We propose a spectral-averaging procedure that enables the computation of bandwidth-integrated local density of states (LDOS) from a single scattering calculation, and exploit it to investigate the minimum extinction achievable from dipolar sources over nonzero bandwidths in structured media. Structure-agnostic extinction bounds are derived, providing analytical insights into scaling laws and fundamental design tradeoffs with implications to bandwidth and material selection. We find that perfect LDOS suppression over a nonzero bandwidth $\mathrm{\Delta }\omega$ is impossible. Inspired by limits which predict nontrivial $\sqrt{\mathrm{\Delta }\omega }$ scaling in systems with material dissipation, we show that the pseudogap edge states of quasi-one-dimensional bullseye gratings can—by simultaneously minimizing material absorption and radiation—yield arbitrarily close to perfect LDOS suppression in the limit of vanishing bandwidth.

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• Light propagation, transmission & absorption
• Local density of states
• Photonics
• Spontaneous emission
• Bragg structures
• Optical fibers