Abstract

A two-substrate method is developed to simultaneously determine emissivity, transmittance, and reflectance of semitransparent materials with a single measurement under the same environment at elevated temperature. The three quantities can be obtained through the emissivities of substrates and the apparent emissivities resulting from the radiance of the sample heated by substrates. The two-substrate method is compared with the conventional method by measuring sapphire samples with various thicknesses, resulting in good agreements for all the samples. The present method will be useful to measure the temperature dependence of optical properties of porous ceramic materials.

© 2010 Optical Society of America

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  1. N. M. Kirby, N. W. Chen-Tan, and C. E. Buckley, J. Eur. Ceram. Soc. 27, 2039 (2007).
    [CrossRef]
  2. J. I. Eldridge, C. M. Spuckler, and J. R. Markham, J. Am. Ceram. Soc. 92, 2276 (2009).
    [CrossRef]
  3. W. Ma, H. Dong, H. Guo, S. Gong, and X. Zheng, Surf. Coat. Technol. 204, 3366 (2010).
    [CrossRef]
  4. B. Rousseau, M. Chabin, P. Echegut, A. Sin, F. Weiss, and P. Odier, Appl. Phys. Lett. 79, 3633 (2001).
    [CrossRef]
  5. T. A. Morris, M. A. Marciniak, G. C. Wollenweber, and J. A. Turk, Infrared Phys. Technol. 48, 130 (2006).
    [CrossRef]
  6. R. Apetz and M. P. B. Van Bruggen, J. Am. Ceram. Soc. 86, 480 (2003).
    [CrossRef]
  7. I. Yamashita, H. Nagayama, and K. Tsukuma, J. Am. Ceram. Soc. 91, 2611 (2008).
    [CrossRef]
  8. B.-N. Kim, K. Hiraga, K. Morita, H. Yoshida, T. Miyazaki, and Y. Kagawa, Acta Mater. 57, 1319 (2009).
    [CrossRef]
  9. O. Rozenbaum, D. De Sousa Meneses, Y. Auger, S. Chermanne, and P. Echegut, Rev. Sci. Instrum. 70, 4020 (1999).
    [CrossRef]
  10. R. M. Sova, M. J. Linevsky, M. E. Thomas, and F. F. Mark, Infrared Phys. Technol. 39, 251 (1998).
    [CrossRef]
  11. B. Rousseau, J. F. Brun, D. De Sousa Menese, and P. Echegut, Int. J. Thermophys. 26, 1277 (2005).
    [CrossRef]
  12. J. Ishii and A. Ono, Meas. Sci. Technol. 12, 2103 (2001).
    [CrossRef]
  13. L. del Campo, R. B. Perez-Saez, X. Esquisabel, I. Fernandez, and M. J. Tello, Rev. Sci. Instrum. 77, 113111 (2006).
    [CrossRef]
  14. J. Manara, M. Arduini-Schuster, H.-J. Ratzer-Scheibe, and U. Schulz, Surf. Coat. Technol. 203, 1059 (2009).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  18. G. W. Lee, S. Jeon, N. J. Yoo, C. Park, S. Park, S. Kweon, and S. Lee are preparing a manuscript to be called “Normal and directional spectral emissivity measurement of alumina.”
  19. G. W. Lee, S. Jeon, S-N. Park, Y. S. Yoo, and C. Park are preparing a manuscript to be called “Temperature and thickness dependence of IR optical properties of sapphire at moderate temperature.”

2010 (1)

W. Ma, H. Dong, H. Guo, S. Gong, and X. Zheng, Surf. Coat. Technol. 204, 3366 (2010).
[CrossRef]

2009 (3)

J. I. Eldridge, C. M. Spuckler, and J. R. Markham, J. Am. Ceram. Soc. 92, 2276 (2009).
[CrossRef]

B.-N. Kim, K. Hiraga, K. Morita, H. Yoshida, T. Miyazaki, and Y. Kagawa, Acta Mater. 57, 1319 (2009).
[CrossRef]

J. Manara, M. Arduini-Schuster, H.-J. Ratzer-Scheibe, and U. Schulz, Surf. Coat. Technol. 203, 1059 (2009).
[CrossRef]

2008 (2)

L. del Campo, R. B. Rerez-Saez, and M. J. Tello, Int. J. Thermophys. 29, 93 (2008).
[CrossRef]

I. Yamashita, H. Nagayama, and K. Tsukuma, J. Am. Ceram. Soc. 91, 2611 (2008).
[CrossRef]

2007 (1)

N. M. Kirby, N. W. Chen-Tan, and C. E. Buckley, J. Eur. Ceram. Soc. 27, 2039 (2007).
[CrossRef]

2006 (2)

T. A. Morris, M. A. Marciniak, G. C. Wollenweber, and J. A. Turk, Infrared Phys. Technol. 48, 130 (2006).
[CrossRef]

L. del Campo, R. B. Perez-Saez, X. Esquisabel, I. Fernandez, and M. J. Tello, Rev. Sci. Instrum. 77, 113111 (2006).
[CrossRef]

2005 (1)

B. Rousseau, J. F. Brun, D. De Sousa Menese, and P. Echegut, Int. J. Thermophys. 26, 1277 (2005).
[CrossRef]

2003 (1)

R. Apetz and M. P. B. Van Bruggen, J. Am. Ceram. Soc. 86, 480 (2003).
[CrossRef]

2001 (2)

B. Rousseau, M. Chabin, P. Echegut, A. Sin, F. Weiss, and P. Odier, Appl. Phys. Lett. 79, 3633 (2001).
[CrossRef]

J. Ishii and A. Ono, Meas. Sci. Technol. 12, 2103 (2001).
[CrossRef]

1999 (2)

A. Shimota, H. Kobayashi, and S. Kadokura, Appl. Opt. 38, 571 (1999).
[CrossRef]

O. Rozenbaum, D. De Sousa Meneses, Y. Auger, S. Chermanne, and P. Echegut, Rev. Sci. Instrum. 70, 4020 (1999).
[CrossRef]

1998 (1)

R. M. Sova, M. J. Linevsky, M. E. Thomas, and F. F. Mark, Infrared Phys. Technol. 39, 251 (1998).
[CrossRef]

1988 (1)

Apetz, R.

R. Apetz and M. P. B. Van Bruggen, J. Am. Ceram. Soc. 86, 480 (2003).
[CrossRef]

Arduini-Schuster, M.

J. Manara, M. Arduini-Schuster, H.-J. Ratzer-Scheibe, and U. Schulz, Surf. Coat. Technol. 203, 1059 (2009).
[CrossRef]

Auger, Y.

O. Rozenbaum, D. De Sousa Meneses, Y. Auger, S. Chermanne, and P. Echegut, Rev. Sci. Instrum. 70, 4020 (1999).
[CrossRef]

Brun, J. F.

B. Rousseau, J. F. Brun, D. De Sousa Menese, and P. Echegut, Int. J. Thermophys. 26, 1277 (2005).
[CrossRef]

Buckley, C. E.

N. M. Kirby, N. W. Chen-Tan, and C. E. Buckley, J. Eur. Ceram. Soc. 27, 2039 (2007).
[CrossRef]

Buijs, H.

Chabin, M.

B. Rousseau, M. Chabin, P. Echegut, A. Sin, F. Weiss, and P. Odier, Appl. Phys. Lett. 79, 3633 (2001).
[CrossRef]

Chen-Tan, N. W.

N. M. Kirby, N. W. Chen-Tan, and C. E. Buckley, J. Eur. Ceram. Soc. 27, 2039 (2007).
[CrossRef]

Chermanne, S.

O. Rozenbaum, D. De Sousa Meneses, Y. Auger, S. Chermanne, and P. Echegut, Rev. Sci. Instrum. 70, 4020 (1999).
[CrossRef]

De Sousa Menese, D.

B. Rousseau, J. F. Brun, D. De Sousa Menese, and P. Echegut, Int. J. Thermophys. 26, 1277 (2005).
[CrossRef]

De Sousa Meneses, D.

O. Rozenbaum, D. De Sousa Meneses, Y. Auger, S. Chermanne, and P. Echegut, Rev. Sci. Instrum. 70, 4020 (1999).
[CrossRef]

del Campo, L.

L. del Campo, R. B. Rerez-Saez, and M. J. Tello, Int. J. Thermophys. 29, 93 (2008).
[CrossRef]

L. del Campo, R. B. Perez-Saez, X. Esquisabel, I. Fernandez, and M. J. Tello, Rev. Sci. Instrum. 77, 113111 (2006).
[CrossRef]

Dong, H.

W. Ma, H. Dong, H. Guo, S. Gong, and X. Zheng, Surf. Coat. Technol. 204, 3366 (2010).
[CrossRef]

Echegut, P.

B. Rousseau, J. F. Brun, D. De Sousa Menese, and P. Echegut, Int. J. Thermophys. 26, 1277 (2005).
[CrossRef]

B. Rousseau, M. Chabin, P. Echegut, A. Sin, F. Weiss, and P. Odier, Appl. Phys. Lett. 79, 3633 (2001).
[CrossRef]

O. Rozenbaum, D. De Sousa Meneses, Y. Auger, S. Chermanne, and P. Echegut, Rev. Sci. Instrum. 70, 4020 (1999).
[CrossRef]

Eldridge, J. I.

J. I. Eldridge, C. M. Spuckler, and J. R. Markham, J. Am. Ceram. Soc. 92, 2276 (2009).
[CrossRef]

Esquisabel, X.

L. del Campo, R. B. Perez-Saez, X. Esquisabel, I. Fernandez, and M. J. Tello, Rev. Sci. Instrum. 77, 113111 (2006).
[CrossRef]

Fernandez, I.

L. del Campo, R. B. Perez-Saez, X. Esquisabel, I. Fernandez, and M. J. Tello, Rev. Sci. Instrum. 77, 113111 (2006).
[CrossRef]

Gong, S.

W. Ma, H. Dong, H. Guo, S. Gong, and X. Zheng, Surf. Coat. Technol. 204, 3366 (2010).
[CrossRef]

Guo, H.

W. Ma, H. Dong, H. Guo, S. Gong, and X. Zheng, Surf. Coat. Technol. 204, 3366 (2010).
[CrossRef]

Hiraga, K.

B.-N. Kim, K. Hiraga, K. Morita, H. Yoshida, T. Miyazaki, and Y. Kagawa, Acta Mater. 57, 1319 (2009).
[CrossRef]

Howell, H. B.

Ishii, J.

J. Ishii and A. Ono, Meas. Sci. Technol. 12, 2103 (2001).
[CrossRef]

Jeon, S.

G. W. Lee, S. Jeon, N. J. Yoo, C. Park, S. Park, S. Kweon, and S. Lee are preparing a manuscript to be called “Normal and directional spectral emissivity measurement of alumina.”

G. W. Lee, S. Jeon, S-N. Park, Y. S. Yoo, and C. Park are preparing a manuscript to be called “Temperature and thickness dependence of IR optical properties of sapphire at moderate temperature.”

Kadokura, S.

Kagawa, Y.

B.-N. Kim, K. Hiraga, K. Morita, H. Yoshida, T. Miyazaki, and Y. Kagawa, Acta Mater. 57, 1319 (2009).
[CrossRef]

Kim, B.-N.

B.-N. Kim, K. Hiraga, K. Morita, H. Yoshida, T. Miyazaki, and Y. Kagawa, Acta Mater. 57, 1319 (2009).
[CrossRef]

Kirby, N. M.

N. M. Kirby, N. W. Chen-Tan, and C. E. Buckley, J. Eur. Ceram. Soc. 27, 2039 (2007).
[CrossRef]

Kobayashi, H.

Kweon, S.

G. W. Lee, S. Jeon, N. J. Yoo, C. Park, S. Park, S. Kweon, and S. Lee are preparing a manuscript to be called “Normal and directional spectral emissivity measurement of alumina.”

LaPorte, D. D.

Lee, G. W.

G. W. Lee, S. Jeon, N. J. Yoo, C. Park, S. Park, S. Kweon, and S. Lee are preparing a manuscript to be called “Normal and directional spectral emissivity measurement of alumina.”

G. W. Lee, S. Jeon, S-N. Park, Y. S. Yoo, and C. Park are preparing a manuscript to be called “Temperature and thickness dependence of IR optical properties of sapphire at moderate temperature.”

Lee, S.

G. W. Lee, S. Jeon, N. J. Yoo, C. Park, S. Park, S. Kweon, and S. Lee are preparing a manuscript to be called “Normal and directional spectral emissivity measurement of alumina.”

Linevsky, M. J.

R. M. Sova, M. J. Linevsky, M. E. Thomas, and F. F. Mark, Infrared Phys. Technol. 39, 251 (1998).
[CrossRef]

Ma, W.

W. Ma, H. Dong, H. Guo, S. Gong, and X. Zheng, Surf. Coat. Technol. 204, 3366 (2010).
[CrossRef]

Manara, J.

J. Manara, M. Arduini-Schuster, H.-J. Ratzer-Scheibe, and U. Schulz, Surf. Coat. Technol. 203, 1059 (2009).
[CrossRef]

Marciniak, M. A.

T. A. Morris, M. A. Marciniak, G. C. Wollenweber, and J. A. Turk, Infrared Phys. Technol. 48, 130 (2006).
[CrossRef]

Mark, F. F.

R. M. Sova, M. J. Linevsky, M. E. Thomas, and F. F. Mark, Infrared Phys. Technol. 39, 251 (1998).
[CrossRef]

Markham, J. R.

J. I. Eldridge, C. M. Spuckler, and J. R. Markham, J. Am. Ceram. Soc. 92, 2276 (2009).
[CrossRef]

Miyazaki, T.

B.-N. Kim, K. Hiraga, K. Morita, H. Yoshida, T. Miyazaki, and Y. Kagawa, Acta Mater. 57, 1319 (2009).
[CrossRef]

Morita, K.

B.-N. Kim, K. Hiraga, K. Morita, H. Yoshida, T. Miyazaki, and Y. Kagawa, Acta Mater. 57, 1319 (2009).
[CrossRef]

Morris, T. A.

T. A. Morris, M. A. Marciniak, G. C. Wollenweber, and J. A. Turk, Infrared Phys. Technol. 48, 130 (2006).
[CrossRef]

Nagayama, H.

I. Yamashita, H. Nagayama, and K. Tsukuma, J. Am. Ceram. Soc. 91, 2611 (2008).
[CrossRef]

Odier, P.

B. Rousseau, M. Chabin, P. Echegut, A. Sin, F. Weiss, and P. Odier, Appl. Phys. Lett. 79, 3633 (2001).
[CrossRef]

Ono, A.

J. Ishii and A. Ono, Meas. Sci. Technol. 12, 2103 (2001).
[CrossRef]

Park, C.

G. W. Lee, S. Jeon, N. J. Yoo, C. Park, S. Park, S. Kweon, and S. Lee are preparing a manuscript to be called “Normal and directional spectral emissivity measurement of alumina.”

G. W. Lee, S. Jeon, S-N. Park, Y. S. Yoo, and C. Park are preparing a manuscript to be called “Temperature and thickness dependence of IR optical properties of sapphire at moderate temperature.”

Park, S.

G. W. Lee, S. Jeon, N. J. Yoo, C. Park, S. Park, S. Kweon, and S. Lee are preparing a manuscript to be called “Normal and directional spectral emissivity measurement of alumina.”

Park, S-N.

G. W. Lee, S. Jeon, S-N. Park, Y. S. Yoo, and C. Park are preparing a manuscript to be called “Temperature and thickness dependence of IR optical properties of sapphire at moderate temperature.”

Perez-Saez, R. B.

L. del Campo, R. B. Perez-Saez, X. Esquisabel, I. Fernandez, and M. J. Tello, Rev. Sci. Instrum. 77, 113111 (2006).
[CrossRef]

Ratzer-Scheibe, H.-J.

J. Manara, M. Arduini-Schuster, H.-J. Ratzer-Scheibe, and U. Schulz, Surf. Coat. Technol. 203, 1059 (2009).
[CrossRef]

Rerez-Saez, R. B.

L. del Campo, R. B. Rerez-Saez, and M. J. Tello, Int. J. Thermophys. 29, 93 (2008).
[CrossRef]

Revercomb, H. E.

Rousseau, B.

B. Rousseau, J. F. Brun, D. De Sousa Menese, and P. Echegut, Int. J. Thermophys. 26, 1277 (2005).
[CrossRef]

B. Rousseau, M. Chabin, P. Echegut, A. Sin, F. Weiss, and P. Odier, Appl. Phys. Lett. 79, 3633 (2001).
[CrossRef]

Rozenbaum, O.

O. Rozenbaum, D. De Sousa Meneses, Y. Auger, S. Chermanne, and P. Echegut, Rev. Sci. Instrum. 70, 4020 (1999).
[CrossRef]

Schulz, U.

J. Manara, M. Arduini-Schuster, H.-J. Ratzer-Scheibe, and U. Schulz, Surf. Coat. Technol. 203, 1059 (2009).
[CrossRef]

Shimota, A.

Sin, A.

B. Rousseau, M. Chabin, P. Echegut, A. Sin, F. Weiss, and P. Odier, Appl. Phys. Lett. 79, 3633 (2001).
[CrossRef]

Smith, W. L.

Sova, R. M.

R. M. Sova, M. J. Linevsky, M. E. Thomas, and F. F. Mark, Infrared Phys. Technol. 39, 251 (1998).
[CrossRef]

Spuckler, C. M.

J. I. Eldridge, C. M. Spuckler, and J. R. Markham, J. Am. Ceram. Soc. 92, 2276 (2009).
[CrossRef]

Sromovsky, L. A.

Tello, M. J.

L. del Campo, R. B. Rerez-Saez, and M. J. Tello, Int. J. Thermophys. 29, 93 (2008).
[CrossRef]

L. del Campo, R. B. Perez-Saez, X. Esquisabel, I. Fernandez, and M. J. Tello, Rev. Sci. Instrum. 77, 113111 (2006).
[CrossRef]

Thomas, M. E.

R. M. Sova, M. J. Linevsky, M. E. Thomas, and F. F. Mark, Infrared Phys. Technol. 39, 251 (1998).
[CrossRef]

Tsukuma, K.

I. Yamashita, H. Nagayama, and K. Tsukuma, J. Am. Ceram. Soc. 91, 2611 (2008).
[CrossRef]

Turk, J. A.

T. A. Morris, M. A. Marciniak, G. C. Wollenweber, and J. A. Turk, Infrared Phys. Technol. 48, 130 (2006).
[CrossRef]

Van Bruggen, M. P. B.

R. Apetz and M. P. B. Van Bruggen, J. Am. Ceram. Soc. 86, 480 (2003).
[CrossRef]

Weiss, F.

B. Rousseau, M. Chabin, P. Echegut, A. Sin, F. Weiss, and P. Odier, Appl. Phys. Lett. 79, 3633 (2001).
[CrossRef]

Wollenweber, G. C.

T. A. Morris, M. A. Marciniak, G. C. Wollenweber, and J. A. Turk, Infrared Phys. Technol. 48, 130 (2006).
[CrossRef]

Yamashita, I.

I. Yamashita, H. Nagayama, and K. Tsukuma, J. Am. Ceram. Soc. 91, 2611 (2008).
[CrossRef]

Yoo, N. J.

G. W. Lee, S. Jeon, N. J. Yoo, C. Park, S. Park, S. Kweon, and S. Lee are preparing a manuscript to be called “Normal and directional spectral emissivity measurement of alumina.”

Yoo, Y. S.

G. W. Lee, S. Jeon, S-N. Park, Y. S. Yoo, and C. Park are preparing a manuscript to be called “Temperature and thickness dependence of IR optical properties of sapphire at moderate temperature.”

Yoshida, H.

B.-N. Kim, K. Hiraga, K. Morita, H. Yoshida, T. Miyazaki, and Y. Kagawa, Acta Mater. 57, 1319 (2009).
[CrossRef]

Zheng, X.

W. Ma, H. Dong, H. Guo, S. Gong, and X. Zheng, Surf. Coat. Technol. 204, 3366 (2010).
[CrossRef]

Acta Mater. (1)

B.-N. Kim, K. Hiraga, K. Morita, H. Yoshida, T. Miyazaki, and Y. Kagawa, Acta Mater. 57, 1319 (2009).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

B. Rousseau, M. Chabin, P. Echegut, A. Sin, F. Weiss, and P. Odier, Appl. Phys. Lett. 79, 3633 (2001).
[CrossRef]

Infrared Phys. Technol. (2)

T. A. Morris, M. A. Marciniak, G. C. Wollenweber, and J. A. Turk, Infrared Phys. Technol. 48, 130 (2006).
[CrossRef]

R. M. Sova, M. J. Linevsky, M. E. Thomas, and F. F. Mark, Infrared Phys. Technol. 39, 251 (1998).
[CrossRef]

Int. J. Thermophys. (2)

B. Rousseau, J. F. Brun, D. De Sousa Menese, and P. Echegut, Int. J. Thermophys. 26, 1277 (2005).
[CrossRef]

L. del Campo, R. B. Rerez-Saez, and M. J. Tello, Int. J. Thermophys. 29, 93 (2008).
[CrossRef]

J. Am. Ceram. Soc. (3)

R. Apetz and M. P. B. Van Bruggen, J. Am. Ceram. Soc. 86, 480 (2003).
[CrossRef]

I. Yamashita, H. Nagayama, and K. Tsukuma, J. Am. Ceram. Soc. 91, 2611 (2008).
[CrossRef]

J. I. Eldridge, C. M. Spuckler, and J. R. Markham, J. Am. Ceram. Soc. 92, 2276 (2009).
[CrossRef]

J. Eur. Ceram. Soc. (1)

N. M. Kirby, N. W. Chen-Tan, and C. E. Buckley, J. Eur. Ceram. Soc. 27, 2039 (2007).
[CrossRef]

Meas. Sci. Technol. (1)

J. Ishii and A. Ono, Meas. Sci. Technol. 12, 2103 (2001).
[CrossRef]

Rev. Sci. Instrum. (2)

L. del Campo, R. B. Perez-Saez, X. Esquisabel, I. Fernandez, and M. J. Tello, Rev. Sci. Instrum. 77, 113111 (2006).
[CrossRef]

O. Rozenbaum, D. De Sousa Meneses, Y. Auger, S. Chermanne, and P. Echegut, Rev. Sci. Instrum. 70, 4020 (1999).
[CrossRef]

Surf. Coat. Technol. (2)

W. Ma, H. Dong, H. Guo, S. Gong, and X. Zheng, Surf. Coat. Technol. 204, 3366 (2010).
[CrossRef]

J. Manara, M. Arduini-Schuster, H.-J. Ratzer-Scheibe, and U. Schulz, Surf. Coat. Technol. 203, 1059 (2009).
[CrossRef]

Other (2)

G. W. Lee, S. Jeon, N. J. Yoo, C. Park, S. Park, S. Kweon, and S. Lee are preparing a manuscript to be called “Normal and directional spectral emissivity measurement of alumina.”

G. W. Lee, S. Jeon, S-N. Park, Y. S. Yoo, and C. Park are preparing a manuscript to be called “Temperature and thickness dependence of IR optical properties of sapphire at moderate temperature.”

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Figures (3)

Fig. 1
Fig. 1

Radiation sources entering a detector when a sample is heated; measured signals include a heated sample signal ( S s ), a substrate signal ( T s S sub ) passing through the sample, and a sample signal ( S s R sub T s ) passing through itself after being reflected by the substrate.

Fig. 2
Fig. 2

Apparent emissivities of sapphire with (a) black paint and (b) gold substrates at 300 ° C . The emissivities of substrates at 300 ° C is shown in the inset.

Fig. 3
Fig. 3

Emissivity, transmittance, and reflectance of sapphire with different thicknesses at 300 ° C measured by the two-substrates method (curves). The symbols represent the emissivity (solid), transmittance (half solid), and reflectance (hollow) measured by the conventional method.

Equations (5)

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

S * = S s + T s S sub + S s R sub T s = R f [ ( ε s L BB + r s L sur ) + T s ( ε sub L BB sub + r sub L sur ) + R sub T s ( ε s L BB + r s L sur ) + L o ] R f ( ε * L BB + r * L sur + L o ) ,
ε * = ε s + T s ε sub + ε s R sub T s .
ε 2 * = ε s + T s ε sub 2 + ε s R sub 2 T s , ε 1 * = ε s + T s ε sub 1 + ε s R sub 1 T s ,
2 T s 2 + T s [ 2 C D B ( 2 A ) ] 2 D B = 0 , ( A = ε sub 2 + ε sub 1 , B = ε sub 2 ε sub 1 , C = ε 2 * + ε 1 * , D = ε 2 * ε 1 * , X = 2 C D B ( 2 A ) ) .
T s = X + X 2 + 16 D B 4 , ε s = 1 D T s B , r s = 1 T s ε s .

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