Abstract

An emissivity-invariant condition for a silicon wafer was determined by simulation modeling and it was confirmed experimentally. The p-polarized spectral emissivity at a wavelength of 900nm and at temperatures over 900K was constant at 0.83 at an angle of about 55.4° irrespective of large variations in the oxide layer thickness and the resistivity due to the different impurity doping concentrations of the silicon wafer. The expanded uncertainty, Uc=kuc (k=2), of the temperature measurement is estimated to be 4.9K. This result is expected to significantly enhance the accuracy of radiometric temperature measurements of silicon wafers in actual manufacturing processes.

© 2011 Optical Society of America

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References

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  1. B. E. Adams, C. W. Schietinger, and K. G. Kerider, “Radiation thermometry in the semiconductor industry,” in Radiometric Temperature Measurements, II. Applications, A.M.Zhang, B.K.Tsai, and G.Machin, eds. (Elsevier, 2010), pp. 137–216.
    [CrossRef]
  2. R. L. Anderson, “Review of temperature measurement in the semiconductor industry,” in Proceedings of Temperature: Its Measurement and Control in Science and Industry, J.F.Schooley, ed. (American Institute of Physics, 1992), pp. 1117–1122.
  3. B. E. Adams, “The challenges of temperature measurement in the semiconductor industry,” in Proceedings of 7th International Symposium on Temperature and Thermal Measurements in Industry and Science, J.F.Dubbeldam and M.J.deGroot, eds. (Nederlands Meetinstitute, 1999), pp. 1–10.
    [PubMed]
  4. F. Roozeboom and N. Parekh, “Rapid thermal processing systems: A review with emphasis on temperature control,” J. Vac. Sci. Technol. B 8, 1249–1259 (1990).
    [CrossRef]
  5. A. T. Fiory, “Recent developments in rapid thermal processing, J. Electron. Mater. 31, 981–987 (2002).
    [CrossRef]
  6. C. W. Schietinger, B. E. Adams, and C. Yarling, “Ripple technique: A novel non-contact wafer emissivity and temperature measurement method for RTP,” in Rapid Thermal and Integrated Processing, Vol. 224 of Materials Research Society Proceedings, J.C.Gelpey, M.L.Green, R.Singh, and J.J.Wortman, eds. (Materials Research Society, 1991), pp. 23–31.
  7. B. K. Tsai, “A summary of lightpipe radiation thermometry research at NIST,” J. Res. Natl. Inst. Stand. Technol. 111, 9–30 (2006).
  8. R. J. Schreuutelkamp, P. Vandenabeele, B. Deweerdt, and W. Coppye, “In-situ emissivity measurements to probe the phase transformations during rapid thermal processing cosilicidation,” Appl. Phys. Lett. 61, 2296–2298 (1992).
    [CrossRef]
  9. W. A. Kimes, K. G. Kreider, D. C. Ripple, and B. K. Tsai, “Emissivity compensated pyrometry for specular silicon surfaces on the NIST RTP test bed,” in Proceedings of the 12th IEEE International Conference on Advanced Thermal Processing—RTP 2004, J.Gelpey, B.Lojek, Z.Nenyei, and R.Singh, eds. (IEEE, 2004), pp. 156–161.
  10. T. Sato, “Spectral emissivity of silicon,” Jpn. J. Appl. Phys. 6, 339–347 (1967).
    [CrossRef]
  11. P. Vandenabeele and K. Maex, “Emissivity of silicon wafers during rapid thermal processing,” Proc. SPIE 1393, 316–336(1991).
    [CrossRef]
  12. P. J. Timans, “Emissivity of silicon at elevated temperatures,” J. Appl. Phys. 74, 6353–6364 (1993).
    [CrossRef]
  13. B. J. Lee, Z. M. Zhang, E. A. Early, D. P. DeWitt, and B. K. Tsai, “Modeling radiative properties of silicon with coatings and comparison with reflectance measurements,” J. Thermophys. Heat Transfer 19, 558–569 (2005).
    [CrossRef]
  14. H. R. Philipp, “Silicon dioxide (SiO2),” in Handbook of Optical Constants of Solids, E.D.Palik, ed. (Elsevier, 1985), pp. 719–763.
  15. D. F. Edwards, “Silicon (Si),” in Handbook of Optical Constants of Solids, E.D.Palik, ed. (Elsevier, 1985), pp. 547–569.
  16. Z. M. Zhang and B. J. Lee, “Theory of thermal radiation and radiative properties,” in Radiometric Temperature Measurements, I. Fundamentals, Z.M.Zhang, B.K.Tsai, and G.Machin, eds. (Elsevier, 2010), pp. 73–132.
  17. S. M. Sze, “Carrier transport phenomena,” in Semiconductor Devices, Physics, and Technology, 2nd ed. (Wiley, 2002), pp. 47–83.
  18. T. Iuchi and A. Gogami, “Uncertainty of a hybrid surface temperature sensor for silicon wafers and comparison with an embedded thermocouple,” Rev. Sci. Instrum. 80, 126109 (2009).
    [CrossRef]
  19. S. M. Sze and K. K. Ng, “Physics and properties of semiconductors:—A review,” in Physics and Semiconductor Devices, 3rd ed. (Wiley, 2006), pp. 19–21.
  20. G. D. Nutter, “Radiation thermometry—The measurement problem,” in Applications of Radiation Thermometry, J.C.Richmond and D.P.DeWitt, eds. (ASTM, 1984), pp. 3–23.

2009 (1)

T. Iuchi and A. Gogami, “Uncertainty of a hybrid surface temperature sensor for silicon wafers and comparison with an embedded thermocouple,” Rev. Sci. Instrum. 80, 126109 (2009).
[CrossRef]

2006 (1)

B. K. Tsai, “A summary of lightpipe radiation thermometry research at NIST,” J. Res. Natl. Inst. Stand. Technol. 111, 9–30 (2006).

2005 (1)

B. J. Lee, Z. M. Zhang, E. A. Early, D. P. DeWitt, and B. K. Tsai, “Modeling radiative properties of silicon with coatings and comparison with reflectance measurements,” J. Thermophys. Heat Transfer 19, 558–569 (2005).
[CrossRef]

2002 (1)

A. T. Fiory, “Recent developments in rapid thermal processing, J. Electron. Mater. 31, 981–987 (2002).
[CrossRef]

1993 (1)

P. J. Timans, “Emissivity of silicon at elevated temperatures,” J. Appl. Phys. 74, 6353–6364 (1993).
[CrossRef]

1992 (1)

R. J. Schreuutelkamp, P. Vandenabeele, B. Deweerdt, and W. Coppye, “In-situ emissivity measurements to probe the phase transformations during rapid thermal processing cosilicidation,” Appl. Phys. Lett. 61, 2296–2298 (1992).
[CrossRef]

1991 (1)

P. Vandenabeele and K. Maex, “Emissivity of silicon wafers during rapid thermal processing,” Proc. SPIE 1393, 316–336(1991).
[CrossRef]

1990 (1)

F. Roozeboom and N. Parekh, “Rapid thermal processing systems: A review with emphasis on temperature control,” J. Vac. Sci. Technol. B 8, 1249–1259 (1990).
[CrossRef]

1967 (1)

T. Sato, “Spectral emissivity of silicon,” Jpn. J. Appl. Phys. 6, 339–347 (1967).
[CrossRef]

Adams, B. E.

B. E. Adams, C. W. Schietinger, and K. G. Kerider, “Radiation thermometry in the semiconductor industry,” in Radiometric Temperature Measurements, II. Applications, A.M.Zhang, B.K.Tsai, and G.Machin, eds. (Elsevier, 2010), pp. 137–216.
[CrossRef]

B. E. Adams, “The challenges of temperature measurement in the semiconductor industry,” in Proceedings of 7th International Symposium on Temperature and Thermal Measurements in Industry and Science, J.F.Dubbeldam and M.J.deGroot, eds. (Nederlands Meetinstitute, 1999), pp. 1–10.
[PubMed]

C. W. Schietinger, B. E. Adams, and C. Yarling, “Ripple technique: A novel non-contact wafer emissivity and temperature measurement method for RTP,” in Rapid Thermal and Integrated Processing, Vol. 224 of Materials Research Society Proceedings, J.C.Gelpey, M.L.Green, R.Singh, and J.J.Wortman, eds. (Materials Research Society, 1991), pp. 23–31.

Anderson, R. L.

R. L. Anderson, “Review of temperature measurement in the semiconductor industry,” in Proceedings of Temperature: Its Measurement and Control in Science and Industry, J.F.Schooley, ed. (American Institute of Physics, 1992), pp. 1117–1122.

Coppye, W.

R. J. Schreuutelkamp, P. Vandenabeele, B. Deweerdt, and W. Coppye, “In-situ emissivity measurements to probe the phase transformations during rapid thermal processing cosilicidation,” Appl. Phys. Lett. 61, 2296–2298 (1992).
[CrossRef]

Deweerdt, B.

R. J. Schreuutelkamp, P. Vandenabeele, B. Deweerdt, and W. Coppye, “In-situ emissivity measurements to probe the phase transformations during rapid thermal processing cosilicidation,” Appl. Phys. Lett. 61, 2296–2298 (1992).
[CrossRef]

DeWitt, D. P.

B. J. Lee, Z. M. Zhang, E. A. Early, D. P. DeWitt, and B. K. Tsai, “Modeling radiative properties of silicon with coatings and comparison with reflectance measurements,” J. Thermophys. Heat Transfer 19, 558–569 (2005).
[CrossRef]

Early, E. A.

B. J. Lee, Z. M. Zhang, E. A. Early, D. P. DeWitt, and B. K. Tsai, “Modeling radiative properties of silicon with coatings and comparison with reflectance measurements,” J. Thermophys. Heat Transfer 19, 558–569 (2005).
[CrossRef]

Edwards, D. F.

D. F. Edwards, “Silicon (Si),” in Handbook of Optical Constants of Solids, E.D.Palik, ed. (Elsevier, 1985), pp. 547–569.

Fiory, A. T.

A. T. Fiory, “Recent developments in rapid thermal processing, J. Electron. Mater. 31, 981–987 (2002).
[CrossRef]

Gogami, A.

T. Iuchi and A. Gogami, “Uncertainty of a hybrid surface temperature sensor for silicon wafers and comparison with an embedded thermocouple,” Rev. Sci. Instrum. 80, 126109 (2009).
[CrossRef]

Iuchi, T.

T. Iuchi and A. Gogami, “Uncertainty of a hybrid surface temperature sensor for silicon wafers and comparison with an embedded thermocouple,” Rev. Sci. Instrum. 80, 126109 (2009).
[CrossRef]

Kerider, K. G.

B. E. Adams, C. W. Schietinger, and K. G. Kerider, “Radiation thermometry in the semiconductor industry,” in Radiometric Temperature Measurements, II. Applications, A.M.Zhang, B.K.Tsai, and G.Machin, eds. (Elsevier, 2010), pp. 137–216.
[CrossRef]

Kimes, W. A.

W. A. Kimes, K. G. Kreider, D. C. Ripple, and B. K. Tsai, “Emissivity compensated pyrometry for specular silicon surfaces on the NIST RTP test bed,” in Proceedings of the 12th IEEE International Conference on Advanced Thermal Processing—RTP 2004, J.Gelpey, B.Lojek, Z.Nenyei, and R.Singh, eds. (IEEE, 2004), pp. 156–161.

Kreider, K. G.

W. A. Kimes, K. G. Kreider, D. C. Ripple, and B. K. Tsai, “Emissivity compensated pyrometry for specular silicon surfaces on the NIST RTP test bed,” in Proceedings of the 12th IEEE International Conference on Advanced Thermal Processing—RTP 2004, J.Gelpey, B.Lojek, Z.Nenyei, and R.Singh, eds. (IEEE, 2004), pp. 156–161.

Lee, B. J.

B. J. Lee, Z. M. Zhang, E. A. Early, D. P. DeWitt, and B. K. Tsai, “Modeling radiative properties of silicon with coatings and comparison with reflectance measurements,” J. Thermophys. Heat Transfer 19, 558–569 (2005).
[CrossRef]

Z. M. Zhang and B. J. Lee, “Theory of thermal radiation and radiative properties,” in Radiometric Temperature Measurements, I. Fundamentals, Z.M.Zhang, B.K.Tsai, and G.Machin, eds. (Elsevier, 2010), pp. 73–132.

Maex, K.

P. Vandenabeele and K. Maex, “Emissivity of silicon wafers during rapid thermal processing,” Proc. SPIE 1393, 316–336(1991).
[CrossRef]

Ng, K. K.

S. M. Sze and K. K. Ng, “Physics and properties of semiconductors:—A review,” in Physics and Semiconductor Devices, 3rd ed. (Wiley, 2006), pp. 19–21.

Nutter, G. D.

G. D. Nutter, “Radiation thermometry—The measurement problem,” in Applications of Radiation Thermometry, J.C.Richmond and D.P.DeWitt, eds. (ASTM, 1984), pp. 3–23.

Parekh, N.

F. Roozeboom and N. Parekh, “Rapid thermal processing systems: A review with emphasis on temperature control,” J. Vac. Sci. Technol. B 8, 1249–1259 (1990).
[CrossRef]

Philipp, H. R.

H. R. Philipp, “Silicon dioxide (SiO2),” in Handbook of Optical Constants of Solids, E.D.Palik, ed. (Elsevier, 1985), pp. 719–763.

Ripple, D. C.

W. A. Kimes, K. G. Kreider, D. C. Ripple, and B. K. Tsai, “Emissivity compensated pyrometry for specular silicon surfaces on the NIST RTP test bed,” in Proceedings of the 12th IEEE International Conference on Advanced Thermal Processing—RTP 2004, J.Gelpey, B.Lojek, Z.Nenyei, and R.Singh, eds. (IEEE, 2004), pp. 156–161.

Roozeboom, F.

F. Roozeboom and N. Parekh, “Rapid thermal processing systems: A review with emphasis on temperature control,” J. Vac. Sci. Technol. B 8, 1249–1259 (1990).
[CrossRef]

Sato, T.

T. Sato, “Spectral emissivity of silicon,” Jpn. J. Appl. Phys. 6, 339–347 (1967).
[CrossRef]

Schietinger, C. W.

C. W. Schietinger, B. E. Adams, and C. Yarling, “Ripple technique: A novel non-contact wafer emissivity and temperature measurement method for RTP,” in Rapid Thermal and Integrated Processing, Vol. 224 of Materials Research Society Proceedings, J.C.Gelpey, M.L.Green, R.Singh, and J.J.Wortman, eds. (Materials Research Society, 1991), pp. 23–31.

B. E. Adams, C. W. Schietinger, and K. G. Kerider, “Radiation thermometry in the semiconductor industry,” in Radiometric Temperature Measurements, II. Applications, A.M.Zhang, B.K.Tsai, and G.Machin, eds. (Elsevier, 2010), pp. 137–216.
[CrossRef]

Schreuutelkamp, R. J.

R. J. Schreuutelkamp, P. Vandenabeele, B. Deweerdt, and W. Coppye, “In-situ emissivity measurements to probe the phase transformations during rapid thermal processing cosilicidation,” Appl. Phys. Lett. 61, 2296–2298 (1992).
[CrossRef]

Sze, S. M.

S. M. Sze, “Carrier transport phenomena,” in Semiconductor Devices, Physics, and Technology, 2nd ed. (Wiley, 2002), pp. 47–83.

S. M. Sze and K. K. Ng, “Physics and properties of semiconductors:—A review,” in Physics and Semiconductor Devices, 3rd ed. (Wiley, 2006), pp. 19–21.

Timans, P. J.

P. J. Timans, “Emissivity of silicon at elevated temperatures,” J. Appl. Phys. 74, 6353–6364 (1993).
[CrossRef]

Tsai, B. K.

B. K. Tsai, “A summary of lightpipe radiation thermometry research at NIST,” J. Res. Natl. Inst. Stand. Technol. 111, 9–30 (2006).

B. J. Lee, Z. M. Zhang, E. A. Early, D. P. DeWitt, and B. K. Tsai, “Modeling radiative properties of silicon with coatings and comparison with reflectance measurements,” J. Thermophys. Heat Transfer 19, 558–569 (2005).
[CrossRef]

W. A. Kimes, K. G. Kreider, D. C. Ripple, and B. K. Tsai, “Emissivity compensated pyrometry for specular silicon surfaces on the NIST RTP test bed,” in Proceedings of the 12th IEEE International Conference on Advanced Thermal Processing—RTP 2004, J.Gelpey, B.Lojek, Z.Nenyei, and R.Singh, eds. (IEEE, 2004), pp. 156–161.

Vandenabeele, P.

R. J. Schreuutelkamp, P. Vandenabeele, B. Deweerdt, and W. Coppye, “In-situ emissivity measurements to probe the phase transformations during rapid thermal processing cosilicidation,” Appl. Phys. Lett. 61, 2296–2298 (1992).
[CrossRef]

P. Vandenabeele and K. Maex, “Emissivity of silicon wafers during rapid thermal processing,” Proc. SPIE 1393, 316–336(1991).
[CrossRef]

Yarling, C.

C. W. Schietinger, B. E. Adams, and C. Yarling, “Ripple technique: A novel non-contact wafer emissivity and temperature measurement method for RTP,” in Rapid Thermal and Integrated Processing, Vol. 224 of Materials Research Society Proceedings, J.C.Gelpey, M.L.Green, R.Singh, and J.J.Wortman, eds. (Materials Research Society, 1991), pp. 23–31.

Zhang, Z. M.

B. J. Lee, Z. M. Zhang, E. A. Early, D. P. DeWitt, and B. K. Tsai, “Modeling radiative properties of silicon with coatings and comparison with reflectance measurements,” J. Thermophys. Heat Transfer 19, 558–569 (2005).
[CrossRef]

Z. M. Zhang and B. J. Lee, “Theory of thermal radiation and radiative properties,” in Radiometric Temperature Measurements, I. Fundamentals, Z.M.Zhang, B.K.Tsai, and G.Machin, eds. (Elsevier, 2010), pp. 73–132.

Appl. Phys. Lett. (1)

R. J. Schreuutelkamp, P. Vandenabeele, B. Deweerdt, and W. Coppye, “In-situ emissivity measurements to probe the phase transformations during rapid thermal processing cosilicidation,” Appl. Phys. Lett. 61, 2296–2298 (1992).
[CrossRef]

J. Appl. Phys. (1)

P. J. Timans, “Emissivity of silicon at elevated temperatures,” J. Appl. Phys. 74, 6353–6364 (1993).
[CrossRef]

J. Electron. Mater. (1)

A. T. Fiory, “Recent developments in rapid thermal processing, J. Electron. Mater. 31, 981–987 (2002).
[CrossRef]

J. Res. Natl. Inst. Stand. Technol. (1)

B. K. Tsai, “A summary of lightpipe radiation thermometry research at NIST,” J. Res. Natl. Inst. Stand. Technol. 111, 9–30 (2006).

J. Thermophys. Heat Transfer (1)

B. J. Lee, Z. M. Zhang, E. A. Early, D. P. DeWitt, and B. K. Tsai, “Modeling radiative properties of silicon with coatings and comparison with reflectance measurements,” J. Thermophys. Heat Transfer 19, 558–569 (2005).
[CrossRef]

J. Vac. Sci. Technol. B (1)

F. Roozeboom and N. Parekh, “Rapid thermal processing systems: A review with emphasis on temperature control,” J. Vac. Sci. Technol. B 8, 1249–1259 (1990).
[CrossRef]

Jpn. J. Appl. Phys. (1)

T. Sato, “Spectral emissivity of silicon,” Jpn. J. Appl. Phys. 6, 339–347 (1967).
[CrossRef]

Proc. SPIE (1)

P. Vandenabeele and K. Maex, “Emissivity of silicon wafers during rapid thermal processing,” Proc. SPIE 1393, 316–336(1991).
[CrossRef]

Rev. Sci. Instrum. (1)

T. Iuchi and A. Gogami, “Uncertainty of a hybrid surface temperature sensor for silicon wafers and comparison with an embedded thermocouple,” Rev. Sci. Instrum. 80, 126109 (2009).
[CrossRef]

Other (11)

S. M. Sze and K. K. Ng, “Physics and properties of semiconductors:—A review,” in Physics and Semiconductor Devices, 3rd ed. (Wiley, 2006), pp. 19–21.

G. D. Nutter, “Radiation thermometry—The measurement problem,” in Applications of Radiation Thermometry, J.C.Richmond and D.P.DeWitt, eds. (ASTM, 1984), pp. 3–23.

H. R. Philipp, “Silicon dioxide (SiO2),” in Handbook of Optical Constants of Solids, E.D.Palik, ed. (Elsevier, 1985), pp. 719–763.

D. F. Edwards, “Silicon (Si),” in Handbook of Optical Constants of Solids, E.D.Palik, ed. (Elsevier, 1985), pp. 547–569.

Z. M. Zhang and B. J. Lee, “Theory of thermal radiation and radiative properties,” in Radiometric Temperature Measurements, I. Fundamentals, Z.M.Zhang, B.K.Tsai, and G.Machin, eds. (Elsevier, 2010), pp. 73–132.

S. M. Sze, “Carrier transport phenomena,” in Semiconductor Devices, Physics, and Technology, 2nd ed. (Wiley, 2002), pp. 47–83.

W. A. Kimes, K. G. Kreider, D. C. Ripple, and B. K. Tsai, “Emissivity compensated pyrometry for specular silicon surfaces on the NIST RTP test bed,” in Proceedings of the 12th IEEE International Conference on Advanced Thermal Processing—RTP 2004, J.Gelpey, B.Lojek, Z.Nenyei, and R.Singh, eds. (IEEE, 2004), pp. 156–161.

C. W. Schietinger, B. E. Adams, and C. Yarling, “Ripple technique: A novel non-contact wafer emissivity and temperature measurement method for RTP,” in Rapid Thermal and Integrated Processing, Vol. 224 of Materials Research Society Proceedings, J.C.Gelpey, M.L.Green, R.Singh, and J.J.Wortman, eds. (Materials Research Society, 1991), pp. 23–31.

B. E. Adams, C. W. Schietinger, and K. G. Kerider, “Radiation thermometry in the semiconductor industry,” in Radiometric Temperature Measurements, II. Applications, A.M.Zhang, B.K.Tsai, and G.Machin, eds. (Elsevier, 2010), pp. 137–216.
[CrossRef]

R. L. Anderson, “Review of temperature measurement in the semiconductor industry,” in Proceedings of Temperature: Its Measurement and Control in Science and Industry, J.F.Schooley, ed. (American Institute of Physics, 1992), pp. 1117–1122.

B. E. Adams, “The challenges of temperature measurement in the semiconductor industry,” in Proceedings of 7th International Symposium on Temperature and Thermal Measurements in Industry and Science, J.F.Dubbeldam and M.J.deGroot, eds. (Nederlands Meetinstitute, 1999), pp. 1–10.
[PubMed]

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Figures (9)

Fig. 1
Fig. 1

Simulation model consisting of a silicon wafer and an oxide layer ( SiO 2 ).

Fig. 2
Fig. 2

Simulated results of polarized emissivities as a function of angle, θ 1 . (a) p-polarized emissivity, ε p ( θ 1 ) and (b) s-polarized emissivity, ε s ( θ 1 ) , respectively (d: oxide film thickness, nm).

Fig. 3
Fig. 3

Simulated relationship between oxide film thickness, d, and polarized emissivity, ε p ( s ) ( θ 1 ) , at θ 1 = 55.4 ° .

Fig. 4
Fig. 4

Experimental system used for polarized emissivity measurements.

Fig. 5
Fig. 5

Experimental and simulation results for directional, polarized emissivity, ε p ( s ) ( θ 1 ) , of a silicon wafer ( d = 30 nm ).

Fig. 6
Fig. 6

Experimental results for directional polarized emissivities of silicon wafers with different oxide film thickness, d, at T = 1000 K and λ = 900 nm . (a) p-polarized emissivity, ε p ( θ 1 ) and (b) s-polarized emissivity, ε s ( θ 1 ) , respectively.

Fig. 7
Fig. 7

Experimental relationship between oxide film thickness, d, and polarized emissivity, ε p ( s ) ( θ 1 ) , at θ 1 = 55.4 ° .

Fig. 8
Fig. 8

Experimental results for p-polarized emissivity at θ 1 = 55.4 ° for silicon wafers with wide ranges of resistivities and oxide film thicknesses and at various temperatures.

Fig. 9
Fig. 9

Example of a measurement system in a manufacturing process.

Tables (1)

Tables Icon

Table 1 Experimental Measurement Conditions

Equations (21)

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

ε p ( s ) ( θ 1 ) = 1 + r 12 p ( s ) 2 r 23 p ( s ) 2 r 12 p ( s ) 2 r 23 p ( s ) 2 1 + r 12 p ( s ) 2 r 23 p ( s ) 2 + 2 r 12 p ( s ) r 23 p ( s ) cos δ ,
δ = 2 π λ · 2 n 2 · d · cos θ 2 ,
r 12 p = cos θ 1 n 1 cos θ 2 n 2 cos θ 1 n 1 + cos θ 2 n 2 ,
r 23 p = cos θ 2 n 2 cos θ 3 n 3 cos θ 2 n 2 + cos θ 3 n 3 ,
r 12 s = n 1 cos θ 1 n 2 cos θ 2 n 1 cos θ 1 + n 2 cos θ 2 ,
r 23 s = n 2 cos θ 2 n 3 cos θ 3 n 2 cos θ 2 + n 3 cos θ 3 ,
n 1 sin θ 1 = n 2 sin θ 2 = n 3 sin θ 3 ,
θ 1 = tan 1 ( n 2 n 1 ) .
ε p ( θ 1 ) = 1 r 23 p 2 .
E p = k p ε p ( θ 1 ) L λ , b ( T ) ,
E s = k s ε s ( θ 1 ) L λ , b ( T ) ,
E p , b ( T ) = k p L λ , b ( T ) ,
E s , b ( T ) = k s L λ , b ( T ) .
ε p ( s ) ( θ 1 ) = E p ( s ) E p ( s ) , b ( T ) .
u t = 1 n u ε ε av T ,
n = 14388 λ T .
u c = u h 2 + u r 2 + u t 2 .
E 1 = k p ε p ( θ 1 ) L λ , b ( T 1 ) + k p { 1 ε p ( θ 1 ) } L λ , b ( T 2 ) .
E 2 = k L λ , b ( T 2 ) ,
E 2 = k p L λ , b ( T 2 ) .
k p L λ , b ( T 1 ) = E 1 { 1 ε p ( θ 1 ) } E 2 ε p ( θ 1 ) .

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