© Erwin Puts
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Chapter 3:

Part 1: A gentle introduction to optical design and aberrations.

3.1 Optics and optical designers: ray bending.

The basic property of a lens is to converge light rays to a common focus. A burningglass will converge the energy from the sun to a very small and very hot spot. In fact 'focus' is Latin and means 'hearth', a nice reference to the heat that will burn a hole in a piece of paper. Never leave your Leica rangefinder-body in the sun with the lens uncapped: the sun could burn a hole in the shutter curtain! The light-bending property of a curved piece of glass has been known for ages. The single lens (singlet) was for ages the only instrument to be used for changing the path of the light ray.

Many substances, like ice, diamonds, quartz, turpentine and glass, have this property of ray bending or diverting the path of the ray. We can experimentally establish the amount of deflection and scientists have given it a name: index of refraction (n).

Already in 1621, the Dutch scientist Snell discovered a Law, which defines accurately how light is bent when it passes from one substance to another with different indices of refraction. The law is a very simple one: n1.sinè1 = n2.sinè2 It tells you in simple words that the angle (sinè1) with which a ray travels through a substance with a certain index of refraction (n1) is changed into a new angle (sinè2) which is directly related to the index of refraction of the second substance (n2). And in fact that is all there is to know about optics. Look at figure 1.1, which shows the paths of some rays when they pass through an early Summicron. At every air-glass boundary the path of the ray is diverted from its original course. Quite miraculously however all rays from the same object point do focus at the same image point. Or do they? In order to answer that question I have to introduce a few conventions that will help to understand the basics of optical design. Any lens is a piece of glass with a curved or plane surface. Most often the curvature is positive, that is outward. The front lens of the Summicron is an example. We also have plane surfaces, like the front lens of the current Elmarit-M 2.8/28mm or inwardly curving surfaces like the Summilux -M 1:1.4/35mm ASPH. Most lenses are circular too, which might be puzzling at first as the negative format is a rectangle. In fact you could produce a rectangular lens, matching the negative format, but it would be expensive to produce. Circular lenses with spherical or flat surfaces are the most cost effective pieces to manufacture. The curvature of the surface of a lens is part of a circle: the word 'spherical' means 'part of a sphere' and for a lens with two spherical surfaces, we have two centres of curvature. The line connecting these two points is called the optical axis. It is also the midpoint of the lens itself when viewed from the front.

 

Figure 56: 3.1.A: ray path

The Summicron lens, we will use as example has a focal length of inscribed 50mm, but in reality is 52mm. The focal length is the distance from the optical centre of the lens to a point on the optical axis where the rays converge that arrive from a very distant object point. On the distance ring of a lens you will notice the infinity setting ( 8). This location is theoretically at an infinite distance, which is a bit too far away for photographic purposes. There is some confusion what the real distance is of this indication. The optical definition is simple: when all the rays from a distant object point are parallel to each other ('collimated') when they enter the lens, we have what may be called optical infinity. It is more correct to say that all the rays enter the lens with the same angle of incidence, which is in fact the same. (but see later!).

In photographic practice, we can use this thumb of rule: at 100 x the focal length we have practical infinity and for wide angles we can use 5 meters as a rule. Image height of the negative. A lens projects a circular image of the object onto the focal plane.

The Leica format, as it was originally referred to, has dimensions of 24 x 36mm. A rectangle with these dimensions has a diameter of 43.2mm or a radius of 21.6mm. As a lens is symmetrical around the axis, it is customary to discuss only the area from centre to edge, which in this case has a distance from 0 to 21.6mm. As we will see later, the performance of a lens differs from centre to corner and so we need to establish a way to define or locate the positions of an object point on the image.

 

Figure 57: 3.1.B image height on CD2

The image height tells us how far from the centre an object point is located on the negative format. Leica uses increments of 3mm to locate image positions and the 0, 3, 6, 9, 12, 15, 18 and 21.6mm distances from the axis (centre) define the most important areas. The 12mm position as example defines the negative area at the 24mm (horizontal) line and the 18mm position gives the limit of the negative format at the vertical line (the negative area in the horizontal direction is 36mm). Zones of a lens. With these image heights we can define zones on the lens if we draw circles with a specific length of the radius. The on-axis position is evidently the '0' distance.

All other locations are off-axis or field positions. The zone from 18 to 21.6 mm is often called the marginal zone, the 9 to 15mm is the outer zone and the 3 to 9 is the most important zone, as it covers the portion of the negative area , where the important picture elements are normally located. Surface height of a lens While the diameter of the negative area has a fixed value (21.6mm) we can assign definite numbers to any position, as we did when discussing the image height. The diameter of a lens, however is not fixed and mostly smaller than the diameter of the negative or image. The front lens of the Summicron 2/50 has a radius of +18mm, but the Elmar 2.8/50 of 10mm.

 

Figure 58: 3.1.C surface height

Only lenses like the Noctilux-M 1/50 or the Apo-Summicron-R 2/180 have a radius larger than that of the image. (25mm and 45mm). If we need to specify the location on the surface of a lens where a certain ray of light enters the lens, we can only use relative positions, measured from the optical axis, or absolute numbers, like 6mm from the centre (op optical axis). If we want to define the position of a ray on the surface of the lens, we use the notion of surface height as a fraction of the total radius. A ray located at surface height 0.5 will be positioned halfway between the axis and the edge.

 

3.2 The origin of aberrations.

The rays from a distant object point strike the surface of the lens parallel to the optical axis and all rays should converge to one point only, as this is the exact image of the object point and at a location that is the geometrical equivalent of the location of the object point. The law of refraction tells us that any ray that arrives at a certain angle at the lens surface will be deflected. The angle is measured with respect to a strange thing called the normal, which in the case of a lens is the line drawn between the centre of the curvature and the surface height.

 

Figure 3.1.C shows that parallel rays striking the surface at different heights will have different angles of incidence and so also different angles of deflection.

One has to realize that we are used to seeing a few rays drawn from an object point through the lens. In reality rays of light are fantasy constructs: they do exist only in the mind of the designer. Light is a form of energy, consisting of photons. The object point that emits or reflects light is radiating a stream of photons in all directions, like an inflating balloon. The surface of the lens is bombarded by a barrage of photons from axis (centre) to rim (edge) and if we would draw the trajectory path of every photon stream, we get the familiar ray path. So the light energy of one single object point fills the whole surface area of the lens. Tracing rays would be hopeless as there are millions of rays that can be traced from one single object point. Happily we can simplify a bit. The surface of the lens is part of a sphere and the curvature is the same at every location. If we imagine a ray entering the lens at an image height of 0.9, that is at the outer edge of the lens, we only have to draw one ray. If we rotate the lens, the image height does not change, nor does the curvature and most importantly the angle of incidence. One ray is enough for all rays that enter the lens at that specific surface height. And all rays from the object point entering the surface of the lens at that location will converge to the same image point as all angles of incidence are the same. We do have to pay attention to a few rays only, that is rays that enter the lens at different surface heights in the same plane. If we draw a lens on a piece of paper, we say that the plane of the paper is the main section for which we will draw and calculate rays. Any ray will intersect (pass through) the optical axis and this plane is called the meridional plane.

 

Figure 59: 3.2.A angles (Figure to show different angles at different heights)

If we draw a second ray at a surface height of 0.5, we will notice that now the angle of incidence has changed and so will the amount of deflection of the ray. This ray will strike the optical axis at a different location and now we are in trouble! Very big trouble indeed. We have defined the image plane as the plane upon which the lens will project its image of the object points. It is located at a distance from the lens where the rays close to the optical axis converge. All other rays that cross the optical axis before this plane will intersect the image plane above and below the central point. Instead of a very small point of light, we see on the image plane a patch of light with a higher core intensity and surrounded by rings of d nishing intensity. This representation of the point light source by a bright spot surrounded by flare is the basic aberration of a spherical lens. It is known as spherical aberration or Offnungsfehler.

 

Figure 60: 3.2.B spot diagram

This behavior of a lens was known to everybody engaged in optical constructions, but the craftsmen building telescopes and other instruments, lacked the knowledge to do anything about it. As long as the angle of view was quite small, as it can be with a telescope, only the rays close to the axis were used and an acceptable image could be produced. When Daguerre in 1839 invented the first useful photographic process, he needed a lens with a much wider angle of view. In 1812, Wollaston had, by experiment, found a solution to the demand for a lens with a wide field of view, by employing a singlet meniscus lens and an aperture stop in front of the lens, but at a substantial distance. He may be the first person to use the location of the aperture stop as a design element. Berek did the same a century later with his Elmar design.

So Daguerre at first used a lens of the Wollaston type. But he ran into another problem. In those days, the photographer would focus by moving the lens closer to or farther away from the film plane and observing the sharpness of the image on the ground glass screen. The best focus as established visually departed significantly from the best focus needed for the Daguerro-type plates. The emulsion was sensitive to the blue-violet region of the spectrum, but the human eye is more sensitive to the green-yellow region. This phenomenon is the well-known dispersion: the refractive index changes with the wavelength. We have all seen the rainbow which is a clear example of the fact that white light is decomposed into the several colours of the spectrum from ultraviolet and blue to red and infrared. We have monochromatic rays, that are refracted (deflected) depending on the surface height, but now we have to accept that rays with a different wavelength, but at the same surface height will be deflected at different angles, due to different wavelengths. The red rays are refracted less than the blue rays. And if the image plane would be located (as a compromise) between the blue and red extremes, the white point now is visible as a spot with a green-yellow small core and a magenta soft ring (red + blue). The ideal of an image point becomes more and more an enigma. The bending of the different colours (wavelengths) to different focal points is a characteristic of the glass, some glass types do deflect colours more than others do. The amount of this change in direction is indicated by the index of refraction and because this index depends on the wavelength (ë) of light, we can describe a certain glass with a series of numbers.

Visible light has a spectral range from 400 nm (violet light) to 760nm (red light) and in fact there are an infinite number of wavelengths in this range. It is customary to use a selection of wavelengths to indicate the bandwidth of refractive indices. See table in Chapter 1.2, apochromats A certain glass type (as example Schott BK7) will have a refractive index for the 435 wavelength of 1.52668 and for the 706 wavelength an index of 1.51289. How close these rays will focus on the optical axis can be seen from the picture which shows an actual computation with these data.

Optical designers are indeed working with very small dimensions in the micrometer space where a millimetre is a vast distance. Compare the values for these wavelengths with another Schott glass, like SF2: 1.67249 and 1.63902. This variation of refractive index with wavelength is called dispersion and we saw that the range is different for each type of glass. The amount of change can be as big as 4%. Ernst Abbe found a formula to indicate the magnitude of this change and the Abbe-Number is the result.

A number higher than 50 (or 55) indicates a low dispersion and glass that has a number is this range is called flint glass. A lower number (below 55 or 50) indicates high dispersion and this glass is called crown glass. There are, as usual, exceptions to this classification. The correction of chromatic errors then is first and for all a matter of suitable selection of glass types. When we combine two glass types with dispersion characteristics that complement each other, we can design a lens with a good colour correction. Such lenses are called achromatic (achromat = a-chromat = non-colour) doublets and the first one was made in 1729 already. It is a very important class of lenses and we can find them as telescopes, microscopes, magnifiers, eye-pieces and several other types of optical instruments. Two lens-elements however do not correct all of the chromatic errors. There is a small rest of errors left and this is called the secondary spectrum.

 

Figure 61: 3.2.C monochrome

The designer can choose two wavelengths to focus to the same location, but all other wavelengths (colours) will be closer or farther away from that point. This focal shift is smaller than the focal shift produced by the spherical aberration. Remember that in that case the rays from the outer zones focus closer to the image plane than rays near the optical axis. Stopping down will exclude the rays from the outer zones from image formation and so the focus plane will shift. Here we note one of the most troublesome aspects of optical designers. You can correct one aberration to a high degree and another one pops up. Telescopes and other instruments were mainly employed to look at objects close to the centre and so another disturbing aberration did not get much attention. It is clear that a curved surface of a lens will produce an image that is curved too. The classical box camera acknowledged this fact by using a curved negative to counter this effect. But any flat image plane will be plagued by this curvature of field. Let us return to Daguerre, who asked many professors and craftsmen to design for him a lens for his camera with the required flat image and wide angle of field. No one, however could, and now experimental knowledge and tradition failed. Optical theory and mathematical methods were not en vogue then, even if the laws of optics (refraction, dispersion etc) were known for some time. The first person who computed a lens on paper, without any experimentation with the real glass, was Joseph Petzval, and his lens was clearly superior to anything then available. With Snell's law and the knowledge of the refractive index of a glass, we can compute the angle of deflection of a ray that strikes the lens at a certain surface height. Knowing this we also can locate the surface height where the ray will hit the next surface and so on till the image plane. The computation of angles however needs a lot of trigonometric number juggling and as the numbers are very small a high accuracy of maybe 3 to 5 positions behind the decimal point is required. We can compute to any required accuracy the exact location on the image plane of a ray traced through the lens system. But an awful lot of equations and logarithmic tables is a must. And then we may imagine that an experimental approach, trying this glass and that shape in order to get a decent result, was very tempting. But while Petzval could compute the path of the ray through a lens, he did not exactly understand why this path sometimes deviated from the expected course. He had a good understanding of the optical aberrations and he specifically gave attention to this curvature of field. The theoretical 'curvature' for a flat image plane is of course zero and we call this construct still the Petzval curvature.

 

3.3 The primary aberrations.

In 1856, Ludwig von Seidel made the first comprehensive study of the aberrations and he was the first to establish a theory of image formation. Aberration is from the Latin words áb' (from) and érrare' (to stray), meaning 'to stray from the right path'.

Seidel formulated his theory, based on an analysis of the principles and problems of image formation, as aberrations are fundamental design shortcomings. He identified 5 monochromatic aberrations and two chromatic ones. The monochromatic ones are the familiar spherical aberration, coma, astigmatism, field curvature and distortion.

The chromatic ones are the longitudinal and lateral chromatic aberrations. These seven aberrations all work together in an optical system do degrade the image quality.

They can be grouped in a different way to make them more understandable. We have sharpness errors: SA, C and A, positioning errors: FC and D and chromatic errors: longitudinal and lateral CA. We should note here already that image degradation in an optical system is the sum of optical and mechanical errors, the latter ones being quite nasty. The ideal (theoretical) image quality will be found on or near the optical axis, as in this region the curvature of the lens is not a problem. All rays that enter the lens close to the axis will have almost identical angles of incidence and so their refraction will be identical too and all rays from one object point will converge to one and only one corresponding image point. These two points, the object point and the image point, that are intimately related are called 'conjugates'. The plane of focus where all rays close to the axis, converge is known as the Gaussian plane or par-axial plane. Gauss studied the theory of image formation from the ideal perspective, assuming that there are no aberrations involved. Indeed, around the axis we have surprisingly few optical errors. This may explain, by the way, why older Leica lenses, and certainly the first ones (Elmax, Elmar, Summar) had such good performance in the centre of the image.

 

3.3.1 Spherical aberration

We already have met the spherical aberration. Rays that pass the lens at the outer zones, all focus closer to the lens than do the rays that pass along the axis. From the figure we can infer that all rays from an object point enter the lens as a cone of light, with the tip pointed at the focus plane. Due to spherical aberration, we do not have a really sharp pointed tip of the cone, but we have a small bundle of rays with a certain diameter, that passes the image plane.

 

Figure 62: 3.3.1. A and B caustics

We now that the ideal plane is the Gaussian plane, but in this case only the rays close to the optical axis are sharply focused. All the others will be unfocused and produce the familiar rims of light of diminishing brightness around the central core. If we would locate the image plane closer to the lens, the rays from the outer zones will be sharply focused, but now the axial rays will produce blurred spots. A compromise has to be made and here the designer has to be very careful. If the photographer stops down the lens, the outer rays are cut off and now the cone of light is more like the ideal shape, but the focus plane has more depth. Incidentally, this is the effect of extending the depth of field when stopping down.

Figure : colour slide 1a: spherical aberration.

So if the designer would select a location for the image plane close to the lens, to assure that the outer rays (for wide open performance) will be focused correctly, he might find that stopped down the result s unsatisfactory. The skill of the designer is needed here. As a numerical example: assume that the core of the image point at the Gaussian plane has a diameter of 0.02mm, with a halo rim around it of 0.08mm. The total dimension for the spot would be 0.1mm. At another location, very slightly shifted away from the original, we have a core of 0.025mm, a bit larger, but now the rim is reduced to 0.04mm, the total now being 0.065mm. In the first situation, the resolution would be a slightly higher, in the second one the contrast would be better.

Overall, the second location would be optimized for photographic purposes.

Mechanically the demands are very high here. The image plane is the film plane which is defined by the register between the film guide rails and the bayonet flange.

A difference of a few hundreds of a millimetre in the location of the actual film plane might reduce the theoretical image performance substantially. Leica indeed uses an accuracy of 0.01mm when machining and assembling the Leica bodies and an even higher one in the assembly of lenses. The manufacturing tolerances need to be as small as to support the quality of the lens design.

 

3.3.2 Coma Spherical aberration

Coma spherical aberration affects the object points around the optical axis (the centre of the image). Object points farther away from the axis, will enter the lens in a skewed (asymmetrical) fashion. The bundle of rays has still the shape of a cone, but the rays will transverse the lens obliquely. This generates the second monochromatic error: coma. Coma is in fact the skew version of the spherical aberration and is also called asymmetry error. The rays that strike the glass surface will have different angels of incidence, because the bundle of rays is skewed. In the illustration you can see that the rays coming in from below are bent more strongly than the upper rays.

 

Figure 63: 3.3.2.A coma (picture of coma)

Note also that the rays from below are fanned out over a larger area than the rays from above, producing the familiar coma-shape of a bright central core and a fanned out triangular tail. This aberration becomes larger in the outer zones of the image. In a picture, coma is not only visible when bright light spots are being imaged, but also as a reduction of contrast in the outer areas of the image.

 

Figure 64: 3.3.2.B coma spots

Figure to show coma over the area.

 

Figure 65: colour slide 2 (grey slide): coma

 

3.3.3 Astigmatism

This cone of light that strikes the lens asymmetrically, introduces yet another aberration. It is related to the field curvature, that we already encountered. It is astigmatism. The Greek word 'stigma' means 'point' and a-stigmatism means notpoint.

If a lens has no astigmatism and also a flat field, we call it an Anastigmat (notnot- point) and this type of lens was fitted to the first Leica bodies. The oblique cone of light will be intersected by the curvature of the surface of the lens. If you look at a circular lens from the front, you will see s full circle. If you now rotate the lens in a vertical direction, it will become an ellipse, like a cat's eye. It is clear that rays entering the lens in the vertical direction, the longer one, will have different angles of inclination than rays entering in the horizontal (the shorter direction). Geographers who had measured the earth and put a grid on its surface for navigation purposes, call the lines from pole to pole 'meridional' lines and the lines parallel to the equator do not have a particular designation. In optics, these 'equator' lines, which run perpendicular to the meridional lines, are called sagittal lines and the meridional lines are alternatively called tangential (but often meridional as well). The bundle of rays in the meridional plane focus to a different location than do the rays in the sagittal plane. They do not form a point of light, but a line, one laying in front of the other and mutually perpendicular. All object points that have a tangential focus, are located on a tangential plane, which is not flat but has the shape of a parabola or ellipsoid. In the same pattern, we also have a sagittal plane, which is curved as well. At the optical axis, these surfaces coincide, but they diverge quite significantly in the outer zones.

This aberration is difficult to correct and progress had to wait till 1890, when Schott introduced new glass types.

 

Figure 66: colour slide 3a: astigmatism

 

Figure 67: 3.3.3.A astigmatism

 

3.3.4 Field curvature

Field curvature is an easily observed phenomenon. Most projector lenses can demonstrate it. Focus the slide in the middle of the image and the edges will be blurred or when you focus sharply at the edges, the centre will be blurred. Any lens has one focal length, as the focal point is related to the curvature. Assume that a lens has a focal length of 50mm. If we have an object point, close to the optical axis it will be focused at 50mm distance from the lens. If we now take an object point at the periphery of the image, this point will also focus at a distance of 50mm. But the distance from the lens to the edge of the negative is longer than to the centre of the negative, as some simple geometry will show. The image space is not flat, as is the emulsion plane, but an arc of a circle (in this case with radius 50mm) and the resulting image surface has the shape of a saucer, or more scientifically a paraboloid shape. The fact that the rays from the lens to the edge of the negative area have to travel a longer distance, is also illustrated by the phenomenon of light fall-off or vignetting: the darkening of the edges of an image is due to the lesser amount of light reaching the corners of the image. The sum of both astigmatism and field curvature is the occurrence of three different shapes where the mage is formed and it results in severe blurring of the overall picture in the zones of the image. In a laboratory setting these three may be made visible and the optical designer has to battle with them all. Photographically we only see one result: loss of contrast, and blurring of details. If only field curvature would be present and we use an R-Leica, you should be able to focus on a subject in the centre of the image and note a blurring of the edges. If you now focus on the outer zones, you see the image there progressively becoming sharper, with the centre now becoming blurred. Such an extreme manifestation you will not encounter easily, but with some very wide angle lenses it may be observable.

 

3.3.5 Distortion

All of these aberrations are sharpness errors that diminish the sharpness and contrast of the image. But there are other aberrations that only affect the shape of the image, even if the image points were absolutely sharp. This, the fifth aberration is called distortion. An optical system always depicts an object in a specific size. A 50 mm lens focused at 10 meters (32'10') reduces every object by a factor of 200. But one can expect that the reduced image is geometrically accurate and that the ratio of reduction remains constant across the entire image. This is called scale fidelity.

Unfortunately this is not the case with most lenses, because the reduction scale varies within the image area. When the scale increases as the distance from the centre of the image increases, the result is pincushion distortion. When the scale is reduced towards he edges of the image, we get barrel-shaped distortion.

 

3.3.6 Chromatic aberrations

These five aberrations are called monochromatic aberrations because they act on a single wavelength. Because light diffraction (colour dispersion) is not the same for blue light as that for red light, these colours are refracted differently. Blue, for instance, is bent more sharply than red light and they converge on different focal points. If we place the image plane in the middle between these two focal points, we will see a green (or yellow) core with a purple fringe (red plus blue). If we shift the image plane, the colour of the fringe will change from blue to red or vice-versa. This imaging error is called longitudinal chromatic aberration and just like spherical aberration, it causes the image to appear flat because it reduces contrast. With this chromatic error the image plane will be in a different place for each wavelength. The dispersion of the glass will also cause a change in the size of the colour image in each wavelength. Because short-wave light (blue) is refracted more strongly, blue rays will converge at a closer focal point. The effect is similar to that of a lens with a short focal length, which depicts objects at a reduced scale. The focal length is linked to the magnification factor, and that is why a variation in the refractive index also causes a variation in magnification. This error is called lateral chromatic aberration and it mostly affects the reproduction of fine structures. A white image point is separated into its component colours and reproduced as a stretched rainbow. A dark point with a light background is reproduced with a colour fringe that appears in blue on the upper rim and in red on the lower rim. Aberrations are often reduced when the lens is stopped down, because marginal rays no longer contribute to the imaging errors. Lateral chromatic aberration is not diminished by stopping down the aperture and it is very difficult to correct. The chromatic aberrations (lateral and longitudinal) increase from the centre of the image towards its edges. (Here the colour plates from Solms)

Figure 68: colour slides 4a, 4b, 4c 4a: achromatic correction type 1. 4b;: achromatic correction type b, 4c: apochromatic correction

 

3.4 Higher Aberrations.

I noted earlier that rays close to the optical axis are, basically, aberration free and that form an image in the Gaussian space. As soon as the angle of view becomes larger, as with the Daguerre lens, a first group of aberrations emerges, which are called the Seidel aberrations. There is a kind of law here, which tells you that the wider the angle of view and/or the greater the aperture, there will be more and more severe aberrations. You will see, when I discuss the evolution of the Leica lens, that high aperture lenses and wide angle lenses, or worse a combination of wide angle and high aperture, as in the case of the Summilux 1:1.4/35mm are quite challenging for the optical designer. There is a logical order in the levels of aberrations. Remember that the index of refraction is at the bottom of all aberrations and calculations are based on the sine of the refracted angle. The sine function can also be expressed as a geometric series: sin p = p - p3/3! + p5/5! - p7/7! + . Each term in this series is related to a group of aberrations of a certain order. The first term represents an error-free image, as it occurs in the centre of a picture. This very small area around the optical axis is called the paraxial region. It is customary to associate it with aberrations of the first order or Gaussian errors (two Seidel aberrations). The next term incorporates the number '3' and it is therefore related to the aberrations of the third order (the five Seidel aberrations). Because this series only contains uneven numbers, the next term involves aberrations of the fifth order (the nine Schwarzschild aberrations), the seventh order (14 different aberrations without names), and so on. For many photographic lenses, the correction of the third-order aberrations, will result in very fine imagery. Many Leica lenses however are now on such a level of sophistication, that fifth- and seventh- aberrations need to be corrected, or more accurate need to be balanced.

 

3.5 The analytical approach in designing lenses.

Now that we have identified the aberrations or image degrading components of an optical system, I will give a short sketch how a lens designer tries to correct and eliminate them. For all rays close to the axis (the Gaussian space) the equations are relatively simple and the calculations can be based on the two-dimensional trigonometric concepts of sine and cosine. While tedious and cumbersome with log tables , theses calculations are accurate. As soon as we leave this space and widen our field or aperture, the oblique rays are becoming more important and we need solid geometry and three dimensional trigonometric formulae to calculate the rays with good accuracy. Such calculations however were beyond the abilities of the designers, and if they were able to trace an oblique ray, it took too much time. So these rays were not calculated at all or only a few or approximations were used. The Seidel theory of aberrations used approximations to approach the exact values. The value of this method was its relative simplicity and good accuracy. The designer had incomplete knowledge of the exact state of the correction of the aberrations. He employed approximation formulae and in addition to this, he had to use his full experience, and knowledge and creativity to find the desired solution. In those days the chief designer at Leitz would employ a whole group of (up to twenty) calculators, often women, because of their great accuracy and reliability, who would do part of the calculations and pass the intermediate results on the next in the line. At the end of the session, the chief designer would evaluate the results and decide on the next step. The design of a lens could take years to complete and naturally one would be reluctant to start all over again, if the end-result was not satisfactory. The final test for the performance of a lens, was the finished prototype and the actual photographs taken with this lens. Leitz used test equipment too for the analysis of the lens performance. While the designer had a very clear notion of the image quality of the lens, he was not sure of its real performance before the prototype was finished. And then some unpleasant surprises might be discovered. The performance was less than expected or the mechanical department might send a message that they were unable to build the lens, because of the too small tolerances for the lens elements or the mounting. In such cases a new design was developed or some compromise solution had to be found. Before the introduction of computers and, even more importantly, the use of optical design programs, the analytical method was the only viable option.

An experienced computer, (as these persons doing calculations were called in the past) needed two to three months to calculate a sufficient number of ray traces through an only mildly complex optical system, like a triplet. It is understandable that approximations were used and that very complicated calculations were simply omitted. The resulting optical design showed inadequate knowledge of the exact extent of optical aberrations. Still, one has to recognize that these approximations helped the designers to determine the characteristics of many aberrations, and their experience constitutes valuable background for today's optical designers at Leica. It was not easy to optimize a design. The algebraic equations that describe the aberrations are non-linear, which means that they cannot be predicted by solving the equation. The exact calculation of all ray traces through the lens system, is the proper way to analyze and optimize a lens design. A successful design required much creativity and a very sensitive grasp at the effects of aberrations. When one looks at some of the older designs today, one is compelled to admire the achievements. An unbiased evaluation with modern instruments shows that many of these famous designs often lack refinements, but that they do have a worthy character. This method of lens design, using calculations, approximation formulae and experience, was the only one that could be used until the widespread introduction of the computer and design programs, which could calculate faster and also handle the oblique rays.

 

3.6 The numerical approach in designing lenses.

With the introduction of computers, the limitations of optical calculations were lifted, so that the (more exact) numerical method could now be employed to full advantage. Numerical methods can be used to achieve better control of important aberrations and they can also be used to optimize an optical system. This wealth of information can also entail its own problems. Did anyone ever tell you that the task of an optical designer nowadays is easy? The magnitude of the optical designer's task can be illustrated quite forcefully. A lens element is characterized by a few basic properties, as glass type, curvature of surface, thickness, and the distance to the next elements. They are also known as parameters, that is properties that can change in value and magnitude. As the designer is free to change any one of these values, they are aptly referred to as degrees of freedom. It can be shown that every degree of freedom can be used to correct only one aberration. As a simple example: the degree of spherical aberration can be changed by the bending of the lens, that is changing its surface curvature. And chromatic aberrations can be addressed by selection of suitable glass compositions. A six-element 50 mm f/2 Summicron lens has 10 air-toglass surfaces and curvatures, six thicknesses (one per lens element) and four distances between elements. In addition, each type of glass has a refractive index and a dispersion number. The exact position of the iris diaphragm must also be determined. With these 36 degrees of freedom, the designer has to correct more than 60 (!) different aberrations. Every parameter can have approximately 10,000 distinct values and more than 6,000 different ray paths have to be computed for every change in a parameter. The 36 degrees of freedom also are not fully independent.

Some need to be combined, and some are tightly constrained by other parameters.

Thus the 36 degrees of freedom are in fact reduced to only 20, making the task even more complicated. Given the specified conditions and considerations, it is not surprising that hundreds, if not thousands of designs can be generated that are very close to the desired solution. It has been estimated that a complete evaluation of all possible variants of the six-element Summicron design, using high speed computers that calculate ray traces at a rate of 100,000 surfaces per second, would require 1099 years! That is obviously impossible. In order to select the best design from this virtually infinite number of possibilities, the designer needs a lot of creativity and insight in the true nature and character of a design. The dilemma is now clear. The analytical method offers understanding and insight about a design, but its solution is an approximation of the required state of correction. The numerical method gives the true state of the correction, but the computer program has to consider thousands of possibilities without any guidance. And even an optimization program will be fooled.

 

Figure 69: slides 5 and 6 show aberrations together: 5 = colour astigmatism and 6 is coma + astigmatism.

Berek was very well aware of this 'conflict of interests' and in his book from 1930 he presents both approaches and expresses his preference for the analytical method, as it fosters understanding. He is aware of course of the limitations and gives many rules for the analysis of the magnitude of aberrations. He also proposes to do some exact ray trace calculations, when the problems of a design do require more study.

 

3.7 The designer at work.

Berek notes in his book, that the triplet, as designed by Taylor (1895), is a very interesting design, as it will correct all seven Seidel aberrations with a minimum of effort. This example is very important, because it demonstrates how an optical designer goes about his task and why creativity still plays such a large and decisive role in that task. The seven aberrations can be corrected with a minimum of eight independent system parameters (degrees of freedom). (The focal length also has to be taken into account). A triplet (a three-element lens) normally consists of two collective outside elements (crown glass) and one inside dispersive element (flint glass). That results in six curvatures and two separating distances between the three elements, giving eight degrees of freedom. At the beginning, the designer selects basic system parameters, such as type of glass, element thickness, distances between elements, and curvatures. The designer now has six surfaces to play with, and he or she can now calculate the amount and kind of aberrations that each surface contributes. As an example, we can establish (in a very simplified manner) that in the case of the triplet, the radius of the second surface (of the first lens element) contributes spherical aberration and chromatic aberration, and that the radius of the third surface contributes coma and astigmatism. The optical designer must now decide how to correct these aberrations. He might try to change the curvature of the first lens in such a way as to reduce spherical aberration. But the curvature also determines the focal length, which should not be changed. It may also happen that a change in the curvature will reduce spherical aberration, but that the amount of coma will simultaneously increase. The designer may also choose to distribute the correction over several system parameters in order to reduce the likelihood of increasing other aberrations. It is dangerous to use one single system parameter for the correction of one particular aberration as fully as possible. It might happen that the construction department cannot manufacture this very parameter within established tolerances. And then the whole system will be out of balance. But let us return to the correction of aberrations. The optical designer will continue to alter system parameters until the correction of the seven aberrations has reached a level where residual imaging errors are very small. The designer will also strive to correct each aberration by using several degrees of freedom at the same time. The 'burden' of correction will then be distributed over several surfaces and the entire system will appear more balanced. The designer can select the types of glass and the curvatures within certain limits, but each combination will result in a different kind of overall correction. Berek remarks that even the simple triplet has so many possibilities, that is would be almost impossible for two designers to find exactly the same solution. If we reflect on a classical seven-element system, like the first Summicron or the Summilux, we may get a feeling for the enormous complexity of the task of a designer to find the solution that is desired. This state of affairs explains the large variety of characteristics of lenses, that look superficially the same. Here I want to caution the reader not to try to extract too much information from published lens diagrams, as they lack vital information for a true appreciation of the art. Two lens diagrams may be very different, yet quite close in performance and character and two others may be superficially almost identical, but with markedly different characteristics. The Elmar lens, is in fact a triplet, where the last element is replaced by a cemented doublet. Using the available standard glass, the designer may encounter restrictions in required dispersion characteristics of a glass type. When cementing two different glasses, a new glass type is constructed. A cemented doublet might also be employed to get a wider aperture or a slightly better correction.

 

3.8 Max Berek (1886 - 1949).

The success story of the Leica camera (at first introduced as Leica-camera), had not been possible without the genius and persistence of Oskar Barnack and the daring decision of Ernst Leitz to go ahead with the production of the Leica, against the strong negative advice of his managers. Without the optical computations of Max Berek however, the potential of the Leica might have been impossible to exploit. He was a modest man, who loved to work at night in the quiet of his room, where he would sit at his desk with a pot of tea and smoking cigars, pondering questions of optical nature. Max was born on August 16, 1886 in the small town Ratibor as son of a mill worker. As so many of his contemporaries in the late part of the 19th century, when Germany experienced a cultural and scientific explosion, he went to the university to expand his knowledge. He started his study in mathematics and mineralogy in Berlin in 1907 and finished there in 1911 with a famous crystallographic research. He worked from 1912 till his death in the Leitz company.

He made important contributions to the design of and measurement techniques for polarization microscopes. The 'Berek-compensator'and the '-prism' are well known concepts even today in their field, as is the formula to compute depth of field of microscopic vision). The first Leitz lens for the Leica, the Anastigmat, had been designed by , and a long list of lenses for the Leica followed, 23 in total, including the Elmar 1:3.5/35 and the Elmar 1:4.5/135 and his last one, the Summarex 1.5/85. He received a personal price, the Grand Prix in 1937 at the Paris World Fair for his accomplishments. Up till now Leica has produced for the rangefinder system about 65 different lenses. Berek alone accounts for more than 35% of all Leica rf-lenses and his design considerations still can be noted in today's designs. He wrote a major study about optical design, called 'Grundlagen der praktischen Optik' (Untertitel 'Analyse und Synthese optischer Systeme') or 'Fundamentals of practical optics (subtitle: Analysis and Synthesis of optical systems). It was published in 1930 and many reprints were made till 1986, when the last version was printed. The book is still very interesting for its approach and contents. Berek was very well versed in the flute and played in many chamber music sessions. Given the close relationship between optical and sound-waves, his accomplishments in both areas are not surprising.

 

3.9 Erwin Lihotzky (1887 - 1941)

From the same generation as Berek, Lihotzky was born in Vienna within a upperclass family. He studied engineering and specifically railway design. In his spare time he studied optics and made some remarkable observations. He discovered, that with wide aperture lenses the rays from a point will not converge to a common point (a focus) , but they will be many foci, that will describe a surface, known as the caustic surface. Imagine that the converging rays form a tunnel of light, that becomes quite narrow over a small length and then widens again, just as when the light is passing through a funnel. At some point where the mouth of the funnel is very narrow, the designer will locate the focal plane, as the converging rays are close together, and will represent the object point as a small circle of light. Lihotzky designed a method with the designer could always determine the status of the caustic surface. In 1919 he made improvements on the theories described by Petzval and Fraunhofer with his seminal study: 'Generalisation of the Sine -condition of Abbe (as a prerequisite for the disappearance of coma close to the optical axis) for optical systems with longitudinal aberrations'. In this work he formulated the so-called isoplanatic condition, which even today is part of the designer's toolbox. did read the book (long titles in those days were not a deterrent) and invited Lihotzky to join him at Leitz, what happened in 1920. At Leitz he improved the viewing systems for microscopes, designed the illumination system for the Leitz enlargers, projectors and became in 1934 the head of the department for micro-optics. Here he systematized the design of optical systems in several optical groups, and the designation of all new microscope lenses starts with an 'L', which stands for Lihotzky. All camera lenses start with a 'B' for Berek.

 

3.10 The current design team and design method.

Some snapshots from the past will help you to understand the vast differences in design environment since Berek's days. Some aspects have not changed surprisingly.

The basics of ray tracing, the study of the aberrations, the general approach to the design process are comparable, but as usual, the most far-reaching differences are can only be detected in the subtle details. Berek designed his Elmax, working alone, often at night with a pot of tea and smoking a cigar, on a piece of paper, tracing rays with log tables and using the Seidel coefficients to optimize his creation. In the thirties, mechanical calculating machines were introduced and the designer now employed a staff of people to work out calculations, while he could concentrate on the work to correct the aberrations. In the fifties the electronic calculators speeded up the calculations with a factor 2 and the chief designer would delegate more of the work and check the progress. With the advent of the computer and the optical design program, both methods will work. One team can design a lens, or one individual person can. Leica has opted for the latter approach.

If you look at what they are doing, you might think you are in a average computer department. Every person sits in front of a very large monitor and on the desk you will see a number of print outs of long rows of figures and graphs and diagrams.

Looking a bit more closely you will note a most remarkable phenomenon. Every person is studying and analyzing his own lens. If you would need to single out one important characteristic of Leica lens design, it would be this: a Leica lens is the work of one creative individual. Nowadays we have computer programs that can design a lens almost without human intervention. This is a bit exaggerating of course, but you get the point. Consider this: when we take a photograph of an object (let us say a portrait) this face will be recorded on film and every point in the face (eyelashes, pupils etc.) will have an equivalent point on the film. In a lens without aberrations, we have an exact replica to the smallest detail. In a real lens we have some aberrations, which means that the position and shape of an image point is not recorded in the same position and with the same shape on the film. We can calculate these differences and we can even identify the causes for these differences. This is not a new science. The designers from the 'thirties could do it too. They just lacked the time and the computing power. If we start the computer program we can specify the location of a point in object space and let the computer calculate the corresponding position of that point on the film plane. Repeat this a number of times and we get a clear view of the image. Then we ask the computer to give us the values of the aberrations detected. This is a list of figures and specifies that spherical aberration has a value of 0.0003 and coma has a value of 0.0457. (Ideally all these values would be zero). You could then tell the computer: please, re-arrange the lens elements in the design such that the coma value is halved, meaning of course less coma in the design. So a team of designers could produce a lens design in shared tasks. Everyone could do a part of the design and later it could be put together. This is how (as example) some Japanese design teams function. A lens design has a character of its own. Some designs are more promising, some are more elegant in its choice of glass or shape of glass elements and more. It is the difference between a novel written by one writer or a novel written by a collective. The latter one is certainly readable and enjoyable, the former one shows more genius and cohesion, because its concept is linked to that individuals creativity. When a Leica designer starts with a lens design, (s)he often starts with a blank piece of paper. You may not have thought about it, but the physical dimensions (diameter of bayonet throat, length of the lens, thickness of the lens) already determine many characteristics of the lens. (S)he will study the potential of the design, its requirements, its imaging capabilities etc., partly from scratch, partly from own experience and partly from the archives of Leica, where many designs and studies are kept that have been done in the past.

Why is this initial stage of free creativity so important.? In lens design we encounter any number of aberrations we need to correct to make sure we get good image quality on the film. These aberrations can be grouped into several classes: the third order aberrations and fifth order aberrations are the ones the designers will encounter when creating lenses for photographic purposes. The wide aperture of current Leica lenses make the correction of these fifth order aberrations mandatory to get excellent image quality. When a designer has corrected the third order aberrations, (s)he will encounter the fifth-order ones and while you can not correct these fully, the designer has to allow for some third order aberrations to be used as a balance against the higher order errors. So any designer will study a design to see how well it can cope with the corrections needed for fifth order aberrations. At the start however (s)he has not a good idea of how these aberrations will materialize.

Now this looks a bit hopeless: you have to account for trouble you do not know about. In real life the computer will help you, as by adding lens elements you can correct part of the higher order aberrations. But then you have more lens elements that will also influence your earlier calculations etc.

So a designer needs to study the image quality that is needed at the end and (s)he should have a very firm idea of what design (initial layout) is promising enough to make the requirements fit. That is why Leica insists that one person should study one design. Of course colleagues will help when needed, but the responsibility of a design is with the designer. This approach has advantages too. The design of the 70-180 took more than 18 months, because it was new territory to Leica. But after studying the true character of such a design, the second one (80-200/4) could be accomplished in about half a year. You know the basics of a design and its optical characteristics and then a simpler version is not that difficult. The same with the 1.4/35 asph. The first design took two years, the 2/35 asph less than a year. But both lenses, while obviously closely related, show a different fingerprint. I will tell you more about this. The Leica design team is not grouped into R or M segments.

Anyone will in principle design any lens. The young woman designer who computed the 70-180 lens, also designed the 2/35 asph. The designer of the 2.8/24 M also produced the 2.8/35-70. As examples!

3.11 What about this fingerprint of a lens?

Looking at the pictures (see colour print section) the reader will notice that the aberrations have a different shape and magnitude when you see them in front of or behind the plane of sharpness. This phenomenon will translate in unsharpness areas that will be slightly different before and after the object of sharpness. But the choices here are very critically related to the sharpness plane too. The choice of glass types, the sequence in which aberrations are corrected (first spherical aberration, then chromatic aberrations, then coma or any other sequence) the magnitude of corrections, the balance between the aberrations left over in the system, all these design components will influence the image quality we get on film. The best lenses of other companies may not differ that much in the factual correction of aberrations (that is computer stuff partly), but they will certainly differ in image quality if Leica meets their target with a 7 element design and some one else needs 11 glass-elements for the same kind of target. Leica designers, backed up by a vast pool of knowledge about photographic and microscopic optical designs, and supported by a high level creativity in design, and some unique insights into the intricacies of the light rays flowing through a design are able to generate remarkable optics today. The graphs generated by the computer which give an accurate view of the aberration correction are in fact an excellent substitute for a prototype glass lens. These graphs are, again, not new. In the Leica archives you will find the same diagrams, now being generated by the computer, and then drawn by hand!. It must have taken months to compute and draw all these lines! It also shows that the link with the past is still strong. Leica designs now may be created by a new and young generation of designers, the roots are in the Berek days and we are very fortunate that the original creativity of lens design is still very much alive in Solms. A Leica lens often exhibits a character as does a good wine or a good novel. Because it is designed by a strong minded and very creative individual we might consider it as a work of art. But without the help of the computer it would not be possible to redefine the state of the art of lens design.

We may be very fortunate that Leica has never abandoned the individually creative side of lens design. In any modern Leica lens there has been invested several years of highly creative opto-mechanical design, that is truly cutting-edge technology. There is evidently a tension between the optical quality of the current Leica lenses and the users of these lenses, who are striving to get this quality on film. But it is up to us, Leica users to honour the work done and to give serious feedback to the designers.

© Erwin Puts
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