1. Introduction

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 [1]–[4] equations, by using phase matching [5, 6] conditions, by transformation to a nonorthonormal coordinate system [7] 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 n_o and n_e , 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 [10].

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 ξ_i =0, and substituting into Eq. (1) from reference [8] we can writing Snell’s law as

where, η_o =sinθ_o and ζ_o =cosθ_o , or

where θ_i is the angle of incidence and θ_o is the refraction angle for the ordinary ray in the first interface[11].

The extraordinary ray S _e=(ξ_e,η_e,ζ_e ) depends on the direction cosines of the ordinary ray S_o=(ξ_o,η_o,ζ_o ), the direction cosines of the crystal axis A=(α,β,γ), and the ordinary and extraordinary refractive indices n_o, n_e 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. [9] we get

 

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.

Download Full Size | PDF

where

3. Conditions for maximum and minimum separation between ordinary and extraordinary rays for normal incidence.

For normal incidence we have θ_i =0, and substituting into Eqs. (3,4) we obtain

where the superscript 0° means normal incidence. Eq. (6) gives what is known as the dispersion angle[4, 5, 12, 13].

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 n_o =1.658, n_e =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, ${\theta }_{e1}^{\mathit{\text{max}}}$=arctan ($n_o^2$-$n_e^2$)/(2n_en_o ), which give us the maximum separation between ordinary and extraordinary rays for normal incidence.

 

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.

Download Full Size | PDF

4. Applications

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 tanθ_o and tanθ_e 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 θ_i . An intent to obtain the condition from Eq. (10) is given by [14] in a vague sense. When n_e →n_o Eq. (9) reduces to Snell’s law Eq. (2).

 

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 n_o 1.658, n_e =1.486, n_i =1.

Download Full Size | PDF

5. Examples

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 θ_e and θ_o for different positions of the crystal axis lying in the incidence plane. for n_i=n _i2=1, and the values for n_o and n_e 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 (θ_i =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 θ_eca =38.9099° this is greater than the critical angle for the ordinary ray θ_oca =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 θ_oca =37.09° is greater than the critical angle for the extraordinary ray θ_eca=33.1217°, see Table 1 and Fig. 5.

 

Fig. 4. Ray tracing for the ordinary and extraordinary rays when the crystal axis is at 30° to the normal of the refracting surface.

Download Full Size | PDF

 

Fig. 5. Ray tracing for the ordinary and extraordinary rays when the crystal axis is at 60° to the normal of the refracting surface.

Download Full Size | PDF

6. Conclusions

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.

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.

View Table