Abstract

Čerenkov radiation is generated in optical fibers immersed in radiation fields and can interfere with signal transmission. We develop a theory for predicting the intensity of Čerenkov radiation generated within the core of a multimode optical fiber by using a ray optic approach and use it to make predictions of the intensity of radiation transmitted down the fiber in propagating modes. The intensity transmitted down the fiber is found to be dominated by bound rays with a contribution from tunneling rays. It is confirmed that for relativistic particles the intensity of the radiation that is transmitted along the fiber is a function of the angle between the particle beam and the fiber axis. The angle of peak intensity is found to be a function of the fiber refractive index difference as well as the core refractive index, with larger refractive index differences shifting the peak significantly toward lower angles. The angular range of the distribution is also significantly increased in both directions by increasing the fiber refractive index difference. The intensity of the radiation is found to be proportional to the cube of the fiber core radius in addition to its dependence on refractive index difference. As the particle energy is reduced into the nonrelativistic range the entire distribution is shifted toward lower angles. Recommendations on minimizing the quantity of Čerenkov light transmitted in the fiber optic system in a radiation field are given.

© 2006 Optical Society of America

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  6. M. R. Arnfield, H. E. Gaballa, R. D. Zwicker, Q. Islam, and R. Schmidt-Ullrich, "Radiation-induced light in optical fibers and plastic scintillators: application to brachytherapy dosimetry," IEEE Trans. Nucl. Sci. 43, 2077-2084 (1996).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  18. M. A. Clift, P. N. Johnston, and D. V. Webb, "A temporal method of avoiding the Cerenkov radiation generated in organic scintillator dosimeters by pulsed mega-voltage electron and photon beams," Phys. Med. Biol. 47, 1421-1433 (2002).
    [CrossRef] [PubMed]
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2004

M. Geso, N. Robinson, W. Schumer, and K. Williams, "Use of water-equivalent plastic scintillator for intravascular brachytherapy dosimetry," Australas. Phys. Eng. Sci. Med. 27, 5-10 (2004).
[CrossRef] [PubMed]

M. C. Aznar, C. E. Andersen, L. Botter-Jensen, S. A. J. Back, S. Mattsson, F. Kjaer-Kristoffersen, and J. Medin, "Real-time optical-fibre luminescence dosimetry for radiotherapy: physical characteristics and applications in photon beams," Phys. Med. Biol. 49, 1655-1669 (2004).
[CrossRef] [PubMed]

B. L. Justus, P. Falkenstein, A. L. Huston, M. C. Plazas, H. Ning, and R. W. Miller, "Gated fiber-optic-coupled detector for in vivo real-time radiation dosimetry," Applied Optics 43, 1663-1668 (2004).
[CrossRef] [PubMed]

2002

J. M. Fontbonne, G. Iltis, G. Ban, A. Battala, J. C. Venhes, J. Tillier, N. Bellaize, C. LeBrun, B. Tamain, K. Mercier, and J. C. Motin, "Scintillating fiber dosimeter for radiation therapy accelerator," IEEE Trans. Nucl. Sci. 49, 2223-2227 (2002).
[CrossRef]

M. A. Clift, P. N. Johnston, and D. V. Webb, "A temporal method of avoiding the Cerenkov radiation generated in organic scintillator dosimeters by pulsed mega-voltage electron and photon beams," Phys. Med. Biol. 47, 1421-1433 (2002).
[CrossRef] [PubMed]

J. C. Polf, S. W. S. McKeever, M. S. Akselrod, and S. Holmstrom, "A real-time, fibre-optic dosimetry system using Al2O3 fibres," Radiat. Prot. Dosim. 100, 301-304 (2002).

2000

F. Pain, P. Laniece, R. Mastrippolito, Y. Charon, D. Comar, V. Leviel, J. F. Pujol, and L. Valentin, "SIC, an intracerebral radiosensitive probe for in vivo neuropharmacology investigations in small laboratory animals: theoretical considerations and practical characteristics," IEEE Trans. Nucl. Sci. 47, 25-32 (2000).
[CrossRef]

M. A. Clift, R. A. Sutton, and D. V. Webb, "Dealing with Cerenkov radiation generated in organic scintillator dosimeters by bremsstrahlung beams," Phys. Med. Biol. 45, 1165-1182 (2000).
[CrossRef] [PubMed]

1998

S. Gripp, F. W. Haesing, H. Bueker, and G. Schmitt, "Clinical in vivo dosimetry using optical fibers," Radiat. Oncol. Invest. 6, 142-149 (1998).
[CrossRef]

1996

E. Auffray, D. Bouttet, I. Dafinei, J. Fay, P. Lecoq, J. A. Mares, M. Martini, G. Maze, F. Meinardi, B. Moine, M. Nikl, C. Pedrini, M. Poulain, M. Schneegans, S. Tavernier, and A. Vedda, "Cerium-doped heavy metal fluoride glasses, a possible alternative for electromagnetic calorimetry," Nucl. Instrum. Methods Phys. Res A 380, 524-536 (1996).
[CrossRef]

M. R. Arnfield, H. E. Gaballa, R. D. Zwicker, Q. Islam, and R. Schmidt-Ullrich, "Radiation-induced light in optical fibers and plastic scintillators: application to brachytherapy dosimetry," IEEE Trans. Nucl. Sci. 43, 2077-2084 (1996).
[CrossRef]

1993

S. F. de Boer, A. S. Beddar, and J. A. Rawlinson, "Optical filtering and spectral measurements of radiation-induced light in plastic scintillation dosimetry," Phys. Med. Biol. 38, 945-958 (1993).
[CrossRef]

1992

A. S. Beddar, T. R. Mackie, and F. H. Attix, "Water-equivalent plastic scintillation detectors for high-energy beam dosimetry. II. Properties and measurements," Phys. Med. Biol. 37, 1901-1913 (1992).
[CrossRef] [PubMed]

A. S. Beddar, T. R. Mackie, and F. H. Attix, "Water-equivalent plastic scintillation detectors for high-energy beam dosimetry. I. Physical characteristics and theoretical considerations," Phys. Med. Biol. 37, 1883-1900 (1992).
[CrossRef] [PubMed]

1984

B. L. Pruett, R. T. Peterson, D. E. Smith, L. D. Looney, and R. N. Shelton, Jr., "Gamma-ray to Cerenkov-light conversion efficiency for pure-silica-core optical fibers," in Proc. SPIE 506, 10-17 (1984).

Akselrod, M. S.

J. C. Polf, S. W. S. McKeever, M. S. Akselrod, and S. Holmstrom, "A real-time, fibre-optic dosimetry system using Al2O3 fibres," Radiat. Prot. Dosim. 100, 301-304 (2002).

Andersen, C. E.

M. C. Aznar, C. E. Andersen, L. Botter-Jensen, S. A. J. Back, S. Mattsson, F. Kjaer-Kristoffersen, and J. Medin, "Real-time optical-fibre luminescence dosimetry for radiotherapy: physical characteristics and applications in photon beams," Phys. Med. Biol. 49, 1655-1669 (2004).
[CrossRef] [PubMed]

Arnfield, M. R.

M. R. Arnfield, H. E. Gaballa, R. D. Zwicker, Q. Islam, and R. Schmidt-Ullrich, "Radiation-induced light in optical fibers and plastic scintillators: application to brachytherapy dosimetry," IEEE Trans. Nucl. Sci. 43, 2077-2084 (1996).
[CrossRef]

Attix, F. H.

A. S. Beddar, T. R. Mackie, and F. H. Attix, "Water-equivalent plastic scintillation detectors for high-energy beam dosimetry. I. Physical characteristics and theoretical considerations," Phys. Med. Biol. 37, 1883-1900 (1992).
[CrossRef] [PubMed]

A. S. Beddar, T. R. Mackie, and F. H. Attix, "Water-equivalent plastic scintillation detectors for high-energy beam dosimetry. II. Properties and measurements," Phys. Med. Biol. 37, 1901-1913 (1992).
[CrossRef] [PubMed]

Auffray, E.

E. Auffray, D. Bouttet, I. Dafinei, J. Fay, P. Lecoq, J. A. Mares, M. Martini, G. Maze, F. Meinardi, B. Moine, M. Nikl, C. Pedrini, M. Poulain, M. Schneegans, S. Tavernier, and A. Vedda, "Cerium-doped heavy metal fluoride glasses, a possible alternative for electromagnetic calorimetry," Nucl. Instrum. Methods Phys. Res A 380, 524-536 (1996).
[CrossRef]

Aznar, M. C.

M. C. Aznar, C. E. Andersen, L. Botter-Jensen, S. A. J. Back, S. Mattsson, F. Kjaer-Kristoffersen, and J. Medin, "Real-time optical-fibre luminescence dosimetry for radiotherapy: physical characteristics and applications in photon beams," Phys. Med. Biol. 49, 1655-1669 (2004).
[CrossRef] [PubMed]

Back, S. A. J.

M. C. Aznar, C. E. Andersen, L. Botter-Jensen, S. A. J. Back, S. Mattsson, F. Kjaer-Kristoffersen, and J. Medin, "Real-time optical-fibre luminescence dosimetry for radiotherapy: physical characteristics and applications in photon beams," Phys. Med. Biol. 49, 1655-1669 (2004).
[CrossRef] [PubMed]

Ban, G.

J. M. Fontbonne, G. Iltis, G. Ban, A. Battala, J. C. Venhes, J. Tillier, N. Bellaize, C. LeBrun, B. Tamain, K. Mercier, and J. C. Motin, "Scintillating fiber dosimeter for radiation therapy accelerator," IEEE Trans. Nucl. Sci. 49, 2223-2227 (2002).
[CrossRef]

Battala, A.

J. M. Fontbonne, G. Iltis, G. Ban, A. Battala, J. C. Venhes, J. Tillier, N. Bellaize, C. LeBrun, B. Tamain, K. Mercier, and J. C. Motin, "Scintillating fiber dosimeter for radiation therapy accelerator," IEEE Trans. Nucl. Sci. 49, 2223-2227 (2002).
[CrossRef]

Beddar, A. S.

S. F. de Boer, A. S. Beddar, and J. A. Rawlinson, "Optical filtering and spectral measurements of radiation-induced light in plastic scintillation dosimetry," Phys. Med. Biol. 38, 945-958 (1993).
[CrossRef]

A. S. Beddar, T. R. Mackie, and F. H. Attix, "Water-equivalent plastic scintillation detectors for high-energy beam dosimetry. I. Physical characteristics and theoretical considerations," Phys. Med. Biol. 37, 1883-1900 (1992).
[CrossRef] [PubMed]

A. S. Beddar, T. R. Mackie, and F. H. Attix, "Water-equivalent plastic scintillation detectors for high-energy beam dosimetry. II. Properties and measurements," Phys. Med. Biol. 37, 1901-1913 (1992).
[CrossRef] [PubMed]

S. Law and A. S. Beddar, "Capture of Cerenkov radiation generated on the axis of an optical fibre when the fibre axis lies on the Cerenkov Cone," presented at the Conference on the Optical Internet/Australian Conference on Optical Fibre Technology 2003, Melbourne, Australia, 13-16 July 2003.

Bellaize, N.

J. M. Fontbonne, G. Iltis, G. Ban, A. Battala, J. C. Venhes, J. Tillier, N. Bellaize, C. LeBrun, B. Tamain, K. Mercier, and J. C. Motin, "Scintillating fiber dosimeter for radiation therapy accelerator," IEEE Trans. Nucl. Sci. 49, 2223-2227 (2002).
[CrossRef]

Botter-Jensen, L.

M. C. Aznar, C. E. Andersen, L. Botter-Jensen, S. A. J. Back, S. Mattsson, F. Kjaer-Kristoffersen, and J. Medin, "Real-time optical-fibre luminescence dosimetry for radiotherapy: physical characteristics and applications in photon beams," Phys. Med. Biol. 49, 1655-1669 (2004).
[CrossRef] [PubMed]

Bouttet, D.

E. Auffray, D. Bouttet, I. Dafinei, J. Fay, P. Lecoq, J. A. Mares, M. Martini, G. Maze, F. Meinardi, B. Moine, M. Nikl, C. Pedrini, M. Poulain, M. Schneegans, S. Tavernier, and A. Vedda, "Cerium-doped heavy metal fluoride glasses, a possible alternative for electromagnetic calorimetry," Nucl. Instrum. Methods Phys. Res A 380, 524-536 (1996).
[CrossRef]

Bueker, H.

S. Gripp, F. W. Haesing, H. Bueker, and G. Schmitt, "Clinical in vivo dosimetry using optical fibers," Radiat. Oncol. Invest. 6, 142-149 (1998).
[CrossRef]

Charon, Y.

F. Pain, P. Laniece, R. Mastrippolito, Y. Charon, D. Comar, V. Leviel, J. F. Pujol, and L. Valentin, "SIC, an intracerebral radiosensitive probe for in vivo neuropharmacology investigations in small laboratory animals: theoretical considerations and practical characteristics," IEEE Trans. Nucl. Sci. 47, 25-32 (2000).
[CrossRef]

Clift, M. A.

M. A. Clift, P. N. Johnston, and D. V. Webb, "A temporal method of avoiding the Cerenkov radiation generated in organic scintillator dosimeters by pulsed mega-voltage electron and photon beams," Phys. Med. Biol. 47, 1421-1433 (2002).
[CrossRef] [PubMed]

M. A. Clift, R. A. Sutton, and D. V. Webb, "Dealing with Cerenkov radiation generated in organic scintillator dosimeters by bremsstrahlung beams," Phys. Med. Biol. 45, 1165-1182 (2000).
[CrossRef] [PubMed]

Comar, D.

F. Pain, P. Laniece, R. Mastrippolito, Y. Charon, D. Comar, V. Leviel, J. F. Pujol, and L. Valentin, "SIC, an intracerebral radiosensitive probe for in vivo neuropharmacology investigations in small laboratory animals: theoretical considerations and practical characteristics," IEEE Trans. Nucl. Sci. 47, 25-32 (2000).
[CrossRef]

Dafinei, I.

E. Auffray, D. Bouttet, I. Dafinei, J. Fay, P. Lecoq, J. A. Mares, M. Martini, G. Maze, F. Meinardi, B. Moine, M. Nikl, C. Pedrini, M. Poulain, M. Schneegans, S. Tavernier, and A. Vedda, "Cerium-doped heavy metal fluoride glasses, a possible alternative for electromagnetic calorimetry," Nucl. Instrum. Methods Phys. Res A 380, 524-536 (1996).
[CrossRef]

de Boer, S. F.

S. F. de Boer, A. S. Beddar, and J. A. Rawlinson, "Optical filtering and spectral measurements of radiation-induced light in plastic scintillation dosimetry," Phys. Med. Biol. 38, 945-958 (1993).
[CrossRef]

Falkenstein, P.

B. L. Justus, P. Falkenstein, A. L. Huston, M. C. Plazas, H. Ning, and R. W. Miller, "Gated fiber-optic-coupled detector for in vivo real-time radiation dosimetry," Applied Optics 43, 1663-1668 (2004).
[CrossRef] [PubMed]

Fay, J.

E. Auffray, D. Bouttet, I. Dafinei, J. Fay, P. Lecoq, J. A. Mares, M. Martini, G. Maze, F. Meinardi, B. Moine, M. Nikl, C. Pedrini, M. Poulain, M. Schneegans, S. Tavernier, and A. Vedda, "Cerium-doped heavy metal fluoride glasses, a possible alternative for electromagnetic calorimetry," Nucl. Instrum. Methods Phys. Res A 380, 524-536 (1996).
[CrossRef]

Fontbonne, J. M.

J. M. Fontbonne, G. Iltis, G. Ban, A. Battala, J. C. Venhes, J. Tillier, N. Bellaize, C. LeBrun, B. Tamain, K. Mercier, and J. C. Motin, "Scintillating fiber dosimeter for radiation therapy accelerator," IEEE Trans. Nucl. Sci. 49, 2223-2227 (2002).
[CrossRef]

Gaballa, H. E.

M. R. Arnfield, H. E. Gaballa, R. D. Zwicker, Q. Islam, and R. Schmidt-Ullrich, "Radiation-induced light in optical fibers and plastic scintillators: application to brachytherapy dosimetry," IEEE Trans. Nucl. Sci. 43, 2077-2084 (1996).
[CrossRef]

Geso, M.

M. Geso, N. Robinson, W. Schumer, and K. Williams, "Use of water-equivalent plastic scintillator for intravascular brachytherapy dosimetry," Australas. Phys. Eng. Sci. Med. 27, 5-10 (2004).
[CrossRef] [PubMed]

Gladney, L. D.

L. D. Gladney, "Cerenkov Radiation," University of Pennsylvania, Department of Physics and Astronomy, 2000, retrieved 2004, http://dept.physics.upenn.edu/balloon/cerenkov_radiation.html.

Gripp, S.

S. Gripp, F. W. Haesing, H. Bueker, and G. Schmitt, "Clinical in vivo dosimetry using optical fibers," Radiat. Oncol. Invest. 6, 142-149 (1998).
[CrossRef]

Haesing, F. W.

S. Gripp, F. W. Haesing, H. Bueker, and G. Schmitt, "Clinical in vivo dosimetry using optical fibers," Radiat. Oncol. Invest. 6, 142-149 (1998).
[CrossRef]

Holmstrom, S.

J. C. Polf, S. W. S. McKeever, M. S. Akselrod, and S. Holmstrom, "A real-time, fibre-optic dosimetry system using Al2O3 fibres," Radiat. Prot. Dosim. 100, 301-304 (2002).

Huston, A. L.

B. L. Justus, P. Falkenstein, A. L. Huston, M. C. Plazas, H. Ning, and R. W. Miller, "Gated fiber-optic-coupled detector for in vivo real-time radiation dosimetry," Applied Optics 43, 1663-1668 (2004).
[CrossRef] [PubMed]

Iltis, G.

J. M. Fontbonne, G. Iltis, G. Ban, A. Battala, J. C. Venhes, J. Tillier, N. Bellaize, C. LeBrun, B. Tamain, K. Mercier, and J. C. Motin, "Scintillating fiber dosimeter for radiation therapy accelerator," IEEE Trans. Nucl. Sci. 49, 2223-2227 (2002).
[CrossRef]

Islam, Q.

M. R. Arnfield, H. E. Gaballa, R. D. Zwicker, Q. Islam, and R. Schmidt-Ullrich, "Radiation-induced light in optical fibers and plastic scintillators: application to brachytherapy dosimetry," IEEE Trans. Nucl. Sci. 43, 2077-2084 (1996).
[CrossRef]

Johnston, P. N.

M. A. Clift, P. N. Johnston, and D. V. Webb, "A temporal method of avoiding the Cerenkov radiation generated in organic scintillator dosimeters by pulsed mega-voltage electron and photon beams," Phys. Med. Biol. 47, 1421-1433 (2002).
[CrossRef] [PubMed]

Justus, B. L.

B. L. Justus, P. Falkenstein, A. L. Huston, M. C. Plazas, H. Ning, and R. W. Miller, "Gated fiber-optic-coupled detector for in vivo real-time radiation dosimetry," Applied Optics 43, 1663-1668 (2004).
[CrossRef] [PubMed]

Kjaer-Kristoffersen, F.

M. C. Aznar, C. E. Andersen, L. Botter-Jensen, S. A. J. Back, S. Mattsson, F. Kjaer-Kristoffersen, and J. Medin, "Real-time optical-fibre luminescence dosimetry for radiotherapy: physical characteristics and applications in photon beams," Phys. Med. Biol. 49, 1655-1669 (2004).
[CrossRef] [PubMed]

Laniece, P.

F. Pain, P. Laniece, R. Mastrippolito, Y. Charon, D. Comar, V. Leviel, J. F. Pujol, and L. Valentin, "SIC, an intracerebral radiosensitive probe for in vivo neuropharmacology investigations in small laboratory animals: theoretical considerations and practical characteristics," IEEE Trans. Nucl. Sci. 47, 25-32 (2000).
[CrossRef]

Law, S.

S. Law and A. S. Beddar, "Capture of Cerenkov radiation generated on the axis of an optical fibre when the fibre axis lies on the Cerenkov Cone," presented at the Conference on the Optical Internet/Australian Conference on Optical Fibre Technology 2003, Melbourne, Australia, 13-16 July 2003.

LeBrun, C.

J. M. Fontbonne, G. Iltis, G. Ban, A. Battala, J. C. Venhes, J. Tillier, N. Bellaize, C. LeBrun, B. Tamain, K. Mercier, and J. C. Motin, "Scintillating fiber dosimeter for radiation therapy accelerator," IEEE Trans. Nucl. Sci. 49, 2223-2227 (2002).
[CrossRef]

Lecoq, P.

E. Auffray, D. Bouttet, I. Dafinei, J. Fay, P. Lecoq, J. A. Mares, M. Martini, G. Maze, F. Meinardi, B. Moine, M. Nikl, C. Pedrini, M. Poulain, M. Schneegans, S. Tavernier, and A. Vedda, "Cerium-doped heavy metal fluoride glasses, a possible alternative for electromagnetic calorimetry," Nucl. Instrum. Methods Phys. Res A 380, 524-536 (1996).
[CrossRef]

Leviel, V.

F. Pain, P. Laniece, R. Mastrippolito, Y. Charon, D. Comar, V. Leviel, J. F. Pujol, and L. Valentin, "SIC, an intracerebral radiosensitive probe for in vivo neuropharmacology investigations in small laboratory animals: theoretical considerations and practical characteristics," IEEE Trans. Nucl. Sci. 47, 25-32 (2000).
[CrossRef]

Looney, L. D.

B. L. Pruett, R. T. Peterson, D. E. Smith, L. D. Looney, and R. N. Shelton, Jr., "Gamma-ray to Cerenkov-light conversion efficiency for pure-silica-core optical fibers," in Proc. SPIE 506, 10-17 (1984).

Love, J. D.

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

Mackie, T. R.

A. S. Beddar, T. R. Mackie, and F. H. Attix, "Water-equivalent plastic scintillation detectors for high-energy beam dosimetry. II. Properties and measurements," Phys. Med. Biol. 37, 1901-1913 (1992).
[CrossRef] [PubMed]

A. S. Beddar, T. R. Mackie, and F. H. Attix, "Water-equivalent plastic scintillation detectors for high-energy beam dosimetry. I. Physical characteristics and theoretical considerations," Phys. Med. Biol. 37, 1883-1900 (1992).
[CrossRef] [PubMed]

Mares, J. A.

E. Auffray, D. Bouttet, I. Dafinei, J. Fay, P. Lecoq, J. A. Mares, M. Martini, G. Maze, F. Meinardi, B. Moine, M. Nikl, C. Pedrini, M. Poulain, M. Schneegans, S. Tavernier, and A. Vedda, "Cerium-doped heavy metal fluoride glasses, a possible alternative for electromagnetic calorimetry," Nucl. Instrum. Methods Phys. Res A 380, 524-536 (1996).
[CrossRef]

Martini, M.

E. Auffray, D. Bouttet, I. Dafinei, J. Fay, P. Lecoq, J. A. Mares, M. Martini, G. Maze, F. Meinardi, B. Moine, M. Nikl, C. Pedrini, M. Poulain, M. Schneegans, S. Tavernier, and A. Vedda, "Cerium-doped heavy metal fluoride glasses, a possible alternative for electromagnetic calorimetry," Nucl. Instrum. Methods Phys. Res A 380, 524-536 (1996).
[CrossRef]

Mastrippolito, R.

F. Pain, P. Laniece, R. Mastrippolito, Y. Charon, D. Comar, V. Leviel, J. F. Pujol, and L. Valentin, "SIC, an intracerebral radiosensitive probe for in vivo neuropharmacology investigations in small laboratory animals: theoretical considerations and practical characteristics," IEEE Trans. Nucl. Sci. 47, 25-32 (2000).
[CrossRef]

Mattsson, S.

M. C. Aznar, C. E. Andersen, L. Botter-Jensen, S. A. J. Back, S. Mattsson, F. Kjaer-Kristoffersen, and J. Medin, "Real-time optical-fibre luminescence dosimetry for radiotherapy: physical characteristics and applications in photon beams," Phys. Med. Biol. 49, 1655-1669 (2004).
[CrossRef] [PubMed]

Maze, G.

E. Auffray, D. Bouttet, I. Dafinei, J. Fay, P. Lecoq, J. A. Mares, M. Martini, G. Maze, F. Meinardi, B. Moine, M. Nikl, C. Pedrini, M. Poulain, M. Schneegans, S. Tavernier, and A. Vedda, "Cerium-doped heavy metal fluoride glasses, a possible alternative for electromagnetic calorimetry," Nucl. Instrum. Methods Phys. Res A 380, 524-536 (1996).
[CrossRef]

McKeever, S. W. S.

J. C. Polf, S. W. S. McKeever, M. S. Akselrod, and S. Holmstrom, "A real-time, fibre-optic dosimetry system using Al2O3 fibres," Radiat. Prot. Dosim. 100, 301-304 (2002).

Medin, J.

M. C. Aznar, C. E. Andersen, L. Botter-Jensen, S. A. J. Back, S. Mattsson, F. Kjaer-Kristoffersen, and J. Medin, "Real-time optical-fibre luminescence dosimetry for radiotherapy: physical characteristics and applications in photon beams," Phys. Med. Biol. 49, 1655-1669 (2004).
[CrossRef] [PubMed]

Meinardi, F.

E. Auffray, D. Bouttet, I. Dafinei, J. Fay, P. Lecoq, J. A. Mares, M. Martini, G. Maze, F. Meinardi, B. Moine, M. Nikl, C. Pedrini, M. Poulain, M. Schneegans, S. Tavernier, and A. Vedda, "Cerium-doped heavy metal fluoride glasses, a possible alternative for electromagnetic calorimetry," Nucl. Instrum. Methods Phys. Res A 380, 524-536 (1996).
[CrossRef]

Mercier, K.

J. M. Fontbonne, G. Iltis, G. Ban, A. Battala, J. C. Venhes, J. Tillier, N. Bellaize, C. LeBrun, B. Tamain, K. Mercier, and J. C. Motin, "Scintillating fiber dosimeter for radiation therapy accelerator," IEEE Trans. Nucl. Sci. 49, 2223-2227 (2002).
[CrossRef]

Miller, R. W.

B. L. Justus, P. Falkenstein, A. L. Huston, M. C. Plazas, H. Ning, and R. W. Miller, "Gated fiber-optic-coupled detector for in vivo real-time radiation dosimetry," Applied Optics 43, 1663-1668 (2004).
[CrossRef] [PubMed]

Moine, B.

E. Auffray, D. Bouttet, I. Dafinei, J. Fay, P. Lecoq, J. A. Mares, M. Martini, G. Maze, F. Meinardi, B. Moine, M. Nikl, C. Pedrini, M. Poulain, M. Schneegans, S. Tavernier, and A. Vedda, "Cerium-doped heavy metal fluoride glasses, a possible alternative for electromagnetic calorimetry," Nucl. Instrum. Methods Phys. Res A 380, 524-536 (1996).
[CrossRef]

Motin, J. C.

J. M. Fontbonne, G. Iltis, G. Ban, A. Battala, J. C. Venhes, J. Tillier, N. Bellaize, C. LeBrun, B. Tamain, K. Mercier, and J. C. Motin, "Scintillating fiber dosimeter for radiation therapy accelerator," IEEE Trans. Nucl. Sci. 49, 2223-2227 (2002).
[CrossRef]

Nikl, M.

E. Auffray, D. Bouttet, I. Dafinei, J. Fay, P. Lecoq, J. A. Mares, M. Martini, G. Maze, F. Meinardi, B. Moine, M. Nikl, C. Pedrini, M. Poulain, M. Schneegans, S. Tavernier, and A. Vedda, "Cerium-doped heavy metal fluoride glasses, a possible alternative for electromagnetic calorimetry," Nucl. Instrum. Methods Phys. Res A 380, 524-536 (1996).
[CrossRef]

Ning, H.

B. L. Justus, P. Falkenstein, A. L. Huston, M. C. Plazas, H. Ning, and R. W. Miller, "Gated fiber-optic-coupled detector for in vivo real-time radiation dosimetry," Applied Optics 43, 1663-1668 (2004).
[CrossRef] [PubMed]

Pain, F.

F. Pain, P. Laniece, R. Mastrippolito, Y. Charon, D. Comar, V. Leviel, J. F. Pujol, and L. Valentin, "SIC, an intracerebral radiosensitive probe for in vivo neuropharmacology investigations in small laboratory animals: theoretical considerations and practical characteristics," IEEE Trans. Nucl. Sci. 47, 25-32 (2000).
[CrossRef]

Panofsky, W. K. H.

W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism, 2nd ed. (Addison-Wesley, 1962), p. 494.

Pedrini, C.

E. Auffray, D. Bouttet, I. Dafinei, J. Fay, P. Lecoq, J. A. Mares, M. Martini, G. Maze, F. Meinardi, B. Moine, M. Nikl, C. Pedrini, M. Poulain, M. Schneegans, S. Tavernier, and A. Vedda, "Cerium-doped heavy metal fluoride glasses, a possible alternative for electromagnetic calorimetry," Nucl. Instrum. Methods Phys. Res A 380, 524-536 (1996).
[CrossRef]

Peterson, R. T.

B. L. Pruett, R. T. Peterson, D. E. Smith, L. D. Looney, and R. N. Shelton, Jr., "Gamma-ray to Cerenkov-light conversion efficiency for pure-silica-core optical fibers," in Proc. SPIE 506, 10-17 (1984).

Phillips, M.

W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism, 2nd ed. (Addison-Wesley, 1962), p. 494.

Plazas, M. C.

B. L. Justus, P. Falkenstein, A. L. Huston, M. C. Plazas, H. Ning, and R. W. Miller, "Gated fiber-optic-coupled detector for in vivo real-time radiation dosimetry," Applied Optics 43, 1663-1668 (2004).
[CrossRef] [PubMed]

Polf, J. C.

J. C. Polf, S. W. S. McKeever, M. S. Akselrod, and S. Holmstrom, "A real-time, fibre-optic dosimetry system using Al2O3 fibres," Radiat. Prot. Dosim. 100, 301-304 (2002).

Poulain, M.

E. Auffray, D. Bouttet, I. Dafinei, J. Fay, P. Lecoq, J. A. Mares, M. Martini, G. Maze, F. Meinardi, B. Moine, M. Nikl, C. Pedrini, M. Poulain, M. Schneegans, S. Tavernier, and A. Vedda, "Cerium-doped heavy metal fluoride glasses, a possible alternative for electromagnetic calorimetry," Nucl. Instrum. Methods Phys. Res A 380, 524-536 (1996).
[CrossRef]

Pruett, B. L.

B. L. Pruett, R. T. Peterson, D. E. Smith, L. D. Looney, and R. N. Shelton, Jr., "Gamma-ray to Cerenkov-light conversion efficiency for pure-silica-core optical fibers," in Proc. SPIE 506, 10-17 (1984).

Pujol, J. F.

F. Pain, P. Laniece, R. Mastrippolito, Y. Charon, D. Comar, V. Leviel, J. F. Pujol, and L. Valentin, "SIC, an intracerebral radiosensitive probe for in vivo neuropharmacology investigations in small laboratory animals: theoretical considerations and practical characteristics," IEEE Trans. Nucl. Sci. 47, 25-32 (2000).
[CrossRef]

Rawlinson, J. A.

S. F. de Boer, A. S. Beddar, and J. A. Rawlinson, "Optical filtering and spectral measurements of radiation-induced light in plastic scintillation dosimetry," Phys. Med. Biol. 38, 945-958 (1993).
[CrossRef]

Robinson, N.

M. Geso, N. Robinson, W. Schumer, and K. Williams, "Use of water-equivalent plastic scintillator for intravascular brachytherapy dosimetry," Australas. Phys. Eng. Sci. Med. 27, 5-10 (2004).
[CrossRef] [PubMed]

Schmidt-Ullrich, R.

M. R. Arnfield, H. E. Gaballa, R. D. Zwicker, Q. Islam, and R. Schmidt-Ullrich, "Radiation-induced light in optical fibers and plastic scintillators: application to brachytherapy dosimetry," IEEE Trans. Nucl. Sci. 43, 2077-2084 (1996).
[CrossRef]

Schmitt, G.

S. Gripp, F. W. Haesing, H. Bueker, and G. Schmitt, "Clinical in vivo dosimetry using optical fibers," Radiat. Oncol. Invest. 6, 142-149 (1998).
[CrossRef]

Schneegans, M.

E. Auffray, D. Bouttet, I. Dafinei, J. Fay, P. Lecoq, J. A. Mares, M. Martini, G. Maze, F. Meinardi, B. Moine, M. Nikl, C. Pedrini, M. Poulain, M. Schneegans, S. Tavernier, and A. Vedda, "Cerium-doped heavy metal fluoride glasses, a possible alternative for electromagnetic calorimetry," Nucl. Instrum. Methods Phys. Res A 380, 524-536 (1996).
[CrossRef]

Schumer, W.

M. Geso, N. Robinson, W. Schumer, and K. Williams, "Use of water-equivalent plastic scintillator for intravascular brachytherapy dosimetry," Australas. Phys. Eng. Sci. Med. 27, 5-10 (2004).
[CrossRef] [PubMed]

Shelton, R. N.

B. L. Pruett, R. T. Peterson, D. E. Smith, L. D. Looney, and R. N. Shelton, Jr., "Gamma-ray to Cerenkov-light conversion efficiency for pure-silica-core optical fibers," in Proc. SPIE 506, 10-17 (1984).

Smith, D. E.

B. L. Pruett, R. T. Peterson, D. E. Smith, L. D. Looney, and R. N. Shelton, Jr., "Gamma-ray to Cerenkov-light conversion efficiency for pure-silica-core optical fibers," in Proc. SPIE 506, 10-17 (1984).

Snyder, A. W.

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

Sutton, R. A.

M. A. Clift, R. A. Sutton, and D. V. Webb, "Dealing with Cerenkov radiation generated in organic scintillator dosimeters by bremsstrahlung beams," Phys. Med. Biol. 45, 1165-1182 (2000).
[CrossRef] [PubMed]

Tamain, B.

J. M. Fontbonne, G. Iltis, G. Ban, A. Battala, J. C. Venhes, J. Tillier, N. Bellaize, C. LeBrun, B. Tamain, K. Mercier, and J. C. Motin, "Scintillating fiber dosimeter for radiation therapy accelerator," IEEE Trans. Nucl. Sci. 49, 2223-2227 (2002).
[CrossRef]

Tavernier, S.

E. Auffray, D. Bouttet, I. Dafinei, J. Fay, P. Lecoq, J. A. Mares, M. Martini, G. Maze, F. Meinardi, B. Moine, M. Nikl, C. Pedrini, M. Poulain, M. Schneegans, S. Tavernier, and A. Vedda, "Cerium-doped heavy metal fluoride glasses, a possible alternative for electromagnetic calorimetry," Nucl. Instrum. Methods Phys. Res A 380, 524-536 (1996).
[CrossRef]

Tillier, J.

J. M. Fontbonne, G. Iltis, G. Ban, A. Battala, J. C. Venhes, J. Tillier, N. Bellaize, C. LeBrun, B. Tamain, K. Mercier, and J. C. Motin, "Scintillating fiber dosimeter for radiation therapy accelerator," IEEE Trans. Nucl. Sci. 49, 2223-2227 (2002).
[CrossRef]

Valentin, L.

F. Pain, P. Laniece, R. Mastrippolito, Y. Charon, D. Comar, V. Leviel, J. F. Pujol, and L. Valentin, "SIC, an intracerebral radiosensitive probe for in vivo neuropharmacology investigations in small laboratory animals: theoretical considerations and practical characteristics," IEEE Trans. Nucl. Sci. 47, 25-32 (2000).
[CrossRef]

Vedda, A.

E. Auffray, D. Bouttet, I. Dafinei, J. Fay, P. Lecoq, J. A. Mares, M. Martini, G. Maze, F. Meinardi, B. Moine, M. Nikl, C. Pedrini, M. Poulain, M. Schneegans, S. Tavernier, and A. Vedda, "Cerium-doped heavy metal fluoride glasses, a possible alternative for electromagnetic calorimetry," Nucl. Instrum. Methods Phys. Res A 380, 524-536 (1996).
[CrossRef]

Venhes, J. C.

J. M. Fontbonne, G. Iltis, G. Ban, A. Battala, J. C. Venhes, J. Tillier, N. Bellaize, C. LeBrun, B. Tamain, K. Mercier, and J. C. Motin, "Scintillating fiber dosimeter for radiation therapy accelerator," IEEE Trans. Nucl. Sci. 49, 2223-2227 (2002).
[CrossRef]

Webb, D. V.

M. A. Clift, P. N. Johnston, and D. V. Webb, "A temporal method of avoiding the Cerenkov radiation generated in organic scintillator dosimeters by pulsed mega-voltage electron and photon beams," Phys. Med. Biol. 47, 1421-1433 (2002).
[CrossRef] [PubMed]

M. A. Clift, R. A. Sutton, and D. V. Webb, "Dealing with Cerenkov radiation generated in organic scintillator dosimeters by bremsstrahlung beams," Phys. Med. Biol. 45, 1165-1182 (2000).
[CrossRef] [PubMed]

Weisstein, E. W.

E. W. Weisstein, "Ellipse" (MathWorld-A Wolfram Web Resource, 1999), retrieved 2004, http://mathworld.wolfram.com/Ellipse.html.References.

Williams, K.

M. Geso, N. Robinson, W. Schumer, and K. Williams, "Use of water-equivalent plastic scintillator for intravascular brachytherapy dosimetry," Australas. Phys. Eng. Sci. Med. 27, 5-10 (2004).
[CrossRef] [PubMed]

Zwicker, R. D.

M. R. Arnfield, H. E. Gaballa, R. D. Zwicker, Q. Islam, and R. Schmidt-Ullrich, "Radiation-induced light in optical fibers and plastic scintillators: application to brachytherapy dosimetry," IEEE Trans. Nucl. Sci. 43, 2077-2084 (1996).
[CrossRef]

Applied Optics

B. L. Justus, P. Falkenstein, A. L. Huston, M. C. Plazas, H. Ning, and R. W. Miller, "Gated fiber-optic-coupled detector for in vivo real-time radiation dosimetry," Applied Optics 43, 1663-1668 (2004).
[CrossRef] [PubMed]

Australas. Phys. Eng. Sci. Med.

M. Geso, N. Robinson, W. Schumer, and K. Williams, "Use of water-equivalent plastic scintillator for intravascular brachytherapy dosimetry," Australas. Phys. Eng. Sci. Med. 27, 5-10 (2004).
[CrossRef] [PubMed]

IEEE Trans. Nucl. Sci.

F. Pain, P. Laniece, R. Mastrippolito, Y. Charon, D. Comar, V. Leviel, J. F. Pujol, and L. Valentin, "SIC, an intracerebral radiosensitive probe for in vivo neuropharmacology investigations in small laboratory animals: theoretical considerations and practical characteristics," IEEE Trans. Nucl. Sci. 47, 25-32 (2000).
[CrossRef]

M. R. Arnfield, H. E. Gaballa, R. D. Zwicker, Q. Islam, and R. Schmidt-Ullrich, "Radiation-induced light in optical fibers and plastic scintillators: application to brachytherapy dosimetry," IEEE Trans. Nucl. Sci. 43, 2077-2084 (1996).
[CrossRef]

J. M. Fontbonne, G. Iltis, G. Ban, A. Battala, J. C. Venhes, J. Tillier, N. Bellaize, C. LeBrun, B. Tamain, K. Mercier, and J. C. Motin, "Scintillating fiber dosimeter for radiation therapy accelerator," IEEE Trans. Nucl. Sci. 49, 2223-2227 (2002).
[CrossRef]

Nucl. Instrum. Methods Phys. Res A

E. Auffray, D. Bouttet, I. Dafinei, J. Fay, P. Lecoq, J. A. Mares, M. Martini, G. Maze, F. Meinardi, B. Moine, M. Nikl, C. Pedrini, M. Poulain, M. Schneegans, S. Tavernier, and A. Vedda, "Cerium-doped heavy metal fluoride glasses, a possible alternative for electromagnetic calorimetry," Nucl. Instrum. Methods Phys. Res A 380, 524-536 (1996).
[CrossRef]

Phys. Med. Biol.

A. S. Beddar, T. R. Mackie, and F. H. Attix, "Water-equivalent plastic scintillation detectors for high-energy beam dosimetry. II. Properties and measurements," Phys. Med. Biol. 37, 1901-1913 (1992).
[CrossRef] [PubMed]

A. S. Beddar, T. R. Mackie, and F. H. Attix, "Water-equivalent plastic scintillation detectors for high-energy beam dosimetry. I. Physical characteristics and theoretical considerations," Phys. Med. Biol. 37, 1883-1900 (1992).
[CrossRef] [PubMed]

S. F. de Boer, A. S. Beddar, and J. A. Rawlinson, "Optical filtering and spectral measurements of radiation-induced light in plastic scintillation dosimetry," Phys. Med. Biol. 38, 945-958 (1993).
[CrossRef]

M. C. Aznar, C. E. Andersen, L. Botter-Jensen, S. A. J. Back, S. Mattsson, F. Kjaer-Kristoffersen, and J. Medin, "Real-time optical-fibre luminescence dosimetry for radiotherapy: physical characteristics and applications in photon beams," Phys. Med. Biol. 49, 1655-1669 (2004).
[CrossRef] [PubMed]

M. A. Clift, R. A. Sutton, and D. V. Webb, "Dealing with Cerenkov radiation generated in organic scintillator dosimeters by bremsstrahlung beams," Phys. Med. Biol. 45, 1165-1182 (2000).
[CrossRef] [PubMed]

M. A. Clift, P. N. Johnston, and D. V. Webb, "A temporal method of avoiding the Cerenkov radiation generated in organic scintillator dosimeters by pulsed mega-voltage electron and photon beams," Phys. Med. Biol. 47, 1421-1433 (2002).
[CrossRef] [PubMed]

Proc. SPIE

B. L. Pruett, R. T. Peterson, D. E. Smith, L. D. Looney, and R. N. Shelton, Jr., "Gamma-ray to Cerenkov-light conversion efficiency for pure-silica-core optical fibers," in Proc. SPIE 506, 10-17 (1984).

Radiat. Oncol. Invest.

S. Gripp, F. W. Haesing, H. Bueker, and G. Schmitt, "Clinical in vivo dosimetry using optical fibers," Radiat. Oncol. Invest. 6, 142-149 (1998).
[CrossRef]

Radiat. Prot. Dosim.

J. C. Polf, S. W. S. McKeever, M. S. Akselrod, and S. Holmstrom, "A real-time, fibre-optic dosimetry system using Al2O3 fibres," Radiat. Prot. Dosim. 100, 301-304 (2002).

Other

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

E. W. Weisstein, "Ellipse" (MathWorld-A Wolfram Web Resource, 1999), retrieved 2004, http://mathworld.wolfram.com/Ellipse.html.References.

S. Law and A. S. Beddar, "Capture of Cerenkov radiation generated on the axis of an optical fibre when the fibre axis lies on the Cerenkov Cone," presented at the Conference on the Optical Internet/Australian Conference on Optical Fibre Technology 2003, Melbourne, Australia, 13-16 July 2003.

W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism, 2nd ed. (Addison-Wesley, 1962), p. 494.

L. D. Gladney, "Cerenkov Radiation," University of Pennsylvania, Department of Physics and Astronomy, 2000, retrieved 2004, http://dept.physics.upenn.edu/balloon/cerenkov_radiation.html.

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Figures (11)

Fig. 1
Fig. 1

(Color online) (a) Meridional and (b) (c) skew rays in a multimode fiber and their defining angles, ϕ z , ϕ a z , and α.

Fig. 2
Fig. 2

(Color online) Geometry of Čerenkov radiation generated within the fiber core: s = the closest approach of the particle track to the fiber axis; γ = the fiber and∕or particle track angle; P = the position of the point of origin of the Čerenkov cone.

Fig. 3
Fig. 3

(Color online) Geometry for off-axis Čerenkov generation.

Fig. 4
Fig. 4

Projection of Fig. 3 onto the cross section of the fiber. Points P , B , D , and angle θ are the projections of P, B, D, and θ in Fig. 3.

Fig. 5
Fig. 5

(Color online) Longitudinal propagation angle (vertical axis) as a function of fiber and∕or particle angle, γ, and angle around the Čerenkov cone, θ, for a fiber of core refractive index 1:47. The reference plane shows the critical longitudinal propagation angle for the standard fiber considered.

Fig. 6
Fig. 6

(Color online) Incident angle (vertical axis) as a function of fiber and∕or particle angle, γ, and angle around the Čerenkov cone, θ for the standard fiber. The reference plane shows the critical incident angle.

Fig. 7
Fig. 7

Diagram of the cross section of the fiber core showing points on lines of constant sin - 1 ( f s / f r ) , the origins of the Čerenkov cones used in the calculation of the fraction of tunneling modes.

Fig. 8
Fig. 8

(Color online) The fraction of rays in the Čerenkov cone transmitted in bound (×) and tunneling modes as a function of the fractional radial position, f r , of the origin of the Čerenkov cone. The fraction of bound rays is constant with f r while the fraction of tunneling rays varies with f r and f s . The tunneling modes are shown for a range of angles relative to the particle track.

Fig. 9
Fig. 9

(Color online) Intensity of bound Čerenkov radiation as a function of fiber∕particle angle and Δ n for a silica-cored fiber ( n c o = 1.47 ) .

Fig. 10
Fig. 10

(Color online) Relative intensity of bound radiation at the angle of maximum transmission as a function of n c o and Δ n .

Fig. 11
Fig. 11

(Color online) Angular dependence of Čerenkov transmission as a function of energy for nonrelativistic particle energies.

Tables (1)

Tables Icon

Table 1 Effect of Fiber Parameters on Angular Dependence of Čerenkov Radiation Transmission a

Equations (34)

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

cos   α = sin   ϕ z sin   ϕ a z .
0 ϕ z < ϕ z c ,
0 α < α c ,
ϕ z c ϕ z π / 2 ,
α c α π / 2 .
κ = cos - 1 ( 1 / n β ) ,
( 1 1 / β 2 n 2 ) .
C D 2 = B P 2 tan 2 γ + B P 2 tan 2 κ 2 B P 2   tan   γ   tan   κ × cos   θ .
C D 2 = B P 2 cos 2 γ + B P 2 cos 2 κ 2 B P 2 cos   γ   cos   κ   cos   ϕ z .
cos   ϕ z = cos   γ   cos   κ + sin   γ   sin   κ   cos   θ ,
ϕ z = cos - 1 ( cos   γ   cos   κ + sin   γ   sin   κ   cos   θ ) .
sin ( 90 - ϕ a z ) r = sin ( ε + θ + ζ ) ρ .
ϕ a z = cos - 1 ( r ρ  sin ( ε + θ + ζ ) ) .
ε = sin - 1 ( s r ) .
x = B P   tan   κ   cos   θ
y = B P   tan   κ   sin   θ .
x = x   cos   γ = B P   tan   κ   cos   θ   cos   γ = r   cos   θ
y = y = B P   tan   κ   sin   θ = r   sin   θ .
sin   θ = sin   θ sin 2 θ + cos 2 θ cos 2 γ
cos   θ = cos   θ   cos   γ sin 2 θ + cos 2 θ cos 2 γ .
sin   ζ B P = sin ( 180 θ ζ ) B D .
B P = B P   sin   γ ,
B D = B P   tan   κ sin 2 θ + cos 2 θ cos 2 γ .
tan   ζ = sin   γ   sin   θ tan   κ sin 2 θ + cos 2 θ cos 2 γ sin   γ   cos   θ .
ζ = tan - 1 ( sin   θ   sin   γ tan   κ ( sin 2 θ + cos 2 θ cos 2 γ ) cos   θ cos 2 γ ) .
ε + θ + ζ for ε + θ + ζ < 180 °
360 ° ( ε + θ + ζ ) for ε + θ + ζ > 180 ° .
ϕ a z = cos - 1 ( r ρ | sin ( sin - 1 ( s r ) + sin - 1 ( sin   θ sin 2 θ + cos 2 θ cos 2 γ ) + tan - 1 ( sin   θ   sin   γ tan   κ ( sin 2 θ + cos 2 θ cos 2 γ ) cos   θ cos 2 γ ) ) | ) .
θ B = cos - 1 ( n c l v c   cos   γ sin   γ n c o 2 v 2 c 2 ) = cos - 1 ( v ( n c o Δ n ) c   cos   γ sin   γ n c o 2 v 2 c 2 ) ,
F cap = 2 θ B 2 π = 1 π cos - 1 ( v ( n c o Δ n ) c   cos   γ sin   γ v 2 n c o 2 c 2 ) ,
I cap v 2 n c o 2 c 2 π v 2 n c o 2 cos - 1 ( v ( n c o Δ n ) c   cos   γ sin   γ v 2 n c o 2 c 2 ) .
I B 2 π ρ 3 sin   γ v 2 n c o 2 c 2 π v 2 n c o 2 cos - 1 ( v ( n c o Δ n ) c   cos   γ sin   γ v 2 n c o 2 c 2 ) .
I B 2 π ρ 3 sin   γ n c o 2 1 π n c o 2 cos - 1 ( n c o Δ n cos   γ sin   γ n c o 2 1 ) .
α = cos - 1 ( sin ( cos - 1 ( cos   γ   cos   κ + sin   γ   sin   κ   cos   θ ) ) sin ( cos - 1 ( f r | sin ( sin - 1 ( f s f r ) + sin - 1 ( sin   θ sin 2 θ + cos 2 θ cos 2 γ ) + tan - 1 ( sin   θ   sin   γ tan   κ ( sin 2 θ + cos 2 θ cos 2 γ ) cos   θ cos 2 γ ) ) | ) ) ) .

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