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

Abstract

<sec>Nonimaging optical design aiming at energy control has wide 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 will make some suitable morphs when the target spot translates on the screen, and these morph operators can well be described by the group theory.</sec><sec>The basic LTE is established for a normal nonimaging optical system, which is to design an optical element for redirecting the light from the source so that a prescribed light distribution is generated on a given target. A translation light taming equation(T-LTE) is derived for the case of only spot translating on the target screen, and an optical translation group(OTG) is 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 is deduced, and the OTG can be discussed in K-space. A deep neural network(DNN) is introduced to fit the KE mapping and spot translating operators to obtain the required optical surface. Taking the uniform square spot for example, the SQM generates the sample data of spot translation to train the 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 on the order of 10⁻³, which indicates that the DNN and KE mapping method have successfully realized the function of the OTG. The results are of guiding significance in implementing the intelligent nonimaging optical design.</sec>