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超表面是一种亚波长尺度下的超薄二维阵列平面,由超材料结构单元组成。超材料是在亚波长尺度下按照某种规律排列的人工微结构,可实现任意地等效介电常数和磁导率[12]。超材料比自然材料具有更丰富的电磁调控能力,其独特的电磁响应主要来自于设计的结构。超表面可视为二维超材料,其厚度远小于入射波长。相比于超材料,超表面的制作相对简单、损耗相对较低、集成度高,通过人工微结构的改变可以灵活地控制电磁波的振幅和相位、传播模式、偏振态等特性[13]。
2011年哈佛大学Yu等人[14]首次设计了在硅上的光学薄金属天线阵列,观察到沿界面线相位变化的异常反射和折射现象,这与由费马原理导出的广义斯涅耳定律一致。如图1中红色和蓝色的两条光路所示,在两种介质之间的界面引入突变线性相位变化,根据费马原理(光线传播的路径是需时最少的路径),入射角为
${\rm{\theta }}_{{i}}$ 的平面波入射,两条光路都无限接近实际光路,两者总相位变化相同。省略相同的光程引起的相位变化,可以得到公式(1):$$ \begin{split} {{{k}}}_{0}{{{n}}}_{{i}}{{{\rm{s}}}}{{{\rm{i}}}}{{{\rm{n}}}}\left({{\theta }}_{{i}}\right){{{\rm{d}}}}{{x}}+&\left({{\phi }}+{{{\rm{d}}}}{{\phi }}\right)=\\ &{{{k}}}_{0}{{{n}}}_{{{t}}}{{{\rm{s}}}}{{{\rm{i}}}}{{{\rm{n}}}}\left({{\theta }}_{{{t}}}\right){{{\rm{d}}}}{{x}}+{{\phi }} \end{split} $$ (1) 式中:
${{{k}}}_{0}=2\pi /{{\lambda}}_0$ ,$ {\rm{\lambda}} _0 $ 是真空中的波长;${{\theta }}_{{{t}}}$ 是折射角;${{{rm{{\rm{d}}}}}}{{x}}$ 是两光路与界面交点之间的距离;${{{n}}}_{{i}}$ 和${{{n}}}_{{{t}}}$ 分别是两种介质的折射率,等式两边的第一项代表光程引入的相位差,$ {\rm{\phi }} $ 和${\rm{\phi }}+{{{\rm{d}}}}{\rm{\phi }}$ 分别为两光路在界面上的相位突变,改变公式(1)的形式得到公式(2):$$ {{{n}}}_{{{t}}}{{{\rm{s}}}}{{{\rm{i}}}}{{{\rm{n}}}}\left({{\theta }}_{{{t}}}\right)-{{{n}}}_{{i}}{{{\rm{s}}}}{{{\rm{i}}}}{{{\rm{n}}}}\left({{\theta }}_{{i}}\right)=\dfrac{{{{\lambda }}}_{0}}{2\pi }\dfrac{{{{\rm{d}}}}{{\phi }} }{{\rm{d}}x} $$ (2) 同理,反射情况下可以得到公式(3):
$$ {{{\rm{s}}}}{{{\rm{i}}}}{{{\rm{n}}}}\left({{\theta }}_{{{r}}}\right)-{{{\rm{s}}}}{{{\rm{i}}}}{{{\rm{n}}}}\left({{\theta }}_{{i}}\right)=\dfrac{{{{\lambda }}}_{0}}{2\pi {{{n}}}_{{i}}}\dfrac{{{{\rm{d}}}}{{\phi }} }{{\rm{d}}x} $$ (3) 式中:
${{\theta }}_{{{r}}}$ 是反射角。公式(2)和公式(3)合称广义斯涅耳定律。由此,可以设计相位变化梯度
$\dfrac{{{{\rm{d}}}}{{\phi }} }{{\rm{d}}x}$ ,实现光束任意角度的折射和反射。 -
通常用哈密顿量H来描述系统的总能量,H
包含系统的所有特性。宇称−时间(PT)对称性指系统的哈密顿量经过一次宇称变换(parity)和时间反演变换(time reversal)后形式不变, $ \left[ {PT,H} \right]= PTH - HPT=0$ ,其中P和T操作算符满足$ {P^2}=1, P={P^ + }, {T^2}=1, T={T^ + }, \left[ {P,T} \right]=0$ 。宇称变换利用空间反转算符$ \hat P$ 表示:$ {\rm{}}\hat p \to - \hat p$ 、$ \hat x \to - \hat x$ , i→i,时间反演变换利用时间反演算符$ {\hat T}$ 表示:$ \hat p \to - \hat p$ 、$ \hat x \to \hat x$ 、i→-i,其中$ {\hat x}$ 和$ {\hat p}$ 分别是位置算符和动量算符,满足$ \left[ {x,p} \right]= xp - px=i\hbar $ ,i是虚数单位[15]。在经典量子力学中,哈密顿量一般要求为厄米形式,
$ H=H^+$ ,其中+代表厄米共轭,此时系统不具有增益或者损耗,且不与外界发生能量交换,即封闭系统,其满足宇称时间对称性且能量本征值为实数;如果系统具有增益或者损耗,或者与外界发生能量交换,即开放系统,系统的能量不再守恒,$ \hat H$ 为非厄米哈密顿量,$ H \ne {H^ + }$ 。1998年,Carl M. Bender和Stefan Boettcher提出非厄米哈密顿量在满足宇称时间对称性时本征值为实数,并发现非厄米哈密顿量的宇称时间对称性被破坏时会导致本征值由实数变为复数[16]。满足PT对称的一个必要条件(但不是充分条件)是系统哈密顿量中的势函数满足
$ {V}^{*}\left(-x\right)=V\left(x\right) $ ,将势函数写成复数的形式,$ V\left(x\right)={V}_{r}\left(x\right)+i{V}_{i}\left(x\right) $ ,$ {V}^{*}\left(-x\right)={V}_{r}\left(-x\right)-i{V}_{i}(-x) $ ,其中$ {V}_{r} $ 和$ {V}_{i} $ 分别代表势函数的实部和虚部,从而可以推导出$ {V}_{r}\left(x\right)={V}_{r}\left(-x\right) $ ,$ {V}_{i}\left(x\right)=-{V}_{i}(-x) $ 。在这种伪厄米构型中,本征函数不再正交,$ <m|n>\ne {\delta }_{mn} $ ,矢量空间是扭曲的,在奇异点处系统会发生明显的对称性破坏的现象。在破坏机制下,系统的哈密顿量和PT操作算符不再具有相同的本征函数,系统的本征值不再是实数,转变为共轭复数。量子力学中的薛定谔方程(4)与亥姆霍兹方程(5)具有同构等价性[17],所以光学可作为探索PT对称的理想平台[18]。
$$ i\dfrac{\partial \varPsi \left(\overrightarrow{r},t\right)}{\partial t}=[-\dfrac{{\hslash }^{2}}{2m}{\nabla }^{2}+V\left(\overrightarrow{r}\right)]\varPsi \left(\overrightarrow{r},t\right) $$ (4) $$ \begin{split} i\dfrac{\partial E\left(x,z\right)}{\partial z}=&-\dfrac{1}{2n{k}_{0}}\frac{{\partial }^{2}E\left(x,z\right)}{\partial {x}^{2}}+\\ &{k}_{0}n\left(x\right)E(x,z) \end{split} $$ (5) 式中:
${{{k}}}_{0}=2\pi /{{\lambda}}_0$ ,$ {\rm{\lambda}}_0$ 是真空中的波长;$ n\left(x\right)={n}_{{\rm{R}}}\left(x\right)+ i{n}_{{\rm{I}}}\left(x\right) $ 是复折射率,又称为系统的光学势能,容易看出,$ V\left(\overrightarrow{r}\right) $ 和$ {k}_{0}n\left(x\right) $ 等价,所以PT对称的光学系统的复折射率满足$ {n}^{*}\left(x\right)=n(-x) $ ,更具体的形式是$ {n}_{{\rm{R}}}\left(x\right)={n}_{{\rm{R}}}\left(-x\right) $ 、$ {n}_{{\rm{I}}}\left(x\right)=-{n}_{{\rm{I}}}\left(-x\right) $ , 复折射率实部偶对称,虚部奇对称,函数图像如图2所示。在光学材料和结构中,折射率实部
$ {n}_{{\rm{R}}}\left(x\right) $ 表示色散,折射率虚部$ {n}_{{\rm{I}}}\left(x\right) $ 表示光能的增益和损耗,PT对称系统就是具有完全平衡的增益和损耗的开放物理系统[19-20]。 -
具有非厄米哈密顿量的系统存在奇异点,特点是两个或多个本征值简并,并且对应的本征态也合并成一个量子态。以具有增益或损耗的两个耦合谐振单元组成的二能级系统为研究对象,如图3所示,它的
$ 2\times 2 $ 非厄米哈密顿量为:$$ \hat H=\left[ {\begin{array}{*{20}{c}} {{\omega _1} - i{\gamma _1}}&\mu \\ \mu &{{\omega _2} - i{\gamma _2}} \end{array}} \right] $$ (6) 式中:
$ {\omega }_{1} $ 、$ {\omega }_{2} $ 分别是两个耦合谐振单元的共振频率;$ \mu $ 是两者之间的耦合系数;$ {\gamma }_{1} $ 、$ {\gamma }_{2} $ 是增益或者损耗系数。哈密顿量的本征值为:$$ \begin{split} {\omega _ \pm }=&\dfrac{{{\omega _1} + {\omega _2}}}{2} - i\dfrac{{{\gamma _1} + {\gamma _2}}}{2}\pm\\ &\sqrt {{\mu ^2} + {{({\omega _{{\rm{dif}}}} + i{\gamma _{{\rm{dif}}}})}^2}} \end{split} $$ (7) 式中:
${\omega }_{{\rm{dif}}}=\dfrac{{\omega }_{1}-{\omega }_{2}}{2}$ ,${\gamma }_{{\rm{dif}}}=\dfrac{{\gamma }_{1}-{\gamma }_{2}}{2} {\text{。}} $ 当
$\sqrt{{\mu }^{2}+{({\omega }_{{\rm{dif}}}+i{\gamma }_{{\rm{dif}}})}^{2}}=0$ 时,两本征值相同,本征态共线,出现奇异点,解为(${\omega }_{{\rm{dif}}}$ =0,μ=$\pm {\gamma }_{{\rm{dif}}}$ )或者$({\gamma }_{{\rm{dif}}}$ =0,μ=$\pm i{\omega }_{{\rm{dif}}})。$ 当
$ {\gamma }_{1}{+\gamma }_{2}=0 $ 且${\mu }^{2}+{({\omega }_{{\rm{dif}}}+i{\gamma }_{{\rm{dif}}})}^{2}\ge 0$ 时,本征值为实数,是PT对称的,解为$ {(\gamma }_{1}=-{\gamma }_{2}=\gamma ,{\omega }_{1}={\omega }_{2}=\omega ) $ ,此时具有PT对称性的哈密顿量为:$$ \hat H=\left[ {\begin{array}{*{20}{c}} {{\omega _1} - i{\gamma _1}}&\mu \\ \mu &{{\omega _2} + i{\gamma _2}} \end{array}} \right] $$ (8) 本征值为:
$$ {\omega }_{\pm }=\omega \pm \sqrt{{\mu }^{2}-{\rm{\gamma }}^{2}} $$ (9) 在具有相同增益和损耗平行放置的双波导系统中[22],如图4所示,
${{{g}}}_{1}$ 是波导的增益/损耗系数,$ \kappa $ 是两波导间的耦合系数,也称为增益或损耗的阈值。可以改变增益/损耗的大小改变系统的本征值${\omega }_{\pm }= \omega \pm \sqrt{{\kappa }^{2}-{{{{g}}}_{1}}^{2}}$ ,如图5所示,当${{{g}}}_{1} < \kappa$ 时,本征值为两个不同的实数,系统具有PT对称性,在双波导系统中能量守恒;当${{{g}}}_{1} > \kappa$ 时,本征值为两个不同的复数,系统PT对称性被打破,光在双波导系统中能量指数增长或衰减;当${{{g}}}_{1}=\kappa$ 时,本征值和本征态都简并,该点为系统由 PT对称性到PT对称打破的相变点,称为奇异点(Exceptional Points)。在此点会发生很多新奇独特的物理性质,比如奇异点附近灵敏度会大大增强,可应用于探测器[23]和传感器[25]的研究,利用奇异点处发生自发PT对称性打破的特点,产生非互易性,可应用于隔离器[25]和二极管[20]的研究,此外还可实现拓扑手性[26]、激光模式选择[27-28]等。图 5 具有相同增益和损耗的耦合双波导系统和本征值随增益/损耗系数变化情况。两个本征值的实部(“Re”实线)和虚部(“Im”虚线)以及EP点的位置[29]
Figure 5. Eigenvalues of coupled dual waveguide systems with the same gain/loss and the eigenvalue varies with the gain/loss coefficient. Real parts (“Re”, solid lines) and imaginary parts (“Im”, dashed lines) of the two normalized eigenvalues, and the position of the exceptional points[29]
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2017年,Hossein等人[24]发现在以上介绍的二阶奇异点附近,扰动强度
$ {\rm{\varepsilon }} $ 引起的本征频率分裂正比于$ {{\rm{\varepsilon }}}^{1/2} $ ,N阶奇异点附近,ε引起的本征频率分裂正比于$ {{\rm{\varepsilon }}}^{1/N} $ ,在外界扰动强度足够小时,高阶奇异点的灵敏性明显增强。一般是通过构建多组元结构实现高阶奇异点的,如图6的结构是由损耗−中性−增益的三个环形谐振腔组成的非厄米系统,当损耗和增益相等,且增益/损耗与耦合的比例${{g}}/{\rm{\kappa }}=\sqrt{2}$ ,外界扰动强度ε=0时,出现三阶EP点,此时三个本征值都简并到一点,在${{g}}/{\rm{\kappa }} < \sqrt{2}$ 、ε=0时,系统处于PT对称的状态,本征值为实数;在$ {\rm{g}}/{\rm{\kappa }}>\sqrt{2} $ 、ε=0时,系统处于PT对称破缺的状态,本征值成为复数,如图7所示。图 7 关于增益/损耗与耦合强度的比例g/κ和扰动强度与耦合强度的比例ε/κ的具有宇称−时间对称性的三微环系统的本征值的实部(左)和虚部(右)[24]
Figure 7. Real parts (left) and the imaginary parts (right) of the eigenfrequencies of the ternary parity-time-symmetric system as a function of the normalized gain/loss contrast g/κ and the detuning ε/κ
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超表面为探索PT对称和奇异点的物理学提供了平台,因为既可以精确控制结构参数,调整谐振性能来制造超表面,也能使用光学反射或透射测量进行探测[30]。PT对称性一般通过系统完全平衡的增益和损耗实现,但也可以从本征值中提取一个定值,来实现等效的PT对称系统,此类PT对称性可以发生在无源系统中,由纯损耗构成的系统,没有增益,使得制造和表征的PT结构更简单[31]。
以具有各向异性吸收特性,具有镜面对称分布的开口谐振环(Split Ring Resonators ,SRR)为例介绍超表面中出现EP点的原理,在硅衬底上两个正交方向的开口谐振环组成的阵列如图8所示,开口谐振环的共振频率由其几何结构决定,两个开口谐振环的结构参数完全相同,两种金属分别是铅(青绿色或深灰色)和银(黄色或浅灰色),铅的欧姆损耗比银大,所以两个环有相同的共振频率和不同的线宽。每个开口谐振环都可以被入射电磁场
${\hat E_{x,y}} = \left( {\widetilde {{E_x}},\widetilde {{E_y}}} \right){{\rm{e}}^{i\omega t}}$ 激发,等效成两个垂直方向的电偶极矩${\hat p_{x,y}} = {\tilde p_{x,y}}\left( \omega \right){{\rm{e}}^{i\omega t}}$ ,电偶极矩和入射电磁场之间由极化矩阵$ \mathop P\limits^ \leftrightarrow $ 联系,如公式(10)所示:$$ \begin{split} g\left( {\begin{array}{*{20}{c}} {{{\tilde E}_x}}\\ {{{\tilde E}_y}} \end{array}} \right)=& \\ & \left( {\begin{array}{*{20}{c}} {\delta + i{\gamma _x} + {G_{xx}}}&{{G_{xy}}}\\ {{G_{xy}}}&{\delta + i{\gamma _y} + {G_{yy}}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{{\tilde p}_x}}\\ {{{\tilde p}_y}} \end{array}} \right) \end{split} $$ (10) 式中:
$ {G}_{xx} $ =$ {G}_{xy} $ 是同一方向的偶极子中其他偶极子对其中一个偶极子作用的延迟耦合总和;$ {G}_{xy} $ 是所有y(x)方向的偶极子对一个x(y)方向的偶极子作用的延迟场总和。两个开口谐振环通过近场耦合相互作用,相互作用强度与两个开口谐振环之间的距离s有关。如图9所示,在y=-x平面遵循PT对称,通过改变两个开口谐振环之间的距离改变两个谐振环的耦合强度,就会产生相变。透过超表面传输的电磁场可以用具有等效PT对称哈密顿量形式的传输矩阵描述,如公式(11)所示:
图 9 关于y=−x镜面对称的带有较少损耗的偶极子(蓝色)和较多损耗的偶极子(红色)的PT对称超表面的原理图
Figure 9. Schematic of PT symmetric metasurface with less lossy dipoles (blue) and more lossy dipoles (red) that is symmetric about y equals minus x
$$ \left( {\begin{array}{*{20}{c}} {{{\tilde E}_{tx}}}\\ {{{\tilde E}_{ty}}} \end{array}} \right) = \mathop M\limits^ \leftrightarrow \left( {\begin{array}{*{20}{c}} {{{\tilde E}_x}}\\ {{{\tilde E}_y}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {{{\tilde E}_x}}\\ {{{\tilde E}_y}} \end{array}} \right) + \dfrac{{I\omega {\eta _0}}}{{2{a^2}}}\left( {\begin{array}{*{20}{c}} {{{\tilde p}_x}}\\ {{{\tilde p}_y}} \end{array}} \right) $$ (11) 从以上两式可以看出传输矩阵和极化矩阵
$ \mathop M\limits^ \leftrightarrow \mathop {,P}\limits^ \leftrightarrow $ 相同,公式(11)中可以把极化矩阵$\mathop {P}\limits^ \leftrightarrow $ 拆成两项,右边两项分别是各向异性PT对称部分和各向同性损耗部分,如公式(12)所示:$$ \begin{split} &g\left( {\begin{array}{*{20}{c}} {{{\tilde E}_x}}\\ {{{\tilde E}_y}} \end{array}} \right)=\\ &\left( {\begin{array}{*{20}{c}} {\delta + {G_{xx}} + i\dfrac{{{\gamma _x} + {\gamma _y}}}{2}}&0\\ 0&{\delta + {G_{yy}} + i\dfrac{{{\gamma _x} + {\gamma _y}}}{2}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{{\tilde p}_x}}\\ {{{\tilde p}_y}} \end{array}} \right) +\\ & \left( {\begin{array}{*{20}{c}} { - i\dfrac{{{\gamma _y} - {\gamma _x}}}{2}}&{{G_{xy}}}\\ {{G_{xy}}}&{i\dfrac{{{\gamma _y} - {\gamma _x}}}{2}} \end{array}} \right)\left( {\begin{array}{*{20}{c}} {{{\tilde p}_x}}\\ {{{\tilde p}_y}} \end{array}} \right) \end{split} $$ (12) 这种两个偶极子都带有损耗的等效PT对称结构也可以看作具有平衡损耗和增益的二元模型(左边第二项)嵌入在带有损耗的背景介质中(左边第一项),具有平衡损耗和增益的等效系统哈密顿量是:
$$ \left( {\begin{array}{*{20}{c}} { - i\dfrac{{{\gamma _y} - {\gamma _x}}}{2}}&{{G_{xy}}}\\ {{G_{xy}}}&{i\dfrac{{{\gamma _y} - {\gamma _x}}}{2}} \end{array}} \right) $$ (13) 如图10所示,当
$ 2{G_{xy}} > \left| {{\gamma _x} - {\gamma _y}} \right|$ 时满足PT对称,此时距离s小,两个偶极子强耦合,有两个实数本征值,通过透射率发现共振频率分裂现象;有两个本征态,超表面的本征极化态为$\hat x \pm {{\rm{e}}^{ \pm i\theta }}\hat y$ ,对应于沿±45°方向旋转的两个椭圆,其中$\theta ={\rm{arcsin}}\left[ {\left( {{\gamma _x} - {\gamma _y}} \right)/2{G_{xy}}} \right]$ 。图 10 Gxy变化时通过理想的PT对称超表面传输的本征极化态
Figure 10. Eigenpolarization states of transmission through an ideal PT symmetric metasurface when Gxy changes
当
$ 2{G_{xy}} < \left| {{\gamma _x} - {\gamma _y}} \right|$ 时PT对称破缺,此时两个开口谐振环的距离s较大,两个偶极子弱耦合,有两个复数本征值,即两个模态具有相同共振频率,不同衰减率;有两个本征态,超表面的本征极化态为$ \hat x \mp i{e^\theta }\hat y$ ,对应于沿着0°和90°方向旋转的两个椭圆,其中$\theta ={\rm{cos}}{h^{ - 1}}\left[ {\left( {{\gamma _x} - {\gamma _y}} \right)/2{G_{xy}}} \right]$ 。当
$ 2{G}_{xy}=\left|{\gamma }_{x}-{\gamma }_{y}\right| $ 出现EP点,本征值简并,只存在一个左旋圆偏振的本征态。
Exceptional points in metasurface
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摘要: 奇异点是非厄米系统中的特殊点,奇异点附近的参数空间会出现很多新奇的物理现象。超表面是物理学近年来兴起的一个研究热点,人们基于超表面的平台已经设计实现了大量性能优越的器件。超表面的出现为研究奇异点提供了一个易操作的平台,通过精确控制超表面的结构参数,可以方便地研究奇异点周围的参数空间。研究超表面中的奇异点也为研究新的物理规律提供了基础的平台,文中首先介绍了奇异点和超表面中的奇异点的基本理论,之后介绍了超表面中奇异点的最新研究进展,最后对目前该领域亟待解决的问题进行了分析总结,对该领域的发展进行了展望。Abstract: Exceptional points are special points in non-Hermitian systems, and there are many novel physical phenomena in the parameter space near the exceptional points. In recent years, metasurface has been a popular topic in physics. A large number of devices with superior performance have been designed based on metasurface platform. The appearance of the metasurface provides an easy platform for the study of exceptional points. By precisely controlling the structural parameters of the metasurface, it is convenient to study the parameter space of the exceptional points. The research on exceptional points in non-Hermitian metasurface also provides a foundation platform for studying new laws of physics. Firstly, the basic theory of exceptional points and the exceptional point in metasurface was introduced. Secondly, the recent research on the exceptional points in metasurface was introduced. Finally, the current problems needing to be solved were analyzed and the development of the field in the future was prospected.
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Key words:
- exceptional points /
- non-Hermitian systems /
- metasurface
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图 5 具有相同增益和损耗的耦合双波导系统和本征值随增益/损耗系数变化情况。两个本征值的实部(“Re”实线)和虚部(“Im”虚线)以及EP点的位置[29]
Figure 5. Eigenvalues of coupled dual waveguide systems with the same gain/loss and the eigenvalue varies with the gain/loss coefficient. Real parts (“Re”, solid lines) and imaginary parts (“Im”, dashed lines) of the two normalized eigenvalues, and the position of the exceptional points[29]
图 7 关于增益/损耗与耦合强度的比例g/κ和扰动强度与耦合强度的比例ε/κ的具有宇称−时间对称性的三微环系统的本征值的实部(左)和虚部(右)[24]
Figure 7. Real parts (left) and the imaginary parts (right) of the eigenfrequencies of the ternary parity-time-symmetric system as a function of the normalized gain/loss contrast g/κ and the detuning ε/κ
图 11 超表面设计原理图。每个单超包含两个面外方向厚度为t=30 nm、长度为a1=401.75 nm、长度为a2=435 nm、宽度为w1=50 nm、w2=100 nm的条形天线。两个天线都是独立的,晶格周期d=600 nm
Figure 11. Schematic of the metasurface design. Each unit cell contains two strip antennas with thickness t=30 nm in the out-of-plane direction, lengths a1=401.75 nm and a2=435 nm, and widths w1=50 nm and w2=100 nm. Both antennas are free-standing, with a lattice period d=600 nm
图 13 原理图(a)基于一对放大和衰减超表面的PT对称系统和(b)它允许互易性和单向无反射透明性。PT对称满足的条件:G2=−G1=−γY0, B2=B1=χY0,其中Gi和Bi为第i个超表面的表面电导和电纳;如果增益−损失参数γ>0,超表面1和2分别提供损失和增益,反之亦然。在太赫兹状态下,光学泵浦石墨烯超表面代表有源超表面(G2<0),而金属丝代表电阻片(G1>0)
Figure 13. Schematics of (a) the PT-symmetric system based on a pair of amplifying and attenuating metasurfaces, and (b) its enabled reciprocal and unidirectional reflectionless transparency. PT symmetry is satisfied with constrains: G2=−G1=−γY0 and B2=B1=χY0,where Gi and Bi are the surface conductance and susceptance of the ith metasurface; if the gain-loss parameter γ>0, metasurface 1 and 2 provide loss and gain, respectively, and vice versa. In the THz regime, the optically pumped graphene metasurface represents an active metasurface (G2<0), while the metallic filament represents a resistive sheet (G1>0)
图 14 PT对称的超表面的示意图。右边的图像显示了超表面的顶部和侧面。两种颜色分别代表两种金属(紫色:钛,黄色:金),从而获得高的电导率损失对比。SRR的外径和谐振环的开口间距的大小分别用dm和gm(m=1、2)表示,弧的宽度用w表示, s表示两个外弧之间的距离,其中一个SRR被投影到另一个SRR所在的平面上
Figure 14. Schematic illustration of a PT-symmetric metasurface. The images on the right show the top and side views of the metasurface. Two colors represent, respectively, two kinds of metal (purple: titanium, yellow: gold) to obtain a high loss-contrast of conductivity. The outer diameter and the gap size of the SRR is denoted by dm and gm (m=1, 2), respectively, and the width of the arc is denoted by w. s represents the distance between the two outer arcs as one SRR is projected in the plane where the other SRR is
图 16 非理想PT超表面结构的单位单元示意图。参数分别为h=20 nm, r1=61 nm, R1=122 nm, r2=63 nm, R2=170 nm, t=640 nm。距离s是可变的。入射波在x - z平面上,与+z轴成反射角θ
Figure 16. Schematic of unit cell of the non-ideal PT metasurface structure. The parameters are h=20 nm, r1=61 nm, R1=122 nm, r2=63 nm, R2=170 nm and t=640 nm, respectively. The distance s is variable. The incident wave is in x − z plane and has an angle θ with +z axis
图 17 超表面单个结构的示意图,在基板上用黑色实线表示。金层、PMMA层和ITO层的厚度分别为45 nm、180 nm和65 nm。s表示长度l1=140 nm, l2=170 nm,宽度w1=98 nm, w2=110 nm的两个正交槽/杆的间距
Figure 17. Schematic diagram of the unit cell, highlighted by black dotted line, on a glass substrate. Thicknesses of gold, PMMA, and ITO layers are 45 nm, 180 nm, and 65 nm, respectively. s denotes the separation between the two orthogonal slots/bars with lengths l1=140 nm, l2=170 nm and widths w1=98 nm, w2=110 nm
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