Abstract

If w denotes an interface Fresnel reflection or transmission coefficient for s - or p -polarized light at an oblique angle of incidence ϕ, and z denotes the same coefficient at normal incidence, we find that w is an analytic function of z, w = f(z), that depends on ϕ 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 lumiére polarisée,” in Oeuvres Complétes 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 ν 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).
    [CrossRef]
  5. It is important to observe that any pair of Fresnel coefficients can be directly interrelated by the elimination of their common variable ν.
  6. In the Nebraska (Muller) conventions, the ejωt 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 ν → 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 (ν 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 ϕ. The oblique reflection is total (i.e., |w| =1) if the refractive index ratio ν⩾sinϕ or z⩾(1−sinϕ)/(1+sinϕ).
  9. Total reflection at normal incidence (|z| = 1) leads, as expected, to total reflection at oblique incidence (|w| = 1). Notice that |z| → 1 as ν → ∞ 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).
    [CrossRef]
  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).
    [CrossRef]
  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).
    [CrossRef]

1969 (1)

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).
[CrossRef]

1965 (1)

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. R. Soc. Lond. 77, 949–957 (1961).
[CrossRef]

1951 (1)

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 lumiére polarisée,” in Oeuvres Complétes de Fresnel, Vol. 1, H. Senarmont, É. Verdet, and L. Fresnel, 1866, pp. 767–775 (Johnson Reprint Corporation, New York, 1965).

Fresnel, L.

A. Fresnel, “Mémoire sur la loi des modifications que la réflection imprime a la lumiére polarisée,” in Oeuvres Complétes 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).
[CrossRef]

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).
[CrossRef]

Senarmont, H.

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

Šimon, I.

Verdet, É.

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

J. Opt. Soc. Am. (2)

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

Surf. Sci. (1)

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).
[CrossRef]

Other (12)

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

In the Nebraska (Muller) conventions, the ejωt 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 ν → 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 (ν 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 ϕ. The oblique reflection is total (i.e., |w| =1) if the refractive index ratio ν⩾sinϕ or z⩾(1−sinϕ)/(1+sinϕ).

Total reflection at normal incidence (|z| = 1) leads, as expected, to total reflection at oblique incidence (|w| = 1). Notice that |z| → 1 as ν → ∞ 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 lumiére polarisée,” in Oeuvres Complétes 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 ν as a conformal mapping between two complex planes.

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Figures (10)

FIG. 1
FIG. 1

The reflection and transmission (refraction) of a plane wave of light at the planar interface between two media with refractive indices n and N. ϕ is the angle of incidence and p and s are the two principal polarization directions parallel and perpendicular to the plane of incidence, respectively.

FIG. 2
FIG. 2

Mapping between the complex plane of the refractive index ratio ν and the complex plane of the normal-incidence s-reflection coefficient z.

FIG. 3
FIG. 3

Mapping of the Fresnel reflection coefficient for s-polarized light between normal incidence (z plane) and 45° oblique incidence (w plane). The straight lines through the origin a, b, c,… l in the z plane correspond to argz = (0, 180°), (15°, 195°), (30°, 210°),…, (165°, 345°), respectively, and their images in the w plane are marked by the corresponding capital letters A, B, C,…, L. As z scans line d from d1 to d2, w scans its image D from D1 to D2.

FIG. 4
FIG. 4

Mapping of the Fresnel reflection coefficient for s-polarized light between normal incidence (z plane) and 45° oblique incidence (w plane). The circles centered on the origin a, b, c,…, j in the z plane correspond to |z| = 0.1, 0.2, 0.3,…, 1 and their images in the w plane are marked by A, B, C,…, J, respectively. As the circle d in the z plane is traced from d1 to d2, its image D in the w plane is traced from D1 to D2, in the directions of the indicated arrows. All circles 0 < | z | 0.1716 in the z plane are imaged into closed contours in the w plane [0.1716 = (1 − sinϕ)/(1 + sinϕ) where ϕ = 45° is the angle of incidence].

FIG. 5
FIG. 5

The superposition of Figs. 3 and 4 produces orthogonal sets of contours in the z and w planes.

FIG. 6
FIG. 6

Inverse mapping of the Fresnel reflection coefficient for s-polarized light between oblique incidence at 45° (w plane) and normal incidence (z plane). The straight lines through the origin of the w plane A, B, C,…, L correspond to arg w = (0, 180°), (15°, 195°), (30°, 210°),…, (165°, 345°) and their images in the z plane are denoted by a, b, c, …, l, respectively.

FIG. 7
FIG. 7

Inverse mapping of the Fresnel reflection coefficient for s-polarized light between oblique incidence at 45° (w plane) and normal incidence (z plane). The circles A, B, C,…, J in the w plane correspond to |w| = 0.1, 0.2, 0.3,…, 1 and their images in the z plane are a, b, c,…, j, respectively.

FIG. 8
FIG. 8

The superposition of Figs. 6 and 7 yields orthogonal sets of curves in the w and z planes.

FIG. 9
FIG. 9

Geometrical construction to show how the complex Fresnel reflection coefficient for s-polarized light at normal incidence (z or z*) can be determined from two intensity reflectances measured at normal and 45° oblique incidence (shown here to be |z|2 = 0.09 and |w|2 = 0.25, respectively).

FIG. 10
FIG. 10

Nomogram for applying the two-reflectance method (TRM), using s-polarized light at normal and 45° incidence, to determine the complex normal-incidence reflection coefficient (hence the optical properties of isotropic and aniostropic materials, see text).

Equations (21)

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ν = N / n
r ϕ s = cos ϕ ( ν 2 sin 2 ϕ ) 1 / 2 cos ϕ + ( ν 2 sin 2 ϕ ) 1 / 2 ,
r ϕ p = ν 2 cos ϕ ( ν 2 sin 2 ϕ ) 1 / 2 ν 2 cos ϕ + ( ν 2 sin 2 ϕ ) 1 / 2 ,
t ϕ s = 2 cos ϕ cos ϕ + ( ν 2 sin 2 ϕ ) 1 / 2 ,
t ϕ p = 2 ν cos ϕ ν 2 cos ϕ + ( ν 2 sin 2 ϕ ) 1 / 2 .
r 0 s = r 0 p = ( 1 ν ) / ( 1 + ν ) ,
t 0 s = t 0 p = 2 / ( 1 + ν ) .
w = ( 1 + z ) ( z 2 a z + 1 ) 1 / 2 ( 1 + z ) + ( z 2 a z + 1 ) 1 / 2 ,
w = ( 1 + z ) 2 ( 1 z ) ( 1 + a z + z 2 ) 1 / 2 ( 1 + z ) 2 + ( 1 z ) ( 1 + a z + z 2 ) 1 / 2 ,
a = 2 ( 2 tan 2 ϕ + 1 ) .
w = 2 z / [ z + ( z 2 + 4 c z + 4 c ) 1 / 2 ] ,
w = 2 z ( 2 z ) ( 2 z ) 2 + z ( z 2 + 4 c z + 4 c ) 1 / 2 ,
c = sec 2 ϕ .
z = ( a + 2 p ) ± [ ( a 2 4 ) + 4 ( a + 2 ) p ] 1 / 2 2 ( 1 p ) ,
p = [ ( 1 w ) / ( 1 + w ) ] 2 ,
α 4 z 4 + α 3 z 3 + α 2 z 2 + α 1 z + α 0 = 0 ,
α 0 = α 4 = p 1 , α 1 = α 3 = 4 p a + 2 , α 2 = 6 p + 2 a 2 ,
z = 2 c ± 2 [ c 2 + c ( q 2 1 ) ] 1 / 2 q 2 1 ,
q = ( 2 w ) / w ,
β 4 z 4 + β 3 z 3 + β 2 z 2 + β 1 z + β 0 = 0 ,
β 0 = 4 w 2 , β 1 = 8 w 2 8 w , β 2 = ( 6 c ) w 2 + 12 w + 4 , β 3 = ( c 2 ) w 2 6 w 4 , β 4 = w + 1 ,