Sub-luminal travelling wave

Laser-driven terahertz sources based on photoconductive antennas (PCA) or optical rectification provide a conversion of the optical pulse (group) arrival time to a terahertz carrier phase. For the PCA the optical energy deposition is responsible for carrier generation, setting the time (phase) of the radiating surface current surge43. For optical rectification, the coherent convolution of difference frequency mixing combinations can be shown to produce a terahertz carrier field following the time-derivative of the optical envelope44. To achieve a sub-luminal carrier wave, we exploit this group-to-phase conversion together with tilted optical pulse-fronts giving a controllable arrival-time delay exciting a planar terahertz source. Tilted pulse-fronts are generated through imaging of the pulse propagated from an optical diffraction grating. The optical wavefronts of the pulse remain orthogonal to the propagation direction, while there is a transverse time delay in the local energy content of the pulse. As shown in Fig. 1 when impinging on a source plane (a nonlinear χ (2) material or PCA) the delayed optical energy arrival produces an effective travelling wave source for the terahertz radiation. In the case of optical rectification, the preservation of wavefront direction orthogonal to propagation allows the one-dimensional pencil-beam χ (2) phase-matching conditions to be maintained independent of the pulse-front-tilt angle. The use of tilted pulse-fronts is central to obtaining an effective sub-luminal terahertz phase velocity. With pulse-front-tilt spanning the range \(0\leqslant {\phi _{{\rm{tilt}}}} < \pi {\rm{/}}2\) the effective surface optical group velocity (and hence terahertz phase velocity) can span \(0 < v_{{\rm{group}}}^{{\rm{opt,eff}}}\le \infty \). As the optical carrier wavefront remains travelling with normal angle of incidence with effective surface velocity \(v_\phi ^{{\rm{opt,eff}}} = \infty \) it cannot however be exploited directly for acceleration. Although a simpler non-normal incident planar pulse front can also provide effective velocity control, such an arrangement is limited to velocities greater than c; due to Snell’s law refraction of the pulse front on entry to the velocity reducing medium, this limitation of planar pulse-fronts holds even when the source is embedded in a material of refractive index n > 1.

Fig. 1 Effective travelling wave concept, for both a nonlinear material and a photoconductive antenna. a For a tilted pulse front the effective velocity is unbounded, while for an obliquely incident plane wave the effective source velocity is constrained to be greater than c. Tilt and incident angles, θ tilt and θ inc , respectively, are those external to the source material. PCA: photoconductive antenna. b, c Schematic of Huygens waves (black) generating the propagating field in the vacuum region above a dielectric medium. The shaded regions represent the dielectric medium. b For a super-luminal source velocity the Huygens waves interfere to give a plane wave, while c for a single-cycle sub-luminal source interference and cancellation is absent Full size image

Propagating evanescent waves

The excitation of the terahertz sources through tilted optical pulse fronts allows coupling of the optical energy into the source material at sub-luminal velocities. The outcoupling of the terahertz pulse from inside the source material is likewise subject to conditions of boundary continuity and refraction, and for conventional many-cycle electromagnetic waves the transition to sub-luminal source propagation is equivalent to meeting conditions for the critical angle of total internal reflection and post-boundary evanescent wave propagation45. For the single-cycle pulses generated by optical rectification the classification of the post-boundary fields as non-propagating and exponentially decaying in amplitude is no longer appropriate. Conceptually describing the travelling wave source as a superposition of time delayed Huygens spherical waves the sub-luminal and super-luminal velocities, and single-cycle vs. multi-cycle pulses, give rise to qualitatively different solutions for the propagating waves. For the super-luminal source (Fig. 1b) a superposition of waves gives rise to a solution with a far-field distribution approximating an obliquely propagating plane-wave satisfying Snell’s law of refraction. For the sub-luminal case the Huygens waves of a single-cycle pulse cannot coincide, with the exception being v ϕ = c for which they add coherently along the surface. For single-cycle pulses the process of constructive or destructive interference for Δϕ > 2π does not arise and the exponential decay of the field with distance from the surface associated with evanescent waves is supplanted with a slower 1/r decay of the Huygens fields and a temporal stretching arising from the superposition of wavelets retarded in time by the sub-luminal source velocity.

To provide a detailed and quantitative picture of the single-cycle propagation from a sub-luminal source, finite difference time domain (FDTD) simulations have been undertaken (Methods section). An example of finite difference time domain calculation of emission from (and into) the surface under sub-luminal conditions is presented in Fig. 2a. In this example the terahertz radiation is generated only within the top 50 μm layer of the dielectric; such a situation arises, for example, with a (110) oriented and χ (2) active ZnTe layer on a thicker inactive (100) oriented ZnTe substrate. Thinner sub-micron source layers are expected for photoconductive antenna. For thicker source regions spatio-temporal shaping of the source may also be required for high-efficiency generation, as discussed below for LiNbO 3 sources. After an initial stage of propagation where the field is established in the region above the source plane, a stable pulse is obtained, travelling with a wavefront normal to the surface and with a velocity set by the effective source velocity. The normal-to-surface extention of the field structure, which evolves over a finite time, is responsible for an apparent instananeous propagagtion of the field across a gap in frustrated total internal reflection45, 46. The physical connection to total internal reflection is described schematically in Fig. 2b, and is apparent in the numerically calculated field profiles of Fig. 2a. A surface sub-luminal travelling source is analogous to that arising from a virtual planar wavefront arriving at an angle of incidence exceeding the critical angle. This virtual incident beam establishes a real beam propagating as if reflected, together with the sub-luminal fields propagating across the boundary into the vacuum region. The use of total internal refection and evanescent fields has been previously proposed for particle acceleration47, 48. While the resulting field structures and evolution is analogous to that expected in total internal reflection, our scheme does not rely on a real (terahertz) incident beam and avoids the associated obstacles of in-coupling of a pulse at a sub-critical angle.

Fig. 2 The field evolution for a nonlinear source within a dielectric. a, b Finite difference time domain calculations of the electric field for a travelling source within a dielectric layer on a dielectric substrate. The terahertz source is only present within a 50 μm active layer. The source velocity is greater than the speed of light within the dielectric, but less than the speed of light in vacuum. Time snap-shots are shown for a 0.5 ps and b 4.5 ps after the start of the nonlinear interaction. c The travelling source can be viewed as a surface excitation driven by a virtual incident plane wave. For a sub-luminal (in vacuum) effective surface velocity the angle of incidence for the virtual wave will exceed the critical angle for total internal reflection. For thicker source regions the virtual source has a real component propagating within the dielectric and spatio-temporal shaping will be required for maximum terahertz generation efficiency Full size image

Observation of single-cycle sub-luminal propagation

Sub-luminal dispersionless terahertz pulses have been generated using the concepts of Fig. 1, in both a large-area interdigitated photoconductive antenna and through optical rectification in a (110)-cut ZnTe single crystal (see Methods section). The experimental configuration is shown schematically in Fig. 3a. In both cases the optical pulse (group) front tilts were produced by a diffraction grating, with the diffracted pulse imaged onto the generation PCA or ZnTe crystal. The optical pump beam was incident normal to the generation plane. Temporal and spatial characterisation of the field was undertaken with electro-optic sampling with a 50 fs optical probe retro-reflected from the internal boundary of a separate ZnTe electro-optic detection crystal. The terahertz electric field emitted from the source and propagated through air into the detection material effectively creates a birefringence in the detection crystal that temporally and spatially replicates the incident terahertz pulse, and this birefringence is observed through the polarisation change of the optical probe (see Methods section). Figure 3c shows pulse propagation measurements for a normal-incidence planar pulse front optical pump on a ZnTe emitter, with \(v_{{\rm{eff}}}^{{\rm{THz}}} = \infty \), and measurements of \(v_{{\rm{eff}}}^{{\rm{THz}}} \approx c\) pulse propagation in ZnTe and PCA sources. The tuneability from sub- to super-luminal propagation has been observed with temporal-spatial mapping of the pulse evolution undertaken for a range of pulse-front tilts, corresponding to effective velocities from 1.75c down to 0.77c. Measured world lines in ZnTe are shown in Fig. 3b. We observe no significant pulse broadening or reshaping as a function of wave propagation along the surface; variations in intensity are attributable principally to the transverse intensity profile of the optical pulse providing the terahertz excitation.

Fig. 3 Experimental measurements of the terahertz emission from a travelling source. a Experimental arrangement for the spatial and temporal characterisation of the terahertz pulse emitted from a ZnTe or interdigitated photoconductive antenna (iPCA) travelling wave source. b World lines of the terahertz pulse measured for a ZnTe source with differing pulse-front tilts. The data points are the time of the peak electric field. The lines are linear fits to the data, and correspond to phase and group velocities from 0.77c to 1.75c. c Examples of the single-cycle terahertz pulse temporal profiles measured at positions in a plane parallel to the emitting surface. The longitudinal direction is defined as the direction of propagation of the travelling wave source. From top to bottom, the pulses were generated by a planar wavefront optical pulse exciting a ZnTe emitter (v eff = ∞), and θ tilt ~ 45° pulse front exciting an PCA and ZnTe emitters Full size image

Application to particle acceleration and manipulation

Both PCAs and appropriately oriented χ (2) optical rectification sources will produce terahertz electric fields polarised in the plane of the source. With the field polarisation aligned in the direction of propagation of the travelling wave (the longitudinal direction) co-propagating charged-particle acceleration becomes possible. The longitudinal accelerating electric fields are accompanied by significant transverse electric and magnetic fields which arise through the spatial and temporal gradients of the terahertz source. For a relativistic particle with β ≲ c the magnetic deflecting force becomes comparable to the accelerating force. Additionally the transverse electric field necessary to satisfy the condition of vanishing electric field divergence in the vacuum gap becomes comparable to the accelerating field temporally ahead and behind the region of acceleration. As in laser-dielectric acceleration schemes, and in conventional RF cavity acceleration, these deflection forces can be eliminated or reduced through imposing additional symmetry around the particle beam-axis. Here, we consider a two-dimension arrangement with sources located in a pair of z−y planes. This two-dimensional (2D) symmetric arrangement and the resultant field structure is similar to the total internal reflection scheme proposed by Frandsen et al.47 and Pálfalvi et al.48. Figure 4 shows the electric and magnetic fields, calculated with FDTD modelling, in the vacuum gap between a pair of opposing sources embedded in dielectric media separated by 400 μm (see Methods section). The longitudinal electric field sources E z (x = ±250 μm, z−β s ct, t) are localised in z−y planes with x = ±250 μm, and propagate with source velocity cβ s . The source velocity in these calculations was β s = 0.995, corresponding to velocity matching for a 5 MeV electron beam. Due to cancellation of the transverse electric and magnetic fields in the symmetry plane of the source-pair, the deflection forces are minimal in a region of ≈100 μm around the central beam-axis, providing an acceleration potential bucket sufficient for injection of particle beams from conventional electron guns. The field structure produced from the opposing pair travelling sources is similar to that found within conventional RF accelerating structures, with the exception that the accelerating fields and electromagnetic energy are co-propagating in synchrony with the charged particles rather than stored over the longer times and spatial extent in an accelerating cavity structure.

Fig. 4 Particle acceleration in a paired travelling source structure. a–c longitudinal electric field (E z ) and transverse electric and magnetic field components (E x , H y ) at t = 18 ps for a 500 fs, bipolar terahertz source with peak longitudinal field strength of 10 MV m−1 generated in each crystal. Nonlinear dielectric media are present at x < 2.0 mm and x > 2.4 mm, with a vacuum region in between. The vacuum/dielectric interfaces are marked with dotted lines. The nonlinear interaction in each crystal generates identical longitudinal field E z and opposite polarity transverse fields E x , B y . d–f Transverse position, angle and energy respectively for 1000 electrons after acceleration in a 15 mm structure as a function of particle position in the bunch; particles enter the structure at 5.2 MeV/c, with 500 fs rms duration with 100 μm rms diameter. The blue, yellow and green particle distributions correspond to those at times 0, 8.8 and 17.7 ps, respectively. For a 5 MeV beam energy with particle velocity β/c = 0.995, there is negligible velocity spread within the bunch, allowing the behaviour of shorter duration bunches to be inferred directly from the corresponding temporal slice in the particle distributions shown Full size image

The results of modelling particle acceleration of a 5 MeV electron beam injected into the travelling source paired structure are shown in Fig. 4d–f. The particle dynamics has been determined through numerical solution of the relativistic Lorentz force equation, and include the electric and magnetic deflection forces (see Methods section). For the 2D arrangement of the source, the acceleration and deflection occur in the z and x axes, with E y , H x , H z = 0 by symmetry. The injected beam has a σ t = 1 ps root mean square (r.m.s.) pulse duration, and a transverse phase-space emittance of 0.3 mm mrad, consistent with that obtainable from state-of-the-art RF photo-injectors1, 49. We show the evolution of the particle distribution as the particles interact over ~7 mm with a peak accelerating gradient of 10 MV m−1 and a 500 fs bipolar terahertz field representative of our LiNbO 3 source to be described in the following section. Net energy gain of ~100 keV is obtained when injecting close to the peak phase of the accelerating field. In contrast to acceleration schemes using optical wavelengths, the approximately 100 μm longitudinal dimension of the acceleration bucket ensures negligible phase slippage of electron bunches with a typical 10−3 energy spread, while the 100 μm transverse dimensions of the accelerating field allow for the entire injected electron bunch to be captured and accelerated without transverse phase-space degradation.

MV m−1 field strengths for particle acceleration

The above measurements with interdigitated PCA and ZnTe optical rectification sources serve to illustrate the concept of a single-cycle source with longitudinal polarisation and sub-luminal dispersionless propagation. Neither of these optical to terahertz mechanisms or media are however capable of generating the multi-MV m−1 field strengths sought for high-gradient relativistic particle acceleration. To achieve high field strengths the same travelling source concept has been adapted to work with a LiNbO 3 nonlinear medium, a crystal that is capable of greater than 100 MV m−1 field strengths50. A significant hurdle is encountered in using this established material for high-field terahertz generation, in that non-colinear propagation of terahertz and optical frequency waves is required to satisfy the terahertz and optical phase-matching51. For LiNbO 3 the optical group velocity exceeds the phase velocity of the generated terahertz radiation by a factor of approximately 2.5. As a consequence the terahertz radiation is generated in an obliquely propagating Cherenkov cone centred around the laser propagation axis. To allow efficient generation from a transversely extended optical beam a temporal-spatial correlation is introduced in the optical excitation pulse, with the correlation such that the locally produced Cherenkov cones add constructively to produce plane-wave emission in a specific Cherenkov direction51, 52. To provide a \(v_\phi ^{{\rm{eff}}} < c\) travelling source in LiNbO 3 , we have developed a scheme that maintains the standard Cherenkov pulse-front tilt in one plane, while adding spatially dependent time delays in the orthogonal direction. As shown in Fig. 5, the LiNbO 3 crystal is oriented with the standard Cherenkov tilt in the x,y plane, with the additional time delays in the x–z plane giving rise to a propagating source in the z-direction. While the z-direction delays could in principle be introduced by an optical staircase in the laser beam, here we achieve the continuous rather than discretised propagation through introduction of an additional component of the pulse-front tilt in the x–z plane. An x,y tilt of 24° corresponds to a propagation velocity of \(v_\phi ^{\rm{z}} = c\). We refer to this arrangement as a ‘double-tilt’ configuration, although clearly each of the tilts are actually orthogonal components of a single (larger) pulse-front tilt projected into the exit plane of the LiNbO 3 crystal.

Fig. 5 Laser pulse-front tilt for terahertz generation in LiNbO 3 . a Traditional configuration, with a pulse-front tilt in the x,y direction matched to the Cherenkov angle. b Modified configuration which maintains the Cherenkov tilt angle in the x,y plane, with an additional tilt component in the x–z plane, resulting in an effective travelling source in the z-direction Full size image

Fig. 6 Measured terahertz pulse propagation in a plane parallel to the LiNbO 3 emitting surface. a Electric field temporal profiles as a function of position along the LiNbO 3 surface. The field strength is normalised to the peak strength at each measurement position. b Example terahertz electric field profiles taken from the same data set shown in a, without normalisation. c Arrival time of the peak electric field as a function of longitudinal position. The red line is linear fit to the peak positions and corresponds to a phase and group velocity of 0.98c Full size image

As constructed, the double-tilt arrangement seeks to maintain the terahertz generation efficiency enhancement that comes from transverse velocity matching to the Cherenkov angle. While such an enhancement will be expected to be maintained for a propagation delays introduced by a sufficiently coarse staircase delay optical element, the effect of a continuous delay from the orthogonal tilt component is less clear. We have carried out three dimensional FDTD simulations of the terahertz generation with single-tilt and double-tilt optical propagation and have confirmed that for our LiNbO 3 parameters the source-propagation tilt has a negligible effect on the efficiency of the terahertz field strength in the emission surface.

The double-tilt approach has been demonstrated through an optical arrangement similar to that employed for the ZnTe and PCA measurements. The combined pulse-front tilt was maintained in the horizontal lab-frame horizontal plane, with the consequence that the transverse and longitudinal (effective propagation) directions, and the emission surface of the LiNbO 3 were oriented out of the laboratory frame horizontal or vertical planes. As for the ZnTe and PCA measurements, a temporal and spatial mapping of the fields near the surface was obtained with EO sampling. The transverse intensity profile of the optical beam was also shaped with cylindrical telescopes so that following oblique incidence on the grating and projection onto the out-of-plane oriented crystal surface, the terahertz source dimensions were ~15 mm in the propagating source longitudinal direction, and 2 mm in the orthogonal transverse (Cherenkov matched) direction.