Exact mode volume and Purcell factor of open optical systems

Abstract

The Purcell factor quantifies the change of the radiative decay of a dipole in an electromagnetic environment relative to free space. Designing this factor is at the heart of photonics technology, striving to develop ever smaller or less lossy optical resonators. The Purcell factor can be expressed using the electromagnetic eigenmodes of the resonators, introducing the notion of a mode volume for each mode. This approach allows an analytic treatment, reducing the Purcell factor and other observables to sums over eigenmode resonances. Calculating the mode volumes requires a correct normalization of the modes. We introduce an exact normalization of modes, not relying on perfectly matched layers. We present an analytic theory of the Purcell effect based on this exact mode normalization and the resulting effective mode volume. We use a homogeneous dielectric sphere in vacuum, which is analytically solvable, to exemplify these findings. We furthermore verify the applicability of the normalization to numerically determined modes of a finite dielectric cylinder. DOI: 10.1103/PhysRevB.94.235438 In his short communication [1] published in 1946, Purcell introduced a factor of enhancement of the spontaneous emission rate of a dipole of frequency ω resonantly coupled to a mode in an optical resonator, which is now known as the Purcell factor (PF). He estimated this factor aswith the speed of light c, the quality factor Q n of the optical mode n, and its effective volume V n , the latter being evaluated as simply the volume of the resonator. This rough estimate of V n has subsequently been refined [2,3] towhere r d is the position of the dipole and e the unit vector of its polarization. In this expression, the electric field of the modewhere ε(r) is the permittivity of the resonator. The integration is performed over the "quantization volume" V. However, for an open system this volume is not defined, and simply extending V over the entire space leads to a diverging normalization integral since eigenmodes of an open system grow exponentially outside of the system due to their leakage. This issue was mostly ignored in the literature and patched by phenomenologically choosing a finite integration volume. Such an approach can result in relatively small errors when dealing with modes of high Q n , as we will see later. However, the fundamental problem of calculating the exact mode normalization and thus of the mode volume remained.Recently, a solution to this problem has been suggested. Kristensen et al. [4,5] have used the normalization which was introduced by Leung et al. [6] for one-dimensional (1D) optical systems and later applied [7] to three dimensions. In this approach, the volume integral in Eq. (3) is complemented by a surface term and the limit of infinite volume V is taken:where ω n is the complex eigenfrequency of the mode and S V is the boundary of V. It was found [4] that for high-Q modes, the surface term was leading to an approximately converging value of the normalization with increasing V, for the limited ...

Mode Volume, Energy Transfer, and Spaser Threshold in Plasmonic Systems with Gain

“…Equation (1) is similar to the standard laser condition [19] that determines the threshold value of N 21 , but with the cavity mode quality factor and volume replaced by their plasmon counterparts in metaldielectric system characterized by dispersive dielectric function ε(ω, r). While the plasmon quality factor Q is well-defined in terms of the metal dielectric function ε(ω) = ε ′ (ω) + iε ′′ (ω), there is an active debate on mode volume definition in plasmonic systems [25][26][27][28][29][30][31][32][33][34][35]. Since QEs are usually distributed outside the plasmonic structure, the standard expression for cavity mode volume, dV ε(r)|E(r)| 2 /max[ε(r)|E(r)| 2 ], where E(r) is the mode electric field, is ill-defined for open systems [27,28,31].…”

How to calculate the pole expansion of the optical scattering matrix from the resonant states

“…It should be noted that this formulation of the normalization is slightly different than in most of our previous works on the resonant state expansion [10,[15][16][17][18][19][20][21]25,27,28], which is valid also for magnetic and bianisotropic materials [26]. This formulation can be reduced to our previous results for nonmagnetic materials that are solely described by the electric field, the electric permittivity, and the electric current as a special case.…”

“…(15), it is possible to calculate, for a given incident field, the expansion coefficients of the outgoing field as the elements of the vector O in Eq. (19):…”

“…When normalizing the resonant states appropriately [10,[15][16][17][18][19][20][21]26,27], they can be used together with possible cuts to expand the Green's dyadic with outgoing boundary conditions as follows:…”

“…Several approaches have been suggested for the normalization of resonant states [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26]. This includes approximate formulations for highquality modes [23][24][25], the utilization of perfectly matched layers [14] or, equivalently, complex coordinates [11] in the exterior of the system, as well as numerical approaches [20,22].…”