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In this dissertation, we address the burgeoning fields of diffractive optics, metals-optics and plasmonics, and computational inverse problems in the engineering design of electromagnetic structures. We focus on the application of the optical nano-focusing system that will enable Heat-Assisted Magnetic Recording (HAMR), a higher density magnetic recording technology that will fulfill the exploding worldwide demand of digital data storage. The heart of HAMR is a system that focuses light to a nano- sub-diffraction-limit spot with an extremely high power density via an optical antenna. We approach this engineering problem by first discussing the fundamental limits of nano-focusing and the material limits for metal-optics and plasmonics. Then, we use efficient gradient-based optimization algorithms to computationally design shapes of 3D nanostructures that outperform human designs on the basis of mass-market product requirements.In 2014, the world manufactured ~1 zettabyte (ZB), ie. 1 Billion terabytes (TBs), of data storage devices, including ~560 million magnetic hard disk drives (HDDs) [1]. Global demand of storage will likely increase by 10x in the next 5-10 years, and manufacturing capacity cannot keep up with demand alone. We discuss the state-of-art HDD and why industry invented Heat- Assisted Magnetic Recording (HAMR) [2][3] to overcome the data density limitations. HAMR leverages the temperature sensitivity of magnets, in which the coercivity suddenly and non- linearly falls at the Curie temperature. Data recording to high-density hard disks can be achieved by locally heating one bit of information while co-applying a magnetic field.The heating can be achieved by focusing 100 μW of light to a ~30nm diameter spot on the hard disk. This is an enormous light intensity, roughly ~100,000,000x the intensity of sunlight on the earth’s surface! This power density is ~1,000x the output of gold-coated tapered optical fibers used in Near-field Scanning Optical Microscopes (NSOM), which is the incumbent technology allowing the focus of light to the nano-scale. Even in these lower power NSOM probe tips, optical self-heating and deformation of the nano- gold tips are significant reliability and performance bottlenecks [4][5]. Hence, the design and manufacture of the higher power optical nano-focusing system for HAMR must overcome great engineering challenges in optical and thermal performance.There has been much debate about alternative materials for metal-optics and plasmonics to cure the current plague of optical loss and thermal reliability in this burgeoning field. We clear the air. For an application like HAMR, where intense self-heating occurs, refractory metals and metals nitrides with high melting points but low optical and thermal conductivities are inferior to noble metals. This conclusion is contradictory to several claims and may be counter-intuitive to some, but the analysis is simple, evident and relevant to any engineer working on metal-optics and plasmonics. Indeed, the best metals for DC and RF electronics are also the best at optical frequencies.We also argue that the geometric design of electromagnetic structures (especially sub- wavelength devices) is too cumbersome for human designers, because the wave nature of light necessitates that this inverse problem be non-convex and non-linear. When the computation for one forward simulation is extremely demanding (hours on a high-performance computing cluster), typical designers constrain themselves to only 2 or 3 degrees of freedom. We attack the inverse electromagnetic design problem using gradient-based optimization after leveraging the adjoint-method to efficiently calculate the gradient (ie. the sensitivity) of an objective function with respect to thousands to millions of parameters. This approach results in creative computational designs of electromagnetic structures that human designers could not have conceived yet yield better optical performance.After gaining key insights from the fundamental limits and building our Inverse Electromagnetic Design software, we finally attempt to solve the challenges in enabling HAMR and the future supply of digital data storage hardware. In 2014, the hard disk industry spent ~$200 million dollars in R&D but poor optical and thermal performance of the metallic nano-transducer continues to prevent commercial HAMR product. Via our design process, we successfully computationally-generated designs for the nano-focusing system that meets specifications for higher data density, lower adjacent track interference, lower laser power requirements and, most notably, lower self-heating of the crucial metallic nano-antenna. We believe that computational design will be a crucial component in commercial HAMR as well as many other commercially significant applications of micro- and nano- optics.If successful in commercializing HAMR, the hard disk industry may sell 1 billion HDDs per year by 2025, with an average of 6 semiconductor diode lasers and 6 optical chips per drive. The key players will become the largest manufacturers of integrated optical chips and nano- antennas in the world. This industry will perform millions of single-mode laser alignments per day. All academic and industrial players in micro- and nano- optics should excitingly watch what Seagate, Western Digital, HGST and TDK accomplish in the next 5-10 years.