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光纤中传输OAM模式的基本原理在于同阶本征模式TE模、TM模、HE模以及EH模的叠加,具体叠加方式如公式(1)所示[16]:
$$ \begin{gathered} \left\{ {\begin{array}{*{20}{l}} {{\text{OAM}}_{ \pm l,m}^ \pm = {\text{HE}}_{l + 1,m}^{{\text{even }}} \pm {\text{jHE}}_{l + 1,m}^{{\text{odd }}}} \\ {{\text{OAM}}_{ \pm l,m}^ \mp = {\text{EH}}_{l - 1,m}^{{\text{even }}} \pm {\text{jEH}}_{l - 1,m}^{{\text{odd }}}} \end{array}} \right\}(l \gt 1) \\ \left\{ {\begin{array}{*{20}{l}} {{\text{OAM}}_{ \pm l,m}^ \pm = {\text{HE}}_{2,m}^{{\text{even }}} \pm {\text{jHE}}_{2,m}^{{\text{odd }}}} \\ {{\text{OAM}}_{ \pm l,m}^ \mp = {\text{T}}{{\text{M}}_{0,m}} \pm {\text{jT}}{{\text{E}}_{0,m}}} \end{array}} \right\}(l = 1) \\ \end{gathered} $$ (1) 式中:HEeven和EHeven分别为HE和EH模式的偶模;HEodd和EHodd分别为HE和EH模式的奇模;m为径向阶数;OAM模式由同阶本征模式的奇偶模叠加而成。j表示π/2的相位差,OAM的上标符号“+”表示右旋圆偏振,“−”表示左旋圆偏振,下标符号“±”表示螺旋相位旋转方向。当m>1时,会出现“意外退化”而导致不支持OAM模式[17]。当l=1时,由于TM模式和TE模式具有不同的有效折射率,在传播过程中会造成模式走离现象。因此两种模式叠加生成的OAM模式不稳定,不做考虑,故此时同一拓扑荷数下包含两个OAM模式。当l>1时,同一拓扑荷数下包含4个OAM模式。
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根据传输OAM模式光纤的设计要求[18]:首先,光纤需要具有环形传输区域,这与OAM模式的模态分布有关;其次,光纤结构设计要能够满足OAM传输基本要求,即相邻本征模式之间的有效折射率差应大于10−4以防止耦合成不能承载OAM模式的LP模式。Bai等[13]通过分析比较得出结论:可支持的OAM模式数量随包层空气填充率的增加而增加,环形传输区域与包层间较大的折射率差有利于支持更多OAM模式的传输。同时较大的折射率差也能够有效防止环形传输区域内的光强泄露进包层中,从而保证了高质量传输。空气折射率小于SiO2的折射率,因此可通过提高光纤包层空气填充率进而降低包层折射率以增加环形传输区域与包层之间的折射率差。对于圆形空气孔和矩形空气孔,其相关充气分数分别如公式(2)所示[13]:
$$ {f_{cs}} = \frac{{\text{π }}}{4}{\left( {\frac{d}{\varLambda }} \right)^2};\;\;{f_{ss}} = \frac{{ab}}{{{\varLambda ^2}}} $$ (2) 式中:fcs为圆形空气孔的相关充气分数;d为圆的直径;Λ为孔间距;fss为矩形空气孔的相关充气分数;a和b分别为矩形的长和宽。将两种空气孔的相关充气分数进行对比,当a和b均等于d时,矩形空气孔具有更大的相关充气分数。
因此,矩形空气孔有更高的空气填充率,为了进一步降低包层的折射率,采用矩形空气孔进行排列;同时环形区域填充折射率为1.56的高折射率材料进一步提高了二者的折射率差。为使相邻本征模式间实现较大的有效折射率差以防止LP模式的合成,包层中第二层空气孔采用圆形空气孔以正六边形排列。具体的光子晶体光纤横截面如图1所示,包括中间大空气孔及环形传输区域,分别由18个大小相同的矩形空气孔以及36个圆形空气孔在包层中按照图1方式均匀排列一周。综合考虑性能和加工难度,经过反复计算确定r1为20 μm,r2为35 μm,b为4 μm,d为5 μm,d1为22.5 μm,d2为31.2 μm,d3为24 μm,d4为5.2 μm。
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利用有限元软件Comsol Multiphysics对设计的光子晶体光纤进行建模计算,为了提高计算精度、保证结果的可靠性,采用较细化网格剖分且在光纤外层添加一层完美匹配层作为边界条件。为了实现光纤结构的最佳性能,对包层中矩形空气孔的长宽比、环形传输区域的厚度进行调整以分析对结果的影响。确定最优结构后,对所提出光子晶体光纤支持传输的OAM模式数目进行了分析,为了保证OAM模式的高质量传输,对光纤的传输特性,主要包括有效折射率、相邻本征模式间的有效折射率差、有效模场面积、非线性系数、限制性损耗、色散分布以及模式纯度进行了分析。
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环形传输区域周围规律排列矩形空气孔,矩形空气孔的长宽比直接影响到包层空气填充率,进而影响包层与环形传输区域间的折射率差,对OAM模式的传输性能具有一定的影响。通过调整矩形空气孔的长度a来改变长宽比,分析了其大小对支持传输OAM数目及传输质量的影响,选择典型模式HE15,1在1 550 nm处的传输性能进行了分析。综合考虑传输性能以及制造难度,同时考虑到光纤结构的限制,为防止气孔间出现接触现象,设置a在7~7.8 μm范围进行分析。结果表明,在参数变化范围内,光纤在1 550 nm处均支持150种OAM模式的传输,说明a对OAM传输数目影响不大。图2为HE15,1在1 550 nm处的模式纯度及HE15,1和EH13,1间的有效折射率差随a的变化,结果表明,a越大,模式纯度越高、有效折射率差越大,传输性能越好。综上,综合考虑OAM模式传输数目、传输质量以及光纤制造难度,选择a为7.8 μm。
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光纤环形传输区域的厚度r也对光纤性能具有较大的影响,r等于r1与r3的差值,通过调整r3的大小实现传输区域厚度的变化来分析其对光纤的影响。图3为1 550 nm处r在1.5~2.5 μm范围内变化对传输OAM数目以及传输质量的影响。结果表明,随着r的增大,支持OAM数目呈现出先增加后减小的趋势,在1.9 μm、2.0 μm及2.1 μm处能够实现最大150种OAM模式的传输。通过对典型模式HE15,1在1 550 nm处的性能分析发现,随着r的增大,模式纯度呈增大趋势,原因是环形传输区域厚度的增加使得OAM模式所占功率在所有功率中的比例增大。但模间有效折射率差有所减小,原因是传输区域厚度的增大降低了光纤结构的不对称性。综合考虑两个参数及可支持的OAM数目,选择r为2 μm,即r3为18 μm。
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为了验证2.1.1及2.1.2节所选取的结构参数,考虑到a和r的交叉影响,设置a在7~7.8 μm范围,r在1.5~2.5 μm范围组成55种结构组合方式。分析了在这55种结构参数组合方式下在1 550 nm 处所支持的OAM传输数目及传输性能,具体的组合方式及结果如图4及表1所示。结果表明,无论r取值为多少,有效折射率差和模式纯度随a的变化趋势均为正相关,a=7.8 μm时性能均为最佳,验证了2.1.1节的结论。另外,无论a如何取值,有效折射率差及模式纯度随r的变化趋势分别为负相关和正相关,与2.1.2节的结论一致。首先考虑了所支持的OAM数目,然后综合考虑了有效折射率差和模式纯度。当a=7.8 μm、r=2 μm时,虽然有效折射率差和模式纯度不是最优,但却是在支持OAM数目最多条件下,唯一同时满足有效折射率差大于0.002、模式纯度大于0.96的参数组合方式。因此,所选取的a和r的取值能够兼顾OAM模式传输数目及传输质量。
图 4 结构组合及其对应有效折射率差和模式纯度。 (a)有效折射率差;(b)模式纯度
Figure 4. Structure combination and corresponding effective refractive index difference and mode purity. (a) Effective refractive index difference; (b) Mode purity
表 1 结构组合对应支持的传输OAM数目
Table 1. The supported number of transmitted OAM corresponding to the structure combination
Ring thickness/μm a=7.0 a=7.2 a=7.4 a=7.6 a=7.8 r=1.5 138 138 138 138 138 r=1.6 142 142 142 142 142 r=1.7 146 146 146 146 146 r=1.8 150 150 150 150 146 r=1.9 150 150 150 150 150 r=2.0 150 150 150 150 150 r=2.1 150 150 150 150 150 r=2.2 150 150 150 146 146 r=2.3 142 142 146 146 146 r=2.4 142 146 146 142 146 r=2.5 142 142 142 142 142 -
通过有限元法能够得到各个波长下本征模式对应的z方向上的电场分布,通过z方向上的电场分布能够清楚分辨出本征模式的类型、阶数及奇偶。图5为1 550 nm处一些典型本征模式对应的z方向电场分布图。图6为该光子晶体光纤支持的OAM数目随波长的变化以及不同波长对应的最高阶HE模式,图中的模式依次为HE42,1、HE41,1、HE40,1、HE39,1、HE38,1以及HE37,1,可以看出,支持传输的OAM模式数随着波长的增加而逐渐减少。在S+C+L+U波段,该光纤能够支持的OAM数目达到142。
图 5 z方向电场分布图。 (a) HE9,1; (b) HE20,1; (c) HE30,1; (d) HE37,1; (e) EH9,1; (f) EH20,1; (g) EH30,1; (h) EH35,1
Figure 5. Diagram of electric field distribution in z direction. (a) HE9,1; (b) HE20,1; (c) HE30,1; (d) HE37,1; (e) EH9,1; (f) EH20,1; (g) EH30,1; (h) EH35,1
较高的有效折射率差有利于保证轨道角动量模式的稳定传输,在外界微扰情况下,能够有效避免简并矢量模式间的耦合,相邻本征模式间有效折射率差大于10−4是光纤支持OAM模式稳定传输的基本要求[9]。有效折射率差Δneff能够通过公式(3)得到:
$$ \Delta {n_{{\text{eff}}}} = \left| {{n_{{\text{eff}}}}({\text{H}}{{\text{E}}_{l + 1,m}}) - {n_{{\text{eff}}}}({\text{E}}{{\text{H}}_{l - 1,m}})} \right| $$ (3) 式中:neff为有效折射率。
为实现相邻本征模式间有效折射率差的增大,文中设计的光纤包层中圆形空气孔呈现六边形分布。图7为HE模式和EH模式在1 450~1 650 nm范围内有效折射率随波长的变化,可以看出HE模式和EH模式对应的有效折射率均随波长的增加呈减小趋势,模式阶数越高,减小趋势越明显。
图 7 有效折射率随波长的变化。 (a) HE模式有效折射率随波长的变化;(b) EH模式有效折射率随波长的变化
Figure 7. Change of effective refractive index with wavelength. (a) Change of effective refractive index of HE mode with wavelength; (b) Change of effective refractive index of EH mode with wavelength
通过HE模式和EH模式的有效折射率能够计算出相邻本征模式间的有效折射率差,图8为相应模间有效折射率差随波长的变化情况,能够看出有效折射率差随波长的增加而增大。大部分相邻模式间的有效折射率差大于10−3,所有模间有效折射率均大于10−4,满足OAM模式传输的基本条件[18],能够有效防止简并矢量模式之间耦合成LP模式,保证了OAM模式的稳定传输。
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限制性损耗是光通信中的一个重要参数,直接影响到OAM模式的传输质量,是判断传输距离的重要参数。限制性损耗主要由光泄露引起,泄露部分的光能量即为光纤的限制性损耗,光纤对光束的约束能力越强,限制性损耗越小。设计出具有低限制性损耗的光纤对于OAM模式的长距离传输具有重要意义,具体的限制性损耗L计算方法如公式(4)所示[19]:
$$ L(\lambda ) = \frac{{40{\text{π }}}}{{\ln (10)\lambda }} \cdot {{\rm{Im}}} ({n_{{\rm{eff}}}}) $$ (4) 式中:λ为波长;Im(neff)为有效折射率的虚部。
图9(a)和图9(b)分别为1 450~1 650 nm范围内HE模式和EH模式对应限制性损耗随波长的变化,可以看出在波长范围内限制性损耗变化无规则,但所有模式限制性损耗均保持在10−12~10−9 dB/m量级上。其中,EH18,1在1 450 nm处能够达到最低限制性损耗为1.7×10−12 dB/m,HE26,1在1 500 nm处达到最高限制性损耗为9.8×10−9 dB/m。因此,该光纤所有本征模式限制性损耗均维持在较低水平,有利于OAM模式的长距离传输。
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有效模场面积能够反映光纤内功率密度的大小,直接影响到光纤的非线性系数,光纤中的非线性效应对光纤的容量和性能具有不利影响。较小的非线性系数有利于限制光纤中的非线性效应,有效模场面积Aeff和非线性系数γ的计算方法如公式(5)[20]和公式(6)所示[21]:
$$ {A_{{\text{eff}}}} = \frac{{{{\left( {\iint {{{\left| {E(x,y)} \right|}^2}{{\mathrm{d}}} x{{\mathrm{d}}} y}} \right)}^2}}}{{\iint {{{\left| {E(x,y)} \right|}^4}{{\mathrm{d}}} x{{\mathrm{d}}} y}}} $$ (5) $$ \gamma {\text{ = }}\frac{{2{\text{π }}{n_2}}}{{\lambda {A_{{\text{eff}}}}}} $$ (6) 式中:E(x, y)为光传播的横向电场分布;n2为背景材料的非线性折射率。
图10分别为HE模式和EH模式对应有效模场面积在1 450~1 650 nm范围内随波长的变化情况,能够看出随波长的增大,有效模场面积平稳增加,变化趋势不明显。有效模场面积随波长增大的原因可解释为随波长的增加,光纤对光强的限制能力逐渐减弱,导致部分光场泄露进包层中,引起模场面积的增加。其中,HE37,1在1 650 nm处的有效模场面积最大,能够达到206.18 μm2。
图 10 有效模场面积随波长的变化。 (a) HE模式有效模场面积随波长的变化;(b) EH模式有效模场面积随波长的变化
Figure 10. Change of effective mode field area with wavelength. (a) Change of effective mode field area of HE mode with wavelength; (b) Change of effective mode field area of EH mode with wavelength
图11为非线性系数随波长的变化情况,与有效模场面积随波长的变化趋势相反,非线性系数随波长的增加呈减小趋势,与公式(6)吻合。与有效模场面积对应,HE37,1在1 650 nm处获得最小非线性系数为0.397 W−1∙km−1,所有模式的非线性系数均小于1.426 W−1∙km−1。因此,所提出的光纤在抑制非线性效应方面表现较好,有助于OAM模式的长距离传输。
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色散是光通信中重要的参数,是造成脉冲展宽以及光纤模式传输不稳定的重要原因。在描述光纤的传输特性时,色散变化越平坦,光纤性能越好。光纤的色散系数D与有效折射率随波长的变化有关,具体的计算方式如公式(7)所示[22]:
$$ D(\lambda ) = - \frac{\lambda }{{{{{{c}}}}} } \cdot \frac{{{{d} ^2}{{\rm{Re}}} ({n_{{{\rm{eff}}} }})}}{{{\text{d}}{\lambda ^2}}} $$ (7) 式中:Re(neff)为有效折射率的实部;c为真空中的光速。
图12分别为HE模式和EH模式在1 450~1 650 nm范围内色散系数随波长的变化情况。可看出色散系数随波长的增加线性增大,基本趋势是,模式阶数越高,变化趋势越明显,这与有效折射率随波长的变化趋势相对应。其中,在1 450~1 650 nm范围内最小色散变化为1.457 8 ps/(nm∙km),因此该光纤具有较为平坦的色散系数变化。
图 12 色散随波长的变化。 (a) HE模式色散随波长的变化;(b) EH模式色散随波长的变化
Figure 12. Change of dispersion with wavelength. (a) Change of HE mode dispersion with wavelength; (b) Change of EH mode dispersion with wavelength
模式纯度表示OAM模式所占功率在所有功率中的比例,它是评估OAM模式传输性能的重要指标,影响着复用和解复用,模式纯度越高,表示轨道角动量模式的传输质量越好,且较高的模式纯度有利于防止模式间串扰。模式纯度η具体的计算方法如公式(8)所示[23]:
$$ \eta = \frac{{\iint_{{\text{ring}}} {{{\left| {E(x,y)} \right|}^2}{\text{d}}x{\text{d}}y}}}{{\iint_{{\text{cross - section}}} {{{\left| {E(x,y)} \right|}^2}{\text{d}}x{\text{d}}y}}} $$ (8) 图13分别为HE和EH模式对应模式纯度随波长的变化,能够看出所有模式的纯度均随波长的增加而减小。这与有效模场面积随波长的变化相对应,波长越大,光纤对光束的限制能力越弱,导致了模式纯度的下降。HE模式的纯度在94.2%~96.8%范围内,EH模式的纯度在93.4%~95.8%范围内,所有模式的纯度均大于93.4%。因此,所提出光纤所有本征模式均具有较高的模式纯度,这保证了OAM模式的高质量传输。
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表2为文中光子晶体光纤与典型OAM传输光子晶体光纤在包层空气孔形状、环形传输区域所用填充材料、传输OAM数目N、最小非线性系数γmin、最大限制性损耗Lmax、最小色散变化$\Delta {D_{\min }}$以及模式纯度方面的比较。结果表明,文中提出光纤在S+C+L+U波段能够支持142种OAM的传输,与其它典型光子晶体光纤相比处于较高水平。虽然该光纤的模式纯度不是处于最高水平,但光纤的最大限制性损耗处于最低水平,有利于OAM模式的长距离传输,最小非线性系数相较于其他光纤维持在较低水平、色散变化较平坦。因此,文中所提出光子晶体光纤不仅能够支持传输更多轨道角动量模式,并且具有更加优良的传输性能,在大容量光纤通信系统中具有一定的应用前景。
表 2 与典型光子晶体光纤性能比较
Table 2. Compared with typical photonic crystal fiber
Ref. Shape & Material N ${\gamma _{\min }}$/W−1∙km−1 Lmax/dB∙m−1 $\Delta {D_{\min }}$/ps∙nm−1∙km−1 Mode quality [14] Circle & Schott SF6 56 - 10−8 - >80% [28] Elliptical & Silica 80 0.55 10−8 23.1 - [29] Bezier & SiO2 38 1.04 10−6 4.75 >85% [30] Circle & GeO2-SiO2 48 0.93 10−3 - - [31] Circle & SiO2 30 2.178 10−7 3.59 >90.67% [32] Circle & Schott SF6 48 - 10−8 - >95% Proposed Rectangle & 1.56 high-index material 142 0.397 10−9 1.46 >93.4%
Design and transmission characteristics of high-order orbital angular momentum transmission fiber (inside back cover paper)
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摘要: 为解决一般轨道角动量传输光纤传输轨道角动量模式数量少、质量差的问题,提出了一种基于正六边形空气孔排列的新型光子晶体光纤结构。该光纤引入了空气填充率高的矩形空气孔以及采用高折射率材料填充环形传输区域,能够有效提高环形传输区域和包层间的折射率差,且正六边形排列空气孔有利于提高模间有效折射率差。经过结构优化得到最优光纤结构,有限元法的分析结果表明,最优结构下,该光纤在常用波段S+C+L+U波段上能够支持142种轨道角动量模式的传输,最高阶数达到36阶。且所提出光纤具有良好的传输特性,本征模式的最高限制性损耗为10−9 dB/m量级,与典型轨道角动量传输光子晶体光纤相比至少降低了一个数量级;最大有效模场面积能够达到206.18 μm2,最小非线性系数低至0.397 W−1∙km−1;色散平坦且最小色散变化低至1.457 8 ps/(nm∙km);所有本征模式纯度均在93.4%~96.8%范围内。且该光纤具有较好的制备可行性,对制造精度要求不高。因此,该光纤在基于轨道角动量光纤的复用系统中具有广阔的应用前景,为提高通信容量提供了一种有效手段。Abstract:
Objective Orbital angular momentum based multiplexing is a special form of space division multiplexing, different OAM modes are orthogonal to each other, based on which different modes can carry different information, and multiple OAM beams with different topological loads can be used as carriers for information transmission, which can greatly improve the channel capacity of the communication system without the need for additional bandwidth. Compared with the transmission of OAM in free space, the transmission of OAM modes in optical fibers can effectively avoid the interference of external factors, and ordinary optical fibers are unable to meet the requirements for the transmission of OAM modes. Photonic crystal optical fibers, as a kind of special optical fibers with high structural designability, offer the possibility of realizing the transmission of OAM modes. In order to achieve high quality transmission of more OAM modes, it is necessary to design photonic crystal optical fibers with suitable structures that can support the transmission of OAM modes. Methods In this paper, a novel photonic crystal fiber structure based on a positive hexagonal arrangement of air holes is proposed. The fiber introduces rectangular air holes with high air filling rate and high refractive index materials to fill the ring transmission region, which can effectively improve the refractive index difference between the ring transmission region and the cladding, and the hexagonal arrangement of the air holes is conducive to the improvement of the effective refractive index difference between the modes. Structure optimization and optimal structure verification take into account the number of OAMs that the fiber can support as well as the effective refractive index difference between modes and mode purity. The optimal fiber structure is obtained through structural optimization, and the performance of the fiber is analyzed using finite element analysis. Results and Discussions The results of the finite element method analyses show that the optimal optical fiber structure is optimized to support 142 OAM modes in the commonly used S+C+L+U band, with the topological charge ordering up to 36. Moreover, the proposed fiber has good transmission characteristics. The confinement loss is below 10-9 for all eigenmodes, which is at least one order of magnitude lower than the typical photonic crystal fibers; The maximum effective mode field area can reach 206.18 μm2 and the minimum nonlinearity coefficient is as low as 0.397 W−1∙km−1; Flat dispersion and minimum dispersion variation are as low as 1.457 8 ps/(nm∙km); And purity are 93.4%-96.8% for all eigenmodes. Based on the effect of manufacturing errors on the performance of the optical fiber, it can be seen that the optical fiber does not require high manufacturing accuracy. Conclusions In order to achieve high-quality transmission of more number of OAM modes, this paper proposes a design method of photonic crystal fiber based on hexagonal structure by combining the two ways of rectangular air holes and the filling of annular transmission region with high refractive index materials. The introduction of rectangular air holes and high refractive index materials increases the refractive index difference between the ring transmission region and the cladding, which in turn facilitates the stable transmission of a larger number of OAM modes. The performance analysis of the optical fiber by Comsol Multiphysics finite element analysis software shows that the photonic crystal fiber can not only support a larger number of OAM modes, but also has the characteristics of low confinement loss, low nonlinear coefficient, flat dispersion change, and high mode purity, which is valuable for high-capacity optical fiber communication. -
表 1 结构组合对应支持的传输OAM数目
Table 1. The supported number of transmitted OAM corresponding to the structure combination
Ring thickness/μm a=7.0 a=7.2 a=7.4 a=7.6 a=7.8 r=1.5 138 138 138 138 138 r=1.6 142 142 142 142 142 r=1.7 146 146 146 146 146 r=1.8 150 150 150 150 146 r=1.9 150 150 150 150 150 r=2.0 150 150 150 150 150 r=2.1 150 150 150 150 150 r=2.2 150 150 150 146 146 r=2.3 142 142 146 146 146 r=2.4 142 146 146 142 146 r=2.5 142 142 142 142 142 表 2 与典型光子晶体光纤性能比较
Table 2. Compared with typical photonic crystal fiber
Ref. Shape & Material N ${\gamma _{\min }}$/W−1∙km−1 Lmax/dB∙m−1 $\Delta {D_{\min }}$/ps∙nm−1∙km−1 Mode quality [14] Circle & Schott SF6 56 - 10−8 - >80% [28] Elliptical & Silica 80 0.55 10−8 23.1 - [29] Bezier & SiO2 38 1.04 10−6 4.75 >85% [30] Circle & GeO2-SiO2 48 0.93 10−3 - - [31] Circle & SiO2 30 2.178 10−7 3.59 >90.67% [32] Circle & Schott SF6 48 - 10−8 - >95% Proposed Rectangle & 1.56 high-index material 142 0.397 10−9 1.46 >93.4% -
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