Probability of radiation of twisted photons in the infrared domain
The infrared asymptotics of probability of radiation of twisted photons in an arbitrary scattering process of quantum electrodynamics (QED) in a vacuum is investigated. This asymptotics is universal and corresponds to the radiation produced by a classical current. Such a radiation is known as the edge radiation. We represent it in terms of the twisted photons: the exact analytical formulas for the average number of radiated twisted photons are derived. We find the average projection of the total angular momentum of the edge radiation and the angular momentum per photon. It is shown that the edge radiation can be used as a source of twisted photons with large angular momentum. Moreover, this radiation can be employed as a superradiant coherent source of twisted photons in the infrared domain, in particular, in the THz part of the electromagnetic spectrum. Several general selection rules for the radiation and absorbtion of twisted photons are proved. These selection rules allow one, in particular, to modulate the one-particle radiation probability by means of scattering of charged particles on symmetrically arranged crystals.
The infrared asymptotics of radiation of photons in a vacuum is known to be universal for any quantum electrodynamic (QED) process (see, e.g., [, , , , , , ]). 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 [, , , , , , , , , , ]. The edge radiation can be routinely generated on the acceleration facilities (see, e.g., [, , , , , , ]). It is used now as a brilliant source of the infrared radiation [, , , ] and applied to the direct control of numerous low-energy excitations in solids and molecules [, ], to the study of mechanisms of the high-temperature superconductivity [], to THz spectroscopy and microscopy, medicine, biology, and even to the preservation of cultural heritage [, ].
Surprisingly, this type of radiation has not been examined as a possible source of the vortex electromagnetic radiation, which also finds a wide range of applications in fundamental science and technology [, , , ]. The vortex radiation, i.e., the electromagnetic waves carrying an angular momentum, can be rigorously described in terms of the so-called twisted photons [, , , , , , , ]. These photons are the excitations of a quantum electromagnetic field with definite the energy, the longitudinal projection of momentum, the projection of the total angular momentum, and the helicity. The angular momentum can be regarded as an additional degree of freedom of the field and can be used for the high-density information transfer or for the manipulation of the rotational degrees of freedom of an irradiated object. The appreciation of the utility of such an instrument resulted in a rapid growth in research on the generation and application of the vortex fields, not only the electromagnetic ones but with the spin one-half as well (see, for review, [, ]). Using recently developed general formalism [], we shall give the complete description of the edge radiation in terms of the twisted photons and find the conditions when the edge radiation can be regarded as the source of the electromagnetic waves with a considerable angular momentum. Since the edge radiation is infrared, it provides an exceptional opportunity for creation of a superradiant coherent source of twisted photons. Its intensity and, consequently, the angular momentum of radiation produced by it are proportional to the square of the particle number in the bunch of charged particles. As for the plane-wave photons, their coherent edge radiation was already observed experimentally [, , , ].
In [], 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 [, , , , , , , , ]. 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.
The list of the main results obtained in this paper is presented in Conclusion. So we do not dwell on it here. In Sec. 2, we recapitulate the main formulas of [] regarding the probability of radiation of twisted photons by classical currents and prove the general selection rules for symmetric sources. Notice that the probability of radiation of twisted photons not only reveals the content of radiation in terms of the twisted photons but it can be directly observed [, , , ]. In Sec. 3, we find the spectrum domain where the edge radiation dominates and show that the edge radiation is a universal infrared asymptotics of any radiation of charged particles in a vacuum, the effects of a chamber [, , ] where the process is evolving being supposed to be negligible. Section 4 is devoted to the edge radiation of twisted photons. In Sec. 4.1, we consider the edge radiation produced by the charged particles with the trajectories possessing the break on the detector axis. In this case, we derive the exact formula for the probability of radiation of twisted photons and find the main characteristics of the twist of radiation: the differential asymmetry, the average projection of the total angular momentum of radiation, and the angular momentum per photon. In particular, we show that the photons with large angular momentum can be generated only by the scattering of ultrarelativistic charged particles with the sufficiently large velocity component transversal to the detector axis. We also establish there the symmetry property of the probability of radiation of twisted photons. This symmetry property holds for the radiation of twisted photons in any QED process in the far infrared. In particular, we obtain the infrared asymptotics of radiation of twisted photons produced in scattering of ultrarelativistic electrons by crystals positioned exactly on the detector axis. In Sec. 4.2, we investigate the edge radiation of twisted photons by the charged particles moving along the trajectories with the break located out of the detector axis. We find the probability of radiation of twisted photons and the angular momentum per photon of the edge radiation. The edge radiation of twisted photons for several processes of scattering of electrons by the crystals positioned out of the detector axis is studied numerically. We show that the optimal situation for the production of twisted photons with large angular momentum in this configuration occurs when the scattered electrons possess a large angular momentum with respect to the detector axis. As another example, we consider in this section the scattering of electrons by the crystal located on the detector axis in the magnetic field of a solenoid, the axis of the solenoid coinciding with the detector axis. For such a configuration, the electrons initially moving along the detector axis escape the magnetic field of the solenoid with a large angular momentum with respect to this axis. Therefore, one should expect that the radiation produced possesses a large angular momentum too. We show that this is indeed the case and find the main characteristics of the twist of the radiation created. We also find the general bound on the maximal magnitude of the projection of the angular momentum of a twisted photon radiated by a classical current and verify this bound on several examples.
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 []. Consider the theory of a quantum electromagnetic field interacting with a classical current . We suppose that is the current density of a point charge
where and , and is the time period when the particle moves with acceleration. The expression (1) can be cast into the standard form
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,
the last two terms in (1) correspond to the boundary terms in
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., [, , , , , , , , , , , , , , , , , , , , , , , ]). 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., [, , , , , , , ]). 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
The basis vector is directed along the axis of the detector that records the twisted photons. It is clear that
and arbitrary vector can be decomposed in this basis as
where and .
In the presence of a classical current, the process of radiation of photons is possible
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
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.
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
A similar property is known to be fulfilled for the scattering of twisted photons by microstructures [, ]. In order to prove this statement, we employ the pictorial representation of the radiation amplitude of twisted photons (see, for details, []) 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 [], the contribution to the radiation amplitude of every point on this circle comes with the common phase factor
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 []). 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
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 [, , , , ]. Consequently, the average number of twisted photons is nonzero only when (see []).
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
where is the number of particles. In particular, if
where and are some fixed parameters, then
where . This interference factor modulates the one-particle radiation probability. It possesses the sharp global maxima at
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
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
we see that
i.e., we have exactly the spectrum of the forward radiation of twisted photons by the helical undulator []. The case formally corresponds to the scattering of particles on the spiral phase plate commonly used to produce the twisted electrons [, ] and photons [] or to the helical trajectory of a charged particle in the undulator, for example, (see [Fig. 5, []]). In this case, we obtain the selection rule
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
where , , , and are constant vectors, characterizes the time scale of variations of the trajectory, and
The parameters , , , and specify the asymptotes of the trajectory in the future and in the past. More precisely, we suppose that
when , and the same estimates hold for . Then, we partition the integral over in (10) into two,
and stretch the integration variable
in the first and the second integrals, respectively. Hereinafter, , , and so
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 :
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., []). 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 []. 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,
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 [] 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.
4.1 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
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
Taking into account that
Using this formula, we obtain
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)
where is the phase difference, which is brought to the interval .
The structure of the expression (37) implies the symmetry property
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
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 []. 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., [, , , , , , , , , , , ]), 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
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
Here the characteristic strength of the electromagnetic field is measured in the units of the critical field
The typical crystalline field strengths are of the order , where is the Bohr radius in the Compton wavelengths (see, e.g., [, , ]). The reflection angle in this case is of the order of the Lindhard critical angle
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
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
Hereinafter we use the approximate expressions
It is not difficult to find the main parameters characterizing the twist of the radiation (see []). 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
where, by definition, . In that case, we find the differential asymmetry
the projection of the total angular momentum
and the average number of twisted photons
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
The magnitude of this expression reaches its maximal value at , where
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
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
Since , we can replace
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
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
and the total average number of radiated twisted photons is found to be
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),
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 [] in studying the scattering of twisted electrons on the screened Coulomb potential. However, their properties were not investigated there.
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
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
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., [])
For , the function goes exponentially fast to zero.
Now we turn to the general case. Employing the integral representation
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