Positive and negative lenses. Optical lenses (physics): definition, description, formula and solution. The principle of constructing an image with a converging lens

A lens is an optical part bounded by two refractive surfaces, which are the surfaces of bodies of revolution, one of which may be flat. Lenses are usually round shape, but may also have a rectangular, square, or some other configuration. As a rule, the refractive surfaces of a lens are spherical. Aspherical surfaces are also used, which can be in the form of surfaces of revolution of an ellipse, hyperbola, parabola and curves. higher order. In addition, there are lenses whose surfaces are part of the lateral surface of the cylinder, called cylindrical. Also used are toric lenses with surfaces having different curvature in two mutually perpendicular directions.

As individual optical parts, lenses are almost never used in optical systems, with the exception of simple magnifiers and field lenses (collectives). They are usually used in various complex combinations, such as glued two or three lenses and sets of a number of single and glued lenses.

Depending on the shape, there are collective (positive) and divergent (negative) lenses. The group of converging lenses usually includes lenses, in which the middle is thicker than their edges, and the group of diverging lenses is lenses, the edges of which are thicker than the middle. It should be noted that this is true only if the refractive index of the lens material is greater than that of environment. If the refractive index of the lens is less, the situation will be reversed. For example, an air bubble in water is a biconvex diffusing lens.

Lenses are characterized, as a rule, by their optical power (measured in diopters), or focal length, as well as aperture. For the construction of optical devices with corrected optical aberration (primarily chromatic aberration due to light dispersion, achromats and apochromats), other properties of lenses / their materials are also important, for example, the refractive index, the dispersion coefficient, the transmittance of the material in the selected optical range.

Sometimes lenses/lens optical systems(refractors) are specifically designed for use in media with a relatively high refractive index.

Types of lenses

Collective:

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 a meniscus and can be converging (thickens towards the middle) or divergent (thickens towards the edges). The meniscus, whose surface radii are equal, has optical power, zero(used for dispersion correction or as a cover lens). So, the lenses of myopic glasses are usually negative menisci. A distinctive property of a converging 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 - the main optical axis - a straight line passing through the centers of spherical surfaces limiting the lens; O - optical center - a point that, for biconvex or biconcave (with the same surface radii) lenses, is located on the optical axis inside the lens (in its center).

If a luminous point S is placed at some distance in front of the converging lens, then a beam 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 and will be the image of the point S. This point is called the conjugate focus, or simply the focus.

If light from a very distant source falls on the lens, the rays of which can be represented as traveling in a parallel beam, then upon exiting the lens, the rays will be refracted at a large angle and the point F will move closer to the lens on the optical axis. Under these conditions, the point of intersection of the rays emerging from the lens is called the main focus F, and the distance from the center of the lens to the main focus is called the main focal length.

Rays incident on a diverging lens, upon exiting it, will be refracted towards the edges of the lens, that is, they will be scattered. If these rays continue in the opposite direction as shown in the figure by the dotted line, then they will converge at one point F, which will be the focus of this lens. This focus will be imaginary.

What has been said about the focus on the main optical axis applies equally to those cases when the image of a point is located on a secondary or inclined optical axis, i.e., 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 the main focal plane, and at the conjugate focus, simply the focal plane.

Collecting lenses can be directed to the object by any side, as a result of which the rays passing through the lens can be collected from one or the other side of it. Thus, the lens has two focuses - front and rear. They are located on the optical axis on both sides of the lens.

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

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

In modern photographic lenses, they are widely used, also, aspherical lenses whose surface shape is different 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.

Lenses can be made from various varieties optical glass, as well as transparent plastics.

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

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

ill. 1. Collecting and diverging lenses.

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

According to the shape of the cross section, six main types of lenses are distinguished:

biconvex converging lenses;
plano-convex converging lenses;
concave-convex converging lenses (menisci);
biconcave diffusing lenses;
plano-concave diffusing lenses;
convex-concave diffusing lenses.

ill. 2. Six types of spherical lenses.

The spherical surfaces of the lens can 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 the formation of a biconvex lens.

C1 and C2 are the centers of the spherical surfaces bounding 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 line connecting points C1 and C2 is called main optical axis lenses.
The points of intersection of the main optical axis with the surfaces of the lens (A and B) are called lens vertices.
Distance from point A to the point B called axial lens thickness.

If a parallel beam of light rays is directed to the lens from a point lying on the main optical axis, then after passing through it, they will gather at the point F, which is also 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 greater the bulge) - the shorter the focal length.

Lenses. Optical devices

Lens is called a transparent body, which is bounded by two curved surfaces.

The lens is called thin if its thickness is much less than the radii of curvature of its surfaces.

The straight line passing through the centers of curvature of the lens surfaces is called the main optical axis of the lens. If one of the lens surfaces is a plane, then the optical axis runs perpendicular to it (Fig. 1).

Fig.1.

The point on a thin lens through which rays pass without changing their direction is called optical center lenses. The main optical axis passes through the optical center.

Any other straight line passing through the optical center of the lens is called secondary axis lenses. The point at which the rays of light converge, running parallel to the main optical axis, is called focus.

The plane passing through the focus perpendicular to the main optical axis is called focal plane.

Thin lens formula (Fig. 2):

In formula (1), the quantities a 1 , a 2 , r 1 and r 2 are considered positive if their counting directions from the optical center of the lens coincide with the direction of light propagation; otherwise, these values are considered negative.

Lenses are the main element of many optical devices.

The eye, for example, is an optical device, where the cornea and lens act as lenses, and the image of the object is obtained on the retina of the eye.

angle of view called the angle formed by the rays that pass from extreme points object or its image through the optical center of the lens of the eye.

Many optical devices are designed to obtain images of objects on screens, on light-sensitive films, or in the eye.

Apparent magnification of the optical device:

Lens in optical instrument facing the object (object) is called the lens; the lens facing the eye is called the eyepiece. In technical instruments, the objective and the eyepiece consist of several lenses. This partially eliminates errors in the images.

Magnifier magnification (Fig. 3):

The reciprocal of the focal length is called optical power lenses: AT = 1/f. The unit of optical power of a lens is the diopter ( D) equal to the optical power of a lens with a focal length of 1 m.

The optical power of two thin lenses put together is equal to the sum of their optical powers.

There are objects that are capable of changing the density of the electromagnetic radiation flux incident on them, that is, either increasing it by collecting it at one point, or decreasing it by scattering it. These objects are called lenses in physics. Let's consider this question in more detail.

What are lenses in physics?

This concept means absolutely any object that is capable of changing the direction of propagation of electromagnetic radiation. it 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 bounded by 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.

The principle of the lens

The essence of this simple optical object is the phenomenon of refraction of sunlight. 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 last name. The formulation 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 clarify the above, let's give an example: let the light fall on the surface of the water, while 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 the water already at an angle θ 2 to the normal to the surface. According to Snell's law, we get: sin (θ 1) * n 1 \u003d sin (θ 2) * n 2, here n 1 and n 2 are the refractive indices for air and water, respectively. What is the refractive index? This is a value showing how many times the propagation speed of electromagnetic waves in vacuum is greater than that for an optically transparent medium, that is, n = c/v, where c and v are the speeds of light in vacuum and in the medium, respectively.

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

The type of optical lens in physics is determined solely by the shape of the surfaces that form it. The direction of refraction of the beam incident on them 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). Conversely, if the curvature of the surface is negative (concave), then passing through the optical glass, the beam will move away from its central axis.

We note once again that the surface of any curvature refracts rays in the same way (according to Stella's law), but the normals to them have a different slope relative to the optical axis, as a result, different behavior refracted beam.

A lens 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 views are associated with a combination of the indicated surfaces, to which a plane is also added. What property the combined lens will have (diffusing or converging) depends on the total curvature of the radii of its surfaces.

Lens elements and ray properties

To build in lenses in imaging physics, it is necessary to get acquainted with the elements of this object. They are listed 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 beam does not experience refraction.
Focal length and focus - the distance between the center and a point on the optical axis, in which all rays incident on the lens parallel to this axis are collected. This definition is true for collecting optical glasses. In the case of divergent lenses, it is not the rays themselves that will converge to 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 \u003d 1 / f. It is measured in diopters (diopters), 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 falling 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 (as they are called, because it does not matter what spheres they are formed and how thick they are, only the optical properties of the object matter) are used to build images in them.

Images in optical glasses: how to build?

The figure below 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, it follows important findings:

Any image is built on only 2 rays (passing through the center and parallel to the main optical axis).
Converging lenses (denoted with arrows at the ends pointing outward) can give both an enlarged and reduced image, which in turn can be real (real) or imaginary.
If the object is in focus, then the lens does not form its image (see the lower diagram on the left in the figure).
Scattering optical glasses (denoted by arrows at their ends pointing inward) always give a reduced and imaginary 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 give 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 is the main focus. If a we are talking about the collecting optical glass, then the f-number will be positive. Conversely, for a diverging lens, f is negative.

Let's use this formula and solve a simple problem: 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 condition of the problem we have: 1/f = 1/(2*f)+1/d i . From: 1/d i = 1/f - 1/(2*f) = 1/(2*f), i.e. d i = 2*f. Thus, the image will appear at a distance of two foci from the lens, but on the other side than the object itself (this is indicated by the positive sign of the value d i).

Short story

It is curious to give the etymology of the word "lens". It originates from Latin words lens and lentis, which means "lentil", since optical objects in their shape really look like the fruit of this plant.

The 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 made 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 (at the beginning of this century, Galileo invented the first telescope). Note that the mathematical formulation of Stella's law of refraction, without knowing which it is impossible to manufacture lenses with desired 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 objects. An example of the former are magnetic lenses in electron microscope, a vivid example of the latter is the distortion of the direction of the light flux when it passes near massive space bodies(stars, planets).

Unlike prismatic and other diffusers, lenses in lighting fixtures are almost always used for spot lighting. As a rule, optical systems using lenses consist of a reflector (reflector) and one or more lenses.

Converging lenses direct light from a source located at the focal point into a parallel beam of light. As a rule, they are used in lighting structures together with a reflector. The reflector directs the light flux in the form of a beam in the right direction, and the lens concentrates (collects) the light. The distance between the converging lens and the light source is usually variable, allowing the angle to be obtained to be adjusted.

A system of both a light source and a converging lens (left) and a similar system of a source and a Fresnel lens (right). The angle of the light flux can be changed by changing the distance between the lens and the light source.

Fresnel lenses consist of separate concentric ring-shaped segments adjacent to each other. They got their name in honor of the French physicist Augustin Fresnel, who first proposed and put into practice such a design in lighthouse lighting fixtures. The optical effect of such lenses is comparable to that of traditional lenses of similar shape or curvature.

However, Fresnel lenses have a number of advantages due to which they find wide application in lighting designs. In particular, they are much thinner and cheaper to manufacture than converging lenses. Designers Francisco Gomez Paz and Paolo Rizzatto did not fail to take advantage of these features in their work on a bright and magical model range.

Made from lightweight and thin polycarbonate, the "sheets" of Hope, as Gomez Paz calls them, are nothing more than thin and large diffusing Fresnel lenses that create a magical, sparkling and voluminous glow by coating with a polycarbonate film textured with microprisms.

Paolo Rizzatto described the project as follows:
“Why have crystal chandeliers lost their relevance? Because they are too expensive, very difficult to handle and manufacture. We have decomposed the idea itself into components and modernized each of them.”

Here is what a colleague had to say about it:
“A few years ago, the marvelous possibilities of Fresnel lenses caught our attention. Their geometric features make it possible to obtain the same optical properties as those of conventional lenses, but on a completely flat surface of the petals.

However, the use of Fresnel lenses to create such unique products, combining a great design project with modern technological solutions, is still rare.

Such lenses are widely used in stage lighting with spotlights, where they allow you to create an uneven light spot with soft edges, blending perfectly with the overall light composition. Nowadays, they have also become widespread in architectural lighting schemes, in cases where individual adjustment of the angle of light is required, when the distance between the illuminated object and the lamp can change.

The optical performance of a Fresnel lens is limited by the so-called chromatic aberration that forms at the junctions of its segments. Because of it, a rainbow border appears on the edges of the images of objects. The fact that a seemingly flawed feature of a lens has been turned into a virtue in again emphasizes the power of the authors' innovative thought and their attention to detail.

Lighting design of a lighthouse using Fresnel lenses. The ring structure of the lens is clearly visible in the picture.

Projection systems consist of either an elliptical reflector or a combination of a parabolic reflector and a condenser directing light onto a collimator, which can also be supplemented optical accessories. After that, the light is projected onto a plane.

Spotlight systems: A uniformly illuminated collimator (1) directs the light through a lens system (2). Left - parabolic reflector, with high rate light output, on the right - a condenser that allows you to achieve high resolution.

The size of the image and the angle of light is determined by the features of the collimator. Simple curtains or iris diaphragms form light beams different sizes. Contour masks can be used to create different contours of the light beam. You can project logos or images using a gobo lens with drawings printed on them.

Different angles of light or image size can be selected depending on the focal length of the lenses. Unlike lighting fixtures using Fresnel lenses, here it is possible to create light rays with clear contours. Soft contours can be achieved by shifting the focus.

Examples of optional accessories (from left to right): a lens to create a wide beam of light, a sculpted lens to give the beam an oval shape, a grooved deflector and a "honeycomb lens" to reduce glare.

Stepped lenses convert light rays in such a way that they are somewhere between the "flat" light of a Fresnel lens and the "hard" light of a plano-convex lens. The convex surface is preserved in stepped lenses, however, stepped recesses are made on the side of the flat surface, forming concentric circles.

The front parts of the steps (risers) of concentric circles are often opaque (either painted over or have a chipped matte surface), which makes it possible to cut off the scattered radiation of the lamp and form a beam of parallel rays.

Fresnel spotlights form an uneven light spot with soft edges and a slight halo around the spot, making it easy to mix with other light sources, creating a natural light pattern. That is why Fresnel spotlights are used in cinema.

Projectors with a plano-convex lens, compared to projectors with a Fresnel lens, form a more uniform spot with a less pronounced transition at the edges of the light spot.

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