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4 The optics of simple lens systems 4.1 Aperture lenses One might almost say that aperture lenses are the ‘traditional’ particle lens. A great variety of detailed shapes and proportions have been used which makes it difficult to prepare tables of focal data for general application. In...

4 The optics of simple lens systems 4.1 Aperture lenses One might almost say that aperture lenses are the ‘traditional’ particle lens. A great variety of detailed shapes and proportions have been used which makes it difficult to prepare tables of focal data for general application. In the context of low energy beams there is no need for smoothly rounded electrodes: simple thin discs with circular holes can be used and, provided that there is some agreement about the spacing of the discs, it should be possible to make systems of two or three apertures which behave in a predictable fashion. Figure 3.10 shows the important dimensions of such systems. For the purposes of scaling, it is usual to express all the dimensions in terms of the diameter, D1 , of the apertures themselves. The significance of the outer diameter of the discs, D3 , was discussed earlier in the context of establishing a simple boundary condition to use in the relaxation calculation, but it has a further, practical, significance in that the lens system does not exist in a vacuum, but rather in a vacuum system which has walls which will either be of metal and therefore have some definite potential, or of an insulating material and have a rather indefinite potential. In both cases it is important that this should not influence the potential distribution within the lens and it is as well to ensure that (D3 − D1 )/A > 3. A similar argument requires that the first and last apertures be mounted on cylinders to guarantee uniform potentials in the object and image spaces, and the diameter, D2 , of these cylinders will affect the potential distribution and, therefore, to some extent the focusing properties. Figure 4.1 shows the axial potential distribution in a two aperture lens with A = D1 /2 and a number of values of D2 /D1. Near the centre of the lens the distributions are very similar, but with increasing distance from the centre the potential tends towards the simple exponential decay characteristic of the dominant term in the Bessel function expansion, V (z, 0) ∝ exp(−k0 z/ 21 D2 ). Most of the calculations were made with the length of the D2 -cylinder great enough that the potential fell smoothly, but one example is included of a cylinder of length 4D1 , and therefore only 4D2 /3, to illustrate the way in which the end-plate can influence the potential distribution. 61 62 Electrostatic Lens Systems Figure 4.1 Axial potential distributions for two aperture lenses having A = D1 /2. Values of the ratio D2 /D1 are shown on the diagram. The potentials all have the value 0.5 at the lens centre, but fall with increasing axial distance, z, as exp(−4.8096(z/D2 )). The potentials approach these asymptotes from above except in the case of the cylinder lens. Is there a value of D2 /D1 for which the potential falls exponentially for all z? The different potential distributions away from the apertures themselves affect the properties of the lens as illustrated in figure 4.2. This shows the first focal length, f1 , mid-focal distance, F1 , and spherical aberration coefficient, Cs0 , for a voltage ratio, V2 /V1 = 10, as functions of the ratio D2 /D1 , and includes the case of the cylinder lens of the same spacing, which one might consider to be a rather extreme aperture lens having D2 /D1 = 1. The overall effect on the focal properties is not large, but the lower aberration of the cylinder lens is quite noticeable. Changes of a similar scale result if the separation of the apertures is doubled, but whereas this is a fairly obvious effect, the effect of changes in D2 is a little more subtle. The thickness of the aperture discs also plays a part, but not on the same scale. The data presented in the LENSYS program are for D2 /D1 = 3 which represents a compromise between reduced dependence at large values of the ratio and sheer physical size of the electrodes. 4.1.1 Three-aperture lenses As is the case with all lenses with three elements, the principal function is the decoupling of the focal properties of the lens, in particular the behaviour between conjugates, from the ratio of the potentials of the image and object spaces. Data for this sort of application are generally presented in the form of plots of the ratio V2 /V1 against V3 /V1. These plots are usually called ‘focal loci’ or ‘zoom  The optics of simple lens systems 63 Figure 4.2 Focal lengths (
), mid-focal distances ( ) and spherical aberration coefficients () for two aperture lenses having A = D1 /2 and V2 /V1 = 10. curves’ and lenses used in this fashion are referred to as ‘zoom lenses’, but the behaviour is more restricted than the use of this term in light optics would imply. We shall return to this topic in section 4.6.1. 4.1.1.1 The Einzel lens. Nowadays this term is understood to refer to a three element lens operated symmetrically, with V3 = V1 and with V2 taking any appropriate value. Historically the Einzel, or unipotential, lens signified a system with V2 held at the potential of the cathode. The major advantage of this is that the focal properties, depending as they do on voltage ratios, are the same for all values of V1 = V3. The disadvantage is that the aberrations of this lens are greater than those of the symmetric lens with V2 > V1,3. The potential distribution in an Einzel lens is different in character from that shown for the two cylinder lens in figure 2.1(b) and we illustrate this in figure 4.3. Apart from the bowing out of the equipotentials which occurs for any aperture lens, the potential close to the centre forms a saddle where the potential rises (say) in the r-direction and falls in the ±z-direction. This general form will apply to any three aperture lens for which V2 is outside the range spanned by V1 and V3. One of the equipotentials crosses the axis at the centre and it is simple to show that the angle between this equipotential (the value of which will depend on the thickness of the aperture plates) and the axis is ±54.7◦. Expanding the axial potential as V (z, 0) = V0 (z, 0) + zV0 (z, 0) + 21 z2 V0 (z, 0) + · · · and substituting this into equation (2.2) gives V (z, r) = V0 (z, 0) + zV0 (z, 0) + 21 z2 V0 (z, 0) − 41 r 2 V0 (z, 0) + · · ·. 64 Electrostatic Lens Systems Figure 4.3 Equipotentials in an Einzel lens. Near the centre of the lens the equipotentials are at 0.300, 0.294 and 0.288 of the potential difference between the outer and centre electrodes. Close to the axis the equipotentials are hyperbolæ, and at the centre they form a pair of straight lines which cross the axis at angles of ±54.7◦. V0 (z, 0) = 0 at the saddle point and V (z, r) = V0 (z, 0) along the equipotential, so the equation of the equipotential close to the saddle point is 21 z2 = 41 r 2 from which the result for the angle follows immediately. 4.2 Cylinder lenses Cylinder lenses offer a number of advantages over aperture lenses of the same inner diameter. For a given voltage ratio they are a little stronger and have smaller aberration coefficients, but there are two quite important mechanical advantages as well. They are easy to make and to mount: it is easier to turn cylinders on a lathe than to punch clean-edged holes in metal sheet. It is also simple to make lens systems with rather small inter-electrode gaps. This is an important factor in keeping the potential distribution in the gap at the inner surface of the electrodes close to that assumed in the calculation of the entire potential distribution and also ensures good screening from any stray electric fields around the lens structure. The potential and field distributions in a system of several coaxial cylinders can be found quite adequately by the superposition of those for contiguous pairs, provided that the gaps between the cylinders are small (say, > 0.2 D) and the cylinders themselves are rather longer than the gaps. In practice it is common  The optics of simple lens systems 65 to use cylindrical elements no shorter than D/2 (gap centre to gap centre) with gaps of D/10. This allows lens systems having a large number of cylinders to be designed and analysed, procedures which would be much more difficult with aperture electrodes. If the gaps were large, it would be necessary to solve each geometry with appropriate symmetric and antisymmetric potentials on all electrodes, as in the case of the three aperture lens. 4.3 The calculation of ray paths There are no purely analytic solutions to this problem, only various procedures for the numerical integration of the path starting from a known position and slope at some point. The choice of method depends on the extent and quality of the data. If only the axial potential is known, then integration of the Picht equation can be done, and there is one procedure which is particularly well suited to this. On the other hand, if the potential and its derivatives are known quite accurately throughout the space, a direct application of the laws of simple dynamics will give good results. If the potential is known everywhere, but to limited accuracy, various standard procedures are available. However, care should be taken to distinguish between the precision of the integration procedure and the accuracy of the trajectory. Ray tracing in the correct potential will give results which show the effects of aberrations, while integration of the Picht equation will give only the paraxial, Gaussian, behaviour though corrections can be applied and aberration coefficients calculated even in this case. 4.3.1 Ray tracing in a known potential If the potential is not known really well throughout the space, the best approach is to use some standard expression for the integration of the differential equations, but if it and its derivatives are well known an analysis by direct application of the laws of dynamics using expansions of the coordinates and velocities about each point will give good results. 4.3.1.1 Integration of the equation of motion. Taking the time, t, to be the independent variable we write the coordinates as  dz  t 2  d2 z  t 3  d3 z  z(t) = z0 + t + + + ··· (4.1) dt 0 2 dt 2 0 6 dt 3 0 with a similar expression for r(t), where z(t) and r(t) are the axial and radial positions of a particle, having initial coordinates denoted by the ‘0’subscripts, after a short time interval t. The components of the velocity follow from differentiation of these equations dz(t)  dz   d2 z  t 2  d3 z  = +t +. (4.2) dt dt 0 dt 2 0 2 dt 3 0 66 Electrostatic Lens Systems The accelerations are given by  d2 z  e ∂V (z, r) = dt 2 0 m ∂z and the third time-derivatives of the coordinates are therefore  d3 z  e  ∂ 2 V   dz  e  ∂ 2 V   dz  = +. dt 3 0 m ∂z2 0 dt 0 m ∂z∂r 0 dt 0 Substituting these expressions √ into equations (4.1) and (4.2), and introducing a ‘reduced time’ τ = t 2e/m to avoid the ‘ me ’ which occurs in nearly every term, we construct the following recurrence relationships.  dz  τ 2  ∂V  z(t) = z0 + τ + dτ 0 4 ∂z 0 τ 3  ∂ 2 V   dz   ∂ 2 V   dr   + + 12 ∂z2 0 dτ 0 ∂z∂r 0 dτ 0 dz(t)  dz   τ ∂V  = + dτ dτ 0 2 ∂z 0 τ 2  ∂ 2 V   dz   ∂ 2 V   dr   + + 4 ∂z 0 dτ 0 2 ∂z∂r 0 dτ 0 with two similar expressions for the radial coordinate and velocity. Only expressions for the potential which can be evaluated and differentiated analytically at any point, such as those expressed in terms of Bessel functions or elliptic integrals, are likely to be good enough to yield accurate values for the second derivative required for this type of analysis. The calculations are surprisingly simple and we illustrate this using the potential in the gap region of a two cylinder lens developed in section 3.3.6 (equation (3.26b)) ∞ V1 + V2 V2 − V 1 V2 − V1  VI I (Z, R) = + Z+ Qn J0 (Kn R)(1/a1n − a2n ). 2 G G n=1 The five differential expressions can be written as ∞ ∂V (V2 − V1 ) 2(V2 − V1 )  = − Qn Kn J0 (Kn R)(1/a1n + a2n ) ∂Z G G n=1 ∞ ∂V 2(V2 − V1 )  =− Qn Kn J1 (Kn R)(1/a1n − a2n ) ∂R G n=1 ∞ ∂ 2V 4(V2 − V1 )  = Qn Kn2 J0 (Kn R)(1/a1n − a2n ) ∂Z 2 G n=1 ∂ 2V 2 ∂V ∂ 2V = − − ∂R 2 R ∂R ∂Z 2 ∞ ∂ V 2 4(V2 − V1 )  = Qn Kn2 J1 (Kn R)(1/a1n − a2n ). ∂Z∂R G n=1  The optics of simple lens systems 67 Very similar expressions apply in the outer regions, the principal differences arising from the different exponents. Simple modifications of the BesselPot function of section 3.6.3 may be used to calculate these differentials. A Pascal procedure for the calculation of the J1 Bessel function required for the R-derivatives is given in. The accuracy of the ray integration depends on the interval in reduced time, Bτ , which is related to the step length, Bl, by Bτ = Bl/V 1/2 and it is important to choose a suitable value. Too short an interval will waste computer time, but the optimum interval will depend on the strength of the field and so it needs to be adjusted dynamically. The accuracy of the calculations is open to a simple test: the difference in potential between the initial and final points of each step should be equal to the difference in the squares of the velocities of the particle multiplied by m/2e. If the fractional difference found after a step is greater than some chosen value, then the step should be recalculated with Bτ halved. If the difference exceeds some smaller criterion, Bτ may be doubled for the next step. 4.3.1.2 Runge–Kutta methods. Unless we have a good enough potential we cannot make proper use of terms beyond the first order of the Taylor expansion of equation (4.1) and need to find some other way to make allowance for the higher order terms. The Runge–Kutta methods are based on the replacement of these terms by first order expansions at nearby points. Figure 4.4 illustrates the steps required to apply these methods. The function describing some aspect of the trajectory is represented by a curve starting at the point A, (xn , yn ), and we wish to find the y-coordinate at C, for which xn+h = xn + h. We can write this as yn+1 = yn + k where k is h times some mean gradient. The very first approximation would be to write k = k1 = hy  (xn , yn ) where the gradient of the function is evaluated at the point A(xn , yn ). This would take us to the point D1 and is plainly not a good approximation. However, common sense and the mean value theorem tell us that at some point along the curve the tangent is parallel to the chord AC joining the initial and the true final points. The next stage of approximation is therefore to evaluate k at the mid-point, B1 , of the chord AD1 and write k = k2 = hy  (xn + h2 , yn + k21 ), which will take us to the point D2. This is a significantly better approximation and the errors are of order h3 : it is therefore referred to as the second order Runge–Kutta method. It is worth making two further steps of this sort, because the gain in precision outweighs the complexity of the calculation, but beyond that point one should really question whether the potential function is good enough to justify the effort. The four successive values for k can be summarized as k1 = hy  (xn , yn ) at A k2 = hy  (xn + h/2, yn + k1 /2) at B1 k3 = hy  (xn + h/2, yn + k2 /2) at B2 k4 = hy  (xn + h, yn + k3 ) at D3. 68 Electrostatic Lens Systems D1 D3 C D2 B1 B2 A Figure 4.4 A schematic illustration of the steps in the fourth order Runge–Kutta method. AC represents the real trajectory and AD1 is the tangent to the trajectory at A. The gradient of the function at B1 is used to find D2 , the second approximation to C, then the gradients at B2 and D3 are evaluated and a weighted sum of the four gradients is then a good approximation to that of the chord AC. These expressions have a nice symmetry and we write k1 k2 k3 k4 k= + + + 6 3 3 6 where the weights given to the ki have been chosen such that the third and fourth order terms in the Taylor expansions cancel and the errors are now of order h5 giving the fourth order Runge–Kutta method. To apply this to the tracing of ray paths, we write the axial, zn , and radial, rn , positions and the corresponding velocities, un and vn respectively, in terms of the reduced time, τ , as before (i.e. un = dτ dz etc) and apply the recurrence relations to all four variables. zn+1 = zn + j rn+1 = rn + k un+1 = un + l vn+1 = vn + m where j, k, l and m are the appropriate weighted sums. The individual approximations are l1 = −hEz (zn , rn )/2 j1 = hun

 l1 j1 l1 l2 = −hEz zn + , rn + /2 j 2 = h un + 2 2 2

 l2 j2 l2 l3 = −hEz zn + , rn + /2 j3 = h un + 2 2 2
 j4 = h(un + l3 ) l4 = −hEz zn + l3 , rn + j3 /2  The optics of simple lens systems 69 with a similar set for the radial motion. h is an interval of reduced time, and Ez and Er are the axial and radial components of the electric field strength which must be found at all the indicated points by numerical differentiation of the potential. 4.3.2 Integration of the Picht equation The Picht equation describes the evolution of a reduced radius defined by R(z) = r(z)V 1/4 (z, 0) as a function of z. It involves the axial potential, V (z, 0) and its first derivative and contains only the second derivative of R(z). There is a standard procedure for the integration of such equations which is faster than fourth order Runge–Kutta with the same step size and is effectively of higher order. We write the Picht equation as d2 R(z) = −T ∗ (z)R(z) dz2 where 3  V  (z, 0) 2 T ∗ (z) =. 16 V (z, 0) T ∗ (z) = 16 3 T (z)2 where T (z) was introduced in chapter 2. Some authors use T (z) where we use T ∗ (z), but it is necessary to distinguish the two symbols because T (z) itself will appear in expressions for the aberration coefficients. For compactness we shall omit the explicit references to z in R and T ∗ in what follows. We shall integrate the equation using equal steps, h, in z and so write for the values of R at adjacent steps h2  h3  h4 (4) Rn±1 = Rn ± hRn + R ± Rn + Rn ± · · · 2 n 6 24 and rearrange this to obtain Rn+1 − 2Rn + Rn−1  h2 (4) = R n + R + ···. h2 12 n Rn = −(T ∗ R)n and the fourth differential of Rn can be written in terms of the second differential of (T ∗ R)n as d2 (−T ∗ R)n Rn(4) = dz2 (T ∗ R)n+1 − 2(T ∗ R)n + (T ∗ R)n−1 =− h2 using an exactly similar expansion of T ∗ R, but neglecting the fourth order term in the expansion, giving Rn+1 − 2Rn + Rn−1 = −(T ∗ R)n h2 h2 [(T ∗ R)n+1 − 2(T ∗ R)n + (T ∗ R)n−1 ] − 12 h2 70 Electrostatic Lens Systems and, rearranging this equation, we finally have h2 2Rn − Rn−1 − [(T ∗ R)n−1 + 10(T ∗ R)n ] Rn+1 = 12 h2 ∗ (4.3) [1 + 12 Tn+1 ] This equation is known as Numerov’s algorithm† and it is very effective within its limitations. Two values of R(z) are required to start the process, but this is not a problem if the starting point is in a field free region. If this is not the case, some other method has to be used to start the process, and a Runge–Kutta method would normally be used. The step size has to be chosen carefully as there is no easy way to adjust it dynamically, since the particle velocity is not calculated at any stage. This means that a step size suitable for the worst part of the trajectory has to be used throughout. 4.3.2.1 The Runge–Kutta method applied to the Picht equation. As usual, the Runge–Kutta method requires parameters to be calculated in the middle of the integration steps. If the values of T ∗ are known at intervals (h/2) the Runge– Kutta method must use a step size of h while the Numerov method can use the smaller step giving it a 16-fold advantage in precision. The expressions for the reduced radius R and its gradient R  are
 h2 ∗ ∗ h2 ∗ ∗ Rn+1 = Rn − Rn Tn + 2Tn+1/2 − Tn Tn+1/2 6 4
 h 2 + Rn 1 − Tn+1/2 ∗ 6
  h h2 Rn+1 = −Rn Tn∗ + 4Tn+1/2 ∗ − Tn∗ Tn+1/2 ∗ 6 2
 h ∗ 2 h ∗ + Rn 1 − Tn+1/2 − Tn+1 Rn+1. 3 6  Notice that it is necessary to calculate Rn+1 before Rn+1. 4.4 Real and asymptotic cardinal points Particle lenses differ from those for light in having no abrupt boundaries between regions of different refractive index, so there is no clear cut point at which the lens begins or ends. Though the ray paths are asymptotically straight, away from the lens, a particle emerging from the lens field may cross the axis before this asymptotic region is reached. If the particle had approached the lens parallel to the axis, this crossing point will be a focal point and the mid-focal distance will † It is also known as the Manning–Millman and the Fox–Goodwin methods.  The optics of simple lens systems 71 be well defined, but how should the focal length be defined? In chapter 1 the focal length was defined as the distance of the focal point from a principal plane which was itself defined by the intersection of asymptotes. This is not appropriate to the present case and we must define a real principal plane in terms of the incident asymptote and a tangent to the ray path at the focal point. These real cardinal points are of limited use in the analysis of lens behaviour because for an object not at infinity the image will be formed beyond the focal point and therefore closer to the asymptotic region, if not fully in it. Fortunately, the distinction is necessary only for lenses which are stronger than are commonly used except, perhaps, for microscope objectives. 4.4.1 The determination of the cardinal points by ray tracing If the path of a ray starting some three to four diameters from the lens is traced, using any of the methods outlined in section 3 the position of the focal point can be found by noting the values of z for which the radius, r, or reduced radius, R, changes sign and interpolating to find the position of the crossing point. This will be the true focal point, but, unless the lens is very strong indeed, it will be very close to the asymptotic focal point. If the ray path has not crossed the axis by the end of the integration, the focal point can be located by extrapolating the last few points calculated. The corresponding principal point is found from the slope of either the tangent to the path at the focal point or of the asymptote. With a good potential the effects of spherical aberration can be seen by changing the off-axis distance of the incident ray, and the spherical aberration coefficient can be determined. If the trace is done by integrating the Picht equation, it is not necessary to convert back to real radius as both r and R are zero at the crossing point and the asymptote is only reached in a region of constant potential. Nonetheless, it is interesting and instructive to make the conversion as the approach to the asymptotes has a rather different appearance. 4.5 Windows and pupils While it is the paths of individual rays which are calculated, measurements are made with beams of particles and these have finite lateral and angular extent. These beams must be defined by suitable apertures, and in figure 4.5 we illustrate some of the terms and parameters involved. The beam is defined by two apertures, called conventionally the window and the pupil. The window is regarded as the object on which the lens acts to produce an image and the distinction is sometimes made between the object window and the image window. The pupil, or angle stop, defines the range of angles which may enter the lens from any point in the window. Naturally the lens produces an image of the pupil also, and the position and size of the image pupil are important parameters of an optical system. In figure 4.5 the lens is represented as a thin lens with coincident principal planes purely to avoid complexity in the diagram. The lateral dimensions are grossly exaggerated. The 72 Electrostatic Lens Systems Figure 4.5 Diagram to illustrate the formation of the image of an entrance, or object, window W0 , by a bundle of rays limited by an entrance pupil, P0. The semi-angular divergence of the bundle of rays is specified by the pencil angle, θ0 , and the maximum inclination of the central ray of the bundle to the axis in the plane of the entrance pupil is the beam angle, α0. The lens forms images of the object-side window and pupil, and the lower diagram shows the special case where the object-side pupil is at the first focal point, giving an image pupil at infinity and zero beam angle on the image side. upper part of the diagram shows the paths of a number of rays from the bottom of the object window and a point within it which pass through the object pupil at its upper and lower edges and also through the centre of the pupil. In the image space the rays from each point in the object window cross at the corresponding point in the image window and those rays which passed through particular points of the object pupil pass through the corresponding parts of the image pupil. Two sets of angles are indicated: α and θ. α is known as the beam angle and is the symbol used for the angle between the principal rays in the construction of figure 1.3. It is the angle subtended at the centre of the pupil by the radius of the window and is therefore the angle at which the central ray from the edge of the window crosses the axis in the plane of the pupil. θ is the pencil angle, describing the angular spread of a pencil of rays from a point in the window, and is the angle subtended at the centre of the window by the radius of the pupil. Suffices distinguish the object  The optics of simple lens systems 73 and image space values of these angles. If the radii of the window and pupil are written as rw and rp respectively and the distance between them is l then the angles can be expressed as rw rp α= θ=. l l The lower part of figure 4.5 shows a special case. The object pupil is at the first focal point of the lens and is therefore imaged at infinity. The image side beam angle is now zero and the total angular divergence of the beam, which can be written as (|α| + |θ |), is a minimum. 4.5.1 The energy-add lens If a real image of a window is formed at the first principal plane of a lens, the emergent rays appear to come from a virtual image of the same size at the second principal plane. These rays may be focused to give a final, real image. The simplest practical system consists of four cylinders. The central two cylinders form the energy-add lens proper, and exploit the fact that the positions of the principal planes of a two cylinder lens do not change much over a fairly wide range of voltage ratios. The lens formed by the first two cylinders produces the image at the first principal plane of this central lens, and that formed by the last two cylinders gives the final image. The voltage ratios V2 /V1 and V4 /V3 are held constant, but the ratio V3 /V2 is changed, adding energy e(V3 − V2 ) to the particle. This apparently simple method of maintaining the position of the final image over a range of overall voltage ratio, V4 /V1 , has major drawbacks for all but the crudest applications. While the imaging of the entrance window may be quite well controlled, there is no control over the position and size of the pupil image formed by the first lens. The action of the second and third lenses will only make matters worse, most probably leading to excessive beam angles at the final image. The presence of an intermediate image is also likely to cause increased aberrations. The simple multi-element lenses described in the next section are much better behaved. 4.6 Multi-element lenses 4.6.1 Zoom lenses A lens having two elements allows a conjugate focus condition to be met in just two ways, with an accelerating or a decelerating voltage ratio. The addition of a third element allows the condition to be met over a wide range of overall voltage ratios by a suitable choice of the potential of the third, centre, electrode. The focal locus for one such lens is shown in figure 4.6. The magnification of the lens can only be controlled in the most general fashion by, for example, selecting the lengths of the cylinders to give particular values at a limited number (three, at the most) of points on the locus. In order to make a true zoom lens, one in which the magnification for a given pair of conjugates can be controlled independently of the overall voltage 74 Electrostatic Lens Systems Figure 4.6 Focal locus for a three cylinder lens having elements of length 3.0, 0.75 and 2.5 diameters. The variation of the magnification and of the spherical aberration round the locus are also shown. A identifies corresponding points on the three loci. ratio, a fourth element is necessary. It is difficult to obtain much flexibility with aperture lenses because the electrodes have to be comparatively close together but, with cylindrical electrodes, the distance between the two outer electrodes, which determine the overall voltage ratio, can be usefully large. The two central electrodes must have potentials which combine in some way to satisfy the focus condition, but this can be done in many ways and one might consider the two potentials to have some average magnitude (though not a simple arithmetic mean) and also a ‘centre of action’ which can be moved to the left or right by changing their ratio. Figure 4.7 illustrates this for two four cylinder lenses with symmetric conjugates separated by six diameters and with their central two elements of length D in one case and D/2 in the other. This figure shows the magnifications which may be obtained for an overall voltage ratio of unity. The lens with the longer central elements offers a range of magnifications from 0.67 to 1.5 while the other will only cover from 0.87 to 1.15. Notice that the extreme magnifications are obtained with both centre electrodes at higher potentials than the outer ones and not, as one might have expected, with one of the centre elements at the outer potential. It is left to the reader to investigate (using the LENSYS program) other  The optics of simple lens systems 75 Figure 4.7 The potentials of the two centre elements of four cylinder lenses of total length six diameters required to give a range of magnifications. The upper curves relate to a lens with centre elements each of length one diameter, and the lower curves to one with lengths of 0.5 diameter. The potentials of the first and fourth elements are each equal to 1. possible pairs of values giving the same magnifications and to deduce why they are not good combinations. The greater range of magnification obtained with the longer central elements raises the question of further increase. However, just extending the lengths of these two elements will not help a great deal. A much better approach is to place a fifth electrode in the middle of the system, and for the moment one can think of this merely as a ‘spacer’ keeping the two magnification-controlling electrodes a good distance apart! Before discussing this way of obtaining zoom behaviour on a par with that obtainable with quite modest cameras, it will be helpful to examine a somewhat specialized five element cylinder lens. 4.6.2 The afocal lens We consider in this section a combination of two lenses which has an interesting and useful property. Figure 4.8 shows the optics of this combination schematically and illustrates the construction, using principal rays, of the image of an object to the left of the first lens of the pair. The essential point of the lens combination is that the second focal point of the first (left hand) lens coincides with the first focal point of the second (right hand) lens. This means that the first principal ray, drawn from the top of the object parallel to the axis, passes through this common focal point and emerges from the second lens again parallel to the axis. The immediate 76 Electrostatic Lens Systems Figure 4.8 The construction of the image of a given object by two lenses having a common focal point in the region between them. consequence of this is that, wherever the object may be, the image will have the same size: in other words, the magnification of the lens combination does not depend on the position of the object. The second principal ray is drawn through the first focal point of the left hand lens, emerges from that lens parallel to the axis and is then deviated to emerge from the right hand lens as though from the second focal point of that lens. The focal lengths and the object and image distances are indicated on the figure and we use Newton’s relation to examine their relationship. The first lens would produce an intermediate image I of O a distance q from the common focal point such that q = f1 f2 /p where p is the distance of the object from the first focal point of the left hand lens. This image lies beyond the first focal point of the second lens and is imaged by that lens as I  , a distance q  from the second focal point of this lens such that q  = f1 f2 /q. In terms of the position of the original object q  = (f1 f2 /f1 f2 )p. If the two lenses are identical then we have the interesting result that q  = p and the final image is the same distance from the second focal point of the right hand lens as the object is from the first focal point of the left hand lens. The fact that rays entering parallel to the axis emerge parallel to the axis means that the lens system has no focal points—it is afocal or telescopic. Remembering that F1 is a negative quantity, the distance between the object and the final image is 2(F2 − F1 ), and this must be held constant, independent of the overall voltage ratio, if the special property of this lens is to be maintained. The values of F1 and of F2 will change, of course, and if the difference is not to change, each lens must consist of three electrodes. The last element of the left hand lens and the first element of the right hand lens are at the same potential (otherwise there would be further lens action in the middle of the system) so the entire lens consists of five elements. Because the two lenses are identical  The optics of simple lens systems 77 V2 V4 V3 V5 = and = V1 V3 V1 V3 and therefore  V 1/2 V3 5 =. V1 V1 The magnification of the lens can be calculated from the figure and is M = f1 /f2 = f1 /f2 for identical lenses. The angular magnification is given by α  h  h  f f1 1 Mα = = =  = = M. α f2 f1 f2 f2 The magnification, angular magnification and the overall voltage ratio are related by the law of Helmholtz and Lagrange  V 1/2  V −1/4 5 5 MMα =1 so M = Mα =. V1 V1 Apart from the trivial case of a lens which is electrically and mechanically symmetric, this is the only lens for which the magnification can be calculated without any knowledge of the potential distribution; it depends only on the identity of the two component lenses. An important application of this lens is in the production of a well defined particle beam. The images of an object side window and pupil will have the same axial separation and be magnified by the same factor, so if the object side apertures are of equal diameter, the particle beam will be bounded by a cylindrical surface in a region of the image space though no apertures are present there, reducing the chance of unwanted collisions of the beam with the metalwork. The beam angle and the pencil angle will be equal in this situation. 4.6.3 The zoom afocal lens If we remove the constraint on the two component lenses of figure 4.8, but maintain the same distance between conjugates, the lens system will not, in general, remain afocal. It does then, however, have the desirable property of the zoom lens. The magnification is controlled by the relative strengths of the first and second lenses in just the same way as is done by the two centre electrodes of a four cylinder lens. For the moment we keep the potential of the centre electrode at the geometric mean of those of the first and fifth electrodes and show in figure 4.9 (by broken curves) the values of the potentials of the second and fourth electrodes required to give a range of magnifications. The particular lens geometry consists of two lenses, each of three cylinders having lengths of 1.25, 0.5 and 1.25 diameters joined end to end to give a five element lens with a centre element 2.5 diameters long and a total length of six diameters, the same as the two four cylinder lenses shown in figure 4.7. The very substantial increase in the range of magnifications is apparent. 78 Electrostatic Lens Systems Figure 4.9 The electrode potentials required to give a range of magnifications for a five cylinder lens of total length six diameters with V5 = V1 = 1. Two sets of data are shown: the broken curves are for the case of V3 = 1, and the solid curves are for V3 adjusted to give a pupil magnification equal to the window magnification. Only four elements are necessary for a lens to have a zoom capability over a range of voltage ratios, and the present lens has, therefore, an extra element which might be used to control another parameter of the lens. There is more than one possibility, but in the present context the control of a second conjugate pair would allow the lens to be again afocal. The effect of adjusting the potential of the centre electrode (with compensating adjustments of those of the second and fourth) is illustrated in figure 4.10 for this same six diameter length lens. The potentials have been adjusted to give a window magnification of −1 and the figure shows the magnification of a pupil located D/2 to the right of the window as a function of the voltage ratio V3 /V1 for a number of values √ of the overall voltage ratio, V5 /V1. The points marked correspond to V3 /V1 = V5 /V1. Notice that, for both accelerating and retarding lenses, V3 has to be increased to make the pupil magnification equal that of the window and so make the lens afocal. For an overall  The optics of simple lens systems 79 Figure 4.10 The pupil magnification of a five cylinder lens as a function of the potential of the centre element for various values of the overall voltage ratio, V5 /V1. The lens is operated to give a window magnification of −1 and the object pupil is at a distance D/2 from the object window. voltage ratio of unity, the potentials of the three centre electrodes are shown also in figure 4.9 (by full curves) for this zoom afocal configuration. 5 Aberrations So far we have considered only the focusing of paraxial rays of charged particles homogeneous in energy. The true situation is more complex and we should consider rays which are non-paraxial and particles which have a small spread in energy. Particle lenses are subject to the same optical aberrations as photon lenses and the effects can be much worse because it is not possible to use materials of different dispersion to reduce chromatic aberration or to grind surfaces of special forms to reduce spherical and other aberrations. 5.1 Spherical aberration The handling of particle beams is usually an axis-centred problem so spherical aberration is normally much more serious than the off-axis aberrations, such as coma, astigmatism and distortion. We shall therefore consider only rays which cross the axis at some point. Figure 5.1 is a schematic diagram showing the emergent asymptote corresponding to a ray incident from an axial object point. For meridional rays, the relationship of the radial positions and slopes of the rays at the first and second focal planes of the lens can be expressed, following Verster as r1 r1  r 2 r2 = − + m13 r13 + m14 r12 + m15 r1 1 f2 f2 f2  r 3 r1  r 2 + q11 r15 + q12 r14 + q13 r13 1 1 + m16 f2 f2 f2  r 3  r 4  r 5 + q14 r12 + q15 r1 1 1 1 + q16 + ··· (5.1a) f2 f2 f2 r2 r1  r 2 − = r1 + m23 r13 + m24 r12 + m25 r1 1 f1 f2 f2  r 3 r1  r 2 + q21 r15 + q22 r14 + q23 r13 1 1 + m26 f2 f2 f2  r 3  r 4  r 5 + q24 r12 + q25 r1 1 1 1 + q26 + ··· (5.1b) f2 f2 f2 80  Aberrations 81 r’ 1 l r r’ 1 r 2 o 2 i F H H F r 1 2 1 2 p q Figure 5.1 Definition of the ray parameters for the description of the meridional aberration coefficients. remembering that f1 is negative†. The coefficients, mij and qij , are dimensionless and also negative. The terms of fifth order are not often used, but are included here as we shall later show how some of them may be deduced from fairly simple ray traces. The coefficient of spherical aberration for a lens operated with magnification M, Cs (M), is defined by Br = −MCs (M)αo3 where αo = −r1. From figure 5.1 we can write r1 = −pr1 , and Br = r2 + qr2 for the distance off-axis at which the ray crosses the Gaussian image plane. The object and image distances can be written in terms of the focal lengths and the magnification as p = −f1 /M and q = −Mf2. Substitution of these expressions into equation (5.1), ignoring the fifth order terms, gives the following relationship between the coefficient of spherical aberration and the meridional coefficients: f1  f 2 1 Cs (M) = −m13 f2 − (m14 + m23 ) − (m15 + m24 ) /f2 M M  f 3  f 4 1 1 − (m16 + m25 ) /f22 − m26 /f23. (5.2) M M Each term of equation (5.2) has a positive value so no cancellation is possible. Cs has the dimensions of Br, but it is a common convention to express both in terms of the diameter of the lens, and all the values presented in this book and in the LENSYS program are strictly of Cs /D. Equation (5.2) shows that Cs (M) could be expressed as a fourth order polynomial in the object distance, p = −f1 /M, but it is more usual to write it in terms of the magnification, M, Cs (M) = Cs0 + Cs1 /M + Cs2 /M 2 + Cs3 /M 3 + Cs4 /M 4 (5.3) where the Csi are properties of the lens itself. † A number of the equations in this chapter differ from their counterparts in other texts as a consequence of our Cartesian sign convention. 82 Electrostatic Lens Systems There are certain well established relationships between some of the eight meridional aberration coefficients m14 = 3m23 + 1.5 m15 = m24 m25 = 3m16 + 1.5 (5.4) and so the five spherical aberration coefficients are Cs0 = −m13 f2 Cs1 = −(m14 + m23 )f1 = −(4m23 + 1.5)f1 Cs2 = −(m15 + m24 )f12 /f2 = −2m24 f12 /f2 (5.5) Cs3 = −(m16 + m25 )f13 /f22 = −(4m16 + 1.5)f13 /f22 Cs4 = −m26 f14 /f23. If the object is at infinity, r1 = 0 and the magnification is therefore zero. It is then necessary to use equation (5.1b) directly. Including the relevant fifth order term, q26 (r1 /f2 )5 , we have  r 3  r 5 1 1 Br = r2 = −m26 f1 − q26 f1. f2 f2 The third order term can be written as  r 3 1 Br = Cs4. (5.6) f1 5.1.1 The axial displacement The aberrated ray crosses the axis a distance Bl before the Gaussian image point where Br αo3 M Bl = −  = −MC s =− Cs αo2 r2 αi Mα f2  f 3 2 = M 2 Cs αo2 = M 4 Cs αi2 (5.7) f1 f1 and where αi = −r2 is the beam angle in the image space. For parallel input M = 0 so M 4 Cs = Cs4 and, using αi = r1 /f2 ,  f 3 f2  r1 2 2 Bl = Cs4 αi2 = Cs. (5.8) f1 f1 4 f1 If we wish to include the fifth order term without excessive algebra, we can write r2 = −r2 /Bl = −r1 /(f2 + Bl)  Aberrations 83 zg h g G m l zm Figure 5.2 Aberrated rays in the region of the disc of minimum confusion. giving r1 Bl r2 = f2 + Bl from which we find the following working expression: f  Bl  r 2  r 4 2 1 1 G(Bl) = = −m26 − q26 (5.9) f1 f2 + Bl f2 f2 which we shall use later. 5.1.2 The disc of minimum confusion If the object is a point source of particles, the ‘image’ in the Gaussian plane will be a round spot with a radius determined by the pencil angle and the aberration coefficients of the lens. Rays which have passed close to the limits of the entrance pupil will contribute to the outer parts of this spot, having crossed the axis somewhat closer to the lens. Between the Gaussian plane and the point on the axis where these marginal rays are focused, there will be some rays which cross the axis going upwards and some going downwards, with the former being more steeply inclined, and the spot formed by the envelope of all the rays will have a minimum radius somewhere between these points. This spot is known as ‘the disc of minimum confusion’. Figure 5.2 shows the marginal ray and a general ray which intersect a distance h from the axis; the axial position of the intersection is a distance l from the Gaussian focus, G. We can express h in two ways h = αg (zg − l) = αm (zm − l) Aαg3 − lαg = Aαm 3 − lαm 84 Electrostatic Lens Systems so l = A(αm 2 + αm αg + αg2 ) and h = −A(αm 2 αg + αm αg2 )  3 where A = Cs4 f2 /f1 by equation (5.8), and is negative. We find the minimum value of h by differentiation dh αm = −A(αm 2 + 2αm αg ) = 0 if αg = − dαg 2 and this leads to the radius of the disc of minimum confusion as αm3 hmin = A 4 and its position as 3 2 l= Aα. 4 m The radius is a quarter of that of the Gaussian plane spot and it is three quarters of the way from the Gaussian image to the most aberrated image point. 5.2 Chromatic aberration Figure 5.3 illustrates the passage of two pairs of rays through a lens system ignoring the effects of spherical aberration. One ray of each pair has the correct energy to form an image at I. The other has a slightly higher energy. We consider first the pair of rays which start from a point on the axis. The ‘correct’ ray crosses the axis at the image, but the higher energy ray crosses the axis a distance δl beyond this and, at the image plane, is a distance δr above the axis. These distances are related by the angle, αi in the image space and can be referred to the corresponding angle at the object using the Helmholtz–Lagrange relation. 1  Vo  21 δr = αi δl = Mα αo δl = αo δl. M Vi A coefficient of chromatic aberration, Cc , has historically been defined by δr = −MCc αo δV Vo and we may, therefore, express this coefficient in terms of either observable by 1 δr Cc = (5.10a) Mαo (δV /Vo ) 1  Vo  21 δl = 2. (5.10b) M Vi (δV /Vo )  Aberrations 85 I r (z ) δl δr r 2 (z ) Figure 5.3 Two pairs of rays traced to illustrate chromatic aberration. The lower ray of each pair has the correct energy to form an image at I and the other rays, which have a slightly higher energy, cross the image plane at points which are displaced from the true image points. The second of these equations illustrates the non-dependence of the aberration coefficient on the angle αo. This coefficient is properly referred to as the coefficient of axial chromatic aberration to distinguish it from the effect illustrated by the other pair of rays in figure 5.3 which are launched parallel to the axis from a point above the axis. The ‘correct’ ray crosses the image plane and defines the magnification of the lens. The higher energy ray crosses the image plane closer to the axis, with a lower magnification. This effect is described as chromatic aberration of magnification and the coefficient, CD , which quantifies the effect is sometimes referred to as the coefficient of lateral chromatic aberration. It is defined in terms of the radial change, δrI , in the ray at the image plane by the following equations.
 δV δrI = −MCD ro Vo −1 δr CD = (5.11) Mro (δV /Vo ) where ro is the initial off-axis distance of the ray and, for the case of the general ray, r2 (z), which crosses the axis at an angle of 45◦ , is equal to f1. This aberration is an off-axis effect and we discuss, in the next section, the off-axis effects due to the angle dependent behaviour. 86 Electrostatic Lens Systems r q 1 r 0 r 2 F1 H 2 H 1 F2 p r’ Mr 2 0 r Figure 5.4 Definition of the ray parameters for the description of the off-axis aberrations of a lens. 5.3 Off-axis aberrations Figure 5.4 is similar to figure 5.1, but indicates the asymptotes to a ray from an object point, a distance r0 from the axis, which crosses the first focal plane at a radial distance r1. The path of the ray is still in a plane containing the axis; we shall not be considering skew rays. Using equations (5.1) again, we write the position of the ray at the image plane as the sum of the geometric, Gaussian, distance, Mr0 , and an aberration term which can be written as the sum of terms involving r03 , r02 r1 , r0 r12 and r13.   f2 1 Br = Mr0 + r03 M 4 m13 3 + M 3 m23 2 f1 f1   f2 1 1 − r02 r1 3M 4 m13 3 + M 3 (m14 + 3m23 ) 2 + M 2 m24 f1 f1 f1 f2  f2 1 + r0 r12 3M 4 m13 3 + M 3 (2m14 + 3m23 ) 2 f1 f1  1 1 + M (m15 + 2m24 ) 2 + Mm25 2 f1 f2 f2  f 2 1 1 − r13 M 4 m13 3 + M 3 (m14 + m23 ) 2 + M 2 (m15 + m24 ) f1 f1 f1 f2  1 f1 + M(m16 + m25 ) 2 + m26 3. f2 f2 The terms in r03 describe the distortion of the image, because they imply a magnification which depends on the position of the point in the object but not on the lens aperture. From equation (5.5) we see that Cs0 and Cs1 describe this aberration. The terms in r02 r1 describe the combined effects of astigmatism and curvature of field and those in r0 r12 describe coma. The terms in r13 describe spherical aberration  Aberrations 87 f1 and transform to the form of equation (5.2) on writing r1 = − M αo. If r0 = r1 , which implies M = 0, only the term in m26 , equivalent to Cs4 , remains and equation (5.6) is recovered. Notice that m26 appears only in the coefficient of r13. While the terms in r02 r1 determine the curvature of field for a specific magnification—and the curvature is usually dominated by the effect of astigmatism—there is a limiting value which obtains even in the absence of astigmatism. This minimum curvature is expressed by the Petzval integral, which is never zero for a particle lens. The integral can only be zero if there are contributions from divergent lens elements and, as we saw in chapter 2, immersion lenses, with which we are exclusively concerned, are always convergent. It is a property of the given lens and depends only on the potential distribution and not on the choice of conjugates. It is, therefore, most readily calculated along with the basic lens parameters. 5.4 Interrelations of the spherical aberration coefficients By considering a lens to be ‘thin’ in the sense that R does not change over the distance for which T is significant, it is possible to factor a term out of the aberration integral and hence develop simple relationships between the mij coefficients. It has been demonstrated that these remain reasonably valid even when the ‘thin’ approximation does not appear to be appropriate. The relationships are of the form m2j /m1j = σ where the value of the ratio has been determined empirically as σ = (V2 /V1 )1/4 for a two cylinder lens. Taken in conjunction with the well established relationships between three pairs of mij (equation (5.4)) and ignoring the added ‘1.5’ terms, which is a reasonable approximation except for very strong lenses, there appears to be a pattern of interrelations between all the mij coefficients, mi6  σ mi5 /3  σ 2 mi4 /3  σ 3 mi3 m2j  σ m1j. With these factors, equation (5.2) becomes  σ f1  σ 2  f 2 1 Cs (M) = − m13 f2 1 + 4 +6 M f2 M f2  σ 3  f 3  σ 4  f 4  1 1 +4 +. (5.12) M f2 M f2 If σ is indeed (V2 /V1 )1/4 , then σ 2 = −f2 /f1 so σf1 /f2 = −1/σ and, writing Cs0 for −m13 f2 , we can express the spherical aberration coefficient for a given magnification as
 1 4 Cs (M) = Cs0 1 −. (5.13) σM 88 Electrostatic Lens Systems H(r) R 2 H(r) 1 r r’ 2 r’ r r 1 1 2 F 2 F 1 Figure 5.5 Diagram to illustrate the curved principal surfaces. A careful study of more complex lenses shows that this is indeed rather approximate, but another result based on the same relationships does seem to have a wider validity. We define a function, Y , by  −σf 4  −σf 3  −σf 2  −σf  1 1 1 1 Y = C s0 + C s1 + C s2 + C s3 + C s4 f2 f2 f2 f2 (5.14) and substitute for the Csi the appropriate terms of equation (5.12), giving  −σf 4 1 Y = Cs0 (1 − 4 + 6 − 4 + 1) = 0. f2 If we write Y+ to represent the sum of the first, third and fifth terms of equation (5.14) and Y− to represent the sum of the second and fourth (which is usually negative), then the ratio YY++ +Y −Y− − will be a fair measure of the accuracy of the statement Y = 0. For a wide range of lenses (and the reader is invited to test this using LENSYS) this ratio is less than 0.01 and usually much less. 5.4.1 The principal surfaces We noted in chapter 1 that the principal ‘planes’ were not in fact flat and we are now in a position to examine their true form. Figure 5.5 shows the principal surfaces as defined by the intersection of the object and image asymptotes of rays incident at a radial distance, r, parallel to the axis. We can express the distance from the second Gaussian focal point to the projection of the intersection on to the axis, H2 (r), as F2 − H2 (r) = −(r2 + r)/r2  Aberrations 89 using equations (5.1)  r 2 F2 − H2 (r) = f2 + (f1 m26 + f2 m16 ) f2 and writing σ 2 = −f2 /f1   r 2  F2 − H2 (r) = f2 1 + m13 σ 2 (σ − 1). f2 A similar analysis gives, for the first principal surface, F1 − H1 (r) = −(r1 + r)/r1  r 2 = f1 + (f1 m23 + f2 m13 ) f1   r 2  = f1 1 − m13 σ (σ − 1). f1 Note that m13 is negative and σ > 1 for an accelerating lens, so both principal surfaces are concave towards the high potential side of the lens. 5.5 The determination of aberration coefficients If the potential distribution throughout the lens is known with sufficient accuracy, rays will follow their correct paths and, if they are not paraxial, will show the effects of the lens aberrations. In principle, then, all that is needed is careful ray tracing to find the intercept of the ray at the Gaussian image plane. This raises the first problem: where is the Gaussian image plane? It is necessary to trace more than one ray and extrapolate to zero αo or r1 as appropriate and it is tempting to use the axial displacement directly. If only the axial potential is known, then calculations must be based on the effect of the terms omitted from the axial equation of motion. 5.5.1 Direct ray tracing Figure 5.6 shows rays traced in an accelerating and a decelerating lens. The much greater effect of spherical aberration in the retarding lens is apparent. To illustrate a procedure for the determination of the aberration coefficients appropriate to parallel input rays we shall analyse some data obtained with the ‘student’ version of the CPO-2D program. For the purposes of the present analysis, the important parameters of the trajectory are those indicated in figure 5.7. The position of the focal point, F2 , is not known a priori and we cannot, therefore, determine r2 directly, but,from measurements made at the reference plane, the program provides values for the axial position, zi , at which a ray launched parallel to the axis at a radial distance r1 crosses the axis and the slope, r2 , of the ray at that point. From these values 90 Electrostatic Lens Systems Figure 5.6 Rays traced in a two cylinder lens having voltage ratios of (a) 1 : 10, and (b) 10 : 1. The aberrations of the retarding lens are so great that the outermost ray of (a) would strike the inner surface of the electrode well before the end of the diagram. Gaussian plane r reference plane ∆l F z r Figure 5.7 The path of an aberrated ray which crosses the Gaussian plane at a distance r2 having crossed the axis at a position zi a distance Bl before. we can determine the focal parameters f2 and F2 , and the aberration coefficients m16 , m26 , q16 and q26. If we trace rays in a second lens, having the reciprocal  Aberrations 91 G(∆l) 0.50 0.25 0 r' –0.25 0 0.05 0.10 0.15 0.20 r /f Figure 5.8 Data plots used to determine the spherical aberration coefficients of the lenses shown in figure 5.6. The curves represent r2 = − fr12 + m16 ( fr12 )3 + q16 ( fr12 )5 and G(Bl) = m26 ( fr12 )2 + q26 ( fr12 )4. The solid data points refer to the accelerating lens and the open data points to the decelerating lens. voltage ratio, we also find f1 , F1 , m13 , m23 , q11 and q21. The first step is to find F2 by writing zi = F2 + Bl, where Bl is negative and is given by equation (5.8). A graph of zi against r12 is almost a straight line and a quadratic fit will give a very good value for F2. We next plot r2 against r1 and fit an expression in odd powers of r1. From equation (5.1a) the coefficient of the first term is −1/f2 and those of the other terms will give the aberration coefficients. It is a little neater to repeat this with r1 /f2 as the abscissa and we show such a plot in figure 5.8. Now that we have a value for f2 it is possible to use equation (5.9), remembering that the ratio f2 /f1 = −(V2 /V1 )1/2 , to find the remaining coefficients. In figure 5.8 we show the results for both the accelerating (1 : 10) and decelerating (10 : 1) lenses. The values of the coefficients are found to be 1 : 10 10 : 1 m16 = −6.41 −2.27 m26 = −12.1 −1.42 q16 = −100 −10.8 q26 = −134 −10.6 Figure 5.9 shows values of the chromatic aberration coefficient, Cc , determined by measuring the image plane ‘miss distance’, δrI , for a ray launched with an 92 Electrostatic Lens Systems Cc 1/M Figure 5.9 The coefficient of chromatic aberration for a two cylinder lens with a voltage ratio of 1 : 10. The data can be represented by Cc (M) = 0.834 − 1.005/M + 0.314/M 2. energy 2% greater than nominal in a two cylinder lens with a potential ratio of 1 : 10. The coefficients of a power law fit to the data agree very well with values calculated directly using expressions which we develop later. 5.5.2 Perturbation of the paraxial solution We can modify the equation of motion to take account of terms omitted in our earlier analysis or to allow for a spread in energy and, provided that the effects are small, we can write the solution to the modified equations as the simple sum of the paraxial solution and an additional term. The details of the calculation depend on the problem and are expanded in the next two sections. 5.5.2.1 Spherical aberration. In chapter 2, the higher order terms in the expansion of the axial potential were ignored as was the contribution made to the total energy of the particle by its radial velocity. These simplifications led to the paraxial equation of motion, equation (2.3), and then to the Picht equation, (2.4). In order to describe the non-paraxial behaviour, it is necessary to replace these terms. The equation of radial motion becomes d2 r r r 3 (4)   m = eE r = e V − V dt 2 2 16 and the total energy of the particle has to be written as 1  dz 2  dr 2   r2  m + + e V − V  = 0. 2 dt dt 2  Aberrations 93 The equation of motion is then d2 r 1 V  dr r V   1 V (4) 1  V  2  3 + + = − r dz2 2 V dz 4 V 32 V 16 V 1  V  V  V   2  1 V  2 1 V  3 + − r r − rr − r. (5.15) 8 V V2 4 V 2V To solve this equation, we write r = rp +ra where rp is the solution to the equation of paraxial motion and ra is a small perturbation representing the aberration due to the use of better approximations to the real potential. ra is small enough that only first order terms need to be retained. Each term of the left hand side of equation (5.15) splits into one containing rp and one containing ra. Those in rp sum to zero, because they are just the terms of the paraxial equation. The terms on the right hand side involve the third power of r and so only the rp part survives, giving an equation for the perturbation, ra , d 2 ra 1 V  dra ra V   1 V (4) 1  V  2  3 + + = − rp dz 2 2 V dz 4 V 32 V 16 V 1  V  V  V   2  1 V  2 1 V  3 + − r r p p − rp rp − r. (5.16) 8 V V2 4 V 2V p Solutions of the homogeneous equation, obtained by setting the right hand side of equation (5.16) equal to zero, are known: they are just the general solutions of the Gaussian ray equation discussed in chapter 1. We can therefore write the perturbation as ζ (z)r1 (z) + ξ(z)r2 (z) where we stress the variability of ζ and ξ , reflecting the non-zero right hand side. The problem reduces to the calculation of ξ(zi ) where zi is the position of the Gaussian image. After a lot of algebra, two integrations by parts and transformation of the radial variable to the reduced radius, R = rV 1/4 , the coefficient of spherical aberration can be written as 1 ∞ R 4  4 11  R  Cs (M) = 3T + 5T 2 − T 2 T  + 4T T  dz −∞ V R 1/2 1/2 2 64V0 (5.17) where V0 is the potential at the lower limit of integration†. In this equation, R(z) corresponds to the ray which passes through the axial object point with unit slope. Every variable, bar one, in this equation will have been determined in the course of calculating the paraxial image position. The missing variable is T  and it is a trivial matter to find this. We noted earlier that the sign of T is not important to the calculation of the paraxial focus as it appears as T 2 in the Picht equation, but the presence of T  here shows that it does matter for the calculation of the aberrations. † The use of the reduced radius has transformed a term in (V /V0 )1/2 into (1/V0 V )1/2. Almost always, the potentials will be expressed in terms of V0. 94 Electrostatic Lens Systems Figure 5.10 The integrands of the aberration integrals of equa- tion (5.17) for three aperture and Afocal8 lenses. Though the limits of integration are marked as ±∞, in practice a much shorter interval will suffice. Figure 5.10 shows the integrand of equation (5.17) for the cases of a three aperture lens, used asymmetrically, and a five cylinder lens used in the afocal mode. The range of axial position over which the integrand is significantly greater than zero is appreciably less than the distance between the conjugate points. For a two element lens, we might expect the integrand to show two maxima, one where R is large and the other where R  is large. In the three element lens, especially with aperture electrodes, two of the four possible maxima run into each other, but the cylinder lens shows two maxima associated with each gap. The ray paths shown in this figure represent the true radius, r(z). The ratio of the reduced and true radii has the same value at any point for all trajectories, so we can write R(z) = R1 (z) + M1 R2 (z) and expand the radial terms  Aberrations 95 Figure 5.11 The integrands of the integrals analogous to equa- tion (5.17) for the five spherical aberration coefficients for a two aper- ture lens. The curves can readily be associated with particular coeffi- cients as these are markedly different in magnitude. as R 4 = R14 + 4R13 R2 /M + 6R12 R22 /M 2 + 4R1 R23 /M 3 + R24 /M 4 and similarly for R 3 R . This allows us to separate the integral of equation (5.17) into the sum of five integrals, one for each of the Csi. Each integral is written as the product of terms involving the T -parameters,   1 2 11 c1 = 3T + 5T 4 − T 2T  V 1/2 2 4T T  c2 = 1/2 V which, being functions of only the electrode geometry and potentials, are the same for all applications of the lens, and terms specific to each coefficient I0 = c1 R14 + c2 R13 R1 I1 = c1 4R13 R2 + c2 (3R12 R2 R1 + R13 R2 I2 = c1 6R12 R22 + c2 (3R1 R22 R1 + 3R12 R2 R2 ) (5.18) I3 = c1 4R1 R23 + c2 (R23 R1 + 3R1 R22 R2 ) I4 = c1 R24 + c2 R23 R2. r1 and r2 are asymptotic to ±f1 , which is not known a priori so the focal properties must be calculated using arbitrary horizontal asymptotes (usually 1) and then scaled to provide the values required for the aberration calculation. However, 96 Electrostatic Lens Systems to calculate the coefficient Cs (M) between conjugates, the ray R(z) can be used with no further scaling. Figure 5.11† shows the integrands used to calculate the five Csi. The individual curves can easily be associated with particular coefficients and for this lens they are, from the top, Cs2 , Cs0 , Cs4 , Cs3 , Cs1. 5.5.2.2 The Petzval integral. The Petzval sum describes the curvature of the image surface in the absence of astigmatism. In photon optics, where one has sharp boundaries separating regions of different refractive index, the radius of curvature can be expressed in terms of the radii of the boundaries and the refractive indices and, using equations  (1.3) to introduce the focal lengths of each boundary, reduces to 1/rp = (1/nj fj ). The  surface is a paraboloid of revolution and is concave towards the lens elements if (1/nj fj ) is positive. This will be the case for a single lens, but it is quite possible for the Petzval sum to be zero for discrete lens combinations. Indeed there is a well known photographic lens, the Petzval lens, which has a slight telephoto action as it consists of a convergent element followed by a divergent element. Notice that the details of the ray path make no contribution to the sum. For charged particle optics we have the Petzval integral, a direct analogue, in which the square root of the axial potential has the rôle of the refractive index and T  accounts for the bending of the rays. The full expression is ∞ 1 Vo 1/2  P tz = T dz. (5.19) 8 −∞ V Some examples of the integrand are shown in the appendix and, though the changes of sign are apparent, the integral as a whole is never negative. 5.5.2.3 Chromatic aberration. In this case, the perturbing factor can be expressed as a constant change in the potential, and we can write the equation of motion as d2 r 1 V dr r V  + + =0 dz2 2 (V + δV ) dz 4 (V + δV ) and, as δV is small in comparison with V , we can multiply top and bottom by V − δV to give d2 r 1 V  dr r V   1 V  dr 1 V   + + = δV +. dz2 2 V dz 4 V 2 V 2 dz r V 2 We continue, as before, by writing r = rp +rc where rc is a change in the trajectory, small enough that the product rc δV can be neglected. d 2 rc 1 V  drc rc V   1 V  dr rp V   p + + = δV +. dz2 2 V dz 4 V 2 V 2 dz 4 V2 † Figures 5.10 and 5.11 have been prepared with a version of LENSYS modified to print larger dots.  Aberrations 97 We again write the perturbation as ζ (z)r1 (z) + ξ(z)r2 (z) where r1 and r2 are the general solutions of the ray equation. In this case we are interested in the values of both ζ (zi ) and ξ(zi ) Using the expressions for the chromatic aberration coefficients (equations (5.10) and (5.11)) and then changing the radial variable to the reduced radius we find ∞ 1 3 V 2 2 Cc = r dz V0 1/2 −∞ 8 V 5/2 ∞ 1 3 V 2 2 = R dz V0 1/2 −∞ 8 V2 ∞ 1 3 2 2 = 1/2 T R dz (5.20) V0 −∞ 8 and ∞ 1 3 V 2 CD = r2 r dz 1/2 V0 f1 −∞ 8 V 5/2 ∞ 1 3 V 2 = R2 R dz 1/2 V0 f1 −∞ 8 V2 ∞ 1 3 2 = 1/2 T R2 R dz. (5.21) V0 f1 −∞ 8 We have also substituted T for the ratio of the potential gradient to the potential. In these expressions R(z) is the ray which passes through the axial object point with unit slope and we can expand it in terms of the general solutions by R(z) = R1 (z)+(1/M)R2 (z). This enables us to find the magnification dependence of the chromatic aberration coefficients. As in the case of spherical aberration, these coefficients have to be calculated using arbitrarily normalized representations of r1 and r2 and then scaled by multiplying by f12. Only in the calculation of Cc (M) is this scaling not necessary, because R(z) is the correctly normalized reduced ray. ∞
 1 3 2 2 2 1 2 Cc = T R1 + R1 R2 + 2 R2 dz 1/2 V0 −∞ 8 M M = Cc0 + Cc1 /M + Cc2 /M 2 (5.22) ∞
 1 3 2 1 CD = 1/2 T R1 R2 + R22 dz V0 f1 −∞ 8 M = CD0 + CD1 /M. (5.23) It follows from these equations that CD0 = Cc1 /2f1 and CD1 = −Cc2 /f 1. Note that both Cc1 and CD1 are negative quantities. 98 Electrostatic Lens Systems 5.6 The aberrations of retarding lenses 5.6.1 Spherical aberration The equations (5.1) which describe the ray position and slope at the second focal plane in terms of the values of these parameters at the first may be inverted algebraically to describe the effect of reversing the direction of the rays r2 r2  r 2  r 3 r1 = − + m26 r23 + m25 r22 + m24 r2 2 2 + m23 + ··· f1 f1 f1 f1  r 2  r 3 (5.24) r1 r2 = −r2 − m16 r23 − m15 r22 − m14 r2 2 2 − m13 + ··· f2 f1 f1 f1 or equations (5.1) may be written in terms of the coefficients of the lens having the reciprocal voltage ratio r1  2  r 3 ∗ 2 r1 ∗  r1 r2 = − ∗ 3 ∗ 1 + m r + m r + m r + m + ··· f2∗ 13 1 14 1 f2∗ 15 1 f2∗ 16 f2∗ r2 r1  r 2  r 3 − ∗ = r1 + m∗23 r13 + m∗24 r12 ∗ + m∗25 r1 ∗ 1 1 + m + ··· f1 f2 f2∗ 26 f2∗ (5.25) where the asterisks (∗ ) denote the coefficients and focal lengths of the reversed lens. A term by term comparison of equations (5.24) and (5.25) shows that each coefficient for the reversed lens is the same as one for the forward lens, with an obvious symmetry m∗13 = m26 m∗14 = m25 m∗15 = m24 m∗16 = m23 and also f1∗ = −f2. Notice that as we have not specified which lens is the accelerating one, this set of equations can be read either way. Using equation (5.5) we can develop the relation between the coefficients of spherical aberration: Cs0 = −m13 f2 so Cs∗0 = −m∗13 f2∗ = −m26 f2∗ substituting Cs4 = −m26 (f14 /f23 ) gives Cs∗0 = Cs4 f2∗ (f23 /f14 ) = −Cs4 (f2 /f1 )3 as f2∗ = −f1 = Cs4 (Vn /V1 )3/2 where Vn is the potential of the last electrode and the ratio is that for the unasterisked  V 3/2 n direction. It is easy to show, by similar arguments, that Cs∗1 = Cs3 V1  Aberrations 99  V 3/2 n and Cs∗2 = Cs2. Knowing that the reverse magnification M ∗ = 1/M V1 we can write equation (5.3) for the reversed direction as  V 3/2   n Cs∗ (M ∗ ) = Cs4 + Cs3 /M ∗ + Cs2 /M ∗2 + Cs1 /M ∗3 + Cs0 /M ∗4 V1  V 3/2   n = Cs4 + MCs3 + M 2 Cs2 + M 3 Cs1 + M 4 Cs0 V1  V 3/2   n = M4 Cs4 /M 4 + Cs3 /M 3 + Cs2 /M 2 + Cs1 /M + Cs0 V1  V 3/2 n = M4 Cs (M). (5.26) V1 In the case of the afocal five cylinder lens discussed in chapter 4, the magnification is given simply by Maf = (V5 /V1 )−1/4 , so the ratio of the spherical aberration coefficients in the forward and reverse directions is (V5 /V1 )1/2. 5.6.2 Fifth order effects In the preceding section, we ignored the fifth order terms of equation (5.1) because, in the reverse direction, the coefficients, qij∗ are not so simply related to those for the forward direction as are the third order coefficients. These can be evaluated in a similar way and are: ∗ q21 = q16 + 3m216 − m15 m26 ∗ q22 = q15 + m15 (5m16 − m25 ) − 2m14 m26 ∗ q23 = q14 + 2m14 (2m16 − m25 ) + 2m215 − 3m13 m26 − m15 m24 ∗ q24 = q13 3m13 (m16 − m25 ) + m14 (3m15 − 2m24 ) − m15 m23 ∗ q25 = q12 + m13 (2m15 − 3m24 ) + m214 − 2m14 m23 ∗ q26 = q11 + m13 (m14 − 3m23 ) ∗ q11 = q26 + 3m16 m26 − m25 m26 ∗ q12 = q25 + 3m15 m26 + 2m16 m25 − 2m24 m26 − m225 ∗ q13 = q24 + 3m14 m26 + 2m15 m25 + m16 m24 − 3m23 m26 − 3m24 m25 ∗ q14 = q23 + 3m13 m26 + 2m14 m25 + m15 m24 − 4m23 m25 − 2m224 ∗ q15 = q22 + 2m13 m25 + m14 m24 − 5m23 m24 ∗ q16 = q21 + m13 m24 − 3m223 where the mij coefficients are those for the forward direction. 5.6.2.1 Spherical aberration of the afocal lens. While the magnification of an afocal lens does not depend on the position of the object, the coefficient of spherical 100 Electrostatic Lens Systems r r' O I r r' d d F*F dr r r' F F* Figure 5.12 The path of a ray which leaves the axis a distance, d, to the left of the first focal plane of an afocal lens and crosses the image plane at a distance dr below the axis as a consequence of spherical aberration. aberration does so depend. In figure 5.12 we show a schematic path for a ray which originates at a point, O, on the axis a distance, d, to the left of the first focal plane of the system. There are three focal planes to consider. The first crosses the axis at F1 , the first focal point of the first lens of the pair which constitutes the afocal combination. The second is at the second focal point, F2 , of this lens and the first focal point, F∗1 , of the second lens. The third focal plane is at the second focal point, F∗2 , of the second lens. The afocal lens produces an image of the object at a point at this same distance, d, to the left of this