Astigmatism and field curvature

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Ideally, a photographic lens images the world in a plane, where it is recorded by a sensor. This sensor is typically either an approximately flat film or a strictly flat digital array. Departures from a flat image surface are associated with astigmatism and field curvature, and lead to a spatial mismatch between the image and the sensor. As a result, the sensor samples a part of space in front of or behind the sharp image, and its representation of the image will thus be blurred. Owing to the closely connected natures of astigmatism and field curvature, it is convenient to treat these Seidel aberrations together.


Spoked wheel

For a single color of light; chromatic aberrations lead to blurring too.

In the absence of spherical aberration and coma, a lens that is additionally free of astigmatism offers stigmatic imaging, i.e. points in object space are imaged as true points somewhere in image space. A lens that suffers from astigmatism, however, does not offer stigmatic imaging. In the presence of astigmatism the rendering of an object detail depends on the orientation of that detail. For instance, a (short) line oriented towards the image center is called a sagittal (radial) detail, whereas a detail perpendicular to the radial direction is called a tangential detail. The astigmatic lens may be focussed to yield a sharp image of either the sagittal or the tangential detail, but not simultaneously. This is illustrated in Fig. 1 with the archetypal example of astigmatism: a spoked wheel. A well-corrected lens delivers an all-sharp image (left wheel). On the other hand, an astigmatically aberrated lens may be focussed to yield a sharp image of the spokes (middle wheel), but at the expense of blurring of the rims, which have a tangential orientation. Vice versa, when the rim is in focus the spokes are blurred. It is customary to speak of the sagittal focus and tangential focus, respectively, as indicated in Fig. 1. Note that these names are potentially confusing, because a "sagittal focus" actually implies blurring in the sagittal direction and a "tangential focus" implies blurring in the tangential direction. Together with lateral color, astigmatism is a leading cause of differences between the sagittal and tangential modulation transfer functions (MTF).

A spoked wheel

Figure 1. Classic example of astigmatism. Left wheel: no astigmatism. In the presence of astigmatism (middle and right wheels) one discriminates between the sagittal and tangential foci.

Note that the astigmatism of a photographic lens or a telescope is different from ophthalmic astigmatism. The latter arises from an uneven curvature of the cornea and destroys rotational symmetry [1]. With an astigmatic eye the perceived sharpness of the spokes in Fig. 1 would depend on their orientation.


Curved surfaces

Although the wheels in Fig. 1 are instructive, they are an oversimplification of astigmatism as it occurs with photographic lenses. Where the figure suggests that the amount of blurring in either the sagittal or radial direction is constant across the field, this is not the case in practice. Unless a lens is poorly assembled, there will be no astigmatism near the image center. The aberration occurs off-axis. With a real lens, the sagittal and tangential focal surfaces are in fact curved. Fig. 2 displays the astigmatism of a simple lens. Here, the sagittal (S) and tangential (T) images are paraboloids which curve inward to the lens. As a consequence, when the image center is in focus the image corners are out of focus, with tangential details blurred to a greater extent than sagittal details. Although off-axis stigmatic imaging is not possible in this case, there is a surface lying between the S and T surfaces that can be considered to define the positions of "best focus".

Primary astigmatism of a simple lens

Figure 2. A simple lens with un(der)corrected astigmatism. T = tangential surface; S = sagittal surface; P = Petzval surface.

Lens designers have a few degrees of freedom, such as the position of the aperture stop and the choice of glass types for individual lens elements, to reduce the amount of astigmatism, and, most desirably, to manoeuvre the S and T surfaces closer to the sensor plane. A complete elimination of astigmatism is illustrated in the left sketch of Fig. 3. Although astigmatism is fully absent, i.e., the S and T surfaces coincide, there is a penalty in the form of a pronouncedly curved field. When the image center is in focus on the sensor the corners are far out of focus, and vice versa. In the late nineteenth century, Paul Rudolf coined the word anastigmat to describe a lens for which the astigmatism at one off-axis position could be reduced to zero [2]. The right sketch in Fig. 3 depicts a typical photographic anastigmat. As a slight contradiction in terms, the anastigmat has some residual astigmatism, but more importantly, the S and T surfaces are more flat than those in the uncorrected scheme of Fig. 2 and the strictly stigmatic left scheme in Fig. 3. As such, the anastigmat offers an attractive compromise between astigmatism and field curvature.

Reduced astigmatism

Figure 3. Two cases of reduced astigmatism. Scheme B is known as an anastigmat.

Note

The surface P in Fig. 2 and Fig. 3 is the Petzval surface, named after the mathematician Joseph Mikza Petzval. It is a surface that is defined for any lens, but that does not relate directly to the image quality—unless astigmatism is completely absent (e.g., scheme A in Fig. 3). In the presence of astigmatism the image is always curved (whether it concerns S, T, or both) even if P is flat as a pancake. In third order aberration theory S, T, and P obey the relationship TP = 3×SP [3]. Here, TP is the longitudinal (horizontal in the sketches) separation between T and P, and SP is the separation between S and P.


Astigmatism in practice

Astigmatism and field curvature are not usually obvious with modern photographic lenses. Sure, when a lens is used at full aperture the corner definition is often noticeably worse than the center definition and the above aberrations can be (partly) responsible, but it's no trivial matter to tell them apart from other oblique aberrations. A different situation arises when a lens is used in a scenario for which it was not designed. As a case in point, let us consider a standard lens, meant to be used for objects at intermediate and long ranges, as it performs in the macro regime. The idea is that a lens that is well corrected for infinity use is not necessarily corrected for use at close range, and vice versa.
With the help of bellows, a white target with a series of black crosses was reproduced at unit magnification with the Carl Zeiss Planar 1.4/50. Starting from center focus, and with the target and lens positions fixed, the camera was moved towards the lens with 0.5-mm increments. In Fig. 2, this would correspond to the sensor (a film in this case) moving to the left. At center focus (Fig. 4) the image quality distressingly deteriorates with an increasing distance from the image center. Crosses number 2 and 3 are noticeably aberrated, with the tangential cross bars blurred more than the sagittal bars.

Target at center focus

Figure 4. The target photographed at 1:1 with the center cross in sharp focus. (Planar 1.4/50 @ f/1.4)

When the film is 1.5 mm closer to the lens, the image center is out of focus—as one would normally expect: Fig 5. However, to some extent the off-axis crosses have improved. The sagittal bars 3S of the outermost crosses, which were not well defined in Fig. 4, are now nicely resolved and the same can be said about the tangential bars 2T of cross number 2.

Sensor 1.5 mm in front of center focus

Figure 5. Compared with Fig. 4, a sensor displacement of 1.5 mm towards the lens shifts the tangential focus to 2T and the sagittal focus to 3S.

Finally, when the film is 4.5 mm in front of center focus, the only structure that is resolved are the tangential bars 3T of the outermost crosses (Fig. 6). Everything else is blurred.

Sensor 4.5 mm in front of center focus

Figure 6. A displacement of 4.5 mm puts the tangential focus at 3T. There are no crosses left with a sagittal focus.

The three snapshots shown in Figs. 4–6 are consistent with the sagittal and tangential focal surfaces of the uncorrected scheme in Fig. 2. This can be seen by realizing that, when the sensor is moved to the left, the sagittal focus shifts relatively quickly to the image periphery. The tangential focus also moves out from the center, but at a slower pace. When the tangential focus reaches the image periphery, the sagittal focal surface S is already behind the sensor, and, consequently, all radial bars are out of focus. Just like the situation in Fig. 6.

It should be remarked that a detailed study of the test images reveals that there is also some spherical and chromatic aberration going on. The dominant aberrations, however, are astigmatism and the accompanying curved fields. From the previously mentioned relationship TP = 3×SP it can be deduced that the Petzval surface is actually quite flat for this 1.4/50 at unit magnification, but that is little consolation when there is a substantial amount of astigmatism.

As a last illustration, a different type of target is shown in Fig. 7. It consists of white dots against a black background. The blur patches of these dots are a rough indication of the so-called point spread function of the lens. The configuration of Fig. 7 is exactly the same as that in Fig. 4—only the target differs. The peculiar elongation of the blur patches towards the image corners can be directly related to the astigmatism sketched in Fig. 2. Since the tangential focal surface is further away from the sensor than the sagittal focal surface, off-axis tangential details will be blurred more at the position of the sensor. In other words, there will be more blur in the sagittal (i.e. radial) direction than in the tangential direction, which is indeed the case in Fig. 7. The blur patches in Fig. 7, which are mainly due to astigmatism, should not be confused with the characteristic blur shape of a lens that suffers from coma.

Target with white dots at center focus

Figure 7. Astigmatism as evidenced by the blurring of white dots. (Planar 1.4/50 at f/1.4 and at center focus)


Final remarks

A trusty method to mitigate image impairment by astigmatism and field curvature is to stop down the lens. The curved S and T surfaces themselves are not affected by the F-number, but the proportions of the blur at the position of the sensor will decrease. Or, differently phrased, the increased depth of focus helps to mask the worse effects. Of course, the Planar 1.4/50 is not designed for use in the macro regime. A far more elegant solution to overcome field curvature is found in the use of a dedicated macro lens. Such lenses are aberration-corrected for use at close range, including astigmatism, field curvature and distortion, to enable copying work. Yet another remedy is the use of floating elements in the lens design. Floating elements (the differential movement of one or more lens groups) expand the working range of a lens by controlling aberrations over an extended range of object distances. Short-focus lenses for SLRs with floating elements are more expensive than their unit-focus competitors, but the performance gain at close range can be truly remarkable. Just keep in mind that floating elements are only instrumental when the lens is focussed by means of its focussing ring. Elements do not float with added extensions.

© Paul van Walree 2004–2014



References


[1]   Eugene Hecht, Optics, 3rd ed., Addison Wesley (1998).
[2]   Born and Wolf, Principles of Optics, 7th ed., Cambridge University Press (1999).
[3]   A.E. Conrady, Applied optics and optical design, part one, Dover Publications (1985).