Scattering of strong electromagnetic wave by relativistic electrons: Thomson and Compton regimes

Nuclear Instruments and Methods in Physics Research A 851 (2017) 82–91

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Nuclear Instruments and Methods in Physics Research A journal homepage: www.elsevier.com/locate/nima

Scattering of strong electromagnetic wave by relativistic electrons: Thomson and Compton regimes

MARK

⁎

A.P. Potylitsyna,b, , A.M. Kolchuzhkinc a b c

National Research Tomsk Polytechnic University, 634050 Tomsk, Russia National Research Nuclear University MEPhI, 115409 Moscow, Russia Moscow State University of Technology “STANKIN”, 127994 Moscow, Russia

A R T I C L E I N F O

A BS T RAC T

MSC: 65C05

The processes of the nonlinear Compton and the nonlinear Thomson scattering in a field of intense plane electromagnetic wave in terms of photon yield have been considered. The quantum consideration of the Compton scattering process allows us to calculate the probability of a few successive collisions k of an electron with laser photons accompanied by the absorption of n photons (nonlinear regime) when the number of collisions and the number of absorbed photons are of random quantities. The photon spectrum of the nonlinear Thomson scattering process was obtained from the classical formula for intensity using the Planck's law. The conditions for which the difference between the classical and the quantum regimes is manifested was obtained. Such a condition is determined by a discrete quantum radiation mechanism, namely, by the mean number of photons k emitted by an electron passing through the laser pulse.

Keywords: Compton Scattering Thomson scattering Nonlinear processes

1. Introduction During the last years, a few accelerator facilities have been developing new sources of mono-chromatic X- and γ-rays which are based on the Compton/Thomson backscattering (CBS/TS) processes [1–17]. One of the main purposes of the ELI-NP project is to produce an intense γ-beam with the energy =ω0 ∼ 10 MeV and a monochromaticity much less than 1% [18] using tight collimation with an opening angle θc ⪡γ0−1 (γ0 is the Lorentz-factor of initial electrons). In order to get the intensity of such a collimated γ-beam which can also be used for different applications, there should be used an intense laser pulse or, in other words, “the strong wave” which can be characterized by a value of the dimensionless laser strength parameter a 0 :

a0 =

e E0 . mc ω0

(1)

In (1), e and m are the charge and the mass of an electron, c is the speed of light, E0 is the laser field, ω0 is the radiation frequency. For the strong wave, such a parameter can achieve the value compared with unity (or more). The well-known process of the intense laser radiation scattering by free electrons in the classical electrodynamics is described as the nonlinear Thomson scattering (TS). The same process in the quantum electrodynamics is treated as the nonlinear Compton scattering (CS).

⁎

In the former case, the electron oscillates in the laser wave and continuously emits radiation consisting of n harmonics (n ≥ 1). In the latter case there is a discrete process of the photon emission in which an electron “absorbs” n ≥ 1 laser photons with an energy ℏω0 and emits “hard” photon only with an energy =ω ∼ n=ω0 . Many authors have considered both radiation mechanisms (see, for instance, [19–22] and cited papers there). The drastic difference between TS and CS regi- mes occurs for ultrashort strong laser wave a 0 ≥ 1, N0 ∼ 101, N0 is the number of cycles in the laser pulse (see, for instance, [19,20]). For “long” laser pulses (a 0 ∼ 1, N0 ≥ 10 2) the spectral distributions of emitted photons for both mechanisms are similar and the difference is in the so-called “redshift”. Such a red-shift characterizes the decrease of the energies of the spectral maximum for the nonlinear Compton scattering in comparison with the Thomson scattering due to the quantum recoil effect. The authors of the paper [21] compared both nonlinear mechanisms and obtained the following conditions providing a coincidence of results from the Compton and the Thomson mechanisms:

n

4γ0 =ω0 mc 2 (1 + a 02 )

⪡1.

(2)

Here, n is the number of absorbed photon, =ω0 is the energy of the laser photon. It means that an energy of the laser photon in the rest system

Corresponding author at: National Research Tomsk Polytechnic University, 634050 Tomsk, Russia. E-mail address: [email protected] (A.P. Potylitsyn).

http://dx.doi.org/10.1016/j.nima.2017.01.055 Received 6 July 2016; Received in revised form 29 December 2016; Accepted 24 January 2017 Available online 26 January 2017 0168-9002/ © 2017 Elsevier B.V. All rights reserved.

Nuclear Instruments and Methods in Physics Research A 851 (2017) 82–91

A.P. Potylitsyn, A.M. Kolchuzhkin

(where the electron is in a rest in average) is much less than the electron mass and, accordingly, a quantum recoil effect does not change the energy of scattered photons practically. In the paper [21] the authors claimed that for ultrashort femtosecond laser pulses and γ0 ≤ 10 3, a 0 ∼ 1 “the classical and quantum results are practically the same” if the condition γ0 =ω0 / mc 2⪡1 is fulfilled. Seipt and Kämpfer in the work [19] found that spectral distributions for radiation at a fixed angle consisting of a set of sub-peaks are determined by the temporal shape of the ultrashort laser pulse and differ for both mechanisms. The energy of scattered photons will be strictly defined by the emission angle and the parameter a0 only for long pulses (N0 ⪢10 2 ). They showed that Compton and Thomson crosssections practically coincide if the condition (2) is fulfilled. Also, they noticed that for a 0 ⪢1 these cross-sections will be different although the above-mentioned condition is kept. In our work, we concentrated our attention on the discrete process of photon emission which can't be considered in the classical approach. Below we shall show that the process of emission of a few photons by each electron during successive interactions with laser photons (multiphoton emission [24] or multiple Compton backscattering [25]) can only be considered in the quantum approach. Such a multiple backscattering process leads to a significant difference between the Compton and the Thomson spectra even if the condition Eq. (2) is fulfilled. We consider the multiple Compton backscattering process as the independent emission of photons because the formation length of the emitted photon ∼γ 2λ [26] is much less than a length of the laser pulse cτ . For an electron energy ∼103 MeV and an energy of the emitted photon ∼10 MeV (λ ∼ 10−7 um ) the formation length achieves macroscopic size 0.5 µm but for a laser pulse duration τ ∼ 1 ps the pulse length is much longer than the formation length. The paper is organized as follows: the Section 1 is devoted to the Introduction. In Section 2, we consider the nonlinear Thomson scattering process in the classical electrodynamics frame and obtain the semi-classical emitted photon spectrum from the classical expression for the intensity spectrum using the Planck's law. The consideration of the nonlinear Compton backscattering process as the typical quantum process was performed in Section 3 on the base of formulas for total cross-sections and for the luminosity. The differential crosssections characterizing the nonlinear Compton scattering were obtained in Section 4 for small values of the laser strength parameter a0 and by taking into account the number of absorbed laser photons n = 1, 2, 3, …. In Section 5 we describe the Monte-Carlo algorithm for simulation of the nonlinear Compton scattering process where an electron absorbs n laser photons and in each collision emits k = 0, 1, 2, … hard quanta (multiple Compton backscattering). The main results obtained after comparing both approaches considered in the paper, are discussed in Section 6. The final conclusions are presented in Section 7.

ω(n) = n

4γ02 ω0 1 + (γ0 θ )2 + a 02 /2

.

(3)

It is convenient to use a dimensionless spectral variable:

S (n ) =

ω(n) n = , 4γ02 ω0 1 + (γ0 θ )2 + a 02 /2

(n ) 0 ≤ S (n) ≤ Smax ,

(n ) Smax = n /(1 + a 02 /2).

(4)

The spectral distribution of emitted photons of an electron passing the laser field for the fixed harmonic number n can be obtained from the classical intensity spectrum dividing one of the emitted photon energy =ω [22]. 2 ⎧ ⎫ [n − S (n) (2 + a 02 )] dN (n) = 2παa 02 N0 × ⎨ (n) 2 Jn2 (nz ) + Jn2′ (nz ) ⎬. 2 ( n ) n ( ) dS ⎩ 2S a 0 [n − S (1 + a 0 /2)] ⎭ ⎪

⎪

⎪

⎪

(5) In the formula (5) N0 is the number of electron oscillations along its trajectory governed by a laser field, the Bessel function and its derivative of order n, are denoted by Jn(x), J ′n (x ), were argument nz is

nz =

2

2 a 0 S (n) n − S (n) (1 + a 02 /2) .

The number of photons emitted at the n-th harmonic can be calculated after integration of the spectrum (5): (n )

N (n ) =

∫0

Smax

dN (n) (n) dS . dS (n)

(6)

It is evidently that such a value is determined by the field strength parameter a0 and the number of cycles only. In Fig. 1 we show the dependence of the emitted photons number N (n) per an electron and per an oscillation (N0 = 1) on the parameter a0 (for n=1, 2, 3). The total number of emitted photons can be found by summing up the contributions from the n-th order harmonic number (6). ∞

Ntot =

∑ N (n ) .

(7)

n =1

In order to realize the summation procedure, the maximal harmonic number n max , which depends on the parameter a0, should be chosen. For this purpose we suggest the following criterion:

N (nmax +1) < 0.0005, Ntot

(8)

where n max

Ntot =

∑

N (n ) . (9)

n =1

Such a choice provides a relative accuracy in the calculation of the value Ntot not more than 1–2%. The dependence of the total number of emitted photons Ntot on the laser field strength is shown in Fig. 2, where

2. Photon flux from the nonlinear Thomson scattering process Let us write the known formulas for the nonlinear TS for a strong circularly-polarized laser wave scattering on a counterpropagated relativistic electron. The incoherent sum of two circularly polarized waves with equal intensities and opposite helicities gives the result, coinciding with an unpolarized beam. So our consideration can be extended for unpolarized radiation because a spectral-angular distribution of scattered photons does not depend on a helicity of the initial wave. Below, we consider the radiation of an electron in the field of the strong monochromatic wave only. In this approach, one can use the well-known relation connecting the frequency of n-th harmonic and emission angle [20,22].

0

Fig. 1. Dependences of the emitted photon number N (n) for the different harmonic number (n = 1, 2, 3) on the strength field parameter a 0 per a cycle of the wave.

83

Nuclear Instruments and Methods in Physics Research A 851 (2017) 82–91

A.P. Potylitsyn, A.M. Kolchuzhkin

Fig. 2. Dependence of the total number of emitted photons vs parameter a 0 (points) and the quadratic approximation of this dependence (solid curve) calculated per a cycle of the wave.

Fig. 4. Spectral distributions for the Thomson radiation regime with a different nonlinear parameter a 0 . Dependencies on dimensionless spectral variable S (left axis) and on energy units (right axis), calculated for E0 = 360 MeV ; λ 0 = 0.52 um .

the approximation of such a dependence is also presented: app Ntot (a 0 ) = 0.654παa 02.

(10)

For the range a 0 ≤ 1 there is a quadratic rise but with an increasing of the parameter a0 the growth of the photon number Ntot becomes slower. For a large value of a0 (much higher than unity) the trajectory of an electron under a circularly polarized wave will be a helix. Keeping the analogy between radiation generated in such a wave and in a helical wiggler one can estimate the number of emitted photons per turn:

Nw (a 0 ) =

5 5 πα 1 + a 02 /2 ≈ παa 0 . 3 6

(11)

As one can see from Figs. 2, 3 for a 0 ≤ 1 there is the quadratic dependence of the total photon number of this parameter but for a 0 ≥ 2 such a dependence is closed to the linear one. It means that the number of photons grows linearly with increasing of a0 if a 0 ⪢1. The comparison of results of calculation using the formula (9) with the estimation (11) is given in Fig. 3, where one can see the reasonable agreement. The spectral distribution summed over all harmonics can be found using the following expression:

dNtot = dS

n max

∑ n =1

dN (n) , dS (n)

0≤S≤

n max . 1 + a 02 /2

Fig. 5. Spectral distributions (12) (upper curves) calculated for a 0 = 0.5 and a 0 = 0.25 in comparison with approximation (13) (lower curves).

{} ≈

1 [1 − 2S (1) + 2(S (1) )2 ]. 2

(13)

Fig. 5 shows spectra obtained from Eqs. (5), (13) for a0=0.25; 0.5. Evidently, that for a 0 ⪡1 one can use the simple approximation (13) instead of Eq. (5) to describe the spectral distributions.

(12)

The spectral distribution of the photon beam after collimation

(n ) dNcoll

dS (n)

with aperture θc ∼ γ0−1 can be calculated using the same formula (5) for the following limits for variable S (n):

Spectral distributions calculated from Eq. (12) for different parameters a0 are presented in Fig. 4. Instead of the dependence on dimensionless variable S we show, as an example, the spectra, depending on the photon energy =ω(n) obtained from Eq. (3) for the following parameters: the electron energy E0=360 MeV, laser wavelength λ 0 = 0.52 um . The maximal photon energy for the linear case =ω lin = 4γ02 =ω0 = 4.798 MeV is shown by an arrow. For small parameters a 0 ⪡1, the expression in the brackets of Eq. (5) can be expanded on the power a 02n and keeping the lowest term we get:

(n ) (n ) Smin ≤ S (n) ≤ Smax ,

(n ) Smin =

n 1 + (γ0 θc )2 + a 02 /2

(14)

and, accordingly, for photon energy: (n ) (n ) =ωmin ≤ =ω(n) ≤ =ωmax ,

(n ) =ωmin =n

4γ02 =ω0 1 + (γ0 θc )2 + a 02 /2

.

(15)

on the aperture angle θc The dependencies of photon number for different values a0 are shown in Fig. 6. The effect of collimation with an aperture θc ∼ γ0−1 are shown in Fig. 7. One can see that the intensity of collimated beams is increased very close to a linear law with a0 parameter rise. However, for the range a 0 ⪡1, such a behavior can be approximated by the quadratic dependence (see Fig. 8). Spectra of collimated radiation are shown in Fig. 9. Even for strong collimation, there are the contributions from higher harmonics. Such contributions lead to an appearance of additional peaks in a “hard” part spectrum but not in a “soft” part. With the increasing of a nonlinearity parameter there occurs the red-shift of the spectral line with a bandwidth of the main line: (n ) Ncoll

(γ0 θc )2 ▵=ω(1) ∼ . (1) =ω 1 + (γ0 θc )2 + a 02 /2

(16)

The total number of emitted photons is determined by the squared

Fig. 3. Comparison of the linear approximation Ntot (a 0 ) with simulation results.

84

Nuclear Instruments and Methods in Physics Research A 851 (2017) 82–91

A.P. Potylitsyn, A.M. Kolchuzhkin

Fig. 6. Dependence of photon yield Ncoll in aperture angle θc on parameter a 0 .

Fig. 9. Contributions to the collimated spectrum from different harmonics (n = 1, 2, 3) for E0 = 360° MeV ; λ 0 = 0.52 um ; a 0 = 1; γθc = 1.0 . The spectrum for n = 2 is multiplied by 4, for n = 3 by 8.

intensity I0 (power per unit area) as [22]:

a 02 = 2I0 λ 02 re / πmc 3,

(17)

where λ 0 is the laser wavelength, re is the classical electron radius. The quantum consideration gives the following expression for the same parameter:

a 02 = 4αƛe2λ 0 n 0 .

(18)

In the Eq. (18) α = 1/137 is the fine structure constant, ƛ e = ℏ/ mc is the Compton wavelength of electron, n 0 is the concentration of laser photons in an interaction point. The concentration n 0 can be estimated for a laser pulse with the total energy W0 , duration τL , focused to the area with the radius w0 in the following manner:

Fig. 7. Dependence of photon yield Ncoll on aperture angle θc for different parameter a 0 .

n0 ≈

W0 1 . =ω0 πw02 cτL

(19)

The mean number of emitted photons by each electron k when passing through an intense laser pulse can be found by calculating the luminosity L . In our case, this quantity characterizes the interaction between counterpropagating electron and laser beams, each of which is described by its four-dimensional distribution in the vicinity of a collision point. The luminosity for the head-on electron-photon collision is determined by the expression [9,24]: Fig. 8. Dependence of photon yield Ncoll on aperture angle θc in comparison with quadratic approximation of its (a 0 = 0.25).

L = c (1 + β0 ) Ne NL

a 02

×

∫ dVdtFe (x, z, y, t ) FL (x, y, z, t ).

(20)

nonlinearity parameter and number of oscillations N0 (see Eqs. (5), (10)), because in this case there is no dependence on the electron energy. Such a consideration in the classical electrodynamics frame is performing for the constant energy of the initial electron (neglecting by an energy loss in the emission process). In order to compare the results obtained from Eqs. (5), (10) with calculations based on quantum consideration of the interaction of relativistic electrons with counterpropagated circularly polarized wave, one has to use the additional factor 2 to take into account the difference between a number of the laser pulse periods and a number of oscillations [23]. As was shown in the work [22] such a multiplier is connected with “… the single electron-laser interaction time …cT = λ 0 N0 /(1 + β0 )…”. Below we consider the emission process in the frame of the quantum electrodynamics.

In Eq. (20) the speed of electrons of the monoenergetic beam is expressed as β0 c , Ne (NL ) is the total number of electrons (photons) in a bunch, Fe (FL ) is the distribution of electrons (photons) in bunches normalized to unity. For the sake of simplicity let's consider the simplest distributions of azimuthally symmetric colliding beams – 3D Gaussian distributions describing an electron bunch and uniform distribution of laser photons in a cylinder with radius w0 and length ℓL (see, for instance, [17,18]):

3. Quantum characteristics of interaction of electron bunch with intense laser pulse

Here ρb is the radius of the electron bunch at focus, σez is the rms length of the bunch. For head-on colliding beams integration over z, t in Eq. (20) can be done easily using substitution:

The field strength parameter a 0 can be determined using the laser 85

Fe (ρ , z ) =

1 2πρb2

FL (ρ , z ) =

1 1 (ρ ≤ w0, z πw02 ℓL

× exp{−ρ2 / ρb2 } ×≤

1 π σ 2 ez

⎛ z2 ⎞ exp ⎜ − 2 ⎟ , ⎝ 2σez ⎠

⎞ ℓL , ℓL = cτL ⎟ . ⎠ 2

(21)

Nuclear Instruments and Methods in Physics Research A 851 (2017) 82–91

A.P. Potylitsyn, A.M. Kolchuzhkin

⎧ z2 ⎫ ⎧ (z − β c t)2 ⎫ 0 ⎬. Fe (z, t ) ∼ exp ⎨− 2 ⎬ ⇒ exp ⎨− 2σez2 ⎩ 2σez ⎭ ⎩ ⎭ ⎪

1 FL (z, t ) ⇒ , lL

⎪

⎪

In the simplest case ρb ⪡w0 , one can obtain:

ℓ ℓ − L ≤ z + ct ≤ . 2 2

k =

Changing the variable z by z1 from the condition, z + ct = z1 after integration over t and then over z1 in the limits −(lL /2 ≤ z1 ≤ ℓL /2), one can obtain ∫ Fe (z, t ) FL (z, t ) dzdt = 1. It means that for the case under consideration the luminosity does not depend on the lengths of colliding bunches. The final radial integration in Eq. (20) gives the following result:

⎛ ⎛ w2 ⎞⎞ NN L = 2 e 2L ⎜⎜1 − exp ⎜⎜ − 02 ⎟⎟ ⎟⎟ . πw0 ⎝ ⎝ 2ρb ⎠ ⎠

1 = 2Ne n 0 ℓL , πw02

where n 0 =

NL πw02 ℓL

k = (22)

(23)

is the concentration of laser photons in the cylindrical

Lσ = 2n 0 ℓL σ . Ne

(24)

Evidently, formulas (23), (24) are valid if the electron energy decrease along its trajectory is negligible. In order to take into account such a decrease (and, subsequently, the cross-section changing) we introduced the probability of interaction per path unit ∑ = 2n 0 σ . In real experiments, the transverse distribution of a laser beam is described through Gauss-Hermite polynoms, and longitudinal distribution is determined by time structure of an optical system. Transverse size of an electron beam is determined by emittance and beta-function βf of accelerating section upstream a collision point, whereas longitudinal distribution is determined by a source and accelerator systems. In the work [24], the authors carried out the scattered photons intensity calculations for the Thomson linear scattering in a more realistic model where radial distributions of both beams are defined by expressions:

⎡ ⎤ ρ2 ⎥, Fe (ρ , z ) ∼ exp ⎢ − 2 2 2 ⎢⎣ ρb (1 + z / βf ) ⎥⎦

a 02 α σ (a ) σN0 = a 02 N0 2 0 . 2 2αλe2 re

(28)

4. Differential cross-section and kinematics of the nonlinear Compton scattering The cross-section of the nonlinear Compton scattering process is defined by the kinematics of the process (Lorentz-factor of electrons γ0 , the energy of laser photons =ω0 , and collision geometry, furthermore, the head-on geometry only) and its dynamics (laser strength field parameter a 0 ). The following invariant kinematical variable x 0 for headon collisions will be used [27]:

x0 =

⎡ ⎤ ρ2 ⎥. FL (ρ , z ) ∼ exp ⎢ −2 2 2 2 ⎣ w0 (1 + z / zR ) ⎦

2(1 + β0 ) γ0 =ω0 4γ =ω0 ≈ 0 2 . mc 2 mc

(29)

For a monochromatic wave, the energy of the scattering photon =ω(n) determined by the photon outgoing angle θ (relative to the electron momentum) and depends on the number of “absorbed” laser photons n and parameter a 0 as following:

(25)

1/ e 2

of focal radius, zR is the Rayleigh length. Here w0 is the Longitudinal distributions of the beams were approximated by Gaussians with parameters ℓL and ℓe . In this approximation, the mean number of scattered photons per an electron is determined by the following expression:

k =

(27)

Here N0 = ℓL / λ 0 is the number of cycles in a laser pulse. Generally speaking, in the expression (28) one should use the crosssection σ averaged over the electron energy distribution which occurs due to electron energy losses during each interaction. However, for many important cases (for instance, k ≤ 1 ) the crosssection remains practically unchanged (σ ≈ σ (a 0 )). For picosecond laser pulses with the number of cycles N0 ⪢10 2 , the mean number of collisions can exceed unity (k > 1), even for a weak wave a 0 ⪡1. It means that each electron can emit a few “hard” photons (k > 1) with probability P (k , k ) > 0 . For a strong wave, the process of emission of a number of photons by an electron becomes much more is probable. Such a process (the multiple Compton backscattering process) has to be treated as a discrete quantum process only [25]. In this paper we'll consider the scattering process in the field of the “long” laser pulse only (picosecond laser pulses) for which the approximation of monochromatic wave can be used with high accuracy.

pulse. The mean number of photons, emitted by each electron (a mean number of collisions), can be found from the known luminosity:

k =

r2 8/3πre2 16 NL 2 e 2 ≈ 2NL = 2n 0 σT ℓL , 3 ρL + ρb πρL2

where σT = 8/3πre2 is the Thomson cross-section. The obtained results coincide with Eq. (24), however, instead of the Thomson cross-section, one has to use the cross-section of the nonlinear Compton scattering σ depending on the parameter a 0 . Using Eq. (18), it is possible to obtain the following expression for the quantity k from (24):

In the case ρb ≤ w0 , it is possible to neglect by the dependence on the transversal electron beam size:

L ≈ 2Ne NL

1 exp(−x 2 ) . π x

Φ (x ) ≈ 1 −

⎪

=ω(n) = n

γ0 mc 2x 0 1 + nx 0 + (γ0 θ )2 + a 02 /2

.

(30)

Maximal positions on the energy scale are determined by the relation Eq. (30) for θ = 0 . As the spectral variable we will use the dimensionless one:

16 π re2 NL f (ℓe, ℓL , r ), 3 ρ02

=ω(n) . γ0 mc 2

where

y (n ) =

f (ℓe, ℓL , r ) = exp[(1 + 2r 2 )/(μ2 + 2r 2η2 )]/[(μ2 + 2r 2η2 (1 + 2r 2 )]1/2 ℓL × {1 − Φ [(1 + 2r 2 )1/2 /(μ2 + 2r 2η2 )1/2 ]}, μ = , 2 2 zR ρ ℓe η= , r= b, w0 2 2 βf (26)

The “partial” differential cross-section of the CBS process depends on x 0 , y(n) and can be obtained in the QED frame [27]:

4πr02 dσ (n) = dy(n) x 0 a 02

⎧ ⎞ a2 ⎛ 1 × ⎨−4Jn2 + 0 ⎜1 − y(n) + ⎟ 2 ⎝ 1 − y (n ) ⎠ ⎩

×(Jn2−1 + Jn2+1 − 2Jn2 )}.

Φ (x ) is the error function. In the limit zR → ∞, βf → ∞ (neglecting by beam divergences) and using the asymptotic expression for the error function

(31)

In Eq. (31), Jm is the Bessel function of order m = n − 1, n , n + 1 depending on the argument 86

Nuclear Instruments and Methods in Physics Research A 851 (2017) 82–91

A.P. Potylitsyn, A.M. Kolchuzhkin

zn =

2 na 0

⎡ y(n) (1 + a 02 /2) ⎤ ⎥. × ⎢1 − (1 − y(n) ) nx 0 ⎦ ⎣

y (n ) (1 − y(n) ) nx 0

of powers of a 02n and x n . Keeping the lowest terms we get: (32)

⎧⎛ 8 ⎛ 28 8 52 2⎞ 58 σ (1) = πre2 ⎨ ⎜ − x 0 + x 0 ⎟ +a 02 ⎜ − + x0 − ⎝ 15 ⎠ ⎝ 3 15 15 ⎩ 3 ⎫ ⎧ ⎛8 16 x0 + + a 04 (−x 0 + 4x 02 ) ⎬, σ (2) = πre2 ⎨a 02 ⎜ − 5 ⎭ ⎩ ⎝5

y (n )

After integration of Eq. (32) over in the limits (n ) 0 ≤ y(n) ≤ ymax = nx 0 /(1 + nx 0 + a 02 /2), one can obtain the partial crosssection of the nonlinear CBS process describing the process with “absorption” of n laser photons: (n )

σ (n ) =

∫0

ymax

dσ (n) (n) dy . dy(n)

∞ n =1

For this case, the difference between cross-section σ (1) and crosssection of the linear CBS process σlin = σtot (a 0 → 0) ≈ σT is less than 3%. For such a process if x 0 ⪡1 instead of Eq. (28) one can use the most simple formula [29]:

k =

(34)

We will perform the calculations for the maximal number n max defined by the same criterion as Eq. (8):

σ (nmax +1) / σtot < 0.0005.

4 απa 02 N0 = 1.33παa 02 N0 . 3

(38)

The numerical factor in Eq. (38) coincides with the doubled one from Eq. (10) as expected. The differential cross-section

σ (n).

n =1

(37)

⎛ 81 243 6561 2⎞ σ (3) = πre2 a 04 ⎜ − x0 + x 0 ⎟. ⎝ 70 70 560 ⎠

n max

∑ σ (n ) ≈ ∑

264 2⎞ x0 ⎟ 35 ⎠

⎛ 248 664 5872 2⎞ ⎫ +a 04 ⎜ − + x0 − x0 ⎟ ⎬, ⎝ 105 105 315 ⎠ ⎭

(33)

Below we show that for moderately strong wave (a 0 ∼ 1) spectral photon distributions from the nonlinear Compton scattering simulated for the number of absorbed laser photons n = 1, 2, 3 quantitatively coincide with the same distributions obtained for the harmonic number n = 1, 2, 3 from the Thomson scattering with accuracy better than 10−2 . We suppose there is a close analogy between both considered processes and we used the same index n considering the Thomson and the Compton processes. The total CBS cross-section can be obtained by summation of Eq. (33) over all possible numbers of absorbed photons:

σnon lin = σtot =

248 2⎞ x0 ⎟ 35 ⎠

dσtot = dy

(35)

nmax =8

∑ n =1

dσ (n) dy(n)

(39)

In this case, the final accuracy for calculations of the total cross-section doesn't exceed one percent. Knowing the total cross-section σtot in Eq. (34) and partial ones σ (n) Eq. (33), one can calculate probabilities of the CBS process with the absorption of the fixed number of laser photons n :

is shown in Fig. 11 for conditions: E0 = 50 MeV ; =ω0 = 1.17 eV ; x 0 = 0.0009; a 0 = 1. The number of terms in the sum (39) was chosen from the criterion (35). In order to obtain the differential cross-section (39) depending on the scattered photon energy =ω(n), one has to use the relation:

Pnon lin (n ) = σ (n) / σtot .

=ω(n) = γ0 mc 2y(n) .

(36)

σ (n )

Fig. 10 presents the dependence of partial cross-sections on a number of absorbed photons n = 1, 2, 3… calculated for the initial electron energy E0 = 50 MeV and the laser photon energy =ω0 = 1.17 eV (x 0 = 0.0009) for different parameters a 0 . As one can see from the Fig. 10 contribution of cross-sections σ (n) (n ≥ 3) in the total cross-section does not exceed 1% for a 0 = 0.2 . Up to now all projects to design Thomson/ Compton sources and its constructing based on electron accelerators are used laser systems with a weak nonlinearity (a 02 ⪡1). In such projects, electron and laser photon energies correspond to the condition x 0 ⪡1. In this case it is possible to obtain the simple formulas for partial cross-sections instead cumbersome expressions Eq. (33) where the integrand was expanded in series

(40)

With the parameter a 0 increasing, the number of summands in Eq. (34) is growing very quickly. Fig. 12 presents the ratio of the cross-section σtot to the Thomson one depending on a 0 (x 0 = 0.0009; x 0 = 0.05; x 0 = 0.2). For instance, the criterion (35) is fulfilled for a 0 = 3.0 and x 0 = 0.05 if n max = 54 . The relative red-shift of the first maximum due to recoil effect can be estimated from the simple formula (x 0 ⪡1)

Δ=

(1) (1) =ωmax Thom − =ωmax Comp (1) =ωmax Thom

≈

x0 , 1 + a 02 /2

(41)

which gives value Δ = 0.03 for x 0 = 0.05 and a 0 = 1.

Fig. 10. Dependence of the nonlinear Compton partial cross-sections on the number of “absorbed” photons n for different parameters a 0 (E0 = 50 MeV ; =ω0 = 1.17 eV ; x 0 = 0.0009 ).

Fig. 11. The energy dependence of the Compton cross-section for the same parameters as in Fig. 10 for a 0 = 1 (per unit energy interval – right axis).

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Fig. 13. Photon spectra from Thomson scattering (solid curve) and Compton scattering (points). Both spectra were obtained for parameters: E0 = 360 MeV ; λ 0 = 520 nm ; a 0 = 1; N0 = 36 . Calculations of the Thomson spectrum and the Compton cross-section were performed for n ≤ n max = 11.

Fig. 12. The total cross-section of Compton scattering depending on the parameter a 0 for different kinematical cases (■ – E0 = 50 MeV ; =ω0 = 1.17 eV ; ● – E0 = 2044 MeV ; =ω0 = 1.6 eV ; ▴ – E0 = 5600 MeV ; =ω0 = 2.34 eV ).

6. Results and discussion 5. Simulating the multiple nonlinear Compton scattering by the Monte Carlo technique

The photon spectrum calculated using analytical formulas for the Thomson regime Eqs. (5), (12) in comparison with results of MonteCarlo simulation for the Compton regime is presented in Fig. 13. The spectrum was found for the following parameters:

The passage of a relativistic electron through an intense laser pulse (light target) is a stochastic process, where the electron free path between successive collisions with laser photons, the number of photons absorbed in each collision, and the energy of emitted quantum are random quantities. The probabilistic characteristics of these random quantities can be expressed in terms of known cross sections and this enables us to simulate the random collisions along the electron path by the Monte Carlo technique. At the first stage we determined the maximal number of absorbed laser photons n max from the criterion Eq. (35) for the fixed parameters x 0 and a 0 , calculate the cross-section σtot (x 0 , a 0 ) Eq. (34) and probabilities Pnonlim (n ) of absorption of n ≤ n max photons (see, Eq. (36)). The electron free path ℓ between successive collisions with laser photons obeys the probability distribution of exponential form

w (ℓ) =

E0 = 360 MeV;

λ 0 = 520 nm; a 0 = 1.0;

τL = 0.06 ps;

N0 = 36.

Calculation of the total Compton cross-section for such a process using Eq. (34) for x 0 = 0.013 giving the result:

σtot = 2.404 πre2, which allows to estimating the mean number of emitted photons k (see Eq. (28)): k = 1.00 . The criterion (35) is empirical which was chosen as compromise between accuracy of calculations and the required time. We took into account the absorption of n photons in each collision (n ≤ n max = 8). We calculated probabilities of the emission of k photons Psim (k , k ) by an electron for k = 1 which coincide with the Poisson law with accuracy better than 10−3. It should be mentioned that there is the non-zero probability for an electron passage through a laser pulse without interaction with laser photons (Psim (k = 0, k = 1) = 0.368). Calculating the number of photons for the Thomson spectrum from Eqs. (5), (9) for the number of oscillations 2N0 we get Ntot = 1.007. The coincidence of the simulated value of k and the calculated quantity Ntot with accuracy better than 10−2 confirms the adequacy of Thomson and Compton approaches for calculation of the total photon yield for conditions x 0 ≤ 1, a 0 ∼ 1 and k ∼ 1. However, the more detailed comparison of both approaches for collimated γ -beams should be done because of the required monochromaticity ▵=ω / =ωmax of such beams would be about a few percent level at least. In order to achieve the required value, a beam collimation with angular aperture θc ⪡γ0−1 should be used. For instance, the aperture angle θc = 0.15γ0−1 will determine the bandwidth around 2% (see Eq. (16)). Collimated spectra calculated for the Thomson regime are shown in Fig. 9. For the case E0 = 360 MeV , a 0 = 1.0 , the simulation of the Compton collimated spectrum with aperture angle θc = 0.001 (γ0 θc = 0.7) is shown in Fig. 14. We simulated the spectrum for n = 1, 2 only. Despite a weak discrepancy with the Thomson spectrum shape (essentially for n = 2 ), both regimes give practically the same (2) (1) (2) (1) result (Ncoll = 0.045; kcoll = 0.199; kcoll = 0.044 ). = 0.200 ; Ncoll Fig. 15 presents results of simulation and calculations of Thomson spectra for γ0 = 300 , θc = 5·10−4 (γ0 θc = 0.15) for different values of the parameter a 0 . For all spectra we keep the mean number of scattered photons k = 1.00 . Accordingly to Eq. (28) for the condition k = const there was chosen the duration of the laser pulse from τL = 0.5 ps (N0 = 300) for a 0 = 0.35 and higher. The number of oscillations to calculate the Thomson spectrum of the same parameters was chosen as

∑ Exp(− ∑ l),

where ∑ = 2n 0 σtot is the macroscopic cross section of interaction. Simulating the random quantity ℓ is carried out using the wellknown formula ℓ = −1/ ∑ ln η , where η is the random number uniformly distributed within the interval (0, 1). Then we simulate the number of absorbed laser photons n knowing probabilities Pnonlim (n ). Where the number of absorbed photons is fixed, the simulation of a random energy of emitted quantum is made up by the rejection method [28] using a known differential cross section (31). The energy of a quantum is subtracted from the electron energy and this takes into account the electron energy loss along the trajectory and, subsequently, a growth of the total cross-section. The angle with which the quantum is emitted is determined by the kinematics of the process (30). It can be shown (see [27]), a relativistic electron is scattered in each collision at the angle ∼x 0 / γ0 . It means that in the most interesting case kx 0 ⪡1 one can neglect the angular deflection of the electron and use so-called straight ahead approximation where the electron trajectory is a straight line along its passing through a laser pulse. The phase coordinates of an electron obtained in such a way can be used as the previous ones for simulation of the next step along the trajectory. The detailed description of the algorithm used can be found in [25]. According to this algorithm, the electron is followed until it goes out of the laser pulse with the length ℓL . In a thick light target, each electron may undergo the multiple Compton scattering and while the randomness of all elements of the trajectory reveals in the number of collisions and in the energy of emitted quanta. The developed code enables us to calculate the mean values of these characteristics as well as the energy spectra of electrons. 88

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a) 1 2 3

4

Fig. 16. Collimated photon spectra simulated for the different mean number of emitted photons (k = 1.09 , curve 1; k = 0.75, curve 2; k = 0.5, curve 3; k = 0.25, curve 4). The remained parameters are the same as in Fig. 15.

b)

after the first scattering. Thus, the multiple Compton backscattering processes (the multiphoton emission) leads to an unavoidable the worsening of the bandwidth (see, [25]). It is expected that such an effect has to be manifested with an increasing k and γ0 . In the former case a probability of emission of k ≥ 2 photons will be higher and in the latter one an energy loss in each scattering will grow as ∼γ02 . A distortion of the collimated spectra determined by the growth of the mean number of emitted photons k is illustrated by Fig. 16. Probabilities of multiphoton emission Psim (k ≥ 2, k ) obtained from simulation and sums of Poisson probabilities ∞ Pp (k ≥ 2, k ) = ∑k =2 e−k (k n / k!) coincided with accuracy better than 10−2 . Having in mind the expressions (18) and (19) one can obtain the relation connecting the parameter a 0 and a laser pulse energy W0 :

Fig. 14. Comparison of Monte-Carlo simulation results (Compton scattering) with analytical calculations (Thomson scattering) for the collimated spectrum, presented in Fig. 9b).

a 02 = 2

α ƛ2e λ 02 W0 . π 2 ℏc 2 ω02 τL

Histograms in Fig. 17 were obtained for different energies W0 for fixed remained parameters (=ω0 = 2.34 eV , τL = 0.5 ps, γθc = 0.15). In order to estimate the number of photons with energy out of the spectral line we simulated its shape for an energy spread of the initial electron beam ▵γ0 / γ0 = 0.2%. The average energy of the spectral line and its bandwidth FWHM can be estimated from (15) as

=ω = (=ωmax + =ωmin )/2

≈ 4γ02 =ω0 (1 − a 02 /2 − (γ0 θc )2 /2), FWHM

≈ 4γ02 =ω (γ0 θc )2 , if a 02 ⪡1, (γ0 θc )2 ⪡1. The results of simulations are presented in Table 1. A number of photons with the energy less than the minimal energy of the spectral line n ph (=ω < =ωmin ) was calculated for each value of k and is shown there too. For all values of the parameter a 0 we have obtained the same value of the full width of half maximum (∼0.44 MeV ) as expected. About 3% of the emitted photons only (k coll / k ≈ 0.03) accepted by the chosen aperture (γθc = 0.15). The shape of spectral lines is worsening with an increase of the mean number of emitted photons k . Even for a small multiplicity k ∼ 0.5 about 13% of emitted photons have energies out of the spectral line.

Fig. 15. The same for collimated spectra obtained for different parameters a 0 . The Compton Thomson difference between =ωmax and =ωmax occurs due to recoil effect (see, Eq. (41)). Parameters: γ0 = 300 ; =ω0 = 2.4 eV ; θc = 0.0005; k = 1.

2N0 = 600 . The number of photons in both spectra was almost the same (1) (1) = 0.032 ). At the same figure we present the simulated (kcoll = 0.033; Ncoll Compton spectral line for the “linear case” in comparison with Thomson one. The simulation was done using the procedure for the nonlinear scattering with the nonlinearity parameter a 0 = 0.001. Calculations of the Thomson spectrum were performed using the same Eqs. (9), (15) for the number of oscillations 2N0 = 7.2·107 (determined by the small value of a 0 ). As one can see for all spectra, there is the red-shift due to recoil effect, proportional to x 0 = 0.0058 the quantity of which agrees with estimation (41) good (▵ est = 0.0054 ; ▵sim = 0.0045) and such a value is comparable with the bandwidth. Another important feature which characterizes the spectral line is connected with the appearance of photons out of the spectral line in the soft part of the spectral distribution. This effect occurs due to multiphoton emission. It is evident that the maximal energy of the photon emitted during the second scattering will be less than the same energy

7. Conclusions We showed the one-to-one correspondence between the nonlinear Thomson and Compton photon spectra with a small difference due to such quantum effects as the recoil effect and the multiphoton emission for the case x 0 ⪡1, a 0 ≤ 1. Knowing the parameters of colliding an electron and laser beams, it is possible to estimate the mean number of emitted photons by an 89

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Table 1 Characteristics of the collimated γ -spectra.

a0 W0, J k kcoll

0.05 0.125 0.02 0.00055

0.1 0.5 0.08 0.0022

0.25 3.125 0.5 0.015

nph (=ω < =ωmin ) k coll

0.006

0.03

0.13

photons with the energy 10 MeV via interaction of 150 MeV electrons with an intense laser (energy of photons 1.5 eV, a 0 ≈ 2 ) and observed the “…clear evidence of the onset of non-linear Thomson scattering”. From our point of view such a process has to be treated as a non-linear Compton backscattering process. Our comparison of the nonlinear Thomson and Compton scattering processes allows making the following conclusions. (1) We used the classical approach to calculating the Thomson photon spectra based on the transition from the continuous frequency spectrum to the photon on using the Planck law [22]. The results of such analytical semi-classical calculations coincide with the quantum simulation based on the nonlinear Compton cross-section in general if the conventional condition is fulfilled:

x0 =

4γ0 =ω0 ⪡1. mc 2

(2) The red-shift of the Compton spectrum relative to the Thomson one for collimated γ -beams can be neglected in the case x 0 /(γ0 θc )2 ⪡1 only. In the opposite case such a red-shift is compared with the bandwidth of the beam even for a weak nonlinearity (a 0 < 1). (3) The bandwidth of the collimated photon spectra can be worsened if the condition kx 0 /(γ0 θc )2 ⪡1 will be violated. In this case the continuous distribution of photons with energy =ω < =ωmin arises because of the effect of multiple Compton backscattering (or multiphoton emission). The number of photons with energy out of the spectral line is increasing with the k rise and may achieve a few ten percent from the expected intensity of the spectral line. Acknowledgments The work was partially supported by the Russian Ministry of Education and Science within the program “Nauka” Grant No 3.709.2014/K. References [1] I.V. Pogorelsky, I. Ben-Zvi, T. Hirose, et al., Demonstration of 8 × 1018 photons/ second peaked at 1.8 Å in a relativistic Thomson scattering experiment, Phys. Rev. Spec. Top. - Accel. Beams 3 (2000) 090702. [2] F.V. Hartemann, A.M. Tremaine, S.G. Anderson, et al., Characterization of a bright, tunable, ultrafast Compton scattering X-ray source, Laser Part. Beams 22 (2004) 221. [3] M. Babzien, I. Ben-Zvi, K. Kusche, et al., Observation of the second harmonic in thomson scattering from relativistic electrons, Phys. Rev. Lett. 96 (2006) 054802. [4] G. Priebe, D. Laundy, M.A. MacDonald, et al., Inverse Compton backscattering source driven by the multi-10 TW laser installed at Daresbury, Laser Part. Beams 26 (2008) 649. [5] O. Williams, G. Andonian, M. Babzien, et al., Characterization results of the BNL ATF Compton X-ray source using K-edge absorbing foils, Nucl. Instrum. Methods Phys. Res. Sect. A 608 (2009) S18. [6] I. Sakai, T. Aoki, K. Dobashi, et al., Production of high brightness γ -rays through backscattering of laser photons on high-energy electrons, Phys. Rev. Spec. Top. Accel. Beams 6 (2003) 091001. [7] M. Bech, O. Bunk, C. David, et al., Hard X-ray phase-contrast imaging with the compact light source based on inverse compton X-rays, J. Synchrotron Radiat. 16 (2009) 43. [8] J. Oliva, A. Bacci, U. Bottigli, et al., Start-to-end simulation of a Thomson source for mammography, Nucl. Instrum. Methods Phys. Res. Sect. A 615 (2010) 93. [9] J. Yang, M. Washio, A. Endo, T. Hori, Evaluation of femtosecond X-rays produced by Thomson scattering under linear and nonlinear interactions between a low-

Fig. 17. The same for higher Lorentz-factor γ0 = 1500 .

electron k calculating the luminosity or using the expression (28). Using the approximation for connection between k and a number of cycles in a laser pulse N0 , it is possible to use the Thomson formulas for analytical calculation of the photon spectra. Authors of the work [30] investigated the process of generation of

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