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Photonic-Crystal Fibers  光子晶体光纤

Philip St.J. Russell, Member, IEEE
菲利普·圣·J·拉塞尔,IEEE 会员

Invited Paper 邀请论文

Abstract 摘要

The history, fabrication, theory, numerical modeling, optical properties, guidance mechanisms, and applications of photonic-crystal fibers are reviewed.
对光子晶体光纤的历史、制造、理论、数值建模、光学特性、引导机制和应用进行了回顾。

Index Terms-Microstructured optical fibers, nonlinear optics, periodic structures, photonic band gaps (PBGs), photonic-crystal fibers (PCFs), waveguides.
索引词条-微结构光纤、非线性光学、周期结构、光子带隙(PBGs)、光子晶体光纤(PCFs)、波导。

LIST OF ACRONYMS 首字母缩略词列表

AS Anti-Stokes.
ESM Endlessly single mode. 无尽的单模式。
FDTD Finite-difference time domain.
有限差分时域。
GVD Group velocity dispersion.
群速色散。
IR Infrared.
MCVD Modified chemical vapor deposition.
改性化学气相沉积。
PBG Photonic band gap. 光子带隙。
PCF Photonic-crystal fiber. 光子晶体光纤。
S Stokes.
SC Supercontinuum.
SMF Single-mode fiber. 单模光纤。
TIR Total internal reflection.
全内反射。
WDM Wavelength division multiplexing.
波分复用。

I. Introduction I. 介绍

HOTONIC-CRYSTAL fibers (PCFs)-fibers with a periodic transverse microstructure-have been in practical existence as low-loss waveguides since early 1996 [1], [2]. The initial demonstration took four years of technological development, and since then, the fabrication techniques have become more and more sophisticated. It is now possible to manufacture the microstructure in air-glass PCF to accuracies of on the scale of , which allows remarkable control of key optical properties such as dispersion, birefringence, nonlinearity, and the position and width of the PBGs in the periodic "photonic-crystal" cladding. PCF has, in this way, extended the range of possibilities in optical fibers, both by improving well-established properties and introducing new features such as low-loss guidance in a hollow core.
光子晶体光纤(PCFs)-具有周期性横向微结构的光纤自 1996 年初以来一直作为低损耗波导存在[1],[2]。最初的演示经历了四年的技术发展,自那时起,制造技术变得越来越复杂。现在可以将微结构制造到空气-玻璃 PCF 中,精度达到 尺度,这使得对色散、双折射、非线性和周期性“光子晶体”包层中 PBGs 的位置和宽度等关键光学特性具有显著控制。PCF 以这种方式扩展了光纤的可能性范围,既通过改善已经建立的特性,又引入了新特性,例如在中空芯中低损耗引导。
Standard SMF, with a normalized core-cladding refractiveindex difference , a core diameter of ,
标准 SMF,具有归一化的芯-包层折射率差 ,芯直径为
and, of course, very high optical clarity (better than at ), is actually quite limiting for many applications. Two major factors contribute to this. The first is the smallness of , which causes bend loss at in Corning SMF-28 for one turn around a mandrel in diameter [3]) and limits the degree to which GVD and birefringence can be manipulated. Although much higher values of can be attained (MCVD yields an index difference of 0.00146 per up to a maximum for [4]), the singlemode core radius becomes very small, and the attenuation rises through increased absorption and Rayleigh scattering. The second factor is the reliance on TIR so that guidance in a hollow core is impossible, however useful it would be in fields requiring elimination of glass-related nonlinearities or enhancement of laser interactions with dilute or gaseous media. PCF has made it possible to overcome these limitations, and as a result, many new applications of optical fibers are emerging.
当然,非常高的光学透明度(比 ),实际上对许多应用是相当有限的。有两个主要因素导致了这一点。第一个是 的小,导致了在 Corning SMF-28 中在直径为 的曲率半径上绕一圈时的弯曲损耗 处[3],并限制了 GVD 和双折射可以被操纵的程度。尽管可以获得更高的 值(MCVD 每 产生 0.00146 的折射率差异,最大 [4]),但单模芯半径变得非常小,通过增加吸收和瑞利散射导致衰减增加。第二个因素是依赖于全反射,因此在空心芯中无法进行引导,无论在需要消除与玻璃相关的非线性或增强与稀薄或气态介质的激光相互作用的领域中有多么有用。PCF 已经使得克服这些限制成为可能,因此许多新的光纤应用正在出现。

A. Outline of the Article
文章概要

In Section II, a brief history of PCFs is given, and in Section III, fabrication techniques are reviewed. Numerical modeling and analysis are covered in Section IV and the optical properties of the periodic photonic-crystal cladding in Section V. The characteristics of guidance are discussed in Section VI (A: resonance and antiresonance; B and C: PCFs with positive and negative core-cladding index differences, respectively; D: birefringence; E: GVD; F: attenuation mechanisms; G: Kerr nonlinearities). In Section VII, many of the applications of PCFs are discussed (A: high-power transmission; B: lasers and amplifiers; C: intrafiber devices, cleaving, and splicing; D: Kerr-related nonlinear effects; E: Brillouin scattering; F: gas-based devices; G: telecommunications; H: laser tweezers; I: optical sensors), and some final remarks are made in Section VIII.
在第二部分中,简要介绍了 PCF 的历史,在第三部分中回顾了制备技术。数值建模和分析涵盖在第四部分,周期性光子晶体包层的光学特性在第五部分讨论。在第六部分讨论了引导特性(A:共振和反共振;B 和 C:具有正和负核包层折射率差异的 PCF,分别;D:双折射;E:色散;F:衰减机制;G:Kerr 非线性)。在第七部分,讨论了 PCF 的许多应用(A:高功率传输;B:激光和放大器;C:纤内器件,切割和拼接;D:Kerr 相关的非线性效应;E:布里渊散射;F:基于气体的器件;G:电信;H:激光夹持器;I:光学传感器),并在第八部分做出一些最终的评论。

II. BRIEF HISTORY 二、简史

The original motivation for developing PCFs was the creation of a new kind of dielectric waveguide-one that guides light by means of a two-dimensional (2-D) PBG. In 1991, the idea that the well-known "stopbands" in periodic structures [5] could be extended to prevent propagation in all directions [6] was leading to attempts worldwide to fabricate three-dimensional PBG materials. At that time, the received wisdom was that the refractive-index difference needed to create a PBG in two dimensions was large-of order . It was not widely recognized that the refractive-index difference requirements for
开发 PCFs 的最初动机是创建一种新型介电波导-通过二维(2-D)PBG 引导光线的波导。 1991 年,周期结构中众所周知的“禁带”可以扩展以阻止在所有方向传播的想法,导致全球范围内尝试制造三维 PBG 材料。 那时,人们普遍认为在二维中创建 PBG 所需的折射率差异很大-约为 的数量级。 当时并未广泛认识到在两个维度中创建 PBG 所需的折射率差异要求
PBG formation in two dimensions are greatly relaxed if, as in a fiber, propagation is predominantly along the third axis-the direction of invariance.
如果在光纤中,传播主要沿着第三轴-不变方向-的方向,则二维中 PBG 的形成要求得到极大放宽。

A. PCFs

My idea, then, was to trap light in a hollow core by means of a 2-D "photonic crystal" of microscopic air capillaries running along the entire length of a glass fiber [7]. Appropriately designed, this array would support a PBG for incidence from air, preventing the escape of light from a hollow core into the photonic-crystal cladding and avoiding the need for TIR.
我的想法是通过沿着玻璃纤维整个长度运行的微观空气毛细管的二维“光子晶体”,将光困在中空芯中。适当设计,这个阵列将支持从空气入射的 PBG,防止光从中空芯逃逸到光子晶体包层中,并避免需要 TIR。
A question sometimes asked is whether these developments were really "new" or whether some aspects could be traced to previous work. While it is clear that no previous attempt had been made to produce photonic-crystal lattices of air holes in fiber form, or to create PBGs, there had been previous work on microstructured fibers. For example, in the 1970s, "single-component" fibers were investigated in which a central glass strand was held in place by two thin webs of glass [8]. This technology was rapidly abandoned with the introduction of MCVD. In the 1980s, in-fiber polarizers were developed at Southampton by drawing fibers with hollow side-channels ( in diameter) for introducing metal wires [9].
有时会有人问这些发展是否真的“新颖”,或者某些方面是否可以追溯到以前的工作。虽然很明显以前没有尝试过在纤维形式中制造空气孔的光子晶体晶格,或者创建 PBG,但以前确实有关于微结构光纤的工作。例如,在 1970 年代,曾研究过“单组分”光纤,其中一个中央玻璃细丝由两根薄玻璃网固定在原位。这项技术在 MCVD 引入后迅速被放弃。在 1980 年代,南安普敦开发了在绘制带有中空侧通道(直径为 )的光纤以引入金属丝的光纤偏振器。
The first four years of work on understanding and fabricating PCF were a journey of exploration. Our first plan was to drill holes in a short rod of silica glass and then draw it down to fiber. Silica is a mechanically very hard material, and machining a large array of small holes in it proved beyond the capabilities of the tools available. In 1993, with funding from the Defence Research Agency in Malvern, U.K., work began in earnest, the initial idea being to adapt techniques widely used for the fabrication of multichannel image intensifier plates. These are made by stacking individual cylindrical elements into a 2-D close-packed array, the end result after drawing being a honeycomb of diameter waveguide "pixels," each surrounded by black glass to reduce crosstalk. We also considered adapting techniques developed at the Naval Research Laboratory, Washington, DC, where cores from soluble glass, surrounded by insoluble glass, are arranged in a close-packed lattice and drawn down to microscopic dimensions. The soluble glass can then be dissolved out, leaving an array of tiny hollow channels in plates as thick as [10], [11].
对于理解和制造 PCF 的前四年是一次探索之旅。我们最初的计划是在硅玻璃短棒上钻孔,然后将其拉制成光纤。硅是一种机械性能非常硬的材料,用现有工具加工大量小孔的能力被证明是不可能的。1993 年,在英国马尔文的国防研究机构的资助下,工作正式开始,最初的想法是改编广泛用于制造多通道图像增强器板的技术。这些板是通过将单个圆柱形元素堆叠成二维紧密排列阵列制成的,拉制后的最终结果是直径为 的波导“像素”蜂窝结构,每个像素周围都被黑色玻璃包围以减少串扰。我们还考虑了在美国华盛顿特区海军研究实验室开发的技术,该技术是将可溶玻璃芯包围在不溶玻璃中,排列成紧密堆积的晶格,然后将其拉制至微观尺寸。然后可溶玻璃可以被溶解掉,留下厚度为 的板中一组微小空心通道。
At this time, the parallel task of solving Maxwell's equations numerically was making good progress, culminating in a 1995 paper that showed that PBGs did indeed exist in 2-D silica-air structures for "conical" incidence from vacuum-this being an essential prerequisite for hollow-core guidance [12].
此时,通过数值方法解决麦克斯韦方程的并行任务取得了良好的进展,最终在 1995 年的一篇论文中表明,对于从真空中“锥形”入射的 2-D 硅-空气结构确实存在 PBGs,这是中空芯引导的一个基本前提[12]。
The first successful silica-air PCF structure was made in late 1995 by stacking 217 silica capillaries (eight layers outside the central capillary), specially machined with hexagonal outer cross sections and a circular inner cross section. The diameterto-pitch ratio of the holes in the final stack was , which theory showed was too small for PBG guidance in a hollow core, so we decided to make a PCF with a solid central core surrounded by 216 air channels [Fig. 1(a)] [1], [13], [14]. This led to the discovery of ESM PCF, which, if it guides at all, only supports the fundamental guided mode [15]. The success of these initial experiments led rapidly to a whole series of new
第一个成功的硅-空气 PCF 结构是在 1995 年底制成的,通过堆叠 217 根硅毛细管(中央毛细管外有八层),这些毛细管具有六边形外横截面和圆形内横截面。最终堆叠中孔的直径与间距比 ,理论表明这对于中空芯的 PBG 引导来说太小了,因此我们决定制作一个中心实心芯被 216 个空气通道包围的 PCF [图 1(a)] [1], [13], [14]。这导致了 ESM PCF 的发现,如果它有引导,只支持基本引导模式[15]。这些最初实验的成功迅速导致了一系列新的

Fig. 1. Selection of scanning electron micrographs of PCF structures. (a) First working PCF-the solid glass core is surrounded by a triangular array of 300-nm-diameter air channels, spaced apart [13], [14]. (b) Detail of a low-loss solid-core PCF (interhole spacing ). (c) First hollow-core PCF [19]. (d) PCF extruded from Schott SF6 glass with a core in diameter .
图 1. PCF 结构的扫描电子显微镜选择。(a) 第一个工作的 PCF-固体玻璃芯被 300 纳米直径的空气通道三角阵列包围,间隔 [13],[14]。(b) 低损耗固芯 PCF 的细节(间隔 )。(c) 第一个中空芯 PCF [19]。(d) 从 Schott SF6 玻璃挤出的 PCF,芯部直径 ,外径
types of PCF-large mode area [16], dispersion controlled [17], [18] hollow core [19], birefringent [20], and multicore [21].
PCF 的类型-大模式区域[16],色散控制[17],[18],中空芯[19],双折射[20],和多芯[21]。
These initial breakthroughs led quickly to applications, perhaps the most celebrated being the report in 2000 of SC generation from unamplified Ti:sapphire femtosecond laser pulses in a PCF with a core small enough to give zero dispersion at wavelength (Section VII-D1) [22].
这些最初的突破迅速导致了应用,也许最值得庆贺的是 2000 年报告的 SC 生成,来自未放大的钛宝石飞秒激光脉冲在一个 PCF 中,其芯部足够小以在 波长处产生零色散(第 VII-D1 节)[22]。

B. Bragg Fibers B. 布拉格光纤

In the late 1960s and early 1970s, theoretical proposals were made for another kind of fiber with a periodically structured cross section [23], [24]. This was a cylindrical "Bragg" fiber that confines light within an annular array of rings of high and low refractive index arranged concentrically around a central core. A group in France has made a solid-core version of this structure using MCVD [25]. Employing a combination of polymer and chalcogenide glass, researchers in the United States have realized a hollow-core version of a similar structure [26], reporting a loss at the wavelength (the losses at telecom wavelengths have yet to be specified). This structure guides light in the mode, which is used in microwave telecommunications because of its extremely low loss; the field moves away from the attenuating waveguide walls as the frequency increases, resulting in very low losses, although the guide must be kept very straight to avoid the fields entering the cladding and experiencing high absorption.
在 20 世纪 60 年代末和 70 年代初,有关另一种具有周期结构横截面的光纤的理论提议被提出[23],[24]。这是一种圆柱形的“布拉格”光纤,它将光束限制在高折射率和低折射率环形阵列的环绕中心核心同心排列的环形阵列中。法国的一个团队使用 MCVD 制造了这种结构的实心核心版本[25]。美国的研究人员利用聚合物和硫化物玻璃的组合实现了类似结构的空心核心版本[26],报道了在 波长处的 损耗(在电信波长处的损耗尚未指定)。这种结构引导光束进入 模式,该模式在微波通信中使用,因为其损耗极低;随着频率的增加,场远离衰减波导壁,导致非常低的损耗,尽管必须保持波导非常直以避免场进入包层并经历高吸收。

III. Fabrication Techniques
III. 制造技术

PCF structures are currently produced in many laboratories worldwide using a variety of different techniques (see Fig. 1
目前,全球许多实验室使用各种不同的技术生产 PCF 结构(见图 1)。
Fig. 2. Preform stack containing (a) birefringent solid core, (b) seven-cell hollow core, (c) solid isotropic core, and (d) doped core. The capillary diameters are -large enough to ensure that they remain stiff for stacking.
图 2。预制块堆叠包含(a)双折射固体芯,(b)七腔中空芯,(c)固体各向同性芯和(d)掺杂芯。毛细管直径为 -足够大,以确保它们在堆叠时保持坚挺。
for some example structures). The first stage is to produce a "preform"-a macroscopic version of the planned microstructure in the drawn PCF. There are many ways to do this, including stacking of capillaries and rods [13], [14] extrusion [27]-[30], sol-gel casting [31], injection molding, and drilling.
一些示例结构)。第一阶段是制备“预制块”-计划中微结构的宏观版本,用于拉制 PCF。有许多方法可以做到这一点,包括毛细管和棒材的堆叠[13],[14]挤压[27]-[30],溶胶-凝胶铸造[31],注塑和钻孔。
The most widely used technique is stacking of circular capillaries (Fig. 2). Typically, meter-length capillaries with an outer diameter of are drawn from a starting tube of high-purity synthetic silica with a diameter of . The inner/outer diameter of the starting tube, which typically lies in the range from 0.3 up to beyond 0.9 , largely determines the value in the drawn fiber. The uniformity in diameter and circularity of the capillaries must be controlled to at least of the diameter. They are stacked horizontally in a suitably shaped jig to form the desired crystalline arrangement. The stack is bound with wire before being inserted into a jacketing tube, and the whole assembly is then mounted in the preform feed unit for drawing down to fiber. Judicious use of pressure and vacuum during the draw allows some limited control over the final structural parameters, for example, the value.
最广泛使用的技术是圆形毛细管的堆叠(图 2)。通常,从直径为 的高纯度合成二氧化硅起始管中拉出米长的毛细管,其外径为 。起始管的内/外径通常在 0.3 至 0.9 以上的范围内,主要决定了拉伸纤维中的 值。毛细管的直径和圆度必须至少控制到直径的 。它们水平堆叠在适当形状的夹具中,形成所需的晶体排列。在将堆叠体插入护套管之前,用线束绑扎,然后整个组件安装在预制进给单元中以拉伸成纤维。在拉伸过程中,适度使用压力和真空可以在一定程度上控制最终的结构参数,例如 值。
Extrusion offers an alternative route to making PCF, or the starting tubes, from bulk glass; it permits formation of structures that are not readily made by stacking. While not suitable for silica (no die material has been identified that can withstand the processing temperatures without contaminating the glass), extrusion is useful for making PCFs from compound silica glasses, tellurites, chalcogenides, and polymers-materials that melt at lower temperatures. Fig. 1(d) shows the cross section of a fiber extruded, through a metal die, from a commercially available glass (Schott SF6) [28]. PCF has also been extruded from tellurite glass, which has excellent IR transparency out to beyond , although the reported fiber losses (a few decibels per meter) are as yet rather high [30]. Polymer PCFs, which were first developed in Sydney, have been successfully made using many different approaches, for example, extrusion, casting, molding, and drilling [32].
挤出提供了制作 PCF 或起始管道的另一种途径,它允许形成通过堆叠难以制造的结构。虽然不适用于二氧化硅(尚未确定能够承受 加工温度而不污染玻璃的模具材料),但挤出对于利用化合物二氧化硅玻璃、碲酸盐、硫化物和聚合物等在较低温度下熔化的材料制作 PCF 非常有用。图 1(d) 显示了通过金属模具从一种商用玻璃(Schott SF6)挤出的光纤的横截面 [28]。PCF 也已从碲酸盐玻璃挤出,该玻璃在红外透明度方面表现出色,尽管报告的光纤损耗(每米几分贝)目前仍然相当高 [30]。最初在悉尼开发的聚合物 PCF 已成功地使用许多不同的方法制造,例如挤出、铸造、成型和钻孔 [32]。

A. Design Approach A. 设计方法

The successful design of a PCF for a particular application is not simply a matter of using numerical modeling (see Section IV) to calculate the parameters of a structure that yields the required performance. This is because the fiberdrawing process is not lithographic but introduces its own highly reproducible types of distortion through the effects of viscous flow, surface tension, and pressure. As a result, even if the initial preform stack precisely mimics the theoretically required structure, several modeling and fabrication iterations are usually needed before a successful design can be reached.
对于特定应用的 PCF 的成功设计并不仅仅是使用数值建模(见第四节)来计算产生所需性能的结构参数的问题。这是因为光纤拉拔过程并不是光刻过程,而是通过粘性流动、表面张力和压力的影响引入了自己高度可重复的扭曲类型。因此,即使初始预制坯堆精确地模仿了理论上所需的结构,通常需要进行多次建模和制造迭代,才能达到成功的设计。

IV. Modeling And Analysis
IV. 建模与分析

The complex structure of PCF-in particular, the large refractive-index difference between glass and air-makes its electromagnetic analysis challenging. Maxwell's equations must usually be solved numerically using one of a number of specially developed techniques [12], [33]-[36]. Although standard optical fiber analyses and a number of approximate models are occasionally helpful, these are only useful as rough guidelines to the exact behavior unless checked against accurate numerical solutions.
PCF 的复杂结构,特别是玻璃和空气之间的大折射率差使得其电磁分析具有挑战性。通常必须使用一种特殊开发的技术[12],[33]-[36]来数值求解麦克斯韦方程。尽管标准光纤分析和一些近似模型偶尔会有所帮助,但除非与准确的数值解进行核对,否则这些只能作为粗略指导,无法准确反映实际行为。

A. Maxwell's Equations 麦克斯韦方程组

In most practical cases, a set of equal frequency modes is more useful than a set of modes of different frequency sharing the same value of axial wavevector component . It is, therefore, convenient to arrange Maxwell's equations with as eigenvalue, i.e.,
在大多数实际情况下,一组相同频率模式比一组不同频率但共享相同轴波矢分量 的模式更有用。因此,方便起见,将麦克斯韦方程组排列为 作为特征值,即,
where all the field vectors are taken in the form is the dielectric constant, is the position in the transverse plane, and is the vacuum wavevector. This form allows material dispersion to be easily included, which is something that is not possible if the equations are set up with as eigenvalue. Written out explicitly in Cartesian coordinates, (1) yields two equations relating and as follows:
其中所有场矢量采用形式 是介电常数, 是横向平面上的位置, 是真空波矢。这种形式允许轻松地包含材料色散,如果方程组设置 作为特征值,则不可能实现这一点。在笛卡尔坐标中明确写出,(1)得到两个关于 的方程如下:
and a third differential equation relating , and , which is, however, not required to solve (2).
和第三个微分方程,涉及 ,然而,不需要解决(2)。

B. Scalar Approximation B. 标量近似

In the paraxial scalar approximation, the second term inside the operator in (1), which gives rise to the middle terms in (2) that couple between the vector components of the field, can be neglected, yielding the following scalar wave equation:
在偏轴标量近似中,(1)中算子内的第二项,导致(2)中耦合场的矢量分量之间的中间项,可以忽略,得到以下标量波动方程:
This leads to a scaling law similar to the one used in standard SMF analyses [38], which can be used to parameterize the fields
这导致了类似于标准 SMF 分析中使用的缩放定律 [38],可用于参数化场
[39]. Defining as the interhole spacing and and as the refractive indices of the two materials used to construct a particular geometrical shape of photonic crystal, the mathematical forms of the fields and the dispersion relations are identical, provided the generalized -parameter defined by
[39]。将 定义为间隙距离,将 定义为用于构建光子晶体特定几何形状的两种材料的折射率,只要保持广义 -参数不变,场的数学形式和色散关系是相同的
is held constant. This has the interesting (though in the limit not exactly practical) consequence that band gaps can exist for vanishingly small index differences, provided the structure is made sufficiently large (see Section VI-C4).
。这具有有趣的(尽管在极限情况下并非完全实用)结果,即即使折射率差异趋近于零,只要结构足够大(见第 VI-C4 节),就可以存在带隙。

C. Numerical Techniques C. 数值技术

A common technique for solving (1) employs a Fourier expansion to create a basis set of plane waves for the fields, which reduces the problem to the inversion of a matrix equation suitable for numerical computation [34]. In contrast to the versions of Maxwell's equations with as eigenvalue [40], (1) is non-Hermitian, which means that standard matrix inversion methods cannot straightforwardly be applied. An efficient iterative scheme can, however, be used to calculate the inverse of the operator by means of fast Fourier transform steps. This method is useful for accurately finding the modes guided in a solid-core PCF, which are located at the upper edge of the eigenvalue spectrum of the inverted operator. In hollow-core PCF, however (or other fibers relying on a cladding band gap to confine the light), the modes of interest lie in the interior of the eigenvalue spectrum. A simple transformation can, however, be used to move the desired interior eigenvalues to the edge of the spectrum, greatly speeding up the calculations and allowing as many as a million basis waves to be incorporated [41], [42].
一种常见的解决方法是利用傅立叶展开来创建一组平面波作为场的基础集,从而将问题简化为适合数值计算的矩阵方程的求逆[34]。与具有 作为特征值的麦克斯韦方程的版本相比[40],(1)是非厄米的,这意味着标准矩阵求逆方法不能直接应用。然而,可以使用高效的迭代方案通过快速傅立叶变换步骤计算算子的逆。这种方法对于准确找到固体芯 PCF 中引导的模式非常有用,这些模式位于反转算子的特征值谱的上边缘。然而,在空芯 PCF 中(或其他依赖包层带隙来限制光的光纤中),感兴趣的模式位于特征值谱的内部。然而,可以使用简单的转换将所需的内部特征值移动到谱的边缘,大大加快计算速度,并允许包含多达一百万个基础波[41],[42]。
To treat PCFs with a central guiding core in an otherwise periodic lattice, a supercell is constructed, its dimensions being large enough so that once tiled, the guided modes in adjacent cores do not significantly interact. The planewave expansion method uses a Fourier series to represent the discontinuous dielectric function, so it suffers from the Gibb's phenomenon-the inability to accurately represent step changes. This introduces inaccuracy in the solutions that can usually (though not always, e.g., in high-index glass-air structures) be reduced by retaining more and more plane-wave components. A successful and accurate approach to alleviating this problem is to smooth out the sharp edges using superGaussian functions, though careful checks of the accuracy of the solutions must be carried out.
为了处理具有中心引导核心的周期晶格中的 PCFs,构建了一个超胞,其尺寸足够大,以便一旦铺设,相邻核心中的引导模式不会显著相互作用。平面波展开方法使用傅里叶级数来表示不连续的介电函数,因此它受到吉布斯现象的影响-无法准确表示阶跃变化。这会在解决方案中引入不准确性,通常(尽管并非总是如此,例如在高折射率玻璃-空气结构中)可以通过保留更多平面波分量来减少。缓解这一问题的成功和准确方法是使用超高斯函数平滑尖锐边缘,尽管必须仔细检查解决方案的准确性。
Other numerical techniques include expanding the field in terms of Hermite-Gaussian functions [42], [43], the use of FDTD analysis (a simple and versatile tool for exploring waveguide geometries [44]) and the finite-element approach [45]. If the PCF structure consists purely of circular holes, the multipole or Rayleigh method is a particularly fast and efficient method [36], [37]. It uses Mie theory to evaluate the scattering of the field incident on each hole. Yet another approach is a source-model technique that uses two sets of fictitious elementary sources to approximate the fields inside and outside circular cylinders .
其他数值技术包括用 Hermite-Gaussian 函数展开场[42],[43],使用 FDTD 分析(一种探索波导几何形状的简单多功能工具[44])和有限元方法[45]。如果 PCF 结构纯粹由圆形孔组成,多极或瑞利方法是一种特别快速高效的方法[36],[37]。它使用 Mie 理论评估入射到每个孔上的场的散射。另一种方法是源模型技术,它使用两组虚拟基本源来近似圆柱内外的场
Fig. 3. Propagation diagram for a PCF with air-filling fraction. Note the different regions where light is (1) able to propagate in all regions, (2) able to propagate also in the photonic-crystal cladding, (3) able to propagate only in silica glass, and (4) cut off completely. The "fingers" indicate the positions of full 2-D photonic band gaps .
图 3. 具有 空气填充分数的 PCF 的传播图。请注意光能够传播的不同区域,即(1)能够在所有区域传播,(2)也能够在光子晶体包层中传播,(3)只能在二氧化硅玻璃中传播,以及(4)完全截止。 “手指”指示完整的 2-D 光子带隙的位置

V. Characteristics of Photonic-Crystal Cladding
V. 光子晶体包层的特性

The simplest photonic-crystal cladding is a biaxially periodic defect-free composite material with its own well-defined dispersion and band structure. These properties determine the behavior of the guided modes that form at cores (or "structural defects" in the jargon of photonic crystals). A convenient graphical tool is the propagation diagram - a map of the ranges of frequency and axial wavevector component where light is evanescent in all transverse directions regardless of its polarization state (Fig. 3) [12]. The vertical axis is the normalized frequency , and the horizontal axis is the normalized axial wavevector component . Light is unconditionally cut off from propagating (due to either TIR or a PBG) in the black regions.
最简单的光子晶体包层是一种具有自身明确定义的色散和带结构的双轴周期无缺陷复合材料。这些特性决定了在核心处形成的引导模式的行为(或者说是光子晶体行话中的“结构缺陷”)。一个方便的图形工具是传播图 - 一个频率范围和轴向波矢分量 的地图,在这些范围内,光在所有横向方向上都是耗散的,而不管其偏振状态如何(图 3)[12]。垂直轴是归一化频率 ,水平轴是归一化轴向波矢分量 。光在黑色区域中被无条件地切断传播(由于全反射或者光子带隙)。
In any subregion of isotropic material (i.e., glass or air) at fixed optical frequency, the maximum possible value of is given by , where is the refractive index (at that frequency) of the region under consideration. For , light is free to propagate, for , it is evanescent, and at , the critical angle is reached, denoting the onset of TIR for light incident from a medium of index larger than .
在各向同性材料的任何子区域(即玻璃或空气)中,在固定的光学频率下, 的最大可能值由 给出,其中 是考虑区域的折射率(在该频率下)。对于 ,光可以自由传播,对于 ,它是耗散的,而在 处,达到了临界角,表示从折射率大于 的介质入射的光的全反射的开始。
The slanted guidelines (Fig. 3) denote the transitions from propagation to evanescence for air, the photonic crystal, and glass. At fixed optical frequency for , light propagates freely in every subregion of the structure. For , in which is the index of the glass, light propagates in the glass substrands and is evanescent in the hollow regions. Under these conditions, the "tight binding" approximation holds, and the structure may be viewed as an array of coupled glass waveguides.
斜向指导线(图 3)表示空气、光子晶体和玻璃从传播到消逝的过渡。对于 的固定光学频率,光在结构的每个子区域中自由传播。对于 ,其中 是玻璃的折射率,光在玻璃基片中传播并在空心区域中消失。在这些条件下,“紧束缚”近似成立,结构可以看作是一组耦合的玻璃波导阵列。

A. Maximum Refractive Index
A. 最大折射率

The maximum axial refractive index in the photonic-crystal cladding lies in the range as expected of a composite glass-air material. This value coincides with the -pointing "peaks" of the dispersion surfaces in
光子晶体包层中的最大轴向折射率 位于 范围内,符合复合玻璃-空气材料的预期。该值与色散曲面的 -指向“峰值”相一致。

Fig. 4. (a) Real-space schematic of the tight-binding picture of modes guided in the glass substrands in a triangular array of air holes. Each substrand couples to three neighbors, and by applying Bloch's theorem, the allowed values of transverse wavevector may be calculated at fixed and . (b) Reciprocal space picture of the dispersion surfaces for increasing values of . At a particular value of , a passband opens up (the dark-shaded triangular features) and the field amplitudes change sign between adjacent substrands. As increases, the passband is traversed, terminating at a maximum value (corresponding to ) where the field amplitudes in all substrands have the same sign.
图 4. (a) 紧束缚模型的实空间示意图,显示在三角形排列的空气孔中引导的玻璃基底中的模式。每个基底与三个邻居耦合,并通过应用布洛赫定理,可以在固定 时计算横向波矢的允许值。(b) 递归空间中的色散曲面图,显示不断增加的 值。在特定的 值处,一个通带打开(深色阴影的三角形特征),并且场振幅在相邻基底之间改变符号。随着 的增加,通过通带,终止于最大值(对应于 ),在所有基底中场振幅具有相同符号。
reciprocal space, where multiple values of transverse wavevector are allowed, one in each tiled Brillouin zone. For a constant value of slightly smaller than , these wavevectors lie on small approximately circular loci, with a transverse component of group velocity that points normal to the circles in the direction of increasing frequency (Fig. 4). Thus, light can travel in all directions in the transverse plane, even though its wavevectors are restricted to values lying on the circular loci.
逆空间,其中允许横向波矢的多个值,每个瓦格布里温区域中有一个。对于一个略小于 的常数 ,这些波矢位于小的近似圆形轨迹上,其横向分量的群速度指向圆圈的法线方向,即频率增加的方向(图 4)。因此,光线可以在横向平面的所有方向传播,即使其波矢被限制在圆形轨迹上的值上。
Thus, depends strongly on frequency, even though neither the air nor the glass is assumed dispersive in the analysis; microstructuring itself creates dispersion through a balance between transverse energy storage and energy flow that is highly dependent upon frequency.
因此, 强烈依赖于频率,即使在分析中假定空气和玻璃都不是色散的;微结构本身通过横向能量存储和高度依赖于频率的能量流之间的平衡创造了色散。
By averaging the square of the refractive index in the photonic-crystal cladding, it is simple to show that
通过对光子晶体包层中折射率的平方进行平均,可以简单地表明
in the long-wavelength limit for a scalar approximation, where is the air-filling fraction, and is the index in the holes, which we take to be equal to 1 in what follows.
在标量近似的长波极限下,其中 是空气填充分数, 是孔洞中的折射率,在接下来的讨论中我们取其为 1。
As the wavelength of the light falls, the optical fields are better able to distinguish between the glass regions and the air. The light piles up more and more in the glass, causing the effective "seen" by it to change. In the limit of small wavelength , light is strongly excluded from the air holes by TIR, and the field profile "freezes" into a shape that is independent of wavelength. The variation of with frequency may be estimated by expanding fields centered on the air holes in terms of Bessel functions and applying symmetry [15]. Defining the normalized parameters
随着光的波长减小,光学场能够更好地区分玻璃区域和空气。光在玻璃中堆积越来越多,导致其所看到的有效 发生变化。在小波长 的极限下,光被全反射强烈排斥出空气孔洞,场的轮廓“冻结”成与波长无关的形状。通过将以贝塞尔函数为中心的场展开并应用对称性[15],可以估计 随频率的变化。定义归一化参数。
Fig. 5. Maximum axial refractive index in the photonic-crystal cladding as a function of the normalized frequency parameter for and 1.444. For this filling fraction of air ( , the value at long wavelength is , which is in agreement with (5). The horizontal dashed gray line represents the case when the core is replaced with a glass of refractive index (below that of silica) when guidance ceases for .
图 5. 光子晶体包层中最大轴向折射率随归一化频率参数 的变化。对于 1.444。对于空气的这个填充分数( ),长波长处的值 ,与(5)一致。水平虚线灰线代表当核心被折射率为 (低于二氧化硅)的玻璃替换时,当 时导引停止。
the analysis yields the following polynomial fit (see the Appendix):
分析得出以下多项式拟合(见附录):
for and . This polynomial is accurate to better than in the range . The resulting expression for is plotted in Fig. 5 against the parameter .
对于 。这个多项式在范围 内的精度优于 。得到的 表达式针对参数 绘制在图 5 中。

B. Transverse Effective Wavelength
B. 横向有效波长

The transverse effective wavelength in the th material is defined as follows:
个材料中的横向有效波长定义如下:
where is its refractive index. This wavelength can be many times the vacuum value, tends to infinity at the critical angle , and is imaginary when . It is a measure of whether or not the light is likely to be resonant within a particular feature of the structure, for example, a hole or a strand of glass, and defines PCF as a wavelength-scale structure.
其中 是其折射率。这个波长可以是真空值的许多倍,在临界角 处趋于无穷大,并且在 时是虚数。它衡量了光线是否可能在结构的特定特征内共振,例如孔洞或玻璃丝,并将 PCF 定义为波长尺度结构。

C.

Full 2-D PBGs exist in the black finger-shaped regions in Fig. 3. Some of these extend into the region , where light is free to propagate in vacuum, confirming the feasibility of trapping light within a hollow core.
完整的二维 PBG 存在于图 3 中的黑色手指形状区域中。其中一些延伸到区域 ,在那里光可以在真空中传播,证实了在中空核心内捕获光的可行性。
The band gap edges coincide with points where resonances in the cladding unit cells switch on and off, i.e., the eigenvalues of the unitary inter-unit-cell field transfer matrices change from (propagation) to (evanescence). At these transitions, depending on the band edge, the light is to a greater or lesser degree preferentially redistributed into the low- or high-index subregions. For example, at fixed optical frequency and small values of , leaky modes peaking in the low-index channels form a passband that terminates when the standing wave pattern has visibility. For the high-index strands
带隙边缘与包层单元格中的谐振点重合,即单元间场传输矩阵的特征值从 (传播)变为 (消失)的点。在这些转变点上,根据带边,光会在低或高折射率子区域中被更多或更少地重新分配。例如,在固定光频率和 的小值时,峰值在低折射率通道中的泄漏模式形成一个通带,当驻波模式具有 的可见度时终止。对于高折射率的细丝。
(Fig. 4), on the other hand, the band of real states is bounded by a lower value of where the field amplitude changes sign between selected pairs of adjacent strands, depending on the lattice geometry, and an upper bound where the field amplitude does not change sign between the strands (this field distribution yields .
(图 4), 另一方面, 真实状态的带是由一个较低值的 界限制的, 在选定的相邻绞线对之间, 根据晶格几何形状, 场振幅改变符号, 以及一个上限, 在这个上限内, 场振幅在绞线之间不改变符号 (这个场分布产生

VI. Characteristics OF GUIDANCE
VI. 引导特性

In SMF, guided modes form within the range of axial refractive indices when light is evanescent in the cladding ; core and cladding indices are represented by and ). In PCF, three distinct guidance mechanisms exist, namely 1) a modified form of TIR [15], [47], 2) PBG guidance [19], [48], and 3) a leaky mechanism based on a low density of photonic states in the cladding [49]. In the following subsections, we explore the role of resonance and antiresonance and discuss chromatic dispersion, attenuation mechanisms, and guidance in cores with refractive indices raised and lowered relative to the "mean" cladding value.
在 SMF 中, 当光在包层 中是腐蚀的时候, 引导模式在轴向折射率 范围内形成; 核心和包层折射率分别由 表示). 在 PCF 中, 存在三种不同的引导机制, 即 1) 修改的全反射形式[15], [47], 2) PBG 引导[19], [48], 和 3) 基于包层中光子态密度较低的泄漏机制[49]. 在接下来的小节中, 我们探讨共振和反共振的作用, 并讨论色散、衰减机制, 以及在相对于“平均”包层值升高和降低的核心中的引导。

A. Resonance and Antiresonance
A. 共振和反共振

It is helpful to view the guided modes as being confined (or not confined) by resonance and antiresonance in the unit cells of the cladding crystal. If the core mode finds no states in the cladding with which it is phase-matched, light cannot leak out. This is a familiar picture in many areas of photonics. What is perhaps not so familiar is the use of the concept in two dimensions, where a repeating unit is tiled to form a photoniccrystal cladding. This allows the construction of an intuitive picture of "cages," "bars," and "windows" for light and actually leads to a blurring of the distinction between guidance by modified TIR and PBG effects.
有助于将引导模式视为受限于(或不受限于)包层晶体的单元格中的共振和反共振。如果核心模式在包层中找不到与之相位匹配的状态,光就无法泄漏出来。这在许多光子学领域是一个熟悉的图景。也许不那么熟悉的是在二维中使用这个概念,其中一个重复单元被平铺以形成光子晶体包层。这允许构建一个关于光的“笼子”、“条形物”和“窗户”的直观图像,并实际上导致了通过改进的全反射和 PBG 效应进行引导之间的区别模糊。

B. Positive Core-Cladding Index Difference
B. 正核心-包层折射率差异

This type of PCF may be defined as one where the mean cladding refractive index in the long-wavelength limit (5) is lower than the core index (in the same limit). Under correct conditions (high air-filling fraction), PBG guidance may also occur in this case, although experimentally, the TIR-guided modes will dominate.
这种类型的 PCF 可以被定义为在长波长极限下,包层折射率的平均值 (5)低于核心折射率(在相同极限下)。在正确的条件下(高空气填充率),在这种情况下也可能发生 PBG 引导,尽管在实验中,TIR 引导模式将占主导地位。
  1. Controlling the Number of Modes: A striking feature of this type of PCF is that it is "ESM," i.e., the core does not become multimode in the experiments no matter how short the wavelength of the light [15]. Although the guidance in some respects resembles conventional TIR, it turns out to have some interesting and unique features that distinguish it markedly from step-index fiber. These are due to the piecewise discontinuous nature of the core boundary-sections where, for , air holes that strongly block the escape of light are interspersed with regions of barrier-free glass. In fact, the cladding operates in a regime where the transverse effective wavelength (8) in silica is comparable with the glass substructures in the cladding. The zone of operation in Fig. 3 is (point A).
    控制模式数量:这种类型的 PCF 的一个显著特点是它是“ESM”,即在实验中,无论光的波长多短,核心都不会变成多模[15]。尽管在某些方面引导类似于传统的 TIR,但事实证明它具有一些有趣且独特的特征,使其与阶跃折射率光纤明显区别开来。这些特征是由核心边界部分的分段不连续性所致,在这些部分中, ,强烈阻止光逃逸的空气孔与无障碍玻璃区域交替出现。事实上,包层在硅中的横向有效波长(8)与包层中玻璃亚结构相当。图 3 中的操作区域是 (点 A)。
In a solid-core PCF, taking the core radius and using the analysis in [15], the effective -parameter can be calcu-
在固芯 PCF 中,取芯半径 并使用[15]中的分析,可以计算有效 -参数
Fig. 6. -parameter for solid-core PCF (triangular lattice) plotted against the ratio of hole spacing to vacuum wavelength for different values of . Numerical modeling shows that ESM behavior is maintained for or
图 6. 固芯 PCF(三角晶格)的 -参数针对孔间距与真空波长比值绘制,对不同 值进行了数值建模,显示 ESM 行为在 时保持不变

Fig. 7. Modal filtering in a solid-core PCF. (a) Fundamental mode is trapped. (b) Higher order modes leak away through the gaps between the air holes.
图 7. 固芯 PCF 中的模式滤波。(a)基模被困住。(b)高阶模式通过空气孔之间的间隙泄漏出去
lated. This yields the plot in Fig. 6, where the full behavior from very low to very high frequency is predicted. As expected, the number of guided modes is almost independent of wavelength at high frequencies; the single-mode behavior is determined solely by the geometry. Numerical modeling shows that if , the fiber never supports any higher order guided modes, i.e., it is ESM.
这导致了图 6 中的情节,预测了从非常低到非常高频率的完整行为。如预期,在高频率下,引导模式的数量 几乎与波长无关;单模行为仅由几何形状决定。数值建模表明,如果 ,光纤永远不支持任何更高阶的引导模式,即它是 ESM。
This behavior can be understood by viewing the array of holes as a modal filter or "sieve" (Fig. 7). The fundamental mode in the glass core has a transverse effective wavelength . It is, thus, unable to "squeeze through" the glass channels between the holes, which are wide and, thus, below the Rayleigh resolution limit . Provided the relative hole size is small enough, higher order modes are able to escape-their transverse effective wavelength is shorter, so they have higher resolving power. As the holes are made larger, successive higher order modes become trapped.
这种行为可以通过将孔阵列视为模式滤波器或“筛子”(图 7)来理解。玻璃芯中的基本模式具有横向有效波长