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 (5)

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 (1)

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 (1)

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 (2)

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 (2)

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 (2)

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 (1)

R. Szweda, “Lasers at the cutting edge,” III-Vs Rev. 12,28–31 (1999).
[Crossref]

1997 (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]

S. Tryka, “Angular distribution of the solid angle at a point subtended by a circular disk,” Optics Comm. 137,317–333 (1997).
[Crossref]

1996 (1)

D.H. Sliney, “Laser effect on vision and ocular exposure limit,” App. Occup. Environ. Hyg. 11,313–319 (1996).
[Crossref]

1995 (1)

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]

Becherer, R.J.

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

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]

Cess, R.

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

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]

Fasol, G.

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

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]

Grum, F

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

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]

Nakamura, S.

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

Paez, G

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.

Pearton, S.

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

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]

Sparrow, E.

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

Strojnik, M.

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.

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]

Wolfram, S.

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

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. (1)

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 (1)

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. (1)

D.H. Sliney, “Laser effect on vision and ocular exposure limit,” App. Occup. Environ. Hyg. 11,313–319 (1996).
[Crossref]

Appl. Phys. Lett. (1)

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. (1)

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. (1)

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. (1)

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. (2)

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. (1)

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. (1)

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. (1)

J.M. Gaines, “Modeling of multichip LED packages for illumination,” Lighting Res. Technol. 38,152–165 (2006).
[Crossref]

Opt. Lasers Tech. (1)

C. Gardner, “The use of misuse of coloured light in the urban environment,” Opt. Lasers Tech. 38,366–376 (2006).
[Crossref]

Optics Comm. (1)

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. (1)

V.P. Gribkovskii, “Injection lasers,” Prog. Quant. Electr. 23,41–88 (1995).
[Crossref]

Radiat. Meas. (1)

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. (1)

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. (1)

T.R. Fry, “Laser safety,” Vet. Clin. Small Anim. 32,535–547 (2002).
[Crossref]

Other (5)

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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