Optical Cherenkov radiation of moderately relativistic ions in a CVD-diamond plate with frequency dispersion is considere

Alekseev, B. A.

Kubankin, A. S.

Potylitsyn, A. P.

Shevelev, M. V.

Vukolov, A. V.

DSpace at Belgorod State University

ISSN 0021-3640, JETP Letters, 2022, Vol. 115, No. 8, pp. 439–443. © Pleiades Publishing, Inc., 2022.Russian Text © The Author(s), 2022, published in Pis’ma v Zhurnal Eksperimental’noi i Teoreticheskoi Fiziki, 2022, Vol. 115, No. 8, pp. 474–478.OPTICS AND LASER PHYSICSMonochromatic Optical Cherenkov Radiation of Moderately Relativistic Ions in Radiators with Frequency DispersionA. P. Potylitsyna, *, B. A. Alekseeva, A. V. Vukolova, M. V. Sheveleva, A. A. Baldinb, c,V. V. Blekob, P. V. Karataevd, and A. S. Kubankine, fa National Research Tomsk Polytechnic University, Tomsk, 634050 Russiab Joint Institute for Nuclear Research, Dubna, Moscow region, 141980 Russiac Institute for Promising Research OMEGA, Dubna, Moscow region, 141980 Russiad Department of Physics, John Adams Institute at Royal Holloway, University of London,Egham, Surrey, TW20 0EX, United Kingdome Belgorod National Research University, Belgorod, 308015 Russiaf Lebedev Physical Institute, Russian Academy of Sciences, Moscow, 119991 Russia*e-mail: potylitsyn@tpu.ruReceived February 17, 2022; revised March 15, 2022; accepted March 15, 2022Optical Cherenkov radiation of moderately relativistic ions in a CVD-diamond plate with frequency disper-sion is considered. It has been shown that Cherenkov radiation extracted from the inclined diamond plate tovacuum at a fixed observation angle becomes monochromatic. The wavelength of the spectral line dependson the energy of an ion and on the geometry of an experiment (observation angle and plate inclination angle).An experiment has been proposed to study the monochromatization of Cherenkov radiation on the beam ofthe JINR Nuclotron for the purpose of its subsequent use in the diagnostics of ion beams. The method canbe applied to monitor the NICA ion beams energy.DOI: 10.1134/S0021364022100393Optical Cherenkov radiation (CR) [1] is widelyused in various fields such as the development of ele-mentary-particle detectors [2–5], diagnostics of mod-ern accelerator beams [6–8], infrastructure of nuclearfusion devices [9], and measurements of dose fields ofradiotherapy facilities [10].In almost all cited works, CR was characterized bythe directivity(1)where θCh is the emission angle of Cherenkov photonsin a medium through which a charged particle passesat the velocity  = βc (c is the speed of light in vac-uum), and n(λ) is the refractive index of the mediumat the wavelength λ.It is clear that Eq. (1) also determines the “thresh-old” character of the Cherenkov effect appearingwhen the velocity of the particle exceeds the phasevelocity of light in the medium:(2)The spectral composition of CR in the medium isdetermined by the condition(3)i.e., the spectrum of CR for almost all insulators iscontinuous from the ultraviolet to infrared range.Formula (1) is strictly speaking valid for aninfinitely thick emitter. In the real case, radiationformed in the finite segment of the trajectory L prop-agates in the form of a cone of Cherenkov photonswith the finite “width” [11](4)In the traditional geometry of generation of CR bythe moderately relativistic charge in the dielectricplate (see Fig. 1а), the radiation cone is extracted tovacuum if the velocity of the charge satisfies the con-dition(5)In the far-field region, where the length of the tra-jectory in which radiation is generated can beneglected and radiation can be considered as that froma point source, inequality (5) corresponds to the emis-sion angle to vacuum θvac, which is measured from themomentum of the electron (see Fig. 1а):(6)( )( )θ = λ βChcos 1/ ,nv( )β > λ1/ .n( )λ > 1;nλΔθπ θch ch2 .sinL∼β λ β β λ −2thr max= 1/ ( ) < < = 1/ ( ) 1.n nθ πvac0 < < /2.439440 POTYLITSYN et al.Fig. 1. (Color online) Schematic of the generation of Cherenkov radiation in the dielectric plate: (а) perpendicular f light of thecharge and (b) f light through the inclined plate.''θchθchTo extract the radiation of the charge with from the radiator to vacuum, its “output” surfaceshould be conical [12]. An alternative simpler way is touse an inclined dielectric plate as the emitter [13] (seeFig. 1b). In this case, only a part of the Cherenkovcone is extracted to vacuum. If the medium has fre-quency dispersion, because of refraction on the outputsurface of the emitter at a fixed θ = const (measuredfrom the normal, see Fig. 1b) in the plane specified bythe momentum of the electron and the normal to theoutput surface (coplanar geometry), the emitted radi-ation is monochromatic with the wavelength  deter-mined by the relation(7)Here, ψ is the angle between the plate and themomentum of the charged particle (see Fig. 1b).This effect of monochromatization of CR wasrecently observed in the experiment with an 855-MeVelectron beam and a 200-μm quartz plate used as theemitter [14]. The number of Cherenkov photons thatare emitted at the angle θvac by the electron from suchan inclined emitter and are observed in the far-fieldβ βmax>λ    θ λ − ψ   λ β    1= arcsin ( )sin arccos .( )nnregion was calculated in [14] within the polarized cur-rent model [15, 16]:(8)Here, α ≈ 1/137 is the fine structure constant, L is thethickness of the emitter, θ is the polar emission angleof a Cherenkov photon to vacuum, sincx = sinx/x,, and λ is the wavelength of CR.This expression, being simpler, is generally in good( )θ ε −αλ Ω ε− β − β θ2 2222 2 2cos 1= 4(1 ) cosy y zd Nd d n− β − β × π λ βλ  2231sinc z y yzZ nL L( )( ε× β β ϕ × + θ θ +22 2 2 2 2sin | | sincosy zZZε+ε θ +2cos Z)× β + β + β − θ − β β ϕ 22( 1)sin cos .z y y z y zn Z Zε λ − θ2= ( ) sinZJETP LETTERS  Vol. 115  No. 8  2022MONOCHROMATIC OPTICAL CHERENKOV RADIATION 441agreement with Pafomov’s formula given by Eq. (1)in [17].In the coordinate system  associated withthe emitter, the components of the velocity of thecharge measured in the speed of light are given by theexpressions (see Fig. 1b)and the components of the unit vector in the directionof the wave vector have the formRelation (7) directly follows from the conditionthat the function sincx in Eq. (8) is maximal, i.e., itsargument is zero:(9)In the general case (for a noncoplanar geometry of theprocess), Eq. (9) specifies a two-dimensional region ofangles  that describes the part of the Cherenkovcone emitted to vacuum:(10)The solution of Eq. (10) for  has the form(11)For the coplanar geometry with  (see Fig. 1),Eq. (7), which is a consequence of Snell’s law, can eas-ily be obtained from Eq. (11) by straightforward alge-bra.The azimuth region limiting the part of the Che-renkov cone in vacuum(12)is determined from Eq. (11) under the boundary con-ditionUnder the condition Δϕ < 1, one can obtain the esti-mate(13)In particular, for 165-MeV/nucleon ions and the565-nm Cherenkov spectral line (see below), the exactsolution of Eq. (10) gives ,whereas Eq. (13) yields an overestimated value of.We note that, when the velocity of ions increases,the angular range Δϕ is narrowed and Eq. (13) pro-vides not too large an error.The vacuum angles θvac and ϕvac defined in a moreconvenient coordinate system  with the  axisdirected along the velocity of the charge (see Fig. 1b){ , , }x y zβ = β ψ β = β ψsin , cos ,y zθ ϕ θ ϕ θ= sin sin ; = sin cos ; = cos .x y zn n n− β − β1 = 0.z y yZ nθ ϕ{ , }− β − β − β ψ ε − θ− θ ϕβ ψ21 = 1 cos sinsin cos sin = 0.z y yZ nθsinθ ψ ϕ + ψ× εβ − ψ ϕ − β − ψ ϕ2 2 2 2 2sin = (sin cos cos(1 ) 1)/ (1 ).sin sin sin sinϕ π=π + Δϕ ≤ ϕ ≤ π − Δϕθ =sin 1.Δϕβ − λ θ − ψ ψ θ − ψ2ch ch( )= 4 [1 ( )sin( )]/ sin sin 2( ).nΔϕ ≈ °= 0.764 43.8Δϕ ≈ ≈ °1.11 63.6{ ', ', '}x y z 'zJETP LETTERS  Vol. 115  No. 8  2022are related to the angles θ and ϕ through the rotationby the angle ψ:(14)(15)For the coplanar geometry , Eqs. (14) and(15) giveBecause of frequency dispersion, CR with differentwavelength is emitted to vacuum at different anglesθvac. If Δθ ≪ ΔθCh, where Δθ is the aperture of thedetector and ΔθCh given by Eq. (4) is the “natural”angular width of the Cherenkov cone emitted by thecharge from the finite trajectory L, the quasimono-chromatic spectrum of radiation should be expected.This expectation was confirmed in the experimentreported in [14].When the emitter is made from a material with ahigh refractive index , e.g., diamond with ,CR is generated by the charge moving at the velocity, i.e., at . The Lorentz factorfor the ion beam with such a velocity is.Synthetic diamond (CVD diamond) has a fre-quency dispersion described by the Sellmeier formula[18, 19](16)where λ is the wavelength in nanometers.Using Eq. (8), one can calculate the spectrum ofCherenkov photons from the moderately relativisticion beam passing through the diamond plate after themultiplication by z2, where z is the charge number ofthe ion.Figure 2 presents the dependences of the yield ofCherenkov photons with various wavelengths on theobservation angle in the geometry shown in Fig. 1а.The calculations were carried out by Eq. (8) for thevelocity of ions   passing throughthe 100-μm diamond target. According to Fig. 2, ≈ 1.5° = 0.026 at the wavelength λ =0.4 μm. The velocity at the wavelength λ = 0.5 μm is, corresponding to the Lorentz factor, which is only 1.7% larger than thethreshold value .To extract CR from more strongly relativistic ionsto vacuum, it is necessary to use the inclined geome-try. Figure 3 shows the spectra of CR emitted at theangle θvac = 79° from ions with energies less than0.2 GeV/nucleon passing through the 100-μm dia-mond target inclined at an angle of 17°.Figure 4 shows the spectra of CR for the samegeometry obtained after the integration of Eq. (8) overθ θ ψ + θ ϕ ψvaccos = cos cos sin cos sin ,θ ϕϕθ ψ − θ ϕ ψvacsin sintan = .sin cos sin cos sinϕ π( = )θ = θ + ψ ϕ =vac vac, 0.λ( )n > 2.4nv > /2.4c β > 0.417γ ≥ − β ≥21/ 1 1.09ε λ λ + λ λ −2 2 2 2( ) = ( ) = 1 4.658 /( 112.5 ),nβ = 0.4166 γ( = 1.1)Δθ ≈vac FWHMβmax = 0.45116γmax = 1.119γ − β2thr thr= 1/ 1442 POTYLITSYN et al.Fig. 2. (Color online) Angular dependence of the intensityof Cherenkov radiation of the charge passing at the velocity  through the 100-μm perpendiculardiamond plate .β = 0.4166 γ( = 1.1)ψ( = 0)Fig. 3. (Color online) Spectrum of monochromatic Che-renkov radiation of moderately relativistic ions passingthrough the 100-μm inclined diamond plate (ψ = 17°) atthe observation angle θvac = 79° (θ = 62°).Fig. 4. (Color online) Spectrum of Cherenkov radiation ofions under the same condition as in Fig. 3 at detection bya spectrometer with finite apertures (solid lines) Δθap =Δϕap = 0.76° and (dashed lines) Δθap = 0.76° and Δϕap =1.52°.the solid angle (θvac ± Δθap = 79° ± 0.76°, ϕ ± Δϕap =±0.76°), where (Δθap, Δϕap) is the angular aperture ofthe detector corresponding to the -mm collima-tor placed at a distance of 150 mm from the emitter, aswell as for the aperture (Δθap = 0.76°, Δϕap = 1.52°).The broadening of the line is due primarily to “cap-ture” in the polar angle Δθap, whereas broadeningcaused by finite capture in the azimuth is muchsmaller; therefore, a collimator with an increased azi-muth aperture can be used to increase the number ofcollected events.The yield of Cherenkov photons from165-MeV/nucleon ions into the aperture (Δθap =0.76°, Δϕap = 1.52°) in the geometry under consider-ation is estimated as  photons/ion,where z is the charge number of the ion.To conclude, we note that monochromatic opticalCR from a moderately relativistic ion beam propagat-ing in the CVD-diamond emitter can be detected inthe appropriately chosen geometry of the experiment.Since the spectrum of this monochromatic optical CR“carries” information on the energy of the ion beam,this technique can be used as a new approach to spec-trometry.The average energy of the ion beam was measuredin [6] using the photometric method based on thedirectivity of optical CR, but the achieved accuracywas low. The proposed method based on spectral mea-surements is apparently more accurate.The reported results can provide a foundation for anew method of measurement of the energy of weaklyrelativistic ions. The reported calculations will beverified at the Laboratory of High-Energy Physics,Joint Institute for Nuclear Research, where a systemwill be developed to measure the energy dispersion ofions from carbon to gold with energies up to4 GeV/nucleon. The corresponding experiments areplanned on the beams extracted from the NICA at theMARUSYA setup in the SPD test region, where thenecessary infrastructure exists [20, 21].FUNDINGThis work was supported by the Government of the Rus-sian Federation (project no. Prioritet-2030-NIP/IZ-005-0000-2030, Program of Strategic Academic Leadership“Prioritet 2030”).CONFLICT OF INTERESTThe authors declare that they have no conflicts of interest.REFERENCES1. P. A. Cherenkov, Usp. Fiz. Nauk 68, 373 (1959).×4 4Δ 2= 0.00174N zJETP LETTERS  Vol. 115  No. 8  2022MONOCHROMATIC OPTICAL CHERENKOV RADIATION 4432. D. F. Anderson, M. Kobayash, Y. Yoshimura, andC. L. Woody, Nucl. Instrum. Methods Phys. Res.,Sect. A 290, 385 (1990).3. Yu. Barnyakov, M. Yu. Barnyakova, J. B. Bähr, et al.,Nucl. Instrum. Methods Phys. Res., Sect. A 553, 125(2005).4. K. Fohl, Nucl. Instrum. Methods Phys. Res., Sect. A595, 88 (2008).5. P. Krizan, Nucl. Instrum. Methods Phys. Res., Sect. A876, 272 (2017).6. J. Ruzicka, S. Saro, V. P. Zrelov, et al., Nucl. Instrum.Methods Phys. Res., Sect. A 369, 23 (1996).7. O. V. Bogdanov, E. I. Fiks, and Yu. L. Pivovarov,J. Exp. Theor. Phys. 115, 392 (2012).8. H. Liu, G.-Q. Liao, Y.-H. Zhang, B.-J. Zhu, Z. Zhang,Y. T. Li, G. C. Scott, D. R. Rusby, C. Armstrong,E. Zemaityte, L. C. Carroll, S. Astbury, P. Bradford,N. C. Woolsey, and P. McKenna, Rev. Sci. Instrum.89, 083302 (2018).9. M. Shaikh, A. D. Lad, D. Sarkar, K. Jana, G. RavindraKumar, and P. P. Rajeev, Rev. 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Monochromatic optical cherenkov radiation of moderately relativistic lons in radiators with frequency dispersion

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Monochromatic optical cherenkov radiation of moderately relativistic lons in radiators with frequency dispersion

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