Abstract

We propose simple and good approximate equations for the reflectance of the perpendicular and parallel polarizations of light upon an isotropic absorbing medium deduced from the Fresnel coefficients of reflection. We also calculate the principal incidence and the pseudo-Brewster angle by using approximate expressions. These formulas are valid when the optical constants n and k of the surface are such that n2 + k2 ≫ 1 and are more accurate than those used so far in the literature. The excellent agreement between approximate and exact values verifies the high accuracy of the given equations.

© 1988 Optical Society of America

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References

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  1. J. M. Bennett, H. E. Bennett, “Polarization,” in Handbook of Optics, W. G. Driscoll, W. Vaughan, eds. (McGraw-Hill, New York, 1978), in particular, p. 10-7.
  2. J. Figueira, S. Thomas, “Damage thresholds at metal surfaces for short pulse IR lasers,” IEEE J. Quantum Electron. QE-18, 1381–1386 (1982).
    [CrossRef]
  3. R. M. A. Azzam, “Direct relation between Fresnel’s interface reflection coefficients for the parallel and perpendicular polarizations,” J. Opt. Soc. Am. 69, 1007–1016 (1979).
    [CrossRef]
  4. R. M. A. Azzam, “Relationship between the p and s Fresnel reflection coefficients of an interface independent of angle of incidence,” J. Opt. Soc. Am. A 3, 928–929 (1986).
    [CrossRef]
  5. W. D. Kimura, D. H. Ford, “Absorptance measurements of metal mirrors at glancing incidence,” Appl. Opt. 25, 3740–3750 (1986).
    [CrossRef] [PubMed]
  6. F. Abelès, “Un théorème relatif à la réflexion métallique,” C. R. Acad. Sci. 230, 1942–1943 (1950).
  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. 77, 949–957 (1960).
    [CrossRef]
  8. S. Y. Kim, K. Vedam, “Analytical solution of the pseudo-Brewster angle,” J. Opt. Soc. Am. A 3, 1772–1773 (1986).
    [CrossRef]
  9. T. E. Darcie, M. S. Whalen, “Determination of optical constants using pseudo-Brewster angle and normal incidence reflectance measurements,” Appl. Opt. 23, 1130–1131 (1984).
    [CrossRef] [PubMed]
  10. W. R. Hunter, “Measurements of optical constants in the vacuum ultraviolet spectral region,” in Handbook of Optical Constants of Solids, E. B. Palik, ed. (Academic, New York, 1985), Chap. 4, pp. 69–87.
  11. S. Wong, G. Krauss, J. Bennett, “Optical and metallurgical characterization of molybdenum laser mirrors,” in Laser-Induced Damage in Optical Materials, Natl. Bur. Stand. (U.S.) Spec. Publ. 541, 132–163 (1978).

1986 (3)

1984 (1)

1982 (1)

J. Figueira, S. Thomas, “Damage thresholds at metal surfaces for short pulse IR lasers,” IEEE J. Quantum Electron. QE-18, 1381–1386 (1982).
[CrossRef]

1979 (1)

1978 (1)

S. Wong, G. Krauss, J. Bennett, “Optical and metallurgical characterization of molybdenum laser mirrors,” in Laser-Induced Damage in Optical Materials, Natl. Bur. Stand. (U.S.) Spec. Publ. 541, 132–163 (1978).

1960 (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. 77, 949–957 (1960).
[CrossRef]

1950 (1)

F. Abelès, “Un théorème relatif à la réflexion métallique,” C. R. Acad. Sci. 230, 1942–1943 (1950).

Abelès, F.

F. Abelès, “Un théorème relatif à la réflexion métallique,” C. R. Acad. Sci. 230, 1942–1943 (1950).

Azzam, R. M. A.

Bennett, H. E.

J. M. Bennett, H. E. Bennett, “Polarization,” in Handbook of Optics, W. G. Driscoll, W. Vaughan, eds. (McGraw-Hill, New York, 1978), in particular, p. 10-7.

Bennett, J.

S. Wong, G. Krauss, J. Bennett, “Optical and metallurgical characterization of molybdenum laser mirrors,” in Laser-Induced Damage in Optical Materials, Natl. Bur. Stand. (U.S.) Spec. Publ. 541, 132–163 (1978).

Bennett, J. M.

J. M. Bennett, H. E. Bennett, “Polarization,” in Handbook of Optics, W. G. Driscoll, W. Vaughan, eds. (McGraw-Hill, New York, 1978), in particular, p. 10-7.

Darcie, T. E.

Figueira, J.

J. Figueira, S. Thomas, “Damage thresholds at metal surfaces for short pulse IR lasers,” IEEE J. Quantum Electron. QE-18, 1381–1386 (1982).
[CrossRef]

Ford, D. H.

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. 77, 949–957 (1960).
[CrossRef]

Hunter, W. R.

W. R. Hunter, “Measurements of optical constants in the vacuum ultraviolet spectral region,” in Handbook of Optical Constants of Solids, E. B. Palik, ed. (Academic, New York, 1985), Chap. 4, pp. 69–87.

Kim, S. Y.

Kimura, W. D.

Krauss, G.

S. Wong, G. Krauss, J. Bennett, “Optical and metallurgical characterization of molybdenum laser mirrors,” in Laser-Induced Damage in Optical Materials, Natl. Bur. Stand. (U.S.) Spec. Publ. 541, 132–163 (1978).

Thomas, S.

J. Figueira, S. Thomas, “Damage thresholds at metal surfaces for short pulse IR lasers,” IEEE J. Quantum Electron. QE-18, 1381–1386 (1982).
[CrossRef]

Vedam, K.

Whalen, M. S.

Wong, S.

S. Wong, G. Krauss, J. Bennett, “Optical and metallurgical characterization of molybdenum laser mirrors,” in Laser-Induced Damage in Optical Materials, Natl. Bur. Stand. (U.S.) Spec. Publ. 541, 132–163 (1978).

Appl. Opt. (2)

C. R. Acad. Sci. (1)

F. Abelès, “Un théorème relatif à la réflexion métallique,” C. R. Acad. Sci. 230, 1942–1943 (1950).

IEEE J. Quantum Electron. (1)

J. Figueira, S. Thomas, “Damage thresholds at metal surfaces for short pulse IR lasers,” IEEE J. Quantum Electron. QE-18, 1381–1386 (1982).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Laser-Induced Damage in Optical Materials (1)

S. Wong, G. Krauss, J. Bennett, “Optical and metallurgical characterization of molybdenum laser mirrors,” in Laser-Induced Damage in Optical Materials, Natl. Bur. Stand. (U.S.) Spec. Publ. 541, 132–163 (1978).

Proc. Phys. Soc. (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. 77, 949–957 (1960).
[CrossRef]

Other (2)

W. R. Hunter, “Measurements of optical constants in the vacuum ultraviolet spectral region,” in Handbook of Optical Constants of Solids, E. B. Palik, ed. (Academic, New York, 1985), Chap. 4, pp. 69–87.

J. M. Bennett, H. E. Bennett, “Polarization,” in Handbook of Optics, W. G. Driscoll, W. Vaughan, eds. (McGraw-Hill, New York, 1978), in particular, p. 10-7.

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Figures (10)

Fig. 1
Fig. 1

Differences, in absolute values, between exact reflectances Rs and the approximate values given by relation (7), Rs ≅ (ρ0 − cos θ0)/(ρ0 + cos θ0), with respect to the index n of the substrate. The isoreflectance R0 at normal incidence is taken as a parameter varying from 0.5 to 0.99, and ρ0 = (1 + R0)/(1 − R0). The incident angle is θ0= 45°, 60°. The entrance medium is air (n0 = 1). The minimum in Rs continues to zero, but these curves are omitted for clarity.

Fig. 2
Fig. 2

Same as for Fig. 1 but with θ0 = 85° in air.

Fig. 3
Fig. 3

Differences, in absolute values, between exact reflectances Rp and the approximate values given by relation (12), Rp ≅ (ρ0 cos θ0 − 1)/(ρ0 cos θ0 + 1), with respect to the index n of the substrate. The incident angle is θ0 = 45° in air.

Fig. 4
Fig. 4

Same as for Fig. 3 but with θ0 = 60° in air.

Fig. 5
Fig. 5

Differences, in absolute values, between exact reflectances Rp and the approximate values given by relation (11) and valid for high incidence: Rp ≅ [n0 sin4θ0 + 2n cos θ0(ρ0 cos θ0 − 1)]/[n0 sin4θ0 + 2n cos θ0(ρ0 cos θ0 + 1)]. The incident angle is θ0 = 60° in air. In comparison with Fig. 4, the accuracy is about an order of magnitude better.

Fig. 6
Fig. 6

Same as for Fig. 5 but with θ0 = 85° in air.

Fig. 7
Fig. 7

Absolute values of the differences, in degrees, between the exact principal incidence θp [Eq. (14)] and the approximate value given by relation (15), tan2θP ≅ 2ρ0n/n0, with respect to the index n of the substrate. The entrance medium is air.

Fig. 8
Fig. 8

Absolute values of the difference, in degrees, between the exact pseudo-Brewster angle θB [Eq. (16)] and the approximate value given by relation (17), tan2θB ≅ 2ρ0n/n0 − 3, with respect to the index n of the substrate. The ambient medium is air.

Fig. 9
Fig. 9

Differences, in absolute values, between the exact reflectance Rs and the approximate value [relation (19)] at the pseudo-Brewster angle, Rs ≅ (ρ0p − 1)/(ρ0p + 1), with p = [2(ρ0n/n0 − 1)]1/2.

Fig. 10
Fig. 10

Differences, in absolute values, between the exact reflectance Rp and the approximate value [relation (20)] at the pseudo-Brewster angle, Rp ≅ (pn/n0)/(p + n/n0), p = [2(ρ0n/n0 − 1)]1/2.

Equations (69)

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R 0 = ( n 0 n ) 2 + k 2 ( n 0 + n ) 2 + k 2 ,
n 2 2 n n 0 ρ 0 + k 2 + n 0 2 = 0 ,
ρ 0 = ( 1 + R 0 ) / ( 1 R 0 ) .
n 0 [ ρ 0 ( ρ 0 2 1 ) 1 / 2 ] n n 0 [ ( ρ 0 + ( ρ 0 2 1 ) 1 / 2 ] ,
r ν = ( y 0 y N y 0 + y N ) ν .
ν = s y 0 = n 0 cos θ 0 and y N = N cos θ N ; ν = p y 0 = n 0 / cos θ 0 and y N = N / cos θ N .
R s = ( cos θ 0 u ) 2 + υ 2 ( cos θ 0 + u ) 2 + υ 2 ,
R p = [ ( n 2 k 2 ) cos θ 0 u ] 2 + ( 2 n k cos θ 0 υ ) 2 [ ( n 2 k 2 ) cos θ 0 + u ] 2 + ( 2 n k cos θ 0 + υ ) 2 ,
2 u 2 = n 2 k 2 sin 2 θ 0 + [ ( n 2 k 2 sin 2 θ 0 ) 2 + 4 n 2 k 2 ] 1 / 2
2 υ 2 = k 2 n 2 + sin 2 θ 0 + [ ( n 2 k 2 sin 2 θ 0 ) 2 + 4 n 2 k 2 ] 1 / 2 .
sin θ N = n 0 sin θ 0 N * N N *
cos 2 θ N = 1 ( n 2 k 2 ) u 2 + i 2 n k u 2 ,
u = n 0 sin θ 0 n 2 + k 2 .
x 2 y 2 = 1 ( n 2 k 2 ) u 2 , x y = n k u 2 .
x = 1 ½ ( n 2 k 2 ) u 2 , y = n k u 2 ,
υ = ½ ( n 2 + k 2 ) u 2 2 k 2 u 2 , w = ½ ( n 2 + k 2 ) u 2 2 n 2 u 2 ;
R s = [ n 0 cos θ 0 n ( 1 υ ) ] 2 + k 2 ( 1 + w ) 2 [ n 0 cos θ 0 + n ( 1 υ ) ] 2 + k 2 ( 1 + w ) 2 .
A = [ n 0 2 cos 2 θ 0 + n 2 ( 1 2 υ ) 2 n n 0 ( 1 υ ) cos θ 0 ] + k 2 ( 1 + 2 w ) .
n 2 + k 2 = 2 n n 0 ρ 0 n 0 2
2 ( k 2 w n 2 υ ) = n 0 2 sin 2 θ 0 2 n 2 ( n 2 + k 2 ) u 2 ,
A = 2 n n 0 ( ρ 0 cos θ 0 + α u 2 ) ,
α = 1 2 ( n 2 3 k 2 ) cos θ 0 n n 0 ( n 2 + k 2 ) .
B = 2 n n 0 ( ρ 0 + cos θ 0 β u 2 ) ,
β = 1 2 ( n 2 3 k 2 ) cos θ 0 + n n 0 ( n 2 + k 2 ) ,
R s = ρ 0 cos θ 0 + α u 2 ρ 0 + cos θ 0 β u 2
R s = ρ 0 cos θ 0 ρ 0 + cos θ 0 { 1 + u 2 cos θ 0 ρ 2 cos 2 θ 0 × [ ρ 0 ( n 2 3 k 2 ) 2 n n 0 ( n 2 + k 2 ) ] } .
R s ρ 0 cos θ 0 ρ 0 + cos θ 0 .
n 0 = 1 , n = 1.2 , k = 2.18174 θ 0 = 45 ° .
R s = 0.6211708 ,
R s = 0.5969630 ,
R s = 0.6185129 ,
y N = ( n i k ) / ( x + i y ) .
y N = n ( 1 + υ ) i k ( 1 w ) .
R p = [ n 0 n ( 1 + υ ) cos θ 0 ] 2 + k 2 ( 1 w ) 2 cos 2 θ 0 [ n 0 + n ( 1 + υ ) cos θ 0 ] 2 + k 2 ( 1 w ) 2 cos 2 θ 0 .
C = [ n 0 2 2 n n 0 ( 1 + υ ) cos θ 0 + n 2 ( 1 + 2 υ ) cos 2 θ 0 ] + k 2 ( 1 2 w ) cos 2 θ 0 , C = [ n 0 2 2 n n 0 cos θ 0 + ( n 2 + k 2 ) cos 2 θ 0 ] [ 2 ( k 2 w n 2 υ ) cos 2 θ 0 + 2 n n 0 cos θ 0 υ ] , C = [ n 0 2 sin 2 θ 0 + 2 n n 0 cos θ 0 ( ρ 0 cos θ 0 1 ) ] { n 0 2 sin 2 θ 0 cos 2 θ 0 + n n 0 cos θ 0 u 2 × [ 2 n n 0 ( n 2 + k 2 ) cos θ 0 + ( n 2 3 k 2 ) ] } .
C = n 0 2 sin 4 θ 0 + 2 n n 0 cos θ 0 [ ( ρ 0 cos θ 0 1 ) + γ u 2 ] ,
γ = n n 0 ( n 2 + k 2 ) cos θ 0 1 2 ( n 2 3 k 2 ) .
D = n 0 2 sin 4 θ 0 + 2 n n 0 cos θ 0 [ ( ρ 0 cos θ 0 + 1 ) + δ u 2 ] ,
δ = n n 0 ( n 2 + k 2 ) cos θ 0 + 1 2 ( n 2 3 k 2 ) ,
R p n 0 sin 4 θ 0 + 2 n cos θ 0 ( ρ 0 cos θ 0 1 + γ u 2 ) n 0 sin 4 θ 0 + 2 n cos θ 0 ( ρ 0 cos θ 0 + 1 + δ u 2 ) .
R p n 0 sin 4 θ 0 + 2 n cos θ 0 ( ρ 0 cos θ 0 1 ) n 0 sin 4 θ 0 + 2 n cos θ 0 ( ρ 0 cos θ 0 + 1 ) .
R p ρ 0 cos θ 0 1 ρ 0 cos θ 0 + 1 ,
ρ ν = 1 + R ν 1 R ν ,
ρ s ρ 0 cos θ 0 ,
ρ p ρ 0 cos θ 0 .
ρ s ρ p ρ 0 2 , ρ p ρ s cos 2 θ 0 .
R s ( 1 + cos 2 θ 0 ) R p + sin 2 θ 0 R p sin 2 θ 0 + 1 + cos 2 θ 0 . R p ( 1 + cos 2 θ 0 ) R s sin 2 θ 0 R s sin 2 θ 0 + 1 + cos 2 θ 0
R p 3 R s 1 3 R s ,
R p = R s 2 .
2 ( q + 1 ) sin 6 θ p ( p 2 + 4 q + 1 ) sin 4 θ p + 2 ( p 2 + q ) sin 2 θ p p 2 = 0 ,
r p r s = cos ( θ 0 + θ N ) cos ( θ 0 θ N )
r p r s = 1 tan θ 0 tan θ N 1 + tan θ 0 tan θ N ,
r p r s = N cos θ N n 0 sin θ 0 tan θ 0 N cos θ N + n 0 sin θ 0 tan θ 0 .
N cos θ N ( N cos θ N ) * n 0 2 sin 2 θ p tan 2 θ p = 0
n 2 + k 2 n 0 2 sin 2 θ p tan 2 θ p 2 ( n 2 υ k 2 w ) = 0 .
n 2 + k 2 n 0 2 sin 2 θ p tan 2 θ p = 0
n 0 cos 4 θ p ( 2 n ρ 0 + n 0 ) cos 2 θ p + n 0 = 0 .
cos 2 θ p = 1 2 n 0 { 2 n ρ 0 + n 0 ( 2 n ρ 0 + n 0 ) × [ 1 4 n 0 2 ( 2 n ρ 0 + n 0 ) 2 ] 1 / 2 } .
cos 2 θ p n 0 / ( 2 n ρ 0 + n 0 )
tan θ p ( 2 n n 0 ρ 0 ) 1 / 2 .
cotan 2 θ B = α [ cos ( γ / 3 ) + 3 sin ( γ / 3 ) ] ,
α = [ n 0 2 / ( n 2 + k 2 ) ] 2 + 1 / 9 , β = [ n 0 2 / ( n 2 + k 2 ) ] 3 ( n 2 k 2 ) / ( n 2 + k 2 ) + 1 / 27 , cos γ = β / α .
n 0 sin 2 θ B ( tan 2 θ B + 4 ) 2 n ρ 0 = 0 .
tan 4 θ B + 2 tan 2 θ B ( 2 n n 0 ρ 0 ) 2 n n 0 ρ 0 = 0 ,
tan 2 θ B = n n 0 ρ 0 2 + ( n n 0 ρ 0 1 ) [ 1 + 3 ( n n 0 ρ 0 1 ) 2 ] 1 / 2 .
tan θ B ( 2 n n 0 ρ 0 3 ) 1 / 2 .
tan 2 θ B tan 2 θ p 3 .
R s ρ 0 [ 2 ( n n 0 ρ 0 1 ) ] 1 / 2 1 ρ 0 [ 2 ( n n 0 ρ 0 1 ) ] 1 / 2 + 1 .
R p [ 2 ( n n 0 ρ 0 1 ) ] 1 / 2 n n 0 [ 2 ( n n 0 ρ 0 1 ) ] 1 / 2 + n n 0 .

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