Abstract

We have found a significant relation, rp = rs(rs − cos2ϕ)/(1 − rs cos2ϕ), between Fresnel’s interface complex-amplitude reflection coefficients rp and rs for the parallel (p) and perpendicular (s) polarizations at the same angle of incidence ϕ. This relation is universal in that it applies to reflection at all interfaces between homogeneous isotropic media collectively and, of course, throughout the electromagnetic spectrum. We investigate the properties of this function, rp = f(rs), and its inverse, rs = g(rp) conformal mappings between the complex planes of rs and rp. A related function, ρ = (rs − cos2ϕ)/(1 − rs cos2ϕ), which is a bilinear transformation, is also studied, where ρ = rp/rs 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 rp is real, negative, and in the range −1 ≤ rp ≤ −tan2(ϕ − 45°), rs 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 |rs| = |rp|1/2.

© 1979 Optical Society of America

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Corrections

F. A. Hopf, A. Tomita, K. H. Womack, J. L. Jewell, and R. M. A. Azzam, "Errata," J. Opt. Soc. Am. 70, 261_1-261 (1980)
https://www.osapublishing.org/josa/abstract.cfm?uri=josa-70-2-261_1

References

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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).
    [Crossref]
  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).
    [Crossref]
  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. Acta26(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).
    [Crossref]
  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).
    [Crossref]
  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).
    [Crossref]
  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).
    [Crossref]
  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).
    [Crossref]
  15. When ϕ= 45°, we have cos2ϕ= 0 and Rp=Rs2. This reduces Eq. (18) to cosδs= 0/0, hence δs becomes indeterminate, as is expected in this special case.

1979 (2)

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).
[Crossref]

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).
[Crossref]

1978 (1)

1969 (2)

R. H. Muller, “Definitions and conventions in ellipsometry,” Surf. Sci. 16, 14–33 (1969).
[Crossref]

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

1966 (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. Phys. Soc. Lond. 77, 949–957 (1961).
[Crossref]

Azzam, R. M. A.

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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

R. M. A. Azzam, “On the reflection of light at 45° angle of incidence,” Opt. Acta26(1979), (in press).

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

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).
[Crossref]

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).
[Crossref]

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. (4)

J. Opt.Soc. Am. (1)

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).
[Crossref]

Proc. Phys. 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. Phys. Soc. Lond. 77, 949–957 (1961).
[Crossref]

Surf. Sci. (1)

R. H. Muller, “Definitions and conventions in ellipsometry,” Surf. Sci. 16, 14–33 (1969).
[Crossref]

Other (8)

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.

R. M. A. Azzam, “On the reflection of light at 45° angle of incidence,” Opt. Acta26(1979), (in press).

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=Rs2. 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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Figures (14)

FIG. 1
FIG. 1

Domains of all permissible values of Fresnel’s interface reflection coefficients rs (left) and rp (right) for the s and p polarizations.

FIG. 2
FIG. 2

Mapping of rs onto rp according to the complex analytic function rp = rs (rs − cos2ϕ)/(1 − rs cos2ϕ) when ϕ = 30°. The angularly equispaced straight lines through the origin 0, 1, 2, …, 12 in the rs plane represent lines of equal reflection phase shift for the s polarization, δs = argrs = 0, 15°, 30°, …, 180° respectively and their images in the rp plane are marked by the same numbers. Points that are images of one another are also marked by the same letters. (This notation applies to the following figures as well.) As rs scans straight line 3, for example, from the origin O to U on the unit circle in the rs plane, rp scans curve 3 from the origin O to U on the unit circle in the rp plane, in the direction of the indicated arrows.

FIG. 3
FIG. 3

Mapping of rs onto rp according to the complex analytic function rp = rs (rs − cos2ϕ)/(1 − rs cos2ϕ) when ϕ = 30°. The equispaced semicircles centered on the origin 1, 2, 3 …,10 in the rs plane represent lines of equal amplitude reflectance for the s polarization, |rs| =0.1, 0.2, 0.3 …,1.0 respectively and their images in the rp are marked by the same numbers. The images in the rp plane of semicircles 1, 2 lie entirely below the real axis, while the images of semicircles 3 to 10 are one-full-revolution spirals. As rs traces semicircle 6, for example, from S to F in the rs plane, rp traces spiral 6 from S to F in the rp plane, in the direction of the indicated arrows.

FIG. 4
FIG. 4

Mapping of rs onto rp according to the complex analytic function rp = rs(rs − cos2ϕ)/(1 − rs cos2ϕ) when ϕ = 15°, 30°, 45°, 60°, 75°. The orthogonal families of straight lines and semicircles through and around the origin in the rs plane are mapped onto orthogonal sets of curves in the rp plane. The orthogonal sets that correspond to ϕ = 30° are obtained from the superposition of Figs. 2 and 3. To identify individual curves use Figs. 2 and 3 as a guide. (Continued on p. 1010.)

FIG. 5
FIG. 5

Inverse mapping of rp onto rs according to the complex analytic function rs = (½) cos2ϕ(1 − rp) + [rp + (¼) cos22ϕ(1 − rp)2]1/2 when ϕ = 30°. The angularly equispaced straight lines through the origin 0, 1, 2, …, 23 in the rp plane represent lines of equal reflection phase shift for the p polarization, δp = arg rp = 0, 15°, 30° …, 345° respectively and their images in the rs plane are marked by the same numbers. The images in the rs plane of straight lines, 1,2, …, 11 in the upper half of the rp plane all originate from the point B(rs = cos2ϕ = 0.5), while the images of straight lines 13, 14, …,23 in the lower half of the rp plane all pass through the origin O. The segment MA2 of the negative real axis of the rp plane is imaged onto circle arc MA2 in the rs plane. MA2 divides the upper half of the unit circle in the rs plane into two domains, one to its right and the other to its left, that correspond to the upper and lower halves of the unit circle in the rp plane respectively.

FIG. 6
FIG. 6

Inverse mapping of rp onto rs according to the complex analytic function rs = (½) cosϕ(1 − rp) + [rp + (¼) cos22ϕ(1 − rp)2]1/2 when ϕ = 30°. The equispaced circles centered on the origin 1, 2, 3, …,10 in the rp plane represent lines of equal amplitude reflectance for the p polarization |rp| =0.1, 0.2, 0.3,…, 1.0 respectively and their images in the |rs|plane are marked by the same numbers.

FIG. 7
FIG. 7

Inverse mapping of rp onto rs according to the complex analytic function rs = (½) cos2ϕ(1 − rp) + [rp + (¼) cos22ϕ(1 − rp)2]1/2 when ϕ = 15°, 30°, 45°, 60°, 75°. The orthogonal families of straight lines and circles through and around the origin in the rp plane are mapped onto orthogonal sets of curves in the rs plane. The orthogonai sets that correspond to ϕ = 30° are obtained from the superposition of Figs. 5 and 6. To identify individual curves use Figs. 5 and 6 as a guide.

FIG. 8
FIG. 8

Mapping of rs onto ρ = rp/rs according to the bilinear transformation ρ = (rs − cos2ϕ)/(1 − rscos2ϕ) when ϕ = 30°. Straight lines through the origin in the rs plane, δs = argrs = 0, 15°, 30°, …,180°, are mapped onto arcs of coaxial circles that pass through the points ρ = −cos2ϕ and −sec2ϕ (not shown) in the ρ plane.

FIG. 9
FIG. 9

Mapping of rs onto ρ = rp/rs according to the bilinear transformation ρ = (rs − cos2ϕ)/(1 − rscos2ϕ) when ϕ = 30°. Semicircles centered on the origin in the rs plane, |rs| =0.1, 0.2, …,1.0, are mapped onto semicircles of coaxial circles that enclose the point ρ = cos2ϕ in the ρ plane.

FIG. 10
FIG. 10

Mapping of rs onto ρ = rp/rs according to the bilinear transformation ρ = (rs − cos2ϕ)/(1 − rs cos2ϕ) when ϕ = 15°, 30°, 45°, 60°, 75°. The orthogonal families of straight lines and semicircles through and around the origin in the rs are mapped onto orthogonal circle arcs and semicircles through and around the point ρ = − cos2ϕ in the ρ plane. The orthogonal sets that correspond to ϕ = 30° are obtained from the superposition of Figs. 8 and 9. To identify individual curves use Figs. 8 and 9 as a guide.

FIG. 11
FIG. 11

A nomogram in the complex rs plane for the graphical determination of the reflection phase shift δs from the measured reflectances Rs and Rp of the s and p polarizations at the same angle of incidence ϕ = 60°. The semicircles centered on the origin represent lines of equal amplitude reflectance for the s polarization | r s | = R s 1 / 2 = 0.1 , 0.2 , 0.3 , , 1.0, while the semicircles that enclose the point rs = cos2ϕ represent lines of equal ratio of p and s amplitude reflectances |ρ| = (Rp/Rs)1/2 = 0.1, 0.2, 0.3, …,1.0. The measured reflectances Rs and Rp specify one semicircle from each family and their point of intersection gives the complex reflection coefficient rs while its angular polar coordinate gives δs = argrs.

FIG. 12
FIG. 12

A graphical construction that can be readily made and used in lieu of the nomogram of Fig. 11.

FIG. 13
FIG. 13

The loci of the complex reflection coefficient for the s polarization when the reflection coefficient for the p polarization is real and negative in the range −1 ≤ rp ≤ −tan2(ϕ − 45°) at several different angles of incidence ϕ = 0, 5°, 10°,…, 90° marked by each curve. Each locus or curve at a given ϕ is an arc of a circle with center on the real axis at sec2ϕ and radius of | tan2ϕ|. Circle arcs that correspond to ϕ = 45° + θ and 45° − θ are mirror images of one another with respect to the imaginary axis which represents ϕ = 45°.

FIG. 14
FIG. 14

The reflection phase shift for the s polarization δs, when the reflection phase shift for the p polarization δp is equal to π, as a function of the absolute value of rp, with the angle of incidence ϕ as a parameter marked by each curve. Mirror reflection with respect to the line δs − 90°, which represents ϕ = 45°, relate δs (|rp|) at any pair of angles of incidence ϕ = 45° ± 0.

Equations (30)

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r p = cos ϕ ( sin 2 ϕ ) 1 / 2 cos ϕ + ( sin 2 ϕ ) 1 / 2 ,
r s = cos ϕ ( sin 2 ϕ ) 1 / 2 cos ϕ + ( sin 2 ϕ ) 1 / 2 ,
x = ( sin 2 ϕ ) 1 / 2 ,
r p = cos ϕ ( x 2 + sin 2 ϕ ) x cos ϕ ( x 2 + sin 2 ϕ ) + x ,
r s = ( cos ϕ x ) / ( cos ϕ + x ) .
r p = r s ( r s cos 2 ϕ ) / ( 1 r s cos 2 ϕ ) .
r s = cos 2 ϕ ,
r p r s = ( r s 2 cos 2 ϕ + 2 r s cos 2 ϕ ) / ( 1 r s cos 2 ϕ ) 2 ,
r s = sec 2 ϕ tan 2 ϕ = tan ( ϕ 45 ° ) .
r p = ( sec 2 ϕ tan 2 ϕ ) 2 = tan 2 ( ϕ 45 ° ) = r s 2 .
r s = 1 2 cos 2 ϕ ( 1 r p ) + ( r p + 1 4 cos 2 2 ϕ ( 1 r p ) 2 ) 1 / 2 .
ρ = r p / r s
ρ = ( r s cos 2 ϕ ) / ( 1 r s cos 2 ϕ ) .
r s = ( ρ + cos 2 ϕ ) / ( 1 + ρ cos 2 ϕ ) ,
r s = [ ρ cos 2 ( 90 ° ϕ ) / [ 1 ρ cos 2 ( 90 ° ϕ ) ] .
R = | r l | 2 , l = p , s .
R p = R s R s + cos 2 2 ϕ 2 R s 1 / 2 cos 2 ϕ cos δ s 1 + R s + cos 2 2 ϕ 2 R s 1 / 2 cos 2 ϕ cos δ s ,
cos δ s = ( R s 2 r p ) + R s ( 1 R p ) cos 2 2 ϕ 2 R s 1 / 2 ( R s R p ) cos 2 ϕ .
= sin 2 ϕ + cos 2 ϕ [ ( 1 r s ) / ( 1 + r s ) ] 2 ,
tan Δ = R s 1 / 2 sin δ s sin 2 2 ϕ R s 1 / 2 cos δ s ( 1 + cos 2 2 ϕ ) R s cos 2 ϕ ,
| ρ cos 2 ϕ | = | r s cos 2 ϕ | / | r s sec 2 ϕ | .
1 r p tan 2 ( ϕ 45 ° ) ,
Q = r p + ( 1 / 4 ) cos 2 2 ϕ ( 1 r p ) 2 ,
Q = 0 .
r p = ( sec 2 ϕ tan 2 ϕ ) 2 = tan 2 ( ϕ 45 ° ) .
r s = x + j y ,
x = ( 1 / 2 ) cos 2 ϕ ( 1 r p ) , y = ( Q ) 1 / 2 .
( x sec 2 ϕ ) 2 + y 2 = tan 2 2 ϕ ,
| r s | = | r p | 1 / 2 .
R p = R s 2 ,