Ray tracing in monoaxial crystals; are the exact formulas necessary? Maria C. Simon

University of Buenos Aires, Physics Department, Optics Laboratory, Ciudad Universitaria, Pabellon I, 1428 Bue­nos Aires, Argentina. Received 2 April 1987. 0003-6935/87/163187-03$02.00/0. © 1987 Optical Society of America. In an earlier paper1 we developed the formulas that allow

exact calculation of the ray paths through monoaxial bire-fringent media. The formulas we obtained were used to study the aberrations produced by a Wollaston prism when it is introduced in a convergent beam.2 These calculations showed that the most important aberrations are astigmatism and anamorphic distortion. Optimizing the design of the Wollaston prism, it was found that when these aberrations are small, the effect of coma is notorious. It is well known that when a wedge of an isotropic material is introduced in a convergent beam, similar aberrations appear. Therefore, the question arises whether the exact formulas are really necessary or whether it is easier and quicker to perform an approximate calculation considering the Wollaston prism to be composed of two wedges of isotropic materials with differ­ent indices of refraction. With this approximation it would also be possible to use analytical expressions for the aberra­tions, such as those obtained by Howard3 for wedges,. for example. To study this, we performed the exact calculations and the corresponding approximate calculations for a Wol-laston prism of angle a = 45°, which is introduced in a convergent beam of aperture ƒ/24. Figure 1 shows the ray paths and formation of the astigmatic images. We also rotated the Wollaston through an angle α1 = —28.8° with respect to the incident beam (Fig. 1) to consider a case where the aberrations are practically minimal (see Fig. 6 of Ref. 2). Performing the exact calculations as we have explained in

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Fig. 2. Spot diagrams for α = 45°, α1 = -28.8°, and = 0; (a) exact calculation; (b) approximate calculation with the same foci.

Fig. 3. Spot diagrams obtained with the foci corresponding to the approximate calculation ( = 45°, α1 = -28.8°, and = 0).

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which gives a value of n'' = 1.505 for α1 = -28.8°. The index of the second element is n0 = 1.66 as the rays are ordinary there. An exact calculation of n" is also performed by the ray tracing program, thus providing control.

When the ray tracing is performed with the above values for the indices, the spot diagrams shown in Fig. 2(b) are obtained, and it may be observed that the forms of the images are very different. Figure 3 shows the spot diagrams obtained when one seeks the new foci, and it may be observed that these are also different to the exact calculations [Fig. 2(a)]. When the longitudinal astigmatisms A'L and A'L are computed, the exact calculation gives the values A'L = 3.8 mm and A'L = 4.2 mm, while the approximate method yields A'L = 5.3 mm and AL = -6 .5 mm. Thus we observe that the approximate calculation gives an inversion of astigmatism for the images formed by the rays that are extraordinary in the first element of the prism.

When the lateral displacement Zmc of the images is calcu­lated, a difference of 0.8 mm is obtained for the images formed by the rays that are extraordinary in the first ele-

Fig. 1. Wollaston prisms: ray paths and formation of astigmatic images

earlier works,1,2 the spot diagrams represented in Fig. 2(a) are obtained. For the approximate calculation a ray tracing was performed considering the two prisms that make up the Wollaston as wedges of isotropic material with the following indices:

(1) For the ordinary rays in the first element and the extraordinary rays in the second, the index of the first ele­ment is the ordinary index (no = 1.66), and the index of the second element is the one corresponding to the principal ray, i.e., ne = 1.49.

(2) The rays are extraordinary in the first element and ordinary in the second one. In this case, the average index, which is that of the principal ray, is obtained from Eq. (9) of Ref. 2:

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ment. This difference is caused by the deviation of the rays with respect to the normal to the wavefront, as it may not be attributed to an error in the index, as the chosen value is exact for the principal ray. No difference is obtained for the images formed by the rays that are extraordinary in the second element, as the principal ray is perpendicular to the optical axis in that element. When the same calculations are carried out for a Wollaston that has been modified so that the optical axis of the first prism forms an angle m with the surface, similar results are obtained. The angle m is deter­mined by the expression

[Eq. (1) of Ref. 2]; i.e., the extraordinary ray of the first element is perpendicular to the optical axis. In this case, the same result is obtained for the lateral displacement of the images in both situations as the principal ray is always per­pendicular to the optical axis. However, the same results are not obtained for the forms of the images, and the longitudi­nal astigmatisms are different. The exact calculation yields an astigmatism A"L = —5.7 mm, while the approximate calcu­lation gives A"L = -7 .1 mm.

As the ray tracing calculation was performed for a beam with small aperture (//24), and astigmatism is the main aberration, the analysis of aberrations with the isotropic prism approximation is no longer possible. This applies not only to ray tracing but also to formulas of third order or higher. From these results we conclude that the exact for­mulas are necessary to analyze the images with precision and obtain a trustworthy optimal design of the elements made of birefringent materials.

My thanks to Marta Pedernera for preparing the illustra­tions and the Centre de Tecnologia y Ciencia de Sistemas for use of the computer. This work was done with the support of CONICET and a grant from the University of Buenos Aires.

References 1. M. C. Simon, "Ray Tracing Formulas for Monoaxial Optical

Components," Appl Opt. 22, 354 (1983). 2. M. C. Simon, "Wollaston Prisms with Large Split Angle," Appi.

Opt. 25, 369 (1986). 3. J. W. Howard, "Formulas for the Coma and Astigmatism of

Wedge Prisms Used in Converging Light," J. Opt. Soc. Am. 1, 1219 (1984).