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This article reviews the discovery of the Higgs boson, discusses the studies of its properties, and introduces the physical prospects of the future Higgs factories. The greatest goal of particle physics is to understand the fundamental particles of the universe and how they interact with each other (or more generally, how the universe operates). In the standard model of particle phyiscs, the Higgs mechanism is proposed to explain the origin of elementary particle mass and predict the existence of the Higgs boson. Higgs physics is one of the most important research areas in particle physics. The Large Hadron Collider (LHC) at CERN (Geneva, Switzerland) accelerates proton beams to collide at center-of-mass energy of 13 TeV, thus defining the world’s energy frontier. The ATLAS and CMS detectors are two general-purpose detectors at the LHC for studying the debris from the collisions. The Higgs boson was discovered in the ATLAS and CMS experiments in 2012. This discovery completed the fundamental particle spectrum of the standard model and was an important milestone for particle physics. Since then, many studies have been conducted on the properties of Higgs boson, including spin, mass and couplings, to deepen our understanding of the Higgs mechanism. In particular, the Higgs boson couplings to fermions and to themselves present new kinds of fundamental interactions with paramount significance, which have not been fully confirmed. Additionally, the Higgs bosons has become an important tool to search for dark matter, heavy resonance, and other new physical phenomena. So far, there has been no deviation from the predictions of the standard model. Looking forward to the future, it is proposed to use the electron-positron collisions to study the Higgs boson in more depth. Physics studies have shown that these Higgs factories can significantly improve the accuracy of many properties of the Higgs boson, including width and couplings, and provide great physics prospects.
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Keywords:
- standard model /
- electroweak symmetry breaking /
- Higgs boson
1. 引 言
光既可以具有自旋角动量, 又可携带轨道角动量. 自旋角动量与光的偏振有关, 如左右旋圆偏振光子分别携带
$ \pm \hbar $ 的自旋角动量. 轨道角动量有两类[1,2], 一类为内禀(intrinsic)的轨道角动量, 与涡旋光场有关, 每个光子携带$l\hbar $ 的轨道角动量, 其中l为涡旋相位的拓扑荷数; 另一类为外禀(extrinsic)的轨道角动量, 和光束传播的轨迹有关, 定义为坐标原点到光束中心的距离与线动量的叉乘, 与经典粒子的机械角动量类似. 光的自旋角动量和轨道角动量之间的相互转换和耦合被称为自旋-轨道相互作用(spin-orbit interaction, SOI)或耦合[1,2]. 它是光学中的一种基本效应, 广泛存在于界面的反射和折射、非均匀各向异性介质、强聚焦、粒子散射、表面波和消逝波等体系中, 在光学、纳米光子学和等离子光学等领域扮演越来越重要的角色, 并在精密测量与探测、信息存储与处理、微粒操纵以及各种功能光子器件设计等方面显示出巨大的应用潜力[1-9]. 在旋转对称的系统中, 光的SOI表现为自旋可控的涡旋相位的产生(内禀轨道角动量); 在旋转对称性破缺的系统中, 它表现为自旋霍尔效应(外禀轨道角动量)[1,2].光的自旋霍尔效应存在于很多体系中, 如光束在突变界面的斜入射[1,10-17]、一维的潘查拉特南-贝里(Pancharatnam-Berry, PB)相位元件[18-21]、各向同性的非均匀材料[21-23]等; 自旋可控的涡旋相位也在方位变化的PB相位元件[24-27]、强聚焦[28,29]、单轴晶体中的传输[30,31]等体系中出现. 然而, 有趣的是, 当光束正入射至均匀的、各向同性的突变界面时, 也能产生自旋相关的涡旋相位[32-34]. 光束正入射时, 极小的一部分入射光束发生自旋反转(左旋变为右旋或者右旋变为左旋), 并获得拓扑荷数为±2的涡旋相位(图1(a)). 其内在机制被认为是SOI, 但这种相位的物理来源、为什么拓扑荷数为±2以及界面在其中究竟扮演何种角色等一系列的问题, 目前并不清楚. 另外, 该SOI与光束通过方位变化的各向异性PB相位元件[24-27]时产生涡旋相位的过程极为相似. 光束入射到方位PB相位元件时, 一部分入射光束发生自旋反转并获得2倍于元件拓扑荷数(q)的涡旋相位因子2qϕ, 其中ϕ是PB相位元件的局部的光轴方向, 是坐标位置的函数. 也就是说这种相位因子来源于PB相位元件的非均匀的各向异性. 而前文所提到的界面是各向同性且均匀的, 这与PB相位元件的情况又有何联系和区别?
图 1 光束正入射至各向同性的突变界面时SOI的示意图 (a) 左旋圆偏振光束正入射至界面后, 部分光束发生自旋反转变成右旋光, 并获得拓扑荷数为2的涡旋相位(两个小图分别表示一种典型的涡旋光束的强度和相位分布); 注意, 未发生SOI的那部分光束并没有在图中画出;$\left| + \right\rangle $ $\left| - \right\rangle $ ${\varOmega _\xi }$ Fig. 1. Schematic illustration of the SOI for a light beam normally impinging onto a sharp isotropic interface. (a) When a left-circularly polarized beam normally passes through the interface, part of the incident beam converts into a right-circularly polarized beam, and carries a vortex phase with a topological charge of 2. Note that the spin-maintained portion is not shown in the picture.$\left| + \right\rangle $ $\left| - \right\rangle $ ${\varOmega _\xi }$ 首先建立一个菲涅耳琼斯矩阵来描述光束正入射至突变界面的透(反)射光束, 并发现其产生的涡旋相位因子来源于光束本身的拓扑结构, 具有几何性, 是一种自旋重构的贝里(spin-redirection Berry)相位. 界面的性质影响光束中各平面波分量的菲涅耳系数, 并决定发生了SOI的那部分光束的转换效率. 而对于PB相位元件, 光束也是部分发生自旋反转, 经历SOI, 并获得PB涡旋相位, 但这种相位来源于外部材料的各向异性. 进一步研究发现, 光束正入射至突变界面的SOI的转换效率取决于光束中各斜入射的平面波的TM和TE分量的菲涅耳系数之差. 对于传统材料来说, 这种效应非常弱, 转换效率极低, 这限制了它的应用, 目前也没有这方面的实验见诸报道. Ciattoni等[33]从理论上提出, 采用各向同性的、介电常数近零的薄层来增强这种效应, 但因各向同性材料可调的自由度非常有限, 其转换效率最高也只能达到20%左右. 由于各向异性材料具有更多调控的自由度, 因此提出用光轴方向平行于界面法线方向的单轴晶体薄层, 来极大地增强正入射时的SOI, 使转换效率在某些条件下可达100%.
2. 涡旋相位的物理来源: 理论与模型
2.1 建立透射与入射光束之间的菲涅耳琼斯矩阵
考虑一个单色的有限宽傍轴光束, 正入射至一个由各向同性的、均匀的、无吸收材料构成的突变界面, 如图1(a)所示. 由于反射和透射光束具有相似的行为, 因此本文以透射为例来分析其中的SOI过程. 由角谱理论可知, 有限宽的光束可以看成由许多具有略微不同传播方向的平面波相干叠加而成. 众所周知, 平面波在界面的透射可以由菲涅耳公式来描述, 因此, 光束在界面的透射场由其所有平面波分量的透射场相加(积分)而成. 根据角谱理论, 可以把入射(i)和透射(t)电场统一地写成如下傅里叶积分形式(a = i, t)[35]:
$${ U}_ \bot ^a\left( {{ r}_ \bot } \right) = \int {{{\rm{d}}^2}{ k}_ \bot {{\rm{e}}^{{\rm{i}}{ k}_ \bot \cdot { r}_ \bot + {\bf{i}}k_z^az}}{\tilde {{U}}}_ \bot ^a\left( {{ k}_ \bot } \right)},$$ (1) 其中,
${ k}_ \bot = k_x{\hat {{x}}} + k_y{\hat {{y}}}$ 和${ r}_ \bot = x{\hat {{x}}} + y{\hat {{y}}}$ 分别表示横向的波矢和位置矢量(其中$\left\{ {{\hat {{x}}},\; {\hat {{y}}},\; {\hat {{z}}}} \right\}$ 为对应于$\left\{ {x, y, z} \right\}$ 的各方向的单位矢量),$k_z^a = \sqrt {{{\left( {{k^a}} \right)}^2} - k_ \bot ^2} $ 为纵向波矢分量(${k^a} = {n^a}2{\text{π}} /\lambda $ 为波数,${n^a}$ 为折射率),${\tilde {{U}}}_ \bot ^a\left( {{ k}_ \bot } \right)$ 是垂直于中心平面波矢的横向场分布(动量空间中的角谱). 在圆偏振基下考虑自旋、轨道角动量及它们之间的SOI会使问题变得更为简便, 因此将${\tilde{ U}}_ \bot ^a\left( {{{k}}_ \bot } \right)$ 表示为两个圆偏振分量相加的形式, 即${\tilde{ U}}_ \bot ^a\left( {{{k}}_ \bot } \right) = {\tilde U}_ + ^a({{k}}_ \bot ){\hat{ V}}_ + + {\tilde U}_ - ^a({{k}}_ \bot ){\hat{ V}}_ - $ , 其中${\hat{ V}}_ \pm = \left( {{\hat{ x}} \pm {\rm{i}}{\hat{ y}}} \right)/\sqrt 2 $ 为圆偏振单位矢量, 下标+和–分别表示左、右旋圆偏振.假设入射光
${{U}}_ \bot ^{\rm{i}}\left( {{{r}}_ \bot } \right)$ 为左旋圆偏振光束, 且具有旋转不变的特征(如高斯光束和贝塞尔光束), 即在光束中心轴的横截面上, 其偏振态为均匀的左旋圆偏振分布. 这也意味着, 光束角谱中每一支平面波在其各自的传播方向的横向平面内并不一定是圆偏振, 而是椭圆偏振. 通过(1)式的傅里叶逆积分, 可以求出入射光束在动量空间的角谱${\tilde{ U}}_ + ^{\rm{i}}({{k}}_ \bot )$ . 现在, 只要解出透射光束的角谱${\tilde{ U}}_ \bot ^{\rm{t}}({{k}}_ \bot )$ , 就可以通过(1)式计算透射光的电场${{U}}_ \bot ^{\rm{t}}({{r}}_ \bot )$ .实际上, 可以用一个2 × 2的矩阵将
${\tilde{ U}}_ \bot ^{\rm{t}}({{k}}_ \bot )$ 与${\tilde{ U}}_ \bot ^{\rm{i}}({{k}}_ \bot )$ 联系起来. 首先, 光束横向角谱${\tilde{ U}}_ \bot ^a\left( {{{k}}_ \bot } \right)$ 中任意平面波的电场可写为${\tilde{ u}}_ \bot ^{{a}}({{{k}}^{{a}}}) = {\tilde u}_{\rm{TM}}^{{a}}{\hat{ v}}_{\rm{TM}}^{{a}} +$ $ {\tilde u}_{\rm{TE}}^{{a}}{\hat{ v}}_{\rm{TE}}^{{a}} $ 形式, 其中$$\begin{split}{\hat{ v}}_{\rm{TE}}^{{a}} \; &= {\hat{ z}} \times {{{k}}^{{a}}}/|{\hat{ z}} \times {{{k}}^{{a}}}| = \left( { - {k_y}{\hat{ x}} + {k_x}{\hat{ y}}} \right)/{k_ \bot } \\ &= - \sin \varphi {\hat{ x}} + \cos \varphi {\hat{ y}}\end{split}$$ 与
$$ \begin{split} {\hat{ v}}_{\rm{TM}}^{{a}}\;& = {\hat{ v}}_{\rm{TE}}^{{a}} \times {{{k}}^{{a}}}/{k^{{a}}}\\ &=\cos {\vartheta ^{{a}}}\left( {\cos \varphi {\hat{ x}} + \sin \varphi {\hat{ y}}} \right) - \sin {\vartheta ^{{a}}}{\hat{ z}} \end{split}$$ 为任意平面波的TE和TM分量的单位矢量, 且
${k_ \bot } = {\left( {k_x^2 + k_y^2} \right)^{1/2}}$ ,${\vartheta ^{{a}}} = {\sin ^{ - 1}}({k_ \bot }/{k^{{a}}})$ 为任意平面波的波矢与z轴的夹角(图1(b)),$\varphi = {\tan ^{ - 1}}({k_y}/{k_x})$ 为方位角. 各平面波的透射场与入射场之间通过菲涅耳透射系数${t_{{\rm{TM}}, {\rm{TE}}}}({\vartheta ^{\rm{i}}})$ 联系起来:$$\left( \begin{aligned} {\tilde u}_{\rm{TM}}^{\rm{t}} \\ {\tilde u}_{\rm{TE}}^{\rm{t}} \end{aligned} \right) \!=\! {{\hat{ T}}_ \bot }\left( \begin{aligned} {\tilde u}_{\rm{TM}}^{\rm{i}} \\ {\tilde u}_{\rm{TE}}^{\rm{i}} \end{aligned} \right) \!=\! \left[\!\!{\begin{array}{*{20}{c}} {{t_{\rm{TM}}}({\vartheta ^{\rm{i}}})}&{} \\ {}&{{t_{\rm{TE}}}({\vartheta ^{\rm{i}}})} \end{array}}\!\!\right]\left( \begin{aligned} {\tilde u}_{\rm{TM}}^{\rm{i}} \\ {\tilde u}_{\rm{TE}}^{\rm{i}} \end{aligned} \right).$$ (2) 为了使表达式看起来简洁, 以下将
${t_{{\rm{TM}}, {\rm{TE}}}}({\vartheta ^{\rm{i}}})$ 简写为${t_{{\rm{TM}}, {\rm{TE}}}}$ . 由于透射光和入射光的观察面上的场均是指垂直于中心波矢的横向场分布, 因此将非中心平面波的偏振矢量投影到中心平面波的偏振矢量上, 并忽略纵向z分量得$$\begin{split} & {\tilde U}_ + ^{{a}} = {\hat{ v}}_{\rm{TM}}^{{a}} \cdot {\left( {{\hat{ V}}_ + } \right)^ * }{\tilde u}_{\rm{TM}}^{{a}} + {\hat{ v}}_{\rm{TE}}^{{a}} \cdot {\left( {{\hat{ V}}_ + } \right)^ * }{\tilde u}_{\rm{TE}}^{{a}}, \\ & {\tilde U}_ - ^{{a}} = {\hat{ v}}_{\rm{TM}}^{{a}} \cdot {\left( {{\hat{ V}}_ - } \right)^ * }{\tilde u}_{\rm{TM}}^{{a}} + {\hat{ v}}_{\rm{TE}}^{{a}} \cdot {\left( {{\hat{ V}}_ - } \right)^ * }{\tilde u}_{\rm{TE}}^{{a}}. \end{split} $$ (3) 将上式写成矩阵形式, 并计算得
$$\begin{split} & \left[ \begin{aligned} {\tilde U}_ + ^{{a}}({{{k}}_ \bot }) \\ {\tilde U}_ - ^{{a}}({{{k}}_ \bot }) \\ \end{aligned} \right] = {{\hat{ P}}^{{a}}}\left[ \begin{aligned} {\tilde u}_{\rm{TM}}^{{a}}({{{k}}^{{a}}}) \\ {\tilde u}_{\rm{TE}}^{{a}}({{{k}}^{{a}}}) \\ \end{aligned} \right],\\ \qquad & {{\hat{ P}}^{{a}}} = \frac{1}{{\sqrt 2 }}\left(\!\!\!{\begin{array}{*{20}{c}} {\cos {\vartheta ^{{a}}}{{\rm{e}}^{ - {\rm{i}}\varphi }}}&{ - {\rm{i}}{{\rm{e}}^{ - {\rm{i}}\varphi }}} \\ {\cos {\vartheta ^{{a}}}{{\rm{e}}^{{\rm{i}}\varphi }}}&{{\rm{i}}{{\rm{e}}^{{\rm{i}}\varphi }}} \end{array}}\!\!\!\right).\qquad \end{split}$$ (4) $$\left[ \begin{aligned} {\tilde U}_ + ^{\rm{t}}({{{k}}_ \bot }) \\ {\tilde U}_ - ^{\rm{t}}({{{k}}_ \bot }) \end{aligned} \right] = {{\hat{ P}}^{\rm{t}}}{{\hat{ T}}_ \bot }{\left( {{{{\hat{ P}}}^{\rm{i}}}} \right)^{ - 1}}\left[ \begin{aligned} {\tilde U}_ + ^{\rm{i}}({{{k}}_ \bot }) \\ {\tilde U}_ - ^{\rm{i}}({{{k}}_ \bot }) \end{aligned} \right],$$ (5) 通过矩阵相乘得:
$$\begin{split} \; & {\hat{ M}} = {{\hat{ P}}^{\rm{t}}}{{\hat{ T}}_ \bot }{\left( {{{{\hat{ P}}}^{\rm{i}}}} \right)^{ - 1}} \\ &= \frac{1}{2}\!\left[\!\!\!{\begin{array}{*{20}{c}} {\left( {\zeta {t_{\rm{TM}}} + {t_{\rm{TE}}}} \right)}\!&\!{{{\rm{e}}^{ - {\rm{i}}2\varphi }}\left( {\zeta {t_{\rm{TM}}} - {t_{\rm{TE}}}} \right)} \\ {{{\rm{e}}^{{\rm{i}}2\varphi }}\left( {\zeta {t_{\rm{TM}}} - {t_{\rm{TE}}}} \right)}\!&\!{\left( {\zeta {t_{\rm{TM}}} + {t_{\rm{TE}}}} \right)} \end{array}}\!\!\!\right]\!,\end{split}$$ (6) 其中
$\zeta = \cos {\vartheta ^{\rm{t}}}/\cos {\vartheta ^{\rm{i}}}$ . 至此, 在圆偏振基下, 用菲涅耳琼斯矩阵${\hat{ M}}$ 来建立${\tilde{ U}}_ \bot ^{\rm{t}}({{k}}_ \bot )$ 与${\tilde{ U}}_ \bot ^{\rm{i}}({{k}}_ \bot )$ 之间的关系. 很显然, (6)式的矩阵中, 反对角元是自旋反转的项, 而对角元是自旋不变的项, 可分别称之为反常(abnormal)模式和寻常(normal)模式. 这种矩阵不仅比已有的方法[32,33]简洁, 而且清晰地展现了光束本身拓扑结构(${{\hat{ P}}^{{a}}}$ 矩阵)与界面性质 (${{\hat{ T}}_ \bot }$ 矩阵)的不同的物理贡献. 也就是说, 光束本身的拓扑结构贡献了涡旋相位因子${{\rm{e}}^{ \pm {\rm{i}}2\varphi }}$ ; 而与界面性质有关的菲涅耳系数贡献了反常模式和寻常模式的振幅$\left( {\zeta {t_{\rm{TM}}} \pm {t_{\rm{TE}}}} \right)/2$ , 并决定了透射光束中反常模式和寻常模式的“比重”(即SOI中的转换效率). 在左旋圆偏振光束(其圆偏基下的角谱分布为${(1, 0)^{\rm{T}}}{\tilde U}_ + ^{\rm{i}}$ )入射下, 根据(6)式可得$${\tilde{ U}}_ \bot ^{\rm{t}}({{k}}_ \bot ) = \left[ {\frac{{\zeta {t_{\rm{TM}}} + {t_{\rm{TE}}}}}{2}{\hat{ V}}_ + + {{\rm{e}}^{2{\rm{i}}\varphi }}\frac{{\zeta {t_{\rm{TM}}} - {t_{\rm{TE}}}}}{2}{\hat{ V}}_ - } \right]{\tilde U}_ + ^{\rm{i}}.$$ (7) 2.2 Berry相位: 涡旋相位的物理来源
下面分析涡旋相位因子的物理来源. 很显然, (7)式中的反常模式携带一个拓扑荷数为2的涡旋相位. 由(4)式可知, 它来源于非中心平面波与中心平面波的投影操作. 任意非中心平面波的偏振矢量投影到中心平面波的偏振矢量上后, 产生了一个自旋相关的涡旋相位因子
${{\rm{e}}^{ \pm {\rm{i}}\varphi }}$ . 这种相位因子与各平面波的入射面的方位角有关, 而中心平面波的入射面无法确定, 即TM和TE分量的偏振矢量也无法确定, 因此是涡旋相位的奇点. 对于旋转不变的光束(如高斯光束和贝塞尔光束), 该相位因子都存在. 从本质上看, 该相位因子来源于光束本身的拓扑结构, 是几何性的, 本身是一个不可观测的量, 只有透射和入射光束的相位差才是可观测量. 对于反常模式, 由于发生了自旋反转, 相位差为$ \pm2\varphi $ ; 对于寻常模式, 相互抵消, 相位差为0.本文用自旋角动量与坐标旋转之间的耦合[1]或者光学科里奥利(Coriolis)效应[36]来解释这种相位的物理来源. 由于要求入射光束在横截面上是均匀的圆偏振(实验中容易产生), 即每个非中心平面波的偏振矢量在投影到中心平面波的横向面之后都是圆偏振的(图1(b)), 因此光束在z方向上的平均光子自旋角动量为
${{J}} = \sigma {\hat{ z}}$ (其中σ = +1和–1分别表示左旋和右旋圆偏振). 以每个非中心平面波各自的TM和TE偏振矢量构成的局部坐标框架, 投影到实验室坐标上后, 相对于实验室坐标的空间旋转率为${\varOmega _\xi } = {\rm{d}}\varphi /{\rm{d}}\xi $ , 其中$\xi $ 是坐标框架的旋转路径, 旋转轴为z轴. 因此, 推导出一个几何相位[1,36]:$$\varPhi _{\rm{B}}^{{a}} = - \int {{{J}} \cdot } {{{\varOmega}} _\xi }{\rm{d}}\xi = - \int {{\sigma ^{{a}}}} {\rm{d}}\varphi = - {\sigma ^{{a}}}\varphi,$$ (8) 它体现为自旋(
${\sigma ^{{a}}}$ )与坐标旋转($\varphi $ )之间的耦合, 只与坐标旋转的路径有关, 因此是几何性的. 而$\varphi $ 又是螺旋相位因子, 与内禀轨道角动量有关, 因此$\varPhi _{\rm{B}}^{{a}}$ 又体现为自旋与内禀轨道角动量之间的耦合, 即SOI. 这里的坐标旋转, 是指各非中心平面波所在的局部坐标系在实验室坐标上的投影, 相对于实验室坐标系的旋转. 这种几何相位与光束中各平面波的传播方向的SO(3)旋转有关, 因此它是自旋重构的贝里相位[1,36-38].实际上, 最终透射光束的寻常和反常模式的相位是透射光束的几何相位与入射光束的几何相位之差
$\varPhi _{\rm{B}}^{\rm{t}} - \varPhi _{\rm{B}}^{\rm{i}}$ . 由于反常模式是自旋反转的结果, 即${\sigma ^{\rm{t}}} = - {\sigma ^{\rm{i}}}$ , 因此反常模式的几何相位为$\varPhi _{\rm{B}}^{\rm{t}} - \varPhi _{\rm{B}}^{\rm{i}} =$ $ - {\sigma ^{\rm{t}}}\varphi - \left( { - {\sigma ^{\rm{i}}}\varphi } \right) = 2{\sigma ^{\rm{i}}}\varphi $ ; 而寻常模式的几何相位为0. 这个结果与(7)式的计算结果是一致的, 也与文献[32-34]中的结果相同. 这种相位因子还与PB相位元件中产生的涡旋相位在形式上极为相似. 在PB相位元件中, PB相位来源于材料外部的各向异性; 而这里的几何相位来源于光束本身的拓扑结构以及光束中各平面波分量由于斜入射造成的${t_{\rm{TM}}}$ 与${t_{\rm{TE}}}$ 之间的不同(也可看作是一种“各向异性”). 这两种情况在原理上是不同的, 但在形式上又是一致的, 可以用上述的自旋角动量与坐标旋转的耦合模型来统一地理解[1,36].还可以从角动量守恒的角度来考虑. 由于本文中研究的界面是关于z轴旋转对称的、无吸收的体系, 因此参与SOI的那部分光束在z方向上的总角动量必须守恒(诺特定理). 入射光束中各平面波分量投影到z方向上的自旋角动量(光子的平均自旋角动量)为
$\sigma \hbar $ , 且不携带轨道角动量. 透射后, 部分光束发生自旋反转, 其自旋角动量变为$ - \sigma \hbar $ , 且同时获得与入射自旋相关的、$2\sigma \hbar $ 的额外轨道角动量. 此时这部分光束的总的角动量还是$\sigma \hbar $ , 并无增减. 因此, z方向的总角动量是守恒的.3. 增强光束正入射时的SOI效率的方法
3.1 现有方法的SOI效率
上文建立了一个由各向同性的、均匀的、无吸收材料构成的界面的菲涅耳琼斯矩阵, 分析了光束本身的拓扑结构和构成界面的材料性质的各自贡献, 即涡旋相位来源光束本身的拓扑结构, 而界面性质影响SOI的转换效率. 现以左旋圆偏振贝塞尔光束的正入射为例来具体讨论. 零阶贝塞尔光束的横向的电场可写为
$$U_ + ^{\rm{i}}\left( {{{r}}_ \bot } \right) = {A_0}\exp ({\rm{i}}k_z^{\rm{t}}z){{\rm{J}}_0}\left( {\Delta k{r_ \bot }} \right),$$ 其中A0为任意振幅,
$\Delta k = 2{\text{π}} /{w_0}$ 为横向的谱半宽度, w0为光束束腰半宽度,$k_z^{\rm{t}} =\! \sqrt {{{\left( {{k^{\rm{t}}}} \right)}^2} - \!\Delta k_{}^2} $ ,${{\rm{J}}_n}\left( \xi \right)$ 表示第一类n阶贝塞尔函数. 其角谱分布为一个冲激函数的形式:$${\tilde U}_ + ^{\rm{i}}({k_ \bot }) = \dfrac{{{A_0}{w_0}}}{{{2^{5/2}}{{\text{π}} ^2}}}\delta ({k_ \bot } - \Delta k),$$ 这意味着, 贝塞尔光束的角谱实际上呈旋转不变的、空心圆锥状分布(图1(b)), 中心轴垂直于界面, 且所有平面波分量均为斜入射, 虽方位角不同, 但入射角均为
${\vartheta ^{\rm{i}}} = {\sin ^{ - 1}}(\Delta k/{k^{\rm{i}}})$ , 即${\vartheta ^{\rm{i}}}$ 由w0决定. 由此, 将${\tilde U}_ + ^{\rm{i}}({k_ \bot })$ 代入(7)式并联立(1)式, 可得透射光束的电场为$$ \begin{split}{{U}}_ \bot ^{\rm{t}}\left( {{{r}}_ \bot } \right) =\; &\exp ({\rm{i}}k_z^{{t}}z)\left[\frac{{\zeta {t_{\rm{TM}}} + {t_{\rm{TE}}}}}{2}{{\rm{J}}_0}\left( {\Delta k{r_ \bot }} \right){\hat{ V}}_ +\right. \\ &-\left. {{\rm{e}}^{{\rm{i}}2\varphi }}\frac{{\zeta {t_{\rm{TM}}} - {t_{\rm{TE}}}}}{2}{{\rm{J}}_2}\left( {\Delta k{r_ \bot }} \right){\hat{ V}}_ - \right],\end{split} $$ (9) 式中透射光束分为两部分, 一部分与入射光束相同, 是寻常模式; 另一部分表现出自旋反转现象并携带拓扑荷数为2的涡旋相位, 是反常模式. 图2给出了寻常模式和反常模式的光强和相位分布. 反常模式(图2(a))光斑中心是光强为0的空心区域, 相位在方位方向变化4π, 即拓扑荷数为2的涡旋相位; 寻常模式(图2(b)) 光斑中心是实心区域, 不携带方位方向的涡旋相位. 当寻常模式强度为0, 即只有反常模式时, 该SOI过程中的转换效率为100%. 然而一般情况下, 对于传统材料构成的界面,
$\left| {{t_{\rm{TM}}} - {t_{\rm{TE}}}} \right|$ 实际上是一个非常小的值, 尤其在入射角较小时. 这种情况意味着SOI的转换效率是极低的, 常规的实验精度难以被观察到. 这也是这种效应迄今为止没有在实验上被观察到的原因之一.
图 2 左旋圆偏振贝塞尔光束正入射至一个界面时, 透射光束的反常模式(a)和寻常模式(b)的归一化光强分布, 其中两个小图分别表示为对应的相位分布, 在计算中, 取入射光束的波长$\lambda = 1$ ${w_0} = 20\lambda $ Fig. 2. Normalized intensity distribution of the abnormal mode (a) and normal mode (b) of transmitted light beam under the normal incidence of a left-handed circularly polarized Bessel beam at a sharp interface. The insets represent the phase distribution of corresponding modes. Here, we take the working wavelength as$\lambda = 1$ ${w_0} = 20\lambda $ SOI中的转换效率可定义为透射光束的反常模式的功率与入射光束的功率之比. 当考虑贝塞尔光束正入射时, SOI的转换效率为[33]
$$\eta = \frac{{{{\left| {\zeta {t_{\rm{TM}}} - {t_{\rm{TE}}}} \right|}^2}}}{4}.$$ (10) 因此, 转换效率取决于贝塞尔光束中各平面波的TM和TE分量菲涅耳系数之差. 对傍轴光束(
$\Delta k \ll k$ )来讲,$\zeta = \cos {\vartheta ^{\rm{t}}}/\cos {\vartheta ^{\rm{i}}}$ 是一个接近于1的值. 若考虑入射介质和出射介质折射率相等, 如一个放置于自由空间中的薄层,$\zeta \equiv 1$ . 此时, SOI的转换效率$\eta = {\left| {{t_{\rm{TM}}} - {t_{\rm{TE}}}} \right|^2}/4$ . 由于贝塞尔光束中的平面波分量的入射角${\vartheta ^{\rm{i}}}$ 取决于光束束腰半宽度w0的大小, 且具有一个确定的值, 因此, 只要找到合适的材料, 使$\left| {{t_{\rm{TM}}} - {t_{\rm{TE}}}} \right|$ 在感兴趣的${\vartheta ^{\rm{i}}}$ 范围内具有尽可能大的值, 就能获得尽可能增强的SOI. 实际理想情况下, 如果能够使${t_{\rm{TM}}}$ 和${t_{\rm{TE}}}$ 的值一个为1, 另一个为–1, 则能获得100%的效率.然而, 传统的材料(如空气、玻璃等)构成的界面, SOI的转换效率极低. 考虑一个放置于自由空间、介电常数为ε = 2.25、厚度为h (波长量级)的非磁性(磁导率为1)各向同性薄膜, 其TM和TE平面波分量的透射系数分别为[39]
$${t_{\rm{TM}}}({\vartheta ^{\rm{i}}}) = \dfrac{1}{{\cos (k_z^{(2)}h) - \dfrac{\rm{i}}{2}\left( {\dfrac{{k_z^{(1)}\varepsilon }}{{k_z^{(2)}}} + \dfrac{{k_z^{(2)}}}{{k_z^{(1)}\varepsilon }}} \right)\sin (k_z^{(2)}h)}},\tag{11a}$$ $${t_{\rm{TE}}}({\vartheta ^{\rm{i}}}) = \dfrac{1}{{\cos (k_z^{(2)}h) - \dfrac{\rm{i}}{2}\left( {\dfrac{{k_z^{(1)}}}{{k_z^{(2)}}} + \dfrac{{k_z^{(2)}}}{{k_z^{(1)}}}} \right)\sin (k_z^{(2)}h)}},\tag{11b}$$ 其中,
$k_z^{(1, 2)}$ 表示平面波法线方向(z方向)的波矢分量,$k_z^{(1)} = {k^{\rm{i}}}\cos {\vartheta ^{\rm{i}}}$ ,$k_z^{(2)} = {k^{\rm{t}}}\cos {\vartheta ^{\rm{t}}}$ . 由(11)式可知${t_{\rm{TM}}}$ 和${t_{\rm{TE}}}$ 相差极小, 转换效率极低(图3(a)).
图 3 三种放置于自由空间的单层薄膜材料的透射系数, 以及SOI的转换效率$\eta = {\left| {{t_{\rm{TM}}} - {t_{\rm{TE}}}} \right|^2}/4$ $\varepsilon = 2.25$ $\varepsilon = 0.01$ ${\varepsilon _x} = {\varepsilon _y} = 1$ ${\varepsilon _z} = 0.01$ $\lambda = 1$ $h = 2\lambda $ ${\vartheta ^{\rm{i}}}$ ${\varepsilon _z}$ ${\vartheta ^{\rm{i}}}$ ${\varepsilon _x} = {\varepsilon _y} = 1$ $h = 1\lambda $ ${\varepsilon _x} = {\varepsilon _y} = 1$ ${\varepsilon _z} = 0.01$ Fig. 3. Transmission coefficients and conversion efficiency ($\eta = {\left| {{t_{\rm{TM}}} - {t_{\rm{TE}}}} \right|^2}/4$ $\varepsilon = 2.25$ $\varepsilon = 0.01$ ${\varepsilon _x} = {\varepsilon _y} = 1$ ${\varepsilon _z} = 0.01$ $\lambda = 1$ $h = 2\lambda $ ${\vartheta ^{\rm{i}}}$ ${\varepsilon _z}$ ${\varepsilon _x} = {\varepsilon _y} = 1$ $h = 1\lambda $ ${\vartheta ^{\rm{i}}}$ ${\varepsilon _x} = {\varepsilon _y} = 1$ ${\varepsilon _z} = 0.01$ 鉴于传统材料的SOI极弱, Ciattoni等[33]在2017年从理论上提出采用介电常数近零的各向同性薄层来增强这种效应(图3(b)). 对于介电常数近零材料薄层, TM波在入射角很小时就可以满足法布里-珀罗共振, 使
$\left| {{t_{\rm{TM}}}} \right|$ 达到1, 并同时使$\left| {{t_{\rm{TM}}} - {t_{\rm{TE}}}} \right|$ 达到较大的值. 然而, 其转换效率最高也只可达20%左右. 其原因是, 对于各向同性材料,${t_{\rm{TM}}}$ 和${t_{\rm{TE}}}$ 同时受到介电常数的影响, 无法独立地调控, 很难使$\left| {{t_{\rm{TM}}} - {t_{\rm{TE}}}} \right|$ 接近于2, 因此转换效率难以达到100%.3.2 通过单轴薄层增强SOI的效率至100%
各向异性材料比各向同性材料具有更多的自由度, 有望获得接近100%的效率. 考虑厚度为h的非磁性(磁导率为1)单轴晶体薄层, 其介电常数张量为
$$ {{\varepsilon}} = \left( {\begin{array}{*{20}{c}} {{\varepsilon _x}}&{}&{} \\ {}&{{\varepsilon _y}}&{} \\ {}&{}&{{\varepsilon _z}} \end{array}} \right),\;{\text{且}} {\varepsilon _x} = {\varepsilon _y} \ne {\varepsilon _z}. $$ (12) 即此单轴层的光轴与z轴平行, 也就是光束的传输方向. 这种情况下, 体系依然具有旋转不变性, 上文的所有理论仍然适用. 由于该材料具有
${\varepsilon _{x, y}}$ 和${\varepsilon _z}$ 两个可以独立调控的介电常数, 所以相比于各向同性材料多了一个调控的自由度. 写出其透射系数为[40]$${t_{\rm{TM}}}({\vartheta ^{\rm{i}}}) = \dfrac{1}{{\cos ({q_{\rm{e}}}h) - \dfrac{\rm{i}}{2}\left( {\dfrac{{{q_{\rm{e}}}}}{{{\varepsilon _x}{q_1}}} + \dfrac{{{\varepsilon _x}{q_1}}}{{{q_{{e}}}}}} \right)\sin ({q_{\rm{e}}}h)}}, \tag{13a}$$ $${t_{\rm{TE}}}({\vartheta ^{\rm{i}}}) = \dfrac{1}{{\cos ({q_{\rm{o}}}h) - \dfrac{\rm{i}}{2}\left( {\dfrac{{{q_{\rm{o}}}}}{{{q_1}}} + \dfrac{{{q_1}}}{{{q_{\rm{o}}}}}} \right)\sin ({q_{\rm{o}}}h)}},\tag{13b}$$ 注意, 其中TM和TE波的交叉偏振透射系数
$$\begin{split}&{t_{{\rm{TM}} \to {\rm{TE}}}} = {t_{{\rm{TE}} \to {\rm{TM}}}} = 0,\\ & {q_1} = k\cos \vartheta _{}^{\rm{i}},\\ &{q_{\rm{o}}} = \sqrt {{\varepsilon _x}} k\cos \vartheta _{}^o,\\ &{q_{\rm{e}}} = \sqrt {{\varepsilon _x}{k^2} - \frac{{{\varepsilon _x}}}{{{\varepsilon _z}}}k_{//}^2},\\ &{k_{//}} = k\sin \vartheta _{}^{\rm{i}},\\ &\vartheta _{}^o = {\sin ^{ - 1}}\left[ {\sin (\vartheta _{}^{\rm{i}})/\sqrt {{\varepsilon _x}} } \right],\\ &k = 2{\text{π}} {\rm{/}}\lambda \end{split}$$ 当
${\varepsilon _x} = {\varepsilon _y} = {\varepsilon _z}$ 时, (13)式回到各向同性材料的情况, 即(11)式.要想获得100%的转换效率, 必定要使
${t_{\rm{TM}}}$ 和${t_{\rm{TE}}}$ 的值一个为1, 另一个为–1, 当然也就没有反射. 首先, 令${\varepsilon _x} = {\varepsilon _y} = 1$ , 即平行于界面方向的介电常数与自由空间相同, 这保证了TE波的透射系数为${t_{\rm{TE}}} = \exp ({\rm{i}}{q_{\rm{o}}}h)$ , 其模值$\left| {{t_{\rm{TE}}}} \right| \equiv 1$ (图3(c)), 即任意入射角下, TE波均全部透射. 然后, 令${\varepsilon _z} \to 0$ (远小于自由空间的介电常数), 使TM波的全反射临界角(即满足${\varepsilon _z} = {\sin ^2}\vartheta _{}^{\rm{i}}$ 条件时)变得很小. 大于临界角时,${t_{\rm{TM}}} = 0$ ,$\left| {{t_{\rm{TM}}} - {t_{\rm{TE}}}} \right| = 1$ , 转换效率恒为25%; 小于临界角时, 由于TM波可能满足法布里-珀罗共振条件$2{q_{\rm{e}}}h = 2 m{\text{π}} $ (m为整数), 而出现全透射的情况(${t_{\rm{TM}}} = \pm 1$ ), 使得$\left| {{t_{\rm{TM}}} - {t_{\rm{TE}}}} \right|$ 的值可能为2, 也可能为0, 以及它们之间任意的中间值(两个共振峰之间). 也即, 在TM波满足法布里-珀罗共振时, 有可能实现100%的效率.下面具体计算
${\varepsilon _x} = {\varepsilon _y} = 1$ 且${\varepsilon _z} = 0.01$ 时, 厚度h = 2λ的单轴薄层的透射系数和转换效率. 如图3(c)所示,$\left| {{t_{\rm{TE}}}} \right|$ 在任意入射角度时恒为1; 而TM波在$\vartheta _{}^{\rm{i}} > {5.8^ \circ }$ 时发生全反射($\left| {{t_{\rm{TM}}}} \right| = 0$ ), 当$\vartheta _{}^{\rm{i}} < $ 5.8°时, 在某些角度发生法布里-珀罗共振, 出现${t_{\rm{TM}}}\! =$ ±1的情况, 使SOI的转换效率$\eta = {\left| {{t_{\rm{TM}}} - {t_{\rm{TE}}}} \right|^2}/4$ 在$\vartheta _{}^{\rm{i}} = {3.8^ \circ }$ 和${5.6^ \circ }$ 附近达到100%. 当${\varepsilon _z}$ 和厚度h发生改变时, 影响法布里-珀罗共振出现的角度, 因此出现100%效率的角度$\vartheta _{}^{\rm{i}}$ 也随之改变, 如图3(d)和图3(e)所示.虽然具有这种极端的介电常数的单轴材料很难在自然界中找到, 但近年来随着微纳光学, 特别是超构材料和超构表面领域的飞速发展, 具有上述等效介电常数的人工合成材料有望通过合适地设计超构材料或超构表面来实现, 比如双曲超构材料(hyperbolic metamaterials)[41,42]. 最后还需要指出的是, 本文所建立的全波理论适合于旋转不变的体系, 比如由各向同性材料构成的界面、各向同性材料与光轴平行于界面法线方向的单轴晶体构成的界面. 当系统不具有旋转不变性时, 比如光轴方向平行于界面时的单轴晶体, 需对本文理论作较大修正才能适用.
4. 结 论
首先建立了菲涅耳琼斯矩阵来描述光束正入射至突变界面时的SOI, 分析和澄清了光束本身的拓扑结构和界面性质在SOI中所扮演的不同角色, 并揭示了其中所产生的涡旋相位的物理根源. 简而言之, 这种涡旋来源于光束本身的几何拓扑结构, 在本质上是一种自旋重构的贝里相位; 而界面的性质影响SOI的转换效率. 在形式上, 这种SOI与PB相位元件中的SOI极为相似, 但PB相位来源于材料外部的各向异性. 由于该效应在一般情况下极弱, 限制了其应用. 因此提出用光轴平行于界面法线方向的单轴晶体薄层来有效地增强它, 使之在一定条件下达到100%的转换效率. 本文的研究不但为这种SOI建立了简洁明晰的理论框架, 而且揭示了现象背后的物理机理, 并进一步给出了增强这种效应的可行方案, 为未来的潜在应用指明了方向.
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图 6 希格斯玻色子胶子-胶子融合产生机制以及希格斯衰变为光子对的费曼图
Fig. 6. The Feynman diagram of ggF and Higgs decay into a pair of photons
图 7 希格斯玻色子伴随一对顶夸克对产生模式的费曼图
Fig. 7. The Feynman diagram of $ {\rm{t}}{\bar {\rm{t}}}{\rm{H}} $ production model
图 8 ATLAS实验组获得的$ {\rm{t}}{\bar {\rm{t}}}{\rm{H}} $产生过程候选事例三维展示图, 在事例中, 探测器下部有两个孤立的光子(绿色), 共产生六个喷注(黄色锥形), 包括一个B-标记喷柱(蓝色锥形)[19]
Fig. 8. Three-dimensional (3D) display of a candidate event of $ {\rm{t}}{\bar {\rm{t}}}{\rm{H}} $ production mode from ATLAS. The event has the two isolated photons (green line below the detector) and six jets (yellow cone), including one B-tagged jet (blue cone).
图 10 标准模型中, 发生自发性破缺前的希格斯势场的示意图, 形状酷似“墨西哥帽”. 可以看出, 势能的最低点并不处于原点
Fig. 10. Diagram of the Higgs potential field before spontaneous breaking in the standard model, shaped like a “Mexican hat”. As you can see from the diagram, the lowest point of the potential energy is not at the origin.
图 11 希格斯玻色子对和三希格斯玻色子末态的费曼图, 左图为希格斯玻色子对过程, 蓝色圈代表三希格斯玻色子自耦合顶点$ \lambda_3 $, 右图为三希格斯玻色子产生过程, 红色圈代表四希格斯玻色子自耦合顶点$ \lambda_4 $
Fig. 11. The Feynman diagram of the Higgs boson pair and triple Higgs final state, the left picture is the Higgs boson pair process, the blue circle represents the three Higgs Boson self-coupling vertex $ \lambda_3 $, the right picture is the triple Higgs boson production process, the red circle represents the quartic Higgs Boson self-coupling vertex $ \lambda_4 $.
图 14 希格斯玻色子的矢量玻色子融合产生(左)和矢量玻色子伴随产生(右)两种模式的领头阶费曼图
Fig. 14. Leading-order Feynman diagrams for Higgs boson production via vector boson fusion (left) and vector boson associated production (right) modes.
图 18 二期运行期间ATLAS和CMS组统计合并测量希格斯玻色子衰变到Z玻色子和一个光子的信号强度结果[33]
Fig. 18. Combined measurement of the signal strength for Higgs boson decay into a Z boson and a photon by the ATLAS and CMS collaborations during Run II.
图 21 LHC实验对暗物质粒子的探测示意图[46], 暗物质对探测器不可见, LHC实验通过计算所有可见物质的横向动量和来推算缺失动量, 从而搜寻潜在的暗物质信号
Fig. 21. An illustration of the LHC experiment’s search for dark matter particles[46], which are invisible to the detector. The search for potential dark matter signals is conducted by calculating the transverse momentum of all visible matter and inferring the missing transverse momentum.
图 23 ATLAS合作组分析mono-Higgs信号得到的排除轮廓[48], 合并了$ {\rm{b}}\bar{\rm{b}} $和$ \gamma\gamma $希格斯玻色子衰变道的分析结果. 黑色虚线表示仅有标准模型背景假设下的预期轮廓, 绿带为1σ误差范围, 黑色实线为观测轮廓. 灰色虚线为动力学限制, 即$ m_{{\mathrm{Z}}'_{\mathrm{B}}} = 2 m_{\text{χ}}$
Fig. 23. Exclusion contours obtained by the ATLAS collaboration analyzing the mono-Higgs signal[48], combining results from the Higgs boson decay channels of $ {\rm{b}}\bar{\rm{b}} $ and $ \gamma\gamma $. The black dashed line represents the expected contour under the assumption of only the standard model background, with the green band indicating the 1σ error range, and the black solid line representing the observed contour. The grey dashed line represents a kinematic constraint, namely $ m_{{\mathrm{Z'_B}}} = 2 m_{\text{χ}} $.
图 24 胶子融合产生希格斯玻色子对的领头阶费曼图[49]. 左图和中图: 非共振态粒子产生希格斯玻色子的三角图和盒形图, 其符合标准模型. 右图: 通过一个新共振粒子产生的希格斯玻色子费曼图, 其中新共振粒子用X表示
Fig. 24. Leading order Feynman diagrams of Higgs boson pair production via gluon fusion[49]. Left and middle: the triangle and box diagrams, respectively for nonresonant H production, as expected from the SM. Right: diagram for H boson production through a new resonance of labeled as X
图 25 自旋为0的共振态粒子$ {\rm{X}}\rightarrow {\rm{HH}} $的产生截面与衰变分支比乘积$ {\sigma {\cal{B}}} $在95%置信水平下的上限结果[49]. 其中实线表示观察到的结果, 虚线表示的是期望结果
Fig. 25. Search for $ {\rm{X}}\rightarrow {\rm{HH}} $: Observed and expected 95% CL upper limits on the product of the cross section for the production of a spin-0 resonance X and the branching fraction for the corresponding HH decay[49]. The observed limits are indicated by markers connected with solid lines and the expected limits by dashed lines.
图 26 hMSSM模型对$ {\rm{X}}\rightarrow {\rm{HH}} $的寻找结果的诠释[49]. 图中展示了在$ (m_{\rm{A}}, \, \tan\beta) $平面上, HH的联合分析在95%置信水平下观察到的和预期的排除区域, 并将其与hMSSM模型中重标量粒子衰变为$ \tau\tau $[64], tt[65]和WW[66]的寻找结果进行了对比. 此外, 还给出了$ {\rm{A}}\rightarrow {\rm{ZH}} $的一个代表性寻找结果[67], 以及通过测量希格斯玻色子耦合强度给出的间接约束[68]
Fig. 26. Interpretation of the results from the searches for the $ {\rm{X}}\rightarrow {\rm{HH}} $ decay, in the hMSSM model[49]. The observed and expected exclusion contours at 95% CL, in the $ (m_{\rm{A}}, \tan \beta) $ plane from the combined likelihood analysis of HH analyses are shown. A comparison of the region excluded by the combined likelihood analysis shown in this panel with selected results from other searches for the production of heavy scalar bosons in the hMSSM, in $ \tau\tau $[64], tt[65] and WW[66] decays is shown. Also shown, are the results from one representative search for A → ZH[67] and indirect constraints obtained from measurements of the coupling strength of the observed H boson[68].
图 32 用喷注本源鉴别算法得到的Higgs稀有衰变以及奇异衰变的分支比上限, 如绿色柱状图所示[76]. CEPC (蓝色柱状图)[75]和高亮度大型强子对撞机(橙色柱状图)[80]预期的Higgs耦合的相对不确定度
Fig. 32. Expected upper limits on the branching ratios of rare Higgs boson decays (green bars)[76] and the relative uncertainties of Higgs couplings anticipated at CEPC (blue)[75] and HL-LHC (orange)[80].
图 29 经过CEPC基线探测器[73]模拟的对撞能量为$ 240\, $ GeV的$ { {\rm{e}}^+{\rm{e}}^-} \to\nu\bar{\nu} {{\rm{H}}} \to\nu\bar{\nu} {\rm{gg}} $事例样本[76]
Fig. 29. Event display of an $ { {\rm{e}}^+{\rm{e}}^-} \to\nu\bar{\nu} {{\rm{H}}} \to\nu\bar{\nu} {\rm{gg}} $ ($ \sqrt{s} = $$ 240\; {\rm{GeV}} $) event[76] simulated and reconstructed with the CEPC baseline detector[73].
图 30 正负电子对撞能量为240 GeV时, 利用CEPC基线探测器全模拟的$ \nu\bar{\nu} {\rm{H}}, {\rm{H}}\to {\rm{jj}} $样本, 基于ParticleNet深度学习模型训练, 在测试数据上得到的喷注味道鉴别矩阵[76]
Fig. 30. Confusion matrix $ M_{11} $ based on full simulated $ \nu\bar{\nu}{\rm{H}} $, $ {\rm{H}}\to {\rm{jj}} $ at 240 GeV center-of-mass energy at CEPC baseline detector[76].
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