Abstract

A method for solving the optical constants of a thin film on a slightly absorbing substrate is proposed. Using only a single-reflectance measurement together with the interference fringes, one can calculate with good accuracy the thickness and the real and imaginary parts of the refractive index of the film. Besides its simplicity, this method also avoids the problem of multiple solutions. We show that this technique is effective in solving for the optical constants of thin films with weak or moderate absorption characteristics, and in particular we applied it to obtain the optical properties of amorphous C thin films on Si.

© 1995 Optical Society of America

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References

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  1. J. A. Dobrowolski, F. C. Ho, A. Waldorf, “Determination of optical constants of thin-film coating material based on inverse synthesis,” Appl. Opt. 22, 3191–3200 (1983).
    [CrossRef] [PubMed]
  2. D. P. Arndt, R. M. A. Azzam, J. M. Bennett, J. P. Borgogno, C. K. Carniglia, W. E. Case, J. A. Dobrowolski, U. J. Gibson, T. T. Hart, F. C. Ho, V. A. Hodgkin, W. P. Klapp, H. A. Macleod, E. Pelletier, M. K. Purvis, D. M. Quinn, D. H. Strome, R. Swenson, P. A. Temple, T. F. Thonn, “Multiple determination of the optical constants of thin-film coating material,” Appl. Opt. 23, 3571–3596 (1984).
    [CrossRef] [PubMed]
  3. O. Stenzel, V. Hopfe, P. Klobes, “Determination of optical parameters for amorphous thin-film materials on semitransparent substrates from transmittance and reflectance measurement,” J. Phys. D 24, 2088–2094 (1991).
    [CrossRef]
  4. G. Amaratunga, W. Milne, A. Putnis, “Heterojunction diodes formed using thin-film C containing polycrystalline diamond and Si,” IEEE Electron Device Lett. 11, 33–35 (1990).
    [CrossRef]
  5. K. Okano, T. Kurosu, M. Lida, T. Eickhoff, H. Wilhelm, D. R. T. Zahn, “An optical investigation of diamond thin films on silicon,” Vacuum 41, 1387–1389 (1990).
    [CrossRef]
  6. Rusli, S. R. P. Silva, G. Amaratunga, “The optical properties of band-gap-modulated diamond-like carbon thin films,” Diamond Related Mater. 3, 817–820 (1994).
    [CrossRef]
  7. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  8. A. A. Khan, D. Mathine, J. A. Woollam, “Optical properties of diamondlike carbon films: an ellipsometric study,” Phys. Rev. B 28, 7229–7235 (1983).
    [CrossRef]
  9. G. Leveque, Y. Villachon-Renard, “Determination of optical constants of thin film from reflectance spectra,” Appl. Opt. 29, 3207–3212 (1990).
    [CrossRef] [PubMed]
  10. D. L. Windt, W. C. Cash, J. M. Scott, P. Arendt, B. Newnam, R. F. Fisher, A. B. Swartzlander, P. Z. Takacs, L. M. Pinneo, “Optical constants for thin films of C, diamond, Al, Si, and CVD SiC from 24 Å to 1216 Å,” Appl. Opt. 27, 279–295 (1988).
    [CrossRef] [PubMed]
  11. O. Stenzel, R. Petrich, S. Roth, B. Mainz, W. Scharff, “Determination of the optical constants of fine grained diamond layers on silicon substrates using curve fitting procedures,” Diamond Related Mater. 2, 704–707 (1993).
    [CrossRef]
  12. 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. 9, 1002–1004 (1976).
    [CrossRef]
  13. R. Swanepoel, “Determination of the thickness and optical constants of amorphous silicon,” J. Phys. E. 16, 1214–1222 (1983).
    [CrossRef]
  14. D. A. Minkov, “Calculation of the optical constants of a thin layer upon a transparent substrate from the reflection spectrum,” J. Phys. D 22, 1157–1161 (1989).
    [CrossRef]
  15. I. An, H. V. Nguyen, A. R. Heyd, R. W. Collins, “Simultaneous real-time spectroscopic ellipsometry and reflectance for monitoring thin-film preparation,” Rev. Sci. Instrum. 65, 3489–3500 (1994).
    [CrossRef]
  16. E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, New York, 1985).
  17. A. R. Forouhi, I. Bloomer, “Optical dispersion relations for amorphous semiconductors and amorphous dielectrics,” Phys. Rev. B 34, 7018–7026 (1986).
    [CrossRef]
  18. G. Amaratunga, A. Putnis, K. Clay, W. Milne, “Crystalline diamond growth in thin film deposited from a CH4/Ar plasma,” Appl. Phys. Lett. 55, 634–635 (1989).
    [CrossRef]
  19. L. Stichauer, G. Gavoille, “A new method for the determination of the optical constants of thin films,” Phys. Status Solidi A 133, 547–553 (1992).
    [CrossRef]

1994 (2)

I. An, H. V. Nguyen, A. R. Heyd, R. W. Collins, “Simultaneous real-time spectroscopic ellipsometry and reflectance for monitoring thin-film preparation,” Rev. Sci. Instrum. 65, 3489–3500 (1994).
[CrossRef]

Rusli, S. R. P. Silva, G. Amaratunga, “The optical properties of band-gap-modulated diamond-like carbon thin films,” Diamond Related Mater. 3, 817–820 (1994).
[CrossRef]

1993 (1)

O. Stenzel, R. Petrich, S. Roth, B. Mainz, W. Scharff, “Determination of the optical constants of fine grained diamond layers on silicon substrates using curve fitting procedures,” Diamond Related Mater. 2, 704–707 (1993).
[CrossRef]

1992 (1)

L. Stichauer, G. Gavoille, “A new method for the determination of the optical constants of thin films,” Phys. Status Solidi A 133, 547–553 (1992).
[CrossRef]

1991 (1)

O. Stenzel, V. Hopfe, P. Klobes, “Determination of optical parameters for amorphous thin-film materials on semitransparent substrates from transmittance and reflectance measurement,” J. Phys. D 24, 2088–2094 (1991).
[CrossRef]

1990 (3)

G. Amaratunga, W. Milne, A. Putnis, “Heterojunction diodes formed using thin-film C containing polycrystalline diamond and Si,” IEEE Electron Device Lett. 11, 33–35 (1990).
[CrossRef]

K. Okano, T. Kurosu, M. Lida, T. Eickhoff, H. Wilhelm, D. R. T. Zahn, “An optical investigation of diamond thin films on silicon,” Vacuum 41, 1387–1389 (1990).
[CrossRef]

G. Leveque, Y. Villachon-Renard, “Determination of optical constants of thin film from reflectance spectra,” Appl. Opt. 29, 3207–3212 (1990).
[CrossRef] [PubMed]

1989 (2)

G. Amaratunga, A. Putnis, K. Clay, W. Milne, “Crystalline diamond growth in thin film deposited from a CH4/Ar plasma,” Appl. Phys. Lett. 55, 634–635 (1989).
[CrossRef]

D. A. Minkov, “Calculation of the optical constants of a thin layer upon a transparent substrate from the reflection spectrum,” J. Phys. D 22, 1157–1161 (1989).
[CrossRef]

1988 (1)

1986 (1)

A. R. Forouhi, I. Bloomer, “Optical dispersion relations for amorphous semiconductors and amorphous dielectrics,” Phys. Rev. B 34, 7018–7026 (1986).
[CrossRef]

1984 (1)

1983 (3)

A. A. Khan, D. Mathine, J. A. Woollam, “Optical properties of diamondlike carbon films: an ellipsometric study,” Phys. Rev. B 28, 7229–7235 (1983).
[CrossRef]

J. A. Dobrowolski, F. C. Ho, A. Waldorf, “Determination of optical constants of thin-film coating material based on inverse synthesis,” Appl. Opt. 22, 3191–3200 (1983).
[CrossRef] [PubMed]

R. Swanepoel, “Determination of the thickness and optical constants of amorphous silicon,” J. Phys. E. 16, 1214–1222 (1983).
[CrossRef]

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

Amaratunga, G.

Rusli, S. R. P. Silva, G. Amaratunga, “The optical properties of band-gap-modulated diamond-like carbon thin films,” Diamond Related Mater. 3, 817–820 (1994).
[CrossRef]

G. Amaratunga, W. Milne, A. Putnis, “Heterojunction diodes formed using thin-film C containing polycrystalline diamond and Si,” IEEE Electron Device Lett. 11, 33–35 (1990).
[CrossRef]

G. Amaratunga, A. Putnis, K. Clay, W. Milne, “Crystalline diamond growth in thin film deposited from a CH4/Ar plasma,” Appl. Phys. Lett. 55, 634–635 (1989).
[CrossRef]

An, I.

I. An, H. V. Nguyen, A. R. Heyd, R. W. Collins, “Simultaneous real-time spectroscopic ellipsometry and reflectance for monitoring thin-film preparation,” Rev. Sci. Instrum. 65, 3489–3500 (1994).
[CrossRef]

Arendt, P.

Arndt, D. P.

Azzam, R. M. A.

Bashara, N. M.

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

Bennett, J. M.

Bloomer, I.

A. R. Forouhi, I. Bloomer, “Optical dispersion relations for amorphous semiconductors and amorphous dielectrics,” Phys. Rev. B 34, 7018–7026 (1986).
[CrossRef]

Borgogno, J. P.

Carniglia, C. K.

Case, W. E.

Cash, W. C.

Clay, K.

G. Amaratunga, A. Putnis, K. Clay, W. Milne, “Crystalline diamond growth in thin film deposited from a CH4/Ar plasma,” Appl. Phys. Lett. 55, 634–635 (1989).
[CrossRef]

Collins, R. W.

I. An, H. V. Nguyen, A. R. Heyd, R. W. Collins, “Simultaneous real-time spectroscopic ellipsometry and reflectance for monitoring thin-film preparation,” Rev. Sci. Instrum. 65, 3489–3500 (1994).
[CrossRef]

Dobrowolski, J. A.

Eickhoff, T.

K. Okano, T. Kurosu, M. Lida, T. Eickhoff, H. Wilhelm, D. R. T. Zahn, “An optical investigation of diamond thin films on silicon,” Vacuum 41, 1387–1389 (1990).
[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. 9, 1002–1004 (1976).
[CrossRef]

Fisher, R. F.

Forouhi, A. R.

A. R. Forouhi, I. Bloomer, “Optical dispersion relations for amorphous semiconductors and amorphous dielectrics,” Phys. Rev. B 34, 7018–7026 (1986).
[CrossRef]

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. 9, 1002–1004 (1976).
[CrossRef]

Gavoille, G.

L. Stichauer, G. Gavoille, “A new method for the determination of the optical constants of thin films,” Phys. Status Solidi A 133, 547–553 (1992).
[CrossRef]

Gibson, U. J.

Hart, T. T.

Heyd, A. R.

I. An, H. V. Nguyen, A. R. Heyd, R. W. Collins, “Simultaneous real-time spectroscopic ellipsometry and reflectance for monitoring thin-film preparation,” Rev. Sci. Instrum. 65, 3489–3500 (1994).
[CrossRef]

Ho, F. C.

Hodgkin, V. A.

Hopfe, V.

O. Stenzel, V. Hopfe, P. Klobes, “Determination of optical parameters for amorphous thin-film materials on semitransparent substrates from transmittance and reflectance measurement,” J. Phys. D 24, 2088–2094 (1991).
[CrossRef]

Khan, A. A.

A. A. Khan, D. Mathine, J. A. Woollam, “Optical properties of diamondlike carbon films: an ellipsometric study,” Phys. Rev. B 28, 7229–7235 (1983).
[CrossRef]

Klapp, W. P.

Klobes, P.

O. Stenzel, V. Hopfe, P. Klobes, “Determination of optical parameters for amorphous thin-film materials on semitransparent substrates from transmittance and reflectance measurement,” J. Phys. D 24, 2088–2094 (1991).
[CrossRef]

Kurosu, T.

K. Okano, T. Kurosu, M. Lida, T. Eickhoff, H. Wilhelm, D. R. T. Zahn, “An optical investigation of diamond thin films on silicon,” Vacuum 41, 1387–1389 (1990).
[CrossRef]

Leveque, G.

Lida, M.

K. Okano, T. Kurosu, M. Lida, T. Eickhoff, H. Wilhelm, D. R. T. Zahn, “An optical investigation of diamond thin films on silicon,” Vacuum 41, 1387–1389 (1990).
[CrossRef]

Macleod, H. A.

Mainz, B.

O. Stenzel, R. Petrich, S. Roth, B. Mainz, W. Scharff, “Determination of the optical constants of fine grained diamond layers on silicon substrates using curve fitting procedures,” Diamond Related Mater. 2, 704–707 (1993).
[CrossRef]

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. 9, 1002–1004 (1976).
[CrossRef]

Mathine, D.

A. A. Khan, D. Mathine, J. A. Woollam, “Optical properties of diamondlike carbon films: an ellipsometric study,” Phys. Rev. B 28, 7229–7235 (1983).
[CrossRef]

Milne, W.

G. Amaratunga, W. Milne, A. Putnis, “Heterojunction diodes formed using thin-film C containing polycrystalline diamond and Si,” IEEE Electron Device Lett. 11, 33–35 (1990).
[CrossRef]

G. Amaratunga, A. Putnis, K. Clay, W. Milne, “Crystalline diamond growth in thin film deposited from a CH4/Ar plasma,” Appl. Phys. Lett. 55, 634–635 (1989).
[CrossRef]

Minkov, D. A.

D. A. Minkov, “Calculation of the optical constants of a thin layer upon a transparent substrate from the reflection spectrum,” J. Phys. D 22, 1157–1161 (1989).
[CrossRef]

Newnam, B.

Nguyen, H. V.

I. An, H. V. Nguyen, A. R. Heyd, R. W. Collins, “Simultaneous real-time spectroscopic ellipsometry and reflectance for monitoring thin-film preparation,” Rev. Sci. Instrum. 65, 3489–3500 (1994).
[CrossRef]

Okano, K.

K. Okano, T. Kurosu, M. Lida, T. Eickhoff, H. Wilhelm, D. R. T. Zahn, “An optical investigation of diamond thin films on silicon,” Vacuum 41, 1387–1389 (1990).
[CrossRef]

Pelletier, E.

Petrich, R.

O. Stenzel, R. Petrich, S. Roth, B. Mainz, W. Scharff, “Determination of the optical constants of fine grained diamond layers on silicon substrates using curve fitting procedures,” Diamond Related Mater. 2, 704–707 (1993).
[CrossRef]

Pinneo, L. M.

Purvis, M. K.

Putnis, A.

G. Amaratunga, W. Milne, A. Putnis, “Heterojunction diodes formed using thin-film C containing polycrystalline diamond and Si,” IEEE Electron Device Lett. 11, 33–35 (1990).
[CrossRef]

G. Amaratunga, A. Putnis, K. Clay, W. Milne, “Crystalline diamond growth in thin film deposited from a CH4/Ar plasma,” Appl. Phys. Lett. 55, 634–635 (1989).
[CrossRef]

Quinn, D. M.

Roth, S.

O. Stenzel, R. Petrich, S. Roth, B. Mainz, W. Scharff, “Determination of the optical constants of fine grained diamond layers on silicon substrates using curve fitting procedures,” Diamond Related Mater. 2, 704–707 (1993).
[CrossRef]

Rusli,

Rusli, S. R. P. Silva, G. Amaratunga, “The optical properties of band-gap-modulated diamond-like carbon thin films,” Diamond Related Mater. 3, 817–820 (1994).
[CrossRef]

Scharff, W.

O. Stenzel, R. Petrich, S. Roth, B. Mainz, W. Scharff, “Determination of the optical constants of fine grained diamond layers on silicon substrates using curve fitting procedures,” Diamond Related Mater. 2, 704–707 (1993).
[CrossRef]

Scott, J. M.

Silva, S. R. P.

Rusli, S. R. P. Silva, G. Amaratunga, “The optical properties of band-gap-modulated diamond-like carbon thin films,” Diamond Related Mater. 3, 817–820 (1994).
[CrossRef]

Stenzel, O.

O. Stenzel, R. Petrich, S. Roth, B. Mainz, W. Scharff, “Determination of the optical constants of fine grained diamond layers on silicon substrates using curve fitting procedures,” Diamond Related Mater. 2, 704–707 (1993).
[CrossRef]

O. Stenzel, V. Hopfe, P. Klobes, “Determination of optical parameters for amorphous thin-film materials on semitransparent substrates from transmittance and reflectance measurement,” J. Phys. D 24, 2088–2094 (1991).
[CrossRef]

Stichauer, L.

L. Stichauer, G. Gavoille, “A new method for the determination of the optical constants of thin films,” Phys. Status Solidi A 133, 547–553 (1992).
[CrossRef]

Strome, D. H.

Swanepoel, R.

R. Swanepoel, “Determination of the thickness and optical constants of amorphous silicon,” J. Phys. E. 16, 1214–1222 (1983).
[CrossRef]

Swartzlander, A. B.

Swenson, R.

Takacs, P. Z.

Temple, P. A.

Thonn, T. F.

Villachon-Renard, Y.

Waldorf, A.

Wilhelm, H.

K. Okano, T. Kurosu, M. Lida, T. Eickhoff, H. Wilhelm, D. R. T. Zahn, “An optical investigation of diamond thin films on silicon,” Vacuum 41, 1387–1389 (1990).
[CrossRef]

Windt, D. L.

Woollam, J. A.

A. A. Khan, D. Mathine, J. A. Woollam, “Optical properties of diamondlike carbon films: an ellipsometric study,” Phys. Rev. B 28, 7229–7235 (1983).
[CrossRef]

Zahn, D. R. T.

K. Okano, T. Kurosu, M. Lida, T. Eickhoff, H. Wilhelm, D. R. T. Zahn, “An optical investigation of diamond thin films on silicon,” Vacuum 41, 1387–1389 (1990).
[CrossRef]

Appl. Opt. (4)

Appl. Phys. Lett. (1)

G. Amaratunga, A. Putnis, K. Clay, W. Milne, “Crystalline diamond growth in thin film deposited from a CH4/Ar plasma,” Appl. Phys. Lett. 55, 634–635 (1989).
[CrossRef]

Diamond Related Mater. (2)

Rusli, S. R. P. Silva, G. Amaratunga, “The optical properties of band-gap-modulated diamond-like carbon thin films,” Diamond Related Mater. 3, 817–820 (1994).
[CrossRef]

O. Stenzel, R. Petrich, S. Roth, B. Mainz, W. Scharff, “Determination of the optical constants of fine grained diamond layers on silicon substrates using curve fitting procedures,” Diamond Related Mater. 2, 704–707 (1993).
[CrossRef]

IEEE Electron Device Lett. (1)

G. Amaratunga, W. Milne, A. Putnis, “Heterojunction diodes formed using thin-film C containing polycrystalline diamond and Si,” IEEE Electron Device Lett. 11, 33–35 (1990).
[CrossRef]

J. Phys. D (2)

O. Stenzel, V. Hopfe, P. Klobes, “Determination of optical parameters for amorphous thin-film materials on semitransparent substrates from transmittance and reflectance measurement,” J. Phys. D 24, 2088–2094 (1991).
[CrossRef]

D. A. Minkov, “Calculation of the optical constants of a thin layer upon a transparent substrate from the reflection spectrum,” J. Phys. D 22, 1157–1161 (1989).
[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 thin 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).
[CrossRef]

Phys. Rev. B (2)

A. A. Khan, D. Mathine, J. A. Woollam, “Optical properties of diamondlike carbon films: an ellipsometric study,” Phys. Rev. B 28, 7229–7235 (1983).
[CrossRef]

A. R. Forouhi, I. Bloomer, “Optical dispersion relations for amorphous semiconductors and amorphous dielectrics,” Phys. Rev. B 34, 7018–7026 (1986).
[CrossRef]

Phys. Status Solidi A (1)

L. Stichauer, G. Gavoille, “A new method for the determination of the optical constants of thin films,” Phys. Status Solidi A 133, 547–553 (1992).
[CrossRef]

Rev. Sci. Instrum. (1)

I. An, H. V. Nguyen, A. R. Heyd, R. W. Collins, “Simultaneous real-time spectroscopic ellipsometry and reflectance for monitoring thin-film preparation,” Rev. Sci. Instrum. 65, 3489–3500 (1994).
[CrossRef]

Vacuum (1)

K. Okano, T. Kurosu, M. Lida, T. Eickhoff, H. Wilhelm, D. R. T. Zahn, “An optical investigation of diamond thin films on silicon,” Vacuum 41, 1387–1389 (1990).
[CrossRef]

Other (2)

E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, New York, 1985).

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

Cited By

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


Figures (9)

Fig. 1
Fig. 1

System of an absorbing thin film on a thick absorbing substrate.

Fig. 2
Fig. 2

Continuous curve, reflectance of the hypothetical diamondlike C film on Si; dashed curves, envelopes for the interference fringes; dashed-dot curve, bare Si reflectance with k 2 = 0.

Fig. 3
Fig. 3

Data points +, ○, two solutions for n 1 obtained in the transparent, weak, and medium absorption regions; continuous curve, values of n 1 of the hypothetical film.

Fig. 4
Fig. 4

Data points +, ○, two solutions for x obtained in the transparent, weak, and medium absorption regions; continuous curve, values of x of the hypothetical film.

Fig. 5
Fig. 5

Dashed curve, measured transmittance; continuous curve, measured reflectance spectra of the diamondlike C film grown on glass substrate.

Fig. 6
Fig. 6

Continuous curve, measured reflectance of diamondlike C film grown on a Si substrate; dashed curves, envelopes for the constructive and destructive interference fringes; dashed–dot curve, reflectance of a bare Si substrate.

Fig. 7
Fig. 7

Data points × and +, two first trial solutions for n 1 obtained for the film grown on the Si substrate; ○, the final n 1 obtained after selection of the correct solution and correction for the order number.

Fig. 8
Fig. 8

n 1 of the diamond C film grown obtained by three different methods: ×, that obtained by our proposed method; ○, that obtained from the conventional R, T method; +, that obtained from the Swanepoel method.

Fig. 9
Fig. 9

k 1 of the diamond C film grown obtained by three different methods: ×, that obtained by our proposed method; ○, that obtained from the conventional R, T method; +, that obtained from the Swanepoel method.

Tables (2)

Tables Icon

Table 1 Values of the Optical Constants and Thickness Solved by Use of Constructive Interference for the Hypothetical Filma

Tables Icon

Table 2 Values of the Optical Constants and Thickness Solved by Use of Destructive Interference for the Hypothetical Filma

Equations (31)

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

r = r 01 + r 12 exp ( - i 2 δ 1 ) 1 + r 01 r 12 exp ( - i 2 δ 1 ) ,
δ 1 = 2 π d 1 n 1 λ ,
r 01 = 1 - n 1 1 + n 1 ,
r 12 = n 1 - n 2 n 1 + n 2 .
n 1 2 k 1 2 , n 2 2 k 2 2 .
2 n 1 d 1 = { m λ constructive ( m + ½ ) λ destructive ,
R c = F x 2 + G x + E I x 2 + G x + H ,
R d = F x 2 - G x + E I x 2 - G x + H ,
x = exp ( - α d 1 ) , α = 4 π k 1 λ , E = ( 1 - n 1 ) 2 ( n 1 + n 2 ) 2 , F = ( 1 + n 1 ) 2 ( n 1 - n 2 ) 2 , G = 2 ( 1 - n 1 2 ) ( n 1 2 - n 2 2 ) , H = ( 1 + n 1 ) 2 ( n 1 + n 2 ) 2 , I = ( 1 - n 1 ) 2 ( n 1 - n 2 ) 2 .
R c = ( 1 - n 2 ) 2 ( 1 + n 2 ) 2 ,
R d = ( n 1 2 - n 2 ) 2 ( n 1 2 + n 2 ) 2 .
x = ( n 1 2 - 1 ) ( 1 - R c ) - 4 n 1 R c ( 1 - n 1 ) 2 R c - ( 1 + n 1 ) 2 n 2 + n 1 n 2 - n 1 ,
x = - ( n 1 2 - 1 ) ( 1 - R d ) ± 4 n 1 R d ( 1 - n 1 ) 2 R d - ( 1 + n 1 ) 2 n 2 + n 1 n 2 - n 1 .
C 1 y 4 + C 2 y 3 + C 3 y 2 + C 4 y + C 5 = 0 ,
y = 1 - n 1 1 + n 1 , C 1 = ( R c R d ) 1 / 2 ( R c - R d ) , C 2 = R c + R d - 2 R c R d , C 3 = [ ( R c R d ) 1 / 2 ± 1 ] ( ± R c + R d ) , C 4 = R c + R d - 2 , C 5 = R d - R c .
d 1 = λ i λ j 2 ( λ i n j - λ j n i ) ,
R E H .
n th ( ) = n ( ) + B 0 + C 0 2 - B + C ,
k th ( ) = A ( - E g ) 2 2 - B + C ,
B 0 = A Q ( - B 2 2 + E g B - E g 2 + C ) ,
C 0 = A Q [ ( E g 2 + C ) B 2 - 2 E g C ] ,
Q = 1 2 ( 4 C - B 2 ) 1 / 2 ,
E r r = region A ( n th - n 1 ) 2 + ( k th - k 1 ) 2 + region B ( R th - R ) 2 ,
Δ n 1 | d n 1 d y | Δ y ,
Δ y | y R c | Δ R c + | y R d | Δ R d ,
y R j = - C 1 R j y 4 + C 2 R j y 3 + C 3 R j y 2 + C 4 R j y + C 5 R j 4 C 1 y 3 + 3 C 2 y 2 + 2 C 3 y + C 4
C 1 R d - C 2 R d + C 3 R d - C 4 R d + C 5 R d ,
Δ x | x n 1 | Δ n 1 + | x R c | Δ R c .
R = F x 2 + G x + E I x 2 + J x + H ,
φ = 4 π n 1 d 1 λ , α = 4 π k 1 λ , x = exp ( - α d 1 ) , A = ( 1 - n 1 2 - k 1 2 ) ( n 1 2 - n 2 2 + k 1 2 - k 2 2 ) + 4 k 1 ( n 1 k 2 - n 2 k 1 ) , B = 2 ( 1 - n 1 2 - k 1 2 ) ( n 1 k 2 - n 2 k 1 ) - 2 k 1 ( n 1 2 - n 2 2 + k 1 2 - k 2 2 ) , C = ( 1 - n 1 2 - k 1 2 ) ( n 1 2 - n 2 2 + k 1 2 - k 2 2 ) - 4 k 1 ( n 1 k 2 - n 2 k 1 ) , D = 2 ( 1 - n 1 2 - k 1 2 ) ( n 1 k 2 - n 2 k 1 ) + 2 k 1 ( n 1 2 - n 2 2 + k 1 2 - k 2 2 ) , E = [ ( 1 - n 1 ) 2 + k 1 2 ] [ ( n 1 + n 2 ) 2 + ( k 1 + k 2 ) 2 ] , F = [ ( 1 + n 1 ) 2 + k 1 2 ] [ ( n 1 - n 2 ) 2 + ( k 1 - k 2 ) 2 ] , G = 2 [ A cos ( φ ) + B sin ( φ ) ] , H = [ ( 1 + n 1 ) 2 + k 1 2 ] [ ( n 1 + n 2 ) 2 + ( k 1 + k 2 ) 2 ] , I = [ ( 1 - n 1 ) 2 + k 1 2 ] [ ( n 1 - n 2 ) 2 + ( k 1 - k 2 ) 2 ] , J = 2 [ C cos ( φ ) + D sin ( φ ) ] .
C 1 R c = R d - R d 2 R c , C 2 R c = 1 - 2 R d , C 3 R c = ± R d + R d + 1 2 R c , C 4 R c = 1 , C 5 R c = - 1 2 R c , C 1 R d = R c 2 R d - R c , C 2 R d = 1 - 2 R c , C 3 R d = R c ± R c + 1 2 R d , C 4 R d = 1 , C 5 R d = 1 2 R d .

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