Calculation of a lightweight parabolic mirror design. Off-axis parabolic mirrors Fermat's principle predicts a number of new facts

Fermat's principle predicts a number of new facts. Let there be

environment and call it n i (for example, n i for air is the ratio

Rice. 2.8. Passage of radio waves through a narrow slit

Lecture 3. Laws of geometric optics: Spherical surfaces. Prisms. Lenses

Ideally, the mirror should have a parabolic shape, but if the deviation of the sphere from the paraboloid does not exceed 1/8 of the wavelength of light, then such a sphere works exactly like a paraboloid. A paraboloid has less curvature at the edges than at the center. This means that when tested with a shadow device, when the “star” and the knife are located in the center of curvature, the shadow pattern for the paraboloid should have the same appearance as for a mirror with a blockage at the edge (see, Fig. 29, c). This blockage is not just any one, but absolutely precisely calculated. The difference in the positions of the centers of curvature of the central and outer zones is equal to

where D is the diameter of the mirror in millimeters, and R is the radius of curvature. For our mirror these values are 150 mm and 2400 mm, respectively. The longitudinal aberration of this paraboloid when tested from the center of curvature is 2.3 mm. In the prefocal critical position of the knife, a “hillock” with a flat top is visible in the shadow picture - the right side is occupied by shadow in all zones, and penumbra in the central zone. As the knife moves further from the mirror, a blockage resembling a donut becomes visible. This “donut” is best seen when the knife is between two critical positions, exactly in the middle. Its “top,” however, is clearly shifted from the middle zones closer to the edge of the mirror. Calculations show that when the knife is positioned exactly in the middle between the critical positions, the “top” of the “donut” is located at a distance of 0.7 of the radius of the mirror blank; in our case, for a 150 mm mirror, the “top” is located at a distance of 53 mm from its center. Finally, when the knife approaches the back-focal critical position, all of the shadow except the penumbra rim at the edge of the mirror will take a position on the left side of the mirror.

If we manage to artificially distort a flat relief so that it takes the shape of a smooth “donut” without “fractures” (sharply defined zones), then this will mean that we have succeeded in obtaining a paraboloid from a sphere. Let us remind you once again that not any blockage, but only a smooth “donut” with a “top” at a distance of 0.7 radii from the center of the mirror blank and with a given longitudinal aberration, is a paraboloid.

Rice. 30. Shadow reliefs of the same parabolic mirror at different positions of the knife. The letter designations are the same as in Fig. 29.

To get a smooth pit in the center and “lower” the edges, you need to increase the curvature in the center of the mirror so that it gradually decreases when moving from the center to the edge (Fig. 30). In order to get such a hole, there are several ways.

1. Find a square on the polishing pad, the center of which lies approximately in the 0.7r zone. Scrape it to a thickness of 0.5 mm. Every 10 minutes we check the mirror on the shadow device (Fig. 31, a).

2. We will widen the grooves at the edge, but leave them intact in the center up to the 0.3 zone, as shown in Fig. 31, b. We check the mirror every 10 minutes.

3. Scrape off a thin layer (0.5 mm) of resin in small sections on average 1-2 cm2 in such a way that the polishing pad is most weakened in zone 0.7. In the central zone and in the outermost zone we leave the polishing pad untouched (Fig. 31, c). We polish on a trimmed polishing pad and control the mirror with a shadow device every 15 minutes.

4. In a paper circle, the outer diameter of which is 15-20 mm larger than the diameter of the polishing pad, cut out a star, as shown in Fig. 31, g. Wet the circle with water and place it on a polishing pad heated in water. After this, we shape the polishing pad with a mirror, placing the mirror on the resin and a weight on the mirror. After 3-5 minutes of this forming, remove the weight and “polish” for 5-10 minutes without the crocus, without removing the circle. After this, remove the circle. A star will be extruded onto the surface of the polishing pad. She will make a recess in the center of the mirror.

When polishing on a trimmed or shaped polishing pad, zonal errors are possible.

Rice. 31. Methods of applying a polishing pad to a mirror during parabolization.

a) Trimming a square in the 70% zone, b) widening the grooves on the edge, c) trimming the 70% zone, d) forming a star.

If it is a “roller”, we will polish it with local retouching. If there is a “ditch”, we will increase the trimming of this zone.

When examining a mirror with a tone device, you need to carefully monitor the edge, since now it is easy to see the unintended obstruction of the edge, which looks like a narrow strip that sharply increases the radius of curvature of the outer zone. In order to prevent it, we will widen the grooves in a zone 3-5 mm wide on the edge of the polishing pad, as indicated earlier.

parabolic mirror- paraboliškasis veidrodis statusas T sritis radioelektronika atitikmenys: engl. parabolic mirror vok. Parabolspiegel, m rus. parabolic mirror, n pranc. miroir parabolique, m... Radioelektronikos terminų žodynas

parabolic mirror- parabolinis veidrodis statusas T sritis fizika atitikmenys: engl. parabolic mirror vok. Parabolspiegel, m rus. parabolic mirror, n pranc. miroir parabolique, m … Fizikos terminų žodynas

parabolic mirror with a central location of the irradiator- An axisymmetric parabolic mirror in which the feed is located at its focus F. With this design, partial shading of the antenna mirror, the feed system and its supports located in the main beam of the antenna occurs (Fig. C 4). Wed... ...

parabolic mirror with offset feed- Non-axisymmetric parabolic mirror (parabola segment) with an irradiator located outside the main direction of radiation (Fig. O 2). With this design, shadowing of the surface of the antenna mirror is eliminated and the radiation level is reduced by... ... Technical Translator's Guide

parabolic mirror (solar installation)- - [A.S. Goldberg. English-Russian energy dictionary. 2006] Energy sector topics in general EN dish ... Technical Translator's Guide

multi-section mirror- A collapsible mirror (usually parabolic), consisting of a large number of sections. Used to create large antennas deployed in space (Fig. M 5). [L.M. Nevdyaev. Telecommunication technologies. English Russian Dictionary… … Technical Translator's Guide

A device for emitting and receiving radio waves. The transmitting antenna converts the energy of high-frequency electromagnetic oscillations concentrated in the output oscillatory circuits of the radio transmitter into the energy of emitted radio waves. Transformation... ...

Archaeologists have found abundant evidence that in prehistoric times people showed great interest in the sky. The most impressive are the megalithic structures built in Europe and other continents several thousand years ago.… … Collier's Encyclopedia

This table presents the main astronomical instruments that are used in domestic research. Abbreviation Full name Manufacturer Optical system Aperture diameter (mm) Focal length (mm) Observatories in ... Wikipedia

- (from Latin reflecto I turn back, I reflect) a telescope equipped with a mirror Lens. Radars are used primarily for photographing the sky, photoelectric and spectral studies, and less often for visual observations. IN… … Great Soviet Encyclopedia

into focus R. To do this, you need to find a curved mirror surface for which the sum of the distances XX" + X"P" will be constant, regardless of the choice of point X, the geometric locus of all points equidistant from the line and some given point. Such a curve is called a parabola. The telescope mirror is made in parabola shape (Fig. 2.7).

The given examples illustrate the principle of design of optical systems. Exact curves can be calculated by using the rule that the times on all paths leading to the focal point are equal, and by requiring that the travel times on all adjacent paths be large.

three media - glass, water and air, and we observe the phenomenon

refraction and measure the index n

to move from one environment

to another.

Let's denote

index

refraction for

transition from air (1) to water (2), and through n 13

– to move from

air (1) into the glass (3). By measuring the refraction in the water -

glass, let's find another refractive index n 23. Based on

from the principle of least time, then the exponent n is 12

the ratio of the speed of light in air to the speed of light in water;

indicator n 13 is the ratio of speed in air to speed in glass, and

n is the ratio of the speed in water to the speed in glass. That's why

we get

In other words, the refractive index for transition from one material to another can be obtained from the refractive index of each material with respect to some medium, say air or vacuum. By measuring the speed of light in all media, we will determine the refractive index for the transition from vacuum to

Such a connection exists, and this served as an argument in favor of the principle of least time.

Another prediction of the principle of least time is that the speed of light in water, when measured, should be less than the speed of light in air. This prediction is theoretical in nature and has nothing to do with the observations from which Fermat derived the principle of least time (until now we have dealt only with angles). The speed of light in water is indeed less than the speed in air, and just enough to give the correct refractive index.

Fermat's principle says that light takes the path with the shortest, or extreme, time. This ability of light cannot be explained within the framework of geometric optics. It is related to the concept of wavelength, roughly speaking, that

a segment of the path ahead that light can “feel” and compare with neighboring paths. This fact is difficult to demonstrate experimentally with light, since the wavelength of light is extremely short. But radio waves with a wavelength of, say, 3 cm “see” much further. Suppose we have a radio wave source, a detector and a screen with a slit, as shown in Fig. 2.8; under these conditions, the rays will pass from S to D, since this is a straight path, and even if the slit is narrowed, the rays will still pass. But if we now move the detector to point D", then

with a wide slit, the waves will not go from S to D" because they will compare nearby paths and say: “all these paths require a different time." On the other hand, if you leave only a narrow slit and thus prevent the waves from choosing a path, they will turn out to be suitable There are already several paths, and the waves will follow them! If the slit is narrow, more radiation will get to point D" than through a wide slit!

speed in air to speed in vacuum, etc.). Index

refraction for any two materials i and j is equal to

3.1. Focal length of a spherical surface

Let's study the basic properties of optical systems based on Fermat's principle of least time.

To calculate the time difference on two different paths of light, we obtain a geometric formula: let us be given a triangle whose height h is small and whose base d is large (Fig. 3.1); then the hypotenuse s is greater than the base. Let's find how much larger the hypotenuse is

bases: = s – d. By the Pythagorean theorem s 2 – d 2 = h 2 or
	But s – d = , and s + d ~ 2s. Thus,
(s – d)(s + d) = h

Rice. 3.1. A triangle whose height h is less than the base d and whose hypotenuse s is greater than the base

This relationship is useful for studying images produced using curved surfaces. Let's consider a refractive surface separating two media with different refractive indices (Fig. 3.2). Let the speed of light on the left be c, and on the right c/n, where n is the refractive index. Let's take a point O at a distance s from the front surface of the glass and another point O" at a distance s" inside the glass and try to choose a curved surface so that each ray leaving O and entering

Rice. 3.2. Focusing on a refractive surface

on the surface at P, came to point O" (Fig. 3.2). To do this, you need to give the surface such a shape that the sum of the time of passage of light on the way from O to P (i.e. the distance OP divided

to the speed of light) plus n c O P , i.e. travel time from R to O",

was constant value, independent of the position of point P. This condition gives the equation for determining the surface of a fourth-order surface.

Assuming that P is close to the axis, we lower the perpendicular PQ of length h (Fig. 3.2). If the surface were a plane passing through P, then the time spent on the path from O to P would exceed the time on the path from O to Q, and the time on the path from P to O" would exceed the time from Q to O." The surface of the glass should be curved. In this case, the excess time on the path OV is compensated by the delay in passing the path from V to Q. The excess time on the path OR is equal to h 2 /2sc, the excess time on the segment O"P is equal to nh 2 /2s "c. The travel time VQ is n times greater than the corresponding time in vacuum, and therefore the extra time on the segment VQ is equal to (n – 1)VQ /C. If C is the center of a sphere with radius R, then the length VQ is h 2 /2R. The law that connects the lengths s and s" and determines the radius of curvature R of the desired surface follows from the condition that the travel times of light from O to O along any path are equal:



		2 s

This lens formula allows you to calculate the required radius of curvature of the surface that focuses light at point O when it is emitted at O.

The same lens with a radius of curvature R will focus at other distances, i.e. it is focusing for any pair of distances for which the sum of the reciprocal of one distance and the reciprocal of the other, multiplied by n, is a constant number - 1/s + n/s = constant.

An interesting special case is s - a parallel beam of light. As s increases, the distance s" decreases. When point O moves away, point O" approaches, and vice versa. If point O goes to infinity, point O" moves inside the glass up to a distance called the focal length f". If a parallel beam of rays falls on a lens, it will be collected in the lens at a distance f. You can ask the question in another way. If the source

light is inside the glass, where will the rays come into focus? In particular, if the source inside the glass is at infinity (s =), then where is the focus outside the lens? This distance is denoted by f. You can, of course, say it differently.

If the source is located at a distance f, then the rays passing through

the surface of the lens will enter the glass in a parallel beam. It's easy to define f and f :

If we divide each focal length by its corresponding refractive index, we get the same result. This is a general theorem. It's true for any complex lens system, so it's worth remembering. It turns out that in general the two focal lengths of a certain system are related in a similar way. Sometimes

Hi all! Vitaly Solovey is with you. Today my article will be on the topic of parabolic mirrors and solar energy in general. A couple of years ago, on the Internet in the USA, I came across a device that was unique at that time - a parabolic mirror, which is also called a concentrator of direct sunlight. Visually, it resembles a satellite dish with a mirror surface inside.

The principle of operation of this plate is such that when sunlight hits a mirror surface, the rays are reflected and accumulate at one point. This happens due to the parabolic shape of the plate and the light beam is reflected at exactly the same angle at which it hit the mirror surface.

With the correct design of the so-called convex mirror, the temperature at the point where the rays accumulate can reach 2,000 degrees Celsius.

Here's a video to prove this:

The surface of a parabolic mirror can be either solid, that is, without seams, or made from pieces of mirrors or reflective film. In the video above, the mirror was made up of 5,800 individual small mirrors. But the difficulty is to place them all correctly. Place all 5800 mini mirrors at the correct angle.

The surface can also be covered with pieces of reflective silver film, which is also not good, since due to the numerous seams, Sun rays dissipate slightly and the effect will be much weaker.

A solution in this situation may be if the convex plate itself is made from several longitudinal parts onto which a reflective film is evenly glued.

In this case, the reflected rays at the most correct angle will be focused at the point of accumulation. But most effective way production is still a natural glass mirror of a parabolic shape, which, of course, will cost enormously for using the mirror in everyday life.

The simplest and most effective option that I have found is the method of vacuum forming a parabolic mirror.

During gluing, it is better to spread the film with the mirror side to the tabletop, and cover it with the glued dish and press it a little.

Now, in order to form a parabolic shape for the film, you will need to pump out the air from the resulting vessel. To do this, drill a hole in any part of the plastic container and insert a bicycle spool there.

Important! The spool needs to be installed reverse side inside out, since we will be pumping out air rather than pumping it inside the vessel.

And this is what should happen ideally:

That’s all for now; in subsequent articles I’ll tell you about other, equally important applications of a parabolic mirror. And finally, a video on how to start a fire using toilet paper and a tablespoon:

In practice, mainly four types of parabolic reflecting mirrors are used (Fig. 41).

The first type of reflector (Fig. 41, A) is a parabolic cylinder, along the focal line of which linear emitters are located. As a result, the directivity of the antenna system is in the plane of the focal line (plane XOZ) depends on the number of irradiating elements, as in planar antennas.

The directivity of this antenna is in a perpendicular plane YOZ is determined mainly by the dimensions of the parabolic cylinder related to the wavelength.

So, if half-wave vibrators with reflectors are used as the irradiator of a parabolic cylinder (to eliminate confusion, the reflector at the irradiator is called counter-reflector), (Fig. 41, a), then the opening angle of the radiation pattern between the points of half the power value in the plane YOZ is equal to 51°, and the radiation pattern itself is expressed by curve a shown in Fig. eleven.

Another type are antennas with reflectors in the form of paraboloids of rotation (Fig. 41,b). Antennas of this type are used in cases where it is necessary to obtain a “needle” radiation pattern, that is, a narrow pattern, both in the vertical and horizontal planes.

In Fig. 41c, shows an antenna with a truncated paraboloid of rotation, and in Fig. 41 G- a paraboloid bounded by an ellipse-shaped contour. The latter type of reflector is sometimes called a “lemon drop” paraboloid due to some external resemblance to the latter.

The antennas shown in Fig. 41v and G, are used to create fan and sector radiation patterns with a small opening angle in one plane and a wide opening in a plane perpendicular to it.

To create fan diagrams, segmented parabolic antennas are also used, one of the varieties of which is shown in Fig. 42. This antenna is a parabolic cylinder of small height, closed at the ends with metal plates. Radiation pattern of a segmented parabolic antenna in a plane YOZ similar to that of a sector horn. In the same plane XOZ it is significantly narrower, due to the fact that in the opening of a segmented parabolic antenna there appears plane wave(due to reflection from a parabolic surface), whereas in the aperture of sector horn antennas the wave front is cylindrical.

Segmented parabolic antennas are used both independently and as feeds for parabolic cylindrical antennas.

In properly designed segmented parabolic antennas, the surface utilization factor 7 is slightly greater than 0.8.