Abstract

Reflection by anisotropic media is characterized by the four reflection amplitudes rpp, rss, rps, and rsp. We show that rpp can be zero at angles θpp, the anisotropic Brewster angles, and that a quantity related to θpp satisfies a quartic equation. When the refractive index of the medium of incidence lies between the ordinary and the extraordinary indices of the crystal, it is possible for rss to be zero at an angle θss, and there exist four equivalent orientations of the crystal optic axis for which rpp, rss, and either rps or rsp are simultaneously zero, at angle of incidence equal to arctan (no/n1).

© 1993 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Born, E. Wolf, Principles of Optics, 3rd ed. (Pergamon, Oxford, 1965), Sec. 1.5.3.
  2. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Secs. 4.2 and 4.7.3.1.
  3. J. Lekner, Theory of Reflection (Nijhoff/Kluwer, Dordrecht, The Netherlands, 1987), Secs. 1-2 and 7-1.
  4. M. Elshazly-Zaghloul, R. M. 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. A 6, 607 (1989).
    [CrossRef]
  5. J. Lekner, “Reflection and refraction by uniaxial crystals,”J. Phys. Condens. Matter 3, 6121–6133 (1991).
    [CrossRef]
  6. J. Lekner, “Bounds and zeros in reflection and refraction by uniaxial crystals,”J. Phys. Condens. Matter 4, 9459–9468 (1992).
    [CrossRef]
  7. M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), Sec. 3.8.
  8. J. Lekner, “Optical properties of an isotropic layer on a uniaxial crystal substrate,”J. Phys. Condens. Matter 4, 6569–6586 (1992).
    [CrossRef]

1992 (2)

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

J. Lekner, “Optical properties of an isotropic layer on a uniaxial crystal substrate,”J. Phys. Condens. Matter 4, 6569–6586 (1992).
[CrossRef]

1991 (1)

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

1982 (1)

Azzam, R. M.

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Secs. 4.2 and 4.7.3.1.

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Secs. 4.2 and 4.7.3.1.

Born, M.

M. Born, E. Wolf, Principles of Optics, 3rd ed. (Pergamon, Oxford, 1965), Sec. 1.5.3.

Elshazly-Zaghloul, M.

Lekner, J.

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

J. Lekner, “Optical properties of an isotropic layer on a uniaxial crystal substrate,”J. Phys. Condens. Matter 4, 6569–6586 (1992).
[CrossRef]

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

J. Lekner, Theory of Reflection (Nijhoff/Kluwer, Dordrecht, The Netherlands, 1987), Secs. 1-2 and 7-1.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 3rd ed. (Pergamon, Oxford, 1965), Sec. 1.5.3.

J. Opt. Soc. Am. (1)

J. Phys. Condens. Matter (3)

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]

J. Lekner, “Optical properties of an isotropic layer on a uniaxial crystal substrate,”J. Phys. Condens. Matter 4, 6569–6586 (1992).
[CrossRef]

Other (4)

M. Born, E. Wolf, Principles of Optics, 3rd ed. (Pergamon, Oxford, 1965), Sec. 1.5.3.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), Secs. 4.2 and 4.7.3.1.

J. Lekner, Theory of Reflection (Nijhoff/Kluwer, Dordrecht, The Netherlands, 1987), Secs. 1-2 and 7-1.

M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), Sec. 3.8.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

Exact and approximate Brewster angles for calcite in air at 633 nm (no = 1.655, ne = 1.485). The curves show θpp for β = 0 (optic axis in the plane of incidence) as a function of γ2, the square of the cosine of the angle between the optic axis and the normal to the reflecting surface. The solid curve is from Eq. (14), the dashed curve from expression (28).

Fig. 2
Fig. 2

Brewster angles for calcite with optic axis in the plane of incidence, immersed in a medium of refractive index 1.48, as a function of γ2. The notation is as for Fig. 1, but note the change of vertical scale. The curves cross at γ2 = o/(1 + o).

Fig. 3
Fig. 3

Brewster angles for calcite immersed in a medium of index n1 = 1.51, for the optic axis in the plane of incidence. Note that a zero of rpp exists only for a range of inclinations of the optic axis to the surface normal (here between 22° and 66°) as given by expressions (35). These bounding values of the square of the direction cosine γ are shown by vertical dashed lines.

Fig. 4
Fig. 4

Reflection amplitude rpp for calcite immersed in a liquid of refractive index 1.51 and direction cosines of the optic axis given by α2 = 0.02,β2 = 0.67,γ2 = 0.31. Beyond θce there is an imaginary part of rpp (dashed curve). The θpp values are 54.03° and 80.27°; the value of θce is just 1 mdeg more than the larger θpp.

Fig. 5
Fig. 5

Reflection amplitude rss for calcite immersed in a liquid of refractive index 1.51 and direction cosines of the optic axis given by α2 = 0.02, β2 = 0.67, γ2 = 0.31. Beyond θce = 80.27° there is an imaginary part of rss (dashed curve). The reflection amplitude is zero at θss = 75.12°.

Equations (48)

Equations on this page are rendered with MathJax. Learn more.

K = k 1 sin θ = n 1 ω c sin θ .
q 1 2 = k 1 2 - K 2 = k 1 2 cos 2 θ .
k o = n o ω c ,
q o 2 = k o 2 - K 2 .
q e = q ¯ - α γ K Δ / γ ,
q ¯ 2 = o [ e γ ω 2 / c 2 - K 2 ( e - β 2 Δ ) ] / γ 2 ,
γ = o + γ 2 Δ .
r s s = a ( q 1 - q o ) + b ( q 1 - q e ) a ( q 1 + q o ) + b ( q 1 + q e ) ,
- r p p = a ( q 1 + q o ) + b ( q 1 + q e ) a ( q 1 + q o ) + b ( q 1 + q e ) ,
r s p , p s = 2 β ( α q o ± γ K ) ( q o - q e ) k 1 k o 2 a ( q 1 + q o ) + b ( q 1 + q e )
a = ( α q o - γ K ) [ α ( k o 2 q e + q t q o 2 ) - γ K ( k o 2 + q t q e ) ] , a = ( α q o - γ K ) [ α ( k o 2 q e - q t q o 2 ) - γ K ( k o 2 - q t q e ) ] , b = β 2 k o 2 ( k o 2 + q t q o ) ,             b = β 2 k o 2 ( k o 2 - q t q o ) ,
r s s ( β = 0 ) = q 1 - q o q 1 + q o ,             r p p ( β = 0 ) = Q - Q 1 Q + Q 1 ,
q γ 2 = γ ω 2 / c 2 - K 2 .
tan 2 θ p p ( β = 0 ) = o e - 1 γ 1 ( γ - 1 ) .
tan 2 θ p p ( α 2 = 1 ) = o ( e - 1 ) 1 ( o - 1 ) ,
tan 2 θ p p ( γ 2 = 1 ) = e ( o - 1 ) 1 ( e - 1 ) .
q ¯ 2 = q o 2 + [ α 2 q o 2 + β 2 k o 2 + γ 2 K 2 + ( α γ ) 2 K 2 Δ / γ ] Δ / γ .
I + V I + V = ( 1 + V / I 1 + V / I ) I I = ( I / V + 1 I / V + 1 ) V V .
r s s = ( q 1 - q o ) [ ( q o + q ¯ ) ( k o 2 + q t q o ) + ( α 2 k o 2 q o + γ 2 K 2 q t ) Δ / γ ] - β 2 k o 2 ( k o 2 + q t q o ) Δ / γ ( q 1 + q o ) [ ( q o + q ¯ ) ( k o 2 + q t q o ) + ( α 2 k o 2 q o + γ 2 K 2 q t ) Δ / γ ] + β 2 k o 2 ( k o 2 + q t q o ) Δ / γ , - r p p = ( q 1 - q o ) [ ( q o + q ¯ ) ( k o 2 - q t q o ) + ( α 2 k o 2 q o - γ 2 K 2 q t ) Δ / γ ] + β 2 k o 2 ( k o 2 - q t q o ) Δ / γ ( q 1 + q o ) [ ( q o + q ¯ ) ( k o 2 + q t q o ) + ( α 2 k o 2 q o + γ 2 K 2 q t ) Δ / γ ] + β 2 k o 2 ( k o 2 + q t q o ) Δ / γ .
ω 2 / c 2 = q t q o             or             / 1 = q o / q 1 .
( c K ω ) 2 = 1 ( 2 - 1 o ) 2 - 1 2 ,             ( c q 1 ω ) 2 = 1 2 ( o - 1 ) 2 - 1 2 , ( c q o ω ) 2 = 2 ( o - 1 ) 2 - 1 2 .
( o - ) ( - 1 ) ( q 1 + q o ) q ¯ c 2 / ω 2 + ( o - ) ( o - 1 ) + [ α 2 o ( o - 1 ) + β 2 o ( o - ) ( - 1 ) - γ 2 1 ( 2 - 1 o ) ] Δ / γ .
( q 1 + q o ) 2 = ( 1 + ) ( o - 1 ) - 1 ω 2 c 2 .
a 0 + a 1 + a 2 2 + a 3 3 + a 4 4 = 0 , a 0 = - 1 2 o { o 2 ( o - 1 ) ( β 2 + γ 2 ) + [ ( β 2 o - γ 2 1 ) 2 + o ( o - 1 ) γ 2 ] Δ } , a 1 = 2 β 2 1 o ( o ( o 2 - 1 2 ) + { o [ o + 1 ( 2 β 2 - 1 ) ] - 2 γ 2 1 2 } Δ ) , a 2 = ( o - 1 ) o [ 1 o ( 1 - 5 β 2 ) + 1 2 ( γ 2 - β 2 ) - o 2 ( 1 - γ 2 ) ] + ( ( o - 1 ) 2 ( o + 1 ) γ 2 - o { 2 1 ( β 2 o - γ 2 1 ) ( β 2 + γ 2 ) + [ o + 1 ( 2 β 2 - 1 ) ] 2 } ) Δ , a 3 = 2 β 2 o { o 2 - 1 2 + [ 1 ( 2 β 2 + 2 γ 2 - 1 ) + o ] Δ } , a 4 = ( o - 1 ) [ o ( 1 - 2 β 2 - γ 2 ) - 1 ( 1 - β 2 ) ] + [ ( o - 1 ) γ 2 - o ( β 2 + γ 2 ) 2 ] Δ .
γ 2 1 o 1 - ( 1 - γ 2 ) o ,             o e ( o - 1 ) + 1 2 ( γ - o ) γ - 1 .
tan 2 θ = 2 - 1 o 1 ( o - 1 ) ,
tan 2 θ p p ( β 2 = 1 ) = o 1 .
o + ( α 2 o - γ 2 1 ) Δ / 2 o .
tan 2 θ p p o 1 - ( α 2 o - γ 2 1 ) Δ 1 ( o - 1 ) .
tan 2 θ p p o α γ - 1 γ 1 ( γ - 1 ) ,
α γ = o + ( α 2 + γ 2 ) Δ = e - β 2 Δ .
sin θ c o = n o n 1 ,             sin θ c e = n e n γ n 1 n α γ ,
sin θ c e ( β = 0 ) = n γ n 1 .
sin 2 θ p p ( β = 0 ) = o e - 1 γ o e - 1 2 .
γ 2 > o e ( 1 - o ) 1 2 ( e - o ) = γ c 2
| o - 1 o - e |             and             o 1 | 1 - e o - e | .
a 2 + b + c = 0 : a = ( o 2 - 1 2 ) [ 1 2 + β 2 ( o 2 + 1 o - 1 2 ) ] + o [ 1 2 + β 2 ( o 2 + 2 1 o - 1 2 + 1 2 β 2 ) ] Δ , b = 2 1 o [ 1 ( 2 β 2 - 1 ) + β 2 o ] [ o 2 - 1 2 + ( o + β 2 1 ) Δ ] , c = o 1 2 { ( o 2 - 1 2 ) ( o + β 2 1 ) + [ o 2 ( 1 - β 2 + β 4 ) + 2 1 o β 2 ( 2 β 2 - 1 ) + β 2 1 2 ( 4 β 2 - 3 ) ] Δ } .
β d 2 = ( o e - 1 2 ) [ ( o + 1 ) 2 + ( 3 1 + o ) Δ ] Δ [ o 3 + 2 o 2 1 - 3 o 1 2 - 4 1 3 + o ( 3 1 + o ) Δ ] .
2 = 1 o ( β 2 o - γ 2 1 ) 1 ( 2 β 2 - 1 ) + o α 2 .
( a 0 + a 2 2 + a 4 4 ) 2 = ( a 1 + a 3 2 ) 2 .
α 2 o = γ 2 1 .
tan 2 θ j = o 1 .
γ 2 = ( 1 - β 2 ) o 1 + o ,
α j 2 = ( 1 + o ) ( e - 1 ) / 4 1 Δ , β j 2 = ( 1 + o ) [ ( 1 + o ) 2 + ( 3 1 + o ) Δ ] / 4 1 2 Δ , γ j 2 = o ( 1 + o ) ( e - 1 ) / 4 1 2 Δ .
Δ > ( 1 + o ) 2 / ( 3 1 + o ) .
( c K ω ) 2 1 o 1 + o ,             ( c q o ω ) 2 o 2 1 + o ,
α 2 q o 2 - γ 2 K 2 = 0.
( o - 1 ) ( 1 - e ) 1 2 - 1 o + 3 1 e + o e .

Metrics