Detecting transition radiation from a magnetic moment

Igor P. Ivanov1, 2, ∗ and Dmitry V. Karlovets3, † 1IFPA, Universit´ede Li`ege,All´eedu 6 Aoˆut17, bˆatimentB5a, 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, `~ω/Ee, becomes much larger than the spin-induced contribution ~ω/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, ∝ `~ω/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 ε(ω). 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 ~ω/Ee  1, where ~ω 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 ε(ω), 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, ε(ω). It stays roughly flat up to √ plex ε, but if they have equal phases, dW vanishes. γω [4], with the plasma frequency ω 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 ε = 0) and of an ideal conductor (Im ε = ∞). making the ratio ω/E ≈ ω /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 ε, 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 iω/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 ~ω 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. Large ` par- rection. Following the suggestion [9], vortex electrons tially compensates this suppression, but it still remains with Ee = 200 − 300 keV and ` up to 100 were recently prohibitively difficult to detect. created in experiments by several groups [10]. 3

The simplest example of a vortex state for a spinless ted light wavelength λ (focusing vortex electrons to an particle is given by the Bessel beam state [12, 13], de- Angstr¨omsize˚ spot was achieved in [17]). The same ap- scribed by a coordinate wave function ψ(r⊥, φr, z) ∝ plicability condition requires also that the longitudinal ikz z i`φr e e J`(k⊥r⊥). At large `, it has a narrow radial extent of the individual-electron wave function is much distribution located around r⊥ ≈ `/k⊥, confirming the shorter than λ. This extent can be quantified by the quasiclassical picture of such an electron as a rotating self-correlation length of the electron beam, which is re- ring of electronic density. The spin degree of freedom lated to the monochromaticity of the electron beam and for a vortex electron was accurately treated in [13, 14]. can be measured experimentally by counting the number Both spin and OAM induce magnetic moment [14], see of fringes in a diffraction experiment. The longitudinal [15] for a recent theoretical and experimental investiga- compactness condition implies that the monochromatic- tion of these contributions, but at large ` the spin contri- ity should not be too good. bution and spin-orbital coupling can be neglected leading Turning to the calculation of the radiation fields, we to µ ≈ `µB (in the electron rest frame). use the geometric set-up of Fig. 1 and write the elec- As explained above, large ` allows for a self-consistent tron velocity as u = u(sin α, 0, cos α). We start with the quasi-classical treatment of TR from an OAM-induced currents je and jµ of the two sources, find their Fourier magnetic moment, in which the magnetic moment effects components, calculate the partial Fourier transforms of of the order of `~ω/Ee are retained while quantum and the electric fields they generate, Ee and Eµ, and finally spin effects of the order of ~ω/Ee are neglected. One extract the radiation field in the wave zone: can then approximate a vortex electron with large ` by a √  2 i εrω/c pointlike particle with charge e and an intrinsic magnetic R 2πω ε − 1 e H (r, ω) = [ek × J ] , (4) moment µ, and calculate TR from both sources, without c 4π r the need to discern the microscopic origin of µ. The only assumption we make is that, in the absence of magnetic where monopoles in nature, the magnetic moment arises from Z 0 0 −iz kz 0 0 a closed charge current loop, see discussion on this issue J = dz e [Ee(k⊥, z , ω) + Eµ(k⊥, z , ω)] . (5) in [4]. To control the validity of this approach, we devised an- Explicit expressions for the fields and a detailed discus- other quasi-classical model, in which we treat the vortex sion can be found in [3]. We introduced here the “on- electron beam as a very short bunch of a large number shell” wave vector in the medium k = ekω/c, where of electrons, N  1, carrying no intrinsic magnetic mo-     ment and uniformly moving along straight rays passing √ sin θm cos φ sin θ cos φ e = ε sin θ sin φ = sin θ sin φ , (6) through a ring of microscopic size R  λ at a fixed skew k  m   √  angle. The calculation is then the standard one of coher- cos θm ± εθ ent TR from a compact bunch with the only exception √ p 2 that the total charge of the bunch is just e instead of Ne. and εθ ≡ ε − sin θ. The two expressions in (6) re- Using the quasiclassical estimate of the effective emergent late the emission polar angle in the medium θm with the OAM `eff = Rp sin ξ/~, where p is the electron momen- emission angle θ in the vacuum. The latter is connected tum and ξ is the skew angle, we checked that the two with the “flat” angles θ1,2 by cos θ = cos θ2 cos(α + θ1). models lead to quantitative agreement, see the details in The integration in (5) is carried out from 0 to ∞ for [16]. Below we focus only on the first model. backward TR (in this case ek,z < 0) and from −∞ to 0 These models can be applicable to a realistic experi- for forward TR (ek,z > 0). mental set-up with vortex electrons, if certain coherence The radiation field can be conveniently written in the conditions are satisfied. First, the quasiclassical treat- coordinates related with the photon production plane ment of the electrons as point-like particles in the trans- (ek, z). The radiation field (4) is orthogonal to ek and verse space is valid only if the vortex electrons are fo- therefore has two components: one that lies in the pro- R R cused in a spot of a much smaller size than the emit- duction plane, Hin, and one out of that plane, Hout:

h √ ω  √ √ i HR = N s (1 − β2c2 − β · e ) ± β2s c c ε + iµ s s βc s2 ∓ βs s c ε ± ε , (7) out θ α k α α φ θ eγc α φ α θ α θ φ θ θ √  ω  HR = N ε β2s c s + iµ βs (1 − s2 s2 ) − s c  , (8) in α α φ eγc θ α φ α φ

where we used obvious shorthand notations for sines and cosines and introduced a common kinematical factor N , 4 which we omit here. Note that at normal incidence, A ,% α = 0, or for in-plane radiation, at φ = 0, the charge con- 1.4 R 1.2 tributes only to Hout, the magnetic moment contributes È È R only to Hin, so that there is no interference. On the 1.0 other hand, the magnetic moment makes TR elliptically 0.8 polarized, which is another subtle effect to be explored 0.6 [18, 19]. The upper and lower signs in these expressions 0.4 correspond to forward and backward radiation, respec- 0.2 tively. The spectral-angular distributions of the radiated ÑΩ, eV energy can be found from the reciprocity theorem and 2 4 6 8 10 reads [3]: FIG. 3: The value of the asymmetry A defined in (3) as a d2W HR cos θ 2 HR cos θ 2 function of the emitted photon energy. The solid and dashed out in ∝ √ + √ √ .(9) lines correspond to forward and backward TR, respectively. dωdΩ ε cos θ + εθ ε(cos θ + εθ) ◦ ◦ Parameters are α = 70 , θ1 = −40 , ` = 1000. Substituting here the explicit expressions for the radia- tion field and sorting out the charge and magnetic mo- ment contributions, one can break the expression for the energy into the three parts introduced in Eq. (2). ` = 25 has been achieved; a tenfold increase of this value Numerical results. — In Figs. 2 and 3 we show nu- is highly desirable. Manufacturing such diffraction grat- merical results for 300-keV electrons incident on an alu- ings is challenging but seems to be within technological minium foil (aluminium permittivity data were taken limits. Alternatively, one can use a novel method for from [20]). For non-vortex beams, the angular depen- vortex electron generation [21] via the passage of ring- shaped non-vortex electrons through the tip of a mag- dence of TR is θ2-symmetric, see black curve in Fig. 2. Non-zero ` induces a left-right asymmetry, which be- netic whisker. Note also that we do not require the vortex comes huge for ` = 104. For smaller `, this asymmetry electrons to be in a state of definite `; the effect remains can be extracted via Eq. (3). In Fig. 3 we show its magni- even if OAM is spread over a broad range of values. tude as a function of the photon energy. The initial rise, Detecting a small asymmetry necessitates large count- −4 ∝ x` ∝ ~ω, slows down above 5 eV due to dispersion, ing statistics. Our calculations give nγ ∼ O(10 ) TR which makes the UV-range optimal for detecting the ef- photons per incident electron, which can be seen from fect, see details in [16]. We emphasize that the values Fig. 2. With a current of 1 nA, easily achievable in vor- of the asymmetry depend rather weakly on the target tex electron experiments, and a photon detector with a medium (provided it is a metal) and the emission angle quantum efficiency of 10%, one can expect about 105 θ1. photons per second. With a sufficient integration time, a left-right asymmetry of order A ∼ 0.1% can be reliably d2 W ´104 detected. dÑΩdW In summary, we showed that by studying UV transi- 0.8 tion radiation from vortex electrons with large OAM, one can detect for the first time the magnetic moment contri- 0.6 bution to polarization radiation. For ` = 100 − 1000 we 0.4 predict an asymmetry of the order of 0.1% − 1%, which could be measurable with existing technology. Simulta- 0.2 neously, it gives a novel method to measure large OAM in electron vortex beams. Θ2, deg -50 0 50 The authors are grateful to J. Verbeeck and members FIG. 2: (Color online.) Distribution of the forward TR over of his team for discussions on the experimental feasibil- θ2 for ` = 0 (black solid line), ` = 1000 (red dashed line), ity of the proposed measurement and to J.-R. Cudell and ` = 10000 (blue dotted line). Parameters are α = 70◦, for his remarks on the manuscript. I.P.I. acknowledges ◦ θ1 = −40 , ~ω = 5 eV. grants RFBR 11-02-00242-a and RF President grant for scientific schools NSc-3802.2012.2. D.V.K. acknowledges Experimental feasibility. — Let us briefly comment on grants of the Russian Ministry for Education and Sci- the feasibility of the proposed observation. The state-of- ence within the program “Nauka” Nos. 14.B37.21.0911, the-art experiments with vortex electron beams already 14.B37.21.1298, and the RFBR grant No.12-02-31071- satisfy the coherence requirements. The key issue is to mol a. He also wishes to thank the IFPA group at the obtain large OAM in the first diffraction peak. So far, University of Li`egefor hospitality during his visit. 5

301 (2010); B. J. McMorran et al, Science 331, 192 (2011); K. Saitoh, Y. Hasegawa, N. Tanaka, and M. Uchida, J. Electron. Microsc. (Tokyo) 61, 171 (2012). ∗ E-mail: [email protected] [11] V. L. Ginzburg and I. M. Frank, J. Phys. (USSR) 9, 353 † E-mail: [email protected] (1945) [Zh. Eksp. Teor. Fiz. 16, 15 (1946)]. [1] M. Ya. Amusia, Radiat. Phys. Chem. 75, 1232 (2006). [12] U. D. Jentschura and V. G. Serbo, Phys. Rev. Lett. 106, [2] A. P. Potylitsyn, M. I. Ryazanov, M. N. Strikhanov, 013001 (2011); U. D. Jentschura and V. G. Serbo, Eur. A. A. Tishchenko, Diffraction radiation from relativis- Phys. J. C71, 1571 (2011). tic particles, STMP 239 (Springer, Berlin, Heidelberg, [13] D. V. Karlovets, Phys. Rev. A 86, 062102 (2012). 2010). [14] K. Yu. Bliokh, M. R. Dennis, F. Nori, Phys. Rev. Lett. [3] D. V. Karlovets, J. Exp. Theor. Phys. 113, 27 (2011). 107, 174802 (2011). [4] V. L. Ginzburg and V. N. Tsytovich, Transition Radi- [15] K. Bliokh, P. Schattschneider, J. Verbeeck, F. Nori, Phys. ation and Transition Scattering (Nauka, Moscow, 1984; Rev. X 2, 041011 (2012); G. Guzzinati, P. Schattschnei- Adam Hilger, New York, 1990); Phys. Rept. 49, 1 (1979). der, K. Bliokh, F. Nori, J. Verbeeck, Phys. Rev. Lett. [5] I. M. Frank, Sov. Phys. Usp. 27, 772 (1984). 110, 093601 (2013). [6] S. A. Belomestnykh, A. E. Bondar, M. N. Egorychev et [16] I. P. Ivanov, D. V. Karlovets, arXiv:1305.6592 al., Nucl. Instrum. Meth. A 227, 173 (1984). [physics.optics]. [7] K. Kirsebom, U. Mikkelsen, E. Uggerhøj et al., Phys. [17] J. Verbeeck et al, Appl. Phys. Lett. 99, 203109 (2011). Rev. Lett. 87, 054801 (2001). [18] A. S. Konkov, A. P. Potylitsin, and V. A. Serdyutskii, [8] P. Ouyang, V. Mohta, R. L. Jaffe, Ann. Phys. (New York) Russian Physics Journal 54, 1249 (2012). 275, 297 (1999); V. Bagrov, D. Gitman, Ann. Phys. [19] A. S. Konkov, A. P. Potylitsyn and M. S. Polonskaya, (Leipzig) 14, 467 (2005). arXiv:1304.7363 [physics.acc-ph]. [9] K. Yu. Bliokh, Yu. P. Bliokh, S. Savel’ev, F. Nori, Phys. [20] A. D. Raki´c,Appl. Optics 34, 4755 (1995). Rev. Lett. 99, 190404 (2007). [21] A. B´ech´e,R. Van Boxem, G. Van Tendeloo, J. Verbeeck, [10] M. Uchida and A. Tonomura, Nature 464, 737 (2010); arXiv:1305.0570 [physics.ins-det]. J. Verbeeck, H. Tian, P. Schattschneider, Nature 467,