 Hello and welcome to lecture 6 of module 2 of this course on Accelerative Physics. Today we will learn about traveling wave, Linux and periodic accelerating structures. So in the last lecture, we learned about the modes in a pillbox cavity and we saw that how they can be used for acceleration of charge particles, so they were standing waves. We will try to see whether using the traveling wave in a waveguide we can accelerate the charge particles. So we saw yesterday that TE and TM MNP modes are excited in a pillbox cavity. The variations of the fields in the theta and z direction they have sinusoidal dependence whereas in the r direction the fields have vessel function type of variation. We also saw that TEMN 0 mode cannot exist in a pillbox cavity because by boundary condition this mode is not allowed. We also saw that in a good conductor the fields penetrate up to a distance equal to the skin depth and due to this there is some RF surface resistance and power is dissipated on the cavity walls. In the TM010 mode in the pillbox cavity we have only the Ez field that is constant along the length of the cavity. So in the TM010 mode we have a field Ez field along the z direction and the v theta field and these fields are varying with time, both Ez and v theta are varying with time. So this Ez field since it has a time variation like this and we know how to accelerate using a small portion of this field, using the small portion of this field for acceleration this Ez field can be used for acceleration. Drift tubes can be put inside this cavity. If you make a long pillbox cavity you can put drift tubes inside this and the resulting structure is known as the drift tube linac or the DTL and this can be used for acceleration of charge particles. Then we also studied about various figure of merits of the cavity like the quality factor shunt impedance. So the aim of the cavity design is to maximize the quality factor and shunt impedance. So today let us see whether we can use traveling waves for acceleration. So we saw in the beginning that when electromagnetic waves propagate inside conducting boundaries. So for example inside a waveguide then there are two cases possible, okay the TM wave and the TE wave. In the TM wave there is a component of electric field along the direction of propagation, okay. So here we have E parallel to V. So it may be possible, so at that time we had discussed that it may be possible to use the TM waves inside the space bounded by conducting boundaries for example in a waveguide for acceleration. So let us see whether we can use this traveling wave for acceleration. So let us consider the, let us consider a lossless uniform waveguide. So this is a hollow waveguide of circular cross section and we have already studied the fields in this structure. So here since the waves, since there is no boundary in the z direction it is a propagating wave in the z direction with a propagation vector Kg. So the electric field component Ez for the TM mode, TM01 mode can be written like this. So here m is equal to 0. So the theta component is equal to 1. So there is this cos m theta which has gone to 1. So we have Ez written as Ej0kr and it is a propagating wave in the z direction. So the wave propagates in the z direction with a propagation constant Kg which is given as under root of K0 square minus Kc square, okay. So Kg is 2 pi by lambda g where lambda is the wavelength of the propagating wave. So originally the electromagnetic wave had a propagation constant K0 in free space when it entered the waveguide it is now propagating with a propagation constant Kg. In the direction in which boundaries are applied that is the r and theta it has formed a standing wave and the propagation constant so or the wave number of associated with this standing wave is given by Kc and it is given as K0 square minus Kg square where Kc is equal to 2 pi by lambda c. So here lambda c is the cutoff wavelength of the waveguide which depends upon the transverse dimensions of the waveguide in this case the radius. So we can write K0 square is equal to Kc square plus Kg square or in other words K0 square can be written as omega by c whole square. So this wave propagates inside the waveguide with a propagation constant Kg if Kg is real that means K0 should be greater than Kc. Now we define the phase velocity of the resultant wave so that is the velocity with which a particular phase appears to be moving. So this is the phase velocity is given by v phase is equal to omega by Kg, okay. So Kg is the propagation constant in the waveguide. Now in real life if you see there are no truly monochromatic waves in nature. So there is some variation in the frequency of the waves propagating in nature. So a real wave exists in the form of a wave group which consists of superposition of waves of different frequencies and wave numbers. So if the spread in the phase velocities of individual waves is small the envelope of the wave pattern will tend to maintain its shape as it moves with a velocity which is called the group velocity. So let us see what is the phase velocity and group velocity for a group of waves. So let us say we have two individual waves of slightly different wave numbers and slightly different frequencies. So the first wave represented here has a wave number k1 and the second wave has a wave number k2. These are slightly different from each other and the first wave has a frequency omega1 and the second wave has a frequency omega2. Again these are slightly different from each other. The phase velocity of the first wave is given by omega1 by k1 and the phase velocity of the second wave is given by omega2 by k2. Both of these waves are propagating in the z direction. So the resultant wave can be calculated by taking the superposition of the two waves. So we simply add up the two waves and this is the net resultant that we get. So we see that here the exponential factor this describes a traveling wave with mean frequency. So the frequency now is omega1 plus omega2 by 2 and a mean wave number. So the wave number is k1 plus k2 by 2. And the first factor represents a slowly varying modulation of the wave amplitude. So the wave amplitude is now modulated. So this is how the resultant wave looks like and it is again propagating in the z direction. It is now propagating as a wave group. So this is the addition of the two waves and this is what we get. And so now we can calculate the phase velocity of the resultant wave. So we see that the phase velocity is given by omega1 plus omega2 by 2 divided by k1 plus k2 by 2. So it is now omega1 plus omega2 divided by k1 plus k2. So average of the frequency divided by the average of the wave number. The group velocity is defined as the velocity of the amplitude modulation envelope. So this is defined as the velocity with which the wave packet or the wave group is moving. And we can write this as, so this is coming from the amplitude modulated part. So here the group velocity is omega1 minus omega2 divided by k1 minus k2. So which is d omega by d2. So the group velocity represents the velocity of the whole wave packet or wave group whereas the phase velocity represents the velocity of a phase. So it is the apparent velocity with which the phase of any part of the wave appears to be moving along in the set direction. Now in general the main phase velocity and the group velocity are not necessarily equal. Let us see for a monochromatic plane wave. So as you know electromagnetic wave in free space is, if you take a single wave it is like a monochromatic plane wave. Now you can take the dispersion diagram or a preload diagram. So that is a graphic relationship between omega and k. For a monochromatic plane wave you know that omega is equal to ck. So it is a wave that is propagating with propagation constant k frequency omega with velocity c and the equation of the wave is given like this which you are very familiar now. So if you draw this omega versus k you see that it is a straight line which represents the velocity c with which it is moving. So here in this case the phase velocity is simply omega by k which is equal to c. If you calculate the group velocity it is del omega by del k which is also equal to c. So in this case the phase velocity is equal to the group velocity is equal to c. Now let us see the dispersion diagram for a hollow waveguide. Now the relation between the frequency and wave number for wave portion inside a uniform hollow waveguide is given by this expression. So omega is equal to c under root kc square plus kg square. So you know that for a waveguide any frequency below the cutoff frequency. So this is the cutoff frequency, any frequency below the cutoff frequency is attenuated and any frequency that is above the cutoff frequency will be propagated. So above the cutoff frequency you can choose any frequency here it will propagate with a propagation constant kg corresponding to this. So this curve has been drawn from this expression. So we see that for omega is equal to omega c the propagation constant kg goes to 0. So you can calculate from here for omega equal to omega c the propagation constant goes to 0 and the phase velocity becomes infinite. So there is no propagation of waves for omega less than omega c. The wave propagates through the hollow waveguide only for omega greater than or equal to omega c. For each value of frequency there is a certain phase velocity and group velocity. So let us say the, let us take a frequency here. So there is a corresponding to this point on the curve there is a phase velocity and a group velocity. What is the phase velocity? The phase velocity as you know it is omega by k. So it is the, so phase velocity at any point on the curve it is the slope of the line from the origin to that point. So at this point, so for this frequency at this point you take a line from the origin to this point. The phase velocity is defined as the slope of this line. Okay, what about the group velocity? The group velocity is given by the slope of the dispersion curve at that point. So the slope of the curve at this point this gives you the group velocity. So in this way from the dispersion diagram you can calculate the phase velocity and group velocity. These dotted lines show for omega is equal to kgc. So this, the slope of this line is c. So in general you can make out from this figure here that the phase velocity is greater than c whereas the group velocity is less than c. Now again the relation between the frequency and wave number is given by this. So you can from here you can calculate the phase velocity. So phase velocity is omega by kg, we take kg because kg is the propagation constant or the propagation vector with which the wave is propagating in the waveguide. So omega by kg is you simply, you can take omega from here and divide it by kg. So you can take kg out of the under root and so it will get cancelled with this kg. So you are left with c under root 1 plus kc by kg the whole square. Now this quantity is greater than 1. So here if you see c into some quantity greater than 1. So this number or this value whatever it comes out will be greater than c. So the phase velocity is greater than c which we can also see from here. This slope corresponds to c. If you see the phase velocity here it is greater than the value of c. So wave propagates in this empty waveguide with the phase velocity greater than the velocity of light. Now however it does not violate any laws of physics because this is the phase velocity is the apparent velocity with which a phase of the wave appears to be moving. It does not carry any energy. The energy is carried by the group velocity. So it is not violating any laws of physics. However, it is not possible to accelerate a charged particle with this wave because the particle velocity is always lower than the velocity of light. Now you know that no particle can move with the velocity greater than the velocity of light. So the particle velocity will always be less than the velocity of light and the phase velocity is greater than the velocity of light. So particle velocity is now less than the phase velocity. Now while accelerating the charged particle has to see the same phase at all times only then it will get accelerated. So here it is not possible since the particle velocity is less than the phase velocity it is not possible to synchronize it with the wave. So synchronism between wave and particle that is necessary for particle acceleration does not occur. So you cannot accelerate using the waves in a hollow waveguide because the phase velocity of the wave in the waveguide is greater than the velocity of light. So even though there is a TM mode there is an electric field along the direction of the charged particles it is not possible to accelerate using a hollow waveguide. Now let us calculate the group velocity in a waveguide. So the group velocity is given by del omega by del kg. So we differentiate omega with respect to kg from here and we get c divided by under root of 1 plus kc square by kg square. Now this number is greater than 1. So you divide c by a number greater than 1. So this will come out to be less than c. So energy propagates along the waveguide with group velocity. So no laws of physics are violated. So that is for a uniform waveguide if you see phase velocity and group velocity you multiply the two of them you get equal to c square where the group velocity is less than the velocity of light and phase velocity is greater than the velocity of light. So you can also see from here for this value of omega. So the phase velocity is given by the slope of this line whereas the group velocity is given by the slope of the curve at that point. So you can see here the phase velocity is always greater than c whereas the group velocity is always less than c. So the group velocity rather than the phase velocity is used to characterize the motion of the wave packet. So we have just seen in a hollow uniform waveguide the phase velocity of the wave is always greater than the velocity of light. Hence the uniform waveguide is not suitable for synchronous acceleration. So we need to modify the structure to slow down the phase velocity. If we could slow down or if we could bring the phase velocity less than the velocity of light to match it to the velocity of the charged particles then we can use a waveguide for acceleration. So what is the solution? We load the uniform waveguide periodically with obstacles. So this is the waveguide and now you have loaded it with periodic obstacles. So these are conducting plates and at periodic intervals n. So there is a hole for the beam to pass through. This is known as a disc loaded waveguide. So the waveguide is loaded periodically with metal discs with period length l. As you can see here so these are metal discs and they are separated by some uniform distance l. So this here l is known as the cell length and there is an aperture here for the beam to pass through. Now loading the uniform waveguide periodically and thus converting it into a periodic structure we expect that the field distribution will get perturbed by introducing a Z periodic modulation of the amplitude of the wave. So giving the TM01 propagating wave solution of the form like this. So because now you have loaded the waveguide with periodic metal discs. So now your electric field in the Z direction will also get modified. So your amplitude has now got modified. Now from Floke's theorem which is also analogous to Bloch's theorem in solid state physics. It states that in a given mode. So in this case we are talking of let us say TM01 mode in the cavity. In a given mode of an infinite periodic structure. So we have a structure now that is periodic. The field differs from one period to the next only by a constant factor. So that means if you take the field at this point and you take the field at this point and they are separated by l. So the field at the two points will differ only by a constant factor which is in general a complex number or a phase term. So EZ at Z plus l. So at Z plus l. So the value of the electric field in the Z direction at Z plus l is same as at Z except for an additional factor constant factor which is complex. Here the sign depends upon the direction of propagation. So now this term AI it has to be periodic and it can be developed into Fourier series. So you can write AI which is a function of r and Z as summation from n is equal to minus infinity to infinity AI nr e to the power of 2 pi n Z by l. And then if you substitute this in the expression for the electric field this is what you get. So here n will take values from minus infinity to infinity so n is an integer. So now you see that this wave it has n space harmonics. n can take values different integral values so it has there will be n space harmonics here and the propagation constant for each space harmonic is given by this value. So you can write KN which is the propagation constant for the nth space harmonic as KG plus 2 pi n by l for n is equal to 0 K0 is equal to KG. So the phase velocity of the nth space harmonic will now be given by omega by KN. So you can write omega divided by KN from here. So now you divide the numerator and denominator by KG you get omega by KG which is the phase velocity in a hollow waveguide uniform hollow waveguide. So here you get omega by KG and here you get 1 plus 2 pi n by KG l. Okay so this is a number which is greater than 1. So this will be less than the phase velocity because the phase velocity is omega KG where V phase is the phase velocity of the hollow waveguide. So now you have by loading this with periodic obstacles with metal discs you have been able to reduce the phase velocity as compared to the phase velocity in a hollow waveguide. And now you can choose your n sufficiently large and then you will you can reduce the phase velocity of the nth space harmonic. So by choosing n sufficiently large we can obtain an arbitrary low phase velocity. So we get a slow wave which can be used for particle acceleration. So you have slowed down the velocity, the phase velocity of the travelling wave now and using that you can now synchronize it with the charged particle and use it for acceleration of the charged particle. So now if you plot the dispersion curve for a periodically loaded structure this is what you get. So this is what n is equal to 0, this is what n is equal to 1 and so on. So here you notice that earlier when there was no loading the dispersion curve was something like this. Now you see that here it is taking a time. So frequencies above this frequency is not allowed. So there was one cutoff frequency here and now there is another frequency here above which the wave is not allowed. So the waves now will propagate in only a narrow band of frequency which is known as the pass band. So the above equation represents an infinite number of travelling waves which are known as space harmonic. So each of these waves are known as space harmonics and each of them are denoted by the index n. The travelling wave inside the periodically loaded waveguide is sum of all the spatial harmonics. The principal wave is given by n is equal to 0. Each harmonic has a propagation constant kn. So there is a propagation constant kn associated with each. So with each of the harmonics, so you can put in the values of n is equal to 0, 1, 2, so on and you will get the value of propagation constant and it has a phase velocity and a group velocity. Now at a given frequency, let us say you choose this frequency, all space harmonics. So at this frequency n is equal to 1 corresponds to this value of k, sorry n is equal to 0 corresponds to this value of k, n is equal to 1 corresponds to this value of k and so on. Now if you see their phase velocity, the phase velocity of the principal wave is the slope of this curve. The phase velocity of n is equal to 1 is the slope of this line. So each of them has a different phase velocity. However, if you see the group velocity, group velocity is simply the slope of this curve at this point. So the group velocity remains the same, however the phase velocity varies. So at a given frequency, all space harmonics have the same group velocity. So they have the same group velocity and they have different phase velocity. So they have different phase velocities. So space harmonics have same frequency but different wave numbers and each has a constant amplitude En which is independent of z. The waves for n greater than 0, they travel in the positive z direction and those for n less than 0 travel in the negative z direction. So this is n greater than 0, these are travelling in the forward direction, these are travelling in the reverse direction. The wave number of the nth space harmonic is shifted from the wave number k0 of the principal wave by 2 pi n by L. So you can see here the wave number of each harmonic is shifted from the wave number of the principal harmonic by 2 pi n by L. The phase velocity for the nth space harmonic is given by omega by kn which is omega by kg divided by 1 plus 2 pi n by kg L. So by choosing n sufficiently large, one can obtain an arbitrary low phase velocity to synchronize it with the velocity of the charge particles. The resultant group velocity for the principal wave at a given frequency is obtained from the slope of the dispersion curve. So now we have seen that the addition of the disc inside the cylindrical waveguide space by a distance L, it induces multiple reflections between the disc and this causes a change in the dispersion curve. So waves now propagate in limited frequency interval. So we have seen that this wave is now propagating in a limited frequency interval. Earlier it was propagating for this particular mode, let us say TM01 mode it was propagating. There was no cutoff in the upper direction. But now we see that there is a stop band. So above a certain frequency again the waves do not propagate. So these intervals are known as pass bands. The group velocity is equal to 0 at the end of pass band and at omega 0 and omega. So here if you see the group velocity, the group velocity is the slope of this curve. So at the end of the pass bands at this location and this location the group velocity is equal to 0 and it is maximum at the center here. At low frequencies there is normally no wave propagation between two adjacent pass bands. We have a stop band. So this is mode 1, let us say this is TM01 mode, this is TM next mode. So the two modes are now separated by a pass band.