 Hello and welcome to the lecture 4 of module 2 on this course on accelerator physics. So today we learn about the behavior of electromagnetic waves in waveguides and cavities. So before that just let us quickly revise what we have learned so far. If you want to accelerate charge particles to high energies using time varying fields or RF accelerators. So Alvarez had proposed that to use the electric fields associated with the electromagnetic waves in a high Q cavity. So in order to understand the behavior of electromagnetic waves we learnt about the electromagnetic waves their propagation in free space and in between parallel conducting planes. So we have seen that so far. So we saw that the electromagnetic waves in free space they are TEM waves that means the electric field and magnetic field are perpendicular to each other and they are perpendicular to the direction of propagation. So they are propagating with a propagation constant k and velocity c. The electromagnetic waves in free space cannot be used for acceleration because here the electric fields are always perpendicular to the direction of the velocity of the beam. So in this case we have E dot V is equal to 0. So there is no acceleration using electromagnetic waves in free space. When electromagnetic waves propagate in regions bounded by conducting boundaries they form standing waves known as modes in the direction in which the boundaries are applied. So and the amplitude has a sinusoidal variation in the direction in which the boundaries are applied in order to satisfy this boundary condition. The wavelength of the standing wave depends upon the dimensions of the system. So we call this as the cutoff wavelength and it is equal to 2 pi by kc. In the direction in which there are no conducting boundaries the electromagnetic wave continues to travel as a propagating wave and now with a new propagation constant kg. We also learnt about the waveguide. So the waveguide is a hollow pipe of infinite extent so it can have an arbitrary cross section. So it has boundaries in two directions. So let us say for example it is bounded in x and y direction and it is free to the electromagnetic wave is free to propagate in the z direction. So no boundary in the z direction in the x and y direction there are boundaries. So we saw that it is sufficient to determine EZ and BZ as solutions of the two dimensional wave equation and the other transverse components can be calculated from the above equations. So that means Ex, Py, Vx and Py are all functions of EZ and BZ only. So if you calculate EZ and BZ it is sufficient you can then find out all the other components. Also in a waveguide TEM waves cannot propagate the electromagnetic wave will propagate inside the waveguide either as a TM wave or a TE wave. So this is because in a TEM wave by definition the both the electric field and the magnetic field the z component is 0. Now if both these go to 0 then we know that Ex, Py, Bx and Py will also go to 0 because they depend upon EZ and BZ only. So all fields are 0 hence no fields and no modes. So TEM waves cannot propagate in a waveguide. Only TM and TE modes will propagate in the waveguide. So in a TM mode the magnetic field is transverse to the direction of propagation. So here we have BZ is 0. So magnetic field will be only in the x direction or y direction it cannot be along the direction of propagation. So we have BZ is equal to 0 and in TE mode the electric field is transverse to the direction of propagation. So here we will have EZ is equal to 0. So Ex and Py component will exist. Also one of the two either EZ or BZ definitely should exist otherwise all fields will go to 0. So today let us study about a rectangular waveguide. A rectangular waveguide has a rectangular cross section. So you can see here it has a rectangular cross section in x and y. So we now have conducting boundaries at x is equal to 0 and x is equal to A. Also at y is equal to 0 and y is equal to B. There is no boundary in the z direction. So the wave is free to propagate in the z direction. So we will see how the electromagnetic wave propagates inside this rectangular waveguide. So we know that now since there is a boundary in the x direction and the y direction the amplitude of the electric field will now have dependence on x and y. The wave unlike in free space where the amplitude was constant will now have dependence on both x and y and it will now propagate in the z direction with the propagation constant Kg. The propagation constant was K in free space. Similarly the magnetic field also the amplitude will have x and y dependence and it will propagate in the z direction with a propagation constant Kg. Now let us consider for the Te mode, for the Te mode Ez is equal to 0 so Bz will exist. So we need to solve the wave equation for Bz. So the wave equation is written here del 2 by del x square plus del 2 by del y square plus Kc square times Bz is equal to 0. Now the boundary conditions for magnetic field are that the normal component of magnetic field should be 0 at the boundary. So that means wherever you have a conducting boundary the normal component of magnetic field should be equal to 0. So this is the boundary condition. So del Vz by del n is equal to 0 at the boundary. So the boundaries are at x is equal to 0 and x is equal to a. So the boundary condition says that del Vz by del x is 0 at x is equal to 0 and x is equal to a. Similarly del Vz by del y is equal to 0 at y is equal to 0 and y is equal to 0. Now we can solve this equation by the method of separation of variables. So we can write Bz which is a function of x and y as x capital X which is a function of x only and capital Y which is a function of y only. So x, y. Okay, so now substituting Bz is equal to x, y in this wave equation we get this expression. So now this double derivative is with respect to x only. So we can, so y will come outside and this will be derivative with respect to x. Similarly here this double derivative is with respect to y only. So x can be taken out and here it is kc square multiplied by x square. Now dividing throughout by x, y we will get 1 by x. So y will get cancelled here, we will get 1 by x here and similarly 1 by y here and then here both x and y get cancelled. So now if you see this equation it can be written in this form where we have taken just the x component and the kc square together on the left hand side and the y part is taken on the right hand side. Now notice that the left hand side depends only on x and the right hand side depends only on y. So both these equations are satisfied when they are independently equal to a constant. So let us say that they are, both of these equations are separately equal to a constant which is p square. So we can write this equation on the left hand side as del 2x by del x square plus kc square minus p square times x is equal to 0 and let us call this kc square minus p square is equal to q square. Now here both p and q are integers and the equation for y can be written as minus 1 by y del 2y by del y square is equal to p square. So let us solve the first equation. This is the equation of a simple harmonic oscillator and it has solutions in the form of sin and cos. So we can write x as a cos qx plus b sin qx. Now the boundary condition is that del bz by del x is equal to 0 at x is equal to 0 and x is equal to a because the normal component of the magnetic field is low at the boundaries. So bz is a function of x and y. So we can write del x by del x is equal to 0 at both x is equal to 0 and x is equal to a. So let us calculate from this expression the derivative of capital X with respect to x. So we get minus aq sin qx plus bq cos qx. Now applying the boundary condition at x is equal to 0. So at x is equal to 0 we can put 0 here and here del x by del x is equal to 0. So we get 0 is equal to b into q or in other words b is equal to 0. So we are applying the second boundary condition that means at x is equal to a del x by del x is equal to 0. So we get 0 is equal to bq sin qa. So this expression is satisfied if qa is equal to m pi where m is m integer or in other words q is equal to m pi by a. So now we have b is equal to 0 and q is equal to m pi by a. So substituting in the value of x we get x is equal to capital A cos qx. So qx can be written as m pi a into x. Similarly we can solve for y. So this again is the equation of a simple harmonic oscillator. So the solution can again be written as c cos p y plus d sin p y. Now again equating the normal component of magnetic field as equal to 0 at the boundaries we get del vz by del y is equal to 0 at y is equal to 0 and y is equal to p. And since vz is a function of x and y we can write del y by del y is equal to which is equal to minus cp sin p y plus dp cos p y this is equal to 0 at the boundaries y is equal to 0 and y is equal to p. So simplifying this here we now get p is equal to n pi by d where now n is another integer and we get d is equal to 0. So substituting here the values of d and p we get y is equal to c cos n pi y by b. And we can substitute now the values of capital X and capital Y in vz. So we get b, b is some constant cos n pi x by a cos n pi y by b e to the power of ikgz minus omega t. So we see that it has a cosine variation in x and a cosine variation in y. These are the directions in which the boundaries have been applied and it is a propagating wave in the z direction. So this is the solution for T n wave. Now here we have kc square is equal to q square plus p square. So this is how we have defined q, q is equal to kc square minus p square. So we have kc square is equal to q square plus p square and we know that q is equal to m pi by a and p is equal to n pi by b. So we can write kc square as equal to m pi by a the whole square plus n pi by b the whole square and from this we can find out the value of the cutoff frequency. So the cutoff frequency for mn mode is given by pi under root m square by a square plus m square by b square. So this is the cutoff frequency for the mn mode and we see that it depends only on the values of the dimensions of the system. It depends only upon the length of the system in the x direction and the length of the system in the y direction. So this is analogous to the case of the stretched string, the modes in a stretched string where the frequencies of the modes depend only on the dimensions of the system. So here m is an integer and it is a number of half period variations in x. Similarly n is a, it is an integer and it is a number of half period variations in y. So n and n can take values from 0, 1, 2, 3 so on. Now similarly we can solve for tm mode, for tm mode bz is equal to 0 and similarly and applying the boundary conditions we get ez is equal to e sin m pi x by a sin n pi y by b e to the power of ik z minus omega t. So now here we see that the variation in the variation in x and y has a sinusoidal form. So here we have solved by similar method and then applying the boundary condition that the tangential component of electric field is 0 at the boundary, is 0 at the boundary. So again if you calculate the cutoff frequency for the mn mode we see that the formula is the same as it was for the te mode. So the cutoff frequency is the same and here again m and n are integers and they are the number of half period variations in x and y respectively. So summarizing the results we see that for the te mode ez is equal to 0, bz has to exist, bz has a form where the amplitude is a function of x and y and x and y are the directions in which the boundaries are applied and it is a propagating wave in the z direction. So if we solve this by putting the value of bz in the wave equation and applying the boundary conditions we see that bz has a form where it has cosine dependence on x and y. Similarly for tm mode bz is equal to 0 and ez has a form like this where the amplitude has a sinusoidal variation in x and y and it is again a propagating wave in the z direction in which there is no boundaries. So there is sinusoidal variation in both x and y. The cutoff frequency formula for the mn mode for both tm and te mode is the same and it depends upon the dimensions of the system. Now if a is greater than b that means the dimension in x is greater than the dimension in b then te10 mode is the lowest mode or the fundamental mode for a rectangular wave. So it has the lowest frequency. The fundamental mode for the tm mode is the tm11 mode. So here for the tm mode m and n will take values from 1, 2, 3 so on they cannot take values 0 because if either m or n is equal to 0 then the electric field variation is sinusoidal. So if either m or n is 0 then ez will go to 0 and bz is already 0. So if both ez and bz are 0 all other field components will go to 0 and we will have no field. So the fundamental tm mode is the tm11 mode. So tm01 and tm10 mode do not exist since if either m or n is 0 ez is equal to 0. So now let us see the field patterns for the te10 mode in a rectangular waveguide. As we have already seen the te10 mode is the fundamental mode in the rectangular waveguide. So here we have ez is equal to 0. So since it is the te mode ez is equal to 0 so the electric field is along x and y direction only. So it is the transverse electric mode so the electric field will be in the transverse direction only and now since ez is equal to 0 bz has to exist because we see that if both ez and bz go to 0 then there are no fields. So bz definitely has to be there and electric field is along x and y direction only. So now we have this is the te10 mode so m is equal to 1 and m is equal to 0. So m is equal to 1 means there is one half period variation in x of any field. So if you see any field component and see the variation in the x direction there is one half period variation. If you see the variation in the z direction since n is equal to 0 there is no variation in the field. So this is the rectangular waveguide here so this is the x direction, this is the y direction and this is the z direction. So let us see the variation of the fields in the various direction. So we have now one half period variation in x of any of the fields. So let us draw the variation in x from 0 to a we can take any field component so here it is convenient to take so we have here the e y component so we see that this electric field is in the y component so it is convenient to take this the e y component. So let us take e y and see the variation in x direction from 0 to a. So we see that at this boundary here this is the tangential component of the electric field similarly at x is equal to a this is the tangential component of electric field and by boundary conditions the tangential component of electric field has to be 0. So the electric field has to be 0 at x is equal to 0 and at x is equal to a and there is one half period variation in x of any of the field. So there has to be one half period variation so the field will look like this so we see that the e y field is 0 at x is equal to 0 and x is equal to a and it is maximum at the centre this is at a particular instant of time the fields are varying or oscillating with time. Now we can see similarly the variation of e y with y from 0 to b. So at a fixed value of x and z we can see the variation of e y with y so from 0 to b. So let us say we take at this location of x and z so we see that e y field with y from 0 to b it is constant there is no variation so there is no variation in y of any of the fields. We can also see the magnetic field component the magnetic field lines as we can see here they form closed loop lines around the electric field lines. So here we can draw here with x the b x component ok. So we see here at x is equal to 0 at this boundary the magnetic field component the b x component is the normal component so again by boundary condition it has to go to 0. Similarly at x is equal to a it is the normal component of the magnetic field so it has to go to 0. So if we plot the variation of b x with x we will get a variation like this from 0 to a because in order to satisfy the boundary condition the b x component has to go to 0 at x is equal to 0 and x is equal to a. So the field pattern or the modes are formed such that the waves now satisfy the boundary conditions inside the waveguide. So this is how the field patterns look like. So this picture shows another view of the TE10 mode so this is the waveguide here so this is again the x direction this is the y direction and this is the z direction. So in the z direction since there are no boundaries the wave is free to propagate the red lines show the electric field and the blue lines show the magnetic field. So if you see the front view here so we can see that again the there is one half period variation in the electric field and the magnetic field lines they form closed loop lines around the electric field lines. So we can see the side view in the central plane here so this picture shows this so this is the electric field lines these are all at a particular instant of time. So this is the electric field lines and these are the magnetic field lines. So these magnetic field lines are shown by the dots here. The top view is shown here so in the top view you can see that the magnetic field lines are forming closed loop lines around the electric field lines. So here this distance corresponds to lambda g. So here we can see the electric field lines for the TE10 mode in a rectangular waveguide so this is the so we see that it is a propagating wave in the z direction and in the x and y direction where the boundaries have been applied it is a standing wave. So you can see that the field here is always a 0 or a minimum and here it is always a maximum and varying with time there is variation with time. So we have already calculated this expression before so here we have 1 by lambda 0 square is equal to 1 by lambda c square plus 1 by lambda g square. So where lambda 0 is the free space wavelength, lambda c is the cutoff wavelength and lambda g is the guided wavelength. So you can write it in terms of frequency and the wave numbers like this. Now for a given waveguide the frequency of the TE10 mode is fixed or the wavelength of the TE10 mode is fixed it depends only on the dimension. So lambda c10 is equal to pi c by a calculated by this formula so this is fixed. So for a given waveguide lambda c is fixed. Now as lambda 0, so lambda 0 is the free space wavelength. So as lambda 0 is increased lambda g increases so as this is fixed as you increase lambda 0 lambda g will also increase and this will increase the velocity of propagation of this wave inside the waveguide until lambda 0 is equal to lambda c. So when lambda 0 becomes equal to lambda c then lambda g and vg both become imaginary and the wave gets attenuated. So for lambda 0 greater than lambda c the wave is not propagated in the waveguide so it is now attenuated. So the frequency corresponding to the lambda c so omega c that is known as the cutoff frequency of the waveguide. So let us see that some other TE modes in a rectangular waveguide. So we have already seen the TE10 mode so where we have one half period variation in the field in the x direction. Similarly we can have a TE20 mode so here we have two half period variations in the field in the x direction. So the fields will always satisfy the boundary condition here so you see that the field will so the field will be maximum here in this direction and maximum here in the opposite direction and at the boundary here since it is a tangential component it will go to 0 so the field here is 0 and again as before the magnetic field lines will form closed loop lines around the electric field lines. The direction of these two loops will now be opposite to each other because the field is in the opposite direction here the electric field is in this direction and here the electric field is in this direction at a particular instant of time. Similarly for the TE30 mode so there are three half period variations of the field in the x direction. So here the field is in the forward direction here it is in the opposite direction and here again in the this direction and then the magnetic field lines will form closed loop lines around the electric field lines like this. Now we can also have the TM11 mode so for the TM mode this is the fundamental mode in the rectangular wave kite because we have seen that a TM01 or a TM10 mode cannot exist in the field kite. So since it is a TM mode BZ is equal to 0 and we will have magnetic field along x and y direction only. So it is a transverse magnetic field so magnetic field will be in the transverse direction only and EZ should definitely exist. And N and N are equal to 1 here so there is one half period variation in x of any of the field and one half period variation in y of any of the field. So this figure shows the field pattern in the front view this shows the top view and this shows the side view in the central plane. So these dotted lines are the magnetic field lines so we can draw for example let us say Bx with x from 0 to A so this is Bx this is the x component of the magnetic field again you see that the lines here this is the normal component at this boundary the magnetic field line this will be the normal component. So this has to go to 0 here it has to go to 0 at this boundary so that is why in order to satisfy the boundary conditions the magnetic field lines they take a turn and they form a closed loop line like this such that at none of the boundaries they form the normal component of the magnetic field. So if you see the variation of Bx with x from 0 to A it will be like this so it is 0 here and it is 0 here and in between it is maximum. Similarly you can also plot the variation in y from 0 to B of let us say dy and you will have one half period variation so m is equal to 1 and n is equal to 1 here. So for a given waveguide now we can have various modes so we can have the Te mn mode and the Te mn modes and the frequencies of these modes are fixed for fixed dimensions of the waveguide. So let us say we have a waveguide and we can calculate the frequencies of the various modes. So let us say FC1 is the frequency of the Te10 mode, FC2 is the frequency of the Te01 mode, FC3 is the frequency of Te11 and Te11 mode so these are degenerate modes and so on. So let us say F0 is the free space frequency so this is the frequency of a wave that you are trying to propagate through this waveguide. Now you know that when F0 is less than FC1 so if F0 is less than FC1 there will be no propagation the wave will not be able to propagate through the waveguide it will get attenuated. Now if F0 lies between FC1 and FC2 if it lies in this region so it is above the cutoff frequency for the Te10 mode. So it will now propagate in the Te10 mode. Similarly now if let us say F0 lies in this region between FC2 and FC3 now the wave inside the waveguide will propagate with a combination of Te10 mode and Te01 mode because it is above the both these cutoff frequencies so both these modes are possible so it will now propagate inside the waveguide in a combination of the Te10 and the Te01 modes.