Abstract

For a given pseudo-Brewster angle ϕpB of minimum reflectance |rp| of p-polarized light at a dielectric-conductor interface, the second-Brewster angle ϕ2B of minimum reflectance ratio |ρ|=|rp||rs| of the p and s polarizations is determined for all possible values of the complex relative dielectric function ϵ that lead to the same ϕpB. The difference ϕ2BϕpB is considered as a function of ϕpB and θ=arg(ϵ). For any given ϕpB, the difference ϕ2BϕpB=0 at θ=0(ϵr>0,ϵi=0) increases monotonically as a function of θ and reaches maximum value {ϕ2BϕpB}max in the limit as θ180° (ϵr<0,ϵi=0). This maximum difference {ϕ2BϕpB}max has an upper limit of 15.701° when ϕpB=28.195°.

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References

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    [CrossRef]
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2008

1999

M. Born and E. Wolf, Principles of Optics (Cambridge Univ. Press, 1999).

1989

1988

1987

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987).

J. Lekner, Theory of Reflection (Martinus, 1987).

1983

1978

J. M. Bennett and H. E. Bennett, “Polarization,” in Handbook of Optics, W.G.Driscoll and W.Vaughan, eds. (McGraw-Hill, 1978), pp. 10-11–10-12.

1967

Alsamman, A.

Azzam, R. M. A.

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987).

Bennett, H. E.

J. M. Bennett and H. E. Bennett, “Polarization,” in Handbook of Optics, W.G.Driscoll and W.Vaughan, eds. (McGraw-Hill, 1978), pp. 10-11–10-12.

Bennett, J. M.

J. M. Bennett and H. E. Bennett, “Polarization,” in Handbook of Optics, W.G.Driscoll and W.Vaughan, eds. (McGraw-Hill, 1978), pp. 10-11–10-12.

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge Univ. Press, 1999).

Holl, H. B.

Lekner, J.

J. Lekner, Theory of Reflection (Martinus, 1987).

Ugbo, E.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge Univ. Press, 1999).

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Figures (6)

Fig. 1
Fig. 1

| ρ | , | r p | , and | r s | plotted as functions of the angle of incidence ϕ in degrees for ϵ = 0.5183 j 0.2992 . The pseudo-Brewster angle of minimum | r p | ( ϕ p B = 30 ° ) and the second-Brewster angle of minimum | ρ | ( ϕ 2 B = 44.9 ° ) are indicated.

Fig. 2
Fig. 2

| ρ | as a function of the angle of incidence ϕ in degrees for different values of complex ϵ that are calculated for θ values from 0° to 180° in increments of 10° using Eqs. (5), for pseudo-Brewster angle ϕ p B = 30 ° .

Fig. 3
Fig. 3

3-D rendering of | ρ | as a function of ϕ and θ in degrees at constant pseudo-Brewster angle ϕ p B = 30 ° .

Fig. 4
Fig. 4

Difference ϕ 2 B ϕ p B plotted as a function of θ (all angles in degrees) for different values of the pseudo-Brewster angle ϕ p B : (a) ϕ p B assumes values from 2.5° to 27.5° in equal increments of 2.5°, and (b) ϕ p B takes values from 30° to 80° in equal steps of 5°.

Fig. 5
Fig. 5

3-D plot of ϕ 2 B ϕ p B as a function of ϕ p B and θ. All angles are in degrees.

Fig. 6
Fig. 6

Maximum difference { ϕ 2 B ϕ p B } max is plotted as a function of ϕ p B , with all angles in degrees. The maximum difference reaches an upper limit of 15.701° at ϕ p B = 28.195 ° .

Tables (1)

Tables Icon

Table 1 Maximum Difference between the Second-Brewster Angle ϕ 2 B and Pseudo-Brewster Angle ϕ p B for Selected Values of ϕ p B a

Equations (23)

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r p = ϵ cos ϕ ( ϵ sin 2 ϕ ) 1 2 ϵ cos ϕ + ( ϵ sin 2 ϕ ) 1 2 ,
r s = cos ϕ ( ϵ sin 2 ϕ ) 1 2 cos ϕ + ( ϵ sin 2 ϕ ) 1 2 ,
ϵ = N 1 2 n 0 2 = ( n j k ) 2 = ϵ r j ϵ i .
ρ = r p r s = sin ϕ tan ϕ ( ϵ sin 2 ϕ ) 1 2 sin ϕ tan ϕ + ( ϵ sin 2 ϕ ) 1 2 .
ϵ r = | ϵ | cos θ , ϵ i = | ϵ | sin θ ,
| ϵ | = l cos ( ζ 3 ) ,
l = 2 u ( 1 2 3 u ) 1 2 ( 1 u ) ,
ζ = cos 1 [ ( 1 u ) cos θ ( 1 2 3 u ) 3 2 ] ,
u = sin 2 ϕ p B ,
0 θ 180 ° .
Im [ ( u ϵ ) ( u ϵ ϵ + 1 ) 2 ( u 2 ϵ ϵ + 1 ) 2 ] = 0 ,
ϕ 2 B = arcsin u ,
0 u 1 .
a 4 u 4 + a 3 u 3 + a 2 u 2 + a 1 u + a 0 = 0 ,
a 0 = β 0 r γ 0 i β 0 i γ 0 r ,
a 1 = β 0 r γ 1 i + β 1 r γ 0 i β 0 i γ 1 r β 1 i γ 0 r ,
a 2 = β 2 r γ 0 i + β 1 r γ 1 i β 0 i β 1 i γ 1 r β 2 i γ 0 r ,
a 3 = β 2 r γ 1 i + γ 0 i β 1 i β 2 i γ 1 r ,
a 4 = γ 1 i β 2 i ;
β 0 = ϵ ( ϵ ¯ ) 2 , β 1 = ( ϵ ¯ ) 2 + 2 ϵ ( ϵ ¯ ) 2 , β 2 = ϵ 2 ( ϵ ¯ ) ,
γ 0 = 4 ( ϵ ¯ ) 2 , γ 1 = 4 ( ϵ ¯ ) ,
( ϵ ¯ ) = ϵ ( ϵ + 1 ) ,
β k = β k r + j β k i , γ k = γ k r + j γ k i , k = 0 , 1 , 2 .

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