4) When calculating the freeform surface profile, except the condition
k = 1, the position of the
k-th point (facet)
is geometrically determined by
and
(the geometrical construction way has been described in Ref. [
6]). The way to ascertain the normal vector
for the
k-th point (facet) by ECER method is illustrated in the dashed box in Fig. 4, which includes the numeric iterations. Preliminarily, taking
as
, then an
ECER1 can be calculated as a preliminary incident ray, and its emergent ray’s direction is determined by the source-to-target mapping, then a new normal vector
can be obtained by Snell’s Law. Then based on
, an
ECER2 can be newly calculated as the incident ray. If the emitting positions of
ECER1 and
ECER2 are not consistent with each other, substituting
ECER2 for
ECER1, and a new normal
would be determined again in the same way, and the old one would be substituted. The iteration would be finished until the emitting positions of
ECER1 and
ECER2 are consistent with each other, then the normal
would be the eventual
for the
k-th point (facet) of the freeform surface profile. In this process, the calculation only needs a small number of iterations, and the calculating time would usually be only hundreds milliseconds.