Realization of translation group in optical design with deep nerual network under eikonal-energy mapping

Abstract

Nonimaging optical design aiming at energy control has a wide range of applications in optoelectronics. A nonimaging optical system is composed of a light source, optical components, and a target screen, and can be described by an equation named light taming equation(LTE). Given the light source and prescribed target spot, the required freeform surfaces of the optical component can be obtained by solving the LTE. If the light source profile does not change, the optical surface has to make some suitable morphs when the target spot translates on the screen, and the group theory can describe these morph operators well.<br>The basic LTE was established for a normal nonimaging optical system, which is to design an optical element for redirecting the lights from the source so that a prescribed light distribution is generated on a given target. A translation light taming equation(T-LTE) was derived for the case of only spot translating on the target screen, and an optical translation group(OTG) was introduced for describing all of the morph operators of the optical surface caused by light spot translation. There are multiple solutions for the same T-LTE, but the uniqueness of the T-LTE solution is necessary for OTG. Fortunately, the eikonal-energy(KE) mapping method can guarantee the uniqueness of the T-LTE solution, where K is the optical path length. The supporting quadric method(SQM) is one of the KE mapping methods when the nonimaging optical system has only one optical surface to be resolved. The LTE with SQM was deduced, and the OTG can be discussed in K-space. A deep neural network(DNN) was introduced to fit the KE mapping and spot translating operators to obtain the required optical surface. Taking the uniform square spot as an example, the SQM generated the sample data of spot translation to train DNN. The optical simulation results show that the error between the light distribution generated by the DNN and the standard uniform square spot is small, all within the order of 10<sup>-3</sup>, which indicates that the DNN and KE mapping method successfully realized the function of the OTG. There is a guiding significance for intelligent nonimaging optical design.

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