Distortion (optical)

A lens with distortion forms a sharp image in the wrong place. Each object point still maps to a point, but the magnification changes with distance from the axis, so the image of a square grid has bowed lines. When magnification falls toward the edge of the field the grid bulges outward, barrel distortion; when it rises the corners are pulled out, pincushion distortion. Because it does not blur the image, distortion does not affect resolution or the MTF, and it can be removed afterwards in software; for measurement and machine vision it is often the aberration that matters most.

It is quoted as the percentage difference between the real image height $y$ and the ideal, paraxial height $y_0$:

Negative values are barrel and positive values pincushion. In third-order (Seidel) theory the displacement grows as the cube of the field height, so the percentage grows as its square, and a lens with 1% at the corner of its field has about 0.25% at half that height. The main cause is the position of the aperture stop: a stop in front of a positive lens gives barrel distortion and a stop behind it pincushion, and a lens symmetric about its stop is nearly free of it, which is why symmetric designs are used for copying and measurement.

Designed distortion: f-theta lenses

In laser scanning, a mirror deflects the beam by an angle $\theta$, and a lens with no distortion places the spot at $y = f\tan\theta$, so equal steps in angle give unequal steps on the work surface. An f-theta scan lens is designed with barrel distortion chosen so that $y = f\theta$ instead, making spot position proportional to scan angle. At a 20° scan angle the f-theta height is 4.1% smaller than $f\tan\theta$, which is the distortion such a lens builds in on purpose. Fisheye lenses go much further, mapping a hemisphere onto a disk.

Measurement

Distortion is measured by imaging a precise grid or dot target and comparing the measured positions of the dots with those expected from the central magnification. Camera calibration software fits a radial polynomial to the displacements, and the fitted model then corrects images; the same fit gives the percentage distortion at each field height.

References: W. J. Smith, Modern Optical Engineering, 4th ed. (McGraw-Hill, 2008); W. T. Welford, Aberrations of Optical Systems (Adam Hilger, 1986); Z. Zhang, IEEE Trans. Pattern Anal. Mach. Intell. 22, 1330 (2000).