Let W be the wavefront passing through the center of the exit pupil and associated with the beam that forms the image and exits the point. If there are no aberrations, then W coincides with the sphere S, the center of which lies at the point of the paraxial image, and it itself passes through the point, S is called the Gaussian reference sphere (Fig. 2.2).

Let and be the points of intersection of the ray with the reference sphere and the wave front W, respectively.

Rice. 2.2.Wave and radiation aberrations

The optical path length Ф = can be called the aberration of the wave element at point Q or simply wave aberration and considered positive if and are located on opposite sides of Q. In conventional devices, wave aberrations reach 40-50 wavelengths, however, in devices used for more precise research (for example, in astronomical telescopes or microscopes), they should be much smaller, on the order of fractions of a wavelength.

Expressions for wave aberration are easily obtained using the Hamilton point characteristic function of the system.

If we use square brackets to denote the optical path length, then

Here we used the fact that the points and lie on the same wave front, i.e. .

Let us introduce two rectangular coordinate systems with mutually parallel axes, the origins of which are located at the axial points and planes of the object and image, and the Z axes coincide with the axis of the system. Points in object space will be considered in the first system, and in image space - in the second. Z-coordinates of the planes in which the pupils lie are indicated by and (in Fig. 2.1 ) .

According to (1), wave aberration is expressed through the point characteristic V as follows:

where () are the coordinates of point , and (X,Y,Z) are the coordinates of point Q. The coordinates (X,Y,Z) are no longer independent; they are connected by a relationship taking into account that point Q lies on the reference sphere, i.e.

Coordinates of the paraxial image point, M - Gaussian transverse magnification and R - radius of the Gaussian reference sphere

The value of Z in expression (2) can be eliminated using (3), as a result of which Ф becomes a function of only , , and , i.e.

Radiation aberrations are associated with the aberration function Ф (, ; X, Y) simple relations. From (2) we have

(6)

If , and are the angles that form the ray with the axes, and (X, Y, Z) and () are the coordinates of the points and then, in Fig. 2.2, we get

is the distance from to , and is the refractive index of the medium in image space. Further from (3) we have

(9)

Substituting (7) and (9) into relation (6), we find for the components of the beam aberration

(10)

The last relations are exact, but the quantity on the right is

itself depends on the coordinates of the point, i.e., on ray aberrations. However, for most practical purposes it can be replaced by the radius of the reference sphere R or another approximate expression (see below, equation (15)). It is easy to show that, due to the symmetry of the problem, the value of Ф depends on four variables included in only three combinations, namely: , and . In fact, if we introduce polar coordinates in the XY planes, i.e., put

then it turns out that Ф depends only on , , and , or, which is the same, Ф depends on , , and 0. Let us now assume that the X and Y axes of systems with origins at And rotates at the same angle and in the same direction relative to the axis of the system.

In this case, , , do not change, and the angle 0 increases by the angle of rotation. Since the function Ф is invariant under such rotations, it should not depend on the last variable, that is, it depends only on , , and . Consequently, the aberration function Ф is a function of three scalar products

two vectors and .

It follows from this that when decomposing F in a series in powers of four coordinates, odd powers will be absent. Because F(0, 0; 0, 0) = 0, then there will be no terms of degree zero either. Moreover, there will be no terms of the second degree, since, according to (10), they correspond to ray aberrations that linearly depend on the coordinates, and this contradicts the fact that , is a paraxial image of the point . Thus, our expansion has the form

where c is a constant, a is a polynomial of degree 2k in coordinates and contains them only in the form of three scalar invariants (12). The 2k degree term is said to describe wave aberration of order 2k. The lowest order aberrations (2k = - 4) are usually called primary aberrations or Seidel aberrations.

To estimate the order of magnitude of some expressions and the accuracy of our calculations, it is convenient to introduce the parameter . This parameter can be any first-order quantity, say, the angular aperture of the system. Then we can assume that all rays passing through the system make angles O() with the optical axis, where the symbol O() means that the angle is of the order of .

Let us estimate the error that arises when replacing in the main equation (10) with values independent of and . From (3) and (5) we have

then instead of (8) we can write

Relations (10) for the components of beam aberration take the form

(16)

(17)

3. Primary aberrations (Seidel aberrations)

Using arguments completely similar to those related to the aberration function, it can be shown that the power series expansion of the perturbed Schwarzschild eikonal has the following form, due to the symmetry of the problem:

Where is a polynomial of degree 2 k by four variables; Moreover, these variables come in only three combinations:

In relation (1) there is no term of the second degree, since otherwise this would contradict the fact that, , , and in the approximation of paraxial optics.

Since variables only appear in combinations (2), the term must be of the form

where A, B,... are constants. The signs and numerical factors in (3) are generally accepted; expressions for ray aberrations in this case take a simple form.

Of course, the power series expansion of a function has the same form as (1), but it does not contain a zero-order term (), and the main term differs from it in that it does not contain the term . Thus, the general expression for the lowest (fourth) order wave aberration is written as follows:

where B, C,. - the same coefficients as in (3).

The general expression for the lowest (third) order components of ray aberration is in the form

(5)

Coefficient A is not included in expressions (4) and (5), i.e., there are only five types of aberration of the lowest order, characterized by five coefficients B, C, D, E and F. As mentioned above, these aberrations are called primary aberrations or aberrations Seidel.

When studying Seidel aberrations, it is convenient to choose the axes in such a way that the plane yz passed through a point of the object; Then . If you then enter polar coordinates

then (4) will take the form

In the particular case of equality to zero of all coefficients in (7), the wave front passing through the exit pupil coincides (in the considered approximation) with the Gaussian reference sphere (see Fig. 2.2). In general, these coefficients are different from zero. Then each term in (7) describes certain type we deviate the new front from the correct spherical shape; in Fig. 3.1 shows five various types aberrations.

The importance of ray aberrations associated with a specific point on an object can be illustrated graphically using so-called aberration (or characteristic) curves. These curves are the geometric location of the intersection points of the rays emerging from the fixed zone =const of the exit pupil with the image plane. Then the surface formed by aberration curves. corresponding to all possible values of , represents a non-ideal image.

Fig. 3.1 Primary wave aberrations.

A) spherical. B) coma. B) astigmatism. D) field curvature. D) distortion

Let us consider separately each of the Seidel aberrations

3.1 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.

Lens aberrations. Aberrations of optical systems

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 things arise? 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 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:

Comatic aberration.
Distortion.
Astigmatism.
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 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 Its significant compensation is the use of aspherical elements in the optical design.

3. Comatic aberration, or “Coma”

This private view spherical aberration for side beams. 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.

Aberrations of optical systems(from Latin aberratio– deviation) – distortions, errors, or inaccuracies of images formed by optical systems. The reason for their occurrence is that the beam deviates from the direction in which it should go in a close-to-ideal optical system. Various violations of homocentricity (distinctness, correspondence or color) in the structure of beams of rays emerging from the optical system characterize aberrations.

The most common types of aberrations in optical systems can be considered:

1. Spherical aberration. It is characterized by a lack of image. With it, light rays emitted by one point of the object, passing near the axis of the optical system, and rays passing through parts of the system distant from the axis, are not collected at one point.

2. To whom. This is the name for aberration that occurs during the oblique passage of light rays through an optical system. As a result of this, a violation of the symmetry of the beam of rays relative to its axis is observed and the image of the point (which is created by the system) takes the form of an asymmetric scattering spot.

3. Astigmatism. ABOUT b This aberration is said to occur when a light wave experiences deformation while passing through an optical system. As a result of this, a deformation is observed in which beams of rays emanating from one point of the object do not intersect at one point, but are located in two mutually perpendicular segments at a certain distance from each other. Such beams are called astigmatic.

4. Distortion. This is the name of aberration, characterized by a violation of the geometric similarity between the object and the image of the object. It is caused by the difference in linear optical magnification in different areas of the image.

5. Curvature of the image field. With this aberration, a process is observed when the image of a flat object turns out to be sharp on a curved surface, and not on a plane, as it should be.

All of the above types of aberrations in optical systems are called geometric or Seidel aberrations. In real systems, certain types of geometric aberrations can be found extremely rarely. Much more often we can observe the symbiosis of all aberrations. And the selection method individual species aberration is an artificial technique designed to facilitate the analysis of a phenomenon.

At the same time, chromatic aberration also exists. There is a connection between this about the type of aberration and the dependence of the refractive index of optical media on the wavelength of light. Manifestations of this aberration are observed in optical systems that include elements made of refractive materials. As an example, lenses. We also note that mirrors are characterized by achromaticity.

The manifestation of chromatic aberrations can be observed when the appearance of extraneous coloring of the image, as well as when the image of an object appears colored contours that were not previously observed in the object. Chromatic aberrations are caused by the dispersion of optical media (the dependence of the refractive index of optical materials on the length of the transmitted light wave). It is from them that the optical system is formed

These aberrations include chromatic aberration or position chromatism (sometimes called "longitudinal chromatism") and chromatic aberration or magnification chromatism.

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In this article with a scary title, we will look into the features of optical distortion of lenses. Have you noticed that when shooting at a wide angle, the edges of the frame are distorted? And when you try to take a photo in backlight, a pink, blue or greenish fringing appears around objects? If you haven't noticed, take another look. In the meantime, let's figure out why this happens.

First, you need to understand and accept the fact that ideal optical systems (i.e., in our case, lenses) do not exist. Each optical system has inherent distortions that it introduces into the projection of reality onto the image (photography). Distortions of optical systems are scientifically called aberrations, i.e. deviations from the norm or from the ideal.

Aberrations of various optical systems can take different shapes and be more noticeable or virtually invisible. Typically, the more expensive the lens, the better the quality of its optical system, which means the fewer aberrations it has.

Types of aberrations

Most often, the word “aberration” itself in photography is used in the combination “chromatic aberration.” As you might have already guessed, chromatic aberration- this is one of the types of distortions caused by the characteristics of the optical system of the lens, which is expressed in the form of color deviations. A typical example of chromatic aberration is unnatural colored outlines at the edges of subjects. Chromatic aberrations appear most clearly on contours in high-contrast areas of the image. For example, on the border of tree branches shot against a bright sky, or along the contour of hair when shooting a portrait in .

The cause of chromatic aberration is an optical phenomenon such as dispersion of the glass from which the lenses are made. Glass dispersion is that light waves different lengths(of different color spectrum) when passing through the lens they are refracted at different angles. White light (which contains a whole spectrum of light waves of different lengths, i.e. different colors), passing through the objective lens, first breaks up into a color spectrum, which is then reassembled into a beam for projecting the image onto the camera matrix. As a result, due to the difference in the refraction angles of colored rays, deviations occur during image formation. This is reflected in errors in the color distribution in the image. This is why colored outlines, colored spots or stripes may appear in a photograph that were not present on the subject.

Chromatic aberrations to one degree or another are inherent in almost all lenses. Cheap optics are much more lame than elite series lenses. During the design phase of an optical system, manufacturers can minimize chromatic aberration by using achromatic lenses. Secret achromatic lens is that its design consists of two types of glass: one with a low and the other with a high refractive index of light. Selecting the proportion of a combination of materials with different refractive indices of light makes it possible to reduce the deviations of light waves at the moment of splitting white light.

Don't be too upset if your lens doesn't contain achromatic lenses - chromatic aberration occur mainly when shooting in difficult conditions lighting, and are very noticeable only when viewing a photograph at 80-100% magnification. In addition, no one has canceled processing in graphic editors, which can eliminate such optical errors. To learn how to do this, read the next article, “Correcting Lens Errors” (coming soon).

Another type of lens aberration includes geometric distortion, which is commonly called lens distortion. Lens distortion manifests itself in distortion of the proportions of objects located closer to the edges of the frame. Speaking scientific language, with distortion, the linear increase in objects in the field of view occurs unevenly. As a result, objects at the edges of the frame appear unnaturally flattened or elongated.

Based on the nature of distortions, there are two types: type of distortion: positive ( concave or cushion-shaped) and negative ( convex or barrel-shaped). If no geometric distortions are observed in the frame, then they say that there is no distortion. In this case, the image looks smooth and flat; pay attention to the perfectly straight horizon line in the image below. Typically, it is along the horizon line that you can easily notice geometric distortions in landscape photography.

Distortion is most pronounced when used. Moreover, the larger the viewing angle of the lens (the shorter the focal length), the more pronounced geometric aberrations . You've probably noticed that vertical and horizontal lines, when shooting wide, become curved as they approach the edges of the frame. The most striking example lens distortion- These are photographs taken with an ultra-wide-angle fisheye lens. But in the case of fisheye, distortion is not an error or defect in the optics. Rather, it is its feature that allows you to expand the viewing angle of the lens to 180 degrees (and even more).

When using wide-angle lenses (FR<24 мм) можно наблюдать бочкообразную (вогнутую) дисторсию, при использовании длиннофокусных объективов (ФР>200 mm) pincushion (convex) distortion may appear. Lenses with average values focal lengths Geometric distortions across the frame field are usually not typical.

This is why they say that a wide-angle lens distorts proportions, and lenses with a focal length of 70-200 mm smooth out any distortions. And that is why it is customary to shoot portraits with 70-200 mm lenses, which do not distort the proportions of the face and figure. But portraits shot wide open look comical and are used only to create a special caricature effect. Moreover, the smaller the distance between the shooting point and the subject, the stronger the distortion of proportions. For example, as in the famous portrait of Bill Clinton (photo below), the head looks disproportionately small compared to big hands and knees. But in this case, this is precisely the creative idea, the author’s style of the photographer. By using a wide-angle lens, he was able to create a vivid visual image - an association with a person former president USA.

Just like chromatic aberration, distortion can be corrected when designing the lens. For this purpose, a aspherical lens, and lenses with corrected distortion are called aspherical. You may have seen such names (ASP) in the description technical characteristics to the lens. Such lenses are usually more expensive than their spherical counterparts, but when shooting they convey the proportions of objects in the frame without distortion. However, there is a relatively inexpensive Sigma 10-20 mm F4-5.6 EX DC HSM lens, which gives a smooth picture even at a maximum viewing angle of 102 degrees.

If your wide angle lens gives geometric aberrations, so there are two ways to fix this:

If you're using a zoom lens, you can simply set it to a longer focal length and take a couple of steps back. So, you will have the same composition in your frame, but by changing the focal length you will get rid of distortions.
Geometric aberrations can be corrected using graphic editors (primarily Photoshop). But at the same time, be prepared to lose some of the objects in the photo, because when correcting curvatures, cropping occurs at the edges of the frame. Read the next article to learn how to do this.

We acquire the idea of the eye as a perfect optical device from school when studying the section of physics “Optics”. When studying relevant sciences in higher or secondary specialized educational institutions This idea of the eye is consolidated, acquiring additional information. Therefore, the statement of S.N. Fedorov that the eye is an imperfect instrument and the task of the ophthalmologist is to improve it, was perceived with skepticism by many doctors for a long time.

When designing even the simplest telescope, it is necessary not only to focus the optical system at one point (to exclude myopia, farsightedness and astigmatism of the telescope), but also to ensure the quality of the resulting image. The lenses from which the telescope is made must be made of good glass, almost ideal shape and with a well-finished surface. Otherwise, the image will be unclear, distorted and blurry. That's when the study of aberrations began - the smallest roughness and unevenness of refraction. And with the advent of devices for identifying and measuring eye aberrations, a new dimension entered ophthalmology - aberrometry.

Aberrations can be of different orders. The simplest and most well-known aberrations are actually myopia, farsightedness and astigmatism. They are called defocus or aberrations of the second, lower order. Aberrations higher order and are the same roughness and unevenness of refraction.

Higher order aberrations are also divided into several orders. It is generally accepted that the quality of vision is affected by aberrations mainly up to the seventh order. For ease of perception, there is a set of Zernike polynomials that display the types of monochromatic aberrations as a three-dimensional model of refractive unevenness. A set of these polynomials can more or less accurately display any unevenness in the refraction of the eye.

Aberrations are divided into three main groups:

Higher order monochromatic aberrations:

spherical aberration,
coma,
astigmatism of oblique beams,
field curvature, distortion,
irregular aberrations.

To describe a complex of higher order monochromatic aberrations, polynomials of the Zernike (Zernike) mathematical formalism are used. It’s good if they are close to zero, and the root mean square deviation of the wavefront RMS (root mean square) is less than the wavelength or equal to 0.038 μm (Marechal criterion). However, these are the subtleties of refractive surgery.

Standard table of Zernike polynomialsis a kind of set of three-dimensional illustrations of aberrations up to the seventh order:

defocus,
astigmatism,
astigmatism of oblique beams,
coma,
spherical aberration,
shamrock,
quatrefoil and so on, up to octafoil (trefoil, tetrafoil, pentafoil, hexafoil...).

"Trefoils" are from three to eight uniform sectors of a circle with increased optical power. Their occurrence may be associated with the main centripetal directions of stroma fibrils, a kind of stiffening ribs of the cornea.

The aberration pattern of the eye is very dynamic. Monochromatic aberrations mask chromatic ones. When the pupil dilates in a darker room, spherical aberrations increase, but diffraction aberrations decrease, and vice versa. With age-related decline in accommodation abilities, higher-order aberrations, which were previously a stimulus and increased the accuracy of accommodation, begin to reduce the quality of vision.

Therefore, it is currently difficult to determine the significance of positive and negative influence each type of aberration on the vision of each person.

Causes of aberrations

Everyone has them. These make up the individual refractive map of the eye. Modern devices detect higher order aberrations that somehow affect the quality of vision in 15% of people. But everyone has individual characteristics of refraction.

Suppliers of aberrations are the cornea and lens.

The causes of aberrations may be:

congenital anomaly(very small irregularities that have little effect on vision, lenticonus);
corneal injury(the corneal scar tightens the surrounding tissue, depriving the cornea of sphericity);
operation(radial keratotomy, lens removal through a corneal incision, laser correction, thermokeratoplasty and other operations on the cornea);
corneal diseases(consequences of keratitis, cataract, keratoconus, keratoglobus).

The reason ophthalmologists pay attention to aberrations isophthalmic surgery. Ignoring aberrations and not taking into account their impact on the quality of vision, ophthalmology existed for quite a long time. Before this, aberrations were studied and fought against them. negative impact only manufacturers of telescopes, telescopes and microscopes.

Surgeries on the cornea or lens (meaning a corneal incision) increases higher order aberrations by several orders of magnitude, which can sometimes lead to a decrease in postoperative visual acuity. Therefore, the widespread introduction of artificial lens implantation, keratotomy and laser correction into ophthalmological practice contributed to the development of diagnostic equipment: keratotopographs appeared that analyze the refractive map of the cornea, and now aberrometers that analyze the entire wavefront from the anterior surface of the cornea to the retina.

Aberrations caused by LASIK

By correcting defocus (myopia, farsightedness), the refractive surgeon adds high-order aberrations to the patient.
Formation of a corneal flap by a microkeratome leads to an increase in higher order aberrations.
Complications during LASIK lead to an increase in higher order aberrations.
The healing process leads to an increase in higher order aberrations.

It was not possible to remove micro-roughness and unevenness using an excimer laser with a slit beam. A device with the possibility of point ablation was invented and put into production, that is, the diameter of the laser beam in some models is less than a millimeter. Using Zernike polynomials were introduced into practice computer programs, allowing you to automatically convert the data obtained from the aberrometer individual card refraction in a laser installation into an algorithm that controls the beam, eliminating not only residual defocus, but also higher-order aberrations.Zernike polynomials become a set of tools, each designed to remove a specific component in the aberration complex.

When performing such personalized laser ablation, the cornea should approach in shape to the level of an optically ideal sphere.

Higher order aberrations

Chromatic, astigmatism of oblique beams, coma, etc. All together they form an image of the surrounding world on the retina, the perception of which is strictly individual for each person.

Spherical aberration. Light passing through the periphery of a biconvex lens is refracted more than at the center. The main “supplier” of spherical aberration in the eye is the lens, and secondarily the cornea. The wider the pupil, that is, the larger part of the lens takes part in the visual act, the more noticeable the spherical aberration.
In refractive surgery, spherical aberration is most often induced by artificial lenses, LASIK andlaser thermokeratoplasty.
Aberrations of inclination angles of optical beams. Asphericity of refractive surfaces is a mismatch between the centers of images of luminous points located outside the axis of the optical system. They are divided into aberrations of large angles of inclination (astigmatism of oblique beams) and small angles of inclination (coma).
Coma has nothing to do with the known diagnosis of resuscitators. Its aberrometric pattern is similar to a circle located in the optical center of the cornea and divided by a line into two even halves. One of the halves has high optical power, and the other has low optical power. With such an aberration, a person sees a luminous point as a comma. When describing objects, people with such aberration use the words “tail”, “shadow”, “additional contour”, “double vision”. The direction of these optical effects (the aberration meridian) can be different. The cause of coma may be a congenital or acquired imbalance of the optical system of the eye. The optical axis (on which the focus of the lens is located) of the cornea does not coincide with the axis of the lens and the entire optical system is not focused in the center of the retina, in the macula. Coma may also be one of the components of refractive unevenness in keratoconus. During LASIK, coma may appear as a result of decentering of the laser ablation zone or the healing characteristics of the cornea during laser correction of farsightedness.
Distortion- violation of geometric similarity between an object and its image - distortion. Points of an object at different distances from the optical axis are depicted with different magnifications.

Laser correction is not a monopolist in the correction of aberrations. Artificial lenses and contact lenses have already been developed that compensate for some types of higher order aberrations.

Ministry of Education

RUSSIAN FEDERATION

TYUMEN STATE UNIVERSITY

Coursework

"Aberrations of optical systems"

Completed by: 2nd student

course gr. 473

…………….

Checked:

Tyumen 2009

Introduction

1. Chromatic aberration

2. Wave and radiation aberrations; aberration functions

3. Primary aberrations (Seidel aberrations)

3.1 Spherical

3.3 Astigmatism and field curvature

3.4 Distortion

References

Introduction

Aberrations of optical systems (from the Latin Aberratio - deviation), distortions, image errors formulated by optical systems. Aberrations in optical systems mean that optical images are not entirely clear, do not accurately match objects, or appear colored. The most common types of aberrations in optical systems are: spherical– a defect in the image in which the light rays emitted by one point of the object, passing near the optical axis of the system, and the rays passing through parts of the system distant from the axis, are not collected at one point: coma– aberration that occurs when light rays pass obliquely through an optical system. If, during the passage of an optical system, a spherical light wave is deformed so that beams of rays emanating from one point of the object do not intersect at one point, but are located in two mutually perpendicular segments at some distance from each other, then such beams are called astigmatic, and this beam itself aberration – astigmatism. An aberration called distortion, leads to a violation of the geometric similarity between the object and its image. Aberrations of optical systems also include the curvature of the image field.

Optical systems can simultaneously exhibit several types of aberrations. Their elimination is carried out in accordance with the purpose of the system; it is often a difficult task. The aberrations of optical systems listed above are called geometric. There is also chromatic aberration associated with the dependence of the refractive index of optical media on the wavelength of light.

1. Chromatic aberration

If a beam of non-monochromatic light falls on a refractive surface, it is split into several rays, each of which has a specific wavelength. Therefore, when crossing an optical system, rays of light with different wavelengths will propagate after the first refraction along not exactly the same paths. As a result, the image will appear blurry, in which case the system is said to have chromatic aberration.

Rice. 1. Longitudinal and transverse chromatic aberration.

We will limit ourselves to considering points and rays located near the axis, i.e., we will assume that for each wavelength the mapping obeys the laws of paraxial optics. In this case, we talk about first-order chromatic aberration, or primary aberration. Let and be point mappings R at different wavelengths (Fig. 1); then the projections on the directions parallel and perpendicular to the axis determine the longitudinal and transverse chromatic aberrations, respectively.

Consider the change in the focal length of a thin lens depending on the change in the refractive index. Magnitude (n - 1)f for such a lens does not depend on the wavelength. Hence

(1)

Magnitude

(2)

Fig.2. Typical dispersion curves for different types of glass

I – heavy flint; II – heavy barium crown; III – light flint; IV – heavy crown; V – borosilicate crown.

where , and are the refractive indices corresponding to the Fraunhofer lines F, D and C (4861, 5893 and 6563), serves as a rough measure of glass dispersion and is called relative dispersion. From (1) it is clear that this value is approximately equal to the distance between the red and blue images divided by the focal length of the lens. In Fig. Figure 2 shows the change in refractive index values with wavelength for several types of glass commonly used in optical systems. The corresponding values range from 1/60 to 1/30.

Rice. 3. Achromatic doublet

To obtain an image good quality It is necessary that both monochromatic and chromatic aberrations be small. Usually some compromise solution is chosen, since in general it is impossible to eliminate all types of aberrations simultaneously. It is often sufficient to eliminate chromatic aberration for two selected wavelengths. The choice of these wavelengths depends, naturally, on the purpose of a particular optical system; For example, photographic lenses, unlike instruments used for visual observation, are usually “achromatized” for colors close to the blue end of the spectrum, since a regular photographic plate is more sensitive to the blue region of the spectrum than the human eye. Of course, achromatization for two wavelengths does not completely eliminate color error. The remaining chromatic aberration is called the secondary spectrum.

Let us now consider the conditions under which two thin lenses form a combination free from chromatic focal length. The reciprocal of the focal length of a combination of two thin lenses located at a distance l from each other, equal

(3)

As we see, when

(4)

If achromatization is performed for lines C and F, then using (1) and (2) we obtain

(5)

Where and are the relative dispersions of both lenses.

One method of reducing chromatic aberration is to use two touching thin lenses (Fig. 3), one made of crown and the other of flint. In this case, since l = 0, we obtain from (5)

(6)

or, using(3),

, (7)

relations (7) for given types of glass and a given focal length uniquely determine , and . But , and depend on three radii of curvature, therefore, the value of one of them can be chosen arbitrarily. This additional degree of freedom can sometimes reduce spherical aberration to a minimum.

Another way to create an achromatic system is to use two race lenses made of the same glass (), and located from each other at a distance equal to half the sum of their focal lengths, i.e.

(8)

The achromatic nature of such a combination of lenses follows directly from (5).

In a device consisting of several parts, in general, it is impossible to simultaneously eliminate position chromatism and magnification chromatism unless this is done for each of its parts. Let us prove the last statement for the case of two centered thin lenses separated by a distance l .

The thin lens' image is a central projection from its center; therefore (Fig. 4),

Fig.4. Achromatization of a system of two thin lenses

Since , we find to increase

If the wavelength changes, the magnitude will remain the same, and the magnitude will also be the same if we assume that there is no chromaticity in position. Consequently, the condition for the absence of chromaticism in the increase of the system can be written in the form

(11)

Since , , then (11) is satisfied only when , i.e. if each of these lenses is achromatized.

2. Wave and ray aberrations, functions of aberrations

Let us consider a rotationally symmetric optical system. Let , and , be the points of intersection of the ray emerging from the object point , respectively, with the plane of the entrance pupil, the plane of the exit pupil and the plane of the paraxial image. If the is a paraxial image of a point, then the vector is called ray aberration or simply ray aberration (Fig. 2.1).

Rice. 2.1. Radiation aberration

Fig.3.2. 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.

3.2 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

Fig.3.3. Coma.

As we see, for fixed and zone radius, the 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 to , and its center is located at a distance from the paraxial focus to the side negative values at. Consequently, this circle touches two lines passing through the paraxial image, and components with the axis at angles of 30°. If everyone comes running possible values, 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.

3.3 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. and ).

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)

(12)

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

(14)

Likewise

(15)

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

(16)

and similarly

(17)

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, is 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

(20)

Half-difference

(21)

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.

3.4 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. Distortion A) subject. B) barrel-shaped. B) cushion-shaped

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 curvature in the distortion field is relatively harmless, since they can be eliminated using appropriate calculations.

optical aberration chromatic astigmatism distortion

References:

1. Savelyev I.V. Course of general physics, volume 3, optics, atomic physics.

2. Landsberg G. S. Optics.

3. Sivukhin D.V. General course of physics, vol. 4, optics.

4. Born M., Wolf E. Fundamentals of optics

5. Physical encyclopedic dictionary, ed. A. M. Prokhorova.