The measuring device used is shown in Figure 31. During the experiments the ‗focus‘
position of beam changes. Two scans are made – along X-axis and after that along Y-axis (see Figure 31). The measured current distributions represent a set of linear integrals of the current distributions along the other axis. They are presented on a single bitmap for different focus positions on Figure 32. There, each line corresponds to the integral current distributions for different cross-sections and for different focus positions (see the distance a2 in Figure 30)
The empiric formula, which gives the connection between the distance of the focus position from the main axis of the focusing lens f and the corresponding number of the bitmap line NL for the studied EBW gun is:
Figure 31. Experimental measuring device
Figure 32. Two measurements of the integral current distributions along X- and Y-axes
e) f)
g) h)
Figure 33. (Continued)
a) b)
c) d)
Design of High Brightness Welding Electron Guns and Characterization… 45
i)
Figure 33. Experimental data(curves 1) and (curves 2) fitted to normal integral beam current distributions at different ―focus‖ positions from X-axis scans (see Table 4)
0.1714 HV4.142*10 N 200
/ HV
f 6 L (76)
where HV = 60 keV is the energy of the electrons, NL is the number of the line, presenting in the bitmap, shown on Figure 32, the beam. This bitmap shows the converted into light current distribution transferred through the slit at beam scan at two slits, as this is displayed at the insertion of Figure 31. The beam current is 10 mA.
The integral current distributions in nine cross-sections from the X-axis and respectively the Y-axis distributions, corresponding to equal focus positions, are investigated.
On Figure 33 are presented the results, obtained on the base of experimental data (curves 1), from fitting the measured integral current distributions to normal distributions (curves 2) at nine ―focus‖ plane positions from the X-axis scans. They are fitted using the least squares method. The calculated values of the focus position, the number of the bitmap line and the estimated standard deviation are given in Table 4.
It can be observed, that with the decrease of the beam diameter the accuracy of the approximation of the current distribution with a normal one increases. The deformations (deviations of measured current distributions from the normal one) are a result of aberrations and beam ion generation as well as non-uniformities in the beam transport track.
Then, using this formula the distance a1 (Figure 30), which is constant, can be calculated from the measurement, when distance to the image of the crossover coincides with the
Then the distance a2 to the image s2 (Figure 30) for different focal length of the lens can be calculated by (see Table 4):
a2
One can write the following relations based on the optical theory (Figure 30):
1 2 1
2 1 2
a M a s
s
(see Table 4) (79)
The value of the standard deviation of the image in the focus 2 =sx0= 0.1661 [mm] is estimated from the experimental data (Table 4-e)), consequently the variance 1=2 /M=0.1908.
Calculations are made for the respective cross-sections from the Y-axis. The estimated normal integral beam current distributions together with the experimental ones are presented on Figure 34. The values of the focus position f are the same as the ones given in Table 4 for the X-axsis cross-sections. The values of a2 and M are also the same. The value of the standard deviation of the image in the focus 2 =sy0= 0.1713 [mm] is estimated from the experimental data (Table 5)), consequently the variance 1=2/M=0.1968.
Table 4. The parameters of the beam current distribution along X-axis Figure
33 NL f [mm] sxi [mm] s2xi[mm2] a2 [mm] M=a2/a1 2I0=1M a) 0 493.0318 1.2828 1.6456 1450.6383 1.9423 0.3706 b) 50 447.3539 0.9749 0.9504 1115.5091 1.4936 0.2850 c) 100 409.4222 0.6727 0.4525 906.1652 1.2133 0.2315 d) 150 377.4209 0.3837 0.1472 762.9821 1.0216 0.1949 e) 205 347.5388 0.1661 0.0276 650.0000 0.8703 0.1661 f) 250 326.3956 0.2897 0.0839 579.7600 0.7762 0.1481 g) 300 305.7294 0.5705 0.3255 517.6114 0.6930 0.1322 h) 350 287.5243 0.8510 0.7242 467.4968 0.6259 0.1194 i) 415 266.8662 1.2121 1.4692 415.2339 0.5560 0.1061
a) b)
Design of High Brightness Welding Electron Guns and Characterization… 47
c) d)
e) f)
g) h)
i)
Figure 34. Experimental (curves 1) and fitted to normal (curves 2) integral beam current distributions at different ―focus‖ positions from Y-axis scans
Table 5. The parameters of the beam-current distribution along Y-axis
To characterize the beam quality through the values of beam emittance could be used the equation:
(ζxi)2=(ζx0i)2+(z0i-z0)2(ζx‘0i)2, (80) written at a condition of zero value of the co-variance between x and x in the canonic position of the emittance diagram, one can find ζx0i at measured ζxi. The parameters of the beam current distribution, calculated on the base of experimental data, for the mentioned nine positions of the ―focus‖ plane along X-axis are given in Table 6.
Since i220i20i'2i0, then:
From the obtained results is concluded, that the current distribution of the beam is very close to an axis-symmetrical one, which reveals its good adjustment. Contour plots of the canonical view of the emittance is calculated for the investigated 8 cross-sections (without the beam focus) by finding the mean values of ζx and ζx from the data for x and y assuming the case that in a rotation symmetric beam they are identical. In this way a transition is made to rr‘ coordinate system (instead of xx‘ and yy‘). The mean values are given in Table 8.
Design of High Brightness Welding Electron Guns and Characterization… 49
Table 6. The beam emittance along X-axis from the investigated cross-sections
NL 0-i=ai0-a650 0-i2 [mm2] i20 [mm2] '02i [mm2] '0i [mm] [mm.mrad]
0 800.6383 641021.6874 0.1373 2.3529*10-6 1.5339*10-3 0.5685 50 465.5091 216698.7222 0.0812 4.0111*10-6 2.0028*10-3 0.5708 100 256.1652 65620.6097 0.0536 6.0789*10-6 2.4655*10-3 0.5708 150 112.9821 12764.9549 0.0380 8.5547*10-6 2.9248*10-3 0.5701
205 0 0 0.0276 * * *
250 70.2400 4933.6576 0.0219 12.5667*10-6 3.5450*10-3 0.5250 300 132.3886 17526.7414 0.0175 17.5731*10-6 4.1920*10-3 0.5542 350 182.5032 33307.4180 0.0143 21.3136*10-6 4.6167*10-3 0.5512 415 234.7661 55115.1217 0.0113 26.4519*10-6 5.1431*10-3 0.5457
Table 7. The beam emittance along Y-axis from the investigated cross-sections
NL i20 [mm2] '02i [mm2] '0i [mm] [mm.mrad]
0 0.1461 2.0641*10-6 1.43669*10-3 0.54910380 50 0.0864 3.9415*10-6 1.98533*10-3 0.58348751 100 0.0570 5.7630*10-6 2.40063*10-3 0.57327149 150 0.0404 7.3058*10-6 2.70293*10-3 0.54355978
205 0.0293 * * *
250 0.0233 15.3136*10-6 3.91326*10-3 0.59794645 300 0.0186 19.7410*10-6 4.44308*10-3 0.60603641 350 0.0152 23.1817*10-6 4.81474*10-3 0.59317538 415 0.0120 28.5971*10-6 5.34762*10-3 0.58503005
Table 8. The mean values for X- and Y-axes of , ’ and
NL f [mm] r0 ‘r0 [mm]
[mm.mrad]
0 493.0318 0.37640 0.0014853
0.567994 50 447.3539 0.28945 0.0019967
100 409.4222 0.23515 0.0024331 150 377.4209 0.19800 0.0028139 250 326.3956 0.15045 0.0037291 300 305.7294 0.13430 0.0043175 350 287.5243 0.12130 0.0047157 415 266.8662 0.10775 0.0052454
On Figure 35 are presented the plots of the dependencies between the main axes of the ellipse of the emittance (r0 and ‘r0.100) and the focus position from the main axis of the focusing lens f.
In order to calculate easily the values of these axes as a function of the focus position value, regression equations are estimated:
r0 = - 0.22068 + 0.0024082 f - 0.0000067632 f2 + 0.00000000879 f3; (82)
’r0 = 0.0153310 - 0.000049513 f + 0.00000004367 f2. (83) The continuous curves on Figure 35 represent the functions (82) and (83), while the dots show the calculated emittance ellipse axes values for the investigated cross-sections (Table 8).
The relation between the distance of the focus position from the main axis of the focusing lens f and the corresponding number of the bitmap line NL calculated by eq. (76) is shown on Figure 36.
Figure 35. Dependencies between the main axes of the ellipse of the emittance (r0 and ‘r0.100) and the focus position from the main axis of the focusing lens f
Figure 36. Relation between the distance of the focus position from the main axis of the focusing lens f and the corresponding number of the bitmap line NL - eq. (76)
Design of High Brightness Welding Electron Guns and Characterization… 51
a)
b)
Figure 37. Contour plots of the emittance in canonical view for different focus positions and parts of the beam current p:
a.ellipses: 1 is calculated for p=0.39; 2 – for p= 0.86; 3 – for p=0.99; NL=0, b) p=0.99 and ellipses: 1 – for NL=0; 2 – for NL=50; 3 – for NL=150; 4 – for NL=250; 5 – for NL=300; 6 – for NL=350; 7 – for NL=415. b. ellipses position in r.r' plane for p=0,99
On Figure 37a are presented the contour plots of the emittance in canonical view for the cross-section NL=0. The contours are evaluated for parts of the beam current: p=0.39, 0.86 and 0.99.
On Figure 37b is given the emmitance canonical view of all the investigated cross-sections for p=0.99.
The current density distribution in the phase plane can be defined as particle flow per mmmrad. It is calculated for the first cross-section (Table 8) assuming its normal distribution and asis-symmetrical beam. 2D and 3D view of this distribution is presented on Figure 38 a,b.
Figure 38. 2D and 3D presentation of the calculated current density in the phase plane from the first
Figure 39 shows 2D and 3D view of the calculated current density in the beam focus.
Another invariant, besides the emittance, the beam brightness per volt accelerating voltage is:
(B/U)p=2Ip/(2p2
U). (85)
There B/U is the average value for emittance ellipse, through which the part Ip of the beam current is transferred.
Design of High Brightness Welding Electron Guns and Characterization… 53
Figure 39. 2D and 3D view of the calculated current density in the beam focus (NL=205). Note that the coordinate
The values of (B/U)p calculated for some parts of the electron beam current are given in Table 9. The obtained from experimental data values for the brightness differ slightly for the different cross-sections for different parts of the beam current. Their mean values presenting theoretically invariant (B/U)p are calculated.
The power density distribution is calculated assuming 2D normal distribution for the different focusing positions, corresponding to the explored 9 cross-sections. The obtained results are presented on Figure 40. The formula used is:
P0(x,y)=
On Figure 41 is presented 3D view of the power density distribution, calculated for the beam focus – case e) on Figure 40.
a) b)
c) d)
e) f)
Figure 40. (Continued)
Design of High Brightness Welding Electron Guns and Characterization… 55
g)
e) The contours that are not signed have levels: P0 = 500, 1000, 2000, 3000 [W/mm2]
f) The contours that are not signed have levels: P0 = 800,1300,1800,2300,2900 h) The contours that are not signed have levels: P0 = 2000,3000,4000,5000 [W/mm2]
i) The contours that are not signed have levels:
P0 = 2000,3000,4000,5000, 6000,7000
Figure 40. The power density distribution P0 for the different focusing positions. Signatures a)-i) correspond to NL=0 to NL=415.The contours represent 2D presentation of a given constant level of the function P0 (x,y)
Table 9. Brightness per volt accelerating voltage (B/U)p. The index p determines the calculation in the given part of the beam current p
p=Im/Ib 0.39 0,63 0,78 0,86 0.92 0,99
εp ζx.ζx 2ζx.ζx 3ζx.ζx 4ζx.ζx 5ζx.ζx 9ζx.ζx
(B/U)p
NL = 0 4.2185*1010 1.7036*1010 9.3744*109 5.8139*109 3.9805*109 1.3220*109 NL = 50 3.9474*1010 1.5941*1010 8.7720*109 5.4403*109 3.7247*109 1.2371*109 NL = 100 4.0279*1010 1.6266*1010 8.9508*109 5.5512*109 3.8006*109 1.2623*109 NL = 150 4.2475*1010 1.7153*1010 9.4390*109 5.8540*109 4.0079*109 1.3311*109 NL = 250 4.1888*1010 1.6916*1010 9.3085*109 5.7731*109 3.9525*109 1.3127*109 NL = 300 3.9216*1010 1.5837*1010 8.7147*109 5.4048*109 3.7004*109 1.2290*109 NL = 350 4.0297*1010 1.6274*1010 8.9548*109 5.5537*109 3.8024*109 1.2629*109 NL = 415 4.1275*1010 1.6669*1010 9.1723*109 5.6886*109 3.8947*109 1.2935*109 MEAN 4.0886*1010 1.6511*1010 9.0858*109 5.6349*109 3.8580*109 1.2813*109
Regression equation giving the dependence between the maximum value of the beam power density distribution P0max and any focus position from the main axis of the focusing lens f is estimated:
P0max = 116668 - 934.43 f + 2.9571 f2 -0.0043089 f3 +0.00000241 f4 (87) This function - P0max(f), together with the calculated data from the investigated cross-sections (signed with dots) are presented on Figure 42.
Figure 41. 3D view of the power density distribution in the beam focus (NL=0)
Figure 42. The maximum value of the beam power density distribution P0max and any focus position from the main axis of the focusing lens f