Abstract

Ray tracing formulas in a plane-parallel uniaxial plate bounded by an isotropic medium are analyzed when the crystal axis lies in the incident plane, and when its orientation is arbitrary. We present the behavior of the critical angle for the extraordinary ray as a function of the crystal axis position with respect to the normal to the refracting surface. We give the conditions in order to obtain the incidence angle at which the ordinary and extraordinary ray have the same refraction angle into the uniaxial crystal for particular positions of the optical crystal axis, also we give a condition for normal incidence in order to maximize or minimize the separation between the ordinary and extraordinary ray as a function of the optical crystal axis.

© 2005 Optical Society of America

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References

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  9. M. Avendaño-Alejo, O. Stavroudis, �??Huygens�?? Principle and Rays in Uniaxial Anisotropic Media II. Crystal Axis with Arbitrary Orientation,�?? J. Opt. Soc. Am. A. 19, 1674-1679, (2002).
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  13. M. Avendaño-Alejo, M. Rosete-Aguilar, �??Optical path difference in a plane parallel uniaxial plate,�?? unpublished.
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Appl. Opt. (5)

for refraction and internal reflection i (1)

G. Beyerle and I. S. McDermind, �??Ray-tracing formulas for refraction and internal reflection in uniaxial crystal,�?? Appl. Opt. 37, 7947�??7953 (1998).
[CrossRef]

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

M. Avendaño-Alejo, O. Stavroudis, A. R. Boyain, �??Huygens�?? Principle and Rays in Uniaxial Anisotropic Media I. Crystal Axis Normal to Refracting Surface,�?? J. Opt. Soc. Am. A. 19, 1668-1673, (2002).
[CrossRef]

M. Avendaño-Alejo, O. Stavroudis, �??Huygens�?? Principle and Rays in Uniaxial Anisotropic Media II. Crystal Axis with Arbitrary Orientation,�?? J. Opt. Soc. Am. A. 19, 1674-1679, (2002).
[CrossRef]

M. Avendaño-Alejo, M. Rosete-Aguilar, �??Paraxial Theory of birefringent lenses,�?? J. Opt. Soc. Am. A. 22, No. 5, (2005).
[CrossRef]

J. Opt. Technol. (1)

J. Phys. Condes. Matter (1)

J. Lekner, �??Reflection and refraction by uniaxial crystals,�?? J. Phys. Condes. Matter 3, 6122-6133, (1991).
[CrossRef]

Other (3)

David Park, Classical Dynamics and its Quantum Analogues, second edition, (Springer Verlag, New York, 1990), pp. 18.

J. P. Mathieu, Optics Parts 1 and 2, (Pergamon Press, New York, 1975), pp. 94.

M. Avendaño-Alejo, M. Rosete-Aguilar, �??Optical path difference in a plane parallel uniaxial plate,�?? unpublished.

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Figures (5)

Fig. 1.
Fig. 1.

The Y-Z-plane is the plane of incidence. Z is the normal to the first refracting surface. S i, S o,S e are the incident, ordinary and extraordinary rays respectively.

Fig. 2.
Fig. 2.

Ray tracing for both ordinary and extraordinary rays in uniaxial plane-parallel plate of calcite for ϕ=42° which maximizes the separation between the rays.

Fig. 3.
Fig. 3.

Numerical solution for the incidence angle and crystal axis position in which the directions for both the ordinary and the extraordinary ray can be brought into coincidence. Where no 1.658, ne =1.486, ni =1.

Fig. 4.
Fig. 4.

Ray tracing for the ordinary and extraordinary rays when the crystal axis is at 30° to the normal of the refracting surface.

Fig. 5.
Fig. 5.

Ray tracing for the ordinary and extraordinary rays when the crystal axis is at 60° to the normal of the refracting surface.

Tables (1)

Tables Icon

Table 1. Angle of incidence, ordinary and extraordinary refracted angle for different positions of the crystal axis with respect to the normal to the refracting surface.

Equations (10)

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( ξ o sin θ o cos θ o ) = ( 0 n i n o sin θ i 1 n o n o 2 n i 2 sin 2 θ i ) ,
tan θ o = n i sin θ i n o 2 n i 2 sin 2 θ i ,
( ξ o sin θ e cos θ e ) = ( 0 n e n i n o sin θ i + ( n e 2 n o 2 ) Π cos ϕ sin ϕ n o 2 ( n e 2 cos 2 ϕ + n o 2 sin 2 ϕ ) 2 + ( n e 2 n o 2 ) ( n i n o sin θ i sin ϕ + n e Π cos ϕ ) 2 ( n e 2 cos 2 ϕ + n o 2 sin 2 ϕ ) Π n o 2 ( n e 2 cos 2 ϕ + n o 2 sin 2 ϕ ) 2 + ( n e 2 n o 2 ) ( n i n o sin θ i sin ϕ + n e Π cos ϕ ) 2 ) ,
Π = n e 2 cos 2 ϕ + n o 2 sin 2 ϕ n i 2 sin 2 θ i ,
( ξ o sin θ e 0 cos θ e 0 ) = ( 0 ( n e 2 n o 2 ) cos ϕ sin ϕ n e 4 cos 2 ϕ + n o 4 sin 2 ϕ n e 2 cos 2 ϕ + n o 2 sin 2 ϕ n e 4 cos 2 ϕ + n o 4 sin 2 ϕ ) ,
tan θ e 0 = ( n e 2 n o 2 ) cos ϕ sin ϕ n e 2 cos 2 ϕ + n o 2 sin 2 ϕ
ϕ = 0 or ϕ = π 2 ,
ϕ = arctan n e n o = arccos n o n e 2 + n o 2 = 1 2 arccos n o 2 n e 2 n e 2 + n o 2 ,
tan θ e = n e n i n o sin θ i + ( n e 2 n o 2 ) cos ϕ sin ϕ n e 2 cos 2 ϕ + n o 2 sin 2 ϕ n i 2 sin 2 θ i ( n e 2 cos 2 ϕ + n o 2 sin 2 ϕ ) n e 2 cos 2 ϕ + n o 2 sin 2 ϕ n i 2 sin 2 θ i ,
tan θ o = tan θ e ,

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