Proposal for the experimental observation of twisted photons intransition and Vavilov-Cherenkov radiations

O.V. Bogdanov1),2)∗, P.O. Kazinski1)†, and G.Yu. Lazarenko1),2)‡

1) Physics Faculty, Tomsk State University, Tomsk 634050, Russia2) Division for Mathematics and Computer Sciences,Tomsk Polytechnic University, Tomsk 634050, Russia

Abstract

We propose to use Vavilov-Cherenkov (VC) and transition radiations as a source of twisted photonsin a wide range of energies. The experimental setup to observe the orbital angular momentum of photonsconstituting those radiations is described. The radiation produced by relativistic electrons and ions inpassing dielectric plates is considered. The experimental verification of the addition rule for the totalangular momentum in the radiation of twisted photons by helically microbunched beams is proposed forVC and transition radiations. The parameters of the experiments are presented.

1 Introduction

The transition and Vavilov-Cherenkov (VC) radiations of plane-wave photons are well studied and find variousapplications [1–5]. These radiations allow one to generate the photons from the THz spectral range [6] toX-rays [3, 7, 8]. The properties of VC and transition radiations are employed for elaboration of detectors ofcharged particles [7–9]. The analysis of these radiations is used for diagnostics of the form of particle bunchesand the energy distribution of particles in them [3, 4, 7, 8, 10].

As was shown in [11–19], the radiation by relativistic charges moving in or near the medium possesses newproperties due to the orbital angular momentum of radiating or radiated particles. Usually, the conversionof plane-wave photons to the twisted ones is achieved by the use of phase or holographic plates [16, 20]. Thisapproach cannot be employed for development of the brilliant sources of hard twisted photons. The use oftransition and VC radiations allows one to generate twisted photons in a wide spectral range up to X-rays[17, 18, 21]. The helically microbunched beams of charged particles [22] propagating in a medium producethe coherent transition and VC radiations with large intensity and angular momentum [17, 18, 23], what isnot achievable by using the phase and holographic plates at the present moment.

In comparison with the plane-wave photons, the twisted photons possess an additional degree of freedom– the projection of the total angular momentumm on the propagation axis. This additional degree of freedomwas used for overcoming the diffraction limit [24], to increase the capacity of telecommunication channels[25, 26], and for quantum cryptography [26]. One of the property of twisted light is that its intensity vanishesat the center of the beam. This allows one to design the optical tweezers and to manipulate nanoparticles[27, 28]. The distribution of radiation over m is rather sensitive to the structure of the beam of radiatingparticles. This property can be employed for an improvement of the VC detectors [19] and the detectorsbased on the transition radiation [11, 12].

2 Theory

The twisted photons are the quanta of the electromagnetic field with the definite energy k0, the momentumprojection k3, the projection of the total angular momentum m on the same axis, and the helicity s. Wesuppose that the axis 3 coincides with the detector axis. The absolute value of the momentum componentperpendicular to the detector axis is k⊥ =

√k20 − k23.

∗E-mail: bov@tpu.ru†E-mail: kpo@phys.tsu.ru‡E-mail: laz@phys.tsu.ru

1

arX

iv:2

001.

0522

9v1

[ph

ysic

s.ac

c-ph

] 1

5 Ja

n 20

20

MonochromatorHomogeneous circular polarizer

CCD

Contour for the central slot

F Dielectric plate

z

Gaussian bunches

Helical bunches

Beam

Lens

���

Figure 1: The scheme for the experimental setup.

In the paper [17], the general theory of radiation of twisted photons in the presence of a dispersive mediumwas developed. The probability of radiation of a twisted photon by a charged particle takes the form

dP1(s,m, k⊥, k3) = e2|a|2∣∣∣∣ ∫ ∞−∞

dτe−ik0x0(τ)(x(τ),Φ(s,m, k⊥, k3)

)∣∣∣∣2( k⊥2k0)3dk3dk⊥

2π2, (1)

where xµ(τ) is the particle’s worldline, |a| is the normalization constant, and Φ(s,m, k⊥, k3) are the modefunctions of twisted photons (see for details [17]). We consider the case when the ordinary Gaussian andhelically microbunched beam of identical particles falls normally onto the dielectric plate. In that case, theprobability of radiation of twisted photons can be found from the formula

dPρ(s,m, k⊥, k3) =[Nfm +N(N − 1)|ϕm|2

]dP1(s, 0, k⊥, k3), (2)

where dP1(s, 0, k⊥, k3) is the probability of radiation of a twisted photon by one charged particle movingalong the center of the beam. The functions fm and ϕm are the incoherent and coherent interference factors,respectively (see for details [17, 18, 29]).

3 Experiment and results

The scheme of the experiment is presented in Fig. 1. The beam of relativistic charged particles moves alongthe detector axis (the axis z or 3), traverses the dielectric plate, and then is deflected. The parameters of thebeam and the dielectric plate are chosen such that the created transition and VC radiations are concentratedin a narrow cone. As for transition radiation, the maximum of its intensity is located at n⊥ := k⊥/k0 =

√3/γ.

The VC radiation is concentrated at n⊥ =√ε′(k0)− β−2, where ε′(k0) is the real part of the permittivity.

We have to demand n⊥ � 1, i.e., the paraxial approximation for radiation must be applicable. In thisapproximation, m = l + s, where l is the orbital angular momentum of a twisted photon. The createdtransition and VC radiations pass sequentially through the monochromator separating a narrow spectralband and the homogeneous circular polarizer distinguishing the helicity s = 1 or s = −1. In fact, themonochromator is only needed for the successive circular polarizer as, usually, the latter works only in thenarrow spectral band. Then the radiation falls onto the detector of twisted photons. The simplest detector ofpure states of twisted photons is the triangle aperture [30–32] with the lens. The diffraction pattern formedby this aperture and recorded by the CCD camera, which is located at the focal distance F from the lens, hasa rather peculiar form for the infalling radiation with definite orbital angular momentum. This diffractionpattern represents a collection of spots [30–32] and the number of spots arranged along the perimeter of thetriangle is equal to 3|l|. The size of the aperture should be slightly smaller than the size of the spot of theinfalling radiation. Such a detector of twisted photons operates in different spectral ranges. The experiments[31, 32] show that the lens in this scheme can be removed without a large distortion of the diffraction pattern.

As follows from (2) and the properties of the interference factors, the pure source of twisted photons isobtained when a narrow beam of charged particles radiates. It should be such that k⊥σ⊥ � 1, where σ⊥ is thetypical transverse size of the beam. When this condition is satisfied, the ordinary (uniform) Gaussian beam of

2

0.0 0.4 0.80

2

4

6

8

n⊥

10-21dP

/dk 0dk

⊥

-2 -1 0 1 20

2

4

6

8

m

10-23dP

/dk 0dk

⊥

s = 1, n⊥ = 3 γ

σ⊥ = 125 μm , N = 109

k0=0.132 eV, l =0k0σ3=100k⊥σ⊥=0.09

0.14 0.16 0.18 0.20 0.22 0.24 0.260

2

4

6

8

10

ℏω, eV

10-22dP

/dk 0dk

⊥

0 1 2 30

1

2

3

4

m

10-28dP

/dk 0dk

⊥

0.00 0.04 0.080

1

2

3

4

n⊥

10-28dP

/dk 0dk

⊥

s = 1, n⊥ = 0.0149, k0 = 0.137 eV

n = 2, σ⊥ = 125 μm, N = 109, χ = 1k0σ3=104, k⊥σ⊥=1.3

0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.200

1

2

3

4

ℏω, eV

10-28dP

/dk 0dk

⊥

Figure 2: The transition radiation from charged particles traversing the LiF dielectric plate of the thickness L = 100 µm. Theelectrons move along the detector axis towards the detector. The number of electrons in the beam is N = 109 and their Lorentzfactor is γ = 1673, E = 855 MeV. The beam is supposed to have a Gaussian profile with the longitudinal dimension σ3 = 150

µm (duration 0.5 ps) and the transverse size σ⊥ = 125 µm. Left panel: The transition radiation of the uniform Gaussian beamof particles. The contribution of coherent radiation is strongly suppressed. The total angular momentum per photon ` = 0.The small hump at n⊥ ≈ 0.6 is the VC radiation. Right panel: The transition radiation of the helically microbunched beam ofparticles. The fulfillment of the addition rule on the second coherent harmonic of radiation is clearly seen. The total angularmomentum per photon ` = 2.

relativistic charged particles traversing the dielectric plate radiates the twisted photons with m = 0 (cf. theplots in Figs. 2 and 3). Therefore, in the paraxial regime, the orbital angular momentum of this radiation isl = −s. The use of helically microbunched beams of particles instead of the ordinary Gaussian ones allows oneto shift the total angular momentum of radiated twisted photons. Namely, for the configuration considered,the total angular momentum of twisted photons radiated at the nc-th coherent harmonic is χnc, where χis the chirality of the helical beam (see for details [17, 18]). The shift of the orbital angular momentum inthe radiation produced by the helically microbunched electron beam evolving in the planar undulator wasobserved experimentally in [23]. The technique for production of helical beams is described in [22].

The use of ion beams in production of twisted photons by means of transition and VC radiations is moreadvantaged in comparison with the electron beams so long as the intensity of radiation of ion beams is Z2

times more intense, where Z is the charge number (see Fig. 3). The moderate Lorentz factors of ion beamslead to a small VC angle of radiation and so the paraxial approximation holds. Unfortunately, the design ofhelically microbunched beams of heavy ions is an open problem at present.

4 Conclusion

The prospective results of the proposed experiments:

1. It will be shown for the first time that the transition and VC radiations possess the orbital angularmomentum l = −s for narrow relativistic Gaussian charged particle beams.

2. It will be experimentally verified for the first time that the addition rulem = χnc holds in the transitionand VC radiations produced by narrow helically microbunched beams of charged relativistic particles.

As far as transition radiation is concerned, the same experiments can be conducted with the metallicplate instead of the dielectric one.

Acknowledgments. We are grateful to P. Karataev and D. Karlovets for the fruitful conversations duringthe symposium RREPS-2019. The work was supported by the Russian Science Foundation (project No.17-72-20013).

References

[1] V. L. Ginzburg, V. N. Tsytovich, Transition Radiation and Transition Scattering (Hilger, Bristol, 1990).

3

0.0 0.4 0.80

2

4

6

8

n⊥

10-20dP

/dk 0dk

⊥

-3-2-1 0 1 2 30

1

2

3

4

5

m10

-23dP

/dk 0dk

⊥

s = 1, n⊥ = 3 γ

σ⊥ = 1mm , N = 109

k0=0.132 eV, l =0k0σ3=100k⊥σ⊥=0.69

0.14 0.16 0.18 0.20 0.22 0.24 0.260

1

2

3

4

5

6

7

ℏω, eV

10-22dP

/dk 0dk

⊥

-2 -1 0 1 20.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

m

10-20dP

/dk 0dk

⊥

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

1.2

n⊥

10-21dP

/dk 0dk

⊥

s = 1, n⊥ = 0.157, σ⊥ = 3 μm , N = 64000k0=0.138 eV, l =0, k0σ3=20944, k⊥σ⊥=0.1

0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.260.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

ℏω, eV

10-20dP

/dk 0dk

⊥

Figure 3: Left panel: The same as on the left panel in Fig. 2 but for σ⊥ = 1 mm. The distribution of radiated twisted photonswith respect to m spreads over a wider interval. The total angular momentum per photon ` = 0. Right panel: The VC radiationfrom the bare ions of 238

92 U traversing the LiF dielectric plate of the thickness L = 100 µm with the Lorentz factor γ = 2, thekinetic energy T = 1 GeV per nucleon. The bunch is supposed to have a Gaussian profile with the longitudinal dimension σ3 = 3

cm (the duration 100 ns) and the transverse size σ⊥ = 3 µm. The number of particles in the bunch N = 6.4 × 103. The VCradiation is concentrated at the angle n⊥ = 0.157. The total angular momentum per photon ` = 0.

[2] A. P. Potylitsyn, M. I. Ryazanov, M. N. Strikhanov, A. A. Tishchenko, Radiation from Relativis-tic Particles, in Diffraction Radiation from Relativistic Particles. Springer Tracts in Modern Physics(Springer, Berlin, 2010), Vol. 239, p. 1.

[3] V. A. Bazylev et al., X-ray Čerenkov radiation. Theory and experiment, Zh. Eksp. Teor. Fiz. 81, 1664(1981) [Sov. Phys. JETP 54, 884 (1982)].

[4] R. Kieffer et al., Direct observation of incoherent Cherenkov diffraction radiation in the visible range,Phys. Rev. Let. 121, 054802 (2018).

[5] V. M. Arutyunyan, S. G. Oganesyan, The stimulated Cherenkov effect, Phys. Usp. 37, 1005 (1994).

[6] A. Curcio et al., Beam-based sub-THz source at the CERN linac electron accelerator for researchfacility, Phys. Rev. Accel. Beams 22, 020402 (2019).

[7] M. Shevelev, A. Konkov, A. Aryshev, Soft-x-ray Cherenkov radiation generated by a charged particlemoving near a finite-size screen, Phys. Rev. A 92, 053851 (2015).

[8] A. S. Konkov, A. S. Gogolev, A. P. Potylitsyn, X-ray Cherenkov radiation as a source for relativisticcharged particle beam diagnostics, in Proceedings of IBIC-2013 (Oxford, UK, 2013) p. 910.

[9] L. C. L. Yuan, C. L. Wang, S. Prunster, X-ray transition radiation applied to the detection of superhigh-energy particles, Phys. Rev. Let. 23, 496 (1969).

[10] A. A. Savchenko et al., X-Ray Transition Radiation from a Polypropylene Radiator: Experiment andGeant4, Physics of Atomic Nuclei 81, 1618 (2018).

[11] E. Hemsing et al., Experimental observation of helical microbunching of a relativistic electron beam,Appl. Phys. Lett. 100, 091110 (2012).

[12] E. Hemsing, J. B. Rosenzweig, Coherent transition radiation from a helically microbunched electronbeam, J. Appl. Phys. 105, 093101 (2009).

[13] I. P. Ivanov, V. G. Serbo, V. A. Zaytsev, Quantum calculation of the Vavilov-Cherenkov radiation bytwisted electrons, Phys. Rev. A 93, 053825 (2016).

[14] I. P. Ivanov, D. V. Karlovets, Polarization radiation of vortex electrons with large orbital angularmomentum, Phys. Rev. A 88, 043840 (2013).

4

[15] A. S. Konkov, A. P. Potylitsyn, M. S. Polonskaya, Transition radiation of electrons with orbital angularmomentum, Pis’ma v ZhETF 100, 475 (2014) [JETP Lett. 100, 421 (2014)].

[16] B. A. Knyazev, V. G. Serbo, Beams of photons with nonzero projections of orbital angular momenta:New results, Phys.-Usp. 61, 449 (2018).

[17] O. V. Bogdanov, P. O. Kazinski, G. Yu. Lazarenko, Probability of radiation of twisted photons in theisotropic dispersive medium, Phys. Rev. A 100, 043836 (2019).

[18] O. V. Bogdanov, P. O. Kazinski, G. Yu. Lazarenko, Probability of radiation of twisted photons by coldrelativistic particle bunches, arXiv:1905.07688.

[19] I. Kaminer et al., Quantum Cerenkov radiation: Spectral cutoffs and the role of spin and orbital angularmomentum, Phys. Rev. X 6, 011006 (2016).

[20] F. Seiboth et al., Refractive hard x-ray vortex phase plates, Opt. Lett. 44, 4622 (2019).

[21] O. V. Bogdanov, P. O. Kazinski, G. Yu. Lazarenko, Probability of radiation of twisted photons in theinfrared domain, Annals Phys. 406, 114 (2019).

[22] E. Hemsing et al., Beam by design: Laser manipulation of electrons in modern accelerators, Rev. Mod.Phys. 86, 897 (2014).

[23] E. Hemsing et al., Coherent optical vortices from relativistic electron beams, Nature Phys. 9, 549(2013).

[24] A. Popiolek-Masajada, J. Masajada, P. Kurzynowski, Analytical model of the optical vortex scanningmicroscope with a simple phase object, Photonics 4, 38 (2017).

[25] A. E. Willner, H. Huanget al., Optical communications using orbital angular momentum beams, Adv.Opt. Photon 7, 66 (2015).

[26] S. Gröblacher et al., Experimental quantum cryptography with qutrits, New J. Phys. 8, 75 (2006).

[27] D. G. Grier, A revolution in optical manipulation, Nature 424, 810 (2003).

[28] Y. Ma, G. Rui, B. Gu, Y. Cui, Trapping and manipulation of nanoparticles using multifocal opticalvortex metalens, Sci. Rep. 7, 14611 (2017).

[29] O. V. Bogdanov, P. O. Kazinski, Probability of radiation of twisted photons by axially symmetricbunches of particles, Eur. Phys. J. Plus 134, 586 (2019).

[30] J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, S. Chávez-Cerda, Unveiling a truncated optical latticeassociated with a triangular aperture using light’s orbital angular momentum, Phys. Rev. Lett. 105,053904 (2010).

[31] Y. Goto, T. I. Tsujimura, S. Kubo, Diffraction patterns of the millimeter wave with a helical wavefrontby a triangular aperture, Int. J. Infrared Millimeter Waves 40, 943 (2019).

[32] Y. Taira, Y. Kohmura, Measuring the topological charge of an x-ray vortex using a triangular aperture,J. Opt. 21, 045604 (2019).

5