Since scalar imaging theory treats propagating waves as being scalar, it does not completely describe their physical properties. However, it plays an important role in the field of optical design and applications, such as semiconductor exposure projection optics. Therefore, we establish an accurate and meaningful scalar imaging theory, applicable even for high numerical aperture optics. We call this the consistent scalar imaging theory and consider the following three criteria for consistency. (1) When the wavelength of light becomes close to zero, the point spread function of wave optics coincides with the spot diagram (correspondence principle). (2) When considering point imaging, the total energy incident on the pupil equals that on the image. (3) The reciprocity theorem holds between object and image. These conditions are simultaneously fulfilled by introducing appropriate outgoing and incoming inclination factors. We emphasize that the pupil coordinates should be defined as the direction cosine of the ray.
Structured light beams with extraordinary properties have revolutionized a wide range of applications such as optical manipulation, high-resolution imaging, classical and quantum communications, and so on. These beams can be cleverly shaped with tailored amplitudes, phases, polarizations, and coherences. With advancements in this field, an increasing number of light beams have been uncovered and investigated, including non-diffracting Bessel beams, self-accelerating Airy beams, and abruptly autofocusing circular Airy beams. Pearcey beams, as emerging stars in the field of structured light beams, have garnered considerable attention due to their autofocusing performance and inversion effect. In this review, we provide an overview of the recent advancements in Pearcey beams. We start with the introduction of the prototypical Pearcey beams including the coherent versions that were first proposed and observed in 2012, and their partially coherent counterparts that were reported in 2020. Besides, we describe a virtual source method for generating non-paraxial Pearcey beams. Then, we focus on the diverse Pearcey variants and their remarkable features, including the ring Pearcey, spectrum-engineered Pearcey, chirped Pearcey, and Pearcey pulse, which enrich the Pearcey family. Furthermore, we discuss the practical applications with the Pearcey beams, particularly in particle manipulation and topological charge measurement. These studies serve as a catalyst for our exploration into the realm of higher-dimensional manipulation and broader applications of Pearcey light fields.
Analog-based all-optical computation has recently gained renewed interest due to the growing demand for data processing and the associated need for low-power, high-speed and compact computational platforms. While optical computing platforms based on bulky optical setups—such as 4f systems and pulse shapers—have been studied for decades, their large footprints have prevented their miniaturization and integration. Recently, a new all-optical computing platform has been proposed based on metamaterials—artificially structured materials with morphological features smaller than the radiation wavelength. By accurately tailoring their transfer function, it is possible to design metamaterials that perform computational tasks similar to their macroscopic counterparts, but within subwavelength footprints. In this chapter, we discuss the general working principle of these devices, their advantages and challenges, and the relevant figures of merit to be optimized. We also highlight specific examples and applications, providing a review of recent literature and an outlook on the field.
