Abstract

A general formula and some special integral formulas were presented for calculating radiative fluxes incident on a circular plane from a planar multiple point source within a coaxial cylindrical enclosure perpendicular to the source. These formula were obtained for radiation propagating in a homogeneous isotropic medium assuming that the lateral surface of the enclosure completely absorbs the incident radiation. Exemplary results were computed numerically and illustrated with three-dimensional surface plots. The formulas presented are suitable for determining fluxes of radiation reaching planar circular detectors, collectors or other planar circular elements from systems of laser diodes, light emitting diodes and fiber lamps within cylindrical enclosures, as well as small biological emitters (bacteria, fungi, yeast, etc.) distributed on planar bases of open nontransparent cylindrical containers.

© 2007 Optical Society of America

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References

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  1. V. P. Gribkovskii, "Injection lasers," Prog. Quant. Electr. 23, 41-88 (1995).
    [CrossRef]
  2. A.C. Schuerger, C.S. Brown and E.C. Stryjewski, "Anatomical futures of pepper plants (Capsicum annuum L.) grown under red light emitting diodes supplemented with blue or far-red light," Ann. Botany 79, 273-282 (1997).
    [CrossRef]
  3. R. Szweda, "Lasers at the cutting edge," III-Vs Rev. 12, 28-31 (1999).
    [CrossRef]
  4. L. Botter-Jensen, E. Bulur, G. A. T. Duller and A. S. Murray, "Advances in luminescence instrument systems," Radiat. Meas. 32, 523-528 (2000).
    [CrossRef]
  5. O. Monje, G. W. Stutte, G. D. Goins, D. M. Porterfield and G. E. Bingham, "Farming in space: environmental and biophysical concerns," Adv. Space. Res. 31, 151-167 (2003).
    [CrossRef] [PubMed]
  6. K. T. Lau, W. S. Yerazunis, R. L. Shepherd and D. Diamond, "Quantitative colorimetric analysis of dye mixtures using an optical photometer based on LED array," Sens. Actuators B-Chem. 114, 819-825 (2006).
    [CrossRef]
  7. S. Nakamura, S. Pearton and G. Fasol, The blue laser diode: The complete story (Springer-Verlag, Berlin, 2000).
  8. C. Curachi, A. M. Toboy, D. V. Magalhaes and V.S . Bagnato, "Hardness evaluation of a dental composite polymerized with experimental LED-based devices," Dent. Mat. 17, 309-315 (2001).
    [CrossRef]
  9. G. Glickman, B. Byrne, C. Pineda, W. W. Hauck and G. C. Brainard, "Light therapy for seasonal affective disorder with blue narrow-band light-emitting diodes (LEDs)," Biol. Psych. 59, 502-507 (2006).
    [CrossRef]
  10. A. Juzeniene, P. Juzenas, L.-W. Ma, V. Iani and J. Moan, "Effectiveness of different light sources for 5-aminolevolinic acid photodynamic therapy," Lasers Med. Sci. 19, 139-149 (2004).
    [CrossRef] [PubMed]
  11. B. Link, S. Ruhl, A. Peters, A. Junemann and F. K. Horn, "Pattern reversal ERG and VEP - comparison of stimulation by LED, monitor and a Maxwellian-view system," Doc. Opthalmol. 12,1-11 (2006).
    [CrossRef]
  12. J. M. Gaines, "Modeling of multichip LED packages for illumination," Lighting Res. Technol. 38, 152-165 (2006).
    [CrossRef]
  13. A. Mills, "Trends in HB-LED markets," III-Vs Rev. 14, 38-42 (2001).
    [CrossRef]
  14. C. Gardner, "The use of misuse of coloured light in the urban environment," Opt. Lasers Tech. 38, 366-376 (2006).
    [CrossRef]
  15. A. Zukauskas, R. Vaicekauskas, F. Ivanauskas, R. Gaska and M. S. Shur, "Optimization of white polychromic semiconductors lamps," Appl. Phys. Lett. 80, 234-236 (2002).
    [CrossRef]
  16. F Grum and R.J. Becherer, Radiometry (Academic Press, New York, 1979) pp. 30-52, 81-83.
  17. M. Strojnik and G Paez, "Radiometry" in Handbook of Optical Engineering, D. Malacara and B. J. Thompson eds. (Marcel Dekker, New York, 2001) pp. 649-699.
  18. E. Sparrow and R. Cess, Radiation Heat Transfer, (McGraw-Hill, New York, 1978) pp. 77-136.
  19. D. H. Sliney, "Laser effect on vision and ocular exposure limit," App. Occup. Environ. Hyg. 11, 313-319 (1996).
    [CrossRef]
  20. T. R. Fry, "Laser safety," Vet. Clin. Small Anim. 32, 535-547 (2002).
    [CrossRef]
  21. S. Tryka, "Angular distribution of the solid angle at a point subtended by a circular disk," Optics Comm. 137, 317-333 (1997).
    [CrossRef]
  22. S. Tryka, "Angular distribution of an average solid angle subtended by a circular disc from multiple points uniformly distributed on planar circular surface coaxial to the disc," J. Mod. Opt. 47, 769-791 (2000).
  23. S. Wolfram, Mathematica-A System for Doing Mathematics by Computer (Addison-Wesley, Reading, Mass. 1993), pp. 44-186.

2006

K. T. Lau, W. S. Yerazunis, R. L. Shepherd and D. Diamond, "Quantitative colorimetric analysis of dye mixtures using an optical photometer based on LED array," Sens. Actuators B-Chem. 114, 819-825 (2006).
[CrossRef]

G. Glickman, B. Byrne, C. Pineda, W. W. Hauck and G. C. Brainard, "Light therapy for seasonal affective disorder with blue narrow-band light-emitting diodes (LEDs)," Biol. Psych. 59, 502-507 (2006).
[CrossRef]

B. Link, S. Ruhl, A. Peters, A. Junemann and F. K. Horn, "Pattern reversal ERG and VEP - comparison of stimulation by LED, monitor and a Maxwellian-view system," Doc. Opthalmol. 12,1-11 (2006).
[CrossRef]

J. M. Gaines, "Modeling of multichip LED packages for illumination," Lighting Res. Technol. 38, 152-165 (2006).
[CrossRef]

C. Gardner, "The use of misuse of coloured light in the urban environment," Opt. Lasers Tech. 38, 366-376 (2006).
[CrossRef]

2004

A. Juzeniene, P. Juzenas, L.-W. Ma, V. Iani and J. Moan, "Effectiveness of different light sources for 5-aminolevolinic acid photodynamic therapy," Lasers Med. Sci. 19, 139-149 (2004).
[CrossRef] [PubMed]

2003

O. Monje, G. W. Stutte, G. D. Goins, D. M. Porterfield and G. E. Bingham, "Farming in space: environmental and biophysical concerns," Adv. Space. Res. 31, 151-167 (2003).
[CrossRef] [PubMed]

2002

T. R. Fry, "Laser safety," Vet. Clin. Small Anim. 32, 535-547 (2002).
[CrossRef]

A. Zukauskas, R. Vaicekauskas, F. Ivanauskas, R. Gaska and M. S. Shur, "Optimization of white polychromic semiconductors lamps," Appl. Phys. Lett. 80, 234-236 (2002).
[CrossRef]

2001

A. Mills, "Trends in HB-LED markets," III-Vs Rev. 14, 38-42 (2001).
[CrossRef]

C. Curachi, A. M. Toboy, D. V. Magalhaes and V.S . Bagnato, "Hardness evaluation of a dental composite polymerized with experimental LED-based devices," Dent. Mat. 17, 309-315 (2001).
[CrossRef]

2000

L. Botter-Jensen, E. Bulur, G. A. T. Duller and A. S. Murray, "Advances in luminescence instrument systems," Radiat. Meas. 32, 523-528 (2000).
[CrossRef]

S. Tryka, "Angular distribution of an average solid angle subtended by a circular disc from multiple points uniformly distributed on planar circular surface coaxial to the disc," J. Mod. Opt. 47, 769-791 (2000).

1999

R. Szweda, "Lasers at the cutting edge," III-Vs Rev. 12, 28-31 (1999).
[CrossRef]

1997

A.C. Schuerger, C.S. Brown and E.C. Stryjewski, "Anatomical futures of pepper plants (Capsicum annuum L.) grown under red light emitting diodes supplemented with blue or far-red light," Ann. Botany 79, 273-282 (1997).
[CrossRef]

S. Tryka, "Angular distribution of the solid angle at a point subtended by a circular disk," Optics Comm. 137, 317-333 (1997).
[CrossRef]

1996

D. H. Sliney, "Laser effect on vision and ocular exposure limit," App. Occup. Environ. Hyg. 11, 313-319 (1996).
[CrossRef]

1995

V. P. Gribkovskii, "Injection lasers," Prog. Quant. Electr. 23, 41-88 (1995).
[CrossRef]

Bagnato, V. S.

C. Curachi, A. M. Toboy, D. V. Magalhaes and V.S . Bagnato, "Hardness evaluation of a dental composite polymerized with experimental LED-based devices," Dent. Mat. 17, 309-315 (2001).
[CrossRef]

Bingham, G. E.

O. Monje, G. W. Stutte, G. D. Goins, D. M. Porterfield and G. E. Bingham, "Farming in space: environmental and biophysical concerns," Adv. Space. Res. 31, 151-167 (2003).
[CrossRef] [PubMed]

Botter-Jensen, L.

L. Botter-Jensen, E. Bulur, G. A. T. Duller and A. S. Murray, "Advances in luminescence instrument systems," Radiat. Meas. 32, 523-528 (2000).
[CrossRef]

Brainard, G. C.

G. Glickman, B. Byrne, C. Pineda, W. W. Hauck and G. C. Brainard, "Light therapy for seasonal affective disorder with blue narrow-band light-emitting diodes (LEDs)," Biol. Psych. 59, 502-507 (2006).
[CrossRef]

Brown, C. S.

A.C. Schuerger, C.S. Brown and E.C. Stryjewski, "Anatomical futures of pepper plants (Capsicum annuum L.) grown under red light emitting diodes supplemented with blue or far-red light," Ann. Botany 79, 273-282 (1997).
[CrossRef]

Bulur, E.

L. Botter-Jensen, E. Bulur, G. A. T. Duller and A. S. Murray, "Advances in luminescence instrument systems," Radiat. Meas. 32, 523-528 (2000).
[CrossRef]

Byrne, B.

G. Glickman, B. Byrne, C. Pineda, W. W. Hauck and G. C. Brainard, "Light therapy for seasonal affective disorder with blue narrow-band light-emitting diodes (LEDs)," Biol. Psych. 59, 502-507 (2006).
[CrossRef]

Curachi, C.

C. Curachi, A. M. Toboy, D. V. Magalhaes and V.S . Bagnato, "Hardness evaluation of a dental composite polymerized with experimental LED-based devices," Dent. Mat. 17, 309-315 (2001).
[CrossRef]

Diamond, D.

K. T. Lau, W. S. Yerazunis, R. L. Shepherd and D. Diamond, "Quantitative colorimetric analysis of dye mixtures using an optical photometer based on LED array," Sens. Actuators B-Chem. 114, 819-825 (2006).
[CrossRef]

Duller, G. A. T.

L. Botter-Jensen, E. Bulur, G. A. T. Duller and A. S. Murray, "Advances in luminescence instrument systems," Radiat. Meas. 32, 523-528 (2000).
[CrossRef]

Fry, T. R.

T. R. Fry, "Laser safety," Vet. Clin. Small Anim. 32, 535-547 (2002).
[CrossRef]

Gaines, J. M.

J. M. Gaines, "Modeling of multichip LED packages for illumination," Lighting Res. Technol. 38, 152-165 (2006).
[CrossRef]

Gardner, C.

C. Gardner, "The use of misuse of coloured light in the urban environment," Opt. Lasers Tech. 38, 366-376 (2006).
[CrossRef]

Gaska, R.

A. Zukauskas, R. Vaicekauskas, F. Ivanauskas, R. Gaska and M. S. Shur, "Optimization of white polychromic semiconductors lamps," Appl. Phys. Lett. 80, 234-236 (2002).
[CrossRef]

Glickman, G.

G. Glickman, B. Byrne, C. Pineda, W. W. Hauck and G. C. Brainard, "Light therapy for seasonal affective disorder with blue narrow-band light-emitting diodes (LEDs)," Biol. Psych. 59, 502-507 (2006).
[CrossRef]

Goins, G.D.

O. Monje, G. W. Stutte, G. D. Goins, D. M. Porterfield and G. E. Bingham, "Farming in space: environmental and biophysical concerns," Adv. Space. Res. 31, 151-167 (2003).
[CrossRef] [PubMed]

Gribkovskii, V. P.

V. P. Gribkovskii, "Injection lasers," Prog. Quant. Electr. 23, 41-88 (1995).
[CrossRef]

Hauck, W. W.

G. Glickman, B. Byrne, C. Pineda, W. W. Hauck and G. C. Brainard, "Light therapy for seasonal affective disorder with blue narrow-band light-emitting diodes (LEDs)," Biol. Psych. 59, 502-507 (2006).
[CrossRef]

Horn, F.K.

B. Link, S. Ruhl, A. Peters, A. Junemann and F. K. Horn, "Pattern reversal ERG and VEP - comparison of stimulation by LED, monitor and a Maxwellian-view system," Doc. Opthalmol. 12,1-11 (2006).
[CrossRef]

Iani, V.

A. Juzeniene, P. Juzenas, L.-W. Ma, V. Iani and J. Moan, "Effectiveness of different light sources for 5-aminolevolinic acid photodynamic therapy," Lasers Med. Sci. 19, 139-149 (2004).
[CrossRef] [PubMed]

Ivanauskas, F.

A. Zukauskas, R. Vaicekauskas, F. Ivanauskas, R. Gaska and M. S. Shur, "Optimization of white polychromic semiconductors lamps," Appl. Phys. Lett. 80, 234-236 (2002).
[CrossRef]

Junemann, A.

B. Link, S. Ruhl, A. Peters, A. Junemann and F. K. Horn, "Pattern reversal ERG and VEP - comparison of stimulation by LED, monitor and a Maxwellian-view system," Doc. Opthalmol. 12,1-11 (2006).
[CrossRef]

Juzenas, P.

A. Juzeniene, P. Juzenas, L.-W. Ma, V. Iani and J. Moan, "Effectiveness of different light sources for 5-aminolevolinic acid photodynamic therapy," Lasers Med. Sci. 19, 139-149 (2004).
[CrossRef] [PubMed]

Juzeniene, A.

A. Juzeniene, P. Juzenas, L.-W. Ma, V. Iani and J. Moan, "Effectiveness of different light sources for 5-aminolevolinic acid photodynamic therapy," Lasers Med. Sci. 19, 139-149 (2004).
[CrossRef] [PubMed]

Lau, K. T.

K. T. Lau, W. S. Yerazunis, R. L. Shepherd and D. Diamond, "Quantitative colorimetric analysis of dye mixtures using an optical photometer based on LED array," Sens. Actuators B-Chem. 114, 819-825 (2006).
[CrossRef]

Link, B.

B. Link, S. Ruhl, A. Peters, A. Junemann and F. K. Horn, "Pattern reversal ERG and VEP - comparison of stimulation by LED, monitor and a Maxwellian-view system," Doc. Opthalmol. 12,1-11 (2006).
[CrossRef]

Ma, L.-W.

A. Juzeniene, P. Juzenas, L.-W. Ma, V. Iani and J. Moan, "Effectiveness of different light sources for 5-aminolevolinic acid photodynamic therapy," Lasers Med. Sci. 19, 139-149 (2004).
[CrossRef] [PubMed]

Magalhaes, D. V.

C. Curachi, A. M. Toboy, D. V. Magalhaes and V.S . Bagnato, "Hardness evaluation of a dental composite polymerized with experimental LED-based devices," Dent. Mat. 17, 309-315 (2001).
[CrossRef]

Mills, A.

A. Mills, "Trends in HB-LED markets," III-Vs Rev. 14, 38-42 (2001).
[CrossRef]

Moan, J.

A. Juzeniene, P. Juzenas, L.-W. Ma, V. Iani and J. Moan, "Effectiveness of different light sources for 5-aminolevolinic acid photodynamic therapy," Lasers Med. Sci. 19, 139-149 (2004).
[CrossRef] [PubMed]

Monje, O.

O. Monje, G. W. Stutte, G. D. Goins, D. M. Porterfield and G. E. Bingham, "Farming in space: environmental and biophysical concerns," Adv. Space. Res. 31, 151-167 (2003).
[CrossRef] [PubMed]

Murray, A. S.

L. Botter-Jensen, E. Bulur, G. A. T. Duller and A. S. Murray, "Advances in luminescence instrument systems," Radiat. Meas. 32, 523-528 (2000).
[CrossRef]

Peters, A.

B. Link, S. Ruhl, A. Peters, A. Junemann and F. K. Horn, "Pattern reversal ERG and VEP - comparison of stimulation by LED, monitor and a Maxwellian-view system," Doc. Opthalmol. 12,1-11 (2006).
[CrossRef]

Pineda, C.

G. Glickman, B. Byrne, C. Pineda, W. W. Hauck and G. C. Brainard, "Light therapy for seasonal affective disorder with blue narrow-band light-emitting diodes (LEDs)," Biol. Psych. 59, 502-507 (2006).
[CrossRef]

Porterfield, D. M.

O. Monje, G. W. Stutte, G. D. Goins, D. M. Porterfield and G. E. Bingham, "Farming in space: environmental and biophysical concerns," Adv. Space. Res. 31, 151-167 (2003).
[CrossRef] [PubMed]

Ruhl, S.

B. Link, S. Ruhl, A. Peters, A. Junemann and F. K. Horn, "Pattern reversal ERG and VEP - comparison of stimulation by LED, monitor and a Maxwellian-view system," Doc. Opthalmol. 12,1-11 (2006).
[CrossRef]

Schuerger, A. C.

A.C. Schuerger, C.S. Brown and E.C. Stryjewski, "Anatomical futures of pepper plants (Capsicum annuum L.) grown under red light emitting diodes supplemented with blue or far-red light," Ann. Botany 79, 273-282 (1997).
[CrossRef]

Shepherd, R. L.

K. T. Lau, W. S. Yerazunis, R. L. Shepherd and D. Diamond, "Quantitative colorimetric analysis of dye mixtures using an optical photometer based on LED array," Sens. Actuators B-Chem. 114, 819-825 (2006).
[CrossRef]

Shur, M. S.

A. Zukauskas, R. Vaicekauskas, F. Ivanauskas, R. Gaska and M. S. Shur, "Optimization of white polychromic semiconductors lamps," Appl. Phys. Lett. 80, 234-236 (2002).
[CrossRef]

Sliney, D. H.

D. H. Sliney, "Laser effect on vision and ocular exposure limit," App. Occup. Environ. Hyg. 11, 313-319 (1996).
[CrossRef]

Stryjewski, E. C.

A.C. Schuerger, C.S. Brown and E.C. Stryjewski, "Anatomical futures of pepper plants (Capsicum annuum L.) grown under red light emitting diodes supplemented with blue or far-red light," Ann. Botany 79, 273-282 (1997).
[CrossRef]

Stutte, G. W.

O. Monje, G. W. Stutte, G. D. Goins, D. M. Porterfield and G. E. Bingham, "Farming in space: environmental and biophysical concerns," Adv. Space. Res. 31, 151-167 (2003).
[CrossRef] [PubMed]

Szweda, R.

R. Szweda, "Lasers at the cutting edge," III-Vs Rev. 12, 28-31 (1999).
[CrossRef]

Toboy, A. M.

C. Curachi, A. M. Toboy, D. V. Magalhaes and V.S . Bagnato, "Hardness evaluation of a dental composite polymerized with experimental LED-based devices," Dent. Mat. 17, 309-315 (2001).
[CrossRef]

Tryka, S.

S. Tryka, "Angular distribution of an average solid angle subtended by a circular disc from multiple points uniformly distributed on planar circular surface coaxial to the disc," J. Mod. Opt. 47, 769-791 (2000).

S. Tryka, "Angular distribution of the solid angle at a point subtended by a circular disk," Optics Comm. 137, 317-333 (1997).
[CrossRef]

Vaicekauskas, R.

A. Zukauskas, R. Vaicekauskas, F. Ivanauskas, R. Gaska and M. S. Shur, "Optimization of white polychromic semiconductors lamps," Appl. Phys. Lett. 80, 234-236 (2002).
[CrossRef]

Yerazunis, W. S.

K. T. Lau, W. S. Yerazunis, R. L. Shepherd and D. Diamond, "Quantitative colorimetric analysis of dye mixtures using an optical photometer based on LED array," Sens. Actuators B-Chem. 114, 819-825 (2006).
[CrossRef]

Zukauskas, A.

A. Zukauskas, R. Vaicekauskas, F. Ivanauskas, R. Gaska and M. S. Shur, "Optimization of white polychromic semiconductors lamps," Appl. Phys. Lett. 80, 234-236 (2002).
[CrossRef]

Adv. Space. Res.

O. Monje, G. W. Stutte, G. D. Goins, D. M. Porterfield and G. E. Bingham, "Farming in space: environmental and biophysical concerns," Adv. Space. Res. 31, 151-167 (2003).
[CrossRef] [PubMed]

Ann. Botany

A.C. Schuerger, C.S. Brown and E.C. Stryjewski, "Anatomical futures of pepper plants (Capsicum annuum L.) grown under red light emitting diodes supplemented with blue or far-red light," Ann. Botany 79, 273-282 (1997).
[CrossRef]

App. Occup. Environ. Hyg.

D. H. Sliney, "Laser effect on vision and ocular exposure limit," App. Occup. Environ. Hyg. 11, 313-319 (1996).
[CrossRef]

Appl. Phys. Lett.

A. Zukauskas, R. Vaicekauskas, F. Ivanauskas, R. Gaska and M. S. Shur, "Optimization of white polychromic semiconductors lamps," Appl. Phys. Lett. 80, 234-236 (2002).
[CrossRef]

Biol. Psych.

G. Glickman, B. Byrne, C. Pineda, W. W. Hauck and G. C. Brainard, "Light therapy for seasonal affective disorder with blue narrow-band light-emitting diodes (LEDs)," Biol. Psych. 59, 502-507 (2006).
[CrossRef]

Dent. Mat.

C. Curachi, A. M. Toboy, D. V. Magalhaes and V.S . Bagnato, "Hardness evaluation of a dental composite polymerized with experimental LED-based devices," Dent. Mat. 17, 309-315 (2001).
[CrossRef]

Doc. Opthalmol.

B. Link, S. Ruhl, A. Peters, A. Junemann and F. K. Horn, "Pattern reversal ERG and VEP - comparison of stimulation by LED, monitor and a Maxwellian-view system," Doc. Opthalmol. 12,1-11 (2006).
[CrossRef]

III-Vs Rev.

A. Mills, "Trends in HB-LED markets," III-Vs Rev. 14, 38-42 (2001).
[CrossRef]

R. Szweda, "Lasers at the cutting edge," III-Vs Rev. 12, 28-31 (1999).
[CrossRef]

J. Mod. Opt.

S. Tryka, "Angular distribution of an average solid angle subtended by a circular disc from multiple points uniformly distributed on planar circular surface coaxial to the disc," J. Mod. Opt. 47, 769-791 (2000).

Lasers Med. Sci.

A. Juzeniene, P. Juzenas, L.-W. Ma, V. Iani and J. Moan, "Effectiveness of different light sources for 5-aminolevolinic acid photodynamic therapy," Lasers Med. Sci. 19, 139-149 (2004).
[CrossRef] [PubMed]

Lighting Res. Technol.

J. M. Gaines, "Modeling of multichip LED packages for illumination," Lighting Res. Technol. 38, 152-165 (2006).
[CrossRef]

Opt. Lasers Tech.

C. Gardner, "The use of misuse of coloured light in the urban environment," Opt. Lasers Tech. 38, 366-376 (2006).
[CrossRef]

Optics Comm.

S. Tryka, "Angular distribution of the solid angle at a point subtended by a circular disk," Optics Comm. 137, 317-333 (1997).
[CrossRef]

Prog. Quant. Electr.

V. P. Gribkovskii, "Injection lasers," Prog. Quant. Electr. 23, 41-88 (1995).
[CrossRef]

Radiat. Meas.

L. Botter-Jensen, E. Bulur, G. A. T. Duller and A. S. Murray, "Advances in luminescence instrument systems," Radiat. Meas. 32, 523-528 (2000).
[CrossRef]

Sens. Actuators B-Chem.

K. T. Lau, W. S. Yerazunis, R. L. Shepherd and D. Diamond, "Quantitative colorimetric analysis of dye mixtures using an optical photometer based on LED array," Sens. Actuators B-Chem. 114, 819-825 (2006).
[CrossRef]

Vet. Clin. Small Anim.

T. R. Fry, "Laser safety," Vet. Clin. Small Anim. 32, 535-547 (2002).
[CrossRef]

Other

S. Wolfram, Mathematica-A System for Doing Mathematics by Computer (Addison-Wesley, Reading, Mass. 1993), pp. 44-186.

S. Nakamura, S. Pearton and G. Fasol, The blue laser diode: The complete story (Springer-Verlag, Berlin, 2000).

F Grum and R.J. Becherer, Radiometry (Academic Press, New York, 1979) pp. 30-52, 81-83.

M. Strojnik and G Paez, "Radiometry" in Handbook of Optical Engineering, D. Malacara and B. J. Thompson eds. (Marcel Dekker, New York, 2001) pp. 649-699.

E. Sparrow and R. Cess, Radiation Heat Transfer, (McGraw-Hill, New York, 1978) pp. 77-136.

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Figures (3)

Fig. 1.
Fig. 1.

Definition of some geometrical variables and the perspective view of the planar circular multiple point source of surface σ within the cylindrical enclosure contributing optical radiation to the planar circular coaxial surface S when aR and H > h (a) and the scheme of the σ-S system when a > R and H > h (b). The symbol r denotes the radial distance between the points P(ρi , ϕj ) and P′(ρi , ϕj , r, φ).

Fig. 2.
Fig. 2.

Geometrical dependencies between a, ρi , r and γa, ρi at aR (a) and between R, ρi , r and γR, ρi at a > R (b). The radial distance r is defined as r = h tan θ (a) and as r = H tan θ (b).

Fig. 3.
Fig. 3.

The average spectral radiative flux <Φ λ, σ→S > within wavelength interval Δλ = 1 nm obtained for IP (θ 1) = I 0 cos2 θ 1 as a function of R and H at a = 5 and αat = 0 (a), as a function of R and H at a = 5 and αat = 0.1 (b), as a function of a and H at R = 5 and αat = 0 (c), and as a function of a and H at R = 5 and αat = 0.1 (d). The data were calculated for the relative units of R, a, H, and αat i.e. if R, a, and H are given in m then αat is expressed in m-1. The radiant intensity was taken as I 0 = 1 W∙sr-1, so the flux <Φ λ, σ→S > is given in W∙nm-1.

Tables (1)

Tables Icon

Table 1. The average spectral radiative fluxes <Φ λ, σ→S > calculated for I λ, 0 = 1 W∙sr-1 and Δλ = 1 nm , at h = 1 and selected values of R, a, H, and αλ, at given in the relative units.

Equations (61)

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Φ σ S = λ min λ max Φ λ , σ S d λ ,
Φ λ , σ S = i = 1 N λ , ρ j = 1 N λ , ϕ Φ λ , P S ( ρ i , ϕ j ) .
Φ λ , P S ( ρ i , ϕ j ) = S I λ , P ( ρ i , ϕ j , θ , φ , H , α at ) d ω P d S ( θ , φ ) ,
d ω P d S ( θ , φ ) = sin θ d θ d φ .
α λ , at = α λ , ab + α λ , sc ,
I λ , P ( ρ i , ϕ j , θ , φ , H , α λ , at ) = I λ , P ( ρ i , ϕ j , θ , φ ) τ P ( θ , H , α λ , at ) ,
τ P ( θ , H , α λ , at ) = exp ( H α λ , at cos θ ) , 0 θ < π 2 ,
Φ λ , P S ( ρ i , ϕ j ) = S I λ , P ( ρ i , ϕ j , θ , φ ) τ P ( θ , H , α λ , at ) sin θ d θ d φ ,
Φ λ , P S ( ρ i , ϕ j ) = 0 π 2 sin θ d θ 0 2 π I λ , P ( θ , φ ) d φ , ρ i = 0 ,
Φ λ , P S ( ρ i , ϕ j ) = 0 , ρ i = 0 ,
Φ λ , P S , ( ρ i , ϕ j ) = 0 π 2 sin θ d θ 0 2 π I λ , P ( θ , φ ) d φ , ρ i = 0 ,
Φ λ , P S ( ρ i , ϕ j ) = { 0 θ a τ P ( θ , H , α λ , at ) sin θ d θ 0 2 π I λ , P ( θ , φ ) d φ , 0 = ρ i < a , 0 θ a ρ i τ P ( θ , H , α λ , at ) sin θ d θ 0 2 π I λ , P ( ρ i , ϕ j , θ , φ ) d φ + θ a ρ i θ a + ρ i τ P ( θ , H , α λ , at ) sin θ d θ γ a , ρ i γ a , ρ i I λ , P ( ρ i , ϕ j , θ , φ ) d φ , 0 < ρ i < a , 0 θ 2 a τ P ( θ , H , α λ , at ) sin θ d θ γ 2 a γ 2 a I λ , P ( ρ i , ϕ j , θ , φ ) d φ , 0 < ρ i = a ,
Φ λ , P S ( ρ i , ϕ j ) = { 0 θ a τ P ( θ ; H ; α λ , at ) sin θ d θ 0 2 π I λ , P ( θ , φ ) d φ , 0 = ρ i < a , 0 θ a ρ i τ P ( θ , H , α λ , at ) sin θ d θ 0 2 π I λ , P ( ρ i , ϕ j , θ , φ ) d φ + θ a ρ i θ a + ρ i τ P ( θ , H , α λ , at ) sin θ d θ γ a , ρ i γ a , ρ i I λ , P ( ρ i , ϕ j , θ , φ ) d φ , 0 < ρ i Y a , 0 θ a ρ i τ P ( θ , H , α λ , at ) sin θ d θ 0 2 π I λ , P ( ρ i , ϕ j , θ , φ ) d φ + θ a ρ i θ a + Y τ P ( θ , H , α λ , at ) sin θ d θ γ a , ρ i γ a , ρ i I λ , P ( ρ i , ϕ j , θ , φ ) d φ + θ a + Y θ R + ρ i τ P ( θ , H , α λ , at ) sin θ d θ γ R , ρ i γ R , ρ i I λ , P ( ρ i , ϕ j , θ , φ ) d φ , 0 < Y < ρ i < a ,
Φ λ , P S ( ρ i , ϕ j ) = { 0 θ R τ P ( θ , H , α λ , at ) sin θ d θ 0 2 π I λ , P ( θ , φ ) d φ , 0 = ρ i < a , 0 θ R ρ i τ P ( θ , H , α λ , at ) sin θ d θ 0 2 π I λ , P ( ρ i , ϕ j , θ , φ ) d φ + θ R ρ i θ R + ρ i τ P ( θ , H , α λ , at ) sin θ d θ γ R , ρ i γ R , ρ i I λ , P ( ρ i , ϕ j , θ , φ ) d φ , 0 < Y ρ i a , 0 θ a ρ i τ P ( θ , H , α λ , at ) sin θ d θ 0 2 π I λ , P ( ρ i , ϕ j , θ , φ ) d φ + θ a ρ i θ a + Y τ P ( θ , H , α λ , at ) sin θ d θ γ a , ρ i γ a , ρ i I λ , P ( ρ i , ϕ j , θ , φ ) d φ + θ a + Y θ R + ρ i τ P ( θ , H , α λ , at ) sin θ d θ γ R , ρ i γ R , ρ i I λ , P ( ρ i , ϕ j , θ , φ ) d φ , 0 < ρ i Y a ,
Φ λ , P S ( ρ i , ϕ j ) = { 0 θ R τ P ( θ ; H ; α λ , at ) sin θ d θ 0 2 π I λ , P ( θ , φ ) d φ , 0 = ρ i < R , 0 θ R ρ i τ P ( θ , H , α λ , at ) sin θ d θ 0 2 π I λ , P ( ρ i , ϕ j , θ , φ ) d φ + θ R ρ i θ R + ρ i τ P ( θ , H , α λ , at ) sin θ d θ γ R , ρ i γ R , ρ i I λ , P ( ρ i , ϕ j , θ , φ ) d φ , 0 < ρ i < R , 0 θ 2 R τ P ( θ , H , α λ , at ) sin θ d θ γ 2 R γ 2 R I λ , P ( ρ i , ϕ j , θ , φ ) d φ , 0 < ρ i = R , θ ρ i R θ R + ρ i τ P ( θ , H , α λ , at ) sin θ d θ γ R , ρ i γ R , ρ i I λ , P ( ρ i , ϕ j , θ , φ ) d φ , 0 < R < ρ i ,
γ a , ρ i = arccos [ ( ρ i 2 + h 2 tan 2 θ a 2 ) ( 2 ρ i h tan θ ) ] ,
γ R , ρ i = arccos [ ( ρ i 2 + H 2 tan 2 θ R 2 ) ( 2 ρ i H tan θ ) ] ,
γ 2 a = arccos [ h tan θ ( 2 a ) ] ,
γ 2 R = arccos [ H tan θ ( 2 R ) ] ,
θ a = arctan ( a h ) ,
θ a ρ i = arctan [ ( a ρ i ) h ] ,
θ a + ρ i = arctan [ ( a + ρ i ) h ] ,
θ 2 a = arctan ( 2 a h ) ,
θ a + Y = arctan [ ( a + Y ) h ] ,
θ R + ρ i = arctan [ ( R + ρ i ) H ] ,
θ R = arctan ( R H ) ,
θ R ρ i = arctan [ ( R ρ i ) H ] ,
θ 2 R = arctan ( 2 R H ) ,
θ ρ i R = arctan [ ( ρ i R ) H ] ,
Y = ( hR Ha ) ( H h ) .
Φ λ , σ S = N λ < Φ λ , σ S > ,
< Φ λ , σ S > = 1 N λ i = 1 N λ , ρ j = 1 N λ , ϕ Φ λ , P S ( ρ i , ϕ j ) ,
i = 1 m N λ ( ρ i ) = N λ , i = 1 m ( ρ i ρ i 1 ) = i = 1 m Δ ρ i = a ,
N λ ( ρ i ) ( 2 π ρ i Δ ρ i ) = N λ ( π a 2 )
< Φ λ , σ S > = { 0 π 2 sin θ d θ 0 2 π I λ , P ( θ , φ ) d φ , h = H = 0 , 0 , 0 < h H ,
< Φ λ , σ S > = 1 N i = 1 m N ( ρ i ) Φ λ , P S ( ρ i ) = 2 a 2 i = 1 m Φ λ , P S ( ρ i ) ρ i Δ ρ i , 0 < h H ,
< Φ λ , σ S > = 2 a 2 0 a Φ λ , P S ( ρ ) ρ d ρ , 0 < h H .
< Φ λ , σ S > = 2 a 2 [ 0 arctan ( a h ) τ P ( θ , H , α λ , at ) sin θ d θ 0 a h tan θ ρ d ρ 0 2 π I λ , P ( θ , φ ) d φ + 0 arctan ( a h ) τ P ( θ , H , α λ , at ) sin θ d θ a h tan θ 0 ρ d ρ γ a , ρ γ a , ρ I λ , P ( θ , φ ) d φ + arctan ( a h ) arctan ( 2 a h ) τ P ( θ , H , α λ , at ) sin θ d θ h tan θ a a ρ d ρ γ a , ρ γ a , ρ I λ , P ( θ , φ ) d φ ] ,
< Φ λ , σ S > = 2 a 2 [ 0 arctan ( a h ) τ P ( θ , H , α λ , at ) sin θ d θ 0 a h tan θ ρ d ρ 0 2 π I P ( θ , φ ) d φ + 0 arctan ( a h ) τ P ( θ , H , α λ , at ) sin θ d θ a h tan θ a ρ d ρ γ a , ρ γ a , ρ I λ , P ( θ , φ ) d φ + arctan ( a h ) arctan [ ( a + Y ) h ] τ P ( θ , H , α λ , at ) sin θ d θ h tan θ a a ρ d ρ γ a , ρ γ a , ρ I λ , P ( θ , φ ) d φ + arctan [ ( a + Y ) h ] arctan [ ( R + a ) H ] τ P ( θ , H , α λ , at ) sin θ d θ H tan θ R a ρ d ρ γ R , ρ γ R , ρ I λ , P ( θ , φ ) d φ ] ,
< Φ λ , σ S > = 2 a 2 [ 0 arctan [ ( R + Y ) H ] τ P ( θ , H , α λ , at ) sin θ d θ 0 a h tan θ ρ d ρ 0 2 π I λ , P ( θ , φ ) d φ + arctan [ ( R + Y ) H ] arctan ( R H ) τ P ( θ , H , α λ , at ) sin θ d θ 0 R H tan θ ρ d ρ 0 2 π I λ , P ( θ , φ ) d φ + arctan [ ( R + Y ) H ] arctan ( R H ) τ P ( θ , H , α λ , at ) sin θ d θ R H tan θ Y ρ d ρ γ R , ρ γ , R , ρ I λ , P ( θ , φ ) d φ + arctan [ ( R + Y ) H ] arctan ( R H ) τ P ( θ , H , α λ , at ) sin θ d θ Y a h tan θ ρ d ρ γ a , ρ γ a , ρ I λ , P ( θ , φ ) d φ + 0 arctan ( R H ) τ P ( θ , H , α λ , at ) sin θ d θ a h tan θ a ρ d ρ γ a , ρ γ a , ρ I λ , P ( θ , φ ) d φ + arctan ( R H ) arctan [ ( R + a ) H ] τ P ( θ , H , α λ , at ) sin θ d θ H tan θ R a ρ d ρ γ R , ρ γ R , ρ I λ , P ( θ , φ ) d φ ] ,
< Φ λ , σ S > = 2 a 2 [ 0 arctan ( R H ) τ P ( θ , H , α λ , at ) sin θ d θ 0 R H tan θ ρ d ρ 0 2 π I λ , P ( θ , φ ) d φ + 0 arctan ( R H ) τ P ( θ , H , α λ , at ) sin θ d θ R H tan θ R + H tan θ ρ d ρ γ R , ρ γ R , ρ I λ , P ( θ , φ ) d φ + arctan ( R H ) arctan [ ( a R ) H ] τ P ( θ , H , α λ , at ) sin θ d θ H tan θ R R + H tan θ ρ d ρ γ R , ρ γ R , ρ I λ , P ( θ , φ ) d φ + arctan [ ( a R ) H ] arctan [ ( R + a ) H ] τ P ( θ , H , α λ , at ) sin θ d θ H tan θ R a ρ d ρ γ R , ρ γ R , ρ I λ , P ( θ , φ ) d φ ] ,
< Φ λ , σ S > = 2 a 2 [ 0 arctan ( R H ) τ P ( θ , H , α λ , at ) sin θ d θ 0 R H tan θ ρ d ρ 0 2 π I λ , P ( θ , φ ) d φ + 0 arctan [ ( a R ) H ] τ P ( θ , H , α λ , at ) sin θ d θ R H tan θ R + H tan θ ρ d ρ γ R , ρ γ R , ρ I λ , P ( θ , φ ) d φ + arctan [ ( a R ) H ] arctan ( R H ) τ P ( θ , H , α λ , at ) sin θ d θ R H tan θ a ρ d ρ γ R , ρ γ R , ρ I λ , P ( θ , φ ) d φ + arctan ( R H ) arctan [ ( R + a ) H ] τ P ( θ , H , α λ , at ) sin θ d θ H tan θ R a ρ d ρ γ R , ρ γ R , ρ I λ , P ( θ , φ ) d φ ] ,
< Φ λ , σ S > = { 2 π 0 π 2 I λ , P ( θ ) sin θ d θ , h = H = 0 , 0 , 0 < h H ,
< Φ λ , σ S > = 2 a 2 0 arctan ( 2 a h ) I λ , P ( θ ) τ P ( θ , H , α λ , at ) g a , a ( θ , a , h ) sin θ d θ ,
< Φ λ , σ S > = 2 a 2 [ 0 arctan [ ( a + Y ) h ] I λ , P ( θ ) τ P ( θ , H , α λ , at ) g a , a ( θ , a , h ) sin θ d θ + arctan [ ( a + Y ) h ] arctan [ ( R + a ) H ] I λ , P ( θ ) τ P ( θ , H , α λ , at ) g R , a ( θ , R , a , h ) sin θ d θ ] ,
< Φ λ , σ S > = 2 a 2 [ 0 arctan ( R H ) I λ , P ( θ ) τ P ( θ , H , α λ , at ) g a , a ( θ , a , h ) sin θ d θ + arctan [ ( R + Y ) H ] arctan ( R H ) I λ , P ( θ ) τ P ( θ , H , α λ , at ) g R , Y ( θ , R , a , H ) sin θ d θ arctan [ ( R + Y ) H ] arctan ( R H ) I λ , P ( θ ) τ P ( θ , H , α λ , at ) g a , Y ( θ , a , h ) sin θ d θ + arctan ( R H ) arctan [ ( R + a ) H ] I λ , P ( θ ) τ P ( θ , H , α λ , at ) g R , a ( θ , R , a , H ) sin θ d θ ] ,
< Φ λ , σ S > = 2 a 2 [ π R 2 0 arctan [ ( a R ) H ] I λ , P ( θ ) τ P ( θ , H , α λ , at ) sin θ d θ + arctan [ ( a R ) H ] arctan [ ( R + a ) H ] I λ , P ( θ ) τ P ( θ , H , α λ , at ) g R , a ( θ , R , a , H ) sin θ d θ ] ,
g a , a ( θ , a , h ) = 2 a 2 arccos [ h tan θ ( 2 a ) ] 1 2 h tan θ 4 a 2 h 2 tan 2 θ ,
g a , Y ( θ , R , a , H ) = Y 2 arccos [ a 2 h 2 tan 2 θ Y 2 2 Yh tan θ ] + a 2 arccos [ a 2 + h 2 tan 2 θ Y 2 2 ah tan θ ] 1 2 [ ( a Y ) 2 h 2 tan 2 θ ] [ h 2 tan 2 θ ( a + Y ) 2 ] ,
g R , Y ( θ , R , a , H ) = Y 2 arccos [ R 2 H 2 tan 2 θ Y 2 2 YH tan θ ] + R 2 arccos [ R 2 + H 2 tan 2 θ Y 2 2 RH tan θ ] 1 2 [ ( R Y ) 2 H 2 tan 2 θ ] [ H 2 tan 2 θ ( R + Y ) 2 ] ,
g R , a ( θ , R , a , H ) = a 2 arccos ( a 2 + H 2 tan 2 θ R 2 2 aH tan θ ) + R 2 arccos ( R 2 + H 2 tan 2 θ a 2 2 RH tan θ ) 1 2 [ ( R + a ) 2 H 2 tan 2 θ ] [ H 2 tan 2 θ ( R a ) 2 ] .
< Φ λ , σ S > = { 2 π I λ , 0 , h = H = 0 , 0 , 0 < h H ,
< Φ λ , σ S > = 2 I λ , 0 a 2 0 arctan ( 2 a h ) τ P ( θ , H , α λ , at ) g a , a ( θ , a , h ) sin θ d θ ,
< Φ λ , σ S > = 2 I λ , 0 a 2 [ 0 arctan [ ( a + Y ) h ] τ P ( θ , H , α λ , at ) g a , a ( θ , a , h ) sin θ d θ + arctan [ ( a + Y ) h ] arctan [ ( R + a ) H ] τ P ( θ , H , α λ , at ) g R , a ( θ , R , a , H ) sin θ d θ ] ,
< Φ λ , σ S > = 2 I λ , 0 a 2 [ 0 arctan ( R H ) τ P ( θ , H , α λ , at ) g a , a ( θ , a , h ) sin θ d θ + arctan [ ( R + Y ) H ] arctan ( R H ) τ P ( θ , H , α λ , at ) g R , Y ( θ , R , a , H , h ) sin θ d θ arctan [ ( R + Y ) H ] arctan ( R H ) τ P ( θ , H , α λ , at ) g a , Y ( θ , R , a , H , h ) sin θ d θ + arctan ( R H ) arctan [ ( R + a ) H ] τ P ( θ , H , α λ , at ) g R , a ( θ , R , a , H ) sin θ d θ ] ,
< Φ λ , σ S > = 2 I λ , 0 a 2 [ π R 2 0 arctan [ ( a R ) H ] τ P ( θ , H , α λ , at ) sin θ d θ + arctan [ ( a R ) H ] arctan [ ( R + a ) H ] τ P ( θ , H , α λ , at ) g R , a ( θ , R , a , H ) sin θ d θ ] ,
< Φ λ , σ S > = { 2 π I λ , 0 , h = H = 0 , 0 , 0 < h H ,
< Φ λ , σ S > = 2 I λ , 0 a 2 0 arctan ( 2 a h ) g a , a ( θ , a , h ) sin θ d θ ,
< Φ λ , σ S > = 2 I λ , 0 a 2 [ 0 arctan [ ( a + Y ) h ] g a , a ( θ , R , a , H ) sin θ d θ + arctan [ ( a + Y ) h ] arctan [ ( R + a ) H ] g R , a ( θ , R , a , H ) sin θ d θ ] ,
< Φ λ , σ S > = 2 I λ , 0 a 2 [ 0 arctan ( R H ) g a , a ( θ , a , h ) sin θ d θ + arctan [ ( R + Y ) H ] arctan ( R H ) g R , Y ( θ , R , a , H ) sin θ d θ arctan [ ( R + Y ) H ] arctan ( R H ) g a , Y ( θ , R , a , h ) sin θ d θ + arctan ( R H ) arctan [ ( R + a ) H ] g R , a ( θ , R , a , H ) sin θ d θ ] ,
< Φ λ , σ S > = 2 I λ , 0 a 2 { π R 2 [ 1 H H 2 + ( R a ) 2 ] + arctan [ ( a R ) H ] arctan [ ( R + a ) H ] g R , a ( θ , R , a , H ) sin θ d θ } ,

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