Aberrations in electromagnetic plasma lenses proposed for intense large-aperture ion beam focusing

It is shown, that in electromagnetic plasma lenses moment aberrations obey the conservation law of the moment of the ion generalized momentum; they disappear when an ion source is placed in a zero magnetic field. Geometrical aberrations can be removed by choice of the optimal electric field intensit...

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Datum:	2003
Hauptverfasser:	Butenko, V.I., Ivanov, B.I.
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Sprache:	English
Veröffentlicht:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2003
Schriftenreihe:	Вопросы атомной науки и техники
Schlagworte:	Plasma electronics
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spelling	irk-123456789-1105392017-01-05T03:03:50Z Aberrations in electromagnetic plasma lenses proposed for intense large-aperture ion beam focusing Butenko, V.I. Ivanov, B.I. Plasma electronics It is shown, that in electromagnetic plasma lenses moment aberrations obey the conservation law of the moment of the ion generalized momentum; they disappear when an ion source is placed in a zero magnetic field. Geometrical aberrations can be removed by choice of the optimal electric field intensity distribution along the radius that have been calculated. As a result, the ion current density and beam compression can achieve in the focus a value up to 10³ A/cm², and 10⁵, accordingly. Then influence of discrete distribution of the external (boundary) electrical potential upon ion focusing was studied, and an example of two-lens achromatic system have been investigated as well. Досліджується плазмова лінза Морозова, у котрої магнітні поверхні є еквіпотенціалями електричного поля. Магнітне поле створюється центральним струмовим витком і двома бічними витками, включеними назустріч центральному. У роботі приведені результати комп'ютерного моделювання фокусування іонів з урахуванням їх поздовжнього, радіального й азимутального руху. Розглянуто засоби усунення моментних, геометричних і хроматичних аберацій. Проводиться оптимізація магнітного й електричного полів по величині і розподілу в просторі. Промодельоване вплив дискретного розподілу потенціалу на фокусування іонів і розглянуті пов'язані з цим аберації. Розглянуто комп'ютерну модель двухлінзової ахроматичної системи. Исследуется плазменная линза Морозова, в которой магнитные поверхности являются эквипотенциалями электрического поля. Магнитное поле создается центральным токовым витком и двумя боковыми витками, включенными навстречу центральному. В работе приведены результаты компьютерного моделирования фокусировки ионов с учетом их продольного, радиального и азимутального движения. Рассмотрены способы устранения моментных, геометрических и хроматических аберраций. Проводится оптимизация магнитного и электрического полей по величине и распределению в пространстве. Промоделировано влияние дискретного распределения потенциала на фокусировку ионов и рассмотрены связанные с этим аберрации. Рассмотрена компьютерная модель двухлинзовой ахроматической системы. 2003 Article Aberrations in electromagnetic plasma lenses proposed for intense large-aperture ion beam focusing / V.I. Butenko, B.I. Ivanov // Вопросы атомной науки и техники. — 2003. — № 1. — С. 121-124. — Бібліогр.: 11 назв. — англ. 1562-6016 PACS: 52.40.Mj http://dspace.nbuv.gov.ua/handle/123456789/110539 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Plasma electronics Plasma electronics
spellingShingle	Plasma electronics Plasma electronics Butenko, V.I. Ivanov, B.I. Aberrations in electromagnetic plasma lenses proposed for intense large-aperture ion beam focusing Вопросы атомной науки и техники
description	It is shown, that in electromagnetic plasma lenses moment aberrations obey the conservation law of the moment of the ion generalized momentum; they disappear when an ion source is placed in a zero magnetic field. Geometrical aberrations can be removed by choice of the optimal electric field intensity distribution along the radius that have been calculated. As a result, the ion current density and beam compression can achieve in the focus a value up to 10³ A/cm², and 10⁵, accordingly. Then influence of discrete distribution of the external (boundary) electrical potential upon ion focusing was studied, and an example of two-lens achromatic system have been investigated as well.
format	Article
author	Butenko, V.I. Ivanov, B.I.
author_facet	Butenko, V.I. Ivanov, B.I.
author_sort	Butenko, V.I.
title	Aberrations in electromagnetic plasma lenses proposed for intense large-aperture ion beam focusing
title_short	Aberrations in electromagnetic plasma lenses proposed for intense large-aperture ion beam focusing
title_full	Aberrations in electromagnetic plasma lenses proposed for intense large-aperture ion beam focusing
title_fullStr	Aberrations in electromagnetic plasma lenses proposed for intense large-aperture ion beam focusing
title_full_unstemmed	Aberrations in electromagnetic plasma lenses proposed for intense large-aperture ion beam focusing
title_sort	aberrations in electromagnetic plasma lenses proposed for intense large-aperture ion beam focusing
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
publishDate	2003
topic_facet	Plasma electronics
url	http://dspace.nbuv.gov.ua/handle/123456789/110539
citation_txt	Aberrations in electromagnetic plasma lenses proposed for intense large-aperture ion beam focusing / V.I. Butenko, B.I. Ivanov // Вопросы атомной науки и техники. — 2003. — № 1. — С. 121-124. — Бібліогр.: 11 назв. — англ.
series	Вопросы атомной науки и техники
work_keys_str_mv	AT butenkovi aberrationsinelectromagneticplasmalensesproposedforintenselargeapertureionbeamfocusing AT ivanovbi aberrationsinelectromagneticplasmalensesproposedforintenselargeapertureionbeamfocusing
first_indexed	2025-07-08T00:43:19Z
last_indexed	2025-07-08T00:43:19Z
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fulltext	ABERRATIONS IN ELECTROMAGNETIC PLASMA LENSES PROPOSED FOR INTENSE LARGE-APERTURE ION BEAM FOCUSING V.I. Butenko, B.I. Ivanov NSC KIPT, Kharkov, 61108, Ukraine (E-mail: butenko@kipt.kharkov.ua) It is shown, that in electromagnetic plasma lenses moment aberrations obey the conservation law of the moment of the ion generalized momentum; they disappear when an ion source is placed in a zero magnetic field. Geometrical aber- rations can be removed by choice of the optimal electric field intensity distribution along the radius that have been cal- culated. As a result, the ion current density and beam compression can achieve in the focus a value up to 103 A/cm2 , and 105 , accordingly. Then influence of discrete distribution of the external (boundary) electrical potential upon ion focus- ing was studied, and an example of two-lens achromatic system have been investigated as well. PACS: 52.40.Mj 1. INTRODUCTION The problems of intense ion beam focusing are important for the controlled fusion, nuclear physics, beam technologies, etc. [,]. The essential feature of intense ion beams is that they should be charge compensated during the focusing to prevent their destruction. In this case, the application of plasmaoptic focusing systems which development is initiated by A.I. Morozov and co-workers [-, 8] and recently mainly developed by A.A. Goncharov group [–] is expedient. In the plasma lens of Morozov type the magnetic sur- faces are the equipotentials of the electrical field. In prac- tice the electrical potentials are entered in plasma by a discrete manner, using "basic" ring electrodes. The exper- imental researches [–7] confirm the theoretical model [-, 8] but some problems remain, in particular, aberrations and methods of their elimination. For this aim, computer modeling of plasma-optic focusing devices is considered in the proposed paper. 2. THE PROBLEM DEFINITION, MAIN EQUATIONS Following [], we enter function of a magnetic flow: ( ) ( )zrrAzr ,, ϕ=Ψ , (1) where Aϕ is the azimuth component of vector potential. The equation of magnetic surfaces have the form: constrA =ϕ . (2) The connection between Ψ(r, z) and magnetic flow Ν passing through cross-section (r,z) is (see []): Ψ==⋅=Ν ∫ ππ ϕ 2),(2 zrrAdSnH S  (3) In a Morozov lens equipotentiality of magnetic sur- faces is determined by a relation [-] ( ) ( )Ψ=Φ Fzr , (4) where Φ is a potential of an electric field. Let's express components of an electric and magnetic fields through Ψ: zr B r ∂ Ψ∂−= 1 rr B z ∂ Ψ∂= 1 (5) zr B d dFr rd dF r E Ψ −= ∂ Ψ∂ Ψ −= ∂ Φ∂−= (6) rz B d dFr zd dF z E Ψ = ∂ Ψ∂ Ψ −= ∂ Φ∂−= . (7) Substituting these expressions in the equations of par- ticle motion in the form of Newton, with account of Lorentz force, we have: r V d dFrV crmr q dt dVr 21 ϕ ϕ +     Ψ − ∂ Ψ∂= (8) r VV r V z V mcr q dt dV r rz ϕϕ −     ∂ Ψ∂+ ∂ Ψ∂−= (9)      − Ψ∂ Ψ∂−= ϕV cd dFr zmr e dt dV z 1 (10) In experimental works [–] the configuration of a mag- netic field with counter connection of solenoids is pro- posed, that allows to locate basic electrodes near to the central plane of a lens. In the given work the lens is simu- lated by three coils with opposite currents. The central coil is located at z = 0, lateral ones at z=±5.0 cm. In the central part of the lens (-2.8 cm < z < 2.8 cm) the poten- tials of basic ring electrodes are applied to magnetic sur- faces, and the magnetic surfaces at the left and right of the central area are considered as grounded. The magnetic field of a ring current Jn (at radius of a coil aс and coordinate l on an axis z) is described by az- imuth component of vector potential:     −−= )()()21(4 2 , nn nc n n n kEkKk r a ck JAϕ (11) 22 2 )()( 4 nc c n lzra rak −++= where c is the light velocity, K and E are the complete el- liptic integrals of 1-st and 2 kind, n is a ring number. The sum field is ∑= n nAA ,ϕϕ Further we used the case of Jcenter=-1.5Jside. The equipoten- tial surfaces topography is presented in Fig. 1. Fig. 1. (1-central coil, 2-side coils, 3-electrodes) 3. MOMENT ABERRATIONS Computer modeling and analysis of ion trajectories, carried out in [], show that along the calculated ion trajec- tory, the conservation law for the ion moment of the gen- eralized momentum, and the conservation law for the total ion energy are both satisfied to within five significant decimal digits. Problems of Atomic Science and Technology. 2003. № 1. Series: Plasma Physics (9). P. 121-124 121 mailto:butenko@kipt.kharkov.ua If we are interested not by trajectories but only condi- tions at which moment aberrations are negligible, they can be obtained directly from the initial equations. To this effect, the equation of azimuth motion (9) after some transformations can present as: dt d c q dt rVd m Ψ−=ϕ )( , (12) whence follows the conservation law for the moment Μϕ of the generalized momentum of a particle Pϕ: ( ) constcqAmVrrP =+==Μ ϕϕ /ϕϕ . (13) With account of (3), it is follows from (12), that for Morozov axysimmetric static lenses the Busch theorem is valid as well as for vacuum magnetic ones: 000 22 Ν+=Ν+ ϕϕ mc qVr mc qrV ππ , (14) where Ν corresponds to a magnetic flow which is passing through a circle of radius r, on which at the given moment of time there is a particle. Functions Ν(r,z) and Ψ(r, z) can be measured with help of the electromagnetic induc- tion law and suitable diagnostics. As follows from (13), (14), in order for a parallel par- ticle beam to be focused into the focal point of a lens, it suffices that the initial azimuth velocities of the particles be zero and that the magnetic field vanish in the injection region and in the focal plane. The lenses used in of [5–7] satisfy these conditions. Further we can consider mini- mization of geometrical aberrations. 4. GEOMETRICAL ABERRATIONS AT CON- TINUOUS DISTRIBUTION OF POTENTIAL 4.1. PARALLEL BEAM FOCUSING Initial conditions of particle injection in a lens are: at t = 0 - Vz = V0, Vr = Vϕ = 0, z = zi, r = ri, (15) where zi is the injector coordinate, ri is the radius at which an ion is injected; ri is varied from zero to a value some- what smaller than the radius R of the basic electrodes, which, in turn, is smaller than the radius ac of the current- carrying coils. The calculations of ions trajectory were made at pa- rameters comparable to the Kiev lens [–]: energy of pro- tons W=20 keV, initial beam radius r0=3.5 cm, beam is parallel, radii of current rings ra=6.5 cm, coordinate of protons injector zi=-30 cm, current of protons 1 A. The distribution of potential 2)0,( rr ∝Φ is applied to minimization geometrical aberrations in [5]. But, as it is shown in [10], at focusing large-aperture beams it does not give satisfactory results. In works [-] the distribution Φ∝Ψ is considered also. In Fig. 2 at distribution Φ = KrA ϕ, where K = 4.8⋅10-4 V/Gs cm2 the trajectories of protons are submitted. Thus maximal density of a current of pro- tons reaches only 5 A/cm2 at zf =17.5 cm. r, cm 1.510.50 j, A /c m ^2 4 3 2 1 0 Fig. 2. For reduction geometrical aberrations we will set the optimized distribution of potential on radius (in GS) as polynomial in which denominate factors are picked up such that the focusing will be best: Φ(r,0) = 0.75 r 2 – 0.0143 r 4 + 0.00014 r 6 (16) For this case the trajectories of protons and distribution of current density of protons on radius in focal plane are cal- culated. As a result of optimization the current density has increased up to 9 kA/cm2 (see Fig. 3), and compression factor of a beam (i.e., ratio between initial current density to final) - up to 3.6·105. In work [] best focusing (with compression factor about 30 at a total ion current 0.24 А) was experimentally obtained at distribution of potential proportional to intensity of a longitudinal magnetic field on the lens axis. r, cm 0.020.0150.010.005 j, A /c m ^2 8,000 6,000 4,000 2,000 0 Fig. 3. For such distribution the trajectories of protons and cur- rent density distribution on radius in focal plane have been modeling by us. In this case the current density in focus Jmax=4.2 A/cm2 is obtained at average beam radius 0.5 cm and compression factor about 50. These values considerably concede the calculated results, see above. 4.2. FOCUSING OF IONS EMERGING FROM POINT SOURCE As an example, it have been simulated the focusing of protons emerging from a point source at zi = -30 cm. Beam divergence is 0.1 rad, so at the lens center the beam radius not exceeds 3.0 cm. Optimum distribution of po- tential is Ф (r, 0) =1.20r2-0.0180r4+0.000228r6. The tra- jectories of protons and current density in focal plane zf = 27.18 cm are designed up to 75 кА/cm2 at the spot aver- age radius 0.002 cm (see Fig.4). So, in this case spherical aberrations are negligible (see also [2]). Fig. 4. 122 5. GEOMETRICAL ABERRATIONS AT DISCRETE DISTRIBUTION OF FOCUSING POTENTIALS Till now in calculations we accepted continuous distri- bution of potential on coordinates. As against it, in experi- ments [–] potentials in plasma are entered with the help of finite number (5 or 9) cylindrical electrodes. Let's consider a case of 9-electrode lens, that corre- sponds to the preset of 6 discrete values of potential on half of lens (6-th potential corresponds to zero potential on an axis). If these 6 values to set on the optimized curve corresponded to Fig. 3, and then to smooth by the splines, the satisfactory concurrence of these curves will turn out. However in experiments the electrodes of finite length specifying step distribution of potential were applied which in plasma smoothed out. Characteristics of this smoothing are not yet investigated experimentally. In cal- culations this smoothing was simulated by B-splines of 3- rd order, and the degree of smoothing was defined by a ratio of an effective length of electrodes and an effective gap between them. At the effective gap between elec- trodes of 2 mm (that is close to the real experimental val- ue) the distribution with smoothed stairs is obtained (Fig. 5). The trajectories of protons which correspond to this case are submitted on Fig. 6 and current density in the fo- cus region on Fig. 7. Current density (about 0.1 A/cm2) and half width of a focal spot (about 1 cm) under the or- der coincide with the experimental results [–]. In this case because of step distribution of potential in plasma bad fo- cusing of a beam takes place. The current density (∼ 0.3 A/cm2) and the focal spot half length (∼1 cm, see Fig. 7) are correspond to the experimental results by the order of the values. More details for this problem see in [11]. z, cm 2.521.510.50 Φ , G S 8 7 6 5 4 3 2 1 Fig. 5. z, cm 26242220181614121086420 r, cm 6 4 2 0 Fig. 6. r, cm 21.510.50 j, A /c m ^2 0.3 0.2 0.1 0 Fig. 7. 6. CHROMATIC ABERRATIONS Accordingly to [4], in Fig. 8 it is presented the system of two Morozov lenses for study and removing chromatic aberrations at a tube ion beam focusing. (It is used the rule: before joining, it's necessary to separate). z, cm 9080706050403020100-10 r, c m 6 4 2 0 Fig. 8. The first lens parameters: current circle radii 6.5 cm, center circle current 30 kA, electrodes radii 5 cm, maxi- mal potential 3 kV, central circle coordinate z = 0, half length of the lens is 5 cm. The second lens parameters: current circle radii 4 cm, center circle current 30 kA, elec- trodes radii 2 cm, maximal potential 1.5 kV, central circle coordinate z = 3.5 cm, half length of the lens is 5 cm. In- jector coordinate z = -70 cm, initial radius of the tube pro- ton beam 3.5 cm. In Fig. 9 it is presented the potential dis- tribution in 2-nd lens that was selected from the condition of minimum dependence of focal distance on proton ener- gy in the range 16–21 keV. In this case it was used the ap- proximation of potential distribution by B-splanes of third order in the left half length of the electrodes system with control points amount n = 11. Trajectories of protons at fo- cus region are shown on Fig. 10. The authors would like to thank A.A. Goncharov and I.M. Protsenko for consultations on their lenses. z, cm 0.80.70.60.50.40.30.20.10 U , G S 0.5 0.4 0.3 0.2 0.1 0 Fig. 9. z, cm 848382818079 r, cm 0.08 0.06 0.04 0.02 Fig. 10. REFERENCES 1.M.D.Gabovich, N.V.Pleshivtsev, N.N.Semashko, Ion and Atom Beams for Controlled Thermonuclear Fusion and Technology. M.: Energoatomizdat, 1986. 2.A.I.Morozov // Encyclopedia of Low-temperature Plas- mas /Ed. V.E. Fortov, Entry Volume, Book III,M.: Nau- ka, 2000, p.435-443. 3.A.I.Morozov // DAN SSSR. 1965. V.163. P.1363. 123 4.A.I.Morozov, S.V.Lebedev// Plasma Theory Problems, M.: Atomizdat, 1974, V. 8, P.247. 5.A.A.Goncharov, A.N.Dobrovolskii, A.N.Kotsarenko e.a. // Plasma Physics. 1994. V.20. p.499. 6.A.Goncharov, A.Dobrovolskii, I.Litovko e.a. // IEEE Trans. Plasma Sci. 1997. V. 25. p.709. 7.A.A.Goncharov, I.M.Protsenko, G.Yu. Yushkov, I.G. Brown // Appl. Phys. Lett. 1999. V.75, p.911. 8.A.I.Morozov, L.S.Solov'ev // Plasma Theory Problems, M.: Gosatomizdat, 1963, V. 2, p. 3. 9.W.R.Smithe, Electrostatics and Electrodynamics, Ch. 7, Oxford, 1950. 10.V.I.Butenko, B.I.Ivanov // Plasma Physics. 2002. V.28. No.7. p.651. 11.V.I.Butenko // Problems of Atomic Science and Tech- nology, 2001. No.5. p.74. АБЕРАЦІЇ У ЕЛЕКТРОМАГНІТНІЙ ПЛАЗМОВІЙ ЛІНЗІ, ЗАПРОПОНОВАНОЇ ДЛЯ ФОКУСУВАННЯ ІНТЕНСИВНИХ ШИРОКОАПЕРТУРНИХ ІОННИХ ПУЧКІВ В.І. Бутенко, Б.І. Іванов Досліджується плазмова лінза Морозова, у котрої магнітні поверхні є еквіпотенціалями електричного поля. Магнітне поле створюється центральним струмовим витком і двома бічними витками, включеними назустріч центральному. У роботі приведені результати комп'ютерного моделювання фокусування іонів з урахуванням їх поздовжнього, радіального й азимутального руху. Розглянуто засоби усунення моментних, геометричних і хроматичних аберацій. Проводиться оптимізація магнітного й електричного полів по величині і розподілу в просторі. Промодельоване вплив дискретного розподілу потенціалу на фокусування іонів і розглянуті пов'язані з цим аберації. Розглянуто комп'ютерну модель двухлінзової ахроматичної системи. АБЕРРАЦИИ В ЭЛЕКТРОМАГНИТНОЙ ПЛАЗМЕННОЙ ЛИНЗЕ, ПРЕДЛОЖЕННОЙ ДЛЯ ФОКУСИ- РОВКИ ИНТЕНСИВНЫХ ШИРОКОАПЕРТУРНЫХ ИОННЫХ ПУЧКОВ В.И. Бутенко, Б.И. Иванов Исследуется плазменная линза Морозова, в которой магнитные поверхности являются эквипотенциалями электрического поля. Магнитное поле создается центральным токовым витком и двумя боковыми витками, включенными навстречу центральному. В работе приведены результаты компьютерного моделирования фоку- сировки ионов с учетом их продольного, радиального и азимутального движения. Рассмотрены способы устра- нения моментных, геометрических и хроматических аберраций. Проводится оптимизация магнитного и элек- трического полей по величине и распределению в пространстве. Промоделировано влияние дискретного рас- пределения потенциала на фокусировку ионов и рассмотрены связанные с этим аберрации. Рассмотрена компьютерная модель двухлинзовой ахроматической системы. 124 ABERRATIONS IN ELECTROMAGNETIC PLASMA LENSES PROPOSED FOR INTENSE LARGE-APERTURE ION BEAM FOCUSING 1. Introduction 2. The problem definition, main equations 3. Moment aberrations 4. Geometrical aberrations at continuous distribution of potential 4.1. Parallel beam focusing 4.2. Focusing of ions emerging from point source 5. Geometrical aberrations at discrete distribution of focusing potentials 6. Chromatic aberrations References АБЕРАЦІЇ У ЕЛЕКТРОМАГНІТНІЙ ПЛАЗМОВІЙ ЛІНЗІ, ЗАПРОПОНОВАНОЇ ДЛЯ ФОКУСУВАННЯ ІНТЕНСИВНИХ ШИРОКОАПЕРТУРНИХ ІОННИХ ПУЧКІВ В.І. Бутенко, Б.І. Іванов АБЕРРАЦИИ В ЭЛЕКТРОМАГНИТНОЙ ПЛАЗМЕННОЙ ЛИНЗЕ, ПРЕДЛОЖЕННОЙ ДЛЯ ФОКУСИРОВКИ ИНТЕНСИВНЫХ ШИРОКОАПЕРТУРНЫХ ИОННЫХ ПУЧКОВ В.И. Бутенко, Б.И. Иванов

Aberrations in electromagnetic plasma lenses proposed for intense large-aperture ion beam focusing

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