Albuquerque, NM, August 21, 2012

Transcription

1 Albuquerque, NM, August 21, EXPLORING DEGENERATE BAND EDGE MODE IN HPM TRAVELING TUBE Alex Figotin and Filippo Capolino University of California at Irvine Supported by AFOSR

2 2 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.

3 TWT with super amplification via DBE Mode 3 TWT with super amplification via the DBE mode. A, B, and C are three different waveguide sections with distinct transverse anisotropy. p(z) Frozen DBE mode magnitude profile RBE mode magnitude profile A B C A B C Injected electron beam u z Electron super bunching as they move along z-axis with velocity u Frozen mode axial electric field phase profile d

4 FROZEN MODE REGIMES 4

5 Stationary points of the dispersion relation. Slow waves. ω vg = = 0, at ω=ω s =ω( ks). k 1. Dramatic increase in density of modes. 2. Qualitative changes in the eigenmode structure (can lead to the frozen mode regime). 5 Examples of stationary points: - Regular band edge (RBE): - Stationary inflection point (SIP): - Degenerate band edge (DBE): ( k k ), v ( k k ) ( ) 2 1/2 ω ω ω ω g g g g g ( k k ) v ( k k ) ( ) 3 2 2/3 ω ω, ω ω. 0 0 g 0 0 ( k k ) v ( k k ) ( ) 4 3 3/4 ω ω, ω ω. d d g d d. a) b) c) frequency ω RBE frequency ω SIP frequency ω DBE ω g ω 0 ω d wavenumber k wavenumber k wavenumber k Each stationary point is associated with slow wave, but there are some fundamental differences between these three cases.

6 6 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).

7 SLOW WAVE RESONANCE Slow-wave phenomena in bounded photonic crystals. 7

8 Cavity Resonator vs. Slow Wave Resonator Examples of Plane-Parallel Open Resonators 8 Simplest uniform resonance cavity with metallic reflectors Uniform resonance cavity with photonic reflectors (DBR) Partial Mirror Cavity Mirror Cavity Single mode photonic cavity Slow wave photonic resonator (no reflectors needed) Defect

9 EM energy density distribution at resonance frequency 9 1 W z E z H z 8π ( ) = 2 2 ε ( ) + µ ( ) Uniform (empty) resonance cavity: Standing wave const Regular slow wave resonance at a RBE: Standing Bloch wave: 2 2 D sin π z D Regular slow wave resonance at a RBE: Standing Bloch wave D sin π z D Giant slow wave resonance at a DBE: NOT a standing Bloch wave 4 D 0 z D 0 z D 0 z D 0 z D Poor confinement Better confinement Best confinement

10 Squared amplitude Squared amplitude Transmittance Transmittance Transmission band edge resonances near a RBE 10 a) N = 16 b) N = Transmission dispersion of periodic stacks with different N. ω g the RBE frequency Frequency ω g Frequency ω g a) N = 16, s = 1 b) N = 32, s = Location z Location z Smoothed energy density distribution at frequency of the first resonance max 2 ( W) WN I

11 Squared amplitude Transmittance Giant transmission band edge resonances near a DBE 11 1 a) N = 16 1 b) N = Transmission dispersion of periodic stacks with different N. ω d the DBE frequency Frequency ω ω d Frequency ω ω d 2500 a) N = 16, s = b) N = 32, s = Location z Location z Smoothed Field intensity distribution at frequency of first transmission resonance max 4 ( W) WN I

12 Summary: RBE resonator vs. DBE resonator Regular Band Edge: ω ωg : max ( ) N W WI m a k k 2 2 ( ) g ω g k Degenerate Band Edge: ω ω max ( ) d N W WI m a4 k k d 4 4 ( ) 4 : ω d k

13 Example: Slow-wave cavity resonance in periodic stacks composed of different number N of unit cells. 13 Energy density distribution inside photonic crystal at frequency of slow wave resonance 2 Regular Band Ed ge: max ( W) WN 4 Degenerate Band E dge: max ( W) WN 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. I I

14 Floquet expansion of fields The electric field in periodic structures (periodic except for an inter-element phase shift): ( + ˆ, ) = (, ) Er dzk Erk e z z ik d A B C A B C A B C z A 1 A 2 F A 1 A 2 F A 1 A 2 F A 1 A 2 F L z A mode is expressed in term of Fourier series expansion, and thus represented as the superposition of Floquet spatial harmonics mode ik z z = p z p= zp, mode (, k ) e ( xyk,, ) E r e kzp, = kz+ 2 π p/ d k = β + iα zp, zp, z

15 Physical modes for coupling A B C A B C u z Forward/Backward k β = β + iα β zp, zp, z zp, zp, α z z > 0 α < 0 Forward waves Backward waves d Slow/Fast (coupling with field produced by electron bunches) Slow Mode: all its Floquet wavenumbers are outside the visible region, or β > zp, k Fast Mode: mode has at least one Floquet wavenumber within the visible region, or β < k zp,

16 Physical waves in open periodic structures Forward Wave β α > zp, z 0 Backward Wave β α < zp, z 0 k = β + iα zp, zp, z 2π p β zp, = kz + d Slow Wave (A) β α zp, > k ρ, p > 0 (proper, bound) (B) β α zp, > k ρ, p > 0 (proper, bound) Fast Wave (C) β α zp, < k ρ, p < 0 (improper, leaky) (D) β α zp, < k ρ, p > 0 (proper, leaky) Theory is complicated, but it can be summarized

17 Methods for complex mode calculations 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: 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

18 Points to be developed 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

19 Modes Waveguide with elliptical sections The elliptical cross sections may act as anisotropic sections

20 Analyzing Modes Vanishing derivatives (up to the third one)