GROUP VELOCITY MATCHING IN DIELECTRIC-LINED WAVEGUIDESAND ITS ROLE IN ELECTRON-TERAHERTZ INTERACTION

A.L. Healy∗, G. Burt, Cockcroft Institute, Lancaster, UKS.P. Jamison, STFC Daresbury Laboratory, UK

AbstractTerahertz(THz)-driven dielectric-lined waveguides have

applications in electron manipulation, particularly accelera-tion, as the use of dielectric allows for phase velocities belowthe speed of light. However matching a single frequency tothe correct velocity does not maximise electron-THz interac-tion; waveguide dispersion typically results in an unmatchedgroup velocity and so the pulse envelope of a short THzpulse changes along the length of the structure. This reducesfield amplitude and therefore accelerating gradient as theenvelope propagates at a different velocity to the electron.Presented here is an analysis of the effect of waveguide dis-persion on THz-electron interaction and its influence onstructure dimensions and choice of THz pulse generation.This effect on net acceleration is demonstrated via an ex-ample of a structure excited by a single-cycle THz pulse,optimised for high field strength but short interaction length,and a structure with a focus on high interaction length. Thisis combined with a comparison of multi-cycle, lower inten-sity THz pulses on net acceleration.

INTRODUCTIONTerahertz frequencies are considered as an alterna-

tive to radio frequencies due to increased acceleratingfield gradients. This is a result of higher breakdownthreshold, which scales with surface electric field, Es,as Es ∝ f 1/2τ−1/4, where f is the operating frequencyand τ is the pulse length [1]. Therefore short pulsedurations and high frequencies are desirable. The use ofTHz over higher frequencies allows for larger structureswhich can be conventionally machined, and an electronbunch of higher charge can be confined within a singleacceleration period. Dielectric-lined waveguides (DLWs)have been experimentally verified for THz-driven electronacceleration at 60 keV [2], in which an accelerating gradientof 2.5 MeV m−1 was achieved for a 10 μJ THz pulseenergy. Recent relativistic experiments of beam-drivenwakefield structures have found accelerating gradients of320 MeV m−1 [3].

DISPERSION IN DIELECTRIC-LINEDWAVEGUIDES

The accelerating modes, LSMm,n=odd, of a rectangularDLW, such as in Fig. 1 are described by the dispersionrelation [4]

∗ alisa.healy@cockcroft.ac.uk

Figure 1: Half cross-section of the waveguide design. Thewaveguide is symmetric about the x-axis.

Figure 2: Dispersion relation for the TM11 mode of a hollowrectangular waveguide and the LSM11 mode of a dielectric-lined waveguide. The dashed line correspond to ω = cβ,the speed-of-light line.

k1y tan(k1y (b − a)

)= �r k0y cot

(k0ya

), (1)

where k0y =√(ω0

c

)2 − β2 − (mπw )2 and k1y =√�r

(ω0c

)2 − β2 − (mπw )2. ω0 is the free-space fre-quency, β is the propagation constant inside the DLW and cis the speed of light. b, a, and w are defined in Fig. 1, and�r is the relative permittivity of the dielectric. The modesare described as longitudinal section magnetic/electric(LSM/LSE), with LSM11 being the first acceleratingmode. These are a hybrid of TM/TE modes due to thedielectric-vacuum interface. Throughout the paper parame-ters a = 100 μm, w = 500 μm, �r = 5.68 (corresponding toCVD diamond) and dielectric thickness t = b − a = 60 μmwill be used. The dispersion relation is shown in Fig. 2 andcompared to a corresponding hollow rectangular waveguide.The phase velocity vp and group velocity vg are given by

vp =ω

β (ω) ,

vg =

(dβ (ω)

dω

)−1.

(2)

The use of dielectric reduces vp to below the speed of light,making continuous acceleration of charged particles possible.

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03 Novel Particle Sources and Acceleration TechniquesA15 New Acceleration Techniques

vp and vg are shown in Fig. 3 for the DLW. vp can bematched to electron velocity, ve, which is only possible forthe synchronous frequency fs . It is assumed ve = c.

Figure 3: vp and vg for the LSM11 DLW. At vp = vec,vg = 0.24 c.

Pulse SlippageThe THz pulse envelope propagates at vg. If vg � ve,

fs (and the electron) propagates at a different velocity tothe pulse and so the field amplitude experienced by a co-propagating electron decreases along the length of the struc-ture. For a given interaction length L the temporal pulselength necessary for continuous acceleration is described by

τp ≥(

1vg

− 1ve

)L . (3)

For a transform-limited Gaussian pulse, the bandwidth Δν ≈0.44/τp , corresponding to

Δν ≤ 0.44L(

1vg

− 1ve) . (4)

This assumes that vg is frequency-independent. As vg =vg(ω), as the pulse propagates the envelope is temporallybroadened due to group velocity dispersion. If vg

Figure 4: Peformance parameters of the DLW as a function of w and t for fixed a and �r .

Figure 5: Energy gain of an electron co-propagating with asingle cycle, 10-cycle and 20-cycle THz pulse of the energy.The solid curves represent the highest possible energy gainfor a pulse with off-centre maximum. The dashed curverepresents energy gain for an electron injected at the peakof the THz pulse, with pulse maximum in the centre.

in Fig. 5. Note that, unless stated, the THz pulse maximumis not in the centre of the pulse. A 20-cycle pulse is includedas the bandwidth does not overlap with the cut-off frequency,which is 0.44 THz for the given DLW dimensions. Thereis an increase in energy gain with number of cycles for thesame THz pulse energy. There is not constant accelerationfor multiple cycles and the effect is more pronounced forincreasing pulse length. An electron passing through severalcycles experiences decelerating field. Included is the energygain for an electron injected on the peak of a 10-cycle THzpulse where the pulse maximum is in the centre. This elec-tron does not experience deceleration but net energy gain islower. The temporal broadening of each pulse after 1 mmis shown in Figs. 6 and 7. The single-cycle pulse has beenhighly dispersed and is no longer recognisable, unlike the20-cycle pulse.

CONCLUSIONSGroup velocity matching is not achievable for ultrarela-

tivistic electrons in the range 0.2-0.8 THz. For a single-cycleinput THz pulse it is also not possible to achieve interactionover the entire length of a structure, with a sub-mm interac-tion length. The interaction length is larger than predicted

Figure 6: Field amplitude of a single-cycle pulse, 1 mm intothe DLW. Inset is the pulse 1 μm into the DLW, showingbroadening due to loss of lower-frequency components.

Figure 7: Field amplitude of a 20-cycle pulse, 1 mm intothe DLW. Inset is the pulse 1 μm into the DLW, showingthe broadening of the pulse due to loss of lower-frequencycomponents.due to the effect of the cut-off frequency of the waveguide,which causes temporal broadening, and the effect of groupvelocity dispersion. A longer pulse results in greater electronacceleration for the same pulse energy despite the lower fieldamplitude. This is a result of the rapid dispersion, whichreduces field amplitude.

REFERENCES[1] Loew, G.A & Wang J.W., "RF breakdown studies in room

temperature electron linac structures", SLAC-PUB-4647, 1988[2] Nanni, E.A. et al., "Terahertz-driven linear electron accelera-

tion", Nature Communications, vol.6, 2015[3] O’Shea, B.D. et al., "Observation of acceleration and deceler-

ation in gigaelectron-volt-per-metre gradient dielectric wake-field accelerators", Nature Communications, vol.7, 2016

[4] Xiao, L. et al., "Field analysis of a dielectric-loaded rectan-gular waveguide accelerating structure", Phys Rev. E, vol. 65pp.016505, 2001

WEPVA019 Proceedings of IPAC2017, Copenhagen, Denmark

ISBN 978-3-95450-182-33298Co

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03 Novel Particle Sources and Acceleration TechniquesA15 New Acceleration Techniques