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Optical nonreciprocity has been a popular research topic in recent years. Semiconductor quantum wells (SQWs) play a key role in many high-performance optoelectronic devices. In this paper, we propose a theoretical scheme to achieve nonmagnetic optical nonreciprocity based on the four-wave mixing effect in SQW nanostructures. Using the experimentally available parameters, the nonreciprocal behavior of the probe field in forward direction and backward direction is achieved through this SQW, where both nonreciprocal transmission and nonreciprocal phase shift have high transmission rates. Furthermore, by embedding this SQW nanostructure into a Mach-Zender interferometer, a reconfigurable nonreciprocal device based on high transmission nonreciprocal phase shift that can be used as an isolator or a circulator, is designed and analyzed. The device can be realized as a two-port optical isolator with an isolation ratio of 92.39 dB and an insertion loss of 0.25 dB, and as a four-port optical circulator with a fidelity of 0.9993, a photon survival probability of 0.9518 and a low insertion loss with suitable parameters. Semiconductor media have the advantages of easier integration and tunable parameters, and this scheme can provide theoretical guidance for implementing nonreciprocal and nonreciprocal photonic devices based on semiconductor solid-state media.
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Keywords:
- optical nonreciprocity /
- semiconductor quantum well /
- four-wave mixing
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Google Scholar
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图 1 本文考虑的理论模型 (a)被$ {\mathrm{Al}}_{0.33}{\mathrm{Ga}} _{0.67}{\mathrm{As}} $势垒隔开的GaAs调制掺杂量子阱结构的单周期能带示意图, 图中还显示了电子能级的位置及相应的波函数. (b)能级排列示意图. 前向传输时光场$ \varOmega_{{\mathrm{p}}}^{{\mathrm{f}}} $(红色), $ \varOmega_{{\mathrm{c}}} $(深蓝色), $ \varOmega_{{\mathrm{d}}} $(浅蓝色)与相应能级跃迁相互作用, 产生混频场$ \varOmega_{{\mathrm{m}}}^{{\mathrm{f}}} $(绿色); 后向传输时, 三个光场$ \varOmega_{{\mathrm{p}}}^{{\mathrm{b}}} $(玫红色), $ \varOmega_{{\mathrm{c}}} $, $ \varOmega_{{\mathrm{d}}} $耦合相应的能级跃迁. (c) SDQW样品与光场相互作用示意图
Figure 1. Theoretical model considered in this paper: (a) Schematic band diagram of a single period of the GaAs modulation-doped quantum wells structure separated by a $ {\mathrm{Al}}_{0.33}{\mathrm{Ga}} _{0.67}{\mathrm{As}} $ barrier. The positions of the calculated energy subbands and the corresponding modulus squared of the electronic wave functions are also displayed. (b) Schematic of the energy level arrangement. During forward transmission, the light fields $ \varOmega_{{\mathrm{p}}}^{{\mathrm{f}}} $ (red), $ \varOmega_{{\mathrm{c}}} $ (dark blue), $ \varOmega_{{\mathrm{d}}} $ (light blue) interact with the corresponding energy level transitions to generate a mixing field $ \varOmega_{{\mathrm{m}}}^{{\mathrm{f}}} $ (green); during backward transmission, the three light fields $ \varOmega_{{\mathrm{p}}}^{{\mathrm{b}}} $ (rose red), $ \varOmega_{{\mathrm{c}}} $, $ \varOmega_{{\mathrm{d}}} $ are coupled with corresponding energy level transitions. (c) Schematic diagram of the SDQW sample interacting with all optical fields.
图 2 前向探测场和后向探测场不同光学深度时的仿真图 (a)前向探测场$ \varOmega_{{\mathrm{p}}}^{{\mathrm{f}}} $和后向探测场$ \varOmega_{{\mathrm{p}}}^{{\mathrm{b}}} $在光学深度范围0—800内变化的相位图; (b)前向探测场和后向探测场的振幅随$ \alpha $的变化; (c)前向探测场和后向探测场的相移随$ \alpha $的变化. 其他参数取值为$ \gamma_{21}=0.2\; {\mathrm{meV }}$, $ \gamma_{31}=\gamma_{41}=2\; {\mathrm{meV }} $, $ \varOmega_{{\mathrm{c}}}=\varOmega_{{\mathrm{d}}}=24.5\varGamma $, $ \delta_{{\mathrm{p}}}=\delta_{{\mathrm{c}}}=0 $, $ \delta_{{\mathrm{d}}}=100.25\varGamma $, $ \varGamma=4\; {\mathrm{meV }} $
Figure 2. Simulation diagrams of the forward probe field and the backward probe field at different optical depths: (a) Phase diagram of the forward probe field $ \varOmega_{{\mathrm{p}}}^{{\mathrm{f}}} $ and the backward probe field $ \varOmega_{{\mathrm{p}}}^{{\mathrm{b}}} $ in the optical depth range 0 to 800; (b) diagram of the amplitude of the forward probe field and the backward probe field changing with $ \alpha $; (c) diagram of the phase shift of the forward probe field and the backward probe field changing with $ \alpha $. Other parameters are $ \gamma_{21}=0.2\; {\mathrm{meV }} $, $ \gamma_{31}=\gamma_{41}=2\; {\mathrm{meV }} $, $ \varOmega_{{\mathrm{c}}}=\varOmega_{{\mathrm{d}}}=24.5\varGamma $, $ \delta_{{\mathrm{p}}}=\delta_{{\mathrm{c}}}=0 $, $ \delta_{{\mathrm{d}}}=100.25\varGamma $, and $ \varGamma=4 \; {\mathrm{meV }}$.
图 3 不同探测场的振幅和相移在不同驱动失谐$ \delta_{{\mathrm{d}}} $时的仿真结果 (a)前向探测场和后向探测场的振幅随$ \delta_{{\mathrm{d}}} $的变化; (b)前向探测场和后向探测场的相移随$ \delta_{{\mathrm{d}}} $的变化. 其他参数取值与图2 相同, 除了$ \alpha $ = 630
Figure 3. Simulation results of the amplitude and phase shift of different probe fields under different detuning of driving fields $ \delta_{{\mathrm{d}}} $: (a) Change graph of the amplitude of the forward probe field and the backward probe field with $ \delta_{{\mathrm{d}}} $; (b) change graph of the phase shift of the forward probe field and the backward probe field with $ \delta_{{\mathrm{d}}} $. Other parameters are the same as in Fig. 2, except for $ \alpha $ = 630.
图 4 光隔离器和环行器装置的简单示意图. 对于光隔离器, 只考虑嵌入SDQW纳米结构的上分支. 为了实现光环行器, 下分支与上分支使用分束器BS 1和BS 2组成MZI
Figure 4. A simple schematic of the optical isolator and circulator devices. For the optical isolator, we consider only the upper branch embedded in the SDQW nanostructure. To implement the optical circulator, the lower branch and the upper branch use beam splitters BS 1 and BS 2 to form the MZI.
图 5 用作光隔离器时的仿真结果 (a)传输系数$ T_{12} $和$ T_{21} $随探测失谐$ \delta_{{\mathrm{p}}} $的变化图; (b)隔离比IR和插入损耗IL随探测失谐$ \delta_{{\mathrm{p}}} $的变化图. 除了$ \alpha=630 $, 其他参数取值与图2 相同
Figure 5. Simulation results when used as an optical isolator: (a) Graph of transmission coefficients $ T_{12} $ and $ T_{21} $ with probe detuning $ \delta_{{\mathrm{p}}} $; (b) graph of isolation ratio IR and insertion loss IL with probe detuning $ \delta_{{\mathrm{p}}} $. Parameters are the same as in Fig. 2, except that $ \alpha=630 $.
图 6 用作光环行器时的仿真结果 (a), (b)前向循环和后向循环传输系数与探测失谐$ \delta_{{\mathrm{p}}} $的关系; (c)光环行器的保真度F和平均光子存活率η与探测失谐$ \delta_{{\mathrm{p}}} $的关系. 除了$ \alpha=630 $, 其他参数取值与图2 相同
Figure 6. Simulation results when used as an optical circulator: (a), (b) Relationship between the transmission coefficients of the forward cycle and the backward cycle and the probe detuning $ \delta_{{\mathrm{p}}} $; (c) fidelity F and average survival probability η of the optical circulator versus probe detuning $ \delta_{{\mathrm{p}}} $. Parameters are the same as in Fig. 2 except for $ \alpha=630 $.
表 1 4个光隔离器性能指标的比较
Table 1. Comparing of the performance metrics for four optical isolators
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[1] Malz D, Tóth L D, Bernier N R, Feofanov A K, Kippenberg T J, Nunnenkamp A 2018 Phys. Rev. Lett. 120 023601
Google Scholar
[2] Hu Y, Qi Y, You Y, Zhang S, Lin G, Li X, Gong J, Gong S, Niu Y 2021 Phys. Rev. Appl. 16 014046
Google Scholar
[3] Muñoz H A, Carusotto I 2022 Phys. Rev. A 106 063523
Google Scholar
[4] Singh N, Kärtner F X 2022 Opt. Express 30 5973
Google Scholar
[5] Bi L, Hu J, Jiang P, Kim D H, Dionne G F, Kimerling L C, Ross C A 2011 Nat. Photonics 5 758
Google Scholar
[6] Gao J, Wang X, Han F, Wan J, Gu W 2023 Micromachines-basel 14 33
Google Scholar
[7] Yang Z, Cheng Y, Wang N, Chen Y, Wang S 2022 Opt. Express 30 27993
Google Scholar
[8] Liang J, Li Y, Dai T, Zhang Y, Zhang X, Liu H, Wang P 2023 Opt. Express 31 8375
Google Scholar
[9] Wang X, Ptitcyn G, Asadchy V S, Díaz R A, Mirmoosa M S, Fan S, Tretyakov S A 2020 Phys. Rev. Lett. 125 266102
Google Scholar
[10] Shah M, Briggs I, Chen P K, Hou S, Fan L 2023 Opt. Lett. 48 1978
Google Scholar
[11] 张利巍, 李贤丽, 杨柳 2019 物理学报 68 170701
Google Scholar
Zhang L W, Li X L, Yang L 2019 Acta Phys. Sin. 68 170701
Google Scholar
[12] Tang J S, Nie W, Tang L, Chen M, Su X, Lu Y, Nori F, Xia K 2022 Phys. Rev. Lett. 128 203602
Google Scholar
[13] Lan Y T, Su W J, Wu H, Li Y, Zheng S B 2022 Opt. Lett. 47 1182
Google Scholar
[14] 刘妮, 马硕, 梁九卿 2023 物理学报 72 060702
Google Scholar
Liu N, Ma S, Liang J Q 2023 Acta Phys. Sin. 72 060702
Google Scholar
[15] Xia K, Nori F, Xiao M 2018 Phys. Rev. Lett. 121 203602
Google Scholar
[16] You Y, Hu Y, Lin G, Qi Y, Niu Y, Gong S 2021 Phys. Rev. A 103 063706
Google Scholar
[17] Liang C, Liu B, Xu A N, Wen X, Lu C, Xia K, Tey M K, Liu Y C, You L 2021 Phys. Rev. Lett. 125 123901
Google Scholar
[18] Hu X X, Wang Z B, Zhang P, Chen G J, Zhang Y L, Li G, Zou X B, Zhang T, Tang H X, Dong C H 2021 Nat. Commun. 12 2389
Google Scholar
[19] 李鑫, 解舒云, 李林帆, 周海涛, 王丹, 杨保东 2022 物理学报 71 184202
Google Scholar
Li X, Xie S Y, Li L F, Zhou H T, Wang D, Yang B D 2022 Acta Phys. Sin. 71 184202
Google Scholar
[20] Zheng Y, Yang J, Shen Z, Cao J, Chen X, Liang X, Wan W 2015 Light-Sci. Appl. 5 e16072
Google Scholar
[21] Shui T, Yang W X, Cheng M T, Lee R K 2022 Opt. Express 30 6284
Google Scholar
[22] Rashidi S, Entezar S R, Rashidi A 2021 Appl. Opt. 60 8651
Google Scholar
[23] Mai J, Cheah K W 2022 Opt. Express 30 46357
Google Scholar
[24] Al M M, Xiang Q, Miura Y, Belmoubarik M, Masuda K, Kasai S, Sukegawa H, Mitani S 2021 Phys. Rev. B 103 L180408
Google Scholar
[25] Bouscher S, Panna D, Balasubramanian K, Cohen S, Ritter D, Hayat A 2022 Phys. Rev. Lett. 128 127701
Google Scholar
[26] Kinoshita K, Moriya R, Okazaki S, Zhang Y, Masubuchi S, Watanabe K, Taniguchi T, Sasagawa T, Machida T 2022 Nano Lett. 22 4640
Google Scholar
[27] Liu S, Yang W X, Chuang Y L, Chen A X, Liu A, Huang Y, Lee R K 2014 Opt. Express 22 29179
Google Scholar
[28] Hao X, Li J, Liu J, Song P, Yang X 2008 Phys. Lett. A 372 2509
Google Scholar
[29] Zhang Y, Wang Z, Qiu J, Hong Y, Yu B 2019 Appl. Phys. Lett. 115 171905
Google Scholar
[30] Qiu J, Wang Z, Ding D, Li W, Yu B 2020 Opt. Express 28 2975
Google Scholar
[31] Zhang S, Lin G, Hu Y, Qi Y, Gong S 2020 Phys. Rev. Appl. 14 024032
Google Scholar
[32] Faist J, Sirtori C, Capasso F, Pfeiffer L, West K W 1994 Appl. Phys. Lett. 64 872
Google Scholar
[33] Wang Z, Zhang Y, Paspalakis E, Yu B 2020 Phys. Rev. A 102 063509
Google Scholar
[34] Kok P, Munro W J, Nemoto K, Ralph T C, Dowling J P, Milburn G J 2007 Rev. Mod. Phys. 79 135
Google Scholar
[35] Li E Z, Ding D S, Yu Y C, Dong M X, Zeng L, Zhang W H, Ye Y H, Wu H Z, Zhu Z H, Gao W, Guo G C, Shi B S 2020 Phys. Rev. Res. 2 033517
Google Scholar
[36] Hsiao Y F, Chen H S, Tsai P J, Chen Y C 2014 Phys. Rev. A 90 055401
Google Scholar
[37] Cong G W, Akimoto R, Akita K, Hasama T, Ishikawa H 2007 Appl. Phys. Lett. 90 181919
Google Scholar
[38] Zhang C, Yang W, Song X, Xu Z 2009 J. Phys. B: At. Mol. Opt. Phys. 42 125604
Google Scholar
[39] 武香莲, 赵珂, 贾海洪, 王富青 2015 物理学报 64 233301
Google Scholar
Wu X L, Zhao K, Jia H H, Wang F Q 2015 Acta Phys. Sin. 64 233301
Google Scholar
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