Any deviation of an image from its ideal shape is called distortion. For example, if the image of a rectangle is not a rectangle, the lens that produced it is said to suffer from distortion. If you have arrived here from our Optical Aberrations page, you know where this aberration fits in the grand scheme of things. So let’s just get down to a description of the various types of distortion.
First, the magnification of an optical system is defined as the ratio of the image height to the object height, and the magnification is assumed constant with respect to field. In other words, in a distortion-free optical system, the distance from the optical axis to any image point (the image height) is proportional to the distance from the optical axis to the corresponding object point (the object height).
For example, with a pinhole lens, the optical axis, the line between an object and its image and the line segments from the optical axis to the object point and from the optical axis to the image point form similar triangles. This means that object height and image height are always proportional and a pinhole lens can have no distortion.
The complexity comes in when we introduce angles. Again referring to the pinhole camera, let’s define the angle between the optical axis and a line through the object point, the center of the lens and the image point as Θ. If the distance from the pinhole to the image plane is f, then the image height, y, is given by y = f*tan(Θ). A pinhole lens is distortion-free, so this equation always holds. In other types of lenses, there can be distortion, which is defined as the difference between the actual image height and the ideal image height calculated from y = f*tan(Θ). This is, of course, f*tan(Θ) distortion.
With some lenses, such as the conoscopic lenses we use for our scatterometers, we are not attempting to map an object height to an image height. We are trying to measure, for example, light scattering as a function of angle. We could do this for a lens that is corrected for f*tan(Θ) distortion, but we’d be forever taking arctangents. Our lives become much simpler when the image height is proportional to the object angle rather than the object height. In this case, the ideal image height is given by y = f*Θ. Any deviation from this ideal is referred to as f*Θ distortion.
A more obscure type of this aberration is stereoscopic distortion. This is the type of distortion you need to correct for if you want to image a sphere onto a plane and make sure that the image is always circular. For this type of distortion, the ideal image height is given by y = 2*f*tan(Θ/2). When plotted, stereoscopic lies between f*tan(Θ) and f*Θ distortion.
Distortion is defined as any deviation from ideal image geometry, typically measured as the difference between actual image height and the expected value from a distortion-free model like y = f·tan(θ).
Different distortion models serve different imaging goals:
f·tan(θ) models assume constant magnification and are used when mapping object height to image height.
f·θ models are preferred for angular measurements (e.g., in scatterometers) because they simplify the math and better align image height with object angle.
Stereoscopic distortion is a specialized model used for projecting curved surfaces (like spheres) onto flat image planes while preserving circular symmetry, with the ideal defined by y = 2·f·tan(θ/2).
We have designed and built lenses with all of the above types of distortion correction. If your project needs distortion-corrected lenses, please give us a call at (651)315-8249, or fill out our contact us form.
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