Astigmatism and field curvature

A photographic lens ideally casts a sharp image on a sensor. For a long time, this sensor used to be an approximately flat film, but nowadays most cameras are equipped with a strictly flat array of photosites. 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. The sensor then samples a part of space which is partially in front of (or behind) the sharp image, resulting in a blurred capture. Owing to the intimate relationship between astigmatism and field curvature, it is convenient to treat these two Seidel aberrations together.

Spoked wheel

In the absence of spherical aberration, coma, and chromatic aberration, a lens that is additionally free of astigmatism offers stigmatic imaging. That is, points in object space are imaged as true points somewhere in image space. A lens that suffers from astigmatism does not image a point as a point. The rendering of an object detail then depends on the orientation of that detail. A line detail 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 focused to yield a sharp image of either the sagittal or the tangential detail, but not simultaneously. This is illustrated in Fig. 1 with the classical example of a spoked wheel. A well-corrected lens delivers an all-sharp image (left wheel). An astigmatic lens may be focused to yield a sharp image of the spokes, but the rim is then out of focus. Alternatively the lens can be focused to obtain a sharp rim, at the expense of blurred spokes. It is customary to speak of the sagittal focus and tangential focus, respectively, as indicated in Fig. 1. These names are potentially confusing, because a “sagittal focus” implies blurring in the sagittal direction, and a “tangential focus” implies blurring in the tangential direction. Together with lateral color, astigmatism is at the basis 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 in photography. The figure suggests that the amount of blur in either the sagittal or radial direction is constant across the field, but this is not the case. There is no astigmatism near the image center, unless a lens is poorly assembled. The aberration occurs off-axis only. With an actual photographic lens, the sagittal and tangential focal surfaces are curved. Fig. 2 displays the astigmatism of a simple, uncorrected lens, where 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 more than sagittal details. Off-axis imaging is not stigmatic, but there is a surface of “best focus”, situated somewhere between the S and T surfaces.

Primary astigmatism of a simple lens
Figure 2. A simple lens with undercorrected astigmatism. T = tangential surface; S = sagittal surface; P = Petzval surface. (Curvatures are exaggerated for clarity.)

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 maneuver the S and T surfaces closer to the sensor plane. Complete elimination of astigmatism is illustrated in the left sketch of Fig. 3. The S and T surfaces coincide, implying stigmatic imaging where each point in object space yields a true point in image space (it is still assumed that there is no spherical aberration, coma, and chromatic aberration). However, there is a penalty in the form of a dramatically curved image surface. When the image center is in focus on the sensor the corners are far out of focus, and vice versa. This curved surface is known as field curvature, the fourth of the Seidel aberrations. It only makes sense to discuss field curvature as a separate aberration when astigmatism is well corrected.

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. Despite its name, an anastigmat has some residual astigmatism, but the S and T surfaces are flatter than those of the uncorrected lens of Fig. 2 and the stigmatic lens at the left. The image quality is better. As such, the anastigmat offers an attractive compromise between astigmatism and field curvature.

Reduced astigmatism
Figure 3. Two cases of reduced astigmatism. Scheme A has zero astigmatism, but suffers from field curvature. Scheme B is known as an anastigmat. (Curvatures are exaggerated for clarity.)

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. S, T, and P obey the relationship TP = 3×SP in third order aberration theory [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. 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. Let us consider a standard 50/1.4 lens of the double Gauss type, meant to be used for objects at intermediate and long ranges, and assess its performance in the macro regime. 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 pattern of black double crosses was reproduced at unit magnification (1:1) on a sensor measuring 24 x 36 mm. Starting from center focus, and with the target and lens positions fixed, the camera was moved towards the lens in 0.5-mm increments. In Fig. 2, this corresponds to the sensor moving to the left. At center focus (Fig. 4), the image quality distressingly deteriorates with an increasing image height (the distance from the center). Crosses number 2 and 3 are noticeably aberrated, with the tangential cross bars being blurred more than the sagittal bars. The rendering suggests that cross 2 has only a single tangential bar; this is a case of spurious resolution.

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

When the sensor 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 sensor 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 images in Figs. 4–6 are consistent with the sagittal and tangential focal surfaces of the uncorrected scheme in Fig. 2. This is understood 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 completely behind the sensor, and, consequently, all radial bars are out of focus. Just like the situation in Fig. 6.

A detailed study of the test images (not shown) reveals that there is also some spherical and chromatic aberration. The dominant aberrations, however, are astigmatism and the accompanying curved fields. It can be deduced from the previously mentioned relationship TP = 3×SP 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.

The reproduction of another 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 the same as that of Fig. 4; only the target differs. The peculiar elongation of the blur patches towards the image corners follows directly from the astigmatism sketched in Fig. 2. Since the tangential focal surface is further from the sensor than the sagittal surface, there is more blur in the radial direction.

Target with white dots at center focus
Figure 7. With the 50/1.4 at center focus, astigmatism blurs off-axis white dots mostly in the radial direction.

Figure 8 shows the dots after moving the sensor 3.0 mm closer to the lens. The blur patch is circular in the image center, but its character changes with an increasing image height. At an image height of about 12 mm, the combination of tangential focus and sagittal blur yields a tangential blur stripe. Further out towards the corners, the sensor is located between the S and T surfaces, which results in peculiarly shaped blur patches. Similar flying saucers may be encountered in astrophotography, with lenses at infinity focus. Although these patches are much smaller than in the present example, they are due to the same cause. The point spread function of astigmatism should not be confused with the characteristic blur shape of a lens that suffers from coma. To be sure, there are several aberrations at play with a 50/1.4 lens at 1:1. There is a bit of coma visible in the corners, but astigmatism is the dominant cause of the variety of blur shapes in Figs. 7 and 8.

Sensor 3.0 mm in front of center focus
Figure 8. A displacement of 3.0 mm gives rise to a peculiar variation of blur patches over the field.

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 amount of blur at the sensor will decrease. Or put differently, the increased depth of focus helps to mask the worst effects. The lens used for the above images was not designed for use in the macro regime, and a more elegant solution to overcome field curvature at close focus is the use of a dedicated macro lens. For an infinity application like astrophotography, where lenses are often used at their largest aperture, a well-corrected ‘infinity lens’ is required. Lens designs with floating elements, i.e., the differential movement of one or more lens groups during focusing, may have an expanded working range where aberrations are controlled over an extended range of object distances. Just keep in mind that floating elements are only instrumental when the lens is focused by means of its focusing ring or autofocus mechanism. Elements do not float with added extensions.

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).