Abstract

I present a Monte Carlo model for predicting the performance of integrating spheres as a function of incident flux direction. The model was developed specifically to aid in the design of integrating spheres used as cosine collectors but is of general applicability. I discuss a method of generating uncorrelated random numbers. The probability density functions associated with uniform irradiance over a circular entrance port and Lambertian reflectors or emitters are presented. A comparison of the model with analytic equations predicting performance for an unbaffled integrating sphere is included. The average of the data generated by the model agrees with the analytic solution for sphere throughput to better than 0.25% (σ = 8.3%).

© 1996 Optical Society of America

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References

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  1. P. N. Slater, Remote Sensing: Optics and Optical Systems (Addison-Wesley, Reading, Mass., 1980).
  2. L. Harrison, J. Michalsky, J. Berndt, “Automated multi-filter rotation shadow-band radiometer: an instrument for optical depth and radiation measurements,” Appl. Opt. 33, 5118–5125 (1994).
    [CrossRef] [PubMed]
  3. R. M. Schotland, J. D. Copp, “Optical properties of a plastic pyranometer head,” J. Appl. Meteorol. 21, 735–739 (1982).
    [CrossRef]
  4. S. F. Biggar, P. N. Slater, D. I. Gellman, “Uncertainties in the in-flight calibration of sensors with reference to measured ground sites,” Remote Sensing Environ. 48, 245–252 (1994).
    [CrossRef]
  5. W. Budde, “Integrating sphere for the photometry of the sky,” Appl. Opt. 3, 939–941 (1964).
    [CrossRef]
  6. J. A. Jacquez, H. F. Kuppenheim, “Theory of the integrating sphere,” J. Opt. Soc. Am. 45, 460–470 (1955).
    [CrossRef]
  7. D. G. Goebel, “Generalized integrating-sphere theory,” Appl. Opt. 6, 125–128 (1967).
    [CrossRef] [PubMed]
  8. M. W. Finkel, “Integrating sphere theory,” Opt. Commun. 2, 25–28 (1970).
    [CrossRef]
  9. R. Frieden, Probability, Statistical Optics and Data Testing: a Problem Solving Approach (Springer-Verlag, New York, 1991).
    [CrossRef]
  10. SunOS Reference Manual, Sun Release 4.1, 1990, Vol. II, Sect. 3, p 3–255.
  11. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, New York, 1988), Chap. 7.
  12. B. R. Frieden, Optical Sciences Center, University of Arizona, Tucson, Arizona 85721 (personal communication, 1993).
  13. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1989), Chap. 1, pp. 36–47.
  14. R. K. Wangsness, Electromagnetic Fields, 2nd ed. (Wiley, New York, 1986), Chap. 1.
  15. K. F. Carr, “Integrating sphere calibration sources for remote sensing imaging radiometers,” in Optical Radiation Measurements II, J. M. Palmer, ed., Proc. SPIE1109, 99–113 (1989).

1994 (2)

S. F. Biggar, P. N. Slater, D. I. Gellman, “Uncertainties in the in-flight calibration of sensors with reference to measured ground sites,” Remote Sensing Environ. 48, 245–252 (1994).
[CrossRef]

L. Harrison, J. Michalsky, J. Berndt, “Automated multi-filter rotation shadow-band radiometer: an instrument for optical depth and radiation measurements,” Appl. Opt. 33, 5118–5125 (1994).
[CrossRef] [PubMed]

1982 (1)

R. M. Schotland, J. D. Copp, “Optical properties of a plastic pyranometer head,” J. Appl. Meteorol. 21, 735–739 (1982).
[CrossRef]

1970 (1)

M. W. Finkel, “Integrating sphere theory,” Opt. Commun. 2, 25–28 (1970).
[CrossRef]

1967 (1)

1964 (1)

1955 (1)

Berndt, J.

Biggar, S. F.

S. F. Biggar, P. N. Slater, D. I. Gellman, “Uncertainties in the in-flight calibration of sensors with reference to measured ground sites,” Remote Sensing Environ. 48, 245–252 (1994).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1989), Chap. 1, pp. 36–47.

Budde, W.

Carr, K. F.

K. F. Carr, “Integrating sphere calibration sources for remote sensing imaging radiometers,” in Optical Radiation Measurements II, J. M. Palmer, ed., Proc. SPIE1109, 99–113 (1989).

Copp, J. D.

R. M. Schotland, J. D. Copp, “Optical properties of a plastic pyranometer head,” J. Appl. Meteorol. 21, 735–739 (1982).
[CrossRef]

Finkel, M. W.

M. W. Finkel, “Integrating sphere theory,” Opt. Commun. 2, 25–28 (1970).
[CrossRef]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, New York, 1988), Chap. 7.

Frieden, B. R.

B. R. Frieden, Optical Sciences Center, University of Arizona, Tucson, Arizona 85721 (personal communication, 1993).

Frieden, R.

R. Frieden, Probability, Statistical Optics and Data Testing: a Problem Solving Approach (Springer-Verlag, New York, 1991).
[CrossRef]

Gellman, D. I.

S. F. Biggar, P. N. Slater, D. I. Gellman, “Uncertainties in the in-flight calibration of sensors with reference to measured ground sites,” Remote Sensing Environ. 48, 245–252 (1994).
[CrossRef]

Goebel, D. G.

Harrison, L.

Jacquez, J. A.

Kuppenheim, H. F.

Michalsky, J.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, New York, 1988), Chap. 7.

Schotland, R. M.

R. M. Schotland, J. D. Copp, “Optical properties of a plastic pyranometer head,” J. Appl. Meteorol. 21, 735–739 (1982).
[CrossRef]

Slater, P. N.

S. F. Biggar, P. N. Slater, D. I. Gellman, “Uncertainties in the in-flight calibration of sensors with reference to measured ground sites,” Remote Sensing Environ. 48, 245–252 (1994).
[CrossRef]

P. N. Slater, Remote Sensing: Optics and Optical Systems (Addison-Wesley, Reading, Mass., 1980).

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, New York, 1988), Chap. 7.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, New York, 1988), Chap. 7.

Wangsness, R. K.

R. K. Wangsness, Electromagnetic Fields, 2nd ed. (Wiley, New York, 1986), Chap. 1.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1989), Chap. 1, pp. 36–47.

Appl. Opt. (3)

J. Appl. Meteorol. (1)

R. M. Schotland, J. D. Copp, “Optical properties of a plastic pyranometer head,” J. Appl. Meteorol. 21, 735–739 (1982).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

M. W. Finkel, “Integrating sphere theory,” Opt. Commun. 2, 25–28 (1970).
[CrossRef]

Remote Sensing Environ. (1)

S. F. Biggar, P. N. Slater, D. I. Gellman, “Uncertainties in the in-flight calibration of sensors with reference to measured ground sites,” Remote Sensing Environ. 48, 245–252 (1994).
[CrossRef]

Other (8)

P. N. Slater, Remote Sensing: Optics and Optical Systems (Addison-Wesley, Reading, Mass., 1980).

R. Frieden, Probability, Statistical Optics and Data Testing: a Problem Solving Approach (Springer-Verlag, New York, 1991).
[CrossRef]

SunOS Reference Manual, Sun Release 4.1, 1990, Vol. II, Sect. 3, p 3–255.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, New York, 1988), Chap. 7.

B. R. Frieden, Optical Sciences Center, University of Arizona, Tucson, Arizona 85721 (personal communication, 1993).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1989), Chap. 1, pp. 36–47.

R. K. Wangsness, Electromagnetic Fields, 2nd ed. (Wiley, New York, 1986), Chap. 1.

K. F. Carr, “Integrating sphere calibration sources for remote sensing imaging radiometers,” in Optical Radiation Measurements II, J. M. Palmer, ed., Proc. SPIE1109, 99–113 (1989).

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Figures (7)

Fig. 1
Fig. 1

Errors in the predicted reflectance at the air–fiber interface with and without shuffling. Computations were done at normal incidence with a fiber index of refraction of 1.46.

Fig. 2
Fig. 2

Photon entrance points resulting from randomly generating x and y coordinates and throwing away those points outside the entrance aperture. A total of 1000 points were generated, and an entrance aperture of radius 1 unit was used.

Fig. 3
Fig. 3

Photon entrance points resulting from the use of transformation Eqs. (7) and (8) to generate the x and y coordinates. A total of 1000 points were generated, and an entrance aperture of radius 1 unit was used.

Fig. 4
Fig. 4

Unbaffled integrating sphere with an incorporated fiber-optic cable at the exit port. The axes of the entrance and exit ports correspond to the z and x axis, respectively.

Fig. 5
Fig. 5

Photons entering the fiber-optic cable of the unbaffled sphere as a function of azimuth angle at zenith angles of 10°, 20°, and 30°. 106 photons were traced at each input direction.

Fig. 6
Fig. 6

Photons entering the fiber-optic cable of the unbaffled sphere as a function of azimuth angle at zenith angles of 40°, 50°, and 60°. 106 photons were traced at each input direction.

Fig. 7
Fig. 7

Photons entering the fiber-optic cable of the unbaffled sphere as a function of azimuth angle at zenith angles of 70° and 80°. 106 photons were traced at each input direction.

Tables (1)

Tables Icon

Table 1 Statistics on the Number of Photons Entering the Fiber-Optic Cablea

Equations (26)

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u = F ( c ) - c p c ( c ) d c .
I ( θ ) = I 0 cos ( θ ) .
N ( r ) = π r 2 E / h ν ,
F ( r ) = r 2 / R 2 ,
p ( r ) = 2 r / R 2 .
r = R u .
x = R u cos ( 2 π v ) ,
y = R u sin ( 2 π v ) .
Φ ( θ ) = ϕ = 0 2 π θ = 0 θ I 0 cos ( θ ) sin ( θ ) d θ d ϕ = I 0 π sin 2 ( θ ) .
F ( θ ) = sin 2 ( θ ) .
p ( θ ) = 2 sin ( θ ) cos ( θ ) .
u = F ( θ ) = sin 2 ( θ ) .
θ = a sin ( u ) .
r p = n t cos ( θ i ) - n i cos ( θ t ) n t cos ( θ i ) + n i cos ( θ t ) ,
r s = n i cos ( θ i ) - n t cos ( θ t ) n i cos ( θ i ) + n t cos ( θ t ) .
R = r p 2 cos 2 ( θ p ) + r s 2 sin 2 ( θ p ) .
x 1 = x 0 + α t ,
y 1 = y 0 + β t ,
z 1 = z 0 + γ t ,
x ^ = - θ ^ = - cos ( θ α x ) cos ( ϕ α x ) x ^ - cos ( θ α x ) sin ( ϕ α x ) y ^ + sin ( θ α x ) z ^ ,
y ^ = ϕ ^ = - sin ( ϕ α x ) x ^ + cos ( ϕ α x ) y ^ ,
z ^ = - r ^ = - sin ( θ α x ) cos ( ϕ α x ) x ^ - sin ( θ α x ) sin ( ϕ α x ) y ^ - cos ( θ α x ) z ^ .
α = x ^ · x ^ α ´ + x ^ · y ^ β ´ + x ^ · z ^ γ ´ ,
β = y ^ · x ^ α ´ + y ^ · y ^ β ´ + y ^ · z ^ γ ´ ,
γ = z ^ · x ^ α ´ + z ^ · y ^ β ´ + z ^ · z ^ γ ´ .
Φ p f = ρ Φ p e p A f ( NA ) 2 ( 1 - R ) A s { 1 - ρ [ 1 - ( A f + A e p A s ) ] } ,

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