Fly-eyes illumination analysis

The Fly-eyes element has been widely applied to acquire uniform illumination in microscope, printing, projection, and so on. With large number of fly-eyes pieces and specified aperture shape, illumination of high efficiency, high uniformity, and specified area shape can be acquired. In this paper, mathematical expressions for fly-eyes illumination are derived for free form incoming light distribution. With these expressions, the reason why uniform illumination can be acquired with fly-eyes is given, and the defects of fly-eyes illumination is also discovered. According to traditional design experience, the illumination uniformity acquired by fly-eyes illumination will be increased with the increment of fly-eyes number. But with the derived expressions, it can be found that there are some limitations for this experience. For some odd symmetrical illumination cases, uniform illumination can not be acquired even with large number of fly-eyes pieces. This feature of fly-eyes illumination should be pay good attentions when dealing with non-symmetrical illumination cases. Another feature of fly-eyes illumination is that discrete angular distribution instead of continuous angular distribution of output light is acquired, and this discrete angular pattern varies with numerical aperture of the incoming light. This may be a bad feature for microscope with fly-eyes illumination, but this is a good feature for projection. With these derived mathematical expressions, guidelines for the application of fly-eyes illumination can be found and a deep view into the fly-eyes illumination can be acquired.