Study on differential algebraic aberration method for electrostatic electron lenses

Study on differential algebraic aberration method for electrostatic electron lenses* Min Cheng, Tiantong Tang, Zhenhua Yao Department of Electronics Science and Technology, School of Electronics and Information Engineering, Xi’an Jiaotong University, Xi’an, 710049, P. R. China

Abstract: Differential algebraic method is a powerful technique in computer numerical analysis. It presents a straightforward method for computing arbitrary order derivatives of functions with extreme high accuracy limited only by the machine error. When applied to nonlinear dynamics systems, the arbitrary high order transfer properties of the system can be derived directly. In this paper, the principle of differential algebraic method is applied to calculate high order aberrations of electrostatic electron lenses. Gaussian properties and third order geometric aberration coefficients of an electrostatic lens with analytical expressions have been calculated. Relative errors of the Gaussian properties and spherical aberration coefficient of the lens compared with the analytic solutions are of the order 10– 11 or smaller. It is proved that differential algebraic aberration method is very helpful for high order aberration analysis and computation of electrostatic electron lenses Key words: Differential algebra – electrostatic electron lenses – aberration analysis

tical systems. It has been successfully employed in magnetic focusing and deflection systems [4]. Yet its application to electrostatic electron lens systems has not been reported. In this paper, differential algebraic aberration method of electrostatic lenses is presented and is applied to describe Gaussian optical properties and high order aberrations. As an example, the Gaussian properties and the third order aberrations have been calculated for Schiske’s model electrostatic lens, which is an extensively studied model [5]. Relative errors of the Gaussian properties compared with the analytic solutions are on the scale of 10– 11 or small. It is shown that differential algebraic method is very effective for high order aberration analysis of electron optical systems.

2. Principle of differential algebra 1. Introduction With the increasing development of high definition display devices and electron beam lithography techniques, it has become of great importance to improve the aberration performance of high-resolution electron optical systems. Then it is necessary to investigate higher order aberrations of the systems. Various theoretical tools have been developed to deal with the high order aberration analysis and correction, such as approximately analytical method [1], canonical theory [2], and Lie algebra [3]. These methods simplify the derivation of high order aberrations, but they have little advantage in numerical calculation and computer programming. What is more, the complexity of the expressions of aberration coefficients increases dramatically with the order of aberrations. In contrast, differential algebraic method provides a powerful technique for high order aberration analysis and numerical calculation of electron op* Supported by the National Science Foundation of P. R. China (69971019) Received 29 November 2000; accepted 3 March 2001. Correspondence to: T. T. Tang Fax: ++86-29-2668643 E-mail: [email protected] Optik 112, No. 6 (2001) 250–254 © 2001 Urban & Fischer Verlag http://www.urbanfischer.de/journals/optik

Differential algebra is based on the theories of non-standard analysis and formal series [6]. It exploits the classic theorems of differential calculus to propagate information about derivatives through arithmetic operations. In this way, derivatives of a function can be calculated using the same program that calculates the function itself. For any smooth function f (x1, x2, … , xn), its 0 ~ n order derivatives can be expressed by nDn structure in differential algebra. All the derivatives are arranged in a special sequence to generate a differential algebraic vector a = (a1, a2, … , aN). According to the series theory, a function at some point can be expanded as:  ∂f   ∂f   ∂2 f  2 f = f0 +   x1 +   x 2 + … +  2  x1 + …  ∂x1   ∂x 2   ∂x1    ∂n f +  x1 x 2 … xn + …  ∂x1∂x 2 … ∂xn 

(1)

where the coefficient of each monomial is relate to a component of the differential algebraic vector. Therefore, the dimension of nDn is the number of the monomials in n variables through order n, that is, N (n, n) = (n + n)!/(n!n!) = C (n + n, n). Now assume the I-th monomial MI = x1i1 x2i2… xnin, we define FI = i1! i2!…in!. Then an addition, a scalar multiplication and a vector multiplication in 0030-4026/01/112/06-250 $ 15.00/0

Min Cheng et al., Study on differential algebraic aberration method for electrostatic electron lenses n Dn

can be defined as follows: ( a1, a2, …, a N ) + ( b1 , b2, …, bN ) = ( a1 + b1 , a2 + b2 , …, a N + bN ) t ⋅ ( a1, a2, …, a N ) = ( t ⋅ a1 , t ⋅ a2 , …, t ⋅ a N ) ( a1, a2, …, a N ) ⋅ ( b1 , b2, …, bN ) = ( c1 , c2, …, c N )

(2)

where t is an arbitrary real number, the coefficient cI is: c I = FI ⋅

∑

0 ≤ J, K ≤ N MJ ⋅ MK = MI

a J ⋅ bK FJ ⋅ FK

( I = 1, 2, …, N ) .

(3)

Differential calculus ∂j can also be defined in nDn , for example, the first order derivative is defined as: ∂ j ( a1, a2, …, a N ) = ( d1 , d2, …, d N )

( j = 1, 2, …, n ) , (4)

where dI (I = 1, 2, …, N) is equal to aJ (J is the ordinal number of the monomial MI · xj) while the order of MI is less than n; otherwise, dI is equal to 0. With the existence of ∂j operation as a kind of arithmetic operations, nDn becomes a differential algebra. For any differential algebra vector of the form (0, q2, …, qN) Î nDn , that is, with a zero in the component belonging to the zero-order monomial, we have the following nilpotent property: (0, q2, …, qN)m = (0, 0, …, 0)

for m > n

(5)

which follows directly from the definition of the multiplication in nDn defined in eq. (2). The fundamental functions, such as exponent, logarithm, trigonometric functions, can be extented into nDn by series expansion to finite order without truncation error and rounding error because of the nilpotent property. For instance: log [( a1, a2, …, a N )]    a a a   = log  a1 ⋅ 1 +  0, 2 , 3 , …, N    a1      a1 a1  = log ( a1) +

∞



a

∑ ( – 1)i + 1 1i  0, a2

i =1

1

,

(6)

a3 a  , …, N  a1 a1 

i

( a1 > 0 ) = log ( a1) +

i

 a a a  ∑ ( – 1)i + 1 1i  0, a2 , a3 , …, aN  . 1 1 1 i =1 n

3. Differential algebraic aberration method for electrostatic electron lenses The focusing and imaging properties of a charged particle optical system can be described by a transfer map as follows [7]: rf = R (r0, d)

(7)

251

where rf denotes the final coordinates of a particle to its initial coordinates r0, and d contains other systemic parameters of interest. ∂R/∂r is corresponding to systemic aberrations, while ∂R/∂d is corresponding to systemic sensitivities. Except for the most trivial cases, it is impossible to find a closed analytic solution for the map R. Expanding R in a power series yields a set of differential equations for the expansion coefficients that in many cases can be solved analytically up to some order. The complexity of the resulting differential equations increases dramatically with the order of the expansion coefficients. Therefore, this procedure is limited to lower medium orders. However, differential algebraic method presents a straightforward way to compute nonlinearity to arbitrary orders. Here no analytic formulas for derivatives must be derived; on the other hand, the method is always accurate to machine precision independent of the order of the derivative, which is in sharp contrast to methods of numerical differentiation. In the laboratory Cartesian coordinates (x, y, z), electron trajectory equations in electric field can be expressed as [8]:  x ′′ = 1 (1 + x ′ 2 + y ′ 2 )  ∂u – x ′ ∂u    ∂x 2u ∂z    ∂u  1 ∂ 2 2 (8)  y ′′ = (1 + x ′ + y ′ )  – y′ u  .   ∂y 2u ∂z  In a rationally symmetric system, electric field u (x, y, z) can be determined by axial potential f [9]: u ( x , y, z ) = f ( z ) – +

( x 2 + y2) f ′′ ( z ) 4

( x 2 + y 2 )2 ( 4 ) f (z) 64 n

–…+

 x 2 + y2  (– 1) n ( 2 n ) z f ( )   + ….  4  ( n!) 2

(9)

The above-mentioned series is truncated to some limited order without bringing errors, because of the nilpotent property of differential algebra. Introducing eq. (9) in eq. (8) and using the present numerical integrating methods, the electron trajectory equations are solved. Then the relation between the initial coordinates r0 and the final coordinates rf are gained. In nD4, the results from the differential algebraic method take the form:  Aijkl  x f   Bijkl  yf  i + j + k + l = n .  = x 0i y0j x 0′ k y0′ l  ∑  Cijkl   x ′f  i , j , k , l = 0 ~ n  Dijkl   y′  n f 

(10)

This denotes Gaussian properties expression (while n = 1) and arbitrary order aberration expressions (while n > 1, n is an integer) by differential algebraic method. For example, the third order aberration in x- and y-direction are expressed as follows:

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Min Cheng et al., Study on differential algebraic aberration method for electrostatic electron lenses

3xf

= A0030 x 0′ 3 + A0021 x 0′ 2 y0′ + A0012 x 0′ y0′ 2 + A0003 y0′ 3 ⇒ spherical aberration + A1020 x 0 x 0′ 2 + A1011 x 0 x 0′ y0′ + A1002 x 0 y0′ 2   + A0120 y0 x 0′ 2 + A0111 y0 x 0′ y0′ + A0102 y0 y0′ 2 

⇒ coma

+ A2010 x 02 x 0′ + A1110 x 0 y0 x 0′ + A0210 y02 x 0′   + A2001 x 02 y0′ + A1101 x 0 y0 y0′ + A0201 y02 y0′ 

⇒ field curvature and astigmatism

+ A3000 x 03 + A2100 x 02 y0 + A1200 x 0 y02 + A0300 y03

⇒ distortion .

The expression of 3 yf has the similar form with 3 xf , only replacing Aijkl with Bijkl.

1 =1 fi a

Here we introduce an example of electrostatic electron lens to show the advantages of differential algebra used in high order aberration analysis. Schiske’s model is a widely studied model of electrostatic electron lenses, which axial electric field distribution is described by an analytic expression:

∫

y

∫

3

(12)

0

dq 2

(1 – k sin

2

1 q)2

 sin 2y 0 sin 2y i ⋅ 1 – 1   (1 – k 2 sin 2 y 0 ) 2 (1 – k 2 sin 2 y i ) 2

= F (y , k ) ,

2 w := 1 – k 2 2

E0 ( k ) =

where F ( y, k) denotes an elliptic integral of the first kind. After theoretic derivation, the image plane conjugate to z = z0 plane locates at [5]:

Ei ( k ) =

∫

1 – k 2 sin 2 q d q ,

0 yi

∫

1 – k 2 sin 2 q d q .

(18)

Using a rotational coordinate system, the Gaussian properties are described by a first order transfer map [10]:

(14)

sin y 0 sin y i

y0

0

 M  xg   M  yg   1 Ms   = –  x g′   fi – 1  yg′    fi

and the magnification M, the parameter fi are respectively shown below: M = (– 1) n

    (17)   

where E(k) denotes the elliptic integral of the second kind:

(13)

zi = z 0 – n π w

(16)

(1 – k 2 sin 2 y 0 ) 2  E0 ( k ) – Ei ( k ) Cs = Ma  2 2 2 sin 4 y 0  k (1 – k )  8 – k2  1 – nπ  12 + – 16 w 4  2 (1 – k 2 ) w k

z = : a cot y , dy = f

)

For third order aberrations, the expressions of the aberration coefficients are very complex. Therefore, for instance, the coefficient Cs of the third order spherical aberration is given by:

Substituting this expression in the paraxial trajectory equations and introducing:

z : = f0

(

f0 sin y 0 cos y i 1 – k 2 sin 2 y i fi

– cos y 0 sin y i 1 – k 2 sin 2 y 0 .

4. Application: Schiske’s model electrostatic electron lens

  k2 f ( z ) = f0 1 – .  1 + ( z / a )2 

(11)

(15)

   x0    y0      x 0′  Ms   y0′  

(19)

Table 1. The comparison of the Gaussian optical properties and Cs between differential algebraic results and the analytic solutions for Schiske’s model electrostatic lens ( f0 = 5V, k2 = 0.5, a = 0.025 m, z0 = – 0.5 m). M

Ms

– 1/fi

Cs

Analytic solutions

– 1.633299180136

– 0.612018921815

– 3.217556194266

– 1198.665836584

Results of differential algebraic method

– 1.633299180119

– 0.612018921804

– 3.217556194246

– 1198.665836548

Relagive errors

1.04084´ 10– 11

1.79733 ´10– 11

6.21590 ´10– 12

3.00334´ 10– 11

Min Cheng et al., Study on differential algebraic aberration method for electrostatic electron lenses

253

Table 2. Results of the third order geometric aberration coefficients for Schiske’s model electrostatic lens ( f0 = 5V, k2 = 0.5, a = 0.025 m, z0 = – 0.5 m). a) The third order geometric aberration coefficients in x-direction Spherical aberration coefficients

A 0030 – 1198.665836548

A 0021 0.0

A 0012 – 1198.665836548

A 0003 0.0

Coma coefficients

A1020 – 7174.059869613

A1011 0.0

A1002 – 2391.353289871

A 0120 0.0

A 0111 – 4782.706579742

A 0102 0.0

Field curvature and astigmatism coefficients

A 2010 – 14323.92239355

A1110 0.0

A 0210 – 4779.181961961

A 2001 0.0

A1101 – 9544.740431594

A 0201 0.0

Distortion coefficients

A 3000 – 9540.879159025

A 2100 0.0

A1200 – 9540.879159025

A 0300 0.0

b) The third order geometric aberration coefficients in y-direction Spherical aberration coefficients

B0030 0.0

B0021 – 1198.665836548

B0012 0.0

B0003 – 1198.665836548

Coma coefficients

B1020 0.0

B1011 – 4782.706579742

B1002 0.0

B0120 – 2391.353289871

B0111 0.0

B0102 – 7174.059869613

Field curvature and astigmatism coefficients

B2010 0.0

B1110 – 9544.740431594

B0210 0.0

B2011 – 4779.181961961

B1101 0.0

B0201 – 14323.92239355

Distortion coefficients

B3000 0.0

B2100 – 9540.879159025

B1200 0.0

B0300 – 9540.879159025

where (xg, yg, x¢g, y¢g) is the vector containing positions and slope on the Gaussian image plane, (x0, y0, x¢0, y¢0) is the vector containing positions and slope on the object plane and Ms is the reciprocal magnification: Ms = 1 M

f0 . fi

(20)

Now we use differential algebraic method to calculate Gaussian properties and third order aberrations if Schiske’s model electrostatic lens. The variables x, y, x¢, y¢ are set to be differential algebraic vectors and the axial electric field distribution expression (12) is substituted in eq. (9) to determibe the electric field u (x, y, z) in eq. (8). We can solve the trajectory equations (8) by performing Runge-Kutta method and gain the differential algebraic vercots xg, yg, x¢g, y¢g in the Gaussian imaging plane. Therefore, the Gaussian optical properties and arbitrary high order aberrations can be obtained by the differential algebraic method shown in eq. (10). We calculate a real Schiske’s model electrostatic lens with the parameters: f0 = 5V, k2 = 0.5, a = 0.025 m, the object plane locates at z0 = – 0.5 m. The comparison of the Gaussian optical properties and Cs between differential algebraic results and the analytic solutions are shown in table 1. From the relative errors of the two methods, it is proved that the differential algebraic method has very high accuracy.

Fig. 1. Diagram of third order geometric aberration distribution of Schiske’s model electrostatic lens. a) Spherical aberration (x 0 = 0, y 0 = 0, r 0¢ = 0 ~ 5 mrad); b) coma (r 0 = 5 mm, j0 = p/4, r 0¢ = 0 ~ 5 mrad); c) field curvature (r 0 = 5 mm, j0 = p/4, r 0¢ = 0 ~ 5 mrad); d) distortion: The solid lines denote Gaussian image, and the dashed lines denote superposition of Gaussian image and third order distortion).

254

Min Cheng et al., Study on differential algebraic aberration method for electrostatic electron lenses

All the coefficients of the third order geometric aberrations are calculated by differential algebraic method shown in table 2. Since all the third order geometric aberration coefficients have been successfully calculated, the diagrams of the aberration distribution are subsequently given in fig. 1. Using the mentioned differential algebraic method, it is easily to calculate fifth order or higher order aberrations of the electrostatic electron lenses.

5. Conclusions In this paper, differential algebraic aberration method for electrostatic electron lenses is presented. By employing the effective tool, the arbitrary high order aberrations can be calculated with extreme high accuracy up to the machine precision. As an example, an important analytical model of electrostatic lenses named Schiske’s model lens has been studied, and the Gaussian properties and third order geometric aberration coefficients have been calculated. The results show that differential algebraic method is an effective tool with excellent accuracy for the aberration analysis and calculation of electrostatic electron lenses. This developed method can be of great utility in

high order aberration analysis and computation for charged particle optical systems.

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