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Barrel and Pincushion distortion are due to the fact that the magnification of the lens changes as you get further off axis.

Let's consider a single thin lens with an aperture stop. These two pictures shows that it is affected by Barrel or Pincushion distortion depending on the aperture and lens relative positions.

enter image description here

enter image description here

The single lens equation shows a magnification equal to f/(d-f). It does not depend on the axial position of the subject (hence of the image). Is it the aperture the cause of the magnification dependence on such a distance? I can't visualize it on the previous pictures.

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    \$\begingroup\$ The single lens equation is a simplification of reality. \$\endgroup\$
    – xenoid
    Dec 27, 2022 at 8:48
  • \$\begingroup\$ @xenoid Of course, but I'd like to know the cause of the magnification dependence on off-axis distance. If it does depend! Some sources like those picture say it is due to the aperture and not to the lens per se! \$\endgroup\$
    – Kinka-Byo
    Dec 27, 2022 at 11:31

4 Answers 4

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Look at it this way: Without any aperture in front of or behind the single element lens, both distortions make it through and are superimposed upon one another. That's a major part of why an uncorrected single lens is so blurry at the edges. It also should be apparent why the effect is much worse with lenses of higher refractive power (wider angle lenses) than lenses of lower refractive power. The more the lens bends incoming light, the greater the divergence between the rays from any point on the edge of the field that strike opposite sides of the lens' surface.

The aperture does not cause any distortion. Both pincushion and barrel distortion are both already there before the aperture is placed in the optical path.

By putting an aperture stop in front of or behind the lens, it attenuates one but not the other (barrel or pincushion) distortion. This allows the lens to be sharper at the edges - and the remaining distortion to be more noticeable in the absence of the counterpart.

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  • \$\begingroup\$ Thanks for the explanation. Could you tell me which is the cause of both pincushion and barrel distortion in a single lens element? Its behaviour being soft ad the edges is the spherical aberration, why is it related to distortion? \$\endgroup\$
    – Kinka-Byo
    Dec 28, 2022 at 7:03
  • \$\begingroup\$ Aberration is a wavefront error, what does it have to do with distortion, which is an alteration of perspective? \$\endgroup\$
    – Kinka-Byo
    Dec 28, 2022 at 7:04
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    \$\begingroup\$ From each point of light in the field rays strike the entire front surface of the lens. Especially from points on the edge of the field, the rays that strike the nearer edges of the front of the lens and the rays that strike the further edges of the front of the lens will be refracted differently with the result that the rays striking one side are magnified more while the rays striking the other side are magnified less. Rays striking intermediate positions are magnified at factors between the two extremes. Thus the image is blurred. \$\endgroup\$
    – Michael C
    Dec 28, 2022 at 7:23
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    \$\begingroup\$ From the parts of the lens where the rays are magnified slightly more, the image is more spread out. On the parts of the lens where the rays are magnified slightly less, the image is more compact. The most severe differences in both directions are the rays that strike opposite edges of the front of the lens. \$\endgroup\$
    – Michael C
    Dec 28, 2022 at 7:27
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"d" in the single lens formula stands for distance... i.e. magnification IS dependent on distance, axial or otherwise. A simple lens with uncorrected spherical aberrations will create an undistorted (but less sharp) image. The difference is that the single lens formula assumes the aperture/stop is the lens itself, and therefore does not affect the magnification (aberrations) and cause (reveal) distortion.

When the lens has an additional restrictive aperture to correct spherical aberrations it "causes" the distortion because:

  • In the case of pincushion distortion, it is preventing light rays from the source, which is leaving from all areas of the lens element, from reaching the image plane. It allows the rays with shorter travel distance and more magnification pass, resulting in positive displacement.
  • In the case of barrel distortion, it is preventing light rays from reaching all areas of the lens element, and therefore the image. It allows the rays with greater travel distance and less magnification pass, resulting in negative displacement.

Perhaps these drawings are clearer...

enter image description here

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  • \$\begingroup\$ Thanks for the drawings. But I still do not understand the point. The aperture blocks some rays. Less rays means less light being focused on the same point. Distortion is about geometry. I miss the concept. \$\endgroup\$
    – Kinka-Byo
    Dec 27, 2022 at 16:43
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    \$\begingroup\$ I added the ideal ray placement to the drawings. \$\endgroup\$ Dec 27, 2022 at 16:53
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    \$\begingroup\$ Aperture is also about removing spherical aberrations, which primarily come from the periphery of the lens element due to the greater amount to which the light rays must be bent... particularly for off axis sources. Adding a restrictive aperture blocks some peripheral rays, thereby reducing chromatic aberrations and increasing image sharpness. i.e. a more distinct/clearly rendered geometry; the side effect is that it is distorted. \$\endgroup\$ Dec 27, 2022 at 16:59
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    \$\begingroup\$ Yes, it moves the circle of confusion location/center; and makes it smaller as well. How far it moves depends on how far off axis it is (distance difference) and how much the rays must be bent (lens curvature). In the case of a lens which renders significant curvilinear distortion; without the restrictive aperture the image would otherwise be unusable/indistinct (in those regions). \$\endgroup\$ Dec 27, 2022 at 17:22
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    \$\begingroup\$ There are some explanations and formulas for calculating the lens distortion here, but I don't think they are really practical from an observer's point of view. personal.math.ubc.ca/~cass/courses/m309-03a/m309-projects/le/…. \$\endgroup\$ Dec 27, 2022 at 17:37
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The aperture limits the light to a subset that shows a reasonable amount of convergence for the given lens, at least in its widest setting. As the aperture gets smaller, the subset tends to be one of better convergence, but that makes a noticeable difference only for about one stop (typically). At very narrow apertures, diffraction governs an increasing amount of the involved light. But the geometric features of the principal image are identical across all apertures.

So barrel/pincushion distortion is not really related to aperture or the aperture effect since the bulk of the light actually coming in is almostly always nowhere near the aperture blades.

One indicator that this is not aperture-related is that with a zoom lens, it is typical to have barrel distortion at the wide end and pincushion distortion at the long end. This is due to spherical aberration (look it up, it's really a bit involved to explain in detail here) that modern lens assemblies tend to correct with some aspherical lens elements, but correcting for all focal lengths and focusing distances always tends to end up a compromise.

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  • \$\begingroup\$ You're talking about compound lenses with many elements, typically some in front of and others behind the physical aperture diaphragm. At least one of those elements will be there to correct spherical aberration, which is what causes geometric distortion with single element lenses. Thus the effect of the diaphragm with multi-element compound lenses is different for the elements on either side of it. The question does not address such lenses, but only single element lenses. \$\endgroup\$
    – Michael C
    Dec 28, 2022 at 3:24
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As you know, the camera lens projects an image of the outside world onto the surface of film / digital sensor. The size of the image of objects is intertwined with their actual size – their distance from the camera – the focal length of the lens (magnification). We expect a faithful reproduction, i.e., one without distortion. Two types of distortion are common. We are talking pincushion and barrel. These are variations of magnification, center of image to edge. Barrel distortion results when the magnification decreases with distance away from axis, while pincushion results is from increasing magnification. Both are induced by a slight misshape in the figure of the lens.

The countermeasure is carful placement for the location to the lens stop (iris) or the inclusion of two symmetrical lens elements fore and aft of the lens stop.

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