Abstract

The range of incidence angle, 0<φ<φe, over which p-polarized light is reflected at interfaces between transparent and absorbing media with reflectance below that at normal incidence is determined. Contours of constant φe in the complex plane of the relative dielectric constant ε are presented. A method for determining the real and imaginary parts of the complex refractive index, ε1/2=n+jk, which is based on measuring φe and the pseudo-Brewster angle φpB, is viable in the domain of fractional optical constants, n, k<1.

© 2002 Optical Society of America

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References

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  1. See, for example, R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987), Chap. 4.
  2. S. P. F. Humphreys-Owen, “Comparison of reflection methods for measuring optical constants without polarimetric analysis, and proposal for new methods based on the Brewster angle,” Proc. Phys. Soc. London 77, 949–957 (1961).
    [CrossRef]
  3. R. M. A. Azzam, “Maximum minimum reflectance of parallel-polarized light at interfaces between transparent and absorbing media,” J. Opt. Soc. Am. 73, 959–962 (1983).
    [CrossRef]
  4. S. Y. Kim, K. Vedam, “Analytic solution of the pseudo-Brewster angle,” J. Opt. Soc. Am. A 3, 1772–1773 (1986).
    [CrossRef]
  5. R. M. A. Azzam, E. E. Ugbo, “Contours of constant pseudo-Brewster angle in the complex ε plane and an analytical method for the determination of optical constants,” Appl. Opt. 28, 5222–5228 (1989).
    [CrossRef] [PubMed]
  6. R. M. A. Azzam, “Equalization of reflectance of parallel-polarized electromagnetic waves at normal and oblique incidence of interfaces between transparent media and its use for measurement of the dielectric constant,” Appl. Phys. 20, 193–195 (1979).
    [CrossRef]
  7. R. M. A. Azzam, “Analytical determination of the complex dielectric function of an absorbing medium from two angles of incidence of minimum parallel reflectance,” J. Opt. Soc. Am. A 6, 1213–1216 (1989).
    [CrossRef]
  8. W. R. Hunter, “Measurement of optical constants in the vacuum ultraviolet spectral region,” in Handbook of Optical Constants of Solids, E. Palik, ed. (Academic, New York, 1985).

1989 (2)

1986 (1)

1983 (1)

1979 (1)

R. M. A. Azzam, “Equalization of reflectance of parallel-polarized electromagnetic waves at normal and oblique incidence of interfaces between transparent media and its use for measurement of the dielectric constant,” Appl. Phys. 20, 193–195 (1979).
[CrossRef]

1961 (1)

S. P. F. Humphreys-Owen, “Comparison of reflection methods for measuring optical constants without polarimetric analysis, and proposal for new methods based on the Brewster angle,” Proc. Phys. Soc. London 77, 949–957 (1961).
[CrossRef]

Azzam, R. M. A.

Bashara, N. M.

See, for example, R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987), Chap. 4.

Humphreys-Owen, S. P. F.

S. P. F. Humphreys-Owen, “Comparison of reflection methods for measuring optical constants without polarimetric analysis, and proposal for new methods based on the Brewster angle,” Proc. Phys. Soc. London 77, 949–957 (1961).
[CrossRef]

Hunter, W. R.

W. R. Hunter, “Measurement of optical constants in the vacuum ultraviolet spectral region,” in Handbook of Optical Constants of Solids, E. Palik, ed. (Academic, New York, 1985).

Kim, S. Y.

Ugbo, E. E.

Vedam, K.

Appl. Opt. (1)

Appl. Phys. (1)

R. M. A. Azzam, “Equalization of reflectance of parallel-polarized electromagnetic waves at normal and oblique incidence of interfaces between transparent media and its use for measurement of the dielectric constant,” Appl. Phys. 20, 193–195 (1979).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Proc. Phys. Soc. London (1)

S. P. F. Humphreys-Owen, “Comparison of reflection methods for measuring optical constants without polarimetric analysis, and proposal for new methods based on the Brewster angle,” Proc. Phys. Soc. London 77, 949–957 (1961).
[CrossRef]

Other (2)

See, for example, R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1987), Chap. 4.

W. R. Hunter, “Measurement of optical constants in the vacuum ultraviolet spectral region,” in Handbook of Optical Constants of Solids, E. Palik, ed. (Academic, New York, 1985).

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Figures (8)

Fig. 1
Fig. 1

Reflection of p-polarized light at an angle φ by the planar interface of two media with dielectric constants ε1 and ε2.

Fig. 2
Fig. 2

Family of reflectance-versus-angle (rprp*-versus-φ) curves that share the same pseudo-Brewster angle φpB=50°. The associated values of complex ε=|ε|exp(jθ) are obtained from Eqs. (5)–(7) by allowing θ to assume values from 0 to 180° in steps of 15°.

Fig. 3
Fig. 3

Minimum reflectance at the pseudo-Brewster angle φpB,(rprp*)min, as a function of θ for constant values of φpB from 5 to 80° in steps of 5°.

Fig. 4
Fig. 4

Angle difference φe-φpB as a function of θ, for constant values of φpB from 5 to 80° in steps of 5°. φe defines the upper limit of the range of incidence angle for which the p reflectance at oblique incidence is less than that at normal incidence.

Fig. 5
Fig. 5

Family of contours of constant φe=45 to 80° in steps of 5°, and φe=80 to 85° in steps of 1°. φe defines the upper limit of the range of incidence angle for which the p reflectance at oblique incidence is less than that at normal incidence.

Fig. 6
Fig. 6

Family of constant-φe contours in the nk complex refractive index plane for the same values of φe as in Fig. 5.

Fig. 7
Fig. 7

Families of constant-φpB and constant-φe contours in the nk plane in the domain of fractional optical constants (n, k<1).

Fig. 8
Fig. 8

Families of constant-φpB and constant-φe contours in the nk plane for n, k>1.

Equations (7)

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rp=[ε cos φ-(ε-sin2 φ)1/2]/[ε cos φ+(ε-sin2 φ)1/2],
ε=ε2/ε1
rp(φ)rp*(φ)=rp(0)rp*(0).
tan φe=(ε2+ε)1/2.
|ε|=ι cos(ζ/3),
ι=2 tan2 φpB(1-23sin2 φpB)1/2,
ζ=arccos[-cos θ cos2 φpB(1-23sin2 φpB)-3/2],

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