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Polarization-maintaining low-loss porous-core spiral photonic crystal fiber for terahertz wave guidance
用于太赫兹波引导的保偏低损耗多孔芯螺旋光子晶体光纤

Md. Rabiul Hasan, 1 , 2 , ∗ 1 , 2 , ∗ ^(1,2,**){ }^{1,2, *} Md. Shamim Anower, 2 2 ^(2){ }^{2} Md. Ariful Islam, 3 3 ^(3){ }^{3} and S. M. A. Razzak 2 2 ^(2){ }^{2}
Md. Rabiul Hasan、 1 , 2 , ∗ 1 , 2 , ∗ ^(1,2,**){ }^{1,2, *} Md. Shamim Anower、 2 2 ^(2){ }^{2} Md. Ariful Islam 3 3 ^(3){ }^{3} 和 S. M. A. Razzak 2 2 ^(2){ }^{2} 1 1 ^(1){ }^{1} Department of Electronics & Telecommunication Engineering, Rajshahi University of Engineering & Technology, Rajshahi 6204, Bangladesh
1 1 ^(1){ }^{1} 拉杰沙希工程与技术大学电子与电信工程系，孟加拉国拉杰沙希6204 2 2 ^(2){ }^{2} Department of Electrical & Electronic Engineering, Rajshahi University of Engineering & Technology, Rajshahi 6204, Bangladesh
2 2 ^(2){ }^{2} 拉杰沙希工程与技术大学电气与电子工程系，拉杰沙希6204，孟加拉国 3 3 ^(3){ }^{3} Department of Physics, Rajshahi University of Engineering & Technology, Rajshahi 6204, Bangladesh
3 3 ^(3){ }^{3} 拉杰沙希工程与技术大学物理系，孟加拉国拉杰沙希6204*Corresponding author: ruetrabu@gmail.com
*通讯作者：ruetrabu@gmail.com

Received 29 February 2016; revised 25 April 2016; accepted 25 April 2016; posted 25 April 2016 (Doc. ID 260243); published 17 May 2016
2016 年 2 月 29 日接收;修订于 2016 年 4 月 25 日;2016 年 4 月 25 日接受;发布于 2016 年 4 月 25 日（文档 ID 260243）;发布时间 17 May 2016

Abstract 抽象

A polarization-maintaining porous-core spiral photonic crystal fiber is proposed for efficient transmission of polarization-maintaining terahertz (THz) waves. The finite element method with perfectly matched layer boundary conditions is used to characterize the guiding properties. We demonstrate that by creating artificial asymmetry in the porous core, an ultrahigh birefringence of 0.0483 can be obtained at the operating frequency of 1.0 THz. Moreover, a low effective material loss of 0.085 cm − 1 0.085 cm − 1 0.085cm^(-1)0.085 \mathrm{~cm}^{-1} and very small confinement loss of 1.91 × 1.91 × 1.91 xx1.91 \times 10 − 3 dB / cm 10 − 3 dB / cm 10^(-3)dB//cm10^{-3} \mathrm{~dB} / \mathrm{cm} are achieved for the y y yy-polarization mode with optimal design parameters. This article also focuses on some crucial design parameters such as power fraction, bending loss, and dispersion for usability in the THz regime. © 2016 Optical Society of America
该文提出一种保偏多孔芯螺旋光子晶体光纤，用于高效传输保偏太赫兹 （THz） 波。具有完美匹配层边界条件的有限元方法用于表征导向特性。我们证明，通过在多孔核心中产生人工不对称性，可以在 1.0 THz 的工作频率下获得 0.0483 的超高双折射。此外，具有最优设计参数的极化模式实现了低有效材料损耗 0.085 cm − 1 0.085 cm − 1 0.085cm^(-1)0.085 \mathrm{~cm}^{-1} 和非常小的约束损耗 1.91 × 1.91 × 1.91 xx1.91 \times 10 − 3 dB / cm 10 − 3 dB / cm 10^(-3)dB//cm10^{-3} \mathrm{~dB} / \mathrm{cm} 。 y y yy 本文还重点介绍了一些关键的设计参数，例如功率分数、弯曲损耗和色散，以便在太赫兹范围内的可用性。© 2016 美国光学学会

OCIS codes: ( 260.1440 ) ( 260.1440 ) (260.1440)(260.1440) Birefringence; ( 060.5295 ) ( 060.5295 ) (060.5295)(060.5295) Photonic crystal fibers.
OCIS 代码： ( 260.1440 ) ( 260.1440 ) (260.1440)(260.1440) 双折射; ( 060.5295 ) ( 060.5295 ) (060.5295)(060.5295) 光子晶体光纤。
http://dx.doi.org/10.1364/AO.55.004145

1. INTRODUCTION 1. 引言

The terahertz (THz) radiation spectrum, which lies between optical and microwave frequency bands, is extensively used in various applications ranging from security-sensitive areas to medical imaging, sensing, and spectroscopy [1-4]. Significant progress has been achieved over the last decade in designing efficient THz sources and detectors. Conventional THz systems based on free-space line-of-sight communications that suffer from high absorption loss due to water vapor. Moreover, misalignment between the transmitter and the receiver creates improper reception of THz waves. To eliminate these shortcomings, various forms of waveguide structures such as hollow metallic waveguides [5,6], parallel-plate waveguides [7,8], bare metal wires [ 9 , 10 ] [ 9 , 10 ] [9,10][9,10], and metallic slot waveguides [11] have been proposed. However, metallic waveguides suffer from high ohmic losses and strong dispersion near the cut-off frequency of the mode. Parallel-plate waveguides have no dispersion issues; however, these types of waveguides suffer from ohmic and divergence losses. On the other hand, the limitation of bare metal wires is high radiation losses. The shortcoming of metallic slot waveguides is higher attenuation losses compared to metal wires. Various forms of polymer fibers have been reported such as Bragg fiber [12], plastic fiber [13], subwavelength porous fiber [14], and hollow core fiber [15] for efficient transmission of THz waves. Among them, porous photonic
太赫兹 （THz） 辐射光谱位于光频段和微波频段之间，广泛用于从安全敏感领域到医学成像、传感和光谱学的各种应用 [1-4]。在过去十年中，在设计高效的太赫兹源和探测器方面取得了重大进展。基于自由空间视距通信的传统太赫兹系统，由于水蒸气而遭受高吸收损失。此外，发射器和接收器之间的错位会导致对太赫兹波的接收不当。为了消除这些缺点，人们提出了各种形式的波导结构，如空心金属波导[5,6]、平行板波导[7,8]、裸金属线 [ 9 , 10 ] [ 9 , 10 ] [9,10][9,10] 和金属缝隙波导[11]。然而，金属波导在模式的截止频率附近存在高欧姆损耗和强色散。平行板波导没有色散问题;然而，这些类型的波导会受到欧姆和发散损耗的影响。另一方面，裸金属丝的局限性是高辐射损耗。金属缝隙波导的缺点是与金属线相比，衰减损耗更高。已经报道了各种形式的聚合物纤维，如布拉格纤维 [12]、塑料纤维 [13]、亚波长多孔纤维 [14] 和空心纤维 [15]，用于高效传输太赫兹波。其中，多孔光子
crystal fibers have attracted increased attention due to their low material absorption loss, low confinement loss, very small dispersion, and flexibility in fabrication [16,17].
晶体纤维因其低材料吸收损耗、低约束损耗、非常小的色散和制造灵活性而受到越来越多的关注[16,17]。
It is known that dry air is the most transparent medium for THz propagation since it does not absorb THz waves [14]. Under this supposition, an array of smaller air holes also known as subwavelength porous air holes is placed in the solid material core of a photonic crystal fiber (PCF). Thus, by transmitting most of the mode power through the porous air core, it is possible to propagate THz waves with minimum absorption losses.
众所周知，干燥空气是太赫兹传播最透明的介质，因为它不吸收太赫兹波[14]。在这个假设下，在光子晶体光纤 （PCF） 的固体材料芯中放置了一组较小的气孔，也称为亚波长多孔气孔。因此，通过多孔空气核心传输大部分模式功率，可以以最小的吸收损耗传播太赫兹波。
One of the most important properties showed by porouscore PCFs is the controllability of birefringence. Birefringence is induced in polarization-maintaining PCFs by deliberately breaking the symmetry of either core or cladding [18,19]. Highly birefringent THz PCFs have significant applications in optical sensing [20], coherent heterodyne time-domain spectrometry [21], and measurements of biomaterials in THz frequency bands [22]. Researchers have carried out distinct works to investigate highly birefringent THz fibers. A rectangular PCF with square lattice subwavelength air holes has been proposed that exhibits a very low birefringence of 1.26 × 10 − 3 1.26 × 10 − 3 1.26 xx10^(-3)1.26 \times 10^{-3} [23]. Asymmetrical subwavelength air-hole PCF has been proposed by Atakaramians et al. that shows a low birefringence of 1.2 × 10 − 2 1.2 × 10 − 2 1.2 xx10^(-2)1.2 \times 10^{-2} at 0.65 THz [24]. Moreover, this PCF suffers from a high effective material loss of > 0.25 cm − 1 > 0.25 cm − 1 > 0.25cm^(-1)>0.25 \mathrm{~cm}^{-1} at frequencies
多孔核 PCF 显示的最重要特性之一是双折射的可控性。在保持极化的 PCF 中，通过故意打破纤芯或包层的对称性来诱导双折射 [18\u201219]。高双折射太赫兹 PCF 在光学传感 [20]、相干外差时域光谱法 [21] 和太赫兹频段生物材料测量中具有重要应用[22]。研究人员已经开展了不同的工作来研究高度双折射的太赫兹纤维。已经提出了一种具有方形晶格亚波长气孔的矩形 PCF，其双折射非常低 1.26 × 10 − 3 1.26 × 10 − 3 1.26 xx10^(-3)1.26 \times 10^{-3} [23]。Atakaramians 等人提出了不对称的亚波长气孔 PCF，在 0.65 THz 1.2 × 10 − 2 1.2 × 10 − 2 1.2 xx10^(-2)1.2 \times 10^{-2} 处显示出低双折射 [24]。此外，这种 PCF 在 > 0.25 cm − 1 > 0.25 cm − 1 > 0.25cm^(-1)>0.25 \mathrm{~cm}^{-1} AT 频率下具有很高的有效材料损失
above 0.8 THz . A hexagonal lattice PCF with asymmetry in both core and cladding has been proposed in [25] that offers a high birefringence of 4.5 × 10 − 2 4.5 × 10 − 2 4.5 xx10^(-2)4.5 \times 10^{-2} at a frequency of 1 THz . However, the authors have not reported any information about confinement loss, which is crucial for practical implementation. Besides, a high birefringence of 4.45 × 10 − 2 4.45 × 10 − 2 4.45 xx10^(-2)4.45 \times 10^{-2} has been reported in [26] by using porous fiber with elliptical air holes. However, fabrication of such type of polymer fiber is too difficult because of its structural complexity. Most recently, an ultrahigh birefringence THz PCF has been proposed in [27], where the highest birefringence of 7.5 × 10 − 2 7.5 × 10 − 2 7.5 xx10^(-2)7.5 \times 10^{-2} has been achieved to date. This structure contains slotted air holes, which are almost impractical to fabricate.
高于 0.8 THz 。[25] 中提出了一种在磁芯和包层中均不对称的六方晶格 PCF，它在 1 THz 的频率 4.5 × 10 − 2 4.5 × 10 − 2 4.5 xx10^(-2)4.5 \times 10^{-2} 下具有很高的双折射率。然而，作者尚未报告任何有关月子损失的信息，这对于实际实施至关重要。此外，在 [26] 中通过使用带有椭圆形气孔的多孔纤维报道了高双折射 4.45 × 10 − 2 4.45 × 10 − 2 4.45 xx10^(-2)4.45 \times 10^{-2} 。然而，由于其结构复杂性，制造这种类型的聚合物纤维太困难了。最近，在 [27] 中提出了一种超高双折射 THz PCF，迄今为止实现了最高的双折射 7.5 × 10 − 2 7.5 × 10 − 2 7.5 xx10^(-2)7.5 \times 10^{-2} 。这种结构包含开槽的气孔，制造起来几乎不切实际。
In this paper, we analyze a porous-core spiral PCF in order to achieve ultrahigh birefringence by intentionally creating asymmetry in the core. Other important modal properties such as effective material loss, bending loss, power fraction, dispersion, and confinement loss are thoroughly discussed with the variation of different structural parameters. The structure and design of the proposed porous-core spiral PCF are presented in Section 2. The simulation results and discussion are given in Section 3. Finally, conclusions are summarized in Section 4.
在本文中，我们分析了多孔芯螺旋 PCF，以便通过故意在芯中产生不对称性来实现超高双折射。通过不同结构参数的变化，对其他重要的模态属性，如有效材料损耗、弯曲损耗、功率分数、色散和约束损耗进行了深入讨论。第 2 节介绍了所提出的多孔芯螺旋 PCF 的结构和设计。模拟结果和讨论在第 3 节中给出。最后，第 4 节总结了结论。

2. STRUCTURE AND DESIGN OF THE PROPOSED POROUS-CORE SPIRAL PCF
2. 所提出的多孔芯螺旋 PCF 的结构与设计

The cross section of the proposed porous-core spiral PCF is shown in Fig. 1 along with the enlarged view of the core. Spiral topology has been chosen since it shows ultralow bending loss and excellent modal confinement properties compared to conventional hexagonal PCF [28]. Spiral symmetry consists of nine circular rings with ten spiral arms, where each arm consists of nine air holes. The first air hole in each spiral ring is placed at a distance of r 0 r 0 r_(0)r_{0}. The distance of the second air hole of each ring from the center is r 1 = r 0 + ( 0.48 × Λ ) r 1 = r 0 + ( 0.48 × Λ ) r_(1)=r_(0)+(0.48 xx Lambda)r_{1}=r_{0}+(0.48 \times \Lambda), where Λ Λ Lambda\Lambda is the hole-to-hole distance between two adjacent rings. In our proposed structure, we have chosen r 0 = Λ r 0 = Λ r_(0)=Lambdar_{0}=\Lambda for the simplicity of the design. Throughout the analysis, the value of Λ Λ Lambda\Lambda is selected as 0.53 D core 0.53 D core  0.53D_("core ")0.53 D_{\text {core }}, where D core D core  D_("core ")D_{\text {core }} is the core diameter. The diameter of first air-hole ring is d 1 = 0.60 × Λ d 1 = 0.60 × Λ d_(1)=0.60 xx Lambdad_{1}=0.60 \times \Lambda, and the diameters of the second and third air-hole rings are identical at d = 0.90 × Λ d = 0.90 × Λ d=0.90 xx Lambdad=0.90 \times \Lambda. The diameter of the circular air holes in the cladding of the proposed structure is selected as large as possible to ensure better light confinement. The size of air holes should not be enlarged because the extension might result in overlapping among air holes. The number of air-hole rings is selected as maximum (here N r = 9 N r = 9 N_(r)=9N_{r}=9 ) as possible to obtain low confinement loss since increasing the number of rings in the outer cladding results in tight light confinement in the core. The distance of n n nnth air hole from the center is r n = r n − 1 + ( 0.48 × Λ ) r n = r n − 1 + ( 0.48 × Λ ) r_(n)=r_(n-1)+(0.48 xx Lambda)r_{n}=r_{n-1}+(0.48 \times \Lambda) with an angular displacement of θ n = ( n × 360 ∘ ) / ( 2 × N ) θ n = n × 360 ∘ / ( 2 × N ) theta_(n)=(n xx360^(@))//(2xx N)\theta_{n}=\left(n \times 360^{\circ}\right) /(2 \times N), where N N NN is the number of rings. For example, the angular displacement of the first ring is θ 1 = ( 360 ∘ ) / ( 2 × N ) θ 1 = 360 ∘ / ( 2 × N ) theta_(1)=(360^(@))//(2xx N)\theta_{1}=\left(360^{\circ}\right) /(2 \times N). In the core region, circular air holes are arranged vertically, which is shown in the enlarged view of Fig. 1. The hole-to-hole distance between two adjacent vertical axes is related to D core D core  D_("core ")D_{\text {core }} and is given by Λ c = 0.175 D core Λ c = 0.175 D core  Lambda_(c)=0.175D_("core ")\Lambda_{c}=0.175 D_{\text {core }}. Throughout the simulation, the core airfilling ratio ( d c / Λ c ) d c / Λ c (d_(c)//Lambda_(c))\left(d_{c} / \Lambda_{c}\right) is kept as large as possible, which is
图 1 显示了所提出的多孔芯螺旋 PCF 的横截面以及芯的放大视图。之所以选择螺旋拓扑结构，是因为与传统的六边形 PCF 相比，它表现出超低的弯曲损耗和出色的模态约束性能 [28]。螺旋对称由九个圆环和十个螺旋臂组成，其中每个臂由九个气孔组成。每个螺旋环中的第一个气孔放置在 的距离处。 r 0 r 0 r_(0)r_{0} 每个环的第二个气孔距中心的距离为 r 1 = r 0 + ( 0.48 × Λ ) r 1 = r 0 + ( 0.48 × Λ ) r_(1)=r_(0)+(0.48 xx Lambda)r_{1}=r_{0}+(0.48 \times \Lambda) ，其中 Λ Λ Lambda\Lambda 是两个相邻环之间的孔到孔的距离。在我们提议的结构中，我们选择了 r 0 = Λ r 0 = Λ r_(0)=Lambdar_{0}=\Lambda 简单的设计。在整个分析过程中， 的值 Λ Λ Lambda\Lambda 被选为 0.53 D core 0.53 D core  0.53D_("core ")0.53 D_{\text {core }} ，其中 D core D core  D_("core ")D_{\text {core }} 是型芯直径。第一个气孔环的直径为 d 1 = 0.60 × Λ d 1 = 0.60 × Λ d_(1)=0.60 xx Lambdad_{1}=0.60 \times \Lambda ，第二个和第三个气孔环的直径在 d = 0.90 × Λ d = 0.90 × Λ d=0.90 xx Lambdad=0.90 \times \Lambda 处相同。建议结构的覆层中的圆形气孔直径选择得尽可能大，以确保更好的光线限制。不应放大气孔的大小，因为扩展可能会导致气孔之间重叠。气孔环的数量选择为最大值（此处 N r = 9 N r = 9 N_(r)=9N_{r}=9 ），以获得低约束损耗，因为增加外覆层中的环数量会导致芯中的紧密光限制。 n n nn 气孔距中心的距离的 r n = r n − 1 + ( 0.48 × Λ ) r n = r n − 1 + ( 0.48 × Λ ) r_(n)=r_(n-1)+(0.48 xx Lambda)r_{n}=r_{n-1}+(0.48 \times \Lambda) 角位移为 θ n = ( n × 360 ∘ ) / ( 2 × N ) θ n = n × 360 ∘ / ( 2 × N ) theta_(n)=(n xx360^(@))//(2xx N)\theta_{n}=\left(n \times 360^{\circ}\right) /(2 \times N) ，其中 N N NN 是环的数量。例如，第一个环的角位移为 θ 1 = ( 360 ∘ ) / ( 2 × N ) θ 1 = 360 ∘ / ( 2 × N ) theta_(1)=(360^(@))//(2xx N)\theta_{1}=\left(360^{\circ}\right) /(2 \times N) 。在核心区域，圆形气孔垂直排列，如图 1 的放大图所示。 两个相邻垂直轴之间的孔到孔距离与 D core D core  D_("core ")D_{\text {core }} 相关，并由 Λ c = 0.175 D core Λ c = 0.175 D core  Lambda_(c)=0.175D_("core ")\Lambda_{c}=0.175 D_{\text {core }} 给出。在整个仿真过程中，核心充气率 ( d c / Λ c ) d c / Λ c (d_(c)//Lambda_(c))\left(d_{c} / \Lambda_{c}\right) 保持尽可能大，即
Fig. 1. Cross section of the proposed porous-core spiral THz fiber with TOPAS as the background material. An enlarged view of the porous core is shown in the inset.
图 1.以 TOPAS 为背景材料的所提出的多孔芯螺旋太赫兹纤维的横截面。插图显示了多孔芯的放大视图。
0.24. Selecting the core air-filling ratio this way offers maximum birefringence. Moreover, the value of d c / Λ c d c / Λ c d_(c)//Lambda_(c)d_{c} / \Lambda_{c} should not be increased further since this will result in overlapping among air holes. In such a situation, unrealistic fabrication will arise. Note that there are more air holes in the vertical axis direction compared to the horizontal axis direction, which creates asymmetry in the core. The formation of such asymmetry is responsible for inducing a high level of birefringence. The proposed PCF uses TOPAS as its background material. The refractive index of TOPAS is 1.5258 and it remains constant over a 0.1 − 2 THz 0.1 − 2 THz 0.1-2THz0.1-2 \mathrm{THz} frequency range [12]. Moreover, it shows low loss and low dispersion at THz bands [29]. In addition, it is insensitive to environmental aspects such as humidity and water vapor absorption [30].
0.24. 以这种方式选择核心充气比可提供最大的双折射。此外，不应进一步增加 的值 d c / Λ c d c / Λ c d_(c)//Lambda_(c)d_{c} / \Lambda_{c} ，因为这将导致气孔之间重叠。在这种情况下，将出现不切实际的制造。请注意，与水平轴方向相比，垂直轴方向上的气孔更多，这在核心中产生了不对称性。这种不对称性的形成导致了高水平的双折射。拟议的 PCF 使用 TOPAS 作为其背景材料。TOPAS 的折射率为 1.5258，在一定 0.1 − 2 THz 0.1 − 2 THz 0.1-2THz0.1-2 \mathrm{THz} 频率范围内保持不变 [12]。此外，它在太赫兹波段表现出低损耗和低色散 [29]。此外，它对湿度和水蒸气吸收等环境因素不敏感 [30]。

3. SIMULATION RESULTS AND DISCUSSION
3. 仿真结果和讨论

The finite-element-method (FEM)-based commercially available COMSOL v.5.0 software has been used to design and simulate the proposed porous-core spiral PCF. A circular perfectly
基于有限元法 （FEM） 的商用 COMSOL v.5.0 软件已被用于设计和模拟所提出的多孔芯螺旋 PCF。完美的圆形
matched layer boundary condition has been imposed to calculate the leakage loss. The complete mesh contains 93,722 triangular elements, 7469 edge elements, and 804 vertex elements. The average element quality is 0.9577 , and the element area ratio is 4.38 × 10 − 5 4.38 × 10 − 5 4.38 xx10^(-5)4.38 \times 10^{-5}. The minimum length of the triangular mesh is 0.783 μ m 0.783 μ m 0.783 mum0.783 \mu \mathrm{~m}. Finer element size has been used so that the triangular mesh can accurately map the different air holes. The material abortion loss of 0.20 cm − 1 0.20 cm − 1 0.20cm^(-1)0.20 \mathrm{~cm}^{-1} has been included in the simulation. This value can be compared with the experimental result in Ref. [12], where material abortion loss of less than 1 dB / cm 1 dB / cm 1dB//cm1 \mathrm{~dB} / \mathrm{cm} or less than 0.23 cm − 1 0.23 cm − 1 0.23cm^(-1)0.23 \mathrm{~cm}^{-1} was reported. The mode field profile of the proposed PCF for different operating frequencies is shown in Fig. 2. It is evident that mode fields are tightly bounded in the core region.
已施加匹配的层边界条件来计算泄漏损耗。完整的网格包含 93,722 个三角形单元、7469 个边缘单元和 804 个顶点单元。平均单元质量为 0.9577 ，单元面积比为 4.38 × 10 − 5 4.38 × 10 − 5 4.38 xx10^(-5)4.38 \times 10^{-5} 。三角形网格的最小长度为 0.783 μ m 0.783 μ m 0.783 mum0.783 \mu \mathrm{~m} 。使用了更精细的单元尺寸，以便三角形网格可以精确地映射不同的气孔。的材料中止损失 0.20 cm − 1 0.20 cm − 1 0.20cm^(-1)0.20 \mathrm{~cm}^{-1} 已包含在模拟中。该值可以与参考文献 [12] 中的实验结果进行比较，其中物质流产损失小于 1 dB / cm 1 dB / cm 1dB//cm1 \mathrm{~dB} / \mathrm{cm} 或小于 0.23 cm − 1 0.23 cm − 1 0.23cm^(-1)0.23 \mathrm{~cm}^{-1} 报道。图 2 显示了所提出的 PCF 在不同工作频率下的模场分布。很明显， mode fields 在 core 区域中是紧密绑定的。
To operate as an effective polarization-maintaining THz PCF, the level of birefringence should be as high as possible. With this in mind, we first investigate the birefringence property of the proposed PCF. The birefringence has been calculated using the following formula [27]:
为了作为有效的保偏太赫兹 PCF 运行，双折射水平应尽可能高。考虑到这一点，我们首先研究了所提出的 PCF 的双折射特性。双折射使用以下公式计算 [27]：
B = | n x − n y | B = n x − n y B=|n_(x)-n_(y)|B=\left|n_{x}-n_{y}\right|
where B B BB stands for birefringence, n x n x n_(x)n_{x} and n y n y n_(y)n_{y} are the effective refractive indices of the x x xx-polarization and y y yy-polarization modes, respectively.
其中 B B BB 代表双折射， n x n x n_(x)n_{x} n y n y n_(y)n_{y} 分别是 x x xx -polarization 和 y y yy -polarization 模式的有效折射率。
Fig. 2. Mode field distributions of the proposed PCF for D core = D core  = D_("core ")=D_{\text {core }}= 410 μ m 410 μ m 410 mum410 \mu \mathrm{~m} and different operating frequencies of (a) 0.8 THz x x xx-polarization, (b) 0.8 THz y 0.8 THz y 0.8THzy0.8 \mathrm{THz} y-polarization, © 1 THz x 1 THz x 1THzx1 \mathrm{THz} x-polarization, (d) 1 THz y 1 THz y 1THzy1 \mathrm{THz} y-polarization, (e) 1.2 THz x 1.2 THz x 1.2THzx1.2 \mathrm{THz} x-polarization, and (f) 1.2 THz y y yy-polarization.
图 2.所提出的 PCF D core = D core  = D_("core ")=D_{\text {core }}= 的模场分布 410 μ m 410 μ m 410 mum410 \mu \mathrm{~m} 以及 （a） 0.8 THz x x xx 极化、（b） 0.8 THz y 0.8 THz y 0.8THzy0.8 \mathrm{THz} y 极化、© 1 THz x 1 THz x 1THzx1 \mathrm{THz} x 极化、（d） 1 THz y 1 THz y 1THzy1 \mathrm{THz} y 极化、（e） 1.2 THz x 1.2 THz x 1.2THzx1.2 \mathrm{THz} x 极化和 （f） 1.2 THz y y yy 极化的不同工作频率。
Birefringence as a function of the core air-filling ratio ( d c / Λ c ) d c / Λ c (d_(c)//Lambda_(c))\left(d_{c} / \Lambda_{c}\right) for different D core D core  D_("core ")D_{\text {core }} is shown in Fig. 3. As seen in the figure, birefringence is an increasing function with increasing d c / Λ c d c / Λ c d_(c)//Lambda_(c)d_{c} / \Lambda_{c} for the same value of D core D core  D_("core ")D_{\text {core }}. The reason can be understood due to the fact that when d c / Λ c d c / Λ c d_(c)//Lambda_(c)d_{c} / \Lambda_{c} is increased, the diameter of the porous air holes also increases. Therefore, induced asymmetry becomes stronger and as a result, birefringence increases. Obviously, this structure exhibits nearly zero birefringence without porous air holes. It is important to note that the maximum value of d c / Λ c d c / Λ c d_(c)//Lambda_(c)d_{c} / \Lambda_{c} is set to be equal to or less than 0.24 because further extending the value of d c / Λ c d c / Λ c d_(c)//Lambda_(c)d_{c} / \Lambda_{c} results in overlap among air holes along the vertical axis. Figure 3 also shows birefringence for different D core D core  D_("core ")D_{\text {core }}, where it is evident that birefringence increases with increasing D core D core  D_("core ")D_{\text {core }}. When D core D core  D_("core ")D_{\text {core }} is increased, the air holes become larger. Therefore, asymmetry between the x x xx-polarization mode and the y y yy-polarization mode enhances and consequently, birefringence increases. An ultrahigh birefringence of 0.0483 has been achieved for f = 1 THz , D core = 410 μ m f = 1 THz , D core  = 410 μ m f=1THz,D_("core ")=410 mumf=1 \mathrm{THz}, D_{\text {core }}=410 \mu \mathrm{~m}, and d c / Λ c = 0.24 d c / Λ c = 0.24 d_(c)//Lambda_(c)=0.24d_{c} / \Lambda_{c}=0.24.
双折射作为不同 D core D core  D_("core ")D_{\text {core }} 纤芯充气比 ( d c / Λ c ) d c / Λ c (d_(c)//Lambda_(c))\left(d_{c} / \Lambda_{c}\right) 的函数如图 3 所示。如图所示，双折射是一个递增函数，当 的值相同时，会随着 d c / Λ c d c / Λ c d_(c)//Lambda_(c)d_{c} / \Lambda_{c} D core D core  D_("core ")D_{\text {core }} 的增加而增加。原因可以理解为，当增加时 d c / Λ c d c / Λ c d_(c)//Lambda_(c)d_{c} / \Lambda_{c} ，多孔气孔的直径也会增加。因此，诱导的不对称性变得更强，因此，双折射增加。显然，这种结构的双折射几乎为零，没有多孔气孔。请务必注意，最大值 of d c / Λ c d c / Λ c d_(c)//Lambda_(c)d_{c} / \Lambda_{c} 设置为等于或小于 0.24，因为进一步扩展 的值 d c / Λ c d c / Λ c d_(c)//Lambda_(c)d_{c} / \Lambda_{c} 会导致气孔沿垂直轴重叠。图 3 还显示了不同 D core D core  D_("core ")D_{\text {core }} 的双折射，其中很明显，双折射随着 的增加 D core D core  D_("core ")D_{\text {core }} 而增加。当 增加时 D core D core  D_("core ")D_{\text {core }} ，气孔会变大。因此， x x xx 偏振模式和 y y yy 偏振模式之间的不对称性增强，因此双折射增加。已经实现了 f = 1 THz , D core = 410 μ m f = 1 THz , D core  = 410 μ m f=1THz,D_("core ")=410 mumf=1 \mathrm{THz}, D_{\text {core }}=410 \mu \mathrm{~m} 、 和 d c / Λ c = 0.24 d c / Λ c = 0.24 d_(c)//Lambda_(c)=0.24d_{c} / \Lambda_{c}=0.24 的 0.0483 的超高双折射。
Birefringence as a function of frequency for different D core D core  D_("core ")D_{\text {core }} is depicted in Fig. 4. As seen in the figure, birefringence is above 0.05 for frequencies ranging from 1.1 to 2 THz with D core = 410 μ m D core  = 410 μ m D_("core ")=410 mumD_{\text {core }}=410 \mu \mathrm{~m}. The most interesting characteristic shown by this structure is the nearly constant birefringence over a wide band of frequencies. The highest birefringence of ∼ 0.577 ∼ 0.577 ∼0.577\sim 0.577 can be obtained for D core = 460 μ m D core  = 460 μ m D_("core ")=460 mumD_{\text {core }}=460 \mu \mathrm{~m} at f = 2 THz f = 2 THz f=2THzf=2 \mathrm{THz}, where birefringence is constant over f = 1.2 − 2 THz f = 1.2 − 2 THz f=1.2-2THzf=1.2-2 \mathrm{THz}. To the best of our knowledge, this is the highest birefringence value reported to date that shows constant ultrahigh birefringence over such wide bands [23-27].
双折射作为不同 D core D core  D_("core ")D_{\text {core }} 频率的函数如图 4 所示。如图所示，对于 1.1 至 2 THz 的频率范围，双折射高于 0.05，其中 D core = 410 μ m D core  = 410 μ m D_("core ")=410 mumD_{\text {core }}=410 \mu \mathrm{~m} 。这种结构显示的最有趣的特征是在宽频带上几乎恒定的双折射。在 D core = 460 μ m D core  = 460 μ m D_("core ")=460 mumD_{\text {core }}=460 \mu \mathrm{~m} 处 f = 2 THz f = 2 THz f=2THzf=2 \mathrm{THz} ∼ 0.577 ∼ 0.577 ∼0.577\sim 0.577 可以获得最高的双折射，其中双折射在 上是恒定的 f = 1.2 − 2 THz f = 1.2 − 2 THz f=1.2-2THzf=1.2-2 \mathrm{THz} 。据我们所知，这是迄今为止报道的最高双折射值，在如此宽的条带上显示出恒定的超高双折射[23-27]。
Effective material loss (EML) is an important parameter in designing PCFs used for THz guidance and is quantified by the following expression [14]:
有效材料损耗 （EML） 是设计用于太赫兹制导的 PCF 的一个重要参数，可通过以下表达式 [14] 进行量化：
α eff = ( ε 0 / μ 0 ) 1 / 2 ∫ A mar n α mat | E | 2 d A 2 ∫ All S z d A cm − 1 α eff = ε 0 / μ 0 1 / 2 ∫ A mar    n α mat | E | 2 d A 2 ∫ All   S z d A cm − 1 alpha_(eff)=((epsi_(0)//mu_(0))^(1//2)int_(A_("mar "))nalpha_(mat)|E|^(2)(d)A)/(2int_(All)S_(z)(d)A)cm^(-1)\alpha_{\mathrm{eff}}=\frac{\left(\varepsilon_{0} / \mu_{0}\right)^{1 / 2} \int_{A_{\text {mar }}} n \alpha_{\mathrm{mat}}|E|^{2} \mathrm{~d} A}{2 \int_{\mathrm{All}} S_{z} \mathrm{~d} A} \mathrm{~cm}^{-1}
where ε 0 ε 0 epsi_(0)\varepsilon_{0} and μ 0 μ 0 mu_(0)\mu_{0} are the permittivity and permeability of vacuum, respectively, n n nn is the refractive index of TOPAS, α mat α mat  alpha_("mat ")\alpha_{\text {mat }} is the bulk material absorption loss, E E EE is the electric field component, and S z S z S_(z)S_{z} is the z z zz-component of the Poynting vector. Integration of the numerator of Eq. (2) is performed for the
其中 ε 0 ε 0 epsi_(0)\varepsilon_{0} 和 μ 0 μ 0 mu_(0)\mu_{0} 是真空的介电常数和磁导率， n n nn 分别是 TOPAS 的折射率， α mat α mat  alpha_("mat ")\alpha_{\text {mat }} 是块状材料的吸收损失， E E EE 是电场分量， S z S z S_(z)S_{z} 是坡印廷矢量的 z z zz -分量。对方程 （2） 的分子进行积分，以便
Fig. 3. Birefringence of the proposed PCF as a function of core air-filling ratio ( d c / Λ c ) d c / Λ c (d_(c)//Lambda_(c))\left(d_{c} / \Lambda_{c}\right) for different D core D core  D_("core ")D_{\text {core }} at f = 1 THz f = 1 THz f=1THzf=1 \mathrm{THz}.
图 3.对于不同的 D core D core  D_("core ")D_{\text {core }} at f = 1 THz f = 1 THz f=1THzf=1 \mathrm{THz} ，所提出的 PCF 的双折射作为核心充气比 ( d c / Λ c ) d c / Λ c (d_(c)//Lambda_(c))\left(d_{c} / \Lambda_{c}\right) 的函数。