Complete analysis of the reduced aberration Luneburg lens

The modified Luneburg lens, otherwise known as the Gutman lens, is a gradient-index spherical lens whose focus occurs inside the lens in the stigmatic case (aberration-free). When the focus is set to lie outside the lens, spherical aberration is introduced. Recently, a scaled version of Gutman's equation has been proposed to minimize this aberration and the corresponding lens was referred to as the reduced aberration acoustic Luneburg lens. The minimization process was based on numerical optimization methods. The term "acoustic" is used since it was originally designed with dimensions suitable for propagating acoustic waves. However, this lens is general enough to be analyzed in terms of its maximum radius, allowing it to be scaled to any size and capable of propagating any wavelength. In other words, this lens can be more broadly referred to as the reduced aberration Luneburg lens (RALL). In this paper, we present the complete formalism of the RALL, conducting a detailed and comprehensive analysis of its performance. Establishing the physical characteristics of the proposed scaled Gutman's equation, we perform an aberration analysis using ray tracing based on the Fermat's invariants. Our analysis reveals that the RALL is essentially a Gutman lens with a larger radius. The reduction of spherical aberration is due to the fact that the rays impinge on a trimmed anterior effective area of the lens. In any optical system it is common that the major contribution for spherical aberration comes from rays at the periphery.