Geometric optics, the limits of its application. Basic principle of geometric optics

The basic laws of geometric optics have been known since ancient times. So, Plato (430 BC) established the law rectilinear propagation Sveta. Euclid's treatises formulate the law of rectilinear propagation of light and the law of equality of the angles of incidence and reflection. Aristotle and Ptolemy studied the refraction of light. But the exact wording of these laws of geometric optics Greek philosophers could not find. geometric optics is the limiting case wave optics, when the wavelength of light tends to zero. Protozoa optical phenomena, such as the appearance of shadows and the acquisition of images in optical instruments, can be understood within the framework of geometric optics.

The formal construction of geometric optics is based on four laws established empirically: the law of rectilinear propagation of light; the law of independence of light rays; the law of reflection; the law of refraction of light. To analyze these laws, H. Huygens proposed a simple and intuitive method, later called Huygens principle .Each point to which the light excitation reaches is ,in its turn, center of secondary waves;the surface that envelopes these secondary waves at a certain moment of time indicates the position at that moment of the front of the actually propagating wave.

Based on his method, Huygens explained straightness of light propagation and brought laws of reflection and refraction .The law of rectilinear propagation of light :· light travels in a straight line in an optically homogeneous medium.The proof of this law is the presence of a shadow with sharp boundaries from opaque objects when illuminated by small sources. Careful experiments have shown, however, that this law is violated if light passes through very small holes, and the deviation from straightness of propagation is greater, the smaller the holes. .

The shadow cast by an object is caused by rectilinear propagation of light rays in optically homogeneous media. Fig 7.1 Astronomical illustration rectilinear propagation of light and, in particular, the formation of a shadow and penumbra can serve as the shading of some planets by others, for example lunar eclipse , when the Moon falls into the shadow of the Earth (Fig. 7.1). Due to the mutual motion of the Moon and the Earth, the shadow of the Earth moves over the surface of the Moon, and moon eclipse passes through several partial phases (Fig. 7.2).

The law of independence of light beams :· the effect produced by a single beam does not depend on whether,whether other beams act simultaneously or they are eliminated. By splitting the light flux into separate light beams (for example, using diaphragms), it can be shown that the action of the selected light beams is independent. Law of reflection (Fig. 7.3): the reflected ray lies in the same plane as the incident ray and the perpendicular,drawn to the interface between two media at the point of incidence;· angle of incidenceα equal to the angle of reflectionγ: α = γ

To derive the law of reflection Let's use the Huygens principle. Let us assume that a plane wave (wave front AB with, falls on the interface between two media (Fig. 7.4). When the wave front AB reaches the reflective surface at a point BUT, this point will begin to radiate secondary wave .· For the wave to travel the distance Sun time required Δ t = BC/ υ . During the same time, the front of the secondary wave will reach the points of the hemisphere, the radius AD which is equal to: υ Δ t= sun. The position of the reflected wave front at this moment of time, in accordance with the Huygens principle, is given by the plane DC, and the direction of propagation of this wave is ray II. From the equality of triangles ABC and ADC follows law of reflection: angle of incidenceα equal to the angle of reflection γ . Law of refraction (Snell's law) (Fig. 7.5): the incident beam, the refracted beam and the perpendicular drawn to the interface at the point of incidence lie in the same plane;· the ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant value for given media.

Derivation of the law of refraction. Let us assume that a plane wave (wave front AB) propagating in vacuum along the direction I with a velocity with, falls on the interface with the medium, in which the velocity of its propagation is equal to u(Fig. 7.6). Let the time taken by the wave to travel the path Sun, equals D t. Then sun=s D t. During the same time, the front of the wave excited by the point BUT in an environment with speed u, reaches the points of a hemisphere, the radius of which AD = u D t. The position of the refracted wave front at this moment of time, in accordance with the Huygens principle, is given by the plane DC, and the direction of its propagation - beam III . From fig. 7.6 shows that , i.e. .This implies Snell's law : A somewhat different formulation of the law of light propagation was given by the French mathematician and physicist P. Fermat.

Physical studies include for the most part to optics, where he established in 1662 the basic principle of geometric optics (Fermat's principle). The analogy between Fermat's principle and the variational principles of mechanics has played a significant role in the development of modern dynamics and the theory of optical instruments. According to Fermat's principle , light travels between two points along a path that requires least time. We will show the application of this principle to the solution of the same problem of light refraction. A beam from a light source S located in vacuum goes to the point AT located in some medium outside the interface (Fig. 7.7).

In each environment, the shortest path will be direct SA and AB. Point A characterize by the distance x from the perpendicular dropped from the source to the interface. Determine the time taken to complete the path SAB:.To find the minimum, we find the first derivative of τ with respect to X and equate it to zero: from here we come to the same expression that was obtained on the basis of the Huygens principle: Fermat's principle has retained its significance to this day and served as the basis for the general formulation of the laws of mechanics (including the theory of relativity and quantum mechanics). From Fermat's principle has several consequences. Reversibility of light rays : if you reverse the beam III (Fig. 7.7), causing it to fall on the interface at an angleβ, then the refracted beam in the first medium will propagate at an angle α, i.e. will go to reverse direction along the beam I . Another example is a mirage , which is often observed by travelers on sun-hot roads. They see an oasis ahead, but when they get there, there is sand all around. The essence is that we see in this case the light passing over the sand. The air is very hot above the most expensive, and in the upper layers it is colder. Hot air, expanding, becomes more rarefied and the speed of light in it is greater than in cold air. Therefore, light does not travel in a straight line, but along a trajectory with the shortest time, wrapping in warm layers of air. If light propagates from media with a high refractive index (optically denser) into a medium with a lower refractive index (optically less dense) ( > ) , for example, from glass to air, then, according to the law of refraction, the refracted ray moves away from the normal and the angle of refraction β is greater than the angle of incidence α (Fig. 7.8 a).

With an increase in the angle of incidence, the angle of refraction increases (Fig. 7.8 b, in), until at a certain angle of incidence () the angle of refraction is equal to π / 2. The angle is called limiting angle . At angles of incidence α > all incident light is completely reflected (Fig. 7.8 G). As the angle of incidence approaches the limit, the intensity of the refracted beam decreases, and the reflected beam increases. If, then the intensity of the refracted beam goes to zero, and the intensity of the reflected beam is equal to the intensity of the incident (Fig. 7.8 G). · Thus,at angles of incidence ranging from to π/2,the beam is not refracted,and fully reflected on the first Wednesday,and the intensities of the reflected and incident rays are the same. This phenomenon is called complete reflection. The limiting angle is determined from the formula: ; .The phenomenon of total reflection is used in total reflection prisms (Fig. 7.9).

The refractive index of glass is n » 1.5, so limit angle for glass-air border \u003d arcsin (1 / 1.5) \u003d 42 °. When light falls on the glass-air interface at α > 42° there will always be total reflection. In fig. 7.9 shows total reflection prisms that allow you to: a) rotate the beam by 90 °; b) rotate the image; c) wrap the rays. Total reflection prisms are used in optical devices (for example, in binoculars, periscopes), as well as in refractometers, which allow determining the refractive indices of bodies (according to the law of refraction, by measuring, we determine the relative refractive index of two media, as well as the absolute refractive index of one of the media, if the refractive index of the second medium is known).

The phenomenon of total reflection is also used in light guides , which are thin, randomly bent filaments (fibers) made of an optically transparent material. Fig. 1. 7.10 In fiber parts, glass fiber is used, the light-guiding core (core) of which is surrounded by glass - a shell of another glass with a lower refractive index. Light incident on the end of the light guide at angles greater than the limit , undergoes at the interface between the core and the cladding total reflection and spreads only along the light-guiding core. Light guides are used to create high-capacity telegraph and telephone cables . The cable consists of hundreds and thousands of optical fibers as thin as a human hair. Up to eighty thousand telephone conversations can be simultaneously transmitted over such a cable, the thickness of an ordinary pencil. purposes of integrated optics.

Some optical laws were already known before the nature of light was established. The basis of geometric optics is formed by four laws: 1) the law of rectilinear propagation of light; 2) the law of independence of light rays; 3) the law of reflection of light; 4) the law of refraction of light.

The law of rectilinear propagation of light: Light travels in a straight line in an optically homogeneous medium. This law is approximate, since when light passes through very small holes, deviations from straightness are observed, the larger the smaller the hole.

The law of independence of light beams: the effect produced by a single beam does not depend on whether the other beams act simultaneously or are eliminated. The intersections of the rays do not prevent each of them from propagating independently of each other. By splitting the light beam into separate light beams, it can be shown that the action of the selected light beams is independent. This law is valid only for not too high light intensities. At intensities achieved with lasers, the independence of the light beams is no longer respected.

Law of reflection: the beam reflected from the interface between two media lies in the same plane as the incident beam and the perpendicular drawn to the interface at the point of incidence; the angle of reflection is equal to the angle of incidence.

Law of refraction: the incident beam, the refracted beam and the perpendicular drawn to the interface at the point of incidence lie in the same plane; the ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant value for given media

sin i 1/sin i 2 \u003d n 12 \u003d n 2 / n 1, obviously sin i 1/sin i 2 \u003d V 1 / V 2, (1)

where n 12 - relative refractive index second environment relative to the first. The relative refractive index of two media is equal to the ratio of their absolute refractive indices n 12 = n 2 / n 1 .

The absolute refractive index of the medium is called. the value n, equal to the ratio of the speed C of electromagnetic waves in vacuum to their phase speed V in the medium:

A medium with a large optical refractive index is called. optically denser.

The symmetry of expression (1) implies reversibility of light rays, the essence of which is that if a light beam is directed from the second medium to the first at an angle i 2 , then the refracted beam in the first medium will come out at an angle i one . When light passes from an optically less dense medium to a denser one, it turns out that sin i 1 > sin i 2 , i.e. the angle of refraction is less than the angle of incidence of light, and vice versa. In the latter case, as the angle of incidence increases, the angle of refraction increases to a greater extent, so that at a certain limiting angle of incidence i pr angle of refraction becomes equal to π/2. Using the law of refraction, you can calculate the value of the limiting angle of incidence:

sin i pr / sin (π / 2) = n 2 / n 1, whence i pr \u003d arcsin n 2 / n 1. (2)

In this limiting case, the refracted beam slides along the interface between the media. At angles of incidence i > i Since light does not penetrate into the depths of an optically less dense medium, the phenomenon takes place total internal reflection. Injection i pr is called limiting angle total internal reflection.

Phenomenon total internal reflection used in total reflection prisms, which are used in optical instruments: binoculars, periscopes, refractometers (devices that allow you to determine the optical refractive indices), in light guides, which are thin, bending threads (fibers) of an optically transparent material. Light incident on the end of the fiber at angles greater than the limiting one undergoes complete internal reflection and propagates only along the light-guiding core. With the help of light guides, you can bend the path of the light beam as you like. Multi-core optical fibers are used for image transmission. Discuss the use of light guides.

To explain the law of refraction and bending of rays as they pass through optically inhomogeneous media, the concept is introduced optical path length

L = nS or L = ∫ndS,

for homogeneous and inhomogeneous media, respectively.

In 1660, the French mathematician and physicist P. Fermat established extremity principle(Fermat's principle) for the optical path length of a beam propagating in inhomogeneous transparent media: the optical path length of a beam in a medium between two given points minimal, or in other words, Light travels along a path that has the shortest optical length.

Photometric quantities and their units. Photometry is a branch of physics dealing with the measurement of the intensity of light and its sources. 1. Energy quantities:

radiation fluxФ e - a value numerically equal to the ratio of energy W radiation by the time t during which the radiation occurred:

F e = W/ t, watt (W).

Energy luminosity(radiance) R e - a value equal to the ratio of the radiation flux Ф e emitted by the surface to the area S of the section through which this flux passes:

R e \u003d F e / S, (W / m 2)

those. is the surface radiation flux density.

Energy power of light (radiant power) I e is determined using the concept of a point source of light - a source whose dimensions, compared with the distance to the observation point, can be neglected. The energy intensity of light I e is a value equal to the ratio of the radiation flux Ф e of the source to the solid angle ω, within which this radiation propagates:

I e \u003d F e / ω, (W / sr) - watts per steradian.

The intensity of light often depends on the direction of radiation. If it does not depend on the direction of radiation, then such source called isotropic. For an isotropic source, the luminous intensity is

I e \u003d F e / 4π.

In the case of an extended source, we can talk about the luminous intensity of an element of its surface dS.

Energy brightness (radiance) AT e is a value equal to the ratio of the energy intensity of light ΔI e of the element of the radiating surface to the area ΔS of the projection of this element on a plane perpendicular to the direction of observation:

AT e = ∆I e / ∆S. (W/sr.m 2)

Energy illumination(irradiance) E e characterizes the degree of illumination of the surface and is equal to the magnitude of the radiation flux incident on a unit of the illuminated surface. (W/m2.

2. Light values. In optical measurements, various radiation receivers are used, the spectral characteristics of the sensitivity of which to light of different wavelengths are different. The relative spectral sensitivity of the human eye V(λ) is shown in fig. V(λ)

400 555 700 λ, nm

Therefore, light measurements, being subjective, differ from objective, energy ones, and light units are introduced for them, which are used only for visible light. The basic unit of light in SI is the intensity of light - candela(cd), which is equal to the luminous intensity in a given direction of a source emitting monochromatic radiation with a frequency of 540 10 12 Hz, the luminous intensity of which in this direction is 1/683 W/sr.

The definition of light units is similar to energy units. To measure light quantities, special devices are used - photometers.

Light flow. The unit of luminous flux is lumen(lm). It is equal to the luminous flux emitted by an isotropic light source with a power of 1 cd within a solid angle of one steradian (with a uniform radiation field inside a solid angle):

1 lm \u003d 1 cd 1 sr.

It has been experimentally established that a luminous flux of 1 lm, formed by radiation with a wavelength of λ = 555 nm, corresponds to an energy flux of 0.00146 W. A luminous flux of 1 lm, formed by radiation with a different λ, corresponds to an energy flux

Ф e \u003d 0.00146 / V (λ), W.

1 lm = 0.00146 W.

illumination E- the value wound by the ratio of the luminous flux Ф, incident on the surface, to the area S of this surface:

E\u003d F / S, lux (lx).

1 lux is the illumination of the surface, on 1 m 2 of which a luminous flux of 1 lm falls (1 lux \u003d 1 lm / m 2).

Brightness R C (luminosity) of a luminous surface in a certain direction φ is a value equal to the ratio of the luminous intensity I in this direction to the area S of the projection of the luminous surface onto a plane perpendicular to this direction:

R C \u003d I / (Scosφ). (cd / m 2).

Chapter 3 Optics

Optics- a branch of physics that studies the properties and physical nature of light, as well as its interaction with matter. The doctrine of light is usually divided into three parts:

• geometric or ray optics , which is based on the concept of light rays;
• wave optics , which studies phenomena in which the wave properties of light are manifested;
• quantum optics , which studies the interaction of light with matter, in which the corpuscular properties of light are manifested.

This chapter deals with the first two parts of optics. Corpuscular properties light will be considered in Chap. v.

geometric optics

Basic laws of geometric optics

The basic laws of geometric optics were known long before the establishment of the physical nature of light.

The law of rectilinear propagation of light: Light travels in a straight line in an optically homogeneous medium. An experimental proof of this law can be sharp shadows cast by opaque bodies when illuminated by light from a source of sufficiently small dimensions ("point source"). Another proof is the well-known experiment on the passage of light from a distant source through a small hole, as a result of which a narrow light beam is formed. This experience leads to the idea of ​​a light beam as a geometric line along which light propagates. It should be noted that the law of rectilinear propagation of light is violated and the concept of a light beam loses its meaning if the light passes through small holes, the dimensions of which are comparable to the wavelength. Thus, geometric optics based on the idea of ​​light rays is the limiting case of wave optics at λ → 0. The limits of applicability of geometric optics will be considered in the section on light diffraction.

At the interface between two transparent media, light can be partially reflected in such a way that part of the light energy will propagate after reflection in a new direction, and part will pass through the interface and continue to propagate in the second medium.

Law of light reflection: the incident and reflected beams, as well as the perpendicular to the interface between two media, restored at the point of incidence of the beam, lie in the same plane ( plane of incidence ). The angle of reflection γ is equal to the angle of incidence α.

Law of refraction of light: the incident and refracted beams, as well as the perpendicular to the interface between two media, restored at the point of incidence of the beam, lie in the same plane. The ratio of the sine of the angle of incidence α to the sine of the angle of refraction β is a constant value for two given media:

The laws of reflection and refraction are explained in wave physics. According to wave concepts, refraction is a consequence of a change in the speed of wave propagation during the transition from one medium to another. The physical meaning of the refractive index is the ratio of the wave propagation speed in the first medium υ 1 to the speed of their propagation in the second medium υ 2:

Figure 3.1.1 illustrates the laws of reflection and refraction of light.

A medium with a lower absolute refractive index is called optically less dense.

When light passes from an optically denser medium to an optically less dense one n 2 < n 1 (for example, from glass to air) can observe the phenomenon total reflection , that is, the disappearance of the refracted beam. This phenomenon is observed at angles of incidence exceeding a certain critical angle α pr, which is called limiting angle of total internal reflection (see fig. 3.1.2).

For the angle of incidence α = α pr sin β = 1; sin valueα pr \u003d n 2 / n 1 < 1.

If the second medium is air ( n 2 ≈ 1), it is convenient to rewrite the formula in the form

The phenomenon of total internal reflection finds application in many optical devices. The most interesting and practically important application is the creation fiber light guides , which are thin (from a few micrometers to millimeters) arbitrarily curved filaments made of an optically transparent material (glass, quartz). Light falling on the end of the fiber can propagate along it over long distances due to total internal reflection from the side surfaces (Fig. 3.1.3). Scientific and technical direction engaged in the development and application of optical light guides, is called fiber optics .

Mirrors

The simplest optical device capable of creating an image of an object is flat mirror . The image of an object given by a flat mirror is formed by rays reflected from the mirror surface. This image is imaginary, since it is formed by the intersection not of the reflected rays themselves, but of their continuations in the "mirror" (Fig. 3.2.1).

Due to the law of light reflection, the imaginary image of an object is located symmetrically with respect to the mirror surface. The size of the image is equal to the size of the object itself.

spherical mirror called a specularly reflective surface having the shape of a spherical segment. The center of the sphere from which the segment is cut is called optical center of the mirror . The top of a spherical segment is called pole . The straight line passing through the optical center and the pole of the mirror is called main optical axis spherical mirror. The main optical axis is distinguished from all other straight lines passing through the optical center only by the fact that it is the axis of symmetry of the mirror.

Spherical mirrors are concave and convex . If a beam of rays parallel to the main optical axis falls on a concave spherical mirror, then after reflection from the mirror the rays will intersect at a point called main focus F mirrors. The distance from the focus to the pole of the mirror is called focal length and denoted by the same letter F. A concave spherical mirror has a real focus. It is located in the middle between the center and the pole of the mirror (Figure 3.2.2).

It should be borne in mind that the reflected rays intersect at approximately one point only if the incident parallel beam was sufficiently narrow (the so-called paraxial bundle ).

The main focus of a convex mirror is imaginary. If a beam of rays parallel to the main optical axis falls on a convex mirror, then after reflection at the focus, not the rays themselves will intersect, but their continuations (Fig. 3.2.3).

The focal lengths of spherical mirrors are assigned a certain sign: for a concave mirror, for a convex one, where R is the radius of curvature of the mirror.

Image of any point A An object in a spherical mirror can be constructed using any pair of standard rays:

• Ray AOC passing through the optical center of the mirror; reflected beam COA goes along the same straight line;
• Ray AFD, going through the focus of the mirror; the reflected beam goes parallel to the main optical axis;
• Ray AP incident on the mirror at its pole; the reflected beam is symmetrical with the incident beam about the main optical axis.
• Ray AE, parallel to the main optical axis; reflected beam EFA 1 passes through the focus of the mirror.

In Figure 3.2.4, the standard beams listed above are shown for the case of a concave mirror. All these rays pass through the point A", which is the image of the point A. All other reflected rays also pass through the point A". The course of the rays, in which all Rays leaving one point are collected at another point, called stigmatic . Line segment A"B" is an image of the subject AB. The constructions for the case of a convex mirror are similar.

Image position and size can also be determined using spherical mirror formulas :

Here d is the distance from the object to the mirror, f is the distance from the mirror to the image. Quantities d and f obey a certain sign rule:

• d> 0 and f> 0 - for real objects and images;
• d < 0 и f < 0 – для мнимых предметов и изображений.

For the case shown in Figure 3.2.4, we have:

F> 0 (mirror is concave); d = 3F> 0 (real item).

According to the formula of a spherical mirror, we get: therefore, the image is real.

If instead of a concave mirror there was a convex mirror with the same focal length modulo, we would get the following result:

F < 0, d = –3F> 0, – the image is imaginary.

The linear magnification of a spherical mirror Γ is defined as the ratio of the linear dimensions of the image h"and subject h.

size h" it is convenient to attribute a certain sign depending on whether the image is direct ( h"> 0) or inverted ( h" < 0). Величина h always considered positive. With this definition, the linear magnification of a spherical mirror is expressed by a formula that can be easily obtained from Figure 3.2.4:

In the first of the examples discussed above, therefore, the image is inverted, reduced by 2 times. In the second example, the image is straight, reduced by 4 times.

Thin lenses

Lens A transparent body bounded by two spherical surfaces is called. If the thickness of the lens itself is small compared to the radii of curvature of spherical surfaces, then the lens is called thin .

Lenses are part of almost all optical devices. Lenses are gathering and scattering . The converging lens in the middle is thicker than at the edges, the diverging lens, on the contrary, is thinner in the middle part (Fig. 3.3.1).

Straight line passing through the centers of curvature O 1 and O 2 spherical surfaces, called main optical axis lenses. In the case of thin lenses, we can approximately assume that the main optical axis intersects with the lens at one point, which is commonly called optical center lenses O. A beam of light passes through the optical center of the lens without deviating from its original direction. All lines passing through the optical center are called side optical axes .

If a beam of rays parallel to the main optical axis is directed to the lens, then after passing through the lens the rays (or their continuation) will gather at one point F, which is called main focus lenses. A thin lens has two main foci located symmetrically on the main optical axis relative to the lens. Converging lenses have real foci, diverging lenses have imaginary foci. Beams of rays parallel to one of the secondary optical axes, after passing through the lens, are also focused to a point F", which is located at the intersection of the side axis with focal planeФ, that is, a plane perpendicular to the main optical axis and passing through the main focus (Fig. 3.3.2). Distance between the optical center of the lens O and main focus F called the focal length. It is denoted by the same letter F.

The main property of lenses is the ability to give images of objects . Images are direct and upside down , valid and imaginary ,enlarged and reduced .

The position of the image and its nature can be determined using geometric constructions. To do this, use the properties of some standard rays, the course of which is known. These are rays passing through the optical center or one of the foci of the lens, as well as rays parallel to the main or one of the secondary optical axes. Examples of such constructions are shown in Figs. 3.3.3 and 3.3.4.

Note that some of the standard beams used in Fig. 3.3.3 and 3.3.4 for imaging do not pass through the lens. These rays do not really participate in the formation of the image, but they can be used for constructions.

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 through f, then the thin lens formula can be written as:

The formula for a thin lens is similar to that for a spherical mirror. It can be obtained for paraxial rays from the similarity of triangles in Fig. 3.3.3 or 3.3.4.

It is customary to attribute certain signs to the focal lengths of lenses: for a converging lens F> 0, for scattering F < 0.

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 continuations of rays converging behind the lens) and images;
d < 0 и f < 0 – для мнимых источников и изображений.

For the case shown in Fig. 3.3.3, we have: F> 0 (converging lens), d = 3F> 0 (real item).

According to the thin lens formula, we get: therefore, the image is real.

In the case shown in Fig. 3.3.4, F < 0 (линза рассеивающая), d = 2|F| > 0 (real object), that is, the image is imaginary.

Depending on the position of the object in relation to the lens, the linear dimensions of the image change. Linear zoom lens Γ is the ratio of the linear dimensions of the image h" and subject h. size h", as in the case of a spherical mirror, it is convenient to assign plus or minus signs depending on whether the image is upright or inverted. Value h always considered positive. Therefore, for direct images Γ > 0, for inverted images Γ< 0. Из подобия треугольников на рис. 3.3.3 и 3.3.4 легко получить формулу для линейного увеличения тонкой линзы:

In the considered example with a converging lens (Fig. 3.3.3): d = 3F> 0, therefore, the image is inverted and reduced by 2 times.

In the diverging lens example (Figure 3.3.4): d = 2|F| > 0, ; therefore, the image is straight and reduced by 3 times.

optical power D lens depends on both the radii of curvature R 1 and R 2 of its spherical surfaces, and on the refractive index n the material from which the lens is made. In optics courses, the following formula is proved:

The radius of curvature of a convex surface is considered positive, and that of a concave surface is negative. This formula is used in the manufacture of lenses with a given optical power.

In many optical instruments, light passes sequentially through two or more lenses. The image of the object given by the first lens serves as the object (real or imaginary) for the second lens, which builds the second image of the object. This second image can also be real or imaginary. The calculation of an optical system of two thin lenses is reduced to applying the lens formula twice, with the distance d 2 from the first image to the second lens should be set equal to the value l – f 1 , where l is the distance between the lenses. The value calculated from the lens formula f 2 determines the position of the second image and its character ( f 2 > 0 – real image, f 2 < 0 – мнимое). Общее линейное увеличение Γ системы из двух линз равно произведению линейных увеличений обеих линз: Γ = Γ 1 · Γ 2 . Если предмет или его изображение находятся в бесконечности, то линейное увеличение утрачивает смысл.

A special case is the telescopic path of rays in a system of two lenses, when both the object and the second image are at infinite distances. The telescopic path of the rays is realized in spotting scopes - Kepler astronomical tube and Galileo's earth tube (see § 3.5).

Thin lenses have a number of disadvantages that do not allow obtaining high-quality images. Distortions that occur during image formation are called aberrations . The main ones are spherical and chromatic aberrations. Spherical aberration manifests itself in the fact that in the case of wide light beams, rays that are far from the optical axis cross it out of focus. The thin lens formula is valid only for rays close to the optical axis. The image of a distant point source, created by a wide beam of rays refracted by a lens, is blurred.

Chromatic aberration occurs because the refractive index of the lens material depends on the wavelength of light λ. This property of transparent media is called dispersion. The focal length of the lens is different for light with different wavelengths, which leads to blurring of the image when using non-monochromatic light.

In modern optical devices, not thin lenses are used, but complex multi-lens systems in which various aberrations can be approximately eliminated.

The formation of a real image of an object by a converging lens is used in many optical devices, such as a camera, a projector, etc.

Camera is a closed light-tight chamber. The image of photographed objects is created on photographic film by a lens system called lens . A special shutter allows you to open the lens during exposure.

A feature of the operation of the camera is that on a flat photographic film, sufficiently sharp images of objects located at different distances should be obtained.

In the plane of the film, only images of objects that are at a certain distance are sharp. Focusing is achieved by moving the lens relative to the film. Images of points that do not lie in the sharp pointing plane are blurred in the form of circles of scattering. The size d these circles can be reduced by aperture of the lens, i.e. decrease relative borea / F(Fig. 3.3.5). This results in an increase in the depth of field.

Figure 3.3.5. Camera

projection apparatus designed for large scale imaging. Lens O projector focuses the image of a flat object (transparency D) on the remote screen E (Fig. 3.3.6). Lens system K called condenser , designed to concentrate the light source S on a diapositive. Screen E creates a truly enlarged inverted image. The magnification of the projection apparatus can be changed by zooming in or out of the screen E while changing the distance between the transparencies D and lens O.

Similar information.

Application limits:

The laws of geometric optics are satisfied accurately only if the dimensions of the obstacles in the path of light propagation are much larger than the wavelength of the light.

The basic principle:

The basic principle of geometric optics is the concept of a light beam. This definition assumes that the direction of flow radiant energy(path of the light beam) does not depend on the transverse dimensions of the light beam.

Because light is wave phenomenon, interference occurs, as a result of which a limited beam of light does not propagate in any one direction, but has a finite angular distribution, i.e. diffraction occurs. However, in those cases where the characteristic transverse dimensions of light beams are sufficiently large compared to the wavelength, one can neglect the divergence of the light beam and assume that it propagates in one single direction: along the light beam.

Laws of geometric optics:

"The law of rectilinear propagation of light" In a transparent homogeneous medium, light travels in straight lines. In connection with the law of rectilinear propagation of light, the concept of a light beam appeared, which has geometric sense like a line along which light travels.

"The law of independent propagation of rays"- the second law of geometric optics, which states that light rays propagate independently of each other.

"The Law of Light Reflection"- sets a change in the direction of the light beam as a result of a meeting with a reflecting (mirror) surface: the incident and reflected rays lie in the same plane with the normal to the reflecting surface at the point of incidence, and this normal divides the angle between the rays into two equal parts.

"The Law of Light Refraction (Snell's Law, or Snell)"- when light reaches the interface between two transparent media, part of it is reflected, and the rest passes through the boundary. The refraction of light is the change in the direction of propagation of light when it passes through the interface between two media.

"The law of reversibility of a light beam"- according to him, a ray of light propagating along a certain trajectory in one direction will repeat its course exactly when propagating in the opposite direction.

called

5.2. THE LAW OF REFRACTION OF LIGHT. ABSOLUTE AND RELATIVE REFRACTIVE INDICATORS. TOTAL AND INTERNAL REFLECTION End of refraction - when light passes from one transparent medium to another transparent medium at the interface, the light rays deviate from their direction, and the ratio of the sine of incidence to the sine of the angle of refraction is a constant value for these media and

is called at the point of incidence, and this normal divides the angle between the rays into two equal parts Angle of incidence = angle of reflection, specular, perfectly smooth surface)

Geometric optics is a branch of optics that studies the propagation of light in transparent media and develops rules for constructing images during the passage of light rays in optical systems (without taking into account wave properties light). Light is seen as a beam. In the case of radiation with wavelengths small compared to the size of obstacles and details of the optical system and characteristic distances, light can be considered as corpuscular motion - the limiting case of wave motion.

The main simplification of geometric optics is the concept of a light beam. It is assumed that the direction of the light flux does not depend on the transverse dimensions of the light beam.

Basic law of geometric optics : "Light, when propagating from one point to another, chooses such a path, which corresponds to the extreme (minimum or extreme) time for propagation between two points among an infinite number all possible closest paths. "(The basic principle of geometric optics was formed by the French physicist Farm)

Laws of geometric optics:

1) the law of rectilinear propagation of light (In an optically homogeneous medium (vacuum), light rays propagate rectilinearly).

2) the law of independence of light rays.

3) the law of refraction (The incident beam, the refracted beam and the perpendicular to the interface lie in the same plane. When light passes from one transparent medium to another at the interface between the media, the light rays deviate from their direction. Moreover, the ratio sin of the angle of incidence to sin of the angle of refraction is constant for 2 media and called the relative refractive index).

Reversibility of light rays:

Absolute refractive index - refractive index obtained when light from a vacuum falls on a medium.

Relative refractive index - the ratio of the absolute refractive indices of the second and first media.

Conversely, when moving from the second environment to the first:

A medium with a higher index is called optically denser.

4) the law of reflection (the law of reflection (At the boundary of two media, a reflected ray occurs, lying in the plane of incidence, i.e. in the plane containing the incident ray and the normal of the boundary of two media, restored at the point of incidence, and the angle of incidence is equal to the angle of reflection).

Limits of applicability of geometric optics:
The laws of geometrical optics are fulfilled accurately enough only if the size of the obstacle in the path of light propagation is much greater than the wavelength of the light.

Law of refraction of light

Refraction of light is a phenomenon in which a ray of light, passing from one medium to another, changes direction at the boundary of these media.

The refraction of light occurs according to the following law:
The incident and refracted rays and the perpendicular drawn to the interface between two media at the point of incidence of the beam lie in the same plane. The ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant value for two media:
,
where α is the angle of incidence,
β - angle of refraction,
n- constant, independent of the angle of incidence.

When the angle of incidence changes, the angle of refraction also changes. The larger the angle of incidence, the larger the angle of refraction.
If a the light is coming from an optically less dense medium to a denser medium, then the angle of refraction is always less than the angle of incidence: β< α.
A beam of light directed perpendicular to the interface between two media passes from one medium to another without refraction.

absolute refractive index of a substance - a value equal to the ratio of the phase velocities of light (electromagnetic waves) in vacuum and in a given medium n \u003d c / v
The quantity n included in the law of refraction is called relative indicator refraction for a pair of media.

The value n is the relative refractive index of medium B with respect to medium A, and n" = 1/n is the relative refractive index of medium A with respect to medium B.

This value, with other equal conditions more than unity when the beam passes from a denser medium to a less dense one, and less than unity when the beam passes from a less dense medium to a denser medium (for example, from a gas or from a vacuum to a liquid or solid). There are exceptions to this rule, and therefore it is customary to call a medium optically more or less dense than another.

A beam falling from airless space onto the surface of some medium B is refracted more strongly than when falling on it from another medium A; The refractive index of a ray incident on a medium from airless space is called its absolute refractive index.

(Absolute - relative to vacuum.
Relative - relative to any other substance (the same air, for example).
The relative index of two substances is the ratio of their absolute indices.)

Total internal reflection

Light propagating in a medium falls on the interface between this medium and the medium less dense(i.e., the absolute refractive index is less). An increase in the proportion of reflected energy also occurs as the angle of incidence increases, BUT:

Starting from a certain angle of incidence, all light energy is reflected from the interface. The angle of incidence, starting from which all light energy is reflected from the interface, is called the limiting angle of total internal reflection.

When light falls on the interface at the limiting angle, the angle of refraction is 90 degrees:

refraction angle sin = 1/n

At angles of incidence, large angles of refraction, the refracted beam does not exist.

Example: total internal reflection can be observed at the boundary of air bubbles in water. They shine because the sunlight falling on them is completely reflected without passing through the bubbles.

Types of reflections:

The reflection of light can be specular (that is, as observed when using mirrors) or diffuse (in this case, the reflection does not preserve the path of the rays from the object, but only the energy component of the light flux) depending on the nature of the surface.

Mirror reflection

Specular reflection of light is distinguished by a certain relationship between the positions of the incident and reflected rays: 1) the reflected ray lies in a plane passing through the incident ray and the normal to the reflecting surface, restored at the point of incidence; 2) the angle of reflection is equal to the angle of incidence. The intensity of the reflected light (characterized by the reflection coefficient) depends on the angle of incidence and polarization of the incident beam of rays, as well as on the ratio of the refractive indices n 2 and n 1 of the 2nd and 1st media. Quantitatively, this dependence (for a reflecting medium - a dielectric) is expressed by the Fresnel formulas. From them, in particular, it follows that when light is incident along the normal to the surface, the reflection coefficient does not depend on the polarization of the incident beam and is equal to

In an important special case of normal incidence from air or glass to their interface (refractive index of air = 1.0; glass = 1.5), it is 4%.

Total internal reflection

Observed for electromagnetic or sound waves at the interface between two media, when the wave is incident from the medium with slower speed propagation (in the case of light rays, this corresponds to a higher refractive index).

With an increase in the angle of incidence, the angle of refraction also increases, while the intensity of the reflected beam increases, and that of the refracted beam decreases (their sum is equal to the intensity of the incident beam). At a certain critical value, the intensity of the refracted beam becomes zero and total reflection of the light occurs. The value of the critical angle of incidence can be found by setting the angle of refraction equal to 90° in the law of refraction:

Diffuse reflection of light

Scattering of light in all directions. There are two main D. forms of light scattering: light scattering on surface microroughness (surface scattering) and scattering in the volume of a body associated with the presence of finely dispersed particles (volume scattering). The properties of diffusely reflected light depend on the lighting conditions, optically. properties of the scattering substance and the microrelief of the reflecting surface (see Reflection of light). An ideally diffused surface has the same brightness in all directions, regardless of the lighting conditions. To estimate the light-scattering characteristics of real objects, the coefficient is introduced. D. O., which is defined as the ratio of the light flux reflected from a given surface to the flux reflected by an ideal diffuser. Spectral composition, coefficient Before. and the brightness indicatrix D. o. the light of real objects depends on both forms of scattering - surface and volume.

Light

1) If an object encounters a transparent body, then it passes through him, but less reflected and absorbed.

2) If the object is opaque - reflection and absorption of light.

1. Reflection coefficient- dimensionless physical quantity characterizing the ability of a body to reflect radiation incident on it. Greek or Latin is used as a letter designation.

Quantitatively, the reflection coefficient is equal to the ratio of the radiation flux reflected by the body to the flux incident on the body:

2.Transmittance - dimensionless physical quantity equal to the ratio of the radiation flux passed through the medium to the radiation flux that fell on its surface:

3. Absorption coefficient- a dimensionless physical quantity characterizing the ability of a body to absorb radiation incident on it. The Greek [

Numerically, the absorption coefficient is equal to the ratio of the radiation flux, absorbed by the body, to the radiation flux, incident on the body:

4.Scattering factor- a dimensionless physical quantity characterizing the ability of a body to scatter radiation incident on it. Greek is used as a letter designation.

Quantitatively, the scattering coefficient is equal to the ratio of the radiation flux scattered by the body to the flux incident on the body:

Conclusion: The sum of the absorption coefficient and the coefficients of reflection, transmission and scattering is equal to one. This statement follows from the law of conservation of energy.

Optical density is a measure of the attenuation of light by transparent objects (such as crystals, glasses, photographic film) or the reflection of light by opaque objects (such as photographs, metals, etc.).

Calculated as decimal logarithm the ratio of the radiation flux incident on an object to the radiation flux that has passed through it (reflected from it), that is, it is the logarithm of the reciprocal of the transmittance (reflection):

(D = - lg T = lg (1/ T)

TICKET #6

white light and Colorful temperature

6.1. WHITE LIGHT. REFRACTIVE INDEX DEPENDENCE ON THE RADIATION SPEED (LIGHT DISPERSION) The dependence of the refractive index in a transparent medium on the wavelength of the transmitted light is the dispersion of light. The measure of dispersion is the difference between the refractive indices of wavelengths. Light passes through a Newtonian prism ....... red - the speed of propagation in the medium is maximum, and the degree of refraction is minimum, the light purple the speed of propagation in the medium is minimal, and the degree of refraction is maximum.

Light dispersion- The dependence of the refractive index on the oscillation frequency (or light wavelength) is called the dispersion of light. In most cases, as the wavelength increases, the refractive index decreases. Such a dispersion is called normal.

White light - electromagnetic radiation in the visible range, which causes in the normal human eye a light sensation that is neutral with respect to color. (Or when all the colors of the spectrum come together). The dispersion of light is the dependence of the refractive index in a transparent medium on the wavelength. Ray white light refracted as it passes through the crystal. Refraction occurs due to the different densities of the 2 media, due to which the light changes.

Light dispersion (light decomposition) is a phenomenon due to the dependence absolute indicator refraction of a substance on the frequency (or wavelength) of light (frequency dispersion), or, the same thing, the dependence of the phase velocity of light in a substance on the wavelength (or frequency). Experimentally discovered by Newton around 1672, although theoretically well explained much later. due to the dependence of the refraction of light on the speed of its propagation, a beam of white light (since it is complex), passing through a crystal, is refracted, since it passes from one medium to another with different densities and the speed of light changes. Decomposition of white light into a spectrum. A beam of white light, passing through a trihedral prism, is not only deflected, but also decomposed into component colored rays. This phenomenon was established by Isaac Newton. Newton directed a beam of sunlight through a small hole on glass prism. Getting on the prism, the beam was refracted and gave a spectrum on the opposite wall.

6.2. COLOR TRIANGLE. BASIC AND ADDITIONAL COLORS. THREE-COMPONENT VISION. (Clockwise arrangement of colors from 12 o'clock: k, g, h, g, s, p) Primary colors: Blue, green, red - form White color Additional colors: yellow, magenta, cyan. K+G=B;z+p=B;s+g=B. K+Z=W, Z+S=G, S+K=p The three-eyed eye has three types of radiant energy receivers (cones) that perceive the red (long-wavelength), yellow (medium-wavelength) and blue (short-wavelength) parts of the visible spectrum. Red perceives better than purple 6.3. ABSOLUTELY BLACK BODY. ITS STANDARD AND RADIATION SPECTRUM. COLORFUL TEMPERATURE. UNIT OF COLOR TEMPERATURE. A. The model of an ideal radiation source, does not absorb or transmit anything at a given t. Emits a large amount of any monochromatic radiation than any other source. B. The radiation spectrum of an absolutely black body is determined only by its temperature. In this case, the body completely absorbs all radiation incident on it. If the absorption coefficient is equal to unity (max) for all wavelengths, then such a body is called a completely black body. An absolutely black body radiates more energy in any region of the spectrum than any other body with the same temperature. For pretty large area spectrum - from infrared to ultraviolet radiation The properties of an absolutely black body are possessed by a surface covered with a layer of soot (hot tungsten metal). real body. It is measured in kelva and mired.

6.4 THE IMPORTANCE OF COLOR TEMPERATURE IN PHOTOGRAPHY. GRAY BODY RADIATION. REAL RADIATION SOURCES THAT HAVE A SPECTRAL ENERGY DISTRIBUTION EQUAL TO THE BLACK BODY RADIATION. RADIATION SOURCES TO WHICH THE CONCEPT OF COLOR T IS NOT APPLICABLE. To select bb. The gray body, the radiation is identical to the gray body, close to the black body. A body whose absorption coefficient is less than 1 and does not depend on the radiation wavelength and abs. t. Gray radiation - thermal radiation, the same spectrum. composition with the radiation of a completely black body, but differing from it in a lower energy. brightness.

(Gray bodies: candle flame, incandescent lamps, hot metal). The concept is not applicable: laser, LED, vapor, fluorescent, gas discharge tube. Photodetectors

7.1 PHOTOELECTRIC EFFECT. LAWS OF THE PHOTOEFECT. EFFECT EXTERNAL AND INTERNAL. PHOTOELECTRIC EFFECT - knocking out electrons from the surface of conductive materials by light.

Order of photoelectric effect 1.dependence of photoemission. The strength of the photo radiation current is directly proportional to the incident radiation flux (illuminance) 2. The speed of the radiation current. Directly proportional to the incident radiation flux (illuminance) The speed of the electrons released under the action, the speed of the emitted electrons does not depend on the illumination, but is determined by the frequency of the radiation. (Blue prints are registered faster) The higher the frequency, the shorter the wavelength, the sooner the electron will fly 3. The red border corresponds to the maximum wavelength that can cause the photoelectric effect. E=h*v - total energy. Receipt from an electron with a frequency v, equals the product of this frequency by the post. Planck-6.6 * 10 in the 36th \u003d h

external photoelectric effect(photoelectronic emission) is called the emission of electrons by a substance under the action of electromagnetic radiation. Internal photoelectric effect called the redistribution of electrons energy states in solid and liquid semiconductors and dielectrics, which occurs under the action of radiation. Semiconductors in a matrix of silicon, carbon, selenium (not metal) SiO2 (sand, polycrystalline silicon) Current does not flow, the potential barrier is not overcome, if the conductor is heated, then the conductivity will be / additional occurrence of charges. P type - more holes N type - more electrons But if we have not + -, but - +, then if we heat the current will overcome the barrier. + protons - electrons Silver halide (yellow)

The street starts to get dark, turn brown, smell like chlorine

Geometric optics uses the concept of light rays propagating independently of each other, rectilinear in a homogeneous medium, reflected and refracted at the boundaries of media with different optical properties. Along the rays, the energy of light vibrations is transferred.

The refractive index of the medium. The optical properties of a transparent medium are characterized by the refractive index, which determines the speed (more precisely, the phase speed) of light waves:

where c is the speed of light in vacuum. The refractive index of air is close to unity (for water, its value is 1.33, and for glass, depending on the grade, it can range from 1.5 to 1.95. The refractive index of diamond is especially high - approximately 2.5.

The value of the refractive index, generally speaking, depends on the wavelength R (or on the frequency: This dependence is called the dispersion of light. For example, in crystal (lead glass), the refractive index smoothly changes from 1.87 for red light with a wavelength to 1.95 for blue light from

The refractive index is related to permittivity medium (for a given wavelength or frequency) by the relation Medium with great value refractive index is called optically denser.

Laws of geometric optics. The behavior of light rays obeys the basic laws of geometric optics.

1. In a homogeneous medium, light rays are rectilinear (the law of rectilinear propagation of light).

2. At the boundary of two media (or at the boundary of a medium with a vacuum), a reflected beam arises, lying in the plane formed by the incident beam and the normal to the boundary, i.e., in the plane of incidence, and the angle of reflection is equal to the angle of incidence (Fig. 224):

(law of reflection, light).

3. The refracted beam lies in the plane of incidence (when light is incident on the boundary of an isotropic medium) and forms an angle with the normal to the boundary (angle of refraction) determined by the relation

(the law of refraction of light or Snell's law).

When light passes into an optically denser medium, the beam approaches the normal. The ratio is called the relative refractive index of two media (or the refractive index of the second medium relative to the first).

Rice. 224. Reflection and refraction sung on a flat boundary of two media

When light falls from vacuum onto the boundary of a medium with a refractive index, the law of refraction takes the form

For air, the refractive index is close to unity, therefore, when light falls from air onto a certain medium, formula (4) can be used.

When light passes into an optically less dense medium, the angle of incidence cannot exceed the limit value, since the angle of refraction cannot exceed (Fig. 225):

If the angle of incidence is complete reflection, i.e., all the energy of the incident light returns to the first, optically denser medium. For glass-air border

Rice. 225. Limiting angle of total reflection

Huygens principle and laws of geometrical optics. The laws of geometric optics were established long before the nature of light was elucidated. These laws can be derived from wave theory based on the Huygens principle. Their applicability is limited by diffraction phenomena.

Let us dwell in more detail on the transition from the wave representations of the propagation of light to the representations of geometric optics. Using the Huygens principle, given the wave surface of the incident wave, one can construct the wave surfaces of the refracted and reflected waves. In this case, it should be taken into account that the light rays are perpendicular to the wave surfaces.

Consider a plane light wave incident from medium 1 (with a refractive index onto a flat interface with medium 2 (with a refractive index at an angle (Fig. 226). The angle of incidence is the angle between the incident beam and the normal to the interface.

Rice. 226. Huygens construction for reflection and refraction of light

At the same time, is the angle between the interface and the wave surface of the incident wave. Let at some moment this wave surface occupies a position After a while, it will reach point B of the interface. During the same time, the secondary wave from point A, propagating in the medium X, will expand to a radius Substituting here we get From here it is clear that the wave surface of the reflected wave, which is the envelope of all secondary spherical waves with centers on the segment, is inclined to the interface at an angle which is equal to ( equality of angles and follows from the equality right triangles and having a common hypotenuse and equal legs and Thus, the reflected beam, perpendicular to the front of the reflected wave, forms an angle with the normal equal to the angle fall

Similarly, from this construction of Huygens one can obtain the law of refraction. In medium 2, secondary waves propagate with speed, and therefore the spherical wave emerging from point A after a time has a radius Substituting here we find Dividing both parts of this equality by we arrive at the relation

which, obviously, coincides with the law of refraction (3), since the angle of inclination of the wave surface of the wave in medium 2 is at the same time the angle between the refracted beam and the normal to the interface (the angle of refraction, Fig. 226).

Reflection and refraction on a curved surface. plane wave is characterized by the property that its wave surfaces are unlimited planes, and the direction of its propagation and amplitude are the same everywhere. Often electromagnetic waves that are not plane can be roughly considered as being plane over a small region of space. For this, it is necessary that the amplitude and direction of propagation of the wave hardly change over distances of the order of the wavelength. Then it is also possible to introduce the concept of rays, i.e., lines, the tangent to which at each point coincides with the direction of wave propagation. If, in this case, the interface between two media, for example, the surface of a lens, can be considered approximately flat at distances of the order of a wavelength, then the behavior of light rays at such an interface will be described by the same laws of reflection and refraction.

The study of the laws of propagation of light waves in this case is the subject of geometric optics, since in this approximation the optical laws can be formulated in the language of geometry. Many optical phenomena, such as, for example, the passage of light through optical systems that form an image, can be considered in terms of light rays, completely abstracting from the wave nature of light. Therefore, the representations of geometric optics are valid only to the extent that the phenomena of diffraction of light waves can be neglected. Diffraction is the weaker, the shorter the wavelength. This means that geometric optics corresponds to the limiting case of short wavelengths:

A physical model of a beam of light rays can be obtained by passing light from a source of negligible size through a small hole in an opaque screen. The light emerging from the hole fills a certain area, and if the wavelength is negligible compared to the size of the hole, then at a small distance from it we can speak of a beam of light rays with a sharp boundary.

Intensity of reflected and refracted light. The laws of reflection and refraction allow us to determine only the direction of the corresponding light rays, but say nothing about their intensity. Meanwhile, experience shows that the ratio of the intensities of the reflected and refracted beams, into which the original beam is split at the interface, strongly depends on the angle of incidence. For example, when light is normally incident on the glass surface, about 4% of the energy of the incident light beam is reflected, and when it falls on the water surface, only 2%. But during grazing incidence, the surfaces of glass and water reflect almost all of the incident radiation. Thanks to this, we can admire the mirror reflections of the shores in the calm clear water of mountain lakes.

Rice. 227. In a natural spell, sector E fluctuations occur in all possible directions in a plane perpendicular to the beam

natural light. light wave, like any electromagnetic wave, is transverse: the vector E lies in a plane perpendicular to the direction of propagation. The light emitted by ordinary sources (for example, incandescent bodies) is unpolarized light. This means that in a light beam, the oscillations of the vector E occur in all possible directions in a plane perpendicular to the direction of the beam (Fig. 227). Such unpolarized light is called natural light. It can be represented as an incoherent mixture of two light waves of the same intensity, linearly polarized in two mutually perpendicular directions. These directions can be chosen arbitrarily.

Polarization of light upon reflection. When studying the reflection of unpolarized light from the interface between media, it is convenient to choose one of two independent directions of the vector E in the plane of incidence, and the second direction is perpendicular to it. The conditions for reflection of these two waves turn out to be different: a wave whose vector E is perpendicular to the plane of incidence (i.e., parallel to the interface) at all angles of incidence (except 0 and 90°) is reflected more strongly. Therefore, the reflected light turns out to be partially polarized, and when reflected at a certain specific angle (for glass, about 56 °), it is completely polarized.

This circumstance is used to eliminate glare, for example, when photographing a landscape with water surface. By properly choosing the orientation of a polarizing filter that allows light vibrations to pass only a certain polarization, you can almost completely eliminate glare in a photograph.

Fermat's principle. The basic laws of geometric optics - the law of rectilinear propagation of light in a homogeneous medium, the laws of reflection and refraction of light at the interface between two media - can be obtained using Fermat's principle. According to this principle, the actual path of propagation of a monochromatic light beam is the path for which the light takes an extreme (usually minimal) time compared to any other conceivable path between the same points that is close to it.

Rice. 228. To the derivation of the law of light reflection from Fermat's principle

Let's take the law of reflection of light as an example. It is immediately clear that it follows directly from Fermat's principle. Let a ray of light coming out of point A be reflected from a mirror at some point C and come to a given point B (Fig. 228). According to Fermat's principle, traversed by light the path must be shorter than any other path along a close trajectory, for example, to find the position of the reflection point C, set aside an equal segment on the perpendicular to the mirror lowered from point A and connect points A and B with a straight line segment.

The intersection of this segment with the surface of the mirror gives the position of point C. Indeed, it is easy to see that, therefore, the path of light from point A to point B is equal to the segment. The path of light from A to B through any other point equal will be longer, since the straight line is shortest distance between two points A and B. From fig. 228 it is immediately clear that it is precisely this position of the point C that corresponds to the equality of the angles of incidence and reflection:

Rice. 229. Imaginary image of point A in flat mirror

Image in a flat mirror. Point A, located symmetrically to point A relative to the surface of a flat mirror, is the image of point A in this mirror. Indeed, a narrow beam of rays emerging from

A, reflected in the mirror and falling into the eye of the observer (Fig. 229), will seem to come out of point A. The image created by a flat mirror is called imaginary, since at point A it is not the reflected rays themselves that intersect, but their extensions back. Obviously, the image of an extended object in a flat mirror will be equal in size to the object itself.

What are light rays? How does this concept relate to the concept of a wave surface? What does the rays have to do with the direction of propagation of light vibrations?

Under what conditions can the concept of light rays be used?

What is the refractive index of a medium? How is it related to the speed of light?

Formulate the basic laws of geometric optics. What is a plane of incidence? Explain, on the basis of symmetry considerations, why the beam, both during reflection and refraction, does not leave this plane.

Under what conditions will the reflection of light at the interface be complete? What is the limiting angle of total reflection?

Explain how the laws of rectilinear propagation, reflection and refraction can be obtained based on Huygens' principle.

Why can the laws of reflection and refraction of light formulated for a flat interface be applied to curved surfaces (lenses, water drops, etc.)?

Give examples of the phenomena you have observed that indicate the dependence of the intensity of reflected light on the angle of incidence.

Why on reflection natural light Is it partially polarized light?

Formulate Fermat's principle and show that the law of reflection of light follows from it.

Prove that the image of an object in a plane mirror is equal in size to the object itself.

Fermat's principle and lens formula. The speed of light in a medium with a refractive index is Therefore, Fermat's principle can be formulated as a requirement for the minimum optical length of the beam when light propagates between two given points. The optical length of a beam is understood as the product of the refractive index and the length of the beam path. In an inhomogeneous medium, the optical length is the sum of the optical lengths by separate sections. Using this principle allows us to consider some problems from a slightly different point of view than with the direct application of the laws of reflection and refraction. For example, when considering a focusing optical system, instead of applying the law of refraction, one can simply require that the optical lengths of all rays be equal.

Using Fermat's principle, we obtain the formula for a thin lens without resorting to the law of refraction. For definiteness, we will consider a biconvex lens with spherical refractive surfaces, the radii of curvature of which are equal (Fig. 230).

It is well known that a converging lens can be used to obtain a real image of a point. Let the subject, its image. All rays emanating from and passing through the lens are collected at one point Let lies on the main optical axis of the lens, then the image also lies on the axis. What does it mean to get a lens formula? This means to establish a relationship between the distances from the object to the lens and from the lens to the image and the quantities that characterize this lens: the radii of curvature of its surfaces and the refractive index

It follows from Fermat's principle that the optical lengths of all rays leaving the source and converging at a point that is its image are the same. Let's consider two of these beams: one going along the optical axis, the second - through the edge of the lens (Fig. 230a).

Rice. 230. To the output of the thin lens formula

Despite the fact that the second beam travels a greater distance, its path through the glass is shorter than that of the first one, so the propagation time of the light is the same for them. Let's express this mathematically. The designations of the values ​​of all segments are indicated in the figure. Let us equate the optical lengths of the first and second beams:

We express by the Pythagorean theorem:

Now we use an approximate formula that is valid for up to terms of the order. Assuming small in comparison with up to terms of the order, we have

Similarly for we get

We substitute expressions (8) and (9) into the main relation (7) and give similar terms:

In this formula, in the case of a thin lens, one can neglect the values ​​in the denominators of the right side compared to and it is obvious that the left side of the expression should be kept, because this term is a multiplier.

With the same accuracy as in formulas (8) and (9), using the Pythagorean theorem, it can be represented as (Fig. 230b)

Now it remains only to substitute these expressions into the left side of formula (10) and reduce both sides of the equality by :

This is the desired formula for a thin lens. Introducing the notation

it can be rewritten in the form

Focal length of the lens. From formula (12) it is easy to understand what is the focal length of the lens: if the source is at infinity (i.e., a parallel beam of rays falls on the lens), its image is in focus. Assuming we get

aberrations. The obtained property of focusing a parallel beam of monochromatic rays is, as can be seen from the derivation made, approximate and is valid only for a narrow beam, i.e., for rays that are not too far from the optical axis. For wide beams of rays, spherical aberration takes place, which manifests itself in the fact that rays far from the optical axis cross it out of focus (Fig. 231). As a result, the image of an infinitely distant point source, created by a wide beam of rays refracted by the lens, turns out to be somewhat blurred.

In addition to spherical aberration, the lens as an optical device that forms an image has a number of other disadvantages.

For example, even a narrow parallel beam of monochromatic rays, forming a certain angle with the optical axis of the lens, after refraction is not collected at one point. When using non-monochromatic light, the lens also exhibits chromatic aberration, due to the fact that the refractive index depends on the wavelength. As a result, as can be seen from formula (11), a narrow parallel beam of white light rays intersects after refraction in the lens at more than one point: the rays of each color have their own focus.

In the design of optical instruments, these shortcomings can be eliminated to a greater or lesser extent by using specially designed complex multi-lens systems. However, it is impossible to eliminate all the shortcomings at the same time. Therefore, one has to compromise and, by designing optical devices designed for a specific purpose, seek to eliminate some shortcomings and put up with the presence of others. For example, lenses designed to observe objects of low brightness must transmit as much light as possible, which forces one to put up with some aberrations that are inevitable when using wide beams of light.

Rice. 231. Spherical lens aberration

For telescope lenses, where the studied objects are stars - point sources located near the optical axis of the device, it is especially important to eliminate spherical and chromatic aberration for wide beams parallel to the optical axis. The easiest way to eliminate chromatic aberration is to use reflection instead of refraction in the optical system. Since rays of all wavelengths are reflected equally, a reflecting telescope, unlike a refractor, is completely devoid of chromatic aberration. If, at the same time, the shape of the surface of the reflecting mirror is properly chosen, then spherical aberration for beams parallel to the optical axis can also be completely eliminated. To obtain a point axial image, the mirror must be parabolic.

Squaring both sides and quoting like terms, we find

This is the equation of a parabola.

Rice. 232. All parallel rays after reflection from a parabolic mirror are collected at a point

Parabolic mirrors are used in all largest telescopes. Spherical and chromatic aberrations have been eliminated in these telescopes; however, parallel beams propagating even at small angles to the optical axis do not intersect at one point after reflection and produce highly distorted off-axis images. Therefore, the field of view suitable for work turns out to be very small, on the order of several tens of arc minutes,

Explain why, as applied to a focusing optical system, Fermat's principle is formulated as the condition for the equality of the optical lengths of all rays from the point of the object to its image.

Use Fermat's principle to derive the law of refraction of light at the interface between two media.

Formulate approximations under which the thin lens formula is valid.

What are spherical and chromatic aberrations of a lens?

What are the advantages and disadvantages of a parabolic mirror compared to a spherical one?

Show that an elliptical mirror reflects all the rays that emerge from one focus of the ellipsoid to another focus.