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.

© 2013 Optical Society of America

1. INTRODUCTION

A salient feature of the reflection of collimated monochromatic p (TM)-polarized light at a planar interface between a transparent medium of incidence (dielectric) and an absorbing medium of refraction (conductor) is the appearance of a reflectance minimum at the pseudo-Brewster angle (PBA) ϕpB. If the medium of refraction is also transparent, the minimum reflectance is zero and ϕpB reverts back to the usual Brewster angle ϕB=tan1n=tan1εr. The PBA ϕpB is determined by the complex relative dielectric function ε=ε1/ε0=εrjεi, εi>0, where ε0 and ε1 are the real and complex permittivities of the dielectric and conductor, respectively, by solving a cubic equation in u=sin2ϕpB [15]. Measurement of ϕpB and of reflectance at that angle or at normal incidence enables the determination of complex ε [1,69]. It is also possible to determine ε of an optically thick absorbing film from two PBAs measured in transparent ambient and substrate media that sandwich the thick film [10]. Reflection at the PBA has also had other interesting applications [11,12].

For light reflection at any angle of incidence ϕ the complex-amplitude Fresnel reflection coefficients (see, e.g., [13]) of the p and s polarizations are given by

rp=εcosϕ(εsin2ϕ)1/2εcosϕ+(εsin2ϕ)1/2,
rs=cosϕ(εsin2ϕ)1/2cosϕ+(εsin2ϕ)1/2.
All possible values of complex ε=(εr,εi) that share the same ϕpB are generated by using the following algorithm [8,14]:
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°.
As θ is increased from 0° to 180°, the minimum reflectance |rp|min at a given ϕpB increases monotonically from 0 to 1 [15] and also as is evident in Fig. 1 of Section 2.

 

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°.

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In this paper, loci of all possible values of complex rp=|rp|exp(jδp), rs=|rs|exp(jδs), and ρ=rp/rs=|ρ|exp(jΔ) at the PBA are determined at discrete values of ϕpB from 5° to 85° in equal steps of 5° and as θ=arg(ε) covers the full range 0°θ180°. These results are presented in Sections 2, 3, and 4, respectively, and lead to interesting conclusions. In particular, questions related to phase shifts that accompany the reflection of p- and s-polarized light at the PBA (e.g., [12]) are settled. Section 5 summarizes the essential conclusions of this paper.

2. COMPLEX REFLECTION COEFFICIENT OF THE p POLARIZATION AT THE PBA

Figure 1 shows the loci of complex rp as θ increases from 0° to 180° at constant values of ϕpB from 5° to 85° in equal steps of 5°. All constant-ϕpB contours begin at the origin O (θ=0) as a common point, that represents zero reflection at an ideal Brewster angle, and end on the 90° arc of the unit circle in the third quadrant (shown as a dotted line) that represents total reflection |rp|=1 at θ=180° (εi=0,εr<0). A quick conclusion from Fig. 1 is that for ϕpB>70° (high-reflectance metals) rp at the PBA is essentially pure negative imaginary, and δp90°.

In Fig. 1 the constant-ϕpB contours of rp for 0<ϕpB<45° spill over into a limited range of the second quadrant of the complex plane and each contour intersects the negative real axis. In Appendix A it is shown that θ at the point of intersection, where δp=arg(rp)=π, is given by the remarkably simple formula

θ(δp=π)=tan1(cos(2ϕpB)).
A graph of this function of Eq. (5) is shown in Fig. 2.

 

Fig. 2. Graph of the function of Eq. (5). Both ϕpB and θ are in degrees.

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The locus of complex ε such that δp=arg(rp)=π at the PBA [as determined by Eqs. (3)–(5)] falls in the domain of fractional optical constants and is shown in Fig. 3. The end points (0, 0) and (1, 0) of this trajectory correspond to ϕpB=0 and 45°, respectively. At ε=(0.6,0.3), a point that falls exactly on the curve very near to its peak, ϕpB=37.761°.

 

Fig. 3. Locus of all possible values of complex ε such that δp=arg(rp)=π at the PBA.

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For small PBAs, ϕpB5°, the upper limit on |ε| is calculated from of Eq. (4), |ε|=0.0153, and represents the domain of so-called epsilon-near-zero (ENZ) materials [16].

Negative real values of ε at θ=180° [14] are given by

ε=εr=12tan2ϕpB[1+(98sin2ϕpB)1/2]
and represent light reflection at an ideal dielectric–plasmonic medium interface. The corresponding total reflection phase shift δp as θ180° (at the end point of each contour in Fig. 1) is obtained from Eqs. (1) and (6) and is plotted as a function of ϕpB in Fig. 4. In Fig. 4 δp increases monotonically from 180° to 90° as ϕpB increases from 0° to 90°. The initial rise of δp with respect to ϕpB is linear for ϕpB<20° and then transitions to saturation at ϕpB>70°, in accord with Fig. 1.

 

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.

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In Fig. 5 δp is plotted as a function of θ for ϕpB from 10° to 40° in equal steps of 10°. Vertical transitions from +180° to 180° are located at θ values that agree with Eq. (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.

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Another family of δp-versus-θ curves for ϕpB from 45° to 85° in equal steps of 5° is shown in Fig. 6. For ϕpB>45° the δp-versus-θ curve first exhibits a minimum then reaches saturation as θ180°. The saturated value of δp is a function of ϕpB and is shown in Fig. 4.

 

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.

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3. COMPLEX REFLECTION COEFFICIENT OF THE s POLARIZATION AT THE PBA

Figure 7 shows the loci of complex rs as θ increases from 0° to 180° at discrete values of ϕpB from 5° to 85° in equal steps of 5°. All curves start on the real axis at θ=0, rs=cos(2ϕpB), which is the s amplitude reflectance at the Brewster angle of a dielectric–dielectric interface [17], and terminate on the upper half of the unit circle (dotted line) that represents total reflection rs=exp(jδs) at θ=180° (εi=0,εr<0). The associated total reflection phase shift δs along the dotted semicircle is a function of ϕpB as shown in Fig. 4.

 

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°.

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Although we are locked on the PBA, all possible values of complex rs (within the upper half of the unit circle) are generated at that angle. This is not the case of complex rp at the PBA (Fig. 1) which is squeezed mostly in the third quadrant of the unit circle. Recall that the unconstrained domain of rp for light reflection at all dielectric–conductor interfaces is on and inside the full unit circle [17].

4. RATIO OF COMPLEX REFLECTION COEFFICIENTS OF THE p AND s POLARIZATIONS AT THE PBA

The ratio of complex p and s reflection coefficients, also known as the ellipsometric function ρ=tanψexp(jΔ) [13], is obtained from Eqs. (1) and (2) as

ρ=rp/rs=sinϕtanϕ(εsin2ϕ)1/2sinϕtanϕ+(εsin2ϕ)1/2.

Figure 8 shows loci of complex ρ as θ increases from 0° to 180° at constant values of ϕpB from 5° to 85° in equal steps of 5°. All contours begin at the origin O (as a common point that represents the ideal Brewster-angle condition of rp=0 at θ=0), then fan out and terminate on the 90° arc of the unit circle in the second quadrant of the complex plane (dotted line), so that ρ=exp(jΔ) at θ=180° (εi=0,εr<0). The differential reflection phase shift Δ=δpδs+360° at θ=180° decreases monotonically from 180° to 90° as ϕpB increases from 0° to 90° as shown in Fig. 4.

 

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°.

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5. SUMMARY

The Fresnel complex reflection coefficients rp, rs and their ratio ρ=rp/rs are evaluated at the PBA ϕpB of a dielectric–conductor interface for all possible values of the complex relative dielectric function ε=|ε|exp(jθ)=εrjεi, εi>0. Complex-plane loci of rp, rs, and ρ at the PBA are obtained at discrete values of ϕpB from 5° to 85° in equal steps of 5° and as θ increases from 0° to 180°; these are presented in Figs. 1, 7, and 8, respectively. The reflection phase shift δp of the p polarization at the PBA is plotted as function of θ in Figs. 5 and 6 for two different sets of ϕpB. For ϕpB>70° (e.g., high-reflectance metals in the IR), rp at the PBA is essentially pure negative imaginary and 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) at θ=tan1(cos(2ϕpB)). The associated locus of complex ε is shown in Fig. 3. Finally, the total reflection phase shifts δp, δs, Δ=arg(ρ) at an ideal dielectric–plasmonic medium interface (εi=0,εr<0), are shown as functions of ϕpB in Fig. 4.

APPENDIX A

By setting εr=x and εi=y, the Cartesian equation of a constant-ϕpB contour (a cardioid [8]) takes the form [10]

y2=a+(a2bx)1/2x2,
a=u2(1.5u)/(1u)2,b=u3/(1u)2,u=sin2ϕpB.
The locus of complex ε such that δp=arg(rp)=π at a given angle of incidence ϕ=sin1u is a circle [18]
y2=2uxx2.
Equations (A1) and (A3) are satisfied simultaneously if their right-hand sides are equal; this gives
(a2bx)1/2=2uxa.
By squaring both sides of Eq. (A4) we obtain
4u2x2=(4aub)x.
Equation (A5) is obviously satisfied when x=0, and from Eq. (A3) one gets y=0 and ε=0. The more significant solution of Eq. (A5) is
x=(4aub)/(4u2).
Substitution of a and b from Eq. (A2) in Eq. (A6) leads to the simple result
x=u/(1u).
The associated value of y is then obtained from Eq. (A3) as
y=u12u/(1u).
The angle θ=arg(ε) is determined from Eqs. (A7) and (A8) by
tanθ=y/x=12u.
Finally, substitution of u=sin2ϕpB in Eq. (A9) gives
θ(δp=π)=tan1(cos(2ϕpB)).
This completes the proof of Eq. (5).

REFERENCES

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]  

2. H. B. Holl, “Specular reflection and characteristics of reflected light,” J. Opt. Soc. Am. 57, 683–690 (1967). [CrossRef]  

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

4. 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]  

5. S. Y. Kim and K. Vedam, “Analytic solution of the pseudo-Brewster angle,” J. Opt. Soc. Am. A 3, 1772–1773 (1986). [CrossRef]  

6. T. E. Darcie and M. S. Whalen, “Determination of optical constants using pseudo-Brewster angle and normal-incidence reflectance,” Appl. Opt. 23, 1130–1131 (1984). [CrossRef]  

7. Y. Lu and A. Penzkofer, “Optical constants measurements of strongly absorbing media,” Appl. Opt. 25, 221–225 (1986). [CrossRef]  

8. 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]  

9. M. A. Ali, J. Moghaddasi, and S. A. Ahmed, “Optical properties of cooled rhodamine B in ethanol,” J. Opt. Soc. Am. B 8, 1807–1810 (1991). [CrossRef]  

10. 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]  

11. I. H. Campbell and P. M. Fauchet, “Temporal reshaping of ultrashort laser pulses after reflection from GaAs at Brewster’s angle,” Opt. Lett. 13, 634–636 (1988). [CrossRef]  

12. Y. Lv, Z. Wang, Y. Jin, M. Cao, L. Han, P. Zhang, H. Li, H. Gao, and F. Li, “Spin polarization separation of light reflected at Brewster angle,” Opt. Lett. 37, 984–986 (2012). [CrossRef]  

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

14. 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]  

15. 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]  

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). [CrossRef]  

17. 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]  

18. 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]  

References

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  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]
  2. H. B. Holl, “Specular reflection and characteristics of reflected light,” J. Opt. Soc. Am. 57, 683–690 (1967).
    [Crossref]
  3. G. P. Ohman, “The pseudo-Brewster angle,” IEEE Trans. Antennas Propag. 25, 903–904 (1977).
    [Crossref]
  4. 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]
  5. S. Y. Kim and K. Vedam, “Analytic solution of the pseudo-Brewster angle,” J. Opt. Soc. Am. A 3, 1772–1773 (1986).
    [Crossref]
  6. T. E. Darcie and M. S. Whalen, “Determination of optical constants using pseudo-Brewster angle and normal-incidence reflectance,” Appl. Opt. 23, 1130–1131 (1984).
    [Crossref]
  7. Y. Lu and A. Penzkofer, “Optical constants measurements of strongly absorbing media,” Appl. Opt. 25, 221–225 (1986).
    [Crossref]
  8. 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]
  9. M. A. Ali, J. Moghaddasi, and S. A. Ahmed, “Optical properties of cooled rhodamine B in ethanol,” J. Opt. Soc. Am. B 8, 1807–1810 (1991).
    [Crossref]
  10. 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]
  11. I. H. Campbell and P. M. Fauchet, “Temporal reshaping of ultrashort laser pulses after reflection from GaAs at Brewster’s angle,” Opt. Lett. 13, 634–636 (1988).
    [Crossref]
  12. Y. Lv, Z. Wang, Y. Jin, M. Cao, L. Han, P. Zhang, H. Li, H. Gao, and F. Li, “Spin polarization separation of light reflected at Brewster angle,” Opt. Lett. 37, 984–986 (2012).
    [Crossref]
  13. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987).
  14. 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]
  15. 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]
  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).
    [Crossref]
  17. 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]
  18. 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]

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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