Sharp Phase Variations from the Plasmon Mode Causing the Rabi-Analogue Splitting

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. Based on Eq. by the plasmon mode on the metallic nanohole array has (1), the calculated reflection spectrum of the proposed a great influence on the FP cavity proposed in Figure 1A. FP cavity (cavity length of L = 4415 nm) is shown by the

When the cavity length of the FP cavity is L = 4415 nm, the red line in Figure 2A. For the reference FP cavity, r1(λ) and simulated reflection spectrum of the proposed FP cavity r2(λ) are the reflection coefficients of the two gold films, is shown by the red line in Figure 1F. The reflection spec- and t1(λ) is the transmission coefficient of the thinner trum of a reference FP cavity with two gold films (one gold film. The reflection spectrum of the reference FP being 20 nm thick and the other being 250 nm thick) is cavity is given by the black line in Figure 2A. In Figure 2A, also given by the black line in Figure 1F. In the reference it is also observed that the photonic mode (black line) is FP cavity, only the photonic mode is supported because split into two modes (red line); thus, the Rabi-analogue no plasmon modes are excited on the two flat gold films splitting (31.5 meV) occurs. This phenomenon matches [24]. For the proposed FP cavity, it is observed that the well with the simulation results in Figure 1F, verifying photonic mode (black line) is split into two modes (red the validity of the analytic model. To further understand line), and the splitting energy is 31.5 meV (corresponding the Rabi-analogue splitting, we calculated the reflection to Δλ = 14 nm in the spectral range). Hence, the Rabi-ana- spectra of the FP cavity at different cavity lengths using logue splitting is achieved in the plasmonic-photonic cou- the analytic model, as shown in Figure 2B. Herein, the pling system. This phenomenon is similar to the previous white dashed line denotes the resonant wavelength of the results [14–23]. Correspondingly, Figure 1G gives the field plasmon mode on the metallic nanohole array (Figure 1C), distributions (|E | 2) at the wavelengths of λ = 740, 747, and and the blue circles present the resonant wavelengths of 754 nm, as denoted by the blue circles in Figure 1F. From the photonic mode in the reference FP cavity. In Figure Figure 1G and I, the strong standing fields in the FP cavity 2B, it is observed that the photonic modes of the refer- are obviously observed at the reflectance valleys of λ = 740 ence FP cavity overlap with the plasmon mode at the and 754 nm. Moreover, we can get that the number of the cavity lengths of L = 4.04, 4.41, and 4.78 μm. At the over- antinodes is 11 for both Figure 1G and I. This indicates that lapping points, the anticrossing is observed (Figure 2B); the resonant orders in Figure 1G and I are the same. For thus, the photonic mode is split. This is according well 1104 Y. Wang et al.: Sharp phase variations from the plasmon mode causing the Rabi-analogue splitting

Figure 2: (A) Analytic reflection spectra of the proposed FP cavity (red line) and reference FP cavity (black line) as well as the accumulated phase delay per round trip (blue solid line) in the proposed FP cavity with L = 4415 nm. (B) Analytic reflection spectra of the FP cavity at differ- ent cavity lengths. The white dashed line denotes the resonant wavelength of the plasmon mode on the metallic nanohole array (Figure 1C), and the blue circles denote the reference FP photonic mode. with the previous work [14–23], in which the strong cou- modes is attributed to the sharp kink of the accumulated pling between the FP photonic modes and the plasmon phase delay ϕ(λ,L), as shown in Figure 2A. Hence, the mode is expected when the two modes overlap with each mode splitting originates from the sharp phase variation other. In addition, the FP photonic modes in the proposed brought by the plasmon mode (λ = 747 nm) of the metal- FP cavity nearly remain as that in the reference FP cavity lic nanohole array. In addition, the accumulated phase when the strong coupling condition is not satisfied, as delay is also equal to 22π at λ = 747 nm. However, there shown in Figure 2B. is not an FP photonic mode at this wavelength, and a To give a clearer physical picture to understand the reflection peak is observed in Figure 2A. This is attributed working mechanism of the mode splitting, we make a to the zero reflectance of the metallic nanohole array at phase analysis. It is easy to get that the accumulated λ = 747 nm; thus, there are no standing waves in the FP phase delay per round trip in the analytic model of the FP cavity, as shown in Figure 1H. The proposed FP cavity cavity can be written as with the missing resonance is different from the conven- tional FP cavity. Based on the above discussion, the phase ϕ(,λπLn)4=+L /(λθλθ)(+ λ). (2) 12 analysis matches well with the simulation results. Under

In the proposed FP cavity, θ1(λ) and θ2(λ) are the the strong coupling condition, the FP photonic mode is reflection phases of the gold film and the metallic nano- split into two modes, and this splitting originates from hole array, respectively. The simulations show that θ1(λ) is the sharp variation of the reflection phase brought by the nearly a constant with the increase of the wavelength λ. plasmon mode. The phase analysis based on the analytic

However, θ2(λ) exhibits a sharp variation near the resonant model is generally applicable, and it can also been used wavelength of the plasmon mode, as shown in Figure 1E. to explain the Rabi-analogue splitting phenomena in the Usually, the constructive interferences will occur when previous structures [14–23]. ϕ(λ,L) is equal to 2mπ (m = 0, 1, 2, …), and strong stand- To further test our proposal experimentally, the nano- ing waves can be observed in the FP cavity. However, for hole array is fabricated using focused ion beams (FIB) in a mirror with a wavelength-dependent reflectance (such a 250-nm-thick gold film, which is evaporated on a glass as the metallic nanohole array), the phenomenon may be substrate with a 30-nm-thick titanium adhesion layer. different. Based on Eq. (2), the dependence of the accu- Figure 3A shows the scanning electron microscopy (SEM) mulated phase delay ϕ(λ,L) on the wavelength is shown image of the metallic nanohole array. Figure 3B gives the by the blue line in Figure 2A, which exhibits a sharp kink detail of the metallic nanohole array. The measured geo- near the resonant wavelength of the plasmon mode. The metrical parameters of the metallic nanohole array are as accumulated phase delays of the two split modes are follows: the period is p = 700 nm and the nanohole dia- ­ϕ(λ = 740 nm) = ϕ(λ = 754 nm) = 22π, which reveals that the meter is d = 200 nm. To form an FP cavity, a 19-nm-thick two modes are both the 11th-order (m = 11) resonant mode gold film evaporated on a glass substrate is covered on the in the proposed FP cavity. The resonant order (m = 11) metallic nanohole array. The gap between the 19-nm-thick agrees well with the number of the antinodes in Figure 1G gold film and metallic nanohole array can be continually and I. The same resonant order for the two splitting tuned from 3 to 6 μm with different pressures. Y. Wang et al.: Sharp phase variations from the plasmon mode causing the Rabi-analogue splitting 1105

Figure 3: (A) SEM image of the metallic nanohole array. (B) Zoomed-in SEM image of the nanohole array. (C–G) Measured reflection spectra of the FP cavity (red solid lines) at different cavity lengths. The black solid line is the measured reflection spectrum of the metallic nanohole array. Simulation and analytic results of the reflection spectra of the FP cavity at different cavity lengths of (H) L = 4500 nm, (I) L = 4445 nm, (J) L = 4413 nm, (K) L = 4370 nm, and (L) L = 4285 nm. The black dashed lines denote the reflection spectrum of the metallic nanohole array. The red solid lines are the analytic results, and the blue dots are the simulation results.

In the experiment, the experimental sample is nor- splitter. One beam is used for real-time monitoring, and mally illuminated with a super-continuum white light the other is coupled to a fiber that connects a spectrograph source (Fianium). The white light is first polarized to be a (Andor). The measured reflection spectra of the proposed p-polarized light by a Glan-Taylor prism and then focused FP cavity with different cavity lengths are displayed by the on the sample by a microscope objective (Mitutoyo 20×, red solid line in Figure 3C–G, where the black line is the NA = 0.4). The radius of the focused spot is about 15 μm. measured reflection spectrum of the nanohole array. Due Next, the reflected light is collected by the same micro- to the coupling of the plasmon mode on the metallic nano- scope objective and then split into two beams by a beam hole array, the photonic mode in the FP cavity is split, as 1106 Y. 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