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Detecting Transition Radiation from a Magnetic Moment

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Detecting Transition Radiation from a Magnetic Moment Detecting transition radiation from a magnetic moment Igor P. Ivanov1, 2, ∗ and Dmitry V. Karlovets3, y 1IFPA, Universit´ede Li`ege,All´eedu 6 Ao^ut17, b^atimentB5a, 4000 Li`ege,Belgium 2Sobolev Institute of Mathematics, Koptyug avenue 4, 630090, Novosibirsk, Russia 3Tomsk Polytechnic University, Lenina 30, 634050 Tomsk, Russia (Dated: November 13, 2018) Electromagnetic radiation can be emitted not only by particle charges but also by magnetic moments and higher electric and magnetic multipoles. However experimental proofs of this funda- mental fact are extremely scarce. In particular, the magnetic moment contribution has never been observed in any form of polarization radiation. Here, we propose to detect it using vortex electrons carrying large orbital angular momentum (OAM) `. The relative contribution of the OAM-induced magnetic moment, `~!=Ee, becomes much larger than the spin-induced contribution ~!=Ee, and it can be observed experimentally. As a particular example, we consider transition radiation from vortex electrons obliquely incident on an interface between a vacuum and a dispersive medium, in which the magnetic moment contribution manifests itself via a left-right angular asymmetry. For electrons with Ee = 300 keV and ` = 100−1000, we predict an asymmetry of the order of 0:1%−1%, which could be measured with existing technology. Thus, vortex electrons emerge as a new tool in the physics of electromagnetic radiation. PACS numbers: 41.60.Dk, 42.50.Tx Introduction. | Radiation of electromagnetic (EM) only in [9] that freely propagating vortex electrons were waves is an inherent property of charges. In gen- discussed in detail and practical methods for their cre- eral, there exist two broad classes of radiation: ation were proposed. Three years later, this proposal bremsstrahlung and polarization radiation (PR). The for- was brought to life by several experimental groups [10]. mer is produced by accelerating charges, while the latter Vortex electrons carry an intrinsic orbital angular mo- can be emitted by a uniformly moving charge but only in mentum (OAM) ` with respect to their average propaga- the presence of a medium. Depending on the medium or tion direction, and values of ` ∼ 100 have already been target geometry, one distinguishes different forms of PR: achieved. The magnetic moment associated with OAM is Cherenkov radiation, transition radiation, diffraction ra- correspondingly large, µ ≈ `µB, where µB = e~=2mc is diation, Smith-Purcell radiation, etc. (see e.g. [1{3]). the Bohr magneton. One then enters the regime in which EM radiation can obviously be produced not only by the OAM-induced magnetic moment contribution to PR charges but also by neutral particles carrying higher mul- is only moderately attenuated, / `~!=Ee . 1, remaining tipoles: electric or magnetic dipoles, quadrupoles, etc. much larger than quantum effects. This improves chances For example, transition radiation from these multipoles to detect this elusive effect and, at the same time, makes was studied theoretically in detail e.g. in [4], while its quasiclassical calculation self-consistent. This contri- Cherenkov radiation of a magnetic moment was consid- bution can be predicted, and its observation would be ered e.g. in [5]. It is therefore remarkable that experi- the first clear evidence of PR by a multipole. mental observations of the influence of the magnetic mo- In this Letter, we propose to measure this contribu- ment or of any higher multipole on the EM radiation are tion in transition radiation (TR) of vortex electrons with very scarce and are limited to very few cases of spin- ` 1 obliquely incident on an interface between a vac- induced effects in bremsstrahlung (\spin light") [6, 7]. In uum and a medium with arbitrary (complex) permittiv- particular, the contribution of the magnetic moment to ity "(!). We show that the magnetic-moment contribu- any kind of PR has never been detected, and there are tion manifests itself as a left-right asymmetry of the emit- not only technological but also fundamental reasons for ted radiation with respect to the incidence plane, and we arXiv:1304.0359v2 [physics.optics] 20 Jul 2013 that. Compared with radiation from charge, the relative predict for electrons with Ee = 300 keV and ` ∼ O(1000) contribution of the spin-induced magnetic moment to PR an asymmetry of the order of 1%. is attenuated by ~!=Ee 1, where ~! and Ee are the TR from \charge + magnetic dipole": Qualitative fea- photon and electron energies, respectively. But the quan- tures. | Transition radiation occurs when a uniformly tum effects in radiation are of the same order. Therefore, moving charge crosses an interface separating two me- this contribution simply cannot be self-consistently cal- dia with different permittivities [11]. The accompanying culated within the standard quasi-classical treatment of electromagnetic field reorganizes itself when it crosses the PR, in which one neglects quantum effects. interface, and it is partly \shaken off" in the form of Recently created vortex electrons put a dramatic twist electromagnetic radiation, see [4] for many details of the on this problem. Although solutions of Dirac equation theoretical description of this process. with helical wave fronts were known before [8], it was Consider first a point-like charge e with no magnetic 2 → Now, in the case of an electron, we deal with both e –, μ charge and magnetic moment contributions to TR. Fields → from both sources add up, and the radiated energy can u contain three terms dW = dWe + dWeµ + dWµ ; (2) θ 1 → θ2 k describing the radiation energy of charge dWe, that of magnetic moment dWµ, and their interference dWeµ. Since x` is very small, one can only hope to detect the x α magnetic moment contribution via dWeµ. This task turns out to be tricky due to a number of y → reasons. First, µ is a pseudovector, therefore dW must n eµ z contain the triple product ek · [µ n], where ek is the di- rection of the emitted photon and n is the normal to the FIG. 1: Our angle conventions at an oblique incidence with interface. This triple product vanishes for normal inci- the example of backward TR. The direction of specular re- dence, while for oblique incidence it changes sign under flection is shown by the gray dashed line. θ2 ! −θ2. Therefore, the interference can be observed only at oblique incidence and only in a differential distri- moment obliquely incident on a flat interface separating bution, not in the total energy. It will manifest itself in a vacuum from a medium of permittivity "(!), Fig. 1. the form of a left-right asymmetry Z The angle between the particle trajectory and the nor- WL − WR dW A = ;WL;R = dΩL;R ; (3) mal to the interface is α. The direction of the emitted WL + WR dΩ photons can be described by two “flat” angles: θ lying 1 where dΩ and dΩ refer to two hemispheres lying to in the incidence plane and measured from the direction L R the left and to the right of the incidence plane. Alterna- of specular reflection, and θ describing an out-of-plane 2 tive definitions of this asymmetry using a weight function deviation. antisymmetric in θ can also be used. TR is mostly emitted into two prominent lobes near 2 Next, the curl in j produces an extra i factor in the the \forward" (along the particle velocity) and \back- µ Fourier-components. As a result, the radiation field HR ward" (i.e. specular) directions, which are symmetric in contains the charge and magnetic moment contributions θ and have an angular spread of ∼ 1/γ = p1 − β2. The 2 with a relative phase: HR = H + H = a + ix b. spectrum of TR photons is mostly shaped by the disper- e µ ` These two quantities a and b are complex due to the com- sion of the medium, "(!). It stays roughly flat up to p plex ", but if they have equal phases, dW vanishes. γ! [4], with the plasma frequency ! around 10 − 30 eV eµ p p This happens, in particular, in the cases of a transparent for many materials, and rapidly decreases above it, thus medium (Im " = 0) and of an ideal conductor (Im " = 1). making the ratio !=E ≈ ! =m ∼ 10−5. ~ e ~ p e Furthermore, it means that this interference is absent TR from a pointlike magnetic moment has also been for Cherenkov radiation in a transparent medium. Ob- studied in detail, see e.g. [4]. The main change of the servation of a non-zero asymmetry requires, therefore, a TR from a longitudinally oriented pointlike magnetic mo- real medium with a sizable (but not asymptotically large) ment µ = `µB with respect to the TR from a charge can Im ", which is the case, for instance, for any real metal. be anticipated from the comparison of the respective cur- If all these conditions are satisfied, we can expect, very rents: jµ = c rot[µδ(r − ut)]/γ vs. je = e u δ(r − ut) roughly, the asymmetry (3) to be of the order of A ∼ x`. (here and below µ denotes the magnetic moment in the For the typical experiments with vortex electrons in mi- particle rest frame; in the lab frame it is equal to µ/γ). croscopes, this amounts to A ∼ O(1%) for optical/UV Curl leads to an extra factor i!=c in the Fourier com- TR from electrons with ` ∼ O(1000), and a proportion- ponents of the radiation field, and the relative strength ally weaker asymmetry for smaller `. of the magnetic moment PR always bears the following TR from vortex electrons: quantitative description. | small factor A vortex electron state is a freely propagating electron ~! described by a wave function containing phase singular- x` = ` : (1) ities with non-zero winding number `. Such an elec- Ee tron state is characterized, simultaneously, by an average The radiated energy contains this factor squared, mak- propagation direction and by an intrinsic orbital angular ing the radiation of pure magnetic moments many orders momentum (OAM) with a projection L = ~` on this di- of magnitude weaker than that of charges.
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