Analysis and Synthesis of Quasi-Optical Launchers for High Power

Gyrotrons

Jeff Neilson

Calabazas Creek Research, Inc

Funded by US DOE SBIR Program

Outline

• Gyrotrons and QO launchers

• Motivation

• QO Launcher analysis

• Launcher synthesis

• Impact

Gyrotrons• High output power

vacuum tubes

• Primarily used for heating plasmas in fusion reactors

• Other applications

• Ceramic sintering

• Millimeter wave radar

• Non-lethal weapon systems (“pain ray”)

110 GHz, 1 MW CW Gyrotron

QO Launcher in Gyrotron Cavity Mode Field (TE22,6)

Gaussian Field at Window

Whispering Gallery Mode to Gaussian Mode Conversion*

• Ray picture shows alternate approach

• Combine 9 modes so amplitude can be described as

f(φ,z) = ( 1 + ½ exp(j3φ) + ½ exp(-j3φ) ) ( 1 + ½ exp(j2πz/Lc) + ½ exp(-j2πz/Lc) )

• Produces Gaussian like beam in ray propagation direction

*G.G. Denisov, A.N. Kuftin, V.I. Malygin, N.P. Venediftov, D.V. Vinogradov and V.E. Zapevalov, “110 GHz gyrotron with built-in high-efficiency converter,". J. Electronics, vol. 72, nos. 5 and 6, pp. 1079-1091, 1992.

Original Analysis Approach

• Codes predicted Gaussian conversion efficiencies to low 90% increasing to near 100% with phase correcting mirrors in transmission line

• Large variation in reported Gaussian conversion efficiency, typically 85%-95% range

• Measured internal diffraction losses always significantly higher than code prediction

• Coupled-mode theory used to determine mode content in waveguide converter and Stratton Chu integral to evaluate radiated fields

Motivation• Why large variations in results, how to increase

efficiency?• Potential sources of discrepancies:

– Analysis code (coupled-mode, Stratton-Chu) (CMSC) approximations

– Perturbation theory use to calculate coupling coefficients of azimuthal wall deformations in converter section

– Edge currents along axial cut ignored – Aperture field assumed to be that of closed waveguide

– Machining tolerances, waveguide converter deformations small fractions of wavelength (0.2mm)

– Measurement uncertainties– Generation of high purity whispering gallery mode– Difficulties in obtaining accurate measurement at mm wavelengths

An exact analysis of launcher field would be useful tool for reducing some of the design uncertainties in QO system

Computational Problem

• QO launcher very large compared to wavelength

• Intractable problem from memory and computation time using available commercial codes (e.g. HFSS) on desktop PC

• Solution desired with better than 1% accuracy

Ei,Hi

60-170 λ

15-25λ

Surface Electric Field Integral Equation

On the surface, requirement of E tangential equal zero yields the following integral equation for the unknown surface current Js(r)

Js(r)Ei,Hi

Mi = -n x Ei, Ji = n x HiE(r) = Ei(Ji,Mi) + E(Js)

Original problem Equivalent problem(equivalence theorem)

' )'('

))'(exp()(

4)(E ' dsrJ

rr

rrjkI

jkr s

i

Method of Moments

• Integral operator is discretized by representing unknown Js(r) as basis functions on triangular mesh*

N

ins JrfrJ

1

)()(

ijj

ji

ji dsdsrfrr

rrjkI

jk )(

))(exp()(

4Z '

ij

Z J = Ei

• 6 to 10 basis points per wavelength typically sufficient for accurate solution

*S.Rao,D.Wilton,A.Glisson,”Electromagnetic Scattering by Surfaces of Arbitrary Shapes,”IEEE Trans.APP, Vol 30 ,No 3,May 1982

Solution of Matrix Equation

• A direct solution for unknown J is an intractable problem for typical QO launchers with 200K – 1M unknowns– O(N2) memory requirements, many 100’s of GB– O(N3) computation time requires month/years CPU time

• Iterative solution of ZJ = Ei

– O(N2) memory requirements, many 100’s of GB– Evaluation of ZJ product for iterative solution has an

O(N2) operational count times M iterations – For M<<N , weeks/months of CPU time for solution

Fast Multipole Method*

• Developed in 1985 to speed up the calculation of long-range forces in the N-body simulations

• Cited as one of the top 10 algorithms of the 20th century

Application to MOM Solution*

*R.Coifman,V. Rokhlin, S.Wandzura,”The Fast Multipole Method: A Pedestrian Prescription,”,IEEE Ant and Prop. Mag.,V 35,No. 3, June 1993

• FMM accelerates evaluation of ZJ– O(N3/2)execution– O(N) memory – only ‘near’ interaction Z stored– Days/weeks of CPU time

• Multi-Level FM algorithm (MLFMA)– O(N Log(N)) execution, O(N) memory– Calculation time reduced to minutes/hours

Telephone Switchboard Analogy to FMM

Direct connections O(N2) links

One level switching O(N3/2) links)

Plane Wave Representation of a Group of Basis Functions

• Basis functions are grouped and their common field is represented as a plane wave expansion (PWE)

center

mrjm

s jmj

rjk rfedSkV jm )()ˆ(

FMM Implementation

• A translation operator translates the outgoing PWE of group M to the center of group N

)ˆˆ()()12(),( )2(

04 krPkrhlirkT mnlmnl

L

l

likmn

),( mnrkT

center

m

dm

center

ndn

mnr

Calculation of ZJ product

• Product ZJ is now calculated as

ZJ =V* T V J + Znear J

where Znear is the interaction between basis

• ZJ product is a O(N3/2) calculation using FMM

functions in the same group

MLFM Algorithm

MLFMA calculation (N Log(N) links)

FMM calculation (N3/2 links)

MLFMA*

• Groups of basis functions are grouped together• The PWE of the “parent” groups is obtained by

summing the shifted and interpolated PWEs of the “children” groups

• Communication between large groups using translation operator

• Results in a O(N Log(N)) calculation for ZJ

),( mnrkT

*W.Chew,et.al.,”Fast and Efficient Algorithms in Computational Electromagnetics”,Artech House,2001, ISBN 1-58053-152-0

Example Calculations

Constant diameter 4 Lb 110 GHz TE22,6 Launcher*

-1 0 1 2 3 4

12

14

16

18

20

22

Azimuth (radians) Azimuth (radians)

-1 0 1 2 3 4

12

14

16

18

20

22

-30

-25

-20

-15

-10

-5

0

EFIE Calculation Overlay of EFIE and CMSC Calculation

*M. Blank, K. Kreischer, and R.J. Temkin, “Theoretical and Experimental Investigation of a QO Mode Converter for a 110-GHz Gyrotron”, IEEE Trans. On Plasma Science, Vol. 24, No. 3, June 1996

Axi

s (c

m)

Tapered 3 Lb 110 GHz TE22,6 launcher (CMSC calculation)

-1 0 1 2 3 4

12

14

16

18

20

22

-30

-25

-20

-15

-10

-5

0

-1 0 1 2 3 4

12

14

16

18

20

22

-30

-25

-20

-15

-10

-5

0

Measured*

Azimuth (radians) Azimuth (radians)

CMSC Calculation

Axi

s (c

m)

*Measurement done at Univ of Wisconsin

Tapered 3 Lb 110 GHz TE22,6 launcher

-1 0 1 2 3 4

12

14

16

18

20

22

-30

-25

-20

-15

-10

-5

0

-1 0 1 2 3 4

12

14

16

18

20

22

-30

-25

-20

-15

-10

-5

0

Measured*

Azimuth (radians) Azimuth (radians)

EFIE Calculation

*Measurement done at Univ of Wisconsin

Axi

s (c

m)

Analysis Summary

• Primary error in CMSC calculation is neglected field on axial edge cut

– For very good converter designs (low edge fields) good agreement between SIE and CMSC

• For converters with large wall perturbations, coupled-mode perturbation theory in CMSC codes breaks down

• Primary reason for large variations in reported QO launcher efficiencies due to optimistic predictions of CMSC code for poor converter designs

• Good agreement between measurement and EFIE code shows launchers with small deformations can be built as designed

Launcher Synthesis

Better Performance Desired

• Even small amounts of diffraction losses may cause internal heating and prevent long pulse operation in MW level gyrotrons

• Synthesized mirrors typically used to maximize Gaussian content of output beam– Expensive to design and construct– Critical alignment necessary to achieve best

performance

Analytic Design Method

• 9 TE modes are combined to form a Gaussian like field distribution along axis and azimuthally

Field on Virtual Aperture

• Significant sidelobes, field profile not symmetric about centroid• Not free space Gaussian; eigenmode of open transmission line

Modification to Analytic Design Method

• Generate Gaussian field profile along axis and radially • Optimal coupling to free space Gaussian

Modification to Analytic Design Method- Continued

• Standard analytic two sinusoidal variation design approach

1 1 2 3( , ) ( ) cos( ( ) ) ( ) cos( ( ) 3 ) oR z r z z H z z z H z z

• Numerical optimization of generalized surface given

( , ) ( ) cos( ) ( )sin( ) , 1, 2,3,...o l ll

R z r z a z l b z l l al(z) and bl(z) modeled as cubic splines with spline points as free parameters

usually modeled as constant or linear variation( )z

Optimization Code

• Coupled-mode Stratton-Chu calculation used to get wall and radiated fields

• Goal function to maximize aperture field Gaussian content

• Optimization done via Quasi-Newton method with finite-difference gradient

Example Optimization

• 140GHz TE28,8 FZK design

• Highest performance design based on analytic design method; used in gyrotron with internal diffraction RF loss ~ 2%

• Converter optimized using modified design approach

Launcher Wall Field

Analytic Design Optimized Design

Launcher Aperture Field

Z (mm)

R (

mm

)

160 170 180 190 200 2106

8

10

12

14

16

18

20

22

-30

-25

-20

-15

-10

-5

0

Z (mm)

R (

mm

)

160 170 180 190 200 2106

8

10

12

14

16

18

20

22

-30

-25

-20

-15

-10

-5

0

Analytic Design Optimized Design

Radiated Fields on Cylinder Surrounding Launcher

Analytic Design Optimized Design

0 1 2 3 4 5 6160

180

200

220

240

260

280

300

320

Azimuthal Position (radians)

Z (

mm

)

-30

-25

-20

-15

-10

-5

0

0 1 2 3 4 5 6160

180

200

220

240

260

280

300

320

Azimuthal Position (radians)

Z (

mm

)

-30

-25

-20

-15

-10

-5

0

Gaussian coupling factor 0.92

Azimuthal beam divergence angle 41o

Gaussian coupling factor 0.99

Azimuthal beam divergence angle 25o

Impact• SIE and launcher optimization codes in use at CPI,

KIT, JAEA, MIT and UW for gyrotron launcher design• SIE code considered the benchmark code for launcher

analysis• CPI ,JAEA and MIT have built gyrotrons with launcher

designed using synthesis code– Gyrotron with record output frequency-power-pulse width

product, 170 GHz-0.6 MW-3600sec

• Results presented at IRMMW 2006 as invited keynote talk