Non-Linear Compton Scattering of Ultrashort and Ultraintense Laser Pulses

# Non-Linear Compton Scattering of Ultrashort and Ultraintense Laser Pulses

D. Seipt    B. Kämpfer Forschungszentrum Dresden-Rossendorf, POB 51 01 19, 01314 Dresden, Germany
###### Abstract

The scattering of temporally shaped intense laser pulses off electrons is discussed by means of manifestly covariant quantum electrodynamics. We employ a framework based on Volkov states with a time dependent laser envelope in light-cone coordinates within the Furry picture. An expression for the cross section is constructed, which is independent of the considered pulse shape and pulse length. A broad distribution of scatted photons with a rich pattern of subpeaks like that obtained in Thomson scattering is found. These broad peaks may overlap at sufficiently high laser intensity, rendering inappropriate the notion of individual harmonics. The limit of monochromatic plane waves as well as the classical limit of Thomson scattering are discussed. As a main result, a scaling law is presented connecting the Thomson limit with the general result for arbitrary kinematics. In the overlapping regions of the spectral density, the classical and quantum calculations give different results, even in the Thomson limit. Thus, a phase space region is identified where the differential photon distribution is strongly modified by quantum effects.

high-intensity laser pulses, Compton scattering, Volkov states
###### pacs:
12.20.Ds, 41.60.-m

## I Introduction

The use of chirped-pulse amplification mourou1985 () has led to a prodigious advance in available laser power. The current records reach several petawatts, and accompanying interest in strong-field physics culminates in planned large-scale laser facilities such as the anticipated “Extreme Light Infrastructure” (ELI) eli (). The pioneering theoretical studies in strong-field physics considered both pair creation in a strong field Reiss:1962 () and the cross channel process, electron photon scattering Nikishov:1963 (); Nikishov:1964a (); Nikishov:1964b (); Goldman:1964 (); Narozhnyi:1964 (); Brown:1964zz (); Kibble:1965zz (), dubbed non-linear Compton scattering, where the use of laser beams has already been suggested. Since then there has been a wealth of theoretical papers and we refer the reader to the reviews McDonald:1986zz (); Fernow (); Lau:2003 (); Mourou:2006zz (); Salamin:2006ff (); Marklund:2006my (). In non-linear Compton scattering

 e(p)+ℓγL(k)→e′(p′)+γ(k′) (1)

a number of photons with momentum from a high intensity laser, scatter off an electron with momentum . A convenient measure of laser intensity is the dimensionless laser amplitude , with being the root-mean-square electric field and the laser frequency. The parameter is a purely classical quantity, representing the work performed by the field on the electron in one wavelength. Thus, is the classical nonlinearity parameter ritus_doctorskaya () and it is related to the ponderomotive potential . The definition of can be made explicitly Lorentz and gauge invariant Heinzl:2008rh (). When becomes of order unity the quiver motion of the electron in the laser beam becomes relativistic in a classical picture.

The spectrum of non-linear Compton scattering has been observed in several experiments colliding laser and electron beams, such as low-intensity laser photons () with low-energy () electrons from an electron gun Englert:1983zz (), photons with plasma electrons from a gas jet Umstadter () and, more recently, sub-terawatt photons () from a laser with MeV electrons from a linac at the BNL-ATF Babzien:2006zz (). Using linearly polarized photons the latter two experiments Umstadter (); Babzien:2006zz () have analyzed the characteristic azimuthal intensity distributions confirming quadrupole and sextupole patterns for the second and third harmonics, respectively. Recently, the energy spectrum of the scattered radiation has been measured in an all-optical setup using laser accelerated electrons schwoerer:2006 ().

Probably the best known experiment is SLAC E-144 probing strong-field QED using a terawatt Nd:glass laser () in conjunction with high-energy ( GeV) electrons Bamber:1999zt (). The observation of non-linear Compton scattering has been reported Bula:1996st () as well as the observation of the crossed process of non-linear pair creation, due to the interaction of a Compton scattered high-energy photon with a second laser beam Burke:1997ew ().

The low-energy limit (in terms of laser frequency) of Compton scattering is Thomson scattering which is described completely classically Schappert (); Esarey (). This classical picture is used as theoretical framework for many applications of laser Compton scattering such as X-ray sources schoenlein (); chouffani (); debus () or diagnostic tools leemans (). A convenient parameter to distinguish the two regimes is the quantity

 yℓ = sℓ−m2m2=2ℓk⋅pm2, (2)

where expresses the center of mass energy squared for the generation of the th harmonic in a Lorentz invariant manner111 an are four-vectors, thus denotes a scalar product; we employ units with .. The process is kinematically equivalent to the scattering of one photon with momentum off an electron with momentum , thus it appears as a (pseudo) process. The Thomson regime is recovered for , while for one finds striking differences to the Thomson scattering. The electron recoil during the scattering may be quantified by the Lorentz-invariant quantity which is in the range , i.e. in the Thomson regime holds.

A quantity measuring non-linear quantum effects,

 χR = e√(Fμνpν)2m3=12ay1, (3)

has been introduced in Nikishov:1963 (); Narozhnyi:1964 (). It measures the work done by the field over the Compton wavelength in the rest frame of the initial electron where the four-vector . Introducing the critical field strength sauter () this may also be written as , where is the rms electric field strength in the electron’s rest frame. The parameter combines nonlinearity and quantum effects. is of the order of unity if both and are of the order of unity. Thus, the corrections to the classical description (Thomson scattering) are important if either () an ultraintense high-energy photon pulse, e.g. produced by an X-ray free electron laser, interacts with low-energy electrons, or () a multi-GeV electron beam is brought to collision with an optical high-intensity laser. The latter scenario is similar to the SLAC E-144 experiment but with a higher value of . For the SLAC beam in conjunction with a counterpropagating optical laser () one has . These parameters will be used mainly below for numerical calculations. The FACET project facet () at SLAC envisages investigations within such kinematics in line with ().

Ultraintense lasers use short pulses (few fs, few laser cycles) requiring a proper treatment of the laser pulse structure. Rich substructures of the scattered photon spectra were predicted within the classical picture Gao (); Krafft (); HartemannPRE (); SeiptHeinzl () of Thomson scattering. These substructures have not yet been confirmed experimentally. The effect of radiation back reaction on the spectra was studied in hartemannprl () and found to be important. Only a few publications address quantum calculations in pulsed fields for scalar particles Neville () and for spinor particles fofanov (); Boca (). In Piazza (), a connection between the emitted angular spectrum in non-linear Compton scattering and the carrier envelope phase in few-cycle laser pulses was established. In a related field, electron wave-packet dynamics in strong laser fields has been studied in e.g. Keitel1 (); Keitel2 ().

In our paper we calculate the emitted photon spectrum in non-linear Compton scattering using a generalized Volkov solution with temporal shape in light-cone coordinates. For this purpose we first focus on the structure of the Volkov wavefunction in a pulsed laser field. The aim of our study is to compare the QED calculations for non-linear Compton scattering with results from a classical calculation, i.e. Thomson scattering.

Our paper is organized as follows. In section II we present Volkov states in pulsed laser fields. Section III continues with the calculation of the matrix element and the transition probability. A slowly varying envelope approximation is discussed. Numerical results are presented in section IV. We discuss various limiting cases of our general results, including monochromatic plane waves and the Thomson limit. As a main result, a scaling law is presented, connecting the Thomson spectrum with the Compton spectrum. In the Appendix we summarize the kinematics in light cone coordinates and present the Fourier transformation of the Volkov state.

## Ii Temporally Shaped Volkov States

A strong laser field may be considered as a coherent state of photons , characterized by the polarization and momentum distribution , if the depletion of the laser photons from by an interaction process with electrons is negligible, i.e. for any relevant scattering process with is valid, where in and out are particle number states without coherent parts HarveyHeinzl (). Then, it is possible to work within the Furry picture Furry (), where the interaction of an electron with the classical background field , which is the Fourier transform of , is treated nonperturbatively and solutions of the Dirac equation

 (i⧸∂−e⧸A−m)ψ(x) = 0 (4)

are utilized as basic in and out states for the perturbative expansion of the matrix. For background fields in the form of plane waves, closed solutions of (4) can be found,

 ψp,s(x)=(1+e2k⋅p⧸k⧸A)exp{iSp(x)}up,s√2Ep, (5)

where the free Dirac spinor for momentum and spin fulfills and is normalized to . The phase is the classical Hamilton Jacobi action

 Sp(x) =−p⋅x+f(k⋅x), (6)

with , and . Equation (5) represents the famous Volkov states, whose perturbative expansion in terms of interactions with the laser field is depicted in figure 1 of HarveyHeinzl () (The expansion parameter, i.e. the coupling strength at the vertices, is defined below).

For the vector potential we use a real transverse plane wave

 Aμ=A0g(k⋅x)(ϵμ1cosξcosk⋅x+ϵμ2sinξsink⋅x), (7)

modified by an envelope function and fulfilling and . The parameter determines the polarization of the laser: It is linearly polarized for and circularly polarized for . For other values of , the laser is elliptically polarized Panek (). The vector potential is normalized such that the mean energy density or the energy flux , where means averaging over the fast oscillations of the carrier wave, is independent of , but the dimensionless laser amplitude , as defined in the introduction, is time dependent. A time independent laser strength parameter may be defined by the normalized peak value of the vector potential (with this definition, for ). The vector potential (7) can also be cast into a complex form

 Aμ=A0g(k⋅x)2(ϵμ+e−ik⋅x+ϵμ−e+ik⋅x)=A0g2Bμ, (7′)

with the complex polarization vectors with , and the definition .

In what follows, the temporal pulse shape will often be chosen as a cosh pulse

 g(k⋅x) (8)

with width , or a Gaussian

 g(k⋅x)=exp{−(k⋅x)22σ2}. (9)

Besides these special cases, any other smooth function which depends solely on is possible.

Since the vector potential depends only on , it is convenient to work in light-cone coordinates with (see Appendix A). In these coordinates, the Volkov wavefunction (5) reads

 ψp,s(x) =Cp(x⊥,x−,x+)up,s√2Ep, (10) Cp(x⊥,x−,x+) =[1+dpg(x+)(⧸k⧸ϵ−eiωx++⧸k⧸ϵ+e−iωx+)]e−i2(p+x−+p−x+)+ip⊥⋅x⊥+if(x+) (11)

with . For many purposes it is sufficient to consider only the function since it contains the relevant information on the interaction of the electron with the laser pulse. The real part222The imaginary part gives no further information; it has a shifted phase as compared to the real part. of the scalar projection is visualized in Fig. 1 in the frame where the electron is initially at rest. The scalar projection is essentially equivalent to probing the state with and an average over the spins

 12∑s¯up,s√2Epψp,s(x)=mEpS. (12)

In that frame the (free) electron wavefunction outside the laser pulse behaves as . The effect of the laser pulse is a local deformation of the electron wavefronts due to the build-up of an effective, time dependent momentum with . The momentum achieves its maximum at the center of the laser pulse , depicted as diagonal straight line in Fig. 1, where it coincides with the usual quasi momentum in monochromatic plane waves. Thus, inside the laser pulse, especially for , the fully dressed electron wavefunction behaves as , i.e. the electron wavelength changes and the wavefronts become tilted. Both effects are proportional to the ponderomotive potential, i.e. , where is the laser frequency in the initial electron rest frame and . In Fig. 1, is chosen which can be achieved, for instance, with laser photons of energy colliding head-on with electrons, or a X-ray laser beam with electrons. The electron wavefunction changes its behavior from the free case to the fully dressed case over oscillations of the free electron wavefunction, i.e.  for the parameters employed in Fig. 1. For lower values of , the behavior of the electron wavefunction changes slowly over many oscillations, e.g. for ( electrons colliding with photons) and one finds . For a linearly polarized laser, additional ripples appear in the interaction region due to the oscillating terms in the phase. For non-head-on geometries, similar ripples are also present for circular polarization. The smoothness of the pulse envelope ensures the smoothness of the Volkov wavefunction in the transition from the field-free regions to the laser pulse.

Vector projections with an odd number of Dirac matrices, such as or , vanish identically and are, therefore, not useful for characterizing . The pseudoscalar also vanishes. The antisymmetric tensor projections , where is the spin tensor, allow for a further characterization by for circular laser polarization and for for linear polarization. These tensor projections are nonzero but only inside the laser pulse. They mix contributions with different spin orientation and are therefore proportional to a combination of and , i.e. the spin-up wavefunction contains contributions with spin-down and vice versa. From the structure of the Pauli interaction term , where is the electromagnetic field strength tensor, one can infer that corresponds to the interaction of the electron with the component of the electric field, and corresponds to the component of the magnetic field. From this correspondence and by inspecting (7) it is easy to understand why some projections are zero for linear polarization while they are nonzero for circular polarization. As an example, the tensor projection is shown in Fig. 2.

## Iii Calculation of the Matrix Element

### iii.1 The S matrix

The interaction of the Volkov electron with photon modes different from the laser field are treated by perturbative matrix expansion. The Born approximation of the matrix element for the emission of one photon, i.e. non-linear Compton scattering , is depicted in Fig. 3. Using Feynman rules Landau4 (), the matrix element for such a process is given by

 Sfi=⟨p′,s′;k′,ϵ′λ′|S[A]|p,s⟩ =−ie∫d4x¯ψp′,s′(x)eik′⋅x√2ω′⧸ϵ′λ′ψp,s(x), (13)

which reads in light-cone coordinates, suppressing spin and polarization indices from now on,

 Sfi = N0∫d4x¯up′(1+dp′⧸B⧸k)⧸ϵ′(1+dp⧸k⧸B)upei(Sp−Sp′+ik′⋅x) (14) = N02∫d2x⊥dx+dx−Γ(x+)eiH(x+,x−,x⊥) (15)

with and

 Γ(x+) = \mathrsfsT00+geiωx+\mathrsfsT11+ge−iωx+\mathrsfsT1−1+g2\mathrsfsT20+g2e2iωx+\mathrsfsT22+g2e−2iωx+\mathrsfsT2−2, (16)

where

 \mathrsfsT00 =¯up′⧸ϵ′up, (17) \mathrsfsT1±1 =¯up′(dp′⧸ϵ∓⧸k⧸ϵ′+dp⧸ϵ′⧸k⧸ϵ∓)up, (18) \mathrsfsT20 =4(k⋅ϵ′)dpdp′¯up′⧸kup, (19) \mathrsfsT2±2 =dpdp′¯up′(⧸ϵ∓⧸k⧸ϵ′⧸k⧸ϵ∓)up. (20)

Due to , one finds for circular polarization. Furthermore,

 H(x+,x−,x⊥) = Sp−Sp′+ik′⋅x (21) = (p′+k′−p)⋅x+f(x+)−f′(x+)

with and

 f1(x+;p) =ma0k⋅pk⋅x∫ϕ0dϕg(ϕ)[p⋅ϵ1cosξcosϕ+p⋅ϵ2sinξsinϕ], (22) f2(x+;p) =−m2a202k⋅pk⋅x∫ϕ0dϕg2(ϕ)[cos2ξcos2ϕ+sin2ξsin2ϕ]. (23)

Inspecting Eq. (21), it is obvious that the dependence of on and is trivial and the integrations over these variables in Eq. (15) can be done analytically. As a result, energy-momentum conservation is imposed on the components and , and the exponent

 H+(x+) = 12(k′−+p′−−p−)x++f(x+)−f′(x+) (24)

remains. Due to the non-trivial pulse dependent structure of , the integration does not yield another conservation law. Thus, the frequency of scattered photons is not fixed by energy and momentum conservation as a function of scattering angle and remains as independent parameter. Including the dependence of , some rather complicated functions of emerge

 \mathrsfsAMN =∞∫−∞dx+gM(x+)expi{H+(x+)+Nωx+}. (25)

With these definitions, the matrix element can be written as

 Sfi =(2π)3δ2(k′⊥+p′⊥−p⊥)δ(k′++p′+−p+)N0\mathrsfsM (26)

with

 \mathrsfsM =\mathrsfsT00\mathrsfsA00+\mathrsfsT11\mathrsfsA11+\mathrsfsT1−1\mathrsfsA1−1+\mathrsfsT20\mathrsfsA20+\mathrsfsT22\mathrsfsA22+\mathrsfsT2−2\mathrsfsA2−2. (27)

The integrals are numerically convergent for due to the presence of the pulse function in the integrand, rendering the range of integration practically finite. The integral , however, contains a divergent part and must be regularized. A possible method has been proposed in Boca (), where one multiplies the integrand with a convergence factor , and performs an integration by parts. The result is

 \mathrsfsA00 = −2P−∞∫−∞dx+d(f−f′)dx+exp{iH+(x+)}+4ei[f(0)−f′(0)]limε→0+εP2−+ε2, (28)

with . In (28), the first part is now convergent and the second part is proportional to a distribution with support at . The latter contribution can be neglected in our analysis for . This regularized version of will be used in the subsequent numeric calculations.

### iii.2 Slowly Varying Envelope Approximation

The calculations are simplified upon utilizing the slowly varying envelope approximation (SVEA) of the phase of the functions. This approximation scheme is suitable for long pulses with . Typically is proportional to the number of laser oscillations under the envelope. For , which is proportional to (see Eq. (22)), an integration by parts is performed, yielding

 ∫dϕg(ϕ)sinϕ = −g(ϕ)cosϕ+∫dϕdgdϕcosϕ, (29) ∫dϕg(ϕ)cosϕ = g(ϕ)sinϕ−∫dϕdgdϕsinϕ. (30)

SVEA basically means neglecting the second terms containing the derivative of the pulse shape because it is smaller than the first term. For (, cf. Eq. (23)) we use

 ∫dϕg2(ϕ)cos2ϕ ≈ 12∫dϕg2(ϕ)+12g2(ϕ)sinϕcosϕ, (31) ∫dϕg2(ϕ)sin2ϕ ≈ 12∫dϕg2(ϕ)−12g2(ϕ)sinϕcosϕ, (32)

which becomes particularly handy if is known analytically, such as for the sech pulse (8), where , or the Gaussian pulse (9), , where is the normalized error function. Finally, the SVEA result for the phase reads

 f1 = ma0k⋅pg(x+)[p⋅ϵ1cosξsinωx+−p⋅ϵ2sinξcosωx+], (33) f2 = −m2a204k⋅p[ωx+∫dϕg2(ϕ)+g2(x+)cosωx+sinωx+(cos2ξ−sin2ξ)], (34)

generalizing the approximation scheme of fofanov () to linear laser polarization.

Even for short pulses, such as for meaning that there are about laser oscillations in the pulse, i.e. the pulse length is for , SVEA is quite a good approximation, see Fig. 4 for selected examples.

### iii.3 The spectral distribution of scattered photons and the cross section

In the standard formalism, scattering experiments are thought of as constant streams of particles interacting. Consequently, the square of the matrix contains a factor which originates from the square of the energy-momentum conservation which is interpreted as , with the volume and interaction time which are both put to infinity. On the purpose of rendering this quantity finite, usually the differential rate per unit time and unit volume is considered, where denotes the final state phase space. Here, however, the interaction is happening only within a finite time interval. Because of lacking one distribution, the square of the matrix now reads

 |Sfi|2=(2π)3Vδ(p′⊥+k′⊥−p⊥)δ(p′++k′+−p+)|N0\mathrsfsM|2, (35)

where the dependence on the interaction time is contained in and is finite. Thus, it is not necessary to define a differential rate per unit time. An appropriate observable is the emission probability of photons per unit volume and laser pulse

 dN=|Sfi|2VdΠ, (36)

which has as classical analog the spectral density of scattered photons in Thomson scattering (cf. e.g. SeiptHeinzl (); Schappert ())

 d2Nclassicaldω′dΩ =−ω′16π3j∗(k′)⋅j(k′), (37) jμ(k′) =e∫dτuμ(τ)eik′⋅x(τ), (38)

where are the classical velocity and orbit from a solution of the Lorentz force equation for a spinless pointlike charge, and is the retarded Fourier transform of the electron current. The notion of Thomson scattering is specified to mean this particular calculation scheme. A quantum spectral density is given by the Lorentz invariant expression

 d2Nss′λ′dω′dΩ =e2ω′16π314p+p′+|\mathrsfsMss′λ′|2, (39)

which depends on spin and polarization indices. Averaging over the spin of the incoming electron and summing over the spin of the outgoing electron and the polarization of the outgoing photon yields a quantity which is directly comparable to the classical spectral density

 d2Nquantumdω′dΩ=122∑s,s′=1∑λ′d2Nss′λ′dω′dΩ. (40)

Now we construct an invariant cross section by dividing Eq. (40) by the normalized number of photons in the laser pulse, i.e.

 d2σdω′dΩ=1NLd2Nquantumdω′dΩ (41)

with , where is the Poynting vector of the laser field, derived from the vector potential (7), yielding

 NL=ω2a20m2e2∞∫−∞dtg(t)2 (42)

with and for the pulse shapes (8) and (9), respectively. Using this definition, the total cross section is independent of the pulse shape function and the pulse length . (A different definition of the cross section without this property has been proposed in Boca ().) This has been checked numerically for by a comparison of with the differential Klein-Nishina cross section Landau4 (), or with the total Klein-Nishina cross section. In particular, in the limit we obtain the total Thomson cross section accurately, as exhibited in Fig. 5 for three different pulse shapes and pulse lengths.

## Iv Discussion

### iv.1 Monochromatic Limit

In the famous case of monochromatic Compton scattering, the frequency of the scattered photon is uniquely defined by the scattering angle. For a finite temporal laser pulse, however, this tight relation is lost. As outlined in subsection III.3, there is a distribution of the emitted photons for a fixed angle. As an example, we exhibit in the left top panel of Fig. 6, the spectral density as a function of for fixed . The vertical thin lines depict the positions of the harmonics for a monochromatic plane wave with infinite duration and the same value of , given by Landau4 ()

 ω′ℓ=ω′ℓ,quantum =ℓk⋅q(q+ℓk)⋅n′, (43)

introducing the intensity dependent quasi-momentum of the electron with and the dressed mass-shell relation . Note that only , the conjugate momentum to , is modified by an intensity dependent contribution, i.e.  and . The integer labels the individual harmonics, which are not equidistant in general.

In a pulsed laser field, each harmonic consists of a bunch of spectral “lines” (or subpeaks) visible in the top panels of Fig. 6 with a certain width determined by the minimum and maximum values of intensity in the laser pulse. The high-energy tail of each harmonic bunch is given by , and is produced at the edges of the laser pulse. The low-energy edge is given by and accounts for the maximum red-shift at the center of the pulse. Thus, the spectral width of each harmonic is given by

 Δω′ℓ=ω′ℓ(a0→0)−ω′ℓ(a0)=ω′ℓ(a0)ω′ℓ(a0→0)bpk⋅n′ℓk⋅p. (44)

The number of subpeaks in a bunch is proportional to the pulse length and the intensity . The highest subpeak takes its maximum value at a higher frequency than predicted by (43), thus, at a smaller intensity-dependent red-shift than the monochromatic harmonics due to a lower average . Hence, one could say that this maximum is blue-shifted as compared to the monochromatic plane wave.

Increasing from to does not lead to an accumulation of spectral weight at the non-linear Compton frequencies as could be expected naively. The number of subpeaks increases but the average shape of the harmonic bunch is more or less the same for and with the same spectral width. In fact, to obtain the monochromatic limit, it is not efficient to take simply the limit . A method with better convergence is to introduce a flat-top area in the pulse. This however, introduces a second pulse length parameter: The total pulse length now consists of the rise ”time” and the flat-top ”time” . The flat-top part of the pulse is parametrized as , where a factor is introduced so that is comparable to the Gaussian and cosh widths in terms of laser oscillations under the envelope, and is the Heaviside step function. Then, the complete pulse is parametrized as

 gtot(ϕ;σ,τ) = Θ(ϕ+πτ)Θ(πτ−ϕ)+g(ϕ−πτ)Θ(ϕ−πτ)+g(ϕ+πτ)Θ(−ϕ−πτ). (45)

The spectrum converges rather fast to sharp peaks centered at the non-linear Compton frequencies upon increasing from to while keeping constant, as seen in the bottom panel of Fig. 6: The strengths are located at the sharp non-linear Compton energies. The remaining wiggles around the non-linear Compton energies vanish upon increasing further.

In the monochromatic limit , the rising and trailing edges of the pulse shape function become unimportant, i.e. , and the function in (24) reduces to

 H+=12(k′−+q′−−q−)x++α1sinωx+−α2cosωx+−bp−bp′2(cos2ξ−sin2ξ)sin2ωx+ (46)

with and re-identifying the electron quasi-momenta . Upon plugging (46) into (15) and expanding into a Fourier series, one obtains a fourth energy-momentum conservation by integrating over , yielding . The four energy-momentum constraints together lead again to Eq. (43).

The individual harmonics, consisting of a multitude of subpeaks, begin to overlap if the lower edge of the st harmonic coincides with the upper edge of the th harmonic, i.e.

 ω′ℓ(a0→0) ≥ ω′ℓ+1(a0). (47)

This happens always for sufficiently large values of and . The notion of individual harmonics becomes inappropriate, as one rather observes a continuous spectral distribution.

### iv.2 Comparison with Thomson scattering

There are different bookkeeping parameters for the characterization of the Thomson regime as limiting case of the presently considered scenario. One parameter is introduced in Eq. (2). An alternative would be to employ the outgoing momenta instead of the incoming ones, defining with . When four-momentum conservation holds, both definitions coincide (since and both depend on ) and coincides with the usual Mandelstam variable . However, this is not the case here. These recoil parameters are compared in Fig. 7. The parameter is a function of , as it depends on through

 ^y = 2p′⋅k′m2=2m2(p⋅k′)(p⋅k)(p⋅k−k′⋅k)=2γ2ω′(1−n′⋅β)mγ−ω′1−n⋅n′1−n⋅β, (48)

which

diverges at , defining the boundary of phase space. Thus the physical phase space is given by , and An interpretation of the phase space boundary will be given in subsection IV.3.

To relate the Compton amplitude with the classical Thomson counterpart it is instructive to consider the phase exponential, e.g.  in , cf. Eq. (25). For the sake of simplicity, a backscattering head-on geometry with a circularly polarized laser is assumed in this subsection. Then, after using some light-cone algebra, the phase reads

 H+±ωx+ = [(k′−+p′−−p−)±ω]x+−(m2a202k⋅p−m2a202k⋅p′)ωx+∫dϕg2(ϕ) (49) = [k′⋅pn