Abstract

Optical interference fringe measurements of the thickness of transparent layers can be rapid, accurate, and nondestructive. If the refractive index n of the layer being measured is known, it may be combined directly with interference fringe information to yield the layer thickness t. If, however, n is unknown, the measurement procedure necessarily becomes more complicated. In this paper, a new and simpler optical interference method is presented for the approximate determination of both n and t of a transparent layer on a transparent substrate. The required experimental information is obtained from a single spectrophotometric recording of either the reflectance or transmittance of the layer and its substrate. The theory of the method is presented, and an application of the method to measurements of layers of SIPOS (Semi-Insulating POlycrystalline Silicon) is described. The method requires that the layer being measured must be uniformly deposited on a flat substrate, and it must neither absorb nor scatter the light passing through it. The major approximation inherent in the method is that both the layer and the substrate are assumed to be nondispersive over the wavelength region of interest.

© 1978 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, 1965), Chap. 4, 5.
  2. W. A. Pliskin, “Nondestructive Optical Techniques for Thin-Film Thickness Measurements,” in Physical Measurements and Analysis of Thin Films, E. M. Murt, W. G. Gulder, Eds. (Plenum, New York, 1969), Chap. 1.
  3. K. Blodgett, I. Langmuir, Phys. Rev. 51, 964 (1937).
    [CrossRef]
  4. W. A. Pliskin, E. E. Conrad, IBM J. Res. Dev. 8, 43 (1964).
    [CrossRef]
  5. E. Q. Adams, L. S. Ickis, Gen. Electr. Rev. 42, 450 (1939).
  6. K. M. Greenland, Vacuum 2, 216 (1952).
    [CrossRef]
  7. P. Bousquet, Opt. Acta 3, 153 (1956).
    [CrossRef]
  8. E. P. Brightwell, ASTM Bull. 237, 67 (April1959).
  9. A-F. Bogenschütz, F. Bergmann, J. Jentsch, Z. Angew. Phys. 14, 469 (1962).
  10. E. A. Corl, H. Wimpfheimer, Solid State Electron. 7, 755 (1964).
    [CrossRef]
  11. F. Reizmann, J. Appl. Phys. 36, 3804 (1965).
    [CrossRef]
  12. J. C. Banter, J. Electrochem. Soc. 112, 388 (1965).
    [CrossRef]
  13. N. Goldsmith, L. A. Murray, Solid State Electron. 9, 331 (1966).
    [CrossRef]
  14. E. A. Corl, K. Kosanke, Solid State Electron. 9, 943 (1966).
    [CrossRef]
  15. L. A. Murray, N. Goldsmith, E. L. Jordan, Electrochem. Technol. 4, 508 (1966).
  16. F. Lukes, E. Schmidt, Solid State Electron. 10, 264 (1967).
    [CrossRef]
  17. F. Reizmann, W. Van Gelder, Solid State Electron. 10, 625 (1967).
    [CrossRef]
  18. B. Celustka, A. Persin, D. Bidjin, J. Appl. Phys. 41, 813 (1970).
    [CrossRef]
  19. M. Ruiz-Urbieta, E. M. Sparrow, E. R. G. Eckert, J. Opt. Soc. Am. 61, 351 (1971).
    [CrossRef]
  20. B. Jordanov, L. Popova, Solid State Electron. 14, 753 (1971).
    [CrossRef]
  21. N. J. Harrick, Appl. Opt. 10, 2344 (1971).
    [CrossRef] [PubMed]
  22. C. J. Dell’Oca, J. Electrochem. Soc. 119, 108 (1972).
    [CrossRef]
  23. F. Schwidefsky, F. Holtz, Thin Solid Films 17, 93 (1973).
    [CrossRef]
  24. T. Nakagawa, N. Kono, K. Asama, IEEE Trans. Magnetics Mag-9, 1397 (1975).
    [CrossRef]
  25. F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1950), p. 255.
  26. T. Matsushita, T. Aoki, T. Otsu, H. Yamoto, H. Hayashi, M. Okayama, Y. Kawana, Jpn. J. Appl. Phys. Suppl. 15, 35 (1976).
  27. H. Mochizuki, T. Aoki, H. Yamoto, M. Okayama, M. Abe, T. Ando, Jpn. J. Appl. Phys. Suppl. 15, 41 (1976).
  28. M. Hamasaki, T. Adachi, S. Wakayama, M. Kikuchi, Solid State Commun. 21, 591 (1977).
    [CrossRef]
  29. I. H. Malitson, J. Opt. Soc. Am. 52, 1377 (1962).
    [CrossRef]
  30. H. B. Briggs, Phys. Rev. 77, 287 (1950).
    [CrossRef]
  31. C. D. Salzberg, J. J. Villa, J. Opt. Soc. Am. 47, 244 (1957).
    [CrossRef]
  32. B. H. Billings, “Optics,” in American Institute of Physics Handbook, D. E. Gray, Ed. (McGraw-Hill, New York, 1963), Chap. 6.
  33. M. E. Hills, A. L. Olsen, L. W. Nichols, Appl. Opt. 7, 1437 (1968).
    [CrossRef] [PubMed]
  34. A. K. E. Hagopian, Chem. Phys. Lett. 12, 327 (1971).
    [CrossRef]

1977 (1)

M. Hamasaki, T. Adachi, S. Wakayama, M. Kikuchi, Solid State Commun. 21, 591 (1977).
[CrossRef]

1976 (2)

T. Matsushita, T. Aoki, T. Otsu, H. Yamoto, H. Hayashi, M. Okayama, Y. Kawana, Jpn. J. Appl. Phys. Suppl. 15, 35 (1976).

H. Mochizuki, T. Aoki, H. Yamoto, M. Okayama, M. Abe, T. Ando, Jpn. J. Appl. Phys. Suppl. 15, 41 (1976).

1975 (1)

T. Nakagawa, N. Kono, K. Asama, IEEE Trans. Magnetics Mag-9, 1397 (1975).
[CrossRef]

1973 (1)

F. Schwidefsky, F. Holtz, Thin Solid Films 17, 93 (1973).
[CrossRef]

1972 (1)

C. J. Dell’Oca, J. Electrochem. Soc. 119, 108 (1972).
[CrossRef]

1971 (4)

B. Jordanov, L. Popova, Solid State Electron. 14, 753 (1971).
[CrossRef]

A. K. E. Hagopian, Chem. Phys. Lett. 12, 327 (1971).
[CrossRef]

M. Ruiz-Urbieta, E. M. Sparrow, E. R. G. Eckert, J. Opt. Soc. Am. 61, 351 (1971).
[CrossRef]

N. J. Harrick, Appl. Opt. 10, 2344 (1971).
[CrossRef] [PubMed]

1970 (1)

B. Celustka, A. Persin, D. Bidjin, J. Appl. Phys. 41, 813 (1970).
[CrossRef]

1968 (1)

1967 (2)

F. Lukes, E. Schmidt, Solid State Electron. 10, 264 (1967).
[CrossRef]

F. Reizmann, W. Van Gelder, Solid State Electron. 10, 625 (1967).
[CrossRef]

1966 (3)

N. Goldsmith, L. A. Murray, Solid State Electron. 9, 331 (1966).
[CrossRef]

E. A. Corl, K. Kosanke, Solid State Electron. 9, 943 (1966).
[CrossRef]

L. A. Murray, N. Goldsmith, E. L. Jordan, Electrochem. Technol. 4, 508 (1966).

1965 (2)

F. Reizmann, J. Appl. Phys. 36, 3804 (1965).
[CrossRef]

J. C. Banter, J. Electrochem. Soc. 112, 388 (1965).
[CrossRef]

1964 (2)

W. A. Pliskin, E. E. Conrad, IBM J. Res. Dev. 8, 43 (1964).
[CrossRef]

E. A. Corl, H. Wimpfheimer, Solid State Electron. 7, 755 (1964).
[CrossRef]

1962 (2)

A-F. Bogenschütz, F. Bergmann, J. Jentsch, Z. Angew. Phys. 14, 469 (1962).

I. H. Malitson, J. Opt. Soc. Am. 52, 1377 (1962).
[CrossRef]

1959 (1)

E. P. Brightwell, ASTM Bull. 237, 67 (April1959).

1957 (1)

1956 (1)

P. Bousquet, Opt. Acta 3, 153 (1956).
[CrossRef]

1952 (1)

K. M. Greenland, Vacuum 2, 216 (1952).
[CrossRef]

1950 (1)

H. B. Briggs, Phys. Rev. 77, 287 (1950).
[CrossRef]

1939 (1)

E. Q. Adams, L. S. Ickis, Gen. Electr. Rev. 42, 450 (1939).

1937 (1)

K. Blodgett, I. Langmuir, Phys. Rev. 51, 964 (1937).
[CrossRef]

Abe, M.

H. Mochizuki, T. Aoki, H. Yamoto, M. Okayama, M. Abe, T. Ando, Jpn. J. Appl. Phys. Suppl. 15, 41 (1976).

Adachi, T.

M. Hamasaki, T. Adachi, S. Wakayama, M. Kikuchi, Solid State Commun. 21, 591 (1977).
[CrossRef]

Adams, E. Q.

E. Q. Adams, L. S. Ickis, Gen. Electr. Rev. 42, 450 (1939).

Ando, T.

H. Mochizuki, T. Aoki, H. Yamoto, M. Okayama, M. Abe, T. Ando, Jpn. J. Appl. Phys. Suppl. 15, 41 (1976).

Aoki, T.

H. Mochizuki, T. Aoki, H. Yamoto, M. Okayama, M. Abe, T. Ando, Jpn. J. Appl. Phys. Suppl. 15, 41 (1976).

T. Matsushita, T. Aoki, T. Otsu, H. Yamoto, H. Hayashi, M. Okayama, Y. Kawana, Jpn. J. Appl. Phys. Suppl. 15, 35 (1976).

Asama, K.

T. Nakagawa, N. Kono, K. Asama, IEEE Trans. Magnetics Mag-9, 1397 (1975).
[CrossRef]

Banter, J. C.

J. C. Banter, J. Electrochem. Soc. 112, 388 (1965).
[CrossRef]

Bergmann, F.

A-F. Bogenschütz, F. Bergmann, J. Jentsch, Z. Angew. Phys. 14, 469 (1962).

Bidjin, D.

B. Celustka, A. Persin, D. Bidjin, J. Appl. Phys. 41, 813 (1970).
[CrossRef]

Billings, B. H.

B. H. Billings, “Optics,” in American Institute of Physics Handbook, D. E. Gray, Ed. (McGraw-Hill, New York, 1963), Chap. 6.

Blodgett, K.

K. Blodgett, I. Langmuir, Phys. Rev. 51, 964 (1937).
[CrossRef]

Bogenschütz, A-F.

A-F. Bogenschütz, F. Bergmann, J. Jentsch, Z. Angew. Phys. 14, 469 (1962).

Bousquet, P.

P. Bousquet, Opt. Acta 3, 153 (1956).
[CrossRef]

Briggs, H. B.

H. B. Briggs, Phys. Rev. 77, 287 (1950).
[CrossRef]

Brightwell, E. P.

E. P. Brightwell, ASTM Bull. 237, 67 (April1959).

Celustka, B.

B. Celustka, A. Persin, D. Bidjin, J. Appl. Phys. 41, 813 (1970).
[CrossRef]

Conrad, E. E.

W. A. Pliskin, E. E. Conrad, IBM J. Res. Dev. 8, 43 (1964).
[CrossRef]

Corl, E. A.

E. A. Corl, K. Kosanke, Solid State Electron. 9, 943 (1966).
[CrossRef]

E. A. Corl, H. Wimpfheimer, Solid State Electron. 7, 755 (1964).
[CrossRef]

Dell’Oca, C. J.

C. J. Dell’Oca, J. Electrochem. Soc. 119, 108 (1972).
[CrossRef]

Eckert, E. R. G.

Goldsmith, N.

N. Goldsmith, L. A. Murray, Solid State Electron. 9, 331 (1966).
[CrossRef]

L. A. Murray, N. Goldsmith, E. L. Jordan, Electrochem. Technol. 4, 508 (1966).

Greenland, K. M.

K. M. Greenland, Vacuum 2, 216 (1952).
[CrossRef]

Hagopian, A. K. E.

A. K. E. Hagopian, Chem. Phys. Lett. 12, 327 (1971).
[CrossRef]

Hamasaki, M.

M. Hamasaki, T. Adachi, S. Wakayama, M. Kikuchi, Solid State Commun. 21, 591 (1977).
[CrossRef]

Harrick, N. J.

Hayashi, H.

T. Matsushita, T. Aoki, T. Otsu, H. Yamoto, H. Hayashi, M. Okayama, Y. Kawana, Jpn. J. Appl. Phys. Suppl. 15, 35 (1976).

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1965), Chap. 4, 5.

Hills, M. E.

Holtz, F.

F. Schwidefsky, F. Holtz, Thin Solid Films 17, 93 (1973).
[CrossRef]

Ickis, L. S.

E. Q. Adams, L. S. Ickis, Gen. Electr. Rev. 42, 450 (1939).

Jenkins, F. A.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1950), p. 255.

Jentsch, J.

A-F. Bogenschütz, F. Bergmann, J. Jentsch, Z. Angew. Phys. 14, 469 (1962).

Jordan, E. L.

L. A. Murray, N. Goldsmith, E. L. Jordan, Electrochem. Technol. 4, 508 (1966).

Jordanov, B.

B. Jordanov, L. Popova, Solid State Electron. 14, 753 (1971).
[CrossRef]

Kawana, Y.

T. Matsushita, T. Aoki, T. Otsu, H. Yamoto, H. Hayashi, M. Okayama, Y. Kawana, Jpn. J. Appl. Phys. Suppl. 15, 35 (1976).

Kikuchi, M.

M. Hamasaki, T. Adachi, S. Wakayama, M. Kikuchi, Solid State Commun. 21, 591 (1977).
[CrossRef]

Kono, N.

T. Nakagawa, N. Kono, K. Asama, IEEE Trans. Magnetics Mag-9, 1397 (1975).
[CrossRef]

Kosanke, K.

E. A. Corl, K. Kosanke, Solid State Electron. 9, 943 (1966).
[CrossRef]

Langmuir, I.

K. Blodgett, I. Langmuir, Phys. Rev. 51, 964 (1937).
[CrossRef]

Lukes, F.

F. Lukes, E. Schmidt, Solid State Electron. 10, 264 (1967).
[CrossRef]

Malitson, I. H.

Matsushita, T.

T. Matsushita, T. Aoki, T. Otsu, H. Yamoto, H. Hayashi, M. Okayama, Y. Kawana, Jpn. J. Appl. Phys. Suppl. 15, 35 (1976).

Mochizuki, H.

H. Mochizuki, T. Aoki, H. Yamoto, M. Okayama, M. Abe, T. Ando, Jpn. J. Appl. Phys. Suppl. 15, 41 (1976).

Murray, L. A.

L. A. Murray, N. Goldsmith, E. L. Jordan, Electrochem. Technol. 4, 508 (1966).

N. Goldsmith, L. A. Murray, Solid State Electron. 9, 331 (1966).
[CrossRef]

Nakagawa, T.

T. Nakagawa, N. Kono, K. Asama, IEEE Trans. Magnetics Mag-9, 1397 (1975).
[CrossRef]

Nichols, L. W.

Okayama, M.

H. Mochizuki, T. Aoki, H. Yamoto, M. Okayama, M. Abe, T. Ando, Jpn. J. Appl. Phys. Suppl. 15, 41 (1976).

T. Matsushita, T. Aoki, T. Otsu, H. Yamoto, H. Hayashi, M. Okayama, Y. Kawana, Jpn. J. Appl. Phys. Suppl. 15, 35 (1976).

Olsen, A. L.

Otsu, T.

T. Matsushita, T. Aoki, T. Otsu, H. Yamoto, H. Hayashi, M. Okayama, Y. Kawana, Jpn. J. Appl. Phys. Suppl. 15, 35 (1976).

Persin, A.

B. Celustka, A. Persin, D. Bidjin, J. Appl. Phys. 41, 813 (1970).
[CrossRef]

Pliskin, W. A.

W. A. Pliskin, E. E. Conrad, IBM J. Res. Dev. 8, 43 (1964).
[CrossRef]

W. A. Pliskin, “Nondestructive Optical Techniques for Thin-Film Thickness Measurements,” in Physical Measurements and Analysis of Thin Films, E. M. Murt, W. G. Gulder, Eds. (Plenum, New York, 1969), Chap. 1.

Popova, L.

B. Jordanov, L. Popova, Solid State Electron. 14, 753 (1971).
[CrossRef]

Reizmann, F.

F. Reizmann, W. Van Gelder, Solid State Electron. 10, 625 (1967).
[CrossRef]

F. Reizmann, J. Appl. Phys. 36, 3804 (1965).
[CrossRef]

Ruiz-Urbieta, M.

Salzberg, C. D.

Schmidt, E.

F. Lukes, E. Schmidt, Solid State Electron. 10, 264 (1967).
[CrossRef]

Schwidefsky, F.

F. Schwidefsky, F. Holtz, Thin Solid Films 17, 93 (1973).
[CrossRef]

Sparrow, E. M.

Van Gelder, W.

F. Reizmann, W. Van Gelder, Solid State Electron. 10, 625 (1967).
[CrossRef]

Villa, J. J.

Wakayama, S.

M. Hamasaki, T. Adachi, S. Wakayama, M. Kikuchi, Solid State Commun. 21, 591 (1977).
[CrossRef]

White, H. E.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1950), p. 255.

Wimpfheimer, H.

E. A. Corl, H. Wimpfheimer, Solid State Electron. 7, 755 (1964).
[CrossRef]

Yamoto, H.

T. Matsushita, T. Aoki, T. Otsu, H. Yamoto, H. Hayashi, M. Okayama, Y. Kawana, Jpn. J. Appl. Phys. Suppl. 15, 35 (1976).

H. Mochizuki, T. Aoki, H. Yamoto, M. Okayama, M. Abe, T. Ando, Jpn. J. Appl. Phys. Suppl. 15, 41 (1976).

Appl. Opt. (2)

ASTM Bull. (1)

E. P. Brightwell, ASTM Bull. 237, 67 (April1959).

Chem. Phys. Lett. (1)

A. K. E. Hagopian, Chem. Phys. Lett. 12, 327 (1971).
[CrossRef]

Electrochem. Technol. (1)

L. A. Murray, N. Goldsmith, E. L. Jordan, Electrochem. Technol. 4, 508 (1966).

Gen. Electr. Rev. (1)

E. Q. Adams, L. S. Ickis, Gen. Electr. Rev. 42, 450 (1939).

IBM J. Res. Dev. (1)

W. A. Pliskin, E. E. Conrad, IBM J. Res. Dev. 8, 43 (1964).
[CrossRef]

IEEE Trans. Magnetics (1)

T. Nakagawa, N. Kono, K. Asama, IEEE Trans. Magnetics Mag-9, 1397 (1975).
[CrossRef]

J. Appl. Phys. (2)

B. Celustka, A. Persin, D. Bidjin, J. Appl. Phys. 41, 813 (1970).
[CrossRef]

F. Reizmann, J. Appl. Phys. 36, 3804 (1965).
[CrossRef]

J. Electrochem. Soc. (2)

J. C. Banter, J. Electrochem. Soc. 112, 388 (1965).
[CrossRef]

C. J. Dell’Oca, J. Electrochem. Soc. 119, 108 (1972).
[CrossRef]

J. Opt. Soc. Am. (3)

Jpn. J. Appl. Phys. Suppl. (2)

T. Matsushita, T. Aoki, T. Otsu, H. Yamoto, H. Hayashi, M. Okayama, Y. Kawana, Jpn. J. Appl. Phys. Suppl. 15, 35 (1976).

H. Mochizuki, T. Aoki, H. Yamoto, M. Okayama, M. Abe, T. Ando, Jpn. J. Appl. Phys. Suppl. 15, 41 (1976).

Opt. Acta (1)

P. Bousquet, Opt. Acta 3, 153 (1956).
[CrossRef]

Phys. Rev. (2)

K. Blodgett, I. Langmuir, Phys. Rev. 51, 964 (1937).
[CrossRef]

H. B. Briggs, Phys. Rev. 77, 287 (1950).
[CrossRef]

Solid State Commun. (1)

M. Hamasaki, T. Adachi, S. Wakayama, M. Kikuchi, Solid State Commun. 21, 591 (1977).
[CrossRef]

Solid State Electron. (6)

B. Jordanov, L. Popova, Solid State Electron. 14, 753 (1971).
[CrossRef]

E. A. Corl, H. Wimpfheimer, Solid State Electron. 7, 755 (1964).
[CrossRef]

F. Lukes, E. Schmidt, Solid State Electron. 10, 264 (1967).
[CrossRef]

F. Reizmann, W. Van Gelder, Solid State Electron. 10, 625 (1967).
[CrossRef]

N. Goldsmith, L. A. Murray, Solid State Electron. 9, 331 (1966).
[CrossRef]

E. A. Corl, K. Kosanke, Solid State Electron. 9, 943 (1966).
[CrossRef]

Thin Solid Films (1)

F. Schwidefsky, F. Holtz, Thin Solid Films 17, 93 (1973).
[CrossRef]

Vacuum (1)

K. M. Greenland, Vacuum 2, 216 (1952).
[CrossRef]

Z. Angew. Phys. (1)

A-F. Bogenschütz, F. Bergmann, J. Jentsch, Z. Angew. Phys. 14, 469 (1962).

Other (4)

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, New York, 1965), Chap. 4, 5.

W. A. Pliskin, “Nondestructive Optical Techniques for Thin-Film Thickness Measurements,” in Physical Measurements and Analysis of Thin Films, E. M. Murt, W. G. Gulder, Eds. (Plenum, New York, 1969), Chap. 1.

F. A. Jenkins, H. E. White, Fundamentals of Optics (McGraw-Hill, New York, 1950), p. 255.

B. H. Billings, “Optics,” in American Institute of Physics Handbook, D. E. Gray, Ed. (McGraw-Hill, New York, 1963), Chap. 6.

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Figures (10)

Fig. 1
Fig. 1

Sample arrangement for reflectance measurement.

Fig. 2
Fig. 2

Sample arrangement for transmittance measurement: (a) actual structure; (b) equivalent structure.

Fig. 3
Fig. 3

Reflectance ratio ρR vs transparent layer refractive index n1 for three substrates assuming an angle of incidence of 10° (see text for further explanation).

Fig. 4
Fig. 4

Transmittance ratio ρT vs transparent layer refractive index n1 for three substrates.

Fig. 5
Fig. 5

Variation of log10 reflectance with wavelength of a SIPOS layer on an N+–Si substrate.

Fig. 6
Fig. 6

Variation of log10 transmittance with wavelength of a SIPOS layer on a sapphire substrate

Fig. 7
Fig. 7

Refractive index nr of SIPOS layers on Si substrates determined from reflectance measurements vs refractive index nt determined from transmittance measurements of similar layers on sapphire.

Fig. 8
Fig. 8

Thickness tr of SIPOS layers on Si substrates determined from reflectance measurements vs thickness tt determined from transmittance measurements of similar layers on sapphire.

Fig. 9
Fig. 9

Comparison of thickness values of SIPOS layers on sapphire substrates determined from transmittance measurements with Talysurf measurements of the same layers.

Fig. 10
Fig. 10

Comparison of the thickness values of SIPOS layers determined from transmittance measurements using two different sampling areas with Talysurf measurements of the same layers.

Tables (1)

Tables Icon

Table I Comparison of Measurement Results on SIPOS Layers Produced by Atmospheric Pressure Chemical Vapor Deposition

Equations (30)

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Δ = n 1 ( a b c ¯ ) n 0 ( a g ¯ ) = n 1 ( a b f ¯ ) , Δ = 2 n 1 t cos θ 1 .
R = I I 0 = r 1 2 + 2 r 1 r 2 cos 2 δ + r 2 2 1 + 2 r 1 r 2 cos 2 δ + r 1 2 r 2 2 ,
δ = ( Δ / λ ) π = ( 2 n 1 t π cos θ 1 ) / λ ,
r 1 = r 1 p = tan ( θ 1 θ 0 ) tan ( θ 1 + θ 0 ) ,
r 2 = r 2 p = tan ( θ 2 θ 1 ) tan ( θ 2 + θ 1 ) .
r 1 = r 1 s = sin ( θ 1 θ 0 ) sin ( θ 1 + θ 0 )
r 2 = r 2 s = sin ( θ 2 θ 1 ) sin ( θ 2 + θ 1 ) .
R s = I s / I 0 s = R | r 1 = r 1 s r 2 = r 2 s ,
R p = I p / I 0 p = R | r 1 = r 1 p r 2 = r 2 p .
M λ = 2 n 1 t cos θ 1 = 2 t ( n 1 2 sin 2 θ 0 ) 1 / 2 ,
R MAX = r 1 2 + 2 r 1 r 2 + r 2 2 1 + 2 r 1 r 2 + r 1 2 r 2 2 = ( r 1 + r 2 ) 2 ( 1 + r 1 r 2 ) 2 .
( M + 1 / 2 ) λ = 2 n 1 t cos θ 1 = 2 t ( n 1 2 sin 2 θ 0 ) 1 / 2 ,
R MIN = r 1 2 2 r 1 r 2 + r 2 2 1 2 r 1 r 2 + r 1 2 r 2 2 = ( r 1 r 2 ) 2 ( 1 r 1 r 2 ) 2 .
ρ R = [ ( r 1 ± r 2 ) 2 ( 1 ± r 1 r 2 ) 2 ] λ=λ MAX × [ ( 1 r 1 r 2 ) 2 ( r 1 r 2 ) 2 ] λ=λ MIN ,
ρ R ( r 1 ± r 2 ) 2 ( 1 r 1 r 2 ) 2 ( r 1 r 2 ) 2 ( 1 ± r 1 r 2 ) 2 .
t = M a b λ a λ b 2 ( λ a λ b ) ( n 1 2 sin 2 θ 0 ) 1 / 2 ,
T 1 = 8 n 0 n 1 2 n 2 ( n 0 2 + n 1 2 ) ( n 1 2 + n 2 2 ) + 4 n 0 n 1 2 n 2 + ( n 0 2 n 1 2 ) ( n 1 2 n 2 2 ) cos ( 2 π n 1 t 1 λ ) .
T 2 = 4 n 0 n 2 ( n 0 + n 2 ) 2 .
I = I 0 [ T 1 T 2 + T 1 T 2 R 1 R 2 + T 1 T 2 ( R 1 R 2 ) 2 + ] .
I = I 0 [ ( T 1 T 2 ) / ( 1 R 1 R 2 ) ] .
T = I / I 0 = T 1 T 2 [ 1 / ( T 1 + T 2 T 1 T 2 ) ]
T MAX = 4 n 0 n 1 2 n 2 ( n 0 2 + n 1 2 ) ( n 1 2 + n 2 2 ) ,
T MIN = 2 n 0 n 2 n 0 2 + n 2 2 ,
ρ T 1 = T MAX T MIN = [ 4 n 0 n 1 2 n 2 ( n 1 2 + n 0 2 ) ( n 1 2 + n 2 2 ) ] λ=λ MAX × ( n 0 2 + n 2 2 2 n 0 n 2 ) λ=λ MIN .
ρ T 1 [ 2 n 1 2 ( n 0 2 + n 2 2 ) ( n 1 2 + n 0 2 ) ( n 1 2 + n 2 2 ) ] ,
n 1 = { ( n 0 2 + n 2 2 ) ( 1 2 ρ T 1 ) + [ ( n 0 2 + n 2 2 ) 2 ( 1 2 ρ T 1 ) 2 4 n 0 2 n 2 2 ] 1 / 2 2 } 1 / 2 .
ρ T 2 = T MAX T MIN = ( 2 n 0 n 2 n 0 2 + n 2 2 ) λ=λ MAX × [ ( n 1 2 + n 0 2 ) ( n 1 2 + n 2 2 ) 4 n 0 n 1 2 n 2 ] λ=λ MIN ,
ρ T 2 [ ( n 1 2 + n 0 2 ) ( n 1 2 + n 2 2 ) 2 n 1 2 ( n 0 2 + n 2 2 ) ] .
n 1 = { ( n 0 2 + n 2 2 ) ( 1 2 ρ T 2 ) + [ ( n 0 2 + n 2 2 ) 2 ( 1 2 ρ T 2 ) 2 4 n 0 2 n 2 2 ] 1 / 2 2 } 1 / 2 .
ρ R = ( R p MAX + R s MAX ) / ( R p MIN + R s MIN ) .

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