December, 2013

Plasma Science and Fusion Center Massachusetts Institute of Technology

Cambridge MA 02139 USA This work was supported by AFOSR MURI under Grant No. FA9550-12-1-0489. Reproduction, translation, publication, use and disposal, in whole or in part, by or for the United States government is permitted.

PSFC/JA-13-56

Design of a metamaterial-based backward-wave oscillator

Hummelt, J.S., Lewis, S.M., Shapiro, M.A, Temkin, R.J.

1

Design of a Metamaterial Based Backward-WaveOscillator

J. S. Hummelt, Student Member, IEEE, S. M. Lewis, M. A. Shapiro, Member, IEEE,and R. J. Temkin Fellow, IEEE,

Abstract—In this paper we present the design of a microwavegenerator using metamaterials (MTMs) in a negative indexwaveguide interacting with a high power electron beam. Themicrowave structure is constructed by inserting two MTM platesmade of complementary split-ring-resonators (CSRRs) into arectangular waveguide. Electromagnetic simulations using theHFSS code confirm the presence of a negative index TM-likemode suitable for use in a backward-wave oscillator. Particle incell (PIC) simulations using the CST Particle Studio code areperformed to evaluate the efficiency of the metamaterial BWO(MTMBWO) and the performance of the device in use with a 500keV, 80 A electron beam. Efficiency of >14% is observed withsaturated output at 5.75 MW. The MTMBWO requires about250 ns to reach saturation of the output power. The MTMBWOis also modeled by representing the MTM plates, which consist ofCSRRs, as dielectric slabs whose effective permittivity is given bya Lorentzian model. The dielectric slab model is also simulatedwith the CST PIC code and shows good qualitative agreementwith the more exact model using the actual CSRRs. A structurewas fabricated from brass to test the theoretical predictions ofthe microwave transmission vs. frequency of the negative indexwaveguide. Test results at low power testing using a vectornetwork analyzer showed very good agreement with simulationfor the excitation of the negative index TM-like mode near 2.6GHz. The proposed structure appears to be promising for use ina MTMBWO high power microwave generator.

Index Terms—Metamaterial, BWO, Vacuum Electronics,Plasma Waves

I. INTRODUCTION

METAMATERIALS (MTMs) have unique electromag-netic properties with the potential to open new pos-

sibilities in the design of high power microwave devices. Oneunique property of MTMs is the ability to support negativeindex modes due to a simultaneous negative effective per-mittivity and permeability [1], [2]. Sub-wavelength resonantstructures are used to create macroscopic negative effectiveparameters. There has been much experimental and theoreticalwork to investigate the application of these so called double-negative metamaterials in various engineering applications,including perfect lens design, electromagnetic cloaking, ad-vanced antenna design, accelerator applications, and coherentmicrowave generation [3], [4]. Of interest to this paper is theuse of MTMs in microwave generation, in which a limitedamount of theoretical and experimental work has already beenperformed [5], [6], [7], [8], [9]. The challenges faced in the

The authors are with the Plasma Science and Fusion Center,Massachusetts Institute of Technology, Cambridge, MA 02139 USA(email: hummelt@mit.edu; lewissamim@gmail.com; shapiro@psfc@mit.edu;temkin@mit.edu).

construction of most vacuum electron devices (high powermicrowave fields, vacuum environment, frequency tunability,etc.) place unique design considerations on a MTM which arenot present in optics or low power microwave electronics.

The use of high power electron beams to produce or amplifycoherent microwave radiation is a well established yet activearea of research [10]. Traveling wave tubes (TWTs) andbackward wave oscillators (BWOs) are both common typesof microwave generators that rely on Cerenkov or Smith-Purcell radiation from an electron beam interacting with aslow wave (phase velocity vph < c) to produce coherentradiation. While the physical description of both devices isquite similar, the TWT amplifies a microwave signal travelingwith the electron beam and the BWO generates a backwardwave (group velocity vgr < 0) traveling in the oppositedirection of the beam.

Arrays of split-ring resonators (SRRs) were introduced asa means of achieving an effective negative permeability in abulk material [11]. When used in conjunction with an array ofmetallic posts, one can create a negative index medium, hav-ing simultaneous negative permeability and permittivity. Theelectric analog of the SRR, the complementary-SRR (CSRR)has been shown to produce a negative permittivity [12]. Inaddition, a TM mode in a below-cutoff waveguide is equivalentto a negative effective permeability medium. In order to createa waveguide that supports a negative index mode, we proposethe use of CSRRs in a below-cutoff waveguide. Since a TMmode is of interest for interacting axially with an electronbeam, two parallel CSRR plates running along the electronbeam trajectory are used. The geometrical arrangement of theMTM layers is shown to affect the properties of the modesthey support, which has also been discussed in Ref. [13].

In a mode with negative dispersion, power flows anti-parallel to the electron beam’s motion. While the interactionis similar in nature to a traditional BWO, it is unique in that aconventional BWO relies upon the interaction of an electronbeam with spatial harmonics of a slow wave structure. In anegative index guide, a mode with negative group velocityis supported by the simultaneous negative permittivity andpermeability of the guide. A related approach in creatingstructures that support modes with negative group velocitymakes use of plasmonic waveguides [14], [15], [16], [17], [18],[19]. The use of MTMs in a waveguide provides a means toartificially duplicate the dispersive effect of the plasma withoutcreating the problem of actually producing and sustaining aplasma in a microwave tube.

In addition to the novel physical motivation for their study,

2

MTM based vacuum electron devices may have several keyadvantages over conventional devices from a purely practicaland engineering standpoint. MTMs are frequency selective,and thus will be useful for applications that require precisefrequency control. It is also possible that MTM based vacuumelectron devices could be easier to manufacture than conven-tional tubes. This paper examines the use of a completelyplanar MTM structure created from a metal plate.

II. STRUCTURE DESIGN AND ELECTROMAGNETICSIMULATIONS

In Fig. 1 we show a specific design for a proposed MTMbased BWO (MTMBWO) structure that will operate in S-Band(2-4 GHz). Other frequency devices can also be designed byscaling the structure shown. The MTMBWO is constructed byplacing two periodic MTM plates into a rectangular waveg-uide. Schematics of one period of the MTM plate and ofthe structure are shown in Fig. 1a and Fig. 1b, respectively.The resonant frequency of the MTM plates is designed tobe below cutoff for TM modes in the waveguide. The MTMplates are created by machining CSRRs with period p = 8 mmand width h = 41 mm along two metal plates of thickness t= 1 mm. The resonators have a slot width of a = 2.5 mm.These plates are then placed with a separation of d = 42 mmin a standard size WR340 waveguide (inner dimensions 86mm x 43 mm). In Fig. 1c, a structure is shown with outputcoupling included, which is WR284 (inner dimensions 76 mmx 34 mm) waveguide mated to the MTMBWO structure. Theoverall length of the structure is 440 mm. The cutoff frequency,fc, of the lowest order TM mode in a rectangular waveguideof transverse dimensions l1 x l2 is given by the relation

fc =1

2π√ε0µ0

√π2

l21+π2

l22(1)

where ε0 and µ0 are the free space permittivity and permeabil-ity, respectively. For an empty 43 mm x 86 mm rectangularwaveguide fc = 3.90 GHz.

The eigenmode solver of the High Frequency StructureSimulator (HFSS) code is used to simulate the eigenmodes ofone period of the total structure. The eigenmode simulationsalso generate 3D E and H field vectors in the structurewhich can be used to estimate BWO performance parameters(coupling impedance, start current, etc) [20], [21]. Becauseof symmetry along the center of the structure in the planeparallel to the CSRR plates, a perfect-H boundary was usedto reduce simulation time. In Fig. 2 the dispersion relationcalculated from the HFSS simulation is shown, along withboth the beam line 2πf = kzv0 and the light line 2πf = kzc.Here 2πf is the angular frequency, c is the speed of light, kz isthe wavenumber, and v0 = 0.86 c is the velocity of a 500 keVelectron. The structure supports two negative index modes, onewith Ez ≈ 0 at the axis at ∼ 2 GHz and a TM-like (Hz = 0at the axis) at ∼ 2.5 GHz. Because the axial field is small,the negative index mode at 2 GHz is not expected to interactstrongly with the electron beam and is also not observed in PICsimulation. The beam line intersects the dispersion relation forthe negative index TM-like mode at approximately 2.65 GHz,

h = 41 mm

p =

8 m

m

a=2.5 mm

86 m

m

d=42

mm

43 mm

t=1 mm

Output Ports

c)

a)

CSRR Plates

b)

Fig. 1. a) Simplified schematic of one period of CSRR MTM plate. b)Simplified schematic of MTMBWO waveguide. CSRRs are machined intotwo metal plates of thickness t = 1 mm and inserted into a standard sizeWR340 waveguide with a separation of d = 42 mm. The CSRRs haveperiod p = 8 mm and width h = 41 mm. c) Semi-transparent view oftotal structure including output coupling, created by mating wr284 waveguideto the MTMBWO.

0 10 20 30 40 50 602

2.5

3

3.5

4

4.5

5

kzp (degrees)

Freq

uenc

y (G

Hz)

Dispersion

neg. index TM-like

pos. index TM-like

neg. index Ez ≈ 0

Beam L

ine

Ligh

t Line

Fig. 2. Dispersion relation for one period of the MTMBWO shown in Fig. 1.The frequency of the BWO interaction for the negative index TM-like modepredicted by HFSS is 2.65 GHz.

where the group velocity vgr ≡ ∂ω∂k < 0 and the phase velocity

vph ≡ ωk > 0. The magnitude of the group velocity of the

negative index TM-like mode at the point of intersection withthe beam line is |vgr| ≈ 0.075c.

Of particular interest is the performance of the MTMBWOwhen compared against a more conventional BWO such as arippled wall device. In such a microwave generator there is astart current for oscillation, Ist, above which the device willoscillate with zero input signal. The start current depends onthe device geometry, mode of interaction, and beam param-eters. Traditionally one operates the BWO at ∼ 3× the startcurrent. Increasing the current well beyond the start current(> 7×) can result in auto-modulation of the output power asthe device enters a stochastic regime. This type of behaviorhas been observed in both theory and experiment as shown,for example, in Ref. [22]. In order to estimate the start currentfor the MTMBWO, we use a loss-less linear theory outlined

3

350. 375. 400. 425. 450. 475. 500. 525.550.

10

20

15

Length HmmL

Sta

rtC

urr

ent

HAL

Start Current

Fig. 3. Start current as a function of total structure length in mm for thedesign shown in Fig. 1.

in Ref. [21]. The coupling impedance is given by the relation

Z =|Ew|2

2k2z0P(2)

where |Ew| is the component of the electric field parallel tothe direction of and in phase with the electron beam, P isthe power flux, and kz0 is the wavenumber. The numericalvalue of the coupling impedance is calculated from the electricand magnetic field profiles generated by the HFSS eigenmodesimulation for the negative index TM-like mode and is foundto be Z = 46 Ω. The start current is then readily calculatedfrom the equation

Ist = 4U0(CN)3stλ

3z

ZL3(3)

where U0 is the beam energy, λz = 2πkz

is the longitudinalwavelength (9.6 cm), L is the total length of the structure,N = L/λz is the number of longitudinal wavelengths, and Cis the Pierce parameter given by the equation

C3 =I0Z

4U0(4)

Here I0 is the beam current. From Table 8.1 in Ref. [21]the start condition (CN)st is taken to be 0.314 for zerothorder axial harmonic operation. From Eq. 3 the start currentis plotted in Fig. 3 as a function of structure length.

III. PIC SIMULATION RESULTS AND DISCUSSION

We use the PIC solver of CST Particle Studio to investigatethe performance of the MTMBWO utilizing a relativisticelectron beam. A variable current (60, 80, and 100 A) anda 500 keV (v0 = 0.86 c) beam of radius 2.2 mm is used forthe simulations. The electron beam simulated was a DC beamwith a 4 ns rise time to reach full current from zero initialstarting current. A uniform and axial magnetic field of 1.5 kGis used for all simulations. Output ports are shown in Fig. 1cand were used to record the power generated and coupled outof the structure.

Fig. 4 shows the output power as a function of time fora simulation of a 438 mm long MTMBWO structure using

0 100 200 300 400 500 600103

104

105

106

107

tsa

Pst

Output Power

Pow

er (W

)

Time (ns)

105

104

103

2.6 2.72.5f (GHz)

Power Spectrum

Fig. 4. Output power as a function of time for 80 A in a structure of length438 mm. Here tsa = 258ns is the saturation time of the structure, andPst = 5.75MW defines the stationary power output. The Fourier transformof the output signal after saturation is reached is shown in the inset.

3001500 4500

1000

2000

3000

4000El

ectro

nD

ensi

ty(m

m_ 1)

Length (mm )

Electron Bunching

Fig. 5. Plot of electron density showing the formation of axial bunches atλz ≈ 9 cm. The electron density shown is taken 400 ns after the injection ofthe electron beam, i.e. the device has reached saturation.

an 80 A electron beam. The device takes 260 ns to reach astationary output power regime where 5.75 MW average powerwas produced at 2.595 GHz. Shown in Fig. 5 is a plot of theelectron density for the same simulation taken at 400 ns whichclearly demonstrates the formation of electron bunches in thedevice. As the bunches travel along the axis, they start to breakup as they lose energy to the wave and due to space charge.

A plot of the electron trajectories is shown in Fig. 6a. Thisplot is also from the same simulation and taken at 400 ns.The energy of individual particles is indicated by their color,and distinct electron bunches are visible. The beam size isaffected by the RF field and a slight increase in the beamradius is visible as the beam exits the MTMBWO structure.The electric field magnitude at this time is shown in Fig. 6b.The field grows from zero at the electron beam exit to its peakvalue at the beam entrance, indicating the backwardness of themode. The field strength is greatest close to the MTM plates.

In Fig. 7 the output power and spectrum are shown for a60, 80, and 100 A beam for a fixed length of 462 mm. Using

4

-45-450

700 300

E field magnitude (dB)

E-beam direction

Particle Energy (keV)a)

b)

Fig. 6. a) Cross section of the MTMBWO showing particle trajectories for a438 mm long structure. The individual energy of each particle is indicated bythe color scale. b) Electric field magnitude in the MTMBWO. Both snapshotsare taken 400 ns after the injection of the electron beam, i.e. the device hasreached saturation.

60 A

80 A

100 A

0 300 600

0.01

0.1

10

1

Output Power (MW) Power Spectra

2.4 2.5 2.6 2.7

103

105

102

104

Time (ns) f (GHz)

2.4 2.5 2.6 2.7

103

105

102

104

2.4 2.5 2.6 2.7

103

105

102

104

0 300 600

0.01

0.1

10

1

0 300 600

0.01

0.1

10

1

Current

Fig. 7. Output powers and power spectra of stationary outputs for currentsof 60, 80, and 100 A. The structure length for all three simulations was 462mm.

the results of Eq. 3 and Fig. 3, the nominal start current forthe MTMBWO is 12 A for a 462 mm long structure. A clearchange in the behavior of the MTMBWO is visible as thecurrent is increased from 80 to 100 A, corresponding to 6.7and 8.3 times the start current respectively. At 100 A thereis auto-modulation of the output power, which is indicated bythe multi-frequency operation in the output spectra. Note thatfor an interaction length of 438 mm and a beam current of 80A the auto-modulation instability is absent, which is shown inFig. 4

Output efficiency is an important figure of merit whencomparing the MTMBWO against more traditional BWOdesigns. The output efficiency, calculated by taking the averagesteady state power output at the output ports divided by thebeam power is shown in Fig. 8 as a function of structure lengthfor three different beam currents. A peak efficiency of 14.5 %

400 420 440 460 480 500 520

5

10

15

20

Length (mm)

Out

put

Effic

ienc

y(%

)

Output Efficiency

60 A80 A100 A

Fig. 8. Output efficiency vs. structure length for I = 60, 80, and 100A.Dashed lines are included as a guide for the eye.

is predicted for the 438 mm long structure using 80 A.

IV. EFFECTIVE MEDIUM PIC SIMULATION RESULTS ANDDISCUSSION

A standard and convenient way to interpret the dispersivequalities of a MTM is by use of an effective medium model,which treats the collective electromagnetic behavior of indi-vidual MTM resonators as a bulk material with an effectivepermittivity or permeability [23]. Comparison between sucha model and actual MTM resonators can verify that the res-onators are indeed acting as a MTM (i.e. providing an effectivenegative permittivity or permeability at the frequency of inter-est). More practically, the simulation of a MTM using a PICcode can be computationally intensive, as accurately meshingthe sub-wavelength resonators can make for extremely largemesh sizes. Thus substituting the actual MTM resonators witha bulk material can save simulation time. A similar approachto simulate a MTM interacting with an electron beam usingan effective medium was used in Ref. [5].

We again use the PIC solver in CST Particle Studio tosimulate a MTM based BWO similar to the MTMBWOinvestigated in the previous section. Instead of using CSRRbased MTM plates, an isotropic effective medium is insertedinto the waveguide which models the dispersive qualitiesof the CSRRs. The model is shown in Fig. 9, where thesame outer waveguide as the MTMBWO (WR340) is used toprovide a negative permeability. Using the Drude model, thepermeability of a waveguide follows the expression [3], [24]

µeff = 1− ω2c

ω2(5)

where ωc = 2π · 3.90GHz is the cutoff frequency of the fun-damental TM mode, and the numerical value was calculatedfor the WR340 waveguide using Eq. 1. For ω < ωc, thepermeability of the waveguide is negative.

Coupling power out of the structure is identical in both theeffective medium model and MTMBWO. Two dielectric slabs

5

Effective Medium Structure

E-beam direction

Copperεeff

w

Fig. 9. Effective medium BWO. The device is identical to the MTMBWOintroduced in the previous section, but the CSRR plates have been replacedby an isotropic dielectric slab of width w = 12 mm and permittivity modeledby Eq. 6

of width w = 12 mm are placed in the same location as theCSRR loaded plates in the MTMBWO. The overall lengthof the structure is L = 438 mm long. The electron beamparameters used, I = 80 A, U0 = 500 keV and beam radiusrb = 2.2 mm, are consistent with those used in the simulationsof the MTMBWO. A Lorentz model for the dispersion of theeffective medium is used, which is given by the relation [3],[24]

εeff = 1−ω2p

ω2 − ω20

(6)

where ωp is the plasma frequency and ω0 is the resonantfrequency of the CSRRs. The particular numerical values arechosen for these two frequencies from the dispersion relationobtained by the HFSS eigenmode solver in order to mimic thedispersion of the MTMBWO. Since the permittivity is givenby ε = c2k2

ω2 , the resonance and plasma frequencies in Eq. 6are determined from the dispersion relation (shown in Fig. 2)by letting ε → ∞ and ε → 0, which corresponds to lettingkzp → ∞ and kzp → 0, respectively. We find the resonanceto be at ω0 = 2π · 2.3GHz and the plasma frequency isdetermined to be ωp = 2π · 1.7GHz.

Fig. 10a shows the particle trajectories and formation ofelectron bunches due to the BWO interaction. The magnitudeof the electric field is displayed in Fig. 10b. Comparing thiswith the fields shown for the CSRR based MTMBWO inFig. 6b we see a similar field structure in the region of theguide outside of the dielectric slab (i.e. in the beam tunnel andabove and below the slab). The output power as a function oftime and the Fourier transform of this signal is given in Fig.11. The device saturates at 3 MW and the frequency is 2.62GHz, which is near the output frequency of the MTMBWOof the same length at 2.595 GHz and approximately half ofthe saturated power. Different slab widths w were investigated,with 3 mm to 20 mm having a similar frequency response withsome variation in output power and saturation time. For w = 1mm the slab was too thin for the beam to couple to the guide,and as the slab was made very large (> 20 mm) the presence ofthe slab interfered with power coupling out of the guide. As the

E field magnitude (dB)

E-beam direction

Particle Energy (keV)a)

b) -45-450

700 300

Fig. 10. a) Cross section of the effective medium BWO showing particletrajectories for a 438 mm long structure. The individual energy of each particleis indicated by the color scale. Note the qualitative similarity to Fig. 6a. b)Electric field magnitude in the effective medium BWO. Both snapshots aretaken 700 ns after the injection of the electron beam, i.e. the device hasreached saturation.

0 200 40010-1

10

103

105

107

tsa

Output Power

Pow

er (W

)

Time (ns)800600

Out

put (

a.u.

)

f (GHz)2.62.5 2.7

104

103

Fig. 11. Output power and power spectrum of the full output signal for thestructure in Fig 9. The structure simulated was 438 mm long. The effectivemedium operation frequency of 2.62 GHz is close to the actual MTMBWOstructure of the same length at 2.595 GHz.

slab thickness has been shown to be important in the overallresponse of MTM layers, a more detailed analysis would benecessary to determine exact slab parameters to replicate theCSRR loaded plates [25]. Although the results of the analysisprovided by this model are not in exact agreement with theexact MTMBWO structure, they do provide valuable physicalinsight. [26]

V. STRUCTURE COLD TEST RESULTS

To validate the electromagnetic response of the MTMBWOdesign, measurements of microwave transmission in a testMTMBWO structure made of brass were performed using avector network analyzer (VNA). Experimental measurementsof various coupling schemes to MTM devices has been inves-tigated due to the importance for real-world MTMs [27], [28].The test structure and schematics are shown in Fig. 12. Thestructure is identical to that shown in Fig. 1 except with p = 7mm, d = 16 mm, and a = 2 mm. The overall structure lengthwas 20 periods (160 mm). The MTM plates were aligned in

6

Fig. 12. a) Schematics and b) photograph of test structure and CSRR loadedthin brass plate. The photo is taken looking into one of the identical WR284input ports for coupling in/out of the structure. The negative index mode in thetest CSRR structure is excited by the fundamental mode of the input WR284waveguide.

the WR340 waveguide by end supporting plates, and the MTMresonators were created by machining out two brass plates.Coupling to the negative index mode was non-trivial sincethe electric field topology of the negative index mode in theMTMBWO is unlike the fundamental waveguide mode.

Several coupling designs were investigated to try and min-imize reflection near the operation frequency. In this testdesign, coupling was accomplished through WR284 waveg-uide mated to the side of the MTMBWO test structure. Thismatched the polarization of electric field in the fundamentalTE mode of the WR284 waveguide and the polarization ofthe axial field in the negative index TM-like mode of theMTMBWO test structure. This coupling scheme is differentthan that used in the PIC simulations of the MTMBWO, andwas chosen for simplicity of design. Coax to WR284 waveg-uide adapters are used to excite the fundamental TE modein the WR284 waveguide. The transmission measurement isshown in Fig. 13 in blue, along with a simulation of thesame setup using CST Microwave Studio in red. Excitationof the negative index mode is demonstrated by non-zerotransmission where the negative index mode was predicted byHFSS eigenmode simulation. The planned operation frequencyof the MTMBWO near 2.6 GHz is highlighted in light blue.

2.42.3 2.5 32.6 2.7 2.8 2.92.2-80

-60

-40

-20

0

f HGHzL

Tra

nsm

issi

on

HdBL

Fig. 13. Transmission measurement of the test MTMBWO structure (blue)compared with CST Microwave Studio simulation (red). The planned opera-tion frequency band is highlighted in light blue.

VI. CONCLUSION

The design of a microwave generator using MTMs in a neg-ative index waveguide has been presented. HFSS eigenmodesimulations have been presented to confirm the presence ofthe negative index mode and estimate coupling to the beam.The device operates as a backward wave oscillator with thenegative index mode supported by the MTM plates. CST PICsimulations have been used to demonstrate the performance ofthe device in use with a 500 keV and 60-100 A electron beam.Efficiency of >14% is observed with saturated output at 5.75MW. An effective medium model is presented which modelsthe MTM as a dielectric with a Lorentzian response. Themodel structure is also simulated with the CST PIC code andthe output power and frequency are in qualitative agreementwith the full MTM PIC simulations. Finally, to verify theelectromagnetic design of the MTM device a test structurewas designed and tested using a VNA, and transmissionmeasurements confirmed the presence and excitation of thenegative index mode.

ACKNOWLEDGMENT

This research is supported by AFOSR MURI grant FA9550-12-1-0489 administered through the University of New Mex-ico.

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