Abstract

If w denotes an interface Fresnel reflection or transmission coefficient for s- or p-polarized light at an oblique angle of incidence ø, and z denotes the same coefficient at normal incidence, we find that w is an analytic function of z, w = ƒ(z), that depends on ø but not on the specific optical properties of the two media on both sides of the interface. All four functions that correspond to the four distinct Fresnel coefficients and their inverses are determined. We single out for detailed examination, as an example, the relationship between the reflection of s -polarized light at 45° angle of incidence and at normal incidence for any transparent medium/absorbing medium interface by considering the mapping properties of the associated transformation and its inverse between the z and w planes. A useful byproduct of this investigation is a technique for the determination of the optical properties of isotropic and uniaxially and biaxially anisotropic absorbing media from measurement of reflectance at normal and oblique incidence.

© 1979 Optical Society of America

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References

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  1. A. Fresnel, "Mémoire sur la loi des modifications que la réflection imprime a la lumière polarisée," in Oeuvres Complètes de Fresnel, Vol. 1, H. Senarmont, É. Verdet, and L. Fresnel, 1866, pp. 767–775 (Johnson Reprint Corporation, New York, 1965).
  2. See, for example, R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Sec. 4.2.
  3. It will be significant to study the relationship between each Fresnel coefficient and the refractive index ratio v as a conformal mapping between two complex planes.
  4. This choice of the p and s directions follows the Nebraska (Muller) conventions. See, R.H. Muller, "Definitions and conventions in ellipsometry," Surf. Sci. 16, 14–33 (1969).
  5. It is important to observe that any pair of Fresnel coefficients can be directly interrelated by the elimination of their common variable v.
  6. In the Nebraska (Muller) conventions, the ejwt time dependence is chosen (see Ref. 4).
  7. Zero reflection at normal incidence (z = 0) leads, as expected, to zero reflection at oblique incidence (w = 0). Notice that z → 0 as v → 1, i.e., when the difference between the refractive indices of the two media that surround the interface tends to zero.
  8. Points along the real axis of the z plane represent normal-incidence reflection of s-polarized light at a dielectric/dielectric interface (v real, hence z real) and the image points in the w plane represent the reflection of that light from the same interface at an angle ø. The oblique reflection is total (i.e., |w| = 1) if the refractive index ratio v ≥ sinø or z ≤ (1 - sinø)/(1 + sinø).
  9. Total reflection at normal incidence (|z| = 1) leads, as expected, to total reflection at oblique incidence (|w| = 1). Notice that |z| → 1 as v → ∞ which happens when light is reflected from the interface between a medium with a finite refractive index and another which is perfectly conducting.
  10. These lines are the loci of constant normal-incidence phase shift.
  11. These circles are the loci of constant normal-incidence (amplitude or intensity) reflectance.
  12. O. S. Heavens, Optical Properties of Thin Solid Films. (Dover, New York, 1965), p. 85.
  13. Various two reflectance methods for determining the optical properties of materials have been previously described. See, for example, Refs. 14–16. Although we assume s-polarized light, identical procedures apply for p-polarized light. The sensitivity of the TRM is known to be poorer with the s polarization than with the p polarization, but straightforward extension of the method to anisotropic media is possible only with s-polarized light (as is illustrated in this section).
  14. I. Šimon, "Spectroscopy in the infrared by reflection and its use for highly absorbing substances," J. Opt. Soc. Am. 41, 336–345 (1951).
  15. 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. R. Soc. Lond. 77, 949–957 (1961).
  16. W. R. Hunter, "Errors in using the reflectance vs angle of incidence method for measuring optical constants," J. Opt. Soc. Am. 55, 1197–1204 (1965).

1969

This choice of the p and s directions follows the Nebraska (Muller) conventions. See, R.H. Muller, "Definitions and conventions in ellipsometry," Surf. Sci. 16, 14–33 (1969).

1965

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. R. Soc. Lond. 77, 949–957 (1961).

1951

Azzam, R. M. A.

See, for example, R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Sec. 4.2.

Bashara, N. M.

See, for example, R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Sec. 4.2.

Fresnel, A.

A. Fresnel, "Mémoire sur la loi des modifications que la réflection imprime a la lumière polarisée," in Oeuvres Complètes de Fresnel, Vol. 1, H. Senarmont, É. Verdet, and L. Fresnel, 1866, pp. 767–775 (Johnson Reprint Corporation, New York, 1965).

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films. (Dover, New York, 1965), p. 85.

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. R. Soc. Lond. 77, 949–957 (1961).

Hunter, W. R.

Muller, R.H.

This choice of the p and s directions follows the Nebraska (Muller) conventions. See, R.H. Muller, "Definitions and conventions in ellipsometry," Surf. Sci. 16, 14–33 (1969).

Šimon, I.

J. Opt. Soc. Am.

Proc. R. 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. R. Soc. Lond. 77, 949–957 (1961).

Surf. Sci.

This choice of the p and s directions follows the Nebraska (Muller) conventions. See, R.H. Muller, "Definitions and conventions in ellipsometry," Surf. Sci. 16, 14–33 (1969).

Other

It is important to observe that any pair of Fresnel coefficients can be directly interrelated by the elimination of their common variable v.

In the Nebraska (Muller) conventions, the ejwt time dependence is chosen (see Ref. 4).

Zero reflection at normal incidence (z = 0) leads, as expected, to zero reflection at oblique incidence (w = 0). Notice that z → 0 as v → 1, i.e., when the difference between the refractive indices of the two media that surround the interface tends to zero.

Points along the real axis of the z plane represent normal-incidence reflection of s-polarized light at a dielectric/dielectric interface (v real, hence z real) and the image points in the w plane represent the reflection of that light from the same interface at an angle ø. The oblique reflection is total (i.e., |w| = 1) if the refractive index ratio v ≥ sinø or z ≤ (1 - sinø)/(1 + sinø).

Total reflection at normal incidence (|z| = 1) leads, as expected, to total reflection at oblique incidence (|w| = 1). Notice that |z| → 1 as v → ∞ which happens when light is reflected from the interface between a medium with a finite refractive index and another which is perfectly conducting.

These lines are the loci of constant normal-incidence phase shift.

These circles are the loci of constant normal-incidence (amplitude or intensity) reflectance.

O. S. Heavens, Optical Properties of Thin Solid Films. (Dover, New York, 1965), p. 85.

Various two reflectance methods for determining the optical properties of materials have been previously described. See, for example, Refs. 14–16. Although we assume s-polarized light, identical procedures apply for p-polarized light. The sensitivity of the TRM is known to be poorer with the s polarization than with the p polarization, but straightforward extension of the method to anisotropic media is possible only with s-polarized light (as is illustrated in this section).

A. Fresnel, "Mémoire sur la loi des modifications que la réflection imprime a la lumière polarisée," in Oeuvres Complètes de Fresnel, Vol. 1, H. Senarmont, É. Verdet, and L. Fresnel, 1866, pp. 767–775 (Johnson Reprint Corporation, New York, 1965).

See, for example, R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Sec. 4.2.

It will be significant to study the relationship between each Fresnel coefficient and the refractive index ratio v as a conformal mapping between two complex planes.

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