Nanophotonics 2017; 6(5): 1101–1107 Research article Open Access Yujia Wang, Chengwei Sun, Fengyuan Gan, Hongyun Li, Qihuang Gong and Jianjun Chen* Sharp phase variations from the plasmon mode causing the Rabi-analogue splitting DOI 10.1515/nanoph-2016-0163 Keywords: Rabi-analogue splitting; phase variation; Received September 26, 2016; revised November 30, 2016; accepted phase analysis; plasmonic-photonic coupling system; December 17, 2016 wavelength-dependent mirror. Abstract: The Rabi-analogue splitting in nanostructures resulting from the strong coupling of different resonant Photonic cavities provide a route to confining the elec- modes is of importance for lasing, sensing, switching, tromagnetic fields into ultra-small volumes with a large modulating, and quantum information processes. To give field enhancement, which has important applications in a clearer physical picture, the phase analysis instead of the areas of lasing, sensing, switching, modulating, and the strong coupling is provided to explain the Rabi-ana- quantum information processes [1–7]. When quantum logue splitting in the Fabry-Pérot (FP) cavity, of which one emitters (such as atom [8, 9], molecule [10], or quantum end mirror is a metallic nanohole array and the other is dot [11–13]) are placed into a photonic cavity, the Rabi- a thin metal film. The phase analysis is based on an ana- analogue splitting can be observed if the strong coupling lytic model of the FP cavity, in which the reflectance and between the emitter and the cavity is satisfied [1]. In the the reflection phase of the end mirrors are dependent on past decades, a lot of classical analogues that can also the wavelength. It is found that the Rabi-analogue split- achieve the Rabi-analogue splitting have been demon- ting originates from the sharp phase variation brought by strated [14–23]. When the photonic cavities combine with the plasmon mode in the FP cavity. In the experiment, the the metal particles [15, 16], nanowires [17–19], nanosand- Rabi-analogue splitting is realized in the plasmonic-pho- wich [20], and the patterned metal surface [17, 19, 21–23], tonic coupling system, and this splitting can be continu- the Rabi-analogue splitting is observed in the plasmonic- ally tuned by changing the length of the FP cavity. These photonic coupling systems [14–23]. In these structures, it is experimental results agree well with the analytic and sim- concluded that the strong coupling between the photonic ulation data, strongly verifying the phase analysis based modes and the plasmon modes results in the splitting on the analytic model. The phase analysis presents a clear [14–23]. However, the underlying mechanism and physical picture to understand the working mechanism of the Rabi- picture of the Rabi-analogue splitting are unclear. analogue splitting; thus, it may facilitate the design of the In the letter, a Fabry-Pérot (FP) cavity, of which one plasmonic-photonic and plasmonic-plasmonic coupling end mirror is a metallic nanohole array and the other is systems. a thin metal film, is designed to investigate the under- lying mechanism of the Rabi-analogue splitting in the plasmonic-photonic coupling system. The phase analysis *Corresponding author: Jianjun Chen, State Key Laboratory for based on an analytic model is provided to give a clearer Mesoscopic Physics, Collaborative Innovation Center of Quantum physical picture of the Rabi-analogue splitting. It is con- Matter, Department of Physics, Peking University, Beijing 100871, firmed that the Rabi-analogue splitting originates from China; and Collaborative Innovation Center of Extreme Optics, the sharp variation of the reflection phase brought by the Shanxi University, Taiyuan, Shanxi 030006, China, plasmon mode on the metallic nanohole array. In addi- e-mail: [email protected] Yujia Wang, Chengwei Sun and Hongyun Li: State Key Laboratory for tion, it is discovered that the resonance of the proposed Mesoscopic Physics, Collaborative Innovation Center of Quantum FP cavity does not always occur when the accumulated Matter, Department of Physics, Peking University, Beijing 100871, phase shift is equal to integral multiple of 2π. Experimen- China tally, the Rabi-analogue splitting is also observed in the FP Fengyuan Gan and Qihuang Gong: State Key Laboratory for cavity. By varying the cavity length of the FP cavity with a Mesoscopic Physics, Collaborative Innovation Center of Quantum Matter, Department of Physics, Peking University, Beijing 100871, pressure, the Rabi-analogue splitting can be continuously China; and Collaborative Innovation Center of Extreme Optics, tuned. The experimental results agree well with the ana- Shanxi University, Taiyuan, Shanxi 030006, China lytic and simulation data. ©2017, Jianjun Chen et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. 1102 Y. Wang et al.: Sharp phase variations from the plasmon mode causing the Rabi-analogue splitting The proposed FP cavity (cavity length of L) is sche- film with a 30-nm-thick titanium adhesion layer on a glass matically shown in Figure 1A. The two end mirrors of this substrate, as schematically shown in Figure 1B. The trans- cavity are a 20-nm-thick gold film and a metallic nano- mission and reflection spectra of the metallic nanohole hole array, respectively. A normal p-polarized incident array are calculated with the finite element method (FEM) beam impinges the thin gold film (20 nm thick) and then of COMSOL Multiphysics. In the simulations, the period of goes into the FP cavity, as shown in Figure 1A. It should be the nanohole array and nanohole diameter are set to be pointed out that the plasmon mode on the thin gold film p = 700 nm and d = 260 nm, respectively. The permittivities (20 nm thick) could not be excited by the free-space inci- of gold (εm) and titanium as a function of wavelength are dent beam [24]. The nanohole array (period of p and nano- taken from the experiment results [25, 26]. The permittivity hole diameter of d) is fabricated in a 250-nm-thick gold of air is εd = 1.0. Under a normal p-polarized incident light Figure 1: (A) Schematic and geometric parameters of the FP cavity. (B) Schematic and geometric parameters of the nanohole array in the gold film. Simulation results of the (C) transmission spectrum, (D) reflection spectrum, and (E) reflection phase of the nanohole array normally illuminated by the p-polarized incident light. (F) Simulated reflection spectra of the proposed FP cavity and the reference FP cavity when the cavity length is L = 4415 nm. Simulated field distributions (|E | 2) of the proposed FP cavity at (G) λ = 740 nm, (H) λ = 747 nm, and (I) λ = 754 nm. Y. Wang et al.: Sharp phase variations from the plasmon mode causing the Rabi-analogue splitting 1103 (magnetic vector paralleling to the y axis), the transmis- the wavelength of λ = 747 nm (the resonant wavelength of sion and reflection spectra of the metallic nanohole array the plasmon mode), no field is observed in the FP cavity, are simulated, and the results are displayed in Figure 1C as shown in Figure 1H. and D. In Figure 1C, it can be observed that the transmit- To explore the underlying physics of the Rabi-ana- tance from the metallic nanohole array reaches to a peak logue splitting in the proposed FP cavity, an analytic at the resonant wavelength (λ = 747 nm) of the plasmon model is established. When a normal p-polarized inci- mode because of the extraordinary optical transmission dent beam transmits through the thin gold film (20 nm (EOT) effect [27, 28]. Correspondingly, the reflectance thick) into the FP cavity, the light can be reflected back becomes a valley at this wavelength, as shown in Figure and forth off the two end mirrors. Based on the multibeam 1D. For the metallic nanohole array, the resonant wave- interference, the reflectance of the FP photonic cavity is 2 2 1/2 length is approximately determined by λ = p/(i + j ) [εdεm/ expressed as 1/2 (εd + εm)] [27, 28]. Here, i and j are integers. Hence, the 2 λλ2 ⋅ πλ resonant wavelength is calculated to be λ = 721 nm, which rt21()() exp(in4/L ) R()λλ=+r1() , (1) agrees with the simulation result (Figure 1C). Near the 1(−⋅rrλλ)() exp(in4/πλL ) 12 resonant wavelength of the plasmon mode, the reflection phase of the metallic nanohole array exhibits a sharp vari- where ri(λ) = | ri(λ) | exp[iθi(λ)] (i = 1, 2) is the reflection coef- ation, as depicted in Figure 1E. Based on the microscopic ficient of the end mirrors in the FP cavity, and θi(λ) (i = 1, 2) model (Supplementary Material), the reflection light of is the reflection phase of the end mirrors in the FP cavity. the nanohole array has two contributions (the directly t1(λ) = | t1(λ) | exp[iψ1(λ)]) is the transmission coefficient of reflected light and the scattered plasmon mode). The Mirror 1, and ψ1(λ) is the transmission phase of Mirror 1. 1/2 interference of the directly reflected light and the scat- n =(εd) = 1.0 is the refractive index of the dielectric in the tered plasmon mode results in the sharp variations of the cavity, and L is the cavity length. reflection phase, and the decrease of the reflection phase For the proposed FP cavity, r1(λ) and r2(λ) are the with the wavelength is attributed to that the phase of the reflection coefficients of the gold film and the metallic scattered plasmon mode decreases with the wavelength nanohole array, and t1(λ) is the transmission coefficient (Figure S1d in Supplementary Material). of the gold film. r1(λ), r2(λ), and t1(λ) can be obtained by The sharp variation of the reflection phase brought the simulations with COMSOL Multiphysics.