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# Nanophotonics

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# Sharp phase variations from the plasmon mode causing the Rabi-analogue splitting

Yujia Wang
• State Key Laboratory for Mesoscopic Physics, Collaborative Innovation Center of Quantum Matter, Department of Physics, Peking University, Beijing 100871, China
• Other articles by this author:
/ Chengwei Sun
• State Key Laboratory for Mesoscopic Physics, Collaborative Innovation Center of Quantum Matter, Department of Physics, Peking University, Beijing 100871, China
• Other articles by this author:
/ Fengyuan Gan
• State Key Laboratory for Mesoscopic Physics, Collaborative Innovation Center of Quantum Matter, Department of Physics, Peking University, Beijing 100871, China
• Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, Shanxi 030006, China
• Other articles by this author:
/ Hongyun Li
• State Key Laboratory for Mesoscopic Physics, Collaborative Innovation Center of Quantum Matter, Department of Physics, Peking University, Beijing 100871, China
• Other articles by this author:
/ Qihuang Gong
• State Key Laboratory for Mesoscopic Physics, Collaborative Innovation Center of Quantum Matter, Department of Physics, Peking University, Beijing 100871, China
• Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, Shanxi 030006, China
• Other articles by this author:
/ Jianjun Chen
• Corresponding author
• State Key Laboratory for Mesoscopic Physics, Collaborative Innovation Center of Quantum Matter, Department of Physics, Peking University, Beijing 100871, China
• Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, Shanxi 030006, China
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Published Online: 2017-06-24 | DOI: https://doi.org/10.1515/nanoph-2016-0163

## Abstract

The Rabi-analogue splitting in nanostructures resulting from the strong coupling of different resonant modes is of importance for lasing, sensing, switching, modulating, and quantum information processes. To give a clearer physical picture, the phase analysis instead of the strong coupling is provided to explain the Rabi-analogue splitting in the Fabry-Pérot (FP) cavity, of which one end mirror is a metallic nanohole array and the other is a thin metal film. The phase analysis is based on an analytic model of the FP cavity, in which the reflectance and the reflection phase of the end mirrors are dependent on the wavelength. It is found that the Rabi-analogue splitting originates from the sharp phase variation brought by the plasmon mode in the FP cavity. In the experiment, the Rabi-analogue splitting is realized in the plasmonic-photonic coupling system, and this splitting can be continually tuned by changing the length of the FP cavity. These experimental results agree well with the analytic and simulation data, strongly verifying the phase analysis based on the analytic model. The phase analysis presents a clear picture to understand the working mechanism of the Rabi-analogue splitting; thus, it may facilitate the design of the plasmonic-photonic and plasmonic-plasmonic coupling systems.

This article offers supplementary material which is provided at the end of the article.

Photonic cavities provide a route to confining the electromagnetic fields into ultra-small volumes with a large field enhancement, which has important applications in the areas of lasing, sensing, switching, modulating, and quantum information processes [1], [2], [3], [4], [5], [6], [7]. When quantum emitters (such as atom [8], [9], molecule [10], or quantum dot [11], [12], [13]) are placed into a photonic cavity, the Rabi-analogue splitting can be observed if the strong coupling between the emitter and the cavity is satisfied [1]. In the past decades, a lot of classical analogues that can also achieve the Rabi-analogue splitting have been demonstrated [14], [15], [16], [17], [18], [19], [20], [21], [22], [23]. When the photonic cavities combine with the metal particles [15], [16], nanowires [17], [18], [19], nanosandwich [20], and the patterned metal surface [17], [19], [21], [22], [23], the Rabi-analogue splitting is observed in the plasmonic-photonic coupling systems [14], [15], [16], [17], [18], [19], [20], [21], [22], [23]. In these structures, it is concluded that the strong coupling between the photonic modes and the plasmon modes results in the splitting [14], [15], [16], [17], [18], [19], [20], [21], [22], [23]. However, the underlying mechanism and physical picture of the Rabi-analogue splitting are unclear.

In the letter, a Fabry-Pérot (FP) cavity, of which one end mirror is a metallic nanohole array and the other is a thin metal film, is designed to investigate the underlying mechanism of the Rabi-analogue splitting in the plasmonic-photonic coupling system. The phase analysis based on an analytic model is provided to give a clearer physical picture of the Rabi-analogue splitting. It is confirmed that the Rabi-analogue splitting originates from the sharp variation of the reflection phase brought by the plasmon mode on the metallic nanohole array. In addition, it is discovered that the resonance of the proposed FP cavity does not always occur when the accumulated phase shift is equal to integral multiple of 2π. Experimentally, the Rabi-analogue splitting is also observed in the FP cavity. By varying the cavity length of the FP cavity with a pressure, the Rabi-analogue splitting can be continuously tuned. The experimental results agree well with the analytic and simulation data.

The proposed FP cavity (cavity length of L) is schematically shown in Figure 1A. The two end mirrors of this cavity are a 20-nm-thick gold film and a metallic nanohole array, respectively. A normal p-polarized incident beam impinges the thin gold film (20 nm thick) and then goes into the FP cavity, as shown in Figure 1A. It should be pointed out that the plasmon mode on the thin gold film (20 nm thick) could not be excited by the free-space incident beam [24]. The nanohole array (period of p and nanohole diameter of d) is fabricated in a 250-nm-thick gold film with a 30-nm-thick titanium adhesion layer on a glass substrate, as schematically shown in Figure 1B. The transmission and reflection spectra of the metallic nanohole array are calculated with the finite element method (FEM) of COMSOL Multiphysics. In the simulations, the period of the nanohole array and nanohole diameter are set to be p=700 nm and d=260 nm, respectively. The permittivities of gold (εm) and titanium as a function of wavelength are taken from the experiment results [25], [26]. The permittivity of air is εd=1.0. Under a normal p-polarized incident light (magnetic vector paralleling to the y axis), the transmission and reflection spectra of the metallic nanohole array are simulated, and the results are displayed in Figure 1C and D. In Figure 1C, it can be observed that the transmittance from the metallic nanohole array reaches to a peak at the resonant wavelength (λ=747 nm) of the plasmon mode because of the extraordinary optical transmission (EOT) effect [27], [28]. Correspondingly, the reflectance becomes a valley at this wavelength, as shown in Figure 1D. For the metallic nanohole array, the resonant wavelength is approximately determined by λ=p/(i2+j2)1/2[εdεm/(εd+εm)]1/2 [27], [28]. Here, i and j are integers. Hence, the resonant wavelength is calculated to be λ=721 nm, which agrees with the simulation result (Figure 1C). Near the resonant wavelength of the plasmon mode, the reflection phase of the metallic nanohole array exhibits a sharp variation, as depicted in Figure 1E. Based on the microscopic model (Supplementary Material), the reflection light of the nanohole array has two contributions (the directly reflected light and the scattered plasmon mode). The interference of the directly reflected light and the scattered plasmon mode results in the sharp variations of the reflection phase, and the decrease of the reflection phase with the wavelength is attributed to that the phase of the scattered plasmon mode decreases with the wavelength (Figure S1d in Supplementary Material).

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.

The sharp variation of the reflection phase brought by the plasmon mode on the metallic nanohole array has a great influence on the FP cavity proposed in Figure 1A. When the cavity length of the FP cavity is L=4415 nm, the simulated reflection spectrum of the proposed FP cavity is shown by the red line in Figure 1F. The reflection spectrum of a reference FP cavity with two gold films (one being 20 nm thick and the other being 250 nm thick) is also given by the black line in Figure 1F. In the reference FP cavity, only the photonic mode is supported because no plasmon modes are excited on the two flat gold films [24]. For the proposed FP cavity, it is observed that the photonic mode (black line) is split into two modes (red line), and the splitting energy is 31.5 meV (corresponding to Δλ=14 nm in the spectral range). Hence, the Rabi-analogue splitting is achieved in the plasmonic-photonic coupling system. This phenomenon is similar to the previous results [14], [15], [16], [17], [18], [19], [20], [21], [22], [23]. Correspondingly, Figure 1G gives the field distributions (|E|2) at the wavelengths of λ=740, 747, and 754 nm, as denoted by the blue circles in Figure 1F. From Figure 1G and I, the strong standing fields in the FP cavity are obviously observed at the reflectance valleys of λ=740 and 754 nm. Moreover, we can get that the number of the antinodes is 11 for both Figure 1G and I. This indicates that the resonant orders in Figure 1G and I are the same. For the wavelength of λ=747 nm (the resonant wavelength of the plasmon mode), no field is observed in the FP cavity, as shown in Figure 1H.

To explore the underlying physics of the Rabi-analogue splitting in the proposed FP cavity, an analytic model is established. When a normal p-polarized incident beam transmits through the thin gold film (20 nm thick) into the FP cavity, the light can be reflected back and forth off the two end mirrors. Based on the multibeam interference, the reflectance of the FP photonic cavity is expressed as

$R(λ)=|r1(λ)+r2(λ)t1(λ)2 exp(i⋅4πnL/λ)1−r1(λ)r2(λ) exp(i⋅4πnL/λ)|2,$(1)

where ri(λ)=|ri(λ)|exp[i(λ)] (i=1, 2) is the reflection coefficient of the end mirrors in the FP cavity, and θi(λ) (i=1, 2) is the reflection phase of the end mirrors in the FP cavity. t1(λ)=|t1(λ)|exp[1(λ)]) is the transmission coefficient of Mirror 1, and ψ1(λ) is the transmission phase of Mirror 1. n=(εd)1/2=1.0 is the refractive index of the dielectric in the cavity, and L is the cavity length.

For the proposed FP cavity, r1(λ) and r2(λ) are the reflection coefficients of the gold film and the metallic nanohole array, and t1(λ) is the transmission coefficient of the gold film. r1(λ), r2(λ), and t1(λ) can be obtained by the simulations with COMSOL Multiphysics. Based on Eq. (1), the calculated reflection spectrum of the proposed FP cavity (cavity length of L=4415 nm) is shown by the red line in Figure 2A. For the reference FP cavity, r1(λ) and r2(λ) are the reflection coefficients of the two gold films, and t1(λ) is the transmission coefficient of the thinner gold film. The reflection spectrum of the reference FP cavity is given by the black line in Figure 2A. In Figure 2A, it is also observed that the photonic mode (black line) is split into two modes (red line); thus, the Rabi-analogue splitting (31.5 meV) occurs. This phenomenon matches well with the simulation results in Figure 1F, verifying the validity of the analytic model. To further understand the Rabi-analogue splitting, we calculated the reflection spectra of the FP cavity at different cavity lengths using the analytic model, as shown in Figure 2B. Herein, the white dashed line denotes the resonant wavelength of the plasmon mode on the metallic nanohole array (Figure 1C), and the blue circles present the resonant wavelengths of the photonic mode in the reference FP cavity. In Figure 2B, it is observed that the photonic modes of the reference FP cavity overlap with the plasmon mode at the cavity lengths of L=4.04, 4.41, and 4.78 μm. At the overlapping points, the anticrossing is observed (Figure 2B); thus, the photonic mode is split. This is according well with the previous work [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], in which the strong coupling between the FP photonic modes and the plasmon mode is expected when the two modes overlap with each other. In addition, the FP photonic modes in the proposed FP cavity nearly remain as that in the reference FP cavity when the strong coupling condition is not satisfied, as shown in Figure 2B.

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 different 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.

To give a clearer physical picture to understand the working mechanism of the mode splitting, we make a phase analysis. It is easy to get that the accumulated phase delay per round trip in the analytic model of the FP cavity can be written as

$φ(λ,L)=4πnL/λ+θ1(λ)+θ2(λ).$(2)

In the proposed FP cavity, θ1(λ) and θ2(λ) are the reflection phases of the gold film and the metallic nanohole array, respectively. The simulations show that θ1(λ) is nearly a constant with the increase of the wavelength λ. However, θ2(λ) exhibits a sharp variation near the resonant wavelength of the plasmon mode, as shown in Figure 1E. Usually, the constructive interferences will occur when φ(λ,L) is equal to 2 (m=0, 1, 2, …), and strong standing waves can be observed in the FP cavity. However, for a mirror with a wavelength-dependent reflectance (such as the metallic nanohole array), the phenomenon may be different. Based on Eq. (2), the dependence of the accumulated phase delay φ(λ,L) on the wavelength is shown by the blue line in Figure 2A, which exhibits a sharp kink near the resonant wavelength of the plasmon mode. The accumulated phase delays of the two split modes are φ(λ=740 nm)=φ(λ=754 nm)=22π, which reveals that the two modes are both the 11th-order (m=11) resonant mode in the proposed FP cavity. The resonant order (m=11) agrees well with the number of the antinodes in Figure 1G and I. The same resonant order for the two splitting modes is attributed to the sharp kink of the accumulated phase delay φ(λ,L), as shown in Figure 2A. Hence, the mode splitting originates from the sharp phase variation brought by the plasmon mode (λ=747 nm) of the metallic nanohole array. In addition, the accumulated phase delay is also equal to 22π at λ=747 nm. However, there is not an FP photonic mode at this wavelength, and a reflection peak is observed in Figure 2A. This is attributed to the zero reflectance of the metallic nanohole array at λ=747 nm; thus, there are no standing waves in the FP cavity, as shown in Figure 1H. The proposed FP cavity with the missing resonance is different from the conventional FP cavity. Based on the above discussion, the phase analysis matches well with the simulation results. Under the strong coupling condition, the FP photonic mode is split into two modes, and this splitting originates from the sharp variation of the reflection phase brought by the plasmon mode. The phase analysis based on the analytic model is generally applicable, and it can also been used to explain the Rabi-analogue splitting phenomena in the previous structures [14], [15], [16], [17], [18], [19], [20], [21], [22], [23].

To further test our proposal experimentally, the nanohole array is fabricated using focused ion beams (FIB) in a 250-nm-thick gold film, which is evaporated on a glass substrate with a 30-nm-thick titanium adhesion layer. Figure 3A shows the scanning electron microscopy (SEM) image of the metallic nanohole array. Figure 3B gives the detail of the metallic nanohole array. The measured geometrical parameters of the metallic nanohole array are as follows: the period is p=700 nm and the nanohole diameter is d=200 nm. To form an FP cavity, a 19-nm-thick gold film evaporated on a glass substrate is covered on the metallic nanohole array. The gap between the 19-nm-thick gold film and metallic nanohole array can be continually tuned from 3 to 6 μm with different pressures.

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 normally illuminated with a super-continuum white light source (Fianium). The white light is first polarized to be a p-polarized light by a Glan-Taylor prism and then focused on the sample by a microscope objective (Mitutoyo 20×, NA=0.4). The radius of the focused spot is about 15 μm. Next, the reflected light is collected by the same microscope objective and then split into two beams by a beam splitter. One beam is used for real-time monitoring, and the other is coupled to a fiber that connects a spectrograph (Andor). The measured reflection spectra of the proposed FP cavity with different cavity lengths are displayed by the red solid line in Figure 3C–G, where the black line is the measured reflection spectrum of the nanohole array. Due to the coupling of the plasmon mode on the metallic nanohole array, the photonic mode in the FP cavity is split, as shown by the red line in Figure 3E. The splitting energy is 23.6 meV (corresponding to Δλ=10.5 nm in the spectral range). Moreover, the two splitting modes can be continually tuned by varying the cavity length with a pressure, as shown in Figure 3C–G. The corresponding simulation data and analytic data based on Eq. (1) for the cavity length of L=4500, 4445, 4413, 4370, and 4285 nm are depicted in Figure 3H–L. It is observed that the experimental results match well with the simulation and analytic data. The slight deviations are attributed to the large loss of the plasmon mode and the lower reflectance of gold in the experiment (black line in Figure 3C–G).

In summary, the phase analysis based on an analytic model was provided to investigate the underlying mechanism of the Rabi-analogue splitting in the plasmonic-photonic coupling system. From both the analytic results and the simulation results, it was observed that the photonic mode was split into two modes in the overlapping area of the photonic mode and plasmon mode. Using the phase analysis instead of the strong coupling, it was confirmed for the first time that this Rabi-analogue splitting originated from the sharp variation of the reflection phase brought by the plasmon mode on the metallic nanohole array. In the experiment, the Rabi-analogue splitting was demonstrated in the proposed FP cavity. The experimental results agreed well with the analytic and simulation data. This strongly verified the phase analysis based on the analytic model. The phase analysis based on the analytic model is generally applicable, and it can also been used to explain the Rabi-analogue splitting phenomena in the previous structures [14], [15], [16], [17], [18], [19], [20], [21], [22], [23]. The phase analysis presents a clear picture in the plasmonic-photonic coupling system; thus, it could facilitate the design of the plasmonic-photonic and plasmonic-plasmonic coupling systems.

## Acknowledgments

This work was supported by the National Basic Research Program of China (2013CB328704 and 2016YFA0203500) and the National Natural Science Foundation of China (11674014, 11204018, 61475005, and 11134001).

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## Supplemental Material:

The online version of this article (DOI: https://doi.org/10.1515/nanoph-2016-0163) offers supplementary material, available to authorized users.

Revised: 2016-11-30

Accepted: 2016-12-17

Published Online: 2017-06-24

Citation Information: Nanophotonics, ISSN (Online) 2192-8614,

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