[펌] Spherical aberration (and focus shift)

Most photographic lenses are composed of elements with spherical surfaces. Such elements are relatively easy to manufacture, but their shape is not ideal for the formation of a sharp image. Spherical aberration (SA) is an image imperfection that is due to the spherical lens shape. Fig. 1 illustrates the aberration for a single, positive element. Light that hits the lens close to the optical axis is focused at position c. The light that traverses the margins of the lens comes to a focus at a position a closer to the lens. In this manner the focus position depends on the zone of the lens that is considered. When the marginal focus is closer to the lens than the axial focus, such as exhibited by the positive element in Fig. 1, one speaks of undercorrected spherical aberration. Conversely, when the marginal focus is located beyond the axial focus the lens is said to suffer from overcorrected spherical aberration.

The image of a point formed by a lens with SA is usually a bright dot surrounded by a halo of light. The effect of SA on an extended image is to soften the contrast and to blur its details [1]. Spherical aberration is uniform over the field, in the sense that the longitudinal focus difference between the lens margins and center does not depend on the obliquity of the incident light.

Figure 1. Illustration of undercorrected spherical aberration. The margins of the lens have a shorter focal length than the center.

From Fig. 1 it appears that a spherically aberrated lens has no well-defined focus. At any position behind the lens a sensor will be confronted with a finite circle of confusion (blur disk) rather than a true image point. However, there is a geometrically "best" focus [2], which corresponds to the circle of least confusion at position b. This is just the place where the ensemble of light cones has its minimum cross section.

Focus shift

An interesting phenomenon occurs when an aperture stop is placed next to the lens in Fig. 1. If the aperture is closed so as to block the marginal rays, it is observed that the best focus shifts to the right. At small lens apertures the best focus is found at position c and it will also be a better focus, i.e., the circle of least confusion is a smaller circle than the circle of least confusion at full aperture. To profit fully from the improved performance, the sensor should ideally be located at position c. This cleary presents a risk of underachievement, since many photographic systems are focused with the lens at full aperture. The unsuspecting photographer focuses with the lens wide open to place the circle of least confusion at the sensor position b, and, taking the actual picture at a reduced aperture, he is unaware of the ensuing focus shift that prevents him from getting the best out of the lens. Surely, the reduced aperture mitigates the effect of spherical aberration also for a sensor fixed at b, but it is possible to do better. SLR users may stop down the lens with the DOF preview button to focus at the actual working aperture. An automatic compensation mechanism for the focus shift is proposed by Goldberg [3]. Zeiss launch a line of rangefinder lenses for the return of the Zeiss Ikon legend, for which they mention that the minimization of focus shifts with aperture changes was a specific design goal [4]. Apparently spherical aberration is noticeably reduced, which makes the reader wonder how serious the focus shift is with existing (rangefinder) lenses. Values of order 100 µm are reported for the Noctilux-M 1.0/50 [5].

Out-of-focus character

The influence of spherical aberration on in-focus parts of the image can be difficult to recognize or tell apart from other image degrading factors such as a slight misfocus. In contrast, SA may leave a marked fingerprint on out-of-focus image parts. Reverting to Fig. 1, it is remarked that the light intensity distribution within the circle of confusion is not uniform in the presence of SA. At position c the blur disk is characterized by a bright core surrounded by a faint halo, whereas the blur disk at position a has a darker core surrounded by a bright ring of light. Such anomalous light distributions can be manifest in the out-of-focus parts of a photograph. As an example, Fig. 2 shows the center dot of this target reproduced at 1:1 with an 85/1.4 lens on bellows. When the film is 5 mm behind the "best" focus the blur disk shows the bright-ring effect, in a similar fashion to the circle of confusion produced by mirror lenses. Then, when the film is 5 mm in front of the best focus (i.e., closer to the lens) the blur characteristics have noticeably changed towards a brighter core surrounded by a faint halo. Apparently this lens suffers from overcorrected spherical aberration at unit magnification, i.e., it behaves opposite to the example in Fig. 1.

Figure 2. Through focus blur changes of a white dot in the image center. (85/1.4 @ f/1.4)
Another compilation illustrates the joint effect of two aberrations on out-of-focus images. Fig. 3 shows the center cross of this target as reproduced with the same 85/1.4 lens that is responsible for the phenomena observed in Fig. 2. The bellows extension was approximately 85 mm, which gives a magnification of about 1:1. The camera, and the film with it, was moved in 1-mm increments from 4 mm behind, to 4 mm in front of the best focus. The camera movement allows for a direct survey of the image space at a fixed separation between the lens and the target. The target, which is a more complex structure than a dot, and the color registration give rise to a remarkable series of blurred crosses.

Figure 3. Through focus blur changes of a cross with the film position indicated in millimeters. The film is moved in 1-mm steps from 4 mm behind (-4) to 4 mm in front (+4) of the best focus position (0).
Spherical aberration is responsible for the rugged blur character at negative distances, and the transition to a smoother appearance at positive distances. Equally fascinating are the color effects, which are due to the longitudinal chromatic aberration (axial color). Indeed, unless a lens is poorly assembled spherical aberration and axial color are the only aberrations that affect the image center. More often than not the amount, and sometimes the sign, of spherical aberration depends on the wavelength of the light. In that case the combined effects of SA and LCA are known as spherochromatism. So much is clear, the striking phenomena in Fig. 3 evidence that the lens is not designed for use as a macro lens. Most lenses are optimized for use at or near infinity focus, not 1:1. Such lenses will perform less than genuine macro lenses when they are used at close range. Nonetheless, even when a lens is used for its intended application residues of both spherical and chromatic aberrations can pop up in the out-of-focus parts of normal photographic images and influence the bokeh.

Concluding remarks

Of course, the illustration in Fig. 1 is a gross exaggeration. The amount of residual spherical aberration in photographic lenses is small. The effects are mitigated by the combination of lens elements with opposite amounts of SA, the use of high-index glasses, clever shaping of elements, and the use of aspherical elements [6]. Moreover, floating elements may be employed to establish a reduced SA over an extended range of working distances. For a given lens with residual spherical aberration, a very effective way to improve the image quality is to decrease the size of the aperture. The diameter of the blur disks diminishes with the cube of the aperture diameter - for the uncorrected element in Fig. 1. This dependence may be different for the residual SA in compound lenses, but generally an aperture reduction by one stop already presents a noticeable improvement to the image.

Alternatively, instead of combatting spherical aberration a photographer can decide to introduce it deliberately into his creations. The Zeiss softars are filters that feature tiny lenses on their otherwise flat surface, which inject spherical aberration into the image. They are popular among portrait photographers for their associated soft-focus quality and impressionistic character.

© PA van Walree 2004-2007


[1] Warren J. Smith, Modern optical engineering, 3rd ed., McGraw-Hill (2000).
[2] A.E. Conrady, Applied optics and optical design, part one, Dover Publications (1985).
[3] Norman Goldberg, Camera technology - the dark side of the lens, Academic Press (1992).
[4] Zeiss Ikon, http://www.zeissikon.com/lenses.htm
[5] Erwin Puts, http://www.imx.nl/photosite/leica/mseries/testm/M10-50.html
[6] J. Mukai, Y. Matsui, and I. Harumoto, "Effects of aspheric surfaces on optical performance and their application to lenses for 35mm Cinematography," J. SMPTE 88, 542-545 (1979).


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