Abstract

In this paper we present a detailed theoretical study that describes the generation of scattered light in step-index polymer optical fibers by using the side-illumination scattering measurement technique. A detailed analysis of the variation of the maximum angle of acceptance within the fiber has been carried out in order to calculate the scattered light as a function of different launching conditions. The theoretical model has been developed by using the Mie theory for spheres in the independent-scatterer approximation.

© 2012 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. Zubia and J. Arrue, “Plastic optical fibers: an introduction to their technological processes and applications,” Opt. Fiber Technol. 7, 101–140 (2001).
    [CrossRef]
  2. T. Kaino, “Polymer optical fibers,” in Polymers for Lightwave and Integrated Optics (Marcel Dekker, 1992), Chap. 1.
  3. O. Ziemann, J. Krauser, P. E. Zamzow, and W. Daum, POF Handbook: Optical Short Range Transmission Systems, 2nd ed. (Springer, 2008).
  4. D. Kalymnios, P. Scully, J. Zubia, and H. Poisel, “POF sensors overview,” in Proceedings of the 13th International Plastic Optical Fibres Conference, (Nürnberg, Germany, 2004), 237–244.
  5. M. C. J. Large, D. Blacket, and C. A. Bunge, “Microstructured polymer optical fibers compared to conventional POF: novel properties and applications,” IEEE Sensors Journal 10, 1213–1217 (2010).
    [CrossRef]
  6. J. Clark and G. Lanzani, “Organic photonics for communications,” Nature Photonics 4, 438–446 (2010).
    [CrossRef]
  7. 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 fiebers doped with organic dyes: generation and amplification of coherent radiation,” Quantum Electron. 37, 53–59 (2007).
    [CrossRef]
  8. 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, 8089–8094 (2007).
    [CrossRef]
  9. 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).
    [CrossRef]
  10. D. Gloge, “Optical power flow in multimode fibers,” Bell Syst. Tech. J. 51, 1767–1783 (1972).
  11. A. W. Snyder and J. D. Love, Optical Waveguide Theory(Chapman and Hall, 1983).
  12. Y. Koike, N. Tanio, and Y. Ohtsuka, “Light scattering and heterogeneities in low-loss poly(methyl methacrylate) glasses,” Macromolecules 22, 1367–1373 (1989).
    [CrossRef]
  13. Y. Koike, S. Matsuoka, and H. E. Bair, “Origin of excess light scattering in poly(methyl methacrylate) glasses,” Macromolecules 25, 4807–4815 (1992).
    [CrossRef]
  14. C. A. Bunge, R. Kruglov, and H. Poisel, “Rayleigh and Mie scattering in polymer optical fibers,” J. Lightwave Technol. 24, 3137–3146 (2006).
  15. M. G. Kuzyk, Polymer Fiber Optics: Materials, Physics, and Applications (Taylor and Francis, 2007).
  16. G. Aldabaldetreku, I. Bikandi, M. A. Illarramendi, G. Durana, and J. Zubia, “A comprehensive analysis of scattering in polymer optical fibers,” Opt. Express 18, 24536–24555 (2010).
    [CrossRef]
  17. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).
  18. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles(Cambridge University, 2002).
  19. M. J. Adams, An Introduction to Optical Waveguide, (John Wiley, 1981).
  20. M. Kerker, The Scattering of Lght and Other Electromagnetic Radiation (Academic, 1969).
  21. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley, 1983).

2010 (3)

M. C. J. Large, D. Blacket, and C. A. Bunge, “Microstructured polymer optical fibers compared to conventional POF: novel properties and applications,” IEEE Sensors Journal 10, 1213–1217 (2010).
[CrossRef]

J. Clark and G. Lanzani, “Organic photonics for communications,” Nature Photonics 4, 438–446 (2010).
[CrossRef]

G. Aldabaldetreku, I. Bikandi, M. A. Illarramendi, G. Durana, and J. Zubia, “A comprehensive analysis of scattering in polymer optical fibers,” Opt. Express 18, 24536–24555 (2010).
[CrossRef]

2009 (1)

2007 (2)

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, 8089–8094 (2007).
[CrossRef]

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 fiebers doped with organic dyes: generation and amplification of coherent radiation,” Quantum Electron. 37, 53–59 (2007).
[CrossRef]

2006 (1)

2001 (1)

J. Zubia and J. Arrue, “Plastic optical fibers: an introduction to their technological processes and applications,” Opt. Fiber Technol. 7, 101–140 (2001).
[CrossRef]

1992 (1)

Y. Koike, S. Matsuoka, and H. E. Bair, “Origin of excess light scattering in poly(methyl methacrylate) glasses,” Macromolecules 25, 4807–4815 (1992).
[CrossRef]

1989 (1)

Y. Koike, N. Tanio, and Y. Ohtsuka, “Light scattering and heterogeneities in low-loss poly(methyl methacrylate) glasses,” Macromolecules 22, 1367–1373 (1989).
[CrossRef]

1972 (1)

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

Adams, M. J.

M. J. Adams, An Introduction to Optical Waveguide, (John Wiley, 1981).

Aldabaldetreku, G.

Arrue, J.

J. Zubia and J. Arrue, “Plastic optical fibers: an introduction to their technological processes and applications,” Opt. Fiber Technol. 7, 101–140 (2001).
[CrossRef]

Bair, H. E.

Y. Koike, S. Matsuoka, and H. E. Bair, “Origin of excess light scattering in poly(methyl methacrylate) glasses,” Macromolecules 25, 4807–4815 (1992).
[CrossRef]

Bazzana, L.

Bikandi, I.

Blacket, D.

M. C. J. Large, D. Blacket, and C. A. Bunge, “Microstructured polymer optical fibers compared to conventional POF: novel properties and applications,” IEEE Sensors Journal 10, 1213–1217 (2010).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley, 1983).

Bunge, C. A.

M. C. J. Large, D. Blacket, and C. A. Bunge, “Microstructured polymer optical fibers compared to conventional POF: novel properties and applications,” IEEE Sensors Journal 10, 1213–1217 (2010).
[CrossRef]

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

Clark, J.

J. Clark and G. Lanzani, “Organic photonics for communications,” Nature Photonics 4, 438–446 (2010).
[CrossRef]

Daum, W.

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

Dolotov, S. M.

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 fiebers doped with organic dyes: generation and amplification of coherent radiation,” Quantum Electron. 37, 53–59 (2007).
[CrossRef]

Durana, G.

Gloge, D.

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

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley, 1983).

Illarramendi, M. A.

Kaino, T.

T. Kaino, “Polymer optical fibers,” in Polymers for Lightwave and Integrated Optics (Marcel Dekker, 1992), Chap. 1.

Kalymnios, D.

D. Kalymnios, P. Scully, J. Zubia, and H. Poisel, “POF sensors overview,” in Proceedings of the 13th International Plastic Optical Fibres Conference, (Nürnberg, Germany, 2004), 237–244.

Kerker, M.

M. Kerker, The Scattering of Lght and Other Electromagnetic Radiation (Academic, 1969).

Koike, Y.

Y. Koike, S. Matsuoka, and H. E. Bair, “Origin of excess light scattering in poly(methyl methacrylate) glasses,” Macromolecules 25, 4807–4815 (1992).
[CrossRef]

Y. Koike, N. Tanio, and Y. Ohtsuka, “Light scattering and heterogeneities in low-loss poly(methyl methacrylate) glasses,” Macromolecules 22, 1367–1373 (1989).
[CrossRef]

Kopylova, T. N.

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 fiebers doped with organic dyes: generation and amplification of coherent radiation,” Quantum Electron. 37, 53–59 (2007).
[CrossRef]

Krauser, J.

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

Kruglov, R.

Kuzyk, M. G.

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

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles(Cambridge University, 2002).

Lanzani, G.

J. Clark and G. Lanzani, “Organic photonics for communications,” Nature Photonics 4, 438–446 (2010).
[CrossRef]

Large, M. C. J.

M. C. J. Large, D. Blacket, and C. A. Bunge, “Microstructured polymer optical fibers compared to conventional POF: novel properties and applications,” IEEE Sensors Journal 10, 1213–1217 (2010).
[CrossRef]

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory(Chapman and Hall, 1983).

Maier, G. V.

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 fiebers doped with organic dyes: generation and amplification of coherent radiation,” Quantum Electron. 37, 53–59 (2007).
[CrossRef]

Matsuoka, S.

Y. Koike, S. Matsuoka, and H. E. Bair, “Origin of excess light scattering in poly(methyl methacrylate) glasses,” Macromolecules 25, 4807–4815 (1992).
[CrossRef]

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles(Cambridge University, 2002).

Monich, A. E.

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 fiebers doped with organic dyes: generation and amplification of coherent radiation,” Quantum Electron. 37, 53–59 (2007).
[CrossRef]

Monich, E. A.

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 fiebers doped with organic dyes: generation and amplification of coherent radiation,” Quantum Electron. 37, 53–59 (2007).
[CrossRef]

Nampoori, V. P. N.

Ohtsuka, Y.

Y. Koike, N. Tanio, and Y. Ohtsuka, “Light scattering and heterogeneities in low-loss poly(methyl methacrylate) glasses,” Macromolecules 22, 1367–1373 (1989).
[CrossRef]

Podgaetskii, V. M.

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 fiebers doped with organic dyes: generation and amplification of coherent radiation,” Quantum Electron. 37, 53–59 (2007).
[CrossRef]

Poisel, H.

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

D. Kalymnios, P. Scully, J. Zubia, and H. Poisel, “POF sensors overview,” in Proceedings of the 13th International Plastic Optical Fibres Conference, (Nürnberg, Germany, 2004), 237–244.

Ponomareva, O. V.

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 fiebers doped with organic dyes: generation and amplification of coherent radiation,” Quantum Electron. 37, 53–59 (2007).
[CrossRef]

Radhakrishnan, P.

Rajesh, M.

Sarasua, J. R.

Scully, P.

D. Kalymnios, P. Scully, J. Zubia, and H. Poisel, “POF sensors overview,” in Proceedings of the 13th International Plastic Optical Fibres Conference, (Nürnberg, Germany, 2004), 237–244.

Sheeba, M.

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory(Chapman and Hall, 1983).

Svetlichnyi, V. A.

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 fiebers doped with organic dyes: generation and amplification of coherent radiation,” Quantum Electron. 37, 53–59 (2007).
[CrossRef]

Tanio, N.

Y. Koike, N. Tanio, and Y. Ohtsuka, “Light scattering and heterogeneities in low-loss poly(methyl methacrylate) glasses,” Macromolecules 22, 1367–1373 (1989).
[CrossRef]

Thomas, K. J.

Travis, L. D.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles(Cambridge University, 2002).

Vallabhan, C. P. G.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

Zamzow, P. E.

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

Ziemann, O.

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

Zubia, J.

G. Aldabaldetreku, I. Bikandi, M. A. Illarramendi, G. Durana, and J. Zubia, “A comprehensive analysis of scattering in polymer optical fibers,” Opt. Express 18, 24536–24555 (2010).
[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, 3220–3226 (2009).
[CrossRef]

J. Zubia and J. Arrue, “Plastic optical fibers: an introduction to their technological processes and applications,” Opt. Fiber Technol. 7, 101–140 (2001).
[CrossRef]

D. Kalymnios, P. Scully, J. Zubia, and H. Poisel, “POF sensors overview,” in Proceedings of the 13th International Plastic Optical Fibres Conference, (Nürnberg, Germany, 2004), 237–244.

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

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

IEEE Sensors Journal (1)

M. C. J. Large, D. Blacket, and C. A. Bunge, “Microstructured polymer optical fibers compared to conventional POF: novel properties and applications,” IEEE Sensors Journal 10, 1213–1217 (2010).
[CrossRef]

J. Lightwave Technol. (2)

Macromolecules (2)

Y. Koike, N. Tanio, and Y. Ohtsuka, “Light scattering and heterogeneities in low-loss poly(methyl methacrylate) glasses,” Macromolecules 22, 1367–1373 (1989).
[CrossRef]

Y. Koike, S. Matsuoka, and H. E. Bair, “Origin of excess light scattering in poly(methyl methacrylate) glasses,” Macromolecules 25, 4807–4815 (1992).
[CrossRef]

Nature Photonics (1)

J. Clark and G. Lanzani, “Organic photonics for communications,” Nature Photonics 4, 438–446 (2010).
[CrossRef]

Opt. Express (1)

Opt. Fiber Technol. (1)

J. Zubia and J. Arrue, “Plastic optical fibers: an introduction to their technological processes and applications,” Opt. Fiber Technol. 7, 101–140 (2001).
[CrossRef]

Quantum Electron. (1)

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 fiebers doped with organic dyes: generation and amplification of coherent radiation,” Quantum Electron. 37, 53–59 (2007).
[CrossRef]

Other (10)

A. W. Snyder and J. D. Love, Optical Waveguide Theory(Chapman and Hall, 1983).

T. Kaino, “Polymer optical fibers,” in Polymers for Lightwave and Integrated Optics (Marcel Dekker, 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 Fibres Conference, (Nürnberg, Germany, 2004), 237–244.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles(Cambridge University, 2002).

M. J. Adams, An Introduction to Optical Waveguide, (John Wiley, 1981).

M. Kerker, The Scattering of Lght and Other Electromagnetic Radiation (Academic, 1969).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley, 1983).

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (16)

Fig. 1.
Fig. 1.

Scattering generated in a fiber with a monochromatic laser beam by side-illumination technique. The scattered light that propagates a distance z within the fiber to one of the fiber ends is detected. (a) Angular scan. α i is the angle made by the incident beam with the normal to the fiber surface at y = 0 μm . The incident beam always lies in the x y -plane. (b) Lateral scan. y P refers to the y -coordinate of the point of incidence relative to y = 0 μm . The incident beam always lies in the x y -plane (parallel to the x -axis).

Fig. 2.
Fig. 2.

Contour plots of sin ( θ z ) c as a function of both x Q and ϕ x corresponding to two fibers with the same core material, n core = 1.49 and radius ρ core = 490 μm , but different cladding material. Black solid lines correspond to sin ( θ z ) c = ± 1 and green solid ones x Q = ± n ρ core . Red painted areas indicate the forbidden region for x Q and ϕ x giving no real critical angle. (Top plot) n clad = 1 . Blue dashed lines correspond to sin ( θ z ) c = ± 0.75 . (Bottom plot) n clad = 1.4036 . Blue dashed lines correspond to sin ( θ z ) c = ± 0.35 .

Fig. 3.
Fig. 3.

Contour plots of sin ( θ z ) c for a scattering center placed at the core-cladding interface [Eq. (25)] as a function of both y P and ϕ x corresponding to a fiber with n core = 1.49 , n clad = 1.4036 , and ρ core = 490 μm . Black solid lines correspond to the limit sin ( θ z ) c = ± 1 . Filled areas indicate the range of values of y P and ϕ x leading to no real critical angle.

Fig. 4.
Fig. 4.

Scattering quality factor plotted versus the particle size of an air sphere (solid line) and a cladding sphere (dashed line) enclosed by core material for a radiation of wavelength λ = 633 nm . For ϕ λ (geometrical optics limit) both scattering quality factors converge to 2.

Fig. 5.
Fig. 5.

Theoretical simulations of I ( α i ) curves for two polarizations of light at 633 nm corresponding to three different particle sizes. Scattering is caused by one air scatterer surrounded by core material placed at the core-cladding interface ( m = 1 / n core ). Solid lines: parallel polarization (TM); dashed lines: vertical polarization (TE). Calculations have been done for a POF with n core = 1.49 , n clad = 1.4036 , and ρ core = 490 μm .

Fig. 6.
Fig. 6.

Theoretical simulations of I ( α i ) curves for two polarizations of light at 633 nm corresponding to three different particle sizes. Scattering is caused by one cladding scatterer surrounded by core material placed at the core-cladding interface ( m = n clad / n core ). Solid lines: parallel polarization (TM); dashed lines: vertical polarization (TE). Calculations have been done for a POF with n core = 1.49 , n clad = 1.4036 and ρ core = 490 μm .

Fig. 7.
Fig. 7.

Theoretical simulations of I ( y P ) curves for two polarizations of light at 633 nm corresponding to three different particle sizes. Scattering is caused by one air scatterer surrounded by core material placed at the core-cladding interface ( m = 1 / n core ). Solid lines: parallel polarization (TM); dashed lines: vertical polarization (TE). Calculations have been done for a POF with n core = 1.49 , n clad = 1.4036 , and ρ core = 490 μm .

Fig. 8.
Fig. 8.

Theoretical simulations of I ( y P ) curves for two polarizations of light at 633 nm corresponding to three different particle sizes. Scattering is caused by one cladding scatterer surrounded by core material placed at the core-cladding interface ( m = n clad / n core ). Solid lines: parallel polarization (TM); dashed lines: vertical polarization (TE). Calculations have been done for a POF with n core = 1.49 , n clad = 1.4036 , and ρ core = 490 μm .

Fig. 9.
Fig. 9.

Theoretical simulations of I ( α i ) curves for TM polarization of light at 633 nm corresponding to different particle sizes. Scattering is caused by one cladding scatterer surrounded by core material placed at the core-cladding interface ( m = n clad / n core ). Calculations have been done for a POF with n core = 1.49 , n clad = 1.4036 , and ρ core = 490 μm . The experimental result corresponds to the curve for parallel polarization of Fig. 5(a) in [16].

Fig. 10.
Fig. 10.

Theoretical simulations of I ( y P ) curves for TE polarization of light at 633 nm corresponding to different particle sizes. Scattering is caused by one cladding scatterer surrounded by core material placed at the core-cladding interface ( m = n clad / n core ). Calculations have been done for a POF with n core = 1.49 , n clad = 1.4036 , and ρ core = 490 μm . The experimental result corresponds to the curve for vertical polarization of Fig. 4(a) in [16].

Fig. 11.
Fig. 11.

Theoretical simulations of I ( α i ) curves for two polarizations of light at 633 nm corresponding to two different particle sizes. Scattering is caused by air scatterers within the core ( m = 1 / n core ). Calculations have been done for a POF with n core = 1.49 , n clad = 1.4036 , and ρ core = 490 μm . Lines with symbols: vertical polarization (TE). Lines without symbols: parallel polarization (TM). Solid lines: using Eq. (5). Dashed lines: using Eq. (12).

Fig. 12.
Fig. 12.

Theoretical simulations of I ( α i ) curves for two polarizations of light at 633 nm corresponding to two different particle sizes. Scattering is caused by cladding material scatterers within the core ( m = n clad / n core ). Calculations have been done for a POF with n core = 1.49 , n clad = 1.4036 , and ρ core = 490 μm . Lines with symbols: vertical polarization (TE). Lines without symbols: parallel polarization (TM). Solid lines: using Eq. (5). Dashed lines: using Eq. (12).

Fig. 13.
Fig. 13.

Theoretical simulations of I ( y P ) curves for two polarizations of light at 633 nm corresponding to two different particle sizes. Scattering is caused by air scatterers within the core ( m = 1 / n core ). Calculations have been done for a POF with n core = 1.49 , n clad = 1.4036 , and ρ core = 490 μm . Lines with symbols: vertical polarization (TE). Lines without symbols: parallel polarization (TM). Solid lines: using Eq. (9). Dashed lines: using Eq. (13).

Fig. 14.
Fig. 14.

Theoretical simulations of I ( y P ) curves for two polarizations of light at 633 nm corresponding to two different particle sizes. Scattering is caused by cladding material scatterers within the core ( m = n clad / n core ). Calculations have been done for a POF with n core = 1.49 , n clad = 1.4036 , and ρ core = 490 μm . Lines with symbols: vertical polarization (TE). Lines without symbols: parallel polarization (TM). Solid lines: using Eq. (9). Dashed lines: using Eq. (13).

Fig. 15.
Fig. 15.

(a) and (b) Geometrical arrangement of the POF with respect to the incident beam for angular scanning. α i is the angle made by the incident beam with the normal to the fiber surface at y = 0 μm . The incident beam i always lies in the x z -plane. s represents the scattered beam, while r the refracted beam. (c) Definition of angles used for theoretical analysis.

Fig. 16.
Fig. 16.

(a) Geometrical arrangement of the POF with respect to the incident beam for lateral scanning. y P refers to the y -coordinate of the point of incidence relative to y = 0 μm . The incident beam always lies in the x y -plane (parallel to the x -axis). s represents the scattered beam, while i and r , the incident and refracted beams. (b) Definition of angles used for theoretical analysis.

Tables (4)

Tables Icon

Table 1. Principal Expressions Used in the Theoretical Calculations of the Scattered Intensity as a Function of the Incident Angle ( α i )a

Tables Icon

Table 2. Principal Equations Used in the Theoretical Calculations of the Scattered Intensity from Scatterers Placed within the Core as a Function of Incident Lateral Height ( y P )a

Tables Icon

Table 3. Principal Equations Used in the Theoretical Calculations of the Scattered Intensity from One Scatterer at the Core-Cladding Interface ( ρ core = x Q 2 + y Q 2 = x P 2 + y P 2 ) as a Function of Incident Lateral Height ( y P )a

Tables Icon

Table 4. Expressions Obtained for the Angular Distribution Function F ( θ z , ϕ x ) in the Rayleigh Approximation for Two Launching Conditions and Two Polarizationsa

Equations (32)

Equations on this page are rendered with MathJax. Learn more.

x = 2 π r λ n sr , m = n scatt n sr ,
sin ( θ z ) c ( 0 , ϕ x ) = 1 ( n clad n core ) 2 = 1 n 2 ,
1 > n 2 ( x Q ρ core sin ϕ x ) 2 for sin ( θ z ) c ( x Q , ϕ x ) = 1 n 2 1 ( x Q ρ core sin ϕ x ) 2 .
F ( θ IS , ϕ PS ) = [ | S 1 ( θ IS ) | 2 sin 2 ϕ PS + | S 2 ( θ IS ) | 2 cos 2 ϕ PS ] .
I TE , TM ( α i ) T TE , TM ( α i ) [ 2 0 arcsin ( n ) d ϕ x ρ core ρ core d x Q cos α r 0 ( θ z ) c ( x Q , ϕ x ) F ( θ z , ϕ x ) | TE , TM sin θ z d θ z + 2 π arcsin ( n ) π d ϕ x ρ core ρ core d x Q cos α r 0 ( θ z ) c ( x Q , ϕ x ) F ( θ z , ϕ x ) | TE , TM sin θ z d θ z + 2 arcsin ( n ) π arcsin ( n ) d ϕ x n ρ core / sin ϕ x + n ρ core / sin ϕ x d x Q cos α r 0 ( θ z ) c ( x Q , ϕ x ) F ( θ z , ϕ x ) | TE , TM sin θ z d θ z ] ,
I TE , TM ( α i ) T TE , TM ( α i ) [ 2 0 arcsin ( n ) d ϕ x 0 ( θ z ) c ( ρ core , ϕ x ) F ( θ z , ϕ x ) | TE , TM sin θ z d θ z d ϕ x + 2 π arcsin ( n ) π d ϕ x 0 ( θ z ) c ( ρ core , ϕ x ) F ( θ z , ϕ x ) | TE , TM sin θ z d θ z d ϕ x ] ,
1 > n 2 ( x Q ) 2 + ( y P n core ) 2 ρ core 2 sin 2 [ ϕ x + arctan ( y P n core x Q ) + arctan ( y P ρ core 2 y P 2 ) arcsin ( y P ρ core n core ) ] .
1 > n 2 sin 2 [ ϕ x + arctan ( y P ρ core 2 y P 2 ) ] .
I TE , TM ( y P ) T TE , TM ( y P ) R d x Q R d ϕ x 0 ( θ z ) c ( x Q , ϕ x , , y P ) F ( θ z , ϕ x ) | TE , TM sin θ z d θ z .
I TE , TM ( y P ) T TM , TE ( y P ) R d ϕ x 0 ( θ z ) c ( ϕ x , y P ) F ( θ z , ϕ x ) | TM , TE sin θ z d θ z ,
arcsin ( n ) arctan ( y P ρ core 2 y P 2 ) or 0 ϕ x arcsin ( n ) arctan ( y P ρ core 2 y P 2 ) , arcsin ( n ) arctan ( y P ρ core 2 y P 2 ) + π ϕ x arcsin ( n ) arctan ( y P ρ core 2 y P 2 ) + π , arcsin ( n ) arctan ( y P ρ core 2 y P 2 ) + 2 π ϕ x arcsin ( n ) arctan ( y P ρ core 2 y P 2 ) + 2 π or 2 π .
I TE , TM ( α i ) T TE , TM ( α i ) 0 2 π d ϕ x n ρ core n ρ core d x Q cos α r 0 ( θ z ) c ( x Q , ϕ x ) F ( θ z , ϕ x ) | TE , TM sin θ z d θ z .
I TE , TM ( y P ) T TE , TM ( y P ) ρ core 2 n 2 ( y P / n core ) 2 ρ core 2 n 2 ( y P / n core ) 2 d x Q 0 2 π d ϕ x 0 ( θ z ) c ( x Q , ϕ x , , y P ) F ( θ z , ϕ x ) | TE , TM sin θ z d θ z .
i = cos α i x + sin α i z , r = cos α r x + sin α r z , n = x .
sin α r = sin α i n clad ,
T TE ( α i ) = 4 cos α i n clad 2 sin 2 α i ( cos α i + n clad 2 sin 2 α i ) 2 ,
T TM ( α i ) = 4 cos α i n clad 2 sin 2 α i ( n clad cos α i + 1 n clad n clad 2 sin 2 α i ) 2 .
s = sin θ z cos ϕ x x sin θ z sin ϕ x y + cos θ z z .
sin α c = n clad n core .
sin ( θ z ) c = cos α c sin ϕ ϕ ,
cos ϕ ϕ = x Q ρ core sin ϕ x .
sin ( θ z ) c ( x Q , ϕ x ) = 1 ( n clad n core ) 2 sin ( arccos ( x Q ρ core sin ϕ x ) ) = 1 n 2 1 ( x Q ρ core sin ϕ x ) 2 .
sin ( θ z ) c ( ρ core , ϕ x ) = 1 n 2 1 ( sin ϕ x ) 2 = 1 n 2 cos ϕ x .
i = x , r = cos ( α i α r ) x sin ( α i α r ) y .
sin α r = sin α i n clad , with α i = arctan ( y P x P ) and ρ core = x P 2 + y P 2 .
T TM ( y P ) = 4 ρ core 2 y P 2 ρ core 2 n clad 2 y P 2 ( n clad ρ core 2 y P 2 + 1 n clad ρ core 2 n clad 2 y P 2 ) 2 ,
T TE ( y P ) = 4 ρ core 2 y P 2 ρ core 2 n clad 2 y P 2 ( ρ core 2 y P 2 + ρ core 2 n clad 2 y P 2 ) 2 .
cos ϕ ϕ = r ρ core sin [ ϕ x + arctan ( y Q x Q ) ] , with r = x Q 2 + y Q 2 .
arctan ( y Q x Q ) = arctan ( y Q x Q ) + α i α r = arctan ( y Q x Q ) + arctan ( y P x P ) arcsin ( sin α i n core ) ,
sin ( θ z ) c ( x Q , ϕ x , y P ) = 1 n 2 sin Δ ,
sin Δ = 1 ( x Q ) 2 + ( y P n core ) 2 ρ core 2 sin 2 [ ϕ x + arctan ( y P n core x Q ) + arctan ( y P ρ core 2 y P 2 ) arcsin ( y P ρ core n core ) ] .
sin ( θ z ) c ( ϕ x , y P ) = 1 n 2 cos [ ϕ x + arctan ( y P ρ core 2 y P 2 ) ] ,

Metrics