## View abstract

### Session S02 - Diverse Aspects of Elliptic PDEs and Related Problems

Thursday, July 15, 18:00 ~ 18:30 UTC-3

## Bloch waves in 3-dimensional high-contrast photonic crystals

### Robert Viator

#### Swarthmore College, United States of America   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak111e5ee6e35bb8a19745da0a03cc6d97').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy111e5ee6e35bb8a19745da0a03cc6d97 = 'rv&#105;&#97;t&#111;r1' + '&#64;'; addy111e5ee6e35bb8a19745da0a03cc6d97 = addy111e5ee6e35bb8a19745da0a03cc6d97 + 'sw&#97;rthm&#111;r&#101;' + '&#46;' + '&#101;d&#117;'; var addy_text111e5ee6e35bb8a19745da0a03cc6d97 = 'rv&#105;&#97;t&#111;r1' + '&#64;' + 'sw&#97;rthm&#111;r&#101;' + '&#46;' + '&#101;d&#117;';document.getElementById('cloak111e5ee6e35bb8a19745da0a03cc6d97').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy111e5ee6e35bb8a19745da0a03cc6d97 + '\'>'+addy_text111e5ee6e35bb8a19745da0a03cc6d97+'<\/a>';

We investigate the Bloch eigenvalues of a 3-dimensional high-contrast photonic crystal. The Bloch eigenvalues (for a fixed quasi-momentum) can be expanded in a power series in the material contrast parameter $k$ about $k=\infty$. We achieve this power series, together with a radius of convergence, by decomposing an appropriate vector-valued Sobolev space into three mutually orthogonal subspaces which are curl-free in certain subdomains of the period cell. We will also identify the limit spectral problem as contrast becomes large, and (time permitting) we will describe a class of crystal geometries which permit the power series structure of Bloch eigenvalues described above.

Joint work with Silvia Jimenez (Colgate University) and Robert Lipton (Louisiana State University).