Using TE021 s = 2 mode to simulation the possible competition mode and operate in the specific voltage and magnetic field condition that can stationary operation

1. TE0,2,1 mode S =2, Vb = 30 kV, alpha = 2.57, velocity spread = 0%
Rc / Rw = 0.3268, Rw = 0.3571 cm
L
rw
2 3 4 5 6 7 8 9 10
0
1x10
5
2x10
5
3x10
5
4x10
5
5x10
5
L / rw
beamefficiency(%)
0
10
20
30
40
QPb(kW)

2. Rw = 0.3571 cm
Vb = 30 kV, alpha = 2.57, Bz0 = 17.4 kG
-20 -10 0 10 20
0
20
40
60
80
100
120
140
160
TE13
TE32
TE61
TE02 (operating mode)
TE22
TE51
TE12
TE31
TE01TE21
S = 2
frequency(GHz)
kz
(cm
-1
)
S = 1
TE11

3. (1) Do QPVSB runs with the ideal cavity parameters you used to run for Larry's TE02, s=2 design
Previous parameters:
Rw = 0.3571 cm, L/ Rw = 6, Rc / Rw = 0.3268
Vb = 30 kV, alpha = 2.57, delta = -10 ~ 20
Operation mode TE021 S=2
-10 -5 0 5 10 15 20
0.0
2.0x10
4
4.0x10
4
6.0x10
4
8.0x10
4
1.0x10
5 TE(2)
022
TE(2)
025
TE(2)
023
TE(2)
024 TE(1)
115
TE(2)
022
TE(1)
214
TE(1)
213
TE(1)
212
delta
QP(kW)
TE(2)
021
16.8 17.2 17.6 18.0
0.0
2.0x10
4
4.0x10
4
6.0x10
4
8.0x10
4
1.0x10
5
TE(1)
115
TE(2)
025
TE(2)
024
TE(2)
023
TE(2)
022
TE(2)
022
TE(1)
214
TE(1)
213
QP(kW)
B (kG)
TE(2)
021TE(1)
212

4. (2) Let Vb =30 kV, Rc= 1.26 mm, alpha= 2.0, dVz/Vz = 5%, ismain =2, immain=0, inmain =2,
ilmain=1, dmin=-10, dmax= 20
Rw = 0.3571 cm, L/ Rw = 6, Rc / Rw = 0.3528, velocity spread = 5%
Vb = 30 kV, alpha = 2.0, delta = -10 ~ 20
Operation mode TE021 S=2
-10 -5 0 5 10 15 20
0.0
2.0x10
4
4.0x10
4
6.0x10
4
8.0x10
4
1.0x10
5 TE(2)
022
TE(1)
215
TE(2)
023
TE(2)
024
TE(1)
115
TE(2)
022 TE(1)
214
TE(1)
213 TE(1)
212
delta
QP(kW)
TE(2)
021
16.8 17.2 17.6 18.0
0.0
2.0x10
4
4.0x10
4
6.0x10
4
8.0x10
4
1.0x10
5
TE(1)
115
TE(1)
215
TE(2)
024
TE(2)
023
TE(2)
022
TE(2)
022
TE(1)
214
TE(1)
213
QP(kW)
B (kG)
TE(2)
021TE(1)
212

5. Electron-wave resonance lines for the S=1 ~ S=3
Rw = 0.3571 cm, Vb = 30 kV, alpha = 2.0, Bz0 = 17.5 kG
Operation mode TE021 S=2
-20 -10 0 10 20
20
40
60
80
100
120
140
160
180 TE14
TE33TE91
TE52TE03
TE23TE81
TE42
TE71TE13
TE32
TE61
TE02TE22
TE51
TE12TE41
S = 3
S = 2
S = 1
TE31
TE01
TE21
TE11
kz
(cm-1
)
frequency(GHz)

6. (1) Do QPVSB runs with the ideal cavity parameters you used to run for Larry's TE02, s=2 design
Previous parameters:
Rw = 0.3571 cm, L/ Rw = 6, Rc / Rw = 0.3268, velocity spread = 0%
Vb = 30 kV, alpha = 2.57, delta = -10 ~ 20
Operation mode TE021 S=2
16.8 17.2 17.6 18.0
0.0
5.0x10
5
1.0x10
6
1.5x10
6
2.0x10
6
TE(1)
115TE(1)
214
TE(1)
213
TE(1)
212
TE(2)
226
TE(2)
225
TE(2)
224
TE(2)
224
TE(2)
223
TE(2)
223
TE(2)
222TE(2)
222
TE(2)
221 TE(2)
516
TE(2)
515
TE(2)
515
TE(2)
514
TE(2)
513
TE(2)
512
TE(2)
025
TE(2)
024
TE(2)
023
TE(2)
022
TE(2)
022
TE(2)
021
TE(3)
426
TE(3)
425
TE(3)
236
TE(3)
236
TE(3)
235TE(3)
235 TE(3)
234
TE(3)
234
TE(3)
233
TE(3)
233
TE(3)
232
TE(3)
232TE(3)
231
TE(3)
036
TE(3)
035
TE(3)
035
TE(3)
034
TE(3)
034 TE(3)
033
TE(3)
033
TE(3)
032 TE(3)
032TE(3)
031
TE(3)
526
TE(3)
525TE(3)
524
TE(3)
523
TE(3)
523
TE(3)
522
TE(3)
521
B (kG)
QP(kW)
17.0 17.2 17.4 17.6 17.8 18.0
0.0
2.0x10
5
4.0x10
5
6.0x10
5
8.0x10
5
1.0x10
6
TE(1)
115
TE(1)
214
TE(1)
213
TE(2)
226
TE(2)
225
TE(2)
224
TE(2)
223
TE(2)
222
TE(2)
516
TE(2)
514
TE(2)
513
TE(2)
025
TE(2)
024
TE(2)
023
TE(2)
022
TE(2)
022
TE(2)
021
TE(3)
426
TE(3)
235
TE(3)
234
TE(3)
233
TE(3)
232
TE(3)
036TE(3)
035
TE(3)
034
TE(3)
034
TE(3)
033
TE(3)
033
TE(3)
032
TE(3)
032
TE(3)
031
TE(3)
526
TE(3)
525
TE(3)
524
TE(3)
522
TE(3)
522
TE(3)
523
TE(3)
521
B (kG)
QP(kW)

7. 16.8 17.2 17.6 18.0
0
1x10
5
2x10
5
3x10
5
4x10
5
5x10
5
TE(2)
223
TE(2)
223
TE(2)
222
TE(2)
222
TE(2)
221
TE(2)
515TE(2)
514
TE(2)
022
TE(2)
426
TE(3)
526
TE(3)
523
TE(3)
036
TE(3)
035
TE(3)
035
TE(3)
034
TE(3)
034 TE(3)
033
TE(3)
033
TE(3)
032
TE(1)
115TE(1)
214 TE(1)
213
TE(1)
212
TE(2)
425
TE(2)
516
TE(2)
515
TE(2)
513
TE(3)
031
TE(2)
512
TE(2)
025
TE(2)
024TE(2)
023
TE(2)
022
TE(3)
032
TE(3)
524
TE(3)
523
TE(3)
522
TE(3)
525
TE(3)
521
TE(2)
021
B (kG)
QP(kW)
16.8 17.2 17.6 18.0
0.0
2.0x10
4
4.0x10
4
6.0x10
4
8.0x10
4
1.0x10
5
TE(1)
115
TE(2)
025
TE(2)
024
TE(2)
023
TE(2)
022
TE(2)
022
TE(1)
214
TE(1)
213
QP(kW)
B (kG)
TE(2)
021TE(1)
212

8. (2) Let Vb =30 kV, Rc= 1.26 mm, alpha= 2.0, dVz/Vz = 5%, ismain =2, immain=0, inmain =2,
ilmain=1, dmin=-10, dmax= 20
Rw = 0.3571 cm, L/ Rw = 6, Rc / Rw = 0.3528, velocity spread = 5%
Vb = 30 kV, alpha = 2.0, delta = -10 ~ 20
Operation mode TE021 S=2
16.4 16.8 17.2 17.6 18.0
0.0
5.0x10
5
1.0x10
6
1.5x10
6
2.0x10
6 TE(1)
115
TE(1)
215
TE(1)
214
TE(1)
213
TE(1)
212
TE(2)
416
TE(2)
126
TE(2)
516
TE(2)
515
TE(2)
515
TE(2)
514
TE(2)
513
TE(2)
512
TE(2)
026
TE(2)
025
TE(2)
024
TE(2)
023
TE(2)
022
TE(2)
022
TE(2)
021
TE(3)
424
TE(3)
036
TE(3)
035
TE(3)
035
TE(3)
034
TE(3)
034 TE(3)
033
TE(3)
033
TE(3)
032
TE(3)
032TE(3)
031
TE(3)
526
TE(3)
525
TE(3)
524
TE(3)
523
TE(3)
523
TE(3)
522
TE(3)
521
B (kG)
QP(kW)
17.0 17.2 17.4 17.6 17.8 18.0
0.0
2.0x10
5
4.0x10
5
6.0x10
5
8.0x10
5
1.0x10
6
TE(1)
115
TE(1)
215
TE(1)
214
TE(1)
213
TE(2)
514
TE(2)
513
TE(2)
026
TE(2)
025TE(2)
024
TE(2)
023
TE(2)
022
TE(2)
022
TE(3)
036 TE(3)
034TE(3)
033TE(3)
032
TE(3)
031
TE(3)
526
TE(3)
525
TE(3)
524TE(3)
523
TE(3)
522
TE(3)
522
TE(3)
521
TE(2)
021
B (kG)
QP(kW)

9. 16.5 17.0 17.5 18.0
0
1x10
5
2x10
5
3x10
5
4x10
5
5x10
5
TE(1)
115
TE(1)
215
TE(1)
214
TE(1)
213
TE(1)
212
TE(2)
416
TE(2)
126
TE(2)
516
TE(2)
515
TE(2)
515
TE(2)
514
TE(2)
513
TE(2)
512
TE(2)
026
TE(2)
025
TE(2)
024
TE(2)
023
TE(2)
022
TE(2)
022
TE(3)
032
TE(3)
524
TE(3)
523 TE(3)
523
TE(3)
522
TE(3)
522
TE(3)
521
TE(2)
021
B (kG)
QP(kW)
16.8 17.2 17.6 18.0
0.0
2.0x10
4
4.0x10
4
6.0x10
4
8.0x10
4
1.0x10
5
TE(1)
115
TE(1)
215
TE(2)
024
TE(2)
023
TE(2)
022
TE(2)
022
TE(1)
214
TE(1)
213
QP(kW)
B (kG)
TE(2)
021TE(1)
212

10. Structure (all dimension in cm)
0 1 2 3 4
0.0
0.1
0.2
0.3
0.4
0.5
3.9855 cm
0.4077r1 = 0.35 0.3571
12.3
0.1355
0.3
0.15
Rw(cm)
Z (cm)
0.1
Parameters Table.1. (cavity dimensions shown above)
Vb = 35kV, alpha = 2.0, dVz/Vz = 6 %, Rc/Rw = 0.3192, and / 1Cu   in all structure section.
Operation mode: (2)
021TE

11. Use the stationary self-consistent code(parameters as in Table.1) to estimate the start oscillation current vs
B(uniform) for each competition mode.
8 10 12 14 16 18 20 22
0
10
20
30
40
50
TE(2)
5,1
TE(2)
2,2
TE(1)
1,1
TE(1)
2,1
TE(2)
0,2
B (kG)
Ist(A)
16 17 18 19 20
0
2
4
6
8
10
TE(2)
5,1
TE(2)
2,2
TE(1)
1,1
TE(1)
2,1
TE(2)
0,2
B (kG)
Ist(A)

12. The corresponding hot f and hot Q vs B (Hot Q defined in Sec. III-B of S. H. Kao, C. C. Chiu, P. C. Chang, K.
L. Wu, and K. R. Chu, “Harmonic Mode Competition in a THz Gyrotron Backward-Wave Oscillator,” Phys.
Plasmas 19, 103103 (2012).)
17.5 18.0 18.5 19.0 19.5 20.0
90
92
94
96
98
100
102
104
B (kG)
 = 3
 = 2
 = 1
0
1000
2000
3000
4000
5000
TE(2)
0,2
Hotf(GHz)
HotQ
10 12 14 16 18 20
20
25
30
35
40
45
50
Hotf(GHz)
20
40
60
80
100
120
TE(1)
1,1
B (kG)
HotQ

13. 15 16 17 18 19 20
40
42
44
46
48
50
TE(1)
2,1
B (kG)
Hotf(GHz)
HotQ
0
100
200
300
400
17 18 19 20
88
90
92
94
96
98
100
0
1000
2000
3000
4000
5000
TE(2)
2,2
HotQ
Hotf(GHz)
B (kG)
16 17 18 19 20
85
90
95
100
0
500
1000
1500
2000
2500
3000
TE(2)
5,1
HotQ
B (kG)
Hotf(GHz)

14. Change r1 to see the trend of the Ist vs B
r1 = 0.35 cm
15 16 17 18 19 20
0
2
4
6
8
10
TE(2)
5,1
TE(2)
2,2
TE(1)
1,1
TE(1)
2,1
TE(2)
0,2
B (kG)
Ist(A)
r1 = 0.3 cm
15 16 17 18 19 20
0
2
4
6
8
10
TE(2)
5,1
TE(1)
1,1
TE(1)
2,1
TE(2)
0,2
B (kG)
Ist(A)
r1 = 0.25 cm
15 16 17 18 19 20
0
2
4
6
8
10
TE(2)
5,1
TE(1)
1,1
TE(1)
2,1
TE(2)
0,2
B (kG)
Ist(A)

15. Using stationary self-consistent code to determined efficiency and forward wave power
choose r1 = 0.35 cm, B = 17.78 kG (the magnetic field corresponding to the lowest Ist)
other parameters are the same as in Table.1.
0 2 4 6 8 10 12
93.90
93.92
93.94
93.96
93.98
94.00
94.02
94.04
94.06 TE(2)
0,2
Ib (A)
Frequency(GHz)
0 1 2 3 4 5 6 7 8 9 10 11 12
0
5
10
15
20
25
30
35
40
TE(2)
0,2
backward wave power
Ib (A)
power(kW)
forward wave power
TE
(2)
02
Ist
0.93 A 1.51 A
TE
(1)
21
Ist TE
(1)
11
Ist
2.76 A
0 1 2 3 4 5 6 7 8 9 10 11 12
0
1
2
3
4
5
6
7
8
9
10
11
TE(2)
0,2
TE
(2)
02
Ist
0.93 A 1.51 A
TE
(1)
21
Ist TE
(1)
11
Ist
backward wave efficiency
forward wave efficiency
Efficiency(%)
Ib (A)2.76 A

16. choose r1 = 0.35 cm, Ib = 5A, tune B-field, other parameters are the same as in Table.1.
17.6 17.8 18.0 18.2 18.4 18.6
93.8
94.0
94.2
94.4
94.6
94.8
95.0
95.2
95.4
95.6
Frequency(GHz)
B (kG)
17.4 17.6 17.8 18.0 18.2 18.4 18.6
0
10
20
30
40
50
60
70
backward wave power
forward wave power
power(kW)
B (kG)
17.4 17.6 17.8 18.0 18.2 18.4 18.6
0
5
10
15
20
25
30
35
40
backward wave efficiency
forward wave efficiency
Efficiency(%)
B (kG)

17. Using stationary self-consistent code to determined efficiency and forward wave power
choose r1 = 0.3 cm, B = 17.78 kG (the magnetic field corresponding to the lowest Ist)
other parameters are the same as in Table.1.
0 2 4 6 8 10 12
93.94
93.96
93.98
94.00
94.02
94.04
94.06
94.08
TE(2)
0,2
Ib (A)
Frequency(GHz)
0 1 2 3 4 5 6 7 8 9 10 11 12
0
5
10
15
20
25
30
35
40
TE(2)
0,2
backward wave power~0
Ib (A)
power(kW)
forward wave power
TE
(2)
02
Ist
1.25 A 1.78 A
TE
(1)
21
Ist TE
(1)
11
Ist
2.75 A
0 1 2 3 4 5 6 7 8 9 10 11 12
0
1
2
3
4
5
6
7
8
9
10
11
TE(2)
0,2
TE
(2)
02
Ist
1.25 A 1.78 A
TE
(1)
21
Ist TE
(1)
11
Ist
backward wave efficiency~0
forward wave efficiency
Efficiency(%)
Ib (A)2.75 A

18. choose r1 = 0.3 cm, Ib = 5A, tune B-field, other parameters are the same as in Table.1.
17.4 17.6 17.8 18.0 18.2 18.4 18.6
93.8
94.0
94.2
94.4
94.6
94.8
95.0
TE(2)
0,2
Frequency(GHz)
B (kG)
17.4 17.6 17.8 18.0 18.2 18.4 18.6
0
10
20
30
40
50
60
70
TE(2)
0,2
backward wave power
forward wave power
power(kW)
B (kG)
17.4 17.6 17.8 18.0 18.2 18.4 18.6
0
5
10
15
20
25
30
35
40
TE(2)
0,2
backward wave efficiency
forward wave efficiency
Efficiency(%)
B (kG)

19. Mode Competition Criteria
Ref.: S. H. Kao, C. C. Chiu, P. C. Chang, K. L. Wu, and K. R. Chu, “Harmonic Mode Competition in a
THz Gyrotron Backward-Wave Oscillator,” Phys. Plasmas 19, 103103 (2012).
The mode competition processes examined in the above reference (Sec. IV-C) consistently follow three
criteria:
(1) The presence of the s = 2 mode enhances the Ist of the competing modes. For example, in Fig. 4(d-f), the s
= 2 mode enhances the linear Ist of the lowest-kz, s = 1 mode by a factor of 2.63, 2.14, and 1.63, respectively. As
can be seen from the figures, the enhancement factor is larger for a higher-amplitude s = 2 mode. As shown in [30],
the enhancement factor can be as large as 15 if the competing mode is a high-kz, s = 1 mode. Clearly, this criterion
also applies to an early-starting mode of any cyclotron harmonic number.
(2) The early-starting s = 2 mode is eventually suppressed by an s = 1 mode because of the unfavorable
evolution of the s = 2 coupling coefficient. This is a criterion that gives a lower-s mode a dominant advantage over
a higher-s mode.
(3) Among the s = 1 modes, the one with the lowest kz (instead of the lowest Ist) has a competitive advantage.
This criterion plays an insignificant role in an s = 1 gyrotron because one can always tune the magnetic field to
favor a low-kz (e.g. 1 ) mode. However, it governs the competition among the s = 1 modes when the magnetic
field is tuned in favor of an s>1 mode as in the present case. For the reason discussed at the end of Sec. IV-A, this
criterion can be generalized to the competition between any two modes, with the same or different cyclotron
harmonic numbers.
In multiple-mode competitions, more than one criterion may be at work. In this case, all three criteria could
work in favor of one mode (e.g. an early-starting, low-kz, s = 1 mode) or two criteria play opposing roles [e.g. an
early-starting, low-kz, s = 2 mode competing with a low-kz, s = 1 mode with a higher Ist, as in Fig. 4(d-f)]. One
criterion does not necessarily override an opposing one. The outcome of the competition depends on the relative
weight of the criteria, which in turn depends on the relative magnitude of kz, the separation of Ist of the modes
involved, as well as the peak Ib value. Hence, a clearer picture lies in the details in the Ist versus B chart.
Although criteria (1) and (3) do not bias a particular cyclotron harmonic number, criterion (2) is inherently in
favor of a lower-s mode. Thus, overall, a higher-s mode is much more likely to be suppressed by a lower-s mode,
as in Fig. 4(d-f) and [13, 14, 16, 19, 20, 30, 31].

20. Analysis of the case inTable I
(r1 = 0.35 cm, Vb=35 kV, α=2, Δvz/vz=6%, Rc/Rw=0.3192, uniform B-field, etc.)
1. At B = 17.78 kG, as Ib rises to 0.93 A, the TE02 (s=2) mode will be the first mode excited. It will remain in
single-mode operation until Ib=1.51 A (Pout ~3 kW, η~5.7%) which is the linear Ist of the TE21 (s=1) mode.
As Ib rises further to Ib=2.76 A, it hits the linear Ist of the TE11 (s=1) mode. We have assumed α=2 for all Ib.
2. In the competition between the early-starting, low-kz,TE02 (s=2) mode and the two, higher-kz, s=1 modes,
Criteria (1) and (3) favor the TE02 (s=2) mode, while Criterion (2) favors the s=1 modes. By Criteria (1) and
(3), it is likely that the TE02 (s=2) mode can survive up to Ib~2.5 A (Pout~7.5 kW, η~8.7%) before it is
eventually suppressed by the TE21 (s=1) mode. By Criterion (3), the highest-kz, TE11 (s=1) is less
competitive than the TE21 (s=1) mode.
3. If Criteria (1) and (3) dominate over Criterion (2) at still higher Ib (e.g. 5A), the TE02 (s=2) mode may
remain in single mode operation with significantly higher Pout (e.g. Pout ~18 kW and η~10.3% at Ib~5 A).
4. From cases studied in the above reference, it is unlikely that the TE02 (s=2) mode can operate at a more
optimal B-field (e.g. 17.6 kG) by suppressing the early-starting TE21 (s=1) mode.
Scenario 2 is reasonable, but Scenario 3 may be too optimistic. Unfortunately, no one here knows how
to run our time-dependent, multi-mode code to verify Scenarios 2 and 3. It will take ~2 months to get one
data point anyway.