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EXPLORING DEGENERATE BAND EDGE MODE IN HPM TRAVELING TUBE

Albuquerque, NM, August 21, 2012

Alex Figotin and Filippo Capolino University of California at Irvine

Supported by AFOSR

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MAIN OBJECTIVES FOR THE FIRST YEAR

- Explore degenerate band edge (DBE) modes for multidimensional transmission lines and waveguides.

- DBE mode with alternating axial electric field .

- Transmission line model of TWT that can account for significant feature of the amplification.

- Suggested design of realistic waveguide for HPM TWT supporting DBE.

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z

u

Electron super bunching as theymove along z-axis with velocity u

d

Frozen DBE mode magnitude profile

p(z)

TWT with super amplification via the DBE mode. A, B, and C are three different waveguide sections with distinct transverse anisotropy.

Injected electron beam

Frozen mode axial electric field phase profile

+- -

A B C A B C

RBE mode magnitude profile

TWT with super amplification via DBE Mode

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FROZEN MODE REGIMES

5 ( )

1. Dramatic increase in density of modes.2. Qualitativ

0, at .

e ch

Stationary points of the dispersion relation. Slow waves.

g s sv kk∂ω

= = ω = ω = ω∂

( ) ( ) ( )2 1/ 2

anges in the eigenmode structure (can lead to the frozen mode regime).

- Regular band edge (RBE):

- Stationar

,

y inf

.

Examples of stationary points:

g g g g gk k v k kω−ω ∝ − ∝ − ∝ ω−ω

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

3 2 2/30 0 0 0

4 3 3/ 4

lection point (SIP):

- Degenerate band edge (DBE):

, .

, .

g

d d g d d

k k v k k

k k v k k

ω−ω ∝ − ∝ − ∝ ω−ω

ω−ω ∝ − ∝ − ∝ ω−ω

ωg

RBE

wavenumber k

frequ

ency

ω

a)

ω0

SIP

wavenumber k

frequ

ency

ω

b)

ωd

DBE

wavenumber k

frequ

ency

ω

c)

Each stationary point is associated with slow wave, but there are some fundamental differences between these three cases.

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BASIC CHARACTERISTIC OF THE FROZEN MODE REGIME

- The frozen mode regime is not a conventional resonance – it is not particularly sensitive to the shape and dimensions of the structure.

- The frozen mode regime is much more robust than a common resonance.

- The frozen mode regime persists even for relatively short pulses (bandwidth advantage).

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SLOW WAVE RESONANCE Slow-wave phenomena in bounded photonic crystals.

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Uniform resonance cavity with photonic reflectors (DBR)

Cavity

Slow wave photonic resonator (no reflectors needed)

Single mode photonic cavity

Simplest uniform resonance cavity with metallic reflectors

Mir

ror

Part

ial

Mir

ror

Cavity

Def

ect

Cavity Resonator vs. Slow Wave Resonator Examples of Plane-Parallel Open Resonators

9 EM energy density distribution at resonance frequency

( ) ( ) ( )2 21 8

W z E z H zε µπ = +

Regular slow wave resonance at a RBE:

Standing Bloch wave:

Giant slow wave resonance at a DBE:

NOT a standing Bloch wave

Regular slow wave resonance at a RBE: Standing Bloch wave

2 2sinD zDπ

2 2 2sinD zDπ

0 0 0 0 D D D D z z z z

4D

Poor confinement Better confinement Best confinement

Uniform (empty) resonance cavity:

Standing wave

const

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Smoothed energy density distribution at frequency of the first resonance

Transmission dispersion of periodic stacks with different N. ωg – the RBE frequency

1.6 1.65 1.70

1

Frequency

Tran

smitt

ance

ωg

a) N = 16

1.6 1.65 1.70

1

ωg

Frequency

Tran

smitt

ance

b) N = 32

1 2 3 4 5 1 2

0 4 8 12 160

20

40

60

80

100

Location z

Squa

red

ampl

itude

a) N = 16, s = 1

0 8 16 24 320

20

40

60

80

100

Location z

Squa

red

ampl

itude

b) N = 32, s = 1

( ) 2max IW W N∝

Transmission band edge resonances near a RBE

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Smoothed Field intensity distribution at frequency of first transmission resonance

Giant transmission band edge resonances near a DBE

0 4 8 12 160

500

1000

1500

2000

2500

Location z

Squa

red

ampl

itude

a) N = 16, s = 1

0 8 16 24 320

10000

20000

30000

40000

Location z

b) N = 32, s = 1

( ) 4max IW W N∝

Transmission dispersion of periodic stacks with different N. ωd – the DBE frequency

1.006 1.0080

0.2

0.4

0.6

0.8

1

Frequency ω ω

d

Tran

smitt

ance

a) N = 16

1.0092 1.00930

0.2

0.4

0.6

0.8

1

Frequency ω ω

d

b) N = 32

1 2

2 1

12

( )

( )

( )

( )

22

2

44

4

Regular Band Edge: :

Degenerate Band Edge

2

:

max

4

x

:

ma

g g

I

d d

I

a k k

NW Wm

a k k

NW Wm

ω ≈ ω − −

∝

ω ≈ ω − −

∝

Summary: RBE resonator vs. DBE resonator

k

ω g

d

ω

k

13 Example: Slow-wave cavity resonance in periodic stacks composed of different number N of unit cells.

Energy density distribution inside photonic crystal at frequency of slow wave resonance

( ) 4Degenerate Band E maxdge: IW W N∝

( ) 2Regular Band Ed maxge: IW W N∝

A DBE slow-wave resonator composed of N layers performs similar to a standard RBE resonator composed of N 2 layers, which implies a huge size reduction.

• The electric field in periodic structures (periodic except for an inter-element phase shift):

( ) ( ), ,ˆ zik d

z zd k k e+ =E r z E r

( ) ( ),mode mode, , ,z pik zz p z

pk e x y k

∞

=−∞= ∑E r e

, 2 /z p zk k p dπ= +

, ,z p z p zk iβ α= +

Floquet expansion of fields

A1 A2 F A1 A2 F A1 A2 F A1 A2 F

L

• A mode is expressed in term of Fourier series expansion, and thus represented as the superposition of Floquet spatial harmonics

A B C A B C A B C

z

• Forward/Backward ,

, ,,

0

0z p z

z p z p zz p z

k iβ α

β αβ α

⇒>= + <

Forward waves

Backward waves

Physical modes for coupling

• Slow/Fast (coupling with field produced by electron bunches) Slow Mode: all its Floquet wavenumbers are outside the “visible” region, or

Fast Mode: mode has at least one Floquet wavenumber within the “visible”

region, or

,z p kβ >

,z p kβ <

z

u

d

A B C A B C

Physical waves in open periodic structures

Forward Wave , 0z p zβ α >

Backward Wave , 0z p zβ α <

Slow Wave

(A) ,

, 0z p

p

k

ρ

β

α

>

> (proper, bound) (B) ,

, 0z p

p

k

ρ

β

α

>

> (proper, bound)

Fast Wave

(C) ,

, 0z p

p

k

ρ

β

α

<

< (improper, leaky) (D) ,

, 0z p

p

k

ρ

β

α

<

> (proper, leaky)

Theory is complicated, but it can be summarized

, ,z p z p zk iβ α= +

,2

z p zpk

dπβ = +

Methods for complex mode calculations

• Green’s function methods, combined with method of moments (MoM) • Mode matching (field expansions) • Commercial software is not able to determine complex modes, but it can be combined with properties of complex modes (i.e., moving around constraints of commercial software, HFSS, CST, FEKO, NEC) • Analytic and physical properties

Peculiar modes investigated here need some fine determination: • complex wavenumber or complex frequency descriptions • pairing of modes (long discussion in literature) • spectral points with vanishing derivative • time domain description of polarization

Methods:

• Field in periodic structures • Complex modes in periodic structures • Peculiar spectral points (RBE, SIP, DBE) • Possible structures exhibiting peculiar points • Excitation of complex modes in periodic structures and in

truncated periodic structures • Coupling of modes with fields produced by electron

bunches • Understanding complex modes in the time domain,

including polarization evolution

Points to be developed

• Waveguide with elliptical sections

Modes

The elliptical cross sections may act as anisotropic sections

Vanishing derivatives (up to the third one)

Analyzing Modes