Abstract

We have found a significant relation, <i>r<sub>p</sub></i> = <i>r<sub>s</sub></i> (<i>r<sub>s</sub></i> -cos2ϕ)/(1-<i>r<sub>s</sub></i> cos2ϕ), between Fresnel’s interface complex-amplitude reflection coefficients <i>r<sub>p</sub></i> and <i>r<sub>s</sub></i> for the parallel (<i>p</i>) and perpendicular (<i>s</i>) polarizations at the same angle of incidence ϕ. This relation is universal in that it applies to reflection at <i>all</i> interfaces between homogeneous isotropic media <i>collectively</i> and, of course, throughout the electromagnetic spectrum. We investigate the properties of this function, <i>r<sub>p</sub></i> = <i>f</i> (<i>r<sub>s</sub></i>), and its inverse, <i>r<sub>s</sub></i> = g (<i>r<sub>p</sub></i>), as conformal mappings between the complex planes of <i>r<sub>s</sub></i> and <i>r<sub>p</sub></i>. A related function, ρ = (<i>r<sub>s</sub></i> -cos2ϕ)/(1-<i>r<sub>s</sub></i> cos2ϕ), which is a bilinear transformation, is also studied, where ρ = <i>r<sub>p</sub></i> / <i>r<sub>s</sub></i> is the (ellipsometric) ratio of reflection coefficients. Several previously described reflection characteristics come out readily as specific results of this work. Simple explicit analytical and graphical solutions are provided to determine reflection phase shifts and the dielectric function from measured p and s reflectances at the same angle of incidence. We also show that when <i>r<sub>p</sub></i> is real, negative, and in the range -1 ≤ <i>r<sub>p</sub></i> ≤ -tan<sup>2</sup>(ϕ -45°), <i>r<sub>s</sub></i> is complex and its locus in the complex plane is an arc of a circle with center on the real axis at sec2ϕ and radius of |tan2ϕ|. Under these conditions, we also find the interesting result that | <i>r<sub>s</sub></i> | = | <i>r<sub>p</sub></i> |½.

© 1979 Optical Society of America

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  1. See, for example, M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975), 5th edition, p. 40.
  2. R. H. Muller, "Definitions and conventions in ellipsometry," Surf. Sci. 16, 14–33 (1969).
  3. R. M. A. Azzam, "Transformation of Fresnel's interface reflection and transmission coefficients between normal and oblique incidence," J. Opt. Soc. Am. 69, 590–596 (1979).
  4. Equation (6) can be cast in the alternative form (rs2 - rp)/(rs - rsrp) = cos2ϕ. The left-hand-side function of rs and rp is therefore real and invariant, at a given angle of incidence ϕ, with respect to changes of media and/or wavelength.
  5. R. M. A. Azzam, "On the reflection of light at 45° angle of incidence," Opt. Acta 26 (1979), (in press).
  6. D. W. Berreman, "Simple relation between reflectances of polarized components of a beam when the angle of incidence is 45°," J. Opt. Soc. Am. 56, 1784 (1966).
  7. 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. Lond. 77, 949–957 (1961).
  8. The second root of the quadratic equation that results from setting the numerator of the right-hand side of Eq. (8) equal to zero has an absolute value greater than 1, hence is physically unacceptable.
  9. R. M. A. Azzam, "Consequences of light reflection at the interface between two transparent media such that the angle of refraction is 45°," J. Opt. Soc. Am. 68, 1613–1615 (1978).
  10. See, for example, A. Kyrala, Applied Functions of a Complex Variable (Wiley-Interscience, New York, 1972), Ch. 8.
  11. The mapping is conformal at all points in the complex plane except where ∂rp/∂rs = 0 or ∞. This excludes the point specified by rs and rp in Eqs. (9) and (10).
  12. R. M. A. Azzam, "Reflection of an electromagnetic plane wave with 0 or π phase shift at the surface of an absorbing medium," J. Opt. Soc. Am. 69, 487–488 (1979).
  13. R. M. A. Azzam and N. M. Bashara, Ellipsometry and polarized Light (North-Holland, Amsterdam, 1977).
  14. M. R. Querry, "Direct solution of the generalized Fresnel reflectance equations," J. Opt. Soc. Am. 59, 876–877 (1969).
  15. When ϕ = 45°, we have cos2ϕ = 0 and Rp = R2s. This reduces Eq. (18) to cosδs = 0/0, hence δs becomes indeterminate, as is expected in this special case.

1979

1978

1969

R. H. Muller, "Definitions and conventions in ellipsometry," Surf. Sci. 16, 14–33 (1969).

M. R. Querry, "Direct solution of the generalized Fresnel reflectance equations," J. Opt. Soc. Am. 59, 876–877 (1969).

1966

1961

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. Lond. 77, 949–957 (1961).

Azzam, R. M. A.

Bashara, N. M.

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

Berreman, D. W.

Born, M.

See, for example, M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975), 5th edition, p. 40.

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. Lond. 77, 949–957 (1961).

Kyrala, A.

See, for example, A. Kyrala, Applied Functions of a Complex Variable (Wiley-Interscience, New York, 1972), Ch. 8.

Muller, R. H.

R. H. Muller, "Definitions and conventions in ellipsometry," Surf. Sci. 16, 14–33 (1969).

Querry, M. R.

Wolf, E.

See, for example, M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975), 5th edition, p. 40.

J. Opt. Soc. Am.

Opt. Acta

R. M. A. Azzam, "On the reflection of light at 45° angle of incidence," Opt. Acta 26 (1979), (in press).

Proc. Phys. Soc. Lond.

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. Lond. 77, 949–957 (1961).

Surf. Sci.

R. H. Muller, "Definitions and conventions in ellipsometry," Surf. Sci. 16, 14–33 (1969).

Other

See, for example, M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975), 5th edition, p. 40.

Equation (6) can be cast in the alternative form (rs2 - rp)/(rs - rsrp) = cos2ϕ. The left-hand-side function of rs and rp is therefore real and invariant, at a given angle of incidence ϕ, with respect to changes of media and/or wavelength.

The second root of the quadratic equation that results from setting the numerator of the right-hand side of Eq. (8) equal to zero has an absolute value greater than 1, hence is physically unacceptable.

See, for example, A. Kyrala, Applied Functions of a Complex Variable (Wiley-Interscience, New York, 1972), Ch. 8.

The mapping is conformal at all points in the complex plane except where ∂rp/∂rs = 0 or ∞. This excludes the point specified by rs and rp in Eqs. (9) and (10).

When ϕ = 45°, we have cos2ϕ = 0 and Rp = R2s. This reduces Eq. (18) to cosδs = 0/0, hence δs becomes indeterminate, as is expected in this special case.

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

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