The paraxial theory of image formation discussed above is necessarily an idealization of the behavior of real optical systems. The goal of stigmatic imaging—recombining the light passing through a finite aperture from an extended object to project all the energy from each object point to a single image point simultaneously—is never perfectly achieved in practice. Even at the best possible focus, light from a single object point is distributed over a small area of the image. Each image point receives light predominantly from one object point but also receives some light from neighboring object points. The image point resembles but does not duplicate the object point. Because rays do not focus perfectly stigmatically, the image does not contain as much detail as the original object. This discrepancy is referred to in general as the aberration of an optical system.
Point Spread Function
The region over which light from a single object point is spread in a pinhole camera is a “blur circle” (see the Quick-Start Guide). Strictly, the word circle is somewhat misleading. Even with simple ray tracing through a small circular aperture, the image region is typically an ellipse, not a circle. Light from any single object point tends to focus to an irregularly shaped smudge. Moreover, within the smudge light is usually not evenly distributed; some areas are brighter than others. The distribution of light (from a single object point) in the image is aptly named the point spread function (PSF), because it describes how light from a single object point spreads out in the image.
An image is therefore composed of multiple partially overlapping smudges, one smudge for each object point. The point spread function is a quantitative description of the smudge and is quite useful because we can deduce all the imaging characteristics of an optical system from the PSF. (The process for extracting image information from the PSF requires advanced mathematics beyond the scope of clinical practice.)
As we have seen, imaging for incident rays near the optic axis is, to an excellent approximation, stigmatic, but rays outside the paraxial region do not focus stigmatically. The size of the paraxial region depends on several factors and varies from lens to lens. The paraxial region can be quite large but typically is small—a few millimeters or less. Nevertheless, the paraxial region is very important because paraxial rays focus stigmatically. Rays outside the paraxial region account for most aberrations.
The vergence equation and transverse magnification equation are based entirely on paraxial rays, completely ignoring rays outside the paraxial region. Paraxial rays alone are enough to calculate image location, size, and orientation to sufficient accuracy. However, to understand image quality, brightness, and depth of focus, we must consider nonparaxial rays. For instance, image quality is diminished because rays do not focus stigmatically. Because paraxial rays do focus stigmatically, it is the nonparaxial rays that determine image quality. We can say nothing about image quality based on paraxial rays alone.
Moreover, the notion of optical power applies to the paraxial region only. There is no such thing as refractive power outside the paraxial region. This is in stark contrast to corneal topographic power maps that assign a “power” to every point on the cornea, even the periphery. However, corneal power maps define power in 2 ways (axial and tangential), and neither is consistent with the correct definition of refractive power as we have used it in this chapter. In the early 1950s, lens designers developed wavefront theory, the appropriate way to analyze the optical properties of imaging systems beyond the paraxial regime, especially aberrations.
Excerpted from BCSC 2020-2021 series : Section 3 - Clinical Optics. For more information and to purchase the entire series, please visit https://www.aao.org/bcsc.