Not long ago we discussed in some detail the properties of light waves and their interference, that is, the effect of superposition of two waves from different sources. But it was assumed that the frequencies of the sources were the same. In this chapter we will dwell on some phenomena that arise when two sources with different frequencies interfere.
It is not difficult to guess what will happen. Proceeding as before, let us assume that there are two identical oscillating sources with the same frequency, and their phases are selected so that at some point the signals arrive with the same phase. If it is light, then at this point it is very bright, if it is sound, then it is very loud, and if it is electrons, then there are a lot of them. On the other hand, if the incoming waves differ in phase by 180°, then there will be no signals at the point, because the total amplitude will have a minimum here. Now suppose that someone turns the “phase adjustment” knob of one of the sources and changes the phase difference at a point here and there, let’s say first he makes it zero, then equal to 180°, etc. In this case, of course, it will change and the strength of the incoming signal. It is now clear that if the phase of one of the sources changes slowly, constantly and evenly compared to the other, starting from zero, and then increases gradually to 10, 20, 30, 40°, etc., then at the point we will see a series of weak and strong “pulsations”, because when the phase difference passes through 360°, a maximum appears in the amplitude again. But the statement that one source changes its phase with respect to another at a constant speed is equivalent to the statement that the number of oscillations per second for these two sources is somewhat different.
So, now we know the answer: if you take two sources whose frequencies are slightly different, then the addition results in oscillations with a slowly pulsating intensity. In other words, everything said here is actually relevant!
This result is easy to obtain mathematically. Suppose, for example, that we have two waves and forget for a minute about all spatial relationships, and just look at what comes to the point. Let a wave come from one source, and a wave come from another, and both frequencies are not exactly equal to each other. Of course, their amplitudes can also be different, but first let's assume that the amplitudes are equal. We will consider the general problem later. The total amplitude at a point will be the sum of two cosines. If we plot amplitude versus time as shown in Fig. 48.1, it turns out that when the crests of two waves coincide, a large deviation is obtained, when the crest and trough coincide - practically zero, and when the crests coincide again, a large wave is again obtained.
Fig. 48.1. Superposition of two cosine waves with a frequency ratio of 8:10. Exact repetition of oscillations within each beat is not typical for the general case.
Mathematically, we need to take the sum of two cosines and somehow rearrange it. This will require some useful relationships between cosines. Let's get them. You know, of course, that
and that the real part of the exponent is equal to , and the imaginary part is equal to . If we take the real part 
, then we get , and for the product
we get plus some imaginary addition. For now, however, we only need the real part. Thus,
If we now change the sign of the quantity , then, since the cosine does not change the sign, but the sine changes the sign to the opposite, we obtain a similar expression for the cosine of the difference
After adding these two equations, the product of the sines cancels, and we find that the product of two cosines is equal to half the cosine of the sum plus half the cosine of the difference
Now you can wrap this expression around and get a formula for if you simply put , a, i.e., a:
But let's return to our problem. The sum and is equal to
Let now the frequencies be approximately the same, so that it is equal to some average frequency, which is more or less the same as each of them. But the difference is much smaller than and , since we assumed that and are approximately equal to each other. This means that the result of the addition can be interpreted as if there is a cosine wave with a frequency more or less equal to the original, but that its "sweep" is slowly changing: it pulsates with a frequency equal to . But is this the frequency with which we hear beats? Equation (48.0) says that the amplitude behaves as 
, and this must be understood in such a way that high-frequency oscillations are contained between two cosine waves with opposite signs (dashed line in Fig. 48.1). Although the amplitude does change with frequency, however, if we are talking about the intensity of the waves, then we must imagine the frequency to be twice as high. In other words, amplitude modulation in the sense of its intensity occurs with a frequency, although we multiply by the cosine of half the frequency.
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 issues.
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 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 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.
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.
Interference is a redistribution of the flow of electromagnetic energy in space, resulting from the superposition of waves arriving in a given region of space from different sources. If a screen is placed in the area of interference of light waves, then there will be
light and dark areas, such as stripes, are observed.
They can only interfere coherent waves. Sources (waves) are called coherent if they have the same frequencyand a time-constant phase difference of the waves they emit.
Only point monochromatic sources can be coherent. Lasers have similar properties to them. Conventional radiation sources are incoherent, since they are non-monochromatic and are not point-like.
The non-monochromatic nature of the radiation from conventional sources is due to the fact that their radiation is created by atoms emitting wave trains of length L=c =3m over a period of time of the order of =10 -8 s. The emissions from different atoms are not correlated with each other.
However, wave interference can also be observed using conventional sources if, using some technique, two or more sources similar to the primary source are created. There are two methods for producing coherent light beams or waves: wavefront division method And wave amplitude division method. In the wavefront splitting method, a beam or wave is split by passing through closely spaced slits or holes (diffraction grating) or by reflecting and refractive obstacles (mirror and Fresnel biprism, reflective diffraction grating).
IN 
In the dividing method, the wave amplitude of the radiation is divided on one or more partially reflective, partially transmitting surfaces. An example is the interference of rays reflected from a thin film.
Points A, B and C in Fig. are the division points of the wave amplitude
Quantitative description of wave interference.
Let two waves arrive at point O from sources S 1 and S 2 along different optical paths L 1 =n 1 l 1 and L 2 =n 2 l 2 .
The resulting field strength at the observation point is equal to
E=E 1 +E 2 . (1)
The radiation detector (eye) registers not the amplitude, but the intensity of the wave, so let’s square relation (1) and move on to the wave intensities
E 2 =E 1 2 +E 2 2 +E 1 E 2 (2)
Let's average this expression over time
Last term in (3) 2
2<E 1 E 2 >=2
where is the angle between vectors E 1 and E 2. If /2, then cos=0 and the interference term will be equal to zero. This means that waves polarized in two mutually perpendicular planes cannot interfere. If the secondary sources from which interference is observed are received from one primary source, then the vectors E 1 and E 2 are parallel and cos = 1. In this case, (3) can be written in the form
where the time-averaged functions have the form
E 1 =E 10 cos(t+), E 2 =E 20 cos(t+), (6)
=-k 1 l 1 + 1 , =-k 2 l 2 + 2 .
Let us first calculate the time-average value of the interference term
(7)
whence at =:
Denoting I 1 =E 2 10, I 2 =E 2 20 and 
, formula (5) can be written in terms of wave intensity. If the sources are incoherent, then
I=I 1 +I 2 , (9)
and if they are coherent, then
I=I 1 +I 2 +2 
cos (10)
k 2 l 2 -k 1 l 1 + - (11)
is the phase difference of the added waves. For sources. received from one primary source 1 = 2, therefore
=k 2 l 2 -k 1 l 1 =k 0 (n 2 l 2 -n 1 l 1)=(2/ ) (12)
where K 0 =2 is the wave number in vacuum, is the optical difference in the path of rays 1 and 2 from S 1 and S 2 to the interference observation point 0. We got
(13)
From formula (10) it follows that at point 0 there will be a maximum interference if cos = 1, whence
m, or=m (m=0,1,2,…) (14)
The minimum interference condition will be at cos = -1, whence
=2(m+½), or=(m+½) (m=0,1,2,…) (14)
Thus, the waves at the point of overlap will strengthen each other, if their optical path difference is equal to an even number of half-waves they will weaken each other
if it is equal to an odd number of half-waves.
The degree of coherence of the source radiation. Interference of partially coherent waves.
Real light beams arriving at the interference observation point are partially coherent, i.e. contain coherent and incoherent light. To characterize partially coherent light, we introduce degree of coherence 0< < 1 which represents the fraction of incoherent light in the light beam. With the interference of partially coherent beams we obtain
I= nekog +(1-)I cos =(I 1 +I 2)+(1-)(I 1 +I 2 +2I 1 I 2 cos
From whereI=I 1 +I 2 +2I 1 I 2 cos (17)
If =0 or =1, then we come to the cases of incoherent and coherent addition of wave interference.
Young's experiment (wavefront division)
P 
The first experiment in observing interference was carried out by Jung (1802). Radiation from a point source S passed through two point holes S 1 and S 2 in the diaphragm D and at point P on the screen E, interference of rays 1 and 2 passing along the geometric paths SS 1 P and SS 2 P was observed.
Let's calculate the interference pattern on the screen. The geometric difference in the path of rays 1 and 2 from source S to point P on the screen is equal to
l=(l` 2 +l 2) (l` 1 +l 1)= (l` 2 1` 1)+(l 2 l 1) (1)
Let d be the distance between S 1 and S 2 , b be the distance from the source plane S to the diaphragm D, a be the distance from the diaphragm D to the screen E, x be the coordinate of point P on the screen relative to its center, ax` the coordinate of source S relative to the center of the source plane. Then, according to the figure using the Pythagorean theorem, we obtain
The expressions for l` 1 and l` 2 will be similar if we replace ab, xx`. Suppose d and x< Likewise Taking into account (3) and (4), the geometric difference in the path of rays 1 and 2 will be equal to If rays 1 and 2 pass through a medium with refractive index n, then their optical path difference is equal to The conditions for maximum and minimum interference on the screen have the form Where do the coordinates of the maxima x=x m and minima x=x"m of the interference pattern on the screen come from? If the source has the form of a strip with coordinate x" perpendicular to the plane of the picture, then the image on the screen will also have the form of strips with coordinate x" perpendicular to the plane of the picture. The distance between the nearest interference maxima and minima or the width of the interference fringes (dark or light) will be, according to (8), equal to x=x m+1 -x m =x` m+1 -x` m = where = /n – wavelength in a medium with refractive index n. Spatial coherence (incoherence) of source radiation A distinction is made between spatial and temporal coherence of source radiation. Spatial coherence is related to the finite (non-point) dimensions of the source. It leads to a broadening of the interference fringes on the screen and, at a certain source width D, to the complete disappearance of the interference pattern. Spatial incoherence is explained as follows. If the source has a width D, then each luminous stripe of the source with coordinate x" will give its own interference pattern on the screen. As a result, different interference patterns on the screen shifted relative to each other will overlap each other, which will lead to smearing of the interference fringes and at a certain width source D to the complete disappearance of the interference pattern on the screen. It can be shown that the interference pattern on the screen will disappear if the angular width of the source, =D/l, visible from the center of the screen, is greater than the ratio /d: The method of obtaining secondary sources S 1 and S 2 using a Fresnel biprism is reduced to Young's scheme. Sources S 1 and S 2 lie in the same plane as the primary source S. It can be shown that the distance between sources S 1 and S 2 obtained using a biprism with a refractive angle and index n is equal to d=2a 0 (n-1), (2) and the width of the interference fringes on the screen The interference pattern on the screen will disappear when the condition is met If l = 0.5 m, and 0 = 0.25 m, n = 1.5 - glass, = 6 10 -7 - wavelength of green light, then the width of the source at which the interference pattern on the screen disappears is D = 0, 2mm. Temporal coherence of source radiation. Time and length of coherence. Temporal coherence associated with the non-monochromatic nature of the source radiation. It leads to a decrease in the intensity of the interference fringes with distance from the center of the interference pattern and its subsequent break. For example, when observing an interference pattern using a non-monochromatic source and a Fresnel biprism, from 6 to 10 bands are observed on the screen. When using a highly monochromatic laser radiation source, the number of interference fringes on the screen reaches several thousand. Let us find the condition for interruption of interference due to the non-monochromatic nature of the source emitting in the wavelength range (). The position of the m-th maximum on the screen is determined by the condition where 0 /n is the wavelength with refractive index n. It follows that each wavelength has its own interference pattern. As increases, the interference pattern shifts, the greater the greater the order of interference (interference fringe number) m. As a result, it may turn out that the m-th maximum for the wavelength is superimposed on the (m+1)-th maximum for the length waves.In this case, the interference field between the m-th and (m+1)-th maxima for the wavelengthwill be uniformly filled with interference maxima from the interval () and the screen will be uniformly illuminated, i.e. The IR will cut out. Interference pattern termination condition X max (m,+)=X max (m+1,) (2) From where according to (1) (m+1)=m(, (3) which gives for the order of interference (number of interference fringe) at which the IR will break The condition of interference maxima is associated with the optical difference in the path of rays 1 and 2 arriving at the interference observation point on the screen by the condition Substituting (4) into (5), we find the optical difference in the path of rays 1 and 2, at which the interference disappears on the screen When >L cog the interference pattern is not observed. The quantity L cog = is called length of (longitudinal) coherence, and the value t cog =L cog /c (7) -coherence time. Let us reformulate (6) in terms of radiation frequency. Considering that c, we get |d|= Then according to (6) L cog = And according to (7) We obtained a relationship between the coherence time t coh and the width of the frequency interval of the source radiation. For the visible range (400-700) nm with an interval width = 300 nm at an average wavelength = 550 nm, the coherence length is of the order of L cog =10 -6 m, and the coherence time of the order of t cog =10 -15 s. The coherence length of laser radiation can reach several kilometers. Note that the emission time of an atom is of the order of 10 -8 s, and the lengths of the wave trains are of the order of L = 3 m. Huygens and Huygens-Fresnel principles. IN Huygens' principle is purely geometric and allows one to derive. for example, the laws of reflection and refraction of light, explains the phenomena of light propagation in anisotropic crystals (birefringence). But it cannot explain most optical phenomena caused by wave interference. Fresnel supplemented Huygens' principle with the condition for the interference of secondary waves emanating from the wave front. This extension of Huygens' principle is called the Huygens-Fresnel principle. Fresnel zones. Fresnel proposed a simple method for calculating the result of the interference of secondary waves. coming from the wave front to an arbitrary point P lying on a straight line passing through the source S and point P. Let's consider Fresnel's idea using the example of a spherical wave emitted by a point source S. Let the wave front from the source S at some instant of time be at a distance a from S and at a distance b from point P. Let us divide the wave front into ring zones so that the distance from the edges of each zone to point P differs by /l. With this construction, the oscillations in neighboring zones are shifted in phase by, i.e. occur in antiphase. If we denote the amplitudes of oscillations in the zones E 1, E 2, ... with E 1 > E 2 >..., then the amplitude of the resulting oscillation at point P will be equal to E=E 1 -E 2 +E 3 -E 4 +… (1) Here there is an alternation of signs (+) and (-), since oscillations in adjacent zones occur in antiphase. Let us represent formula (1) in the form where it is set E m = (E m-1 + E m+1)/2. We found that the amplitude of oscillations at point P, if oscillations from the entire wave front arrive at it, is equal to E = E 1 /2, i.e. equal to half the amplitude of the wave arriving at point P from the first Fresnel zone. If you close all even or odd Fresnel zones using special plates called zone plates, then the amplitude of oscillations at point P will increase and will be equal to E=E 1 +E 3 +E 5 +…+E 2m+1 , E=|E 2 +E 4 +E 6 +…+E 2m +…| (3) If a screen with a hole is placed in the path of the wave front, which would open a finite even number of Fresnel zones, then the light intensity at point P will be equal to zero E=(E 1 -E 2)+(E 3 -E 4)+(E 5 -E 6)=0 (4) those. in this case there will be a dark spot at point P. If you open an odd number of Fresnel zones, then at point P there will be a bright spot: E=E 1 -E 2 +E 3 -E 4 +E 5 =E 1 (4) To overlap fresnel zones using screens or zone plates, it is necessary to know the radii of the fresnel zones. According to Fig. We get r r 2 m =(b+m / 2) 2 -(b+h m) 2 =bm-2bh m (7) where terms with 2 and h m 2 were neglected. Equating (5) and (6), we get Substituting formula (8) into (6), the radius of the m-th Fresnel zone where m=1,2,3,... is the number of the Fresnel zone, is the wavelength of the radiation emitted by the source. If the water front is flat (a ->), then For a fixed radius of the hole in the screen placed in the path of the wave, the number m of Fresnel zones opened by this hole depends on the distances a and b from the hole to the source S and point P. Diffraction of waves (light). Diffraction call a set of interference phenomena observed in media with sharp inhomogeneities commensurate with the wavelength, and associated with the deviation of the laws of light propagation from the laws of geometric optics. Diffraction, in particular, leads to waves bending around obstacles and the penetration of light into the region of a geometric shadow. The role of inhomogeneities in the medium can be played by slits, holes and various obstacles: screens, atoms and molecules of matter, etc. There are two types of diffraction. If the source and the observation point are located so far from the obstacle that the rays incident on the obstacle and the rays going to the observation point are practically parallel, then we talk about Fraunhofer diffraction (diffraction in parallel rays), otherwise we talk about Fresnel diffraction (diffraction in converging rays) Fresnel diffraction by a circular hole. Let a spherical wave from a source in fall on a round hole in the diaphragm. In this case, a diffraction pattern in the form of light and dark rings will be observed on the screen. If the hole opens an even number of Fresnel zones, then there will be a dark spot in the center of the diffraction pattern, and if it opens an odd number of Fresnel zones, then there will be a light spot. When moving a diaphragm with a hole between the source and the screen, either an even or an odd number of Fresnel zones will fit within the hole, and the appearance of the diffraction pattern (either with a dark or with a light spot in the center) will constantly change. Fraunhofer diffraction by a slit. Let a spherical wave propagate from a source S. With the help of lens L 1, it turns into a plane wave, which falls on a slit of width b. The rays diffracted into the slits at an angle are collected on the screen located in the focal plane of lens L 2, at point F The intensity of the diffraction pattern at point P of the screen is determined by the interference of secondary waves emanating from all elementary sections of the slit and propagating to point P in the same direction . Since a plane wave is incident on the slit, the phases of oscillations at all points of the slit are the same. The intensity at point P of the screen, caused by waves propagating in the direction , will be determined by the phase shift between waves emanating from the flat front of wave AB, perpendicular to the direction of wave propagation (see figure), or by waves. emanating from any plane parallel to the direction AB. The phase shift between the waves emitted by strip 0 at the center of the slit and the strip with coordinate x measured from the center of the slit is kxsin (Fig.). If the slit has a width b and emits a wave with amplitude E 0, then a strip with coordinate x and width dx emits a wave with amplitude (Eo/b)dx. From this strip a wave with amplitude will arrive at point P of the screen in the direction The factor it, which is the same for all waves arriving at point P of the screen, can be omitted, since it will disappear when calculating the intensity of the wave at point P. The amplitude of the resulting oscillation at point P, due to the superposition of secondary waves arriving at point P from the entire slit, will be equal to where u=(k b / 2)sin=( b / )sin, is the wavelength emitted by the source. Wave intensity I=E 2 at point P of the screen will be equal to where I 0 is the intensity of the wave emitted by the slit in the direction=0, when (sin u/u)=1. At point P there will be a minimum intensity if sin u=0 or This is the condition for diffraction minima of dark bands on the screen). We find the condition for diffraction maxima by taking the derivative of I() but u and equating it to zero, which leads to the transcendental equation tg u=u. You can solve this equation graphically According to Fig. the straight line y=u intersects the curves y=tg u approximately at points with a coordinate along the abscissa axis equal to u=(2m+1) / 2 =(m+½), and u=0 =0, (5) which allows us to write an approximate, but quite accurate solution to the equation tg u=u in the form ABOUT bsinm+½) (m=1,2,…). (7) The central maximum at =0 is not included in condition (7) The intensity distribution on the screen during light diffraction at one slit is shown in Fig. Diffraction grating and its use for decomposing non-monochromatic radiation from a source into a spectrum. Diffraction grating can be considered any device that provides spatial periodic modulation of the light wave incident on it in amplitude and phase. An example of a diffraction grating is a periodic system. Nparallel slits separated by opaque spaces lying in the same plane, the distance d between the midpoints of adjacent slits is called period or constant lattice. A diffraction grating has the ability to decompose non-monochromatic radiation from a source into a spectrum, creating on the screen diffraction patterns shifted relative to each other, corresponding to different wavelengths of the source radiation. Let us first consider the formation of a diffraction pattern for radiation from a source with a fixed wavelength . Let a plane monochromatic wave with wavelength be normally incident on the grating, and the diffraction pattern is observed in the focal plane of the lens L. The diffraction pattern on the screen is a multi-beam interference of coherent light beams of equal intensity going to the observation point P from all slits in the direction . To calculate the interference pattern (IR), we denote by E 1 () the amplitude of the wave (formula (2) of the previous section) arriving at the observation point P from the first structural element of the array, the amplitude of the wave from the second structural element E 2 =E 1 e i , from third E 2 =E 1 e 2i etc. Where =kasin= The phase shift of waves arriving at point P from adjacent slits with a distance d between them. The total amplitude of oscillations created at point P by waves arriving at it from all N slits of the diffraction grating is represented by the sum of the geometric progression E P =E 1 ()(1+e i +e 2i +…+e i(N-1) )=E 1 () The intensity of the wave at point P is equal to I()=E p E * p, where E * p is the complex conjugate amplitude. We get I()=I 1 () where indicated It follows that the intensity distribution on the screen I(), created by radiation from N 12 slits, is modulated by the intensity function of one slit I 1 () = I 0 (sin(u)/u) 2. The intensity distribution on the screen, determined by the formula (3) is shown in Fig. It can be seen from the figure that there are sharp maxima in the IR, called main, between which low-intensity maxima and minima are observed, called side effects. The number of side minima is N-1, and the number of side maxima is N-2. Points at which I 1 () = 0 are called main minima. Their location is the same as in the case of one slit. Let's look at the formation of the main highs. They are observed in directions determined by the condition sin/2=0 (but at the same time sin N/2=0, which leads to uncertainty I()=0/00. The condition sin/2=0 gives / 2=k or dsin=k, k=0,1,2,… (5) where k is the order of the main maximum. Let's look at the formation of lows. The first condition sin u=0 at u0 leads to the condition of main minima, the same as in the case of one slit bsin=m, m=0,1,2,… (6) The second condition sin N/2=0at sin/20 determines the position of side minima at values N
, (N+1), … (2N-1); (7) 2
N
, (2N+1),… (3N-1); The underlined values are multiples of N and lead to the condition of main maxima N=Nkor /2=k. These valuesshould be excluded from the list of secondary minima. The remaining values can be written as whence we obtain the condition for side minima dsin=(k+ P / N), P=0,1,2,…N-1 (9) where k is the fixed order of the main maximum. You can allow negative values p = -1, -2, ...-(N-1), which will give the position of the side minima to the left of the k-th main maximum. From the conditions of the main and secondary maxima and minima it follows that radiation with a different wavelength will correspond to a different angular arrangement of minima and maxima in the diffraction pattern. This means that the diffraction grating decomposes the non-monochromatic radiation of the source into a spectrum. Characteristics of spectral devices: angular and linear dispersion and resolution of the device. Any spectral device decomposes radiation into monochromatic components by spatially separating them using a dispersing element (prism, diffraction grating, etc.) To extract the necessary information from the observed spectra, the device must provide good spatial separation of spectral lines, and also provide the ability to separate observations of close spectral lines. In this regard, to characterize the quality of a spectral device, the following quantities are introduced: angular D =ddor linear D l =dld variances device and its resolution R=/, where is the minimum difference in the wavelengths of the spectral lines that the device allows you to see longitudinally. The smaller the difference “visible” by the device, the higher its resolution R. Angular dispersion D determines the angle = D by which the device separates two spectral lines whose wavelengths differ by one (for example, in optics it is assumed = 1 nm). Linear dispersion D l determines the distance l =D l between spectral lines on the screen, the wavelengths of which differ by one ( = 1 nm). The higher the values of Dand D l the ability of the spectral device to spatially separate spectral lines. Specific expressions for the dispersions of the device D and D l and its resolution R depend on the type of device used to record the emission spectra of various sources. In this course, the issue of calculating the spectral characteristics of a device will be considered using the example of a diffraction grating. Angular and linear dispersion of a diffraction grating. The expression for the angular dispersion of the diffraction grating can be found by differentiating the condition of the main maxima d sin =kby. We obtain dcos d=kd, from where Instead of angular dispersion, you can use linear Considering that the position of the spectral line, measured from the center of the diffraction pattern, is equal to l=Ftg, where F is the focal length of the lens in the focal plane of which the spectrum is recorded, we obtain Resolution of the diffraction grating. Large angular dispersion is a necessary but not sufficient condition for the separate observation of close spectral lines. This is explained by the fact that spectral lines have width. Any detector (including the eye) registers the envelope of spectral lines, which, depending on their width, can be perceived as either one or two spectral lines. In this regard, an additional characteristic of a spectral device is introduced - its resolution: R = , where is the minimum difference in the wavelengths of the spectral lines that the device allows to see separately. To obtain a specific expression for R for a given device, it is necessary to specify a resolution criterion. It is known that the eye perceives two lines separately if the depth of the “dip” in the envelope of the spectral lines is at least 20% of the intensity at the maxima of the spectral lines. This condition is satisfied by the criterion proposed by Rayleigh: two spectral lines of the same intensity can be observed separately if the maximum of one of them coincides with the “edge” of the other. The position of the side minima closest to it can be taken as the “edges” of the line. In Fig. two spectral lines are depicted, corresponding to radiation with wavelength < The coincidence of the “edge” of one line with the maximum of another is equivalent to the same angular position , for example, of the maximum, the left line corresponding to the wavelength , and the left “edge” of the line corresponding to the wavelength . The position of the kth maximum of the spectral line with wavelength is determined by the condition dsin=k (1) The position of the left "edge" of the line with wavelength is determined by the angular position of its first left side minimum (p = -1) dsin=(k- 1 / N) 2 (2) Equating the right-hand sides of formulas (1) and (2), we obtain K 1 =(k- 1 / N) 2, ork( - 1)= /N, (3) It was found that the resolution R=kN of the diffraction grating increases with increasing number N of grooves on the grating, and at a fixed N with increasing order k of the spectrum. Thermal radiation. Thermal radiation (RT) is the emission of EM waves by a heated body due to its internal energy. All other types of luminescence of bodies, excited by types of energy, in contrast to thermal energy, are called luminescence. Absorption and reflectivity of the body. Absolutely black, white and gray bodies. In general, any body reflects, absorbs and transmits radiation incident on it. Therefore, for the radiation flux incident on a body we can write: Where
,
A,t-reflection, absorption and transmission coefficients, also called its reflective, absorption and transmittance abilities. If a body does not transmit radiation, then t=
0
, And
+a=1. In general, the coefficients
And A depend on the radiation frequency
and body temperature: If a body completely absorbs radiation of any frequency incident on it, but does not reflect it ( A T
=
1
, Energy characteristics of radiation. The radiation field is usually characterized by the radiation flux F
(W). Flow is the energy transferred by radiation through an arbitrary surface per unit time. Radiation flux emitted per unit area. body is called the energetic luminosity of the body and denotes R T
(W/m 3
)
.
Energy luminosity of a body in the frequency range Dimension The energetic luminosity of a body over the entire range of emitted radiation frequencies is equal to Relationship between the spectral characteristics of radiation by frequency and wavelength. Frequency-dependent emission characteristics
or wavelength
radiation is called spectral. Let's find the connection between these characteristics in terms of wavelength and frequency. Considering, dR
=
dR
,we get: Thermal radiation. Wien's and Stefan-Boltzmann's laws. Thermal radiation is EM radiation emitted by a substance due to its internal energy. TI has a continuous spectrum, i.e. its emissivity r
or r
depending on the frequency or wavelength of the radiation, it changes continuously, without jumps. TI is the only type of radiation in nature that is equilibrium, i.e. is in thermodynamic or thermal equilibrium with the body emitting it. Thermal equilibrium means that the radiating body and the radiation field have the same temperature. TI is isotropic, i.e. the probabilities of emitting radiation of different wavelengths or frequencies and polarizations in different directions are equally probable (the same). Among emitting (absorbing) bodies, a special place is occupied by absolutely black bodies (ABB), which completely absorb the radiation incident on it, but do not reflect it. If the black body is heated, then, as experience shows, it will shine brighter than a gray body. For example, if you paint a pattern on a porcelain plate with yellow, green and black paint, and then heat the plate to a high temperature, the black pattern will glow brighter, the green pattern will glow weaker, and the yellow pattern will glow very weakly. An example of a hot black body is the Sun. Another example of a blackbody is a cavity with a small hole and specularly reflective internal walls. External radiation, having entered the hole, remains inside the cavity and practically does not come out of it, i.e. the absorption capacity of such a cavity is equal to unity, and this is the black body. For example, an ordinary window in an apartment, open on a sunny day, does not let out the radiation that gets inside, and it appears black from the outside, i.e. behaves like a black hole. Experience shows that the dependence of the emissivity of the black body Schedule Wien's first law states: position of the maximum emissivity of the black body (r o
)
m
inversely proportional to its temperature: Where b
=
2,9
10
-3
m
TO
-the first constant of Guilt. Wien's second law states: the maximum emissivity of the black body is proportional to the fifth power of its temperature: Where With
= 1,3
10
-5
W/m 3
TO 5
-the second constant of Guilt. If we calculate the area under the graph of the emissivity of the black body, we will find its energetic luminosity R o T. It turns out to be proportional to the fourth power of the temperature of the black body. Thus This Stefan-Boltzmann law,
= 5,67
10
-8
W/m 2
TO 4
- Stefan-Boltzmann constant. Kirchhoff's law. Kirchhoff proved the following property of thermal emitters: body emissivity ratio r
to its absorption capacity a
at the same temperature T does not depend on the nature of the emitting body, for all bodies the same and equal to the emissive ability of the black body r o
:
r
/a
=
r o
.
This is the basic law of thermal radiation. To prove it, consider a thermally insulated cavity A with a small hole, inside of which there is a body B. Cavity A is heated and exchanges heat with body B through the radiation field of cavity C. In a state of thermal equilibrium, the temperatures of cavity A, body B and radiation field C are the same and equal to T In the experiment it is possible to measure the flow Radiation flux
, falling from a heated cavity A onto body B is absorbed by this body and reflected, and body B itself emits energy. In a state of thermal equilibrium, the flow emitted by the body r
and the stream reflected by it (1-a
)
must be equal to the flow
thermal radiation of the cavity where This is Kirchhoff's law. In its derivation, the nature of body B was not taken into account, therefore it is valid for any body and, in particular, for the black body, for which the emissivity is equal to r o
, and the absorption capacity a
=1
. We have: We found that the ratio of the emissivity of a body to its absorption capacity is equal to the emissivity of the black body at the same temperature T.Equality r o
=
indicates that according to the radiation flux leaving the cavity
it is possible to measure the emissivity of the black body r o
.
Planck's formula and proof of experimental laws using itGuiltand Stefan-Boltzmann. For a long time, various scientists tried to explain the patterns of blackbody radiation and obtain an analytical form of the function r o
.
In attempting to solve the problem, many important laws of thermal radiation were derived. Yes, in particular. Win, based on the laws of thermodynamics, showed that the emissivity of the black body r o
is a function of the radiation frequency ratio
and its temperature T, coinciding with the temperature of the black body: r o
=
f (
/
T) First explicit form for a function r o
was obtained by Planck (1905). At the same time, Planck assumed that TI contains 3M waves of various frequencies (wavelengths) in the interval ( Where
0
(
)
- the minimum quantum (portion) of energy that a frequency field oscillator can possess
. Based on this assumption, Planck obtained the following expression for the emissivity of the black body (see any textbook): Where With
= 3
10
8
m/s
- speed of light, k=1.38
10
-23
J/C- Boltzmann constant. According to Wien's theorem r o
=f(
/T) it is necessary to assume that the energy quantum of a field oscillator is proportional to its frequency
: where is the proportionality coefficient h=
6,62
10
-34
J
With or
For practical calculations it is convenient to substitute the values of the constants c,k,h and write Planck's formula in the form Where a 1
= 3,74
10
-16
W.m 2
,
a 2
=
1,44
10
-2
mK. The resulting expression for r o
gives a correct description of the law of radiation of the black body, corresponding to the experiment. The maximum of the Planck function can be found by calculating the derivative dr o
/d
and setting it equal to zero, which gives This is Wien's first law. Substituting
=
m into the expression for the Planck function, we get This is Wien's second law. The integral energetic luminosity (the area under the graph of the Planck function) is found by integrating the Planck function over all wavelengths. As a result we get (see textbook): This is the Stefan-Boltzmann law. Thus, Planck's formula explains all the experimental laws of black body radiation. Gray body radiation. A body for which the absorption capacity a
=a
<1
and does not depend on the frequency of radiation (its wavelength) is called gray. For a gray body according to Kirchhoff's law: For non-gray bodies (selective absorbers), for which a
depends on
or
,connection R
=a
R 0
does not hold, and we need to calculate the integral:
(4)
(5)
(7)
(9)
(1)
(3)
or with a source width equal to 
, i.e. width of the interference fringe. We obtain, taking into account (3)
(4)
(1)
(4)
(6)
or= 
(8)
(9)
or 
(10)
There are two principles in wave optics: the Huygens principle and the Huygens-Fresnel principle. Huygens' principle postulates that every point on the wave front is a source of secondary waves. By constructing the envelope of these waves, one can find the position of the wave front at subsequent times.
2 m =a 2 -(a-h m) 2 =2ah m (6)
(8)
(9)
(10)
(1)
(2)
(3)
whence bsin=m, (m=1,2,…) (4)
(6)
where we find that the condition for diffraction maxima (light stripes on the screen) has the form
(1)
(2)
(3)
,
(4)
, … (N-1);
,where p is an integer not a multiple of N (8)
(1)
(2)
, what gives 
(3)
(4)
(2)
And 
.
), then the body is called absolutely black, and if a body completely reflects radiation but does not absorb it, then the body is called white, if A T <1
, then the body is called gray. If the absorptive capacity of a body depends on the frequency or wavelength of the incident radiation and a
<1
, then the body is called selective absorber.
denote dR
,
and if it depends on body temperature T, then dR
.Energetic luminosity is proportional to width d
frequency interval of radiation: 
.Proportionality factor 
called emissivity of the body or spectral energy luminosity.
.
. From communication
=s/
should |d
|=(c/
2
)d
.
Then
on the radiation wavelength
has the form:
has a maximum. With increasing body temperature, the maximum dependence 
from
shifts towards shorter wavelengths (higher frequencies), and the body begins to shine brighter. This circumstance is reflected in two experimental Wien's laws and the Stefan-Boltzmann law.
(1)
(2)
(3)
radiation emerging from an opening, the properties of which are similar to those of radiation C inside the cavity.
(1)
(2)
).Fixed frequency wave
called EM field oscillator. According to Planck's assumption, the energy of each oscillator of the frequency field
It is quantized, that is, it depends on an integer parameter, which means it changes in a discrete way (jump):
(1)
(2)
(3)
=1,
02
10
-34
called Planck's constant
=
2
-cyclic frequency of radiation (field oscillator). Substituting (3) into formula (2), we get
(4)
(5)
(6)
(7)
(8)
(9)
, Where r o
-
Planck function
, Where 
(1)
(2)