Abstract

A general multidomain integral formula is presented for calculating fluxes of radiation striking a circular disk from various off-axis point source types embedded in an attenuating or nonattenuating medium. This formula is expressed by double line integrals of radiant intensity and sine functions with respect to the polar and horizontal angles determining the angular distribution of the emitted radiation. The formula reduces to single line integral expressions when radiation does not depend on the horizontal angle and is directly applicable for calculating fluxes of revolutional symmetry around the optical axis of the source perpendicular to the disk. The applicability of this reduced formula is tested by computing radiant fluxes from Lambertian and Gaussian point sources using a simple numerical procedure for single integrals. The computed data are illustrated graphically, tabulated, and validated using OSLO. Finally, the accuracy, similarity, and applicability of the results provided by the integral formula and the OSLO program are analyzed. Numerical results have shown the effectiveness of the presented formulas for calculating radiant fluxes from various on- and off-axis point sources passing through a nonattenuating or attenuating homogeneous isotropic media and incident on a circular disk perpendicular to optical axes of these sources. Practical applications of these formulas include optical sensing and metrology, optical coupling, fiber optic for biomedical measurements, and creative lighting design.

© 2013 Optical Society of America

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References

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    [CrossRef]
  38. P. Manninen, J. Hovila, P. Kärhä, and E. Ikonen, “Method for analysing luminous intensity of light-emitting diodes,” Meas. Sci. Technol. 18, 223–229 (2007).
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  44. S. Tryka, “Spherical object in radiation field from Gaussian source,” Opt. Express 13, 5925–5938 (2005).
    [CrossRef]
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    [CrossRef]
  46. E. M. Sparrow and R. D. Cess, Radiation Heat Transfer (Hemisphere, 1978).
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    [CrossRef]
  51. C. C. Sun, T. X. Lee, S. H. Ma, Y. L. Lee, and S. M. Huang, “Precise optical modeling for LED lighting verified by cross correlation in the midfield region,” Opt. Lett. 31, 2193–2195 (2006).
    [CrossRef]
  52. S. Tryka, “The flux of radiation incident on a circular planar detector from a coaxial circular plane extended source,” J. Mod. Opt. 53, 365–380 (2006).
    [CrossRef]

2013 (2)

X. Liu, W. Cai, X. Lei, X. Du, and W. Chen, “Far-field distance for surface light source with different luminous area,” Appl. Opt. 52, 1629–1635 (2013).
[CrossRef]

Y. S. Kim, A. S. Choi, and J. W. Jeong, “Applying microgenetic algorithm to numerical model for luminous intensity distribution of planar prism LED luminaire,” Opt. Commun. 293, 22–30 (2013).
[CrossRef]

2012 (1)

Ch. Li, N. Li, and J. X. Chen, “The research of LED arrays for uniform illumination,” Adv. Inf. Sci. Service Sci. 4, 174–182 (2012).

2010 (1)

T.-M. Chung and S. Dai, “A study of the spatial intensity distribution of LED for general lighting,” J. Light Visual Environ. 34, 170–175 (2010).
[CrossRef]

2009 (2)

2008 (4)

I. Moreno and C. C. Sun, “Modeling the radiation pattern of LEDs,” Opt. Express 16, 1808–1819 (2008).
[CrossRef]

I. Moreno and C. C. Sun, “LED array: where does far-field begin?” Proc. SPIE 7058, 70580R (2008).
[CrossRef]

R. D. Dupuis and M. R. Krames, “History, development, and applications of high-brightness visible light emitting diodes,” J. Lightwave Technol. 26, 1154–1171 (2008).
[CrossRef]

J. T. Conway, “Calculations for a disk source and a general detector using a radiation vector potential,” Nucl. Instrum. Methods Phys. Res. A 589, 20–33 (2008).
[CrossRef]

2007 (3)

P. Manninen, J. Hovila, P. Kärhä, and E. Ikonen, “Method for analysing luminous intensity of light-emitting diodes,” Meas. Sci. Technol. 18, 223–229 (2007).
[CrossRef]

D. V. Kisevetter, “Approximating the angular transfer responses of fiber lightguides,” J. Opt. Technol. 74, 592–599 (2007).
[CrossRef]

J. T. Conway, “Geometric efficiency for a parallel-surface source and detector system with at least one axisymmetric surface,” Nucl. Instrum. Methods Phys. Res. A 583, 382–393 (2007).
[CrossRef]

2006 (2)

C. C. Sun, T. X. Lee, S. H. Ma, Y. L. Lee, and S. M. Huang, “Precise optical modeling for LED lighting verified by cross correlation in the midfield region,” Opt. Lett. 31, 2193–2195 (2006).
[CrossRef]

S. Tryka, “The flux of radiation incident on a circular planar detector from a coaxial circular plane extended source,” J. Mod. Opt. 53, 365–380 (2006).
[CrossRef]

2005 (1)

2004 (1)

2003 (1)

S. Pommé, L. Johansson, G. Sibbens, and B. Denecke, “An algorithm for the solid angle calculation applied in alpha-particle counting,” Nucl. Instrum. Methods Phys. Res. A 505, 286–289 (2003).
[CrossRef]

1999 (1)

S. Tryka, “A method for calculating the average solid angle subtended by a circular disk from uniformly distributed points within a coaxial circular plane,” Rev. Sci. Instrum. 70, 3915–3920 (1999).
[CrossRef]

1997 (1)

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

1993 (1)

P. Olivier and D. Gagnon, “Mathematical modeling of the solid angle function, part I: approximation in homogeneous medium,” Opt. Eng. 32, 2261–2265 (1993).
[CrossRef]

1988 (1)

J. L. Rodgers and W. A. Nicewander, “Thirteen ways to look at the correlation coefficient,” Am. Statist. 42, 59–66 (1988).
[CrossRef]

1984 (1)

C. J. Bland, “Tables of the geometrical factor for various source detector configurations,” Nucl. Instrum. Methods Phys. Res. 223, 602–606 (1984).
[CrossRef]

1973 (1)

J. P. Dakin, W. A. Gambling, H. Matsumara, D. N. Payne, and H. R. D. Sunak, “Theory of dispersion in lossless multimode optical fibres,” Opt. Commun. 7, 1–5 (1973).
[CrossRef]

1971 (1)

R. P. Gardner and K. Verghesse, “On the solid angle subtended by a circular disc,” Nucl. Instrum. Methods 93, 163–167 (1971).
[CrossRef]

1966 (2)

A. Feingold, “Radiant-interchange configuration factors between various selected plane surfaces,” Proc. R. Soc. A 292, 51–60 (1966).
[CrossRef]

H. Kogelnik and T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966).
[CrossRef]

1961 (2)

1959 (1)

F. Paxton, “Solid angle calculation for a circular disk,” Rev. Sci. Instrum. 30, 254–258 (1959).
[CrossRef]

1957 (2)

A. V. Masket, “Solid angle contour integrals, series, and tables,” Rev. Sci. Instrum. 28, 191–197 (1957).
[CrossRef]

M. Naito, “A method of calculating the solid angle subtended by a circular aperture,” J. Phys. Soc. Jpn. 12, 1122–1129 (1957).
[CrossRef]

1954 (3)

M. W. Garrett, “Solid angle subtended by a circular aperture,” Rev. Sci. Instrum. 25, 1208–1211 (1954).
[CrossRef]

H. Jaffy, “Solid angle subtended by a circular aperture at point and spread sources: formulas and some tables,” Rev. Sci. Instrum. 25, 349–354 (1954).
[CrossRef]

J. H. Smith and M. L. Storm, “Generalized off-axis distribution from disk sources of radiation,” J. Appl. Phys. 25, 519–527 (1954).
[CrossRef]

1951 (1)

E. Berne, “The calculation of the geometrical efficiency of end-window Geiger-Müller tubes,” Rev. Sci. Instrum. 22, 509–512 (1951).
[CrossRef]

Becherer, R. J.

F. Grum and R. J. Becherer, Optical Radiation Measurements, Vol. 1—Radiometry (Academic, 1979).

Berne, E.

E. Berne, “The calculation of the geometrical efficiency of end-window Geiger-Müller tubes,” Rev. Sci. Instrum. 22, 509–512 (1951).
[CrossRef]

Bland, C. J.

C. J. Bland, “Tables of the geometrical factor for various source detector configurations,” Nucl. Instrum. Methods Phys. Res. 223, 602–606 (1984).
[CrossRef]

Bouguer, P.

P. Bouguer, Essai d’optique sur la gradation de la lumière (Chez Claude Jombert, 1729).

Cai, W.

Capellaro, D. F.

Cess, R. D.

E. M. Sparrow and R. D. Cess, Radiation Heat Transfer (Hemisphere, 1978).

Chares, J.

J. Chares, Introduction to Nonimaging Optics (CRC Press, 2008), Chap. 16.

Chen, J. X.

Ch. Li, N. Li, and J. X. Chen, “The research of LED arrays for uniform illumination,” Adv. Inf. Sci. Service Sci. 4, 174–182 (2012).

Chen, W.

Chien, W. T.

Chien, Y.

Choi, A. S.

Y. S. Kim, A. S. Choi, and J. W. Jeong, “Applying microgenetic algorithm to numerical model for luminous intensity distribution of planar prism LED luminaire,” Opt. Commun. 293, 22–30 (2013).
[CrossRef]

Chung, T.-M.

T.-M. Chung and S. Dai, “A study of the spatial intensity distribution of LED for general lighting,” J. Light Visual Environ. 34, 170–175 (2010).
[CrossRef]

Conway, J. T.

J. T. Conway, “Calculations for a disk source and a general detector using a radiation vector potential,” Nucl. Instrum. Methods Phys. Res. A 589, 20–33 (2008).
[CrossRef]

J. T. Conway, “Geometric efficiency for a parallel-surface source and detector system with at least one axisymmetric surface,” Nucl. Instrum. Methods Phys. Res. A 583, 382–393 (2007).
[CrossRef]

Dai, S.

T.-M. Chung and S. Dai, “A study of the spatial intensity distribution of LED for general lighting,” J. Light Visual Environ. 34, 170–175 (2010).
[CrossRef]

Dakin, J. P.

J. P. Dakin, W. A. Gambling, H. Matsumara, D. N. Payne, and H. R. D. Sunak, “Theory of dispersion in lossless multimode optical fibres,” Opt. Commun. 7, 1–5 (1973).
[CrossRef]

Denecke, B.

S. Pommé, L. Johansson, G. Sibbens, and B. Denecke, “An algorithm for the solid angle calculation applied in alpha-particle counting,” Nucl. Instrum. Methods Phys. Res. A 505, 286–289 (2003).
[CrossRef]

Du, X.

Dupuis, R. D.

Dushkina, N.

N. Dushkina, “Light sources,” in Handbook of Optical Metrology. Principles and Applications, T. Yoshizawa, ed. (CRC Press, 2009), Chap. 1.

Feingold, A.

A. Feingold, “Radiant-interchange configuration factors between various selected plane surfaces,” Proc. R. Soc. A 292, 51–60 (1966).
[CrossRef]

Gagnon, D.

P. Olivier and D. Gagnon, “Mathematical modeling of the solid angle function, part I: approximation in homogeneous medium,” Opt. Eng. 32, 2261–2265 (1993).
[CrossRef]

Gambling, W. A.

J. P. Dakin, W. A. Gambling, H. Matsumara, D. N. Payne, and H. R. D. Sunak, “Theory of dispersion in lossless multimode optical fibres,” Opt. Commun. 7, 1–5 (1973).
[CrossRef]

Gardner, R. P.

R. P. Gardner and K. Verghesse, “On the solid angle subtended by a circular disc,” Nucl. Instrum. Methods 93, 163–167 (1971).
[CrossRef]

Garrett, M. W.

M. W. Garrett, “Solid angle subtended by a circular aperture,” Rev. Sci. Instrum. 25, 1208–1211 (1954).
[CrossRef]

Grum, F.

F. Grum and R. J. Becherer, Optical Radiation Measurements, Vol. 1—Radiometry (Academic, 1979).

Held, G.

G. Held, Introduction to Light Emitting Diode Technology and Applications (CRC Press, 2009).

Hovila, J.

P. Manninen, J. Hovila, P. Kärhä, and E. Ikonen, “Method for analysing luminous intensity of light-emitting diodes,” Meas. Sci. Technol. 18, 223–229 (2007).
[CrossRef]

Hsieh, C. C.

Huang, S. M.

Ikonen, E.

P. Manninen, J. Hovila, P. Kärhä, and E. Ikonen, “Method for analysing luminous intensity of light-emitting diodes,” Meas. Sci. Technol. 18, 223–229 (2007).
[CrossRef]

Issa, N. A.

Ivanov, R.

Jaffy, H.

H. Jaffy, “Solid angle subtended by a circular aperture at point and spread sources: formulas and some tables,” Rev. Sci. Instrum. 25, 349–354 (1954).
[CrossRef]

Jeong, J. W.

Y. S. Kim, A. S. Choi, and J. W. Jeong, “Applying microgenetic algorithm to numerical model for luminous intensity distribution of planar prism LED luminaire,” Opt. Commun. 293, 22–30 (2013).
[CrossRef]

Johansson, L.

S. Pommé, L. Johansson, G. Sibbens, and B. Denecke, “An algorithm for the solid angle calculation applied in alpha-particle counting,” Nucl. Instrum. Methods Phys. Res. A 505, 286–289 (2003).
[CrossRef]

Kapany, N. S.

Kärhä, P.

P. Manninen, J. Hovila, P. Kärhä, and E. Ikonen, “Method for analysing luminous intensity of light-emitting diodes,” Meas. Sci. Technol. 18, 223–229 (2007).
[CrossRef]

Kim, Y. S.

Y. S. Kim, A. S. Choi, and J. W. Jeong, “Applying microgenetic algorithm to numerical model for luminous intensity distribution of planar prism LED luminaire,” Opt. Commun. 293, 22–30 (2013).
[CrossRef]

Kisevetter, D. V.

Kogelnik, H.

Krames, M. R.

Lambert, J. H.

J. H. Lambert, Photometria sive de mensura et gradibus luminis colorum et umbrae (Augsburg, 1760). German translation by E. Anding (Leipzig, Verlag von Wilhelm Engelmann, 1892).

Lee, T. X.

Lee, Y. L.

Lei, X.

Li, Ch.

Ch. Li, N. Li, and J. X. Chen, “The research of LED arrays for uniform illumination,” Adv. Inf. Sci. Service Sci. 4, 174–182 (2012).

Li, N.

Ch. Li, N. Li, and J. X. Chen, “The research of LED arrays for uniform illumination,” Adv. Inf. Sci. Service Sci. 4, 174–182 (2012).

Li, T.

Liu, X.

Lynos, L.

L. Lynos, A Practical Guide to Data Analysis for Physical Science Students (Cambridge University, 1991).

Ma, S. H.

Manninen, P.

P. Manninen, J. Hovila, P. Kärhä, and E. Ikonen, “Method for analysing luminous intensity of light-emitting diodes,” Meas. Sci. Technol. 18, 223–229 (2007).
[CrossRef]

P. Manninen, “Characterization of diffusers and light-emitting diodes using radiometric measurements and mathematical modeling,” Doctoral dissertation (Helsinki University of Technology, Finland, 2008).

Masket, A. V.

A. V. Masket, “Solid angle contour integrals, series, and tables,” Rev. Sci. Instrum. 28, 191–197 (1957).
[CrossRef]

Matsumara, H.

J. P. Dakin, W. A. Gambling, H. Matsumara, D. N. Payne, and H. R. D. Sunak, “Theory of dispersion in lossless multimode optical fibres,” Opt. Commun. 7, 1–5 (1973).
[CrossRef]

Moreno, I.

Naito, M.

M. Naito, “A method of calculating the solid angle subtended by a circular aperture,” J. Phys. Soc. Jpn. 12, 1122–1129 (1957).
[CrossRef]

Nicewander, W. A.

J. L. Rodgers and W. A. Nicewander, “Thirteen ways to look at the correlation coefficient,” Am. Statist. 42, 59–66 (1988).
[CrossRef]

Olivier, P.

P. Olivier and D. Gagnon, “Mathematical modeling of the solid angle function, part I: approximation in homogeneous medium,” Opt. Eng. 32, 2261–2265 (1993).
[CrossRef]

Padden, W. E.

Palais, J. C.

J. C. Palais, “Optical communications,” in Engineering Electromagnetic Applications, B. Rajeev, ed. (CRC Press, 2006), Chap. 4, pp. 121–160.

Paxton, F.

F. Paxton, “Solid angle calculation for a circular disk,” Rev. Sci. Instrum. 30, 254–258 (1959).
[CrossRef]

Payne, D. N.

J. P. Dakin, W. A. Gambling, H. Matsumara, D. N. Payne, and H. R. D. Sunak, “Theory of dispersion in lossless multimode optical fibres,” Opt. Commun. 7, 1–5 (1973).
[CrossRef]

Pommé, S.

S. Pommé, L. Johansson, G. Sibbens, and B. Denecke, “An algorithm for the solid angle calculation applied in alpha-particle counting,” Nucl. Instrum. Methods Phys. Res. A 505, 286–289 (2003).
[CrossRef]

Quimby, R. S.

R. S. Quimby, Photonics and Lasers: An Introduction. (Wiley, 2006), Chap. 4.

Rodgers, J. L.

J. L. Rodgers and W. A. Nicewander, “Thirteen ways to look at the correlation coefficient,” Am. Statist. 42, 59–66 (1988).
[CrossRef]

Rowlands, G.

G. Rowlands, “Solid angle calculations,” Int. J. Appl. Radiat. Isot. 10, 86–93 (1961).
[CrossRef]

Sibbens, G.

S. Pommé, L. Johansson, G. Sibbens, and B. Denecke, “An algorithm for the solid angle calculation applied in alpha-particle counting,” Nucl. Instrum. Methods Phys. Res. A 505, 286–289 (2003).
[CrossRef]

Smith, J. H.

J. H. Smith and M. L. Storm, “Generalized off-axis distribution from disk sources of radiation,” J. Appl. Phys. 25, 519–527 (1954).
[CrossRef]

Sparrow, E. M.

E. M. Sparrow and R. D. Cess, Radiation Heat Transfer (Hemisphere, 1978).

Storm, M. L.

J. H. Smith and M. L. Storm, “Generalized off-axis distribution from disk sources of radiation,” J. Appl. Phys. 25, 519–527 (1954).
[CrossRef]

Sun, C. C.

Sunak, H. R. D.

J. P. Dakin, W. A. Gambling, H. Matsumara, D. N. Payne, and H. R. D. Sunak, “Theory of dispersion in lossless multimode optical fibres,” Opt. Commun. 7, 1–5 (1973).
[CrossRef]

Tryka, S.

S. Tryka, “The flux of radiation incident on a circular planar detector from a coaxial circular plane extended source,” J. Mod. Opt. 53, 365–380 (2006).
[CrossRef]

S. Tryka, “Spherical object in radiation field from Gaussian source,” Opt. Express 13, 5925–5938 (2005).
[CrossRef]

S. Tryka, “A method for calculating the average solid angle subtended by a circular disk from uniformly distributed points within a coaxial circular plane,” Rev. Sci. Instrum. 70, 3915–3920 (1999).
[CrossRef]

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

Verghesse, K.

R. P. Gardner and K. Verghesse, “On the solid angle subtended by a circular disc,” Nucl. Instrum. Methods 93, 163–167 (1971).
[CrossRef]

Wolfram, S.

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

Adv. Inf. Sci. Service Sci. (1)

Ch. Li, N. Li, and J. X. Chen, “The research of LED arrays for uniform illumination,” Adv. Inf. Sci. Service Sci. 4, 174–182 (2012).

Am. Statist. (1)

J. L. Rodgers and W. A. Nicewander, “Thirteen ways to look at the correlation coefficient,” Am. Statist. 42, 59–66 (1988).
[CrossRef]

Appl. Opt. (3)

Int. J. Appl. Radiat. Isot. (1)

G. Rowlands, “Solid angle calculations,” Int. J. Appl. Radiat. Isot. 10, 86–93 (1961).
[CrossRef]

J. Appl. Phys. (1)

J. H. Smith and M. L. Storm, “Generalized off-axis distribution from disk sources of radiation,” J. Appl. Phys. 25, 519–527 (1954).
[CrossRef]

J. Light Visual Environ. (1)

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Figures (3)

Fig. 1.
Fig. 1.

Geometry of an off-axis point source P at the radial distance ρ separated by a single homogeneous isotropic medium at the axial distance H from the circular disk of the radius R.

Fig. 2.
Fig. 2.

Radiant fluxes from the Lambertian source as dependent on distance ρ and H determined (a) by Eqs. (8) and (14) for the radiation passing through the nonattenuating homogeneous isotropic medium and (b) by Eqs. (5) and (6) for the radiation passing through the attenuating homogeneous isotropic medium. The data were computed at I0,λ=1W·sr1 for ρ and H in relative dimensional units with respect to R=1. The attenuating effect of the medium was calculated at αat,λ=0.1 in the reciprocal unit of R, so that if R is expressed in m then αat,λ is given in 1/m.

Fig. 3.
Fig. 3.

Radiant fluxes from the Gaussian source as dependent on distances ρ and H determined (a) by Eqs. (13) and (14) for the radiation passing through the nonattenuating homogeneous isotropic medium and (b) by Eqs. (5) and (12) for the radiation passing through the attenuating homogeneous isotropic medium. The data were computed for ρ and H in the relative dimensional units with respect to R=1, at I0,λ=1W·sr1, Θe=0.1°, and λ=400nm. The attenuating effect of the medium was calculated at αat,λ=0.001 in the reciprocal unit of R, so that if R is expressed in m, then αat,λ is given in 1/m

Tables (2)

Tables Icon

Table 1. Comparison of Factors FPS,λ Calculated from Combined Eqs. (8), (14), (17), and (18) for the Radiation from the Lambertian Source Passing through the Nonattenuating Homogeneous Isotropic Medium to Factors FPS,λ,O Simulated by the OSLO Program for 1108 Sampling Probesa

Tables Icon

Table 2. Comparison of Factors FPS,λ Calculated from Combined Eqs. (13), (14), (17), and (18) for the Radiation from the Gaussian Source Passing through the Nonattenuating Homogeneous Isotropic Medium to the Factors FPS,λ,O Simulated by the OSLO Program for 1108 Sampling Probesa

Equations (21)

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ΦPS,λ=0π/2sinθdθ02πIP1,at,λ(H,θ,φ)dφ,0ρ<R,H=0,=0arctan(R/H)sinθdθ02πIP1,at,λ(H,θ,φ)dφ,0=ρ<R,H>0,=0arctan[(Rρ)/H]sinθdθ02πIP1,at,λ(H,θ,φ)dφ+arctan[(Rρ)/H]arctan[(R+ρ)/H]sinθdθarccos[(ρ2+H2tan2θR2)/(2ρHtanθ)]arccos[(ρ2+H2tan2θR2)/(2ρHtanθ)]IP1,at,λ(H,θ,φ)dφ,0<ρ<R,H>0,=0π/2sinθdθ0πIP1,at,λ(H,θ,φ)dφ,0<ρ=R,H=0,=0arctan(2R/H)sinθdθarccos[Htanθ/(2R)]arccos[Htanθ/(2R)]IP1,at,λ(H,θ,φ)dφ,0<ρ=R,H>0,=0,0<R<ρ,H=0,=arctan[(ρR)/H]arctan[(R+ρ)/H]sinθdθarccos[(ρ2+H2tan2θR2)/(2ρHtanθ)]arccos[(ρ2+H2tan2θR2)/(2ρHtanθ)]IP1,at,λ(H,θ,φ)dφ,0<R<ρ,H>0,
IP1,at,λ(H,θ,φ)=IP1,λ(H,θ,φ)TPP1,λ(H,θ,φ),
TPP1,λ(H,θ)=exp(αat,λH/cosθ),if0θ<π/2,
IP1,at,λ(H,θ,φ)=IP1,at,λ(H,θ)=IP1,λ(H,θ)TPP1,λ(H,θ).
ΦPS,λ=2π0π/2IP1,λ(H,θ)TPP1,λ(H,θ)sinθdθ,0ρ<R,H=0,=2π0arctan(R/H)IP1,λ(H,θ)TPP1,λ(H,θ)sinθdθ,0=ρ<R,H>0,=2π0arctan[(Rρ)/H]IP1,λ(H,θ)TPP1,λ(H,θ)sinθdθ+2arctan[(Rρ)/H]arctan[(R+ρ)/H]IP1,λ(H,θ)TPP1,λ(H,θ)arccos(ρ2+H2tan2θR22ρHtanθ)sinθdθ,0<ρ<R,H>0,=π0π/2IP1,λ(H,θ)TPP1,λ(H,θ)sinθdθ,0<ρ=R,H=0,=20arctan(2R/H)IP1,λ(H,θ)TPP1,λ(H,θ)arccos(Htanθ2R)sinθdθ,0<ρ=R,H>0,=0,0<R<ρ,H=0,=2arctan[(ρR)/H]arctan[(R+ρ)/H]IP1,λ(H,θ)TPP1,λ(H,θ)arccos(ρ2+H2tan2θR22ρHtanθ)sinθdθ,0<R<ρ,H>0.
IP1,λ(H,θ)=IP1,λ(θ)=I0,λ·cosmθ,
m=ln2/ln(cosθ1/2),
IP1,at,λ(H,θ,φ)=I0,λ·(cosθ)ln2/ln(cosθ1/2)×exp(αat,λHcosθ),if0θ<π/2.
IP1,λ(H,θ)=IP,λexp[2H2tan2θwλ2(H)],
wλ(H)=w0,λ1+(Hλπw0,λ2)2,
w0,λ=λπθe,
IP1,λ(H,θ)=IP,λexp(2π2θe2H2tan2θπ2θe4H2+λ2).
IP1,at,λ(H,θ,φ)=IP,λexp(2π2θe2H2tan2θπ2θe4H2+λ2)×exp(αat,λHcosθ),if0θ<π/2.
ΦPS,λ=2π0π/2IP1,λ(H,θ)sinθdθ,0ρ<R,H=0,=2π0arctan(R/H)IP1,λ(H,θ)sinθdθ,0=ρ<R,H>0,=2π0arctan[(Rρ)/H]IP1,λ(H,θ)sinθdθ+2arctan[(Rρ)/H]arctan[(R+ρ)/H]IP1,λ(H,θ)arccos(ρ2+H2tan2θR22ρHtanθ)sinθdθ,0<ρ<R,H>0,=π0π/2IP1,λ(H,θ)sinθdθ,0<ρ=R,H=0,=20arctan(2R/H)IP1,λ(H,θ)arccos(Htanθ2R)sinθdθ,0<ρ=R,H>0,=0,0<R<ρ,H=0,=2arctan[(ρR)/H]arctan[(R+ρ)/H]IP1,λ(H,θ)arccos(ρ2+H2tan2θR22ρHtanθ)sinθdθ,0<R<ρ,H>0.
ΦPS,λ=πI0,λ2×R2H2ρ2+[(R+ρ)2+H2][(Rρ)2+H2][(R+ρ)2+H2][(Rρ)2+H2].
ΦPS,λ=I0,λΩPS,
FPS,λ=ΦPS,λ/ΦP,λ.
ΦP,λ=0πIP,λ(θ,φ)sinθdθ02πdφ,
FPS,λ,O=NS,λ/NP,λ,
RMSD=1ni=1n(FPS,λ,iFPS,λ,O,i)2
PCC=i=1n(FPS,λ,iF¯PS,λ)(FPS,λ,O,iF¯PS,λ,O)i=1n(FPS,λ,iF¯PS,λ)2j=1n(FPS,λ,O,iF¯PS,λ,O)2,

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