Abstract

In the previous paper [J. Opt. Soc. Am. A 28, 126 (2011)], an analytical formula was presented for calculating radiative fluxes from arbitrarily distributed and arbitrarily radiating multiple-point emitters of bioluminescent or chemiluminescent sources within cylindrical reactors, when the radiation from these point emitters propagates through two homogeneous isotropic media and reaches a planar-circular coaxial detector. This formula was based on two assumptions. The first is that radiation passes across a planar boundary interface between the two media. The second is that the surface reflections on the lateral surface and on the reactor base opposite the detector may be neglected. In this paper, the formula obtained previously was simplified for the case of uniformly distributed point emitters of bioluminescent or chemiluminescent sources emitting an identical rotationally symmetric radiation. The simplified formula is suitable for optimizing and calibrating the analyzed reactor–detector sys tem, which is most commonly used to study the bioluminescence emitted by small biological objects and the chemiluminescence from chemical reactions. Representative data were calculated, illustrated graphically, and tabulated.

© 2011 Optical Society of America

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2011 (1)

2009 (1)

S. Tryka, “A planar-circular detector based on multiple point chemi- or bio-luminescent source within coaxial cylindrical reactor,” J. Quant. Spectrosc. Radiat. Transfer 110, 1864–1878(2009).
[CrossRef]

2007 (1)

2005 (1)

O. Hofmann, P. Miller, P. Sullivan, T. S. Jones, J. C. deMello, D. C. Bradley, and A. J. deMello, “Thin-film organic photodiodes as integrated detectors for microscale chemiluminescence assays,” Sens. Actuators B 106, 878–884 (2005).
[CrossRef]

2004 (1)

Z. R. Xu and Z. L. Fang, “Composite poly(dimethylsiloxane)/glass microfluidic system with an immobilized enzymatic particle-bed reactor and sequential sample injection for chemiluminescence determinations,” Anal. Chim. Acta 507, 129–135(2004).
[CrossRef]

2003 (4)

H. Qiu, J. Yan, X. Sun, J. Liu, W. Cao, X. Yang, and E. Wang, “Microchip capillary electrophoresis with an integrated indium tin oxide electrode-based electrochemiluminescence detector,” Anal. Chem. 75, 5435–5440 (2003).
[CrossRef]

A. M. Jorgensen, K. B. Mogensen, J. P. Kutter, and O. Geschke, “A biochemical microdevice with an integrated chemiluminescence detector,” Sens. Actuators B 90, 15–21 (2003).
[CrossRef]

M. Havaux, “Spontaneous and thermoinduced photon emission: new method to detect and quantify oxidative stress in plants,” Trends Plant Sci. 8, 409–413 (2003).
[CrossRef] [PubMed]

A. Roda, M. Guardigli, E. Michelini, M. Mirasoli, and P. Pasini, “Analytical bioluminescence and chemiluminescence,” Anal. Chem. 75, 462A–470A (2003).
[CrossRef]

2002 (3)

P. J. Worsfold, E. P. Achterberg, A. R. Bowie, V. Cannizzaro, S. Charles, J. M. Costa, F. Dubois, R. Pereiro, B. San Vicente, A. Sanz-Medel, R. Vandeloise, E. Vander Donckt, P. Wollast, and S. Yunus, “Integrated luminometer for the determination of trace metals in seawater using fluorescence, phosphorescence and chemiluminescence detection,” J. Autom. Methods Manag. Chem. 24, 41–47 (2002).

N. Pizà, M. Miró, G. De Armas, E. Becerr, J. M. Estela, and V. Cerdà, “Implementation of chemiluminescence detection in the multisyringe flow injection technique,” Anal. Chim. Acta 467, 155–166 (2002).
[CrossRef]

S. Tryka, “Optical radiation flux illuminating a circular disk from an off-axis point source through two different homogeneous refractive media,” Opt. Commun. 211, 15–30 (2002).
[CrossRef]

2001 (1)

H. Nakamura, Y. Murakami, K. Yokoyama, E. Tamiya, I. Karube, M. Suda, and S. Uchiyama, “A compactly integrated flow cell with a chemiluminescent FIA system for determining lactate concentration in serum,” Anal. Chem. 73, 373–378 (2001).
[CrossRef] [PubMed]

2000 (1)

E. L’Hostis, P. E. Michel, G. C. Fiaccabrino, D. J. Strike, N. F. de Rooij, and M. Koudelka-Hep, “Microreactor and electrochemical detectors fabricated using Si and EPON SU-8,” Sens. Actuators A, Phys. 64B, 156–162 (2000).

1999 (1)

A. W. Knight, “A review of recent trends in analytical applications of electrogenerated chemiluminescence,” Trends Anal. Chem. 18, 47–62 (1999).
[CrossRef]

1997 (3)

F. James, J. Hoogland, and R. Klerss, “Multidimensional sampling for simulation and integration: measures, discrepancies, and quasi-random numbers,” Comput. Phys. Commun. 99, 180–220 (1997).
[CrossRef]

T. Yeh, “Trapezoidal rule for multiple integrals over hyperquadrilaterals,” Appl. Math. Comput. 87, 227–246 (1997).
[CrossRef]

B. Devaraj, M. Usa, and H. Inaba, “Biophotons: ultraweak emission from living systems,” Curr. Opin. Solid State Mater. Sci. 2, 188–193 (1997).
[CrossRef]

1993 (1)

W. Mueller-Klieser and S. Walenta, “Geographical mapping of metabolites in biological tissue with quantitative bioluminescence single photon imaging,” Histochem. J. 25, 407–420 (1993).
[CrossRef] [PubMed]

1988 (1)

H. Inaba, “Super-high sensitivity systems for detection and spectral analysis of ultraweak photon emission from biological cells and tissues,” Experientia 44, 550–559 (1988).
[CrossRef] [PubMed]

1984 (1)

A. Boveris, S. A. Puntarulo, A. H. Roy, and R. A. Sanchez, “Spontaneous chemiluminescence embryonic axes during imbibition,” Plant Physiol. 76, 447–451 (1984).
[CrossRef] [PubMed]

1982 (1)

H. Inaba, Y. C. Takyu, B. Yoda, Y. Goto, T. Miyazawa, T. Keneda, and A. Saeki, “Development of an ultra-high sensitive photon counting system and its application to biomedical measurements,” Opt. Lasers Eng. 3, 125–130 (1982).
[CrossRef]

1981 (1)

E. S. RichJr., C. H. Groover, and J. E. Wampler, “The spatial distribution of light emission from liquid phase bio- and chemiluminescence: variations with container types, turbidity and container frosting,” Photochem. Photobiol. 33, 727–736 (1981).
[CrossRef]

1980 (1)

F. James, “Monte Carlo theory and practice,” Rep. Prog. Phys. 43, 1145–1189 (1980).
[CrossRef]

1979 (1)

H. Inaba, Y. Shimizu, Y. Tsuji, and A. Yamagishi, “Photon counting spectral analyzing system of extra-weak chemi- and bioluminescence for biochemical applications,” Photochem. Photobiol. 30, 169–175 (1979).
[CrossRef]

1973 (1)

Y. Shimizu, H. Inaba, K. Kumaki, K. Mizuno, S. I. Hata, and S. Tomioka, “Measuring methods for ultra-low light intensity and their application to extra-weak spontaneous bioluminescence from living tissues,” IEEE Trans. Instrum. Meas. 22, 153–157 (1973).
[CrossRef]

1965 (1)

J. Lee and H. H. Seliger, “Absolute spectral sensitivity of phototubes and the application of the measurement of the absolute quantum yields of chemiluminescence and bioluminescence,” Photochem. Photobiol. 4, 1015–1048 (1965).
[CrossRef] [PubMed]

Achterberg, E. P.

P. J. Worsfold, E. P. Achterberg, A. R. Bowie, V. Cannizzaro, S. Charles, J. M. Costa, F. Dubois, R. Pereiro, B. San Vicente, A. Sanz-Medel, R. Vandeloise, E. Vander Donckt, P. Wollast, and S. Yunus, “Integrated luminometer for the determination of trace metals in seawater using fluorescence, phosphorescence and chemiluminescence detection,” J. Autom. Methods Manag. Chem. 24, 41–47 (2002).

Becerr, E.

N. Pizà, M. Miró, G. De Armas, E. Becerr, J. M. Estela, and V. Cerdà, “Implementation of chemiluminescence detection in the multisyringe flow injection technique,” Anal. Chim. Acta 467, 155–166 (2002).
[CrossRef]

Boveris, A.

A. Boveris, S. A. Puntarulo, A. H. Roy, and R. A. Sanchez, “Spontaneous chemiluminescence embryonic axes during imbibition,” Plant Physiol. 76, 447–451 (1984).
[CrossRef] [PubMed]

Bowie, A. R.

P. J. Worsfold, E. P. Achterberg, A. R. Bowie, V. Cannizzaro, S. Charles, J. M. Costa, F. Dubois, R. Pereiro, B. San Vicente, A. Sanz-Medel, R. Vandeloise, E. Vander Donckt, P. Wollast, and S. Yunus, “Integrated luminometer for the determination of trace metals in seawater using fluorescence, phosphorescence and chemiluminescence detection,” J. Autom. Methods Manag. Chem. 24, 41–47 (2002).

Bradley, D. C.

O. Hofmann, P. Miller, P. Sullivan, T. S. Jones, J. C. deMello, D. C. Bradley, and A. J. deMello, “Thin-film organic photodiodes as integrated detectors for microscale chemiluminescence assays,” Sens. Actuators B 106, 878–884 (2005).
[CrossRef]

Cannizzaro, V.

P. J. Worsfold, E. P. Achterberg, A. R. Bowie, V. Cannizzaro, S. Charles, J. M. Costa, F. Dubois, R. Pereiro, B. San Vicente, A. Sanz-Medel, R. Vandeloise, E. Vander Donckt, P. Wollast, and S. Yunus, “Integrated luminometer for the determination of trace metals in seawater using fluorescence, phosphorescence and chemiluminescence detection,” J. Autom. Methods Manag. Chem. 24, 41–47 (2002).

Cao, W.

H. Qiu, J. Yan, X. Sun, J. Liu, W. Cao, X. Yang, and E. Wang, “Microchip capillary electrophoresis with an integrated indium tin oxide electrode-based electrochemiluminescence detector,” Anal. Chem. 75, 5435–5440 (2003).
[CrossRef]

Cerdà, V.

N. Pizà, M. Miró, G. De Armas, E. Becerr, J. M. Estela, and V. Cerdà, “Implementation of chemiluminescence detection in the multisyringe flow injection technique,” Anal. Chim. Acta 467, 155–166 (2002).
[CrossRef]

Charles, S.

P. J. Worsfold, E. P. Achterberg, A. R. Bowie, V. Cannizzaro, S. Charles, J. M. Costa, F. Dubois, R. Pereiro, B. San Vicente, A. Sanz-Medel, R. Vandeloise, E. Vander Donckt, P. Wollast, and S. Yunus, “Integrated luminometer for the determination of trace metals in seawater using fluorescence, phosphorescence and chemiluminescence detection,” J. Autom. Methods Manag. Chem. 24, 41–47 (2002).

Costa, J. M.

P. J. Worsfold, E. P. Achterberg, A. R. Bowie, V. Cannizzaro, S. Charles, J. M. Costa, F. Dubois, R. Pereiro, B. San Vicente, A. Sanz-Medel, R. Vandeloise, E. Vander Donckt, P. Wollast, and S. Yunus, “Integrated luminometer for the determination of trace metals in seawater using fluorescence, phosphorescence and chemiluminescence detection,” J. Autom. Methods Manag. Chem. 24, 41–47 (2002).

Davis, P. J.

P. J. Davis and I. Polonsky, Numerical interpolation, differentiation, and integration in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 10th ed., M.Abramowicz and J.A.Stegum, eds., Applied Mathematics Series (National Bureau of Standards, 1972), Vol. 55, pp. 875–899.

De Armas, G.

N. Pizà, M. Miró, G. De Armas, E. Becerr, J. M. Estela, and V. Cerdà, “Implementation of chemiluminescence detection in the multisyringe flow injection technique,” Anal. Chim. Acta 467, 155–166 (2002).
[CrossRef]

de Rooij, N. F.

E. L’Hostis, P. E. Michel, G. C. Fiaccabrino, D. J. Strike, N. F. de Rooij, and M. Koudelka-Hep, “Microreactor and electrochemical detectors fabricated using Si and EPON SU-8,” Sens. Actuators A, Phys. 64B, 156–162 (2000).

deMello, A. J.

O. Hofmann, P. Miller, P. Sullivan, T. S. Jones, J. C. deMello, D. C. Bradley, and A. J. deMello, “Thin-film organic photodiodes as integrated detectors for microscale chemiluminescence assays,” Sens. Actuators B 106, 878–884 (2005).
[CrossRef]

deMello, J. C.

O. Hofmann, P. Miller, P. Sullivan, T. S. Jones, J. C. deMello, D. C. Bradley, and A. J. deMello, “Thin-film organic photodiodes as integrated detectors for microscale chemiluminescence assays,” Sens. Actuators B 106, 878–884 (2005).
[CrossRef]

Devaraj, B.

B. Devaraj, M. Usa, and H. Inaba, “Biophotons: ultraweak emission from living systems,” Curr. Opin. Solid State Mater. Sci. 2, 188–193 (1997).
[CrossRef]

Dubois, F.

P. J. Worsfold, E. P. Achterberg, A. R. Bowie, V. Cannizzaro, S. Charles, J. M. Costa, F. Dubois, R. Pereiro, B. San Vicente, A. Sanz-Medel, R. Vandeloise, E. Vander Donckt, P. Wollast, and S. Yunus, “Integrated luminometer for the determination of trace metals in seawater using fluorescence, phosphorescence and chemiluminescence detection,” J. Autom. Methods Manag. Chem. 24, 41–47 (2002).

Estela, J. M.

N. Pizà, M. Miró, G. De Armas, E. Becerr, J. M. Estela, and V. Cerdà, “Implementation of chemiluminescence detection in the multisyringe flow injection technique,” Anal. Chim. Acta 467, 155–166 (2002).
[CrossRef]

Fang, Z. L.

Z. R. Xu and Z. L. Fang, “Composite poly(dimethylsiloxane)/glass microfluidic system with an immobilized enzymatic particle-bed reactor and sequential sample injection for chemiluminescence determinations,” Anal. Chim. Acta 507, 129–135(2004).
[CrossRef]

Fiaccabrino, G. C.

E. L’Hostis, P. E. Michel, G. C. Fiaccabrino, D. J. Strike, N. F. de Rooij, and M. Koudelka-Hep, “Microreactor and electrochemical detectors fabricated using Si and EPON SU-8,” Sens. Actuators A, Phys. 64B, 156–162 (2000).

Geschke, O.

A. M. Jorgensen, K. B. Mogensen, J. P. Kutter, and O. Geschke, “A biochemical microdevice with an integrated chemiluminescence detector,” Sens. Actuators B 90, 15–21 (2003).
[CrossRef]

Goto, Y.

H. Inaba, Y. C. Takyu, B. Yoda, Y. Goto, T. Miyazawa, T. Keneda, and A. Saeki, “Development of an ultra-high sensitive photon counting system and its application to biomedical measurements,” Opt. Lasers Eng. 3, 125–130 (1982).
[CrossRef]

Groover, C. H.

E. S. RichJr., C. H. Groover, and J. E. Wampler, “The spatial distribution of light emission from liquid phase bio- and chemiluminescence: variations with container types, turbidity and container frosting,” Photochem. Photobiol. 33, 727–736 (1981).
[CrossRef]

Guardigli, M.

A. Roda, M. Guardigli, E. Michelini, M. Mirasoli, and P. Pasini, “Analytical bioluminescence and chemiluminescence,” Anal. Chem. 75, 462A–470A (2003).
[CrossRef]

Hata, S. I.

Y. Shimizu, H. Inaba, K. Kumaki, K. Mizuno, S. I. Hata, and S. Tomioka, “Measuring methods for ultra-low light intensity and their application to extra-weak spontaneous bioluminescence from living tissues,” IEEE Trans. Instrum. Meas. 22, 153–157 (1973).
[CrossRef]

Havaux, M.

M. Havaux, “Spontaneous and thermoinduced photon emission: new method to detect and quantify oxidative stress in plants,” Trends Plant Sci. 8, 409–413 (2003).
[CrossRef] [PubMed]

Hofmann, O.

O. Hofmann, P. Miller, P. Sullivan, T. S. Jones, J. C. deMello, D. C. Bradley, and A. J. deMello, “Thin-film organic photodiodes as integrated detectors for microscale chemiluminescence assays,” Sens. Actuators B 106, 878–884 (2005).
[CrossRef]

Hoogland, J.

F. James, J. Hoogland, and R. Klerss, “Multidimensional sampling for simulation and integration: measures, discrepancies, and quasi-random numbers,” Comput. Phys. Commun. 99, 180–220 (1997).
[CrossRef]

Inaba, H.

B. Devaraj, M. Usa, and H. Inaba, “Biophotons: ultraweak emission from living systems,” Curr. Opin. Solid State Mater. Sci. 2, 188–193 (1997).
[CrossRef]

H. Inaba, “Super-high sensitivity systems for detection and spectral analysis of ultraweak photon emission from biological cells and tissues,” Experientia 44, 550–559 (1988).
[CrossRef] [PubMed]

H. Inaba, Y. C. Takyu, B. Yoda, Y. Goto, T. Miyazawa, T. Keneda, and A. Saeki, “Development of an ultra-high sensitive photon counting system and its application to biomedical measurements,” Opt. Lasers Eng. 3, 125–130 (1982).
[CrossRef]

H. Inaba, Y. Shimizu, Y. Tsuji, and A. Yamagishi, “Photon counting spectral analyzing system of extra-weak chemi- and bioluminescence for biochemical applications,” Photochem. Photobiol. 30, 169–175 (1979).
[CrossRef]

Y. Shimizu, H. Inaba, K. Kumaki, K. Mizuno, S. I. Hata, and S. Tomioka, “Measuring methods for ultra-low light intensity and their application to extra-weak spontaneous bioluminescence from living tissues,” IEEE Trans. Instrum. Meas. 22, 153–157 (1973).
[CrossRef]

James, F.

F. James, J. Hoogland, and R. Klerss, “Multidimensional sampling for simulation and integration: measures, discrepancies, and quasi-random numbers,” Comput. Phys. Commun. 99, 180–220 (1997).
[CrossRef]

F. James, “Monte Carlo theory and practice,” Rep. Prog. Phys. 43, 1145–1189 (1980).
[CrossRef]

Jones, T. S.

O. Hofmann, P. Miller, P. Sullivan, T. S. Jones, J. C. deMello, D. C. Bradley, and A. J. deMello, “Thin-film organic photodiodes as integrated detectors for microscale chemiluminescence assays,” Sens. Actuators B 106, 878–884 (2005).
[CrossRef]

Jorgensen, A. M.

A. M. Jorgensen, K. B. Mogensen, J. P. Kutter, and O. Geschke, “A biochemical microdevice with an integrated chemiluminescence detector,” Sens. Actuators B 90, 15–21 (2003).
[CrossRef]

Karube, I.

H. Nakamura, Y. Murakami, K. Yokoyama, E. Tamiya, I. Karube, M. Suda, and S. Uchiyama, “A compactly integrated flow cell with a chemiluminescent FIA system for determining lactate concentration in serum,” Anal. Chem. 73, 373–378 (2001).
[CrossRef] [PubMed]

Keneda, T.

H. Inaba, Y. C. Takyu, B. Yoda, Y. Goto, T. Miyazawa, T. Keneda, and A. Saeki, “Development of an ultra-high sensitive photon counting system and its application to biomedical measurements,” Opt. Lasers Eng. 3, 125–130 (1982).
[CrossRef]

Klerss, R.

F. James, J. Hoogland, and R. Klerss, “Multidimensional sampling for simulation and integration: measures, discrepancies, and quasi-random numbers,” Comput. Phys. Commun. 99, 180–220 (1997).
[CrossRef]

Knight, A. W.

A. W. Knight, “A review of recent trends in analytical applications of electrogenerated chemiluminescence,” Trends Anal. Chem. 18, 47–62 (1999).
[CrossRef]

Koudelka-Hep, M.

E. L’Hostis, P. E. Michel, G. C. Fiaccabrino, D. J. Strike, N. F. de Rooij, and M. Koudelka-Hep, “Microreactor and electrochemical detectors fabricated using Si and EPON SU-8,” Sens. Actuators A, Phys. 64B, 156–162 (2000).

Kumaki, K.

Y. Shimizu, H. Inaba, K. Kumaki, K. Mizuno, S. I. Hata, and S. Tomioka, “Measuring methods for ultra-low light intensity and their application to extra-weak spontaneous bioluminescence from living tissues,” IEEE Trans. Instrum. Meas. 22, 153–157 (1973).
[CrossRef]

Kutter, J. P.

A. M. Jorgensen, K. B. Mogensen, J. P. Kutter, and O. Geschke, “A biochemical microdevice with an integrated chemiluminescence detector,” Sens. Actuators B 90, 15–21 (2003).
[CrossRef]

L’Hostis, E.

E. L’Hostis, P. E. Michel, G. C. Fiaccabrino, D. J. Strike, N. F. de Rooij, and M. Koudelka-Hep, “Microreactor and electrochemical detectors fabricated using Si and EPON SU-8,” Sens. Actuators A, Phys. 64B, 156–162 (2000).

Lee, J.

J. Lee and H. H. Seliger, “Absolute spectral sensitivity of phototubes and the application of the measurement of the absolute quantum yields of chemiluminescence and bioluminescence,” Photochem. Photobiol. 4, 1015–1048 (1965).
[CrossRef] [PubMed]

Liu, J.

H. Qiu, J. Yan, X. Sun, J. Liu, W. Cao, X. Yang, and E. Wang, “Microchip capillary electrophoresis with an integrated indium tin oxide electrode-based electrochemiluminescence detector,” Anal. Chem. 75, 5435–5440 (2003).
[CrossRef]

Michel, P. E.

E. L’Hostis, P. E. Michel, G. C. Fiaccabrino, D. J. Strike, N. F. de Rooij, and M. Koudelka-Hep, “Microreactor and electrochemical detectors fabricated using Si and EPON SU-8,” Sens. Actuators A, Phys. 64B, 156–162 (2000).

Michelini, E.

A. Roda, M. Guardigli, E. Michelini, M. Mirasoli, and P. Pasini, “Analytical bioluminescence and chemiluminescence,” Anal. Chem. 75, 462A–470A (2003).
[CrossRef]

Miller, P.

O. Hofmann, P. Miller, P. Sullivan, T. S. Jones, J. C. deMello, D. C. Bradley, and A. J. deMello, “Thin-film organic photodiodes as integrated detectors for microscale chemiluminescence assays,” Sens. Actuators B 106, 878–884 (2005).
[CrossRef]

Mirasoli, M.

A. Roda, M. Guardigli, E. Michelini, M. Mirasoli, and P. Pasini, “Analytical bioluminescence and chemiluminescence,” Anal. Chem. 75, 462A–470A (2003).
[CrossRef]

Miró, M.

N. Pizà, M. Miró, G. De Armas, E. Becerr, J. M. Estela, and V. Cerdà, “Implementation of chemiluminescence detection in the multisyringe flow injection technique,” Anal. Chim. Acta 467, 155–166 (2002).
[CrossRef]

Miyazawa, T.

H. Inaba, Y. C. Takyu, B. Yoda, Y. Goto, T. Miyazawa, T. Keneda, and A. Saeki, “Development of an ultra-high sensitive photon counting system and its application to biomedical measurements,” Opt. Lasers Eng. 3, 125–130 (1982).
[CrossRef]

Mizuno, K.

Y. Shimizu, H. Inaba, K. Kumaki, K. Mizuno, S. I. Hata, and S. Tomioka, “Measuring methods for ultra-low light intensity and their application to extra-weak spontaneous bioluminescence from living tissues,” IEEE Trans. Instrum. Meas. 22, 153–157 (1973).
[CrossRef]

Mogensen, K. B.

A. M. Jorgensen, K. B. Mogensen, J. P. Kutter, and O. Geschke, “A biochemical microdevice with an integrated chemiluminescence detector,” Sens. Actuators B 90, 15–21 (2003).
[CrossRef]

Mueller-Klieser, W.

W. Mueller-Klieser and S. Walenta, “Geographical mapping of metabolites in biological tissue with quantitative bioluminescence single photon imaging,” Histochem. J. 25, 407–420 (1993).
[CrossRef] [PubMed]

Murakami, Y.

H. Nakamura, Y. Murakami, K. Yokoyama, E. Tamiya, I. Karube, M. Suda, and S. Uchiyama, “A compactly integrated flow cell with a chemiluminescent FIA system for determining lactate concentration in serum,” Anal. Chem. 73, 373–378 (2001).
[CrossRef] [PubMed]

Nakamura, H.

H. Nakamura, Y. Murakami, K. Yokoyama, E. Tamiya, I. Karube, M. Suda, and S. Uchiyama, “A compactly integrated flow cell with a chemiluminescent FIA system for determining lactate concentration in serum,” Anal. Chem. 73, 373–378 (2001).
[CrossRef] [PubMed]

Pasini, P.

A. Roda, M. Guardigli, E. Michelini, M. Mirasoli, and P. Pasini, “Analytical bioluminescence and chemiluminescence,” Anal. Chem. 75, 462A–470A (2003).
[CrossRef]

Pereiro, R.

P. J. Worsfold, E. P. Achterberg, A. R. Bowie, V. Cannizzaro, S. Charles, J. M. Costa, F. Dubois, R. Pereiro, B. San Vicente, A. Sanz-Medel, R. Vandeloise, E. Vander Donckt, P. Wollast, and S. Yunus, “Integrated luminometer for the determination of trace metals in seawater using fluorescence, phosphorescence and chemiluminescence detection,” J. Autom. Methods Manag. Chem. 24, 41–47 (2002).

Pizà, N.

N. Pizà, M. Miró, G. De Armas, E. Becerr, J. M. Estela, and V. Cerdà, “Implementation of chemiluminescence detection in the multisyringe flow injection technique,” Anal. Chim. Acta 467, 155–166 (2002).
[CrossRef]

Polonsky, I.

P. J. Davis and I. Polonsky, Numerical interpolation, differentiation, and integration in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 10th ed., M.Abramowicz and J.A.Stegum, eds., Applied Mathematics Series (National Bureau of Standards, 1972), Vol. 55, pp. 875–899.

Puntarulo, S. A.

A. Boveris, S. A. Puntarulo, A. H. Roy, and R. A. Sanchez, “Spontaneous chemiluminescence embryonic axes during imbibition,” Plant Physiol. 76, 447–451 (1984).
[CrossRef] [PubMed]

Qiu, H.

H. Qiu, J. Yan, X. Sun, J. Liu, W. Cao, X. Yang, and E. Wang, “Microchip capillary electrophoresis with an integrated indium tin oxide electrode-based electrochemiluminescence detector,” Anal. Chem. 75, 5435–5440 (2003).
[CrossRef]

Rich, E. S.

E. S. RichJr., C. H. Groover, and J. E. Wampler, “The spatial distribution of light emission from liquid phase bio- and chemiluminescence: variations with container types, turbidity and container frosting,” Photochem. Photobiol. 33, 727–736 (1981).
[CrossRef]

Roda, A.

A. Roda, M. Guardigli, E. Michelini, M. Mirasoli, and P. Pasini, “Analytical bioluminescence and chemiluminescence,” Anal. Chem. 75, 462A–470A (2003).
[CrossRef]

Roy, A. H.

A. Boveris, S. A. Puntarulo, A. H. Roy, and R. A. Sanchez, “Spontaneous chemiluminescence embryonic axes during imbibition,” Plant Physiol. 76, 447–451 (1984).
[CrossRef] [PubMed]

Saeki, A.

H. Inaba, Y. C. Takyu, B. Yoda, Y. Goto, T. Miyazawa, T. Keneda, and A. Saeki, “Development of an ultra-high sensitive photon counting system and its application to biomedical measurements,” Opt. Lasers Eng. 3, 125–130 (1982).
[CrossRef]

San Vicente, B.

P. J. Worsfold, E. P. Achterberg, A. R. Bowie, V. Cannizzaro, S. Charles, J. M. Costa, F. Dubois, R. Pereiro, B. San Vicente, A. Sanz-Medel, R. Vandeloise, E. Vander Donckt, P. Wollast, and S. Yunus, “Integrated luminometer for the determination of trace metals in seawater using fluorescence, phosphorescence and chemiluminescence detection,” J. Autom. Methods Manag. Chem. 24, 41–47 (2002).

Sanchez, R. A.

A. Boveris, S. A. Puntarulo, A. H. Roy, and R. A. Sanchez, “Spontaneous chemiluminescence embryonic axes during imbibition,” Plant Physiol. 76, 447–451 (1984).
[CrossRef] [PubMed]

Sanz-Medel, A.

P. J. Worsfold, E. P. Achterberg, A. R. Bowie, V. Cannizzaro, S. Charles, J. M. Costa, F. Dubois, R. Pereiro, B. San Vicente, A. Sanz-Medel, R. Vandeloise, E. Vander Donckt, P. Wollast, and S. Yunus, “Integrated luminometer for the determination of trace metals in seawater using fluorescence, phosphorescence and chemiluminescence detection,” J. Autom. Methods Manag. Chem. 24, 41–47 (2002).

Seliger, H. H.

J. Lee and H. H. Seliger, “Absolute spectral sensitivity of phototubes and the application of the measurement of the absolute quantum yields of chemiluminescence and bioluminescence,” Photochem. Photobiol. 4, 1015–1048 (1965).
[CrossRef] [PubMed]

Shimizu, Y.

H. Inaba, Y. Shimizu, Y. Tsuji, and A. Yamagishi, “Photon counting spectral analyzing system of extra-weak chemi- and bioluminescence for biochemical applications,” Photochem. Photobiol. 30, 169–175 (1979).
[CrossRef]

Y. Shimizu, H. Inaba, K. Kumaki, K. Mizuno, S. I. Hata, and S. Tomioka, “Measuring methods for ultra-low light intensity and their application to extra-weak spontaneous bioluminescence from living tissues,” IEEE Trans. Instrum. Meas. 22, 153–157 (1973).
[CrossRef]

Strike, D. J.

E. L’Hostis, P. E. Michel, G. C. Fiaccabrino, D. J. Strike, N. F. de Rooij, and M. Koudelka-Hep, “Microreactor and electrochemical detectors fabricated using Si and EPON SU-8,” Sens. Actuators A, Phys. 64B, 156–162 (2000).

Stroud, A. H.

A. H. Stroud, Approximate Calculation of Multiple Integrals (Prentice-Hall, 1971).

Suda, M.

H. Nakamura, Y. Murakami, K. Yokoyama, E. Tamiya, I. Karube, M. Suda, and S. Uchiyama, “A compactly integrated flow cell with a chemiluminescent FIA system for determining lactate concentration in serum,” Anal. Chem. 73, 373–378 (2001).
[CrossRef] [PubMed]

Sullivan, P.

O. Hofmann, P. Miller, P. Sullivan, T. S. Jones, J. C. deMello, D. C. Bradley, and A. J. deMello, “Thin-film organic photodiodes as integrated detectors for microscale chemiluminescence assays,” Sens. Actuators B 106, 878–884 (2005).
[CrossRef]

Sun, X.

H. Qiu, J. Yan, X. Sun, J. Liu, W. Cao, X. Yang, and E. Wang, “Microchip capillary electrophoresis with an integrated indium tin oxide electrode-based electrochemiluminescence detector,” Anal. Chem. 75, 5435–5440 (2003).
[CrossRef]

Takyu, Y. C.

H. Inaba, Y. C. Takyu, B. Yoda, Y. Goto, T. Miyazawa, T. Keneda, and A. Saeki, “Development of an ultra-high sensitive photon counting system and its application to biomedical measurements,” Opt. Lasers Eng. 3, 125–130 (1982).
[CrossRef]

Tamiya, E.

H. Nakamura, Y. Murakami, K. Yokoyama, E. Tamiya, I. Karube, M. Suda, and S. Uchiyama, “A compactly integrated flow cell with a chemiluminescent FIA system for determining lactate concentration in serum,” Anal. Chem. 73, 373–378 (2001).
[CrossRef] [PubMed]

Tomioka, S.

Y. Shimizu, H. Inaba, K. Kumaki, K. Mizuno, S. I. Hata, and S. Tomioka, “Measuring methods for ultra-low light intensity and their application to extra-weak spontaneous bioluminescence from living tissues,” IEEE Trans. Instrum. Meas. 22, 153–157 (1973).
[CrossRef]

Tryka, S.

S. Tryka, “Radiative flux from a multiple point bio- or chemi-luminescent source within a cylindrical reactor incident on a planar-circular coaxial detector. I. Arbitrary radiation field,” J. Opt. Soc. Am. A 28, 126–135 (2011).
[CrossRef]

S. Tryka, “A planar-circular detector based on multiple point chemi- or bio-luminescent source within coaxial cylindrical reactor,” J. Quant. Spectrosc. Radiat. Transfer 110, 1864–1878(2009).
[CrossRef]

S. Tryka, “Radiative flux from a planar multiple point source within a cylindrical enclosure reaching a coaxial circular plane,” Opt. Express 15, 3777–3790 (2007).
[CrossRef] [PubMed]

S. Tryka, “Optical radiation flux illuminating a circular disk from an off-axis point source through two different homogeneous refractive media,” Opt. Commun. 211, 15–30 (2002).
[CrossRef]

Tsuji, Y.

H. Inaba, Y. Shimizu, Y. Tsuji, and A. Yamagishi, “Photon counting spectral analyzing system of extra-weak chemi- and bioluminescence for biochemical applications,” Photochem. Photobiol. 30, 169–175 (1979).
[CrossRef]

Uchiyama, S.

H. Nakamura, Y. Murakami, K. Yokoyama, E. Tamiya, I. Karube, M. Suda, and S. Uchiyama, “A compactly integrated flow cell with a chemiluminescent FIA system for determining lactate concentration in serum,” Anal. Chem. 73, 373–378 (2001).
[CrossRef] [PubMed]

Usa, M.

B. Devaraj, M. Usa, and H. Inaba, “Biophotons: ultraweak emission from living systems,” Curr. Opin. Solid State Mater. Sci. 2, 188–193 (1997).
[CrossRef]

Vandeloise, R.

P. J. Worsfold, E. P. Achterberg, A. R. Bowie, V. Cannizzaro, S. Charles, J. M. Costa, F. Dubois, R. Pereiro, B. San Vicente, A. Sanz-Medel, R. Vandeloise, E. Vander Donckt, P. Wollast, and S. Yunus, “Integrated luminometer for the determination of trace metals in seawater using fluorescence, phosphorescence and chemiluminescence detection,” J. Autom. Methods Manag. Chem. 24, 41–47 (2002).

Vander Donckt, E.

P. J. Worsfold, E. P. Achterberg, A. R. Bowie, V. Cannizzaro, S. Charles, J. M. Costa, F. Dubois, R. Pereiro, B. San Vicente, A. Sanz-Medel, R. Vandeloise, E. Vander Donckt, P. Wollast, and S. Yunus, “Integrated luminometer for the determination of trace metals in seawater using fluorescence, phosphorescence and chemiluminescence detection,” J. Autom. Methods Manag. Chem. 24, 41–47 (2002).

Walenta, S.

W. Mueller-Klieser and S. Walenta, “Geographical mapping of metabolites in biological tissue with quantitative bioluminescence single photon imaging,” Histochem. J. 25, 407–420 (1993).
[CrossRef] [PubMed]

Wampler, J. E.

E. S. RichJr., C. H. Groover, and J. E. Wampler, “The spatial distribution of light emission from liquid phase bio- and chemiluminescence: variations with container types, turbidity and container frosting,” Photochem. Photobiol. 33, 727–736 (1981).
[CrossRef]

J. E. Wampler, “Instrumentation: seeing the light and measuring it,” in Chemi- and Bioluminescence, J.G.Burr, ed. (Marcel Dekker, 1985), pp. 1–44.

Wang, E.

H. Qiu, J. Yan, X. Sun, J. Liu, W. Cao, X. Yang, and E. Wang, “Microchip capillary electrophoresis with an integrated indium tin oxide electrode-based electrochemiluminescence detector,” Anal. Chem. 75, 5435–5440 (2003).
[CrossRef]

Wolfram, S.

S. Wolfram, Mathematica—A System for Doing Mathematics by Computer (Addison-Wesley, 1993), pp. 683–707.

Wollast, P.

P. J. Worsfold, E. P. Achterberg, A. R. Bowie, V. Cannizzaro, S. Charles, J. M. Costa, F. Dubois, R. Pereiro, B. San Vicente, A. Sanz-Medel, R. Vandeloise, E. Vander Donckt, P. Wollast, and S. Yunus, “Integrated luminometer for the determination of trace metals in seawater using fluorescence, phosphorescence and chemiluminescence detection,” J. Autom. Methods Manag. Chem. 24, 41–47 (2002).

Worsfold, P. J.

P. J. Worsfold, E. P. Achterberg, A. R. Bowie, V. Cannizzaro, S. Charles, J. M. Costa, F. Dubois, R. Pereiro, B. San Vicente, A. Sanz-Medel, R. Vandeloise, E. Vander Donckt, P. Wollast, and S. Yunus, “Integrated luminometer for the determination of trace metals in seawater using fluorescence, phosphorescence and chemiluminescence detection,” J. Autom. Methods Manag. Chem. 24, 41–47 (2002).

Xu, Z. R.

Z. R. Xu and Z. L. Fang, “Composite poly(dimethylsiloxane)/glass microfluidic system with an immobilized enzymatic particle-bed reactor and sequential sample injection for chemiluminescence determinations,” Anal. Chim. Acta 507, 129–135(2004).
[CrossRef]

Yamagishi, A.

H. Inaba, Y. Shimizu, Y. Tsuji, and A. Yamagishi, “Photon counting spectral analyzing system of extra-weak chemi- and bioluminescence for biochemical applications,” Photochem. Photobiol. 30, 169–175 (1979).
[CrossRef]

Yan, J.

H. Qiu, J. Yan, X. Sun, J. Liu, W. Cao, X. Yang, and E. Wang, “Microchip capillary electrophoresis with an integrated indium tin oxide electrode-based electrochemiluminescence detector,” Anal. Chem. 75, 5435–5440 (2003).
[CrossRef]

Yang, X.

H. Qiu, J. Yan, X. Sun, J. Liu, W. Cao, X. Yang, and E. Wang, “Microchip capillary electrophoresis with an integrated indium tin oxide electrode-based electrochemiluminescence detector,” Anal. Chem. 75, 5435–5440 (2003).
[CrossRef]

Yeh, T.

T. Yeh, “Trapezoidal rule for multiple integrals over hyperquadrilaterals,” Appl. Math. Comput. 87, 227–246 (1997).
[CrossRef]

Yoda, B.

H. Inaba, Y. C. Takyu, B. Yoda, Y. Goto, T. Miyazawa, T. Keneda, and A. Saeki, “Development of an ultra-high sensitive photon counting system and its application to biomedical measurements,” Opt. Lasers Eng. 3, 125–130 (1982).
[CrossRef]

Yokoyama, K.

H. Nakamura, Y. Murakami, K. Yokoyama, E. Tamiya, I. Karube, M. Suda, and S. Uchiyama, “A compactly integrated flow cell with a chemiluminescent FIA system for determining lactate concentration in serum,” Anal. Chem. 73, 373–378 (2001).
[CrossRef] [PubMed]

Yunus, S.

P. J. Worsfold, E. P. Achterberg, A. R. Bowie, V. Cannizzaro, S. Charles, J. M. Costa, F. Dubois, R. Pereiro, B. San Vicente, A. Sanz-Medel, R. Vandeloise, E. Vander Donckt, P. Wollast, and S. Yunus, “Integrated luminometer for the determination of trace metals in seawater using fluorescence, phosphorescence and chemiluminescence detection,” J. Autom. Methods Manag. Chem. 24, 41–47 (2002).

Anal. Chem. (3)

A. Roda, M. Guardigli, E. Michelini, M. Mirasoli, and P. Pasini, “Analytical bioluminescence and chemiluminescence,” Anal. Chem. 75, 462A–470A (2003).
[CrossRef]

H. Nakamura, Y. Murakami, K. Yokoyama, E. Tamiya, I. Karube, M. Suda, and S. Uchiyama, “A compactly integrated flow cell with a chemiluminescent FIA system for determining lactate concentration in serum,” Anal. Chem. 73, 373–378 (2001).
[CrossRef] [PubMed]

H. Qiu, J. Yan, X. Sun, J. Liu, W. Cao, X. Yang, and E. Wang, “Microchip capillary electrophoresis with an integrated indium tin oxide electrode-based electrochemiluminescence detector,” Anal. Chem. 75, 5435–5440 (2003).
[CrossRef]

Anal. Chim. Acta (2)

Z. R. Xu and Z. L. Fang, “Composite poly(dimethylsiloxane)/glass microfluidic system with an immobilized enzymatic particle-bed reactor and sequential sample injection for chemiluminescence determinations,” Anal. Chim. Acta 507, 129–135(2004).
[CrossRef]

N. Pizà, M. Miró, G. De Armas, E. Becerr, J. M. Estela, and V. Cerdà, “Implementation of chemiluminescence detection in the multisyringe flow injection technique,” Anal. Chim. Acta 467, 155–166 (2002).
[CrossRef]

Appl. Math. Comput. (1)

T. Yeh, “Trapezoidal rule for multiple integrals over hyperquadrilaterals,” Appl. Math. Comput. 87, 227–246 (1997).
[CrossRef]

Comput. Phys. Commun. (1)

F. James, J. Hoogland, and R. Klerss, “Multidimensional sampling for simulation and integration: measures, discrepancies, and quasi-random numbers,” Comput. Phys. Commun. 99, 180–220 (1997).
[CrossRef]

Curr. Opin. Solid State Mater. Sci. (1)

B. Devaraj, M. Usa, and H. Inaba, “Biophotons: ultraweak emission from living systems,” Curr. Opin. Solid State Mater. Sci. 2, 188–193 (1997).
[CrossRef]

Experientia (1)

H. Inaba, “Super-high sensitivity systems for detection and spectral analysis of ultraweak photon emission from biological cells and tissues,” Experientia 44, 550–559 (1988).
[CrossRef] [PubMed]

Histochem. J. (1)

W. Mueller-Klieser and S. Walenta, “Geographical mapping of metabolites in biological tissue with quantitative bioluminescence single photon imaging,” Histochem. J. 25, 407–420 (1993).
[CrossRef] [PubMed]

IEEE Trans. Instrum. Meas. (1)

Y. Shimizu, H. Inaba, K. Kumaki, K. Mizuno, S. I. Hata, and S. Tomioka, “Measuring methods for ultra-low light intensity and their application to extra-weak spontaneous bioluminescence from living tissues,” IEEE Trans. Instrum. Meas. 22, 153–157 (1973).
[CrossRef]

J. Autom. Methods Manag. Chem. (1)

P. J. Worsfold, E. P. Achterberg, A. R. Bowie, V. Cannizzaro, S. Charles, J. M. Costa, F. Dubois, R. Pereiro, B. San Vicente, A. Sanz-Medel, R. Vandeloise, E. Vander Donckt, P. Wollast, and S. Yunus, “Integrated luminometer for the determination of trace metals in seawater using fluorescence, phosphorescence and chemiluminescence detection,” J. Autom. Methods Manag. Chem. 24, 41–47 (2002).

J. Opt. Soc. Am. A (1)

J. Quant. Spectrosc. Radiat. Transfer (1)

S. Tryka, “A planar-circular detector based on multiple point chemi- or bio-luminescent source within coaxial cylindrical reactor,” J. Quant. Spectrosc. Radiat. Transfer 110, 1864–1878(2009).
[CrossRef]

Opt. Commun. (1)

S. Tryka, “Optical radiation flux illuminating a circular disk from an off-axis point source through two different homogeneous refractive media,” Opt. Commun. 211, 15–30 (2002).
[CrossRef]

Opt. Express (1)

Opt. Lasers Eng. (1)

H. Inaba, Y. C. Takyu, B. Yoda, Y. Goto, T. Miyazawa, T. Keneda, and A. Saeki, “Development of an ultra-high sensitive photon counting system and its application to biomedical measurements,” Opt. Lasers Eng. 3, 125–130 (1982).
[CrossRef]

Photochem. Photobiol. (3)

H. Inaba, Y. Shimizu, Y. Tsuji, and A. Yamagishi, “Photon counting spectral analyzing system of extra-weak chemi- and bioluminescence for biochemical applications,” Photochem. Photobiol. 30, 169–175 (1979).
[CrossRef]

J. Lee and H. H. Seliger, “Absolute spectral sensitivity of phototubes and the application of the measurement of the absolute quantum yields of chemiluminescence and bioluminescence,” Photochem. Photobiol. 4, 1015–1048 (1965).
[CrossRef] [PubMed]

E. S. RichJr., C. H. Groover, and J. E. Wampler, “The spatial distribution of light emission from liquid phase bio- and chemiluminescence: variations with container types, turbidity and container frosting,” Photochem. Photobiol. 33, 727–736 (1981).
[CrossRef]

Plant Physiol. (1)

A. Boveris, S. A. Puntarulo, A. H. Roy, and R. A. Sanchez, “Spontaneous chemiluminescence embryonic axes during imbibition,” Plant Physiol. 76, 447–451 (1984).
[CrossRef] [PubMed]

Rep. Prog. Phys. (1)

F. James, “Monte Carlo theory and practice,” Rep. Prog. Phys. 43, 1145–1189 (1980).
[CrossRef]

Sens. Actuators A, Phys. (1)

E. L’Hostis, P. E. Michel, G. C. Fiaccabrino, D. J. Strike, N. F. de Rooij, and M. Koudelka-Hep, “Microreactor and electrochemical detectors fabricated using Si and EPON SU-8,” Sens. Actuators A, Phys. 64B, 156–162 (2000).

Sens. Actuators B (2)

A. M. Jorgensen, K. B. Mogensen, J. P. Kutter, and O. Geschke, “A biochemical microdevice with an integrated chemiluminescence detector,” Sens. Actuators B 90, 15–21 (2003).
[CrossRef]

O. Hofmann, P. Miller, P. Sullivan, T. S. Jones, J. C. deMello, D. C. Bradley, and A. J. deMello, “Thin-film organic photodiodes as integrated detectors for microscale chemiluminescence assays,” Sens. Actuators B 106, 878–884 (2005).
[CrossRef]

Trends Anal. Chem. (1)

A. W. Knight, “A review of recent trends in analytical applications of electrogenerated chemiluminescence,” Trends Anal. Chem. 18, 47–62 (1999).
[CrossRef]

Trends Plant Sci. (1)

M. Havaux, “Spontaneous and thermoinduced photon emission: new method to detect and quantify oxidative stress in plants,” Trends Plant Sci. 8, 409–413 (2003).
[CrossRef] [PubMed]

Other (4)

J. E. Wampler, “Instrumentation: seeing the light and measuring it,” in Chemi- and Bioluminescence, J.G.Burr, ed. (Marcel Dekker, 1985), pp. 1–44.

P. J. Davis and I. Polonsky, Numerical interpolation, differentiation, and integration in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 10th ed., M.Abramowicz and J.A.Stegum, eds., Applied Mathematics Series (National Bureau of Standards, 1972), Vol. 55, pp. 875–899.

A. H. Stroud, Approximate Calculation of Multiple Integrals (Prentice-Hall, 1971).

S. Wolfram, Mathematica—A System for Doing Mathematics by Computer (Addison-Wesley, 1993), pp. 683–707.

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Figures (2)

Fig. 1
Fig. 1

Perspective view of the volume multiple-point source V c within a cylindrical reactor in axial symmetry with respect to the planar-circular detector of surface S 2 , as well as some definitions of geometrical variables.

Fig. 2
Fig. 2

Average fluxes Φ λ , V c S 2 computed for h = 4 cm and α λ , a t 2 = 0 cm 1 , (a) as a function of R and H 1 at a = 3 cm , H 2 = 4 cm , and α λ , a t 1 = 0 cm 1 ; (b) as a function of R and H 1 at a = 3 cm , H 2 = 4 cm , and α λ , a t 1 = 0.1 cm 1 ; (c) as a function of a and H 1 at R = 3 cm , H 2 = 4 cm , and α λ , a t 1 = 0 cm 1 ; (d) as a function of a and H 1 at R = 3 cm , H 2 = 4 cm , and α λ , a t 1 = 0.1 cm 1 ; (e) as a function of a and R at H 1 = 3 cm , H 2 = 4 cm , and α λ , a t 1 = 0 cm 1 ; (f) as a function of a and R at H 1 = 3 cm , H 2 = 4 cm , and α λ , a t 1 = 0.1 cm 1 ; (g) as a function of a and H 2 at R = 3 cm , H 1 = 3 cm , and α λ , a t 1 = 0 cm 1 ; and (h) as a function of a and H 2 at R = 3 cm , H 1 = 3 cm , and α λ , a t 1 = 0.1 cm 1 . The radiative fluxes Φ λ , V c S 2 were calculated within the spectral bandwidth Δ λ = 1 nm at n 1 = 1.33 and n 2 = 1.0 for the radiant intensity I λ , P 1 = I λ , 0 = 1 nW · sr 1 · nm 1 .

Tables (2)

Tables Icon

Table 1 Average Fluxes Φ λ , V c s 2 Computed for I λ , 0 = 1 nW · sr 1 · nm 1 at h = 1.0 cm , H 2 = 1.0 cm , R = 1.0 cm , n 1 = 1.33 , n 2 = 1.00 , and α λ , a t 2 = 0 a

Tables Icon

Table 2 Average Fluxes Φ λ , V c S 2 Δ λ = 1 nm Computed for I λ , 0 = 1 nW · sr 1 · nm 1 at h = 1.0 cm , H 1 = 0.75 cm , R = 1.0 cm , n 1 = 1.33 , n 2 = 1.00 , and α λ , a t 2 = 0 a

Equations (31)

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Φ λ , V c S 2 = i = 1 N λ , ρ j = 1 N λ , ϕ k = 1 N λ , z Φ λ , P 1 S 2 ( ρ i , ϕ j , z k ) , [ Φ λ , V c S 2 ] = W · m 1 ,
Φ λ , V c S 2 = i = 1 N λ , ρ j = 1 N λ , ϕ k = 1 N λ , z Φ λ , P 1 S 2 ( ρ i , ϕ j , z k ) = N λ , ϕ i = 1 N λ , ρ k = 1 N λ , z Φ λ , P 1 S 2 ( ρ i , z k ) = 2 N λ a 2 H 1 i = 1 N λ , ρ k = 1 N λ , z Φ λ , P 1 S 2 ( ρ i , z k ) ,
Φ λ , V c S 2 = i = 1 N λ , ρ j = 1 N λ , ϕ k = 1 N λ , z Φ λ , P 1 S 2 ( ρ i , ϕ j , z k ) = N λ π a 2 H 1 V c Φ λ , P 1 S 2 ( ρ , ϕ , z ) d V c = N λ π a 2 H 1 V c Φ λ , P 1 S 2 ( ρ , ϕ , z ) ρ d ρ d ϕ d z = 2 N λ a 2 H 1 0 H 1 d z 0 a Φ λ , P 1 S 2 ( ρ , z ) ρ d ρ ,
Φ λ , V c S 2 = Φ λ , V c S 2 N λ = 2 a 2 H 1 i = 1 N λ , ρ k = 1 N λ , z Φ λ , P 1 S 2 ( ρ i , z k ) ,
Φ λ , V c S 2 = Φ λ , V c S 2 N λ = 2 a 2 H 1 0 H 1 d z 0 a Φ λ , P 1 S 2 ( ρ , z ) ρ d ρ ,
Φ λ , P 1 S 2 ( ρ i , z k ) = { 0 , ρ i = 0 , 0 z k < H 1 h , 0 , ρ i = 0 , 0 z k = H 1 < h , 2 π 0 arctan ( R / H 2 ) I λ , P 1 τ λ , P 1 P 2 , z k = H 1 sin θ 1 d θ 1 , ρ i = 0 , 0 z k = H 1 = h ,
Φ λ , P 1 S 2 ( ρ i , z k ) = { 2 π 0 θ 1 , a , z k < H 1 I λ , P 1 A λ τ λ , P 1 P 2 , z k < H 1 d θ 1 , 0 = ρ i < a , 2 π 0 θ 1 , a ρ i , z k < H 1 I λ , P 1 A λ τ λ , P 1 P 2 , z k < H 1 d θ 1 + 2 θ 1 , a ρ i , z k < H 1 θ 1 , a + ρ i , z k < H 1 I λ , P 1 A λ τ λ , P 1 P 2 , z k < H 1 arccos [ ρ i 2 + r 2 , a , z k < H 1 2 a 2 2 ρ i r 2 , a , z k < H 1 ] d θ 1 , 0 < ρ i < a , 2 0 θ 1 , 2 a , z k < H 1 I λ , P 1 A λ τ λ , P 1 P 2 , z k < H 1 arccos [ r 2 , a , z k < H 1 2 a ] d θ 1 , 0 < ρ i = a ,
Φ λ , P 1 S 2 ( ρ i , z k ) = { 2 π 0 θ 1 , a , z k < H 1 I λ , P 1 A λ τ λ , P 1 P 2 , z k < H 1 d θ 1 , 0 = ρ i < a , 2 π 0 θ 1 , a ρ i , z k < H 1 I λ , P 1 A λ τ λ , P 1 P 2 , z k < H 1 d θ 1 + 2 θ 1 , a ρ i , z k < H 1 θ 1 , a + ρ i , z k < H 1 I λ , P 1 A λ τ λ , P 1 P 2 , z k < H 1 arccos [ ρ i 2 + r 2 , a , z k < H 1 2 a 2 2 ρ i r 2 , a , z k < H 1 ] d θ 1 , 0 < ρ i Y 1 a , 2 π 0 θ 1 , a ρ i , z k < H 1 I λ , P 1 A λ τ λ , P 1 P 2 , z k < H 1 d θ 1 + 2 θ 1 , a ρ i , z k < H 1 θ 1 , a + Y 1 I λ , P 1 A λ τ λ , P 1 P 2 , z k < H 1 arccos [ ρ i 2 + r 2 , a , z k < H 1 2 a 2 2 ρ i r 2 , a , z k < H 1 ] d θ 1 + 2 θ 1 , a + Y 1 θ 1 , R 2 + ρ i , z k < H 1 I λ , P 1 A λ τ λ , P 1 P 2 , z k < H 1 arccos [ ρ i 2 + r 2 , R , z k < H 1 2 R 2 2 ρ i r 2 , R , z k < H 1 ] d θ 1 , 0 < Y 1 < ρ i a ,
Φ λ , P 1 S 2 ( ρ i , z k ) = { 2 π 0 θ 1 , R , z k < H 1 I λ , P 1 A λ τ λ , P 1 P 2 , z k < H 1 d θ 1 , 0 = ρ i < a , 2 π 0 θ 1 , R ρ i , z k < H 1 I λ , P 1 A λ τ λ , P 1 P 2 , z k < H 1 d θ 1 + 2 θ 1 , R ρ i , z k < H 1 θ 1 , R + ρ i , z k < H 1 I λ , P 1 A λ τ λ , P 1 P 2 , z k < H 1 arccos [ ρ i 2 + r 2 , R , z k < H 1 2 R 2 2 ρ i r 2 , R , z k < H 1 ] d θ 1 , 0 < ρ i Y 1 a , 2 π 0 θ 1 , a ρ i , z k < H 1 I λ , P 1 A λ τ λ , P 1 P 2 , z k < H 1 d θ 1 + 2 θ 1 , a ρ i , z k < H 1 θ 1 , R , z k < H 1 I λ , P 1 A λ τ λ , P 1 P 2 , z k < H 1 arccos [ ρ i 2 + r 2 , a , z k < H 1 2 a 2 2 ρ i r 2 , a , z k < H 1 ] d θ 1 + 2 θ 1 , R , z k < H 1 θ 1 , R + ρ i , z k < H 1 I λ , P 1 A λ τ λ , P 1 P 2 , z k < H 1 arccos [ ρ i 2 + r 2 , R , z k < H 1 2 R 2 2 ρ i r 2 , R , z k < H 1 ] d θ 1 , 0 < Y 1 < ρ i a ,
Φ λ , P 1 S 2 ( ρ i , z k ) = { 2 π 0 θ 1 , R , z k < H 1 I λ , P 1 A λ τ λ , P 1 P 2 , z k < H 1 d θ 1 , 0 = ρ i < R , 2 π 0 θ 1 , R ρ i , z k < H 1 I λ , P 1 A λ τ λ , P 1 P 2 , z k < H 1 d θ 1 + 2 θ 1 , R ρ i , z k < H 1 θ 1 , R + ρ i , z k < H 1 I λ , P 1 A λ τ λ , P 1 P 2 , z k < H 1 arccos [ ρ i 2 + r 2 , R , z k < H 1 2 R 2 2 ρ i r 2 , R , z k < H 1 ] d θ 1 , 0 < ρ i < R , 2 0 θ 1 , 2 R , z k < H 1 I λ , P 1 A λ τ λ , P 1 P 2 , z k < H 1 arccos [ r 2 , R , z k < H 1 2 R ] d θ 1 , 0 < ρ i = R , 2 θ 1 , ρ i R , z k < H 1 θ 1 , R + ρ i , z k < H 1 I λ , P 1 A λ τ λ , P 1 P 2 , z k < H 1 arccos [ ρ i 2 + r 2 , R , z k < H 1 2 R 2 2 ρ i r 2 , R , z k < H 1 ] d θ 1 , 0 < R < ρ i ,
Φ λ , P 1 S 2 ( ρ i , z k ) = { 2 π 0 arctan [ a / ( h H 1 ) ] I λ , P 1 τ λ , P 1 P 2 , z k = H 1 sin θ 1 d θ 1 , 0 = ρ i < a , 2 π 0 arctan [ ( a ρ i ) / ( h H 1 ) ] I λ , P 1 τ λ , P 1 P 2 , z k = H 1 sin θ 1 d θ 1 + 2 arctan [ ( a ρ i ) / ( h H 1 ) ] arctan [ ( a + ρ i ) / ( h H 1 ) ] I λ , P 1 τ λ , P 1 P 2 , z k = H 1 arccos [ ρ i 2 + ( h H 1 ) 2 tan 2 θ 1 a 2 2 ρ i ( h H 1 ) tan θ 1 ] sin θ 1 d θ 1 , 0 < ρ i < a , 2 0 arctan [ 2 a / ( h H 1 ) ] I λ , P 1 τ λ , P 1 P 2 , z k = H 1 arccos [ ( h H 1 ) tan θ 1 2 a ] sin θ 1 d θ 1 , 0 < ρ i = a ,
Φ λ , P 1 S 2 ( ρ i , z k ) = { 2 π 0 arctan [ a / ( h H 1 ) ] I λ , P 1 τ λ , P 1 P 2 , z k = H 1 sin θ 1 d θ 1 , 0 = ρ i < a , 2 π 0 arctan [ ( a ρ i ) / ( h H 1 ) ] I λ , P 1 τ λ , P 1 P 2 , z k = H 1 sin θ 1 d θ 1 + 2 arctan [ ( a ρ i ) / ( h H 1 ) ] arctan [ ( a + ρ i ) / ( h H 1 ) ] I λ , P 1 τ λ , P 1 P 2 , z k = H 1 arccos [ ρ i 2 + ( h H 1 ) 2 tan 2 θ 1 a 2 2 ρ i ( h H 1 ) tan θ 1 ] sin θ 1 d θ 1 , 0 < ρ i Y 2 a , 2 π 0 arctan [ ( a ρ i ) / ( h H 1 ) ] I λ , P 1 τ λ , P 1 P 2 , z k = H 1 sin θ 1 d θ 1 + 2 arctan [ ( a ρ i ) / ( h H 1 ) ] arctan [ ( R a ) / ( H 1 + H 2 h ) ] I λ , P 1 τ λ , P 1 P 2 , z k = H 1 arccos [ ρ i 2 + ( h H 1 ) 2 tan 2 θ 1 a 2 2 ρ i ( h H 1 ) tan θ 1 ] sin θ 1 d θ 1 + 2 arctan [ ( R a ) / ( H 1 + H 2 h ) ] arctan [ ( R + ρ i ) / H 2 ] I λ , P 1 τ λ , P 1 P 2 , z k = H 1 arccos [ ρ i 2 + H 2 2 tan 2 θ 1 R 2 2 ρ i H 2 tan θ 1 ] sin θ 1 d θ 1 , 0 < Y 2 < ρ i a ,
Φ λ , P 1 S 2 ( ρ i , z k ) = { 2 π 0 arctan ( R / H 2 ) I λ , P 1 τ λ , P 1 P 2 , z k = H 1 sin θ 1 d θ 1 , 0 = ρ i < a , 2 π 0 arctan [ ( R ρ i ) / H 2 ] I λ , P 1 τ λ , P 1 P 2 , z k = H 1 sin θ 1 d θ 1 + 2 arctan [ ( R ρ i ) / H 2 ] arctan [ ( R + ρ i ) / H 2 ] I λ , P 1 τ λ , P 1 P 2 , z k = H 1 arccos [ ρ i 2 + H 2 2 tan 2 θ 1 R 2 2 ρ i H 2 tan θ 1 ] sin θ 1 d θ 1 , 0 < ρ i Y 2 a , 2 π 0 arctan [ ( a ρ i ) / ( h H 1 ) ] I λ , P 1 τ λ , P 1 P 2 , z k = H 1 sin θ 1 d θ 1 + 2 arctan [ ( a ρ i ) / ( h H 1 ) ] arctan ( R / H 2 ) I λ , P 1 τ λ , P 1 P 2 , z k = H 1 arccos [ ρ i 2 + ( h H 1 ) 2 tan 2 θ 1 a 2 2 ρ i ( h H 1 ) tan θ 1 ] sin θ 1 d θ 1 + 2 arctan ( R / H 2 ) arctan [ ( R + ρ i ) / H 2 ] I λ , P 1 τ λ , P 1 P 2 , z k = H 1 arccos [ ρ i 2 + H 2 2 tan 2 θ 1 R 2 2 ρ i H 2 tan θ 1 ] sin θ 1 d θ 1 , 0 < Y 2 < ρ i a ,
Φ λ , P 1 S 2 ( ρ i , z k ) = { 2 π 0 arctan ( R / H 2 ) I λ , P 1 τ λ , P 1 P 2 , z k = H 1 sin θ 1 d θ 1 , 0 = ρ i < R , 2 π 0 arctan [ ( R ρ i ) / H 2 ] I λ , P 1 τ λ , P 1 P 2 , z k = H 1 sin θ 1 d θ 1 + 2 arctan [ ( R ρ i ) / H 2 ] arctan [ ( R + ρ i ) / H 2 ] I λ , P 1 τ λ , P 1 P 2 , z k = H 1 arccos [ ρ i 2 + H 2 2 tan 2 θ 1 R 2 2 ρ i H 2 tan θ 1 ] sin θ 1 d θ 1 , 0 < ρ i < R , 2 0 arctan ( 2 R / H 2 ) I λ , P 1 τ λ , P 1 P 2 , z k = H 1 arccos [ H 2 tan θ 1 2 R ] sin θ 1 d θ 1 , 0 < ρ i = R , 2 arctan [ ( ρ i R ) / H 2 ] arctan [ ( R + ρ i ) / H 2 ] I λ , P 1 τ λ , P 1 P 2 , z k = H 1 arccos [ ρ i 2 + H 2 2 tan 2 θ 1 R 2 2 ρ i H 2 tan θ 1 ] sin θ 1 d θ 1 0 < R < ρ i ,
A λ = n 1 2 sin θ 1 cos θ 1 / [ n 2 ( n 2 2 n 1 2 sin 2 θ 1 ) 1 / 2 ] , if 0 θ 1 < π / 2 n 1 n 2 0 θ 1 < arcsin ( n 2 / n 1 ) n 1 > n 2 ,
τ λ , P 1 P 2 , z k < H 1 = τ λ , 12 ( θ 1 ) exp [ α λ , a t 1 ( H 1 z k ) cos θ 1 α λ , a t 2 H 2 n 2 ( n 2 2 n 1 2 sin 2 θ 1 ) 1 / 2 ] , if 0 θ 1 < π / 2 n 1 n 2 0 θ 1 < arcsin ( n 2 / n 1 ) n 1 > n 2 ,
τ λ , P 1 P 2 , z k = H 1 = exp ( α λ , a t 2 H 2 cos θ 1 ) , if 0 θ 1 < π / 2 ,
a ( H 1 z k ) sin θ 1 , a , z k < H 1 ( 1 sin 2 θ 1 , a , z k < H 1 ) 1 / 2 ( h H 1 ) n 1 sin θ 1 , a , z k < H 1 ( n 2 2 n 1 2 sin 2 θ 1 , a , z k < H 1 ) 1 / 2 = 0 , if 0 θ 1 , a , z k < H 1 < π / 2 n 1 n 2 0 θ 1 , a , z k < H 1 < arcsin ( n 2 / n 1 ) n 1 > n 2 ,
a ρ i ( H 1 z k ) sin θ 1 , a ρ i , z k < H 1 ( 1 sin 2 θ 1 , a ρ i , z k < H 1 ) 1 / 2 ( h H 1 ) n 1 sin θ 1 , a ρ i , z k < H 1 ( n 2 2 n 1 2 sin 2 θ 1 , a ρ i , z k < H 1 ) 1 / 2 = 0 , if 0 θ 1 , a ρ i , z k < H 1 < π / 2 n 1 n 2 0 θ 1 , a ρ i , z k < H 1 < arcsin ( n 2 / n 1 ) n 1 > n 2 ,
a + ρ i ( H 1 z k ) sin θ 1 , a + ρ i , z k < H 1 ( 1 sin 2 θ 1 , a + ρ i , z k < H 1 ) 1 / 2 ( h H 1 ) n 1 sin θ 1 , a + ρ i , z k < H 1 ( n 2 2 n 1 2 sin 2 θ 1 , a + ρ i , z k < H 1 ) 1 / 2 = 0 , if 0 θ 1 , a + ρ i , z k < H 1 < π / 2 n 1 n 2 0 θ 1 , a + ρ i , z k < H 1 < arcsin ( n 2 / n 1 ) n 1 > n 2 ,
2 a ( H 1 z k ) sin θ 1 , 2 a , z k < H 1 ( 1 sin 2 θ 1 , 2 a , z k < H 1 ) 1 / 2 ( h H 1 ) n 1 sin θ 1 , 2 a , z k < H 1 ( n 2 2 n 1 2 sin 2 θ 1 , 2 a , z k < H 1 ) 1 / 2 = 0 , if 0 θ 1 , 2 a , z k < H 1 < π / 2 n 1 n 2 0 θ 1 , 2 a , z k < H 1 < arcsin ( n 2 / n 1 ) n 1 > n 2 ,
θ 1 , a + Y 1 = arctan [ n 2 ( R a ) / n 1 2 [ ( H 1 + H 2 h ) 2 + ( R a ) 2 ] n 2 2 ( R a ) 2 ] ,
R ( H 1 z k ) sin θ 1 , R , z k < H 1 ( 1 sin 2 θ 1 , R , z k < H 1 ) 1 / 2 H 2 n 1 sin θ 1 , R , z k < H 1 ( n 2 2 n 1 2 sin 2 θ 1 , R , z k < H 1 ) 1 / 2 = 0 , if 0 θ 1 , R , z k < H 1 < π / 2 n 1 n 2 0 θ 1 , R , z k < H 1 < arcsin ( n 2 / n 1 ) n 1 > n 2 ,
R ρ i ( H 1 z k ) sin θ 1 , R ρ i , z k < H 1 ( 1 sin 2 θ 1 , R ρ i , z k < H 1 ) 1 / 2 H 2 n 1 sin θ 1 , R ρ i , z k < H 1 ( n 2 2 n 1 2 sin 2 θ 1 , R ρ i , z k < H 1 ) 1 / 2 = 0 , if 0 θ 1 , R ρ i , z k < H 1 < π / 2 n 1 n 2 0 θ 1 , R ρ i , z k < H 1 < arcsin ( n 2 / n 1 ) n 1 > n 2 ,
R + ρ i ( H 1 z k ) sin θ 1 , R + ρ i , z k < H 1 ( 1 sin 2 θ 1 , R + ρ i , z k < H 1 ) 1 / 2 H 2 n 1 sin θ 1 , R + ρ i , z k < H 1 ( n 2 2 n 1 2 sin 2 θ 1 , R + ρ i , z k < H 1 ) 1 / 2 = 0 , if 0 θ 1 , R + ρ i , z k < H 1 < π / 2 n 1 n 2 0 θ 1 , R + ρ i , z k < H 1 < arcsin ( n 2 / n 1 ) n 1 > n 2 ,
2 R ( H 1 z k ) sin θ 1 , 2 R , z k < H 1 ( 1 sin 2 θ 1 , 2 R , z k < H 1 ) 1 / 2 H 2 n 1 sin θ 1 , 2 R , z k < H 1 ( n 2 2 n 1 2 sin 2 θ 1 , 2 R , z k < H 1 ) 1 / 2 = 0 , if 0 θ 1 , 2 R , z k < H 1 < π / 2 n 1 n 2 0 θ 1 , 2 R , z k < H 1 < arcsin ( n 2 / n 1 ) n 1 > n 2 ,
ρ i R ( H 1 z k ) sin θ 1 , ρ i R , z k < H 1 ( 1 sin 2 θ 1 , ρ i R , z k < H 1 ) 1 / 2 H 2 n 1 sin θ 1 , ρ i R , z k < H 1 ( n 2 2 n 1 2 sin 2 θ 1 , ρ i R , z k < H 1 ) 1 / 2 = 0 , if 0 θ 1 , ρ i R , z k < H 1 < π / 2 n 1 n 2 0 θ 1 , ρ i R , z k < H 1 < arcsin ( n 2 / n 1 ) n 1 > n 2 .
Y 1 = R ( h H 1 ) a H 2 H 1 + H 2 h + n 2 ( R a ) ( H 1 z k ) n 1 2 [ ( R a ) 2 + ( H 1 + H 2 h ) 2 ] n 2 2 ( R a ) 2 , if h < H 1 + H 2 n 1 > n 2 ( R a ) / ( R a ) 2 + ( H 1 + H 2 h ) 2 ,
Y 2 = R ( h H 1 ) a H 2 H 1 + H 2 h , if h < H 1 + H 2 ,
r 2 , a , z k < H 1 = ( H 1 z k ) sin θ 1 / ( 1 sin 2 θ 1 ) 1 / 2 + ( h H 1 ) n 1 sin θ 1 / ( n 2 2 n 1 2 sin 2 θ 1 ) 1 / 2 , if 0 θ 1 < π / 2 n 1 n 2 0 θ 1 < arcsin ( n 2 / n 1 ) n 1 > n 2 ,
r 2 , R , z k < H 1 = ( H 1 z k ) sin θ 1 / ( 1 sin 2 θ 1 ) 1 / 2 + H 2 n 1 sin θ 1 / ( n 2 2 n 1 2 sin 2 θ 1 ) 1 / 2 , if 0 θ 1 < π / 2 n 1 n 2 0 θ 1 < arcsin ( n 2 / n 1 ) n 1 > n 2 .

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