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非线性差频产生(difference frequency generation, DFG)是实现太赫兹(terahertz, THz)源的重要方式之一. 利用微纳结构的DFG产生THz源可以不考虑相位匹配, 同时是器件小型化、可集成化的重要研究方向. 借助微纳结构的共振模式增强的局域电场在宽波段范围内实现高效的、可调谐的THz源是该领域的研究重点. 本文研究了宽波段范围内具有高Q因子的光栅-波导结构中的DFG产生高效可调谐的THz辐射. 理论上, 通过调控相邻光栅中其中一个的位置扰动, 从而实现光栅周期的加倍, 进而使得布里渊区发生折叠, 光线下方波导层中导模色散曲线折叠到光锥上方, 形成超高Q因子的导模共振, 可以实现在宽光谱范围内增强的THz 产生. 以硫化镉(cadmium sulfide, CdS)光栅-波导为例, 数值研究表明, 在两束基频光光强均为100 kW/cm2时, THz的转换效率可达到10–8 W –1的量级, 为相同厚度CdS薄膜转换效率的109倍. 通过改变两束基频光入射角, 可实现不同共振基频组合, 实现任意频率THz波产生, 从而实现了在宽光谱范围内高效可调谐的THz源.Nonlinear difference frequency generation (DFG) is a key mechanism for realizing terahertz (THz) sources. Utilization of DFG within micro- and nano-structures can circumvent the phase-matching limitations while supporting device miniaturization and integrability, thus the DFG is made a significant area of research. Enhancing the local electric fields through resonant modes in micro- and nano-structures has become a promising approach to achieving efficient and tunable THz sources across a broad wavelength range. In this work, the mechanism of DFG in high-Q-factor grating-waveguide structures for efficiently tuning THz radiation over a wide spectral range is investigated by using numerical simulations based on the finite element method (COMSOL Multiphysics). Theoretical analysis reveals that modulating the positional perturbation of one of the adjacent gratings effectively doubles the grating period, causing Brillouin zone to fold. This folding shifts the dispersion curve of the guided mode (GM) within the waveguide layer above the light cone, forming a guided mode resonance (GMR) with an ultra-high Q-factor, thereby significantly enhancing THz generation in a broad spectral range. Taking a cadmium sulfide (CdS) grating-waveguide structure for example, numerical simulations demonstrate that the THz conversion efficiency reaches an order of 10–8 W–1 when both fundamental frequency beams have an intensity of 100 kW/cm2, which is 109 times higher than the conversion efficiency of a CdS film of the same thickness. Moreover, the fundamental frequency resonance wavelength can be widely tuned by adjusting the incident angle. High-Q-factor resonance modes enable various fundamental frequency combinations by changing the incident angles of the two fundamental frequency beams, facilitating the generation of THz waves with arbitrary frequencies. This approach ultimately enables a highly efficient and tunable THz source in a wide spectral range, providing valuable insights for generating THz sources on micro- and nanophotonic platforms.
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图 1 四部分光栅-波导结构和光配置示意图, 其中Λ是光栅-波导结构的周期, wg是 CdS 的宽度, wa和wb是空气的宽度, hg和hw 分别是光栅层和波导层的高度
Fig. 1. Schematic diagram of grating waveguide structure and light configuration. Λ is the periodicity of grating waveguide structure, wg is the width of CdS, wa and wb are the width of air, hg and hw are the height of grating layer and waveguide layer, respectively.
图 2 (a) TE偏振光照射下周期为P = Λ/2的未扰动光栅-波导结构(绿点)和几何扰动δ = 0.1、周期为Λ的光栅结构-波导结构(红点和蓝点)的能带, 插图显示了箭头所示位置的GMR模式的电场(TE)分布; (b)波段A和波段B的Q因子与kx的关系; (c)波导层中TE0导波模式的色散关系(黑色实线), 以及kx = kx, i (i = –1, –2)在不同入射角θ下的色散关系, 分别为θ = 1° (酒红色虚线)、2° (红色虚线)、4° (绿色虚线)、6° (蓝色虚线)、8° (青色虚线)、10° (品红色虚线)
Fig. 2. (a) Band structure of the unperturbated grating-waveguide nanostructure of period P = Λ/2 (green dots) and geometrical perturbated δ = 0.1 grating-waveguide nanostructure of period Λ (red and blue dots). The inset shows the electric field (TE) distribution of the GMR mode at the kx as the arrow given. (b) Dependence of Q-factors of band A and B on kx. (c) Dispersion relations of the TE0 guide mode in the waveguide layer (black solid line), and kx = kx, i (i = –1, –2) under different angle of incidence θ = 1° (Wine red dashed line), 2° (red dashed lines), 4° (green dashed lines), 6° (blue dashed lines), 8° (cyan dashed lines), 10° (Magenta dashed line), respectively.
图 3 (a)不同参数δ的光栅-波导结构在入射角θ = 6°时的透射光谱, 插图分别显示了 δ = 0.2和1.0 结构中共振模式处的Ez分布; (b) δ = 0.1 的光栅-波导结构的透射率与入射角的关系, 插图显示了θ = 6°结构中共振模式处的电场Ez分布; (c) 在TE偏振光照射下, 光栅-波导结构的Q因子与δ的关系, 插图显示了Q因子与δ–2之间的线性关系, 虚线为线性拟合; (d) δ = 0.1 时, 共振波长(黑色实线)和品质因数(黑色虚线)与入射角的关系
Fig. 3. (a) Transmittance spectra of grating waveguide structure of different parameter δ at the incidence angle θ = 6°. The inset shows the electric field Ez distribution at the resonance modes in the structure of δ = 0.2 and 1.0, respectively. (b) The dependence of transmittance of grating waveguide structure of δ = 0.1 on the incidence angle. The inset shows the electric field Ez distribution at the resonance modes in the structure of θ = 6°. (c) Dependence of Q-factor of the grating waveguide structure on δ under TE-polarized light irradiation. The inset shows the linear relationship between Q-factor and δ–2, and the dash line is a linear fitting. (d) The relation of resonance wavelength (solid black line) and quality factor (black dashed line) with the angle of incidence at the grating waveguide structure of δ = 0.1.
图 4 不同入射角下DFG产生的THz转换效率与入射波长的关系, 其中入射角分别为(a) 1°和2°, (b) 3°和4°, (c) 5°和6°及(d) 7°和8°
Fig. 4. Generated THz conversion efficiency (CE) from DFG as a function of incident wavelength for different incidence angles. The incident angles are (a) 1° and 2°, (b) 3° and 4°, (c) 5° and 6°, and (d) 7° and 8°.
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[1] Tonouchi M 2007 Nat. Photonics 1 97
Google Scholar
[2] Huang Y, Shen Y C, Wang J Y 2023 Engineering 22 106
Google Scholar
[3] Koch M, Mittleman D M, Ornik J, Castro-Camus E 2023 Nat. Rev. Methods Primers 3 48
Google Scholar
[4] Rubano A, Mou S, Marrucci L, Paparo D 2019 ACS Photonics 6 1515
Google Scholar
[5] Li X R, Li J X, Li Y H, Ozcan A, Jarrahi M 2023 Light Sci. Appl. 12 233
Google Scholar
[6] Lewis R A 2014 J. Phys. D: Appl. Phys. 47 374001
Google Scholar
[7] Li H T, Lu Y L, He Z G, Jia Q K, Wang L 2016 J. Infrared, Millimeter, Terahertz Waves 37 649
Google Scholar
[8] Li Q, Li Y D, Ding S H, Wang Q 2012 J. Infrared Millim. Te. 33 548
Google Scholar
[9] 曹俊诚, 韩英军 2024 中国激光 51 0114001
Google Scholar
Cao J C, Han Y J 2024 Chin. J. Lasers 51 0114001
Google Scholar
[10] Lai R K, Hwang J R, Norris T B, Whitaker J F 1998 Appl. Phys. Lett. 72 3100
Google Scholar
[11] Upadhya P C, Fan W H, Burnett A, Cunningham J, Davies A G, Linfield E H, Lloyd-Hughes J, Castro-Camus E, Johnston M B, Beere H 2007 Opt. Lett. 32 2297
Google Scholar
[12] Fan W H 2011 Chin. Opt. Lett. 9 110008
Google Scholar
[13] Bakunov M I, Bodrov S B 2014 J. Opt. Soc. Am. B 31 2549
Google Scholar
[14] 柴路, 牛跃, 栗岩锋, 胡明列, 王清月 2016 物理学报 65 070702
Google Scholar
Chai L, Niu Y, Li Y F, Hu M L, Wang Q Y 2016 Acta Phys. Sin. 65 070702
Google Scholar
[15] 黄敬国, 陆金星, 周炜, 童劲超, 黄志明, 褚君浩 2013 物理学报 62 120704
Google Scholar
Huang J G, Lu J X, Zhou W, Tong J C, Huang Z, Chu J H 2013 Acta Phys. Sin. 62 120704
Google Scholar
[16] 刘欢, 徐德刚, 姚建铨 2008 物理学报 57 5662
Google Scholar
Liu H, Xu D G, Yao J Q 2008 Acta Phys. Sin. 57 5662
Google Scholar
[17] 钟凯, 姚建铨, 徐德刚, 张会云, 王鹏 2011 物理学报 60 034210
Google Scholar
Zhong K, Yao J Q, Xu D G, Zhang H Y, Wang P 2011 Acta Phys. Sin. 60 034210
Google Scholar
[18] Bakunov M I, Efimenko E S, Gorelov S D, Abramovsky N A, Bodrov S B 2020 Opt. Lett. 45 3533
Google Scholar
[19] Lu Y, Wang X, Miao L, Zuo D, Cheng Z 2011 Appl. Phys. B 103 387
Google Scholar
[20] Tochitsky S Y, Ralph J E, Sung C, Joshi C 2005 J. Appl. Phys. 98 026101
Google Scholar
[21] Zhong K, Yao J Q, Xu D G, Wang Z, Li Z Y, Zhang H Y, Wang P 2010 Opt. Commun. 283 3520
Google Scholar
[22] Jiang Y, Ding Y J 2007 Appl. Phys. Lett. 91 091108
Google Scholar
[23] Shi W, Ding Y J 2005 Opt. Lett. 30 1861
Google Scholar
[24] Brenier A 2018 Appl. Phys. B 124 194
Google Scholar
[25] Liu P X, Xu D G, Li J Q, Yan C, Li Z X, Wang Y Y, Yao J Q 2014 IEEE Photonics Technol. Lett. 26 494
Google Scholar
[26] Wu F, Wu J J, Guo Z W, Jiang H T, Sun Y, Li Y H, Ren J, Chen H 2019 Phys. Rev. Appl. 12 014028
Google Scholar
[27] Ning T Y, Li X, Zhao Y, Yin L Y, Huo Y Y, Zhao L N, Yue Q Y 2020 Opt. Express 28 34024
Google Scholar
[28] Wu F, Qin M B, Xiao S Y 2022 J. Appl. Phys. 132 193101
Google Scholar
[29] Wu F, Liu T T, Long Y, Xiao S Y, Chen G Y 2023 Phys. Rev. B 107 165428
Google Scholar
[30] Wu F, Qi X, Luo M, Liu T T, Xiao S Y 2023 Phys. Rev. B 108 165404
Google Scholar
[31] Wu F, Qi X, Qin M B, Luo M, Long Y, Wu J J, Sun Y, Jiang H T, Liu T T, Xiao S Y, Chen H 2024 Phys. Rev. B 109 085436
Google Scholar
[32] 闫梦, 孙珂, 宁廷银, 赵丽娜, 任莹莹, 霍燕燕 2023 物理学报 72 044202
Google Scholar
Yan M, Sun K, Ning T Y, Zhao L N, Ren Y Y, Huo Y Y 2023 Acta Phys. Sin. 72 044202
Google Scholar
[33] Sun K L, Wei H, Chen W J, Chen Y, Cai Y J, Qiu C W, Han Z H 2023 Phys. Rev. B 107 115415
Google Scholar
[34] Boyd R W 2020 Nonlinear Optics (London: Academic Press
[35] Jiang H, Han Z H 2022 J. Phys. D: Appl. Phys. 55 385106
Google Scholar
[36] Sutherland R L 2003 Handbook of Nonlinear Optics (New York: Marcel Dekker
[37] Amnon Yariv, Yeh P 1984 Optical Waves in Crystals (New York: Wiley
[38] Lu J, Ding B Y, Huo Y Y, Ning T Y 2018 Opt. Commun. 415 146
Google Scholar
[39] Liu W X, Li Y H, Jiang H T, Lai Z Q, Chen H 2013 Opt. Lett. 38 163
Google Scholar
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