Software for designing multipole aberration correctors

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Physics Procedia001 (2008) (2008)000–000 339–353 Physics Procedia www.elsevier.com/locate/procedia www.elsevier.com/locate/XXX

Proceedings of the Seventh International Conference on Charged Particle Optics

Software for designing multipole aberration correctors Haoning Liu∗, John Rouse, Liping Wang, Eric Munro Munro’s Electron Beam Software Ltd., 14 Cornwall Gardens, London SW7 4AN, England Elsevier only:received Receivedindate here; form revised date here; date here 2008 Received 9 Julyuse 2008; revised 9 July 2008;accepted accepted 9 July

Abstract A set of formulae has been derived for computing paraxial rays, geometrical and chromatic aberrations of multipole systems containing electrostatic and magnetic round, quadrupole, hexapole and octopole lenses, and deflectors. Using these formulae, we have developed a software package for simulating and designing such systems. This new software, using the previously computed field functions as input, computes and plots paraxial rays, geometrical aberrations up to the third-order and the firstorder chromatic aberrations for the analyzed systems. It has an optimization module where a weighted set of aberrations can be minimized by the automatic adjustment of a set of user-defined system variables. We have also added a graphical user interface which affords a better overview of the design process and an interactive control of the system. An example of a hexapole spherical aberration corrector is presented to illustrate the functionality of this software package. © 2008 Elsevier B.V. All rights reserved. PACS: 41.85.-p; 41.85.Gy; 02.70.-c Keywords: Multipole system, Quadrupole, Hexapole, Octopole, Aberration corrector; Computer software

1. Introduction Due to the recent advances in computer-controlled lens systems for electron microscopes, the use of multipole aberration correctors, which were first suggested by Scherzer [1] almost sixty years ago, have become a practical reality. These aberration correctors, in general, contain electrostatic and magnetic multipole lenses. Numerical simulation of systems including multipole elements is very important and useful for designing these aberration correctors. The theory for analyzing individual multipole lenses was well established by Hawkes [2-6] and Kanaya [7-9] in the 1960’s. Smith [10] presented a comprehensive aberration theory for designing multipole systems containing electrostatic round lenses, electrostatic and magnetic quadrupole and octopole lenses, as well as deflectors. We previously reported a software package based on an extension of Smith’s theory, which computed paraxial rays, and primary and secondary aberrations for multipole systems without magnetic round lenses [11]. Many practical electron optical systems for which an aberration corrector is desirable, however, involve magnetic round lenses. Consequently, there has been an increasing need for a software package to analyze and optimize ∗ Corresponding author: Tel./Fax.: +44-20-7581-4479 E-mail address: [email protected]

doi:10.1016/j.phpro.2008.07.114

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systems containing both magnetic round lenses and multipole lenses. The methodology used in our previous software is no longer practicable when the image twisting caused by magnetic round lenses is to be taken into account. We have been working on a new theory and software package for analyzing such systems. To compute the aberrations of such systems, it is essential to calculate the field functions and their derivatives in the round and multipole lenses and deflectors to a high accuracy. The power of modern computers makes these field calculations possible using our SOFEM and 3D software packages [12]. Using the aberration integral method, we have derived a set of formulae for computing the first order optical properties, second and third order geometrical aberrations and first order chromatic aberrations of multipole systems containing electrostatic and magnetic round, quadrupole, hexapole and octopole lenses and deflectors. A new software package has been developed by implementing these formulae with several new features that enhance the suitability of the software for design of multipole correctors. This new software uses the field functions computed by the SOFEM and 3D programs as input, and computes paraxial rays, geometrical aberrations up to the third-order and the first-order chromatic aberrations for the analyzed systems. The software has an optimization module where a weighted set of aberrations can be minimized by the automatic adjustment of a set of user-defined system variables. We have also added a graphical user interface which affords a better overview of the design process and an interactive control of the system. In this paper, we will present our theory for multipole systems containing electrostatic and magnetic round lenses, multipole lenses and deflectors. We will then demonstrate the capabilities of the new software by considering an illustrative example of a hexapole system for the correction of primary spherical aberration.

2. Expressions for the electrostatic and magnetic scalar potentials 2.1. Lens potentials The scalar potentials at any point near the optical axis of electrostatic and magnetic round, quadrupole, hexapole and octopole lenses are as follows: 1 1 Φ 0 ( r , θ , z ) = φ ( z ) − φ '' ( z ) r 2 + φ '''' ( z ) r 4 + 64 4 1 '' f 2 ( z )r 4 cos 2(θ − β 2 ) + Φ 2 (r ,θ , z ) = f 2 ( z )r 2 − 12 1 '' f 3h ( z )r 5 cos 3(θ − β 3 ) + Φ 3 (r ,θ , z ) = f 3h ( z )r 3 − 16 Φ 4 (r , θ , z ) = f 4 j ( z )r 4 cos 4(θ − β 4 j ) + 1 1 Ψ0 (r ,θ , z ) = ψ ( z ) − ψ '' ( z )r 2 + ψ '''' ( z )r 4 + 4 64 1 Ψ2 (r ,θ , z ) = d 2 ( z ) r 2 − d 2'' ( z )r 4 cos 2(θ − γ 2 ) + 12 1 3 Ψ3 (r ,θ , z ) = d 3h ( z ) r − d 3''h ( z )r 5 cos 3(θ − γ 3 ) + 16 Ψ4 (r , θ , z ) = d 4 ( z )r 4 cos 4(θ − γ 4 ) +

(1)

where φ (z ) , f2(z), f3h(z) and f4(z) are the electrostatic potentials at the axis of the electrostatic round, quadrupole, hexapole and octopole lenses, and 2, 3, and 4 are the rotation angles of the electrostatic quadrupole, hexapole and octopole lenses relative to the fixed reference axes, respectively. Similarly, (z), d2(z), d3h(z) and d4(z) are the magnetic scalar potentials at the axis of the magnetic round, quadrupole, hexapole and octopole lenses, and 2, 3, and 4 are the rotation angles of the magnetic quadrupole, hexapole and octopole lenses relative to the fixed reference axes, respectively. The primes denote the derivatives with respect to z.

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2.2. Deflection potentials Chu and Munro presented the expression for the electrostatic and magnetic deflection potential for an individual deflector as follows [13]: 1 Φ 1 (r , θ , z ) = −V x f1 ( z )r − f 1'' ( z )r 3 cos(θ − β 1 ) + f 3 ( z ) r 3 cos 3(θ − β1 ) + ⋅ ⋅ ⋅ 8 1 '' − V y f1 ( z ) r − f1 ( z )r 3 sin(θ − β1 ) + f 3 ( z ) r 3 sin 3(θ − β1 ) + ⋅ ⋅ ⋅ (2) 8 1 Ψ1 (r , θ , z ) = − I y d1 ( z )r − d1'' ( z )r 3 cos(θ − γ 1 ) + d 3 ( z )r 3 cos 3(θ − γ 1 ) + ⋅ ⋅ ⋅ 8 1 '' − I x d1 ( z )r − d1 ( z )r 3 sin(θ − γ 1 ) + d 3 ( z )r 3 sin 3(θ − γ 1 ) + ⋅ ⋅ ⋅ 8 where f1(z) and f3(z) are the first and third harmonic field functions for the electrostatic deflector, and d1(z) and d3(z) are the first and third harmonic field functions for the magnetic deflector. (Vx, Vy) are the deflection voltages applied to the electrostatic deflector, and (Ix, Iy) are the deflection currents applied to the magnetic deflector. Ix is defined to be the current which deflects the beam along the x-axis of the deflector, and Iy is defined to be the current which deflects the beam along the y-axis of the deflector. 1 and 1 are the rotation angles of the electrostatic and magnetic deflectors relative to the fixed reference axes, respectively. 2.3. The overall potential and field components We now define a complex coordinate w and its complex conjugate w : iθ

w = x + iy = re ,

w = x − iy = re

− iθ

Eq. (1) can then be written as follows: 1 1 Φ 0 ( w, w , z ) = φ − φ '' ww + φ '''' w 2 w 2 + 4 64 1 1 1 '' 1 '' 3 Φ 2 ( w, w , z ) = F2 w 2 + F2 w 2 − F2 ww 3 − F2 w w + 2 2 24 24 1 1 1 1 Φ 3 ( w, w , z ) = F3h w 3 + F3h w 3 − F3'h' ww 4 − F3'h' w 4 w + 2 2 32 32 1 1 Φ 4 ( w, w , z ) = F4 w 4 + F4 w 4 + 2 2 1 '' 1 Ψ0 ( w, w , z ) = ψ − ψ ww + ψ '''' w 2 w 2 + 4 64 1 1 1 '' 1 '' 3 Ψ2 ( w, w , z ) = D2 w 2 + D2 w 2 − D 2 ww 3 − D2 w w + 2 2 24 24 1 1 1 1 Ψ3 ( w, w , z ) = D3h w 3 + D3h w 3 − D3''h ww 4 − D3''h w 4 w + 2 2 32 32 1 1 Ψ4 ( w, w , z ) = D4 w 4 + D4 w 4 + 2 2

(3)

where F2 = f 2 ( z )e i 2 β 2 , D2 = d 2 ( z )e i 2 γ 2 ,

F3h = f 3h ( z )e i 3β3 , D3h = d 3h ( z )e i 3γ 3 ,

F4 = f 4 ( z )e i 4 β 4 D4 = d 4 ( z )e i 4 γ 4

(4)

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The complex form of Eq. (2) was given by Chu and Munro [13]: 1 1 1 1 1 1 Φ 1 ( w, w , z ) = − VF1 w − V F1 w + VF1'' ww 2 + V F1'' w 2 w − VF3 w 3 − V F3 w 3 + 2 2 16 16 2 2 i i i i i i '' 2 '' 2 3 Ψ1 ( w, w , z ) = ID1 w − I D1 w − ID1 ww + I D1 w w + ID3 w − I D3 w 3 + 2 2 16 16 2 2

(5)

where V = V x + iV y ,

F1 = f 1 ( z )e iβ 3 ,

F3 = f 3 ( z )e i 3 β 4

I = I x + iI y ,

D1 = d1 ( z )e iγ 3 ,

D3 = d 3 ( z )e i 3γ 4

(6)

For the system containing round and multipole lenses and deflectors, the overall electrostatic potential is obtained by adding 0, 1, 2, 3 and 4; while the overall magnetic scalar potential is obtained by adding 0, 1, 2, 3 and 4. Using the same sign convention as that used by Chu and Munro [13], the electric field E and magnetic flux density B are E = −grad ,

B=

0grad

Using complex notation, the components of E and B are given by ∂Φ , ∂w ∂Ψ , Bw = B x + iB y = 2 0 ∂w E w = E x + iE y = −2

∂Φ ∂z ∂Ψ Bz = 0 ∂z Ez = −

(7)

By applying these relations to Eqs. (3) and (5), we obtain the following expressions for the electrostatic potential , the electrostatic field components (Ew, Ez) and the magnetic flux density components (Bw, Bz): 1 1 Φ( w, w , z ) = φ − φ '' ww + φ '''' w 2 w 2 4 64 1 1 1 1 1 1 − VF1 w − V F1 w + VF1'' ww 2 + V F1'' w 2 w − VF3 w 3 − V F3 w 3 2 2 16 16 2 2 (8) 1 1 1 '' 1 '' 3 2 2 3 F2 ww − F2 w w + F2 w + F2 w − 2 2 24 24 1 1 1 '' 1 3 3 4 + F3h w + F3h w − F3h ww − F3'h' w 4 w 2 2 32 32 1 1 + F4 w 4 + F4 w 4 + 2 2 1 '' 1 '''' 2 1 1 E w ( w, w , z ) = φ w − φ w w + VF1 − VF1'' ww − V F1'' w 2 + 3V F3 w 2 (9) 2 16 4 8 1 '' 1 '' 3 2 2 3 − 2 F2 w + F2 ww + F2 w − 3F3h w − 4 F4 w + 4 12 1 ''' 1 1 1 1 ' (10) E z ( w, w , z ) = −φ + φ ww + VF1' w + V F1' w − F2' w 2 − F2' w 2 + 4 2 2 2 2 1 1 i i Bw ( w, w , z ) = − B ' w + B ''' w 2 w + iID1 − ID1'' ww + I D1'' w 2 − 3iI D3 w 2 (11) 2 16 4 8 1 '' 1 + 2 D2 w − D2 ww 2 − D2'' w 3 + 3D3h w 2 + 4 D4 w 3 + 4 12 1 '' i i 1 1 ' (12) B z ( w, w , z ) = B − B ww + ID1 w − I D1' w + D2' w 2 + D2' w 2 + 4 2 2 2 2 where B = 0ψ ' has been used to denote the magnetic round lens flux density at the z-axis.

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3. Ray equation in multipole systems

The complex form of general ray equation for electrons in the combined focusing and deflection field was given by Chu and Munro [13]: Ew η ' d ' ' −1 / 2 ' ' ' 1/ 2 (13) dz

[

where

]

Φ (1 + w w )

w =−

2 Φ

(1 + w w )

+i

2

( w B z − Bw )

is the charge/mass ratio of the electron (absolute value).

To derive the ray equation up to the third-order, we substitute Eqs. (8)-(12) into Eq. (13), retaining only terms of up to third-order in w; I and V; this yields the ray equation up to the third-order as follows: F VF φ' ' φ η η η 1 w '' + w + w−i ( Bw ' + B ' w) − 2 − 2i D2 w = − 1 + ID1 (14) φ 2φ 4φ 2φ 2 2φ 2φ 2φ + P2 ( z ) + P3 ( z )

where P2 ( z ) =

3F3h η − 3i D3h w 2 2φ 2φ

P3 ( z ) =

1 d φ dz −

φ w'

(15)

VF VF F F φ '' 1 ww + 1 w + 1 w − 2 w 2 − 2 w 2 + w ' w ' 8φ 4φ 4φ 4φ 4φ 2

VF F φ '' w+ 1 − 2 w φ 4φ 2φ

VF VF F F φ '' 1 ww + 1 w + 1 w − 2 w 2 − 2 w 2 + w ' w ' 8φ 4φ 4φ 4φ 4φ 2

(16)

3V F3 2 F2'' VF '' V F '' F '' 2F φ '''' 2 + w w + 1 ww + 1 w 2 − w − ww 2 − 2 w 3 + 4 w 3 φ 32φ 8φ 16φ 2φ 8φ 24φ +

ID ' ID ' ID '' I D '' η iB '' iB ''' 2 ww ' w − w w − 1 w ' w + 1 ww ' − 1 ww + 1 w 2 w − 2φ 4 16 2 2 4 8

− 3ID3 w 2 +

iD2' ' 2 iD2' ' 2 iD2'' i D '' ww + ww + ww 2 + 2 w 3 − 4iD4 w 3 2 2 4 12

4. First order optical properties

Retaining only terms of up to first-order in w, I and V in Eq. (14), yields the paraxial ray equation as follows: w '' +

F VF φ' ' φ η 1 η η w + w−i ( Bw ' + B ' w) − ( 2 − 2i D2 ) w = − 1 + ID1 2φ 4φ 2φ 2 φ 2φ 2φ 2φ

(17)

For an electron leaving the object plane zo with initial complex slope wo' = xo' + iyo' and initial complex position wo= xo + iyo, the paraxial ray calculated from Eq. (17) can be expressed as w( z ) = xo' wax ( z ) + xo wbx ( z ) + I x wmx ( z ) + V x wex ( z )

(18)

+ y o' way ( z ) + y o wby ( z ) + I y wmy ( z ) + V y wey ( z )

The rays wax(z), wbx(z), way(z) and wby(z) are the fundamental rays for the lens system, while wmx(z), wex(z), wmy(z) and wey(z) are the fundamental rays for the magnetic and electrostatic deflection respectively. These rays’ initial conditions are given in Table 1.

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Table 1 The initial conditions of eight fundamental rays. xo'

xo

Ix

Vx

yo'

yo

Iy

wax(z)

1

0

0

0

0

0

0

0

wbx(z)

0

1

0

0

0

0

0

0

Vy

wmx(z)

0

0

1

0

0

0

0

0

wex(z)

0

0

0

1

0

0

0

0 0

way(z)

0

0

0

0

1

0

0

wbv(z)

0

0

0

0

0

1

0

0

wmy(z)

0

0

0

0

0

0

1

0

wey(z)

0

0

0

0

0

0

0

1

By the following definitions w ' + wo' w + wo , , xo' = o xo = o 2 2 ' ' w − wo w − wo y o' = o yo = o , , 2i 2i The Eq. (18) can be re-written as follows:

I+I , 2 I −I Iy = , 2i

Ix =

w( z ) = wo' wa ( z ) + wo wb ( z ) + Iwm ( z ) + Vwe ( z )

V +V 2 V −V Vy = 2i

Vx =

(19)

(20)

+ wo' wac ( z ) + wo wbc ( z ) + I wmc ( z ) + V wec ( z )

where

[

]

[

]

1 wax ( z ) − iway ( z ) 2 1 wb ( z ) = wbx ( z ) − iwby ( z ) 2 1 wm ( z ) = wmx ( z ) − iwmy ( z ) 2 1 we ( z ) = wex ( z ) − iwey ( z ) 2 1 wac ( z ) = wax ( z ) + iway ( z ) 2 1 wbc ( z ) = wbx ( z ) + iwby ( z ) 2 1 wmc ( z ) = wmx ( z ) + iwmy ( z ) 2 1 wec ( z ) = wex ( z ) + iwey ( z ) 2

wa ( z ) =

[

[

]

[

]

[

]

[

[

] (21)

]

]

The lens rays, wax(z), wbx(z), way(z) and wby(z), are computed first, by setting the right-hand side of Eq. (17) to zero (I = V = 0), and solving the resulting homogeneous equation. This involves evaluating two coupled equations in both real and imaginary parts of the homogeneous equation, subject to the initial conditions in Table 1. Consequently, the rays wa(z), wb(z), wac(z) and wbc(z) are then obtained. The deflection rays, wm(z), we(z), wmc(z) and wec(z), are then computed by solving the inhomogeneous Eq. (17) by the method of variation of parameters; the resulting expressions for wm(z), we(z), wmc(z) and wec(z) are

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z

z

z

z

zo z

zo

zo

zo

z

z

z

zo

zo

zo

zo

wm ( z ) = wa ( z ) ∫ Pm1Q1 dz + wb ( z ) ∫ Pm1Q2 dz + wac ( z ) ∫ Pm1Q3 dz + wbc ( z ) ∫ Pm1Q4 dz we ( z ) = wa ( z ) ∫ Pe1Q1 dz + wb ( z ) ∫ Pe1Q2 dz + wac ( z ) ∫ Pe1Q3 dz + wbc ( z ) ∫ Pe1Q4 dz z

z

z

zo z

zo z

z

z

zo

zo

zo

zo

(22)

z

wmc ( z ) = wa ( z ) ∫ Pm1T1 dz + wb ( z ) ∫ Pm1T2 dz + wac ( z ) ∫ Pm1T3 dz + wbc ( z ) ∫ Pm1T4 dz zo

zo

wec ( z ) = wa ( z ) ∫ Pe1T1 dz + wb ( z ) ∫ Pe1T2 dz + wac ( z ) ∫ Pe1T3 dz + wbc ( z ) ∫ Pe1T4 dz

where Pm1 =

η 2φ

D1 ,

Q1 =

Q2 = −

Q3 =

Q4 = −

=

Pe1 = −

F1 2φ

(23)

wb wbc wbc'

wac wa wa'

wbc wb wb'

T1 = −

,

wb wbc wb'

wac wa ' wac

wbc wb wbc'

wa

wac

wbc

wa

wac

wbc

wac wac'

wa wa'

wb wb'

wac wa'

wa ' wac

wb wbc'

wa wac

wb wbc

wbc wb

wa wac

wb wbc

wbc wb

wac'

wbc'

wb'

wa'

wb'

wbc'

wa wac wac'

wb wbc wbc'

wac wa wa'

wa wac wa'

wb wbc wb'

wbc wa ' wac

wa wac wa'

wb wbc wb'

wac wa ' wac

wbc wb wbc'

wac'

wbc'

wa'

wb'

T2 =

,

T3 = −

,

T4 =

,

(24)

When the fundamental rays have been computed, their values at the image plane zi give us the first order optical properties: Magnification in x Magnification in y Image rotation angle in x Image rotation angle in y

= wbx ( z i ) = wby ( z i ) = Arg [wbx ( z i )] = Arg [wby ( z i ) ]

Magnetic deflection sensitivity in x

= wmx ( z i )

Magnetic deflection sensitivity in y

= wmy ( z i )

Magnetic deflection direction in x

= Arg [wmx ( z i )] = Arg [wmy ( z i ) ]

Magnetic deflection direction in y Landing slope per unit magnetic deflection in x

' = wmx ( z i ) / wmx ( z i )

Landing slope per unit magnetic deflection in y

' = wmy ( z i ) / wmy ( z i )

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Electrostatic deflection sensitivity in x

= wex ( z i ) = wey ( z i )

Electrostatic deflection sensitivity in y Electrostatic deflection direction in y

= Arg [wex ( z i )] = Arg [wey ( z i ) ]

Landing slope per unit electrostatic deflection in x

= wex' ( z i ) / wex ( z i )

Landing slope per unit electrostatic deflection in y

= wey' ( zi ) / wey ( zi )

Electrostatic deflection direction in x

5. Second and third order geometrical aberrations

To solve the ray equation up to the third-order, Eq. (14), we define the second and third order geometrical aberrations δw2 ( z ) and δw3 ( z ) as the differences between the solutions of the ray equation up to the third-order, Eq. (14) and the paraxial ray equation, Eq. (17). The equations for δw2 ( z ) and δw3 ( z ) can then be written as

δw2'' +

F φ' ' φ η 1 η δw2 + δw2 − i ( Bδw2' + B 'δw2 ) − 2 − 2i D2 δw2 = P2 ( z ) 2φ 4φ 2φ 2 φ 2φ

δw3'' +

∂P F 1 φ' ' φ η η ( Bδw3' + B 'δw3 ) − 2 − 2i D2 δw3 = 2 δw2 δw3 + δw3 − i ∂w 2φ 4φ 2φ 2 2φ φ ∂P2 δw2 ∂w + P3 ( z ) +

(25)

(26)

It can be found from Eq. (15) that ∂P2 is equal to zero, so Eq. (26) can be written as ∂w

F ∂P 1 φ' φ η η ( Bδw3' + B 'δw3 ) − 2 − 2i D2 δw3 = 2 δw2 δw3'' + δw3' + δw3 − i 2φ 2φ 4φ 2φ 2 φ ∂w

(27)

+ P3 ( z )

The initial conditions for δw2 ( z ) and δw3 ( z ) at the object plane zo are

δw2 ( z o ) = δw2' ( z o ) = 0 δw3 ( z o ) = δw3' ( z o ) = 0

(28)

The left-hand sides of Eqs. (25) and (27) are same as the left-hand side of the paraxial ray equation (17), for which the general solution is already known, therefore, Eqs. (25) and (27) can also be solved by the method of variation of parameters. The resulting expressions for the second and third order aberrations δw2 ( z i ) and δw3 ( z i ) at the image plane zi are

δw2 ( z i ) = wb ( z i )

∫

zi

zo

+ wbc ( z i )

δw3 ( z i ) = wb ( z i )

∫

zi

zo

+ wbc ( z i )

zi

P2 ( z )Q2 ( z )dz + ∫ P2 ( z )T2 ( z )dz zo

∫

zi

zo

P2 ( z )Q4 ( z )dz + ∫ P2 ( z )T4 ( z )dz zo

z i ∂P ∂P2 2 δw2 + P3 ( z ) Q2 ( z ) dz + ∫ δw2 + P3 ( z ) T2 ( z ) dz z o ∂z ∂z

∫

zi

zo

(29)

zi

(30)

z i ∂P ∂P2 2 δw2 + P3 ( z ) Q4 ( z ) dz + ∫ δw2 + P3 ( z ) T4 ( z )dz zo ∂z ∂z

Considering a system containing either electrostatic or magnetic deflection, if we substitute Eq. (20) for a general paraxial ray w(z) into Eqs. (29) and (30), and separate out terms according their dependence on wo', wo, I and V and

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their complex conjugates, we can obtain 21 second-order complex geometrical aberration coefficients if hexapole lenses exist, and 56 third-order complex geometrical aberration coefficients, as shown in Tables 2 and 3. All the second and third order complex geometrical aberration coefficients have been derived in detail and programmed in the software. The detailed formulae are too long to present in full here, but the general forms of the second and third order aberrations can be seen in the above Tables. 6. First order chromatic aberrations

If the electron’s energy is perturbed by a small amount, (electron volts), this is equivalent to changing the electrostatic focusing potential φ (z ) by (volts). Let the corresponding change in the paraxial ray w(z) be δwc (z ) . Thus the perturbed values of the electrostatic focusing potential and the paraxial ray are φˆ( z ) = φ ( z ) + ϕ ,

wˆ ( z ) = w( z ) + δwc1 ( z )

Substituting these into the paraxial ray equation (17) and neglecting terms higher than first order in the equation for the first order chromatic aberration δwc (z ) :

δwc''1 +

F φ' ' φ η 1 η δwc1 + δwc1 − i ( Bδwc' 1 + B 'δwc1 ) − 2 − 2i D2 δwc1 = Pc1 ( z ) 2φ 4φ 2φ 2 φ 2φ

, we obtain

(31)

Where ϕ φ

Pc1 ( z ) =

−

φ ' ' φ '' i η 1 w + w− Bw ' + B ' w 2φ 4φ 2 2φ 2 F2

φ

−i

η 2φ

D2 w +

(32)

VF1 1 η ID1 − 2φ 2 2φ

Table 2 The second-order complex geometrical aberration coefficients. Axial Aperture Aberration

Shaped Beam ' o ' o ' o

C21 w w C22 w w C23 w w

Beam Blur Aberration

Distortion

' o ' o ' o

Deflection

Hybrid

-

-

-

-

-

-

-

-

-

-

C24 w ' w

-

C25 w ' w

-

C26 w ' w

-

C27 w w ' o o

C211 w w ' d o

-

-

C212 w w o o

C215 w w d d

C218 w w d o

-

C213 w w o o

C216 w w d d

C219 w w d o

-

C214 w w o o

C217 w w d d

C220 w w o d

-

-

-

C221 w w d o

o

o o

o

C28 w ' w

o

C29 w ' w

d

-

o

C210 w ' w

-

o

o

o

d

d

-

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Table 3 The third-order complex geometrical aberration coefficients.

Aperture Aberration

Axial C31 w ' w ' w ' o ' o

o ' o

o ' o

' o

' o

' o

C32 w w w

C33 w w w

C34 w ' w ' w ' o o o -

Coma

Field Curvature

-

-

-

-

-

-

-

-

C35 w ' w ' w o o o

C311 w ' w ' w o o d

-

' o

-

C313 w ' w ' w o o d

-

C37 w ' w ' w o o o

-

' o

C38 w w w o

-

C39 w w ' w ' o o o

C315 w w ' w ' d o o

-

-

C310 w ' w ' w o o o

C316 w ' w ' w o o d

-

-

C317 w ' w w o o o

C320 w ' w w o d d

C323 w ' w w o d o

-

C318 w ' w w o o o

C321 w ' w w o d d

C324 w ' w w o o d

' o

C325 w ' w w o d o

' o

' o

' o

C312 w w w d

' o

' o

' o

C314 w w w d

' o

-

C319 w w w o o -

C322 w w w d d

-

C327 w w w ' o o o

C330 w w w ' d d o

C333 w w w ' d o o

-

C328 w w w o o

' o

' o

C334 w w w ' d o o

-

C329 w w w ' o o o -

C332 w w w ' d d o -

C335 w w w ' d o o

-

C337 w w w o o o

C341 w w w d d d

C345 w w w d d o

-

C338 w w w o o o

C342 w w w d d d

C346 w w w d o d

-

C339 w w w o o o

C343 w w w d d d

C347 w w w d d o

-

C340 w w w o o o -

C344 w w w d d d -

C348 w w w d d o

-

C326 w ' w w o d o C331 w w w d d

C336 w w w ' o d o

C349 w w w o d d

-

-

-

C350 w w w d d o

-

-

-

C351 w w w o o d

-

-

-

C352 w w w d o o

-

-

-

C353 w w w d o o

-

-

-

C354 w w w d o o

-

-

-

C355 w w w d o o

-

-

-

C356 w w w o d o

In Tables 2 and 3, I V

-

C36 w w w o

-

wd =

Hybrid

-

-

Distortion

Deflection -

-

-

Astigmatism

Shaped Beam -

for magnetic deflection for electrosta tic deflection

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Now Eq. (31) is identical to Eq. (25), except that P2(z) is replaced by Pc1(z). Furthermore, the initial conditions for δwc (z ) are δwc ( z o ) = δwc' ( z o ) = 0 at the object plane zo. Therefore, we can deduce, by analogy with Eq. (25), that the first order chromatic aberration δwc1 ( z i ) at the image plane zi is

δwc1 ( z i ) = wb ( z i )

∫

zi

zo

+ wbc ( z i )

zi

Pc1 ( z )Q2 ( z )dz + ∫ Pc1 ( z )T2 ( z )dz

(33)

zo

∫

zi

zo

zi

Pc1 ( z )Q4 ( z )dz + ∫ Pc1 ( z )T4 ( z )dz zo

We now insert the expression for the paraxial ray w(z), Eq. (20), into the above equation, we then find that the expression for the first order chromatic aberrations has the general form shown in Table 4. Table 4 The first-order complex chromatic aberration coefficients. Axial Axial Chromatic Aberration

Shaped Beam

Cc1 w '

o

Cc2 w '

o

Transverse Chromatic Aberration

ϕ / φi ϕ / φi

Deflection

-

-

-

-

Cc3 w

-

Cc4 w

o o

ϕ / φi ϕ / φi

Cc5 w

d

Cc6 w

d

ϕ / φi ϕ / φi

Again, all these 6 first order complex chromatic aberration coefficients have been derived in detail and programmed in the software.

7. Design of multipole aberration correctors

A software package has been developed by implementing the above theory with Borland C++ Builder. The software now has the ability to include magnetic round lenses and also has an optimisation module where a weighted set of aberrations can be minimised by the automatic adjustment of a set of user-defined system variables. We have also added a graphical user interface which affords a better overview of the design process and an interactive control of the system. We will demonstrate the capabilities of the new software by considering an illustrative example of a hexapole system for the correction of primary spherical aberration. The system consists of three main parts, shown in Fig. 1: the magnetic objective lens whose spherical aberration we wish to reduce; a transfer lens to produce an image outside the objective lens (see Fig. 2); and a corrector. The corrector consists of four magnetic round lenses and two hexapole lenses, arranged as shown in Fig. 1. The same geometry was used for the transfer lens and the four round lenses in the corrector. The axial flux density distributions are shown in Fig. 3. After setting up the magnetic round lenses, as shown in Fig. 1 with the ray paths as shown in Fig. 2, we then introduce the magnetic hexapoles and adjust their strengths to reduce the spherical aberration of the system. The axial hexapole field function is shown in Fig. 4. If we include the first hexapole in the system, we can see (Table 5) that there are second-order aberrations due to the first hexapole (Ac Ac). Also there are third order aberrations due to the first hexapole that are of the same form as those due to the round lenses (A A Ac). Similarly, if we include only the second hexapole in the system, there are second-order aberrations which are equal in magnitude and opposite in sign to those when the first hexapole was energised. Also there are again third order aberrations due to the second hexapole that are of the same form as those due to the round lenses. Our aim is to energise both hexapoles together so that their combined second-order aberrations cancel and to adjust their strengths to cancel the third-order aperture aberration of the round lenses.

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Fig. 1. Schematic of spherical aberration corrector system

Fig. 2. Focus ray (above) and field rays (below) in the system

Fig. 3. Enlarged view of axial magnetic flux density distribution in objective lens(left) and transfer/ auxiliary lens (right)

Fig. 4. Enlarged view of axial hexapole field function

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Table 5 Selected aberration coefficients when each hexapole is energised in turn. (A is the aperture angle and Ac its conjugate) Aberration

Functional Dependence

1st hexapole only energised

2nd hexapole only energised

Isotropic

Anisotropic

Isotropic

Anisotropic

Second-order aperture aberrations due to hexapoles

AA

0.00000e+00

0.00000e+00

0.00000e+00

0.00000e+00

Ac Ac

-5.25834e-04

9.81826e-05

5.25831e-04

-9.81832e-05

A Ac

0.00000e+00

0.00000e+00

0.00000e+00

0.00000e+00

Third-order aperture aberrations due to hexapoles

A A Ac

1.52193e-04

-2.84484e-05

1.52192e-04

-2.84481e-05

Third-order aperture aberrations due to round lenses

AAA

0.00000e+00

0.00000e+00

0.00000e+00

0.00000e+00

Ac Ac Ac

3.12073e-21

4.03835e-22

3.74065e-21

5.89806e-21

A Ac Ac

0.00000e+00

0.00000e+00

0.00000e+00

0.00000e+00

A A Ac

1.43168e-02

-6.56171e-03

1.43168e-02

-6.56171e-03

AAA

0.00000e+00

0.00000e+00

0.00000e+00

0.00000e+00

Ac Ac Ac

0.00000e+00

0.00000e+00

0.00000e+00

0.00000e+00

A Ac Ac

0.00000e+00

0.00000e+00

0.00000e+00

0.00000e+00

We now show some screen shots from the program to illustrate this procedure. Fig. 5 shows the main screen of the program, which contains all the data about the optical elements and which systems parameters can be used as variables for autofocusing the system, which parameters can be used for optimising the aberrations and which are fixed. Similar variables can be linked together, so that they vary in a well-defined and co-ordinated fashion, if required. Fig. 6 shows the screen that controls the weighting factors for targeting the optical properties to be achieved or optimised. Fig. 7 shows the optimisation window, which contains two parts: the upper part is the state of the system at the start of the optimisation process; the lower half of the window is the state of the system at the current optimisation cycle. As can be seen from Fig. 7, the spot size has been significantly reduced after two optimisation cycles, and the spherical aberration coefficient has been reduced by a factor of 4.

Fig. 5. Screen shot showing selection of optimization (refine) variables for hexapole strengths

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Fig. 6. Screen shot showing weighting factors for minimising spherical aberration

Fig. 7. Optimisation Process window, after two refine cycles

8. SUMMARY

A software package based on the method described in this paper has been developed. The input data of the software are the field functions of round and multipole lenses as well as the imaging conditions of the multipole system. The output results contain the first-order optical properties, geometrical aberration coefficients up to the third order, and the first order chromatic aberration coefficients. The overall aberration effects are visually shown in spot diagram format. The graphical user interface (GUI) affords a better overview of the design process and an interactive control of the system. An example of a hexapole spherical aberration corrector is presented in this paper to illustrate how the software can handle magnetic and electrostatic round lenses and multipole lenses, including an optimisation engine to minimise chosen aberrations.

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ACKNOWLEDGEMENT

We would like to express our thanks to Mr. Xieqing Zhu (H. Chu) for his help in the work presented in this paper. We have benefited greatly from numerous personal discussions during the development of this software package.

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