Global phase and frequency comb structures in nonlinear Compton and Thomson scattering
aa r X i v : . [ h e p - ph ] M a r Global phase and frequency comb structures in nonlinear Compton and Thomsonscattering
K. Krajewska † , ∗ M. Twardy ‡ , and J. Z. Kami´nski † † Institute of Theoretical Physics, Faculty of Physics,University of Warsaw, Ho˙za 69, 00-681 Warszawa, Poland ‡ Faculty of Electrical Engineering, Warsaw University of Technology, Plac Politechniki 1, 00-661 Warszawa, Poland (Dated: August 8, 2018)The Compton and Thomson radiation spectra, generated in collisions of an electron beam witha powerful laser beam, are studied in the framework of quantum and classical electrodynamics,respectively. We show that there are frequency regimes where both radiation spectra are nearlyidentical, which for Compton scattering relates to the process which preserves the electron spin.Although the radiation spectra are nearly identical, the corresponding probability amplitudes exhibitdifferent global phases. This has pronounced consequences, which we demonstrate by investigatingtemporal power distributions in both cases. We show that, contrary to Thomson scattering, it isnot always possible to synthesize short laser pulses from Compton radiation. This happens whenthe global phase of the Compton amplitude varies in a nonlinear way with the frequency of emittedphotons. We also demonstrate that while the Compton process driven by a non-chirped laser pulsecan generate chirped bursts of radiation, this is not the case for the Thomson process. In principle,both processes can lead to a generation of coherent frequency combs when single or multiple drivinglaser pulses collide with electrons. Once we synthesize these combs into short bursts of radiation,we can control them, for instance, by changing the time delay between the driving pulses.
PACS numbers: 12.20.Ds, 12.90.+b, 42.55.Vc, 13.40.-f
I. INTRODUCTION
Owing to the rapid development of high-power lasertechnology, in recent years we observe a renaissanceof theoretical interest in studying strong-field quantumelectrodynamics (QED) processes [3–5]. Note that pro-ceeding theoretical works were based on the monochro-matic plane wave approximation [6–10]. However, withthe parallel development of computational technology,it is possible now to extend these explorations and tostudy fundamental QED processes in multichromaticlaser fields [11, 12] or in short laser pulses [13–18].New aspects of strong-field QED such as the electron(positron) polarization effects [17, 19], the energy andangular correlations [20–24], the bremsstrahlung pro-cess at relativistically intense laser radiation [25], orthe electron-positron cascades [26], are of great interestnowadays. Usually, different types of cross sections orprobability distributions are analyzed, leaving out prob-lems related to the phase of probability amplitudes. How-ever, in many cases, it is the global phase that plays asignificant role. For instance, it is a very important pa-rameter when studying coherence of high-order harmon-ics and the synthesis of attosecond pulses [27–30].In this paper, we shall study the important role playedby the global phase of probability amplitudes in nonlin-ear Compton scattering [16, 17] and its classical approx-imation – nonlinear Thomson scattering [31–44]. For theclassical theory we discuss the conditions of its applicabil-ity. We show that, in order to get comparable temporal ∗ E-mail address: [email protected] power distributions from both theories, an extra condi-tion on the Compton phase has to be imposed. We alsopropose the method of controlling the energy distributionof emitted radiation by properly modulated laser pulsessuch that it leads to the generation of coherent high-orderharmonics combs. In addition, we investigate propertiesof the synthesized short radiation pulses.The paper is organized as follows. In Sec. II, we definetemporal and polarization properties of the laser pulsesconsidered. Supplementary definitions of mutually or-thogonal and normalized triad of vectors for two (in gen-eral elliptic) polarizations and for the direction of pulsepropagation are discussed in Appendix A. The basic the-oretical scheme of the laser-induced QED Compton scat-tering is presented in Sec. III, together with the deriva-tion of the global phase for the Compton amplitude. Weshow that the total phase can be split into the kinematicand the dynamic parts. The kinematic phase is inde-pendent of the electron spin degrees of freedom (hence,it is applicable also to the Compton scattering of spin-0 particles). While it can be derived analytically (see,Eq. (25)), the dynamic phase can be determined only nu-merically. For the pulses considered in this paper, the dy-namic phase appears to be independent of the frequencyof photons generated during the process. The reader canfind more details in Appendix C. The analogous analysisfor nonlinear classical Thomson scattering is presented inSec. IV and Appendix B. In Secs. III and IV, we also com-pare the predictions of quantum and classical approachesfor the energy distributions of emitted radiation, draw-ing particular attention to the different dependence ofthe global phase on the frequency of generated radiation.As we show, this is related to the quantum recoil of elec-trons during the emission of photons. We demonstratein Sec. V that the frequency dependence of the globalphase plays a vital role in the temporal synthesis of gen-erated radiation. We conclude that the quantum recoileffects result in a broader temporal distribution of radi-ation power for Compton scattering as compared to thepredictions drawn from the classical theory. The inter-ference of photons generated in Compton scattering bya modulated laser pulse (which consists of subpulses) isinvestigated in Sec. VI. In Sec. VI A, we show how thedistance between the peaks in the energy spectrum canbe controlled by the time delay of such subpulses. Al-though the presented results are for Compton scattering,we remark that a similar pattern is expected also forclassical Thomson scattering provided that the frequen-cies are much smaller than the cut-off frequency for thequantum process. The analysis of the frequency combs inthe laboratory frame is presented in Sec. VII, with spe-cial emphasis on the partially angular-integrated energydistributions. This sort of investigation is related to thefact that the frequency-comb structure is very sensitiveto the direction of emission of generated radiation. Be-cause quantum calculations are numerically demanding,in this analysis we choose frequencies much smaller thanthe cut-off frequency, which ensures that the classical cal-culations provide similar results. By doing this, we showthat the comb structure survives the partial angular inte-gration and, in principle, can be detected experimentally.Finally, in Sec. VIII we draw some conclusions.In analytical formulas we put ~ = 1. Hence, the fine-structure constant is α = e / (4 πε c ). We use this con-stant also in expressions derived from classical electro-dynamics, where it is meant to be multiplied by ~ whenrestoring the physical units. Unless stated otherwise, innumerical analysis we use relativistic units (rel. units)such that ~ = m e = c = 1 where m e is the electron mass. II. LASER PULSE
As in our previous investigations [11, 16, 45], the laserpulse is described by the vector potential A ( φ ) = A B [ ε f ( φ ) + ε f ( φ )] , (1)where the shape functions f j ( φ ) vanish for φ < φ > π . The duration of the laser pulse, T p , introducesthe fundamental frequency, ω = 2 π/T p , such that φ = k · x = ω (cid:16) t − n · r c (cid:17) , (2)in which the unit vector n points in the direction of prop-agation of the pulse. In a given reference frame, this di-rection is determined by the polar and azimuthal angles, θ L and ϕ L , respectively. This, according to Appendix A,settles the real polarization vectors ε j = a j and n = a ,Eq. (A1). The constant B > µ = | eA | m e c , (3)where e = −| e | is the electron charge. With these nota-tions, the electric and magnetic components of the laserpulse are equal to E ( φ ) = ωm e cµe B (cid:2) ε f ′ ( φ ) + ε f ′ ( φ ) (cid:3) , (4)and B ( φ ) = ωm e µe B (cid:2) ε f ′ ( φ ) − ε f ′ ( φ ) (cid:3) , (5)where ’ prime ’ means the derivative with respect to φ .The shape functions are always normalized such that h f ′ i + h f ′ i = 12 , (6)where h F i = 12 π Z π F ( φ )d φ. (7)Note that, with such a normalization, the phase-averagedPoynting vector equals h S i = 12 ε c (cid:16) Bωm e cµe (cid:17) n . (8)Laser pulses are also characterized by the number ofoscillations of its electric or magnetic components, N osc .Together with the fundamental frequency ω (or, the pulseduration T p ), they define the carrier frequency (or, thecentral frequency of a pulse), ω L = N osc ω . For a pulsegenerated by a given laser device, the carrier frequencyis fixed whereas the remaining parameters can change.Therefore, it is useful to express the averaged intensityof the laser field, I , which is the modulus of the averagedPoynting vector, Eq. (8), in terms of ω L . Moreover, whencomparing results for different shapes of laser pulses wehave to impose extra conditions. For instance, by as-suming that the flow of laser radiation per unit surfaceand unit time (i.e., the intensity I ) for different dura-tions of laser pulses is independent of N osc , we have tokeep B = N osc . This leads to I = m c πα (cid:16) ω L m e c (cid:17) µ = A I (cid:16) ω L m e c (cid:17) µ , (9)with A I ≈ . × W / cm . On the other hand, if weassume that the averaged energy flow per unit surface(i.e., W = T p I ) is fixed, we have to put B = √ N osc , and W = m c α ω L m e c µ = A W ω L m e c µ , (10)with A W ≈ . × J / cm . The latter situation couldbe met in experiments, as the energy of a laser pulse anda size of the laser focus are quite often given whereas thetime-duration is changed.In our numerical illustrations, we shall choose the lin-early polarized laser pulse, f ( φ ) = f ( φ ) and f ( φ ) = 0,and use the shape functions with the sin -type envelope, f ′ ( φ ) ∝ sin (cid:16) N rep φ (cid:17) sin( N rep N osc φ + χ ) , (11)where the proportionality constant is determined by thenormalization condition (6). Here, χ denotes the carrier-envelope phase, N rep determines the number of modu-lations in the pulse (or, the number of subpulses), and N osc sets the number of cycles within the subpulse. N osc also establishes the central frequency ω L = N osc ω , whichis considered to be fixed and equal to ω L = 1 . λ L = 800nm. Let us also re-mark that, while for φ = 0 we have f (0) = 0, for φ = 2 π the vector potential has to vanish as well. This is au-tomatically satisfied if N osc >
2, whereas for N osc = 1the envelope phase χ must be equal to 0 or π . We alsoassume that B = N rep N osc , so that the averaged laserfield intensity is independent of integers N rep and N osc . III. COMPTON SCATTERING
When scattering a laser pulse off a free electron, a non-laser photon is detected. It is described by the wavefour-vector K and, in general, the elliptic polarizationfour-vector ε K σ ( σ = 1 ,
2) such that K · ε K σ = 0 , ε K σ · ε ∗ K σ ′ = − δ σσ ′ . (12)The wave four-vector K satisfies the on-shell mass rela-tion K · K = 0 as well as it defines the photon frequency ω K = cK = c | K | . As shown in Ref. [45], ε K σ can bechosen as the space-like vector, i.e., ε K σ = (0 , ε K σ ). Thescattering is accompanied by the electron transition fromthe initial (i) to the final (f) state, each characterized bythe four-momentum and the spin projection; ( p i , λ i ) and( p f , λ f ). While moving in a laser pulse, the electron ac-quires additional momentum shift [16] (see, also Ref. [45])which leads to a notion of the laser-dressed momentum:¯ p = p − µm e c (cid:16) p · ε p · k h f i + p · ε p · k h f i (cid:17) k + 12 ( µm e c ) h f i + h f i p · k k. (13)It was discussed in Refs. [16, 45] that the laser-dressedmomenta (13) are gauge-dependent, and therefore theydo not have clear physical meaning. Nevertheless, allformulas derived in Ref. [16] depend on the quantity P N = ¯ p i − ¯ p f + N k − K, (14)where the difference ¯ p i − ¯ p f is already gauge-invariant. We take from our previous paper [16] the derivationof the Compton photon spectra. Hence, the frequency-angular distribution of energy of scattered photons for anunpolarized electron beam is given by the formulad E C d ω K d Ω K = 12 X σ =1 , X λ i = ± X λ f = ± d E C ,σ ( λ i , λ f )d ω K d Ω K , (15)where d E C ,σ ( λ i , λ f )d ω K d Ω K = α (cid:12)(cid:12) A C ,σ ( λ i , λ f ) (cid:12)(cid:12) , (16)and the scattering amplitude equals A C ,σ ( λ i , λ f ) = m e cK p p k ( k · p f ) A C ,σ ( λ i , λ f ) , (17)with A C ,σ ( λ i , λ f ) = X N D N − e − π i P N /k π i P N /k . (18)The scattering amplitude is expressed as a Fourier series;for the coefficients D N , the reader is referred to Eqs. (23)and (44) in Ref. [16]. P N is obtained from Eq. (14).Eq. (17) allows one to define the phase − π < Φ C ,σ π of the Compton scattering amplitude,Φ C ,σ ( ω K , λ i , λ f ) = arg( A C ,σ ( λ i , λ f ))= arg( A C ,σ ( λ i , λ f )) , (19)which depends on the electron spin degrees of freedom.This phase is gauge and relativistically invariant, and canbe split into two parts, if we rewrite A C ,σ ( λ i , λ f ) as A C ,σ ( λ i , λ f ) = e i πN eff X N ( − N D N sinc[ π ( N − N eff )] , (20)where sinc( x ) = sin( x ) /x , and N eff = ( K + ¯ p − ¯ p ) /k . (21)With this factorization, the Compton phase becomesΦ C ,σ ( ω K , λ i , λ f ) = Φ kinC ( ω K ) + Φ dynC ,σ ( ω K , λ i , λ f ) , (22)whereΦ kinC ( ω K ) = arg (cid:0) e i πN eff (cid:1) = πN eff (mod 2 π ) (23)andΦ dynC ,σ ( ω K , λ i , λ f ) = arg nX N ( − N D N sinc[ π ( N − N eff )] o . (24)The former phase we call kinematic , as it depends onlyon the kinematics of the process and it is independentof the spin degrees of freedom (it remains the same forspin-0 particles). The latter phase we call dynamic . In ω K /m e c ene r g y d i s t r . (r e l . un i t s ) ω K /m e c ( ω L / N o s c ) Φ ′ C , σ ( ω K , λ i , λ f ) FIG. 1. (Color online) The energy distribution, Eq. (16), forthe non-spin flipping Compton process (upper panel; in theinset an enlarged portion of the distribution is presented),and the derivative of the phase as a function of the frequency ω K (lower panel), Eq. (19). The laser pulse is linearly po-larized in the x -direction ( ε = e x ) and it propagates inthe z -direction. The electron beam propagates in the op-posite direction. The calculation is performed in the refer-ence frame of electrons for the laser pulse carrier frequency ω L = 4 . × − m e c , and for µ = 2, N osc = 16, N rep = 1,and χ = 0. The scattered Compton photon is emitted inthe direction θ K = 0 . π and ϕ K = 0. In the laboratoryframe, for the Ti-Sapphire laser pulse of the central frequency1.548eV, these parameters correspond to the electron beam’senergy 35MeV, whereas the Compton photon of frequency0 . m e c relates to ω LAB K = 0 . θ LAB K = 0 . π .In the electron beam reference frame, the cut-off frequency is ω cut ≈ . m e c . general, the dynamic phase can be determined only nu-merically. However, for linearly polarized laser pulsesand linearly polarized emitted photons (both consideredin this paper) this phase is frequency-independent.On the other hand, by applying the momenta conser-vation laws (i.e., k · P N = 0 and ε j · P N = 0 for j = 1 , kinC ( ω K ) = F ω Th K ω (mod 2 π ) , (25) ω K /m e c ene r g y d i s t r . (r e l . un i t s ) ω K /m e c ( ω L / N o s c ) Φ ′ C , σ ( ω K , λ i , λ f ) FIG. 2. (Color online) The same as in Fig. 1, but for ω L =4 . × − m e c , µ = 1, and θ K = 0 . π . In the laboratoryframe, for the Ti-Sapphire laser pulse of the central frequency1.548eV, these parameters correspond to the electron beam’senergy 35GeV. In the lower panel, the thick blue (dark) linerepresents the derivative of the Compton phase, Eq. (19), onwhich the thin cyan (gray) line is overprinted, representing thekinematic Compton phase, Eq. (23). In the electron frame,the cut-off frequency is ω cut ≈ . m e c . where the Thomson (i.e., classical) frequency ω Th K [47] isequal to ω Th K = N eff ck · p i n K · [¯ p i + µm e c ( h f i ε + h f i ε )] , (26)or ω Th K = ω K − ω K /ω cut , (27)where ω cut = cp i · nn K · n (28)is the cut-off frequency for the Compton spectra, ω K <ω cut [47]. In Eq. (25), F is independent of ω K , F = π p i · n K p i · n (1 + F + F + F sq ) , (29)with F j = µm e c h f j i (cid:16) n K · ε j n K · p i − p i · ε j p i · n n K · nn K · p i (cid:17) , j = 1 , , (30)and F sq = 12 ( µm e c ) ( h f i + h f i ) 1 p i · n n K · nn K · p i . (31)In Figs. 1 and 2, we show the results for the spin-conserved ( λ i λ f = 1) Compton process in the electronbeam reference frame. In Fig. 1, we choose the frequencyrange much smaller than the cut-off frequency, Eq. (28).One can see that, for this range of frequencies, the deriva-tive of the Compton phase linearly depends on the emit-ted photon frequency, ω K . Since ω K ≪ ω cut , one canexpect that the classical theory will give almost an iden-tical result. On the other hand, Fig. 2 presents the re-sults for frequencies comparable to the cut-off value. Aswe see, when ω K is approaching ω cut , the derivative ofthe Compton phase (and the phase itself) starts to de-pend nonlinearly on the Compton photon frequency andtends to infinity when ω K → ω cut . Moreover, as men-tioned above, for the considered linear polarizations ofthe laser and scattered radiation, the Compton phase,up to a constant term (i.e., independent of ω K ), is equalto the kinematic one. IV. THOMSON SCATTERING
As one can check in Ref. [51], the acceleration a of anelectron in arbitrary electric and magnetic fields, E and B , is given by the formula a = em e p − β (cid:2) E − β ( β · E ) + c β × B (cid:3) . (32)Hence, the relativistic Newton-Lorentz equations whichdetermine the classical trajectory of accelerated electronscan be rewritten in the formd t ( φ )d φ = 1 ω (1 − n · β ( φ )) , (33)d r ( φ )d φ = cω β ( φ )1 − n · β ( φ ) , d β ( φ )d φ = µ p − β ( φ )1 − n · β ( φ ) × h(cid:0) ε − β ( φ )( β ( φ ) · ε ) + β ( φ ) × ε (cid:1) f ′ ( φ )+ (cid:0) ε − β ( φ )( β ( φ ) · ε ) − β ( φ ) × ε (cid:1) f ′ ( φ ) i . Here, the phase, φ = ω ( t − n · r ( t ) /c ), is used as anindependent variable, instead of time t . The frequency-angular distribution of emitted radiation of polarization ε K ,σ is given by the Thomson formula [50] (we use thesame notation for the radiation emitted during this pro-cess as for the Compton scattering)d E Th ,σ d ω K d Ω K = α (cid:12)(cid:12) A Th ,σ (cid:12)(cid:12) , (34) where A Th ,σ = 12 π π Z Υ σ ( φ ) exp h i ω K ℓ ( φ ) c i d φ, (35)withΥ σ ( φ ) = ε ∗ K ,σ · n K × [( n K − β ( φ )) × β ′ ( φ )] (cid:0) − n K · β ( φ ) (cid:1) , (36)and ℓ ( φ ) = ct ( φ ) − n K · r ( φ ) . (37)Here, ’ prime ’ means again the derivative with respect tothe phase φ .Let us further define the position four-vector x ( φ ) = ( ct ( φ ) , r ( φ )) . (38)After some algebraic manipulations, we show thatΥ σ ( φ ) = K ( ε K ,σ · x ′′ )( K · x ′ ) − ( ε K ,σ · x ′ )( K · x ′′ )( K · x ′ ) . (39)Now, we can present the Thomson formula in a mani-festly relativistic form. To do so, we define the relativis-tically invariant quantities: Υ inv σ ( φ ) = Υ σ ( φ ) /K and A invTh ,σ = 12 π Z π Υ inv σ ( φ )e i K · x ( φ ) d φ. (40)Hence, the frequency-angular distribution of radiated en-ergy equals d E Th ,σ d ω K d Ω K = α ω K c (cid:12)(cid:12) A invTh ,σ (cid:12)(cid:12) . (41)The advantage of the above formulation is that the in-variant amplitude A invTh ,σ can be calculated in the mostconvenient reference frame (for instance, in the referenceframe of initial electrons), and afterwards transformed toanother reference frame. It also leads to the simplifica-tions for the invariant amplitude. Indeed, integrating byparts, we get A invTh ,σ = 12 π h ε K ,σ · x ′ K · x ′ e i K · x (cid:12)(cid:12)(cid:12) π − i Z π ( ε K ,σ · x ′ )e i K · x d φ i , (42)or, in a particular reference frame, A invTh ,σ = 12 π cω K h − ε K ,σ · β − n K · β e i ω K ( t − n K · r /c ) (cid:12)(cid:12)(cid:12) π + i ω K ω Z π ε K ,σ · β − n · β e i ω K ( t − n K · r /c ) d φ i . (43)This is an analogue of Jackson’s formula (Ref [50],Eq. 14.67), except that the integral now is finite and ω K /m e c ene r g y d i s t r . (r e l . un i t s ) FIG. 3. (Color online) The same as in Fig. 2, but for Thom-son scattering. The derivative of the phase Φ Th ,σ ( ω K ) is notpresented, as it is constant in the entire range of consideredfrequencies. presented in the relativistically invariant form. Also, wehave checked that the integration over φ can be effectivelycarried out even with the simplest trapezoid or Simpsonformulas. Let us note that the two expressions for theThomson amplitude [i.e., Eqs. (35) and (43)] can be alsoused as a test when determining classical trajectories andevaluating the integral over φ . We define next the phaseof the complex Thomson amplitudeΦ Th ,σ ( ω K ) = arg( A Th ,σ ) = arg( A invTh ,σ ) , (44)which is relativistically invariant but, in contrast to theCompton scattering, independent of spin degrees of free-dom.In order to compare predictions of the classical andthe quantum theories, let us study now the Thomsonprocess for the same parameters as in Figs. 1 and 2. Forthe parameters relevant to Fig. 1, the classical energydistribution is identical to the quantum energy distribu-tion for the spin-conserved process. The difference be-tween these two approaches shows up if we compare thecorresponding phases, which for the classical theory lin-early depends on the frequency of the generated radiation(meaning that its derivative is constant). The same hap-pens for the parameters relevant to Fig. 2, for which theenergy distribution is shown in Fig. 3. Let us remark thatthe energy distributions for the Compton (Fig. 2) and theThomson (Fig. 3) processes, although not identical, arestill comparable in the sense that every peak or zero inthese two distributions can unambiguously be related toeach other [47]. However, the corresponding phases de-pend on ω K differently. We would like to emphasize thatthe nonlinear dependence of the Compton phase on thefrequency of emitted photons (contrary to the Thomsonphase, which linearly depends on ω K ) is of the quantumorigin. Such a qualitative difference between the classical and quantum results can be associated with a change ofthe electron final momentum in the Compton scattering,which introduces decoherence in the process. This fact,although in some cases unnoticed for the energy distri-bution of emitted photons, has far-reaching consequencesfor the temporal behavior of radiation generated by thesetwo processes. This will be demonstrated in the next sec-tion.Our main interest in this paper is nonlinear Comp-ton scattering rather than its classical analogue, whichis nonlinear Thomson scattering. The point being thatthe classical approach is an approximation of the com-plete quantum theory which takes into account the elec-tron spin and the quantum recoil of electrons during thescattering. The complication being, that the quantumtheory does now allow for as detailed description of thedriving laser beam as the classical theory does. Also, it ismore demanding computationally. For these reasons, weinvestigate Thomson scattering for temporarily shapedlaser pulses and for parameters for which both classicaland quantum theories give either the same or different re-sults. Our aim is to compare both theories in the contextof short pulse generation. V. SYNTHESIS OF SHORT PULSES
It is well-known that the energy distribution of gener-ated radiation can be converted into the temporal powerdistribution. Currently, this is the standard techniqueused for the synthesis of attosecond pulses from the co-herent combs of high-order harmonics. Here, let us con-sider the Thomson process and assume that the radia-tion is emitted in a given space direction, n K . In thiscase, the temporal power distribution in the far radiationzone, which is remote from the scattering region by thedistance R , is given by the formula (see, e.g. [49])d P Th ,σ ( φ r )d Ω K = απ (cid:0) Re ˜ A (+)Th ,σ ( φ r ) (cid:1) . (45)Here, ˜ A (+)Th ,σ ( φ r ) = Z ∞ d ω A Th ,σ ( ω )e − i ωφ r /ω (46)is related to the electric field of the scattered radiation E σ ( φ r ) = e πε cR A (+)Th ,σ ( φ r ) , (47)where the symbol “Re“ means the real value and σ la-bels the polarization properties of emitted radiation. Thequantity φ r , which we call the retarded phase, is definedas φ r = ω (cid:16) t − n K · r c (cid:17) = ω (cid:16) t − Rc (cid:17) , (48)with a priori an arbitrary real and positive ω that intro-duces the time-scale for the process. The retarded phase, −5 φ r / π po w e r d i s t r . ( a r b . un i t s ) φ r / π po w e r d i s t r . ( a r b . un i t s ) FIG. 4. (Color online) Temporal power distribution, Eq. (45),synthesized from the energy distribution for the Thomson pro-cess, when ω = ω L . The energy distribution is almost iden-tical to that presented in Fig. 1 in the upper panel, exceptthat the derivative of the Thomson phase is independent ofthe frequency. In the upper panel, the power distribution ispresented in the logarithmic scale and it embraces the entiretime-domain of the generated radiation. In the lower panel,the enlarged part of the central peak is shown in the linearscale. Since the temporal power distribution is proportionalto the electric field of the emitted radiation squared, we seethat the individual peaks form practically one-cycle pulses,which are well separated from each other. The distributionsare scaled to their maximum value. for a given distance R and ω , determines the arrival-timeof a light signal to the detector.All the formulas presented in this section for the tem-poral power distributions also apply to the Compton pro-cess if in Eq. (46) the Thomson amplitude is replaced bythe corresponding Compton amplitude. In this case, thepower distribution depends not only on the polarizationof emitted radiation but also on the spin degrees of free-dom of the initial and final electrons.For long laser pulses, the temporal power distributioncould be a very rapidly oscillating function of time. Forthis reason, it is sometimes more convenient to considerthe temporal power distribution averaged over such rapid −5 φ r / π po w e r d i s t r . ( a r b . un i t s )
19 19.1 19.200.51 φ r / π po w e r d i s t r . ( a r b . un i t s ) FIG. 5. (Color online) The same as in Fig. 4, but for theCompton process such that λ i λ f = 1. The nonlinear depen-dence of the Compton phase on the frequency ω K leads to amore complex structure of a given peak in the temporal powerdistribution. Instead of one-cycle pulses, as generated fromThomson scattering, now we observe many-cycle and chirpedpulses of emitted radiation. oscillations,d h P Th ,σ i ( φ r )d Ω K = α π | ˜ A (+)Th ,σ ( φ r ) | , (49)and similarly for the Compton process.In Figs. 4 and 5, we show the synthesis of the energydistribution from Thomson and Compton scattering, re-spectively. We see that temporal power distributions forboth classical and quantum processes are qualitativelysimilar, as both consist of a sequence of very sharp peaks.However, individual peaks in each sequence are quite dif-ferent, as presented in lower panels. For Thomson scat-tering (Fig. 4), a peak consists practically of a singleoscillation of the electric field. For Compton scattering(Fig. 5), on the other hand, the structure of an individualpeak is more complex. Namely, it represents a pulse of afew electric field oscillations with decreasing period. Theorigin of such a chirp is the nonlinear dependence of theCompton phase on ω K . Note that this nonlinearity isthe genuine quantum effect [47]. Therefore, the chirp ap-pearing in the generated radiation can be considered as φ r / π po w e r d i s t r . ( a r b . un i t s ) FIG. 6. (Color online) Temporal power distribution, Eq. (45),synthesized from the energy distribution of the Thomson pro-cess for the laser- and electron-beam parameters specified inFig. 3, and for ω = ω L . Since the phase of the Thom-son amplitude linearly depends on ω K we obtain the trainof well-separated half-cycle pulses of emitted radiation. Thedistribution is normalized to its maximum value. a quantum signature in collisions of a non-chirped laserpulse with free electrons.Frequently, only a part of the spectrum of emitted ra-diation is used for the composition or detection of shortlaser pulses (see, for instance, the FROG technique [52]).To account for this fact a window function (in the FROGit is called the gate function), W ( ω ), is introduced, whichpicks up a part of the frequency spectrum. The window-ing of the emitted spectrum could also be related to theproperties of detectors of radiation, that can be sensitiveto frequencies from a particular range. In such a case,we define the window-selected amplitude˜ A (+)Th ,σ ( φ r ; W ) = Z ∞ d ωW ( ω ) A Th ,σ ( ω )e − i ωφ r /ω , (50)so that the corresponding temporal power distributionsare equal tod P Th ,σ ( φ r ; W )d Ω K = απ (cid:0) Re ˜ A (+)Th ,σ ( φ r ; W ) (cid:1) , (51)d h P Th ,σ i ( φ r ; W )d Ω K = α π | ˜ A (+)Th ,σ ( φ r ; W ) | , (52)and similarly for the Compton scattering.Fig. 5 presents synthesized pulses in the case whenthe nonlinear dependence of the Compton phase on thefrequency of scattered photons is small. To complementthese results, we consider now the case of a strong depen-dence of the Compton phase on ω K , i.e., for parametersspecified in Fig. 2. Again, for these laser- and electron-beam parameters the derivative of the Thomson phaseover ω K is constant, which leads to a very regular tem-poral power distribution of the generated radiation (see,Fig. 6). Now, the individual peaks represent half-cyclepulses. We meet a completely different situation for theCompton process, for which the synthesis does not lead φ r / π po w e r d i s t r . ( a r b . un i t s ) φ r / π po w e r d i s t r . ( a r b . un i t s ) FIG. 7. (Color online) Temporal power distribution synthe-sized from the energy distribution of the Compton process forthe laser- and electron-beam parameters specified in Fig. 2,and for ω = ω L (upper panel). Since the phase of theCompton amplitude nonlinearly depends on ω K the emit-ted radiation does not form a train of short pulses. In thelower panel, the window-selected temporal power distribution[i.e., Eq. (51) with the window function (53)] is presented for ω wmax = 0 . m e c . Although the window function selects onlya ’regular’ part from the energy distribution, nevertheless thecorresponding temporal power distribution also does not ex-hibit a train of very short pulses. It rather represents a longpulse with many electric field oscillations. Both distributionsare scaled to their maximum values. to a sequence of well-separated short pulses as it is inthe case of the classical process. Instead, we obtain abroad and irregular signal of emitted radiation, as shownin the upper panel of Fig. 7. We want to emphasize thatthe reason for such a qualitative discrepancy between theclassical and the quantum processes is the highly nonlin-ear dependence of the Compton phase on the frequencyof emitted photons.A question arises: Can the window-selecting help inproducing trains of short pulses? To answer this questionwe consider the window function, W ( ω ) = , ω < (cid:0) πω/ω wmax ) (cid:1) , ω ω wmax , ω > ω wmax (53)with ω wmax = 0 . m e c , such that it removes the ir-regular high-frequency part of the energy distributionshown in Fig. 2. The synthesized window-averaged tem-poral power distribution is presented in the lower panelof Fig. 7. Indeed, we removed an irregular part of thepower distribution for large retarded phases. However,instead of a sequence of sharp spikes observed classically,we obtain the single pulse consisting of many regular os-cillations of the electric field. The reason being that, forfrequencies in the domain defined by the window func-tion, the nonlinear terms in the Compton phase are stillsignificant.The great advantage of the classical approach is thatcalculations can be carried out quite easily, even foran arbitrary space and time dependent laser field. Forthis reason, the classical approach is extensively usedin plasma physics and also in the context of ultra-shortpulse generation [39–44]. Even though Thomson theoryhas some important shortcomings. For instance, it doesnot account for the spin of electrons, which for the high-frequency part of the spectrum starts to play a signif-icant role [17, 47], especially for very short and intenselaser pulses. Another defect of the classical theory, whichappears to be crucial for the extremely short pulse gen-eration, is that it neglects the recoil of electrons duringthe emission of high-frequency photons [31]. It has beennoted that the electron recoil effects are small if ω K ≪ ω cut = c n · p i n · n K , (54)independently of the laser field intensity, I , and also ofthe laser field carrier frequency, ω L . On the other hand,it is well-known from the Fourier analysis that in order togenerate the shorter radiation pulses the broader energyspectra have to be used for the pulse synthesis. There aretwo possibilities to increase the bandwidth of the energydistribution in Thomson or Compton processes. Namely,one can either increase the energy of electron beams orincrease the intensity of the laser beam. Mostly, the sec-ond scenario is used [39, 41–44]. The results presentedin this section show that this scenario does not work forsufficiently intense laser pulses such that photons of fre-quencies comparable to ω cut are created with significantprobabilities. Thus, conclusions drawn from the classi-cal theory concerning the generation of extremely shortradiation pulses, which are synthesized from frequenciesclose to the cut-off values, generally cannot be trusted. VI. FREQUENCY COMB STRUCTURES
Discovery of the high-order harmonics in the interac-tion of laser pulses with atoms [29] and their subsequenttheoretical analysis in terms of the three-step model [30]has stimulated a number of investigations. In particu-lar, the coherent properties of the harmonics led Farkasand T´oth [27] to the idea of composing attosecond pulsesfrom at least a part of the high-order harmonics comb. e - e - KK K T sub T d FIG. 8. (Color online) Schematic diagram of Compton andThomson scattering induced by a single (upper cartoon) anda double laser pulse with a time delay T d (lower cartoon).The radiation emitted from each subpulse interfere leading tothe formation of frequency combs in the energy distribution.The separation between peaks in the comb can be controlledby the time delay. This is a routine method used currently in attosecondphysics [28]. It was also shown that the three-step modelis not the only mechanism responsible for the high-orderharmonics generation and that such a comb of frequenciescan be effectively generated by the channeling of initiallyunbounded electrons through crystal structures [54]. Inthis case the emergence of multiple plateaus in the har-monics spectrum is due to resonance transitions betweenthe laser-modified Floquet-Bloch states of electrons [55](very recently the Floguet-Bloch states have been de-tected experimentally [56]). A similar situation is met forthe Thomson and Compton scattering, when the electronbeam traverses the periodic structure of a laser beam (ifapproximated by a plane wave). This problem was ex-tensively studied by Salamin and Faisal [35–37] withinclassical theory.For short laser pulses the situation is different. Insteadof sharp peaks, as the ones observed for long pulses, weobserve broad coherent peak structures [49] extending toa few MeV. In our recent paper we demonstrated that,within such broad structures, it is possible to create co-herent frequency combs for both the electromagnetic andthe matter waves [48]. The idea is to use a modulatedlaser pulse, as illustrated in Fig. 8. For instance, if wecollide a sequence of two subpulses of duration T sub each,and delayed by T d , with a nearly monochromatic electronbeam (see, e.g., Ref. [48]), then the photons generated byeach of these subpulses can interfere with each other. Asa result, one might observe an interference pattern inthe energy distribution of emitted radiation. This is, ofcourse, only the motivation and a priori it is not obviousthat the generated comb structures have similar coher-ent properties as the high-order harmonics combs. Only0
35 35.5 3600.511.522.5 x 10 −7 ω K /ω L ene r g y d i s t r . (r e l . un i t s ) FIG. 9. (Color online) Compton energy distribution, Eq. (16),as a function of frequency ω K for λ i λ f = 1. The laser beam,linearly polarized in the x -direction, propagates in the z -direction and collides with the electron beam in the head-on geometry. The distribution is calculated in the referenceframe of electrons with the laser pulse parameters such that ω L = 4 . × − m e c , µ = 1, N osc = 16, χ = 0. The emittedradiation is calculated for θ K = 0 . π and ϕ K = 0. The thinblack line (the envelope) corresponds to N rep = 1, the thickdashed red line to N rep = 2, the thick blue line to N rep = 3,and the distributions are divided by N . The correspondingenergy distribution for the Thomson process looks identicalexcept that the classical one is blue-shifted by 0 . ω L . a numerical analysis of the Compton process can provideinformation about the phases of peaks within the comband whether the Compton amplitudes can be synthesizedto the finite and well-separated pulses; this is indeed thecase for the high-order harmonics combs. Note that thecorresponding analysis of the classical Thomson processis insufficient. First of all, because it is only an approx-imation of the quantum process. Secondly, as it followsfrom our discussion presented above, the phase propertiesof these two processes are in general different.In Fig. 9, we present the Compton energy distributionfor a particular range of frequencies of emitted photonsand for the undelayed subpulses, T d = 0. In this case,we obtain a broad structure which does not resemble thefrequency comb. However, for N rep > N rep , but they appear for the same frequencies inde-pendent of N rep . Moreover, the height of the individualpeak scales as N , which already indicates the coher-ence of the generated comb. The numerical analysis ofthe phase of the Compton amplitude shows that at thepeak frequencies phases are equal to 0 modulo π [48]. Inaddition, the derivative of the Compton phase with re-spect to ω K is almost constant (in the considered domainof ω K ). This proves that the separation between the con-secutive peaks is nearly the same; hence, a coherent andequally spaced frequency comb is created.In the upper panel of Fig. 10, we present the power
15 20 2500.51 φ r / π po w e r d i s t r . ( a r b . un i t s )
15 20 2500.51 φ r / π po w e r d i s t r . ( a r b . un i t s ) FIG. 10. (Color online) Temporal power distribution (upperpanel; Eq. (45) for the Compton scattering) for an unmodu-lated laser pulse, N rep = 1. The remaining parameters are thesame as in Fig. 9. The power distribution is synthesized fromthe energy distribution represented by the thin black line inFig. 9. While this distribution shows very rapid oscillations,in the lower panel it is averaged over these oscillations. Bothdistributions are normalized to their maximum values. distribution generated by a single pulse. As we see,the broad structure represented in Fig. 9 by the enve-lope curve is converted into the rapidly oscillating andmodulated pulse of radiation. The power distribution,averaged over these rapid oscillations, is shown in thelower panel of Fig. 10. Note that the emitted pulse has amarginal chirp, which is the consequence of a very smallnonlinearity in the dependence of the Compton phase onthe frequency of created photons in the considered rangeof energies. Next, we synthesize the power distributionfrom the frequency comb generated by a sequence of N rep pulses. As a result, we obtain nearly identical, well sepa-rated, and equally spaced in time N rep copies of the samesignal which was obtained for a single pulse. This is pre-sented in Fig. 11 for N rep = 3. This proves the coherentproperties of the frequency comb generated from nonlin-ear Compton (Thomson) process for this particular rangeof frequencies.In Appendices B and C we derive the diffraction formu-las for the Thomson and Compton amplitudes that prove1 φ r / π po w e r d i s t r . ( a r b . un i t s ) FIG. 11. (Color online) Temporal power distribution averagedover the fast oscillations in the case of Compton scattering,for the same parameters as in Fig. 10 but for three subpulses, N rep = 3. The synthesis of the corresponding energy distri-bution, represented in Fig. 9 by the thick blue line, leads toa train of three identical pulses. The distribution is scaled toits maximum value. the ’phase-matching’ conditions for the peaks in the en-ergy distributions at which the global phases change by π .We also show there that, although for classical theory thiscan happen for the equally spaced frequencies, for quan-tum theory this is not the case. The individual harmonicsin frequency combs are approximately equally separatedfrom each other only within finite frequency intervals, inwhich the nonlinear dependence of the Compton phaseon the emitted photon frequency can be neglected. A. Combs for delayed subpulses
The distance between peaks in the comb can be madesmaller or, equivalently, the separation between the syn-thesized pulses of scattered radiation can be made larger,if subpulses are delayed with respect to each other. Toillustrate this, we have to properly define the shape func-tion (we denote it by f d ( φ ) for 0 φ π and 0 other-wise) for such a situation. Hence, we divide the durationof the pulse T p into three pieces and, for simplicity, weassume that the outermost time intervals are equal. Sucha situation is described by the following choice f d ( φ ) = , φ ξπ ¯ f ( φ ) , ξπ < φ < π − ξπ , π − ξπ φ π (55)and 0 otherwise, where 0 ξ <
1. This shape functionis illustrated in the upper cartoon of Fig. 8. If the pulselasts for T p , then the time when it does not vanish isequal to T sub = (1 − ξ ) T p . For the function ¯ f ( φ ) wechoose ¯ f ( φ ) = p − ξ f (cid:16) φ − ξπ − ξ (cid:17) , (56)
35 35.5 3600.511.522.5 x 10 −7 ω K /ω L ene r g y d i s t r . (r e l . un i t s ) FIG. 12. (Color online) The same as in Fig. 9, but for thedelayed sequence of driving subpulses with T sub = T d (cf.,Fig. 8). The delay between subpulses leads to a denser distri-bution of peaks in the frequency comb. where f ′ ( φ ) ∝ sin (cid:16) φ (cid:17) sin( N osc φ + χ ) , (57)as defined by Eq. (11) for N rep = 1. Hence, the normal-ization condition, Eq. (6), remains the same. Moreover,the central frequency of the laser field, ω L , is related tothe fundamental frequency, ω = 2 π/T p , such that ω L = N osc ω − ξ . (58)In order to form a sequence of N rep subpulses, as illus-trated in the lower cartoon of Fig. 8 for N rep = 2, wehave to repeat N rep times the function (55); this way weobtain subpulses with a time delay T d = ξT p . Then, weneed to compress them back to the interval [0 , π ] re-membering to divide the fundamental frequency and tomultiply the laser central frequency by N rep .We remark that for a single pulse ( N rep = 1) the physi-cal situation stays the same independently of which valuefor ξ we choose. The change of ξ only means that wechange the outermost time intervals, at which the electro-magnetic field is 0. This means that all physical quanti-ties including the energy distribution of emitted radiation(and, hence, the structure and the width of synthesizedpulses) have to be the same. Only the time of creation ofthose pulses is shifted. This is a strong test for the cor-rectness of the numerical analysis presented here. It hasto be stressed, however, that for a nonzero ξ the numer-ical calculations become much longer. The reason beingthat more Fourier components of the shape function haveto be accounted for in order to properly approximate thevanishing parts of the driving pulse. The same applies tothe sequence of driving subpulses.In Fig. 12, we present the energy distribution of gener-ated Compton radiation for the same laser and electronbeam parameters as in Fig. 9. This time, however, the2driving subpulses are delayed by T d = T sub (cf., Fig. 8); inother words ξ = 0 .
5. As we see, the results for N rep = 1are identical. On the other hand, the time delay be-tween subpulses leads to a denser distribution of peaksin the frequency comb. Specifically, for the consideredtime delay the number of peaks doubles. The temporalpower distribution also looks similar to the one shownin Fig. 11, except that the first pulse is delayed and thetime distance to the next one is doubled. A very similarpattern is observed for Thomson scattering. VII. COMBS IN LABORATORY FRAME
The discussion above concerned Thomson and Comp-ton processes when analyzed in the rest frame of elec-trons. This is a convenient reference frame for funda-mental theoretical investigations, as most of geometricaldegrees of freedom are eliminated and the analysis canfocus mainly on dynamical aspects of these processes.From an experimental point of view, it is also not a seri-ous limitation as the radiation generated during the col-lision of laser and electron beams interacts directly withthe same electron beam. This was the case in the SLACexperiment [53] in which electron-positron pairs had beengenerated by means of the Breit-Wheeler process (see,e.g., [6, 7, 45]). This takes place in the cascade problemsas well [26]. This means that properties of the generatedradiation (such as chirping of the scattered radiation orthe generation of frequency combs) can be detected indi-rectly by analyzing their consequences.Apart from this, it is interesting to investigate proper-ties of nonlinear Thomson and Compton scattering in thelaboratory frame. It was shown [39–44, 49], for instance,that in the laboratory frame the synthesis of generatedradiation leads to zepto- or even yoctosecond pulses. Thissignificantly extends the already well developed tech-nique for attosecond pulse generation, which is basedon the synthesis of coherent high-order harmonics combs[27]. The aim of this section is to investigate the possibil-ity of direct detection of the frequency comb structuresin the laboratory frame. In our analysis, we consider theThomson scattering for the laser- and electron-beam pa-rameters such that classical and quantum theories givesimilar results for the energy distribution of generatedradiation. The reason for this limitation is that, fromthe numerical point of view, the classical calculation ismuch faster. A similar analysis for the Compton pro-cess is much more time-consuming and is going to bepresented elsewhere in due course.In order to obtain a significant signal of the emit-ted high-frequency radiation from Thomson or Comptonscattering, when analyzed in the laboratory frame, theenergy of the electron beam has to be sufficiently large.On the other hand, the central frequency of very intenselaser pulses is much smaller than the rest mass of elec-trons. It follows from these two facts that the majorityof Thomson (Compton) radiation is emitted into a very
FIG. 13. (Color online) Color mappings of the Thomson en-ergy distribution produced in a head-on geometry of a laserbeam and an electron beam. The electric field of a drivingpulse, linearly polarized in the x -direction, is described by theshape function (11) with N osc = 17, and N rep = 1 (upper leftpanel), N rep = 2 (upper right panel), N rep = 3 (lower leftpanel), N rep = 4 (lower right panel). Its central frequency inthe laboratory frame equals ω L = 1 . ≈ × − m e c and the averaged intensity is determined by µ = 5 /
16. Elec-trons move with momentum p i = 1000 m e c e z and the scat-tering process occurs in the plane Φ K = π/
2. The emittedradiation is linearly polarized in the ( xz )-plane (or, equiva-lently, in the ( x ′ y ′ )-plane). narrow cone. For the head-on collision of the laser andelectron beams, this radiation is emitted mostly in thedirection of the electron beam propagation. For this rea-son, it is better to parametrize the angular distribution ofemitted radiation by a new set of angles. Let us changethe Cartesian coordinates such that( x, y, z ) → ( x ′ , y ′ , z ′ ) = ( z, x, y ) , (59)which is still a right-handed system of coordinates. Next,in the primed coordinates we introduce the polar, 0 Φ K < π , and azimuthal, 0 Θ K π , angles. Hence,we find the following equations:sin Φ K cos Θ K = cos θ K , sin Φ K sin Θ K = sin θ K cos ϕ K , cos Φ K = sin θ K sin ϕ K , (60)which uniquely define a transformation between two pairsof angles. The scattering plane ( xz ), which was definedbefore by two conditions, ϕ K = 0 and ϕ K = π , nowis defined by a single condition, Φ K = π/
2. The sameparametrization was applied in our previous analysis ofCompton scattering [16]. Note that now the measure ofthe solid angle isd Ω K = sin Φ K dΦ K dΘ K , (61)where, for the considered head-on geometry, we can ap-proximate sin Φ K by 1 if integrating over a narrow an-gular cone.3 ω K /m e c ene r g y d i s t r . ( a r b . un i t s ) ω K /m e c ene r g y d i s t r . ( a r b . un i t s ) ω K /m e c ene r g y d i s t r . ( a r b . un i t s ) ω K /m e c ene r g y d i s t r . ( a r b . un i t s ) FIG. 14. (Color online) The same as in Fig. 13, but energydistributions are integrated over the angle Θ K . They are nor-malized to the maximum value of the energy distribution foran unmodulated pulse (upper left panel). The comb peakslocated at the same frequencies are clearly visible. Due to theintegration over the polar angle, which introduces incoher-ence into the distribution, the maxima of the comb peaks donot scale as N . Nevertheless, the visibility of these peaksincrease with increasing the number of subpulses. In Fig. 13, we present color mappings of the energy dis-tribution of radiation generated in the scattering plane,Φ K = π/
2, for up to four repetitions ( N rep = 1 , , T d = 0. The re-sults are for such frequencies ω K and angles Θ K for whichmost of the energy is emitted during the process. As ex-pected, the energy is radiated in the very close vicinityof Θ K = π . For a single pulse ( N rep = 1), we observe theformation of a broad hill for frequencies between 8 and9 m e c (i.e., around 4MeV). The coherent properties ofsuch structures (which in photonic physics are called thesupercontinua [57]) were considered elsewhere [49]. If,instead of a single pulse, we consider a sequence of suchpulses then these broad structures are sliced into stripesand the coherent frequency comb is formed for a givenangle (see, the discussion in the previous section). Thesedistributions integrated over the angle Θ K ,d E C sin Φ K d ω K dΦ K = Z π dΘ K d E C d ω K d Ω K , (62)are presented in Fig. 14. We clearly see the formation ofthe comb peaks, whose positions stay the same for dif-ferent number of subpulses. The maxima of these peaksincrease with increasing N rep , although they do not scalelike N . This is the signature of the incoherence causedby the integration over the angle, which also leads to thedecrease of the visibility of the comb peaks in the inte-grated distribution. However, due to the large separationof these peaks and their comparable intensities, we areconvinced that they could be detected experimentally.We remark that the survival of comb structures, even af- ter integrating over angles, is due to significant collima-tion of the generated Thomson and Compton radiation,which happens for highly energetic electron beams. VIII. CONCLUSIONS
In this paper, we studied the nonlinear Thomson (clas-sical theory) and the Compton (quantum theory) scatter-ing of free electrons with temporarily finite laser pulses.We showed that, for the Compton spin-conserved pro-cess, the energy distribution of emitted photons can bewell described by the classical Thomson theory providedthat frequencies of generated photons are much smallerthan the characteristic cut-off frequency. However, thephases of the corresponding classical and quantum am-plitudes differ from each other. This results in differ-ent temporal power distributions for these two cases, al-though the corresponding energy distributions are nearlyidentical. Our analysis showed that, contrary to the clas-sical theory, it is not always possible to synthesize shortpulses from nonlinear Compton scattering. The point isthat one has to choose the range of Compton photon fre-quencies in which nonlinear (or, equivalently, quantum)corrections to the Compton phase play a marginal role.This statement can be roughly quantified by the condi-tion that (∆ ω K ) d d ω K Φ C ,σ ( ω K , λ i , λ f ) ≪ , (63)for λ i λ f = 1, where ∆ ω K is the frequency bandwidthused for the synthesis of generated pulses of radiation or,in other words, the nonlinear corrections to the Comp-ton phase within the frequency bandwidth are very small.The condition above is violated, for instance, for parame-ters specified in Fig. 1, although the frequencies are muchsmaller than ω cut and the classical and quantum energydistributions are almost identical.In addition, we investigated a possibility of generatingcoherent frequency combs from Thomson and Comptonscattering in the presence of a sequence of short sub-pulses. This was motivated by the celebrated high-orderharmonic generation and by the resulting synthesis of at-tosecond pulses out of the frequency spectrum of thoseharmonics combs. We showed that the separation ofpeaks in the Compton-based (Thomson-based) frequencycomb can be controlled by a time delay of subpulses.Note that such a control is not possible for the high-order harmonics, for which the distance between peaksis not smaller than the central frequency of the drivingpulse, ω L . The possible generation of a sequence of shortpulses has also been investigated. In this context, asfollows from our previous considerations [49], the nonlin-ear Thomson and Compton processes provide the uniquemechanism for the generation of zepto- or even yoctosec-ond pulses. Moreover, by analyzing nonlinear Thomsonscattering in the laboratory frame, we presented a clearsignature of the frequency comb in the angle-integrated4energy distribution of emitted radiation, which could bedetected experimentally.We studied here the generation of frequency-combstructures for the ideal situation when all subpulses areidentical. Such a situation can be well-modeled by com-posing laser pulses from a few monochromatic ones. Infact, the laser pulse shapes considered in this paper arecomposed from three monochromatic components withappropriately chosen amplitudes, and from only two ofsuch components one can build the sequence of identicalsubpulses for N osc = 2. This fact raises the question:How sensitive is the formation of frequency combs if wechange relative phases of these monochromatic compo-nents? This and similar problems are currently investi-gated and are going to be presented in due course. ACKNOWLEDGEMENTS
This work is supported by the Polish National ScienceCenter (NCN) under Grant No. 2012/05/B/ST2/02547.
Appendix A: Triads of unit vectors
The aim of this appendix is to settle the conventionfor the polarization vectors for both the laser pulse andthe radiation emitted during Thomson or Compton scat-tering. Let us define three normalized and mutually or-thogonal real vectors, a j , j = 1 , ,
3, such that a = cos θ cos ϕ cos θ sin ϕ − sin θ , a = − sin ϕ cos ϕ , a = sin θ cos ϕ sin θ sin ϕ cos θ , (A1)where θ and ϕ are the polar and azimuthal angles in anarbitrary chosen reference frame. These vectors consti-tute a right-handed basis of vectors, since a i = ε ijl a j × a l , (A2)where ε ijl is the antisymmetric tensor such that ε = 1.Moreover, an arbitrary vector V can be decomposed as V = a ( a · V ) + a ( a · V ) + a ( a · V ) . (A3)Usually, we shall assume that, if the radiation propa-gates in the a direction, then two real vectors, a and a , describe two linear polarizations of radiation. In or-der to account for elliptic polarizations, we should con-sider two linear combinations, a δ, = cos δ a + i sin δ a , (A4) a δ, = i sin δ a + cos δ a , (A5)such that the orthogonality condition reads a δ,j · a ∗ δ,l = δ jl . In this case, the right-handed condition, Eq. (A2),remains valid and, for an arbitrary vector V , the follow-ing decomposition is fulfilled: V = a δ, ( a ∗ δ, · V ) + a δ, ( a ∗ δ, · V ) + a ( a · V ) . (A6) In particular, for δ = π/ a δ, and a δ, cor-respond to the right-handed and left-handed circular po-larizations.Note that the choice of vectors a and a in Eq. (A1) isnot unique. We can use this freedom to define another setof vectors which determines the polarization properties ofa beam of photons propagating in different directions. If,for instance, we have a polarizer which does not transmitradiation polarized perpendicular to the unit vector N pol ,then it is sometimes more convenient to introduce a triadof vectors ( a k , a ⊥ , n ) such that a k = v a + v a , a ⊥ = − v a + v a , a k × a ⊥ = n , (A7)where v i = a i · N pol p ( a · N pol ) + ( a · N pol ) , i = 1 , . (A8) Appendix B: Diffraction and global phase forThomson scattering
We derive here the diffraction formula for classicalThomson scattering that resembles very much the diffrac-tion grating formula for angular distributions. For thispurpose let us consider an arbitrary pulse defined by twoshape functions f j ( φ ) ( j = 1 , , π/N rep ] together with their first derivatives,and for N rep = 1 , , . . . . If we define now the shape func-tions f j ( φ ) in Eq. (1) such that f j ( φ ) = ( f j ( φ ) , φ ∈ [0 , π/N rep ] , , otherwise , (B1)then the Thomson formula, Eq. (35), defines the energydistribution for a single pulse. Since the acceleration ofelectrons for φ > π/N rep vanishes, therefore the upperlimit of the integration over the phase φ can be shrunkto 2 π/N rep . On the other hand, the shape functions f j ( φ + 2 π ( L − /N rep ) = f j ( φ ) , for L = 1 , , . . . , N rep , (B2)define the pulse consisting of N rep copies of the samesubpulse. In this situation, A Th ,σ ( ω K ) = 12 π Z π d φ Υ σ ( φ )e i ω K ℓ ( φ ) /c = 12 π N rep X L =1 Z π/N rep d φ Υ σ (cid:16) φ + 2 π L − N rep (cid:17) × exp h i ω K c ℓ (cid:16) φ + 2 π L − N rep (cid:17)i . (B3)For 0 φ π/N rep ,Υ σ (cid:16) φ + 2 π L − N rep (cid:17) = Υ σ ( φ ) (B4)5and ℓ (cid:16) φ + 2 π L − N rep (cid:17) = ( L − ℓ (cid:16) πN rep (cid:17) + ℓ ( φ ) . (B5)Hence, after some algebraic manipulations, we arrive atthe diffraction formula for the Thomson amplitude, A Th ,σ ( ω K ) = exp h i ω K c ( N rep − ℓ (cid:16) πN rep (cid:17)i × sin h ω K N rep c ℓ (cid:16) πN rep (cid:17)i sin h ω K c ℓ (cid:16) πN rep (cid:17)i A (1)Th ,σ ( ω K ) (B6)where (see, Eq. (34) with the comments below (B1)) A (1)Th ,σ ( ω K ) = 12 π Z π/N rep d φ Υ σ ( φ )e i ω K ℓ ( φ ) /c (B7)is the Thomson amplitude for the single subpulse.For particular frequencies ω K ,L that fulfill the condi-tion ω K ,L c ℓ (cid:16) πN rep (cid:17) = 2 πL, L = 1 , , . . . , (B8)we have the diffraction enhancement of the energy distri-bution generated by Thomson scattering (similar to thediffraction grating pattern for the angular distribution),as |A Th ,σ ( ω K ,L ) | = N |A (1)Th ,σ ( ω K ,L ) | . (B9)Moreover, for N rep >
1, the Thomson amplitude vanishfor ω K such that ω K N rep c ℓ (cid:16) πN rep (cid:17) = πL, L = 1 , . . . , N rep − , (B10)and, for N rep >
2, it has minor maxima if ω K N rep c ℓ (cid:16) πN rep (cid:17) = πL + π , L = 1 , . . . , N rep − . (B11)This pattern is exactly observed in our numerical analysisand is very well-known for the angular distribution ofradiation passing through the diffraction grating.The global phase of Thomson amplitude equalsarg A Th ,σ ( ω K ) =( N rep − h π + ω K c ℓ (cid:16) πN rep (cid:17)i + arg A (1)Th ,σ ( ω K ) , (B12)and the determination of the phase for a single subpulsefor a general form of the shape functions and arbitrarypolarizations of emitted radiation can be done only nu-merically. However, for special types of pulses consideredin this paper the analytical formula for this phase can beprovided. Indeed, by inspecting Fig. 15, together with φ / π Υ σ ( φ )( r e l. un i t s ) φ / π ℓ ( φ )( r e l. un i t s ) FIG. 15. (Color online) Functions Υ σ ( φ ) and ℓ ( φ ) for theThomson amplitude. The parameters are the same as in Fig. 2except that N rep = 2. These functions, for the consideredlaser pulse shapes, satisfy the symmetry conditions (B13) and(B14). We draw horizontal and vertical lines to emphasize theimportant symmetries of these functions. the comments made below Eq. (B3), one can notice thefollowing symmetry properties, valid for φ ∈ [0 , π/N rep ],Υ σ ( φ ) = − Υ σ (2 π/N rep − φ ) (B13)and ℓ ( φ ) + ℓ (2 π/N rep − φ ) = 2 ℓ ( π/N rep ) . (B14)These relations allow us to write down the Thomson am-plitude A (1)Th ,σ ( ω K ) as follows: A (1)Th ,σ ( ω K ) = 1 π exp h i (cid:16) π ℓ ( π/N rep ) c ω K (cid:17)i × Z π/N rep d φ Υ σ ( φ ) sin h ω K c (cid:0) ℓ ( φ ) − ℓ ( π/N rep ) (cid:1)i , (B15)and, since ℓ (2 π/N rep ) = 2 ℓ ( π/N rep ), we finally arrive atthe global phase for Thomson amplitude,Φ Th ,σ ( ω K ) = (cid:16) N rep ∓ (cid:17) π + N rep ω K c ℓ (cid:16) πN rep (cid:17) , (B16)where “ − “ is if the integral in (B15) is positive, and “+“if negative. Therefore, we see that for laser pulses con-sidered in this paper the global phase is a linear function6of the frequency of emitted radiation and, moreover, forthe peak frequencies ω K ,L , Eq. (B8), we obtain,Φ Th ,σ ( ω K ,L ) = (cid:16) N rep ∓ (cid:17) π + N rep Lπ. (B17)Hence, up to the same constant term, the phase is 0modulo π , which proves the coherent properties of theThomson combs. Moreover, the peak frequencies ω K ,L are equally separated from each other, which is not thecase for Compton scattering.We remark that, in order to derive the diffraction for-mula (B6), one has to assume that for each individualsubpulse all necessary conditions imposed on a laser pulsehave to be preserved; namely, the electromagnetic fieldstrength and vector potential in the beginning and at theend of a subpulse has to vanish. Otherwise, the symmetryrelations (B4) and (B5) would not be satisfied. The sameapplies to the quantum case, as it follows from analysispresented in Appendix C. This, in particular, excludesthe case of a plane wave as for the single oscillation theseconditions are not satisfied. Appendix C: Diffraction and global phase forCompton scattering
A similar analysis as in Appendix B, can be also car-ried out for Compton scattering. Since in this case theformulas are much longer, we first introduce simplifiednotations. In this Appendix the integers j, j ′ = 1 , µ i = µ m e c p i · k , µ f = µ m e c p f · k , (C1) S (+) p ( x ) = p · x + Z k · x h eA ( φ ) · pk · p − e A ( φ )2 p · k i d φ, (C2)and the four-vector Q = p i − p f − K. (C3)Then the probability amplitude for Compton scatteringcan be written as [16] A ( e − p i λ i → e − p f λ f + γ K σ ) = i s παc ( m e c ) E p i E p f ω K V A , (C4)where V is the quantization volume and A = Z d x e − i( S (+) p i ( x ) − S (+) p f ( x ) − K · x ) ¯ u (+) p i λ i ˆ C ( k · x ) u (+) p f λ f , (C5)with the 4 × C ( k · x ) = /ε K σ − µ i f j ( k · x ) /ε K σ /k/ε j − µ f f j ( k · x ) /ε j /k/ε K σ + µ i µ f f j ( k · x ) f j ′ ( k · x ) /ε j /k/ε K σ /k/ε j ′ . (C6) For finite laser pulses this expression, although finite,is not convenient for numerical and analytical analysis.Therefore, we apply the transformation defined in Ap-pendix B in Ref. [16], and originally introduced by Bocaand Florescu in Ref. [14]. This transformation leads tothe change of ˆ C ( k · x ),ˆ C ( k · x ) = (cid:0) ˜ a j f j ( k · x ) + ˜ b [ f ( k · x ) + f ( k · x )] (cid:1) /ε K σ − µ i f j ( k · x ) /ε K σ /k/ε j − µ f f j ( k · x ) /ε j /k/ε K σ + µ i µ f f j ( k · x ) f j ′ ( k · x ) /ε j /k/ε K σ /k/ε j ′ . (C7)Here, ˜ a j = 2 k Q ( µ i p i · ε j − µ f p f · ε j ) , (C8)and ˜ b = − k Q µm e c ( µ i − µ f ) . (C9)Now, accounting for the laser pulse-dressed electron mo-mentum, Eq. (13), we introduce the following decompo-sition (this is in fact the definition of G ( k · x )), S (+) p i ( x ) − S (+) p f ( x ) − K · x = ¯ Q · x + G ( k · x ) , (C10)where ¯ Q = ¯ p i − ¯ p f − K. (C11)The purpose of this decomposition is such that the func-tions G ( φ ) and ˆ C ( φ ) for the laser pulse consisting of N rep copies of identical subpulses satisfy, for φ ∈ [0 , π/N rep ]and L = 1 , . . . , N rep −
1, the symmetry conditions G ( φ + 2 πL/N rep ) = G ( φ ) , (C12)and ˆ C ( φ + 2 πL/N rep ) = ˆ C ( φ ) , (C13)similar to Eq. (B4) for Thomson scattering. Further, foran arbitrary four-vector a , we define the light-cone vari-ables ( n is the propagation direction of the laser beam) a k = n · a , a − = a − a k , a + = a + a k , a ⊥ = a − a k n . (C14)Since ( x − = k · x/k = φ/k )¯ Q · x = ( ¯ Q + /k ) φ + ¯ Q − x − − ¯ Q ⊥ · x ⊥ , (C15)and d x = 1 k d φ d x + d x ⊥ , (C16)we rewrite the Compton amplitude (C5) as A =(2 π ) δ ( Q − ) δ (2) ( Q ⊥ ) 1 k × Z π d φ e − i( ¯ Q + /k ) φ e − i G ( φ ) ¯ u (+) p i λ i ˆ C ( φ ) u (+) p f λ f (C17)7Applying now the decomposition (B3) to the integral over φ and accounting for the symmetries (C12) and (C13), wearrive finally at the diffraction formula for the Comptonamplitude, A = exp (cid:16) − i π ¯ Q + ( N rep − k N rep (cid:17) sin( π ¯ Q + /k )sin( π ¯ Q + /k N rep ) A (1) , (C18)where A (1) is the Compton amplitude for a single pulse.For frequencies of emitted photons, ω K ,L with integer L ,that satisfy the condition π ¯ Q + /k N rep = − πL, (C19)we have the coherent enhancement of the Compton am-plitude, which leads to the quadratic, N , enhancementof probability distributions. However, contrary to theThomson case, these frequencies are not exactly equallyseparated from each other on the whole interval of al-lowed frequencies, i.e. [0 , ω cut ]. When ω K approachesthe cut-off value ω cut the spectrum of ω K ,L becomes in-creasingly denser. This means that one can get the fre-quency comb for Compton scattering with approximatelyequally spaced peak frequencies, only over some limitedfrequency intervals. Since for a single subpulse (see, discussion in Sec. III)arg A (1) = − π ¯ Q + k N rep + Φ dynC ,σ ( ω K , λ i , λ f ) , (C20)where Φ dynC ,σ ( ω K , λ i , λ f ) is the dynamic phase of a singlesubpulse, therefore the global phase of the Compton am-plitude equalsarg A = Φ C ,σ ( ω K , λ i , λ f ) = − π ¯ Q + k + Φ dynC ,σ ( ω K , λ i , λ f ) . (C21)For arbitrary laser pulses and polarizations of emittedphotons the dynamic phase can only be calculated nu-merically. 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