What is a lens in physics. Concave-convex lens

Types of lenses

Reflection and refraction of light are used to change the direction of rays or, as they say, to control light beams. This is the basis for the creation of special optical instruments, such as a magnifying glass, telescope, microscope, camera and others. The main part of most of them is the lens. For example, glasses are lenses enclosed in a frame. This example alone shows how important the use of lenses is for a person.

For example, in the first picture the flask is as we see it in life,

and on the second, if we look at it through a magnifying glass (the same lens).

In optics, spherical lenses are most often used. Such lenses are bodies made of optical or organic glass, limited by two spherical surfaces.

Lenses are transparent bodies bounded on both sides by curved surfaces (convex or concave). The straight line AB passing through the centers C1 and C2 of the spherical surfaces limiting the lens is called the optical axis.

This figure shows cross sections of two lenses with centers at point O. The first lens shown in the figure is called convex, the second is called concave. Point O, lying on the optical axis at the center of these lenses, is called the optical center of the lens.

One of the two bounding surfaces may be flat.

On the left the lenses are convex,

on the right - concave.

We will consider only spherical lenses, that is, lenses bounded by two spherical surfaces.
Lenses limited to two convex surfaces, are called biconvex; lenses bounded by two concave surfaces are called biconcave.

By directing a beam of rays parallel to the main optical axis of the lens at a convex lens, we will see that after refraction in the lens, these rays are collected at a point called the main focus of the lens

- point F. The lens has two main foci, on both sides at the same distance from optical center. If the light source is in focus, then after refraction in the lens the rays will be parallel to the main optical axis. Every lens has two focal points - one on each side of the lens. The distance from a lens to its focus is called the focal length of the lens.
Let us direct a beam of diverging rays from a point source lying on the optical axis to a convex lens. If the distance from the source to the lens is greater than the focal distance, then the rays, after refraction in the lens, will intersect the optical axis of the lens at one point. Consequently, a convex lens collects rays coming from sources located from the lens at a distance greater than its focal length. Therefore, a convex lens is otherwise called a converging lens.
When rays pass through a concave lens, a different picture is observed.
Let us send a beam of rays parallel to the optical axis onto a biconcave lens. We will notice that the rays will emerge from the lens in a diverging beam. If this diverging beam of rays enters the eye, then it will seem to the observer that the rays are coming out from point F. This point is called the imaginary focus of a biconcave lens. Such a lens can be called diverging.

Figure 63 explains the action of converging and diverging lenses. Lenses can be represented as a large number of prisms. Since prisms deflect rays, as shown in the figures, it is clear that lenses with thickening in the middle collect the rays, and lenses with thickening at the edges scatter them. The middle of the lens acts like a plane-parallel plate: it does not deflect rays in either the collecting or diverging lens

In the drawings, converging lenses are designated as shown in the figure on the left, and diverging lenses - in the figure on the right.

Among convex lenses there are: biconvex, plano-convex and concave-convex (respectively in the figure). All convex lenses have a wider middle cut than the edges. These lenses are called converging lenses.

Among concave lenses there are biconcave, plano-concave and convex-concave (respectively in the figure). All concave lenses have a narrower middle section than the edges. These lenses are called diverging lenses.

Light is electromagnetic radiation perceived by the eye through visual sensation.

Law of rectilinear propagation of light: light propagates rectilinearly in a homogeneous medium
A light source whose dimensions are small compared to the distance to the screen is called a point light source.

The incident beam and the reflected beam lie in the same plane with the perpendicular restored to the reflecting surface at the point of incidence. Angle of incidence equal to angle reflections.
If a point object and its reflection are swapped, the path of the rays will not change, only their direction will change.

flat mirror

A lens whose thickness is much less than the radii of curvature of its surfaces is called a thin lens.
A lens that converts a beam of parallel rays into a converging one and collects it into one point is called a converging lens.
A lens that converts a beam of parallel rays into a diverging one - diverging.

For a collecting lens

For a diverging lens:

In all positions of the object, the lens gives a reduced, virtual, direct image lying on the same side of the lens as the object.

Properties of the eye:

accommodation (achieved by changing the shape of the lenses);
adaptation (adaptation to different conditions illumination);
visual acuity (the ability to separately distinguish two close points);
visual field (the space observed when the eyes move but the head remains stationary)

Visual impairments

myopia (correction - diverging lens);

farsightedness (correction - converging lens).

A thin lens represents the simplest optical system. Simple thin lenses are used mainly in the form of glasses for glasses. In addition, the use of a lens as a magnifying glass is well known.

The action of many optical instruments - a projection lamp, a camera and other devices - can be schematically likened to the action thin lenses. However, a thin lens gives a good image only in that relatively in a rare case, when you can limit yourself to a narrow single-color beam coming from the source along the main optical axis or at a large angle to it. In the majority practical problems, where these conditions are not met, the image given by a thin lens is rather imperfect.
Therefore, in most cases they resort to building more complex optical systems that have large number refractive surfaces and not limited by the requirement of proximity of these surfaces (a requirement that a thin lens satisfies). [4]

4.2 Photographic apparatus. Optical instruments.

All optical instruments can be divided into two groups:

1) devices with which optical images are obtained on a screen. These include projection devices, cameras, movie cameras, etc.

2) devices that operate only in conjunction with through human eyes and do not form images on the screen. These include a magnifying glass, a microscope and various instruments of the telescope system. Such devices are called visual.

Camera.

Modern cameras have complex and varied structure, we will look at what basic elements a camera consists of and how they work.

The image of a lens formed by an optical system or part of an optical system. Used in the calculation of complex optical systems.

Encyclopedic YouTube

Story

The oldest lens is more than 3000 years old, the so-called Nimrud lens. It was found during excavations of one of the ancient capitals of Assyria in Nimrud by Austin Henry Layard in 1853. The lens has a shape close to an oval, roughly ground, one side is convex and the other is flat, and has a 3x magnification. The Nimrud Lens is on display in the British Museum.

First mention of lenses can be found in the ancient Greek play "The Clouds" by Aristophanes (424 BC), where fire was produced using convex glass and sunlight.

Characteristics of simple lenses

Depending on the forms there are collecting(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 the refractive index of the lens material is greater than that of 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 typically characterized by their optical power (measured in diopters) and focal length.

To build optical devices with corrected optical aberration (primarily chromatic, caused by the dispersion of light - achromats and apochromats), other properties of lenses and their materials are also important, for example, refractive index, dispersion coefficient, absorption index and scattering index of the material in the selected optical range .

Sometimes lenses/lens optical systems (refractors) are specifically designed for use in environments with relatively high rate refraction (see immersion microscope, immersion liquids).

A convex-concave lens is called meniscus and can be collective (thickens towards the middle), diffuse (thickens towards the edges) or telescopic (focal length is infinity). So, for example, the lenses of glasses for myopia are, as a rule, negative menisci.

Contrary to popular misconception, the optical power of a meniscus with equal radii is not zero, but positive, and depends on the refractive index of the glass and the thickness of the lens. A meniscus, the centers of curvature of the surfaces of which are located at one point, is called a concentric lens (optical power is always negative).

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.

The main elements of the lens: NN - optical axis - a straight line passing through the centers of the spherical surfaces limiting the lens; O - optical center - the point that for biconvex or biconcave (with the same surface radii) lenses is located on the optical axis inside the lens (at its center).
Note. The path of rays is shown as in an idealized (thin) lens, without indicating refraction at the real interface between the media. 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 being 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 imagined 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 focus F’, and the distance from the center of the lens to the focus is the focal length.

Rays incident on a diverging lens will be refracted toward 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.

1 u + 1 v = 1 f (\displaystyle (1 \over u)+(1 \over v)=(1 \over f))

Where u (\displaystyle u)- distance from the lens to the object; v (\displaystyle v) f (\displaystyle f)- the main focal length of the lens. In the case of a thick lens, the formula remains unchanged with the only difference that the distances are measured not from the center of the lens, but from the main planes.

To find one or another unknown quantity with two known ones, use the following equations:

f = v ⋅ u v + u (\displaystyle f=((v\cdot u) \over (v+u))) u = f ⋅ v v − f (\displaystyle u=((f\cdot v) \over (v-f))) v = f ⋅ u u − f (\displaystyle v=((f\cdot u) \over (u-f)))

It should be noted that the signs of the quantities u (\displaystyle u), v (\displaystyle v), f (\displaystyle 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 to be negative; if the object is imaginary, the distance to it is negative; if the lens is diverging, the focal length is negative.

Images of black letters through a thin convex lens with a focal length f(in red). Showing rays for letters E, I And K(blue, green and orange respectively). Letter image E(located at a distance of 2 f) real and inverted, the same size. Image I(on f) - at infinity. Image TO(on f/2) imaginary, direct, doubled

Linear increase

Linear increase m = a 2 b 2 a b (\displaystyle m=((a_(2)b_(2)) \over (ab)))(for the drawing from the previous section) is called the ratio of the dimensions of the image to the corresponding dimensions of the object. This ratio can also be expressed as a fraction m = a 2 b 2 a b = v u (\displaystyle m=((a_(2)b_(2)) \over (ab))=(v \over u)), Where v (\displaystyle v)- distance from the lens to the image; u (\displaystyle u)- distance from the lens to the object.

Here m (\displaystyle m) is the coefficient of linear magnification, that is, a number showing how many times the linear dimensions of the image are smaller (larger) than the actual linear dimensions of the object.

In the practice of calculations, it is much more convenient to express this relationship in values u (\displaystyle u) or f (\displaystyle f), Where f (\displaystyle f)- focal length of the lens.

M = f u − f ; m = v − f f (\displaystyle m=(f \over (u-f));m=((v-f) \over f)).

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, for example, 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. Optical power a system of two lenses can be found as the simple sum of the optical powers of each lens (provided that both lenses can be considered thin and they are located close to each other on the same axis):

1 F = 1 f 1 + 1 f 2 (\displaystyle (\frac (1)(F))=(\frac (1)(f_(1)))+(\frac (1)(f_(2)) )).

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:

1 F = 1 f 1 + 1 f 2 − L f 1 f 2 (\displaystyle (\frac (1)(F))=(\frac (1)(f_(1)))+(\frac (1) (f_(2)))-(\frac (L)(f_(1)f_(2)))),

Where L (\displaystyle L)- the distance between the main planes of the lenses.

Disadvantages of a simple lens

Modern optical devices place 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 of the shortcomings is achieved by appropriate selection of a number of lenses into a centered optical system - lens. 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 of the optical system and has nothing to do with the lens itself).

Lens called a transparent body bounded by two curved (most often spherical) or curved and flat surfaces. Lenses are divided into convex and concave.

Lenses whose middle is thicker than the edges are called convex. Lenses whose middle is thinner than the edges are called concave.

If the refractive index of the lens is greater than the refractive index of the surrounding medium, then in a convex lens a parallel beam of rays after refraction is converted into a converging beam. Such lenses are called collecting(Fig. 89, a). If a parallel beam in a lens is converted into a divergent beam, then these lenses are called scattering(Fig. 89, b). Concave lenses, which external environment serves air, are dissipative.

O 1, O 2 - geometric centers of the spherical surfaces limiting the lens. Straight O 1 O 2 connecting the centers of these spherical surfaces is called the main optical axis. We usually consider thin lenses whose thickness is small compared to the radii of curvature of its surfaces, so points C 1 and C 2 (vertices of the segments) lie close to each other; they can be replaced by one point O, called the optical center of the lens (see Fig. 89a). Any straight line drawn through the optical center of a lens at an angle to the main optical axis is called secondary optical axis(A 1 A 2 B 1 B 2).

If a beam of rays parallel to the main optical axis falls on a collecting lens, then after refraction in the lens they are collected at one point F, which is called main focus of the lens(Fig. 90, a).

At the focus of the diverging lens, continuations of the rays intersect, which before refraction were parallel to its main optical axis (Fig. 90, b). The focus of a diverging lens is imaginary. There are two main focuses; they are located on the main optical axis at the same distance from the optical center of the lens on opposite sides.

Reciprocal value focal length lenses, it's called optical power . Lens optical power - D.

The SI unit of optical power for a lens is the diopter. Diopter is the optical power of a lens whose focal length is 1 m.

The optical power of a converging lens is positive, while that of a diverging lens is negative.

The plane passing through the main focus of the lens perpendicular to the main optical axis is called focal(Fig. 91). A beam of rays incident on the lens parallel to some secondary optical axis is collected at the point of intersection of this axis with the focal plane.

Constructing an image of a point and an object in a converging lens.

To construct an image in a lens, it is enough to take two rays from each point of the object and find their point of intersection after refraction in the lens. It is convenient to use rays whose path after refraction in the lens is known. Thus, a ray incident on a lens parallel to the main optical axis, after refraction in the lens, passes through the main focus; the beam passing through the optical center of the lens is not refracted; the ray passing through the main focus of the lens, after refraction, goes parallel to the main optical axis; a ray incident on the lens parallel to the secondary optical axis, after refraction in the lens, passes through the point of intersection of the axis with the focal plane.

Let the luminous point S lie on the main optical axis.

We choose a beam at random and draw a secondary optical axis parallel to it (Fig. 92). The selected ray will pass through the point of intersection of the secondary optical axis with the focal plane after refraction in the lens. The point of intersection of this ray with the main optical axis (the second ray) will give a valid image of the point S - S`.

Let's consider constructing an image of an object in a convex lens.

Let the point lie outside the main optical axis, then the image S` can be constructed using any two rays shown in Fig. 93.

If the object is located at infinity, then the rays will intersect at the focus (Fig. 94).

If the object is located behind the double focus point, then the image will be real, inverse, reduced (camera, eye) (Fig. 95).

Everyone knows that a photographic lens consists of optical elements. Most photographic lenses use lenses as such elements. The lenses in a photographic lens are located on the main optical axis, forming the optical design of the lens.

Optical spherical lens - is a transparent homogeneous element bounded by two spherical or one spherical and the other flat surfaces.

In modern photographic lenses we have widespread, Also, aspherical lenses whose surface shape differs from a sphere. In this case, there may be parabolic, cylindrical, toric, conical and other curved surfaces, as well as surfaces of revolution with an axis of symmetry.

The materials used to make lenses can be various types of optical glass, as well as transparent plastics.

The entire variety of spherical lenses can be reduced to two main types: Collecting(or positive, convex) and Scattering(or negative, concave). Converging lenses in the center are thicker than at the edges, on the contrary, diverging lenses in the center are thinner than at the edges.

In a converging lens, parallel rays passing through it are focused at one point behind the lens. In diverging lenses, rays passing through the lens are scattered to the sides.

Ill. 1. Converging and diverging lenses.

Only positive lenses can produce images of objects. IN optical systems giving a real image (in particular lenses), diverging lenses can only be used together with collective ones.

There are six main types of lenses based on their cross-sectional shape:

biconvex converging lenses;
plano-convex converging lenses;
concave-convex collecting lenses (menisci);
biconcave diverging lenses;
flat-concave diverging lenses;
convex-concave diverging lenses.

Ill. 2. Six types of spherical lenses.

The spherical surfaces of the lens may have different curvature(degree of convexity/concavity) and different axial thickness.

Let's look at these and some other concepts in more detail.

Ill. 3. Elements of a biconvex lens

In Figure 3 you can see a diagram of the formation of a biconvex lens.

C1 and C2 are the centers of the spherical surfaces limiting the lens, they are called centers of curvature.
R1 and R2 are the radii of the spherical surfaces of the lens or radii of curvature.
The straight line connecting points C1 and C2 is called main optical axis lenses.
The points where the main optical axis intersects the lens surfaces (A and B) are called the vertices of the lens.
Distance from point A to the point B called axial lens thickness.

If a parallel beam of light rays is directed at a lens from a point lying on the main optical axis, then after passing through it they will converge at a point F, which is also located on the main optical axis. This point is called main focus lenses, and the distance f from the lens to this point - main focal length.

Ill. 4. Main focus, main focal plane and focal length of the lens.

Plane MN perpendicular to the main optical axis and passing through the main focus is called main focal plane. This is where the photosensitive matrix or photosensitive film is located.

The focal length of a lens directly depends on the curvature of its convex surfaces: the smaller the radii of curvature (i.e., the larger the convexity), the shorter the focal length.

There are objects that are capable of changing the density of the flux incident on them electromagnetic radiation, that is, either increase it by collecting it at one point, or reduce it by dispersing it. These objects are called lenses in physics. Let's take a closer look at this issue.

What are lenses in physics?

This concept means absolutely any object that is capable of changing the direction of propagation of electromagnetic radiation. This general definition lenses in physics, which includes optical glasses, magnetic and gravitational lenses.

In this article, the main attention will be paid to optical glasses, which are objects made of transparent material, and limited to two surfaces. One of these surfaces must necessarily have curvature (that is, be part of a sphere of finite radius), otherwise the object will not have the property of changing the direction of propagation of light rays.

Lens operating principle

The essence of the operation of this simple optical object lies in the phenomenon of refraction sun rays. At the beginning of the 17th century, the famous Dutch physicist and astronomer Willebrord Snell van Rooyen published the law of refraction, which currently bears his name. The wording of this law is as follows: when sunlight passes through the interface between two optically transparent media, then the product of the sine between the beam and the normal to the surface and the refractive index of the medium in which it propagates is a constant value.

To explain the above, let us give an example: let light fall on the surface of water, and the angle between the normal to the surface and the beam is equal to θ 1. Then, the light beam is refracted and begins its propagation in water at an angle θ 2 to the normal to the surface. According to Snell's law, we obtain: sin(θ 1)*n 1 = sin(θ 2)*n 2, here n 1 and n 2 are the refractive indices for air and water, respectively. What is refractive index? This is a value showing how many times the speed of propagation electromagnetic waves in a vacuum is greater than that for an optically transparent medium, that is, n = c/v, where c and v are the speed of light in a vacuum and in a medium, respectively.

The physics of refraction lies in the fulfillment of Fermat's principle, according to which light moves in such a way that least time overcome the distance from one point to another in space.

The appearance of an optical lens in physics is determined solely by the shape of the surfaces that form it. The direction of refraction of the incident beam depends on this shape. So, if the curvature of the surface is positive (convex), then upon exiting the lens the light beam will propagate closer to its optical axis (see below). On the contrary, if the curvature of the surface is negative (concave), then after passing through the optical glass, the beam will begin to move away from its central axis.

Let us note again that a surface of any curvature refracts rays equally (according to Stell's law), but the normals to them have a different inclination relative to the optical axis, resulting in different behavior of the refracted ray.

A lens that is bounded by two convex surfaces is called a converging lens. In turn, if it is formed by two surfaces with negative curvature, then it is called scattering. All other types are associated with a combination of the specified surfaces, to which a plane is also added. What property the combined lens will have (divergent or converging) depends on the total curvature of the radii of its surfaces.

Lens elements and ray properties

To construct images in lenses in physics, you need to become familiar with the elements of this object. They are given below:

Main optical axis and center. In the first case, they mean a straight line passing perpendicular to the lens through its optical center. The latter, in turn, is a point inside the lens, passing through which the ray does not experience refraction.
Focal length and focus - the distance between the center and the point on the optical axis into which all rays incident on the lens parallel to this axis are collected. This definition is true for those who collect optical glasses. In the case of diverging lenses, it is not the rays themselves that will be collected into a point, but their imaginary continuation. This point is called the main focus.
Optical power. This is the name of the reciprocal of the focal length, that is, D = 1/f. It is measured in diopters (dopters), that is, 1 diopter. = 1 m -1 .

The following are the main properties of rays that pass through a lens:

the beam passing through the optical center does not change the direction of its movement;
rays incident parallel to the main optical axis change their direction so that they pass through the main focus;
Rays incident on optical glass at any angle, but passing through its focus, change their direction of propagation in such a way that they become parallel to the main optical axis.

The above properties of rays for thin lenses in physics (they are called so because it does not matter what spheres they are formed of and how thick they are, only the optical properties of the object matter) are used to construct images in them.

Images in optical glasses: how to build?

Below is a figure that shows in detail the schemes for constructing images in the convex and concave lenses of an object (red arrow) depending on its position.

From the analysis of the circuits in the figure, important conclusions follow:

Any image is built on only 2 rays (passing through the center and parallel to the main optical axis).
Converging lenses (indicated by arrows at the ends pointing outward) can produce either a magnified or a reduced image, which in turn can be real (real) or virtual.
If an object is in focus, then the lens does not form its image (see the lower diagram on the left in the figure).
Diffusing optical glasses (indicated by arrows at their ends directed inward) always give a reduced and virtual image, regardless of the position of the object.

Finding the distance to an image

To determine at what distance the image will appear, knowing the position of the object itself, we present the lens formula in physics: 1/f = 1/d o + 1/d i, where d o and d i are the distance to the object and to its image from the optical center, respectively, f - main focus. If we're talking about about collecting optical glass, then the number f will be positive. On the contrary, for a diverging lens f is negative.

Let's use this formula and solve simple task: let the object be at a distance d o = 2*f from the center of the collecting optical glass. Where will his image appear?

From the problem conditions we have: 1/f = 1/(2*f)+1/d i . From: 1/d i = 1/f - 1/(2*f) = 1/(2*f), that is, d i = 2*f. Thus, the image will appear at a distance of two focal points from the lens, but on the other side than the object itself (this is indicated by positive sign values d i).

Brief history

It is interesting to give the etymology of the word “lens”. It comes from the Latin words lens and lentis, which mean “lentil”, since optical objects in their shape are really similar to the fruit of this plant.

Refractive power of spherical transparent bodies was known to the ancient Romans. For this purpose they used round glass vessels filled with water. Glass lenses themselves began to be manufactured only in the 13th century in Europe. They were used as a reading tool (modern glasses or a magnifying glass).

The active use of optical objects in the manufacture of telescopes and microscopes dates back to the 17th century (Galileo invented the first telescope at the beginning of this century). Note that the mathematical formulation of Stell's law of refraction, without knowledge of which it is impossible to produce lenses with given properties, was published by a Dutch scientist at the beginning of the same 17th century.

Other types of lenses

As noted above, in addition to optical refractive objects, there are also magnetic and gravitational ones. An example of the former are magnetic lenses in electron microscope, shining example The second is to distort the direction of the light flow when it passes near massive cosmic bodies (stars, planets).