Confinement Factors and Modal Volumes of Micro- and Nanocavities Invariant to Integration Regions

We present a convenient and self-consistent approach to calculate confinement factors and modal volumes of micro- and nanocavities, which are important for ultrasmall lasers and cavity quantum electrodynamics. This scheme does not rely on the numerical integrations related to optical fields and can avoid the indefinite dependence of physical quantities on integration regions. As a result of this built-in invariance to integration regions, the field representation of the confinement factor, in additional to its conventional expression, contains counter terms of volume and surface integrals, which cancel the effect of arbitrary integration volumes. This procedure is useful for small open cavities or those without sharp boundaries that distinguish cavity regions from free spaces. The uncertainty from different choices of integration regions can be thus eliminated.