Reference: Moore, T.E., C. R. Chappell, M. O. Chandler, S. A. Fields, C. J. Pollock, D. L. Reasoner, D. T. Young, J. L. Burch, N. Eaker, J. H. Waite, Jr., D. J. McComas, J. E. Nordholt, M. F. Thomsen, J. J. Berthelier, and R. Robson, The Thermal Ion Dynamics Experiment and Plasma Source Instrument, Space Sci. Rev., 1995.
Appendix 1: Mirror Shape:
Ideal optical mirrors are parabolic. This is the shape required to bring a parallel incoming photon beam through a focal line or point, in the cases of 1 or 2-D focusing, respectively. When this condition is met, the angular distribution of the external flux of photons is mapped by the mirror to a spatial image at the focal plane. The fact that the photons penetrate the mirror to a depth which is negligible in comparison to the system dimensions is essential to the simplicity of the parabolic focusing shape. Owing to the variation of charged particle penetration with angle of incidence onto an electrostatic mirror, it is clear that the form of an electrostatic mirror required for focusing departs from parabolic. However, the shape and placement of the mirror, with respect to the desired focal line or point, is not intuitively obvious.
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Figure 17. Geometry for the derivation of the TIDE mirror shape required to bring a parallel beam, incident from the right, to a focus at the desired point (0, f/2).
Consider the geometry shown in Figure 17. The tangent to the front (grounded) grid, at an arbitrary point along it, is indicated along with a charged particle trajectory. The rear (biased) grid is assumed to be located at a fixed distance along the normal to the front (grounded) grid surface. The electric field between the grids points from the rear toward the front grid. Particles incident from the right, moving along the negative "y" direction, feel no force until they pass through the front grid, apart from that due to grid fringe fields, which introduces some scattering. While between the two grids, these particles are under the influence of the applied electric field. Those particles with large enough velocity components along the field direction pass through the rear grid and are lost to the system. Those with smaller velocity along the field direction are turned by the field and directed toward the focal point, located at (0,f/2). We calculate the mirror geometry by determining the functional form of the front grid as y(x) under the following simplifying assumptions:
(a) The grid is considered to be locally flat. That is, the slope of the curve y(x) is considered to be constant over the region between a reflected ion's entry and exit points. This gives rise to a uniform electric field between the grids
where Vm is the commandable applied mirror voltage, D is the distance between the front and back grids, and n is the local normal to the front grid, pointing from back to front grids.
(b) The curve y(x) is considered to be near enough to parabolic that a perturbation from the parabolic case provides a solution to the derived differential equation.
Assumption (1) above, in combination with the focusing condition and simple particle dynamics within the electric field region, leads to the following differential equation for the curve y(x):
where R is the ratio of the nominally selected ion kinetic energy per charge to applied mirror voltage (E/(qVm)), and y' is the slope of the curve y(x) between the points of particle entry and exit. We have chosen a boundary condition such that, in the limit as x approaches zero, the slope y'(x) should approach zero in the same manner as the parabolic solution does. That is,
This nonlinear first order boundary value problem for y(x) is solved through a perturbation technique, using the second approximation above, as described by Pollock et al. [1994].
In the TIDE instrument, the following parameters were selected for mirror construction:
f = 56.7 mm, D = 5 mm, R = 1.60
The basis for these choices is as follows: In order that the intergrid spacing be large compared with the grid wire spacing of 0.036 mm (necessary for small fringe field scattering effects), D was chosen to be 14 times as large. In order that the focusing direction angular response of TIDE be a nominal 10û, and in view of the 10 mm aperture of the TOF section, the focal length was chosen as 10 mm x tan(10û) = 56.7 mm. R is chosen to provide optimal focusing at an energy "typical" of TIDE operation, which may extend from approximately one to two times the mirror potential.
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