# Broadband Performance of Lenses Designed with Quasi-Conformal Transformation Optics

Jogender Nagar, Sawyer D. Campbell, Donovan E. Brocker, Xiande Wang, Kenneth L. Morgan, and Douglas H. Werner

*Department of Electrical Engineering, Penn State University,*

*University Park, PA 16802, USA *
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## Introduction

Gradient-index (GRIN) lenses have become an increasingly popular topic in modern optical system design. GRIN lenses possess spatially varying indices of refraction throughout the bulk material of the lens. By choosing a proper GRIN distribution, the power of a lens can be increased. It has also been shown that carefully selected GRIN distributions within a lens can also reduce aberrations including, but not limited to, chromatic and spherical aberrations [39, 16]. When applied to entire optical systems, GRIN technology could potentially lead to smaller and lighter optical devices by reducing the

*Broadband Metamaterials in Electromagnetics: Technology and Applications *Edited by Douglas H. Werner

Copyright © 2017 Pan Stanford Publishing Pte. Ltd.

ISBN 978-981-4745-68-0 (Hardcover), 978-1-315-36443-8 (eBook) www.panstanford.com number of optical components needed to reach diffraction-limited performance. While GRIN lenses are currently hailed as an advanced and modern technology, the concept of GRIN lenses has actually existed for over a century [40, 61, 62]. One of the earliest GRIN lens examples was a spherical fish-eye lens proposed by J. C. Maxwell in 1854 [40], which is similar to the more modern Luneburg lens [61]. Interestingly, GRIN lenses have been found to exist naturally in the eyes of many living creatures. For example, the nucleus of the lens in the human eye has an index distribution with a An of approximately 0.03 [43, 19, 22] and whose origin is attributed to varying protein densities within the lens [43]. For the case of humans, the curvature and the GRIN of the eye aid in the focusing of light onto the retina so that humans can see a focused image of their surroundings. In order to make use of the potential of GRIN in human-made lenses, the index distribution of the GRIN must be carefully designed to obtain the desired focusing performance. When including traditional lens parameters such as radii of curvature, aspherical curvature terms, thickness, bandwidth, field of view (FOV), and other constraints of the system, it becomes apparent that a sophisticated methodology is needed to effectively design GRIN lenses. Transformation optics (TO) or transformation electromagnetics (TEM) is a method of designing electromagnetic and optical devices that exploit the form-invariance of Maxwells equations under coordinate transformations [54]. The method involves mapping a given geometric space onto a second, more desired geometric space via coordinate transformations, then realizing this geometric transformation through spatially varying material parameters. In order to physically implement the design, exotic materials are often required, including inhomogeneous and anisotropic materials. The necessary refractive indices can be zero or even negative, which require the use of certain types of metamaterials to physically realize in the design. Unfortunately, these metamaterials can be very difficult to manufacture and are often very narrowband and lossy [34]. However, if the transformation is a conformal map, the medium parameters will be strictly isotropic and inhomogeneous (graded-index), thus improving the design manufacturing feasibility. The application of a conformal mapping will yield an entirely dielectric material if we only consider a single polarization, resulting in potentially broadband designs with low loss [6]. However, it is impossible to analytically define a conformal map for any general transformation. A numerical method called quasi-conformal transformation optics (qTO) is a generalization of conformal mapping that uses grid generation tools to create maps between two bounded regions by solving Laplace’s equations [31]. Without any specified boundary conditions, there are an infinite number of transformations that are not strictly conformal. However, through a judicious selection of boundary conditions [6, 31], we can generate a map with negligible anisotropy, which can be accurately realized through an inhomogeneous isotropic all-dielectric medium [27]. Some applications of qTO include electromagnetic cloaks, flat focusing lenses, virtual conformal arrays, and right-angle bends [28].

This chapter will focus on broadband lens designs enabled or inspired by qTO by exploring a wide variety of applications. Here, the term “broadband” is used to describe lenses that possess desirable performance over large FOVs and/or wide frequency ranges. First, the mathematics of TO and the numerical algorithm of qTO will be described. Refractive index profiles engineered through qTO will be referred to as qTO-derived inhomogeneous metamaterials, while those developed using TO will be called anisotropic metamaterials. Next, a series of broadband qTO-derived inhomogeneous metamaterials created from transforming the geometry of classical designs, including the Luneburg and Fresnel lenses, will be presented. Then, qTO will be used to transform wavefronts and design a multibeam lens antenna. One unfortunate downside of these designs is that there could be significant reflections due to discontinuities at the interfaces. To deal with this, the next section will describe a systematic design procedure using qTO for anti-reflective coatings. Another downside of these designs is that there could be significant dispersion in the optical regime and the traditional qTO procedure cannot be directly used to correct for this. An alternative method that can correct for dispersion, called wavefront matching (WFM), will then be described. While this method is extremely general, it relies on the performance of a numerical optimizer to be effective. The final section will discuss the effects of material dispersion on the performance of GRIN lenses and a set of classical as well as qTO- inspired corrections will be presented.