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. ρ, 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.
纳米间隙光学天线谐振行为的直观解析描述
图片说明：(I) (a) 纳米间隙光学天线QNM的SPP模型图示，天线由形成纳米间隙的两根金纳米线组成。a1, a2, b1, b2是SPP的复振幅系数。(b)-(c) 模型中SPP散射系数ρ, τ, r和散射场ΨSPP,+的定义。(II) 天线QNM的场分布（天线臂长L=0.6 μm）。左列、右列分别给出了非周期Fourier模式法(a-FMM)数值计算结果和SPP模型的结果。(a)-(c) 显示了M=1, 2, 3级次的对称QNM。(d)-(f) 显示了N=1, 2, 3级次的反对称QNM。
纳米间隙光学天线作为射频天线概念在光学波段的延伸，是由相互靠近而发生强耦合的金属纳米线组成。它能够把远场能量有效聚焦在纳米尺度的间隙内，或者增强光频辐射源（如分子或量子点）的辐射场并实现定向发射。这些特性使其在增强光谱传感、近场显微镜、纳米催化、光电探测、光辐射等诸多领域有广泛应用。
为成功实现上述应用，确保纳米间隙光学天线器件的精确设计是关键。基于准简正模式（QNM，也被普遍称为局域表面等离激元共振，LSPR）的方法由于能够给出天线频率响应的解析描述，因而成为天线设计的一个强有力工具。利用该频率响应的解析描述能够得出，当激发光频率和天线QNM复数谐振频率的实部匹配时，天线的电磁场会发生共振激发（即LSPR被激发）。然而，目前天线的QNM往往是通过数值求解Maxwell方程组得到，缺乏解析层面的、具有物理直观的描述，这就阻碍了对QNM的深入理解和器件的有效设计。
此外，另一类基于传播的表面等离激元（SPP）的Fabry-Perot模型也被提出，此类模型能够给出描述纳米间隙光学天线的一个直观的物理图像：SPP被入射场激发后在天线两臂上往返传播，在到达天线纳米间隙或端面时发生透射、反射或散射，由此形成天线的辐射场或散射场。但是此类模型无法给出天线频率响应的解析描述。
为了兼顾Fabry-Perot模型和QNM方法在理解天线谐振特性方面的上述优势，南开大学现代光学研究所刘海涛教授课题组提出了纳米间隙光学天线QNM的一个半解析SPP模型。该工作发表在Photonics Research 2016年第4卷第6期（H. Jia, et al., Understanding localized surface plasmon resonance with propagative surface plasmon polaritons in optical nanogap antennas）。
在这项工作中，纳米间隙光学天线QNM的电磁场和复数本征频率能够被SPP模型精确地复现，这样就为天线的LSPR（即QNM）来自SPP的Fabry-Perot共振这样一个直观的物理图像提供了一个严格的电磁学基础。该研究通过求解模型导出的两个超越方程，证明了导致天线产生谐振的低损耗QNM的存在性。利用该模型，纳米间隙光学天线附近的点辐射源激发的辐射场能够用QNM展开，由此给出了天线辐射特性的频率响应的一个解析描述。对于描述天线引起辐射源辐射增强的Purcell因子的计算，该模型给出了QNM模式体积采用SPP散射系数表达的一个新的解析表达式。该模型在LSPR和传播的SPP这两个概念之间建立了明确的联系，可以作为设计各类天线器件的有效工具。
刘海涛教授表示，这一工作实现了局域表面等离激元共振和传播的表面等离激元这两个不同概念在理解纳米光学天线谐振行为方面的统一。
下一步的工作目标是将该模型推广到其它的天线结构（例如交叉天线、开口环天线或天线阵列）或者其它类型的纳米谐振结构的设计和物理描述当中去。