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Properties of an ultrarelativistic charged particle radiation in a constant homogeneous crossed electromagnetic field
Accepted Manuscript Properties of an ultrarelativistic charged particle radiation in a constant homogeneous crossed electromagnetic field O.V. Bogdanov, P.O. Kazinski, G.Yu. Lazarenko PII: DOI: Reference:
S0003-4916(17)30085-4 http://dx.doi.org/10.1016/j.aop.2017.03.010 YAPHY 67355
To appear in:
Annals of Physics
Received date: 17 August 2016 Accepted date: 12 March 2017 Please cite this article as: O.V. Bogdanov, P.O. Kazinski, G.Y. Lazarenko, Properties of an ultrarelativistic charged particle radiation in a constant homogeneous crossed electromagnetic field, Annals of Physics (2017), http://dx.doi.org/10.1016/j.aop.2017.03.010 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Highlights
>Properties of an electron radiation in a crossed electromagnetic field are studied.> >Spectral angular distribution of the synchrotron entrance radiation is described.> >Spectral angular distribution of the de-excited radiation is described.> >De-excited radiation is almost completely circularly polarized.> >Photon energy at the maximum of the de-excited radiation decreases with increasing the initial energy of an electron.>
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Properties of an ultrarelativistic charged particle radiation in a constant homogeneous crossed electromagnetic field O.V. Bogdanov,1, 2, ∗ P.O. Kazinski,1, 2, † and G.Yu. Lazarenko1, ‡ 1 Physics
Faculty, Tomsk State University, Tomsk, 634050 Russia
2 Department
of Higher Mathematics and Mathematical Physics,
Tomsk Polytechnic University, Tomsk, 634050 Russia (Dated: February 23, 2017)
Abstract The properties of radiation created by a classical ultrarelativistic scalar charged particle in a constant homogeneous crossed electromagnetic field are described both analytically and numerically with radiation reaction taken into account in the form of the Landau-Lifshitz equation. The total radiation naturally falls into two parts: the radiation formed at the entrance point of a particle into the crossed field (the synchrotron entrance radiation), and the radiation coming from the late-time asymptotics of a particle motion (the deexcited radiation). The synchrotron entrance radiation resembles, although does not coincide with, the ultrarelativistic limit of the synchrotron radiation: its distribution over energies and angles possesses almost the same properties. The de-excited radiation is soft, not concentrated in the plane of motion of a charged particle, and almost completely circularly polarized. The photon energy delivering the maximum to its spectral angular distribution decreases with increasing the initial energy of a charged particle, while the maximum value of this distribution remains the same at the fixed photon observation angle and entrance angle of a charged particle. The ultraviolet and infrared asymptotics of the total radiation are also described. Keywords: Spectral angular radiation distribution, polarization, Landau-Lifshitz equation, crossed electromagnetic field.
∗ † ‡
E-mail:[email protected] E-mail:[email protected] E-mail:[email protected]
1
I.
INTRODUCTION
The form of radiation of an ultrarelativistic charged particle moving in a crossed electromagnetic field is discussed in many textbooks and papers (see, e.g., [1–4]). However, according to these approaches to the problem, the radiation reaction acting on a charged particle is completely neglected or taken into account only as a small perturbation. This is not always a justified approximation, especially when the external electromagnetic field is strong enough or the particle spends a sufficiently long time in the field [5–7]. We describe (to our knowledge, for the first time) in detail the properties of radiation created by a classical charged scalar particle entering a constant homogeneous crossed electromagnetic field and moving in it for an infinite time with radiation reaction taken into account. The infinite time condition can be relaxed under certain restrictions we shall specify. Our investigation shows that, loosely speaking, this radiation consists of two parts: the first one is an intense hard radiation created by a charged particle at the entrance point (we shall call it, for brevity, the synchrotron entrance radiation), and the second one is a comparatively weak soft radiation formed on the late-time asymptotics of particle trajectory (the de-excited radiation) [6]. The properties of the synchrotron entrance radiation resemble the properties of an ultrarelativistic asymptotics of the synchrotron radiation. In particular, the maximum of the spectral density is reached at the photon energy the same as for the synchrotron radiation, and the most part of the radiation is concentrated in the cone opening of the order 2γ −1 , where γ is the Lorentz factor. It is this radiation which is subject to considerable quantum corrections at high external field intensities. Notice that this is not the so-called edge radiation as it is standardly defined (see the definition, e.g., in [8, 9]). The de-excited radiation is soft, its spectrum stretches from far-infrared to hard X-rays (for γ ∼ 104 and the field strengths corresponding to the intensity I ∼ 1022 W/cm2 ), and the quantum corrections to it are negligible. We find the region of observation angles and energies of photons where these two types of radiation can be discerned and observed. Besides that the synchrotron entrance and de-excited radiations are distinguished by the ranges of the photon energies and observation angles where these radiations are concentrated, it turns out that they possess distinct polarization properties. The de-excited radiation is almost completely circularly polarized, while the synchrotron entrance radiation has mostly a linear polarization. We present a detailed description of the polarization properties of these two types of radiations both analytically and numerically. The study of properties of an ultrarelativistic charged particle radiation in a crossed electro2
magnetic field is important for two reasons, at least. First, in the ultrarelativistic limit, every electromagnetic field becomes crossed in the momentary comoving frame. Therefore, one may expect that, in a certain approximation, the properties of radiation we discussed in this paper should be inherent to any electromagnetic field, which can be considered as constant and homogeneous on the radiation formation length scale. Second, the crossed field approximation is the standard one [1, 10] in considering the radiation of charged particles in strong laser fields. The intensity of the laser radiation, which will become accessible in the nearest future [11, 12], allows one to observe the radiation reaction effects we discuss [5–7, 10, 13–15]. The study of classical radiation is relevant even in that domain of parameters where one expects considerable quantum corrections. In this case, the classical results can be used to distinguish clearly the quantum corrections. This idea was employed, for example, in a recent proposal [16]. Notice that the numerical simulations of dynamics of electrons and their radiation in a strong laser wave and in crystals with radiation reaction taken into account already present in the literature (see, e.g., [13–15, 17–24]). The exact solution to the Landau-Lifshitz equation [25] in the crossed electromagnetic field is also known for a long time (see [26] and references therein). However, the problem of radiation coming from the whole electron trajectory in a crossed field has somehow escaped an in-depth study. The aim of the present paper is to fill this gap and to provide an analytical and numerical investigation of the radiation properties with radiation reaction taken into account.
II.
NOTATION AND SOLUTION TO THE EQUATIONS OF MOTION
The action functional of a charged particle with the charge e and the mass m interacting with the electromagnetic field Aµ on the Minkowski spacetime R1,3 with the metric ηµν = diag(1, −1, −1, −1) has the form S[x(τ ), A(x)] = −m
Z
dτ
√
x˙ 2
−e
Z
1 dτ Aµ x˙ − 16π µ
Z
d4 xFµν F µν ,
(1)
where Fµν := ∂[µ Aν] is the strength tensor of the electromagnetic field, and the speed of light c = 1. The Landau-Lifshitz equation in the natural parametrization x˙ 2 = 1 is written as [25] 2 e ˙ e2 e2 m¨ xµ = eFµν x˙ ν + e2 Fµν x˙ ν + 2 Fµν F νρ x˙ ρ − 2 x˙ λ Fλν F νρ x˙ ρ x˙ µ . 3 m m m
(2)
For a thorough discussion of its applicability to the problem at hand see, e.g., [6, 7] and references therein. Let us choose the Compton wavelength lC := ~/mc as the length unit: xµ → lC xµ , 3
τ → lC τ.
(3)
Then Eq. (2) is reduced to x ¨µ = fµν x˙ ν + λ(f˙µν x˙ ν + fµν f νρ x˙ ρ − x˙ λ fλν f νρ x˙ ρ x˙ µ ),
(4)
where xµ , τ , fµν , and λ are dimensionless quantities, and fµν = sgn(e)Fµν /E0 . It is useful to bear in mind that the electromagnetic field strength is measured in the units of the critical field E0 , and the unit of energy is the electron rest energy: lC ≈ 3.86 × 10−11 cm, E0 =
tC ≈ 1.29 × 10−21 s,
m2 ≈ 4.41 × 1013 G = 1.32 × 1016 V/cm, |e|~
m ≈ 5.11 × 105 eV, λ=
2α 2 ≈ , 3 411
(5)
The modern accelerator facilities are able to accelerate electrons up to 25 GeV and higher, and the intensities of the achievable at the present moment laser fields [27] are of the order 1022 W/cm2 . These data correspond to γ ≈ 105 ,
ω ≈ 1.47 × 10−4 ,
λω ≈ 7.14 × 10−7 ,
(6)
where ω is the electromagnetic field strength in the laser wave. The field strength for the constant homogeneous crossed electromagnetic field is [µ ν]
f µν = ωe− e1 ,
2 − fµν = ω 2 e− µ eν ,
ex = −ω,
hz = ω,
x1 ≥ 0,
(7)
where ω is a constant, ex and hz are the non-zero fµν components, x− = x0 − x2 , and the 4-vectors eµ− = (1, 0, 1, 0), eµ1 = (0, 1, 0, 0) were introduced. Any 4-vector can be represented in the form 1 + 1 3 jµ = (j+ e− µ + j− eµ ) − j1 eµ − j3 eµ , 2
(8)
where eµ+ = (1, 0, −1, 0), eµ3 = (0, 0, 0, 1), and ja := eµa jµ . In particular, υ− υ+ − υ12 − υ32 = 1,
∗ jµ∗ j µ = Re(j+ j− ) − |j1 |2 − |j3 |2 ,
jµ ∈ C,
(9)
where υa := x˙ a , and the dot denotes the derivative with respect to the natural parameter. The solution to the Landau-Lifshitz equation for such a field configuration can be cast into the form [6, 26, 28–30] h i2 1 −1 −1 r+ = r32 (0) + r¯(0) + (υ− (0) + λω 2 x− )2 + (υ− (0) + λω 2 x− )2 , 2λω 1 −1 r1 = r¯(0) + (υ −1 (0) + λω 2 x− )2 , υ− = (υ− (0) + λω 2 x− )−1 , r3 = r3 (0), 2λω −
(10)
2 (0))−1 . The initial data can be written as where ra := υa /υ− and r¯(0) := r1 (0) − (2λωυ−
r1 (0) = −
β(0) sin θ0 sin ϕ0 , 1 − β(0) sin θ0 cos ϕ0
r3 (0) = −
β(0) cos θ0 sin ϕ0 , 1 − β(0) sin θ0 cos ϕ0
υ− (0) = γ(0)(1 − β(0) sin θ0 cos ϕ0 ), 4
(11)
where β(0) is the initial 3-velocity modulus, and θ0 , ϕ0 determine the entrance angle. The definition of angles θ, ϕ and the schematic sketch of electron’s dynamics are presented in Fig. 1. The trajectory is r¯(0) + λω r2 (0) + r¯2 (0) 1 x+ = x+ (0) − 2 3 3 − 3 2 − 5 (0) + 3 λω υ− (0) 3λ ω υ− (0) 20λ ω 4 υ− r 2 −3/2 h 2 2 s5 i + |ω| r3 (0) + r¯2 (0) s + sgn(ω)(¯ r(0) + λω)s3 + , λ 3 5 r r¯(0) 1 2 −3/2 h s3 i x1 = x1 (0) − − |ω| r ¯ (0)s + sgn(ω) , + 3 (0) λω 2 υ− (0) 6λ2 ω 3 υ− λ 3 r 2 −3/2 r3 (0) + |ω| r3 (0)s, x3 = x3 (0) − 2 λω υ− (0) λ r 2 −3/2 1 x− = |ω| s− , 2 λ λω υ− (0)
(12)
where the last equality is the definition of s, and we put x− (0) = 0. The velocity components (10) in terms of this new variable become r+ = r¯2 (0) + r32 (0) + 2 sgn(ω)(¯ r(0) + λω)s2 + s4 ,
r1 = r¯(0) + sgn(ω)s2 ,
r3 = r3 (0). (13)
We suppose that the charged particle enters the electromagnetic field at the instant τ = 0 and then moves in it for an infinite time. The last assumption can be relaxed if the radiation formation time is much shorter than the time that the particle spends in the electromagnetic field. Further, we shall give the estimates for the radiation formation time and length. The case of particles reflected from the electromagnetic field [7] will be considered elsewhere.
III.
a.
RADIATION
General formulas. The spectral angular distribution of a one charged particle radiation
summed over the photon polarizations is written as dE(k) = |E(k)|2
dk R2 dk = −e2 jµ∗ (k)j µ (k) 2 , 4π 4π 2 k02
k 2 = 0,
(14)
k µ jµ (k) = 0.
(15)
where jµ (k) =
Z
τ2
τ1
dτ x˙ µ e−ikν x
ν (τ )
−
ix˙ µ −ikν xν τ2 e , kλ x˙ λ τ1
Notice that if one calculates the contribution of the boundary points in the integral (15) by the WKB method (when this approximation is applicable) then the leading WKB contribution is canceled by 5
the out of the integral term in (15). The projections of the radiation electric field in the wave zone take the form Eα (k) = −iek0
e−ik0 R µ b(α) jµ (k), R
α = 1, 2,
(16)
where bµ(α) are the physical photon polarization vectors and R is a distance from the source of radiation to the observation point. The Stokes parameters are ξ1 = 2
Re(E1 E2∗ ) , |E|2
ξ2 = 2
Im(E1 E2∗ ) , |E|2
ξ3 =
|E1 |2 − |E2 |2 . |E|2
(17)
In particular, ξ12 + ξ32 = 1 − ξ22 =
|jµ j µ |2 |E2 |2 = . |E|4 (jν∗ j ν )2
(18)
In the case of circularly polarized wave, ξ2 = ±1. At ξ2 = 0, the electromagnetic wave is linearly polarized and ξ1 = sin(2ϑ),
ξ3 = cos(2ϑ),
(19)
where ϑ is the angle between the polarization plane and the axis with the unit vector b(1) . Thus the problem is reduced to the evaluation of integrals (15), where, in our case, τ1 = 0 and τ2 → ∞. The out of the integral term corresponding to τ2 → ∞ is absent in (15). It turns out that the main characteristics of the radiation created by a charged particle in the crossed field with radiation reaction taken into account can be described analytically with a rather good accuracy. b.
De-excited radiation. The expression standing in the exponent in (15) can be cast into the
form kµ xµ = kµ xµ (0) −
i k− h |z|2 z 0 − λω 1 − − 3 (0) 5 (0) 2 λω 2 υ− (0) 3λ2 ω 3 υ− 20λ3 ω 4 υ− r i 2 k− h 10 2 0 3 5 5|z| s − sgn(ω)(z − λω)s + s , (20) + λ 10|ω|3/2 3
where z ≡ z 0 + iz 00 := ζ 0 − r¯(0) + i(ζ 00 − r3 (0)),
ζ ≡ ζ 0 + iζ 00 :=
k1 + i|k3 | , k−
k− =
2k0 . (21) 1 + |ζ|2
The coordinates ζ 0 , ζ 00 are coordinates of the stereographic projection of the sphere |k| = 1 to the plane (k 1 , k 3 ) from the pole (0, 1, 0). In the ultrarelativistic limit, r1 and r3 are also coordinates of the stereographic projection of the sphere |υ| = 1 to the plane (υ 1 , υ 3 ) from the pole (0, 1, 0). 6
It is convenient to introduce the notation φ := φ0 + iφ00 , ρ := ρ0 + iρ00 , where φ0 := sgn(ω)(z 0 − λω),
|φ|2 := |z|2 ⇔ (φ00 )2 = (z 00 )2 + 2λωz 0 − λ2 ω 2 ,
ρ0 := − sgn(ω)(¯ r(0) + λω), r 2 k− , ε := λ 10|ω|3/2
|ρ|2 := r¯2 (0) + r32 (0),
(22)
and ψ0 is the expression in the first line on the right-hand side of equality (20). In particular, the following relation holds (φ0 − ρ)(φ0 − ρ∗ ) − (φ0 − ρ0 − λ|ω|)2 − r32 (0) = 2λ|ω|φ0 .
(23)
Using the new notation, r+ = (s2 − ρ)(s2 − ρ∗ ),
r1 = −λω + sgn(ω)(s2 − ρ0 ).
(24)
This notation is useful for the evaluation of integrals (15). The integrals determining the Fourier transform of jµ are of the same type as considered in [6] (see also App. A) and are calculated analogously. It is useful to deform the integration contour as depicted in Fig. 1. The radiation created by a de-excited charged particle corresponds to the case (i) in (A4). Let us change the integration variable y = (20εφ0 )1/3 (s −
p φ0 ).
(25)
Then 8
+ 5(φ00 )2 (φ0 )1/2 + By + y 3 /3 + hy 4 /4 + h2 y 5 /20, ε2 1/3 h : = (20ε)−1/3 (φ0 )−5/6 , B := 5 (φ00 )2 . 20φ0
kµ xµ = ψ0 + ε
3 (φ
0 5/2
)
(26)
Having neglected the out of the integral terms in (15), the contributions coming from the contour C0 , and assuming that (A6) is fulfilled, we obtain up to the terms of the order h2 : 0 0 0 0 ∗ 0 0 0 ∗ iI4 c−1 0 j+ = (φ − ρ)(φ − ρ )I0 + h 4φ (φ − ρ )I1 − (φ − ρ)(φ − ρ ) 4 h iI5 i I 8 + h2 2φ0 (3φ0 − ρ0 )I2 − φ0 (φ0 − ρ0 )iI5 − (φ0 − ρ)(φ0 − ρ∗ ) + , 32 20 ih I8 iI5 c−1 I4 − h2 + , j3 = r3 (0)j− , 0 j− = I0 − 4 32 20 n iI4 c−1 j = sgn(ω) (φ0 − ρ0 − λ|ω|)I0 + h 2φ0 I1 − (φ0 − ρ0 − λ|ω|) 1 0 4 io h iI I iI 5 8 5 ) − (φ0 − ρ0 − λ|ω|) + . + h2 φ0 (I2 − 2 32 20 7
(27)
It is the order h2 where the leading contribution to the de-excited radiation comes from. Here Z r 8 2 −3/2 3 −iψ0 −iε 3 (φ0 )5/2 +5(φ00 )2 (φ0 )1/2 0 −1/3 dttn e−i(Bt+t /3) . (28) |ω| (20εϕ ) , In (B) := c0 := e λ C The integrals In (B), n ∈ N, are expressed through the Airy functions (see [6]). As a result, ∗ Re(j+ j− ) − |j1 |2 − |j3 |2 = n h o iI1 I4 3iI0 I5 I0 I8 I42 i 2 0 2 2 2 0 2 2 = |c0 | 2λ|ω|φ I0 + h I0 I2 + + 4h (φ ) (I1 + I0 I2 ) . (29) − − + 2 5 16 16
In the case when the applicability conditions (39) of the approximation considered are fulfilled, the last term in the curly brackets dominates over the first terms in the square brackets. Thus, according to (9), (14), the spectral angular distribution of the de-excited radiation has the form (cf. [6]) dk0 dζ 0 dζ 00 dk 2 2 0 2 dE(k) ≈ 4e2 B B[Ai(B)]2 + [Ai0 (B)]2 = 4e B B[Ai(B)] + [Ai (B)] 2 (φ00 )2 . (30) (φ00 )2 k− The maximum of this radiation is reached at the photon energy 0 k0m = (Bext |ω|)3/2 (2λ)1/2 (1 + |ζ|2 )
(φ0 )1/2 , |φ00 |3
(31)
0 where Bext ≈ 0.8, and it is supposed that φ0 ≥ 0. If 2 |r1 (0)| (2λωυ− (0))−1 ,
|r1 (0)| λ|ω|,
(32)
φ00 ≈ |ζ 00 − r3 (0)|.
(33)
which are satisfied as a rule, then φ0 ≈ sgn(ω)(ζ 0 − r1 (0)),
Hence, the formulas from [6] for the spectral angular distribution are obtained when one puts r1 (0) = r3 (0) = 0. These initial data were implicitly assumed in [6]. In contrast to the synchrotron radiation, the right-hand member of (31) depends rather weakly on the energy of the incident particle. In increasing the energy, the magnitude of k0m slightly declines due to decreasing of φ0 and tends to a finite value independent of the incident particle energy (see Fig. 2). At that, according to the approximate formula (30), the value of the spectral density at the maximum does not virtually change. The numerical simulations show (see Fig. 2) that there exists a region of observation angles where formula (30) provides a quite good approximation for the spectral density of radiation for all 8
the photon energies. In this case, one can derive the angular distribution of radiation integrating (30) over k0 . Since Z
0
we deduce
∞
1 dtt3/2 t[Ai(t)]2 + [Ai0 (t)]2 = , 16
dE(θ, ϕ) =
3e2 (φ0 )1/2 (2λ|ω|3 )1/2 (1 + |ζ|2 )3 00 5 dΩ. 32 |φ |
(34)
(35)
This expression tends to infinity for |ζ| → +∞ or φ00 → 0. The first case corresponds to the
radiation along the x2 axis. The second case refers to the photon observation angles lying in the plane of motion of a charged particle. In the both cases, formula (30) is not valid (see the applicability domain (39) and Fig. 3). The angular distribution (35) depends on the energy of the incident particle in the same way as (31) at the fixed photon observation angle and entrance angle of a charged particle, viz., it diminishes slightly with increase of γ and tends to a finite value independent of the incident particle energy. Employing (18) and (27), it is not difficult to find the expression for the Stokes parameter ξ2 of the de-excited radiation. Up to the terms of the order h2 , we have h n o ih iI1 I4 3iI0 I5 I0 I8 I42 i jµ j µ = c20 2λ|ω|φ0 I02 − I0 I4 +4iI1 +h2 I0 I2 − −4h2 (φ0 )2 (I12 −I0 I2 ) . − − − 2 2 5 16 16 (36) In the parameter space we consider, the leading contribution gives the last term in the curly brackets. Then, in accordance with (18), we derive s r (B[Ai(B)]2 − [Ai0 (B)]2 )2 2 1/3 1 B 36Γ3 (7/6) 3/2 |ξ2 | ≈ 1 − − 2 + B + O(B 5/2 ) = Γ 3 6 π (B[Ai(B)]2 + [Ai0 (B)]2 )2 π 3/2 1 3 =1− + + O(B −6 ). 3 32B 64B 9/2 (37) 0 , we have The series for B large is asymptotic. In particular, at B = Bext
|ξ2 | ≈ 0.98,
(38)
i.e., the radiation is almost completely circularly polarized. In general, the Stokes parameter at the maximum may depend on the two observation angle variables, the field strength, and the initial momenta of a charged particle, but in our case it is universal. The dependence of ξ2 on the angles and radiated photon energy is plotted in Figs. 2, 3. Roughly, one can explain the presence of circular polarization of the de-excited radiation by the fact that this radiation is not concentrated in the 9
plane of motion of a charged particle. Looking to the particle trajectory from the point where the de-excited radiation is maximal, one sees this trajectory as an arc (see Fig. 1). Therefore, the radiation detected there must possess a circular polarization. However, this reasoning neither explains so high degree of its polarization nor gives the universal value (38). One can find a rough estimate for the applicability conditions of expressions (27). These conditions define the domain of energies and photon observation angles where the de-excited radiation dominates. The derivation of these conditions is obvious but rather awkward. We present only the final answer: 0 2/3
a) 2(λk− |φ |)
1,
b)
λ1/3 |ω| 2/3
2k− |φ0 |5/3
1 d) 2 (0)|φ0 | 1, 2λ|ω|υ−
e)
2/3
k− (φ00 )2 . 1, c) 2(λ|φ0 |)1/3 |ω|
1,
16λ1/3 ω 2 5/3
πυ− (0)k− (φ00 )2
(39)
1.
Here a) is the condition that the last term in (29) is much greater than the first one; b) is the condition h2 1; c) is B . 1 what is necessary since (30) is exponentially suppressed for B large; d) is the radiation formation condition (the saddle points are located sufficiently far from the integration contour boundary); e) implies that the contribution (30) dominates over the contribution coming from the contour C0 . Graphically, these conditions are depicted in Fig. 3. Despite the fact that we consider the radiation of a particle that enters the electromagnetic field and then moves in it for an infinite time, let us estimate the formation time of the de-excited radiation. In terms of the variable (25), the de-excited radiation is formed when yf
√
B, B & 1;
yf & 1, 0 < B 1.
(40)
2|φ0 | 1/2
(41)
Hence, keeping in mind (39), we obtain xf−
2|φ0 | 1/2 λ|ω|3
xf− &
, B & 1;
λ|ω|3
, 0 < B 1.
Eventually, employing formula (11) of [7], we come to (cf. (17) of [7]) |φ0 |
λ|ω| 6d 2/3 , B & 1; 2 λ
|φ0 | .
λ|ω| 6d 2/3 , 0 < B 1, 2 λ
(42)
where the notation of [7] was used. It also follows from d) and (41) that λω 2 υ− (0)xf− 1.
(43)
The inequalities (41), (43) are very close to the condition (i) in (10) of [7] specifying the asymptotic regime of particle dynamics. Combining d) with (42), one can deduce the minimal depth of the 10
field needed for the de-excited radiation to be formed: d (6λ2 χ3 )−1 ≈ 7.0 × 103 χ−3 ≈ 2.7 × 10−7 χ−3 cm,
(44)
where χ := |ω|υ− (0) is the parameter characterizing a relevance of quantum effects at the entrance point. As follows from (12), (43), the radiation formation length is `d ≈ x0 &
105 10 −1 −4 ≈ 0.84γ −1 χ−4 cm. 5 (0) ≈ 2.2 × 10 γ χ 40λ3 ω 4 υ−
(45)
On these scales, the field must be approximately crossed, constant, and homogenous along the particle trajectory for the formulas above can be applied. To minimize (44), (45), it can be taken χ ∼ 1 which is a rather mild assumption since χ decreases during the evolution for both classical and quantum considerations. c.
Synchrotron entrance radiation. The ultrarelativistic particle entering the crossed electro-
magnetic field produce an intense hard electromagnetic radiation. This radiation resembles in many respects the ultrarelativistic limit of the synchrotron radiation. In particular, we shall see that the maximum of the spectral density of radiation is reached at the photon energy which is described quite well by the formula for the synchrotron radiation, and the most part of the radiation is concentrated in the cone opening 2/γ. However, the form of the spectral angular distribution deviates from the ultrarelativistic limit of the synchrotron radiation due to the fact that the acceleration of a charged particle changes stepwise when it enters the electromagnetic field. This is not the edge radiation, which is a long-wavelength radiation as defined in [8, 9]. Though, of course, the distortion of the synchrotron spectrum is the “edge effect”, i.e., the result of an abrupt change of particle acceleration. A similar radiation was discussed in [31, 32] (see also [33]). The derivation of the radiation spectral angular distribution of an ultrarelativistic particle entering the electromagnetic field is analogous to the derivation of an ultrarelativistic asymptotics of the synchrotron radiation (see, e.g., [25, 32, 34]). It is convenient to parameterize the particle worldline by the laboratory time t. If the modulus of the radiation observation angle counted from the entrance direction of a particle is less than or of order γ −1 , and the incoming particle is ultrarelativistic, then 1 − βk ≈
2 1 + γ 2 β⊥ , 2γ 2
β¨k ≈ −|a⊥ |2 ≈ −|a|2 ,
β ⊥ := β − βk n,
(46)
˙ βk := (βn), and n := k/k0 . Also where a := β, h t2 t3 i , kµ xµ ≈ kµ xµ (0) + k0 (1 − βk )t − β˙ k − β¨k 2 6 11
β ⊥ ≈ β ⊥ (0) + a⊥ (0)t.
(47)
Therefore, up to an irrelevant common phase factor, the electric field strength of radiation created by a charged particle entering the electromagnetic field is Z ∞ k0 β ⊥ (0) µ µ dtβ ⊥ eikµ [x (0)−x (t)] RE = e − iek0 kµ x˙ µ (0) 0 n β a⊥ (2D)1/2 I˜1 (x, D) o ⊥ ≈e , 1 − iDI˜0 (x, D) − i 1 − βk a⊥ (1 − βk )1/2
(48)
where it is assumed that β⊥ . γ −1 , all the kinematic quantities in the approximate equality are taken at the initial instant of time, and the following notation was introduced: Z ∞ 2k 2 1/3 k 2/3 2 3 0 2 dttn e−i(Dt+xt +t /3) , D := (1 − βk ) 20 I˜n (x, D) := ≈ (1 + γ 2 β⊥ ) , 3 a 2aγ 0 (49) ak 2D 1/2 x := − . 2a 1 − βk
The integration contour in I˜n (x, D) is shifted slightly below the real axis at t → +∞ so that the
integral converges for all Re n > −1. Some properties of the functions In (x, D) are presented in App. B. The spectral angular distribution of radiation is obtained from (48) by formula (14), the Stokes parameters being given by expressions (17). The comparison with numerical results is presented in Fig. 4. The fact that the radiation (48) is not the edge radiation (in the narrow sense) is simply seen by comparison of (48) with formula (25) of [8] and formula (15.5) of [9]. In particular, the spectral angular distribution of the edge radiation is independent of the photon energy in contrast to (48). For the forward radiation, β ⊥ = 0 and x ≈ 0. Then it follows from (48) that the radiation is linearly polarized in this case for all the photon energies. The maximum of radiation power is reached at the photon energy 3/2
k0m = 2Dext aγ 3 ,
Dext ≈ 0.81,
(50)
where Dext provides the maximum to the function D|I˜1 (D)|2 . Formula (50) is in a good agreement with the formula for the synchrotron radiation maximum (see, e.g., [25, 32, 34]). The expression for the radiation formation length of this type of radiation is derived similarly to the case of synchrotron radiation (see, e.g., [1]). So we have ` (aγ)−1 ≈ |ω|−1 , k0 & k0m ;
` & κ−1/3 (aγ)−1 ≈ κ−1/3 |ω|−1 , k0 = κk0m , 0 < κ 1, (51)
where k0m is given by (50). Recall that |ω|−1 ≈ 6.8 × 103 ≈ 2.6 × 10−7 cm for (6). Of course, a sharp edge of the electromagnetic field assumed in (7) is just an approximation. However, if the radiation formation length ` is much bigger than the characteristic length scale of the electromagnetic field 12
variation then this approximation is valid and the formulas above are applicable ([35, 36], for more details see App. C). On the other hand, since we do not take into account the contribution of the boundary at the exit point of a particle from the electromagnetic field, the length of particle trajectory passing in the field must be much larger than `. As a rule, in the classical domain and for not very large energies, viz., γ 2 χ3 . 2.2 × 1010 ,
(52)
the radiation formation length (51) of the synchrotron entrance radiation is much smaller than the formation length (45) of the de-excited radiation. In this case, one can distinguish the three cases in considering the radiation of a charged particle penetrating a region filled with the crossed electromagnetic field of a finite width: 1. The width is sufficiently large so that the particle becomes de-excited; 2. The width is so small that neither the synchrotron entrance nor de-excited radiations are formed; 3. The intermediate case when the synchrotron entrance radiation is formed while the de-excited radiation is not. In the first case, the contribution of the boundary at the exit point to the total radiation is negligible since the charged particle is de-excited at the exit point and radiates much less energy than at the entrance point and on the trajectory in the field. The second case is the case of a “short magnet” (see [31, 32] for the properties of this radiation), and the spectral angular distribution is of the form analogous to (C7). In the third case, a separate study is needed, and we leave it for a future research. In the case when the inequality (52) is violated, a charged particle is either de-excited, and then the contribution from the exit point can be neglected, or the radiation is not formed, and we have the case of a “short magnet”. d.
Ultraviolet asymptotics. In order to find the ultraviolet asymptotics of radiation, one can
use the WKB method for the boundary point in evaluating the integrals (15). As we have already noted, the leading WKB contribution is canceled by the out of the integral term in (15). The next to leading contribution gives dE = −e2
d x˙ µ d 1 d x˙ µ x ¨2 (k x) ˙ 2 − 2(x¨ ˙ x)(k x)(k¨ ˙ x) + x˙ 2 (k¨ x)2 dk for , 6 2 (k x) ˙ 4π dτ (k x) ˙ dτ (k x) ˙ dτ (k x) ˙ 13
(53)
where the dot denotes the derivative with respect to an arbitrary time parameter, 4-vectors are contracted with the aid of the Minkowski metric, and all the quantities are taken at the initial instant of time. Choosing the laboratory time as the time variable, we arrive at dE = e2
a2 (1 − βk )2 + 2(aβ)ak (1 − βk ) − a2k (1 − β 2 ) dk0 dΩ (1 − βk )6
4π 2 k02
,
(54)
where ak := (an) and βk := (βn). In increasing the radiated photon energy, the spectral density of radiation decreases as k0−2 [31, 32]. The applicability condition of the WKB expansion for the boundary point can be roughly written as a2 1. k04 (1 − βk )4
(55)
This radiation is linearly polarized with the polarization vector independent of the photon energy e=
a(1 − βk ) + (β − n)ak . |a(1 − βk ) + (β − n)ak |
(56)
If eπ and eσ are the polarization vectors of the π and σ components, respectively, then the Stokes parameters in this basis are ξ1 = 2(eπ e)(eσ e),
ξ2 = 0,
ξ3 = (eπ e)2 − (eσ e)2 .
(57)
The comparison of the ultraviolet asymptotics of the spectral angular distribution of radiation with its exact values is given in Fig. 4. e.
Infrared limit. It is not difficult to find the infrared asymptotics, k0 → 0, of the spectral
angular distribution of radiation. The Fourier transform of the current density components in this case looks as follows r
2 −3/2 iυ+ (0) |ω| (20εφ0 )−1/3 h2 φ02 + , λ kµ x˙ µ (0) r 20 1/5 iυ− (0) Γ(1/5) 2 −3/2 |ω| (20εφ0 )−1/3 + , j− ≈ 2 5 λ ih kµ x˙ µ (0) r 20 1/5 Γ(1/5) 2 −3/2 iυ3 (0) j3 ≈ r3 (0) |ω| (20εφ0 )−1/3 , + 2 5 λ ih kµ x˙ µ (0) r 20 3/5 iυ1 (0) Γ(3/5) 2 −3/2 + j1 ≈ − sgn(ω) |ω| (20εφ0 )−1/3 h2 φ0 . 5 λ ih2 kµ x˙ µ (0) j+ ≈ −4i
(58)
Consequently, in the leading order at k0 → 0, we have jµ∗ j µ ≈
(k− x˙ µ − e− ˙ 2 k− x˙ 2 − 2(k x)υ ˙ − µ (k x)) = , 2 (k x) k− (k x) ˙ 2 k− ˙ 2 14
(59)
where all the quantities on the right-hand side are taken at the initial instant of time. It is also easy to check that, in this approximation,
jµ j µ = −jµ∗ j µ ,
(60)
i.e., the radiation is linearly polarized. The spectral angular distribution of radiation and the polarization vector are written as
dE = e e=
2 2(1
k
− βk )(1 − (βe− )) − (1 − e− )(1 − β 2 ) dk0 dΩ , k 4π 2 (1 − e )(1 − βk )2
β(1 − |β(1 −
k e− ) k e− )
−
k
− e− (1 − βk ) − n(βk − e− ) k
− e− (1 − βk ) − n(βk − e− )|
(61)
,
k
where e− := (e− n). The Stokes parameters are found according to (57). Notice that formula (61) is exact in the limit k0 → 0.
IV.
CONCLUSION
We have described in detail both analytically and numerically the properties of radiation created by a classical scalar charged particle in a crossed field with radiation reaction taken into account employing the exact solution to the Landau-Lifshitz equation. The results of numerical simulations illustrating and confirming the analytical formulas are summarized in Figs. 2, 3, and 4. These results clearly show the existence of the two types of electromagnetic radiation, which we call the de-excited and synchrotron entrance ones. These two types of radiation possess distinct characteristic energy scales, polarization properties, and angles of observation. All these characteristics were described in the paper, and simple analytical formulas were obtained for them. In particular, it turns out that the value of the Stokes parameter ξ2 at the maximum of the spectral angular distribution of the de-excited radiation is universal: |ξ2 | ≈ 0.98. So, the de-excited radiation is almost completely circularly polarized. We found the conditions when the radiations mentioned can be observed and estimated their radiation formation lengths. The minimal length scale (44) needed for the deexcited radiation to be formed was also determined. The both types of radiation can be observed in the forthcoming experiments [11, 12] with scattering of electrons on intense laser pulses as it was discussed in [6, 7]. 15
ACKNOWLEDGMENTS
We are thankful to V.G. Bagrov and D.V. Karlovets for fruitful conversations. We also appreciate the anonymous referee for valuable comments. The work is supported by the RFBR grants No. 16-02-00284 and No. 16-32-00464-mol-a, and by the Russian Federation President grant No. MK 5202.2015.2.
Appendix A: Asymptotics of the function F (φ)
Let us consider the integral Z
dye−i(y
5 +ay 3 +by)
C
,
a, b ∈ C,
(A1)
where the integration contour C is depicted in Fig. 1. This integral is an entire function of a and b. We are interested in the case when a=−
10 0 φ, 3
φ = φ0 + iφ00 ,
b = 5|φ|2 ,
φ0 , φ00 ∈ R.
(A2)
Let us define F (φ) :=
Z
dye−i(y
5 − 10 φ0 y 3 +5|φ|2 y) 3
.
(A3)
C
We give the expansion of the function F (ϕ) in the following regions (i) |φ0 |5/6 20−1/3 , |φ00 |4 4|φ0 |3/2 ;
(ii) 25|φ00 |3/2 |φ|1/4 ;
(iii) |φ| 1.
(A4)
In the case (i), we have the asymptotic expansion n 2π −i[ 38 (φ0 )5/2 +5(φ00 )2 (φ0 )1/2 ] F (φ) = e Ai(B)− (20φ0 )1/3 h B 4 27B i o 13B 2 0 ih 2 B Ai(B) + 2 Ai0 (B) − h2 + Ai(B) + Ai (B) + · · · , (A5) − 4 32 40 40
where h := 20−1/3 (ϕ0 )−5/6 , B := 5(φ00 )2 (20φ0 )−1/3 . The condition (i) means that h 1,
B 2 h 1.
(A6)
For φ0 < 0, it is necessary to put φ0 → φ0 − i0 in (A5) and take the principal branches of the multi-valued functions. In the case (ii), the standard WKB method gives in the leading order r √ ∗ 2 1 ∗ 2 π −i 10 φ |φ| − 5 (φ ) 3 √ ∗e . F (φ) ≈ 00 10φ φ 16
(A7)
The condition (ii) guarantees the applicability of the WKB method. In the case (iii), we deduce π
F (φ) = (i + e−i 10 )Γ
6
4 2 h i π 7π 2 4 − (i − ei 10 )Γ φ0 − (i − e−i 10 )Γ |φ|2 − (φ0 )2 + 5 3 5 5 9 3 h i 7π 5 4 32 + (i + ei 10 )Γ |φ|4 − (φ0 )2 |φ|2 + (φ0 )4 + o(|φ|4 ). (A8) 2 5 3 81
Appendix B: Functions I˜n (x, D)
The function I˜n (x, D) (see (49)) is an entire function of the complex variables x and D. The following representation holds I˜0 (x, D) =
∞ X
(−ix)n
n=0
I˜2n (D) , n!
I˜n (D) := I˜n (0, D).
(B1)
An analogous expansion is valid for I˜1 (x, D) with the replacement of I˜2n (D) by I˜2n+1 (D). In virtue of the relations, (n) I˜n (D) = in I˜0 (D),
I˜000 (D) = DI˜0 (D) + i,
(B2)
all I˜n (D), n ∈ N, are expressed through I˜0 (D) and I˜1 (D). For example, I˜2 (D) = −DI˜0 (D) − i,
I˜3 (D) = −DI˜1 (D) − iI˜0 (D),
I˜6 (D) = −(D3 + 4)I˜0 (D) + 6iDI˜1 (D) − iD2 ,
I˜7 (D) = −(D3 + 10)I˜1 (D) − 9iD2 I˜0 (D) + 8D. (B3)
I˜4 (D) = D2 I˜0 (D) − 2iI˜1 (D) + iD,
I˜5 (D) = D2 I˜1 (D) + 4iDI˜0 (D) − 3,
The functions I˜0 (D), I˜1 (D) are reduced to the known special functions iπ iD2 4 5 D3 I˜0 (D) = π Ai(D) − Bi(D) + 1 F2 1; 3 , 3 ; 9 , 3 2 π 3D4 3 7 8 D3 I˜1 (D) = iπ Ai0 (D) + Bi0 (D) − D1 F2 1; 43 , 53 ; D9 − 1 F2 2; 3 , 3 ; 9 . 3 40
(B4)
The function I˜0 (D) (up to the common factor π) is also known as the Scorer function [37]. The hypergeometric function entering into I˜0 (D) is an entire function of D and can be expressed through the Anger functions [38] 3
4 5 z 1 F2 1; 3 , 3 ; 9
4π h = √ J1/3 3 −3z 3
2 3
p −z 3 − J−1/3
2 3
p i −z 3 .
(B5)
In the ultrarelativistic limit and for β⊥ . γ −1 , the magnitude of x is rather small (see (49)). For |x| 1, the expansion (B1) is rapidly converging and so one may keep only a few first terms in 17
it. In the case |x| . 1, it is useful to employ the other representation of the functions I˜0 (x, D) and
I˜1 (x, D):
2 h i 1 ∂ 3 e−ix(D−x ) − 1 2 I˜0 (x, D) = eix(D−2x /3) I˜0 (D − x2 ) + e− 3 ∂D3 , i(D − x2 ) 2 h 1 ∂3 ∂ e−ix(D−x ) − 1 i 2 I˜1 (x, D) = eix(D−2x /3) I˜1 (D − x2 ) + e− 3 ∂D3 − xI˜0 (x, D). ∂D D − x2
(B6)
If |x| . 1 then the expansion of exp[− 31 ∂ 3 /∂D3 ] can be terminated with a few first terms left.
The function I˜0 (x, D) as a function of D satisfies the linear inhomogeneous ordinary differential
equation I˜000 (x, D) − 2ixI˜00 (x, D) − DI˜0 (x, D) = i,
(B7)
where the prime denotes the derivative with respect to D. After the substitution I˜0 (x, D) = eixD f (D − x2 ), this equation is reduced to the inhomogeneous Airy equation for the function f (t). The solution of this equation is found by the standard means. As a result, we deduce the integral representations: I˜0 (x, D) = πc1 (x)eixD Ai(D − x2 ) − πc2 (x)eixD Bi(D − x2 ) Z D Z 2 ix(D−s) 2 2 + iπ Bi(D − x ) dse Ai(s − x ) − iπ Ai(D − x ) 0
D
0
dseix(D−s) Bi(s − x2 ), (B8)
where c1 = e−2ix
3 /3
e−2ix c2 = i 3
−x
3 /3
−x
3 Bi0 (−x2 ) + ix Bi(−x2 ) 2 F2 ( 21 , 1; 23 , 43 ; − 4ix 3 )−
3ix 2
3 Ai0 (−x2 ) + ix Ai(−x2 ) 2 F2 ( 21 , 1; 23 , 43 ; − 4ix 3 )−
3 Bi(−x2 )2 F2 (1, 32 ; 43 , 35 ; − 4ix 3 ) ,
3ix 2
3 Ai(−x2 )2 F2 (1, 32 ; 43 , 53 ; − 4ix 3 ) ,
(B9)
and also 2 2 I˜0 (x, D) = d1 (x)eix(D−x )+iπ/6 Ai(D − x2 ) − d2 (x)eix(D−x )+iπ/6 Bi(D − x2 ) Z D−x2 Z D−x2 2 2 ix(D−x2 −s) 2 + iπ Bi(D − x ) dse Ai(s) − iπ Ai(D − x ) dseix(D−x −s) Bi(s), (B10)
0
0
where d1 =
e
ix3 3
2
ix3
1 3 3 Γ( 3 )Γ( 23 , ix3 )−e2πi/3 Γ( 23 )Γ( 13 , ix3 ) ,
e 3 3 3 d2 = √ Γ( 13 )Γ( 32 , ix3 )+e2πi/3 Γ( 23 )Γ( 13 , ix3 ) . 2 3 (B11) 18
Appendix C: Radiation from the transition layer
In this appendix, we consider the contribution to the spectral angular radiation distribution of the part of particle trajectory passing trough a thin transition layer where the field strength gradually increases from zero to its value (7). The radiation coming from this part of the worldline is actually a particular case of the radiation from a “short magnet”. The properties of this radiation were thoroughly investigated (see, e.g., [31–33] and references therein). Therefore, we give here only the main formulas adapted to our case. In the thin layer described above, the 4-momentum of a charged particle can be approximated by x˙ 0 = x˙ 0 (0) cosh
wτ 2 wτ 2 − x˙ 1 (0) sinh , 2T 2T x ¨20
−
x˙ 1 = −x˙ 0 (0) sinh
x ¨21
wτ 2 wτ 2 + x˙ 1 (0) cosh , 2T 2T
τ2 = −w 2 , T
(C1)
2
where w is the acceleration modulus in the momentary comoving frame at the proper time T . Further, we assume that wT 1.
(C2)
Then x0 ≈ x˙ 0 (0)τ −
wτ 3 x˙ 1 (0), 6T
x1 ≈ x˙ 1 (0)τ −
wτ 3 x˙ 0 (0), 6T
(C3)
where we have set x0 (0) = x1 (0) = 0. In the ultrarelativistic case, we obtain i h wτ 3 wτ 3 τ− (1 − βk ) − (1 + β ) k , 6T 12γ 2 T h h wτ 2 wτ 2 i wτ 2 1 i x˙ 0 ≈ γ 1 − + 2 , x˙ 1 ≈ γ 1 − − 2 . 2T 4γ T 2T 2γ
(C4)
k0 T γ
(C5)
kµ xµ ≈ k0 γ
If
the exponent in (15) can be approximated by the first two terms of its Taylor series. For the photon energy at the maximum of synchrotron radiation (50), the requirement (C5) moves to (C2). Hence, in the leading order j0 ≈
2) iwT (2γ 2 − 1 − γ 2 β⊥ , 2 )2 k0 (1 + γ 2 β⊥
19
j1 ≈
iwT (2γ 2 − 1) 2 )2 , k0 (1 + γ 2 β⊥
(C6)
where it is supposed that β⊥ ∼ γ −1 . We see that, in this approximation, the radiation formed is linearly polarized. Besides, dE =
e2 γ 2 w 2 T 2 2 2 2 )4 γ β⊥ dk0 dΩ. π 2 (1 + γ 2 β⊥
(C7)
The spectral angular distribution does not depend on the photon energy in the interval (C5) [31, 32] as distinct from the synchrotron entrance radiation described by (48). Comparing by the order of magnitude (C7) with the spectral angular distribution following from (48), one can see that the synchrotron entrance radiation dominates over (C7) provided (wT )2 1,
(C8)
i.e., whenever (C2) is satisfied. The condition (C2) can be written as ∆ γχ−1 ≈ |ω|−1 ,
(C9)
where ∆ is the width of the transition layer. This condition coincides with that deduced from the general considerations given below formula (51).
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21
FIGURES
1.0
x3 q 0.5
k j
E
1
x2
x
s0
0.0
C0
2
x
C
0.5
H 1.0 0.0
0.5
1.0
1.5
2.0
FIG. 1. Left panel: Schematic sketch of electron’s dynamics and the system of coordinates. Right panel: Deformation 2 of the integration contour in the s plane. The point s0 corresponds to (2λ|ω|υ− (0))−1/2 .
1.0 12 102 dE dWdk 0
5
10
0.5
102 dE dWdk 0
Ξ2
8 6
0.0
a)
4 3 2 1 0 0.0
1.0
2.0 106 k0
0.6
0.8
4
3.0
2
-0.5
0 0.0
0.2
0.4
104 k0
-1.0 0
1
1.0
2
3
4
5
104 k0 80
12
Γ=104 Γ=103 Γ=5´102 Γ=2´102 Γ=102
0.12
b)
102 dEdWdk0
60
40
20
0
0.10
8 6
0.08
4
c)
2
0.38
0.40
0.42
0.06 0.04
0 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 k0=0 k0=10-9´k0m k0=10-7´k0m k0=k0m k0=10´k0m
dEdWdk0
102 dE dWdk 0
10
j
0.02
0.44 Θ
0.46
0.48
0.50
22
0.00 0.0
0.5
1.0 104 k0
1.5
2.0
FIG. 2. De-excited radiation for γ = 5 × 103 , ω = −1.47 × 10−4 , and the entrance angle θ0 = ϕ0 = π/2. The observation angles θ, ϕ are counted in the units of π on the plots. a) The dependence of the Stokes parameter ξ2 on the radiated photon energy at the observation angles θ = 7π/15, ϕ = π/5. The blue line is the numerical result. The dashed line is (37). The de-excited radiation is almost completely circularly polarized. The radiation at extremely low and high frequencies is linearly polarized in compliance with (56), (61). The inset: The spectral angular distribution at θ = 7π/15, ϕ = π/5. The orange line is the numerical result. The dashed line is (30). The small plot is the same for θ = 9π/20, ϕ = π/10. We see that formula (30) can describe the radiation quite well even out of the applicability domain (39). b) The spectral angular distribution for different photon energies at ϕ = π/5. Here k0m ≈ 1.20 × 10−5 ≈ 6.13 eV is given by (31) at θ = 7π/15, ϕ = π/5. The vertical dashed line is θ = 7π/15. The de-excited radiation is not concentrated in the plane of particle motion. The inset: The same for θ = 7π/15. The vertical dashed line is ϕ = π/5. The two peaks correspond to the de-excited and synchrotron entrance radiations. The curve for k0 = 0 coincides with (61). c) The dependence of the spectral angular distribution on the initial energy of a charged particle at θ = 7π/15, ϕ = π/5. The maximum of the curve is well described by (30), (31) and its magnitude does not change in a wide range of γ’s.
0.6
0
0.3
-1 -1 -2 -2 -3 -3
j
-5
0.2
0.5 0.4 -4
-4 H715, 15L
H920, 110L
0.3 0.2
-6
-6
0.1
-5
j
0.4
1.0 0.5 0.0 -0.5 -1.0
0.1 0.0
0.44
0.46
0.48
0.50 Θ
0.52
0.54
0.0
0.56
0.35
0.40
0.45
0.50 Θ
0.55
0.60
0.65
FIG. 3. The same as in Fig. 2. Left panel: The applicability domain of (30) as it follows from (39) for the radiated photon energy taken at the maximum (31). The level lines are the lines of a constant lg k0m . Right panel: The Stokes parameter ξ2 at the photon energy k0m ≈ 1.20 × 10−5 ≈ 6.13 eV. The plateaus |ξ2 | ≈ 1 of the circularly polarized de-excited radiation are clearly seen.
23
5
5
2.0 2.0
3
1.5
1.0
1
2
0
0.6
2 Γ
Π 2
0.8
10-4 dEdWdk0
1 Γ
2
3 10-4 dE dWdk0
10-3 dEdWdk0
4
Θ
0.4
1
1.5 1.0
1.0
0.5 0.0
0.4996 0.4998 0.5000 j
0.5
0.2 1.5 2.0
0
10-4 dE dWdk 0
10-3 dE dWdk 0
4
0
1
3.0 -4 5.0 7.0 10.0 15.0 10 k0
2
3
4
0.0
5
10-4 k0
0.45
0.46
0.47
0.48 j
0.49
0.50
0.51
FIG. 4. Synchrotron entrance radiation for γ = 5 × 103 , ω = −1.47 × 10−4 , r3 (0) = 0, and the entrance angle θ0 = ϕ0 = π/2. The orange line is the numerical result, the dashed line follows from (48), and the dot-dashed line is (54). Left panel: The spectral angular distribution of the forward synchrotron entrance radiation. The inset: The spectral angular distribution of the synchrotron entrance radiation for ϕ = π/2 and the photon energy k0m ≈ 5358 ≈ 2.74 GeV given by (50). The small plot: The ultraviolet asymptotics of the spectral angular distribution. Right panel: The spectral angular distribution of the synchrotron entrance radiation for θ = π/2 and the photon energy k0m ≈ 5358. On the plots, ϕ is counted in the units of π. The inset: The fine structure of the main peak. Formula (48) leads to a spectral angular distribution describing with a good accuracy the actual distribution in the cone opening less than or equal to 50γ −1 .
24
S0003-4916(17)30085-4 http://dx.doi.org/10.1016/j.aop.2017.03.010 YAPHY 67355
To appear in:
Annals of Physics
Received date: 17 August 2016 Accepted date: 12 March 2017 Please cite this article as: O.V. Bogdanov, P.O. Kazinski, G.Y. Lazarenko, Properties of an ultrarelativistic charged particle radiation in a constant homogeneous crossed electromagnetic field, Annals of Physics (2017), http://dx.doi.org/10.1016/j.aop.2017.03.010 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Highlights
>Properties of an electron radiation in a crossed electromagnetic field are studied.> >Spectral angular distribution of the synchrotron entrance radiation is described.> >Spectral angular distribution of the de-excited radiation is described.> >De-excited radiation is almost completely circularly polarized.> >Photon energy at the maximum of the de-excited radiation decreases with increasing the initial energy of an electron.>
*Manuscript Click here to view linked References
Properties of an ultrarelativistic charged particle radiation in a constant homogeneous crossed electromagnetic field O.V. Bogdanov,1, 2, ∗ P.O. Kazinski,1, 2, † and G.Yu. Lazarenko1, ‡ 1 Physics
Faculty, Tomsk State University, Tomsk, 634050 Russia
2 Department
of Higher Mathematics and Mathematical Physics,
Tomsk Polytechnic University, Tomsk, 634050 Russia (Dated: February 23, 2017)
Abstract The properties of radiation created by a classical ultrarelativistic scalar charged particle in a constant homogeneous crossed electromagnetic field are described both analytically and numerically with radiation reaction taken into account in the form of the Landau-Lifshitz equation. The total radiation naturally falls into two parts: the radiation formed at the entrance point of a particle into the crossed field (the synchrotron entrance radiation), and the radiation coming from the late-time asymptotics of a particle motion (the deexcited radiation). The synchrotron entrance radiation resembles, although does not coincide with, the ultrarelativistic limit of the synchrotron radiation: its distribution over energies and angles possesses almost the same properties. The de-excited radiation is soft, not concentrated in the plane of motion of a charged particle, and almost completely circularly polarized. The photon energy delivering the maximum to its spectral angular distribution decreases with increasing the initial energy of a charged particle, while the maximum value of this distribution remains the same at the fixed photon observation angle and entrance angle of a charged particle. The ultraviolet and infrared asymptotics of the total radiation are also described. Keywords: Spectral angular radiation distribution, polarization, Landau-Lifshitz equation, crossed electromagnetic field.
∗ † ‡
E-mail:[email protected] E-mail:[email protected] E-mail:[email protected]
1
I.
INTRODUCTION
The form of radiation of an ultrarelativistic charged particle moving in a crossed electromagnetic field is discussed in many textbooks and papers (see, e.g., [1–4]). However, according to these approaches to the problem, the radiation reaction acting on a charged particle is completely neglected or taken into account only as a small perturbation. This is not always a justified approximation, especially when the external electromagnetic field is strong enough or the particle spends a sufficiently long time in the field [5–7]. We describe (to our knowledge, for the first time) in detail the properties of radiation created by a classical charged scalar particle entering a constant homogeneous crossed electromagnetic field and moving in it for an infinite time with radiation reaction taken into account. The infinite time condition can be relaxed under certain restrictions we shall specify. Our investigation shows that, loosely speaking, this radiation consists of two parts: the first one is an intense hard radiation created by a charged particle at the entrance point (we shall call it, for brevity, the synchrotron entrance radiation), and the second one is a comparatively weak soft radiation formed on the late-time asymptotics of particle trajectory (the de-excited radiation) [6]. The properties of the synchrotron entrance radiation resemble the properties of an ultrarelativistic asymptotics of the synchrotron radiation. In particular, the maximum of the spectral density is reached at the photon energy the same as for the synchrotron radiation, and the most part of the radiation is concentrated in the cone opening of the order 2γ −1 , where γ is the Lorentz factor. It is this radiation which is subject to considerable quantum corrections at high external field intensities. Notice that this is not the so-called edge radiation as it is standardly defined (see the definition, e.g., in [8, 9]). The de-excited radiation is soft, its spectrum stretches from far-infrared to hard X-rays (for γ ∼ 104 and the field strengths corresponding to the intensity I ∼ 1022 W/cm2 ), and the quantum corrections to it are negligible. We find the region of observation angles and energies of photons where these two types of radiation can be discerned and observed. Besides that the synchrotron entrance and de-excited radiations are distinguished by the ranges of the photon energies and observation angles where these radiations are concentrated, it turns out that they possess distinct polarization properties. The de-excited radiation is almost completely circularly polarized, while the synchrotron entrance radiation has mostly a linear polarization. We present a detailed description of the polarization properties of these two types of radiations both analytically and numerically. The study of properties of an ultrarelativistic charged particle radiation in a crossed electro2
magnetic field is important for two reasons, at least. First, in the ultrarelativistic limit, every electromagnetic field becomes crossed in the momentary comoving frame. Therefore, one may expect that, in a certain approximation, the properties of radiation we discussed in this paper should be inherent to any electromagnetic field, which can be considered as constant and homogeneous on the radiation formation length scale. Second, the crossed field approximation is the standard one [1, 10] in considering the radiation of charged particles in strong laser fields. The intensity of the laser radiation, which will become accessible in the nearest future [11, 12], allows one to observe the radiation reaction effects we discuss [5–7, 10, 13–15]. The study of classical radiation is relevant even in that domain of parameters where one expects considerable quantum corrections. In this case, the classical results can be used to distinguish clearly the quantum corrections. This idea was employed, for example, in a recent proposal [16]. Notice that the numerical simulations of dynamics of electrons and their radiation in a strong laser wave and in crystals with radiation reaction taken into account already present in the literature (see, e.g., [13–15, 17–24]). The exact solution to the Landau-Lifshitz equation [25] in the crossed electromagnetic field is also known for a long time (see [26] and references therein). However, the problem of radiation coming from the whole electron trajectory in a crossed field has somehow escaped an in-depth study. The aim of the present paper is to fill this gap and to provide an analytical and numerical investigation of the radiation properties with radiation reaction taken into account.
II.
NOTATION AND SOLUTION TO THE EQUATIONS OF MOTION
The action functional of a charged particle with the charge e and the mass m interacting with the electromagnetic field Aµ on the Minkowski spacetime R1,3 with the metric ηµν = diag(1, −1, −1, −1) has the form S[x(τ ), A(x)] = −m
Z
dτ
√
x˙ 2
−e
Z
1 dτ Aµ x˙ − 16π µ
Z
d4 xFµν F µν ,
(1)
where Fµν := ∂[µ Aν] is the strength tensor of the electromagnetic field, and the speed of light c = 1. The Landau-Lifshitz equation in the natural parametrization x˙ 2 = 1 is written as [25] 2 e ˙ e2 e2 m¨ xµ = eFµν x˙ ν + e2 Fµν x˙ ν + 2 Fµν F νρ x˙ ρ − 2 x˙ λ Fλν F νρ x˙ ρ x˙ µ . 3 m m m
(2)
For a thorough discussion of its applicability to the problem at hand see, e.g., [6, 7] and references therein. Let us choose the Compton wavelength lC := ~/mc as the length unit: xµ → lC xµ , 3
τ → lC τ.
(3)
Then Eq. (2) is reduced to x ¨µ = fµν x˙ ν + λ(f˙µν x˙ ν + fµν f νρ x˙ ρ − x˙ λ fλν f νρ x˙ ρ x˙ µ ),
(4)
where xµ , τ , fµν , and λ are dimensionless quantities, and fµν = sgn(e)Fµν /E0 . It is useful to bear in mind that the electromagnetic field strength is measured in the units of the critical field E0 , and the unit of energy is the electron rest energy: lC ≈ 3.86 × 10−11 cm, E0 =
tC ≈ 1.29 × 10−21 s,
m2 ≈ 4.41 × 1013 G = 1.32 × 1016 V/cm, |e|~
m ≈ 5.11 × 105 eV, λ=
2α 2 ≈ , 3 411
(5)
The modern accelerator facilities are able to accelerate electrons up to 25 GeV and higher, and the intensities of the achievable at the present moment laser fields [27] are of the order 1022 W/cm2 . These data correspond to γ ≈ 105 ,
ω ≈ 1.47 × 10−4 ,
λω ≈ 7.14 × 10−7 ,
(6)
where ω is the electromagnetic field strength in the laser wave. The field strength for the constant homogeneous crossed electromagnetic field is [µ ν]
f µν = ωe− e1 ,
2 − fµν = ω 2 e− µ eν ,
ex = −ω,
hz = ω,
x1 ≥ 0,
(7)
where ω is a constant, ex and hz are the non-zero fµν components, x− = x0 − x2 , and the 4-vectors eµ− = (1, 0, 1, 0), eµ1 = (0, 1, 0, 0) were introduced. Any 4-vector can be represented in the form 1 + 1 3 jµ = (j+ e− µ + j− eµ ) − j1 eµ − j3 eµ , 2
(8)
where eµ+ = (1, 0, −1, 0), eµ3 = (0, 0, 0, 1), and ja := eµa jµ . In particular, υ− υ+ − υ12 − υ32 = 1,
∗ jµ∗ j µ = Re(j+ j− ) − |j1 |2 − |j3 |2 ,
jµ ∈ C,
(9)
where υa := x˙ a , and the dot denotes the derivative with respect to the natural parameter. The solution to the Landau-Lifshitz equation for such a field configuration can be cast into the form [6, 26, 28–30] h i2 1 −1 −1 r+ = r32 (0) + r¯(0) + (υ− (0) + λω 2 x− )2 + (υ− (0) + λω 2 x− )2 , 2λω 1 −1 r1 = r¯(0) + (υ −1 (0) + λω 2 x− )2 , υ− = (υ− (0) + λω 2 x− )−1 , r3 = r3 (0), 2λω −
(10)
2 (0))−1 . The initial data can be written as where ra := υa /υ− and r¯(0) := r1 (0) − (2λωυ−
r1 (0) = −
β(0) sin θ0 sin ϕ0 , 1 − β(0) sin θ0 cos ϕ0
r3 (0) = −
β(0) cos θ0 sin ϕ0 , 1 − β(0) sin θ0 cos ϕ0
υ− (0) = γ(0)(1 − β(0) sin θ0 cos ϕ0 ), 4
(11)
where β(0) is the initial 3-velocity modulus, and θ0 , ϕ0 determine the entrance angle. The definition of angles θ, ϕ and the schematic sketch of electron’s dynamics are presented in Fig. 1. The trajectory is r¯(0) + λω r2 (0) + r¯2 (0) 1 x+ = x+ (0) − 2 3 3 − 3 2 − 5 (0) + 3 λω υ− (0) 3λ ω υ− (0) 20λ ω 4 υ− r 2 −3/2 h 2 2 s5 i + |ω| r3 (0) + r¯2 (0) s + sgn(ω)(¯ r(0) + λω)s3 + , λ 3 5 r r¯(0) 1 2 −3/2 h s3 i x1 = x1 (0) − − |ω| r ¯ (0)s + sgn(ω) , + 3 (0) λω 2 υ− (0) 6λ2 ω 3 υ− λ 3 r 2 −3/2 r3 (0) + |ω| r3 (0)s, x3 = x3 (0) − 2 λω υ− (0) λ r 2 −3/2 1 x− = |ω| s− , 2 λ λω υ− (0)
(12)
where the last equality is the definition of s, and we put x− (0) = 0. The velocity components (10) in terms of this new variable become r+ = r¯2 (0) + r32 (0) + 2 sgn(ω)(¯ r(0) + λω)s2 + s4 ,
r1 = r¯(0) + sgn(ω)s2 ,
r3 = r3 (0). (13)
We suppose that the charged particle enters the electromagnetic field at the instant τ = 0 and then moves in it for an infinite time. The last assumption can be relaxed if the radiation formation time is much shorter than the time that the particle spends in the electromagnetic field. Further, we shall give the estimates for the radiation formation time and length. The case of particles reflected from the electromagnetic field [7] will be considered elsewhere.
III.
a.
RADIATION
General formulas. The spectral angular distribution of a one charged particle radiation
summed over the photon polarizations is written as dE(k) = |E(k)|2
dk R2 dk = −e2 jµ∗ (k)j µ (k) 2 , 4π 4π 2 k02
k 2 = 0,
(14)
k µ jµ (k) = 0.
(15)
where jµ (k) =
Z
τ2
τ1
dτ x˙ µ e−ikν x
ν (τ )
−
ix˙ µ −ikν xν τ2 e , kλ x˙ λ τ1
Notice that if one calculates the contribution of the boundary points in the integral (15) by the WKB method (when this approximation is applicable) then the leading WKB contribution is canceled by 5
the out of the integral term in (15). The projections of the radiation electric field in the wave zone take the form Eα (k) = −iek0
e−ik0 R µ b(α) jµ (k), R
α = 1, 2,
(16)
where bµ(α) are the physical photon polarization vectors and R is a distance from the source of radiation to the observation point. The Stokes parameters are ξ1 = 2
Re(E1 E2∗ ) , |E|2
ξ2 = 2
Im(E1 E2∗ ) , |E|2
ξ3 =
|E1 |2 − |E2 |2 . |E|2
(17)
In particular, ξ12 + ξ32 = 1 − ξ22 =
|jµ j µ |2 |E2 |2 = . |E|4 (jν∗ j ν )2
(18)
In the case of circularly polarized wave, ξ2 = ±1. At ξ2 = 0, the electromagnetic wave is linearly polarized and ξ1 = sin(2ϑ),
ξ3 = cos(2ϑ),
(19)
where ϑ is the angle between the polarization plane and the axis with the unit vector b(1) . Thus the problem is reduced to the evaluation of integrals (15), where, in our case, τ1 = 0 and τ2 → ∞. The out of the integral term corresponding to τ2 → ∞ is absent in (15). It turns out that the main characteristics of the radiation created by a charged particle in the crossed field with radiation reaction taken into account can be described analytically with a rather good accuracy. b.
De-excited radiation. The expression standing in the exponent in (15) can be cast into the
form kµ xµ = kµ xµ (0) −
i k− h |z|2 z 0 − λω 1 − − 3 (0) 5 (0) 2 λω 2 υ− (0) 3λ2 ω 3 υ− 20λ3 ω 4 υ− r i 2 k− h 10 2 0 3 5 5|z| s − sgn(ω)(z − λω)s + s , (20) + λ 10|ω|3/2 3
where z ≡ z 0 + iz 00 := ζ 0 − r¯(0) + i(ζ 00 − r3 (0)),
ζ ≡ ζ 0 + iζ 00 :=
k1 + i|k3 | , k−
k− =
2k0 . (21) 1 + |ζ|2
The coordinates ζ 0 , ζ 00 are coordinates of the stereographic projection of the sphere |k| = 1 to the plane (k 1 , k 3 ) from the pole (0, 1, 0). In the ultrarelativistic limit, r1 and r3 are also coordinates of the stereographic projection of the sphere |υ| = 1 to the plane (υ 1 , υ 3 ) from the pole (0, 1, 0). 6
It is convenient to introduce the notation φ := φ0 + iφ00 , ρ := ρ0 + iρ00 , where φ0 := sgn(ω)(z 0 − λω),
|φ|2 := |z|2 ⇔ (φ00 )2 = (z 00 )2 + 2λωz 0 − λ2 ω 2 ,
ρ0 := − sgn(ω)(¯ r(0) + λω), r 2 k− , ε := λ 10|ω|3/2
|ρ|2 := r¯2 (0) + r32 (0),
(22)
and ψ0 is the expression in the first line on the right-hand side of equality (20). In particular, the following relation holds (φ0 − ρ)(φ0 − ρ∗ ) − (φ0 − ρ0 − λ|ω|)2 − r32 (0) = 2λ|ω|φ0 .
(23)
Using the new notation, r+ = (s2 − ρ)(s2 − ρ∗ ),
r1 = −λω + sgn(ω)(s2 − ρ0 ).
(24)
This notation is useful for the evaluation of integrals (15). The integrals determining the Fourier transform of jµ are of the same type as considered in [6] (see also App. A) and are calculated analogously. It is useful to deform the integration contour as depicted in Fig. 1. The radiation created by a de-excited charged particle corresponds to the case (i) in (A4). Let us change the integration variable y = (20εφ0 )1/3 (s −
p φ0 ).
(25)
Then 8
+ 5(φ00 )2 (φ0 )1/2 + By + y 3 /3 + hy 4 /4 + h2 y 5 /20, ε2 1/3 h : = (20ε)−1/3 (φ0 )−5/6 , B := 5 (φ00 )2 . 20φ0
kµ xµ = ψ0 + ε
3 (φ
0 5/2
)
(26)
Having neglected the out of the integral terms in (15), the contributions coming from the contour C0 , and assuming that (A6) is fulfilled, we obtain up to the terms of the order h2 : 0 0 0 0 ∗ 0 0 0 ∗ iI4 c−1 0 j+ = (φ − ρ)(φ − ρ )I0 + h 4φ (φ − ρ )I1 − (φ − ρ)(φ − ρ ) 4 h iI5 i I 8 + h2 2φ0 (3φ0 − ρ0 )I2 − φ0 (φ0 − ρ0 )iI5 − (φ0 − ρ)(φ0 − ρ∗ ) + , 32 20 ih I8 iI5 c−1 I4 − h2 + , j3 = r3 (0)j− , 0 j− = I0 − 4 32 20 n iI4 c−1 j = sgn(ω) (φ0 − ρ0 − λ|ω|)I0 + h 2φ0 I1 − (φ0 − ρ0 − λ|ω|) 1 0 4 io h iI I iI 5 8 5 ) − (φ0 − ρ0 − λ|ω|) + . + h2 φ0 (I2 − 2 32 20 7
(27)
It is the order h2 where the leading contribution to the de-excited radiation comes from. Here Z r 8 2 −3/2 3 −iψ0 −iε 3 (φ0 )5/2 +5(φ00 )2 (φ0 )1/2 0 −1/3 dttn e−i(Bt+t /3) . (28) |ω| (20εϕ ) , In (B) := c0 := e λ C The integrals In (B), n ∈ N, are expressed through the Airy functions (see [6]). As a result, ∗ Re(j+ j− ) − |j1 |2 − |j3 |2 = n h o iI1 I4 3iI0 I5 I0 I8 I42 i 2 0 2 2 2 0 2 2 = |c0 | 2λ|ω|φ I0 + h I0 I2 + + 4h (φ ) (I1 + I0 I2 ) . (29) − − + 2 5 16 16
In the case when the applicability conditions (39) of the approximation considered are fulfilled, the last term in the curly brackets dominates over the first terms in the square brackets. Thus, according to (9), (14), the spectral angular distribution of the de-excited radiation has the form (cf. [6]) dk0 dζ 0 dζ 00 dk 2 2 0 2 dE(k) ≈ 4e2 B B[Ai(B)]2 + [Ai0 (B)]2 = 4e B B[Ai(B)] + [Ai (B)] 2 (φ00 )2 . (30) (φ00 )2 k− The maximum of this radiation is reached at the photon energy 0 k0m = (Bext |ω|)3/2 (2λ)1/2 (1 + |ζ|2 )
(φ0 )1/2 , |φ00 |3
(31)
0 where Bext ≈ 0.8, and it is supposed that φ0 ≥ 0. If 2 |r1 (0)| (2λωυ− (0))−1 ,
|r1 (0)| λ|ω|,
(32)
φ00 ≈ |ζ 00 − r3 (0)|.
(33)
which are satisfied as a rule, then φ0 ≈ sgn(ω)(ζ 0 − r1 (0)),
Hence, the formulas from [6] for the spectral angular distribution are obtained when one puts r1 (0) = r3 (0) = 0. These initial data were implicitly assumed in [6]. In contrast to the synchrotron radiation, the right-hand member of (31) depends rather weakly on the energy of the incident particle. In increasing the energy, the magnitude of k0m slightly declines due to decreasing of φ0 and tends to a finite value independent of the incident particle energy (see Fig. 2). At that, according to the approximate formula (30), the value of the spectral density at the maximum does not virtually change. The numerical simulations show (see Fig. 2) that there exists a region of observation angles where formula (30) provides a quite good approximation for the spectral density of radiation for all 8
the photon energies. In this case, one can derive the angular distribution of radiation integrating (30) over k0 . Since Z
0
we deduce
∞
1 dtt3/2 t[Ai(t)]2 + [Ai0 (t)]2 = , 16
dE(θ, ϕ) =
3e2 (φ0 )1/2 (2λ|ω|3 )1/2 (1 + |ζ|2 )3 00 5 dΩ. 32 |φ |
(34)
(35)
This expression tends to infinity for |ζ| → +∞ or φ00 → 0. The first case corresponds to the
radiation along the x2 axis. The second case refers to the photon observation angles lying in the plane of motion of a charged particle. In the both cases, formula (30) is not valid (see the applicability domain (39) and Fig. 3). The angular distribution (35) depends on the energy of the incident particle in the same way as (31) at the fixed photon observation angle and entrance angle of a charged particle, viz., it diminishes slightly with increase of γ and tends to a finite value independent of the incident particle energy. Employing (18) and (27), it is not difficult to find the expression for the Stokes parameter ξ2 of the de-excited radiation. Up to the terms of the order h2 , we have h n o ih iI1 I4 3iI0 I5 I0 I8 I42 i jµ j µ = c20 2λ|ω|φ0 I02 − I0 I4 +4iI1 +h2 I0 I2 − −4h2 (φ0 )2 (I12 −I0 I2 ) . − − − 2 2 5 16 16 (36) In the parameter space we consider, the leading contribution gives the last term in the curly brackets. Then, in accordance with (18), we derive s r (B[Ai(B)]2 − [Ai0 (B)]2 )2 2 1/3 1 B 36Γ3 (7/6) 3/2 |ξ2 | ≈ 1 − − 2 + B + O(B 5/2 ) = Γ 3 6 π (B[Ai(B)]2 + [Ai0 (B)]2 )2 π 3/2 1 3 =1− + + O(B −6 ). 3 32B 64B 9/2 (37) 0 , we have The series for B large is asymptotic. In particular, at B = Bext
|ξ2 | ≈ 0.98,
(38)
i.e., the radiation is almost completely circularly polarized. In general, the Stokes parameter at the maximum may depend on the two observation angle variables, the field strength, and the initial momenta of a charged particle, but in our case it is universal. The dependence of ξ2 on the angles and radiated photon energy is plotted in Figs. 2, 3. Roughly, one can explain the presence of circular polarization of the de-excited radiation by the fact that this radiation is not concentrated in the 9
plane of motion of a charged particle. Looking to the particle trajectory from the point where the de-excited radiation is maximal, one sees this trajectory as an arc (see Fig. 1). Therefore, the radiation detected there must possess a circular polarization. However, this reasoning neither explains so high degree of its polarization nor gives the universal value (38). One can find a rough estimate for the applicability conditions of expressions (27). These conditions define the domain of energies and photon observation angles where the de-excited radiation dominates. The derivation of these conditions is obvious but rather awkward. We present only the final answer: 0 2/3
a) 2(λk− |φ |)
1,
b)
λ1/3 |ω| 2/3
2k− |φ0 |5/3
1 d) 2 (0)|φ0 | 1, 2λ|ω|υ−
e)
2/3
k− (φ00 )2 . 1, c) 2(λ|φ0 |)1/3 |ω|
1,
16λ1/3 ω 2 5/3
πυ− (0)k− (φ00 )2
(39)
1.
Here a) is the condition that the last term in (29) is much greater than the first one; b) is the condition h2 1; c) is B . 1 what is necessary since (30) is exponentially suppressed for B large; d) is the radiation formation condition (the saddle points are located sufficiently far from the integration contour boundary); e) implies that the contribution (30) dominates over the contribution coming from the contour C0 . Graphically, these conditions are depicted in Fig. 3. Despite the fact that we consider the radiation of a particle that enters the electromagnetic field and then moves in it for an infinite time, let us estimate the formation time of the de-excited radiation. In terms of the variable (25), the de-excited radiation is formed when yf
√
B, B & 1;
yf & 1, 0 < B 1.
(40)
2|φ0 | 1/2
(41)
Hence, keeping in mind (39), we obtain xf−
2|φ0 | 1/2 λ|ω|3
xf− &
, B & 1;
λ|ω|3
, 0 < B 1.
Eventually, employing formula (11) of [7], we come to (cf. (17) of [7]) |φ0 |
λ|ω| 6d 2/3 , B & 1; 2 λ
|φ0 | .
λ|ω| 6d 2/3 , 0 < B 1, 2 λ
(42)
where the notation of [7] was used. It also follows from d) and (41) that λω 2 υ− (0)xf− 1.
(43)
The inequalities (41), (43) are very close to the condition (i) in (10) of [7] specifying the asymptotic regime of particle dynamics. Combining d) with (42), one can deduce the minimal depth of the 10
field needed for the de-excited radiation to be formed: d (6λ2 χ3 )−1 ≈ 7.0 × 103 χ−3 ≈ 2.7 × 10−7 χ−3 cm,
(44)
where χ := |ω|υ− (0) is the parameter characterizing a relevance of quantum effects at the entrance point. As follows from (12), (43), the radiation formation length is `d ≈ x0 &
105 10 −1 −4 ≈ 0.84γ −1 χ−4 cm. 5 (0) ≈ 2.2 × 10 γ χ 40λ3 ω 4 υ−
(45)
On these scales, the field must be approximately crossed, constant, and homogenous along the particle trajectory for the formulas above can be applied. To minimize (44), (45), it can be taken χ ∼ 1 which is a rather mild assumption since χ decreases during the evolution for both classical and quantum considerations. c.
Synchrotron entrance radiation. The ultrarelativistic particle entering the crossed electro-
magnetic field produce an intense hard electromagnetic radiation. This radiation resembles in many respects the ultrarelativistic limit of the synchrotron radiation. In particular, we shall see that the maximum of the spectral density of radiation is reached at the photon energy which is described quite well by the formula for the synchrotron radiation, and the most part of the radiation is concentrated in the cone opening 2/γ. However, the form of the spectral angular distribution deviates from the ultrarelativistic limit of the synchrotron radiation due to the fact that the acceleration of a charged particle changes stepwise when it enters the electromagnetic field. This is not the edge radiation, which is a long-wavelength radiation as defined in [8, 9]. Though, of course, the distortion of the synchrotron spectrum is the “edge effect”, i.e., the result of an abrupt change of particle acceleration. A similar radiation was discussed in [31, 32] (see also [33]). The derivation of the radiation spectral angular distribution of an ultrarelativistic particle entering the electromagnetic field is analogous to the derivation of an ultrarelativistic asymptotics of the synchrotron radiation (see, e.g., [25, 32, 34]). It is convenient to parameterize the particle worldline by the laboratory time t. If the modulus of the radiation observation angle counted from the entrance direction of a particle is less than or of order γ −1 , and the incoming particle is ultrarelativistic, then 1 − βk ≈
2 1 + γ 2 β⊥ , 2γ 2
β¨k ≈ −|a⊥ |2 ≈ −|a|2 ,
β ⊥ := β − βk n,
(46)
˙ βk := (βn), and n := k/k0 . Also where a := β, h t2 t3 i , kµ xµ ≈ kµ xµ (0) + k0 (1 − βk )t − β˙ k − β¨k 2 6 11
β ⊥ ≈ β ⊥ (0) + a⊥ (0)t.
(47)
Therefore, up to an irrelevant common phase factor, the electric field strength of radiation created by a charged particle entering the electromagnetic field is Z ∞ k0 β ⊥ (0) µ µ dtβ ⊥ eikµ [x (0)−x (t)] RE = e − iek0 kµ x˙ µ (0) 0 n β a⊥ (2D)1/2 I˜1 (x, D) o ⊥ ≈e , 1 − iDI˜0 (x, D) − i 1 − βk a⊥ (1 − βk )1/2
(48)
where it is assumed that β⊥ . γ −1 , all the kinematic quantities in the approximate equality are taken at the initial instant of time, and the following notation was introduced: Z ∞ 2k 2 1/3 k 2/3 2 3 0 2 dttn e−i(Dt+xt +t /3) , D := (1 − βk ) 20 I˜n (x, D) := ≈ (1 + γ 2 β⊥ ) , 3 a 2aγ 0 (49) ak 2D 1/2 x := − . 2a 1 − βk
The integration contour in I˜n (x, D) is shifted slightly below the real axis at t → +∞ so that the
integral converges for all Re n > −1. Some properties of the functions In (x, D) are presented in App. B. The spectral angular distribution of radiation is obtained from (48) by formula (14), the Stokes parameters being given by expressions (17). The comparison with numerical results is presented in Fig. 4. The fact that the radiation (48) is not the edge radiation (in the narrow sense) is simply seen by comparison of (48) with formula (25) of [8] and formula (15.5) of [9]. In particular, the spectral angular distribution of the edge radiation is independent of the photon energy in contrast to (48). For the forward radiation, β ⊥ = 0 and x ≈ 0. Then it follows from (48) that the radiation is linearly polarized in this case for all the photon energies. The maximum of radiation power is reached at the photon energy 3/2
k0m = 2Dext aγ 3 ,
Dext ≈ 0.81,
(50)
where Dext provides the maximum to the function D|I˜1 (D)|2 . Formula (50) is in a good agreement with the formula for the synchrotron radiation maximum (see, e.g., [25, 32, 34]). The expression for the radiation formation length of this type of radiation is derived similarly to the case of synchrotron radiation (see, e.g., [1]). So we have ` (aγ)−1 ≈ |ω|−1 , k0 & k0m ;
` & κ−1/3 (aγ)−1 ≈ κ−1/3 |ω|−1 , k0 = κk0m , 0 < κ 1, (51)
where k0m is given by (50). Recall that |ω|−1 ≈ 6.8 × 103 ≈ 2.6 × 10−7 cm for (6). Of course, a sharp edge of the electromagnetic field assumed in (7) is just an approximation. However, if the radiation formation length ` is much bigger than the characteristic length scale of the electromagnetic field 12
variation then this approximation is valid and the formulas above are applicable ([35, 36], for more details see App. C). On the other hand, since we do not take into account the contribution of the boundary at the exit point of a particle from the electromagnetic field, the length of particle trajectory passing in the field must be much larger than `. As a rule, in the classical domain and for not very large energies, viz., γ 2 χ3 . 2.2 × 1010 ,
(52)
the radiation formation length (51) of the synchrotron entrance radiation is much smaller than the formation length (45) of the de-excited radiation. In this case, one can distinguish the three cases in considering the radiation of a charged particle penetrating a region filled with the crossed electromagnetic field of a finite width: 1. The width is sufficiently large so that the particle becomes de-excited; 2. The width is so small that neither the synchrotron entrance nor de-excited radiations are formed; 3. The intermediate case when the synchrotron entrance radiation is formed while the de-excited radiation is not. In the first case, the contribution of the boundary at the exit point to the total radiation is negligible since the charged particle is de-excited at the exit point and radiates much less energy than at the entrance point and on the trajectory in the field. The second case is the case of a “short magnet” (see [31, 32] for the properties of this radiation), and the spectral angular distribution is of the form analogous to (C7). In the third case, a separate study is needed, and we leave it for a future research. In the case when the inequality (52) is violated, a charged particle is either de-excited, and then the contribution from the exit point can be neglected, or the radiation is not formed, and we have the case of a “short magnet”. d.
Ultraviolet asymptotics. In order to find the ultraviolet asymptotics of radiation, one can
use the WKB method for the boundary point in evaluating the integrals (15). As we have already noted, the leading WKB contribution is canceled by the out of the integral term in (15). The next to leading contribution gives dE = −e2
d x˙ µ d 1 d x˙ µ x ¨2 (k x) ˙ 2 − 2(x¨ ˙ x)(k x)(k¨ ˙ x) + x˙ 2 (k¨ x)2 dk for , 6 2 (k x) ˙ 4π dτ (k x) ˙ dτ (k x) ˙ dτ (k x) ˙ 13
(53)
where the dot denotes the derivative with respect to an arbitrary time parameter, 4-vectors are contracted with the aid of the Minkowski metric, and all the quantities are taken at the initial instant of time. Choosing the laboratory time as the time variable, we arrive at dE = e2
a2 (1 − βk )2 + 2(aβ)ak (1 − βk ) − a2k (1 − β 2 ) dk0 dΩ (1 − βk )6
4π 2 k02
,
(54)
where ak := (an) and βk := (βn). In increasing the radiated photon energy, the spectral density of radiation decreases as k0−2 [31, 32]. The applicability condition of the WKB expansion for the boundary point can be roughly written as a2 1. k04 (1 − βk )4
(55)
This radiation is linearly polarized with the polarization vector independent of the photon energy e=
a(1 − βk ) + (β − n)ak . |a(1 − βk ) + (β − n)ak |
(56)
If eπ and eσ are the polarization vectors of the π and σ components, respectively, then the Stokes parameters in this basis are ξ1 = 2(eπ e)(eσ e),
ξ2 = 0,
ξ3 = (eπ e)2 − (eσ e)2 .
(57)
The comparison of the ultraviolet asymptotics of the spectral angular distribution of radiation with its exact values is given in Fig. 4. e.
Infrared limit. It is not difficult to find the infrared asymptotics, k0 → 0, of the spectral
angular distribution of radiation. The Fourier transform of the current density components in this case looks as follows r
2 −3/2 iυ+ (0) |ω| (20εφ0 )−1/3 h2 φ02 + , λ kµ x˙ µ (0) r 20 1/5 iυ− (0) Γ(1/5) 2 −3/2 |ω| (20εφ0 )−1/3 + , j− ≈ 2 5 λ ih kµ x˙ µ (0) r 20 1/5 Γ(1/5) 2 −3/2 iυ3 (0) j3 ≈ r3 (0) |ω| (20εφ0 )−1/3 , + 2 5 λ ih kµ x˙ µ (0) r 20 3/5 iυ1 (0) Γ(3/5) 2 −3/2 + j1 ≈ − sgn(ω) |ω| (20εφ0 )−1/3 h2 φ0 . 5 λ ih2 kµ x˙ µ (0) j+ ≈ −4i
(58)
Consequently, in the leading order at k0 → 0, we have jµ∗ j µ ≈
(k− x˙ µ − e− ˙ 2 k− x˙ 2 − 2(k x)υ ˙ − µ (k x)) = , 2 (k x) k− (k x) ˙ 2 k− ˙ 2 14
(59)
where all the quantities on the right-hand side are taken at the initial instant of time. It is also easy to check that, in this approximation,
jµ j µ = −jµ∗ j µ ,
(60)
i.e., the radiation is linearly polarized. The spectral angular distribution of radiation and the polarization vector are written as
dE = e e=
2 2(1
k
− βk )(1 − (βe− )) − (1 − e− )(1 − β 2 ) dk0 dΩ , k 4π 2 (1 − e )(1 − βk )2
β(1 − |β(1 −
k e− ) k e− )
−
k
− e− (1 − βk ) − n(βk − e− ) k
− e− (1 − βk ) − n(βk − e− )|
(61)
,
k
where e− := (e− n). The Stokes parameters are found according to (57). Notice that formula (61) is exact in the limit k0 → 0.
IV.
CONCLUSION
We have described in detail both analytically and numerically the properties of radiation created by a classical scalar charged particle in a crossed field with radiation reaction taken into account employing the exact solution to the Landau-Lifshitz equation. The results of numerical simulations illustrating and confirming the analytical formulas are summarized in Figs. 2, 3, and 4. These results clearly show the existence of the two types of electromagnetic radiation, which we call the de-excited and synchrotron entrance ones. These two types of radiation possess distinct characteristic energy scales, polarization properties, and angles of observation. All these characteristics were described in the paper, and simple analytical formulas were obtained for them. In particular, it turns out that the value of the Stokes parameter ξ2 at the maximum of the spectral angular distribution of the de-excited radiation is universal: |ξ2 | ≈ 0.98. So, the de-excited radiation is almost completely circularly polarized. We found the conditions when the radiations mentioned can be observed and estimated their radiation formation lengths. The minimal length scale (44) needed for the deexcited radiation to be formed was also determined. The both types of radiation can be observed in the forthcoming experiments [11, 12] with scattering of electrons on intense laser pulses as it was discussed in [6, 7]. 15
ACKNOWLEDGMENTS
We are thankful to V.G. Bagrov and D.V. Karlovets for fruitful conversations. We also appreciate the anonymous referee for valuable comments. The work is supported by the RFBR grants No. 16-02-00284 and No. 16-32-00464-mol-a, and by the Russian Federation President grant No. MK 5202.2015.2.
Appendix A: Asymptotics of the function F (φ)
Let us consider the integral Z
dye−i(y
5 +ay 3 +by)
C
,
a, b ∈ C,
(A1)
where the integration contour C is depicted in Fig. 1. This integral is an entire function of a and b. We are interested in the case when a=−
10 0 φ, 3
φ = φ0 + iφ00 ,
b = 5|φ|2 ,
φ0 , φ00 ∈ R.
(A2)
Let us define F (φ) :=
Z
dye−i(y
5 − 10 φ0 y 3 +5|φ|2 y) 3
.
(A3)
C
We give the expansion of the function F (ϕ) in the following regions (i) |φ0 |5/6 20−1/3 , |φ00 |4 4|φ0 |3/2 ;
(ii) 25|φ00 |3/2 |φ|1/4 ;
(iii) |φ| 1.
(A4)
In the case (i), we have the asymptotic expansion n 2π −i[ 38 (φ0 )5/2 +5(φ00 )2 (φ0 )1/2 ] F (φ) = e Ai(B)− (20φ0 )1/3 h B 4 27B i o 13B 2 0 ih 2 B Ai(B) + 2 Ai0 (B) − h2 + Ai(B) + Ai (B) + · · · , (A5) − 4 32 40 40
where h := 20−1/3 (ϕ0 )−5/6 , B := 5(φ00 )2 (20φ0 )−1/3 . The condition (i) means that h 1,
B 2 h 1.
(A6)
For φ0 < 0, it is necessary to put φ0 → φ0 − i0 in (A5) and take the principal branches of the multi-valued functions. In the case (ii), the standard WKB method gives in the leading order r √ ∗ 2 1 ∗ 2 π −i 10 φ |φ| − 5 (φ ) 3 √ ∗e . F (φ) ≈ 00 10φ φ 16
(A7)
The condition (ii) guarantees the applicability of the WKB method. In the case (iii), we deduce π
F (φ) = (i + e−i 10 )Γ
6
4 2 h i π 7π 2 4 − (i − ei 10 )Γ φ0 − (i − e−i 10 )Γ |φ|2 − (φ0 )2 + 5 3 5 5 9 3 h i 7π 5 4 32 + (i + ei 10 )Γ |φ|4 − (φ0 )2 |φ|2 + (φ0 )4 + o(|φ|4 ). (A8) 2 5 3 81
Appendix B: Functions I˜n (x, D)
The function I˜n (x, D) (see (49)) is an entire function of the complex variables x and D. The following representation holds I˜0 (x, D) =
∞ X
(−ix)n
n=0
I˜2n (D) , n!
I˜n (D) := I˜n (0, D).
(B1)
An analogous expansion is valid for I˜1 (x, D) with the replacement of I˜2n (D) by I˜2n+1 (D). In virtue of the relations, (n) I˜n (D) = in I˜0 (D),
I˜000 (D) = DI˜0 (D) + i,
(B2)
all I˜n (D), n ∈ N, are expressed through I˜0 (D) and I˜1 (D). For example, I˜2 (D) = −DI˜0 (D) − i,
I˜3 (D) = −DI˜1 (D) − iI˜0 (D),
I˜6 (D) = −(D3 + 4)I˜0 (D) + 6iDI˜1 (D) − iD2 ,
I˜7 (D) = −(D3 + 10)I˜1 (D) − 9iD2 I˜0 (D) + 8D. (B3)
I˜4 (D) = D2 I˜0 (D) − 2iI˜1 (D) + iD,
I˜5 (D) = D2 I˜1 (D) + 4iDI˜0 (D) − 3,
The functions I˜0 (D), I˜1 (D) are reduced to the known special functions iπ iD2 4 5 D3 I˜0 (D) = π Ai(D) − Bi(D) + 1 F2 1; 3 , 3 ; 9 , 3 2 π 3D4 3 7 8 D3 I˜1 (D) = iπ Ai0 (D) + Bi0 (D) − D1 F2 1; 43 , 53 ; D9 − 1 F2 2; 3 , 3 ; 9 . 3 40
(B4)
The function I˜0 (D) (up to the common factor π) is also known as the Scorer function [37]. The hypergeometric function entering into I˜0 (D) is an entire function of D and can be expressed through the Anger functions [38] 3
4 5 z 1 F2 1; 3 , 3 ; 9
4π h = √ J1/3 3 −3z 3
2 3
p −z 3 − J−1/3
2 3
p i −z 3 .
(B5)
In the ultrarelativistic limit and for β⊥ . γ −1 , the magnitude of x is rather small (see (49)). For |x| 1, the expansion (B1) is rapidly converging and so one may keep only a few first terms in 17
it. In the case |x| . 1, it is useful to employ the other representation of the functions I˜0 (x, D) and
I˜1 (x, D):
2 h i 1 ∂ 3 e−ix(D−x ) − 1 2 I˜0 (x, D) = eix(D−2x /3) I˜0 (D − x2 ) + e− 3 ∂D3 , i(D − x2 ) 2 h 1 ∂3 ∂ e−ix(D−x ) − 1 i 2 I˜1 (x, D) = eix(D−2x /3) I˜1 (D − x2 ) + e− 3 ∂D3 − xI˜0 (x, D). ∂D D − x2
(B6)
If |x| . 1 then the expansion of exp[− 31 ∂ 3 /∂D3 ] can be terminated with a few first terms left.
The function I˜0 (x, D) as a function of D satisfies the linear inhomogeneous ordinary differential
equation I˜000 (x, D) − 2ixI˜00 (x, D) − DI˜0 (x, D) = i,
(B7)
where the prime denotes the derivative with respect to D. After the substitution I˜0 (x, D) = eixD f (D − x2 ), this equation is reduced to the inhomogeneous Airy equation for the function f (t). The solution of this equation is found by the standard means. As a result, we deduce the integral representations: I˜0 (x, D) = πc1 (x)eixD Ai(D − x2 ) − πc2 (x)eixD Bi(D − x2 ) Z D Z 2 ix(D−s) 2 2 + iπ Bi(D − x ) dse Ai(s − x ) − iπ Ai(D − x ) 0
D
0
dseix(D−s) Bi(s − x2 ), (B8)
where c1 = e−2ix
3 /3
e−2ix c2 = i 3
−x
3 /3
−x
3 Bi0 (−x2 ) + ix Bi(−x2 ) 2 F2 ( 21 , 1; 23 , 43 ; − 4ix 3 )−
3ix 2
3 Ai0 (−x2 ) + ix Ai(−x2 ) 2 F2 ( 21 , 1; 23 , 43 ; − 4ix 3 )−
3 Bi(−x2 )2 F2 (1, 32 ; 43 , 35 ; − 4ix 3 ) ,
3ix 2
3 Ai(−x2 )2 F2 (1, 32 ; 43 , 53 ; − 4ix 3 ) ,
(B9)
and also 2 2 I˜0 (x, D) = d1 (x)eix(D−x )+iπ/6 Ai(D − x2 ) − d2 (x)eix(D−x )+iπ/6 Bi(D − x2 ) Z D−x2 Z D−x2 2 2 ix(D−x2 −s) 2 + iπ Bi(D − x ) dse Ai(s) − iπ Ai(D − x ) dseix(D−x −s) Bi(s), (B10)
0
0
where d1 =
e
ix3 3
2
ix3
1 3 3 Γ( 3 )Γ( 23 , ix3 )−e2πi/3 Γ( 23 )Γ( 13 , ix3 ) ,
e 3 3 3 d2 = √ Γ( 13 )Γ( 32 , ix3 )+e2πi/3 Γ( 23 )Γ( 13 , ix3 ) . 2 3 (B11) 18
Appendix C: Radiation from the transition layer
In this appendix, we consider the contribution to the spectral angular radiation distribution of the part of particle trajectory passing trough a thin transition layer where the field strength gradually increases from zero to its value (7). The radiation coming from this part of the worldline is actually a particular case of the radiation from a “short magnet”. The properties of this radiation were thoroughly investigated (see, e.g., [31–33] and references therein). Therefore, we give here only the main formulas adapted to our case. In the thin layer described above, the 4-momentum of a charged particle can be approximated by x˙ 0 = x˙ 0 (0) cosh
wτ 2 wτ 2 − x˙ 1 (0) sinh , 2T 2T x ¨20
−
x˙ 1 = −x˙ 0 (0) sinh
x ¨21
wτ 2 wτ 2 + x˙ 1 (0) cosh , 2T 2T
τ2 = −w 2 , T
(C1)
2
where w is the acceleration modulus in the momentary comoving frame at the proper time T . Further, we assume that wT 1.
(C2)
Then x0 ≈ x˙ 0 (0)τ −
wτ 3 x˙ 1 (0), 6T
x1 ≈ x˙ 1 (0)τ −
wτ 3 x˙ 0 (0), 6T
(C3)
where we have set x0 (0) = x1 (0) = 0. In the ultrarelativistic case, we obtain i h wτ 3 wτ 3 τ− (1 − βk ) − (1 + β ) k , 6T 12γ 2 T h h wτ 2 wτ 2 i wτ 2 1 i x˙ 0 ≈ γ 1 − + 2 , x˙ 1 ≈ γ 1 − − 2 . 2T 4γ T 2T 2γ
(C4)
k0 T γ
(C5)
kµ xµ ≈ k0 γ
If
the exponent in (15) can be approximated by the first two terms of its Taylor series. For the photon energy at the maximum of synchrotron radiation (50), the requirement (C5) moves to (C2). Hence, in the leading order j0 ≈
2) iwT (2γ 2 − 1 − γ 2 β⊥ , 2 )2 k0 (1 + γ 2 β⊥
19
j1 ≈
iwT (2γ 2 − 1) 2 )2 , k0 (1 + γ 2 β⊥
(C6)
where it is supposed that β⊥ ∼ γ −1 . We see that, in this approximation, the radiation formed is linearly polarized. Besides, dE =
e2 γ 2 w 2 T 2 2 2 2 )4 γ β⊥ dk0 dΩ. π 2 (1 + γ 2 β⊥
(C7)
The spectral angular distribution does not depend on the photon energy in the interval (C5) [31, 32] as distinct from the synchrotron entrance radiation described by (48). Comparing by the order of magnitude (C7) with the spectral angular distribution following from (48), one can see that the synchrotron entrance radiation dominates over (C7) provided (wT )2 1,
(C8)
i.e., whenever (C2) is satisfied. The condition (C2) can be written as ∆ γχ−1 ≈ |ω|−1 ,
(C9)
where ∆ is the width of the transition layer. This condition coincides with that deduced from the general considerations given below formula (51).
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FIGURES
1.0
x3 q 0.5
k j
E
1
x2
x
s0
0.0
C0
2
x
C
0.5
H 1.0 0.0
0.5
1.0
1.5
2.0
FIG. 1. Left panel: Schematic sketch of electron’s dynamics and the system of coordinates. Right panel: Deformation 2 of the integration contour in the s plane. The point s0 corresponds to (2λ|ω|υ− (0))−1/2 .
1.0 12 102 dE dWdk 0
5
10
0.5
102 dE dWdk 0
Ξ2
8 6
0.0
a)
4 3 2 1 0 0.0
1.0
2.0 106 k0
0.6
0.8
4
3.0
2
-0.5
0 0.0
0.2
0.4
104 k0
-1.0 0
1
1.0
2
3
4
5
104 k0 80
12
Γ=104 Γ=103 Γ=5´102 Γ=2´102 Γ=102
0.12
b)
102 dEdWdk0
60
40
20
0
0.10
8 6
0.08
4
c)
2
0.38
0.40
0.42
0.06 0.04
0 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 k0=0 k0=10-9´k0m k0=10-7´k0m k0=k0m k0=10´k0m
dEdWdk0
102 dE dWdk 0
10
j
0.02
0.44 Θ
0.46
0.48
0.50
22
0.00 0.0
0.5
1.0 104 k0
1.5
2.0
FIG. 2. De-excited radiation for γ = 5 × 103 , ω = −1.47 × 10−4 , and the entrance angle θ0 = ϕ0 = π/2. The observation angles θ, ϕ are counted in the units of π on the plots. a) The dependence of the Stokes parameter ξ2 on the radiated photon energy at the observation angles θ = 7π/15, ϕ = π/5. The blue line is the numerical result. The dashed line is (37). The de-excited radiation is almost completely circularly polarized. The radiation at extremely low and high frequencies is linearly polarized in compliance with (56), (61). The inset: The spectral angular distribution at θ = 7π/15, ϕ = π/5. The orange line is the numerical result. The dashed line is (30). The small plot is the same for θ = 9π/20, ϕ = π/10. We see that formula (30) can describe the radiation quite well even out of the applicability domain (39). b) The spectral angular distribution for different photon energies at ϕ = π/5. Here k0m ≈ 1.20 × 10−5 ≈ 6.13 eV is given by (31) at θ = 7π/15, ϕ = π/5. The vertical dashed line is θ = 7π/15. The de-excited radiation is not concentrated in the plane of particle motion. The inset: The same for θ = 7π/15. The vertical dashed line is ϕ = π/5. The two peaks correspond to the de-excited and synchrotron entrance radiations. The curve for k0 = 0 coincides with (61). c) The dependence of the spectral angular distribution on the initial energy of a charged particle at θ = 7π/15, ϕ = π/5. The maximum of the curve is well described by (30), (31) and its magnitude does not change in a wide range of γ’s.
0.6
0
0.3
-1 -1 -2 -2 -3 -3
j
-5
0.2
0.5 0.4 -4
-4 H715, 15L
H920, 110L
0.3 0.2
-6
-6
0.1
-5
j
0.4
1.0 0.5 0.0 -0.5 -1.0
0.1 0.0
0.44
0.46
0.48
0.50 Θ
0.52
0.54
0.0
0.56
0.35
0.40
0.45
0.50 Θ
0.55
0.60
0.65
FIG. 3. The same as in Fig. 2. Left panel: The applicability domain of (30) as it follows from (39) for the radiated photon energy taken at the maximum (31). The level lines are the lines of a constant lg k0m . Right panel: The Stokes parameter ξ2 at the photon energy k0m ≈ 1.20 × 10−5 ≈ 6.13 eV. The plateaus |ξ2 | ≈ 1 of the circularly polarized de-excited radiation are clearly seen.
23
5
5
2.0 2.0
3
1.5
1.0
1
2
0
0.6
2 Γ
Π 2
0.8
10-4 dEdWdk0
1 Γ
2
3 10-4 dE dWdk0
10-3 dEdWdk0
4
Θ
0.4
1
1.5 1.0
1.0
0.5 0.0
0.4996 0.4998 0.5000 j
0.5
0.2 1.5 2.0
0
10-4 dE dWdk 0
10-3 dE dWdk 0
4
0
1
3.0 -4 5.0 7.0 10.0 15.0 10 k0
2
3
4
0.0
5
10-4 k0
0.45
0.46
0.47
0.48 j
0.49
0.50
0.51
FIG. 4. Synchrotron entrance radiation for γ = 5 × 103 , ω = −1.47 × 10−4 , r3 (0) = 0, and the entrance angle θ0 = ϕ0 = π/2. The orange line is the numerical result, the dashed line follows from (48), and the dot-dashed line is (54). Left panel: The spectral angular distribution of the forward synchrotron entrance radiation. The inset: The spectral angular distribution of the synchrotron entrance radiation for ϕ = π/2 and the photon energy k0m ≈ 5358 ≈ 2.74 GeV given by (50). The small plot: The ultraviolet asymptotics of the spectral angular distribution. Right panel: The spectral angular distribution of the synchrotron entrance radiation for θ = π/2 and the photon energy k0m ≈ 5358. On the plots, ϕ is counted in the units of π. The inset: The fine structure of the main peak. Formula (48) leads to a spectral angular distribution describing with a good accuracy the actual distribution in the cone opening less than or equal to 50γ −1 .
24