Probability of radiation of twisted photons in the infrared domain

# Probability of radiation of twisted photons in the infrared domain

## Abstract

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## 1 Introduction

The infrared asymptotics of radiation of photons in a vacuum is known to be universal for any quantum electrodynamic (QED) process (see, e.g., [[1], [2], [3], [4], [5], [6], [7]]). It is described by the radiation produced by the classical current of free charged particles in the in- and out-states, i.e., by the current of charged particles moving uniformly along straight lines with a break. This radiation is called the edge radiation [[8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18]]. The edge radiation can be routinely generated on the acceleration facilities (see, e.g., [[19], [20], [21], [22], [23], [24], [25]]). It is used now as a brilliant source of the infrared radiation [[24], [25], [23], [21]] and applied to the direct control of numerous low-energy excitations in solids and molecules [[26], [27]], to the study of mechanisms of the high-temperature superconductivity [[28]], to THz spectroscopy and microscopy, medicine, biology, and even to the preservation of cultural heritage [[18], [29]].

In [[43]], the general formula for the probability of radiation of a twisted photon by a classical current was obtained. The replacement of the quantum current operator by a classical quantity is justified only when the quantum recoil experienced by the source in the process of radiation can be neglected. The infrared limit described by the edge radiation provides an optimal situation for such an approximation works. The definition of the infrared domain where the edge radiation dominates depends on the process at hand. Loosely speaking, the so-defined infrared domain corresponds to the photon energies much less than the minimal characteristic energy scale of the process. As we shall see, the edge radiation may extend to the X-ray range for certain processes. Of course, the coherence properties of radiation become worse in increasing the photon energy. As the particular case for application of general formulas, we shall consider the edge radiation created in the process of scattering of electrons (positrons) by crystals: the so-called volume reflection and the volume capture [[44], [45], [46], [47], [48], [49], [50], [51], [52]]. In this case, the electrons with the energies of the order GeV produce the edge radiation with the photon energies up to eV with large angular momentum.

We use the system of units such that and , where is the fine structure constant.

## 2 General formulas

Let us briefly recall in this section the general formulas describing the radiation of twisted photons by classical currents [[43]]. Consider the theory of a quantum electromagnetic field interacting with a classical current . We suppose that is the current density of a point charge

 Unknown environment '%' (1)

where and , and is the time period when the particle moves with acceleration. The expression (1) can be cast into the standard form

 jμ(x)=e∫∞−∞dτ˙xμ(τ)δ4(x−x(τ)), (2)

where it is assumed that, for , the particle moves with the constant velocity , while, for , it moves with the constant velocity . On performing the Fourier transform,

 jμ(x)=:∫d4k(2π)4eikνxνjμ(k), (3)

the last two terms in (1) correspond to the boundary terms in

 jμ(k)=e(∫τ2τ1dτ˙xμe−ikνxν(τ)−i˙xμkλ˙xλe−ikνxν∣∣τ2τ1). (4)

These boundary contributions are responsible for the radiation created by a particle when it enters to and exits from the external field (see, e.g., [[58], [59], [60], [57], [19], [20], [21], [22], [23], [24], [25], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [4], [7]]). It is the so-called edge radiation. These terms gives the leading contribution to the probability of radiation of photons with small energies (see, e.g., [[1], [2], [3], [4], [5], [6], [7], [15]]). In fact, we shall study the contribution of these terms to the radiation of twisted photons. The model of a classical source producing the twisted photons is a good one when the quantum recoil experienced by the source can be neglected. The infrared regime of radiation that we are going to investigate represents an ideal situation where such an approximation works.

Let us introduce the right-handed orthonormal triple and

 e±:=e1±ie2. (5)

The basis vector is directed along the axis of the detector that records the twisted photons. It is clear that

 (e±,e±)=0,(e±,e∓)=2,e∗±=e∓, (6)

and arbitrary vector can be decomposed in this basis as

 x=12(x−e++x+e−)+x3e3, (7)

where and .

In the presence of a classical current, the process of radiation of photons is possible

 0→γα+X, (8)

where is the vacuum state, denotes the photon in the state recorded by the detector, and is for the other created photons. The probability of the inclusive process (8) is given by

 wincl(α;0)=1−e−n(α;0)≈n(α;0), (9)

where is the average number of photons in the state created by the current during the whole observation period, and it is supposed in the approximate equality that the population of the state is small. It is always small provided the volume of the chamber where the photons are created is sufficiently large.

The density of the average number of twisted photons created by the current (1) reads [[43]]

 Missing or unrecognized delimiter for \big (10)

where is the photon energy, all the vectors components are defined as in (7), and the notation has been introduced [[40], [39], [38], [37], [36], [35], [43]]

 a3(m,k⊥;x)=xm/2+xm/2−Jm(k⊥x1/2+x1/2−)=:jm(k⊥x+,k⊥x−),a±(s,m,k3,k⊥;x)=ik⊥sk0±k3jm±1(k⊥x+,k⊥x−), (11)

where is the photon helicity and is the projection of the total angular momentum of a photon onto the detector axis. The functions are entire functions of the complex variables and . It is also assumed in (10) that the origin of the reference frame is taken on the line passing through the detector axis. The expression (10) is invariant under the translations of the origin of the system of coordinates along the detector axis.

Formula (10) is obviously written for the current density of charged particles and for distributed currents. We shall also need the following general statement. Let the distributed current density be invariant under the rotation by an angle of , , around the detector axis for all . Then

 dP(s,m,k3,k⊥)=0,m≠lr,l∈Z. (12)

A similar property is known to be fulfilled for the scattering of twisted photons by microstructures [[61], [62]]. In order to prove this statement, we employ the pictorial representation of the radiation amplitude of twisted photons (see, for details, [[43]]) in terms of the plane-wave ones. Let us single out the circle that is obtained by the rotation of some point around the detector axis. In accordance with the prescription described in [[43]], the contribution to the radiation amplitude of every point on this circle comes with the common phase factor

 eimφ+eim(φ+2π/r)+eim(φ+4π/r)+⋯+eim(φ+2π(r−1)/r)=reimφδm,lr, (13)

which was to be proved. Another way to prove this statement is to see that the radiation amplitude entering into (10) acquires the phase factor when the rotation by an angle of is performed around the detector axis (see [[43]]). This immediately implies that either or the amplitude vanishes. In particular, if the current density is invariant with respect to the rotations by an arbitrary angle, which formally corresponds to , then the average number of radiated twisted photons is nonzero for only. Here, of course, we assume that the radiation fields created by add up coherently.

The transformation property of the amplitude can be used to show another one statement. Let the current density of identical charged particles be obtained from the current density of one charged particle by rotating its trajectory by an angle of , , around the detector axis. Then

 dP(s,m,k3,k⊥)=r2dP1(s,m,k3,k⊥)δm,lr,l∈Z, (14)

where is the average number of twisted photons created by the current density of one charged particle. For example, if one considers the ideal situation when electrons move in the helical undulator along the ideal circular helix such that their trajectories pass to each other under the rotation around the detector axis by an angle of , then the average number of photons is proportional to , . On the other hand, the forward radiation of twisted photons obeys in this case the selection rule , where is the harmonic number and the sign is determined by the handedness of the helix [[63], [64], [65], [66], [43]]. Consequently, the average number of twisted photons is nonzero only when (see [[15]]).

Formula (14) is the particular case of a more general statement on the form of the average number of twisted photons produced by the system of identical charged particles moving along the trajectories that are obtained from each other by the rotation around the detector axis, the translation along it, and the translation in time. Namely, consider a set of trajectories of the identical charged particles that are obtained from one trajectory by the rotation by an angle of , the translation along by , and the translation in time . Then, as follows from the transformation properties of the integrand of (10), the average number of twisted photons radiated by such a system of particles can be cast into the form

 dP(s,m,k3,k0)=∣∣r∑k=1eimφk+ik3xk3−ik0xk0∣∣2dP1(s,m,k3,k0)=:I(m,k3,k0)dP1(s,m,k3,k0), (15)

where is the number of particles. In particular, if

 φk=2πkr,xk3=λ02πφk,xk0=λ02πβ∥φk, (16)

where and are some fixed parameters, then

 I(m,k3,k0)=sin2(πδ)sin2(πδ/r),δ:=m+k0λ02π(β−1∥−n3), (17)

where . This interference factor modulates the one-particle radiation probability. It possesses the sharp global maxima at

 δ=lr,l∈Z, (18)

where , and the lateral local maxima, where . The function vanishes at when . Therefore, for , we reproduce (14). In the general case, we have the selection rule

 m=sgn(λ0)n+lr,k0=kn0:=2πn|λ0|(β−1∥−n3),n∈Z∖{0}, (19)

at the maxima. For the photon energy , the average number of radiated twisted photons vanishes for those that do not satisfy (19). If then the harmonic number . Introducing the notation

 ω0:=2πβ∥/|λ0|, (20)

we see that

 kn0=ω0n1−β∥n3, (21)

i.e., we have exactly the spectrum of the forward radiation of twisted photons by the helical undulator [[43]]. The case formally corresponds to the scattering of particles on the spiral phase plate commonly used to produce the twisted electrons [[41], [42]] and photons [[67]] or to the helical trajectory of a charged particle in the undulator, for example, (see [Fig. 5, [[43]]]). In this case, we obtain the selection rule

 m=sgn(λ0)n, (22)

i.e., all the twisted photons radiated at the th harmonic possess the total angular momentum (22). We should mention once again that all these selection rules imply the coherent addition of radiation fields of charged particles.

The above general statements are valid for the inverse process too, i.e., the system possessing such a current density does not absorb the twisted photons that do not obey these selection rules within the bounds of the approximations made in replacing the current operator by the classical quantity. This property follows from unitarity of a quantum evolution.

## 3 Infrared asymptotics

In order to find the infrared asymptotics of (10), we parameterize the worldline by the laboratory time, , and assume that the trajectory of a charged particle has the form

 x(t)=x0+vt+δx(ωt)=y0+ut+δy(ωt), (23)

where , , , and are constant vectors, characterizes the time scale of variations of the trajectory, and

 δx(τ)→τ→+∞0,δy(τ)→τ→−∞0,δ˙x(τ)→τ→+∞0,δ˙y(τ)→τ→−∞0. (24)

The parameters , , , and specify the asymptotes of the trajectory in the future and in the past. More precisely, we suppose that

 k3|δx3(τ)|≪1,k⊥|δx+(τ)|≪1, (25)

when , and the same estimates hold for . Then, we partition the integral over in (10) into two,

 ∫∞−∞dt⋯=∫0−∞dt⋯+∫∞0dt⋯, (26)

and stretch the integration variable

 t→tk0(1−n3v3),t→tk0(1−n3u3), (27)

in the first and the second integrals, respectively. Hereinafter, , , and so

 n23+n2⊥=1. (28)

Having performed such a transform, we see from (25) that, when [[4]]

 ωk0(1−n3v3)≳2π,ωk0(1−n3u3)≳2π, (29)

and the estimates (25) are satisfied, the particle trajectory in the partitioned integral (26) can be replaced, with good accuracy, by the corresponding asymptotes in the future and in the past. The whole trajectory in this approximation is discontinuous both in the velocity and the position of a charged particle with the discontinuity point :

 x(t)={x0+vt,t>0;y0+ut,t<0. (30)

For such a trajectory, the second integral in (26) is obtained from the first one by a change of sign and the replacement , . Hence, in evaluating the integrals it is sufficient to consider the first integral.

This integral possesses the physical meaning in itself (see, e.g., [[13]]). It describes the amplitude of the twisted photon production in the processes of the “instantaneous” stopping of a charged particle in a target and the “instantaneous” acceleration of a charged particle from a state of rest [[4]]. The latter process is realized, for example, in the production of charged particles in nuclear reactions. In these cases, the contribution of one of the integrals in (26) is zero.

In the ultrarelativistic regime, the main part of radiation is concentrated in the cone with the opening of the order . Therefore,

 n3≈1−n2⊥2,v3≈1−1+β2⊥γ22γ2, (31)

where . In this case, the conditions (25), (29) look as

 k0δx3(2π)≪1,nkk0Kγ|δx+(2π)|≪1,2πk0≲2ωγ21+K2(1+n2k), (32)

where and . Notice that the right-hand side of the last inequality is the energy of twisted photons at the first harmonic of the forward undulator radiation [[43]] provided that is the circular oscillation frequency of the electron in the undulator. We see that the domain of the infrared radiation we are investigating enlarges with increasing and .

Mention should be made that if the conditions (25), (29), or (32), hold simultaneously for the trajectories of several particles then the trajectories of these particles can be replaced by the straight lines with break of the form (30), and the radiation amplitudes corresponding to these particles ought to be added up. Since the asymptotes (30) are much easier to control than the complicated dynamics in between them, it is much easier to create a coherent radiation of twisted photons in the infrared regime. The probability of such a radiation is proportional to , where is the number of identical charged particles (see, e.g., (14)). The condition when the radiation of different charged particles add up coherently is the same as for the radiation of plane-wave photons. In particular, the bunch of charged particles radiates coherently at a given wavelength if the bunch size is less than that wavelength.

## 4 Processes

### 4.1 Trajectories with a break on the detector axis

We begin with the integrals determining the contribution to the radiation amplitude of the part of the trajectory (30) for with . This class of trajectories describes, in particular, the far infrared asymptotics of the average number of radiated twisted photons (10). In this limit, not only are the conditions (25), (29) met, but also

 k3|x3(0)|≪1,k⊥|x+(0)|≪1, (33)

and the same estimates should be valid for . Notice that, in contrast to the plane-wave photon radiation probability, the expression (10) is not invariant under the translations of the origin of the reference frame that are perpendicular to the detector axis. Therefore, one cannot vanish arbitrary by a proper choice of the system of coordinates.

Let us introduce the notation

 I3:=∫∞0dtv3e−ik0t(1−n3v3)jm(k⊥v+t,k⊥v−t),I±:=in⊥s∓n3∫∞0dtv±e−ik0t(1−n3v3)jm∓1(k⊥v+t,k⊥v−t). (34)

Taking into account that

 jm(k⊥v+t,k⊥v−t)=eimδJm(k⊥|v+|t), (35)

where , the integrals (34) are reduced to the Laplace transform of the Bessel function [[68]]

 Missing or unrecognized delimiter for \big (36)

Using this formula, we obtain

 I3+12(I++I−) =i−1−mk0n2⊥eimδ(v3−n3κ(v)−ssgn(m))q|m|(v), for |m|>1; (37) I3+12(I++I−) =i−1k0n2⊥(v3−n3κ(v)+n3), for m=0;

where

 κ(v):=[(1−n3v3)2−n2⊥β2⊥]1/2,q(v):=n⊥β⊥1−n3v3+κ(v)=1−n3v3−κ(v)n⊥β⊥,0≤q<1. (38)

Notice that formula (37) is exact. If a charged particle moves along the detector axis, i.e., , then the contribution at only survives in (37). Taking into account the contribution of the second part of the trajectory () with , we have from (37)

 Missing or unrecognized delimiter for \Big (39)

for all . Further we ought to square the modulus of (39) and substitute the result into (10). This leads to

 Missing or unrecognized delimiter for \Big (40)

where is the phase difference, which is brought to the interval .

The structure of the expression (37) implies the symmetry property

 dP(s,m,k3,k⊥)=dP(−s,−m,k3,k⊥) (41)

for the average number of twisted photons radiated by an arbitrary number of charged particles with the trajectories of the form (30) with and the arbitrary velocities , . Indeed, it follows from (37) that the average number of twisted photons produced in this case is of the form

 dP(s,m)=k∑l=1al(|m|,ssgn(m))eimδl[k∑l=1al(|m|,ssgn(m))eimδl]∗, (42)

where , , are real-valued quantities. Therefore, we obtain (41). The symmetry property (41) also holds for the radiation produced by a charged particle moving along an arbitrary planar trajectory provided that the detector axis belongs to the orbit plane [[43]]. Contrary to that, in the case we consider here, the trajectories do not lie on one plane and the direction of the detector axis is arbitrary. The only restriction is that the breaks of the trajectories (30) are located at one point and this point lies on the detector axis. In particular, the symmetry property (41) is fulfilled in the far infrared for the radiation produced by charged particles moving along the arbitrary trajectories with asymptotes (23).

The trajectory of the form (30) does not possess a distinguished length scale when . This results in a simple dependence of (40) on the photon energy . Of course, this property is a consequence of the approximations made in replacing the exact particle trajectory by the two straight lines (30). Such a simple dependence on disappears when the conditions (25), (29), and (33) become violated. In particular, when one considers the reflection of ultrarelativistic electrons (or positrons) from a crystal (see, e.g., [[71], [72], [69], [70], [45], [46], [47], [48], [49], [50], [51], [52]]), the trajectory of the particle can be replaced by (30) in evaluating the radiation of twisted photons provided that the photon energy satisfies the estimate

 k0≲α2γπ(1+(1+n2k)K2), (43)

where the photon energy is measured in the rest energies of the electron, MeV. This crude estimate follows from the last inequality in (32), where

 ω∼Eγ−1≈α2γ−1. (44)

Here the characteristic strength of the electromagnetic field is measured in the units of the critical field

 E0=m2|e|ℏ≈4.41×1013G=1.32×1016V/cm. (45)

The typical crystalline field strengths are of the order , where is the Bohr radius in the Compton wavelengths (see, e.g., [[73], [7], [74]]). The reflection angle in this case is of the order of the Lindhard critical angle

 θc∼(α/γ)1/2. (46)

Therefore,

 K≲(αγ)1/2. (47)

Even larger can be achieved not for the volume reflected but for the volume captured electrons in a bent crystal (see, e.g., [[52], [51], [50], [49], [46], [47]]).

For the processes we consider, the twisted photons with large total angular momentum can be generated only when . This occurs in the ultrarelativistic case, , and for . Then

 q≈K2(n2k+1)+1−[K4(n2k−1)2+2K2(n2k+1)+1]1/22K2nk≈1−[δn2k+K−2]1/2, (48)

where the absolute value of must be much less than unity. It follows from (40) that the average number of twisted photons produced in this process drops by a factor of in comparison with for

 |m|=mmax≈[δn2k+K−2]−1/2. (49)

Hereinafter we use the approximate expressions

 1−n3v3≈1+K2(1+n2k)2γ2,n3−v3≈1+K2(1−n2k)2γ2,κ2(v)≈(1+K2(nk−1)2)(1+K2(nk+1)2)4γ4, (50)

where only the leading terms at large ’s are kept. The typical dependence is given in Figs. 1, 2.

It is not difficult to find the main parameters characterizing the twist of the radiation (see [[43]]). For the amplitude (37), i.e., for the process of the instantaneous acceleration or stopping, the average number of photons produced is given by (40) with . Employing the approximate expressions (50), we can write

 dP(s,m,k3,k⊥)≈e216π2(K2(n2k−1)−1[K4(n2k−1)2+2K2(n2k+1)+1]1/2−ssgn(m))2q2|m|(v)dk3dk⊥k0k⊥, (51)

where, by definition, . In that case, we find the differential asymmetry

 A(s,m,k3,k⊥)=−s2(v3−n3)κ(v)(v3−n3)2+κ2(v)≈sK1−2K2δnk1+2K2δn2k[1+K2δn2k]1/2, (52)

the projection of the total angular momentum

 dJ3(s,k3,k⊥)=e216π2v3−n3κ(v)−4sq2(1−q2)2dk3dk⊥k0k⊥, (53)

and the average number of twisted photons

 dP(s,k3,k⊥)=e216π2[(1+(v3−n3)2κ2(v))2q21−q2+(v3−n3κ(v)+n3)2]dk3dk⊥k0k⊥. (54)

The last expression does not depend on the photon helicity. The second term in the square brackets in (54) can be neglected in comparison with the first one when . Then the projection of the angular momentum per photon is given by

 ℓ(s,k3,k⊥)=dJ3(s,k3,k⊥)dP(s,k3,k⊥)=A1−q2≈s21−2K2δnk1+2K2δn2k. (55)

The magnitude of this expression reaches its maximal value at , where

 ℓ(s,k3,k⊥)≈∓sK2√2. (56)

Notice that if one considers charged particles with the trajectories of the form (30) for with such that these trajectories pass to each other under the rotation by an angle of then the property (14) is valid. In this case, the projection of the angular momentum per photon remains the same as in (55) for because

 ℓ(s,k3,k⊥)=rA1−q2r≈A1−q2,r≪mmax. (57)

Nonetheless, the number of radiated photons increases by the factor (see Fig. 2).

Let us find a rough estimate for the average number of radiated twisted photons per unit energy interval for , . For these projections of the total angular momentum, the average number of twisted photons is peaked near (see Fig. 1). From (49) we conclude that the characteristic width of this peak is

 δn⊥=Kγ−1[m−2−K−2]1/2. (58)

Since , we can replace

 dk3dk⊥k0k⊥→[m−2−K−2]1/2dk0k0 (59)

in (51). The factor in the parenthesis in (51) is approximately equal to unity at the peak. Hence,

 dP(s,m,k0)∼α4πe−2|m|K−1[m−2−K−2]1/2dk0k0. (60)

The average number of photons diverges in the infrared as it should be for the edge radiation.

As for the average number of twisted photons (40), we see that it oscillates with the period

 Tm=2π/|δ12|. (61)

In particular, in case of the elastic reflection from the target located on the detector axis, when , , and , the average number of twisted photons vanishes for even (see Fig. 1). The projection of the total angular momentum reads

 dJ3(s,k3,k⊥)=−2se216π2{v3−n3κ(v)2q2(v)(1−q2(v))2+u3−n3κ(u)2q2(u)(1−q2(u))2−−(v3−n3κ(v)+u3−n3κ(u))(q(v)q(u)eiδ12(1−q(v)q(u)eiδ12)2+c.c.)}dk3dk⊥k0k⊥, (62)

and the total average number of radiated twisted photons is found to be

 dP(s,k3,k⊥)=e216π2[(1+(v3−n3)2κ2(v))2q2(v)1−q2(v)+(1+(u3−n3)2κ2(u))2q2(u)1−q2(u)−−2(v3−n3κ(v)u3−n3κ(u)+1)(q(u)q(v)eiδ121−q(u)q(v)eiδ12+c.c.)+(v3−n3κ(v)−u3−n3κ(u))2]dk3dk⊥k0k⊥. (63)

Obviously, it is independent of the photon helicity. The projection of the angular momentum per photon is given by the ratio of (62) and (63).

The plots of the average number of twisted photons and the projection of the total angular momentum per photon are presented in Fig. 2 for several processes of scattering of electrons by the target lying on the detector axis.

### 4.2 Trajectories with a break out of the detector axis

Now we investigate the production of twisted photons by charged particles moving along the trajectories (30) with the break out of the detector axis. In contrast to the case of scattering by the target lying on the detector axis, the problem at issue possesses a distinguished length scale . Therefore, the average number of radiated twisted photons will depend nontrivially on the energy of a radiated photon, which, of course, will complicate the analysis.

As we have discussed above, it is sufficient to consider the radiation amplitude corresponding to the part of the trajectory with positive . In a general case, the integrals entering into (10),

 I3:=∫∞0dtv3e−ik0t(1−n3v3)+ik3x3jm(k⊥(v+t+x+),k⊥(v−t+x−)),I±:=in⊥s∓n3∫∞0dtv±e−ik0t(1−n3v3)+ik3x3jm∓1(k⊥(v+t+x+),k⊥(v−t+x−)), (64)

seem not to be expressible in a closed form in terms of the known special functions. Notice that the integrals of the same type arose in [[75]] in studying the scattering of twisted electrons on the screened Coulomb potential. However, their properties were not investigated there.

#### Exact formulas

In the particular case when the particle accelerates instantaneously from a state of rest and then moves parallel to the detector axis, the integrals (64) are readily evaluated

 I3+12(I++I−)=−iv3jm(k⊥x+,k⊥x−)k0(1−n3v3)eik3x3. (65)

A similar expression is obtained for the amplitude of the twisted photon production in the processes of the instantaneous stopping of a charged particle (see the remark after formula (30)). This expression also describes, with good accuracy, the amplitude of radiation of twisted photons with in the case . The corresponding average number of twisted photons is written as

 dP(s,m,k3,k⊥)=e2v23n3⊥(1−n3v3)2|jm(k⊥x+,k⊥x−)|2dk3dk⊥16π2k20=e2v23n3⊥(1−n3v3)2J2m(k⊥|x+|)dk3dk⊥16π2k20. (66)

We see that it is proportional to the modulus squared of the third component of the twisted photon “wave function”. It is independent of the photon helicity, symmetric with respect to , and possesses a maximum at (see, e.g., [[76]])

 k⊥|x+|≈|m|+0.81|m|1/3+⋯,|m|≥1, (67)

where

 J2m(k⊥|x+|)≈0.46|m|−2/3. (68)

For , the function goes exponentially fast to zero.

Now we turn to the general case. Employing the integral representation

 jm(p,q)=∫|z|=1dz2πiz−m−1e12(pz−qz), (69)

we find

 I3=2im−1v3k⊥β⊥eimδ+ik3x3∫|z|=1dz2πiz−m(z+q)(z+q−1)exp[k⊥x⊥z−z−12−ik⊥x∥z+z−12], (70)

where

 x⊥=|x+|sin(ψ−δ),x∥=|x+|cos(ψ−δ),ψ:=argx+. (71)

The quantity is the shortest distance from the detector axis to the trajectory of a charged particle and is the component of the vector parallel to the vector . Since

 z(z+q)(z+q−1)=1q