EntranceWhat is the angular diameter of a telescope's diffraction disk. Corrections introduced by diffraction theory into the geometric theory of images
The case of light diffraction with an obstacle having an open small part of the 1st Fresnel zone is of particular interest for practice. The diffraction pattern in this case m = R 2 L λ ≪ 1 or R 2 ≪ L λ is observed at large distances. When R = 1 m m, λ = 550 n m, then the distance L will be more than two meters. Such rays drawn to a distant point are considered parallel. This case is considered as diffraction in parallel rays or Fraunhofer diffraction.
Definition 1The main condition for Fraunhofer diffraction– this is the presence of Fresnel zones passing through the point of the wave, which are flat relative to each other.
When a collecting lens is located behind an obstacle to the passage of rays at an angle θ, they converge at some point in the plane. This is shown in Figure 3. 9 . 1 . It follows that any point in the focal plane of a lens is equivalent to a point at infinity in the absence of a lens.
Figure 3. 9 . 1 . Diffraction in parallel rays. The green curve is the intensity distribution in the focal plane (the scale is increased along the x-axis).
A Fraunhofer diffraction pattern located at the focal plane of the lens is now available. Based geometric optics, the focus must have a lens with a point image of a distant object. The image of such an object is blurred due to diffraction. This is a manifestation of the wave nature of light.
An optical illusion does not produce a point-by-point image. If a Fraunhofer diffraction with a circular hole of diameter D has a diffraction image consisting of an Airy disk, then it accounts for about 85% of the light energy with surrounding light and dark rings. This is shown in Figure 3. 9 . 2. The resulting spot is taken as the image of a point source and is considered as Fraunhofer diffraction by an aperture.
Definition 2
To determine the radius of the central spot of the focal plane of the lens, the formula r = 1.22 λ D F is used.
The lens frame has the property of diffraction of light if rays fall on it, that is, it acts as a screen. Then D is denoted as the diameter of the lens.
Figure 3. 9 . 2. Diffraction image of a point source (circular hole diffraction). About 85% of the light energy falls into the central spot.
Diffraction images are very small in size. The central bright spot in the focal plane with a lens diameter D = 5 cm, focal length F = 50 cm, wavelength in monochromatic light λ = 500 nm has a value of about 0.006 mm. Strong distortion is masked in cameras, projectors by due to imperfect optics. Only high-precision astronomical instruments can realize the diffraction limit of image quality.
Diffraction blurring of two closely spaced points can give the result of observing a single point. When an astronomical telescope is set to observe two nearby stars with an angular distance ψ, defects and aberrations are eliminated, causing the focal plane of the lens to produce diffraction images of the stars. This is considered to be the diffraction limit of the lens.
Figure 3. 9 . 3. Diffraction images of two nearby stars in the focal plane of a telescope lens.
The above figure explains that the distance Δ l between the centers of diffraction images of stars exceeds the value of the radius r of the central bright spot. This case allows you to perceive the image separately, which means it is possible to see two closely located stars at the same time.
If you decrease the angular distance ψ, then overlap will occur, which will not allow you to see two close stars at once. IN late XIX century, J. Rayleigh proposed to consider the resolution conditionally complete when the distance between the centers of the images Δ l is equal to the radius r of the Airy Disk. Figure 3. 9 . 4 . shows this process in detail. The equality Δ l = r is considered the Rayleigh solution criterion. It follows that Δ l m i n = ψ m i n ċ F = 1, 22 λ D F or ψ m i n = 1, 22 λ D.
If the telescope has an objective diameter D = 1 m, then it is possible to resolve two stars when located at an angular distance ψ m i n = 6, 7 ċ 10 – 7 rad (for λ = 550 n m). Since the resolution cannot be greater than the value ψ m i n , the limitation is made using the diffraction limit of the space telescope, and due to atmospheric distortions.
Figure 3. 9 . 4 . Rayleigh solution limit. The red curve is the distribution of total light intensity.
Since 1990, the Hubble Space Telescope has been launched into orbit with a mirror having a diameter of D = 2.40 m. The maximum angular resolution of the telescope at a wavelength λ = 550 nm is considered to be ψ m i n = 2.8 ċ 10 – 7 r a d. The operation of a space telescope does not depend on atmospheric disturbances. You should enter the value R, which is the reciprocal of the limit angle ψ m i n.
Definition 3
In other words, the quantity is called telescope power and is written as R = 1 ψ m i n = D 1, 22 λ.
To increase the resolving power of the telescope, increase the size of the lens. These properties apply to the eyes. Its operation is similar to that of a telescope. The pupil diameter d z r acts as D. From here we assume that d з р = 3 mm, λ = 550 n m, then for the maximum angular resolution of the eye we accept the formula ψ g l = 1, 22 λ d з р = 2, 3 ċ 10 − 4 r a d = 47 " " ≈ 1 " .
The result is assessed using the resolution of the eye, which is performed taking into account the size of the light-sensitive elements of the retina. We conclude: a light beam with diameter D and wavelength λ, due to the wave nature of light, experiences diffraction broadening. The angular half-width φ of the beam is of the order of λ D, then the recording of the full width of the beam d at a distance L will take the form d ≈ D + 2 λ D L.
In Figure 3. 9 . 5 . It is clearly visible that when moving away from the obstacle, the light beam transforms.
Figure 3. 9 . 5 . A beam of light that expands due to diffraction. Area I – the concept of a ray of light, the laws of geometric optics. Region II – Fresnel zones, Poisson spot. Region III – diffraction in parallel rays.
The image shows the angular divergence of the beam and its decrease with increasing transverse dimension D. This judgment applies to waves of any physical nature. It follows that in order to send a narrow beam to the Moon, you first need to expand it, that is, use a telescope. When the laser beam is directed into the eyepiece, it travels the entire distance inside the telescope with a diameter of D.
Figure 3. 9 . 6. Resolution of the laser beam using a telescopic system.
Only under such conditions will the beam reach the surface of the Moon, and the radius of the spot will be written as
R ≈ λ D L, where L is denoted as the distance to the Moon. We take the value D = 2.5 m, λ = 550 n m, L = 4 ċ 10 6 m, we obtain R ≈ 90 m. If a beam with a diameter of 1 cm was directed, its “exposure” on the Moon would be in the form of a spot with with a radius 250 times greater.
A microscope is used to observe closely located objects, so the resolution depends on the linear distance between close points. The location of the subject should be near the front focus of the lens. There is a special liquid that is used to fill the space in front of the lens, which is clearly shown in Figure 3. 9 . 7. A geometrically conjugate object located in the same plane with its enlarged image is viewed using an eyepiece. Each point is blurred due to light diffraction.
Figure 3. 9 . 7. Immersion liquid in front of the microscope lens.
Definition 4
Microscope lens resolution limit was defined in 1874 by G. Helmholtz. This formula is written:
l m i n = 0.61 λ n · sin α .
The sign λ is required to indicate the wavelength, n - for the refractive index of the immersion liquid, α - to indicate the aperture angle. The quantity n · sin α is called the numerical aperture.
High-quality microscopes have an aperture angle α, which is close to the limit value α ≈ π 2. According to the Helmholtz formula, the presence of immersion allows one to improve the resolution limit. Suppose that sin α ≈ 1, n ≈ 1, 5, then l m i n ≈ 0, 4 λ.
It follows that a microscope does not provide the full ability to view any details with dimensions much smaller than the light wavelength. The wave properties of light affect the limit of image quality of an object that can be obtained using any optical system.
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P. P. Dobronravin
At the beginning of 1610, Galileo pointed the telescope he had just built at the sky. In the very first nights of observations, he saw a lot of interesting things: he saw that the Moon has mountains and plains, that the planets have noticeable disks, discovered four satellites of Jupiter, was able to distinguish the phases of Mercury and Venus, similar to the phases of the Moon, and could even notice on the disks of Jupiter and Mars some components. But when Galileo pointed his telescope at the stars, he was probably somewhat disappointed. True, the stars in the telescope were visible brighter, there were more of them, but each star remained the same point as it was visible to the eye, and even vice versa: the bright stars seemed to become smaller, they lost the rays that surrounded them when viewed with the naked eye .
Observatory in Barcelona.
Rice. 1. Diffraction of waves on water. The waves go around the obstacle.
Rice. 3. The simplest stellar interferometer-telescope, the lens of which has a cover with two holes.
Rice. 4. Path of rays in a 6-meter stellar interferometer.
Figure 5. Large telescope at Mount Wilson Observatory.
Rice. 6, 2.5-meter mirror of the Mount Wilson Observatory.
Rice. 7. View of the diffraction disk of a star and the stripes on it at different distances between the interferometer mirrors. The streaks are weakest in medium images, when the distance between the mirrors is close to that corresponding to the apparent diameter of the star
Rice. 8. Arrangement of mirrors in a 15-meter stellar interferometer.
Rice. 9. Comparative diameters of some stars and the orbits of Earth and Mars.
Science and life // Illustrations
Rice. 10. Mount Wilson Observatory.
300 years have passed since then. Modern telescopes are immeasurably superior both in size and in the quality of optics to Galileo's first telescope, but until now no one has seen the disk of a star through a telescope. True, a star, when viewed through a telescope, especially with high magnification, appears to be a circle, but the diameters of these circles are the same for all stars, which could not be the case if we saw the real disk of the star - after all, the stars are different in size and located at different distances from U.S. In addition, as the diameter of the telescope lens increases, the diameter of these circles decreases, the stars become brighter, but smaller.
In optics, it is proven that the stellar disks we see have nothing in common with the actual sizes of stars and are a consequence of the very nature of light, resulting from the “diffraction” of light. The limit of visibility in a telescope is set by the light itself.
But, as often happens in science, the same properties of light, skillfully used, made it possible to measure the actual diameters of stars.
A little about the properties of light
The electromagnetic theory of light teaches that a light beam can be considered as a set of electromagnetic oscillations - waves propagating in space at a colossal speed of 300,000 km/sec. Oscillations have a certain periodicity in time and space. This means, firstly, that they occur with a certain frequency - about 600 billion times per second for visible light, Secondly. that there are points along the ray at some certain distance from each other that are in the same state. The distance between two such points is called the wavelength and for visible light is about 0.0005 mm. Frequency and wavelength determine the color of the beam.
To better understand further phenomena, let's imagine waves on the surface of water. They hit the shore a certain number of times per minute - this is their frequency; ridge after ridge goes at a certain constant distance - this is the wavelength. And just as there is a depression in the middle between two ridges on the water, between two points of the ray, separated by a distance of one wavelength, there will be a point whose deviation from the state of equilibrium will be opposite to the deviation of the first two points. It is customary to say that two points at a distance of a wavelength are in identical phases, and at a distance of half a wave - in opposite phases, like the crest and trough of waves on water (a phase is a quantity that characterizes the state of an oscillating point in this moment). It must be remembered that the similarity of snow will and waves on water refers only to the laws that determine both phenomena, and not to try to imagine a light beam as a mechanical “trembling” of some substance - such an extension of the analogy was illegal and incorrect.
If there is some obstacle in the way of the water will, for example a stone, then you can notice (Fig. 1) that the waves seem to go around its edges and go behind the stone. The same thing happens with light waves. When encountering any obstacle, light waves bend around its edges, deviating from straight-line propagation; however, since the magnitude of the obstacle is always many times greater than the wavelength, it is not so easy to notice these “bent” rays. They give rise to the phenomenon of light diffraction - the appearance of light where it could not exist if the beam were a geometric straight line. So, looking through a microscope at the shadow from the sharp edge of the screen, you can see light and dark stripes, in the center of the shadow from a small circle you can see a light point formed by light waves that go around the edges of the circle, etc.
Diffraction also occurs with rays of starlight entering the telescope lens. The outer rays of the beam experience deflection (“bending”) at the edge of the lens frame and produce a small disk at the focus of the telescope, the smaller the larger the lens diameter at a given focal length. Consequently, if the light source is even a geometric point in the full sense of the word, then the telescope, due to diffraction, will always show it in the form of a small circle. And these “diffraction disks” do not make it possible to see the actual disks of stars.
The second phenomenon that is significant for us is the interference of light. Let's imagine that two systems of waves of equal strength and the same frequency hit the shore, for example, waves scattering from two stones irrigated into water. At some points on the shore, the crests of both waves will arrive simultaneously, the waves will add up, and the water vibration will be strong; in others, on the contrary, the crest of one wave will arrive simultaneously with the trough of another, the waves will destroy each other, and the water will remain calm. IN intermediate points the waves will intensify and weaken to varying degrees.
The same phenomenon, only more complicated, will occur with light waves. Under some specific conditions, shining two beams of the same color on a white screen can cause “interference” of light. At those points where the vibrations come in the same phases, they should add up and the brightness of the light should increase; at other points of the screen, where the waves of both rays arrive in opposite phases, with a difference of half a wave, they will cancel each other out, and the two rays, when combined, will give darkness.
Such an experiment was made around 1820 by the French physicist Fresnel. He placed a glass prism P (Fig. 2) with a very obtuse angle between the light source S and the white screen E. Instead of even illumination, the screen produced a picture consisting of alternating light and dark stripes. This happened because the prism divided the beam of rays into two beams of identical composition, as if coming from two imaginary sources, S1 and S2. Point a is at an equal distance from both of these sources, the “ridges” and “valleys” (speaking purely conventionally, using the analogy with water waves) coincide in both rays, the vibrations add up and reinforce each other; a bright light will be observed. The situation is different at point b: it is half a wavelength closer to S2 than to S1, the oscillations come in opposite phases, the “ridges” superimposing on the “valleys” cancel each other out, there are no oscillations, and a dark stripe is observed. Reasoning in the same way, we find that on both sides of the light central stripe a there will be alternating light and dark stripes, which is confirmed by experiment.
This is how the phenomenon will be observed if all the rays of the light source have the same wavelength. Ordinary white light consists of a mixture of rays of different colors, i.e., with different wavelengths. The rays of each color will give their own system of light and dark stripes, these systems will overlap each other, and on the screen on both sides of the central white stripe there will be stripes painted in different colors.
What are the diameters of the stars?
Imagine that you are looking at a ball with a diameter of 1 mm from a distance of 206 m. Of course, you cannot examine it; the diameter of the ball will be visible at an angle of one second of arc.
Modern large telescopes can, at high magnification, show separately two luminous points at an angular distance of tenths of a second. It can be calculated that the diameter of the diffraction disk of a star at the world's largest 2.5-meter reflector (a reflecting telescope with a main mirror diameter of 2.5 m), located at the Mount Wilson Observatory (USA, California) is theoretically equal to O''45. And since even through this telescope all the stars appear the same, their real angular disks are obviously even smaller.
The angular diameter of stars can be estimated by indirect methods. There are stars that change their brightness strictly periodically, due to the fact that these stars are double and the brighter one is eclipsed by a less bright companion with each revolution of the pair around a common center of gravity. The study of the law of change in the brightness of these stars in combination with spectroscopic observations of the speeds of their movement makes it possible to determine the linear dimensions of both stars, and from here, if the distance to the star is known, to calculate its angular diameter.
By studying the distribution of energy in the stellar spectrum, one can find out the temperature of the star; By measuring the total radiation coming from a star to Earth, you can calculate the angle at which the diameter of the star is visible, even without knowing its distance.
It turned out that the apparent diameters of even the largest stars are only about 0.05, the same size as the diffraction disk of a 2.5-meter reflector. That is why, even in the greatest telescope in the world, all the stars appear the same. Only with the new giant With the telescope, which is now being built in America and will have a main mirror with a diameter of 5 m, it will be possible to see that some stars are larger than others, to see the real disks of stars.
The diffraction disk of this telescope will have a diameter of 0.022.
But 70 years ago, in 1868, Fizeau pointed out the possibility of applying the phenomenon of light interference to measuring the diameters of stars. The main idea of the method is very simple. Let's imagine that in front of the Fresnel prism (Fig. 2) there are not one, but two light sources. Each of them gives its own system of light and dark stripes on the screen. By moving the light sources, you can arrange them so that light stripes from one source will fall on dark stripes from another, and vice versa. The screen will have even lighting. Knowing the data of the installation taken for the experiment, it is possible to calculate the angle at which the distance between the sources is visible from the center of the screen at the moment the stripes disappear.
You can do the same with a telescope. If you put a cover with two holes on the telescope lens (Fig. 3), then the light rays passing through the lens will first of all give the usual image of a star, a diffraction disk. But, in addition, the rays coming from both holes, meeting at the main focus of the telescope, will interfere, like rays behind a Fresnel prism, and will give stripes on the disk of the star. Having closed one of the holes, we will see that the disk will remain, but the stripes on it will disappear. The farther apart the holes in the diaphragm are, the smaller the distances between the strips. Such a device is called a stellar interferometer.
Let us now assume that the star is double, that is, in fact there are two located so close that even in a telescope they are visible as one. Each of the stars will give its own system of stripes on the disk; These systems will overlap one another. By changing the distance between the holes in the diaphragm, you can select it so that the stripes on the disk will cease to be visible: the light stripes given by one star will coincide with the dark ones given by the other, and the disk will be illuminated evenly. Knowing the distance between the holes in the aperture and the focal length of the telescope, it will be possible to calculate the angle at which the distance between the components of the double star is visible, although it will not be possible to distinguish them separately.
Fiso took the next step. His reasoning, which is in fact somewhat more complex, can be simplified as follows: if a star is not a point, but a small disk, then it can be imagined as consisting of two “half-disks” and further consider each of them as an independent source of light, giving its own strip system. Then, by changing the distance between the holes in the telescope diaphragm, it is possible to achieve the disappearance of the fringes and uniform illumination of the diffraction disk of the star. From the distance of the holes in the diaphragm, you can calculate the distance between the “centers of gravity” of both “half-disks,” and from here, using the geometry formulas, find the diameter of the star.
Fizeau's ideas were used by Stephen.
Using the 80-centimeter refractor of the Marseille Observatory, he observed interference fringes from many stars, but was never able to make them disappear. Then the works of Fizeau and Stephen were forgotten.
These ideas were expressed again in 1890 by the famous American physicist Michelson. Using various telescopes, he showed that with the help of interference it was possible to measure the distances between the components of very close double stars, the diameters of the satellites of Jupiter, etc. The results agreed well with the results of conventional measurements with a precision micrometer. However, astronomers did not immediately pay attention to Michelson's results. Only around 1920 were these experiments repeated at the Mount Wilson Observatory, first on a one and a half meter and then on a 2.5 meter reflector. It was possible to measure the distances in some very close star pairs, for example the distance between the components of the Capella binary star, equal to only 0"",045.
But it was discovered that even when the diaphragm holes are located at the edges of a 2.5-meter mirror, the stripes on the diffraction disks of stars do not disappear - this distance is still too short. A lens or mirror with a diameter of more than 2.5 m did not exist then, and still does not exist now, and, it would seem, there is nowhere to go further.
However, Michelson solved the problem extremely simply and ingeniously, as if artificially increasing the size of the 2.5-meter mirror by another 2.5 times. In Fig. Figure 4 shows the path of rays in the Michelson stellar interferometer located on the main telescope of the Mount Wilson Observatory. On a steel beam 6 m long, mounted at the end of the reflector, there are two flat mirrors 1 at an angle of 45° to the axis of the telescope. The rays from these mirrors go to two flat mirrors 2, the main concave mirror of the reflector 3 and after reflection from the convex mirror 4 and flat 5 into the eyepiece 6. Meeting at the focus of the telescope, the rays give the same picture as with two holes in the cover on the lens , i.e., a diffraction disk and a system of stripes on it. The distance between the mirrors can vary from 2.5 to 6 m.
On December 13, 1920, the long-set goal was achieved. The first star for which it was possible to achieve the disappearance of stripes (Fig. 7) at a distance between the interferometer mirrors of 3 m was Alpha Orionis (Betelgeuse). For its diameter, the value obtained was 0.047, in good agreement with theoretical calculations. The same interferometer measured the apparent diameters of several more stars.
But even a distance of 6 m between the interferometer mirrors is too small for the vast majority of stars. Since to measure the diameters of stars it is not important that the main mirror of the telescope has a maximum diameter, but the distance between the moving mirrors is important, a new interferometer was built in 1930 with a main mirror with a diameter of 100 cm and a beam 15 m long (Fig. 8). This interferometer is no longer an attachment to a telescope, but a completely independent instrument. With it, using an improved observation technique (not only the distance at which the stripes disappeared was observed, but the degree of visibility of the stripes was also assessed at other distances between the mirrors by comparison with artificial stripes), it was possible to measure the diameters of quite large number stars Some of the results of these measurements are given in the table. It can be noted that the agreement between the observed and theoretically calculated stellar diameters is very good.
Of course, the diameters of only the very large stars closest to us can now be measured - the diameters of the remaining stars are much smaller and are inaccessible even to a 15-meter interferometer. The last row of the table shows Vega, one of the brightest stars in our northern sky. To measure its diameter, the interferometer mirrors would have to be moved apart by 50 m.
The last column of the tablet shows the actual diameters of stars, with the diameter of the Sun taken as one. The actual size of a star is easy to calculate if its angular diameter and distance to it are known. This column shows how huge some stars are. If, for example, Antares were in the place of our Sun, then not only the Earth’s orbit, but also the orbit of Mars would lie inside it (Fig. 9); Mars, whose average distance from the Sun is 228 million km, would move inside Antares. Knowing the size of Antares and its mass, we can calculate the average density of its substance. And it turns out that this density is three million times less than the density of the matter of our Sun.
The use of mirrors in a stellar interferometer on a telescope. The angular diameter of Betelgeuse turned out to be equal to 0.05, which corresponds to a diameter of 400,000,000 km.
The angular diameter of Betelgeuse turned out to be equal to 0.05, which corresponds to a diameter of 400,000,000 km. IN Lately An interferometer was built at the Mount Wilson Observatory, which makes it possible to move the mirrors apart up to 18 m and, therefore, measure angles in thousandths of a second.
Michelson interferometer diagram. Si i Si - mirrors. Pi - separation plate. Рг - compensation plate. The angular diameter of the rings, depending on the difference in the lengths of the interferometer arms and the order of interference, is determined from the relation 2d cos r m K. Obviously, the movement of the mirror by a quarter of the wavelength will correspond, at small values of the angle r, to the transition of a light ring to the place of a dark one in the field of view, and vice versa , dark instead of light.
Spherical aberration. The angular diameter of the scattering circle is usually expressed in milliradians. In Fig. Figure 3.15 shows the dependence of the angular size of spherical aberration on the size of the relative hole for thin lenses made of various materials and a spherical mirror.
The Sun (the angular diameter of the Sun is equal to 3G 0 01 rad.
A When the angular diameter of the Moon is larger: when it is near the zenith or near the horizon.
Sometimes the angular diameter of the scattering circle is used.
As is well known, the angular diameters at which stars are visible from Earth are so small that no existing telescope can resolve them. At the focal plane of a telescope, starlight produces a diffraction pattern that is indistinguishable from that which would be produced by light from a point source diffracted at the telescope aperture and degraded as it passed through the Earth's atmosphere.
Illustration of the concept of coherence volume. There are many stars whose angular diameter is much smaller than Betelgeuse's, so the high degree of correlation in light from these stars occurs over much larger areas.
Unlike the Sun, whose angular diameter is 30, the indicated sources of the Galaxy have angular dimensions no more than 3 - m - 37 and can be considered as pointlike.
In this way, the angular diameter of the source can be measured by gradually increasing the interval between the two holes until the interference fringes disappear.
Great Oppositions of Mars from 1830 to 2035 The distance from Earth to Mars is given in astronomical units (AU and kilometers). For observers of a planet, the main factor is the angular diameter of its disk.
Scheme of the Fizeau-Michelson method for determining the angular distance between stars or the angular diameter of stars. So, the method also allows you to determine the angular diameter of the light source (cf.
Scheme of experiments to measure the diameter of the stars proposed. So, the method also allows you to determine the angular diameter of the light source (cf.
Most typical example Stars of this type are those whose angular diameter is small fractions of a second.
There are many stars whose angular diameter is much smaller than Betelgeuse's, so the high degree of correlation in light from these stars occurs over much larger areas.
The angular diameter 2v of the central diffraction spot is also called the angular diameter of the diffraction pattern.
Processing flat images of areas of the starry sky is advisable when the angular diameter of the machine frame is small. In this case, projective distortions during the formation of the frame slightly distort the positions of the stars on the celestial sphere. Since the probability of correct identification increases with the number of star images, the small angular dimensions of the machine frame lead to the need to expand the range of luminosities of the analyzed stars. As a result, the probability of missing faintly luminous stars increases significantly, and a low brightness threshold also leads to an increase in the probability of false marks. Ultimately, the small angular dimensions of the machine frame lead to low identification efficiency of the star sighted by the spacecraft's astro sensor.
Illustration of the diagram and notation for the formula (James and Wolf, 1991a.| Changes created by interference at the axial point PQ in the Planck spectrum for different values of d. The source was assumed to be at a temperature T of 3000 K and subtend the angular half-diameter a x 10 - rad. at point O. The units of measurement on the vertical axis are arbitrary (James and Wolf, 199 la. Bessel of the first kind and first order, 2a is the angular diameter that the source subtends at the midpoint O between two holes and d is the distance between them, c is the speed light in a vacuum.
Twice the magnitude, or 41, is comparable to the magnitude of 40 5 angular diameter of the apparent orbit of the star observed by Bradley.
If instead of two sources (double star) we have a source with an angular diameter of 8, then it gives the interference pattern shown in Fig. 9.14, where the observed stripe is shaded, and dotted and solid lines indicate stripes caused by the edges of the source separately; The shaded area gives an approximate idea of the appearance of the stripes.
Electronic density Ne and temp - pa T, solar atmosphere. Exactly in the center of the Galaxy is the radio source Strelts-A, consisting of a central bright source with an angular diameter of 3 (linear size, like Andromeda, 8 ps), immersed in the conceptrich. The central source has a complex spectrum containing a non-thermal component.
The size of the Sun (or Moon) can simply be related to its distance to us by measuring its angular diameter.
From this expression it is clear that to determine T, it is necessary to know only the temperature of the surface of the Sun and the angular diameter of the Sun 2Rc / r, visible from the Earth. This diameter is 0.01 radians, and the temperature of the Sun's surface is approximately 6000 K.
From this expression it is clear that to determine T, it is necessary to know only the temperature of the surface of the Sun and the angular diameter of the Sun 2Rc / r, visible from the Earth. This diameter is equal to 0 01 radians, and the temperature of the surface of the Sun is approximately 6000 K - Using formula (7.5) we find G 300 K.
Jupiter and Saturn are visible in the form of disks in a telescope with high magnification, which made it possible to measure their angular diameters, and then calculate their linear values.
Grimaldi described the phenomenon of alternation of light and shadow he observed when two adjacent slits were illuminated by the light of the Sun (the angular diameter of the Sun is 31 - 0 01 rad.
Mj and M2) with a diameter of 1 56 m and with a variable base of up to 14 m was used for the first time to measure the angular diameter of Sirius.
He notes that since the afterimage is localized at the leading edge of the background against which it is observed, and since its apparent angular diameter is preserved, it usually changes size significantly during movement. When the background is removed, the afterimage also appears more distant and therefore (due to the preservation of the angular diameter) significantly increased in size. As the background approaches, the opposite happens. Fluctuations in size can be large.
Heliometers, which consist of a telescope whose lens is divided along its diameter and the two halves can move; they are used to measure the angular diameter of the Sun and the angular distance between two celestial bodies.
The reader may wonder why the Fizeau stellar interferometer, which uses only part of the telescope's aperture, is more suitable for measuring the angular diameter of a distant object than methods that use the full aperture. The point is that it is necessary to take into account the effects of random spatial and temporal fluctuations in the earth's atmosphere (vision through the atmosphere), which is discussed in detail in Chapter.
The simplest possible application of a Michelson stellar interferometer is to determine the interval s0 at which the interference fringes begin to disappear, and therefore the angular diameter of a distant source.
Visibility curve and radial distribution of radio brightness across the solar disk (the arrow marks the edge of the Sun in optics. During the appearance of a large sunspot in 1946, when solar radiation increased significantly, Ryle and Vonberg used their instrument to determine the angular diameter of a radio source on the Sun. For various distances between the antennas, they measured the ratio of the maximum to the minimum of the lobes forming the interference curve. Based on these results, they concluded that the angular diameter of the source was 1 (V. Since this value did not significantly exceed the diameter of the visually observed sunspot, they concluded that the radio source belonged to to the visual spot or at least associated with it.
Intensity distribution in interference rings. In the case of a glass plate 0 5 mm thick with a refractive index n 1 5, the first bright ring has an angular diameter of 21, 8 times the angular diameter of the Sun. Some differences can be noted between these rings and the rings localized at infinity, which are observed in the Michelson interferometer.
The literature also describes discharge tubes designed specifically for excitation of the spectra of substances present in very small quantities, and high-aperture discharge tubes with a large angular diameter of the observation window. To service the discharge tube, a simple vacuum installation is used, consisting of a rotary forevacuum and diffusion mercury or oil pumps (with a forevacuum pump giving a vacuum of up to 10 - 3 mm Hg, the use of a diffusion pump is not necessary), a discharge tube, a pressure gauge (usually a U-shaped oil or thermocouple vacuum gauge) and a gas cylinder. In addition, continuous gas purification is often used, which is provided by a special circulation system.
Prism has the property of giving a distorted image of infinitely distant objects; the angular diameter of an object in a direction parallel to the edge of the prism, naturally, does not change, if only the object is depicted by rays, parallel to the plane main section of the prism; but the angular diameter in the direction perpendicular to the edge can vary. Let dij (Fig. VII.4) be the angle at which an infinitely distant object is visible; Let's determine at what angle di 2 the same object will be visible after the prism.
The creation of a coherent optical installation at the institute was associated with an attempt to apply the idea of signal accumulation to determine the figure of Mercury by analyzing images obtained during the passage of Mercury across the disk of the Sun on May 9, 1970. As is known, when observing astronomical objects through a telescope, the inhomogeneities of the earth’s atmosphere are usually not allow you to achieve a resolution better than I-2, even if the diffraction resolution of the telescope is much better. The angular diameter of Mercury when observed from Earth is about 10, therefore, in order to notice a deviation of the shape of Mercury's disk from a circle of less than 10%, it is necessary to overcome the interfering influence of the Earth's atmosphere.
Attention should be paid to the decrease in amplitude in the case of an extended source. The angular diameter w is related to the value of P by the ratio w P / (V2d) / 2, where K is the wavelength, ad is the distance to the Moon: v is proportional to time, v 0 corresponds to the geometric value; / o - relative flux density at the edge of the geometric theine. The diffraction pattern of ZS 273 observed on August 5, 1962 at a frequency of 410 MHz is shown in Fig. 3, c. The immersion diffraction pattern from October 26, 1962 at a frequency of 1420 MHz is reproduced in Fig. 3, d. It can be seen that ZS 273 is resolved for a point source and an extended region.
Knowing the distance to Betelgeuse, calculated from parallax, you can find the linear diameter of the star. The angular diameters of several stars were measured in this way. All of them, like Betelgeuse, are giants, many times larger than the Sun. The vast majority of stars differ little in diameter from the Sun. Building an interferometer with such a base (the distance between the external mirrors) is an extremely difficult technical task. In addition, with a large base, observations are complicated by atmospheric turbulence, although this affects the operation of the interferometer less than when observing through a telescope. Changes in the refractive index of the air in front of the mirrors affect the phase difference of the rays and only shift the interference pattern without affecting its visibility, so that the fringes remain distinguishable if these changes occur slowly.
In table 2 - 20 presents data on the angular dimensions of the Sun. As follows from this table, the average angular diameter of the Sun in relation to orbital spacecraft can be taken equal to 32, the solid angle of the solar disk is approximately 7 - 10 - 5 sr.
Such a concentrator is used to increase the temperature in the working area by increasing the density of solar energy incident on it. In this case, sections of the curve are determined by the magnitude of the angular diameter of the sun, and the roundings at points a and c are determined by the uneven brightness of the solar disk.
Here it is time to remember that so far we have dealt, in essence, only with the inclinations of the fronts of partial plane waves; taking into account diffraction, the divergence of each of them is not at all infinitesimal and is equal to 20D / D. For this reason, it makes sense to monitor the process of reducing the angular diameters of the spots only until they are compared with diffraction width divergence. On subsequent rounds, the real distribution pattern no longer changes, and the loss of light from the diffraction core due to light scattering is compensated by the arrival of spots formed on previous rounds due to compression.
The Michelson stellar interferometer makes it possible to determine not only the angular distance between the components of double stars, but also the angular diameters of not too distant single stars. The first star for which Michelson was able to measure the angular diameter was Betelgeuse, which belongs to the so-called red giants.
Mikelson's method makes it possible to determine not only the angular distance between the components of double stars, but also the angular diameters of not too distant single stars. The first star for which Michelson was able to measure the angular diameter was Betelgeuse, which belongs to the so-called red giants.
The image that actually appears during the refraction and reflection of light is noticeably different from the geometric image that exists only in our imagination.
Examining the image of a star formed by the lens through a strong eyepiece, we notice that it is not a point, as required by the geometric diagram just discussed, but looks like a circle surrounded by several concentric rings, the brightness of which quickly decreases towards the periphery (Fig. 2.20).
Rice. 2.20. The appearance of images of luminous points of varying brightness when viewed at the focal point of the lens using a strong eyepiece.
But this bright circle is not the true disk of the star, but the visible result of the phenomenon of light diffraction.
The light central circle is called diffraction disk, and the rings surrounding it are called diffraction rings . As the theory shows, the apparent angular diameter of the diffraction disk depends on the wavelength of the light (i.e., on the color of the incident rays) and on the diameter of the lens. This dependence is expressed by the following formula:
Where ρ - angular radius of the diffraction disk (when observing it from the center of the lens), D- diameter of the free lens opening (in centimeters) and λ - wavelength of light (in centimeters). This expression gives the angular radius of the disk in radians; to convert it to degrees (arcseconds), it must be multiplied by the radian value in seconds. Hence,
D 206 265 arc seconds.
At this angle, the radius of the diffraction disk is visible from the center of the lens; at the same angle it is projected from the center of the lens onto celestial sphere. Its angular diameter will, of course, be twice as large. This is equivalent to as if the true disk of the observed star had such an angular diameter.
The linear radius of the diffraction disk is found by the formula
r = ρ f , whence r =l,22λƒ/ D.
Thus, the angular dimensions of the diffraction pattern of the image are determined by the diameter of the lens and the wavelength of light (the color of the rays) and do not depend on f, and the linear dimensions depend on the relative focus and wavelength of light, but do not depend on D. In the same way, the sizes of the diffraction rings surrounding the central disk depend on the same quantities. From the fact that the size of the rings depends on the wavelength of light, it is clear that in the case of white light they should be colored in rainbow colors; in fact, you can see that the inner edges of the rings. have a blue color, and the outer ones are red (since the wavelength of blue rays is less than the wavelength of red ones).
From these few pieces of information we can draw conclusions; having great importance for working with a telescope: 1) the larger the diameter of the lens, the finer the details discernible with its help; 2) for each lens there is the smallest angular distance between two luminous points (for example, stars), which can still be distinguished separately using this lens; this smallest angular distance is called limit angle permissions I or; resolved angle and is a fundamental characteristic of a lens by which it is evaluated resolving power . The smaller the limiting angle of resolution, the higher the resolving power of the lens.
The real value of the resolving power will become quite clear to us if we observe double stars with small angular distances between the components. If the images of stars at the focus of the lens were points, then at an arbitrarily small distance they would be observed as separate; in a strong enough eyepiece we would be able to see two separate points. But in reality due to diffraction;
images of stars are not dots, but circles; and if so, then at a certain minimum distance their images will touch each other, and with a further decrease in the distance between the components, they, more and more overlapping each other, will merge into one slightly oblong spot (Fig. 2.21.). Really
Rice. 2.21. Images of two Stars merge if the angular distance between them is less than the resolving power of the telescope.
the existing two separate stars will appear as one, and no eyepiece will be able to see two images. The only way to see two such close stars separately is to use a lens with a large free aperture, as it will depict them in the form of circles of a smaller angular size.
Let us now substitute the wavelength of light into the formula expressing the angular radius of the diffraction disk, taking green-yellow rays (to which the eye is most sensitive) with an average wavelength λ = 0.00055 mm
ρ = 1.22 λ/D 206265 = 1.22 0.00055/ D 206265= 138/ D(arcseconds)
or, rounding,
D (arc seconds),
Where D expressed in millimeters.
Using the same substitution, we obtain the value for the linear radius of the diffraction disk (for the same rays)
r = 1.22 0.00055 ƒ/ D = 0.00067 ƒ/ D mm= 0.67 ƒ/D µm.
These numbers speak for themselves. No matter how small the luminous point is, its angular radius when viewed through a lens with a free hole diameter of 140 mm. cannot be less than 1"; it will therefore be represented by a circle with a diameter of 2". If we remember that the true angular diameter of stars rarely exceeds thousandths of a second, then it will become clear how far from the truth the idea of the object given by such a lens , although a telescope with a lens diameter of 140 mm is already one of the rather powerful tools. It is appropriate to point out here that the angular radius of the diffraction disk given by a 200-inch reflector (D== 5000mm), equal to 140/5000 ~ 0",03 - exactly the value of the largest known true angular diameter of the star.
The angular diameter of the diffraction disk does not depend on the focal length, and its linear diameter is determined by the relative aperture of the lens. With the same 140 mm lens at a relative aperture of 1:15, the linear diameter of the diffraction disk will be
2r= 2 0.00067 15 ~ 0.02 mm ~ 20 µm .
Without going into the details of the theory, which would take us too far, let's say that the actual value of the limiting resolution angle is somewhat less than the angular radius of the diffraction disk. Studying this issue leads to the conclusion that the fraction 120/ can practically be taken as a measure of the resolvable angle D(assuming equal brightness of the components of the binary star). Thus, a lens with a free aperture diameter of 120 mm can, to the limit, separate a double star with a distance of components of equal magnitude. On the surface of Mars during the era of great oppositions (the angular diameter of the disk is about 25"), with the help of such a lens it is still possible to distinguish two objects lying from each other at a distance of 1/25 of the visible diameter of the planet's disk, which corresponds to approximately 270 km; On the Moon, objects located at a distance of two kilometers from each other can be separately visible.
The resolving power of a telescope is usually understood as the resolving power of its lens. Telescopes are designed to observe distant objects (stars). Let us use a telescope, the lens of which has a diameter D, to examine two close stars located at an angular distance θ .The image of each star in the focal plane of the lens has a linear size (Airy spot radius) equal to 1.22 λF/D. In this case, the centers of the images are located at a distance y*F. As in the case of spectral instruments, when determining the diffraction limit of resolution, the conditional Rayleigh criterion is used (Fig. 2.22). The difference is that in the case of spectral instruments we are talking about resolving two close spectral lines from their images, and in the case of optical instruments we are talking about resolving two close points of an object.
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(2.6) It should be noted that in the center of the curve of the total intensity distribution (Fig. 2.24.) there is a dip of the order of 20% and therefore the Rayleigh criterion only approximately corresponds to the capabilities of visual observation. An experienced observer can confidently resolve two close points of an object located at a distance several times smaller than y min. The numerical estimate gives for a lens with a diameter of D = 10 cm, y min = 6.7*10 -6 rad = 1.3”, and for D=10 2 cm, y min = 0.13". This example shows how important large astronomical instruments are. The world's largest operating reflecting telescope has a mirror diameter of D = 6 m. The theoretical value of the resolution limit of such a telescope is y min = 0.023" . For the second largest reflecting telescope at Mount Palomar Observatory with D = 5 m, the theoretical value is y min = 0.028”. However, non-stationary processes in the atmosphere make it possible to approach the theoretical value of the resolution limit of such giant telescopes only during those rare short-term observation periods. Large telescopes are built primarily to increase the amount of light entering the lens from distant celestial objects. Parameters of the Hubble telescope located in Earth orbit at an altitude of 570 km. with a circulation period of 96 minutes. the following: D =2.4m, ƒ=57.6m, ƒ/D= 24, refractor of the Ritchie-Chritien system with an optical resolution of 0.05 sec. Surface shape tolerance 1/20λ, mirror coating Al (d=75nm) and MgF 2 protection (d=25nm). 2.4.2. Resolution power of the eye.  |
(2.7)  |
The resolution limit of a microscope lens was first determined in 1874 by the German physicist G. Helmholtz, formula (2.9) is called the Helmholtz formula
Here λ is the wavelength, n is the refractive index of the immersion liquid, α is the so-called aperture angle (Fig. 2.20). Magnitude n sinα is called numerical aperture .
| Rice. 2.24. |
Immersion liquid in front of the microscope lens
In good microscopes, the aperture angle α is close to its limit: α ≈π/2. As can be seen from the Helmholtz formula, the use of immersion slightly improves the resolution limit. Assuming sinα≈1 for estimates, n≈1.5, we get:
l min ≈0.4λ.
Thus, using a microscope it is fundamentally impossible to examine any details whose size is significantly smaller than the wavelength of light. The wave properties of light determine the quality limit of the image of an object obtained using any optical system.
2.4.4. A note on normal magnification for optical instruments. In both a telescope and a microscope, the image obtained through the lens is viewed by the eye through the eyepiece. In order to realize the full resolution of the lens, the eyepiece-eye system should not introduce additional diffraction distortions. This is achieved by an appropriate choice of magnification of the optical instrument (telescope or microscope). With a given lens, the task comes down to selecting an eyepiece. Based on general considerations of wave theory, the following condition can be formulated under which the resolution of the lens will be fully realized: the diameter of the beam of rays emerging from the eyepiece should not exceed the diameter of the pupil of the eye d 3p. Thus, the eyepiece of the optical instrument must be sufficiently short-focus.
. Rice. 2.24 Telescopic path of rays Let us explain this statement using the example of a telescope. In Fig. 2.24 shows the telescopic path of rays.  |
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(2.11) In the case of normal magnification, the diameter of the beam of rays emerging from the eyepiece is equal to the diameter of the pupil d 3p. At g >
g N in the telescope-eye system, the resolution of the lens is fully used. The issue of magnifying the microscope is resolved in a similar way. The magnification of a microscope is understood as the ratio of the angular size of an object observed through a microscope to the angular size of the object itself, observed with the naked eye at the distance of best vision d, which for a normal eye is assumed to be 25 cm. Calculation of the normal magnification of a microscope leads to the expression: (2.12) Derivation of the formula (2.12) is a useful exercise for students. As with a telescope, the normal magnification of a microscope is the lowest magnification at which the full resolution of the lens can be used. It should be emphasized that using magnifications larger than normal may not reveal new details of the object. However, for physiological reasons, when working at the limit of instrument resolution, it is sometimes advisable to select a magnification that is 2–3 times higher than normal. Conclusion The practical importance of optics and its influence on other branches of knowledge are extremely great. The invention of the telescope and spectroscope opened up to man a most amazing and rich world of phenomena occurring in the vast Universe. The invention of the microscope revolutionized biology. Photography has helped and continues to help almost all branches of science. One of the most important elements of scientific equipment is the lens. Without it there would be no microscope, telescope, spectroscope, camera, cinema, television, etc. there would be no glasses, and many people over 50 years of age would be unable to read and do many jobs related to vision. The area of phenomena studied by physical optics is very extensive. Optical phenomena are closely related to phenomena studied in other branches of physics, and optical research methods are among the most subtle and accurate. Therefore, it is not surprising that optics has for a long time played a leading role in many basic research and the development of basic physical views. Suffice it to say that both main physical theories of the last century - the theory of relativity and the theory of quantum - originated and developed to a large extent on the basis of optical research. The invention of lasers has opened up vast new possibilities not only in optics, but also in its applications in various branches of science and technology. 1. Determine the magnification factor of a magnifying glass with a focus of 50mm.
2. Determine the focal length of the lens with a magnification of 30 x.
3. Determine the total optical power of two lenses with a magnification factor of 5 x and 15 x.
4. Create an optical design of a microscope with a magnification of 1500 x using microlenses from a number of focal lengths ƒ = 5;10;20;25;30;35mm and eyepieces with a magnification factor Г =15;20;25;30;40. Determine the length of the tube.
6. Determine the linear size of the aberration spot for a telescope with an aperture of 300 mm. and a focal length of 2.4 m from the star.
8. What do stars look like when observed through a telescope? Does their appearance change depending on magnification?
9. What is largest diameter lenses for modern refractors?
10. What causes the greatest interference when observing stars under terrestrial conditions?
11. What is the largest lens diameter of modern reflectors?
12. What is the lens of a reflecting telescope? Who was the first to build a refracting telescope?
13. Draw a diagram of a meniscus telescope.
14. What determines the aperture ratio of a telescope?
15. Name the three brightest objects in the earth's sky.
16. Why is a meniscus needed in a meniscus telescope?
17. Draw a diagram of the reflector.
18. What determines the magnification of a telescope?
19. What is the purpose of the eyepiece?
20. Draw a diagram of a refractor.
21. What are telescopes used for when observing the Moon and planets?
22. Who was the first to build a reflecting telescope?
23. What are telescopes used for when observing stars?
24. What characteristics visually distinguish stars from each other?
25. What characteristics visually distinguish stars from planets?
26. Give the names of any three stars.
27. Give the names of any three constellations.
28. What curvature of the mirror is installed on reflectors?
29. Who was the first to build a meniscus telescope?
30. What other telescopes, besides optical ones, do you know?
31. Why, when observing the Moon and planets through a telescope, do they use a magnification of no more than 500-600 times? What is the purpose of the lens
32. What lens parameters determine resolution.
33. What lens parameter determines the linear diameter of the diffraction disk.
34. Microscope resolution limit.
35. What is the width of the beam when illuminated by a gas laser with a divergence of 1` (one arc min.) at a distance of 10 km.
36. What is the Huygens-Fresnel principle and the phenomenon of diffraction? electromagnetic waves
37. What is the Fresnel zone method? How to split a wavefront into Fresnel zones?
38. What happens to the illumination of the central point of the screen when an opaque plane with a hole approaches or moves away from it?
39. Knowing the diameter of the hole, the wavelength of light and the distance from the point source of light S to the screen, determine what is the minimum integer number of Fresnel zones that the hole can be divided into in the Fresnel experiment?
40. How to determine the size of the diffraction image of a circular hole in a converging wave? How does this size depend on the size of the hole? From the distance to the screen?