In a perfect imaging system, light exists as a spherical wave that converges to form a point image. However, in practice wavefront aberrations act to perturb the wavefront from its ideal spherical shape, which can degrade image quality. The appropriate use of ashperical lenses in an optical system can improve performance with a minimum addition of optical elements.

Figure 1. A sphere and an asphere are defined by their sag equations.
High performance optical imaging systems require good “image quality”, a loose term that refers to the ability to resolve fine image detail. Optical engineers quantify this ability by using metrics such as MTF (modulation transfer function), Strehl ratio, spot size or wave-front error. The highest possible image quality occurs when the light exiting the optical system has a perfectly spherical wavefront. Deviations from that spherical wavefront are called “aberrations” and virtually all practical optical systems have them.

Advanced optical systems with large fields of view and “fast” apertures are especially prone to having significant optical aberrations. (Fast means a low F/# or F-stop, which is the ratio of focal length to collecting diameter.) Before the advent of aspheric manufacturing, optical engineers used many spherical surfaces to balance aberrations, and the literature is filled with many such multi-element design forms that utilize all spherical elements, such as the Cooke Triplet, Double Gauss, etc. Computer controlled manufacturing coupled with computerized metrology, however, has now enabled the fabrication of aspheric surfaces that allow aberrations to be balanced with fewer optical surfaces.

Figure 2. Wavefront error comparison for a multi element spherical system to a multi order aspheric polynomial. The aspheric element has one spherical side.
A fundamental definition of an asphere is a surface that does not have a spherical shape. A sphere is simply defined by its radius of curvature, R. Familiar aspheric forms come from conic sections such as the ellipse, parabola, or hyperbola, and are characterized by the conic constant (k) or eccentricity (ε) of the conic. It is convenient to express the shape of an aspheric surface in terms of its “sag” (deviation from a plane at its vertex) and its aperture radius, ρ (Figure 1). By including even ordered polynomials (C2, C4,.... CN) to the surface shape, optical engineers have garnered considerable power to eliminate aberrations from the optical system.

Asphere vs Multiple Spheres

The power of aspheres can be demonstrated by comparing their ability to correct aberrations with spherical elements, as shown in Figure 2. The comparison contrasts an aspheric singlet with varying polynomial degree to a set of spherical lenses in an F/1.25 system with a 4° full field of view using Schott N-BK7 and monochromatic light. The axial or central beam’s wavefront error is shown plotted versus number of elements and the aspheric polynomial. A rule of thumb derived from the Rayleigh Criterion states that a diffraction limited spot can be achieved for wavefront error that is smaller than a quarter of the wavelength of light. As shown in the figure, it takes five spherical elements to achieve quarter wave performance while it can be done with a single element with a 10th order aspheric element.

There are standard locations for placing aspheres in an optical system. Theoretically only two aspheres are required for good imagery for fast, wide field of view systems1.

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