STUDY ON AN OPEN COLUMN SLOW-WAVE GRATING STRUCTURE WITH ARBITRARILY-SHAPED SLOTS ∗

Beiran Chen ξ, Yanyu Wei, Yubin Gong, Juan Lin, Guoqing Zhao, and Wenxiang Wang Vacuum Electronics National Laboratory, University of Electronic Science and Technology of China,

Chengdu,610054,China

∗ Work supported by National Science Fund of China (Grant No.60401005, 60532010). ξ email: cbr19414@126.com

Abstract

The open column slow-wave grating structure, a novel type slow-wave system for Smith-Purcell FEL and relativistic TWT, is presented in this paper. The cold and “hot” dispersion equations of the structure with arbitrarily-shaped slots are derived by means of an approximate field-theory analysis, in which the continuous profile of the groove is approximately replaced by a series of rectangular steps, and the field continuity at the interface of two neighboring steps and the matching conditions at the interface between the groove region and central region are employed. The dispersion characteristics, interaction impedances and the small signal gain of the fundamental wave are numerically calculated and the effects of slot profile are analyzed. .

I. Introduction For the generation of high power millimeter wave even

Terahertz wave in a relatively wide frequency range, the vacuum electron devices based on the Cherenkov effect, such as Orotrons [1],Smith-Purcell FEL, Traveling Wave Tube(TWT)[2] or Backward Wave Oscillator (BWO) and Relativistic multi-wave Cherenkov generators[3] etc, are renewedly paid attention. And much effort has been devoted to study the beam-wave interactions in these devices. As the key component, the interaction system plays a vital role in the performance of the tube.

The open column slow-wave grating structure, a novel type slow-wave system for Smith-Purcell FEL and relativistic TWT, has a series of advantages, such as good dissipation ability, high power capacity, well integrated structure. It is known that as the grating’s cross-section is changed, which causes the boundary conditions of the electromagnetic field to be different, the distribution of the EM would vary. Therefore, it may lead to an influence on the beam-wave filed interactions. In this regards, an open column grating structure with arbitrarily-shaped slots is studied here. The cold and “hot” dispersion equations of this structure are derived, respectively. The dispersion characteristics, interaction

impedances and the small signal gain of the fundamental wave are numerically calculated and the effects of slot profile and other geometrical dimensions of the grating are also analyzed.

II. Dispersion Equation A. The field distribution

Figure 1. The profile of an open grating with a thin annular beam

A thin annular electron beam with radius er propagates through the interaction region of this structure. The structure is divided into three regions: region I ( a br r r≤ ≤ ), region II ( er r≤ ), and region III ( b er r r≤ ≤ ), as shown in Fig.1. The current density of the electron beam is described as 0 ( ) ( )e e en r n r r rδ= ∆ − , Where, en represents the beam density; er∆ indicate the thickness of the annular beam, respectively. The beam is assumed to propagate uniformly along the longitudinal direction of the system, and its velocity is e zv v e= . The expressions of each region’s fields can be written as:

In region I ( a br r r≤ ≤ ):

' '

0

0

[ ( ) ( ) ( ) ( )]

[ ( ) ( ) ( ) ( )]

[ ( ) ( ) ( ) ( )]

0

j t jNz N N a N N a N

j t jNN N a N N a N

j t jNr N N a N N a N

r z

E E J kr Y kr Y kr J kr ejH E J kr Y kr Y kr J kr e

ZNH E J kr Y kr Y kr J kr e

Z kr

E E H

ω ϕ

ω ϕϕ

ω ϕ

ϕ

Ι +

Ι +

Ι +

Ι Ι Ι

= −= − −= − −

= = =

(1) In region II ( er r≤ ):

1-4244-0914-4/07/$25.00 ©2007 IEEE. 1517

2

2

'0

'

0

( )

( )

( )

( )

( )

( )

m

m

m

m

j zj t jNz m N m m

m

j zj t jNz m N m m

m

m m N mm

j t jN j zm m N m

m m N mm

j t jN j zm m N m

r m m mm

E e A K r e

H je B K r e

NE A K rr

kZ B K r ekH j A K rZ

N B K r er

E j A

βω ϕ

βω ϕ

ϕ

ω ϕ β

ϕ

ω ϕ β

γ γ

γ γ

β γ

γ γ

γ γ

β γ

β γ

+∞−Π +

=−∞+∞

−Π +

=−∞+∞

Π

=−∞+ −

+∞Π

=−∞

+ −

Π

=

=

= − +

= −

=

∑

∑

∑

∑

'

0

0

'

( )

( )

( )

( )

m

m

N m

j t jN j zm N m

r m N mm

j t jN j zm m m N m

K r

N kZ B K r er

N kH A K rr Z

B K r e

ω ϕ β

ω ϕ β

γ

γ

γ

β γ γ

+∞

=−∞

+ −

+∞Π

=−∞

+ −

− = −

∑

∑

(2)

In region III ( b er r r≤ ≤ ):

2

2

'0

[ ( ) ( )]

[ ( ) ( )]

[ ( ) ( )]

[ ( ) ( )]

m

m

m

j zj t jNz m N m m N m m

m

j zj t jNz m N m m N m m

m

m m N m m N mm

j t jN j zm m N m m N m

m

E e C I r D K r e

H je X I r Y K r e

NE C I r D K rrkZ X I r Y K r e

H j

βω ϕ

βω ϕ

ϕ

ω ϕ β

ϕ

γ γ γ

γ γ γ

β γ γ

γ γ γ

+∞−ΙΙΙ +

=−∞

+∞−ΙΙΙ +

=−∞

+∞ΙΙΙ

=−∞

+ −

+∞ΙΙΙ

=−∞

= +

= +

= − +

+ +

=

∑

∑

∑

∑ ' '

0

' '

0

0

[ ( ) ( )]

[ ( ) ( )]

[ ( ) ( )]

[ ( ) ( )]

[ ( ) ( )]

[

N

m

N

m

m m m m N m

j t jN j zm m N m m N m

r m m m m m N mm

j t jN j zm N m m N m

r m N m m N mm

m m m

k C I r D K rZN X I r Y K r er

E j C I r D K r

N kZ X I r Y K r er

N kH C I r D K rr Z

X I

ω ϕ β

ω ϕ β

γ γ γ

β γ γ

β γ γ γ

γ γ

γ γ

β γ

+ −

+∞ΙΙΙ

=−∞

+ −

+∞ΙΙΙ

=−∞

+

− +

= +

− +

= +

−

∑

∑' '( ) ( )] m

N

j t jN j zm m N mr Y K r e ω ϕ βγ γ + −

+ (3)

Where, NnE , mA , mB denote the field amplitude factors, respectively; k indicates the wave number in the free space; 0Z represents the wave impedance in vacuum.

0 2m m Lβ β π= + ( 0, 1, 2,...m = ± ± ) represents the axial propagation constant of mth spatial harmonic. NJ , NY represent the regular Bessel and Neumann functions respectively; NI , NK denote the modified Bessel

functions of the first and second kinds,

respectively. 2 2m m kγ β= − denotes the radial propagation

constant of the m-order spatial harmonic in region II and III. B. Boundary Conditions and the Dispersion Equation • The boundary conditions at the plane br r= are

( ) ( )b bE r E rϕ ϕΙ ΙΙΙ=

0z

zE

EΙΙΙ

Ι =

2 2

( 1)2

b b

b

W WTL z TL

WTL z T L

− ≤ ≤ +

+ ≤ ≤ +

2 2

2 2

b b

b b

W WTL TL

W WTL TL

H dz H dzϕ ϕ

+ +Ι ΙΙΙ

− −=∫ ∫

(4) The continuous profile of the groove is approximately

replaced by a series of rectangular steps, as shown in Fig.2.

Figure 2. The staircase representation of an arbitrary groove of open column slow-wave grating structure

By utilizing the admittance matching conditions at the

interface of two successive steps [4], and the recurrence relationship of admittance is derived.

2

1

1

1( ) 2 1

1 ( ) 1 2

1 ( ) 1 2

( , )

( ) ( , )[ ]

( ) ( , )

m

m

n

m r N m m

m m m r N m m

m m m r N m m

Y P kr kr

W W Y Q kr krW W Y P kr kr

−

−

−

− − −

− − −

− − −

= ⋅

+−

(5) Where

' '

' ' ' '

' '

( ) ( ) ( ) ( )( , )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( , )

( ) ( ) ( ) ( )

N N N NN

N N N N

N N N NN

N N N N

J x Y y Y x J yP x y

J x Y y Y x J y

J x Y y Y x J yQ x y

Y x J y J x Y y

−=

−

−=

−

When m= M, ' '

1 11

1 1

( ) ( ) ( ) ( )( )

( ) ( ) ( ) ( )N M N a N a N M

M MN M N a N a N M

J kr Y kr J kr Y krY r

J kr Y kr J kr Y kr− −

−− −

−=

−（6）

The expression of 1( )bY r can be obtained by using the recurrence relationship given in (5) and the initial condition (6).

At the join of region II and region III ( er r= ), zE , Hϕ , Eϕ must match at the join:

1518

( ) ( ); ( ) ( )b b z b z bE r E r E r E rϕ ϕΙΙ ΙΙΙ ΙΙ ΙΙΙ= = ,

( ) ( )b bH r H rϕ ϕΙΙ ΙΙΙ= (7)

Since the beam is assumed to be a thin annulus and contributes only at err = , the solution for the axial field can be expressed in terms of a jump condition [5] in the radial derivative of the electric field, which expresses the discontinuity in the derivative of the axial electric field.

22

3 2 ( )ee

pz zm e z e

r r r r

dE dE r E rdr dr

ωγ

γ ω+ −

Π ΙΙΙΙΙΙ

= =

− = ∆∆

(8)

Therein, emvβωω −=∆ describes the frequency

shifting of the mth space harmonics, 022 / εω eep mne=

is the square of the electron plasma frequency. With the help of the field matching conditions and the

field expressions (1-8), we can obtain the ‘hot’ dispersion equation of this structure,

2

1

22

'4 3 7

2 2 '4 3 2

sin2( )

2

( ) ( ) ( ) 0( ) ( ) ( )

m b

bb

m bm m

mbN m b N m e N m b

N m b N m e m N m b

WW kY r

WL

NrC K r CC K r K r

C K r CC K r k K r

β

β γ

βγ γ γγ γ γ γ

+∞

=−∞

+ ⋅

− − =−

∑ (9)

2 ( ) ( ) ( ) ( )N m b N m e N m e N m bC I r K r I r K rγ γ γ γ= −

2

3 3 2p e

m

rC

ωγ

γ ω∆

=∆

' '4 ( ) ( ) ( ) ( )

N NN m e m e N m e m eC I r K r K r I rγ γ γ γ= − ' '

7 ( ) ( ) ( ) ( )N N N Nm b m e m b m eC I r K r K r I rγ γ γ γ= −

Where, 1 ( )bY r is defined as the surface admittance at the mouth of groove ( br r= ). The factor 2 2/p erω ω∆ ⋅ ∆ in

equation (8) and coefficient 3C can be expressed as

0 02 3

2 3 2

2p e e e

em

I Zr r v c

vkc

ηω π γ

ωγβ

∆=

∆ −

(10)

2 20

11(1 / )ev c

V cη= −

+ ;

( )2

1

1 ev cγ =

− (11)

Where, 0I ， 0V indicate the electron beam current and

acceleration voltage, respectively; 0 377Z = Ω is the wave impedance in vacuum; c is the velocity of light; γ is

the relativity coefficient. η denotes the ratio of the

electron charge to its mass. When the electron current 0I equal zero, the ‘hot’ dispersion equation (9) leads to the cold dispersion equation:

2

1

22

'

2 2 '

sin2( )

2

( ) ( ) 0( ) ( )

s b

bb

s bs

sbN s b N s b

s N s b s N s b

WW

Y r WL

NrK r K rk

K r k K r

β

β

βγ γγ γ γ γ

+∞

=−∞

+

⋅ − =

∑ (12)

According to Pierce’s definition, the interaction impedances of mth space harmonics is given by

*

22zm zm

cm

E EK

Pβ

Π Π

= (13)

Where P denotes the power flows in interaction region. C. Numerical Calculation And Analysis

The ‘hot’ dispersion equation (9) is a complex transcendental equation for k and 0β . It is convenient to regard 0I , 0V , w , nW , ar and br as independent variables, respectively, when carrying out the calculations.

The small-signal gain per period of the fundamental space harmonic of this type of slow-wave structure can be expressed by

)Im(686.8 0 LG β= (14) Where ‘Im’ implies the imaginary part of a complex

number. In a practical calculation of the dispersion equations, and the space harmonic number m takes values from –3 to +3.

In the following analysis, four different groove shapes (triangle shaped groove, trapezium shaped groove, rectangular shaped groove and swallow-tailed shaped groove) are chosen to be calculated. We will compare their influence on the small-signal gain, interaction impedances and ‘hot’ phase velocity by numerical calculation.

0 1 2 3 4 50.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1: swallow-tailed shaped groove Wn/rb=0.35 V0=302kv2: rectangular shaped groove Wn/r

b=0.2 V

0=375kv

3: trapezium shaped groove Wn/rb=0.1 V0=445kv4:triangle shaped groove Wn/rb=0 V0=545kvra/rb=0.75, w/rb=0.2, L/rb=0.37I0=800A

G/(d

B/pe

riod

)

Krb

1

2

3

4

Figure 3. The small-signal gain curves for four different groove shapes

1519

0 1 2 3 4 5

0.5

0.6

0.7

0.8

0.9

1.0

1: Wn/rb=0.35 V

0=302kv

2: Wn/rb=0.2 V

0=375kv

3: Wn/rb=0.1 V0=445kv4: Wn/r

b=0 V

0=545kv

ra/r

b=0.75, w/r

b=0.2, L/r

b=0.37

I0=800A

V p/c

Krb

43

21

Figure 4. Phase velocities for four different groove shapes (Solid line: ‘hot’ dispersion; Dash line: cold dispersion).

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.00

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

1: Wn/rb=0.35 2: Wn/rb=0.2 3: Wn/r

b=0.1

4: Wn/rb=0

ra/rb=0.75, w/rb=0.2, L/rb=0.37

Kc/

Ohm

s

Krb

234

1

Figure 5. Interaction impedances for four different groove shapes.

Fig.4 shows ‘hot’ phase velocity of those four different structures. The bandwidth decreases from the triangle shaped groove to the swallow-tailed shaped groove sequentially, while the small-signal gain (Fig.3) and interaction impedances increase (Fig.5). This indicates the triangle shaped groove can be employed when broad operating band is required; and if the high gain is needed, the swallow-tailed shaped groove structure should be considered.

III. SUMMARY

By means of a self-consistent field theory, the expression of the cold and ‘hot’ dispersion equation of an open column slow-wave grating structure is obtained. The influence of various beam parameters and groove’s geometric parameters on the dispersion relation and the

small signal gain is discussed. The numerical results reveal that among the four different grooves, the triangle swallow-tailed shaped groove has the widest bandwidth and the smallest gain, while the swallow-tailed shaped groove has the narrowest bandwidth and the biggest gain. These results obtained in this paper can provide a theoretical basis for designing the open column slow-wave grating structure.

IV. REFERENCES [1] Marshall E M, Phillips P M and Walsh J E, “Planar orotron experiments in the millimeter wavelength band”, IEEE Trans. Plasma Sci., vol.16, no.1, pp.199-205,1975. [2] Collin R E, “Foundation of Microwave Engineering”, New York: McGraw-Hill, 1996,PP.315. [3] McVey B D, Basten M A, Booske J H, et al., “Analysis of rectangular waveguide-gratings for amplifier applications”, IEEE Trans. on MTT,vol. 42, no. 4, pp. 995-1003,1994. [4] Y Y Wei, W X Wang and J H Sun, “An approach to the analysis of arbitrarily shaped helical groove waveguides”, IEEE Microwave and Wave Lett, vol. 10, no. 4, pp.4-6, 2000. [5] H. P. Freund, M. A. Kodis and N. R. Vanderplats, “Self-consistent field theory of a helix traveling wave tube amplifier,” IEEE Trans. Electron Devices , vol. 20, pp. 543-553, 1992.

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