Semiclassical probability of radiation of twisted photons in the ultrarelativistic limit
SSemiclassical probability of radiation of twisted photons in theultrarelativistic limit
O.V. Bogdanov , ∗ , P.O. Kazinski † , and G.Yu. Lazarenko ‡ Physics Faculty, Tomsk State University, Tomsk 634050, Russia Tomsk Polytechnic University, Tomsk 634050, Russia
Abstract
The semiclassical general formula for the probability of radiation of twisted photons by ultrarelativisticscalar and Dirac particles moving in the electromagnetic field of a general form is derived. This formulais the analog of the Baier-Katkov formula for the probability of radiation of one plane wave photon withthe quantum recoil taken into account. The derived formula is used to describe the radiation of twistedphotons by charged particles in undulators and laser waves. Thus, the general theory of undulator radia-tion of twisted photons and radiation of twisted photons in the nonlinear Compton process is developedwith account for the quantum recoil. The explicit formulas for the probability to record a twisted photonare obtained in these cases. In particular, we found that the quantum recoil and spin degrees of freedomincrease the radiation probability of twisted photons in comparison with the formula for scalar particleswithout recoil. In the range of applicability of the semiclassical formula, the selection rules for undulatorradiation established in the purely classical framework are not violated. The manifestation of the blossom-ing out rose effect in the nonlinear Compton process in a strong laser wave with circular polarization andin the wiggler radiation is revealed. Several examples are studied: the radiation of MeV twisted photonsby
GeV electrons in the wiggler; the radiation of twisted photons by
MeV electrons in strongelectromagnetic waves produced by the CO and Ti:Sa lasers; and the radiation of MeV twisted photonsby . MeV electrons in the electromagnetic wave generated by the FEL with photon energy keV. Nowadays the Baier-Katkov (BK) semiclassical method [1–4] is a standard tool to describe radiation ofplane wave photons by ultrarelativistic charged particles in external electromagnetic fields of a general form.This method effectively includes the quantum recoil experienced by a charged particle in radiating one hardphoton and is applicable for the energies of radiated photons right up to the energy of the radiating particle(for other semiclassical methods see, e.g., [5–10]). The BK method is realized in several computer codes[11–14] and proved to be very successful [4, 15–22]. A comparison of the radiation probability obtained bythis method with the exact QED results, when they are obtainable, reveals a spectacular agreement [2–4, 16].We use this method to derive the radiation probability of one twisted photon [23–32] by an ultrarelativisticcharged particle with account for the quantum recoil. In the case of negligible quantum recoil, the obtainedgeneral formula reduces to the one derived in [33].According to the BK method, in the ultrarelativistic limit, the one-photon radiation probability can becalculated by means of a formula resembling the classical formula for the intensity of radiation [34, 35]. Oneneed not solve the Dirac or Klein-Gordon equations in the given external fields but just find the solution ofthe Lorentz equations in these fields. The spin degrees of freedom are characterized by the spin vector andits evolution is governed by the Bargmann-Michel-Telegdi equation. If the radiation probability is summedover spin polarizations of the escaping particle and averaged over spin polarizations of the incoming one,the dynamics of the spin vector are irrelevant for evaluation of the radiation probability. All that drasticallysimplifies the calculations of radiation probability. Of course, there are certain limitations of this semiclassicalmethod. The complete list of them is presented in Sec. 2. The main idea standing behind our derivation of theradiation probability of twisted photons is to find the approximate expression for the product of radiation ∗ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected] a r X i v : . [ phy s i c s . acc - ph ] M a r mplitudes using the procedure developed in [1–4] and to integrate it over the azimuth angles of photonmomenta with the corresponding weights. The last step creates the twisted photon in the out-state.Having derived the general formula, we employ it to investigate the radiation probability of twistedphotons by electrons in undulators (Sec. 3) and strong laser pulses (Sec. 4). Currently, the twisted photonsof different spectral ranges are used in fundamental science and technology [27–32]. In the optical range andbelow, the detectors are designed allowing to decompose an arbitrary electromagnetic radiation into twistedphotons [36–40]. The formulas we obtain are aimed to describe correctly the radiation of hard twisted photonswith MeV energies and above that are not accessible for a direct observation. The hard twisted photons canbe employed to study the properties of nuclear matter by exciting higher multipole transitions in nuclei andhadrons (see, e.g., the discussion in [24, 25, 41, 42]). Rather recently, it was shown that hard twisted photonscan be generated in channeling radiation [43, 44] and strong laser pulses [41, 45–48]. In Sec. 3, we show thatMeV photons can be produced by GeV electrons in helical wigglers. Besides, we develop a general theoryof undulator radiation of twisted photons with the quantum recoil taken into account. In particular, we showthat the selection rules for the forward radiation of twisted photons by helical undulators [33, 41, 45, 49–55]are not affected by the quantum recoil, at least, in the domain of applicability of the semiclassical method.Then, in Sec. 4, we revisit the problem of radiation of twisted photons by electrons in laser waves studied in[41, 45, 46]. We generalize the results of [41, 45, 46] to the case of laser waves with an arbitrary amplitudeenvelope and include the influence of quantum recoil. Thus, we develop a general theory of twisted photonproduction in the nonlinear Compton process. Of course, it is just another description of the nonlinearCompton process usually formulated in terms of plane wave photons [2–4, 15, 17, 19–21, 56–66]. In Sec.4, we apply this general theory and describe the radiation of twisted photons by electrons in strong laserpulses produced by the free electron laser (FEL), CO and Ti:Sa lasers. The obtained general formulas canbe implemented in computer codes to describe the radiation of twisted photons by ultrarelativistic chargedparticles in electromagnetic fields of a rather general configuration, in particular, in channeling.The paper is organized as follows. In Sec. 2, we derive the general semiclassical formulas for the one-photon radiation probability of twisted photons by scalar and Dirac particles and discuss their applicabilityconditions. In Sec. 3, we elaborate a general theory of undulator radiation of twisted photons with accountfor the quantum recoil undergone by a charged particle in radiating a photon. We also obtain a simpleestimate for the number of radiated twisted photons and specialize the general applicability conditions tothe case of undulator radiation. As expected, the wiggler radiation of lower harmonics where the main part oftwisted photons is radiated has a clear imprint of the “blossoming out rose” effect [9]. In Sec. 4, we considerthe radiation of twisted photons by charged particles in an intense laser wave of a circular polarization.In particular, we trace the manifestation of blossoming out rose effect in this radiation. Several examplesare presented there. Some cumbersome calculations are removed to Appendices A, B. In Conclusion, wesummarize the results.We use the system of units such that (cid:126) = c = 1 and e = 4 πα , where α is the fine structure constant.Besides, we vastly use the notation introduced in [33]. Let us begin with the case of a stationary external magnetic field. The generalization to the case of a generalelectromagnetic field will be given below. In the presence of the electromagnetic field, the following processis possible e − i → γ α + X, (1)where e − i is the initial electron in the state i , γ α is the twisted photon [23–27, 67–71] recorded by the detectorin the state α , and X denotes the rest of particles that are not recorded by the detector. The probability ofsuch an inclusive process equals w ( α ; i ) = (cid:88) X (cid:104) i | ˆ U −∞ , ∞ | X ; γ α (cid:105)(cid:104) γ α ; X | ˆ U ∞ , −∞ | i (cid:105) , (2)where ˆ U is the evolution operator of QED in the presence of the external field (see, e.g., [58, 72]). In the firstBorn approximation with the exact account for the external electromagnetic field, the final state containsonly one electron w ( α ; i ) ≈ (cid:88) f (cid:104) i | ˆ U −∞ , ∞ | f ; γ α (cid:105)(cid:104) γ α ; f | ˆ U ∞ , −∞ | i (cid:105) , (3)2here f characterizes the final electron state. In this approximation, using the completeness relation, weobtain w ( α ; i ) = (cid:104) i | ˆ U −∞ , ∞ | γ α (cid:105)(cid:104) γ α | ˆ U ∞ , −∞ | i (cid:105) , (4)i.e., the probability of process (1) is equal to the average number of photons in the final state α . Accordingto (181), the twisted photon state can be decomposed into the plane wave ones [23–27, 67–71]. Then w ( α ; i ) = (cid:88) k , k Λ ∗ α ; s, k Λ α ; s, k (cid:104) i | ˆ U −∞ , ∞ | s, k (cid:105)(cid:104) s, k | ˆ U ∞ , −∞ | i (cid:105) , (5)where Λ α ; s, k are the coefficients of expansion (181). Therefore, in the first Born approximation, w ( α ; i ) canbe found from the matrix element for the plane wave photons (see the notation in [4]) C ( k , k ) := e (2 π ) V k (cid:104) i | (cid:90) ∞−∞ dt dt e ik ( t − t ) ˆ M † ( t , k ) ˆ M ( t , k ) | i (cid:105) , (6)which should be integrated over the azimuth angles of the vectors k , with the corresponding phase factorsas in (181). Notice that we work in the coordinate system adapted to the detector of twisted photons withthe axis directed along the unit vector e (see for details [33]). The unit vectors { e , e , e } of this coordinatesystem constitute a right-hand triple. The perpendicular components of a vector are those lying in the planespanned by { e , e } . The photons are supposed to lie on the mass shell k = (cid:113) k ⊥ + k = (cid:113) k ⊥ + k , (7)and k ⊥ = k ⊥ , k = k , (8)i.e., the photon momenta k , differ only by the azimuth angles. The helicities of photons with momenta k , are the same.We will evaluate (6) using the approximation introduced in [1–4]. Recall that it is assumed in thisapproximation that1. The charged particle is ultrarelativistic, i.e., the Lorentz factor γ (cid:29) and κ /γ (cid:28) , where κ =max(1 , K ) , K = (cid:104) β ⊥ (cid:105) γ is the undulator strength parameter, and (cid:104) β ⊥ (cid:105) is a characteristic value of thevelocity component perpendicular to the detector axis;2. The particle wave packet is sufficiently narrow in the momentum space;3. The size of the wave packet in the configuration space is small in comparison with the characteristicscale of variation of the external electromagnetic field in space;4. n , := k , /k lie in the cone directed along the detector axis (the axis ) with the opening of order κ /γ , i.e., we consider the region where the main part of radiation is concentrated.Below, there will appear the additional restriction related to the fact that we consider the radiation of twistedphotons.If the above conditions are satisfied, then, in evaluating average (6) in the leading order in /γ , we canuse the analog of the Thomas-Fermi approximation. Namely, the commutator |(cid:104) [ ˆ P µ , ˆ A ν ] (cid:105)| / (cid:104) ˆ P (cid:105) ∼ HH γ (cid:28) , γ (cid:29) , (9)is small for large particle energies. Here H is the characteristic magnitude of the electromagnetic field, H is the critical (Schwinger) field, and ˆ P µ = ˆ p µ − eA µ (ˆ x ) . Hence, the noncommutativity of operators ˆ x and ˆ P entering into the operators ˆ M † ( t , k ) , ˆ M ( t , k ) can be neglected. At the same time, the noncommutativityof ˆ P and e − i k ˆ x in ˆ M cannot be ignored because the exponent is a rapidly varying function for | k | ∼ ε , where ε is the energy of a charged particle. 3 calar particle. For a scalar charged particle, we have [4] ˆ M ( t , k ) = ˆ P − / ( f ∗ ˆ P ( t )) e − i k ˆ x ( t ) ˆ P − / , ˆ M † ( t , k ) = ˆ P − / ( f ˆ P ( t )) e i k ˆ x ( t ) ˆ P − / , (10)where f , = f ( k , ) are the polarization vectors of physical photons. All the operators in (10) are given inthe Heisenberg representation with the Hamiltonian ˆ H = (cid:113) ˆ P + m + eA (ˆ x ) , (11)Notice that A = 0 in the case of time-independent magnetic field. However, we leave A intact since itspresence influences the derivation only in one point (Eq. (21)), which we shall discuss separately below. Inorder to avoid additional complications with the vacuum definition (see, e.g., [58, 72–74]), we suppose that | eA | (cid:46) m, (12)which is fulfilled for all the electromagnetic fields achievable at the present moment in laboratories. Therefore, P = mγ ≈ ε up to the terms of order /γ .Let us carry the exponent entering into ˆ M † ( t , k ) to the right and the one entering into ˆ M ( t , k ) tothe left: ˆ M † ( t , k ) ˆ M ( t , k ) = ˆ P − / ( f ˆ P ) ˆ P (cid:48)− / e i k ˆ x e − i k ˆ x ˆ P (cid:48)− / ( f ∗ ˆ P ) ˆ P − / , (13)where ˆ P (cid:48) , = P ( ˆ P (cid:48) , ) , ˆ P (cid:48) , := ˆ P , − k , , (14)and ˆ P , ≡ ˆ P ( t , ) . Then we transform the operator expression e i k ˆ x e − i k ˆ x = e i k ⊥ ˆ x ⊥ e ik ˆ x e − ik ˆ x e − i k ⊥ ˆ x ⊥ = e i k ⊥ ˆ x ⊥ e i ˆ Hτ ( e − i ˆ Hτ ) ˆ P → ˆ P − k e − i k ⊥ ˆ x ⊥ , (15)where τ := t − t . In order to proceed, we assume that e − i k ⊥ , ˆ x ⊥ , | i (cid:105) ≈ e − i k ⊥ , x ⊥ , | i (cid:105) , (16)where x , is the average value of the corresponding operator with respect to the state | i (cid:105) . The approximateequality takes place, if k ⊥ σ ⊥ (cid:28) , (17)where σ ⊥ is the characteristic transverse (with respect to the detector axis) size of the wave packet in theconfiguration space.Notice that condition (17) also arises in considering the radiation of twisted photons by a bunch ofcharged particles [75, 76]. When condition (17) is satisfied, the probability of radiation of a twisted photonby the bunch of N particles is the same as for one charged particle, multiplied by N (incoherent radiation)or by N (coherent radiation). To put it differently, the form of the wave packet does not affect the radiationspectrum of twisted photons in this case. It is clear that if (17) is violated, the approximation we use cannotbe employed. In that case, the probability of radiation of twisted photons depends severely on the form ofthe wave packet and is not determined only by the average values of the particle momentum and coordinate.In particular, all the fine effects stemming from the form of the wave packet of a charged particle (see,e.g., [26, 77–82]) cannot be reproduced by the semiclassical approach considered here. However, condition(17) cannot be strongly violated. It was shown in [75, 76] that, in a general position, the coherent radiationproduced by a smooth wave packet of a charged particle is strongly suppressed for k ⊥ σ ⊥ (cid:38) due to destructiveinterference of the radiation amplitudes coming from different points of the wave packet.Furthermore, we have |(cid:104) [ e − i k ⊥ , ˆ x ⊥ , , ˆ P ] (cid:105)| / |(cid:104) ˆ P (cid:105)| / ∼ k / ( εγ ) (cid:46) /γ, (18)i.e., up to the terms of order /γ , we can carry these exponents through the operators entering into ˆ M † and ˆ M and make use of (16). Now take into account that (cid:113) ( P − k ) + P ⊥ + m ≈ ( P − k ) (cid:16) k P − k P ( P − k ) − k ⊥ P − k ) (cid:17) . (19)4he approximation (19) is valid provided P k (1 − n ˙ x )( P − k ) ∼ k /P (1 − k /P ) κ γ (cid:28) . (20)If κ /γ is small, estimate (20) holds up to k (cid:46) P . Then, repeating the calculations presented in [4], we canwrite e − ik τ e i ˆ Hτ ( e − i ˆ Hτ ) ˆ P → ˆ P − k ≈ e − i ˆ P P − k [ k τ − k ( x − x ) − k ⊥ τ/ (2 ˆ P )] , (21)where we have used the fact that, in time-independent magnetic fields, P = mγ is an integral of motion ofthe Lorentz equations. It is argued in [4] that approximation (21) is valid in the leading order in κ /γ forgeneral nonstationary external electromagnetic fields as well, provided π/ ( T ε ) (cid:28) , (22)where T is the characteristic time- or length-scale of changing of the external electromagnetic fields. In thatcase, ˆ P should be replaced by ˆ P i in (21), i.e., by the particle energy in the initial state | i (cid:105) where the externalfields are absent. This prescription ensures, in particular, that the right-hand side of (21) is invariant undertranslations in the spacetime.Disentangling expression (15) and taking into account the conditions 1-4 and (9), (17), we can replacethe operators in matrix element (6) by their average values. In particular, ˆ P (cid:48) , → P (cid:48) , = P ( P , − k , ) ≈ P , − k . (23)Then, in the scalar case, C ( k , k ) = e (2 π ) V k c ( k ) c ∗ ( k ) , (24)where c ( k ) := (cid:90) ∞−∞ dte − iq i [ k − k ⊥ / (2 P i )] t + iq i k x + i k ⊥ x ⊥ q / ( f ˙ x ) , (25)and ˙ x ( t ) = P ( t ) /P ( t ) , q ( t ) = P ( t ) /P (cid:48) ( t ) , (26)and q i = P i /P (cid:48) i . Recall that P i is the energy of particle in the initial state and P (cid:48) i = P i − k . Theapproximate expression for probability (5) derived from (24) is nonnegative. Furthermore, matrix element(24) with the photon momenta satisfying (8) transforms properly under translations in the spacetime, x µ → x µ + a µ , viz., C ( k , k ) → C ( k , k ) e i ( k ⊥ − k ⊥ ) a ⊥ , (27)where a ⊥ is the translation four-vector component perpendicular to the detector axis.Employing formula (181), we find the leading contribution to the probability of radiation of a twistedphoton by a charged scalar particle with the quantum recoil taken into account dP ( s, m, k ⊥ , k ) = e (cid:12)(cid:12)(cid:12) (cid:90) ∞−∞ dte − iq i [ k − k ⊥ / (2 P i )] t + iq i k x q / (cid:0) (cid:2) ˙ x + a − + ˙ x − a + (cid:3) + ˙ x a (cid:1)(cid:12)(cid:12)(cid:12) (cid:16) k ⊥ k (cid:17) dk dk ⊥ π , (28)where a ± ≡ a ± ( s, m, k , k ⊥ ; x ) , a ≡ a ( s, m, k ⊥ ; x ) . (29)Recall that we use the system of units such that e = 4 πα . In the case q ≈ , i.e., when the energy of radiatedphoton is negligibly smaller than the energy of charged particle, formula (28) passes into formula [(36), [33]].As a rule, q ≈ const in the ultrarelativistic limit and so q / can be removed from the integrand of (28).Since q > , we see that the quantum recoil tends to increase the radiation probability in comparison withthe answer which does not include it.The estimate (17) is most stringent condition on the range of applicability of formula (28). It is absent inthe semiclassical description of radiation of plane wave photons [4]. The narrower transverse size of a particlewave packet, the better this condition is satisfied. Currently, there are techniques allowing to produce theelectron wave packets with the waist of order ˚A [83]. Theoretically, for a given instant of time, the electronwave packet can be focused into a region much smaller than the Compton wavelength. However, the problemis how to create such wave packets and keep them in this state for a sufficiently long time so that the5lectron will have time to radiate a twisted photon. For a particle, which is at rest on average, the width ofthe wave packet cannot be smaller than the Compton wavelength. Otherwise, the momentum uncertainty islarger than m and the height of a potential barrier confining such a particle must be larger than m . Suchfields create electron-positron pairs, and the problem becomes essentially multiparticle. Nevertheless, if theelectron is moving on average, then its wave packet can be squeezed stronger in the transverse directionswith the aid of the fields not exceeding the critical (Schwinger) field in the laboratory frame. In the comovingreference frame, the potential well depth can become larger than the critical one but this does not result inpair creation. In passing to the comoving frame, the magnetic field arises and the respective invariants of theelectromagnetic field remain the same as in the laboratory frame.This situation is naturally realized for axial channeling of particles in crystals (see for details, e.g.,[4, 8, 9, 84, 85]). Suppose that the particle momentum component p (cid:29) m and | p , | (cid:28) p . Then thesolution of the stationary Dirac or Klein-Gordon equations is reduced approximately to the solution of theSchr¨odinger equation with a certain effective potential U ( x ⊥ ) : (cid:16) p ⊥ m eff + U (cid:17) ψ = δEψ, (30)where δE = E − m eff , m eff := m + p , and it is assumed that | U | (cid:28) m eff and | δE | (cid:28) m eff . The effectivemass is well approximated by m eff ≈ γm. (31)The potential barrier of the height U can hold a particle with typical momentum uncertainty | ∆ p ⊥ | (cid:46) (cid:112) m eff U . (32)Then from the uncertainty relation, we obtain the minimum transverse size σ m ⊥ ∼ (2 m eff U ) − / ≈ / ( θ c ε ) , (33)where the critical Lindhard angle has been introduced θ c := (cid:114) U ε , U ∼ Zα m. (34)Since β ⊥ ≈ θ c , then σ m ⊥ ∼ / ( mK ) . (35)As we see, the transverse size of the particle wave packet can be much smaller than the Compton wavelengthfor axal channeling in the wiggler regime [86, 87]. The maximum transverse squeezing of a particle wavepacket is achieved at σ m ⊥ ∼ / ( mγ ) . (36)However, this is possible only in the external fields of the same order as the critical one in the laboratoryframe. Such maximally localized wave packets were investigated in [88, 89].Consider the fulfillment of condition (17) for axial channeling with the wave packet transverse size (35).The respective undulator frequency is estimated as ω ∼ πθ c d , (37)where d is the channel width, d ∼ ˚A ∼ m − . (38)Then, in the dipole regime, at the n -th harmonic k ∼ ωγ n. (39)Hence, k ⊥ σ m ⊥ ∼ π n ⊥ γmd n (cid:28) . (40)Bearing in mind that n ⊥ γ (cid:46) , we see that (17) is satisfied. This condition is fulfilled even for σ ⊥ = d in thedipole regime for not too large harmonic numbers. In the wiggler case for n ⊥ γ ≈ K , we have k ∼ ωγ n/K . (41)6hen k ⊥ σ m ⊥ ∼ πnmdK (cid:28) . (42)In fact, condition (17) is already satisfied for σ ⊥ n (cid:46) d/ . The energy shift due to quantum recoil wasneglected in these formulas. The inclusion of quantum recoil only improves the estimate.The dynamical picture looks as follows. The wide particle wave packet falls onto the crystal surface and,in the channeling regime, is split into wave packets with transverse sizes from (35) up to d . Because ofinteraction with the crystalline potential and other electrons, the different parts of the wave packet movingin different channels do not almost interfere. In order to describe the radiation of twisted photons producedby each part of such dispersed wave packet, formula (28) can be used or, rather, its analog for Dirac particlesthat will be obtained below. As the particle escaped the crystal, the parts of the wave packet spread in thetransverse directions with the characteristic velocity β ⊥ ≈ θ c . Dirac particle.
In case of radiation of twisted photons by the Dirac particles, the considerations areanalogous but more cumbersome. Using the notation of [4], we have approximately M ( t , k ) = (cid:16) mP (cid:17) / ¯ u s (cid:48) ( P )ˆ f ∗ e − i k x u s ( P ) (cid:16) mP (cid:17) / = e − i k x (cid:16) mP (cid:48) (cid:17) / ¯ u s (cid:48) ( P (cid:48) )ˆ f ∗ u s ( P ) (cid:16) mP (cid:17) / ,M † ( t , k ) = (cid:16) mP (cid:17) / ¯ u s ( P )ˆ f e i k x u s (cid:48) ( P ) (cid:16) mP (cid:17) / = (cid:16) mP (cid:17) / ¯ u s ( P )ˆ f u s (cid:48) ( P (cid:48) ) (cid:16) mP (cid:48) (cid:17) / e i k x , (43)where s and s (cid:48) characterize the initial and final electron spin states. For brevity, we do not write hats overthe operators P and x anymore. Further, we employ formulas (15), (21), substitute all the operators bytheir averages, sum over spin polarizations of the escaping electron and average over spin polarizations ofthe incoming electron. As a result, we come to (cid:88) spins M † ( t , k ) M ( t , k ) →
12 Sp (cid:2) ( A ∗ − i ( B ∗ σ ))( A + i ( B σ )) (cid:3) = A ∗ A + ( B ∗ B ) , (44)where A , = − ( f ∗ , P , )2 (cid:113) P (cid:48) , P , (cid:18)(cid:115) P (cid:48) , + mP , + m + (cid:115) P , + mP (cid:48) , + m (cid:19) , B , = − (cid:113) P (cid:48) , P , (cid:18)(cid:115) P (cid:48) , + mP , + m [ f ∗ , , P , ] − (cid:115) P , + mP (cid:48) , + m [ f ∗ , , P (cid:48) , ] (cid:19) . (45)In the ultrarelativistic limit, we can neglect the mass in expressions (45) since it gives the contributions oforder /γ as compared to the main contribution. Then A ∗ A + ( B ∗ B ) = 14 P (cid:48) P (cid:48) (cid:2) ( P + P (cid:48) )( P + P (cid:48) )( f ˙ x )( f ∗ ˙ x ) −− k ( f , ˙ x − n )( f ∗ , ˙ x − n ) + k ( f f ∗ )( ˙ x − n , ˙ x − n ) (cid:3) . (46)This expression turns into formula [(2.41), [4]] for k , = k .The integration over azimuth angles of the vectors k , is considered in Appendix B. With the aid of thenotation introduced there, we can write dP ( s, m, k ⊥ , k ) = e (cid:90) ∞−∞ dt dt P (cid:48) P (cid:48) e − i ( k − k ⊥ / (2 P i )) q i ( t − t )+ ik q i ( x − x ) ×× (cid:110) ( P + P (cid:48) )( P + P (cid:48) )( [ ˙ x − a ∗− + ˙ x a ∗ + ] + ˙ x a ∗ )( [ ˙ x a − + ˙ x − a + ] + ˙ x a )++ k (cid:2) ( ˙ x a ∗ + − in ⊥ a ∗ + ( m − x − a + + in ⊥ a + ( m − x − a ∗− + in ⊥ a ∗− ( m + 1))( ˙ x a − − in ⊥ a − ( m + 1)) (cid:3)(cid:111)(cid:16) k ⊥ k (cid:17) dk dk ⊥ π (47)7n the leading order in κ /γ . Here a ± ≡ a ± ( s, m, k , k ⊥ ; x ) , a ≡ a ( s, m, k ⊥ ; x ) ,a ∗± ≡ a ∗± ( s, m, k , k ⊥ ; x ) , a ∗ ≡ a ∗ ( s, m, k ⊥ ; x ) . (48)The approximate expression (47) for probability density (5) is nonnegative. In the case k (cid:28) ε , , formula(47) passes into expression [(36), [33]] without the quantum recoil due to photon radiation. Just as for thescalar particle case, we see that the quantum recoil tends to increase the radiation probability as comparedto the formula without recoil. As long as P + P (cid:48) P (cid:48) = 1 + q , k P (cid:48) = q − δq, (49)the second and third terms in (47) are proportional to ( δq ) , while the first term is proportional to (1+ δq/ .Thus, in the limit of small recoil, q ≈ , the second and third terms can be neglected and the contributionof the first term is the same as in the case of a scalar particle (28) since q / ≈ δq/ . The spin effectsbecome irrelevant in this limit within the bounds of the approximations made in deriving (47).Contrary to the classical formula for radiation probability, expression (47) does not factorize into c ∗ c ,where c is determined by the particle trajectory. This is a consequence of the fact that the averaging overspin polarizations was performed and the quantum recoil was taken into account. If one neglects the recoil,the created radiation will be described by a coherent state in the Fock space [33, 67, 90–92] and nontrivialquantum correlations will be absent. In the case of a scalar charged particle, formula (28) does factorize intothe amplitude and its complex conjugate, and the nontrivial quantum correlations are absent within thebounds of the approximations made.Notice that formulas (28), (47) describe the radiation produced by one charged particle. As a rule, inreal experiments, the bunch of charged particles radiates. Under usual conditions, radiation amplitudes ofhard photons produced by different particles in the bunch add up incoherently, i.e., expressions (28) or (47)should be summed over different particles in the bunch. In the papers [75, 76], the simple formulas wereobtained allowing to find the radiation of twisted photons by a bunch of charged particles using the one-particle radiation probability distribution. These formulas can be applied to (28) and (47) as the initial planewave matrix elements (27) transform correctly under translations. Below, we shall employ the formulas from[75] for axially symmetric bunches. Such bunches are created, for example, in the electron-positron colliderVEPP-2000, Novosibirsk [93]. Let us apply the general formulas derived in the preceding section to the forward radiation of charged particlesin undulators with the quantum recoil taken into account. We shall investigate the undulator radiation inthe dipole regime for an arbitrary periodic trajectory of a charged particle. As for the wiggler case, we shallobtain the exact expression for the probability of radiation of twisted photons by charged particles movingalong an ideal helix (the helical wiggler).
Scalar particle.
The general formula for the radiation probability of a twisted photon by a scalar chargedparticle has the form (28). The radiation of twisted photons by undulators without quantum recoil wasdescribed in ([33], Sec. 5). Comparing [(82), [33]] with (28), we see that the account for quantum recoil leadsonly to a change of the energy spectrum of radiated photons and to the appearance of the common factor q when the forward radiation produced by a scalar particle in the undulator in the dipole regime and in theideal helical wiggler is considered. Notice that, according to the Lorentz equations, q = q i = const (50)for the motion of charged particles in undulators. As a result, in the dipole approximation, dP ( s, m, k ⊥ , k ) = e n ⊥ ∞ (cid:88) n =1 δ N (cid:104) qk (cid:16) − n υ − n ⊥ k ⊥ P (cid:17) − nω (cid:105) ×× q (cid:110) δ m, (cid:16) k ⊥ υ + ωnn ⊥ n − s (cid:17) | r + ( n ) | + δ m, − (cid:16) k ⊥ υ + ωnn ⊥ n + s (cid:17) | r − ( n ) | (cid:111) dk dk ⊥ , (51)8here the notation introduced in [33] was used. As in the case of radiation without recoil, the main part offorward radiation consists of twisted photons with m = ± .The energy spectrum is found from the equation P k n (cid:16) − n υ − n ⊥ k n P (cid:17) = nω ( P − k n ) . (52)In virtue of condition (17) with minimum σ ⊥ taken from (35), the last term in the round brackets on theleft-hand side is small and should be taken into account perturbatively. Indeed, − n υ − n ⊥ k n P ≈ K + n ⊥ γ − n ⊥ γ k n /P γ = 1 + K + n ⊥ γ /q γ . (53)For k ∼ ε , it follows from (17), (35) that n ⊥ γ (cid:28) K , and the contribution of this term can be neglected ascompared to the contributions of the first terms. For k (cid:28) ε , obviously, the contribution of this term is alsonegligibly small in comparison with the contributions of the first terms in this expression. Then the physicalsolution to (52) takes the form k n = P (1 − n υ ) n ⊥ (cid:110) k n P − (cid:104)(cid:16) k n P (cid:17) − k n n ⊥ P (1 − n υ ) (cid:105) / (cid:111) ≈≈ P (1 + K + n ⊥ γ )2 n ⊥ γ (cid:110) k n P − (cid:104)(cid:16) k n P (cid:17) − k n n ⊥ γ P (1 + K + n ⊥ γ ) (cid:105) / (cid:111) , (54)where ¯ k n := ωn − n υ (55)is the energy of radiated twisted photon without quantum recoil. If condition (17) with minimum σ ⊥ takenfrom (35) is fulfilled, the last term under the square root in (54) is small. Developing (54) as a series, weobtain in the leading order k n = nω − n υ + nω/P = ¯ k n k n /P ⇔ k n = 1¯ k n + 1 P . (56)In fact, this is the BK prescription for the shift of the energy spectrum due to quantum recoil. Formulas(54), (56) imply that k n < P ≈ ε .Using formulas from [33], the radiation of twisted photons by a scalar charged particle moving along anideal helix can readily be derived. Denoting by χ = ± the helicity of the particle trajectory, we find from(28) that dP ( s, m, k ⊥ , k ) = e n ⊥ δ N (cid:104) qk (cid:16) − n υ − n ⊥ k ⊥ P (cid:17) − χmω (cid:105) ×× q (cid:104)(cid:16) υ − χ n ωmn ⊥ k ⊥ (cid:17) J m (cid:16) k ⊥ Kωγ (cid:17) − χ sKn ⊥ γ J (cid:48) m (cid:16) k ⊥ Kωγ (cid:17)(cid:105) dk dk ⊥ . (57)Thus the selection rule m = χn for the forward radiation produced by an ideal helical wiggler survives evenwhen the quantum recoil is taken into account, within the bounds of the approximations made in derivingformula (28). The energy spectrum of radiated twisted photons is given by (54), (56). Dirac particle.
In the dipole approximation, it is necessary to expand the integrand of (47) into series in K and take into account only the leading contribution (see the estimates in Sec. 5.A of [33]). Then, up to acommon factor, the first term in the curly brackets in (47) turns into the same expression as that appearingwhen the quantum recoil is neglected,
14 ( P + P (cid:48) ) (cid:104) − δ m, (cid:16) in ⊥ n − s ˙ r + k ⊥ υ r (cid:17)(cid:16) in ⊥ n − s ˙ r − + k ⊥ υ r − (cid:17) −− δ m, − (cid:16) in ⊥ s + n ˙ r − − k ⊥ υ r − (cid:17)(cid:16) in ⊥ s + n ˙ r + k ⊥ υ r (cid:17)(cid:105) . (58)9 = γ = × n ⊥ = K / γ K = = - - η - - d P / d k d k ⊥
31 32 33 3401234 k - d P / d k d k ⊥ s =
1, n ⊥ = K / γ , K = γ = × k = σ = μ mk ⊥ σ = l = A - m - - - - m - d P / N d k d k Figure 1:
The radiation of twisted photons by
GeV electrons in the helical wiggler. The seventh harmonic is presented.The wiggler period . cm, the number of periods N = 15 , and the magnetic field strength in the wiggler . kG. Theapplicability conditions (93) are satisfied for σ ⊥ (cid:46) m − . The photon energy is measured in the electron rest energies. Leftpanel: The probability to record a twisted photon produced by one electron in the wiggler against the photon energy. Leftinset: The probability distribution over m . Right inset: The relative change of radiation probability due to quantum recoil: η := ( dP cl − dP ) /dP cl , where dP cl is the radiation probability without quantum recoil. Right panel: The distribution over m of probability per particle to record a twisted photon produced by an incoherent axially symmetric bunch of particles in thewiggler. The bunch width is σ = 125 µ m. The angular momentum per photon is the same as for radiation of one electron. Inset:The asymmetry of distribution over m . The second term in the curly brackets in (47) standing at k / is written as n ⊥ ( s + n ) (cid:104) ˙ r δ m, − − in ⊥ (cid:0) δ m, + δ m, k ⊥ r − − δ m, − k ⊥ r (cid:1)(cid:105) ×× (cid:104) ˙ r − δ m, − + in ⊥ (cid:0) δ m, + δ m, k ⊥ r − δ m, − k ⊥ r − (cid:1)(cid:105) ++ n ⊥ ( s − n ) (cid:104) ˙ r − δ m, + in ⊥ (cid:0) δ m, + δ m, k ⊥ r − − δ m, − k ⊥ r (cid:1)(cid:105) ×× (cid:104) ˙ r δ m, − in ⊥ (cid:0) δ m, + δ m, k ⊥ r − δ m, − k ⊥ r − (cid:1)(cid:105) . (59)Expanding r , into a Fourier series and integrating over time, we have dP ( s, m, k ⊥ , k ) = e n ⊥ ∞ (cid:88) n =1 δ N (cid:104) qk (cid:16) − n υ − n ⊥ k ⊥ P (cid:17) − nω (cid:105) ×× (cid:110) δ m, (cid:104) ( P + P (cid:48) ) (cid:16) k ⊥ υ + ωnn ⊥ n − s (cid:17) + k ⊥ ( n − s ) (cid:16) ωn − n ⊥ k ⊥ (cid:17) + n ⊥ k ⊥ n + s ) (cid:105) | r + ( n ) | ++ δ m, − (cid:104) ( P + P (cid:48) ) (cid:16) k ⊥ υ + ωnn ⊥ n + s (cid:17) + k ⊥ ( n + s ) (cid:16) ωn − n ⊥ k ⊥ (cid:17) + n ⊥ k ⊥ n − s ) (cid:105) | r − ( n ) | (cid:111) dk dk ⊥ P (cid:48) . (60)As in the case of a scalar particle, only the twisted photons with m = ± are radiated. The photon energyspectrum has the form (54), (56).Performing calculations along the same lines as in the case of a scalar particle, we find for an ideal helicalwiggler dP ( s, m, k ⊥ , k ) = e n ⊥ δ N (cid:104) qk (cid:16) − n υ − n ⊥ k ⊥ P (cid:17) − χmω (cid:105) ×× (cid:110) ( P + P (cid:48) ) (cid:104)(cid:16) υ − χ n ωmn ⊥ k ⊥ (cid:17) J m − χ sKn ⊥ γ J (cid:48) m (cid:105) + k ⊥ n + s ) (cid:16) Kγ J m +1 − χn ⊥ J m (cid:17) ++ k ⊥ n − s ) (cid:16) Kγ J m − − χn ⊥ J m (cid:17) (cid:111) dk dk ⊥ P (cid:48) . (61)The argument of the Bessel functions is the same as in formula (57). Employing the recurrence relationsfor the Bessel functions and keeping in mind that n ≈ , the last two terms in the curly brackets can be10rought to k (cid:104)(cid:16) ωmn ⊥ k ⊥ − (cid:17) J m + sKn ⊥ γ J (cid:48) m (cid:105) , (62)to the accuracy we work. Setting υ = n = 1 in the first term in the curly brackets in (61), we obtain dP ( s, m, k ⊥ , k ) ≈ e n ⊥ δ N (cid:104) qk (cid:16) − n υ − n ⊥ k ⊥ P (cid:17) − χmω (cid:105) ×× (1 + q ) (cid:104)(cid:16) − χ ωmn ⊥ k ⊥ (cid:17) J m − χ sKn ⊥ γ J (cid:48) m (cid:105) dk dk ⊥ . (63)The forward radiation of twisted photons in helical wigglers obeys the selection rule m = χn within thebounds of the approximations made (see Fig. 1). Comparing (57) with (63), we see that the probability ofradiation of twisted photons by Dirac particles is always bigger than the same quantity for scalar particlessince (1 + q ) / − q > for q > (see Fig. 2). For q ≈ , the respective radiation probabilities are almostequal. Number of radiated twisted photons.
Let us find the number of radiated twisted photons with a givenprojection of the total angular momentum m per energy interval k and the total number of radiated twistedphotons. Further, we set χ = 1 since χ = − can be obtained by the substitution m → − m , s → − s .To shorten formulas, we suppose that the last term in the round brackets in (52) is small and the energyspectrum is given by (56). Then δ N [ qk (1 − n υ ) − mω ] dk dk ⊥ = δ N ( x ) dk dxυ qn ⊥ ω ≈ δ N ( x ) dk dxqn ⊥ ω . (64)As long as the function δ N ( x ) is localized near the point x = 0 for N (cid:38) , the integral over x can easily befound (cid:90) ∞−∞ dxδ N ( x ) = N. (65)Substituting (64), (65) into (57), (63), we obtain dP ( s, m, k ) /dk . The analogous formulas can be obtainedin the dipole case as well. However, in that case, the overlapping of harmonics has to be taken into account.Namely, the integration over x results in ∞ (cid:88) n =1 δ N [ qk (1 − n υ ) − nω ] dk dk ⊥ → N ∞ (cid:88) n =1 θ k (cid:104) nω nω/P , nω − υ + nω/P (cid:105) dk qn ⊥ ω , (66)where θ x [ a, b ] := (cid:26) , x ∈ [ a, b ] ; , x (cid:54)∈ [ a, b ] , (67)and n ⊥ is to be expressed through k n by using (56).The twisted photons with large projections m of the total angular momentum can be produced byundulators only in the wiggler regime. When m (cid:38) , the Bessel functions entering into (57), (63) areexpressed through the Airy functions [9] with (see the notation in [(122), [33]]) x ≈ − n ⊥ γ K (1 + K + n ⊥ γ ) q . (68)For the probability of radiation of twisted photons with the total angular momentum projection m , m (cid:38) ,not to be exponentially suppressed, quantity (68) must be small. This is achieved when K (cid:38) and n k := n ⊥ γ/K ≈ . (69)Then, it follows from (63), (64), (65) that dP ( s, m, k ) dk ≈ πN α K γ q + q − ω (cid:16) m (cid:17) / (cid:104) Ai (cid:48) ( y ) + sn k (cid:16) − q n k n k (cid:17)(cid:16) m (cid:17) / Ai( y ) (cid:105) , (70)where y ≈ (cid:16) m (cid:17) / (cid:104) − n k (1 + n k ) q (cid:105) , (71)11nd n k should be expressed in terms of k m from (56), (69). The Airy function and its derivative dropexponentially to zero for y (cid:38) / . Therefore, the radiation of twisted photons is exponentially suppressedwhen m (cid:38) m c , m c := min((1 − q − ) − / , K ) / √ , (72)The second quantity in the min function appearing in the definition of m c comes from the estimate ofsubsequent terms of the expansion of y in K − (see [(120), [33]]).The distribution (70) reaches the maximum at n k ≈ q − sc (2 /m ) / , (73)where c ≈ . is found from the equation (Ai (cid:48) ( c ) − c Ai( c )) (cid:48) = 0 . (74)If m < m c , then dP ( m, k ) dk = (cid:88) s = ± dP ( s, m, k ) dk (75)possesses the maxima at points (73) with s = ± and the local minimum at the point n k ≈ q . For m (cid:38) m c ,the maxima coalesce in the point n k ≈ q .Such a behavior of radiation maxima is expectable, if one bears in mind that the wiggler radiation is justthe synchrotron one in the Lorentz frame where the electron is at rest on average. The intensity profiles oflower synchrotron harmonics, m (cid:28) m c , were thoroughly investigated in Sec. 1.3.4 of [9] where the effect of ablossoming out rose was revealed. The maxima of intensity of these harmonics do not lie in the orbit plane inthe ultrarelativistic limit. In the orbit plane, the intensity of these harmonics possesses a local minimum. Forlarge harmonics, m (cid:38) m c , this minimum disappears. The maxima and minima of intensities of synchrotronharmonics in the laboratory frame are found from the standard transformation law for angles sin θ = sin θ (cid:48) − (1 + K ) /γ ] / cos θ (cid:48) √ K γ , (76)where θ (cid:48) is the polar angle counted in the “synchrotron” frame and θ is the same angle in the laboratoryframe. In the synchrotron frame, the electron has the Lorentz factor γ s = (cid:112) K , (77)and is ultrarelativistic in the wiggler case. The orbit plane, θ (cid:48) = π/ , is seen in the laboratory frame at theangle θ = arcsin( √ K /γ ) ≈ K/γ . We shall return to the effect of blossoming out rose in Sec. 4.Now we find a loose estimate for the number of twisted photons radiated by the right-handed helicalwiggler at the harmonic n = m . The function (70) is peaked at n k ≈ . The width of this peak can be foundfrom the equation − n k (1 + n k ) q = b (cid:16) m (cid:17) / , (78)where b ∼ . Solving this equation, we obtain ∆ n k ≈ q (cid:2) b (2 /m ) / + q − − (cid:3) / . (79)Take into account that dk = − υ n k ωm K γ n k dn k . (80)Then, assuming ( q − m / (cid:28) and multiplying the value of dP/dn k at the point n k = 1 by ∆ n k , we arriveat ∆ P ( s, m ) ≈ παN (2 /m ) / (cid:2) b (2 /m ) / + q − − (cid:3) / Ai (cid:48) (cid:16)(cid:16) m (cid:17) / (1 − q − ) (cid:17) . (81)This quantity is independent of K . In the classical limit, q = 1 , we have ∆ P ( s, m ) ≈ . × − N m − / , (82)12or b = 1 . (in [(135), [33]], the other quantity was estimated). Such a value of b is taken for concordanceof the estimate with the numerical calculations. This estimate shows, in particular, that, in describing theleading contribution to radiation of twisted photons produced in wigglers, the one-photon approximation isjustified when N m − / (cid:46) . (83)If this condition is violated, the trajectory can be partitioned into pieces such that condition (83) is satisfiedfor each part of the trajectory. The probabilities of radiation from different parts of the trajectory shouldbe summed with account for a change of the electron energy-momentum due to radiation reaction on eachpart of the trajectory. In the classical regime, q − (cid:28) , the Landau-Lifshitz equation can be employed todescribe the effective electron dynamics [15, 19–21, 34, 35, 84, 94–101]. Applicability conditions.
Let us find the domain of applicability of the above formulas for the radiationof twisted photons in undulators with the quantum recoil taken into account. To shorten formulas, we supposethat all the dimensional quantities are measured in the units of the electron rest energy or in the electronCompton wavelengths.Formula (72) implies that the quantum recoil diminishes the maximum attainable value of m for thetwisted photons generated in the forward undulator radiation. Another restriction on the maximum m follows from the requirement (17). If n ⊥ γσ ⊥ (cid:28) , (84)then (17) holds. The condition (84) can be satisfied only in the dipole regime. In the wiggler case, the mainpart of radiation is produced with n ⊥ γ ≈ K (cid:38) and even for the wave packet waist (35) estimate (84) isnot fulfilled.If n ⊥ γσ ⊥ (cid:38) , (85)then (17), (56) imply ¯ k m /γ = q − (cid:46) / (10 n ⊥ γσ ⊥ ) ⇒ q − (cid:28) , (86)i.e., in radiating a photon, the quantum recoil experienced by the electron should be small. In the wigglercase, for n k ≈ , we obtain m (cid:46) K/ (10 ωγσ ⊥ ) = R/ (10 σ ⊥ ) , (87)where K = ωγR and R is the radius of the spiral turn along which the electron is moving. Taking intoaccount (35), we deduce the upper estimate m (cid:46) RK/ , (88)where recall that R is measured in the Compton wavelengths λ C . The estimates (87), (88) are the necessarycondition for the approximation of a point particle can be used. In this case, the localized wave packet ofa particle radiating twisted photons in a wiggler can be characterized only by the average coordinate andmomentum. As we see, the quantum recoil should be small for that to be the case.It is useful to write restrictions (72), (87) as a system of inequalities specifying the admissible region onthe plane ( k m /ε, m ) : k m γ (cid:46) (cid:16) m (cid:17) / , m (cid:46) K √ , k m γ (cid:46) Kσ ⊥ . (89)Note that k m ≈ ¯ k m . One can distinguish two cases i ) 5 σ ⊥ > K, ii ) 5 σ ⊥ < K. (90)In the case (i), the region (89) is reduced to a rectangle k m γ (cid:46) Kσ ⊥ , m (cid:46) K √ . (91)The twisted photons with the maximum energy and projection of the total angular momentum are producedwhen inequalities (91) turn into the equalities. In the case (ii), the region (89) has nontrivial angular pointsat k m = γ Kσ ⊥ , m = (5 Kσ ⊥ ) / √ k m = γ K , m = K √ , (92)13hich coalesce for σ ⊥ = K . In terms of m , the system of inequalities (89) is written as m (cid:46) (cid:16) K γω (cid:17) / , m (cid:46) K √ , m (cid:46) K σ ⊥ γω , (93)respectively. Let us apply the above general formulas for description of radiation of twisted photons by an ultrarelativisticcharged particle in the laser wave of a circular polarization. We suppose that the one-photon radiation givesthe leading contribution to radiation of twisted photons. Then we can employ the formulas from the precedingsections to describe this radiation. The strength tensor of the electromagnetic field reads as follows (we usethe notation borrowed from [98, 99, 102]) eF µν = a ( ξ ) h [ µ − (cid:104) h ν ]1 cos ϕ ( ξ ) + h ν ]2 sin ϕ ( ξ ) (cid:105) , (94)where h µ − = (1 , , , ζ ) , h µ , = δ µ , , the function a ( ξ ) characterizes the amplitude of the electromagnetic fieldand ϕ ( ξ ) is the phase, where ξ = h µ − x µ = x − ζx . We consider the situation when the electromagnetic wavepropagates along the axis of the detector of twisted photons. The quantity ζ = ± , where the upper signcorresponds to the case when the wave moves towards the detector and the lower sign is for the case whenthe wave moves from the detector.It is useful to convert all the quantities to the dimensionless ones using the Compton wavelength as aunit length [(5), [102]]. Then, for example, the laser wave with intensity W/cm and photon energy . eV [103] corresponds to | a | ≈ . × − , | Ω | ≈ . × − . (95)The Lorentz equations are easily solved with arbitrary function a ( ξ ) (see, e.g., [(51), [98]] for λ = 0 and [34]): x ( ξ ) = x (0) + r (0) ξ − υ − − (cid:90) ξ dx (cid:90) x dya ( y ) cos ϕ ( y ) ,x ( ξ ) = x (0) + r (0) ξ − υ − − (cid:90) ξ dx (cid:90) x dya ( y ) sin ϕ ( y ) ,x ( ξ ) + ζx ( ξ ) = (cid:90) ξ dx (cid:104) υ − − + (cid:16) r (0) − υ − − (cid:90) x dya ( y ) cos ϕ ( y ) (cid:17) ++ (cid:16) r (0) − υ − − (cid:90) x dya ( y ) sin ϕ ( y ) (cid:17) (cid:105) ,ξ = υ − τ, (96)where r µ := υ µ /υ − , υ µ is the -velocity, υ µ υ µ = 1 , υ − := υ − ζυ = const , τ is the proper time, and itis assumed that x = x = 0 at the initial instant of time. It is clear this assumption does not destroy thegenerality of our considerations. Upon shifting x and x by constants, the amplitude of radiation of a twistedphoton changes by an overall phase, which does not affect the probability to record the twisted photon by adetector. It is convenient to pass in formulas (28), (47) and [(36), [33]] from the integration variable t to thevariable ξ . The corresponding derivatives take the form r ( ξ ) = r (0) − υ − − (cid:90) ξ dxa ( x ) cos ϕ ( x ) ,r ( ξ ) = r (0) − υ − − (cid:90) ξ dxa ( x ) sin ϕ ( x ) ,r ( ξ ) + ζr ( ξ ) = υ − − + (cid:16) r (0) − υ − − (cid:90) ξ dxa ( x ) cos ϕ ( x ) (cid:17) + (cid:16) r (0) − υ − − (cid:90) ξ dxa ( x ) sin ϕ ( x ) (cid:17) . (97)In order to obtain analytic formulas, we assume that the phase ϕ = Ω ξ + ϕ , (98)14here Ω is the frequency of the electromagnetic wave and ϕ is the initial phase. The amplitude is chosen as a ( ξ ) = const, ξ ∈ [0 , πN ] , (99)where N is the number of periods of the electromagnetic wave. The amplitude vanishes outside this interval,i.e., it is assumed that the laser wave pulse possesses sharp rising and descending edges. In that case,employing the notation from (28), (47), and [(36), [33]], we have r ± = ¯ r ± ± iKυ − − e ± iϕ , r = 12 υ − (cid:2) υ − + 1 + K + ¯ υ ⊥ − K ¯ υ ⊥ sin( ϕ − ρ ) (cid:3) ,r = ζ υ − (cid:2) K + ¯ υ ⊥ − υ − − K ¯ υ ⊥ sin( ϕ − ρ ) (cid:3) , (100)where K := a/ Ω , ¯ r ± := r ± (0) ∓ iKυ − − e ± iϕ , ¯ υ ⊥ := υ − | ¯ r ± | (101)and ρ = arg ¯ r + . The solution to the Lorentz equations is given by x ± = ¯ x ± + ¯ r ± ξ + Kυ − Ω e ± iϕ , x = 12 υ − (cid:2) ( υ − + 1 + K + ¯ υ ⊥ ) ξ + 2¯ υ ⊥ K Ω (cos( ϕ − ρ ) − cos( ϕ − ρ )) (cid:3) ,x = ζ υ − (cid:2) (1 + K + ¯ υ ⊥ − υ − ) ξ + 2¯ υ ⊥ K Ω (cos( ϕ − ρ ) − cos( ϕ − ρ )) (cid:3) , (102)where ¯ x ± := x ± (0) − Kυ − Ω e ± iϕ . (103)Notice that if the charged particle moves initially along the axis of the twisted photon detector, viz., r ± (0) = 0 ,then ¯ r ± (cid:54) = 0 . Radiation without recoil.
Let us consider the radiation of twisted photons in the case when the quantumrecoil is negligible. In order to find the probability of radiation [(36), [33]], it is necessary to evaluate theamplitudes I = (cid:90) T N dξr ( ξ ) e − ik ( x − n x ) j m ( k ⊥ x + , k ⊥ x − ) ,I ± = in ⊥ s ∓ n (cid:90) T N dξr ± ( ξ ) e − ik ( x − n x ) j m ∓ ( k ⊥ x + , k ⊥ x − ) . (104)Then the probability to record the twisted photon is dP ( s, m, k ⊥ , k ) = e (cid:12)(cid:12)(cid:12) I + 12 I + + 12 I − (cid:12)(cid:12)(cid:12) n ⊥ dk dk ⊥ π . (105)In formula (104), the contributions from the parts of particle trajectory with ξ (cid:54)∈ [0 , πN ] are neglected.These contributions correspond to the edge radiation. They can be ignored when the energies of recordedphotons are sufficiently large. A thorough description of the edge radiation in terms of twisted photons isgiven in [104].The evaluation of integrals (104) is performed analogously to the case of undulator radiation studied in[33]. First of all, we shift the integration variable ξ → ξ + T N/ and make use of the addition theorem [(A6),[33]] for the Bessel functions (see also [105]): j m ( k ⊥ x + , k ⊥ x − ) = ∞ (cid:88) l = −∞ j m − l (cid:0) k ⊥ z + , k ⊥ z − (cid:1) j l ( k ⊥ y + , k ⊥ y − ) , (106)where z ± := ¯ x ± + ¯ r ± T N/ , y ± := ¯ r ± ξ + Kυ − Ω e ± iϕ . (107)15ow the phase ϕ entering into ϕ includes πN . We denote this phase by ϕ N . Notice that, on shifting thevariable ξ , the phase ϕ appearing explicitly in formulas (101), (102), and (103) does not change. Substitutethe integral representation [(A8), [33]] j l ( k ⊥ y + , k ⊥ y − ) = i − l (cid:90) π − π dψ π e ilψ e ik ⊥ ( y sin ψ + y cos ψ ) (108)into (106) and then (106) into (104). As a result, the expression standing in the exponent in the integrandof I becomes − i k υ − ξ (cid:2) (1 − ζn )(1 + K + ¯ υ ⊥ ) + (1 + ζn ) υ − − n ⊥ ¯ υ ⊥ υ − cos( ψ − ρ ) (cid:3) + iη sin( ϕ + δ ) + ilψ, (109)up to a constant term that does not influence the probability of radiation. Here η cos δ := Kk υ − Ω ( n ⊥ sin ψ − (1 − ζn )¯ r ) , η sin δ := Kk υ − Ω ( n ⊥ cos ψ − (1 − ζn )¯ r ) . (110)Using the Fourier series expansion e iη sin( ϕ + δ ) = ∞ (cid:88) n = −∞ e in ( ϕ + δ ) J n ( η ) , (111)the integral over ξ is reduced to the delta-like sequence (cid:90) T N/ − T N/ dξ π e − ix n ξ = δ N ( x ) , δ N ( x n ) := sin( T N x n / πx n . (112)The argument of the regularized delta function reads x n = k υ − (cid:2) (1 − ζn )(1 + K + ¯ υ ⊥ ) + (1 + ζn ) υ − − n ⊥ ¯ υ ⊥ υ − cos( ψ − ρ ) (cid:3) − Ω n. (113)For N large, the main contribution to the integral over ψ comes from the points where the argument of theregularized delta function vanishes.Below we shall assume that Ω > and, at the end, shall discuss how the results change for Ω < . Thecondition x n = 0 can be conveniently written as x n = Ω2 ( b n + a n )[cos ξ n − cos( ψ − ρ )] = 0 , (114)where the notation has been introduced [33] a n := n − k ω − , b n := k ω − − − n, ξ n := arccos b n − a n b n + a n , (115)and ω ± := 2Ω υ − (1 − ζn )(1 + K + ¯ υ ⊥ ) + (1 + ζn ) υ − ∓ n ⊥ ¯ υ ⊥ υ − . (116)If N is so large that δ N ( x n ) removes the integration over ψ , then the energy spectrum of radiated twistedphotons consists of the intervals k ∈ n [ ω − , ω + ] , n = 1 , ∞ . (117)The radiation is suppressed outside these intervals. These intervals become overlapping starting from theharmonic number n = ω − ω + − ω − . (118)When k belongs to the intervals, a n (cid:62) , b n (cid:62) , and ξ n ∈ [0 , π ] .As a result, neglecting the terms at nonpositive n , we have16 ≈ ∞ (cid:88) n =1 ∞ (cid:88) l = −∞ j m − l ( k ⊥ z + , k ⊥ z − ) (cid:90) π − π dψδ N ( x n ) e in ( δ + ϕ N )+ ilψ i − l ×× ζ υ − (cid:104) (1 + K + ¯ υ ⊥ − υ − ) J n ( η ) + iK ¯ υ ⊥ (cid:0) e − i ( δ + ρ ) J n − ( η ) − e i ( δ + ρ ) J n +1 ( η ) (cid:1)(cid:105) . (119)As for the rest integrals, we obtain similarly I ± ≈ ∞ (cid:88) n =1 ∞ (cid:88) l = −∞ j m − l ( k ⊥ z + , k ⊥ z − ) (cid:90) π − π dψδ N ( x n ) e in ( δ + ϕ N )+ i ( l ∓ ψ i − l ∓ n ⊥ s ∓ n (cid:104) ¯ r ± J n ± iKυ − − J n ∓ e ∓ iδ (cid:105) , (120)where the arguments of the Bessel functions are the same as in (119). In order to obtain (120), one needsto shift the summation index l → l ∓ in the series (106). Taking into account estimates (188), the totalcontribution to the radiation amplitude takes the form I + 12 ( I + + I − ) ≈ ∞ (cid:88) n =1 ∞ (cid:88) l = −∞ j m − l ( k ⊥ z + , k ⊥ z − ) (cid:90) π − π dψ δ N ( x n ) e in ( δ + ϕ N )+ ilψ i − l g n ( ψ ) , (121)where g n ( ψ ) := ζ K + ¯ υ ⊥ − υ − υ − J n + ζ iK ¯ υ ⊥ υ − (cid:0) e − i ( δ + ρ ) J n − − e i ( δ + ρ ) J n +1 (cid:1) −− n ⊥ e − isψ (cid:0) ¯ r s J n + s iKυ − J n − s e − isδ (cid:1) . (122)The last expression can be rewritten in terms of the Bessel function and its derivative with the same index g n ( ψ ) = 2 (cid:104) ζ K + ¯ υ ⊥ − υ − υ − − isKnn ⊥ υ − η e − is ( δ + ψ ) + ζ K ¯ υ ⊥ nυ − η sin( δ + ρ ) − ¯ r s n ⊥ e − isψ (cid:105) J n −− iKn ⊥ υ − (cid:104) e − is ( δ + ψ ) − ζ n ⊥ ¯ υ ⊥ υ − cos( δ + ρ ) (cid:105) J (cid:48) n . (123)Further, we suppose that N is so large that δ N ( x n ) removes the integration over ψ . The solution of (114) is,evidently, ψ = ρ ± ξ n . (124)As in the case of undulator radiation, the three cases occur [33]: (a) the regular case ξ n (cid:54) = { , π } ; (b) theweakly degenerate case ξ n = { , π } ; and (c) the strongly degenerate case a n = b n = 0 .Let us begin with the regular case. For N large, in the leading order, we deduce δ N ( x n ) ≈ θ ( a n ) θ ( b n )Ω √ a n b n [ δ ( ψ − ρ − ξ n ) + δ ( ψ − ρ + ξ n )] . (125)The delta functions remove integration in (121). The remaining sum over l can be performed by using therelation ∞ (cid:88) k = −∞ t k j k ( p, q ) = e ( pt − q/t ) / , (126)which follows from [(A7), [33]]. Then, up to an irrelevant phase, I + 12 ( I + + I − ) ≈ ∞ (cid:88) n =1 θ ( a n ) θ ( b n )2Ω √ a n b n e inϕ N (cid:110) e inδ + imψ + ik ⊥ | z + | cos( ψ − arg z + ) g n ( ψ ) (cid:12)(cid:12)(cid:12) ψ = ρ + ξ n ++ e inδ + imψ + ik ⊥ | z + | cos( ψ − arg z + ) g n ( ψ ) (cid:12)(cid:12)(cid:12) ψ = ρ − ξ n (cid:111) . (127)In the photon energy range where the harmonics do not overlap, we obtain dP ( s, m, k ⊥ , k ) ≈ e ∞ (cid:88) n =1 θ ( a n ) θ ( b n )4Ω a n b n (cid:12)(cid:12)(cid:12)(cid:12) e inδ + imψ + ik ⊥ | z + | cos( ψ − arg z + ) g n ( ψ ) (cid:12)(cid:12)(cid:12) ψ = ρ + ξ n ++ e inδ + imψ + ik ⊥ | z + | cos( ψ − arg z + ) g n ( ψ ) (cid:12)(cid:12)(cid:12) ψ = ρ − ξ n (cid:12)(cid:12)(cid:12)(cid:12) n ⊥ dk dk ⊥ π . (128)17he dependence on m for the energies k belonging to the spectral band with number n is periodic with theperiod [33] T m = (cid:26) π/ξ n , ξ n ∈ (0 , π/ π/ ( π − ξ n ) , ξ n ∈ [ π/ , π ) . (129)Of course, this periodicity holds only for those quantum numbers m where δ N ( x n ) can be replaced by deltafunction (125).Now we turn to the weakly degenerate case. Let a n = 0 , b n > , i.e., k = nω + . Then b n = n ( ω + ω − − − , ξ n = 0 , ψ = ρ. (130)We assume that N is so large that all the integrand functions in (121), apart form δ N ( x n ) , can be taken atthe point ψ = ρ and be removed from the integrand. In that case, the integral arises (cid:90) π − π dψδ N ( x n ) ≈ (cid:90) ∞−∞ dψ sin (cid:2) πN n ( ω + ω − − − ψ / (cid:3) π Ω n ( ω + ω − − − ψ / − (cid:115) Nn ( ω + ω − − − . (131)The probability to record a twisted photon becomes dP ( s, m, k ⊥ , k ) ≈ e N | g n ( ρ ) | n ⊥ Ω n ( ω + ω − − − dk dk ⊥ π . (132)For b n = 0 , a n > , i.e., for k = nω − , the similar calculations lead to dP ( s, m, k ⊥ , k ) ≈ e N | g n ( π − ρ ) | n ⊥ Ω n (1 − ω − ω − ) dk dk ⊥ π . (133)In the domain where the applicability conditions of the approximations made are fulfilled, the explicit de-pendence of the twisted photon radiation probability on m disappears.In the strongly degenerate case, the spectral bands (117) turn into narrow lines k = nω + = nω − . Thishappens when ¯ υ ⊥ ≈ . Then η = Kk ⊥ υ − Ω , δ = π/ − ψ, (134)The functions, x n = k υ − (cid:2) (1 − ζn )(1 + K ) + (1 + ζn ) υ − (cid:3) − Ω n,g n = ζ K − υ − υ − J n − Kn ⊥ υ − J n − s = (cid:104) ζ K − υ − υ − − nn ⊥ k ⊥ (cid:105) J n − sKn ⊥ υ − J (cid:48) n , (135)do not depend on ψ , and the integral over ψ in (121) is readily performed. As a result, I + 12 ( I + + I − ) ≈ πδ N ( x n ) j m − n ( k ⊥ z + , k ⊥ z − ) e inϕ ( − nN g n . (136)The probability to record a twisted photon is given by dP ( s, m, k ⊥ , k ) = e δ N ( x n ) J m − n ( k ⊥ | z + | ) g n n ⊥ dk dk ⊥ . (137)If k ⊥ | z + | (cid:28) , then the selection rule m = n is fulfilled [33, 41, 45, 49–55].Let us show how the above results are modified for Ω < . The sign change of Ω corresponds to a changeof polarization of the incident electromagnetic wave (94). This, in turn, leads to a change of handedness ofthe helix along which the charged particle is moving. Upon changing the sign of Ω , formulas (115)-(118)remain valid with the replacement Ω → | Ω | . Since the substitution Ω → − Ω results in K → − K, n → − n, (138)we have g n ( ψ ) → ( − n g n ( ψ ) . (139)18herefore, on substituting Ω → −| Ω | in (128), (132), (133), and (137), one must set e inδ → e − inδ , (140)in formula (128), formulas (132), (133) remain intact, and, in formula (137), one needs to replace J m − n ( k ⊥ | z + | ) → J m + n ( k ⊥ | z + | ) . (141)For k ⊥ | z + | (cid:28) , the selection rule in the strongly degenerate case looks as m = − n .Consider in more detail the cases when the electromagnetic wave propagates towards the detector oftwisted photons or from it. In these cases, the electron bunch moves approximately along the direction ofpropagation of the electromagnetic wave or in the opposite direction, respectively. In the case when theelectromagnetic wave propagates toward the detector, we have ζ = 1 and υ − ≈ υ ⊥ γ ∼ κ γ , υ ⊥ ∼ κ , ¯ υ ⊥ (cid:46) κ , n ⊥ (cid:46) κ γ . (142)The bounds of the spectral bands (117) are expressed through ω ± ≈ Ω (cid:104)(cid:0) ∓ n ⊥ ¯ υ ⊥ υ − (cid:1) + n ⊥ υ − (1 + K ) (cid:105) − ∼ Ω . (143)In the strongly degenerate case, the radiation spectrum becomes k ≈ Ω n (cid:104) n ⊥ υ − (1 + K ) (cid:105) − , n = 1 , ∞ . (144)For n (cid:62) , the Bessel functions entering into g n can be expressed through the Airy functions (see [(122),[33]] and also [9]) with x = 1 − K υ − /n ⊥ (cid:0) K + 4 υ − /n ⊥ (cid:1) . (145)For the radiation probability not to be exponentially suppressed, this quantity should be small x (cid:46) / .This occurs for K (cid:38) and n ⊥ ≈ υ − K ≈ υ ⊥ γK ∼ κ γ . (146)In that case, the radiation probability drops exponentially to zero at the harmonic numbers [33] n (cid:38) K √ . (147)If the electromagnetic wave moves from the detector, i.e., the head-on collision of the laser wave with thebunch of charged particles is considered, then ζ = − and υ − ≈ γ, ¯ υ ⊥ (cid:46) κ , n ⊥ (cid:46) κ /γ, (148)and also ω ± ≈ Ω υ − K + (¯ υ ⊥ ∓ n ⊥ υ − / ∼ γ κ . (149)The analysis in this case is completely analogous to the analysis of the undulator radiation [33] at theobservation angle θ := ¯ υ ⊥ /γ and the undulator frequency ω := 2Ω (cf. [(85), [33]]). In the strongly degeneratecase, the radiation spectrum looks as k ≈ Ω nυ − K + n ⊥ υ − / , n = 1 , ∞ . (150)For n (cid:62) , the radiation probability is not exponentially suppressed at K (cid:38) if n ⊥ ≈ K/γ and n (cid:46) K / √ .19 d P / d k d k ⊥ - - - - - - d P / d k d k ⊥ s =
1, n ⊥ = / ( σ ⊥ γ ) k = = = × - γ = l = ClassicalScalarDirac - - - - d P / d k d k s =
1, n ⊥ = / γ k =
499 eVK = × - γ = l = k - d P / d k d k Figure 2:
The radiation of twisted photons by . MeV electrons evolving in the circularly polarized electromagnetic waveproduced by the free electron laser with the photon energy keV, intensity . × W/cm , and amplitude envelope (172)with N = 20 . These data correspond to Ω ≈ . × − and a ≈ . × − . The applicability conditions (93) are satisfied for σ ⊥ (cid:46) m − . The photon energy is measured in the electron rest energies. Left panel: The head-on collision. The first harmonic(175) for radiation without recoil (classical), scalar, and Dirac particles is depicted. For the parameters chosen, the quantumrecoil halves the energy of radiated photons in comparison with formula without recoil. The probability of radiation of twistedphotons by Dirac particles is bigger than by the one by scalar particles which, in turn, is bigger than the probability of radiationof twisted photons without quantum recoil (see the discussion after Eq. (63)). Right panel: The laser wave is overtaking theelectron. The first harmonic (175) is shown. The quantum recoil is negligible in this case. Insets: The distributions over m atthe main maxima of harmonics. Radiation with recoil.
Now we take the quantum recoil into account. In the case ζ = 1 , estimate (143)holds. Therefore, the quantum recoil can be neglected for reasonable photon energies of the laser wave (see(95)). In that case, the probability to record a twisted photon radiated by both scalar and Dirac particles isdescribed by formula [(36), [33]]. Thus, formulas obtained above remain intact with good accuracy.In the case ζ = − , the quantum recoil can be significant. Since the approximate equality (148) is valid,the quantity q = P /P (cid:48) = υ /υ (cid:48) = γ/ ( γ − k ) ≈ υ − / ( υ − − k ) = const, (151)up to the terms of order κ /γ . Hence, the probability to detect a twisted photon radiated by a chargedscalar particle with the quantum recoil taken into account (28) is obtained from the formulas above, wherethe recoil was ignored, by multiplying the probability by q and substituting k → ¯ k := qk (152)in the definitions of a n , b n (115) and the radiation spectrum (117). As was discussed in the precedingsections, the term standing in the exponent and proportional to k ⊥ / (2 P i ) can be safely neglected. As far asthe strongly degenerate case (137) is concerned, substitution (152) has to be done in formula (135) for x n ,and, of course, (137) must be multiplied by q .The treatment of the Dirac particle case is a bit more complex. The probability to record a twistedphoton equals dP ( s, m, k , k ⊥ ) = dP ( s, m, k ⊥ , k ) + dP a ( s, m, k ⊥ , k ) , (153)where dP ( s, m, k ⊥ , k ) is the contribution of the first term in (47) and dP a ( s, m, k ⊥ , k ) is the contributionof the second and third terms in (47). As long as (151) holds, the contribution of the first term in (47) isevaluated as in the case of negligible quantum recoil: the probability to record a twisted photon withoutrecoil must be multiplied by (1 + q ) / , (154)and substitution (152) must be performed in the definitions of a n , b n (115) and the radiation spectrum (117).The contribution of the last two terms in (47) has to be evaluated from scratch. It follows from theexplicit expressions for the mode functions [(13), [33]] that for s = − the third term in (47) can be omittedwhile for s = 1 the second term can be thrown out. Let I a := k υ (cid:48) (cid:90) T N dξe − i ¯ k ( x − n x ) [ r s a − s − isn ⊥ r a − s ( m + s )] . (155)20 = ⊥ = K /( γ ) K = γ = - - - - - - - - - - d P / d k d k ⊥ k = = = = = =
10 15 20 25 30 3501234 10 k - d P / d k d k ⊥ s = ⊥ = K /( γ ) K = γ = - - - - - - - - - - d P / d k d k ⊥ k = = = = = = k - d P / d k d k ⊥ - - - - - - - - - - d P / d k d k ⊥ s = ⊥ = K / γ K = γ = = = = = = = k - d P / d k d k ⊥ - - - - - - - - - - d P / d k d k ⊥ s = ⊥ = / γ K = γ = = = = = = = k - d P / d k d k ⊥ Figure 3:
The radiation of twisted photons in head-on collision of
MeV electrons with the circularly polarized electromag-netic wave produced by the CO laser with the wavelength µ m, intensity W/cm , and amplitude envelope (172) with N = 20 . These data correspond to Ω ≈ . × − and a ≈ . × − . The first three harmonics (175) are depicted. Theapplicability conditions (93) are satisfied for σ ⊥ (cid:46) m − . The photon energy is measured in the electron rest energies. Inset:The distributions over m at the main maxima of harmonics. The contribution of the last two terms in (47) is proportional to the modulus squared of this integral. Forthe head-on collision, we have r ≈ / , (156)Then, performing the calculations completely analogous to the case of negligible recoil, we find I a ≈ k υ (cid:48) ∞ (cid:88) n =1 ∞ (cid:88) l = −∞ j m − l ( k ⊥ z + , k ⊥ z − ) (cid:90) π − π dψδ N ( x n ) e in ( δ + ϕ N )+ ilψ i − l g an ( ψ ) , (157)where g an ( ψ ) := (cid:16) − r s n ⊥ e − isψ (cid:17) J n ( η ) − isKn ⊥ υ − J n − s ( η ) e − is ( δ + ψ ) , (158)and x n has the form (114) with a n := n − ¯ k ω − , b n := ¯ k ω − − − n, ξ n := arccos b n − a n b n + a n . (159)The approximate expressions for ω ± are written in (149).Let us consider separately the regular, weakly degenerate, and strongly degenerate cases. In the regularcase, under the same assumptions that was made in considering the first term in (47), we deduce up to anirrelevant phase I a ≈ k υ (cid:48) ∞ (cid:88) n =1 θ ( a n ) θ ( b n )Ω √ a n b n e inϕ N (cid:110) e inδ + imψ + ik ⊥ | z + | cos( ψ − arg z + ) g an ( ψ ) (cid:12)(cid:12)(cid:12) ψ = ρ + ξ n ++ e inδ + imψ + ik ⊥ | z + | cos( ψ − arg z + ) g an ( ψ ) (cid:12)(cid:12)(cid:12) ψ = ρ − ξ n (cid:111) . (160)21 = ⊥ = K /( γ ) K = γ = - - - - - - - - - - d P / d k d k ⊥ k =
659 eVk = = = = = k - d P / d k d k ⊥ s = ⊥ = K /( γ ) K = γ = - - - - - - - - - - d P / d k d k ⊥ k =
557 eVk = = = = = k - d P / d k d k ⊥ s = ⊥ = K / γ K = γ = =
342 eVk =
669 eVk =
991 eV n = = = - - - - - - - - - - d P / d k d k ⊥ k - d P / d k d k ⊥ - - - - - - - - - - d P / d k d k ⊥ k =
132 eVk =
261 eVk =
389 eV s = ⊥ = / γ K = γ = = = = k - d P / d k d k ⊥ Figure 4:
The radiation of twisted photons in head-on collision of
MeV electrons with the circularly polarized electro-magnetic wave produced by the Ti:Sa laser with parameters (95) and amplitude envelope (172) with N = 20 . The first threeharmonics (175) are depicted. The applicability conditions (93) are satisfied for σ ⊥ (cid:46) m − . The photon energy is measuredin the electron rest energies. Inset: The distributions over m at the main maxima of harmonics. The respective contribution to the probability, in the region of photon energies where the harmonics do notoverlap, becomes dP a ( s, m, k ⊥ , k ) ≈ e k υ (cid:48) ∞ (cid:88) n =1 θ ( a n ) θ ( b n )Ω a n b n (cid:12)(cid:12)(cid:12)(cid:12) e inδ + imψ + ik ⊥ | z + | cos( ψ − arg z + ) g an ( ψ ) (cid:12)(cid:12)(cid:12) ψ = ρ + ξ n ++ e inδ + imψ + ik ⊥ | z + | cos( ψ − arg z + ) g an ( ψ ) (cid:12)(cid:12)(cid:12) ψ = ρ − ξ n (cid:12)(cid:12)(cid:12)(cid:12) n ⊥ dk dk ⊥ π . (161)This expression, just as the contribution of the first term in (47), is a periodic function of m with the periodgiven in (129).In the weakly degenerate case, for a n = 0 , b n > , i.e., ¯ k = nω + , we obtain dP a ( s, m, k ⊥ , k ) ≈ e k υ (cid:48) N | g an ( ρ ) | n ⊥ Ω n ( ω + ω − − − dk dk ⊥ π . (162)If b n = 0 , a n > , i.e., ¯ k = nω − , then dP a ( s, m, k ⊥ , k ) ≈ e k υ (cid:48) N | g an ( π − ρ ) | n ⊥ Ω n (1 − ω − ω − ) dk dk ⊥ π . (163)These expressions are independent of m .In the strongly degenerate case a n = 0 , b n = 0 , we suppose that ¯ υ ⊥ = 0 . Then x n does not depend on ψ , g an = J n ( η ) − Kn ⊥ υ − J n − s ( η ) = (cid:16) − n Ω n ⊥ k ⊥ (cid:17) J n ( η ) − sKn ⊥ υ − J (cid:48) n ( η ) = g n , (164)and relations (134) take place. The contribution to the probability to detect a twisted photon is dP a ( s, m, k ⊥ , k ) = e k υ (cid:48) δ N ( x n ) J m − n ( k ⊥ | z + | ) g n n ⊥ dk dk ⊥ . (165)22 - -
20 0 20 40 60012345 m - d P / d k d k ⊥ s = ⊥ = K / γ K = γ = =
639 eV l = k - d P / d k d k ⊥ s = ⊥ = K / γ K = γ = =
996 eV l = - - - - - - d P / d k d k ⊥ k - d P / d k d k ⊥ Figure 5:
The radiation of twisted photons in head-on collision of
MeV electrons with the circularly polarized electro-magnetic wave produced by the Ti:Sa laser. The parameters are the same as in Fig. 4 but the laser pulse amplitude envelopesare different. Left panel: The amplitude envelope has the form (99). The initial laser wave phase ϕ = 0 . The maximum near k = 3 ω + is shown. Right panel: The amplitude envelope is a sin (Ω ξ/ (2 N )) . The third harmonic (175) is depicted. Insets: Thedistributions over m in the main maxima. The total probability (153) becomes (cf. (63)) dP ( s, m, k ⊥ , k ) = e (1 + q ) δ N ( x n ) J m − n ( k ⊥ | z + | ) g n n ⊥ dk dk ⊥ . (166)For k ⊥ | z + | (cid:28) , the selection rule m = n is fulfilled. As in the case of forward undulator radiation, theapplicability conditions (83), (93) must be satisfied. The number of twisted photons recorded by the detectoris approximately given by (82). Examples.
As is seen from (137), (166), the electrons moving in the laser wave represent a pure source oftwisted photons only in the strongly degenerate case when ¯ υ ⊥ ≈ and k ⊥ | z + | ≈ . For this to be the case,the initial data must be taken in the form (see (101), (103), (107)) r + (0) = iKυ − − e iϕ , x + (0) = Kυ − − Ω − e iϕ . (167)Since it is very hard to control the initial phase of a laser wave, in the wiggler regime, K (cid:38) , it is practicallyimpossible to launch the electron to the electromagnetic wave so that the radiation produced by it wouldcorrespond to the strongly degenerate case. For an arbitrarily chosen phase, equalities (167) are stronglyviolated, harmonics (117) spread violently, and the twisted photon detector records a wide distribution over m (see Fig. 5). In this case, the twisted photons escape the laser wave at large angles to the detector axisrather than move along it (see Fig. 6).It turns out that this situation can be improved if one takes into account that, usually, the laser wavepulses generated in experiments have no sharp rising and descending edges. The amplitude envelope a ( ξ ) is asmooth function vanishing at ξ = { , T N } and a (cid:48) ( ξ ) /a ( ξ ) ∼ /N . For N (cid:38) , this entails that the radiationprobability ceases to depend virtually on the initial phase ϕ . As a result, it is possible to choose the initialvelocity and entrance point of the electron to the electromagnetic wave so that the corresponding radiationwill be a sufficiently pure source of twisted photons.Indeed, under the restrictions on the form of the envelope mentioned above, the integrals entering intogeneral solution (96), (97) of the Lorentz equations, can be approximately evaluated for N (cid:38) . Integratingonce by parts and keeping only the integrated term, we obtain (cf. (100), (102)) x ± ≈ x ± (0) + r ± (0) ξ + Kυ − Ω e ± iϕ , x ≈ υ − (cid:2) ( υ − + 1 + K + υ ⊥ (0)) ξ + 2 υ ⊥ (0) K Ω cos( ϕ − ρ ) (cid:3) ,x ≈ ζ υ − (cid:2) (1 + K + υ ⊥ (0) − υ − ) ξ + 2 υ ⊥ (0) K Ω cos( ϕ − ρ ) (cid:3) , (168)23 - - - - - - x - y - - - - r x r y - - - - - - - - - -
50 10 - x - y - - - - - - -
20 10 r x r y - - - - - - - x - y - - - - r x r y a b c Figure 6:
The coordinates x , y and “velocities” r x , r y for head-on collision of an electron with the circularly polarized electro-magnetic wave produced by the Ti:Sa laser. The parameters are the same as in Fig. 5. The electron moves initially along thedetector axis. The initial laser wave phase ϕ = 0 . The lengths are measured in the Compton wavelengths. a) The amplitudeenvelope has the form (99). b) The amplitude envelope is given in (172). c) The amplitude envelope is a sin (Ω ξ/ (2 N )) . where K := a ( ξ ) / Ω and r ± ≈ r ± (0) ± iKυ − − e ± iϕ , r ≈ υ − (cid:2) υ − + 1 + K + υ ⊥ (0) − Kυ ⊥ (0) sin( ϕ − ρ ) (cid:3) ,r ≈ ζ υ − (cid:2) K + υ ⊥ (0) − υ − − Kυ ⊥ (0) sin( ϕ − ρ ) (cid:3) . (169)The accuracy of this approximation increases as N increases. As is seen, the form of the trajectory is almostthe same as in the case of a laser wave with constant amplitude but without strong dependence on the initialphase. Now the dependence on the initial phase is contained only in ϕ .Substituting approximate trajectory (168), (169) into (104), it is easy to see that the pure source oftwisted photons can be obtained when x + (0) ≈ , υ ⊥ (0) ≈ , (170)i.e., in the strongly degenerate case. In this paper we will investigate only this case. The plots of typicaltrajectories in this case are presented in Fig. 6. The calculations are performed along the same lines as thosemade above, except that the shift ξ → ξ + T N/ is unnecessary. In particular, formulas (134), (135) holdand the radiation amplitude is proportional to I + 12 ( I + + I − ) = 12 ∞ (cid:88) n = −∞ j m − n ( k ⊥ x + (0) , k ⊥ x − (0)) e inϕ (cid:90) T N dξg n ( ξ ) e − iξx n ( ξ ) . (171)On stretching the variable ξ → T N ξ , it is clear that the integral over ξ can be approximately evaluated bythe WKB method. For the envelope with one maximum as, for example, a ( ξ ) = a sin (Ω ξ/ (2 N )) , (172)24he function ξx n ( ξ ) has two extrema, ξ ± ( k ) , on the interval (0 , T N ) , in a general position. At these extrema, k n = 2Ω nυ − (1 − ζn )(1 + K + 2 ξKK (cid:48) ) + (1 + ζn ) υ − > . (173)Therefore, we have I + 12 ( I + + I − ) ≈ (cid:114) π ∞ (cid:88) n = −∞ j m − n ( k ⊥ x + (0) , k ⊥ x − (0)) e inϕ (cid:104) g n ( ξ ) e − iξx n ( ξ ) (cid:112) i ( ξx n ( ξ )) (cid:48)(cid:48) (cid:12)(cid:12)(cid:12) ξ = ξ + + g n ( ξ ) e − iξx n ( ξ ) (cid:112) i ( ξx n ( ξ )) (cid:48)(cid:48) (cid:12)(cid:12)(cid:12) ξ = ξ − (cid:105) , (174)where the principal branch of the square root is taken. The contribution of the boundaries is suppressed since,in the leading order in /N , the same contribution but with opposite sign comes from the edge radiation (seefor details, e.g., [102, 104]). As a result, the contributions of the internal stationary points are only relevant.One of the extremum points, ξ + , is close to the point where a (cid:48) ( ξ ) = 0 , i.e., K ( ξ ) is close to its maximumvalue at this point. Taking into account the form of g n ( ξ ) , we see that this stationary point gives the leadingcontribution to (174). The main maximum is located approximately at k n ≈ nυ − (1 − ζn )(1 + K max ) + (1 + ζn ) υ − , Ω n > . (175)In fact, the maximum is slightly shifted to the right since the stationary point ξ + is displaced a little fromthe extremum of K ( ξ ) . If k ⊥ | x + (0) | (cid:28) , then j m − n ( k ⊥ x + (0) , k ⊥ x − (0)) = δ mn and dP ( s, m, k ⊥ , k ) = e (cid:12)(cid:12)(cid:12)(cid:12) g m ( ξ ) e − iξx m ( ξ ) (cid:112) i ( ξx m ( ξ )) (cid:48)(cid:48) (cid:12)(cid:12)(cid:12) ξ = ξ + + g m ( ξ ) e − iξx m ( ξ ) (cid:112) i ( ξx m ( ξ )) (cid:48)(cid:48) (cid:12)(cid:12)(cid:12) ξ = ξ − (cid:12)(cid:12)(cid:12)(cid:12) n ⊥ dk dk ⊥ π . (176)At the extremum points ξ ∼ N , x (cid:48) n ( ξ ) ∼ /N , and x (cid:48)(cid:48) n ( ξ ) ∼ /N . Therefore, the radiation probability isproportional to N .The dependence of the probability density on k differs from the profile δ N ( x m ) and is depicted in Figs.3, 4, 7. As is seen, the radiation probability dP ( m ) is nearly zero for k < k m , then it rapidly grows inthe vicinity of k = k m , and for k > k m it declines performing oscillations to zero. When n ⊥ γ (cid:46) K max / ,this decrease is quite slow. So, for these values of n ⊥ , the radiation probability dP ( m, k ) taken in theneighborhood of the point k = k m can contain a considerable contribution of photons with the projectionof the total angular momentum m − . For very small n ⊥ , the contribution of photons with all the lowerprojections of the angular momentum are relevant (see Figs. 3, 4). When n ⊥ γ ≈ K max , the peaks of dP ( m, k ) with different m are virtually not overlapping. In that case, the radiation at k ≈ k m consists of twistedphotons with the projection of the angular momentum m , i.e., the selection rule m = χn is fulfilled, where n is the number of harmonic (175) and χ = ± is the handedness of the helix along which the electron ismoving.As follows from (82), the most part of twisted photons is radiated at lower harmonics (175). In thewiggler case, K (cid:29) , these harmonics are fairly well described as the Lorentz boosted lower harmonics ofsynchrotron radiation (see the discussion after (75)). They would perfectly coincide if the charged particlemoved along an ideal helix. Those lower harmonics were studied in [9] where the effect of blossoming outrose was established: in the ultrarelativistic limit, even for β = 1 , these harmonics do not drive to the orbitplane, and the maximum intensity of radiation of every harmonic is achieved at some finite angle to the orbitplane. These angles for β = 1 are given in Sec. 1.3.4 of [9]. The maxima of the first harmonic are located atthe angles θ (cid:48) = { , π } .In the laboratory frame, the imprint of this effect on the properties of radiation is as follows. Oneobserves the maxima of radiation of twisted photons at these harmonics at the angles taken from [9] andsubstituted into (76). The orbit plane is seen in the laboratory frame as a cone with opening K/γ . Since,in the synchrotron frame, the front lobes are mostly right-handed polarized (if, in the laboratory frame,the particle moves along a right-handed helix) and the back lobes are mostly left-handed polarized, andthis property is Lorentz invariant, in the laboratory frame, the twisted photons with s = 1 dominate for n ⊥ < K/γ while, for n ⊥ > K/γ , the twisted photons with s = − prevail. The first harmonic with s = − does not die out for large n ⊥ and even for n ⊥ ≈ . The plots of lower harmonics are presented in Fig. 7.Strictly speaking, formula (47) is not applicable for so large n ⊥ . However, in the case of small quantum recoil,we can use exact formula [(36), [33]]. Numerical calculation shows that formulas (47) and [(36), [33]] give thesame results in this case. 25 k - d P / d k d k ⊥ - - - - - d P / d k d k ⊥ - - - - d P / d k d k ⊥ s = - ⊥ =
10 K / γ K = γ = =
51 eVk =
102 eVk =
153 eV n = = =
10 15 20 25 30 3502468 10 k - d P / d k d k ⊥ k - d P / d k d k ⊥ - - - - d P / d k d k ⊥ - - - - - d P / d k d k ⊥ k = =
16 eVk =
24 eV s = - ⊥ = / γ K = γ = = = = k - d P / d k d k ⊥ Figure 7:
The imprint of blossoming out rose effect [9] on the radiation of twisted photons in head-on collision of electronswith circularly polarized electromagnetic wave produced by the CO (left panel) and Ti:Sa (right panel) lasers. The parametersare the same as in Figs. 3, 4. The first harmonic (175) with s = − dominates. This is just a Lorentz boosted back lobe of thefirst harmonic of synchrotron radiation (see Sec. 1.3.4 of [9]). It is mostly left-handed polarized, and this property is preservedby the Lorentz transformations. This harmonic does not die out even for n ⊥ ≈ . Upper left inset: The distributions over m at the main maxima of the first two harmonics. Upper right inset: The distribution over m at the main maximum of the thirdharmonic. Lower inset: The second and third harmonics are separately depicted. Let us summarize the results. Using the BK semiclassical approach [1–4], we derived the general formulafor the one-photon radiation probability of a twisted photon by scalar (28) and Dirac (47) ultrarelativisticparticles moving in the electromagnetic field of a general configuration. This formula takes into account thequantum recoil undergone by a charged particle in radiating the twisted photon and, in the case of negligiblerecoil, turns into the formula given in [33]. Then we applied this formula to radiation of charged particlesin helical undulators and in circularly polarized laser waves with a plane wavefront. The explicit formulasfor the probability to record the twisted photon by a detector were obtained in these cases. The inclusionof quantum recoil forbids radiation of twisted photons with energies larger than the initial particle energy.We established that, as a rule, the quantum recoil increases the total yield of radiation in comparison withclassical formulas (see Fig. 2) and, at the same time, diminishes the energy of radiated photons. The spindegrees of freedom of a radiating particle increase the probability of radiation of twisted photons.The conditions when the developed semiclassical approach is justified were found and analyzed. The moststringent among these conditions is (17). It guarantees that, in describing the radiation of twisted photons, itis sufficient to characterize the particle wave packet by its average coordinate and momentum. In particular,it turns out that the radiation of twisted photons with large projection m of the total angular momentumproduced by electrons in helical wigglers and strong laser waves can be described semiclassically only inthe case of a small quantum recoil. In the dipole regime, the quantum recoil can be substantial and still bedescribed semiclassically (see Fig. 2). We found estimate (82) for the number of twisted photons with largeprojections of the total angular momentum produced in the forward radiation. We also described the effectof blossoming out rose [9] in the radiation of twisted photons by electrons evolving in strong laser waves withcircular polarization and wigglers (see Fig. 7).As an example, we considered the radiation of twisted photons with large angular momentum in thehelical wiggler (see Fig. 1) and in the circularly polarized strong laser waves produced by the CO andTi:Sa lasers (Figs. 3, 4, 5, 7). The parameters are given in these figures. In particular, we showed that MeVtwisted photons with m ∼ can be generated in helical wigglers. As for lasers, we found that the design of asufficiently pure source of twisted photons based on the nonlinear Compton process is only possible for longlaser pulses, N (cid:38) , with a smooth amplitude envelope. For short pulses, the escape direction of a twistedphoton depends severely on the initial phase of a laser wave that is virtually uncontrollable. Therefore, thedetector (the atom, for example) will feel the radiation consisting of twisted photons with wide spread of thetotal angular momentum projections m (see Fig. 5). For the systems concerned, we also described the effectof a finite width of a particle bunch on the incoherent radiation of twisted photons [75].26 cknowledgments. We are thankful to Yu.L. Pivovarov and D.V. Karlovets for fruitful conversations.This work is supported by the Russian Science Foundation (project No. 17-72-20013).
A Twisted photons in terms of plane waves
For the convenience of the reader, we shall provide the representation of the states of twisted photons interms of the plane wave ones (see the detailed exposition, e.g., in [25]). The states describing the photonswith plane wave front, | s, k , k , k (cid:105) , (177)and the states of twisted photons, | s, m, k ⊥ , k (cid:105) , (178)constitute the complete sets in the Hilbert space of one-particle states: (cid:88) s = ± (cid:90) V d k (2 π ) k | s, k , k , k (cid:105)(cid:104) s, k , k , k | == (cid:88) s = ± ∞ (cid:88) m = −∞ (cid:90) ∞−∞ L dk π (cid:90) ∞ Rdk ⊥ π k | s, m, k ⊥ , k (cid:105)(cid:104) s, m, k ⊥ , k | = 1 . (179)The states are normalized as (cid:104) s, k , k , k | s, k , k , k (cid:105) = (cid:104) s, m, k ⊥ , k | s, m, k ⊥ , k (cid:105) = (2 k ) − (180)where k = | k | = (cid:113) k ⊥ + k . One can decompose the state (178) in terms of the states (177). Carrying outrather simple calculations, we come to | s, m, k ⊥ , k (cid:105) = √ k V √ RL (cid:16) k ⊥ k (cid:17) / (cid:90) π − π dϕ π i − m e imϕ | s, k ⊥ cos ϕ, k ⊥ sin ϕ, k (cid:105) . (181) B Evaluation of integrals over the azimuth angle
It is convenient to evaluate the integrals over the azimuth angles of the vectors k , in expression (46) asfollows. Up to a common factor, which can be restored easily from (181), we have the correspondence f ∗ → ( [ a ∗ + e + + a ∗− e − ] + a ∗ e ) e − ik q x =: a ∗ , f → ( [ a + e − + a − e + ] + a e ) e ik q x =: a . (182)Hereinafter, for brevity, we write only those arguments of the mode functions a ± , , a ∗± , that differ fromthose written in formula (48). The basis vectors e i are defined in [(9), [33]]. Notice that q , = q i in (46) butwe keep q , different. Then, for example, ( f ∗ ˙ x )( f ˙ x ) → e ik ( q x − q x ) ( [ ˙ x − a ∗− + ˙ x a ∗ + ]+ ˙ x a ∗ )( [ ˙ x a − + ˙ x − a + ]+ ˙ x a ) = ( a ∗ ˙ x )( a ˙ x ) . (183)The additional powers of k , can be obtained by differentiation of the expression with respect to b , := q , x , e + x ⊥ , . (184)For example, k ( f f ∗ )( ˙ x k ) → ik (cid:16) ˙ x ∂∂b + ˙ x ∂∂b + ˙ x − ∂∂b − (cid:17) ( a ∗ a ) . (185)The derivatives of the mode functions are calculated with the aid of relations [(A3), [33]]: ∂ a ∂b = k ⊥ a ( m − , ∂ a ∂b − = − k ⊥ a ( m + 1) , ∂ a ∂b = ik a ,∂ a ∗ ∂b = − k ⊥ a ∗ ( m + 1) , ∂ a ∗ ∂b − = k ⊥ a ∗ ( m − , ∂ a ∗ ∂b = − ik a ∗ . (186)27pplying these relations to the expression in the square brackets in (46), we arrive at rather bulky formula ( P + P (cid:48) )( P + P (cid:48) )( a ∗ ˙ x )( a ˙ x )++ k (cid:2) ( a ∗ a )( ˙ x − n e , ˙ x − n e ) − ( a ∗ , ˙ x − n e )( a , ˙ x − n e ) (cid:3) ++ ik k ⊥ (cid:2) ( a ∗ ( m − , ( ˙ x − n e ) a − − ˙ x − a ) − ( a ∗ ( m + 1) , ( ˙ x − n e ) a − ˙ x a ) −− (( ˙ x − n e ) a ∗ − − ˙ x a ∗ , a ( m − x − n e ) a ∗ − ˙ x − a ∗ , a ( m + 1)) (cid:3) ++ k ⊥ (cid:2) a ∗ − ( m + 1) a ( m −
1) + a ∗ ( m − a − ( m + 1) − a ∗ ( m + 1) a ( m + 1) −− a ∗ − ( m − a − ( m −
1) + 2( a ∗ ( m + 1) , a ( m + 1)) + 2( a ∗ ( m − , a ( m − (cid:3) , (187)where n = k /k . Now, take into account that, in the region where the radiation of an ultrarelativisticparticle is concentrated, k /ε (cid:46) , | ˙ x ± | ∼ κ /γ, | ˙ x | ≈ , | ˙ x − n | (cid:46) κ /γ , | n ⊥ | (cid:46) κ /γ, n ≈ , (188)where n ⊥ = k ⊥ /k . It follows from the explicit expressions for the mode functions [(13), [33]] that | a ∗± | ∼ | a ± | (cid:46) γ/ κ , | a | ∼ | a ∗± a ∓ | ∼ . (189)Expanding the scalar products in (187) and neglecting the terms of order κ /γ (cid:28) in comparison with themain contribution, we obtain e ik ( b − b ) (cid:110) ( P + P (cid:48) )( P + P (cid:48) )( [ ˙ x − a ∗− + ˙ x a ∗ + ] + ˙ x a ∗ )( [ ˙ x a − + ˙ x − a + ] + ˙ x a )++ k (cid:2) ( ˙ x a ∗ + − in ⊥ a ∗ + ( m − x − a + + in ⊥ a + ( m − x − a ∗− + in ⊥ a ∗− ( m + 1))( ˙ x a − − in ⊥ a − ( m + 1)) (cid:3)(cid:111) . (190)Setting q , = q i and taking into account the common factor in (46), we deduce (47). References [1] V. N. Baier, V. M. Katkov, Quasiclassical theory of bremsstrahlung by relativistic particles, Zh. Eksp.Teor. Fiz. , 1542 (1968) [J. Exp. Theor. Phys. , 807 (1969)].[2] V. N. Baier, V. M. Katkov, A. I. Mil’shtein, V. M. Strakhovenko, The theory of quantum processes inthe field of a strong electromagnetic wave, Zh. Eksp. Teor. Fiz. , 783 (1975) [J. Exp. Theor. Phys. , 400 (1975)].[3] V. N. Baier, V. M. Katkov, V. M. Strakhovenko, Semiclassical theory of electromagnetic processes ina plane wave and a constant field, Zh. Eksp. Teor. Fiz. , 1713 (1991) [J. Exp. Theor. Phys. , 945(1991)].[4] V. N. Baier, V. M. Katkov, V. M. Strakhovenko, Electromagnetic Processes at High Energies in Ori-ented Single Crystals (World Scientific, Singapore, 1998).[5] A. I. Akhiezer, N. F. Shul’ga, Quasiclassical theory of radiation emission from high-energy particlesin an external field and the problem of boundary conditions, Zh. Eksp. Teor. Fiz. , 791 (1991) [J.Exp. Theor. Phys. , 437 (1991)].[6] V. G. Bagrov, V. V. Belov, A. Yu. Trifonov, Theory of spontaneous radiation by electrons in atrajectory-coherent approximation, J. Phys. A: Math. Gen. , 6431 (1993).[7] V. V. Belov, D. V. Boltovskiy, A. Yu. Trifonov, Theory of spontaneous radiation by bosons in quasi-classical trajectory-coherent approximation, Int. J. Mod. Phys. B , 2503 (1994).[8] A. I. Akhiezer, N. F. Shulga, High-Energy Electrodynamics in Matter (Gordon and Breach, New York,1996). 289] V. G. Bagrov, G. S. Bisnovatyi-Kogan, V. A. Bordovitsyn, A. V. Borisov, O. F. Dorofeev, V. Ya.Epp, V. S. Gushchina, V. C. Zhukovskii,
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