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Optics of conical electrostatic analysing and focusing systems
Nuclear Instruments and Methods in Physics Research A298 (1990) 421-425 North-Holland
421
Optics of conical electrostatic analysing and focusing systems S.Ya. Yavor and L.A . Baranova A.F. Ioffe Physico-Technical Institute of the Academy of Sciences, Polytekhnicheskaya 26, 194021 Leningrad, USSR
Energy analysis and focusing of hollow beams of charged particles may be performed with the aid of conical electrode structures . Two analyser geometries and two lens designs are described and the behaviour of some typical configurations is studied numerically. Some advantages of comcal lenses over round lenses of more conventional design are indicated .
Traditionally electron optics deals with rather narrow beams of charged particles, moving close to the optic axis . Such beams allow high focusing quality and are widely used in various devices. There are well developed mathematical procedures for calculating such devices with series expansions of fields and trajectories around the optic axis being used . Narrow beams provide comparatively low intensity; however, an intensity rise at the expense of beam widening inevitably results in degradation of focusing . A possible solution of the problem is to widen the beam in the direction of one of the coordinates . In this way, by using a proper lens system it is possible to increase considerably the intensity while preserving sharp focusing. Recently there has appeared a number of problems requiring the application of hollow conical beams with charged particles moving far from the symmetry axis . Due to the increase of one of the angular dimensions up to 360', such beams possess higher intensity than narrow ones . Moreover, the use of conical beams in materials investigations provides a large amount of information and allows some additional data to be obtained, for example, angular distribution of charged particles. The focusing and analysis of hollow beams require the design of special electron-optical systems whose electrodes best match the beam shape. In our opinion, the systems of coaxial conical electrodes embracing the beam from inside and outside are most suitable for this purpose. Electron-optical systems formed by coaxial conical electrodes may be used for energy analysis of charged particles or focusing depending on the prevalent field direction, determined by the mode of the potential supply. Schematic diagrams of conical analysers are shown in figs . 1 and 2. The generating lines of their electrodes can be either parallel or emerging from the same point. The advantages of such analysers are high gathering power and the possibility of simultaneous energy and angular analysis . It should be pointed out that the angular spectrum can be obtained without moving the sample or the analyser. Unlike the cylindrical mirror analyser, the conical mirror allows us to obtain a charged particle distribution depending on both azimuth and polar angles. In the latter case, the beams are disk-shaped. The description of an analyser with conical electrodes is given in ref. [1]. Such instruments are applied for energy and angular electron spectroscopy, which now is one of the main techniques for investigating surface electronic structure. For focusing hollow beams and for changing their energy if necessary, lenses with conical electrode shape should be used . Various schemes of such lenses are presented in figs. 3 and 4. If the potential supply is as shown in figs . 3 and 4, the electrical field within the lens is mainly longitudinal and in this sense the lens is analogous to round or two-dimensional lenses . Conical lenses form circular images of point or circular sources . By changing the potential supply, it is possible to create an additional radial field which curves the axial trajectory, changes the image radius and contributes to the optical power. The double conical lens shown in fig. 3 was used for focusing the beam coming from a toroidal analyser onto a circular detector [2]. 0168-9002/90/$03 .50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
422
S Ya Yaoor, L.A . Baranooa / Conical electrostatic analysing andfocusing systems
Fig. 1. Conical analyser with parallel electrodes .
Fig. 2. Conical analyser with concurrent electrodes.
We have investigated the electron-optical properties of both energy analysers and lenses, formed by conical electrodes . The characteristics of the conical analysers shown in figs. 1 and 2 have been studied . Analytical methods have been developed for their approximate calculations . This approach allows us to establish the basic features of the analyser behavior. The potential distribution in the analyser with parallel electrodes was found in the adiabatic approximation, that is, the potential variation along the x-coordinate was assumed to be small . Then the solution of the Laplace equation has the form
(VZ - Vl ) ln(1+y/(x tan eo)) ' ln(1 + g/(x tan eo)) 12
Fig. 3. Double-cone lens.
Fig 4. Triple-cone lens .
S. Ya. Yavor, L .A . Baranova / Conical electrostatic analysing andfocusing systems
423
where x, y denote local coordinates, shown in fig. 1. We get the approximate analytic expressions for trajectories by using the energy conservation law, by passing over from time differentiation to x-coordinate differentiation and then by performing a single integration: - arccosh
x - x0 = Ag cos fx
p tan 00 +
'2 -
B
(y/g)
+ arccosh
p tan 00 + i B ~
1
arccosh p
tan 00
B
tan00 + .l - (ylg) + arccosh p
1
x _< x x
~
xm,
1
where xm corresponds to the point of trajectory maximum, A = (2E/eV) p tan 00 ,
B=
V
- Vp tan 00 sin2ß + (p tan 00 + z
)2
,
V = VZ - VI
.
Here a large parameter p = x0/g is introduced, x0 is the coordinate of the point where a particle enters the field and g denotes the distance between the electrodes . Such a definition of the large parameter means that the analyser entrance slit is placed rather far from the cone top. Let us consider a general case in which a circular source and a circular detector are situated outside the field, their distances from the inner electrode being h 0 and h, respectively and H = h 0 + h,. We have found the distance L between the initial and final point of the trajectory measured along the x-axis to be L
Ag cos ,ß I arccosh p tan ß +
tan 000 + -
1-
Eq. (3) makes it possible to find easily the conditions of the first- and second-order focusing of charged particles. For this purpose, the first and second derivatives of L with respect to the entrance angle ß must be equal to zero : dL
âß
= 0,
d2L dß2 = 0.
From eqs. (4) we can find the value of the angle ß which provides the first-order and the second-order focusing . Substituting it into eq. (3) we obtain the distance d between the source and detector . Using these formulae we carried out some calculations, resulting in the following conclusions. If the source and detector are arranged on the inner electrode, only first-order focusing is possible . Unlike the homogeneous field, the focusing angle ß may be not 45' . For the given cone angle 200 , the focusing angle ß increases with the ratio of particle energy to the analyser potential (EleV) . Some results of the calculations are given in table 1 for H = 0 and p = 9.55. Second-order focusing occurs when the source or/and detector are placed outside the field. In this case the values of the focusing angle ß do not coincide with the value ß = 30' specific to the homogeneous field . With increasing parameter p, the characteristics of the conical analyser approach those of the homogeneous field. In particular, the first-order focusing angle becomes 45 ° and the second-order focusing angle becomes equal to 30'. Calculation of the relative analyser dispersion has shown that the approximation used leads to the following formula: D/d =1/(2 cos 2,8) .
The conical analyser dispersion is somewhat higher than that of a cylindrical or flat mirror due to the larger value of the focusing angle ß . It should be pointed out that the equation ß + 00 = iT/2 must hold when beams of a disk shape perpendicular to the analyser symmetry axis are used. The focusing onto the z-axis is possible only if ,ß > 00. When ß = 00 the beam leaving the analyser is parallel to the z-axis . IV . ELECTRON/ION OPTICS
424
S. Ya. Yavor, L.A . Baranova / Conical electrostatic analysing and focusing systems
Table 1 Properties of a conical analyser 80 [deg]
E/eV
/8 [deg]
d/g
D/d
30 30 30 45 45 45 60 60 60
1 .7 1 .2 0.5 1 .7 1 .2 0.5 1 .7 1 .2 0.5
48.3 47.0 45 .8 46 .8 46 .2 45 .5 46 .0 45 .7 45 .3
3.46 2 .36 0.94 3 .43 2.38 0.97 3 .42 2 .39 0 .98
1 .13 1 .07 1 .02 1 .07 1 .04 1 .02 1 .04 1 .03 1 .01
We have carried out the calculations of the analyser, formed by two cones with a common top. In this case the potential distribution may be written in the analytical form 0/2 _ In tan - In tan 00/2 In tan 01 /2 - In tan 00/2 ' where 200 is the inner cone angle and 20, stands for the outer cone angle. We have obtained trajectory equations in analytical form assuming that (01 - 00 ) << 00 . The analyser in question is found to possess higher relative dispersion but larger linear magnification than the analyser with parallel electrodes . The data obtained agree with the results of the numerical calculations within 10% if 0 1 - 00 < 10 °. Let us consider now the electron-optical properties of the conical lenses shown in figs . 3 and 4. The lens given in fig. 3 consists of two cones and a charged particle beam moves between them . Each of the cones possesses two transversal cuts and the lens can be either einzel or immersion depending on the applied potentials . In the system given in fig. 4, axial trajectories of the beam are perpendicular to the cone-generating lines. In the former lens the number of electrodes is determined by the number of cuts. In the latter every additional electrode is formed by an additional cone . Here both einzel and immersion lenses can be built as well. The development of the focusing theory in such lenses presents an important problem. We have found the paraxial trajectory equation in a meridional plane to have the form y"+ 2$y'+
2~(~if +
x$, )Y = 2 ,
where O(x) is the potential distribution along the x-axis, the primes denote differentiation with respect to x, 01(z) is the potential derivative with respect to y on the x-axis. We assume $1(z) to be of the same order as y and y' . Consequently the paraxial trajectories are described by means of an inhomogenous linear second-order differential equation. When -01 = 0 it coincides with the paraxial trajectory equation of a transaxial lens in a vertical plane. It should be noted that eq. (7) does not depend on the cone angle 200 . However, when 00 = 0, which means that conical lenses transform into coaxial cylindrical ones, the paraxial equation takes another form (since the second term in the brackets, O'/x, vanishes), namely 20
2ss'
20'
The left part of eq . (8) coincides with the trajectory equation in two-dimensional lenses . As is well known, the general solution of the inhomogeneous linear differential equation is a sum of a general solution of the homogeneous equation and a particular solution of the inhomogeneous one. The general solution of the homogeneous equation describes the focusing properties of the lens. The particular solution of the inhomogeneous equation does not depend on the initial trajectory conditions and therefore describes the deflection of the beam as a whole. Hence the conical lenses converge charged particles, thus
S. Ya. Yavor, L.A. Baranova / Conical electrostatic analysing andfocusing systems
425
Table 2 Properties of a double-cone einzel lens
-0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.5 4.0 5.0
f,
X(A)
X(fo)
0.52 0.68 0.89 1.19 1.62 2.27 3.27 4.92 7.90 13.97 29.15 86.05 48.19 15.92 8.44 5.49 4.00 3.13 2.58 2.20 1.93 1.73 1.41 1.23 1.06
0.62 0.79 1.01 1.32 1.77 2.43 3.46 5.16 8.20 14.38 29.77 87.24 47.54 15.48 8.08 5.16 3.69 2.82 2.27 1.88 1.60 1.39 1.02 0.79 0.49
-0.56 -0.70 -0.90 -1 .18 -1 .59 -2.20 -3 .16 -4.77 -7.67 -13.63 -28.58 -84.90 -48.83 -16.32 -8 .73 -5 .72 -4.18 -3 .28 -2 .69 -2 .28 -1 .99 -1 .76 -1 .38 -1 .14 -0.84
X~
0.26 0.31 0.37 0.46 0.59 0.79 1 .13 1 .84 4.13 -45.51 -4.00 -2.25 -1 .04 -0 .92 -0.85 -0.83 -0.82 -0 .83 -0.85 -0.88 -0 .92 -0 .97 -1 .11 -1 .32 -1 .97
tan a -0 .015 -0 .011 -0 .008 -0 .006 -0 .004 -0 .002 -0 .001 -0 .001 0 0 0 0 0 -0 .001 -0 .002 -0 .002 -0 .003 -0 .004 -0 .004 -0 .005 -0 .005 -0 .006 -0 .006 -0 .007 -0 .006
forming circular images of point or circular sources. The term on the right-hand side of the trajectory equation causes the deflection of a charged particle beam and so changes the circular image diameter. The calculation of the potential distribution inside the conical lens shown in fig. 3 has been performed by an asymptotic method, which gives an approximate analytical result . The distance 2b between conical surfaces is assumed to be small compared with the radius rc, which is equal to the distance from the lens center to the axis z (e = b/r < 1). Numerical integration of the trajectory equation has been made on the basis of the potential distribution obtained . A set of two-electrode immersion and three-electrode einzel lenses has been studied. The cardinal elements of conical lenses as well as the deflection angle and the position of the deflection center are calculated. It is shown that the angle between the x-axis and the reference trajectory remains small for a wide range of lens potentials and as a rule is negative . Thus the lens slightly deflects the beam towards the symmetry axis, reducing the circular image radius. The position of the deflection center essentially differs from the positions of the principal planes . Due to geometrical asymmetry of the conical lenses the principal planes and the focal points are situated asymmetrically relative to the lens center C even in the einzel lenses . The main parameters of the einzel conical lenses with parallel electrodes for a wide range of the potential ratio are given in table 2. It should be noted that the electron-optical properties of such lenses and those with the common cone top are similar. Comparison of the lenses under study with ordinary round lenses shows that they are about two times stronger than the latter. Furthermore, round lenses strongly curve the axial trajectories of a hollow beam while coaxial conical lenses keep them approximately straight. References [1]
D.F.C. Brewer, W.R . Newell and A.C.H . Smith, J. Phys . E13 (1980) 114. [2] H.A. Engelhardt, W. Back and D. Menzel, Rev. Sci. Instr. 52 (1981) 835 .
421
Optics of conical electrostatic analysing and focusing systems S.Ya. Yavor and L.A . Baranova A.F. Ioffe Physico-Technical Institute of the Academy of Sciences, Polytekhnicheskaya 26, 194021 Leningrad, USSR
Energy analysis and focusing of hollow beams of charged particles may be performed with the aid of conical electrode structures . Two analyser geometries and two lens designs are described and the behaviour of some typical configurations is studied numerically. Some advantages of comcal lenses over round lenses of more conventional design are indicated .
Traditionally electron optics deals with rather narrow beams of charged particles, moving close to the optic axis . Such beams allow high focusing quality and are widely used in various devices. There are well developed mathematical procedures for calculating such devices with series expansions of fields and trajectories around the optic axis being used . Narrow beams provide comparatively low intensity; however, an intensity rise at the expense of beam widening inevitably results in degradation of focusing . A possible solution of the problem is to widen the beam in the direction of one of the coordinates . In this way, by using a proper lens system it is possible to increase considerably the intensity while preserving sharp focusing. Recently there has appeared a number of problems requiring the application of hollow conical beams with charged particles moving far from the symmetry axis . Due to the increase of one of the angular dimensions up to 360', such beams possess higher intensity than narrow ones . Moreover, the use of conical beams in materials investigations provides a large amount of information and allows some additional data to be obtained, for example, angular distribution of charged particles. The focusing and analysis of hollow beams require the design of special electron-optical systems whose electrodes best match the beam shape. In our opinion, the systems of coaxial conical electrodes embracing the beam from inside and outside are most suitable for this purpose. Electron-optical systems formed by coaxial conical electrodes may be used for energy analysis of charged particles or focusing depending on the prevalent field direction, determined by the mode of the potential supply. Schematic diagrams of conical analysers are shown in figs . 1 and 2. The generating lines of their electrodes can be either parallel or emerging from the same point. The advantages of such analysers are high gathering power and the possibility of simultaneous energy and angular analysis . It should be pointed out that the angular spectrum can be obtained without moving the sample or the analyser. Unlike the cylindrical mirror analyser, the conical mirror allows us to obtain a charged particle distribution depending on both azimuth and polar angles. In the latter case, the beams are disk-shaped. The description of an analyser with conical electrodes is given in ref. [1]. Such instruments are applied for energy and angular electron spectroscopy, which now is one of the main techniques for investigating surface electronic structure. For focusing hollow beams and for changing their energy if necessary, lenses with conical electrode shape should be used . Various schemes of such lenses are presented in figs. 3 and 4. If the potential supply is as shown in figs . 3 and 4, the electrical field within the lens is mainly longitudinal and in this sense the lens is analogous to round or two-dimensional lenses . Conical lenses form circular images of point or circular sources . By changing the potential supply, it is possible to create an additional radial field which curves the axial trajectory, changes the image radius and contributes to the optical power. The double conical lens shown in fig. 3 was used for focusing the beam coming from a toroidal analyser onto a circular detector [2]. 0168-9002/90/$03 .50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
422
S Ya Yaoor, L.A . Baranooa / Conical electrostatic analysing andfocusing systems
Fig. 1. Conical analyser with parallel electrodes .
Fig. 2. Conical analyser with concurrent electrodes.
We have investigated the electron-optical properties of both energy analysers and lenses, formed by conical electrodes . The characteristics of the conical analysers shown in figs. 1 and 2 have been studied . Analytical methods have been developed for their approximate calculations . This approach allows us to establish the basic features of the analyser behavior. The potential distribution in the analyser with parallel electrodes was found in the adiabatic approximation, that is, the potential variation along the x-coordinate was assumed to be small . Then the solution of the Laplace equation has the form
(VZ - Vl ) ln(1+y/(x tan eo)) ' ln(1 + g/(x tan eo)) 12
Fig. 3. Double-cone lens.
Fig 4. Triple-cone lens .
S. Ya. Yavor, L .A . Baranova / Conical electrostatic analysing andfocusing systems
423
where x, y denote local coordinates, shown in fig. 1. We get the approximate analytic expressions for trajectories by using the energy conservation law, by passing over from time differentiation to x-coordinate differentiation and then by performing a single integration: - arccosh
x - x0 = Ag cos fx
p tan 00 +
'2 -
B
(y/g)
+ arccosh
p tan 00 + i B ~
1
arccosh p
tan 00
B
tan00 + .l - (ylg) + arccosh p
1
x _< x x
~
xm,
1
where xm corresponds to the point of trajectory maximum, A = (2E/eV) p tan 00 ,
B=
V
- Vp tan 00 sin2ß + (p tan 00 + z
)2
,
V = VZ - VI
.
Here a large parameter p = x0/g is introduced, x0 is the coordinate of the point where a particle enters the field and g denotes the distance between the electrodes . Such a definition of the large parameter means that the analyser entrance slit is placed rather far from the cone top. Let us consider a general case in which a circular source and a circular detector are situated outside the field, their distances from the inner electrode being h 0 and h, respectively and H = h 0 + h,. We have found the distance L between the initial and final point of the trajectory measured along the x-axis to be L
Ag cos ,ß I arccosh p tan ß +
tan 000 + -
1-
Eq. (3) makes it possible to find easily the conditions of the first- and second-order focusing of charged particles. For this purpose, the first and second derivatives of L with respect to the entrance angle ß must be equal to zero : dL
âß
= 0,
d2L dß2 = 0.
From eqs. (4) we can find the value of the angle ß which provides the first-order and the second-order focusing . Substituting it into eq. (3) we obtain the distance d between the source and detector . Using these formulae we carried out some calculations, resulting in the following conclusions. If the source and detector are arranged on the inner electrode, only first-order focusing is possible . Unlike the homogeneous field, the focusing angle ß may be not 45' . For the given cone angle 200 , the focusing angle ß increases with the ratio of particle energy to the analyser potential (EleV) . Some results of the calculations are given in table 1 for H = 0 and p = 9.55. Second-order focusing occurs when the source or/and detector are placed outside the field. In this case the values of the focusing angle ß do not coincide with the value ß = 30' specific to the homogeneous field . With increasing parameter p, the characteristics of the conical analyser approach those of the homogeneous field. In particular, the first-order focusing angle becomes 45 ° and the second-order focusing angle becomes equal to 30'. Calculation of the relative analyser dispersion has shown that the approximation used leads to the following formula: D/d =1/(2 cos 2,8) .
The conical analyser dispersion is somewhat higher than that of a cylindrical or flat mirror due to the larger value of the focusing angle ß . It should be pointed out that the equation ß + 00 = iT/2 must hold when beams of a disk shape perpendicular to the analyser symmetry axis are used. The focusing onto the z-axis is possible only if ,ß > 00. When ß = 00 the beam leaving the analyser is parallel to the z-axis . IV . ELECTRON/ION OPTICS
424
S. Ya. Yavor, L.A . Baranova / Conical electrostatic analysing and focusing systems
Table 1 Properties of a conical analyser 80 [deg]
E/eV
/8 [deg]
d/g
D/d
30 30 30 45 45 45 60 60 60
1 .7 1 .2 0.5 1 .7 1 .2 0.5 1 .7 1 .2 0.5
48.3 47.0 45 .8 46 .8 46 .2 45 .5 46 .0 45 .7 45 .3
3.46 2 .36 0.94 3 .43 2.38 0.97 3 .42 2 .39 0 .98
1 .13 1 .07 1 .02 1 .07 1 .04 1 .02 1 .04 1 .03 1 .01
We have carried out the calculations of the analyser, formed by two cones with a common top. In this case the potential distribution may be written in the analytical form 0/2 _ In tan - In tan 00/2 In tan 01 /2 - In tan 00/2 ' where 200 is the inner cone angle and 20, stands for the outer cone angle. We have obtained trajectory equations in analytical form assuming that (01 - 00 ) << 00 . The analyser in question is found to possess higher relative dispersion but larger linear magnification than the analyser with parallel electrodes . The data obtained agree with the results of the numerical calculations within 10% if 0 1 - 00 < 10 °. Let us consider now the electron-optical properties of the conical lenses shown in figs . 3 and 4. The lens given in fig. 3 consists of two cones and a charged particle beam moves between them . Each of the cones possesses two transversal cuts and the lens can be either einzel or immersion depending on the applied potentials . In the system given in fig. 4, axial trajectories of the beam are perpendicular to the cone-generating lines. In the former lens the number of electrodes is determined by the number of cuts. In the latter every additional electrode is formed by an additional cone . Here both einzel and immersion lenses can be built as well. The development of the focusing theory in such lenses presents an important problem. We have found the paraxial trajectory equation in a meridional plane to have the form y"+ 2$y'+
2~(~if +
x$, )Y = 2 ,
where O(x) is the potential distribution along the x-axis, the primes denote differentiation with respect to x, 01(z) is the potential derivative with respect to y on the x-axis. We assume $1(z) to be of the same order as y and y' . Consequently the paraxial trajectories are described by means of an inhomogenous linear second-order differential equation. When -01 = 0 it coincides with the paraxial trajectory equation of a transaxial lens in a vertical plane. It should be noted that eq. (7) does not depend on the cone angle 200 . However, when 00 = 0, which means that conical lenses transform into coaxial cylindrical ones, the paraxial equation takes another form (since the second term in the brackets, O'/x, vanishes), namely 20
2ss'
20'
The left part of eq . (8) coincides with the trajectory equation in two-dimensional lenses . As is well known, the general solution of the inhomogeneous linear differential equation is a sum of a general solution of the homogeneous equation and a particular solution of the inhomogeneous one. The general solution of the homogeneous equation describes the focusing properties of the lens. The particular solution of the inhomogeneous equation does not depend on the initial trajectory conditions and therefore describes the deflection of the beam as a whole. Hence the conical lenses converge charged particles, thus
S. Ya. Yavor, L.A. Baranova / Conical electrostatic analysing andfocusing systems
425
Table 2 Properties of a double-cone einzel lens
-0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.5 4.0 5.0
f,
X(A)
X(fo)
0.52 0.68 0.89 1.19 1.62 2.27 3.27 4.92 7.90 13.97 29.15 86.05 48.19 15.92 8.44 5.49 4.00 3.13 2.58 2.20 1.93 1.73 1.41 1.23 1.06
0.62 0.79 1.01 1.32 1.77 2.43 3.46 5.16 8.20 14.38 29.77 87.24 47.54 15.48 8.08 5.16 3.69 2.82 2.27 1.88 1.60 1.39 1.02 0.79 0.49
-0.56 -0.70 -0.90 -1 .18 -1 .59 -2.20 -3 .16 -4.77 -7.67 -13.63 -28.58 -84.90 -48.83 -16.32 -8 .73 -5 .72 -4.18 -3 .28 -2 .69 -2 .28 -1 .99 -1 .76 -1 .38 -1 .14 -0.84
X~
0.26 0.31 0.37 0.46 0.59 0.79 1 .13 1 .84 4.13 -45.51 -4.00 -2.25 -1 .04 -0 .92 -0.85 -0.83 -0.82 -0 .83 -0.85 -0.88 -0 .92 -0 .97 -1 .11 -1 .32 -1 .97
tan a -0 .015 -0 .011 -0 .008 -0 .006 -0 .004 -0 .002 -0 .001 -0 .001 0 0 0 0 0 -0 .001 -0 .002 -0 .002 -0 .003 -0 .004 -0 .004 -0 .005 -0 .005 -0 .006 -0 .006 -0 .007 -0 .006
forming circular images of point or circular sources. The term on the right-hand side of the trajectory equation causes the deflection of a charged particle beam and so changes the circular image diameter. The calculation of the potential distribution inside the conical lens shown in fig. 3 has been performed by an asymptotic method, which gives an approximate analytical result . The distance 2b between conical surfaces is assumed to be small compared with the radius rc, which is equal to the distance from the lens center to the axis z (e = b/r < 1). Numerical integration of the trajectory equation has been made on the basis of the potential distribution obtained . A set of two-electrode immersion and three-electrode einzel lenses has been studied. The cardinal elements of conical lenses as well as the deflection angle and the position of the deflection center are calculated. It is shown that the angle between the x-axis and the reference trajectory remains small for a wide range of lens potentials and as a rule is negative . Thus the lens slightly deflects the beam towards the symmetry axis, reducing the circular image radius. The position of the deflection center essentially differs from the positions of the principal planes . Due to geometrical asymmetry of the conical lenses the principal planes and the focal points are situated asymmetrically relative to the lens center C even in the einzel lenses . The main parameters of the einzel conical lenses with parallel electrodes for a wide range of the potential ratio are given in table 2. It should be noted that the electron-optical properties of such lenses and those with the common cone top are similar. Comparison of the lenses under study with ordinary round lenses shows that they are about two times stronger than the latter. Furthermore, round lenses strongly curve the axial trajectories of a hollow beam while coaxial conical lenses keep them approximately straight. References [1]
D.F.C. Brewer, W.R . Newell and A.C.H . Smith, J. Phys . E13 (1980) 114. [2] H.A. Engelhardt, W. Back and D. Menzel, Rev. Sci. Instr. 52 (1981) 835 .