Topics of the USE codifier: light interference.

In the previous leaflet devoted to the Huygens principle, we said that the overall picture of the wave process is created by the superposition of secondary waves. But what does it mean - "overlay"? What is the specific physical meaning of superposition of waves? What generally happens when several waves simultaneously propagate in space? It is to these questions that this leaflet is devoted.

Addition of vibrations.

Now we will consider the interaction of two waves. The nature of wave processes does not play a role - it can be mechanical waves in an elastic medium or electromagnetic waves (in particular, light) in a transparent medium or in a vacuum.

Experience shows that waves add to each other in the following sense.

The principle of superposition. If two waves are superimposed on each other in a certain area of ​​space, then they give rise to a new wave process. In this case, the value of the oscillating quantity at any point in this area is equal to the sum of the corresponding oscillating quantities in each of the waves separately.

For example, when two mechanical waves are superimposed, the displacement of a particle of an elastic medium is equal to the sum of the displacements created separately by each wave. When two electromagnetic waves are superimposed, the electric field strength at a given point is equal to the sum of the strengths in each wave (and the same for the magnetic field induction).

Of course, the principle of superposition is valid not only for two, but in general for any number of superimposed waves. The resulting oscillation at a given point is always equal to the sum of the oscillations generated by each wave individually.

We confine ourselves to considering the superposition of two waves of the same amplitude and frequency. This case is most often encountered in physics and, in particular, in optics.

It turns out that the amplitude of the resulting oscillation is strongly affected by the phase difference of the folding oscillations. Depending on the phase difference at a given point in space, two waves can either reinforce each other or completely cancel each other out!

Suppose, for example, that at some point the phases of oscillations in superimposed waves coincide (Fig. 1).

We see that the maxima of the red wave fall exactly on the maxima of the blue wave, the minima of the red wave - on the minima of the blue one (left side of Fig. 1). Adding up in phase, the red and blue waves amplify each other, generating oscillations of double amplitude (on the right in Fig. 1).

Now let's shift the blue sinusoid relative to the red one by half a wavelength. Then the maxima of the blue wave will coincide with the minima of the red wave and vice versa - the minima of the blue wave will coincide with the maxima of the red wave (Fig. 2, left).

The vibrations created by these waves will occur, as they say, in out of phase- the phase difference of the oscillations will be equal to . The resulting fluctuation will be equal to zero, i.e., the red and blue waves will simply destroy each other (Fig. 2, right).

coherent sources.

Let there be two point sources that create waves in the surrounding space. We believe that these sources are consistent with each other in the following sense.

coherence. Two sources are said to be coherent if they have the same frequency and a constant, time-independent phase difference. The waves generated by such sources are also called coherent.

So, we consider two coherent sources and . For simplicity, we assume that the sources emit waves of the same amplitude, and the phase difference between the sources is zero. In general, these sources are "exact copies" of each other (in optics, for example, a source serves as an image of a source in some optical system).

The superposition of the waves emitted by these sources is observed at some point . Generally speaking, the amplitudes of these waves at a point will not be equal to each other - after all, as we remember, the amplitude of a spherical wave is inversely proportional to the distance to the source, and at different distances the amplitudes of the arriving waves will turn out to be different. But in many cases, the point is located far enough from the sources - at a distance much greater than the distance between the sources themselves. In such a situation, the difference in the distances and does not lead to a significant difference in the amplitudes of the incoming waves. Therefore, we can assume that the amplitudes of the waves at the point also coincide.

Maximum and minimum condition.

However, the quantity called stroke difference, is of paramount importance. What decisively depends on it is what result of the addition of incoming waves we will see at the point .

In the situation in Fig. 3 the path difference is equal to the wavelength . Indeed, three full waves fit on a segment, and four on a segment (this, of course, is only an illustration; in optics, for example, the length of such segments is about a million wavelengths). It is easy to see that the waves at a point add up in phase and create oscillations of double amplitude - it is observed, as they say, interference maximum.

It is clear that a similar situation will arise for a path difference equal not only to the wavelength, but also to any integer number of wavelengths.

Maximum condition . When coherent waves are superimposed, oscillations at a given point will have a maximum amplitude if the path difference is equal to an integer number of wavelengths:

(1)

Now let's look at fig. four . Two and a half waves fit on the segment, and three waves on the segment. The path difference is half the wavelength (d=\lambda /2 ).

Now it is easy to see that the waves at the point add up in antiphase and cancel each other - it is observed interference minimum. The same will happen if the path difference turns out to be equal to half the wavelength plus any integer number of wavelengths.

Minimum condition .
Coherent waves, adding up, cancel each other if the path difference is equal to a half-integer number of wavelengths:

(2)

Equality (2) can be rewritten as follows:

Therefore, the minimum condition is also formulated as follows: the path difference must be equal to an odd number of half-wavelengths.

interference pattern.

But what if the path difference takes some other value, not equal to an integer or half-integer number of wavelengths? Then the waves arriving at this point create oscillations in it with some intermediate amplitude located between zero and the doubled value 2A of the amplitude of one wave. This intermediate amplitude can take on all values ​​from 0 to 2A as the path difference changes from half an integer to an integer number of wavelengths.

Thus, in the region of space where the waves of coherent sources and are superimposed, a stable interference pattern is observed - a fixed time-independent distribution of oscillation amplitudes. Namely, at each point of a given region, the oscillation amplitude takes on its own value, determined by the difference in the path of the waves arriving here, and this amplitude value does not change with time.

Such stationarity of the interference pattern is ensured by the coherence of the sources. If, for example, the phase difference of the sources is constantly changing, then no stable interference pattern will arise.

Now, finally, we can say what interference is.

Interference - this is the interaction of waves, as a result of which a stable interference pattern arises, that is, a time-independent distribution of the amplitudes of the resulting oscillations at the points of the region where the waves overlap each other.

If the waves, overlapping, form a stable interference pattern, then they simply say that the waves interfere. As we found out above, only coherent waves can interfere. When, for example, two people are talking, we do not notice alternations of highs and lows of loudness around them; there is no interference, since in this case the sources are incoherent.

At first glance, it may seem that the phenomenon of interference contradicts the law of conservation of energy - for example, where does the energy go when the waves completely cancel each other out? But, of course, there is no violation of the energy conservation law: the energy is simply redistributed between different parts of the interference pattern. The largest amount of energy is concentrated in the interference maxima, and no energy enters the points of interference minima at all.

On fig. 5 shows the interference pattern created by superimposing the waves of two point sources and . The picture is built on the assumption that the region of observation of the interference is far enough from the sources. The dotted line marks the symmetry axis of the interference pattern.

The colors of the dots of the interference pattern in this figure change from black to white through intermediate shades of gray. Black color - interference minima, white color - interference maxima; gray color is an intermediate value of the amplitude, and the greater the amplitude at a given point, the brighter the point itself.

Notice the straight white stripe that runs along the painting's axis of symmetry. Here are the so-called central highs. Indeed, any point of this axis is equidistant from the sources (the path difference is zero), so that at this point an interference maximum will be observed.

The remaining white stripes and all black stripes are slightly curved; it can be shown that they are branches of hyperbolas. However, in the region located at a great distance from the sources, the curvature of the white and black stripes is hardly noticeable, and these stripes look almost straight.

The interference experience shown in Fig. 5 , together with the corresponding method for calculating the interference pattern is called Young's scheme. This scheme underlies the famous
Young's experience (which will be discussed in the topic Diffraction of light). Many experiments on the interference of light in one way or another are reduced to the Young scheme.

In optics, the interference pattern is usually observed on a screen. Let's take a look at Fig. 5 and imagine a screen placed perpendicular to the dotted axis.
On this screen we will see the alternation of light and dark interference fringes.

On fig. 6 sinusoid shows the distribution of illumination along the screen. At the point O, located on the axis of symmetry, there is a central maximum. The first maximum at the top of the screen, adjacent to the central one, is at point A. Above are the second, third (and so on) maximums.

Rice. 6. Interference pattern on the screen

The distance equal to the distance between any two adjacent highs or lows is called fringe width. Now we are going to find this value.

Let the sources be at a distance from each other, and the screen is located at a distance from the sources (Fig. 7). Screen replaced by axis ; the origin, as above, corresponds to the central maximum.

The points and serve as projections of the points and onto the axis and are located symmetrically with respect to the point . We have: .

The point of observation can be anywhere on the axis (on the screen). point coordinate
we denote . We are interested at what values ​​at the point the interference maximum will be observed.

The wave emitted by the source travels the distance:

. (3)

Now remember that the distance between the sources is much less than the distance from the sources to the screen: . In addition, in such interference experiments, the coordinate of the observation point is also much smaller. This means that the second term under the root in expression (3) is much less than one:

If so, you can use the approximate formula:

(4)

Applying it to expression (4) , we get:

(5)

In the same way, we calculate the distance that the wave travels from the source to the observation point:

. (6)

Applying the approximate formula (4) to expression (6), we obtain:

. (7)

Subtracting expressions (7) and (5) , we find the path difference:

. (8)

Let be the wavelength emitted by the sources. According to condition (1), an interference maximum will be observed at a point if the path difference is equal to an integer number of wavelengths:

From here we get the coordinates of the maxima in the upper part of the screen (in the lower part the maxima are symmetrical):

At , we obtain, of course, (central maximum). The first maximum near the central corresponds to the value and has the coordinate . The width of the interference fringe will be the same.

Let us now consider a situation where there is not one, but several sources of waves (oscillators). The waves emitted by them in a certain region of space will have a cumulative effect. Before starting an analysis of what can happen as a result, let us first dwell on a very important physical principle, which we will repeatedly use in our course - principle of superposition. Its essence is simple.

Let's assume that there is not one, but several sources of perturbation (they can be mechanical oscillators, electric charges, etc.). What will be noted by a device that simultaneously registers disturbances of the medium from all sources? If the components of a complex impact process do not mutually influence each other, then the resulting effect will be the sum of the effects caused by each impact separately, regardless of the presence of the others - this is the principle of superposition, i.e. overlays. This principle is the same for many phenomena, but its mathematical notation may be different depending on the nature of the phenomena under consideration - vector or scalar.

The principle of superposition of waves is not fulfilled in all cases, but only in the so-called linear media. The medium, for example, can be considered linear, if its particles are under the action of an elastic (quasi-elastic) restoring force. Environments in which the superposition principle does not hold are called non-linear. Thus, when waves of high intensity propagate, a linear medium can become nonlinear. Extremely interesting and technically important phenomena arise. This is observed during the propagation of high-power ultrasound (in acoustics) or laser beams in crystals (in optics) in a medium. The scientific and technical areas involved in the study of these phenomena are called nonlinear acoustics and nonlinear optics, respectively.

We will consider only linear effects. As applied to waves, the principle of superposition states that each of them?, (x, t) propagates regardless of whether there are sources of other waves in the given medium or not. Mathematically, in the case of propagation N waves along the axis X, it is expressed like this

where c(x, 1)- total (resulting) wave.

Consider the superposition of two monochromatic waves of the same frequency w and polarization propagating in the same direction (axis X) from two sources

We will observe the result of their addition at a certain point M, those. fix the coordinate x = x m in the equations describing both waves:

At the same time, we eliminated the double periodicity of the process and turned the waves into oscillations occurring at one point. M with one time period T= 2l/co and differing in the initial phases Ф, = to g x m and f 2 = krs m, those.

and

Now to find the resulting process t(t) at the point M we have to add 2,! and q2: W)= ^i(0 + с 2 (0- We can use the results obtained earlier in subsection 2.3.1. Using formula (2.21), we obtain the amplitude of the total oscillation BUT, expressed through BUT, f! and A 2 , fg like

Meaning A m(the amplitude of the total oscillation at the point M) depends on the difference in the phases of the oscillations Af = f 2 - f). What happens in the case of different values ​​of Φ is discussed in detail in subsection 2.3.1. In particular, if this difference Af remains constant all the time, then, depending on its value, it may turn out that in the case of equality of the amplitudes BUT = A 2 \u003d A resulting amplitude A m will be zero or 2 BUT.

In order for the phenomenon of an increase or decrease in amplitude when waves are superimposed (interference) to be observed, it is necessary, as already mentioned, that the phase difference Df \u003d f 2 - f! remained constant. This requirement means that fluctuations must be coherent. The sources of fluctuations are called coherent", if the phase difference of the oscillations excited by them does not change over time. The waves generated by such sources are also coherent. In addition, it is necessary that the combined waves be equally polarized, i.e. so that the displacements of particles in them occur, for example, in one plane.

It can be seen that the implementation of wave interference requires compliance with several conditions. In wave optics, this means the creation of coherent sources and the implementation of a method for combining the waves excited by them.

1 Distinguish between coherence (from lat. cohaerens- “in connection”) temporal, associated with the monochromaticity of waves, which is discussed in this section, and spatial coherence, the violation of which is typical for extended radiation sources (heated bodies, in particular). We do not consider the features of spatial coherence (and incoherence).

Standing wave equation.

As a result of the superposition of two opposite plane waves with the same amplitude, the resulting oscillatory process is called standing wave . Practically standing waves arise when reflected from obstacles. Let's write the equations of two plane waves propagating in opposite directions (initial phase):

Let's add the equations and transform according to the formula of the sum of cosines: . Because , then we can write: . Considering that , we get standing wave equation : . The expression for the phase does not include the coordinate, so you can write: , where the total amplitude .

Wave interference- such an imposition of waves, in which their mutual amplification, stable in time, occurs at some points in space and attenuation at others, depending on the ratio between the phases of these waves. The necessary conditions to observe interference:

1) the waves must have the same (or close) frequencies so that the picture resulting from the superposition of the waves does not change in time (or does not change very quickly so that it can be registered in time);

2) waves must be unidirectional (or have a similar direction); two perpendicular waves will never interfere. In other words, the added waves must have the same wave vectors. Waves for which these two conditions are satisfied are called coherent. The first condition is sometimes called temporal coherence, second - spatial coherence. Consider as an example the result of adding two identical unidirectional sinusoids. We will vary only their relative shift. If the sinusoids are located so that their maxima (and minima) coincide in space, their mutual amplification will occur. If the sinusoids are shifted relative to each other by half a period, the maxima of one will fall on the minima of the other; sinusoids will destroy each other, that is, their mutual weakening will occur. We add two waves:

here x 1 and x 2- distances from the wave sources to the point in space where we observe the result of the overlay. The square of the amplitude of the resulting wave is given by:

The maximum of this expression is 4A2, minimum - 0; everything depends on the difference in the initial phases and on the so-called wave path difference D:

At at a given point in space, an interference maximum will be observed, at - an interference minimum. If we move the observation point away from the straight line connecting the sources, we will find ourselves in a region of space where the interference pattern changes from point to point. In this case, we will observe the interference of waves with equal frequencies and close wave vectors.

Electromagnetic waves. Electromagnetic radiation is a perturbation (change of state) of the electromagnetic field (that is, electric and magnetic fields interacting with each other) that propagates in space. Among electromagnetic fields in general, generated by electric charges and their movement, it is customary to attribute to radiation that part of alternating electromagnetic fields that is capable of propagating the farthest from its sources - moving charges, fading most slowly with distance. Electromagnetic radiation is divided into radio waves, infrared radiation, visible light, ultraviolet radiation, x-rays and gamma rays. Electromagnetic radiation can propagate in almost all environments. In a vacuum (a space free from matter and bodies that absorb or emit electromagnetic waves), electromagnetic radiation propagates without attenuation over arbitrarily large distances, but in some cases it propagates quite well in a space filled with matter (although somewhat changing its behavior). The main characteristics of electromagnetic radiation are considered to be frequency, wavelength and polarization. Wavelength is directly related to frequency through the (group) velocity of radiation. The group velocity of propagation of electromagnetic radiation in vacuum is equal to the speed of light, in other media this speed is less. The phase velocity of electromagnetic radiation in vacuum is also equal to the speed of light, in various media it can be either less or more than the speed of light.

What is the nature of light. Light interference. Coherence and monochromaticity of light waves. Application of light interference. Diffraction of light. Huygens-Fresnel principle. Fresnel zone method. Fresnel diffraction by a circular hole. dispersion of light. Electronic theory of light dispersion. polarization of light. Natural and polarized light. The degree of polarization. Polarization of light during reflection and refraction at the interface of two dielectrics. Polaroids

What is the nature of light. The first theories about the nature of light - corpuscular and wave - appeared in the middle of the 17th century. According to the corpuscular theory (or the theory of expiration), light is a stream of particles (corpuscles) that are emitted by a light source. These particles move in space and interact with matter according to the laws of mechanics. This theory well explained the laws of rectilinear propagation of light, its reflection and refraction. The founder of this theory is Newton. According to the wave theory, light is elastic longitudinal waves in a special medium that fills all space - the luminiferous ether. The propagation of these waves is described by the Huygens principle. Each point of the ether, to which the wave process has reached, is a source of elementary secondary spherical waves, the envelope of which forms a new front of the ether oscillations. The hypothesis about the wave nature of light was put forward by Hooke, and it was developed in the works of Huygens, Fresnel, and Young. The concept of elastic ether has led to irresolvable contradictions. For example, the phenomenon of light polarization showed. that light waves are transverse. Elastic transverse waves can propagate only in solids where shear deformation takes place. Therefore, the ether must be a solid medium, but at the same time not impede the movement of space objects. The exotic properties of the elastic ether was a significant shortcoming of the original wave theory. The contradictions of the wave theory were resolved in 1865 by Maxwell, who came to the conclusion that light is an electromagnetic wave. One of the arguments in favor of this statement is the coincidence of the speed of electromagnetic waves, theoretically calculated by Maxwell, with the speed of light, determined experimentally (in the experiments of Roemer and Foucault). According to modern concepts, light has a dual corpuscular-wave nature. In some phenomena, light reveals the properties of waves, and in others, the properties of particles. Wave and quantum properties complement each other.

Wave interference.
is the superposition of coherent waves
- characteristic of waves of any nature (mechanical, electromagnetic, etc.

coherent waves are waves emitted by sources having the same frequency and constant phase difference. When coherent waves are superimposed at any point in space, the amplitude of oscillations (displacement) of this point will depend on the difference in distances from the sources to the point under consideration. This distance difference is called the path difference.
When coherent waves are superimposed, two limiting cases are possible:
1) Maximum condition: The path difference is equal to an integer number of wavelengths (otherwise an even number of half-wavelengths).
where . In this case, the waves at the considered point come with the same phases and reinforce each other - the amplitude of oscillations of this point is maximum and equal to the doubled amplitude.

2) Minimum condition: The path difference is equal to an odd number of half-wavelengths. where . The waves arrive at the point under consideration in antiphase and cancel each other out. The oscillation amplitude of this point is equal to zero. As a result of the superposition of coherent waves (wave interference), an interference pattern is formed. When waves interfere, the amplitude of oscillations of each point does not change in time and remains constant. When incoherent waves are superimposed, there is no interference pattern, because the amplitude of oscillations of each point changes with time.

Coherence and monochromaticity of light waves. The interference of light can be explained by considering the interference of waves. A necessary condition for the interference of waves is their coherence, i.e., the coordinated flow in time and space of several oscillatory or wave processes. This condition is satisfied monochromatic waves- Unlimited in space waves of one definite and strictly constant frequency. Since no real source produces strictly monochromatic light, the waves emitted by any independent light sources are always incoherent. In two independent light sources, the atoms radiate independently of each other. In each of these atoms, the radiation process is finite and lasts a very short time ( t » 10–8 s). During this time, the excited atom returns to its normal state and the emission of light ceases. Excited again, the atom again begins to emit light waves, but with a new initial phase. Since the phase difference between the radiation of two such independent atoms changes with each new act of emission, the waves spontaneously emitted by atoms of any light source are incoherent. Thus, the waves emitted by atoms have approximately constant amplitude and phase of oscillations only during a time interval of 10–8 s, while both amplitude and phase change over a longer period of time.

Application of light interference. The phenomenon of interference is due to the wave nature of light; its quantitative patterns depend on the wavelength l 0 . Therefore, this phenomenon is used to confirm the wave nature of light and to measure wavelengths. The phenomenon of interference is also used to improve the quality of optical devices ( enlightenment of optics) and obtaining highly reflective coatings. The passage of light through each refractive surface of the lens, for example, through the glass-air interface, is accompanied by a reflection of ≈4% of the incident flux (with a refractive index of glass ≈1.5). Since modern lenses contain a large number of lenses, the number of reflections in them is large, and therefore the loss of light flux is also large. Thus, the intensity of the transmitted light is attenuated and the luminosity of the optical device decreases. In addition, reflections from lens surfaces lead to glare, which often (for example, in military technology) unmasks the position of the device. To eliminate these shortcomings, the so-called illumination of the optics. To do this, thin films with a refractive index lower than that of the lens material are applied to the free surfaces of the lenses. When light is reflected from the air-film and film-glass interfaces, interference of coherent rays occurs. Film thickness d and refractive indices of glass n c and film n can be chosen so that the waves reflected from both surfaces of the film cancel each other out. To do this, their amplitudes must be equal, and the optical path difference is equal to . The calculation shows that the amplitudes of the reflected rays are equal if Since n With, n and the refractive index of air n 0 satisfy the conditions n c > n>n 0 , then the loss of the half-wave occurs on both surfaces; therefore, the minimum condition (we assume that the light falls normally, i.e. i= 0), , where nd-optical film thickness. Usually accepted m=0, then

Diffraction of light. Huygens-Fresnel principle.Diffraction of light- deviation of light waves from rectilinear propagation, rounding of encountered obstacles. Qualitatively, the phenomenon of diffraction is explained on the basis of the Huygens-Fresnel principle. The wave surface at any moment of time is not just an envelope of secondary waves, but the result of interference. Example. A plane light wave incident on an opaque screen with a hole. Behind the screen, the front of the resulting wave (the envelope of all secondary waves) is bent, as a result of which the light deviates from the original direction and enters the region of the geometric shadow. The laws of geometric optics are satisfied accurately only if the dimensions of the obstacles in the path of light propagation are much greater than the wavelength of the light wave: Diffraction occurs when the dimensions of the obstacles are commensurate with the wavelength: L ~ L. The diffraction pattern obtained on a screen located behind various obstacles, is the result of interference: the alternation of light and dark bands (for monochromatic light) and multi-colored bands (for white light). Diffraction grating - an optical device, which is a collection of a large number of very narrow slits separated by opaque gaps. The number of strokes in good diffraction gratings reaches several thousand per 1 mm. If the width of the transparent gap (or reflective stripes) is a, and the width of the opaque gaps (or light-scattering stripes) is b, then the value d = a + b is called lattice period.

Stronger evidence is needed that light propagates like a wave as it propagates. Any wave motion is characterized by the phenomena of interference and diffraction. In order to be sure that light has a wave nature, it is necessary to find experimental evidence of the interference and diffraction of light.

Interference is a rather complex phenomenon. To better understand its essence, we will first dwell on the interference of mechanical waves.

The addition of waves. Very often several different waves simultaneously propagate in the medium. For example, when several people are talking in a room, the sound waves are superimposed on each other. What is happening?

The easiest way to follow the superposition of mechanical waves is to observe the waves on the surface of the water. If we throw two stones into the water, thus creating two annular waves, it is easy to see that each wave passes through the other and behaves further as if the other wave did not exist at all. Similarly, any number of sound waves can simultaneously propagate through the air without interfering with each other. Many musical instruments in an orchestra or voices in a choir create sound waves that are simultaneously picked up by our ear. Moreover, the ear is able to distinguish one sound from another.

Now let's take a closer look at what happens in places where the waves overlap. Observing the waves on the surface of the water from two stones thrown into the water, one can notice that some parts of the surface are not disturbed, while in other places the disturbance has intensified. If two waves meet in one place with crests, then in this place the perturbation of the water surface increases.

If, on the contrary, the crest of one wave meets the trough of another, then the surface of the water will not be perturbed.

In general, at each point of the medium, the oscillations caused by two waves simply add up. The resulting displacement of any particle of the medium is an algebraic (i.e., taking into account their signs) sum of displacements that would occur during the propagation of one of the waves in the absence of the other.

Interference. The addition in space of waves, in which a time-constant distribution of the amplitudes of the resulting oscillations is formed, is called interference.

Let us find out under what conditions the interference of waves takes place. To do this, consider in more detail the addition of waves formed on the surface of the water.

You can simultaneously excite two circular waves in the bath with the help of two balls mounted on a rod that performs harmonic oscillations (Fig. 118). At any point M on the surface of the water (Fig. 119), oscillations caused by two waves (from sources O 1 and O 2) will add up. The amplitudes of the oscillations caused at the point M by both waves will, generally speaking, be different, since the waves travel different paths d 1 and d 2 . But if the distance l between the sources is much less than these paths (l « d 1 and l « d 2), then both amplitudes
can be considered almost the same.

The result of the addition of waves arriving at point M depends on the phase difference between them. After passing various distances d 1 and d 2 , the waves have a path difference Δd = d 2 -d 1 . If the path difference is equal to the wavelength λ, then the second wave is delayed compared to the first by exactly one period (just in a period the wave travels a distance equal to the wavelength). Consequently, in this case, the crests (as well as the troughs) of both waves coincide.

Maximum condition. Figure 120 shows the time dependence of the displacements X 1 and X 2 caused by two waves at Δd= λ. The phase difference of the oscillations is equal to zero (or, what is the same, 2n, since the period of the sine is 2n). As a result of the addition of these oscillations, a resulting oscillation with a doubled amplitude arises. The fluctuations of the resulting displacement in the figure are shown in color (dotted line). The same will happen if not one, but any integer number of wavelengths fits on the segment Δd.

The amplitude of oscillations of the medium at a given point is maximum if the difference between the paths of two waves that excite oscillations at this point is equal to an integer number of wavelengths:

where k=0,1,2,....

Minimum condition. Let now half of the wavelength fit on the segment Δd. Obviously, in this case, the second wave lags behind the first by half a period. The phase difference turns out to be equal to n, i.e., the oscillations will occur in antiphase. As a result of the addition of these oscillations, the amplitude of the resulting oscillation is equal to zero, that is, there are no oscillations at the considered point (Fig. 121). The same thing will happen if any odd number of half-waves fits on the segment.

The amplitude of oscillations of the medium at a given point is minimal if the difference between the paths of two waves that excite oscillations at this point is equal to an odd number of half-waves:

If the stroke difference d 2 - d 1 takes an intermediate value
between λ and λ/2, then the amplitude of the resulting oscillation takes on some intermediate value between the doubled amplitude and zero. But most importantly, the amplitude of oscillation at any point he changes with time. On the water surface, a certain, time-invariant distribution of oscillation amplitudes occurs, which is called an interference pattern. Figure 122 shows a drawing from a photograph of the interference pattern of two circular waves from two sources (black circles). The white areas in the middle of the photo correspond to the swing highs, while the dark areas correspond to the lows.

coherent waves. For the formation of a stable interference pattern, it is necessary that the wave sources have the same frequency and the phase difference of their oscillations be constant.

Sources that satisfy these conditions are called coherent. Waves created by them are also called coherent. It is only when coherent waves are added that a stable interference pattern is formed.

If the difference in the phases of the oscillations of the sources does not remain constant, then at any point in the medium the difference in the phases of the oscillations excited by two waves will change. Therefore, the amplitude of the resulting oscillations changes over time. As a result, the maxima and minima move in space and the interference pattern is blurred.

Distribution of energy during interference. Waves carry energy. What happens to this energy when the waves are canceled by each other? Maybe it turns into other forms and heat is released in the minima of the interference pattern? Nothing like this. The presence of a minimum at a given point in the interference pattern means that the energy does not enter here at all. As a result of interference, the energy is redistributed in space. It is not distributed evenly over all particles of the medium, but is concentrated in the maxima due to the fact that it does not enter the minima at all.

INTERFERENCE OF LIGHT WAVES

If light is a stream of waves, then the phenomenon of light interference should be observed. However, it is impossible to obtain an interference pattern (alternating maxima and minima of illumination) using two independent light sources, for example, two electric light bulbs. Turning on another bulb only increases the illumination of the surface, but does not create an alternation of minima and maxima of illumination.

Let us find out what is the reason for this and under what conditions it is possible to observe the interference of light.

Condition of coherence of light waves. The reason is that the light waves emitted by different sources are not coordinated with each other. To obtain a stable interference pattern, matched waves are needed. They must have the same wavelengths and a constant phase difference at any point in space. Recall that such matched waves with the same wavelengths and a constant phase difference are called coherent.

Almost exact equality of wavelengths from two sources is not difficult to achieve. To do this, it is sufficient to use good light filters that transmit light in a very narrow wavelength range. But it is impossible to implement the constancy of the phase difference from two independent sources. Atoms of sources radiate light independently of each other in separate "snatches" (trains) of sinusoidal waves, having a length of about a meter. And such trains of waves from both sources are superimposed on each other. As a result, the amplitude of oscillations at any point in space changes chaotically with time, depending on how the trains of waves from different sources are shifted relative to each other in phase at a given time. Waves from different light sources are incoherent due to the fact that the phase difference of the waves does not remain constant. No stable picture with a certain distribution of illumination maxima and minima in space is observed.

Interference in thin films. Nevertheless, the interference of light can be observed. The curiosity is that it was observed a very long time ago, but they just did not realize it.

You, too, have seen the interference pattern many times when, as a child, you had fun blowing soap bubbles or watching the iridescent overflow of colors of a thin film of kerosene or oil on the surface of the water. “A soap bubble floating in the air... lights up with all the shades of colors inherent in the surrounding objects. The soap bubble is perhaps the most exquisite miracle of nature” (Mark Twain). It is the interference of light that makes the soap bubble so admirable.

The English scientist Thomas Young was the first to come up with a brilliant idea about the possibility of explaining the colors of thin films by adding waves 1 and 2 (Fig. 123), one of which (1) is reflected from the outer surface of the film, and the second (2) from the inner one. In this case, the interference of light waves occurs - the addition of two waves, as a result of which a stable pattern of amplification or weakening of the resulting light vibrations at various points in space is observed in time. The result of interference (strengthening or weakening of the resulting vibrations) depends on the angle of incidence of light on the film, its thickness and wavelength. Amplification of light will occur if the refracted wave 2 lags behind the reflected wave 1 by an integer number of wavelengths. If the second wave lags behind the first by half a wavelength or by an odd number of half-waves, then the light will be attenuated.

The coherence of the waves reflected from the outer and inner surfaces of the film is ensured by the fact that they are parts of the same light beam. The train of waves from each emitting atom is divided by the film into two, and then these parts are brought together and interfere.

Jung also realized that the difference in color is due to the difference in the wavelength (or frequency) of the light waves. Light beams of different colors correspond to waves of different lengths. For mutual amplification of waves differing from each other in length (the angles of incidence are assumed to be the same), different film thicknesses are required. Therefore, if the film has an unequal thickness, then when it is illuminated with white light, different colors should appear.

A simple interference pattern occurs in a thin layer of air between a glass plate and a plano-convex lens placed on it, the spherical surface of which has a large radius of curvature. This interference pattern has the appearance of concentric rings, called Newton's rings.

Take a plano-convex lens with a small curvature of the spherical surface and put it on a glass plate. Carefully examining the flat surface of the lens (preferably through a magnifying glass), you will find a dark spot at the point of contact between the lens and the plate and a set of small iridescent rings around it. The distances between adjacent rings rapidly decrease with increasing their radius (Fig. 111). These are Newton's rings. Newton observed and studied them not only in white light, but also when the lens was illuminated with a single-color (monochromatic) beam. It turned out that the radii of rings of the same serial number increase when moving from the violet end of the spectrum to the red; red rings have a maximum radius. All this you can check with the help of independent observations.

Newton could not satisfactorily explain why rings arise. Jung succeeded. Let's follow the course of his reasoning. They are based on the assumption that light is waves. Let us consider the case when a wave of a certain length falls almost perpendicularly onto a plano-convex lens (Fig. 124). Wave 1 appears as a result of reflection from the convex surface of the lens at the glass-air interface, and wave 2 - as a result of reflection from the plate at the air-glass interface. These waves are coherent: they have the same length and a constant phase difference, which occurs due to the fact that wave 2 travels a longer distance than wave 1. If the second wave lags behind the first by an integer number of wavelengths, then, adding up, the waves amplify each friend. The vibrations they cause occur in one phase.

On the contrary, if the second wave lags behind the first by an odd number of half-waves, then the oscillations caused by them will occur in opposite phases and the waves will cancel each other out.

If the radius of curvature R of the lens surface is known, then it is possible to calculate at what distances from the point of contact of the lens with the glass plate the path differences are such that waves of a certain length λ cancel each other out. These distances are the radii of Newton's dark rings. After all, the lines of constant thickness of the air gap are circles. By measuring the radii of the rings, the wavelengths can be calculated.

The length of the light wave. For red light, measurements give λcr = 8 10 -7 m, and for violet - λ f = 4 10 -7 m. Wavelengths corresponding to other colors of the spectrum take intermediate values. For any color, the wavelength of light is very small. Imagine an average sea wave a few meters long, which has increased so much that it occupied the entire Atlantic Ocean from the coast of America to Europe. The wavelength of light at the same magnification would only slightly exceed the width of this page.

The phenomenon of interference not only proves that light has wave properties, but also allows you to measure the wavelength. Just as the pitch of a sound is determined by its frequency, the color of light is determined by its frequency or wavelength.

Outside of us in nature there are no colors, there are only waves of different lengths. The eye is a complex physical device capable of detecting a difference in color, which corresponds to a very small (about 10 -6 cm) difference in the length of light waves. Interestingly, most animals are unable to distinguish colors. They always see a black and white picture. Color blind people also do not distinguish colors - people suffering from color blindness.

When light passes from one medium to another, the wavelength changes. It can be found like this. Let us fill the air gap between the lens and the plate with water or another transparent liquid with a refractive index. The radii of the interference rings will decrease.

Why is this happening? We know that when light passes from vacuum to some medium, the speed of light decreases by n times. Since v = λv, then either the frequency or the wavelength should decrease by n times. But the radii of the rings depend on the wavelength. Therefore, when light enters a medium, it is the wavelength that changes n times, not the frequency.

Interference of electromagnetic waves. In experiments with a microwave generator, interference of electromagnetic (radio) waves can be observed.

The generator and receiver are placed opposite each other (Fig. 125). Then a metal plate is brought from below in a horizontal position. Gradually raising the plate, alternately attenuation and amplification of sound are found.

The phenomenon is explained as follows. Part of the wave from the generator horn directly enters the receiving horn. The other part of it is reflected from the metal plate. By changing the location of the plate, we change the path difference between the direct and reflected waves. As a result, the waves either strengthen or weaken each other, depending on whether the path difference is equal to an integer number of wavelengths or an odd number of half-waves.

Observation of the interference of light proves that light, when propagating, exhibits wave properties. Interference experiments make it possible to measure the wavelength of light: it is very small, from 4 10 -7 to 8 10 -7 m.

Interference of two waves. Fresnel biprism - 1

Wave interference(from lat. inter- mutually, among themselves and ferio- I hit, I hit) - mutual strengthening or weakening of two (or more) waves when they are superimposed on each other while simultaneously propagating in space.

Usually under interference effect understand the fact that the resulting intensity at some points in space is greater, at others - less than the total intensity of the waves.

Wave interference- one of the main properties of waves of any nature: elastic, electromagnetic, including light, etc.

Interference of mechanical waves.

The addition of mechanical waves - their mutual superposition - is easiest to observe on the surface of the water. If you excite two waves by throwing two stones into the water, then each of these waves behaves as if the other wave does not exist. Sound waves from different independent sources behave similarly. At each point in the medium, the oscillations caused by the waves simply add up. The resulting displacement of any particle of the medium is an algebraic sum of displacements that would occur during the propagation of one of the waves in the absence of the other.

If at the same time at two points About 1 and About 2 excite two coherent harmonic waves in the water, then ridges and troughs will be observed on the surface of the water that do not change with time, i.e. there will be interference.

The condition for the occurrence of the maximum intensity at some point M located at distances d 1 and d 2 from wave sources About 1 and About 2, the distance between which l ≪ d 1 and l ≪d2(Figure below) will be:

Δd = kλ,

where k = 0, 1 , 2 , a λ — wavelength.

The amplitude of oscillations of the medium at a given point is maximum if the difference between the paths of two waves that excite oscillations at this point is equal to an integer number of wavelengths and provided that the phases of the oscillations of the two sources coincide.

Under the travel difference Δd here they understand the geometric difference in the paths that waves travel from two sources to the point in question: Δd =d2- d 1 . With a travel difference Δd = kλ the phase difference of the two waves is equal to an even number π , and the oscillation amplitudes will add up.

Minimum condition is:

Δd = (2k + 1)λ/2.

The amplitude of the oscillations of the medium at a given point is minimal if the difference between the paths of the two waves that excite oscillations at this point is equal to an odd number of half-waves and provided that the phases of the oscillations of the two sources coincide.

The phase difference of the waves in this case is equal to an odd number π , i.e., oscillations occur in antiphase, therefore, they are extinguished; the amplitude of the resulting oscillation is zero.

Distribution of energy during interference.

As a result of interference, the energy is redistributed in space. It concentrates in the highs due to the fact that it does not enter the lows at all.