At any location between the optical concentrator and the target of the concentrator or behind the target point (focus) it is possible to define an orientation of a surface which yields the maximum illumination for this location.

Fig. 1 shows the distribution of the maximum achievable concentration of a solar beam with the parameters described by Buie et al. (2003) for a paraboloidal dish which has a diameter of four times the focal length, which means that the focal point lies in the same plane as the rim of the dish. This plane shall be called the focal plane for which in Fig. 1 z = 0. For almost all positions the maximum concentration can be achieved for a receiver surface element orientation normal to the radiation reflected from the dish. Due to end effects, this is only for such positions well below or above the focal plane. In the following, we will show that the isolines in Fig. 1 form a parabola. z

0.00 -0.01 -0.02 -0.03 -0.04 -0.05 -0.06 -0.07 -0.08 -0.09 -0.10

-0.11 A /’ /

Fig. 1 shows the position and orientation of several area elements with an concentration of about 400 suns. Each element is normal to the radiation reflected from the dish and therefore would be tangential to a sphere with a radius equal to the distance of the area element to the focus. It can be seen that the area elements with maximum equal concentration do not add up to a smooth, continuously differentiable surface. Therefore, it is not possible to find a continuously differentiable surface with uniform illumination, which achieves the maximum concentration at each point in space.

At each position in space the illumination on a surface can be reduced to any arbitrary concentration value below the maximum concentration at this position by changing the orientation of the surface. The relation between orientation of the surface and concentration can be described by following equation:

C( x, y) = Cmax( x)cos(y) Eq.(1)

with Cmax(x) as the maximum achievable concentration at the location X and у the angle between the normal of the orientated surface and the normal of the surface of maximum concentration.

Therefore, for a specific desired illumination level there must be a region that is defined by the fact that, along the perimeter of said region, the maximum illumination is equal to the desired illumination. The perimeters for specific concentration values are shown in Fig. 1. Within such a perimeter, at every point a surface can be constructed that will have at least one orientation with the wanted illumination. An almost symmetrical region lies behind the target.

If the concentrator or radiation source creates an illumination field which is continuously differentiable throughout the aforesaid region, there must logically exist at least one continuous surface which has the property of uniform illumination. This surface can be analytically or numerically calculated if the amount and direction of incoming radiation is known at every point.