Mathematical and computational methods in photonics and phononics. (English) Zbl 1420.78001

Mathematical Surveys and Monographs 235. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-4800-4/hbk; 978-1-4704-4909-4/ebook). viii, 509 p. (2018).
This book provides a very through description of recent analytical and computational techniques and results in photonics and phonics. A large emphasis is placed on layer potential techniques. The book is divided into five main parts. The first part describes the main mathematical and computational tools for photonics and phononics. The starting point is a generalized argument principle and Rouché’s theorem (the Gohberg-Sigal theory). This serves as foundation for the rest of the analytical results that follow. Then, a very comprehensive description of layer potentials is provided. Applications are given to integral representation formulas for the full Maxwell system as well as for the Lamé system. Cavities and resonators are discussed in the last section of Part I. Next, in Part II, scattering by periodic structures is considered. In particular, tools for studying bandgap structures of such problems are provided via layer potential methods.
Part III addresses direct and inverse scattering problems for sub-wavelength resonators. It is shown that certain nanoparticles can provide an opportunity for super-resolution imaging. Again, layer potentials play a fundamental role in this part of the book. In this section, it is also discussed how to mathematically approach super resolution by resonant media. Green’s function plays a key role here.
Part IV involves cloaking of waves and metasurfaces. Near-cloaking is analyzed for the Helmholtz equation, the Maxwell system as well as for the elasticity system. Anomalous resonance is further addressed, again with the help of layer potentials.
Finally, the last part is devoted to sub-wavelength acoustic resonators. It is shown that bubbles in particular can be used as a building block for acoustic metamaterials. Minnaert resonances for bubbles is then addressed. Using these Minnaert resonances, it is shown that super focusing properties (as well as other properties) can be obtained with a system of bubbles.
The mathematical analysis throughout the book is quite detailed, and many references are provided in the bibliography. This analysis is paired nicely with a number of numerical results as well, and frequently tables are provided with results predicted from the theory as well as the actual numerical results. Another helpful aspect is that links to many of the associated Matlab codes are provided throughout the book. Overall, this book may be recommended for graduate students and researchers studying partial differential equations in electromagnetism, elasticity, and acoustics.


78-02 Research exposition (monographs, survey articles) pertaining to optics and electromagnetic theory
78-04 Software, source code, etc. for problems pertaining to optics and electromagnetic theory
78A45 Diffraction, scattering
78A46 Inverse problems (including inverse scattering) in optics and electromagnetic theory
76Q05 Hydro- and aero-acoustics
74J20 Wave scattering in solid mechanics
78M35 Asymptotic analysis in optics and electromagnetic theory
35B34 Resonance in context of PDEs
00A79 Physics
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