Aberrations
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To determine the precise performance of a lens system, we can trace the path of light rays through it,
using Snell's law at each optical interface to determine the subsequent ray direction. This process,
called ray tracing, is usually accomplished on a computer. When this process is completed, it is
typically found that not all the rays pass through the points or positions predicted by paraxial
theory. These deviations from ideal imaging are called lens aberrations. |
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The direction of a light ray after refraction at the interface between two homogeneous, isotropic media
of differing index of refraction is given by Snell's law: |
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where q1 is the angle of incidence, q2 is the angle of refraction, and both angles are measured from the surface normal as shown in the figure below. |
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| Refraction of light at a dielectric boundary |
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Even though tools for precise analysis of an optical system are becoming easier to use and are readily
available, it is still quite useful to have a method for quickly estimating lens performance. This not
only saves time in the initial stages of system specification, but can also help achieve a better
starting point for any further computer optimization. The first step in developing these rough guidelines is to realize that the sine functions in Snell's law can be expanded in an infinite Taylor series: |
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The first approximation we can make is to replace all sine functions with their arguments (i.e., replace sinq1 with q1 itself and so on). This is called first-order or paraxial theory because only the first terms of the sine expansions are used. Design of any optical system generally starts with this approximation. The assumption that sinq = q is reasonably valid for q close to zero (i.e., high f-number lenses). With more highly curved surfaces (and particularly marginal rays), paraxial theory yields increasingly large deviations from real performance because sinq ≠ q. These deviations are known as aberrations. Because a perfect optical system (one without any aberrations) would form its image at the point and to the size indicated by paraxial theory, aberrations are really a measure of how the image differs from the paraxial prediction. As already stated, exact ray tracing is the only rigorous way to analyze real lens surfaces. Before the advent of computers, this was excessively tedious and time consuming. Seidel addressed this issue by developing a method of calculating aberrations resulting from the q3/3! term. The resultant third-order lens aberrations are therefore called Seidel aberrations. To simplify these calculations, Seidel put the aberrations of an optical system into several different classifications. In monochromatic light they are spherical aberration, astigmatism, field curvature, coma, and distortion. In polychromatic light there are also chromatic aberration and lateral color. Seidel developed methods to approximate each of these aberrations without actually tracing large numbers of rays using all the terms in the sine expansions. In actual practice, aberrations occur in combinations rather than alone. This system of classifying them, which makes analysis much simpler, gives a good description of optical system image quality. In fact, even in the era of powerful ray-tracing software, Seidel's formula for spherical aberration is still widely used. |
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| Optics Guide Copyright 2002 Melles Griot Inc. |