FIR Center Report

FIR FU-99 January 2010

PHYSICAL MODELS AND COMPUTER

CODES OF THE GYRSIM (GYRotron

SIMulation) SOFTWARE PACKAGE

S. Sabchevski, T. Idehara, T. Saito,

I. Ogawa, S. Mitsudo, Y. Tatematsu,

Research Center for Development of

Far-Infrared Region

University of Fukui

Bunkyo 3-9-1, Fukui 910-8507, Japan

Tel 81 776 27 8657

Fax 81 776 27 8770

PHYSICAL MODELS AND COMPUTER

CODES OF THE GYRSIM (GYRotron SIMulation)

SOFTWARE PACKAGE

S. Sabchevski1,2

, T. Idehara1, T. Saito

1,

I. Ogawa1, S. Mitsudo

1, Y. Tatematsu

1,

1Research Center for Development of Far Infrared Region

University of Fukui, 3-9-1 Bunkyo, 910-8507 Fukui, Japan

2Institute of Electronics of the Bulgarian Academy of Sciences,

1784 Sofia, Bulgaria

Abstract

In this FIR FU Center Report we present the current status and the capabilities of the

GYRSIM software package (which stands for GYRotron SIMulation), that has been

developed here and is used by the researchers of Center for analysis, optimization and

computer aided design (CAD) of powerful, high-frequency gyrotrons through numerical

experiments. We depict briefly the physical models implemented in the codes and give

some working notes that are intended to facilitate the researchers in finding the necessary

information (references, test examples, prerequisites) about the main characteristics of the

available software tools for modeling and simulation of gyrotrons. Therefore the Report

is intended to be both an inventory of the available software tools and a guide for finding

more detailed information about the models and the computer codes.

Key words: GYRSIM package, physical models, modeling and simulation of gyrotrons,

computer aided design (CAD)

Physical Models and Computer Codes of the GYRSIM

(GYRotron SIMulation) Software Package

S. Sabchevski, T. Idehara1, T. Saito

1,

I. Ogawa1, S. Mitsudo

1, Y. Tatematsu

1,

1. Introduction

In order to assist both the computer aided design (CAD) and the study of powerful

high-frequency gyrortons that are being developed in the Research Center for

Development of Far-Infrared Region (FIR FU Research Center) at the University of

Fukui a number of physical models were formulated and implemented in computer codes

during the recent years. These simulation tools were used successfully for analysis,

optimization and CAD of two series of devices, namely Gyrotron FU Series and Gyrotron

FU CW Series as well as for education of our students. Their computational modules are

under a continuous adaptation to the changing computational infrastructure (operating

systems, computing platforms, numerical libraries) and are undergoing numerous

modifications aiming better performance (increased execution speed, debugging, more

convenient structure of the input/output data etc.) as well as inclusion of new physical

models. A work on the development of novel computer codes and on the integration of

numerous already available stand alone programs, numerical libraries and subroutines in

a problem oriented software package is in progress now. We call this union of tools for

numerical simulation of gyrotrons GYRSIM package (which stands for GYRotron

SIMulation).

Recently, on a Seminar of the Research Center the current status of the software

package has been reviewed and discussed together with some directions for its further

development. It has been suggested to prepare a short description and inventory of the

available tools in order to facilitate their use. This motivated us to write the current short

report with a hope that it will be helpful and could be the first step toward the preparation

of a more complete documentation (user guide, reference guide and so on).

The report is organized as follows. In Sect. 2 we present the physical models and the

program implementation of a sub-package used for simulation of the beam formation in

the magnetron injection gun (MIG) and in the entire electron-optical system (EOS) of the

gyrotron tube. This material is focused mainly on the most important component of the

package, namely the GUN-MIG/CUSP suite of codes. In the Sect. 3 we give similar

information about the computer codes used for analysis and design of the resonant

structure (resonant cavity) of the gyrotron. In Sect. 4 we mention briefly some additional

software package used in the course of CAD of gyrotrons. Some conclusions and outlook

are presented in Sect. 5. A collection of illustrative examples that elucidates both the

input and the output of the codes is presented in several appendices.

The flowchart of the process of CAD is shown in Fig. 1. Usually it begins with a

specification of the desired parameters (notably the output frequency and the output

power, tunability range etc.). During this stage some set of specific requirements (for

example pulse length or CW mode of operation, stability of the output parameters,

coupling of the radiation to the output window and the transmission line, constrains to the

weight and linear dimensions, as well as constrains imposed by the available power

supplies and so on) has to be formulated as well. Very often some previously developed

tube is being selected as a prototype and initially its parameters are scaled using well

known physical relations (and usually approximate ones, e.g. based on the adiabatic

theory). This being done, the CAD starts with the so called conceptual design. During

this step one usually selects the types of the components (for example diode or triode

magnetron injection gun, conventional or coaxial resonant cavity, demountable or sealed-

off tube, radial or axial coupling of the radiation and so on).

CAD OF

THE

EOS

TARGET

PARAMETERS

CAD OF

THE

CAVITY

CONCEPTUAL

DESIGN

EXPERIMENT

CHANGE

EOS AND /

OR CAVITY

CAD OF EOS

AND/OR

CAVITY

DESIGN

OTHER

COMPONENTS

MECHANICAL DESIGN

Meet

target?

YES

NO

FABRICATION

INITIAL DESIGN

OPTIMIZED DESIGN

Fig. 1 Flow chart of the computer aided design (CAD) of a gyrotron

The conceptual design is followed by a CAD of the cavity resonator performing a

series of numerical experiments using a set of physical models and computer codes. We

describe them briefly in Sect. 3. It is clear that each cavity design is performed assuming

some parameters of the electron beam. Therefore, the next step, namely the initial design

of the electron-optical system, is an iterative process during which the main goal is to

find a configuration (MIG, magnet, beam tunnel, collector etc.), which provides such

desired beam parameters or at least parameters close to them. Both the physical models

and the computer programs used for CAD of the EOS we present briefly in Sec. 2. The

initial design is then followed by another iterative process in which the configuration and

the parameters of both the cavity and the gun or only of one of these most essential

components are changed until a satisfactory solution is obtained. Thus the optimized

design is the one which satisfied the target parameters. Based on it, a design of the

remaining components (e.g. the output window, quasi optical system) is carried out. The

final stage of the CAD is the mechanical design of the tube. It includes also a selection of

appropriate materials and technologies for manufacturing of the device.

2. Computer codes for trajectory analysis (ray tracing) of magnetron injection

guns (MIG) and electron-optical systems (EOS) generating helical and axis-

encircling electron beams

2.1 General description of the GUN-MIG/CUSP software package

The beginning of the GYRSIM package can be traced back to the development of

its first component, namely the trajectory analysis (ray tracing) code GUN-MIG [A1,A2].

It is based on a self consistent physical model which takes into account the relativistic

and space-charge effects and is formulated in a six-dimensional phase space of three

cylindrical coordinates and their corresponding momenta (velocities) but the electrostatic

field is considered axially-symmetric, i.e. the potential distribution depends only on the

radial and the longitudinal (axial) coordinates but not on the azimuthal angle. Therefore,

the physical model implemented in this code could be classified as a two and a half

dimensional (2.5D) one. It consists of a system of equations that includes: (i) Poisson

equation for the electrostatic potential with Neumann type boundary condition along the

axis of the meridional cross-section of and Dirichlet boundary conditions along the rest of

the boundary contour; (ii) normalized relativistic equations of motion in the electrostatic

field and an ambient magnetic field; (iii) a set of equations describing the extraction of

the beam current from the emitter in different regimes (see below). The computational

modules (functions, subroutines, auxiliary programs) are written in Fortran 77. An

inventory of the main components of the software package is presented in Table 1.

The boundary value problem for the electrostatic potential (Poisson equation) is solved

on a sequence of grids with increasing finesse (gradually decreasing mesh step) by

means of an iterative procedure based on the finite difference method (FDM) with a

successive over-relaxation (SOR) and automatic selection of the an optimal relaxation

parameter. The electric field (gradient of the potential) is calculated using a Lagrange

interpolation on a sub-grid of 4x4 nodes. The convergence of the solution can be

increased additionally through a gradually increasing the temperature of the emitter

during the first several iterations. Their number is specified by the user in the input file.

The code uses a magnetic field profile which is specified by a polynomial

approximation or is read from an input file. It should be noted that the package GUN-

MIG/CUSP consists in fact of two modifications. The second one, namely GUN-CUSP

was developed to study and design MIG forming axis encircling electron beams for large

orbit gyrotrons (LOG). In such systems there is always present a magnetic field reversal

which, generally speaking, could be quite abrupt (like in a magnetic cusp). In order to

treat with an increased precision the approximation of the magnetic field higher

derivatives in the paraxial approximation must be used. For their calculation a high order

polynomial approximation is performed on the axial magnetic field distribution (obtained

experimentally or calculated by another code for example POISSON/ SUPERFISH,

FEMM or by our own program COILS) using the preprocessing program MAGFI (which

is also a part of the package).

The geometry of the simulated EOS is described by the boundary contour which

is defined as a sequence of points connected by simple elements such as straight lines and

arcs of circles. This information is read from an ASCII file which contains also the rest

of the parameters of the simulation which can be divided in two groups, namely physical

parameters (e.g. the accelerating voltage, temperature of the cathode, saturation current

density of the emitter and so on), and computational parameters that characterize the

simulation (for example the accuracy of the solution of the boundary value problem;

maximum number of iterations, the initial value of the relaxation parameter, the scope of

the and form of the printed output etc.). A typical file with input data is shown in the

Appendix 1. The user can simply modify the corresponding parameters and keys

according to her/his case using as a reference the comments that explain each variable. In

more details the physical nodel is depicted below.

Table 1

GUN-MIG/CUSP software package

Module/Code Functionality Implementation

GUN-MIG Main program Fortran 77

GUN-CUSP Main program Fortran 77

GUN-LIB Library of functions and

subroutines

Fortran 77

COILS Calculates the magnetic

field produced by a set of

coils

Fortran 77

MAGFI, SCALLING Preprocessing and

polynomial approximation

of the magnetic field

Fortran 77; requires the

package DPOLFT

GUN-VIEW

CUSP-VIEW

Post-processing programs

for visualization of the

results

Fortran 77; Lahey F77-

EM/32 graphic library or its

replacement, e.g. PGPLOT

EGUN A legacy code for reference

and benchmarking

Fortran 77 (various

modifications)

2.2 Main equations of the physical model implemented in the GUN-

MIG /CUSP package

As already mentioned, the physical model is formulated in a cylindrical

coordinate system ( R Z, , ), where the electron trajectories can be found integrating the

relativistic equation of motion:

d mv

dte E v B

( )( )

, (1)

ZkRjRiv ZR

. (2)

Here m and e are the mass and charge of an electron, E and

B are the electric and

magnetic fields. The velocity vector v

is represented by its components along the unit

vectors iR ,

j and

kZ . In a component form (1) can be written as [1,2]

( ) ( ) /Rm

fe

cR RE R E ZE R

11

0

2 1 2

2

2 R R Z

(3)

( ) ( )

/

11

2

0

2 1 2

2m Rf

e

cR RE R E ZE

R

RR Z

, (4)

( ) ( )/Zm

fe

cZ RE R E ZE

1

10

2 1 2

2 Z R Z

, (5)

where f eE e R B ZBR R Z ( ) , f eE e ZB RB ( )R Z

, f eE e RB R BZ Z R ( ) , and

2 2 2 2 2 [ ( ) ] /R R Z c .

The axial and the radial components of the electric field ( EZ

z

, E

RR

,

0E ) are calculated from the electrostatic potential distribution ),( ZR , which obeys

the Poisson’s equation

1 2

2

0R RR

R Z

, (6)

as well as the Dirichlet ( i on a segment i of the boundary i ) and Neumann

( 0/ z along Z at R =0) boundary conditions as mentioned above. Here is the

space charge density and 0 is the permittivity of vacuum. The electric field (E ) in an

arbitrary point at the current particle position is calculated as a gradient of the

electrostatic potential on the computational grid using the following Lagrange

interpolation and 4x4 nodes sub-mesh [3]

( , )R Z a bj

ii

j ji

1

4

1

4

, E adb

dRR i

j

jiji

1

4

1

4

, E bda

dZZ j

i

jiji

1

4

1

4

. (7)

Here

a Z Z Z Z Zi i kk k i

kk k i

( ) ( ) ( ), ,

1

41

1

4, b R R R R Rj j k

k k jk

k k

( ) ( ) ( ), ,

1

41

1

4 (8)

The off-axis components of the magnetic field are computed from the following

paraxial expansions [4]

B R Z BR d B

dZ

R d B

dZ

R d B

dZZ ( , )

2 2

2

2 4

4

4 6

64 16 576 , (9)

5

54

3

32

19282),(

dZ

BdR

dZ

BdR

dZ

BdRZRBR , (10)

where ),0( ZBB is the axial magnetic field. In the current version of the code the self-

magnetic field of the beam current is neglected.

The space-charge distribution is computed using a combination of the particle-in-

cell method (PICM) with the area-weighted algorithm (AWA) for allocation of the

charges to the mesh [5]. The cells are defined by the same mesh which is used for the

solution of the boundary-value problem. The electron beam is represented by a finite

number of N rays, each carrying a fraction Ii of the total beam current I Ib ii

N

1

. For

each time step t inside a given cell, the charge Q I td i . , is deposited to the four

adjacent vortices of the cell (mesh nodes) according to the relation

QH

H

H

HQn

R Zd

1 1 , (11)

where HR and HZ are the distances along R and Z axes between the corresponding n-th

node and the position of the particle in the middle of the ray region considered. The

space-charge density in the node with radial coordinate Rn is then given by

nn

n

Q

H R

2 2 . (12)

Allocating n and summing up the contribution of all rays the charge density in each

node is obtained. For calculation of the current extracted from the emitter the region in the vicinity of

the cathode is divided into a number of small virtual diodes in which the current is

governed by the potential distribution and the initial velocities of the thermoelectrons. It

is assumed that in each virtual diode the Langmuir’s theory holds. This approach is

grounded on the fact that the thermal velocities of the electrons in the beam are

determined by the temperature of the emitter and usually correspond to energies less than

1 eV. As a result, the space-charge density in the vicinity of the cathode is high and

significantly affects the potential distribution. Moreover, usually a space-charge cloud

forms which produces a potential minimum in the proximity of the emitter. A fraction of

the emitted electrons does not have sufficient energy to surmount the potential barrier and

returns back to the cathode. It is important to note that in reality various cathode regions

can function under different operating conditions. For instance, if there is no potential

minimum in front of a particular cathode area all emitted electrons will be extracted by

the accelerating field and will take part in the beam formation. This regime is called the

saturation or temperature limited mode of operation and is the most frequently used in the

MIG for gyrotrons. At the same time, other cathode regions in front of which emitted

electrons are facing a retarding electric field will work in the space charge limited mode.

In the former case the current density is given by

satJJ , (13)

while in the latter

)/exp( Bminsat TkeJJ , (14)

where satJ is the temperature limited current density (also called “saturated current

density”) calculated from the Richardson–Dushman equation and min is the potential

minimum.

In order to take into account both the initial thermal velocities of the electrons

leaving the cathode and the variations of the potential near the emitter, the cathode

surface is divided into a number of small annular regions. The extracted current density

in each annular region is calculated by considering it as a small planar diode and by

applying the Langmuir’s theory. Langmuire’s solution for planar diode relates the

dimensionless potential

)( minB

Tk

e (15)

and the dimensionless distance

)( minzz , (16)

where minz is the distance between the potential minimum and the emitting surface, 2/14/3 JcT , and c is a constant. The dependence between and is available in

tabular form as well as in the form of approximations [ )(1 F and )(2 F ], adapted

to represent the tabular data with the required accuracy.

In the calculations there are three known quantities for each elementary diode:

cathode temperature, anode potential a and anode-to-cathode distance ad . In order to

obtain the extracted current density aJ , an iterative procedure is used to solve the

transcendental equation

0)( aa J , (17)

taking advantage of the following relations:

)ln(a

Bmin

J

J

e

Tk , (18)

Tke Bminc / , (19)

),( c2c F (20)

aca d , (21)

))(/()( caBa eTkJ . (22)

Using the technique described above, the current density as well as the location

and depth of the potential minimum for each cathode region is calculated. The entire area

of the boundary-value problem is divided in two partially overlapping regions. The first

one spans between the cathode and the anodes of the elementary diodes. The second

region begins at the potential minimum and spreads to the end of the area. Thus, the

potential minimum is used as a boundary condition for the solution of Poisson’s equation

in the second region in order to obtain a self consistent solution.

2.3 Notes regarding the current status of the GUN-MIG/CUSP package and

its usage

The code calculates the electron trajectories and the main beam quality

parameters like the pitch factor (velocity ratio), velocity spread, beam radius and outputs

their average values and statistical estimates. If the electron orbits are traced till a cross-

section which is before the resonant cavity it is possible to extrapolate beam parameters

for the interaction region. The calculations in this case are performed on the basis of the

adiabatic theory.

The results of the simulation can be visualized on a sequence of frames (screens)

by post-processing program GUN_VIEW (CUSP_VIEW). It was developed using the

graphical library of the Lahey F77-EM/32 Fortran compiler which was operational under

DOS but is not available to us under the novel operating systems. It is possible however

to modify the source code and replace the above mentioned graphic library with pgplot

[6], which is portable to various platforms. Another, easy way around this problem is to

use the data files produced by the code and visualize them using any preferred plotting

program (for example gnuplot [7]).

Regrettably, as it is frequently the case with the research codes (in contrast to

their commercial cousins), the software package described above is not well documented.

Therefore, the user is referred to the papers in which it has been described [A1-A6] as

well as to papers where some results of its usage have been published [B1-B9]. The latter

group of articles provides good examples demonstrating the capabilities of the codes and

the scope of the output data. An illustrative example which characterizes the output of

simulations is presented in the Appendix 4.

It should be mentioned also that the package described above exists in many

slightly different versions that are a result of numerous modifications made in order to

experiment with alternative algorithms, various forms of input/output or simply to adapt

the code to someone’s preferences. This means that the user should be careful for

distinguishing between different editions. One way to avoid confusion is to keep different

variants in different directories (with short notes about the status, origin of the code and

changes made) but always together with appropriate test examples that can be easily

reproduced after recompilation (build) of the library.

And, finally, there are several similar programs (mainly open source and/or public

domain) but also some closed source (available only in a binary form) programs that are

available to us and can be used as a reference, for benchmarking and comparison as well

as for education. Among them is the famous EGUN electron optics and gun design code

[1]. One should be warned however that this code also exists (available in source code) in

many different versions and it is the user’s responsibility to be aware of the differences

between them.

3. Computer codes for simulation of the interaction of the electron beam with

the high-frequency electromagnetic field in the resonant cavity

The main codes of GYRSIM software package that are being used for simulation

of the resonant cavity are: (i) a set of cold cavity codes (CCC) that are based basically on

one and the same physical model but differ in their implementation; (ii) a single mode

time independent code (CS-SMTI); (iii) multimode time dependent code (CS-MMTD),

and, (iv) a library of auxiliary programs, functions and subroutines). An inventory of

these modules is presented in Table.2

3.1 Collection of cold cavity codes (CCC) included in GYRSIM. Physical

model and program implementation.

As already mentioned, the GYRSIM software package includes several codes that

represent different implementation of one and the same so called cold cavity model in

Table 2

Inventory of the cavity codes and modules of GYRSIM package

Module/Code Functionality Implementation

CCC_FG (aka CAV)

Calculates the cold cavity

field (amplitude and phase

along the axis), frequency

and Q factor

Fortran 77;

Fox-Goodwin method for

Integration of the

Helmholtz equation

CCC_EB (aka GYROS)

Calculates the cold cavity

field, frequency and Q

factor. The output is used to

integrate the equations of

motion in other codes.

Fortran 77; Standard

Vlasov model. Wave

equation integrated by

Numerov’s method.

Requires MINUIT package

for finding the eigen-

frequency.

CCC_CAV

Calculates the field,

frequency, Q factor and

energy balance tracing

electrons in the cold cavity

field

Fortran 77. The

implementation is similar to

GYROS but without

MINUIT

CS-SMTI

Calculates the efficiency

of the interaction

Fortran 77. Implements a

modified Fliflet’s model

CS-MMTD

Simulates mode interaction

in an oversized cavity

Fortran 77 and Fortran 90

for SX-8 available. The F77

version visualizes electron

bunching using pgplot

graphic library.

EIGEN, BMOD, IS, QFAC

Calculation/search/scan of

resonances and

corresponding magnetic

fields, starting currents

calculation, Q- factors etc.

Fortran 77. A collection of

small programs

(calculators) for simple

estimates of the cyclotron

frequency, resonant

frequency, magnetic field

etc.

BESSLIB

Subroutines for calculation

of various Bessel functions

Fortran 77. Well tested

implementations from

various repositories of

numerical libraries with

some minor modifications

Vlassov’s formulation [8] (see Table 2). According to this model (aka “slightly irregular

waveguide model” or “a model of a slightly irregular string”) the transverse structure of

the electromagnetic field remains the same along the cavity axis and only its amplitude

f changes according to the equation [8]

(23) )()(,0),( 2222

2

2

zc

hfzhdz

fdssss

s

Here and sh are the circular frequency and the longitudinal wave-number, respectively,

c is the speed of light in vacuum and s an index that designates a given TEmn mode. The

transverse wave-number )(zR

xmns (where mnx is the eigenvalue of the mode TEmn, and

)(zR is the radius of the cavity) changes slightly so that the following condition is

fulfilled

. (24)

The wave equation is supplemented by the following boundary conditions

(25)

Here the 1z and 2z correspond to the “entrance” and the “exit” of the cavity. It is

important to note that the above mentioned boundary value problem has a discrete

spectrum of complex frequencies . They can be found in an iterative procedure

integrating (23) for different initial guesses and searching for values that satisfy the

boundary condition (25). Such approach is realized in the codes GYROS and

CCC_CAV. The iterative procedure in GYROS is performed as an optimization

problem carried out using the package MINUIT, while in CCC_CAV a fastest

gradient method is used.

Another approach is realized in CCC_FG (CAV) code. It is based on the

following considerations. The waveguide is cut-off ( 2

sh <0) in the regions I and III

(see the Fig. 2) so the microwave energy is kept (contained) in the cavity (region II).

Fig. 2. Schematics of a cavity treated as a slightly irregular waveguide.

Since the cavity is open-ended, however some fraction of the energy escapes along

the output waveguide (region IV).

2

ss h

dz

dh

0;0

21

zz

sss

zz

sss fih

dz

dffih

dz

df

In the regions I and IV provided we are away from points where 0sh , the WKB

solution (after G. Wentzel, H.A. Kramers, and L. Brillouin) is an appropriate

approximation of the problem under consideration. The general WKB solution has the

form [8]

hdzihdzi

eh

Be

h

Af . (26)

At the electron gun end of region I, the waveguide is well and truly cut-off. The wave is

evanescent there and its amplitude decreases exponentially towards the electron gun,

since here

hdzi

ehAf )/( , (27)

assuming 0}Im{ h . At the output end of the region IV the solution gives an out-going

wave which is travelling in the positive direction of the z axis,

hdzi

ehBf )/( , (28)

assuming 0}Re{ h .

The method implemented in the CCC_FG code was realized by Dr. G. F. Brand.

It solves the equation (23) numerically using the following algorithm: (i) Initially a value

of the complex frequency is chosen and the computation begins at a point in region IV;

(ii) f is given an initial value of 1 and from the equation (28) dzdf / has an initial value

of ih , i.e. the boundary condition there is satisfied; (iii) then the program steps in the

left direction (towards the gun end) computing f and dzdf / on each step using the Fox-

Goodman algorithm; (iv) At the electron gun end, the stepping halts and the final values

of f and dzdf / are tested to see if the second boundary condition, namely

ihfdzdf / is satisfied. If the agreement is not sufficiently good, a new value of the

complex frequency is selected and the whole process is repeated iteratively till a

satisfactory solution is found. This being done the program prints the obtained resonance

frequency, diffractive Q factor of the resonator, field amplitude and phase along the axis

of the cavity.

3.2 Single mode time-independent cavity simulation code (CS-SMTI) –

Physical model and program implementation

The physical model realized in CS-SMTI code is one proposed by Fliflet et al.

[9]. In accordance with this model the electron beam is represented as an ensemble of

particles and their trajectories are traced in an ambient magnetic field zBB

00 and

high-frequency electric field

ErEE r of a TEmn mode in a cylindrical coordinate

system with unit vectors r

,

and z

. The field components are given by

)exp()()(),,,( tizFztzrE mnrmnr

; (29)

)exp()()(),,,( tizFztzrE mnmn

, (30)

where the membrane function )exp()( imrkJC mnmmnmn is a solution of the

corresponding wave equation for a cylindrical waveguide. Here )(xJm is the m -th order

Bessel function of the first kind and the transverse wave number cavmnmn Rk / is

determined by the boundary condition 0)( mnmJ on the cavity wall of radius cavR . The

normalization coefficient is given by 122 )]()([ mnmmnmn JmC . The axial

dependence of the field intensity is described by the complex profile function

)](exp[)()( zizFzFmn with amplitude )(zF and phase )(z . As usually, harmonic

time dependence with an angular frequency of all field components is assumed. It is

presumed also that the rotation of the TEmn modes is in the same direction as the gyro

motion of the electrons (co rotation). To the co-rotation corresponds a minus sign in the

exponent, which depends on the azimuthal angle. The individual electron orbits are

described by the relativistic equation of motion

)( 00 Bp

Eedt

pdm

, (31)

where vp is the normalized momentum expressed by the velocity of the electron

v

and the relativistic factor 2/122 )/1( cp . Here 0m is the rest mass of an electron,

and c is the speed of light. In a slow-time-scale formulation Eq. (31) is reduced to the

following two equations for the normalized transverse momentum tp and the slow time

scale variable [9]

)cos()()(,, )( szFpRsBRsAdt

dptLmngcmn

t ; (32)

otLmngcmn

t sspRsBRsA

pdt

d

)sin()(,,1 )( , (33)

where s is the harmonic number, gcR is the radius of the guiding center of the electron

orbit, 0/ tL pR is the Larmor radius, 000 /meB is the cyclotron frequency and,

)(),( gcmnsmmnmngcmn RkJCkRsA , )()(2

, 11

0

)(

tLmnstLmnsLmm pRkJpRkJm

eRsB

.

In the framework of the weakly irregular waveguide theory the Maxwell’s curl

equations yield the following equation for the field profile function

dSJEiFk

cdz

Fdmnmn

mn

.0

2

2

2

2

2

, (34)

where 0 is the vacuum permeability, J

is the high frequency beam current density

obtained by summing the contributions of all particles and the integration is performed

over the cavity cross-section. For a thin annular beam, the transverse current density is

given by

t

z

b vv

IJ

, (35)

where bI is the total current, tv and zv being the transverse and the axial velocity,

respectively. Using the same technique as in the derivation of Eqs (32)-(33) the right-

hand side of Eq. (34) takes the form [9]

0

2

0 0

1

2

2

2

)exp(2

1,

dis

pkJRsAIiFk

dz

Fd tmnsgcmnbmnz

mn

, (36)

where zt vv / is the orbital-to-axial velocity ratio (pitch factor). Here the longitudinal

wave number 2/12222 )](//[ zRck mnz depends on the axial coordinate through the

cavity radius )(zR . If the cavity radius changes slowly along the axis we can consider a

weakly irregular waveguide as one where the fields at any z have one and the same

transverse structure and differ only in their amplitude through the function ).(zFmn

Although we have already discussed this topic in the previous section (See Sect.

3.1) it will be repeated here again because it is essential for the physical model of the CS-

SMTI code.

Indeed, the gyrotron interaction cavities are typically cylindrical, with tapers of

various sizes and lengths on either end to aid in mode selection and to improve the

efficiency of interaction. Such cavities are capable of supporting a large number of

modes, but gyrotrons are typically operated near cutoff ( zmn kk ). Therefore, at the gun

end the waveguide is well and truly cut-off. The wave there is evanescent and its

amplitude becomes smaller the closer it goes towards the electron gun. On the contrary,

at the output end there is an outgoing wave traveling in the z direction. This situation is

described by the following so-called radiation boundary conditions

0)()(

zFikdz

zdFmnz

mn , (37)

where the minus sign corresponds to the entrance ( inzz ) and plus to the exit ( outzz )

end of the cavity (please recall that the boundary conditions here are identical with (25)

in Sect. 2.1).

Calculation of the efficiency involves an average over the electrons in the

beam. The electronic efficiency (efficiency of the interaction between beam electrons and

the field) is given by

10

0

i

el , (38)

where 0 and i are initial and final values for the electron respectively and the average

is taken over the ensemble of particles representing the beam.

The physical model outlined above is implemented in a computer code CS-TISM.

The source code, written in FORTRAN, is well structured and allows to be extended

easily by addition of new computational modules. The input dada include the parameters

of the electron beam (current, accelerating voltage, average values of the velocity ratio

and beam ripple as well as their relative standard deviations), the configuration of the

cavity, the intensity of the constant magnetic field and a set of parameters (steps of

integration, number of particles, criteria for self-consistent solution etc.) that are related

to the numerical routines. An example of the input file of the code is shown in the

Appendix 2.

3.3 Multimode time-dependent cavity simulation code CS-MMTD – Physical

model and program implementation

In the CS-MMTD code a well-known and widely used multimode physical model

is realized (see, e.g. [10,11]). It consists of a system of equations that describe self-

consistently both the dynamics of the electrons and the excitation of the resonant cavity.

Representing the high-frequency field as a superposition of modes labeled by the index s

the equation of electron motion takes the form

s

ssss

nnifFpippi

d

dps )(exp)(1 0

12

, (39)

where )](exp[)/( 00 ippp tt is the normalized orbital momentum, 0 is the

initial value of the slowly varying phase ta , is the azimuthal coordinate of

the electron in the frame of the guiding center, a is the frequency over which averaging

is performed, sn is the harmonic number of the s -th mode with longitudinal

profile )(sf and the normalized axial coordinate is given by

zc

a

0||

2

0 . (40)

Here cv /00 and cv /0||0|| are the transverse and the axial velocities normalized

to the speed of light in vacuum c , respectively. The normalized frequency mismatch

that characterizes the difference between the initial cyclotron frequency 0c and the

fundamental frequency a (which is close to 0c ) is computed from the relation

)1(2 0

2

0 a

c

. (41)

The normalized mode amplitudes sF and phases s obey the system [10,11]

'

2s

s

s

s

s

Q

nF

d

dF

; ''

'

ss

a

ss nd

d

, (42)

where ta is the dimensionless time, '''

sss i , sss Q2/''' , sQ is the

quality factor of the resonant cavity for the s -th mode . The complex value '''

sss i which characterizes interaction between the electron beam and s -th

mode, is given by

dddnifpF

Ii

out

s

sss

n

s

ss

2

0

0

2

0 0

0 )](exp[)()(2

1

2

1, (43)

Where is the azimuthal coordinate (angle) of the guiding centers of the electrons and

sI is the dimensionless current [10,11]. Therefore the averaging is performed over 0 and

. It should be noticed that Eq. (43) is obtained for a thin hollow electron beam in which

both the velocity spread and the spread of the radii of the cyclotron orbits can be

neglected. The dimensionless current is defined as follows 1

0

2222

221

00||

)2(2

03 )())((2!2

1047.0

dfmJR

Jn

nII

end

ssss

ss

ssssm

s

nm

s

n

n

s

n

s , (44)

Where I is the beam current in amperes, 0

is the Lorentz relativistic factor, R is the

radius of the electron beam, sm is the azimuthal index and s is the wavelength of the s -

th mode with an eigenvalue s (defined for a TEmn mode as the n-th root of the equation

0)(' mJ , where '

mJ is the derivative of the Bessel function, i.e. ddJvJ mm /)()(' ).

The code CS-MM exists in two versions, namely for PC (Fortran 77) and SX-5

computer (Fortran90). This code has been used successfully to study mode interactions in

gyrotrons developed at FIR FU Research center.

An example of a file with input data for simulations by CS-MMTD code is

presented in Appendix 3. A general description of the code can be found in [C2,C3],

while examples of its use are presented in [D5,D7]. Recently the novel version of the

code CS-MMTD for SX-8 supercomputer has been used to study the interaction between

modes in the first tube (FU CW I) of the Gyrotron FU CW Series. Alongside with the

single mode operation several multi-mode phenomena observed experimentally have

been simulated [D7].

4. Additional programs and numerical libraries

In the preceding sections of this Report we described briefly the basic components

of the GYRSIM package. Alongside with its codes we use a number of stand-alone

programs and numerical libraries for: (i) debugging and optimization of the

computational modules; (ii) development of novel modules; (iii) preparing the input data

for the simulations; (iv) post-processing and visualization of the results of the simulations

and so on. Here we mention only some of them.

Although the simulation package for modeling of the EOS contains its own codes

for calculation of the magnetic field produced by a system of solenoids, we also use for

this purpose the POISSON/SUPERFIS [12] and FEMM [13].

POISSON/SUPERFIS is a collection of programs for calculating static magnetic

and electric fields and radio-frequency electromagnetic fields in either 2-D Cartesian

coordinates or axially symmetric cylindrical coordinates. The programs generate a

triangular mesh fitted to the boundaries of different materials in the problem geometry.

Plotting programs and other postprocessor codes present the results in various forms. We

are using this package for checking our calculations of the magnetic field as well as for

preparing the file containing the magnetic field distribution for the trajectory analysis

code EGUN. We have also some experience using POISSON/SUPERFISH for

calculation of the microwave field and eigen-frequencies of multi-vane peniotron

cavities.

FEMM is a suite of programs for solving lowfrequency electromagnetic problems

on two-dimensional planar and axisymmetric domains. The program currently addresses

linear and nonlinear magnetostatic problems, linear and nonlinear time harmonic

magnetic problems, linear electrostatic problems as well as and steady-state heat flow

problems. We use this package in the same manner as POISSON/SUPERFISH.

Results of the numerical simulation carried out using the codes of GYRSIM are

outputted and stored in ASCII files. There structure is self-explanatory and is suitable for

post-processing and visualization of a number of graphical systems or plotting programs.

Our choice is usually gnuplot [13], gmsh [14], grace [15], and SciLab [16] as the most

suitable and adequate for the data we obtain in the simulations.

SciLab Scilab is a scientific software package for numerical computations

providing a powerful open computing environment for engineering and scientific

applications. We are using it not only for visualization but also for performing

calculations that are not (yet) implemented in GIRSIM package. For example, recently

SciLab was used in order to write a small toolbox which then was applied to the CAD of

an open resonator of Fabry-Perot type for measurements of the energy levels of

positronium.

As it has already been mentioned, some of the simulation tools of the GYRSIM

package have its own visualization tools (e.g. the post-processors GUN-VIEW and

CUSP-VIEW), which make use of the Lahey’s graphical fortran library and/or on pgplot

[6].

Among the numerical libraries we will mention only the most frequently used

ones, notably GSL (Gnu Scientific Library) [17], port [18] as well as the Netlib

Repository of Mathematical Software [19].

5. Conclusions and outlook

In this Report the current status of the GYRSIM software package developed in the

FIR FU Research Center has been presented. It should be mentioned however that it is

under continuous development and improvement. Some of the recent changes were just

adaptations to the ever changing computational infrastructure (e.g. new operating

systems, novel versions of the used compilers and numerical libraries as well as new

hardware) while the other modifications were focused on the debugging and optimization

of the program realization. Performing these changes the main focus is always on the

portability and extensibility of the codes that are being developed. Most of them are

operational on different workstations and in different environments: DOS, Windows,

Cygwin [20], and Ubuntu Linux [21]. Recently, we have gained some positive experience

from the modern tools for virtualization (like VirtualBox [22]) which give an opportunity

to take advantage of a simultaneous access to several operating systems.

It should be mentioned that some of the simulations are very computing intensive

and require significant resources (CPU time, memory). This motivated us to adapt some

of the codes for execution on the most powerful supercomputer, which is available to us,

namely the SX-8 at the National Institute for Fusion Science. We intend to continue this

work further as well as to consider parallelization of the codes.

A constant driving force for the further improvement of the package is our desire

to use more adequate physical models and to take into accounts more physical factors and

phenomena in the simulations. Since many of them are inherently three dimensional we

consider development of 3D modules. Such codes will also require a parallel

implementation of the basic algorithms. A work in this direction has already been started

and currently we are studying and testing various available 3D solvers based on fast and

efficient numerical algorithms.

Finally, we would like to mention again that the GYRSIM package can be used

for CAD of two of the most essential systems of the gyrotron, namely the EOS and the

resonant cavity. The performance of the device however depends on the synergy of all

the systems that form the tube (including the output window, collector etc.) and also the

quasi-optical transmission line which delivers the radiation to the region in the space

where it is used. Without going into details we will only stress that for many applications

an internal mode converter is a must. In order to respond to such demands we intend to

start a development of novel tools for simulation of the quasi-optical system of the

gyrotron and their integration in the GYRSIM package.

Acknowledgements

This work was partly supported by a Special Fund for Education and Research

from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) and

Project Allocation Fund of the University of Fukui and was carried out in the framework

of the collaboration and Agreement for Scientific Exchange between the Institute of

Electronics of the Bulgarian Academy of Sciences (IE-BAS) and the Research Center for

Development of the Far-Infrared Region (FIR FU) at the University of Fukui.

Part of the numerical calculations were made on SX-8 computer at the Computer

Center of the National Institute for Fusion Science (NIFS), Japan. The cooperation of the

NIFS is greatly acknowledged.

REFERENCES

General references

1. Herrmannsfeldt W B 1979 Electron Trajectory Program SLAC-226, last version

SLAC-331 (1998), Stanford University

2. Bleivas I.M. On the mathematical formulation of the self-consistent problem in

electron (ion) optics.- In: Problems of physical electronics, Ed. by N.D.

Deviatkov (Nauka, Moscow) 1982,p p. 85-98 (in Russian).

3. Ximen J. Aberration theory in electron and ion optics.- In: Advances in

Electronics and Electron Physics, Ed. by P.W.Hawkes (Academic Press) Suppl.

17, (1986).

4. Hawkes P.W., Kasper E. Principles of Electron Optics, Vol.1 Basic

Geometrical Optics, Academic Press (1989).

5. M.A.J. van der Stam, Space-charge in electron guns (Modelling and

simulation), D2-report, Delft University of Technology, The Netherlands

(1991) pp. 43-47.

6. PGPLOT - Fortran or C-callable Graphics Subroutine Library

(http://www.astro.caltech.edu/~tjp/pgplot/)

7. Gnuplot - command-line driven graphing utility (http://www.gnuplot.info/)

8. S.N. Vlasov, G.M. Zhisllin, I.M. Orlova, M.I. Petelin, G.G. Rogacheva,

Irregular Waveguides as Open Resonators, Izv. VUZ, Radiofizika, Vol. 12, N 8

(1969) 1236-1244 (in Russian)

9. Fliflet A.W., Read M.E., Chu K.R., Seeley R. Int. J. Electronics, 53 (1982)

505-521.

10. Nusinovich G.S. Mode interaction in gyrotrons, Int. J. Electronics 51 (1981)

457-474.

11. Dumbrajs O. Review of the theory of mode competition in gyrotrons.-In:

Gyrotron oscillators, Ed by C.J. Edgcombe (Taylor&Francis, London, 1993)

82-125.

12. POISSON/SUPERFISH

(http://laacg1.lanl.gov/laacg/services/serv_codes.phtml#ps)

13. FEMM–Finite Element Method Magnetics

(http://www.femm.info/wiki/HomePage)

14. Gmsh – A 3D finite element grid generator with a build-in CAD engine and

post-processor (http://www.geuz.org/gmsh/)

15. Grace - WYSIWYG 2D plotting tool (http://plasma-gate.weizmann.ac.il/Grace)

16. (http://www.scilab.org/)

17. GSL- The GNU Scientific Library (http://www.gnu.org/software/gsl/ )

18. PORT – (http://www.netlib.org/port/)

19. Netlib Repository of Mathematical Software (http://www.netlib.org)

20. Cygwin (http://www.cygwin.com/)

21. Ununtu Linux (http://www.ubuntu.com/)

22. VirtualBox (http://www.virtualbox.org/)

References related to the GYRSIM package

In order to facilitate the search for the corresponding references we divide them in

several groups (e.g. references to papers on ray-tracing codes, cavity codes, additional

programs etc.). Since some of the papers however contain a description of more than

one code or topic they will appear in several groups of references.

A. References to papers on the ray tracing codes of GYRSIM package

A1. Sabchevski S., Mladenov G., Idehara T. Modeling and Simulation of Magnetron

Injection Guns for Gyrotrons.- Proc. Of Int. Conf. Electronika’98 (10-15 Oct. 1998,

Botevgrad) 1998, 23-28.

A2. Sabchevski S., Mladenov G., Idehara T. Modeling and Simulation of Magnetron

Injection Guns for Submillimeter Wave Gyrotrons.- Intern. Journal of Infrared and

Millimeter Waves, Vol. 20, N 6 (1999) 1019-1035.

A3. Sabchevski S., Idehara T., Glyavin M., Mitsudo S., Ogawa I., Ohashi K., Kobayashi

H. Design of a Large Orbit Gyrotron with a Permanent Magnet System.- Vacuum, Vol.

62, N2-3, 2001, (Special issue: The Sixth International Conference on Electron Beam

Technologies EBT-2000) pp. 133-142.

A4. Sabchevski S., Idehara T., Glyavin M., Ogawa I., Mitsudo S. Modelling and

simulation of gyrotrons. Vacuum, Vol. 77, N 4 (2005) pp. 519-525.

A5. Sabchevski S. Modeling and Simulation of Gyrotrons (invited talk). Collections of

Viewgraphs of the 2nd

International Workshop on Far Infrared Technologies (September

12-13, 2002 Fukui University, Fukui, Japan) pp. 72-120.

A6. Sabchevski S. Modeling and Simulation of Gyrotrons (invited talk). Proc. Of the

2nd

International Workshop on Far Infrared Technologies (September 12-13, 2002

Fukui University, Fukui, Japan) pp. 43-58.

A7. S. Sabchevski, T. Idehara, T. Saito, I. Ogawa, S. Mitsudo, Y. Tatematsu, La Agusu,

Numerical Investigations of the Gyrotron FU and Gyrotron FU CW Series Proc. Mini-

symposium on Infrared Science and Technology-Satellite Meeting of ISMASM2008

(May 30, 2008, Osaka Sangyo University) pp. 31-36 (Invited talk).

B. References to papers in which results obtained by the ray tracing codes of

GYRSIM package are used

B1. Sabchevski S., Idehara T., Ogawa I., Glyavin M., Mitsudo S., Ohashi K., Kobayasi

H. Computer Simulation of Axis-Encircling Beams Generated by an Electron Gun with a

Permanent Magnet System.- Intern. Journal of Infrared and Millimeter Waves, Vol. 21, N

8 (2000) 1191-1209.

B2. Sabchevski S., Idehara T., Glyavin M., Mitsudo S., Ogawa I., Ohashi K., Kobayashi

H. Design of a Large Orbit Gyrotron with a Permanent Magnet System.- Vacuum, Vol.

62, N2-3, 2001, (Special issue: The Sixth International Conference on Electron Beam

Technologies EBT-2000) pp. 133-142.

B3. Zapevalov V., Idehara T., Sabchevski S., Ohashi K., Manuilov V., Glyavin M.,

Kornishin S., Kuftin A., Lygin V., Malygin O., Moiseev M., Pavel’ev A., Tzalolikhin V.,

Zavolski N., Kobayashi H., Yokoyama T., Ogawa I., Mitsudo S., Kanemaki T., Iwata Y.

and Hoshizuki H. Design of a Large Orbit Gyrotron with a Permanent Magnet System.-

Int. Journal of Infrared and Millimeter Waves, Vol. 24, N 3 (2003) pp. 253-260.

B4. Idehara T. Ogawa I. Mitsudo S. Iwata Y. Watanabe S. Itakura Y. Ohashi

K. Kobayashi H. Yokoyama T. Zapevalov V. E. Glyavin M. Y. Kuftin A. N.

Malygin O. V. Sabchevski S. P. A High Harmonic Gyrotron With an Axis-Encircling

Electron Beam and a Permanent Magnet.- IEEE Trans. Plasma Sci., Vol. 32, N 3, pp.

903-909 (2004).

B5. Sabchevski S., Idehara T., Glyavin M., Ogawa I., Mitsudo S. Modelling and

simulation of gyrotrons. Vacuum, Vol. 77, N 4 (2005) pp. 519-525.

B6. Sabchevski, S.; Idehara, T. ; Mitsudo, S. ; Fujiwara, T. Conceptual Design Study of a

Novel Gyrotron for NMR/DNP Spectroscopy.- Intern. J. Infrared and Millimeter Waves,

Vol. 26, N 24 (2005) pp. 1241-1264

B7. Sabchevski S., Idehara T., Ogawa I., Glyavin M., Mitsudo S., Ohashi K. Computer

Simulation of Axis-Encircling Beams Generated by Electron Gun with Permanent

Magnet System.- Proc. Of 13th Internat. Conf. On High-Power Particle Beams (25-30

June 2000, Nagaoka, Japan) Vol.1, p. 889-892.

B8. Idehara T., Sabchevski S., Glyavin M., Mitsudo S., Ogawa I., Ohashi K. Design of

Weakly Relativistic Large Orbit Gyrotron with Permanent Magnet System.- Proc. Of

13th Internat. Conf. On High-Power Particle Beams (25-30 June 2000, Nagaoka, Japan)

Vol. 1, pp. 1008-1011.

B9. Zapevalov V., Idehara T., Sabchevski S., Ohashi K., Manuilov V., Glyavin M.,

Kornishin S., Kuftin A., Lygin V., Malygin O., Moiseev M., Pavel’ev A., Tsalolikhin V.,

Zavolski N., Kobayashi H., Yokoyama T., Ogawa I., Mitsudo S., Kanemaki T., Iwata Y.,

Hoshizuki H. Simulation and Design of High Harmonic Gyrotron with a Permanent

Magnet System. In: Proceedings of the Intern. Workshop on Strong Microwaves in

Plasmas (N. Novgorod, 1-9 Aug 2002) Ed. By A.G.Litvak, Vol. 1 (2003) pp. 129-134.

C. References to papers on the cavity codes of GYRSIM package

C1. Sabchevski S., Idehara T., Ogawa I., Glyavin M., Ohashi K. Simulation of a High Harmonic

Gyrotron with Axis-Encircling Electron Beam and Permanent Magnet. Int. Journal of Infrared

and Millimeter Waves, Vol. 23, N 5 (2002) pp. 675-692.

C2. Sabchevski S., Idehara T., Glyavin M., Ogawa I., Mitsudo S. Modelling and

simulation of gyrotrons. Vacuum, Vol. 77, N 4 (2005) pp. 519-525.

C3. Sabchevski S. Modeling and Simulation of Gyrotrons (invited talk). Collections of

Viewgraphs of the 2nd

International Workshop on Far Infrared Technologies (September

12-13, 2002 Fukui University, Fukui, Japan) pp. 72-120.

C4. Sabchevski S. Modeling and Simulation of Gyrotrons (invited talk). Proc. Of the

2nd

International Workshop on Far Infrared Technologies (September 12-13, 2002

Fukui University, Fukui, Japan) pp. 43-58.

C5. S. Sabchevski, T. Idehara, T. Saito, I. Ogawa, S. Mitsudo, Y. Tatematsu, La Agusu,

Numerical Investigations of the Gyrotron FU and Gyrotron FU CW Series Proc. Mini-

symposium on Infrared Science and Technology-Satellite Meeting of ISMASM2008

(May 30, 2008, Osaka Sangyo University) pp. 31-36 (Invited talk).

D. References to papers on the use of the cavity codes of GYRSIM package

D1. Glyavin M., Sabchevski S., Idehara T., Ogawa I., Mitsudo S., Ohashi K., Kobayashi

H. Numerical Analysis of a Weakly Relativistic Large Orbit Gyrotron with Permanent

Magnet System.- Intern. Journal of Infrared and Millimeter Waves, Vol. 21, N 8 (2000)

1211-1221.

D2. Sabchevski S., Idehara T., Ogawa I., Glyavin M., Ohashi K. Simulation of a High

Harmonic Gyrotron with Axis-Encircling Electron Beam and Permanent Magnet. Int.

Journal of Infrared and Millimeter Waves, Vol. 23, N 5 (2002) pp. 675-692.

D3. Zapevalov V., Idehara T., Sabchevski S., Ohashi K., Manuilov V., Glyavin M.,

Kornishin S., Kuftin A., Lygin V., Malygin O., Moiseev M., Pavel’ev A., Tzalolikhin V.,

Zavolski N., Kobayashi H., Yokoyama T., Ogawa I., Mitsudo S., Kanemaki T., Iwata Y.

and Hoshizuki H. Design of a Large Orbit Gyrotron with a Permanent Magnet System.-

Int. Journal of Infrared and Millimeter Waves, Vol. 24, N 3 (2003) pp. 253-260.

D4. Idehara T. Ogawa I. Mitsudo S. Iwata Y. Watanabe S. Itakura Y. Ohashi

K. Kobayashi H. Yokoyama T. Zapevalov V. E. Glyavin M. Y. Kuftin A. N.

Malygin O. V. Sabchevski S. P. A High Harmonic Gyrotron With an Axis-Encircling

Electron Beam and a Permanent Magnet.- IEEE Trans. Plasma Sci., Vol. 32, N 3, pp.

903-909 (2004).

D5. Sabchevski S., Idehara T., Glyavin M., Ogawa I., Mitsudo S. Modelling and

simulation of gyrotrons. Vacuum, Vol. 77, N 4 (2005) pp. 519-525.

D6. Sabchevski, S.; Idehara, T. ; Mitsudo, S. ; Fujiwara, T. Conceptual Design Study of a

Novel Gyrotron for NMR/DNP Spectroscopy.- Intern. J. Infrared and Millimeter Waves,

Vol. 26, N 24 (2005) pp. 1241-1264.

D7. Sabchevski S., Saito T., Idehara T., Nakano T., Tatematsu Y. Simulation of Mode

Interaction in the Gyrotron FU CW I.- Int. Journal of Infrared and Millimeter Waves

Vol. 28 (2007) pp. 1079-1093.

DOI 10.1007/s10762-007-9294-9; ISSN0195-9271 (Print) 1572-9559 (Online))

6. Appendixes

Apendix 1

An example of an ASCII input file for GUN-MIG/CUSP

DESCRIPTION OF THE INPUT DATA (Please keep these comments in each input file before the actual NAMELIST) LG - number of points along the boundary contour

H - step of the mesh

T - temperature of the cathode

PTN - saturation current density of the emitter

EPS - criterium for the solution of the boundary

value problem

EPSS - criterium for the solution of the self-consistent

problem

KEY(1) =T –presence of curved electrodes

KEY(2) =T –computation ot the loses of current (not used in GUN-MIG)

KEY(3) =T –variation of the potential of one of the electrodes

If KEY(3)=T: NVARU - number of the variations

NU1,NU2- number of the first and the last points

DELU - change of the potential (+ or -)

KEY(4) =T –variation of the geometry of one of the electrodes

If KEY(4)=T: NVARG - number of the variations

NG1,NG2- number of the first and the last points

DELZ - shift of the coordinates along Z axis (+ or -) DELR - shift of the coordinates along R axis (+ or -) KEY(5) =T –presence of cutting diaphragms (not used in GUN-MIG)

KEY(6) =T –partition of an axis-adjacent region

If KEY(6)=T: RMAX - radius of the region

KEY(7) =T –limited emitting earia of the cathode

KEY(8) =T –analysis in the equipotential space (not used in GUN-MIG)

KEY(9) =T –(not used in GUN-MIG)

KEY(10)=T –increase cathode temperature gradually

OUTPUT OF:

IOUT(1) = T – potential distribution (in a table) If zooming windows are used LSEL is the number of the windows;

ZN(I),ZK(I) – beginning/ending Z coordinates of I-th window

RN(I),RK(I) – beginning/endinq R coordinates of I-th window

IDSZ(I),IDSR(I)-multiplicity factor for augmentation of the step in Z and R directions.

If these parameters are not specified the output includes

the full array. IOUT(2) = T – current density distribution

IOUT(3) = T – beam envelope (not used in GUN-MIG)

IOUT(4) = T – cathode current density distribution (not used in GUN-MIG)

IOUT(5) = T – current density distribution in the crossover (not in GUN-

MIG) IOUT(6) = T – current density distribution in selected cross-sections (not yet implemented ) IOUT(7) = T – output of the cut-off potential of the system (not used in MIG) IOUT(8) = T – output of information about the iterative process

IOUT(9) = T – plot of the geometry of the system

IOUT(10)= T – solution of self-consistent problem

GEOM: consists of axial coorditate Z, radial coordinate R and potential U of all LG boundary point along the boundary contour. Potential of the cathode must be zero.

KRIV: consists of the codes of the boundary segment (1-line; 2-arc of circle) PRMT: consists of the cardinal elements of the boundary segments THE ACTUAL NAMELIST STARTS HERE &GUN

KEY = t F F F F F F F F F

LG = 20

EPS = 1e-05

H = 0.020

T = 1350.0

PTN = 1.1284084

EPSS = 0.01

IOUT = t F F F F F F t F t

GEOM = 3.00000 0.00000 0.00

3.00000 0.12000 0.00

2.92000 0.20000 0.00

2.16500 0.20000 0.00

1.62500 0.36500 0.00

1.30000 0.46500 0.00

1.12000 0.58731 0.00

0.83000 0.68000 0.00

0.00000 0.68000 0.00

0.00000 1.36000 15000.00

1.16000 1.36000 15000.00

1.45000 1.03000 15000.00

2.10000 0.71000 15000.00

3.05000 0.71000 15000.00

3.37000 1.03000 15000.00

3.37000 1.36000 15000.00

8.00000 1.36000 40000.00

8.50000 0.86000 40000.00

12.00000 0.86000 40000.00

12.00000 0.00000 40000.00

KRIV = 1 2 1 1 1 1

2 1 1 1 1 1

1 2 1 1 2 1

1

PRMTR = 0.00000e+00 8.00000e-02 0.00000e+00 0.00000e+00

0.00000e+00 0.00000e+00 5.00000e-01 0.00000e+00

0.00000e+00 0.00000e+00 0.00000e+00 0.00000e+00

0.00000e+00 3.20000e-01 0.00000e+00 0.00000e+00

5.00000e-01 0.00000e+00 0.00000e+00

ZE1=1.625 RE1=0.365 ZE2=1.3 RE2=0.465 NEM=40 NRAY=20 IORB=10 TSH=1.0e-07

NMF=2106 NDEG=100 MAGFI=c7123co2.DAT

/

ZE1,RE1 and ZE2,RE2 – coordinates of the points that specify the

emitting ring

NEM – number of emitting anular regions

NRAY – number of rays used for trajectory plot

IORB – number of the selected ray plotted in X-Y plane

TSH - step of integration

CBZZ: coefficiens in the polinomial expansion of the magnetic field

Bz=B0+B1*Z+B2*(Z**2)+B3*(Z**3)+B4*(Z**4)+B5*(Z**5)+B6*(Z**6)

_

Appendix 2

Example input file for the code CS-SMTI

&CAVMODEB

M=4, N=1, NS=4, RCAV=2.3074, UAC=37.0, BCU=1.1, B0=1.06,

WL1=0.01, WL2=0.05, WL3=0.05, R1=0.002, RN=0.005,NF=400,

NPHY=36, ALFA0=1.6, ALFASP=0.0, NALFA=10,

AVRIPL=7.0, RIPLSP=0.0, NRIPL=10, DELT=0.01,

EPS=0.005, MAXNBIT=100, FMISI=0.0000, STFM=0.0001, NMIS=90

/

M,N - mode indeces

NS - harmonic number

RCAV - cavity radius

B0 - magnetic field in the cavity [T]

UAC - accelerating loltage [kV]

ALFA - velocity ratio

ALFASP - relative standard deviation of velocity ratio [%]

NALFA - number of initial values of velocity ratio

AVRIPL - average beam ripple [%]

RIPLSP - relative standard deviation of the ripple

NRIPL - number of initial values of the ripple

WL1 - length of the first cavity section

WL2 - lenght of the central (active) cavity section

WL3 - length of the output cavity section

ZN - initial coordinate of the cavity

NF - number of points in the longitudinal field prfile

NPHY - number of initial condiions for azimuthal angle

NRAY - No of orbit for the output (NRAY.LE.NPHY)

DELT - Time step measured in cyclotron peropds (DT=DELT*Tc)

MAXNBIT- maximum number of iterations

EPS - criteria for self-consistent solution

FMIS - frequency mismatch (W-Wc)/Wc

NMIS - number of values for the frequency mismatch

_

Appendix 3

An example of a file with input data for the CS-MMTD code

&MULTIMOD

NM=2,

MM(1)=2, NN(1)=3, MM(2)=2, NN(2)=6,

NS(1)=1, NS(2)=2, MOI=1,

AS0(1)=0.001,AS0(2)=0.001,

QSDIF(1)=6307.0, QSDIF(2)=33743.0,

RCAV=2.425, CAVL=14.5, UAC=40.0, BCU=1.20,

ALFA=1.5, NP=200, DEL=0.68, RBE=1.25,

H=0.20, TS=300.0, TLIM=1800000.,NAA=16, ICH=1, ITYP=2, STPA=-0.04

/

NM - number of modes considered

MM(I) - azimuthal index of I-th mode (I=1,..NM)

NN(I) - radial index of I-th mode (I=1,..NM)

NS(I) - harmonic number of I-th mode (I=1,..NM)

AS(I) - initial ampltude of I-th mode

QSDIF(I) - diffractive Q-factor for I-th mode

MOI - number of mode which frequency is used for averaging

RCAV - cavity radius

CAVL - cavity length

UAC - accelerating loltage [kV]

BCU - beam current [A]

ALFA - velocity ratio

NP - number of particles

DEL - frequency mismatch

RBE - beam radius

H - axial step

TS - time step

NAA - number of azimuthal angles for averaging over phases

of the modes

ICH - number of the variations of the parameter (BCU or DEL)

ITYP - ITYP=1 - beam current is varied

ITYP=2 - frequency mismatch is varied

STPA - step (+ or -) for change of the parameter

_

Appendix 4

Visualization of the result of trajectory analysis of MIG performed using the ray

tracing codes of the GYRSIM package – illustrative examples

0.0 2.0 4.0 6.0 8.0 10.0 12.0Z (cm)

0.0

0.5

1.0

1.5

R

(cm

)

Ua= -15 kV Ub= 0

Uc=-40 kV

GUN-MIG: GEOMETRY OF THE GUN AND ELECTRON TRAJECTORIES Ib= 1.0 A , B1=0.222 T

-0.6 -0.3 0.0 0.3 0.6

X (cm)

-0.6

-0.3

0.0

0.3

0.6

Y

(cm

)

Projection of the central orbit of the beam (shown in the left figure) on the X-Y plane and the azimuthal angle vs longitudinal coordinate Z

0 2 4 6 8 10 12

Z (cm)

0.0

0.5

1.0

1.5

2.0

O (

rad

)

Emitting ring of

the cathode

Appendix 5

Visualization of the result of trajectory analysis of MIG for LOG performed using

the ray tracing codes of the GYRSIM package – illustrative examples

0.0 2.0 4.0 6.0 8.0 10.0 12.0Z (cm)

0.0

0.5

1.0

1.5

2.0

R

(cm

)

UaUw

Uc

GUN-MIG/CUSP: GEOMETRY OF THE GUN, AXIAL MAGNETIC FIELD AND ELECTRON TRAJECTORIES

Ua= 0, Uc = Uw= -36 kV , Ib= 1.0 A , Js= 5.0 A /cm2,

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Bz (

T)

Configuration of the gun, profile of the magnetic field produced by the permanent

magnet and electron trajectories of the beam

Projection of the central orbit on the X-Y plane (left) and its isometric view (right).

0 2 4 6 8 10 12 14 16 18 20 22

Z (cm)

0.0

0.5

1.0

1.5

2.0

R (c

m)

-0.4

0.0

0.4

0.8

1.2

B (

T)

GUN-MIG/CUSP (Ver. 6.00) RAY TRACING OF THE ELECTRON BEAM FORMATION

ANODE

CAVITY

Electron trajectories

Axial magnetic field

Ua=0, Uc=Uw=-37 kV , Ib=1 A

(MAGNETIC FIELD PRODUCED BY PM AND ADITIONAL COILS)

Ua

Uw

Uc

Configuration of the EOS, magnetic field profile and electron trajectories.

30 32 34 36 38 40Accelerating voltage (kV)

1.2

1.6

2.0

2.4

Velo

cit

y ra

tio

AVERAGE VELOCITY RATIO IN THE CAVITY REGION(simulation performed using the measured magnetic field)

30 32 34 36 38 40Accelerating voltage (kV)

0

2

4

6

8

Sp

read

(%

)

Spread of velocity ratio and axial velocityin the cavity region for the measured magnetic

field of the PM circuit

Spread of velocity ratio

Spread of axial velocity

30 32 34 36 38 40Accelerating voltage (kV)

5

10

15

20

25

Rip

ple

(

%)

AVERAGE BEAM RIPPLE IN THE CAVITY REGION(simulation performed using the measured magnetic field)

Appendix 6

Typical illustrative results obtained using the CCC_EB code

Appendix 7

Typical illustrative results obtained using CS-SMTI code

0 2 4 6 8 10 12

Z (cm)

0.0

0.2

0.4

0.6

0.8

R

(cm

)

0.0

0.2

0.4

0.6

0.8

1.0

Norm

alized

fie

ld p

rofi

le

Configuration of the resonant cavity and field profile obtained for: aU =37 kV, bI =1.5

A, =1.6, rA =8 %, 0B =1.06 T, =0.54 %, cavR =2.307 mm (LOG operating at 4TE401

mode)

Convergence of the iterative process to a self-consistent solution

Appendix 8

Typical illustrative results obtained using the CS-MMTD code

Temporal evolution of mode excitation and bunching of the electrons in the case of single

mode operation at second harmonic

Mode competition between TE23 and TE26.

0 100000 200000 300000 400000 500000 600000DIMENSIONLESS TIME

0.0

0.5

1.0

1.5

2.0

OU

TP

UT

PO

WE

R

(kW

)

A

B

C

D

E F

A B

C

D E F

EXCITED MODE: 2 TE26

U b = 40 kV ; I

b = 0.6 A

R b = 1.25 mm ; ALFA=1.5

R cav = 2.425 mm ; L

cav = 14.5 mm

B = 7.337 T (DELTA=0.625)

0 200000 400000 600000 800000 1000000 1200000 1400000DIMENSIONLESS TIME

0.0

0.5

1.0

1.5

2.0

OU

TP

UT

PO

WE

R

(kW

)

U b = 40 kV ; I

b = 0.6 A

R b = 1.25 mm ; ALFA=1.5

R cav = 2.425 mm ; L

cav = 14.5 mm

B = 7.356 T (DELTA=0.575)

2 TE26

1 TE23

Mode cooperation (TE26 mode facilitates the excitation of the TE23 mode).

Competition between the modes TE22,8 and TE3,16 in the Gyrotron FU CW I, which ends

up with suppression of the design mode by the competing mode which oscillates at the

frequency of 298.8 GHz.

0 150000 300000 450000 600000DIMENSIONLESS TIME

0.0

2.5

5.0

7.5

10.0

12.5

15.0O

UT

PU

T P

OW

ER

(k

W)

U b = 40 kV ; I

b = 0.7 A

R b = 1.25 mm ; ALFA=1.5

R cav = 2.425 mm ; L

cav = 14.5 mm

B = 7.365 T (DELTA=0.55)

2 TE26

1 TE23