Controlling Beam Wave InteractionInside Traveling Wave Tube Using

MetamaterialsExploiting Plasmon Resonances associated with Sub-wavelength

Geometry

Lokendra Thakurwith Robert Lipton and Anthony Polizzi

Overview

Objective

Control beam-wave interaction in high power microwave amplifiers.

Design metallic sub-wavelength structures that mimic the response

of a dielectric for generating slow electromagnetic waves.

D.Shiffler, J.Luginsland, D.M.French, L.Watrous, IEEE Trans.

Plasma Sci.2010.

Methodology

Influence beam wave interaction frequencies by controlling intrinsic

plasmon resonances of metallic sub-wavelength structure. Seek

gain over a sufficiently broad band of frequencies.

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TWT Amplifier Geometry

Creat slow wave structure using metallic concentric rings

beam

Electrongun Collector

concentric ringgeometry

Goal

Design ring geometry for amplifiers operating in the S- & K- band

regimes.

For high power application, we adopt a periodic ring structure.

R

Rd

Rbd

d

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We follow [Shiffler, Lunginsland, French, Watrous, IEEE Plasma

Sci.] and use a metamaterials approach to design.

When the periodic geometry of the rings are sub-wavelenght.

We replace geometry with an effective dielectric constant �eff .

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Fundamental Theoretical Issue

How do we choose the effective property �eff so that the

approximation provides an adequate representation of the actual

electromagnetic fields inside the TWT?

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Asymptotic Approach

Two scale expansions and metamaterials

λ= wavelength, Sub-wavelength geometry d <λ2π . For S-band

d <7.5cm2π . Write solutions of Maxwell’s equations inside TWT as

power series in powers of d.

� �E =�∞

n=0 dn �En

� �H =�∞

n=0 dn �Hn

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� The leading order terms in the power series solutions identify

the sub-wavelength physics usefull for controlling beam wave

interaction for the regime d <λ2π .

� We shall see that the plasmon resonances of the subwave

length ring structure control the interaction frequencies to

leading order.

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Geometry and Problem statement� Infinitely long TWT loaded with sub-wavelength ring

structure.

� Beam is modeled using the hydrodynamics approximation.

Look for TM modes exhibiting gain inside the TWT of the form

�B = �eθBθ(r, z) = �eθψ(r, z)eikz

,

Goal

We would like to find and control frequency bands ωl < ω < ωu,

such that k is negative and imaginary.

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Here �B = �eθBθ(r, z) solves the Maxwell’s equations.

For 0 < r < R,

∇× (�−1m (r, z)(∇× �B)) = ω2

c2�B.

On boundary of TWT,

[�n× (∇×Bθ�eθ)]R+

R− = 0.

On metal ring interface,

[c2�−1m (r, z)�n× (∇× �B)]metal

host = 0, [Bθ]metalhost = 0.

On beam host interface, r = Rb,

[c2�−1(r, z)�n× (∇× �B)]R+

b

R−b

= 0, [Bθ]R+

b

R−b

= 0.

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with

�−1(r, z) =

�−1m (r, z), Rd < r < R,

1, Rb < r < Rd,

�er�er + �eθ�eθ +�1 + ωp

2

γ3(ω−v0k)2

��ez�ez, 0 < r < Rb,

Where,

�−1m (r, z) =

1, in Host,

,

�−1p (ω), in metal ring,

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Expand solution �B = �eθψ(r, z)eikz in Taylor series.

ψ(r, z) = ψ0(r) + dψ1(ρ, y, r) + d2ψ2(ρ, y, r) + d

3ψ3(ρ, y, r) + · · ·

ω

c= k(ξ0 + dkξ1 + (dk)2ξ2 + (dk)3ξ3 + · · · )

where

ρ =r

d, y =

z

d.

are “fast variables” .

Substitution of Taylor series into Maxwell’s equations and equating

like powers gives a leading order theory. This corresponds to the

TWT loaded with metamaterial with an effective dielectric

constant �−1eff = �

−1eff (kξ0), that is frequency dependent.

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The poles and zeros of the effective dielectric constant are

determined by the plasmon resonances of the sub-wavelength

structure.

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The leading order solution is given by the pair

Bθ(r, z) = ψ(r, z)eikz�eθ,ω

c= kξ0

and solves Maxwell’s equations for the metamaterial loaded TWT

given by

�

(∇+ ik�ez)× (�−1eff (kξ0)(∇+ ik�ez))× �eθψ0(r) = k

2ξ0ψ0(r)�eθ.

In the micro structural region (Rd < r < R), with frequency

dependent effective property �−1eff (kξ0)

�(∇+ ik�ez)× (∇+ ik�ez)× �eθψ0(r) = k

2ξ0ψ0(r)�eθ.

In the vacuum (Rb < r < Rd).

�

(∇+ ik�ez)× (�−1beam(kξ0)(∇+ ik�ez))× �eθψ0(r) = k

2ξ0ψ0(r).

In the electron beam (0 < r < Rb)

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With

�

�−1beam(kξ0) = �er�er + �eθ�eθ +

�1 +

ωp2

γ3(ω − v0k)2

��ez�ez.

and

�−1eff (kξ0) = [(1− θp) + θp�

−1p (kξ0)](�er�er + �ez�ez)

−∞�

j=1

(1− �−1p (kξ0)) �αj �αj

1�−1p (kξ0)−1

+ (12 − λj)

� where θp=Area fraction occupied by P .

� And −12 < λj <

12 are the plasmon resonances of the

sub-wavelength structure.

� �αj are weights associated with the plasmons.

�αj = αρj�eρ + α

zj�ez

The formula for �−1eff (ω) holds for any form of frequency dependent

dielectric constant �−1p (ω) describing the metal rings.

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What a plasmon is

� The plasmon is a source free magnetic field that permeates

both the metal ring and surrounding host material

� It is controlled by the sub-wavelength geometry.

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The plasmon mode and resonance are the pairs ωj�eθ, λj that

solves the source free problem.

�∇×∇× (ωj�eθ) = 0

�

−1

2[�n× (∇× ωj�eθ) |∂P out + �n× (∇× ωj�eθ) |∂P in ]

= λj [�n× (∇× ωj�eθ) |∂P in − �n× (∇× ωj�eθ) |∂P out ]

The geometry of P controls ωj and λj .16 / 25

Review of effective dielectric constant

�−1eff (kξ0) = [(1− θp) + θp�

−1p (kξ0)](�er�er + �ez�ez)

−∞�

j=1

(1− �−1p (kξ0)) �αj �αj

1(�−1

p (kξ0)−1)+ (12 − λj)

Observation

� (1− �−1p (kξ0)) is controlled by dielectric property of metal

rings.

� �αj �αj and (12 − λj) are controlled by the geometry.

Here

�αj = αrj�er + α

zj�ez

αrj =

�

P(∇× �eθωj).�er

αzj =

�

P(∇× �eθωj).�ez

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Plasmon resonance frequencies for the metamaterial

�−1eff (ω)

ω

ωjplasmon ω

j+1plasmon

ωjplasmon is the root of

1�−1p (ωj

plasmon)−1+ (12 − λj) = 0

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Approach to Design

Given that we know the behavior of the dielectric constant �−1p (ω)

of the metal across S-band frequencies then we can change the

shape of the sub-wavelength geometry to manipulate plasmon

resonance frequencies ωjplasmon.

P

λj

�−1eff (ω)

P +∆P

λj +∆λj

�−1eff (ω) +∆�−1

eff (ω)for ω in S-band

ωjplasmon ω

jplasmon +∆ω

jplasmon

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Incorporate this strategy within Pierce theory for TWT loaded with

a metamaterial.

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We design for gain of TM modes given by the solutions of the

leading order theory (ω, ψ0(r)eikz, �eθ) with dispersion relation

Dleading order(ω, k) = 0.

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We apply the assumptions of a Pierce Theory∗to write

k = k0 + q.

Where k0 satisfies the dispersion relation in the cold structure

DNobeam(ω, k0),

and q satisfies the Pierce-like relation

q(q −∆k)2 = −k03.

∆k (known as slip) is equal to ω( 1vbeam

− 1vph

)

vbeam is electron beam velocity

vph = ωk0

is phase velocity of electromagnetic wave.

k0 is the coupling wave number.

∗ L.Schacter, J.Natron and G.kerslick Journal of applied physics

88(11), 1990 pp5874-5882.

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The coupling wave number is a function of beam current, beam

radius and �−1eff and is given by

k03 =

beam parameters� �� �1

2

eIη0

mc2

1

(γβ)21

πR2[I0

2(θb)− I12(θb)

����k=k0

×

slow wave structure� �� �

[(R

Rd)2χdT0(χd)

I0(θd){∂DNo−beam(ω, k0)

∂k}−1

����k=k0

β=vbeam

c , I =Average beam current, m=electron mass,

γ = (1− β2)−1

, η0=vacuum characteristic impedance,

Γ = k02 − (ωc )

2, θb = ΓRd,

κ = k02 − �eff(

ωc )(

ωc )

2

χd = kRd,

I0 is a zero-order Bessel function of 1st kind,I1 is 1st order Bessel function,T0(κr) = J0(κr)Y0(κR)− Y0(κr)J0(κR)

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Under suitable assumptions∗maximum growth (gain) occurs for

the root

q = k0(1

2+ i

√3

2)

and the growth rate is

Im(q) =

√3

2k0

∗ See L.Schachter Beam-Wave Interaction in periodic and Quasi

periodic structure Springer 2011.

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Summary

� We have a generalization of Pierce Theory for TWT’s loaded

with transversely symmetric metallic metamaterials.

� Provide an explicit connection between metamaterial loaded

TWTs and metal-ring-loaded TWTs as a power series

representation of solutions.

� The generalized Pierce theory for metamaterial loaded TWT’s

will be used as a theoretical platform to quickly test and

design sub-wavelength geometries for gain vs. bandwidth.

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