radiation from a wakefield dielectric structure …...saint petersburg state university *saint...

RADIATION FROM A WAKEFIELD DIELECTRIC STRUCTURE WITH OPEN END

Sergey N. Galyamin, Andrey V. Tyukhtin, Victor V. Vorobev, Aleksandra A. Grigoreva

Saint Petersburg State University

*Saint Petersburg Electrotechnical University "LETI"

Stanislav S. Baturin*, Alexander M. Altmark*

2

Recent results 1

0a

b

x

z

x′

z′α

d

q

S.N. Galyamin, A.V. Tyukhtin, S. Antipov, and S.S. Baturin // Opt. Express, 22(8), 008902 (2014)

adS

x′

y′a b

sinb αsina α

avS

High-order TM0m mode, ultrarelativistic bunch

Vacuum channel: Kirchhoff approximation

Dielectric part of the aperture: approximate decomposition of a

waveguide mode into two plane waves, refraction with Fresnel coefficients

z

1

0o

90o

90− o

30o

60o

30− o

60− o

1θ

Eθ

vEθ

dEθ

planexz −

1

310−

Eθ1

90o90− o 0o 30o 60o60− o 30− o1θ

planeyz−90α = o

( )0 0 exp 2mAk g i tω−

( )0 0expg ik R R′ ′=

01R k′>>

sinR a α′>>

( )2sin

m

aR

αλ

′ >>

02 2 1m k d mλ π εµ= = −

2mma ≈ 1mmb d≈ ≈ 10ε ≈10m =

10 10 (2 ) 500GHzν ω π= =

15mmR′ >>

2(0) 11

1sinm

d nbψθ

χ−=

Direction of maximum:

1 5.14ψ ≈ 2 1( ) 0J ψ =

Stratton-Chu formulas

3

Recent results 2 S.N. Galyamin, A.V. Tyukhtin, V.V. Vorobev Days on Diffraction 2015

(2)

(1)

(2)

a0lTM z

(1)a εx

y

(2)

(2)x ρ ϕ

1 ,mpm m pW NwN∞= =∑

2 10 1 0

1 0

( )( )

( ) ( ) ( )( ) ( ),

2 ( )( )

p lp p p zp p

p m l zl l p

l l p l

w ia j J j k

G G J j k

δ α εα α α ε κ α

α κ α α α

−

+ + +

−

= − +

++

+

1 02 10 1 0

( ) ( ) ( )( ) ( )( )( )

2 ( )( )p m m zm m p

mp mp p p zp pm m p m

G G J j kW ia j J j k

α α α ε κ αδ α ε

α κ α α α+ + +−

−

−= + +

+

( )0 0( / ) ,zlik zi

z lE NJ j a eω ρ= ( )0 01 ( / ) ,zmik zr

z mmmE J j a eNω ρ∞ −==∑

2 2 20 0 ,zl lk k j aε −= − 0 0( ) 0lJ j =0 / ,k cω=

0( / ),p pj aα κ=

mNN

1

0 10 15

10(616GHz)l =( )i

( )r

1 m5

mNN

1 ( )i

m

( )r

1 (48GHz)l =

0 10 151 5

2ε = 1

0 20 301 10 m

mNN

10 (640GHz)l =( )i

( )r

mNN

1 ( )i

1 10 m

( )r

1 (50GHz)l =

0 5 15

10ε =

4

dL vL

a

b

x

z(1)

(2)σ =∞

(1)(2)

(3)

(3)

ε

σ =∞

σ =∞

σ =∞

y

xab

0TM l

Open-ended waveguide with uniform filling inside regular waveguide

0 0( ) 0.lJ j =

0 / ,k cω=

(1)( )1 0( / ) ,zl zi

lH J j b e κωϕ ρ −=

(1)(1)1 01 ( / ) ,zmz

m mmH B J j b eκωϕ ρ∞==∑

(3)(3)1 01 ( / ) ,zm z

m mmH A J j a e γωϕ ρ∞ −

==∑

(2) (2)0(2) 1

0 1 ( ) ,z zmz zm m mmH C e C Z eγ γ

ωϕ ρ ρχ∞−== +∑

(3) 2 2 20 0 .zm mj a kγ −= −

(2)00 ,z ikγ = − (2) 2 2

0 ,zm m kγ χ= −

11 1 0 0( ) ( ) ( ) ( ) ( ),m m mZ J N J a N aξ ξ ξ χ χ−= − 0 0 0 0( ) ( ) ( ) ( ) 0.m m m mJ b N a J a N bχ χ χ χ− =

(1) 2 2 20 0 ,lzl j b kκ ε−= −

:mχ

R. Mittra, S.W. Lee, Analytical Techniques in the Theory of Guided Waves (Macmillan, 1971)

5


Boundary conditions for z = 0 { } 0,Hωϕ = { } 0.Eωρ =

( )2

0 201 1 1 0

0 2

bpm

p pmjj bJ J d J j

b bρ ρ ρ ρ δ

⎛ ⎞⎛ ⎞ =⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠∫

( )1 0 0 000

1 1 2 2(3) (1)0

,b p m

pm

zm zp

bbJ j J jjj aJ J da b

ρ ρ ρ ργ γ

⎛ ⎞− ⎜ ⎟⎛ ⎞⎛ ⎞ ⎝ ⎠=⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎡ ⎤ ⎡ ⎤−⎣ ⎦ ⎣ ⎦∫

0 00

10

a mm

mb

bJ jj aJ da j a

ρ ρ

⎛ ⎞⎜ ⎟⎛ ⎞ ⎝ ⎠=⎜ ⎟⎝ ⎠∫

( ) ( ) ,a

m m p p pp pmb

Z Z d Iρ χ ρ χ ρ ρ δ=∫

11 1 0 0( ) ( ) ( ) ( ) ( ),m m mZ J N J a N aξ ξ ξ χ χ−= −

( ) ( )2 2

2 2

2 2pp p p p pa bI Z a Z bχ χ= −

( )( )0 0 0

01 2 2(2) (3)

a m m p pm

p pb zp zm

b bj J j Z bj a aZ J da

χρ χ ρ ρ ρ

γ γ

⎛ ⎞− ⎜ ⎟⎛ ⎞ ⎝ ⎠=⎜ ⎟⎝ ⎠ ⎡ ⎤ ⎡ ⎤−⎣ ⎦ ⎣ ⎦∫

0 0 0 0( ) ( ) ( ) ( ) 0.m m m mJ b N a J a N bχ χ χ χ− =

(1) 2 2 20 0zp pj b kγ −= −

:mχ

6

(1) (1)1 0

(3) (1) (3) (1) (1) (1)1

2 ( ),m p lp p zp zpm

m zm zp zm zp zp zp

A R bJ jA δ γ κγ γ γ γ κ εγ

∞

=

⎛ ⎞ −⎜ ⎟+ =⎜ ⎟− + +⎝ ⎠

∑%%

(3) (2)1

0,m

m zm zn

Aγ γ

∞

==

−∑

%

(1) (1)

(3) (1) (3) (1) (1) (1)1

4,m p p zp zpm

m zm zp zm zp zp zp

A R BA γ κγ γ γ γ κ εγ

∞

=

⎛ ⎞⎜ ⎟+ =⎜ ⎟− + +⎝ ⎠

∑% %% (2)

(3) (2)1

2 ,mn zn

m zm zn

A C γγ γ

∞

== −

+∑

% %

0,1,...n =1,2,...p =

(1) 2 2 20 0 ,zm mj b kγ −= −

0 00 ,m m

m mbj jA A Ja a

⎛ ⎞= ⎜ ⎟⎝ ⎠% 1 0( )

,2p

p pbJ j

B B=% 0 0 ln ,aC Cb

⎛ ⎞= ⎜ ⎟⎝ ⎠%

22 ( ) ( ) .2 ( ) 2

n nn n n n

n n

Z aa bC C Z bb Z b

χ χχ

⎡ ⎤= −⎢ ⎥

⎢ ⎥⎣ ⎦%

( )( ) 1(1) (1) (1) (1) ,p zp zp zp zpR εγ κ εγ κ−

= − +

sin( ) ( 1) (2 2).πτ ε ε= − +


1/21~zE z τ−

x

z

(2)σ =∞

(1)εσ =∞

Meixner edge condition:

11 ,m m

Am τ+→∞

⎯⎯⎯→⇒

7

Known sets

10,i

i ji

Ax y

∞

==

−∑1

,ijk

i ji

A Fx y

δ∞

==

+∑ 1,2,...j =

{ }ix { }iy F const=1,2,...i =

Residue calculus technique

{ } ?iA −

1 ( ) 0,2 jC

f w dwi w yπ

∞

=−∫—( ) :f w ( ) 0 for | |f w w→ →∞ ⇒

1 ( ) 02 jC

f w dwi w yπ

∞

=+∫—

On the other hand

( ) :f w poles ,iw x= zeros ,iw y=

1

Res ( )0 ,ii ji

f xx y

∞

==

−∑1

Res ( )0 ( )ik

i ji

f x f yx y

∞

== +

+∑

⇒zeros for ,iw y i k= − ≠

( )

1 10

1

( ) ( )( )

( )

ki i

i i

ii

w y w yf w C

w x

∞ ∞

= =∞

=

− +=

−

∏ ∏

∏

0( ) ,kf y F C= ⇒ Res ( )i iA f x= R. Mittra, S.W. Lee, Analytical Techniques in the Theory of Guided Waves (Macmillan, 1971)

8

(1) (1)

( )1 ( ) 0,2

p

zp zpC

R f wf w dwi w wπ γ γ

∞

⎛ ⎞⎜ ⎟+ =⎜ ⎟− +⎝ ⎠

∫— (2)1 ( ) 0,2 znC

f w dwi wπ γ

∞

=−∫—

(1) (1)

( )1 ( ) 0,2

p

zp zpC

R f w f w dwi w wπ γ γ

∞

⎛ ⎞⎜ ⎟+ =⎜ ⎟− +⎝ ⎠

∫— (2)1 ( ) 0.2 znC

f w dwi wπ γ

∞

=+∫—

(3)Res ( ),m zmA f γ=% (2) (2) 1( )(2 )n zn znC f γ γ −= −%( ) ( )(1) (1)

(1) (1)(1) (1)

( ) ( ),

4p zp zp

p zp zpzp zp

R f fB

γ γεγ κ

γ κ

+ −= − +%

(1) ,s zs s bγ πΓ = + Δ (1) 1~4s zss

sb bπ πγ τ τ

→∞⎛ ⎞Γ ⎯⎯⎯→ + − +⎜ ⎟⎝ ⎠


( ) :f w (3)poles ,zmw γ= (2)zeros ,zmw γ=

(1) (1)( ) ( ) 0,zp p zpf R fγ γ+ − =

( 1/2)| |

( ) ,w

f w w τ− +→∞

⎯⎯⎯→

,p l≠(1) (1)

(1) (1) 1 0(1)(1)

2 ( )( ) ( ) .l zl zl

lzl zlzp zl

bJ jf R f

γ κγ γ

κ εγ+ − =

+

sin( ) ( 1) (2 2).πτ ε ε= − +

Shifted zeros:

Meixner edge condition:

9

(1)(1)(1) (1) (3)(2)(1)(1) (2)

( , ) 2 (1)0(2) (1) (1)(1)(1) (1)1 1 10

(2) (3)

112 ( ),

1 1

zszszm zs s zql s zszs z zn

s s s zsm n qzs zszs zzm zs s

zn zq

b bR G

b

γγπγ γ γγ γγ γ γππ γ γγ γγ γγ γ

∞ ∞ ∞

= = =

−++ + Δ⎛ ⎞ +Δ = + Δ −⎜ ⎟⎜ ⎟ +⎝ ⎠ − + Δ − +

∏ ∏ ∏

(2)0 (2)

1 ( )0

1(3)

1

( ) 1( ) 1 ( )

1

zn zn l

ss

m zn

wwwf w P G w

w

γγ

γ

∞

∞=

∞=

=

⎛ ⎞− −⎜ ⎟⎜ ⎟ ⎛ ⎞⎝ ⎠= −⎜ ⎟Γ⎛ ⎞ ⎝ ⎠−⎜ ⎟⎜ ⎟⎝ ⎠

∏∏

∏

( ) exp ln ln ,w b a bG w b aa b aπ

⎡ ⎤−⎛ ⎞⎛ ⎞ ⎛ ⎞= − +⎜ ⎟ ⎜ ⎟⎢ ⎥⎜ ⎟−⎝ ⎠ ⎝ ⎠⎝ ⎠⎣ ⎦


Infinite nonlinear system for shifted zeros:

(1) (1)(1) (1) 1 0

(1)(1)2 ( )

( ) ( ) .l zl zllzl zl

zp zl

bJ jf R f

γ κγ γ

κ εγ+ − =

+0 :P

s sτ

→∞Δ ⎯⎯⎯→

10

Numerical results

321 4

0

0321 m4 5

1

1

1

l=2

l=

88.5% 5.7%5.8%

66.2% 2.0%31.8%

99.8% 0.0%0.2%

99.9% 0.0%0.1%

99.9% 0.0%0.1%

321 4 55

( )i

87.4% 6.3%6.3%

37GHz10ε =

2

ab

=

4

ab

=

8

ab

=

8a

b=

4

ab

=

2

ab

=

evan.−prop.−

( )i ( )i

( )i( )i( )i

b=2.4mm @ 37GHz

10ε =

11

Numerical results b=2.4mm @ 37GHz

50mmvL =30mmdL =

15.372mmz = −

30mmz = −

50mmz =

0z =

CST simulation

•  2 orthogonal H-planes: (xz), (yz).

•  Time-domain solver (36-38GHz wave packet)

•  3 axial symmetric modes (15 CST modes) at each port

12

2a b=

4a b=


CST simulation

Mode (area) CST PTC Mathcad 15 1 (1) 87.9% 87.3% 2 (1) 0.1% 0.1% 0 (2) 6.0% 6.3% 1 (3) 6.0% 6.3%

Total: 100% 100.0% Time: 23 min* 30 sec*

2a b=

Mode (area) CST PTC Mathcad 15 1 (1) 88.4% 88.4% 2 (1) 0.05% 0.1% 0 (2) 2.8% 2.6% 1 (2) 2.7% 3.2% 1 (3) 2.9% 0.4% 2 (3) 5.2% 5.3%

Total: 102% 100.0% Time: - 34 sec*

4a b=

13


1

l=

0

11

l=

0531 m7 9

1

1311531 7 9 1311 15531 7 9 1311 15

prop.−evan.−

2

ab

=

71.7% 27.5%0.8%100GHz10ε =

9.8% 88.7%1.5%

2a

b=

( )i

( )i

( )i

100GHz2ε =

( )i

4

ab

=

4a

b=

72.1% 27.1%0.8%

8.6% 90.0%1.4%

( )i

( )i

8

ab

=

8a

b=

70.7% 28.2%1.1%

10.2% 87.6%2.2%

2a b=Hωϕ

100GHz2ε = 4a b= 8a b=

60o45o

30o

15o

z

L

θR

0%531 m7 9

20%

1311

40%

Power reflected by the second open end

max3L λ=1l =

2a b=Hωϕ

100GHz10ε = 4a b= 8a b=

60o45o

30o

15o1.6× 1.6× 1.6×

10ε =

2ε =

14

Thank you for your attention!

radiation from a wakefield dielectric structure …...saint petersburg state university *saint...

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