Often several waves propagate in a substance at the same moment in time. In this case, any particle of matter that falls into this complex wave field undergoes vibrations that are the result of each of the wave processes under consideration. The total displacement of a particle of matter at an arbitrary moment in time is the geometric sum of the displacements that are caused by each of the individual oscillation processes. Each wave propagates through matter as if other wave processes did not exist. The law of addition of waves (oscillations) is called the principle of superposition or the principle of independent superposition of waves on each other. An example of independent addition of oscillations is the addition of oscillations of sound waves when an orchestra plays. By listening to it, you can distinguish the sound of individual instruments. If the principle of superposition were not fulfilled, then music would not be possible.
Determination of wave interference
DEFINITION
The addition of oscillations in which they mutually strengthen or weaken each other is called interference.
Translated from French, interferer means to interfere.
Interference of waves occurs when oscillations in waves occur at the same frequencies, the same directions of particle displacement and a constant phase difference. Or, in other words, with coherence of wave sources. (Translated from Latin cohaerer - to be in connection). In the event that one stream of traveling waves, which consistently create identical oscillations at all points of the studied part of the wave field, is superimposed on a coherent flow of similar waves, creating wave oscillations with the same amplitude, then the interference of the oscillations leads to a time-invariant division of the wave field into:
- Areas of amplification of oscillations.
- Areas of weakening oscillations.
The geometric location of the site of interference amplification of oscillations determines the difference in the wave paths (). The greatest amplification of oscillations is located where:
where n is an integer; - wavelength.
The maximum attenuation of vibrations occurs where:
The phenomenon of interference can be observed in any type of wave. This phenomenon, for example, can be observed for light waves. For a certain value of the difference between the paths of a direct and reflected ray of light, when hitting one point, the rays in question are able to completely cancel each other out.
Examples of problem solving
EXAMPLE 1
| Exercise | Two oscillations occur in accordance with the equations: and . Show how to obtain the maximum and minimum intensity conditions when two given waves are superimposed. |
| Solution | If the addition of oscillations in one direction is considered, then the displacement that a point receives in each oscillation will add algebraically. And the resulting offset is: Let us depict a vector diagram of the addition of two oscillations of the same frequency (those specified according to our condition (Fig. 1)). The total displacement x (1.1) is obtained by projecting the vectors amplitudes and onto the vertical diameter. For any moment in time, the displacement x is the projection of the vector, which is equal to: Therefore, we have: From Fig. 1 it follows that: The energy of the total harmonic vibration is equal to the sum of the vibration energies if: Expression (1.6) is satisfied if (in accordance with (1.5)) the phases of the summed oscillations differ by the amount , where If the phase difference is: Then they consider that the oscillations are in antiphase, then: In the case in which: |
The wave nature of light is most clearly manifested in the phenomena of interference and diffraction of light, which are based on wave addition . The phenomena of interference and diffraction have, in addition to their theoretical significance, wide application in practice.
This term was proposed by the English scientist Jung in 1801. Literally translated, it means intervention, collision, meeting.
To observe interference, conditions for its occurrence are necessary, there are two of them:
interference occurs only when the superposing waves have the same length λ (frequency ν);
immutability (constancy) of the oscillation phase difference.
Examples of wave addition:
Sources that provide the phenomenon of interference are called coherent , and the waves – coherent waves .
To clarify the question of what will happen at a given point max or min, you need to know in what phases the waves will meet, and to know the phases you need to know wave path difference. What it is?
at (r 2 –r 1) =Δr, equal to an integer number of wavelengths or an even number of half-waves, at point M there will be an increase in oscillations;
with d equal to an odd number of half-waves at point M there will be a weakening of oscillations.
The addition of light waves occurs in a similar way.
The addition of electromagnetic waves of the same oscillation frequency coming from different light sources is called interference of light .
For electromagnetic waves, when superimposed, we apply the principle of superposition, actually first formulated by the Italian Renaissance scientist Leonardo da Vinci:
Emphasize that the principle of superposition is strictly valid only for waves of infinitesimal amplitude.
A monochromatic light wave is described by the harmonic vibration equation:
,
where y – tension values 
And 
, whose vectors oscillate in mutually perpendicular planes.
If there are two waves of the same frequency:
And 
;
arriving at one point, then the resulting field is equal to their sum (in the general case, geometric):
If ω 1 = ω 2 and (φ 01 – φ 02) = const, the waves are called coherent .
The value of A, depending on the phase difference, lies within the limits:
|A 1 – A 2 | ≤ A ≤ (A 1 + A 2)
(0 ≤ A ≤ 2A, if A 1 = A 2)
If A 1 = A 2, (φ 01 – φ 02) = π or (2k+ 1)π, cos(φ 01 – φ 02) = –1, then A = 0, i.e. interfering waves completely cancel each other out (min illumination, if we take into account that E 2 J, where J is intensity).
If A 1 = A 2, (φ 01 – φ 02) = 0 or 2kπ, then A 2 = 4A 2, i.e. interfering waves reinforce each other (maximum illumination occurs).
If (φ 01 – φ 02) changes chaotically over time, with a very high frequency, then A 1 = 2A 1, i.e. is simply the algebraic sum of both wave amplitudes emitted by each source. In this case, the provisions max And min quickly change their position in space, and we will see some average illumination with an intensity of 2A 1. These sources are incoherent .
Any two independent light sources are incoherent.
Coherent waves can be obtained from a single source by splitting a beam of light into several beams that have a constant phase difference.
Topics of the Unified State Examination codifier: interference of light.
In the previous leaflet on Huygens' principle, we talked about the fact that the overall picture of the wave process is created by the superposition of secondary waves. But what does this mean - "overlay"? What is the specific physical meaning of wave superposition? What actually happens when several waves propagate in space simultaneously? This leaflet is dedicated to these questions.
Addition of vibrations.
Now we will consider the interaction of two waves. The nature of the wave processes does not matter - these 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 the waves add to each other in the following sense.
Superposition principle. If two waves overlap each other in a certain region 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 region 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 generally for any number of overlapping waves. The resulting oscillation at a given point is always equal to the sum of the oscillations created by each wave separately.
We will limit 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 influenced by the phase difference of the resulting oscillations. Depending on the phase difference at a given point in space, two waves can either enhance each other or completely cancel each other out!
Let us assume, for example, that at some point the phases of oscillations in overlapping waves coincide (Fig. 1).
We see that the highs of the red wave fall exactly on the highs of the blue wave, and the lows of the red wave coincide with the lows of the blue wave (left side of Fig. 1). When added in phase, the red and blue waves reinforce each other, generating oscillations of double amplitude (on the right in Fig. 1).
Now let's shift the blue sine wave relative to the red one by half the wavelength. Then the highs of the blue wave will coincide with the lows of the red wave and vice versa - the lows of the blue wave will coincide with the highs of the red wave (Fig. 2, left).
The oscillations created by these waves will occur, as they say, in antiphase- the phase difference of the oscillations will become equal to . The resulting oscillation will be equal to zero, that is, 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. Waves excited 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 overlap of waves emitted by these sources is observed at a certain 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 be different. But in many cases the point is located quite far from the sources - at a distance much greater than the distance between the sources themselves. In such a situation, the difference in distances does not lead to a significant difference in the amplitudes of incoming waves. Therefore, we can assume that the amplitudes of the waves at the point also coincide.
Maximum and minimum conditions.
However, the quantity called stroke difference, is of utmost importance. It most decisively determines what result of the addition of incoming waves we will see at point .
In the situation in Fig. 3 the path difference is equal to the wavelength. Indeed, there are three full waves on a segment, and four on a segment (this, of course, is just 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 when the path difference is equal not only to the wavelength, but to any integer number of wavelengths.
Maximum condition . When coherent waves are superimposed, the 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. 4 . There are two and a half waves on a segment, and three waves on a segment. The path difference is half the wavelength (d=\lambda /2).
Now it is easy to see that the waves at a 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-wave lengths.
Interference pattern.
But what if the path difference takes on some other value, not equal to an integer or half-integer number of wavelengths? Then the waves arriving at a given point create oscillations in it with a certain intermediate amplitude located between zero and double the 2A value of the amplitude of one wave. This intermediate amplitude can take on anything from 0 to 2A as the path difference changes from a half-integer to an integer number of wavelengths.
Thus, in the region of space where 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 in a given region, the amplitude of the oscillations 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 between 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 points in 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 alternating maximums and minimums of volume 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 law of conservation of energy: the energy is simply redistributed between different parts of the interference pattern. The greatest amount of energy is concentrated in the interference maxima, and no energy is supplied to the interference minima points at all.
In Fig. Figure 5 shows the interference pattern created by the superposition of waves from two point sources and . The picture is constructed under the assumption that the interference observation region is located sufficiently far from the sources. The dotted line marks the axis of symmetry of the interference pattern.
The colors of the interference pattern dots in this figure vary from black to white through intermediate shades of gray. Black color - interference minima, white color - interference maxima; gray color is an intermediate amplitude value, and the greater the amplitude at a given point, the lighter the point itself.
Pay attention to the straight white stripe that runs along the axis of symmetry of the picture. Here are the so-called central maxima. Indeed, any point on a given axis is equidistant from the sources (the path difference is zero), so that an interference maximum will be observed at this point.
The remaining white stripes and all the black stripes are slightly curved; it can be shown that they are branches of hyperbolas. However, in an area located at a great distance from the sources, the curvature of the white and black stripes is little noticeable, and these stripes look almost straight.
The interference experiment 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 experiment (which will be discussed in the topic Diffraction of Light). Many experiments on the interference of light in one way or another come down to Young’s scheme.
In optics, the interference pattern is usually observed on a screen. Let's look at Fig. again. 5 and imagine a screen placed perpendicular to the dotted axis.
On this screen we will see alternating light and dark interference fringes.
In Fig. 6 sinusoid shows the distribution of illumination along the screen. At 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 located at point A. Above are the second, third (and so on) maximums.
 |
| Rice. 6. Interference pattern on the screen |
A distance equal to the distance between any two adjacent maximums or minimums is called interference fringe width. Now we will start finding this value.
Let the sources be at a distance from each other, and the screen located at a distance from the sources (Fig. 7). The screen is replaced by an axis; the reference point, as above, corresponds to the central maximum.
Points and serve as projections of points and onto the axis and are located symmetrically relative to the point. We have: .
The observation point can be anywhere on the axis (on the screen). Point coordinate
we will denote . We are interested in at what values an interference maximum will be observed at a point.
A wave emitted by a 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 an 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 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 next to the central one corresponds to the value and has the coordinate. The width of the interference fringe will be the same.
More convincing evidence is needed that light behaves like a wave when it travels. 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 interference and diffraction of light.
Interference is a rather complex phenomenon. To better understand its essence, we will first focus on the interference of mechanical waves.
Addition of waves. Very often, several different waves simultaneously propagate in a medium. For example, when several people are talking in a room, the sound waves overlap each other. What happens?
The easiest way to observe the superposition of mechanical waves is by observing waves on the surface of the water. If we throw two stones into the water, thereby creating two annular waves, then it is easy to notice that each wave passes through the other and subsequently behaves as if the other wave did not exist at all. In the same way, any number of sound waves can simultaneously propagate through the air without interfering with each other in the least. Many musical instruments in an orchestra or voices in a choir create sound waves that are simultaneously detected by our ears. 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 each other. Observing waves on the surface of the water from two stones thrown into the water, you can notice that some areas of the surface are not disturbed, but in other places the disturbance has intensified. If two waves meet in one place with crests, then in this place the disturbance of the water surface intensifies.
If, on the contrary, the crest of one wave meets the trough of another, then the surface of the water will not be disturbed.
In general, at each point in 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 of waves in space, 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 wave interference occurs. To do this, let us consider in more detail the addition of waves formed on the surface of the water.
It is possible to simultaneously excite two circular waves in a bath using two balls mounted on a rod, which 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 oscillations caused at point M by both waves will, generally speaking, differ, 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 identical.
The result of the addition of waves arriving at point M depends on the phase difference between them. Having traveled 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 during the period the wave travels a path 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 zero (or, which 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 double amplitude appears. Fluctuations in the resulting displacement are shown in color (dotted line) in the figure. The same thing will happen if the segment Δd contains not one, but any integer number of wavelengths.
The amplitude of oscillations of the medium at a given point is maximum if the difference in the paths of the two waves exciting oscillations at this point is equal to an integer number of wavelengths:
where k=0,1,2,....
Minimum condition. Let now the segment Δd fit half the wavelength. It is obvious that the second wave lags behind the first by half the 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 zero, that is, there are no oscillations at the point under consideration (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 in the paths of the two waves exciting oscillations at this point is equal to an odd number of half-waves:
If the path 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 double the amplitude and zero. But the most important thing is that the amplitude of oscillations at any point he changes over time. On the surface of the water, a certain, time-invariant distribution of vibration amplitudes appears, 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 part of the photograph correspond to the swing maxima, and the dark areas correspond to the swing minima.
Coherent waves. To form a stable interference pattern, it is necessary that the wave sources have the same frequency and the phase difference of their oscillations is constant.
Sources that satisfy these conditions are called coherent. The waves they create are also called coherent. Only when coherent waves are added together does a stable interference pattern form.
If the phase difference between the oscillations of the sources does not remain constant, then at any point in the medium the phase difference between 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.
Energy distribution during interference. Waves carry energy. What happens to this energy when the waves cancel 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 energy does not flow here at all. Due to interference, 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 light bulbs. Turning on another light bulb only increases the illumination of the surface, but does not create an alternation of minimums and maximums of illumination.
Let's find out what is the reason for this and under what conditions the interference of light can be observed.
Condition for coherence of light waves. The reason is that the light waves emitted by different sources are not consistent with each other. To obtain a stable interference pattern, consistent waves are needed. They must have the same wavelengths and a constant phase difference at any point in space. Recall that such consistent waves with identical 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 enough to use good light filters that transmit light in a very narrow wavelength range. But it is impossible to realize the constancy of the phase difference from two independent sources. Atoms of the sources emit light independently of each other in separate “scraps” (trains) of sine waves, about a meter long. And such wave trains from both sources overlap each other. As a result, the amplitude of oscillations at any point in space changes chaotically with time, depending on how, at a given moment in time, wave trains from different sources are shifted relative to each other in phase. Waves from different light sources are incoherent because the phase difference between the waves does not remain constant. No stable pattern with a specific distribution of maxima and minima of illumination in space is observed.
Interference in thin films. Nevertheless, the interference of light can be observed. The curious thing is that it was observed for a very long time, but they just did not realize it.
You, too, have seen an interference pattern many times when, as a child, you had fun blowing soap bubbles or watched the rainbow colors of a thin film of kerosene or oil on the surface of water. “A soap bubble floating in the air... lights up with all the shades of colors inherent in the surrounding objects. A soap bubble is perhaps the most exquisite miracle of nature" (Mark Twain). It is the interference of light that makes a soap bubble so admirable.
The English scientist Thomas Young was the first to come up with the brilliant idea of 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. In this case, interference of light waves occurs - the addition of two waves, as a result of which a time-stable pattern of amplification or weakening of the resulting light vibrations is observed at different points in space. The result of interference (amplification or weakening of the resulting vibrations) depends on the angle of incidence of light on the film, its thickness and wavelength. Light amplification 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 an odd number of half-waves, then the light will weaken.
The coherence of 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 wave train from each emitting atom is divided into two by the film, and then these parts are brought together and interfere.
Jung also realized that differences in color were due to differences in wavelength (or frequency of light waves). Light beams of different colors correspond to waves of different lengths. For mutual amplification of waves that differ 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 unequal thickness, then when 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 plane-convex lens placed on it, the spherical surface of which has a large radius of curvature. This interference pattern takes the form of concentric rings, called Newton's rings.
Take a plano-convex lens with a slight curvature of a spherical surface and place 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 collection of small rainbow rings around it. The distances between adjacent rings quickly decrease as their radius increases (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 the maximum radius. You can check all this through independent observations.
Newton was unable to satisfactorily explain why rings appear. 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 plane-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 arises due to the fact that wave 2 travels a longer path than wave 1. If the second wave lags behind the first by an integer number of wavelengths, then, adding up, the waves reinforce each other friend. The oscillations 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 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.
Light wavelength. For red light, measurements give λ cr = 8 10 -7 m, and for violet light - λ f = 4 10 -7 m. The wavelengths corresponding to other colors of the spectrum take intermediate values. For any color, the wavelength of light is very short. Imagine an average sea wave several meters long, which grew so large that it occupied the entire Atlantic Ocean from the shores of America to Europe. The wavelength of light at the same magnification would be only slightly longer than the width of this page.
The phenomenon of interference not only proves that light has wave properties, but also allows us to measure the wavelength. Just as the pitch of a sound is determined by its frequency, the color of light is determined by its vibrational frequency or wavelength.
Outside of us, there are no colors in nature, there are only waves of different lengths. The eye is a complex physical device capable of detecting differences in color, which correspond to a very slight (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. Colorblind people - people suffering from color blindness - also do not distinguish colors.
When light passes from one medium to another, the wavelength changes. It can be detected like this. 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 a vacuum into some medium, the speed of light decreases by a factor of n. Since v = λv, then either the frequency or the wavelength must decrease 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, one can observe the interference of electromagnetic (radio) waves.
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, an alternating weakening and strengthening of the sound is detected.
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 difference between the paths of 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 exhibits wave properties when propagating. 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
Standing wave equation.
As a result of the superposition of two counter-propagating plane waves with the same amplitude, the resulting oscillatory process is called standing wave
.
Almost standing waves arise when reflected from obstacles. Let us write the equations of two plane waves propagating in opposite directions (initial phase): 
Let's add the equations and transform using the sum of cosines formula: . Because , then we can write: . Considering that , we get standing wave equation
: 
. The expression for the phase does not include the coordinate, so we can write: , where the total amplitude 
.
Wave interference- such a superposition of waves in which their mutual amplification, stable over time, occurs at some points in space and weakening at others, depending on the relationship 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 waves does not change over time (or does not change very quickly so that it can be recorded in time);
2) the waves must be unidirectional (or have a similar direction); two perpendicular waves will never interfere. In other words, the waves being added must have identical wave vectors. Waves for which these two conditions are met are called coherent. The first condition is sometimes called temporal coherence, second - spatial coherence. Let us consider as an example the result of adding two identical unidirectional sinusoids. We will only vary their relative shift. If the sinusoids are located so that their maxima (and minima) coincide in space, they will be mutually amplified. 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; the sinusoids will destroy each other, that is, their mutual weakening will occur. Add two waves:
Here x 1 And x 2- the distance from the wave sources to the point in space at which we observe the result of the superposition. The squared amplitude of the resulting wave is given by:
The maximum of this expression is 4A 2, minimum - 0; everything depends on the difference in the initial phases and on the so-called difference in the wave path D:
When at a given point in space, an interference maximum will be observed, and when - 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 disturbance (change in state) of an electromagnetic field propagating in space (that is, electric and magnetic fields interacting with each other). Among electromagnetic fields in general, generated by electric charges and their movement, it is customary to classify as radiation that part of alternating electromagnetic fields that is capable of propagating farthest from its sources - moving charges, attenuating most slowly with distance. Electromagnetic radiation is divided into radio waves, infrared radiation, visible light, ultraviolet radiation, x-rays and gamma radiation. Electromagnetic radiation can propagate in almost all environments. In a vacuum (a space free of 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 (while slightly 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 speed of propagation of electromagnetic radiation in a vacuum is equal to the speed of light; in other media this speed is less. The phase speed of electromagnetic radiation in a vacuum is also equal to the speed of light; in different media it can be either less or greater than the speed of light.
What is the nature of light. Interference of light. 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. Degree of polarization. Polarization of light during reflection and refraction at the boundary of two dielectrics. Polaroids
What is the nature of light. The first theories about the nature of light - corpuscular and wave - appeared in the mid-17th century. According to the corpuscular theory (or outflow theory), 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 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 vibrations of the ether. 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 led to insoluble contradictions. For example, the phenomenon of polarization of light has shown. that light waves are transverse. Elastic transverse waves can propagate only in solids where shear deformation occurs. Therefore, the ether must be a solid medium, but at the same time not interfere with the movement of space objects. The exotic properties of the elastic ether were a significant drawback 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 exhibits the properties of waves, and in others, the properties of particles. Wave and quantum properties complement each other.
Wave interference.
is the phenomenon of superposition of coherent waves
- characteristic of waves of any nature (mechanical, electromagnetic, etc.
Coherent waves- These are waves emitted by sources having the same frequency and a constant phase difference. When coherent waves are superimposed at any point in space, the amplitude of the oscillations (displacement) of this point will depend on the difference in distances from the sources to the point in question. This distance difference is called the stroke difference.
When superposing coherent waves, two limiting cases are possible:
1) Maximum condition: The difference in wave path is equal to an integer number of wavelengths (otherwise an even number of half-wavelengths). 
Where 
. In this case, the waves at the point under consideration arrive with the same phases and reinforce each other - the amplitude of the oscillations of this point is maximum and equal to double the amplitude.
2) Minimum condition: The difference in wave path is equal to an odd number of half-wave lengths. 
Where 
. The waves arrive at the point in question in antiphase and cancel each other out. The amplitude of oscillations of a given point is zero. As a result of the superposition of coherent waves (wave interference), an interference pattern is formed. With wave interference, the amplitude of the oscillations of each point does not change over time and remains constant. When incoherent waves are superimposed, there is no interference pattern, because the amplitude of oscillations of each point changes over 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 occurrence in time and space of several oscillatory or wave processes. This condition is satisfied monochromatic waves- waves unlimited in space of one specific 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, atoms emit 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 its emission of light stops. Having become 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 the 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 over a longer period of time both the amplitude and phase change.
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 instruments ( optics clearing) 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 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 large. Thus, the intensity of the transmitted light is weakened and the aperture ratio of the optical device decreases. In addition, reflections from lens surfaces lead to glare, which often (for example, in military equipment) reveals the position of the device. To eliminate these shortcomings, the so-called enlightenment of 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 s and films n can be chosen so that the waves reflected from both surfaces of the film cancel each other. To do this, their amplitudes must be equal, and the optical path difference must be equal to . The calculation shows that the amplitudes of the reflected rays are equal if n With, n and refractive index of air n 0 satisfy the conditions n from > n>n 0, then the loss of 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 taken m=0, then
Diffraction of light. Huygens–Fresnel principle.Diffraction of light- deviation of light waves from rectilinear propagation, bending around encountered obstacles. Qualitatively, the phenomenon of diffraction is explained on the basis of the Huygens-Fresnel principle. The wave surface at any moment in 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 quite accurately only if the size of the obstacles in the path of light propagation is much greater than the light wavelength: Diffraction occurs when the size of the obstacles is commensurate with the wavelength: L ~ L. The diffraction pattern obtained on a screen located behind various obstacles, is the result of interference: alternation of light and dark stripes (for monochromatic light) and multi-colored stripes (for white light). Diffraction grating - an optical device consisting of a large number of very narrow slits separated by opaque spaces. The number of lines of 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 quantity d = a + b is called lattice period.
