Giant Terahertz Pulses Generated by Relativistic Beam in a Dielectric Channel
GGiant Terahertz Pulses Generated by Relativistic Beam in a Dielectric Channel ∗ G. Stupakov and S. GessnerSLAC National Accelerator Laboratory, Menlo Park, CA 94025, USA
We analyze the electromagnetic field of a short relativistic electron beam propagating in a round,hollow dielectric channel. We show that if the beam propagates with an offset relative to the axisof the channel, in a steady state, its electromagnetic field outside of the channel extends to largeradii and carries an energy that scales as the Lorentz factor γ squared (in contrast to the scalingln γ without the channel). When this energy is converted into a terahertz pulse and focused on atarget, the electric field in the focus can greatly exceed typical values of the field that are currentlyachieved by sending beams through thin metallic foils. Terahertz radiation defined as the range between 100GHz and 30 THz finds applications in fields as diverseas chemical and biological imaging, material science,telecommunication, semiconductor and superconductorresearch [1]. In addition to the established laser-basedsources of such radiation [2, 3], beam-based sources uti-lizing short, relativistic electron bunches [4, 5] show agreat promise. To generate ultra-fast pulses of THz radi-ation, a short electron bunch is sent onto a thin metallicfoil to emit coherent transition radiation (CTR). Experi-ments that use this method at the Linac Coherent LightSource (LCLS) [6] have obtained single-cycle pulses ofradiation that is broad-band, centered on 10 THz, andcontain > . µ J.In the method that utilizes the CTR, it is the electro-magnetic energy of the Coulomb field carried by the elec-tron bunch in free space that is converted by the metallicfoil to radiation. Considering as an example a sphericallysymmetric electron bunch with a Gaussian charge distri-bution moving with a relativistic velocity v , it is easy tocalculate its electromagnetic energy, W ≈ ( Q / √ πσ ) ln γ, (1)where Q is the bunch charge, σ is the rms size of thebunch, and γ = (1 − v /c ) − / is the Lorentz factor. Forthe parameters of the experiment [6] where σ = 10 µ m, Q = 0 . γ = 3 × , we find W ≈ .
23 mJ, ina reasonable agreement with the experiment. The fre-quency range of the CTR radiation can be estimated as ω ∼ c/σ ∼ π × . dW/df ∼ πW/ω ∼ . ∗ Work supported in part by the U.S. Department of Energy undercontracts No. DE-AC02-76SF00515 and DE-AC02-05CH11231. from the exit of the pipe in the direction of beam propa-gation. The radiation energy is proportional to the lengthof the pipe and, in principle, can be larger than in thecase of the foil. Unfortunately, the duration of the radi-ation pulse is long, so the energy density is low.In both methods, the THz radiation is radially polar-ized, and when focussed onto a target, produces an illu-minated spot with zero field at the center, which is notoptimal for most applications.In this paper, we show how the energy of the THz pulsegenerated by a short, relativistic bunch can be consider-ably increased, replacing the scaling W ∝ ln γ in Eq. (1)with W ∝ γ .We consider the setup shown in Fig. 1. A dielectric �� �� Δ FIG. 1. A point charge traveling off-axis in a hollow dielectricchannel shown by red color. channel has an inner radius a , an outer radius b , and thedielectric permittivity (cid:15) . Outside and inside the channel (cid:15) = 1. A short bunch of charge Q , which we treat asa point charge, propagates in the channel with the ve-locity v parallel to its axis and an offset ∆. Assuming∆ (cid:28) a , the electromagnetic field of the beam can berepresented as a combination of the monopole axisym-metric field (corresponding to the limit ∆ = 0) and thedipole field proportional to ∆, which is the main interestof this paper. In the cylindrical coordinate system r, θ, z centered on the axis of the channel and the angle θ mea-sured from the direction of the offset, the dipole electricfield can be represented as E ( r, θ, z, t ) = Re ˜ E ( r, z, t ) e iθ ,with a similar expression for the magnetic field B . Ourgoal is to calculate these fields.In a steady state, the fields depend on z and t in the a r X i v : . [ phy s i c s . acc - ph ] D ec combination ξ = z − vt , i.e., ˜ E ( r, ξ ). Making the Fouriertransformationˆ E ( r, k ) = (cid:90) ∞−∞ ˜ E ( r, ξ ) e − ikξ dξ π , (2)with a similar expression for the magnetic field, we sub-stitute the fields into the Maxwell equations. The equa-tions for ˆ E ( r, k ) and ˆ B ( r, k ) are then solved using thecontinuity of the tangential components of the fields, aswell as (cid:15) ˆ E r and ˆ B r , at r = a and r = b . These fields havea singularity at r = 0 due to the presence of the chargesand currents near the axis. The singularity is the sameas in free space:lim r → r ˆ E r = 2 p, lim r → r ˆ E θ = − ip, lim r → r ˆ B r = 2 ipv, lim r → r ˆ B θ = 2 pv, (3)where p = Q ∆ is the dipole moment. Using Eqs. (3)for the calculation of the dipole components of the fields,we actually treat the moving off-axis point charge as acombination of electric and magnetic moments localizedon the axis of the channel, which is a valid approxima-tion when ∆ (cid:28) a . The resulting equations constitute a4 × | ˆ E r | as a function of radius obtainedwith this solution is shown in Fig. 2 for the following pa- �� �� �� �� ��� ��� ������������������ � / � | � � | � � ������� � / � | � � | FIG. 2. Plot of the absolute value of the radial electric field | ˆ E r | versus radius outside of the channel. The inset shows theradial distribution of | ˆ D r | at small radii r/a <
4; the regioninside the dielectric is colored in green. The dashed curvenear the origin shows the dipole field in free space. The fieldsare normalized by p/a . rameters: b/a = 2, (cid:15) = 3 . k = 2 /a and γ = 200. The field is normalized bythe factor p/a . Note a remarkable feature of this field—it extends in the radial direction far beyond the channel boundaries, reaching r ∼ γ/k ∼ a . For comparison,the dashed line shows the dipole field, | ˆ E r | = 2 p/r infree space (in the absence of the dielectric).We will now focus on the structure of the field outsideof the channel, at r > b . The electric field in this regionis given by the following equationsˆ E r = iAK ( ρ ) + iA − Cβρ K ( ρ ) , ˆ E θ = iCβK ( ρ ) + A + iCβρ K ( ρ ) , (4)where ρ = | k | r/γ , β = v/c , and A and C are constantfactors. The magnetic field in the limit γ (cid:29) B r ≈ − E θ , B θ ≈ E r .In the general case, the coefficients A and C are ex-tremely complicated expressions involving Bessel func-tions. To make the problem tractable analytically, here,we consider a simplified limiting case assuming, in addi-tion to γ (cid:29)
1, that b − a (cid:28) a and keeping only the lowestorders in ( b − a ) and 1 /γ . The coefficients A and C inthis limit are given by the following formulas: A = − ik pγ − iC, C = − k p ( (cid:15) − ( b − a ) a(cid:15) . (5)The longitudinal components of the fields are negligiblysmall.A remarkable property of this solution is that it showsthat the fields outside of the channel extend in radial di-rection over the distance r ∼ γ/k (as we have alreadyobserved in Fig. 2) and, because of the large volume thatthe field occupies, it carries a considerable amount of theelectromagnetic energy. To calculate the spectral energy(per unit frequency interval dω = c dk ) we need to com-pute the following integral dWdω = πc (cid:90) ∞ b rdr ( | ˆ E r | + | ˆ E θ | ) (6)where we took into account that in the relativistic limitthe magnetic energy is equal to the electric one. In thelimit γ → ∞ , we have ( b − a ) /a (cid:29) γ − , and A + iC ∝ γ − .In this case, the second terms on the right-hand side ofEqs. (4) are small and can be neglected in the calculationof the spectral energy, reducing (4) toˆ E r ≈ − i ˆ E θ ≈ CK ( ρ ) . (7)The electric field (7) has a linear polarization along the x axis (the direction of the offset) with a slow fall off in theradial direction, ˆ E x = CK ( ρ ) and ˆ E y = 0. In the limit γ → ∞ it approaches a linearly polarized plane wave thatextends to infinity. Substituting Eq. (7) into (6) gives dWdω = πγ ck C . (8)We see that the spectral energy indeed scales as dW/dω ∝ γ . Using this formula we can calculate the electromag-netic energy carried by a bunch of rms length σ z : thespectrum dW/dω should be integrated over the frequen-cies with the weight e − ω σ z /c . As an illustrative exam-ple, consider the following set of parameters: Q = 0 . σ z = 200 µ m, γ = 100, a = 0 . b = 0 .
75 mm, (cid:15) = 3 .
8, ∆ /a = 0 .
3; the calculation gives W ≈
17 mJ.While this number is to some extent an over-estimate forthe reasons that are explained below, comparing it withthe estimate after Eq. (1) one sees a remarkable potentialof the proposed technique.Using Eqs. (7) we can make the inverse Fourier trans-form and find the electric field E x ( r, ξ ) of the point chargein physical space r , ξ : E x ( r, ξ ) = − π p ( (cid:15) − ( b − a ) a(cid:15) r γ − − ξ ( r γ − + ξ ) / . (9)The plot of E x (in arbitrary units) is shown in Fig. 3; we FIG. 3. Electric field E x (in arbitrary units) in r − ξ planearound the moving dipole. Note that the radial coordinate isscaled by γa while the longitudinal one is scaled by a . Thefield has a singular point at the origin which is truncated inthe graph. Note that the radial coordinate is scaled by aγ and the longitudinal one by a . see that the field is localized in the vicinity of the charge, r = ξ = 0, in a pancake-like region ξ ∼ a and r ∼ γa .We will now explain the physical mechanism behindthe giant (from ∝ ln γ in free space to ∝ γ in the di-electric channel) increase in the electromagnetic energy.The total field can be considered as a superposition ofthe vacuum field of the moving dipole (that is the fieldin the absence of the dielectic) and the field generated bycharges and currents in the dielectric. The vacuum fieldof the dipole at small distances, kr (cid:28) γ , is ˆ E r ≈ p/r ,ˆ E θ ≈ − ip/r (see Eqs. (3)). This field induces a dipolemoment dp x ( z ) (in the direction of the beam offset) ineach slice dz of the dielectric tube. In the limit of a smalltube thickness, b − a (cid:28) a , one can neglect the interac-tion between different slices taking into account only the vacuum field of the beam and integrating the polariza-tion induced by this field, ( (cid:15) − E / π (where E is thefield inside the dielectric), over the volume of the slice.A straightforward calculation yields dp x = De ikξ dz, (10)where D = − p ( b − a )( (cid:15) − (cid:15)a . (11)In the next step, we calculate the radiation of thesedipoles at small angles to the axes using the standardformulas of the dipole radiation with the frequency ω , E rad = ω c R e iωR/c ( n × d p ) × n , (12)where n = R /R , and R is the vector from the locationof the dipole to the observation point. Recalling that ξ = z − vt , it follows from Eq. (10) that ω = kv . With d p directed along x , the radiation field near the axis z is alsodirected along x . Using R = (cid:112) ( z − z (cid:48) ) + r ≈ | z − z (cid:48) | + r / | z − z (cid:48) | , where r, z refer to the observation point and z (cid:48) is the coordinate of the slice, we arrive at the followingexpression for the radiation field of the dielectric tube: E rad x ≈ ω Dc e − iωt + ikz (cid:90) z −∞ dz (cid:48) | z − z (cid:48) | e ikr z − z (cid:48) ) − ik ( z − z (cid:48) )2 γ = 2 ω Dc e − iωt + ikz K ( ρ ) . (13)With D given by Eq. (11) this expression is the sameas Eqs. (7) and (6). This simple calculation corrobo-rates our analysis based on the the direct solution of theMaxwell equations with the boundary conditions at theinterfaces with the dielectric.The mechanism described above also explains why anon-axis beam does not show the same amplification of thefield energy as an off-axis one. For an on-axis beam, itselectric field is axisymmetric and it does not induce dipolemoments in the slices of the dielectric. It does inducethe quadrupole moments, but the quadrupole radiation issuppressed in the forward direction (along z ), in contrastto the dipole one.With the understanding that the field outside of thechannel is generated by the radiation of the dielectricslices from the preceding part of the trajectory we candraw an important conclusion from Eq. (13) about the formation length of the steady state field. It is easy tosee that for r ∼ γ/k , the integral (13) converges onthe distance z − z (cid:48) ∼ γ /k . For 1 THz frequency, wehave k − ≈ µ m and for γ = 100 in our numericalexample above it means that the length of the channelshould be comparable, or longer, than about 0.5 m. If thechannel is shorter than the formation length, while onecan still achieve a considerable amplification of the elec-tromagnetic energy in comparison with the free space,the effect will be smaller than predicted by the steady-state analysis of our paper. This is especially true whenelectron bunches of GeV energy ( γ ∼ γ . In ad-dition, for very large values of γ , the radial extension ofthe field γ/k can exceed the transverse size of the vac-uum chamber in which such an experiment would be con-ducted, and the model of free space outside of the channelused in this work becomes invalid.One more conclusion can be drawn from the previousanalysis. The dipole momenta induced in the dielectricdue to the offset of the beam, can be alternatively pro-duced if the beam propagates on axis of the pipe, butthe dielectric constant of the tube varies with the angle θ , (cid:15) ( θ ). For example, the pipe can be manufactured insuch a way that its top and bottom parts have differ-ent values of the dielectric constant. Such a setup mayhave some advantages in practice, because it facilitatesthe beam transport through the tube. A sketch of theexperimental setup is shown in Fig. 4. FIG. 4. A sketch of an experiment that generates a THz pulsefrom the electromagnetic field of a short relativistic bunchmoving off-axis in a hollow dielectric channel. The yellowcolor shows the beam field extending outside of the channel.This field is intercepted by a tilted at 45 degrees metal foil andis converted into THz radiation. The radiation propagates at90 degrees to the direction of the beam orbit.
The beam propagating with an offset inside the di-electric channel experiences a strong deflecting transverseforce in the direction of the offset. An estimate of thisforce can be easily obtained assuming b − a ∼ a , (cid:15) − ∼ ∼ a , F ⊥ ∼ Q a . (14)For the parameters of the illustrative example afterEq. (8), this force would deflect the beam into the wallof the dielectric at the beginning of the channel. Onecan try to keep the beam on the straight path applying astrong external focussing, however, a more practical ap-proach would be to use a beam of much larger energy, on the order of several GeV. Of course, Eq. (8), as wasdiscussed above, is not applicable for very large γ , butan approximate energy of dW/dω can be obtained not-ing that γ /k in this equation is the formation lengthof the fields. In the limit when the the channel length l is shorter than the formation length, γ /k should bereplaced by l with the result dWdω ≈ πl ck C . (15)Note that this formula does not involve γ any more.If we drop the assumption of the smallness of b − a used in our analysis, numerical calculations show that,in addition to the field localized in the pancake regionaround the beam (as illustrated in Fig. 3), there are res-onant monochromatic modes excited by the beam andpropagating behind it. For example, for b = 2 a , (cid:15) = 3 . γ = 100, the frequencies of the lowest four reso-nant modes are ωa/c = 0 . , . , . , .
76. The ex-citation of such modes, and the electromagnetic energythat goes into them, depends on their coupling to thebeam, calculations of which is beyond the scope of thispaper. Conceptually, the mechanism of the excitation ofthese modes is the same and the experiments with thedielectric-metallic pipe of Ref. [7], although we are deal-ing with the dipole modes instead of the monopole ones.Instead of the dielectric pipe, one can use a hollowplasma channel [8]. Assuming a cold plasma, its dielec-tric constant is (cid:15) = 1 − ω p /ω , where ω p is the plasmafrequency. In principle, the diameter of the plasma chan-nel can be made smaller than the dielectric one, however,it may be a challenge to have a uniform, stable plasmaconfiguration of tens of centimeters in length. In contrastto a solid dielectric, under the effect of strong relativisticfields of the bunch, the plasma response can quickly be-come nonlinear [9–11]. This nonlinear plasma response,in addition to amplification of the electromagnetic en-ergy, can, in principle, demonstrate an up-conversion ofthe radiation frequency when compared with the linearresponse of the dielectric tube.In conclusion, in this paper we presented analysis thatshows that a short relativistic beam propagating withan offset inside a dielectric channel, possesses a pancake-like electromagnetic field that extends in radial directionseveral orders of magnitude of the size of the channel.With the help of a standard setup using a metal foil inthe path of the beam, this field can be converted intoan intense, short, linearly polarized pulse of terahertzradiation. Focusing this pulse on a sample would allowto achieve record electric fields for many applications.G.S. thanks Max Zolotorev for fruitful discussions.This work was supported by the Department of Energy,contract DE-AC03-76SF00515. [1] M. Tonouchi, Nat Photon , 97 (2007). [2] D. Auston et al. , Phys. Rev. Lett. , 1555 (1984). [3] D. You et al. , Opt. Lett. , 290 (1993).[4] T. Nakazato et al. , Phys. Rev. Lett. , 1245 (1989).[5] G. L. Carr et al. , Nature , 153 (2002).[6] D. Daranciang et al. , Appl. Phys. Lett. , 141117(2011).[7] A. M. Cook et al. , Phys. Rev. Lett. , 095003 (2009).[8] S. Gessner, E. Adli, J. M. Allen, W. An, C. I. Clarke,C. E. Clayton, S. Corde, J. P. Delahaye, J. Frederico,S. Z. Green, C. Hast, M. J. Hogan, C. Joshi, C. A. Lind-strom, N. Lipkowitz, M. Litos, W. Lu, K. A. Marsh, W. B. Mori, B. O/’Shea, N. Vafaei-Najafabadi, D. Walz,V. Yakimenko, and G. Yocky, Nat Commun (2016).[9] I. Kostyukov, A. Pukhov, and S. Kiselev, Physics ofPlasmas , 5256 (2004).[10] W. Lu, C. Huang, M. Zhou, M. Tzoufras, F. S. Tsung,W. B. Mori, and T. Katsouleas, Physics of Plasmas ,056709 (2006).[11] S. A. Yi, V. Khudik, C. Siemon, andG. Shvets, Physics of Plasmas20