Abstract

This method was developed to determine the complex infrared optical constant of a single free-standing partially absorbing plate as well as a thin solid film deposited on it. The method is based on exact formulas normal transmittance T and near-normal reflectance R of the substrate as well as the film–substrate double layer. Coherent multiple reflections throughout the film and incoherent multiple reflections in the substrate as well as the intensity losses on the rough surface are taken into account. The influence of various data on solution of the inverse problem is discussed by a contour map study. The method is explained using examples of both- and single-side-polished silicon wafers where the transmission and reflection roughness factor functions HT,HR are determined for the rough surface. The thin-film example has been the silicon oxide formed on the single-side-polished silicon substrate by chemical vapor deposition.

© 1991 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E. D. Palik, ed. Handbook of Optical Constants of Solids (Academic, Orlando, Fla., 1985).
  2. D. M. Roessler, “Kramers–Kronig analysis of reflection data,” Br. J. Appl. Phys. 16, 1119–1123 (1965).
    [CrossRef]
  3. D. M. Roesseler, “Kramers–Kronig analysis of non-normal incidence reflection,” Br. J. Appl. Phys. 16, 1359–1366 (1965).
    [CrossRef]
  4. P. O. Nilsson, L. Munkby, “Investigation of errors in the Kramers–Kronig analysis of reflectance data,” Phys. Kondens. Mater. 10, 290–298 (1969).
  5. R. T. Graf, J. L. Koenig, H. Ishida, “Optical constant determination of thin polymer films in the infrared,” Appl. Spectrosc. 9, 405–408 (1985).
    [CrossRef]
  6. 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]
  7. R. Swanepoel, “Determination of the thickness and optical constants of amorphous silicon,” J. Phys. E 16, 1214–1222 (1983).
    [CrossRef]
  8. B. Bouvard, F. J. Milligen, M. J. Messerly, S. G. Saxe, H. A. Macleod, “Optical constants derivation for an inhomogeneous thin film from in situ transmission measurements,” Appl. Opt. 24, 1803–1807 (1985).
    [CrossRef]
  9. K. A. Epstein, D. K. Misemer, G. D. Vernstrom, “Optical parameters of absorbing semiconductors from transmission and reflection,” Appl. Opt. 26, 294–299 (1987).
    [CrossRef] [PubMed]
  10. J. E. Nestell, R. W. Christy, “Derivation of optical constants of metals from thin-film measurements at oblique incidence,” Appl. Opt. 11, 643–651 (1972).
    [CrossRef] [PubMed]
  11. R. E. Denton, R. D. Campbell, S. G. Tomlin, “The determination of the optical constants of thin films from measurements of reflectance and transmittance at normal incidence,” J. Phys. D 5, 852–863 (1972).
    [CrossRef]
  12. R. T. Phillips, “A numerical method for determining the complex refractive index from reflectance and transmittance of supported thin films,” J. Phys. D 16, 489–497 (1983).
    [CrossRef]
  13. A. Hjortsberg, “Determination of optical constants of absorbing materials using transmission and reflection of thin films on partially metallized substrates: analysis of the new (T, Rm) technique,” Appl. Opt. 20, 1254–1263 (1981).
    [CrossRef] [PubMed]
  14. R. C. McPhedran, L. C. Botten, D. R. McKenzie, R. P. Netterfield, “Unambiguous determination of optical constants of absorbing films by reflectance and transmittance measurements,” Appl. Opt. 23, 1197–1205 (1984).
    [CrossRef] [PubMed]
  15. T. C. Paulick, “Inversion of normal-incidence (R,T) measurements to obtain n + ik for thin films,” Appl. Opt. 25, 562–564 (1986).
    [CrossRef] [PubMed]
  16. T. Buffeteau, B. Desbat, “Thin-film optical constants determined from infrared reflectance and transmittance measurements,” Appl. Spectrosc. 43, 1027–1032 (1989).
    [CrossRef]
  17. J. M. Pawlikowski, “Determination of the absorption coefficient of a real semiconductor film: application to ZnSe,” Thin Solid Films 125, 213–220 (1985).
    [CrossRef]
  18. D. L. Windt, W. C. Cash, M. Scott, P. Arendt, B. Newnam, R. F. Fisher, A. B. Swartzlander, P. Z. Takacs, J. 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]
  19. C. L. Nagendra, G. K. M. Thutupalli, “Optical constants of absorbing materials: a new approach,” Appl. Opt. 20, 2747–2753 (1981).
    [CrossRef] [PubMed]
  20. P. Stallhofer, D. Huber, “Oxygen and carbon measurements on silicon slices by the IR method,” Solid State Technol. 26, 233–237 (1983).
  21. F. Schomann, K. Graff, “Correction factors for the determination of oxygen in silicon by IR spectrometry,” J. Electrochem. Soc. 136, 2025–2031 (1989).
    [CrossRef]
  22. O. S. Heavens, Optical Properties of Thin Solid Films (Butterworth, London, 1955), Chap. 4.
  23. H. E. Bennett, J. O. Porteus, “Relation between surface roughness and specular reflectance at normal incidence,” J. Opt. Soc. Am. 51, 123–129 (1961).
    [CrossRef]
  24. J. P. Hawranek, P. Neelakantan, R. P. Young, R. N. Jones, “The control of errors in I.R. spectrophotometry. III. Transmission measurements using thin cells,” Spectrochim. Acta Part A 32, 75–84 (1976).
    [CrossRef]
  25. D. F. Edwards, “Silicon (Si),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 547–570.
  26. H. T. Kinasewitz, D. Senitzky, “Investigation of the complex permittivity of n-type silicon at millimeter wavelengths,” J. Appl. Phys. 54, 3394–3398 (1983).
    [CrossRef]
  27. A. Roos, M. Bergkvist, C. G. Ribbing, “Determination of the SiO2/Si interface roughness by diffuse reflectance measurements,” Appl. Opt. 27, 4314–4317 (1988).
    [CrossRef] [PubMed]
  28. H. R. Philipp, “Silicon Dioxide (SiO2) (Glass),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 749–764.
  29. M. K. Gunde, B. Aleksandrov, “Thickness-dependent frequency shift in infrared spectral absorbance of silicon oxide film on silicon,” Appl. Spectrosc. 44, 970–974 (1990).
    [CrossRef]

1990 (1)

1989 (2)

F. Schomann, K. Graff, “Correction factors for the determination of oxygen in silicon by IR spectrometry,” J. Electrochem. Soc. 136, 2025–2031 (1989).
[CrossRef]

T. Buffeteau, B. Desbat, “Thin-film optical constants determined from infrared reflectance and transmittance measurements,” Appl. Spectrosc. 43, 1027–1032 (1989).
[CrossRef]

1988 (2)

1987 (1)

1986 (1)

1985 (3)

R. T. Graf, J. L. Koenig, H. Ishida, “Optical constant determination of thin polymer films in the infrared,” Appl. Spectrosc. 9, 405–408 (1985).
[CrossRef]

J. M. Pawlikowski, “Determination of the absorption coefficient of a real semiconductor film: application to ZnSe,” Thin Solid Films 125, 213–220 (1985).
[CrossRef]

B. Bouvard, F. J. Milligen, M. J. Messerly, S. G. Saxe, H. A. Macleod, “Optical constants derivation for an inhomogeneous thin film from in situ transmission measurements,” Appl. Opt. 24, 1803–1807 (1985).
[CrossRef]

1984 (1)

1983 (4)

R. T. Phillips, “A numerical method for determining the complex refractive index from reflectance and transmittance of supported thin films,” J. Phys. D 16, 489–497 (1983).
[CrossRef]

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

P. Stallhofer, D. Huber, “Oxygen and carbon measurements on silicon slices by the IR method,” Solid State Technol. 26, 233–237 (1983).

H. T. Kinasewitz, D. Senitzky, “Investigation of the complex permittivity of n-type silicon at millimeter wavelengths,” J. Appl. Phys. 54, 3394–3398 (1983).
[CrossRef]

1981 (2)

1976 (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]

J. P. Hawranek, P. Neelakantan, R. P. Young, R. N. Jones, “The control of errors in I.R. spectrophotometry. III. Transmission measurements using thin cells,” Spectrochim. Acta Part A 32, 75–84 (1976).
[CrossRef]

1972 (2)

J. E. Nestell, R. W. Christy, “Derivation of optical constants of metals from thin-film measurements at oblique incidence,” Appl. Opt. 11, 643–651 (1972).
[CrossRef] [PubMed]

R. E. Denton, R. D. Campbell, S. G. Tomlin, “The determination of the optical constants of thin films from measurements of reflectance and transmittance at normal incidence,” J. Phys. D 5, 852–863 (1972).
[CrossRef]

1969 (1)

P. O. Nilsson, L. Munkby, “Investigation of errors in the Kramers–Kronig analysis of reflectance data,” Phys. Kondens. Mater. 10, 290–298 (1969).

1965 (2)

D. M. Roessler, “Kramers–Kronig analysis of reflection data,” Br. J. Appl. Phys. 16, 1119–1123 (1965).
[CrossRef]

D. M. Roesseler, “Kramers–Kronig analysis of non-normal incidence reflection,” Br. J. Appl. Phys. 16, 1359–1366 (1965).
[CrossRef]

1961 (1)

Aleksandrov, B.

Arendt, P.

Bennett, H. E.

Bergkvist, M.

Botten, L. C.

Bouvard, B.

Buffeteau, T.

Campbell, R. D.

R. E. Denton, R. D. Campbell, S. G. Tomlin, “The determination of the optical constants of thin films from measurements of reflectance and transmittance at normal incidence,” J. Phys. D 5, 852–863 (1972).
[CrossRef]

Cash, W. C.

Christy, R. W.

Denton, R. E.

R. E. Denton, R. D. Campbell, S. G. Tomlin, “The determination of the optical constants of thin films from measurements of reflectance and transmittance at normal incidence,” J. Phys. D 5, 852–863 (1972).
[CrossRef]

Desbat, B.

Edwards, D. F.

D. F. Edwards, “Silicon (Si),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 547–570.

Epstein, K. A.

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.

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]

Graf, R. T.

R. T. Graf, J. L. Koenig, H. Ishida, “Optical constant determination of thin polymer films in the infrared,” Appl. Spectrosc. 9, 405–408 (1985).
[CrossRef]

Graff, K.

F. Schomann, K. Graff, “Correction factors for the determination of oxygen in silicon by IR spectrometry,” J. Electrochem. Soc. 136, 2025–2031 (1989).
[CrossRef]

Gunde, M. K.

Hawranek, J. P.

J. P. Hawranek, P. Neelakantan, R. P. Young, R. N. Jones, “The control of errors in I.R. spectrophotometry. III. Transmission measurements using thin cells,” Spectrochim. Acta Part A 32, 75–84 (1976).
[CrossRef]

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworth, London, 1955), Chap. 4.

Hjortsberg, A.

Huber, D.

P. Stallhofer, D. Huber, “Oxygen and carbon measurements on silicon slices by the IR method,” Solid State Technol. 26, 233–237 (1983).

Ishida, H.

R. T. Graf, J. L. Koenig, H. Ishida, “Optical constant determination of thin polymer films in the infrared,” Appl. Spectrosc. 9, 405–408 (1985).
[CrossRef]

Jones, R. N.

J. P. Hawranek, P. Neelakantan, R. P. Young, R. N. Jones, “The control of errors in I.R. spectrophotometry. III. Transmission measurements using thin cells,” Spectrochim. Acta Part A 32, 75–84 (1976).
[CrossRef]

Kinasewitz, H. T.

H. T. Kinasewitz, D. Senitzky, “Investigation of the complex permittivity of n-type silicon at millimeter wavelengths,” J. Appl. Phys. 54, 3394–3398 (1983).
[CrossRef]

Koenig, J. L.

R. T. Graf, J. L. Koenig, H. Ishida, “Optical constant determination of thin polymer films in the infrared,” Appl. Spectrosc. 9, 405–408 (1985).
[CrossRef]

Macleod, H. A.

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]

McKenzie, D. R.

McPhedran, R. C.

Messerly, M. J.

Milligen, F. J.

Misemer, D. K.

Munkby, L.

P. O. Nilsson, L. Munkby, “Investigation of errors in the Kramers–Kronig analysis of reflectance data,” Phys. Kondens. Mater. 10, 290–298 (1969).

Nagendra, C. L.

Neelakantan, P.

J. P. Hawranek, P. Neelakantan, R. P. Young, R. N. Jones, “The control of errors in I.R. spectrophotometry. III. Transmission measurements using thin cells,” Spectrochim. Acta Part A 32, 75–84 (1976).
[CrossRef]

Nestell, J. E.

Netterfield, R. P.

Newnam, B.

Nilsson, P. O.

P. O. Nilsson, L. Munkby, “Investigation of errors in the Kramers–Kronig analysis of reflectance data,” Phys. Kondens. Mater. 10, 290–298 (1969).

Paulick, T. C.

Pawlikowski, J. M.

J. M. Pawlikowski, “Determination of the absorption coefficient of a real semiconductor film: application to ZnSe,” Thin Solid Films 125, 213–220 (1985).
[CrossRef]

Philipp, H. R.

H. R. Philipp, “Silicon Dioxide (SiO2) (Glass),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 749–764.

Phillips, R. T.

R. T. Phillips, “A numerical method for determining the complex refractive index from reflectance and transmittance of supported thin films,” J. Phys. D 16, 489–497 (1983).
[CrossRef]

Pinneo, J. M.

Porteus, J. O.

Ribbing, C. G.

Roesseler, D. M.

D. M. Roesseler, “Kramers–Kronig analysis of non-normal incidence reflection,” Br. J. Appl. Phys. 16, 1359–1366 (1965).
[CrossRef]

Roessler, D. M.

D. M. Roessler, “Kramers–Kronig analysis of reflection data,” Br. J. Appl. Phys. 16, 1119–1123 (1965).
[CrossRef]

Roos, A.

Saxe, S. G.

Schomann, F.

F. Schomann, K. Graff, “Correction factors for the determination of oxygen in silicon by IR spectrometry,” J. Electrochem. Soc. 136, 2025–2031 (1989).
[CrossRef]

Scott, M.

Senitzky, D.

H. T. Kinasewitz, D. Senitzky, “Investigation of the complex permittivity of n-type silicon at millimeter wavelengths,” J. Appl. Phys. 54, 3394–3398 (1983).
[CrossRef]

Stallhofer, P.

P. Stallhofer, D. Huber, “Oxygen and carbon measurements on silicon slices by the IR method,” Solid State Technol. 26, 233–237 (1983).

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.

Takacs, P. Z.

Thutupalli, G. K. M.

Tomlin, S. G.

R. E. Denton, R. D. Campbell, S. G. Tomlin, “The determination of the optical constants of thin films from measurements of reflectance and transmittance at normal incidence,” J. Phys. D 5, 852–863 (1972).
[CrossRef]

Vernstrom, G. D.

Windt, D. L.

Young, R. P.

J. P. Hawranek, P. Neelakantan, R. P. Young, R. N. Jones, “The control of errors in I.R. spectrophotometry. III. Transmission measurements using thin cells,” Spectrochim. Acta Part A 32, 75–84 (1976).
[CrossRef]

Appl. Opt. (9)

B. Bouvard, F. J. Milligen, M. J. Messerly, S. G. Saxe, H. A. Macleod, “Optical constants derivation for an inhomogeneous thin film from in situ transmission measurements,” Appl. Opt. 24, 1803–1807 (1985).
[CrossRef]

K. A. Epstein, D. K. Misemer, G. D. Vernstrom, “Optical parameters of absorbing semiconductors from transmission and reflection,” Appl. Opt. 26, 294–299 (1987).
[CrossRef] [PubMed]

J. E. Nestell, R. W. Christy, “Derivation of optical constants of metals from thin-film measurements at oblique incidence,” Appl. Opt. 11, 643–651 (1972).
[CrossRef] [PubMed]

A. Hjortsberg, “Determination of optical constants of absorbing materials using transmission and reflection of thin films on partially metallized substrates: analysis of the new (T, Rm) technique,” Appl. Opt. 20, 1254–1263 (1981).
[CrossRef] [PubMed]

R. C. McPhedran, L. C. Botten, D. R. McKenzie, R. P. Netterfield, “Unambiguous determination of optical constants of absorbing films by reflectance and transmittance measurements,” Appl. Opt. 23, 1197–1205 (1984).
[CrossRef] [PubMed]

T. C. Paulick, “Inversion of normal-incidence (R,T) measurements to obtain n + ik for thin films,” Appl. Opt. 25, 562–564 (1986).
[CrossRef] [PubMed]

D. L. Windt, W. C. Cash, M. Scott, P. Arendt, B. Newnam, R. F. Fisher, A. B. Swartzlander, P. Z. Takacs, J. 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]

C. L. Nagendra, G. K. M. Thutupalli, “Optical constants of absorbing materials: a new approach,” Appl. Opt. 20, 2747–2753 (1981).
[CrossRef] [PubMed]

A. Roos, M. Bergkvist, C. G. Ribbing, “Determination of the SiO2/Si interface roughness by diffuse reflectance measurements,” Appl. Opt. 27, 4314–4317 (1988).
[CrossRef] [PubMed]

Appl. Spectrosc. (3)

Br. J. Appl. Phys. (2)

D. M. Roessler, “Kramers–Kronig analysis of reflection data,” Br. J. Appl. Phys. 16, 1119–1123 (1965).
[CrossRef]

D. M. Roesseler, “Kramers–Kronig analysis of non-normal incidence reflection,” Br. J. Appl. Phys. 16, 1359–1366 (1965).
[CrossRef]

J. Appl. Phys. (1)

H. T. Kinasewitz, D. Senitzky, “Investigation of the complex permittivity of n-type silicon at millimeter wavelengths,” J. Appl. Phys. 54, 3394–3398 (1983).
[CrossRef]

J. Electrochem. Soc. (1)

F. Schomann, K. Graff, “Correction factors for the determination of oxygen in silicon by IR spectrometry,” J. Electrochem. Soc. 136, 2025–2031 (1989).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Phys. D (2)

R. E. Denton, R. D. Campbell, S. G. Tomlin, “The determination of the optical constants of thin films from measurements of reflectance and transmittance at normal incidence,” J. Phys. D 5, 852–863 (1972).
[CrossRef]

R. T. Phillips, “A numerical method for determining the complex refractive index from reflectance and transmittance of supported thin films,” J. Phys. D 16, 489–497 (1983).
[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. Kondens. Mater. (1)

P. O. Nilsson, L. Munkby, “Investigation of errors in the Kramers–Kronig analysis of reflectance data,” Phys. Kondens. Mater. 10, 290–298 (1969).

Solid State Technol. (1)

P. Stallhofer, D. Huber, “Oxygen and carbon measurements on silicon slices by the IR method,” Solid State Technol. 26, 233–237 (1983).

Spectrochim. Acta Part A (1)

J. P. Hawranek, P. Neelakantan, R. P. Young, R. N. Jones, “The control of errors in I.R. spectrophotometry. III. Transmission measurements using thin cells,” Spectrochim. Acta Part A 32, 75–84 (1976).
[CrossRef]

Thin Solid Films (1)

J. M. Pawlikowski, “Determination of the absorption coefficient of a real semiconductor film: application to ZnSe,” Thin Solid Films 125, 213–220 (1985).
[CrossRef]

Other (4)

O. S. Heavens, Optical Properties of Thin Solid Films (Butterworth, London, 1955), Chap. 4.

D. F. Edwards, “Silicon (Si),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 547–570.

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

H. R. Philipp, “Silicon Dioxide (SiO2) (Glass),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, Orlando, Fla., 1985), pp. 749–764.

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 (14)

Fig. 1
Fig. 1

Intensities of the beams reflected and transmitted successively at the front and rear surfaces (a) of the substrate and (b) of the film–substrate double layer at oblique incidence. All the surfaces are specular.

Fig. 2
Fig. 2

The T(n2, k2) and R(n2, k2) planes (arbitrary unit) for a 0.5-mm thick plate at λ = 0.01 mm (1000 cm−1).

Fig. 3
Fig. 3

Projections of Rexp = 0.45 (solid line) and Texp = 0.53 (dashed line) contours on the (n2, k2) plane with a reticle of 10 × 10 equidistant n and k isolines. Example of a 0.5-mm thick both-side-polished silicon wafer for λ = 0.01 mm. The width of the contours represents the experimental error of 1 ± 0.005.

Fig. 4
Fig. 4

Variation of the contour map by the value of (d/λ) as obtained for 1-μm thick film on a transparent substrate (n2 = 3.4, k2 = 0). All the surfaces are specular. The reflection and transmission contours are represented by solid and dashed lines, respectively. The spectral positions for (d/λ) are a, 0.4; b, 0.2; and c, 0.04.

Fig. 5
Fig. 5

The R(n1, k1 = 0) curve for 1-μm thick film on a substrate with n2 = 3.4 and of (d/λ): a, three-dimensional k2 = 0 for variation representation b, 0.4; c, 0.3; d, 0.2; e, 0.1; and f, 0.04. Data for local extrema are given in Table I.

Fig. 6
Fig. 6

Dependence of the contour map on the extinction coefficient of substrate k2 for a 1-μm thick film at λ = 10 μm. Substrate data are n2 = 3.4 and D = 0.5 mm. All the surfaces are specular. The reflection and transmission contours are represented by solid and dashed lines, respectively. The k2 values are a, 0; b, 10−3; and c, 4.

Fig. 7
Fig. 7

IR spectra of both-side-polished 0.491-mm thick FZ silicon wafer: normal-incidence transmittance (solid line) and near-normal-incidence reflectance (dashed line).

Fig. 8
Fig. 8

Calculated optical constants for FZ silicon wafer: extinction coefficient k2 (solid line) and refractive index n2 (dashed line).

Fig. 9
Fig. 9

IR spectra of the single-side-polished 0.55-mm thick CZ-grown silicon wafer: normal-incidence transmittance (solid line) and near-normal reflectance (dashed line).

Fig. 10
Fig. 10

Roughness factor functions as calculated (solid line) for reflection from the rough rear surface of the silicon wafer (HR), and transmission through the same rough surface (HT). The points represent the smoothed curves (see text for details).

Fig. 11
Fig. 11

Optical constants of CZ-grown silicon wafer as obtained by roughness correction: k2, solid line; n2, dashed line. Only the phonon region is shown.

Fig. 12
Fig. 12

IR spectra of silicon oxide film formed on single-side-polished CZ silicon wafer: near-normal reflectance (solid line) and normal-incidence transmittance (dashed line).

Fig. 13
Fig. 13

All n1 solutions for the silicon oxide film in the 4000–400-cm−1 spectral range with 0 < n1 < 4 values. The results were obtained with d = 1186 ± 25 nm and by taking into account the experimental error of 1 ± 0.005 on transmittance and 1 ± 0.01 on reflectance. The A,B,C,D,E,F,G,H refer to Table III.

Fig. 14
Fig. 14

Optical constants of chemical vapor deposition silicon oxide film: refractive index (solid line) and extinction coefficient (dashed line).

Tables (3)

Tables Icon

Table I The (n1d/λ) Values of Extrema of the R(n1,0) Curves Shown In Fig. 5

Tables Icon

Table II Computation Precision According to Input Parameters and the Number of Recursion Stepsa

Tables Icon

Table III The (n1d/λ) Values of Contour Tangency Points as Labeled In Fig. 13

Equations (21)

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

T = T 02 F T 23 1 - F 2 R 23 R 20 ,
R = R 02 + T 02 F 2 R 23 T 20 1 - F 2 R 23 R 20 ,
T i , j = | n ^ j n ^ i | t ^ i , j t ^ i , j * ,
R i , j = r ^ i , j r ^ i , j * ,
F = exp ( - 4 π k 2 D / λ )
n ^ i ( λ ) = n i ( λ ) - i k i ( λ )
T = T [ n 2 ( λ ) , k 2 ( λ ) , D ] , R = R [ n 2 ( λ ) , k 2 ( λ ) , D ] .
T = T 012 F T 23 1 - F 2 R 23 R 210 ,
R = R 012 + T 012 F 2 R 23 T 210 1 - F 2 R 23 R 210 .
T i j l = | n ^ l n ^ i | τ ^ i j l τ ^ i j l * ,
R i j l = ρ ^ i j l ρ ^ i j l * .
τ ^ i j l = t ^ i j t ^ j l exp ( - i δ ^ ) 1 + r ^ i j r ^ j l exp ( - 2 i δ ^ ) ,
ρ ^ i j l = r ^ i j r ^ j l exp ( - 2 i δ ^ ) 1 + r ^ i j r ^ j l exp ( - 2 i δ ^ ) .
δ ^ = 2 π λ n ^ 1 d ;
T = T [ n 1 ( λ ) , k 1 ( λ ) , d , n 2 ( λ ) , k 2 ( λ ) , D ] , R = R [ n 1 ( λ ) , k 1 ( λ ) , d , n 2 ( λ ) , k 2 ( λ ) , D ] .
R r ( λ ) = R s ( λ ) H R ( λ , ) ,
T r ( λ ) = T s ( λ ) H T ( λ , ) ,
T = T 02 F H T T 23 1 - F 2 R 23 H R R 20 ,
R = R 02 + T 02 F 2 R 23 H R T 20 1 - F 2 R 23 H R R 20 .
T = T [ n 2 ( λ ) , k 2 ( λ ) , D , H R ( λ ) , H T ( λ ) ] , R = R [ n 2 ( λ ) , k 2 ( λ ) , D , H R ( λ ) ] .
n 1 d / λ = m / 16 ,

Metrics