Abstract:
An ElectroMagnetic-Mechanical Pulser can generate electron pulses at rates up to 50 GHz, energies up to 1 MeV, duty cycles up to 10%, and pulse widths between 100 fs and 10 ps. A modulating Transverse Deflecting Cavity (“TDC”) imposes a transverse modulation on a continuous electron beam, which is then chopped into pulses by an adjustable Chopping Collimating Aperture. Pulse dispersion due to the modulating TDC is minimized by a suppressing section comprising a plurality of additional TDC&#39;s and/or magnetic quadrupoles. In embodiments the suppression section includes a magnetic quadrupole and a TDC followed by four additional magnetic quadrupoles. The TDC&#39;s can be single-cell or triple-cell. A fundamental frequency of at least one TDC can be tuned by literally or virtually adjusting its volume. TDC&#39;s can be filled with vacuum, air, or a dielectric or ferroelectric material. Embodiments are easily switchable between passive, continuous mode and active pulsed mode.

Description:
RELATED APPLICATIONS 
       [0001]    This application claims the benefit of U.S. Provisional Application No. 62/143,667, filed Apr. 6, 2015, which is herein incorporated by reference in its entirety for all purposes. 
     
    
     FIELD OF THE INVENTION 
       [0002]    The invention relates to apparatus and methods for generating electron beams, and more particularly, to apparatus and methods for generating and controlling low and medium energy electron beams at very high rates. 
       BACKGROUND OF THE INVENTION 
       [0003]    Generation and precise control of low and medium energy pulsed electron beams is required for many industrial, medical, and research applications, including scanning electron microscopy (SEM), transmission electron microscopy (TEM), and horizontal/vertical accelerator-based beamlines (HAB/VAB), as well as relevant experimental analytical methods that use electron beams in SEM or TEM, or HAB/VAB as probes. 
         [0004]    In research, pulsed electron beams with ultrashort pulse durations are used for investigating dynamic processes in a variety of materials. Frequently, the electron beams are combined with other primary excitation probes such as laser beams or other photon-based probes such as X-ray beams. An example would be the “pump-probe” class of experiments. 
         [0005]    One approach for generating electron beam pulses of a specific length and charge (i.e. intensity) in a periodic sequence is to create electron pulses directly on the surface of an electron source (cathode) by exciting the electrons using either a laser or heat combined with an external electric field. 
         [0006]    If a laser is used as the excitation method, the sequences of electron pulses are controlled by adjusting the wavelength, power, and/or temporal structure (pulse length and repetition frequency) of the laser photon pulses. For example, if a combination of femtosecond lasers and photocathode electron emitters is used, the electron pulse lengths are strictly determined by the pulse lengths of the fs-laser and the response time of the photocathode. Using this approach, it is possible to routinely obtain pulse lengths as short as 100 femtoseconds (“fs”) or less. 
         [0007]    However, high repetition rates, defined herein as being repetition rates of at least 1 GHz or higher, are simply not available for laser-excited electron beams, because modern lasers are only capable of repetition rates on the order of 100 MHz or less (0.1 GHz or less). 
         [0008]    In addition, it is often important in experimental systems to provide flexible and simple solutions for switching between continuous and pulsed beam modes. If the combination of a photocathode and an fs-laser is used for pulsed beam generation, then the required continuous beam must be generated using a separate thermionic or field emission source. 
         [0009]    On the other hand, if heat combined with an external electric field is used as the excitation method, then the sequences of electron pulses are controlled by the electric field strengths and the temporal structure (pulse length and repetition frequency) of the electric field pulses. 
         [0010]    Still another approach is to generate a continuous electron beam, and then to mechanically or electromagnetically block and unblock (i.e. “chop”) the beam with a desired periodicity, according to the desired electron pulse timing in the beam sequence. Approaches that use deflecting cavity technology for chopping electron beams of tens of kV in the GHz frequency range have been known since the 1970′s. However, these approaches, which typically employ just one single-cell deflecting cavity, are generally limited to pulse lengths of 1 picosecond (“ps”) at best and repetition rates of 1 GHz or less. Furthermore, these approaches are only applicable for generating low energy electron beams having energies of less than 100 kilo-electron Volts (“keV”). Perhaps even more importantly, these approaches typically result in very extensive electron beam quality deterioration in both the transverse direction (beam diameter and divergence) and longitudinal direction (temporal coherence). 
         [0011]    What is needed, therefore, is an apparatus for generating electron beams that can be pulsed at a high duty cycle with pulsing rates greater than 1 GHz and pulse length less than 1ps, and with minimal transverse and longitudinal dispersion. 
       SUMMARY OF THE INVENTION 
       [0012]    A combined ElectroMagnetic-Mechanical Pulser (“EMMP”) is disclosed for generating electron beams that can be pulsed at a high duty cycle with pulsing rates greater than 1 GHz and with minimal transverse and longitudinal dispersion. The EMMP uses a continuous input electron beam derived from any source, a Transverse Deflecting Cavity (“TDC”), an adjustable Chopping Collimating Aperture (“CCA”), and a dispersion suppressing section comprising a plurality of pillbox cavity resonators, cavity resonators, and/or magnetic quadrupoles. 
         [0013]    In embodiments, the number of electrons per pulse and the pulse repetition frequency (repetition rate) are determined by the original continuous electron beam current, the power driving the TDC, the fundamental TDC frequency (i.e. cavity size), and the aperture size. In embodiments, the disclosed EMMP device is easily switchable between passive, continuous mode and active pulsed mode, by making sure that the downstream TDCs and/or quadrupoles do not cause any deterioration of the beam quality in terms of electron coordinate-momentum space (i.e. the phase space defined by the combined locations and momentums of the electrons in the beam). 
         [0014]    Accordingly, the disclosed EMMP produces short electron pulses (in the range of 100 fs to 10 ps) in sequences with pulse repetition rates in the range of 0.1 to 50 GHz and duty cycles up to 10%, and with medium or low energy, defined herein as between 10 keV (kilo-electron-volts) and 1 MeV (mega-electron-volts). The disclosed device is an enabling technology for SEMs, TEMs and HABs/VABs to be operated in a fundamentally different GHz stroboscopic mode. When slicing the continuous electron beam at the input, this device preserves the original beam quality at the output, such that the transverse and longitudinal phase-spaces of the input continuous beam are nearly identical to that of exiting electron pulses. 
         [0015]    Advantages of the present invention include:
       The disclosed device does not require any specialty photocathode.   No specialty laser system or heater is required.   The standard continuous electron beam is sliced into short electron pulses with high repetition rates by only allowing electrons in the continuous beam to pass through the device which have specific temporal and spatial characteristics that match a frequency required by a specific experiment.   The repetition rates of the resulting pulsed electron beam can be adjusted over a range of approximately 0.1-50 GHz, which is not possible using laser-excited beams.   Beam phase space degradation is controlled and minimized.       
 
         [0021]    The advantages of the present invention include, without limitation, that it is versatile, compact, and can be used to form the basis for a number of devices to generate electron pulses with varied length and repetition rate, and excellent phase-space quality. An RF signal empowering the TDC can be used for synchronization of the disclosed EMMP device with other experimental system components, including an object or sample under an experimental study, if needed, or other pumping or probing beams. 
         [0022]    In embodiments, the disclosed system comprises:
       (i) An input through which a continuous, direct current (“dc”) electron beam of low or medium energy axially enters the device.   (ii) A metallic single-cell TDC, referred to herein as the first or “modulating” TDC, operated in any of TE11n or TM11n modes, where n can be any integer, n=0, 1, 2, 3 and higher, and configured to modulate the incoming continuous electron beam transversely, i.e. perpendicularly its line of propagation, into a sinusoid according to a magnetic component (for TM modes) or electric component (for TE modes) of an electromagnetic field introduced into the TDC by an external RF generator at a fundamental frequency of the TDC resonator. Since the magnetic field oscillates with a radial frequency, the modulation force depends on the time at which electrons arrive in the TDC resonator. The amplitude of the resulting sinusoid grows as the modulated beam propagates.   (iii) A chopping collimating aperture (“CCA”) with adjustable diameter located on the longitudinal axis of the electron beam and configured to chop the entering beam and reformat it into pulses. Because the aperture chops electrons on both sides of the sinusoid-modulated beam, the actual repetition rate of the resulting pulses is twice the fundamental frequency of the modulating TDC. The temporal lengths of the resulting electron pulses can be varied between about 100 femto-seconds (“fs) and about 10 pico-seconds (“ps”) by adjusting the aperture diameter and/or the power of the driving RF source.   (iv) A dispersion suppressing section comprising a plurality of magnetic quadrupoles and/or TDCs (identical to the first TDC) positioned a certain distance from the CCA and configured to demodulate the pulses and reduce their emittance (angle divergence) growth, energy spread growth, and spatial lengthening due to drifting.       
 
         [0027]    In embodiments, at least one of the TDC&#39;s is empty or vacuum-filled. In various embodiments, at least one of the TDC&#39;s is at least partially filled with an axially symmetric dielectric having permittivity greater than 1 and operated in any of HEM11n or HEM12n modes, where n is an integer, i.e., n=0, 1, 2, 3 or higher. For these embodiments, the continuous input electron beam is modulated by both electric and magnetic components of the electromagnetic field. 
         [0028]    In still other embodiments at least one of the TDC&#39;s is at least partially filled with an axially symmetric ferroelectric material having permittivity greater than 1, and operated in any of HEM11n or HEM12n modes, where n is an integer, i.e., n=0, 1, 2, 3 and higher. In these embodiments the continuous input electron beam is modulated by both electric and magnetic components of the electromagnetic field. In some of these embodiments, the fundamental frequency of the TDC (hence the repetition rate of the first TDC) can be controlled by changing the permittivity of the ferroelectric layer, either by adjusting its temperature and/or by applying a dc electric potential difference across the ferroelectric material. 
         [0029]    For lower fundamental frequencies, in some of these embodiments where TDC&#39;s are at least partially filled with a dielectric or ferroelectric materials, the size of these TDC&#39;s is the same as the size of comparable vacuum-filled TDC&#39;s, thereby allowing the EMMP device to be equally compact and yet rendered more energy efficient than the comparable empty or vacuum filled TDC&#39;s. 
         [0030]    In some embodiments, the device includes at least one TDC having three cells, so as to remove additional off-axis displacements of the propagating electron beam. Such three-cell TDC&#39;s can be used as tunable TDC&#39;s, in place of single-cell TDC&#39;s used in other embodiments. 
         [0031]    In still other embodiments, at least one TDC has an adjustable volume and can thereby function as a tunable resonator cavity, whereby the fundamental frequency of the TDC is tuned so as to tune the repetition rate of the EMMP device. 
         [0032]    In yet other embodiments, the electron pulse repetition rate can be changed by replacing the first TDC with a replacement TDC having a primary resonance frequency that differs from the first TDC. 
         [0033]    In some embodiments, the dispersion suppressing section comprises a pair of TDC&#39;s, while in similar embodiments the dispersion suppressing section comprises a magnetic quadrupole followed by a TDC. In still other embodiments, the dispersion suppressing section comprises two magnetic quadrupoles with a TDC sandwiched in between, and in yet other embodiments the dispersion suppressing section includes a magnetic quadrupole, a TDC, and at least four additional magnetic quadrupoles following the TDC. In embodiments, all TDC&#39;s included in the EMMP are identical, or at least have the same fundamental frequencies. 
         [0034]    The features and advantages described herein are not all-inclusive and, in particular, many additional features and advantages will be apparent to one of ordinary skill in the art in view of the drawings, specification, and claims. Moreover, it should be noted that the language used in the specification has been principally selected for readability and instructional purposes, and not to limit the scope of the inventive subject matter. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0035]      FIG. 1  is a conceptual diagram that illustrates the fundamental concepts underlying embodiments of the present invention; 
           [0036]      FIG. 2A  is a graphical illustration indicating the functioning and resulting beam dispersion of the first, modulating TDC in an embodiment where the first TDC is a single-cell TDC having a cavity with the magnetic field Hx of TM  110  mode on-axis oriented transversely with respect to the beam propagation direction through the pipes on both ends of the cavity; 
           [0037]      FIG. 2B  is a graphical illustration of the functioning and resulting beam dispersion in an embodiment similar to  FIG. 2A , but where the first TDC is a three-cell TDC; 
           [0038]      FIG. 3  is a block diagram of an embodiment in which the dispersion suppressing section includes two TDC&#39;s; 
           [0039]      FIG. 4  is a block diagram of an embodiment in which the dispersion suppressing section includes a magnetic quadrupole followed by a TDC; 
           [0040]      FIG. 5  is a block diagram of an embodiment in which the dispersion suppressing section includes two magnetic quadrupoles with a TDC sandwiched in between; and 
           [0041]      FIG. 6  is a block diagram of an embodiment in which the dispersion suppressing section includes a magnetic quadrupole and a TDC, followed by four additional magnetic quadrupoles. 
       
    
    
     DETAILED DESCRIPTION 
       [0042]    Referring to  FIG. 1 , a conceptual diagram is shown that illustrates the fundamental concepts underlying embodiments of the present invention. In the illustrated embodiment, an initially continuous, “dc” electron beam  100  is transversely modulated into a sinusoid  110  by a pair of vacuum-filled TDC&#39;s  102 , which are operated at a frequency within a range that extends from below 1 GHz to above 10 GHz. The amplitude of the sinusoid  110  grows as the modulated beam propagates, and then the beam  110  impinges upon a chopping, collimating aperture, or “CCA”  104 , having an opening  106  that is adjustable between 10 and 200 μm. The CCA “chops” the beam into pulses  108  that emerge from the CCA at an ultrahigh repetition rate that is twice the TDC modulation rate, because the pulses  108  are produced by cutting the sinusoid  110  of the beam modulation on both the up-swing and the down-swing. The aperture opening  106  and the modulating field of the TDC tune the pulse lengths to between 100 fs and 10 ps, resulting in duty cycles of the EMMP device of less than or equal to 20%. 
         [0043]    After the beam  100  has been chopped into pulses  108 , both the beam size and the divergence of the stream of pulses  108  will increase. As shown in  FIG. 1 , additional components  112 ,  114  are included in a divergence suppressing section downstream of the CCA  110  that reverses and suppresses this divergence. In the embodiment of  FIG. 1 , the divergence is partially suppressed by two additional components  112 ,  114  which can be two additional TDC&#39;s  112  and  114 , or a magnetic quadrupole  112  followed by an additional TDC  114 . This basic design removes energy spread and significantly reduces transverse-longitudinal correlations (i.e. x-z and y-z correlations) introduced by first TDC  102 . But it does not restore the correct relation between the two transverse spatial components (i.e. x and y). In similar embodiments, as discussed in more detail below, various other combinations of TDC&#39;s and magnetic quadrupoles are utilized to more effectively demodulate the beam and reduce the spatial distortions, the emittance growth, and the energy spread. In embodiments, all TDC&#39;s included in the EMMP are identical, or at least have the same fundamental frequencies. 
         [0044]    At the last step, matching schemes after  114  included in the dispersion suppressing section bring the two transverse (with respect to the optical beam axis z) spatial components into the correct relation with each other. Namely, the two transverse beam components x and y of the pulsed beam  108  are made to be approximately equal, a state which is referred to herein as a “round beam.” Ideally, the continuous input beam  100  is round, but the electron pulses  108  emerging from the CCA  104  are not generally round anymore. The goal of having a matching scheme  114  is to make the pulsed beam  108  round again. 
         [0045]    For example, a vacuum TDC  102 , externally driven by an RF source, operated in TM 110  mode at f 0 =10 GHz (corresponding to a TDC diameter of 39 mm) and a CCA  104  can be used to form ultrahigh repetition rate pulse sequences having a repetition rate of 20 GHz (because pulses are produced by cutting both sides of the 10 GHz sinusoid). At the fixed fundamental TDC frequency of 10 GHz, the pulse length can be continuously changed between 100 fs and 10 ps by varying the CCA diameter and/or RF power in the TDC. The exact range of duty cycle depends on the ratio of the diameters of the TDC (determining f 0 ) and the CCA, and the power fed by the RF source into the TDC. For the TM 110  mode in a pillbox, a general relation between all the parameters involved is described as 
         [0000]    
       
         
           
             
               
                 
                   
                     P 
                     ∝ 
                     
                       B 
                       2 
                     
                   
                   = 
                   
                     
                       r 
                       × 
                       
                         m 
                         e 
                       
                     
                     
                       d 
                       × 
                       e 
                       × 
                       Δ 
                        
                       
                           
                       
                        
                       t 
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where P and B are power and magnetic component of the electromagnetic field in the TDC  102 , respectively; m e  and e are the electron mass and charge, respectively; r is the radius of the CCA  104 ; d is the free-drifting distance between the TDC  102  and the CCA  104 ; and Δt is the electron pulse length. This leads to duty cycles of up to 2×10 −1  (or 20%). 
         [0046]    Note that the TDC technology is downwards compatible to sampling rates (or strobe rates) below 1 GHz by replacing vacuum in the TDC with a high permittivity dielectric. The general relation linking the TDC diameter (D), the fundamental TDC frequency (f 0 ) and the permittivity (ε) is 
         [0000]    
       
         
           
             
               
                 
                   D 
                   ∼ 
                   
                     1 
                     
                       
                         f 
                         0 
                       
                       × 
                       
                         ɛ 
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
         [0000]    With a high permittivity ferroelectric, the TDC can be continuously tunable too in a specific frequency range. 
         [0047]    With reference to  FIGS. 2A, 2B, 3, and 4 , additional magnetic quadrupoles and TDCs and/or TDC design modifications from a single-cell design to a three-cell design can further demodulate the beam  100  and reduce spatial distortions, the emittance growth and the energy spread.  FIG. 2A  illustrates the field distribution  200  within a single cell TDC  102 , and the resulting dispersion  202  of the electron beam  100 .  FIG. 2B  illustrates the field distribution  204  within a three cell TDC  102 , and the resulting dispersion  206  of the electron beam  100 . 
         [0048]      FIG. 3  is a simplified block diagram illustrating the embodiment of  FIG. 1 , whereas  FIG. 4  is a simplified block diagram illustrating a similar embodiment in which the dispersion suppression section includes a magnetic quadrupole  400  in lieu of the second TDC&#39;s  112 . 
         [0049]      FIG. 5  illustrates an embodiment that is similar to  FIG. 4  but includes a second identical magnetic quadrupole  402 . This design is able to restore the ideal relation between the x and y components of the electron pulses  108 , making the pulsed beam  108  round. While  FIG. 5  illustrates a “simplest matching” solution, the alignment of the entire EMMP for this embodiment must be changed every time there is a change to the continuous input beam  100 . If the continuous input beam is fixed, the EMMP alignment remains fixed once its best alignment has been found and set. 
         [0050]      FIG. 6  illustrates an embodiment similar to  FIG. 5  that includes an additional quadrupole triplet  404 ,  406 ,  408  of magnetic quadrupoles. This quadrupole triplet  404 ,  406 ,  408  functions as a matching section that is able to preserve the “roundness” of the pulsed beam  108 , regardless of any variations of the continuous input beam parameters. Hence, while this design of the matching scheme in  FIG. 6  is more complex than for  FIG. 5 , the optimal EMMP alignment, once established and set, remains the same even if the parameters of the input beam  100  are changed. 
         [0051]    The design of specific solutions for removing the post-TDC distortions of the resulting electron pulses can be facilitated through the use of generalized matrix calculations in thin lens approximation. Matrix components depend on the type of the components, allowing the strength of various effects on electron dynamics in the phase space to be crudely predicted and evaluated. The matrix methodology disclosed herein relies on three basic assumptions:
       (1) electron optics elements are approximated as thin lenses;   (2) a single particle/electron is considered; and   (3) only linear matrix transformations are considered.       
 
         [0055]    These three assumptions are intertwined. When combined, they establish the basis for the geometrical optics framework in which the problem is solved. This idealized framework provides a good first-order model for rapid progress in the design, to be followed up with full ray-trace calculations including space charge effects to determine the effects of aberrations and undesired couplings on the electron phase space. An initial and a final state of an electron at input and at the output of the EMMP are linked in the momentum-coordinate phase space via a beam transport matrix as follows: 
         [0000]    
       
         
           
             
               
                 
                   
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                               41 
                             
                           
                           
                             
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                               43 
                             
                           
                           
                             
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                               44 
                             
                           
                           
                             
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                               45 
                             
                           
                           
                             
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                               46 
                             
                           
                         
                         
                           
                             
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                               51 
                             
                           
                           
                             
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                               52 
                             
                           
                           
                             
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                               53 
                             
                           
                           
                             
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                               54 
                             
                           
                           
                             
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                               55 
                             
                           
                           
                             
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                               56 
                             
                           
                         
                         
                           
                             
                               R 
                               61 
                             
                           
                           
                             
                               R 
                               62 
                             
                           
                           
                             
                               R 
                               63 
                             
                           
                           
                             
                               R 
                               64 
                             
                           
                           
                             
                               R 
                               65 
                             
                           
                           
                             
                               R 
                               66 
                             
                           
                         
                       
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                      
                     
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                                   i 
                                 
                               
                               
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                                 0 
                               
                             
                           
                         
                       
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                   3 
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         [0000]    where x is the relative horizontal beam position, x′ is the horizontal divergence, y is the relative vertical beam position, y′ is the vertical divergence, z is the relative longitudinal position or time, and Δp/p 0  is the relative longitudinal P 0  momentum. In Eq. 3, the matrix R(6×6) is called the “transport” matrix. It is a result of multiplication of all matrices describing every single component of an EMMP design, including the drifting matrix, which describes empty gaps/pipes between hardware components. The perfect case is when the matrix R has only diagonal elements, indicating that an electron beam transformation took place, yet cross-correlations, described by off-diagonal elements resulting in pulse size change in transverse and longitudinal directions and energy spread, are absent. 
         [0056]    A number of combinations and designs were analyzed, and the main conclusion was that at least 2 active elements in the divergence suppression section must be present, after the CCA, to minimize the off-diagonal elements in the transport matrix. While the first element is always a TDC  102 , second and third elements are shown as blank squares  112 ,  114  in  FIG. 1 , and should be determined from the matrix analysis. In general, these elements can be combinations of TDC&#39;s and/or magnetic quadrupoles (MQ&#39;s). Two such designs are illustrated in  FIGS. 3 and 4 . Thus, the resulting matrix is R(6×6) for the embodiments of  FIG. 3  and  FIG. 4 , and the R-matrix is the product of 5-fold multiplication of 3 key matrices. 
         [0057]    A free drift beam pipe of length d (empty space between either pair of optical components in the EMMP) [measured in meters] is described by the following matrix 
         [0000]    
       
         
           
             
               
                 
                   
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                   , 
                 
               
               
                 
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                   4 
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         [0000]    where  65   is the Lorentz factor. Its value depends on the electron energy. The magnetic quadrupole with a focal length f [measured in meters] is described as 
         [0000]    
       
         
           
             
               
                 
                   
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         [0058]    The TDC has a matrix 
         [0000]    
       
         
           
             
               
                 
                   
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                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           k 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                         
                           0 
                         
                       
                       
                         
                           k 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ) 
                   
                   , 
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where k is the transverse momentum acquired by an electron in the TDC, measured in reciprocal meters or eV. In what follows, k will be referred to as the “kick.” Resulting transport matrices for the designs sketched in  FIGS. 3 and 4  can be optimized in order to zero as many off-diagonal elements as possible. 
         [0059]    As an example, the following parameters of the continuous input beam can be considered: (1) beam energy (E 0 ) 200 keV; (2) energy spread (ΔE) 0.5 eV; (3) emittance 1.5 nm×rad which is a product of a beam diameter of 10 μm and a divergence angle of 0.15 mrad. For the 3TDC ( FIG. 3 ) design, the R-matrix is 
         [0000]    
       
         
           
             
               
                 
                   ( 
                   
                     
                       
                         
                           1 
                           - 
                           
                             
                               
                                 
                                   d 
                                   1 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       d 
                                       1 
                                     
                                     + 
                                     
                                       d 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 k 
                                 1 
                                 2 
                               
                             
                             
                               γ 
                               2 
                             
                           
                         
                       
                       
                         
                           
                             d 
                             1 
                           
                           + 
                           
                             d 
                             2 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           - 
                           
                             
                               
                                 
                                   d 
                                   1 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       d 
                                       1 
                                     
                                     + 
                                     
                                       d 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 k 
                                 1 
                               
                             
                             
                               γ 
                               2 
                             
                           
                         
                       
                     
                     
                       
                         
                           - 
                           
                             
                               
                                 
                                   d 
                                   1 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       d 
                                       1 
                                     
                                     + 
                                     
                                       d 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 k 
                                 1 
                                 2 
                               
                             
                             
                               
                                 d 
                                 2 
                               
                                
                               
                                 γ 
                                 2 
                               
                             
                           
                         
                       
                       
                         
                           1 
                           + 
                           
                             
                               
                                 
                                   d 
                                   1 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       d 
                                       1 
                                     
                                     + 
                                     
                                       d 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 k 
                                 1 
                                 2 
                               
                             
                             
                               
                                 d 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         d 
                                         1 
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             d 
                                             1 
                                           
                                           + 
                                           
                                             d 
                                             2 
                                           
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       k 
                                       1 
                                       2 
                                     
                                   
                                   - 
                                   
                                     γ 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           - 
                           
                             
                               
                                 
                                   
                                     d 
                                     1 
                                     2 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         d 
                                         1 
                                       
                                       + 
                                       
                                         d 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                                 2 
                               
                                
                               
                                 k 
                                 1 
                                 2 
                               
                             
                             
                               
                                 d 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       d 
                                       1 
                                       2 
                                     
                                      
                                     
                                       k 
                                       1 
                                       2 
                                     
                                   
                                   + 
                                   
                                     
                                       d 
                                       1 
                                     
                                      
                                     
                                       d 
                                       2 
                                     
                                      
                                     
                                       k 
                                       1 
                                       2 
                                     
                                   
                                   - 
                                   
                                     γ 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           
                             d 
                             1 
                           
                           + 
                           
                             d 
                             2 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         
                           - 
                           
                             
                               
                                 
                                   d 
                                   1 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       d 
                                       1 
                                     
                                     + 
                                     
                                       d 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 k 
                                 1 
                               
                             
                             
                               γ 
                               2 
                             
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           1 
                           - 
                           
                             
                               
                                 
                                   d 
                                   1 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       d 
                                       1 
                                     
                                     + 
                                     
                                       d 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 k 
                                 1 
                                 2 
                               
                             
                             
                               γ 
                               2 
                             
                           
                         
                       
                       
                         
                           
                             
                               d 
                               1 
                             
                             + 
                             
                               d 
                               2 
                             
                           
                           
                             γ 
                             2 
                           
                         
                       
                     
                     
                       
                         0 
                       
                       
                         
                           - 
                           
                             
                               
                                 
                                   
                                     d 
                                     1 
                                     2 
                                   
                                    
                                   
                                     ( 
                                     
                                       
                                         d 
                                         1 
                                       
                                       + 
                                       
                                         d 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                                 2 
                               
                                
                               
                                 k 
                                 1 
                                 3 
                               
                             
                             
                               
                                 d 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       d 
                                       1 
                                       2 
                                     
                                      
                                     
                                       k 
                                       1 
                                       2 
                                     
                                   
                                   + 
                                   
                                     
                                       d 
                                       1 
                                     
                                      
                                     
                                       d 
                                       2 
                                     
                                      
                                     
                                       k 
                                       1 
                                       2 
                                     
                                   
                                   - 
                                   
                                     γ 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           - 
                           
                             
                               
                                 
                                   d 
                                   1 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       d 
                                       1 
                                     
                                     + 
                                     
                                       d 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 k 
                                 1 
                                 2 
                               
                             
                             
                               d 
                               2 
                             
                           
                         
                       
                       
                         
                           1 
                           + 
                           
                             
                               
                                 
                                   d 
                                   1 
                                   2 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       d 
                                       1 
                                     
                                     + 
                                     
                                       d 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 k 
                                 1 
                                 2 
                               
                             
                             
                               
                                 d 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       
                                         d 
                                         1 
                                       
                                        
                                       
                                         ( 
                                         
                                           
                                             d 
                                             1 
                                           
                                           + 
                                           
                                             d 
                                             2 
                                           
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       k 
                                       1 
                                       2 
                                     
                                   
                                   - 
                                   
                                     γ 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                   ) 
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where k 2 =(d 1 +d 2 )/d 2  and k 3 =y 2  d 1 k 1 /(y 2 −d 1 (d 1 +d 2 )k 1   2 ) are found optimal for the overall system design, i.e. maximum off-diagonal elements are zeros. 
         [0060]    The TDC+MQ+TDC design ( FIG. 4 ) has a transport matrix 
         [0000]    
       
         
           
             
               
                 
                   ( 
                   
                     
                       
                         
                           - 
                           
                             
                               d 
                               2 
                             
                             
                               d 
                               1 
                             
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           - 
                           
                             
                               
                                 ( 
                                 
                                   
                                     γ 
                                     2 
                                   
                                   - 
                                   
                                     
                                       d 
                                       1 
                                       2 
                                     
                                      
                                     
                                       k 
                                       1 
                                       2 
                                     
                                   
                                 
                                 ) 
                               
                                
                               
                                 ( 
                                 
                                   
                                     d 
                                     1 
                                   
                                   + 
                                   
                                     d 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             
                               
                                 d 
                                 1 
                               
                                
                               
                                 d 
                                 2 
                               
                                
                               
                                 γ 
                                 2 
                               
                             
                           
                         
                       
                       
                         
                           - 
                           
                             
                               d 
                               1 
                             
                             
                               d 
                               2 
                             
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           
                             
                               d 
                               1 
                             
                              
                             
                               
                                 k 
                                 1 
                               
                                
                               
                                 ( 
                                 
                                   
                                     d 
                                     1 
                                   
                                   + 
                                   
                                     d 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                           
                           
                             
                               d 
                               2 
                             
                              
                             
                               γ 
                               2 
                             
                           
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           2 
                           + 
                           
                             
                               d 
                               2 
                             
                             
                               d 
                               1 
                             
                           
                         
                       
                       
                         
                           2 
                            
                           
                             ( 
                             
                               
                                 d 
                                 1 
                               
                               + 
                               
                                 d 
                                 2 
                               
                             
                             ) 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         
                           
                             
                               d 
                               1 
                             
                             + 
                             
                               d 
                               2 
                             
                           
                           
                             
                               d 
                               1 
                             
                              
                             
                               d 
                               2 
                             
                           
                         
                       
                       
                         
                           2 
                           + 
                           
                             
                               d 
                               1 
                             
                             
                               d 
                               2 
                             
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                     
                     
                       
                         
                           
                             
                               k 
                               1 
                             
                              
                             
                               ( 
                               
                                 
                                   d 
                                   1 
                                 
                                 + 
                                 
                                   d 
                                   2 
                                 
                               
                               ) 
                             
                           
                           
                             γ 
                             2 
                           
                         
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                       
                         
                           
                             
                               d 
                               1 
                             
                             + 
                             
                               d 
                               2 
                             
                           
                           
                             γ 
                             2 
                           
                         
                       
                     
                     
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         0 
                       
                       
                         1 
                       
                     
                   
                   ) 
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where d 1  and d 2  are the drift distances between the first TOC  102  and the MQ  400 , and between the MQ  400  and the second TOC  114  respectively, and k 1  is the kick strength of the first TOC  102 . The focal length of the MQ  400  is f=−d 1 d 2 /(d 1 +d 2 ) and the kick strength of the second deflecting cavity is k 2 =d 1 /d 2 k 1 . 
         [0061]    From the matrix (8) describing the TDC+MQ+TDC case, it can be seen that two block sub-matrices for the two transverse beam components x and y are different (namely, R 11 , R 12 , R 21 , R 22  which are related to x and R 33 , R 34 , R 43 , R 44  which are related to y). To make the beam round, one needs to make R 11 =R 33 , R 12 =R 34 , R 21 =R 43 , R 22 =R 44 . This is performed through additional MQ&#39;s (1 or 4) after the second TDC  114 . Once these conditions are satisfied, x and y are equal at the output, meaning the pulsed beam is round (assuming that the continuous input beam is round). 
         [0062]    From the matrices presented in (7) and (8) above, it can be seen that in transverse directions both designs ( FIG. 3  and  FIG. 4 ) may lead to satisfactory results, such that sufficient spatial coherence in the beam is conserved upon EMMP installation. The main problem here is the matrix term R 65  which is responsible for energy spread growth. In this idealized geometrical optics framework, R 65  is zero for the TDC+MQ+TDC design of  FIG. 4 , but is finite for the 3 TDC design of  FIG. 3 . That is, in the 3 TDC case, the additional energy spread at E 0 =200 keV is higher than 1 eV on top of the default/intrinsic energy spread of 0.5 eV. 
         [0063]    The foregoing description of the embodiments of the invention has been presented for the purposes of illustration and description. Each and every page of this submission, and all contents thereon, however characterized, identified, or numbered, is considered a substantive part of this application for all purposes, irrespective of form or placement within the application. 
         [0064]    This specification is not intended to be exhaustive. Although the present application is shown in a limited number of forms, the scope of the invention is not limited to just these forms, but is amenable to various changes and modifications without departing from the spirit thereof. One or ordinary skill in the art should appreciate after learning the teachings related to the claimed subject matter contained in the foregoing description that many modifications and variations are possible in light of this disclosure. Accordingly, the claimed subject matter includes any combination of the above-described elements in all possible variations thereof, unless otherwise indicated herein or otherwise clearly contradicted by context. In particular, the limitations presented in dependent claims below can be combined with their corresponding independent claims in any number and in any order without departing from the scope of this disclosure, unless the dependent claims are logically incompatible with each other.