Light dispersion

Electromagnetic waves can propagate not only in vacuum, but also in various media. But only in vacuum the speed of propagation of waves is constant and does not depend on frequency. In all other media, the propagation velocities of waves of different frequencies are not the same. Since the absolute refractive index depends on the speed of light in a substance (), then the dependence of the refractive index on the wavelength is experimentally observed - the dispersion of light.

The absence of light dispersion in vacuum is confirmed with great certainty by observations of astronomical objects, since interstellar space is the best approximation to vacuum. The average density of matter in interstellar space is 10 -2 atoms per 1 cm 3 , while in the best vacuum devices it is not less than 10 4 atoms per 1 cm 3 .

Convincing evidence for the absence of dispersion in space comes from studies of the eclipse of distant binary stars. The light pulse emitted by a star is not monochromatic. Suppose it consists of red and blue rays, and the red rays travel faster than the blue ones. Then, at the beginning of the eclipse, the light of the star should change from normal to blue, and when it leaves it, from red to normal. With the huge distances that light travels from a star, even an insignificant difference in the speeds of red and blue rays could not go unnoticed. Nevertheless, the results of the experiments showed that there were no changes in the spectral composition of the radiation before and after the eclipse. Arago, observing the binary star Algol, showed that the difference in the speeds of red and blue waves cannot exceed one hundred thousandth the speed of light. These and other experiments convince us that the absence of light dispersion in interstellar space should be recognized (with the accuracy that modern experiment achieves).

In all other media dispersion takes place. Media with dispersion are called dispersive. In dispersive media, the speed of light waves depends on the wavelength or frequency.

Thus, the dispersion of light is the dependence of the refractive index of a substance or the dependence of the phase velocity of light waves on frequency or wavelength. This dependence can be characterized by the function

,

(4.1)

where is the wavelength of light in vacuum.

For all transparent colorless substances, function (4.1) in the visible part of the spectrum has the form shown in Fig. 4.1. As the wavelength decreases, the refractive index increases at an ever-increasing rate. In this case, the dispersion is called normal.

If a substance absorbs part of the rays, then in the absorption region and near it, the behavior of the dispersion reveals an anomaly. Over a certain range of wavelengths, the refractive index increases with increasing wavelength. Such course of dependence on is called anomalous dispersion.

On fig. 4.2 sections 1-2 and 3-4 correspond to normal dispersion. In section 2–3, the dispersion is anomalous.

First experimental studies dispersions of light are due to Newton (1672). They were made according to the method of refraction of the sun's ray in a prism.

Rice. 4.2

A beam of light from the sun passed through a hole in the shutter and, refracted in a prism, gave an image on a sheet of white paper. In this case, the image of a round hole was stretched into a colored strip from red to purple. In his Optics, Newton described his research as follows: I placed in a very dark room at a round hole about a third of an inch wide in the window shutter a glass prism, whereby the beam of sunlight entering through this hole could be refracted upwards to the opposite wall of the room and form there a color image of the sun ... A spectacle of vivid and bright colors, The result was a very pleasant experience for me.».

Newton called the color band resulting from the refraction of light in a prism a spectrum. In the spectrum, seven main colors are conditionally distinguished, gradually passing from one to another, occupying sections of various sizes in it (Fig. 4.3).

Rice. 4.3

This is due to the fact that the colored rays that make up white light are refracted differently by a prism. The red part of the spectrum has the smallest deviation from the original direction, the violet part has the largest, therefore, the lowest refractive index is for red rays, the largest for violet, that is, light with different wavelengths propagates in a medium with different speeds: violet - with the lowest, red - with the most.

The color rays of the spectrum emerging from the prism can be collected by a lens or a second prism and a spot of white light can be obtained on the screen. If, however, a colored beam of rays of any one color, for example, red, is selected from the spectrum and passed through a second prism, then the beam will deviate due to refraction, but no longer decomposing into composite tones and without changing colors. It follows that the prism does not change the white light, but decomposes it into its component parts. Beams of various colors can be distinguished from white light, and only their combined action gives us the feeling of white light.

Newton's method is still good method research and demonstration of variance. When comparing the spectra obtained using prisms with equal refractive angles, but from different substances, one can see the difference in the spectra, which consists not only in the fact that the spectra are deflected at a different angle due to a different refractive index for the same wavelength, but they are also stretched unequally due to different dispersion, that is, different dependence of the refractive index on the wavelength.

Rice. 4.4

An illustrative method that makes it possible to investigate the dispersion in prisms of various materials is the method of crossed prisms, which was also first used by Newton. In this method, light passes successively through two prisms. R 1 And R 2, whose refracting edges are perpendicular to each other (Fig. 4.4). With lenses L1 And L2 light is collected on screen AB. If there was only one prism R 1, then a colored horizontal stripe would appear on the screen. In the presence of a second prism, each beam will be deflected downward and the stronger, the greater its refractive index in the prism R 2. The result is a curved strip. The red end will be shifted the least, the purple end the most. The entire strip will visually represent the course of dispersion in the prism R 2.

On fig. Figure 4.5 shows the refraction of white light at a flat interface between a vacuum and a transparent substance with a very high refractive index. For clarity, the spectrum resulting from the dispersion is represented by separate rays corresponding to the primary colors of the spectrum. The calculation allows you to see which of the rays will deviate to large, and which - to smaller angles.

Rice. 4.5

In 1860, the French physicist Leroux, while measuring the refractive index for a number of substances, unexpectedly discovered that iodine vapor refracts blue rays to a lesser extent than red ones. Leroux called the phenomenon he discovered anomalous dispersion of light. If with normal dispersion the refractive index decreases with increasing wavelength, then with anomalous dispersion the refractive index, on the contrary, increases. The phenomenon of anomalous dispersion was studied in detail by the German physicist Kundt in 1871–1872. At the same time, Kundt used the method of crossed prisms, which was proposed by Newton in his time.

Systematic experimental studies of anomalous dispersion by Kundt showed that the phenomenon of anomalous dispersion is associated with absorption, that is, an anomalous course of dispersion is observed in the wavelength region in which light is strongly absorbed by matter.

The anomalous dispersion is most clearly observed in gases (vapours) with sharp absorption lines. All substances absorb light, however, for transparent substances, the absorption region, and hence the region of anomalous dispersion, lies not in the visible, but in the ultraviolet or infrared region.

According to the electromagnetic theory of light, the phase velocity of an electromagnetic wave is related to the speed of light in vacuum by the relation

where is the permittivity and is the magnetic permeability. In the optical region of the spectrum for all substances it is very close to 1. Therefore, the refractive index of the substance will be equal to

and hence the dispersion of light is explained as a function of frequency. This dependence is related to the interaction electromagnetic field light wave with atoms and molecules of matter.

From the classical point of view, the dispersion of light arises as a result of forced oscillations of charged particles - electrons and ions - under the action of an alternating field of an electromagnetic wave. The alternating field of an electromagnetic wave periodically accelerates numerous microscopic charges of matter. Charges accelerated by the field lose their excess energy in two ways. Firstly, they transfer energy to the medium, and secondly, like any accelerated charges, they radiate new waves. In the first case, radiation is absorbed, and in the second, radiation propagates in the medium due to continuous absorption and re-emission electromagnetic waves charges of matter.

All electrons entering an atom can be divided into peripheral, or optical, and electrons of inner shells. Only optical electrons affect the emission and absorption of light. The natural frequencies of the electrons in the inner shells are too high, so that their oscillations are practically not excited by the field of the light wave. Therefore, in the theory of dispersion, one can confine oneself to consideration of optical electrons alone.

The dispersion of light in matter is explained by the fact that optical electrons in atoms perform forced oscillations with the frequency of the incident waves under the action of the electric field of electromagnetic waves. Oscillating electrons emit secondary electromagnetic waves of the same frequency. These waves, adding up with the incoming wave, form the resulting wave propagating in the medium, which propagates in the medium with a phase velocity different from the speed of light in vacuum.

The wave behaves in a special way in the region of frequencies close to the natural frequency of electron oscillations. In this case, the phenomenon of resonance takes place, as a result of which the phase shift of the primary wave and secondary waves is equal to zero, the amplitude of the forced oscillations of electrons increases sharply, and a significant absorption of the energy of the incident waves by the medium is observed.

Far from resonance, the phase velocity decreases with increasing frequency, and the refractive index increases, and hence normal dispersion is observed. In the frequency range close to natural oscillations of optical electrons, the phase velocity increases with increasing frequency, and the refractive index decreases, that is, anomalous dispersion is observed.

Rice. 4.6

Dispersion of light in a prism. Consider the dispersion of light in a prism. Let a monochromatic beam of light fall on a prism with a refractive angle BUT and refractive index n. After a double refraction on the faces of the prism, the beam deviates from the original direction by an angle (Fig. 4.6). From fig. 4.6 shows that . Since then . If the angle of incidence of the beam on the left side is small and the refractive angle of the prism is also small, then the angles will also be small. Then, writing the law of refraction for each face of the prism, you can use their value instead of the sines of the angles, therefore, . It follows that the refractive angle of the prism , and the angle of deflection of the rays by the prism.

Since the refractive index depends on the wavelength, the rays of different wavelengths after passing through the prism will deviate to different angles, which was observed by Newton.

By decomposing light into a spectrum using a prism, one can determine its spectral composition, just as with a diffraction grating. The colors in the spectra obtained with a prism and with a diffraction grating are located differently. The diffraction grating, as follows from the condition for the main maximum, deflects rays with a longer wavelength more strongly. A prism, on the other hand, decomposes light into a spectrum in accordance with the refractive index, which in the region of normal dispersion decreases with increasing wavelength. Therefore, red rays are deflected by the prism less than violet ones.

A schematic diagram of the simplest spectral device, the operation of which is based on the phenomenon of dispersion, is shown in fig. 4.7. Radiation source S is in the focal plane of the lens. A parallel beam of light exiting the lens is incident on a prism. Due to the dispersion of light in the substance of the prism, rays corresponding to different wavelengths exit the prism at different angles. In the focal plane of the lens there is a screen on which the spectrum of the incident radiation is displayed.

This is interesting!

Rainbow

Rainbow

A rainbow is a beautiful celestial phenomenon that occurs during rain - has always attracted the attention of man. The rainbow has seven primary colors that smoothly transition from one to another. The shape of the arc, the brightness of the colors, the width of the stripes depend on the size of the water droplets and their number.

The rainbow theory was first given in 1637 by Rene Descartes. He explained the appearance of the rainbow by the reflection and refraction of light in raindrops. The formation of colors and their sequence were explained later, after unraveling the complex nature of white light and its dispersion in a medium. Getting inside the drop, the sun's ray is refracted and, due to dispersion, decomposes into a spectrum; the colored rays of the solar radiation spectrum reflected from the rear hemisphere of the drop exit back through the front surface of the drop. Therefore, you can see a rainbow only when the Sun is on one side of the observer, and the rain is on the other side.

Due to dispersion, each color in the reflected rays gathers at its own angle, so the rainbow forms an arc in the sky. The colors in the rain rainbow are not very clearly separated, since the drops have different diameters, and on some drops the dispersion is more pronounced, on others it is weaker. Large drops create a narrower rainbow, with sharply prominent colors, small drops create an arc that is vague and dim. Therefore, in summer, after a thunderstorm, during which large drops fall, a particularly bright and narrow rainbow is visible.

Halo

Halo

Halo is a group of optical phenomena in the atmosphere. They arise due to the refraction and reflection of light by ice crystals that form cirrus clouds and fogs. The term comes from the French halo and the Greek halos, a ring of light around the sun or moon. The halo usually appears around the Sun or Moon, sometimes around other powerful light sources such as street lights. The manifestations of the halo are very diverse: in the case of refraction, they look like iridescent stripes, spots, arcs and circles on the vault of heaven, and when reflected, the stripes are white.

The shape of the observed halo depends on the shape and location of the crystals. The light refracted by ice crystals decomposes into a spectrum due to dispersion, which makes the halo look like a rainbow.

The halo should be distinguished from the crowns, which are outwardly similar to it, but have a different, diffractive, origin.

green beam

green beam

A green beam is a rare optical phenomenon, which is a flash of green light at the moment the solar disk disappears under the horizon or appears from behind the horizon. To observe the green beam, three conditions are necessary: ​​an open horizon (in the steppe or at sea in the absence of waves), clean air and a cloud-free side of the horizon where the sunset or sunrise occurs. The normal duration of the green beam is only a few seconds. The reason for this phenomenon is the refraction (refraction) of sunlight in the atmosphere, accompanied by their dispersion, that is, decomposition into a spectrum.

Refraction of light in the atmosphere is an optical phenomenon caused by the refraction of light rays in the atmosphere and manifests itself in the apparent displacement of distant objects, and sometimes in the apparent change in their shape. Some manifestations of refraction, for example, the oblate shape of the disks of the Sun and Moon near the horizon, the twinkling of stars, the trembling of distant earthly objects on a hot day, were already noticed in antiquity. The reason for this is that the atmosphere is an optically inhomogeneous medium, the rays of light propagate in it not in a straight line, but along a certain curved line. Therefore, the observer sees objects not in the direction of their actual position, but along a tangent to the ray path at the point of observation. In this case, the power of refraction depends on the wavelength of the beam: the shorter the wavelength of the beam, the more it will rise due to refraction. Due to the difference in refraction for rays with different wavelengths, especially large near the horizon, a colored border can be observed near the disk of the rising or setting Sun (blue-green above, red below). This explains the phenomenon of the green ray.

The red and orange parts of the Sun's disk set below the horizon before the green and blue parts. The dispersion of the sun's rays manifests itself most clearly at the very last moment of sunset, when a small upper segment remains above the horizon, and then only the very top of the solar disk. When the Sun plunges below the horizon, the last ray we should see is purple. However, the shortest-wavelength rays - violet, blue, blue - are scattered so strongly that they do not reach the earth's surface. In addition, human eyes are less sensitive to the rays of this part of the spectrum. Therefore, at the last moment of sunset, there is a rapid change of colors from red through orange and yellow to green, and the last ray of the setting Sun turns out to be a bright emerald color. This phenomenon is called the green beam.

At sunrise, the reverse color change takes place. The first ray of the rising Sun - green - is replaced by yellow, orange, and, finally, the red edge of the rising luminary is shown from behind the horizon.

light absorption

When electromagnetic waves pass through matter, part of the wave energy is spent on excitation of electron oscillations in atoms and molecules. In an ideal homogeneous medium, periodically oscillating dipoles radiate coherent secondary electromagnetic waves of the same frequency and, at the same time, completely give up the absorbed fraction of energy. The corresponding calculation shows that, as a result of interference, the secondary waves completely cancel each other in all directions, except for the direction of propagation of the primary wave, and change its phase velocity. Therefore, in the case of an ideal homogeneous medium, light absorption and redistribution of light in directions, that is, light scattering, does not occur.

In a real substance, not all the energy of oscillating electrons is emitted back in the form of an electromagnetic wave, but part of it goes into other forms of energy and, mainly, into heat. Excited atoms and molecules interact and collide with each other. During these collisions, the energy of oscillations of electrons inside atoms can be converted into the energy of external chaotic motions of atoms as a whole. In metals, an electromagnetic wave sets free electrons in oscillatory motion, which then, during collisions, give off the accumulated excess energy to the ions of the crystal lattice and thereby heat it. In some cases, the energy absorbed by a molecule can be concentrated on a specific chemical bond and completely spent on breaking it. These are the so-called photochemical reactions, that is, reactions that occur due to the energy of a light wave.

Therefore, the intensity of light when passing through ordinary matter decreases - light is absorbed in matter. The absorption of light can be described from an energetic point of view.

Consider a wide beam of parallel rays propagating in an absorbing medium (Fig. 4.8). Let us denote the initial intensity of the radiant flux in the plane as . Having passed the path z in the medium, the radiant beam is attenuated as a result of light absorption, and its intensity becomes less.

Let's select in the medium a section with thickness . The intensity of the light that has traveled a path equal to will be less than , that is, . The quantity represents the decrease in the intensity of the incident radiation due to absorption in the area . This value is proportional to the thickness of the area and the intensity of the light incident on this area, that is, where is the absorption coefficient, which depends both on the nature of the substance (its chemical composition, state of aggregation, concentration, temperature) and on the wavelength of the light interacting with the substance . The function that determines the dependence of the absorption coefficient on the wavelength is called the absorption spectrum.

Expression for the intensity of light passing through a medium of a certain thickness z, is called Bouguer's law:

where is the light intensity at , is the base of the natural logarithm.

For all substances, absorption is selective. For liquid and solid substances, the dependence has a form similar to that shown in Fig. 4.9. In this case, strong absorption is observed in a wide range of wavelengths. The presence of such absorption bands underlies the action of light filters - plates containing additives of salts or organic dyes. The filter is transparent to those wavelengths that it does not absorb.

Metals are practically opaque to light. This is due to the presence of free electrons in them, which, under the action of the electric field of a light wave, begin to move. According to the Joule–Lenz law, the rapidly alternating currents that arise in the metal are accompanied by heat release. As a result, the energy of the light wave rapidly decreases, turning into internal energy metal.

Rice. 4.10

In the case of gases or vapors at low pressure, only for very narrow spectral intervals (Fig. 4.10). In this case, the atoms practically do not interact with each other, and the maxima correspond to the resonant frequencies of electron oscillations inside the atoms. Inside the absorption band, anomalous dispersion is observed, that is, the refractive index decreases with decreasing wavelength.

In the case of polyatomic molecules, absorption is also possible at frequencies corresponding to vibrations of atoms inside molecules. But since the masses of atoms are tens of thousands of times greater than the mass of electrons, these frequencies correspond to the infrared region of the spectrum. Therefore, many substances that are transparent to visible light have absorption in the ultraviolet and infrared regions of the spectrum. So, ordinary glass absorbs ultraviolet rays and infrared rays with high frequencies. Quartz glasses are transparent to ultraviolet rays.

The selective absorption of glass or polyethylene film is due to the so-called greenhouse effect: infrared radiation emitted by the heated earth is absorbed by the glass or film and, therefore, is retained inside the greenhouse.

Biological tissues and some organic molecules strongly absorb ultraviolet radiation, which is detrimental to them. The living nature on Earth is protected from ultraviolet radiation by the ozone layer in the upper atmosphere, which intensively absorbs ultraviolet radiation. That is why humanity is so concerned about the appearance of the ozone hole in the South Pole.

Rice. 4.12

The dependence of the absorption coefficient on the wavelength is explained by the coloration of the absorbing bodies. So, rose petals (Fig. 4.11) when illuminating it sunlight weakly absorb red rays and strongly absorb rays corresponding to other lengths of the solar spectrum, so the rose is red. The petals of the white orchid (Figure 4.12) reflect all wavelengths of the solar spectrum. And the leaves of both flowers are green, which means that from the entire range of waves they reflect mainly the waves of the green part of the spectrum, and the rest absorb.

light scattering

From the classical point of view, the process of light scattering consists in the fact that light, passing through a substance, excites vibrations of electrons in atoms. The oscillating electrons become sources of secondary waves. The secondary waves are coherent and therefore must interfere. In the case of a homogeneous medium, the secondary waves cancel each other in all directions, except for the direction of propagation of the primary wave. Therefore, there is no scattering of light, that is, its redistribution in different directions. In the direction of the primary wave, the secondary waves, interfering with the primary wave, form the resulting wave, the phase velocity of which is different from the speed of light in vacuum. This explains the dispersion of light.

Rice. 4.13

Consequently, light scattering occurs only in an inhomogeneous medium. Such media are called turbid. Smokes (suspensions of tiny particles in gases) can be examples of turbid media; fogs (suspensions of liquid droplets in gases); suspensions formed by small solid particles floating in a liquid; emulsions, that is, suspensions of particles of one liquid in another (for example, milk is a suspension of fat droplets in water).

If the inhomogeneities were arranged in a certain order, then during the propagation of the wave, a diffraction pattern would be obtained with its characteristic alternation of intensity maxima and minima. However, most often their coordinates are not only random, but also change over time. Therefore, the secondary radiation arising from inhomogeneities gives a fairly uniform intensity distribution in all directions. This phenomenon is called light scattering. As a result of scattering, the energy of the primary beam of light gradually decreases, as in the case of the transition of the energy of excited atoms into other forms of energy. So the light of a street lamp in the fog does not propagate in a straight line, but is scattered in all directions, and its intensity decreases rapidly with distance from the lamp, both due to absorption and scattering (Fig. 4.13)

Rayleigh's law. Scattering of light in turbid media by inhomogeneities whose dimensions are small compared to the wavelength can be observed, for example, when sunlight passes through a vessel with water to which a little milk is added. When viewed from the side in scattered light, the medium appears blue, that is, the scattered radiation is dominated by waves corresponding to the short-wavelength part of the solar radiation spectrum. The light that has passed through a thick layer of a turbid medium appears reddish.

This can be explained by the fact that electrons performing forced oscillations in atoms are equivalent to a dipole, which oscillates with the frequency of the light wave incident on it. The intensity of the light it emits is proportional to the fourth power of the frequency, or inversely proportional to the fourth power of the wavelength:

This statement is the content of Rzley's law.

It follows from Rayleigh's law that the short-wavelength part of the spectrum is scattered much more strongly than the long-wavelength part. Since the frequency of blue light is about 1.5 times greater than that of red, it scatters 5 times more intensely than red. This explains the blue color of the scattered light and the red light of the past.

Electrons that are not bound in atoms, but free - for example, in plasma - also sway with light and scatter it to the sides. In particular, it is due to this effect that we can observe the glow of the solar corona and, therefore, obtain information about the solar stratosphere.

Molecular scattering. Even liquids and gases purified from impurities scatter light. The role of optical inhomogeneities in this case is played by density fluctuations. Density fluctuations are understood as density deviations within small volumes from its average value, arising in the process of chaotic thermal motion of medium molecules. Scattering of light due to density fluctuations is called molecular scattering

Rice. 4.14

Rice. 4.15

That's why the sky looks blue and the Sun yellowish! Enjoying the sight of a cloudless sky, we are hardly inclined to remember that the blue of the sky is one of the manifestations of light scattering. The continuous density fluctuations in the atmosphere, in accordance with Rayleigh's law, cause the blue and blue components of sunlight to scatter more strongly than the yellow and red ones. When we look at the sky, we see scattered sunlight there, where the short waves of the blue part of the spectrum predominate (Fig. 4.14). When you look at the Sun, we observe the spectrum of its radiation, from which, due to scattering, part of the blue rays has been removed. This effect is especially well manifested at a low position of the Sun above the horizon. Well, who has not admired the bright red rising or setting Sun! At sunset, when the sun's rays make a much longer journey through the atmosphere, the Sun seems to us especially red, because in this case, not only blue, but also green and yellow rays scatter and disappear from its spectrum (Fig. 4.15).

This is interesting!

blue sun

How often do you see "blue sun" in fantasy novels! Is such a phenomenon possible?

We have already found out that due to Rayleigh scattering in the atmosphere, the Sun should be reddish. However, Rayleigh scattering takes place only when the wavelength of the light passing through the medium is much larger than the inhomogeneities on which the scattering occurs. In the case of larger particles, scattering is practically independent of the wavelength of light. That is why fog, clouds are white, and on a hot day with high humidity, the sky turns from blue to whitish.

It turns out that the Sun can also sometimes, very rarely, be seen blue. In September 1950, such a phenomenon was observed over the North American continent. The sky over southern Canada, over Ontario and other great lakes, over the east coast of the United States on a clear cloudless day took on a reddish-brown tint. And a hazy blue Sun shone in the sky! And at night the blue moon rose into the sky.

However, nothing mystical actually happened. It is connected with optical effects in the earth's atmosphere. If there are many particles in the atmosphere about a micron (millionth of a meter) in size, then the air begins to play the role of a blue filter. It doesn't matter what kind of particles they are: water droplets, ice crystals, particles of smoke from a burning forest, volcanic ash, or just wind-blown dust. It is important that they are the same, micron size.

The reason for the blue sun over Canada was that peat bogs had been smoldering in Alberta for many years. Suddenly, the fire broke out and became extremely intensified. A strong wind carried the products of combustion to the south, covering vast areas. During the fire, a large number of oil droplets arose, which hung in the atmosphere for more than one day. They are guilty of an unusual celestial phenomenon. If the dimensions of the scattering particles are close to the wavelength of the incident light, a resonance occurs, and the scattering at this wavelength increases sharply. In the autumn of 1950, the size of the droplets was just about the wavelength of red-orange light. That is why the sky turned from blue to red, and the Moon and the Sun turned from reddish to blue.

Similar strange optical phenomena were observed in the 19th century. after the eruption of the Krakatoa volcano. So blue moon and the Sun is a very rare phenomenon, but not unique, and even more so not impossible.

light and color

The world around us is always full of various colors. How does this color richness come about? Why is each substance a different color? Emerald green meadows, golden dandelion flowers, bright plumage of birds, butterfly wings, drawings and illustrations - all this is created by the peculiarities of the interaction of light with matter and human color vision. The objects around us, being illuminated by the same white sunlight, appear to our eyes to be differently colored.

Falling on an illuminated object, the wave is usually divided into three parts: one part is reflected from the surface of the object and scattered in space, the other part is absorbed by the substance, and the third part passes through it.

Rice. 4.16

Rice. 4.17

If the reflected and transmitted components are absent, that is, the substance absorbs the radiation that has fallen on it, then the observer's eye will not perceive anything, and the substance in question will look black. In the absence of a passed component, it will be opaque. It is clear that in this case the color of the substance is determined by the balance between the absorption and reflection of the rays incident on it. For example, a blue cornflower absorbs red and yellow rays, and reflects blue - this is the reason for its color. Sunflower flowers are yellow, which means that from the entire wavelength range they mainly reflect the waves of the yellow part of the spectrum, and absorb the rest.

The top of the apple shown in Fig. 4.16 is red. This means that it reflects the wavelengths corresponding to the wavelength of the red part of the spectrum. The lower part of the apple is not illuminated, and therefore its surface appears black. But the apple in Fig. 4.17, illuminated by light with the same spectral composition, reflects the green part of the spectrum, so we see it as green.

Thus, if we say that an object has some color, this means that the surface of this object has the property of reflecting waves of a certain length, and the reflected light is perceived as the color of the object. If an object completely absorbs the incident light, it will appear black to us, and if it reflects all the incident rays, it will appear white. True, the last statement will be true only if the incident light is white. If the incident light acquires a certain shade, then the reflecting surface will also have the same shade. This can be observed in the setting sun, which makes everything around crimson (Fig. 4.18), or on a twilight winter evening, when the snow looks blue (Fig. 4.19).

And how will the color of a substance change if we replace solar radiation, for example, with the radiation of an ordinary electric light bulb?

In the spectrum of an incandescent lamp, compared to the solar spectrum, the proportion of yellow and red rays is noticeably larger. Therefore, in reflected light, their proportion will also increase in comparison with what is obtained with sunshine. This means that objects illuminated by a light bulb will look “yellow” than in sunlight. The leaf of the plant will already turn yellow-green, and the blue cornflower will turn blue-green or even completely green.

Thus, the concept of "substance color" is not absolute, the color depends on the illumination. Therefore, reports about the ability of some people to recognize the color of an object placed in an opaque cassette are meaningless. The concept of color in the dark is meaningless.

The mechanism of color formation is subject to very specific laws, which were discovered relatively recently - about 150 years ago. The dispersion of light causes when white light passes through a prism, it is decomposed into seven primary spectral colors - red, orange, yellow, green, cyan, indigo. Conversely, if you mix the colors of the spectrum, you get a beam of white light. The seven primary spectral colors make up that rather narrow range of electromagnetic waves (from about 400 to 700 nanometers) that our eye can capture, but even these three hundred nanometers are enough to give rise to the color variety of the world around us.

Light waves enter the retina of the eye, where they are perceived by light-sensitive receptors that transmit signals to the brain, and already there a sensation of color is formed. This sensation depends on the wavelength and intensity of the radiation. The wavelength forms the sensation of color, and the intensity - its brightness. Each color corresponds to a certain range of wavelengths.

Rice. 4.20. Formation of a shade from three basic colors

The most important law of color creation is the law of three-dimensionality, which states that any color can be created by three linearly independent colors. The most striking practical use of this law is color television. The entire plane of the screen is a tiny cell, each of which has three beams - red, green and blue. The color of the image on the screen is formed using these three independent colors. This principle of color synthesis is also used in scanners and digital cameras. The mechanism of color formation is shown in fig. 4.20.

The colors with which a color image is reproduced are called primary colors. The most varied combinations of three independent colors can be chosen as primary colors. However, in accordance with the spectral sensitivity of the eye, either blue, green and red, or yellow, magenta and cyan are most often accepted as primary colors. Colors that, when mixed, produce white are called complementary colors. In a mixed color, we cannot see its individual components.

Rice. 4.21

You can experimentally observe the effect of color mixing using Newton's disk. Newton's color disc is a glass disc divided into sectors, which are colored in different colors (from red to purple) (Fig. 4.21).

We will rotate the disk around its axis. As the rotation speed increases, we will notice that the boundaries between the sectors are blurred, the colors become mixed and faded. And at a certain speed of disk rotation, our eyes perceive the light passing through it as white, that is, they cease to distinguish colors.

It can be explained like this. Receptors are located on the retina of the eye, which perceive light signals. Let the eye first perceive, for example, the color blue. In this case, the receptors are in the corresponding excited state. Turn off the blue light. The receptors will go into the ground state in a certain time interval. The color sensation will disappear. If we now turn on, for example, red light, then the receptors will perceive it as one color. If blue and red light alternate after a very short time interval, then the receptors will perceive these colors simultaneously. Therefore, by rotating Newton's disk at a speed at which the eye ceases to distinguish individual colors of the sectors, we "force" the eye to sum up all these colors, and we see white light.

Thus, with the joint action of two or more light waves of different frequencies corresponding to different colors on the eye, a qualitatively new subjectively perceived color is obtained. The sensation of color is formed in the human brain, where the signal from the eye goes. Light enters the eye, penetrating through the cornea and pupil, "registering" on the retina, on which the nerve cells are located. Upon receiving a signal, neurons send electrical impulses to the brain, where information about the proportions and intensity of the primary colors forms a full-color picture of the world with a huge number of shades.

POLARIZATION OF LIGHT

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3.2.6 Dispersion of electromagnetic waves. Refractive index of air

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Monochromatic waves with different frequencies (wavelengths) propagate in the environment, strictly speaking, at different speeds. The dependence of the speed of electromagnetic waves on frequency is called dispersion .

Speed ​​of electromagnetic waves in a real environment is related to the speed of light in vacuum through one of the most important characteristics of the medium - the refractive index :

(3.30)

The refractive index in electrodynamics is determined from the relationship

(3.31)

where is the permittivity of the medium;

is the magnetic permeability of the medium.

Based on the foregoing, we can say that the dispersion of light is the phenomenon caused by the dependence of the refractive index of a substance from the wavelength

(4.30)

For radio waves, the lower layer of the atmosphere, up to about 11 km, is a non-dispersive medium. For the optical and VHF bands, the atmosphere is a dispersive medium.

For most transparent substances, the refractive index increases with increasing wavelength. This type of dispersion is called normal .

The dependence on in the region of normal dispersion is described by the Cauchy formula

(4.31)

where , , are constant coefficients that are found experimentally for each substance.

If a substance absorbs part of the light flux, then anomalous dispersion can be observed in the absorption region, i.e. decrease in the refractive index with decreasing wavelength.

In transparent media, as a result of a change in the direction of light propagation during refraction, the dispersion of light leads to the decomposition of light into a spectrum. Experience shows that if a beam of white light is passed through a refracting prism - a transparent body bounded by flat intersecting surfaces, then on the screen behind the prism we get a colored strip in the following sequence of colors: red, orange, yellow, green, blue, indigo, violet.

The nature of the dispersion for different transparent media, including different types of glass, is different.

For waves of the ultrashort and light ranges, the refractive index depends on the meteorological parameters of the atmosphere: temperaturet, pressure Pand air humiditye. In combination with the above dependence of the refractive index on the wavelength or frequency , in general, the dependence of the refractive index on the specified parameters can be written as

. (4.31)

In this regard, to determine the refractive index or, what is the same, the propagation velocity of an electromagnetic wave with a wavelength , it is necessary to determine the temperature, pressure and humidity of the air. The last parameter affects the speed of EMW propagation in the optical range to a much lesser extent than temperature and pressure. Therefore, the main determinable parameters for rangefinders operating on the waves of the optical range are only temperature and pressure.

All modern rangefinders provide for the input of a correction for atmospheric parameters. The formulas by which the indicated correction is calculated are hardwired into the instrument software.

(For independent study: Bolshakov V.D., Deimlikh F., Golubev A.N., Vasiliev V.P. Radio geodetic and electro-optical measurements. - M .: Nedra, 1985. - 303 p. - Paragraph 8. The speed of propagation of electromagnetic waves, pp. 68-78).

Bibliography

1. V. D. Bol’shakov, F. Deimlikh, A. N. Golubev, and V. P. Vasiliev, Russ. Radio geodetic and electro-optical measurements. - M.: Nedra, 1985. - 303 p.

2. Gorelik G.S. Vibrations and waves. Introduction to acoustics, radiophysics and optics. – M.: Ed. Phys.-Math. liters. 1959. - 572 p.

3. Detlaf A.A., Yavorsky B.M. Physics course. Volume 3. Wave processes. Optics. Atomic and nuclear physics. – M.: high school. 1979. - 511 p.

4. Zisman G.A., Todes O.M. Course of general physics. T. III .. Optics. Physics of atoms and molecules. Physics of the atomic nucleus and microparticles - M.: Nauka. 1970 - 495 p.

5. Landsberg G.S. Elementary textbook of physics. Volume III. Vibrations, waves. Optics. The structure of the atom. – M.: Science. 1970 - 640 p.

6. Schroeder G., Treiber H. Technical optics. – M.: Technosfera, 2006. – 424 p.

Today, quantitative knowledge of the electronic structure of atoms and molecules, as well as solids built from them, is based on experimental studies of optical reflection, absorption, and transmission spectra and their quantum mechanical interpretation. The band structure and defectiveness of various types of solids (semiconductors, metals, ionic and atomic crystals, amorphous materials) are being studied very intensively. Comparison of the data obtained in the course of these studies with theoretical calculations made it possible to reliably determine for a number of substances the features of the structure of energy bands and the values ​​of interband gaps (band gap E g) in the vicinity of the main points and directions of the first Brillouin zone. These results, in turn, make it possible to reliably interpret such macroscopic properties of solids as electrical conductivity and its temperature dependence, the refractive index and its dispersion, the color of crystals, glasses, ceramics, glass-ceramics and its variation under radiation and thermal effects.

2.4.2.1. Dispersion of electromagnetic waves, refractive index

Dispersion is a phenomenon of the relationship between the refractive index of a substance, and, consequently, the phase velocity of wave propagation, with the wavelength (or frequency) of radiation. Thus, the transmission of visible light through a glass trihedral prism is accompanied by decomposition into a spectrum, and the violet short-wavelength part of the radiation is deviated most strongly (Fig. 2.4.2).

The dispersion is called normal if, as the frequency n(w) increases, the refractive index n also increases dn/dn>0 (or dn/dl<0). Такой характер зависимости n от n наблюдается в тех областях спектра, где среда прозрачна для излучения. Например, силикатное стекло прозрачно для видимого света и обладает в этом интервале частот нормальной дисперсией.

The dispersion is called anomalous if, with increasing radiation frequency, the refractive index of the medium decreases (dn/dn<0 или dn/dl>0). Anomalous dispersion corresponds to frequencies corresponding to optical absorption bands; the physical content of the absorption phenomenon will be briefly discussed below. For example, for sodium silicate glass, the absorption bands correspond to the ultraviolet and infrared regions of the spectrum, quartz glass in the ultraviolet and visible parts of the spectrum has normal dispersion, and in the infrared - anomalous.

Rice. 2.4.2. Dispersion of light in glass: a - decomposition of light by a glass prism, b - graphs n = n (n) and n = n (l 0) for normal dispersion, c - in the presence of normal and anomalous dispersion In the visible and infrared parts of the spectrum, normal dispersion is characteristic for many alkali-halide crystals, which determines their wide use in optical devices for the infrared part of the spectrum.

The physical nature of normal and anomalous dispersion of electromagnetic waves becomes clear if we consider this phenomenon from the standpoint of classical electron theory. Let us consider a simple case of normal incidence of a plane electromagnetic wave of the optical range on a flat boundary of a homogeneous dielectric. The electrons of a substance associated with atoms under the action of an alternating field of a wave with strength perform forced oscillations with the same circular frequency w, but with a phase j that differs from the phase of the waves. Taking into account the possible attenuation of the wave in a medium with a natural frequency of electron oscillations w 0 , the equation of forced transverse oscillations in the direction - the direction of propagation of a plane polarized wave - has the form

(2.4.13)

known from the course of general physics (q and m - the charge and mass of the electron).

For the optical region, w 0 » 10 15 s -1 , and the attenuation coefficient g can be determined in an ideal medium under the condition of a non-relativistic electron velocity (u<

(2.4.14)

At w 0 = 10 15 s -1 the value g » 10 7 s -1 . Neglecting the relatively short stage of unsteady oscillations, let us consider a particular solution of the inhomogeneous equation (2.4.13) at the stage of steady oscillations. We are looking for a solution in the form

(2.4.15)

Then from equation (2.4.13) we obtain

or , where the oscillation amplitude is equal to

(2.4.16)

here

Then the solution for the coordinate (2.4.15) can be rewritten as

(2.4.17)

Thus, the forced harmonic oscillations of an electron occur with amplitude A and are ahead in phase of the oscillations in the incident wave by an angle j. Near the resonance value w = w 0 , the dependence of A and j on w/w 0 is of special interest.

On fig. 2.4.3 shows the graphs of the dependences of the amplitude and phase near the resonant frequency.

Rice. 2.4.3. Graphs of the amplitude (a) and phase (b) of electron oscillations near the resonant frequency (for g » 0.1w 0)

In real cases, usually g is less than g » 0.1 w 0 , chosen for clarity in Fig. 2.4.3, the amplitude and phase change more sharply. If the light incident on the dielectric is not monochromatic, then near the resonance, at frequencies w®w 0, it is absorbed, the electrons of the substance dissipate this energy in the volume. This is how absorption bands appear in the spectra. The line width of the absorption spectrum is determined by the formula

It follows from Maxwell's macroscopic electromagnetic theory that the absolute refractive index of the medium

where is the dielectric constant of the medium, is the magnetic permeability. In the optical region of the spectrum for all substances 1, therefore

From this formula, some contradictions with experience are revealed: the value n, being a variable, remains at the same time equal to a certain constant - . In addition, the values ​​of n obtained from this expression do not agree with the experimental values. Difficulties in explaining the dispersion of light from the point of view of Maxwell's electromagnetic theory are eliminated by Lorentz's electron theory. In Lorentz's theory, the dispersion of light is considered as the result of the interaction of electromagnetic waves with charged particles that are part of the substance and perform forced oscillations in the alternating electromagnetic field of the wave.

Let us apply the electronic theory of light dispersion for a homogeneous dielectric, assuming formally that the dispersion of light is a consequence of the dependence on the frequency of light waves. The permittivity of a substance is

where w is the dielectric susceptibility of the medium, 0 is the electrical constant, P is the instantaneous value of the polarization. Consequently,

i.e., it depends on R. In this case, the electron polarization is of primary importance, i.e., forced oscillations of electrons under the action of the electric component of the wave field, since for the orientational polarization of molecules, the frequency of oscillations in a light wave is very high (v 10 15 Hz) .

In the first approximation, we can assume that forced oscillations are performed only by external electrons, which are most weakly connected with the nucleus - optical electrons. For simplicity, let us consider oscillations of only one optical electron. The induced dipole moment of an electron performing forced oscillations is p = ex, where e is the charge of the electron, x is the displacement of the electron under the action of the electric field of the light wave. If the concentration of atoms in the dielectric is n 0 then the instantaneous value of the polarization

Consequently, the problem is reduced to determining the displacement x of an electron under the action of an external field E. The field of a light wave will be considered a function of the frequency co, i.e., changing according to the harmonic law: E = E 0 cost.

The equation of forced oscillations of an electron for the simplest case (without taking into account the resistance force that determines the absorption of the energy of the incident wave) can be written as

where F 0 = eE 0 is the amplitude value of the force acting on the electron from the wave field, is the natural oscillation frequency of the electron, m is the mass of the electron. Having solved the equation, we find = n 2 depending on the constants of the atom (e, m, 0) and the frequency of the external field, i.e. we will solve the dispersion problem. The solution of the equation can be written as

If there are different charges eh in the substance that perform forced oscillations with different natural frequencies ea0|, then

where m 1 is the mass of the i-th charge.

It follows from the expressions and that the refractive index n depends on the frequency of the external field, i.e., the dependences obtained really confirm the phenomenon of light dispersion, although under the above assumptions, which must be eliminated in the future. It follows from the expressions and that in the region from = 0 to = 0 n 2 is greater than one and increases with increasing (normal dispersion); at = 0 n 2 = ± ; in the region from = 0 to = n 2 is less than one and increases from - to 1 (normal dispersion). Passing from n 2 to n, we obtain that the dependence of n on has the form shown in Fig. 3.

This behavior of n near 0 is the result of the assumption that there are no resistance forces during electron oscillations. If this circumstance is also taken into account, then the graph of the function l(co) near too is given by the dashed line AB. The AB region is the region of anomalous dispersion (n decreases with increasing), the remaining sections of the dependence of n on describe normal dispersion (n increases with increasing).

Russian physicist D.S. Rozhdestvensky (1876-1940) belongs to the classic work on the study of anomalous dispersion in sodium vapor. He developed an interference method for very accurate measurement of the refractive index of vapors and experimentally showed that the formula

correctly characterizes the dependence of n on, and also introduced a correction into it that takes into account the quantum properties of light and atoms.

In optics, the phenomenon of light dispersion is well known, i.e., the dependence of the speed of propagation of light in a medium on its frequency. Since [see. (38.4)]

then the refractive index of the medium also depends on the frequency. A similar dependence is observed not only in the optical range, but also for electromagnetic waves of any other frequencies. The first satisfactory explanation of the phenomenon of dispersion and simultaneous absorption of electromagnetic waves in media was given in the framework of Lorentz's electronic theory.

It is obvious that the phenomenon of dispersion is primarily associated with the influence of the electromagnetic field of a wave propagating in a medium on the dipole moments of molecules: For simplicity, we assume that the molecules are quite massive, and the frequency is large enough, so the change with time can be neglected. Thus, we will take into account only the induced dipole moment

As a model of a molecule, consider an individual electron with charge and mass me, displaced on a relatively positively charged core. If the speed of the electron is small compared to the speed of light, i.e., then in the expression for the Lorentz force, the contribution of the magnetic induction B of the wave can be neglected, since B Assuming also that the electron is held in the molecule by a quasi-elastic force - and taking into account the force of the radiation reaction, we write the equation of motion electron in the form

The solution of this equation can be used to calculate the total current density in the medium, assuming that electrons make the main contribution to it. In particular, assuming that the medium is homogeneous with an electron density, we have

We now write the averaged Maxwell-Lorentz equations (57.6):

Considering that, according to the law of conservation of charge,

Polarization, we write the Maxwell equations in the form

To find the polarization, we use equations (61.1) and (61.2). Namely: considering only the steady motion of the electron, i.e., assuming

and assuming that the intensity varies little within the molecule, from (61.2) we deduce

Finally, taking the strength of the acting field equal to

and taking into account (61.6) and (61.7), we find from (61.1)

Here where y is the coefficient of radiative friction; natural frequency of oscillations of an electron in an isolated atom; the natural frequency of electronic oscillations in an atom in a medium (i.e., changed under the influence of the fields of surrounding atoms); plasma frequency corresponding to oscillations of free electrons in a quasi-neutral medium (plasma or Lang-Mur oscillations).

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Having the expression (61.8) for the polarization, it is not difficult to find the electric induction vector:

where the complex permittivity is introduced

Here it is appropriate to note that y in (61.10) can be regarded as the coefficient of radiative friction only on the assumption that collisions of molecules with each other and with free electrons are unlikely. Indeed, as a result of collisions, part of the energy of the electrons is converted into the energy of motion of the molecules themselves, i.e., into heat. These energy losses by electrons must be added to the purely electromagnetic radiation losses. Phenomenologically, this is done by adding to y some independent part.

The above expression for is typical for a single-resonance oscillatory model of matter, in which it is assumed that the eigenfrequencies of all electrons are the same and equal. In fact, this is not the case, especially since it is also necessary to take into account oscillations of ions, whose eigenfrequencies usually lie in the infrared region. In order to take into account all electronic frequencies, the frequency distribution function of dispersive electrons is usually introduced. Normalizing it to unity, i.e., setting

Can be interpreted as the concentration of electrons whose natural frequencies lie in the interval In this case, expression (61.10) takes the form

Interestingly, the same expression is obtained in quantum theory, where it is called the strength of the oscillator.

What is the physical meaning of the complex permittivity? To clarify this, we separate the real and imaginary parts. Then

From (61.12) it follows that is an even, and is an odd function of the frequency:

and, in addition, the inequality

As it was shown in § 50, it is connected with thermal losses. In order to make sure that this is indeed the case and that the heat losses are proportional to a clearly positive value, we calculate the average power of the “friction” force acting on a single electron over a period:

The released thermal power is obtained by multiplying this expression by the electron concentration and integrating over

Taking into account the expressions for following from (61.7) and (61.8), we obtain

Comparing (61.15) with the expression for Joule losses

we come to the conclusion that the electrical conductivity of the medium and are interconnected:

In particular, for metals, in which the main contribution to the conductivity is made by free electrons with we have

This ratio is called the Drude-Zener formula and expresses the dependence of the electrical conductivity of metals on frequency.

Note that with the help of (61.16) the expression for в is reduced to the form

whence it follows that for metals in the static limit β has a pole singularity of the type

where a is the static electrical conductivity.

Of particular interest is the structure in for plasma, in which free electrons play the main role, i.e., according to (61.11), we can set

It is obvious that this behavior of the permittivity is characteristic of any medium in the limit of extremely high frequencies, since at , all electrons can be considered free. If in (61.20) we neglect losses, i.e. put then we get

Let us now study the propagation of electromagnetic waves in a dispersive medium. Let's start with the simplest plane monochromatic waves, i.e., put in equations (61.4)

where are constant vectors. Then, taking into account (61.9), we have:

Eliminating from these equations, we arrive at the wave equation

which allows two types of solutions corresponding to transverse and longitudinal waves.

Transverse waves satisfy the condition i.e. the vectors k form a right orthogonal triple (Fig. 61.1). In this case, from the wave equation (61.23) we deduce that

i.e., the wave vector k is complex. Assuming that the wave propagates along the axis i.e. assuming we have

complex refractive index.

To clarify the physical meaning, consider a plane electromagnetic wave:

where is the wavelength in vacuum. From this it follows that determines the attenuation of the wave amplitude at a distance of the order of the wavelength and is therefore called the absorption coefficient. As for this, this is the usual refractive index, which determines the speed of movement of the surface of the constant phase, i.e., the phase velocity of the wave

Dividing the real and imaginary parts in the ratio, we find:

The dependence in the simplest case, when there is only one isolated natural frequency near the frequency and, therefore, we can restrict ourselves to the single-resonance approximation, is given in Fig. 61.2 [-curve curve 2]. Dependence analysis shows that the coefficient y, which usually satisfies the condition, has the meaning of the absorption line width.

In particular, in the transparency region of a substance, i.e., far from the absorption line, when and can be put in the single-resonance approximation

Recalling that and resolving (61.29) relatively, we arrive at the relation

(Lorentz-Lorentz formula). It was developed independently of each other in 1869 by the Dane Lorentz, in 1873 by J.K. Maxwell, and in 1879 by G.A. Lorentz (the result

Maxwell remained unnoticed at the same time). According to (61.30), at a given frequency, it turns out to be proportional to the electron concentration. Obviously, the Lorentz - Lorentz formula is a generalization of the Clausius - Mosotti relation (58.26).

Let us pass to consideration of the second type of plane waves in the medium - longitudinal ones. In this case, therefore, it follows from equations (61.22) that

i.e. these waves are purely electrical and can only exist for those frequencies that are the roots of the equation

If c is sufficiently large, then, neglecting losses, we can use the simplified expression (61.21), from which it follows that Thus, in accordance with the result of problem 61.1, longitudinal waves are associated with polarization oscillations of electrons in the medium and therefore are often called polarization waves or Bohr waves, who first used them to calculate the energy loss of a charged particle moving in a medium.

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In real physical problems, it is often necessary to investigate the propagation in a medium not only of plane electromagnetic waves, but also of wave packets. A wave packet in a dispersive medium can be constructed by analogy with (39.11) and (39.13). Restricted to transverse waves, we have:

where is the solution of the dispersion equation (61.24).

Let us consider sufficiently narrow wave packets, i.e., we assume that the function has a sharply pronounced maximum at some point. To describe the behavior of such a wave packet, it is convenient to introduce the concept of its center, which can be

where the average is over the period

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Thus, for practically any time, the group velocity can be calculated using formula (61.36) only in the transparent region, in which In this case, differentiating relation (61.24) with respect to k, we find

This shows that in the region of normal dispersion, when the group velocity does not exceed the phase velocity, i.e., however, in the region of anomalous dispersion, when and since values ​​are possible, the group velocity can exceed the speed of light. Meanwhile, as can be seen, for example, from Fig. 61.2, the region of anomalous dispersion coincides with the region of absorption, in which formula (61.36) cannot be used and conclusions from it are invalid.

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In addition to the phase and group velocities, the concepts of signal speed and signal front speed are often used. A signal is usually understood as a wave packet with sharply limited edges. Its leading edge is called the front. It can be shown that the speed of the signal front in any medium is equal to the speed of light in vacuum [theorem of T. Levi-Civita (1913)]. The reason for this is not difficult to understand, if we note that the field experiences sharp changes in the front region, and this, in turn, is associated with the presence of infinitely high frequencies in the Fourier expansion of the field. But, according to (61.21), therefore, the medium behaves in relation to such changes in the field as vacuum. Obviously, this is due to the inertia of charged particles.

The structure of the signal front in a dispersive medium was studied in detail by A. Sommerfeld and L. Brillouin in 1914. They found that in a medium with absorption in the gap between the front and the main group, two regions with a noticeably increased field intensity can be distinguished. Brillouin called them the first and second harbingers. As expected, their speeds do not exceed c, and the speed of the main group, or the speed of the signal, differs from the group speed calculated by formula (61.36), only in the absorption region. The dependence of the signal speed on frequency is schematically shown in Fig. 3. 61.3 (on the example of a single-resonance model).

An interesting phenomenon related to the influence of matter on an electromagnetic field was discovered in 1934 by Soviet physicists P. A. Cherenkov and S. I. Vavilov. They observed a narrow cone of radiation emitted by fast electrons in a medium provided that their velocity exceeded the phase velocity of light, i.e., at