Abstract

Silicon nitride thin films play an important role in the realization of sensors, filters, and high-performance circuits. Estimates of the dielectric function in the far- and mid-IR regime are derived from the observed transmittance spectra for a commonly employed low-stress silicon nitride formulation. The experimental, modeling, and numerical methods used to extract the dielectric parameters with an accuracy of approximately 4% are presented.

© 2012 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. J. Goldie, A. V. Velichko, D. M. Glowacka, and S. Withington, Appl. Phys. 109, 084507 (2011).
    [CrossRef]
  2. G. Wang, V. Yefremenko, V. Novosad, A. Datesman, J. Pearson, R. Divan, C. L. Chang, L. Bleem, A. T. Crites, J. Mehl, T. Natoli, J. McMahon, J. Sayre, J. Ruhl, S. S. Meyer, and J. E. Carlstrom, IEEE Trans. Appl. Supercond. 21, 232 (2011).
    [CrossRef]
  3. J. M. Martinis, K. B. Cooper, R. McDermott, M. Steffen, M. Ansmann, K. D. Osborn, K. Cicak, S. Oh, D. P. Pappas, R. W. Simmonds, and C. C. Yu, Phys. Rev. Lett. 95, 210503 (2005).
    [CrossRef]
  4. H. Paik and K. D. Osborn, Appl. Phys. Lett. 96, 072505 (2010).
    [CrossRef]
  5. T. Eriksson, S. Jiang, and C. Granqvist, Appl. Opt. 24, 745 (1985).
    [CrossRef]
  6. E. A. Taft, J. Electrochem. Soc. 118, 1341 (1971).
    [CrossRef]
  7. E. D. Palik, Handbook of Optical Constants of Solids(Elsevier, 1998), Vol. 1, pp. 771–774.
  8. M. Sekimoto, H. Yoshihara, and T. Ohkubo, J. Vac. Sci. Technol. 21, 1017 (1982).
    [CrossRef]
  9. T. Makino, J. Electrochem. Soc. 130, 450 (1983).
    [CrossRef]
  10. Addison Engineering, 150 Nortech Parkway, San Jose, California 95134 (Orientation 〈100〉, Czochralski, p-type B doped, bulk resistivity <0.005  Ω cm).
  11. F. Gervais, in Infrared and Millimeter Waves, K. J. Button, ed. (Academic, 1983), Vol. 8, Part I, pp. 284–287.
  12. C. C. Kim, J. W. Garland, H. Abad, and P. M. Raccah, Phys. Rev. B 45, 11749 (1992).
    [CrossRef]
  13. P. Yeh, Optical Waves in Layered Media (Wiley, 1988), pp. 102–111.
  14. M. C. Biggs, in Towards Global Optimization, L. C. W. Dixon and G. P. Szergo, eds. (North-Holland, 1975), pp. 341–349.
  15. M. J. D. Powell, in Mathematical Programming: The State of the Art, A. Bachem, M. Grotschel, and B. Korte, eds. (Springer Verlag, 1983), pp. 288–311.
  16. L. D. Landau and E. M. Lifshitz, Electrodyamics of Continuous Media (Pergamon, 1960), Vol. 8, pp. 253–262.
  17. M. Mori and T. Ooura, in Applicable Analysis (World Scientific Series, 1993), Vol. 2, pp. 301–308.
  18. R. Nitsche and T. Fritz, Phys. Rev. B 70, 195432 (2004).
    [CrossRef]
  19. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes—The Art of Scientific Computing (Cambridge Univ., 2007), pp. 799–806.

2011 (2)

D. J. Goldie, A. V. Velichko, D. M. Glowacka, and S. Withington, Appl. Phys. 109, 084507 (2011).
[CrossRef]

G. Wang, V. Yefremenko, V. Novosad, A. Datesman, J. Pearson, R. Divan, C. L. Chang, L. Bleem, A. T. Crites, J. Mehl, T. Natoli, J. McMahon, J. Sayre, J. Ruhl, S. S. Meyer, and J. E. Carlstrom, IEEE Trans. Appl. Supercond. 21, 232 (2011).
[CrossRef]

2010 (1)

H. Paik and K. D. Osborn, Appl. Phys. Lett. 96, 072505 (2010).
[CrossRef]

2005 (1)

J. M. Martinis, K. B. Cooper, R. McDermott, M. Steffen, M. Ansmann, K. D. Osborn, K. Cicak, S. Oh, D. P. Pappas, R. W. Simmonds, and C. C. Yu, Phys. Rev. Lett. 95, 210503 (2005).
[CrossRef]

2004 (1)

R. Nitsche and T. Fritz, Phys. Rev. B 70, 195432 (2004).
[CrossRef]

1992 (1)

C. C. Kim, J. W. Garland, H. Abad, and P. M. Raccah, Phys. Rev. B 45, 11749 (1992).
[CrossRef]

1985 (1)

1983 (1)

T. Makino, J. Electrochem. Soc. 130, 450 (1983).
[CrossRef]

1982 (1)

M. Sekimoto, H. Yoshihara, and T. Ohkubo, J. Vac. Sci. Technol. 21, 1017 (1982).
[CrossRef]

1971 (1)

E. A. Taft, J. Electrochem. Soc. 118, 1341 (1971).
[CrossRef]

Abad, H.

C. C. Kim, J. W. Garland, H. Abad, and P. M. Raccah, Phys. Rev. B 45, 11749 (1992).
[CrossRef]

Ansmann, M.

J. M. Martinis, K. B. Cooper, R. McDermott, M. Steffen, M. Ansmann, K. D. Osborn, K. Cicak, S. Oh, D. P. Pappas, R. W. Simmonds, and C. C. Yu, Phys. Rev. Lett. 95, 210503 (2005).
[CrossRef]

Biggs, M. C.

M. C. Biggs, in Towards Global Optimization, L. C. W. Dixon and G. P. Szergo, eds. (North-Holland, 1975), pp. 341–349.

Bleem, L.

G. Wang, V. Yefremenko, V. Novosad, A. Datesman, J. Pearson, R. Divan, C. L. Chang, L. Bleem, A. T. Crites, J. Mehl, T. Natoli, J. McMahon, J. Sayre, J. Ruhl, S. S. Meyer, and J. E. Carlstrom, IEEE Trans. Appl. Supercond. 21, 232 (2011).
[CrossRef]

Carlstrom, J. E.

G. Wang, V. Yefremenko, V. Novosad, A. Datesman, J. Pearson, R. Divan, C. L. Chang, L. Bleem, A. T. Crites, J. Mehl, T. Natoli, J. McMahon, J. Sayre, J. Ruhl, S. S. Meyer, and J. E. Carlstrom, IEEE Trans. Appl. Supercond. 21, 232 (2011).
[CrossRef]

Chang, C. L.

G. Wang, V. Yefremenko, V. Novosad, A. Datesman, J. Pearson, R. Divan, C. L. Chang, L. Bleem, A. T. Crites, J. Mehl, T. Natoli, J. McMahon, J. Sayre, J. Ruhl, S. S. Meyer, and J. E. Carlstrom, IEEE Trans. Appl. Supercond. 21, 232 (2011).
[CrossRef]

Cicak, K.

J. M. Martinis, K. B. Cooper, R. McDermott, M. Steffen, M. Ansmann, K. D. Osborn, K. Cicak, S. Oh, D. P. Pappas, R. W. Simmonds, and C. C. Yu, Phys. Rev. Lett. 95, 210503 (2005).
[CrossRef]

Cooper, K. B.

J. M. Martinis, K. B. Cooper, R. McDermott, M. Steffen, M. Ansmann, K. D. Osborn, K. Cicak, S. Oh, D. P. Pappas, R. W. Simmonds, and C. C. Yu, Phys. Rev. Lett. 95, 210503 (2005).
[CrossRef]

Crites, A. T.

G. Wang, V. Yefremenko, V. Novosad, A. Datesman, J. Pearson, R. Divan, C. L. Chang, L. Bleem, A. T. Crites, J. Mehl, T. Natoli, J. McMahon, J. Sayre, J. Ruhl, S. S. Meyer, and J. E. Carlstrom, IEEE Trans. Appl. Supercond. 21, 232 (2011).
[CrossRef]

Datesman, A.

G. Wang, V. Yefremenko, V. Novosad, A. Datesman, J. Pearson, R. Divan, C. L. Chang, L. Bleem, A. T. Crites, J. Mehl, T. Natoli, J. McMahon, J. Sayre, J. Ruhl, S. S. Meyer, and J. E. Carlstrom, IEEE Trans. Appl. Supercond. 21, 232 (2011).
[CrossRef]

Divan, R.

G. Wang, V. Yefremenko, V. Novosad, A. Datesman, J. Pearson, R. Divan, C. L. Chang, L. Bleem, A. T. Crites, J. Mehl, T. Natoli, J. McMahon, J. Sayre, J. Ruhl, S. S. Meyer, and J. E. Carlstrom, IEEE Trans. Appl. Supercond. 21, 232 (2011).
[CrossRef]

Eriksson, T.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes—The Art of Scientific Computing (Cambridge Univ., 2007), pp. 799–806.

Fritz, T.

R. Nitsche and T. Fritz, Phys. Rev. B 70, 195432 (2004).
[CrossRef]

Garland, J. W.

C. C. Kim, J. W. Garland, H. Abad, and P. M. Raccah, Phys. Rev. B 45, 11749 (1992).
[CrossRef]

Gervais, F.

F. Gervais, in Infrared and Millimeter Waves, K. J. Button, ed. (Academic, 1983), Vol. 8, Part I, pp. 284–287.

Glowacka, D. M.

D. J. Goldie, A. V. Velichko, D. M. Glowacka, and S. Withington, Appl. Phys. 109, 084507 (2011).
[CrossRef]

Goldie, D. J.

D. J. Goldie, A. V. Velichko, D. M. Glowacka, and S. Withington, Appl. Phys. 109, 084507 (2011).
[CrossRef]

Granqvist, C.

Jiang, S.

Kim, C. C.

C. C. Kim, J. W. Garland, H. Abad, and P. M. Raccah, Phys. Rev. B 45, 11749 (1992).
[CrossRef]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodyamics of Continuous Media (Pergamon, 1960), Vol. 8, pp. 253–262.

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodyamics of Continuous Media (Pergamon, 1960), Vol. 8, pp. 253–262.

Makino, T.

T. Makino, J. Electrochem. Soc. 130, 450 (1983).
[CrossRef]

Martinis, J. M.

J. M. Martinis, K. B. Cooper, R. McDermott, M. Steffen, M. Ansmann, K. D. Osborn, K. Cicak, S. Oh, D. P. Pappas, R. W. Simmonds, and C. C. Yu, Phys. Rev. Lett. 95, 210503 (2005).
[CrossRef]

McDermott, R.

J. M. Martinis, K. B. Cooper, R. McDermott, M. Steffen, M. Ansmann, K. D. Osborn, K. Cicak, S. Oh, D. P. Pappas, R. W. Simmonds, and C. C. Yu, Phys. Rev. Lett. 95, 210503 (2005).
[CrossRef]

McMahon, J.

G. Wang, V. Yefremenko, V. Novosad, A. Datesman, J. Pearson, R. Divan, C. L. Chang, L. Bleem, A. T. Crites, J. Mehl, T. Natoli, J. McMahon, J. Sayre, J. Ruhl, S. S. Meyer, and J. E. Carlstrom, IEEE Trans. Appl. Supercond. 21, 232 (2011).
[CrossRef]

Mehl, J.

G. Wang, V. Yefremenko, V. Novosad, A. Datesman, J. Pearson, R. Divan, C. L. Chang, L. Bleem, A. T. Crites, J. Mehl, T. Natoli, J. McMahon, J. Sayre, J. Ruhl, S. S. Meyer, and J. E. Carlstrom, IEEE Trans. Appl. Supercond. 21, 232 (2011).
[CrossRef]

Meyer, S. S.

G. Wang, V. Yefremenko, V. Novosad, A. Datesman, J. Pearson, R. Divan, C. L. Chang, L. Bleem, A. T. Crites, J. Mehl, T. Natoli, J. McMahon, J. Sayre, J. Ruhl, S. S. Meyer, and J. E. Carlstrom, IEEE Trans. Appl. Supercond. 21, 232 (2011).
[CrossRef]

Mori, M.

M. Mori and T. Ooura, in Applicable Analysis (World Scientific Series, 1993), Vol. 2, pp. 301–308.

Natoli, T.

G. Wang, V. Yefremenko, V. Novosad, A. Datesman, J. Pearson, R. Divan, C. L. Chang, L. Bleem, A. T. Crites, J. Mehl, T. Natoli, J. McMahon, J. Sayre, J. Ruhl, S. S. Meyer, and J. E. Carlstrom, IEEE Trans. Appl. Supercond. 21, 232 (2011).
[CrossRef]

Nitsche, R.

R. Nitsche and T. Fritz, Phys. Rev. B 70, 195432 (2004).
[CrossRef]

Novosad, V.

G. Wang, V. Yefremenko, V. Novosad, A. Datesman, J. Pearson, R. Divan, C. L. Chang, L. Bleem, A. T. Crites, J. Mehl, T. Natoli, J. McMahon, J. Sayre, J. Ruhl, S. S. Meyer, and J. E. Carlstrom, IEEE Trans. Appl. Supercond. 21, 232 (2011).
[CrossRef]

Oh, S.

J. M. Martinis, K. B. Cooper, R. McDermott, M. Steffen, M. Ansmann, K. D. Osborn, K. Cicak, S. Oh, D. P. Pappas, R. W. Simmonds, and C. C. Yu, Phys. Rev. Lett. 95, 210503 (2005).
[CrossRef]

Ohkubo, T.

M. Sekimoto, H. Yoshihara, and T. Ohkubo, J. Vac. Sci. Technol. 21, 1017 (1982).
[CrossRef]

Ooura, T.

M. Mori and T. Ooura, in Applicable Analysis (World Scientific Series, 1993), Vol. 2, pp. 301–308.

Osborn, K. D.

H. Paik and K. D. Osborn, Appl. Phys. Lett. 96, 072505 (2010).
[CrossRef]

J. M. Martinis, K. B. Cooper, R. McDermott, M. Steffen, M. Ansmann, K. D. Osborn, K. Cicak, S. Oh, D. P. Pappas, R. W. Simmonds, and C. C. Yu, Phys. Rev. Lett. 95, 210503 (2005).
[CrossRef]

Paik, H.

H. Paik and K. D. Osborn, Appl. Phys. Lett. 96, 072505 (2010).
[CrossRef]

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids(Elsevier, 1998), Vol. 1, pp. 771–774.

Pappas, D. P.

J. M. Martinis, K. B. Cooper, R. McDermott, M. Steffen, M. Ansmann, K. D. Osborn, K. Cicak, S. Oh, D. P. Pappas, R. W. Simmonds, and C. C. Yu, Phys. Rev. Lett. 95, 210503 (2005).
[CrossRef]

Pearson, J.

G. Wang, V. Yefremenko, V. Novosad, A. Datesman, J. Pearson, R. Divan, C. L. Chang, L. Bleem, A. T. Crites, J. Mehl, T. Natoli, J. McMahon, J. Sayre, J. Ruhl, S. S. Meyer, and J. E. Carlstrom, IEEE Trans. Appl. Supercond. 21, 232 (2011).
[CrossRef]

Powell, M. J. D.

M. J. D. Powell, in Mathematical Programming: The State of the Art, A. Bachem, M. Grotschel, and B. Korte, eds. (Springer Verlag, 1983), pp. 288–311.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes—The Art of Scientific Computing (Cambridge Univ., 2007), pp. 799–806.

Raccah, P. M.

C. C. Kim, J. W. Garland, H. Abad, and P. M. Raccah, Phys. Rev. B 45, 11749 (1992).
[CrossRef]

Ruhl, J.

G. Wang, V. Yefremenko, V. Novosad, A. Datesman, J. Pearson, R. Divan, C. L. Chang, L. Bleem, A. T. Crites, J. Mehl, T. Natoli, J. McMahon, J. Sayre, J. Ruhl, S. S. Meyer, and J. E. Carlstrom, IEEE Trans. Appl. Supercond. 21, 232 (2011).
[CrossRef]

Sayre, J.

G. Wang, V. Yefremenko, V. Novosad, A. Datesman, J. Pearson, R. Divan, C. L. Chang, L. Bleem, A. T. Crites, J. Mehl, T. Natoli, J. McMahon, J. Sayre, J. Ruhl, S. S. Meyer, and J. E. Carlstrom, IEEE Trans. Appl. Supercond. 21, 232 (2011).
[CrossRef]

Sekimoto, M.

M. Sekimoto, H. Yoshihara, and T. Ohkubo, J. Vac. Sci. Technol. 21, 1017 (1982).
[CrossRef]

Simmonds, R. W.

J. M. Martinis, K. B. Cooper, R. McDermott, M. Steffen, M. Ansmann, K. D. Osborn, K. Cicak, S. Oh, D. P. Pappas, R. W. Simmonds, and C. C. Yu, Phys. Rev. Lett. 95, 210503 (2005).
[CrossRef]

Steffen, M.

J. M. Martinis, K. B. Cooper, R. McDermott, M. Steffen, M. Ansmann, K. D. Osborn, K. Cicak, S. Oh, D. P. Pappas, R. W. Simmonds, and C. C. Yu, Phys. Rev. Lett. 95, 210503 (2005).
[CrossRef]

Taft, E. A.

E. A. Taft, J. Electrochem. Soc. 118, 1341 (1971).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes—The Art of Scientific Computing (Cambridge Univ., 2007), pp. 799–806.

Velichko, A. V.

D. J. Goldie, A. V. Velichko, D. M. Glowacka, and S. Withington, Appl. Phys. 109, 084507 (2011).
[CrossRef]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes—The Art of Scientific Computing (Cambridge Univ., 2007), pp. 799–806.

Wang, G.

G. Wang, V. Yefremenko, V. Novosad, A. Datesman, J. Pearson, R. Divan, C. L. Chang, L. Bleem, A. T. Crites, J. Mehl, T. Natoli, J. McMahon, J. Sayre, J. Ruhl, S. S. Meyer, and J. E. Carlstrom, IEEE Trans. Appl. Supercond. 21, 232 (2011).
[CrossRef]

Withington, S.

D. J. Goldie, A. V. Velichko, D. M. Glowacka, and S. Withington, Appl. Phys. 109, 084507 (2011).
[CrossRef]

Yefremenko, V.

G. Wang, V. Yefremenko, V. Novosad, A. Datesman, J. Pearson, R. Divan, C. L. Chang, L. Bleem, A. T. Crites, J. Mehl, T. Natoli, J. McMahon, J. Sayre, J. Ruhl, S. S. Meyer, and J. E. Carlstrom, IEEE Trans. Appl. Supercond. 21, 232 (2011).
[CrossRef]

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, 1988), pp. 102–111.

Yoshihara, H.

M. Sekimoto, H. Yoshihara, and T. Ohkubo, J. Vac. Sci. Technol. 21, 1017 (1982).
[CrossRef]

Yu, C. C.

J. M. Martinis, K. B. Cooper, R. McDermott, M. Steffen, M. Ansmann, K. D. Osborn, K. Cicak, S. Oh, D. P. Pappas, R. W. Simmonds, and C. C. Yu, Phys. Rev. Lett. 95, 210503 (2005).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. (1)

D. J. Goldie, A. V. Velichko, D. M. Glowacka, and S. Withington, Appl. Phys. 109, 084507 (2011).
[CrossRef]

Appl. Phys. Lett. (1)

H. Paik and K. D. Osborn, Appl. Phys. Lett. 96, 072505 (2010).
[CrossRef]

IEEE Trans. Appl. Supercond. (1)

G. Wang, V. Yefremenko, V. Novosad, A. Datesman, J. Pearson, R. Divan, C. L. Chang, L. Bleem, A. T. Crites, J. Mehl, T. Natoli, J. McMahon, J. Sayre, J. Ruhl, S. S. Meyer, and J. E. Carlstrom, IEEE Trans. Appl. Supercond. 21, 232 (2011).
[CrossRef]

J. Electrochem. Soc. (2)

E. A. Taft, J. Electrochem. Soc. 118, 1341 (1971).
[CrossRef]

T. Makino, J. Electrochem. Soc. 130, 450 (1983).
[CrossRef]

J. Vac. Sci. Technol. (1)

M. Sekimoto, H. Yoshihara, and T. Ohkubo, J. Vac. Sci. Technol. 21, 1017 (1982).
[CrossRef]

Phys. Rev. B (2)

C. C. Kim, J. W. Garland, H. Abad, and P. M. Raccah, Phys. Rev. B 45, 11749 (1992).
[CrossRef]

R. Nitsche and T. Fritz, Phys. Rev. B 70, 195432 (2004).
[CrossRef]

Phys. Rev. Lett. (1)

J. M. Martinis, K. B. Cooper, R. McDermott, M. Steffen, M. Ansmann, K. D. Osborn, K. Cicak, S. Oh, D. P. Pappas, R. W. Simmonds, and C. C. Yu, Phys. Rev. Lett. 95, 210503 (2005).
[CrossRef]

Other (9)

E. D. Palik, Handbook of Optical Constants of Solids(Elsevier, 1998), Vol. 1, pp. 771–774.

Addison Engineering, 150 Nortech Parkway, San Jose, California 95134 (Orientation 〈100〉, Czochralski, p-type B doped, bulk resistivity <0.005  Ω cm).

F. Gervais, in Infrared and Millimeter Waves, K. J. Button, ed. (Academic, 1983), Vol. 8, Part I, pp. 284–287.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes—The Art of Scientific Computing (Cambridge Univ., 2007), pp. 799–806.

P. Yeh, Optical Waves in Layered Media (Wiley, 1988), pp. 102–111.

M. C. Biggs, in Towards Global Optimization, L. C. W. Dixon and G. P. Szergo, eds. (North-Holland, 1975), pp. 341–349.

M. J. D. Powell, in Mathematical Programming: The State of the Art, A. Bachem, M. Grotschel, and B. Korte, eds. (Springer Verlag, 1983), pp. 288–311.

L. D. Landau and E. M. Lifshitz, Electrodyamics of Continuous Media (Pergamon, 1960), Vol. 8, pp. 253–262.

M. Mori and T. Ooura, in Applicable Analysis (World Scientific Series, 1993), Vol. 2, pp. 301–308.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1.
Fig. 1.

Room-temperature transmission of a silicon nitride sample 0.5 μm thick: measured (grey), model (black dotted), and residual (red). The shaded band’s width delimits the estimated 3σ measurement uncertainty. A 30 GHz (1cm1) resolution is employed for the measurement. The insert depicts the geometry of the SiNx membrane and micromachined silicon frame.

Fig. 2.
Fig. 2.

Measured (solid grey) and model (black dotted) transmission for a three-layer stack of silicon nitride samples 2.3 μm in thickness with 998 μm intermembrane delays that complements the data shown in Fig. 1. The sample response in the far-IR was acquired with a resolution of 3 GHz (0.1cm1).

Fig. 3.
Fig. 3.

Real and imaginary parts (solid red curves) of the dielectric function as extracted from the data shown in Fig. 1. The line thickness is indicative of the propagated 4% error band. The numerical Hilbert transform of the modeled εr(ω) is indicated in the upper panel (dashed blue line) to facilitate comparison with εr(ω). The filled symbols indicate the parameters derived from the data presented in Fig. 2.

Tables (1)

Tables Icon

Table 1. Fit Parameter Summary

Equations (3)

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

ε^r(ω)=ε^+j=1MΔε^j·ωTj2ωTj2ω2iωΓj(ω),
Γj(ω)=Γjexp[αj(ωTj2ω2ωΓj)2],
minDOFχ2=minDOFk=1N[T(ε^r(ω),h)TFTSk]2,

Metrics