Laser Focus World - December 2007 - Industry Reports (Page 63)

_Merit0.036 function _0.024

0.012

-1000 -507.5 - 15

Parameter 2 on object 2

0.060

0.048

ing the first angular moment of the detector data at (0, 0)—so that the beam of light is centered on-axis— and targeting the second angular moment to zero—so that the beam has a narrow distribution around zero—an even better merit function can be defined (see figure, bottom). Not only is this merit function smooth, but it has only a single minimum compared to the multiple minima of the previous functions, which greatly simplifies optimization.

_Merit0.036 function _0.024

0.012

-1000 -507.5

Parameter 2 on object 2

Merit ⁹function ₆

-1000

-507.5
Parameter 2 on object 2

In the design of a nonsequential system, the merit function curve is very choppy when just the base radius of curvature of a mirror is varied, so it is difficult to optimize (top). Pixel-interpolation routines improve the definition of the merit function by spreading the energy of a single ray to multiple pixels. The resulting merit function is smoother and indicates regions of minimized merit function and maximum on-axis brightness (center). Using the moment-of-illumination data, any slight change to the optical design that affects any ray is accounted for. The resulting merit function is far smoother (bottom).

location. The resulting merit function is far smoother, with gradients that point to the region with minimized merit function and maximum on-axis brightness (see figure, center). This is accomplished using the same number of rays as the previous example.

Even more improvement can be obtained by making use of the moment-of-illumination data. The first moment of a grid of data is its centroid, and the second is its effective width. By defin-

The moment-of-illumination
data is so useful because all the
rays that hit the detector contrib-
ute to the merit function. In the
first merit function, only those
that land within a cone of a few
^{- 15}degrees contribute. Pixel interpo-
lation allows rays over a slightly
larger angular range to contrib-
ute. But with the moment data, all
rays contribute. For example, if a
ray lands at 30°, a system change
during optimization can make
that ray now land at 29. 9°. This
change will contribute to the first
and second moment data. Even in
the absence of pixel interpolation,
^{- 15}the moment data is smooth and
has a well-defined gradient that
points in the direction of the de-
sired performance.
We have used the ZEMAX op-
tical-system design program to
perform an optimization over 22
variables in a free-form mirror.
Using the first merit function, we
improved the brightness of the
LED from 25 to 250 cd, a tenfold
increase. But the optimization
took many hours because of the
complexity and noisiness of the
22-variable merit function space.
Using the third merit function,
a better improvement was obtained
from the same starting point in just a
few minutes. The use of pixel interpo-
lation and moment-of-illumination al-
gorithms, such as those built into the
latest release by ZEMAX, dramatically
improves how optimization can be
used in a wide range of illumination
problems. ❏

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