Abstract

The energy transfer integral between radiating rectangular and detecting circular parallel plates having nonideal angular characteristics is solved for modeling the distance dependence of the irradiance signal. The equation derived for the irradiance signal, which is called the modified inverse-square law, depends on the position, shape, size, and angular characteristics of the light source and the detector. We apply the new model equation to the calibration of a spectroradiometer to determine accurately the distance offsets, which fix the positions of the effective receiving apertures of diffusers used in the entrance optics of spectroradiometers. Earlier measurement results, e.g., for solar UV irradiance, may include uncorrected effects and can be corrected reliably as diffuser offsets and other correction factors are determined with the modified inverse-square law. Simplifications of the modified inverse-square law for analyzing the distance offsets and the correction factors are studied. Simplified equations for the diffuser offset analysis may be used without losing the accuracy when the cosine response of the diffuser is reasonably good. However, for diffusers whose angular responsivities deviate much from the cosinusoidal angular responsivity, large approximation errors in the diffuser offset values may appear if the angular effects are not properly taken into account.

© 2008 Optical Society of America

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References

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  1. P. Manninen, J. Hovila, L. Seppälä, L. Ylianttila, P. Kärhä, and E. Ikonen, “Determination of distance offsets of diffusers for accurate radiometric measurements,” Metrologia 43, S120-S124 (2006).
    [CrossRef]
  2. J. Hovila, M. Mustonen, P. Kärhä, and E. Ikonen, “Determination of the diffuser reference plane for accurate illuminance responsivity calibrations,” Appl. Opt. 44, 5894-5898 (2005).
    [CrossRef] [PubMed]
  3. J. Gröbner and M. Blumthaler, “Experimental determination of the reference plane of shaped diffusers by solar ultraviolet measurements,” Opt. Lett. 32, 80-82 (2007).
    [CrossRef]
  4. J. Metzdorf, K. D. Stock, P. Sperfeld, A. Sperling, S. Winter, and T. Wittchen, “Aspects of quality assurance in monitoring solar UV irradiance,” Metrologia 40, S66-S69 (2003).
    [CrossRef]
  5. J. Gröbner, M. Blumthaler, S. Kazadzis, A. Bais, A. Webb, J. Schreder, G. Seckmeyer, and D. Rembges, “Quality assurance of spectral solar UV measurements: results from 25 UV monitoring sites in Europe, 2002 to 2004,” Metrologia 43, S66-S71 (2006).
    [CrossRef]
  6. G. Bernhard, R. L. McKenzie, M. Kotkamp, S. Wood, C. R. Booth, J. C. Ehramjian, P. Johnston, and S. E. Nichol, “Comparison of ultraviolet spectroradiometers in Antarctica,” J. Geophys. Res. 113, D14310 (2008).
    [CrossRef]
  7. R. R. Cordero, G. Seckmeyer, D. Pissulla, L. DaSilva, and F. Labbe, “Uncertainty evaluation of spectral UV irradiance measurements,” Meas. Sci. Technol. 19, 045104 (2008).
    [CrossRef]
  8. G. Xu and X. Huang, “Characterization and calibration of broadband ultraviolet radiometers,” Metrologia 37, 235-242 (2000).
    [CrossRef]
  9. K. Lakkala, “A decade of spectral UV measurements at Sodankylä,” Ph.D. dissertation (University of Helsinki, 2001).
  10. D. I. Lawson and D. Hird, “A photometric method of determining configuration factors,” Br. J. Appl. Phys. 5, 72-74 (1954).
    [CrossRef]
  11. J. W. T. Walsh, Photometry (Dover, 1965).
  12. D. S. Goodman, “Linear systems methods in some problems of radiative transfer between parallel plates,” Appl. Opt. 24, 3240-3248 (1985).
    [CrossRef] [PubMed]
  13. 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]
  14. E. Ikonen, P. Manninen, and P. Kärhä, “Modeling distance dependence of LED illuminance,” Light Eng. 15, 57-61 (2007).
  15. Y. Ohno, “Photometric standards,” in OSA/AIP Handbook of Applied Photometry, C. DeCusatis, ed. (Optical Society of America, 1997), pp. 55-90.
  16. J. G. Schreder, M. Blumthaler and M. Huber, “Design of an input optic for solar UV-measurements” (Internet Photochemistry and Photobiology, 1998), http://www.photobiology.com/UVR98/schreder/index.htm.

2008 (2)

G. Bernhard, R. L. McKenzie, M. Kotkamp, S. Wood, C. R. Booth, J. C. Ehramjian, P. Johnston, and S. E. Nichol, “Comparison of ultraviolet spectroradiometers in Antarctica,” J. Geophys. Res. 113, D14310 (2008).
[CrossRef]

R. R. Cordero, G. Seckmeyer, D. Pissulla, L. DaSilva, and F. Labbe, “Uncertainty evaluation of spectral UV irradiance measurements,” Meas. Sci. Technol. 19, 045104 (2008).
[CrossRef]

2007 (2)

E. Ikonen, P. Manninen, and P. Kärhä, “Modeling distance dependence of LED illuminance,” Light Eng. 15, 57-61 (2007).

J. Gröbner and M. Blumthaler, “Experimental determination of the reference plane of shaped diffusers by solar ultraviolet measurements,” Opt. Lett. 32, 80-82 (2007).
[CrossRef]

2006 (3)

J. Gröbner, M. Blumthaler, S. Kazadzis, A. Bais, A. Webb, J. Schreder, G. Seckmeyer, and D. Rembges, “Quality assurance of spectral solar UV measurements: results from 25 UV monitoring sites in Europe, 2002 to 2004,” Metrologia 43, S66-S71 (2006).
[CrossRef]

P. Manninen, J. Hovila, L. Seppälä, L. Ylianttila, P. Kärhä, and E. Ikonen, “Determination of distance offsets of diffusers for accurate radiometric measurements,” Metrologia 43, S120-S124 (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)

2003 (1)

J. Metzdorf, K. D. Stock, P. Sperfeld, A. Sperling, S. Winter, and T. Wittchen, “Aspects of quality assurance in monitoring solar UV irradiance,” Metrologia 40, S66-S69 (2003).
[CrossRef]

2000 (1)

G. Xu and X. Huang, “Characterization and calibration of broadband ultraviolet radiometers,” Metrologia 37, 235-242 (2000).
[CrossRef]

1985 (1)

1954 (1)

D. I. Lawson and D. Hird, “A photometric method of determining configuration factors,” Br. J. Appl. Phys. 5, 72-74 (1954).
[CrossRef]

Bais, A.

J. Gröbner, M. Blumthaler, S. Kazadzis, A. Bais, A. Webb, J. Schreder, G. Seckmeyer, and D. Rembges, “Quality assurance of spectral solar UV measurements: results from 25 UV monitoring sites in Europe, 2002 to 2004,” Metrologia 43, S66-S71 (2006).
[CrossRef]

Bernhard, G.

G. Bernhard, R. L. McKenzie, M. Kotkamp, S. Wood, C. R. Booth, J. C. Ehramjian, P. Johnston, and S. E. Nichol, “Comparison of ultraviolet spectroradiometers in Antarctica,” J. Geophys. Res. 113, D14310 (2008).
[CrossRef]

Blumthaler, M.

J. Gröbner and M. Blumthaler, “Experimental determination of the reference plane of shaped diffusers by solar ultraviolet measurements,” Opt. Lett. 32, 80-82 (2007).
[CrossRef]

J. Gröbner, M. Blumthaler, S. Kazadzis, A. Bais, A. Webb, J. Schreder, G. Seckmeyer, and D. Rembges, “Quality assurance of spectral solar UV measurements: results from 25 UV monitoring sites in Europe, 2002 to 2004,” Metrologia 43, S66-S71 (2006).
[CrossRef]

J. G. Schreder, M. Blumthaler and M. Huber, “Design of an input optic for solar UV-measurements” (Internet Photochemistry and Photobiology, 1998), http://www.photobiology.com/UVR98/schreder/index.htm.

Booth, C. R.

G. Bernhard, R. L. McKenzie, M. Kotkamp, S. Wood, C. R. Booth, J. C. Ehramjian, P. Johnston, and S. E. Nichol, “Comparison of ultraviolet spectroradiometers in Antarctica,” J. Geophys. Res. 113, D14310 (2008).
[CrossRef]

Cordero, R. R.

R. R. Cordero, G. Seckmeyer, D. Pissulla, L. DaSilva, and F. Labbe, “Uncertainty evaluation of spectral UV irradiance measurements,” Meas. Sci. Technol. 19, 045104 (2008).
[CrossRef]

DaSilva, L.

R. R. Cordero, G. Seckmeyer, D. Pissulla, L. DaSilva, and F. Labbe, “Uncertainty evaluation of spectral UV irradiance measurements,” Meas. Sci. Technol. 19, 045104 (2008).
[CrossRef]

Ehramjian, J. C.

G. Bernhard, R. L. McKenzie, M. Kotkamp, S. Wood, C. R. Booth, J. C. Ehramjian, P. Johnston, and S. E. Nichol, “Comparison of ultraviolet spectroradiometers in Antarctica,” J. Geophys. Res. 113, D14310 (2008).
[CrossRef]

Goodman, D. S.

Gröbner, J.

J. Gröbner and M. Blumthaler, “Experimental determination of the reference plane of shaped diffusers by solar ultraviolet measurements,” Opt. Lett. 32, 80-82 (2007).
[CrossRef]

J. Gröbner, M. Blumthaler, S. Kazadzis, A. Bais, A. Webb, J. Schreder, G. Seckmeyer, and D. Rembges, “Quality assurance of spectral solar UV measurements: results from 25 UV monitoring sites in Europe, 2002 to 2004,” Metrologia 43, S66-S71 (2006).
[CrossRef]

Hird, D.

D. I. Lawson and D. Hird, “A photometric method of determining configuration factors,” Br. J. Appl. Phys. 5, 72-74 (1954).
[CrossRef]

Hovila, J.

P. Manninen, J. Hovila, L. Seppälä, L. Ylianttila, P. Kärhä, and E. Ikonen, “Determination of distance offsets of diffusers for accurate radiometric measurements,” Metrologia 43, S120-S124 (2006).
[CrossRef]

J. Hovila, M. Mustonen, P. Kärhä, and E. Ikonen, “Determination of the diffuser reference plane for accurate illuminance responsivity calibrations,” Appl. Opt. 44, 5894-5898 (2005).
[CrossRef] [PubMed]

Huang, X.

G. Xu and X. Huang, “Characterization and calibration of broadband ultraviolet radiometers,” Metrologia 37, 235-242 (2000).
[CrossRef]

Huber, M.

J. G. Schreder, M. Blumthaler and M. Huber, “Design of an input optic for solar UV-measurements” (Internet Photochemistry and Photobiology, 1998), http://www.photobiology.com/UVR98/schreder/index.htm.

Ikonen, E.

E. Ikonen, P. Manninen, and P. Kärhä, “Modeling distance dependence of LED illuminance,” Light Eng. 15, 57-61 (2007).

P. Manninen, J. Hovila, L. Seppälä, L. Ylianttila, P. Kärhä, and E. Ikonen, “Determination of distance offsets of diffusers for accurate radiometric measurements,” Metrologia 43, S120-S124 (2006).
[CrossRef]

J. Hovila, M. Mustonen, P. Kärhä, and E. Ikonen, “Determination of the diffuser reference plane for accurate illuminance responsivity calibrations,” Appl. Opt. 44, 5894-5898 (2005).
[CrossRef] [PubMed]

Johnston, P.

G. Bernhard, R. L. McKenzie, M. Kotkamp, S. Wood, C. R. Booth, J. C. Ehramjian, P. Johnston, and S. E. Nichol, “Comparison of ultraviolet spectroradiometers in Antarctica,” J. Geophys. Res. 113, D14310 (2008).
[CrossRef]

Kärhä, P.

E. Ikonen, P. Manninen, and P. Kärhä, “Modeling distance dependence of LED illuminance,” Light Eng. 15, 57-61 (2007).

P. Manninen, J. Hovila, L. Seppälä, L. Ylianttila, P. Kärhä, and E. Ikonen, “Determination of distance offsets of diffusers for accurate radiometric measurements,” Metrologia 43, S120-S124 (2006).
[CrossRef]

J. Hovila, M. Mustonen, P. Kärhä, and E. Ikonen, “Determination of the diffuser reference plane for accurate illuminance responsivity calibrations,” Appl. Opt. 44, 5894-5898 (2005).
[CrossRef] [PubMed]

Kazadzis, S.

J. Gröbner, M. Blumthaler, S. Kazadzis, A. Bais, A. Webb, J. Schreder, G. Seckmeyer, and D. Rembges, “Quality assurance of spectral solar UV measurements: results from 25 UV monitoring sites in Europe, 2002 to 2004,” Metrologia 43, S66-S71 (2006).
[CrossRef]

Kotkamp, M.

G. Bernhard, R. L. McKenzie, M. Kotkamp, S. Wood, C. R. Booth, J. C. Ehramjian, P. Johnston, and S. E. Nichol, “Comparison of ultraviolet spectroradiometers in Antarctica,” J. Geophys. Res. 113, D14310 (2008).
[CrossRef]

Labbe, F.

R. R. Cordero, G. Seckmeyer, D. Pissulla, L. DaSilva, and F. Labbe, “Uncertainty evaluation of spectral UV irradiance measurements,” Meas. Sci. Technol. 19, 045104 (2008).
[CrossRef]

Lakkala, K.

K. Lakkala, “A decade of spectral UV measurements at Sodankylä,” Ph.D. dissertation (University of Helsinki, 2001).

Lawson, D. I.

D. I. Lawson and D. Hird, “A photometric method of determining configuration factors,” Br. J. Appl. Phys. 5, 72-74 (1954).
[CrossRef]

Manninen, P.

E. Ikonen, P. Manninen, and P. Kärhä, “Modeling distance dependence of LED illuminance,” Light Eng. 15, 57-61 (2007).

P. Manninen, J. Hovila, L. Seppälä, L. Ylianttila, P. Kärhä, and E. Ikonen, “Determination of distance offsets of diffusers for accurate radiometric measurements,” Metrologia 43, S120-S124 (2006).
[CrossRef]

McKenzie, R. L.

G. Bernhard, R. L. McKenzie, M. Kotkamp, S. Wood, C. R. Booth, J. C. Ehramjian, P. Johnston, and S. E. Nichol, “Comparison of ultraviolet spectroradiometers in Antarctica,” J. Geophys. Res. 113, D14310 (2008).
[CrossRef]

Metzdorf, J.

J. Metzdorf, K. D. Stock, P. Sperfeld, A. Sperling, S. Winter, and T. Wittchen, “Aspects of quality assurance in monitoring solar UV irradiance,” Metrologia 40, S66-S69 (2003).
[CrossRef]

Mustonen, M.

Nichol, S. E.

G. Bernhard, R. L. McKenzie, M. Kotkamp, S. Wood, C. R. Booth, J. C. Ehramjian, P. Johnston, and S. E. Nichol, “Comparison of ultraviolet spectroradiometers in Antarctica,” J. Geophys. Res. 113, D14310 (2008).
[CrossRef]

Ohno, Y.

Y. Ohno, “Photometric standards,” in OSA/AIP Handbook of Applied Photometry, C. DeCusatis, ed. (Optical Society of America, 1997), pp. 55-90.

Pissulla, D.

R. R. Cordero, G. Seckmeyer, D. Pissulla, L. DaSilva, and F. Labbe, “Uncertainty evaluation of spectral UV irradiance measurements,” Meas. Sci. Technol. 19, 045104 (2008).
[CrossRef]

Rembges, D.

J. Gröbner, M. Blumthaler, S. Kazadzis, A. Bais, A. Webb, J. Schreder, G. Seckmeyer, and D. Rembges, “Quality assurance of spectral solar UV measurements: results from 25 UV monitoring sites in Europe, 2002 to 2004,” Metrologia 43, S66-S71 (2006).
[CrossRef]

Schreder, J.

J. Gröbner, M. Blumthaler, S. Kazadzis, A. Bais, A. Webb, J. Schreder, G. Seckmeyer, and D. Rembges, “Quality assurance of spectral solar UV measurements: results from 25 UV monitoring sites in Europe, 2002 to 2004,” Metrologia 43, S66-S71 (2006).
[CrossRef]

Schreder, J. G.

J. G. Schreder, M. Blumthaler and M. Huber, “Design of an input optic for solar UV-measurements” (Internet Photochemistry and Photobiology, 1998), http://www.photobiology.com/UVR98/schreder/index.htm.

Seckmeyer, G.

R. R. Cordero, G. Seckmeyer, D. Pissulla, L. DaSilva, and F. Labbe, “Uncertainty evaluation of spectral UV irradiance measurements,” Meas. Sci. Technol. 19, 045104 (2008).
[CrossRef]

J. Gröbner, M. Blumthaler, S. Kazadzis, A. Bais, A. Webb, J. Schreder, G. Seckmeyer, and D. Rembges, “Quality assurance of spectral solar UV measurements: results from 25 UV monitoring sites in Europe, 2002 to 2004,” Metrologia 43, S66-S71 (2006).
[CrossRef]

Seppälä, L.

P. Manninen, J. Hovila, L. Seppälä, L. Ylianttila, P. Kärhä, and E. Ikonen, “Determination of distance offsets of diffusers for accurate radiometric measurements,” Metrologia 43, S120-S124 (2006).
[CrossRef]

Sperfeld, P.

J. Metzdorf, K. D. Stock, P. Sperfeld, A. Sperling, S. Winter, and T. Wittchen, “Aspects of quality assurance in monitoring solar UV irradiance,” Metrologia 40, S66-S69 (2003).
[CrossRef]

Sperling, A.

J. Metzdorf, K. D. Stock, P. Sperfeld, A. Sperling, S. Winter, and T. Wittchen, “Aspects of quality assurance in monitoring solar UV irradiance,” Metrologia 40, S66-S69 (2003).
[CrossRef]

Stock, K. D.

J. Metzdorf, K. D. Stock, P. Sperfeld, A. Sperling, S. Winter, and T. Wittchen, “Aspects of quality assurance in monitoring solar UV irradiance,” Metrologia 40, S66-S69 (2003).
[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]

Walsh, J. W. T.

J. W. T. Walsh, Photometry (Dover, 1965).

Webb, A.

J. Gröbner, M. Blumthaler, S. Kazadzis, A. Bais, A. Webb, J. Schreder, G. Seckmeyer, and D. Rembges, “Quality assurance of spectral solar UV measurements: results from 25 UV monitoring sites in Europe, 2002 to 2004,” Metrologia 43, S66-S71 (2006).
[CrossRef]

Winter, S.

J. Metzdorf, K. D. Stock, P. Sperfeld, A. Sperling, S. Winter, and T. Wittchen, “Aspects of quality assurance in monitoring solar UV irradiance,” Metrologia 40, S66-S69 (2003).
[CrossRef]

Wittchen, T.

J. Metzdorf, K. D. Stock, P. Sperfeld, A. Sperling, S. Winter, and T. Wittchen, “Aspects of quality assurance in monitoring solar UV irradiance,” Metrologia 40, S66-S69 (2003).
[CrossRef]

Wood, S.

G. Bernhard, R. L. McKenzie, M. Kotkamp, S. Wood, C. R. Booth, J. C. Ehramjian, P. Johnston, and S. E. Nichol, “Comparison of ultraviolet spectroradiometers in Antarctica,” J. Geophys. Res. 113, D14310 (2008).
[CrossRef]

Xu, G.

G. Xu and X. Huang, “Characterization and calibration of broadband ultraviolet radiometers,” Metrologia 37, 235-242 (2000).
[CrossRef]

Ylianttila, L.

P. Manninen, J. Hovila, L. Seppälä, L. Ylianttila, P. Kärhä, and E. Ikonen, “Determination of distance offsets of diffusers for accurate radiometric measurements,” Metrologia 43, S120-S124 (2006).
[CrossRef]

Appl. Opt. (2)

Br. J. Appl. Phys. (1)

D. I. Lawson and D. Hird, “A photometric method of determining configuration factors,” Br. J. Appl. Phys. 5, 72-74 (1954).
[CrossRef]

J. Geophys. Res. (1)

G. Bernhard, R. L. McKenzie, M. Kotkamp, S. Wood, C. R. Booth, J. C. Ehramjian, P. Johnston, and S. E. Nichol, “Comparison of ultraviolet spectroradiometers in Antarctica,” J. Geophys. Res. 113, D14310 (2008).
[CrossRef]

J. Mod. Opt. (1)

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]

Light Eng. (1)

E. Ikonen, P. Manninen, and P. Kärhä, “Modeling distance dependence of LED illuminance,” Light Eng. 15, 57-61 (2007).

Meas. Sci. Technol. (1)

R. R. Cordero, G. Seckmeyer, D. Pissulla, L. DaSilva, and F. Labbe, “Uncertainty evaluation of spectral UV irradiance measurements,” Meas. Sci. Technol. 19, 045104 (2008).
[CrossRef]

Metrologia (4)

G. Xu and X. Huang, “Characterization and calibration of broadband ultraviolet radiometers,” Metrologia 37, 235-242 (2000).
[CrossRef]

J. Metzdorf, K. D. Stock, P. Sperfeld, A. Sperling, S. Winter, and T. Wittchen, “Aspects of quality assurance in monitoring solar UV irradiance,” Metrologia 40, S66-S69 (2003).
[CrossRef]

J. Gröbner, M. Blumthaler, S. Kazadzis, A. Bais, A. Webb, J. Schreder, G. Seckmeyer, and D. Rembges, “Quality assurance of spectral solar UV measurements: results from 25 UV monitoring sites in Europe, 2002 to 2004,” Metrologia 43, S66-S71 (2006).
[CrossRef]

P. Manninen, J. Hovila, L. Seppälä, L. Ylianttila, P. Kärhä, and E. Ikonen, “Determination of distance offsets of diffusers for accurate radiometric measurements,” Metrologia 43, S120-S124 (2006).
[CrossRef]

Opt. Lett. (1)

Other (4)

Y. Ohno, “Photometric standards,” in OSA/AIP Handbook of Applied Photometry, C. DeCusatis, ed. (Optical Society of America, 1997), pp. 55-90.

J. G. Schreder, M. Blumthaler and M. Huber, “Design of an input optic for solar UV-measurements” (Internet Photochemistry and Photobiology, 1998), http://www.photobiology.com/UVR98/schreder/index.htm.

J. W. T. Walsh, Photometry (Dover, 1965).

K. Lakkala, “A decade of spectral UV measurements at Sodankylä,” Ph.D. dissertation (University of Helsinki, 2001).

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Figures (10)

Fig. 1
Fig. 1

Geometric parameters between rectangular and circular parallel plates.

Fig. 2
Fig. 2

Illustration of the rectangular approximation of the FEL lamp. The parameter values are x 0 = 5 mm and y 0 = 18 mm .

Fig. 3
Fig. 3

Angular responsivities of the Bentham D3 diffuser at four wavelength bands. The inset shows the fit of the angular responsivity curve at the 350 nm band.

Fig. 4
Fig. 4

Angular responsivities of the Bentham D5 diffuser at four wavelength bands. The inset shows the fits of the angular responsivity curves.

Fig. 5
Fig. 5

Angular responsivities of the Bentham D7 diffuser at three wavelength bands. The inset shows the fit of the angular responsivity curve at the 700 nm band.

Fig. 6
Fig. 6

Angular responsivities of the Schreder J1002-01 diffuser at two wavelength bands. The inset shows the fit of the angular responsivity curve at the 350 nm band.

Fig. 7
Fig. 7

Spatial responsivities of the spectroradiometer with the diffusers studied. The effective radii were estimated from the FWHM of the responses. The physical radii of the D3, D5, D7, and J1002 diffusers are 12.5, 11.5, 5, and 9 mm , respectively.

Fig. 8
Fig. 8

Distance offsets of diffusers as a function of wavelength determined by the method in Subsection 2B. The offsets fix the position of the effective receiving surface of diffusers relative to the outermost surface of the diffuser.

Fig. 9
Fig. 9

Deviations between Eq. (11) and approximation Cases 1–3 for diffuser D5 distance offsets (see text for details).

Fig. 10
Fig. 10

Correction factors for a spectroradiometer with different diffusers at three calibration distances. The solid symbols refer to Eq. (12) and the open symbols to the corresponding approximation case.

Tables (3)

Tables Icon

Table 1 Values of Parameter m Fitted to the Angular Responsivity Data of the Diffusers at Four Wavelengths a

Tables Icon

Table 2 Modified Radius Parameters r S and r D for the Diffusers

Tables Icon

Table 3 Approximations of the Modified Radius Parameters r S and r D in Three Cases and the Corresponding Characterization Requirements a

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

d S = R ( r , φ ; θ ) L e ( x , y ; θ ) d A d Ω ,
L e ( x , y ; α , β ) = I 0 x 0 y 0 I ( α , β ) cos α cos β ,
I ( α , β ) ( 1 1 2 p α 2 ) ( 1 1 2 q β 2 ) cos p α cos q β ,
R ( r , φ ; θ ) = R 0 π r 0 2 K ( θ ) cos θ ,
K ( θ ) 1 1 2 m θ 2 cos m θ ,
d S = R 0 I 0 x 0 y 0 π r 0 2 cos m θ cos p α cos q β D S 2 d A D d A S .
S ( D ) = R 0 I 0 D m + p + q x 0 y 0 π r 0 2 y 0 2 y 0 2 x 0 2 x 0 2 0 r 0 0 2 π 1 [ D 2 + r 2 + x 2 + y 2 2 r ( x cos φ + y sin φ ) ] ( m + 2 ) / 2 × r d φ d r d x d y [ D 2 + x 2 2 r x cos φ + r 2 cos 2 φ ] p / 2 [ D 2 + y 2 2 r y sin φ + r 2 sin 2 φ ] q / 2 .
[ 1 + ( r D ) 2 + ( x D ) 2 + ( y D ) 2 2 r D 2 ( x cos φ + y sin φ ) ] ( m + 2 ) / 2 1 m + 2 2 D 2 [ r 2 + x 2 + y 2 2 r ( x cos φ + y sin φ ) ] ,
S ( D ) = R 0 I 0 D 2 ( 1 r S 2 + r D 2 D 2 ) ,
[ r S r D ] = 1 2 [ m + p + 2 6 x 0 2 + m + q + 2 6 y 0 2 r 0 m + p + q 2 + 2 ] .
S ( D ) R 0 I 0 D 2 + r S 2 + r D 2 ,
C = ( d + Δ d S ) 2 + r S 2 + r D 2 ( d + Δ d S + Δ d D ) 2 + r S 2 + r D 2 ,

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