Spherical aberration can be corrected. spherical aberration. Elimination of spherical aberration

The word aberration denotes a set of optical effects associated with the distortion of an object during observation. In this article, we will talk about several types of aberration that are most relevant for astronomical observations.

aberration of light in astronomy, it is the apparent displacement of a celestial object due to the finite speed of light combined with the motion of the observed object and the observer. The action of aberration leads to the fact that the apparent direction to the object does not coincide with the geometric direction to it at the same time.

The effect is that due to the motion of the Earth around the Sun and the time it takes for the light to propagate, the observer sees the star in a different place than where it is. If the Earth were stationary, or if light propagated instantaneously, then there would be no light aberration. Therefore, when determining the position of a star in the sky with a telescope, we must not count the angle at which the star is tilted, but slightly increase it in the direction of the Earth's movement.

The aberration effect is not great. Its greatest value is achieved under the condition that the earth moves perpendicular to the direction of the beam. At the same time, the deviation of the position of the star is only 20.4 seconds, because the earth travels only 30 km in 1 second of time, and a ray of light - 300,000 km.

There are also several types geometric aberration. Spherical aberration- an aberration of a lens or lens, which consists in the fact that a wide beam of monochromatic light emanating from a point lying on the main optical axis of the lens, when passing through the lens, intersects not at one, but at many points located on the optical axis at different distances from the lens , resulting in an unsharp image. As a result, such a point object as a star can be seen as a small ball, taking the size of this ball as the size of the star.

Curvature of the image field- aberration, as a result of which the image of a flat object, perpendicular to the optical axis of the lens, lies on a surface that is concave or convex to the lens. This aberration causes uneven sharpness across the image field. Therefore, when the center of the image is sharply focused, the edges of the image will lie out of focus and the image will be blurry. If the sharpness setting is made along the edges of the image, then its central part will be unsharp. This kind of aberration is not essential for astronomy.

And here are some more types of aberration:

Diffractive aberration occurs due to the diffraction of light by the aperture and barrel of a photographic lens. Diffractive aberration limits the resolution of a photographic lens. Due to this aberration, the minimum angular distance between points allowed by the lens is limited by the value of lambda / D radians, where lambda is the wavelength of the light used (the optical range usually includes electromagnetic waves with a wavelength from 400 nm to 700 nm), D is the diameter of the lens . Looking at this formula, it becomes clear how important the diameter of the lens is. It is this parameter that is key for the largest and most expensive telescopes. It is also clear that a telescope capable of seeing in x-rays compares favorably with a conventional optical telescope. The fact is that the wavelength of X-rays is 100 times less than the wavelength of light in the optical range. Therefore, for such telescopes, the minimum distinguishable angular distance is 100 times smaller than for conventional optical telescopes with the same objective diameter.

The study of aberration made it possible to significantly improve astronomical instruments. In modern telescopes, the effects of aberration are minimized, but it is aberration that limits the capabilities of optical instruments.

1. Introduction to the theory of aberrations

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

Why are there such different opinions? 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 aberrations of a photographic lens are. We start with theory and some definitions.

In general usage, the term Aberration (lat. ab- “from” + lat. errare “wander, err”) - this is a deviation from the norm, an error, some kind of violation 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 medium there can be a significant deviation of the rays 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, strong blurring at the edges, distortion of geometry and space, color halos, etc.

The main aberrations characteristic of photographic lenses are as follows:

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

Before getting to know each of them better, let's recall from the article how rays pass through a lens in an ideal optical system:

ill. 1. The passage of rays in an ideal optical system.

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

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

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

ill. 2. Spherical aberration.

In photographs, the effect of spherical aberration appears as a softened image. Especially often the effect is noticeable at open apertures, and lenses with a larger aperture are more susceptible to this aberration. As long as the edges are sharp, this soft effect can be very useful for some types of photography, such as portraits.

Fig.3. 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 super-aperture lenses, the only effective way to significantly compensate for it is to use aspherical elements in the optical design.

3. Coma aberration, or "Coma"

This is a particular type of spherical aberration for side beams. Its action lies in the fact that the rays coming 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. A coma can also cause areas of the image in the blur zone to be blown out.

ill. 4. Coma.

ill. 5. Coma on a photo image

It is a direct consequence of the dispersion of light. Its essence lies in the fact that a beam of white light, passing through the lens, decomposes into its constituent colored rays. Short-wavelength 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. Ф - 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 blurry circle / disk.

In photographs, chromatic aberration appears as ghosting and colored outlines on subjects. The effect of aberration is especially noticeable in contrasting subjects. Currently, XA is quite easily corrected in RAW converters if the shooting was done in RAW format.

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

5. Distortion

Distortion is manifested 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 are curved towards the center or towards the edges.

Distinguish barrel-shaped or negative(most typical for a wide angle) and pillow-shaped or positive distortion (more often manifested at a long focus).

ill. 8. Pincushion and barrel distortion

Distortion is usually much more pronounced with zoom lenses than with prime lenses. Some spectacular lenses, such as Fish Eye, deliberately do not correct and even emphasize distortion.

ill. 9. Pronounced barrel lens distortionZenitar 16mmfish eye.

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

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 the point is outside this axis - as a blackout, crossed lines, etc.

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

ill. 10. Manifestation of astigmatism

7. Curvature of the image field

Curvature of the image field- 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 that is concave or convex to the lens. This aberration causes uneven sharpness across the image field. When the center of an image is sharply focused, the edges of the image will lie out of focus and not appear sharp. If the sharpness setting is made along the edges of the image, then its central part will be unsharp.

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

Most photographic lenses are made up of elements with spherical surfaces. Such elements are relatively easy to manufacture, but their shape is not ideal for imaging.

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

Rays that pass through the lens farther from the optical axis are focused at position With. Rays that pass closer to the optical axis are focused in position a, they are closer to the surface of the lens. Thus, the position of the focus depends on the location where 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 say spherical aberration undercorrected. Conversely, if the edge focus is behind the axial focus, then the spherical aberration is said to be recorrected.

The image of a dot made by a lens with spherical aberrations is usually obtained by dots surrounded by a halo of light. Spherical aberration usually shows up 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 matrix), instead of a clear point, a blur disk will be projected.

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

Focus shift

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

To get the best sharpness on a covered 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 work with an open aperture.

The photographer focuses at full aperture, and projects the disk of least blur at position b, then when shooting, the aperture automatically closes to the set value, and he does not suspect anything about the subsequent at this moment focus shift, which does not allow him to achieve the best sharpness.

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

SLR users can close the preview aperture to focus at the actual aperture.

Automatic focus shift compensation was proposed by Norman Goldberg. Zeiss has launched a line of rangefinder lenses for the Zeiss Ikon cameras, which have a specially designed circuitry to minimize focus shift with aperture changes. At the same time, spherical aberrations in lenses for rangefinder cameras are significantly reduced. How important is focus shift for rangefinder lenses, you ask? According to the manufacturer of the LEICA NOCTILUX-M 50mm f/1 lens, this value is about 100 µm.

The nature of the blur out of focus

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 blur zone.

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 The 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 distributions of light may appear in the blurred area of ​​the image.

Rice. 2 Blur changes before and after the focus point

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

And when the sensor is 5 mm ahead of the best focus (ie closer to the lens), the nature of the blur has changed towards a bright center surrounded by a weak halo. As you can see, the lens has been corrected for spherical aberration because it behaves in the opposite way to the example in Fig. one.

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

On Fig. 3 shows a cross, which was photographed in the center of the frame with the same 85 / 1.4 lens. The macrofur is extended by about 85 mm, which gives an increase of about 1:1. The camera (matrix) was moved in increments of 1 mm in both directions from the maximum focus. The cross is a more complex image than the dot, and the color indicators give 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. camera moves from -4 to +4 mm in 1 mm increments from best focus position (0)

Spherical aberration is responsible for the harsh nature of the blur at negative distances and the transition to soft blur at positive distances. Also of interest are color effects that occur due to longitudinal chromatic aberration (axial color). If the lens is poorly assembled, then spherical aberration and axial color are the only aberrations that show up 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 indicates that this lens is not intended for use as a macro lens. Most lenses are optimized for near field focusing and infinity focusing, but not for 1:1 macro. At this zoom, conventional lenses will behave worse than macro lenses, which are used specifically at close range.

However, even if the lens is used for standard applications, spherochromatism can show up in the out-of-focus area in normal shooting and affect 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 greatly 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 operating distance range.

In the case of lenses with undercorrected spherical aberration, an effective way to improve image quality is to stop down 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 schemes, but, as a rule, closing the aperture by one stop already gives a noticeable improvement in the image.

Alternatively, instead of fighting spherical aberration, the photographer may intentionally exploit it. Zeiss softening filters, despite the flat surface, add spherical aberrations to the image. They are popular with portrait photographers for their soft effect and impressive character.

© Paul van Walree 2004–2015
Translation: Ivan Kosarekov

Of all types of aberrations, spherical aberration is the most significant and in most cases the only 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 due to 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 pupil diameter decreases, therefore, in bright light, the resolution of the eye is higher than in low light, when the pupil diameter increases and the size of the spot, which is an 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 - to change the shape of the refractive surface. Such a possibility exists, in principle, in the surgical correction of the curvature of the cornea and in the replacement of a natural lens that has lost its optical properties, for example, due to a cataract, with an artificial one. An artificial lens can have refractive surfaces of any form accessible to modern technologies. Investigation of the influence of the shape of refractive surfaces on spherical aberration can most effectively and accurately be performed using computer simulations. Here we consider a rather simple computer simulation algorithm that allows such a study to be carried out, as well as the main results obtained using this algorithm.

The simplest way is 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 suffices to perform such a calculation in a two-dimensional approximation. The light beam is located in the main plane and is directed to the refractive surface parallel to the main optical axis. The course of this ray after refraction can be described using the circle equation, 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 beam with the main optical axis, i.e. refractive surface focus coordinates. This expression contains the surface parameters (radius), refractive indices and the distance between the main optical axis and the point where the beam hits the surface. The dependence of the focus coordinate on the distance between the optical axis and the point of incidence of the beam is spherical aberration. This dependence is easy to calculate and represent graphically. For a single spherical surface that deflects rays towards the main optical axis, the focal coordinate always decreases with increasing distance between the optical axis and the incident beam. The farther from the axis the beam falls on the refracting surface, the closer to this surface it crosses the axis after refraction. This is positive spherical aberration. As a result, the 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 a finite diameter in this plane, which leads to a decrease in the image contrast, i.e. to the deterioration of its quality. At one point, only those rays intersect that fall on the surface very close to the main optical axis (paraxial rays).

If a converging 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 a positive spherical aberration, i.e. rays falling parallel to the main optical axis farther from it cross this axis closer to the lens than rays going closer to the axis. Spherical aberration is practically absent also only for paraxial beams. If both surfaces of the lens are convex (like the lens), then the spherical aberration is greater than when the second refractive surface of the lens is concave (like the cornea).

Positive spherical aberration is due to excessive curvature of the refractive surface. As you move away from the optical axis, the angle between the tangent to the surface and the perpendicular to the optical axis increases faster than is necessary in order 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 should decrease with distance from the optical axis, i.e. the surface should not be spherical, in which the curvature is the same at all its points. In other words, the reduction of spherical aberration can only be achieved by using lenses with aspherical refractive surfaces. These can be, for example, the surfaces of an ellipsoid, a paraboloid, and a hyperboloid. In principle, other surface shapes can also be used. The attractiveness of elliptical, parabolic and hyperbolic forms is only in the fact 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 investigated theoretically using the method described above.

It is always possible to choose the parameters of spherical, elliptical, parabolic and hyperbolic surfaces so that their curvature in 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 surface the slowest, the parabolic surface even slower, and the hyperbolic surface the slowest of all (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 the lens farther from the optical axis will cross it farther from the lens than rays incident on the lens closer to the optical axis. For a hyperbolic lens, you can even choose such 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 - 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 action on the refractive power (the position of the paraxial focus) and spherical aberration is due to the presence of two geometric parameters, two semiaxes, in aspherical surfaces of revolution, the selection of which can ensure a reduction in spherical aberration without changing the refractive power. A spherical surface does not have such an opportunity, 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 no such possibility either, since a paraboloid of revolution also has only one parameter - the focal parameter. Thus, of the three aspherical surfaces mentioned, only two are suitable for controlled independent action 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 of 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 coordinate 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, one 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. It can be easily verified, for example, 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 far beams is about 3% closer than for paraxial beams. The entire optical system of the eye with this lens and with this cornea gathers the far rays about 4% closer than the paraxial rays. It can be said 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, set as a lens, gives a spherical aberration, approximately the same as the cornea alone, i.e. minimizing spherical aberration of the lens alone is not sufficient to minimize the entire optical system of the eye.

Thus, in order to minimize the 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 a minimum spherical aberration, but one that minimizes aberration in interaction with the cornea. If the refractive surfaces of the cornea are considered spherical, then in order to almost completely eliminate 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% at any pupil diameter. Almost the same neutralization of the spherical aberration of the optical system of the eye (up to approximately 0.3%) can be obtained even with the help of a lens, in which the first refractive surface is spherical and the second is hyperbolic.

Thus, the use of an artificial lens with aspherical, in particular, 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 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) that have left one source point and hit different points. retina as a result of exposure to all geometric aberrations and possible inaccurate focusing of the system. By summing up all the rays at all points of the retina that came there from all points of the source, such a model makes it possible 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 that the size of the scattering spot can be reduced to almost 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 the reduction in the size of the scattering spot. This limit is set by the wave nature of light. According to wave-based diffraction theory, the minimum diameter of a light spot in the image plane due to the diffraction of light by a circular hole is proportional (with a proportionality factor of 2.44) to the product of the focal length and the wavelength of the 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 for 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 reduce all rays to one point. Diffraction limits the improvement in image quality provided by any refractive optical system, even the ideal one. At the same time, light diffraction, which is no worse than refraction, can be used to obtain an image, which is successfully used in diffractive-refractive IOLs. But that is another topic.

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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 about the optical axis, it is sufficient to restrict ourselves to the choice of rays lying in the meridional plane. On 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 off-axis point

The ideal image of the object point A is determined by the paraxial beam that intersects the optical axis at a distance from the last surface. Rays that form end angles with the optical axis do not come to the point of an ideal image. For a single positive lens, the greater the absolute value of the angle, the closer to the lens the beam crosses the optical axis. This is due to the unequal optical power of the lens in its various zones, which increases with distance from the optical axis.

The indicated 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 plane of an ideal image, a circle of scattering is obtained, the diameter of which is equal to twice the value. The latter is related to the longitudinal spherical aberration by the relation

and is called transverse spherical aberration.

It should be noted that in the case of spherical aberration, symmetry is preserved in the beam of rays that has left the system. Unlike other monochromatic aberrations, spherical aberration takes place at all points of the field of the optical system, and in the absence of other aberrations for off-axis points, the beam of rays leaving the system will remain symmetrical with respect to the main beam (Fig. 114).

The approximate value of spherical aberration can be determined from the formulas for third-order aberrations through

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

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

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

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

Accordingly, for the 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 a real focal length.

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

if calculated under normalization conditions (256).

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

It follows from the above formulas that, for a given, the third-order spherical aberration is the greater, the larger the beam coordinate at the entrance pupil.

Since spherical aberration is present at all points in the field, when aberration correction of an optical system, priority is given 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. Both in positive and negative lenses, the extreme zones refract rays more strongly than the zones located near the axis (Fig. 115). The negative lens has positive spherical aberration. Therefore, the combination of a positive lens having negative spherical aberration with a negative lens results in a system with corrected spherical aberration. Unfortunately, spherical aberration can be eliminated only for some beams, 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 a residual spherical aberration. The residual aberrations of an optical system are usually presented in the form of tables and illustrated with graphs. For an object point located on the optical axis, plots of longitudinal and transverse spherical aberrations are given, presented as functions of coordinates, or

The curves of the longitudinal and the corresponding transverse spherical aberration are shown in Figs. 116. Graphs in fig. 116a 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. 116b correspond to the optical system, in which the spherical aberration is corrected for the beam passing through the edge of the entrance pupil, and the graphs in Fig. 116, c - optical system with redirected spherical aberration. Correction or recorrection of spherical aberration can be obtained, for example, by combining positive and negative lenses.

Transverse spherical aberration characterizes a circle of scattering, which is obtained instead of an ideal image of a point. The diameter of the circle of scattering for a given optical system depends on the choice of the image plane. If this plane is displaced relative to the ideal image plane (the Gaussian plane) by a value (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