An optical element not often described in depth in textbooks is the toroid. A toroidal surface is a special case of an aspheric surface. It features two different curvatures perpendicular to each other. A lens or a mirror with such a shape therefore has two focal points, separated by an amount dependent on the difference of the two curvatures. This characteristic of focus separation is not only used to correct astigmatism of the eye, it is also applied in the design of anamorphic systems with different magnifications in the two azimuths. The motion picture wide-screen projection is a typical application, in which an extra-large horizontal field is employed.

**Figure 1.** Toroidal components can be classed as doughnut-type (right) or barrel-shaped (left). The power of such surfaces can be positive or negative.

There are two forms of toroids: one is part of a ring (doughnut) and the other is barrel-shaped; the power of such surfaces can be positive or negative (see figure 1). Special cases of a toroidal surface are the cylinder lens or cylinder mirror, which are curved in one direction only.

As an example of the benefits of toroids, consider a concave spherical mirror used to image an off-axis point. The minimum image blur spot (circle of least confusion) is located approximately halfway between the sagittal and tangential foci (see figure 2). To correct for the astigmatism, the surface radius in the sagittal plane has to be decreased, which makes the mirror a toroid. Applying Coddington's equations, we find the ratio between the two radii *R*_{s}/*R *_{t} = cos^{2 }*u*_{p}, where *u*_{p} is the angle of incidence.^{1}

**Figure 2.** Note that when the angle of incidence *u*_{p} = 0, for a concave spherical mirror with rays entering obliquely from infinity, the sagittal and tangential foci coincide at *F*_{0} and no astigmatism is present.^{2,3} Astigmatism increases with the increase of *u*_{p}, and the difference of the radii of the correcting toroid will grow accordingly.

With a tangential radius of *R *_{t} = 200 mm and an angle *u*_{p} = 5°, the sagittal *R*_{s} = 198.480775 mm. Note that the ratio of the two radii is independent of the object distance; however, optimum image quality is achieved at unit magnification, i.e., when object and image distances are equal and approximately *R *_{t}. The resultant mirror is barrel shaped with *R*_{s} being the radius of rotation.

Toroidal surfaces lend themselves conveniently to single-point diamond turning, as long as the material is suitable for that process and the swing radius (radius of rotation) is within a reasonable dimension (see **oe**magazine, July 2004). The latter depends on the size of the machine. Note that with such a setup, the contour of the curve perpendicular to the radius of rotation does not have to be circular. If the optical component to be machined is a mirror, and the object to be imaged with that mirror is located at a finite distance, the shape will be an ellipse. In this case, the toroidal surface has become an off-axis segment of an ellipsoid. If the object point is located at the first focus of the ellipsoid, the image will be formed at the second focal point without aberrations. If the object is located at infinity, the surface turns into a paraboloid. *oe*

*References *

*W. Smith, Modern Optical Engineering, Third Edition, McGraw-Hill, p. 317 (2000). **M. Riedl, Optical Design Fundamentals for Infrared Systems, Second Edition, SPIE Press, p. 65 (2001). **M. Riedl, Astigmatism and the Spherical Mirror, Electro-Optical Systems Design, Kiver Publications, p. 27 (September 1977).*

**Max Riedl**

*Max Riedl serves as technical adviser to Precitech Inc., Keene, NH.*