How to find the limiting angle of total reflection. Limit angle of total reflection

Total internal reflection

Internal reflection- the phenomenon of reflection of electromagnetic waves from the interface between two transparent media, provided that the wave falls from a medium with a higher refractive index.

Incomplete internal reflection- internal reflection, provided that the angle of incidence is less than the critical angle. In this case, the beam splits into refracted and reflected.

Total internal reflection- internal reflection, provided that the angle of incidence exceeds a certain critical angle. In this case, the incident wave is completely reflected, and the value of the reflection coefficient exceeds its highest values ​​for polished surfaces. In addition, the reflection coefficient for total internal reflection does not depend on the wavelength .

This optical phenomenon is observed for a wide spectrum of electromagnetic radiation including the X-ray range.

Within the framework of geometric optics, the explanation of the phenomenon is trivial: based on Snell's law and taking into account that the angle of refraction cannot exceed 90 °, we obtain that at an angle of incidence whose sine is greater than the ratio of the smaller refractive index to the larger coefficient, an electromagnetic wave should be completely reflected into the first medium .

In accordance with the wave theory of the phenomenon, the electromagnetic wave nevertheless penetrates into the second medium - the so-called "non-uniform wave" propagates there, which decays exponentially and does not carry away energy with it. The characteristic depth of penetration of an inhomogeneous wave into the second medium is of the order of the wavelength.

Total internal light reflection

Consider internal reflection using the example of two monochromatic rays incident on the interface between two media. Rays fall from a zone of denser medium (indicated in darker blue) with a refractive index to the boundary with a less dense medium (indicated in light blue) with a refractive index.

The red beam falls at an angle , that is, at the boundary of the media, it bifurcates - it is partially refracted and partially reflected. Part of the beam is refracted at an angle.

The green beam falls and is completely reflected src="/pictures/wiki/files/100/d833a2d69df321055f1e0bf120a53eff.png" border="0">.

Total internal reflection in nature and technology

Reflection of x-rays

The refraction of x-rays in grazing incidence was first formulated by M. A. Kumakhov, who developed the x-ray mirror, and theoretically substantiated by Arthur Compton in 1923.

Other wave phenomena

Demonstration of refraction, and hence the effect of total internal reflection, is possible, for example, for sound waves on the surface and in the bulk of a liquid during the transition between zones of different viscosity or density.

Phenomena similar to the effect of total internal reflection of electromagnetic radiation are observed for beams of slow neutrons.

If a vertically polarized wave falls on the interface at the Brewster angle, then the effect of complete refraction will be observed - there will be no reflected wave.

Notes

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See what "Total internal reflection" is in other dictionaries:

TOTAL INTERNAL REFLECTION- reflection email. magn. radiation (in particular, light) when it falls on the interface between two transparent media from a medium with a high refractive index. P. in. about. is carried out when the angle of incidence i exceeds a certain limiting (critical) angle ... Physical Encyclopedia

Total internal reflection- Total internal reflection. When light passes from a medium with n1 > n2, total internal reflection occurs if the angle of incidence a2 > apr; at an angle of incidence a1 Illustrated Encyclopedic Dictionary

Total internal reflection- reflection of optical radiation (See Optical radiation) (light) or electromagnetic radiation of a different range (for example, radio waves) when it falls on the interface between two transparent media from a medium with a high refractive index ... ... Great Soviet Encyclopedia

TOTAL INTERNAL REFLECTION- electromagnetic waves, occurs when they pass from a medium with a high refractive index n1 to a medium with a lower refractive index n2 at an angle of incidence a exceeding the limiting angle apr, determined by the ratio sinapr=n2/n1. Complete… … Modern Encyclopedia

TOTAL INTERNAL REFLECTION- TOTAL INTERNAL REFLECTION, REFLECTION without light refraction at the boundary. When light passes from a denser medium (such as glass) to a less dense medium (water or air), there is a zone of refraction angles in which light does not pass through the boundary ... Scientific and technical encyclopedic dictionary

total internal reflection- Reflection of light from an optically less dense medium with complete return to the medium from which it falls. [Collection of recommended terms. Issue 79. Physical optics. USSR Academy of Sciences. Committee of Scientific and Technical Terminology. 1970] Topics… … Technical Translator's Handbook

TOTAL INTERNAL REFLECTION- electromagnetic waves occur when they fall obliquely on the interface between 2 media, when radiation passes from a medium with a high refractive index n1 to a medium with a lower refractive index n2, and the angle of incidence i exceeds the limiting angle ... ... Big Encyclopedic Dictionary

total internal reflection- electromagnetic waves, occurs with oblique incidence on the interface between 2 media, when radiation passes from a medium with a high refractive index n1 to a medium with a lower refractive index n2, and the angle of incidence i exceeds the limiting angle ipr ... encyclopedic Dictionary

First, let's fantasize a little. Imagine a hot summer day BC, a primitive man hunts fish with a spear. He notices her position, aims and strikes for some reason not at all where the fish was visible. Missed? No, the fisherman has the prey in his hands! The thing is that our ancestor intuitively understood the topic that we will study now. In everyday life, we see that a spoon dipped into a glass of water appears crooked, when we look through a glass jar, objects appear crooked. We will consider all these questions in the lesson, the theme of which is: “Refraction of light. The law of refraction of light. Total internal reflection.

In previous lessons, we talked about the fate of a ray in two cases: what happens if a ray of light propagates in a transparently homogeneous medium? The correct answer is that it will spread in a straight line. And what will happen when a beam of light falls on the interface between two media? In the last lesson we talked about the reflected beam, today we will consider that part of the light beam that is absorbed by the medium.

What will be the fate of the beam that has penetrated from the first optically transparent medium into the second optically transparent medium?

Rice. 1. Refraction of light

If a beam falls on the interface between two transparent media, then part of the light energy returns to the first medium, creating a reflected beam, while the other part passes inward to the second medium and, as a rule, changes its direction.

The change in the direction of propagation of light in the case of its passage through the interface between two media is called refraction of light(Fig. 1).

Rice. 2. Angles of incidence, refraction and reflection

In Figure 2 we see an incident beam, the angle of incidence will be denoted by α. The beam that will set the direction of the refracted beam of light will be called the refracted beam. The angle between the perpendicular to the interface between the media, restored from the point of incidence, and the refracted beam is called the angle of refraction, in the figure this is the angle γ. To complete the picture, we also give an image of the reflected beam and, accordingly, the reflection angle β. What is the relationship between the angle of incidence and the angle of refraction, is it possible to predict, knowing the angle of incidence and from which medium the beam passed into which, what will be the angle of refraction? It turns out you can!

We obtain a law that quantitatively describes the relationship between the angle of incidence and the angle of refraction. Let us use the Huygens principle, which regulates the propagation of a wave in a medium. The law consists of two parts.

The incident ray, the refracted ray and the perpendicular restored to 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 two given media and is equal to the ratio of the speeds of light in these media.

This law is called Snell's law, after the Dutch scientist who first formulated it. The reason for refraction is the difference in the speeds of light in different media. You can verify the validity of the law of refraction by experimentally directing a beam of light at different angles to the interface between two media and measuring the angles of incidence and refraction. If we change these angles, measure the sines and find the ratios of the sines of these angles, we will be convinced that the law of refraction is indeed valid.

Evidence of the law of refraction using the Huygens principle is another confirmation of the wave nature of light.

The relative refractive index n 21 shows how many times the speed of light V 1 in the first medium differs from the speed of light V 2 in the second medium.

The relative refractive index is a clear demonstration of the fact that the reason for the change in the direction of light when passing from one medium to another is the different speed of light in two media. The term "optical density of a medium" is often used to characterize the optical properties of a medium (Fig. 3).

Rice. 3. Optical density of the medium (α > γ)

If the beam passes from a medium with a higher speed of light to a medium with a lower speed of light, then, as can be seen from Figure 3 and the law of refraction of light, it will be pressed against the perpendicular, that is, the angle of refraction is less than the angle of incidence. In this case, the beam is said to have passed from a less dense optical medium to a more optically dense medium. Example: from air to water; from water to glass.

The reverse situation is also possible: the speed of light in the first medium is less than the speed of light in the second medium (Fig. 4).

Rice. 4. Optical density of the medium (α< γ)

Then the angle of refraction will be greater than the angle of incidence, and such a transition will be said to be made from an optically denser to a less optically dense medium (from glass to water).

The optical density of two media can differ quite significantly, so the situation shown in the photograph (Fig. 5) becomes possible:

Rice. 5. The difference between the optical density of media

Pay attention to how the head is displaced relative to the body, which is in the liquid, in a medium with a higher optical density.

However, the relative refractive index is not always a convenient characteristic for work, because it depends on the speeds of light in the first and second media, but there can be a lot of such combinations and combinations of two media (water - air, glass - diamond, glycerin - alcohol , glass - water and so on). The tables would be very cumbersome, it would be inconvenient to work, and then one absolute environment was introduced, in comparison with which the speed of light in other environments is compared. Vacuum was chosen as the absolute and the speeds of light are compared with the speed of light in vacuum.

Absolute refractive index of the medium n- this is a value that characterizes the optical density of the medium and is equal to the ratio of the speed of light With in vacuum to the speed of light in a given medium.

The absolute refractive index is more convenient for work, because we always know the speed of light in vacuum, it is equal to 3·10 8 m/s and is a universal physical constant.

The absolute refractive index depends on external parameters: temperature, density, and also on the wavelength of light, so tables usually indicate the average refractive index for a given wavelength range. If we compare the refractive indices of air, water and glass (Fig. 6), we see that the refractive index of air is close to unity, so we will take it as a unit when solving problems.

Rice. 6. Table of absolute refractive indices for different media

It is easy to get the relationship between the absolute and relative refractive index of media.

The relative refractive index, that is, for a beam passing from medium one to medium two, is equal to the ratio of the absolute refractive index in the second medium to the absolute refractive index in the first medium.

For example: = ≈ 1,16

If the absolute refractive indices of the two media are almost the same, this means that the relative refractive index when passing from one medium to another will be equal to one, that is, the light beam will not actually be refracted. For example, when passing from anise oil to a gem, beryl will practically not deviate light, that is, it will behave as it does when passing through anise oil, since their refractive index is 1.56 and 1.57, respectively, so the gem can be how to hide in a liquid, it simply will not be visible.

If you pour water into a transparent glass and look through the wall of the glass into the light, then we will see a silvery sheen of the surface due to the phenomenon of total internal reflection, which will be discussed now. When a light beam passes from a denser optical medium to a less dense optical medium, an interesting effect can be observed. For definiteness, we will assume that light goes from water to air. Let us assume that there is a point source of light S in the depth of the reservoir, emitting rays in all directions. For example, a diver shines a flashlight.

Beam SO 1 falls on the surface of the water at the smallest angle, this beam is partially refracted - beam O 1 A 1 and partially reflected back into the water - beam O 1 B 1. Thus, part of the energy of the incident beam is transferred to the refracted beam, and the remaining part of the energy is transferred to the reflected beam.

Rice. 7. Total internal reflection

Beam SO 2, whose angle of incidence is larger, is also divided into two beams: refracted and reflected, but the energy of the original beam is distributed between them in a different way: the refracted beam O 2 A 2 will be dimmer than the beam O 1 A 1, that is, it will receive a smaller fraction of energy, and the reflected beam O 2 V 2, respectively, will be brighter than the beam O 1 V 1, that is, it will receive a larger share of energy. As the angle of incidence increases, the same regularity is traced - an increasing share of the energy of the incident beam goes to the reflected beam and an ever smaller share to the refracted beam. The refracted beam becomes dimmer and at some point disappears completely, this disappearance occurs when the angle of incidence is reached, which corresponds to a refraction angle of 90 0 . In this situation, the refracted beam OA would have to go parallel to the water surface, but there is nothing to go - all the energy of the incident beam SO went entirely to the reflected beam OB. Naturally, with a further increase in the angle of incidence, the refracted beam will be absent. The described phenomenon is total internal reflection, that is, a denser optical medium at the considered angles does not emit rays from itself, they are all reflected inside it. The angle at which this phenomenon occurs is called limiting angle of total internal reflection.

The value of the limiting angle is easy to find from the law of refraction:

= => = arcsin, for water ≈ 49 0

The most interesting and popular application of the phenomenon of total internal reflection is the so-called waveguides, or fiber optics. This is exactly the way of signaling that is used by modern telecommunications companies on the Internet.

We got the law of refraction of light, introduced a new concept - relative and absolute refractive indices, and also figured out the phenomenon of total internal reflection and its applications, such as fiber optics. You can consolidate knowledge by examining the relevant tests and simulators in the lesson section.

Let's get the proof of the law of refraction of light using the Huygens principle. It is important to understand that the cause of refraction is the difference in the speeds of light in two different media. Let us denote the speed of light in the first medium V 1 , and in the second medium - V 2 (Fig. 8).

Rice. 8. Proof of the law of refraction of light

Let a plane light wave fall on a flat interface between two media, for example, from air into water. The wave surface AC is perpendicular to the rays and , the interface between the media MN first reaches the beam , and the beam reaches the same surface after a time interval ∆t, which will be equal to the path SW divided by the speed of light in the first medium .

Therefore, at the moment when the secondary wave at point B only begins to be excited, the wave from point A already has the form of a hemisphere with radius AD, which is equal to the speed of light in the second medium by ∆t: AD = ∆t, that is, the Huygens principle in visual action . The wave surface of a refracted wave can be obtained by drawing a surface tangent to all secondary waves in the second medium, the centers of which lie on the interface between the media, in this case it is the plane BD, it is the envelope of the secondary waves. The angle of incidence α of the beam is equal to the angle CAB in the triangle ABC, the sides of one of these angles are perpendicular to the sides of the other. Therefore, SW will be equal to the speed of light in the first medium by ∆t

CB = ∆t = AB sin α

In turn, the angle of refraction will be equal to the angle ABD in the triangle ABD, therefore:

AD = ∆t = AB sin γ

Dividing the expressions term by term, we get:

n is a constant value that does not depend on the angle of incidence.

We have obtained the law of refraction of light, the sine of the angle of incidence to the sine of the angle of refraction is a constant value for the given two media and equal to the ratio of the speeds of light in the two given media.

A cubic vessel with opaque walls is located in such a way that the observer's eye does not see its bottom, but completely sees the wall of the vessel CD. How much water must be poured into the vessel so that the observer can see the object F, located at a distance b = 10 cm from the corner D? Vessel edge α = 40 cm (Fig. 9).

What is very important in solving this problem? Guess that since the eye does not see the bottom of the vessel, but sees the extreme point of the side wall, and the vessel is a cube, then the angle of incidence of the beam on the surface of the water when we pour it will be equal to 45 0.

Rice. 9. The task of the exam

The beam falls to point F, which means that we clearly see the object, and the black dotted line shows the course of the beam if there were no water, that is, to point D. From the triangle NFC, the tangent of the angle β, the tangent of the angle of refraction, is the ratio of the opposite leg to the adjacent or, based on the figure, h minus b divided by h.

tg β = = , h is the height of the liquid that we poured;

The most intense phenomenon of total internal reflection is used in fiber optic systems.

Rice. 10. Fiber optics

If a beam of light is directed to the end of a solid glass tube, then after multiple total internal reflection the beam will emerge from the opposite side of the tube. It turns out that the glass tube is a conductor of a light wave or a waveguide. This will happen whether the tube is straight or curved (Figure 10). The first light guides, this is the second name of wave guides, were used to illuminate hard-to-reach places (during medical research, when light is supplied to one end of the light guide, and the other end illuminates the right place). The main application is medicine, defectoscopy of motors, however, such waveguides are most widely used in information transmission systems. The carrier frequency of a light wave is a million times the frequency of a radio signal, which means that the amount of information that we can transmit using a light wave is millions of times greater than the amount of information transmitted by radio waves. This is a great opportunity to convey a huge amount of information in a simple and inexpensive way. As a rule, information is transmitted over a fiber cable using laser radiation. Fiber optics is indispensable for fast and high-quality transmission of a computer signal containing a large amount of transmitted information. And at the heart of all this lies such a simple and common phenomenon as the refraction of light.

Bibliography

1. Tikhomirova S.A., Yavorsky B.M. Physics (basic level) - M.: Mnemozina, 2012.
2. Gendenstein L.E., Dick Yu.I. Physics grade 10. - M.: Mnemosyne, 2014.
3. Kikoin I.K., Kikoin A.K. Physics - 9, Moscow, Education, 1990.

1. edu.glavsprav.ru ().
2. Nvtc.ee ().
3. Raal100.narod.ru ().
4. Optika.ucoz.ru ().

Homework

1. Define refraction of light.
2. Name the reason for the refraction of light.
3. Name the most popular applications of total internal reflection.

If n 1 >n 2, then >α, i.e. if light passes from an optically denser medium to an optically less dense medium, then the angle of refraction is greater than the angle of incidence (Fig. 3)

Limit angle of incidence. If α=α p,=90˚ and the beam will slide along the air-water interface.

If α'>α p, then the light will not pass into the second transparent medium, because will be fully reflected. This phenomenon is called full reflection of light. The angle of incidence α p, at which the refracted beam slides along the interface between the media, is called the limiting angle of total reflection.

Total reflection can be observed in an isosceles rectangular glass prism (Fig. 4), which is widely used in periscopes, binoculars, refractometers, etc.

a) Light falls perpendicular to the first face and therefore does not undergo refraction here (α=0 and =0). The angle of incidence on the second face α=45˚, i.e.>α p, (for glass α p =42˚). Therefore, on this face, the light is completely reflected. This is a rotary prism that rotates the beam 90˚.

b) In this case, the light inside the prism experiences already twofold total reflection. This is also a rotary prism that rotates the beam by 180˚.

c) In this case, the prism is already inverted. When the rays leave the prism, they are parallel to the incident ones, but in this case the upper incident beam becomes lower, and the lower one becomes upper.

The phenomenon of total reflection has found wide technical application in light guides.

The light guide is a large number of thin glass filaments, the diameter of which is about 20 microns, and each is about 1 m long. These threads are parallel to each other and are located close (Fig. 5)

Each filament is surrounded by a thin shell of glass, the refractive index of which is less than that of the filament itself. The light guide has two ends, the mutual arrangement of the ends of the threads on both ends of the light guide is strictly the same.

If an object is placed at one end of the light guide and illuminated, then an image of this object will appear at the other end of the light guide.

The image is obtained due to the fact that light from some small area of ​​​​the object enters the end of each of the threads. Experiencing many total reflections, the light emerges from the opposite end of the filament, transmitting the reflection of a given small area of ​​the object.

Because the location of the threads relative to each other is strictly the same, then the corresponding image of the object appears at the other end. The clarity of the image depends on the diameter of the threads. The smaller the diameter of each thread, the clearer the image of the object will be. The loss of light energy along the path of the light beam is usually relatively small in bundles (light guides), since with total reflection the reflection coefficient is relatively high (~0.9999). Energy loss are mainly due to the absorption of light by the substance inside the fiber.

For example, in the visible part of the spectrum in a fiber 1 m long, 30-70% of the energy is lost (but in the bundle).

Therefore, in order to transmit large light fluxes and maintain the flexibility of the light-guiding system, individual fibers are assembled into bundles (bundles) - light guides.

Light guides are widely used in medicine for illuminating internal cavities with cold light and transmitting images. endoscope- a special device for examining internal cavities (stomach, rectum, etc.). With the help of light guides, laser radiation is transmitted for a therapeutic effect on tumors. Yes, and the human retina is a highly organized fiber-optic system consisting of ~ 130x10 8 fibers.

When waves propagate in a medium, including electromagnetic ones, to find a new wave front at any time, use Huygens principle.

Each point of the wave front is a source of secondary waves.

In a homogeneous isotropic medium, the wave surfaces of secondary waves have the form of spheres of radius v × Dt, where v is the speed of wave propagation in the medium. By conducting the envelope of the wave fronts of the secondary waves, we obtain a new wave front at a given time (Fig. 7.1, a, b).

Law of reflection

Using the Huygens principle, one can prove the law of reflection of electromagnetic waves at the interface between two dielectrics.

The angle of incidence is equal to the angle of reflection. The incident and reflected rays, together with the perpendicular to the interface between two dielectrics, lie in the same plane.Ð a = Ð b. (7.1)

Let a plane light wave fall on a flat SD interface between two media (beams 1 and 2, Fig. 7.2). The angle a between the beam and the perpendicular to the LED is called the angle of incidence. If at a given time the front of the incident wave OB reaches point O, then, according to the Huygens principle, this point

Rice. 7.2

begins to radiate a secondary wave. During the time Dt = IN 1 /v the incident beam 2 reaches t. O 1 . During the same time, the front of the secondary wave, after reflection in point O, propagating in the same medium, reaches the points of the hemisphere, radius OA \u003d v Dt \u003d BO 1. The new wave front is depicted by the plane AO ​​1, and the direction of propagation is represented by the beam OA. The angle b is called the angle of reflection. From the equality of triangles OAO 1 and OBO 1, the law of reflection follows: the angle of incidence is equal to the angle of reflection.

Law of refraction

Optically homogeneous medium 1 is characterized by , (7.2)

Ratio n 2 / n 1 \u003d n 21 (7.4)

called

(7.5)

For vacuum n = 1.

Due to dispersion (light frequencies n » 10 14 Hz), for example, for water n = 1.33, and not n = 9 (e = 81), as follows from electrodynamics for low frequencies. If the speed of propagation of light in the first medium is v 1, and in the second - v 2,

Rice. 7.3

then during the time Dt of the incident plane wave passing the distance AO 1 in the first medium AO 1 = v 1 Dt. The front of the secondary wave, excited in the second medium (in accordance with the Huygens principle), reaches the points of the hemisphere, the radius of which is OB = v 2 Dt. The new front of the wave propagating in the second medium is depicted by the plane BO 1 (Fig. 7.3), and the direction of its propagation is represented by the rays OB and O 1 C (perpendicular to the wave front). Angle b between the beam OB and the normal to the interface between two dielectrics at the point O called the angle of refraction. From the triangles OAO 1 and OBO 1 it follows that AO 1 \u003d OO 1 sin a, OB \u003d OO 1 sin b.

Their attitude expresses law of refraction(law Snell):

. (7.6)

The ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to the relative refractive index of the two media.

Total internal reflection

Rice. 7.4

According to the law of refraction, at the interface between two media, one can observe total internal reflection, if n 1 > n 2, i.e. Рb >Рa (Fig. 7.4). Therefore, there is such a limiting angle of incidence Ða pr when Ðb = 90 0 . Then the law of refraction (7.6) takes the following form:

sin a pr \u003d, (sin 90 0 \u003d 1) (7.7)

With a further increase in the angle of incidence Ða > Ða pr, the light is completely reflected from the interface between two media.

Such a phenomenon is called total internal reflection and widely used in optics, for example, to change the direction of light rays (Fig. 7. 5, a, b).

It is used in telescopes, binoculars, fiber optics and other optical instruments.

In classical wave processes, such as the phenomenon of total internal reflection of electromagnetic waves, phenomena similar to the tunnel effect in quantum mechanics are observed, which is associated with the corpuscular-wave properties of particles.

Indeed, during the transition of light from one medium to another, refraction of light is observed, associated with a change in the speed of its propagation in various media. At the interface between two media, a beam of light is divided into two: refracted and reflected.

A beam of light falls perpendicularly on face 1 of a rectangular isosceles glass prism and, without being refracted, falls on face 2, total internal reflection is observed, since the angle of incidence (Ða = 45 0) of the beam on face 2 is greater than the limiting angle of total internal reflection (for glass n 2 = 1.5; Ða pr = 42 0).

If the same prism is placed at a certain distance H ~ l/2 from face 2, then the light beam passes through face 2 * and exits the prism through face 1 * parallel to the beam incident on face 1. The intensity J of the transmitted light flux decreases exponentially with increasing gap h between prisms according to the law:

,

where w is some probability of the beam passing into the second medium; d is a coefficient depending on the refractive index of the substance; l is the wavelength of the incident light

Therefore, the penetration of light into the "forbidden" region is an optical analogy of the quantum tunneling effect.

The phenomenon of total internal reflection is indeed complete, since in this case all the energy of the incident light is reflected at the interface between two media than when reflected, for example, from the surface of metal mirrors. Using this phenomenon, one can trace another analogy between the refraction and reflection of light, on the one hand, and Vavilov-Cherenkov radiation, on the other hand.

WAVE INTERFERENCE

7.2.1. The role of vectors and

In practice, several waves can propagate simultaneously in real media. As a result of the addition of waves, a number of interesting phenomena are observed: interference, diffraction, reflection and refraction of waves etc.

These wave phenomena are characteristic not only for mechanical waves, but also for electric, magnetic, light, etc. All elementary particles also exhibit wave properties, which has been proven by quantum mechanics.

One of the most interesting wave phenomena, which is observed when two or more waves propagate in a medium, is called interference. Optically homogeneous medium 1 is characterized by absolute refractive index , (7.8)

where c is the speed of light in vacuum; v 1 - the speed of light in the first medium.

Medium 2 is characterized by the absolute refractive index

where v 2 is the speed of light in the second medium.

Ratio (7.10)

called the relative refractive index of the second medium relative to the first. For transparent dielectrics, where m = 1, using Maxwell's theory, or

where e 1 , e 2 are the permittivities of the first and second media.

For vacuum, n = 1. Due to dispersion (light frequencies n » 10 14 Hz), for example, for water, n = 1.33, and not n = 9 (e = 81), as follows from electrodynamics for low frequencies. Light is electromagnetic waves. Therefore, the electromagnetic field is determined by the vectors and , which characterize the strengths of the electric and magnetic fields, respectively. However, in many processes of interaction of light with matter, such as the effect of light on the organs of vision, photocells and other devices, the decisive role belongs to the vector, which in optics is called the light vector.

At a certain angle of incidence of light $(\alpha )_(pad)=(\alpha )_(pred)$, which is called limiting angle, the angle of refraction is equal to $\frac(\pi )(2),\ $in this case, the refracted beam slides along the interface between the media, therefore, there is no refracted beam. Then, from the law of refraction, we can write that:

Picture 1.

In the case of total reflection, the equation is:

has no solution in the region of real values ​​of the angle of refraction ($(\alpha )_(pr)$). In this case, $cos((\alpha )_(pr))$ is purely imaginary. If we turn to the Fresnel Formulas, then it is convenient to represent them in the form:

where the angle of incidence is denoted by $\alpha $ (for brevity), $n$ is the refractive index of the medium where the light propagates.

Fresnel formulas show that the modules $\left|E_(otr\bot )\right|=\left|E_(otr\bot )\right|$, $\left|E_(otr//)\right|=\ left|E_(otr//)\right|$ which means the reflection is "full".

Remark 1

It should be noted that the inhomogeneous wave does not disappear in the second medium. Thus, if $\alpha =(\alpha )_0=(arcsin \left(n\right),\ then\ )$ $E_(pr\bot )=2E_(pr\bot ).$ no case. Since the Fresnel formulas are valid for a monochromatic field, that is, for a steady process. In this case, the law of conservation of energy requires that the average change in energy over the period in the second medium be equal to zero. The wave and the corresponding fraction of energy penetrate through the interface into the second medium to a shallow depth of the order of the wavelength and move in it parallel to the interface with a phase velocity that is less than the phase velocity of the wave in the second medium. It returns to the first environment at a point that is offset from the entry point.

The penetration of the wave into the second medium can be observed in the experiment. The intensity of the light wave in the second medium is noticeable only at distances smaller than the wavelength. Near the interface on which the light wave falls, which experiences total reflection, on the side of the second medium, the glow of a thin layer can be seen if there is a fluorescent substance in the second medium.

Total reflection causes mirages to occur when the earth's surface is at a high temperature. So, the total reflection of light that comes from the clouds leads to the impression that there are puddles on the surface of the heated asphalt.

Under normal reflection, the relations $\frac(E_(otr\bot ))(E_(pad\bot ))$ and $\frac(E_(otr//))(E_(pad//))$ are always real. Under total reflection they are complex. This means that in this case the phase of the wave suffers a jump, while it is different from zero or $\pi $. If the wave is polarized perpendicular to the plane of incidence, then we can write:

where $(\delta )_(\bot )$ is the desired phase jump. Equating the real and imaginary parts, we have:

From expressions (5) we obtain:

Accordingly, for a wave that is polarized in the plane of incidence, one can obtain:

Phase jumps $(\delta )_(//)$ and $(\delta )_(\bot )$ are not the same. The reflected wave will be elliptically polarized.

Application of total reflection

Let us assume that two identical media are separated by a thin air gap. A light wave falls on it at an angle that is greater than the limit. It may happen that it will penetrate into the air gap as an inhomogeneous wave. If the gap thickness is small, then this wave will reach the second boundary of the substance and will not be very weakened. Having passed from the air gap into the substance, the wave will turn again into a homogeneous one. Such an experiment was carried out by Newton. The scientist pressed another prism, which was polished spherically, to the hypotenuse face of a rectangular prism. In this case, the light passed into the second prism not only where they touch, but also in a small ring around the contact, in the place where the gap thickness is comparable to the long wavelength. If the observations were made in white light, then the edge of the ring had a reddish color. This is as it should be, since the penetration depth is proportional to the wavelength (for red rays it is greater than for blue ones). By changing the thickness of the gap, it is possible to change the intensity of the transmitted light. This phenomenon formed the basis of the light telephone, which was patented by Zeiss. In this device, a transparent membrane acts as one of the media, which oscillates under the action of sound incident on it. Light that passes through the air gap changes intensity in time with changes in the strength of the sound. Getting on the photocell, it generates an alternating current, which changes in accordance with changes in the strength of the sound. The resulting current is amplified and used further.

The phenomena of wave penetration through thin gaps are not specific to optics. This is possible for a wave of any nature, if the phase velocity in the gap is higher than the phase velocity in the environment. This phenomenon is of great importance in nuclear and atomic physics.

The phenomenon of total internal reflection is used to change the direction of light propagation. For this purpose, prisms are used.

Example 1

Exercise: Give an example of the phenomenon of total reflection, which is often encountered.

Decision:

One can give such an example. If the highway is very hot, then the air temperature is maximum near the asphalt surface and decreases with increasing distance from the road. This means that the refractive index of air is minimal at the surface and increases with increasing distance. As a result of this, rays having a small angle with respect to the highway surface suffer total reflection. If you focus your attention, while driving in a car, on a suitable section of the surface of the highway, you can see a car going upside down quite far ahead.

Example 2

Exercise: What is the Brewster angle for a beam of light that falls on the surface of a crystal if the limiting angle of total reflection for this beam at the air-crystal interface is 400?

Decision:

\[(tg(\alpha )_b)=\frac(n)(n_v)=n\left(2.2\right).\]

From expression (2.1) we have:

We substitute the right side of expression (2.3) into formula (2.2), we express the desired angle:

\[(\alpha )_b=arctg\left(\frac(1)((sin \left((\alpha )_(pred)\right)\ ))\right).\]

Let's do the calculations:

\[(\alpha )_b=arctg\left(\frac(1)((sin \left(40()^\circ \right)\ ))\right)\approx 57()^\circ .\]

Answer:$(\alpha )_b=57()^\circ .$