The problem is intrinsically hierarchic, allowing the global optimisation to be broken down into three partial optimisations. The outermost lamp pair sees the smallest silhouette of the susceptor, so that the focussing of as much light as possible is most complicated here. The optimisation of these lamps should not be constrained by the properties of the neighbouring lamps. To start, the initial parabolic design is kept for the two inner pairs of lamps, their filaments are turned off, and only the outermost reflectors are optimised with the right lamp burning at 1000W. 5000 or 10000 rays are used. The only relevant constraint is that the reflector surface must not touch the reactor tube.

As the homogeneity of the illumination does not matter, and only the total power transmistted matters, the merit function is quite simple. Please see the file optimized result.zmx in the zip archive at the end of this article, and open the merit fuction editor. First, some NT** operands constain the total tilt so that the lamp does not hit the reactor wall or the susceptor. Then detectors are cleared, a non-sequential ray-trace is performed, and then the total power on all detectors is targeted to 1000W, this being the lamp power.

I fixed the width of the ellipse manually and let the remaining parameters (position, radius of curvature, conic constant and r^2 aspheric coefficient) be optimised by ZEMAX, using default settings. Thus I nested an inner, automatic optimisation in an outer, manual optimisation where I increased the width of the reflector until the ellipse scratched the reactor tube. At this point, the outermost lamp pair was done.

These reflectors were now fixed and their SOURCE FILAMENTs switched off, and the same procedure as above was conducted with the adjacent lamps. Importantly,  the outer sidewalls of the outer lamps serve as prolongation to the elliptic surface which allows the ellipse to be intentionally tilted away from the direct incident. By this, the angular closure of this reflecting element could be raised in total, yielding a maximum of power transmission. This, honestly unexpected, solution was found by ZEMAX, not me! Again, the final design was fixed, the lamp switched off, and the innermost lamps were optimised straight-forwardly.

The toroidal surfaces' surface function was transformed into CNC milling data files using Mathematica (http://www.wolfram.com/products/mathematica/index.html), and that was it!