Modal description of optical nanoresonators

Photonic and plasmonic resonators are dielectric or metallic optical devices that confine light at a scale smaller than the wavelength. They have many applications, especially in sensing,¹ nonlinear nano-optics,² or quantum optics.³ More generally, micro- and nanoresonators play an important role in the conversion of energy from localized fields to radiating waves. They are therefore studied by different communities interested in wave physics, including optics, microwave, and acoustics researchers.

With today's numerical tools, it is possible to design and analyze micro- and nanoresonators directly by solving Maxwell's equations. However, such numerical studies need to repeat many independent computations, as the polarization, incidence, and frequency of the excitation fields are varied. The numerical load may be heavy, and, above all, the computed data obtained with these brute-force calculations may still hide much knowledge about the physical mechanisms at play. To bypass such difficulties, it is tempting to rely on mode theory, which allows us to represent the resonator response as a weighted sum of mode contributions. Each mode is computed only once, because it is independent of the excitation.

Eigenstates of a system are often a particularly useful type of mode to consider. Introducing such eigenstates to study a system's response is a very common framework in many areas of fundamental and applied physics, as one can lean on some important mathematical properties of these modes, such as completeness and orthogonality. The latter has a physical meaning of fundamental importance: once a mode is excited, it cannot transfer its energy to another mode. Such modes are usually referred to as normal modes in the literature.

However, normal-mode theories rely on the energy dissipation in the system being small enough for the theory to be accurate. This key assumption may remain approximately valid for microresonators, but completely breaks down for metallic nanoresonators that confine light at a deep-subwavelength scale, for which radiative leakage and absorption losses are generally large. Studying such systems with the usual normal-mode theories does not give satisfying predictions.

The issue of defining a proper modal theory for dissipative systems (in particular open systems that lose energy by radiation) has been emphasized since the early days of quantum mechanics,⁵ but until now it has been solved only for one-dimensional systems by introducing the concept of quasinormal modes (QNMs), for example, in the context of optical cavities.⁶ We hope that our recent theoretical progress⁷ will permit the concept of QNMs to be extended to virtually any three-dimensional system. This theoretical advance provides solid ground for development of a modal formalism of practical importance in nano-optics. Recently, we built a user-friendly method to compute QNMs easily,⁴ which can be further used to analytically calculate many important physical quantities of nanoresonators, such as the absorption or scattering cross-sections, or the Purcell factor (the enhancement of spontaneous emission) of a quantum emitter located near the resonator.

In our formalism, the first step is to find the QNMs of the resonator and normalize them. We have developed a simple iterative method for this step that may be implemented with any numerical tool, such as the commercial software COMSOL Multiphysics, which we used in our recent work.⁴ This flexibility renders the approach accessible to the greatest number of researchers from different communities. Material dispersion can be taken into account, and normalized QNMs are calculated in a few minutes for 3D systems of size ∼λ³, where λ is the relevant wavelength.

Figure 1. (a) Plot of the intensity distribution, |E_z|, for a normalized quasinormal mode of a 30nm-diameter and 100nm-long cylindrical gold nanorod (dashed line). (b) Purcell factor (enhancement of spontaneous emission rate) of a dipole located on the rod axis 10nm from the rod and polarized along its axis. (c) Extinction cross-section of the rod under illumination by a plane wave polarized along the rod axis. In (b) and (c), black circles are fully vectorial numerical results obtained with the COMSOL Multiphysics software. The red solid curves represent the predictions of our analytical modal method.⁴

In the second step, the resonator response is obtained analytically for any frequency, polarization, or location of the exciting source. As an example, Figure 1(a) shows the intensity distribution of the fundamental electric-dipole-like QNM of a gold nanorod. The solid red curves in Figure 1(b) and (c) show, respectively, the normalized decay rate (Purcell factor) of an electric dipole located on the rod axis and the extinction cross-section of the nanorod. Note the very good agreement between the analytical modal predictions and the numerical data (black circles) obtained with fully vectorial numerical computations.

In summary, we have introduced a method based on quasinormal modes that enables analytical calculations of many important properties of nanoresonators. Three-dimensional open resonators with any complex geometry, including those exhibiting strong absorption and radiative leakage, can now be described by the knowledge of a few modes, whose excitation coefficients are expressed analytically. The results agree quantitatively with numerical calculations, and one can now tackle the problem of quasinormal mode design, so as to exploit the best of the fascinating properties of nano-optics. In particular, the next phase of our work will be to consider the coupling between a QNM and a realistic source of finite spectral width.