Abstract

The use of the biased estimator in the fitting of spectroscopic ellipsometry data is examined and applied to data from two-channel polarization modulation ellipsometry experiments. It is pointed out that the use of the biased estimator, as opposed to the unbiased estimator that is usually found in the literature, allows the experimentalist to weight properly the more accurate parts of the spectrum, to switch among different representations of the data, and to calculate a goodness of fit. The fit to data taken on a 59-nm SiO2 film on Si is examined with both the biased and the unbiased estimators.

© 1991 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. E. Jellison, F. A. Modine, “Optical absorption of silicon between 1.6 and 4.7 eV at elevated temperatures,” Appl. Phys. Lett. 41, 180–182 (1982); “Optical functions of silicon between 1.7 and 4.7 eV at elevated temperatures,” Phys. Rev. B 27, 7466–7472 (1983).
    [CrossRef]
  2. D. E. Aspnes, A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
    [CrossRef]
  3. D. E. Aspnes, J. B. Theeten “Optical properties of the interface between si and its thermally grown oxide,” Phys. Rev. Lett. 43, 1046–1050 (1979); “Spectroscopic analysis of the interface between Si and its thermally grown oxide,” J. Electrochem. Soc. 127, 1359–1365 (1980).
    [CrossRef]
  4. P. J. McMarr, K. Vedam, J. Narayan, “Spectroscopic ellipsometry: a new tool for nondestructive depth profiling and characterization of interfaces,” J. Appl. Phys. 59, 694–701 (1986).
    [CrossRef]
  5. D. E. Aspnes, J. B. Theeten, F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20, 3292–3302 (1979).
    [CrossRef]
  6. R. W. Collins, J. M. Cavese, “Influence of substrate structure on the growth of hydrogenated amorphous silicon studied by in situ ellipsometry,” J. Appl. Phys. 60, 4169–4176 (1986).
    [CrossRef]
  7. P. G. Snyder, M. C. Rost, G. H. Bu-Abbud, J. A. Woolam, S. A. Alterovitz, “Variable angle of incidence spectroscopic ellipsometry: application to GaAs–AlxGa1−xAs multiple heterostructures,” J. Appl. Phys. 60, 3293–3301 (1986).
    [CrossRef]
  8. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  9. S. Y. Kim, K. Vedam, “Proper choice of the error function in modeling spectroellipsometric data,” Appl. Opt. 25, 2013–2021 (1986).
    [CrossRef] [PubMed]
  10. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1986).
  11. P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969).
  12. F. A. Modine, G. E. Jellison, G. R. Gruzalski, “Errors in ellipsometry measurements made with a photoelastic modulator,” J. Opt. Soc. Am. 73, 892–900 (1983).
    [CrossRef]
  13. Aspnes, A. Studna, “High precision scanning ellipsometer,” Appl. Opt. 14, 220–228 (1975).
    [PubMed]
  14. S. N. Jasperson, S. E. Schnatterly, “An improved method for high reflectivity ellipsometry based on a new polarization modulation technique,” Rev. Sci. Instrum. 40, 761–767 (1969); Rev. Sci. Instrum. 41, 152 (1970).
    [CrossRef]
  15. G. E. Jellison, F. A. Modine, “Two-channel polarization modulation ellipsometer,” Appl. Opt. 29, 959–974 (1990).
    [CrossRef] [PubMed]
  16. G. E. Jellison, F. A. Modine, “Accurate calibration of a photoelastic modulator in polarization modulation ellipsometry,” in Polarizaiton Considerations for Optical Systems II, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1166, 231–241 (1989).
  17. I. H. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. 55, 1205–1209 (1965).
    [CrossRef]
  18. D. A. G. Bruggeman, “Berechnung verschiedener physikalisher Konstanten vor heterogenen Substanzen,” Ann. Phys. (Leipzig) 24, 636 (1935).
  19. G. E. Jellison, “Examination of thin SiO2 films on Si using spectroscopic polarization modulation ellipsometry,” J. Appl. Phys.69 (to be published).
  20. C. R. Helms, B. E. Deal, eds., The Physics and Chemistry of SiO2 and the Si-SiO2 Interface (Plenum, New York, 1988).
  21. Ref. 10, p. 499.

1990 (1)

1986 (4)

S. Y. Kim, K. Vedam, “Proper choice of the error function in modeling spectroellipsometric data,” Appl. Opt. 25, 2013–2021 (1986).
[CrossRef] [PubMed]

P. J. McMarr, K. Vedam, J. Narayan, “Spectroscopic ellipsometry: a new tool for nondestructive depth profiling and characterization of interfaces,” J. Appl. Phys. 59, 694–701 (1986).
[CrossRef]

R. W. Collins, J. M. Cavese, “Influence of substrate structure on the growth of hydrogenated amorphous silicon studied by in situ ellipsometry,” J. Appl. Phys. 60, 4169–4176 (1986).
[CrossRef]

P. G. Snyder, M. C. Rost, G. H. Bu-Abbud, J. A. Woolam, S. A. Alterovitz, “Variable angle of incidence spectroscopic ellipsometry: application to GaAs–AlxGa1−xAs multiple heterostructures,” J. Appl. Phys. 60, 3293–3301 (1986).
[CrossRef]

1983 (2)

D. E. Aspnes, A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
[CrossRef]

F. A. Modine, G. E. Jellison, G. R. Gruzalski, “Errors in ellipsometry measurements made with a photoelastic modulator,” J. Opt. Soc. Am. 73, 892–900 (1983).
[CrossRef]

1982 (1)

G. E. Jellison, F. A. Modine, “Optical absorption of silicon between 1.6 and 4.7 eV at elevated temperatures,” Appl. Phys. Lett. 41, 180–182 (1982); “Optical functions of silicon between 1.7 and 4.7 eV at elevated temperatures,” Phys. Rev. B 27, 7466–7472 (1983).
[CrossRef]

1979 (2)

D. E. Aspnes, J. B. Theeten “Optical properties of the interface between si and its thermally grown oxide,” Phys. Rev. Lett. 43, 1046–1050 (1979); “Spectroscopic analysis of the interface between Si and its thermally grown oxide,” J. Electrochem. Soc. 127, 1359–1365 (1980).
[CrossRef]

D. E. Aspnes, J. B. Theeten, F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20, 3292–3302 (1979).
[CrossRef]

1975 (1)

1969 (1)

S. N. Jasperson, S. E. Schnatterly, “An improved method for high reflectivity ellipsometry based on a new polarization modulation technique,” Rev. Sci. Instrum. 40, 761–767 (1969); Rev. Sci. Instrum. 41, 152 (1970).
[CrossRef]

1965 (1)

1935 (1)

D. A. G. Bruggeman, “Berechnung verschiedener physikalisher Konstanten vor heterogenen Substanzen,” Ann. Phys. (Leipzig) 24, 636 (1935).

Alterovitz, S. A.

P. G. Snyder, M. C. Rost, G. H. Bu-Abbud, J. A. Woolam, S. A. Alterovitz, “Variable angle of incidence spectroscopic ellipsometry: application to GaAs–AlxGa1−xAs multiple heterostructures,” J. Appl. Phys. 60, 3293–3301 (1986).
[CrossRef]

Aspnes,

Aspnes, D. E.

D. E. Aspnes, A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
[CrossRef]

D. E. Aspnes, J. B. Theeten “Optical properties of the interface between si and its thermally grown oxide,” Phys. Rev. Lett. 43, 1046–1050 (1979); “Spectroscopic analysis of the interface between Si and its thermally grown oxide,” J. Electrochem. Soc. 127, 1359–1365 (1980).
[CrossRef]

D. E. Aspnes, J. B. Theeten, F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20, 3292–3302 (1979).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Bevington, P. R.

P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969).

Bruggeman, D. A. G.

D. A. G. Bruggeman, “Berechnung verschiedener physikalisher Konstanten vor heterogenen Substanzen,” Ann. Phys. (Leipzig) 24, 636 (1935).

Bu-Abbud, G. H.

P. G. Snyder, M. C. Rost, G. H. Bu-Abbud, J. A. Woolam, S. A. Alterovitz, “Variable angle of incidence spectroscopic ellipsometry: application to GaAs–AlxGa1−xAs multiple heterostructures,” J. Appl. Phys. 60, 3293–3301 (1986).
[CrossRef]

Cavese, J. M.

R. W. Collins, J. M. Cavese, “Influence of substrate structure on the growth of hydrogenated amorphous silicon studied by in situ ellipsometry,” J. Appl. Phys. 60, 4169–4176 (1986).
[CrossRef]

Collins, R. W.

R. W. Collins, J. M. Cavese, “Influence of substrate structure on the growth of hydrogenated amorphous silicon studied by in situ ellipsometry,” J. Appl. Phys. 60, 4169–4176 (1986).
[CrossRef]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1986).

Gruzalski, G. R.

Hottier, F.

D. E. Aspnes, J. B. Theeten, F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20, 3292–3302 (1979).
[CrossRef]

Jasperson, S. N.

S. N. Jasperson, S. E. Schnatterly, “An improved method for high reflectivity ellipsometry based on a new polarization modulation technique,” Rev. Sci. Instrum. 40, 761–767 (1969); Rev. Sci. Instrum. 41, 152 (1970).
[CrossRef]

Jellison, G. E.

G. E. Jellison, F. A. Modine, “Two-channel polarization modulation ellipsometer,” Appl. Opt. 29, 959–974 (1990).
[CrossRef] [PubMed]

F. A. Modine, G. E. Jellison, G. R. Gruzalski, “Errors in ellipsometry measurements made with a photoelastic modulator,” J. Opt. Soc. Am. 73, 892–900 (1983).
[CrossRef]

G. E. Jellison, F. A. Modine, “Optical absorption of silicon between 1.6 and 4.7 eV at elevated temperatures,” Appl. Phys. Lett. 41, 180–182 (1982); “Optical functions of silicon between 1.7 and 4.7 eV at elevated temperatures,” Phys. Rev. B 27, 7466–7472 (1983).
[CrossRef]

G. E. Jellison, “Examination of thin SiO2 films on Si using spectroscopic polarization modulation ellipsometry,” J. Appl. Phys.69 (to be published).

G. E. Jellison, F. A. Modine, “Accurate calibration of a photoelastic modulator in polarization modulation ellipsometry,” in Polarizaiton Considerations for Optical Systems II, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1166, 231–241 (1989).

Kim, S. Y.

Malitson, I. H.

McMarr, P. J.

P. J. McMarr, K. Vedam, J. Narayan, “Spectroscopic ellipsometry: a new tool for nondestructive depth profiling and characterization of interfaces,” J. Appl. Phys. 59, 694–701 (1986).
[CrossRef]

Modine, F. A.

G. E. Jellison, F. A. Modine, “Two-channel polarization modulation ellipsometer,” Appl. Opt. 29, 959–974 (1990).
[CrossRef] [PubMed]

F. A. Modine, G. E. Jellison, G. R. Gruzalski, “Errors in ellipsometry measurements made with a photoelastic modulator,” J. Opt. Soc. Am. 73, 892–900 (1983).
[CrossRef]

G. E. Jellison, F. A. Modine, “Optical absorption of silicon between 1.6 and 4.7 eV at elevated temperatures,” Appl. Phys. Lett. 41, 180–182 (1982); “Optical functions of silicon between 1.7 and 4.7 eV at elevated temperatures,” Phys. Rev. B 27, 7466–7472 (1983).
[CrossRef]

G. E. Jellison, F. A. Modine, “Accurate calibration of a photoelastic modulator in polarization modulation ellipsometry,” in Polarizaiton Considerations for Optical Systems II, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1166, 231–241 (1989).

Narayan, J.

P. J. McMarr, K. Vedam, J. Narayan, “Spectroscopic ellipsometry: a new tool for nondestructive depth profiling and characterization of interfaces,” J. Appl. Phys. 59, 694–701 (1986).
[CrossRef]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1986).

Rost, M. C.

P. G. Snyder, M. C. Rost, G. H. Bu-Abbud, J. A. Woolam, S. A. Alterovitz, “Variable angle of incidence spectroscopic ellipsometry: application to GaAs–AlxGa1−xAs multiple heterostructures,” J. Appl. Phys. 60, 3293–3301 (1986).
[CrossRef]

Schnatterly, S. E.

S. N. Jasperson, S. E. Schnatterly, “An improved method for high reflectivity ellipsometry based on a new polarization modulation technique,” Rev. Sci. Instrum. 40, 761–767 (1969); Rev. Sci. Instrum. 41, 152 (1970).
[CrossRef]

Snyder, P. G.

P. G. Snyder, M. C. Rost, G. H. Bu-Abbud, J. A. Woolam, S. A. Alterovitz, “Variable angle of incidence spectroscopic ellipsometry: application to GaAs–AlxGa1−xAs multiple heterostructures,” J. Appl. Phys. 60, 3293–3301 (1986).
[CrossRef]

Studna, A.

D. E. Aspnes, A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
[CrossRef]

Aspnes, A. Studna, “High precision scanning ellipsometer,” Appl. Opt. 14, 220–228 (1975).
[PubMed]

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1986).

Theeten, J. B.

D. E. Aspnes, J. B. Theeten “Optical properties of the interface between si and its thermally grown oxide,” Phys. Rev. Lett. 43, 1046–1050 (1979); “Spectroscopic analysis of the interface between Si and its thermally grown oxide,” J. Electrochem. Soc. 127, 1359–1365 (1980).
[CrossRef]

D. E. Aspnes, J. B. Theeten, F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20, 3292–3302 (1979).
[CrossRef]

Vedam, K.

P. J. McMarr, K. Vedam, J. Narayan, “Spectroscopic ellipsometry: a new tool for nondestructive depth profiling and characterization of interfaces,” J. Appl. Phys. 59, 694–701 (1986).
[CrossRef]

S. Y. Kim, K. Vedam, “Proper choice of the error function in modeling spectroellipsometric data,” Appl. Opt. 25, 2013–2021 (1986).
[CrossRef] [PubMed]

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1986).

Woolam, J. A.

P. G. Snyder, M. C. Rost, G. H. Bu-Abbud, J. A. Woolam, S. A. Alterovitz, “Variable angle of incidence spectroscopic ellipsometry: application to GaAs–AlxGa1−xAs multiple heterostructures,” J. Appl. Phys. 60, 3293–3301 (1986).
[CrossRef]

Ann. Phys. (Leipzig) (1)

D. A. G. Bruggeman, “Berechnung verschiedener physikalisher Konstanten vor heterogenen Substanzen,” Ann. Phys. (Leipzig) 24, 636 (1935).

Appl. Opt. (3)

Appl. Phys. Lett. (1)

G. E. Jellison, F. A. Modine, “Optical absorption of silicon between 1.6 and 4.7 eV at elevated temperatures,” Appl. Phys. Lett. 41, 180–182 (1982); “Optical functions of silicon between 1.7 and 4.7 eV at elevated temperatures,” Phys. Rev. B 27, 7466–7472 (1983).
[CrossRef]

J. Appl. Phys. (3)

P. J. McMarr, K. Vedam, J. Narayan, “Spectroscopic ellipsometry: a new tool for nondestructive depth profiling and characterization of interfaces,” J. Appl. Phys. 59, 694–701 (1986).
[CrossRef]

R. W. Collins, J. M. Cavese, “Influence of substrate structure on the growth of hydrogenated amorphous silicon studied by in situ ellipsometry,” J. Appl. Phys. 60, 4169–4176 (1986).
[CrossRef]

P. G. Snyder, M. C. Rost, G. H. Bu-Abbud, J. A. Woolam, S. A. Alterovitz, “Variable angle of incidence spectroscopic ellipsometry: application to GaAs–AlxGa1−xAs multiple heterostructures,” J. Appl. Phys. 60, 3293–3301 (1986).
[CrossRef]

J. Opt. Soc. Am. (2)

Phys. Rev. B (2)

D. E. Aspnes, J. B. Theeten, F. Hottier, “Investigation of effective-medium models of microscopic surface roughness by spectroscopic ellipsometry,” Phys. Rev. B 20, 3292–3302 (1979).
[CrossRef]

D. E. Aspnes, A. Studna, “Dielectric functions and optical parameters of Si, Ge, GaP, GaAs, GaSb, InP, InAs, and InSb from 1.5 to 6.0 eV,” Phys. Rev. B 27, 985–1009 (1983).
[CrossRef]

Phys. Rev. Lett. (1)

D. E. Aspnes, J. B. Theeten “Optical properties of the interface between si and its thermally grown oxide,” Phys. Rev. Lett. 43, 1046–1050 (1979); “Spectroscopic analysis of the interface between Si and its thermally grown oxide,” J. Electrochem. Soc. 127, 1359–1365 (1980).
[CrossRef]

Rev. Sci. Instrum. (1)

S. N. Jasperson, S. E. Schnatterly, “An improved method for high reflectivity ellipsometry based on a new polarization modulation technique,” Rev. Sci. Instrum. 40, 761–767 (1969); Rev. Sci. Instrum. 41, 152 (1970).
[CrossRef]

Other (7)

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1986).

P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969).

G. E. Jellison, “Examination of thin SiO2 films on Si using spectroscopic polarization modulation ellipsometry,” J. Appl. Phys.69 (to be published).

C. R. Helms, B. E. Deal, eds., The Physics and Chemistry of SiO2 and the Si-SiO2 Interface (Plenum, New York, 1988).

Ref. 10, p. 499.

G. E. Jellison, F. A. Modine, “Accurate calibration of a photoelastic modulator in polarization modulation ellipsometry,” in Polarizaiton Considerations for Optical Systems II, R. A. Chipman, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1166, 231–241 (1989).

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 (5)

Fig. 1
Fig. 1

Plot of the N, S, and C data for a 59-nm SiO2 film in Si as a function of incident photon energy [see Eqs. (6)]. The angle of incidence is 69.92°.

Fig. 2
Fig. 2

Plot of the ellipsometric angles ψ and Δ determined from the N, S, and C data shown in Fig. 1.

Fig. 3
Fig. 3

Plot of the real (ρr) and imaginary (ρi) parts of the Fresnel reflection coefficient ratio ρ for the data plotted in Fig. 1.

Fig. 4
Fig. 4

Plot of the differences between the experimental and calculated spectra of the real part of the Fresnel reflection coefficient ratio ρ. The calculated spectra were determined from fit 5, using both the biased and unbiased estimators as a figure of merit (see text and Table I). The dots represent the wavelength-dependent error estimations, plotted as + and − from the zero line.

Fig. 5
Fig. 5

Plot of the differences between the experimental and calculated spectra of the imaginary part of the Fresnel reflection coefficient ratio ρ.

Tables (2)

Tables Icon

Table I Nine Different Fits to the Data Using the Four-Media Model Described In the Texta

Tables Icon

Table II Results from the Fitting Procedure for Nine Different Fits to the Data Made with Both the Biased and Unbiased Estimatorsa

Equations (32)

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

ρ ( λ , ϕ ) = r p / r s = tan ψ ( λ , ϕ ) exp i Δ ( λ , ϕ ) ,
χ u 2 = [ 1 / ( n - m - 1 ) ] i = 1 n ρ exp ( λ i ) - ρ calc ( λ i , z ) 2 .
I = I 0 ( 1 + γ cos 2 α + ζ sin 2 α ) ,
χ b 2 = [ 1 / ( n - m - 1 ) ] i n [ ρ exp ( λ i ) - ρ calc ( λ i z ) 2 / δ ρ ( λ i ) 2 ] ,
GOF = Q [ ( 2 n - m ) / 2 , χ 2 / 2 ] ,
N = cos 2 ψ ,
S = sin 2 ψ sin Δ ,
C = sin 2 ψ cos Δ .
β = ( N 2 + S 2 + C 2 ) 1 / 2 = 1.
= 1 + 2 ,
1 = sin 2 ϕ [ 1 + tan 2 ϕ ( N 2 - S 2 ) / ( 1 + C ) 2 ] ,
2 = sin 2 ϕ [ tan 2 ϕ 2 N S / ( 1 + C ) 2 ] .
R i ( ω ) = I i ( ω ) / I i ( dc ) ± 2 J 1 ( A ) S ,
R i ( 2 ω ) = I i ( 2 ω ) / I i ( dc ) ± 2 J 2 ( A ) [ N ± C ] / 2 ,
δ w l = ( R i / λ ) δ λ .
δ R = ( δ l 2 + δ m 2 + δ f 2 + δ w l 2 ) 1 / 2 ,
δ N 1 = δ R ( 2 ω ) / J 2 ( A ) ,
δ C 1 = δ R ( 2 ω ) / J 2 ( A ) ,
δ S 1 = δ R ( ω ) / 2 J 1 ( A ) .
δ β f = [ ( N δ N 1 ) 2 + ( C δ C 1 ) 2 + ( S δ S 1 ) 2 ] / β .
δ β N = ( 1 - N 2 ) β - 1 ,
δ β C = ( 1 - C 2 ) β - 1 ,
δ β S = ( 1 - S 2 ) β - 1 .
δ N = [ δ R ( 2 ω ) 2 + δ β N 2 ] 1 / 2 ,
δ C = [ δ R ( 2 ω ) 2 + δ β C 2 ] 1 / 2 ,
δ S = [ δ R ( ω ) 2 + δ β S 2 ] 1 / 2 .
ψ = 0.5 tan - 1 [ ( S 2 + C 2 ) 1 / 2 / N ]
Δ = tan - 1 ( S / C ) .
χ 2 = [ N m - cos 2 ψ ] 2 / δ N 2 + [ C m - sin 2 ψ cos Δ ] 2 / δ C 2 , + [ S m - sin 2 ψ sin Δ ] 2 / δ S 2 ,
δ ρ r = ( 4 ρ r 2 δ ψ 2 / sin 2 2 ψ + ρ i 2 δ Δ 2 ) 1 / 2 ,
δ ρ i = ( 4 ρ i 2 δ ψ 2 / sin 2 2 ψ + ρ r 2 δ Δ 2 ) 1 / 2 .
n 2 - 1 = A λ 2 / ( λ 2 - λ 0 2 ) .

Metrics