Ivanov image quality by 2IY3h22

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									       2.13. The image quality evaluation

        The final goal for the design of any specific electron optical system (EOS)
forming the images of the objects is the determination of the resolution over the image
area, the modulation-transfer-function (MTF) of the optical tract and the other
parameters, which give the objective estimation for the image quality.
    The worsening of the image clearness can be conditioned by the following main
causes:
    1) The existence of non linear effects (aberrations) in the image transformation by
        electron optical tract as a result of the difference of optical length for the
        trajectories started from different points of the object to the screen cross-section
        by these trajectories;
    2) Normally, minimal difference for optical length is achieved for curvilinear
        surfaces of the cathode and screen. The image transport from a flat surface to
        curvilinear one without substantial geometry aberrations can be done with using
        the fiber optics, but this optics has limited resolution because of small but finite
        diameter of single optical waveguide;
    3) The electrons coming to the screen surface exite the molecules of special coating,
        which gives up the excitation energy as the light emission in optical range.
        However a portion of secondary electrons comes out in this process. These
        electrons cause of parasitic screen illumination, which worses the image contrast;
    4) In manufacturing and adjustment of separate parts of the device the technology
        deviations appear inevitably from the ideal sample with minimal aberrations.
        These may be the deviations of the part dimensions, shift or tilt of the axes,
        elliptic and other deformations of the electrodes. Another kind of deviation is the
        lack of coincidence for the screen position with the surface of the best focusing.
        Further we will give the full analysis of the influence for all mentioned causes to
the quality of the forming image. The image perception is determined also by the features
of human eye, which is an optical system with its specific parameters. In order to present
this process in more details we will describe the mathematical model of the image
forming. The object image is characterized by the 1st-order parameters (magnification
factor, Gauss plane and cross-over positions, angle of the image rotation) called cardinal
elements of Gauss optics, and by high-order aberrations, which classified onto geometry,
chromatic, time-of-flight and combined aberrations. As they play different roles in
different devices, it makes sense to mark out 3 independent sets in describing the
estimation methodics for the image quality: physical temporal resolution, spatial
parameters and spatial-temporal parameters, determine the technical temporal resolution.
We will follow the methodics presented in the publication by Yu. Kulikov [255].

       2.13.1. Computation of the transfer function and physical temporal
       resolution.

        The integral transfer function Wtp is a temporal signal, which is formed in the
image space by the electron optical system as a response for the action of the unit signal
of infinitesimal duration (time point), forming by a small area of the emitter (Figure 2.2).




                                                                                        105
The transfer function depends on the coordinates of the emitting area and on the
coordinates of the image receiving surface.




       Fig. 2.2. The apparatus funstion Wtp, t0 – start moment for emission impulse
   signal,  0 - time-of-flight for the reference particle from the emittion point to the
   image plane z  zi ,  dist - additional time-of-flight for the reference particle to the
   plane z  zi ,  - additional time-of-flight for the arbitrary particle in compare with
   reference one, which is determined by the initial energy spread;  i ,min , i ,max - the time
   moments of appearance and disappearance of the signal in the image plane,  p -
   most probable value for the time-of-flight spread of the particles,   - width of the
   transfer function.

       In axi-semmetric case with one main trajectory (axis z) the transfer function
depends on two parameters – radius of the initial point r0 and the position of the image
plane zi . Using the aberration expansion for the time-of-flight
          0  A2  0n  A11 t  A13  0t cos 0  A22 0n  A33r02 ,
                  *         *        *                 *         *
                                                                                        (2.367)
we get the results
         (r0 , zi )    0  dist  A2  0n  A13  0t cos 0  A22 0n
                                          *         *                   *
                                                                                 (2.368)
and
         dist  A33 0 ,
                  * 2
                              0n   0 cos 0 ,       0t   0 sin0 .          (2.369)
       Here the asterisk symbol marks the aberration coefficients related to the small
parameter set, which includes the normal and tangential components of the initial energy
components on the emitter surface, in contrast to the axial and radial components
correspond to the coefficients without asteriks.




                                                                                            106
      Fig. 2.3. To the start model of the particle: v - initial velocity, v * - its
      projection onto the plane XY; n - normal vector,  - tangential vector;  -
      angle between the vector projection and the axis X;  - angle between the
      projection radius-vector and the axis X;  - angle between the vector of initial
      velocity and the axis Z;  - angle between the normal and velocity vectors;
       - angle between the projection of velocity vector and tangential vector.

        We will be limited by some set of the transfer functions computed along the
reference trajectories, and we will describe how to compute these trajectories. Let D0 -
diameter of entrance aperture, and N – given number of the reference trajectories. Then
the coordinates of the emitting points are evaluated using the formula
                                                            D0
                r0i  ihr 0 , (i  0,..., N  1), hr 0            .               (2.370)
                                                         2( N  1)
        Further we will construct the reference trajectories by using the aberration
expansion. The trouble is that the aberration expansions are not correct near the emitter
surface because of singularity, which produces the surface-layer solutions. Here one
needs some artificial technique to construct the smooth enough trajectories. This
technique can consist of the following steps.
        If the aberration expansion r ( z ) is known, then the reference trajectory for the
particle emitted from the point with coordinates r0 , z0 can be represented as
r ( z, r0 )  r  z  f ( z, z0 )  z  , where f ( z , z0 ) - surface-layer function, which decreases fast
for z  z0 . Let us write, for example
                  f ( z, z0 )  exp  (1  z / z0 )  ,   z  z0 ,   1,                        (2.371)
then we have




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                                                                                                       r02
                  r ( z, r0 )  r0 w  z  f ( z, z0 )  z   r03 E  z  f ( z, z0 )  z  , z0         ,   (2.372)
                                                                                                      2Rc
where E - the distortion coefficient, and Rc - cathode radius.
       Then we make the variable discretization for the initial energy  and initial
angles ,  of the particle
                i   min  h (i  1), i  1,..., imax ,
                   j  min  h ( j  1),          j  1,..., jmax ,                                         (2.373)
                  k  min  h (k  1), k  1,..., kmax ,
which are the arguments of the correspondent given distribution function for the energies
W ( i ) and initial angles W ( j ), W (k ).
       By moving sequentially the values i, j , k and computing the  using the
formula (2.368), we determine the scope of the transfer function  min ,  max . The we
define the quantity of discretization intervals for this function mmax , and evaluate the
subintervals and the centrum positions for these subintervals
                           min
                h  max                     ,  m   min  h (m  1), m  1,..., mmax 1. (2.374)
                              mmax
       By moving sequentially the values i, j , k , we can evaluate the function
                                            hh h
               Wtp ( m , r0 n , zi )      W ( i )W ( j )W (k ) sin k ,           (2.375)
                                               h i, j,k 
where the set i, j , k  is that all values
                                                   h                h 
                 ( i ,  j , k  )    m   ,  m    .                           (2.376)
                                                    2                2
       Finally we evaluate the transfer function
                                               Wtp ( m , r0 n , zi )
               Wtp ( m , r0 n , zi )                                  .                     (2.377)
                                            h Wtp ( m , r0 n , zi )
                                                m
         The resolution of the optical device is quantitative expression for the spectator
ability to recognize two signals close one to other in time or in space as an image of two
different initial objects. In that way, this parameter is determined not only with the
properties of optical tract transforming the image, but also with the properties of the
perceptive person.
         Physical temporal resolution for the pulse test  ppr (pulse physical resolution) is
defined as the root  of the equation
                R ( )  Q pr ( )  0,                                                                        (2.378)
where
          max Ws ( ,  )  min Ws ( ,  )
R( )                                        , Ws ( ,  )  Wtp ( )  Wtp (   ).                      (2.379)
          max Ws ( ,  )  min Ws ( ,  )




                                                                                                                  108
       Fig. 2.4. To the definition of physical temporal resolution.

        The threshold contrast function Q pr ( ) defines the properties of the perceptive
person. In simplest case it is defined by the constant equals to the value 0.05.
        In addition to the pulse test signal one can consider the sinusoidal modulated
signal defined by the formula
                                   1  cos(2 Nt  )
               Wtm,0 ( Nt ,  )                     ,                             (2.380)
                                           2
where N t - modulation frequency of the signal.
        The image of temporal sinusoidal test in the plane z  zi is given by the formula
                           dist

               Wtm,i        
                           dist
                                       Wtm,0 ( Nt ,  ) Wtp (   )d .            (2.381)

        Now one can evaluate the temporal MTF (modulation transfer function)
                                         max Wtm,i  min Wtm,i
                Rt ( Nt , r0n , zi )                          .                     (2.382)
                                         max Wtm,i  min Wtm,i
        Physical temporal resolution
                                 1
                 mpr                                                              (2.383)
                              N t lim
is defined by the limit frequency N t lim , which corresponds to the threshold contrast Qmr .
This frequency is defined as a root of the equation
                Rt ( N t , r0 n , zi )  Qmr ( N t lim )  0.                        (2.384)




                                                                                         109
        2.13.3. Spatial-temporal parameters and technical temporal resolution

       Test-object in general case is a line, oriented at the angle of  0 to the axis ОХ,
which pass through the point М. The line image is dissected onto the image-reciever
surface with a speed vsc in the direction  x (Figure 2.5). The transfer dispersion
function for the spatial-temporal line Wtsl (x , r0 n ,  0 n ,  0 , zi , vsc ) is evaluated using the
same procedure as for the function Wsl . One need mark, when the line has a finite width
l , the line-dispersion function Wtsw is evaluated by integration
                                                                l / 2
                 Wtsw (x , r0 n , 0 n ,  0 , zi , vsc )      
                                                                l / 2
                                                                          Wtsl (xl   )d .   (2.449)

        Technical temporal resolution is defined by a formula
                              1
                tch                 .                                                         (2.450)
                       N tsi ,max vsc
        In MTF computation the formula is used for the value  x in general case
                           w( zs )  0 p
                 x                     sin 20   sin 0  cos 0 m   cos 0 ,        (2.451)
                                                             
                                 0
where
                       vsc A2 ( zs )
                                   0
                                       .                                                        (2.452)
                          0 p w( zs )

       Modulation Transfer Functions for the entrance fiber optics and luminescent
screen are shown in Figure 2.11.




        Fig. 2.11. MTF for the fiber optic plate -1, and for the screen – 2.




                                                                                                    110
       2.13. Micro-channel amplifiers

       2.13.1. Review of the preliminary investigations

        The quality of the image transfer by electron optical converter (EOC) is
characterized by the main parameters like the device sensitivity, luminosity magnification
factor, noisy level, geometry and chromatic aberrations, resolution of the device. For
complex many-electrode devices the aberrarions of electron optical tract can be reduced
greatly, so the image resolution is determined mostly by the MTF of the entrance fiber
optics and by the screen features. For the other equal conditions the minimal distortion of
the image can be obtained easier for more long devices then for short ones. Longer
system permits to reach bigger values of the luminosity magnification factor due to
increasing of the accelerating potentials on the electrodes. In some cases considerable
magnification factor can be reached easier in multi-cascade EOC, but this case the
resolution is decreasing, because of the MTF of single cascades are multiplied.
        In principle new features of the devices, which were called 3rd –generation EOC,
were obtained by using the micro-channell amplifiers (MCA), or signal amplifiers with
micro-channell plate (MCP). First examples of these devices appeared the early 60th, but
imperfect technology of their production and scantily explored the physical phenomena
in cascade amplification of electron flow in the micro channel permit to create the
methodics of their design in 10 years, which can give the satisfactory agreement with
experimental data.
        We do not make the aim to reflect the completeness of historical review of
publications for the early stage of investigation, which is provided in the papers [392]-
[393]. We point only that the most important yield to the theory implementation for the
channel amplification was done by Linder [394], Frant [395] and Gest [396]. For the
works devoted to the implementation of numerical models for MCA design we should
emphasise the papers by Eudokimov and coauthors [397], [398], where the computation
of the noisy parameters of MCA is doing with the Monte-Carlo method.
        In compare with the vacuum devices, the MCAs have the following advantages:
the signal amplification factor of a few millions can be reached for the MCP width of a
few millimeters. Since the radius of micro channels is a hundredth part of a millimeter,
then the geometry aberrations are practically absent in amplification. Main MCA
disadvantage is a noisy factor, which is a factor of 3 more than that factor for the vacuum
EOS.
        As the using of Monte-Carlo method for MCA simulation leads to the significant
spending of the computer run time, we use the mathematical methods based on the
transformations for the current-density functions [399].

       2.13.2. The model of micro-channel amplifier

       The methodics suggested by Yu. Kulikov is putted into the basis of the MCA
model described below. This methodics was realized the first time by Chestnov [400].
That model examined the amplification process in the inner area of MCP only. In our
case we will examine the 3rd-generation EOC as a whole, which is shown in Figure 2.12.
The device consists of the photo-emitter, domain I with uniform field of strength E1 , ion-



                                                                                       111
barrier film, domain II of micro channel plate with a field E2 in the channell, domain III
with a field strength E3 and the screen.




                   Рис. 2.12. Schematic picture of the 3rd-generation EOS.

        Neglecting the fringe fields we can consider that the electron trajectories are
parabolas in each of these 3 domains. The algorithm of taking into account the fringe
fields for the intermediate zones will be described separately. All values at the entrance
of any domain will be denotes by subindex with the number of that domain, but the
values at the exit of the domain have same index and a prime symbol. Figure 2.13 shows
the coordinate system and used designations for the 1st domain.
        The elementary current of photo-emitter can be represented by the expressions
                 dI 01  I 01xy ( x01 , y01 ) K01 ( 01 ,01 , 01 )d 01d01d01dx01dy01 ,
                                                                                             (2.453)
                 K01 ( 01 ,01 ,  01 )   ( 01 ) (01 ) ( 01 )sin 01 ,
but the distribution function for the angles and energies are
                                                           2 01
                               01   01 
                 ( 01 )  1   , A    ( 01 )d 01 ,                                (2.454)
                               A   01                    0
                                                /2
                             cosn 01
                 (01 )                B     cos       01d01 ,
                                                        n
                                      ,                                                    (2.455)
                                B               0

                              1
                 ( 01 )     .                                                          (2.456)
                             2




                                                                                               112
         Fig. 2.13. Coordinate system for the gap photoemitter-MCP.

       Current distribution on the photoemitter surface for the sinusoidal mira                    is given
by formula
                              1
                  j ( x01 )  j0 1  cos(2 N01 x01 )  .                                         (2.457)
                              2
       After the variable discretization the line-dispersion function x  x01                       can be
represented by a sum
                             hhh
        x ( x01  x01 )       (n h ) (n h ) (n h ) sin(n h ),
                                                                                                  (2.458)
                               hx n ,n ,n
where hk - discretization steps,but nk - quantity of intervals for the variable discretization.
       In order to evaluate the dispersion function we should know the trajectory
coordinates in the plane of ion-barrier film. These coordinates for the uniform field are
given by formulas
                                              2              E                                   
                r ( z )  r01ei (01  01 )   01 sin 01  1 ( z  z01 )  cos 2 01  cos 01  , (2.459)
                                              E1              01                                
                                              2              E                                   
                r ( z )  r01ei (01  01 )   01 sin 01  1 ( z  z01 )  cos 2 01  cos 01  , (2.460)
                                              E1              01                                
       Current-density distribution in the image of sinusoidal mira is
                                 
                   j( x01 ) 
                                
                                 
                                      j ( x01 )x (x01  x01 )dx01.
                                                                                                  (2.461)

         In order to evaluate the MTF we use the correlation
                             j j        j (0)  j (2a)        1
                 R( N 01 )  max min                     , a        .                            (2.462)
                             jmax  jmin j (0)  j (2a)      4 N 01



                                                                                                        113
                                                  *                                     *
       The resolution is defined by the value N 01 , for which the contrast value R ( N 01 )
becomes equal to the threshold level of perception vy the human eye, normally is 5%.
       Second computational domain consists of the ion-barrier film as an emitter of
primary particles to the cylindrical domain of the micro channel of radius R0 , and length
L2 .




        Fig. 2.14. Sketch of the gap “ion-barrier film-MCP”.

         There is a spray area at the end of micro channell, where secondary-emission
ability is much less than for the channel surface. The necessity of this area is conditioned
on the exponential increase of the secondary current along the channel axis, so the exit
area plays a role of point with maximal luminocity, which emits in the large angle range.
This leads to the decreasing of the image contrast on the screen. In the existence of the
spray area the position of maximal current density is deepen relatively to the exit hole,
and the polar pattern of the emission is more favorable to create the high-quality image.
The sketch of second domain is presentedin Figure 2.14, but the coordinate system is
shown in Figure 2.15.
         The elementary current from the ion-barrier film is given by an expression
                dI 02  I 02r (r02 ) K 02 ( 02 ,02 ,  02 )d  02 d02 d 02 r02 dr02 , (2.463)




                                                                                             114
        Fig. 2.15. Coordinate system is used in the micro channel area.

But the elementary current coming to the channel surface is
                dI 02  I 02z 0 ( z02 ) K 02 ( 02 ,  ,  02 )d  02 d  d  02 r02 dr02 .
                                                  02
                                                                         02
                                                                                            (2.464)
        The current per unit length of the channel
                dI 02  I 02z 0 ( z02  z02 )
                                                                                             (2.465)
creates the secondary particles emitted by the channel wall. The surface part subjected to
the bombarding by the primary electrons is the 1st cascade of amplification, which is
characterized by the current distribution
                                     W (z  z )
                 z1 ( z12  z02 )  z1 12 02                                                  (2.466)
                                              k1
and by the amplification ratio
                         L1  L2

                  k1      
                           L1
                                   Wz1 ( z12  z02 )dz12 .                                        (2.467)

         There is simple correlation between the film current and the 1st cascade current -
I12  k1 I 02 , but the current coming to the surface of 2nd cascade is I12  I12 . The           
                                       nd
elementary current before the 2 cascade is
                   dI12  I12z1 ( z12  z02 ) K12 (12 , 12 ,  12 )d 12 d 12 d  12 dz12 .
                                                                                        (2.468)
                                                                                                nd
         The current-distribution function on the surface before the 2 cascade is
described by a formula
                                            L1  L2
                    
                  z1 ( z12  z02 )          
                                              L1
                                                                        
                                                      z1 ( z12  z02 )z ( z12  z12 )dz12 ,     (2.469)

but the current-distribution function on the surface after the 2nd cascade is



                                                                                                      115
         Fig. 2.16. The current-density distribution from the elementary ring
         emitter.

                    dI 22  I 22z 2 ( z22  z02 ).                                                             (2.470)
         As the result we obtain the recurrent correlations
         I n 2  I 02 k1k2 ...kn ,                                                                              (2.471)
         zn ( zn 2  z02 )  
                                   z1   ( z12  z02 )z ( z12  z12 )
                                                                          z ( zn2  zn2 )dz12
                                                                                                     dzn2 ,   (2.472)
         Wzn ( zn 2  z02   )      z1   ( z12  z02 )z ( z22  z12 )   z ( zn 2  zn1,2 )dz12     dzn1,2 ,
                                                                                                                (2.473)
               Wzn
         zn       , kn   Wzn ( zn 2  z02 )dzn 2 .                                                          (2.474)
                kn
         Total elementary current is described by a sum
                 dI   dI 02  dI12   dI n 2,                                                                (2.475)
where
                dI n 2  I n 2zn ( zn 2  z02 ) K n 2 ( n 2 , n 2 ,  n 2 )d  n 2 d n 2 d  n 2 dzn 2 . (2.476)
        Total current from the channel surface is
                I   I 02 k ,                                                                              (2.477)
but a total amplification factor of MCP is given by a formula
                k  k1  k1k2   k1k2 kn .                                                                 (2.478)
        Total distribution function for the current density is described using partial
distribution functions for the cascades and their amplification factors
                        k   k k    k1k2 kn zn
                 z   1 z1 1 2 z 2                                         .                               (2.479)
                                              k
        The effective amplification factor of MCP is defined by a formula




                                                                                                                     116
                                               
                keff  k 1    z ( z )dz  .                                   (2.480)
                            L L                
                                1   2           
       We will use the formula for the particle trajectory in the uniform field to evaluate
the coordinate z of the cross-section the channel surface by the trajectory
                                                                                  2
                          E2          R 2  r 2 sin 2 (   )  r cos(   )  
         
        z02  z02 
                    4 02 sin 2 02                                            
                                        0     02         02  02    02    02  02
                                                                                    (2.481)
                  R 2  r 2 sin 2 (   )  r cos(   )  ctg .
                  0 02                 02    02    02             02    02
                                                                             02

        The distribution function of the current density after discretization is
                                              hh hh
         z0 ( z02  z02 )   z0 (nz hz )  r        r sin(n h ),
                                                                                                 (2.482)
                                                 hz    hr , h , h , h

                                    hr h h h
        Wz1 ( z1  z02 )  Wz1 (nz hz ) 
                                         hz
                                                        (02 ,  02 )   r sin(n h ),
                                                hr , h , h , h
                                                                                                (2.483)

where we use the specific distribution functions for the initial angles and energy
                              1
               (n h )        ,                                                               (2.484)
                             2
               (n h )  cos(n h ),                                                          (2.485)
                             n h exp(n h /  02 )
               (n h )  4 02                             ,                                  (2.486)
                                 
                                 0
                                     exp( 02 /  02 )d  02

                                   2
                   r (nr hr )  2 .                                                     (2.487)
                                  R0
         Secondary emission ratio  in general case consists of three components
                         ,                                                        (2.488)
where  - emission ratio for the true secondary electrons,  - ratio of non elastic and  -
elastic reflection of electrons.
         According the known data [400], the coefficients  and  normally are less than
unit, so one can neglect with the reflection phenomena mostly, but   1 10. One can
use the Guest’s formula for true secondary electrons [396]
                                             
                                                                             
          ( ,  )   max         cos   exp  (1  cos  )          cos    , (2.489)
                              max                                  max       
where  max - maximal value for the secondary emission ration in normal fall the electrons
on the surface,  max - correspondent value for the collision energy,  - angle between ve
velocity vector of primary electron and the normal vector on the surface,  - absorbing
ratio of incoming electrons by the wall,  - parameter of the emission model.
         From the experimental data [401] the absorbing ratio was equal   0.62, but we
use an approximation for the parameter 




                                                                                                     117
                     0.55    0.65,    max ,
                     
                                                                                     (2.490)
                     
                          0.25,        max .
       At the point where the electron crosses the surface, it has the energy
                02   02  E2 ( z02  z02 )
                                                                                      (2.491)
and the hade
                              02              2
                                             r02
                cos            sin 02 1  2 sin 2 ( 02  02 ).                    (2.492)
                     01
                               
                              02            R0
        The computation of the line-dispersion function and MTF for the MCP area is
similar to the computations for the 1st domain we described before. At the exit from the
channel the electrons come to the gap MCP-screen with uniform field. The MTF for each
domain are multiplied, and the total resolution of the device is doing by using the total
MTF.




            Fig. 2.17. Current-density modulation by the micro-channell walls
            for a sinusoidal mira.

       In computation of the line-dispersion function for MCP one need to take into
account that the micro channels of radius R0 are situated at the distance D one to other,
so the current-density distribution for sinusoidal mira with taking into account the
channel walls has a shape shown in Figure 2.17. It is defined by the formulas
                           1
                j ( x02 )  F ( x02 ) 1  cos(2 N02 x02 )  ,
                           2
                             j0 , (2n  1) R0  2 D  x02  (2n  1) R0  2 D,           (2.493)
                            
                F ( x02 )  
                            0, (2n  1) R0  2 D  x02  (2n  1) R0  4 D, n  1, 2,...
                            




                                                                                            118
       2.13.3. MCP model with fringe effects

        In taking into account the fringe effects one can not suppose that the field strength
in the channel E2 ( z ) is a constant, which changes stepwise to the constant E3 for the gap
MCP-screen. Since the field at the channel end and in the domain III is a complex shape
function, which is computed numerically, the expression for the trajectories can be
integrated numerically with using the aberration theory.




               Fig. 2.18. The profile of the micro-channell end.

        The geometry of the micro-channell end is shown in Figure 2.18. That domain is
divided onto 5 zones. In the 1st zone the field strength E2 is constant, but the emission
properties of the channel are characterized by the constants  max ,  max ,  n . Second zone
is beginning at the point z b . It has the length h , radius of channell R0  , where  - the
thickness of spay layer. The emission properties of this zone are characterized by the
constants  max ,  max ,  max . Starting the coordinate z c the channel has a cone expansion
with the angle  , which is made by the etching method. Fourth zone begins at the
channel end, which is smoothly passes at z d to the domain of the uniform field with
strength E3 .
        General expression for arbitrary trajectory emitted in zi by the channel wall is
                                                       sin 2 
       r ( z, zi )   sin  ei v( z, zi )  r0ei             w( z, zi ).        (2.494)
                                                       ( zi ) 
       The paraxial trajectories v and w are given by the expressions



                                                                                          119
                                      z  zi                         1
                    v( z, zi )  2           ,   v( z, zi )                  ,
                                        E2                       E2 ( z  zi )                           (2.495)
                    w( z, zi )  1,    w( z, zi )  0
         st
in the 1 zone. At the point z a the passage is doing, and the trajectories in 2nd zone are
               v( z, zi )  v( za , zi )v ( z, za )  v( za , zi ) w( z, za ),
                                                                                      (2.496)
               w( z, zi )  w( za , zi )v ( z, za )  w( za , zi )w( z, za ),
where the auxiliary trajectories v , w are computed with using the formulas
                                                                          2
                                                                                 1  
                                                                 2 1   
          v ( z , za )  ( z  za ) 1  a
                                           ( z  za )  ( z  za )       a
                                                                              
                                                                                      a
                                                                                        
                                     a
                                                                   8   a  4 Ra  a 
                                                                                       
                                                                                                         (2.497)
                             1                           5     
                                                          2            3
                                       1                              
                ( z  za )3      a
                                                 
                                                      a
                                                          
                                                                   a
                                                                       ,
                             24  a 4 Ra
                                                   a  64   a    
                                                         3  3    2 
          v( z, za )  1    a
                                 ( z  za )  ( z  za )  
                                                        2           a
                                                                        a  
                            2 a                            4 Ra  a 8   a  
                                                                               
                                                                                                         (2.498)
                              1  1   2 5   3       
                ( z  za )  
                          3         a
                                         a    a  ,
                              6  a Ra   a  16   a   
                                                                
                                                      1  1   2 
                            2 1 
w( z, za )  1  ( z  za )          a
                                        ( z  za )  
                                                   3         a
                                                                   
                                                                       a
                                                                          
                              4 Ra  a                 24  a 8Ra   a  
                                                                          
                        1           1             3   
                                                                     2
                                                                         5   
                                                                                        3
                                                                                            5     
      ( z  za )  
                3                a
                                      3        a
                                                          2 
                                                                 a
                                                                             
                                                                                    a
                                                                                             
                                                                                                     a a
                                                                                                          ,
                   12 Ra  a 4 Ra  a 32 Ra   a  64 Ra   a  192   a  
                                                                                                       2
                                                                                                         
                                                                                                            (2.499)
                            1                       1  3         2

w( z, za )  ( z  za )           a
                                       ( z  za ) 2       a
                                                                      
                                                                          a
                                                                             
                         8Ra  a                        8  a 8Ra   a  
                                                                              
                                                                                                            (2.500)
                   1  1                       3   
                                                               2
                                                                       5   
                                                                                  3
                                                                                        5     
      ( z  za )3            a
                                    3 a 2 a                         
                                                                              a
                                                                                   
                                                                                             a a
                                                                                                     ,
                   3Ra  a Ra  a 8Ra   a  16 Ra   a  48   a  
                                                                                               2
                                                                                                     
where
                              
                    Ra  2 a ,  a   ( za )   ( zi ).                                                   (2.501)
                              a

         These trajectories are satisfied to the initial data
                    v ( za )  0, v( za )  1, w( za )  1, w( za )  0.                                  (2.502)
        Computing the values v ( zb ), v( zb ), w( zb ), w( zb ), one need to make passage to the
algorithm of the trajectory evaluation in 3rd zone. Here the trajectories are evaluated using
the formulas


                                                                                                              120
                                  z  zi
                             z  zi
                                                            11      
         v ( z , zi )  2        1          ( z  zi ) 2          i 
                              i
                                       2 Ri                 40 Ri  
                                                                   2

                                                                                                     (2.503)
                             101         17  
                                                  i 
                ( z  zi )3                         ,  i   ( zi ),
                             560 Ri 420 Ri   
                                    3
                                                      
                                1         
                                              3                             11  
         v( z , zi )                    1    ( z  zi )  ( z  z i ) 2  2  i  
                             ( z  zi )  2 Ri
                             i                                              8Ri 3 
                                                                                                 (2.504)
                             101        17  
                                                  i 
                ( z  zi )3                        ,
                             80 Ri 60 Ri   
                                   3
                                                     
                            z  zi                1                 1         1 
         w( z, zi )  1            ( z  zi )2  2  i   ( z  zi )3                 i
                                                                                              , (2.505)
                              Ri                  2 Ri 12             10 Ri 60 Ri  
                                                                                3


                           1                R  1                     3  
         w( z, zi )   1  ( z  zi )  i i    ( z  zi ) 2  i              2 
                                                                                         .       (2.506)
                          Ri                6  Ri             20 10 Ri      
        The parameter evaluation for 3rd zone are doing with using the formulas
                                                                                  cos 
        r0  R0    ( L2  zi )  , cos   sin  sin  , cos                       ,              (2.507)
                                                                                   sin 
then the passage to 4th zone is doing at z c , and the passage to 5th zone is doing in z d .
        Axial values for the potential and its derivatives for 2nd -4th zones are evaluated
using a formula
        ( k ) ( z)  E2 (L2  h  2d )1(k ) ( z)  U2 2k ) ( z)   E3 (L2  2d )  U2  3k ) ( z), (2.508)
                                                         (                                   (


where  j ( z ) - unit functions, shown in Figure 2.19, which are the solutions of the
specific boundary problems for each zone.
        In the existence of the spray area, in addition to the main function of the ring
source W1 , the ring function for the spray area is evaluated also, which depends
вычисляется также функция кольцевого источника зоны подпыления, зависящая от
on the constants  max ,  max ,  max . In addition to the before introduced criterion for the
electron run out from the channell r  r0 in z  L2 , the criterion r  R0   is adding
in z  L2  h , where r0  R0   h   . In particular cases the spray area or etching area
can be absent. Then correspondent parameters h,  or  should be equal zero.
        In addition to the mentioned MCA parameters it is reasonable to evaluate the
distribution functions for the energies
                  hh hh
        WU (U )  z     W ( zi )  sin                                     (2.509)
                     hU     hz , h , h , h

and the angles
                      hz h h h
         W ( ) 
                          h sin 
                                           
                                      hz , h , h , h
                                                          W ( zi )  sin                       (2.510)




                                                                                                          121
               Fig. 2.19. Unit functions for the separate zones.

at the channel exit, and it is reasonable to divide the whole energy interval onto 3
subintervals 0 10V , 10 100V and 100 1000V with individual scanning step hU for
each interval to increase the accuracy of computations.

       2.13.4. The results of numerical modeling

        The transfer function W1 for the initial data defined by default and 10
discretization intervals for each variable is shown in Figure 2.20, but the current-
distribution function for the amplification cascades is presented in Figure 2.21. Total
distribution of secondary current over the channel surface is given in Figure 2.22, but the
dependence of the total amplification factor on the micto channel gauge R0 / L2 is shown
in Figure 2.23.
        We can mark that the results of our simulations are in good agreement with the
data by Guest [396], who evaluated the optimal value for the gauge   45.5. The
current-density distribution at the MCP exit is presented in Figure 2.24, but the line-
dispersion function – in Figure 2.25. The point-dispersion function is given in Figure
2.26, angual and energy distributions for the electrons at the exit are shown in Figures
2.27 and 2.28. Figure 2.29 demonstrates the yields of MCP and screen to the MTF, and
the total MTF of the device.




                                                                                       122
       Fig. 2.20. Transfer function for the channel wall from the unit current.




Fig. 2.21. Secondary-emission distribution for the individual cascades.




                                                                                  123
   Fig. 2.22. Total distribution of secondary emission over the channel wall
   for all cascades.




Fig. 2.23. Dependence of the amplification factor on the gauge for the
constant voltage U 2  1000V .



                                                                               124
Fig. 2.24. Current-density distribution at the end of MCP.




      Fig. 2.25. Line-dispersion function.



                                                             125
       Fig. 2.26. Point-dispersion function.




Fig. 2.27. Angular distribution of electrons at the MCP end.



                                                               126
Fig. 2.28. Energy distribution of electrons at the MCP end.




Fig. 2.29. MTF for the parts: 1 – MCP+screen, 2- MCP+photocathode,
3 – total MTF of the device.




                                                                     127

								
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