Chapter 31

HOW THE REFRACTIVE INDEX ARISES

§ 1. Index of refraction

§ 2. Field emitted by the medium

§ 3. Dispersion

§ 4. Absorption

§ 5. Energy of a light wave

§ 1. Index of refraction

We have already said that light travels slower in water than in air, and slightly slower in air than in vacuum. This fact is taken into account by introducing the refractive index n. Let us now try to understand how the decrease in the speed of light arises. In particular, it is especially important to trace the connection of this fact with some physical assumptions or laws that were previously stated and boil down to the following:

a) the total electric field under any physical conditions can be represented as the sum of fields from all charges in the Universe;

b) the radiation field of each individual charge is determined by its acceleration; the acceleration is taken into account the delay arising from the finite velocity of propagation, always equal to c. But you will probably immediately cite a piece of glass as an example and exclaim: “Nonsense, this provision is not suitable here. We must say that the delay corresponds to the speed c/n. However, this is wrong; Let's try to figure out why this is wrong. It seems to the observer that light or any other electric wave propagates through a substance with a refractive index n at a speed c/n. And this is true to some extent. But in fact, the field is created by the movement of all charges, including charges moving in the medium, and all components of the field, all its terms propagate with a maximum speed c. Our task is to understand how the apparent lower speed arises.

Fig. 31.1. Passage of electric waves through a layer of a transparent substance.

Let's try to understand this phenomenon with a very simple example. Let the source (let's call it "external source") be placed at a great distance from a thin transparent plate, say glass. We are interested in the field on the other side of the plate and quite far from it. All this is shown schematically in Fig. 31.1; points S and P here are assumed to be remote at a great distance from the plane. According to the principles we have formulated, the electric field away from the plate is represented by the (vector) sum of the fields of the external source (at point S) and the fields of all charges in the glass plate, each field being taken with a delay at speed c. Recall that the field of each charge does not change from the presence of other charges. These are our basic principles. Thus, the field at the point P

can be written as

where E s is the field of an external source; it would coincide with the desired field at the point P, if there were no plate. We expect that in the presence of any moving charges, the field at P will be different from E r

Where do moving charges in glass come from? It is known that any object consists of atoms containing electrons. An electric field from an external source acts on these atoms and swings the electrons back and forth. The electrons in turn create a field; they can be considered as new emitters. The new emitters are coupled to the source S, since it is the source field that causes them to oscillate. The total field contains not only the contribution from the source S, but also additional contributions from the radiation of all moving charges. This means that the field changes in the presence of glass, and in such a way that its propagation velocity seems to be different inside the glass. It is this idea that we use in the quantitative consideration.

However, the exact calculation is very difficult, because our statement that the charges only experience the action of the source is not entirely correct. Each given charge “feels” not only the source, but, like any object in the Universe, it also feels all other moving charges, in particular charges vibrating in glass. Therefore, the total field acting on a given charge is a combination of fields from all other charges, the movement of which in turn depends on the movement of this charge! You see that deriving the exact formula requires solving a complex system of equations. This system is very complex and you will learn it much later.

And now let us turn to a very simple example in order to clearly understand the manifestation of all physical principles. Let us assume that the action of all other atoms on a given atom is small compared to the action of the source. In other words, we are studying a medium in which the total field changes little due to the motion of the charges in it. This situation is typical for materials with a refractive index very close to unity, for example, for rarefied media. Our formulas will be valid for all materials with a refractive index close to unity. In this way we can avoid the difficulties associated with solving the complete system of equations.

You may have noticed along the way that the movement of the charges in the plate causes another effect. This movement creates a wave propagating backwards in the direction of the source S. Such a backward moving wave is nothing but a beam of light reflected by a transparent material. It comes not only from the surface. Reflected radiation is generated at all points within the material, but the net effect is equivalent to reflection from the surface. Accounting for reflection is beyond the limits of applicability of the present approximation, in which the refractive index is assumed to be so close to unity that the reflected radiation can be neglected.

Before proceeding to the study of the refractive index, it should be emphasized that the phenomenon of refraction is based on the fact that the apparent speed of wave propagation is different in different materials. The deflection of a light beam is a consequence of the change in the effective speed in different materials.

Fig. 31.2. Relationship between refraction and velocity change.

To clarify this fact, we have noted in Fig. 31.2 a series of successive maxima in the amplitude of a wave incident from vacuum onto glass. The arrow perpendicular to the indicated maxima marks the direction of wave propagation. Everywhere in the wave, oscillations occur with the same frequency. (We have seen that the forced oscillations have the same frequency as the oscillations of the source.) It follows that the distances between the wave maxima on both sides of the surface coincide along the surface itself, since the waves here must be matched and the charge on the surface oscillates with the same frequency. The smallest distance between wave crests is the wavelength equal to the speed divided by the frequency. In vacuum, the wavelength is l 0 =2pс/w, and in glass l=2pv/w or 2pс/wn, where v=c/n is the wave speed. As can be seen from FIG. 31.2, the only way to "sew" the waves at the boundary is to change the direction of the wave in the material. Simple geometric reasoning shows that the "stitching" condition reduces to the equality l 0 /sin q 0 =l/sinq, or sinq 0 /sinq=n, and this is Snell's law. Don't worry now about the deflection of the light itself; it is only necessary to find out why, in fact, the effective speed of light in a material with a refractive index n is equal to c/n?

Let us return again to Fig. 31.1. From what has been said it is clear that it is necessary to calculate the field at the point P from the oscillating charges of the glass plate. Let us denote this part of the field, which is represented by the second term in equality (31.2), by E a. Adding to it the source field E s , we obtain the total field at the point P.

The task before us here is perhaps the most difficult of those that we will deal with this year, but its complexity lies only in the large number of terms that are added; each member is very simple in itself. Unlike other times when we used to say: “Forget the conclusion and look only at the result!”, Now for us the conclusion is much more important than the result. In other words, you need to understand the whole physical “kitchen” with which the refractive index is calculated.

To understand what we are dealing with, let's find what the "correction field" E a should be, so that the total field at the point P looks like the source field slowed down when passing through a glass plate. If the plate had no effect on the field, the wave would propagate to the right (along the axis

2) by law

or, using exponential notation,

What would happen if the wave passed through the plate at a slower speed? Let the plate thickness be Dz. If there were no plate, then the wave would travel the distance Dz in the time Dz/c. And since the apparent velocity of propagation is c/n, then the time nDz/c will be required, i.e. more by some additional time equal to Dt=(n-l) Dz/c. Behind the plate, the wave again moves with speed c. We take into account the additional time to pass through the plate, replacing t in equation (31.4) with (t-Dt), i.e. . Thus, if you put the plate, then the formula for the wave should acquire

This formula can also be rewritten in another way:

whence we conclude that the field behind the plate is obtained by multiplying the field that would be in the absence of the plate (ie, E s) by exp[-iw(n-1)Dz/c]. As we know, the multiplication of an oscillating function of the type e i w t by e i q means a change in the phase of the oscillations by an angle q, which occurs due to a delay in the passage of the plate. The phase lags by w(n-1)Dz/c (it lags precisely because the exponent has a minus sign).

We said earlier that the plate adds a field E a to the original field E S = E 0 exp, but instead we found that the effect of the plate is to multiply the field by a factor that shifts the phase of the oscillations. However, there is no contradiction here, since the same result can be obtained by adding an appropriate complex number. This number is especially easy to find for small Dz, since e x for small x equals (1 + x) with great accuracy.

Fig. 31.3. Construction of the field vector of the wave passed through the material at certain values ​​of t and z.

Then one can write

Substituting this equality into (31 6), we obtain

The first term in this expression is simply the source field, and the second should be equated with E a - the field created by the oscillating charges of the plate to the right of it. The field E a is expressed here in terms of the refractive index n; it, of course, depends on the strength of the source field.

The meaning of the transformations made is easiest to understand with the help of the complex number diagram (see Fig. 31.3). Let's set aside E s first (z and t are chosen in the figure such that E s lies on the real axis, but this is not necessary). The delay in the passage of the plate leads to a delay in the phase of E s , i.e. turns E s by a negative angle. It's like adding a small vector E a, directed almost at right angles to E s . This is the meaning of the factor (-i) in the second term (31.8). It means that for real E s the value of E a is negative and imaginary, and in the general case E s and E a form a right angle.

§ 2. Field emitted by the medium

We must now find out whether the field of oscillating charges in the plate has the same form as the field E a in the second term (31.8). If this is so, then we will thereby also find the refractive index n [since n is the only factor in (31.8) that is not expressed in terms of fundamental quantities]. Let us now return to the calculation of the field E a created by the charges of the plate. (For convenience, we have written out in Table 31.1 the notation that we have already used and those that we will need in the future.)

WHEN CALCULATED _______

E s field generated by the source

E a the field created by the charges of the plate

Dz plate thickness

z distance along the normal to the plate

n index of refraction

w frequency (angular) radiation

N is the number of charges per unit volume of the plate

h number of charges per unit area of ​​the plate

q e electron charge

m is the electron mass

w 0 resonant frequency of an electron bound in an atom

If the source S (in Fig. 31.1) is at a sufficiently large distance to the left, then the field E s has the same phase along the entire length of the plate, and near the plate it can be written as

On the plate itself at the point z=0 we have

This electric field affects every electron in the atom, and they will oscillate up and down under the influence of the electric force qE (if e0 is directed vertically). To find the nature of the motion of electrons, let us represent the atoms as small oscillators, i.e., let the electrons be elastically connected to the atom; this means that the displacement of electrons from their normal position under the action of a force is proportional to the magnitude of the force.

If you have heard of a model of an atom in which electrons orbit around the nucleus, then this model of an atom will seem simply ridiculous to you. But this is just a simplified model. A precise theory of the atom, based on quantum mechanics, states that in processes involving light, electrons behave as if they were attached to springs. So, let's assume “that a linear restoring force acts on the electrons, and therefore they behave like oscillators with a mass m and a resonant frequency w 0 . We have already studied such oscillators and we know the equation of motion to which they obey:

(here F is an external force).

In our case, the external force is created by the electric field of the source wave, so we can write

where q e is the electron charge, and as E S we took the value of E S = E 0 e i w t from equation (31.10). The equation of electron motion takes the form

The solution to this equation, found by us earlier, is as follows:

We found what we wanted - the movement of electrons in the plate. It is the same for all electrons, and only the average position (“zero” of motion) is different for each electron.

We are now in a position to determine the field E a produced by the atoms at the point P, since the field of the charged plane was found even earlier (at the end of Chapter 30). Turning to equation (30.19), we see that the field E a at the point P is the charge velocity retarded in time by z/c times a negative constant. Differentiating x from (31.16), we get the speed and, introducing a delay [or simply substituting x 0 from (31.15) into (30.18)], we arrive at the formula

As expected, the forced oscillation of the electrons resulted in a new wave propagating to the right (this is indicated by the factor exp); the wave amplitude is proportional to the number of atoms per unit area of ​​the plate (multiplier h), as well as to the amplitude of the source field (E 0). In addition, there are other quantities that depend on the properties of atoms (q e , m , w 0).

The most important point, however, is that the formula (31.17) for E a is very similar to the expression for E a in (31.8), which we obtained by introducing a delay in a medium with a refractive index n. Both expressions are the same if we put

Note that both sides of this equation are proportional to Dz, since h - the number of atoms per unit area - is equal to NDz, where N is the number of atoms per unit volume of the plate. Substituting NDz for h and canceling by Dz, we get our main result - the formula for the refractive index, expressed in terms of constants depending on the properties of atoms, and the frequency of light:

This formula "explains" the refractive index, which is what we were striving for.

§ 3. Dispersion

Our result is very interesting. It gives not only the refractive index expressed in terms of atomic constants, but also indicates how the refractive index changes with the frequency of light w. With the simple statement "light travels at a slower speed in a transparent medium" we could never arrive at this important property. Of course, it is also necessary to know the number of atoms per unit volume and the natural frequency of atoms w 0 . We are not yet able to determine these quantities, since they are different for different materials, and we cannot now present a general theory on this issue. General theory of the properties of various substances - their natural frequencies and

etc. - is formulated on the basis of quantum mechanics. In addition, the properties of various materials and the magnitude of the refractive index vary greatly from material to material, and therefore one can hardly hope that it will be possible to obtain a general formula suitable for all substances at all.

Nevertheless, let's try to apply our formula to different environments. First of all, for most gases (for example, for air, most colorless gases, hydrogen, helium, etc.), the natural frequencies of electron oscillations correspond to ultraviolet light. These frequencies are much higher than the frequencies of visible light, i.e. w 0 is much greater than w, and in the first approximation w 2 can be neglected compared to w 0 2 . Then the refractive index is almost constant. So, for gases, the refractive index can be considered a constant. This conclusion is also valid for most other transparent media, such as glass. Looking more closely at our expression, we can see that as the co denominator increases, the denominator decreases, and, consequently, the refractive index increases. Thus, n slowly increases with increasing frequency. Blue light has a higher refractive index than red light. That is why blue rays are more strongly deflected by a prism than red ones.

The very fact of the dependence of the refractive index on frequency is called dispersion, since it is precisely because of dispersion that the light “disperses”, decomposes into a spectrum by a prism. The formula expressing the refractive index as a function of frequency is called the dispersion formula. So, we have found the dispersion formula. (Over the past few years, "dispersion formulas" have come into use in the theory of elementary particles.)

Our dispersion formula predicts a number of new interesting effects. If the frequency w 0 lies in the visible light region, or if the refractive index of a substance, such as glass, is measured for ultraviolet rays (where w is close to w 0), then the denominator tends to zero, and the refractive index becomes very large. Further, let w be greater than w 0 . Such a case arises, for example, if substances such as glass are irradiated with x-rays. In addition, many substances that are opaque to ordinary light (say, coal) are transparent to X-rays, so we can talk about the refractive index of these substances for X-rays. The natural frequencies of carbon atoms are much less than the frequency of X-rays. The refractive index in this case is given by our dispersion formula if we put w 0 =0 (ie we neglect w 0 2 compared to w 2).

A similar result is obtained when a gas of free electrons is irradiated with radio waves (or light). In the upper atmosphere, ultraviolet radiation from the sun knocks electrons out of atoms, resulting in a gas of free electrons. For free electrons w 0 =0 (there is no elastic restoring force). Assuming w 0 =0 in our dispersion formula, we obtain a reasonable formula for the refractive index of radio waves in the stratosphere, where N now means the density of free electrons (a number per unit volume) in the stratosphere. But, as can be seen from the formula, when a substance is irradiated with X-rays or an electron gas with radio waves, the term (w02-w2) becomes negative, which implies that n is less than one. This means that the effective speed of electromagnetic waves in matter is greater than c! Could it be?

Maybe. Although we said that signals cannot travel faster than the speed of light, nevertheless, the refractive index at a certain frequency can be either greater or less than unity. It simply means that the phase shift due to light scattering is either positive or negative. In addition, it can be shown that the signal speed is determined by the refractive index not at one frequency value, but at many frequencies. The refractive index indicates the speed of the wave crest. But the wave crest does not yet constitute a signal. A pure wave without any modulations, that is, consisting of infinitely repeating regular oscillations, has no "beginning" and cannot be used to send time signals. To send a signal, the wave needs to be modified, to make a mark on it, that is, to make it thicker or thinner in some places. Then the wave will contain not one frequency, but a number of frequencies, and it can be shown that the speed of signal propagation does not depend on one value of the refractive index, but on the nature of the change in the index with frequency. We will put this question aside for now. In ch. 48 (Issue 4), we calculate the speed of propagation of signals in glass and make sure that it does not exceed the speed of light, although the crests of the wave (purely mathematical concepts) move faster than the speed of light.

A few words about the mechanism of this phenomenon. The main difficulty here is related to the fact that the forced motion of charges is opposite in sign to the direction of the field. Indeed, in expression (31.16) for the displacement of the charge x, the factor (w 0 -w 2) is negative for small w 0 and the displacement has the opposite sign with respect to the external field. It turns out that when the field acts with some force in one direction, the charge moves in the opposite direction.

How did it happen that the charge began to move in the opposite direction to the force? Indeed, when the field is turned on, the charge does not move in the opposite direction to the force. Immediately after the field is turned on, a transitional regime occurs, then the oscillations are established, and only after this oscillation the charges are directed opposite to the external field. At the same time, the resulting field begins to outstrip the source field in phase. When we say that the “phase speed”, or the speed of the wave crests, is greater than c, then we mean exactly the phase advance.

In FIG. 31.4 shows an approximate view of the waves that arise when the source wave is turned on abruptly (ie, when a signal is sent).

Fig. 31.4. Wave "signals".

Fig. 31.5. Refractive index as a function of frequency.

It can be seen from the figure that for a wave passing through a medium with a phase advance, the signal (i.e., the beginning of the wave) does not lead the source signal in time.

Let us now turn again to the dispersion formula. It should be remembered that our result somewhat simplifies the true picture of the phenomenon. To be accurate, some adjustments need to be made to the formula. First of all, damping must be introduced into our model of the atomic oscillator (otherwise the oscillator, once started, will oscillate ad infinitum, which is implausible). We have already studied the motion of a damped oscillator in one of the previous chapters [see. equation (23.8)]. Accounting for damping leads to the fact that in formulas (31.16), and therefore

in (31.19), instead of (w 0 2 -w 2) appears (w 0 2 -w 2 +igw)" where g is the damping factor.

The second correction to our formula arises because each atom usually has several resonant frequencies. Then, instead of one type of oscillators, it is necessary to take into account the action of several oscillators with different resonant frequencies, the oscillations of which occur independently of each other, and add up the contributions from all oscillators.

Let a unit volume contain N k electrons with natural frequency (w k and damping coefficient g k . Our dispersion formula will eventually take the form

This final expression for the refractive index is valid for a large number of substances. An approximate course of the refractive index with frequency, given by formula (31.20), is shown in Fig. 31.5.

You see that everywhere, except for the region where w is very close to one of the resonant frequencies, the slope of the curve is positive. This dependence is called the "normal" variance (because this case occurs most often). Near resonant frequencies, the curve has a negative slope, and in this case one speaks of "anomalous" dispersion (meaning "abnormal" dispersion), because it was observed long before electrons were known, and seemed unusual at that time, C From our point of view, both slopes are quite "normal"!

§ 4 Takeover

You have probably already noticed something strange in the last form (31.20) of our dispersion formula. Because of the attenuation term ig, the refractive index has become a complex quantity! What does this mean? We express n in terms of the real and imaginary parts:

where n" and n" are real. (In" ​​is preceded by a minus sign, and n" itself, as you can easily see, is positive.)

The meaning of the complex refractive index is most easily understood by returning to equation (31.6) for a wave passing through a plate with a refractive index n. Substituting here the complex n and rearranging the terms, we get

The factors denoted by the letter B have the same form and, as before, describe a wave whose phase, after passing through the plate, lags by an angle w (n "-1) Dz / c. The factor A (an exponent with a real exponent) represents something new. The exponent exponential is negative, therefore, A is real and less than unity. The factor A reduces the amplitude of the field; with increasing Dz, the value of A, and, consequently, the entire amplitude decreases. When passing through the medium, the electromagnetic wave decays. The medium "absorbs" part of the wave. The wave leaves the medium , losing part of its energy. This should not be surprising, because the damping of oscillators introduced by us is due to the force of friction and inevitably leads to energy loss. We see that the imaginary part of the complex refractive index n" describes the absorption (or "attenuation") of an electromagnetic wave. Sometimes n" is also called the "absorption coefficient".

Note also that the appearance of the imaginary part of n deflects the arrow representing E a in FIG. 31.3, to the origin.

From this it is clear why the field weakens when passing through the medium.

Usually (as, for example, with glass), the absorption of light is very small. This is exactly what happens according to our formula (31.20), because the imaginary part of the denominator ig k w is much less than the real part (w 2 k -w 2). However, when the frequency w is close to w k , the resonant term (w 2 k -w 2 ) is small compared to ig k w and the refractive index becomes almost purely imaginary. Absorption in this case determines the main effect. It is absorption that produces dark lines in the solar spectrum. Light emitted from the surface of the Sun passes through the solar atmosphere (as well as the Earth's atmosphere), and frequencies equal to the resonant frequencies of the atoms in the Sun's atmosphere are strongly absorbed.

Observation of such spectral lines of sunlight makes it possible to establish the resonant frequencies of atoms, and hence the chemical composition of the solar atmosphere. In the same way, the composition of stellar matter is known from the spectrum of stars. Using these methods, they found that the chemical elements in the Sun and stars do not differ from those on Earth.

§ 5. Energy of a light wave

As we have seen, the imaginary part of the refractive index characterizes absorption. Let us now try to calculate the energy carried by a light wave. We have put forward arguments in favor of the fact that the energy of a light wave is proportional to E 2 , the time average of the square of the electric field of the wave. The weakening of the electric field due to the absorption of the wave should lead to a loss of energy, which turns into some kind of friction of electrons and, ultimately, as you might guess, into heat.

Taking the part of the light wave incident on a single area, for example, on a square centimeter of the surface of our plate in Fig. 31.1, we can write the energy balance in the following form (we assume that energy is conserved!):

Falling energy in 1 sec = Outgoing energy in 1 sec + Work done in 1 sec. (31.23)

Instead of the first term, you can write aE2s, where a is a proportionality factor relating the average value of E 2 to the energy carried by the wave. In the second term, it is necessary to include the radiation field of the atoms of the medium, i.e., we must write

a (Es + E a) 2 or (expanding the square of the sum) a (E2s + 2E s E a + -E2a).

All our calculations were carried out under the assumption that

the thickness of the material layer is small and its refractive index

differs slightly from unity, then E a turns out to be much less than E s (this was done for the sole purpose of simplifying the calculations). Within our approximation, the term

E2a should be omitted, neglecting it in comparison with E s E a . You can object to this: "Then you must also discard E s E a, because this term is much less than El." Indeed, E s E a

much less than E2s, but if we drop this term, we get an approximation in which the effects of the environment are not taken into account at all! The correctness of our calculations within the framework of the approximation made is verified by the fact that we left everywhere the terms proportional to -NDz (density of atoms in the medium), but discarded the terms of order (NDz) 2 and higher powers in NDz. Our approximation could be called the "low-density approximation".

Note, by the way, that our energy balance equation does not contain the energy of the reflected wave. But it should be so, because the amplitude of the reflected wave is proportional to NDz, and the energy is proportional to (NDz) 2 .

To find the last term in (31.23), you need to calculate the work done by the incident wave on electrons in 1 second. Work, as you know, is equal to force multiplied by distance; hence the work per unit time (also called power) is given by the product of force and velocity. More precisely, it is equal to F v, but in our case, the force and speed have the same direction, so the product of vectors is reduced to the usual one (up to sign). So, the work done in 1 second on each atom is equal to q e E s v. Since there are NDz atoms per unit area, the last term in equation (31.23) turns out to be equal to NDzq e E s v. The energy balance equation takes the form

The terms aE 2 S cancel out and we get

Returning to equation (30.19), we find E a for large z:

(recall that h=NDz). Substituting (31.26) into the left side of equality (31.25), we obtain

Ho E s (at the point z) is equal to E s (at the point of the atom) with a delay of z/c. Since the average value does not depend on time, it will not change if the time argument lags by z/c, i.e. it is equal to E s (at the point of the atom) v, but exactly the same average value is on the right side of (31.25 ). Both parts of (31.25) will be equal if the relation holds

Thus, if the law of conservation of energy is valid, then the amount of electric wave energy per unit area per unit time (what we call intensity) should be equal to e 0 sE 2 . Denoting the intensity by S, we get

where the bar means the time average. From our theory of the refractive index, a wonderful result has turned out!

§ 6. Diffraction of light on an opaque screen

The moment has now come to apply the methods of this chapter to the solution of a problem of a different kind. In ch. 30 we said that the distribution of light intensity - the diffraction pattern that occurs when light passes through holes in an opaque screen - can be found by evenly distributing sources (oscillators) over the area of ​​\u200b\u200bthe holes. In other words, the diffracted wave looks like the source is a hole in the screen. We must find out the reason for this phenomenon, because in fact it is in the hole that there are no sources, there are no charges moving with acceleration.

Let's answer the question first: what is an opaque screen? Let there be a completely opaque screen between the source S and the observer P, as shown in Fig. 31.6, a. Since the screen is "opaque", there is no field at point P. Why? According to general principles, the field at the point P is equal to the field E s taken with some delay, plus the field of all other charges. But, as was shown, the E s field sets the screen charges in motion, and they in turn create a new field, and if the screen is opaque, this field of charges should exactly extinguish the E s field from the back of the screen. Here you can object: “What a miracle they will exactly be extinguished! What if the repayment is incomplete? If the fields were not completely suppressed (recall that the screen has a certain thickness), the field in the screen near the rear wall would be nonzero.

Fig. 31.6. Diffraction on an opaque screen.

But then it would set in motion the other electrons of the screen, thereby creating a new field that would tend to compensate for the original field. If the screen is thick, there are enough possibilities in it to reduce the residual field to zero. Using our terminology, we can say that an opaque screen has a large and purely imaginary refractive index, and therefore the wave in it decays exponentially. You probably know that thin layers of most opaque materials, even gold, are transparent.

Let us now see what kind of picture emerges if we take such an opaque screen with a hole as shown in Fig. 31.6, b. What will be the field at point P? The field at the point P is composed of two parts - the source field S and the screen field, i.e., the field from the movement of charges in the screen. The movement of the charges in the screen is apparently very complex, but the field they create is quite simple.

Let us take the same screen, but close the holes with covers, as shown in Fig. 31.6, c. Let the covers be made of the same material as the screen. Note that the covers are placed where in Fig. 31.6, b shows the holes. Let us now calculate the field at point P. The field at point P in the case shown in FIG. 31.6, in, of course, is equal to zero, but, on the other hand, it is also equal to the field of the source plus the field of electrons of the screen and caps. We can write the following equality:

The dashes refer to the case when the holes are closed with lids; the value of E s is, of course, the same in both cases. Subtracting one equality from the other, we get

If the apertures are not too small (for example, many wavelengths wide), then the presence of caps should not affect the screen field, except perhaps a narrow region near the edges of the apertures. Neglecting this small effect, we can write

E walls \u003d E "walls and, therefore,

We come to the conclusion that the field at point P with open holes (case b) is equal (up to sign) to the field created by that part of the solid screen that is located in the place of the holes! (We are not interested in the sign, since one usually deals with intensity proportional to the square of the field.) This result is not only valid (in the approximation of not very small apertures), but also important; among other things, he confirms the validity of the usual theory of diffraction:

The field E "of the cover is calculated under the condition that the movement of charges everywhere in the screen creates exactly such a field that extinguishes the field E s on the back surface of the screen. Having determined the movement of charges, we add the radiation fields of charges in the covers and find the field at point P.

We recall once again that our theory of diffraction is approximate and is valid in the case of not too small apertures. If the size of the holes is small, the term E"of the lid is also small, and the difference E"of the wall -E of the wall (which we considered equal to zero) can be comparable and even much greater than the e"of the lid. Therefore, our approximation is invalid.

* The same formula is obtained with the help of quantum mechanics, but its interpretation in this case is different. In quantum mechanics, even a one-electron atom, such as hydrogen, has several resonant frequencies. Therefore, instead of the number of electrons N k with frequency w k multiplier Nf appears k where N is the number of atoms per unit volume, and the number f k (called oscillator strength) indicates how much weight a given resonant frequency enters w k .

Substances - a value equal to the ratio of the phase velocities of light (electromagnetic waves) in vacuum and in a given medium. They also talk about the refractive index for any other waves, for example, sound waves.

The refractive index depends on the properties of the substance and the wavelength of the radiation, for some substances the refractive index changes quite strongly when the frequency of electromagnetic waves changes from low frequencies to optical and beyond, and can also change even more sharply in certain areas of the frequency scale. The default is usually the optical range, or the range determined by the context.

There are optically anisotropic substances in which the refractive index depends on the direction and polarization of the light. Such substances are quite common, in particular, these are all crystals with a sufficiently low symmetry of the crystal lattice, as well as substances subjected to mechanical deformation.

The refractive index can be expressed as the root of the product of the magnetic and permittivities of the medium

(it must be taken into account that the values ​​of the magnetic permeability and permittivity for the frequency range of interest, for example, the optical one, can differ greatly from the static values ​​of these quantities).

To measure the refractive index, manual and automatic refractometers .

The ratio of the refractive index of one medium to the refractive index of the second is called relative refractive index the first environment in relation to the second. For running:

where and are the phase velocities of light in the first and second media, respectively. Obviously, the relative refractive index of the second medium with respect to the first is a value equal to .

This value, ceteris paribus, is usually less than unity when the beam passes from a denser medium to a less dense medium, and more than unity when the beam passes from a less dense medium to a denser medium (for example, from a gas or from vacuum to a liquid or solid ). There are exceptions to this rule, and therefore it is customary to call the environment optically more or less dense than the other (not to be confused with optical density as a measure of the opacity of a medium).

A beam falling from airless space onto the surface of some medium is refracted more strongly than when falling on it from another medium; the refractive index of a ray incident on a medium from airless space is called its absolute refractive index or simply the refractive index of a given medium, this is the refractive index, the definition of which is given at the beginning of the article. The refractive index of any gas, including air, under normal conditions is much less than the refractive indices of liquids or solids, therefore, approximately (and with relatively good accuracy) the absolute refractive index can be judged from the refractive index relative to air.

Ticket 75.

Law of light reflection: the incident and reflected beams, as well as the perpendicular to the interface between two media, restored at the point of incidence of the beam, lie in the same plane (the plane of incidence). The angle of reflection γ is equal to the angle of incidence α.

Law of refraction of light: the incident and refracted beams, as well as the perpendicular to the interface between two media, restored at the point of incidence of the beam, lie in the same plane. The ratio of the sine of the angle of incidence α to the sine of the angle of refraction β is a constant value for two given media:

The laws of reflection and refraction are explained in wave physics. According to wave concepts, refraction is a consequence of a change in the speed of wave propagation during the transition from one medium to another. The physical meaning of the refractive index is the ratio of the speed of wave propagation in the first medium υ 1 to the speed of their propagation in the second medium υ 2:

Figure 3.1.1 illustrates the laws of reflection and refraction of light.

A medium with a lower absolute refractive index is called optically less dense.

When light passes from an optically denser medium to an optically less dense one n 2< n 1 (например, из стекла в воздух) можно наблюдать total reflection phenomenon, that is, the disappearance of the refracted beam. This phenomenon is observed at angles of incidence exceeding a certain critical angle α pr, which is called limiting angle of total internal reflection(see fig. 3.1.2).

For the angle of incidence α = α pr sin β = 1; value sin α pr \u003d n 2 / n 1< 1.

If the second medium is air (n 2 ≈ 1), then it is convenient to rewrite the formula as

The phenomenon of total internal reflection finds application in many optical devices. The most interesting and practically important application is the creation of fiber light guides, which are thin (from several micrometers to millimeters) arbitrarily bent filaments from an optically transparent material (glass, quartz). Light falling on the end of the fiber can propagate along it over long distances due to total internal reflection from the side surfaces (Fig. 3.1.3). The scientific and technical direction involved in the development and application of optical light guides is called fiber optics.

Dispe "rsiya light" that (decomposition of light)- this is a phenomenon due to the dependence of the absolute refractive index of a substance on the frequency (or wavelength) of light (frequency dispersion), or, the same thing, the dependence of the phase velocity of light in a substance on the wavelength (or frequency). Experimentally discovered by Newton around 1672, although theoretically well explained much later.

Spatial dispersion is the dependence of the tensor of the permittivity of the medium on the wave vector. This dependence causes a number of phenomena called spatial polarization effects.

One of the clearest examples of dispersion - decomposition of white light when passing it through a prism (Newton's experiment). The essence of the phenomenon of dispersion is the difference in the propagation speeds of light rays with different wavelengths in a transparent substance - an optical medium (whereas in vacuum the speed of light is always the same, regardless of the wavelength and hence the color). Usually, the higher the frequency of a light wave, the greater the refractive index of the medium for it and the lower the wave speed in the medium:

Newton's experiments Experiment on the decomposition of white light into a spectrum: Newton directed a beam of sunlight through a small hole onto a glass prism. Getting on the prism, the beam was refracted and gave on the opposite wall an elongated image with iridescent alternation of colors - the spectrum. Experiment on the passage of monochromatic light through a prism: Newton placed red glass in the path of the sun's ray, behind which he received monochromatic light (red), then a prism and observed on the screen only a red spot from the ray of light. Experience in the synthesis (obtaining) of white light: First, Newton directed the sun's beam at a prism. Then, having collected the colored rays emerging from the prism with the help of a converging lens, Newton received a white image of a hole on a white wall instead of a colored strip. Newton's conclusions:- the prism does not change the light, but only decomposes it into components - light rays that differ in color differ in the degree of refraction; violet rays are most strongly refracted, red light is less strongly refracted - red light, which is less refracted, has the highest speed, and violet has the lowest, therefore the prism decomposes the light. The dependence of the refractive index of light on its color is called dispersion.

Findings:- a prism decomposes light - white light is complex (composite) - violet rays are refracted more than red ones. The color of a beam of light is determined by its frequency of oscillation. When moving from one medium to another, the speed of light and wavelength change, but the frequency that determines the color remains constant. The boundaries of the ranges of white light and its components are usually characterized by their wavelengths in vacuum. White light is a collection of wavelengths from 380 to 760 nm.

Ticket 77.

Light absorption. Bouguer's law

The absorption of light in a substance is associated with the conversion of the energy of the electromagnetic field of the wave into the thermal energy of the substance (or into the energy of secondary photoluminescent radiation). The light absorption law (Bouguer's law) has the form:

I=I 0 exp(- x),(1)

where I 0 , I- input light intensity (x=0) and exit from the medium layer of thickness X, - absorption coefficient, it depends on .

For dielectrics  =10 -1  10 -5 m -1 , for metals =10 5  10 7 m -1 , therefore metals are opaque to light.

Dependence  ( ) explains the coloration of absorbing bodies. For example, glass that absorbs little red light will appear red when illuminated with white light.

Scattering of light. Rayleigh's law

Diffraction of light can occur in an optically inhomogeneous medium, for example, in a turbid medium (smoke, fog, dusty air, etc.). Diffracting on inhomogeneities of the medium, light waves create a diffraction pattern characterized by a fairly uniform intensity distribution in all directions.

Such diffraction by small inhomogeneities is called scattering of light.

This phenomenon is observed if a narrow beam of sunlight passes through dusty air, scatters on dust particles and becomes visible.

If the dimensions of the inhomogeneities are small compared to the wavelength (no more than 0,1  ), then the scattered light intensity is inversely proportional to the fourth power of the wavelength, i.e.

I rass ~ 1/  4 , (2)

this relationship is called Rayleigh's law.

Scattering of light is also observed in pure media that do not contain foreign particles. For example, it can occur on fluctuations (random deviations) of density, anisotropy, or concentration. Such scattering is called molecular. It explains, for example, the blue color of the sky. Indeed, according to (2), blue and blue rays are scattered more strongly than red and yellow, because have a shorter wavelength, thus causing the blue color of the sky.

Ticket 78.

Light polarization- a set of phenomena of wave optics, in which the transverse nature of electromagnetic light waves is manifested. transverse wave- particles of the medium oscillate in directions perpendicular to the direction of wave propagation ( fig.1).

Fig.1 transverse wave

electromagnetic light wave plane polarized(linear polarization), if the directions of oscillation of the vectors E and B are strictly fixed and lie in certain planes ( fig.1). A plane polarized light wave is called plane polarized(linearly polarized) light. non-polarized(natural) wave - an electromagnetic light wave in which the directions of oscillation of the vectors E and B in this wave can lie in any planes perpendicular to the velocity vector v. unpolarized light- light waves, in which the directions of oscillations of the vectors E and B randomly change so that all directions of oscillations in planes perpendicular to the beam of wave propagation are equally probable ( fig.2).

Fig.2 unpolarized light

polarized waves- in which the directions of the vectors E and B remain unchanged in space or change according to a certain law. Radiation, in which the direction of the vector E changes randomly - unpolarized. An example of such radiation can be thermal radiation (randomly distributed atoms and electrons). Plane of polarization- this is a plane perpendicular to the direction of oscillation of the vector E. The main mechanism for the occurrence of polarized radiation is the scattering of radiation by electrons, atoms, molecules, and dust particles.

1.2. Types of polarization There are three types of polarization. Let's define them. 1. Linear Occurs if the electric vector E retains its position in space. It sort of highlights the plane in which the vector E oscillates. 2. Circular This is the polarization that occurs when the electric vector E rotates around the direction of wave propagation with an angular velocity equal to the angular frequency of the wave, while maintaining its absolute value. This polarization characterizes the direction of rotation of the vector E in the plane perpendicular to the line of sight. An example is cyclotron radiation (a system of electrons rotating in a magnetic field). 3. Elliptical Occurs when the magnitude of the electric vector E changes so that it describes an ellipse (rotation of the vector E). Elliptical and circular polarization is right (the rotation of the vector E occurs clockwise, if you look towards the propagating wave) and left (the rotation of the vector E occurs counterclockwise, if you look towards the propagating wave).

In fact, the most common partial polarization (partially polarized electromagnetic waves). Quantitatively, it is characterized by a certain quantity called degree of polarization R, which is defined as: P = (Imax - Imin) / (Imax + Imin) where Imax,imin- the highest and lowest electromagnetic energy flux density through the analyzer (Polaroid, Nicol prism…). In practice, radiation polarization is often described by Stokes parameters (radiation fluxes with a given polarization direction are determined).

Ticket 79.

If natural light falls on the interface between two dielectrics (for example, air and glass), then part of it is reflected, and part is refracted and propagates in the second medium. By placing an analyzer (for example, tourmaline) in the path of the reflected and refracted beams, we make sure that the reflected and refracted beams are partially polarized: when the analyzer is rotated around the beams, the light intensity periodically increases and decreases (complete extinction is not observed!). Further studies showed that in the reflected beam, oscillations perpendicular to the plane of incidence prevail (in Fig. 275 they are indicated by dots), in the refracted beam - oscillations parallel to the plane of incidence (shown by arrows).

The degree of polarization (the degree of separation of light waves with a certain orientation of the electric (and magnetic) vector) depends on the angle of incidence of the rays and the refractive index. Scottish physicist D. Brewster(1781-1868) established law, according to which at the angle of incidence i B (Brewster angle), defined by the relation

(n 21 - refractive index of the second medium relative to the first), the reflected beam is plane polarized(contains only oscillations perpendicular to the plane of incidence) (Fig. 276). The refracted beam at the angle of incidencei B polarized to the maximum, but not completely.

If light is incident on the interface at the Brewster angle, then the reflected and refracted rays mutually perpendicular(tg i B=sin i B/cos i b, n 21 = sin i B / sin i 2 (i 2 - angle of refraction), whence cos i B=sin i 2). Hence, i B + i 2 =  /2, but i’ B= i B (law of reflection), so i’ B+ i 2 =  /2.

The degree of polarization of reflected and refracted light at different angles of incidence can be calculated from Maxwell's equations, if we take into account the boundary conditions for the electromagnetic field at the interface between two isotropic dielectrics (the so-called Fresnel formulas).

The degree of polarization of the refracted light can be significantly increased (by repeated refraction, provided that the light falls each time on the interface at the Brewster angle). If, for example, for glass ( n= 1.53), the degree of polarization of the refracted beam is 15%, then after refraction by 8-10 glass plates superimposed on each other, the light emerging from such a system will be almost completely polarized. This set of plates is called foot. The foot can be used to analyze polarized light both in its reflection and in its refraction.

Ticket 79 (for spur)

As experience shows, during the refraction and reflection of light, the refracted and reflected light turns out to be polarized, and the reflection. light can be completely polarized at a certain angle of incidence, but light is always partially polarized. Based on Frinel's formulas, it can be shown that the reflect. light is polarized in a plane perpendicular to the plane of incidence, and refraction. the light is polarized in a plane parallel to the plane of incidence.

The angle of incidence at which the reflection light is fully polarized is called Brewster's angle. Brewster's angle is determined from Brewster's law: -Brewster's law. In this case, the angle between reflection. and break. rays will be equal. For an air-glass system, the Brewster angle is equal. To obtain good polarization, i.e. , when light is refracted, a lot of broken surfaces are used, which are called Stoletov's Foot.

Ticket 80.

Experience shows that during the interaction of light with matter, the main action (physiological, photochemical, photoelectric, etc.) is caused by oscillations of the vector, which in this connection is sometimes called the light vector. Therefore, to describe the patterns of light polarization, the behavior of the vector is monitored.

The plane formed by the vectors and is called the plane of polarization.

If the vector oscillations occur in one fixed plane, then such light (beam) is called linearly polarized. It is arbitrarily designated as follows. If the beam is polarized in a perpendicular plane (in the plane xz, see fig. 2 in the second lecture), then it is denoted.

Natural light (from ordinary sources, the sun) consists of waves that have different, randomly distributed planes of polarization (see Fig. 3).

Natural light is sometimes conventionally referred to as this. It is also called non-polarized.

If during the propagation of the wave the vector rotates and at the same time the end of the vector describes a circle, then such light is called circularly polarized, and the polarization is circular or circular (right or left). There is also elliptical polarization.

There are optical devices (films, plates, etc.) - polarizers, which emit linearly polarized light or partially polarized light from natural light.

Polarizers used to analyze the polarization of light are called analyzers.

The plane of the polarizer (or analyzer) is the plane of polarization of light transmitted by the polarizer (or analyzer).

Let a polarizer (or analyzer) be incident with linearly polarized light with an amplitude E 0 . The amplitude of the transmitted light will be E=E 0 cos j, and the intensity I=I 0 cos 2 j.

This formula expresses Malus' law:

The intensity of linearly polarized light passing through the analyzer is proportional to the square of the cosine of the angle j between the plane of oscillations of the incident light and the plane of the analyzer.

Ticket 80 (for spurs)

Polarizers are devices that make it possible to obtain polarized light. Analyzers are devices with which you can analyze whether light is polarized or not. Structurally, a polarizer and an analyzer are the same. then all directions of the vector E are equal probable. Each vector can be decomposed into two mutually perpendicular components: one of which is parallel to the polarization plane of the polarizer, and the other is perpendicular to it.

Obviously, the intensity of the light leaving the polarizer will be equal. Let us denote the intensity of the light leaving the polarizer by (). If an analyzer is placed on the path of the polarizer, the main plane of which makes an angle with the main plane of the polarizer, then the intensity of the light leaving the analyzer is determined by the law.

Ticket 81.

Studying the luminescence of a solution of uranium salts under the action of -rays of radium, the Soviet physicist P. A. Cherenkov drew attention to the fact that the water itself glows, in which there are no uranium salts. It turned out that when rays (see Gamma radiation) are passed through pure liquids, they all begin to glow. S. I. Vavilov, under whose direction P. A. Cherenkov worked, hypothesized that the glow is associated with the movement of electrons knocked out by radium quanta from atoms. Indeed, the glow strongly depended on the direction of the magnetic field in the liquid (this suggested that its cause is the movement of electrons).

But why do electrons moving in a liquid emit light? The correct answer to this question was given in 1937 by the Soviet physicists I. E. Tamm and I. M. Frank.

An electron, moving in a substance, interacts with the surrounding atoms. Under the action of its electric field, atomic electrons and nuclei are displaced in opposite directions - the medium is polarized. Polarizing and then returning to the initial state, the atoms of the medium, located along the trajectory of the electron, emit electromagnetic light waves. If the electron speed v is less than the speed of light propagation in the medium (- refractive index), then the electromagnetic field will overtake the electron, and the substance will have time to polarize in space ahead of the electron. The polarization of the medium in front of the electron and behind it is opposite in direction, and the radiations of oppositely polarized atoms, "adding up", "extinguish" each other. When the atoms, to which the electron has not yet reached, do not have time to polarize, and radiation appears, directed along a narrow conical layer with a vertex coinciding with the moving electron, and an angle at the vertex c. The appearance of a light "cone" and the condition of radiation can be obtained from the general principles of wave propagation.

Rice. 1. Mechanism of wave front formation

Let an electron move along the axis OE (see Fig. 1) of a very narrow empty channel in a homogeneous transparent substance with a refractive index (an empty channel is needed in order not to take into account collisions of an electron with atoms in a theoretical consideration). Any point on the OE line successively occupied by an electron will be the center of light emission. Waves emanating from successive points O, D, E interfere with each other and are amplified if the phase difference between them is zero (see Interference). This condition is satisfied for the direction that makes an angle of 0 with the trajectory of the electron. Angle 0 is determined by the ratio:.

Indeed, consider two waves emitted in the direction at an angle of 0 to the electron velocity from two points of the trajectory - point O and point D, separated by a distance . At point B, lying on the straight line BE, perpendicular to OB, the first wave at - in time To point F, lying on the straight line BE, the wave emitted from the point will arrive at the time moment after the emission of the wave from point O. These two waves will be in phase, i.e., the straight line will be a wave front if these times are equal:. That as a condition of equality of times gives. In all directions, for which, the light will be extinguished due to the interference of waves emitted from sections of the trajectory separated by a distance D. The value of D is determined by an obvious equation, where T is the period of light oscillations. This equation always has a solution if.

If , then the direction in which the radiated waves, interfering, amplify does not exist, cannot be greater than 1.

Rice. 2. Distribution of sound waves and formation of a shock wave during body motion

Radiation is observed only if .

Experimentally, electrons fly in a finite solid angle, with a certain spread in velocities, and as a result, radiation propagates in a conical layer near the main direction determined by the angle .

In our consideration, we have neglected the deceleration of the electron. This is quite acceptable, since the losses due to Vavilov-Cherenkov radiation are small and, in the first approximation, we can assume that the energy lost by the electron does not affect its speed and it moves uniformly. This is the fundamental difference and unusualness of the Vavilov-Cherenkov radiation. Usually charges radiate, experiencing significant acceleration.

An electron outrunning its own light is like an airplane flying at a speed greater than the speed of sound. In this case, a conical shock wave also propagates in front of the aircraft (see Fig. 2).