Studying the Properties of the Magnetic Optical Lenses by Using Mathematical Functions

Studying the Properties of the Magnetic Optical Lenses by Using Mathematical Functions

Talib Mohsen

Babylon university

college of education ,department of physics for pure science.iraq

Babylon,Iraq

Lamia Merry

Babylon university

college of education ,department of physics for pure science.iraq

Babylon,Iraq

Abstract A suggested new mathematical function has been used for representing the axial magnetic flux density distribution along the optical axis of the magnetic lens. The main feature of this function is that it has more than one optimization parameters to let more flexible for the designer to test more one parameter that suggested in this function to reach the best axial magnetic flux density which gives optimum objective focal properties with available reconstruction pole pieces.

Keywords-electron optics, Optimization, design of magnetic lens

I. INTRODUCTION

Today, the charged-particle devises are considered to be important scientific and technical tools. This importance come from the rapid growth in various field like material science, semiconductors technology, plasma physics, biology and advanced nano-technology. The branch of physics that deals with the problems of motion of charged particle beams

properties and aberrations are totally determined by the axial distribution of the field. Only the axial distributions and their derivatives appear in the equation of motion of charged particle (i.e. paraxial ray equation) and in the expressions of aberration coefficients. Then instead of analyzing a vast amount of different electrode or pole-piece configurations. One can take the criteria defining an optimum system as initial conditions and try to find the imaging field distribution that would produce it. The final step in the synthesis procedure is to synthesize the electrode or pole-piece profiles that generate this field distribution [3,4].

I1.THE MATHEMATICAL MODEL

The present investigation is based on the optimization by synthesis. Accordingly, the magnetic field distribution of rotationally double polepiece magnetic lens along the optical axis can approximated by the following target function.

in electric or magnetic fields is called the electron and ion optics [1].

In sense of electron and ion optics there are two entirely

B(z) (

sinh(2.636 S )

o NI D

)(

S cosh(2.636 S ) cosh(5.272 Z )

(1)

different optimization approaches which are analysis and D D

synthesis. In the conventional design of electron lenses, , the approach of analysis is taken under consideration. According to this approach, design process of magnetic lens consists of two steps: the first one is the design of electron optics while the second one is the design of magnetic circuits including coils, yokes, and poles. If experimentally obtained electron

It can be seen that the target function represented in eq. 1 has three main control parameters Axial bore Diameter (D), Width of air gap between two polepices (S) and excitation of the lens.

1. Polepiece Reconstruction

   optical properties are far from what is expected, the second part must be checked. The first step of the lens design is the

   With aid of equation

   B(z) (o

   dV (z) )

   dz

   the axial

   optical design to determine the optimum polepiece shape and ampere turns (A-t) under given external conditions. The second step is the coil design. The main task here is the estimation of heat conduction. The third is the design of yokes and poles. It is noted that the conventional design

   symmetric scalar potential distribution corresponding to field given by equation (2) can be determined by the following integral along the optical axis.

   Z

   based on the trial and error until a satisfactory performance is

   achieved according to the constraints taken under

   V (z) 1

   f

   B(z)dz

   (2)

   consideration [2].

   Optimization by synthesis has been one of the most ambitious goals of electron and ion optics which is sometimes called inverse design procedure. This approach is based on the fact that any imaging field, its optical

   o Zs

   Where zs and zf are the axial coordinates of the starting and final of magnetic fields distribution. The shape of the

   equipotential surfaces (polepieces in the present work) can be reconstructed by using analytical solution of the Laplace's equation as given by[4].

   1

   evaluation of the resultant lens in accordance with the present work procedure. However, the resolution distance, for any lens system, in terms of the spherical aberration coefficient may expressed in the following form [6,7].

   =0.61(CS3)1/4

   (3)

   R (z) 2V (z) Vp 2

   (7)

   p V (z)

   Where V(z) is the axial magnetic scalar potential distribution, V zis the second derivative of V(z) with

   respect to z, R z is the radial height of the reconstructed

   Indeed, the constant (0.61) appears in equation (7) is come out according to Rayleigh Criterion. The symbol refers to the charged particle associated wave length, which in terms of the accelerating potential (Vr) given by the formula for the case of electron [8]:

   p

   v

   polepiece shape, and

   p

   NI

   2

   if the field of a double

   1.5 ( n m )

   Vr

   (8)

   polepiece lens is symmetrical about the symmetry plane(z=0), while if the field is asymmetrical, then Vp would be equal to the area under the field curve for each side about the position of the maximum value of the field distribution [3]. However, in the case of asymmetrical lens field, the potential values in

   Concerning with semi-angle aperture, the formula that correlated this parameter with spherical aberration coefficientis as shown in the following expression [7,9].

   the object and image spaces are not equal.

2. Objective lens aberrations

(9)

(rad)=(/Cs)1/4

Obviously equations (7 and 9) show that the spherical

z

When the axial magnetic field distribution along the domain of solution is assigned according to equation (1), the paraxial ray equation [2].

aberration coefficient correlates with the resolution distance and semi-angle aperture through the electron associated wave length. So, they are not a direct relation unless state of wave length being fixed. The focusing power () is another

r

(4)

(

8Vr

) B 2 r 0

important property that characteristic any electron-optical lens system. Thus, it is of more interest to consider such a property to be a figure of merit to evaluate the quality of electron

lens for a specific application. Anyway, this lens property

can be solved in order to deducing electron beam trajectory r and its slope r/ . In this work the Fourth-Order

given by the following equation [10].

Runge-Kutta method has been used to achieve this task. It should be mentioned that the parameters and Vr appears in

(10)

=1/f =[e/8Vr]B2z dz

equation (4) are the electron charge to mass quotient and the relativistically accelerating voltage respectively. The spherical and chromatic aberration coefficients Cs and Cc expressed respectively . [5]

Where f is the focal length for the lens of magnetic field distribution Bz. equation (10) is known as Busch's formula for weak lens. However, the counterpart expression for thick lens being more complicated.

C

z 3

4 4

2 4

2 2 2

1. The Effect Of The Axial Bore Diameter(D)

   s 128V

   V

   Bz r

   • 8Bz r

   8Bz r r

   dz

   The effect of the axial bore diameter on the axial

   r zo r

   (5)

   Cc 8 V Bz r dz

   zi 2 2

   r zo

   (6)

   magnetic field distribution and consequently the polepiece configuration is studied keeping the width of air gap, lens excitation and the length of the lens are constants at the following values: S=2 mm, NI=500 a.t L= 40mm. Figures 1 and 2 show the axial magnetic field distribution Bz and its corresponding axial magnetic scalar potential distribution Vz

   which can be computed using any suitable technique . In the present paper Simpsons rule will be used to computed Cs and Cc.

   V.PHYSICAL PARAMETERS

   The defect of spherical aberration has a special importance overits counterpart defects, since it is affects many of the lens characteristics. Among many of them are the resolution distance and semi-angle aperture . So these two parameters have been considered asa figure of merit for the

   for different values of the axial bore diameter of the lens (D

   =2, 4, 6, 8, mm). From figure 1, it is noted that the axial field distribution extends out of the air -gap region (i.e., the action region) , However, the axial magnetic field distribution will be more localized in the air -gap region for small values of the axial bore diameter. It is seen that the behavior of the field distribution as a function of the axial bore diameter is similar to that with the half width of the field for the conventional magnetic lens models (for example, Glaser model). As the axial bore diameter increases the maximum

   value of the magnetic flux density Bz decreases while the half width of the field distribution increasing as shown in figure 3. The pole piece profiles that can produce each Bz distribution, plot in figure 4. It can be seen that the consequences for increasing D lead to decreasing the pole

   face curvature.

   Figure 5 shows the variation of the aberration

   6

   Bmax (T)

   W(mm)

   5

   Bmax (T )& W(mm)

   4

   3

   2

   coefficients Cs and Cc and the objective focal length fo as a

   function of the axial bore diameter D under zero magnification condition at the excitation parameter NI/SQRT(Vr)=20 are amend. However, this amendment of the optical properties is a consequence for the concentration of the flux density distribution within the air-gap region as the axial bore diameter decreases.

   Figure 6 shown the physical parameters as a function of D, from the figure it is noted that resolution limits increases while the focusing power and angular aperture decreases. Furthermore, the decreasing of indicates that a finer details for material sample can be inspect and explored due to the increases of D. The effect of the axial bore diameter D on the objective optical properties and physical parameters of the lens is constant listed in table 1 at constant excitation parameter under zero magnification condition.

   1

   0

   0 1 2 3 4 5 6 7 8 9

   D (mm)

   Figure 3: The maximum flux density value Bmax and the half width W of the lens as a function of the parameter D.

   10

   D=2mm

   9 D=4 mm

   D=6 mm

   8 D=8 mm

   7

   Rp(mm)

   6

   5

   4

   0.6

   D=2mm 3

   D=4 mm

   0.5 D=6 mm 2

   D=8 mm

   1

   Bz (T)

   0.4

   0.3

   0

   -20 -18 -16 -14 -12 -10 -8 -6 -4 -2 0

   Z(mm)

   0.2

   Figure 4: Polepieces shapes for different values of D

   0.1

   0

   20 15 10

   5 0 -5

   Z (mm)

   -10

   -15

   -20

   2.5

   fo (mm)

   Cs (mm)

   Cc (mm)

   2

   Figure 1: The axial magnetic field distribution curves Bz for different values of axial bore diameter of lens

   fo(mm)&Cs(mm)Cc (mm)

   1.5

   600

   D=2 mm 1

   500 D=4 m

   D=6 mm

   400 D=8 mm

   300

   0.5

   200

   Vz (A.t)

   100

   0

   -100

   -200

   -20 -15 -10 -5 0 5 10 15 20

   0

   0 1 2 3 4 5 6 7 8 9

   D (mm)

   Figure 5 :The objective focal length fo and the aberration coefficients Cs and Cc as a function of D at (NI/Vr1/2=20).

   -300

   -400

   -500

   -600

   Z(mm)

   Figure 2: axial magnetic scalar potential distribution curves Vz for different values of the axial bore diameter of the lens.

   (rad) ×10-2& ×103 (1/m) &

   2.5

   2

   1.5

   1

   0.5

   0

   0 5

   D (mm)

   10 15

   VIII. REFERENCES

   1. Al-Khafaji, F. H. ( 2011),"Synthesis of Magnetic Lenses Using the Calculus of Variations", Ph.D. Thesis, Al- Mustansiriyah University, Baghdad, Iraq.

   2. Szilagyi, M. (1988), "Electron and Ion Optics", (Plenum Press: New York).Phys. Lett., 45, 499-501.

   3. Szilagyi, M. (1984), "Reconstruction of Electron and Polepiece From Optimized Axial Field distribution of electron and ion optical systems", Appl. Phys. Lett., 45, 499-501.

   4. Szilagyi, M. (1985), "Electron Optical Synthesis and Optimization"., Proceeding of the IEEE.73, 412-418.

   5. El-Kareh, A. B. and El-Kareh, J. C. J., "Electron Beams, Lenses, and Optics", vol.1 and 2, 267,(1970), Academic Press: New York and London.

      Figure 6: The physical parameters , , and as a function of D.

      Table (1): Some of variables for symmetrical double polepiece

   6. Hawkes, P. W. (1972), "Electron Optics and Electron Microscopy", (Taylor and Francis Ltd., London).

   7. Orloff, M. (2009) "Hand book of charged particle optics

   8. Hawkes, P. W. (1982), "Magnetic Electron Lenses", (Spreinger-

      magnetic lens for various values of the parameter D when S=2 mm and NI=500 a.t NI/Vr1/2=20

      Verlag, Berlin).

   9. Klemperer, O., " Electron Optics",p.170, (1971), 3 (Cambridge).

      rd ed.,

2. CONCLUSIONS

From the investigation of unsaturated symmetrical magnetic double polepiece lenses with the aid of the axial magnetic field distribution function taken under consideration, one can conclude the following:

1. The optical focal properties for objective magnetic lenses and the polepiece profile are different for various values of the axial bore diameter when the excitation of the lens is kept constant.

2. The behavior of the properties and the polepiece shape with the axial bore diameter is similar to that with the axial bore diameter in both analytical and synthesis optimization procedures. However, all parameters of the lens and their polepiece shapes are unaffected by the variation of the lens excitation when the bore diameter is kept constant.

3. The present investigation has shown that the favorable values of the important design parameter D are the smallest values which are less or equal 6mm. This conclusion coincides with the applicable shapes of the polepieces to be reconstructed.

4. when the parameter D increases, the axial magnetic field distribution extends out of the air -gap region as well as the maximum value of flex density decreases.

5. It can be seen that the consequences for increasing D lead to decreasing the focusing power and angular aperture while the resolution limits increases.

Microscopy", Springer, USA.