Analysis: spherical aberrations and lens design - from “periscope” to apochromat

Material especially for Radozhiva prepared Rodion Eshmakov.

main photo

Characteristic bokeh of the lens Triplet 78 / 2.8, which has pronounced spherical aberrations.


Disclaimer: This text may seem quite complex, but even so, I recommend at least looking at the pictures - they can be perceived intuitively.

Manufacturers of photographic equipment today offer a large selection of excellent quality lenses that provide sharp images. Perfectly corrected modern lenses of complex design perform their task in such a way that their own “interference” in the final result is imperceptible - the quality of the final image depends on the composition, subject and... post-processing. Often, when processing images, there is a deliberate reduction in its “information content”: it is popular to apply effects such as glare, specific filters, or simulating a shallow depth of field (bokeh effect). There is also a well-known analog alternative to such “Photoshop” - artistic optics, vintage optics. Due to imperfect optical design, many old lenses have characteristic bokeh, defects in image sharpness, and distorted color rendition - which is replaced by artificial post-processing techniques.

Perhaps one of the main factors influencing the nature of the image formed by the lens is spherical aberration, or more precisely, spherical aberrations. The essence of this optical distortion is that the lens transforms each infinitesimal point of the object into a symmetrical spot in the form of a point of finite size with a halo. If aberration appears on the optical axis, that is, at the central point of the image, then such aberration is called longitudinal. But in fast lenses (a striking example is Helios-44 и Helios-40) Off-axis aberration also plays an important role - the so-called spherical aberration of inclined beams.

Longitudinal spherical aberration of a single lens.

Longitudinal spherical aberration of a single lens.

Longitudinal spherical aberration is one of the main reasons for the reduction in detail and contrast of the formed image, therefore in modern optics its presence is usually minimal. At the same time, the picture of vintage lenses - especially those of moderate aperture (
This article analyzes the types of longitudinal spherical aberration, typical profiles for its correction, and demonstrates the effect of distortion on image quality and bokeh of different lenses - from a two-lens "periscope» to a diffraction quality apochromat telescope.

Briefly about the theory of aberrations

According to the theory of geometric aberrations developed by the German mathematician Philipp von Seidel back in the 19th century, the magnitude of the deviation of the beam path of monochromatic light from the ideal can be associated with the parameters of the optical system (radii of curvature, thickness, properties of refractive materials). Seidel proposed to expand the resulting dependence into a series so that the coefficients in front of each member of the series are determined only by the optical design, and the remainder reflects the dependence of the aberration on the angle of the field of view and the size of the lens pupil.

Seidel’s expansion included members of the series with a maximum total order of dependence on the angle and pupil of the lens equal to 3, therefore the theory he developed was called the theory of third-order aberrations: S1 (spherical aberration, ~R^3), S2 (coma, ~R^2 •W), S3 (astigmatism, R•W^2), S4 (field curvature, R•W^2), S5 (distortion, ~W^3), where R is the lens pupil size, and W is the field of view angle .

Seidel's theory satisfactorily describes only optical systems with small angular fields and low aperture, which most photographic lenses are not. Thus, the Seidel aberration series is usually supplemented by terms of higher orders - 5, 7, 9 - especially when it comes to high-aperture optics.

It is worth noting that the expansion proposed by Seidel turned out to be inconvenient for calculations carried out using computers, and therefore now they mainly use a somewhat less visual representation in the form of expansion into a series of orthogonal Zernike polynomials, to the point of pain reminiscent view of the wave functions of electron orbitals in an atom. Fortunately, specifically spherical aberrations of different orders in the concepts of Seidel and Zernike are close.

Presentation of members of the Zernike series. 1 – “piston” (aberration-free image), 4 – defocusing, 9 – primary spherical aberration (aka 3rd order), 16 – secondary spherical aberration (aka 5th order), 25 – tertiary spherical aberration (aka 7th order).

Presentation of members of the Zernike series. 1 – “piston” (aberration-free image), 4 – defocusing, 9 – primary spherical aberration (aka 3rd order), 16 – secondary spherical aberration (aka 5th order), 25 – tertiary spherical aberration (aka 7th order).

Longitudinal spherical aberration: orders, chromatism and zones

So, longitudinal spherical aberration comes in different orders (according to Seidel): 3, 5, 7, etc. Spherical aberrations of higher orders must be taken into account when calculating high-aperture systems, since their magnitude extremely strongly (by powers 5, 7, 9) depends on the size of the lens pupil. Correction of longitudinal spherical aberration is achieved through mutual compensation of distortions of all observed orders. In other words, in a lens that exhibits spherical aberrations of the 3rd and 5th orders, it is useless to level out only the 3rd order aberrations, but it is effective to select the magnitude of the distortions so that their sum is minimal.
There are significant difficulties in this. Firstly, due to the phenomenon light dispersion The magnitude of spherical aberration for light of different wavelengths differs, which gives rise to the most malicious distortion for high-aperture optics - spherochromatism, that is, the chromatism of spherical aberration. Those fuzzy purple/green or blue/orange borders that everyone hates. To avoid them, the lens must have similar values ​​of spherical aberration for all wavelengths in the operating range.

Spherochromatism is a chromatic aberration of spherical aberration. I think you understand: all that remains is to attach the monitor.

Spherochromatism is a chromatic aberration of spherical aberration. I think you understand: all that remains is to attach the monitor.

In addition, the refraction of the edge and central rays of the incident beam of light in the lens occurs differently - otherwise there would be no aberrations at all. A single lens or objective pupil can be divided into a system of concentric rings similar to a shooting target, and in each selected ring the ratio of aberrations of different orders will be different. Consequently, it is possible to distinguish the so-called zones of the lens pupil in which one or another aberration dominates.

How can this be useful? First of all, consideration of the distribution of aberrations across the zones of the pupil helps to evaluate the change in the image with aperture of the lens, when the outermost, as a rule, the most “problematic” zones are “turned off” from operation. Adjusting the zonal distribution of spherical aberration is the key to controlling lens bokeh in calculations and is the most important way to balance image quality at full and limited apertures.

Longitudinal aberration of a real lens: classic case - Triplet F/2.8

The longitudinal aberration of a real lens can be represented as a graph of the dependence of the position of the focus of rays of a given wavelength on the pupil zone. Below is such a diagram with some additional explanations for the lens Triplet 78 / 2.8. Let's look at it in detail.

Longitudinal aberration diagram of the Triplet 78/2.8 lens.

Longitudinal aberration diagram of the Triplet 78/2.8 lens.

Drawing of the optical design of the Triplet DM-2 78/2.8 lens.

Drawing of the optical design of the Triplet DM-2 78/2.8 lens.

As you can see, the longitudinal aberration curves are a beam that can be divided into two sections: 1) a section of deviation to the left side of the diagram - pupil zone D from 0 to ~20 mm (F/4), 2) a section of deviation to the right side of the graph – pupil zone from ~20 (F/4) to ~30 mm (F/2.8). Deviation of the focus position towards negative values ​​is associated with undercorrection of distortion, and towards positive values ​​– overcorrection. It is very convenient to distinguish these aberrations by the shape of the defocus spots: blurring of the background in the form of a disk with a border is a sign of overcorrected spherical aberration, and the presence of a bright center in the bokeh disk is evidence of undercorrection of spherical aberration.

If we calculate the area of ​​the first and second zones, it turns out that in this lens the zone with undercorrected spherical aberration is 44% of the pupil area, and the overcorrected one is 56% of the pupil area. Taking into account the prevalence of positive deviation over negative, this means that the contribution of overcorrected aberrations at an open aperture will be significantly higher than undercorrected ones, that is, we can expect that the bokeh disk of a lens at an open aperture will have a bright border and bright, but to a much lesser extent , center. And indeed: at F/2.8 and at F/3, the bokeh disk on the optical axis in out-of-focus has the predicted appearance. When aperture is down to F/4, when the influence of the pupil zone with overcorrected spherical aberration is neutralized, only the bright center remains in the bokeh spot, while the border disappears. Similar changes occur with off-axis spots, although in this case other optical distortions also have a large influence.

The bokeh spot diagrams obtained using simulation (F/2.8-F/3) correspond well to those observed when shooting with this lens.

As follows from the above images, the lens in question suffers from pronounced spherochromatism: the beam width of the longitudinal aberration curves in the paraxial and outermost zones differs by a factor of 4. This does not make the image quality on a conventional camera catastrophically bad only due to the relatively low spectral sensitivity of the matrices in the most problematic range of 400-440 nm. Pronounced spherochromaticity leads to the appearance of uneven coloring of the bright border of the bokeh disk.

 

Due to spherochromatic aberration, the bokeh disc edge may appear multi-colored.

Due to spherochromatic aberration, the bokeh disc edge may appear multi-colored.

It is important to note that the pupil zone with a predominant contribution of undercorrected aberrations covers a larger range of apertures compared to the pupil zone with a predominant contribution of overcorrected aberrations. This means that a slight restriction of the aperture is enough to significantly improve image quality. Indeed: even when aperture is set to 1/3 EV, the size of the lens' aberration spots sharply decreases, and aperture by a full stop provides image quality in the central area close to optimal for this lens.

The decrease in the size of the halo around the spots when stopping down is associated with the elimination of the 20-30 mm pupil zone, which is responsible for the manifestation of pronounced overcorrected spherical aberrations. Thus, limiting the aperture contributes to an increase in image contrast at mid-frequencies of 10 and 30 lines/mm and an increase in the resolution of the lens.

Diagram of the frequency-contrast characteristics of the Triplet 78/2.8 lens at different aperture values.

Diagram of the frequency-contrast characteristics of the Triplet 78/2.8 lens at different aperture values.

Note that at D=0 mm the horizontal coordinate on the longitudinal aberration graph is not zero, that is, defocusing is used to compensate for distortion. In this case, when you aperture the lens, a shift in the focal plane (more precisely, the plane of the best setting) is observed, which is usually called “focus shift”. Lenses that suffer from focus shift are of limited use when shooting with rangefinder cameras and when using a jumping or preset aperture. The problem is especially acute when it comes to cameras with high-resolution sensors or cameras in medium and large formats. So, it was precisely because of the focus shift that in the 1950s the GDR completely abandoned equipping Pentacon Six cameras with lenses Tessar 80 / 2.8 in favor of more complex lenses Biometar 80 / 2.8.

What exactly are the spherical aberrations that play a decisive role in the formation of the pattern of the Triplet F/2.8 lens? To answer this question, it is necessary to calculate the values ​​of the terms of the Zernike series corresponding to longitudinal aberration for different lens apertures: Z4 (defocusing), Z9 (primary spherical aberration), Z16 (secondary spherical aberration) and Z25 (tertiary spherical aberration). Accounting for terms of a number of higher orders is not required for this lens: their values ​​are close to zero. The sum of the calculated values ​​is the total longitudinal aberration. The results are presented in the diagram below.

Visual representation of the expansion of longitudinal aberration into a series of Zernike polynomials.

Visual representation of the expansion of longitudinal aberration into a series of Zernike polynomials.

This lens uses compensation for primary spherical aberration at apertures below F/4 (D~20 mm) Z9 using Z4 defocus. At large values ​​of the relative aperture up to ~F/3 (D~25 mm), the primary spherical aberration is corrected due to the secondary one. Further, however, no mutual compensation of primary and secondary aberrations is observed. Moreover, tertiary distortions are already beginning to manifest themselves. Moving on to the terminology of Seidel aberrations, we can say that the characteristic image of this lens in the central image area is formed by spherical aberrations of 3, 5 and, to a much lesser extent, 7 orders; Moreover, their correction is not achieved for the pupil zone in the aperture range from ~F/3.5 to F/2.8.

Now let's briefly look at some more striking examples.

Lens with uncorrected spherical aberration - F/4.5 “periscope”

The simplest “periscope” lens, consisting of two positive lenses and used as a standard lens for the cheapest cameras at the beginning of the 20th century, is characterized by uncorrected spherical and chromatic aberrations. Like "monocle", lenses of this design are often used as soft drawing.

Drawing of the optical design of the Periscope lens 50/4.5.

Drawing of the optical design of the Periscope lens 50/4.5.

Diagrams of longitudinal aberration, aberration spots and bokeh disks for the lens I calculated with parameters 50/4.5 are shown below.

As can be seen from the longitudinal distortion graph, the lens has undercorrected spherical aberration in any zone of the pupil, as well as completely uncorrected chromatism, which is not surprising: there is not even a single negative lens in the design. The only way to control distortion in this lens is by stopping down. Thus, limiting the aperture by 1 stop leads to a twofold reduction in the size of aberration spots, which means an increase in resolution and contrast. The lens provides diffraction quality at a relative aperture of ~F/22: in this case, a resolution of ~50 lines/mm is achievable, determined by the length of the chromatic spectrum.

The background blur spots of such a lens do not have a clear edge, but have a bright center. The opposite situation is observed in foreground defocus spots, where the disks have a pronounced bright border. The calculated appearance of the bokeh spots of the periscope lens fully corresponds to that observed for a commercially produced projection lens "Glavuchtekhprom" 77/2. Examples of photographs taken with this lens are given below.

Lens with uncorrected spherical aberration of oblique beams - Helios-40 85/1.5

Fast lens Helios-40 85/1.5 very famous for its distinctive, expressive background blur. On the Internet you can find many discussions on the reasons for the origin of “that same” Helios-40 bokeh, and this time a comprehensive explanation will be given.

Drawing of the optical design of the Helios-40 85/1.5 lens.

Drawing of the optical design of the Helios-40 85/1.5 lens.

Diagrams of longitudinal aberration, aberration spots and bokeh disks for the Helios-40 lens are shown below.

Firstly, it is worth noting that on the optical axis, longitudinal distortions in the Helios-40 85/1.5 are corrected even better than in the Triplet 78/2.8: the lens has very well compensated (albeit slightly undercorrected) distortions in the pupil area up to ~F /2, as well as fairly low chromatism for the paraxial zone. In the pupil zone F/2-F/1.5, longitudinal aberrations are overcorrected, and pronounced spherochromatism is present. Therefore, aperture down to F/2 should make the Helios-40 a very sharp lens in the central area of ​​the image - which is confirmed by the calculated aberration spots.

When examining background defocus spots, you can find that for a point on the axis at an aperture of F/1.5, the spot has a small fringing, the appearance of which is due to aberrations of the pupil zone F/2-F/1.5. When the lens apertures down to F/2, the edge of the disk disappears. For off-axis spots, however, this fringing of spots is present at both F/1.5 and F/2, and becomes more pronounced the further the spot is from the center of the image. Despite the fact that due to vignetting (cutting of the pupil in front and behind the aperture diaphragm of the lens by the lens frames) off-axis spots have the shape of lemons, it is obvious that the observed border is completely symmetrical, and therefore is also associated with the manifestation of spherical aberration. Indeed, the aberration spots of the lens at F/1.5 also have an almost symmetrical halo, the larger the further from the axis the spots are.

This optical distortion is a spherical aberration of inclined beams - absent on the optical axis, but rapidly increasing with distance from it. If this aberration were not present in lenses such as Helios-40 85/1.5, Biotar 75/1.5, Helios-44 58/2, ОКС1-75-1 75/2, OKS4-75-1 75/2.8, OF-233 210/2.5, LOMO P-5, then their bokeh would not be “swirled” and expressive in the same manner. Other distortions (such as coma) can cause the background to swirl, but due to asymmetry they gives bokeh the appearance of scales, which is unusual for Helios-40. The spherical aberration of oblique beams has a significant impact on the image formed by the Helios-40 lens at apertures F/1.5 - F/2.8, which means that at F/2.8 the lens will no longer swirl the background.

Below are examples of photographs taken with the lens Helios-40 at different apertures.

Spherical aberrations in a long-focus apochromat of the APO Tair type of diffraction quality

In optical instruments such as telescopes with a large aperture and diffraction quality, the presence of spherical aberrations, especially for the outer zones of the pupil, is unacceptable, since it entails a significant deterioration in image quality. Let's consider a five-element astrograph lens 800/8, designed by me, made according to the “APO Tair” type scheme.

Drawing of the optical design of an APO Tair 800/8 lens.

Drawing of the optical design of an APO Tair 800/8 lens.

Characteristic properties of the graphics lens are given below.

As expected, both spherical and chromatic aberration. Spherochromatism can be said to be absent, and the extent of longitudinal distortion in the range of 400-700 nm at full aperture is 115 microns, which is 7 times less than for the above-mentioned Helios-40 at F/1.5. Due to this, the aberration spot of the lens in the central area of ​​the image fits into the size of the Airy disk, which means that the quality corresponds to diffraction. Lens resolution is ~130 lines/mm.

Almost complete correction of spherical aberrations results in the appearance of defocus spots being almost the same for the foreground and background. But can this be achieved in high-aperture optics?

Correction of spherochromatic aberrations in a particularly fast 150/1.4 telephoto lens

So, controlling longitudinal distortion in high-aperture lenses, as can be seen in the example of Helios-40 85/1.5, is a task that requires the use of more complex optical circuits and modern optical materials. Correction of monochromatic spherical aberration is achieved through the use of highly refractive glasses, and the chromatic difference can only be reduced through the use of materials with anomalous dispersion: fluorophosphate, phosphate and heavy phosphate crowns, heavy niobium and tantalum flints, special flints.

Sophisticated, high-quality, fast lenses are extremely useful in applications where the information content of the image is of primary value, and any optical distortion that degrades the quality is unacceptable. Astrophotography is a prime example of the application of such optics: high aperture useful for quick exposures, and aberrations must be well corrected so that the stars in the frame appear as dots. Thus, modern fast telephoto lenses are well suited for shooting star fields: Samyang 135/2, Canon 200 / 2.

Taking into account this well-known experience, in 2022 I had the idea to create an ultra-fast astrograph lens with parameters 150/1.4, the calculation of which was implemented Vladimir Bogdankov (at that time I did not know how to count optics), and the mechanical design was designed by Artyom Timirev - taking into account all the concepts and requirements I had put into it. The optical design of the lens was patented by us in 2024 (RU 2822998 C1).

Drawing of the optical design of the 150/1.4 astrograph lens (RU 2822998 C1).

Drawing of the optical design of the 150/1.4 astrograph lens (RU 2822998 C1).

The lens has a nine-element design using 4 low dispersion glass (fluorite type) lenses. Let's consider some of its characteristics in the context of the topic of this article.

Based on the graph of the frequency-contrast characteristic (MTF), it is clear that the lens has a high resolution at an open aperture - more than 100 lines/mm, which, for example, is unattainable for the Helios-40 at F/1.5. Aberration spots at full aperture have an indistinct spherochromatic halo in the center of the image and a small comatic tail at the edge of the field. When aperture is down to F/2, image quality becomes excellent both in the central area and across the field: both spherochromatism and coma disappear.

Indeed: the longitudinal aberration diagram of the lens indicates that in the pupil zone ~F/2-F/1.4 one can observe uncompensated spherical aberration, which is different for light of different wavelengths. Moreover, it is worth noting that even in the F/1.4 zone the extent of chromatic aberration is only 300 microns. For comparison: secondary spectrum Tair-3 300/4.5 at F/4.5 it is 1500 microns. Already at F/1.6-F/1.8, spherochromatism becomes insignificant. In the pupil area of ​​the lens up to ~F/1.8, excellent compensation for spherical aberration has been achieved: the bunch of curves on the longitudinal distortion graph does not go to the sides and does not bend anywhere. This is also important because due to this correction the lens does not have a focus shift, which is useful in astrophotography.

A high degree of correction of longitudinal distortion leads to the fact that the bokeh of the lens in the central area is completely neutral: the defocus spots of both the background and the foreground look identical - like those of the Tair APO discussed above. In the field, the same effect is achieved at F/1.6-F/1.8, which is evidence of the elimination of residual coma.

Conclusion

So, longitudinal distortion - a combination of spherical aberration and chromatism - has a decisive influence on the image quality of the lens and its pattern (bokeh). Correction of spherical aberration is carried out through mutual compensation of its components of different orders, which leads to several types of correction in the lens - their curves are presented in the figure below.

View of some probable spherical aberration curves: 1) uncorrected spherical aberration, 2) undercorrection of aberration for small apertures and overcorrection for large apertures, 3) overcorrection for an open aperture and almost complete compensation for any smaller apertures, 4) compensation for all apertures, 5) overcorrection for small apertures and undercorrection for large ones.

View of some probable spherical aberration curves: 1) uncorrected spherical aberration, 2) undercorrection of aberration for small apertures and overcorrection for large apertures, 3) overcorrection for an open aperture and almost complete compensation for any smaller apertures, 4) compensation for all apertures, 5) overcorrection for small apertures and undercorrection for large ones.

Diagrams of defocus spots for the curve of each of the types under consideration, indicated by numbers, are given below for an open aperture and when aperture is stopped by 1 stop.

These graphs take into account only monochromatic spherical aberration. In the case where spherochromatism is strongly expressed, it makes sense to consider the curves for significant wavelengths separately, understanding that the final appearance of the bokeh will be determined by the superposition of all aberration curves.

It should not be forgotten that many modern и the old lenses have an even more complex form of curves with a large number of zones of different types of correction of longitudinal aberration, which, however, does not prevent us from using the approach described in this article to their study.

It also follows from the analysis that in a complex optical system, by balancing aberrations of different orders, it is possible to quite flexibly control longitudinal distortions in each zone of the lens pupil. This means that when performing a calculation, if there are a sufficient number of variable design parameters, you can literally set the required lens pattern and control its bokeh, without greatly compromising the quality of the image as a whole. Thus, calculating the optical pattern is another task, in addition to ensuring the required level of quality, that must be performed by an optical engineer when creating a photographic lens.

 

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Comments: 10, on the topic: Analysis: spherical aberrations and lens design - from “periscope” to apochromat

  • Igor

    Difficult to understand)

  • Ernie

    That is, the notorious soft effect is caused solely by spherical aberration?
    Of course, I would like simpler explanations. Mathematics and optics are too specialized a field of knowledge. 90% of even the local audience do not even know the basics of these disciplines and for them/us it’s all just a bunch of words.

    • Rodion

      The soft effect in the central area of ​​the image is actually caused by SfA. This article provides an explanation of how lenses with software similar in degree of manifestation can have different bokeh patterns, for which the concept of zonal spherical aberration is considered.
      I am sure that you can easily find “simpler explanations” on the Internet almost anywhere. But you won’t even find anything in optics textbooks about the relationship between aberrations and lens design, since optics textbooks consider aberrations only in the context of image quality, nothing more. To make this text understandable, some basic knowledge is certainly needed. Half an hour or an hour of surfing Wikipedia will help.

  • Basil

    It's like about a gopher in the tundra... No one has seen it, but it exists...

  • Alex

    Rodion, thank you very much for the article. Although much is not clear, it is very interesting.
    It seems to me that it would be great if someday it would be possible to make a similar review in general about the design of the most popular modern lenses, with examples and explanations. For many beginning photographers, the term “lens design” remains somewhat incomprehensible. You manage to describe the “picture” in technical language.

    • Rodion

      Thanks for your feedback. I have a little trouble with modern lenses, but the idea is good. As a compromise, I’ll probably try to add to the analyzed examples analogues in the correction profile among more or less modern optics.

      • Alex

        I keep sketching out ideas :-)
        Conventionally, take photos from a cheap Canon RF 50/1.8 and an expensive Olympus 25/1.2 from some test and compare the “pictures” from the point of view of optical circuits. Canon gives less depth of field, Olympus gives “softer” blur. For the average person, it’s a reason to take a shower. Your favorite photo site gets a boost in traffic.

        • Rodion

          Drawing such conclusions from someone else's pictures alone is an activity suitable only as a mental exercise, but not for publication. At a minimum, it is necessary to analyze optical circuits using simulations so that the material is convincing and the conclusions are reliable. Otherwise, it will not be qualitatively different from the topics in the photo.

      • Alex

        And be sure to add the word “humiliation” to the title. Just kidding (not kidding)

  • Alexander

    Great article. The visual examples are simply great. Roman, thank you for your work and experiments! They always inspire.

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