the acceptance angle (θ < θmax) is practically negligible in comparison with the intensity detected at angles above this acceptance angle (θ > θmax). We can also observe in this second region that the scattered intensity decreases from its maximum value as θ increases until the limit of the Hamamatsu LEPAS optical beam measurement system has been reached (at approximately 48.5°).

These results suggest that the scattered light propagates both along the cladding and the core, rather than being guided solely inside the core. As will be seen in Sections 4 and 5, the information provided by the experimental near- and far-field patterns will be decisive when delimiting the placement of the most influential inhomogeneities that give rise to the scattering.

Again, we can observe for both launching angles that, on the one hand, the near-field patterns reveal a maximum of scattered intensity around the core/cladding interface, and that, on the other hand, the maximum intensity in the far-field patterns coincides with the acceptance angle of the fiber.

If we now compare the results obtained for each launching angle, it is clear that there are significant differences in the number of intensity counts detected at the output end of the POF sample. Indeed, both near- and far-field patterns for α = +45° show a stronger scattered intensity in comparison with the case corresponding to α = −45°, a result that confirms the tendency observed in Fig. 5 for the angular scan.

Finally, we have reproduced in Fig. 9 the near- and far-field patterns obtained for the same launching angles used previously (i.e. for α = −45° and +45°), but this time using the SK POF and vertical polarization. The similarities between Figs. 8 and 9 should serve to justify the convenience of omitting the rest of the experimental results.

4. Simulation

4.1. Computational modelling

In order to obtain some more conclusive information about the properties of the inhomogeneities that give rise to scattering in real POFs, the experimental measurements have been complemented by additional computer simulations using the ray-tracing method [24

24. G. Aldabaldetreku, J. Zubia, G. Durana, and J. Arrue, “Numerical implementation of the ray-tracing method in the propagation of light through multimode optical fibres,” in POF Modelling: Theory, Measurement and Application, C.-A. Bunge and H. Poisel, eds. (Books on Demand GmbH, Norderstedt (Germany), 2007), pp. 25–48.

]. It is important to stress on the fact that the simulation results are only intended as a first approach, and that under no circumstances should they be regarded as a substitute for the experimental measurements, because of the following assumptions and simplifying hypotheses we have made in the calculation of the scattered intensity:
  1. the inhomogeneities (and/or particles) that give rise to the scattering are approximated by spheres of equivalent area, so that their diameter would represent the mean size of the scatterer,
  2. the refractive index of each scattering sphere has been set to 1.0 on the assumption that the excess scattering in PMMA fibers is mainly caused by voids [11

    11. Y. Koike, S. Matsuoka, and H. E. Bair, “Origin of excess light scattering in poly(methyl methacrylate) glasses,” Macromolecules25, 4807–4815 (1992), http://dx.doi.org/10.1021/ma00044a049. [CrossRef]

    ],
  3. the scattering by an individual scattering sphere is considered to be independent and incoherent [5

    5. H. C. van de Hulst, Light scattering by small particles (Dover Publications, Inc., New York, 1981).

    ],
  4. the effects of multiple scattering have been neglected, i.e. the radiation to which an individual scattering sphere is exposed is essentially the light of the original laser beam [5

    5. H. C. van de Hulst, Light scattering by small particles (Dover Publications, Inc., New York, 1981).

    ], and
  5. the distribution of scattered intensity produced by a scattering sphere has been calculated by Mie theory [5

    5. H. C. van de Hulst, Light scattering by small particles (Dover Publications, Inc., New York, 1981).

    ]; further details about the principal analytical expressions used in our computer simulations can be found in Ref. [25

    25. I. Bikandi, M. A. Illarramendi, J. Zubia, G. Aldabaldetreku, G. Durana, and L. Bazzana, “Analysis of light scattering in plastic optical fibres by side excitation technique: Theory and experimentation,” in POF 2009 Conference Proceedings (CD-ROM), (Sydney (Australia), 2009). Paper no. 35, http://igigroup.net/osc3/index.php?cPath=23.

    ].

Once this distribution of scattered intensity has been calculated, our computational model generates a distribution of rays using the Monte Carlo method, assigning a weighted value (proportional to this distribution of scattered intensity) to the power of each ray according to its direction. Finally, our computational model calculates the propagation of each ray along the fiber (starting from the position occupied by the scattering sphere towards the output end of the fiber), taking into account both bound and tunnelling rays [26

26. G. Aldabaldetreku, J. Zubia, G. Durana, and J. Arrue, “Power transmission coefficients for multi-step index optical fibres,” Opt. Express14, 1413–1429 (2006), http://dx.doi.org/10.1364/OE.14.001413. [CrossRef] [PubMed]

, 27

27. J. D. Love and C. Winkler, “A universal tunneling coefficient for step- and graded-index multimode fibres,” Opt.Quantum Electron.10, 341–351 (1978), http://dx.doi.org/10.1007/BF00620122. [CrossRef]

].

One of the main advantages of using a computational model based on the ray-tracing method is that we are able to calculate at the output end of the fiber not only the guided fraction of the total scattered intensity but also the corresponding near- and far-field patterns. However, it is worthy of remark that our computational model makes an additional simplification, since it calculates the propagation of rays arising from only one scattering sphere (in accordance with assumption iv). This limitation is necessary in order to preserve the total number of rays; otherwise, the analysis would become intractable.

After having explained the key features of our computational model, we proceed to describe the computer simulations. These simulations have been performed for an SI POF with a straight section of length L = 17 cm having the same characteristics as the experimentally measured PGU and SK POFs. We have used the previously selected wavelength (i.e. λ = 633 nm), though the simulated incident spot on the scattering sphere is infinitesimally small so that it impinges on only one scattering sphere. The total number of rays launched from this scattering sphere is approximately 250000. This number ranges between an upper boundary delimited by the total number of modes that can propagate within the fiber and a lower boundary to ensure sufficiently smooth and accurate results [9

9. A. W. Snyder and J. D. Love, Optical waveguide theory (Chapman and Hall, London, 1983).

].

Our computer simulations cover the same set of measurements carried out experimentally, namely the lateral scan (with a finer step size of 10 μm) and the angular scan, and they take into account both polarizations (horizontal and vertical) in the calculation of the distribution of scattered intensity. These simulations have been performed for scattering spheres of different diameters (denoted by ϕscatt from now on) and placed at different positions inside the fiber, for the purpose of understanding whether scattering in real POFs is caused by large or small inhomogeneities, and of identifying in which region the main source of the scattering is predominant.

As to the diameter, we have investigated scattering spheres of very different sizes, namely ϕscatt = 2000 nm, 200 nm, and 10 nm. We have chosen these extreme values because the diameter, through the size parameter x (a quantity that will be defined shortly), has a very strong effect on the pattern of the scattered intensity [5

5. H. C. van de Hulst, Light scattering by small particles (Dover Publications, Inc., New York, 1981).

]: a large value of the size parameter (x ≫ 1) produces a highly directive distribution of scattered intensity in the direction of the incident beam and practically independent of the polarization, whereas a small one (x ≪ 1), corresponding to the condition for Rayleigh scattering, gives rise to a much more weaker annular distribution of scattered intensity and strongly dependent on the polarization. This is shown in Fig. 10. Thus, and taking into account that the size parameter is defined as [5

5. H. C. van de Hulst, Light scattering by small particles (Dover Publications, Inc., New York, 1981).

]
x=πϕscattλnsr,
(1)
with nsr being the refractive index of the medium surrounding the scattering sphere (so that nsr=(ncore2NA2)1/2 for a scattering sphere placed at the core/cladding interface, or nsr = ncore for a scattering sphere in the core region), a scattering sphere of ϕscatt = 2000 nm would yield x = 13.93 at the core/cladding interface (or x = 14.79 in the core region), leading to a very directive distribution of intensity [see Figs. 10(a) and 10(b)]; on the contrary, for a scattering sphere of ϕscatt = 10 nm we would have x = 0.0697 at the core/cladding interface (or x = 0.0739 in the core region), that is to say, we would have an annular distribution of intensity [as shown in Figs. 10(c) and 10(d)]. For the intermediate value of ϕscatt = 200 nm, leading to x = 1.393 at the core/cladding interface (or x = 1.479 in the core region), the pattern of the scattered intensity shows an intermediate behavior, being quite directive and still dependent on polarization. The intermediate value of 200 nm has been chosen on the basis of the observations made in other studies by Koike et al. in Ref. [10

10. Y. Koike, N. Tanio, and Y. Ohtsuka, “Light scattering and heterogeneities in low-loss poly(methyl methacrylate) glasses,” Macromolecules22, 1367–1373 (1989), http://dx.doi.org/10.1021/ma00193a060. [CrossRef]

] and by Bunge et al. in Ref. [13

13. C.-A. Bunge, R. Kruglov, and H. Poisel, “Rayleigh and Mie scattering in polymer optical fibers,” J. Lightwave Technol.24, 3137–46 (2006), http://dx.doi.org/10.1109/JLT.2006.878077. [CrossRef]

]. Obviously, these differences in the distribution of scattered intensity will have a dramatic effect on the amount of intensity collected at the output end of the simulated POF, especially on the results obtained for the angular scan. Therefore, the comparison of these results with the experimental measurements should in principle allow us to draw some conclusion about the mean size of the inhomogeneities causing the predominant scattering in real POFs.

Fig. 10 Distribution of the scattered intensity for two different sizes of the scattering sphere and for each polarization. The scattering sphere is placed at the core/cladding interface (identical results are obtained for a scattering sphere in the core region). The color scale on each subfigure is related to the values shown in the corresponding axes [navy blue corresponds to the lowest intensity value, i.e. 0.0, and red to the highest intensity value, i.e. 13.5 in (a) and (b), or 2.9 × 10−7 in (c) and (d)]. A sketch showing the actual orientation of the fiber and the direction of the incident beam is superimposed on each plot (notice that the size of the distribution of the scattered intensity is not related to the size of the fiber). The launching conditions for the incident beam are y = 0 μm and α = 0°.

Regarding the position, we have analyzed the influence of having the scattering sphere placed either at the core/cladding interface or in the core region of the fiber. In both cases, it is necessary to calculate first the point of incidence at the core/cladding interface, which is obtained from the direction of the incident beam after it has been refracted from the air to the cladding. Once this point of incidence has been obtained, the placement of the scattering sphere is straightforward in the former case (for a scattering sphere at the core/cladding interface), whereas further calculations are required in the latter case (for a scattering sphere in the core region): these calculations involve determining the direction of the incident beam after having been refracted to the core and, then, choosing randomly the final position of the scattering sphere between all possible cases along the line traced by the incident beam in the core. As will be seen in the following subsection, the effects of changing the placement of the scattering sphere will be particularly noticeable in the near- and far-field patterns. For this reason, from the comparison of the simulated near- and far-field patterns with the experimental ones we should be able to infer some information about the actual position of the inhomogeneities that give rise to the predominant scattering.

4.2. Simulation results

Figures 11 and 12 show first the simulation results obtained for the lateral and angular scans. Again, every plot on each figure shows the resultant mean value of the intensity from a set of three computer simulations. Notice that, however, the vertical bars have been omitted due to the practically negligible uncertainty of the simulation results. For the sake of comparison, the resultant mean value of the intensity I of each plot has been divided by a constant reference value I0.

Fig. 11 Numerical results for lateral scan. Step size is of 10 μm between −500 μm and +500 μm. The numerical intensity I is divided by a constant reference value I0. Legend: CC: scattering sphere placed at the core/cladding interface (at r = ρcore); CR: scattering sphere in the core region (r < ρcore). H: horizontal polarization; V: vertical polarization.
Fig. 12 Numerical results for angular scan. Step size is of 0.5° between −45° and +45°. The numerical intensity I is divided by a constant reference value I0. Legend: CC: scattering sphere placed at the core/cladding interface (at r = ρcore); CR: scattering sphere in the core region (r < ρcore). H: horizontal polarization; V: vertical polarization.

From both figures it is clear that a scattering sphere of larger size leads to a higher scattered intensity, a difference that can be as high as six orders of magnitude if compared to the intensity obtained for a small scattering sphere. It can also be observed that the scattered intensity obtained for a scattering sphere placed inside the core region can be, in some cases, thrice the value obtained for a scattering sphere placed at the core/cladding interface.

In order to understand the reason for such a behavior, we have to turn to the distributions of the scattered intensity corresponding to a scattering sphere of ϕscatt = 2000 nm [Figs. 10(a) and 10(b)]: whereas the principal lobe, directed towards the negative x–axis (and, therefore, hardly contributing to the detected intensity at the output end of the simulated fiber), is practically invariant regardless of the polarization, the secondary lobes, oriented in a more suitable way that contributes to the forward propagation along the fiber, have a different shape depending on the polarization [this is better appreciated from the comparison between the insets of Figs. 10(a) and 10(b)]. It is this variation in the secondary lobes which explains the deviations observed in the case of the lateral scan for a scattering sphere of large size.

In the case of the angular scan, the effects of these different secondary lobes become masked by the more significant principal lobe. Indeed, the higher the launching angle, the more favourable the direction of the principal lobe is in terms of light propagation along the fiber (which explains the higher scattered intensity collected at the output end of the simulated POF), so that the increasing role of the principal lobe leads to results that hardly depend on polarization.

Next, Figs. 13 and 14 summarize the near- and far-field patterns calculated under different launching conditions and for different sizes and placements of the scattering sphere.

Fig. 13 Numerical near- and far-field patterns for different lateral y–positions at α = 0° and using horizontal polarization (H). Simulations carried out for scattering spheres of different diameters (ϕscatt = 2000 nm, 200 nm, and 10 nm) and placed either at the core/cladding interface (denoted by CC) or in the core region (denoted by CR). The color scale denotes intensity counts (notice the variation of the maximum value of the color scale as a function of the size of the scattering sphere).
Fig. 14 Numerical near- and far-field patterns for different launching angles α at y = 0 μm and using vertical polarization (V). Simulations carried out for scattering spheres of different diameters (ϕscatt = 2000 nm, 200 nm, and 10 nm) and placed either at the core/cladding interface (denoted by CC) or in the core region (denoted by CR). The color scale denotes intensity counts (notice the variation of the maximum value of the color scale as a function of the size of the scattering sphere).

We turn first our attention to the results obtained for different lateral y–positions at α = 0° (Fig. 13). Here we reproduce only the results obtained for horizontal polarization, since, despite the quantitative differences discussed earlier in Fig. 11, we have not noticed any qualitative discrepancy between these results and those obtained for vertical polarization, so that the latter have been omitted. Likewise, the results corresponding to y = +475 μm have been omitted on grounds of symmetry considerations (since we are dealing with ideal fibers with cylindrical symmetry, identical results are obtained both for y = −475 μm and for y = +475 μm).

From the near-field patterns we can see that, irrespective of the size, there are significant differences depending on the position of the scattering sphere in the fiber. More specifically, for a scattering sphere placed at the core/cladding interface, a maximum of scattered intensity is detected around the interface between the core and the cladding, whereas for a scattering sphere in the core region, a much larger fraction of scattered intensity is spread over the core region, being the maximum value clearly distinguishable around a perimeter defined by the radial coordinate of the position occupied by this scattering sphere (at r ≈ 318 μm for the case corresponding to y = −475 μm, or at r ≈ 245 μm for the case corresponding to y = 0 μm).

As to the far-field patterns, we observe again variations in the obtained results as a function of the radial position occupied by the scattering sphere in the fiber, even though such variations are more subtle. Thus, for a scattering sphere placed at the core/cladding interface, and regardless of its size, there is a clear division between two angular regions, the limit corresponding to the theoretical acceptance angle θmax = 30°: in the first region, at far-field angles below the acceptance angle (θ < θmax), the fraction of scattered intensity detected at the output end of the simulated fiber is extremely low; in the second region, at angles above the acceptance angle (θ > θmax), most of the scattered intensity is detected, starting with a maximum value at θmax and decreasing as the limit angle of 90° is approached. In contrast, in the case of a scattering sphere in the core region, there is no clear distinction between these two regions. In fact, the maximum of intensity is shifted towards higher angles than the acceptance angle, whereas, at lower angles (θ < θmax), there is still a considerable amount of scattered intensity.

These differences are more obvious, especially on the far-field patterns, if we consider the results obtained in Fig. 14 for different launching angles α at y = 0 μm and using vertical polarization (again, and for the reasons commented above, we omit the results obtained for horizontal polarization). Indeed, when the scattering sphere is in the core region, instead of detecting the maximum of scattered intensity at angles higher than the acceptance angle, we detect it at θ = 0°, and it decreases as θ increases. All these differences are a direct consequence not only of the change in the orientation of the distribution of the scattered intensity with α (which depends, of course, on the size of the scattering sphere), but also of the variation in the cone of acceptance of the fiber as a function of the radial position occupied by the scattering sphere inside the fiber.

With regard to the near-field patterns obtained for different launching angles [Figs. 14(a) and 14(c)], we can reach the same conclusions as those drawn from Figs. 13(a) and 13(c), by taking into account that, now, for a scattering sphere in the core region, the maximum scattered intensity takes place around a contour delimited by the radial position occupied by this scattering sphere at r ≈ 380 μm for the case corresponding to α = −45°, and at r ≈ 400 μm for the case corresponding to α = +45°.

We conclude this subsection with a final remark regarding the agreement shown by Figs. 13 and 14 with Figs. 11 and 12 concerning the dependence of the detected scattered intensity both on the size of the scattering sphere ϕscatt and on the launching angle α, which is a consequence of the distribution of scattered intensity produced by the scattering sphere (refer also to Fig. 10). On the one hand, we can observe that, regardless of the launching conditions, the scattered intensity increases with the size of the scattering sphere. On the other hand, this scattered intensity also increases with the launching angle, provided that we have a scattering sphere of moderate (ϕscatt = 200 nm) or large (ϕscatt = 2000 nm) size; instead, for a scattering sphere of small size (ϕscatt = 10 nm), the amount of detected scattered intensity is practically the same for α = −45° and +45° in Fig. 14, which is in accordance with the symmetry exhibited by Fig. 12(c) for negative and positive values of α.

5. Comparison between the experimental and simulation results

In this section we will compare the experimental measurements with the computer simulations. In the first place our analysis will be devoted to the comparison of the results obtained for the lateral and angular scans. After that, we will focus on the comparison of the near- and far-field patterns.

Let us first focus our attention on the results shown in Figs. 5 and 12 (for reasons that will be clarified later, we will only pay attention to the simulation results corresponding to a scattering sphere placed at the core/cladding interface). The simulation results that match better the experimental ones correspond to scattering spheres of moderate (ϕscatt = 200 nm) and large (ϕscatt = 2000 nm) size. The reasons are twofold: first of all, the experimental results are practically insensitive to the polarization, an effect that, according to our simulations, becomes more evident as the size of the scattering sphere increases; and, secondly, the increase in the scattered intensity with the launching angle shown by the experimental results is only shared by the simulations obtained for scattering spheres of moderate and large size.

Therefore, if we were to attend to the results obtained only from the angular scan (so that some caution should be exercised in the statement that follows), we might conclude that the mean size of the inhomogeneities that lead to the scattering in real POFs has a value of at least 200 nm.

Finally, we compare the experimental near- and far-field patterns represented in Figs. 7 and 9 with the numerical near- and far-field patterns shown in Figs. 13 and 14. We can observe that, regardless of the size of the scattering sphere, the simulation results that match the experimental ones correspond to scattering spheres placed at the core/cladding interface.

In conclusion, we can state that the most significant inhomogeneities that give rise to the scattering are placed at the core/cladding interface. Furthermore, these inhomogeneities have presumably a mean size of at least 200 nm.

6. Conclusions

In this paper we have investigated the light scattering properties of two commercial SI POFs of identical characteristics. Using the side-illumination technique, we have measured the total amount of scattered intensity guided along each POF sample under different launching conditions, either by changing laterally the point of incidence of the laser beam or by rotating angularly the direction of the laser beam. Additionally, we have measured the near- and far-field patterns obtained at the output end of each POF sample for different lateral positions of the point of incidence and for different launching angles. The similarities exhibited by the experimental results obtained for both POFs are consistent with the fact that they share identical physical and optical properties.

The experimental measurements have been complemented by computer simulations with the aim of obtaining additional information about the most influential inhomogeneities that give rise to the light scattering. In these simulations, which reproduce the same launching conditions employed in the experimental set-up, we have calculated the total amount of scattered intensity as well as the near- and far-field patterns that would be obtained for scattering spheres of different sizes and placed at different positions inside the fiber. From the comparison of these computer simulations with the experimental measurements, we have deduced that, on the one hand, the mean size of the most influential inhomogeneities that give rise to the scattering in POFs is likely to be of at least 200 nm, and that, on the other hand, these inhomogeneities are placed at the interface between the core and the cladding.

Acknowledgments

This work was supported by the Ministerio de Ciencia e Innovacion under project TEC2009-14718-C03-01, by the Universidad del País Vasco/Euskal Herriko Unibertsitatea under projects GIU05/03 and UE09+/103, by the Gobierno Vasco/Eusko Jaurlaritza under projects AIRHEM, S-PE09CA03, and AVICOHM, and by the Diputación Foral de Bizkaia/Bizkaiko Foru Aldundia under projects DIPE08/24 and 06-12-TK-2010-0022. The research leading to these results has also received funding from the European Commission’s Seventh Framework Programme (FP7) under grant agreement no. 212912 (AISHA II).

References and links

1.

O. Ziemann, J. Krauser, P. E. Zamzow, and W. Daum, POF Handbook: Optical Short Range Transmission Systems (Springer, Berlin, 2008), 2nd ed.

2.

D. Kalymnios, P. Scully, J. Zubia, and H. Poisel, “POF sensors overview,” in 13th international plastic optical fibres conference 2004: Proceedings, (Nuremberg (Germany), 2004), pp. 237–244.

3.

T. Kaino, “Polymer optical fibers,” in Polymers for lightwave and integrated optics, L. A. Hornak, ed. (Marcel Dekker, Inc., New York, 1992), chap. 1.

4.

J. Zubia and J. Arrue, “Plastic optical fibers: An introduction to their technological processes and applications,” Opt. Fiber Technol.7, 101–140 (2001), http://dx.doi.org/10.1006/ofte.2000.0355. [CrossRef]

5.

H. C. van de Hulst, Light scattering by small particles (Dover Publications, Inc., New York, 1981).

6.

C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (John Wiley & Sons, New York, 1983).

7.

M. Born and E. Wolf, Principles of optics (Pergamon Press, New York, 1990), 6th ed.

8.

D. Gloge, “Optical power flow in multimode fibers,” Bell Syst. Tech. J. 51, 1767–1783 (1972).

9.

A. W. Snyder and J. D. Love, Optical waveguide theory (Chapman and Hall, London, 1983).

10.

Y. Koike, N. Tanio, and Y. Ohtsuka, “Light scattering and heterogeneities in low-loss poly(methyl methacrylate) glasses,” Macromolecules22, 1367–1373 (1989), http://dx.doi.org/10.1021/ma00193a060. [CrossRef]

11.

Y. Koike, S. Matsuoka, and H. E. Bair, “Origin of excess light scattering in poly(methyl methacrylate) glasses,” Macromolecules25, 4807–4815 (1992), http://dx.doi.org/10.1021/ma00044a049. [CrossRef]

12.

H. Poisel, A. Hager, V. Levin, and K.-F. Klein, “Lateral coupling to polymer optical fibres,” in 7th international plastic optical fibres conference 1998: Proceedings, (Berlin (Germany), 1998), pp. 114–116.

13.

C.-A. Bunge, R. Kruglov, and H. Poisel, “Rayleigh and Mie scattering in polymer optical fibers,” J. Lightwave Technol.24, 3137–46 (2006), http://dx.doi.org/10.1109/JLT.2006.878077. [CrossRef]

14.

M. A. Illarramendi, J. Zubia, L. Bazzana, G. Durana, G. Aldabaldetreku, and J. R. Sarasua, “Spectroscopic Characterization of Plastic Optical Fibers Doped with Fluorene Oligomers,” J. Lightwave Technol.27, 3220 –3226 (2009), http://dx.doi.org/10.1109/JLT.2008.2010274. [CrossRef]

15.

I. Bikandi, M. A. Illarramendi, J. Zubia, G. Aldabaldetreku, G. Durana, and L. Bazzana, “Dependence of fluorescence in POFs doped with conjugated polymers on launching conditions,” in POF 2009 Conference Proceedings (CD-ROM), (Sydney (Australia), 2009). Paper no. 32, http://igigroup.net/osc3/index.php?cPath=23.

16.

Toray Industries Inc., “Raytela Plastic Optical Fiber,” http://www.toray.co.jp/english/raytela/index.html.

17.

Mitsubishi Rayon Co., Ltd., “Super ESKA Plastic Optical Fiber,” http://www.pofeska.com/.

18.

M. G. Kuzyk, Polymer Fiber Optics: Materials, Physics, and Applications (Taylor and Francis, Boca Raton, 2007).

19.

G. Jiang, R. F. Shi, and A. F. Garito, “Mode coupling and equilibrium mode distribution conditions in plastic optical fibers,” IEEE Photon. Technol. Lett.9, 1128–1131 (1997), http://dx.doi.org/10.1109/68.605524. [CrossRef]

20.

A. F. Garito, J. Wang, and R. Gao, “Effects of random perturbations in plastic optical fibers,” Science281, 962–967 (1998), http://dx.doi.org/10.1126/science.281.5379.962. [CrossRef] [PubMed]

21.

S. Savović and A. Djordjevich, “Mode coupling in strained and unstrained step-index plastic optical fibers,” Appl. Opt.45, 6775–6780 (2006), http://dx.doi.org/10.1364/AO.45.006775. [CrossRef]

22.

M. A. Losada, J. Mateo, I. Garcés, J. Zubia, J. A. Casao, and P. Pérez-Vela, “Analysis of strained plastic optical fibers,” IEEE Photon. Technol. Lett.16, 1513–1515 (2004), http://dx.doi.org/10.1109/LPT.2004.826780. [CrossRef]

23.

Hamamatsu Photonics K. K., “LEPAS–12 optical beam measurement system,” http://sales.hamamatsu.com/en/products/system-division/laser-fiber-optic-measurement/beam-analysis.php.

24.

G. Aldabaldetreku, J. Zubia, G. Durana, and J. Arrue, “Numerical implementation of the ray-tracing method in the propagation of light through multimode optical fibres,” in POF Modelling: Theory, Measurement and Application, C.-A. Bunge and H. Poisel, eds. (Books on Demand GmbH, Norderstedt (Germany), 2007), pp. 25–48.

25.

I. Bikandi, M. A. Illarramendi, J. Zubia, G. Aldabaldetreku, G. Durana, and L. Bazzana, “Analysis of light scattering in plastic optical fibres by side excitation technique: Theory and experimentation,” in POF 2009 Conference Proceedings (CD-ROM), (Sydney (Australia), 2009). Paper no. 35, http://igigroup.net/osc3/index.php?cPath=23.

26.

G. Aldabaldetreku, J. Zubia, G. Durana, and J. Arrue, “Power transmission coefficients for multi-step index optical fibres,” Opt. Express14, 1413–1429 (2006), http://dx.doi.org/10.1364/OE.14.001413. [CrossRef] [PubMed]

27.

J. D. Love and C. Winkler, “A universal tunneling coefficient for step- and graded-index multimode fibres,” Opt.Quantum Electron.10, 341–351 (1978), http://dx.doi.org/10.1007/BF00620122. [CrossRef]

OCIS Codes
(060.2270) Fiber optics and optical communications : Fiber characterization
(060.2300) Fiber optics and optical communications : Fiber measurements
(060.2310) Fiber optics and optical communications : Fiber optics
(290.0290) Scattering : Scattering
(290.4020) Scattering : Mie theory
(290.5820) Scattering : Scattering measurements

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: August 2, 2010
Revised Manuscript: October 19, 2010
Manuscript Accepted: October 25, 2010
Published: November 10, 2010

Citation
Gotzon Aldabaldetreku, Iñaki Bikandi, María Asunción Illarramendi, Gaizka Durana, and Joseba Zubia, "A comprehensive analysis of scattering in polymer optical fibers," Opt. Express 18, 24536-24555 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-24-24536


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References

  1. O. Ziemann, J. Krauser, P. E. Zamzow, and W. Daum, POF Handbook: Optical Short Range Transmission Systems (Springer, Berlin, 2008), 2nd ed.
  2. D. Kalymnios, P. Scully, J. Zubia, and H. Poisel, "POF sensors overview," in 13th international plastic optical fibres conference 2004: Proceedings, (Nuremberg (Germany), 2004), pp. 237-244.
  3. T. Kaino, "Polymer optical fibers," in Polymers for lightwave and integrated optics, L. A. Hornak, ed. (Marcel Dekker, Inc., New York, 1992), chap. 1.
  4. J. Zubia, and J. Arrue, "Plastic optical fibers: An introduction to their technological processes and applications," Opt. Fiber Technol. 7, 101-140 (2001), http://dx.doi.org/10.1006/ofte.2000.0355. [CrossRef]
  5. H. C. van de Hulst, Light scattering by small particles (Dover Publications, Inc., New York, 1981).
  6. C. F. Bohren, and D. R. Huffman, Absorption and scattering of light by small particles (John Wiley & Sons, New York, 1983).
  7. M. Born, and E. Wolf, Principles of optics (Pergamon Press, New York, 1990), 6th ed.
  8. D. Gloge, "Optical power flow in multimode fibers," Bell Syst. Tech. J. 51, 1767-1783 (1972).
  9. A. W. Snyder, and J. D. Love, Optical waveguide theory (Chapman and Hall, London, 1983).
  10. Y. Koike, N. Tanio, and Y. Ohtsuka, "Light scattering and heterogeneities in low-loss poly(methyl methacrylate) glasses," Macromolecules 22, 1367-1373 (1989), http://dx.doi.org/10.1021/ma00193a060. [CrossRef]
  11. Y. Koike, S. Matsuoka, and H. E. Bair, "Origin of excess light scattering in poly(methyl methacrylate) glasses," Macromolecules 25, 4807-4815 (1992), http://dx.doi.org/10.1021/ma00044a049. [CrossRef]
  12. H. Poisel, A. Hager, V. Levin, and K.-F. Klein, "Lateral coupling to polymer optical fibres," in 7th international plastic optical fibres conference 1998: Proceedings, (Berlin (Germany), 1998), pp. 114-116.
  13. C.-A. Bunge, R. Kruglov, and H. Poisel, "Rayleigh and Mie scattering in polymer optical fibers," J. Lightwave Technol. 24, 3137-3146 (2006), http://dx.doi.org/10.1109/JLT.2006.878077. [CrossRef]
  14. M. A. Illarramendi, J. Zubia, L. Bazzana, G. Durana, G. Aldabaldetreku, and J. R. Sarasua, "Spectroscopic Characterization of Plastic Optical Fibers Doped with Fluorene Oligomers," J. Lightwave Technol. 27, 3220-3226 (2009), http://dx.doi.org/10.1109/JLT.2008.2010274. [CrossRef]
  15. I. Bikandi, M. A. Illarramendi, J. Zubia, G. Aldabaldetreku, G. Durana, and L. Bazzana, "Dependence of fluorescence in POFs doped with conjugated polymers on launching conditions," in POF 2009 Conference Proceedings (CD-ROM), (Sydney (Australia), 2009). Paper no. 32, http://igigroup.net/osc3/index.php?cPath=23.
  16. Toray Industries Inc, "Raytela Plastic Optical Fiber," http://www.toray.co.jp/english/raytela/ index.html.
  17. Mitsubishi Rayon Co, Ltd., "Super ESKA Plastic Optical Fiber," http://www.pofeska.com/.
  18. M. G. Kuzyk, Polymer Fiber Optics: Materials, Physics, and Applications (Taylor and Francis, Boca Raton, 2007).
  19. G. Jiang, R. F. Shi, and A. F. Garito, "Mode coupling and equilibrium mode distribution conditions in plastic optical fibers," IEEE Photon. Technol. Lett. 9, 1128-1131 (1997), http://dx.doi.org/10.1109/68.605524. [CrossRef]
  20. A. F. Garito, J. Wang, and R. Gao, "Effects of random perturbations in plastic optical fibers," Science 281, 962-967 (1998), http://dx.doi.org/10.1126/science.281.5379.962. [CrossRef] [PubMed]
  21. S. Savović, and A. Djordjevich, "Mode coupling in strained and unstrained step-index plastic optical fibers," Appl. Opt. 45, 6775-6780 (2006), http://dx.doi.org/10.1364/AO.45.006775. [CrossRef]
  22. M. A. Losada, J. Mateo, I. Garcés, J. Zubia, J. A. Casao, and P. Pérez-Vela, "Analysis of strained plastic optical fibers," IEEE Photon. Technol. Lett. 16, 1513-1515 (2004), http://dx.doi.org/10.1109/LPT.2004.826780. [CrossRef]
  23. K. K. Hamamatsu Photonics, "LEPAS-12 optical beam measurement system," http://sales. hamamatsu.com/en/products/system-division/laser-fiber-optic-measurement/ beam-analysis.php.
  24. G. Aldabaldetreku, J. Zubia, G. Durana, and J. Arrue, "Numerical implementation of the ray-tracing method in the propagation of light through multimode optical fibres," in POF Modelling: Theory, Measurement and Application, C.-A. Bunge and H. Poisel, eds. (Books on Demand GmbH, Norderstedt (Germany), 2007), pp. 25-48.
  25. I. Bikandi, M. A. Illarramendi, J. Zubia, G. Aldabaldetreku, G. Durana, and L. Bazzana, "Analysis of light scattering in plastic optical fibres by side excitation technique: Theory and experimentation," in POF 2009 Conference Proceedings (CD-ROM), (Sydney (Australia), 2009). Paper no. 35, http://igigroup.net/osc3/index.php?cPath=23.
  26. G. Aldabaldetreku, J. Zubia, G. Durana, and J. Arrue, "Power transmission coefficients for multi-step index optical fibres," Opt. Express 14, 1413-1429 (2006), http://dx.doi.org/10.1364/OE.14.001413. [CrossRef] [PubMed]
  27. J. D. Love, and C. Winkler, "A universal tunneling coefficient for step- and graded-index multimode fibres," Opt. Quantum Electron. 10, 341-351 (1978), http://dx.doi.org/10.1007/BF00620122. [CrossRef]

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