Abstract

A model of the quantum efficiency of a planar silicon photodiode that is useful in connection with high-accuracy optical-radiation measurements is developed. The model is based mostly on macroscopic (phenomenological) optical and electronic properties of the device that must be determined from experiments on the device, but the connection with the microscopic physical properties (band structure) of silicon is made. The predictions of this model differ significantly from recent experimental results for the variation of the internal quantum efficiency with angle for a silicon photodiode as reported by Durnin et al. [ J. Opt. Soc. Am. 71, 115 ( 1981)]. A repetition of these measurements is described. The results do not agree with those reported by Durnin et al. but do agree well with the predictions of the quantum-efficiency model.

© 1982 Optical Society of America

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References

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  1. J. Durnin, C. Reece, and L. Mandel, “Does a photodetector always measure the rate of arrival of photons?” J. Opt. Soc. Am. 71, 115–117 (1981).
    [Crossref]
  2. F. Bassani and G. Pastori Parravicini, Electronic States and Optical Transitions in Solids (Pergamon, Oxford, 1975), Chap. 5.
  3. E. F. Zalewski and J. Geist, “Silicon photodiode absolute spectral response self-calibration,” Appl. Opt. 19, 1214–1216 (1980).
    [Crossref] [PubMed]
  4. Also, for reasons not mentioned in the text, such as the possible noncompatibility of the concept of radiance and electromagnetic theory, e.g., see A. T. Friberg, “On the existence of a radiance function for finite planar sources of arbitrary states of coherence,” J. Opt. Soc. Am. 69, 192–202 (1979).
    [Crossref]
  5. J. Geist, “On the possibility of an absolute radiometric standard based on the quantum efficiency of a silicon photodiode,” Proc. Soc. Photo-Opt. Instrum. Eng. 196, 75–83 (1979), presents an earlier version of the model to be derived here.
  6. Two-millimeter thicknesses of pure borosilicate glasses with comparable boron concentration show no absorption in the visible and near-uv spectral regions.
  7. The accumulation of a dirt film on the oxide could alter this conclusion. Under normal circumstances, this process takes a period of the order of years, and such films can be removed by cleaning; e.g., see A. R. Schaefer, “Reflectance and external quantum efficiency change of a silicon photodiode after surface cleaning,” Appl. Opt. 18, 2531 (1979).
    [Crossref] [PubMed]
  8. M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, Oxford, 1965), p. 632.
  9. I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. 55, 1205–1209 (1965).
    [Crossref]
  10. H. R. Philipp, “Influence of oxide layers on the determination of the optical properties of silicon,” J. Appl. Phys. 43, 2835–2839 (1972); H. R. Philipp, General Electric Research and Development, Schenectady, New York (personal communication).
    [Crossref]
  11. D. E. Aspnes and J. B. Theeten, “Spectroscopic analysis of the interface between Si and its thermally grown oxide,” J. Electrochem. Soc. 127, 1359–1365 (1980).
    [Crossref]
  12. F. Bassani and G. Pastori Parravicini, Electronic States and Optical Transitions in Solids (Pergamon, Oxford, 1975), Chap. 5.
  13. J. R. Chelikowsky and M. L. Cohen, “Electronic structure of silicon,” Phys. Rev. B 10, 5095–5107 (1974).
    [Crossref]
  14. W. Hanke and L. J. Sham, “Many-particle effects in the optical excitations of a semiconductor,” Phys. Rev. Lett. 43, 387–390 (1979).
    [Crossref]
  15. O. Christensen, “Quantum efficiency of the internal photoelectric effect in silicon and germanium,” J. Appl. Phys. 47, 689–695 (1976).
    [Crossref]
  16. E. Antončík and N. K. S. Gaur, “Theory of the quantum efficiency in silicon and germanium,” J. Phys. C. 11, 735–743 (1978).
    [Crossref]
  17. R. C. Alig, S. Bloom, and C. W. Struck, “Scattering by ionization and phonon emission in semiconductors,” Phys. Rev. B 22, 5565–5582 (1980).
    [Crossref]
  18. J. Geist, E. F. Zalewski, and L. T. Bao, “The quantum yield of silicon in the near-ultraviolet” (National Bureau of Standards, Washington, D.C., 1981).
  19. W. Spitzer and H. Y. Fan, “Infrared absorption in n-type silicon,” Phys. Rev. 108, 268–271 (1957).
    [Crossref]
  20. H. J. Hovel, Solar Cells, Vol. 11 of Semiconductors and Semimetals, R. K. Willardson and A. C. Beer, eds. (Academic, New York, 1975), p. 16.
  21. J. Geist and J. R. Lowney, “Effect of band-gap narrowing on the built-in electric field in n-type silicon,” J. Appl. Phys. 52, 1121–1123 (1981).
    [Crossref]
  22. G. D. Mahan, “Energy gap in Si and Ge: impurity dependence,” J. Appl. Phys. 51, 2634–2646 (1980).
    [Crossref]
  23. Ref. 20, Chap. 2.
  24. S. S. Li, “The dopant density and temperature dependence of hole mobility and resistivity in boron doped silicon,” Solid-State Electron. 21, 1109–1117 (1978), describes such an effort for holes.
    [Crossref]
  25. J. Geist, “Silicon photodiode front region collection efficiency models,” J. Appl. Phys. 51, 3993–3995 (1980).
    [Crossref]
  26. J. Geist, E. F. Zalewski, and A. R. Schaefer, “Spectral response self-calibration and interpolation of silicon photodiodes,” Appl. Opt. 19, 3795–3799 (1980).
    [Crossref] [PubMed]
  27. J. Geist, E. Liang, and A. R. Schaefer, “Complete collection of minority carriers from the inversion layer in induced junction diodes,” J. Appl. Phys. 52, 4879–4881 (1981).
    [Crossref]
  28. J. Geist and E. F. Zalewski, “The quantum yield of silicon in the visible,” Appl. Phys. Lett. 35, 503–506 (1979).
    [Crossref]
  29. J. Geist, A. J. D. Farmer, P. J. Martin, F. J. Wilkinson, and S. Collicott, “Elimination of interface recombination in oxide passivated silicon p+n photodiodes by storage of negative charge on the oxide surface,” Appl. Opt. 21, 1130–1135 (1982).
    [Crossref] [PubMed]
  30. Identification of commercial devices is provided for completeness of the experimental procedure. It implies neither endorsement of the National Bureau of Standards nor that the device is the best available for the particular application.

1982 (1)

1981 (3)

J. Geist, E. Liang, and A. R. Schaefer, “Complete collection of minority carriers from the inversion layer in induced junction diodes,” J. Appl. Phys. 52, 4879–4881 (1981).
[Crossref]

J. Durnin, C. Reece, and L. Mandel, “Does a photodetector always measure the rate of arrival of photons?” J. Opt. Soc. Am. 71, 115–117 (1981).
[Crossref]

J. Geist and J. R. Lowney, “Effect of band-gap narrowing on the built-in electric field in n-type silicon,” J. Appl. Phys. 52, 1121–1123 (1981).
[Crossref]

1980 (6)

G. D. Mahan, “Energy gap in Si and Ge: impurity dependence,” J. Appl. Phys. 51, 2634–2646 (1980).
[Crossref]

J. Geist, “Silicon photodiode front region collection efficiency models,” J. Appl. Phys. 51, 3993–3995 (1980).
[Crossref]

J. Geist, E. F. Zalewski, and A. R. Schaefer, “Spectral response self-calibration and interpolation of silicon photodiodes,” Appl. Opt. 19, 3795–3799 (1980).
[Crossref] [PubMed]

R. C. Alig, S. Bloom, and C. W. Struck, “Scattering by ionization and phonon emission in semiconductors,” Phys. Rev. B 22, 5565–5582 (1980).
[Crossref]

E. F. Zalewski and J. Geist, “Silicon photodiode absolute spectral response self-calibration,” Appl. Opt. 19, 1214–1216 (1980).
[Crossref] [PubMed]

D. E. Aspnes and J. B. Theeten, “Spectroscopic analysis of the interface between Si and its thermally grown oxide,” J. Electrochem. Soc. 127, 1359–1365 (1980).
[Crossref]

1979 (5)

W. Hanke and L. J. Sham, “Many-particle effects in the optical excitations of a semiconductor,” Phys. Rev. Lett. 43, 387–390 (1979).
[Crossref]

Also, for reasons not mentioned in the text, such as the possible noncompatibility of the concept of radiance and electromagnetic theory, e.g., see A. T. Friberg, “On the existence of a radiance function for finite planar sources of arbitrary states of coherence,” J. Opt. Soc. Am. 69, 192–202 (1979).
[Crossref]

J. Geist, “On the possibility of an absolute radiometric standard based on the quantum efficiency of a silicon photodiode,” Proc. Soc. Photo-Opt. Instrum. Eng. 196, 75–83 (1979), presents an earlier version of the model to be derived here.

The accumulation of a dirt film on the oxide could alter this conclusion. Under normal circumstances, this process takes a period of the order of years, and such films can be removed by cleaning; e.g., see A. R. Schaefer, “Reflectance and external quantum efficiency change of a silicon photodiode after surface cleaning,” Appl. Opt. 18, 2531 (1979).
[Crossref] [PubMed]

J. Geist and E. F. Zalewski, “The quantum yield of silicon in the visible,” Appl. Phys. Lett. 35, 503–506 (1979).
[Crossref]

1978 (2)

E. Antončík and N. K. S. Gaur, “Theory of the quantum efficiency in silicon and germanium,” J. Phys. C. 11, 735–743 (1978).
[Crossref]

S. S. Li, “The dopant density and temperature dependence of hole mobility and resistivity in boron doped silicon,” Solid-State Electron. 21, 1109–1117 (1978), describes such an effort for holes.
[Crossref]

1976 (1)

O. Christensen, “Quantum efficiency of the internal photoelectric effect in silicon and germanium,” J. Appl. Phys. 47, 689–695 (1976).
[Crossref]

1974 (1)

J. R. Chelikowsky and M. L. Cohen, “Electronic structure of silicon,” Phys. Rev. B 10, 5095–5107 (1974).
[Crossref]

1972 (1)

H. R. Philipp, “Influence of oxide layers on the determination of the optical properties of silicon,” J. Appl. Phys. 43, 2835–2839 (1972); H. R. Philipp, General Electric Research and Development, Schenectady, New York (personal communication).
[Crossref]

1965 (1)

1957 (1)

W. Spitzer and H. Y. Fan, “Infrared absorption in n-type silicon,” Phys. Rev. 108, 268–271 (1957).
[Crossref]

Alig, R. C.

R. C. Alig, S. Bloom, and C. W. Struck, “Scattering by ionization and phonon emission in semiconductors,” Phys. Rev. B 22, 5565–5582 (1980).
[Crossref]

Antoncík, E.

E. Antončík and N. K. S. Gaur, “Theory of the quantum efficiency in silicon and germanium,” J. Phys. C. 11, 735–743 (1978).
[Crossref]

Aspnes, D. E.

D. E. Aspnes and J. B. Theeten, “Spectroscopic analysis of the interface between Si and its thermally grown oxide,” J. Electrochem. Soc. 127, 1359–1365 (1980).
[Crossref]

Bao, L. T.

J. Geist, E. F. Zalewski, and L. T. Bao, “The quantum yield of silicon in the near-ultraviolet” (National Bureau of Standards, Washington, D.C., 1981).

Bassani, F.

F. Bassani and G. Pastori Parravicini, Electronic States and Optical Transitions in Solids (Pergamon, Oxford, 1975), Chap. 5.

F. Bassani and G. Pastori Parravicini, Electronic States and Optical Transitions in Solids (Pergamon, Oxford, 1975), Chap. 5.

Bloom, S.

R. C. Alig, S. Bloom, and C. W. Struck, “Scattering by ionization and phonon emission in semiconductors,” Phys. Rev. B 22, 5565–5582 (1980).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, Oxford, 1965), p. 632.

Chelikowsky, J. R.

J. R. Chelikowsky and M. L. Cohen, “Electronic structure of silicon,” Phys. Rev. B 10, 5095–5107 (1974).
[Crossref]

Christensen, O.

O. Christensen, “Quantum efficiency of the internal photoelectric effect in silicon and germanium,” J. Appl. Phys. 47, 689–695 (1976).
[Crossref]

Cohen, M. L.

J. R. Chelikowsky and M. L. Cohen, “Electronic structure of silicon,” Phys. Rev. B 10, 5095–5107 (1974).
[Crossref]

Collicott, S.

Durnin, J.

Fan, H. Y.

W. Spitzer and H. Y. Fan, “Infrared absorption in n-type silicon,” Phys. Rev. 108, 268–271 (1957).
[Crossref]

Farmer, A. J. D.

Friberg, A. T.

Gaur, N. K. S.

E. Antončík and N. K. S. Gaur, “Theory of the quantum efficiency in silicon and germanium,” J. Phys. C. 11, 735–743 (1978).
[Crossref]

Geist, J.

J. Geist, A. J. D. Farmer, P. J. Martin, F. J. Wilkinson, and S. Collicott, “Elimination of interface recombination in oxide passivated silicon p+n photodiodes by storage of negative charge on the oxide surface,” Appl. Opt. 21, 1130–1135 (1982).
[Crossref] [PubMed]

J. Geist and J. R. Lowney, “Effect of band-gap narrowing on the built-in electric field in n-type silicon,” J. Appl. Phys. 52, 1121–1123 (1981).
[Crossref]

J. Geist, E. Liang, and A. R. Schaefer, “Complete collection of minority carriers from the inversion layer in induced junction diodes,” J. Appl. Phys. 52, 4879–4881 (1981).
[Crossref]

J. Geist, “Silicon photodiode front region collection efficiency models,” J. Appl. Phys. 51, 3993–3995 (1980).
[Crossref]

E. F. Zalewski and J. Geist, “Silicon photodiode absolute spectral response self-calibration,” Appl. Opt. 19, 1214–1216 (1980).
[Crossref] [PubMed]

J. Geist, E. F. Zalewski, and A. R. Schaefer, “Spectral response self-calibration and interpolation of silicon photodiodes,” Appl. Opt. 19, 3795–3799 (1980).
[Crossref] [PubMed]

J. Geist, “On the possibility of an absolute radiometric standard based on the quantum efficiency of a silicon photodiode,” Proc. Soc. Photo-Opt. Instrum. Eng. 196, 75–83 (1979), presents an earlier version of the model to be derived here.

J. Geist and E. F. Zalewski, “The quantum yield of silicon in the visible,” Appl. Phys. Lett. 35, 503–506 (1979).
[Crossref]

J. Geist, E. F. Zalewski, and L. T. Bao, “The quantum yield of silicon in the near-ultraviolet” (National Bureau of Standards, Washington, D.C., 1981).

Hanke, W.

W. Hanke and L. J. Sham, “Many-particle effects in the optical excitations of a semiconductor,” Phys. Rev. Lett. 43, 387–390 (1979).
[Crossref]

Hovel, H. J.

H. J. Hovel, Solar Cells, Vol. 11 of Semiconductors and Semimetals, R. K. Willardson and A. C. Beer, eds. (Academic, New York, 1975), p. 16.

Li, S. S.

S. S. Li, “The dopant density and temperature dependence of hole mobility and resistivity in boron doped silicon,” Solid-State Electron. 21, 1109–1117 (1978), describes such an effort for holes.
[Crossref]

Liang, E.

J. Geist, E. Liang, and A. R. Schaefer, “Complete collection of minority carriers from the inversion layer in induced junction diodes,” J. Appl. Phys. 52, 4879–4881 (1981).
[Crossref]

Lowney, J. R.

J. Geist and J. R. Lowney, “Effect of band-gap narrowing on the built-in electric field in n-type silicon,” J. Appl. Phys. 52, 1121–1123 (1981).
[Crossref]

Mahan, G. D.

G. D. Mahan, “Energy gap in Si and Ge: impurity dependence,” J. Appl. Phys. 51, 2634–2646 (1980).
[Crossref]

Malitson, I. H.

Mandel, L.

Martin, P. J.

Pastori Parravicini, G.

F. Bassani and G. Pastori Parravicini, Electronic States and Optical Transitions in Solids (Pergamon, Oxford, 1975), Chap. 5.

F. Bassani and G. Pastori Parravicini, Electronic States and Optical Transitions in Solids (Pergamon, Oxford, 1975), Chap. 5.

Philipp, H. R.

H. R. Philipp, “Influence of oxide layers on the determination of the optical properties of silicon,” J. Appl. Phys. 43, 2835–2839 (1972); H. R. Philipp, General Electric Research and Development, Schenectady, New York (personal communication).
[Crossref]

Reece, C.

Schaefer, A. R.

Sham, L. J.

W. Hanke and L. J. Sham, “Many-particle effects in the optical excitations of a semiconductor,” Phys. Rev. Lett. 43, 387–390 (1979).
[Crossref]

Spitzer, W.

W. Spitzer and H. Y. Fan, “Infrared absorption in n-type silicon,” Phys. Rev. 108, 268–271 (1957).
[Crossref]

Struck, C. W.

R. C. Alig, S. Bloom, and C. W. Struck, “Scattering by ionization and phonon emission in semiconductors,” Phys. Rev. B 22, 5565–5582 (1980).
[Crossref]

Theeten, J. B.

D. E. Aspnes and J. B. Theeten, “Spectroscopic analysis of the interface between Si and its thermally grown oxide,” J. Electrochem. Soc. 127, 1359–1365 (1980).
[Crossref]

Wilkinson, F. J.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, Oxford, 1965), p. 632.

Zalewski, E. F.

E. F. Zalewski and J. Geist, “Silicon photodiode absolute spectral response self-calibration,” Appl. Opt. 19, 1214–1216 (1980).
[Crossref] [PubMed]

J. Geist, E. F. Zalewski, and A. R. Schaefer, “Spectral response self-calibration and interpolation of silicon photodiodes,” Appl. Opt. 19, 3795–3799 (1980).
[Crossref] [PubMed]

J. Geist and E. F. Zalewski, “The quantum yield of silicon in the visible,” Appl. Phys. Lett. 35, 503–506 (1979).
[Crossref]

J. Geist, E. F. Zalewski, and L. T. Bao, “The quantum yield of silicon in the near-ultraviolet” (National Bureau of Standards, Washington, D.C., 1981).

Appl. Opt. (4)

Appl. Phys. Lett. (1)

J. Geist and E. F. Zalewski, “The quantum yield of silicon in the visible,” Appl. Phys. Lett. 35, 503–506 (1979).
[Crossref]

J. Appl. Phys. (6)

J. Geist, “Silicon photodiode front region collection efficiency models,” J. Appl. Phys. 51, 3993–3995 (1980).
[Crossref]

J. Geist, E. Liang, and A. R. Schaefer, “Complete collection of minority carriers from the inversion layer in induced junction diodes,” J. Appl. Phys. 52, 4879–4881 (1981).
[Crossref]

J. Geist and J. R. Lowney, “Effect of band-gap narrowing on the built-in electric field in n-type silicon,” J. Appl. Phys. 52, 1121–1123 (1981).
[Crossref]

G. D. Mahan, “Energy gap in Si and Ge: impurity dependence,” J. Appl. Phys. 51, 2634–2646 (1980).
[Crossref]

H. R. Philipp, “Influence of oxide layers on the determination of the optical properties of silicon,” J. Appl. Phys. 43, 2835–2839 (1972); H. R. Philipp, General Electric Research and Development, Schenectady, New York (personal communication).
[Crossref]

O. Christensen, “Quantum efficiency of the internal photoelectric effect in silicon and germanium,” J. Appl. Phys. 47, 689–695 (1976).
[Crossref]

J. Electrochem. Soc. (1)

D. E. Aspnes and J. B. Theeten, “Spectroscopic analysis of the interface between Si and its thermally grown oxide,” J. Electrochem. Soc. 127, 1359–1365 (1980).
[Crossref]

J. Opt. Soc. Am. (3)

J. Phys. C. (1)

E. Antončík and N. K. S. Gaur, “Theory of the quantum efficiency in silicon and germanium,” J. Phys. C. 11, 735–743 (1978).
[Crossref]

Phys. Rev. (1)

W. Spitzer and H. Y. Fan, “Infrared absorption in n-type silicon,” Phys. Rev. 108, 268–271 (1957).
[Crossref]

Phys. Rev. B (2)

R. C. Alig, S. Bloom, and C. W. Struck, “Scattering by ionization and phonon emission in semiconductors,” Phys. Rev. B 22, 5565–5582 (1980).
[Crossref]

J. R. Chelikowsky and M. L. Cohen, “Electronic structure of silicon,” Phys. Rev. B 10, 5095–5107 (1974).
[Crossref]

Phys. Rev. Lett. (1)

W. Hanke and L. J. Sham, “Many-particle effects in the optical excitations of a semiconductor,” Phys. Rev. Lett. 43, 387–390 (1979).
[Crossref]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

J. Geist, “On the possibility of an absolute radiometric standard based on the quantum efficiency of a silicon photodiode,” Proc. Soc. Photo-Opt. Instrum. Eng. 196, 75–83 (1979), presents an earlier version of the model to be derived here.

Solid-State Electron. (1)

S. S. Li, “The dopant density and temperature dependence of hole mobility and resistivity in boron doped silicon,” Solid-State Electron. 21, 1109–1117 (1978), describes such an effort for holes.
[Crossref]

Other (8)

H. J. Hovel, Solar Cells, Vol. 11 of Semiconductors and Semimetals, R. K. Willardson and A. C. Beer, eds. (Academic, New York, 1975), p. 16.

Ref. 20, Chap. 2.

Identification of commercial devices is provided for completeness of the experimental procedure. It implies neither endorsement of the National Bureau of Standards nor that the device is the best available for the particular application.

Two-millimeter thicknesses of pure borosilicate glasses with comparable boron concentration show no absorption in the visible and near-uv spectral regions.

F. Bassani and G. Pastori Parravicini, Electronic States and Optical Transitions in Solids (Pergamon, Oxford, 1975), Chap. 5.

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon, Oxford, 1965), p. 632.

F. Bassani and G. Pastori Parravicini, Electronic States and Optical Transitions in Solids (Pergamon, Oxford, 1975), Chap. 5.

J. Geist, E. F. Zalewski, and L. T. Bao, “The quantum yield of silicon in the near-ultraviolet” (National Bureau of Standards, Washington, D.C., 1981).

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Figures (8)

Fig. 1
Fig. 1

A cross section of a typical photodiode illustrating the reflection of some of the incident radiation away from the diode and the transmission of the remainder into the silicon, where it is absorbed and generates electron–hole pairs with an efficiency called the quantum yield. The photogenerated electron–hole pairs are then separated by the junction with an efficiency called the collection efficiency, thereby inducing a current in an external circuit.

Fig. 2
Fig. 2

(a) Typical values of the absorption density A(λ, x) and collection efficiency P(x) are shown as a function of x, the depth in silicon. (b) The integral with respect to x of the product of P(x) and A(λ, x) as a function of wavelength λ.

Fig. 3
Fig. 3

Typical wavelength dependencies of a silicon photodiode’s absorptance 1 − ρ(λ), quantum yield η(λ), and collection-efficiency factor F(λ). Their product gives the external quantum efficiency Q(λ).

Fig. 4
Fig. 4

Reflectance as a function of incidence angle and polarization at 0.633 μm. Calculated reflectance values for TM and TE polarizations are represented by dashed and solid lines, respectively. Experimentally determined reflectance values are represented by open circles. The SDC reflectance curves start at ρ = 20%, the EG&G curves at ρ ~ 18%.

Fig. 5
Fig. 5

Experimental arrangement for measuring the angular response of a detector. A lens and a neutral-density filter are drawn with dashed lines to indicate use when appropriate.

Fig. 6
Fig. 6

Response of EG&G detector as a function of incidence angle. Data normalized to response at smallest angle measured. Relative responses uncorrected and corrected for reflectance are represented by circles and triangles, respectively.

Fig. 7
Fig. 7

Response of SDCI detector as a function of incidence angle. Data normalized to response at smallest angle measured. Relative responses uncorrected and corrected for reflectance are represented by circles and triangles, respectively.

Fig. 8
Fig. 8

Response of SDCII detector as a function of incidence angle. Data normalized to response at smallest angle measured. Relative responses uncorrected and corrected for reflectance are represented by circles and triangles, respectively.

Equations (15)

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

r i , f f A v · p i 2 δ ( h ν + E i - E f ) ,
I = L ( s , n ˆ , λ ) Q ( s , n ˆ , λ ) cos θ d λ d Ω n ˆ d A s
( θ , λ ) Q ( θ , λ ) / ϒ ( θ , λ ) .
ϒ ( θ , λ ) = 1 - ρ s ( θ , λ ) ,
η ( λ ) = α 1 ( λ ) [ ( 1 + β ( λ ) ] + α 0 ( λ ) + α 1 ( λ ) + ,
P ( x ) = - μ ( x 0 ) ξ ( x 0 ) n ( x , x 0 ) - D ( x ) d n ( x , x 0 ) / d x 0 ,
ξ ( x ) = k T q 1 N a d N a d x F 1 ( N a ) F 2 ( N a ) ,
( θ , λ ) = 0 h A ( λ , x ) η ( λ ) P ( x ) d x ,
A ( λ , x ) = α ( λ ) exp [ - α ( λ ) x ] ,
( θ , λ ) = P 0 η ( λ ) .
F ( θ , λ ) = 0 h A ( λ , x ) P ( x ) d x .
Q = ( 1 - ρ ) η F ,
P ( x ) = P 0 + ( 1 - P 0 ) ( x / x 0 ) ,             0 < x < x 0 , = 1 ,             x 0 < x < h ,
F [ θ ( θ ) , λ ] = P 0 + ( 1 - P ) ( 1 - e - α x 0 ) / α x 0 ,
F P 0 .