Energy efficiency of superoscillation

Through superoscillation arbitrarily small optical spots can be generated in the far field without evanescent fields. This can potentially revolutionize many photonic technologies, such as superresolution imaging, lithography, data storage, and optical trapping. However, it is generally believed that superoscillation comes with a price of extremely low energy efficiency, making it impractical for applications. Here we provide a detailed investigation of the energy efficiency of superoscillatory functions. We model a wide array of superoscillatory functions with prolate spheroidal wave functions (PSWF). The results show that extremely low energy efficiency is caused by strict modelling constraints and the intrinsic properties of PSWFs, rather than the fundamental limitation imposed by superoscillation. Energy efficiency can be significantly improved by relaxing the constraints. Using an optimization algorithm based on three PSWFs and setting the constraint conditions to two prescribed points, we show that the energy efficiency (the portion of energy contained within a superoscillatory spot) for a spot of size of 0.1λ can reach 10^-3, which is ∼140 orders-of-magnitude improvement from the strict modelling results and is practically viable in many applications. We also establish a general relationship between energy efficiency and the field-of-view/size of a superoscillatory spot. It is revealed that generally energy efficiency decreases exponentially with field-of-view for a given spot size, and for a given field-of-view the logarithmic scale of energy efficiency decreases with reduced spot size following a double exponential law.