Wave interference. Superposition principle for waves

We are surrounded by objects of certain sizes; we know exactly where our body ends, and we are sure that only one person can sit comfortably in one chair. However, in the world of very small things, or in the microquantum world, everything is not so prosaic: a chair and a table, reduced by about ten billion times, to the size of atoms, will lose their clear boundaries and can even take one place in space without interfering with each other. . The reason is that the objects of the quantum world are more like waves penetrating each other than objects limited in space. Therefore, in the microquantum world, you can sit on the same chair with three or ten people.

Things are like waves

To wave properties could be felt experimentally, objects need to be made not only small, but also very cold, that is, with a greatly reduced speed chaotic movement atoms. So, atoms need to be cooled to a billionth of a degree Kelvin, and the wave properties of a table and a chair from the macrocosm should be noticeable at unthinkably low temperatures - colder than 10–40 K.

A remarkable property of waves is their ability to coherently add up. Coherent means consistent, orderly in time or space. Example of coherent in time sound waves- music. Each sound of the melody, its height, duration and strength are in a strictly defined correspondence with each other.

The conductor of a symphony orchestra closely monitors the coherence of a sound stream of hundreds or even thousands of sounds. We perceive the weakening of coherence as a false sound, and its complete loss as noise. Actually, coherence distinguishes a melody from an incoherent set of sounds. In the same way, in the quantum world, the coherence of the wave properties of objects can give them completely new qualities that are not only very unusual, but also important for creating new materials that can radically change existing technologies. It is no coincidence that almost half of the Nobel Prizes in physics awarded over the past ten years are associated with coherent phenomena: in laser radiation (2005), in cold atoms (1997, 2001), in liquid helium (1996) and in superconductors (2003).

Most domestic Nobel laureates in physics received their prizes for coherent phenomena: Petr Kapitsa (1978), Lev Landau (1962), Nikolai Basov and Alexander Prokhorov (1964), Alexei Abrikosov and Vitaly Ginzburg (2003).

Light coherence

The concept of coherence was formed in early XIX century after the experiments of the English scientist Thomas Young. In them, two light waves from different sources fell on the screen and added up. The light from two ordinary light bulbs, which give incoherent radiation, is added simply: the illumination of the screen is equal to the sum of the illuminations from each lamp. Here is the mechanism. For light waves from light bulbs, the phase difference changes randomly over time. If two wave maxima have now arrived at one point of the screen, then at the next moment a minimum can come from one lamp, and a maximum from the other. The result of the addition of waves will give "ripples on the water" - an unstable interference pattern. The ripples of light waves are so fast that the eyes do not keep up with them and see a uniformly lit screen. By analogy from the world of sounds - this is noise.

The result will be completely different if two coherent waves are added on the screen (Fig. 1). Such waves are easiest to obtain from a single laser beam, splitting it into two parts, and then adding them. Then stripes will appear on the screen. Bright areas are areas of the screen where the maxima of light waves always arrive simultaneously (in phase). A remarkable optical effect is that the illuminance will increase not by a factor of two, as in the case of incoherent waves, but by a factor of four. This happens because the maxima of the waves, that is, their amplitudes, are added all the time in the bright band, and the illumination is proportional to the square of the sum of the wave amplitudes. In dim bands, coherent waves from different sources cancel each other out.

Now imagine many coherent waves arriving at some point in phase. For example, a thousand waves. Then the illumination of the bright area will increase by a million times! Coherent radiation of a huge, about 10 22, number of atoms produces a laser beam. The invention of the principles of its work brought in 1964 the Nobel Prize in Physics to the American Charles Townes and two Soviet physicists Nikolai Basov and Alexander Prokhorov. For 40 years, the laser has penetrated our everyday life, with its help we, for example, store information on compact discs and transmit it over optical fiber over great distances.

Coherent Matter Waves

Our world is arranged in such a way that each particle of matter can exhibit the properties of a wave. Such waves are called matter waves, or de Broglie waves. The remarkable French physicist Louis de Broglie in 1923 proposed a very simple formula relating the wavelength λ (the distance between the maxima) with the mass of the particle m and its speed v: λ = h/mv, where h is Planck's constant.

A fundamental property of waves of any nature is the ability to interfere. However, in order to obtain as a result not a uniform noise, but, as in the case of light, a bright band, it is necessary to ensure the coherence of the de Broglie waves. This is hindered thermal motion- atoms with different speeds differ in their wavelengths. When atoms are cooled, according to the de Broglie formula, the wavelength λ increases (Fig. 2). And as soon as its value exceeds the distance between the particles, the de Broglie waves different particles will give a stable interference pattern, since the wave maxima corresponding to the position of the particles will overlap.

In an optical microscope, the interference pattern of de Broglie waves can be seen if their lengths are about 1 micron. To do this, as follows from the de Broglie formula, the speed of the atom must be approximately 1 cm/s, which corresponds to extremely low temperatures - less than one microkelvin. Such a cooled gas of atoms alkali metals managed to cook, and today it interesting object research. (How to cool atoms down to low temperatures and make ultra-precise clocks based on them, was described in "Chemistry and Life", 2001, No. 10. - Note. ed.) Note that Soviet physicists from the Institute of Spectroscopy of the USSR Academy of Sciences, headed by Vladilen Letokhov, put forward and implemented key ideas in 1979, on the basis of which atoms are now cooled to ultralow temperatures.

What are interfering particles of matter? We are used to the fact that the substance can be represented in the form of solid small balls that do not penetrate each other. Waves, on the contrary, can add up and penetrate each other. By analogy with the interference of light, we should get a "bright point on the screen" - a small area in space where the maxima of matter waves add up in phase. It is unexpected that the coherent waves of many and many atoms can occupy one region in space, forming, as it were, a superatom - a set of a huge number of de Broglie waves. In the language of quantum mechanics, this means that the probability of finding coherent atoms in a "bright point" is maximum. it amazing state substances are called Bose-Einstein condensates. Albert Einstein predicted it in 1925 based on the work of the Indian physicist Shatyendranath Bose. In a condensate, all atoms are in the same quantum state and behave like one big wave.

It was possible to experimentally observe the Bose-Einstein condensate (BEC) only 70 years later: two groups of American scientists published a report on this in 1995. In their experiments, atoms fell into the condensate from a cloud of sodium or rubidium vapors locked in a magnetic trap. These pioneering works were awarded the 2001 Nobel Prize in Physics, awarded to Eric Cornell, Wolfgang Ketterle and Carl Wiemann. A vivid figurative representation of the behavior of supercold atoms falling into the BEC was shown on the cover of the December magazine Science for 1995: a group of identical blue cyborgs is marching in the center - these are BEC atoms with zero temperature, and cyborgs of warmer colors are randomly moving around them - slightly heated atoms above condensate. The coherence of atoms deposited in the BEC was demonstrated in a brilliant 1997 experiment by W. Ketterle and colleagues from the Massachusetts Institute of Technology. To do this, the magnetic trap was divided into two parts by a partition of light (Fig. 3a). Two condensates were prepared from clouds of sodium atoms, and then the trap and partition were turned off: the clouds began to expand and overlap. At the place of their overlap, a clear interference pattern appeared (Fig. 3b), similar to the interference of coherent laser beams (see Fig. 1). It was observed by the shadow cast by a cloud of atoms on the screen - the "zebra" in Fig. 3b is the shadow of the interfering waves of matter; the dark regions correspond to the wave maxima of the atoms. Surprisingly, when we add atoms from different condensates, their sum can give zero - "substance disappears" in the area corresponding to the light "zebra" stripe. Of course, the atoms don't actually disappear - they just concentrate in the areas that cast the shadow.

Is it possible to observe the manifestation of wave properties for more massive objects than atoms? It turns out you can. The group of Anton Zeilinger from Vienna in 2003 managed to observe the interference of fullerenes and biomolecules containing about a hundred atoms. For how large particles of matter it will be possible to observe wave properties - the question is open today.

Atomic laser

From the point of view of quantum physics, atoms and photons are similar in that a large number of these particles can simultaneously be in the same quantum state, that is, be coherent. For example, in laser radiation, all photons are coherent: they have the same color, direction of propagation and polarization. Therefore, it is possible to obtain powerful coherent laser beams consisting of a huge number of photons in one state.

And how to get coherent atomic beams? The idea is simple: you need to carefully remove the trapped coherent atoms from the BEC, just as laser radiation is removed from its resonator using a semitransparent mirror. Such a device was called an atomic laser. The first atomic laser in 1997 was created by the same W. Ketterle. In such a laser, a two-coil magnetic trap traps the sodium atoms that form the BEC. The radio field pulses, applied with a period of 5 milliseconds, turn the spins of the atoms, and they can no longer be held in the trap. A bunch of released atoms - radiation from an atomic laser - falls freely under the influence of gravity, which is visualized using the shadow theater techniques described above. Today, the power of atomic lasers is low: they emit 10 6 atoms per second, which is incomparably less than the power of optical lasers. So, for example, a conventional laser pointer emits about 10 9 times more photons in one second.

Unlike weightless photons, atoms have a rest mass. This means that gravity acts on them much more strongly - the interference of coherent waves of matter will strongly depend on the gravitational field that deflects atomic beams. Let two coherent atomic beams interfere in the region of their intersection similarly to laser beams (see Fig. 1). Let us assume that the gravitational field on the path of one of the atomic beams has changed. Then the length of the path of this beam to the meeting with another beam will also change. As a result, the maxima of the matter waves of the two atomic beams will meet in a different place, which will lead to a shift in the interference pattern. By measuring such a displacement, one can determine the change in the gravitational field. Based on this idea, gravitational field sensors have already been created that can detect the difference in the magnitude of acceleration. free fall less than 10–6%. They can be useful for both fundamental research(testing physical theories, measuring constants), and for important applied developments in navigation (creation of precision gyroscopes), geology (probing of minerals) and other sciences. In science fiction writers, for example, one can find a story when, using an instrument for measuring the slightest changes in gravity, archaeologists read inscriptions carved on obelisks buried in the thickness of the earth.

coherent substance

Particularly interesting effects arise when the properties of coherent waves of matter can be observed as macroscopic properties of condensed matter, that is, a solid or liquid. One of clear examples such properties - superfluidity in liquid helium when cooled below 2.2 K. Soviet physicists carried out pioneering studies of superfluidity: this phenomenon was discovered by Pyotr Kapitsa in 1938, and Lev Landau explained. Superfluid helium can flow through small holes at a tremendous rate: at least 108 times faster than water. If we could fill an ordinary bathtub with superfluid helium, it would flow out of it in less than one second through a hole the size of a tiny needle eye. In 2004, the Americans Yun Sung Kim and Moses Chan reported the discovery of superfluidity in solid helium. Their subtle experiment was as follows: solid cooled helium, under pressure at a temperature of about 0.2 K, was placed on a torsion pendulum. If some of the helium goes into a superfluid state, then the frequency of torsional vibrations should increase, since the superfluid component remains stationary, facilitating the oscillations of the pendulum. According to Kim and Chan, about 1% of solid helium went into the superfluid state. These experiments demonstrate that atoms can move freely through a superfluid solid body, therefore, it is able to pass a mass of matter through itself without hindrance: the prospect of passing through walls in such a world seems quite real!

This amazing phenomenon can explain the wave properties of atoms. Waves, unlike particles, bypass obstacles in their path. Let us explain this by the example of the interference of two beams of light on a screen. Let's cut holes in the screen in the area of ​​the bright stripes of the "zebra" (interference pattern). Coherent light will not feel such an obstacle: after all, the screen has been preserved only in the unlit parts of the "zebra". If the beams are not coherent, then a uniformly illuminated screen with holes will inevitably block part of the light. From here it is possible to understand how coherent waves of matter overcome obstacles without loss.

Another unusual macroscopic quantum phenomenon, similar to superfluidity, is superconductivity, discovered by the Dutchman Heike Kamerling-Ohness in 1911 in mercury when it is cooled to the temperature of liquid helium (Nobel Prize in 1913). Superconducting electrons move without resistance, bypassing obstacles, which are the thermal motion of atoms. For example, a current in a superconductor ring can flow indefinitely, since nothing interferes with it. We can say that superconductivity is the superfluidity of an electron liquid. For such superfluidity, it is necessary that a large number of charges be in the same quantum state, as, for example, photons in a laser beam. This requirement runs into a limitation established by the eminent Swiss physicist Wolfgang Pauli in 1924: if the spin number of a particle is 1/2, like an electron, then only one particle can be in one quantum state. Such particles are called fermions. For an integer value of the spin in one quantum state, an arbitrarily large number of particles can be condensed. Such particles are called bosons. Therefore, for a superconducting current, particles of electric charge with an integer spin are needed. If a pair of electrons (fermions) could form a composite particle, then the spin of the pair would be an integer. And then the composite particles will become bosons capable of forming BEC and giving superconducting current.

However, bound pairs of electrons can indeed occur in conductors, despite the fact that Coulomb forces repel electrons from each other - this idea formed the basis of the theory explaining superconductivity in simple metals (John Bardeen, Leon Cooper, John Schrieffer, Nobel Prize in Physics for 1972 year).

BEC superfluidity

So, in the second half of the 20th century, physicists came to the understanding that the BEC can have the properties of superfluidity. Naturally, after obtaining a gaseous BEC, scientists were captivated by the idea of ​​experiments demonstrating superfluidity in it. In 2005, W. Ketterle's group presented the final proof of the superfluidity of a gaseous BEC. The idea of ​​the experiment is based on the fact that a superfluid liquid behaves unusually during rotation. If we could stir a superfluid liquid with a spoon, like coffee in a cup, then it would not begin to rotate in its entirety, but would break up into many small vortices. Moreover, they would arrange themselves in a strict order, forming the so-called Abrikosov vortex lattice. The scheme of this filigree experiment is as follows (Fig. 4). Gas condensate captured by the laser beam and magnetic field, began to rotate with additional laser beams; they swirled the condensate like a spoon - coffee. Then the trap, that is, the beams and the coil, was turned off, and the condensate was left to itself. It expanded and gave a shadow that resembled Swiss cheese(Fig. 4b). The "holes in the cheese" correspond to superfluid vortices. The most important feature of these experiments lies in the fact that they were carried out not only in a gas of bosons (sodium atoms), but also in a gas of fermions (lithium atoms). Superfluidity in lithium gas was observed only when the lithium atoms formed molecules or weak pairs. This was the first observation of the superfluidity of a fermion gas. It provided a solid experimental foundation for the theory of superconductivity based on the idea of ​​Bose-Einstein condensation.

Physicists succeed in pairing lithium atoms with the help of the so-called Feshbach resonance, which occurs in a trap with the simultaneous action of the fields of magnetic coils and laser beams. The magnetic field is adjusted in the region of the Feshbach resonance so that it greatly changes the forces of interaction between gas atoms. You can make atoms attract each other or repel each other. Physicists have come up with other ways to control the properties of supercold atomic gas. One of the most elegant is to place atoms in an interfering field of laser beams - a kind of optical lattice. In it, each atom will be in the center of one of the fringes of the interference pattern (see Fig. 1), so that the waves of light will hold the waves of matter, like a form for storing eggs. Atoms in an optical lattice serve as an excellent model of a crystal, where the distance between atoms is changed using the parameters of laser beams, and the interaction between them is regulated using the Feshbach resonance. As a result, physicists have realized an old dream - to obtain a sample of matter with controlled parameters. Scientists believe that supercold gas is a model not only of a crystal, but also of more exotic forms of matter, such as neutron stars and quark-gluon plasma of the early Universe. Therefore, some researchers, not without reason, believe that supercold gas will help to understand the early stages of the evolution of the Universe.

Coherent future

The phenomena of superfluidity and superconductivity show that the coherence of de Broglie waves of a large number of particles gives unexpected and important properties. These phenomena were not predicted; moreover, it took almost 50 years to explain superconductivity in simple metals. And the phenomenon of high-temperature superconductivity, discovered in 1986 in metal-oxide ceramics at 35 degrees Kelvin by the German Johannes Bednorz and the Swiss Karl Müller (Nobel Prize in 1987), has not yet received a generally accepted explanation, despite the enormous efforts of physicists around the world.

Another area of ​​research in which coherent quantum states are indispensable is quantum computers: only in such a state is it possible to carry out high-performance quantum calculations that are inaccessible to the most modern supercomputers.

So, coherence means the preservation of the phase difference between the folded waves. The waves themselves can be of a different nature: both light and de Broglie waves. Using the example of a gaseous BEC, we see that a coherent substance is actually new form matter previously inaccessible to man. The question arises: does the observation of coherent quantum processes in matter always require very low temperatures? Not always. There is at least one very successful example - the laser. Ambient temperature for laser operation is usually not significant, since the laser operates in conditions far from thermal equilibrium. The laser is a highly non-equilibrium system, since a flow of energy is supplied to it.

Apparently, we are still at the very beginning of research into coherent quantum processes involving a huge number of particles. One of exciting questions, to which there is no definite answer yet - do macroscopic coherent quantum processes occur in living nature? Perhaps life itself can be characterized as a special state of matter with increased coherence.

COHERENCE(from Latin cohaerentio - connection, adhesion) - a coordinated flow in space and time of several oscillatory or wave processes, in which the difference in their phases remains constant. This means that waves (sound, light, waves on the surface of water, etc.) propagate synchronously, lagging behind one another by a well-defined amount. When adding coherent oscillations, interference; the amplitude of the total oscillations is determined by the phase difference.

Harmonic vibrations are described by the expression

A(t) = A 0 cos( w t + j),

where A 0 – initial oscillation amplitude, A(t) is the amplitude at the moment of time t, w is the oscillation frequency, j is its phase.

Oscillations are coherent if their phases j 1, j 2 ... vary randomly, but their difference D j = j 1 – j 2 ... remains constant. If the phase difference changes, the oscillations remain coherent until it becomes comparable in magnitude to p.

Propagating from the source of oscillations, the wave after some time t can "forget" the initial value of its phase and become incoherent with itself. The phase change usually occurs gradually, and the time t 0, during which the value D j there is less p, is called temporal coherence. Its value is directly related to the reliability of the oscillation source: the more stable it works, the greater the temporal coherence of the oscillation.

During t 0 wave, moving at speed With, passes the distance l = t 0c, which is called the coherence length, or the length of the train, that is, a wave segment that has a constant phase. In a real plane wave, the oscillation phase changes not only along the wave propagation direction, but also in the plane perpendicular to it. In this case, one speaks of the spatial coherence of the wave.

The first definition of coherence was given by Thomas Young in 1801 when describing the laws of interference of light passing through two slits: "two parts of the same light interfere." The essence of this definition is as follows.

Conventional sources of optical radiation are composed of many atoms, ions, or molecules that spontaneously emit photons. Each act of emission lasts 10 -5 - 10 -8 seconds; they follow randomly and with randomly distributed phases both in space and in time. Such radiation is incoherent, the averaged sum of all oscillations is observed on the screen illuminated by it, and there is no interference pattern. Therefore, to obtain interference from an ordinary light source, its beam is split using a pair of slots, biprisms or mirrors set at a small angle to one another, and then both parts are brought together. In fact, here we are talking about consistency, coherence of two beams of one act of radiation that occurs randomly.

The coherence of laser radiation has a different nature. Atoms (ions, molecules) of the active substance of the laser emit stimulated radiation caused by the passage of an extraneous photon, "in time", with the same phases equal to the phase of the primary, forcing radiation ( cm. LASER).

In the broadest interpretation, coherence today is understood as the joint occurrence of two or more random processes in quantum mechanics, acoustics, radiophysics, etc.

Sergey Trankovsky

2.1.1. Conditions for maximum and minimum interference of coherent waves

coherent two waves are called, which have the same frequencies, and the phase difference does not change with time.

Light interference - spatial redistribution of the light flux when two (or several) waves are superimposed, as a result of which maxima appear in some places, and intensity minima in others.

To obtain coherent light waves, the method of dividing a wave emitted by one source into two parts is used, which, after passing through different optical paths, are superimposed on each other and an interference pattern is observed. In practice, this can be done using slits, mirrors, lasers, and screens.

Two coherent waves, arriving at a given point, cause in it harmonic vibrations:

y 1 \u003d y 01 cos (ωt + φ 1),

y 2 \u003d y 02 cos (ωt + φ 2)

If the phase difference of the indicated oscillations satisfies the equation:

∆φ ≡ φ 2 -φ 1 =2m π, (2.1)

then the amplitude of the resulting oscillation is the sum of the amplitudes of the interfering waves (see Fig. 2.1):

If the phase difference is an odd number π, i.e.:

∆φ=(2m+1) π, (2.2)

then the waves weaken each other; the amplitude of the resulting oscillation becomes equal to:

y 0 =|y 02 - y 01 |

If the amplitudes of the interfering oscillations are equal, in the first case we have:

y 0 \u003d 2y 01 \u003d 2y 02,

and in the second - y 0 =0.

Equations for two coherent waves propagating in two various environments with refractive indices n 1 and n 2 have the form:

y 1 \u003d y 01 cos (ωt-k 1 x 1),

y 2 \u003d y 02 cos (ωt-k 2 x 2),

If in the first medium the wave travels the distance x=l 1, and in the second - x=l 2, then ∆φ=k 1 l 1 -k 2 l 2 =2π(l 1 /λ 1 -l 2 /λ 2).

Because n 1 \u003d λ 0 / λ 1, and n 2 \u003d λ 0 / λ 2, where λ 0 is the wavelength in vacuum, then the conditions for maximum and minimum interference take the form:

σ ≡ n 1 l 1 -n 2 l 2 =m (λ 0 /2) 2 (2.3)

σ ≡ n 1 l 1 -n 2 l 2 =(2m+1) (λ 0 /2) (2.4)

l 1 - geometric path length of the 1st wave in the 1st medium,

n 1 l 1 is the optical path length of the 1st wave in the 1st medium,

σ is the optical path difference.

If the optical path difference (n 1 l 1 -n 2 l 2) of two interfering waves is equal to an integer number of wavelengths in vacuum(or an even number of half-waves), then the interference produces a maximum of oscillations. If the optical path difference is equal to an odd number of half-waves, then the minimum of oscillations is obtained during interference.

It is a mistake to think that at the points of the wave field, where a minimum of oscillations is observed, the wave energy disappears without a trace. In reality, there is no violation of the law of conservation of energy in this phenomenon either, since as a result of interference, only a redistribution of the energy of the wave field occurs.

2.1.2. Interference when light is reflected from thin plates

Let a flat monochromatic light wave(See Figure 2.2).

On the upper surface, the light beam is split into reflected and beams transmitted into the plate (1 and 2, respectively). If the plate is surrounded by air, the refractive index of which is assumed to be 1, then the plate, in which n>1, is an optically denser medium. When a light wave is reflected from an optically more dense environment half wave is lost. As a result, the optical path difference between the waves reflected from the bottom 3 and top- 1 plate surface is:

σ 13 \u003d 2n d - (λ 0 / 2)

If the equality σ 13 \u003d mλ 0 is satisfied, then the plate appears to us illuminated in reflected light, but if σ 13 \u003d (2m + 1) (λ 0 /2), then the plate is not visible. This phenomenon has received important practical use in "enlightenment" optical systems.

When using multilens optical systems (camera lenses, television or movie cameras, stereo tubes, binoculars, etc.), the problem arises of attenuation of the light beam that has passed through the glass system, the appearance of glare of reflected light beams. To eliminate this kind of interference, the surfaces of the lenses are covered with a thin layer of a translucent substance (see Fig. 2.3).

In this case, the layer thickness is chosen such that the reflected beams 1 and 3 extinguish each other. The material of the layer has an intermediate refractive index, i.e. n 1

The goal is achieved if:

2n 2 d =λ 0 /2.

From where: d \u003d λ 0 / (4 n 2) \u003d λ in / 4.

The wavelength of green light (the most favorable for perception by the human eye) is 0.55 microns. Therefore, the film thickness is tenths of a micrometer. (Explain on your own why coated optics in reflected light appear to us colored lilac).

2.1.3. Interference in a thin wedge

Imagine that a plane light monochromatic wave is incident on a thin wedge made of an optically transparent substance perpendicular to its base (see Fig. 2.4).

The wedge is so thin that the reflected rays 1 and 3 are almost parallel to each other vertically upwards. Viewed from above in reflected light, the wedge will appear to us as "striped", and the light stripes, alternating with dark stripes, will be parallel to the sharp edge of the wedge and will be at an equal distance from each other - x.

For two adjacent interference maxima (two adjacent fringes), we can write:

2nd - (λ 0 /2) = mλ 0

2n(d+h) - (λ 0 /2) = (m+1)λ 0

Subtracting the other from one equation, we get:

Because h = x tgφ ≈ x φ,

then 2nхφ = λ 0 .

Where does it come from:

x \u003d λ 0 / 2nφ,

consequently, the distance between adjacent light (dark) stripes is the greater, the thinner the wedge. In the limit as φ → 0, the surface of the wedge appears to us as either uniformly illuminated or uniformly darkened.

The phenomenon of interference in an optically transparent wedge has found a very important application in the technology of manufacturing optical lenses. After all, the lens is a kind of wedge (although its surfaces are not flat). Observing the surface of the lens in reflected light, it is possible to detect very slight defects by the curvature of the interference fringes - surface irregularities, glass inhomogeneity.

2.1.4. Michelson interferometer

The record accuracy in measuring the length of linear segments (displacements) is achieved using the Michelson interferometer, the scheme of which is shown in Fig. 2.5.

A beam of light from a source S falls on a translucent plate P 1 covered with a thin layer of silver. Half of the incident light flux is reflected by the plate P 1 in the direction of the beam 1, half passes through the plate and propagates in the direction of the beam 2. Beam 1 is reflected from the mirror Z 1 and returns to P 1 . Beam 2, reflected from the mirror Z 2, also returns to the plate R 1 . Beams 1 / and 2 / passed through the plate P 1 are coherent with each other and have the same intensity. The result of the interference of these beams depends on the optical path difference from the plate P 1 to the mirrors 3 1 and Z 2 and vice versa. Beam 2 passes through the thickness of the plate three times, beam 1 - only 1 time. In order to compensate for the optical path difference that arises due to this, which is different (due to dispersion) for different wavelengths and different temperatures, exactly the same as P 1, but not a silver-plated plate P 2, is placed on the path of beam 1. This equalizes the paths of beams 1 and 2 in glass. The interference pattern is observed with the help of a telescope T. By rotating the micrometer screw B, you can smoothly move the mirror 3 2 , thereby changing the optical path difference between the beams 1 / and 2 / .

2n ∆L=2 N λ 0 /2 (max), where n = 1.

Let, as a result of the rotation of the micrometer screw, the mirror Z 2 move along the measured segment by ∆L, while observing through the telescope, we recorded N interference blinks. It is easy to get ∆L=N·λ 0 /2. Whence it follows that the division value of the measuring device is λ 0 /2, i.e. for green light it is 0.27 µm.

2.1.5. Interference refractometers

Allows you to determine slight changes in the refractive index of transparent bodies depending on pressure, temperature, etc.

Two identical cuvettes of length l. One is filled with gas with a known refractive index n 0, and the other with an unknown one - n x. An additional path difference δ = (n x - n 0) ∙ l, which leads to a shift of the interference fringes. Value shows by what part of the width of the interference fringe the interference pattern has shifted. (Because δ = (n x l– n 0 ∙ l) = mλ)

Measuring m 0 (with known l, n 0 , λ), you can find n x.

coherence the coordinated flow of several oscillatory or wave processes is called. The degree of agreement may vary. Accordingly, the concept degree of coherence two waves.

Let two light waves of the same frequency come to a given point in space, which excite oscillations of the same direction at this point (both waves are polarized in the same way):

E \u003d A 1 cos (wt + a 1),

E \u003d A 2 cos (wt + a 2), then the amplitude of the resulting oscillation

A 2 \u003d A 1 2 + A 2 2 + 2A 1 A 2 cosj, (1)

where j = a 1 - a 2 = const.

If the oscillation frequencies in both waves w are the same, and the phase difference j of the excited oscillations remains constant in time, then such waves are called coherent.

When coherent waves are applied, they give a stable oscillation with constant amplitude A = сonst, determined by expression (1) and, depending on the phase difference of oscillations, lying within |а 1 –А 2 ê £ A £ а 1 +А 2.

Thus, when interfering with each other, coherent waves give a stable oscillation with an amplitude not exceeding the sum of the amplitudes of the interfering waves.

If j \u003d p, then cosj \u003d -1 and a 1 \u003d A 2, and the amplitude of the total oscillation is zero, and the interfering waves completely cancel each other out.

In the case of incoherent waves, j changes continuously, taking any values ​​with equal probability, as a result of which the time-averaged value t = 0. Therefore

A 2 >=<А 1 2 > + <А 2 2 >,

whence the intensity observed when superimposing incoherent waves is equal to the sum of the intensities created by each of the waves separately:

I \u003d I 1 + I 2.

In the case of coherent waves, cosj has a constant value in time (but different for each point in space), so that

I = I 1 + I 2 + 2Ö I 1 × I 2 cosj (2)

At those points in space for which cosj >0, I> I 1 +I 2 ; at the points for which cosj<0, IWhen superimposing coherent light waves there is a redistribution of the light flux inspace, as a result of which maxima appear in some places, and in others -intensity minima. This phenomenon is called interference waves. Interference is especially clearly manifested when the intensities of both interfering waves are the same: I 1 =I 2 . Then, according to (2), at the maxima I = 4I 1 , and at the minima I = 0. For incoherent waves, under the same condition, the same intensity is obtained everywhere I = 2I 1 .

All natural light sources (the Sun, incandescent bulbs, etc.) are not coherent.

The incoherence of natural light sources is due to the fact that the radiation of a luminous body is composed of waves emitted by many atoms. Individual atoms emit wave trains with a duration of about 10 -8 s and a length of about 3 m. train train has nothing to do with the phase of the previous train. In a light wave emitted by the body, the radiation of one group of atoms after a time of the order of 10 -8 s is replaced by the radiation of another group, and the phase of the resulting wave undergoes random changes.

Incoherent and unable to interfere with each other are the waves emitted various natural light sources. Is it even possible to create conditions for light under which interference phenomena would be observed? How, using ordinary incoherent light emitters, to create mutually coherent sources?

Coherent light waves can be obtained by dividing (using reflections or refractions) a wave emitted by one light source into two parts. If you make these two waves go through different optical paths, and then superimpose them on top of each other, interference is observed. The difference in the optical lengths of the paths traversed by the interfering waves should not be very large, since the folding oscillations should belong to the same resulting train of waves. If this difference is ³1m, oscillations corresponding to different trains will be superimposed, and the phase difference between them will continuously change in a chaotic manner.

Let the separation into two coherent waves occur at the point O (Fig. 2).

To point P, the first wave passes through the medium refractive index n 1 path S 1 , the second wave passes through a medium with a refractive index n 2 path S 2 . If at point O the phase of the oscillation is equal to wt, then the first wave will excite at point P the oscillation A 1 cosw (t - S 1 / V 1), and the second wave - the oscillation A 2 cosw (t - S 2 / V 2), where V 1 and V 2 - phase speeds. Consequently, the phase difference of the oscillations excited by the waves at the point P will be equal to

j \u003d w (S 2 / V 2 - S 1 / V 1) \u003d (w / c) (n 2 S 2 - n 1 S 1).

Let us replace w/с through 2pn/с = 2p/lo (lo is the wavelength in), then j = (2p/lo)D, where (3)

D \u003d n 2 S 2 - n 1 S 1 \u003d L 2 - L 1

is a quantity equal to the difference in the optical lengths traversed by the waves of the paths, and is called optical path difference.

From (3) it can be seen that if the optical path difference is equal to an integer number of wavelengths in vacuum:

D = ±mlo (m = 0.1.2), (4)

then the phase difference turns out to be a multiple of 2p and the oscillations excited at the point P by both waves will occur with the same phase. Thus, (4) is the condition of the interference maximum.

If the optical path difference D is equal to a half-integer number of wavelengths in vacuum:

D \u003d ± (m + 1/2)lo (m \u003d 0, 1.2, ...) (5)

then j = ± (2m + 1)p, so that the oscillations at the point P are in antiphase. Therefore, (5) is the interference minimum condition.

The principle of obtaining coherent light waves by dividing a wave into two parts passing through different paths can be practically implemented in various ways - with the help of screens and slots, mirrors and refracting bodies.

For the first time, the interference pattern from two light sources was observed in 1802 by the English scientist Jung. In Young's experiment (Fig. 3), light from a point source (small hole S) passes through two equidistant slots (holes) A ​​1 and A 2 , which are, as it were, two coherent sources (two cylindrical waves). The interference pattern is observed on the screen E, located at some distance l parallel to A 1 A 2. The reference point is chosen at point 0, which is symmetric with respect to the slots.

Amplification and attenuation of light at an arbitrary point P of the screen depends on the optical difference in the path of the rays D =L 2 - L 1 . To obtain a distinguishable interference pattern, the distance between the sources А 1 А 2 =d must be much less than the distance to the screen l. The distance x, within which interference fringes are formed, is much smaller l. Under these conditions, we can put S 2 – S 1 » 2 l. Then S 2 – S 1 » xd/ l. Multiplying by n,

Let's learn D = nxd/ l. (6)

Substituting (6) into (4) we get that the intensity maxima will be observed at x values ​​equal to x max = ± m l l/d (m = 0, 1,2,.,.).(7)

Here l = l 0 /n - wavelength in the medium filling the space between the sources and the screen.

The coordinates of the intensity minima will be:

x min \u003d ± (m + 1/2) ll / d (m \u003d 0,1,2, ...). (eight)

The distance between two adjacent intensity maxima is called distancebetween interference fringes and the distance between adjacent minima - the width of the interference fringe. From (7) and (8) it follows that the distance between the strips and the strip width have the same value equal to Dх = l l/d. (9)

By measuring the parameters included in (9), one can determine the optical radiation wavelength l. According to (9), Dx is proportional to 1/d, therefore, in order for the interference pattern to be clearly distinguishable, the above condition must be met: d<< l. The main maximum corresponding to m = 0 passes through point 0. Up and down from it at equal distances from each other are the maxima (minimums) of the first (m = 1), second (m = 2) orders, etc.

This picture is valid when the screen is illuminated with monochromatic light (l 0 = const). When illuminated with white light, the interference maxima (and minima) for each wavelength will, according to formula (9), be shifted relative to each other and look like rainbow fringes. Only for m = 0, the maxima for all wavelengths coincide, and a bright band will be observed in the middle of the screen, on both sides of which there will be symmetrically colored bands of maxima of the first, second orders, etc. (closer to the central bright band there will be zones of violet colors, then red zones).

The intensity of the interference fringes does not remain constant, but varies along the screen according to the square cosine law.

An interference pattern can be observed using a Fresnel mirror, a Loyd mirror, a Fresnel biprism and other optical devices, as well as when light is reflected from thin transparent films.

14. LIGHT INTERFERENCE DURING REFLECTION FROM THIN PLATES. STRIPS OF EQUAL THICKNESS AND EQUAL SLOPE. Interference in thin plates and films is of great practical interest.

Let a plane light wave fall from air (n air » 1) onto a thin plane-parallel plate of thickness b, made of a transparent substance with a refractive index n, which can be considered as a parallel beam of rays (Fig. 4), at an angle Q 1 to the perpendicular.

On the surface of the plate at point A, the beam is divided into two parallel beams of light, one of which is formed due to reflection from the upper surface of the plate, and the second from the lower surface. The path difference acquired by beams 1 and 2 before they converge at point C is equal to

D \u003d nS 2 - S 1 ± l 0 /2

where S 1 is the length of the segment AB, and S 2 is the total length of the segments AO and OS, and the term ± l 0 /2 is due to the loss of a half-wave when light is reflected from the interface between two media with different refractive indices.

From a geometric consideration, a formula is obtained for the optical path difference between beams1 and2:

D \u003d 2bÖ (n 2 - sin 2 Q 1) \u003d 2bn cosQ 2,

and taking into account the loss of a half-wave for the optical path difference, we obtain

D \u003d 2bÖ (n 2 - sin 2 Q 1) ± l 0 / 2 \u003d 2bn cosQ 2 ± l 0 / 2. (ten)

Due to the limitations imposed by temporal and spatial coherence, interference when the plate is illuminated, for example, by sunlight, is observed only if the thickness of the plate does not exceed a few hundredths of a millimeter. When illuminated with light with a higher degree of coherence (for example, a laser), interference is also observed when reflected from thicker plates or films.

In practice, interference from a plane-parallel plate is observed by placing a lens in the path of the reflected beams, which collects the rays at one of the points of the screen located in the focal plane of the lens (Fig. 5). Illumination at an arbitrary point P of the screen depends on the value of D determined by formula (10). When D = mо, the maxima are obtained, when D = (m + 1/2)lо, the intensity minima are obtained (m is an integer).

Let a thin plane-parallel plate be illuminated by scattered monochromatic light (Fig. 5). Let us place a lens parallel to the plate, in the focal plane of which we place the screen. Scattered light contains rays of various directions. Rays parallel to the plane of the figure and incident on the plate at an angle c), after reflection from both surfaces of the plate, will be collected by the lens at point P and create illumination at this point, determined by the value of the optical path difference.

Rays traveling in other planes, but falling on the plate at the same angle Q 1 ¢, will be collected by the lens at other points that are the same distance from the center of the screen O as the point P. The illumination at all these points will be the same. That. rays falling on the plate at the same angle Q 1 ¢ will create on the screen a set of equally illuminated points located on a circle centered at point O. Similarly, rays falling at a different angle Q "1 will create a set on the screen in the same way (but differently, since And another) of illuminated points located along a circle of a different radius.

As a result, the screen will displaya system of alternating light and dark circular stripes with a common center at a pointO). Each band is formed by rays falling on the plate under the same angle Q 1 . Therefore, the interference fringes obtained under the described conditions are called. stripes of equal slope. If the lens is positioned differently relative to the plate (the screen must always coincide with the focal plane of the lens), the shape of the bands of equal slope will be different. The lens of the eye can play the role of a lens, and the retina of the eye can play the role of a screen.

According to (10), the position of the maxima depends on lo. Therefore, in white light, a set of bands shifted relative to others, formed by rays of different colors, is obtained, and the interference pattern acquires iridescent coloration.

The interference pattern from a thin transparent wedge of variable thickness was studied by Newton. Let a parallel beam of rays fall on such a wedge (Fig. 6).

Fig.6.

Now the rays reflected from different surfaces of the wedge will not be parallel. But even in this case the reflected waves will coherent throughoutspace above the wedge, and at any distance of the screen from the wedge, an interference pattern is observed on it in the form of stripes parallel to the top of wedge 0. Each of these stripes arises as a result of reflection from sections of the wedge with the same thickness, as a result of which they are called stripes of equal thickness. In practice, stripes of equal thickness are observed by placing a lens near the wedge and a screen behind it. The role of the lens can be played by the lens, and the role of the screen can be played by the retina. When observed in white light, the bands will be colored, so that the surface of the plate or film appears to be iridescent. For example, thin films of oil and oil spread over the surface of water, as well as soap films, have such a color. notice, that interference from thin filmscan be observed not only in reflected but also in transmitted light.