Observations of the propagation of waves on the surface of water from two or more sources show that the waves pass through one another without affecting each other at all. In the same way, sound waves do not affect each other. When an orchestra plays, the sounds from each instrument come to us exactly the same as if each instrument were playing separately.

This experimentally established fact is explained by the fact that, within the limits of elastic deformation, compression or stretching of bodies along one direction does not affect their elastic properties when deformed in any other directions. Therefore, at each point that waves from different sources reach, the result of the action of several waves at any given time equal to the sum the results of each wave separately. This pattern is called the principle of superposition.

Wave interference.

To gain a deeper understanding of the content of the superposition principle, let us perform the following experiment.

In a wave bath, using a vibrator with two rods, we will create two point sources of waves with the same frequency

hesitation. Observations show that in this case a special pattern of wave propagation appears in the wave bath. On the water surface there are stripes where there are no vibrations (Fig. 226).

A similar phenomenon can be found in experiments with sound waves. Let's install two dynamic speakers and connect them to the output of one sound generator. Moving short distances in a classroom, you can hear by hearing that the sound is loud at some points in the space, and quiet at others. Sound waves from two sources strengthen each other at some points in space and weaken each other at others (Fig. 227).

The phenomenon of an increase or decrease in the amplitude of the resulting wave when two or more waves with the same oscillation periods are added is called wave interference.

The phenomenon of wave interference does not contradict the principle of superposition. At points with zero amplitude of oscillations, two encountering waves do not “cancel” each other; both of them propagate further without changes.

Conditions for interference minimum and maximum.

The amplitude of oscillations is zero at

those points in space at which waves with the same amplitude and frequency arrive with a phase shift of oscillations by or by half the oscillation period. With the same law of oscillation of two wave sources, the difference will be half the oscillation period, provided that the difference in distances from the wave sources to this point is equal to half the wavelength:

or an odd number of half-waves:

The difference is called the path difference of the interfering waves, and the condition

is called the interference minimum condition.

Interference maxima are observed at points in space where waves arrive with the same oscillation phase. Given the same law of oscillation of two sources, to satisfy this condition, the path difference must be equal to an integer number of waves:

Coherence.

Interference of waves is possible only if the coherence condition is met. The word coherence means consistency. Oscillations with the same frequency and a constant phase difference over time are called coherent.

Interference and the law of conservation of energy.

Where does the energy of two waves disappear in places of interference minima? If we consider only one place where two waves meet, then such a question cannot be answered correctly. Wave propagation is not a set of independent oscillation processes at individual points in space. The essence of the wave process is the transfer of oscillation energy from one point in space to another, etc. When waves interfere in places of interference minima, the energy of the resulting oscillations is actually less than the sum of the energies of two interfering waves. But in the places of interference maxima, the energy of the resulting oscillations exceeds the sum of the energies of the interfering waves by exactly the same amount as the energy in the places of interference minima has decreased. When waves interfere, the oscillation energy is redistributed in space, but at the same time the law of conservation of energy is strictly observed.

Wave fraction.

If you reduce the size of the hole in the obstacle along the path of the wave, then the smaller the size of the hole, the greater the deviations from the rectilinear direction of propagation the waves will experience (Fig. 228, a, b). The deviation of the direction of wave propagation from straight line at the boundary of an obstacle is called wave diffraction.

To observe the diffraction of sound waves, we connect loudspeakers to the output of the sound generator and place a screen made of material in the path of sound waves.

absorbing sound waves. By moving the microphone behind the screen, you can find that sound waves are also recorded behind the edge of the screen. By changing the frequency of sound vibrations and thereby the length of the sound waves, it can be established that the phenomenon of diffraction becomes more noticeable with increasing wavelength.

Diffraction of waves occurs when they encounter an obstacle of any shape and size. Usually, when the size of the obstacle or hole in the obstacle is large compared to the wavelength, wave diffraction is little noticeable. Diffraction manifests itself most clearly when waves pass through an opening with dimensions on the order of the wavelength or when encountering obstacles of the same dimensions. At sufficiently large distances between the wave source, the obstacle and the place where the waves are observed, diffraction phenomena can also occur with large openings or obstacles.

Huygens-Fresnel principle.

A qualitative explanation of the phenomenon of diffraction can be given on the basis of Huygens' principle. However, Huygens' principle cannot explain all the features of wave propagation. Let us place a barrier with a wide hole in the path of plane waves in the wave bath. Experience shows that the waves pass through the hole and propagate along the original direction of the beam. Waves from the hole do not propagate in other directions. This contradicts Huygens' principle, according to which secondary waves should propagate in all directions from the points reached by the primary wave.

Let's put a wide barrier in the path of the waves. Experience shows that waves do not propagate beyond an obstacle, which again contradicts Huygens’ principle. To explain the phenomena observed when waves meet obstacles, the French physicist Augustin Fresnel (1788-1827) in 1815 supplemented Huygens' principle with ideas about the coherence of secondary waves and their interference. The absence of waves away from the direction of the beam of the primary wave behind a wide hole according to the Huygens-Fresnel principle is explained by the fact that secondary coherent waves emitted by different parts of the hole interfere with each other. There are no waves in those places where the conditions of interference minima are met for secondary waves from different areas.

Wave polarization.

Interference and diffraction phenomena

are observed both during the propagation of longitudinal and transverse waves. However, transverse waves have one property that longitudinal waves do not have - the property of polarization.

A polarized wave is a transverse wave in which all particles oscillate in the same plane. A plane-polarized wave in a rubber cord is produced when the end of the cord oscillates in one plane. If the end of the cord vibrates in different directions, then the wave propagating along the cord is not polarized.

Polarization of this wave can be achieved by placing an obstacle in its path with an opening in the form of a narrow slit. The slot allows only vibrations of the cord that occur along it. Therefore, after passing through the slit, the wave becomes polarized in the plane of the slit (Fig. 229). If further on the path of a plane-polarized wave a second slit is placed parallel to the first, then the wave passes freely through it. Rotating the second slit relative to the first by 90° stops the process of wave propagation in the cord.

A device that separates out of all possible vibrations those occurring in one plane (the first slit) is called a polarizer. A device that allows you to determine the plane of polarization of the wave (second slit) is called an analyzer.

As mentioned above, in a natural beam chaotic changes in the direction of the plane occur all the time electric field. Therefore, if we imagine a natural beam as the sum of two mutually perpendicular oscillations, then it is necessary to consider the phase difference of these oscillations to also vary chaotically with time.

In § 16 it was explained that a necessary condition interference is the coherence of the added oscillations. From this circumstance and from the definition of a natural ray, one of the basic laws of interference of polarized rays established by Arago follows: if we receive two rays from the same natural ray, mutually perpendicularly polarized, then these two rays turn out to be incoherent and in the future cannot interfere with each other.

Recently, S.I. Vavilov showed theoretically and experimentally that two natural, seemingly coherent beams that do not interfere with each other can exist. For this purpose, in the interferometer on the path of one of the beams, he placed an “active” substance that rotates the plane of polarization by 90° (rotation of the plane of polarization is discussed in § 39). Then the vertical component of the natural beam oscillations becomes horizontal, and the horizontal component becomes vertical, and the rotated components add up with the components of the second beam that are not coherent with them. As a result, after the introduction of the substance, the interference disappeared.

Let us move on to an analysis of the phenomena of interference of polarized light observed in crystals. The usual scheme for observing interference in parallel beams consists (Fig. 140) of a crystal polarizer k and an analyzer a. For simplicity, let us analyze the case when the crystal axis is perpendicular to the beam. Then

a plane-polarized beam emerging from the polarizer in crystal K will be divided into two coherent beams, polarized in mutually perpendicular planes and traveling in the same direction, but at different speeds.

Rice. 140. Diagram of an installation for observing interference in parallel rays.

Of greatest interest are two orientations of the main planes of the analyzer and polarizer: 1) mutually perpendicular main planes (crossed); 2) parallel main planes.

Let us first consider a crossed analyzer and polarizer.

In Fig. 141 OR means the plane of oscillation of the beam passing through the polarizer; -its amplitude; -direction of the optical axis of the crystal; perpendicular to the axis; OA is the main plane of the analyzer.

Rice. 141. Towards the calculation of the interference of polarized light.

The crystal, as it were, decomposes vibrations along axes and into two vibrations, i.e., into extraordinary and ordinary rays. The amplitude of the extraordinary beam is related to the amplitude a and the angle a as follows:

Amplitude of an ordinary beam

Only the projection onto an equal

and the projection of X to the same direction

Thus, we get two oscillations, polarized in the same plane, with equal but oppositely directed amplitudes. The addition of two such oscillations gives zero, i.e., darkness is obtained, which corresponds to the usual case of a crossed polarizer and analyzer. If we take into account that between the two beams, due to the difference in their velocities in the crystal, an additional phase difference has appeared, which we denote by then the square of the resulting amplitude will be expressed as follows (vol. I, § 64, 1959; in the previous edition § 74) :

that is, light passes through a combination of two crossed nicols if a crystal plate is inserted between them. Obviously, the amount of transmitted light depends on the magnitude of the phase difference associated with the properties of the crystal, its birefringence and thickness. Only in the case or will complete darkness be obtained regardless of the crystal (this corresponds to the case when the crystal axis is perpendicular or parallel to the main Nicol plane). Then only one ray passes through the crystal - either ordinary or extraordinary.

The phase difference depends on the wavelength of the light. Let the thickness of the plate be the wavelength (in void) refractive index Then

Here is the wavelength of the ordinary beam, and is the wavelength of the extraordinary beam in the crystal. The greater the thickness of the crystal and the greater the difference between the greater On the other hand, it is inversely proportional to the wavelength. Thus, if for a certain wavelength is equal to what corresponds to the maximum (since in this case it is equal to unity), then for a wavelength 2 times less , is already equal, which gives darkness (because in this case it is equal to zero). This explains the colors observed when white light passes through the described combination of nicols and a crystal plate. Part of the rays that make up white light is extinguished (these are those that are close to zero or an even number, while the other part passes through, and

Rays that are close to an odd number pass through most strongly. For example, red rays pass through, but blue and green rays are weakened, or vice versa.

Since the formula for enters, it becomes clear that a change in thickness should cause a change in the color of the rays passing through the system. If you place a crystal wedge between the nicols, then stripes of all colors will be observed in the field of view, parallel to the edge of the wedge, caused by the continuous increase in its thickness.

Now let's look at what will happen to the observed picture when the analyzer rotates.

Let's rotate the second nicol so that its main plane becomes parallel to the main plane of the first nicol. In this case, in Fig. 141 lines simultaneously depict both main planes. Just like before

But projections to

We get two unequal amplitudes directed in the same direction. Without taking birefringence into account, the resulting amplitude in this case is simply a, as it should be with a parallel polarizer and analyzer. Taking into account the phase difference arising in the crystal between , leads to the following formula for the square of the resulting amplitude:

Comparing formulas (2) and (4), we see that, i.e., the sum of the intensities of the light rays transmitted in these two cases is equal to the intensity of the incident beam. It follows that the pattern observed in the second case is complementary to the pattern observed in the first case.

For example, in monochromatic light, crossed nicols will give light, since in this case, and parallel ones will give darkness, since in white light, if in the first case red rays pass through, then in the second case, when the nicol is rotated 90°, green rays will pass through. This change of colors to additional ones is very effective, especially when

interference is observed in a crystal plate composed of pieces of different thicknesses, giving a wide variety of colors.

Until now, as we have already indicated, we were talking about a parallel beam of rays. A much more complicated situation occurs with interference in a converging or diverging beam of rays. The reason for the complication is the fact that different rays of the beam pass through different thicknesses of the crystal depending on their inclination. We will dwell here only on the simplest case, when the axis of the conical beam is parallel to the optical axis of the crystal; then only the ray traveling along the axis does not undergo refraction; the remaining rays, inclined to the axis, as a result of double refraction, will each decompose into ordinary and extraordinary rays (Fig. 142). It is clear that rays with the same inclination will travel the same paths in the crystal. The traces of these rays lie on the same circle.

4.4.3 Optical properties of uniaxial crystals. Interference of polarized rays

Optically uniaxial crystals have the simplest optical properties, which are also of greatest practical importance. Therefore, it makes sense to highlight this simplest special case.

Optically uniaxial crystals are those whose properties have symmetry of rotation relative to a certain direction, called the optical axis of the crystal.

1. Let us decompose the electric vectors E and D into components E ║ and D ║ along the optical axis and components E ┴ and D ┴ perpendicular to it. Then

D ║ = ε ║ E ║ and D ┴ , = ε ┴ E ┴ , where ε ║ and ε ┴ are constants, called the longitudinal and transverse dielectric constants of the crystal. Optically uniaxial crystals include all crystals of tetragonal, hexagonal and rhombohedral systems. The plane in which the optical axis of the crystal and the normal lie N to the wave front is called the main cross section of the crystal. The main section is not a specific plane, but a whole family of parallel planes.

Let us now consider two special cases.

Case 1. Vector D perpendicular to the main section of the crystal. In this case D == D ┴ , and therefore D = ε ┴ E. The crystal behaves as an isotropic medium with dielectric constant ε┴. For her D = ε ┴ E from Maxwell's equations we obtain D = -с/v H, H =с/v E or ε ┴ E = c/v H, H = -c/v E, where v = v ┴ =v 0 c/√ ε ┴ .

Thus, if the electric vector is perpendicular to the main section, then the wave speed does not depend on the direction of its propagation. Such a wave is called ordinary.

Case 2. Vector D lies in the main section. Since vector E also lies in the main section (Figure 160), then E = E n + E D , Where E n - component of this vector along n, a E D - along D. From vector product [nE ] component E n falls out. Therefore the formula for H from Maxwell's equations can be written in the form H = s/v [nED ] . Obviously E D = ED /D= (E ║ D ║ + E ┴ D ┴)/D = (D ║ 2ε ║ +D ┴ 2ε ┴) /D or E D = D (sin 2 α/ ε ║ + cos2α/ ε ┴ ) = D(n ┴ 2/ ε ║ + n ║ 2/ ε ┴ ), Where α - the angle between the optical axis and the wave normal.

If you enter the designation 1/ε = (n ┴ 2/ ε ║ + n ║ 2/ ε ┴ ), it will work out D = εED, and we arrive at the relations εED = с/v H, H =с/v ED, formally identical with the relations obtained earlier. The role of magnitude ε ┴ now the quantity ε determined by the expression just obtained for it plays. Therefore, the normal wave speed will be determined by the expression v = c/√ ε = c√ (n ┴ 2/ ε ║ + n ║ 2/ ε ┴ . It changes with a change in the direction of the wave normal n. For this reason, a wave whose electric vector lies in the main section of the crystal is called extraordinary.

The term “optical axis” was introduced to designate a straight line along which both waves in the crystal propagate at the same speeds. If there are two such lines in a crystal, the crystal is called optically biaxial. If the optical axes coincide with each other, merging into one straight line, the crystal is called optically uniaxial.

2. Since Maxwell’s equations in crystals are linear and homogeneous, then in the general case, a wave entering a crystal from an isotropic medium is divided inside the crystal into two linearly polarized waves: an ordinary one, the electric induction vector of which is perpendicular to the main section, and an extraordinary one with the vector electrical induction lying in the main section. These waves propagate in the crystal in different directions and at different speeds. In the direction of the optical axis, the velocities of both waves coincide, so that a wave of any polarization can propagate in this direction.

All the arguments that we used to derive the geometric laws of reflection and refraction are applicable to both waves. But in crystals they refer to wave normals, not light rays. The wave normals of the reflected and both refracted waves lie in the plane of incidence. Their directions formally obey Snell's law sinφ/sinψ ┴ = n ┴ , sinφ/sinψ ║ = n ║ , Where n ┴ And n ║ - refractive indices of ordinary and extraordinary waves, i.e. n ┴ = c/v ┴ = n 0 , n ║ = c/v ║ = (n ┴ 2/ ε ║ + n ║ 2/ ε ┴ )-1/2 . Of them n ┴ = n 0 does not depend, but n ║ : depends on the angle of incidence. Constant n v is called the ordinary refractive index of a crystal. When an extraordinary wave propagates perpendicular to the optical axis ( n ┴ = 1, n ║ = 0), n ║ = √ε ║ = n e . Size P e called the extraordinary refractive index of the crystal. It cannot be mixed with the refractive index n ║ extraordinary wave. Magnitude n e there is a constant, and n ║ - function of the direction of wave propagation. The values ​​are the same when the wave propagates perpendicular to the optical axis.

3. Now it is easy to understand the origin of double refraction. Let us assume that a plane wave is incident on a plane-parallel plate made of a uniaxial crystal. When refracted on the first surface of the plate, the wave inside the crystal will split into ordinary and extraordinary. These waves are polarized in mutually perpendicular planes and propagate inside the plate in different directions and at different speeds. The wave normals of both waves lie in the plane of incidence. An ordinary ray, since its direction coincides with the direction of the wave normal, also lies in the plane of incidence. But the extraordinary ray, generally speaking, comes out of this plane. In the case of biaxial crystals, the division into ordinary and extraordinary waves loses its meaning - inside the crystal, both waves are “extraordinary”. During refraction, the wave normals of both waves, of course, remain in the plane of incidence, but both rays, generally speaking, leave it. If the incident wave is limited by a diaphragm, then the plate will produce two beams of light, which, if the plate is thick enough, will be separated spatially. When refracted at the second boundary of the plate, two beams of light will emerge from it, parallel to the incident beam. They will be linearly polarized in mutually perpendicular planes. If the incident light is natural, then two beams will always come out. If the incident light is linearly polarized in the plane of the main section or perpendicular to it, then double refraction will not occur - only one beam will emerge from the plate, maintaining the original polarization.

Double refraction also occurs when light is normally incident on a plate. In this case, the extraordinary ray undergoes refraction, although the wave normals and wave fronts are not refracted. An ordinary beam of rays does not undergo refraction. The extraordinary ray in the plate is deflected, but upon exiting it it again goes in the original direction.

The rays, ordinary and extraordinary, arising from birefringence from natural light are not coherent. The rays, ordinary and extraordinary, arising from the same polarized ray are coherent. If the oscillations in two such beams are brought to the same plane using a polarizing device, then the beams will interfere in the usual way. If oscillations in two coherent plane-polarized beams occur in mutually perpendicular directions, then they, adding up as two mutually perpendicular oscillations, excite oscillations of an elliptical nature.

Light waves in which the electric vector changes over time so that its end describes an ellipse are called elliptically polarized. In a particular case, an ellipse can turn into a circle, and then we are dealing with light polarized in a circle. The magnetic vector in a wave is always perpendicular to the electric vector and in waves of the type under consideration also changes with time in such a way that its end describes an ellipse or circle.

Let us consider the case of the occurrence of elliptic waves in more detail. When a beam of rays is normally incident on a plate made of a uniaxial crystal, the optical axis of which is parallel to the refractive surface, the ordinary and extraordinary rays travel in the same direction, but at different speeds. Let a plane polarized beam fall on such a plate, the plane of polarization of which makes an angle with the plane of the main section of the plate that is different from zero and from π/2. Then both rays, ordinary and extraordinary, will appear in the plate, and they will be coherent. At the moment of their occurrence in the plate, the phase difference between them is zero, but it will increase as the rays penetrate into the plate. The difference between the refractive indices n0-ne and the thicker the crystal l. If the thickness of the plate is chosen so that ∆ = kπ, Where k is an integer, then both beams, leaving the plate, will again produce a plane polarized beam. At k, equal to an even number, its plane of polarization coincides with the plane of polarization of the beam incident on the plate; when k is odd, the plane of polarization of the beam emerging from the plate will be rotated by π/2 relative to the plane of polarization of the beam incident on the plate (Figure 4.53). For all other values ​​of the phase difference Δ, the oscillations of both rays emerging from the plate, adding up, will give an elliptical oscillation. If ∆ = 2k+1)π/2 then the axes of the ellipse will coincide with the directions of oscillations in the ordinary and extraordinary rays (Figure 4.54). The smallest plate thickness capable of converting a plane-polarized beam into a circularly polarized beam ( ∆ = π/2), is determined by the equality π/2 = 2πl/λ (n 0 -n e ), where we get: l = λ/ 4(n 0 -n e )

Such a plate will give a path difference between the ordinary and extraordinary rays equal to λ/4, hence it is called a quarter wave record for short. It is obvious that a quarter-wave plate will give a path difference between both beams equal to λ/4, only for light of a given wavelength λ. For light of other wavelengths it will give a path difference slightly different from λ/4, both because of the direct dependence of l on λ, and because of the dependence on λ refractive index differences ( n 0 -n e ). It is obvious that, along with a quarter-wave plate, it is also possible to produce a “half-wavelength” plate, i.e., a plate that introduces a path difference between the ordinary and extraordinary rays λ/2, what does the phase difference correspond to? π . Such a plate can be used to rotate the plane of polarization of plane polarized light by π/2. As indicated, using a λ/4 plate, a plane-polarized beam can be converted into an elliptically or circularly polarized beam; conversely, from an elliptically polarized or circularly polarized beam, plane polarized light can be obtained using a λ/4 plate. This circumstance is used to distinguish elliptically polarized light from partially polarized light, or circularly polarized light from natural light.

This analysis of elliptically polarized light can be performed using a plate λ/4 in the case when elliptical polarization occurs as a result of the addition of two mutually perpendicular oscillations of different amplitudes with a phase difference π/2. If elliptical polarization occurs as a result of the addition of two mutually perpendicular oscillations with a phase difference ∆≠π/2, then to transform such light into plane polarized it is necessary to introduce such an additional phase difference ∆", which in sum with ∆ would give a phase difference equal to π (or 2kπ). In these cases, instead of a plate λ/4 devices called compensators are used, which make it possible to obtain any value of the phase difference.

Interference of polarized rays– a phenomenon that occurs when coherent polarized light vibrations are added.

With normal incidence of natural light on the face of a crystal plate parallel to the optical axis, ordinary and extraordinary rays propagate without being separated, but at different speeds. Two rays polarized in mutually perpendicular planes will emerge from the plate, between which there will be optical difference progress

or phase difference

where is the thickness of the plate and is the length of light in vacuum. If you place a polarizer in the path of the rays emerging from the crystal plate, then the oscillations of both rays after passing through the polarizer will lie in the same plane. But they will not interfere, since they are not coherent, although they were obtained by dividing light from one source. Ordinary and extraordinary rays contain vibrations belonging to different trains of waves emitted by individual atoms. If plane-polarized light is directed onto a crystal plate, then the vibrations of each train are divided between the ordinary and extraordinary rays in the same proportion, so the emerging rays turn out to be coherent.

Interference of polarized rays can be observed when linearly polarized light (obtained by passing natural light through a polarizer) passes through a crystal plate, passing through which the beam is divided into two coherent, polarized

in mutually perpendicular planes of the beam. The crystal plate ensures the coherence of the ordinary and extraordinary rays and creates a phase difference between them according to the relationship (6.38.9).

To observe the interference pattern of polarized beams, it is necessary to rotate the plane of polarization of one of the beams until it coincides with the plane of polarization of the other beam or to isolate components from both beams with the same direction of oscillation. This is done using a polarizer, which reduces the oscillations of the rays into one plane. An interference pattern can be observed on the screen.

The intensity of the resulting oscillation where is the angle between the plane of the polarizer and the optical axis of the crystal plate, is the angle between the planes of the polarizers and The intensity and color of the light transmitted through the system depends on the wavelength. When one of the polarizers is rotated, the color of the interference pattern will change. If the thickness of the plate is not the same in different places, then a motley-colored picture is observed on the screen.

Test questions for self-preparation of students:

1. What is light dispersion?

2. By what features can spectra obtained using a prism and a diffraction grating be distinguished?

3. What is natural light? plane polarized? partially polarized light?

4. Formulate Brewster’s law.

5. What causes birefringence in an optically anisotropic uniaxial crystal?

6. Kerr effect.

Literary sources:

1. Trofimova, T.I. Physics course: textbook. manual for universities / T.I. Trofimova. – M.: ACADEMIA, 2008.

2. Savelyev, I.V. General physics course: textbook. manual for colleges: in 3 volumes / I.V. Savelyev. – SPb.: Special. lit., 2005.