Completeness and divergence-free behavior of the quasi-normal modes using causality principle

M Ismail Abdelrahman,B Gralak

doi:10.1364/osac.1.000340

Abstract

A fundamental feature of the quasi-normal modes (QNMs), which describe light interaction with open (leaky) systems like nanoparticles, lies in the question of the completeness of the QNMs representation and in the divergence of their field profile due to their leaky behavior and complex eigenfrequency. In this article, the QNMs expansion is obtained by taking into consideration the frequency dispersion and the causality principle. The derivation based on the complex analysis ensures the completeness of the QNMs expansion and prevents from any divergence of the field profile. The general derivation is tested in the case of a one-dimensional open resonator made of a homogeneous absorptive medium with frequency dispersion given by the Lorentz model. For a harmonic excitation, the result of the QNMs expansion perfectly matches the exact formula for the field distribution outside as well as inside the resonator.

Highlights

Light interaction with nanoparticles enables a wide range of unprecedented applications such as high-resolution spectroscopy, photothermal cancer therapy, optical tweezers, and light steering using metasurfaces [1,2,3,4]
Since the energy can escape towards the surroundings, such open systems are nonconservative and their behavior can be described by quasi-normal modes (QNMs) characterized by complex eigenvalues [5,6,7]
The QNMs expansion suffers from a longstanding limitation: the complex nature of the eigenvalues zq of open systems results in the divergence of the field outside the resonator, i.e. eizq |r | → ∞ when |r | → ∞. This divergence complicates the normalization of the QNMs since the energy of the modes is unbounded and, more importantly, the QNMs expansion appears to fail in describing the radiation outside the open resonator

Summary

Introduction

Light interaction with nanoparticles enables a wide range of unprecedented applications such as high-resolution spectroscopy, photothermal cancer therapy, optical tweezers, and light steering using metasurfaces [1,2,3,4]. The QNMs expansion suffers from a longstanding limitation: the complex nature of the eigenvalues zq of open systems results in the divergence of the field outside the resonator, i.e. eizq |r | → ∞ when |r | → ∞. This divergence complicates the normalization of the QNMs since the energy of the modes is unbounded and, more importantly, the QNMs expansion appears to fail in describing the radiation outside the open resonator. The QNMs expansion is derived taking into consideration the frequency dispersion and the causality principle This approach has recently demonstrated the validity of the modal expansion to represent the transient fields produced by a one-dimensional resonator [21]. A simple example is presented for a one-dimensional resonator made of a homogeneous absorptive dispersive medium, where the Green’s function of the resonator is expanded using QNMs and the obtained results are compared to the exact formulation

Causality of the Green’s function

Test case

Findings

Conclusion