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涡旋光束在双拉盖尔-高斯旋转腔中的非互易传输

朱雪松 刘星雨 张岩

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涡旋光束在双拉盖尔-高斯旋转腔中的非互易传输

朱雪松, 刘星雨, 张岩

Nonreciprocal transmission of vortex beam in double Laguerre-Gaussian rotational cavity system

Zhu Xue-Song, Liu Xing-Yu, Zhang Yan
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  • 通过构造两个线性耦合的拉盖尔-高斯旋转腔系统, 实现携带轨道角动量的涡旋光束的非互易传输现象. 系统中, 两个拉盖尔-高斯旋转腔模通过扭力与中间的旋转镜耦合, 同时两个涡旋腔场通过光纤直接耦合起来. 两个强光场分别驱动不同的腔模, 并利用一个弱探测场从系统一侧入射, 从而对该系统两个传播方向的光响应特性进行研究. 利用该系统哈密顿量和海森伯-郎之万方程, 结合输入-输出关系可得到系统的输出光谱. 结果表明此系统中的涡旋光束的非互易性来源于光旋转相互作用以及涡旋腔场相互作用之间的量子干涉效应. 因此, 可以通过调节非互易相位差来对系统的非互易传输进行调制. 此外, 两个涡旋光束所携带的拓扑荷比值会显著影响传输特性; 在适当的拓扑荷比值下, 该系统可以实现涡旋光束的单向传输. 本研究成果有望用于实现理想的涡旋光隔离器.
    By constructing an optorotational system composed of two linearly coupled Laguerre-Gaussian rotational cavities, we realize the nonreciprocal transmission of the vortex beam with the orbital angular momentum. Two vortex beam cavity modes driven by strong fields are coupled with a rotational mirror via the torsion, and two cavity modes interact with each other via the optical fiber. A weak probe field is incident from one side of the system for examining the optical response along one propagating direction. With the Hamiltonian of the system and the Heisenberg-Langevin equation, we can obtain the transmission of the output light field from the input-output theory. The result shows that the optical nonreciprocity of the vortex beam arises from the quantum interference between the optorotational interaction and the linear coupling interaction between two vortex beam modes, and the phase difference can be used to adjust the optical nonreciprocity. The phase difference can determine not only the occurrence of the nonreciprocity but also the direction of nonreciprocity. Moreover, the ratio of the topological charges carried by the two vortex beam fields has an influence on the transmission. Under an appropriate topological charge ratio, the unidirectional transmission of the vortex beam can be realized in such a system. It is found that whether the topological charge ratio is positive or negative, i.e. whether the vortex beam is left-hand beam or right-hand beam, does not affect the transmission; the influence of the topological charge on the transmission amplitude actually comes from the topological charge number carried by the vortex beam, due to the fact that the coupling strength between the rotating mirror mode and the cavity mode depends on the topological charge number. In addition, we also obtain the condition that the system damping rates should meet for realizing the perfect nonreciprocal propagation of the vortex beam. Finally, we can achieve the nonreciprocal group velocity of the slow light. The direction of the nonreciprocal slow light can be controlled via phase modulation. Our work provides a possible application in manipulating the vortex beam propagation. Furthermore, we extend the nonreciprocity of ordinary beams in the optomechanical system to the nonreciprocity of the vortex beam in the optorotational system. The results are expected to be applied to fabricating the ideal optical isolators for the vortex beam carrying the orbital angular momentum in optical communication.
      通信作者: 张岩, zhangy345@nenu.edu.cn
    • 基金项目: 国家自然科学基金(批准号: 11704064)、吉林省教育厅“十四五”科学技术项目(批准号: JJKH20211279KJ) 和中央高校基本科研业务费(批准号: 2412019FZ045) 资助的课题.
      Corresponding author: Zhang Yan, zhangy345@nenu.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11704064), the Science Foundation of the Education Department of Jilin Province during the 14th Five-Year Plan Period, China (Grant No. JJKH20211279KJ), and the Fundamental Research Fund for the Central Universities, China (Grant No. 2412019FZ045).
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    Andersen M F, Ryu C, Clade P, Natarajan V, Vaziri A, Helmerson K, Phillips W D 2006 Phys. Rev. Lett. 97 170406Google Scholar

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    Volpe G, Petrov D 2006 Phys. Rev. Lett. 97 210603Google Scholar

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    Liu Y M, Bai C H, Wang D Y, Wang T, Zheng M H, Wang H F, Zhu A D, Zhang S 2018 Opt. Express 26 6143Google Scholar

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    Deng Z J, Habraken S J M, Marquardt F 2016 New. J. Phys. 18 063022Google Scholar

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  • 图 1  双拉盖尔-高斯旋转腔示意图, 两L-G腔场通过光旋转相互作用与中间的旋转腔镜耦合, 振幅为$ {\varepsilon _{\text{c}}} $$ {\varepsilon _{\text{d}}} $($ {\varepsilon _{\text{L}}} $$ {\varepsilon _{\text{R}}} $)的强驱动场(弱探测场)分别从两侧入射系统来驱动L-G腔模$ {c_1} $$ {c_{\text{2}}} $, 同时用光纤将两L-G腔模线性耦合. 旋转腔镜的平衡位置为$ {\phi _{\text{0}}} $, 在扭力作用下的角位移用$ \phi $角表示

    Fig. 1.  Schematic diagram of double Laguerre-Gaussian (L-G) rotational-cavity. The two L-G cavity modes are coupled with a rotating cavity mirror in the middle via the optical rotation interaction. Two strong pump fields (weak probe fields) with amplitudes $ {\varepsilon _{\text{c}}} $ and $ {\varepsilon _{\text{d}}} $ ($ {\varepsilon _{\text{L}}} $ and $ {\varepsilon _{\text{R}}} $) are incident on the system from both sides to drive the L-G cavity modes $ {c_1} $ and $ {c_{\text{2}}} $, and the two L-G cavity modes are linearly coupled with an optical fiber. The equilibrium position of the rotational mirror is $ {\phi _{\text{0}}} $, and the angular displacement is indicated by angle $ \phi $ under the action of the torsion.

    图 2  传输振幅$ {T_{12}} $(红色实线)和$ {T_{21}} $(蓝色虚线)在不同非互易相位差$ \theta $下随标准化失谐$ x/{\kappa _1} $的演化 (a) $ \theta = 0 $; (b) $ \theta = - {\text{π }}/4 $; (c) $ \theta = {\text{π }}/4 $; (d) $ \theta = - {\text{π }}/2 $; (e) $ \theta = {\text{π }}/2 $. (f)在标准化失谐为零(x = 0)时, 传输振幅$ {T_{12}} $(红色实线)和$ {T_{21}} $(蓝色虚线)随非互易相位差$ \theta $的演化. 其他参数为$ {\kappa _1} = 1 $, $ {\kappa _2} = 1 $, $ \gamma = 1 $, $ {G_i} = \sqrt {{\kappa _i}\gamma } /2 $, $ J = \sqrt {{\kappa _1}{\kappa _2}} /2 $.

    Fig. 2.  Transmission amplitudes $ {T_{12}} $ (red solid line) and $ {T_{21}} $ (blue dotted line) versus normalized detuning $ x/{\kappa _1} $ under different nonreciprocal phase difference: (a) $ \theta = 0 $; (b) $ \theta = - {\text{π }}/4 $; (c) $ \theta = {\text{π }}/4 $; (d) $ \theta = - {\text{π }}/2 $; (e) $ \theta = {\text{π }}/2 $. (f) Transmission amplitudes $ {T_{12}} $ (red solid line) and $ {T_{21}} $ (blue dotted line) versus nonreciprocal phase difference with $ x = 0 $. Other parameters are $ {\kappa _1} = 1 $, $ {\kappa _2} = 1 $, $ \gamma = 1 $, $ {G_i} = \sqrt {{\kappa _i}\gamma } /2 $, $ J = \sqrt {{\kappa _1}{\kappa _2}} /2 $.

    图 3  (a)传输振幅$ {T_{12}} $和(b)传输振幅$ {T_{{\text{21}}}} $作为标准化失谐$ x/{\kappa _1} $和旋转镜与腔$ {c_1} $耗散比$ \gamma /{\kappa _1} $的函数; (c)传输振幅$ {T_{12}} $和(d)传输振幅$ {T_{{\text{21}}}} $作为标准化失谐$ x/{\kappa _1} $和腔$ {c_2} $与腔$ {c_1} $耗散比$ {\kappa _2}/{\kappa _1} $的函数. 其他参数为$ \theta = {\text{π }}/2 $, $ {\kappa _1} = 1 $, $ {G_1} = {G_2} = J = 1/2 $

    Fig. 3.  (a) Transmission amplitude $ {T_{12}} $ and (b) transmission amplitude $ {T_{21}} $ as a function of the normalized detuning $ x/{\kappa _1} $ and the dissipation ratio $ \gamma /{\kappa _1} $ of rotating mirror to cavity $ {c_1} $; (c) transmission amplitudes $ {T_{12}} $ and (d) transmission amplitudes $ {T_{21}} $ as a function of the normalized detuning $ x/{\kappa _1} $ and the dissipation ratio $ {\kappa _2}/{\kappa _1} $ of cavity $ {c_2} $ to cavity $ {c_1} $. Other parameters are $ \theta = {\text{π }}/2 $, $ {\kappa _1} = 1 $, $ {G_1} = {G_2} = J = 1/2 $.

    图 4  传输振幅$ {T_{12}} $(红色圆圈)和$ {T_{21}} $(蓝色圆圈)在不同非互易相位差$ \theta $下随两个涡旋光场所携带拓扑荷之比$ {\ell _2}/{\ell _1} $的演化 (a) $\theta = 0$; (b) $ \theta = - {\text{π }}/4 $; (c) $ \theta = {\text{π }}/4 $; (d) $ \theta = - {\text{π }}/2 $; (e) $ \theta = {\text{π }}/2 $; (f) $ \theta = {\text{π }} $. 其他参数为$ {\kappa _1} = 1 $, $ {\kappa _2} = 1 $, $ \gamma = 1 $, $ J = 2{G_1}{G_2}/\gamma $

    Fig. 4.  Transmission amplitudes $ {T_{12}} $ (red circle) and $ {T_{21}} $ (blue circle) with ratio of topological charges carried by two vortex optical fields $ {\ell _2}/{\ell _1} $ under different nonreciprocal phase difference: (a) $ \theta = 0 $; (b)$ \theta = - {\text{π }}/4 $; (c)$ \theta = {\text{π }}/4 $; (d) $ \theta = - {\text{π }}/2 $; (e) $ \theta = {\text{π }}/2 $; (f) $ \theta = {\text{π }} $. Other parameters are $ {\kappa _1} = 1 $, $ {\kappa _2} = 1 $, $ \gamma = 1 $, $ J = 2{G_1}{G_2}/\gamma $.

    图 5  群延迟$ {\tau _{12}} $(红色圆圈)和$ {\tau _{21}} $(蓝色圆圈)在不同非互易相位差$ \theta $下随两个涡旋光场所携带拓扑荷比$ {\ell _2}/{\ell _1} $的演化 (a) $\theta = 0$; (b) $ \theta = - {\text{π }}/2 $; (c) $ \theta = {\text{π }}/2 $. 其他参数为$ {\kappa _1} = 1 $, $ {\kappa _2} = 1 $, $ \gamma = 1 $, $ J = 2{G_1}{G_2}/\gamma $

    Fig. 5.  Group delay $ {\tau _{12}} $ (red circle) and $ {\tau _{21}} $ (blue circle) with ratio of topological charges carried by two vortex optical fields $ {\ell _2}/{\ell _1} $ under different nonreciprocal phase difference: (a) $ \theta = 0 $; (b) $ \theta = - {\text{π }}/2 $; (c) $ \theta = {\text{π }}/2 $. Other parameters are $ {\kappa _1} = 1 $, $ {\kappa _2} = 1 $, $ \gamma = 1 $, $ J = 2{G_1}{G_2}/\gamma $.

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    Allen L, Beijersbergen M W, Spreeuw R J, Woerdman J P 1992 Phys. Rev. A 45 8185Google Scholar

    [2]

    Yao A M, Padgett M J 2011 Adv. Opt. Photonics 3 161Google Scholar

    [3]

    He H, Friese M E J, Heckenberg N R, Rubinsztein-Dunlop H 1995 Phys. Rev. Lett. 75 826Google Scholar

    [4]

    Andersen M F, Ryu C, Clade P, Natarajan V, Vaziri A, Helmerson K, Phillips W D 2006 Phys. Rev. Lett. 97 170406Google Scholar

    [5]

    Volpe G, Petrov D 2006 Phys. Rev. Lett. 97 210603Google Scholar

    [6]

    Bhattacharya M, Meystre P 2007 Phys. Rev. Lett. 99 153603Google Scholar

    [7]

    Liu Y M, Bai C H, Wang D Y, Wang T, Zheng M H, Wang H F, Zhu A D, Zhang S 2018 Opt. Express 26 6143Google Scholar

    [8]

    Peng J X, Chen Z, Yuan Q Z, Feng X L 2019 Phys. Rev. A 99 043817Google Scholar

    [9]

    Zhang Z, Wang Y P, Wang X 2021 Front. Phys. 16 32503Google Scholar

    [10]

    Kazemi S H, Mahmoudi M 2020 Phys. Scr. 95 045107Google Scholar

    [11]

    Bhattacharya M, Giscard P L, Meystre P 2008 Phys. Rev. A 77 013827Google Scholar

    [12]

    Cheng H J, Zhou S J, Peng J X, Kundu A, Li H X, Jin L, Feng X L 2021 J. Opt. Soc. Am. B 38 285Google Scholar

    [13]

    Xiong H, Huang Y M, Wu Y 2021 Phys. Rev. A 103 043506Google Scholar

    [14]

    Law C K 1995 Phys. Rev. A 51 2537Google Scholar

    [15]

    Xiong H, Si L G, Zheng A S, Yang X, Wu Y 2012 Phys. Rev. A 86 013815Google Scholar

    [16]

    Tian L 2013 Phys. Rev. Lett. 110 233602Google Scholar

    [17]

    Deng Z J, Habraken S J M, Marquardt F 2016 New. J. Phys. 18 063022Google Scholar

    [18]

    Deng Z J, Yan X B, Wang Y D, Wu C W 2016 Phys. Rev. A 93 033842Google Scholar

    [19]

    Yan X B 2017 Phys. Rev. A 96 053831Google Scholar

    [20]

    张秀龙, 鲍倩倩, 杨明珠, 田雪松 2018 物理学报 67 104203Google Scholar

    Zhang X L, Bao Q Q, Yang M Z, Tian X S 2018 Acta Phys. Sin. 67 104203Google Scholar

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    Agarwal G S, Huang S M 2010 Phys. Rev. A 81 041803Google Scholar

    [22]

    Shu J 2011 Chin. Phys. Lett. 28 104203Google Scholar

    [23]

    陈华俊, 米贤武 2011 物理学报 60 124206Google Scholar

    Chen H J, Mi X W 2011 Acta Phys. Sin. 60 124206Google Scholar

    [24]

    He B, Yang L, Lin Q, Xiao M 2017 Phys. Rev. Lett. 118 233604Google Scholar

    [25]

    Li Y, Wu L A, Wang Z D 2011 Phys. Rev. A 83 043804Google Scholar

    [26]

    Xiong H, Fan Y W, Yang X, Wu Y 2016 Appl. Phys. Lett. 109 061108Google Scholar

    [27]

    Chen Y T, Du L, Zhang Y, Wu J H 2021 Phys. Rev. A 103 053712Google Scholar

    [28]

    Ge Y Q, Chen Y T, Yin K, Zhang Y 2020 Phys. Lett. A 384 126836Google Scholar

    [29]

    Chen Y T, Du L, Liu Y M, Zhang Y 2020 Opt. Express 28 7095Google Scholar

    [30]

    Bi L, Hu J, Jiang P, Kim D H, Dionne G F, Kimerling L C, Ross C A 2011 Nat. Photonics 5 758Google Scholar

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    Aplet L J, Carson J W 1964 Appl. Opt. 3 544Google Scholar

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    Korneeva Y P, Vodolazov D Y, Semenov A V, Florya I N, Simonov N, Baeva E, Korneev A A, Goltsman G N, Klapwijk T M 2018 Phys. Rev. Appl. 9 064037Google Scholar

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    Maayani S, Dahan R, Kligerman Y, Moses E, Hassan A U, Jing H, Nori F, Christodoulides D N, Carmon T 2018 Nature 558 569Google Scholar

    [35]

    Li B, Huang R, Xu X W, Miranowicz A, Jing H 2019 Photonics Res. 7 630Google Scholar

    [36]

    Li Y, Huang Y Y, Zhang X Z, Tian L 2017 Opt. Express 25 018907Google Scholar

    [37]

    Mirza I M, Ge W C, Jing H 2019 Opt. Express 27 25515Google Scholar

    [38]

    Jiang C, Song L N, Li Y 2018 Phys. Rev. A 97 053812Google Scholar

    [39]

    Jiang C, Baowei J I, Cui Y S, Zuo F, Shi J, Chen G 2018 Opt. Express 26 15255Google Scholar

    [40]

    Jiang C, Song L N, Li Y 2019 Phys. Rev. A 99 023823Google Scholar

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    Xu X W, Li Y 2015 Phys. Rev. A 91 053854Google Scholar

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    Xu X W, Li Y, Chen A X, Liu Y X 2016 Phys. Rev. A 93 023827Google Scholar

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    Xu X W, Song L N, Zheng Q, Wang Z H, Li Y 2018 Phys. Rev. A 98 063845Google Scholar

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    Yan X B, Lu H L, Gao F, Yang L 2019 Front. Rhys. 14 52601Google Scholar

    [45]

    Xia C C, Yan X B, Tian X D, Gao F 2019 Opt. Commun. 451 197Google Scholar

    [46]

    张利巍, 李贤丽, 杨柳 2019 物理学报 68 170701Google Scholar

    Zhang L W, Li X L, Yang L 2019 Acta Phys. Sin. 68 170701Google Scholar

    [47]

    Agarwal G S, Huang S 2014 New J. Phys. 16 033023Google Scholar

    [48]

    Yan X B, Cui C L, Gu K H, Tian X D, Fu C B, Wu J H 2014 Opt. Express 22 4886Google Scholar

    [49]

    Safavi-Naeini A H, Mayer Alegre T P, Chan J, Eichenfield M, Winger M, Lin Q, Hill J T, Chang D E, Painter O 2011 Nature 472 69Google Scholar

    [50]

    Weis S, Riviere R, Deleglise S, Gavartin E, Arcizet O, Schliesser A, Kippenberg T J 2010 Science 330 1520Google Scholar

    [51]

    Oemrawsingh S S R, Eliel E R, Woerdman J P, Verstegen E J K, Kloosterboer J G, 't Hooft G W 2004 J. Opt. A: Pure Appl. Opt. 6 S228

    [52]

    Li L, Allen Y Y, Huang C, Grewell D A, Benatar A, Chen Y 2006 Opt. Eng. 45 113401Google Scholar

    [53]

    Shi H, Bhattacharya M 2016 J. Phys. B 49 153001Google Scholar

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出版历程
  • 收稿日期:  2022-01-26
  • 修回日期:  2022-04-19
  • 上网日期:  2022-07-25
  • 刊出日期:  2022-08-05

涡旋光束在双拉盖尔-高斯旋转腔中的非互易传输

  • 东北师范大学物理学院, 物理学师范专业国家级实验教学示范中心, 长春 130024
  • 通信作者: 张岩, zhangy345@nenu.edu.cn
    基金项目: 国家自然科学基金(批准号: 11704064)、吉林省教育厅“十四五”科学技术项目(批准号: JJKH20211279KJ) 和中央高校基本科研业务费(批准号: 2412019FZ045) 资助的课题.

摘要: 通过构造两个线性耦合的拉盖尔-高斯旋转腔系统, 实现携带轨道角动量的涡旋光束的非互易传输现象. 系统中, 两个拉盖尔-高斯旋转腔模通过扭力与中间的旋转镜耦合, 同时两个涡旋腔场通过光纤直接耦合起来. 两个强光场分别驱动不同的腔模, 并利用一个弱探测场从系统一侧入射, 从而对该系统两个传播方向的光响应特性进行研究. 利用该系统哈密顿量和海森伯-郎之万方程, 结合输入-输出关系可得到系统的输出光谱. 结果表明此系统中的涡旋光束的非互易性来源于光旋转相互作用以及涡旋腔场相互作用之间的量子干涉效应. 因此, 可以通过调节非互易相位差来对系统的非互易传输进行调制. 此外, 两个涡旋光束所携带的拓扑荷比值会显著影响传输特性; 在适当的拓扑荷比值下, 该系统可以实现涡旋光束的单向传输. 本研究成果有望用于实现理想的涡旋光隔离器.

English Abstract

    • 由于在空间光通信、量子纠缠、粒子操控等领域的应用, 光学涡旋已成为近年来的研究热点. 与普通光束相比, 涡旋光束可以携带轨道角动量. 拉盖尔-高斯(L-G)光束就是一种典型的涡旋光束, 其具有螺旋相位结构和环形场强分布, 光束中心的强度为零[1]. L-G光束沿其传播方向携带每光子$ \ell \hbar $的轨道角动量, $ \ell $表示L-G光场的拓扑荷数[2].

      实验表明, 通过交换轨道角动量, L-G光束可以对物体施加扭矩, 包括微观吸附颗粒[3]、介观玻色-爱因斯坦凝聚[4]及亚微米布朗粒子[5]. 2007年, 基于此性质, 拉盖尔-高斯旋转腔系统被提出[6], 该系统由两个用作腔镜的螺旋相位元件构成, 分别为一个固定腔镜和一个围绕腔轴旋转的旋转腔镜(可视为扭摆). 在这种光旋转系统中, 腔模是L-G光, 它携带轨道角动量, 可以与螺旋相位元件交换轨道角动量. 近期人们对此类系统的兴趣逐渐增加, 并做了一些相关研究, 包括旋转腔镜的冷却[6,7]、光场轨道角动量的检测[8,9]、二阶边带效应[10]和纠缠[11,12]. 然而, 据我们所知, 关于L-G光束在拉盖尔-高斯旋转腔中的非互易传输现象至今还未被报道过. 由于光旋转相互作用[13]和光机械相互作用[14,15]之间存在类似的哈密顿量, 光机械相互作用产生的许多效应[16-29], 在拉盖尔-高斯旋转腔中通过与L-G光场轨道角动量的交换也会发生[6-12]. 这为利用光的轨道角动量而不是动量提供了可能性.

      光的传播通常是互易的, 不过实现光学非互易对于构建非对称量子网络极为重要. 光学非互易性描述的是光场从一个方向通过光学系统但是不能沿原路返回的特性. 光学非互易是实现定向放大器、隔离器、循环器的基础[30], 并且需要打破时间反演对称性. 传统上, 非互易传输依赖于施加的偏置磁场 [31], 但这些传统器件通常体积庞大, 与超低损耗超导电路不兼容, 并且需要相当大的磁场. 近年来, 有人利用辐射-压力诱导光力耦合来打破时间反转对称性, 实现光的非互易效应[32-46]. 这一领域已经取得了重要成果, 包括非互易传输和放大[32-34]、非互易单光子效应[35]、非互易慢光[37]等. 文献[38]提出了一种基于具有光增益的光机械系统来实现光场和微波场之间光定向放大的方案, 发现放大方向可以通过调节有效光机耦合之间的相对相位(非互易相位差[46])来控制. 文献[43]给出了由两个光学膜和一个机械模组成的三模光机械系统中的光学非互易响应, 并证明了光学非互易响应是通过调节光机械耦合速率之间的相位差来打破系统的时间反转对称性来实现的. 然而, 通过光旋转耦合来实现携带轨道角动量的涡旋光束的非互易传输还有待进一步研究, 且之前工作在利用光力耦合来实现普通光束的非互易传输时, 都是利用光学模式与机械模式交换动量来实现, 而非交换角动量.

      本文通过在单个拉盖尔-高斯旋转腔中额外加一个固定腔镜来构建双拉盖尔-高斯旋转腔, 并通过光纤将两个腔模线性耦合, 并且用两个强驱动场驱动不同的腔模, 进而研究涡旋光束的非互易传输现象. 将以往通过光与机械振子交换动量来实现普通光束的非互易传输推广到光与旋转腔镜交换角动量来实现涡旋光束的非互易传输. 研究发现, 非互易相位差可以决定涡旋光束非互易传输的产生及方向; 在一定的拓扑荷比值下, 该系统可以实现完美的非互易性; 系统耗散对传输振幅的变化有比较大的影响; 此外, 通过调节非互易相位差, 在该系统中可以实现非互易的慢光效应. 在此系统中的涡旋光束传输的非互易性, 来源于光旋转耦合相互作用和腔模耦合相互作用形成的两条路径之间发生的量子干涉效应.

    • 图1所示, 本方案考虑了双拉盖尔-高斯旋转腔光旋转系统, 其中3个腔镜均为螺旋相位元件, 其中两个输入腔镜部分透明且刚性固定, 不改变透过它们光束的轨道角动量而会给反射光束增加$ {{ - 2}}{\ell _{\text{1}}} $$ {{ - 2}}{\ell _{\text{2}}} $的拓扑荷; 处于中间位置的旋转腔镜是完全反射的, 可以给反射光增加$ +2{\ell _{\text{1}}} $$ +2{\ell _{\text{2}}} $的拓扑荷, 并可以在支架上绕腔轴$ z $旋转(角平衡位置$ {\phi _{{0}}}=0 $); 同时两个腔场通过光纤线性耦合在一起. 两个频率均为$ {\omega _{\text{c}}} $, 振幅分别为$ {\varepsilon _{\text{c}}} $$ {\varepsilon _{\text{d}}} $的强驱动场分别从两侧注入系统驱动频率为$ {\omega _0} $的L-G腔模. 利用一束频率为$ {\omega _{\text{p}}} $、振幅为$ {\varepsilon _{\text{L}}} $($ {\varepsilon _{\text{R}}} $)的弱探测场从系统左侧(右侧)入射, 从而检查该光旋转系统相对于左侧(右侧)探测光的响应特性. 在驱动场频率$ {\omega _{\text{c}}} $的旋转框架下, 系统的哈密顿量($ \hbar $ = 1)可写为

      图  1  双拉盖尔-高斯旋转腔示意图, 两L-G腔场通过光旋转相互作用与中间的旋转腔镜耦合, 振幅为$ {\varepsilon _{\text{c}}} $$ {\varepsilon _{\text{d}}} $($ {\varepsilon _{\text{L}}} $$ {\varepsilon _{\text{R}}} $)的强驱动场(弱探测场)分别从两侧入射系统来驱动L-G腔模$ {c_1} $$ {c_{\text{2}}} $, 同时用光纤将两L-G腔模线性耦合. 旋转腔镜的平衡位置为$ {\phi _{\text{0}}} $, 在扭力作用下的角位移用$ \phi $角表示

      Figure 1.  Schematic diagram of double Laguerre-Gaussian (L-G) rotational-cavity. The two L-G cavity modes are coupled with a rotating cavity mirror in the middle via the optical rotation interaction. Two strong pump fields (weak probe fields) with amplitudes $ {\varepsilon _{\text{c}}} $ and $ {\varepsilon _{\text{d}}} $ ($ {\varepsilon _{\text{L}}} $ and $ {\varepsilon _{\text{R}}} $) are incident on the system from both sides to drive the L-G cavity modes $ {c_1} $ and $ {c_{\text{2}}} $, and the two L-G cavity modes are linearly coupled with an optical fiber. The equilibrium position of the rotational mirror is $ {\phi _{\text{0}}} $, and the angular displacement is indicated by angle $ \phi $ under the action of the torsion.

      $\begin{split} \;& H = {\varDelta _{\text{c}}}(c_1^\dagger {c_1} + c_2^\dagger {c_2}) + \frac{{L_z^2}}{{2I}} + \frac{1}{2}I\omega _\phi ^2{\phi ^2} \\ &~~+ {g_{\phi 1}}c_1^\dagger {c_1}\phi + {g_{\phi 2}}c_2^\dagger {c_2}\phi + J(c_1^\dagger {c_2} + c_2^\dagger {c_1}) \\ &~~ + {\text{i}}({\varepsilon _{\text{c}}}c_1^\dagger - \varepsilon _{\text{c}}^ * {c_1})+ {\text{i}}({\varepsilon _{\text{d}}}c_2^\dagger - \varepsilon _{\text{d}}^ * {c_2}) \\ &~~ + {\text{i}}{\varepsilon _{\text{L}}}(c_1^\dagger {{\text{e}}^{ - {\text{i}}\delta t}} - {c_1}{{\text{e}}^{{\text{i}}\delta t}}) + {\text{i}}{\varepsilon _{\text{R}}}(c_2^\dagger {{\text{e}}^{ - {\text{i}}\delta t}} - {c_2}{{\text{e}}^{{\text{i}}\delta t}}) \text{, } \end{split}$

      式中, 第1项表示L-G腔模的自由哈密顿量, 其中${c_1} (c_1^\dagger )$$ {c_2}(c_2^\dagger ) $为两个腔的湮灭(产生)算符, ${\varDelta _{\text{c}}} = {\omega _0} - {\omega _{\text{c}}}$为驱动场与腔模的失谐; 第2项和第3项给出了旋转镜的自由哈密顿量, $ {\omega _\phi } $, $ {L_z} $$ \phi $分别为旋转镜的角频率、角动量和角位移, $ I = M{R^2}/2 $为旋转镜的转动惯量, 其中$ M $$ R $分别为旋转镜的质量和半径; 第4项和第5项表示L-G腔模与旋转镜的光旋转耦合, 单光子耦合强度分别为$ {g_{\phi 1}} = c{\ell _1}/L $$ {g_{\phi 2}} = c{\ell _2}/L $, 其中L为两个腔的长度; 第6项表示两个腔模的线性耦合, 其中$ J $为线性耦合强度; 最后4项分别表示强驱动场和弱探测场对腔模的驱动, $ \delta = {\omega _{\text{p}}} - {\omega _{\text{c}}} $为探测场与驱动场之间的失谐.

      为了方便计算, 设定$b = \sqrt {\dfrac{{I{\omega _\phi }}}{2}} \Big(\phi + \dfrac{{\text{i}}}{{I{\omega _\phi }}}{L_z}\Big)$${b^\dagger } = \sqrt {\dfrac{{I{\omega _\phi }}}{2}} \Big(\phi - \dfrac{{\text{i}}}{{I{\omega _\phi }}}{L_z}\Big)$分别表示转动腔镜的湮灭(产生)算符, 系统的哈密顿量可以写为

      $ \begin{split} H = \;&{\varDelta _{\text{c}}}(c_1^\dagger {c_1} + c_2^\dagger {c_2}) + {\omega _\phi }{b^\dagger }b \\ &+ ({g_{\phi 1}}c_1^\dagger {c_1} + {g_{\phi 2}}c_2^\dagger {c_2})({b^\dagger } + b)\\ & + J(c_1^\dagger {c_2} + c_2^\dagger {c_1}) + {\text{i}}({\varepsilon _{\text{c}}}c_1^\dagger - {\varepsilon _{\text{c}}}^ * {c_1})\\ & + {\text{i}}({\varepsilon _{\text{d}}}c_2^\dagger - \varepsilon _{\text{d}}^ * {c_2}) + {\text{i}}{\varepsilon _{\text{L}}}(c_1^\dagger {{\text{e}}^{ - {\text{i}}\delta t}} - {c_1}{{\text{e}}^{{\text{i}}\delta t}}) \\ &+ {\text{i}}{\varepsilon _{\text{R}}}(c_2^\dagger {{\text{e}}^{ - {\text{i}}\delta t}} - {c_2}{{\text{e}}^{{\text{i}}\delta t}}) .\end{split} $

      由系统的哈密顿量, 可以写出描述系统的动力学的相关算符的海森伯-郎之万方程:

      $\begin{split} &{{\dot c}_{1}}=- \Big[\text{i}{\varDelta }_{\text{c}}+\frac{{\kappa }_{1}}{2}+\text{i}{g}_{\phi 1}({b}^{†}+b)\Big]{c}_{1}\\ &\qquad+{\varepsilon }_{\text{c}}+{\varepsilon }_{\text{L}}{\text{e}}^{-\text{i}\delta t}-\text{i}J{c}_{2} \text{, } \\ & {{\dot c}_{2}}=-\Big[\text{i}{\varDelta }_{\text{c}} + \frac{{\kappa }_{2}}{2}+\text{i}{g}_{\phi 2}({b}^{†}+b)\Big]{c}_{2}\\ &\qquad+{\varepsilon }_{\text{d}}+{\varepsilon }_{\text{R}}{e}^{-\text{i}\delta t}-\text{i}J{c}_{1} \text{, } \\ &\dot{b}=-\text{i}{\omega }_{m}b-\frac{{\gamma }_{\phi }}{2}b-\text{i}({g}_{\phi 1}{c}_{1}^{†}{c}_{1}+{g}_{\phi 2}{c}_{\text{2}}^{†}{c}_{\text{2}}) \text{, } \end{split}$

      其中$ {\kappa _j} $$ {\gamma _\phi } $分别为两腔和转动腔镜的弛豫速率. 在乘积算符的平均值满足分解假设$ \left\langle {b{c_j}} \right\rangle = \left\langle b \right\rangle \left\langle {{c_j}} \right\rangle $且忽略探测场的情况下, 可得到算符的稳态解:

      $ \begin{split} &{b_{\text{s}}} = \frac{{ - {\text{i}}({g_{\phi 1}}{{\left| {{c_{1{\text{s}}}}} \right|}^2} + {g_{\phi 2}}{{\left| {{c_{2{\text{s}}}}} \right|}^2})}}{{\dfrac{{{\gamma _\phi }}}{2} + {\text{i}}{\omega _\phi }}} \text{, } \\ &{c_{1{\text{s}}}} = \frac{{\left(\dfrac{{{\kappa _2}}}{2} + {\text{i}}{\varDelta _2}\right){\varepsilon _{\text{c}}} - {\text{i}}J{\varepsilon _{\text{d}}}}}{{{J^2} + \left(\dfrac{{{\kappa _1}}}{2} + {\text{i}}{\varDelta _1}\right)\left(\dfrac{{{\kappa _2}}}{2} + {\text{i}}{\varDelta _2}\right)}} \text{, } \\ &{c_{{\text{2s}}}} = \frac{{\left(\dfrac{{{\kappa _1}}}{2} + {\text{i}}{\varDelta _1}\right){\varepsilon _{\text{d}}} - {\text{i}}J{\varepsilon _{\text{c}}}}}{{{J^2} + \left(\dfrac{{{\kappa _1}}}{2} + {\text{i}}{\varDelta _1}\right)\left(\dfrac{{{\kappa _2}}}{2} + {\text{i}}{\varDelta _2}\right)}} \text{, }\end{split} $

      其中${\varDelta _{1, 2}} = {\varDelta _{\text{c}}} + {g_{\phi 1, \phi 2}}({b_{\text{s}}} + b_{\text{s}}^{\text{*}})$为L-G腔模和驱动场之间的有效失谐.

      在强驱动场的驱动下, 每个算符可以由其平均值和量子涨落的和表示, 即$ {c_j} = {c_{j{\text{s}}}} + {\text{δ}}{c_j} $, $b = {b_{\text{s}}} + {\text{δ}}b$. 将$ {c_j} $$ b $的表达式代入方程(3), 可以得到线性化的海森伯-郎之万方程:

      $ \begin{split} & {\text{δ}\dot{c}_{1}}=-\left(\frac{{\kappa }_{1}}{2}+\text{i}{\varDelta }_{1}\right)\text{δ}{c}_{1}-\text{i}{g}_{\phi 1}{c}_{1s}(\text{δ}{b}^{†}+\text{δ}b)+{\varepsilon }_{\text{L}}{e}^{-\text{i}\delta t}-\text{i}J\text{δ}{c}_{2} \text{, } \\ & {\text{δ}\dot{c}_{2}}=-\left(\frac{{\kappa }_{2}}{2}+\text{i}{\varDelta }_{2}\right)\text{δ}{c}_{2}-\text{i}{g}_{\phi 2}{c}_{\text{2s}}(\text{δ}{b}^{†}+\text{δ}b)+{\varepsilon }_{\text{R}}{\text{e}}^{-\text{i}\delta t}-\text{i}J\text{δ}{c}_{1} \text{, } \\ & {\text{δ}\dot b}=-\left(\frac{{\gamma }_{\phi }}{2}+\text{i}{\omega }_{\phi }\right)\text{δ}b-\text{i}{g}_{\phi 1}({c}_{1\text{s}}\text{δ}{c}_{1}^{†}+{c}_{1\text{s}}^{\ast }\text{δ}{c}_{1})-\text{i}{g}_{\phi 2}({c}_{2\text{s}}\text{δ}{c}_{2}^{†}+{c}_{2\text{s}}^{\ast }\text{δ}{c}_{2}) \text{, } \end{split}$

      采用变换$ {\text{δ}}b \to {\text{δ}}b{{\text{e}}^{ - {\text{i}}{\omega _\phi }t}} $, ${\text{δ}}{c_1} \to {\text{δ}}{c_1}{{\text{e}}^{ - {\text{i}}{\varDelta _1}t}}$, ${\text{δ}}{c_2} \to {\text{δ}}{c_2}{{e} ^{ - {\text{i}}{\varDelta _2}t}}$, 则方程(5)为

      $ \begin{split} & {\text{δ}\dot{c}_{1}}=-\frac{{\kappa }_{1}}{2}\text{δ}{c}_{1}-\text{i}{G}_{1}\left[\text{δ}{b}^{†}{\text{e}}^{\text{i}({\omega }_{\phi }+{\varDelta }_{1})t}+\text{δ}b{\text{e}}^{-\text{i}({\omega }_{\phi }-{\varDelta }_{1})t}\right]+{\varepsilon }_{\text{L}}{\text{e}}^{-\text{i}(\delta -{\varDelta }_{1})t}-\text{i}J\text{δ}{c}_{2}{\text{e}}^{\text{i}({\varDelta }_{1}-{\varDelta }_{2})t} \text{, } \\ & {\text{δ}\dot{c}_{2}}=-\frac{{\kappa }_{2}}{2}\text{δ}{c}_{2}-\text{i}{G}_{2}{\mathrm{e}}^{\text{i}\theta }\left[\text{δ}{b}^{†}{\mathrm{e}}^{\text{i}({\omega }_{\phi }+{\varDelta }_{2})t}+\text{δ}b{\mathrm{e}}^{-\text{i}({\omega }_{\phi }-{\varDelta }_{2})t}\right]+{\varepsilon }_{\text{R}}{\mathrm{e}}^{-\text{i}(\delta -{\varDelta }_{2})t}-\text{i}J\text{δ}{c}_{1}{\mathrm{e}}^{\text{i}({\varDelta }_{2}-{\varDelta }_{1})t} \text{, } \\ & {\text{δ}\dot b}=-\frac{{\gamma }_{\phi }}{2}\text{δ}b-\text{i}{G}_{1}\left[\text{δ}{c}_{1}{\text{e}}^{\text{i}({\omega }_{\phi }-{\varDelta }_{1})t}+\text{δ}{c}_{1}^{†}{\text{e}}^{\text{i}({\omega }_{\phi }+{\varDelta }_{1})t}\right]-\text{i}{G}_{2}\left[{\text{e}}^{-\text{i}\theta }\text{δ}{c}_{2}{\text{e}}^{\text{i}({\omega }_{\phi }-{\varDelta }_{2})t}+{\text{e}}^{\text{i}\theta }\text{δ}{c}_{2}^{†}{\text{e}}^{\text{i}({\omega }_{\phi }+{\varDelta }_{2})t}\right] \text{, }\end{split} $

      其中$ {G_1} $, $ {G_2} $为两个涡旋腔场与旋转腔镜之间的有效光旋转耦合强度, $ {G_1} = {g_{\phi 1}}{c_{1{\text{s}}}} $, $ {G_2} = {g_{\phi 2}}{c_{2{\text{s}}}}{{\text{e}}^{ - {\text{i}}\theta }} $, $ {g_{\phi 1}}{c_{1{\text{s}}}} $$ {g_{\phi 2}}{c_{2{\text{s}}}} $之间的非互易相位差$ \theta $可以通过驱动场$ {\varepsilon _{\text{c}}} $$ {\varepsilon _{\text{d}}} $的相对相位来调节.

      如果每个强驱动场都处于力学红边带(${\varDelta _1} \approx {\varDelta _2} \approx {\omega _\phi }$), 同时旋转镜频率$ {\omega _\phi } $远大于有效耦合强度$ {G_j} $, 方程(6)可以化简为

      $ \begin{split} & {\text{δ}\dot{c}_{1}}=-\frac{{\kappa }_{1}}{2}\text{δ}{c}_{1}-\text{i}{G}_{1}\text{δ}b-\text{i}J\text{δ}{c}_{2}+{\varepsilon }_{\text{L}}{\text{e}}^{-\text{i}xt} \text{, } \\ & {\text{δ}\dot{c}_{2}}=-\frac{{\kappa }_{2}}{2}\text{δ}{c}_{2}-\text{i}{G}_{2}{e}^{\text{i}\theta }\text{δ}b-\text{i}J\text{δ}{c}_{1}+{\varepsilon }_{\text{R}}{\text{e}}^{-\text{i}xt} \text{, } \\ & {\text{δ}\dot b}=-\frac{{\gamma }_{\phi }}{2}\text{δ}b-\text{i}{G}_{1}\text{δ}{c}_{1}-\text{i}{G}_{2}{\text{e}}^{-\text{i}\theta }\text{δ}{c}_{2} \text{, } \end{split}$

      其中$ x = \delta - {\omega _\phi } $. 由方程(7)的具体形式, 可以假设方程(7)的解具有下面的形式:

      $ \begin{split} &{\text{δ}}{c_1} = {\text{δ}}{c_{1 + }}{{\text{e}}^{ - {\text{i}}xt}} + {\text{δ}}{c_{1 - }}{{\text{e}}^{{\text{i}}xt}} \text{, }\\ &{\text{δ}}{c_2} = {\text{δ}}{c_{2 + }}{{\text{e}}^{ - {\text{i}}xt}} + {\text{δ}}{c_{2 - }}{{\text{e}}^{{\text{i}}xt}} \text{, } \\ &{\text{δ}}b = {\text{δ}}{b_ + }{{\text{e}}^{ - {\text{i}}xt}} + {\text{δ}}{b_ - }{{\text{e}}^{{\text{i}}xt}} . \end{split} $

      将方程(8)代入方程(7)得

      $ \begin{split} &{\text{δ}}{c_{1 + }} = \frac{{(8G_2^2 + 2{\gamma _x}{\kappa _{2x}}){\varepsilon _{\text{L}}} - (8{G_1}{G_2}{{\text{e}}^{ - {\text{i}}\theta }} + 4{\text{i}}J{\gamma _x}){\varepsilon _{\text{R}}}}}{{\varOmega - 16{\text{i}}{G_1}{G_2}J\cos \theta }} \text{, } \\ &{\text{δ}}{c_{2 + }} = \frac{{(8G_1^2 + 2{\gamma _x}{\kappa _{1x}}){\varepsilon _{\text{R}}} - (8{G_1}{G_2}{{\text{e}}^{{\text{i}}\theta }} + 4{\text{i}}J{\gamma _x}){\varepsilon _{\text{L}}}}}{{\varOmega - 16{\text{i}}{G_1}{G_2}J\cos \theta }} \text{, } \\ &{b_ + } = \frac{{(4{\text{i}}{G_1}{\kappa _{2x}} + 8{G_2}J{{\text{e}}^{ - {\text{i}}\theta }}){\varepsilon _{\text{L}}} + (8{G_1}J + 4{\text{i}}{G_2}{\kappa _{1x}}{{\text{e}}^{ - {\text{i}}\theta }}){\varepsilon _{\text{R}}}}}{{16{\text{i}}{G_1}{G_2}J\cos \theta - \varOmega }} \text{, } \end{split}$

      其中$ {\gamma _x} = \gamma - 2{\text{i}}x $, $ {\kappa _{jx}} = {\kappa _j} - 2{\text{i}}x $, $\varOmega = 4 G_2^2{\kappa _{1 x}} + 4 G_1^2{\kappa _{2 x}} + 4{J^2}{\gamma _x} + {\kappa _{1 x}}{\kappa _{2 x}}{\gamma _x}$.

      为了研究涡旋光束传输的非互易性, 需要得到$ \varepsilon _{\text{L}}^{{\text{out}}} $$ \varepsilon _{\text{R}}^{{\text{out}}} $. 根据输入输出关系[47,48]

      $ \begin{split} &{\varepsilon }_{\text{L}}^{\text{out}}+{\varepsilon }_{\text{L}}^{\text{in}}{\text{e}}^{-\text{i}xt}=\sqrt{{\kappa }_{1}}\text{δ}{c}_{1}\text{, }\\ &{\varepsilon }_{\text{R}}^{\text{out}}+{\varepsilon }_{\text{R}}^{\text{in}}{\text{e}}^{-\text{i}xt}=\sqrt{{\kappa }_{2}}\text{δ}{c}_{2}\text{, }\end{split} $

      其中$\varepsilon _{{\text{L, R}}}^{{\text{in}}} = {\varepsilon _{{\text{L, R}}}}/\sqrt {{\kappa _{\text{1}}}{\kappa _{\text{2}}}}$, 输出场具有${\text{δ}}{c_1} = {\text{δ}}{c_{1 + }}{{\text{e}}^{ - {\text{i}}xt}} + {\text{δ}}{c_{1 - }}{{\text{e}}^{{\text{i}}xt}}$$ {\text{δ}}{c_2} = {\text{δ}}{c_{2 + }}{{\text{e}}^{ - {\text{i}}xt}} + {\text{δ}}{c_{2 - }}{{\text{e}}^{{\text{i}}xt}} $的形式, 由方程(9)和方程(10)得

      $ \begin{split} &\varepsilon _{{\text{L + }}}^{{\text{out}}} = \sqrt {{\kappa _1}} {\text{δ}}{c_{1 + }} - {\varepsilon _{\text{L}}}/\sqrt {{\kappa _{\text{1}}}} \text{, } \\ &\varepsilon _{{\text{R + }}}^{{\text{out}}} = \sqrt {{\kappa _2}} {\text{δ}}{c_{2 + }} - {\varepsilon _{\text{R}}}/\sqrt {{\kappa _{\text{2}}}} \text{, }\\ &\varepsilon _{{\text{L}}-}^{{\text{out}}} = 0 \text{, }~~ \varepsilon _{{\text{R }}-}^{{\text{out}}} = 0 . \end{split}$

      本文定义$ {T_{12}} $($ {T_{21}} $)为入射腔$ {c_1}({c_2}) $的探测场通过系统后输出时的传输振幅, 根据(9)式和(11)式计算得

      $ \begin{split} {T}_{12}=\;&{T}_{\rm{L\to R}}={\left|\frac{{\varepsilon }_{\text{R+}}^{\text{out}}}{{\varepsilon }_{\text{L}}^{\text{in}}}\right|}_{{\varepsilon }_{\text{R}}^{\text{in}}=0}\\ =\;&\frac{-\sqrt{{\kappa }_{1}{\kappa }_{2}}(8{G}_{1}{G}_{2}{\text{e}}^{\text{i}\theta }+4\text{i}J{\gamma }_{x})}{\varOmega -16\text{i}{G}_{1}{G}_{2}J\mathrm{cos}\theta } \text{, } \\ {T}_{21}=\;&{T}_{\rm{R\to L}}={\left|\frac{{\varepsilon }_{\text{L+}}^{\text{out}}}{{\varepsilon }_{\text{R}}^{\text{in}}}\right|}_{{\varepsilon }_{\text{L}}^{\text{in}}=0}\\ =\;&\frac{-\sqrt{{\kappa }_{1}{\kappa }_{2}}(8{G}_{1}{G}_{2}{\text{e}}^{-\text{i}\theta }+4\text{i}J{\gamma }_{x})}{\varOmega -16\text{i}{G}_{1}{G}_{2}J\mathrm{cos}\theta } . \end{split}$

    • 本节将详细说明在双拉盖尔-高斯旋转腔系统中非互易相位差$ \theta $、涡旋光束所携带拓扑荷$ \ell $和系统耗散对传输振幅的影响. 首先讨论非互易相位差$ \theta $对传输振幅的影响.

      图2给出传输振幅$ {T_{12}} $(红色实线)和$ {T_{21}} $(蓝色虚线)在不同非互易相位差$ \theta $下随标准化失谐$ x/{\kappa _1} $的演化. 可以看出, 相位差$ \theta $对涡旋光束传输有比较大的影响, 在$ \theta $ = 0时, 探测场在两个方向上的传输是互易的, 但当$ \theta \ne 0 $探测场的传输会呈现明显的非互易性. 观察图2(b)图2(e)可以发现, 非互易现象会出现在共振点及附近, 还可以发现相位差的正负可决定非互易的方向($ \theta < 0 $时, $ {T_{21}} > {T_{12}} $; 而$ \theta > 0 $时, $ {T_{21}} < {T_{12}} $). 当$ \theta = \pm {\text{π }}/4 $时, 非互易性在$ x = - 0.4\kappa $处呈现的最明显. 当$ \theta = \pm {\text{π }}/2 $时, 系统可以在共振处($ x = 0 $)呈现完美的涡旋光束非互易传输($ {T_{12}} = 0 $$ {T_{21}} = 1 $, 或$ {T_{12}} = 1 $$ {T_{21}} = 0 $). 由(12)式也可以看出, 当$ \theta = n{\text{π }} $时, $ {T_{12}} = {T_{21}} $, 涡旋光束传输呈现互易性, 而当$ \theta \ne n{\text{π }} $时, $ {T_{12}} \ne {T_{21}} $, 即涡旋光束的传输是非互易的. 实际上, 这里的涡旋光束的非互易传输主要源于光子不同传输路径之间的量子干涉作用. 为了更清楚地展示更多非互易相位差对传输振幅的影响, 图2(f)给出了传输振幅$ {T_{12}} $(红色实线)和$ {T_{21}} $(蓝色虚线)随非互易相位差$ \theta $的演化.

      图  2  传输振幅$ {T_{12}} $(红色实线)和$ {T_{21}} $(蓝色虚线)在不同非互易相位差$ \theta $下随标准化失谐$ x/{\kappa _1} $的演化 (a) $ \theta = 0 $; (b) $ \theta = - {\text{π }}/4 $; (c) $ \theta = {\text{π }}/4 $; (d) $ \theta = - {\text{π }}/2 $; (e) $ \theta = {\text{π }}/2 $. (f)在标准化失谐为零(x = 0)时, 传输振幅$ {T_{12}} $(红色实线)和$ {T_{21}} $(蓝色虚线)随非互易相位差$ \theta $的演化. 其他参数为$ {\kappa _1} = 1 $, $ {\kappa _2} = 1 $, $ \gamma = 1 $, $ {G_i} = \sqrt {{\kappa _i}\gamma } /2 $, $ J = \sqrt {{\kappa _1}{\kappa _2}} /2 $.

      Figure 2.  Transmission amplitudes $ {T_{12}} $ (red solid line) and $ {T_{21}} $ (blue dotted line) versus normalized detuning $ x/{\kappa _1} $ under different nonreciprocal phase difference: (a) $ \theta = 0 $; (b) $ \theta = - {\text{π }}/4 $; (c) $ \theta = {\text{π }}/4 $; (d) $ \theta = - {\text{π }}/2 $; (e) $ \theta = {\text{π }}/2 $. (f) Transmission amplitudes $ {T_{12}} $ (red solid line) and $ {T_{21}} $ (blue dotted line) versus nonreciprocal phase difference with $ x = 0 $. Other parameters are $ {\kappa _1} = 1 $, $ {\kappa _2} = 1 $, $ \gamma = 1 $, $ {G_i} = \sqrt {{\kappa _i}\gamma } /2 $, $ J = \sqrt {{\kappa _1}{\kappa _2}} /2 $.

      接下来讨论系统耗散对涡旋光束传输非互易性的影响. 为不失一般性, 取非互易相位差$ \theta = {\text{π }}/2 $. 图3(a)图3(b)给出了传输振幅$ {T_{12}} $$ {T_{21}} $在旋转腔镜的不同耗散率$ \gamma $下随标准化失谐$ x/{\kappa _1} $的演化. 可以看出在共振点(x = 0)附近, 随着旋转镜耗散率$ \gamma $增加, 从腔$ {c_1} $到腔$ {c_2} $的传输振幅$ {T_{12}}{\text{ = }}1 $保持不变, 而从腔$ {c_2} $到腔$ {c_1} $的传输振幅$ {T_{21}} $呈现出明显的变化, 显示出先降低后增加的趋势. 当旋转镜耗散率$ \gamma {\text{ = }}{\kappa _{\text{1}}}{\text{ = }}1 $时, 涡旋光束的传输非互易性达到最佳($ {T_{12}} = {\text{1}} $, $ {T_{21}} = {\text{0}} $, 在共振点处).

      图  3  (a)传输振幅$ {T_{12}} $和(b)传输振幅$ {T_{{\text{21}}}} $作为标准化失谐$ x/{\kappa _1} $和旋转镜与腔$ {c_1} $耗散比$ \gamma /{\kappa _1} $的函数; (c)传输振幅$ {T_{12}} $和(d)传输振幅$ {T_{{\text{21}}}} $作为标准化失谐$ x/{\kappa _1} $和腔$ {c_2} $与腔$ {c_1} $耗散比$ {\kappa _2}/{\kappa _1} $的函数. 其他参数为$ \theta = {\text{π }}/2 $, $ {\kappa _1} = 1 $, $ {G_1} = {G_2} = J = 1/2 $

      Figure 3.  (a) Transmission amplitude $ {T_{12}} $ and (b) transmission amplitude $ {T_{21}} $ as a function of the normalized detuning $ x/{\kappa _1} $ and the dissipation ratio $ \gamma /{\kappa _1} $ of rotating mirror to cavity $ {c_1} $; (c) transmission amplitudes $ {T_{12}} $ and (d) transmission amplitudes $ {T_{21}} $ as a function of the normalized detuning $ x/{\kappa _1} $ and the dissipation ratio $ {\kappa _2}/{\kappa _1} $ of cavity $ {c_2} $ to cavity $ {c_1} $. Other parameters are $ \theta = {\text{π }}/2 $, $ {\kappa _1} = 1 $, $ {G_1} = {G_2} = J = 1/2 $.

      图3(c)图3(d)给出了传输振幅$ {T_{12}} $$ {T_{21}} $在腔$ {c_2} $的不同耗散率下随标准化失谐$ x/{\kappa _1} $的演化. 可以看出, 在共振点(x = 0)附近, 随着腔$ {c_2} $耗散率$ {\kappa _2} $增加, 从腔$ {c_2} $到腔$ {c_1} $的传输振幅$ {T_{21}}{\text{ = 0}} $保持不变, 而从腔$ {c_1} $到腔$ {c_2} $的传输振幅$ {T_{12}} $呈现逐渐减小的趋势. 当腔$ {c_2} $耗散率$ {\kappa _2}{\text{ = }}{\kappa _{\text{1}}}{\text{ = }}1 $时, 涡旋光束传输的非互易性达到最佳($ {T_{12}} = {\text{1}} $, $ {T_{21}} = {\text{0}} $, 在共振点处). 所以对于涡旋光束的非互易传输来说, 腔场$ {c_2} $的耗散不是越低越好.

      接下来讨论两个涡旋光场所携带拓扑荷比值$ {\ell _2}/{\ell _1} $对传输振幅的影响. 通常, 涡旋光场所携带拓扑荷实际上会影响旋转腔镜与L-G腔模的耦合强度. 为了方便, 这里通过调节驱动场振幅$ {\varepsilon _{\text{c}}} $$ {\varepsilon _{\text{d}}} $的大小从而使得拓扑荷比$ {\ell _2}/{\ell _1} $与有效耦合强度比$ {G_2}/{G_1} $相等. 图4(a)图4(f)给出了在标准化失谐x = 0时传输振幅$ {T_{12}} $$ {T_{21}} $在不同的非互易相位差下随两个涡旋光场所携带拓扑荷比$ {\ell _2}/{\ell _1} $的演化. 可以发现, 当$ \theta = 0 $$ \theta = {\text{π }} $时, 拓扑荷的比值对传输振幅$ {T_{12}} $$ {T_{21}} $的影响是相同的, 即系统对涡旋光束的传输呈现互易性. 当$ \theta = \pm {\text{π }}/4 $$ \theta = \pm {\text{π }}/2 $时, 系统对涡旋光束传输呈现明显的非互易性. 当$ \theta = \pm {\text{π }}/4 $时, 随着拓扑荷比值的增加, 传输振幅$ {T_{12}} $$ {T_{21}} $的变化趋势是相同的, 且是偶对称, 但相同拓扑荷比值所对应的传输振幅$ {T_{12}} $$ {T_{21}} $是不同的, 在拓扑荷比值为$ \pm 1 $处的非互易性是最好的. 当$ \theta = \pm {\text{π }}/2 $时, 随着拓扑荷比值的增加, 传输振幅$ {T_{12}} $仍呈现关于拓扑荷比值等于$ 0 $处的对称性, 但此时传输振幅$ {T_{21}} $保持为$ 0 $不变. 在拓扑荷比值为$ \pm 1 $处, 系统对涡旋光束传输可以呈现完美的非互易性($ {T_{12}} = 0 $$ {T_{21}} = 1 $, 或$ {T_{12}} = 1 $$ {T_{21}} = 0 $). 此外, 拓扑荷的正负代表涡旋光矢量的旋转方向, 根据涡旋光矢量的旋转方向可以将涡旋光束分为左旋涡旋光束和右旋涡旋光束. 从图4(d)图4(e)可以了解到涡旋光束想要呈现完美的非互易性与涡旋光束的左、右旋无关, 只要保证$ \left| {{\ell _2}/{\ell _1}} \right|{\text{ = }}1 $.

      图  4  传输振幅$ {T_{12}} $(红色圆圈)和$ {T_{21}} $(蓝色圆圈)在不同非互易相位差$ \theta $下随两个涡旋光场所携带拓扑荷之比$ {\ell _2}/{\ell _1} $的演化 (a) $\theta = 0$; (b) $ \theta = - {\text{π }}/4 $; (c) $ \theta = {\text{π }}/4 $; (d) $ \theta = - {\text{π }}/2 $; (e) $ \theta = {\text{π }}/2 $; (f) $ \theta = {\text{π }} $. 其他参数为$ {\kappa _1} = 1 $, $ {\kappa _2} = 1 $, $ \gamma = 1 $, $ J = 2{G_1}{G_2}/\gamma $

      Figure 4.  Transmission amplitudes $ {T_{12}} $ (red circle) and $ {T_{21}} $ (blue circle) with ratio of topological charges carried by two vortex optical fields $ {\ell _2}/{\ell _1} $ under different nonreciprocal phase difference: (a) $ \theta = 0 $; (b)$ \theta = - {\text{π }}/4 $; (c)$ \theta = {\text{π }}/4 $; (d) $ \theta = - {\text{π }}/2 $; (e) $ \theta = {\text{π }}/2 $; (f) $ \theta = {\text{π }} $. Other parameters are $ {\kappa _1} = 1 $, $ {\kappa _2} = 1 $, $ \gamma = 1 $, $ J = 2{G_1}{G_2}/\gamma $.

    • 在腔系统中, 光的群延迟是描述光群速度的一个重要指标, 首先介绍群延迟的定义, 即[49,50]

      $ {\tau _{ij}} = \frac{{{\text{d}}{\varTheta _{21(12)}}}}{{{\text{d}}{\omega _{\text{p}}}}} , $

      其中${\varTheta _{21(12)}}$是在频率为$ {\omega _{\text{p}}} $下输出场$ {c_1} \to {c_2} $($ {c_2} \to {c_1} $)的相位, 群延迟$ {\tau _{ij}} < 0 $对应快光, $ {\tau _{ij}} > 0 $对应慢光.

      图5(a)图5(c)给出了针对涡旋光束所对应的群延迟$ {\tau _{12}} $$ {\tau _{21}} $在不同的非互易相位差下随两个涡旋光场所携带拓扑荷比$ {\ell _{\text{2}}}{\text{/}}{\ell _{\text{1}}} $的演化. 在$ \theta = 0 $时群速度是互易的且关于拓扑荷比值偶对称. 在$ \theta = \pm {\text{π }}/2 $时, 系统可以呈现明显的非互易慢光效应, 且相位的变化可以影响非互易慢光的群延迟的变换. 随着拓扑荷比值的增加, $ {\tau _{12}} $$ {\tau _{21}} $均呈现先增大后减小的趋势且关于拓扑荷比值偶对称, 但群延迟的差值($ {\tau _{12}}-{\tau _{21}} $$ {\tau _{21}}-{\tau _{12}} $)一直保持不变.

      图  5  群延迟$ {\tau _{12}} $(红色圆圈)和$ {\tau _{21}} $(蓝色圆圈)在不同非互易相位差$ \theta $下随两个涡旋光场所携带拓扑荷比$ {\ell _2}/{\ell _1} $的演化 (a) $\theta = 0$; (b) $ \theta = - {\text{π }}/2 $; (c) $ \theta = {\text{π }}/2 $. 其他参数为$ {\kappa _1} = 1 $, $ {\kappa _2} = 1 $, $ \gamma = 1 $, $ J = 2{G_1}{G_2}/\gamma $

      Figure 5.  Group delay $ {\tau _{12}} $ (red circle) and $ {\tau _{21}} $ (blue circle) with ratio of topological charges carried by two vortex optical fields $ {\ell _2}/{\ell _1} $ under different nonreciprocal phase difference: (a) $ \theta = 0 $; (b) $ \theta = - {\text{π }}/2 $; (c) $ \theta = {\text{π }}/2 $. Other parameters are $ {\kappa _1} = 1 $, $ {\kappa _2} = 1 $, $ \gamma = 1 $, $ J = 2{G_1}{G_2}/\gamma $.

    • 最后, 根据目前的实验进展, 对本文使用的系统方案的可行性进行了讨论. 本文讨论的双拉盖尔-高斯旋转腔系统是在文献[6]提出的由两个螺旋相位元件所构成的拉盖尔-高斯旋转腔的基础上, 添加一个固定的螺旋相位元件所实现的. 通过三个螺旋相位元件作为腔镜来组成两个光腔, 并且用光纤将两个光腔线性耦合. 其中, 两个输入腔镜FM1和 FM2为部分透明且被刚性固定, 不会改变透射光束的轨道角动量但会给反射光束增加$ {{ - 2}}{\ell _{\text{1}}} $$ {{ - 2}}{\ell _{\text{2}}} $的拓扑荷, 这一效应已经被实验观测到[51]. 另外, 处于中间位置的旋转腔镜RM是完全反射的, 可以给反射光增加$ +2{\ell _{\text{1}}} $$ +2{\ell _{\text{2}}} $的拓扑荷[6]. 随着纳米技术的发展, 螺旋相位元件可以利用超精密点金刚石车削机床直接加工铝盘表面得到[6,52]. 利用螺旋相位元件的反射和透射可以改变激光光束的方位结构, 进而改变其携带的拓扑荷的值. 拓扑荷$ \ell $依赖于螺旋相位元件的结构, 通过将螺旋相位元件的方位角坡道划分为离散的阶跃并控制阶跃高度和旋向, 可以设计出具有特定拓扑荷$ \ell $的涡旋光场[6,53]. 最近实验已经可以通过螺旋相位元件产生携带拓扑荷高达1000的L-G光束[54]. 对于本工作, 若想实现理想的涡旋光隔离器, 就需要在本系统中实现涡旋光束的完美非互易性, 即涡旋光束的单向传输. 在理论分析中得到, 若要呈现完美的非互易涡旋光束传输, 需要保证$ \left| {{\ell _2}/{\ell _1}} \right|{\text{ = }}1 $, 因此这里的涡旋光束所携带的拓扑荷需是可以调节的. 在实验上, 可以通过调整旋转腔镜两侧的阶跃高度和旋向来实现. 另外, 理论分析中得到非互易相位差对涡旋光束的非互易传输有着重要影响, 而非互易相位差的改变在实验上可以通过调节两个强驱动激光场$ {\varepsilon _{\text{c}}} $$ {\varepsilon _{\text{d}}} $的相对相位来实现[45]. 值得强调的是, 本文所构建的系统在实验实现时需要较高精细度的光腔, 在光腔的精细度比较低时, 在该系统所要探究相关现象会不明显[53].

    • 本文研究了涡旋光束在双拉盖尔-高斯旋转腔系统中的非互易传输. 将以往大多在光机械系统中研究的普通光束的非互易性推广到在光旋转系统中研究携带轨道角动量的涡旋光束传输的非互易性. 系统中, 光旋转耦合作用以及光纤的线性耦合作用提供的两条路径, 从而可以产生量子干涉. 首先, 研究了非互易相位差对传输振幅的影响, 发现相位差可以决定涡旋光束非互易性的发生及方向. 然后, 在相位差$ \theta = {\text{π }}/2 $时, 研究了系统耗散对传输振幅的影响, 分析了涡旋光束实现完美的非互易传输时系统的耗散应该满足的条件. 接下来, 分析了涡旋光束携带的拓扑荷比值对系统非互易性的影响, 探究了实现涡旋光束完美的非互易传输时拓扑荷比值应该满足的条件, 发现涡旋光的左、右旋不会对传输产生影响. 拓扑荷对光传输产生影响, 主要是因为涡旋光束携带的拓扑荷数会影响旋转腔镜与腔模的耦合强度. 最后, 分析了该系统产生的非互易的慢光效应. 这些研究成果可用于设计针对携带轨道角动量的涡旋光束的理想光隔离器, 有望应用于光通信等领域.

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