Generation of quasi-monoenergetic positron beams in chirped laser fields
GGeneration of quasi-monoenergetic positron beams in chirped laser fields
S. Tang ∗ Department of Physics, College of Information Science and Engineering,Ocean University of China, Qingdao, Shandong, 266100, China
High energy photons can decay to electron-positron pairs via the nonlinear Breit-Wheeler pro-cess when colliding with an intense laser pulse. The energy spectrum of the produced particles isbroadened because of the variation of their effective mass in the course of the laser pulse. Applyinga suitable chirp to the laser pulse can narrow the energy distribution of the generated electrons andpositrons. We present a scenario where a high-energy electron beam is collided with a chirped laserpulse to generate a beam of quasi-monoenergetic γ -photons, which then decay in a second chirped,UV pulse to produce a quasi-monoenergetic source of high-energy electrons and positrons. I. INTRODUCTION
When a beam of charged particles collides with an in-tense laser pulse, the spectrum of produced photons, viathe process referred to as the nonlinear Compton scat-tering (NLC) [1, 2], is sensitive to the shape of the pulse.Employing a many-cycle laser pulse will lead to outgoingphoton spectra similar to those in a monochromatic back-ground [3, 4]: well-defined harmonic fringes in lightfrontmomenta and emission angle. Collision with short laserpulses will lead to a broadening of outgoing particle har-monic peaks [5–7] and richer spectral structures: infra-red structure [8, 9], asymmetry in emission angle [10],and pronounced interference phenomena [11, 12]. Thespectral broadening can be attributed to the inhomoge-neous effective mass of the charged particle moving inthe intense laser pulse [13–16]: the variation of the ef-fective mass modifies the velocity of the changed particleduring the scattering [17–20]. It is also known, that ifone can prescribe the chirp of the laser pulse, that is, anonlinear dependency on the phase, then this relativis-tic broadening of particle spectra can be compensatedfor, to generate a narrowband source of high energy pho-tons [20–22].The decay of a probe photon to an electron-positronpair in an intense electromagnetic field, is often referredto as the nonlinear Breit-Wheeler process (NBW) [3, 23,24], and has been measured experimentally in the land-mark E144 experiment over two decades ago [25, 26].The phenomenology of the process has been investigatedtheoretically in various types of laser field. First inmonochromatic [3], and constant crossed fields [3, 27]and more recently in finite laser pulses [28–30], as wellas two-colour [31] and double-pulse fields [32–34]. As inNLC process, harmonic structure also arises in the out-going electron-positron pair spectra operating in a many-cycle laser pulse. However, this structure is only clearlydiscernible when the centre-of-mass energy reaches thethreshold of 2 mc already with only a low number oflaser photons, where m is electron (positron) rest massand c is the speed of light. ∗ [email protected] Analogous to the spectral broadening in the NLC pro-cess, the variation of the electron-positron pair’s effectivemass in the course of the intense laser pulse also inducea broadening in their energy spectra. The current paperis a proof of principle calculation to show that a suitablenonlinear chirp of a laser pulse can also be employed tocounterbalance the spectral broadening in the NBW pro-cess, and proposes a simple two-step scenario to providea quasi-monoenergetic source positrons. The existence ofa quasi-monoenergetic positrons source would be usefulin the electron-positron colliding experiments [35–37].The study of laser chirp’s effect on positron spectra, ispartly motivated by upcoming high-energy experimentsLUXE at DESY [38, 39] and E320 at FACET-II [40, 41],where photons with energies O (10 GeV) are planned tobe generated, either directly in the laser pulse throughCompton scattering of the electrons (LUXE and E320),or from a separated bremsstrahlung and inverse Comptonsource (LUXE). The centre-of-mass energy can be effec-tively increased, and hence the multi-photon harmonicregion of the Breit-Wheeler process approached, by gen-erating higher-order harmonics of the interaction laser,using e.g. relativistic plasmas [42–44].The paper is organised as follows. In Sec. II, we presentthe spectrum of produced positrons in the NBW pro-cess, and investigate the contributions from the station-ary phase points. We then analysis the broadening ofthe positron spectrum and propose a special laser fre-quency chirp to counteract the spectral broadening. InSec. III we demonstrate numerical implementations ofour chirp prescription in narrowing the positron spec-tra from a single high-energy photon and from the γ -rayobtained through the NLC process of a high-energy elec-tron. We conclude in Sec. IV. II. THEORETICAL FRAMEWORK
We consider the scenario in which a high-energy pho-ton with momentum (cid:96) colliding (almost) head-on with alaser pulse produces a pair of electron and positron. Thelaser pulse is simplified as a plane wave with scaled vectorpotential a µ = eA µ ( φ ) and wavevector k µ = ω (1 , , , φ = k · x and ω is the laser frequency at the initial a r X i v : . [ h e p - ph ] F e b phase point φ i at which the laser is turned on. The inter-action energy is characterised by η (cid:96) = k · (cid:96)/m . We usenatural units (cid:126) = c = 1 throughout and the fine structureconstant is α = e ≈ / Pd s d q ⊥ = α | I | + ( SI ∗ + IS ∗ − F · F ∗ ) g (2 π ) η (cid:96) (1 − s ) s , (1)where g ≡ [ s + (1 − s ) ] / [4 s (1 − s )]. We sum over thespin of the outgoing particles and average over the po-larisation of the incoming photon. The spectrum (1) isparameterised by the three components of the positronmomentum p : these are s = k · p/k · (cid:96) , the fraction ofthe photon light-front momentum taken by the positron,and q ⊥ = ( q x , q y ), q x,y = p x,y /m − s(cid:96) x,y /m , positron mo-menta in the plane perpendicular to the laser propagatingdirection. q ⊥ reflects the angular spread of the producedpositron around the photon incident direction. In our pa-rameter region, the angular spread is extremely narrow,and for head-on collisions (cid:96) ⊥ = 0, q ⊥ ≈ γ p θ p (cos ψ, sin ψ )where θ p and ψ are the polar and azimuthal angles of thepositron, and γ p is the positron energy factor.The functions I , F and S are defined as I = (cid:90) d φ (cid:18) − (cid:96) · π p (cid:96) · p (cid:19) e i Φ( φ ) ,F µ = 1 m (cid:90) d φ a µ ( φ ) e i Φ( φ ) ,S = 1 m (cid:90) d φ a ( φ ) · a ( φ ) e i Φ( φ ) , with the exponent:Φ( φ ) = (cid:90) φφ i d φ (cid:48) (cid:96) · π p ( φ (cid:48) ) m η (cid:96) (1 − s ) , (2)where π p is the instantaneous momentum of the positronin the field: π p = p + a − p · ak · p k − a k · p k . (3)The completed derivation of the NBW pair productionprobability could start from an S-matrix element withVolkov wavefunctions [45] and has been well documentedin the literature, see for example [46] for an introductionand [12] for an analogous presentation for NLC. Becauseof the charge symmetry, the spectrum (1) can also beapplied to the produced electron by just changing thecorresponding definitions for the electron. After doingthe transverse integral over q ⊥ , the spectrum (1) showsthe symmetry: P ( s ) = P (1 − s ). A. Quasi-monoenergetic positron beams fromchirped laser pulses
We now choose, as an example, the vector potential a µ with circular polarisation: a µ ( φ ) = mξ [0 , cos Ψ( φ ) , sin Ψ( φ ) , f ( φ ) , (4)in which ξ and f ( φ ) are the normalised pulse ampli-tude and envelope, dΨ( φ ) / d φ = ω ( φ ) /ω is the chirpedfrequency of the pulse. At the initial phase φ i = 0, f ( φ i ) = 0 and ω ( φ i ) = ω .We request (i) that the pulse duration is sufficientlylong and the variation of the pulse local amplitude ismuch slower than the laser frequency, and (ii) that thevariation of the local frequency ω ( φ ) is on the same timescale as the pulse local amplitude (This will be clearlater.): ω (cid:48) ( φ ) , f (cid:48) ( φ ) <<
1. Under these conditions, wecan then apply the slowly-varying approximation thatterms of order f (cid:48) ( φ ) [ ω (cid:48) ( φ )] can be neglected [47].The exponent (2) can be expressed approximately asΦ( φ ) ≈ κ ( φ ) φ − ζ ( φ ) sin [Ψ( φ ) − ψ ] (5)where κ = (cid:96) · pm η (cid:96) (1 − s ) + ξ η (cid:96) (1 − s ) s φ (cid:90) φφ i d ˜ φf ( ˜ φ ) ,ζ = ξf ( φ ) η (cid:96) (1 − s ) s ω ω ( φ ) | q ⊥ | . With the Fourier expansions: e − iζ sin(Ψ − ψ ) = + ∞ (cid:88) n = −∞ J n ( ζ ) e inψ e − in Ψ , (6)where J n ( ζ ) is the Bessel function of the first kind, thefunctions S can be expanded approximately as a seriesof harmonics: S ≈ − ξ ∞ (cid:88) n = −∞ e inψ (cid:90) d φ f ( φ ) J n ( ζ ) e i Ω( φ ) , (7)where Ω( φ ) = κ ( φ ) φ − n Ψ( φ ), the harmonic order n means the net number of the laser photons absorbed fromthe background field. The dependence of the argument ζ on the pulse envelope f ( φ ) indicates that the contri-bution of each harmonic varies during the course of thepulse and the high order harmonics contribute only atthe pulse centre where f ( φ ) →
1. The functions I and F can be calculated in the same way as (7) and obtainedwith the exactly same exponent term (and the differentpre-exponents).From (7), one can see that the main contribution tothe functions I , F and S , and therefore to the final spec-trum (1), comes from the stationary phase point where ∂∂φ Ω( φ ) = q ⊥ + m ∗ /m η (cid:96) (1 − s ) s − n ω ( φ ) ω = 0 . (8)where m ∗ ( φ ) = m [1 + ξ f ( φ )] / denotes the effectivemass of the produced positrons in the laser pulse [14, 15].Let us first consider the standard case with constantfrequency: ω ( φ ) = ω . The stationary condition (8) im-plies a chirp in the positron energy varying with the pulseenvelop, d s/ d φ (cid:54) = 0: For the n th harmonic, the positronenergy is in the spectral region s (cid:48) n, − ( φ ) ≤ s ≤ s (cid:48) n, + ( φ ),where s (cid:48) n, ± ( φ ) = [1 ± (cid:112) − m ∗ / ( nη (cid:96) m ) ] / , (9)shifting between the linear [ f ( φ ) = 0] and nonlinear[ f ( φ ) = 1] BW spectral lines. This shifting stems in-deed from the variation of the positron’s effective mass m ∗ ( φ ) during the course of laser pulse [20, 22].To exclude the energy chirp in the positron spectrum,one simple idea from (9) is to adapt the energy parameter η (cid:96) by prescribing the laser frequency with a specific chirp: η (cid:96) ∼ ω ( φ ) ∼ ξ f ( φ ), to compensate the variation ofthe particle effective mass in the laser pulse. As one cansee, this frequency chirp is modulated by the field inten-sity and on the same time scale as the pulse envelope,satisfying the request before.Based on the stationary condition (8), this frequencychirp can be prescribed by solving the differential equa-tion: d ω d φ = ω nη (cid:96) (1 − s ) s dd φ [ q ⊥ + 1 + ξ f ( φ )] , (10)with the initial conditions: f ( φ i ) = 0 and ω ( φ i ) = ω ,and acquired with its explicit expression: ω ( φ ) = ω (cid:2) ξ f ( φ ) / ( q ⊥ + 1) (cid:3) . (11)From (11), one can see a remarkable fact that this chirpprescription is irrelevant to the harmonic order n , whichindicates this spectral broadening can be removed fromall the harmonic lines at the same time. One should alsonote that the complete counterbalance of the broadeningcan only happen at a particular outgoing angle: | q ⊥ | ∼ γ p θ p specified by the chirp (11). For a realistic detectorwith nonzero angular acceptance ∆ θ >
0, the spectralpeaks of the probed positrons would shift slightly fromthe harmonic lines with a finite energy spread (See theresults later in Fig. 3).Similar discussions can also be appliedto linearly polarised field backgrounds: a µ ( φ ) = mξ [0 , cos Ψ( φ ) , , f ( φ ), and the prescrip-tion of frequency chirp (11) is exactly the same, exceptthat ξ → ξ / III. NUMERICAL RESULT
In this section, we first present a numerical exampleof a head-on collision between a 13 . γ -photons obtainedfrom the NLC process of a 16 . q ⊥ → ω ( φ ) = ω (cid:2) ξ f ( φ ) (cid:3) . (12)Inserting back into the stationary condition (8), one canthen get the unchirped positron spectrum peaked at thelinear BW harmonic line s n, ± = [1 ± (cid:112) − / ( nη (cid:96) ) ] / θ p = 0. To produce a narrow-band positronbeam at the cone angle θ p ∼ | q ⊥ | /γ p , one can employ thefrequency chirp (11).To improve the interaction energy parameter η l , weemploy the laser pulse with the UV frequency ω =15 . f ( φ ) = sin ( φ/ N ) where0 < φ < N π and N = 16. The (areal) energy of aplane laser pulse can be calculated as: E = − ω παλ e (cid:90) d φ (cid:18) m d a d φ (cid:19) , (14)in which λ e = 1 /m = 386 .
16 fm is the electron’s re-duced Compton wavelength. For circularly polarisedlaser pulses with the frequency chirp (12), one can ob-tain E = ω ξ αλ e (cid:18)
34 + ξ ξ (cid:19) N . (15)The first term in the bracket corresponds to the laserpulse with a constant frequency: ω ( φ ) = ω .The choice of particle energy parameters is moti-vated by the upcoming high-energy experiments such asLUXE at DESY [38, 39] and E320 at FACET-II [40,41]. The strong UV laser pulse can be obtainedthrough the plasma harmonic generation driven by anultrahigh-power optical laser pulse [42–44]. The fre-quency chirp (12) can be plugged into the laser spectrumvia the coherent superposition of two linearly and oppo-sitely chirped laser pulses with suitable time delay [22]. A. NBW
Fig. 1 depicts the narrowing of the positron spectrafrom the chirped laser pulse benchmarked with the re-sults from a constant-frequency pulse. As shown in Fig. 1(a) and (b), the frequency chirp (12) can effectively com-pensate the broadening of each harmonic line: For theunchirped case in (a), harmonic lines are broadened withplenty of subsidiary peaks and overlap together to bea continuum spectral domain in the positron angular-energy distribution. However for the chirped case in (b),the angular-energy distribution is comprised of a number
FIG. 1. Upper two panels: Angular-energy distributiond P / [d s d( γ p θ p )] of the produced positrons via the NBW pro-cess from a high-energy photon (13 . θ p = π (c) and a narrow acceptance: ∆ θ p = 32 µ rad (d).In (d), the spectral curve for the unchirped case is multipliedby a factor 20 for visibility. The vertical dashed lines showthe location of the first three harmonics in the NBW process: s , + = 0 . s , + = 0 .
88 and s , + = 0 .
91. The chirped laserpulse ( ξ = 1, ω ( φ ) = ω [1 + ξ f ( φ )]) has the same energy asthe unchirped laser pulse ( ξ = 1 . ω ( φ ) = ω ). of well-separated harmonic lines. Around each harmonicline, the distribution is narrowed to be a single peak atthe small outgoing angle θ p → s n, ± (13), where the harmonic order n must be ≥ θ p < ∆ θ p / µ rad: the positron spectrum from thechirped pulse background spikes around the harmoniclines s n ≥ , + , see the vertical dashed lines in Fig. 1 (d),and has a much higher amplitude than that from theunchirped laser pulse which gives a much lower andbroader spectrum in the on-axis direction. We labelthe location of the first three harmonics: s , + = 0 . s , + = 0 .
88 and s , + = 0 .
91 in the chirped case. As onecan see, the first spectral peak has a much higher am-plitude and narrower energy spread ∆ s/s ≈ .
7% thanother higher-order peaks, where ∆ s means the full widthat the half maximum of the peak.As one can see that the x -axes of the panels in Fig. 1(a)-(c) are plotted in the region 0 . < s < s ) = P(1 − s ), see (1). However, thespectrum in Fig. 1 (d), which is plotted for the wholespectral region 0 < s <
1, shows an asymmetric distribu-tion in the higher energy region s > .
5. This is becausethe lower-energy positrons distribute in a broader angu-lar region: γ p → θ p → π/
2. Therefore, thelower-energy positrons could be simply excluded from thegenerated high-energy positron beam by an angular se-lection.
B. NLC + NBW
GeV γ -rays generated by high-energy electron beamsvia the NLC process has been analysed in detail in [4, 48].The spectrum of the emitted γ -rays is presented in [12]in the same way as (1). With a similar stationaryphase analysis, one can prove that a beam of quasi-monoenergetic γ -photons can be obtained in a well-chirped laser background [20]. Replacing the seed photonused in Fig. 1 with these acquired high-energy γ -photons,a source of narrow-band positrons can be obtained. Thistwo-step scenario can be regarded as a part of the tridentprocess [49–51] in which only real photons contribute.In Fig. 2, we plot the distributions of the emitted pho-tons from a 16 . P γ / d θ (cid:96) d r in Fig. 2 FIG. 2. (a) Angular-energy distribution d P γ / (d r d θ (cid:96) ) of the γ -photons generated through the NLC process from a high-energy electron E e = 16 . γ -photons within the whole angular spread: ∆ θ (cid:96) = π (blue dashed) and a narrow acceptance: ∆ θ (cid:96) = 16 µ rad (redsolid). On the top axis is shown the corresponding change inthe photon energy E (cid:96) ≈ rE e from 7 . r = 0 .
80 and r = 0 .
89 corre-sponding to the photon energy E (cid:96) = 13 . . (a), especially for small angle scatterings θ (cid:96) → r = k · (cid:96)/k · p e is the fraction of the light-frontmomentum taken by the scattered photon from the seedelectron, p e is the electron momentum and θ (cid:96) is the polarangle of the scattered photon.With a narrow acceptance: ∆ θ (cid:96) = 16 µ rad collimatedin the on-axis direction in Fig. 2 (b), the collected pho-tons distribute tightly around the first harmonic line r = 0 .
80, corresponding to the energy E (cid:96) = 13 . r/r ≈ . r = 0 .
89 at E (cid:96) =14 . r v = 2 vη e / (2 vη e + 1), η e = k · p e /m , v ≥ θ (cid:96) = π ,the energy spread of the collected γ -photons would bemuch larger, shown as the blue dashed line in Fig. 2 (b).Making use of the obtained γ -ray spectrum : ρ γ ( r ) = dP γ / d r in Fig. 2 (b), we can calculate the to- FIG. 3. (a) Energy distribution dP / d t of the positronsgenerated by the on-axis γ -photons obtained through theNLC process of a high-energy electron E e = 16 . E p isshown on the top axis. The vertical black dashedlines denote the location of each combined harmonic: t , , , = (0 . , . , . , . E p = (10 . , . , . , .
1) GeV. (b)Energy spread and peak location of the first harmonic peakwith the change of the acceptance ∆ θ p . The horizontal dashedline denotes the theoretical location of the first combined har-monic: t . The energy spectrum of the γ -photons used in thecalculation is plotted as the red solid line in Fig. 2 (b). tal number of the generated positrons:P = α (2 πη e ) (cid:90) d t (cid:90) t d rρ γ ( r ) h ( r, t ) (16)where t = k · p/k · p e denotes the fraction of the light-front momentum transferring from the seed electron tothe produced positron, and h = (cid:90) d q ⊥ | I | + ( SI ∗ + IS ∗ − F · F ∗ ) g ( r − t ) rt (17)in which g ≡ [ t + ( r − t ) ] / t ( r − t ) is re-defined.Here, we ignore the small angular spread of the col-lected γ -photons to simplify the numerical calculations.The selection of small-angle photons can be done in ex-periments by adjusting the separation between the twolaser pulses used in each process. The induced numericalerrors can be offset by using a much broader acceptanceto collect the produced positrons: ∆ θ p (cid:29) ∆ θ (cid:96) , see theresult in Fig. 3.In Fig. 3 (a), we plot the yield of the positrons fromthe on-axis γ -photons (∆ θ (cid:96) = 16 µ rad) obtained throughthe NLC process discussed in Fig. 2. With the accep-tance ∆ θ p = 32 µ rad along the direction of the seedelectron, most of the positrons are collected in a narrowenergy region peaked at t = 0 .
625 with a narrow energyspread ∆ t/t ≈ .
2% and a much higher amplitude thanother subpeaks in the higher energy region. All of thesepeaks can be related to the combined harmonic lines: t u = r v s n, + ( r v ), where s n, + ( r v ) is the value of the NBWharmonic line calculated with the NLC harmonic energy η (cid:96) = η e r v , see the vertical dashed lines in Fig. 3 (a): Thedominant peak is relevant to the first combined harmonicline t = r s , + = 0 .
637 corresponding to the energy E p ≈ . t = 0 .
691 cor-responds to the second combination t = r s , + ≈ . E p ≈ . t = 0 .
719 may come from the sum of the combi-nations : t = r s , + = 0 .
726 and t = r s , + = 0 . θ p and can be reduced by narrowing the ac-ceptance, see the red dotted-dashed line in Fig. 3 (b): thedominant peak moves asymptotically back to the locationof the first combined harmonic line t with a decreasingacceptance.To improve the brilliance of the positron beam, oneneeds to increase the detector acceptance: with a largeracceptance ∆ θ p = 48 µ rad in Fig. 3 (a), the amplitudeof the spectral peak becomes much higher, and at thesame time, its energy spread is broadened to be 3 . θ p . With a broad acceptance ∆ θ p ≈ µ rad,one can acquire a positron beam with the energy spreadaround 10%, and with a relatively narrow acceptance∆ θ p < µ rad, the energy spread of the positron beamcan be simply controlled to be less than 5%. IV. CONCLUSION
We investigated the nonlinear Breit-Wheeler processin a chirped laser background with intensity ξ ∼ γ -photons are obtained from achirped laser pulse via the nonlinear Compton scat-tering process and then are used to produce electron-positron pair in the second chirped laser pulse to providea quasi-monoenergetic source of positrons. The producedpositrons are tightly gathered in a narrow energy regionaround the combined harmonic lines from the relevantprocesses. By adjusting the detector acceptance, the en-ergy spread of the obtained positrons can be well con-trolled.In our calculations, we ignore the energy spread of theseed electron beam, which would be crucial if it is inthe same level of or much broader than the predictedpositron energy spread. To obtain the predicted narrow-band positrons, high-quality electron beams with limitedenergy spread are needed [41]. We employ the high-powerlaser pulse with the UV frequency which is critical in ourdiscussion: the UV laser frequency guarantees the scat-tering of high-energy γ -photons, and then significantlyimprove the yield of the electrons and positrons by open-ing the low-order harmonic channels for the followingnonlinear Breit-Wheeler process.As an outlook, the considered two-step scenario ispotential to generate highly polarised positron beams:high-energy γ -photons scattered by the seed electronsare more probable in the polarisation state parallel tothe background field [48], and polarised γ -photons arelikely to decay to electron-positron pairs in particularspin states [52]. V. ACKNOWLEDGMENTS
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