Law of light refraction

Everyone has probably encountered the phenomenon of light refraction more than once in everyday life. For example, if you lower a tube into a transparent glass of water, you will notice that the part of the tube that is in the water seems to be shifted to the side. This is explained by the fact that at the boundary of the two media there is a change in the direction of the rays, in other words, the refraction of light.

In the same way, if you lower a ruler into water at an angle, it will seem that it is refracted and its underwater part rises higher.

After all, it turns out that rays of light, once at the border of air and water, experience refraction. A ray of light hits the surface of the water at one angle, and then it goes deep into the water at a different angle, at a smaller inclination to the vertical.

If you shoot a return beam from water into the air, it will follow the same path. The angle between the perpendicular to the interface at the point of incidence and the incident beam is called the angle of incidence.

The angle of refraction is the angle between the same perpendicular and the refracted ray. The refraction of light at the boundary of two media is explained by the different speed of light propagation in these media. When light is refracted, two laws will always be fulfilled:

Firstly, the rays, regardless of whether they are incident or refracted, as well as the perpendicular, which is the interface between two media at the break point of the ray, always lie in the same plane;

Secondly, the ratio of the sinus angle of incidence to the sinus angle of refraction is a constant value for these two media.

These two statements express the law of refraction of light.

Sinus of the angle of incidence α is related to the sinus of the angle of refraction β, just as the speed of the wave in the first medium - v1 is to the speed of the wave in the second medium - v2, and is equal to the value n. N is a constant value that does not depend on the angle of incidence. The value n is called the refractive index of the second medium relative to the first medium. And if the first medium was a vacuum, then the refractive index of the second medium is called the absolute refractive index. Accordingly, it is equal to the ratio of the sinus angle of incidence to the sinus angle of refraction when a light beam passes from a vacuum into a given medium.

The refractive index depends on the characteristics of light, on the temperature of the substance and on its density, that is, on the physical characteristics of the medium.

More often we have to consider the transition of light through the air-solid or air-liquid boundary than through the vacuum-definite medium boundary.

It should also be noted that the relative refractive index of two substances is equal to the ratio of the absolute refractive indices.

Let's get acquainted with this law with the help of simple physical experiments that are available to all of you in everyday life.

Experience 1.

Let's put the coin in the cup so that it disappears behind the edge of the cup, and now we'll pour water into the cup. And here’s what’s surprising: the coin appeared from behind the edge of the cup, as if it had floated up, or the bottom of the cup had risen up.

Let's draw a coin in a cup of water and the rays of the sun coming from it. At the interface between air and water, these rays are refracted and exit the water at a large angle. And we see the coin in the place where the lines of refracted rays converge. Therefore, the visible image of the coin is higher than the coin itself.

Experience 2.

Let's place a container filled with water with parallel walls in the path of parallel rays of light. At the entrance from the air into the water, all four rays turned through a certain angle, and at the exit from the water into the air, they turned through the same angle, but in the opposite direction.

Let us increase the inclination of the rays, and at the output they will still remain parallel, but will move more to the side. Because of this shift, the book's lines, when viewed through a transparent plate, appear to be cut. They moved up, just as the coin moved up in the first experiment.

As a rule, we see all transparent objects solely due to the fact that light is refracted and reflected on their surface. If such an effect did not exist, then all these objects would be completely invisible.

Experience 3.

Let's lower the plexiglass plate into a vessel with transparent walls. She is clearly visible. Now let’s pour sunflower oil into the vessel, and the plate has become almost invisible. The fact is that light rays at the interface of oil and plexiglass are almost not refracted, so the plate becomes an invisible plate.

Path of rays in a triangular prism

In various optical instruments, a triangular prism is often used, which can be made of a material such as glass or other transparent materials.

When passing through a triangular prism, rays are refracted on both surfaces. The angle φ between the refractive surfaces of the prism is called the refractive angle of the prism. The deflection angle Θ depends on the refractive index n of the prism and the angle of incidence α.

Θ = α + β1 - φ, f= φ + α1

You all know the famous rhyme for remembering the colors of the rainbow. But why these colors are always arranged in such an order, how they are obtained from white sunlight, and why there are no other colors in the rainbow except these seven, is not known to everyone. It is easier to explain this through experiments and observations.

We can see beautiful rainbow colors on soap films, especially if these films are very thin. The soapy liquid flows down and colored stripes move in the same direction.

Let's take a transparent lid from a plastic box, and now tilt it so that the white computer screen is reflected from the lid. Unexpectedly bright rainbow stains will appear on the lid. And what beautiful rainbow colors are visible when light is reflected from a CD, especially if you shine a flashlight on the disk and throw this rainbow picture on the wall.

The great English physicist Isaac Newton was the first to try to explain the appearance of rainbow colors. He let a narrow beam of sunlight into the dark room, and placed a triangular prism in its path. The light emerging from the prism forms a band of color called a spectrum. The color that deviates the least in the spectrum is red, and the color that deviates the most is violet. All other colors of the rainbow are located between these two without particularly sharp boundaries.

Laboratory experience

We will choose a bright LED flashlight as a white light source. To form a narrow light beam, place one slit immediately behind the flashlight, and the second directly in front of the prism. A bright rainbow stripe is visible on the screen, where red, green and blue are clearly visible. They form the basis of the visible spectrum.

Let's place a cylindrical lens in the path of the colored beam and adjust it to sharpness - the beam on the screen gathers into a narrow strip, all the colors of the spectrum are mixed, and the strip becomes white again.

Why does a prism turn white light into a rainbow? It turns out that the fact is that all the colors of the rainbow are already contained in white light. The refractive index of glass differs for rays of different colors. Therefore, the prism deflects these rays differently.

Each individual color of the rainbow is pure and cannot be split into other colors. Newton proved this experimentally by isolating a narrow beam from the entire spectrum and placing a second prism in its path, in which no splitting occurred.

Now we know how a prism splits white light into individual colors. And in a rainbow, water droplets act like small prisms.

But if you shine a flashlight on a CD, a slightly different principle works, unrelated to the refraction of light through a prism. These principles will be studied further in physics lessons devoted to light and the wave nature of light.

organs without surgical intervention (endoscopes), as well as in production to illuminate inaccessible areas.

5. The principles of operation of various optical devices that serve to set light rays in the desired direction are based on the laws of refraction. For example, consider the path of rays in a plane-parallel plate and in a prism.

1). Plane-parallel plate- a plate made of a transparent substance with two parallel flat edges. Let the plate be made of a substance that is optically denser than the surrounding medium. Let's assume that in the air ( n1 =1) there is a glass

plate (n 2 >1), the thickness of which is d (Fig. 6).

Let the beam fall on the upper face of this plate. At point A it will refract and travel in the glass in the direction AB. At point B the beam will refract again and exit the glass into the air. Let us prove that the beam leaves the plate at the same angle at which it falls on it. For point A, the law of refraction has the form: sinα/sinγ=n 2 /n 1, and since n 1 = 1, then n 2 = sinα/sinγ. For

point B, the law of refraction is as follows: sinγ/sinα1 =n 1 /n 2 =1/n 2. Comparison

formulas gives the equality sinα=sinα1, and therefore α=α1. Consequently, the beam

will come out of a plane-parallel plate at the same angle at which it fell on it. However, the beam emerging from the plate is displaced relative to the incident beam by a distance ℓ, which depends on the thickness of the plate,

refractive index and angle of incidence of the beam on the plate.

Conclusion: a plane-parallel plate does not change the direction of the rays incident on it, but will only mix them up if we consider the refracted rays.

2). Triangular prism is a prism made of a transparent substance, the cross-section of which is a triangle. Let the prism be made of a material optically denser than the surrounding medium

(for example, it is made of glass, and there is air around it). Then the ray that fell on its edge

having refracted, it is deflected towards the base of the prism, since it passes into an optically denser medium and, therefore, its angle of incidence φ1 is greater than the angle

refraction φ2. The path of rays in a prism is shown in Fig. 7.

The angle ρ at the vertex of the prism, lying between the faces at which the ray is refracted, is called refractive angle of the prism; and the side

lying opposite this angle is the base of the prism. Angle δ between the directions of continuation of the ray incident on the prism (AB) and the ray (CD)

who came out of it is called beam deflection angle by prism- it shows how much the prism changes the direction of the rays incident on it. If the angle p and the refractive index of the prism n are known, then from the given angle of incidence φ1 one can find the angle of refraction on the second face

φ4. In fact, the angle φ2 is determined from the law of refraction sinφ1 / sinφ2 =n

(a prism made of a material with refractive index n is placed in air). IN

BCN sides ВN and CN are formed by straight lines perpendicular to the faces of the prism, so that angle CNE is equal to angle p. Therefore φ2 +φ3 =р, whence φ3 =р -φ2

becomes famous. The angle φ4 is determined by the law of refraction:

sinφ3 /sinφ4 =1/n.

In practice, it is often necessary to solve the following problem: knowing the geometry of the prism (angle p) and determining the angles φ1 and φ4, find the indicator

prism refraction n. Applying the laws of geometry, we obtain: angle MSV=φ4 -φ3, angle MSV=φ1 -φ2; angle δ is external to the BMC and, therefore,

is equal to the sum of the angles MVS and MSV: δ=(φ1 -φ2 )+(φ4 -φ3 )=φ1 +φ4 -р , where it is taken into account

equality φ3 +φ2 =р. That's why,

δ = φ1 + φ4 -р.

Therefore, the angle The greater the angle of incidence of the beam and the smaller the refractive angle of the prism, the greater the deviation of the beam by the prism. Using relatively complex reasoning, it can be shown that with a symmetrical beam path

through a prism (the light ray in the prism is parallel to its base) δ takes on the smallest value.

Let us assume that the refractive angle (thin prism) and the angle of incidence of the beam on the prism are small. Let us write down the laws of refraction on the faces of a prism:

sinφ1 /sinφ2 =n, sinφ3 /sinφ4 =1/n. Considering that for small angles sinφ≈ tanφ≈ φ,

we get: φ1 =n φ2, φ4 =n φ3. Substituting φ1 and φ3 into formula (8) for δ we obtain:

δ =(n – 1)р.

We emphasize that this formula for δ is correct only for a thin prism and at very small angles of incidence of the rays.

Principles of optical imaging

The geometric principles of obtaining optical images are based only on the laws of reflection and refraction of light, completely abstracting from its physical nature. In this case, the optical length of the light beam should be considered positive when it passes in the direction of light propagation, and negative in the opposite case.

If a beam of light rays emanating from any point S, at

as a result of reflection and/or refraction converges at point S ΄, then S ΄

is considered an optical image or simply an image of the S point.

An image is called real if the light rays actually intersect at point S ΄. If at point S ΄ the continuations of rays intersect, drawn in the direction opposite to propagation

light, then the image is called virtual. With the help of optical devices, virtual images can be converted into real ones. For example, in our eye, a virtual image is converted into a real one, resulting on the retina. For example, consider obtaining optical images using 1)

flat mirror; 2) a spherical mirror and 3) lenses.

1. A flat mirror is a smooth flat surface that specularly reflects rays . The construction of an image in a plane mirror can be shown using the following example. Let's construct how a point light source is visible in a mirror S(Fig.8).

The rule for constructing the image is as follows. Since different rays can be drawn from a point source, we choose two of them - 1 and 2 and find the point S ΄ where these rays converge. It is obvious that the reflected 1΄ and 2΄ rays themselves diverge, only their continuations converge (see the dotted line in Fig. 8).

The image was obtained not from the rays themselves, but from their continuation, and is imaginary. It is easy to show by simple geometric construction that

the image is located symmetrically with respect to the surface of the mirror.

Conclusion: a plane mirror gives a virtual image of an object,

located behind the mirror at the same distance from it as the object itself. If two plane mirrors are located at an angle φ to each other,

then it is possible to obtain several images of the light source.

2. A spherical mirror is a part of a spherical surface,

specularly reflecting light. If the inner part of the surface is mirrored, then the mirror is called concave, and if the outer part is called convex.

Figure 9 shows the path of rays incident in a parallel beam on a concave spherical mirror.

The top of the spherical segment (point D) is called pole of the mirror. The center of the sphere (point O) from which the mirror is formed is called

optical center of the mirror. The straight line passing through the center of curvature O of the mirror and its pole D is called the main optical axis of the mirror.

Applying the law of light reflection, at each point of incidence of rays on mirrors

restore the perpendicular to the surface of the mirror (this perpendicular is the radius of the mirror - dotted line in Fig. 9) and

receive the course of reflected rays. Rays incident on the surface of a concave mirror parallel to the main optical axis, after reflection, are collected at one point F, called mirror focus, and the distance from the focus of the mirror to its pole is the focal length f. Since the radius of the sphere is directed normal to its surface, then, according to the law of light reflection,

the focal length of a spherical mirror is determined by the formula

where R is the radius of the sphere (ОD).

To construct an image, you need to select two rays and find their intersection. In the case of a concave mirror, such rays can be a ray

reflected from point D (it goes symmetrically with the incident one relative to the optical axis), and the ray passing through the focus and reflected by the mirror (it goes parallel to the optical axis); another pair: a ray parallel to the main optical axis (when reflected, it will pass through the focus), and a ray passing through the optical center of the mirror (it will be reflected in the opposite direction).

For example, let’s construct an image of an object (arrows AB) if it is located from the top of the mirror D at a distance greater than the radius of the mirror

(mirror radius is equal to distance OD=R). Let's consider a drawing made according to the described rule for constructing an image (Fig. 10).

Ray 1 propagates from point B to point D and is reflected in a straight line

DE so that angle ADB is equal to angle ADE. Ray 2 from the same point B propagates through the focus to the mirror and is reflected along the line CB "||DA.

The image is real (formed by reflected rays, and not their continuations, as in a plane mirror), inverted and reduced.

From simple geometric calculations, the relationship between the following characteristics can be obtained. If a is the distance from the object to the mirror, plotted along the main optical axis (in Fig. 10 this is AD), b –

distance from the mirror to the image (in Fig. 10 it is DA "), toa/b =AB/A"B",

and then the focal length f of the spherical mirror is determined by the formula

The magnitude of optical power is measured in diopters (dopters); 1 diopter = 1m-1.

3. A lens is a transparent body bounded by spherical surfaces, the radius of at least one of which must not be infinite . The path of rays in a lens depends on the radius of curvature of the lens.

The main characteristics of a lens are the optical center, foci,

focal planes. Let the lens be limited by two spherical surfaces, the centers of curvature of which are C 1 and C 2, and the vertices of the spherical

surfaces O 1 and O 2.

Figure 11 schematically shows a biconvex lens; The thickness of the lens in the middle is greater than at the edges. Fig. 12 schematically shows a biconcave lens (in the middle it is thinner than at the edges).

For a thin lens, it is considered that O 1 O 2<<С 1 О 2 иО 1 О 2 <<С 2 О 2 , т.е.

practically points O 1 and O 2. merged into one point O, which is called

optical center of the lens. The straight line passing through the optical center of the lens is called the optical axis. The optical axis passing through the centers of curvature of the lens surfaces is calledmain optical axis(C 1 C 2, in Fig. 11 and 12). Rays passing through the optical center do not

refract (do not change their direction). Rays parallel to the main optical axis of a biconvex lens, after passing through it, intersect the main optical axis at point F (Fig. 13), which is called the main focus of the lens, and the distance from this point to the lens is f

there is a main focal length. Construct your own path of at least two rays incident on the lens parallel to the main optical axis

(the glass lens is located in the air, take this into account when constructing) to prove that a lens located in the air is converging if it is biconvex, and diverging if the lens is biconcave.

24-05-2014, 15:06

Description

The effect of glasses on vision is based on the laws of light propagation. The science of the laws of light propagation and the formation of images using lenses is called geometric, or ray, optics.

Great French mathematician XVII V. Fermat formulated the principle underlying geometric optics: light always takes the shortest path between two points in time. From this principle it follows that in a homogeneous medium light propagates rectilinearly: the path of a light ray from a point 81 exactly 82 is a straight line segment. From the same principle, two basic laws of geometric optics are derived - reflection and refraction of light.

LAWS OF GEOMETRIC OPTICS

If on the path of light another transparent medium is encountered, separated from the first smooth surface, then the light ray is partly reflected from this surface, partly passes through it, changing its direction. In the first case they talk about the reflection of light, in the second - about its refraction.

To explain the laws of reflection and refraction of light, it is necessary to introduce the concept of a normal - perpendicular to the reflecting or refractive surface at the point of incidence of the beam. The angle between the incident ray and the normal at the point of incidence is called the angle of incidence, and between the normal and the reflected ray is called the angle of reflection.

The law of light reflection states: the incident and reflected rays lie in the same plane with the normal at the point of incidence; The angle of incidence is equal to the angle of reflection.

In Fig. 1 shows the beam path between points S 1 And S 2 when reflected from the surface A 1 A 2. Let's move the point S 2 V S 2 " located behind the reflective surface. Obviously the line S 1 S 2 " will be the shortest if it is straight. This condition is satisfied when the angle u 1 =u 1 " and therefore u 1 = u 2, and also when straight OS 1,FROM And OS 2 are in the same plane.

The law of light refraction states: the incident and refracted rays lie in the same plane with the normal at the point of incidence; the ratio of the sine of the angle of incidence to the sine of the angle of refraction for given two media and for rays of a given wavelength is a constant value.

Without citing calculations, it can be shown that these are the conditions that provide the shortest time for light to travel between two points located in different media (Fig. 2).

The law of light refraction is expressed by the following formula:

Magnitude n 2,1 called the relative refractive index of the medium 2 in relation to the environment 1 .

The refractive index of a given medium relative to a void (the air medium is practically equated to it) is called the absolute refractive index of a given medium n.

Relative refractive index n 2,1 associated with the absolute indicators of the first ( n 1 ) and second ( n 2 ) environment relation:

The absolute indicator is determined by the optical density of the medium: the higher the latter, the slower the light propagates in this medium.

Hence the second expression of the law of refraction of light: the sine of the angle of incidence is related to the sine of the angle of refraction as the speed of light in the first medium is to the speed of light in the second medium:

Since light has maximum speed in vacuum (and in air), the refractive index of all media is greater 1 . So, for water it is 1,333 , for optical glass of different types - from 1,487 before 1,806 , for organic glass (methyl methacrylate) - 1,490 , for diamond- 2,417 . In the eye, optical media have the following refractive indices: cornea- 1,376 , aqueous humor and vitreous humor - 1,336 , lens - 1,386 .

RAY TRAVEL THROUGH PRISM

Let's consider some special cases of light refraction. One of the simplest is the passage of light through a prism. It is a narrow wedge of glass or other transparent material suspended in the air.

In Fig. Figure 3 shows the path of rays through a prism. It deflects light rays towards the base. For clarity, the prism profile is chosen in the form of a right triangle, and the incident beam is parallel to its base. In this case, the refraction of the beam occurs only on the rear, oblique edge of the prism. The angle w by which the incident ray is deflected is called the deflection angle of the prism. It practically does not depend on the direction of the incident beam: if the latter is not perpendicular to the edge of incidence, then the deflection angle is composed of the angles of refraction on both faces.

The deflection angle of a prism is approximately equal to the product of the angle at its apex and the refractive index of the prism substance minus 1 :

The derivation of this formula follows from Fig. 3. Draw a perpendicular to the second face of the prism at the point of incidence of the beam on it (dash-dotted line). It forms an angle with the incident ray ? . This angle is equal to the angle ? at the top of the prism, since their sides are mutually perpendicular. Since the prism is thin and all the angles under consideration are small, their sines can be considered approximately equal to the angles themselves, expressed in radians. Then from the law of refraction of light it follows:

In this expression, n is in the denominator, since light comes from a denser medium to a less dense one.

Let's swap the numerator and denominator, and also change the angle ? at an angle equal to it ? :

Since the refractive index of glass commonly used for spectacle lenses is close to 1,5 , the deflection angle of the prisms is approximately half the angle at their apex. Therefore, prisms with a deflection angle of more than 5°; they will be too thick and heavy. In optometry, the deflecting effect of prisms (prismatic action) is often measured not in degrees, but in prismatic diopters ( ? ) or in centiradians (srad). Deflection of rays by a prism with a force of 1 prdptr ( 1 srad) at a distance of 1 m from the prism is 1 cm. This corresponds to an angle whose tangent is equal to 0,01 . This angle is equal 34" (Fig. 4).

The same applies to the visual defect itself, strabismus, corrected by prisms. The squint angle can be measured in degrees and in prism diopters.

RAY TRAVEL THROUGH THE LENS

The transmission of light through lenses is of greatest importance for optometry. A lens is a body made of a transparent material, bounded by two refractive surfaces, at least one of which is a surface of rotation.

Let's consider the simplest lens—thin, limited by one spherical and one flat surface. Such a lens is called spherical. It is a segment sawn off from a glass ball (Fig. 5, a). The line AO ​​connecting the center of the ball to the center of the lens is called its optical axis. In cross-section, such a lens can be represented as a pyramid made up of small prisms with an increasing angle at the apex (Fig. 5, b).

Rays entering the lens and parallel to its axis undergo refraction, the greater the further they are from the axis. It can be shown that they will all intersect the optical axis at one point ( F" ). This point is called the focus of the lens (more precisely, the back focus). A lens with a concave refractive surface has the same point, but its focus is on the same side from which the rays enter. The distance from the focal point to the center of the lens is called its focal length ( f" ). The reciprocal of the focal length characterizes the refractive power, or refraction, of the lens ( D):

Where D- refractive power of the lens, diopters; f" - focal length, m;

The refractive power of a lens is measured in diopters. It is the basic unit in optometry. Behind 1 diopter ( D, diopters) the refractive power of a lens with a focal length is taken 1 m. Therefore, a lens with a focal length 0,5 m has refractive power 2,0 diopter, 2 m - 0,5 diopter, etc. The refractive power of convex lenses has a positive value, while concave lenses have a negative value.

Not only rays parallel to the optical axis, passing through a convex spherical lens, converge at one point. Rays emanating from any point to the left of the lens (not closer than the focal point) converge to another point to the right of it. Thanks to this, a spherical lens has the property of forming images of objects (Fig. 6).

Just like plano-convex and plano-concave lenses, lenses limited by two spherical surfaces operate - biconvex, biconcave and convex-concave. In spectacle optics, mainly convex-concave lenses, or menisci, are used. The overall effect of the lens depends on which surface has the greater curvature.

The action of spherical lenses is called stigmatic (from the Greek - point), since they form an image of a point in space in the form of a point.

The following types of lenses are cylindrical and toric. A convex cylindrical lens has the property of collecting a beam of parallel rays incident on it into a line parallel to the axis of the cylinder (Fig. 7). Direct F 1 F 2 analogous to the focal point of a spherical lens is called the focal line.

A cylindrical surface, when intersected by planes passing through the optical axis, forms a circle, ellipses and a straight line in sections. Two such sections are called main: one passes through the axis of the cylinder, the other is perpendicular to it. In the first section a straight line is formed, in the second - a circle. Accordingly, in a cylindrical lens there are two main sections, or meridians, - the axis and the active section. Normal rays incident on the axis of the lens are not subject to refraction, but incident on the active section are collected at the focal line, at the point of its intersection with the optical axis.

More complex is a lens with a toric surface, which is formed by rotating a circle or arc with a radius r around the axis. Radius of rotation R not equal to the radius r(Fig. 8).

The refraction of rays by a toric lens is shown in Fig. 9.

A toric lens consists, as it were, of two spherical ones: the radius of one of them corresponds to the radius of the rotating circle, the radius of the second corresponds to the radius of rotation. Accordingly, the lens has two main sections ( A 1 A 2 And B 1 B 2). A parallel beam of rays incident on it is transformed into a figure called a Sturm conoid. Instead of a focal point, the rays are collected into two straight segments lying in the plane of the main sections. They are called focal lines - anterior ( F 1 F 1 ) and back ( F 2 F 2 ).

The property of transforming a beam of parallel rays or rays coming from a point into a Sturm conoid is called astigmatism (literally “deadness”), and cylindrical and toric lenses are called astigmatic lenses. The measure of astigmatism is the difference in refractive power in the two main sections (in diopters). The greater the astigmatic difference, the greater the distance between the focal lines in the Sturm conoid.

Any spherical lens is characterized by astigmatic action if the rays fall on it at a large angle to the optical axis. This phenomenon is called oblique incidence (or oblique beam) astigmatism.

In optometry we have to deal with another type of lens - afocal lenses. An afocal lens is such a lens, both spherical surfaces of which have the same radius, but one of them is concave and the other is convex (Fig. 10, a).

Such a lens has no focus and therefore cannot form an image. But, being in the path of the light beam carrying the image, it increases it (if the light goes from right to left) or decreases it (if the light goes from left to right). This action of the afocal lens is called eikonic (from the Greek - image). More often, systems of lenses, such as telescopes, are used for this purpose rather than single lenses. In Fig. 10, b, shows a diagram of the simplest telescope, consisting of one negative and one positive lens (Galilean system).

Eikonic action is also inherent in ordinary spherical lenses: positive lenses magnify, and negative lenses reduce the image. This effect is measured as a percentage, and at high magnifications - in “cramps” ( X). So, a magnifying glass that magnifies an image in 2 times is called double ( 2x).

Thus, lenses provide four types of optical action: prismatic, stigmatic, astigmatic and eikonic. Next we will show how they are all used to correct vision defects.

Note that in most cases, lenses are characterized not only by the action for which they are intended: spherical (stigmatic) lenses are also characterized by eikonic action, and on the periphery of the glass, in addition, prismatic and astigmatic. Astigmatic lenses are also characterized by stigmatic, prismatic and eikonic action.

COMPLEX OPTICAL SYSTEMS

Until now we have been talking about ideal lenses, seemingly without thickness (with the exception of afocal ones). In optometry you have to deal with lenses that have real thickness, and even more often with lens systems.

Of particular interest are centered systems, i.e. those that consist of spherical lenses having a common optical axis. To describe such systems and calculate their action, two methods are used: with the introduction of so-called cardinal points and planes; using the concept of ray convergence and vertex refraction.

The first method, developed by the German mathematician Gauss, is as follows. There are four Cardinal points on the optical axis of the system: two nodal and two main (Fig. 11).

Nodal points - anterior and posterior ( N And N" ) - have the following property: a ray entering the front point ( S 1 N), comes out parallel to itself from the rear ( N'S 2 ). They are used in the construction of images formed by an optical system.

The main points ( N And N"). The planes perpendicular to the optical axis drawn through them are called the main planes - front and back. A ray of light entering one of them passes to the other parallel to the optical axis. In other words, the image on the rear main plane repeats the image on the front. All distances on the optical axis are measured from the main planes: to the object - from the front, to the image - from the back. Often these planes lie so close to each other that they can be approximately replaced by one main plane.

For example, in the optical system of the human eye, the front main plane lies in 1,47 mm, and the rear - in 1,75 mm from the apex of the cornea. When calculating, it is assumed that both of them are located approximately 1,6 mm from this point.

The second way to describe centered optical systems assumes that the beam of rays at each point on the optical axis has a special property - convergence. It is determined by the reciprocal of the distance to the point of convergence of this beam, and is measured, like refraction, in diopters. The effect of each refractive surface on the path of the beam is a change in convergence. Convex surfaces increase convergence, concave surfaces decrease convergence. The convergence of a parallel beam of rays is zero.

This method is especially convenient for calculating the total refractive power of a system. A typical complex optical system is a thick lens (Fig. 12), which has two refractive surfaces and a homogeneous medium between them.

Changes in the convergence of a parallel beam of rays incident on the lens are determined by the refractive power of these surfaces, the distance between them and the refractive index of the lens material.

Let us accept the following notation:

• L 0 - convergence of a parallel beam incident on the lens;
• L 1 - convergence of the beam after refraction on the first surface of the lens;
• L 2 - convergence of the beam upon reaching the second surface of the lens;
• L 3 - convergence of the beam after refraction on the second surface, i.e., when leaving the lens;
• D 1 - refractive power of the first surface;
• D 2 - refractive power of the second surface;
• d- distance between lens surfaces;
• n- refractive index of the lens material.

At the same time, the values L And D are measured in diopters, and d- b- in meters.

Beam convergence at the lens entrance L 0 = 0 .

After refraction on the front surface of the LENS it becomes equal L 1 = D 1 . When reaching the back surface it acquires the meaning:

and finally, upon exiting the lens

This expression shows the change in the convergence of the beam as it passes through the lens when measuring distances from its front surface. This is called the anterior vertex refraction of the lens. If we consider the path of rays from the back surface to the front, then in the denominator D 1 will be replaced by D 2 . Expression

represents the value of the posterior apical refraction of a thick lens. The lens power values ​​in trial sets of spectacle glasses represent their posterior apical refractions.

The numerator of this expression is a formula for determining the total refractive power of a system consisting of two elements (surfaces or thin lenses):

Where D- total refractive power of the system;

D 1 And D 2 - refractive power of system elements;

n- refractive index of the medium between the elements;

d- distance between system elements.

Video tutorial 2: Geometric optics: Laws of refraction

Lecture: Laws of light refraction. Path of rays in a prism

At the moment when a ray falls on some other medium, it is not only reflected, but also passes through it. However, due to the difference in densities, it changes its path. That is, the beam, hitting the boundary, changes its propagation trajectory and moves with a displacement by a certain angle. Refraction will occur when the beam falls at a certain angle to the perpendicular. If it coincides with the perpendicular, then refraction does not occur and the beam penetrates the medium at the same angle.

Air-Media

The most common situation when light passes from one medium to another is the transition from air.

So, in the picture JSC- ray incident on the interface, CO And OD- perpendiculars (normals) to the sections of the media, lowered from the point of incidence of the beam. OB- a ray that has been refracted and passed into another medium. The angle between the normal and the incident ray is called the angle of incidence (AOC). The angle between the refracted ray and the normal is called the angle of refraction (BOD).

To find out the refractive intensity of a particular medium, a PV is introduced, which is called the refractive index. This value is tabular and for basic substances the value is a constant value that can be found in the table. Most often, problems use the refractive indices of air, water and glass.

Laws of refraction for air-medium

1. When considering the incident and refracted ray, as well as the normal to the sections of the media, all of the listed quantities are in the same plane.

2. The ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant value equal to the refractive index of the medium.

From this relationship it is clear that the value of the refractive index is greater than unity, which means that the sine of the angle of incidence is always greater than the sine of the angle of refraction. That is, if the beam leaves the air into a denser medium, then the angle decreases.

The refractive index also shows how the speed of propagation of light changes in a particular medium, relative to propagation in a vacuum:

From this we can obtain the following relationship:

When we consider air, we can make some neglects - we will assume that the refractive index of this medium is equal to unity, then the speed of light propagation in the air will be equal to 3 * 10 8 m/s.

Ray reversibility

These laws also apply in cases where the direction of the rays occurs in the opposite direction, that is, from the medium into the air. That is, the path of light propagation is not affected by the direction in which the rays move.

Law of refraction for arbitrary media

Geometric optics

Geometric optics is a branch of optics that studies the laws of propagation of light energy in transparent media based on the concept of a light beam.

A light ray is not a beam of light, but a line indicating the direction of propagation of light.

Basic laws:

1. Law on the rectilinear propagation of light.

Light propagates in a straight line in a homogeneous medium. The straightness of the propagation of light explains the formation of a shadow, that is, a place where light energy does not penetrate. Small-sized sources produce a sharply defined shadow, while large-sized sources create shadows and penumbra, depending on the size of the source and the distance between the body and the source.

2. Law of reflection. The angle of incidence is equal to the angle of reflection.

The incident ray, the reflected ray and the perpendicular to the interface between the two media, reconstructed at the point of incidence of the ray, lie in the same plane

b-angle of incidence c-angle of reflection d-perpendicular lowered to the point of incidence

3. Law of refraction.

At the interface between two media, light changes the direction of its propagation. Part of the light energy returns to the first medium, that is, light is reflected. If the second medium is transparent, then part of the light, under certain conditions, can pass through the boundary of the media, also changing, as a rule, the direction of propagation. This phenomenon is called refraction of light.

b-angle of incidence c-angle of refraction.

The incident ray, the reflected ray, and the perpendicular to the interface between the two media, reconstructed at the point of incidence of the ray, 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 given media.

The constant n is called the relative refractive index or refractive index of the second medium relative to the first.

Path of rays in a triangular prism

Optical instruments often use a triangular prism made of glass or other transparent materials.

Path of rays in the cross section of a triangular prism

A ray passing through a triangular glass prism always tends to its base.

The angle is called the refractive angle of the prism. The angle of deflection of the beam depends on the refraction reading n of the prism and the angle of incidence b. Optical prisms in the form of an isosceles right triangle are often used in optical instruments. Their use is based on the fact that the limiting angle of total reflection for glass is 0 = 45 0