The word aberration refers to many 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 movement of the observed object and the observer. The effect of aberration leads to the fact that the apparent direction to an object does not coincide with the geometric direction to it at the same moment in time.
The effect is that, due to the movement of the Earth around the Sun and the time it takes for light to travel, 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 using a telescope, we must not measure 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. In this case, the deviation of the star’s position is only 20.4 seconds, because the earth travels only 30 km in 1 second of time, and the light beam travels 300,000 km.
There are also several types geometric aberration. Spherical aberration- an aberration of a lens or objective, 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 the image being blurry. As a result, a point object such as a star can be seen as a small ball, taking the size of this ball as the size of the star.
Image field curvature- 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. Therefore, when the central part of the image is sharply focused, its edges will be out of focus and the image will be blurry. If you adjust the sharpness along the edges of the image, then its central part will be blurred. This type of aberration is not significant for astronomy.
Here are a few more types of aberration:
Diffraction aberration occurs due to the diffraction of light on the diaphragm and frame of the photographic lens. Diffraction aberration limits the resolving power of a photographic lens. Because of this aberration, the minimum angular distance between points resolved by the lens is limited by lambda/D radians, where lambda is the wavelength of the light used (the optical range usually includes electromagnetic waves with a length from 400 nm to 700 nm), D is the diameter of the lens . Looking at this formula, it becomes clear how important the lens diameter is. This parameter 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 shorter than the wavelength of light in the optical range. Therefore, for such telescopes, the minimum discernible angular distance is 100 times less than for conventional optical telescopes with the same lens diameter.
The study of aberration has 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.
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, we can note 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 focal point
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
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 the 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
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 eliminate 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
