IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (1) 2011 Investigation of the Projector Properties of the Magnetic lenses Using Analytical Function A. H.H. Al-Batat, M. J. Yaseen, M.M. Majeed* Departme nt of Physics, College of Education, Unive rsity of Mustansiriyah * Department of Biomedical, Al-Khawarizmi College of Engineering, Baghdad Unive rsity Recei ved in Se pt. 29 , 2010 Accepted in Feb. 8 , 2011 Abstract A comp utational invest igation h as been carr ied out to describ e sy nthesis optimization p rocedure of magnetic lenses. The r esearch is concentrated on the determination of the inverse design of the sy mmetrical double p olep iece magnetic lenses whose magnetic field distribution is already defined. M agnetic lenses field model well known in electron op tics have been used as the axial magnetic f ield distribution. This field has been st udied when the halfwidth is variable and the maximum magnetic flux density is kep t constant. The imp ortance of this research lies in the p ossibility of using the present sy nthesis optimization procedure for finding the polep ieces desi gn of sy mmetrical double polep iece magnetic lenses which have the best p rojector focal p rop erties. Key word: Electron opt ics, magn etic electron lenses, op timization of magnetic lenses, inverse desi gn of magnetic lens. Introduction In electron op tics, t he sy nthesis p rocedure of electron lenses op timization is based on the fact that, the first-order p rop erties and aberrations of any ima ging magnetic field can be calcu lated by using mathematical fun ctions to approximate the magnetic field sev eral good mathematical functions exist for assign ing the magnetic field distribution such as Gaussian model, Exp onential model, Cosine model ……. etc. It is imp ortant to note that, the values of optical p rop erties, aberrations and p olep iece shap e depend on the mathematical distribution of field function i.e. the optimum design of magnetic lenses depend on the op timization parameters of p rop osed formula to rep resent the optimum axial magn etic field dist ribution [1]. The Mathematical Model Dep ending on the si milarity between the curve shap es of the axial magnetic sc alar p otential distribution and the hyp erbolic tangent function, this following function can be used for approximatin g the axial magnetic scalar potential distribution Vz given by [2]. Vz(z)=α tanh(Z/β) (1) IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (1) 2011 ZS ≤ Z ≤ ZE along the axial len gth of the lens, where ZS and ZE are the terminals of the axial coordinate, here (α=W.Bm ax / 2μo) and (β=W / 2). Where μo is the sp ace p ermeability and equal 4πx10 -7 H.m -1 , W is the halfwidth of the field distribution, Bm ax is the maximum magnetic flux density . W and Bmax rep resent two optimization parameters. For sy mmetrical charged p article lens, the magn etic field and its axial p otential distribution would be sy mmetric about the mid plane. Therefore, the followin g constraints can be st udied: 1. B(z)=B(-z), V(z)=V(-z) 2. | ZS |= ZE accordin gly the lens length can b e exp ressed as L=2ZE or L=2ZS. In the sy nthesis technique, sy mmetric conditions depend on the p rocedure taken under considerations for a electron lens design. However, these conditions dep end on the target function to be analy sized. For more details see [3,4,5,6]. For iron-free r egion the axial magnetic flux density distribution along the optical axis can be determined with the aid of the followin g eq. [2] : dz dV μ(z)B zoz  (2) Therefore, in the present work t he axial magn etic field of the lens can be approximated by the followin g eq. : Bz(z)=Bmaxsech 2 (2.270(Z/W)) (3) It can be seen that the target function rep resented in eq. 3 has three main control p arameters Bm ax, the maximum magn etic flux density , the halfwidth of the field distribution W, and the lens len gth L(=ZE - ZS). The present invest igation concerns with the more effective design p arameter which is the halfwidth, while the other two p arameters Bmax and L are k ept constant at 1T and 40mm. Polepiece Reconstruction To determine the lens p otential, the region of its axial extension is divided into (n-1) subintervals where n rep resents the points of t he field alon g the z-a xis. By numerical analy sis t he axial magnetic flux density distribution Bz is rep resented by a cubic sp line function in each subinterval [3]. By using the eq. 2 and with the aid of the cubic sp line technique, the axial magnetic scalar potential may be given by: kZk1)(kZ GVV  (4) where:                                                3 2 h k 3 B // 1k 3 2 h k B // k 2 2 h k B / kh kB kμ 1 oG k (5) Here hk=z k+1- zk; it represents the width of each subinterval. When the lens is sy mmetrical VZ(n)= -VZ(1) =0.5 NI. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (1) 2011 Consider the magnetic field model given in eq. 3. The magnetic f ield distribution is determined under sy mmetric conditions along the lens optical axis. Since the lens is sy mmetric, then the absolute value of the potential at the lens terminals are equal in magnitude (see fig. 1). This indicates that the area under the condenser and objective fields are equal. At the p lane where z=0 the p otential is zero as shown in fig. (1). The sy mmetrical lens is achieved by maintainin g this p lane and keep ing the condenser field on its left-hand side equivalent to the objective field on the right-hand side. By using the analytical solution of Lap lace’s equation, the shap e of the p olep iece that would p roduce the desired field can be determined. For axially sy mmetric sy stems the electrost atic or magn etic scalar p otential V(r,z) can be calculated fro m the axial distribution of the same potential V(z) by the following p ower series exp ansion [7]:     2k 2k2k P 0k 2 k P dz V(z)d 2 R k! 1 z),V(R           (6) where RP is the radial height of the p olep iece. Taking the first two terms of equation 6 under consideration, the equipotential surfaces (p olep iece shap es) are given by the following formula:   z Pz P V VV 2(z)R    (7) where VP is the value of the potential at t he iron base of the polep eice. Eq. 7 has been app lied on the two equivalent p arts of the sy mmetrical field. Results and Discussion The effect of halfwidth W on the field distribution and its focal p rop erties can be invest igated by keep ing the remain ing two p arameters constant at t he following valu es Bm ax=1T and L=40mm. The axial flux density Bz and the corresp onding axial magnetic scalar p otential Vz for each value of W ar e shown in fig. 2 and 3. It is clear that t he area under the curve in creases as long as W increases, i.e. the refractive p ower NI increases with the increase of the value of W. Since the excitation p arameter NI/Vr 1/2 =25 is kept constant t he acceler ation voltage Vr increases with resp ect to the increase of NI. Fig. 4 shows the electron beam trajectory inside the magnetic f ield distribution for different values of the halfwidth (W=1, 2, 3, 4, 5 mm) when Bm ax=1T and L=40mm at NI/Vr 1/2 =20 under zero magnification condition. It should be noted that the electron beam enters the magn etic field in the same height. Since the p resent investigation deals with determined the p rojector p rop erties, the electron beam trajectory calculated with aid of the paraxial ray equation alon g the total length of the magnetic electron lens. From the figure, one can see that when the halfwidith w incr eases the refractive p ower of the magn etic field decreases for the electron beams, this means that when the halfwidth increases the lens excitation increases. Therefore, electron beam need ed more acceleration voltage to p enetrate field dist ribution for large halfwidth [6]. Fig. 5 shows the reconst ruction p rocedure of the magnetic lens p olep ieces for different values of the halfwidth (W=1, 2, 3, 4, 5 mm) when Bm ax=1T and L=40mm. It should be noted that the polep iece diameter D and the air gap width S are increased with the increase of the value of W. For high v alues of the h alfwidth, the p olep iece shap e are v ery comp lex in design field, therefore, in order to neglect the manufacturin g difficulties. Desi gner should be considered the lower values of the parameter W as p ossible as [1]. Fig. 6 represents the relationship between the p rojector focal len gth FP with the excitation p arameter NI/Vr 1/2 . It is seen that the value of the projector focal len gth decr eases with t he IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (1) 2011 increase of the excitation p arameter until reach the minimum value ((Fp)m in =0.440808, 0.88117, 1.321719, 1.762284, 2.202852 mm) for the mentioned W v alues taken under consideration. So, we seen that t he values of the minimum p rojector focal len gth (Fp)m in for all the values of h alfwidth p roduce at t he same of excitation p arameter NI/Vr 1/2 =14.2. Also, we saw that when the values of halfwidth increases the values of focal length increases, that’s meaning when the values of halfwidth increases the area under the curve of field distribution increases [6]. Fig. 7 r epresents the relationship between the radial distortion coefficient and excitation p arameter NI/Vr 1/2 . It is seen that for each curve of the radial d istortion coefficient has constant value at min imum excitation p arameter and increases tell reaches maximum value, then decreases until reaches zero app roximately at excitation p arameter NI/Vr 1/2 ≈15 and continue toward negative value with the increae of the excitation p arameter. One can see that from the figure, when the halfwidth of t he field increases the radial distortion coefficient Dr has small values. This means that t he field distributions of small halfwidth have large values of the radial distortions [6]. It should be that the magnetic p rojector lenses in p resent work have no r adial distortion at the excitation p arameter in which the minimu m p rojector focal len gth occurs for all W v alues. This means that these lenses have no radial d istortion at the excitation p arameter in which the maximu m magn ification occurs. The radial distortion p arameter Qr as a function of the excitation p arameter NI/Vr 1/2 for various values of t he halfwidth is shown in fig. 8. Fig. 9 represents the variation of the sp iral distortion coefficient with the excitation p arameter NI/Vr 1/2 . It is seen that the values of sp iral distortion coefficient increase rapidly at small v alues of the h alfwidth with the increase of the excitation p arameter. It should be mentioned that for the lar ge v alues of the h alfwidth of the field distribution the magnetic p rojector lens has small distortion. This behavior can be exp lained as the halfwidth increases the bore radius and, the air gap region of the reconstructed p rojector magnetic lenses are increasin g. Fig. 10 shows the variation of the sp iral d istortion p arameter Qs with the excitation p arameter NI/Vr 1/2 . The curves in the figur e 10 have on e minimu m value equal (Qs)m in =0.893 at excitation p arameter extend from (NI/ Vr 1/2 =9.1) to (NI/Vr 1/2 =9.7), this value less than unity ((Qs)m in =1) is well known at symmetrical projector lenses [8]. The low value of sp iral distortion p arameter Qs gives p ossible reduce the p rojector distance for the electron microscope and then get on the lar ge p rojector angle, this leads to reduce the len gth tube of the electron microscop e. Fig. 11 r epresents the minimu m valu e of focal length, radial distortion p arameter and sp iral distortion p arameter, lens excitation NI and acceleration volta ge Vr as a fun ction of the halfwidth at constant maximum value of the flux density . The relationship between (Fp)m in and NI with W are linear. But, Qr and Qs are not effected by values of W because the value of maximu m flux density is constant. T he value of (Qr= 0.13) at ( 1/2 r NI/V =15) and (Qs=0.893) at ( 1/2 r NI/V =9.1 to 9.7). The variation of the accelerated voltage Vr with halfwidth W) is shown in the same figure (11). It indicates that in order to increase the halfwidth W one should increase the acceleration volta ge, but not linear ly because the area under the curve incr eases with halfwidth increase. Re ferences 1. Warid, H.H. (2006), Ap p roximation of the M agnetic Field distribution for Objective M agnetic Lenses by Using Analytical Function"J.of College of Education, 3:360-381. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (1) 2011 2. Hawkes, P.W. (1982). M agnetic Opt ical Sy nthesis and Optimization "PROCEEDINGS OF THE IEEE, 73:412-418. 3. Al-Obaidi, H. N. (1995),Determination of the design of magnetic electron lenses op erated under preassigned ma gnification conditions"Ph.D. Thesis, University of Baghdad, Iraq. 4. Al-Batat, A. H. H. (1996), Inverse design of magnetic lenses usin g a defined magnetic field,M .Sc. Thesis, University of Mustansiriyah, Baghdad, Iraq. 5. Al-Kadumi,K.S. (2007). Computer-Aided-Design of Opt imized M agnetic Electron Lenses M .Sc. Thesis, University of Mustansiriyah, Baghdad, Iraq. 6. Zangana, H. A. (2005). Using Gray’s model as a target function in the inverse design of magnetic lenses M.Sc. Thesis, College of Education, The University of Mustansiriyah, Baghdad, Iraq. 7. Szilagy i, M . (1984), Reconst ruction of electrodes and p olep ieces from optimized axial field distributions of electron and ion optical systems,Appl. Phys. Lett . 45: 499-501. 8. Lambrakk is, E.; M arai, F.Z . and M ulvey , T.,(1977). Correction of Sp iral Dist ortion In the TEM"Develop ment in Electron M icroscopy and Analysis .Ed.D.l.M isell (Inst .Phys.Conf.Sonf.Ser.NO.36: 35-38. Fig.(1): The axial magnetic scalar potential distribution of symme trical double polepiece magne tic lens. Ax ial mag ne ticscalar po tential dis tribution Object space side Image spa ce s ide Opt ic al axis z Neg ative z-axi s po s itive z-ax is 0 1 2 3 4 5 -2 -1 0 1 2 -20 -10 0 10 20R (m m ) Z( mm) 1234 5 0 5 10 15 20 -20 -15 -10 -5 0 R p (m m ) Z( mm) Fi g. (4): Ele ctron be am traje ctory in side the m agne ti c field fo r di fferen t values of (W =1, 2, 3 , 4, 5 mm ) when B max=1T and L=40 mm at NI/Vr 1/ 2 =20. Fig. (5): Profile of polepieces for different values of (W=1, 2, 3, 4, 5 mm) whe n Bmax=1T and L=40mm. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (1) 2011 Fig.(2): The axial magnetic fiel d distributi on for different values of (W=1, 2, 3,4,5 mm) whe n Bmax=1T and L=40mm. Fig.(3): The axial magnetic scalar potenti al distribution for the field plotted in fig. 2. 1 2 3 4 5 -2000 -1500 -1000 -500 0 500 1000 1500 2000 -20 -10 0 10 20 V z (A .t ) Z(mm) 1 2 3 4 5 0 0.2 0.4 0.6 0.8 1 -20 -10 0 10 20 B z (T ) Z(mm) 1 2 3 4 5 0 5 10 15 0 5 10 15 20 25 F p (m m ) N I/Vr1 /2 0 5 10 15 0 5 10 15 20 25 Q r NI/Vr 1/ 2 1 2 3 4 5 0 5 10 15 0 5 10 15 20 25 D s (m m )- 2 NI/Vr 1/2 Fi g. (9 ): The spi ral di storti on coe ffi cien t as a function of the e xcitati on paramete r for various val ues of (W =1, 2, 3, 4, 5 mm) when Bmax=1T an d L=40mm. Fi g. (8 ): Vari ati on of the radial distorti on parame ter with the exci tati on paramete r for di fferen t values of (W =1, 2, 3, 4, 5 mm) when Bmax =1T and L=40mm. Fig. (6): The projector focal length angst excitation paramete r for different values of (W=1, 2, 3, 4, 5 mm) whe n Bmax=1T and L=40mm. Fig. (7): The radial distortion coefficient excitation paramete r for different values of (W=1, 2, 3, 4, 5 mm) whe n Bmax=1T and L=40mm. 1 2 3 4 5 -6 -4 -2 0 2 4 6 0 5 10 15 20 25D r( m m )- 2 N I/Vr1/2 0 5 1 0 1 5 0 5 10 15 20 25 NI/Vr 1/2 Q s (Fp)m in (Q r)m i n (Q s)min NI(A.t) Vr(V) 0 0.5 1 1.5 2 2.5 3 3.5 4 0 1 2 3 4 5 6 (F p ) m in (m m ) , ( Q r) m in , (Q s ) m in , N I( A .t )x 1 0 3 , V r( V )x 1 0 4 W(mm) Fi g. (11): The minimum proje ctor fo cal length , spi ral distortion paramete r, radi al distortion parame ter, lens exci tati on NI an d accele rate d vol tage Vr as a functi on of the hal fwi dth of the m agne ti c fiel d at constant e xcitation paramete r (NI/Vr 1/2 =25) when B =1T and Fig. (10): Variation of the spiral distortion parameter Qs with NI/Vr 1/2 for different value of W at Bmax=1T and L=40mm. IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (1) 2011 2011) 1( 24جلدالم مجلة ابن الهیثم للعلوم الصرفة والتطبیقیة دالة تحلیلیة مسقطیة لعدسات مغناطیسیة باستعمالدراسة الخواص ال *مجید میادة منذر ،محمد جواد یاسین ،علي هادي حسن البطاط الجامعة المستنصریة ، كلیة التربیة ،قسم الفیزیاء * جامعة بغداد ، الخوارزمي هندسة كلیة ، اتيقسم الطب الحی 2010 ایلول 29 في استلم البحث 2011 شباط 8 في قبل البحث الخالصة وقـد تركـز البحـث علـى . في هذا البحث أجریت دراسة تصف طریقة من طرائق التولیف االمثل للعدسات المغناطیسیة وتمـــت . د توزیــع مجالهــا المغناطیســي مســبقاثنائیــة القطــب المتنـــاظرة التــي حــدّ إیجــاد التصــمیم العكســي للعدســات المغناطیســیة توزیع، حیث درس المجال عندما یكون عرض النصف متغیرا والقیمة العظمـى لكثافـة الفـیض المغناطیسـي النموذج أاالستفادة من العدســة أهمیـة هـذا البحــث تكمـن فـي أمكانیـة االســتفادة مـن الطریقـة التولیفیـة المتبعـة وذلــك بإیجـاد تصـمیم أقطـاب تكمـن . ثابتـة .المغناطیسیة الثنائیة القطب المتناظرة التي لها خواص مسقطیة مقبولة ـــترون,عدسات الكترونیة مغناطیسیة ــــ ــــ ـــریات االلكــــ ــــ ــــ ـــاتیح; بصــــ ــ ــ ــ ــ ــ كلمات المفــ المغناطیسیة امسلیة العدسات , تصمیم عكسي للعدسات المغناطیسیة ,