Abstract:
The present invention is based on a relatively simple mechanism which heretofore has not been tried before. The mechanism depends on modulation of a collimated beam transverse to the beam direction rather than the usual longitudinal modulation. Conversion of the transverse motion into longitudinal bunching in an output cavity is accomplished by means of the difference in path length in a bending magnet. Since the present invention does not depend on longitudinal modulation, it is suitable for pulsed superpower (1 GW) applications, but it can be equally suited for multi-megawatt cw applications. The present invention pertains to an apparatus for bunching relativistic electrons. The apparatus comprises means for imparting a periodic velocity in a first direction in a first region to electrons of an electron beam moving in a second direction. The apparatus also is comprised of means for causing electrons to follow a path length in a second region corresponding to the velocity in the first direction such that the path length is determined by the velocity imparted in the first direction. The differing path length causes beam electrons to be bunched as they exit the second region, allowing microwave power to be extracted from the bunches by conventional means.

Description:
FIELD OF THE INVENTION  
         [0001]    The present invention is related to microwave amplifiers. More specifically, the present invention is related to the bunching of relativistic electrons. Bunching is accomplished by first transversely modulating a collimated electron beam. Once modulated, the beam is allowed to pass through a bending magnet which converts the modulated beam into a bunched beam.  
         BACKGROUND OF THE INVENTION  
         [0002]    The development of high-power microwave sources has proceeded slowly over several decades, motivated by different applications at different times. Immediately after World War II, for example, tubes which had been developed for radar and for high-power transmitters were needed to power high-energy particle accelerators. The most dramatic development took place at Stanford University. It was there that the klystron was rapidly developed from the kilowatt level to peak powers exceeding a megawatt. After further development the klystron rapidly became the accepted power tube for a large number of electron accelerators as well as many other applications. It has been developed to the point where reliable tubes produce 50 megawatts peak power and research devices achieve 200 MW at 11.4 GHz for about 10 nanoseconds. T. G. Lee, G. T. Konrad, Y. Okazaki, Masaru Watanabe, and H. Yonezawa, IEEE Trans. Plasma Sci., PS-13, No. 6, 545 (1985), and M. A. Allen et al., LINAC Proc. 508 (1989) CEBAF Report No. 89-001.  
           [0003]    Klystrons and gridded tubes provide for most high-power microwave needs. However, they have definite drawbacks for particular applications. Gridded tubes are severely limited in frequency. Power density, gain and efficiency problems rapidly get worse above 100 Mhz. High-power klystrons also have limitations: they become very large and expensive for the lower frequency range of interest. One solution advanced by Varian Associates is the klystrode. M. B. Shrader and D. H. Priest, IEEE Trans. Nucl. Sci. NS-32, 2751 (1985); M. B. Shrader, Bull. Am. Phys. Soc. 34, 236 (1989). This device combines some of the features of gridded tubes and klystrons.  
           [0004]    For high-power amplifiers, an awkward frequency region exists between approximately 100 MHz and 2 GHz. Moreover, at any frequency, as the peak power increases, designers are forced to use higher voltage to keep the beam current and resulting space charge effects within limits. This means that they are forced to use increasingly relativistic beams which are difficult to axially modulate. In general, it is difficult to achieve high power, high efficiency, high gain, small size/weight, and low cost simultaneously.  
           [0005]    Interest has increased in recent years in other methods of microwave generation. A group led by V. Granatstein at the University of Maryland is pursuing the cyclotron maser mechanism for use in a gyroklystron amplifier. Victor L. Granatstein, IEEE Cat. No. 87CH2387-9, 1696 (1987). Another group led by J. Pasour, J. A. Pasour and T. P. Hughes, Bull. Am. Phys. Soc. 34, 185 (1989), is experimenting with the negative mass instability mechanism proposed by Y. Y. Lau, Y. Y. Lau, Phys. Rev. Lett. 53, 395 (1984). Groups at the Stanford Linear Accelerator Center (SLAC), Lawrence Berkeley Laboratory (LBL), and Lawrence Livermore National Laboratory (LLNL) are collaborating on a relativistic klystron project, T. L. Lavine et al., Bull. Am. Phys. Soc. 34, 186 (1989); R. F. Koontz et al., Bull. Am. Phys. Soc. 34, 188 (1989). And recently at Novosibirsk, USSR, where Budker invented the gyrocon, impressive results have been obtained with a version of the gyrocon called the magnicon, M. M. Karliner et al., Nucl. Inst. Meth. A269, 459 (1988).  
           [0006]    None of these devices is near commercial production. Further research is required to sort out their relative merits and practical benefits. Reviews by Reid and by Faillon for the accelerator community give summaries of much of the above effort, D. Reid, Proc. 1988 Linac Conf., 514 (1989) CEBAF Report No. 89-001; G. Faillon, IEEE Trans. Nucl. Sci. NS-32, 2945 (1985).  
         SUMMARY OF THE INVENTION  
         [0007]    The present invention is based on a relatively simple mechanism which heretofore has not been tried before. The mechanism depends on modulation of a collimated beam transverse to the beam direction rather than the usual longitudinal modulation. Conversion of the transverse motion into longitudinal bunching in an output cavity is accomplished by means of the difference in path length in a bending magnet. Since the present invention does not depend on longitudinal modulation, it is suitable for pulsed superpower (1 GW) applications, but it can be equally suited for multi-megawatt cw applications.  
           [0008]    The present invention pertains to an apparatus for bunching relativistic electrons. The apparatus comprises means for imparting a periodic velocity in a first direction in a first region to electrons of an electron beam moving in a second direction. The apparatus also is comprised of means for causing electrons to follow a path length in a second region corresponding to the velocity in the first direction such that the path length is determined by the velocity imparted in the first direction. The differing path length causes beam electrons to be bunched as they exit the second region, allowing microwave power to be extracted from the bunches by conventional means. 
       
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0009]    In the accompanying drawings, the preferred embodiments of the invention and preferred methods of practicing the invention are illustrated in which:  
         [0010]    [0010]FIG. 1 is a schematic representation of an embodiment of the present invention.  
         [0011]    [0011]FIG. 2 is an alternative embodiment of the present invention.  
         [0012]    [0012]FIG. 3 is another alternative embodiment of the present invention.  
         [0013]    [0013]FIG. 4 a  is a graph of the drift deflection in a TM210 cavity.  
         [0014]    [0014]FIG. 4 b  is a graph of the drift deflection in a TM210 cavity in a different environment that FIG. 4 a.    
         [0015]    [0015]FIG. 5 is a schematic representation of the field pattern in a cylindrical TM110 cavity.  
         [0016]    [0016]FIG. 6 is a graph of the particle response to TM110 cavity mode for an idealized particle drift motion.  
         [0017]    [0017]FIG. 7 a  is a graph of a particle response to TM110 mode for a nonideal particle drift motion.  
         [0018]    [0018]FIG. 7 b  is a graph of the particle response to TM110 mode for nonideal drift. 
     
    
     DESCRIPTION OF THE PREFERRED EMBODIMENT  
       [0019]    Referring now to the drawings wherein like reference numerals refer to similar or identical parts throughout the several views, and more specifically to FIG. 1 thereof, there is shown a schematic diagram of the present invention is shown in FIG. 1. A preferably small diameter, well collimated beam  20  traverses an input cavity  22  which operates in a mode having a magnetic field perpendicular to the plane of the figure. The amplitude of the field is adjusted to give a maximum deflection angle θ m . After traveling a drift distance L 1 , the oscillating beam enters a bending magnet  24  in the median plane of the magnet and is bent through an angle α, with a radius of curvature R, determined by the kinetic energy and applied magnetic field.  
         [0020]    The magnet  24  of FIG. 1 is designed to be focusing in the plane perpendicular to the plane of the figure as well as in the median plane. For a uniform-field magnet  24  this is done by rotating the input edge  26  and output edge  28  by the angles ν 1  and ν 2 , respectively, in the sense shown in FIG. 1. In the simplest embodiment, ν 1 =ν 2 . Edge rotation has the net effect of reducing the focusing power in the median plane and introducing focusing in the normal plane, Harald A. Enge, Chapter “Deflecting Magnets” in “Focusing of Charged Particles” Vol. II, Academic Press, Ed. A. Septier, (1967); Hermann Wollnik, “Optics of Charged Particles,” Academic Press (1987) and T. F. Godlove and W. L. Bendel, Rev. Sci. Inst. 36, 909 (1965). If the magnet  24  is designed to provide equal object and image distance, L 1 =L 2 =L, then mirror symmetry exists about a plane normal to the median plane through the center of the magnet as shown in FIG. 1. For a monoenergetic, well collimated beam  20  and small space charge, all rays from the center of the input cavity  22  are focused to the center of the output cavity  30 , independent of the deflection angle θ up to the point where magnet aberrations become important.  
         [0021]    It should be noted that the center of curvature of each orbit lies in the symmetry plane as well as in the median plane. This means that the angle traversed by each ray within the magnet is given simply by (α+2θ). The distance from the center of the input cavity  22  to the (rotated) edge of the magnet  24  is L/(cos θ−sin θtan ν). The total path length, S, traversed between cavity centers is then given by: 
           S =(α+2θ) R +2 L (cos θ −  sin θ tan ν) −1   (1) 
         [0022]    To determine the deflection angle, θ m , required for bunching, the change in travel time compared to θ=0 is calculated and set equal to one-quarter of the rf period. For small θ, S and ΔS may be approximated by: 
           S=αR+ 2 L+ 2( R+L  tan ν)θ  (2) 
         [0023]    and 
         Δ S =2( R+L  tan ν)Δθ  (3) 
         [0024]    Dividing Eq. (3) by the electron velocity, v, to obtain the difference in travel time, and setting it equal to a quarter period (=λ/4c), where λ is the operating wavelength, the deflection angle for optimum bunching is obtained and is: 
         θ m =Δθ=(1/8)(βλ/ R )[1+( L/R )tan ν} −1   (4) 
         [0025]    β=v/c.  
         [0026]    Equation (4) can be carried one step further. It turns out that L/R and ν are determined by the choice of bending angle, α, and in fact are correlated in such a way that the quantity 1+(L/R)tan ν always equals two, neglecting fringe field effects (Table I, below, gives this parameter in more detail). Eq. (4) then simplifies to: 
         θ m =βλ/(16 R )  (5) 
         [0027]    These equations determine the basic wavelength scaling of the invention. For example, if a conservative limit on θ m  of 7° is assumed, and the criterion is adopted that the path length should be kept as short as possible to reduce space charge effects, then the bend radius R falls in the range βλ/2 to βλ. Setting it at βλ/2 fixes the path length, Eq. (2), at: 
           S =βλ[(α/2)+( L/R )+0.245]  (6) 
         [0028]    which is 4.2βλ, 3βλ, and 2.5βλ for α=60°, 90°, and 120°, respectively, taking L/R from Table I, below.  
         [0029]    Table I gives magnet design parameters for uniform-field bending magnets  24  with equal rotation of input and output edges, and equal object and image distance.  
                                                           TABLE I                           Uniform Field Magnets            α   fg/R   ν   L/R   1+(L/R)tanν                     60°   0   16.1°   3.47   2.00           0.05   17.5°   3.82   2.20           0.10   18.9°   4.27   2.46           0.15   20.4°   4.86   2.81        90°   0   26.6°   2.00   2.00           0.05   27.9°   2.12   2.12           0.10   29.3°   2.27   2.27           0.15   30.7°   2.47   2.47       120°   0   40.9°   1.16   2.00           0.05   42.0°   1.21   2.09           0.10   43.4°   1.27   2.20           0.15   44.7°   1.35   2.34                  
 
         [0030]    In order to obtain Table I, fringe field effects are included in the parameter fg, where g is the gap spacing and f is a dimensionless constant, related to the location of the (assumed) thin lens which provides focusing in the transverse plane, Harald A. Enge, Chapter “Deflecting Magnets” in “Focusing of Charged Particles” Vol. II, Academic Press, Ed. A. Septier, (1967); Hermann Wollnik, “Optics of Charged Particles,” Academic Press (1987). It is typically 0.4 to 0.5. The values of Table I include the first order effect of the fringe field, but not higher order aberrations. These can be reduced by machining the input edge  26  slightly convex in shape, Harald A. Enge, Chapter “Deflecting Magnets” in “Focusing of Charged Particles” Vol. II, Academic Press, Ed. A. Septier, (1967); Hermann Wollnik, “Optics of Charged Particles,” Academic Press (1987). That is, the required edge rotation angle is not strictly constant, but increases slightly as θ increases.  
         [0031]    The parameter 1+(L/R)tan ν which occurs in Eq. (4) is included in Table I to show the relatively small variation of this parameter due to finite fringe field effects.  
         [0032]    Comparing different values of α, 120° is optimum because it gives the shortest path length. At this point, the edge angle rotation is large, 41°, and should not be increased further because of serious aberrations in the magnet  24 . Also, the decrease in path length is only bout 20% compared to the 90° case. Second, the fringe field changes the magnet edge rotation by typically 2 to 3 degrees, and increases the path length by 5%-15%. These corrections, while important, are quite tolerable. Finally, it is interesting to note that for small gaps the deflection angle for optimum bunching, Eq. (5), is independent of α.  
         [0033]    [0033]FIG. 2 shows an alternative embodiment of the invention. It is identical to using α=120°, up to the plane of symmetry in the magnet. The second above-described preferred embodiment half of the magnet  24 , beyond the plane  23  of symmetry, has been removed so that all electrons emerge from the magnet  24  perpendicular to the magnet edge  28  and parallel to the central ray. The effective bend angle is therefore one-half of 120° or 60°. Upon emerging from the magnet  24 , the electrons are focused on the center of the output cavity  30  by means of a solenoid lens  32 . Finite gap corrections modify the input and output edges of the magnet  24  by the same correction angle, up to 4°, given in Table I.  
         [0034]    The advantage of this alternative embodiment is a shorter, more compact apparatus.  
         [0035]    Another alternative to the magnet of FIG. 1 is a magnet with a nonzero field index, n, defined by B=B b (r/R) −n . For example, with n=½ and perpendicular edges (ν=0), the focal length for the case of equal object and image distance, L, and equal median plane/transverse plane focusing, is obtained from 
           L/R =[2 cot(α/(2{square root}2))  (7) 
         [0036]    neglecting the fringe field, Harald A. Enge, Chapter “Deflecting Magnets” in “Focusing of Charged Particles” Vol. II, Academic Press, Ed. A. Septier, (1967); Hermann Wollnik, “Optics of Charged Particles,” Academic Press (1987).  
         [0037]    Table II gives a summary of focal lengths and the resulting path lengths for this magnet as well as the uniform field magnet of version  1 A.  
                                                               TABLE II                           Magnet Comparison                Uniform Magnet   n = {fraction (1/2 )} Magnet   n = {fraction (1/4 )} Magnet            α   L/R   S/R   L/R   S/R   L/R   S/R                60°   3.47   7.99   3.64   8.33                90°   2.00   5.57   2.28   6.12   *   4.5       120°   1.16   4.40   1.55   5.19       155°   NA   NA   1     4.71                          
 
         [0038]    Table II indicates that a uniform magnet, with edge focusing, has a slightly shorter path length than the nonuniform magnet, for the cases considered. The second alternative embodiment which uses one-half of a 120° uniform magnet, can have an even shorter path length. The latter case depends on the details of the solenoid strength. We estimate its total path length at 4.2 R, which is slightly less than the 120° magnet and the 90°, n=¼ magnets of Table II. Version 2, which uses an axial magnetic field for beam containment and focusing, is sufficiently different from the above geometries that it is considered separately.  
         [0039]    It can be concluded that among the versions of magnet design not involving an axial field, either a 120° magnet or the magnet/solenoid combination are the optimum candidates for practical realization of the invention.  
         [0040]    An extensive theoretical investigation has been done on the transverse modulation klystron [(TMK) Y. Seo and P. Sprangle, NRL memorandum report #6756 (1991)] which we summarize below. Basically, the theory indicates that the TMK can achieve the high modulation density necessary for efficient microwave generation. Furthermore, when the current is increased, the electron bunching is deteriorated by the self-field in a conventional klystron, while the self-field enhances the bunching in the TMK. The bunching enhancement is due to the negative mass effect and only occurs in the bend region. In the drift region between the exit of the magnet and the output cavity (FIG. 1, L 2 ) a longitudinal plasma oscillation sets an upper bound for the drift length. That is, in order to not deteriorate the modulated density achieved we must satisfy, 
         L 2 &lt;&lt;1/k S  where k S  is the plasma wave number 
         [0041]    and is given by  
               k   s     =       (     ω   c     )                       (     I     I   0       )       1   /   2              (   γβ   )         -   5     /   2                 (   8   )                               
 
         [0042]    and  
         [0043]    w=radian rf frequency, I=beam current, I o =17 kA and Y=relativistic mass factor. We will derive eq. (8) in the next section. Evaluating eq. (8) for I=1A, f=1.3 GHz and an energy of 50 kev, then 1/k S =66 cm. This result can easily be satisfied for a real device.  
         [0044]    The transverse modulation klystron has theoretically been shown to have a high electrical efficiency, high gain, is compact and produces high power at high voltage, limited by space charge effects.  
         [0045]    In order to have higher power at a given voltage, we must increase the current. This is not possible in the original version of the TMK since the self-fields expand the beam. By applying a modest axial guide magnetic field, it can be shown that current can be increased by an order of magnitude at the same energy.  
         [0046]    [0046]FIG. 3 shows a schematic of the preferred embodiment axial-field TMK. In FIG. 3, there is shown an apparatus  100  for bunching relativistic electrons. The apparatus  100  is comprised of an electron gun  20  for producing a pin beam of electrons. The apparatus  100  is also comprised of a vacuum chamber  102  at least a portion of which is toroidally shaped. The vacuum chamber  102  is comprised of an input cavity  22  having means for imparting a predetermined drift displacement to each electron as it passes therethrough such that electrons are caused to bunch together at a predetermined location in the vacuum chamber  102 . The input cavity  22  is in alignment with the gun  20  to receive electrons therefrom. The vacuum chamber  102  is also comprised of an output cavity  104  disposed at essentially the opposite end of the toroidal portion. The output cavity  104  has means  106  for extracting RF energy from electrons passing therethrough. The apparatus  100  is also comprised of means for producing an axial magnetic field and at least a toroidal portion of the vacuum chamber  102  to maintain the electrons in the chamber. The axial magnetic field producing means is in electromagnetic communication with the vacuum chamber  102 . The apparatus  100  is also comprised of means for producing a vertical magnetic field in the vacuum chamber  102  to maintain the electrons in the chamber  102 . The vertical field producing means is in electromagnetic communication with the vacuum chamber  102 .  
         [0047]    Preferably, the axial magnetic field, B A , is constant along the entire beam trajectory and increases in the output cavity.  
         [0048]    In FIG. 3, a small diameter beam is produced by a magnetron injection gun (MIG), proposed for the higher voltage cases which was analyzed [R. Palmer, W. Herrmannsfeldt and K. Eppley, SLAC-PUB-5026]. For lower voltage, a conventional Pierce gun is proposed where the magnetic field is zero at the cathode and increases up to a constant value. The beam travels into an input cavity which operates in a TM 210  mode and has a transverse rf magnetic field in the plane of the figure. Note that the magnetic field is rotated by 90 degrees from the basic concept in FIG. 1. The modulated beam just after it enters the bend region begins to bunch by the transit time difference the same as in the non-axial-field TMK. The bunched beam is compressed just prior to entering the output cavity. The output cavity is the same as in the non-axial-field version of the TMK.  
         [0049]    In order to accomplish modulation (still in the plane of the figure), the incoming beam is allowed to drift using the FxB A  mechanism, where F=−ev z B rf , v z =axial velocity and B rf =rf magnetic field. FIGS. 4 a  and  4   b  show two examples of the drift modulation method. In each example, three electrons are injected into a TM 210  mode cavity where the cavity length is half the rf period times the particle velocity. The cavity begins at z=0 and extends to the length L c =B z λ/2. The particles are injected at a phase such that the maximum, minimum and no deflection occurs. The resulting time integrated trajectories are shown. The parameters for FIG. 4 a  are f=0.5 GHz, B A =2 Kg, B rf =0.28 Kg, β z =0.6. For FIG. 4 b  the parameters are f=1.3 GHz, B A =1.5 Kg, B rf =0.194 Kg, β z =0.548. FIG. 4 b  shows an oscillation on the trajectory; this is simply the resulting cyclotron motion of the particle. this motion represents a small fraction of the total particle energy. The axial field relaxes the space charge problem but makes the beam deflection more difficult than without it. In order to accomplish deflection a larger rf field is required which in turn requires more rf power, thus reducing the gain. We propose a solution to the gain problem by adding an intermediate cavity. This is described in later sections.  
         [0050]    The drift deflection and Larmor radius for the TM 210  mode have been derived. For this mode the dominant field near the axis is a constant magnetic field, given by B x =B rf  sin(wt), where the x direction is perpendicular to the beam and in the plane of FIG. 3.  
         [0051]    The equations of motion are:  
                 υ   .     x     =       υ   y          Ω   z               (   9   )                     υ   .     y     =         -     υ   x            Ω   z       +       υ   z          Ω   x                
              Ω   z     =       e                   B   A         γ                 m         ,       Ω   x     =         e                   B   x         γ                 m       .                 (   10   )                               
 
         [0052]    Assuming y and ν z  are constant, equations (9) and (10) can be solved to give  
               υ   y     =           υ   z          Ω   rf        ω         Ω   z   2     -     ω   2              [       cos        (     ω                 t     )       -     cos        (       Ω   z        t     )         ]               (   11   )                   υ   x     =           υ   z          Ω   rf           Ω   z   2     -     ω   2              [         Ω   z          sin        (     ω                 t     )         -     ωsin        (       Ω   z        t     )         ]              
              Ω   rf     =       e                   B   rf         γ                 m         ,             (   12   )                               
 
         [0053]    The maximum drift displacement (ΔR) at t=π/ω and Larmor radius (r L ) can be derived from equations (11) and (12) to give,  
               Δ                 R     =       2   _          υ   z            Ω   rf     /     (     ωΩ   z     )                 (   13   )                 r   L     =     Δ                 R                     cos        (       πΩ   z       2      ω       )       /     [         (       Ω   z     /   ω     )     2     -   1     ]                 (   14   )                               
 
         [0054]    After evaluating equations (13) and (14) for parameters of interest, it has been found that enough deflection can be achieved while keeping the transverse energy small. For example at f=1 GHz, B A =1 kG, B rf =0.278 kG, β z =0.6 a deflection of 1.6 cm can be achieved while the transverse energy is about 8% of the axial energy. At f=0.5 GHz, and all other parameters the same, the deflection is 3.18 cm and the transverse to axial energy is about 0.9%.  
         [0055]    For the modulated beam to bunch in the bend we require that the transit time difference between the non-deflected particle trajectory and the maximum deflected particle trajectory to be equal to one-quarter of the rf period, that is:  
                 Δ                 R     =         β   z        c       2      ω                 N         ,                where                 N                 is                 the                 number             (   15   )                               
 
         [0056]    of 180° bend angles. For example, if N=1 then a 180° bend is needed or if N=1.5 then a 270° bend is required.  
         [0057]    The optimum bend angle turns out to be about 257°. Angles much less than 257° require more rf power and drive the displacement into a non-linear region. For angles larger than 257°, a very small beam radius is required which is not possible to achieve with existing electron guns.  
         [0058]    In order to have most of the beam participate in the modulation process, we require the beam radius (r b ) to be small compared to the drift deflection, 
         Δ R=μr   b   , where μ is a number   (16) 
         [0059]    that will be picked such that the beam size will be smaller than the deflection.  
         [0060]    Two limits are calculated on the beam current. The first is the limit that space charge imposes on transport in a magnetic field and the second limit is on bunching.  
         [0061]    In the absence of emittance, the maximum current that can be transported can be calculated from the envelope equation to be,  
               I   =           I   0          β   z          γ   3       8            (       r   b     c     )     2          Ω   z   2         ,           (   17   )                               
 
         [0062]    Next, it is calculated how the current places a limit on the distance over which bunching can occur.  
         [0063]    Consider an electron beam traveling down a perfectly conducting pipe where the beam nearly fills the pipe diameter. The electric and magnetic field are as follows:  
                   E   ρ     =         -     |   e   |     n   t           2        ε   0                (       r   b       ρ   c       )     2        ρ       ,                where                   n   t                   is                 the                 beam                 density     ,     
            ε   0     =     permittivity                 of                 free                 space              
            ρ   c     =     pipe                   radius   .                 (   18   )                               
 
         [0064]    The factor (r b /ρ c ) 2  corrects for the beam changing its radius after it has been modulated and bunched.  
                 B   8     =           μ   0        J     2            (       r   b       ρ   c       )     2        ρ       ,                  where                 J                =                current                 density               (   19   )                               
 
         [0065]    and  82   0  is the permeability of free space.  
         [0066]    From an axial displacement of charge given by 
         δ( z )= A  cos(ω t−kz ) 
         [0067]    the principal density perturbation can be calculated from Fourier analysis and is given by  
                 δ                 n     =       n   0        k                 A                   sin        (       ω                 t     -     k                 z       )           ,                  where                 k     =     ω     υ   z                 (   20   )                               
 
         [0068]    and A is a displacement amplitude factor and n o =n t −δn. A detailed derivation of Eq. (20) can be found in Ref Y. Seo and P. Sprangle, NRL memorandum report #6756 (1991).  
         [0069]    The axial electric field can be found from  
               E   z     =           ∂               ∂   z                ∫   0       Q   z              E   ρ             ρ           -         ∂               ∂   t                ∫   0     Pc            B   8               ρ     .                     (   21   )                               
 
         [0070]    From equations (18)-(21) it can be calculated the final form for the axial electric field.  
               E   z     =       m   e          (     I     I   0       )          I     β   z   3              (     ω   γ     )     2        A                   cos        (       ω                 t     -     k                 z       )                 (   22   )                               
 
         [0071]    which can be written  
               E   z     =       m   e          (     I     I   0       )            (     ω   γ     )     2            δ        (   z   )         β   z   3                 (   23   )                               
 
         [0072]    Now form a right handed coordinate system (r, z, y) where r is an outward radial coordinate in the plane of the paper, i.e., it is perpendicular to the centerline in FIGS.  1  or  3 , y is into the paper, and z is along the centerline.  
         [0073]    The additional magnetic fields introduced by a bending magnet having a field index, n is considered:  
               B   r     =       -     B   oy          n        z   R               (   24   )                               
 
         [0074]    A change of variables is used (from time t to axial position z using the transformation d/dt=ν z d/dz) and assume small perturbations from the equilibrium are valid. Then r=∂r, y=∂y and z=z 0 +∂z. The equations of motion including equations (23)-(26) then become  
                 δ                   r   ″       -       k   r   2        δ                 r       =       k   z        δ                   y   ′               (   27   )                   δy   ″     -       k   y   2        δ                 y       =       k   z        δ                   r   ′               (   28   )                       δz   ″     -       k   s   2        δ                 z       =         -   δ                     r   ′       R       ,              where          
              k   r   2     =       (     l   -   n     )     /     R   2         ,       k   y   2     =     n   /   R       ,       k   z     =       B   A     /     (     RB   oy     )                   (   29   )                               
 
         [0075]    and k S  is from eq. (8).  
         [0076]    Equations (27)-(29) are time averaged over the axial field frequency, which gives the guiding center equations  
                   r   c   ″     -         (       k   y       k   z       )     2          k   1   2          r   c         =         (       k   y       k   z       )     2          k   2        δ                   z   c   ′              
              z   c   ″     +       k   s   2          z   c         =         -     r   c   ′       R                   where               (   30   )                   k   l   2     =         (       y   2     -     (     1   -   n     )       )     /     R   2                     and            
            k   2     =       y   2     /     R   .                 (   31   )                               
 
         [0077]    The most interesting approximate solution is  
                 z   c          (   z   )       ≡       1     k   s            (       Δ                 R     R     )        sin                       k   s          (     z   -     L   1       )       .               (   32   )                               
 
         [0078]    Equation (32) gives an optimum path length L 0 . Equation (32) also shows no negative mass instability which is consistent with reference [P. Sprangle and J Vomvoridis, 16 Part. Accel. 18, 1 (1985)]. L 0  is determined from the beginning of bunching to the output cavity, i.e.,  
               L   0     =     π     2        k   s                 (   33   )                               
 
         [0079]    Equation (33) limits the bunching length, hence the radius of the device (L 0 =RNπ) to  
             R   =       (     c     2      N                 ω       )            (     γβ   z     )       s   /   2                (       I   0     I     )       1   /   2       .               (   34   )                               
 
         [0080]    The cavity losses and fill time can be obtained from standard texts such as reference [S. Ramo, J. R. Whinnery and T. Van Duzer, “Fields and Waves in Communication Electronics,” Wiley Z. Sous (1965)]. These relationships are written for completeness. The power loss to the input cavity in the TM 210  mode is  
                 P   t     =         R   s     8              (         λ                 c       Z   0            B   rt       )     2          [         3        L   e       a     +   2     ]                     where            
              R   s     =     surface                 resistivity       ,       Z   0     =         μ   0       ε   0                     (   35   )                               
 
         [0081]    and a is the cavity height. The cavity width is assumed to be 2a.  
         [0082]    The fill time is given by  
               τ   =       2      Q     ω       ,                where                                the                 Q                 must                 be             (   36   )                               
 
         [0083]    evaluated first for the TM 210  mode. The electrical efficiency is defined by  
               η   4     =         η   rf          P   beam           (       P   t     /     η   i       )     +     P   beam                 (   37   )                               
 
         [0084]    where P beam  is the electron beam power, η rf  is the conversion efficiency from beam to rf power, P t  is the rf power into the input cavity, and η t  is the electrical efficiency for the input cavity rf source. It has been demonstrated with particle pushing codes that the intrinsic conversion efficiency η rf  is 50 to 60%. It will be assumed that η rf  is 50%. Also, it is assumed that η t =50%.  
         [0085]    Equations (13)-(17), (34)-(37), are the governing equations to evaluate the performance of this device. Table III shows the parameters used in the evaluation and Table IV shows the results for two different cases of wall material in the input cavity. The first case is for 304 Stainless Steel (72 μΩ-cm) and the second is for copper (1.8 μΩ-cm). The output power is acceptable. However, the gain is low at low voltages ( ∠ 100 kV). It is possible to improve the gain somewhat by using a different input cavity mode.  
                                         TABLE III                       Parameters Used for Calculations                                Energy   50   100   200   500   1000       (keV)       Current (Å)   10.8   32.6   104.8   575.6   2493.0       Micro-   0.96   1.03   1.17   1.63   2.49       Perveance       Beam Rad.   0.177   0.235   0.298   0.270   0.403       (cm)       Pipe Rad.   1.24   1.64   2.09   2.59   2.82       (cm)       Bend Rad.   7.1   10.2   15.1   26.6   43.4       (cm)       Axial   1019   1110   1291   1836   2745       Field (G)       RF Field   178   194   226   321   480       (G)*                          
 
         [0086]    [0086]                                                                         TABLE IV                       TM-210 Deflection Cavity                   Stainless Steel            Energy (keV)   50   100   200   500   1000       Fill Time (usec)   0.63   0.83   0.95   1.07   1.12       Power In (W)   2.34E + 05   3.06E + 05   4.55E + 05   1.01E + 06   2.36E + 06       Power Out (W)   2.69E + 05   1.63E + 06   1.05E + 07   1.44E + 08   1.25E + 09       Gain   1.1   5.3   23.0   142.0   527.6       Efficiency   0.267   0.421   0.479   0.497   0.499            Copper            Energy (keV)   50   100   200   500   1000       Fill Time (usec)   4.31   5.20   6.00   6.76   7.07       Power In (W)   3.73E + 04   4.86E + 04   7.23E + 04   1.61E + 05   3.76E + 05       Power Out (W)   2.69E + 05   1.63E + 06   1.05E + 07   1.44E + 08   1.25E + 09       Gain   7   34   145   893   3319       Efficiency   0.439   0.486   0.497   0.499   0.500                    
         [0087]    An alternative cavity mode to consider is the TM 110  rotating mode in a cylindrical cavity, shown in FIG. 5. In FIG. 3, the TM 210  cavity is replaced with a TM 110  cavity. The first advantage with this change is that the TM 110  cavity is smaller thus allowing for lower cavity losses which improves the gain.  
         [0088]    Consider cylindrical (ρ, θ, z) and Cartesian (x, y, z) coordinate systems located at the center and base of a cylindrical cavity (y is now out of the page). The exact electromagnetic fields for the TM 110  mode are given in Cartesian component form by,  
               B   x     =     2          B   rf          [             J   1          (   κρ   )       κρ          cos        (     ω                 t     )         -         J   2          (   κρ   )            sin        (     θ   -     ω                 t       )          sin                 θ       ]                 (   38   )                 B   y     =     2          B   rf          [             J   1          (   κρ   )       κρ          sin        (     ω                 t     )         -         J   2          (   κρ   )            sin        (     θ   -     ω                 t       )          cos                 θ       ]                 (   39   )                 E   z     =     2        B   rf            cJ   1          (   κρ   )            cos        (     θ   -     ω                 t       )                 (   40   )                               
 
         [0089]    where J 1 (θρ) and J 2 (θρ) are Bessel functions and θ=ω/c. Near the axis Equations (38)-(40) reduce to 
           B   x   =B   rf  cos(ω t )  (41) 
           B   y   =B   rf  sin(ω t )  (42) 
           E   z   =B   rf   ω[x  cos(ω t )+ y  sin(ω t )]  (43) 
         [0090]    For the analytical analysis, the effects of the E z  field are ignored but are included later in a particle pushing code. Both y and ν z  are assured to be constant. The equations of motion for the beam centroid or a particle are: 
         {dot over (υ)} x =−υ y Ω z +υ z Ω rf  sin(ω t ).  (44) 
         {dot over (υ)} y =−υ z Ω rf  cos(ω t )+υ x Ω rf .  (45) 
         [0091]    With the definitions ν t =ν x +i ν y  and ζ t =x+iy where i={square root}−1, the solution of Equations (44)-(45) are:  
               υ   t     =           -     υ   z            Ω   rf           Ω   z     -   ω                 iωt          [              t        (       Ω   z     -   ω     )            (     t   -     t   0       )         -   1     ]                 (   46   )                 ζ   t     =         υ   z       (       Ω   z     -   ω     )            (       Ω   rf     ω     )               t        (       ω                 t     -     x   2       )         ×     [     1   -       ω     Ω   z                   t        (       Ω   z     -   ω     )            (     t   -     t   0       )           -       (     1   -     ω     Ω   z         )               -     tω        (     t   -     t   0       )               ]               (   47   )                               
 
         [0092]    where the initial conditions for a particle entering at t=t o  are ζ t =ν t =0.  
         [0093]    In order for the particle or beam centroid orbit to follow the phase of the electromagnetic mode, the imaginary terms inside the brackets in Eq. (47) are required to vanish, i.e.,  
                 sin        [       (       Ω   z     -   ω     )          (     t   -     t   0       )       ]       -       (         Ω   z     ω     -   1     )        sin                   ω        (     t   -     t   0       )           =   0           (   48   )                               
 
         [0094]    for Eq. (48) to be satisfied it is necessary that 
         Ω z =2ω.  (49) 
         [0095]    Although it appears that Eq. (48) is satisfied by letting Ω z =ω, this is not true when the denominator of Eq. (47) is taken into account.  
         [0096]    The factor of two in Eq. (49) may not be immediately transparent. Note that Ω z  is the frequency about the orbit axis. If the Larmor frequency had been used which is twice the cyclotron frequency (Ω z ) then the factor of two would disappear. As the mode rotates, the particle rotates with the mode and always is at a position where the electric field E z =0.  
         [0097]    Using Eq. (49) in Equations (46) and (47) results in the following expressions:  
               υ   t     =       -       υ   z          (       Ω   rf     ω     )                   iωt          [            iω        (     t   -     t   0       )         -   1     ]                 (   50   )                 ζ   t     =         2        υ   z          Ω   rf         ω   2            sin   2          ω   2          (     t   -     t   0       )               i        (       ω                 t     -     π   /   2       )                   (   51   )                               
 
         [0098]    Maximum deflection occurs when the interaction angle  
                 ω        (     t   -     t   0       )       =       π   .     
          υ   tm       =       -   2            υ   z          (       Ω   rf     ω     )                 iωt   0                  
        then           (   52   )                 ζ   tm     =     i2            υ   z          Ω   rf         ω   2                 iωt   0                 (   53   )                               
 
         [0099]    As the particle entrance time t o  changes, the orbit centroid rotates about the z-axis. When the particle leaves the cavity it is left rotating about its center displacement. FIG. 6 shows the results for four particles entering the cavity at ωt o =0, π, 3π/2 and leaving the cavity at maximum displacement. After passing through the cavity the particle or beam centroid is left gyrating and drifting about the axial field. These results were calculated with a relativistic 3D particle pusher which uses the fields from Equations (41)-(43). The parameters were β z =0.99, B rf =0.2 kG, B z =2 kG, f=0.395 GHz.  
         [0100]    The displacement of the orbit center can be calculated to  
               Δ                 R     =           υ   z          Ω   rf         ω   2       =       (       υ   z       Ω   z       )          [     4          Ω   rf       Ω   z         ]                 (   54   )                               
 
         [0101]    This is the same displacement as given by Eq. (13) for the TM 210  mode when Ω z =2ω is taken into account. Thus this mode does not improve the displacement if β rf  is provided by an external rf source.  
         [0102]    The quantity in brackets in Eq. (54) is selected for our performance evaluation to be a value of 0.7. The reason for this selection is to avoid non-linear effects as discussed later.  
         [0103]    The rf power lost to the input cavity walls for the TM 110  mode is given by  
               P   i     =         R   s     8                (         λ                 c       Z   o            B   rf       )     2          [         2        L   o       a     +   1     ]       .               (   55   )                               
 
         [0104]    Equation (55) is the power lost to a square TM 110  cavity which will be used for the evaluation of the power lost. The Q and fill time for the TM 110  square cavity will also be used.  
         [0105]    Equations (15)-(17), (34), (49), (54) and (55) are now used to evaluate the performance of this device.  
         [0106]    The input parameters are again in Table III and the results in Table V. The output power is very acceptable. The gain has improved by about a factor of two over the TM210 mode case considered. Above 200 kV the gain is very respectable. Using a modest guide field has increased the power output capability by at least a factor of ten as compared to the case without a guide field.  
                                                                         TABLE V                       TM-110 Deflection Cavity                   Stainless Steel            Energy (keV)   50   100   200   500   1000       Fill Time (usec)   0.62   0.74   0.84   0.93   0.96       Power In (W)   1.29E + 05   1.72E + 05   2.60E + 05   5.88E + 05   1.38E + 06       Power Out (W)   2.69E + 05   1.63E + 06   1.05E + 07   1.44E + 08   1.25E + 09       Gain   2.1   9.5   40.4   244.9   904.6       Efficiency   0.338   0.452   0.488   0.498   0.499            Copper            Energy (keV)   50   100   200   500   1000       Fill Time (usec)   3.91   4.63   5.26   5.83   6.06       Power In (W)   2.05E + 04   2.73E + 04   4.13E + 04   9.34E + 04   2.19E + 05       Power Out (W)   2.69E + 05   1.63E + 06   1.05E + 07   1.44E + 08   1.25E + 09       Gain   13   60   254   1540   5690       Efficiency   0.465   0.492   0.498   0.500   0.500                  
 
         [0107]    In solving Equations (44)-(45), it is assumed the axial velocity ν z  to be a constant. This approximation is not valid for a large ratio of Ω rf /Ω z . As Ω rf  increases in Equations (44)-(45), ν z  decreases such that the product Ω rf ν z  saturates at some value of Ω rf . When ν z  decreases it increases the interaction time in the cavity. Then the maximum velocity obtained in Equation (52) for an interaction time of π/ω(t o =0) will be reduced and of course the position that the particle leaves the cavity will be changed.  
         [0108]    This effect can be substantially compensated for by be simply shortening the length of the cavity, thus reducing the interaction time. FIG. 7 a  shows the particle response to the rf field when Ω rf /Ω z =33%. The interaction time is π/ω. As can be seen when the particle leaves the cavity and enters the drift space (circular orbit) there is very little displacement of the guiding center.  
         [0109]    If the interaction length is reduced by 35%, all else being the same, it can be seen from FIG. 7 b  that there is a 250% increase in the guiding center displacement. These results were produced using a 3D relativistic particle pushing code that includes all of the rf field components.  
         [0110]    In the operation of the invention, a collimated beam  20  of electrons traverses input cavity  22  as shown in FIG. 1. The input cavity  22  has present therein a transverse periodic magnetic field e with respect to the direction of the electron beam  20  traversing the input cavity  22 . The periodic transverse magnetic field is, for instance, sinusoidal. Isolating a period of 2π of the periodic transverse magnetic field, initially the transverse magnetic field is at its greatest positive strength causing the electron entering input cavity  22  being imparted with the greatest transverse force. This electron continues through the input cavity  22  since it maintains its momentum with respect to the axial or second direction of the input cavity  22 . The next electron that enters the input cavity  22  immediately after the previous electron experiences a transverse magnetic field that is slightly less than the electron before it that is passing through the input cavity  22 . This slightly decreasing transverse magnetic field is experienced by subsequent electrons for a period of 2π resulting in each subsequent electron through the period having respectively less transverse force implied to them. Consequently, as each electron leaves the input cavity  22  traveling to the bending magnet  24  they vary in their transverse momentum corresponding to the transverse force applied to it. The electron that has the most transverse force imparted to it enters the bending magnet the highest distance from the axis of bending magnet  24 . The next electron which has a slightly less transverse magnetic force imparted to it, enters the bending magnet  24  at a slightly lower height from the axis of the bending magnet, and so forth, until the electron enters the bending magnet  24  at the lowest position relative to the axis of the bending magnet  24 .  
         [0111]    Since the path length that the electron must follow is longer, the greater the height of the electron which enters the bending magnet  24 , accordingly, the electrons with the most negative transverse momentum to them follow the shortest path length. This results in the electrons passing through the input cavity  22  during the phase from 0 to π of the periodic transverse magnetic field essentially leaving the bending magnet  24  at the same time. The electrons that leave the bending magnet  24  approximately the same time are then focused [T. F. Godlove and W. L. Bendel, Rev. Sci. Inst. 36, 909 (1965)] and provided to the output cavity where microwaves are produced from the bunched electrons as well known in the art [D. Reid, Proc. 1988 Linac Conf., 514 (1989) CEBAF Report No. 89-001; G. Faillon, IEEE Trans. Nucl. Sci. NS-32, 2945 (1985)].  
         [0112]    Although the invention has been described in detail in the foregoing embodiments for the purpose of illustration, it is to be understood that such detail is solely for that purpose and that variations can be made therein by those skilled in the art without departing from the spirit and scope of the invention except as it may be described by the following claims.