Thanks to previous lessons, we know that light is a collection of rectilinear rays that propagate in space in a certain way. However, to explain the properties of some phenomena, we cannot use the concepts of geometric optics, that is, we cannot ignore the wave properties of light. For example, when sunlight passes through a glass prism, a picture of alternating color bands appears on the screen (Fig. 1), which is called a spectrum; a careful examination of the soap bubble reveals its bizarre color (Fig. 2), constantly changing over time. To explain these and other similar examples, we will use a theory that relies on the wave properties of light, that is, wave optics.
Rice. 1. Decomposition of light into a spectrum
Rice. 2. Soap bubble
In this lesson we will look at a phenomenon called light interference. With the help of this phenomenon, scientists in the 19th century proved that light has a wave nature, not a corpuscular one.
The phenomenon of interference is as follows: when two or more waves superimpose on each other in space, a stable pattern of amplitude distribution appears, while at some points in space the resulting amplitude is the sum of the amplitudes of the original waves, at other points in space the resulting amplitude becomes equal to zero. In this case, certain restrictions must be imposed on the frequencies and phases of the initially folding waves.
Example of adding two light waves
The increase or decrease in amplitude depends on the phase difference with which the two folding waves arrive at a given point.
In Fig. Figure 3 shows the case of the addition of two waves from point sources and located at a distance and from the point M, in which amplitude measurements are made. Both waves have at a point M in the general case, different amplitudes, since before reaching this point they travel different paths and their phases differ.
Rice. 3. Addition of two waves
In Fig. Figure 4 shows how the resulting amplitude of oscillation at a point depends M depends on the phases in which its two sine waves arrive. When the ridges coincide, the resulting amplitude is maximized. When the crest coincides with the trough, the resulting amplitude is reset to zero. In intermediate cases, the resulting amplitude has a value between zero and the sum of the amplitudes of the folding waves (Fig. 4).
Rice. 4. Addition of two sine waves
The maximum value of the resulting amplitude will be observed in the case when the phase difference between the two adding waves is zero. The same should be observed when the phase difference is equal to , since this is the period of the sine function (Fig. 5).
Rice. 5. Maximum value of the resulting amplitude
Amplitude of oscillations at a given point maximum, if the difference in the paths of the two waves exciting the oscillation at this point is equal to an integer number of wavelengths or an even number of half-waves (Fig. 6).
Rice. 6. Maximum amplitude of oscillations at a point M
The amplitude of oscillations at a given point is minimal if the difference in the paths of the two waves exciting the oscillation at this point is equal to an odd number of half-waves or a half-integer number of wavelengths (Fig. 7).
Rice. 7. Minimum amplitude of oscillations at a point M
, Where .
Interference can only be observed in the case of addition coherent waves (Fig. 8).
Rice. 8. Interference
Coherent waves- these are waves that have the same frequencies, a phase difference that is constant over time at a given point (Fig. 9).
Rice. 9. Coherent waves
If the waves are not coherent, then at any observation point two waves arrive with a random phase difference. Thus, the amplitude after the addition of two waves will also be a random variable that changes over time, and the experiment will show the absence of an interference pattern.
Incoherent waves- these are waves in which the phase difference continuously changes (Fig. 10).
Rice. 10. Incoherent waves
There are many situations where interference of light rays can be observed. For example, a gasoline stain in a puddle (Fig. 11), a soap bubble (Fig. 2).
Rice. 11. Gasoline stain in a puddle
The example with soap bubbles refers to the case of so-called interference in thin films. The English scientist Thomas Young (Fig. 12) was the first to come up with the idea of the possibility of explaining the colors of thin films by the addition of waves, one of which is reflected from the outer surface of the film, and the other from the inner.
Rice. 12. Thomas Young (1773-1829)
The result of interference depends on the angle of incidence of light on the film, its thickness and the wavelength of the light. Amplification will occur if the refracted wave lags behind the reflected wave by an integer number of wavelengths. If the second wave lags behind by half a wave or an odd number of half-waves, then the light will weaken (Fig. 13).
Rice. 13. Reflection of light waves from film surfaces
The coherence of waves reflected from the outer and inner surfaces of the film is explained by the fact that both of these waves are parts of the same incident wave.
The difference in colors corresponds to the fact that light can consist of waves of different frequencies (lengths). If light consists of waves with the same frequencies, then it is called monochromatic and our eye perceives it as one color.
Monochromatic light(from ancient Greek μόνος - one, χρῶμα - color) - an electromagnetic wave of one specific and strictly constant frequency from the range of frequencies directly perceived by the human eye. The origin of the term is due to the fact that differences in the frequency of light waves are perceived by humans as differences in color. However, by their physical nature, electromagnetic waves in the visible range do not differ from waves in other ranges (infrared, ultraviolet, x-ray, etc.), and the term “monochromatic” (“one-color”) is also used in relation to them, although these have no sensation of color no waves. Light consisting of waves of different wavelengths is called polychromatic(light from the sun).
Thus, if monochromatic light is incident on a thin film, the interference pattern will depend on the angle of incidence (at some angles the waves will enhance each other, at other angles they will cancel each other). With polychromatic light, to observe the interference pattern, it is convenient to use a film of variable thickness, in which waves with different lengths will interfere at different points, and we can get a color picture (like in a soap bubble).
There are special devices - interferometers (Fig. 14, 15), with which you can measure wavelengths, refractive indices of various substances and other characteristics.
Rice. 14. Jamin Interferometer
Rice. 15. Fizeau interferometer
For example, in 1887, two American physicists, Michelson and Morley (Fig. 16), designed a special interferometer (Fig. 17), with which they intended to prove or disprove the existence of the ether. This experiment is one of the most famous experiments in physics.
Rice. 17. Michelson Stellar Interferometer
Interference is also used in other areas of human activity (to assess the quality of surface treatment, to clear optics, to obtain highly reflective coatings).
Condition
Two translucent mirrors are located parallel to each other. A light wave of frequency falls on them perpendicular to the plane of the mirrors (Fig. 18). What should be the minimum distance between the mirrors in order to observe a minimum of first-order interference of passing rays?
Rice. 18. Illustration for the problem
Given:
Find:
Solution
One beam will pass through both mirrors. The other will pass through the first mirror, be reflected from the second and first, and pass through the second. The difference in the path of these rays will be twice the distance between the mirrors.
The minimum number corresponds to the value of an integer.
The wavelength is:
where is the speed of light.
Let us substitute the value and the value of the wavelength into the path difference formula:
Answer: .
To obtain coherent light waves using conventional light sources, wavefront division methods are used. In this case, the light wave emitted by any source is divided into two or more parts, coherent with each other.
1. Obtaining coherent waves by Young's method
The light source is a brightly illuminated slit, from which the light wave falls on two narrow slits parallel to the original slit S(Fig. 19). Thus, the slits serve as coherent sources. On the screen in the area B.C. an interference pattern is observed in the form of alternating light and dark stripes.
Rice. 19. Obtaining coherent waves by Young's method
2. Obtaining coherent waves using a Fresnel biprism
This biprism consists of two identical rectangular prisms with a very small refractive angle, folded at their bases. Light from the source is refracted in both prisms, as a result of which rays propagate behind the prism, as if emanating from imaginary sources and (Fig. 20). These sources are coherent. Thus, on the screen in the area B.C. an interference pattern is observed.
Rice. 20. Obtaining coherent waves using a Fresnel biprism
3. Obtaining coherent waves using optical path length separation
Two coherent waves are created by one source, but different geometric paths of length and pass to the screen (Fig. 21). In this case, each ray travels through a medium with its own absolute refractive index. The phase difference between the waves arriving at a point on the screen is equal to the following value:
Where and are the wavelengths in media whose refractive indices are equal to and respectively.
Rice. 21. Obtaining coherent waves using optical path length separation
The product of the geometric path length and the absolute refractive index of the medium is called optical path length.
,
– optical difference in the path of interfering waves.
Using interference, you can evaluate the quality of surface treatment of a product with an accuracy of wavelength. To do this, you need to create a thin wedge-shaped layer of air between the surface of the sample and a very smooth reference plate. Then surface irregularities up to cm will cause a noticeable curvature of interference fringes formed when light is reflected from the surfaces being tested and the bottom edge (Fig. 22).
Rice. 22. Checking the quality of surface treatment
A lot of modern photographic equipment uses a large number of optical glasses (lenses, prisms, etc.). Passing through such systems, the light flux experiences multiple reflections, which has a detrimental effect on image quality, since part of the energy is lost during reflection. To avoid this effect, it is necessary to use special methods, one of which is the method of clearing the optics.
Optical clearing is based on the phenomenon of interference. A thin film with a refractive index lower than the refractive index of the glass is applied to the surface of optical glass, such as a lens.
In Fig. Figure 23 shows the path of a beam incident on the interface at a slight angle. To simplify, we perform all calculations for an angle equal to zero.
Rice. 23. Coating of optics
The difference in the path of light waves 1 and 2 reflected from the upper and lower surfaces of the film is equal to twice the thickness of the film:
The wavelength in the film is less than the wavelength in vacuum in n once ( n- refractive index of the film):
In order for waves 1 and 2 to weaken each other, the path difference must be equal to half the wavelength, that is:
If the amplitudes of both reflected waves are the same or very close to each other, then the light extinction will be complete. To achieve this, the refractive index of the film is selected accordingly, since the intensity of the reflected light is determined by the ratio of the refractive indices of the two media.
The nature of the observed interference pattern depends on the relative position of the sources and the observation plane P (Fig. 1.1). Interference fringes can, for example, take the form of a family of concentric rings or hyperbolas. The simplest form is the interference pattern obtained by superposing two plane monochromatic waves, when the sources S1 and S2 are located at a sufficient distance from the screen. In this case, the interference pattern has the form of alternating dark and light rectilinear stripes (interference maxima and minima), located at the same distance from each other. It is this case that is realized in many optical interference schemes. Each interference maximum (light stripe) corresponds to a path difference, where m is an integer called the interference order. In particular, a zero-order interference maximum appears. In the case of interference of two plane waves fringe width l is connected by a simple relationship with the angle of convergence of the interfering rays on the screen (Fig. 1.2).
When the screen is symmetrically positioned with respect to beams 1 and 2, the width of the interference fringes is expressed by the ratio: . The approximation, which is valid at small angles, is applicable to many optical interference schemes.
(Fresnel mirror
Two flat touching mirrors OM and ON (Fig. 2) are positioned so that their reflecting surfaces form an angle that differs from 180 0 by fractions of one degree. Parallel to the line of intersection of the mirrors (point 0 in Fig. 2), at a certain distance r from it, a narrow slit S is placed, through which light falls on the mirrors. The opaque screen E1 blocks the path of light from the source S to the screen E. The mirrors throw two coherent cylindrical waves onto the screen E, propagating as if they came from imaginary sources S1 and S2.
The S1S 2 distance is smaller, which means that the interference pattern is larger, the smaller the angle between the mirrors? . The maximum solid angle within which interfering beams can still overlap is determined by the angle 2?=< KS1T =< RS 2 L . При этом экран располагается достаточно далеко. На основании законов отражения угол 2?= 2? . Таким образом,
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.
that is, it again turns out that the high-frequency wave is modulated by a low frequency.
1. Addition of unidirectional waves. Let on the axis OH there are two sources S 1 and S 2 at points with coordinates X 1 and X 2 (Fig. 81). At a moment in time t = 0 sources began to emit two monochromatic ones of the same frequency w Light waves linearly polarized in one plane.
, (10.1)
, (10.2)
Here v- speed of wave propagation.
Electric and magnetic fields obey the principle of superposition. Therefore, when waves are superimposed at any point A their tensions add up. . (10.3)
Here j = w(x 2 - X 1 )/v- phase shift between waves. In addition to the wave parameters w And v it is influenced by the distance between sources D = X 2 - X 1 .
The phase shift determines the amplitude E and the total wave .(10.4)
If the phase difference at a given point in space is constant, then the amplitude of the resulting oscillation at this point is constant. Depending on the phase difference j at the point, either an increase in light intensity will be observed ( j = 0, E A = E a1 + E a2), or weakening ( j = p, E A = E a1 – E a2). If the amplitudes are equal E a1 = E a2 and at j = p, E A = E a1 – E a2 = 0. The light is completely extinguished.
2. 
Interference pattern. In real cases, the folded waves usually converge at a certain angle to each other (Fig. 82). As a result, at different points in space A 1 , A 2 , A 3...phase difference j turns out to be different. A spatial distribution of light intensity appears in the form of alternating light and dark stripes. This is the so-called interference pattern.
The phenomenon of the addition of waves with the same frequency and a constant phase difference in time sufficient for observation, at which a redistribution of intensity occurs in space, is called interference . The interference pattern is most contrasting when the amplitudes of the added waves are the same.
3. Coherence(from Latin cohaerens – in connection) – consistency in time of several oscillatory or wave processes, manifested when they are added together. Natural light sources consist of a huge number of chaotically flaring and fading emitters - atoms and molecules. Through each point of the optically transparent medium surrounding the source, trains of waves emitted by different atoms and having different amplitudes, phases and frequencies pass one after another. Therefore, it is fundamentally impossible to make two non-laser light sources coherent.
Obtaining coherent beams from natural sources is possible by splitting a beam from one source and creating a constant phase shift between them. In this case, the rays repeat themselves in all details and therefore can interfere with each other.
But when creating a phase difference, we must remember that the wave train emitted by an individual atom has a finite extent along the beam. With an emission duration of 10 -11 ¸ 10 -8 s, this extent does not exceed 1 ¸ 3 m. Therefore, we can say that every 10 -8 s the wave emitted by even one atom changes.
But even a single train is not a segment of a sinusoid. Vector oscillation phase E changes continuously throughout its duration. Therefore, the “head” of the train is not coherent with its “tail”.
Time t, during which the phase of oscillations in a light wave, measured at a constant point in space, changes by p, called coherence time. Distance сt, Where With– the speed of light measured along the direction of wave propagation is called coherence length. Light from different sources has a coherence length from several micrometers to several kilometers:
- sunlight, сt» 1 ¸ 2 µm,
– spectra of rarefied gases, сt» 0.1 m,
– laser radiation, сt» 1 ¸ 2 km.
To describe the coherent properties of a wave in a plane perpendicular to the direction of its propagation, the term is used spatial coherence. It is determined by the area of a circle with diameter l, at all points of which the phase difference does not exceed the value p.
The coherence space at a point source of natural light approaches the volume of a truncated cone with a length of several microns and a base diameter of several mm (Fig. 83). It increases with distance from the source.
4. 
Construction of an interference pattern using Young's method. The first two-beam interference scheme was proposed in 1802 by Thomas Young. He was the first to clearly establish the principles of amplitude addition and explain interference in the wave model of light. The essence of Jung's scheme boils down to the following.
Screen E 1 with a narrow slit is installed normally to the rays from a natural light source S. This slit acts as a point light source S. Spreading from S cylindrical wave excites in slits S 1 and S 2 screens E 2 coherent oscillations. Therefore, the waves propagating from the slits S 1 and S 2, upon interaction, produce an interference pattern on screen E 3 in the form of a system of stripes parallel to the slits (Fig. 84).
Although in practice Young's method is not used due to the low illumination of the E 3 screen, it is convenient for the theoretical study of two-beam interference in order to obtain quantitative estimates. To do this, let us present Young's scheme in the form shown in Fig. 85.
If S 1 and S 2 – coherent light sources emitting in the same phase, then to any arbitrary point A screen E 3 waves will arrive with a path difference D = l 2 –l 1 . Believing in the picture A<
The maximum illumination will be at those points of the screen where D is an integer number of waves, and the minimum illumination will be where D is an odd number of half-waves.
|
|
Fig.85, k= 0, 1, 2, 3, (max), (10.5)
, k= 1, 2, 3,(min), (10.6)
Here k– band number. At small angles j the stripes are spaced evenly. The distance between adjacent dark or adjacent light stripes is
. (10.7)
The smaller the distance, the greater it is. A between sources and the greater the distance L from sources to screen.
At a = 1 mm, L= 1 m, D y = 0.5×10 –6×1 ç 10 –3 = 0.5 mm for green rays.
5. Interference pattern contrast depends on the extent of the light source S and on the degree of monochromaticity of the light.
A. The influence of non-monochromatic light. In the case when non-monochromatic waves interfere, the maxima on the screen for different wavelengths do not coincide. As a result, the interference pattern is blurred. It is completely lubricated when k th maximum of a wave with length l+D l have to k + 1st wave maximum with length l.
All minimum space for a wave l occupied by maxima with lengths from l before l+D l.
The monochromaticity criterion limits the number of observed bands. For example, for sunlight with l from 0.4 to 0.8 µm the entire spectral range can be represented as: l = l 0±D l = 0.6±0.2 µm. Maximum order of observed interference fringe k max = l 0 /
D l = 0,6/
0,2 =
3. This means that 6 dark bands can be observed, corresponding k =–3, –2, –1, +1, +2, +3.
By compressing the spectral range using light filters, you can increase the number and contrast of the observed bands.
b. Effect of source length. Let the slot width S equal to b(Fig. 86). To the cracks S 1 and S 2 emitted in the same phase, it is necessary that the rays arriving at each slit from different points of the source S, had a small path difference D, no more than a quarter of the wavelength. 
. (10.9)
Corner w usually no more than 1°. Therefore, the limitation on the slot width can be written as follows: . But w = аç2d, Where A- distance between slots S 1 and S 2 , d- distance from the slot S before S 1 and S 2. Then b
At a = 1 mm, d = 1m, l = 0.6×10 –6 m, b< 0.6×10 –6×1 ç 2×10 –3 = 0.3×10 –3 m = 0.3 mm. To obtain good contrast, this value should be reduced by another 3-4 times.
6. 
Practical methods for observing interference.
A. Fresnel mirrors, 1816 (Fig. 87). Light from a source enclosed in a light-proof casing, through an opening in it, enters in a diverging beam onto two flat mirrors. Angle between mirrors a» 179°.
|
|
Fig.88The advantage of the method is good illumination, the disadvantage is the difficulty of adjusting the mirrors on the optical bench.
b. Fresnel biprism, 1819 (Fig. 88). Advantages - good illumination and ease of adjustment, disadvantage - a special biprism is required, a product of the optical industry.
Here S 1 and S 2 – virtual images of the light source S.
V. Bilinza Biye, 1845 (Fig. 89). The converging or diverging lens is cut (split) along the diameter, and both halves are slightly moved apart.
The farther the half-lenses are moved apart from each other, the more compressed the interference pattern is, the narrower the stripes. Here S 1 and S 2 – actual images of the light source S.
G. Lloyd's Mirror, 1837 (Fig. 90). Direct beam from source S interferes with the beam reflected from the mirror.
Here S- illuminated slit, S 1 – its virtual image.
The wave properties of light reveal themselves most clearly in interference And diffraction. These phenomena are characteristic of waves of any nature and are relatively easily observed experimentally for waves on the surface of water or for sound waves. Observing the interference and diffraction of light waves is possible only under certain conditions. Light emitted by conventional (non-laser) sources is not strictly monochromatic. Therefore, to observe interference, light from one source must be divided into two beams and then superimposed on each other.
Interference microscope.
Existing experimental methods for obtaining coherent beams from a single light beam can be divided into two classes.
IN wavefront division method the beam is passed, for example, through two closely spaced holes in an opaque screen (Jung's experiment). This method is suitable only for sufficiently small source sizes.
In another method, the beam is divided onto one or more partially reflective, partially transmissive surfaces. This method amplitude divisions can also be used with extended sources. It provides greater intensity and underlies the operation of a variety of interferometers. Depending on the number of interfering beams, two-beam and multi-beam interferometers are distinguished. They have important practical applications in engineering, metrology and spectroscopy.
Let two waves of the same frequency, superimposed on each other, excite oscillations of the same direction at some point in space:
; 
,
where under x understand the electrical tension E and magnetic H wave fields, which obey the principle of superposition (see paragraph 6).
The amplitude of the resulting oscillation when adding oscillations directed along one straight line will be found using formula (2.2.2):
If the phase difference of oscillations,excited by waves at some point in space,remains constant in time, then such waves are called coherent.
When incoherent waves, the phase difference continuously changes, taking on any values with equal probability, as a result of which the time-averaged value is zero (varies from –1 to +1). That's why .
The light intensity is proportional to the square of the amplitude: . From this we can conclude that for incoherent sources, the intensity of the resulting wave is the same everywhere and is equal to the sum of the intensities created by each wave separately:
 .
| (8.1.1) |
When coherent waves 
(for each point in space), so
| . | (8.1.2) |
The last term in this expression 
called interference term
.
At points in space where 
, (at maximum), where 
, intensity (minimum). Consequently, when two (or several) coherent light waves are superimposed, a spatial redistribution of the light flux occurs, resulting in intensity maxima in some places and intensity minima in others. This phenomenon is called interference of light
.
A stable interference pattern is obtained only by adding coherent waves. The incoherence of natural light sources is due to the fact that the radiation of a body is composed of waves emitted by many atoms . Phases of each wave train are not related to each other in any way . Atoms emit chaotically.
Periodic sequence of wave crests and troughs,formed during the act of radiation of one atom,called wave train or wave train.
The process of radiation of one atom lasts approximately s. In this case, the length of the train is .
Approximately wavelengths fit into one train.
Conditions for maximum and minimum interference
Let the separation into two coherent waves occur at the point ABOUT(Fig. 8.1).
To the point R the first wave travels in a medium with an index of distance , and the second in a medium with a refractive index of distance . If at the point ABOUT oscillation phase (), then the first wave excites at the point R hesitation
, and the second 
,