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Light refraction is widely used in various optical instruments: cameras, binoculars, telescopes, microscopes. . . The indispensable and most essential part of such devices is the lens.
A lens is an optically transparent homogeneous body, bounded on both sides by two spherical (or one spherical and one flat) surfaces.
Lenses are usually made of glass or special transparent plastics. Speaking about the lens material, we will call it glass; it does not play a special role.
4.4.1 Biconvex lens
Let us first consider a lens bounded on both sides by two convex spherical surfaces (Fig. 4.16). Such a lens is called biconvex. Our task now is to understand the path of rays in this lens.
Rice. 4.16. Refraction in a biconvex lens
The simplest situation is with a beam traveling along the main optical axis of the symmetry axis of the lens. In Fig. 4.16 this ray comes out from point A0. The main optical axis is perpendicular to both spherical surfaces, so this ray passes through the lens without being refracted.
Now let's take a ray AB running parallel to the main optical axis. At point B of the incidence of the beam on the lens, a normal MN is drawn to the surface of the lens; Since the beam passes from air into optically denser glass, the angle of refraction of CBN is less than the angle of incidence of ABM. Consequently, the refracted ray BC approaches the main optical axis.
At point C the beam exits the lens, a normal P Q is also drawn. The beam passes into optically less dense air, therefore the angle of refraction QCD is greater than the angle of incidence P CB; the beam is refracted again towards the main optical axis and intersects it at point D.
Thus, any ray parallel to the main optical axis, after refraction in the lens, approaches the main optical axis and intersects it. In Fig. Figure 4.17 shows the refraction pattern of a fairly wide light beam parallel to the main optical axis.
Rice. 4.17. Spherical aberration in a biconvex lens
As we can see, a wide beam of light is not focused by the lens: the further the incident beam is located from the main optical axis, the closer to the lens it intersects the main optical axis after refraction. This phenomenon is called spherical aberration and is one of the disadvantages of lenses; after all, one would still like the lens to bring a parallel beam of rays to one point5.
Very acceptable focusing can be achieved if you use a narrow light beam coming near the main optical axis. Then spherical aberration almost invisible look at fig. 4.18.
Rice. 4.18. Focusing a narrow beam with a collecting lens
It is clearly seen that a narrow beam parallel to the main optical axis, after passing through the lens, is collected at approximately one point F. For this reason our lens is called
collecting.
5 Accurate focusing of a wide beam is indeed possible, but for this, the surface of the lens must have a more complex shape rather than a spherical one. Grinding such lenses is labor-intensive and impractical. It’s easier to make spherical lenses and deal with the emerging spherical aberration.
By the way, aberration is called spherical precisely because it arises as a result of replacing an optimally focusing complex non-spherical lens with a simple spherical one.
Point F is called the focus of the lens. In general, a lens has two focuses, located on the main optical axis to the right and left of the lens. The distances from the foci to the lens are not necessarily equal to each other, but we will always deal with situations where the foci are located symmetrically relative to the lens.
4.4.2 Biconcave lens
Now we will consider a completely different lens, bounded by two concave spherical surfaces (Fig. 4.19). Such a lens is called biconcave. Just as above, we will trace the path of two rays, guided by the law of refraction.
Rice. 4.19. Refraction in a biconcave lens
The ray emerging from point A0 and traveling along the main optical axis is not refracted because the main optical axis, being the axis of symmetry of the lens, is perpendicular to both spherical surfaces.
Ray AB, parallel to the main optical axis, after the first refraction begins to move away from it (since when passing from air to glass \CBN< \ABM), а после второго преломления удаляется от главной оптической оси ещё сильнее (так как при переходе из стекла в воздух \QCD >\P CB). A biconcave lens converts a parallel beam of light into a divergent beam (Fig. 4.20) and is therefore called divergent.
Spherical aberration is also observed here: the continuations of the diverging rays do not intersect at one point. We see that the further the incident ray is located from the main optical axis, the closer to the lens the continuation of the refracted ray intersects the main optical axis.
Rice. 4.20. Spherical aberration in a biconcave lens
As with a biconvex lens, spherical aberration will be virtually unnoticeable for a narrow paraxial beam (Fig. 4.21). The extensions of the rays diverging from the lens intersect at approximately one point at the focus of the lens F.
Rice. 4.21. Refraction of a narrow beam in a diverging lens
If such a diverging beam hits our eye, we will see a luminous point behind the lens! Why? Remember how the image appears in flat mirror: Our brain has the ability to continue diverging rays until they intersect and create the illusion of a luminous object at the intersection (the so-called virtual image). This is precisely the virtual image located at the focus of the lens that we will see in this case.
In addition to the biconvex lens known to us, here are depicted: a plano-convex lens, in which one of the surfaces is flat, and a concave-convex lens, combining concave and convex boundary surfaces. Please note that a concave-convex lens convex surface more curved (its radius of curvature is smaller); therefore, the converging effect of the convex refractive surface outweighs the scattering effect of the concave surface, and the lens as a whole is converging.
All possible diverging lenses are shown in Fig. 4.23.
Rice. 4.23. Diffusing Lenses
Along with the biconcave lens, we see a plano-concave (one of the surfaces of which is flat) and a convex-concave lens. The concave surface of a convex-concave lens is curved to a greater extent, so that the scattering effect of the concave boundary prevails over the collecting effect of the convex boundary, and the lens as a whole turns out to be scattering.
Try to independently construct the path of rays in those types of lenses that we have not considered, and make sure that they are really collecting or diverging. This is an excellent exercise, and there is nothing complicated about it, exactly the same constructions that we did above!
Video tutorial 2: Dispersing lens - Physics in experiments and experiments
Lecture: Converging and diverging lenses. Thin lens. Focal length and optical power thin lens
Lens. Types of lenses
As you know, everything physical phenomena and processes are used in the design of machinery and other equipment. Refraction of light is no exception. This phenomenon has been used in the manufacture of cameras, binoculars, and human eye is also a kind of optical device capable of changing the course of rays. A lens is used for this.
Lens- This transparent body, which is bounded on both sides by spheres.
IN school course physicists consider lenses made of glass. However, other materials can also be used.
There are several main types of lenses that perform specific functions.
Biconvex lens
If the lenses are made of two convex hemispheres, then they are called biconvex. Let's look at how rays behave when passing through such a lens.
On the image A 0 D- this is the main optical axis. This is the ray that passes through the center of the lens. The lens is symmetrical relative to this axis. All other rays that pass through the center are called secondary axes; relative symmetry is not observed.
Consider an incident ray AB, which is refracted due to the transition to another medium. After the refracted ray touches the second wall of the sphere, it is refracted again until it intersects the main optical axis.
From this we can conclude that if a certain ray was parallel to the main optical axis, then after passing through the lens it will intersect the main optical axis.
All rays that are located near the axis intersect at one point, creating a beam. Those rays that are far from the axis intersect at a place closer to the lens.
The phenomenon in which rays converge at one point is called focusing, and the focus point is focus.
Focus ( focal length) is indicated in the figure by the letter F.
A lens in which the rays are collected at one point behind it is called a converging lens. That is biconvex lens is collecting.
Any lens has two focal points - they are in front of the lens and behind it.
Biconcave lens
A lens made of two concave hemispheres is called biconcave.
As can be seen from the figure, the rays that hit such a lens are refracted, and at the exit they do not intersect the axis, but, on the contrary, tend away from it.
From this we can conclude that such a lens scatters, and is therefore called dispersive.
If the scattered rays are continued in front of the lens, they will converge at one point, which is called imaginary focus.
Converging and diverging lenses can also take other forms, as shown in the figures.
1 - biconvex;
2 - plano-convex;
3 - concave-convex;
4 - biconcave;
5 - flat-concave;
6 - convex-concave.
Depending on the thickness of the lens, it can refract rays either stronger or weaker. To determine how strongly a lens refracts, a quantity called optical power .
D is the optical power of the lens (or lens system);
F is the focal length of the lens (or lens system).
[D] = 1 diopter. The unit of lens power is diopter (m -1).
Thin Lens
When studying lenses, we will use the concept of a thin lens.
So, let's look at a drawing that shows a thin lens. So, a thin lens is one whose thickness is quite small. However, for physical laws Uncertainty is unacceptable, so using the term “sufficient” is risky. It is believed that a lens can be called thin when the thickness is less than the radii of two spherical surfaces.
1340. The focal length of the lens is 10 cm. What is its optical power?
1341. The focal length of a diverging lens is 12.5 cm. Determine optical power lenses.
1342. The focal length of the largest Pulkovo telescope is about 14 m. What is the optical power of its lens?
1343. What is the focal length of a lens if its optical power is 0.4 diopters?
1344. The focal length of the camera lens is 60 mm. What is the optical power of the camera?
1345. There are two lenses: the first with a focal length of 5 cm, the second with a focal length of 20 cm. Which lens refracts more strongly?
1346. A light source was placed at the main focus of a converging lens. Draw the path of the rays.
1347. Construct an image of a vertical pencil formed by a converging lens for the case when the pencil is located beyond double the focal length.
1348. The pencil stands between the focus and the double focal length of the converging lens. Plot the resulting image.
1349. Construct an image of a pencil standing between the focus of a converging lens and the lens itself.
1350. A converging lens scatters rays incident from a point source of light onto the lens. Draw where the point light source is in this case?
1351. Show by construction the simplest way to determine the main focal length of a converging lens. Demonstrate this experience.
1352. Object AB is at the double focus of a converging lens (Fig. 169). Construct its image. Describe the image.
1353. Construct an image of a point source of light S, which is formed by a collecting lens, for the cases shown in Figure 170.
1354. A diverging lens gives an image of an object AB (Fig. 171). Construct this image and list its properties. How does the size of the image depend on the distance between the object and the lens?
1355. Construct an image of a luminous point S, formed by a diverging lens (Fig. 172). Describe the image.
1356. In Figure 173 OO’ is the main optical axis of the lens, S is a point light source, S’ is its image. Construct the position of the lens and its foci. Determine whether it is a converging or diverging lens?
1357. In one of the boxes in Figure 174 there is a converging lens, in the other there is a diverging lens. Determine by construction which lens is which.
1358. An object is located at a distance of 20 cm from a converging lens, and its image is at a distance of f = 10 cm from the lens. What is the distance of the lens?
1359. From the bottle to the collecting lens, the distance is d=30 cm, and its actual image to the lens is distance f=60 cm. Determine the focal length of the lens.
1360. An object is located at a distance of 40 cm from a converging lens. Its image was obtained at a distance of 120 cm. What is the focal length of the lens?
1361. A pencil stands at a distance of 50 cm from the converging lens. At what distance from the lens is its image? The focal length of the lens is 10 cm. Describe the image of the pencil.
1362. The image of an object formed by a converging lens was obtained at a distance of 22 cm. The focal length of the lens is 20 cm. At what distance from the lens is the object if:
a) his image is real;
b) is its image imaginary?
1363. There is a hollow glass biconvex lens filled with air in water. A parallel beam of light rays falls on the lens. What will this beam be like after passing through the lens? Make a drawing.
What images will such a lens produce in water? Is a biconvex lens always a converging lens?
1364. Analyze a similar problem for a hollow biconcave lens filled with air and located in water. If there are watch glasses in the school physics classroom, make the lenses described above from them and perform experiments with them.
1365. Using the formula of a converging lens:
1/d+1/f=1/F, calculate the position and determine the nature of the image of objects at different distances from the lens for the cases indicated in the table.
For cases d
1366. Write the formula for a diverging lens, taking into account that the distance from the optical center of the lens to the virtual image of a point is taken with a minus sign.
1367. Determine the optical power of a lens whose focal length is 10 cm; - 10 cm.
1368. At what distance from a lens with a focal length F = 10 cm will an image of an object placed at a distance of 50 cm from the lens be obtained?
1369. The image of an object placed at a distance of 40 cm from a biconvex lens was obtained at a distance of 15 cm from the lens. Determine the focal length of the lens and the size of the image if the size of the object itself is 60 cm.
1370. In a photograph taken by a camera with a photographic lens whose focal length is 13.5 cm, with a camera length of 15 cm, an image of an object 2 cm in size is obtained. What is the actual size of the object?
1371. The distance between the light bulb and the screen is L = 150 cm. A converging lens is placed between them, which gives a sharp image of the light bulb filaments on the screen at two positions of the lens. What is the focal length of the lens if the distance between the indicated positions of the lens is l = 30 cm?
1372. An object is at a distance of 20 cm from the lens, and its actual image is at a distance of 5 cm from the lens. Determine the optical power of the lens.
1373. The actual image of a bubble with glue was obtained at a distance of 42 cm from a lens whose optical power is 2.5 diopters. How far is the bubble from the lens?
1374. An object is located at a distance of 30 cm from a diverging lens, its virtual image is at a distance of 15 cm from the lens. Determines the focal length of the lens.
1375. The optical power of the lens is 2.5 diopters. The light source is located on its main optical axis. How far is the light source from the lens?
1376. An object 50 cm high is at a distance d=60 cm from a converging lens with focal length F=40 cm. Determine the height of the image.
1377. A man 2 m tall was photographed with a camera (lens focal length 12 cm). The size of the person in the picture turned out to be 10 mm. determine the distance between the person and the lens.
1378*. The projector lens has a focal length of 15 cm and is located at a distance of 6 m from the screen. Determine the linear magnification of the image on the screen.
1379*. Instead of a lens with a focal length of 15 cm (see the previous problem), a lens with a focal length of 12 cm was installed. What was the magnification of the image on the screen?
1382*. Do you think it is possible to obtain a transparencies image from a projector on a mirror screen?
No. Because all rays will be reflected from the surface.
1383*. Plot the path of rays in a microscope.
1384. Draw the path of rays in the telescope.
Biconvex lens
Plano-convex lens
Characteristics of thin lenses
Depending on the forms there are collective(positive) and scattering(negative) lenses. The group of collecting lenses usually includes lenses whose middle is thicker than their edges, and the group of diverging lenses includes lenses whose edges are thicker than the middle. It should be noted that this is only true if refractive index the lens material has more than environment. If the refractive index of the lens is lower, the situation will be reversed. For example, an air bubble in water is a biconvex diverging lens.
Lenses are usually characterized by their optical power(measured in dioptres), or focal length.
To build optical devices with corrected optical aberration(primarily chromatic, conditioned light dispersion , - achromats and apochromats) other properties of lenses/their materials are also important, for example, refractive index, dispersion coefficient, transmittance of the material in the selected optical range.
Sometimes lenses/lens optical systems (refractors) are specifically designed for use in environments with relatively high coefficient refraction (see immersion microscope, immersion liquids).
Types of lenses:
Collecting:
1 - biconvex
2 - flat-convex
3 - concave-convex (positive meniscus)
Scattering:
4 - biconcave
5 - flat-concave
6 - convex-concave (negative meniscus)
A convex-concave lens is called meniscus and can be collective (thickens towards the middle) or scattering (thickens towards the edges). A meniscus whose surface radii are equal has an optical power equal to zero (used for correction variances or as a cover lens). Thus, the lenses of glasses for myopia are, as a rule, negative menisci.
A distinctive property of a collecting lens is the ability to collect rays incident on its surface at one point located on the other side of the lens.
Main elements of the lens: NN - main optical axis- a straight line passing through the centers of the spherical surfaces delimiting the lens; O- optical center- a point that, for biconvex or biconcave (with identical surface radii) lenses, is located on the optical axis inside the lens (at its center).
Note. The path of the rays is shown as in an idealized (flat) lens, without indicating refraction at the real phase boundary. Additionally, a somewhat exaggerated image of a biconvex lens is shown
If a luminous point S is placed at a certain distance in front of the collecting lens, then a ray of light directed along the axis will pass through the lens without refracted, and rays that do not pass through the center will be refracted towards the optical axis and intersect on it at some point F, which will be the image of point S. This point is called conjugate focus, or simply focus.
If light falls on the lens from a very distant source, the rays of which can be represented as coming in a parallel beam, then upon exiting it the rays will refract at a larger angle and point F will move on the optical axis closer to the lens. Under these conditions, the point of intersection of the rays emerging from the lens is called main focus F’, and the distance from the center of the lens to the main focus is main focal length.
Rays incident on a diverging lens will be refracted towards the edges of the lens upon exiting it, that is, scattered. If these rays are continued in the opposite direction as shown in the figure with a dotted line, then they will converge at one point F, which will be focus this lens. This trick will imaginary.
Imaginary focus of a diverging lens
What has been said about focus on the main optical axis equally applies to those cases when the image of a point is on a secondary or inclined optical axis, that is, a line passing through the center of the lens at an angle to the main optical axis. The plane perpendicular to the main optical axis, located at the main focus of the lens, is called main focal plane, and at the conjugate focus - simply focal plane.
Collective lenses can be directed towards an object from either side, as a result of which rays passing through the lens can be collected from both one and the other side. Thus, the lens has two focuses - front And rear. They are located on the optical axis on both sides of the lens at the focal length from the center of the lens.
Constructing an image with a thin converging lens
When presenting the characteristics of lenses, the principle of constructing an image of a luminous point at the focus of a lens was considered. Rays incident on the lens from the left pass through its rear focus, and rays incident on the right pass through its front focus. It should be noted that with diverging lenses, on the contrary, the back focus is located in front of the lens, and the front focus is behind.
The construction by a lens of an image of objects having a certain shape and size is obtained in the following way: Let's say line AB represents an object located at some distance from the lens, significantly greater than its focal length. From each point of the object, an innumerable number of rays will pass through the lens, of which, for clarity, the figure schematically shows the course of only three rays.
Three rays emanating from point A will pass through the lens and intersect at their respective vanishing points at A 1 B 1 to form an image. The resulting image is valid And upside down.
In this case, the image was obtained at a conjugate focus in a certain focal plane FF, somewhat distant from the main focal plane F’F’, running parallel to it through the main focus.
If an object is at an infinite distance from the lens, then its image is obtained at the rear focus of the lens F' valid, upside down And reduced until it looks like a point.
If an object is close to the lens and is at a distance exceeding twice the focal length of the lens, then its image will be valid, upside down And reduced and will be located behind the main focus in the segment between it and the double focal length.
If an object is placed at double the focal length from the lens, then the resulting image is on the other side of the lens at double the focal length from it. The image is obtained valid, upside down And equal in size subject.
If an object is placed between the front focus and double focal length, then the image will be obtained behind double focal length and will be valid, upside down And enlarged.
If the object is in the plane of the front main focus of the lens, then the rays passing through the lens will go parallel, and the image can only be obtained at infinity.
If an object is placed at a distance less than the main focal length, then the rays will come out of the lens in a diverging beam, without intersecting anywhere. The image is then imaginary, direct And enlarged, i.e. in this case the lens works like a magnifying glass.
It is easy to notice that when an object approaches the front focus of the lens from infinity, the image moves away from the back focus and, when the object reaches the front focus plane, it appears at infinity from it.
This pattern has great importance in practice various types photographic work, therefore, to determine the relationship between the distance from the object to the lens and from the lens to the image plane, you need to know the basic lens formula.
Thin Lens Formula
The distances from the object point to the center of the lens and from the image point to the center of the lens are called conjugate focal lengths.
These quantities are interdependent and are determined by a formula called thin lens formula:
where is the distance from the lens to the object; - distance from the lens to the image; - the main focal length of the lens. In the case of a thick lens, the formula remains unchanged with the only difference being that the distances are measured not from the center of the lens, but from main planes.
To find one or another unknown quantity with two known ones, use the following equations:
It should be noted that the signs of the quantities u , v , f are selected based on the following considerations - for a real image from a real object in a converging lens - all these quantities are positive. If the image is imaginary, the distance to it is taken negative, if the object imaginary- the distance to it is negative, if the lens is diverging - the focal length is negative.
Image scale
The image scale () is the ratio of the linear dimensions of the image to the corresponding linear dimensions of the object. This relationship can be indirectly expressed by the fraction , where is the distance from the lens to the image; - distance from the lens to the object.
There is a reduction factor here, i.e. a number showing how many times the linear dimensions of the image are smaller than the actual linear dimensions of the object.
In the practice of calculations, it is much more convenient to express this relationship in values or , where is the focal length of the lens.
.
Calculation of focal length and optical power of a lens
The lenses are symmetrical, that is, they have the same focal length regardless of the direction of light - left or right, which, however, does not apply to other characteristics, e.g. aberrations, the magnitude of which depends on which side of the lens is facing the light.
Combination of multiple lenses (centered system)
Lenses can be combined with each other to build complex optical systems. The optical power of a system of two lenses can be found as the simple sum of the optical powers of each lens (assuming that both lenses can be considered thin and they are located close to each other on the same axis):
.
If the lenses are located at a certain distance from each other and their axes coincide (a system of an arbitrary number of lenses with this property is called a centered system), then their total optical power can be found with a sufficient degree of accuracy from the following expression:
,
where is the distance between main planes lenses
Disadvantages of a simple lens
Modern photographic equipment places high demands on image quality.
The image produced by a simple lens, due to a number of shortcomings, does not satisfy these requirements. Elimination of most shortcomings is achieved by appropriate selection of a number of lenses into a centered optical system - lens. Images obtained with simple lenses have various disadvantages. The disadvantages of optical systems are called aberrations, which are divided into the following types:
- Geometric aberrations
- Diffraction aberration(this aberration is caused by other elements optical system, and has nothing to do with the lens itself).
Lenses with special properties
Organic polymer lenses
Contact lenses
