تصميم نظري لعدسة كهروستاتيكية احادية الجهد معجلة ومبطئة تعمل تحت ظروف التكبير المختلفة Theoretical Design of Electrostatic Lens Accelerating and Decelerating Operated Under Different Magnification Conditions Bushra H. Hussein Dept. of Physics/ College of Education for Pure Science( Ibn-AL Haitham)/ University of Baghdad Receivd in: 5 November 2012, Accepted in: 1 April 2013 Abstract In this paper, design computationl investigation in the field of charged-particle optics with the aid of numerical analysis methods under the absence of space charge effects. The work has been concentrated on the design of three-electrode einzel electrostatic lens accelerating and decelerating operated under different magnification conditions. The potential field distribution of lens has been represented by exponential function. The paraxial-ray equation has been solved for the proposed field to determine the trajectory of charged-particles traversing in the lens. From The axial potential distribution and its first and second derivatives, the optical properties such as the focal length and the spherical and chromatic aberration coefficients have been computed, the electrode shape of lens has been determined by using SIMION computer program . In this research , design electrostatic einzel lens three-electrode accelerating and decelerating L=2mm, 20mm operated under different magnification conditions (zero, infinite, finite). The electrode shape of the electrostatic lens was then determined from the solution of the Laplace's equation. The results showed low values of spherical and chromatic aberrations which are considered as good criteria for good design. Keywords. Electron Optics, Einzel Lens, Accelerating and Decelerating , Aberrations, SIMION computer program 60 | Physics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@I@ÖÜ»€a2@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (2) 2014 Introduction Electrostatic lenses are the principal components in the overwhelming majority of electron optical devices. Both electrostatic and magnetic lenses are used to focus charged particles. Electrostatic lenses are used with applications in many areas of science and technology. For example, electrostatic devices are used to shape ion microprobe beams for secondary ion mass spectroscopic studies. More recently, with the aid of electrostatic lenses ion probes are employed in ion implantation to change the local properties of semiconductors[1] . Electrostatic cylinder lenses are widely used to control beams of charged particles with different energies and directions in several fields, especially in electron spectroscopy were designed by using the SIMION programs[2] . The development of computer programs was accompanied by the need of more application for such ion and electron instruments in various kinds [3-4] . In the present work ,we have a design of accelerating and decelerating electrostatic einzel lens for different applications in electron optical instruments. The einzel lens has the same constant potential at both the object and image sides i.e. the charged- particle energy remains unchanged [5]. The design and implementation of a purely electrostatic deceleration lens are used to obtain beams of highly charged ions at very low energies is presented [6]. Theory The present computational investigation is aimed to design electrostatic lens using analytic expression (exponential model) that would describe the axial potential distribution of einzel lens with electron optically acceptable aberrations. The following expression is suggested to represent the potential distribution along the optical axis of an einzel lens[7]: ( )[ ]2exp11)( zDUzU ±+= (1) Where U1 is the voltage applied on the outer electrodes, D is a constant affecting the value of voltage applied on the central electrodes, The negative (-) sign denotes for an accelerating mode of operation .And the positive (+) sign is denoted for the decelerating mode. The paraxial-ray equation in rotationally symmetric field is given by[7] 0 )(4 )( )(2 )( 2 2 = ′′ + ∂ ∂′ + ∂ ∂ r zU zU z r zU zU z r (2) SIMION is an electrostatic lens analysis and design program that is capable of modeling charged particle optics problems with electrostatic and/or magnetic potential arrays. This program is used to simulate electrostatic and static magnetic device for accelerating, transporting and otherwise manipulating beams of charged particles . For the purposes of this article, only electrostatic fields were modeled. The shape of the electrodes for electrostatic lens is determined from the solution of Laplace , s equation [7]. 4 )()(),( 2r zUzUzrU ′′−= (3) The spherical aberration is one of the most important geometrical aberrations and can be defined as follows: the beam passing within the lens area at a considerable distance from the axis are more (or less) refracted than the paraxial beams so that they intersect closer to (or farther form) the image plan .The spherical and chromatic aberrations are dominant in an electrostatic lens [8]. The coefficients of spherical aberration and chromatic aberration referred to the object space Cso and Cco expressed in the following form [9]. 61 | Physics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@I@ÖÜ»€a2@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (2) 2014 3 4 42 4 2/1 )( )( 3 14 )( )( )( 24 5 )( )( 4 5 16        ′ +                   ′ +      ′′ ′ = ∫ − Ο zU zU zr zU zU zU zU r U Cs zi zoο } dz (z)U)()( )( )( 2 3 )()( 1/222 2 3 zrzr zU zU zrzr ′      ′ −′ (4) dzzUr zU zU zrzr zU zU r zU Cc zi zo )( )(4 )( )()( )( )( 2 1)( 2/12 2 2/1 − Ο ∫       ′′ +′ ′ ′ = ο ο (5) In the image space , the spherical aberration coefficient Csi and chromatic aberration coefficient Cci is expressed in a similar form of equations (3) and (4) , Where U-1/2(zo) and respectively. )(2/1 izU and 4 ir′ are replaced by 4 οr′ In general, the focal length f of a lens at various values voltage ratio is determined from the gradient of the beam trajectory of the point where the charged particles enter or emerge from the lens field region. The object and image-side focal lengths fo and fi respectively have been computed from the following equation [9]:. (6) )(/)( oi zrzrf ′=Ο (7) The magnifications are calculated form following equation [10]. Ο = r r M i A computer program for computing the beam trajectory , the optical properties[11] and electrode shape by using SIMION computer program [12]. Results and Discussion The axial field distribution U(z) given in equation (1) for an einzel lens is shown in figure (1a) with its first derivative )(ZU ′ . The axial field distribution of an einzel lens whose central electrode is at higher voltage for accelerating mode; the distribution in figure (1b) is for an einzel lens whose central electrode is at voltage lower than the equal voltage applied on the first and the third electrodes for decelerating mode. Since the potential distribution U(z) is constant at the boundaries, then its first derivative )(ZU ′ is zero. This indicates that there is no electric field outside the lens i.e. there is a field free region away from the lens terminals where the trajectory of the charged particles beam is a straight line due to the absence of any force acting on it. The profile of three electrodes forming an electrostatic einzel lens is shown in figure (2). Three-element electrostatic lens systems as a function of the lens voltages and their dimensions. Lens systems, which consist of cylindrical electrodes, each spaced 0.1 diameter apart, were designed by using the SIMION programs studied to form an image at a specific position for use in experimental studies. We also discussed the line-shape profile of a three- element, the lens is symmetrical about its center in addition to the rotational symmetry. The lens geometry is independent of the mode of operation and magnification conditions. The central electrode is hole of a radius equivalent to 0.001L where L is the length of the lens )(/)( io zrzrfi ′= 8)( 62 | Physics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@I@ÖÜ»€a2@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (2) 2014 field. The three electrodes have equal outer radius o about 0.25L. The two outer electrode are geometrically identical having the same shape. Two equal gaps of 0.05L are found to separate each of the outer electrodes from central electrodes. Table (1) shows properties the electrostatic einzel lens accelerating and decelerating operated under different magnification conditions. The spherical and chromatic aberration coefficients have been computed for electrostatic einzel lens accelerating and decelerating with length L=2mm using the equations (4), (5), (6), (7),(8). 1- Zero magnification conditions The relative spherical and chromatic aberration coefficients Cs/fi and Cc/fi respectively in the image side at accelerating and decelerating mode are shown in figure (3) as a function of the electrodes voltage ratio U2/U1 under Zero magnification condition. The Cs/fi has two minima of 0.8 at U2/U1= 1000 (accel) and 0.058 at U2/U1=0.3 (decal). The Cc/fi has two minima of 0.1 at U2/U1= 1000 (accel) and 1.5 at U2/U1=0.9 (decal). In the fig (3) the Cs/fi and Cc/fi decrease with the increase of U2/U1in a accelerating mode , Cs/fi decrease and Cc/fi increase with increase U2/U1 in decelerating mode. The spherical aberration is dominant in zero magnification condition. 2- Infinite magnification conditions. The relative spherical and chromatic aberration coefficients Cs/fo and Cc/fo respectively in the object side as a function of the electrodes voltage ratio U2/U1 in figure (3) , when accelerating lens the Cs/fo decrease with the increase of U2/U1 (U2/U1 =50 ,Cs/fo=0.2) while the Cs/fo increase with the increase of U2/U1. The main reason for chromatic aberration is the fact that particles with higher initial energy are less influenced by the imaging field than the lower energy particles. In decelerating lens the Cs/fo and Cc/fo decrease with the increase of U2/U1. 3- Finite magnification condition In figure, (3) the relative spherical and chromatic aberration coefficients Cs/M and Cc/M respectively as a function of the electrodes voltage ratio U2/U1, In accelerating lens U2/U1 >1 the Csi/M and Cci/M decrease with the increase of U2/U1 in low magnification while the Cso/M and Cco/M increase with increase of U2/U1 in high magnification. In decelerating lens U2/U1 <1 all the aberration coefficients decrease with the increase of value of U2/U1. In the decelerating mode, the lens acts as a series of three lenses, namely, from left (object side) to right (image side), a diverging, a converging, and a diverging lens. Since the charged particles slow down in the central electrode region. However, the trajectory spreads out from the axis initially before entering the central electrode region. For this reason, the spherical aberration coefficient of a lens with U2/U1 is always less than that of he corresponding lens with U2/U1 [13]. The best lens in this research that is high magnification condition where U2/U1 =5, Cs/M=0.011 and Cc/M=0.077 U2/U1 =0.9, Cs/M=0.019 and Cc/M=0.23. Table (1) shows the best optical properties the electrostatic einzel lens accelerating and decelerating operated under different Magnification conditions . Figure(4) and table (2) show a comparison between the values of the relative aberration coefficients for a three electrode einzel lens operated under different magnification conditions with length L=2mm, L=20mm when U2/U1=5. . The effect of lens length L on the relative aberration coefficients increase with the increase of L. 63 | Physics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@I@ÖÜ»€a2@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (2) 2014 Conclusion 1-The spherical aberration coefficient is decreased in zero , infinite and finite (low mag.) with increases of the electrodes voltage ratio, while it increases in finite (high mag.) with the electrodes voltage ratio increases. 2- The chromatic aberration coefficient is decreased in zero, infinite and finite (low mag.) with increases of the electrodes voltage ratio, while it increases in finite (high mag.) with the electrodes voltage ratio increases. 3- The optical properties of the electrostatic einzel lens accelerating and decelerating which are determined from the electrodes voltage ratio. 4- The best optical properties in this research that are in high magnification condition where U2/U1=5, Cs/M=0.011 and Cc/M=0.077 U2/U1=0.9, Cs/M=0.019 and Cc/M=0.23. 5- The aberration coefficients increase when lens length increase. References 1- Baranova, L. A. and Yavor, S. Ya., (1984), Electrostatic Lenses, Sov. Phys. Tech. Phys, 29, 827-848. 2- Sise, O.; Ulu, M.and Dogan, M. (2007), Characterization and modeling of multi-element electrostatic lens systems, Radiation Physics and Chemistry ,76 , 593–598. 3- Abdelrahman ,M.M. (2010) ,Two and three electrostatic lens systems for focusing of charged particles, J. Eng. Technol. Res. 2 (2):219-225. 4- Litovko, I. and OKS , E., (2006), 33rd Eps Conference on plasma phys Rome , 19-23 June , ECA 301, 2.104. 5- Szilagyi, M. (1988), Electron and ion optics, (Plenum press: New York). 6- Rajput, J.; Roy, A.; Kanjilal, D. ,; Ahuja, R. and Safvan, C. P. (2010), An electrostatic deceleration lens for highly charged ions, REV. Sci. Instrum., 81 (4):1-5. 7- Hawkes P. W. and Kasper E., (1989), Principles of electron optics ,1 (Academic Press: London). 8- Zhigarev, A., (1975) , Electron optics and electron-beam devices, (Mir Publisher: Moscow) 9- Kiss, L.(1988), Computerized investigation of electrostatic lens potential distribution, 12th IMACS World Congress, ed. R. Vicheretsky, Paris. 10- Rempfer ,G. F. (1999), The relationship of lens aberrations to image aberrations in electron optics, Optik, 110, 17-24. 11- Ahmad, A. K. (1993), Computerised investigation on the optimum design and properties of the electrostatic lens, Ph.D. Thesis, Nahrain University , Baghdad , Iraq. 12-David, J. Munra and Dahl, D. Daho, (2008), SIMION 3D VESION 8.0 User’s Manual, Scientific Instrument Services. 13- EL-Kareh, A. B. and Sturans, M. A. (1971), analysis of the 3-tube symmetrical electrostatic unipotential Lens, J. Appl. Phys., 42, 1870-1876. 64 | Physics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@I@ÖÜ»€a2@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (2) 2014 Table No. (1) shows the best optical properties the electrostatic einzel lens accelerating and decelerating operated under different Magnification conditions Cc/f U2/U1 Cs/f U2/U1 Magnification 0.1 0.1 1000 0.0025 0.8 0.058 1000 0.3 Zero mag. 2.6 0.37 5 0.6 0.2 0.77 50 0.6 Infinite mag. Cc/M U2/U1 Cs/M U2/U1 0.077 0.024 5 0.6 0.011 0.019 5 0.9 Finite [high] 0.14 0.018 1000 0.9 3.6 0.015 1000 0.9 Finite [low] Table No. (2) Shows the optical properties the electrostatic einzel lens operated under different Magnification conditions when U2/U =5. The chromatic aberration coefficient(mm) The spherical aberration coefficient(mm) Lens of length (mm) Magnification 1.5 2 31 52 L=2 L=20 Zero mag. 2.6 3.3 12 15 L=2 L=20 Infinite mag. 0.077 0.1 0.011 0.03 L=2 L=20 Finite [high] 4.5 14 77.5 24398 L=2 L=20 Finite [low] Figure No. (1) The axial potential distribution U (z) and its first derivatives E(Z) of three electrode einzel lens (a) accelerating mode (b) decelerating mode - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0 Z ( m m ) - 6 - 5 - 4 - 3 - 2 - 1 0 1 2 3 4 5 6 a U ( z ) V E ( z ) V / m m - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0 Z ( m m ) - 1 . 0 - 0 . 8 - 0 . 6 - 0 . 4 - 0 . 2 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 b U ( z ) V E ( z ) = V / m m 65 | Physics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@I@ÖÜ»€a2@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (2) 2014 Figure No.(2) Shape of three electrodes einzel lens with best of optical properties U2/U1=5 66 | Physics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@I@ÖÜ»€a2@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (2) 2014 Figure No.(3)The relative spherical and chromatic aberration coefficients as a function of the electrodes Voltage ratio for einzel lens operated under different Magnification conditions 0.0 0.0 0.1 1.0 10.0 100.0 0.1 1.0 10.0 100.0 1000.0 C so /f & C co /f Infinite Magnifigcation Cso/f Cco/f 0.000 0.001 0.010 0.100 1.000 10.000 100.000 0.0 0.1 1.0 10.0 100.0 C si/ f & C ci /f Zero Magnification Csi/f Cci/f 0.0 0.0 0.1 1.0 10.0 100.0 0.0 0.1 1.0 10.0 100.0 1000.0 10000.0 100000.0 C si/ f & C ci /f Low Mag. Csi/f Cci/f 0.0 0.1 1.0 10.0 100.0 0.0 0.1 1.0 10.0 100.0 1000.0 C so /f & C co /f High Mag. Cso/f Cco/f U2/U1 U2/U1 U2/U1 U2/U1 Figure No. (4)The relative spherical and chromatic aberration coefficients as a function of the electrodes Voltage ratio at various value of the lens length for einzel lens operated under different Magnification conditions 0 1 10 100 1000 0 1 10 100 C so /f & C co /f Infinite Mag. Cso/f (L=2mm) Cco/f (L=2mm) Cso/f(L=20mm) Cco/f(L=20mm) 1 10 100 1000 0.1 1.0 10.0 100.0 C si/ f & C ci /f Zero Mag. Cs/fi (L=2mm) Cc/fi (L=2mm) Cs/fi (L=20mm) Cc/fi (L=20mm) 0 1 10 100 1000 0.1 1.0 10.0 C so /M & C co /M High mag. Cso/f (L=2mm) Cco/f (L=2mm) Cso/f(L=20mm) Cco/f(L=20mm) 1 10 100 1000 0 1 10 100 1000 10000 100000 C si/ M & C ci /M Low Mag. Csi/f (L=2mm) Cci/f (L=2mm) Csi/f(L=20mm) Cci/f(L=20mm) U2/U1 U2/U1 U2/U1 U2/U1 معجلة ومبطئة تعمل تحت ظروف التكبیر كھروستاتیكیةعدسة تصمیم نظري ل المختلفة بشرى ھاشم حسین جامعة بغداد/ )ابن الھیثم(للعلوم الصرفة كلیة التربیة /قسم الفیزیاء 2013نیسان 1قبل البحث في ،2012تشرین الثاني 5استلم البحث خالصةال ق التحلیل العددي بطرائ باالستعانةالمشحونة الجسیمات م بحث حاسوبي في مجال بصریات صمفي ھذه الدراسة معجلة ثالثیة األقطاب أحادیة الجھد . لقد تركز البحث على تصمیم عدسة كھروستاتیكیة عند انعدام تأثیرات شحنة الفراغ ومبطئة تعمل تحت ظروف التكبیر المختلفة. ان توزیع مجال الجھد للعدسة تم تمثیلھ بالدالة األسیة. تم حل معادلة األشعة المحوریة للمجال المقترح ألیجاد مسارالجسیمات المشحونة المارة في العدسة. ومن توزیع الجھد المحوري ومشتقتیھ األولى والثانیة حسبت الخواص امج بأستخدام احد بر ومعامل الزیغین الكروي واللوني. كذلك تم ایجاد شكل األقطاب للعدسة،كالبعد البؤري ،البصریة .المحاكاة المعروفة بأسم (سیمیون) L= 2mm, 20 mm بطول ثالثیة االقطاب معجلة ومبطئة احادیة الجھد عدسة تمفي ھذا البحث صم تم الحصول على شكل االقطاب لھذه العدسة باستخدام حیث تعمل تحت ظروف التكبیر المختلفة (صفري، النھائي، نھائي) ة تصمیم یالتي تعطي مؤشراً على كفاین الكروي واللوني حلول معادلة البالس. وقد بینت نتائج البحث قیم قلیلة للزیغ العدسة. . برنامج المحاكاة (سیمیون) ، الزیغ: بصریات الكترونیة ،عدسة أحادیة الجھد ، معجلة ومبطئة، ة یحتاالمف الكلمات 69 | Physics @a@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@ÚÓ‘Ój�n€a@Î@Úœäñ€a@‚Ï‹»‹€@·rÓ:a@Âig@Ú‹©@Ü‹127@@I@ÖÜ»€a2@‚b«@H2014 Ibn Al-Haitham Jour. for Pure & Appl. Sci. Vol. 27 (2) 2014 10- Rempfer ,G. F. (1999), The relationship of lens aberrations to image aberrations in electron optics, Optik, 110, 17-24.