What is spherical aberration. Spherical aberration in lenses. Elimination of spherical aberration

Let us consider the image of a Point located on the optical axis given by the optical system. Since the optical system has circular symmetry relative to the optical axis, it is sufficient to limit ourselves to the choice of rays lying in the meridional plane. In Fig. 113 shows the ray path characteristic of a positive single lens. Position

Rice. 113. Spherical aberration of a positive lens

Rice. 114. Spherical aberration for an off-axis point

The ideal image of an object point A is determined by a paraxial ray crossing the optical axis at a distance from the last surface. Rays forming finite angles with the optical axis do not reach the ideal image point. For a single positive lens, the greater the absolute value of the angle, the closer to the lens the beam intersects the optical axis. This is explained by the unequal optical power of the lens in its different zones, which increases with distance from the optical axis.

This violation of the homocentricity of the emerging beam of rays can be characterized by the difference in the longitudinal segments for paraxial rays and for rays passing through the plane of the entrance pupil at finite heights: This difference is called longitudinal spherical aberration.

The presence of spherical aberration in the system leads to the fact that instead of a sharp image of a point in the ideal image plane, a scattering circle is obtained, the diameter of which is equal to twice the value. The latter is related to longitudinal spherical aberration by the relation

and is called transverse spherical aberration.

It should be noted that with spherical aberration, symmetry is preserved in the beam of rays emerging from the system. Unlike other monochromatic aberrations, spherical aberration occurs at all points in the field of the optical system, and in the absence of other aberrations for points off the axis, the beam of rays emerging from the system will remain symmetrical relative to the main ray (Fig. 114).

The approximate value of spherical aberration can be determined using third-order aberration formulas through

For an object located at a finite distance, as follows from Fig. 113,

Within the limits of the validity of the theory of third-order aberrations, one can accept

If we put something according to the normalization conditions, we get

Then, using formula (253), we find that the third-order transverse spherical aberration for an object point located at a finite distance is

Accordingly, for longitudinal spherical aberrations of the third order, assuming according to (262) and (263), we obtain

Formulas (263) and (264) are also valid for the case of an object located at infinity, if calculated under normalization conditions (256), i.e., at the real focal length.

In the practice of aberration calculation of optical systems, when calculating third-order spherical aberration, it is convenient to use formulas containing the coordinate of the beam on the entrance pupil. Then, according to (257) and (262), we obtain:

if calculated under normalization conditions (256).

For normalization conditions (258), i.e. for the reduced system, according to (259) and (262) we will have:

From the above formulas it follows that for a given spherical aberration of the third order, the greater the coordinate of the beam on the entrance pupil.

Since spherical aberration is present for all points of the field, when aberration correction of an optical system, primary attention is paid to correcting spherical aberration. The simplest optical system with spherical surfaces in which spherical aberration can be reduced is a combination of positive and negative lenses. For both positive and negative lenses, the extreme zones refract the rays more strongly than the zones located near the axis (Fig. 115). A negative lens has positive spherical aberration. Therefore, combining a positive lens having negative spherical aberration with a negative lens produces a spherical aberration corrected system. Unfortunately, spherical aberration can be corrected only for some rays, but it cannot be completely corrected within the entire entrance pupil.

Rice. 115. Spherical aberration of a negative lens

Thus, any optical system always has residual spherical aberration. Residual aberrations of an optical system are usually presented in tabular form and illustrated with graphs. For an object point located on the optical axis, graphs of longitudinal and transverse spherical aberrations are presented, presented as functions of coordinates, or

The curves of longitudinal and corresponding transverse spherical aberration are shown in Fig. 116. Graphs in Fig. 116, and correspond to an optical system with undercorrected spherical aberration. If for such a system its spherical aberration is determined only by third-order aberrations, then according to formula (264) the longitudinal spherical aberration curve has the form of a quadratic parabola, and the transverse aberration curve has the form of a cubic parabola. Graphs in Fig. 116, b correspond to an optical system in which spherical aberration is corrected for a beam passing through the edge of the entrance pupil, and the graphs in Fig. 116, in - an optical system with redirected spherical aberration. Correction or recorrection of spherical aberration can be achieved, for example, by combining positive and negative lenses.

Transverse spherical aberration characterizes the circle of dispersion, which is obtained instead of an ideal image of a point. The diameter of the scatter circle for a given optical system depends on the choice of the image plane. If this plane is shifted relative to the plane of the ideal image (Gaussian plane) by an amount (Fig. 117, a), then in the displaced plane we obtain transverse aberration associated with transverse aberration in the Gaussian plane by the dependence

In formula (266), the term on the graph of transverse spherical aberration plotted in coordinates is a straight line passing through the origin. At

Rice. 116. Graphical representation of longitudinal and transverse spherical aberrations

Of all types of aberrations, spherical aberration is the most significant and, in most cases, the only one practically significant for the optical system of the eye. Since the normal eye always fixes its gaze on the most important object at the moment, aberrations caused by the oblique incidence of light rays (coma, astigmatism) are eliminated. It is impossible to eliminate spherical aberration in this way. If the refractive surfaces of the optical system of the eye are spherical, it is impossible to eliminate spherical aberration in any way at all. Its distorting effect decreases as the diameter of the pupil decreases, therefore, in bright light, the resolution of the eye is higher than in low light, when the diameter of the pupil increases and the size of the spot, which is the image of a point light source, also increases due to spherical aberration. There is only one way to effectively influence the spherical aberration of the optical system of the eye - by changing the shape of the refractive surface. This possibility exists, in principle, with surgical correction of the curvature of the cornea and with the replacement of a natural lens that has lost its optical properties, for example, due to cataracts, with an artificial one. An artificial lens can have refractive surfaces of any shape accessible to modern technologies. The study of the influence of the shape of refractive surfaces on spherical aberration can most effectively and accurately be performed using computer modeling. Here we discuss a fairly simple computer modeling algorithm that allows such a study to be carried out, as well as the main results obtained using this algorithm.

The simplest way to calculate the passage of a light beam through a single spherical refractive surface separating two transparent media with different refractive indices. To demonstrate the phenomenon of spherical aberration, it is enough to perform such a calculation in a two-dimensional approximation. The light beam is located in the main plane and is directed onto the refractive surface parallel to the main optical axis. The course of this ray after refraction can be described by the equation of the circle, the law of refraction, and obvious geometric and trigonometric relationships. As a result of solving the corresponding system of equations, an expression can be obtained for the coordinate of the point of intersection of this ray with the main optical axis, i.e. coordinates of the focus of the refractive surface. This expression contains surface parameters (radius), refractive indices, and the distance between the main optical axis and the point of incidence of the beam on the surface. The dependence of the focal coordinate on the distance between the optical axis and the point of incidence of the beam is spherical aberration. This relationship is easy to calculate and depict graphically. For a single spherical surface deflecting rays towards the main optical axis, the focal coordinate always decreases as the distance between the optical axis and the incident ray increases. The farther from the axis a ray falls on a refracting surface, the closer to this surface it intersects the axis after refraction. This is positive spherical aberration. As a result, rays incident on the surface parallel to the main optical axis are not collected at one point in the image plane, but form a scattering spot of finite diameter in this plane, which leads to a decrease in image contrast, i.e. to deterioration of its quality. Only those rays that fall on the surface very close to the main optical axis (paraxial rays) intersect at one point.

If a collecting lens formed by two spherical surfaces is placed in the path of the beam, then using the calculations described above, it can be shown that such a lens also has positive spherical aberration, i.e. rays incident parallel to the main optical axis further from it intersect this axis closer to the lens than rays traveling closer to the axis. Spherical aberration is practically absent also only for paraxial rays. If both surfaces of the lens are convex (like a lens), then the spherical aberration is greater than if the second refractive surface of the lens is concave (like the cornea).

Positive spherical aberration is caused by excessive curvature of the refractive surface. As one moves away from the optical axis, the angle between the tangent to the surface and the perpendicular to the optical axis increases faster than necessary to direct the refracted beam to the paraxial focus. To reduce this effect, it is necessary to slow down the deviation of the tangent to the surface from the perpendicular to the axis as it moves away from it. To do this, the curvature of the surface must decrease with distance from the optical axis, i.e. the surface should not be spherical, in which the curvature at all its points is the same. In other words, a reduction in spherical aberration can only be achieved by using lenses with aspherical refractive surfaces. These can be, for example, the surfaces of an ellipsoid, paraboloid and hyperboloid. In principle, it is possible to use other surface forms. The attractiveness of elliptical, parabolic and hyperbolic shapes is only that they, like a spherical surface, are described by fairly simple analytical formulas and the spherical aberration of lenses with these surfaces can be quite easily studied theoretically using the technique described above.

It is always possible to select the parameters of spherical, elliptical, parabolic and hyperbolic surfaces so that their curvature at the center of the lens is the same. In this case, for paraxial rays such lenses will be indistinguishable from each other, the position of the paraxial focus will be the same for these lenses. But as you move away from the main axis, the surfaces of these lenses will deviate from the perpendicular to the axis in different ways. The spherical surface will deviate the fastest, the elliptical one slower, the parabolic one even slower, and the hyperbolic one the slowest (of these four). In the same sequence, the spherical aberration of these lenses will decrease more and more noticeably. For a hyperbolic lens, spherical aberration can even change sign - become negative, i.e. rays incident on a lens further from the optical axis will intersect it further from the lens than rays incident on a lens closer to the optical axis. For a hyperbolic lens, you can even select parameters of the refractive surfaces that will ensure the complete absence of spherical aberration - all rays incident on the lens parallel to the main optical axis at any distance from it, after refraction, will be collected at one point on the axis - an ideal lens. To do this, the first refractive surface must be flat, and the second must be convex hyperbolic, the parameters of which and the refractive indices must be related by certain relationships.

Thus, by using lenses with aspherical surfaces, spherical aberration can be significantly reduced and even completely eliminated. The possibility of separate influence on the refractive force (position of the paraxial focus) and spherical aberration is due to the presence of aspherical surfaces of rotation of two geometric parameters, two semi-axes, the selection of which can ensure a decrease in spherical aberration without changing the refractive force. A spherical surface does not have this possibility; it has only one parameter - the radius, and by changing this parameter it is impossible to change the spherical aberration without changing the refractive power. For a paraboloid of revolution there is also no such possibility, since a paraboloid of revolution also has only one parameter - the focal parameter. Thus, of the three mentioned aspherical surfaces, only two are suitable for controlled independent influence on spherical aberration - hyperbolic and elliptical.

Selecting a single lens with parameters that provide acceptable spherical aberration is not difficult. But will such a lens provide the required reduction in spherical aberration as part of the optical system of the eye? To answer this question, it is necessary to calculate the passage of light rays through two lenses - the cornea and the lens. The result of such a calculation will be, as before, a graph of the dependence of the coordinates of the point of intersection of the beam with the main optical axis (focus coordinates) on the distance between the incident beam and this axis. By varying the geometric parameters of all four refractive surfaces, you can use this graph to study their influence on the spherical aberration of the entire optical system of the eye and try to minimize it. One can, for example, easily verify that the aberration of the entire optical system of an eye with a natural lens, provided that all four refractive surfaces are spherical, is noticeably less than the aberration of the lens alone, and slightly greater than the aberration of the cornea alone. With a pupil diameter of 5 mm, the rays farthest from the axis intersect this axis approximately 8% closer than the paraxial rays when refracted by the lens alone. When refracted by the cornea alone, with the same pupil diameter, the focus for distant rays is approximately 3% closer than for paraxial rays. The entire optical system of the eye with this lens and with this cornea collects distant rays about 4% closer than paraxial rays. We can say that the cornea partially compensates for the spherical aberration of the lens.

It can also be seen that the optical system of the eye, consisting of the cornea and an ideal hyperbolic lens with zero aberration, installed as a lens, gives a spherical aberration approximately the same as the cornea alone, i.e. minimizing the spherical aberration of the lens alone is not sufficient to minimize the entire optical system of the eye.

Thus, to minimize spherical aberration of the entire optical system of the eye by choosing the geometry of the lens alone, it is necessary to select not a lens that has minimal spherical aberration, but one that minimizes aberration in interaction with the cornea. If the refractive surfaces of the cornea are considered spherical, then to almost completely eliminate the spherical aberration of the entire optical system of the eye, it is necessary to select a lens with hyperbolic refractive surfaces, which, as a single lens, gives a noticeable (about 17% in the liquid medium of the eye and about 12% in air) negative aberration . The spherical aberration of the entire optical system of the eye does not exceed 0.2% for any pupil diameter. Almost the same neutralization of the spherical aberration of the optical system of the eye (up to about 0.3%) can be achieved even with the help of a lens in which the first refractive surface is spherical and the second is hyperbolic.

So, the use of an artificial lens with aspherical, in particular, with hyperbolic refractive surfaces makes it possible to almost completely eliminate the spherical aberration of the optical system of the eye and thereby significantly improve the quality of the image produced by this system on the retina. This is shown by the results of computer simulation of the passage of rays through the system within the framework of a fairly simple two-dimensional model.

The influence of the parameters of the optical system of the eye on the quality of the retinal image can also be demonstrated using a much more complex three-dimensional computer model that traces a very large number of rays (from several hundred rays to several hundred thousand rays) emerging from one source point and arriving at different points retina as a result of exposure to all geometric aberrations and possible inaccurate focusing of the system. By adding up all the rays at all points of the retina that arrived there from all source points, such a model allows one to obtain images of extended sources - various test objects, both color and black and white. We have such a three-dimensional computer model at our disposal and it clearly demonstrates a significant improvement in the quality of the retinal image when using intraocular lenses with aspherical refractive surfaces due to a significant reduction in spherical aberration and thereby reducing the size of the scattering spot on the retina. In principle, spherical aberration can be eliminated almost completely and, it would seem, the size of the scattering spot can be reduced almost to zero, thereby obtaining an ideal image.

But one should not lose sight of the fact that it is impossible to obtain an ideal image in any way, even if we assume that all geometric aberrations are completely eliminated. There is a fundamental limit to reducing the size of the scattering spot. This limit is set by the wave nature of light. In accordance with the diffraction theory, based on wave concepts, the minimum diameter of the light spot in the image plane, due to the diffraction of light on a circular hole, is proportional (with a proportionality coefficient of 2.44) to the product of the focal length and the wavelength of light and inversely proportional to the diameter of the hole. An estimate for the optical system of the eye gives a scattering spot diameter of about 6.5 µm with a pupil diameter of 4 mm.

It is impossible to reduce the diameter of the light spot below the diffraction limit, even if the laws of geometric optics bring all rays to one point. Diffraction limits the limit of image quality improvement provided by any refractive optical system, even an ideal one. At the same time, light diffraction, no worse than refraction, can be used to obtain an image, which is successfully used in diffractive-refractive IOLs. But that is another topic.

Bibliographic link

Fig.1 Illustration of undercorrected spherical aberration. The surface at the periphery of the lens has a focal length shorter than at the center.

Most photographic lenses consist of elements with spherical surfaces. Such elements are relatively easy to manufacture, but their shape is not ideal for image formation.

Spherical aberration- this is one of the defects in image formation that occurs due to the spherical shape of the lens. Rice. Figure 1 illustrates spherical aberration for a positive lens.

Rays that pass through the lens further from the optical axis are focused at position With. Rays that pass closer to the optical axis are focused at position a, they are closer to the surface of the lens. Thus, the position of focus depends on the location at which the rays pass through the lens.

If the edge focus is closer to the lens than the axial focus, as happens with a positive lens Fig. 1, then they say that spherical aberration uncorrected. Conversely, if the edge focus is behind the axial focus, then spherical aberration is said to be re-corrected.

The image of a point made by a lens with spherical aberrations is usually obtained by points surrounded by a halo of light. Spherical aberration usually appears in photographs by softening contrast and blurring fine details.

Spherical aberration is uniform across the field, which means that the longitudinal focus between the edges of the lens and the center does not depend on the inclination of the rays.

From Fig. 1 it seems that it is impossible to achieve good sharpness on a lens with spherical aberration. In any position behind the lens on the photosensitive element (film or sensor), instead of a clear point, a blur disk will be projected.

However, there is a geometrically "best" focus that corresponds to the disk of least blur. This unique ensemble of light cones has a minimal cross-section, in position b.

Focus shift

When the diaphragm is behind the lens, an interesting phenomenon occurs. If the diaphragm is closed in such a way that it cuts off rays at the periphery of the lens, then the focus shifts to the right. With a very closed aperture, the best focus will be observed in the position c, that is, the positions of the disks with the least blur when the aperture is closed and when the aperture is open will differ.

To get the best sharpness at a closed aperture, the matrix (film) should be placed in the position c. This example clearly shows that there is a possibility that the best sharpness will not be achieved, since most photographic systems are designed to operate with a wide aperture.

The photographer focuses with the aperture fully open, and projects the disk of least blur at the position onto the sensor. b, then when shooting, the aperture automatically closes to the set value, and he suspects nothing of what follows at this moment focus shift, which prevents it from achieving the best sharpness.

Of course, a closed aperture reduces spherical aberrations also at the point b, but still it will not have the best sharpness.

DSLR users can close down the preview aperture to focus at the actual aperture.

Norman Goldberg proposed automatic compensation for focus shifts. Zeiss has launched a line of rangefinder lenses for Zeiss Ikon cameras that feature a specially designed design to minimize focus shift with changing aperture values. At the same time, spherical aberrations in lenses for rangefinder cameras are significantly reduced. How important is focus shift for rangefinder camera lenses, you ask? According to the manufacturer of the LEICA NOCTILUX-M 50mm f/1 lens, this value is about 100 microns.

Out-of-focus blur pattern

The effect of spherical aberrations on an in-focus image is difficult to discern, but can be clearly seen in an image that is slightly out of focus. Spherical aberration leaves a visible trace in the out-of-focus area.

Returning to Fig. 1, it can be noted that the distribution of light intensity in the blur disk in the presence of spherical aberration is not uniform.

Pregnant c a blur disk is characterized by a bright core surrounded by a faint halo. While the blur dial is in position a has a darker core surrounded by a bright ring of light. Such anomalous light distributions may appear in the out-of-focus area of ​​the image.

Rice. 2 Changes in blur in front of and behind the point of focus

Example in Fig. 2 shows a point in the center of the frame, shot in 1:1 macro mode with an 85/1.4 lens mounted on a macro bellows lens. When the sensor is 5 mm behind the best focus (middle point), the blur disk shows the effect of a bright ring (left spot), similar blur disks are obtained with meniscus reflex lenses.

And when the sensor is 5 mm ahead of the best focus (i.e. closer to the lens), the nature of the blur has changed towards a bright center surrounded by a faint halo. As you can see, the lens has overcorrected spherical aberration, since it behaves opposite to the example in Fig. 1.

The following example illustrates the effect of two aberrations on out-of-focus images.

In Fig. 3 shows a cross, which was photographed in the center of the frame using the same 85/1.4 lens. The macrofur is extended by approximately 85 mm, which gives an increase of approximately 1:1. The camera (matrix) was moved in increments of 1 mm in both directions from maximum focus. A cross is a more complex image than a dot, and color indicators provide visual illustrations of its blurring.

Rice. 3 The numbers in the illustrations indicate changes in the distance from the lens to the matrix, these are millimeters. the camera moves from -4 to +4 mm in 1 mm increments from the best focus position (0)

Spherical aberration is responsible for the hard nature of blur at negative distances and for the transition to soft blur at positive ones. Also of interest are color effects that arise from longitudinal chromatic aberration (axial color). If the lens is poorly assembled, then spherical aberration and axial color are the only aberrations that appear in the center of the image.

Most often, the strength and sometimes the nature of spherical aberration depends on the wavelength of the light. In this case, the combined effect of spherical aberration and axial color is called . From this it becomes clear that the phenomenon illustrated in Fig. 3 shows that this lens is not intended to be used as a macro lens. Most lenses are optimized for near field focusing and infinity focusing, but not for 1:1 macro. At such an approach, regular lenses will behave worse than macro lenses, which are used specifically at close distances.

However, even if the lens is used for standard applications, spherochromatism can appear in the out-of-focus area during normal shooting and affect the quality.

conclusions
Of course, the illustration in Fig. 1 is an exaggeration. In reality, the amount of residual spherical aberrations in photographic lenses is small. This effect is significantly reduced by combining lens elements to compensate for the sum of opposing spherical aberrations, the use of high-quality glass, carefully designed lens geometry and the use of aspherical elements. In addition, floating elements can be used to reduce spherical aberrations over a certain range of working distances.

For lenses with undercorrected spherical aberration, an effective way to improve image quality is to close the aperture. For the undercorrected element in Fig. 1 The diameter of the blur disks decreases in proportion to the cube of the aperture diameter.

This dependence may differ for residual spherical aberrations in complex lens designs, but, as a rule, closing the aperture by one stop already gives a noticeable improvement in the image.

Alternatively, rather than fighting spherical aberration, a photographer can intentionally exploit it. Zeiss softening filters, despite their flat surface, add spherical aberrations to the image. They are popular among portrait photographers to achieve a soft effect and an impressive image.

© Paul van Walree 2004–2015
Translation: Ivan Kosarekov

1. Introduction to the theory of aberrations

When talking about lens performance, one often hears the word aberrations. “This is an excellent lens, all aberrations are practically corrected in it!” - a thesis that can very often be found in discussions or reviews. It is much less common to hear a diametrically opposite opinion, for example: “This is a wonderful lens, its residual aberrations are well expressed and form an unusually plastic and beautiful pattern”...

Why do such different opinions arise? I will try to answer this question: how good/bad is this phenomenon for lenses and for photography genres in general. But first, let's try to figure out what photographic lens aberrations are. We'll start with the theory and some definitions.

In general use the term Aberration (lat. ab- “from” + lat. errare “to wander, to be mistaken”) is a deviation from the norm, an error, some kind of disruption of the normal operation of the system.

Lens aberration- error, or image error in the optical system. It is caused by the fact that in a real environment a significant deviation of rays can occur from the direction in which they go in the calculated “ideal” optical system.

As a result, the generally accepted quality of a photographic image suffers: insufficient sharpness in the center, loss of contrast, severe blurring at the edges, distortion of geometry and space, color halos, etc.

The main aberrations characteristic of photographic lenses are as follows:

1. Comatic aberration.
2. Distortion.
3. Astigmatism.
4. Curvature of the image field.

Before we take a closer look at each of them, let’s recall from the article how rays pass through a lens in an ideal optical system:

Ill. 1. Passage of rays in an ideal optical system.

As we see, all the rays are collected at one point F - the main focus. But in reality, everything is much more complicated. The essence of optical aberrations is that rays incident on a lens from one luminous point are not collected at one point. So, let's see what deviations occur in an optical system when exposed to various aberrations.

Here it should also be immediately noted that in both a simple lens and a complex lens, all the aberrations described below act together.

Action spherical aberration is that rays incident on the edges of the lens are collected closer to the lens than rays incident on the central part of the lens. As a result, the image of a point on a plane appears in the form of a blurry circle or disk.

Ill. 2. Spherical aberration.

In photographs, the effects of spherical aberration appear as a softened image. The effect is especially often noticeable at open apertures, and lenses with larger apertures are more susceptible to this aberration. If the sharpness of the contours is preserved, such a soft effect can be very useful for some types of photography, for example, portraiture.

Ill.3. A soft effect on an open aperture due to the action of spherical aberration.

In lenses built entirely from spherical lenses, it is almost impossible to completely eliminate this type of aberration. In ultra-fast lenses, the only effective way to significantly compensate for this is to use aspherical elements in the optical design.

3. Comatic aberration, or “Coma”

This is a special type of spherical aberration for side rays. Its effect lies in the fact that rays arriving at an angle to the optical axis are not collected at one point. In this case, the image of a luminous point at the edges of the frame is obtained in the form of a “flying comet”, and not in the form of a point. Coma can also cause areas of the image in the out-of-focus area to become overexposed.

Ill. 4. Coma.

Ill. 5. Coma in a photo image

It is a direct consequence of light dispersion. Its essence is that a ray of white light, passing through a lens, is decomposed into its constituent colored rays. Short-wave rays (blue, violet) are refracted in the lens more strongly and converge closer to it than long-focus rays (orange, red).

Ill. 6. Chromatic aberration. F - focus of violet rays. K - focus of red rays.

Here, as in the case of spherical aberration, the image of a luminous point on a plane is obtained in the form of a blurred circle/disk.

In photographs, chromatic aberration appears in the form of extraneous shades and colored outlines in the subjects. The influence of aberration is especially noticeable in contrasting scenes. Currently, CA can be easily corrected in RAW converters if the shooting was carried out in RAW format.

Ill. 7. An example of the manifestation of chromatic aberration.

5. Distortion

Distortion manifests itself in the curvature and distortion of the geometry of the photograph. Those. the scale of the image changes with distance from the center of the field to the edges, as a result of which straight lines bend towards the center or towards the edges.

Distinguish barrel-shaped or negative(most typical for a wide angle) and cushion-shaped or positive distortion (more often seen at long focal lengths).

Ill. 8. Pincushion and barrel distortion

Distortion is usually much more pronounced in lenses with variable focal lengths (zooms) than in lenses with fixed focal lengths (fixes). Some spectacular lenses, such as Fish Eye, deliberately do not correct distortion and even emphasize it.

Ill. 9. Pronounced barrel distortion of the lensZenitar 16mmFish Eye.

In modern lenses, including those with variable focal lengths, distortion is quite effectively corrected by introducing an aspherical lens (or several lenses) into the optical design.

6. Astigmatism

Astigmatism(from the Greek Stigma - point) is characterized by the impossibility of obtaining images of a luminous point at the edges of the field, both in the form of a point and even in the form of a disk. In this case, a luminous point located on the main optical axis is transmitted as a point, but if a point is outside this axis, it is transmitted as a darkening, crossed lines, etc.

This phenomenon is most often observed at the edges of the image.

Ill. 10. Manifestation of astigmatism

7. Image field curvature

Image field curvature- this is an aberration, as a result of which the image of a flat object, perpendicular to the optical axis of the lens, lies on a surface concave or convex to the lens. This aberration causes uneven sharpness across the image field. When the central part of the image is sharply focused, its edges will be out of focus and will not appear sharp. If you adjust the sharpness along the edges of the image, then its central part will be blurred.

Spherical aberration ()

If all coefficients, with the exception of B, are equal to zero, then (8) takes the form

Aberration curves in this case have the form of concentric circles, the centers of which are located at the point of the paraxial image, and the radii are proportional to the third power of the zone radius, but do not depend on the position () of the object in the visual zone. This image defect is called spherical aberration.

Spherical aberration, being independent of, distorts both on-axis and off-axis points of the image. Rays emerging from the axial point of an object and making significant angles with the axis will intersect it at points lying in front of or behind the paraxial focus (Fig. 5.4). The point at which the rays from the edge of the diaphragm intersect with the axis was called the edge focus. If the screen in the image area is placed at right angles to the axis, then there is a position of the screen at which the round spot of the image on it is minimal; this minimal “image” is called the smallest circle of scattering.

Coma()

An aberration characterized by a non-zero F coefficient is called coma. The components of radiation aberration in this case have, according to (8). view

As we see, with a fixed zone radius, a point (see Fig. 2.1) when changing from 0 to twice describes a circle in the image plane. The radius of the circle is equal, and its center is at a distance from the paraxial focus towards negative values at. Consequently, this circle touches two straight lines passing through the paraxial image and components with the axis at angles of 30°. If all possible values ​​are used, then the collection of similar circles forms an area limited by the segments of these straight lines and the arc of the largest aberration circle (Fig. 3.3). The dimensions of the resulting area increase linearly with increasing distance of the object point from the system axis. When the Abbe sines condition is met, the system provides a sharp image of an element of the object plane located in close proximity to the axis. Consequently, in this case, the expansion of the aberration function cannot contain terms that linearly depend on. It follows that if the sinus condition is met, there is no primary coma.

Astigmatism () and field curvature ()

It is more convenient to consider aberrations characterized by coefficients C and D together. If all other coefficients in (8) are equal to zero, then

To demonstrate the importance of such aberrations, let us first assume that the imaging beam is very narrow. According to § 4.6, the rays of such a beam intersect two short segments of curves, one of which (tangential focal line) is orthogonal to the meridional plane, and the other (sagittal focal line) lies in this plane. Let us now consider the light emanating from all points of the finite region of the object plane. Focal lines in image space will transform into tangential and sagittal focal surfaces. To a first approximation, these surfaces can be considered spheres. Let and be their radii, which are considered positive if the corresponding centers of curvature are located on the other side of the image plane from where the light propagates (in the case shown in Fig. 3.4. i).

The radii of curvature can be expressed through the coefficients WITH And D. To do this, when calculating ray aberrations taking into account curvature, it is more convenient to use ordinary coordinates rather than Seidel variables. We have (Fig. 3.5)

Where u- small distance between the sagittal focal line and the image plane. If v is the distance from this focal line to the axis, then

if still neglected And compared to, then from (12) we find

Likewise

Let us now write these relations in terms of Seidel variables. Substituting (2.6) and (2.8) into them, we obtain

and similarly

In the last two relations we can replace by and then, using (11) and (6), we obtain

Size 2C + D usually called tangential field curvature, magnitude D -- sagittal field curvature, and their half-sum

which is proportional to their arithmetic mean, - simply field curvature.

From (13) and (18) it follows that at a height from the axis the distance between the two focal surfaces (i.e., the astigmatic difference of the beam forming the image) is equal to

Half-difference

called astigmatism. In the absence of astigmatism (C = 0) we have. Radius R The total, coincident, focal surface can in this case be calculated using a simple formula, which includes the radii of curvature of the individual surfaces of the system and the refractive indices of all media.

Distortion()

If in relations (8) only the coefficient is different from zero E, That

Since this does not include coordinates and, the display will be stigmatic and will not depend on the radius of the exit pupil; however, the distances of the image points to the axis will not be proportional to the corresponding distances for the object points. This aberration is called distortion.

In the presence of such aberration, the image of any line in the plane of the object passing through the axis will be a straight line, but the image of any other line will be curved. In Fig. 3.6, and the object is shown in the form of a grid of straight lines parallel to the axes X And at and located at the same distance from each other. Rice. 3.6. b illustrates the so-called barrel distortion (E>0), and Fig. 3.6. V - pincushion distortion (E<0 ).

Rice. 3.6.

It was previously stated that of the five Seidel aberrations, three (spherical, coma and astigmatism) interfere with image sharpness. The other two (field curvature and distortion) change its position and shape. In general, it is impossible to construct a system that is free both from all primary aberrations and from higher order aberrations; therefore, we always have to look for some suitable compromise solution that takes into account their relative values. In some cases, Seidel aberrations can be significantly reduced by higher order aberrations. In other cases, it is necessary to completely destroy some aberrations, even though other types of aberrations appear. For example, coma must be completely eliminated in telescopes, because if it is present, the image will be asymmetrical and all precision astronomical position measurements will be meaningless . On the other hand, the presence of some field curvature and distortion is relatively harmless, since it can be eliminated using appropriate calculations.

optical aberration chromatic astigmatism distortion