Understanding localized surface plasmon resonance with propagative surface plasmon polaritons in optical nanogap antennas
(I) Schematic of the SPP model of QNMs for the nanogap optical antenna composed of two gold nano-wire arms separated by a nano-gap. �, τ and r are the SPP scattering coefficients used in the model. (II) Field of QNMs (for arm length L=0.6 μm) obtained with the numerical aperiodic Fourier modal method (a-FMM) and with the SPP model.
Optical nanogap antennas formed by strongly coupled metallic nanowires are the analog of radio-wave antennas and can act as efficient receivers for collecting far-field illuminations into the nanogap, or reciprocally, act as efficient transmitters for enhancing and directing the radiation of optical emitters such as molecules and quantum dots. The antennas have wide applications ranging from enhanced spectra sensing, near field microscopy, nanocatalysis to photodetection and light emission.
For the design of the nanogap antenna devices, the formalism of quasi-normal mode (QNM), also called localized surface plasmon resonance (LSPR), is a powerful tool since it can provide an analytical description of the frequency response of the nanogap antennas. For instance, from the frequency-analyticity one can conclude that a resonant excitation of field (or, of LSPR) will happen if the excitation frequency matches the real part of the complex eigen-frequencies of QNMs. However, at present the QNMs of antennas are commonly solved via numerical solvers of Maxwell’s equations without an analytical (or, physically intuitive) description, which blocks a further understanding of QNMs and thus an efficient design. On the other hand, the Fabry-Perot models of propagative surface plasmon polartons (SPPs) have been developed, which reveal such an intuitive picture that the SPPs excited by external illuminations are bouncing back and forth along antenna arms, and are hopping, reflected or scattered at the nanogap or terminations of the antenna, thus forming the radiated or scattered field. But such models cannot provide an analytical description of the frequency response of the antenna.
To utilize both the intuitive force of the Fabry-Perot model and the frequency-analyticity of the QNM formalism in understanding the antenna resonance, the research group led by Prof. Haitao Liu at the Institute of Modern Optics, Nankai University, proposed a semi-analytical SPP model of QNMs for nanogap antennas. This work was published in Photonics Research, Vol. 4, Issue 6, 2016 (H. Jia, et al., Understanding localized surface plasmon resonance with propagative surface plasmon polaritons in optical nanogap antennas).
In this work, the complex eigen-frequencies as well as the field of QNMs can be accurately predicted by the SPP model, which sets a solid electromagnetic foundation for the intuitive picture that the LSPR (i.e. QNM) of the antenna actually arises from the Fabry-Perot resonance of SPPs. The existence of slightly-damped QNMs that cause the resonance nature of nanogap antennas is demonstrated by seeking the solutions of two transcendental equations derived from the model. With the model, the field of the nanogap antenna excited by a nearby point emitter can be expanded upon the basis of QNMs, thus providing an analytical description of the frequency response of the antenna radiation. For calculating the Purcell factor that characterizes the acceleration of the emitter radiation by the antenna, the model provides a new analytical expression of the mode volume of QNMs in terms of the SPP scattering coefficients. The model establishes explicit relations between the concepts of the LSPR and the propagative SPPs, and may act as an efficient tool for the design of various antenna devices.
“The present work is an important step to unify the concepts of the LSPR and the propagative SPPs for understanding the resonance behaviors of optical nano-antennas.� said Prof. Haitao Liu.
Further work will be focused on extending the present model to other antenna structures (for example, cross antennas, split ring antennas or antenna arrays), or more generally, to other types of nano-resonators.