A Normalization for plane-wave particles

# Compton Upconversion of Twisted Photons: Backscattering of Particles with Non-Planar Wave Functions

## Abstract

Twisted photons are not plane waves, but superpositions of plane waves with a defined projection of the orbital angular momentum onto the propagation axis ( is integer and may attain values ). Here, we describe in detail the possibility to produce high-energy twisted photons by backward Compton scattering of twisted laser photons on ultra-relativistic electrons with a Lorentz-factor . When a twisted laser photon with the energy performs a collision with an electron and scatters backward, the final twisted photon conserves the angular momentum , but its energy is increased considerably: , where . The matrix formalism for the description of scattering processes is particularly simple for plane waves with definite 4-momenta. However, in the considered case, this formalism must be enhanced because the quantum state of twisted particles cannot be reduced to plane waves. This implies that the usual notion of a cross section is inapplicable, and we introduce and calculate an averaged cross section for a quantitative description of the process. The energetic upconversion of twisted photons may be of interest for experiments with the excitation and disintegration of atoms and nuclei, and for studying the photo-effect and pair production off nuclei in previously unexplored regimes.
PACS 12.20.-m, 12.20.Ds, 13.60.Fz, 42.50.-p, 42.65.Ky

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## 1 Introduction

Scattering processes lie at the heart of modern physics and have been studied in detail at the tree- and loop-level for particles with well-defined four momenta. In particular, the well-known Feynman rules of quantum electrodynamics (1); (2) apply to the scattering of planar waves and cannot be readily applied to the scattering of particles with more complex wave functions. Consequently, the literature is more scarce when it comes to the scattering of quantal particles described by non-planar waves.

The related questions are far from being academic. E.g., a description using non-planar waves is necessary in the case of single photon bremsstrahlung discovered in experiments on the collider VEPP-4 (3); (4); (5) and then on the collider HERA (6). In these experiments, a remarkable deviation of the measured bremsstrahlung photons from standard calculational methods has been observed. The decrease in the number of observed photons can be explained by the fact that large impact parameters (of the two bunches relative to each other) give the essential contributions to the cross section. These parameters are larger by several orders of magnitude than the transverse beam size. In that case, the standard definitions for the cross section and the number of events become invalid. In particular, it is possible to calculate the matrix element as a superposition of “planar” scattering processes, but the normalization of the flux of incoming particles still constitutes a problem in the calculation of the modified cross section (beyond the matrix element). Modified calculational schemes for the description of particle production in the interaction of two bunches have to be employed (for details, see the review (7)). In this scheme, the colliding bunches are represented as wave packets, and quantum distribution functions are used. The modified definitions of the cross section and the number of events contain the features of “non–locality” and “interference.”

An analogous problem is studied here for Compton backscattering of so-called “twisted photons.” These are defined superpositions of plane waves and have some interesting physical properties (8); (9), such as wavefronts that rotate about the propagation axis and Poynting vectors that look like corkscrews (see Fig. 1 of Ref. (10)). Also, twisted photons have a defined projection of the orbital angular momentum onto the propagation axis (11); (12) which may be quite large, . Experiments demonstrate that micron-sized Teflon, calcite and other micron-sized “particles” start to rotate after absorbing such photons (13); (14); (15); (16); (17). The observation of orbital angular momentum of light scattered by black holes could be very instructive, as pointed out in Ref. (18). Twisted laser photons may be created from usual laser beams by means of numerically computed holograms. Alternative generation mechanisms (in the visible and infrared part of the optical spectrum) have recently been discussed in Refs. (19); (20).

The electromagnetic vector potential describing a twisted photon state adds the orbital angular momentum of the photon to the spin angular momentum of the vector (spin-) field. In some sense, the twisted wave function interpolates between the plane-wave vector potential of the form and the photon vector potential described by a vector spherical harmonic . Indeed, a plane-wave photon whose vector potential is proportional to , describes a photon propagating in the direction. It has zero expectation value for the projection of the angular momentum onto the propagation axis ( axis). A photon described by a vector spherical harmonic fulfills and , where is the total angular momentum (orbital plus spin) of the photon. However, a photon described by a vector spherical harmonic does not have a defined propagation direction.

Twisted photons are rather interesting objects, as they combine, in some sense, the properties of plane-wave photons and those described by vector spherical harmonics: Namely, they have a defined propagation direction (which we choose to be the axis, here) and still, large angular momentum projections onto that same propagation axis. In constructing vector spherical harmonics, one adds the orbital angular momentum from the spherical harmonics to the spin angular momentum, using Clebsch–Gordan coeffcients (21). However, one can also add the orbital angular momentum to the spin angular momentum via a conical momentum spread (in momentum space) multiplied by an angle-dependent phase, or by a Bessel function in the radial variable (in coordinate space). This leads to the twisted states, which are the subject of the current paper.

All experiments performed with twisted photons so far have been in the range of visible light, i.e., with a photon energy of the order of . In our recent paper (22), we have shown that it is possible to upconvert the frequency of a twisted photon using Compton backscattering, from an energy of the order of to an energy in the GeV range. Here, we present the derivation in more detail, and we also address the question of how to convert the result for the matrix element to a generalized cross section. This is nontrivial for the current case because the initial and final photons are described as twisted states (rather than plane waves).

This paper is organized as follows. In Sec. 2.1, we present basic formulas pertaining to a twisted state of a scalar particle, whereas the full vector particle content of a twisted photon is investigated in Sec. 2.2. The Compton scattering is recalled in Sec. 3 for a plane-wave photon, whereas the same effect is studied for backscattered twisted photons in Sec. 4. Details of the conversion of the matrix element to a generalized cross section are discussed in Sec. 4.4 and Appendices A and B. Finally, conclusions are reserved for Sec. 5. Relativistic Gaussian units with , , , and are used throughout the article. We write the electron mass as and denote the scalar product of 4-vectors and by a dot, i.e., , where is the scalar product of 3-vectors.

## 2 Quantum description of twisted states

### 2.1 Twisted scalar particle

We first recall that the usual plane-wave state of a scalar particle with mass equal to zero has a defined 3-momentum , energy and is described by a wave function of the form

 Ψk(t,r)=e−i(ωt−kr)√2ω, (1)

with the normalization condition

 ∫Ψ∗k′(t,r)Ψk(t,r)d3r=(2π)3δ(k−k′)2ω. (2)

A twisted scalar particle with vanishing mass has the following quantum numbers: longitudinal momentum , absolute value of the transverse momentum , energy

 ω=|k|=√ϰ2+k2z, (3)

and projection of the orbital angular momentum onto the axis. In cylindrical coordinates , , and , this state is described by the wave function which satisfies the Klein-Fock-Gordon equation (with mass equal to zero),

 ∂μ∂μΨϰmkz(t,r)=0, (4)

and it is an eigenfunction of the projection of the momentum and of the orbital angular momentum ,

 ˇpzΨϰmkz=kzΨϰmkz,ˇLzΨϰmkz=mΨϰmkz. (5)

Its evident form is

 Ψϰmkz(r,φr,z,t)= e−i(ωt−kzz)√2ωψϰm(r,φr), ψϰm(r,φr)= eimφr√2π√ϰJm(ϰr), (6)

where is the Bessel function

 Jm(x)=12π∫2π0ei(mφ−xsinφ)dφ. (7)

In Fig. 1, we present the dependence of the square of the absolute value of on for different values of . For small , this function is of order ,

 |ψϰm(r,φr)|≈√ϰ2π(ϰr)m2mm!, (8)

has a maximum at and then drops according to the familiar asymptotics of the Bessel function,

 ψϰm(r,φr)≈eimφrπ√rcos(ϰr−mπ2−π4), (9)

at large values .

The function may be expressed as a superposition of plane waves in the plane ()

 ψϰm(r,φ)=∫aϰm(k⊥)eik⊥rd2k⊥(2π)2, (10)

where the Fourier amplitude is concentrated on the circle with ,

 aϰm(k⊥)=(−i)meimφk√2πϰδ(k⊥−ϰ). (11)

Therefore, the function can be regarded as a superposition of plane waves with defined longitudinal momentum , absolute value of transverse momentum , energy and different directions of the vector given by the angle .

### 2.2 Twisted photon

The wave function of a twisted photon (vector particle) can be constructed as a generalization of the scalar wave function. We start from the plane-wave photon state with a defined 4-momentum and helicity ,

 AμkΛ(t,r)= √4πeμkΛe−i(ωt−kr)√2ω, (12a) ekΛ⋅k= 0,e∗kΛ⋅ekΛ′=−δΛΛ′, (12b)

where is the polarization four-vector of the photon. The twisted photon vector potential

 AμϰmkzΛ(r,φr,z,t)=∫aϰm(k⊥)AμkΛ(t,r)d2k⊥(2π)2 (13) =(−i)m√2πϰ2π∫0dφk∞∫0dk⊥δ(k⊥−ϰ)eimφk(2π)2AμkΛ(t,r)

is given as a two-fold integral over the perpendicular components of the wave vector . Using the well-known identity

 ∫∞0Jm(ϰx)Jm(ϰ′x)xdx=1ϰδ(ϰ−ϰ′), (14)

it is not difficult to prove that this function satisfies the normalization condition [compare with Eq. (2)]

 ∫(A∗ϰ′m′k′zΛ′)μ(t,→r)AμϰmkzΛ(t,r)d3r =−4πδΛΛ′2πδ(kz−k′z)2ωδmm′δ(ϰ−ϰ′). (15)

We would like to stress that the orthogonal functions for different values of but fixed axis constitute a complete set of functions and can be used for the description of initial as well as final twisted photons.

The polarization vector depends on the azimuth as with depending on the helicity [see Eq. (4.1) below]. In view of the identity

 ∫eiℓφkaϰm(k⊥)eik⊥rd2k⊥(2π)2=iℓψϰ,m+ℓ(r,φr), (16)

the vector field describes a photon state with defined , absolute value of the transverse momentum , energy and projection of the orbital angular momentum on the axis equal to [see also Eqs. (53)—(54) below]. Strictly speaking, this state is not a photon state with a defined value of . However, for large , the restriction to means that the twisted state is a state with a very restricted angular momentum projection distribution about the central value equal to . The representation (16) is very convenient as it allows us to considerably simplify the analytic calculations. We call such a state a twisted photon (see Fig. 2) and denote it as .

The usual matrix element for plane-wave (PW) Compton scattering involves an electron being scattered from the state with 4-momentum and helicity to a state and a photon being scattered from the state to the state ,

 S(PW)fi≡⟨k′,Λ′,p′,λ′|S|k,Λ,p,λ⟩. (17)

For head-on collisions, the vectors and are anti-parallel.

Let us consider the Compton effect for the case when an initial plane-wave electron in the state performs a head-on collision with an initial twisted photon propagating along the axis. In the final state, there is a plane-wave electron and a final twisted photon propagating along the axis (a schematic view of the initial and final states is given in Fig. 3). As noted above, we can choose the axis along an arbitrary direction, but below we restrict the discussion to the particular scattering geometry where the axes and are naturally defined as being collinear, namely, the strict backscattering geometry. (Moreover, even in a general case it is convenient to choose the axis along the axis, leading to a potential simplification of the calculation.) In view of Eq. (13), the matrix element for such a scattering,

 S(TW)fi≡⟨ϰ′,m′,k′z,λ′;p′,λ′|S|ϰ,m,kz,Λ;p,λ⟩, (18)

needs to be integrated as follows,

 S(TW)fi≡ ∫d2k⊥(2π)2d2k′⊥(2π)2a∗ϰ′m′(k′⊥) (19) ×⟨k′,Λ′,p′,λ′|S|k,Λ,p,λ⟩aϰm(k⊥) = ∫d2k⊥(2π)2d2k′⊥(2π)2a∗ϰ′m′(k′⊥)S(PWC)fiaϰm(k⊥),

where by PWC we denote the scattering matrix elements for the plane-wave component of the twisted photons, with 4-vector components and .

Based on Eq. (11), we conclude that the integration in Eq. (19) is determined by the dependence of the matrix element on the azimuthal angles and of the vectors and . A numerical integration of Eq. (19) then leads to predictions for arbitrary scattering angle of the final electron. In this paper we consider in detail the important case of strict backward Compton scattering when the scattering angle of the final electron equals zero and the vector is directed along the axis. Such a choice is determined mainly by two reasons. First of all, for usual Compton scattering on ultra-relativistic unpolarized electrons, precisely the backward scattering has the largest probability (see Fig. 4 below). Second, the matrix element does not depend on the azimuthal angles and , and therefore, this case allows for a simple and transparent treatment with analytical calculations for usual as well as for twisted photons.

## 3 Compton scattering of plane-wave photons

### 3.1 General formulas

In principle, Compton backscattering is an established method for the creation of high-energy photons and used successfully in various application areas from the study of photo-nuclear reactions (23) to colliding photon beams of high energy (24); (25). Let us consider the collision of an ultra-relativistic electron with four momentum

 p=(E,0,0,vE),v=|p|E,γ=Eme, (20)

whose spatial momentum component points strictly upward, and a photon of energy and three-momentum

 k=ω(sinα0cosφk,sinα0sinφk,−cosα0). (21)

Here, and are the polar and azimuthal angles of the initial photon. For a downward pointing photon (head-on collision), we have . After the scattering, the four-momentum of the electron is , and the scattered photon has energy and three-momentum

 k′=ω′(sinθ′cosφ′k,sinθ′sinφ′k,cosθ′), (22)

where and are the polar and azimuthal angles of the final photon. Let be the angle between the vectors and . For head-on backscattering, we have . In general,

 kk′=−ωω′cosβ, (23)

and

 cosβ=cosα0cosθ′−sinα0sinθ′cos(φk−φ′k). (24)

From the on-mass-shell condition of the scattered electron, we have , and therefore or

 ω′=m2ex2E(1−vcosθ′)+2ω(1+cosβ), (25)

where

 x=2k⋅pm2e=2ωE(1+vcosα0)m2e. (26)

The matrix element for plane waves (either plane direct incoming and outgoing plane waves or plane-wave components of a twisted photon) is

 ⟨k′,Λ′,p′,λ′|S|k,Λ,p,λ⟩ =i(2π)4δ(p+k−p′−k′)Mfi4√EE′ωω′, (27)

where the scattering amplitude in the Feynman gauge is equal to

 Mfi = 4πα(As−m2e+Bu−m2e), (28a) A = ¯up′λ′^e∗k′Λ′(^p+^k+me)^ekΛupλ, (28b) B = ¯up′λ′^ekΛ(^p′−^k+me)^e∗k′Λ′upλ, (28c) s−m2e = 2k⋅p=m2ex,u−m2e=−2k′⋅p. (28d)

The bispinors and describe the initial and final electrons with helicities and , and and are the polarization vectors of the initial and final photons with helicities and . We denote the Feynman dagger as .

For Compton scattering off incoming ultra-relativistic electrons (), the differential cross section has a maximum in the backscattering region, where the polar angle of the scattered photon is small, , and the photon propagates almost along the direction of momentum of the initial electron. Indeed, for unpolarized electrons, the differential Compton cross section reads (1)

 dσdΩ′ = 2α2γ2m2eF(x,n), (29) F(x,n) = (11+x+n2)2[1+x+n21+n2 +1+n21+x+n2−4n2(1+n2)2],

where , , and

 ω′ω=4γ21+x+n2. (30)

In Fig. 4, we show the angular distribution of the final photons which is concentrated to the region . The value of used, namely , corresponds to the collider parameter as defined in Eq. (26), evaluated for the VEPP-4M collider (Novosibirsk) with and  eV. The maximum energy of the final photon is for . For , the energy of the final photon is independent of the azimuth angle or ,

 ω′=x1+x+(γθ′)2E. (31)

To make calculations in the main region more transparent, it is useful to decompose the scattering amplitude (28a) into dominant and negligible items. To do this, in the term defined in Eq. (28b), we transpose and using the Dirac equation and obtain

 (^p+^k+me)^ekΛupλ=^ekΛ(−^p−^k+me)upλ +2(ekΛ⋅p)upλ=−^ekΛ^kupλ+2(ekΛ⋅p)upλ. (32)

Thus, with

 A1= −¯up′λ′^e∗k′Λ′^ekΛ^kupλ, (33a) A2= 2(ekΛ⋅p)¯up′λ′^e∗k′Λ′upλ. (33b) In full analogy, B=B1+B2 with B1= ¯up′λ′^k^ekΛ^e∗k′Λ′upλ, (33c) B2= 2(ekΛ⋅p′)¯up′λ′^e∗k′Λ′upλ. (33d)

The scattering amplitude as defined in Eq. (28a) can thus be written as

 Mfi= M1+M2, (34a) M1= 4πα(A1s−m2e+B1u−m2e), (34b) M2= 4πα(A2s−m2e+B2u−m2e). (34c)

The term will be shown to play the dominant role in our calculation. For further analysis we also introduce three 4-vectors,

 η(±)=∓1√2(0,1,±i,0),η(z)=(0,0,0,1). (35)

### 3.2 Strict backward Compton scattering

For a head-on collision of a plane-wave photon and a counter-propagating electron, the electron after scattering moves in the same direction as before the collision, but with a smaller energy ,

 p′=(E′,0,0,v′E′),v′=|p′|E′. (36)

For plane waves, strict backward scattering has the largest probability (see Fig. 4), and this case allows for a simplified treatment. Indeed, for strict backward geometry, we have , and the photon polarization vectors can be chosen in the form

 ek′,±1=ek,∓1=η(±). (37)

For the considered head-on collision, we have

 ekΛ⋅p= 0,ekΛ⋅p′=0, (38a) A2= 0,B2=0, (38b)

i.e. the term vanishes for plane-wave strict backward scattering.

In order to calculate as given by Eq. (33a), it is useful to represent the expression as

 2^e∗k′Λ′^ekΛ= ΛΛ′−1+(Λ′−Λ)Σz = −2(1+ΛΣz)δΛ,−Λ′, (39)

where is a matrix vector measuring the electron spin. Substituting this expression into , we find

 A(PW)1=8ω√EE′δλλ′δΛ,−Λ′δ2λ,Λ (40a) and, analogously, B(PW)1=−8ω√EE′δλλ′δΛ,−Λ′δ2λ,−Λ. (40b)

Further important kinematic relations are

 s−m2e=m2ex,u−m2e=−m2exx+1. (41)

As a result, the scattering amplitude for plane-wave strict backward scattering is

 M(PW)fi= M(PW)1=8πα√x+1[δ2λ,Λ +(x+1)δ2λ,−Λ]δλλ′δΛ,−Λ′, (42)

with . We emphasize that in head-on backscattering, the electron does not change its helicity during scattering () while the photon does change its helicity, . All these results are in full agreement with known properties of ordinary Compton scattering (1); (26); (27).

## 4 Compton backscattering of twisted photons

### 4.1 Kinematics

In the case of a twisted photon, the final photon state is a superposition of plane waves with high energy, consistent with the general principle of Compton backscattering. The transverse momentum is conserved,

 k′⊥=k⊥, (43)

as can be seen from the conservation law , which follows from the fact that the transverse components of the vectors and are equal to zero. The scattering angle is very small,

 θ′=k′⊥ω′≲ωω′=1+x+n24γ2≈1+x4γ2. (44)

From Eq. (43), we have , and the energy of the scattered photon is

 ω′=x1+xE, (45)

as evident from Eq. (31) in the limit .

In view of the structure of the plane-wave scattering element recorded in Eq. (3.1), the convoluted matrix element for backward scattering of twisted photons given in Eq. (19) takes the form

 S(TW)fi=i2πim′−mδ(ϰ−ϰ′)14√EE′ωω′ (46) ×δ(E+ω−E′−ω′)δ(pz+kz−p′z−k′z) ×2π∫0ei(m−m′)φk(M(PWC)1+M(PWC)2)dφk,

where we have used the decomposition (34), and and are the plane-wave components of the twisted photon scattering matrix element. Note that and are not equal to their plane-wave counterparts and because of the nonvanishing conical momentum spread of the twisted photon.

The admissible values of are determined by the dependence of and on . In order to carry out the integration over , we have to analyze the dependence of the polarization vectors and on the azimuth angle. To this end, we choose the polarization vector of the final photon in the scattering amplitude in the form

 ek′Λ′=−Λ′√2(e(x′)+iΛ′e(y′)), (47)

where the unit vector is in the scattering plane, defined by the vectors and , while the unit vector is orthogonal to it,

 e(x′)∥(p×k′)×k′,e(y′)∥(p×k′). (48)

As a result, we have in four-vector component notation

 ek′Λ′=−Λ′√2⎛⎜ ⎜ ⎜⎝0cosθ′cosφk−iΛ′sinφkcosθ′sinφk+iΛ′cosφk−sinθ′⎞⎟ ⎟ ⎟⎠. (49)

Omitting small terms of the order of , this vector becomes

 ek′Λ′=−Λ′√2(0,1,iΛ′,0)e−iΛ′φk=η(Λ′)e−iΛ′φk. (50)

The polarization vector of a conical component of the initial twisted photon (as a function of ) is obtained by setting in and reads

 ekΛ=Λ√2⎛⎜ ⎜ ⎜⎝0cosα0cosφk+iΛsinφkcosα0sinφk−iΛcosφksinα0⎞⎟ ⎟ ⎟⎠. (51)

Using the 4-vectors defined in Eq. (35), we may write it in the form

 ekΛ= η(−Λ)eiΛφkcos2(α02)+η(Λ)e−iΛφksin2(α02) +Λ√2η(z)sinα0. (52)

With the help of Eq. (16), we can also write the Fourier transform of the product , which is still a 4-vector, as

 IϰmΛ= ∫ekΛaϰm(k⊥)eik⊥rd2k⊥(2π)2 (53) = iΛη(−Λ)ψϰ,m+Λ(r,φr)cos2(α02) − iΛη(Λ)ψϰ,m−Λ(r,φr)sin2(α02) + Λ√2η(z)ψϰm(r,φr)sinα0,

recalling that the function is given in Eq. (2.1). With this formula for , we can write the twisted photon vector potential (13) as

 AμϰmkzΛ(r,φr,z,t)=√4πe−i(ωt−kzz)√2ωIμϰmΛ. (54)

This 4-vector potential corresponds to the initial twisted photon state , which describes a superposition of states with projections of the orbital angular momentum onto the axis equal to and . If the angle becomes small, we have