Abstract

The polarizing angle θpol is the angle of incidence at which an incident wave of arbitrary polarization becomes linearly polarized on reflection. In terms of the reflection amplitudes it is given by rpprss-rpsrsp=0. We show that it may be obtained by the solution of a quartic equation. This equation is closely related to the quartic that defines the Brewster angle θpp at which rpp is zero, previously obtained. The angles θpol and θpp are compared and contrasted. A method of identifying the physical root or roots of each quartic is given. Index matching enhances the difference between θpol and θpp.

© 1999 Optical Society of America

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References

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  1. M. Malus, “Sur une propriété de la lumière réfléchie,” Mém. Phys. Chim. Soc. d’Arcueil 2, 143–158 (1809).
  2. D. Brewster, “On the laws which regulate the polarisation of light by reflexion from transparent bodies,” Philos. Trans. R. Soc. London 105, 125–130, 158–159 (1815).
    [CrossRef]
  3. W. Swindell, ed., Polarized Light (Halsted, New York, 1975).
  4. G. Green, “On the reflexion and refraction of sound,” Trans. Cambridge Philos. Soc. 6, 403–412 (1838).
  5. J. Lekner, Theory of Reflection (Nijhoff/Kluwer, Dordrecht, 1987).
  6. J. Lekner, “Nonreflecting stratifications,” Can. J. Phys. 68, 738–742 (1990).
    [CrossRef]
  7. M. Elshazly-Zaghloul, R. M. A. Azzam, “Brewster and pseudo-Brewster angles of uniaxial crystal surfaces and their use for determination of optical properties,” J. Opt. Soc. Am. 72, 657–661 (1982);erratum, J. Opt. Soc. Am. A6, 607 (1989).
    [CrossRef]
  8. J. Lekner, “Reflection and refraction by uniaxial crystals,” J. Phys. Condens. Matter 3, 6121–6133 (1991).
    [CrossRef]
  9. J. Lekner, “Bounds and zeros in reflection and refraction by uniaxial crystals,” J. Phys. Condens. Matter 4, 9459–9468 (1992).
    [CrossRef]
  10. J. Lekner, “Brewster angles in reflection by uniaxial crystals,” J. Opt. Soc. Am. A 10, 2059–2064 (1993).
    [CrossRef]
  11. S. Bassiri, C. H. Papas, N. Engheta, “Electromagnetic wave propagation through a dielectric–chiral interface and through a chiral slab,” J. Opt. Soc. Am. A 5, 1450–1459 (1988).
    [CrossRef]
  12. I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, Norwood, Mass., 1994), Sec. 3.5.4.
  13. J. Lekner, “Optical properties of isotropic chiral media,” Pure Appl. Opt. 5, 417–443 (1996), Sec. 3.3.
    [CrossRef]
  14. J. Lekner, “Reflection ellipsometry of uniaxial crystals,” J. Opt. Soc. Am. A 14, 1359–1362 (1997).
    [CrossRef]

1997 (1)

1996 (1)

J. Lekner, “Optical properties of isotropic chiral media,” Pure Appl. Opt. 5, 417–443 (1996), Sec. 3.3.
[CrossRef]

1993 (1)

1992 (1)

J. Lekner, “Bounds and zeros in reflection and refraction by uniaxial crystals,” J. Phys. Condens. Matter 4, 9459–9468 (1992).
[CrossRef]

1991 (1)

J. Lekner, “Reflection and refraction by uniaxial crystals,” J. Phys. Condens. Matter 3, 6121–6133 (1991).
[CrossRef]

1990 (1)

J. Lekner, “Nonreflecting stratifications,” Can. J. Phys. 68, 738–742 (1990).
[CrossRef]

1988 (1)

1982 (1)

1838 (1)

G. Green, “On the reflexion and refraction of sound,” Trans. Cambridge Philos. Soc. 6, 403–412 (1838).

1815 (1)

D. Brewster, “On the laws which regulate the polarisation of light by reflexion from transparent bodies,” Philos. Trans. R. Soc. London 105, 125–130, 158–159 (1815).
[CrossRef]

1809 (1)

M. Malus, “Sur une propriété de la lumière réfléchie,” Mém. Phys. Chim. Soc. d’Arcueil 2, 143–158 (1809).

Azzam, R. M. A.

Bassiri, S.

Brewster, D.

D. Brewster, “On the laws which regulate the polarisation of light by reflexion from transparent bodies,” Philos. Trans. R. Soc. London 105, 125–130, 158–159 (1815).
[CrossRef]

Elshazly-Zaghloul, M.

Engheta, N.

Green, G.

G. Green, “On the reflexion and refraction of sound,” Trans. Cambridge Philos. Soc. 6, 403–412 (1838).

Lekner, J.

J. Lekner, “Reflection ellipsometry of uniaxial crystals,” J. Opt. Soc. Am. A 14, 1359–1362 (1997).
[CrossRef]

J. Lekner, “Optical properties of isotropic chiral media,” Pure Appl. Opt. 5, 417–443 (1996), Sec. 3.3.
[CrossRef]

J. Lekner, “Brewster angles in reflection by uniaxial crystals,” J. Opt. Soc. Am. A 10, 2059–2064 (1993).
[CrossRef]

J. Lekner, “Bounds and zeros in reflection and refraction by uniaxial crystals,” J. Phys. Condens. Matter 4, 9459–9468 (1992).
[CrossRef]

J. Lekner, “Reflection and refraction by uniaxial crystals,” J. Phys. Condens. Matter 3, 6121–6133 (1991).
[CrossRef]

J. Lekner, “Nonreflecting stratifications,” Can. J. Phys. 68, 738–742 (1990).
[CrossRef]

J. Lekner, Theory of Reflection (Nijhoff/Kluwer, Dordrecht, 1987).

Lindell, I. V.

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, Norwood, Mass., 1994), Sec. 3.5.4.

Malus, M.

M. Malus, “Sur une propriété de la lumière réfléchie,” Mém. Phys. Chim. Soc. d’Arcueil 2, 143–158 (1809).

Papas, C. H.

Sihvola, A. H.

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, Norwood, Mass., 1994), Sec. 3.5.4.

Tretyakov, S. A.

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, Norwood, Mass., 1994), Sec. 3.5.4.

Viitanen, A. J.

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, Norwood, Mass., 1994), Sec. 3.5.4.

Can. J. Phys. (1)

J. Lekner, “Nonreflecting stratifications,” Can. J. Phys. 68, 738–742 (1990).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Phys. Condens. Matter (2)

J. Lekner, “Reflection and refraction by uniaxial crystals,” J. Phys. Condens. Matter 3, 6121–6133 (1991).
[CrossRef]

J. Lekner, “Bounds and zeros in reflection and refraction by uniaxial crystals,” J. Phys. Condens. Matter 4, 9459–9468 (1992).
[CrossRef]

Mém. Phys. Chim. Soc. d’Arcueil (1)

M. Malus, “Sur une propriété de la lumière réfléchie,” Mém. Phys. Chim. Soc. d’Arcueil 2, 143–158 (1809).

Philos. Trans. R. Soc. London (1)

D. Brewster, “On the laws which regulate the polarisation of light by reflexion from transparent bodies,” Philos. Trans. R. Soc. London 105, 125–130, 158–159 (1815).
[CrossRef]

Pure Appl. Opt. (1)

J. Lekner, “Optical properties of isotropic chiral media,” Pure Appl. Opt. 5, 417–443 (1996), Sec. 3.3.
[CrossRef]

Trans. Cambridge Philos. Soc. (1)

G. Green, “On the reflexion and refraction of sound,” Trans. Cambridge Philos. Soc. 6, 403–412 (1838).

Other (3)

J. Lekner, Theory of Reflection (Nijhoff/Kluwer, Dordrecht, 1987).

W. Swindell, ed., Polarized Light (Halsted, New York, 1975).

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, Norwood, Mass., 1994), Sec. 3.5.4.

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Figures (3)

Fig. 1
Fig. 1

Definition of the coordinate system and of the direction cosines α, β, and γ, which specify the direction of the optic axis c (long-dashed line). The z axis is the inward normal, the zx plane is the plane of incidence, and θ is the angle of incidence. The cosines of the angle between c and the x, y, and z axes are α, β, and γ, respectively. The angle between the z axis and c is χ, and the plane containing c and the z axis cuts the xy plane at angle ϕ to the z axis, so α=sin χ cos ϕ, β=sin χ sin ϕ, and γ=cos χ.

Fig. 2
Fig. 2

Relationship between the Brewster angle θpp and the polarizing angle θpol, as a function of the orientation of the optic axis. The larger triangular area, bounded by β=0, γ=0 and β2+γ2=1, gives the possible ranges of the squares of the direction cosines β and γ. The angles θpp and θpol are equal on the (doubled) lines β=0 and (1-β2)εo=γ2(ε1+εo), which is drawn for calcite in oil of index 1.48. Reflection from the cleavage faces of calcite corresponds to γ2=0.5068 (see Sec. 4 of Ref. 14); the intersection of the doubled lines occurs at γ2=0.5556.

Fig. 3
Fig. 3

Selection of the physical roots of the quartics that determine θpp and θpol. Both quartics are linear in Δε; the solution for Δε is shown as a function of ε. The physical branch cuts the axis at εo (the solution when Δε=0) and then at the experimental value of Δε (dashed line), which is -0.5338 for calcite. The physical root εpp=3.00654 is shown circled. The physical root εpol=5.31746 is off the right-hand side of the diagram. The curves are drawn for calcite in oil of refractive index 1.48, with β2=12=γ2 (optic axis in the yz plane, at 45° to both the y and z axes).

Tables (1)

Tables Icon

Table 1 Values of θpp and θpol for Calcite in Air and Calcite in Oil (of Refractive Index 1.48)a

Equations (36)

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tan2 θpp(β=0)=εoεe-ε1εγε1(εγ-ε1),
εγ=εo+γ2Δε,Δε=εe-εo.
tan2 θ=ε2-ε1εoε1(εo-ε1).
rpprss-rpsrsp=0.
tan ϕ=rps+rss tan ϕrpp+rsp tan ϕ,
ϕ=arctanrssrsp=arctanrpsrpp.
α2+β2+γ2=1.
εo(ω/c)2=qo2+K2,
(αK+γqo)2-(qo2+K2)
Qpol=n=04(An+anΔε)εn,
Qpp=n=04(Bn+bnΔε)εn.
A0=(β2+γ2)ε12εo3(εo-ε1),
A1=2β2ε1εo2(εo-ε1)2,
A2=εo(εo-ε1)[(β2-γ2)ε12-(1+3β2)ε1εo+(1-γ2)εo2],
A3=-2β2εo(εo-ε1)2,
A4=(εo-ε1)[(1-β2)ε1+(2β2+γ2-1)εo],
B0=A0,
B1=A1-4β2ε1εo3(εo-ε1),
B2=A2+8β2ε1εo2(εo-ε1),
B3=A3-4β2ε1εo(εo-ε1),
B4=A4.
a0=ε12εo[γ4ε12+γ2(2β2-1)ε1εo+(β4+γ2)εo2],
a1=2β2ε1εo2(εo-ε1),
a2=-γ2ε13-[(2β2-1)γ2+2γ4-1]ε12εo
-[2β4+2β2γ2-γ2+2]ε1εo2+(1-γ2)εo3,
a3=-2β2εo(εo-ε1),
a4=γ2ε1+[(β2+γ2)2-γ2]εo,
b0=a0-4β2γ2ε13εo2,
b1=a1+4β2ε1εo[γ2ε12+(1-β2)ε1εo-εo2],
b2=a2+4β2ε1εo[(β2-1)ε1+(1+β2+γ2)εo],
b3=a3-4β2(β2+γ2)ε1εo,
b4=a4.
tan2 θpol(β2=1)=εoε1.
α2εo=γ2ε1orγ2=(1-β2)εoε1+εo.
ε=εo+(α2εo-γ2ε1)×Δε2εo 1+powerseriesin Δεεo-ε1.
εpp-εpol=β2ε1(α2εo-γ2ε1)(Δε)22εo2(εo-ε1)×1+powerseriesinΔεεo-ε1.

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