Abstract

The complex Fresnel reflection coefficients rp and rs of p- and s-polarized light and their ratio ρ=rp/rs at the pseudo-Brewster angle (PBA) ϕpB of a dielectric–conductor interface are evaluated for all possible values of the complex relative dielectric function ε=|ε|exp(jθ)=εrjεi, εi>0 that share the same ϕpB. Complex-plane trajectories of rp, rs, and ρ at the PBA are presented at discrete values of ϕpB from 5° to 85° in equal steps of 5° as θ is increased from 0° to 180°. It is shown that for ϕpB>70° (high-reflectance metals in the IR) rp at the PBA is essentially pure negative imaginary and the reflection phase shift δp=arg(rp)90°. In the domain of fractional optical constants (vacuum UV or light incidence from a high-refractive-index immersion medium) 0<ϕpB<45° and rp is pure real negative (δp=π) when θ=tan1(cos(2ϕpB)), and the corresponding locus of ε in the complex plane is obtained. In the limit of εi=0, εr<0 (interface between a dielectric and plasmonic medium) the total reflection phase shifts δp, δs, Δ=δpδs=arg(ρ) are also determined as functions of ϕpB.

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References

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    [Crossref]
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    [Crossref]
  16. A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75, 155410 (2007).
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2012 (1)

2008 (1)

2007 (1)

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75, 155410 (2007).
[Crossref]

2002 (1)

1991 (1)

1989 (2)

1988 (1)

1986 (2)

1984 (1)

1983 (1)

1979 (2)

1977 (1)

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

Ahmed, S. A.

Ali, M. A.

Alsamman, A.

Alù, A.

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75, 155410 (2007).
[Crossref]

Azzam, R. M. A.

R. M. A. Azzam and A. Alsamman, “Plurality of principal angles for a given pseudo-Brewster angle when polarized light is reflected at a dielectric-conductor interface,” J. Opt. Soc. Am. A 25, 2858–2864 (2008).
[Crossref]

R. M. A. Azzam and E. Ugbo, “Angular range for reflection of p-polarized light at the surface of an absorbing medium with reflectance below that at normal incidence,” J. Opt. Soc. Am. A 19, 112–115 (2002).
[Crossref]

R. M. A. Azzam and E. Ugbo, “Contours of constant pseudo-Brewster angle in the complex ε plane and an analytical method for the determination of optical constants,” Appl. Opt. 28, 5222–5228 (1989).
[Crossref]

R. M. A. Azzam, “Analytical determination of the complex dielectric function of an absorbing medium from two angles of incidence of minimum parallel reflectance,” J. Opt. Soc. Am. A 6, 1213–1216 (1989).
[Crossref]

R. M. A. Azzam, “Maximum minimum reflectance of parallel-polarized light at interfaces between transparent and absorbing media,” J. Opt. Soc. Am. 73, 959–962 (1983).
[Crossref]

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]

R. M. A. Azzam, “Reflection of an electromagnetic plane wave with 0 or π phase shift at the surface of an absorbing medium,” J. Opt. Soc. Am. 69, 487–488 (1979).
[Crossref]

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

Bashara, N. M.

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

Campbell, I. H.

Cao, M.

Darcie, T. E.

Engheta, N.

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75, 155410 (2007).
[Crossref]

Fauchet, P. M.

Gao, H.

Han, L.

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]

Jin, Y.

Kim, S. Y.

Li, F.

Li, H.

Lu, Y.

Lv, Y.

Moghaddasi, J.

Ohman, G. P.

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

Penzkofer, A.

Salandrino, A.

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75, 155410 (2007).
[Crossref]

Silveirinha, M. G.

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75, 155410 (2007).
[Crossref]

Ugbo, E.

Vedam, K.

Wang, Z.

Whalen, M. S.

Zhang, P.

Appl. Opt. (3)

IEEE Trans. Antennas Propag. (1)

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

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (4)

J. Opt. Soc. Am. B (1)

Opt. Lett. (2)

Phys. Rev. B (1)

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75, 155410 (2007).
[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]

Other (1)

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

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Figures (8)

Fig. 1.
Fig. 1. Complex-plane trajectories of rp at discrete values of the PBA ϕpB from 5° to 85° in equal steps of 5° as θ=arg(ε) covers the full range 0°θ180°.
Fig. 2.
Fig. 2. Graph of the function of Eq. (5). Both ϕpB and θ are in degrees.
Fig. 3.
Fig. 3. Locus of all possible values of complex ε such that δp=arg(rp)=π at the PBA.
Fig. 4.
Fig. 4. Total reflection phase shifts δp, δs, and Δ=δpδs+360° at the interface between a dielectric and plasmonic medium in the limit as θ180° (εi=0,εr<0) are plotted as a functions of ϕpB. All angles are in degrees.
Fig. 5.
Fig. 5. Family of δp versus θ curves for ϕpB from 10° to 40° in equal steps of 10°. Both θ and δp are in degrees.
Fig. 6.
Fig. 6. Family of δp versus θ curves for ϕpB from 45° to 85° in equal steps of 5°. Both θ and δp are in degrees.
Fig. 7.
Fig. 7. Complex-plane contours of rs at discrete values of the PBA ϕpB from 5° to 85° in equal steps of 5° as θ=arg(ε) covers the full range from 0° to 180°.
Fig. 8.
Fig. 8. Complex-plane trajectories of the ratio ρ=rp/rs at discrete values of the PBA ϕpB from 5° to 85° in equal steps of 5° as θ=arg(ε) covers the full range 0°θ180°.

Equations (17)

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rp=εcosϕ(εsin2ϕ)1/2εcosϕ+(εsin2ϕ)1/2,
rs=cosϕ(εsin2ϕ)1/2cosϕ+(εsin2ϕ)1/2.
εr=|ε|cosθ,εi=|ε|sinθ,
|ε|=cos(ς/3),=2u(123u)1/2/(1u),ς=cos1[(1u)cosθ/(123u)3/2],u=sin2ϕpB,0θ180°.
θ(δp=π)=tan1(cos(2ϕpB)).
ε=εr=12tan2ϕpB[1+(98sin2ϕpB)1/2]
ρ=rp/rs=sinϕtanϕ(εsin2ϕ)1/2sinϕtanϕ+(εsin2ϕ)1/2.
y2=a+(a2bx)1/2x2,
a=u2(1.5u)/(1u)2,b=u3/(1u)2,u=sin2ϕpB.
y2=2uxx2.
(a2bx)1/2=2uxa.
4u2x2=(4aub)x.
x=(4aub)/(4u2).
x=u/(1u).
y=u12u/(1u).
tanθ=y/x=12u.
θ(δp=π)=tan1(cos(2ϕpB)).

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