Abstract

The electric polarization of an interface between an isotropic medium and a uniaxial transparent crystal is analyzed. The case in which the optical axis lies on the incidence plane is considered. When the incidence angle is Brewster’s, angle is shown that the effective electric polarization of the interface has the direction of the reflected ray.

© 2001 Optical Society of America

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References

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  1. J. Lekner, “Brewster angles in reflection by uniaxial crystals,” J. Opt. Soc. Am. A 10, 2059–2064 (1993).
    [Crossref]
  2. D. J. De Smet, “Brewster’s angle and optical anisotropy,” Am. J. Phys. 62(3), 246–248 (1994).
    [Crossref]
  3. J. Heading, “Generalized investigations into the Brewster angle,” Opt. Acta 33(6), 755–770 (1986).
    [Crossref]
  4. M. C. Simon, K. V. Gottschalk, “About the Brewster angle and the electric polarization in birefringent media,” Pure Appl. Opt. 4, 27–38 (1995).
    [Crossref]
  5. M. C. Simon, K. V. Gottschalk, “Brewster angle in dielectric birefringent media: an explanation by means of dipolar model,” Opt. Commun. 126, 113–122 (1996).
    [Crossref]
  6. M. C. Simon, R. M. Echarri, “Ray-tracing formulas for monoaxial opical components: vectorial formulation,” Appl. Opt. 25(12), 1935–1939 (1986).
    [Crossref]
  7. A. Sommerfeld, Vorlesungen über theoretische Physik, (Akademische Verlagsgesellschaft Geest & K-G Portig, Leipzig, 1964), Vol. III, p. 66.
  8. J. Lekner, “Reflection by uniaxial crystals: polarizing angle and Brewster angle,” J. Opt. Soc. Am. A 16(11), 2763–2766 (1999).
    [Crossref]

1999 (1)

1996 (1)

M. C. Simon, K. V. Gottschalk, “Brewster angle in dielectric birefringent media: an explanation by means of dipolar model,” Opt. Commun. 126, 113–122 (1996).
[Crossref]

1995 (1)

M. C. Simon, K. V. Gottschalk, “About the Brewster angle and the electric polarization in birefringent media,” Pure Appl. Opt. 4, 27–38 (1995).
[Crossref]

1994 (1)

D. J. De Smet, “Brewster’s angle and optical anisotropy,” Am. J. Phys. 62(3), 246–248 (1994).
[Crossref]

1993 (1)

1986 (2)

De Smet, D. J.

D. J. De Smet, “Brewster’s angle and optical anisotropy,” Am. J. Phys. 62(3), 246–248 (1994).
[Crossref]

Echarri, R. M.

Gottschalk, K. V.

M. C. Simon, K. V. Gottschalk, “Brewster angle in dielectric birefringent media: an explanation by means of dipolar model,” Opt. Commun. 126, 113–122 (1996).
[Crossref]

M. C. Simon, K. V. Gottschalk, “About the Brewster angle and the electric polarization in birefringent media,” Pure Appl. Opt. 4, 27–38 (1995).
[Crossref]

Heading, J.

J. Heading, “Generalized investigations into the Brewster angle,” Opt. Acta 33(6), 755–770 (1986).
[Crossref]

Lekner, J.

Simon, M. C.

M. C. Simon, K. V. Gottschalk, “Brewster angle in dielectric birefringent media: an explanation by means of dipolar model,” Opt. Commun. 126, 113–122 (1996).
[Crossref]

M. C. Simon, K. V. Gottschalk, “About the Brewster angle and the electric polarization in birefringent media,” Pure Appl. Opt. 4, 27–38 (1995).
[Crossref]

M. C. Simon, R. M. Echarri, “Ray-tracing formulas for monoaxial opical components: vectorial formulation,” Appl. Opt. 25(12), 1935–1939 (1986).
[Crossref]

Sommerfeld, A.

A. Sommerfeld, Vorlesungen über theoretische Physik, (Akademische Verlagsgesellschaft Geest & K-G Portig, Leipzig, 1964), Vol. III, p. 66.

Am. J. Phys. (1)

D. J. De Smet, “Brewster’s angle and optical anisotropy,” Am. J. Phys. 62(3), 246–248 (1994).
[Crossref]

Appl. Opt. (1)

J. Opt. Soc. Am. A (2)

Opt. Acta (1)

J. Heading, “Generalized investigations into the Brewster angle,” Opt. Acta 33(6), 755–770 (1986).
[Crossref]

Opt. Commun. (1)

M. C. Simon, K. V. Gottschalk, “Brewster angle in dielectric birefringent media: an explanation by means of dipolar model,” Opt. Commun. 126, 113–122 (1996).
[Crossref]

Pure Appl. Opt. (1)

M. C. Simon, K. V. Gottschalk, “About the Brewster angle and the electric polarization in birefringent media,” Pure Appl. Opt. 4, 27–38 (1995).
[Crossref]

Other (1)

A. Sommerfeld, Vorlesungen über theoretische Physik, (Akademische Verlagsgesellschaft Geest & K-G Portig, Leipzig, 1964), Vol. III, p. 66.

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Figures (4)

Fig. 1
Fig. 1

Vectorial diagram for the extraordinary wave. v3 is a vector in the optical axis direction: v3=(uo2-ue2)(N^  z^3)z^3.

Fig. 2
Fig. 2

Reflection and refraction in an isotropic medium–uniaxial crystal interface. The x axis is the inward normal and the (x, z) plane is the incidence plane. The optical axis z3 is at angle θ to z axis. The angle α is the incidence angle and β is the refraction angle. In all figures   ˇ corresponds to   ˆ in text and → corresponds to boldface.

Fig. 3
Fig. 3

Direction of the effective electric polarization relative to the reflected ray: (a) α>0; (b) α<0.

Fig. 4
Fig. 4

σ angle as a function of α, for an isotropic medium–calcite interface (ne=1.4865,no=1.6584,θ=20°,ni=1.1).

Equations (40)

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u2=ue2+(uo2-ue2)(N^  z^3)2,
R^=1fn [ue2N^+ (u02-ue2)(N^  z^3)z^3],
fn2=[1-(N^  z^3)2]ue4+(N^  z^3)2ue4.
z^3=-xˆ sin θ+zˆ cos θ.
P=D-εvE,
Pe=P-Pi,
Pi=εiE-εvE,
Pe=D-εiE.
(P  R^)=(Pe  R^)=(D  R^).
(Pe  N^)=-εi(E  N^);
(D  yˆ)=0,
(D  xˆ)=-(N^  xˆ)(N^  zˆ) (D  xˆ),
(E  xˆ)=μvq1+q2(N^  xˆ)(N^  zˆ)(D  xˆ),
(E  yˆ)=0,
(E  xˆ)=-μvq2+q3(N^  xˆ)(N^  zˆ)(D  xˆ),
q1=ue2sin2 θ+u02cos2 θ,
q2=(ue2-u02)sin θ cos θ,
q3=ue2cos2 θ+u02sin2 θ.
(Pe  xˆ)=1-μvεiq1+q2(N^  xˆ)(N^  zˆ)(D  xˆ),
(Pe  zˆ)=-(N^  xˆ)(N^zˆ)+μvεi×q2+q3(N^  xˆ)(N^  zˆ) (D  xˆ).
tan Γe=(Pe  zˆ)/(Pe  xˆ).
tan Γe=μvεiq2tan β+μvεiq3-1(1-μvεiq1) tan β-μvεiq2,
(D  yˆ)=0,
(D  xˆ)=-(Sˆ  xˆ)(Sˆ  zˆ) (D  xˆ),
(E  xˆ)=μνui2(D  xˆ),
(E  yˆ)=0,
(E  xˆ)=-μνui2(Sˆ  xˆ)(Sˆ  zˆ) (D  xˆ),
(D*  yˆ)=0,
(D*  xˆ)=(Sˆ  xˆ)(Sˆ  zˆ) (D*  xˆ),
(E*  xˆ)=μνui2(D*  xˆ),
(E*  yˆ)=0,
(E*  xˆ)=μνui2(Sˆ  xˆ)(Sˆ  zˆ) (D*  xˆ).
(D*  xˆ)=(ui2-q2tan α) tan β-q3tan α(ui2+q2tan α) tan β+q3tan α (D  xˆ),
(D*  xˆ)=2ui2tan β(ui2+q2tan α) tan β+q3tan α (D  xˆ),
R=(D*  xˆ)/(D  xˆ).
tan βp=q3tan αpui2- q2tan αp.
sin βu = sin αui.
[ui2(1+ tan2 α)-q1tan2 α] tan2 β-2q2tan2 α tan β-q3tan2 α=0.
tan2 αp=ui2(ui2-q3)ui2q3-ue2uo2.
tan Γe=-tan αp,

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