## Side-illumination fluorescence critical angle: theory and application to F8BT-doped polymer optical fibers |

Optics Express, Vol. 20, Issue 4, pp. 4630-4644 (2012)

http://dx.doi.org/10.1364/OE.20.004630

Acrobat PDF (2456 KB)

### Abstract

In this work we have analyzed theoretically and experimentally the critical angle for the emission generated in doped polymer optical fibers as a function of different launching conditions by using the side-illumination fluorescence technique. A theoretical model has been developed in order to explain the experimental measurements. It is shown that both the theoretical and experimental critical angles are appreciably higher than the meridional critical angle corresponding to the maximum acceptance angle for a single source placed at the fiber axis. This increase changes the value of several important parameters in the performance of active fibers. The analysis has been performed in polymer optical fibers doped with a conjugated polymer.

© 2012 OSA

## 1. Introduction

24. R. J. Kruhlak and M. G. Kuzyk, “Side-illumination fluorescence spectroscopy. I. Principles,” J. Opt. Soc. Am. B **16**(10), 1749–1755 (1999). [CrossRef]

25. R. J. Kruhlak and M. G. Kuzyk, “Side-illumination fluorescence spectroscopy. II. applications to squaraine-dye-doped polymer optical fibers,” J. Opt. Soc. Am. B **16**(10), 1756–1767 (1999). [CrossRef]

9. C. Pulido and O. Esteban, “Multiple fluorescence sensing with side-pumped tapered polymer fiber,” Sens. Actuators B Chem. **157**(2), 560–564 (2011). [CrossRef]

## 2. Experimental set-up

## 3. Theoretical modeling

*Q*in our model has to be understood as an infinitesimal volume element (of length d

*x*) containing several atoms or molecules. Taking into account that the molecules of F8BT are distributed inhomogeneously in the polymer matrix [28

_{Q}28. G. E. Khalil, A. M. Adawi, A. M. Fox, A. Iraqi, and D. G. Lidzey, “Single molecule spectroscopy of red- and green-emitting fluorene-based copolymers,” J. Chem. Phys. **130**(4), 044903 (2009). [CrossRef] [PubMed]

*ρ*). The doped fibers have been assumed to be step-index.

_{core}*λ*=

*λ*is refracted at point

_{exc}*P*(

*ρ*,0,

_{core}*z*) of the fiber output surface and then it propagates across the core until it finally exits the fiber. Along its path through the core, the incident light is absorbed by a set of molecules at each point

*Q*, which, in turn, emit light isotropically in all directions at wavelengths

*λ*=

*λ*. Some of the emitted light will refract out of the fiber, and the remaining light lying within the internal critical angle (

_{emi}*θ*)

_{z}*will propagate through the fiber and it will be detected at one of the ends. Since the emission is generated from sources placed along the refracted beam, an analysis of the variation of (*

_{c}*θ*)

_{z}*inside the fiber is needed. By performing calculations similar to those exposed in [29*

_{c}29. C.-A. Bunge, R. Kruglov, and H. Poisel, “Rayleigh and Mie scattering in polymer optical fibers,” J. Lightwave Technol. **24**(8), 3137–3146 (2006). [CrossRef]

*θ*)

_{z}*corresponding to the beam emitting from the point source*

_{c}*Q*(

*x*,0,

_{Q}*z*) is obtained:where

*ϕ*represents the angle of the emitted beam relative to the

_{x}*x*-axis in the

*xy*-plane and

*x*is the

_{Q}*x*-position of the source along the refracting beam (see Fig. 3(a)). The angle

*θ*is measured with respect to the fiber axis (

_{z}*z*-axis).

*n*(

_{ratio}*λ*) represents the quotient

_{emi}*n*/

_{clad}*n*at

_{core}*λ*=

*λ*. We see that the maximum acceptance angle of the fiber (

_{emi}*θ*)

_{z}*depends on*

_{c}*x*and

_{Q}*ϕ*, and also on the refractive indices at

_{x}*λ*=

*λ*, whereas it does not depend on the incident angle

_{emi}*α*. It can be seen that sin(

_{i}*θ*)

_{z}*expressed in Eq. (2) is higher than that corresponding to bound rays (meridional angle), which is a simplified expression of Eq. (2) for the case in which the fluorescence were produced from one source placed just at the center of the fiber (*

_{c}*x*= 0), i.e.

_{Q}24. R. J. Kruhlak and M. G. Kuzyk, “Side-illumination fluorescence spectroscopy. I. Principles,” J. Opt. Soc. Am. B **16**(10), 1749–1755 (1999). [CrossRef]

*θ*)

_{z}*cannot exceed*

_{c}*π*/2 (sin(

*θ*)

_{z}*≤ 1), it is required that*

_{c}*x*and

_{Q}*ϕ*satisfy the following constraint:

_{x}*x*) and the emitted direction (

_{Q}*ϕ*) where guided and tunneling rays are generated within the fiber. As an example, Fig. 4 shows the representation of sin(

_{x}*θ*)

_{z}*as a function of both*

_{c}*x*and

_{Q}*ϕ*corresponding to the doped plastic optical fiber analyzed. This representation has been obtained with the characteristic fiber parameters supplied by the manufacturer (

_{x}*n*(

_{core}*λ*) = 1.495, r

_{emi}*= 490 μm*

_{core}*and NA*= (

_{meriodional}*n*

_{core}^{2}-

*n*

_{clad}^{2})

^{1/2}= 0.5). It can be seen that the curves are symmetrical respect to

*x*= 0 and

_{Q}*ϕ*=

_{x}*π*and that there are some source points (near to the fiber interfaces) emitting light in certain specific directions whose maximum acceptance angle is not defined. Those emitted beams generated very close to the core-cladding interface that yield imaginary values for sin(

*θ*)

_{z}*can be interpreted as evanescent modes along the fiber axis. These modes describe the energy stored in the immediate vicinity of fiber discontinuities [17]. As the quotient*

_{c}*n*(

_{ratio}*λ*) approaches the value 1, the forbidden regions, where (

_{emi}*θ*)

_{z}*is not defined, become smaller and, consequently, the effects of the truncation of the sine function on the emitted intensity are insignificant. The imposition that (*

_{c}*θ*)

_{z}*cannot exceed*

_{c}*π*/2 and its effects on the near-field patterns and on the scattered light curves of step-index fibers have been reported in [23].

*θ*)

_{z}*(Eq. (4)), we can now calculate an average of the sine of (*

_{c}*θ*)

_{z}*for the doped fiber as follows:*

_{c}*θ*)

_{z}*can be calculated by taking all the point sources placed in a reduced region, −*

_{c}*(*

_{core}n_{ratio}*λ*) ≤

_{emi}*x*≤

_{Q}*(*

_{core}n_{ratio}*λ*) and 0 ≤

_{emi}*ϕ*≤ 2

_{x}*π*, i.e.

*θ*)

_{z}*obtained under this assumption (Eq. (8)) does not differ very much from the one calculated exactly (Eq. (6)) [30], since it represents the average of the sine of the critical angle when the contribution of some tunneling rays is neglected, particularly those with a high (*

_{c}*θ*)

_{z}*. Notice that Eq. (6) and Eq. (8) depend only on*

_{c}*n*(

_{ratio}*λ*). By averaging Eq. (2) in the reduced region and by taking into account that the emission is isotropic, we have obtained the dependence of the sine of (

_{emi}*θ*)

_{z}*with the position of the emission source. The average of the sine is calculated as follows:*

_{c}*K*is the complete elliptic integral of the first kind [32]. The dependence of sin(

*θ*)

_{z}*on the position of the emission point source*

_{c}*x*along the path of the refracting beam in the fiber core has been plotted in Fig. 5 . In the same figure we have included the averages of the sine of (

_{Q}*θ*)

_{z}*obtained from Eq. (6) and approximately estimated from Eq. (8) as well as the value of sin(*

_{c}*θ*)

_{z}*for bound rays (see Eq. (3)). The symmetrical curve obtained indicates that the closer to the core-cladding interface the point source is placed, the larger its maximum acceptance angle is. It can be observed that sin(*

_{c}*θ*)

_{z}*increases from the value corresponding to bound rays (i.e. sin(*

_{c}*θ*)

_{z}*for one point source placed at the center of the fiber) to <sin(*

_{c}*θ*)

_{z}*obtained from Eq. (6). The relative difference between sin(*

_{c}>*θ*)

_{z}*obtained from Eq. (3) and from Eq. (6) is around 12%, whereas the relative difference between the values obtained from Eqs. (6) and (8) is only 1.5% (much smaller, as expected).*

_{c}*n*(

_{ratio}*λ*) of the values for sin(

_{emi}*θ*)

_{z}*obtained from Eq. (6), Eq. (8) and Eq. (3). The values obtained from Eq. (6) and those calculated in the reduced region (Eq. (8)) differ in less than 2% at any value of*

_{c}*n*(

_{ratio}*λ*). On the other hand, the difference between the sine of the maximum acceptance angle obtained from Eq. (3) and that from Eq. (6) increases with

_{emi}*n*(

_{ratio}*λ*), particularly from 3% to 17%. It can be verified that, if

_{emi}*n*(

_{ratio}*λ*) tends to 1, the values of the average sin(

_{emi}*θ*)

_{z}*obtained from Eqs. (6) and (8) are equal, and that the quotient between this average value and that obtained from Eq. (3) can be expressed as:where Catalan constant has a numerical value ≅ 0.915966. This expression gives the maximum difference between the average value and the meridional one (17%).*

_{c}*θ*)

_{z}*with the lateral height of the incident beam. The geometrical situation corresponding to this excitation was plotted in Figs. 1(b) and 3(b). The calculation of the sine of (*

_{c}*θ*)

_{z}*for any point source located at the refracting beam (at point*

_{c}*Q*) in the

*xy*-plane gives the following expression [29

29. C.-A. Bunge, R. Kruglov, and H. Poisel, “Rayleigh and Mie scattering in polymer optical fibers,” J. Lightwave Technol. **24**(8), 3137–3146 (2006). [CrossRef]

*x'*, or position of the source along the refracting beam (the

_{Q}*x'*-axis is directed along the path covered by the refracted beam);

*ϕ*, or angle of the emitted beam relative to the

_{x}*x*-axis in the

*xy*-plane; and

*y*

_{P}_{,}or height of the incident point on the fiber. In contrast to the expression of sin(

*θ*)

_{z}*obtained for the angular dependence, sin(*

_{c}*θ*)

_{z}*depends now on the refractive index of the fiber core at the excitation wavelength, in addition to the refractive indices of the core and the cladding at the emission wavelength.*

_{c}*x'*), the direction of the emitted beam (

_{Q}*ϕ*) and the lateral height (

_{x}*y*) is found in order to obtain real-valued arguments for the sine function in Eq. (12). This restriction defines a valid region where bound and tunneling rays are created within the fiber. It can be expressed as:

_{P}*θ*)

_{z}*can be interpreted as non-propagating modes whose energy is stored close to fiber discontinuities. As an example, Fig. 7 shows the representation of sin(*

_{c}*θ*)

_{z}*(*

_{c}*ϕ*,

_{x}*x'*) as a function of

_{Q}*ϕ*and

_{x}*x'*for a lateral height

_{Q}*y*=

_{P}*/2, in which there are some forbidden regions corresponding to specific values of*

_{core}*ϕ*and

_{x}*x'*whose maximum acceptance angle is not defined. The results correspond to a doped POF fiber with

_{Q}*NA*= 0.5,

_{meriodional}*n*(

_{core}*λ*) = 1.495, and

_{emi}*= 490 μm. The refractive index of the fiber core at*

_{core}*λ*has been taken to be the same as that at

_{exc}*λ*. In contrast to Fig. 4, the curves are not symmetrical with respect to

_{emi}*x'*= 0 and to

_{Q}*ϕ*=

_{x}*π*. Notice that if variable

*y*were 0, Eqs. (12) and (13) would be simplified to Eqs. (2) and (4), respectively.

_{P}*y*can be calculated over the region R where Eq. (13) is satisfied, i.e.

_{P}*≤ 2*

_{x}*π*, and

*x'*(min) ≤

_{Q}*x'*≤

_{Q}*x'*(max), where

_{Q}*x’*and

_{Q}(min)*x’*are defined asthe average of the sine of the critical angle at an incidence

_{Q}(max)*y*can be calculated as follows:

_{P}*θ*)

_{z}*for the emitted light in the fiber by neglecting the contribution of some tunneling rays. As we have shown before, the differences between the <sin(*

_{c}*θ*)

_{z}*> from the exact Eq. (14) and the approximated Eq. (16) are expected to be small [30].*

_{c}*θ*)

_{z}*with the incidence position*

_{c}*y*for the fiber analyzed can be observed in Fig. 8 . The obtained symmetrical curves indicate that the closer the incidence point is to the core-cladding interface, the larger the maximum acceptance angle is. By integrating the previous equation with respect to

_{P}*y*we obtain the average value of the sine of the critical angle for all the possible lateral incidences

_{P}*y*, from −

_{P}*ρ*to +

_{core}*ρ*. This value of <sin(

_{core}*θ*)

_{z}*> would correspond to the situation where all the section of the fiber is illuminated laterally, for instance by using a cylindrical lens. The obtained average values have been included in Fig. 8 together with the sine of the maximum acceptance angle for bound rays (Eq. (3)). The values for <sin(*

_{c}*θ*)

_{z}*> obtained from the exact average of sin(*

_{c}*θ*)

_{z}*(Eq. (14)) and from Eq. (16) differ from the value obtained from Eq. (3) in around 12% and 11%, respectively, for the analyzed fiber.*

_{c}## 4. Results and discussion

*α*= −45°,

_{i}*α*= 0° and

_{i}*α*= +45°.

_{i}*NA*have been calculated and plotted in Fig. 10 . In agreement with the theoretical model, we can observe that

*NA*hardly varies with the angle of incidence of the exciting beam. In the same figure we can observe the fitting of the experimental

*NA*to the equation

*NA*= <sin(

*θ*)

_{z}*>*

_{c}*n*(

_{core}*λ*) where <sin(

_{emi}*θ*)

_{z}*> is given by Eq. (6). As can be seen, the agreement obtained is very good. From the fitting, the following values of the parameters have been obtained:*

_{c}*NA*= 0.49 and

_{meridional}*n*(

_{core}*λ*) = 1.495. These are almost identical to the characteristic parameters of the fiber used in the previous section.

_{emi}*y*= −450 μm,

_{P}*y*= 0 μm and

_{P}*y*= +450 μm.

_{P}*NA*of the emission at the output of the fiber as a function of

*y*is shown in Fig. 12 . As expected from theoretical predictions, we detect a noticeable increase in the

_{P}*NA*as the incidence beam moves up and down towards the upper and lower edges of the fiber, with a minimum value when the fiber is excited at

*y*= 0 (

_{P}*NA*= 0.557). The fitting of the experimental values to the equation

*NA*= <sin(

*θ*)

_{z}*>(*

_{c}*y*)

_{P}*n*(

_{core}*λ*), where <sin(

_{emi}*θ*)

_{z}*>(*

_{c}*y*) is given by Eq. (14), has also been included in Fig. 12. The fiber parameters obtained from this fitting are the same as those obtained in the fitting of Fig. 10 (

_{P}*NA*= 0.49 and

_{meridional}*n*(

_{core}*λ*) =

_{emi}*n*(

_{core}*λ*) = 1.495). As can be observed, both curves are in fairly good agreement again. Finally, the experimental value for

_{exc}*NA*obtained by exciting all the lateral surface of the fiber with a cylindrical lens has been included in the figure as a dashed line and it has been compared with the theoretical calculation of the average sin(

*θ*)

_{z}*for all the possible lateral incidences*

_{c}*y*, from −

_{P}*ρ*to +

_{core}*ρ*(dash-dotted line). It can be observed that the measured value is very close to the theoretically predicted average <sin(

_{core}*θ*)

_{z}*> (0.575 and 0.572, respectively). On the other hand, we can also notice that the meridional*

_{c}*NA*(0.49) is 15% lower than the averaged

*NA*.

*NA*measured at the end of the fiber. We can observe that there is a low decrease in the angular distribution of the emission along the doped POF sample. Specifically, the measured

*NA*decreases from 0.571 to 0.558 (2%) in 17 centimeters, which is the propagation distance used in our measurements. This very slight reduction in the value of

*NA*with propagation distance validates the assumption made in the theoretical model, in which propagation losses in the calculation of the critical angle are neglected.

## 5. Conclusions

## Acknowledgments

## References and links

1. | J. Zubia and J. Arrue, “Plastic optical fibers: an introduction to their technological processes and applications,” Opt. Fiber Technol. |

2. | T. Kaino, “Polymer optical fibers,” in |

3. | O. Ziemann, J. Krauser, P. E. Zamzow, and W. Daum, |

4. | D. Kalymnios, P. Scully, J. Zubia, and H. Poisel, “POF sensors overview,” in |

5. | H. Liang, Z. Zheng, Z. Li, J. Xu, B. Chen, H. Zhao, Q. Zhang, and H. Ming, “Fabrication and amplification of rhodamine B-doped step-index polymer optical fiber,” J. Appl. Polym. Sci. |

6. | A. Tagaya, S. Teramoto, E. Nihei, K. Sasaki, and Y. Koike, “High-power and high-gain organic dye-doped polymer optical fiber amplifiers: novel techniques for preparation and spectral investigation,” Appl. Opt. |

7. | J. Clark and G. Lanzani, “Organic photonics for communications,” Nat. Photonics |

8. | C. Pulido and O. Esteban, “Improved fluorescence signal with tapered polymer optical fibers under side-illumination,” Sens. Actuators B Chem. |

9. | C. Pulido and O. Esteban, “Multiple fluorescence sensing with side-pumped tapered polymer fiber,” Sens. Actuators B Chem. |

10. | H. Y. Tam, C.-F. Jeff-Pun, G. Zhou, X. Cheng, and M. L. V. Tse, “Special structured polymer fibers for sensing applications,” Opt. Fiber Technol. |

11. | M. Sheeba, K. J. Thomas, M. Rajesh, V. P. N. Nampoori, C. P. G. Vallabhan, and P. Radhakrishnan, “Multimode laser emission from dye doped polymer optical fiber,” Appl. Opt. |

12. | G. V. Maier, T. N. Kopylova, V. A. Svetlichnyi, V. M. Podgaetskii, S. M. Dolotov, O. V. Ponomareva, A. E. Monich, and E. A. Monich, “Active polymer fibers doped with organic dyes: generation and amplification of coherent radiation,” Quantum Electron. |

13. | J. Clark, L. Bazzana, D. Bradley, J. Gonzalez, G. Lanzani, D. Lidzey, J. Morgado, A. Nocivelli, W. Tsoi, T. Virgili, and R. Xia, “Blue polymer optical fiber amplifiers based on conjugated fluorine oligomers,” J. Nanophoton. |

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. |

15. | R. J. Potter, “Transmission properties of optical fibers,” J. Opt. Soc. Am. |

16. | Y. Xu, A. Cotteden, and N. B. Jones, “A theoretical evaluation of fibre-optic evanescent wave absorption in spectroscopy and sensors,” Opt. Lasers Eng. |

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

18. | W. L. Barnes, S. B. Poole, J. E. Townsend, L. Reekie, D. J. Taylor, and D. N. Payne, “Er3+ -Yb3+ and Er3+ doped fiber lasers,” J. Lightwave Technol. |

19. | C. P. Achenbach and J. H. Cobb, “Computational studies of light acceptance and propagation in straight and curved multimodal active fibres,” J. Opt. A, Pure Appl. Opt. |

20. | I. Ayesta, J. Arrue, F. Jimenez, M. A. Illarramendi, and J. Zubia, “Computational analysis of the amplification features of active plastic optical fibers,” Phys. Status Solidi A |

21. | J. Arrue, F. Jimenez, I. Ayesta, M. A. Illarramendi, and J. Zubia, “Polymer-optical-fiber lasers and amplifiers doped with organic dyes,” Polymers |

22. | Y. Zhao and S. Fleming, “Analysis of the effect of numerical aperture on Pr:ZBLAN upconversion fiber lasers,” Opt. Lett. |

23. | M. J. Adams, |

24. | R. J. Kruhlak and M. G. Kuzyk, “Side-illumination fluorescence spectroscopy. I. Principles,” J. Opt. Soc. Am. B |

25. | R. J. Kruhlak and M. G. Kuzyk, “Side-illumination fluorescence spectroscopy. II. applications to squaraine-dye-doped polymer optical fibers,” J. Opt. Soc. Am. B |

26. | L. Bazzana, G. Lanzani, R. Xia, J. Morgado, S. Schrader, and D. G. Lidzey, “Plastic optical fibers with embedded organic semiconductors for signal amplification,” in |

27. | M. Aslund, S. D. Jackson, J. Canning, A. Teixeira, and K. Lyytikainen-Digweed, “The influence of skew rays on angular losses in air-cladd fibres,” Opt. Commun. |

28. | G. E. Khalil, A. M. Adawi, A. M. Fox, A. Iraqi, and D. G. Lidzey, “Single molecule spectroscopy of red- and green-emitting fluorene-based copolymers,” J. Chem. Phys. |

29. | C.-A. Bunge, R. Kruglov, and H. Poisel, “Rayleigh and Mie scattering in polymer optical fibers,” J. Lightwave Technol. |

30. | M. A. Illarramendi, “Side-illumination scattering theory in step-index polymer optical fibers,” J. Opt. Soc. Am. B (to be published). |

31. | M. G. Kuzyk, |

32. | M. Abramowitz and I. A. Stegun, |

**OCIS Codes**

(060.2300) Fiber optics and optical communications : Fiber measurements

(060.2310) Fiber optics and optical communications : Fiber optics

(060.2400) Fiber optics and optical communications : Fiber properties

(300.2140) Spectroscopy : Emission

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: December 5, 2011

Revised Manuscript: January 23, 2012

Manuscript Accepted: January 23, 2012

Published: February 9, 2012

**Virtual Issues**

Vol. 7, Iss. 4 *Virtual Journal for Biomedical Optics*

**Citation**

Iñaki Bikandi, María Asunción Illarramendi, Joseba Zubia, Jon Arrue, and Felipe Jiménez, "Side-illumination fluorescence critical angle: theory and application to F8BT-doped polymer optical fibers," Opt. Express **20**, 4630-4644 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-4-4630

Sort: Year | Journal | Reset

### References

- J. Zubia and J. Arrue, “Plastic optical fibers: an introduction to their technological processes and applications,” Opt. Fiber Technol.7(2), 101–140 (2001). [CrossRef]
- T. Kaino, “Polymer optical fibers,” in Polymers for Lightwave and Integrated Optics (Marcel Dekker, Inc., 1992), chap. 1.
- O. Ziemann, J. Krauser, P. E. Zamzow, and W. Daum, POF Handbook: Optical Short Range Transmission Systems, 2nd ed. (Springer, 2008).
- D. Kalymnios, P. Scully, J. Zubia, and H. Poisel, “POF sensors overview,” in Proceedings of the 13th international plastic optical fibers conference, (Nürnberg, 2004), pp. 237–244.
- H. Liang, Z. Zheng, Z. Li, J. Xu, B. Chen, H. Zhao, Q. Zhang, and H. Ming, “Fabrication and amplification of rhodamine B-doped step-index polymer optical fiber,” J. Appl. Polym. Sci.93(2), 681–685 (2004). [CrossRef]
- A. Tagaya, S. Teramoto, E. Nihei, K. Sasaki, and Y. Koike, “High-power and high-gain organic dye-doped polymer optical fiber amplifiers: novel techniques for preparation and spectral investigation,” Appl. Opt.36(3), 572–578 (1997). [CrossRef] [PubMed]
- J. Clark and G. Lanzani, “Organic photonics for communications,” Nat. Photonics4(7), 438–446 (2010). [CrossRef]
- C. Pulido and O. Esteban, “Improved fluorescence signal with tapered polymer optical fibers under side-illumination,” Sens. Actuators B Chem.146(1), 190–194 (2010). [CrossRef]
- C. Pulido and O. Esteban, “Multiple fluorescence sensing with side-pumped tapered polymer fiber,” Sens. Actuators B Chem.157(2), 560–564 (2011). [CrossRef]
- H. Y. Tam, C.-F. Jeff-Pun, G. Zhou, X. Cheng, and M. L. V. Tse, “Special structured polymer fibers for sensing applications,” Opt. Fiber Technol.16(6), 357–366 (2010).
- M. Sheeba, K. J. Thomas, M. Rajesh, V. P. N. Nampoori, C. P. G. Vallabhan, and P. Radhakrishnan, “Multimode laser emission from dye doped polymer optical fiber,” Appl. Opt.46(33), 8089–8094 (2007). [CrossRef] [PubMed]
- G. V. Maier, T. N. Kopylova, V. A. Svetlichnyi, V. M. Podgaetskii, S. M. Dolotov, O. V. Ponomareva, A. E. Monich, and E. A. Monich, “Active polymer fibers doped with organic dyes: generation and amplification of coherent radiation,” Quantum Electron.37(1), 53–59 (2007). [CrossRef]
- J. Clark, L. Bazzana, D. Bradley, J. Gonzalez, G. Lanzani, D. Lidzey, J. Morgado, A. Nocivelli, W. Tsoi, T. Virgili, and R. Xia, “Blue polymer optical fiber amplifiers based on conjugated fluorine oligomers,” J. Nanophoton.2(1), 023504 (2008). [CrossRef]
- 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(15), 3220–3226 (2009). [CrossRef]
- R. J. Potter, “Transmission properties of optical fibers,” J. Opt. Soc. Am.51(10), 1079–1089 (1961). [CrossRef]
- Y. Xu, A. Cotteden, and N. B. Jones, “A theoretical evaluation of fibre-optic evanescent wave absorption in spectroscopy and sensors,” Opt. Lasers Eng.44(2), 93–101 (2006). [CrossRef]
- A. W. Snyder and J. D. Love, Optical waveguide theory (Chapman and Hall, 1983).
- W. L. Barnes, S. B. Poole, J. E. Townsend, L. Reekie, D. J. Taylor, and D. N. Payne, “Er3+ -Yb3+ and Er3+ doped fiber lasers,” J. Lightwave Technol.7(10), 1461–1465 (1989). [CrossRef]
- C. P. Achenbach and J. H. Cobb, “Computational studies of light acceptance and propagation in straight and curved multimodal active fibres,” J. Opt. A, Pure Appl. Opt.5(3), 239–249 (2003). [CrossRef]
- I. Ayesta, J. Arrue, F. Jimenez, M. A. Illarramendi, and J. Zubia, “Computational analysis of the amplification features of active plastic optical fibers,” Phys. Status Solidi A208(8), 1845–1848 (2011). [CrossRef]
- J. Arrue, F. Jimenez, I. Ayesta, M. A. Illarramendi, and J. Zubia, “Polymer-optical-fiber lasers and amplifiers doped with organic dyes,” Polymers3(3), 1162–1180 (2011). [CrossRef]
- Y. Zhao and S. Fleming, “Analysis of the effect of numerical aperture on Pr:ZBLAN upconversion fiber lasers,” Opt. Lett.23(5), 373–375 (1998). [CrossRef] [PubMed]
- M. J. Adams, An introduction to optical waveguides (John Wiley & Sons, 1981).
- R. J. Kruhlak and M. G. Kuzyk, “Side-illumination fluorescence spectroscopy. I. Principles,” J. Opt. Soc. Am. B16(10), 1749–1755 (1999). [CrossRef]
- R. J. Kruhlak and M. G. Kuzyk, “Side-illumination fluorescence spectroscopy. II. applications to squaraine-dye-doped polymer optical fibers,” J. Opt. Soc. Am. B16(10), 1756–1767 (1999). [CrossRef]
- L. Bazzana, G. Lanzani, R. Xia, J. Morgado, S. Schrader, and D. G. Lidzey, “Plastic optical fibers with embedded organic semiconductors for signal amplification,” in Proceedings of the 16th international plastic optical fibers conference, (Torino, Italy, 2007), 327–332.
- M. Aslund, S. D. Jackson, J. Canning, A. Teixeira, and K. Lyytikainen-Digweed, “The influence of skew rays on angular losses in air-cladd fibres,” Opt. Commun.262(1), 77–81 (2006). [CrossRef]
- G. E. Khalil, A. M. Adawi, A. M. Fox, A. Iraqi, and D. G. Lidzey, “Single molecule spectroscopy of red- and green-emitting fluorene-based copolymers,” J. Chem. Phys.130(4), 044903 (2009). [CrossRef] [PubMed]
- C.-A. Bunge, R. Kruglov, and H. Poisel, “Rayleigh and Mie scattering in polymer optical fibers,” J. Lightwave Technol.24(8), 3137–3146 (2006). [CrossRef]
- M. A. Illarramendi, “Side-illumination scattering theory in step-index polymer optical fibers,” J. Opt. Soc. Am. B (to be published).
- M. G. Kuzyk, Polymer Fiber Optics: Materials, Physics, and Applications (Taylor and Francis, 2007).
- M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover Publications, 1965).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

### Figures

Fig. 1 |
Fig. 2 |
Fig. 3 |

Fig. 4 |
Fig. 5 |
Fig. 6 |

Fig. 7 |
Fig. 8 |
Fig. 9 |

Fig. 10 |
Fig. 11 |
Fig. 12 |

Fig. 13 |
||

« Previous Article | Next Article »

OSA is a member of CrossRef.