Lens properties.

(This option can also give higher-order derivatives of the axial potential, see below.)

In CPO2D the focal properties (that is, focal lengths and aberration coefficients ) of an electrostatic lens can be calculated directly and very accurately, without tracing any rays. This is done by calculating the potentials and fields along the axis and then evaluating certain integrals. The lens properties obtained in this way are the paraxial (that is, near-axis) properties. They can also be described as third-order properties.

The behaviour of a lens can also be determined by ray tracing, using the 'lens coefficients' option.. The behaviour of non-paraxial rays can then be seen.

In the present option it is assumed that electrons are being focussed, and that the kinetic energy of an electron at a point of potential V is E = eV. It is assumed also that the object and image are both 'asymptotic' -that is, that the object is formed by rays that start asymptotically in a field-free region and point towards the specified object position, while the image is the position from which the final asymptotic rays appear to have come. If the characteristics of an 'objective lens' are required (that is, a lens for which the object is real and is immersed in the field of the lens) then direct ray tracing should be used.

The calculations are fully relativistic (see the 'benchmark' test file test2d05.dat and the 'example' file xmpl2d13.dat). The focal properties of an electrostatic lens are independent of the mass of the particle (if the energies are non-relativistic), but the user should reverse all the voltages if the particle is positively charged.

The focal properties of a lens can all be deduced from the way in which the potential of the lens varies along its axis (which is the z axis here).

The user is asked for the limits zi and zf between which the potential variation is significant, and the number of points (nz) to be used in evaluating the integrals that give the focal properties.

This part of the program is designed to deal only with objects and images that both lie in field-free regions (the properties of 'immersed' lenses can be deduced from ray tracings, see below), and so zi and zf should be in field-free regions.

A sensible choice of nz is usually one that divides the interval zf-zi into steps of length approximately 0.02D, where D is the diameter of the lens, unless very high accuracy is required (in which case the step length should be slightly smaller). If nz is too small the integrations will be inaccurate. If nz is too large numerical inaccuracies will accumulate (and the computing time will increase). Graphs of final results versus nz usually have stable regions at the values of nz recommended above. (Technical note: the program will force nz to be of the form 4*integer+1, in order to use Bode’s rule for integrating aberration coefficients).

To find the 'backward' properties of a lens -that is, to exchange the object and image positions -use reciprocals of the energies or simply exchange the values of zi and zf.

The user is then asked for the required fractional inaccuracy of the potentials and fields and the printing level (a value of 2 for this gives all the comments and the axial potentials, while at the other extreme a value of 0 suppresses everything except the focal lengths). (The option to improve the calculation of potentials near to electrodes is normally suppressed because the axis is usually not near electrodes.)

There is then a choice of the voltages to be used or varied:

'present voltages'

'increment one voltage'

'vary up to 5 voltages'

(but see below for the special case of a single antisymmetric voltage).

Then further information is required:

(1) For 'present voltages' the program calculates the focal properties for the set of voltages already read-in (see note on voltage values), so further data are not needed.

(2) For 'increment one voltage' enter the number of the voltage to be varied, followed by its initial value, final value and increment.

(3) For 'vary up to 5 voltages', give the actual number (n) of voltages that are to be varied. These voltages will be the first n of the voltages specified above (see note on voltage values). Then enter a series of lines, each containing a set of n values for these voltages.

The program uses this information to scale the potential field of the original system and to add a constant voltage if necessary, so that the requested voltages on the electrodes represent the kinetic energies E = eV that electrons would have if they were close to the electrodes.

In the special case of a single antisymmetric voltage, where the z = 0 plane is a plane of voltage antisymmetry, the only option allowed is 'vary up to 5 voltages', which in this case signifies that 2 voltages are to be varied the initial and final voltages. So after a line that contains the number 2, enter a series of lines containing the values of the initial and final voltages.

The program always outputs the focal lengths f1 and f2 (for the object and image spaces respectively) and the mid-focal lengths F1 and F2 (which are the distances of the principal focal points from the reference plane z = 0 -so F1 is positive if the first principal focus has z < 0).

The object distance P (which is taken as positive if the object is at z < 0) and the image distance Q (which is taken as positive if the object is at z > 0) are related by

(P - F1)*(Q - F2) = f1*f2.

The linear magnification is

M = -f1/(P - F1).

For printing levels of 0, 1 and 2 various spherical and chromatic aberration coefficients are also given (and these formulas are also given in the general list of formulas).

For information on how the various aberration coefficients are defined, see aberration coefficients.

2D and 3D aberration coefficients, an alternative way.

Another way of obtaining aberration coefficients (for 2D and 3D systems) is to start by using the CPO programs to output the axial potential (and the axial field, if needed) to the output file, by using either the potential along a line option before ray tracing or the potential on a grid option in the contour drop-down menu after ray tracing. Then an external program (not yet supplied in the CPO package) would be used to deduce the required aberration coefficients from the relevant integrals over the axial potentials and its derivatives.

Higher-order derivatives of the axial potential.

This option is available on the lens properties screen.

Derivatives up to the third order are given, with respect to z, which is in metres for this option.

7-point differentiation is used, except near the ends of the range.

For users who are editing or constructing an 'input data file' without the use of the data-builder -that is, pre-processor:

But Manual editing is certainly not recommended -it is a relic from the time when the databuilder was not available All users are strongly encouraged to use the databuilder, which always gives the correct formats and which has many options for which the formats are not described or easily deduced.

Enter 'y' if focal properties are required. If the program does not find 'y' it will jump over the following lines until it finds one that has the form specified in the note on preliminary potential and field information.

Enter the limits zi and zf between which the potential variation is significant, followed by the number of points (nz) to be used in evaluating the integrals that give the focal properties.

On the next line enter the required fractional inaccuracy of the potentials and fields (which should be between 0.1 and 0.000001) and the printing level (a value of 2 for this gives all the comments and the axial potentials, while at the other extreme a value of 0 suppresses everything except the focal lengths). Make the inaccuracy negative to disable the option to improve the calculation of potentials near to electrodes (and since the axis is rarely near an electrode, the inaccuracy value should normally be negative).

The next line should contain either the letter

'p' for 'present voltages', or

'i' for 'increment one voltage', or

'v' for 'vary up to 5 voltages',

(but see below for the special case of a single antisymmetric voltage). This is followed by other lines specified as follows:

(1) For 'present voltages' the program calculates the focal properties for the set of voltages already read-in (see the note on voltages), so further data lines are not needed.

(2) For 'increment one voltage' enter on the next line the number of the voltage to be varied, followed by its initial value, final value and increment.

(3) For 'vary up to 5 voltages', start with a line that gives the actual number (n) of voltages that are to be varied. These voltages will be the first n of the voltages specified above (see the note on voltages). Then enter a series of lines, each containing a set of n values for these voltages. Terminate the list by entering a negative value for voltage number 1.

In the special case of a single antisymmetric voltage, where the z = 0 plane is a plane of voltage antisymmetry, the only option allowed is 'vary up to 5 voltages', which in this case signifies that 2 voltages are to be varied the initial and final voltages. So after a line that contains the number 2, enter a series of lines containing the values of the initial and final voltages.