Abstract

The analytic solution of the pseudo-Brewster angle, for the case of light incident from a transparent medium onto an absorbing substrate, has been obtained.

© 1986 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), Sec. 10, p. 11.
  2. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959), p. 620.
  3. In order to compare Eq. (4) with the corresponding equations [Eqs. (26a) and (26b) of Ref. 2], eliminate Ψ from Eqs. (26a) and (26b) of Ref. 2 and make substitutions such as ϕi→ ϕp, n→ n/n0, nκ→ k/n0, which are necessary owing to the different notation used by those authors.
  4. H. B. Holl, “The reflection of electromagnetic radiation,” Rep. RF-TR-63-4 (U.S. Army Missile Command, Redstone Arsenal, Ala., 1963), Vols. 1 and 2.

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), Sec. 10, p. 11.

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), Sec. 10, p. 11.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959), p. 620.

Holl, H. B.

H. B. Holl, “The reflection of electromagnetic radiation,” Rep. RF-TR-63-4 (U.S. Army Missile Command, Redstone Arsenal, Ala., 1963), Vols. 1 and 2.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959), p. 620.

Other (4)

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

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1959), p. 620.

In order to compare Eq. (4) with the corresponding equations [Eqs. (26a) and (26b) of Ref. 2], eliminate Ψ from Eqs. (26a) and (26b) of Ref. 2 and make substitutions such as ϕi→ ϕp, n→ n/n0, nκ→ k/n0, which are necessary owing to the different notation used by those authors.

H. B. Holl, “The reflection of electromagnetic radiation,” Rep. RF-TR-63-4 (U.S. Army Missile Command, Redstone Arsenal, Ala., 1963), Vols. 1 and 2.

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Tables (1)

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Table 1 Pseudo-Brewster Angles (in degrees) for a Few Selected n, k Values (n0 = 1)

Equations (26)

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tan ϕ B = n / n 0 ,
r p = n 1 cos ϕ 0 n 0 cos ϕ 1 n 1 cos ϕ 0 + n 0 cos ϕ 1 ,
sin ϕ 1 = ( n 0 / n 1 ) sin ϕ 0 .
tan 4 ϕ p sin 4 ϕ p = [ ( n 2 + k 2 ) / n 0 2 ] 2 2 [ ( n 2 k 2 ) / n 0 2 ] × sin 2 ϕ p + sin 4 ϕ p ,
d d ϕ 0 | r p | 2 = 0 at ϕ 0 = ϕ B .
n 2 3 α η + 2 β = 0 ,
η = cot 2 ϕ B + 1 / 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 .
cot 2 ϕ B = α { cos [ 1 / 3 cos 1 ( β / α 3 / 2 ) ] + 3 sin [ 1 / 3 cos 1 ( β / α 3 / 2 ) ] } 1 / 3 .
( n 1 n 1 * cos 2 ϕ 0 + n 0 2 cos ϕ 1 cos ϕ 1 * ) d d ϕ 0 × ( n 0 n 1 cos ϕ 0 cos ϕ 1 * + n 0 n 1 * cos ϕ 0 cos ϕ 1 ) + ( n 0 n 1 cos ϕ 0 cos ϕ 1 * + n 0 n 1 * cos ϕ 0 cos ϕ 1 ) d d ϕ 0 × ( n 1 n 1 * cos 2 ϕ 0 + n 0 2 cos ϕ 1 cos ϕ 1 * ) = 0 at ϕ 0 = ϕ B ,
cos ϕ 1 = [ 1 ( n 0 / n 1 ) sin 2 ϕ 0 ] 1 / 2 .
d d ϕ 0 cos ϕ 1 = ( n 0 / n 1 ) 2 sin ϕ 0 cos ϕ 0 cos ϕ 1 ,
d d ϕ 0 cos ϕ 1 * = ( n 0 / n 1 * ) 2 sin ϕ 0 cos ϕ 0 cos ϕ 1 * ,
( n 1 2 cos 2 ϕ 0 n 0 2 cos 2 ϕ 1 ) ( n 0 2 cos 2 ϕ 0 n 1 * cos 2 ϕ 1 * n 1 * cos ϕ 1 * ) + ( n 1 * 2 cos 2 ϕ 0 n 0 2 cos 2 ϕ 1 * ( n 0 2 cos 2 ϕ 0 n 1 cos ϕ 1 n 1 cos ϕ 1 ) = 0 at ϕ 0 = ϕ B .
n 0 2 cos 2 ϕ 0 n 1 2 cos 2 ϕ 1 = n 0 2 n 1 2 ,
n 0 2 cos 2 ϕ 0 n 1 * 2 cos 2 ϕ 1 * = n 0 2 n 1 * 2 ,
n 1 2 cos 2 ϕ 0 n 0 2 cos 2 ϕ 1 = ( n 1 2 n 0 2 ) [ cos 2 ϕ 0 ( n 0 / n 1 ) 2 sin 2 ϕ 0 ] ,
n 1 * 2 cos 2 ϕ 0 n 0 2 cos 2 ϕ 1 * = ( n 1 * 2 n 0 2 ) [ cos 2 ϕ 0 ( n 0 / n 1 * ) 2 sin 2 ϕ 0 ] ,
( n 0 2 n 1 * 2 ) ( n 1 2 n 0 2 ) [ cos 2 ϕ 0 ( n 0 / n 1 ) 2 sin 2 ϕ 0 n 1 * cos ϕ 1 * + cos 2 ϕ 0 ( n 0 / n 1 * ) 2 sin 2 ϕ 0 n 1 cos ϕ 1 ] = 0 at ϕ 0 = ϕ B .
ξ 3 + ξ 2 3 aa * ξ + aa * ( a + a * 1 ) = 0 ,
ξ = cot 2 ϕ B , a = ( n 0 / n 1 ) 2 , a * = ( n 0 / n 1 * ) 2 .
η = ξ + 1 / 3 = cot 2 ϕ B + 1 / 3 ,
α = aa * + 1 / 9 = [ n 0 2 / ( n 2 + k 2 ) ] 2 + 1 / 9 ,
β = aa * ( a + a * ) / 2 + 1 / 27 = [ n 0 / ( n 2 + k 2 ) ] 3 [ ( n 2 k 2 ) / ( n 2 + k 2 ) ] + 1 / 27 ,
η 3 3 α η + 2 β = 0.

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