Radiation of twisted photons in elliptical undulators
aa r X i v : . [ phy s i c s . acc - ph ] F e b Radiation of twisted photons in elliptical undulators
P.O. Kazinski ∗ and V.A. Ryakin † Physics Faculty, Tomsk State University, Tomsk 634050, Russia
Abstract
The explicit expressions for the average number of twisted photons radiated by a charged particle in anelliptical undulator in the classical approximation as well as in the approach accounting for the quantumrecoil are obtained. It is shown that radiation emitted by a particle moving along an elliptical helix whichevolves around the axis specifying the angular momentum of twisted photons obeys the selection rule: m + n is an even number, where m is a projection of the total angular momentum of a twisted photonand n is the harmonic number of the undulator radiation. This selection rule is a generalization of thepreviously known selection rules for radiation of twisted photons by circular and planar undulators andit holds for both classical and quantum approaches. The class of trajectories of charged particles thatproduce the twisted photon radiation obeying the aforementioned selection rule is described. Nowadays, the undulators are the standard tool for generating a powerful electromagnetic radiation withspecified properties across a wide range of energies of radiated photons. It is known that the helical andplanar undulators can be used as a bright source of twisted photons [1–7]. The properties of such photonswere thoroughly investigated in [1, 8–10]. In this paper, we generalize the previously known results to thecase of an elliptical undulator where the charged particles move along elliptical helices. To our knowledge,the properties of twisted photons generated by such devices in the non-dipole regime have not been describeduntil now. We will derive the explicit expressions for the average number of twisted photons produced by anelliptical undulator with account for the quantum recoil experienced by the charged particle in emitting ahard photon. In particular, we shall prove the selection rule that m + n is an even number, where m is theprojection of the total angular momentum onto the undulator axis and n is the number of the undulatorradiation harmonic. This rule is in agreement with the selection rules obtained for the twisted photonsemitted by circular and planar undulators [8–10].The twisted photons are the quanta of the electromagnetic field with the definite helicity s , the projection m of the total angular momentum on the propagation axis of the photon, the momentum projection on thesame axis, and the modulus of the perpendicular momentum component k ⊥ [11, 12]. These stationary statesform a complete set, they are the eigenvectors of the projection of total angular momentum operator on theaxis and, in the paraxial approximation, k ⊥ / | k | ≪ , they possess the projection l = m − s of the orbitalangular momentum on this axis. These states of the electromagnetic field are used in optical tweezers; inmicroscopy to increase a contrast of the pictures; in telecommunication and quantum cryptography where theprojection of the angular momentum is used as an additional quantum number carrying information; in thestudies of rotational degrees of freedom of quantum objects where the twisted photons stimulate non-dipoletransitions (see, for review, [13–17]). Moreover, the investigations of the radiation properties in the basis oftwisted photons allow one to establish the properties that are difficult to unveil in the basis of plane-wavephotons, which is commonly used to describe radiation. Currently, there are plenty of designs for detectorsenabling one to register twisted photons, to find the number of them and their quantum numbers in theincident radiation [18–27]. Therefore, the expressions for the average number of twisted photons that arederived in the present paper can be directly verified in experiments.The paper is organized as follows. In Sec. 2, for the reader convenience, the general formulas that areused to describe the radiation of twisted photons in elliptical undulators are given. Section 3 is the mainpart of the paper. It contains the explicit formulas for the average number of radiated twisted photons as ∗ E-mail: [email protected] † E-mail: [email protected] ~ = c = 1 and the fine structure constant α = e / (4 π ) . For consistency of notation, it is always supposed that x ≡ x , y ≡ x , and z ≡ x . In this section, we provide some general formulas that are necessary for evaluation of the average number oftwisted photons radiated by a charged particle in elliptical undulators. The trajectory of the particle withthe charge e in the elliptical undulator with the section length λ takes form (see, e.g., [28], Ch. 5) x = x + b x cos ϕ, y = y + b y sin ϕ, z = z + β t + b sin(2 ϕ ) , (1)where t is the laboratory time, ϕ = ωt − χ , the parameters x , y , z , χ are some constants, β is the averageparticle velocity along the z axis, and ω = 2 πβ /λ > . The amplitudes of the particle oscillations alongdifferent axes are expressed via the magnetic field strength in the undulator b x = λ H y π γ , b y = − λ H x π γ , b = λ ( H y − H x )64 π γ , (2)where γ is the Lorentz-factor of the particle, the magnetic field strength H is measured in the units of thecritical field H = m e / | e | ≈ . × G , (3)and the lengths are measured in the units of the Compton wavelength, /m e ≈ . × − cm. Theundulator strength parameter for the trajectory (1) is written as K := γ h β ⊥ i / = λ q H x + H y √ π . (4)By the order of magnitude, the oscillation amplitudes are estimated as b x,y ≈ K ω γ , | b | ≈ K πωγ . (5)The expression (1) is obtained under the assumption that γ ≫ and K/γ ≪ . If the condition K ≪ issatisfied, the undulator is in the dipole regime which was thoroughly investigated in [8, 9]. Henceforth weassume that K & and so the dipole approximation is inapplicable.The trajectory of the charged particle takes the form (1) for t ∈ [ − T N/ , T N/ , where T := 2 π/ω and N ≫ is the number of undulator sections. If t does not belong to the aforementioned interval, wesuppose that the particle moves along the z axis with the velocity β k = p − /γ . We will investigatethe properties of radiation from the part of the trajectory t ∈ [ − T N/ , T N/ . The radiation formed onthis interval dominates when N ≫ for the photon energies corresponding to the harmonics of undulatorradiation.The average number of twisted photons emitted by a classical point charge is given by [8] dP ( s, m, k , k ⊥ ) = e (cid:12)(cid:12)(cid:12)(cid:12) Z dτ e − i [ k x ( τ ) − k x ( τ )] n (cid:2) ˙ x + ( τ ) a − ( s, m, k , k ⊥ ; x ( τ ))++ ˙ x − ( τ ) a + ( s, m, k , k ⊥ ; x ( τ )) (cid:3) + ˙ x ( τ ) a ( m, k ⊥ ; x ( τ )) o(cid:12)(cid:12)(cid:12)(cid:12) n ⊥ dk dk ⊥ π , (6)where s is the twisted photon helicity, m is the projection of the total angular momentum onto the z axis, k ⊥ and k are the projections of the photon momentum, k = q k + k ⊥ is the photon energy, n ⊥ := k ⊥ /k (see, for more details, [8–12]). We also use the following notation x ± := x ± iy. (7)The explicit expressions for the mode functions of twisted photons a ± , a are given, e.g., in formula (2.5) ofthe paper [10]. The parameter τ in (6) is an arbitrary parameter on the particle worldline. In our case, it isconvenient to choose it as τ = t ≡ x ( t ) . 2 Radiation of twisted photons
In order to find the average number of radiated twisted photons, one has to evaluate the integrals in (6). Letus introduce the notation I := Z T N/ − T N/ dte − ik [ t − n ( z + β t + b sin(2 ϕ ))] ˙ x a ( m, k ⊥ ; x ( t )) ,I ± := Z T N/ − T N/ dte − ik [ t − n ( z + β t + b sin(2 ϕ ))] ˙ x ± a ∓ ( s, m, k , k ⊥ ; x ( t )) , (8)where n := k /k . As we shall see, for N ≫ , these integrals give the main contribution to (6), i.e., thecontributions of the edge radiation to (6) are negligible in this case. Then dP ( s, m, k , k ⊥ ) ≈ e (cid:12)(cid:12) I + ( I + + I − ) / (cid:12)(cid:12) n ⊥ dk dk ⊥ π . (9)The components of the particle trajectory (1) are written as x ± = x ± + Re ± iϕ + De ∓ iϕ , (10)where R := ( b x + b y ) / , D := ( b x − b y ) / , x ± := x ± iy . (11)The components of the velocity become ˙ x ± = ± iω ( Re ± iϕ − De ∓ iϕ ) , ˙ x = β + 2 ωb cos(2 ϕ ) . (12)The contribution of the second term in the last expression is negligible when n ⊥ γ . max(1 , K ) , i.e., in thedomain of parameters where the main part of radiation is concentrated.Let us start with the integral I . It is convenient to use the addition theorem for the Bessel functions(see Eq. (A6) of [8]) in the form j ν ( x + + y + , x − + y − ) = ∞ X n = −∞ j ν − n ( x + , x − ) j n ( y + , y − ) , (13)and the property j m ( ap, q/a ) = a m j m ( p, q ) , m ∈ Z . (14)The definition of the functions j ν ( p, q ) and their properties are given in Appendix A of [8]. Employing theseexpressions, we obtain j m ( k ⊥ x + , k ⊥ x − ) e ik b sin(2 ϕ ) = ∞ X n,k,r = −∞ j m − n − k +2 r J n + k − r ( ρ ) J k ( δ ) J r ( κ ) e inϕ , (15)where the exponent on the left-hand side of the expression is expanded by using the Jacobi-Anger identity.Besides, the shorthand notation has been introduced j m := j m ( k ⊥ x , k ⊥ x − ) , ρ := k ⊥ R, δ := k ⊥ D, κ = k b . (16)The expression (15) depends on t only through the exponent standing on the right-hand side. Hence, theintegral over t is readily evaluated, I = 2 πβ ∞ X n,k,r = −∞ δ N (cid:0) k (1 − n β ) − nω (cid:1) e ik z − inχ j m − n − k +2 r J n + k − r ( ρ ) J k ( δ ) J r ( κ ) , (17)where δ N ( x ) := sin( T N x/ πx . (18)3he integrals I ± are evaluated in a similar way. In this case, we obtain I ± = − πω n ± sn ⊥ ∞ X n,k,r = −∞ δ N (cid:0) k (1 − n β ) − nω (cid:1) e ik z − inχ j m − n − k +2 r ×× (cid:2) RJ n + k − r ∓ ( ρ ) J k ( δ ) − DJ n + k − r ( ρ ) J k ∓ ( δ ) (cid:3) J r ( κ ) . (19)The two terms in the square brackets come from the two terms in the expression for ˙ x ± in (12). For N large,the expressions (17), (19) possess the sharp maxima at k = nω − n β , n = 1 , ∞ , (20)that correspond to the undulator radiation harmonics numerated by n .Thus, the one-particle amplitude of twisted photon radiation is proportional to I + ( I + + I − ) / − π ∞ X n,k,r = −∞ δ N (cid:0) k (1 − n β ) − nω (cid:1) e ik z − inχ j m − n − k +2 r J r ( κ ) ×× n(cid:16) ω ( n − r ) n k ⊥ n ⊥ − β (cid:17) J n + k − r ( ρ ) J k ( δ )++ sωn ⊥ (cid:2) RJ ′ n + k − r ( ρ ) J k ( δ ) − DJ n + k − r ( ρ ) J ′ k ( δ ) (cid:3)o . (21)This expression for the radiation amplitude can be used for description of the both coherent and incoherentradiations of twisted photons by particle beams [29–31]. It is clear now that the second term in ˙ x in Eq.(12) substituted into (9) gives rise to the correction to (9) that is of the order or less than n ⊥ in comparisonwith the other terms in this expression. Therefore, this contribution can be discarded for n ⊥ ≪ .Neglecting the interference terms between different harmonics in (9), we find the average number oftwisted photons (6) emitted by a charged particle in an elliptic undulator dP ( s, m, k , k ⊥ ) ≈ e ∞ X n =1 δ N (cid:0) k (1 − n β ) − nω (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X k,r = −∞ j m − n − k +2 r J r ( κ ) ×× n(cid:16) ω ( n − r ) n k ⊥ n ⊥ − β (cid:17) J n + k − r ( ρ ) J k ( δ )++ sωn ⊥ (cid:2) RJ ′ n + k − r ( ρ ) J k ( δ ) − DJ n + k − r ( ρ ) J ′ k ( δ ) (cid:3)o(cid:12)(cid:12)(cid:12)(cid:12) n ⊥ dk dk ⊥ . (22)In the case k ⊥ | x | ≪ , i.e., when the center line of the elliptical helix trajectory is close to the axis used todefine the angular momentum of twisted photons, the following relation is valid j m − n − k +2 r ≈ δ m,n +2 k − r . (23)Therefore, the radiation of twisted photons obeys the selection rule: m + n must be an even number. Thesame selection rule holds for radiation of twisted photons by charged particles moving in a planar undulator[8–10]. The circular undulator radiation obeys the selection rule m = ± n , the sign being specified by thehelix chirality [1, 8, 9, 32–34]. It is evident that m + n is an even number in this case too. It should be notedthat accounting for the second term in the expression for ˙ x in (12) does not spoil this selection rule. Ingeneral, such a selection rule is valid for the twisted photons radiated by the charged particle moving alongthe trajectory x + = ∞ X k = −∞ c k e i (2 k +1) ϕ , x − = ∞ X k = −∞ c ∗ k e − i (2 k +1) ϕ , z = β t + ∞ X k = −∞ d k e ikϕ , (24)where d ∗ k = d − k . The proof of this statement is postponed to Appendix A.Using the relation (23), the expression (22) is simplified to dP ( s, m, k , k ⊥ ) n ⊥ dk dk ⊥ ≈ e ∞ X n =1 δ N (cid:0) k (1 − n β ) − nω (cid:1)(cid:20) ∞ X r = −∞ J r ( κ ) ×× n(cid:16) ω ( n − r ) n k ⊥ n ⊥ − β (cid:17) J ( m + n ) / − r ( ρ ) J ( m − n ) / r ( δ )++ sωn ⊥ (cid:2) RJ ′ ( m + n ) / − r ( ρ ) J ( m − n ) / r ( δ ) − DJ ( m + n ) / − r ( ρ ) J ′ ( m − n ) / r ( δ ) (cid:3)o(cid:21) . (25)4ne can see that the sign flip of the elliptical helix chirality changes the average number of twisted photonsas follows dP ( s, m, k ⊥ , k ) → dP ( − s, − m, k ⊥ , k ) . (26)This transformation becomes a symmetry for a planar undulator [8–10, 35]. In the cases of planar and circularundulators, formula (25) reproduces exactly the expressions for the average number of twisted photons derivedin [8].Now we take into account the quantum recoil effect. We will employ the method developed in [9, 10].This method is an adaptation of the Baier-Katkov method [36, 37] for description of the quantum radiationproduced by charged ultrarelativistic scalar and Dirac particles in the domain of parameters where the mainpart of radiation is concentrated. As long as P := m e γ is an integral of motion of a charged particle in anundulator, we can use formula (2.13) of [10] to describe the radiation of twisted photons. Let q := P / ( P − k ) , (27)and k ′ := q [ k − k ⊥ / (2 P )] , k ′ = qk , κ ′ := qk b . (28)Employing formula (2.13) of [10], we come to the integrals of the form (8) with the obvious replacements.We denote these integrals as I ′ and I ′± . Then, disregarding the irrelevant common phase, we have I ′ = 2 π ∞ X n,k,r = −∞ δ N (cid:0) k ′ − k ′ β − nω (cid:1) e − inχ j m − n − k +2 r J n + k − r ( ρ ) J k ( δ ) J r ( κ ′ ) ,I ′ s = − πωn ⊥ ∞ X n,k,r = −∞ δ N (cid:0) k ′ − k ′ β − nω (cid:1) e − inχ j m − n − k +2 r ×× (cid:2) RJ n + k − r − s ( ρ ) J k ( δ ) − DJ n + k − r ( ρ ) J k − s ( δ ) (cid:3) J r ( κ ′ ) , (29)where we have taken into account that n ≈ within the approximations made.The energy of photons radiated at the n -th harmonic with account for the quantum recoil is deduced bysetting the argument of δ N ( x ) to zero. If we neglect the second term in the square brackets in the expressionfor k in (28), which is small within the approximation considered, then we find the following spectrum ofundulator radiation k = nω − n β + nω/P , n = 1 , ∞ . (30)Neglecting the interference between different harmonics, we obtain the inclusive probability of radiation ofone twisted photon by a charged Dirac particle in an elliptical undulator in the leading order of perturbationtheory dP ( s, m, k , k ⊥ ) n ⊥ dk dk ⊥ ≈ e q ∞ X n =1 δ N (cid:0) k ′ − k ′ β − nω (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X k,r = −∞ j m − n − k +2 r J r ( κ ′ ) ×× n(cid:2) J n + k − r ( ρ ) − ωRn ⊥ J n + k − r − s ( ρ ) (cid:3) J k ( δ ) + ωDn ⊥ J n + k − r ( ρ ) J k − s ( δ ) o(cid:12)(cid:12)(cid:12)(cid:12) . (31)For k ⊥ | x | ≪ , the relation (23) holds and so does the selection rule m + n is an even number, i.e., thisselection rule is valid in the quantum case as well.When the condition k ⊥ | x | ≪ is met, the expression (31) can be simplified. Let us introduce thefunctions g nm ( κ , ρ, δ ) := 1 + ( − n + m ∞ X r = −∞ J r ( κ ) J ( m + n ) / − r ( ρ ) J ( m − n ) / r ( δ ) . (32)In fact, these functions are the coefficients of the Fourier series (15) when x ± = 0 . Some properties of thefunctions (32) are presented in Appendix B. Then the probability (31) takes the form dP ( s, m, k , k ⊥ ) n ⊥ dk dk ⊥ = e q ∞ X n =1 δ N (cid:0) k ′ − k ′ β − nω (cid:1)(cid:16) g nm − ωRn ⊥ g n − s,m − s + ωDn ⊥ g n + s,m − s (cid:17) , (33)5here g nm ≡ g nm ( κ ′ , ρ, δ ) . In order to find the probability of radiation of twisted photons by scalar particles,one should replace the common factor (1 + q ) / by q (see (2.10) in [10]). In the particular cases of circularand planar undulators, formula (33) reproduces formulas (54) of [9] and (3.7) of [10]. Furthermore, assumingthat the quantum recoil is small and employing the paraxial approximation n ⊥ ≪ , formula (33) turns intothe expression (25). Using the symmetry property (38), it is not difficult to show that the sign flip of thechirality of the elliptic helix trajectory of the particle in the undulator results in the transformation rule (26)for the probability of radiation of twisted photons (33). Let us briefly summarize the results. Employing the general formalism developed in the papers [8–10], weobtained the explicit expressions for the average number of twisted photons radiated by a relativistic chargedparticle moving in an elliptic undulator. To account for the quantum recoil, we used the Baier-Katkov method[36, 37] adapted in [9, 10] to describe the inclusive probability of radiation of twisted photons in the domainof parameters where the main part of radiation is concentrated. In this domain, we derived the explicitexpressions for the inclusive probability of radiation of one twisted photon by charged scalar and Diracparticles. In the limit of negligibly small quantum recoil and in the paraxial approximation, these expressionsreproduce the average number of twisted photons created by the classical current of a point charge movingin an elliptical undulator. The last expression was also obtained in the present paper. In the particular casesof circular and planar undulators, the expressions for the probability of twisted photon radiation in ellipticalundulators derived in this paper turn into the known ones [8–10]. It was proven that radiation of twistedphotons obeys the selection rule: m + n is an even number, where m is a projection of the total angularmomentum of a twisted photon and n is the undulator radiation harmonic number. This rule holds forboth classical and quantum approaches for describing radiation. It is the generalization of previously knownselection rules for radiation of twisted photons by circular and planar undulators and it is valid for the classof trajectories of the form (24).The explicit expressions for the one-particle amplitude of twisted photon radiation that were derived inthe present paper allow one to find the average number of twisted photons emitted by the beam of chargedparticles moving in an elliptical undulator. To this end, one can employ the theory of radiation of twistedphotons by particle bunches developed in [29–31]. Since the motion of charged particles in the plane laserwave with an elliptical polarization is similar to the motion of charged particles in an elliptical undulator,the formulas derived in this paper can be used to describe radiation of twisted photons in both cases [9].Moreover, the developed formalism can be employed for description of radiation of twisted photons in axialand planar channelling [38, 39]. The similar properties of radiated twisted photons are expected for transitionradiation in dispersive medium with the permittivity tensor invariant under translations along the ellipticalhelix [30, 40] and for the elliptic undulators filled with dispersive medium [35]. Acknowledgements.
We are indebted to P. S. Korolev and G. Yu. Lazarenko for useful comments. Thework was supported by the RFBR grant No. 20-32-70023.
A Proof of the selection rule
Consider the radiation of twisted photons by a charged particle moving along the trajectory (24). Then, forthe factor entering the integrand of I , we have exp (cid:18) i ∞ X k = −∞ k d k e ikϕ (cid:19) = e ik d ∞ Y k =1 exp (cid:2) i ( k d k e ikϕ + k d ∗ k e − ikϕ ) (cid:3) == e ik d X { k r } exp (cid:18) i ∞ X r =1 rk r ϕ (cid:19) ∞ Y r =1 i k r j k r (2 k d r , k d ∗ r ) , (34)where, in the last expression, the summation over all the integer-valued sequences { k r } , r = 1 , ∞ , with afinite number of nonzero elements is understood. Analogously, using repeatedly the addition theorem (13)6nd the property (14), the another factor in the integrand of I can be written as j m ( k ⊥ x + , k ⊥ x − ) = X { l q } δ m, P q l q exp (cid:18) imϕ + i ∞ X q = −∞ ql q ϕ (cid:19) ∞ Y q = −∞ j l q ( k ⊥ c q , k ⊥ c ∗ q ) , (35)where the sum runs over all the integer-valued sequences { l q } , q = −∞ , ∞ , with a finite number of nonzeroelements. As a result, the integrand of I becomes ˙ x j m ( k ⊥ x + , k ⊥ x − ) exp (cid:18) i ∞ X k = −∞ k d k e ikϕ (cid:19) = e ik d X { k r } , { l q } ,n ′ iωn ′ d n ′ δ m, P q l q e inϕ ×× ∞ Y r =1 h i k r j k r (2 k d r , k d ∗ r ) i ∞ Y q = −∞ h j l q ( k ⊥ c q , k ⊥ c ∗ q ) i , (36)where iωn ′ d n ′ should be replaced by β for n ′ = 0 and the number of the undulator radiation harmonic is n = m + ∞ X r =1 rk r + ∞ X q = −∞ ql q + 2 n ′ . (37)It is evident that m + n is an even number. The integrands of I ± are expanded in the same way. Theseexpansions also respect the relation that m + n is an even number. Furthermore, following along the lines ofSec. 3, it is not difficult to verify that the quantum recoil does not violate this selection rule, at least, in thecase when P is constant. B Some properties of the functions g nm The functions g nm possess the symmetry properties g nm ( κ , ρ, δ ) = ( − m g n, − m ( κ , δ, ρ ) , g nm ( κ , ρ, δ ) = g − n,m ( − κ , ρ, δ ) . (38)The following recurrence relations hold: mg nm = ρ ( g n +1 ,m +1 + g n − ,m − ) + δ ( g n +1 ,m − + g n − ,m +1 ) , ∂g nm ∂ κ = g n − ,m − g n +2 ,m , ∂g nm ∂ρ = g n − ,m − − g n +1 ,m +1 , ∂g nm ∂δ = g n +1 ,m − − g n − ,m +1 , (39)where g nm ≡ g nm ( κ , ρ, δ ) . The generating function is written as ∞ X n,m = −∞ g nm s n t m = exp h κ (cid:16) s − s (cid:17) + ρ (cid:16) st − st (cid:17) + δ (cid:16) ts − st (cid:17)i ,g nm = Z π − π dϕdψ (2 π ) exp n i (cid:2) κ sin(2 ψ ) + ρ sin( ϕ + ψ ) + δ sin( ϕ − ψ ) − nψ − mϕ (cid:3)o . (40)Employing the generating function, it is not difficult to prove the sum rules ∞ X m = −∞ g nm ( κ , ρ, δ ) = J n ( ρ − δ, κ ) , ∞ X n = −∞ g nm ( κ , ρ, δ ) = J m ( ρ + δ ) , ∞ X n,m = −∞ g nm ( κ , ρ, δ ) = 1 , (41)where J n ( x, y ) is a generalized Bessel function of two arguments [28, 41–44].7 eferences [1] S. Sasaki, I. McNulty, Proposal for generating brilliant X-ray beams carrying orbital angular momentum,Phys. Rev. Lett. , 124801 (2008).[2] A. Afanasev, A. Mikhailichenko, On generation of photons carrying orbital angular momentum in thehelical undulator, arXiv:1109.1603.[3] V. A. Bordovitsyn, O. A. Konstantinova, E. A. Nemchenko, Angular momentum of synchrotron radia-tion, Russ. Phys. J. , 44 (2012).[4] E. Hemsing, A. Marinelli, Echo-enabled X-ray vortex generation, Phys. Rev. Lett. , 224801 (2012).[5] J. Bahrdt et al ., First observation of photons carrying orbital angular momentum in undulator radiation,Phys. Rev. Lett. , 034801 (2013).[6] E. Hemsing et al ., Coherent optical vortices from relativistic electron beams, Nature Phys. , 549 (2013).[7] P. R. Ribič, D. Gauthier, G. De Ninno, Generation of coherent extreme-ultraviolet radiation carryingorbital angular momentum, Phys. Rev. Lett. , 203602 (2014).[8] O. V. Bogdanov, P. O. Kazinski, G. Yu. Lazarenko, Probability of radiation of twisted photons byclassical currents, Phys. Rev. A , 033837 (2018).[9] O. V. Bogdanov, P. O. Kazinski, G. Yu. Lazarenko, Semiclassical probability of radiation of twistedphotons in the ultrarelativistic limit, Phys. Rev. D , 116016 (2019).[10] O. V. Bogdanov, P. O. Kazinski, G. Yu. Lazarenko, Planar wiggler as a tool for generating hard twistedphotons, JINST , C04008 (2020).[11] U. D. Jentschura, V. G. Serbo, Generation of high-energy photons with large orbital angular momentumby Compton backscattering, Phys. Rev. Lett. , 013001 (2011).[12] U. D. Jentschura, V. G. Serbo, Compton upconversion of twisted photons: Backscattering of particleswith non-planar wave functions, Eur. Phys. J. C , 1571 (2011).[13] J. P. Torres, L. Torner (Eds.), Twisted Photons (Wiley-VCH, Weinheim, 2011).[14] D. L. Andrews, M. Babiker (Eds.),
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