Calculations of electric fields for radio detection of Ultra-High Energy particles
Daniel García-Fernández, Jaime Alvarez-Muñiz, Washington R. Carvalho Jr, Andrés Romero-Wolf, Enrique Zas
aa r X i v : . [ a s t r o - ph . H E ] O c t Calculations of electric fields for radio detection of Ultra-High Energy particles
Daniel Garc´ıa-Fern´andez, Jaime Alvarez-Mu˜niz, and Washington R. Carvalho Jr.
Depto. de F´ısica de Part´ıculas & Instituto Galego de F´ısica de Altas Enerx´ıas,Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain
Andr´es Romero-Wolf
Jet Propulsion Laboratory, California Institute of Technology,4800 Oak Grove Drive, Pasadena, California 91109, USA
Enrique Zas
Depto. de F´ısica de Part´ıculas & Instituto Galego de F´ısica de Altas Enerx´ıas,Universidad de Santiago de Compostela, 15782 Santiago de Compostela, Spain
The detection of electromagnetic pulses from high energy showers is used as a means to searchfor Ultra-High Energy cosmic ray and neutrino interactions. An approximate formula has beenobtained to numerically evaluate the radio pulse emitted by a charged particle that instantaneouslyaccelerates, moves at constant speed along a straight track and halts again instantaneously. Theapproximate solution is applied to the particle track after dividing it in smaller subintervals. Theresulting algorithm (often referred to as the ZHS algorithm) is also the basis for most of the sim-ulations of the electric field produced in high energy showers in dense media. In this work, theelectromagnetic pulses as predicted with the ZHS algorithm are compared to those obtained withan exact solution of the electric field produced by a charged particle track. The precise conditionsthat must apply for the algorithm to be valid are discussed and its accuracy is addressed. Thiscomparison is also made for electromagnetic showers in dense media. The ZHS algorithm is shownto describe Cherenkov radiation and to be valid for most situations of interest concerning detec-tors searching for Ultra-High Energy neutrinos. The results of this work are also relevant for thesimulation of pulses emitted from air showers.
I. INTRODUCTION
The study of Ultra High Energy Cosmic Rays(UHECR) and neutrinos (UHE ν s) is currently a high pri-ority in Astroparticle Physics with many experimentalefforts being dedicated to these two related areas of re-search. UHECRs are routinely detected through the Ex-tensive Atmospheric Showers (EAS) they produce wheninteracting in the atmosphere. Despite the recent ad-vances in the measurement of the flux of UHECRs [1, 2],their primary composition remains unknown [3, 4], andthis is one of the main obstacles to extract precise con-clusions on their origin. There are strong reasons to be-lieve that UHE ν s should be produced in the interactionsof UHECRs with the material surrounding the sources,and/or in their propagation through the observed CosmicMicrowave Background radiation [5]. However their de-tection has not yet been achieved. Efforts are being madeto improve the experimental situation in both fields. Thechallenge is to instrument sufficiently large target vol-umes to compensate for the low cross section and lowfluxes for UHE ν detection, and to find measurementsthat provide large aperture at the highest energies andhelp to constrain the primary composition of UHECRs.The radio technique is being explored in both fields.As early as in 1962, G. Askaryan proposed to detectUHECRs and UHE ν s by observing the coherent radiopulse from the excess of electrons in a shower developingin a dense, dielectric and nonabsorptive to radiowavesmedium [6]. Soon after, pulses were observed in coin- cidence with air shower arrays [7, 8]. The emission iscoherent in wavelengths which are large compared to thecharacteristic size of the electric charge and current dis-tributions associated to the induced showers. The tech-nique has been receiving a lot of attention in the field ofAstroparticle Physics in the last decade because of therelatively low cost of the antennas needed for the detec-tion systems. Also, coherence implies that the pulse en-ergy scales with the square of the primary energy whichfavours long range detection, a requirement to achievelarge areas and volumes both for UHECR and UHE ν de-tection. Indeed, quadratic scaling has been confirmedin accelerator experiments [9–12] leading to very strongpulses associated to UHE energy showers.A large number of initiatives have been made or arecurrently in development or planning stages. In densemedia neutrino-induced showers are less than a meter inwidth and full coherence is expected to be mantained upto the GHz range (see for instance [13]). A variety of pastand present experiments search for pulses at those fre-quencies produced by neutrinos in ice [14, 15], the moonregolith [16–21] or salt domes [11]. The largest and mostpromising experiments are in planning stages looking atice [22, 23]. Electromagnetic pulses emitted in the MHz-GHz frequency range by EAS induced by UHECR are be-ing measured in the hope of using this complementary in-formation to constrain its composition and/or to developnew cost effective detection systems [24, 25]. In additionthe ANITA balloon flown antennas, initially devised tosearch for neutrinos, has recorded coherent pulses up tothe GHz range [26] consistent with EAS induced by UHE-CRs and there are plans for future developments [27].The success of the radio technique requires an accu-rate and computationally efficient calculation of the ra-dio emission properties of UHE showers. It is necessaryto perform this in an efficient way since the number ofparticles in a shower at EeV energies is ≥ . Sim-ulation techniques for evaluating pulses in dense mediahave been used for more than 20 years [13, 28–41]. Theshower is simulated to obtain, for every particle track, theinformation needed to calculate its contribution to theelectric field. The formula used for this purpose stemsfrom an approximate solution of Maxwell’s equations inthe Fraunhofer limit [8] adapted for simulation purposesin [13]. The electric field due to a short charged particletrack, assumed to be travelling at a constant speed, canbe approximated by two terms which correspond to thestart and end of the track. The resulting electric fielddepends on the particle speed and the angle between theline of sight from the track to the observer [13, 39]. Tocalculate the emission in a shower, tracks are chosen sothat the particle velocity can be approximated to be con-stant, and all track contributions are added taking intoaccount interference effects.Although the approximate formula is sufficient formany practical applications, its range of validity is lim-ited in frequency and position of the observer with re-spect to the track. It is possible to extend its range of va-lidity by subdividing each track in sub-intervals. The re-sulting algorithm (often referred to as “ZHS algorithm”)is easy to implement and fast enough for the simulationof particle showers. This is convenient since it has al-lowed the simulation of pulses in the Fresnel region forneutrino detection [30] and in measurements of EAS [42].Although the range of applicability of the algorithm is en-hanced when used in this way, it is not obvious that thesum of the subcontributions correclty accounts for all theradiation in regions close to the emission source.The object of this article is to study the conditions forthe ZHS algorithm to be valid and to establish its accu-racy. For this purpose we obtain in Section II an exactsolution to the problem of a charged particle instanta-neously accelerating to a constant speed and stoppingabruptly after a discrete time interval [43]. This allowsus to establish the precise conditions necessary to turnthis solution into the basic formula of the ZHS algorithm.In Section III we compare the exact solutions for an infi-nite and a finite track to the result of the ZHS algorithm,and stress the compatible interpretation of the radiationregime in terms of Cherenkov radiation for both cases.In Section IV we compare the exact solution of the sin-gle track problem in nearby regions with the results ofapplying the ZHS algorithm with track subdivisions totest its validity and accuracy. Section V is devoted todiscussing the ZHS algorithm in relation to other ap-proximations made to calculate the pulses emitted fromshowers. Section VI presents the summary and conclu-sions of our work. II. ELECTROMAGNETIC FIELD OF A SINGLECHARGED PARTICLE TRACK
Let us assume an electron ejected from an atom at time t = t that travels at constant speed v through a mediumalong a finite track until it is absorbed by another atomat time t . Neglecting the movement of the atoms, wecan model the electric current associated to the electronas, J ( x , t ) = − e v δ (3) ( x − x − v t ) Θ( t − t )Θ( t − t ) (1)where e = | e | is the charge of a positron, x ( t ) is its po-sition and x an arbitrary reference position. The stepΘ-functions account for the fact that the electron onlymoves in the time interval ( t , t ).In a dielectric medium with permitivity ǫ and magneticsusceptibility µ , Maxwell’s equations for the vector po-tential in the frequency domain A ( x , ω ) can be writtenas [44]: ∇ A ( x , ω ) + µǫω A ( x , ω ) −∇ [ ∇ · A ( x , ω ) − iǫµω φ ( x , ω )] = − µ J ( x , ω ) , (2)where φ ( x , ω ) is the Fourier transform of the scalarpotential and we use the following convention forthe Fourier-transform of a function f ( t ): f ( ω ) = R ∞−∞ d t e iωt f ( t ). In principle ǫ and µ can depend onfrequency and our results below would be equally valid,but we drop the explicit dependence of ǫ and µ with ω for simplicity.We use the Lorenz gauge condition which implies thatEq. (2) for the vector potential becomes: ∇ A + k A = − µ J , (3)with k = nω/c and n the refractive index of the medium.The Lorenz gauge condition in the frequency domain [44]: ∇ · A ( x , ω ) = iǫµω φ ( x , ω ) , (4)implies that the scalar potential for non-zero frequenciesis entirely determined by the divergence of the vector andonly the vector potential is needed to calculate the field. A. Exact solution
The solution of the Helmholtz equation in Eq. (3) isstandard physics and can be obtained, using Green’smethod, as an integral over source positions [44]: A ( x , ω ) = µ π Z d x ′ e ik | x − x ′ | | x − x ′ | J ( x ′ , ω ) (5)where from now on, x will denote the position of theobserver and x ′ the position of the source.For simplicity we rewrite the current in Eq. (1) as: J ( x , t ) = qv Z ( t ) P ( x , t ) ˆ z (6)with q = − e ; Z ( t ) = Θ( t − t )Θ( t − t ) and P ( x , t ) = δ (3) ( x − vt ˆ z − z ˆ z ), where we assume without loss of gen-erality that the electron travels along the z -axis which isparallel to ˆ z . The equations that follow below are clearlyvalid for any continuous and differentiable functions Z ( t )and P ( x , t ). They are also valid for the case consideredin Eq. (1) which can be obtained as a limiting case ofsuitable continuous and differentiable functions.Transforming this definition of the current to the fre-quency domain and substituting into Eq. (5) for the vec-tor potential we obtain: A ( x , ω ) = µ π qv ˆ z Z d x ′ d t ′ e iωt ′ e ik | x − x ′ | | x − x ′ | Z ( t ′ ) P ( x ′ , t ′ )(7)In the Lorenz gauge, the only non-zero component ofthe vector potential for a charge moving along z is the z -component. The scalar potential φ is obtained throughEq. (4). For this purpose we need to obtain ∇ · A : ∇ · A ( x , ω ) = ∂A z ∂z = µ π qv Z d x ′ d t ′ Z ( t ′ ) P ( x ′ , t ′ ) e iωt ′ e ik | x − x ′ | | x − x ′ | (cid:20) ik − | x − x ′ | (cid:21) ( z − z ′ ) | x − x ′ | The electric field E ( x , ω ) can be obtained from thescalar and vector potentials [44] as: E ( x , ω ) = −∇ φ ( x , ω ) + iω A ( x , ω ) (8)Due to the cylindrical symmetry of a charged particletrack we calculate only the radial ( E ρ ) and z ( E z ) com-ponents of the field. Since A only has z -component, thenthe radial component of the field is given by: E ρ ( x , ω ) = − ∂φ ( x , ω ) ∂ρ = − ∂ ρ ∇ · A ( x , ω ) iµǫω (9)where ρ = p x + y is the radial coordinate, and ∂ ρ denotes ∂/∂ρ .The z − component of the field is given by: E z ( x , ω ) = − ∂ z φ ( x , ω ) + iωA z ( x , ω )= − ∂ z ∇ · A ( x , ω ) iωµǫ + iωA z ( x , ω ) (10)with ∂ z denoting ∂/∂z .Performing the derivatives in Eqs. (9) and (10) we ob-tain: E ρ ( x , ω ) = i qvω πǫ Z t t d t ′ e iωt ′ e ikr r × ρ × ( z − z − vt ′ ) × (cid:20) b (cid:18) b − r (cid:19) + 1 r (cid:21) (11)and, E z ( x , ω ) = i qvω πǫ Z t t d t ′ e iωt ′ e ikr r × " b ( z − z − vt ′ ) r + ( z − z − vt ′ ) r − b (cid:18) ( z − z − vt ′ ) r − (cid:19) + iω µ π qv Z t t d t ′ e iωt ′ e ikr r (12)where r = r ( t ′ ) and b = b ( t ′ ) are both functions of thesource time t ′ which is defined as, r ( t ′ ) = | x − x ′ | = p ρ + ( z − z − vt ′ ) (13)and b ( t ′ ) = ik − r ( t ′ ) (14)Eqs. (11) and (12) provide an exact solution for theelectric field of a finite track. In general they do not havean analytical form and a numerical integration needs tobe performed to obtain the field. For the calculationsin this paper, we have divided the integration intervaland applied Simpson’s rule, increasing the number of di-visions until the integral converged. Under certain con-ditions, however, we can give analytical approximationsfor relevant physical situations.In the following we show that the basic expression usedin the ZHS algorithm is a particular case of Eqs. (11) and(12) under certain approximations. B. The ZHS expression
A simple expression for the approximate calculationof the electric field from a single charged particle trackmoving at constant speed was found in [13]. In this sec-tion we derive the expression used for the ZHS algorithmfrom the exact solution and compare the electric fieldas obtained in both the exact calculation and with theZHS expression. This allows us to establish under whichcircumstances the formula gives a good account of theelectric field.The ZHS algorithm can be obtained from Eqs. (11)and (12) if the following set of conditions are fulfilled:1. The observer is in the “far field” zone i.e. kr ≫ kr = k | x − x ′ | ≈ k [ R − v ( t − t ) cos θ ] (16)where R is the distance from the observation pointto a reference point along the track where the par-ticle is located at a reference time t , and θ is theangle between the particle track and the directionfrom the reference point to the observer. This ap-proximation holds provided the parameter η ≪ η defined as: η ( t ) = k [ v ( t − t )] R sin θ, (17)This condition should be fulfilled at any time t from t to t . A more commonly used and nearly equiv-alent form of this condition [46] is: η ′ = kL R sin θ ≪ , (18)where L = v ( t − t ) is the length of the track. Thiscondition is necessary to ensure that the secondand higher order terms for the phases i ( ωt + kr )in Eqs. (11) and (12) have no relevance, even whenthe sum of the leading and first order terms in theTaylor expansion of the phases is zero, as it occursfor observation at the Cherenkov angle [49] definedas cos θ C = 1 /βn with β = v/c .3. Finally the distance to the observer appearing inthe denominators of several terms of Eqs. (11) and(12) must be approximated as:1 r ( t ) ≈ R (19)over the length L of the track, where R is the dis-tance to a reference point along the track (in theZHS algorithm the mid-point of the track is se-lected). The error when making this approximationis of order L/R .In particular, the condition in Eq. (15) implies: b ( t ′ ) ≈ ik ; b − r ≈ b ; b + 1 r ≈ b (20)With these approximations the radial component ofthe field in Eq. (11) becomes: E ρ ≈ i qvω πǫ e ikR R ( ik ) sin θ cos θ e i k · v t Z t t d t ′ e i ( ω − k · v ) t ′ (21)where we have used: ρr ≈ ρR = sin θ, (22)and z − z − vtr ≈ z − z − vt R = cos θ. (23) Eq. (21) can be easily integrated yielding: E ρ = − iq ω µ π v sin θ cos θ e ikR Re i k · v t (cid:20) e i ( ω − k · v ) t − e i ( ω − k · v ) t i ( ω − k · v ) (cid:21) . (24)If we make t = t this becomes the expression for theradial field as used in the ZHS algorithm [13] except fora factor 2 due to the Fourier transform convention usedin [13].Similarly applying the approximations in Eqs. (16),(19) and (20) to Eq. (12) for the z -component of thefield, and using that kR ≫ ⇒ k ≫ k/R , it is straight-forward to show that: E z ≈ − iω qvω µ π e ikR R e i k · v t cos θ Z t t d t e i ( ω − k · v ) t + iqv ω µ π e ikR R e i k · v t Z t t d t e i ( ω − k · v ) t , (25)which can be cast as: E z = iq ω µ π v sin θ e ikR R e i k · v t Z t t d t e i ( ω − k · v ) t (26)Performing the integral, and taking t = t , the ZHSformula is recovered: E z = iq ω µ π v sin θ e ikR Re i k · v t (cid:20) e i ( ω − k · v ) t − e i ( ω − k · v ) t i ( ω − k · v ) (cid:21) . (27) C. The ZHS algorithm
To calculate the electric field of the pulse emitted froma current distribution, such as that produced in a highenergy shower, the ZHS algorithm uses the ZHS expres-sions in Eqs. (24) and (27) to calculate the emission fromall the charged particle tracks. The final result is ob-tained adding up all the contributions. The value of R used for the phase factor in each particle track is the dis-tance from the first point of the track to the observationpoint. This definition is consistent with the conventionto account for the phase change between emission arisingfrom the start and end points of the track. The actualvalue of R used for the denominator is the distance be-tween the midpoint of the track and the observer whichis for all practical purposes the same value when the ap-proximation is valid. The algorithm used in alternativesimulation programs is similar but different in these tech-nical details [45].Naturally it is possible to divide any charged particletrack into arbitrarily small subtracks in order to makethe computation more accurate. This will extend therange of validity of the approximation. It is interesting tonote that it makes absolutely no difference to subdividethe track of a uniformly moving charge when observedin the Fraunhofer limit because the term associated tothe end of one subtrack cancels the term associated tothe beginning of the next subtrack. This is because inthis limit the differences in R between adjacent subtracksare arbitrarily small. However in practical situations thiscancelation is not exact if R is allowed to change from atrack to the next one according to geometry. In the origi-nal ZHS program [28] the tracks were not subdivided butsoon it was realized that a more accurate result was ob-tained by subdividing all tracks at every point there wasa discrete interaction in the simulation program [32, 48].This was the main modification that was ever made tothe original ZHS Monte Carlo [13, 28] and has been effec-tive since then. With this subdivision the distribution oftracks for a shower in ice has a peak at about L ∼ III. CHERENKOV RADIATION
The exact electric field for a charged particle trackderived in Eqs. (11) and (12) must account for everysingle feature of the electric field, since no approxima-tions have been made. In particular, it must reproduceCherenkov radiation which is the only radiation emittedby a charged particle moving at constant speed, when v > c/n in the limit of an infinite track. An analyticalsolution for the electric field produced by such particlecan be obtained and it is given in [49](chapter 4). Theradial ρ and z − components of the fields given in [49]converted to SI units, and using the Fourier transformconvention adopted in this work, can be written as: E ρ ( ρ, z, ω ) = q πǫv e i ωv z uK ( uρ ) (28) E z ( ρ, z, ω ) = iωµq π e i ωv z (cid:18) − µǫv (cid:19) K ( uρ ) , (29)where ρ is the radial coordinate, K and K are the mod-ified Bessel functions of the second kind, and u = u ( ω )is a function that can take two different values depend-ing on the magnitude of the particle speed, v < c/n or v > c/n (subluminal or superluminal regime). In ourconvention: v < cn ⇒ u ( ω ) = ωv (cid:12)(cid:12)(cid:12)p − n β (cid:12)(cid:12)(cid:12) (30) v > cn ⇒ u ( ω ) = − i ωv (cid:12)(cid:12)(cid:12)p n β − (cid:12)(cid:12)(cid:12) . (31)If the particle travels below the speed of light in themedium, the argument u ( ω ) of the Bessel functions isreal and the particle does not radiate as shown in [49].On the contrary, if the speed of the particle is larger than the speed of light, then u ( ω ) is imaginary and the particleradiates. The latter case corresponds to pure Cherenkovradiation [49].With the help of the asymptotic forms for the Besselfunctions, we can obtain the limits of Eqs. (28) and (29)when | uρ | ≪ | uρ | ≫ | uρ | ≪ K Bessel function dominates over the K and only the ra-dial component of the field E ρ matters. In this case,lim | uρ |→ | E ρ | = | q | πǫv ρ (32)and a 1 /ρ dependence with distances is obtained, as wellas no dependence with frequency.If | uρ | ≫ v > c/n the argument u is imaginary the fields can be written as:lim uρ →± i ∞ | E ρ | = | q | πǫv r π (cid:12)(cid:12)(cid:12)(cid:12) ( n β − / r ωvρ (cid:12)(cid:12)(cid:12)(cid:12) (33)lim uρ →± i ∞ | E z | = µ | q | π (cid:18) − µǫv (cid:19) r π (cid:12)(cid:12)(cid:12)(cid:12) n β − / r vωρ (cid:12)(cid:12)(cid:12)(cid:12) (34)In this case the field is proportional to p ω/ρ . This is inagreement with [46] where the same behavior is deducedusing simple arguments of energy conservation througha cylindrical surface surrounding the track.In Fig. 1 the Fourier components of the modulus ofthe electric field for an infinite track as obtained fromEqs. (28) and (29) are shown, for different frequencies,as a function of ρ , the radial distance to the track. Theparticle speed is v ≃ c > c/n travelling in homogeneousice with refractive index n = 1 .
78. Under these circum-stances the quantity | uρ | can be approximated as: | uρ | ≈ (cid:16) ν
100 MHz (cid:17) (cid:16) ρ (cid:17) (35)At large distances to the track when | uρ | ≫ / √ ρ and withfrequency as √ ω , in agreement with the aysmptotic fieldcomponents in Eqs. (33) and (34). This behavior takesplace when ρ > .
1, 1 and 10 m for frequencies ν = 1GHz, 100 MHz and 10 MHz respectively, in agreementwith Eq. (35), as can be seen in Fig. 1.As the distance to the track decreases and the con-dition | uρ | ≪ /ρ and does not depend on frequency as expected fromEq. (32). As can be clearly seen in Fig. 1, the transitionfrom the 1 /ρ behaviour to p ω/ρ occurs at a distancethat depends on frequency because | uρ | involves the fre-quency (Eq. 35). For instance at distances ρ < .
01 mthe condition | uρ | ≪ ν = 100 and 10MHz. The Fourier component of the field scales with 1 /ρ and has the same value for the two frequencies as seen inFig. 1, while this is not the case for a frequency of ν = 1GHz. -13 -12 -11 -10 -9 E ν [ V m - M H z - ] ρ [m] 1 GHz100 MHz10 MHzInfinite trackExact. Finite long trackZHS. Finite long track FIG. 1. Fourier components of the electric field modulusas a function of distance to the particle track for an infinitetrack as obtained from Eqs. (28) and (29) (solid line), andfor a track of length L = 1200 m as obtained with the exactformulas derived in this work (Eqs. (11) and (12)) (open cir-cles) and with the ZHS algorithm (Eqs. (24) and (27)) (opensquares). From top to bottom, the observation frequencies are1 GHz, 100 MHz and 10 MHz. The 1 /ρ and 1 / √ ρ regimesare apparent. In Fig. 2 the modulus of the field for a charged parti-cle in an infinite track - as obtained from Eqs. (28) and(29) - is shown as a function of frequency for an observerat a fixed radial distance. At large enough frequenciesso that the condition | uρ | ≫ √ ω as expected from Eqs. (33) and (34), while it isconstant with frequency for small enough frequencies sothat | uρ | ≪ ρ ∼
10 m, the field should behave as √ ω for ν > ∼
10 MHz (applying Eq. (35)). This is approximatelythe case as can be seen in Fig. 2.The result of the exact calculation for a track of length L = 1 . ν ∼ ( c/n ) /λ with λ ∼ L = 1 . ν < ∼ . -4 -3 -2 -1 -3 -2 -1 R e l a ti v e d i ff e r e n ce [ % ] ν [MHz]Infinite vs exactInfinite vs ZHS10 -12 -11 -10 E ν [ V m - M H z - ] Exact. Finite long trackInfinite trackZHS. Finite long track
FIG. 2. Top panel: Fourier components of the electric fieldmodulus as a function of frequency for an infinite track asobtained from Eqs. (28) and (29) (solid line), and for a trackof length L = 1200 m as calculated in this work from Eqs. (11)and (12) (open circles). The observer is placed at a lateraldistance to the infinite track ρ = 8 .
27 m. Also shown is themodulus of the field for the same finite track ( L = 1 . √ ω (seetext for explanations). Bottom panel: Relative difference (in%) between the solution for an infinite track and the exactsolution for a finite long track (open circles) and between thesolution for an infinite track and that obtained with the ZHSalgorithm (open squares). IV. COMPARISON OF THE EXACTCALCULATION AND THE ZHS ALGORITHM
We have numerically evaluated the exact expressionsfor the z and ρ components of the electric field inEqs. (11) and (12) at different frequencies and observerdistances, for a single tracks of different lenghts, and forthe tracks constituting a shower in ice as obtained in fullsimulations performed with the ZHS Monte Carlo code[13]. In this section we present the results of this com-parison. A. Fourier components of the electric field for asingle track
The applicability of the ZHS expressions in Eqs. (24)and (27) relies on the conditions 1 − k simply by dividing the particle track in a sufficientlylarge number of subtracks. Once this is guaranteed, theZHS expression is applied to every sub-track and the elec-tric field is obtained adding the corresponding contribu-tions. The validity of this procedure will be numericallyconfirmed below when comparing the exact calculationwith the electric field obtained using the ZHS algorithmin the manner just described. However the condition kr ≫
1, Eq. (15), does not depend on the size of thetrack and cannot be enforced by applying the procedureoutlined above. As a consequence kr ≫ -5 -4 -3 -2 -1
1 10 100 1000 R e l a ti v e d i ff e r e n ce [ % ] ν [MHz]
100 m10 m1 m0.1 m -18 -17 -16 -15 -14 -13 -12 -11 -10 E ν [ V m - M H z - ] ZHSExact calculation
FIG. 3. Top panel: Fourier components of the electric fieldmodulus for a single particle track as obtained with the exactcalculation Eqs. (11) and (12) (solid lines) and with the ZHSalgorithm Eqs. (24) and (27) (open symbols). The lengthof the track is L = 1 . − m and the field is shown forobservers at distances (from top to bottom lines) R = 0 . In Fig. 3 we compare the Fourier components of theelectric field modulus for a single particle track as ob-tained with the exact calculation, Eqs. (11) and (12), tothat obtained with the ZHS algorithm. The length ofthe track is chosen to be small L = 1 . − m (closeto the peak value of the distribution of track lengths inthe standard ZHS code). To test the validity of the ZHSalgorithm, we have calculated the spectra for observersat different distances ( R ) measured with respect to thecenter of the track and placed at the Cherenkov angle.The condition kR ≫ n = 1 .
78 can be cast as: kR ∼ . (cid:16) ν
100 MHz (cid:17) (cid:18) R (cid:19) ≫ R = 100 , , . ν > ∼ , ,
100 MHz and 1 GHz respec-tively. The ZHS algorithm is expected to reproduce the results of the exact calculation in this range. This can beseen in Fig. 3. The relative difference between the ZHSand exact calculations is less than ∼
2% at the fron-tier of the validity range in the explored frequency anddistance space. The accuracy can be however orders ofmagnitude better. If kR >
37 is enforced for instancethe corresponding relative difference is below ∼ . R >
10 m and ν > ∼ L = 1 . -15 -14 -13 -12 -11 -10 -9
1 10 100 1000 E ν [ V m - M H z - ] ν [MHz] 0.1 m1 m10 m100 mZHSExact calculation FIG. 4. Same as top panel of Fig. 3 for a single particle trackof length L = 1 . For fixed frequencies the condition in Eq. (36) only ap-plies at sufficiently large distances R to the track. Thisis illustrated in Fig. 5 for a track of length L = 1 . ν = 10, 100MHz and 1 GHz are in agreement with the exact calcu-lation respectively at R > ∼
10, 1 and 0.1 m as expected.Since the typical distance between antennas in experi-ments such as the Askaryan Radio Array (ARA) [22] is ∼ −
100 m, we expect the results of the ZHS algo-rithm obtained through the procedure outlined above, tobe accurate enough in most practical situations.It has been questioned whether the ZHS algorithm re-produces Cherenkov radiation from a single charged par-ticle track [45]. In Figs. 1 and 2 the algorithm is shown tobe in very good agreement with the exact solution for a1.2 km track as long as kR ≫ -5 -4 -3 -2 -1 R e l a ti v e d i ff e r e n ce [ % ] R [m]10 MHz100 MHz1 GHz10 -15 -14 -13 -12 -11 -10 -9 E ν [ V m - M H z - ] ZHSExact calculation
FIG. 5. Top panel: Fourier components of the electric fieldmodulus at ν = 1 GHz, 100 MHz and 10 MHz (from topto bottom lines) for a single particle track as a function ofdistance to the track, as obtained with the exact calculationEqs. (11) and (12) (solid lines) and with the ZHS algorithmEqs. (24) and (27) (open symbols). The length of the trackis L = 1 .
1. Behaviour of the field with frequency
As can be seen in Fig. 3 (solid lines) the exact solu-tion for the modulus of the electric field scales linearlywith frequency provided that kR ≫
1. As a result theelectric field can be approximated with the ZHS expres-sions, Eqs. (24) and (27). For an observer close to theCherenkov angle as in Fig. 3, the factor ( ω − k · v ) ≪ (cid:20) e i ( ω − k · v ) t − e i ( ω − k · v ) t i ( ω − k · v ) (cid:21) ≈ i ( ω − k · v ) t − − i ( ω − k · v ) t i ( ω − k · v ) = t − t (37)making the dependence of the field with ω apparent.Clearly the field cannot grow indefinetely with frequency.This apparent “ultraviolet divergence” is only an arti-fact of considering the unrealistic medium in which thepermittivity ǫ ( ω ) is constant with frequency. In a physi-cal medium absorption at high frequencies will tame thegrowth with frequency of the electric field.When kR < ω − as can be also seen in Fig. 3. In the model of a charged particle at rest for t ≤ t , moving with a speed v between t = t and t = t , and becoming again at rest for t ≥ t ,the Coulomb field dominates at small distances to thetrack and/or low frequencies. The field can be modeledas: E ∝ r Θ( t ,obs − t ) (38)for t < t and E ∝ r Θ( t − t ,obs ) (39)for t > t , where r and r are respectively the distancesfrom the charge to the observer at times t and t , and t ,obs and t ,obs denote the instants of time at which theCoulomb field arrives at the observer. The Fourier trans-form of a Heaviside function at non-zero frequency is pro-portional to ω − , and the two Coulomb fields interferecoherently at low frequencies, explaining the frequencydependence of the electric field. Naturally the ZHS al-gorithm does not reproduce this behavior which is notassociated to radiation. The growth of the field at lowfrequencies is an artifact of not accounting for screeningof the field by the atoms in the medium.
2. Behavior of the field with distance
In Fig. 5 the dependence of the Fourier componentsof the field modulus with distance is shown for severalfrequencies. At sufficiently large distances to the trackthe electric field behaves as 1 /R for all frequencies. Thisis the radiation zone, the field is expected to behave as1 /R as explained in conventional radiation theory [44]. Ifthe observer is placed at very small distances compared tothe length of the track, the situation resembles that of aninfinite track. In this case the discussion in Section IIIapplies. The field behaves as 1 / √ R at distances muchsmaller than the length of the track provided the fre-quency is high enough to satisfy | uρ | ≫ ν = 1 GHz for R < ∼ . | uρ | ≪
1, the field be-comes proportional to 1 /R and independent of frequency- see Eq. (29). This feature can be appreciated in Fig. 5at distances below 0.1 m for the calculations at 10 and100 MHz. The ZHS algorithm reproduces the calculationprovided the kR ≫ /R and 1 / √ R behav-iors. B. Fourier components of the electric field inelectromagnetic showers
It is possible to test the ZHS algorithm in a more re-alistic situation. In this section we compare the Fouriercomponents of the electric field predicted by the ZHS al-gorithm with those obtained using the exact calculationin a full simulation of electromagnetic showers.For this purpose we have applied the exact solutions ofthe field of a track given in Eqs. (11) and (12) in the ZHSMonte Carlo code [13] for the simulation of electron andphoton-induced showers in ice. The ZHS Monte Carlocalculates the start and end points of small sub-tracksof all charged particles (electrons and positrons) in anelectromagnetic shower down to a kinetic energy thresh-old of ∼
100 keV. With these we can calculate the exactelectric field produced by each single sub-track and addthe fields up accounting for interference between differ-ent tracks. Since Eqs. (11) and (12) are only valid fora charged particle travelling along the z axis (parallel tothe shower axis), we perform the necessary rotations ofEqs. (11) and (12) to obtain the field for a particle trackmoving along an arbitrary direction.Simultaneously with the exact calculation, we also ob-tain the field as predicted by the ZHS algorithm for ex-actly the same shower (i.e. the same set of tracks andsub-tracks). As explained above the subdivisions are suchthat the conditions in Eqs. (16), (17) and (19) are fulfilledfor all the sub-tracks in the shower.The result is qualitatively the same as in the case ofsingle tracks. As long as the condition kR ≫ R = 1 mand frequencies above ν ∼
10 MHz, well in the distanceand frequency ranges relevant for experiments lookingfor particle shower induced radio pulses in dense media[14, 15, 22, 23].We stress here that the accuracy reported above refersto the approximation of using the ZHS formula appliedto the standard subdivision of tracks in the ZHS code,instead of the exact expression for the radiation emittedby the same particle sub-tracks. By comparing the re-sults obtained in the Fraunhofer limit with the standardsubdivision of tracks to those obtained with a much finersubdivison, it was determined that the accuracy of theZHS code is ∼
10% at frequencies ∼ ∼ kR ≫ >
10 MHz to few GHz, theZHS algorithm has been shown to give a very accuraterepresentation of the Fourier components of the electricfield (Fig. 6).
V. COMPARISON TO OTHER CALCULATIONS
Several calculations of the field emitted in showers de-veloping in dense media can be found in the literature.In [47] the Finite Diference Time Domain method is usedfor calculating the field of a pancake-like shower with aGaussian longitudinal development and Gaussian radialprofile in the time-domain which is then transformed tothe frequency-domain. In [46] using the saddle-point ap-proximation, an analytic equation for the calculation ofthe electric field of a charge distribution exhibiting a lon-gitudinal profile with a well-pronounced maximum is de-rived. The result is factorized into an integral accountingfor the longitudinal variation of the charge and a formfactor that accounts for the lateral spread of the shower,a procedure revisited in [41] for realistic showers. Assum-ing a Gaussian longitudinal and lateral development forthe charge distribution both results were directly com-pared and turned out to be in good overall agreement asshown in [47]. Minor differences could be attributed tothe form factors used.With the exact calculation of the electric field per-formed in this work, the field due to a Gaussian profilecan also be obtained and compared to the calculationsmentioned before. The electric current for a shower witha Gaussian profile is given by Eq. (1) with the followingreplacements: Z ( t ) = 1 (40)and P ( x , t ) = N πσ r e − ( x + y ) / σ r e − z / σ l δ ( z − vt ) (41)The shower develops in the longitudinal direction parallelto the z ′ coordinate (shower axis), and radially along the x ′ and y ′ coordinates. N is a normalization constant. σ l characterizes the width of the shower along shower axisand σ r the corresponding lateral width. After substitut-ing this current in Eq. (7), and following the same stepsas in Section II A, the expression for the field is the sameas in Eqs. (11) and (12) with the following change: Z d t ′ → Z d t ′ d x ′ d y ′ N πσ r e − ( x ′ + y ′ ) / σ r e − v t ′ / σ l (42)0 -12 -11 -10 -9 -8 E ν [ V m - M H z - ] ν [MHz]
100 m
ZHS, θ c ZHS, θ c +5 ° ZHS, θ c +10 ° ZHS, θ c +20 ° Exact, θ c Exact, θ c +5 ° Exact, θ c +10 ° Exact, θ c +20 ° -4 -3 -2 -1 R e l a ti v e d i ff e r e n ce [ % ] ν [MHz] θ c θ c +5 °θ c +10 °θ c +20 ° -10 -9 -8 -7 E ν [ V m - M H z - ]
10 m
ZHS, θ c ZHS, θ c +5 ° ZHS, θ c +10 ° ZHS, θ c +20 ° Exact, θ c Exact, θ c +5 ° Exact, θ c +10 ° Exact, θ c +20 ° -10 -9 -8 -7 -6 E ν [ V m - M H z - ] ν [MHz] ZHS, θ c ZHS, θ c +10 ° ZHS, θ c +20 ° Exact, θ c Exact, θ c +10 ° Exact, θ c +20 ° FIG. 6. Fourier components of the electric field modulusas obtained in Monte Carlo simulations of a 10 TeV electron-induced shower in ice, with the exact calculation (lines) andwith the ZHS algorithm (symbols). The field is shown forobservers at distances (from top to bottom panels) R = 100,10, 1 m, placed at different observation angles with respectto shower maximum. In the middle panel corresponding to R = 10 m, we also show the relative difference (in %) betweenthe electric field modulus as obtained with the exact solutionand with the ZHS algorithm for the various observation anglesdepicted. Eqs. (11) and (12) after the changes in Eq. (42) can besolved numerically. The ZHS algorithm can also be ap-plied to this situation as long as the condition kr ≫ r to the shower. The procedure consistson slicing the volume occupied by the bulk of the showerin small cubes and approximating each as a track withconstant charge given by the Gaussian distributions inEq. (41). -13 -13 -13 -13
0 20 40 60 80 100 120 E ν [ V m - M H z - ] ν [MHz] θ c θ c +5 °θ c +10 °θ c +20 ° Saddle-pointExact calculation
FIG. 7. Fourier components of the modulus of the electricfield for a Gaussian charge profile given in Eq. (41) with σ l =20 m and σ r = 1 m for an observer at R = 300 m withrespect to the peak of the Gaussian longitudinal profile. Theobservation angles are, from top to bottom, θ C , θ C + 5 ◦ , θ C +10 ◦ , and θ C + 20 ◦ . Fields are calculated with the saddle-point approach and the exact formula Eqs. (11) and (12).The result obtained with the ZHS algorithm is on top of theexact calculation and it is not plotted in the Fig. for clarity. Comparison of the result of the exact calculation inthis work (or the ZHS algorithm) with the saddle-pointcalculation in [46] requires knowing the form factor F fora Gaussian profile. F is defined in [46] as: F ( q ) = Z d x ′ d y ′ d s ′ e − i q · x ′ f ( s ′ , x ′ , y ′ ) (43)with q = ( ω/v, kρ/R ) with ρ = ( x, y ) the radial positionof the observer. Also, x ′ = ( s ′ , x ′ , y ′ ) with s ′ = z ′ − vt ′ . R is the distance from the maximum of the shower to theobserver. The function f represents a normalized chargedensity of the travelling pancake. Assuming a Gaussianfor f of the form: f ( s ′ , x ′ , y ′ ) = 12 πσ r δ ( s ′ ) e − ( x ′ + y ′ ) / σ r , (44)and substituting f into Eq. (43), the form factor reads, F ( q ) = e − ( nωc ρR σ r ) (45)Setting N = 1, σ l = 20 m, σ r = 1 m and R = 300m with the refractive index of ice n = 1 .
78, the electric1fields for a Gaussian charge profile were calculated us-ing the saddle-point approach as described in [46], andcompared to the exact formula given in this work. Theresults are shown in Fig. 7. Since R is large the con-dition kR ≫ ν ∼ VI. SUMMARY AND CONCLUSIONS
We have obtained the results of an exact calculationof the electric field produced by a charged particle thataccelerates instantaneously moves at constant speed andintantaneously deccelerates. The results are used to ob-tain the approximate expression used in the ZHS algo-rithm to calculate the radio emission from showers indense media using shower simulations. This allows theprecise determination of the conditions necessary for thisapproximation to be valid, namely, the observer must bein the far field zone kR ≫ kL sin θ/R ≪ kR ≫ kR > .
7, corre-sponding to
R >
10 m and ν >
10 MHz. By enforcing kR >
37 the precision improves to better than 0 . R >
10 m and ν >
100 MHz, whichare conditions met in most experimental arrangements trying to detect radio pulses form neutrinos interactingin dense media.The ZHS algorithm is tested for completitude whenapplied to a shower simulation. This is done compar-ing the result of the ZHS algorithm to that obtainedwhen the ZHS expression is replaced by the exact cal-culation for every charged particle sub-track for exactlythe same shower. The results indeed confirm that theaccuracies reported for individual sub-tracks are approx-imately mantained in the final ZHS result.Finally, in order to compare to alternative calculationsthe results are compared to the saddle point approxima-tion. This approximation has been used to test solutionsusing the method of Finite Differences to solve Maxwell’sEquations directly in the Time Domain (FDTD) using asimplified shower front based on gaussian distributionsin the shower plane and in time. The comparison of thesaddle point approximation to both the exact solutionand the ZHS algorithm give compatible results confirm-ing that both approaches reproduce the radiation emittedfrom showers.The results presented here in summary confirm thatthe ZHS algorithm can be used to describe most practi-cal applications to detect pulses emitted from high en-ergy showers produced in dense media by neutrinos. Theapproach only begins to show significant discrepancieswhen the observer is at distances comparable to the lat-eral dimensions of the shower ( . ∼ − VII. ACKNOWLEDGMENTS
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