New types of acoustic and photonic crystals - Mathematically rigrous quantitave estimates for location and size of bandgaps.

A photonic (or phononic) crystal is a periodic lattice of inclusions surrounded by a connected phase. The dielectric (or acoustic) properties of the inclusions are different than that of the connected phase and the difference between the dielectric (or accoustic) properties (contrast) can be large. Here the wavelength is on the order of the period spacing and the physical mechanism for creating gaps in the frequency spectrum is the multiple scattering of the waves within the crystal. We focus on photonic solids and the inclusion and surrounding phase are treated as frequency independent dielectrics. The photonic band gap phenomena gives rise to structural coloration in organisms and is responsible for the iridescent blue of butterfly wings.

A complete band gap is a range of frequencies for which there is no wave propagation in the solid, i.e., Bloch waves of any wavelength or direction can not propagate through the crystal at these frequencies. Defects then can be introduced that locally break the symmetry of the crystal and support localized waves on the defect at frequencies inside the band gap. These localized modes enjoy minimal attenuation and this forms the bases of holey optical fibers used to transmit signals for long distances.

Unfortunately the current state of the art (Ammari, Kang, and Lee, 2009), (Ammari, Kang, et al,, 2003), (Bouchitt'e, Bourel, and Felbacq, 2017), (Figotin and Kuchment, 1994), (Figotin and Kuchment, 1996), (Figotin and Kuchment, 1998), (Hempel and Lienau, 2000) is restricted to the asymptotic theory of band gaps at infinite contrast. In this project we exploit structural resonances associated with the Neumann Poincare operator to develop new techniques for complex operator valued functions which deliver explicit formulas for band gaps at finite contrast. The approach developed in this research is distinct from earlier efforts and provides for the first time mathematically rigorous and explicit formulas for the size of band gaps and pass bands given in terms of contrast, shape and configuration of scatters, and lattice parameters. We apply this theory to the systematic design of pass bands and band gaps in terms of these parameters, see, (Lipton and Viator, 2017a), and (Lipton and Viator, 2017b).

The technical challenge standing in the way of explicit formulas is the lack of the appropriate explicit representation formulas for solution operators associated with wave propagation in photonic crystals. Here we allow the contrast between dielectric properties to be any complex number $z\in\mathbb{C}$. We find the subset of $S\subset \mathbb{C}$ for which the solution operator exists and derive explicit formulas for the solution operators as functions of $z$. The explicit solution operators are rational operator valued functions of $z$ and are given in terms of the spectra of the ``quasi-static'' Neumann Poincare operator supported on the surface of the inclusions. With these formulas in hand we can apply analytic perturbation theory for spectra as in (Kato, 1996) together with a tedious calculation to obtain explicit formulas for the size of location of band gaps in terms of the geometry of inclusions inside the period and lattic parameters. Explicit examples are calculated for dispersions of disks inside a period cell. These results have appeared in (Lipton and Viator, 2017a), and (Lipton and Viator, 2017b). We are currently extending our approach to fully three dimensional photonic crystal lattices.

The work discussed here has appeared in Bloch Waves in Crystals and Periodic High Contrast Media. ESAIM M2AN. June 16, 2016, and Creating Band Gaps in Periodic Media. SIAM Multiscale Modeling and Simulation, 15, 2017, pp. 1612-1650.
This work is supported by NSF Grant DMS-1211066 and Air Force Research Office through award FA9950-12-1-0489

References

1. H. Ammari, H. Kang, and H. Lee. Asymptotic analysis of high-contrast phononic crystals and a criterion for the band gap opening. Arch. Rational Mech. Anal., 193:679 714, 2009..
2. H. Ammari, H. Kang, S. Soussi, and H. Zribi. Layer potential techniques in spectral analysis. part 2: Sensitivity analysis of spectral properties of high contrast band-gap materials. Multiscale Model. Simul., 5:646 663, 2006.
3. G. Bouchitt'e, C. Bourel, and D. Felbacq. Homogenization near resonance and artificial magnetism in 3d dielectric materials. Archive for Rational Mechanics and Analysis, September 2017, Volume 225, Issue 3, pp 1233-1277.
4. A. Figotin and P. Kuchment. Band-gap structure of the spectrum of periodic maxwell operators. Journal of Statistical Physics, 74:447-455, 1994.
5. A. Figotin and P. Kuchment. Band-gap structure of spectra of periodic dielectric and acoustic media. 1. scalar model. SIAM J. Appl. Math., 56:68-88, 1996.
6. A. Figotin and P. Kuchment. Spectral properties of classical waves in high-contrast periodic media. SIAM J. Appl. Math., 58:683-702, 1998.
7. R. Hempel and K. Lienau. Spectral properties of periodic media in the large coupling limit. Communications in Partial Differential Equations, 25:1445-1470, 2000.
8. T. Kato. Perturbation Theory for Linear Operators. Springer, Berlin Heidelberg, Germany, 1995.
9. R. Lipton and R. Viator Jr. Bloch waves in crystals and periodic high contrast media. ESAIM Mathematical Modeling and Numerical Analysis, 51:889-918, 2017a.
10. R. Lipton and R. Viator Jr. Creating band gaps in periodic media. Multiscale Model. Simul., 15(4), 1612-1650, 2017b.