Time-dependent perturbation formalism for calculating the aberrations of systems with large ray gradients

LQL _ +I Nuclear Instruments and Methods in Physics Research A 363 (1995) 301-315 NUCLEAR

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INSTRUMENTS

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ELSEVIER SectIon A

Time-dependent perturbation formalism for calculating the aberrations of systems with large ray gradients

H. Rose *, D. Preikszas

Institut fiir Angewandte Physik, Technische Hochschule Darmstadt, Hochschulstra/3e 6. D-64 289 Darmstadt, Germany

Abstract A time-dependent perturbation formalism is outlined which enables the calculation of aberrations for systems with large

gradients of the particle trajectories in a consistent way. The position of a particle is referred to that of an axial reference particle making the axial coordinate a small quantity. The initial conditions of the trajectory are incorporated by transforming

the Lorentz equations in a set of coupled inhomogeneous integral equations for the three positional coordinates of the particle. The integral equations are solved by an iteration procedure which starts from the paraxial approximation. The number of required iteration steps is equal to the rank of the aberration minus one. As an example, the primary and secondary axial aberrations of mirrors are investigated in detail.

1. Introduction

Large ray gradients occur especially in the vicinity of

turning points at which the axial direction of flight of the particle is reversed. Examples for systems with such tum-

ing points are electron mirrors, ion traps and the magnetic bottle. The conventional theory of aberrations in static fields considers the z-coordinate of a charged particle as the independent variable measured along the optic axis.

Usually the optic axis is chosen to coincide with the central trajectory of a bundle of rays regardless of whether this trajectory is straight or curved. Derivatives with re-

spect to the time in the path equation or in the Lagrange function are replaced by derivatives with respect to the

z-coordinate utilizing the conservation of energy. Accord- ingly, the lateral position coordinates x = x(z) and y =

y(z) are functions of the z-coordinate instead of the time t. Unfortunately, the substitution of z for t also intro-

duces the derivatives

dx x’ = - dY

dz ’ y’= -&

into the path equations in a nonlinear form. As a result the paraxial approximation of these equations describes the motion of particles with a sufficient degree of accuracy

only as long as the condition

XI-=X 1, y’ -=X 1 (2)

* Corresponding author.

is fulfilled. This condition, however, is not necessary if we

linearize the Lorentz equation of motion

&)=q.(E+uXB), because both the electric field E and the magnetic induc- tion B are solely functions of the position vector r; m, q

and v are the mass, the charge and the velocity of the particle, respectively. Hence the paraxial approximation of the Lorentz equation (3) describes the motion of particles which are confined to the region of the optic axis with a sufficient degree of accuracy regardless of the slopes (Eq. (1)) of their trajectories.

Owing to this behaviour the z-coordinate cannot be

used as independent variable if the slopes of the trajectories adopt very large values, as it is the case in cathode

lenses or mirrors. In these cases it is mandatory to choose the time t or the equivalent length

r= 11,t (4)

as the independent variable [l-8]. The choice of the nominal velocity u, is arbitrary. A suitable possibility is the mean velocity u. at the object surface where the acceleration voltage has a fixed value cp = ‘p,,. Since we

shall consider in the following only electrons, we replace in Eq. (3) the charge q by the charge --e of the electron. Moreover, because the velocity of the electrons is small compared with that of light in mirrors or cathode lenses, we can restrict our investigations to the nonrelativistic case.

An extensive study of electron mirrors has been per- formed by Kelman and coworkers [9,10] and by other

016%9002/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved

SSDI 0168-9002(95)00065-8 IV. MICROSCOPES/ENERGY ANALYZERS

302 H. Rose, L?. P~~~~z~s/Nli~I~ Ins&. and Mefk. in Pkys. Rex A 363 ~1995~301-315

Russian authors [11,12]. Unfortunately, their investigations did not completely solve the difficult problem of finding aberration coefficients equivalent to those of standard systems. Moreover, their calculation procedures do not repre-

sent a consistent approach for calculating higher-order aberrations. The approach presented here goes consider-

ably beyond the early Russian work and avoids some weaknesses and inconsistencies of these papers.

The choice of T as independent variable introduces,

however, the difficulty that besides x and y also the

variable z depends on the starting values

of the ray at the object surface which may be planar or curved. In the latter case the initial z-coordinate z = ze of

a particle is a function of x0 and ye:

Z() =:~O(XO~ Yo> = 50 + ho. (6) Here lo = z,(O, 0) denotes the axial position of the vertex, and he = h&x,,, ye) is the height of arc taken at the

starting point. In the case of a planar object surface the height of arc vanishes (he = 0). The height of arc he is

positive if the object surface is convex with respect to the

z-axis. Derivatives with respect to the variable 7 are

denoted by dots.

2. Perturbation procedure

For mathematical convenience we describe the off-axial

position of a particle by its complex lateral distance

w=x+iy, W=x-iy. (7)

This complex coordinate

W = W X,, _Yo> f,, $0, K; 7 ( > w

Fig. 1. Propagation of the wave surfaces of an initially spherical

wave originating at the point zO on the axis. The particle rays are

the orthogonal trajectories to the wave surfaces, thus the actual particle and the reference particle are located at a fixed time f, on the same wave surface at the point P, and at the axial point 5,

respectively. The ideal wave surface S, yielding stigmatic imag-

ing at zi is shown for comparison.

is a function of 7, of the initial parameters (5) and of the relative energy deviation K of a particle from the average energy of the beam at the object surface. The expansion of w in a power series with respect to these parameters has the form

w = 5 w(1)(7), r= I

(9)

where W(‘)(T) is a homogeneous polynomial of rank I in

the expansion parameters. We define the sum r = n + 1 of the expansion parameters as the rank of the deviation. The

order n of the deviation is the sum of the exponents of the four geometrical parameters, while the degree 1 is the exponent of the chromatic parameter K. The first polynomial w(‘)(r) describes the paraxial approximation, while the other polynomials with r 2 2 are the deviations of rank n of the true off-axial position from the Gaussian approximation. However, the position of a particle at a fixed time

is defined by w and the axial coordinate

z=z ( x0, y,, i,, j,, K; r)

=6(T)++,), yo, i,, j’o, K; +

Here

(10)

6 = z(0, 0, 0, 0, 0; T) (11)

can be considered as the position of an axial reference

eiectron which starts from the vertex (centre) of the object surface simultaneously with the actual particle, as shown in Fig. 1. In this figure we have depicted the propagation of the wave surfaces of a homocentric bundle of rays originating from the centre of the object plane together with the trajectories. These trajectories are orthogonal to

the set of wave surfaces

S,=.s,(x,=O, yo=o, zo. x, y, z)=E&fv-l&

(12)

taken at a sequence of times t,, V= 1, 2,. . . , in the case of a purely electric field; En = mocZ is the energy of rest of the electron. Accordingly, all monoenergetic particles, which emanate simultaneously at the time 1= t,, from the point x0 = 0, y, = 0, zo, are located at time t = t, on the

common wave surface

S,=E,(t,--to). (13)

The intersection of this wave surface with the optic axis (X = 0, y = 0) defines the position < of the reference particle. Owing to the curvature of the wave surface the axial position of one of the other particles differs from that of the reference electron by its longitudinal deviation

h = h( Xg, J’o, 20, j’,, K; 7 =Z- 1 <= 2 h(‘)(T). T= 1

(14)

This longitudinal deviation coincides with the height of arc

in the case K = 0, where the reference electron and the

H. Rose, D. Preikszas /Nucl. Instr. and Meth. in Phys. Res. A 363 (1995) 301-315 303

actual electron are located on the same wave surface. The arc between the optic axis and the position of the electron

is part of the curve formed by the intersection of the wave surface with the w, z plane. In the general case each function h(‘)(7) is a homogeneous polynomial of rank r in

the ray parameters xu, y,, &, )‘a and K. All polynomials are zero if the wave surface degenerates either to a plane

perpendicular to the optical axis or to a point representing

an ideal image of the conjugate object point. In the case of a monoenergetic homocentric axial bundle of rays the

first-order height of arc, ho), is zero, because at a given

time all particles are then located on the same wave surface, as demonstrated in Fig. 1. Since this surface can be approximated by a paraboloid in the vicinity of the

optical axis, the nonvanishing second-order height of arc, h(‘)(r), depends on the curvature of these paraboloids along the optic axis and on the square of the initial lateral

velocity 1 tiL, 1 of the actual particle. Accordingly, the deviation h(z)(r) can be considered as a small quantity of order two. This height is negative as long as the wave surfaces

are concave with respect to the direction of flight. After

traversing the focusing electric field the curvature of the

wave surfaces may change its sign. As a result the function h(a)(~) becomes positive.

Employing the complex notation of Eq. (7) together with the modified time variable (4), the nonrelativistic Lorentz equation for an electron moving in a static electric field can be written as

1 acp *=----, ‘Pi aw

(15)

1 39 y=-- 29, a2’

(16)

The gauge of the electric potential cp = cp(x, y, z) is

chosen in such a way that cp = 0 for v = 0. A first integral of these equations is the well-known conservation of en-

ergy

cp AE +~+i’=_+-=~+K.

(PO ecpo (PO (17)

Here AE is the deviation of the kinetic energy of the particular electron from the average kinetic energy of the beam. At the surface of the cathode the latter energy represents the mean escape energy

(18) The surface of the cathode represents a surface of constant

potential cp = ‘pc. To solve the two coupled path equations (15) and (16),

we employ a three-dimensional perturbation procedure. For this purpose we expand the electric potential

‘p= cp(& Y, 2) = cp(X? Y, l+h) (19)

in a power series with respect to the small quantities x, y and h. Furthermore, we consider in the following only

electrostatic systems with rotational symmetry. With the

aid of the Laplace equation, we obtain the expansion

cp= i i (_)” @[2n+m1 “‘i; ‘h”‘, i 1 (n!)‘m! 4

(20) n=O m=O

where

@=@(C)=cp(x=O, y=o, l) (21)

represents the axial electric potential at the position l= z

- h of the reference electron. The upper index [2n + m]

denotes the (2n + m)th derivative with respect to 5. For reasons of simplicity we indicate in the following derivatives with respect to 6 by primes, while those with respect

to T, which is defined in Eq. (4), are denoted by dots. Since the object surface is an equipotential, we have

‘p. = cp(x,, y,, z,) = ao. The reference electron starts

with the mean kinetic energy (AE = 0) from the centre of the object. Hence its motion along the optic axis is gov-

erned by the equation

The first integral of this equation,

@(Cl 42=7-1 0

can be used to determine T as a function of 5:

T-To= qy

(22)

(23)

(24)

The lower signs have to be taken if the particle has been reflected at the turning plane 5 = lT and moves in the

negative z-direction. The direction of the initial velocity of the axial reference electron determines the direction of the optical axis. Accordingly, the z-axis points towards a mirror, yet away from the surface of a cathode lens. The turning point lr is given by @( lr) = 0.

Considering the relation (22) and the expansion (20), we can write the path equations (15) and (16) in the form

1 a” k+--w=u,+,,

4 @o

i;_f$h=uh 0

The perturbations a

uw= + ii c Hb%o%o- 1) 0 n=O m=O

@pn+Z+m] - n

X n!(n + l)!m! ( 1

“” h”, 4

(25)

(26)

IV. MICROSCOPES/ENERGY ANALYZERS

304 H. Rose, D. Preikszas / Nucl. Instr. and Meth. in Phys. Rex A 363 (I 995) 301-315

uh=$ k F (-)‘1(1-6n06,,, - %0%1) 0 n=n m=O

@7n+l+m] - n

X (n!)‘m! ( 1

y h”, (27)

contain all nonlinear terms in w,W and h;

6 = 1 forF=V, &!J

( 0 else,

is the familiar Kronecker symbol. The set of equations (25) for the motion of an electron serves as the basis of our

perturbation procedure. These differential equations are valid for an arbitrary electron. In order to obtain the path

of a distinct electron defined by its initial conditions (Eq. (5)) we must incorporate these conditions into the equation

of motion by transforming Eq. (25) in a set of inhomoge-

neous integral equations. To perform this transformation

the paraxial solutions must be known.

3. Paraxial approximation

The paraxial approximation of the path equations are obtained by neglecting in Eq. (25) the nonlinear terms ow and a,,. The solutions w = w, and h = h, of the resulting

homogeneous equations

@,B+f@“w=o, (29) @,i - @“h = 0, (30)

describe the Gaussian dioptrics and form the basis for the

subsequent calculation of the aberrations which employs the method of successive approximation. To illustrate the

equivalence of the representation (30) with the conventional form of the Gaussian path equations in the case of a rotationally symmetric electrostatic field, we replace in Eq. (29) the time variable r by the axial coordinate 5 of the reference electron. Utilizing the second relation in Eq.

(24), we find for the paraxial motion of a charged particle

@w’I + @‘w’ + +@‘Iw = 0. (31)

Fig. 2. Definition of the initial parameters of a particle trajectory.

According to the familiar procedure of Gaussian dioptrics,

we introduce a pair of linearly independent real solutions of Eq. (29),

w, = w,(r) = Rja(T), WV = w,(r) = WY(r), (32)

which satisfy the initial conditions

u$(ro) = 0, Gm(r,) = 1.

u$(T”) = 1, tiJT”> = 0. (33)

Hence the Wronskian of this pair of solutions is equal to unity:

w,tia - w,$ = 1.

The paraxial solution of a particular trajectory

(34)

w, = ‘ywL, + yw,, (35)

with complex constants (Y and y, is determined by the condition that the position and the direction of the paraxial

ray coincide with those of the actual ray at the object point

wO, zO. The direction of the ray at this point is determined by the lateral normalized velocity component

tiL(~~) = w’( 5”) = sin 0” exp(i80), (36)

where B,, is the starting angle enclosed by the tangent of the ray and a straight line parallel to the optical axis, as depicted in Fig. 2. The azimuthal angle 8” determines the direction of the complex off-axial unit vector exp(i8”). It

should be noted that the occurrence of sin 0, instead of the familiar tan B,, in the initial conditions is a direct

consequence of choosing the time as the independent variable.

The positions 5, of the Gaussian image planes are determined by the zeros 7i = T,( 5,) of the axial fundamen-

tal ray W,(T). Assuming that the actual particle starts at time T= TV from the point wO, zO, we find for the complex constants (Y and y the linear dependence

“=+,=bi’(To),

T= W. = W(To). (37)

Accordingly, the complex lateral component w of the position vector (w, z) of the particle is found in first-order

approximation as

w = wr = w(1) = tiOw, + wOwv. (38)

To localize the particle at a fixed time in space we must also know the z-component of the position vector. Since we expand the ray coordinates with respect to

wo> wo, ti,. i$, and the chromatic parameter K, the solution

h, = zr - < (39)

of the linear differential equation (30) comprises the first- rank deviation ho) and contributions to the higher-rank terms I&‘). This behaviour results from the fact that the solutions of the paraxial Eqs. (29) and (30) must satisfy the

H. Rose, D. Preikszas / Nucl. Instr. and Meth. in Phys. Res. A 363 (1995) 301-315 305

initial starting conditions of the true ray at the starting

point wO, 2,) = &Substituting derivatives with respect to T

for those with respect to 5 in Eq. (30), we eventually obtain the equivalent equation

which can be readily integrated. The result

h = h,(T) = p/z, + v/z, (41)

is a linear combination of the two particular solutions h, and h,, which can be written as

k, = k,(T) = (**@)‘yf , l/2

k,=k,(7)= f ) i ! 0

(42)

if @ > 0. The Wronskian of these fundamental solutions has the form

rz”h, - h/J,, = 1. (43)

The two real integration constants p and v are determined by the initial conditions.

Care must be taken in the case of a mirror. In this case h, changes its sign at the turning point c= lT = <(TV) of

the reference electron where the axial potential is zero

(@P( CT) = 01. A~ordingly, h, becomes an antisymmetric function with respect to the turning time T= TV. In order

that k&T) represents the symmetric solution k, (7) for the mirror, we must determine the lower integration limit r& = TV in such a way that h,(~~) = 0. For this purpose it is more convenient to choose 5 as the integration variable resulting in

The lower sign has to be taken after the reflection (7 > ~~1. It should be noted that the last expression on the right hand side is valid only if the axial field strength @’ does not vanish in front of the mirror between lT and &,. Using this relation, we obtain from the condition A,( CT) = 0 the implicit equation

for the position &. Hence J& defines a plane close to & such that 115’ < 0 in the range 5, I 5 5 &. Expanding the

1 tronsmrssion mode

1 reftection mode .-

,-+- ,’

Fig. 3. Schematic potential distribution of a symmetric three-electrode lens and course of the primary fundamental longitudinal deviations h,, h,, and h,, b, for both transmission and reflection of the particle.

electric potential about the turning point, the integration of

the first expression in Eq. (42) yields for k& JT) the value

regardless of the lower integration Iimits Q in Eq. (42).

This simple result, which can also directly be derived from the Wronskian, Eq. (43), clearly reveals that the mirror cannot reflect an axial particle if both @(YT) and @‘(CT) are zero. In this case lT represents a saddle point of the potential. The axial reference electron, heading

towards the mirror, then comes to rest at this point after an infinitely long time.

To elucidate the behaviour of the ~ndamental Iongitu-

dinal paraxial deviations k,, and k,, we consider a symmetric retarding einzel lens consisting of three cathodes. The outer electrodes are held at the potential cp = Qi,,

while the potential ‘pm of the middle electrode can be chosen arbitrarily. In the case a,,, > 0 the system acts as a conventional round lens. In this case the fundamental deviation k, is symmetric with respect to the midplane f. The other fundamental deviation k, may be chosen as the antisymmetric solution of Eq. (301, as shown in Fig. 3.

If the potential cp, is chosen in such a way that the axial potential @ becomes negative in the region & < 5 2 &., the system acts as a mirror. To demonstrate the ~ansition of the lens from the tr~smi~ion mode to the reflection mode, we have also depicted in Fig. 3 the


306 H. Rose. D. Preik.mn/Nucl. Insw. and Meth. in Phys. RES. A 363 (199.5) 301-315

conjugate deviations & and i,, of electrons moving in the negative i-direction. For the transmission mode these devi-

ations are related to h, and h, by the simple relations

&=hi, and h,= -h,.

When going to the reflection mode the fundamental deviations become imaginary in the region 5, I g< f.r

and, therefore, each deviation splits up into two branches located symmetrically (h,, ia,) or antisymmetrically (hP, h,) about the centre of the midplane [=I &,,. For the

reflection mode the fundamental deviation h, can be con-

sidered to consist of two branches; the upper one (T I or) arises from the h, of the transmission mode, the lower one

(r> 7.r) arises from the i, of the transmission mode.

Hence considering h,, = h(7) as a function of time this

deviation has become an antisymmetric function with re-

spect to the turning time rl. = T( lT), although each branch belongs to a symmetric solution with respect to the sym-

metry plane &,, of the system. The equivalent considerations for h, and & show that the resulting fundamental

deviation h, for the reflection mode consists of the branch h, for T I rr and $, for T 2 rr. Accordingly, h, = h,(7) becomes a symmetric function with respect to the turning

time rr. Owing to these considerations the form of the funda-

mental rays depends on the existence of a turning point. In

the presence of a turning point the fundamental solutions

are

by(r) = (@/@o)“z, for 7171,

-(@/@u)“‘, for T> TT, (47)

where &, is determined by Eq. (45). All other solutions are linear combinations of these fundamental solutions. Hence

these solutions also require the knowledge of 5,. This value need not to be known in the absence of a turning

point, as in the case of ordinary round lenses. For such systems the relation

d&= (@/@,,)l’z dr (48)

holds for any time. Moreover, the solution h, does not change its sign because @ has no zero, apart from the cathode plane. Accordingly, any linear combination of the two fundamental solutions (47) yields another solution hP which can be expressed as

(49)

regardless of the time r. The constants JL and Y of the solution (41) are ob-

tained by considering that the paraxial approximations for the position and the velocity of an electron should coincide with the true values at the starting time r = ~a. In the case

of a curved object surface the initial height of arc differs

from zero:

h(T,,)=htt=Z1,-~o=ho(W~,W”, <a)= c h’,“‘. t* = 2

(50)

The initial velocity /;(~a) = &, is obtained from the conservation of energy, Eq. (17). Considering further the

relation (361, we eventually find

A, = C kg) sit2 1 - sin2eo + E i

l/7

_ 1

r= 1 ecp0 1 =;K-$iJo Go- +K2+ . . . . (51)

where

AE AE K=----=_

ecp, e@, is the chromatic parameter which accounts for the energy deviation AE. The comparison between the second and fourth expression in Eq. (51) yields

/$’ = I, 2 -

jp = _ f+o$o _) .I *K? (53)

The longitudinal deviation (41) with h, = h, must satisfy

the initial conditions (50) and (51). Considering further the

Wronskian (43), we find the solution:

Introducing the fundamental solutions

h, =h,,h, - &,o&,

h, = &ah, - h,ah,, (55)

the paraxial longitudinal deviation (54) adopts the familiar form

h, = h,h, + h&, w

where h, and h, satisfy the standard initial conditions

h,, = h,, = 0, ir,, = h,, = 1. (57)

The relations (50), (51) and (53) demonstrate that 12, and & are functions of the expansion parameters wo, Eo, +,, Go and K. Since he is at least of order two in wn and i?,, this quantity does not contribute to the first-rank approximation

ho’= j@h, = f,& .r (58)

which depends only on the chromatic parameter K. Hence for monochromatic electrons with K = 0, the axial position 6 of the reference electron represents within the frame of validity of Gaussian optics the z-coordinate of all other electrons which start at the same time t = to from the object plane. This behaviour becomes obvious if we con-

H. Rose, D. Preikszas /Nucl. fnstr. and Me&. in Phys. Res. A 363 ff995f3OI-315 307

sider that in paraxial approximation the curved object

surface and the wave surfaces are replaced by their tangen-

tial planes. The curvature of the rays at these planes is proportional to the curvature of the electric potential on the

optic axis.

4 Positional deviations

The higher-rank deviations w(‘) and h”’ of the actual

position of a particle from its paraxial position w(l)(~), h(‘)(7) are obtained most conveniently by transforming the

path equations (25) into integral equations. For this purpose we consider the nonlinear terms a,,, and uh as known functions of the modified time 7. In this case the

path equations (25) represent a set of linear inhomoge-

neous differential equations which can be solved using the

method of variation of parameters. As a result of the somewhat lengthy yet straightforward calculation we even-

tually find

f

7

J

T w = w* + w a a,,wv dT-- wr u,w, d7, (59)

70 To

h = h, + h, jTrr,h,, d,r-- h, jTg,,la, dr. (60) 70 Tll

Since a, and a, are functions of w, W and h, Eqs. (59)

and (60) represent a set of two coupled inhomogeneous nonlinear integral equations. Such integral equations can be solved approximately by an iteration procedure. The inhomogeneous terms are the paraxial solutions wi = w(l)

and k,.

In the first step we substitute in CT, and a,, the first-rank approximations (38) and (58) for the unknown

position coordinates w and h, respectively. For the second-rank deviations w(‘) and h(*’ we need to consider only the resulting second-rank terms

(2) - 1 @” 1 @“’

cr, _ - - _,(l$j”’ + - -/@,

8 @o 4 @o

(61)

(62)

of Eqs. (26) and (27) for the perturbations crW and u,,, Replacing in Eq. (59) gW by Eq. (611, we obtain for the second-rank lateral deviation the relation

(63)

This deviation yields the well-known first-order and first- degree chromatic aberration, because h”’ is proportional to K according to Eq. (58). To find the longitudinal second-

rank deviation ht2) = hc2)(7), we must remember that the

solution h, also contains a second-rank contribution. This

part is obtained from Eq. (56) by considering the relations

(50), (52) and (53) as

h\*‘=h~‘h,- ($$&+ $K*)h,. (64)

The coefficient hg’ is of second order in w0 and W,,. It

considers the curvature of the object surface at the optic axis. Hence hf) is zero if the object surface degenerates to

a plane. Taking into account the contribution (64), the total

longitudinal second-rank deviation has the form

where the function a,(‘)(~) is given by Eq. (62). The second-rank and the third-order geometrical devia-

tions are of prime concern in many fields of charged particle optics, because these deviations produce at the

Gaussian image plane zi the aberrations that limit primar- ily the performance of the instmment. According to the

Scherzer theorem the axial chromatic aberration and the third-order spherical aberration are unavoidable in static rotationally symmetric systems if the axial velocity does not change its direction (i > 0). Since these aberrations

decisively limit the resolution of electron microscopes, one seeks appropriate means to correct for these aberrations.

Although the corresponding path deviations cannot be eliminated everywhere, it suffices to place their zeros at

the Gaussian image plane. In this case the aberrations

vanish and thus a corrected image is formed at this singu-

lar plane. The incorporation of a mirror and a beam separator into an electron microscope is a promising possibility of achieving such a corrected image [13-151.

The lateral third-rank deviation ~(~~(7) from the paraxial trajectory is obtained in the second iteration step by inserting w(l) + w@) for w and h(l) + h(") for h in Eq.

(26) for the perturbation cWi,. Retaining only the third-rank terms, we obtain for the third-rank perturbation the expression

&3, 1 4p’”

= w 32 Fw(l)( ,(‘)-,(l) - 4h”‘2)

0

where w(*) and h(‘) are given by Eqs. (63) and (65), respectively. The third-rank lateral deviation wc3) = W(~)(T) is obtained by substituting Eq. (66) for a, in Eq. (59).

In the case of static fields the time can be considered as an auxiliary variable which may be eliminated in the final result. Owing to the relation 5 = L(T), the axial position of the reference electron can also serve as the independent

IV. MtCROSCOPES/~~RGY ANALYZERS

308 H. Rose, D. Preikszas / Nucl. Ins&. and Meth. in Phys. Res. A 363 (1995) 301-315

partlcle \;trajectory

wove surface

Fig. 4. Illustration of the difference between the complex lateral coordinate w = w(l) = W(T), considered as a function of time 7 or of the position b of the reference electron, and the coordinate w,(z = I), where W,(Z) represents the familiar off-axial coordinate of a particle as a function of its axial position z.

variable instead of the time r = ~/ua. However, care must be taken in the case of a mirror. Due to the reversal of flight two different values of T exist for each given 5, as

illustrated in Fig. 3 and/or Fig. 6. It should be noted that w(r) = w(r( 5 )) := w( 5 ) represents the lateral position of the actual electron when the reference electron is at the axial position 5. Therefore, W( 5) does not represent in

general the off-axial position W,(Z) of the particle at the plane z = 5, as shown in Fig. 4. Accordingly, the lateral

deviations wcr’( 5) of rank r will also differ from the

corresponding deviations w~~‘(z> at the plane z = 5. To derive the relations existing between these different

deviations, we consider that the z-coordinate of the particle is given by the relation

Z=~+h(W,, &, i’,, i&,, K; 6)

= f+ c h”‘(S). (67) r= 1

The inverse function i= ((we, Wa, tia, $a, K; z) depends on the actual z-coordinate of the particle and its initial ray parameters. Unfortunately, the implicit equation (67) for 5 cannot be solved directly. However, we can express the solution as a series by utilizing the Lagrange

inversion formula:

{=z- 5 (-)” l d”‘(V”+ ‘)

m=O (m+l)! dc”’ {=I

=z-h+hh’-$Af-hh’2+.... (68) In the last expression the function h and its derivatives are conceived as functions of the z-coordinate. To obtain the deviations w~~‘(z> we insert the series (68) for 5 into the argument of the terms wcr)( 6) of the representation

oc cc

w,(z) = c wy (z) = c w(‘)( i(z)).

By expanding each term w “I( i(z)) in a Taylor series about the point 5 = z and by reordering the resulting terms

with respect to their rank, the comparison between terms

with equal rank in the two sums yields the following expressions for the first three lateral deviations w!~’ = w(‘)(z): ?

w(r) = wW( 5 = z) = w(1),

w!2) = w(2) _ w”“fp~ (70)

w(3) = w(3) _ W(2)‘/p) _ ,cl”p I

+ Ww/p)jp )’ + +,(I T/+1?

According to Eq. (63) the second-rank deviation w(z)

depends on the first-rank longitudinal deviation ho). Since

h’“, Eq. (58), depends solely on the expansion parameter K, the geometrical deviations adopt the simple form

W,!“( Z, K = 0) = 0,

W,!“‘(Z, K=o)=W@)(c=Z, K=o)-W”“({=Z)

h”‘( l=Z, K= 0). (71)

It should be noted that the derivatives on the right hand side of Eqs. (70) have to be taken with respect to 6.

Subsequently the argument IJ’ is put equal to z, as illustrated in Eq. (71).

To check the validity of our results, we assume that the observation plane is located in the field-free region behind

the lens or mirror. In this case h = h( 5) is a linear function of 5. Hence all derivatives of second and higher

order vanish. Considering this behaviour, we derive from the inversion formula (68) the connexion

lzz- 2 (-)mhh,m=Z- h d5 -=z-h-

l+h dz (72)

m=O

between 5 and z. In the field-free region the trajectory of

a particle is also a straight line. Therefore, the lateral distances w;(z) and w( 5 = z) shown in Fig. 4 are connected with each other as follows:

d%(Z) w(&-=z)=w,(z)+-

dz h(l=z). (73)

On the other hand the relation

w*(z)=W(I(Z))=w(z-h~) (74)

must hold true. Expanding the expression on the right hand side in a Taylor series we find

dw(5) W,(z)=w(S=z)-7

(=I h([=z);

dw,( z) =w(&=z)-- dz h(L=z)>

r= 1 r=l which coincides with Eq. (73).

H. Rose, D. Preikszas / Nucl. Instr. and Meth. in Phys. Res. A 363 (I 995) 301-315 309

5. Axial aberrations

Each aberration of rank r is a monomial of the lateral

deviation wi”( z) taken at the Gaussian image plane 5 = & = ii. At this plane the axial fundamental ray wa vanishes.

The value wrl = wr( <,) = M of the field ray wr deter-

mines the magnification M of the image. The magnification is negative if the object is imaged upside down measured in the frame of the spatially fixed coordinate

system. A special and important class of aberrations are the

axial aberrations formed by a pencil of rays originating at the centre (w, = 0, to = z,) of the object. In rotationally

symmetric systems these aberrations enlarge each image

point to a disc. The most important axial aberrations are

the first-order and first-degree chromatic aberration

W$( Zi) = - Kti’OW& (76)

and the third-order spherical aberration

w:$‘( Zi) = *;i&w,,c,. (77)

The aberration coefficients C, and C, = C, determine the magnitude of the corresponding aberration, apart from the

parameters tie, G,, and K. These coefficients have been

defined in such a way that for ordinary round lenses C, > 0 and C, > 0 according to the Scherzer theorem. As a

consequence the outer zones of round lenses always refract the rays more strongly than the inner zones. The positive

value of C, results from the fact that the faster electrons

are less affected by the focusing action of the lens than the

slower ones. The aberration coefficients are usually expressed as integrals which can be written in numerous

ways by means of partial integrations. Although these representations appear very different, they are equivalent. Accordingly, the formulae for the aberration coefficients derived by using the time-dependent perturbation formalism must yield another equivalent representation for each of these coefficients in the special case of ordinary round

lenses. An important difference between the conventional tra-

jectory and eikonal methods and the time-dependent per-

turbation method exists with respect to the number of

iteration steps needed to derive a certain aberration. For example two iteration steps are necessary for obtaining the third-order aberrations by the time-dependent perturbation method, because two coupled integral equations must be solved.

To derive the integral expression for the axial chromatic aberration coefficient C,, we start from the corresponding second-rank deviation (70) taken at the Gaussian image plane [ = zi. Considering further the relations (63), (38) and (58) we obtain

w!‘,‘( Zi) = wi2’( zi, wa = 0)

Wri -w;h, dr- +w;&

If we multiply the second term in the parenthesis by the

Wronskian wvltiaLoli = 1, the comparison of the resulting

expression with Eq. (76) yields

(79)

This integral expression is valid for arbitrary rotationally

symmetric electrostatic systems which do not contain space charges in the region of the beam. Hence in the special case of a conventional round lens Eq. (79) must be another

equivalent representation for C, of round lenses. To prove

this conjecture we consider that in the absence of a turning

point r is a single valued function of 5 which can directly be replaced by this coordinate using Eq. (48).

In the absence of a turning point the longitudinal

fundamental deviation h,, defined in Eq. (55), adopts the

simple form

For transforming the integral expressions of the aberration coefficients, the relation

/ VW,’ d{= .“w,2 + 4@wL2, (81)

will be repeatedly utilized. This result can be verified by

partial integration with respect to @“’ and by subsequent

use of the paraxial equation (31) in the integrand of the resulting integral, making the integrand a total differential.

Inserting Eq. (80) for h, into the integral (79) and consid-

ering Eq. (81), the resulting double integral can be trans- formed by partial integration:

l/2 =,

h,(@“wz + 4@w;*) 5= 20

- @3/l 0 d5.

(82) Substituting the right hand side of this equation for the

integral in Eq. (79) and considering h,( z,) = 0, we obtain the familiar representation

Here the second aberration integral for C, has been derived from the first by partial integrations.

For deriving the aberration coefficient C, of the third- order spherical aberration, we must perform two iteration steps because the integrand (66) of the lateral third-rank deviation W(~)(T) is a function of the paraxial deviations


310 H. Rose, D. Preik.szas/Nucl. lnstr. and Meth. in Phys. Res. A 363 (1995) 301-315

w(r) and h(r) and of the second-rank deviations &’ and h”‘. For an axial trajectory with u’a = 0 and K = 0 the

deviation #’ vanishes, while h”’ reduces to the monomial

hc2) = r-+a$,h -(T) = sin2f3, h - ua aa, (84)

where

(85)

is the axial secondary fundamental longitudinal deviation.

To obtain this expression we have replaced in Eq. (65)

the solution h, by h,. Such a change is always possible, provided that the constant of the Wronskian does not

change, because the variation of parameters is valid for any pair of linearly independent solutions of the corresponding paraxial equation.

For T< rr, Eq. (85) can be considerably simplified by

partial integrations employing Eq. (81). As a result we

obtain

(86)

for ra < r I or. At the limit 5 + cr the expression on the

right hand side adopts the indefinite value of 0 @ ~0. How-

ever, if we carry out this limit properly by expanding @ in a Taylor series about the turning point lr, we obtain the specific value

(87)

This result becomes rather obvious if we consider that it

represents the axial distance of an electron from the vertex 5-r of the equipotential cp = 0 if the electron is reflected at

the off-axial distance wr = WY) = war. Using the power

series expansion of the equipotential cp = (or = 0 of the mirror about the vertex 4’= CT, w = 0:

(or = @;h, - +@;wTWT + . . = 0, (88)

we realize immediately that

(89)

represents the height of arc of the point wr = WY), z = z$?’ located on the paraboloid of revolution which approxi- mates the equipotential cp = (pr = 0 in the vicinity of the optical axis.

The deviation of an axial ray

l i

wyiwOwO Ts = -/ (8@"'h,, - @‘“w,“)w; dr,

32@,, T0

(90)

is a monomial of third order in JJ,, and i$,,. Considering

further the definition (77) for the third-order spherical aberration coefficient and Eq. (711, we find that

(91)

This formula is also valid for conventional round lenses.

Therefore, it must be possible to transform Eq. (91) for C,

into the conventional forms obtained by means of the

standard trajectory procedure or the eikonal method. As-

suming that no turning point exists between the object and

the image plane, Eq. (86) for the second-order longitudinal fundamental deviation h,, is valid throughout the system CT> ~a). The comparison of Eq. (86) with the integral expression (83) for the coefficient C, yields

(92)

Hence the axial second-order longitudinal deviation at the

Gaussian image plane is directly related with the lateral axial chromatic aberration at this plane.

Employing Eqs. (81) and (861, we can transform the first integral in Eq. (91) as follows:

l /Z’ ( +8

VW; + 4@w:?)*

20 ~3/2 dl. (93)

Inserting this relation into Eq. (91) and considering further the relations w,(zi) = w,(z,) = 0, h,,(z,) = 0, we find that the first term on the right hand side of Eq. (93) cancels out the last term in Eq. (91). The remaining integrals can be put in the form

(94)

which is one of the well-known integral representations for the coefficient of the third-order spherical aberration. The partial integrations of the general Eqs. (79) and (91) for C, and C,, respectively, produce powers of @ in the denomi- nator of the integrands of the corresponding integral Eqs. (83) and (94). Since these partial integrations are allowed only if @ has no zero between the object and the image plane, the resulting constituents of Eqs. (83) and (94)

H. Rose. D. Preikszas/Nucl. Instr. and Meth. in Phys. Rex A 363 (1995) 301-315 311

diverge if this condition is not fulfilled, as is the case for

mirrors.

6. Mirrors

The incorporation of a mirror into an imaging system offers the possibility, at least in principle, of forming a

perfect image. To demonstrate this behaviour we consider

a pencil of rays emanating from the centre (wa = 0, z = z,) of the object plane. In the field-free region in front of the

lens the wave surfaces are spheres, as illustrated in Fig. 5.

After the electrons have passed through the lens their

trajectories do not form any longer a homocentric bundle of rays owing to the lens defects. Correspondingly, the shape of the wave or eikonal surfaces differs in the region behind the lens from that of a sphere. Moreover, continuous eikonal surfaces do not exist in the region of the

caustic where the rays intersect each other. However, continuous eikonal surfaces exist again behind the Gauss-

ian image point 2,. In this region they form a set of concave non-spherical surfaces.

Now let us assume that a retarding electric field slows down the electrons in this region. If we can manipulate the

equipotentials in such a way that the equipotential cp = 0

coincides with the surface S = S, of the eikonal, all electrons come momentarily to rest at this turning surface and reverse their direction of flight. Hence all electrons

travel back along their initial trajectories of incidence, thus forming a perfect image point at the origin w,, = 0, z = ze. To separate this image from the object an ideal magnetic beam splitter must be placed in front of the lens. In

practice the beam splitter is incorporated between the objective lens and the mirror such that the reflected rays

pass through a subsequent projector lens instead of the objective lens [14,X]. In addition the mirror must compen-

sate for the aberrations of all points of the imaged object area. Hence it is generally not possible to obtain simultane-

ously perfect images of all object points. However, the image quality is already considerably improved by correct-

_

t,

Fig. 5. Action of an ideal mirror on a pencil of rays emanating

from the centre aa of the object plane and traversing the field of a

round lens. Ideal reflection is obtained if the equipotential cp = (pr

= 0 of the mirror is exactly matched to the surface S, of constant

eikonal.

Fig. 6. Typical arrangement of the electrodes of a concave mirror

and the schematic course of the symmetric and of the antisymmet-

ric paraxial fundamental ray.

ing the primary aberrations discussed in the preceding chapter. Since mirrors can compensate for C, and C,, these elements have been repeatedly proposed as correc-

tors in the past [4,7,13]. The mirror does not introduce field aberrations, such as

first-order chromatic distortion and third-order coma, if it

operates in the symmetric mode. For this mode an image of the diffraction plane is placed at the turning plane

5 = (r and the object plane is placed in front of the mirror

in such a way that this plane is imaged with unit magnification onto itself. In this case the fundamental axial ray

W,(T) = W,(T) is symmetric with respect to the turning time or, while the field ray wY(r) = W,(T) is antisymmetric. As shown in Fig. 6 the mirror inverses the direction of flight of w,. Accordingly it travels along the same trajectory when heading towards and away from the mirror. The two branches of the field ray are located symmetrically

about the JJ’ axis. They meet each other at the turning point

i= ir. The intersection point 5 = cc of the symmetric funda-

mental ray w, with the optical axis can be considered as

the centre of curvature of the electron mirror. For a convex mirror the symmetric ray does not intersect the optic axis.

In this case the point of intersection of the asymptote of w, with the optic axis can be considered as the centre of

curvature of the convex mirror. The mirror usually acts as a concave mirror if @” > 0 in the region of the turning point. Hence, to achieve a positive curvature of the equipotentials in this region, the last mirror electrode should be curved towards the electron beam, as depicted in Fig. 6.

6.1. Model potential

To obtain an understanding of the imaging properties of a mirror, a potential model yielding analytical solutions for the paraxial path equations (29) and (30) is extremely helpful. An appropriate model is furnished by the axial potential


312 H. Rose. D. Preikszas/ Nucl. Instr. and Meih. in Phys. Res. A 363 (1995) 301-315

which has been studied by Glaser and Schiske [16] for the transmission mode k’ < 1 in the context of electrostatic

einzel lenses. The distribution of this axial potential closely resembles that shown in Fig. 3. The case

ka=l-@J@a>l (96)

describes a mirror. For this reflection mode the axial

potential Q(O) = CD,,, at the midplane l= &,, = 0 is nega-

tive. Since the calculation for k2 > 1 is analogue to that by

Schiske which is also outlined in detail in the recent

textbook by Hawkes and Kasper [17], we do not restate this procedure. It is advantageous to replace the 5 coordinate by the auxiliary angle

I+!J = arccot( {/a). (97)

Substituting $ for 5 in Eq. (95), the potential adopts the simple form

@= @,A2, A= (1 -k2 sin2+)1’2. (98)

The turning angle r,!~r is obtained from the condition A=Oas

I&. = arcsin( 1 /k), (99

which corresponds to lT = a(k2 - 1)‘12. Inserting Eq. (98) into Eq. (47) for the fundamental longitudinal deviations h, and h,, we can perform the integration analytically if we allow for elliptical integrals. After a rather lengthy calculation we eventually find for the symmetric longitudi-

nal deviation

2k2-1 h,=&(e) = -cos $+k2~

k2- 1 cos (I, sin *

+A i

2k2-1 2(F-F,) - ~ k2_l (E-J%) )

1 (100)

where F = F(cY, l/k) and E = E((Y, l/k) are the elliptic

integrals of the first and second kind, respectively [18]. The angle (Y is connected with $ via the relation

CI = arcsin( k sin ‘p). (101)

For the turning angle (99) we obtain err = n/2. Accord- ingly the complete elliptic integrals

(102)

are the values of the elliptic integrals at the turning point. Using Eqs. (98), (100) and (101), one can show that h,( I,&) adopts the correct value of Eq. (46). The antisymmetric longitudinal deviation h,(+) is found immediately as

h,(4) = +A, (103)

where the positive sign has to be taken if the electron is heading towards the mirror. The symmetric and the antisymmetric solution IV,($) and w,(q) of the paraxial path

equation (29) are obtained relatively easily by adopting

Schiske’s procedure outlined in Ref. [ 171. Using the abbre- viations

& = 1 _ $2,

we find

cos on WC_ C sin +!I ’

At the turning derivatives with

(104)

sin ofi w,= +-

- w sin +!I’ (105)

point += $r these solutions and their respect to T adopt the values

1 1 W UT z - =k, GvT=

sin I/I~ sin I+!J~ = i,

(106)

wrT = 0, wvT = 0.

Here we have considered the relation

d d drCrd< d _---=

x d+ dl dr f A sir?+---.

d+ (107)

The axial fundamental ray (32) is a special linear combination of the solutions (10.5):

sin w(RF0,) w, =

w sin Cc, sin I& ’ (108)

The lower sign has to be taken after the reflection; R, =

fl(o,, l/k) and $,, are the values of fl and I,!J, respectively, at the object plane 5 = &,. An image will be formed

after the reflection at a plane where the argument of the sinusoidal function in the numerator of Eq. (108) is equal

to 7~. Employing Eq. (104), we find that the plane l= 5,

= a cot I+!J~ is the image plane if

(109)

We have evaluated this equation for the two cases +a = I++ =

AZ and I&, = cm = 0. The former case corresponds to the image with unit magnification formed at the centre of curvature &_- = a cot t,!~c of the mirror. For I&, = 0 the object plane is at infinity. Hence the corresponding image

angle $= & defines the position lF = a cot I,!+ of the focal plane. Real images and real focal planes exist only in the narrow range

1 < k2 < 1.127, (110)

for which the potential (9.5) has the property of a concave mirror. Since the axial curvature of the equipotentials changes its sign at the plane 5 = 0.577~1, the potential acts as a convex mirror for k2 > 1.127. In this case the electrons cannot penetrate sufficiently deep into the region of positive curvature of the equipotentials shown in Fig. 7. For the convex mirror the images and the foci become virtual. Their positions are determined by the intersections

H. Rose, D. Preilwas / Nucl. Instr. and Meth. in Phys. Res. A 363 (1995) 301-315 313

of the asymptotes of the corresponding diverging reflected trajectories with the optic axis. For the special excitation

k2 = 1.127 the model potential describes a planar mirror in

paraxial approximation. It should be noted that a convex mirror can also be used as a corrector if W’ > 0 and @Iv > 0 in the vicinity of the turning point CT.

6.2. Chromatic and spherical aberration -*

To minimize the off-axial aberrations, the object plane

lo must be placed in the plane cc of the centre of curvature of the mirror. In this case the reflected rays form an image at the same plane & = & = 5”. Moreover, if we split up each integrand of the aberration coefficients (79)

and (91) into a symmetric part and an antisymmetric part with respect to the turning time TV, the contribution of the

antisymmetric part cancels out. The integral over the sym-

metric term is twice the integral taken between the object

plane & = cc and the turning plane LT. To separate the integrand of the aberration integral for

the chromatic coefficient (79), it suffices to replace h, by the linear combination (55). Using this expression, we find

(111)

In the remaining symmetric integral we can directly replace the integration variabel 7 by 5 without the need to distinguish between the incident and reflected path of the

electrons. To investigate the structure of the aberration coeffi-

cients C, and C, and their dependence on the properties of the electrostatic field, it is advantageous to perform the

integrals in Eqs. (79) and (91) with respect to 5. Consider-

ing the symmetry with respect to the turning point and the analytic representation (47) of h,( 6 ), we eventually ob-

tain for the coefficient (79) of the axial chromatic aberration

The term h,, is usually small compared with the integral in the second relation. C, can be made negative most effectively by choosing W’ > 0 in the region of the turning point lT, where Q, is small. This requirement may serve as a useful guide for the design of effective mirror correctors. The integral expression also demonstrates the

Fig. 7. Cross-section p = / w 1, I through the equipotentials of the model potential distribution, Eq. (95).

significance of the symmetric longitudinal deviation h,. The conventional trajectory and eikonal methods cannot

account for the integration limit & defined by Eq. (45). To obtain an insight into the structure of C, for a

mirror operating in the symmetric imaging mode, we aim for a formula which corresponds to the representation (112) for the coefficient of the axial chromatic aberration.

For this purpose we substitute first h,, in Eq. (91) by Eq. (8% and omit the antisymmetric parts of the integral. Then we transform those parts of the integral by partial integra-

tion where we do not conflict with the vanishing electric potential cP( lT). Employing this procedure, we eventually

obtain the following integral expression for the coefficient of the third-order spherical aberration of the mirror


314 H. Rose, D. Preikszas /Nucl. Instr. and Meth. in Phys. Rex A 363 (1995) 301-315

This expression is also valid if the plane SC = co = (,

represents a subsequent conjugate plane of the centre of curvature of the mirror.

The structure of Eq. (113) demonstrates that @‘” should be made positive especially in the range where @ becomes

small. Additionally, a focusing mirror with a negative axial

chromatic aberration already has a positive @” and a”’ in this region, as mentioned above. It should be noted that this behaviour does not necessarily imply that the overall

property of the mirror must correspond to the curvature of the equipotential cp = 0. Unlike a light-optical mirror,

where the reflection occurs at the physical surface, the

electron optical mirror consists of an extended inhomogeneous refracting medium which allows the electrons to

penetrate into the “mirror”. In this case the total reflection

is the result of consecutive refractions on a continuous set

of surfaces (curved equipotentials). The electrons stay a relatively long time in the vicinity of the turning point due

to their small axial velocities. Accordingly, the electric potential strongly affects the course of the electrons in this

region. The special form of the aberration integrals (112) and

(113) yields general rules for the appropriate manipulation of the aberration coefficients. These rules will serve as helpful guides for the actual design of feasible mirror

correctors. The conventional path equation (31) as well as the integrands of the l-dependent representations in Eqs.

(112) and (113) diverge at the turning point and, hence, are

not appropriate for evaluating the paraxial fundamental

rays and the axial aberration coefficients numerically. Therefore, we have to use the corresponding time-depen-

dent representations together with the unfamiliar time-dependent calculation procedure. It should, however, be noted that the time-dependent perturbation formalism is best suited for obtaining reliably the higher-order aberrations. For this purpose we have developed a sophisticated alge-

braic computer program [ 191.

7. Conclusion

The time-dependent perturbation formalism, outlined in the preceding chapters, enables the aberrations of arbitrary electron optical systems to be calculated. For reasons of

simplicity and to point out the basic features of the proposed method, we have restricted our considerations to rotationally symmetric electrostatic systems and neglected relativistic effects. Since the procedure starts from the Lorentz equations, the incorporation of magnetic and mul- tipole fields is straightforward. The consideration of these fields only enlarges the amount of effort and the size of the formulae but does not affect the applicability of the theory.

The perturbation formalism yields an expansion of the position coordinates of a particle in a power series with respect to the initial ray parameters. The origin of its moving coordinate system is given by the axial position of

the reference electron, which starts simultaneously with the actual particle from the centre of the object surface. The

deviations from the paraxial trajectories remain modest as

long as the sine of the starting angle and the relative energy deviation are small compared with unity. This

requirement is fulfilled in most systems of charged-particle

optics including mirrors with the exception of cathode lenses. In the latter case the chromatic parameter will become even larger than 1 for some of the emerging electrons. For cathode lenses it is, therefore, more favourable to calculate a set of rays, each ray leaving the

same point on the cathode surface with a different energy.

The smallest waist of the bundle of all rays emanating

from the cathode surface forms the crossover which can be real or virtual depending on the form of the extraction

field. This field is affected by both the shape of the

electrodes and the form of the cathode surface. The proposed perturbation method fills the gap in

aberration theory, because it allows one to derive integral expressions for the aberration coefficients of systems with

large gradients of the trajectories, notably mirrors. These expressions correspond to those obtained by the trajectory and/or eikonal method for conventional systems. Al- though the use of the time or of a time-like variable as the

independent variable is mandatory, the final expressions for the aberration coefficients can always be expressed in

the Cartesian representation provided that the electromag-

netic field is static. The Cartesian representation yields a good insight into the structure of the aberration integrals

resulting in general rules for manipulating the aberration

coefficients. On the other hand, the time-dependent approach is extremely suitable for writing an efficient computer program. Hopefully, the new method will stimulate further investigations on aberrations.

References

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[3] V.K. Zworykin, A.G. Marton, E.G. Ramberg, J. Hillier and

A.W. Vance, Electron Optics and the Electron Microscope

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[4] E.G. Ramberg, J. Appl. Phys. 20 (1949) 183.

[5] A. Septier, Ann. RadioCl. 9 (1953) 374. [6] W. Glaser, Gnmdlagen der Elektronenoptik (Springer, Wien,

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[7] Z. Shao and X.C. Wu, Appl. Phys. L&t. 55 (1989) 2696.

[8] P. Schiske, Optik 14 (1957) 34. [9] V.M. Kelman, L.M. Sekunova and E.M. Yakushev, Sov.

Phys. Tech. Phys. 17 (1973) 2279.

[lo] V.M. Kelman, L.M. Sekunova and E.M. Yakushev, Sov.

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[12] T.D. Daumenov, A.A. Sapargaliev and E.M. Yakushev, Sov.

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[13] G.F. Rempfer, J. Appl. Phys. 67 (1990) 6027. [14] H. Rose and D. Preikszas, Optik 92 (1992) 31. [15] D. Preikszas and H. Rose, in: Proc. 13th Int. Cong. Electr.

Microscopy (Paris, 1994) vol. 1, p. 197. [16] W. Glaser and P. Schiske, Optik 11 (1954) 422. ]17] P.W. Hawkes and E. Kasper, Principles of Electron Optics

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Documents

Time-dependent perturbation formalism for calculating the aberrations of systems with large ray gradients