Abstract

The pseudo-Brewster angle ϕpB, of minimum reflectance pm for the parallel (p) polarization, of an interface between a transparent and an absorbing medium is determined by Im{(u)[1 − (1 + −1)u]2} = 0, where is the complex ratio of dielectric constants of the media and u = sin2ϕpB. It is shown that, for a given value of the normal-incidence amplitude reflectance |r|, there is an associated normal-incidence phase shift, δ = δmm, that leads to maximum minimum parallel reflectance, pmm. We determine δmm, pmm, ϕpBmm as functions of |r|. We find that, as |r| increases from 0 to 1, δmm decreases from 90° to 0, pmm/|r|2 increases from 0 to 1, and the associated ϕpBmm decreases from 45° to 0, all monotonically.

© 1983 Optical Society of America

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References

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  1. D. Brewster, “On the laws which regulate the polarisation of light by reflexion,” Philos. Trans. 105, 125–130 (1815).
    [Crossref]
  2. See, for example, J. M. Bennett and H. E. Bennett, “Polarization,” in Handbook of Optics, W. G. Driscoll and W. Vaughan, eds. (McGraw-Hill, New York, 1978), p. 10–11.
  3. 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. London 77, 949–957 (1961).
    [Crossref]
  4. H. B. Holl, “Specular reflection and characteristics of reflected light,” J. Opt. Soc. Am. 57, 683–690 (1967); see, in particular, footnote 19.
    [Crossref]
  5. R. H. Muller, “Definitions and conventions in ellipsometry,” Surf. Sci. 16, 14–33 (1969).
    [Crossref]
  6. See, for example, M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975), Sec. 1.5.2.
  7. The other solution of Eq. (16), u= ∊, is unacceptable when ∊ > 1 or ∊ < 0. When 0 < ∊ < 1, it yields the critical angle of total internal reflection, sin−1∊1/2.
  8. See, for example, S. M. Selby, ed., Standard Mathematical Tables, 20th ed. (Chemical Rubber, Cleveland, Ohio, 1972), pp. 103–105.
  9. This is Method F of Ref. 3.

1969 (1)

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

1967 (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. London 77, 949–957 (1961).
[Crossref]

1815 (1)

D. Brewster, “On the laws which regulate the polarisation of light by reflexion,” Philos. Trans. 105, 125–130 (1815).
[Crossref]

Bennett, H. E.

See, for example, J. M. Bennett and H. E. Bennett, “Polarization,” in Handbook of Optics, W. G. Driscoll and W. Vaughan, eds. (McGraw-Hill, New York, 1978), p. 10–11.

Bennett, J. M.

See, for example, J. M. Bennett and H. E. Bennett, “Polarization,” in Handbook of Optics, W. G. Driscoll and W. Vaughan, eds. (McGraw-Hill, New York, 1978), p. 10–11.

Born, M.

See, for example, M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975), Sec. 1.5.2.

Brewster, D.

D. Brewster, “On the laws which regulate the polarisation of light by reflexion,” Philos. Trans. 105, 125–130 (1815).
[Crossref]

Holl, H. B.

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. London 77, 949–957 (1961).
[Crossref]

Muller, R. H.

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

Wolf, E.

See, for example, M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975), Sec. 1.5.2.

J. Opt. Soc. Am. (1)

Philos. Trans. (1)

D. Brewster, “On the laws which regulate the polarisation of light by reflexion,” Philos. Trans. 105, 125–130 (1815).
[Crossref]

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

Surf. Sci. (1)

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

Other (5)

See, for example, M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1975), Sec. 1.5.2.

The other solution of Eq. (16), u= ∊, is unacceptable when ∊ > 1 or ∊ < 0. When 0 < ∊ < 1, it yields the critical angle of total internal reflection, sin−1∊1/2.

See, for example, S. M. Selby, ed., Standard Mathematical Tables, 20th ed. (Chemical Rubber, Cleveland, Ohio, 1972), pp. 103–105.

This is Method F of Ref. 3.

See, for example, J. M. Bennett and H. E. Bennett, “Polarization,” in Handbook of Optics, W. G. Driscoll and W. Vaughan, eds. (McGraw-Hill, New York, 1978), p. 10–11.

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Figures (6)

Fig. 1
Fig. 1

Pseudo-Brewster angle ϕpB as a function of the normal-incidence reflection phase shift δ for different constant values of the normal-incidence amplitude reflectance, |r| = 0.1, 0.2, …, 0.9, as a parameter. Both δ and ϕpB are in degrees.

Fig. 2
Fig. 2

Minimum parallel reflectance at the pseudo-Brewster angle pm as a function of the normal-incidence reflection phase shift δ (in degrees) for different constant values of the normal-incidence amplitude reflectance, |r| = 0.1, 0.2, …, 0.9, as a parameter.

Fig. 3
Fig. 3

Minimum parallel reflectance pm versus the pseudo-Brewster angle ϕpB (in degrees) for different constant values of the normal-incidence amplitude reflectance, |r| = 0.1, 0.2, …, 0.9, as a parameter.

Fig. 4
Fig. 4

Normal-incidence reflection phase shift δmm (in degrees) that leads to maximum minimum parallel reflectance for a given normal-incidence amplitude reflectance |r| plotted here versus |r|.

Fig. 5
Fig. 5

Ratio of maximum minimum parallel reflectance pmm to normal-incidence reflectance |r|2 plotted as a function of |r|.

Fig. 6
Fig. 6

Pseudo-Brewster angle (in degrees) ϕpBmm of maximum minimum parallel reflectance plotted versus the normal-incidence amplitude reflectance |r|.

Equations (27)

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ϕ B = tan - 1 1 / 2 ,
= 1 / 0 .
N = N 1 / N 0 = 1 / 2
r = r e j δ
p = r p 2 .
r p = r p e j δ p
r p / r p = ( r p / r p ) + j δ p .
p = 0 ,
2 r p r p = 0
r p = 0.
Re ( r p / r p ) = 0.
r p = ( 1 - X ) / ( 1 + X ) ,
X = ( - sin 2 ϕ ) 1 / 2 / cos ϕ .
r p / r p = - 2 X / ( 1 - X 2 ) = 2 ( 1 - ) sin ϕ ( - sin 2 ϕ ) 1 / 2 ( 2 cos 2 ϕ - + sin 2 ϕ ) .
u = sin 2 ϕ ,
Re [ ( - u ) 1 / 2 ( 1 - + 1 u ) ] = 0.
ϕ pB = sin - 1 u 1 / 2 .
u = / ( + 1 ) .
= u / ( 1 - u ) = tan 2 ϕ ,
Im [ ( - u ) ( 1 - + 1 u ) 2 ] = 0.
= r + j i ,
1 / = ¯ = ¯ r + j ¯ i ,
¯ r = r / ( r 2 + i 2 ) ,             ¯ i = - i / ( r 2 + i 2 )
α 3 u 3 + α 2 u 2 + α 1 u + α 0 = 0 ,
α 0 = - i , α 1 = 2 i , α 2 = - ( i + 3 ¯ i ) , α 3 = 2 ¯ i ( 1 + ¯ r ) .
α 0 = 4 , α 1 = - 2 4 , α 2 = 4 - 3 2 , α 3 = 2 r + 2 2 ,
= ( 1 - r e j δ 1 + r e j δ ) 2 .