Abstract

We discuss a new method to estimate the absorption coefficient, the index of refraction, and the thickness of thin films using optical transmission data only. To solve the problem we used a pointwise constrained optimization approach, defining a nonlinear programming problem, the unknowns of which are the coefficients to be estimated, with linear constraints that represent prior knowledge about the physical solution. The method applies to all kinds of transmission spectra and does not rely on the existence of fringe patterns or transparency. Results on amorphous semiconductor thin films and gedanken films are reported. They show that the new method is highly reliable.

© 1997 Optical Society of America

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References

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  1. O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1991).
  2. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1980).
  3. 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]
  4. 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 film,” J. Phys. E 9, 1002–1004 (1976).
    [CrossRef]
  5. R. Swanepoel, “Determination of the thickness and optical constants of amorphous silicon,” J. Phys. E 16, 1214–1222 (1983); see also “Determination of surface roughness and optical constants of inhomogeneous amorphous silicon films,” J. Phys. E 17, 896–903 (1984).
    [CrossRef]
  6. J. E. Dennis, R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, N.J., 1983).
  7. N. F. Mott, E. A. Davis, Electronic Processess in Non-Crystalline Materials, 2nd ed. (Oxford U. Press, Oxford, UK, 1979).
  8. R. Fletcher, Practical Methods for Optimization (Wiley, Chichester, UK, 1987).
  9. P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, London, 1981).
  10. R. B. Murtagh, M. A. Saunders, minos User’s Guide, , 1977 (Department of Operations Research, Stanford University, Calif.).
  11. R. B. Murtagh, M. A. Saunders, “Large scale linearly constrained optimization,” Math. Programming 14, 41–72 (1978).
    [CrossRef]
  12. J. Tauc, “Absorption edge and internal electric fields in amorphous semiconductors,” Mater. Res. Bull. 5, 721–730 (1970).
    [CrossRef]
  13. G. A. N. Connell, “Optical properties of amorphous semiconductors,” in Amorphous Semiconductors, M. H. Brodsky, ed., Vol. 36 of Topics in Applied Physics (Springer-Verlag, Berlin, 1979), pp. 73–111.
    [CrossRef]
  14. R. A. Street, Hydrogenated Amorphous Silicon (Cambridge U. Press, Cambridge, UK, 1991).
    [CrossRef]
  15. G. D. Cody, “The optical absorption edge of a-Si:H,” in Hydrogenated Amorphous Silicon, J. Pankove, ed., Vol. 21B of Semiconductors and Semimetals Series, R. K. Willardson, A. C. Beer, eds. (Academic, New York, 1984), pp. 11–82.
  16. W. B. Jackson, N. M. Amer, A. C. Boccara, D. Fournier, “Photothermal deflection spectroscopy and detection,” Appl. Opt. 20, 1333–1344 (1981).
    [CrossRef] [PubMed]
  17. S. H. Wemple, W. DiDomenico, “Behavior of the electronic dielectric constant in covalent and ionic materials,” Phys. Rev. B 3, 1338–1351 (1971).
    [CrossRef]
  18. S. H. Wemple, “Refractive-index behavior of amorphous semiconductors and glasses,” Phys. Rev. B 7, 3767–3777 (1973).
    [CrossRef]
  19. D. T. Pierce, W. E. Spicer, “Electronic structure of amorphous silicon from photoemission and optical studies,” Phys. Rev. B 5, 3017–3029 (1972).
    [CrossRef]
  20. G. A. N. Connell, R. J. Temkin, W. Paul, “Amorphous germanium, III. Optical properties,” Adv. Phys. 22, 643–665 (1973).
    [CrossRef]
  21. W. Paul, G. A. N. Connell, R. J. Temkin, “Amorphous germanium, I. A model for the structural and optical properties,” Adv. Phys. 22, 531–580 (1973).
    [CrossRef]
  22. T. M. Donovan, K. Heinemann, “High resolution electron microscope observation of voids in amorphous Ge,” Phys. Rev. Lett. 27, 1794–1796 (1971).
    [CrossRef]
  23. W. Fuhs, H.-J. Hesse, K. H. Langer, “Substrate specific voids in amorphous germanium,” in Amorphous and Liquid Semiconductors, J. Stuke, W. Brenig eds. (Taylor and Francis, London, 1974), pp. 79–84.
  24. M. Mulato, I. Chambouleyron, I. Torriani, “Hydrogen bonding and void microstructure of a-Ge:H films,” J. Appl. Phys. 76, 4453–4455 (1996).
    [CrossRef]

1996 (1)

M. Mulato, I. Chambouleyron, I. Torriani, “Hydrogen bonding and void microstructure of a-Ge:H films,” J. Appl. Phys. 76, 4453–4455 (1996).
[CrossRef]

1983 (1)

R. Swanepoel, “Determination of the thickness and optical constants of amorphous silicon,” J. Phys. E 16, 1214–1222 (1983); see also “Determination of surface roughness and optical constants of inhomogeneous amorphous silicon films,” J. Phys. E 17, 896–903 (1984).
[CrossRef]

1981 (1)

1978 (2)

1976 (1)

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 film,” J. Phys. E 9, 1002–1004 (1976).
[CrossRef]

1973 (3)

S. H. Wemple, “Refractive-index behavior of amorphous semiconductors and glasses,” Phys. Rev. B 7, 3767–3777 (1973).
[CrossRef]

G. A. N. Connell, R. J. Temkin, W. Paul, “Amorphous germanium, III. Optical properties,” Adv. Phys. 22, 643–665 (1973).
[CrossRef]

W. Paul, G. A. N. Connell, R. J. Temkin, “Amorphous germanium, I. A model for the structural and optical properties,” Adv. Phys. 22, 531–580 (1973).
[CrossRef]

1972 (1)

D. T. Pierce, W. E. Spicer, “Electronic structure of amorphous silicon from photoemission and optical studies,” Phys. Rev. B 5, 3017–3029 (1972).
[CrossRef]

1971 (2)

T. M. Donovan, K. Heinemann, “High resolution electron microscope observation of voids in amorphous Ge,” Phys. Rev. Lett. 27, 1794–1796 (1971).
[CrossRef]

S. H. Wemple, W. DiDomenico, “Behavior of the electronic dielectric constant in covalent and ionic materials,” Phys. Rev. B 3, 1338–1351 (1971).
[CrossRef]

1970 (1)

J. Tauc, “Absorption edge and internal electric fields in amorphous semiconductors,” Mater. Res. Bull. 5, 721–730 (1970).
[CrossRef]

Amer, N. M.

Boccara, A. C.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1980).

Chambouleyron, I.

M. Mulato, I. Chambouleyron, I. Torriani, “Hydrogen bonding and void microstructure of a-Ge:H films,” J. Appl. Phys. 76, 4453–4455 (1996).
[CrossRef]

Cody, G. D.

G. D. Cody, “The optical absorption edge of a-Si:H,” in Hydrogenated Amorphous Silicon, J. Pankove, ed., Vol. 21B of Semiconductors and Semimetals Series, R. K. Willardson, A. C. Beer, eds. (Academic, New York, 1984), pp. 11–82.

Connell, G. A. N.

G. A. N. Connell, R. J. Temkin, W. Paul, “Amorphous germanium, III. Optical properties,” Adv. Phys. 22, 643–665 (1973).
[CrossRef]

W. Paul, G. A. N. Connell, R. J. Temkin, “Amorphous germanium, I. A model for the structural and optical properties,” Adv. Phys. 22, 531–580 (1973).
[CrossRef]

G. A. N. Connell, “Optical properties of amorphous semiconductors,” in Amorphous Semiconductors, M. H. Brodsky, ed., Vol. 36 of Topics in Applied Physics (Springer-Verlag, Berlin, 1979), pp. 73–111.
[CrossRef]

Davis, E. A.

N. F. Mott, E. A. Davis, Electronic Processess in Non-Crystalline Materials, 2nd ed. (Oxford U. Press, Oxford, UK, 1979).

Dennis, J. E.

J. E. Dennis, R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, N.J., 1983).

DiDomenico, W.

S. H. Wemple, W. DiDomenico, “Behavior of the electronic dielectric constant in covalent and ionic materials,” Phys. Rev. B 3, 1338–1351 (1971).
[CrossRef]

Donovan, T. M.

T. M. Donovan, K. Heinemann, “High resolution electron microscope observation of voids in amorphous Ge,” Phys. Rev. Lett. 27, 1794–1796 (1971).
[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 film,” J. Phys. E 9, 1002–1004 (1976).
[CrossRef]

Fletcher, R.

R. Fletcher, Practical Methods for Optimization (Wiley, Chichester, UK, 1987).

Fournier, D.

Fuhs, W.

W. Fuhs, H.-J. Hesse, K. H. Langer, “Substrate specific voids in amorphous germanium,” in Amorphous and Liquid Semiconductors, J. Stuke, W. Brenig eds. (Taylor and Francis, London, 1974), pp. 79–84.

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 film,” J. Phys. E 9, 1002–1004 (1976).
[CrossRef]

Gill, P. E.

P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, London, 1981).

Goodman, A. M.

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1991).

Heinemann, K.

T. M. Donovan, K. Heinemann, “High resolution electron microscope observation of voids in amorphous Ge,” Phys. Rev. Lett. 27, 1794–1796 (1971).
[CrossRef]

Hesse, H.-J.

W. Fuhs, H.-J. Hesse, K. H. Langer, “Substrate specific voids in amorphous germanium,” in Amorphous and Liquid Semiconductors, J. Stuke, W. Brenig eds. (Taylor and Francis, London, 1974), pp. 79–84.

Jackson, W. B.

Langer, K. H.

W. Fuhs, H.-J. Hesse, K. H. Langer, “Substrate specific voids in amorphous germanium,” in Amorphous and Liquid Semiconductors, J. Stuke, W. Brenig eds. (Taylor and Francis, London, 1974), pp. 79–84.

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 film,” J. Phys. E 9, 1002–1004 (1976).
[CrossRef]

Mott, N. F.

N. F. Mott, E. A. Davis, Electronic Processess in Non-Crystalline Materials, 2nd ed. (Oxford U. Press, Oxford, UK, 1979).

Mulato, M.

M. Mulato, I. Chambouleyron, I. Torriani, “Hydrogen bonding and void microstructure of a-Ge:H films,” J. Appl. Phys. 76, 4453–4455 (1996).
[CrossRef]

Murray, W.

P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, London, 1981).

Murtagh, R. B.

R. B. Murtagh, M. A. Saunders, “Large scale linearly constrained optimization,” Math. Programming 14, 41–72 (1978).
[CrossRef]

R. B. Murtagh, M. A. Saunders, minos User’s Guide, , 1977 (Department of Operations Research, Stanford University, Calif.).

Paul, W.

W. Paul, G. A. N. Connell, R. J. Temkin, “Amorphous germanium, I. A model for the structural and optical properties,” Adv. Phys. 22, 531–580 (1973).
[CrossRef]

G. A. N. Connell, R. J. Temkin, W. Paul, “Amorphous germanium, III. Optical properties,” Adv. Phys. 22, 643–665 (1973).
[CrossRef]

Pierce, D. T.

D. T. Pierce, W. E. Spicer, “Electronic structure of amorphous silicon from photoemission and optical studies,” Phys. Rev. B 5, 3017–3029 (1972).
[CrossRef]

Saunders, M. A.

R. B. Murtagh, M. A. Saunders, “Large scale linearly constrained optimization,” Math. Programming 14, 41–72 (1978).
[CrossRef]

R. B. Murtagh, M. A. Saunders, minos User’s Guide, , 1977 (Department of Operations Research, Stanford University, Calif.).

Schnabel, R. B.

J. E. Dennis, R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, N.J., 1983).

Spicer, W. E.

D. T. Pierce, W. E. Spicer, “Electronic structure of amorphous silicon from photoemission and optical studies,” Phys. Rev. B 5, 3017–3029 (1972).
[CrossRef]

Street, R. A.

R. A. Street, Hydrogenated Amorphous Silicon (Cambridge U. Press, Cambridge, UK, 1991).
[CrossRef]

Swanepoel, R.

R. Swanepoel, “Determination of the thickness and optical constants of amorphous silicon,” J. Phys. E 16, 1214–1222 (1983); see also “Determination of surface roughness and optical constants of inhomogeneous amorphous silicon films,” J. Phys. E 17, 896–903 (1984).
[CrossRef]

Tauc, J.

J. Tauc, “Absorption edge and internal electric fields in amorphous semiconductors,” Mater. Res. Bull. 5, 721–730 (1970).
[CrossRef]

Temkin, R. J.

W. Paul, G. A. N. Connell, R. J. Temkin, “Amorphous germanium, I. A model for the structural and optical properties,” Adv. Phys. 22, 531–580 (1973).
[CrossRef]

G. A. N. Connell, R. J. Temkin, W. Paul, “Amorphous germanium, III. Optical properties,” Adv. Phys. 22, 643–665 (1973).
[CrossRef]

Torriani, I.

M. Mulato, I. Chambouleyron, I. Torriani, “Hydrogen bonding and void microstructure of a-Ge:H films,” J. Appl. Phys. 76, 4453–4455 (1996).
[CrossRef]

Wemple, S. H.

S. H. Wemple, “Refractive-index behavior of amorphous semiconductors and glasses,” Phys. Rev. B 7, 3767–3777 (1973).
[CrossRef]

S. H. Wemple, W. DiDomenico, “Behavior of the electronic dielectric constant in covalent and ionic materials,” Phys. Rev. B 3, 1338–1351 (1971).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1980).

Wright, M. H.

P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, London, 1981).

Adv. Phys. (2)

G. A. N. Connell, R. J. Temkin, W. Paul, “Amorphous germanium, III. Optical properties,” Adv. Phys. 22, 643–665 (1973).
[CrossRef]

W. Paul, G. A. N. Connell, R. J. Temkin, “Amorphous germanium, I. A model for the structural and optical properties,” Adv. Phys. 22, 531–580 (1973).
[CrossRef]

Appl. Opt. (2)

J. Appl. Phys. (1)

M. Mulato, I. Chambouleyron, I. Torriani, “Hydrogen bonding and void microstructure of a-Ge:H films,” J. Appl. Phys. 76, 4453–4455 (1996).
[CrossRef]

J. Phys. E (2)

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 film,” J. Phys. E 9, 1002–1004 (1976).
[CrossRef]

R. Swanepoel, “Determination of the thickness and optical constants of amorphous silicon,” J. Phys. E 16, 1214–1222 (1983); see also “Determination of surface roughness and optical constants of inhomogeneous amorphous silicon films,” J. Phys. E 17, 896–903 (1984).
[CrossRef]

Mater. Res. Bull. (1)

J. Tauc, “Absorption edge and internal electric fields in amorphous semiconductors,” Mater. Res. Bull. 5, 721–730 (1970).
[CrossRef]

Math. Programming (1)

R. B. Murtagh, M. A. Saunders, “Large scale linearly constrained optimization,” Math. Programming 14, 41–72 (1978).
[CrossRef]

Phys. Rev. B (3)

S. H. Wemple, W. DiDomenico, “Behavior of the electronic dielectric constant in covalent and ionic materials,” Phys. Rev. B 3, 1338–1351 (1971).
[CrossRef]

S. H. Wemple, “Refractive-index behavior of amorphous semiconductors and glasses,” Phys. Rev. B 7, 3767–3777 (1973).
[CrossRef]

D. T. Pierce, W. E. Spicer, “Electronic structure of amorphous silicon from photoemission and optical studies,” Phys. Rev. B 5, 3017–3029 (1972).
[CrossRef]

Phys. Rev. Lett. (1)

T. M. Donovan, K. Heinemann, “High resolution electron microscope observation of voids in amorphous Ge,” Phys. Rev. Lett. 27, 1794–1796 (1971).
[CrossRef]

Other (11)

W. Fuhs, H.-J. Hesse, K. H. Langer, “Substrate specific voids in amorphous germanium,” in Amorphous and Liquid Semiconductors, J. Stuke, W. Brenig eds. (Taylor and Francis, London, 1974), pp. 79–84.

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1991).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1980).

G. A. N. Connell, “Optical properties of amorphous semiconductors,” in Amorphous Semiconductors, M. H. Brodsky, ed., Vol. 36 of Topics in Applied Physics (Springer-Verlag, Berlin, 1979), pp. 73–111.
[CrossRef]

R. A. Street, Hydrogenated Amorphous Silicon (Cambridge U. Press, Cambridge, UK, 1991).
[CrossRef]

G. D. Cody, “The optical absorption edge of a-Si:H,” in Hydrogenated Amorphous Silicon, J. Pankove, ed., Vol. 21B of Semiconductors and Semimetals Series, R. K. Willardson, A. C. Beer, eds. (Academic, New York, 1984), pp. 11–82.

J. E. Dennis, R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, Englewood Cliffs, N.J., 1983).

N. F. Mott, E. A. Davis, Electronic Processess in Non-Crystalline Materials, 2nd ed. (Oxford U. Press, Oxford, UK, 1979).

R. Fletcher, Practical Methods for Optimization (Wiley, Chichester, UK, 1987).

P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, London, 1981).

R. B. Murtagh, M. A. Saunders, minos User’s Guide, , 1977 (Department of Operations Research, Stanford University, Calif.).

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Figures (11)

Fig. 1
Fig. 1

Typical transmission spectra in the visible and near-infrared wavelength range of thin dielectric films deposited onto a transparent substrate; film (a) is transparent at λ > 1000 nm and thick enough to display a fringe pattern, film (b) absorbs over the whole measured range, and film (c) is thin and does not show any interference pattern.

Fig. 2
Fig. 2

Transmittance against wavelength of computer-made film No. 1 (d = 100 nm) and film No. 2 (d = 600 nm). The transmission data, rounded off to three digits, are used to retrieve their thickness and optical constants.

Fig. 3
Fig. 3

Retrieved values of the (a) absorption coefficient and the (b) index of refraction for the gedanken film No. 2 in Fig. 2 against photon energy; the retrieved values are compared with the true values used to generate the transmission data. Note the overall excellent agreement of both. The initial guesses for α and n are also shown.

Fig. 4
Fig. 4

Transmittance versus wavelength of an a-Si:H film deposited onto a Corning 7059 glass substrate. The noisy data at λ > 1745 nm have not been considered in the optimization process. See the text, Subsection 4.A, for the use of Tcut in the calculation of the film thickness and the optical constants.

Fig. 5
Fig. 5

Minimization process [expression (13)] that leads to the best film thickness: a, calculation with a coarse scan step (10 nm) in the 540–660-nm thickness range; b, deep minimum found at 625 nm with a fine (1-nm) scan step calculation in the 620–630-nm thickness range. It corresponds to the true film thickness. Note that Fig. 5b is an enlargement of the minimum seen in Fig. 5a.

Fig. 6
Fig. 6

a, Absorption coefficient; and b, index of refraction of a 625-nm-thick a-Si:H film versus photon energy retrieved from the transmission data of Fig. 4. The retrieved absorption is compared with PDS data normalized at α = 104 cm-1 and with the absorption deduced from ellipsometry on an identical film deposited onto a c-Si substrate. The overall agreement for both optical constants is very good. Note that the pointwise constrained optimization approach retrieves an exponential absorption tail down to α ≅ 100 cm-1.

Fig. 7
Fig. 7

a, Optical transmission of two a-Ge films deposited onto Corning 7059 glass; the thicknesses of film No. 1 and film No. 2 are estimated from the deposition rate to be approximately 100 and 600 nm thick, respectively; b, same a-Ge films deposited in the same run onto crystalline silicon (c-Si) substrates; the transmission of glass and c-Si is also shown.

Fig. 8
Fig. 8

Optical transmission of the a-Ge/c-Si film No. 2, Fig. 7b, measured in a Perkin Elmer Lambda 9 spectrophotometer (full line) and retrieved transmission (open circles) from the minimization process [expression (13), with d = 634 nm].

Fig. 9
Fig. 9

a, Absorption coefficient, and b, index of refraction of the 634-nm-thick a-Ge film deposited onto c-Si (film No. 2, Fig. 7b); the retrieved optical constants are compared with values from the literature on rf-sputtered a-Ge deposited at 350 °C (Ref. 20).

Fig. 10
Fig. 10

Optical constants of the a-Si:H film of Fig. 4 (dashed line) as retrieved with the approximate method of Swanepoel5 (dashed line). The open circles indicate the retrieval obtained with the present method. Note the accuracy of the retrieval of the index of refraction, b; the approximate methods normally fail in the retrieval of the absorption coefficient, a.

Fig. 11
Fig. 11

Optical constants of the gedanken film No. 2, Fig. 2, calculated with Swanepoel’s method5 (dashed line). For the approximate method to be applicable, optical transmission data were calculated up to awavelength λ = 2000 nm. The optical constants estimated with the present method are displayed for comparison (open circles). The retrieval of n and α from Swanepoel’s method5 is less good than in the case of the a-Si:H film shown in Fig. 10.

Tables (1)

Tables Icon

Table 1 Results of Minimization Processa on Computer-Made Dielectric Films Deposited onto Glass

Equations (16)

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T=AxB-Cx+Dx2,
A=16sn2+k2,
B=n+12+k2n+1n+s2+k2,
C=n2-1+k2n2-s2+k2-2k2s2+12×cos φ-k2n2-s2+k2+s2+1×n2-1+k22 sin φ,
D=n-12+k2n-1n-s2+k2,
φ=4πnd/λ,  x=exp-αd,  α=4πk/λ.
Ts=2ss2+1,  or  s=1Ts+1Ts2-11/2;
Tλi, sλi, d, nλi, αλi=Ti.
αi0 and ni1 for all i=1,,N.
αi+1αi and ni+1ni.
nini-1+ni+1-ni-1λi+1-λi-1λi-λi-1.
αiαi-1+αi+1-αi-1λi+1-λi-1λi-λi-1.
Minimizei=1NTi-Tλi, s, d, ni, αi2,
ni=1+0.09195-12600/λi2-11/2,  αinm-1=10-3×exp24780/λi-34,  si=1+0.7568-7930/λi2-11/2,
i=1NTi-Tλi, s, d, ni, αi2
nλ=4.783-3.72×10-4λ-1722/λ+9.135×105/λ2,

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