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
Anisotropic uniaxial materials are used in many components such as polarizers, birefringent filters and liquid crystal displays. These media exhibit birefringence. One of the rays, called an ordinary ray, satisfies Snell’s law of refraction, and calculations of its propagation are the same as those for an isotropic medium. The second ray is called an extraordinary ray, and doesn’t satisfy Snell’s law of refraction, and calculations of its propagation are slightly more difficult. The problem of ray tracing in uniaxial crystals has been solved in different forms by several authors during the last decades by using Maxwell’s – equations, by using phase matching [5, 6] conditions, by transformation to a nonorthonormal coordinate system  and by using Huygens’ principle [8, 9]. In the majority of these papers only formulas to obtain the extraordinary ray refracted in an interface isotropic-uniaxial medium are given. However the problem of tracing the extraordinary ray out of the crystal is more difficult, because the extraordinary ray doesn’t satisfy Snell’s law as we mentioned above. In the preceding papers [8, 9] the equations for tracing the extraordinary ray in an uniaxial crystal were obtained. Huygens’s principle was used to derive these equations, yielding the extraordinary ray vector in terms of the refractive indices no and ne , the direction vector of the crystal axis, and the direction vector of the ordinary ray determined by Snell’s law. The inverse problem was also solved, yielding formulas for the ordinary ray in terms of the extraordinary ray. The formulas for the inverse problem allow us to trace the extraordinary ray out of the crystal. The reversibility principle is used to obtain the extraordinary ray refracted out of the crystal. In this paper we use the results which were obtained in references [8, 9] for ray tracing in uniaxial crystals, enabling us to trace extraordinary rays both into and out of a plane-parallel uniaxial plate as well as through surfaces that are not planar .
2. Ordinary and extraordinary ray tracing
We suppose an incident ray reaching an uniaxial plane-parallel plate, where the Y-Z plane is the incident plane, and we suppose that the normal to the refracting surface is parallel to the Z axis as is shown in Fig. 1. Then ξi =0, and substituting into Eq. (1) from reference  we can writing Snell’s law as
where, ηo =sinθo and ζo =cosθo , or
where θi is the angle of incidence and θo is the refraction angle for the ordinary ray in the first interface.
The extraordinary ray S e=(ξe,ηe,ζe ) depends on the direction cosines of the ordinary ray So=(ξo,ηo,ζo ), the direction cosines of the crystal axis A=(α,β,γ), and the ordinary and extraordinary refractive indices no, ne respectively. In the particular case of a plane-parallel uniaxial plate, the first normal to the surface Z 1 is parallel to the second normal to the surface Z 2; If we suppose that the crystal axis lies in the incidence plane then the direction cosines for the crystal axes take the values; α=0, β=sinϕ, and γ=cosϕ, where ϕ is the angle between the normal to the refracting surface and the crystal axis as shown in Fig. 1. Substituting these values in Eqs. (47–48) from Ref.  we get
3. Conditions for maximum and minimum separation between ordinary and extraordinary rays for normal incidence.
We take the first derivative of Eq. (5) in any component with respect to ϕ or from Eq. (6) in order to obtain the critical points for the extraordinary ray direction as a function of the crystal’s axis position, thus we have;
gives the minimum separation between the ordinary and extraordinary rays for normal incidence, and
gives the maximum separation between the ordinary and extraordinary rays for normal incidence.
For normal incidence the separation between the ordinary and extraordinary rays is independent of the incident medium. If we consider no =1.658, ne =1.486, which correspond to calcite for λ=632[nm], then from Eq. (8) the angle for the crystal axis which maximizes the separation is approximately ϕ≈42°. A plane-parallel uniaxial plate of calcite for normal incidence with this particular position of the crystal in Fig. 2 is shown. Substituting Eq. (8) into Eq. (6) we obtain the refraction angle for the extraordinary ray, =arctan (-)/(2neno ), which give us the maximum separation between ordinary and extraordinary rays for normal incidence.
In order to calculate the directions along which the ordinary and extraordinary rays can be brought into coincidence for a plane-parallel uniaxial plate with oblique incidence we have that, from Eq. (3) the extraordinary ray direction can be written as
thus for the coincidence between the ordinary and the extraordinary rays the following condition must be satisfied
where tanθo and tanθe are given by Eqs. (2,9) respectively. These equations are a function of the refractive indices, angle of incidence and the crystal’s axis positions. The algebraic solution is too complicated. A numerical solution for the angle of incidence and the crystal axis position is shown in Fig. 3. Where for instance, we know the refractive indices values and for a given crystal axis orientation ϕ Eq. (10) is solved for θi . An intent to obtain the condition from Eq. (10) is given by  in a vague sense. When ne →no Eq. (9) reduces to Snell’s law Eq. (2).
We present examples for different positions of the crystal axis with respect to the normal to the refracting surface, we summarize in Table 1 the refraction angles θe and θo for different positions of the crystal axis lying in the incidence plane. for ni=n i2=1, and the values for no and ne given above.
5.1. Crystal axis at 30° to the normal of the refracting surface
In normal incidence the ordinary and extraordinary ray have a different direction in propagation into the uniaxial crystal. For (θi =55.9962°) the ordinary and extraordinary rays have the same refraction angle into the uniaxial crystal as was predicted by Eq. (10) and graphically is shown in Fig. 3. The critical angle for the extraordinary ray is θeca =38.9099° this is greater than the critical angle for the ordinary ray θoca =37.09°, see Table 1. The ray tracing process is shown in Fig. 4 for several cases for the angle of incidence.
5.2. Crystal axis at 60° to the normal of the refracting surface
There is no angle in which the ordinary and extraordinary rays coincide in the uniaxial crystal. The critical angle for the ordinary ray θoca =37.09° is greater than the critical angle for the extraordinary ray θeca=33.1217°, see Table 1 and Fig. 5.
We have explained the general ray tracing method for uniaxial crystals and we applied it to a plane-parallel uniaxial plate. We showed that the critical angle for the extraordinary ray is a function of the angle of the crystal axis with respect to the normal to the refracting surface. The conditions for bringing the ordinary and extraordinary rays into coincidence in a planeparallel uniaxial plate were given. When the crystal axis lies out of the incidence plane the extraordinary ray is refracted in a different plane. We obtained the angle for the crystal axis which either maximize or minimize the separation between ordinary and extraordinary rays for normal incidence. The work presented here is useful for the design of optical systems that incorporate the use of the extraordinary properties of uniaxial birefringent media.
Part of this work has been partially supported by CONACyT under project J-43919-F. We acknowledge to Martha Rosete for her valued assistance and comments. The author can be reached at the address on the title page or tel: 52 5556228614 ext. 1121.
References and links
3. J. Lekner, “Reflection and refraction by uniaxial crystals,” J. Phys. Condes. Matter 3, 6122–6133, (1991). [CrossRef]
7. G. Beyerle and I. S. McDermind, “Ray-tracing formulas for refraction and internal reflection in uniaxial crystal,” Appl. Opt. 377947–7953 (1998). [CrossRef]
8. M. Avendaño-Alejo, O. Stavroudis, and 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]
9. M. Avendaño-Alejo and 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]
10. M. Avendaño-Alejo and M. Rosete-Aguilar, “Paraxial Theory of birefringent lenses,” J. Opt. Soc. Am. A.22, No. 5, (2005) [CrossRef]
11. David Park, Classical Dynamics and its Quantum Analogues, second edition, (Springer Verlag, New York, 1990), pp. 18.
12. J. P. Mathieu, Optics Parts 1 and 2, (Pergamon Press, New York, 1975), pp. 94.
13. M. Avendaño-Alejo and M. Rosete-Aguilar, “Optical path difference in a plane parallel uniaxial plate,” unpublished.
14. A. A. Muryĭand and V. I. Stroganov, “Conditions for bringing the ordinary and extraordinary rays into coincidence in a plane-parallel plate fabricated from am optical uniaxial crystal,” J. Opt. Technol. 71, 283–285, (2004). [CrossRef]