Abstract

An exact formalism devoted to the determination of dispersion coefficients is described. The method takes into account two frequent experimental configurations: a solid thin layer on a substrate and a fluid, or solid, layer between a substrate and a superstrate. Introducing the concepts of reduction and reduced finesse, this method is based entirely on the fringes’ spectral position of the maxima in the transmittance spectrum. It is found that the chromatic dispersion does not affect the spectral position of the minima in the same way as it does for the maxima. There is no need to get the refractive-index curve, n(λ), to determine the dispersion coefficients nor to work at multiple incidence angles. Bringing together the possible nonrestrictive approximations, the method becomes easy and simple to implement from a spectrophotometer in tandem with a computer. In addition, the spectrometer does not require ordinate-axis calibration, and knowledge of the substrate’s and superstrate’s refractive index is not required. Alternatively, the method can be easily used to accurately determine the thickness of thin layers. A numerical example using a thin layer of 2-methyl-4-nitroaniline (MNA) is given.

© 2002 Optical Society of America

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References

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  1. F. Abelès, “La détermination de l’indice et de l’épaisseur des couches minces transparentes,” J. Phys. Radium 11, 310–314 (1950).
    [CrossRef]
  2. O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths, London, 1955).
  3. G. Hass, R. E. Thun, eds., Physics of Thin Films (Academic, New York, 1964).
  4. M. Hacskaylo, “Determination of the refractive index of thin dielectric films,” J. Opt. Soc. Am. 54, 198–203 (1964).
    [CrossRef]
  5. F. Abelès, “Optical properties of metallic films,” in Physics of Thin Films, G. Hass, R. E. Thun, eds. (Academic, New York, 1971), p. 151.
  6. W. N. Hansen, “Optical characterization of thin films: theory,” J. Opt. Soc. Am. 63, 793–802 (1973).
    [CrossRef]
  7. J. W. Seeser, “Effect of dispersion on the reflection and transmission extrema from a monolayer,” Appl. Opt. 14, 640–642 (1975).
    [CrossRef] [PubMed]
  8. E. Pelletier, P. Roche, B. Vidal, “Détermination automatique des constantes optiques et de l’épaisseur de couches minces: application aux couches diélectriques,” Nouv. Rev. Optique 7, 353–362 (1976).
    [CrossRef]
  9. J. C. Manifacier, J. Gasiot, J. P. Fillard, “A simple method for the determination of the optical constants n, k and the thickness of a weakly absorbing thin film,” J. Phys. E Sci. Inst. 9, 1002–1004 (1976).
    [CrossRef]
  10. J. S. Wei, W. D. Westwood, “A new method for determining thin-film refractive index and thickness using guided optical waves,” Appl. Phys. Lett. 32, 819–821 (1978).
    [CrossRef]
  11. A. M. Goodman, “Optical interference method for the approximate determination of refractive index and thickness of a transparent layer,” Appl. Opt. 17, 2779–2787 (1978).
    [CrossRef] [PubMed]
  12. J. P. Borgogno, B. Lazarides, E. Pelletier, “Automatic determination of the optical constants of inhomogeneous thin films,” Appl. Opt. 21, 4020–4029 (1982).
    [CrossRef] [PubMed]
  13. I. Ohlídal, K. Navrátil, E. Schmidt, “Simple method for the complete optical analysis of very thick and weakly absorbing films,” Appl. Phys. A 29, 157–162 (1982).
    [CrossRef]
  14. J. A. Dobrowolski, F. C. Ho, A. Waldorf, “Determination of optical constants of thin film coating materials based on inverse synthesis,” Appl. Opt. 22, 3191–3200 (1983).
    [CrossRef] [PubMed]
  15. K. F. Palmer, M. Z. Williams, “Determination of the optical constants of a thin film from transmittance measurements of a single film thickness,” Appl. Opt. 24, 1788–1797 (1985).
    [CrossRef] [PubMed]
  16. T. Opara, J. W. Baran, J. Zmija, “Interference method of refractive index dispersion measurements in thin solid layers,” Electr. Technol. 18, 73–88 (1985).
  17. R. Swanepoel, “Determining refractive index and thickness of thin films from wavelength measurements only,” J. Opt. Soc. Am. A 2, 1339–1343 (1985).
    [CrossRef]
  18. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, Orlando, Fla., 1985).
  19. E. D. Palik, ed., Handbook of Optical Constants of Solids II (Academic, Boston, Mass., 1991).
  20. L. Ward, The Optical Constants of Bulk Materials and Films, 2nd ed. (IOP, London, 1994).
  21. M. Nowak, “Determination of optical constants and average thickness of inhomogeneous-rough thin films using spectral dependence of optical transmittance,” Thin Solid Films 254, 200–210 (1995).
    [CrossRef]
  22. T. Fukano, I. Yamaguchi, “Simultaneous measurement of thickness and refractive indices of multiple layers by a low-coherence confocal interference microscope,” Opt. Lett. 21, 1942–1944 (1996).
    [CrossRef] [PubMed]
  23. M. A. Khashan, A. M. El-Naggar, E. Shaddad, “A new method of determining the optical constants of a thin film from its reflectance and transmittance interferograms in a wide spectral range: 0.2–3 µm,” Opt. Commun. 178, 123–132 (2000).
    [CrossRef]
  24. M. A. Khashan, A. M. El-Naggar, “A simple method of measuring and applying the dispersion of thin films,” Opt. Commun. 187, 39–47 (2001).
    [CrossRef]
  25. G. Bruhat, Optique, 6th ed. (Masson, Paris, 1965), Sec. 230.
  26. This condition on refractive index is not necessary when substrate and superstrate are made of the same material.
  27. S. L. Mielke, R. E. Ryan, T. Hilgeman, L. Lesyna, R. G. Madonna, W. C. Van Nostrand, “Measurement of the phase shift on reflectance for low-order infrared Fabry–Perot interferometer dielectric stack mirrors,” Appl. Opt. 36, 8139–8144 (1997).
    [CrossRef]
  28. C. Kittel, Physique de l’État Solide, 7th ed. (Dunod, Paris, 1998), p. 282.
  29. Z. Knittl, Optics of Thin Films (Wiley, London, 1976).
  30. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).
  31. Sh. A. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Frontières, Gif-sur-Yvette, France, 1992).
  32. R. T. Holm, “Convention confusions,” in Handbook of Optical Constants of Solids II, E. D. Palik, ed. (Academic, Boston, Mass., 1991), p. 39.
  33. With use of energy conservation, the absorption spectrum (formally the internal absorptance) can be obtained from the knowledge of transmittance and reflectance spectra when scattering, luminescence, nonlinear processes, and other secondary effects can be neglected. The use of two samples having different, known thicknesses should be used to correct for losses due to reflection at interfaces. Alternatively, α can also be estimated from samples having thickness much larger than the coherence length of the radiation source. In such instances, the reflectance spectrum is not required.
  34. R. Morita, N. Ogasawara, S. Umegaki, R. Ito, “Refractive indices of 2-methyl-4-nitroaniline (MNA),” Jpn. J. Appl. Phys. 26, L1711–L1713 (1987).
    [CrossRef]
  35. Ref. 25, Sec. 236.
  36. Making use of an algorithm that uses partial derivatives relative to fitted parameters in search of the smallest chi square further increases the accuracy.
  37. Some algorithms allow the user to set limits for curve-fit parameters.

2001

M. A. Khashan, A. M. El-Naggar, “A simple method of measuring and applying the dispersion of thin films,” Opt. Commun. 187, 39–47 (2001).
[CrossRef]

2000

M. A. Khashan, A. M. El-Naggar, E. Shaddad, “A new method of determining the optical constants of a thin film from its reflectance and transmittance interferograms in a wide spectral range: 0.2–3 µm,” Opt. Commun. 178, 123–132 (2000).
[CrossRef]

1997

1996

1995

M. Nowak, “Determination of optical constants and average thickness of inhomogeneous-rough thin films using spectral dependence of optical transmittance,” Thin Solid Films 254, 200–210 (1995).
[CrossRef]

1987

R. Morita, N. Ogasawara, S. Umegaki, R. Ito, “Refractive indices of 2-methyl-4-nitroaniline (MNA),” Jpn. J. Appl. Phys. 26, L1711–L1713 (1987).
[CrossRef]

1985

1983

1982

J. P. Borgogno, B. Lazarides, E. Pelletier, “Automatic determination of the optical constants of inhomogeneous thin films,” Appl. Opt. 21, 4020–4029 (1982).
[CrossRef] [PubMed]

I. Ohlídal, K. Navrátil, E. Schmidt, “Simple method for the complete optical analysis of very thick and weakly absorbing films,” Appl. Phys. A 29, 157–162 (1982).
[CrossRef]

1978

J. S. Wei, W. D. Westwood, “A new method for determining thin-film refractive index and thickness using guided optical waves,” Appl. Phys. Lett. 32, 819–821 (1978).
[CrossRef]

A. M. Goodman, “Optical interference method for the approximate determination of refractive index and thickness of a transparent layer,” Appl. Opt. 17, 2779–2787 (1978).
[CrossRef] [PubMed]

1976

E. Pelletier, P. Roche, B. Vidal, “Détermination automatique des constantes optiques et de l’épaisseur de couches minces: application aux couches diélectriques,” Nouv. Rev. Optique 7, 353–362 (1976).
[CrossRef]

J. C. Manifacier, J. Gasiot, J. P. Fillard, “A simple method for the determination of the optical constants n, k and the thickness of a weakly absorbing thin film,” J. Phys. E Sci. Inst. 9, 1002–1004 (1976).
[CrossRef]

1975

1973

1964

1950

F. Abelès, “La détermination de l’indice et de l’épaisseur des couches minces transparentes,” J. Phys. Radium 11, 310–314 (1950).
[CrossRef]

Abelès, F.

F. Abelès, “La détermination de l’indice et de l’épaisseur des couches minces transparentes,” J. Phys. Radium 11, 310–314 (1950).
[CrossRef]

F. Abelès, “Optical properties of metallic films,” in Physics of Thin Films, G. Hass, R. E. Thun, eds. (Academic, New York, 1971), p. 151.

Baran, J. W.

T. Opara, J. W. Baran, J. Zmija, “Interference method of refractive index dispersion measurements in thin solid layers,” Electr. Technol. 18, 73–88 (1985).

Borgogno, J. P.

Bruhat, G.

G. Bruhat, Optique, 6th ed. (Masson, Paris, 1965), Sec. 230.

Dobrowolski, J. A.

El-Naggar, A. M.

M. A. Khashan, A. M. El-Naggar, “A simple method of measuring and applying the dispersion of thin films,” Opt. Commun. 187, 39–47 (2001).
[CrossRef]

M. A. Khashan, A. M. El-Naggar, E. Shaddad, “A new method of determining the optical constants of a thin film from its reflectance and transmittance interferograms in a wide spectral range: 0.2–3 µm,” Opt. Commun. 178, 123–132 (2000).
[CrossRef]

Fillard, J. P.

J. C. Manifacier, J. Gasiot, J. P. Fillard, “A simple method for the determination of the optical constants n, k and the thickness of a weakly absorbing thin film,” J. Phys. E Sci. Inst. 9, 1002–1004 (1976).
[CrossRef]

Fukano, T.

Furman, Sh. A.

Sh. A. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Frontières, Gif-sur-Yvette, France, 1992).

Gasiot, J.

J. C. Manifacier, J. Gasiot, J. P. Fillard, “A simple method for the determination of the optical constants n, k and the thickness of a weakly absorbing thin film,” J. Phys. E Sci. Inst. 9, 1002–1004 (1976).
[CrossRef]

Goodman, A. M.

Hacskaylo, M.

Hansen, W. N.

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths, London, 1955).

Hilgeman, T.

Ho, F. C.

Holm, R. T.

R. T. Holm, “Convention confusions,” in Handbook of Optical Constants of Solids II, E. D. Palik, ed. (Academic, Boston, Mass., 1991), p. 39.

Ito, R.

R. Morita, N. Ogasawara, S. Umegaki, R. Ito, “Refractive indices of 2-methyl-4-nitroaniline (MNA),” Jpn. J. Appl. Phys. 26, L1711–L1713 (1987).
[CrossRef]

Khashan, M. A.

M. A. Khashan, A. M. El-Naggar, “A simple method of measuring and applying the dispersion of thin films,” Opt. Commun. 187, 39–47 (2001).
[CrossRef]

M. A. Khashan, A. M. El-Naggar, E. Shaddad, “A new method of determining the optical constants of a thin film from its reflectance and transmittance interferograms in a wide spectral range: 0.2–3 µm,” Opt. Commun. 178, 123–132 (2000).
[CrossRef]

Kittel, C.

C. Kittel, Physique de l’État Solide, 7th ed. (Dunod, Paris, 1998), p. 282.

Knittl, Z.

Z. Knittl, Optics of Thin Films (Wiley, London, 1976).

Lazarides, B.

Lesyna, L.

Madonna, R. G.

Manifacier, J. C.

J. C. Manifacier, J. Gasiot, J. P. Fillard, “A simple method for the determination of the optical constants n, k and the thickness of a weakly absorbing thin film,” J. Phys. E Sci. Inst. 9, 1002–1004 (1976).
[CrossRef]

Mielke, S. L.

Morita, R.

R. Morita, N. Ogasawara, S. Umegaki, R. Ito, “Refractive indices of 2-methyl-4-nitroaniline (MNA),” Jpn. J. Appl. Phys. 26, L1711–L1713 (1987).
[CrossRef]

Navrátil, K.

I. Ohlídal, K. Navrátil, E. Schmidt, “Simple method for the complete optical analysis of very thick and weakly absorbing films,” Appl. Phys. A 29, 157–162 (1982).
[CrossRef]

Nowak, M.

M. Nowak, “Determination of optical constants and average thickness of inhomogeneous-rough thin films using spectral dependence of optical transmittance,” Thin Solid Films 254, 200–210 (1995).
[CrossRef]

Ogasawara, N.

R. Morita, N. Ogasawara, S. Umegaki, R. Ito, “Refractive indices of 2-methyl-4-nitroaniline (MNA),” Jpn. J. Appl. Phys. 26, L1711–L1713 (1987).
[CrossRef]

Ohlídal, I.

I. Ohlídal, K. Navrátil, E. Schmidt, “Simple method for the complete optical analysis of very thick and weakly absorbing films,” Appl. Phys. A 29, 157–162 (1982).
[CrossRef]

Opara, T.

T. Opara, J. W. Baran, J. Zmija, “Interference method of refractive index dispersion measurements in thin solid layers,” Electr. Technol. 18, 73–88 (1985).

Palmer, K. F.

Pelletier, E.

J. P. Borgogno, B. Lazarides, E. Pelletier, “Automatic determination of the optical constants of inhomogeneous thin films,” Appl. Opt. 21, 4020–4029 (1982).
[CrossRef] [PubMed]

E. Pelletier, P. Roche, B. Vidal, “Détermination automatique des constantes optiques et de l’épaisseur de couches minces: application aux couches diélectriques,” Nouv. Rev. Optique 7, 353–362 (1976).
[CrossRef]

Roche, P.

E. Pelletier, P. Roche, B. Vidal, “Détermination automatique des constantes optiques et de l’épaisseur de couches minces: application aux couches diélectriques,” Nouv. Rev. Optique 7, 353–362 (1976).
[CrossRef]

Ryan, R. E.

Schmidt, E.

I. Ohlídal, K. Navrátil, E. Schmidt, “Simple method for the complete optical analysis of very thick and weakly absorbing films,” Appl. Phys. A 29, 157–162 (1982).
[CrossRef]

Seeser, J. W.

Shaddad, E.

M. A. Khashan, A. M. El-Naggar, E. Shaddad, “A new method of determining the optical constants of a thin film from its reflectance and transmittance interferograms in a wide spectral range: 0.2–3 µm,” Opt. Commun. 178, 123–132 (2000).
[CrossRef]

Swanepoel, R.

Tikhonravov, A. V.

Sh. A. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Frontières, Gif-sur-Yvette, France, 1992).

Umegaki, S.

R. Morita, N. Ogasawara, S. Umegaki, R. Ito, “Refractive indices of 2-methyl-4-nitroaniline (MNA),” Jpn. J. Appl. Phys. 26, L1711–L1713 (1987).
[CrossRef]

Van Nostrand, W. C.

Vidal, B.

E. Pelletier, P. Roche, B. Vidal, “Détermination automatique des constantes optiques et de l’épaisseur de couches minces: application aux couches diélectriques,” Nouv. Rev. Optique 7, 353–362 (1976).
[CrossRef]

Waldorf, A.

Ward, L.

L. Ward, The Optical Constants of Bulk Materials and Films, 2nd ed. (IOP, London, 1994).

Wei, J. S.

J. S. Wei, W. D. Westwood, “A new method for determining thin-film refractive index and thickness using guided optical waves,” Appl. Phys. Lett. 32, 819–821 (1978).
[CrossRef]

Westwood, W. D.

J. S. Wei, W. D. Westwood, “A new method for determining thin-film refractive index and thickness using guided optical waves,” Appl. Phys. Lett. 32, 819–821 (1978).
[CrossRef]

Williams, M. Z.

Yamaguchi, I.

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

Zmija, J.

T. Opara, J. W. Baran, J. Zmija, “Interference method of refractive index dispersion measurements in thin solid layers,” Electr. Technol. 18, 73–88 (1985).

Appl. Opt.

Appl. Phys. A

I. Ohlídal, K. Navrátil, E. Schmidt, “Simple method for the complete optical analysis of very thick and weakly absorbing films,” Appl. Phys. A 29, 157–162 (1982).
[CrossRef]

Appl. Phys. Lett.

J. S. Wei, W. D. Westwood, “A new method for determining thin-film refractive index and thickness using guided optical waves,” Appl. Phys. Lett. 32, 819–821 (1978).
[CrossRef]

Electr. Technol.

T. Opara, J. W. Baran, J. Zmija, “Interference method of refractive index dispersion measurements in thin solid layers,” Electr. Technol. 18, 73–88 (1985).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. E Sci. Inst.

J. C. Manifacier, J. Gasiot, J. P. Fillard, “A simple method for the determination of the optical constants n, k and the thickness of a weakly absorbing thin film,” J. Phys. E Sci. Inst. 9, 1002–1004 (1976).
[CrossRef]

J. Phys. Radium

F. Abelès, “La détermination de l’indice et de l’épaisseur des couches minces transparentes,” J. Phys. Radium 11, 310–314 (1950).
[CrossRef]

Jpn. J. Appl. Phys.

R. Morita, N. Ogasawara, S. Umegaki, R. Ito, “Refractive indices of 2-methyl-4-nitroaniline (MNA),” Jpn. J. Appl. Phys. 26, L1711–L1713 (1987).
[CrossRef]

Nouv. Rev. Optique

E. Pelletier, P. Roche, B. Vidal, “Détermination automatique des constantes optiques et de l’épaisseur de couches minces: application aux couches diélectriques,” Nouv. Rev. Optique 7, 353–362 (1976).
[CrossRef]

Opt. Commun.

M. A. Khashan, A. M. El-Naggar, E. Shaddad, “A new method of determining the optical constants of a thin film from its reflectance and transmittance interferograms in a wide spectral range: 0.2–3 µm,” Opt. Commun. 178, 123–132 (2000).
[CrossRef]

M. A. Khashan, A. M. El-Naggar, “A simple method of measuring and applying the dispersion of thin films,” Opt. Commun. 187, 39–47 (2001).
[CrossRef]

Opt. Lett.

Thin Solid Films

M. Nowak, “Determination of optical constants and average thickness of inhomogeneous-rough thin films using spectral dependence of optical transmittance,” Thin Solid Films 254, 200–210 (1995).
[CrossRef]

Other

C. Kittel, Physique de l’État Solide, 7th ed. (Dunod, Paris, 1998), p. 282.

Z. Knittl, Optics of Thin Films (Wiley, London, 1976).

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

Sh. A. Furman, A. V. Tikhonravov, Basics of Optics of Multilayer Systems (Frontières, Gif-sur-Yvette, France, 1992).

R. T. Holm, “Convention confusions,” in Handbook of Optical Constants of Solids II, E. D. Palik, ed. (Academic, Boston, Mass., 1991), p. 39.

With use of energy conservation, the absorption spectrum (formally the internal absorptance) can be obtained from the knowledge of transmittance and reflectance spectra when scattering, luminescence, nonlinear processes, and other secondary effects can be neglected. The use of two samples having different, known thicknesses should be used to correct for losses due to reflection at interfaces. Alternatively, α can also be estimated from samples having thickness much larger than the coherence length of the radiation source. In such instances, the reflectance spectrum is not required.

Ref. 25, Sec. 236.

Making use of an algorithm that uses partial derivatives relative to fitted parameters in search of the smallest chi square further increases the accuracy.

Some algorithms allow the user to set limits for curve-fit parameters.

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworths, London, 1955).

G. Hass, R. E. Thun, eds., Physics of Thin Films (Academic, New York, 1964).

E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, Orlando, Fla., 1985).

E. D. Palik, ed., Handbook of Optical Constants of Solids II (Academic, Boston, Mass., 1991).

L. Ward, The Optical Constants of Bulk Materials and Films, 2nd ed. (IOP, London, 1994).

G. Bruhat, Optique, 6th ed. (Masson, Paris, 1965), Sec. 230.

This condition on refractive index is not necessary when substrate and superstrate are made of the same material.

F. Abelès, “Optical properties of metallic films,” in Physics of Thin Films, G. Hass, R. E. Thun, eds. (Academic, New York, 1971), p. 151.

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Figures (3)

Fig. 1
Fig. 1

Structure, variable definitions, and conventions.

Fig. 2
Fig. 2

Data used in testing the method: (a) uncorrected transmittance spectrum, (b) corrected transmittance spectrum, (c) phase shift (left scale), refractive index (right scale).

Fig. 3
Fig. 3

Result of the curve fit based on the Briot–Sellmeier dispersion equation. The dots are data extracted from the corrected transmittance spectrum. The line is the curve fit.

Equations (44)

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

n(λ)=As+Bsλ2λ2-Cs+Dsλ2,
n(λ)=Ac+Bcλ2+Ccλ4+Dcλ2,
δ(λ)=-12πP0lnλ+λλ-λd ln[r(λ)]dλdλ.
E2+0=T͡E0+E0-T(11)T(12)T(21)T(22)E0+E0-,
T͡=C͡21P͡1C͡10,
C͡lk=1τkl1ρklρkl1,
P͡1=exp(-iϕ)00exp(+iϕ).
ρlk=-ηl-ηkηl+ηk=rlk exp(iδlk)
τlk=2ηkηl+ηk,
ϕ=2πnsλ-i12αs,
I=I0 exp(-αz),
α=4πκλ.
t20=n2n0(τ20τ20*),
τ20E2+E0+=T(11)T(22)-T(12)T(21)T(22)=τ21τ10exp(iϕ)+ρ21ρ10 exp(-iϕ).
ρlk2+τlkτkl=1,
rkl=rlk,
δ01=δ10+π,δ21=δ12-π,
δ12(δml+δkl)=12(δlm+δlk),
t20=n2n01G202+4π2F202 sin22πnsλ-δ.
F20πr12r104(1-r12)(1-r10)
G20exp(12αs)-r12r10 exp(-12αs)(1-r12)(1-r10).
t20(cor)n0n2G202t20=11+4π2F20G202sin22πnsλ-δ11+4π2H202 sin22πnsλ-δ.
H20F20G20=πr12r104exp(12αs)-r12r10 exp(-12αs).
sin2πnsλ-δ=0,
H20λsin2πnsλ-δ-H202πnsλ2-2πsλnλ+δλcos2πnsλ-δ=0.
cos2πnsλ-δ=0.
2πnsλ-δ=Nπ,
2πnsλ=Nπ+δ=N+1πδπNδπ.
1×10-8AsCs(νmax2)2-CsNδ2s2+As+Bs-CsDsνmax2+1×108Nδ2s2-Ds=0.
νmax2=-12αs(βs+βs2-4αsχs),
αs1×10-8AsCs,
βs=βs(Nδ)-CsNδ2s2+As+Bs-CsDs,
χs=χs(Nδ)1×108Nδ2s2-Ds.
νmax2=5×107AsCsCsNδ2s2+As+Bs-CsDs-CsNδ2s2+As+Bs-CsDs2-4AsCsNδ2s2-Ds.
(νmax2)3+1×108BcCc(νmax2)2+1×1016AcCcνmax2-1×10241CcNδ2s2-Dc=0.
νmax2=[127χc3+14δc2+127βc3δc-1108βc2χc2-16βcχcδc-12δc-127βc3+16βcχc]1/3-[127χc3+14δc2+127βc3δc-1108βc2χc2-16βcχcδc+12δc+127βc3-16βcχc]1/3-13βc,
βc1×108BcCc,
χc1×1016AcCc,
δc=δc(Nδ)1×10241CcNδ2s2-Dc.
κ=12[-(As+Σa)+(As+Σa)2+Σb2],
Σa=Bsλ2(λ2-λ02)(λ2-λ02)2+(Γλλ0)2,
Σb=BsΓλ3λ0(λ2-λ02)2+(Γλλ0)2.
βs-CsNδ4s2+As+Bs-CsDs,
χs1×108Nδ4s2-Ds.

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