Abstract

The pseudo-Brewster angle ϕpB of minimum reflectance for p-polarized light and the principal angle ϕ¯ at which incident linearly polarized light of the proper azimuth is reflected circularly polarized are considered as functions of the complex relative dielectric function ε of a dielectric–conductor interface over the entire complex ε plane. In particular, the spread of ϕ¯ for a given ϕpB is determined, and the maximum difference (ϕ¯ϕpB)max is obtained as a function of ϕpB. The maximum difference (ϕ¯ϕpB)max approaches 45° and 0 in the limit as ϕpB0 and 90°, respectively. For ϕpB<22.666°, multiple principal angles ϕ¯i, i=1,2,3, appear for each ε in a subdomain of fractional optical constants. This leads to an elaborate pattern of multiple solution branches for the difference ϕ¯iϕpB, i=1,2,3, as is illustrated by several examples.

© 2008 Optical Society of America

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References

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  2. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987).
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    [Crossref]
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    [Crossref]
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2008 (1)

2002 (1)

1999 (1)

1989 (1)

1988 (1)

1986 (1)

1983 (1)

1981 (1)

1977 (1)

G. P. Ohman, “The pseudo-Brewster angle,” IEEE Trans. Antennas Propag. AP-25, 903-904 (1977).

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]

Arendt, P.

Azzam, R. M. A.

Bashara, N. M.

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

Bassom, A. P.

Born, M.

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

Cash, W. C.

Fisher, R. F.

Holl, H. B.

Hooper, L. R.

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]

Kim, S. Y.

Lekner, J.

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

Newman, B.

Ohman, G. P.

G. P. Ohman, “The pseudo-Brewster angle,” IEEE Trans. Antennas Propag. AP-25, 903-904 (1977).

Pinneo, J. M.

Sambles, J. R.

Scott, M.

Swartzlander, A. B.

Takacs, P. Z.

Ugbo, E.

Vedam, K.

Windt, D. L.

Wolf, E.

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

Appl. Opt. (2)

IEEE Trans. Antennas Propag. (1)

G. P. Ohman, “The pseudo-Brewster angle,” IEEE Trans. Antennas Propag. AP-25, 903-904 (1977).

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (3)

Opt. Express (1)

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]

Other (4)

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

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

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

E.D.Palik, ed., Handbook of Optical Constants of Solids (Academic, 1985).

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Figures (9)

Fig. 1
Fig. 1

Domain of MPAs, shown highlighted, is bounded by the real axis, ε i = 0 , and the dashed curve described by Eqs. [10]. Cusp point P is located at ε = ( 5 27 , 2 27 ) = ( 0.1852 , 0.0524 ) .

Fig. 2
Fig. 2

Difference of PA and PBA ϕ ¯ ϕ p B plotted as a function of the angle θ of complex ε, 0 θ 180 ° , for constant values of ϕ p B from 25° to 85° in equal steps of 5°.

Fig. 3
Fig. 3

Maximum difference ( ϕ ¯ ϕ p B ) max as a function of ϕ p B over the entire range 0 < ϕ p B < 90 ° .

Fig. 4
Fig. 4

Constant-pseudo-Brewster-angle contour (CPBAC) in the complex ε plane that corresponds to ϕ p B = 20 ° , 21°, 22°, and 22.666°. The CPBAC at ϕ p B = 22.666 ° passes through the cusp point P.

Fig. 5
Fig. 5

Amplitude reflectances r p , r s and differential reflection phase shift Δ plotted as functions of the angle of incidence ϕ when ε = ( 0.1349 , 0.0118 ) . Minimum reflectance r p min is located at ϕ = ϕ p B = 20 ° , and Δ = 90 ° occurs at three distinct PAs: ϕ ¯ 1 = 39.13 ° , ϕ ¯ 2 = 24.01 ° , and ϕ ¯ 3 = 20.49 ° .

Fig. 6
Fig. 6

Multiple solution branches of the difference function ϕ ¯ i ϕ p B , i = 1 , 2 , 3 , plotted versus the angle θ of complex ε, for ϕ p B = 20 ° , 21°, and 22°. For each ϕ p B the solid, thin-dashed, and thick-dashed curves correspond to ϕ ¯ 1 > ϕ ¯ 2 > ϕ ¯ 3 .

Fig. 7
Fig. 7

CPBAC for ϕ p B = 22.35 ° . This curve intersects the boundary of the domain of MPAs at three points, A, B, and C, where θ A = 7.730 ° , θ B = 10.763 ° , θ C = 14.614 ° .

Fig. 8
Fig. 8

Multiple solution branches of the difference function ϕ ¯ i ϕ p B , i = 1 , 2 , 3 , plotted versus the angle θ of complex ε when ϕ p B = 22.35 ° . For this PBA, MPAs exist for 0 θ θ A and θ B θ θ C , whereas one PA appears when θ A < θ < θ B and θ > θ C .

Fig. 9
Fig. 9

Composite plot of multiple solution branches of the difference functions ϕ ¯ i ϕ p B , i = 1 , 2 , 3 , for ϕ p B = 21 ° , 22°, 22.3°, 22.35°, 22.5° and 22.666° in the domain of MPAs.

Equations (24)

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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 .
Δ = δ p δ s = 90 ° ,
δ p = arg ( r p ) , δ s = arg ( r s ) .
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 .
ρ = ρ ¯ = j tan ψ ¯ .
a 3 u 3 + a 2 u 2 + a 1 u + a 0 = 0 ,
a 0 = ε r 2 + ε i 2 , a 1 = 2 ( a 0 + ε r ) ,
a 2 = a 0 + 4 ε r + 1 , a 3 = 2 ( ε r + 1 ) ,
u = sin 2 ϕ ¯ .
ε r = u + u 3 ( u 2 ) ( 1 u ) 3 ,
ε i = ( 2 u 6 4 u 5 + u 4 ) 1 2 ( 1 u ) 3 ,
0 u 1 1 2 = 0.293 .
ε r = ε cos θ , ε i = ε sin θ ,
ε = l cos ( ζ 3 ) ,
l = 2 u [ 1 ( 2 u 3 ) ] 1 2 ( 1 u ) ,
ζ = cos 1 ( ( 1 u ) cos θ [ 1 ( 2 u 3 ) ] 3 2 ) ,
u = sin 2 ϕ p B , 0 θ 180 ° .
( ϕ ¯ ϕ p B ) θ = 0 , θ = 0 , 180 °
ε = ε r = 1 2 tan 2 ϕ p B [ 1 + ( 9 8 sin 2 ϕ p B ) 1 2 ] .
ϕ ¯ max = sin 1 { 1 2 [ ( ε r + 1 ) + ( ε r 2 6 ε r + 1 ) 1 2 ] 1 2 } .
0 < ϕ p B < 22.666 ° .
324 u 3 80 u 2 2 u + 1 = 0 .

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