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The natural hyperbolic phonon polariton material-orthorhombic molybdenum trioxide (α-MoO3) has recently attracted much interest , due to the associated ultra-confinement of light and enhanced light-matter interactions. We theoretically propose and study the in-plane anisotropic phonon polaritons (APhPs) in the Kretschmann structure with monolayer and dual layers α-MoO3. The excitation of phonon polaritons and the corresponding dispersion properties in this multilayer system are studied by using a generalized 4×4 transfer matrix method (TMM). The frequency dispersions with geometrical parameters are also discussed in detail. The results confirm that the interlayer coupling can be modulated by stacking the multilayer films and regulating the thickness of each layer. More interestingly, when the distance between double α-MoO3 layers is much smaller than the propagation length of PhPs, a strong coupling phenomenon occurs, and the photon tunneling probability and intensity can be greatly improved. When the incident angle is greater than the total internal reflection angle, the phase matching condition for SPhP excitation can be satisfied. Within the 40° incident angle, the SPhP blue-shifts rapidly with the increase of incident angle. But then the dispersion curve no longer changes with increase of incidence angle. The enlargement of the interstitial layer can also lead the Fabry-Perot (FP) resonance mode to be excited. The APhP in layered heterostructure is an important part of today's nanophotonic technology, our study can help optimize and design tunable optoelectronic devices based on hyperbolic materials.
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图 2 (a) 单一α-MoO3层色散曲线的研究模型; (b)—(d) 计算得到的面内波矢量和光子能量函数的反射光谱. 亮线代表不同方向上SPhP的色散曲线
Figure 2. (a) Model of the dispersion curve of a single α-MoO3 layer; (b)–(d) the calculated reflectance spectra as a function of in-plane wave vector and photon energy. Bright lines represent the dispersion curves of SPhP in different directions.
图 3 由厚度为1 μm的α-MoO3组成的Kretschmann结构对应的反射谱线
Figure 3. Reflection spectral corresponding to the Kretschmann structure composed of α-MoO3 with a thickness of 1 μm.
图 4 双层α-MoO3耦合的多层膜模型
Figure 4. Stacked film model of double-layer α-MoO3 for coupling research.
图 5 计算得到的在不同t条件下反射率随入射角度的变化图谱 (a) t = 20 nm; (b) t = 100 nm; (c) t = 500 nm; (d) t = 1000 nm
Figure 5. Calculated spectrum of reflectance variation with incident angle under different conditions: (a) t = 20 nm; (b) t = 100 nm; (c) t = 500 nm; (d) t = 1000 nm.
图 6 (a)—(c) 频率为820 cm–1处t = 100 nm, t = 500 nm和t = 1000 nm时数值计算得到的等频曲线; (d)—(f) 频率为900 cm–1处t = 100 nm, t = 500 nm和t = 1000 nm时数值计算得到的等频曲线; (g)—(i) 频率为1000 cm–1处t = 100 nm, t = 500 nm和t = 1000 nm时数值计算得到的等频曲线
Figure 6. (a)–(c) The iso-frequency curves obtained by numerical calculation with t = 100 nm, t = 500 nm and t = 1000 nm at a frequency of 820 cm–1; (d)–(f) the iso-frequency curves obtained by numerical calculation at t = 100 nm, t = 500 nm and t = 1000 nm at a frequency of 900 cm–1; (g)–(i) the iso-frequency curves obtained by numerical calculation at 1000 cm–1 with t = 100 nm, t = 500 nm and t = 1000 nm.
图 7 (a) 在Kretschmann结构中激发的 SPhP 反射谱线, t = 1000 nm, θ = 50°; (b) 多层界面处的|Ex|强度分布
Figure 7. SPhP reflection spectral excited in the Kretschmann structure with t = 1000 nm and θ = 50°; (b) |Ex| intensity distribution within the multilayered structure.
表 1 α-MoO3的介电函数的洛伦兹模型对应的参数
Table 1. The parameters corresponding to the Lorentzian model of the dielectric function of α-MoO3.
主轴 模式序数 ${{\varepsilon _x^\infty }} $ ωTO/cm–1 ωLO/cm–1 γ/cm–1 x 1 5.86 506.7 534.3 49.1 x 2 824.1 963 6 x 3 998.7 999.2 0.35 ${{\varepsilon _y^\infty }} $ ωTO/cm–1 ωLO/cm–1 γ/cm–1 y 1 6.59 544.6 850.1 9.5 ${{\varepsilon _z^\infty }}$ ωTO/cm–1 ωLO/cm–1 γ/cm–1 z 1 4.47 444 508 1.5 z 2 956.7 1006.9 1.5 
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[1] 管福鑫, 董少华, 何琼, 肖诗逸, 孙树林, 周磊 2020 物理学报 69 157804
Google Scholar
Guan F X, Dong S H, He Q, Xiao S Y, Sun S L, Zhou L 2020 Acta Phys. Sin. 69 157804
Google Scholar
[2] 朱旭鹏, 张轼, 石惠民, 陈智全, 全军, 薛书文, 张军, 段辉高 2019 物理学报 68 247301
Google Scholar
Zhu X P, Zhang S, Shi H M, Chen Z Q, Quan J, Xue S W, Zhang J, Duan H G 2019 Acta Phys. Sin. 68 247301
Google Scholar
[3] 束方洲, 范仁浩, 王嘉楠, 彭茹雯, 王牧 2019 物理学报 68 147303
Google Scholar
Shu F Z, Fan R H, Wang J N, Peng R W, Wang M 2019 Acta Phys. Sin. 68 147303
Google Scholar
[4] 马赛群, 邓奥林, 吕博赛, 胡成, 史志文 2022 物理学报 71 127104
Google Scholar
Ma S Q, Deng A L, Lü B S, Hu C, Shi Z W 2022 Acta Phys. Sin. 71 127104
Google Scholar
[5] Jacob Z 2014 Nat. Mater. 13 1081
Google Scholar
[6] Chen M, Lin X, Dinh T H, Zheng Z, Shen J, Ma Q, Chen H, Jarillo-Herrero P, Dai S 2020 Nat. Mater. 19 1307
Google Scholar
[7] Wu X, Fu C J 2021 Int. J. Heat Mass Transfer 168 120908
Google Scholar
[8] Feng K, Streyer W, Zhong Y, Hoffman A J, Wasserman D 2015 Opt. Express 23 A1418
Google Scholar
[9] Passler N C, Ni X, Hu G, Matson J R, Carini G, Wolf M, Schubert M, Alu A, Caldwell J D, Thomas G Folland T G, Paarmann A 2022 Nature 602 595
Google Scholar
[10] Dai S, Quan J, Hu G, Qiu C, Tao T, Li X, Alu A 2019 Nano Lett. 19 1009
Google Scholar
[11] Zhao B, Zhang Z M 2017 Opt. Express 25 7791
Google Scholar
[12] Zheng Z, Xu N, Oscurato S L, Tamagnone M, Sun F, Jiang Y, Ke Y, Chen J, Huang W, Wilson W L, Ambrosio A, Deng S, Chen H 2019 Sci. Adv. 24 eaav8690
[13] Larciprete M C, Dereshgi S A, Centini M, Aydin K 2022 Opt. Express 30 12788
Google Scholar
[14] Ni G, McLeod A S, Sun Z, Matson J R, Lo C F B, Rhodes D A, Ruta F L, Moore S L, Vitalone R A, Cusco R, Artús L, Xiong L, Dean C R, Hone A J, Millis J C, Fogler M M, Edgar J H, Caldwell J D, Basov D N 2021 Nano Lett. 21 5767
Google Scholar
[15] Ma W, Hu G, Hu D, Chen R, Sun T, Zhang X, Dai Q, Zeng Y, Alù A, Qiu C W, Li P 2021 Nature 596 362
Google Scholar
[16] Pavlidis G, Schwartz J J, Matson J, Folland T, Liu S, Edgar J H, Caldwell J D, Centrone A 2021 APL Mater. 9 091109
Google Scholar
[17] Chen M, Sanders S, Shen J, Li J, Harris E, Chen C, Ma Q, Edgar J H, Manjavacas A, Dai S 2022 Adv Opt. Mater. 10 2102723
Google Scholar
[18] Toyin O R, Ge W X, Gao L 2021 Chin. Phys. Lett. 38 016801
Google Scholar
[19] 魏晨崴, 曹暾 2021 物理学报 70 048701
Google Scholar
Wei C W, Cao T 2021 Acta Phys. Sin. 70 048701
Google Scholar
[20] Chen M, Lin X, Dinh T, Zheng Z, Shen J, Ma Q, Chen H, Jarillo-Herrero P, Dai S 2020 Nat. Mater. 19 1372
Google Scholar
[21] Gong Y, Zhao Y, Zhou Z, Li D, Mao H, Bao Q, Zhang Y, Wang G 2022 Adv. Opt. Mater. 10 2200038
Google Scholar
[22] Hajian H, Rukhlenko I D, Ercaglar V, Hanson G, Ozbay E 2022 Appl. Phys. Lett. 120 112204
Google Scholar
[23] Zhang Q, Ou Q, Hu G, Liu J, Dai Z, Fuhrer M S, Bao Q, Qiu C W 2021 Nano Lett. 21 3112
Google Scholar
[24] Passler N C, Paarmann A 2017 J. Opt. Soc. Am. B:Opt. Phys. 34 2128
Google Scholar
[25] Wu X H, Fu C J, Zhang Z M 2020 J. Heat Transfer 142 072802
Google Scholar
[26] Passler N C, Jeannin M, Paarmann A 2020 Phys. Rev. B 101 165425
Google Scholar
[27] Li H H 1980 J. Phys. Chem. Ref. Data 9 161
Google Scholar
[28] Xia S, Zhai X, Wang L, Xiang Y, Wen S 2022 Phys. Rev. B 106 075401
Google Scholar
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