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Abstract Recent progress in microspherical superlens nanoscopy raises a fundamental question about the transition from super-resolution properties of mesoscale microspheres, which can provide a subwavelength resolution$$\sim \lambda /7$$ $\sim \lambda /7$, to macroscale ball lenses, for which the imaging quality degrades because of aberrations. To address this question, this work develops a theory describing the imaging by contact ball lenses with diameters$$30 $30<D/\lambda <4000$covering this transition range and for a broad range of refractive indices$$1.3<2.1$$ $1.3<n<2.1$. Starting from geometrical optics we subsequently proceed to an exact numerical solution of the Maxwell equations explaining virtual and real image formation as well as magnificationMand resolution near the critical index$$n\approx 2$$ $n\approx 2$which is of interest for applications demanding the highestMsuch as cellphone microscopy. The wave effects manifest themselves in a strong dependence of the image plane position and magnification on$$D/\lambda $$ $D/\lambda$, for which a simple analytical formula is derived. It is demonstrated that a subwavelength resolution is achievable at$$D/\lambda \lesssim 1400$$ $D/\lambda \lesssim 1400$. The theory explains the results of experimental contact-ball imaging. The understanding of the physical mechanisms of image formation revealed in this study creates a basis for developing applications of contact ball lenses in cellphone-based microscopy. 

The reconstruction of physical properties of a medium from boundary measurements, known as inverse scattering problems, presents significant challenges. The present study aims to validate a newly developed convexification method for a 3D coefficient inverse problem in the case of buried unknown objects in a sandbox, using experimental data collected by a microwave scattering facility at The University of North Carolina at Charlotte. Our study considers the formulation of a coupled quasilinear elliptic system based on multiple frequencies. The system can be solved by minimizing a weighted Tikhonov-like functional, which forms our convexification method. Theoretical results related to the convexification are also revisited in this work.