Abstract

The functions, w = f(z), that describe the transformation of Fresnel’s reflection coefficients of parallel and perpendicularly polarized light between normal and oblique incidence, as well as their inverses, z = g(w), are studied in detail as conformal mappings between the complex z and w planes for angles of incidence of 15, 30, 45, 60, and 75°. New nomograms are obtained for the determination of optical properties of absorbing isotropic and anisotropic media from measurements of reflectances of s- or p-polarized light at normal and oblique incidence.

© 1980 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. M. A. Azzam, J. Opt. Soc. Am. 69, 590 (1979).
    [Crossref]
  2. R. H. Muller, Surf. Sci. 16, 14 (1969).
    [Crossref]
  3. See, for example, A. Kyrala, Applied Functions of a Complex Variable (Wiley-Interscience, New York, 1972), Chap. 8.
  4. R. M. A. Azzam, J. Opt. Soc. Am. 68, 1613 (1978).
    [Crossref]
  5. R. M. A. Azzam, J. Opt. Soc. Am. 69, 1007 (1979).
    [Crossref]
  6. See, for example, M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), p. 40.
  7. W. R. Hunter, J. Opt. Soc. Am. 55, 1197 (1965).
    [Crossref]

1979 (2)

1978 (1)

1969 (1)

R. H. Muller, Surf. Sci. 16, 14 (1969).
[Crossref]

1965 (1)

Azzam, R. M. A.

Born, M.

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

Hunter, W. R.

Kyrala, A.

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

Muller, R. H.

R. H. Muller, Surf. Sci. 16, 14 (1969).
[Crossref]

Wolf, E.

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

J. Opt. Soc. Am. (4)

Surf. Sci. (1)

R. H. Muller, Surf. Sci. 16, 14 (1969).
[Crossref]

Other (2)

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

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

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (12)

Fig. 1
Fig. 1

Mapping of Fresnel’s interface reflection coefficient for s-polarized light between normal incidence [z plane (top)] and oblique incidence at 30° (w plane (bottom)]. Semicircles 1, 2, 3, …, 10 in the z plane are lines of equal normal-incidence amplitude reflectance, |z| = 0.1, 0.2, 0.3, …, 1. Images in the w plane carry the same numbers. Start and finish of one semicircle and its image are marked by S and F.

Fig. 2
Fig. 2

Same as Fig. 1 for a family of straight lines through the origin 0, 1, 2, …, 12 in the z plane, representing lines of equal normal-incidence phase shift, argz = 0, 15, 30, …, 180°.

Fig. 3
Fig. 3

Superposition of Figs. 1 and 2 gives orthogonal sets of curves in the z and w planes. Results shown are for five angles of incidence including 30° (ϕ = 15, 30, 45, 60, 75°). To identify individual curves, use Figs. 1 and 2 (ϕ = 30°) as a guide. Dashed curves in the graph of ϕ = 75° are images of semicircles |z| = 0.01m, where m = 1, 2, …, 9.

Fig. 4
Fig. 4

Inverse mapping of Fresnel’s interface reflection coefficient for s-polarized light between oblique incidence at 30° [w plane (top)] and normal incidence [z plane (bottom)]. Semicircles 1, 2, 3, …, 10 in the w plane are lines of constant oblique-incidence (30°) amplitude reflectance |w| = 0.1, 0.2, 0.3, …, 1. Their images in the z plane are marked by the same numbers.

Fig. 5
Fig. 5

Same as in Fig. 4 for a family of straight lines through the origin 0, 1, 2, …, 12 in the w plane representing lines of constant oblique-incidence (30°) phase shift, argw = 0, 15, 30, …, 180°.

Fig. 6
Fig. 6

Superposition of Figs. 4 and 5 produces orthogonal families of curves in the w and z planes. Results shown are for five angles of incidence including 30° (ϕ = 15, 30, 45, 60, 75°). To identify individual curves use Figs. 4 and 5 (for ϕ = 30°) as a guide.

Fig. 7
Fig. 7

Mapping of Fresnel’s interface reflection coefficient for p-polarized light between normal incidence [z plane (top)] and oblique incidence at 30° [w plane (bottom)]. Semicircles 1, 2, 3, …, 10 in the z plane are lines of equal normal-incidence amplitude reflectance, |z| = 0.1, 0.2, 0.3, …, 1. Their images in the w plane are marked by the same numbers. Significance of points B, C, and M is discussed in the text.

Fig. 8
Fig. 8

Same as in Fig. 7 for a family of straight lines through the origin 0, 1, 2, …, 12 in the z plane, representing lines of equal normal-incidence phase shift, argz = 180, 195, 210, …, 360°.

Fig. 9
Fig. 9

Superposition of Figs. 7 and 8 gives orthogonal sets of curves in the z and w planes. Results shown are for five angles of of incidence including 30° (ϕ = 15, 30, 45, 60, 75°). To identify individual curves, use Figs. 7 and 8 (for ϕ = 30°) as a guide.

Fig. 10
Fig. 10

Inverse mapping of Fresnel’s interface reflection coefficient for p-polarized light between oblique incidence at 30° [w plane (top)]and normal incidence [z plane (bottom)]. Full circles centered on the origin 1, 2, 3, …, 10 in the w plane are lines of constant oblique-incidence (30°) amplitude reflectance, |w| = 0.1, 0.2, 0.3, …, 1. Their images in the z plane are marked by the same numbers. Significance of points B, C, and M is discussed in the text.

Fig. 11
Fig. 11

Same as in Fig. 10 for a family of straight lines through the origin 0, 1, 2, …, 24 in the w plane (top) representing lines of constant oblique-incidence (30°) phase shift, argw = 0, 15, 30, …, 360°.

Fig. 12
Fig. 12

Superposition of Figs. 10 and 11 gives orthogonal sets of curves in the w and z planes. Results shown are for five angles of incidence including 30° (ϕ = 15, 30, 45, 60, 75°). To identify individual curves use Figs. 10 and 11 as a guide.

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

w = ( 1 z ) ( 1 a z + z 2 ) 1 / 2 ( 1 z ) + ( 1 a z + z 2 ) 1 / 2 ,
a = 4 tan 2 ϕ + 2.
z = ( a + 2 p ) ± [ ( a 2 4 ) + 4 ( a + 2 ) p ] 1 / 2 2 ( 1 p ) ,
p = [ ( 1 w ) / ( 1 + w ) ] 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 ,
w p = w s ( w s cos 2 ϕ ) / ( 1 w s cos 2 ϕ ) ,
w s = 1 2 cos 2 ϕ ( 1 w p ) ± [ w p + 1 4 cos 2 2 ϕ ( 1 w p ) 2 ] 1 / 2 .
w p = cos ϕ ( sin 2 ϕ ) 1 / 2 cos ϕ + ( sin 2 ϕ ) 1 / 2 ,
= [ 1 ± ( 1 Q p sin 2 2 ϕ ) 1 / 2 ] / 2 Q p cos 2 ϕ ,
Q p = [ ( 1 w p ) / ( 1 + w p ) ] 2 .
z p = ( 1 / 2 1 ) / ( 1 / 2 + 1 ) .
N = ± n ( 1 z ) / ( 1 + z ) ,

Metrics