Generation of High–Energy Photons with Large Orbital Angular Momentum by Compton Backscattering
Usually, photons are described by plane waves with a definite 4-momentum. In addition to plane-wave photons, “twisted photons” have recently entered the field of modern laser optics; these are coherent superpositions of plane waves with a defined projection of the orbital angular momentum onto the propagation axis, where is integer. In this paper, we show that it is possible to produce high-energy twisted photons by Compton backscattering of twisted laser photons off ultra-relativistic electrons. Such photons may be of interest for experiments related to the excitation and disintegration of atoms and nuclei, and for studying the photo-effect and pair production off nuclei in previously unexplored experimental regimes.
pacs:13.60.Fz, 42.50.-p, 42.65.Ky, 24.30.-v, 27.70.Jj, 12.20.Ds
Introduction.—An interesting research direction in modern optics is related to experiments with so-called “twisted photons.” These are states of the laser beam whose photons have a defined value of the angular momentum projection on the beam propagation axis where is a (large) integer (1). An experimental realization (2) exists for states with projections as large as . Such photons can be created from usual laser beams by means of numerically computed holograms. The wavefront of such states rotates around the propagation axis, and their Poynting vector looks like a corkscrew (see Fig. 1 in Ref. (1)). It was demonstrated that micron-sized teflon and calcite “particles” start to rotate after absorbing twisted photons (3).
In this Letter, we show that it is possible to convert twisted photons from an energy range of about to a higher energies of up to a hundred GeV using Compton backscattering off ultra-relativistic electrons. In principle, Compton backscattering is an established method for the creation of high-energy photons and is used successfully in various application areas from the study of photo-nuclear reactions (5); (4) to colliding photon beams of high energy (6). However, the central question is how to treat Compton backscattering of twisted photons, whose field configuration is manifestly different from plane waves. Below, we use relativistic Gaussian units with , , . We denote the electron mass by and write the scalar product of 4-vectors and as .
Twisted photon.—We wish to construct a twisted photon state with definite longitudinal momentum , absolute value of transverse momentum and projection of the orbital angular momentum onto the axis (propagation axis). We start from a plane-wave photon state with 4-momentum and helicity ,
where is the polarization 4-vector of the photon ( and , with ). The twisted photon vector potential is obtained after integration over the conical transverse momentum components of the wave vector , with amplitude
Here, is the azimuth angle of , and
Furthermore, . For further analysis we introduce the three four-vectors
The initial twisted photon is composed of wave vectors of the form
which for propagate in the negative direction. Here, and are the polar and azimuth angles of the initial photon, and we have . The polarization vectors can be expressed as
Integration leads to
with the scalar twisted particle wave function
The vector field describes a photon state with projections of the orbital angular momentum on the axis equal to . For large , the restriction to means that the twisted state is a state with “almost defined angular momentum projection ” (see Fig. 1), and we 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 the state and a photon being scattered from the state to the state ,
In view of Eq. (Generation of High–Energy Photons with Large Orbital Angular Momentum by Compton Backscattering), the matrix element for the scattering of a twisted (TW) photon into the state needs to be integrated as follows,
Compton scattering of plane-wave photons.—We investigate the collision of an ultra-relativistic electron with 4-momentum , , and , propagating in the positive direction, and a photon of energy and three-momentum given by Eq. (5). After the scattering, the 4-momentum of the electron is , and the scattered photon has energy and three-momentum , where and are the polar and azimuth angles of the final photon. From the equation , we obtain
where and . The -matrix element for plane waves is
where the amplitude in the Feynman gauge is
and , . The bispinors and describe the initial and final electrons, and and are the polarization vectors of the initial and final photon. We denote the Feynman dagger as . Using Dirac algebra, we may write and with
as defined in Eq. (13a) can thus be written as
For a head-on collision of a plane-wave photon and electron, the relativistic kinematics then imply the following differential cross section, for the unpolarized cross section (summation over the outgoing and averaging over the incoming electron and photon polarizations),
According to Eq. (Generation of High–Energy Photons with Large Orbital Angular Momentum by Compton Backscattering), a twisted photon is a superposition of plane-wave photons with the same energy and conical momentum spread. We thus expect that the twisted and plane-wave photon scattering cross section will be related. Indeed, for the mixed (m) case where the initial photon is twisted but the outgoing one is a plane-wave photon, one finds
and the corresponding cross section is given by Eq. (Generation of High–Energy Photons with Large Orbital Angular Momentum by Compton Backscattering) with the only replacement
For strict backscattering geometry, the differential Compton cross section (Generation of High–Energy Photons with Large Orbital Angular Momentum by Compton Backscattering) and the energy of the scattered photon attain maxima, and additional simplifications are possible because the azimuth angle of the photon is conserved, as discussed in the following.
Compton backscattering of twisted photons.—For twisted photons, the final photon is a superposition of plane waves with small transverse momentum and very small scattering angle (see Fig. 2). In this limit, . For strict backward scattering, several quantum numbers in Eq. (Generation of High–Energy Photons with Large Orbital Angular Momentum by Compton Backscattering) are thus conserved under the scattering for twisted (TW) photons,
where we have used the decomposition (15).
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 write the polarization vector of the final photon in the scattering amplitude in the form , where the unit vector is in the scattering plane, defined by the vectors and , while the unit vector is orthogonal to it: and . As a result, we have in 4-vector component notation
Omitting terms of the order of , we obtain
The polarization vector of a “conical” component of the initial twisted photon (as a function of ) is obtained by setting in and coincides with Eq. (Generation of High–Energy Photons with Large Orbital Angular Momentum by Compton Backscattering). Substituting the expressions for and given in Eqs. (21) and (Generation of High–Energy Photons with Large Orbital Angular Momentum by Compton Backscattering) into the definitions of and given in Eqs. (14a) and (14c), we find for twisted photons,
One may write the neglected contribution as . It is negligible for our relativistic kinematics (), because
Therefore, we have for strict backward scattering, and the matrix element reads
with given in Eqs. (15) and (22). This result states that for strict backscattering, the angular momentum projection and the conical momentum spread of the twisted photons are conserved and confirms the principal possiblity for the frequency upconversion of twisted photons under strict Compton backscattering. A technique for the registration of electrons scattered at small (even zero) angles after the loss of energy in the Compton process is implemented, for example, in the device for backscattered Compton photons installed on the VEPP-4M collider (Novosibirsk) (4).
Certainly, it is interesting to estimate the admixture of different twisted photon states if the electron carries away a small transverse momentum under the scattering. A solution of the relativistic kinematic equations then implies that the azimuthal angle of the scattered twisted photon component is not conserved but acquires a phase slip,
Taking into account this phase slip we obtain a distribution approximately given by Eq. (Generation of High–Energy Photons with Large Orbital Angular Momentum by Compton Backscattering) under the replacement
This yields a distribution where the scattered twisted photon angular momenta are displaced from the initial twisted photon angular momentum by . Finally, as the initial twisted photon is obtained by an integration over a conical angular distribution of plane-wave components, the energy of the final twisted photon for the case of non-strict backscattering can be obtained from Eq. (11).
Conclusions.—The general convoluted invariant matrix element (Generation of High–Energy Photons with Large Orbital Angular Momentum by Compton Backscattering) for Compton scattering of twisted photons, which can be evaluated numerically for arbitrary scattering geometry, is found to take a particularly simple form for strict backscattering (see Fig. 2), according to Eqs. (15) and (22). For that geometry, the energy of the final twisted photon is increased most effectively (). According to Eq. (Generation of High–Energy Photons with Large Orbital Angular Momentum by Compton Backscattering), the magnetic quantum number and the conical momentum spread are preserved under strict backscattering. This implies that the conical angle of the scattered twisted photon is very small, .
High-energy photons with large orbital angular momenta projections can be used for experimental studies regarding the excitation of atoms into circular Rydberg states, and for studying the photo-effect and the ionization of atoms, as well as the pair production off nuclei. As ion traps for highly-charge ions are currently under construction (e.g., Ref. (7)), one of the most interesting experiments would concern the question of whether nuclear fission can be achieved by the absorption of fast rotating nuclei, via the absorption of one or more twisted high- photons at energies below the giant dipole resonances (8) which are typically in the range of 10–30 MeV. Such a study might reveal fundamental insight into the dynamics of a fast rotating quantum many-body system.
The authors are grateful to I. Ginzburg, D. Ivanov, I. Ivanov, G. Kotkin, N. Muchnoi, O. Nachtmann, V. Telnov, A. Voitkiv, V. Zelevinsky and V. Zhilich for useful discussions. This research was supported by the National Science Foundation (PHY-8555454) and by the Missouri Research Board. V.G.S. is supported by the Russian Foundation for Basic Research via grants 09-02-00263 and NSh-3810.2010.2.
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