Spin effects in nonlinear Compton scattering in a plane-wave laser pulse

Nuclear Instruments and Methods in Physics Research B 279 (2012) 12–15

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Spin effects in nonlinear Compton scattering in a plane-wave laser pulse Madalina Boca ⇑, Victor Dinu, Viorica Florescu University of Bucharest, Centre for Advanced Quantum Physics, P.O. Box MG11, 077125 Magurele, Romania

a r t i c l e

i n f o

Article history: Received 14 July 2011 Received in revised form 8 August 2011 Available online 17 November 2011 Keywords: Nonlinear Compton Dirac equation Klein–Gordon equation Spin effects

a b s t r a c t We study theoretically the electron angular and energy distribution in the non-linear Compton effect in a finite plane-wave laser pulse. We first present analytical and numerical results for unpolarized electrons (described by a Volkov solution of the Dirac equation), in comparison with those corresponding to a spinless particle (obeying the Klein–Gordon equation). Then, in the spin 1/2 case, we include results for the spin flip probability. The regime in which the spin effects are negligible, i.e. the results for the unpolarized spin 1/2 particle coincide practically with those for the spinless particle, is the same as the regime in which the emitted radiation is well described by classical electrodynamics. Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction The non-linear Compton effect, the scattering of intense electromagnetic radiation on charged particles is an interesting subject for theorists since almost 50 years. Until a few years ago, practically all the reported calculations used the monochromatic approximation for the external electromagnetic field; for a detailed review of the literature see, for example, the recent paper by Ehlotzky et al. [1]. A recent improvement of the theoretical model consists in the description of the laser field as a plane-wave electromagnetic pulse with finite length [2,3]; also, attempts to introduce in the description the initial electron velocity distribution [4] or a focused laser pulse [5] have been made. The experimental detection of the process was realized in 1993, in an experiment performed at SLAC [6], using 46.6 GeV electrons and terawatt pulses from a Nd:glass laser at 1054 and 527 nm wavelengths. Although the general equation from which both photon and electron distributions can be extracted were written long time ago [7], most of the papers present numerical results only for the radiation distribution. The final electron distribution was less studied, and only in the monochromatic approximation. A paper by Panek et al. [8] presents results for the angular distribution of the emitted radiation, separately taking into account the contribution of the processes with or without spin flip; they also present a comparison with the case where the electron is described by a solution of the Klein–Gordon equation, its spin being neglected. Ivanov [9] has expressed the transition probability of the process, in terms of relativistic invariants, and presented numerical calculation showing the influence of the laser polarization on the final electron spin. ⇑ Corresponding author. Tel.: +40 21 457 4949; fax: +40 21 457 4521. E-mail address: [email protected] (M. Boca). 0168-583X/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2011.10.024

A study of the polarization of the final electron as a function of its energy was presented by Bolshedvorsky and Polityko [10]. Our investigation of the angular and energy electron distribution refers to the case of a plane wave laser pulse of finite duration; we consider the regime of ultrarelativistic electrons and laser fields of intensity of the order of atomic unit, at the wavelength k ¼ 1060 nm. We describe the electron by a solution of the Dirac equation, or, neglecting its spin, by a solution of the Klein–Gordon equation. By comparing the Klein–Gordon results with those corresponding to unpolarized spin 1/2 particles, we identify the regime where the spin effects are important. In the Dirac case, we present separately numerical results for the probability distribution with spin-flip. We discuss the influence of the laser intensity on the spin flip probability and also on the angular and energy distribution. In Section 2 we present the main equations used in our calculation and we discuss a classicality parameter which indicates the regime where the spin effects are negligible; in this regime also the emitted radiation spectrum is well described in the classical formalism. In Section 3 we present numerical results illustrating the qualitative theoretical predictions. 2. Theory The description of the non-linear Compton effect is made in a semiclassical formalism: the laser field, with the propagation direction n chosen along the third axis of the reference frame, is treated as a classical field, characterized by the vector potential Að/Þ ? n; the argument / of the vector potential is / ¼ ct  n  r  n  x with n  ð1; nÞ; x  ðct; rÞ. The electron dressed by the laser pulse is described by a Volkov solution of the Dirac or Klein–Gordon equation, which allows introducing in theory a plane wave laser pulse of arbitrary shape and duration, and the emitted photon is described by a

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quantized field, treated in the first order perturbation theory (for details of calculation see, for example, [8,2]). The probability distribution of the electron has the general form 2

p02 3 0 2 c2  h p1

d pe ¼ dX2 dE2 4p

e20

2

jp2 jðmcÞ P; ðn  p2 Þðn  hk2 Þ

e20 ¼

~¼ y

2

e

4p0

:

ð1Þ

The initial and final electron momenta, p1 and p2 , and the momentum of the emitted photon  hk2 obey conservation rules specific for the case of a laser pulse,

p1? ¼ p2? þ hk2? ;

n  p1 ¼ n  p2 þ n  hk2 ;

ð2Þ

where the subscript ? indicates the components orthogonal on the propagation direction n of the laser. The expression of P, for the Dirac case with spin analysis, is ðD;pÞ

P i1 ;i2 ¼

2 X   ^n ^ ^s2 þ mc ^s2 A hfi ;p jb^s2 þ mc n  ^ ^ i jf A i1 ;p1  :  2 2 2n  p 2n  p 2

s2

ð3Þ

1

In the previous equation fi;p with i ¼ 1 is the free particle spinor, ^  mcÞfi;p ; it describes a particle with solution of the equation ðp the projection of the spin on the third axis in its proper frame equal to  h=2 for i = 1 and, respectively,  h=2 for i ¼ 1. The quantities denoted by A and b are the one-dimensional integrals, defined in Eqs. (25) and (26) of [2] and the photon polarization vector s2 was chosen such that s2  n ¼ 0. The probability distribution with spin flip is ðD;pÞ ðD;pÞ given by Eq. (1) with P replaced by ðP þ1;1 þ P 1;þ1 Þ=2. We have calculated the distribution for a spinless particle and for an unpolarized electron; they have a similar structure



   p1 p ; P ¼ b0 jbj2 þ a0 A  A þ a1 R b A   2 n  p1 n  p2

ð4Þ

where the coefficients a0 ; a1 ; b0 have different expressions for the Dirac or Klein–Gordon case:

aD0 ¼ 1; aD1 ¼ aKG 1 D

b0 ¼ 1  KG

b0

ðhk2  nÞ2 ; 2ðn  p1 Þðn  p2 Þ ðn  p1 Þðn  p2 Þ ¼2 ; mcðn  hk2 Þ ðp1  nÞðp2  hk2 Þ þ ðp1  hk2 Þðp2  nÞ aKG 0 ¼ 1þ

ðmcÞ2 n  hk2 2ðp1  nÞðp1  hk2 Þ ¼1 : ðmcÞ2 nhk2

ð5Þ ð6Þ ð7Þ ð8Þ D=KG

hx2  1; ðmcÞc1

h  x1 c 1  1; mc2 ð1 þ g2 =2Þ

ð10Þ

where x1 is the laser frequency and g is a parameter characterizing the laser field intensity I,

sffiffiffiffiffiffiffiffiffiffiffiffiffi 8p r 0 I g¼ : mcx21

ð11Þ

with r0 the classical electron radius. The behavior of the parameter ~ can be easily understood qualitatively; it increases with c1 , i.e. the y quantum effects are stronger for more energetic electrons. The de~ with the field intensity can be explained using the fact crease of y that an electronp dressed by strong laser field behaves as a particle ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of mass m ¼ m 1 þ g2 =2 [14]; when the dressed mass increases, the recoil decreases and the electron behaves ‘‘more classically’’. We mention here that in the classical description of the motion of a charged particle in a plane-wave laser pulse its final momentum is always identical to the initial one, so a signature of the quantum behavior is, except for the spin effects, a non-negligible probability to detect the final electron with an energy significantly different from the initial one. 3. Numerical results We present in this section numerical results for the electron distribution in non-linear Compton effect; the conditions we consider are close to those in the SLAC experiment [6]. We refer to the case of head-on collision, (the incident electron and the laser pulse move in the opposite direction with respect to each other), and the initial electron Lorentz factor c1 ¼ 105 , corresponding to an energy of 51.1 MeV. We choose a circularly polarized laser field, described by the vector potential

A0 Að/Þ ¼ pffiffiffi f ð/Þ½ex sinðk1 /Þ þ ey cosðk1 /Þ; 2

k1 ¼

x1 c

:

ð12Þ

with the envelope f ð/Þ given by

;

A comparison between the two sets of coefficients aD=KG ; aD=KG ; b0 0 1 shows that the Dirac and Klein–Gordon results become identical in the limit ðp1  p2 Þ2  ðmcÞ2 . This condition can be expressed in terms of the energy of the emitted photon, using the conservation rules (2), as

y¼

and ultrarelativistic electrons the condition (9) can be written in terms of the laser and initial electron parameters as [12]

ð9Þ

where c1 is the Lorentz factor of the initial electron. From the point of view of the radiation scattering theory, Eq. (9) expresses the conditions of validity of the classical description of the process. The relation between the quantum and classical description of the radiation scattering was analyzed in several studies in the literature. In the monochromatic case Goreslavskii et al. [11] have derived the classical energy and angular distribution of radiation as the classical limit ð h ! 0Þ of the corresponding quantum results. They have also obtained the validity limit of the classical theory for describing the scattered radiation. The comparison between the classical (Thomson) and quantum (Compton) expressions of the radiation spectrum for the case of a plane wave laser pulse were presented recently by Hartemann [4], Heinzl et al. [12] and Boca and Florescu [13]. For laser intensities of the order of atomic unit, at near-infrared frequencies

8 2 2 2 2 /60 > < exp½1:386 k1 / =ð4p s Þ; f ð/Þ ¼ 1; 0 6 / 6 Nc cT 1 : > : 2 exp½1:386 k1 ð/  Nc cTÞ2 =ð4p2 s2 Þ; / P Nc cT 1 ð13Þ With this choice of the envelope the parameter s is the full width at half maximum (FWHM) of the Gaussian wings, measured in periods T 1 ¼ 2p=x1 and Nc is the length of the flat region, also measured in units of T 1 ; we take s ¼ 2 cycles and for N c values between 0 and 10 cycles. The central frequency of the laser is chosen x1 ¼ 0:043 au ðk ¼ 1060 nmÞ. The parameter g defined in Eq. (11) can be written in terms of A0 as g ¼ jejA0 =ðmcÞ; for it, we consider the values g ¼ 0:5 ðI ¼ 3 1017 W=cm2 Þ; g ¼ 1 ðI ¼ 1:2 1018 W=cm2 Þ and g ¼ 2 ðI ¼ 4:8 1018 W=cm2 Þ. Due to the geometry chosen and to the circular polarization of the laser the problem has an axial symmetry with respect to the common direction of p1 and n; in the representation of our numerical results we shall use as parameters the scattering angle h  \ðp1 ; p2 Þ and the Lorentz factor c2 of the final electron. It is useful here to indicate also the values of the classical~, defined in Eq. (10), for the intensities we consider: ity parameter y ~ ¼ 0:2 for g ¼ 0:5; y ~ ¼ 0:15 for g ¼ 1 and y ~ ¼ 0:077 for we have y g ¼ 2, so that we can expect to observe non-negligible quantum effects only in the case of the lowest value of the intensity. In Fig. 1 we present, in a logarithmic color scale, the double dif2 ferential probability distribution d pe =dE2 dX2 , as a function of c2 and h, for the case of unpolarized Dirac particles. Both Fig. 1(a)

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M. Boca et al. / Nuclear Instruments and Methods in Physics Research B 279 (2012) 12–15

2

Fig. 1. The probability distribution d pe =dX2 dE2 for unpolarized electrons and g ¼ 0:25. (a): N c ¼ 0; (b): N c ¼ 10.

and (b) refer to the case g ¼ 0:25; in (a) the laser pulse was chosen without a flat region ðN c ¼ 0Þ, while in (b) N c is 10 cycles; the values of the other parameters are those given at the beginning of this section. The general behavior in the two cases is similar; first we notice that the angles h for which the electron distribution has non-negligible values are extremely small h < 5 106 p, but the final Lorentz factor spread in a relatively large range, between 4 104 and 105. The difference between the initial and final electron energy is a measure of the electron recoil, and gives an estimation of the energy of the emitted photon. As discussed in the previous section, the large value of this difference is a signature of quantum behavior, which is present for g ¼ 0:25. At fixed values of E2 and variable h the electron distribution consists in successive maxima which are relatively large for N c ¼ 0 and become very sharp for N c ¼ 10. The same dependence of spectrum on the pulse length is met also in the case of photon distribution [13]; it gives an indication on the validity of monochromatic limit, in which the successive maxima are replaced by discrete lines. Next we discuss the spin effects; in Fig. 2 (a) are represented the energy probability distributions dpe =dE2 as functions of c2 for unpolarized electrons (dashed line) and for a spinless particle (full line) for N c ¼ 4 and three values of the parameter g, marked on each pair of curves. For both Dirac and Klein–Gordon cases the shape of the distribution is similar; it consists in a series of almost constant plateau-like regions. For g ¼ 0:25 the first plateau spreads

backward up to c2 ¼ 6 104 ; when the laser intensity increases, the width of these successive plateaux decreases, which means that for higher intensities the energy of the final electron is getting closer to the initial energy E1 ¼ c1 mc2 , i.e. the quantum effects are less pronounced. The comparison between Dirac and Klein–Gordon results shows that the differences between the two cases are larger for smaller c2 (i.e. for large electron recoil), and decrease when g increases, in agreement with the discussion in the previous section. In Fig. 2 (b) we consider the spin flip effect for the three values of g. The quantity represented, denoted by dpsf =dE2 , is the ratio between the energy distribution of the electrons with spin flip and the total probability of the process. For any intensity the spin flip probability becomes zero for E2 ¼ E1 ; for E2 < E1 one can see a series of maxima whose positions coincide with the limits of the successive plateaux of the total distribution. In this scaled representation it is visible that the spin-flip probability is larger at smaller g. Similar conclusions are obtained from the analysis of the angular distribution: in Fig. 3 (a) is represented, for the unpolarized electrons (dashed line) and for spinless particles (full line) the angular probability distribution dpe =dX2 for the same conditions as in Fig. 2; the variable is the scattering angle of the electron, h. In both Dirac and Klein–Gordon cases, and for all the three values of the parameter g the angular distribution has a very sharp maximum near h ¼ 0, which indicates that most of the electrons are

M. Boca et al. / Nuclear Instruments and Methods in Physics Research B 279 (2012) 12–15

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scattered within a cone with an extremely small opening angle around the incident electron direction; at larger values of h a series of plateau-like structures is present. The differences between the Dirac and Klein–Gordon case are very small, but non-zero at h ¼ 0 and increase with h; the explanation is that, as one can see in the double differential distribution, represented in Fig. 1 the probability for the final electron to have a large recoil (i.e. large value of the difference E1  E2 is greater at large scattering angles). The probability distribution with the spin flip, scaled by the total probability of the process, is presented in Fig. 3 (b). Here one can see that in the limit h ¼ 0 the spin flip probability is very small, but not exactly zero, and for larger angles the oscillatory behavior, with maxima around the limits of the plateaux of the total distribution appear. The differences between unpolarized Dirac and Klein–Gordon results in the limit h ! 0, as well as the non-vanishing value of the spin-flip probability in the same limit, can be understood from the graphic of the double differential distribution, Fig. 1. At very small scattering angles the contribution to the integral over the energy comes from several maxima: one located near E2 ¼ E1 , which gives no spin effects, and others of smaller amplitude, whose position depends on g and which contribute to the spin effects. Acknowledgements

Fig. 2. (a) Energy probability distribution dpe =dE2 for three values of g; dotted lines: unpolarized electrons, full lines: Klein–Gordon particle. (b) The scaled spin-flip energy distribution dpsf =dE for three values of g marked on each curve.

For M. Boca, this work was supported by the Strategic Grant POSDRU/89/1.5/S/58852, Project ‘‘Postdoctoral Programme for Training Scientific Researchers’’ cofinanced by the European Social Fund within the Sectorial Operational Program Human Resources Development 2007–2013. References [1] [2] [3] [4] [5] [6]

[7] [8] [9] [10] [11] [12] [13] [14]

Fig. 3. (a) Angular probability distribution dpe =dX2 for three values of g; dashed line: unpolarized electrons, full line: Klein–Gordon particle. (b) The scaled spin-flip angular distribution dpsf =dX2 for three values of marked on each curveg.

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