The telescope (refractor telescope) is designed for observing 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 is a short focal length lens.
Collecting or diffusing lenses are used as an eyepiece.
Computer model of the telescope
With the help of a computer program, you can create a model demonstrating the operation of Kepler's 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 set up so that the observer's eye is accommodated to an infinite distance. Therefore, in the telescope, a telescopic path of rays is performed, that is, a parallel beam of rays from a distant point, which enters the lens at an angle ψ. It leaves the eyepiece in the same way with a parallel beam, but with respect to the optical axis already at a different angle φ.
Angular magnification
Definition 3Angular magnification of the telescope is the ratio of the 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 objective lens F 1 and the eyepiece F 2:
γ = - F 1 F 2.
A negative sign 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 device displays the values of the angle φ and angular magnification γ.
If you notice an error in the text, please select it and press Ctrl + Enter
Course work
by discipline: Applied optics
On the topic: Calculation of the Kepler pipe
Introduction
Telescopic optical systems
1 Aberrations of optical systems
2 Spherical aberration
3 Chromatic aberration
4 Comatic aberration (coma)
5 Astigmatism
6 Curvature of the image field
7 Distortion (distortion)
Dimensional calculation of the optical system
Conclusion
Literature
Applications
Introduction
Telescopes are astronomical optical instruments designed to observe celestial bodies. Telescopes are used with the use of 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 leaving the eyepiece. In this system, the back focus of the objective is aligned with the front focus of the eyepiece. Its main optical characteristics are: apparent magnification Г, angular field of view 2W, exit pupil diameter D ", resolution and penetrating power.
The apparent magnification of the optical system is the ratio of the angle at which the image provided by the optical system of the device is observed to the angular size of the object when viewed directly with the eye. Visible magnification of the telescopic system:
G = f "about / f" ok = D / D ",
where f "about and f" is the focal length of the lens and eyepiece,
D - inlet diameter,
D "- exit pupil. Thus, increasing the focal length of the lens or decreasing the focal length of the eyepiece, you can achieve higher magnifications. However, the higher the magnification of the telescope, the smaller its field of view and the greater the distortion of images of objects due to imperfections in the optics of the system.
The exit pupil is the smallest section of the light beam exiting the telescope. When observing, 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 enter the eye and will be lost. Typically, the diameter of the entrance pupil (lens barrel) is much larger than the pupil of the eye, and point light sources, particularly stars, appear significantly brighter when viewed through a telescope. Their apparent brightness is proportional to the square of the telescope entrance pupil diameter. Faint stars that are invisible to the naked eye can be clearly seen with a telescope with a large entrance pupil diameter. The number of stars visible with a telescope is much larger than that observed directly with the eye.
telescope optical aberration astronomical
1. Telescopic optical systems
1 Aberrations of optical systems
Aberrations of optical systems (lat. - deviation) - distortions, image errors caused by imperfection of the optical system. Any lenses, even the most expensive ones, are subject to aberrations to varying degrees. It is believed that the wider the focal length range 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 lens is not ideal for sharp images. The spherical aberration effect is manifested in softening the contrast and blurring the details, the so-called "soap".
How does this happen? Parallel rays of light are refracted when passing through a spherical lens, 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 due to which spherical aberrations appear.
Light rays passing through the lens near the optical axis (closer to the center) are focused in region B, farther 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 ray of light into its components. Beams with different wavelengths (different colors) are refracted at different angles, so a rainbow is formed from a white beam.
Chromatic aberration leads to a decrease in image clarity and the appearance of colored "fringe", especially on contrasting objects.
To combat chromatic aberrations, special apochromatic lenses made of low-dispersion glass are used that do not decompose light rays into waves.
1.4 Comatic aberration (coma)
Coma or coma aberration is a phenomenon visible at the periphery of an image that is created by a lens that has been corrected for spherical aberration and causes light rays arriving at the edge of the lens at an angle to converge, in the form of a comet, rather than in the form of a desired point. Hence its name.
The comet's shape is oriented radially, with its tail pointing either towards the center or away from the center of the image. The resulting blurring 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 outside the optical axis and passing through the edges of the lens, and the main light ray from the same point passing through the center of the lens.
The coma increases as the angle of the main beam increases and leads to a decrease in contrast at the edges of the image. A certain degree of improvement can be achieved by stopping the lens. Coma can also blow out blurry areas of the image, creating an unpleasant effect.
The elimination of both spherical aberration and coma for an object located at a certain shooting distance is called aplanatism, and a lens corrected in this way is called aplanat.
5 Astigmatism
With the lens corrected for spherical and comatic aberration, the point of an object on the optical axis will be accurately reproduced as a point in the image, but an object point located outside the optical axis will appear not as a point in the image, but rather as a shadow or a line. This type of aberration is called astigmatism.
You can observe this phenomenon at the edges of the image by slightly shifting the focus of the lens to a position where the point of the object is sharply depicted as a line oriented radially from the center of the image, and again shifting focus to another position in which the point of the object is sharply depicted as a line. oriented in the direction of the concentric circle. (The distance between these two focal positions is called the astigmatic difference.)
In other words, the rays of light in the meridional plane and the rays of light in the sagittal plane are in different positions, so these two groups of rays do not connect at one point. When the lens is in the optimal focal position for the meridional plane, the light beams in the sagittal plane are aligned in the direction of a concentric circle (this position is called meridional focus).
Likewise, when the lens is set at the optimal focal position for the sagittal plane, the light beams 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 look curved, blurry in places, straight lines look curved, blackouts are possible. If the lens suffers from astigmatism, then it is allowed for spare parts, since this phenomenon is not curable.
6 Curvature of the image field
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 as distortion of straight lines. If straight lines are concave, the distortion is called pincushion, if convex, it is barrel-shaped. Varifocal lenses typically produce barrel distortion at “wide” (zoom at minimum) and pincushion at telephoto (zoom at maximum).
2. Dimensional calculation of the optical system
Initial data:
To determine the focal lengths of the lens and eyepiece, we will 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 entrance pupil diameter is calculated by the formula D = D'G
D in = 2.5 * 12 = 30 mm;
The linear field of view of the eyepiece is found by the formula:
; y '= 235.3846 * 1.5 o; y '= 6.163781 mm;
The angular field of view of the eyepiece is found by the formula:
Calculation of the prism system
D 1 is the entrance face of the first prism;
D 1 = (D in + 2y ') / 2;
D 1 = 21.163781 mm;
The length of the path of the rays of the first prism = * 2 = 21.163781 * 2 = 42.327562;
D 2 - the input face of the second prism (derivation of the formula in Appendix 3);
D 2 = D in * ((D in -2y ') / L) * (f' ob / 2 +);
D 2 = 18.91 mm;
The length of the path 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 in the range of 0.5-2 mm;
To calculate the prism system, it is necessary to bring it to air.
Let us bring the length of the path of the rays of the prisms to air:
l 01 - reduced to air length of the first prism
n = 1.5688 (refractive index of BK10 glass)
l 01 = l 1 /n=26.981 mm
l 02 = l 2 /n=24.108 mm
Determination of the amount of eyepiece movement to ensure focusing within ± 5 diopters
first, you need to deduct the price of one diopter f 'ok 2/1000 = 0.384764 (price of one diopter)
Moving the eyepiece to maintain the specified focus: 
mm
Checking whether reflective surfaces are required to be coated with a reflective coating:
(permissible deviation angle of deviation from the axial ray, when the condition of total internal reflection is not yet violated)
(the limiting 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, the required eyepiece type is symmetrical.
f 'ok = 19.6154mm (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)
Removal of the exit pupil is determined by the formula: S ’p = S’ F + z ’p
z ’p is found by 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 telescopes with a prismatic processing system, the lens barrel is usually the aperture diaphragm. 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
Removal of the exit pupil is equal to:
S 'p = 14.71155 + 1.634618 = 16.346168 mm Aberration calculation of the components of the optical system. Aberration calculation includes calculation of eyepiece and prism aberrations for three wavelengths. Eyepiece aberration calculation: The calculation of the aberrations of the eyepiece is carried out in the reverse path of the rays, using the ROSA software package. δy 'ok = 0.0243 Calculation of the aberrations of the prism system: The aberrations of the reflective prisms are calculated using the third-order aberration formulas for 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 ’= arctan (D / 2 * f’ ob) = 3.64627 o d = 2D 1 + 2D 2 = 80.15 mm dS 'pr = 0.061337586 Chromatism of position: (S ’f - S’ c) pr = 0.33054442 Meridian coma: δy "= 3d (n 2 -1) * sin 2 b '* tgω 1 / 2n 3 δy "= -0.001606181 Calculation of lens aberrations: Longitudinal spherical aberration δS 'sp: δS ’sp = - (δS’ pr + δS ’ok) = - 0.013231586 Chromatism of position: (S ’f - S’ c) about = δS ’xp = - ((S’ f - S ’c) pr + (S’ f - S ’c) ok) = - 0.42673442 Meridian coma: δy ’to = δy’ ok - δy ’pr δy ’k = 0.00115 + 0.001606181 = 0.002756181 Determination of structural elements of the lens. Aberrations of a thin optical system are determined by three main parameters P, W, C. The approximate formula of prof. G.G. Slyusareva connects the main parameters P and W: P = P 0 +0.85 (W-W 0) The calculation of a two-lens glued lens is reduced to finding a specific combination of glasses with given values of P 0 and C. Calculation of a two-lens objective by the method of prof. G.G. Slyusareva: ) According to the aberration values of the lens δS ’xp, δS’ sf, δy ’k obtained from the conditions for compensating for the aberrations of the prism system and the eyepiece, the aberration sums are found: S I xp = δS 'xp = -0.42673442 S I = 2 * δS ’sf / (tgb’) 2 S I = 6.516521291 S II = 2 * δy to ’/ (tgb’) 2 * tgω S II = 172.7915624 ) According to the sums, the system parameters are found: S I xp / f 'ob S II / f 'ob ) P 0 is calculated: P 0 = P-0.85 (W-W 0) ) According to the graph-nomogram, the line crosses the 20th cell. Let's check the combinations of K8F1 and KF4TF12 glasses: ) From the table, the values of P 0, φ to and Q 0 are found, corresponding to the specified value for K8F1 (not suitable) φ k = 2.1845528 for KF4TF12 (suitable) ) After finding P 0, φ to, and Q 0, Q is calculated by 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 values of a i are used to determine the radii of curvature of thin lenses: Radius of Thin Lens: ) After calculating the radii of a thin lens, the thicknesses of the lenses are selected based on the following design considerations. The thickness along the axis of the positive lens d1 is the sum of the absolute values of the arrows L1, L2 and the thickness along the edge, which must not be 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, the heights are calculated: 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 ) Radii 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 control of the results is carried out by calculation on a computer using the ROSA program: equalization of lens aberrations
The obtained and calculated aberrations are close in value. alignment of telescope aberrations
The arrangement consists in determining the distance to the prism system from the objective 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 length of the beam in the first prism, the distance between the prisms, the length of the beam 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 objectives, resolution is determined by the smallest angular distance between two stars that can be seen separately through a telescope. The theoretical resolution of a visual telescope (in arc seconds) 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 entrance pupil of the telescope, expressed in millimeters. The permeable power of a telescope is the limiting stellar magnitude of a star that can be observed with a given telescope under good atmospheric conditions. Poor image quality, due to the trembling, absorption and scattering of rays by the earth's atmosphere, reduces the limiting stellar magnitude of actually observed stars, reducing the concentration of light energy on the retina of the eye, photographic plate or other radiation detector in a telescope. The amount of light collected by the entrance pupil of the telescope increases in proportion to its area; in this case, the penetrating power of the telescope also increases. For a telescope with a lens diameter D of millimeters, the penetrating force, expressed in magnitudes during visual observations, is determined by the formula: mvis = 2.0 + 5 lg D. Depending on the optical system, telescopes are divided into lens (refractors), mirror (reflectors) and mirror-lens. If a telescopic lens system has a positive (converging) lens and a negative (diffusing) eyepiece, then it is called a Galileo system. Kepler's 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 (large diameter of the exit pupil). Simplicity of design, short length of the system and the possibility of obtaining a direct image 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 objective and the eyepiece does not allow the use of a reticle. Therefore, the Galilean system cannot be used for measurements in the focal plane. Currently, it is used mainly in theater binoculars, where high magnification and field of view are not required. Kepler's 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. In this case, the length of the telescope tube is equal to the sum of the focal lengths of the objective and the 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 a cross hairs. This system is widely used in combination with a prism system to obtain a direct image of lenses. Keplerian systems are mainly used for visual telescopes. In addition to the eye, which is the receiver of radiation in visual telescopes, images of celestial objects can be recorded on a photographic emulsion (such telescopes are called astrographs); a photomultiplier tube and an image converter make it possible to amplify many times a weak light signal from stars distant at great distances; images can be projected onto a television telescope tube. The object image can also be directed to an astrospectrograph or astrophotometer. The telescope mount (tripod) is used to guide the telescope tube to the desired celestial object. It provides the ability to rotate the pipe around two mutually perpendicular axes. The base of the mount carries an axis about which the 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. Altazimuth (or horizontal) mounts have one axis vertically (azimuth axis) and the other (zenith axis) horizontally. The main disadvantage of the 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 supplied 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 - polar or clock - is directed to the pole of the world, 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 diurnal motion of a star, it is enough to rotate the telescope around only one polar axis. Literature 1. Digital technology. / Ed. E.V. Evreinova. - M .: Radio and communication, 2010 .-- 464 p. Kagan B.M. Optics. - M .: Enerngoatomizdat, 2009 .-- 592 p. Skvortsov G.I. Computer Engineering. - MTUSI M. 2007 - 40 p. Annex 1 Focal length 19.615 mm Aperture ratio 1: 8 Angle of view Moving the eyepiece 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 oblique beam
ω 1 = -1 0 30 ' ω 1 = -1 0 10'30 "
The telescope is an optical device designed for viewing very distant objects with the eye. Like a microscope, it consists of an objective and an eyepiece; both are more or less complex optical systems, although not as complex as in the case of a microscope; however, we will schematically represent them with thin lenses. In telescopes, the objective and eyepiece are positioned so that the rear focus of the objective almost coincides with the front focus of the eyepiece (Fig. 253). The lens provides an actual zoomed-out reverse image of an infinitely distant object in its rear focal plane; This image is viewed through the eyepiece, like a magnifying glass. If the front focus of the eyepiece coincides with the rear focus of the objective, then when viewing a distant object, beams of parallel rays emerge from the eyepiece, which is convenient for observation with the normal eye in a calm state (without accommodation) (cf. § 114). But if the observer's vision is somewhat different from normal, then the eyepiece is moved, setting it "in the eyes". By moving the eyepiece, "aiming" of the telescope is also produced when examining objects located at different, not very large distances from the observer.
Rice. 253. Location of the objective and eyepiece in the telescope: back focus. Lens aligns with the front focus of the eyepiece
The telescope objective must always be a collecting system, while the eyepiece can be either a collecting or a scattering system. A telescope with a collecting (positive) eyepiece is called a Kepler tube (Fig. 254, a), a tube with a scattering (negative) eyepiece is called a Galileo tube (Fig. 254, b). The objective 1 of the telescope gives a real reverse image of a distant object in its focal plane. A diverging beam of rays from a point falls on the eyepiece 2; since these rays come from a point in the focal plane of the eyepiece, a beam emerges from it parallel to the secondary optical axis of the eyepiece at an angle to the main axis. Getting into the eye, these rays converge on its retina and give a real image of the source.
Rice. 254. The path of the rays in the telescope: a) Kepler's tube; b) Galileo's pipe
Rice. 255. Path of rays in prismatic field binoculars (a) and its appearance (b). The change in the direction of the arrow indicates the "reversal" of the image after the passage of the rays through part of the system
(In the case of a Galilean tube (b), the eye is not shown so as not to clutter up the drawing.) Angle is the angle that the rays incident on the lens make with the axis.
Galileo's trumpet, often used in ordinary theater binoculars, gives a direct image of an object, Kepler's trumpet - inverted. As a consequence, if the Kepler tube is to serve for terrestrial observation, then it is equipped with a turning system (an additional lens or a system of prisms), as a result of which the image becomes straight. An example of such a device is prismatic binoculars (Fig. 255). The advantage of the Kepler tube is that it contains a real intermediate image, on the plane of which a measuring scale, photographic plate for taking pictures, etc. can be placed. As a result, the Kepler tube is used in astronomy and in all cases related to measurements.
Items not too distant?
Let's say that we want to get a good look at some relatively close object. With the Kepler tube, this is quite possible. In this case, the image produced by the lens will appear slightly farther 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 conduct observations without straining our eyes).
Task 17.1. Kepler's tube is set to infinity. After the eyepiece of this tube has been moved away from the lens at a distance D l= 0.50 cm, objects located at a distance are 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 to
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 nevertheless designed not by Kepler, but by the Italian scientist, physicist, mechanic and astronomer Galileo Galilei (1564-1642) in 1609. In Galileo's tube, unlike Kepler's tube, the eyepiece is not a collecting, but scattering the lens, therefore, the path of the rays in it is more complex (Fig. 17.10).
Rays emanating from an object AB, pass through the lens - the collecting lens O 1, after which they form converging beams of rays. If the subject AB- infinitely remote, then its actual image ab should have turned out in the focal plane of the lens. Moreover, this image would have turned out reduced and inverted. But an eyepiece stands in the way of the converging beams - a diffusing lens O 2, for which the image ab is an apparent source. The eyepiece turns a converging beam of rays into a divergent one and creates imaginary direct image A¢ V¢.
Rice. 17.10 
Angle of view b at which we see the image A 1 V 1, clearly larger than the angle of view a, under which the object is visible AB with the naked eye.
Reader: It is somehow very tricky ... But how can you calculate the angular increase of the pipe?
Rice. 17.11
The lens gives a real image A 1 V 1 in the focal plane. Now let's remember about the eyepiece - a diffusing lens for which the image A 1 V 1 is an apparent source.
Let's construct an image of this imaginary source (Fig. 17.12).
1. Let's draw a beam V 1 O through the optical center of the lens - this ray is not refracted.
Rice. 17.12
2. Let's draw from the point V 1 beam V 1 WITH parallel to the main optical axis. Before crossing the lens (section CD) Is a very real ray, and in the area DB 1 is a purely "mental" line - to the point V 1 in the reality Ray CD does not reach! It refracts so that continuation of the refracted beam passes through the main front focus of the diffusing lens - the point F 2 .
Crossing the beam 1 with continued beam 2 form a point V 2 - ghost image of an imaginary source V one . Dropping out of the point V 2 perpendicular to the main optical axis, we get a point A 2 .
Now note that the angle at which the image is visible from the eyepiece is A 2 V 2 is the angle A 2 OV 2 = b. From D A 1 OV 1 corner. The value | d| can be found from the eyepiece lens formula: here imaginary source gives imaginary the image is in a diffusing lens, so the lens formula is:
.
If we want observation to be possible without eye strain, the virtual image A 2 V 2 must be "sent" to infinity: | f| ® ¥. Then parallel beams of rays will come out of the eyepiece. And the imaginary source A 1 V 1 for this must be in the rear focal plane of the diffusing lens. Indeed, for | f | ® ¥
.
This "limiting" case is shown schematically in Fig. 17.13.
From D A 1 O 1 V 1
h 1 = F 1 a, (1)
From D A 1 O 2 V 1
h 1 = |F 1 | b, (2)
Equating the right-hand sides of equalities (1) and (2), we obtain
.
So, we got the angular magnification of the Galileo tube
As you can see, the formula is very similar to the corresponding formula (17.2) for the Kepler tube.
The length of the Galileo tube, as can be seen from Fig. 17.13 is equal to
l = F 1 – |F 2 |. (17.14)
Task 17.2. The lens of theatrical binoculars is a converging lens with a focal length F 1 = 8.00 cm, and with the eyepiece a diffusing lens with a focal length F 2 = –4.00 cm . What is the distance between the objective and the eyepiece if the image is viewed with the eye from the best viewing distance? How much do you need to move the eyepiece so that the image can be viewed with an eye accommodated at infinity?
This image plays in relation to the eyepiece the role of an imaginary source located at a distance a behind the plane of the eyepiece. Ghost image S 2 given by the eyepiece is at a distance d 0 in front of the eyepiece plane, where d 0 – the distance of best vision of the normal eye.
Let's write down the lens formula for the eyepiece:
The distance between the objective and the eyepiece, as seen in Fig. 17.14 is 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 offset 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: And can Galileo's trumpet give an image on the screen?
 Rice. 17.15 |
We know that a diverging lens can give a valid image only in one case: if the imaginary source is behind the lens in front of the back focus (Fig. 17.15).
Task 17.3. The Galileo tube lens gives a real image of the Sun in the focal plane. At what distance between the objective and the eyepiece can you get on the screen an image of the Sun with a diameter three times larger than the actual image, which would have been obtained without the eyepiece. Lens focal length F 1 = 100 cm, eyepiece - F 2 = –15 cm.
The diffusing lens creates on the screen valid the image of this imaginary source is a segment A 2 V 2. On the image R 1 is the radius of the actual image of the Sun on the screen, and R- the radius of the actual image of the Sun, created only by the lens (in the absence of an eyepiece).
From the similarity D A 1 OV 1 and D A 2 OV 2 we get:
.
Let us write down the lens formula for the eyepiece, while taking into account that d< 0 – источник мнимый, f> 0 - the image is valid:
|d| = 10 cm.
Then from Fig. 17.16 find the required distance l between eyepiece and objective:
l = F 1 – |d| = 100 - 10 = 90 cm.
Answer: l= 90 cm.
STOP! Decide for yourself: C7, C8.
The path of rays in the Galileo tube.
Hearing about the invention of the telescope, the famous Italian scientist Galileo Galilei wrote in 1610: “Ten months ago, a rumor reached our ears that a certain Belgian had built a perspective (as Galileo called the telescope), with the help of which visible objects far from the eyes , become distinctly distinguishable, as if they were close. " Galileo did not know the principle of operation of the telescope, but being well informed in the laws of optics, he soon guessed about its structure and designed the telescope himself. “First I made a lead tube,” he wrote, “at the ends of which I placed two spectacle glasses, both flat on one side, on the other side one was convex-spherical, the other concave. By placing my eye against the concave glass, I saw objects large and close enough. Namely, they seemed three times closer and ten times larger than when viewed with a natural eye. After that, I developed a more accurate tube that represented objects magnified more than sixty times. For this, sparing no labor and no means, I have achieved that I have built myself an organ so excellent that things seemed through it when looking a thousand times larger and more than thirty times closer than when viewed with the help of natural abilities. " Galileo was the first to understand that the quality of the manufacture of lenses for spectacles and for telescopes must be completely different. Out of ten spectacles, only one was suitable for use in a telescope. He has perfected lens technology to a degree never seen before. This allowed him to make a tube with thirtyfold magnification, while the telescopes of spectacle masters were magnified only threefold.
The Galilean telescope consisted of two glasses, of which the one facing the object (lens) was convex, that is, collecting light rays, and the one facing the eye (eyepiece) was a concave, scattering glass. The rays coming from the object were refracted in the lens, but before giving the image, they fell on the eyepiece, which scattered them. With this arrangement of glasses, the rays did not make a real image, it was already compiled by the eye itself, which here constituted, as it were, the optical part of the tube itself.
It can be seen from the figure that the lens O gave in its focus a real image ba of the observed object (this image is the opposite, which could be verified by taking it to the screen). However, the concave eyepiece O1, installed between the image and the lens, scattered the rays coming from the lens, prevented them from crossing, and thus prevented the formation of a real image ba. The scattering lens formed a virtual image of the object at points A1 and B1, which was at the best viewing distance. As a result, Galileo received an imaginary, enlarged, direct image of the object. The magnification of the telescope is equal to the ratio of the focal lengths of the objective to the focal length of the eyepiece. Based on this, it may seem that you can get arbitrarily large increases. However, the limit to the strong increase is put by technical possibilities: it is very difficult to grind glass with a large diameter. In addition, too long focal lengths required an excessively long tube that was impossible to work with. The study of Galileo's telescopes, which are kept in the Museum of the History of Science in Florence, show that his first telescope gave an increase of 14 times, the second - 19.5 times, and the third - 34.6 times.
Despite the fact that Galileo cannot be considered the inventor of the telescope, he was undoubtedly the first to create it on a scientific basis, using the knowledge that was known to optics by the beginning of the 17th century, and turned it into a powerful tool for scientific research. He was the first person to look at the night sky through a telescope. Therefore, he saw what no one had seen before. First of all, Galileo tried to look at the moon. On its surface were mountains and valleys. The tops of mountains and circuses glittered in the sun's rays, and long shadows blackened in the valleys. Measuring the length of the shadows allowed Galileo to calculate the height of the lunar mountains. In the night sky, he discovered many new stars. For example, in the constellation Pleiades there were more than 30 stars, while before there were only seven. In the constellation Orion - 80 instead of 8. The Milky Way, which was previously considered as luminous pairs, disintegrated in a telescope into a huge number of individual stars. To Galileo's great surprise, the stars in the telescope seemed smaller than when observed with the naked eye, since they had lost their halos. Instead, the planets appeared to be tiny discs like the moon. By directing the tube to Jupiter, Galileo noticed four small luminaries moving in space with the planet and changing their positions relative to it. After two months of observations, Galileo guessed that these were the satellites of Jupiter and suggested that Jupiter was many times larger than Earth in size. Examining Venus, Galileo discovered that it has phases similar to the moon and therefore must revolve around the Sun. Finally, observing the Sun through the violet glass, he discovered spots on its surface, and from their motion he established that the Sun rotates on its axis.
All these amazing discoveries were made by Galileo in a relatively short period of time thanks to the telescope. They made a stunning impression on contemporaries. It seemed that the veil of mystery had fallen from the universe and it was ready to open its innermost depths to man. How great the interest in astronomy was at that time can be seen from the fact that only in Italy Galileo immediately received an order for one hundred instruments of his system. Another outstanding astronomer of that time, Johannes Kepler, was one of the first to appreciate Galileo's discoveries. In 1610, Kepler invented a fundamentally new design of the telescope, consisting of two biconvex lenses. In the same year, he published his major work "Dioptrics", which examined in detail the theory of telescopes and optical instruments in general. Kepler himself could not assemble the telescope - for this he had neither the funds nor qualified assistants. However, in 1613, according to Kepler's scheme, another astronomer, Scheiner, built his telescope.
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