Power--bandwidth limitations of an optical resonance

We present shape-independent upper limits to the power--bandwidth product for a single resonance in an optical scatterer, with the bound depending only on the material susceptibility. We show that quasistatic metallic scatterers can nearly reach the limits, and we apply our approach to the problem of designing $N$ independent, subwavelength scatterers to achieve flat, broadband response even if they individually exhibit narrow resonant peaks.