We develop an analytical framework to derive upper bounds to light-matter interactions in the optical near field, where applications ranging from spontaneous-emission amplification to greaterthan-blackbody heat transfer show transformative potential. Our framework connects the classic complex-analytic properties of causal fields with newly developed energy-conservation principles, resulting in a new class of power-bandwidth limits. These limits demonstrate the possibility of ordersof-magnitude enhancement in near-field optical response with the right combination of material and geometry. At specific frequency and bandwidth combinations, the bounds can be closely approached by canonical plasmonic geometries, with the opportunity for new designs to emerge away from those frequency ranges. Embedded in the bounds is a material "figure of merit," which determines the maximum response of any material (metal/dielectric, bulk/2D, etc.), for any frequency and bandwidth. Our bounds on local density of states (LDOS) represent maximal spontaneous-emission enhancements, our bounds on cross density of states (CDOS) limit electromagnetic-field correlations, and our bounds on radiative heat transfer (RHT) represent the first such analytical rule, revealing fundamental limits relative to the classical Stefan-Boltzmann law. * hyungki.shim@yale.edu † owen.miller@yale.edu icant ongoing debate about whether a plasmonic or an all-dielectric approach is better, and in which scenarios 2D materials might be better than conventional bulk materials. Unlike all previous bounds and sum rules [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43], the material figure of merit we derive here enables general quantitative answers to these questions. In a frequencybandwidth phase space, we map out which materials are optimal and where the critical thresholds, from dielectric to plasmonic and bulk to 2D, occur. The techniques developed herein for LDOS, CDOS, and radiative heat transfer should be extensible to other near-field effects ranging from engineered Lamb shifts [44,45] and Förster resonance energy transfer [46,47] to nonlinear (Raman) or fluctuation-induced (Casimir) phenomena.Near-field electromagnetism, in which localized sources interact with scatterers separated by less than an optical wavelength, offers transformative potential for wideranging applications.Quantum emitters that only weakly couple to the radiation continuum can be dramatically amplified by near-field engineering: optical antennas offer prospects for imaging single molecules [13,14,48,49] or for designing nanoscale light-emitting diodes that are faster than lasers [50]. Nonlinear emitters such as Raman-active molecules [51,52] experience even more dramatic enhancements: surface-enhanced Raman scattering (SERS) [12][13][14], for example, scales with the square of the spontaneous-emission enhancement rate. Thermal emission can be accessed and controlled for the productive transfer of heat energy, at rates many orders of magnitude beyond classical...