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A technique for computing the first order focusing properties of an arbitrary static electric field
Nuclear Instruments and Methods North-Holland, Amsterdam
m
Physics Research A273
(1988) 77-86
77
A TECHNIQUE FOR COMPUTING THE FIRST ORDER FOCUSING PROPERTIES OF AN ARBITRARY STATIC ELECTRIC FIELD T.A. WINCHESTER and R.L. DALGLISH School of Physics, University of New South Wales, Kensington, NSW 2033, Australia
Received 22 January
1988
and
m
revised form 26 May
1988
A technique for computing the first order focusing properties of charged particle rays about a curved ray m a static electric field is described. A set of paraxial equations is derived where the paraxial rays are expressed m the same coordinate system as that used for the optical axis. The techmque is designed specifically for efficient computer solution, and the axis need only be known in the form of a numerical solution. A formula for the determinant of the matrix of paraxial solutions is found. A comparison between the computed solutions of the derived equations and a simple test potential which allows a direct analytic solution are made; precise agreement is found.
1. Introduction In recent years we have been investigating the focusing properties of charged particle lenses using transverse rods [1,2]. A method of computing the electric potential for this geometry has been derived, and an approximate formula has been presented in a prior paper [3] for the deflection of ions passing through this system. This paper addresses a problem encountered in computing the exact focusing properties of systems of these elements. The optical axis cannot be defined by consideration of the symmetries of the system. It is not a straight line, but a curve in the plane of mirror symmetry of the electric field; in fact, the only way that the path corresponding to the optical axis could be found was by a numerical solution of the second order differential equations for a ray. It was necessary, therefore, to devise a technique that could compute the focusing of ions about a path known only in the form of a numerical solution for a ray. The focusing effects due to an electric field about an arbitrary axis can be computed by the Method of Trajectories [4], or through the use of the Characteristic Function [5], by expressing the ray about the optical axis using the Frenet coordinate system based upon the selected optical axis. A numerical solution of the resulting equations requires the curvature, torsion, electric potential, and the first and second order covariant derivatives of potential, all of which must be known as a function of the distance along the optical axis. These requirements present no difficulty for a computer program if the optical axis is known i n the form of a simple curve (for example, an arc of a circle, or a helix). Our problem, however, is one where the only information available for the optical axis is the set of discrete points generated by a numerical solution of a ray. Transforming the data available to express it to the Frenet coordinates about the optical axis, presented a major computational problem. It became obvious that a general technique was needed to provide an efficient solution, by a computer program, of the focusing properties of a general static electric field. We will show that this can be achieved by using one coordinate system to represent the field, the optical axis, and the paraxial rays. In this paper we shall use a Cartesian coordinate system to express both the optical axis, and the paraxial rays about it. It will be assumed that the optical axis is a curve that can be found by a numerical solution in Cartesian coordinates, and that the electric field and its derivatives can be determined numerically in the Cartesian coordinate system. A general method for computing the focusing properties of the field about this optical axis is presented below. The solutions can be written as a four dimensional matrix, and a formula for the determinant, the Wronskian of the paraxial matrix, is derived. The method will be verified, both analytically and numerically, by using a simple field whose properties can be found analytically. 0168-9002/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
78
TA
Winchester, R. L Dalghsh / Computing first order focusing properties of a static electric field
W
Fig. 1. All rays are described by the three coordinates s, x 1 , x". The difference between the optical axis and a paraxial ray is described by the coordinates in the travelling plane in the diagram. This plane is always parallel to the x'-x" plane.
2. The paraxial equations for a ray in Cartesian coordinates Let x l , x" describe the the Cartesian components of a point in the plane normal to an s-axis (see fig. 1). From the principle of least action the equations for the path of an ion may be expressed in the form of a pair of second order differential equations, provided that the path is never orthogonal to the s-axis [6]. In their nonrelativistic form, these are represented by eqs. (1) (the extension required to include relativistic effects are straightforward).
L
_d 2 II __ a 2 a~ II a$ _ II , 2 ~~ax " -u as)-e d s 2x
_d 2 1 __ _a 1 1 _a0 _ 1 2$~x'-u as) -e ' ds 2x
where a = (1 + u I u I + uitu111)]/2 (P _ - gV/mc 2 , and V is the electrostatic potential chosen so that V = 0 when the ion is at rest, u', u" are the derivatives of x l , x" respectively with respect to s. The optical axis is a particular solution of eq. (1). To compute the focusing effect, we must compute the path of a nearby ray. This ray is represented by the Cartesian coordinates *x', *xll, *u', *ull; the optical axis by x', x 11 , u', u 1I ; and the differences between these are expressed in the form of eq. (2): *xI=x1+8x1, *xl1=x1I+8x11, *u 1 =u'+8u 1 , *U il =ull=8ul'.
(2)
The equations that describe the difference parameters 8x 1 , 8x 1 ', 8u', 8u", are derived by expanding the right hand side of eqs. (1) in the form of a multidimensional Taylor series about the optical axis, neglecting terms above the first order. This yields eqs. (3), with e l , e l ', and their partial derivatives evaluated on the optical axis. a d2 ae' ae' +8u11 (x'+Sx')=e'+8x'ae' +8x" el +8u' ds 2 ax' ax' 1 au' au" De" d2 II -- (x + 8x") = eII + 8xI ae" + 8xII ae" + 8ul óe11 + 8u,, ds2 óx 1 óx' 1 óu 1 au ll
These equations will be valid for rays that lie close to the chosen optical axis. Since the optical axis satisfies eqs. (1), we may write eqs. (3) in the form of eqs. (4), which is a system of second order linear differential equations in the difference parameters. It can be shown that the only new information requires to evaluate eqs. (4) are the second order derivatives of J) to Cartesian coordinates. d (8x') = 8x' ds (~
1_
al)
~ ax' ) ((
+8u'((2
e' +
a
_ uI
a
ax , as
25 ( ax I ax I '
u'
a
24 ( a x l
a2
a
ax 1I as )~ u11
1 )e'+-()~ +8u"(2 a'- e'); 20 as a2
T.A. Winchester, R. L Dalghsh / Computing first order focusing properties of a static electric field
d
s
d
a 2o
i a~ + a e° 24 ( $ ax , ~
(8x 11 ) = 8x1 ~~
+8x11((_ 1 $ I
+8u'(2
u a2
- u
45
ax , as
ax 1 2ax 11
a o ) en+
v2 (
a Z$
ax 1 '
2$
a 2 x 11
I 11 e ° ) +óu 1 ' I (2 ue 2 11 )
a
u
79
-u
+
a 2~
ax 11 as ~
l l a( - a ) . J 2
0
ll
These equations have four independent solutions; since the equations are linear, any four independent solutions can span all possible solutions. It is possible therefore to select solutions which provide a direct intepretation of the focusing action. The four solutions chosen correspond to the initial conditions listed in eqs. (5). The first and third express the result of a change in the value of x' and x 11 in the plane s , and the second and the fourth changes in u 1 and u l1 relative to the object plane. The notation for the solutions in eqs. (5) has been chosen to unambiguously identify each of these solutions.
o
_ 8x 1 _ Su l 8x 11 Su' I 8xó , óxó , _, óx„óxó _ 8x 1
8u 1
Su 11
8x 11
_ 8x 1 8u 1 8x 1 ' _ 8u 11 (0, 1, 0, 0), 8uó , 8uó , 8uó , óuó
_ (1, 0, 0, 0),
(0, 0, l
8x' i (8u
' O),
Su'
01 ' u Su o '
8x 11 8u
u
li
x0
0
Su l ' Su II
_ (0, 0, 0, 1).
o 8xo 8X() 8 óx0) The solution for eqs. (4) for any set of initial conditions at the plane s o can be expressed in the matrix u
11
II '
form (6): 8X1
Sx 1
Sx 1
8xó
óuá
8xi
Sx' 1 ( 8xó 8uá1
Su l
Su 8xó
_ 8u Sul
_ Su l 8xi
8u1 Bui
8x11
8x11 8x 01 8u 11
8x11 Su l0 Su 11
8x11 8x 110
8x11 ßu o11
8xá
8uó
Bu ll 8zó1
ßu11 Bui
18x 1 1
(
óu n
Su l °
I
8x 11 0
1, 8U Io
3. The Wronskian formula If eq. (4) is written as a first order matrix differential equation, then the determinant of the solutions, W(s), is given by eq. (7), where W is the determinant of the solutions in the plane defined by s [7]:
0
0
del + ae II )). W(s) = W exp(f ds( s au , au 1 ' s
0
o
The integrand in eq. (7), given explicitly in eq. (8), can be manipulated into eq. (9).
4
2
_de l + ae11 - u l e' + u 11e 11 u le 1 + u 11e 11 + a 2 ~ act(u as) 1 du ll ) a2 ( az de l de ll 2 da l 1d~5(8u+al)-2 d
(8)
ds - q) s
When eq. (9) is substituted into eq. (7), the integration yields eq. (10). a W(s)-Wo(< rol ~~
(lo)
80
TA
Winchester, R.L. Dalglish / Computing first order focusing properties
of a static electrie field
The equivalent formula obtained from the equations for paraxial rays expressed in the Frenet coordinates about the optical axis is given in eq. (11) [4]. W(s) = W°(V°
).
(11)
Unlike the Frenet coordinate system, where the vectors used to describe the paraxial rays are always normal to the direction of the optical axis, the Cartesian coordinate system used in this paper is independent of the ray chosen for the optical axis. If the ray corresponds to a straight symmetry axis, then the value of a will be constant and eq. (10) reduces to eq. (11). Therefore the ratio (a/a0)4 can be intepreted as expressing the change in the inclination of the optical axis to the Cartesian axes used to express the rays. 4. A test for the paraxial equations A simple field distribution which allows a full analytic solution of the rays was used to test the validity of the equations derived. A homogeneous field with no component along the s-axis, with fields P, E lf along the x I , x" axes respectively was chosen. The exact equations for a ray take the form of eq. (12). d 2x1 ds2
_ a2
2V
Ei,
d 2x " _ - _a? E,', ds2_ 2V
(12)
where V = V„ + E 'x I + E "x" is the total electric potential, VA is the energy of the ion in eV when it enters the region. Since there is no field component parallel to the s-axis, the value of a/2V is invariant, and is represented by C. The solutions to eqs. (12), eqs. (13), are two parabolas defined by E l , E li , C, and the i nitial values of x', x I ', u 1 , u". I (J) I ] (13) x (s)=x +u s+CE x1 ' (S) =x,) +uös+CE"
V,
V.
Since the value of C varies with the initial conditions for the ray, the paraxial rays will exhibit a small, but computable form of focusing behaviour. This effect is due to the variation of the total electric potential affecting the magnitude of the velocity of the ion along its path. The paraxial equations are given in eqs. (14).
v
d2 8x t =8x I ( "(ds 2 v'e')+8x dz Sx~ t =8x , - Ve El el'li ) +Sx" d s2 ( (
l e')+8uj( u'y' ) +Su"( u ' V l n uu n e li ) +8u ) +8u ( V ).
V
(14)
(uv
Let us assume that the solutions of eqs. (14) will be parabolas; we shall verify this by substituting this form into eqs. (14) according to the initial conditions (5), and, in doing so, determine the algebraic form which is consistent with eqs. (14). The solution that satisfies the first set of initial conditions of (5) will be of the form s2 s2 8x' =1+a 8x " =b
2,
2.
Substitution of this form into eqs. (14) gives the following conditions for the constants a and b: I a= -El e +a(x I -xá)
V
+b(
X101)
V1 ,
T.A. Winchester, R.L. Dalghsh / Computing first order focusing properties of a static electric field
81
It can then be shown that the values of the constants a and b are ' a= -CE' E-, Vo
E'
b= -CE It Vo
The solution for the third set of initial conditions of (5) follows from the symmetry in the equations. The solutions for the second set of initial conditions of (5) of the form 8x'=s+c
s2
2
,
8x11=d
s2
2
The same procedure gives the following conditions for the constants c and d: c= uoy t +c(x'-xo
d=
V
)
+d(x1'-xö
V1'
Eu E" uo' E" +c(xt-xó~ +d(xtt-xö) . V V V
The values of the constants c and d are 1 E' c=uo V ,
~ E" d=u' V
The solution for the fourth set of initial conditions also follows from the symmetry in the equations. The complete set of solutions is listed in eqs. (15). 8 _ 8x11 8u' Sxó
1-CE' E s , Vo 2 - CE'
E'
s,
Vo
óx , _ E" s2 _CE,- 8xo' Vo 2 ' E1' 8u __ -CE' s, Sx0 Vo 8x 11 - _CE"E' s2 _ 8,1 Vo 2 ' E' 8u" -CE"-,
8xó
VO'
E. 11 s 2 8x" =1-CEI' -_ 8xö Vo 2 '
Su" óx 1
E" -CE " -s, Vo
Sx 8uó
=s+UO E s Vo 2
óu , E' - = 1 +u ó -s, bul Vo óx,_
=uon
8 uó'
E
' s2
Vo
-, 2
Su
= uol E s, Vo 8x11 -u' E" S 2 8uo'
8uó 8u 11
(15)
, o Vo 2 E t'
= uó -s, Vo 8uó
&
E" s2 =s+u~'-2 , 8ui1 Vo El' 8u" X
11
-=1+u -s. 8uä
Vo
It can be seen that these solutions for the homogeneous electric field are identical to the partial derivatives of the exact solution of the optical axis with respect to the initial conditions for the optical axis. Rays connecting points on the object and image planes can be considered as four dimensional functions of the object plane coordinates giving values for x l , x", u l , U I ' in the image plane. The solutions of the paraxial equations given by eqs. (15) are the coefficients of the first order terms in the multidimensional
T. A. Winchester, R L Dalghsh / Computing first order focusing properties of a static electric field
82
Table 1 Results for the axis ray trace
Object I mage
X1
ui
X2
U2
O.OOOOOOOOD + 00 -0.125000001)+00
0.000000001) + 00 -0.250000001)+00
O.OOOOOOOOD + 00 0.12500000D + 00
0 00000000 D + 00 0.25000000D + 00
X10
U10
X20
U20
0 93750000D + 00 - 012500000D + 00 0.62500000D - 01 0.12500000D+00
0.10000000D+01 0.10000000D+01 0.44411586D -12 0.90252923D-12
Entry electric potential Exit electric potential
-0.10000000D+06 -0.112500001)+06
Results for the paraxial rays
X1I Uli U21 U21
0.625000001)- 01 0.12500000D + 00 0.93750000D + 00 - 0.12500000D + 00
0.44411586D- 12 0.90252923D -12 0.100000001) + 0 1 0.1000000013+01
DET
0.1125000013+01 WRK 01125000013+01 DIF 0.16453505D-12
Taylor series expansion of the rays about the initial point xó, xó , uó, u 01. This expansion has been calculated for arbitrary values of the initial conditions. The Wronskian formula, the determinant of the matrix of paraxial solutions, is the Jacobian determinant of the mapping. A computer program was written in double precision FORTRAN that solves the paraxial equations. A predictor-corrector method was used where the number of iterations of the corrector formula could be varied, and it was found that four iterations were sufficient. This program could access a subroutine that would return the total electric potential and the derivatives of the potential. To test the program, a subroutine was written that would return the potential used for the above calculations. A value of 100 keV was selected for the initial energy of the ion, and the solutions were performed over a length of one metre. Values for E t , E 1t were selected that would bend the axis by the order of one radian so that the program could be verified under conditions where the field strongly affects the motion of the ions. The program first solved the optical axis numerically, and then used the stored values for the optical axis to form and solve the paraxial equations.
Table 2 Results for the axis ray trace
Object I mage
O.OOOOOOOOD + 00 0.7500000013+00
0.1000000013+01 0.50000000D+00
O.OOOOOOOOD + 00 0.25000000D+00
0.0000OOOOD + 00 0.50000000D+00
X10
U10
X20
U20
0 87500000D + 00 -025000000D+00 0.12500000D+00 0.25000000D + 00
0.75000000D + 00 0.50000000D+00 0.25000000D+00 0.50000000D + 00
0.12500000D + 00 0.25000000D+00 0.87500000D+00 - 0.25000000D + 00
- 0.84413451 D -12 -0.16345175D-11 0.10000000D+01 0.1000000013+01
Entry electric potential Exit electric potential
-0.1000000013+06 -0.750000001)+05
Results for the paraxial rays
X1I Uil X21 U21
DET 0.7500000013+00 WRK 0.7500000013+00 DIF 0.26869618D-11
T.A. Winchester, R. L. Dalghsh / Computing first order focusing properties of a static electric field
Table 3 Results for the axis ray trace
Obj ect Image
X1
U1
X2
U2
O.000OOOOOD + 00 - 0.25000000D + 00
O.000OOOOOD + 00 -0.50000000131+00
O.000OOOOOD + 00 0.12500000D + 01
0.10000000D + 01 0.15000000D + 01
X10
U10
X20
U20
0.ß7500000D+00 - 0.2500000013 + 00 0.1250000013+00 0.2500000013 + 00
0.10000000D + 01 0.10000000D + 01 - 0.92739655D -12 -0.19120615D-II
0.1250000013 + 00 0.2500000013 + 00 0.87500000D + 00 -0.25000000D+00
- 0.25000000D + 00 - 0.50000000D + 00 0.12500000D + 01 0 15000000D + 01
Entry electric potential Exit electric potential
-0.1000000013+06 -0.1750000013+06
Results for the paraxial rays X11 Ull X21 U21
DET 0.17500000D + 01 WRK 0.17500000D+01 DIF -0.74533713D-11
Table 4 Results for the axis ray trace
Object Image
Xl
Ul
X2
U2
O.000OOOOOD + 00 0.62500000D + 00
0.1000000013 + 01 0.25000000D + 00
O.000OOOOOD + 00 0.13750000D + 01
0.10000000D+01 0.1750000013+01
X10
U10
X20
U20
0.81250000D + 00 - 0.3750000013 + 00 0.18750000D+00 0.3750000013 + 00
0.7500000013 + 00 0.5000000013 + 00 0.2500000013 + 00 0.5000000013 + 00
0.18750000D+00 0.3750000013 + 00 0.81250000D + 00 - 0.3750000013 + 00
- 0.25000000D + 00 - 0.50000000D + 00 0.12500000D+01 0.15000000D+01
X1
U1
X2
U2
O.000OOOOOD + 00 -0.10102051D+00
O.000OOOOOD + 00 - 0.18350342D + 00
O.000OOOOOD + 00 0.10102051D+00
O.000OOOOOD + 00 0.18350342D+00
X10
U10
X20
U20
0.95875855D + 00 - 0.68041382D - 01 0.41241452D-01 0.68041382D-01
0.89897949D + 00 0.81649658D + 00 - 0.12586481 D -11 - 0.22358008D -11
0.41241452D-01 0.68041382D-01 0.95875855D+00 - 0.68041382D - 01
- 0.12586481 D -11 - 0.22358008D - I I 0.89897949D + 00 0.81649658D + 00
Entry electric potential Exit electric potential
-O.10000000D+06 -0.1375000013+06
Results for the paraxial rays
X11 Ull X21 U21
DET 0.13750000D+01 WRK 0.13750000D+01 DIF 0.46487258D-11
Table 5 Results for the axis ray trace
Object Image
Entry electric potential Exit electric potential
-0.1000000013+06 -0.1601020513+06
Results for the paraxial rays X11 Ull X21 U21
DET 0.7115646713 + 00 WRK 0.7115646713 + 00 DIF - 0.925 59294D - 12
84
T A. Winchester, R L Dalgltsh / Computing fist order focusing properties of u static electric field
Table 6 Results for the axis ray trace
Object I mage
XI
U1
X2
U2
O.000OOOOOD + 00 0.65685425D + 00
0 100000001) + 01 0.41421356D+00
O.000OOOOOD + 00 0.17157288D+00
O.000OOOOOD + 00 0.29289322D + 00
X10
U10
X20
U20
0.87867966D + 00 - 0.176776701) + 00 0.606601721)- 01 0.ß8388348D-01
0.585786441) + 00 0.353553391)+00 0.12132034D+00 0.176776701) + 00
0.12132034D + 00 0.17677670D + 00 0.93933983D + 00 -0.88388348D-01
0.32542850D - I I - 034172577D -10 0.82842712D + 00 0.70710678D + 00
Entry electric potential Exit electric potential
-0.1000000013+06 -0.125735931)+06
Results for the paraxial rays
X1I Ull X21 U21 DET WRK DIF
0.31433983D + 00 0.31433983D + 00 -0.56141164D - 09
Table 7 Results for the axis ray trace
Object I mage
X1
U1
X2
U2
O.000OOOOOD + 00 -0.17157288D+00
O.000OOOOOD + 00 - 0.29289322D + 00
O.000OOOOOD + 00 0.100000001) + 01
0.1000000013+01 0.1000000013+01
U10
X20
U20
0 606601721)- 01 0.ß8388348D-01 0.100000001) + 01 0.000000001) + 00
- 0.12132034D + 00 - 0.17677670D + 00 0 82842712D + 00 0.70710678D + 00
Entry electric potential Exit electric potential
-0.1000000013+06 -0.20ß57864D+06
Results for the paraxial rays X10 X11 Ull X21 U21 DET WRK DIF
0.93933983D - 0.88388348D O.000OOOOOD 0.000000001)
+ 00 - 01 + 00 + 00
0.82842712D 0.70710678D 0.000000001) O.000OOOOOD
+ 00 + 00 + 00 + 00
0.52144661 D + 00 0.51144661 D + 00 - 0 24707880D - 09
Table 8 Results for the axis ray trace
Object I mage
X1
U1
X2
U2
O.000OOOOOD+00 0.54970355D+00
0,1000000013+01 0.2649110613 + 00
0.000000001) + 00 0.10000000D + Ol
0.1 OOOOOOOD + 01 0.10000000D + 01
Entry electric potential Exit electric potential
-O.10000000D+06 -0.172514ß2D+06
Results for the paraxial rays
X11 Ull X21 U21
X10
U10
X20
U20
0.8576037613+00 - 0.18973666D + 00 O.000OOOOOD + 00 O.000OOOOOD + 00
0 58499011D+00 0.37947332D + 00 O.000OOOOOD + 00 O.000OOOOOD + 00
0.14239624D + 00 0.18973666D + 00 0.1000000013+01 O.000OOOOOD + 00
- 0 18986166D + 00 - 0.25298222D + 00 0 77485177D + 00 0.632455531)+00
DET 0.27602371 D + 00 WRK 0.27602372D + 00 DIF - 0.20261378D - 08
T.A. Winchester, R.L. Dalghsh / Computing first order focusing properties of a static electric field
85
Table 9 Results for the axis ray trace
Object I mage
X1
U1
X2
U2
0.1234000013+00 - 0.3842441313 + 00
- 0.4567000013 + 00 -0.5498534413+00
0.89010000D+00 0.7713565313 + 00
- 0.23450000D + 00 - 0.22834387D - 01
X10
U10
X20
U20
0.98474380D + 00 -U550503313-01 0.34665515D - 01 0.57953180D- 01
0.93674413D+00 0.8795509513+00 - 0.69330304D - 01 - 0.11590514D + 00
0.15256196D - 01 0.25505033D - 01 0.96533448D + 00 -0.57953180D-01
0.15666917D 0.26191668D 0.87063329D 0.76902805D
Entry electric potential Exit electric potential
-0.1383350013+06 -0.2077800313+06
Results for the paraxial rays
XII U11 X21 U21
- 01 - 01 + 00 + 00
DET 0.70782947D + 00 WRK 0.70782947D + 00 DIF - 0.67230665D -11
The results of these calculations are given in tables 1-4, where 128 points were used for the numerical solution. The values of uá, not ( U1, U2 in the tables) were (0, 0), (1, 0), (0, 1), (1, 1) in tables 1-4 respectively, while the values of xó, xo' (X1, X2 in the tables) were left fixed. The initial and final values of the ray variables are given first (Object, Image in the tables), and then the matrix of paraxial solutions in the format of eq. (6). The difference between the derived and the computed solutions is in all cases less than 2 x 10 -9 , or better than single precision. The determinant of the matrix was calculated by using the Laplace expansion, and given in the entry DET. The Wronskian formula is evaluated, and given in the entry WRK, and the difference, not detectable in the precision of the computer output, is given directly below in entry DIP. In these tables it can be verified that the value of the determinant is V/V,. However, the value of C, which is constant, may be used with the Wronskian formula to show that this should be expected. To verify that the general form of the Wronskian formula is correct, a field component parallel to the s-axis of -50 kV m -1 was added, and the solutions of tables 1-4 recomputed. The results of these computations are given in tables 5-8, and it may be seen that the general form of the Wronskian formula remains valid. As a final check on this formula, a random choice for all the initial values for the ray was used, with the results given in table 9, from which it may be seen that the Wronskian formula is satisfied even under these extreme conditions. The numerical solution of the paraxial equations has been shown to agree with the analytic solutions listed in eqs. (15). This example shows that this technique can be used without any assumptions about the symmetry of the electric field. The computer program only requires a subroutine to compute the numerical values of first and second order derivatives of the electric field in Cartesian coordinates.
5. Conclusion
We have derived a method suitable for solution by a computer program for evaluating the focusing behaviour of a static electric field about an optical axis which is known in the form of a numerical solution for a ray in Cartesian coordinates. This axis need not be known prior to the application of this technique, since a numerical solution for the optical axis provides all the information required. Symmetries of the electric field are not relevant, since the technique assumes that the electric field exists, and that it can be numerically determined in Cartesian coordinates. This technique is ideal therefore, for computing the focusing effects in situations where there is little or no prior knowledge about the optical axis.
86
T.A. Winchester, R L Dalghsh / Computing first order focusing properties of a static electric field
The extension of this method to include magnetic fields is straightforward, requiring the use of the exact equations for an ion in a static electromagnetic field instead of the equations for an electric field used in this paper. The effects of relativity may be included also, by using the relativistically correct equations. In both cases, the procedure outlined in section 2 will yield paraxial equations which may be solved if the optical axis is known only in the form of a numerical solution to the exact equations. Finally we observe that, in the example treated in this paper, the solutions of the paraxial equations were identical to the partial derivatives of the exact solutions, and are the coefficients of the first order terms of a Taylor series expansion of the ray in terms of the initial coordinates. These coefficients are computed about an arbitrary point in the object plane, and the method of calculation makes no assumptions about the form of the symmetry of the electric field. Applications of this technique to the evaluation of transverse rod lenses will be reported in subsequent papers.
References [1]
R. L. Dalghsh, A.M. Swath and S. Hughes, Nucl. Instr and Meth. 215 (1983) 9. [2] R.L. Dalglrsh, T.A. Winchester and A.M. Smith, Nucl. Instr. and Meth 218 (1983) 7. [3] T.A. Winchester and R.L. Dalglish, Nucl. Instr. and Meth. A235 (1985) 1. [4] Iu.V Vandukurov, Sov. Phys. Tech. Phys. 2 (1957) 1719. [5] P.A. Sturrock, Proc. Trans. Roy. Soc. A24 5 (1952) 155. [6] L.A. Macoll, Bell Syst Tech. J. 22 (1943) 153. [7] J.D. Dollard and C.N. Friedman, Product Integration with Applications to Differential Equations (Addison-Wesley, 1979).
m
Physics Research A273
(1988) 77-86
77
A TECHNIQUE FOR COMPUTING THE FIRST ORDER FOCUSING PROPERTIES OF AN ARBITRARY STATIC ELECTRIC FIELD T.A. WINCHESTER and R.L. DALGLISH School of Physics, University of New South Wales, Kensington, NSW 2033, Australia
Received 22 January
1988
and
m
revised form 26 May
1988
A technique for computing the first order focusing properties of charged particle rays about a curved ray m a static electric field is described. A set of paraxial equations is derived where the paraxial rays are expressed m the same coordinate system as that used for the optical axis. The techmque is designed specifically for efficient computer solution, and the axis need only be known in the form of a numerical solution. A formula for the determinant of the matrix of paraxial solutions is found. A comparison between the computed solutions of the derived equations and a simple test potential which allows a direct analytic solution are made; precise agreement is found.
1. Introduction In recent years we have been investigating the focusing properties of charged particle lenses using transverse rods [1,2]. A method of computing the electric potential for this geometry has been derived, and an approximate formula has been presented in a prior paper [3] for the deflection of ions passing through this system. This paper addresses a problem encountered in computing the exact focusing properties of systems of these elements. The optical axis cannot be defined by consideration of the symmetries of the system. It is not a straight line, but a curve in the plane of mirror symmetry of the electric field; in fact, the only way that the path corresponding to the optical axis could be found was by a numerical solution of the second order differential equations for a ray. It was necessary, therefore, to devise a technique that could compute the focusing of ions about a path known only in the form of a numerical solution for a ray. The focusing effects due to an electric field about an arbitrary axis can be computed by the Method of Trajectories [4], or through the use of the Characteristic Function [5], by expressing the ray about the optical axis using the Frenet coordinate system based upon the selected optical axis. A numerical solution of the resulting equations requires the curvature, torsion, electric potential, and the first and second order covariant derivatives of potential, all of which must be known as a function of the distance along the optical axis. These requirements present no difficulty for a computer program if the optical axis is known i n the form of a simple curve (for example, an arc of a circle, or a helix). Our problem, however, is one where the only information available for the optical axis is the set of discrete points generated by a numerical solution of a ray. Transforming the data available to express it to the Frenet coordinates about the optical axis, presented a major computational problem. It became obvious that a general technique was needed to provide an efficient solution, by a computer program, of the focusing properties of a general static electric field. We will show that this can be achieved by using one coordinate system to represent the field, the optical axis, and the paraxial rays. In this paper we shall use a Cartesian coordinate system to express both the optical axis, and the paraxial rays about it. It will be assumed that the optical axis is a curve that can be found by a numerical solution in Cartesian coordinates, and that the electric field and its derivatives can be determined numerically in the Cartesian coordinate system. A general method for computing the focusing properties of the field about this optical axis is presented below. The solutions can be written as a four dimensional matrix, and a formula for the determinant, the Wronskian of the paraxial matrix, is derived. The method will be verified, both analytically and numerically, by using a simple field whose properties can be found analytically. 0168-9002/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
78
TA
Winchester, R. L Dalghsh / Computing first order focusing properties of a static electric field
W
Fig. 1. All rays are described by the three coordinates s, x 1 , x". The difference between the optical axis and a paraxial ray is described by the coordinates in the travelling plane in the diagram. This plane is always parallel to the x'-x" plane.
2. The paraxial equations for a ray in Cartesian coordinates Let x l , x" describe the the Cartesian components of a point in the plane normal to an s-axis (see fig. 1). From the principle of least action the equations for the path of an ion may be expressed in the form of a pair of second order differential equations, provided that the path is never orthogonal to the s-axis [6]. In their nonrelativistic form, these are represented by eqs. (1) (the extension required to include relativistic effects are straightforward).
L
_d 2 II __ a 2 a~ II a$ _ II , 2 ~~ax " -u as)-e d s 2x
_d 2 1 __ _a 1 1 _a0 _ 1 2$~x'-u as) -e ' ds 2x
where a = (1 + u I u I + uitu111)]/2 (P _ - gV/mc 2 , and V is the electrostatic potential chosen so that V = 0 when the ion is at rest, u', u" are the derivatives of x l , x" respectively with respect to s. The optical axis is a particular solution of eq. (1). To compute the focusing effect, we must compute the path of a nearby ray. This ray is represented by the Cartesian coordinates *x', *xll, *u', *ull; the optical axis by x', x 11 , u', u 1I ; and the differences between these are expressed in the form of eq. (2): *xI=x1+8x1, *xl1=x1I+8x11, *u 1 =u'+8u 1 , *U il =ull=8ul'.
(2)
The equations that describe the difference parameters 8x 1 , 8x 1 ', 8u', 8u", are derived by expanding the right hand side of eqs. (1) in the form of a multidimensional Taylor series about the optical axis, neglecting terms above the first order. This yields eqs. (3), with e l , e l ', and their partial derivatives evaluated on the optical axis. a d2 ae' ae' +8u11 (x'+Sx')=e'+8x'ae' +8x" el +8u' ds 2 ax' ax' 1 au' au" De" d2 II -- (x + 8x") = eII + 8xI ae" + 8xII ae" + 8ul óe11 + 8u,, ds2 óx 1 óx' 1 óu 1 au ll
These equations will be valid for rays that lie close to the chosen optical axis. Since the optical axis satisfies eqs. (1), we may write eqs. (3) in the form of eqs. (4), which is a system of second order linear differential equations in the difference parameters. It can be shown that the only new information requires to evaluate eqs. (4) are the second order derivatives of J) to Cartesian coordinates. d (8x') = 8x' ds (~
1_
al)
~ ax' ) ((
+8u'((2
e' +
a
_ uI
a
ax , as
25 ( ax I ax I '
u'
a
24 ( a x l
a2
a
ax 1I as )~ u11
1 )e'+-()~ +8u"(2 a'- e'); 20 as a2
T.A. Winchester, R. L Dalghsh / Computing first order focusing properties of a static electric field
d
s
d
a 2o
i a~ + a e° 24 ( $ ax , ~
(8x 11 ) = 8x1 ~~
+8x11((_ 1 $ I
+8u'(2
u a2
- u
45
ax , as
ax 1 2ax 11
a o ) en+
v2 (
a Z$
ax 1 '
2$
a 2 x 11
I 11 e ° ) +óu 1 ' I (2 ue 2 11 )
a
u
79
-u
+
a 2~
ax 11 as ~
l l a( - a ) . J 2
0
ll
These equations have four independent solutions; since the equations are linear, any four independent solutions can span all possible solutions. It is possible therefore to select solutions which provide a direct intepretation of the focusing action. The four solutions chosen correspond to the initial conditions listed in eqs. (5). The first and third express the result of a change in the value of x' and x 11 in the plane s , and the second and the fourth changes in u 1 and u l1 relative to the object plane. The notation for the solutions in eqs. (5) has been chosen to unambiguously identify each of these solutions.
o
_ 8x 1 _ Su l 8x 11 Su' I 8xó , óxó , _, óx„óxó _ 8x 1
8u 1
Su 11
8x 11
_ 8x 1 8u 1 8x 1 ' _ 8u 11 (0, 1, 0, 0), 8uó , 8uó , 8uó , óuó
_ (1, 0, 0, 0),
(0, 0, l
8x' i (8u
' O),
Su'
01 ' u Su o '
8x 11 8u
u
li
x0
0
Su l ' Su II
_ (0, 0, 0, 1).
o 8xo 8X() 8 óx0) The solution for eqs. (4) for any set of initial conditions at the plane s o can be expressed in the matrix u
11
II '
form (6): 8X1
Sx 1
Sx 1
8xó
óuá
8xi
Sx' 1 ( 8xó 8uá1
Su l
Su 8xó
_ 8u Sul
_ Su l 8xi
8u1 Bui
8x11
8x11 8x 01 8u 11
8x11 Su l0 Su 11
8x11 8x 110
8x11 ßu o11
8xá
8uó
Bu ll 8zó1
ßu11 Bui
18x 1 1
(
óu n
Su l °
I
8x 11 0
1, 8U Io
3. The Wronskian formula If eq. (4) is written as a first order matrix differential equation, then the determinant of the solutions, W(s), is given by eq. (7), where W is the determinant of the solutions in the plane defined by s [7]:
0
0
del + ae II )). W(s) = W exp(f ds( s au , au 1 ' s
0
o
The integrand in eq. (7), given explicitly in eq. (8), can be manipulated into eq. (9).
4
2
_de l + ae11 - u l e' + u 11e 11 u le 1 + u 11e 11 + a 2 ~ act(u as) 1 du ll ) a2 ( az de l de ll 2 da l 1d~5(8u+al)-2 d
(8)
ds - q) s
When eq. (9) is substituted into eq. (7), the integration yields eq. (10). a W(s)-Wo(< rol ~~
(lo)
80
TA
Winchester, R.L. Dalglish / Computing first order focusing properties
of a static electrie field
The equivalent formula obtained from the equations for paraxial rays expressed in the Frenet coordinates about the optical axis is given in eq. (11) [4]. W(s) = W°(V°
).
(11)
Unlike the Frenet coordinate system, where the vectors used to describe the paraxial rays are always normal to the direction of the optical axis, the Cartesian coordinate system used in this paper is independent of the ray chosen for the optical axis. If the ray corresponds to a straight symmetry axis, then the value of a will be constant and eq. (10) reduces to eq. (11). Therefore the ratio (a/a0)4 can be intepreted as expressing the change in the inclination of the optical axis to the Cartesian axes used to express the rays. 4. A test for the paraxial equations A simple field distribution which allows a full analytic solution of the rays was used to test the validity of the equations derived. A homogeneous field with no component along the s-axis, with fields P, E lf along the x I , x" axes respectively was chosen. The exact equations for a ray take the form of eq. (12). d 2x1 ds2
_ a2
2V
Ei,
d 2x " _ - _a? E,', ds2_ 2V
(12)
where V = V„ + E 'x I + E "x" is the total electric potential, VA is the energy of the ion in eV when it enters the region. Since there is no field component parallel to the s-axis, the value of a/2V is invariant, and is represented by C. The solutions to eqs. (12), eqs. (13), are two parabolas defined by E l , E li , C, and the i nitial values of x', x I ', u 1 , u". I (J) I ] (13) x (s)=x +u s+CE x1 ' (S) =x,) +uös+CE"
V,
V.
Since the value of C varies with the initial conditions for the ray, the paraxial rays will exhibit a small, but computable form of focusing behaviour. This effect is due to the variation of the total electric potential affecting the magnitude of the velocity of the ion along its path. The paraxial equations are given in eqs. (14).
v
d2 8x t =8x I ( "(ds 2 v'e')+8x dz Sx~ t =8x , - Ve El el'li ) +Sx" d s2 ( (
l e')+8uj( u'y' ) +Su"( u ' V l n uu n e li ) +8u ) +8u ( V ).
V
(14)
(uv
Let us assume that the solutions of eqs. (14) will be parabolas; we shall verify this by substituting this form into eqs. (14) according to the initial conditions (5), and, in doing so, determine the algebraic form which is consistent with eqs. (14). The solution that satisfies the first set of initial conditions of (5) will be of the form s2 s2 8x' =1+a 8x " =b
2,
2.
Substitution of this form into eqs. (14) gives the following conditions for the constants a and b: I a= -El e +a(x I -xá)
V
+b(
X101)
V1 ,
T.A. Winchester, R.L. Dalghsh / Computing first order focusing properties of a static electric field
81
It can then be shown that the values of the constants a and b are ' a= -CE' E-, Vo
E'
b= -CE It Vo
The solution for the third set of initial conditions of (5) follows from the symmetry in the equations. The solutions for the second set of initial conditions of (5) of the form 8x'=s+c
s2
2
,
8x11=d
s2
2
The same procedure gives the following conditions for the constants c and d: c= uoy t +c(x'-xo
d=
V
)
+d(x1'-xö
V1'
Eu E" uo' E" +c(xt-xó~ +d(xtt-xö) . V V V
The values of the constants c and d are 1 E' c=uo V ,
~ E" d=u' V
The solution for the fourth set of initial conditions also follows from the symmetry in the equations. The complete set of solutions is listed in eqs. (15). 8 _ 8x11 8u' Sxó
1-CE' E s , Vo 2 - CE'
E'
s,
Vo
óx , _ E" s2 _CE,- 8xo' Vo 2 ' E1' 8u __ -CE' s, Sx0 Vo 8x 11 - _CE"E' s2 _ 8,1 Vo 2 ' E' 8u" -CE"-,
8xó
VO'
E. 11 s 2 8x" =1-CEI' -_ 8xö Vo 2 '
Su" óx 1
E" -CE " -s, Vo
Sx 8uó
=s+UO E s Vo 2
óu , E' - = 1 +u ó -s, bul Vo óx,_
=uon
8 uó'
E
' s2
Vo
-, 2
Su
= uol E s, Vo 8x11 -u' E" S 2 8uo'
8uó 8u 11
(15)
, o Vo 2 E t'
= uó -s, Vo 8uó
&
E" s2 =s+u~'-2 , 8ui1 Vo El' 8u" X
11
-=1+u -s. 8uä
Vo
It can be seen that these solutions for the homogeneous electric field are identical to the partial derivatives of the exact solution of the optical axis with respect to the initial conditions for the optical axis. Rays connecting points on the object and image planes can be considered as four dimensional functions of the object plane coordinates giving values for x l , x", u l , U I ' in the image plane. The solutions of the paraxial equations given by eqs. (15) are the coefficients of the first order terms in the multidimensional
T. A. Winchester, R L Dalghsh / Computing first order focusing properties of a static electric field
82
Table 1 Results for the axis ray trace
Object I mage
X1
ui
X2
U2
O.OOOOOOOOD + 00 -0.125000001)+00
0.000000001) + 00 -0.250000001)+00
O.OOOOOOOOD + 00 0.12500000D + 00
0 00000000 D + 00 0.25000000D + 00
X10
U10
X20
U20
0 93750000D + 00 - 012500000D + 00 0.62500000D - 01 0.12500000D+00
0.10000000D+01 0.10000000D+01 0.44411586D -12 0.90252923D-12
Entry electric potential Exit electric potential
-0.10000000D+06 -0.112500001)+06
Results for the paraxial rays
X1I Uli U21 U21
0.625000001)- 01 0.12500000D + 00 0.93750000D + 00 - 0.12500000D + 00
0.44411586D- 12 0.90252923D -12 0.100000001) + 0 1 0.1000000013+01
DET
0.1125000013+01 WRK 01125000013+01 DIF 0.16453505D-12
Taylor series expansion of the rays about the initial point xó, xó , uó, u 01. This expansion has been calculated for arbitrary values of the initial conditions. The Wronskian formula, the determinant of the matrix of paraxial solutions, is the Jacobian determinant of the mapping. A computer program was written in double precision FORTRAN that solves the paraxial equations. A predictor-corrector method was used where the number of iterations of the corrector formula could be varied, and it was found that four iterations were sufficient. This program could access a subroutine that would return the total electric potential and the derivatives of the potential. To test the program, a subroutine was written that would return the potential used for the above calculations. A value of 100 keV was selected for the initial energy of the ion, and the solutions were performed over a length of one metre. Values for E t , E 1t were selected that would bend the axis by the order of one radian so that the program could be verified under conditions where the field strongly affects the motion of the ions. The program first solved the optical axis numerically, and then used the stored values for the optical axis to form and solve the paraxial equations.
Table 2 Results for the axis ray trace
Object I mage
O.OOOOOOOOD + 00 0.7500000013+00
0.1000000013+01 0.50000000D+00
O.OOOOOOOOD + 00 0.25000000D+00
0.0000OOOOD + 00 0.50000000D+00
X10
U10
X20
U20
0 87500000D + 00 -025000000D+00 0.12500000D+00 0.25000000D + 00
0.75000000D + 00 0.50000000D+00 0.25000000D+00 0.50000000D + 00
0.12500000D + 00 0.25000000D+00 0.87500000D+00 - 0.25000000D + 00
- 0.84413451 D -12 -0.16345175D-11 0.10000000D+01 0.1000000013+01
Entry electric potential Exit electric potential
-0.1000000013+06 -0.750000001)+05
Results for the paraxial rays
X1I Uil X21 U21
DET 0.7500000013+00 WRK 0.7500000013+00 DIF 0.26869618D-11
T.A. Winchester, R. L. Dalghsh / Computing first order focusing properties of a static electric field
Table 3 Results for the axis ray trace
Obj ect Image
X1
U1
X2
U2
O.000OOOOOD + 00 - 0.25000000D + 00
O.000OOOOOD + 00 -0.50000000131+00
O.000OOOOOD + 00 0.12500000D + 01
0.10000000D + 01 0.15000000D + 01
X10
U10
X20
U20
0.ß7500000D+00 - 0.2500000013 + 00 0.1250000013+00 0.2500000013 + 00
0.10000000D + 01 0.10000000D + 01 - 0.92739655D -12 -0.19120615D-II
0.1250000013 + 00 0.2500000013 + 00 0.87500000D + 00 -0.25000000D+00
- 0.25000000D + 00 - 0.50000000D + 00 0.12500000D + 01 0 15000000D + 01
Entry electric potential Exit electric potential
-0.1000000013+06 -0.1750000013+06
Results for the paraxial rays X11 Ull X21 U21
DET 0.17500000D + 01 WRK 0.17500000D+01 DIF -0.74533713D-11
Table 4 Results for the axis ray trace
Object Image
Xl
Ul
X2
U2
O.000OOOOOD + 00 0.62500000D + 00
0.1000000013 + 01 0.25000000D + 00
O.000OOOOOD + 00 0.13750000D + 01
0.10000000D+01 0.1750000013+01
X10
U10
X20
U20
0.81250000D + 00 - 0.3750000013 + 00 0.18750000D+00 0.3750000013 + 00
0.7500000013 + 00 0.5000000013 + 00 0.2500000013 + 00 0.5000000013 + 00
0.18750000D+00 0.3750000013 + 00 0.81250000D + 00 - 0.3750000013 + 00
- 0.25000000D + 00 - 0.50000000D + 00 0.12500000D+01 0.15000000D+01
X1
U1
X2
U2
O.000OOOOOD + 00 -0.10102051D+00
O.000OOOOOD + 00 - 0.18350342D + 00
O.000OOOOOD + 00 0.10102051D+00
O.000OOOOOD + 00 0.18350342D+00
X10
U10
X20
U20
0.95875855D + 00 - 0.68041382D - 01 0.41241452D-01 0.68041382D-01
0.89897949D + 00 0.81649658D + 00 - 0.12586481 D -11 - 0.22358008D -11
0.41241452D-01 0.68041382D-01 0.95875855D+00 - 0.68041382D - 01
- 0.12586481 D -11 - 0.22358008D - I I 0.89897949D + 00 0.81649658D + 00
Entry electric potential Exit electric potential
-O.10000000D+06 -0.1375000013+06
Results for the paraxial rays
X11 Ull X21 U21
DET 0.13750000D+01 WRK 0.13750000D+01 DIF 0.46487258D-11
Table 5 Results for the axis ray trace
Object Image
Entry electric potential Exit electric potential
-0.1000000013+06 -0.1601020513+06
Results for the paraxial rays X11 Ull X21 U21
DET 0.7115646713 + 00 WRK 0.7115646713 + 00 DIF - 0.925 59294D - 12
84
T A. Winchester, R L Dalgltsh / Computing fist order focusing properties of u static electric field
Table 6 Results for the axis ray trace
Object I mage
XI
U1
X2
U2
O.000OOOOOD + 00 0.65685425D + 00
0 100000001) + 01 0.41421356D+00
O.000OOOOOD + 00 0.17157288D+00
O.000OOOOOD + 00 0.29289322D + 00
X10
U10
X20
U20
0.87867966D + 00 - 0.176776701) + 00 0.606601721)- 01 0.ß8388348D-01
0.585786441) + 00 0.353553391)+00 0.12132034D+00 0.176776701) + 00
0.12132034D + 00 0.17677670D + 00 0.93933983D + 00 -0.88388348D-01
0.32542850D - I I - 034172577D -10 0.82842712D + 00 0.70710678D + 00
Entry electric potential Exit electric potential
-0.1000000013+06 -0.125735931)+06
Results for the paraxial rays
X1I Ull X21 U21 DET WRK DIF
0.31433983D + 00 0.31433983D + 00 -0.56141164D - 09
Table 7 Results for the axis ray trace
Object I mage
X1
U1
X2
U2
O.000OOOOOD + 00 -0.17157288D+00
O.000OOOOOD + 00 - 0.29289322D + 00
O.000OOOOOD + 00 0.100000001) + 01
0.1000000013+01 0.1000000013+01
U10
X20
U20
0 606601721)- 01 0.ß8388348D-01 0.100000001) + 01 0.000000001) + 00
- 0.12132034D + 00 - 0.17677670D + 00 0 82842712D + 00 0.70710678D + 00
Entry electric potential Exit electric potential
-0.1000000013+06 -0.20ß57864D+06
Results for the paraxial rays X10 X11 Ull X21 U21 DET WRK DIF
0.93933983D - 0.88388348D O.000OOOOOD 0.000000001)
+ 00 - 01 + 00 + 00
0.82842712D 0.70710678D 0.000000001) O.000OOOOOD
+ 00 + 00 + 00 + 00
0.52144661 D + 00 0.51144661 D + 00 - 0 24707880D - 09
Table 8 Results for the axis ray trace
Object I mage
X1
U1
X2
U2
O.000OOOOOD+00 0.54970355D+00
0,1000000013+01 0.2649110613 + 00
0.000000001) + 00 0.10000000D + Ol
0.1 OOOOOOOD + 01 0.10000000D + 01
Entry electric potential Exit electric potential
-O.10000000D+06 -0.172514ß2D+06
Results for the paraxial rays
X11 Ull X21 U21
X10
U10
X20
U20
0.8576037613+00 - 0.18973666D + 00 O.000OOOOOD + 00 O.000OOOOOD + 00
0 58499011D+00 0.37947332D + 00 O.000OOOOOD + 00 O.000OOOOOD + 00
0.14239624D + 00 0.18973666D + 00 0.1000000013+01 O.000OOOOOD + 00
- 0 18986166D + 00 - 0.25298222D + 00 0 77485177D + 00 0.632455531)+00
DET 0.27602371 D + 00 WRK 0.27602372D + 00 DIF - 0.20261378D - 08
T.A. Winchester, R.L. Dalghsh / Computing first order focusing properties of a static electric field
85
Table 9 Results for the axis ray trace
Object I mage
X1
U1
X2
U2
0.1234000013+00 - 0.3842441313 + 00
- 0.4567000013 + 00 -0.5498534413+00
0.89010000D+00 0.7713565313 + 00
- 0.23450000D + 00 - 0.22834387D - 01
X10
U10
X20
U20
0.98474380D + 00 -U550503313-01 0.34665515D - 01 0.57953180D- 01
0.93674413D+00 0.8795509513+00 - 0.69330304D - 01 - 0.11590514D + 00
0.15256196D - 01 0.25505033D - 01 0.96533448D + 00 -0.57953180D-01
0.15666917D 0.26191668D 0.87063329D 0.76902805D
Entry electric potential Exit electric potential
-0.1383350013+06 -0.2077800313+06
Results for the paraxial rays
XII U11 X21 U21
- 01 - 01 + 00 + 00
DET 0.70782947D + 00 WRK 0.70782947D + 00 DIF - 0.67230665D -11
The results of these calculations are given in tables 1-4, where 128 points were used for the numerical solution. The values of uá, not ( U1, U2 in the tables) were (0, 0), (1, 0), (0, 1), (1, 1) in tables 1-4 respectively, while the values of xó, xo' (X1, X2 in the tables) were left fixed. The initial and final values of the ray variables are given first (Object, Image in the tables), and then the matrix of paraxial solutions in the format of eq. (6). The difference between the derived and the computed solutions is in all cases less than 2 x 10 -9 , or better than single precision. The determinant of the matrix was calculated by using the Laplace expansion, and given in the entry DET. The Wronskian formula is evaluated, and given in the entry WRK, and the difference, not detectable in the precision of the computer output, is given directly below in entry DIP. In these tables it can be verified that the value of the determinant is V/V,. However, the value of C, which is constant, may be used with the Wronskian formula to show that this should be expected. To verify that the general form of the Wronskian formula is correct, a field component parallel to the s-axis of -50 kV m -1 was added, and the solutions of tables 1-4 recomputed. The results of these computations are given in tables 5-8, and it may be seen that the general form of the Wronskian formula remains valid. As a final check on this formula, a random choice for all the initial values for the ray was used, with the results given in table 9, from which it may be seen that the Wronskian formula is satisfied even under these extreme conditions. The numerical solution of the paraxial equations has been shown to agree with the analytic solutions listed in eqs. (15). This example shows that this technique can be used without any assumptions about the symmetry of the electric field. The computer program only requires a subroutine to compute the numerical values of first and second order derivatives of the electric field in Cartesian coordinates.
5. Conclusion
We have derived a method suitable for solution by a computer program for evaluating the focusing behaviour of a static electric field about an optical axis which is known in the form of a numerical solution for a ray in Cartesian coordinates. This axis need not be known prior to the application of this technique, since a numerical solution for the optical axis provides all the information required. Symmetries of the electric field are not relevant, since the technique assumes that the electric field exists, and that it can be numerically determined in Cartesian coordinates. This technique is ideal therefore, for computing the focusing effects in situations where there is little or no prior knowledge about the optical axis.
86
T.A. Winchester, R L Dalghsh / Computing first order focusing properties of a static electric field
The extension of this method to include magnetic fields is straightforward, requiring the use of the exact equations for an ion in a static electromagnetic field instead of the equations for an electric field used in this paper. The effects of relativity may be included also, by using the relativistically correct equations. In both cases, the procedure outlined in section 2 will yield paraxial equations which may be solved if the optical axis is known only in the form of a numerical solution to the exact equations. Finally we observe that, in the example treated in this paper, the solutions of the paraxial equations were identical to the partial derivatives of the exact solutions, and are the coefficients of the first order terms of a Taylor series expansion of the ray in terms of the initial coordinates. These coefficients are computed about an arbitrary point in the object plane, and the method of calculation makes no assumptions about the form of the symmetry of the electric field. Applications of this technique to the evaluation of transverse rod lenses will be reported in subsequent papers.
References [1]
R. L. Dalghsh, A.M. Swath and S. Hughes, Nucl. Instr and Meth. 215 (1983) 9. [2] R.L. Dalglrsh, T.A. Winchester and A.M. Smith, Nucl. Instr. and Meth 218 (1983) 7. [3] T.A. Winchester and R.L. Dalglish, Nucl. Instr. and Meth. A235 (1985) 1. [4] Iu.V Vandukurov, Sov. Phys. Tech. Phys. 2 (1957) 1719. [5] P.A. Sturrock, Proc. Trans. Roy. Soc. A24 5 (1952) 155. [6] L.A. Macoll, Bell Syst Tech. J. 22 (1943) 153. [7] J.D. Dollard and C.N. Friedman, Product Integration with Applications to Differential Equations (Addison-Wesley, 1979).