Nanostructures with sizes smaller than or comparable to visible light strongly modify the decay rate of dipole emitters placed in their vicinity. Such modification is usually characterized using the local density of photonic states (LDOS), which quantifies the availability of photonic states at a certain position and frequency in the presence of a nanostructure. Here, we present a detailed analysis of the limits of this quantity through the study of a sum rule that bounds its spectral integral, taking into account both its radiative and nonradiative components. The sum rule studied here relates the integral over the spectrum of the LDOS at a certain point to the field induced by a static dipole placed at that same location. We confirm the validity of this sum rule and investigate its implications for the response of nanostructures by performing rigorous numerical calculations for a variety of systems, including nanospheres, nanodisks, and films, made of different metallic and dielectric materials, as well as graphene. Furthermore, we apply the sum rule to the cross density of photonic states (CDOS), a quantity that characterizes the spatial coherence of light in the presence of a nanostructure and determines, as well, the interaction between two dipole emitters located in its vicinity. We show how this result can be used as a guide to select the most favorable nanostructure geometries and materials to achieve strong values of the LDOS and the CDOS over desired parts of the spectrum, thus helping to engineer strong decay rates and coupling enhancements near nanostructures.

中文翻译：

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纳米结构附近光子态局部密度极限的分析

尺寸小于或等于可见光的纳米结构会极大地改变放置在其附近的偶极子发射器的衰减率。通常使用光子态的局部密度（LDOS）来表征这种修饰，该局部密度量化了在存在纳米结构的情况下在特定位置和频率下光子态的可用性。在这里，我们通过研究限制其光谱积分的求和规则，同时考虑到其辐射分量和非辐射分量，对这一数量的限制进行了详细分析。此处研究的求和规则将LDOS频谱在某个点上的积分与放置在同一位置的静态偶极子感应的场相关。我们通过对各种系统（包括由不同的金属和介电材料以及石墨烯制成的纳米球，纳米盘和薄膜）进行严格的数值计算，确认了该求和规则的有效性，并研究了其对纳米结构响应的影响。此外，我们将求和规则应用于光子态的交叉密度（CDOS），该量表征存在纳米结构时光的空间相干性，并确定位于其附近的两个偶极子发射器之间的相互作用。我们展示了如何将此结果用作选择最有利的纳米结构几何形状和材料的指南，以在所需光谱范围内实现LDOS和CDOS的强值，