17 

 

 

  

 

 

                           Gyrotron Development at High Order Modes                 

  

 

  

      

      

                                                                  

                               
 

 

 

  Abstract 

      The present work has been characterized by higher order modes in the cavities of the 

Gyrotron; they are capable of producing RF plasma by developments of it. It uses for fusion 

systems. We choose the TE31,8 mode in our study. The main problem of gyrotron is the 

device of the thermal cavity loading. The problem of the thermal loading is solved when any 

parasitic modes suppress, absence of desired modes; the thermal loading is increased when 

the high power tube of gyrotron operation is unstable. The mathematical interaction model 

contains equations that describe the electron motion and the field profiles of the transferred 

electric modes of the resonator, these are interacting with electrons based on the finite 

difference method that has been designed to study the starting current, the frequency, quality 

factor and calculates the roots of the Bessel function by the program we designed in Fortran 

language. They are used to calculate the operation frequency. Good agreement is between 

our results and the previous published results both confirm the accuracy of the performance 

of the designed program. 

Keywords: developed gyrotron, high order mode, RF plasma. 

1.Introduction  

   The gyrotron is a microwave source capable of producing high power through the 

interaction between the beam and the wave.[1] convert the kinetic energy of electrons into 

radiation. as shown in Figure(1). The gyrotron oscillator consists of these components: a 

gun oscillator is used as a microwave source. It operates at a frequency of (670) GHz and 

releases a power of approx. (11W), interaction circuit (cavity), magnetic field source 

(magnetic or solenoids), spent electron beam collector, output taper and window. In the 

injection gun, a magnetron is generated annular electron beam; it is focused into an open 

Article history: Received  31  August, 2020, Accepted 31,Augest 2020, Published in October 2021. 

 

Ibn Al Haitham Journal for Pure and Applied Science 

Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index 

 

Doi: 10.30526/34.4.2697 

Luma Y. Abbas 

Department of Physics, College 

of Science, The University of 

Mustansiriyah, Baghdad, Iraq. 

lmaphysics2203@gmail.com 

 

Nedal ali Hussain 

Department of Physics, College of 

Science, The University of 

Mustansiriyah , Baghdad, Iraq. 

nidalali44@yahool.com 

 

Ali N . Mohammed 

Department of Physics,College of 

education, The University of 

Mustansiriyah,Baghdad, Iraq. 

aliphysics2203@gmail.com 

 

lmaphysics2203@gmail.com
nidalali44@yahool.com
aliphysics2203@gmail.com


Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

18 

 

cavity resonator with the axial magnetic field. In the cavity of this tube, the kinetic energy is 

converted into an RF from the beam. Finally it is converted into a Gaussian beam. Then the 

spent beam is collected at the collector. to produce high powers at millimeter wavelengths 

.We choose a high order cavity mode such as TE31,8-TE10,6-TE8,5. To compensate for the high 

wall losses in the cavity at a high frequency the modern gyrotron development during the 

last three or four years One may notice that several principal steps we made in the 

development of the megawatt power gyrotron: 

1. At a  very high order volume modem, solve the problem of thermal cavity loading in 

gyrotron.  

2. Demonstration of the depressed collector to solve gyrotron collector and power 

supply problems. 

3. The windows are made of artificial diamond discs. These three points important to be 

used in the gyrotron in the plasma experiment.[2,3] 

 
Figure 1. Schematic simple of oscillator (gyrotron) [1] 

   In the past years, Danly et al. report using a novel quasi –optical output antenna. The first 

experiment on 100 GHz and power (200 kW) are discussed by Bensimhon et al. and show 

the stable operation is possible with a high – order mode, complex interaction cavity. Saito 

et al. designed a 137 GHz gyrotron for plasma scattering. [4] 

Jagadishwar Sinigin [5] in 1999 presented the theory and design of gyrotron-traveling wave 

amplifiers at millimeter wave frequencies suitable for long distance communication and 

radar. The design goals for the tube were 100 kW peak power at 140 GHz and 30% 

efficiency. 

In 2002, Edith Borie et al. [6] studied the operation of multi-frequency and a 149 GHz 

gyrotron was designed to operate in the high order mode at high frequency in GHz. 

   The design of an 8 mm TE13 mode gyrotron was given by Wenxiany, Wang et al. [7]. 

Tsai, et al. [8] in 2004 presented a theoretical study of fundamental cyclotron harmonic 

gyro-TWT at TE01 mode, with distributed wall losses. The initial theoretical, [8] in 1996 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

19 

 

reported the short-and-long pulse operation of a TE10.4; mode and the experiment 

configuration of the tube was described. 

2.Numerical method 

  We get Bessel‘s equation with Bessel function Jm(r) by using a method of variable 

separation. We considered wave propagation on a cylindrical waveguide, as shown in 

Fig.1[9] 

 The longitudinal electric and magnetic fields 𝐸𝑧  and 𝐻𝑧 , in a circular waveguide are: 

𝐻𝑧 = 𝑐𝑚
ˏ

𝐽𝑚(𝑘𝑐 𝑟)  𝑠𝑖𝑛 𝑠𝑖𝑛 (𝑚ɸ)                            (1) 

𝐸𝑧 = 𝑐𝑚𝐽𝑚(𝑘𝑐 𝑟)  𝑐𝑜𝑠 𝑐𝑜𝑠 (𝑚ɸ)                           (2) 

Where 𝑐𝑚 and 𝑐𝑚
ˏ

 denote the coefficients of fields, r the radial distance, 𝐽𝑚(𝑘𝑐 𝑟) is called the 

Bessel function of the first kind, and m is the order of the Bessel function. Subsitituting these 

equations into the Maxwell equation, we  get field components in terms of  𝐸𝑧   and 𝐻𝑧 in 

circular waveguide [9]. 

𝐸∅ =
1

𝑘𝑐
[
𝑗𝑚𝑘𝑧

𝑟
𝑐𝑚𝐽𝑚(𝑘𝑐 𝑟) 𝑠𝑖𝑛 𝑠𝑖𝑛 (𝑚∅)  + 𝑗𝜔𝜇𝑘𝑐

,
𝐽𝑚

,
𝑐𝑜𝑠 (𝑘𝑐 𝑟)𝑐𝑜𝑠 (𝑚∅) ]               (3) 

𝐻∅ =
1

𝑘𝑐
[
𝑗𝑚𝑘𝑧

𝑟
𝑐𝑚

,
𝐽𝑚(𝑘𝑐 𝑟) 𝑐𝑜𝑠 𝑐𝑜𝑠 (𝑚∅)  + 𝑗𝜔 𝑘𝑐 𝐽𝑚

,
𝑐𝑜𝑠 (𝑘𝑐 𝑟)𝑐𝑜𝑠 (𝑚∅) ]              (4) 

Where Cm, 𝐶𝑚
,

 are the coefficients of longitudinal field, Kc is the wave number, is the 

propagation constant, r is the radial distance, 𝐽𝑚(𝑘𝑐 𝑟) is called the Bessel function, 𝐽𝑚
,

(𝑘𝑐 𝑟) 

is the derivative, and m is the order of the Bessel function. 

𝐸(𝑟, ∅, ᴢ) = 𝐸(𝑟, ∅)𝑒𝑥𝑝[𝑖(𝜔𝑡 − 𝑘𝑍 )]                                                                                (5) 

Where t represents the time and 𝜔 the angular frequency. 

3.Design of cavity 

   In (28-170GHz), the magnetron gun produced an annular beam and transported it to the 

cavity [10, 11]. In the middle cavity reach the interaction between the electron beam and RF 

wave when RF fields peak values. The waveguide is closed by a pinhole diaphragm, through 

which part of the generated microwave power escapes. The interaction efficiency is defined 

as: [12]   

𝜂
𝑖𝑛𝑡

= [
𝛼2

2(1 − 𝛾0
−1)

] 𝜂⟘                                                                                                     (6) 

Where  𝜂
⟘

is the orbital efficiency 𝛼, is the ratio between initial orbital velocity normalized 

and the speed of light, 𝛾
0
is the Lorentz factor. The orbital efficiency is the fraction of the 

energy of electro gyration transformed into electromagnetic radiation.  

𝑃𝑜𝑢𝑡 = (
𝜔

𝑄𝐷
) 𝑊                                                                                                               (7) 

𝑃𝛺 = ((
𝜔

𝑄𝛺
)W                                                                                                                  (8) 

Where W is the energy stored in the cavity, 𝑄𝐷 is the diffractive Q-factor and 𝑄𝛺 is the 

ohmic quality factor of the TE_ modes.   



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

20 

 

From Equ. (7,8) These are power inversely proportional to the diffractive and ohmic quality 

factors.[13] 

𝑄𝛺 = (
𝑅𝑊
𝛿

) (
1 − 𝑚2

𝑋𝑚,𝑛
2

)                                                                                    (9) 

Where 𝑅𝑊 is the cavity radius, m is the azimuthal index of the mode, n is the radial index, 

Xmn is eigen number and 𝛿 is skin depth.𝛿 =
1

√𝜋𝑓𝜎𝜇0
 where 𝜎 is the conductivity of the 

cavity wall. The cut off region, prevents backward waves towards the gun. Plasma will be 

formed in the breakdown region. When the electron density is high enough, a significant part 

of the wave beam should be reflected from dense plasma, where plasma fill mentation in a 

gyrotron focused wave beam was observed [13,14] . The plasma density inside the volume is 

approximate by [15]. 

n= 𝑛𝑜 𝑒𝑥𝑝(𝑣𝑖 , 𝑒𝑓𝑓𝑡)                                                                                   (10) 

Where vi, eff is the effective ionization frequency and is dependent on the power density 

over power threshold density. The quality diffraction is changing with a length of the cavity 

and the wavelength.[16]  

𝑄𝑑𝑖𝑓𝑓 𝛼(
𝐿

𝜆
)2                                                                                               (11) 

The relation (11) leads to increased ohmic losses in the cavity by equ.(12) 

𝜂𝛼
𝑄𝑜𝑚𝑖𝑐

𝑄𝑑𝑖𝑓 + 𝑄𝑜ℎ𝑚
                                                                                      (12) 

Where 𝜂 is the total efficiency. 

4.Results and Discussion  

   The quality factor is depends on cavity sizes and the parameters of the electron beam, it is 

very critical parameters for the thermal loading of the cavity, to get on high gyrotron 

efficiency and reducing the ohm losses for TE31,8- mode at a frequency of 670 GHz the wall 

radius is 4.477mm, power is enhanced with cavity size increase, typical QD-factor for fusion 

gyrotron is 1000-1500, Qohm-factor depend  on the cavity size and wavelength. We describe 

the interaction of wave-particle by the computer program (Gyro).This code uses the forward 

finite difference technique as a numerical method for solving all governing differential 

equations in three dimensions (r,∅,z). The input data of the program that beam voltage, the 

external magnetic field, normalized cavity length, the velocity ratio, energy, the mode 

number (m,n,l), and root of Bessel Xmnl. This code calculates the power, efficiency, quality 

factor and beam current in the tube with a single cavity show this result in a Table.1, 2. 

 

Table1. Reported and calculates results of design parameters of a 670-GHz. 

TE 31,8 Our Results Ref [13] 

Beam voltage (kV) 71 70 

Axial velocity ratio 1.3 1.3 

Normalized length 10 10 

The frequency(GHz) 671 670 

Radius of the cavity (mm) 4.54 4.54 

 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

21 

 

 

Table 2. Reported and calculates results of design parameters of a 670-GHz. 

TE 31,18-Mode Our Results Ref.[10, 11 ] 

Vb (kV) 71 72.5 

L/𝜆 5.5 6.4 
Ib (mA) 70 50-70 

Qdif,min 1123 1200 

Axial velocity ratio 1.4 1.4 

The quality factor  given by equation (11) for TEmnl-mode[18] is represented graphically in 

Figuer2 

𝑄𝛿

𝜆
=

[1 − (
𝑚

𝑋𝑚𝑛
)

2

] [𝑥𝑚𝑛
2 + 𝑃2𝑅2]

3
2

2𝜋 [𝑋𝑚𝑛
2 + 𝑃2𝑅3 + (1 − 𝑅) (

𝑃𝑅𝑚
𝑋𝑚𝑛

)
2

]

                           (11) 

Where R=D/L , P=l𝜋/2 and 
𝑄𝛿

𝜆
 is the quality factor a function of  the cavity dimensions. 

Those curves in fig.2 correspond to simulation results of Gaussian Profiles. In this figure, we 

see a maximum of 
𝑄𝛿

𝜆
 when the diameter equal the length because the minimize losses. 

 
Figure 2. QD/L for TE31,8 mode in the cavity. 

To study the relation between (B0D)
2 and (D/L)2 for TE31,8_ mode, we used Eq.12 [18] as 

shown in fig.3. 

(𝐵0𝐷)
2 = (

𝛾𝑐𝑚𝑒
𝑒

)
2

[4𝑥𝑚𝑛
2 + 𝜋2𝑙2 (

𝐷

𝐿
)

2

]                                   (12) 

0.78

0.79

0.8

0.81

0.82

0 0.5 1 1.5 2 2.5 3 3.5

Q
D

/L

D/L



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

22 

 

 
Figure 3. The relation between(B0D) and (D/L)

2 for TE31,8mode. 

We calculated the starting currents by depending on the equation (13) as[19]  

  𝐼𝑆 = −1.5608 × 10
−21.

𝛾𝜔𝑣𝑧
2

𝑄

𝑙2𝑟𝑤
2

𝐿

(𝑋𝑚𝑛
2 −𝑚2)

𝑋𝑚𝑛
2 [𝐽𝑚+1

2 (𝑋𝑚𝑛𝑟𝑜/𝑟𝑤)+𝐽𝑚−1(
𝑋𝑚𝑛𝑟𝑜

𝑟𝑤
))]

[𝐺−
𝐵

⟘𝜔𝑐
2

2𝐾𝑧𝑉𝑧

𝑑𝐺

𝑑𝑋
]

(𝐴𝑀𝑃)   (13) 

Where  𝐵⟘ = (𝑣𝑟
2 + 𝑣𝜑

2 )
1

2/𝐶      ,X= (𝜔𝑐 − 𝜔)/𝐾𝑧𝑉𝑧 

And  

 G= −
1

2
[(−1)𝑙 𝑐𝑜𝑠 𝑐𝑜𝑠 (𝑙𝜋𝑥) − 1 ]/(𝑥2 − 1)2 

   From fig.4 the starting current of TE-mode was calculated as a function of magnetic field, 

when a minimum starting current for excited a single- mode in the interaction region. It is 

smaller if the magnetic field increased until reaching a point where it rises abruptly.   

 
Figure 4. Start oscillation current of cavity TE31,8. 

   The operating modes are set in the cavity resonator and we could predict the radius of this 

cavity by mode chart Figures (5,6,7). We can estimate that the magnetic field is required to 

excite the TE mnl-mode of interest. 

11.46

11.48

11.50

11.52

11.54

11.56

11.58

0 0.5 1 1.5 2 2.5 3

B
o

D
  
(T

.m
)

(D/L)

2

4

6

8

9.5 10 10.5 11

S
ta

rt
in

g
 c

u
rr

e
n
t

(A
)

Main  magnetic field(T)



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

23 

 

 
Figure 5. The TE31,8 modes in circular cylinder. 

 

 
Figure 6. TE8,5 - modes in circular cylinder. 

  

 
Figure 7. TE10,6-modes in circular cylinder. 

5.Conclusions 

        The present paper studies the status of gyrotron development on average power 

achievements. A plasma heating like magnetic fusion experiments depend on the parameter 

average power to produce electric power. The system is usually desired with  a duty factor 

1E+23

1.05E+23

1.1E+23

1.15E+23

0 2 4 6 8

(F
D

)^
2

G
H

z
.C

m
^

2

D/L

5E+22

6E+22

7E+22

0 2 4 6 8

(F
D

)2
G

H
z

.C
m

^
2

(D/L)2

7E+22

8E+22

9E+22

1E+23

1.1E+23

0 1 2 3 4 5 6 7 8 9

(F
D

)2
(C

m
2
)

(D/L)2



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

24 

 

typically 10% for radar application, the gyrotron oscillator developers in at frequencies in 

GHz and high power in megawatt. 

Acknowledgement 

All thanks and gratitude to prof. Dr. Jamal W. Salman. 

References 

1. Nusinovich, G. S. Introduction to the physics of gyrotrons, Baltimore: Johns 

Hopkins University press, 2004; ISBN: 0801879213 9780801879210. 

2. Yeap, K. H.; Tham, C. Y.; Yeong, K. C.; woo, H. J.  Wave Propagation In Lossy 

And Superconducting circular, RADIOENGINEERING, June 2010,19(2), 320-325, 

doi: 10.5277/oa.160215. 

3. Cheng, Dok.,Field and Wave Electromagnetic Fields, Addison-Wesly, 1983,66-557, 

ISBN 0201012391. 

4. Granatstein, V .L.  ; Robertson, C. W.  Guest Eds. Third special issue on free- 

electron lasers, IEEE J. Quatum Electron.July 1985, QE-21, 7, 804-1119, 

doi:  10.1109/JQE.1985.1072763. 

5. Jagadishwar, Sirigiri R.,Theory and Design study of a novel Quasi-Optical Gyrotron 

Traveling wave Amplifier, M. Sc. Thesis, Massachusetts Institute of Technology, 

Cambridge 1999. 

6. Edith Borie, possibilities for multi-frequency operation of a Gyrotron at FZK’, IEEE 

transaction on plasma sciences, June 2002, 30(3), 825, 

doi: 10.1109/TPS.2002.801561. 

7. Wenxiang Wang, Guofen Yu, Meisheng Xu & Yubin Gong, 8mm TE 13 Mode 

Gyrotron, Int. J. of Infrared and Millimeter Waves, 2003, 24, 661–668, 

doi: 10.1023/A:1023307514266 

8. Tsai, W.C.; Chang, T. H.; Chen, N. C.; Chu K. R. Absolute instabilities in a high-

order- mode Gyrotron traveling-wave amplifier’, physical review E, 2004, 70, 

056402, doi: 10.1103/PhysRevE.70.056402 

9. Gunter Dammertz, ;Oliver Braz,; Christors T. ;Iatrou, M. Kuntze, Arnotd Mobius, 

Bernhard Piosczyk, and Manfred Thumm, Long Pulse Operation of a 0.5 Mw TE10,4 

Gyrotron at 140 GHz, IEEE transaction on plasma Sciences, June 1996, 24, 3, 570, 

doi:  10.1109/27.532939. 

10. Sophocles J. Orfanidis, Electromagnetic Waves and Antennas, 2016, Online book 

at: www.ece.rutgers.edu/~orfanidi/ewa. 

11. Nusinovich, G. S.; Read, M, E,; Dumbrajs, O. ; Kreischer, K. E.Theory of gyrotrons 

with coaxial resonators, IEEE Trans.Elecron Devices, March 1994, 41(3), 433-438, 

doi: 10.1109/16.275231. 

12. Bratman, V. L.; Ginzburg, N. S.; Nusinovich, G. S.; Petelinand, M. I. ; Strelkov, P. 

S. Relativistic gyrotrons andcyclotron autoresonance masers, Int. J. Electron, 1981, 

51(4), 541-567, doi: 10.1080/00207218108901356. 

13. Hidaka, H.; Choi, E. M.; Mastovsky, I.; Shapiro, M. A.; Sirigiriand, J. R.; Temkin, 

Observation of Large Arrays of Plasma Filaments in Air Breakdown by 1.5-MW 

110-GHz Gyrotron Pulses, Phys. Rev. Lett., January 2008,  100,  035003, doi:   

                        doi.org/10.1103/PhysRevLett.100.035003. 

https://www.abebooks.co.uk/products/isbn/9789332535022/30689081600&cm_sp=snippet-_-srp1-_-PLP1
https://doi.org/10.1109/JQE.1985.1072763
https://www.researchgate.net/deref/http%3A%2F%2Fdx.doi.org%2F10.1109%2FTPS.2002.801561
https://www.researchgate.net/deref/http%3A%2F%2Fdx.doi.org%2F10.1023%2FA%3A1023307514266
https://www.researchgate.net/deref/http%3A%2F%2Fdx.doi.org%2F10.1103%2FPhysRevE.70.056402
https://www.researchgate.net/deref/http%3A%2F%2Fdx.doi.org%2F10.1109%2F27.532939
http://www.ece.rutgers.edu/~orfanidi/ewa
https://doi.org/10.1109/16.275231
https://www.researchgate.net/deref/http%3A%2F%2Fdx.doi.org%2F10.1080%2F00207218108901356
https://doi.org/10.1103/PhysRevLett.100.035003


Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

25 

 

14. Nusinovich, G. S.; Milikh, G.M.  ; Levush, B. L. Removal of halocarbons from air 

with high‐power microwaves, Appl. Phys, 1996, 80(7), 4189-4195, doi: 

10.1063/1.363559. 

15. Sinitsyn, O. V.  ; Nusinovich, G. S.  ; Antonsen, T. M.  Jr.,Stability of gyrotron 

operation in very high- order modes,  Phys. Plasmas, 2012, 19(6), 063114, doi: 

10.1063/1.4729329. 

16. Bratman, V. L.; Zorin, V. G.  ; Kalynov, Yu. K.; Koldanov, V. A.; Litvak, A. G.; S. 

Razin, V.; Sidorov, A. V.  ; Skalyga, V. A.  Plasma creation by terahertz 

electromagnetic radiation, Phys. Plasmas, 2011, 18(8), 083507, doi: 

https://doi.org/10.1063/1.3622202. 

17. Hornstein, M. K.; Bajaj, V. S.; Griffin, R. G.; Kreischer, K. E.; Mastovsky, I.; 

Shapiro, M. A.; Sirigiri, J. R.; Temkin, R. J. Second Harmonic Operation at 460 

GHz and Broadband Continuous Frequency Tuning of a Gyrotron Oscillator, IEEE 

Transactions on Electron Devices,May 2005,: 52(5), 798 – 807, 

doi: 10.1109/TED.2005.845818. 

18. Nusinovich, G. S.  Introduction to the physics of Gyrotrons, Baltimore: Johns 

Hopkins University press, 2004,  ISBN: 9780801879210. 

19. Salman, J. W. ; Atlifah, N. ; Hasaani, A. S.  Iraqi J. Sci., 1996, 37, 37, 1163-1183. 

 

https://www.researchgate.net/deref/http%3A%2F%2Fdx.doi.org%2F10.1063%2F1.363559
https://www.researchgate.net/deref/http%3A%2F%2Fdx.doi.org%2F10.1063%2F1.4729329
https://doi.org/10.1063/1.3622202
https://doi.org/10.1109/TED.2005.845818