Construct the path of rays in the lens. Thin lenses

There are two conditionally different types tasks:

construction problems in converging and diverging lenses
formula problems for a thin lens

The first type of problem is based on the actual construction of the path of rays from the source and the search for the intersection of rays refracted in lenses. Let's consider a series of images obtained from a point source, which we will place at various distances from the lenses. For a collecting and scattering lens, there are considered (not by us) trajectories of beam propagation (Fig. 1) from the source.

Fig.1. Converging and diverging lenses (ray path)

For a collecting lens (Fig. 1.1) rays:

blue. A ray traveling along the main optical axis passes through the front focus after refraction.
red. The beam passing through the front focus, after refraction, propagates parallel to the main optical axis.

The intersection of any of these two rays (rays 1 and 2 are most often chosen) gives ().

For a diverging lens (Fig. 1.2) rays:

blue. A beam running parallel to the main optical axis is refracted so that the continuation of the beam passes through the back focus.
green. Ray passing through optical center lenses, does not experience refraction (does not deviate from the original direction).

The intersection of the continuations of the considered rays gives ().

Similarly, we obtain a set of images from an object located at various distances from the mirror. Let us introduce the same notation: let be the distance from the object to the lens, - the distance from the image to the lens, - focal length(distance from focus to lens).

For a collecting lens:

Rice. 2. Converging lens (source at infinity)

Because all rays running parallel to the main optical axis of the lens, after refraction in the lens, pass through the focus, then the focal point is the point of intersection of the refracted rays, then it is the image of the source ( point, real).

Rice. 3. Converging lens (source behind double focus)

Let's use the path of a ray running parallel to the main optical axis (reflected into focus) and going through the main optical center of the lens (not refracted). To visualize the image, enter a description of the item using the arrow. The point of intersection of refracted rays is the image ( diminished, real, inverted). The position is between focus and double focus.

Rice. 4. Converging lens (source at double focus)

same size, real, inverted). The position is exactly at double focus.

Rice. 5. Converging lens (source between double focus and focus)

Let's use the path of a ray running parallel to the main optical axis (reflected into focus) and going through the main optical center of the lens (not refracted). The point of intersection of refracted rays is the image ( magnified, real, inverted). The position is behind the double focus.

Rice. 6. Converging lens (source at focus)

Let's use the path of a ray running parallel to the main optical axis (reflected into focus) and going through the main optical center of the lens (not refracted). In this case, both refracted rays turned out to be parallel to each other, i.e. there is no point of intersection of reflected rays. This suggests that no image.

Rice. 7. Converging lens (source in front of focus)

Let's use the path of a ray running parallel to the main optical axis (reflected into focus) and going through the main optical center of the lens (not refracted). However, the refracted rays diverge, i.e. the refracted rays themselves will not intersect, but the extensions of these rays can intersect. The point of intersection of the extensions of refracted rays is the image ( enlarged, imaginary, direct). Position - on the same side as the object.

For diverging lens the construction of images of objects practically does not depend on the position of the object, so we will limit ourselves to the arbitrary position of the object itself and the characteristics of the image.

Rice. 8. Diffusing lens (source at infinity)

Because all rays running parallel to the main optical axis of the lens, after refraction in the lens, must pass through the focus (focus property), however, after refraction in the diverging lens, the rays must diverge. Then the continuations of the refracted rays converge at the focus. Then the focal point is the point of intersection of the continuations of the refracted rays, i.e. it is also an image of the source ( point, imaginary).

any other source position (Fig. 9).

Light refraction- change in the direction of propagation of optical radiation (light) as it passes through the interface between two media.

Laws of light refraction:

1) The incident ray, the refracted ray and the perpendicular, restored to the point of incidence to the interface between two media, lie in the same plane .

2) The ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant value for a given pair of media. This constant is called the refractive index n 21 of the second medium relative to the first:

The relative refractive index of two media is equal to the ratio of their absolute refractive indices n 21 =n 2 /n 1

The absolute refractive index of a medium is the value n, equal to the ratio of the speed c electromagnetic waves in vacuum to their phase velocity v in a medium n=c/v

3) A ray of light incident on the interface between two media perpendicular to the surface passes into the other medium without being refracted.

4) Incident and refracted rays are reversible: if the incident ray is directed along the path of the refracted ray, then the refracted ray will follow the path of the incident ray.

Complete internal reflection - reflection of light at the interface of two transparent substances, not accompanied by refraction. Total internal reflection occurs when a beam of light is incident on a surface separating a given medium from another, optically less dense medium, when the angle of incidence is greater than the limiting angle of refraction.

Path of rays in the lens.

It's called a lens transparent body, bounded by two spherical surfaces. If the thickness itself

lens is small compared to the radii of curvature of spherical surfaces, then the lens is called thin.

Lenses are either converging or diverging. Collecting(positive) lenses are lenses that convert a beam of parallel rays into a converging one. Scattering(negative) lenses are lenses that convert a beam of parallel rays into a divergent one. Lenses whose centers are thicker than the edges are converging, and those whose edges are thicker are diverging.

A straight line passing through the centers of curvature O1 and O2 of spherical surfaces is called main optical axis of the lens. In the case of thin lenses, we can approximately assume that the main optical axis intersects with the lens at one point, which is usually called optical center of the lens O. The light beam passes through the optical center of the lens without deviating from its original direction. All straight lines passing through the optical center are called secondary optical axes.

If a beam of rays parallel to the main optical axis is directed at a lens, then after passing through the lens the rays (or their continuation) will converge at one point F, which is called the main focus of the lens. A thin lens has two main foci, located symmetrically on the main optical axis relative to the lens. Converging lenses have real foci, while diverging lenses have imaginary foci. Beams of rays parallel to one of the secondary optical axes, after passing through the lens, are also focused at point F", which is located at the intersection of the secondary axis with the focal plane Ф, that is, the plane perpendicular to the main optical axis and passing through the main focus. Distance between the optical center lenses O and the main focus F is called the focal length. It is denoted by the same letter F. For a converging lens, F > 0 is considered, for a diverging lens F< 0.

The value of D, the reciprocal of the focal length, is called optical power lenses. The SI unit of optical power is the diopter (dopter).

Path of rays in lenses

The main property of lenses is the ability to produce images of objects. Images can be upright or inverted, real or imaginary, enlarged or reduced.

The position of the image and its character can be determined using geometric constructions. To do this, they use the properties of some standard rays (remarkable rays), the course of which is known. These are rays passing through the optical center or one of the focal points of the lens, as well as rays parallel to the main or one of the secondary optical axes. Constructing an image in a thin lens:

1. A ray parallel to the main optical axis passes through the main focus point.

2. A beam parallel to the secondary optical axis passes through the secondary focus (a point on the secondary optical axis).

3. The beam passing through the optical center of the lens is not refracted.

4. Real image - intersection of rays. Virtual image - intersection of continuations of rays

Converging lens

1. If the subject is located behind a double focus.

To construct an image of an object, you need to shoot two rays. The first ray comes from top point object parallel to the main optical axis. At the lens, the ray is refracted and passes through the focal point. The second ray must be directed from the top point of the object through the optical center of the lens; it will pass through without refraction. At the intersection of two rays we place point A’. This will be the image of the top point of the object. The image of the lower point of the object is constructed in the same way. As a result of the construction, a reduced, inverted, real image is obtained.

2.If the subject is located at the double focus point.

To construct, you need to use two beams. The first ray passes from the top point of the object parallel to the main optical axis. At the lens, the ray is refracted and passes through the focal point. The second ray must be directed from the top point of the object through the optical center of the lens; it will pass through the lens without being refracted. At the intersection of two rays we place point A1. This will be the image of the top point of the object. The image of the lower point of the object is constructed in the same way. As a result of the construction, an image is obtained whose height coincides with the height of the object. The image is upside down and real

3. If the object is located in the space between focus and double focus

To construct, you need to use two beams. The first ray passes from the top point of the object parallel to the main optical axis. At the lens, the ray is refracted and passes through the focal point. The second beam must be directed from the top point of the object through the optical center of the lens. It passes through the lens without being refracted. At the intersection of two rays we place point A’. This will be the image of the top point of the object. The image of the lower point of the object is constructed in the same way. The result of the construction is an enlarged, inverted, real image

diverging lens

The object is placed in front of the diverging lens.

To construct, you need to use two beams. The first ray passes from the top point of the object parallel to the main optical axis. At the lens, the ray is refracted in such a way that the continuation of this ray goes into focus. And the second ray, which passes through the optical center, intersects the continuation of the first ray at point A’ - this will be the image of the upper point of the object. The image of the lower point of the object is constructed in the same way. The result is a direct, reduced, virtual image. When moving an object relative to a diverging lens, a direct, reduced, virtual image is always obtained. When moving an object relative to a diverging lens, a direct, reduced, virtual image is always obtained.

The position of the image and its nature (real or imaginary) can also be calculated using

thin lens formulas. If the distance from the object to the lens is denoted by d, and the distance from the lens to the image by f, then the formula for a thin lens can be written as:

The quantities d and f also obey a certain sign rule: d > 0 and f > 0 – for real objects

(that is, real light sources, and not extensions of rays converging behind the lens) and images; d< 0 и f < 0 – для мнимых источников и изображений.

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 an image appears in a 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!

Subject. Solving problems on the topic "Lens. Constructing images in a thin lens. Lens formula."

Target:

- consider examples of solving problems using the thin lens formula, the properties of the main rays and the rules for constructing images in a thin lens, in a system of two lenses.

Progress of the lesson

Before starting the task, it is necessary to repeat the definitions of the main and secondary optical axes of the lens, focus, focal plane, properties of the main rays when constructing images in thin lenses, the formula of a thin lens (converging and diverging), determination of the optical power of the lens, and magnification of the lens.

To conduct the lesson, students are offered several calculation problems with an explanation of their solution and problems for independent work.

Qualitative tasks

Using a converging lens, a real image of an object with magnification G 1 is obtained on the screen. Without changing the position of the lens, we swapped the object and the screen. What will be the increase in G 2 in this case?
How to arrange two converging lenses with focal lengths F 1 and F 2 so that a parallel beam of light, passing through them, remains parallel?
Explain why, in order to get a clear image of an object, a nearsighted person usually squints his eyes?
How will the focal length of the lens change if its temperature increases?
The doctor's prescription says: +1.5 D. Decipher what kind of glasses these are and for which eyes?

Examples of solving calculation problems

Task 1. The main optical axis of the lens is specified NN, source position S and his images S´. Find by construction the position of the optical center of the lens WITH and its focuses for three cases (Fig. 1).

Solution:

To find the position of the optical center WITH lens and its focal points F We use the basic properties of the lens and rays passing through the optical center, the focal points of the lens, or parallel to the main optical axis of the lens.

Case 1. Item S and its image are located on one side of the main optical axis NN(Fig. 2).

Let's walk you through S And S´ straight line (side axis) until it intersects with the main optical axis NN at the point WITH. Dot WITH determines the position of the optical center of the lens, located perpendicular to the axis NN. Rays passing through the optical center WITH, are not refracted. Ray S.A., parallel NN, refracts and goes through the focus F and image S´, and through S´ the beam continues S.A.. This means that the image S´ in the lens is imaginary. Item S located between the optical center and the focal point of the lens. The lens is converging.

Case 2. Let's walk you through S And S´ secondary axis until it intersects with the main optical axis NN at the point WITH- optical center of the lens (Fig. 3).

Ray S.A., parallel NN, refracting, goes through the focus F and image S´, and through S´ the beam continues S.A.. This means that the image is imaginary, and the lens, as can be seen from the construction, is scattering.

Case 3. Item S and his image lie on different sides from the main optical axis NN(Fig. 4).

Connecting S And S´, we find the position of the optical center of the lens and the position of the lens. Ray S.A., parallel NN, is refracted through the focus F goes to the point S´. The beam passes through the optical center without refraction.

Task 2. In Fig. 5 shows a beam AB passed through a diverging lens. Construct the path of the incident ray if the position of the focal points of the lens is known.

Solution:

Let's continue the beam AB to intersection with the focal plane RR at the point F´ and draw the secondary axis OO through F And WITH(Fig. 6).

Beam along the side axis OO, will pass without changing its direction, the ray D.A., parallel OO, refracted in the direction AB so that its continuation goes through the point F´.

Task 3. On a converging lens with focal length F 1 = 40 cm a parallel beam of rays falls. Where should a diverging lens with focal length be placed? F 2 = 15 cm so that the beam of rays remains parallel after passing through two lenses?

Solution: According to the condition, a beam of incident rays EA parallel to the main optical axis NN, after refraction in the lenses it should remain so. This is possible if the diverging lens is positioned so that the rear focal points of the lenses F 1 and F 2 matched. Then the continuation of the ray AB(Fig. 7), incident on a diverging lens, passes through its focus F 2, and according to the rule of construction in a diverging lens, the refracted ray BD will be parallel to the main optical axis NN, therefore, parallel to the ray EA. From Fig. 7 it can be seen that the diverging lens should be placed at a distance d=F 1 -F 2 =(40-15)(cm)=25 cm from the collecting lens.

Answer: at a distance of 25 cm from the collecting lens.

Task 4. The height of the candle flame is 5 cm. The lens gives an image of this flame 15 cm high on the screen. Without touching the lens, the candle is moved to l= 1.5 cm further from the lens and, moving the screen, again obtained a sharp image of a flame 10 cm high. Determine the main focal length F lenses and the optical power of the lens in diopters.

Solution: Let us apply the thin lens formula, where d- distance from the object to the lens, f- distance from the lens to the image, for two positions of the object:

. (2)

From similar triangles AOB And A 1 O.B. 1 (Fig. 8) the transverse magnification of the lens will be equal to = , whence f 1 = Γ 1 d 1 .

Similarly for the second position of the object after moving it by l: , where f 2 = (d 1 + l)Γ 2 .
Substituting f 1 and f 2 in (1) and (2), we get:

. (3)
From the system of equations (3), excluding d 1, we find

.
Lens power

Answer: , diopters

Task 5. A biconvex lens made of refractive index glass n= 1.6, has a focal length F 0 = 10 cm in air ( n 0 = 1). What is the focal length? F 1 of this lens if placed in a transparent medium with a refractive index n 1 = 1.5? Determine the focal length F 2 of this lens in a medium with a refractive index n 2 = 1,7.

Solution:

The optical power of a thin lens is determined by the formula

,
Where n l- refractive index of the lens, n avg- refractive index of the medium, F- focal length of the lens, R 1 And R 2- radii of curvature of its surfaces.

If the lens is in the air, then

; (4)
n 1:

; (5)
in a medium with a refractive index n :

. (6)
For determining F 1 and F 2 we express from (4):

.
Let's substitute the resulting value into (5) and (6). Then we get

cm,

cm.
The sign "-" means that in a medium with a refractive index greater than that of the lens (in an optically denser medium), the collecting lens becomes divergent.

Answer: cm, cm.

Task 6. The system consists of two lenses with identical focal lengths. One of the lenses is converging, the other is diverging. The lenses are located on the same axis at a certain distance from each other. It is known that if the lenses are swapped, the actual image of the Moon given by this system will shift by l= 20 cm Find the focal length of each lens.

Solution:

Let's consider the case when parallel rays 1 and 2 fall on a diverging lens (Fig. 9).

After refraction, their continuations intersect at the point S, which is the focus of the diverging lens. Dot S is the “subject” for a converging lens. We obtain its image in a collecting lens according to the construction rules: rays 1 and 2 incident on the collecting lens, after refraction, pass through the intersection points of the corresponding secondary optical axes OO And O´O´ with focal plane RR converging lens and intersect at a point S´ on the main optical axis NN, on distance f 1 from the collecting lens. Let us apply the formula for a converging lens

, (7)
Where d 1 = F + a.

Let the rays now fall on a collecting lens (Fig. 10). Parallel rays 1 and 2 after refraction will converge at a point S(focus of the collecting lens). Falling on a diverging lens, the rays are refracted in the diverging lens so that the continuations of these rays pass through the intersection points TO 1 and TO 2 corresponding side axes ABOUT 1 ABOUT 1 and ABOUT 2 ABOUT 2 with focal plane RR diverging lens. Image S´ is located at the intersection point of the extensions of emerging rays 1 and 2 with the main optical axis NN on distance f 2 from the diverging lens.
For diverging lens

, (8)
Where d 2 = a - F.
From (7) and (8) we express f 1 and - f 2:NN and beam S.A. after refraction going in the direction AS´ according to the rules of construction (through the point TO 1 intersection of secondary optical axis OO, parallel to the incident beam S.A., with focal plane R 1 R 1 converging lens). If you put a diverging lens L 2, then the beam AS´ changes direction at a point TO, refracting (according to the construction rule in a diverging lens) in the direction KS´´. Continuation KS´´ passes through the point TO 2 secondary optical axis intersections 0 ´ 0 ´ with focal plane R 2 R 2 diverging lenses L 2 .

According to the formula for a diverging lens

,
Where d- distance from the lens L 2 to item S´, f- distance from the lens L 2 to image S´´.

From here cm.
The "-" sign indicates that the lens is diverging.

Lens power diopter

Answer: cm, diopters

Tasks for independent work

Kasyanov V.A. Physics. 11th grade: Educational. for general education institutions. - 2nd ed., additional. - M.: Bustard, 2004. - P. 281-306.
Elementary textbook of physics / Ed. G.S. Landsberg. - T. 3. - M.: Fizmatlit, 2000 and previous editions.
Butikov E.I., Kondratiev A.S. Physics. T. 2. Electrodynamics. Optics. - M.: Fizmatlit: Laboratory of basic knowledge; St. Petersburg: Nevsky dialect, 2001. - pp. 308-334.
Belolipetsky S.N., Erkovich O.S., Kazakovtseva V.A. and others. Problem book in physics. - M.: Fizmatlit, 2005. - P. 215-237.
Bukhovtsev B.B., Krivchenkov V.D., Myakishev G.Ya., Saraeva I.M. Problems in elementary physics. - M.: Fizmatlit, 2000 and previous editions.

Take another look at the lens drawings from the previous sheet: these lenses have noticeable thickness and significant curvature of their spherical boundaries. We deliberately drew such lenses so that the basic patterns of the path of light rays would appear as clearly as possible.

4.5.1 Thin Lens Concept

Now that these patterns are clear enough, we will look at a very useful idealization called the thin lens. As an example in Fig. 4.24 shows a biconvex lens; points O1 and O2 are the centers of its spherical surfaces6, R1 and R2 are the radii of curvature of these surfaces.

Rice. 4.24. Toward the definition of a thin lens

So, a lens is considered thin if its thickness MN is very small. It is necessary, however, to clarify: small compared to what?

First, it is assumed that MN R1 and MN R2 . Then the surfaces of the lens, although they will be convex, can be perceived as “almost flat”. This fact will come in handy very soon.

Secondly, MN a, where a is the characteristic distance from the lens to the object of interest to us. Actually, only in this case will we be able to correctly talk about the “distance from the object to the lens”, without specifying to which point of the lens this distance is taken.

We have given the definition of a thin lens, referring to the biconvex lens in Fig. 4.24. This definition is transferred without any changes to all other types of lenses. So: a lens is thin if the thickness of the lens is much less than the radii of curvature of its spherical boundaries and the distance from the lens to the object.

The symbol for a thin converging lens is shown in Fig. 4.25.

Rice. 4.25. Designation of a thin converging lens

6 Recall that straight line O1 O2 is called the main optical axis of the lens.

The symbol for a thin diverging lens is shown in Fig. 4.26.

Rice. 4.26. Designation of a thin diverging lens

In each case, the straight line F F is the main optical axis of the lens, and the points F themselves are its foci. Both foci of a thin lens are located symmetrically relative to the lens.

4.5.2 Optical center and focal plane

Points M and N, indicated in Fig. 4.24, for a thin lens they actually merge into one point. This is point O in Fig. 4.25 and 4.26, called the optical center of the lens. The optical center is located at the intersection of the lens with its main optical axis.

The distance OF from the optical center to the focus is called the focal length of the lens. We will denote focal length by f. The value D, the reciprocal of the focal length, is the optical power of the lens:

D = f 1:

Optical power is measured in diopters (Dopters). So, if the focal length of a lens is 25 cm, then its optical power is:

D = 0; 1 25 = 4 diopters:

We continue to get acquainted with new concepts. Any straight line passing through the optical center of the lens and different from the main optical axis is called a secondary optical axis. In Fig. Figure 4.27 shows the secondary optical axis straight OP.

P (side focus)

(focal plane)

Rice. 4.27. Side optical axis, focal plane and side focus

The plane passing through the focus perpendicular to the main optical axis is called the focal plane. The focal plane is thus parallel to the plane of the lens. Having two foci, the lens accordingly has two focal planes located symmetrically relative to the lens.

The point P at which the secondary optical axis intersects the focal plane is called the secondary focus. Actually, each point of the focal plane (except F) is a side focus; we can always draw a side optical axis by connecting this point with the optical center of the lens. And the point F itself, the focal point of the lens, is therefore also called

main focus.

What is in Fig. 4.27 shows a converging lens; it does not play any role. The concepts of secondary optical axis, focal plane and secondary focus are defined in exactly the same way for a diverging lens, with the converging lens replaced by a diverging one in Fig. 4.27.

We now move on to consider the path of rays in thin lenses. We will assume that the rays are paraxial, that is, they form fairly small angles with the main optical axis. If the paraxial rays emanate from one point, then after passing through the lens the refracted rays or their continuations also intersect at one point. Therefore, images of objects produced by the lens in paraxial rays are very clear.

4.5.3 Beam path through the optical center

As we know from the previous section, a ray traveling along the main optical axis is not refracted. In the case of a thin lens, it turns out that the beam traveling along the secondary optical axis is also not refracted!

This can be explained in the following way. Near the optical center O, both surfaces of the lens are indistinguishable from parallel planes, and the beam in this case seems to go through a plane-parallel glass plate (Fig. 4.28).

Rice. 4.28. Beam path through the optical center of the lens

Beam refraction angle AB equal to angle incidence of the refracted ray BC on the second surface. Therefore, the second refracted ray CD emerges from the plane-parallel plate parallel to the incident ray AB. A plane-parallel plate only shifts the beam without changing its direction, and this displacement is less, the smaller the thickness of the plate.

But for a thin lens we can assume that this thickness is zero. Then points B, O and C will actually merge into one point, and ray CD will be simply a continuation of ray AB. This is why it turns out that the beam traveling along the secondary optical axis is not refracted by a thin lens (Fig. 4.29).

Rice. 4.29. A ray passing through the optical center of a thin lens is not refracted

This is the only thing general property converging and diverging lenses. Otherwise, the path of the rays in them turns out to be different, and further we will have to consider the collecting and scattering lenses separately.

4.5.4 Path of rays in a collecting lens

As we remember, a converging lens is so called because a light beam parallel to the main optical axis, after passing through the lens, is collected at its main focus (Fig. 4.30).

Rice. 4.31. Refraction of a beam coming from the main focus

It turns out that a beam of parallel rays falling obliquely on a collecting lens will also be concentrated at a focus, but at a secondary one. This side focus P corresponds to the ray that passes through the optical center of the lens and is not refracted (Fig. 4.32).

Rice. 4.32. A parallel beam is collected at a side focus

Now we can formulate the rules for the path of rays in a collecting lens. These rules follow from Figures 4.29–4.32.

1. The beam passing through the optical center of the lens is not refracted.