Not too distant objects?
Let's say that we want to get a good look at some relatively nearby object. With the help of a Kepler tube this is quite possible. In this case, the image produced by the lens will be slightly further than the rear focal plane of the lens. And the eyepiece should be positioned so that this image is in the front focal plane of the eyepiece (Fig. 17.9) (if we want to make observations without straining our vision).
Problem 17.1. The Kepler tube is set to infinity. After the eyepiece of this tube is moved away from the lens at a distance D l= 0.50 cm, objects located at a distance became clearly visible through the pipe d. Determine this distance if the focal length of the lens F 1 = 50.00 cm.
after the lens was moved, this distance became equal
f = F 1+D l= 50.00 cm + 0.50 cm = 50.50 cm.
Let's write down the lens formula for the objective:
Answer: d» 51 m.
STOP! Decide for yourself: B4, C4.
Galileo's trumpet
The first telescope was designed not by Kepler, but by the Italian scientist, physicist, mechanic and astronomer Galileo Galilei (1564–1642) in 1609. In Galileo’s telescope, unlike Kepler’s telescope, the eyepiece is not a collecting, but scattering lens, therefore the path of rays in it is more complex (Fig. 17.10).
Rays coming from an object AB, pass through the lens - a collecting lens ABOUT 1, after which they form converging beams of rays. If the item AB– infinitely distant, then its actual image ab would have to be in the focal plane of the lens. Moreover, this image would be reduced and inverted. But in the path of the converging beams there is an eyepiece - a diverging lens ABOUT 2, for which the image ab is an imaginary source. The eyepiece turns a converging beam of rays into a diverging one and creates virtual direct image A¢ IN¢.
Rice. 17.10 
Viewing angle b at which we see the image A 1 IN 1, clearly greater than the visual angle a at which the object is visible AB with the naked eye.
Reader: It’s somehow very tricky... How can we calculate the angular magnification of the pipe?
Rice. 17.11
The lens gives a real image A 1 IN 1 in the focal plane. Now let's remember about the eyepiece - a diverging lens for which the image A 1 IN 1 is an imaginary source.
Let's construct an image of this imaginary source (Fig. 17.12).
1. Let's draw a beam IN 1 ABOUT through the optical center of the lens - this ray is not refracted.
Rice. 17.12
2. Let's draw from the point IN 1 beam IN 1 WITH, parallel to the main optical axis. Until the intersection with the lens (section CD) is a very real beam, and in the area DВ 1 is a purely “mental” line – to the point IN 1 in reality beam CD doesn't reach! It is refracted so that continuation of the refracted ray passes through the main front focus of the diverging lens - the point F 2 .
Beam intersection 1 with beam continuation 2 form a point IN 2 – imaginary image of an imaginary source IN 1. Dropping from a point IN 2 perpendicular to the main optical axis, we get a point A 2 .
Now note that the angle at which the image is seen from the eyepiece A 2 IN 2 is the angle A 2 OB 2 = b. From D A 1 OB 1 corner. Magnitude | d| can be found from the eyepiece lens formula: here imaginary source gives imaginary the image is in a diverging lens, so the lens formula is:
.
If we want observation to be possible without eye strain, a virtual image A 2 IN 2 must be “sent” to infinity: | f| ® ¥. Then parallel beams of rays will emerge from the eyepiece. And the imaginary source A 1 IN To do this, 1 must be in the rear focal plane of the diverging lens. In fact, when | f | ® ¥
.
This “limiting” case is shown schematically in Fig. 17.13.
From D A 1 ABOUT 1 IN 1
h 1 = F 1 a, (1)
From D A 1 ABOUT 2 IN 1
h 1 = |F 1 |b, (2)
Let us equate the right-hand sides of equalities (1) and (2), we obtain
.
So, we got the angular magnification of Galileo's tube
As we can see, the formula is very similar to the corresponding formula (17.2) for the Kepler tube.
The length of Galileo's pipe, as can be seen from Fig. 17.13, equal
l = F 1 – |F 2 |. (17.14)
Problem 17.2. The lens of theater binoculars is a converging lens with a focal length F 1 = 8.00 cm, and the eyepiece is a diverging lens with a focal length F 2 = –4.00 cm . What is the distance between the lens and the eyepiece if the image is viewed by the eye from the distance of best vision? How much do you need to move the eyepiece so that the image can be viewed with an eye adjusted to infinity?
In relation to the eyepiece, this image plays the role of an imaginary source located at a distance A behind the plane of the eyepiece. Virtual image S 2 given by the eyepiece is at a distance d 0 in front of the eyepiece plane, where d 0 – distance of best vision of a normal eye.
Let's write down the lens formula for the eyepiece:
The distance between the lens and the eyepiece, as can be seen from Fig. 17.14, equal
l = F 1 – a= 8.00 – 4.76 » 3.24 cm.
In the case when the eye is accommodated to infinity, the length of the pipe according to formula (17.4) is equal to
l 1 = F 1 – |F 2 | = 8.00 – 4.00 » 4.00 cm.
Therefore, the eyepiece displacement is
D l = l – l 1 = 4.76 – 4.00 » 0.76 cm.
Answer: l» 3.24 cm; D l» 0.76 cm.
STOP! Decide for yourself: B6, C5, C6.
Reader: Can Galileo's trumpet produce an image on a screen?
 Rice. 17.15 |
We know that a diverging lens can produce a real image only in one case: if the imaginary source is located behind the lens in front of the back focus (Fig. 17.15).
Problem 17.3. The Galilean telescope lens produces a true image of the Sun in the focal plane. At what distance between the lens and the eyepiece can an image of the Sun be obtained on the screen with a diameter three times larger than that of the actual image that would be obtained without the eyepiece? Lens focal length F 1 = 100 cm, eyepiece – F 2 = –15 cm.
The diverging lens creates on the screen real the image of this imaginary source is a segment A 2 IN 2. In the picture R 1 is the radius of the actual image of the Sun on the screen, and R– radius of the actual image of the Sun created only by the lens (in the absence of an eyepiece).
From similarity D A 1 OB 1 and D A 2 OB 2 we get:
.
Let us write down the lens formula for the eyepiece, taking into account that d< 0 – источник мнимый, f > 0 – valid image:
|d| = 10 cm.
Then from Fig. 17.16 find the required distance l between eyepiece and lens:
l = F 1 – |d| = 100 – 10 = 90 cm.
Answer: l= 90 cm.
STOP! Decide for yourself: C7, C8.
16.12.2009 21:55 | V. G. Surdin, N. L. Vasilyeva
These days we are celebrating the 400th anniversary of the creation of the optical telescope - the simplest and most effective scientific instrument that opened the door to the Universe for humanity. The honor of creating the first telescopes rightfully belongs to Galileo.
As you know, Galileo Galilei began experimenting with lenses in mid-1609, after he learned that a spotting scope had been invented in Holland for the needs of navigation. It was made in 1608, possibly independently of each other, by the Dutch opticians Hans Lippershey, Jacob Metius and Zechariah Jansen. In just six months, Galileo managed to significantly improve this invention, create a powerful astronomical instrument on its principle, and make a number of amazing discoveries.
Galileo's success in improving the telescope cannot be considered accidental. Italian glass masters had already become thoroughly famous by that time: back in the 13th century. they invented glasses. And it was in Italy that theoretical optics was at its best. Through the works of Leonardo da Vinci, it turned from a section of geometry into a practical science. “Make glasses for your eyes so that you can see the moon large,” he wrote at the end of the 15th century. It is possible, although there is no direct evidence of this, that Leonardo managed to implement a telescopic system.
He carried out original research on optics in the middle of the 16th century. Italian Francesco Maurolicus (1494-1575). His compatriot Giovanni Batista de la Porta (1535-1615) dedicated two magnificent works to optics: “Natural Magic” and “On Refraction.” In the latter, he even gives the optical design of the telescope and claims that he was able to see small objects at a great distance. In 1609, he tries to defend priority in the invention of the telescope, but factual evidence for this was not enough. Be that as it may, Galileo's work in this area began on well-prepared ground. But, paying tribute to Galileo’s predecessors, let us remember that it was he who made a functional astronomical instrument from a funny toy.
Galileo began his experiments with a simple combination of a positive lens as an objective and a negative lens as an eyepiece, giving threefold magnification. Now this design is called theater binoculars. This is the most popular optical device after glasses. Of course, modern theater binoculars use high-quality coated lenses, sometimes even complex ones, made up of several glasses, as lenses and eyepieces. They provide a wide field of view and excellent images. Galileo used simple lenses for both the objective and the eyepiece. His telescopes suffered from severe chromatic and spherical aberrations, i.e. produced an image that was blurry at the edges and unfocused in various colors.
However, Galileo did not stop, like the Dutch masters, with “theater binoculars”, but continued experiments with lenses and by January 1610 created several instruments with magnification from 20 to 33 times. It was with their help that he made his remarkable discoveries: he discovered the satellites of Jupiter, mountains and craters on the Moon, myriads of stars in the Milky Way, etc. Already in mid-March 1610, Galileo’s work was published in Latin in 550 copies in Venice. Starry Messenger”, where these first discoveries of telescopic astronomy were described. In September 1610, the scientist discovered the phases of Venus, and in November he discovered signs of a ring on Saturn, although he had no idea about the true meaning of his discovery (“I observed the highest planet in three,” he writes in an anagram, trying to secure the priority of the discovery). Perhaps not a single telescope in subsequent centuries made such a contribution to science as Galileo’s first telescope.
However, those astronomy enthusiasts who have tried to assemble telescopes from spectacle glasses are often surprised by the small capabilities of their designs, which are clearly inferior in “observational capabilities” to Galileo’s homemade telescope. Often, modern “Galileos” cannot even detect the satellites of Jupiter, not to mention the phases of Venus.
In Florence, in the Museum of the History of Science (next to the famous Uffizi Art Gallery), two of the first telescopes built by Galileo are kept. There is also a broken lens of the third telescope. This lens was used by Galileo for many observations in 1609-1610. and was presented by him to Grand Duke Ferdinand II. The lens was later accidentally broken. After the death of Galileo (1642), this lens was kept by Prince Leopold de' Medici, and after his death (1675) it was added to the Medici collection in the Uffizi Gallery. In 1793, the collection was transferred to the Museum of the History of Science.
Very interesting is the decorative figured ivory frame made for the Galilean lens by the engraver Vittorio Crosten. Rich and intricate floral patterns are interspersed with images of scientific instruments; Several Latin inscriptions are organically included in the pattern. At the top there was previously a ribbon, now lost, with the inscription “MEDICEA SIDERA” (“Medici Stars”). The central part of the composition is crowned with an image of Jupiter with the orbits of 4 of its satellites, surrounded by the text “CLARA DEUM SOBOLES MAGNUM IOVIS INCREMENTUM” (“Glorious [young] generation of gods, great offspring of Jupiter”). To the left and right are the allegorical faces of the Sun and Moon. The inscription on the ribbon weaving a wreath around the lens reads: “HIC ET PRIMUS RETEXIT MACULAS PHEBI ET IOVIS ASTRA” (“He was the first to discover both the spots of Phoebus (i.e. the Sun) and the stars of Jupiter”). On the cartouche below is the text: “COELUM LINCEAE GALILEI MENTI APERTUM VITREA PRIMA HAC MOLE NON DUM VISA OSTENDIT SYDERA MEDICEA IURE AB INVENTORE DICTA SAPIENS NEMPE DOMINATUR ET ASTRIS” (“The sky, open to the keen mind of Galileo, thanks to this first glass object, showed the stars, to this day since then invisible, rightly called by their discoverer Medicean. After all, the sage rules over the stars."
Information about the exhibit is contained on the website of the Museum of the History of Science: link No. 100101; reference #404001.
At the beginning of the twentieth century, Galileo's telescopes stored in the Florence Museum were studied (see table). Even astronomical observations were made with them.
Optical characteristics of the first lenses and eyepieces of Galileo telescopes (dimensions in mm)
It turned out that the first tube had a resolution of 20" and a field of view of 15". And the second one is 10" and 15", respectively. The magnification of the first tube was 14x, and the second 20x. A broken lens of the third tube with eyepieces from the first two tubes would give magnification of 18 and 35 times. So, could Galileo have made his amazing discoveries using such imperfect instruments?
Historical experiment
This is exactly the question that the Englishman Stephen Ringwood asked himself and, in order to find out the answer, he created an exact copy of Galileo’s best telescope (Ringwood S. D. A Galilean telescope // The Quarterly Journal of the Royal Astronomical Society, 1994, vol. 35, 1, p. 43-50) . In October 1992, Steve Ringwood recreated the design of Galileo's third telescope and spent a year making all sorts of observations with it. The lens of his telescope had a diameter of 58 mm and a focal length of 1650 mm. Like Galileo, Ringwood stopped down his lens to an aperture diameter of D = 38 mm to obtain better image quality with a relatively small loss of penetrating power. The eyepiece was a negative lens with a focal length of -50 mm, giving a magnification of 33 times. Since in this telescope design the eyepiece is placed in front of the focal plane of the lens, the total length of the tube was 1440 mm.
Ringwood considers the biggest drawback of the Galileo telescope to be its small field of view - only 10", or a third of the lunar disk. Moreover, at the edge of the field of view, the image quality is very low. Using the simple Rayleigh criterion, which describes the diffraction limit of the resolving power of the lens, one would expect quality images at 3.5-4.0". However, chromatic aberration reduced it to 10-20". The penetrating power of the telescope, estimated using a simple formula (2 + 5lg D), was expected around +9.9 m. However, in reality it was not possible to detect stars weaker than +8 m.
When observing the Moon, the telescope performed well. It was possible to discern even more details than were sketched by Galileo on his first lunar maps. “Perhaps Galileo was an unimportant draftsman, or was he not very interested in the details of the lunar surface?” - Ringwood is surprised. Or maybe Galileo’s experience in making telescopes and observing with them was not yet extensive enough? It seems to us that this is the reason. The quality of the glass, polished by Galileo's own hands, could not compete with modern lenses. And, of course, Galileo did not immediately learn to look through a telescope: visual observations require considerable experience.
By the way, why didn’t the creators of the first telescopes - the Dutch - make astronomical discoveries? Having made observations with theater binoculars (magnification 2.5-3.5 times) and with field binoculars (magnification 7-8 times), you will notice that there is a gap between their capabilities. Modern high-quality 3x binoculars make it possible (when observing with one eye!) to hardly notice the largest lunar craters; Obviously, a Dutch trumpet with the same magnification, but lower quality, could not do this either. Field binoculars, which provide approximately the same capabilities as Galileo's first telescopes, show us the Moon in all its glory, with many craters. Having improved the Dutch trumpet, achieving several times higher magnification, Galileo stepped over the “threshold of discovery.” Since then, this principle has not failed in experimental science: if you manage to improve the leading parameter of the device several times, you will definitely make a discovery.
Of course, Galileo's most remarkable discovery was the discovery of four satellites of Jupiter and the disk of the planet itself. Contrary to expectations, the low quality of the telescope did not greatly interfere with observations of the system of Jupiter satellites. Ringwood saw all four satellites clearly and was able, like Galileo, to mark their movements relative to the planet every night. True, it was not always possible to focus the image of the planet and the satellite well at the same time: the chromatic aberration of the lens was very difficult.
But as for Jupiter itself, Ringwood, like Galileo, was unable to detect any details on the planet’s disk. Low-contrast latitudinal bands crossing Jupiter along the equator were completely washed out as a result of aberration.
Ringwood obtained a very interesting result when observing Saturn. Like Galileo, at 33x magnification he saw only faint swellings (“mysterious appendages,” as Galileo wrote) on the sides of the planet, which the great Italian, of course, could not interpret as a ring. However, further experiments by Ringwood showed that when using other high magnification eyepieces, clearer ring features could still be discerned. If Galileo had done this in his time, the discovery of the rings of Saturn would have taken place almost half a century earlier and would not have belonged to Huygens (1656).
However, observations of Venus proved that Galileo quickly became a skilled astronomer. It turned out that at the greatest elongation the phases of Venus are not visible, because its angular size is too small. And only when Venus approached the Earth and in phase 0.25 its angular diameter reached 45", its crescent shape became noticeable. At this time, its angular distance from the Sun was no longer so great, and observations were difficult.
The most interesting thing in Ringwood's historical research, perhaps, was the exposure of one old misconception about Galileo's observations of the Sun. Until now, it was generally accepted that it was impossible to observe the Sun with a Galilean telescope by projecting its image onto a screen, because the negative lens of the eyepiece could not construct a real image of the object. Only the Kepler telescope, invented a little later, consisting of two positive lenses, made this possible. It was believed that the first time the Sun was observed on a screen placed behind an eyepiece was the German astronomer Christoph Scheiner (1575-1650). He simultaneously and independently of Kepler created a telescope of a similar design in 1613. How did Galileo observe the Sun? After all, it was he who discovered sunspots. For a long time there was a belief that Galileo observed the daylight with his eye through an eyepiece, using clouds as light filters or watching for the Sun in the fog low above the horizon. It was believed that Galileo's loss of vision in old age was partly caused by his observations of the Sun.
However, Ringwood discovered that Galileo's telescope could also produce a quite decent projection of the solar image onto the screen, and sunspots were visible very clearly. Later, in one of Galileo's letters, Ringwood discovered a detailed description of observations of the Sun by projecting its image onto a screen. It is strange that this circumstance was not noted before.
I think that every astronomy lover will not deny himself the pleasure of “becoming Galileo” for a few evenings. To do this, you just need to make a Galilean telescope and try to repeat the discoveries of the great Italian. As a child, one of the authors of this note made Keplerian tubes from spectacle glasses. And already in adulthood he could not resist and built an instrument similar to Galileo’s telescope. An attachment lens with a diameter of 43 mm with a power of +2 diopters was used as a lens, and an eyepiece with a focal length of about -45 mm was taken from an old theater binocular. The telescope turned out to be not very powerful, with a magnification of only 11 times, but its field of view turned out to be small, about 50" in diameter, and the image quality is uneven, deteriorating significantly towards the edge. However, the images became significantly better when the lens aperture was reduced to a diameter of 22 mm, and even better - up to 11 mm The brightness of the images, of course, decreased, but observations of the Moon even benefited from this.
As expected, when observing the Sun in projection onto a white screen, this telescope did indeed produce an image of the solar disk. The negative eyepiece increased the equivalent focal length of the lens several times (telephoto lens principle). Since there is no information on which tripod Galileo installed his telescope on, the author observed while holding the telescope in his hands, and used a tree trunk, fence or open window frame as a handhold. At 11x magnification this was sufficient, but at 30x magnification Galileo obviously might have had problems.
We can consider that the historical experiment to recreate the first telescope was a success. We now know that Galileo's telescope was a rather inconvenient and poor instrument from the point of view of modern astronomy. In all respects, it was inferior even to current amateur instruments. He had only one advantage - he was the first, and his creator Galileo “squeezed” everything that was possible out of his instrument. For this we honor Galileo and his first telescope.
Become Galileo
The current year 2009 was declared the International Year of Astronomy in honor of the 400th anniversary of the birth of the telescope. In addition to the existing ones, many new wonderful sites with amazing photographs of astronomical objects have appeared on the computer network.
But no matter how saturated the Internet sites are with interesting information, the main goal of the MGA was to demonstrate the real Universe to everyone. Therefore, among the priority projects was the production of inexpensive telescopes, accessible to anyone. The most widespread was the “galileoscope” - a small refractor designed by highly professional astronomers-optics. This is not an exact copy of Galileo's telescope, but rather its modern reincarnation. The “galileoscope” has a two-lens achromatic glass lens with a diameter of 50 mm and a focal length of 500 mm. The four-element plastic eyepiece provides 25x magnification, and the 2x Barlow lens brings it up to 50x. The telescope's field of view is 1.5 o (or 0.75 o with a Barlow lens). With such an instrument it is easy to “repeat” all of Galileo’s discoveries.
However, Galileo himself, with such a telescope, would have made them much larger. The tool's price of $15-20 makes it truly affordable. Interestingly, with a standard positive eyepiece (even with a Barlow lens), the "Galileoscope" is really a Kepler tube, but when using only a Barlow lens as an eyepiece, it lives up to its name, becoming a 17x Galilean tube. Repeating the discoveries of the great Italian in such an (original!) configuration is not an easy task.
This is a very convenient and quite widespread tool, suitable for schools and novice astronomy enthusiasts. Its price is significantly lower than that of previously existing telescopes with similar capabilities. It would be highly desirable to purchase such instruments for our schools.
A spotting scope (refractor telescope) is designed to make observations of distant objects. The tube consists of 2 lenses: an objective and an eyepiece.
Definition 1
Lens is a converging lens with a long focal length.
Definition 2
Eyepiece- This is a lens with a short focal length.
Converging or diverging lenses are used as an eyepiece.
Computer model of a telescope
Using a computer program, you can create a model demonstrating the operation of the Kepler telescope from 2 lenses. The telescope is designed for astronomical observations. Since the device displays an inverted image, this is inconvenient for ground-based observations. The program is configured so that the observer's eye is accommodated to an infinite distance. Therefore, a telescopic path of rays is performed in the telescope, that is, a parallel beam of rays from a distant point, which enters the lens at an angle ψ. It exits the eyepiece in exactly the same way as a parallel beam, but with respect to the optical axis at a different angle φ.
Angular magnification
Definition 3Angular magnification of telescope is the ratio of angles ψ and φ, which is expressed by the formula γ = φ ψ.
The following formula shows the angular magnification of the telescope through the focal length of the lens F 1 and eyepiece F 2:
γ = - F 1 F 2 .
The negative sign that appears in the angular magnification formula in front of the F 1 lens means that the image is upside down.
If desired, you can change the focal lengths F 1 and F 2 of the lens and eyepiece and the angle ψ. The values of the angle φ and angular magnification γ are indicated on the device screen.
If you notice an error in the text, please highlight it and press Ctrl+Enter
Coursework
discipline: Applied optics
On the topic: Calculation of the Kepler tube
Introduction
Telescopic optical systems
1 Aberrations of optical systems
2 Spherical aberration
3 Chromatic aberration
4 Comatic aberration (coma)
5 Astigmatism
6 Image field curvature
7 Distortion (distortion)
Dimensional calculation of the optical system
Conclusion
Literature
Applications
Introduction
Telescopes are astronomical optical instruments designed for observing celestial bodies. Telescopes are used using various radiation receivers for visual, photographic, spectral, photoelectric observations of celestial bodies.
Visual telescopes have a lens and an eyepiece and are a so-called telescopic optical system: they convert a parallel beam of rays entering the lens into a parallel beam emerging from the eyepiece. In such a system, the back focus of the lens coincides with the front focus of the eyepiece. Its main optical characteristics: apparent magnification Г, angular field of view 2W, exit pupil diameter D", resolution and penetrating power.
The apparent magnification of an optical system is the ratio of the angle at which the image produced by the optical system of the device is observed to the angular size of the object when observing it directly with the eye. Apparent magnification of the telescopic system:
G=f"rev/f"ok=D/D",
where f"about and f"ok are the focal lengths of the lens and eyepiece,
D - inlet diameter,
D" - exit pupil. Thus, by increasing the focal length of the lens or decreasing the focal length of the eyepiece, greater magnifications can be achieved. However, the greater the magnification of the telescope, the smaller its field of view and the greater the distortion of images of objects due to the imperfections of the system's optics.
The exit pupil is the smallest cross section of the light beam emerging from the telescope. During observations, the pupil of the eye is aligned with the exit pupil of the system; therefore it should not be larger than the pupil of the observer's eye. Otherwise, some of the light collected by the lens will not reach the eye and will be lost. Typically, the diameter of the entrance pupil (lens frame) is much larger than the pupil of the eye, and point light sources, particularly stars, appear much brighter when viewed through a telescope. Their apparent brightness is proportional to the square of the diameter of the telescope's entrance pupil. Faint stars, not visible to the naked eye, can be clearly visible in a telescope with a large entrance pupil. The number of stars visible through a telescope is much greater than that observed directly with the eye.
telescope optical aberration astronomical
1. Telescopic optical systems
1 Aberrations of optical systems
Aberrations of optical systems (Latin - deviation) - distortions, image errors caused by imperfections in the optical system. All lenses, even the most expensive ones, are susceptible to aberrations to varying degrees. It is believed that the larger the range of focal lengths of the lens, the higher the level of its aberrations.
The most common types of aberrations are below.
2 Spherical aberration
Most lenses are designed using lenses with spherical surfaces. These lenses are easy to make, but the spherical shape of the lenses is not ideal for producing sharp images. The effect of spherical aberration manifests itself in a softening of contrast and blurring of details, the so-called “soap”.
How does this happen? Parallel rays of light, when passing through a spherical lens, are refracted; rays passing through the edge of the lens merge at a focal point closer to the lens than light rays passing through the center of the lens. In other words, the edges of the lens have a shorter focal length than the center. The image below clearly shows how a beam of light passes through a spherical lens and what causes spherical aberrations.
Light rays passing through the lens near the optical axis (closer to the center) are focused in area B, further from the lens. Light rays passing through the edge zones of the lens are focused in area A, closer to the lens.
3 Chromatic aberration
Chromatic aberration (CA) is a phenomenon caused by the dispersion of light passing through the lens, i.e. decomposition of a beam of light into its components. Rays of different wavelengths (different colors) are refracted at different angles, so a rainbow is formed from a white beam.
Chromatic aberrations lead to a decrease in image clarity and the appearance of color fringe, especially on contrasting objects.
To combat chromatic aberrations, special apochromatic lenses made of low-disperse glass are used, which do not decompose light rays into waves.
1.4 Comatic aberration (coma)
Coma or comatic aberration is a phenomenon visible at the periphery of the image, which is created by a lens corrected for spherical aberration and causes light rays entering the edge of the lens at some angle to converge into the shape of a comet rather than the desired point. Hence its name.
The comet's shape is oriented radially, with its tail pointing either toward or away from the center of the image. The resulting blur at the edges of the image is called comatic flare. Coma, which can occur even in lenses that accurately reproduce a point as a point on the optical axis, is caused by the difference in refraction between light rays from a point located off the optical axis passing through the edges of the lens, and the main light ray from the same point passing through center of the lens.
Coma increases as the main beam angle increases and leads to decreased contrast at the edges of the image. A certain degree of improvement can be achieved by stopping down the lens. Coma can also cause blurred areas of the image to be blown out, creating an unpleasant effect.
The elimination of both spherical aberration and coma for a subject located at a certain shooting distance is called aplanatism, and a lens corrected in this way is called aplanatism.
5 Astigmatism
With a lens corrected for spherical and comatic aberration, an object point on the optical axis will be accurately reproduced as a point in the image, but an object point located off the optical axis will not appear as a point in the image, but rather as a shadow or line. This type of aberration is called astigmatism.
You can observe this phenomenon at the edges of the image by shifting the lens focus slightly to a position in which the object point is sharply depicted as a line oriented radially from the center of the image, and again shifting the focus to another position in which the object point is sharply depicted as a line oriented in the direction of a concentric circle. (The distance between these two focal positions is called the astigmatic difference.)
In other words, the light rays in the meridional plane and the light rays in the sagittal plane are in different positions, so these two groups of rays do not connect at one point. When the lens is set to the optimal focal position for the meridional plane, the light rays in the sagittal plane are aligned in the direction of a concentric circle (this position is called meridional focus).
Similarly, when the lens is set at the optimal focal position for the sagittal plane, light rays in the meridional plane form a line oriented in the radial direction (this position is called sagittal focus).
With this type of distortion, objects in the image appear curved, blurred in places, straight lines appear curved, and darkening is possible. If the lens suffers from astigmatism, then it is sold for spare parts, since this phenomenon cannot be cured.
6 Image field curvature
With this type of aberration, the image plane becomes curved, so if the center of the image is in focus, then the edges of the image are out of focus and vice versa, if the edges are in focus, then the center is out of focus.
1.7 Distortion (distortion)
This type of aberration manifests itself in the distortion of straight lines. If straight lines are concave, the distortion is called pincushion, if it is convex, it is called barrel-shaped. Variable focal length lenses typically create barrel distortion at wide angle (minimum zoom) and pincushion distortion at telephoto (maximum zoom).
2. Dimensional calculation of the optical system
Initial data:
To determine the focal lengths of the lens and eyepiece, we solve the following system:
f' ob + f' ok = L;
f' ob / f' ok =|Г|;
f' ob + f' ok = 255;
f' ob / f' ok =12.
f’ ob +f’ ob /12=255;
f’ ob =235.3846 mm;
f’ ok =19.6154 mm;
The diameter of the entrance pupil is calculated by the formula D=D'Г
D in =2.5*12=30 mm;
We find the linear field of view of the eyepiece using the formula:
; y’ = 235.3846*1.5 o ; y’=6.163781 mm;
The angular field of view of the eyepiece is determined by the formula:
Prism system calculation
D 1 - input face of the first prism;
D 1 =(D input +2y’)/2;
D 1 =21.163781 mm;
Path length of the rays of the first prism =*2=21.163781*2=42.327562;
D 2 - input face of the second prism (derivation of the formula in Appendix 3);
D 2 =D input *((D input -2y’)/L)*(f’ ob /2+);
D 2 =18.91 mm;
Path length of the rays of the second prism =*2=18.91*2=37.82;
When calculating the optical system, the distance between the prisms is chosen within 0.5-2 mm;
To calculate the prism system, it is necessary to bring it to air.
Let us reduce the path length of the prism rays to air:
l 01 - length of the first prism reduced to air
n=1.5688 (refractive index of glass BK10)
l 01 =l 1 /n=26.981 mm
l 02 = l 2 /n=24.108 mm
Determining the amount of eyepiece movement to ensure focusing within ±5 diopters
first you need to calculate the price of one diopter f’ ok 2 /1000 = 0.384764 (price of one diopter)
Moving the eyepiece to achieve the desired focus: 
mm
Checking for the need to apply a reflective coating to the reflective faces:
(permissible angle of deviation of the deviation from the axial beam, when the condition of total internal reflection is not yet violated)
(the maximum angle of incidence of rays on the input face of the prism, at which there is no need to apply a reflective coating). Therefore: no reflective coating is needed.
Eyepiece calculation:
Since 2ω’ = 34.9 o, the required type of eyepiece is symmetrical.
f’ ok =19.6154 mm (calculated focal length);
K p = S’ F /f’ ok = 0.75 (conversion factor)
S ’ F = K p * f’ ok
S ’ F =0.75* f’ ok (back focal length value)
Eye relief is determined by the formula: S’ p = S’ F + z’ p
z’ p is found using Newton’s formula: z’ p = -f’ ok 2 /z p where z p is the distance from the front focus of the eyepiece to the aperture diaphragm. In spotting scopes with a prism turning system, the aperture diaphragm is usually the lens barrel. As a first approximation, we can take z p equal to the focal length of the lens with a minus sign, therefore:
z p = -235.3846 mm
The exit pupil relief is:
S’ p = 14.71155+1.634618=16.346168 mm Aberration calculation of optical system components. Aberration calculation involves calculating the aberrations of the eyepiece and prisms for three wavelengths. Eyepiece aberration calculation: The calculation of eyepiece aberrations is carried out in the reverse path of rays, using the ROSA application package. δy’ approx =0.0243 Calculation of aberrations of a prism system: Aberrations of reflective prisms are calculated using the formulas for third-order aberrations of an equivalent plane-parallel plate. For BK10 glass (n=1.5688). Longitudinal spherical aberration: δS’ pr =(0.5*d*(n 2 -1)*sin 2 b)/n 3 b’=arctg(D/2*f’ ob)=3.64627 o d=2D 1 +2D 2 =80.15 mm dS’pr =0.061337586 Position chromatism: (S’ f - S’ c) pr =0.33054442 Meridional coma: δy"=3d(n 2 -1)*sin 2 b’*tgω 1 /2n 3 δy" = -0.001606181 Calculation of lens aberrations: Longitudinal spherical aberration δS’ sf: δS’ sf =-(δS’ pr + δS’ ok)=-0.013231586 Position chromatism: (S’ f - S’ c) rev = δS’ хр =-((S’ f - S’ c) pr +(S’ f - S’ c) ok)=-0.42673442 Meridional coma: δy’ k = δy’ ok - δy’ pr δy’к =0.00115+0.001606181=0.002756181 Determination of the structural elements of the lens. Aberrations of a thin optical system are determined by three main parameters P, W, C. Approximate formula of Prof. G.G. Slyusareva connects the main parameters P and W: P = P 0 +0.85(W-W 0) Calculation of a two-lens glued lens comes down to finding a certain combination of glasses with given values of P0 and C. Calculation of a two-lens lens using the method of Prof. G.G. Slyusareva: ) Based on the lens aberration values δS’ xp, δS’ sf, δy’ k, obtained from the conditions for compensating for aberrations of the prism system and eyepiece, the aberration sums are found: S I хр = δS’ хр = -0.42673442 S I = 2*δS’ sf /(tgb’) 2 S I =6.516521291 S II =2* δy к ’/(tgб’) 2 *tgω S II =172.7915624 ) Based on the sums, the system parameters are found: S I хр / f’ ob S II / f'ob ) P 0 is calculated: P 0 = P-0.85(W-W 0) ) According to the nomogram graph, the line intersects the 20th cell. Let's check the combinations of K8F1 and KF4TF12 glasses: ) From the table are the values of P 0 , φ k and Q 0 corresponding to the specified value for K8F1 (not suitable) φk = 2.1845528 for KF4TF12 (suitable) ) After finding P 0 ,φ k, and Q 0, Q is calculated using the formula: ) After finding Q, the values a 2 and a 3 of the first zero ray are determined (a 1 =0, since the object is at infinity, and 4 =1 - from the normalization condition): ) The radii of curvature of thin lenses are determined from the values of a i: Thin Lens Radius: ) After calculating the radii of a thin lens, lens thicknesses are selected based on the following design considerations. The thickness along the axis of the positive lens d1 consists of the absolute values of the arrows L1, L2 and the thickness along the edge, which must be no less than 0.05D. h=D in /2 L=h 2 /(2*r 0) L 1 =0.58818 2 =-1.326112 d 1 =L 1 -L 2 +0.05D ) Based on the obtained thicknesses, calculate the heights: h 1 =f about =235.3846 h 2 =h 1 -a 2 *d 1 h 2 =233.9506 h 3 =h 2 -a 3 *d 2 ) Radius of curvature of a lens with finite thicknesses: r 1 =r 011 =191.268 r 2 = r 02 *(h 1 /h 2) r 2 =-84.317178 r 3 =r 03 *(h 3 /h 1) The results are monitored by calculations on a computer using the ROSA program: Lens aberration alignment
The obtained and calculated aberrations are close in values. alignment of telescope aberrations
The layout involves determining the distance to the prism system from the lens and eyepiece. The distance between the objective and the eyepiece is defined as (S’ F ’ ob + S’ F ’ ok + Δ). This distance is the sum of the distance between the lens and the first prism, equal to half the focal length of the lens, the beam path length in the first prism, the distance between the prisms, the beam path length in the second prism, the distance from the last surface of the second prism to the focal plane and the distance from this plane to eyepiece 692+81.15+41.381+14.777=255
Conclusion For astronomical lenses, resolution is determined by the smallest angular distance between two stars that can be seen separately in a telescope. Theoretically, the resolving power of a visual telescope (in arcseconds) for yellow-green rays, to which the eye is most sensitive, can be estimated by the expression 120/D, where D is the diameter of the telescope's entrance pupil, expressed in millimeters. The penetrating power of a telescope is the maximum stellar magnitude of a star that can be observed with a given telescope under good atmospheric conditions. Poor image quality, due to trembling, absorption and scattering of rays by the earth's atmosphere, reduces the maximum stellar magnitude of actually observed stars, reducing the concentration of light energy on the retina, photographic plate or other radiation receiver in the telescope. The amount of light collected by the telescope's entrance pupil increases in proportion to its area; At the same time, the penetrating power of the telescope also increases. For a telescope with a lens diameter of D millimeters, the penetrating power, expressed in magnitude during visual observations, is determined by the formula: mvis=2.0+5 log D. Depending on the optical system, telescopes are divided into lens (refractor), mirror (reflector) and mirror-lens. If a lens telescopic system has a positive (converging) objective and a negative (diffusing) eyepiece, then it is called a Galilean system. The Kepler telescopic lens system has a positive objective and a positive eyepiece. Galileo's system gives a direct virtual image, has a small field of view and a small aperture ratio (large exit pupil diameter). Simplicity of design, short length of the system and the ability to obtain direct images are its main advantages. But the field of view of this system is relatively small, and the absence of a real image of the object between the lens and the eyepiece does not allow the use of a reticle. Therefore, the Galilean system cannot be used for focal plane measurements. Currently, it is used mainly in theater binoculars, where a large magnification and field of view are not required. The Kepler system gives a real and inverted image of an object. However, when observing celestial bodies, the latter circumstance is not so important, and therefore the Kepler system is most common in telescopes. The length of the telescope tube is equal to the sum of the focal lengths of the lens and eyepiece: L=f"about+f"approx. The Kepler system can be equipped with a reticle in the form of a plane-parallel plate with a scale and crosshairs. This system is widely used in combination with a prism system to produce direct images from the lenses. Keplerian systems are used mainly for visual telescopes. In addition to the eye, which is a radiation receiver in visual telescopes, images of celestial objects can be recorded on a photographic emulsion (such telescopes are called astrographs); a photomultiplier and an electron-optical converter make it possible to amplify a weak light signal from stars located at great distances many times; images can be projected onto a television telescope tube. The image of the object can also be sent to an astrospectrograph or astrophotometer. To point the telescope tube at the desired celestial object, use the telescope mount (tripod). It provides the ability to rotate the pipe around two mutually perpendicular axes. The base of the mount carries an axis about which a second axis with the telescope tube rotating around it can rotate. Depending on the orientation of the axes in space, mounts are divided into several types. In altazimuth (or horizontal) mounts, one axis is located vertically (the azimuth axis), and the second (the zenith distance axis) is horizontal. The main disadvantage of an altazimuth mount is the need to rotate the telescope around two axes to track a celestial object moving due to the apparent daily rotation of the celestial sphere. Many astrometric instruments are equipped with altazimuth mounts: universal instruments, passage and meridian circles. Almost all modern large telescopes have an equatorial (or parallax) mount, in which the main axis - the polar or clock axis - is directed towards the celestial pole, and the second - the declination axis - is perpendicular to it and lies in the equatorial plane. The advantage of a parallax mount is that to track the daily movement of a star, it is enough to rotate the telescope only around one polar axis. Literature 1. Digital technology. /Ed. E.V. Evreinova. - M.: Radio and communication, 2010. - 464 p. Kagan B.M. Optics. - M.: Energoatomizdat, 2009. - 592 p. Skvortsov G.I. Computer technology. - MTUSI M. 2007 - 40 p. Appendix 1 Focal length 19.615 mm Relative aperture 1:8 Field of view angle Move the eyepiece by 1 diopter. 0.4 mm Structural elements
19.615; =14.755;
Axial beam
Δ C Δ F S´ F -S´ C Main beam
Meridional section of an inclined beam
ω 1 =-1 0 30’ ω 1 =-1 0 10’30”
The curiosity and desire to make new discoveries of the great scientist G. Galileo gave the world a wonderful invention, without which it is impossible to imagine modern astronomy - this telescope. Continuing the research of Dutch scientists, the Italian inventor achieved a significant increase in the scale of the telescope in a very short time - this happened in just a few weeks.
Galileo's telescope resembled modern samples only vaguely - it was a simple lead stick, at the ends of which the professor placed biconvex and biconcave lenses.
An important feature and the main difference between Galileo’s creation and previously existing telescopes was the good image quality obtained through high-quality grinding of optical lenses - the professor was involved in all processes personally, and did not trust the delicate work to anyone. The scientist’s hard work and determination bore fruit, although a lot of painstaking work had to be done to achieve a decent result - out of 300 lenses, only a few options had the necessary properties and quality.
The samples that have survived to this day are admired by many experts - even by modern standards, the quality of the optics is excellent, and this is considering the fact that the lenses are several centuries old.
Despite the prejudices that reigned during the Middle Ages and the tendency to consider progressive ideas as “the machinations of the devil,” the spotting scope gained well-deserved popularity throughout Europe.
The improved invention made it possible to obtain a thirty-five-fold magnification, unthinkable at the time of Galileo's life. With the help of his telescope, Galileo made many astronomical discoveries, which paved the way for modern science and aroused the enthusiasm and thirst for research in many inquisitive and inquisitive minds.
The optical system invented by Galileo had a number of disadvantages - in particular, it was susceptible to chromatic aberration, but subsequent improvements carried out by scientists made it possible to minimize this effect. It is worth noting that during the construction of the famous Paris Observatory, telescopes were used that were equipped with Galileo’s optical system.
Galileo's telescope or telescope has a small viewing angle - this can be considered its main drawback. A similar optical system is currently used in theater binoculars, which are essentially two spotting scopes connected together.
Modern theater binoculars with a central internal focusing system usually offer 2.5-4x magnification, sufficient for observing not only theatrical productions, but also sports and concert events, and are suitable for sightseeing trips involving detailed sightseeing.
The small size and elegant design of modern theater binoculars make them not only a convenient optical instrument, but also an original accessory.
