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 1 st
-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 W tp
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 W tp,
t
0
– start moment for emission impulse signal,

0
- time-of-flight for the reference particle from the emittion point to the image plane z
 z i
,
  dist
- additional time-of-flight for the reference particle to the plane z
 z i
,
 
- 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 r
0
and the position of the image plane z i
. Using the aberration expansion for the time-of-flight
 
A
*
2

0 n

A
*
11
 t

A
*
13

0 t cos

0

A
*
 n
 * 2
A r , we get the results
     dist

A
*
2

0 n

A
*
13

0 t cos

0

A
*
 n
(2.367)
(2.368) and
 dist

A
*

2
33 0
,

0 n
 
0 cos

0
,

0 t
 
0 sin .
0
(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, projection onto the plane XY; n - normal vector,
 v *
- tangential vector;
- its

- 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 D -
0 diameter of entrance aperture, and N – given number of the reference trajectories. Then the coordinates of the emitting points are evaluated using the formula r
0 i
 ih r 0
, ( i

0,..., N

1), h r 0

D
0
2( N

1)
.
(2.370)
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 ( ) is known, then the reference trajectory for the particle emitted from the point with coordinates
( , )
0
  
( ,
0
)
 z

, where f z z
0
) r z
0 0
can be represented as
- surface-layer function, which decreases fast for z
 z
0
. Let us write, for example f z z
0
)
 exp
 
(1
 z z
0
 z
 z
0
,
 
1, (2.371) then we have
107

( , )
0

0
 
( ,
0
)

0
3
 
( ,
0
)
 z

, z
0
 r
0
2
2 R c
, (2.372) where E - the distortion coefficient, and R - cathode radius. c angles
Then we make the variable discretization for the initial energy

,

of the particle
 i
  min


(

1), i

1,..., i max
,
    j min h

( j

1),
 k
  min


(

1), j

1,..., j max
, k

1,..., k max
,

and initial
(2.373) 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 , , and computing the formula (2.368), we determine the scope of the transfer function
 
using the
  min
,
  max
. The we define the quantity of discretization intervals for this function m , and evaluate the max subintervals and the centrum positions for these subintervals h


  max
  m max
 min ,
  m
   min


(

1),
By moving sequentially the values , , m

1,..., m .
(2.374)
, we can evaluate the function
W tp
(
  m
, r
0 z n i
 h h h
 h


W

 i
W

(
 j
) W

(
 k
) sin
 k
, (2.375) where the set , ,
   i
is that all values
 j

,
 k

)


 m
 h
2

,
  m
 h

2


.
Finally we evaluate the transfer function
W tp
(
  m
, r
0 z n i
 h

W tp
 m
(
  m
, r
0 z n i
W tp
(
  m
, r
0 z n i
.
(2.376)
(2.377)
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 defined as the root

of the equation
R

Q pr
 
0,
(pulse physical resolution) is
(2.378) where
R
 max max
W
W s s
(
(


 
 

 min
W
W s s
(
(


 
 
, W s
(
   
W tp
(

) W tp
(
 
).
(2.379)
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
W tm ,0
( N t
,

2

N t
 
)
, (2.380) where N t
- modulation frequency of the signal.
The image of temporal sinusoidal test in the plane
W
 z
 z i
is given by the formula
 

  dist dist
W tm ,0
( N t

W tp
(
  
.
(2.381)
Now one can evaluate the temporal MTF (modulation transfer function)
( t
,
0 n
, )
 max max
W
W

 min min
W
W
.
(2.382)
Physical temporal resolution
  mpr

1
N t lim
(2.383) is defined by the limit frequency N , which corresponds to the threshold contrast t lim
Q . mr
This frequency is defined as a root of the equation t
( t
,
0
, ) n i

Q mr
( 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

to the axis ОХ,
0 which pass through the point М. The line image is dissected onto the image-reciever surface with a speed v sc
in the direction
 x

(Figure 2.5). The transfer dispersion function for the spatial-temporal line W tsl
(
 x r
0 n
,
 
0 z v sc
) is evaluated using the same procedure as for the function W sl
. One need mark, when the line has a finite width
 l , the line-dispersion function W tsw
is evaluated by integration
W tsw
(
 x r
0 n
,
 
0 n
,
0 z v i sc
)

 l

 l
/ 2
W tsl
(
  l
/ 2
 
.
(2.449)
Technical temporal resolution is defined by a formula
 tch

1
.
N tsi ,max v sc
In MTF computation the formula is used for the value
 x

in general case
(2.450) x
 w z s
 
0

0 p  sin 2
 sin

0 cos

0 m
  cos

0
, (2.451) where
  v A z sc 2
( ) s
 
0

0 p w z
.
(2.452)
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 3 rd –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 60 th
, 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 3 rd
-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 E , ion-
1
111

barrier film, domain II of micro channel plate with a field E in the channell, domain III
2 with a field strength E
3
and the screen.
Рис. 2.12. Schematic picture of the 3 rd
-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 1 st
domain.
The elementary current of photo-emitter can be represented by the expressions dI
01

I
K ( ,
01

01 01 01 xy
,
( x
01 01
01
)
,
 y )

K
(
01
01
(
)
  

(
01
)
01

)
( d
  
01
) sin

01
, dx dy
01 01 01
,
(2.453) but the distribution function for the angles and energies are
 
01
)


01
A

1



01
01


, A

2


01
0
 
01
) d

01
,
(
01
)
 cos n
B

01 , B



0
/ 2 cos n d
01
,
(2.454)
(2.455)
( )
01

1
2

.
(2.456)
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 j x
01
)

1
2 j
0
 
N x
01 01
After the variable discretization the line-dispersion function
(2.457) x
 x
01
can be represented by a sum
 x
( x
  x
01
)
 h x


,

,



(
 
)


( n h )


( n h ) sin( ), (2.458) where h - discretization steps,but k n - quantity of intervals for the variable discretization. k
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
( )
 r e
01 i (

01
 
01
) 
( )
 r e
01 i (

01
 
01
) 
2
E
1

01
2
E
1

01 sin

01 sin

01








E
1
01
E
1
01
( z
 z
01
 2
( z
 z
01
 2

01

01
 cos

01



, (2.459)
 cos

01



, (2.460)
Current-density distribution in the image of sinusoidal mira is
(
01
)
 

 j x
01
)
 x
( x

01
 x
01
) dx
01
.
In order to evaluate the MTF we use the correlation
R N
01
)
 j max j max

 j min j min
 j j


(0)
 j

(2 )
(0)
 j

(2 )
, a

4
1
N
01
.
(2.461)
(2.462)
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 R
0
, and length
L
2
.
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
( r
02
) K (
  
02 02 02 02
) d
  
02 02 r dr
02 02 02
, (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
( z

02
) K

02
(
 
02
,
 
02
,
 
02
) d
 
02 d
 
02 d
 r dr
02 02 02
.
(2.464)
The current per unit length of the channel dI
02

I
02
  z
( z

0 02
 z
02
) (2.465) creates the secondary particles emitted by the channel wall. The surface part subjected to the bombarding by the primary electrons is the 1 st
cascade of amplification, which is characterized by the current distribution
 z 1
( z
12
 z
02
)

W ( z z 1 12
 z
02 k
1
)
(2.466) and by the amplification ratio k
1

L
1

L
2
L
1

W ( z z 1 12
 z
02
) dz
12
.
(2.467)
There is simple correlation between the film current and the 1 st
I
12
 k I
1 02
, but the current coming to the surface of 2 nd
cascade current -
cascade is I
 
12
I
12
.
The elementary current before the 2 dI

12

I
12
 z 1
( z
12 nd

cascade is z
02
) K
12

12
,
 
12
,
 
12
) d
 
12 d
 
12 d

12 dz
12
.
(2.468)
The current-distribution function on the surface before the 2 nd
cascade is described by a formula
 z z 1 12
 z
02
)

L
1

L
2
L
1
 
( z
 z 1 12
 z
02
)
 z
 
12
 z
12
) dz
12
, but the current-distribution function on the surface after the 2 nd
cascade is
(2.469)
115

Fig. 2.16. The current-density distribution from the elementary ring emitter. dI
22

I
22
 z 2
( z
22
 z
02
).
(2.470)
As the result we obtain the recurrent correlations
I n 2

I k k
02 1 2
... , n
   zn n 2
 z
02
)

W ( z zn n 2
 z
02
)

 
 

( z z 1 12
 z
02

( z z 1 12
 z
02
)
 z
 
12
 z
12
)
 z
( z
22
 z
12
)
)
(2.471)
   z n 2
 z n 2
) dz
12 dz n 2
, (2.472)

( z z n 2
 z n

1,2
) dz
12 dz n

1,2
,
(2.473)
 zn

W zn k n
, k n
 
W ( z zn n 2
 z
02
) dz n 2
.
(2.474)
Total elementary current is described by a sum dI

 dI
02
 dI
12
  dI n 2,
(2.475) where dI n 2

I n 2
 zn
( z n 2
 z
02
) K n 2
(
 n 2
,
 n 2
,
 n 2
) d
 n 2 d
 n 2 d
 n 2 dz n 2
.
(2.476)
Total current from the channel surface is
I


I k
02

, (2.477) but a total amplification factor of MCP is given by a formula k

 
1 k k
1 2
  k k k
1 2 n
.
(2.478)
Total distribution function for the current density is described using partial distribution functions for the cascades and their amplification factors
 z

 k
1
 z 1
  z 2
  k
 k k k n
 zn .
(2.479)
The effective amplification factor of MCP is defined by a formula
116

k eff
 k



1
 

L
1

L
2
 z

( )


.
(2.480)
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 z
  z
02

E
2
4

02 sin
2

02


R
0
2  r
2
02
2 sin (

02
 
02
)
 r
02 cos(

02
 
02
)


2

(2.481)


R
2
0
 r
2
02
2 sin (

02
 
02
)
 r
02 cos(

02
 
02
)

 ctg

02
.
The distribution function of the current density after discretization is
 z

( z

0 02
 z
02
)
   z 0
( n h z z
)
 h h h h r
   h z r

, , ,
    r sin( n h ), (2.482)
W ( z z 1 1
 z
02
)
 z 1
( z z
)
 h h h h r
   h z r

, , ,
      
02 02 r sin( n h ), (2.483) where we use the specific distribution functions for the initial angles and energy



( )

1

,
2

(
 
)
 cos( ),


(
 
)
 exp(

4


0
02 exp(
 
02
/

02
) d

/

02
)
02
,
(2.484)
(2.485)
(2.486) where
 r
( n h r r
)

2
.
(2.487)
R
0
2
Secondary emission ratio
  
,

in general case consists of three components
(2.488)

- 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 use the Guest’s formula for true secondary electrons [396]
  
One can where
     max



 max cos



 exp



    



 max cos






, (2.489)

- maximal value for the secondary emission ration in normal fall the electrons max on the surface,

- correspondent value for the collision energy, max

- angle between ve velocity vector of primary electron and the normal vector on the surface,
 ratio of incoming electrons by the wall,

- parameter of the emission model.
- absorbing
From the experimental data [401] the absorbing ratio was equal
 
0.62, use an approximation for the parameter

but we
117




0.55
0.25,
0.65, max
, max
.
At the point where the electron crosses the surface, it has the energy
 
02
 
02

E z

2 02
 z
02
) and the hade cos

 
02
02 sin

02
1
 r
02
2
R
0
2
2 sin (

02
 
02
).
(2.490)
(2.491)
(2.492)
The computation of the line-dispersion function and MTF for the MCP area is similar to the computations for the 1 st 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 R
0
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 j x
02
)

1
2
F x
02
  
N x
02 j
0
, (2 n

1) R
0

2 D
 x
02
F x
02
)
 0, (2 n

1) R
0

2 D
 x
02

(2 n

1) R
0

2 ,

(2 n

1) R
0

4 ,

1, 2,...
(2.493)
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 E z is a constant, which changes stepwise to the constant
2
( ) E
3
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 1 st zone the field strength E is constant, but the emission
2 properties of the channel are characterized by the constants
 max
,

, max
 n
. Second zone is beginning at the point z . It has the length h , radius of channell b
R
 
0
, where

- the thickness of spay layer. The emission properties of this zone are characterized by the constants

,
 max max
,
 max
.
Starting the coordinate z the channel has a cone expansion c with the angle

, which is made by the etching method. Fourth zone begins at the channel end, which is smoothly passes at z to the domain of the uniform field with d strength E
3
.
General expression for arbitrary trajectory emitted in r z z i
  sin
 i
( , ) i


 r e
0 i
 
 sin 2

  z i

 w z z i z i
by the channel wall is
The paraxial trajectories v and w are given by the expressions
(2.494)
119

v z z i

2 w z z i
 z
 z i , ( , ) i

E
2
( , ) i

0
1
E z
2
(
 z i
)
,
(2.495) in the 1 st
zone. At the point z the passage is doing, and the trajectories in 2 a nd
zone are
( , )
 
( , ) ( , a
)

( , ) ( , a
),
( , ) i
 
( , ) ( , a i a
)

( , ) ( , a i a
),
(2.496) where the auxiliary trajectories , are computed with using the formulas v z z a
) ( z z a

) 1


  a
 a
( z
 z a
) ( z z a
)
2


1
8



 a a
 

2

1
4 R a
( ,
( z
 z a
) 3




1
24 a
)
 
  a
2
 a
 a

 a

1
4 R a
( z
 z a
) (


 
 a a


2 z z a
)
2



5
64


 
 a a


3





,

3
4 R a


 a a

3
8




 a a


2 
  w z z a
( z
 z a
)
3




1
6
 a

 a

1
R a
   a
 2

 a


5
16


 
 a a


3





,
) 1 ( z z a
)
2
4
1
R a


 a a z z a
)
3

 
1
24
 a

 a

1
8 R a




 a a


2 
 
  a
 a

 
(2.497)
(2.498)
 a

 a

1
4 R a
3

3
32 R a
2



 a a
 

2

5
64 R a


 
 a a


3

5
192


  a a
 2 a




, ( z
 z a
)
3




1
12 R a


 a a
(2.499)
( , a
) z z a
1
)
8 R a


 a a z z a
)
2




1
8
 a

 a

3
8 R a
( z
 z a
)
3




1
3 R a where
R a

 a a



2

 a a


,
1
R a
3


 a a

3
8 R a
2



 a a
 

2

5
16 R a
   a
( z a
)
 
( ).
i




 a a


2

 




 a a


3

5
48


  a
 2 a




,
(2.500)
(2.501)
These trajectories are satisfied to the initial data v z a
)

0,

( a
)

1, w z a
)

1, ( a
)

0.
(2.502)
Computing the values ( ), ( ), ( ), b b b

( ), b
one need to make passage to the algorithm of the trajectory evaluation in 3 rd
zone. Here the trajectories are evaluated using the formulas
120

v z z

2
 z
 z i
  i

1
 z
 z i
2 R i z z i
)
2


11
40 R i
2
 i
  
 ,

 i
 
 
 z i
(2.503)
( z
 z i
)
3


101
560 R i
3


17
420 R i
  i
(
1 z
 z i
)


1

3
2 R i
( z
 z i
) ( z z i
)
2


11
8 R i
2

 i

3
 



( z
 z i
)
3


101

17
80 R i
3
60 R i
 i
 
 ,
(2.504) w z z i
  z

R i z i z z i
)
2


1
2 R i
2

 i

12
 

 z z i
)
3


1
10 R i
3

1
60 R i

  i
 
, (2.505)
( , ) i
 
1
R i

1 ( z i
)


R i
6
 i

 

1
R i

 z z i
)
2


 i

20
 

3
10 R i
2



 .
(2.506)
The parameter evaluation for 3 rd r
0

R
0
  
( L
2
 z i
 cos

zone are doing with using the formulas
 sin
 sin , cos
   cos sin


, (2.507) then the passage to 4 th
zone is doing at z , and the passage to 5 c th
zone is doing in z . d
Axial values for the potential and its derivatives for 2 nd
-4 th
zones are evaluated using a formula

( )

2
(
2
 
2 )

1
( )

U
2

2
( )
 
(
3 2

2 )

U
2
 
(2.508)
3 where
 j
( ) - 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 W
1
, 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
 r
0
in z

L
2
, the criterion r

R
0
 
is adding in z

L
2
 h , where r
0

R
0
  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
U
  h
U
  and the angles
W

 h h h h z
  h
 sin
 z

,

,  , 
W z i z

,

,

,

W z i sin
 sin

(2.509)
(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
 
V and 100 1000 V with individual scanning step h
U for each interval to increase the accuracy of computations.
2.13.4. The results of numerical modeling
The transfer function W for the initial data defined by default and 10
1 discretization intervals for each variable is shown in Figure 2.20, but the currentdistribution 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 R
0
/ L
2
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 linedispersion 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

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