The Origin of the efractive Index 31-1 The index of refraction We have said before that light goes slower in water than in air, and slower, 31-1 The index of refraction slightly, in air than in vacuum. This effect is described by the index of refraction n. Now we would like to understand how such a slower velocity could come about 31-2 The field due to the material In particular, we should try to see what the relation is to some physical assumptions, 31-3 Dispersion or statements we made earlier, which were the following 31-4 Absorption (a) That the total electric field in any physical circumstance can always be epresented by the sum of the fields from all the charges in the universe 31-5 The energy carried by an electric wave b)That the field from a single charge is given by its acceleration evaluated with a retardation at the speed c, always(for the radiation field 31-6 Diffraction of light by a screen But, for a piece of glass, you might think: Oh, no, you should modify all this. You should say it is retarded at the speed c/n. That, however, is not right, and we have to understand why it is not It is approximately true that light or any electrical wave does appear to travel at the speed c/n through a material whose index of refraction is n, but the fields are still produced by the motions of all the charges--including the charges moving in he material-and with these basic contributions of the field travelling at the ultimate velocity c. Our problem is to understand how the apparently slower velocity comes about We shall try to understand the effect in a very simple case. A source which we shall call"the external source"is placed a large distance away from a thin plate of transparent material, say glass. We inquire about the field at a large distance on the opposite side of the plate. The situation is illustrated by the diagram of Fig. 31-1, where S and P are imagined to be very far away from the plate. According to the principles we have stated earlier, an electric field anywhere n5 that is far from all moving charges is the(vector) sum of the fields produced by the external source (at S) and the fields produced by each of the charges in the plate c of glass, every one with its proper retardation at the velocity c. Remember that the contribution of each charge is not changed by the presence of the other charges These are our basic principles. The field at P can be written thus Fig. 31-1. Electric waves passing through a layer of transparent materia y Eeach charge (311) E=E+∑ Eeach charges here es is the field due to the source alone and would be precisely the field at P if there were no material present. We expect the field at P to be different from Eg if there are any other moving charges Why should there be charges moving in the glass? We know that all material consists of atoms which contain electrons. When the electric field of the source acts on these atoms it drives the electrons up and down, because it exerts a force on the electrons. And moving electrons generate a field-they constitute new radiators These new radiators are related to the source S, because they are driven by the field of the source. The total field is not just the field of the source S, but it is modified by the additional contribution from the other moving charges. This means that the field is not the same as the one which was there before the glass as there but is modified, and it turns out that it is modified in such a way that 31-1
the field inside the glass appears to be moving at a different speed That is the idea which we would like to work out quantitatively Now this is, in the exact case, pretty complicated, because although we have id that all the other moving charges are driven by the source field, that is not quite true. If we think of a particular charge, it feels not only the source, but like anything else in the world, it feels all of the charges that are moving. It feels, in particular, the charges that are moving somewhere else in the glass. So the total field which is acting on a particular charge is a combination of the fields from th can see that it would take a complicated set of equations to get l es doing!You complete and exact formula. It is so complicated that we postpone this problem until next year. Instead we shall work out a very simple case in order to understand all the physical principles very clearly. We take a circumstance in which the effects from the other atoms is very small relative to the effects from the source. In other words e take a material in which the total field is not modified very much by the motion of the other charges. That corresponds to a material in which the index of refraction is very close to 1, which will happen, for example, if the density of the atoms is very low. Our calculation will be valid for any case in which the index is for any reason very close to 1. In this way we shall avoid the complications of the most general, complete solution Incidentally, you should notice that there is another effect caused by the motion of the charges in the plate. These charges will also radiate waves back toward the source S. This backward-going field is the light we see reflected from the surfaces transparent materials. It does ne t the surface The backw radiation comes from everywhere in the interior, but it turns out that the total effect is equivalent to a reflection from the surfaces. These reflection effects are beyond our approximation at the moment because we shall be limited to a calculation for a material with an index so close to l that very little light is reflected Before we proceed with our study of how the index of refraction comes about we should understand that all that is required to understand refraction is to under stand why the apparent wave velocity is different in different materials. The bending of light rays comes about just because the effective speed of the waves is different in the materials. To remind you how that comes about we have drawn in Fig. 31-2 several successive crests of an electric wave which arrives fre vacuum onto the surface of a block of glass. The arrow perpendicular to the wave crests indicates the direction of travel of the wave. Now all oscillations in the wave must have the same frequency. (We have seen that driven oscillations have the same frequency as the driving source. ) This means, also, that the wave crests for the waves on both sides of the surface must have the same spacing along the surface because they must travel together, so that a charge sitting at the boundary will Fig. 31-2. Relation between refrac- feel only one frequency. The shortest distance between crests of the wave, however, on and velocity change is the wavelength which is the velocity divided by the frequency. On the vacuum side it is o=2Tc/w, and on the other side it is x= 2y/w or 2c/ is the velocity of the wave. From the figure we can see that the only way for the vaves to"fit"properly at the boundary is for the waves in the material to be travelling at a different angle with respect to the surface. From the geometry of the figure you can see that for a"fit"we must have Ao/sin 00=A/sin B, or sin 0o/sin 0= n, which is Snell's law. We shall, for the rest of our discussion consider only why light has an effective speed of c/n in material of index n, and the bending of the We go back now to the situation shown in Fig. 31-1. We see that what we have to do is to calculate the field produced at P by all the oscillating charges in the glass plate. We shall call this part of the field Ea, and it is just the sum written as the second term in Eq (31.2). When we add it to the term Es, due to the source, we will have the total field at P
This is probably the most complicated thing that we are going to do this year, but it is complicated only in that there are many pieces that have to be put to- gether; each piece, however, is very simple. Unlike other derivations where we say, "Forget the derivation, just look at the answer!, "in this case we do not need the answer so much as the derivation In other words, the thing to under stand now is the physical machinery for the production of the index To see where we are going, let us first find out what the"correction field" would have to be if the total field at P is going to look like radiation from the source that is slowed down while passing through the thin plate. If the plate had no effect on it, the field of a wave travelling to the right(along the z-axis) would be E。= Eo cos a(t-z/c) or, using the exponential notation (314 Now what would happen if the wave travelled more slowly in going through the plate? Let us call the thickness of the plate Az. If the plate were not there the rave would travel the distance Az in the time Az/c. But if it appears to travel at the speed c/n then it should take the longer time n Az/c or the additional time At=(n-1)4z/c. After that it would continue to travel at the speed c again We can take into account the extra delay in getting through the plate by replacing t in Eq (31.4)by(t-)or by [! -(n-1)Az/c]. So the wave after insertion of the plate should be written We can also write this equation as Enftor plate e -ia(n-1)az/e Eoe(/e), which says that the wave after the plate is obtained from the wave which could exist without the plate, i, e, from Es, by multiplying by the factor e-i(n-m)42/e Now we know that multiplying an oscillating function like e t by a factor e just says that we change the phase of the oscillation by the angle which is, of course what the extra delay in passing through the thickness 4z has done. It has retarded the phase by the amount w(n- D)Az/c(retarded, because of the minus sign the exponent). e have said earlier that the plate should add a field ea to the original field Es eoe but we have found instead that the effect of the plate is to multiply the field by a factor which shifts its phase. However, that is really all right because we can get the same result by adding a suitable complex number. It is articularly easy to find the right number to add in the case that Az is small, for you will remember that if x is a small number then e is nearly equal to(1 x We can write, therefore Using this equality in Eq.(31.6), we have Eafter plate= eoe ie(t-a/c)_iw(n 1)△z Ee(-21).(31.8) En The first term is just the field from the source, and the second term must just be Rool Axis qual to Ea, the field produced to the right of the plate by the oscillating charges of the plate--expressed here in terms of the index of refraction n, and depending, What we have been doing is easily visualized if we look at the complex number i Fig. 31-3. Diagram for the trans- of course, on the strength of the wave from the source ed wave at a particular t and z. diagram in Fig. 31-3. We first draw the number es (we chose some values for z and t so that es comes out horizontal, but this is not necessary). The delay due to
lowing down in the plate would delay the phase of this number, that is, it would rotate Es through a negative angle. But this is equivalent to adding the small vector ea at roughly right angles to Eg. But that is just what the factor -i means in the second term of Eq.(31.8). It says that if Es is real, then Ea is negative imaginary or that, in general, Es and Ea make a right angle. 31-2 The field due to the material We now have to ask: Is the field ea obtained in the second term of Eq. (31.8) the kind we would expect from oscillating charges in the plate? If we can show that it is, we will then have calculated what the index n should be! [Since n is the only nonfundamental number in Eq(31.8). We turn now to calculating what field Ea the charges in th (To help keep track of many symbols we have used up to now, and will be using in the rest of our calcula tion, we have put them all together in Table 31-1.) Symbols used in the calculations Ea= field produced by charges in the plate plate z= perpendicular distance from the plate index of refraction w= frequency(angular)of the radiation charges per unit volume in the pla m= number of charges per unit area of the plate ge charge on an electron wo- resonant frequency of an electron bound in an atom If the source S(of Fig 31-1)is far off to the left then the field Es will hay the same phase everywhere on the plate, so we can write that in the neighborhood f the plate Es= Ec (31.9) Right at the plate, where z=0, we will have Es (at the plate) Each of the electrons in the atoms of the plate will feel this electric field and will be driven up and down (we assume the direction of Eo is vertical) by the electric force ge. To find what motion we expect for the electrons, we will assume that the atoms are little oscillators, that is, that the electrons are fastened elastically to the atoms, which means that if a force is applied to an electron its displacement from its normal position will be proportional to the force. You may think that this is a funny model of an atom if you have heard about electrons whirling around in orbits. But that is just an oversimplified picture ven by the theory of says that, so far as problems involving light are concerned, the electrons behave as though they were held by springs. So we shall suppose that the electrons have a linear restoring force which, together with their mass m, makes them behave like little oscillators, with a resonant frequency wo. We have already studied such cillators, and we know that the equation of their motion is written this way here F is the driving for
For our problem, the driving force comes from the electric field of the wave from the source, so we should us f= gees- eee, where qe is the electric charge on the electron and for Es we use the expression Es= eoew from(31. 10). Our equation of motion for the electron is then geese We have solved this equation before, and we know that the solution is (31.14) where, by substituting in(31. 13), we find that geO (31.15) geO (31.16) We have what we needed to know-the motion of the electrons in the plate. And it is the same for every electron, except that the mean position(the "zero"of the motion)is, of course, different for each electron Now we are ready to find the field ea that these atoms produce at the point P, because we have already worked out(at the end of Chapter 30)what field is pro- duced by a sheet of charges that all move together. Referring back to Eq. (30.19), we see that the field ea at P is just a negative constant times the velocity of the charges retarded in time the amount z/c. Differentiating x in Eq. ( 31. 16) to get the velocity, and sticking in the retardation [or just putting xo from (31.15) into(30. 18)] yield eoc ia qeEo wo(4-2/e) (31.17 Just as we expected, the driven motion of the electrons produced an extra wave which travels to the right(that is what the factor eiw -z/e)says), and the amplitude of this wave is proportional to the number of atoms per unit area in the plate (the factor n)and also proportional to the strength of the source field(the factor Eo). Then there are some factors which depend on the atomic properties (qe, m, and wo), as we should expect n. The two expressions will, in fact, be identical i rial wha,o The most important thing, however, is that this formula(31. 17) for Ea looks very much like the expression for ea that we got in Eq. (31.8)by saying that the original wave was delayed in passing through a mat (n-1)△z=-m (31.18) Notice that both sides are proportional to Az, since n, which is the number of atoms per unit area, is equal to N AZ, where N is the number of atoms per unit volume of the plate. Substituting N AZ for n and cancelling the Az, we get our main ul, a formula for the index of refraction in terms of the properties of the atoms of the material-and of the frequency of the light 1+ (31.19) This equation gives the"explanation "of the index of refraction that we wished to obt 31-5