2024_05_11_7ef6fa13b277a087de5fg

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

的

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

), is actually quite limiting for many applications. Two major factors contribute to this. The first is the smallness of

, which causes bend loss

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

好

），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

在

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

为

[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

为

，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

和

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

和

，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

和

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

和

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

, 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

和

1.444。对于空气的这个填充分数（

），长波长处的值

为

，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

和

。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:
第

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

和

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。

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

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

或

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

。因此，它无法“挤过”孔之间的玻璃通道，这些孔

宽，因此低于瑞利分辨限

。只要相对孔尺寸

足够小，更高阶模式就能逃逸-它们的横向有效波长更短，因此具有更高的分辨能力。随着孔变大，连续的更高阶模式被困住。

ESM behavior may also be viewed as being caused by strong wavelength dispersion in the photonic-crystal cladding, which forces the core-cladding index step to fall as the wavelength gets shorter (Fig. 5) [15], [47]. This counteracts the usual trend toward increasingly multimode behavior at short wavelengths. In the limit of very short wavelength, the light strikes the glassair interfaces at glancing incidence and is strongly rejected from the air holes. In this regime, the transverse single-mode
ESM 行为也可以被视为由于光子晶体包层中的强波长色散而引起，这迫使核心包层折射率阶跃随着波长变短而下降（图 5）[15]，[47]。这抵消了通常在短波长下趋向于越来越多模行为的趋势。在非常短波长的极限情况下，光线以浅角度入射击中玻璃-空气界面，并被空气孔强烈排斥。在这个范围内，横向单模的
profile does not change with wavelength. Consequently, the angular divergence (roughly twice the numerical aperture) of the emerging light is proportional to wavelength-in SMFs, it is approximately constant due to the appearance of more and more higher order guided modes as the frequency increases.
剖面随波长不变。因此，出射光的角发散度（大约是数值孔径的两倍）与波长成正比-在单模光纤中，由于随着频率增加越来越多的高阶导模的出现，它大致保持不变。

Thus, the refractive index of the photonic-crystal cladding increases with optical frequency, tending toward the index of silica glass in the short-wavelength limit. If the core is made from a glass of refractive index lower than that of silica (e.g., fluorine-doped silica), guidance is lost at wavelengths shorter than a certain threshold value (see Fig. 5) [50]. Such fibers have the unique ability to prevent transmission of short-wavelength light-in contrast to conventional fibers that guide more and more modes as the wavelength falls.
因此，光子晶体包层的折射率随光频增加而增加，趋向于在短波长极限处的二氧化硅玻璃的折射率。如果核心由折射率低于二氧化硅的玻璃制成（例如，掺氟二氧化硅），在短于某一阈值的波长下将失去引导（见图 5）[50]。这种光纤具有独特的能力，可以阻止短波长光的传输，与传统光纤相反，传统光纤在波长降低时会引导更多模式。

Ultra-Large-Area Single Mode: The modal filtering in ESM-PCF is controlled only by the geometry (e.g., for a triangular lattice). A corollary is that the behavior is quite independent of the absolute size of the structure, permitting SMF cores with arbitrarily large areas. A single-mode PCF with a core diameter of at was reported in 1998 [16]. In conventional step-index fibers, where for single-mode operation, this would require uniformity of core refractive index to part in -very difficult to achieve if MCVD is used to form the doped core. Larger mode areas allow higher power to be carried before the onset of intensity-related nonlinearities or damage and have obvious benefits for delivery of high laser power, fiber amplifiers, and fiber lasers. The bend-loss performance of such large-core PCFs is discussed in Section VI-B.
超大面积单模：ESM-PCF 中的模式滤波仅由几何形状控制（例如，三角格子为）。一个推论是，其行为与结构的绝对尺寸相当独立，允许具有任意大面积的 SMF 核心。1998 年报道了一个核心直径为，位于的单模 PCF [16]。在传统的步进折射率光纤中，要实现单模操作，这将需要核心折射率的均匀性达到的部分 - 如果使用 MCVD 形成掺杂的核心，这将非常难以实现。更大的模式面积允许在出现与强度相关的非线性或损伤之前携带更高功率，并对高功率激光的传输、光纤放大器和光纤激光器具有明显的好处。这种大核 PCF 的弯曲损耗性能在第 VI-B 节中讨论。
Fibers With Multiple Cores: The stacking procedure makes it straightforward to produce multicore fiber. A preform stack is built up with a desired number of solid (or hollow) cores and drawn down to fiber in the usual manner [21]. The coupling strength between the cores depends on the sites chosen, because the evanescent decay rate of the fields changes with azimuthal direction. Applications include curvature sensing [51]. More elaborate structures can be built up such as fibers with a central single-mode core surrounded by a highly multimode cladding waveguide and are useful in applications such as high-power cladding-pumped fiber lasers [52] and two-photon fluorescence sensors [53] [see Section VII-B and Fig. 8(j)].
具有多芯的光纤：层叠工艺使得生产多芯光纤变得简单。预制坯料堆叠起来，具有所需数量的实心（或空心）芯，并按照通常的方式拉伸成光纤[21]。芯间的耦合强度取决于所选择的位置，因为场的耦合衰减率随方位角方向变化。应用包括曲率传感[51]。还可以构建更复杂的结构，例如具有中央单模核心，周围包围着高度多模包层波导的光纤，可用于高功率包层泵浦光纤激光器[52]和双光子荧光传感器等应用[53] [见第 VII-B 节和图 8(j)]。
Nanoweb Fibers: The stacking and drawing procedure also makes it possible to produce some quite unusual structures such as a "nanoweb" fiber [Fig. 8(b)]. Somewhat similar to the single-component fibers from the 1970s [8], it does not have a deliberately introduced core but instead relies on a gentle thickness gradient to confine the light. The glass web can be as narrow as and as long as . The extremely low loss in what is essentially a planar glass waveguide with very strong waveguide dispersion makes this an interesting structure for experiments in spatial solitons and nonlinear optics .
纳米网纤维: 堆叠和拉伸过程还使得生产一些非常不寻常的结构成为可能，比如“纳米网”纤维[图 8(b)]。与 20 世纪 70 年代的单组分纤维有些相似[8]，它没有刻意引入的芯部，而是依靠温和的厚度梯度来限制光。玻璃网可以像那样窄，长达。在本质上是一个平面玻璃波导，具有非常强的波导色散，极低的损耗使得这成为空间孤子和非线性光学实验中的有趣结构。

C. Negative Core-Cladding Index Difference
C. 负芯-包层折射率差

Since TIR cannot operate under these circumstances, lowloss waveguiding is only possible if a PBG exists in the range

.
由于全反射不能在这些情况下发挥作用，低损耗波导只有在范围

内存在 PBG 时才可能。

Fig. 8. Representative sketches of different types of PCF. The black regions are hollow, the white regions are pure glass, and the gray regions are doped glass. (a) ESM solid-core PCF. (b) Nanoweb fiber (not a PCF). (c) All-solidglass PCF with raised-index doped glass strands (colored gray) in the cladding. (d) SC PCF (with high air-filling fraction and small core). (e) Dual-core PCF (f) Kagomé hollow-core PCF. (g) Seven-cell hollow-core PCF. (h) Birefringent PCF. (i) Carbon-ring structure for PBG guidance. (j) Double-clad PCF with offset doped lasing core and high numerical aperture inner cladding for pumping (the photonic-crystal cladding is held in place by thin webs of glass).
图 8. 不同类型 PCF 的代表性草图。黑色区域为空心，白色区域为纯玻璃，灰色区域为掺杂玻璃。(a) ESM 实芯 PCF。(b) 纳米网纤维（非 PCF）。(c) 全固玻璃 PCF，包含在包层中的提高折射率的掺杂玻璃细丝（灰色）。(d) SC PCF（具有高空气填充率和小芯）。(e) 双芯 PCF。(f) Kagomé 空芯 PCF。(g) 七腔空芯 PCF。(h) 双折射 PCF。(i) 用于 PBG 引导的碳环结构。(j) 偏移掺杂激光芯和高数值孔径内包层的双包层 PCF（光子晶体包层由薄玻璃网固定）。

Hollow-Core Silica-Air: In a silica-air PCF, larger airfilling fractions and small interhole spacings are necessary to achieve PBGs in the region . The relevant operating region in Fig. 3 is to the left of the vacuum line and inside one of the band gap fingers (point B). These conditions ensure that light is free to propagate and form guided modes within the hollow core while being unable to escape into the cladding. The number of such modes is controlled by the depth and width of the refractive-index "potential well" and is approximately given by
空心硅-空气: 在硅-空气 PCF 中，需要更大的空气填充比和较小的孔间距才能实现在区域内的 PBG。图 3 中的相关操作区域位于真空线的左侧，并且在一个带隙指（点 B）内。这些条件确保光线可以自由传播并在空心芯中形成引导模式，同时无法逃逸到包层中。这种模式的数量由折射率“势阱”的深度和宽度控制，大致由以下公式给出

where

and

are the refractive indices at the edges of the PBG at fixed frequency, and

is the core radius. Since the band gaps are quite narrow

is typically a few percent), the hollow core must be sufficiently large if a guided mode is to exist at all. In the first hollow-core PCF reported in 1999 [19], the core was formed by omitting seven capillaries from the preform stack [Fig. 1(c)]. An electron micrograph of a more recent structure, with a hollow core made by removing 19 missing capillaries from the stack, is shown in Fig. 9 [55].
其中

和

是固定频率下 PBG 边缘处的折射率，

是芯子半径。由于带隙相当窄（通常为几个百分点），如果要存在引导模式，则空心芯必须足够大。在 1999 年报道的第一种空心芯 PCF 中[19]，芯子是通过从预制坯料堆中省略七根毛细管而形成的[图 1(c)]。图 9[55]展示了一个更近期的结构的电子显微镜照片，其中空心芯是通过从堆中去除 19 根缺失的毛细管而制成的。

In hollow-core PCF, guidance can only occur when a PBG coincides with a core resonance. This means that only restricted bands of wavelength are guided. This feature can be very useful for suppressing parasitic transitions by filtering away the unwanted wavelengths, for example, in fiber lasers and in stimulated Raman scattering in gases (see Section VII-F1).
在中空芯光子晶体中，只有当光子带隙与芯部谐振重合时才能发生光导。这意味着只有波长的受限带段才能被引导。这个特性对于通过滤除不需要的波长来抑制寄生跃迁非常有用，例如在光纤激光器和气体中的受激拉曼散射中（见第 VII-F1 节）。

Glass of Higher Refractive Index: Achieving a band gap in glasses of higher refractive index for presents, at first glance, a dilemma. Whereas a larger refractive-index contrast generally yields wider band gaps, the higher "mean" refractive index seems likely to make it more difficult to achieve band
折射率较高的玻璃：在折射率较高的玻璃中实现带隙，乍看之下，似乎存在一个困境。尽管较大的折射率对比通常会产生更宽的带隙，但更高的“平均”折射率似乎会使实现带隙变得更加困难。

Fig. 9. (a) Example of a low-loss hollow-core PCF (fabricated by BlazePhotonics Ltd.). The core diameter is

, and the attenuation in the best cases approaches

wavelength. (b) Detail of the core region.
图 9. (a) 低损耗中空芯光子晶体示例（由 BlazePhotonics Ltd.制造）。芯部直径为

，在最佳情况下的衰减在

波长处接近

。（b）芯部区域的细节。

Fig. 10. (a) PCF with a "carbon-ring" lattice of air holes and an extra central hole to form a low-index core. (b) When white light is launched, only certain bands of wavelength are transmitted in the core-here, a six-lobed green mode emerges from the end-face (near-field image) [48].
图 10。(a) 具有“碳环”气孔晶格和额外中心孔以形成低折射率核心的 PCF。(b) 当白光发射时，只有特定波长的频带在核心中传输-这里，一个六瓣绿色模式从端面出现（近场图像）[48]。

gaps for incidence from vacuum. Although this argument holds in the scalar approximation, the result of calculations show that vector effects become important at higher levels of refractiveindex contrast (e.g.,

or higher), and a new species of band gap appears for smaller filling fractions of air than in silicabased structures. The appearance of this new type of gap means that it is actually easier to make hollow-core PCFs from glasses of higher index such as tellurites or chalcogenides [41].
从真空入射的间隙。尽管这个论点在标量近似中成立，但计算结果表明，在较高折射率对比度水平（例如，

或更高水平）时，矢量效应变得重要，并且对比硅基结构中较小填充分数的气体出现了新的带隙。这种新类型间隙的出现意味着实际上更容易从高折射率玻璃（如碲酸盐或硫化物）制造空芯 PCF [41]。

Surface States on Core-Cladding Boundary: The first PCF that was guided by PBG effects consisted of a lattice of air holes arranged in the same way as the carbon rings in graphite. The core was formed by introducing an extra hole at the center of one of the rings, with its low index precluding the possibility of TIR guidance [48]. When white light was launched into the core region, a colored mode was transmitted-the colors being dependent on the absolute size to which the fiber was drawn. The modal patterns had six equally strong lobes that are disposed in a flower-like pattern around the central hole. Closer examination revealed that the light was guided not in the air holes but in the six narrow regions of glass surrounding the core (Fig. 10). The light remained in these regions, despite the close proximity of large "rods" of silica, full of modes. This is because for particular wavelengths, the phase velocity of the light in the core is not coincident with any of the phase velocities available in the transmission bands created by nearest neighbor coupling between rod modes. Light is, thus, unable to couple over to them and remains trapped in the core. This
核心-包层边界上的表面态：第一个由 PBG 效应引导的 PCF 由一组空气孔组成，排列方式与石墨中的碳环相同。核心是通过在一个环的中心引入额外的孔而形成的，其低折射率排除了全反射引导的可能性[48]。当白光被引入核心区域时，会传输出一种有色模式-颜色取决于光纤拉伸到的绝对尺寸。模式图案有六个等强度的叶瓣，围绕中心孔以花瓣状图案排列。更仔细的观察发现，光不是在空气孔中引导的，而是在包围核心的六个狭窄玻璃区域中引导的（图 10）。尽管硅的大“棒”充满模式并且非常接近，但光仍保留在这些区域中。这是因为对于特定波长，光在核心中的相速度与由最近邻耦合的棒模式之间产生的传输带中的任何相速度不重合。因此，光无法耦合到它们上，而是被困在核心中。

Fig. 11. Near-field end-face images of the light transmitted in hollow-core PCF designed for

transmission. For light launched in the core mode, at

, a pure surface mode is transmitted, at

, a coupled surface-core mode is transmitted, and at

, a pure core mode is transmitted (false-color images courtesy G. Humbert, University of Bath) [59].
图 11。在设计用于

传输的中空芯 PCF 中传输的光的近场端面图像。对于在核心模式中发射的光，在

处，传输纯表面模式，在

处，传输耦合表面-核心模式，在

处，传输纯核心模式（伪彩色图像由巴斯大学的 G. Humbert 提供）[59]。

mechanism of PBG formation can only operate in extended one-dimensional (1-D) or 2-D structures (modes trapped within defect layers in multilayer stacks show the same behavior [56]).
PBG 形成机制只能在扩展的一维（1-D）或二维（2-D）结构中运行（在多层堆叠中受困于缺陷层内的模式显示相同的行为[56]）。

Similar guided modes are commonly seen in hollow-core PCF, where they form surface states (analogous with electronic surface states in semiconductor crystals) on the rim of the core, confined on the air side by TIR and on the cladding side by PBG effects. These surface states become phase-matched to the air-guided mode at certain wavelengths (Fig. 11), creating couplings (anticrossings on the frequency-wavevector diagram) that perturb the GVD and contribute additional attenuation (see Section VI-A) [57]-[59].
在中空芯 PCF 中常见类似的引导模式，在那里它们在核心边缘形成表面态（类似于半导体晶体中的电子表面态），在空气侧通过全反射（TIR）限制，在包层侧通过 PBG 效应限制。这些表面态在某些波长上与空气引导模式相位匹配（图 11），产生耦合（在频率-波矢图上的反交叉），扰动 GVD 并提供额外衰减（见第 VI-A 节）[57]-[59]。

All-Solid Structures: In all-solid band gap guiding fibers, the core is made from low-index glass and is surrounded by an array of high-index glass strands [60], [61]. Since the mean core-cladding index contrast is negative, TIR cannot operate, and PBG effects are the only possible guidance mechanism. These structures have some similarities with 1-D "ARROW" structures, where antiresonance plays an important role [62].
全固体结构：在全固体带隙引导光纤中，核心由低折射率玻璃制成，并被一系列高折射率玻璃纤维环绕[60]，[61]。由于平均核心-包层折射率对比为负，全反射无法发生，而光子带隙效应是唯一可能的引导机制。这些结构与 1-D“ARROW”结构有一些相似之处，其中反共振起着重要作用[62]。

When the cladding strands are antiresonant, light is confined to the central low-index core by a mechanism not dissimilar to the modal filtering picture in Section II-A; the high-index cores act as the "bars of a cage" so that no features in the cladding are resonant with the core mode, resulting in a low-loss guided mode. Guidance is achieved over wavelength ranges that are punctuated with high-loss windows where the cladding "bars" become resonant (Fig. 12). Remarkably, it is possible to achieve PBG guidance by this mechanism even at index contrasts of

[61], [63], with losses as low as

[64].
当包层纤维是反共振的时候，光通过一种与第 II-A 节中的模式滤波图像不太相似的机制被限制在中央低折射率核心中；高折射率核心起到“笼子的杆”的作用，使得包层中的特征与核心模式不共振，从而产生低损耗的引导模式。引导是在波长范围内实现的，这些波长范围被高损耗窗口分隔，其中包层的“杆”变得共振（图 12）。值得注意的是，即使在

的折射率对比下[61]，[63]，也可以通过这种机制实现 PBG 引导，损耗低至

在

时[64]。

Low-Density-of-States Guidance: The transmission bands are greatly widened in hollow-core PCFs with a Kagomé lattice in the cladding [49] [Fig. 8(f)]. Fig. 13 shows the attenuation spectrum of such a fiber, the minimum loss being over a bandwidth of several hundred nanometers. Numerical simulations show that while the cladding structure supports no band gaps, the density of states is greatly reduced near the vacuum line. The consequential poor overlap between the core states, together with the greatly reduced number of cladding states, slows down the leakage of light but does not completely prevent it.
导向低密度态: 在带有 Kagomé晶格的中空 PCF 中，传输带宽大大增加[49] [图 8(f)]。图 13 显示了这种光纤的衰减谱，最小损耗为，带宽为几百纳米。数值模拟表明，虽然包层结构不支持带隙，但在真空线附近，态密度大大降低。核心态之间的重叠差，加上包层态数量大大减少，减缓了光的泄漏，但并未完全阻止。

D. Birefringence D. 双折射

The modes of a perfect sixfold symmetric core and cladding structure are not birefringent [65]. In practice, however, the
完美的六重对称核心和包层结构的模式不具双折射性[65]。然而，在实际应用中，

Fig. 12. Lower: Measured transmission spectrum (using a white-light SC source) for a PCF with a pure silica core and a cladding formed by an array of Ge-doped strands (

, hole spacing

, index contrast

. The transmission is strongly attenuated when the core mode becomes phase-matched to different

"resonances" in the cladding strands. Upper: Experimental images [taken with blue

, green

, and red

filters] of the near-field profiles in the cladding strands at three such wavelengths. The fundamental

resonance occurs at

, and the four-lobed blue resonance lies off the edge of the graph.
图 12. 下图：使用白光 SC 光源测得的 PCF 的传输谱（纯二氧化硅芯和由掺锗纤维组成的包层，

，孔间距

，折射率对比

。当芯模式与包层纤维中不同

“谐振”相匹配时，传输会被强烈衰减。上图：在三个这样的波长处，使用蓝

、绿

和红

滤光片拍摄的包层纤维中近场轮廓的实验图像。基本

谐振发生在

处，而四叶蓝色谐振位于图表边缘之外。

Fig. 13. Measured loss of a hollow-core PCF with a Kagomé cladding lattice (scanning electron micrograph of structure in the inset). This type of photonic crystal provides a low density of states, i.e., an incomplete band gap. The peak at around

is due to absorption by an overtone of the

resonance.
图 13. 使用 Kagomé包层晶格的空芯 PCF 的测量损耗（结构的扫描电子显微镜图像见插图）。这种光子晶体提供低态密度，即不完全带隙。约

处的峰值是由

共振的倍频吸收引起的。

large glass-air index difference means that even slight accidental distortions in the structure yield a degree of birefringence. Therefore, if the core is deliberately distorted so as to become twofold symmetric, extremely high values of birefringence can be achieved. For example, by introducing capillaries with different wall thicknesses above and below a solid glass core [Figs. 8(h) and 14(a)], values of birefringence some ten times larger than in conventional fibers can be obtained [20]. Hollowcore PCF with moderate levels of birefringence

can be realized either by forming an elliptical core or by adjusting the structural design of the core surround [66], [67].
大玻璃-空气折射率差意味着即使结构中发生轻微的意外扭曲也会产生一定程度的双折射。因此，如果核心被故意扭曲以成为二重对称，就可以实现极高的双折射值。例如，通过在固体玻璃核心上下引入不同壁厚的毛细管[Figs. 8(h)和 14(a)]，可以获得比传统光纤大约十倍的双折射值[20]。通过形成椭圆形核心或调整核心周围的结构设计[66]，[67]，可以实现具有适度双折射

的中空 PCF。

Fig. 14. (a) Scanning electron micrographs of two different PCFs. (a) Birefringent PCF. (b) PCF with very small core (diameter

) and zero GVD wavelength

.
图 14。 (a) 两种不同 PCF 的扫描电子显微镜照片。 (a) 双折射 PCF。 (b) 具有非常小核心（直径

）和零 GVD 波长

的 PCF。

Fig. 15. Calculated GVD of two circular strands of silica glass, radii 0.4 and

, compared with the dispersion of bulk glass. The narrower strand has two dispersion zeros within the transparency window of silica.
图 15。计算出的两个硅玻璃圆形细丝的 GVD，半径为 0.4 和

，与大块玻璃的色散进行比较。较窄的细丝在硅的透明窗口内具有两个色散零点。

Experiments show that the birefringence in PCFs is some 100 times less sensitive to temperature variations than in conventional fibers, which is important in many applications [68]. This is because traditional "polarization-maintaining" fibers (bowtie, elliptical core, or Panda) contain at least two different glasses, each with a different thermal expansion coefficient. In such structures, the resulting temperature-dependent stresses make the birefringence a strong function of temperature.
实验表明，与传统光纤相比，PCF 中的双折射对温度变化的敏感性要低大约 100 倍，这在许多应用中非常重要[68]。这是因为传统的“保偏”光纤（蝴蝶结构、椭圆核心或熊猫结构）至少包含两种不同的玻璃，每种玻璃的热膨胀系数不同。在这种结构中，由此产生的温度应力使双折射强烈依赖于温度。

E.

GVD, which causes different frequencies of light to travel at different group velocities, is a factor crucial in the design of telecommunications systems and in all kinds of nonlinear optical experiments. PCF offers greatly enhanced control of the magnitude and sign of the GVD as a function of wavelength. In many ways, this represents an even greater opportunity than a mere enhancement of the effective nonlinear coefficient.
引起光的不同频率以不同的群速度传播的色散，是设计电信系统和各种非线性光学实验中至关重要的因素。PCF 在波长函数中提供了对 GVD 的大小和符号的极大控制。在许多方面，这代表了比仅仅增强有效非线性系数更大的机会。

Solid Core: As the optical frequency increases, the GVD in SMF changes sign from anomalous to normal at . In solid-core PCF, as the holes get larger, the core becomes more and more isolated, until it resembles an isolated strand of silica glass (Fig. 15). If the whole structure is made very small (core diameters less than have been made), the zero-dispersion point of the fundamental guided mode can be shifted to wavelengths in the visible [17], [18]. For example, the PCF in Fig. 14(b) has zero dispersion at .
固体芯：随着光频率的增加，SMF 中的 GVD 在异常到正常的处变号。在实心芯 PCF 中，随着孔的变大，芯变得越来越孤立，直到类似于一根孤立的二氧化硅玻璃（图 15）。如果整个结构被制得非常小（核直径小于已经制成），基本导模的零色散点可以被移至可见光波长[17]，[18]。例如，图 14(b)中的 PCF 在处有零色散。

By careful design, the wavelength dependence of the GVD can also be reduced in PCFs at much lower air-filling fractions. Fig. 16 shows the flattened GVD profiles of three PCFs with
通过精心设计，PCF 中 GVD 的波长依赖性也可以在更低的空气填充分数下减小。图 16 显示了三个 PCF 的扁平化 GVD 曲线。

Fig. 16. GVD profiles, against wavelength, for three different PCFs designed to have low-level ultraflattened GVD [69], [70]. The curve for Corning SMF-28 is included for comparison.
图 16。设计为具有低水平超扁平 GVD 的三种不同 PCF 的 GVD 曲线，针对波长。考林 SMF-28 的曲线用于比较。

Fig. 17. Measured attenuation and GVD spectra for a hollow-core PCF designed for

transmission [71]. The core is slightly elliptical, so the dispersion in each eigenstate of polarization is different. The inset shows the measured near-field distribution at

wavelength (the positions of the airglass boundaries are marked with orange lines).
图 17。设计用于

传输的中空 PCF 的测量衰减和 GVD 谱[71]。核心略呈椭圆形，因此每个偏振本征态中的色散不同。插图显示了

波长处的测量近场分布（空气玻璃边界的位置用橙色线标记）。

cores several micrometers in diameter [69], [70]. These fibers operate in the regime where SMF is multimoded. Although the fundamental modes in both SMF and PCF have similar dispersion profiles, the presence of higher order modes (not guided in the PCF, which is ESM) makes the use of SMF impractical.
直径几微米的多芯心[69]，[70]。这些光纤在 SMF 为多模的情况下运行。虽然 SMF 和 PCF 中的基模具有类似的色散特性，但高阶模式的存在（在 PCF 中不导波，即 ESM）使得使用 SMF 不切实际。

A further degree of freedom in GVD design may be gained by working with multicomponent glasses such as Schott SF6, where the intrinsic zero-dispersion point occurs at

[28]. In a highly nonlinear small-core PCF, this shifts the whole dispersion landscape to longer wavelengths than in a silicabased PCF with the same core size and geometry.
通过使用多组分玻璃（如 Schott SF6）可以获得 GVD 设计的进一步自由度，其中固有的零色散点出现在

处[28]。在高非线性小芯 PCF 中，这将使整个色散景观向比具有相同核心尺寸和几何形状的硅基 PCF 更长的波长偏移。

Hollow Core: Hollow-core fiber behaves in many respects rather like a circular-cylindrical hollow metal waveguide, which has anomalous dispersion (the group velocity increases as the frequency rises). The main difference, however, is that the dispersion changes sign at the high-frequency edge due to the approach of the photonic band edge and the weakening of the confinement (Fig. 17) [71].
空芯：空芯光纤在许多方面表现得像一个圆柱形空心金属波导，具有异常色散（随着频率升高，群速度增加）。然而，主要区别在于，由于光子带边缘的接近和约束的减弱，色散在高频边缘改变符号（图 17）[71]。

F. Attenuation Mechanisms
F. 衰减机制

An advantage common to all fibers is the very large extension ratio from preform to fiber, which has the effect of smoothing out imperfections, resulting in a transverse structure that is extremely invariant with distance along the fiber. This is the chief reason for the ultralow attenuation displayed by fibers compared to other waveguide structures. In PCFs, the losses are governed by two main parameters, namely the fraction of light in glass and the roughness at the glass-air interfaces. The light-in-glass fraction can be controlled by judicious design and ranges from close to

in solid-core fibers to less than

in the best hollow-core fibers.
所有光纤共同的优势是从坯棒到光纤的非常大的延伸比，这样做的效果是平滑不完美，导致横向结构在光纤沿距离非常不变。这是光纤显示出的超低衰减与其他波导结构相比的主要原因。在 PCF 中，损耗由两个主要参数控制，即玻璃中的光比例和玻璃-空气界面的粗糙度。光在玻璃中的比例可以通过慎重设计来控制，范围从接近

的实心芯光纤到最好的空芯光纤的小于

。

Absorption and Scattering: The best loss in solid-core PCF, which was reported by a group in Japan, stands at at , with a Rayleigh-scattering coefficient of . One hundred kilometers of this fiber was used in the first PCF-based penalty-free dispersionmanaged soliton transmission system at [72]. The slightly higher attenuation compared to SMF is due to roughness at the glass-air interfaces [58].
吸收和散射：固芯 PCF 中最佳损耗是由日本一个团队报告的，达到，在处，具有的瑞利散射系数。这种光纤的长度达到一百公里，在第一个基于 PCF 的无惩罚色散管理孤子传输系统中使用[72]。与 SMF 相比，稍高的衰减是由于玻璃-空气界面的粗糙度[58]。

It is hollow-core PCF, however, that has the greatest potential for extremely low loss, since light travels predominantly in empty (or gas-filled) space. Although the best reported attenuation in hollow-core PCF stands at

[55], values below

or even lower seem feasible with further development of the technology. The prospect of improving on conventional fiber while at the same time greatly reducing the nonlinearities associated with a solid glass core is intriguing. By using IR glasses, transmission can be extended into the IR [73], and a recent work shows that silica hollow-core PCF can even be used with acceptable levels of loss in the mid-IR [74] due to the very low overlap between the optical field and the glass.
然而，中空芯 PCF 具有极低损耗潜力，因为光主要在空（或充气）空间中传播。尽管中空芯 PCF 中报告的最佳衰减为

[55]，但通过进一步发展技术，可以实现低于

甚至更低的值。在改善传统光纤的同时，大大减少与固体玻璃芯相关的非线性的前景是令人着迷的。通过使用红外玻璃，传输可以延伸到红外[73]，最近的研究表明，二氧化硅中空芯 PCF 甚至可以在中红外波段中以可接受的损耗水平使用[74]，这是由于光场与玻璃之间的重叠非常低。

In the latest hollow-core silica PCF, with loss levels approaching

, very small effects can contribute significantly to the attenuation floor. The ultimate loss limit in such fibers is determined by surface roughness caused by thermally-driven capillary waves, which are present at all length scales. These interface ripples freeze in when the fiber cools, introducing high scattering losses for modes that are concentrated at the interfaces such as surface modes guided on the edge of the core. The pure core mode does not itself "feel" the ripples very strongly, except at anticrossing wavelengths where it becomes phase-matched to surface modes, causing light to move to the surface and experience enhanced scattering.
在最新的中空核硅 PCF 中，损耗水平接近

在

，非常小的效应可以显著影响衰减底线。在这种光纤中，表面粗糙度由热驱动的毛细波引起，这些波在所有长度尺度上都存在，决定了这种光纤的最终损耗极限。这些界面波纹在光纤冷却时凝固，为集中在界面上的模式（如在核心边缘引导的表面模式）引入高散射损耗。纯核心模式本身并不会强烈“感受”到这些波纹，除非在反交叉波长处，它与表面模式相位匹配，导致光线移动到表面并经历增强的散射。

The result is a transmission spectrum consisting of windows of high transparency punctuated with bands of high attenuation (Fig. 18). This picture has been confirmed by measurements of the surface roughness in hollow-core PCFs, the angular distribution of the power scattered out of the core, and the wavelength dependence of the minimum loss of fibers drawn to different scales [55]. The thin glass shell surrounding the hollow core can be designed to be antiresonant with the core mode, permitting further exclusion of light from the glass [75].
结果是由高透明窗口组成的传输光谱，其中穿插着高衰减带（图 18）。这一图像已通过对空芯 PCF 表面粗糙度的测量、从芯部散射出的功率的角分布以及拉伸到不同尺度的光纤的最小损耗的波长依赖性得到确认[55]。包围空心芯的薄玻璃壳可以设计成与芯模式反共振，进一步排除玻璃中的光[75]。

Ignoring material dispersion, the whole transmission landscape shifts linearly in wavelength in proportion to the overall
忽略材料色散，整个传输景观在波长上线性移动，与整体成比例

Fig. 18. Attenuation spectrum of a typical ultra-low-loss hollow-core PCF designed for operation in the

telecommunications band (see micrograph in Fig. 9).
图 18。典型超低损耗空芯 PCF 的衰减光谱，设计用于

通信波段的操作（请参见图 9 中的显微镜照片）。

size of the structure-a consequence of Maxwell's equations. This means that the smallest loss at a given wavelength will be obtained by drawing a fiber to a particular diameter. The optical overlap with the surface roughness scales inversely with the size with the fiber, and the scattering itself may be regarded as being governed by the density of states into which scattering can occur, which, in three dimensions, scales as

. Thus, the wavelength of minimum loss scales as

in contrast to the

dependence of Rayleigh scattering in bulk glass.
结构的尺寸-麦克斯韦方程的结果。这意味着在给定波长下，最小损耗将通过拉伸光纤到特定直径来获得。与表面粗糙度的光学重叠与光纤的尺寸成反比，而散射本身可能被视为受到散射可能发生的态密度的控制，这在三维中按

缩放。因此，最小损耗的波长按

缩放，与体积玻璃中瑞利散射的

依赖性形成对比。

Bend Loss: Conventional fibers suffer additional loss if bent beyond a certain critical radius , which depends on wavelength, core-cladding refractive-index step, and, most notably, the third power of core radius [38]. For wavelengths longer than a certain value (the "long-wavelength bend edge"), all guidance is effectively lost.
弯曲损耗：传统光纤如果弯曲超过一定的临界半径，将遭受额外损耗，这取决于波长、芯-包层折射率差以及尤其是芯径的三次方 [38]。对于波长长于某个值（“长波长弯曲边缘”），所有的引导实际上都会丢失。

A starting point for understanding bend loss in solid-core ESM-PCF (perhaps the most interesting case) is the longwavelength limit. ESM behavior occurs when

, which sets the highest air-filling fraction at

and yields an area-averaged cladding refractive index of 1.388 (silica index of 1.444)—valid in the limit

. This index step is some ten times higher than in Corning SMF-28, making ESM-PCF relatively much less susceptible to bend loss at long wavelengths. For a step-index fiber with a Ge-doped core,

would be needed to reach the same index step (assuming 0.0014 index change per

[4]). The result is that the long-wavelength bend edge in ESM-PCF is in the IR beyond the transparency window of silica glass, even when the core radius is large [76].
在固芯 ESM-PCF 中理解弯曲损耗的起点（也许是最有趣的案例）是长波极限。当

时，ESM 行为发生，这将最高空气填充分数设置为

，并产生一个面积平均包层折射率为 1.388（硅折射率为 1.444）—在极限

有效。这个折射率步跃比 Corning SMF-28 高出大约十倍，使得 ESM-PCF 在长波长时相对不太容易受到弯曲损耗的影响。对于一个具有 Ge 掺杂核心的阶跃折射率光纤，需要

的

才能达到相同的折射率步跃（假设每

[4]变化 0.0014 折射率）。结果是 ESM-PCF 中的长波长弯曲边缘在 IR 范围内，超出了硅玻璃的透明窗口，即使核心半径很大[76]。

ESM-PCF also exhibits a short-wavelength bend edge caused by bend-induced coupling from the fundamental to higher order modes, which, of course, leak out of the core [15]. The critical bend radius for this loss varies as
ESM-PCF 还表现出短波长弯曲边缘，由于弯曲引起的基模到高阶模式的耦合，当然，这些模式会泄漏出核心[15]。这种损耗的临界弯曲半径因材料而异。

compared to

for SMF. The reciprocal dependence on

makes it inevitable for a short-wavelength bend edge to appear in ESM-PCF. Following [76] in taking the prefactor in (10) as unity,

can be plotted against wavelength for different
与 SMF 相比，ESM-PCF 对

的相互依赖性使得在短波长弯曲边缘中不可避免地出现。根据 [76] 中将 (10) 中的前置因子视为单位，可以绘制

针对不同波长的图表

Fig. 19. Short-wavelength critical bend radii for ESM-PCF with

plotted against vacuum wavelength for different values of hole spacing (approximately equal to the core radius). As the wavelength increases at constant core size, a step-index fiber with the same core-cladding index difference as the PCF in the long-wavelength limit [using (5)] becomes single mode when the curves enter the shaded region. The step-index fiber is multimode over wide parameter ranges where ESM-PCF has negligible short-wavelength bend loss.
图 19. ESM-PCF 的短波长临界弯曲半径，其中

针对不同孔间距（大约等于芯径）的真空波长绘制。随着波长在恒定芯径的情况下增加，当曲线进入阴影区域时，具有与 PCF 相同芯包层折射率差异的阶跃指数光纤在长波长极限下 [使用 (5)] 变为单模。在 ESM-PCF 具有可忽略的短波长弯曲损耗的宽参数范围内，阶跃指数光纤是多模的。

values of core radius (

interhole spacing). This is illustrated in Fig. 19. A step-index fiber, with the same core-cladding index step as ESM-PCF in the long-wavelength limit, is multimode over wide parameter ranges where ESM-PCF has negligible bend loss.
芯径值（

孔间距）。如图 19 所示。在长波长极限下，具有与 ESM-PCF 相同芯包层折射率差异的阶跃指数光纤在宽参数范围内是多模的，在此范围内 ESM-PCF 具有可忽略的弯曲损耗。

In contrast, hollow-core PCF is experimentally very insensitive to bend loss-in many cases, no appreciable drop in transmission is observed until the fiber breaks. This is because the effective depth of "potential well" for the guided light (see Section VI-C), which is given by the distance

between the edges of the PBG, is substantially larger than that in SMF.
相比之下，中空芯光子晶体光纤在实验中对弯曲损耗非常不敏感，在许多情况下，直到光纤断裂，才会观察到传输没有明显下降。这是因为引导光的“势阱”的有效深度（见第 VI-C 节）由 PBG 的边缘之间的距离

给出，远大于 SMF 中的深度。

Confinement Loss: The photonic-crystal cladding in a realistic PCF is, of course, finite in extent. For a guided mode, the Bloch waves in the cladding are evanescent, just like the evanescent plane waves in the cladding of a conventional fiber. If the cladding is not thick enough, the evanescent field amplitudes at the cladding/coating boundary can be substantial, causing attenuation. In the solid core case for small values of , the resulting loss can be large unless a sufficiently large number of periods is used [69].
限制损耗：实际 PCF 中的光子晶体包层当然是有限的。对于引导模式，包层中的布洛赫波是耗散的，就像传统光纤包层中的耗散平面波一样。如果包层不够厚，包层/涂层边界处的耗散场幅值可能很大，导致衰减。在小值的固体芯情况下，除非使用足够大数量的周期，否则产生的损耗可能很大。

Very similar losses are observed in hollow-core fibers, where the "strength" of the PBG (closely related to its width in

) determines how many periods are needed to reduce confinement loss to acceptable levels. Numerical modeling is useful for giving an indication of how many periods are needed to reach a required loss level. The cladding field intensity in the ultra-lowloss PCF reported in [55] falls by

per period, reaching

at the edge of the photonic-crystal region.
空芯光纤中观察到非常相似的损耗，其中 PBG 的“强度”（与其宽度密切相关）决定了需要多少周期才能将束缚损耗降低到可接受的水平。数值建模对于指示需要多少周期才能达到所需的损耗水平是有用的。在[55]中报道的超低损耗 PCF 中，包层场强度每个周期下降

，在光子晶体区域边缘达到

。

G. Kerr Nonlinearities G. Kerr 非线性

The ability to enhance or reduce the effective Kerr nonlinearity and, at the same time, control the magnitude and wavelength dependence of the GVD makes PCF a versatile vehicle for studies of effects such as four-wave mixing, self-phase modulation, modulation instability, soliton formation, and stimulated Raman scattering. To take account of the differing proportions
增强或减少有效 Kerr 非线性的能力，并同时控制 GVD 的幅度和波长依赖性，使 PCF 成为研究四波混频、自相位调制、调制不稳定性、孤子形成和受激拉曼散射等效应的多功能工具。考虑不同比例
of light in glass and air, it is necessary to redefine the

coefficient [77] as follows:
在玻璃和空气中的光传输，需要重新定义

系数[77]如下：

In (11),

is the nonlinear refractive index of material

(i.e.,

for air,

for silica, and an order of magnitude or more higher for multicomponent glasses),

is defined as the effective area for the light in material

, and

is the effective nonlinear index for the fiber (core area

).
在（11）中，

是材料

的非线性折射率（即，空气为

，二氧化硅为

，多组分玻璃的非线性折射率则高一个数量级或更高），

被定义为材料

中光的有效面积，

是光纤的有效非线性折射率（核心面积

）。

The highest nonlinearity available in conventional step-index fibers is

[78]. By comparison, a solid-core PCF similar to the one in Fig. 14(b) but with a core diameter of

has a nonlinearity of

, and values as high as

have been measured for PCFs made from multicomponent glasses

.
传统的阶跃折射率光纤中可获得的最高非线性为

在

[78]。相比之下，类似于图 14(b)中的实心芯 PCF，但核心直径为

的 PCF 在

处的非线性为

，而由多组分玻璃制成的 PCF 的非线性值可高达

在

处已被测量。

In complete contrast, hollow-core PCF has extremely low levels of nonlinearity due to the small overlap between the glass and the light. In a recent example, a fiber was reported with an effective nonlinear refractive index of

(roughly 300 times smaller than in silica glass), and a nonlinear coefficient

(some

smaller than in a typical highly nonlinear solidcore PCF)

.
完全相反的是，空芯 PCF 由于玻璃和光之间的重叠很小，非线性水平极低。最近的一个例子中，报道了一种有效非线性折射率为

的光纤（大约比二氧化硅玻璃小 300 倍），非线性系数

（比典型高非线性固芯 PCF 小约

）

。

Although the level of nonlinearity is clearly important, the actual nonlinear effects that appear in a particular case are also strongly dependent on the magnitude, sign, and wavelength dependence of the GVD (see Section VII-D) as well as on the characteristics of the pump laser [81].
尽管非线性水平显然很重要，但在特定情况下出现的实际非线性效应也强烈依赖于 GVD 的大小、符号和波长依赖性（见第 VII-D 节），以及泵浦激光的特性[81]。

VII. AppliCATIONS 七、应用

The diversity of new or improved features beyond conventional fibers means that PCFs are finding an increasing number of applications in ever-widening areas of science and technology.
新型或改进的功能的多样性超越传统纤维，意味着 PCF 在科学技术的不断扩大领域中找到了越来越多的应用。

A. High-Power and-Energy Transmission
A. 高功率和高能量传输

ESM-PCF's ability to remain single mode at all wavelengths where it guides, and for all scales of structure, suggests that it should have superior power-handling properties-the core area can be increased without the penalty of introducing higher order guided modes [16]. The ability to transmit much higher power in a single mode has a major impact in the field of laser machining and high-power fiber lasers and amplifiers. The key issue is bend loss, and as we have seen, it turns out that PCF offers a wider bandwidth of useful single-mode guidance than high-

SMF, because it can operate in the multimode regime of SMF while remaining single mode (Section VI-B1). This also allows the long-wavelength bend edge to be moved to longer wavelengths.
ESM-PCF 在引导的所有波长和所有结构尺度下始终保持单模的能力，表明它应该具有优越的功率处理特性-核心区域可以增加而不会引入更高阶的引导模式的惩罚[16]。在单模中传输更高功率在激光加工和高功率光纤激光器和放大器领域产生了重大影响。关键问题是弯曲损耗，正如我们所看到的，PCF 提供了比高

SMF 更广泛的有用单模引导带宽，因为它可以在 SMF 的多模区域中运行同时保持单模（第 VI-B1 节）。这也允许将长波长弯曲边缘移至更长波长。

Hollow-core PCF is also an excellent candidate for transmitting high continuous-wave power as well as ultrashort pulses with very high peak powers. Solitons have been reported at
空芯 PCF 也是传输高连续波功率以及超短脉冲具有非常高峰值功率的极佳选择。已经报道了孤子。

Fig. 20. Scanning electron micrograph of an air-clad fiber (Yb-doped core circled). The structural parameters are listed as follows: hole-spacing

;

; core diameter

; 54 webs of width

; inner cladding diameter

across flats;

across corners [87].
图 20。空气包层光纤的扫描电子显微镜图（钇掺杂的芯部已圈出）。结构参数如下所列：孔间距

；

；芯直径

；54 个宽度为

的薄膜；内包层直径

横跨平面；

横跨角点 [87]。

with durations of

and peak powers of

[82] and at

using a Ti:sapphire laser [80]. The soliton energy is, of course, determined by the effective value of

and the magnitude of the anomalous GVD. As mentioned in Section VI-E2, the GVD changes sign across the band gap, permitting choice of normal or anomalous dispersion, depending upon the application [71]. Further studies explore the ultimate power-handling capacity of hollow-core PCF [59], [83], [84].

持续时间为

，峰值功率为

[82]，在

使用钛蓝宝石激光器 [80]。孤子能量当然由

的有效值和反常群速色散的大小决定。如第 VI-E2 节所述，群速色散在带隙中改变符号，允许根据应用选择正常或反常色散 [71]。进一步研究探讨了中空芯光子晶体纤维的最终功率处理能力 [59]，[83]，[84]。

B. Fiber Lasers and Amplifiers
B. 光纤激光器和放大器

PCF lasers can be straightforwardly produced by incorporating a rare-earth-doped cane in the preform stack. Many different designs can be realized such as cores with ultralarge mode areas for high power and structures with multiple lasing cores [85]. Cladding-pumping geometries for ultrahigh power can be fashioned by incorporating a second core (much larger and multimode) around a large off-center ESM lasing core. Using microstructuring techniques, this "inner cladding waveguide" can be suspended by connecting it to an outer glass tube with very thin webs of glass (see Fig. 20) [86]. This results in a very large effective index step and, thus, a high numerical aperture (

), making it easy to launch and guide light from high-power diode-bar pump lasers, which typically have poor beam quality. The multimode pump light is efficiently absorbed by the lasing core, and high-power single-mode operation can be achieved [87]-[89]. In one report, a microchip-laser-seeded Yb-doped PCF amplifier generated diffraction-limited 0.45 -nsduration pulses with a peak power of

and a peak spectral brightness greater than

[90].
PCF 激光器可以通过在预制坯料中加入稀土掺杂的藤条来直接制造。可以实现许多不同的设计，例如具有超大模场的核心用于高功率和具有多个激光核心的结构[85]。通过在大偏心 ESM 激光核心周围加入第二个核心（大得多且多模式），可以制作用于超高功率的包层泵浦几何结构。使用微结构化技术，这个“内包层波导”可以通过用非常薄的玻璃网将其连接到外玻璃管来悬浮（见图 20）[86]。这导致非常大的有效折射率步跃，因此具有高数值孔径（

），使得很容易从典型具有较差光束质量的高功率二极管激光泵发射和引导光线。多模泵浦光被激光核心高效吸收，可以实现高功率单模运行[87]-[89]。在一份报告中，一款由微片激光引发的 Yb 掺杂 PCF 放大器产生了具有 0.45 纳秒持续时间的衍射极限脉冲，峰值功率为

，峰值光谱亮度大于

[90]。

Hollow-core PCF, with its superior power handling and designable GVD, is ideal as the last compression stage in chirped-pulse amplification schemes. This permits operation at power densities that would destroy conventional glass-core fibers [91], [92].
中空芯光子晶体光纤（Hollow-core PCF）以其优越的功率处理能力和可设计的色散特性，在啁啾脉冲放大方案中作为最后的压缩级是理想的。这使得可以在会摧毁传统玻璃芯光纤的功率密度下运行[91]，[92]。

C. Intrafiber Devices-Cutting and Joining
C. 光纤内部器件-切割和连接

As PCF becomes more widely used, there is an increasing need for effective cleaves, low-loss splices, multiport couplers, intrafiber devices, and mode-area transformers. The air holes provide an opportunity not available in standard fibers: the creation of dramatic morphological changes by altering the hole
随着 PCF 的广泛应用，对有效的切割、低损耗的熔接、多端口耦合器、光纤内器件和模场变换器的需求日益增加。空气孔提供了一个在标准光纤中无法实现的机会：通过改变孔的位置来产生显著的形态变化。
size by collapse (under surface tension) or inflation (under internal overpressure) when heating to the softening temperature of the glass. Thus, not only can the fiber be stretched locally to reduce its cross-sectional area, but also, the microstructure can itself be radically altered.
当加热至玻璃软化温度时，光纤可以通过表面张力引起的坍塌或内部过压引起的膨胀来改变尺寸。因此，光纤不仅可以在局部拉伸以减小其横截面积，而且，其微结构本身也可以发生根本性改变。

Cleaving and Splicing: PCFs cleave cleanly using standard tools, showing slight end face distortion only when the core crystal is extremely small (interhole spacing ) and the air-filling fraction is very high ( ). Solid glass end caps can be formed by collapsing the holes (or filling them with sol-gel glass) at the fiber end to form a coreless structure through which light can be launched into the fiber. A solidcore PCF can be fusion spliced successfully both to itself and to step-index fiber using resistive heating elements (electric arcs do not allow sufficient control). The two fiber ends are placed in intimate contact and heated to softening point. With careful control, they fuse together without distortion. Provided the mode areas are well matched, splice losses of can normally be achieved, except when the core is extremely small (i.e., less than ). Fusion splicing of hollow-core fibers is feasible when there is a thick solid glass outer sheath [e.g., as depicted in Fig. 9(a)], although very low splice losses can be obtained simply by placing identical fibers end-to-end and clamping them (the index-matching "fluid" for hollow-core PCF is vacuum).
割裂和拼接：PCFs 使用标准工具割裂干净，只有当核晶体极小（间隙）且充气比例很高（）时才会显示轻微的端面畸变。可以通过在光纤端部折叠孔（或用溶胶-凝胶玻璃填充孔）形成固体玻璃端帽，形成一个无核结构，光可以被引入光纤中。固体核 PCF 可以通过电阻加热元件（电弧无法提供足够控制）成功地与自身和阶跃折射率光纤熔接。将两根光纤端部置于亲密接触并加热至软化点。通过仔细控制，它们可以融合在一起而不会变形。只要模式面积匹配良好，通常可以实现的熔接损耗，除非核心极小（即小于）。当有厚实的固体玻璃外护套时（例如图 9(a)所示），可以实现中空芯光纤的熔接，尽管只需将相同的光纤端对端放置并夹紧即可获得非常低的熔接损耗（中空芯 PCF 的折射率匹配“流体”为真空）。

The ability to hermetically splice a gas-filled hollow-core PCF to SMF has made it possible to produce in-line gas cells for stimulated Raman scattering in hydrogen and frequency measurement and stabilization (using acetylene). These developments may lead for the first time to practical miniature gas-laser devices that could even be coiled up inside a credit card

.
能够将充满气体的中空核光子晶体光纤与单模光纤进行气密拼接，使得可以制造用于氢气受激拉曼散射和频率测量与稳定（使用乙炔）的内置气体池。这些进展可能首次导致实用的微型气体激光器器件，甚至可以卷曲在信用卡内。

Mode Transformers: In many applications, it is important to be able to change the mode area without losing light. This is done traditionally using miniature bulk optics-tiny lenses precisely designed to match to a desired numerical aperture and spot size. In PCFs, an equivalent effect can be obtained by scanning a heat source (flame or carbon dioxide laser) along the fiber. This causes the holes to collapse, with the degree of collapse depending on the dwell time of the heat. Drawing the two fiber ends apart at the same time provides additional control. Graded transitions can fairly easily be made-mode diameter reductions as high as have been realized with low loss.
模式变换器：在许多应用中，重要的是能够改变模式面积而不损失光线。传统上使用微型体积光学元件-精确设计的微小透镜来匹配所需的数值孔径和斑点大小来实现这一点。在中空核光子晶体光纤中，可以通过沿着光纤扫描热源（火焰或二氧化碳激光器）来获得等效效果。这会导致孔隙坍塌，坍塌程度取决于热源停留时间。同时将两个光纤端拉开还提供了额外的控制。可以相当容易地制造渐变过渡-模式直径减小高达的情况已经实现，且损耗较低。

Ferrule methods have been developed for making low-loss interfaces between conventional SMFs and PCFs [94]. Adapted from the fabrication of PCF preforms from stacked tubes and rods, these techniques avoid splicing and are versatile enough to interface to virtually any type of index-guiding silica PCF (Fig. 21). They are effective for coupling light into and out of all of the individual cores of a multicore fiber without input or output crosstalk. The technique also creates another opportunity-the use of taper transitions to couple light between a multimode fiber and several SMFs. When the number of SMFs matches the number of spatial modes in the multimode fiber, the transition can have low loss in both directions. This means that the high performance of SMF devices can be reached in multimode systems, for example, a multimode fiber
已开发了法兰方法，用于在传统 SMF 和 PCF 之间制造低损耗接口[94]。这些技术改编自从堆叠的管和棒制作 PCF 坯料的方法，这些技术避免了拼接，并且足够灵活，可以与几乎任何类型的索引导向硅 PCF 接口（图 21）。它们有效地将光耦合到多芯光纤的所有单个芯中，并且没有输入或输出串扰。该技术还创造了另一个机会-使用锥形过渡器在多模光纤和几根 SMF 之间耦合光。当 SMF 的数量与多模光纤中的空间模式数量相匹配时，过渡器在两个方向上的损耗都很低。这意味着可以在多模系统中实现 SMF 器件的高性能，例如，多模光纤

Fig. 21. Schematic (not to scale) of the ferrule tapering technique for creating low-loss interfaces between standard step-index fiber and solid-core PCF. (a) Finished device. (b) Detail of the transition (after [94]).
图 21。用于在标准阶跃折射率光纤和实心芯 PCF 之间创建低损耗接口的法兰锥形技术示意图（非比例）。（a）成品器件。（b）过渡的细节（参考[94]）。

filter with the transmission spectrum of an SMF Bragg grating [95], which is a device that has applications in earth-based astronomy, where the high throughput of a multimode fiber can be retained while unwanted atmospheric emission lines are filtered out.
使用 SMF Bragg 光栅的传输光谱进行滤波[95]，这是一种在地面天文学中应用广泛的设备，可以在保留多模光纤的高吞吐量的同时，过滤掉不需要的大气发射线。

A further degree of freedom may be gained by pressurizing the holes during the taper process [96]. The resulting hole inflation permits radical changes in the guidance characteristics. It is possible, for example, to transform a PCF, with a relatively large core and small air-filling fraction, into a PCF with a very small core and a large air-filling fraction, the transitions having very low loss

.
在锥形过程中加压孔道可以获得更多的自由度[96]。由此产生的孔膨胀允许引导特性发生根本性变化。例如，可以将具有相对较大芯和小气充分数的 PCF 转变为具有非常小芯和大气充分数的 PCF，这些转变具有非常低的损耗

。

In-Fiber Devices: Precise use of heat and pressure induces large changes in the optical characteristics of PCFs, giving rise to a whole family of new intrafiber components. Microcouplers can be made in a PCF with two optically isolated cores by collapsing the holes to allow the mode fields to expand and interact with each other, creating local coupling [98]. Longperiod gratings, which scatter the core light into cladding modes within certain wavelength bands, can be made by periodic modulation of hole size [99]. By rocking a birefringent PCF to and fro while scanning a carbon dioxide laser along it, socalled "rocking filters" can be made, which transfer power from one polarization state to the other within a narrow band of wavelengths [100]. All these components have one great advantage over equivalent devices made in conventional fiber: Because they are formed by permanent changes in morphology, they are highly stable with temperature and over time.
光纤器件：精确使用热量和压力可引起 PCF 光学特性的巨大变化，从而产生一整套新的光纤内部组件。通过将孔塌陷以允许模场扩展并相互作用，可以在 PCF 中制造具有两个光学隔离核心的微耦合器，从而产生局部耦合[98]。通过周期性调制孔径大小，可以制造将核心光散射到特定波长段内包层模式的长周期光栅[99]。通过摇摆双折射 PCF 并沿其扫描二氧化碳激光，可以制造所谓的“摇摆滤波器”，在窄波长段内将功率从一种偏振态转移到另一种[100]。所有这些组件与在传统光纤中制造的等效器件相比有一个巨大优势：因为它们是通过形态的永久变化形成的，所以它们在温度和时间上非常稳定。

Heating and stretching PCF can result in quite remarkable changes in the scale of the micro/nanostructures without significant distortion. Recently, a solid-core PCF was reduced five times in linear scale, resulting in a core diameter of

(Fig. 22). This permitted the formation of a PCF with a zero-dispersion wavelength that matched the

emission wavelength of a frequency-doubled Nd:YAG laser [101] (this is important for SC generation-see Section IV-A). A further
加热和拉伸 PCF 可以在微/纳米结构的尺度上产生相当显著的变化，而不会出现明显的失真。最近，一个实心芯 PCF 在线性尺度上缩小了五倍，导致核直径为

（图 22）。这使得形成了一个零色散波长的 PCF，与倍频 Nd:YAG 激光器的

发射波长相匹配[101]（这对于 SC 产生很重要-见第四部分-A 节）。进一步

Fig. 22. Scanning electron micrographs (depicted to the same scale) of the fiber cross sections produced by tapering a solid-core PCF. The structures are very well preserved, even down to core diameters of

.
图 22。通过锥形化实心芯 PCF 制备的光纤横截面的扫描电子显微镜照片（按相同比例绘制）。即使到核直径为

，结构也被很好地保留。

compelling advantage of the tapering approach is that it neatly sidesteps the difficulty of launching light into submicrometersized cores; light is launched into the entry port (in this case with a core diameter of

) and adiabatically evolves, with negligible loss, into the mode of the

core.
锥形化方法的一个引人注目的优势是，它巧妙地避开了将光引入亚微米尺寸核心的困难；光被引入入口端口（在这种情况下，核直径为

），并且在无损耗的情况下，绝热演化为

核心的模式。

The nonlinear characteristics are determined by the relative values of the nonlinear length

, where

is the peak power, the dispersion length

, where

is the pulse duration and the effective fiber length

, where

(in units per meter) is the power attenuation coefficient [102]. For a solid-core PCF with

, a peak power of

yields

. For typical values of loss (usually between 1 and

, and the nonlinearity dominates. For dispersion values in the range

and pulse durations

. Since both of these lengths are much longer than the nonlinear length, it is easy to observe strong nonlinear effects.
非线性特性由非线性长度

的相对值决定，其中

是峰值功率，色散长度

，其中

是脉冲持续时间和有效光纤长度

，其中

（以每米为单位）是功率衰减系数[102]。对于具有

的实芯 PCF，峰值功率为

产生

。对于损耗的典型值（通常在 1 和

之间），

，非线性占主导地位。对于色散值在

范围内和脉冲持续时间

。由于这两个长度都比非线性长度长得多，因此很容易观察到强烈的非线性效应。

SC Generation: One of the most successful applications of nonlinear PCF is to SC generation from picosecond and femtosecond laser pulses. When high-power pulses travel through a material, their frequency spectrum can be broadened by a range of interconnected nonlinear effects [103]. In bulk materials, the preferred pump laser is a regeneratively amplified Ti:sapphire system producing high-energy (in millijoules) femtosecond pulses at wavelength and kilohertz repetition rate. SCs have also previously been generated in SMF by pumping at 1064 or [104], with the spectrum broadening out to longer wavelengths mainly due to stimulated Raman scattering (SRS). Then, in 2000, it was observed that highly nonlinear PCFs, which are designed with zero GVD close to , massively broaden the spectrum of low-energy (a few nanojoules) unamplified Ti:sapphire pulses launched into just a few centimeters of fiber [22], [105], [106]. Removing the need for a power amplifier, the hugely increased ( repetition rate as well as the spatial and temporal coherence of the light emerging from the core makes this source unique. The broadening extends to both higher and lower frequencies
SC 产生：非线性 PCF 最成功的应用之一是从皮秒和飞秒激光脉冲中产生 SC。当高功率脉冲穿过材料时，它们的频谱可以通过一系列相互关联的非线性效应而变宽。在体材料中，首选的泵浦激光是再生放大的 Ti:蓝宝石系统，在波长和千赫兹重复率下产生高能量（毫焦耳级）飞秒脉冲。SCs 也曾通过在 1064 或泵浦 SMF 来生成[104]，由于受激拉曼散射（SRS）主要导致频谱向更长波长扩展。然后，在 2000 年，观察到高度非线性的 PCFs，设计为接近的零 GVD，大幅扩展了仅有几厘米光纤中发射的低能量（几纳焦耳）未放大的 Ti:蓝宝石脉冲的频谱。无需功率放大器，大幅增加的重复率以及从核心发出的光的空间和时间相干性使得这个光源独特。这种扩展延伸到更高和更低频率。

Fig. 23. Comparison of the brightness of various broadband light sources (SLED-superluminescent light-emitting diode; ASE-amplified spontaneous emission; SC-supercontinuum). The microchip laser SC spectrum was obtained by pumping at

with 600-ps pulses (updated version of a plot by H. Sabert).
图 23。各种宽带光源亮度的比较（SLED-超辐射光发射二极管；ASE-放大自发辐射；SC-超连续光）。微芯片激光 SC 光谱是通过以 600 皮秒脉冲泵浦获得的（H. Sabert 绘制的更新版本）。

Fig. 24. SC light generated in ESM-PCF from picosecond fiber laser (courtesy of Fianium Ltd.). Total SC power is

for a

pump power (5-ps pulses). Repetition rate is

, and the average spectral power density in the range

is a remarkable

(see also Fig. 23).
图 24。ESM-PCF 中由皮秒光纤激光器产生的 SC 光（Fianium Ltd.提供）。总 SC 功率为

，泵浦功率为

（5 皮秒脉冲）。重复频率为

，在

范围内的平均光谱功率密度为显著的

（另见图 23）。

because four-wave mixing operates more efficiently than SRS when the dispersion profile is appropriately designed. This SC source has applications in optical coherence tomography [107], [108], frequency metrology [109], [110], and all kinds of spectroscopy. It is particularly useful as a bright low-coherence source in measurements of group delay dispersion based on a Mach-Zehnder interferometer.
因为四波混频在适当设计色散曲线时比受激拉曼散射更有效。这种 SC 光源在光学相干断层扫描[107]，[108]，频率计量学[109]，[110]和各种光谱学中有应用。在基于马赫-曾德干涉仪的群时延色散测量中，它特别适用作为明亮的低相干源。

A comparison of the bandwidth and spectrum available from different broadband light sources is shown in Fig. 23; the advantages of PCF-based SC sources are evident. SCs have been generated in different PCFs at 532 [101], 647 [111], 1064 [112], and

[28]. Using inexpensive microchip lasers at 1064 or

with an appropriately designed PCF, compact SC sources are now available with important applications across many areas of science (Fig. 24 shows a picosecond fiber-laser-based system in operation). The use of multicomponent glasses such as Schott SF6 or tellurite glass allows the balance of nonlinearity and dispersion to be adjusted as well as offering extended transparency into the IR [113].
不同宽带光源提供的带宽和频谱的比较显示在图 23 中；基于 PCF 的 SC 光源的优势是显而易见的。在 532 [101]、647 [111]、1064 [112]和

[28]等不同 PCF 中已经产生了 SC。使用廉价的微片激光器在 1064 或

处与设计合适的 PCF，现在可以获得紧凑的 SC 光源，并在科学的许多领域中具有重要应用（图 24 展示了一个基于皮秒光纤激光器的系统正在运行）。使用多组分玻璃，如 Schott SF6 或碲酸玻璃，可以调整非线性和色散的平衡，并提供更广泛的 IR 透明度[113]。

Parametric Amplifiers and Oscillators: In step-index fibers, the performance of optical parametric oscillators and amplifiers is severely constrained owing to the limited scope
参数放大器和振荡器：在步进折射率光纤中，光学参量振荡器和放大器的性能受到严重限制，因为范围有限。
for GVD engineering. In PCFs, these constraints are lifted, permitting flattening of the dispersion profile and control of higher order dispersion terms. The wide range of experimentally available GVD profiles has, for example, allowed studies of ultrashort pulse propagation in the wavelength band with flattened dispersion [69], [70]. The effects of higher order dispersion in such PCFs are subtle [114], [115]. Parametric devices have been designed for pumping at 647, 1064, and , the small effective mode areas offering high gain for a given pump intensity, and PCF-based oscillators synchronously pumped by femtosecond and picosecond pump pulses have been demonstrated at relatively low power levels [116]-[119].
对于 GVD 工程。在 PCFs 中，这些约束被解除，允许调整色散曲线并控制高阶色散项。例如，实验中可用的广泛 GVD 曲线范围已允许对具有平坦色散的波段中的超短脉冲传播进行研究[69]，[70]。在这种 PCFs 中，高阶色散的影响是微妙的[114]，[115]。已经设计了用于在 647、1064 和泵浦的参数器件，小有效模式区域为给定泵浦强度提供高增益，并且基于 PCF 的振荡器已经被演示，同步由飞秒和皮秒泵浦脉冲驱动，且功率水平相对较低[116]-[119]。
Correlated Photon Pairs: The use of self-phase modulation to generate bright sources of correlated photon pairs is unsuccessful in step-index fibers due to high Raman-related noise. This is because for and , the modulational instability sidebands are situated very close to the pump frequency within the Raman gain band of the glass. By flattening the GVD profile and making small, however, higher order GVD terms become important, and gain bands can appear in the normal dispersion regime for and [70]. The relevant expression for the sideband gain (in units per meter), including and , is
相关光子对：利用自相位调制在阶跃指数光纤中产生相关光子对的明亮光源是不成功的，因为高拉曼相关噪声。这是因为对于和，调制不稳定的旁带位于玻璃的拉曼增益带内非常靠近泵浦频率。然而，通过使 GVD 曲线变平并使变小，高阶 GVD 项变得重要，增益带可以出现在和的正常色散区域[70]。旁带增益的相关表达式（每米单位），包括和，是

where

is the nonlinear coefficient (11),

is the pump power, and

is the angular frequency offset from the pump frequency. This expression may be used to show that the sidebands can be widely spaced from the pump frequency, their position and width being controllable by engineering the even-order higher order dispersion terms. It is straightforward to arrange that these sidebands lie well beyond the Raman gain band, reducing the Raman noise and allowing PCF to be used as a compact bright tunable single-mode source of photon pairs with wide applications in quantum communications [120], [121]. In a recent example, a PCF with zero dispersion at

was pumped by a Ti:sapphire laser at

(normal dispersion) [122]. Under these conditions, phase matching is satisfied by signal and idler waves at 587 and

, and 10 million photon pairs per second were generated and delivered via SMF to

avalanche detectors, producing

coincidences per second for a pump power of

. These results point the way to practical and efficient sources of entangled photon pairs that can be used as building blocks in future multiphoton interference experiments.
其中

是非线性系数（11），

是泵浦功率，

是与泵浦频率的角频率偏移。这个表达式可以用来表明侧带可以与泵浦频率相隔甚远，它们的位置和宽度可以通过调节偶数阶高阶色散项来控制。可以很容易地安排这些侧带远远超出拉曼增益带，减少拉曼噪声，并允许 PCF 被用作紧凑明亮可调单模光子对源，在量子通信中有广泛应用[120]，[121]。在最近的一个例子中，一个在

处具有零色散的 PCF 被钛宝石激光器（正常色散）在

处泵浦[122]。在这些条件下，信号和识别波在 587 和

处满足相位匹配，每秒产生并通过 SMF 传输

个瓦伦雪崩探测器，为

的泵浦功率产生

个符合。这些结果指明了实用和高效的纠缠光子对源的途径，这些源可以作为未来多光子干涉实验中的构建模块。

Soliton Self-Frequency Shift Cancellation: The ability to create PCFs with negative dispersion slope at the zerodispersion wavelength (in SMFs, the slope is positive, i.e., the dispersion becomes more anomalous as the wavelength increases) has made it possible to observe Čerenkov-like effects in which solitons (which form on the anomalous side of the dispersion zero) shed power into dispersive radiation at longer wavelengths on the normal side of the dispersion zero. This occurs because higher order dispersion causes the edges of the soliton spectrum to phase-match to linear waves. The result is
孤子自频移消除：在零色散波长处创建具有负色散斜率的 PCF 的能力（在 SMFs 中，斜率为正，即随着波长增加，色散变得更加反常）使得观察到类似Čerenkov 效应成为可能，在这种效应中，孤子（形成在色散零点的反常侧）将功率散射到色散边界上的更长波长处的线性波。这是因为高阶色散导致孤子频谱的边缘与线性波相位匹配。结果是

Fig. 25. (a) Illustration of how a trapped acoustic phonon can phase-match to light at the acoustic cutoff frequency. The result is a quasi-Raman scattering process that is automatically phase-matched. (b) Example of PCF used in studies of Brillouin scattering (core diameter

). (c) Frequencies of full phononic band gaps (in-plane propagation and pure in-plane motion) in the cladding of the PCF in (b) (after [130]).
图 25. (a) 描述了被困的声子如何与声子截止频率相位匹配到光。结果是一种准拉曼散射过程，自动相位匹配。(b) 用于布里渊散射研究的 PCF 示例（芯径

）。(c) PCF 包层中全声子带隙的频率（面内传播和纯面内运动）在（b）中的（参考[130]）。

the stabilization of the soliton self-frequency shift at the cost of gradual loss of soliton energy [123]. The behavior of solitons in the presence of wavelength-dependent dispersion is the subject of many recent studies, for example, [124].
在自频移的稳定化过程中，孤子能量逐渐损失 [123]。在波长相关色散存在的情况下，孤子的行为是许多最近研究的主题，例如，[124]。

E. Brillouin Scattering 布里渊散射

The periodic micro/nanostructuring in ultrasmall core glass-air PCFs strongly alters the acoustic properties compared to conventional SMFs [125]-[128]. Sound can be guided in the core both as leaky and as tightly confined acoustic modes. In addition, the complex geometry and "hard" boundaries cause coupling between all three displacement components (radial, azimuthal, and axial), with the result that each acoustic mode has elements of both shear (S) or longitudinal (L) strain. This complex acoustic behavior strongly alters the characteristics of forward and backward Brillouin scattering.
超小芯玻璃-空气 PCFs 中的周期微/纳米结构与传统 SMFs 相比，强烈改变了声学特性 [125]-[128]。声音可以在芯部以泄漏和紧密限制的声学模式中传导。此外，复杂的几何形状和“硬”边界导致所有三个位移分量（径向、方位角和轴向）之间的耦合，导致每个声学模式都具有剪切（S）或纵向（L）应变的元素。这种复杂的声学行为强烈改变了前向和后向布里渊散射的特性。

Backward Scattering: When a solid-core silica-air PCF has a core diameter of around of the vacuum wavelength of the launched laser light [Fig. 25(b)] and the air-filling fraction in the cladding is very high, the spontaneous Brillouin signal displays multiple bands with Stokes frequency shifts in the range. These peaks are caused by discrete guided acoustic modes, each with different proportions of longitudinal and shear strain strongly localized to the core [129]. At the same time, the threshold power for stimulated Brillouin scattering increases fivefold-a rather unexpected result, since conventionally, one would assume that higher intensities yield lower nonlinear threshold powers. This occurs because the effective overlap between the tightly confined acoustic modes and the optical mode is actually smaller than in a conventional fiber core; the sound field contains a large proportion of shear strain, which does not contribute significantly to changes in refractive index. This is of direct practical relevance to parametric amplifiers, which can be pumped five times harder before stimulated Brillouin scattering appears.
反向散射：当固芯二氧化硅-空气 PCF 的芯直径约为所发射激光光波的真空波长的时[图 25(b)]，且包层中的充气比例非常高时，自发布里渊信号显示出多个带有斯托克斯频移的峰值，位于范围内。这些峰值是由离散的引导声学模式引起的，每个模式具有不同比例的纵向和剪切应变，强烈局部化到芯部[129]。同时，受激布里渊散射的阈值功率增加了五倍-这是一个相当意外的结果，因为传统上，人们会认为更高的强度会导致更低的非线性阈值功率。这是因为紧密限制的声学模式与光学模式之间的有效重叠实际上比传统光纤芯中的要小；声场包含大量的剪切应变，这并不会对折射率的变化产生显著影响。这对于参数放大器具有直接的实际意义，可以在受激布里渊散射出现之前进行五倍强度的泵浦。
Forward Scattering: The very high air-filling fraction in the small-core PCF also permits sound at frequencies of a few gigahertz to be trapped purely in the transverse plane by phononic band gap effects (Fig. 25). The ability to confine acoustic energy at zero axial wavevector means that the ratio of frequency to wavevector becomes arbitrarily large as and, thus, can easily match the value for the light guided in the fiber, i.e., . This permits phase-matched interactions between the acoustic mode and two spatially identical optical modes of different frequencies [130]. Under these circumstances, the acoustic mode has a welldefined cutoff frequency above which its dispersion curve-plotted on an diagram-is flat, which is similar to the dispersion curve for optical phonons in diatomic lattices. The result is a scattering process that is Raman like (i.e., the participating phonons are optical phonon like), even though it makes use of acoustic phonons; Brillouin scattering is turned into Raman scattering, power being transferred into an optical mode of the same order, frequency-shifted from the pump frequency by the cutoff frequency. Used in stimulated mode, this effect may permit the generation of combs of frequencies spaced by at wavelength.
正向散射：小芯光子晶体中非常高的充气分数也允许几个千兆赫频率的声音纯粹被声子带隙效应困住在横向平面中（图 25）。将声能限制在零轴向波矢处的能力意味着频率与波矢的比值会随着变得任意大，因此可以轻松匹配光纤中的导光值，即。这允许声学模式与两个空间相同但频率不同的光学模式之间进行相位匹配交互作用[130]。在这些情况下，声学模式具有明确定义的截止频率，超过该频率，其在图上绘制的色散曲线是平坦的，类似于双原子晶格中的光学声子的色散曲线。结果是一个类似拉曼的散射过程（即，参与的声子类似于光学声子），尽管它利用了声学声子；布里渊散射被转化为拉曼散射，功率被转移到一个与泵浦频率相同阶数的光学模式，其频率由截止频率向下频移。在激励模式下使用，这种效应可能允许以间隔的频率生成频率梳在波长处。

F. Gas-Based Nonlinear Optics
基于气体的非线性光学

It is 100 years since Lord Rayleigh first explained the relationship between depth of focus and focal spot size. The diffraction of light beams in free space presents an apparently insuperable barrier to achieving efficient nonlinear interactions between laser light and low-density media such as gases. The requirements of high intensity, long interaction length, and good-quality (preferably single mode) transverse beam profiles simply cannot be met. A structure conceptually capable of delivering all these requirements simultaneously would be a perfectly guiding hollow-core waveguide supporting a single transverse mode with low attenuation losses. Although, theoretically, this could be realized using a perfect metal, the attenuation in real metals at optical frequencies is much too high, especially when the bore is small enough to yield single-mode operation. A number of conventional approaches have been used to circumvent this problem, including focusing a laser beam into a gas with suitable optics, using a

bore fiber capillary to confine the gas and provide some degree of guidance for the light [131], and employing a gas-filled high-finesse Fabry-Pérot cavity to increase the interaction length [132]. None of these approaches comes close, however, to the performance offered by hollow-core PCF [49]. At a bore diameter of

, for example, a focused free-space laser beam is marginally preferable to a capillary, whereas a hollow-core PCF with a

attenuation is 1 million

more effective. Such huge enhancements are rare, and are leading to dramatic improvements in all sorts of nonlinear laser-gas interactions.
自 Lord Rayleigh 首次解释焦深与焦斑大小之间的关系已经过去了 100 年。在自由空间中光束的衍射似乎构成了一个无法逾越的障碍，以实现激光与低密度介质（如气体）之间的高效非线性相互作用。高强度、长相互作用长度和良好质量（最好是单模）横向光束剖面的要求根本无法满足。一个在概念上能够同时满足所有这些要求的结构将是支持低衰减损耗的单横向模式的完美导波中空芯波导。尽管在理论上，这可以通过使用完美的金属来实现，但在光频率下，实际金属的衰减太高，特别是当孔径足够小以产生单模操作时。一些常规方法已被用于规避这个问题，包括将激光束聚焦到具有合适光学特性的气体中，使用

孔纤维毛细管来限制气体并为光提供一定程度的引导[131]，以及使用充满气体的高精度法布里-珀罗腔来增加相互作用长度[132]。然而，这些方法中没有一种能够接近中空芯光子晶体纤维所提供的性能[49]。例如，在

的孔径下，一个聚焦的自由空间激光束略优于毛细管，而具有

衰减的中空芯光子晶体纤维则比后者有效率高出 100 万

。这种巨大的增强是罕见的，并且正在导致各种非线性激光-气体相互作用的显著改进。

Stimulated Raman Scattering: In 2002, stimulated Raman scattering was reported in a hydrogen-filled hollow-core PCF at threshold pulse energies lower than previously possible [49]. More recently, the threshold power for rotational Raman scattering in hydrogen was reduced by more than a
刺激拉曼散射：2002 年，在一个充满氢气的中空芯光子晶体纤维中，刺激拉曼散射的阈值脉冲能量比以前可能的要低 [49]。最近，氢气中旋转拉曼散射的阈值功率降低了超过一倍。

Fig. 26. Attenuation spectrum of a hollow-core PCF designed for low-loss transmission of

light. The pump, Stokes, and anti-Stokes Raman frequencies for rotational scattering in hydrogen are marked in. The vibrational Raman band lies well outside the low-loss window.
图 26。设计用于低损耗传输

光的中空芯光子晶体的衰减光谱。氢气中旋转散射的泵浦、斯托克斯和反斯托克斯拉曼频率被标记出来。振动拉曼带远远超出了低损耗窗口。

million times in a single-pass geometry, and near-perfect quantum efficiency was achieved [133]. Such gas cells have been hermetically spliced to standard all-solid glass SMFs [93].
在单程几何中，气体细胞的透射次数可达百万次，并且实现了接近完美的量子效率[133]。这些气体细胞已经与标准全固体玻璃 SMF 进行了气密拼接[93]。

The limited wavelength ranges of guidance in hollow core are used to advantage here, making it possible to suppress the normally dominant vibrational Raman signal from hydrogen, and enhance the rotational Raman signal, using the PCF in Fig. 26.
中空芯中的有限波长范围的引导在这里被利用，使得可以抑制氢气中通常占主导地位的振动拉曼信号，并使用图 26 中的光子晶体增强旋转拉曼信号。

High Harmonic Generation: Hollow core PCF is likely to have a major impact in other areas of nonlinear optics, such as -ray generation in noble gases pumped by fs Ti:sapphire laser pulses [135]. The conversion efficiency of this process could be further enhanced by modulating the bore diameter of the core so as to phase-match the light and the -rays (this was recently demonstrated in simple glass capillaries [136]). In a hollow core PCF this could be implemented for example by heat treatment with carbon dioxide laser light.
高次谐波产生: 空芯 PCF 可能会在非线性光学的其他领域产生重大影响，例如由飞秒钛蓝宝石激光脉冲泵浦的惰性气体中产生射线[135]。通过调制芯部的孔径以使光和射线相位匹配，可以进一步提高该过程的转换效率（最近在简单玻璃毛细管中已经证明了这一点[136]）。在空芯 PCF 中，例如可以通过二氧化碳激光光照热处理来实现这一点。
Electromagnetically Induced Transparency: Hollowcore PCF filled with acetylene vapor at low pressure has recently been used to demonstrate electromagnetically induced transparency (for both -type and V-type interactions) over several lines of the -branch of the ro-vibrational overtone band [136], [137]. The well-controlled single-mode environment provided by hollow-core PCF makes effects of this kind much easier to control, monitor, and engineer into a practical devices.
电磁诱导透明: 最近使用填充低压乙炔蒸气的空芯 PCF 来演示电磁诱导透明（对于型和 V 型相互作用）跨越的 -分支的多条线[136]，[137]。空芯 PCF 提供的良好控制的单模环境使得这类效应更容易控制、监测并集成到实际设备中。

G. Telecommunications G. 电信

There are many potential applications of PCFs or PCFbased devices in telecommunications, although whether these will be adopted remains an open question. One application that seems quite close to being implemented is the use of solid-core PCF or "hole-assisted" SMF for fiber-to-the-home, where the lower bend loss is the attractive additional advantage offered by the holey structure [138]. Other possibilities include dispersion-compensating fiber and hollow-core PCF for long-haul transmission. Additional opportunities exist in producing bright sources of correlated photon pairs for quantum cryptography, parametric amplifiers with improved
PCF 或基于 PCF 的设备在电信领域有许多潜在的应用，尽管它们是否会被采纳仍然是一个悬而未决的问题。一个似乎非常接近实现的应用是使用实芯 PCF 或“孔辅助”SMF 进行光纤到家庭的传输，其中较低的弯曲损耗是孔状结构提供的吸引人的额外优势[138]。其他可能性包括用于长距离传输的色散补偿光纤和空芯 PCF。在生产用于量子密码学的相关光子对的明亮光源和具有改进的参数放大器方面存在额外的机会

Fig. 27. Realized (real) and plausible minimum attenuation spectra for hollow-core PCF. The scattering floor is proportional to

, with its overall level depending on the amplitude of the surface roughness. Higher air-filling percentages push the IR absorption edge out to longer wavelengths. As a result, the predicted low-loss window sits at

, with an attenuation of

.
图 27。空芯 PCF 的实现（实际）和可能的最小衰减光谱。散射底部与

成正比，其整体水平取决于表面粗糙度的幅度。较高的充气百分比将红外吸收边推向更长波长。因此，预测的低损耗窗口位于

，衰减为

。

characteristics, highly nonlinear fiber for all-optical switching and amplification, acetylene-filled hollow-core PCF for frequency stabilization at

, and the use of sliced SC spectra as WDM channels. There are also many possibilities for ultrastable in-line devices based on permanent morphological changes in the local holey structure induced by heating, collapse, stretching, or inflation.
特征，用于全光开关和放大的高非线性光纤，用于频率稳定的乙炔填充空芯 PCF，以及使用切片 SC 光谱作为 WDM 通道。还有许多基于加热、坍塌、拉伸或膨胀引起的局部孔结构永久形态变化的超稳定内联设备的可能性。

New Telecommunications Window?: Hollow-core PCF is radically different from solid-core SMF in many ways. This makes it difficult to predict whether it could be successfully used in long-haul telecommunications as a realistic competitor for SMF-28. The much lower Kerr nonlinearities mean that WDM channels can be much more tightly packed without nonlinear crosstalk, and the higher power-handling characteristics mean that more overall power can be transmitted. The effective absence of bend losses is also a significant advantage, particularly for short-haul applications. On the other hand, work still needs to be done to reduce the losses to or lower, and to understand-and control-effects such as polarization mode dispersion, differential group delay, and multipath interference. It is interesting that the low-loss window of a plausible hollow-core PCF is centered at , because light travels predominantly in the hollow regions, completely changing the balance between scattering and IR absorption (Fig. 27) [55].
新的电信窗口？：空芯 PCF 在许多方面与实心芯 SMF 截然不同。这使得难以预测它是否能成功地用于长途电信，作为 SMF-28 的现实竞争对手。更低的克尔非线性意味着 WDM 通道可以更紧密地打包，而不会出现非线性串扰，更高的功率处理特性意味着可以传输更多的总功率。有效消除弯曲损耗也是一个重要优势，特别是对于短距离应用。另一方面，仍需努力将损耗降至或更低，并理解和控制极化模色散、差分群时延和多径干扰等效应。有趣的是，一个可行的空芯 PCF 的低损耗窗口位于，因为光主要在空心区域传播，完全改变了散射和红外吸收之间的平衡（图 27）[55]。
Dispersion Compensation: The large glass-air refractive-index difference makes it possible to design and fabricate PCFs with high levels of GVD. A PCF version of the classical W-profile dispersion compensating fiber was recently reported, offering slope-matched dispersion compensation for SMF-28 fiber at least over the entire C-band (Fig. 28). Dispersion values of imply that only of fiber is needed to compensate for of SMF-28. The fiber was made deliberately birefringent to allow control of polarization mode dispersion .
色散补偿：大的玻璃-空气折射率差使得设计和制造具有高 GVD 水平的 PCFs 成为可能。最近报道了经典 W 型色散补偿光纤的 PCF 版本，为 SMF-28 光纤提供了匹配斜率的色散补偿，至少在整个 C 波段内（图 28）。的色散值意味着只需要的光纤来补偿的 SMF-28。该光纤被故意制成双折射，以允许控制偏振模色散。

H. Laser Tweezers in Hollow-Core PCF
H. 空芯 PCF 中的激光光镊

A focused light beam produces both a longitudinal (accelerating) and a transverse (trapping) force on dielectric microparti-
聚焦光束对介电微粒产生纵向（加速）和横向（捕获）力。

Fig. 28. Performance of a PCF designed to provide slope-matched dispersion compensation for Corning SMF-28 over the C-band.
图 28. 为康宁 SMF-28 在 C 波段提供匹配斜率色散补偿设计的 PCF 性能。

cles [140]. For maximum trapping force, the intensity gradient of the light must be as high as possible. This can be achieved by focusing a high-power laser beam with a high numerical aperture lens. The Rayleigh length for such a tightly focused beam is rather short; furthermore, the high intensity will quickly accelerate the particle out of the trapping zone. Stable trapping of a particle in free space requires the longitudinal force to be balanced either by gravity or by a second (or reflected) beam. It has long been recognized that an alternative configuration offering stability would be a nondiffracting or guided beam. To guide a dielectric particle, the beam would have to be trapped in air rather than in glass, which until very recently was not possible. That did not prevent the demonstration of guidance of both solid particles [141] and atoms [142] along lengths of hollow capillary. Of course, use of a capillary prevented the full exploitation of the possibilities of a hollow waveguide; small core sizes are needed for strong transverse confinement, so that capillary losses are very high, even on length scales of a few millimeters. The same is not true in hollow-core PCF, where core diameters of

can be realized with losses as low as

. Such a small spot size gives a strong transverse trapping force for a given longitudinal force so that it becomes possible to envisage guiding particles even along tightly coiled fibers at reasonably high speeds. The first steps toward achieving this goal were recently taken [143].
为了获得最大的捕获力，光的强度梯度必须尽可能高。这可以通过使用高数值孔径透镜聚焦高功率激光束来实现。对于这样一个紧密聚焦的光束，瑞利长度相当短；此外，高强度会迅速将粒子加速出捕获区域。在自由空间中稳定地捕获粒子需要通过重力或第二个（或反射的）光束来平衡纵向力。长期以来，人们已经认识到，提供稳定性的另一种配置将是非衍射或引导光束。为了引导一个介电粒子，光束必须被困在空气中而不是玻璃中，直到最近才有可能。这并没有阻止展示固体粒子和原子沿着空心毛细管长度的引导。当然，使用毛细管阻止了充分利用空心波导的可能性；需要小的芯径尺寸来实现强横向约束，因此即使在几毫米的长度尺度上，毛细管损耗也非常高。空芯 PCF 中并非如此，其中可以实现

的核直径，损耗低至

。这样小的光斑尺寸为给定的纵向力提供了强大的横向束缚力，因此可以设想即使在非常高速的紧密卷曲光纤上引导粒子。最近已经迈出了实现这一目标的第一步[143]。

I. Optical Sensors I. 光学传感器

Sensing is thus far a relatively unexplored area for PCFs, although the opportunities are myriad, spanning many fields including environmental monitoring, biomedical sensing, and structural monitoring [144]. Multicore PCF has been used in bend and shape sensing [51], [145] and Doppler difference velocimetry [146], double-clad PCF in multiphoton fluorescence measurements in medicine [53], and solid-core PCF for hydrostatic pressure sensing [147].
到目前为止，传感在 PCF 中仍然是一个相对未被探索的领域，尽管机遇多种多样，涵盖许多领域，包括环境监测、生物医学传感和结构监测[144]。多芯 PCF 已被用于弯曲和形状传感[51]，[145]以及多普勒差速测速法[146]，双包层 PCF 用于医学中的多光子荧光测量[53]，实芯 PCF 用于静水压力传感[147]。

VIII. Final Remarks 第八部分总结

In writing this review, I have not been able to cover every topic, nor mention every publication, in a field that is growing by the day. Nevertheless, I hope that my subjective selection
在撰写这篇评论时，我无法涵盖每个主题，也无法提及日益增长的领域中的每一篇出版物。尽管如此，我希望我的主观选择
makes clear that, by moving away from constraints of conventional fiber optics, PCFs have created new opportunities in diverse areas of science and technology. Already there is take-up of PCFs in many fields, including compact SC sources, frequency comb systems for frequency metrology, constrained photo- or biochemistry and microfluidics (making use of the hollow channels [148]), biophotonic and biomedical devices, medical imaging, astronomy, particle delivery, high-power fiber lasers, fiber delivery of high-power laser light in manufacturing, and gas-based fiber devices. The next decade should see many of these applications mature into commercial products.
明确表明，通过摆脱传统光纤的限制，PCFs 在科学技术的各个领域创造了新的机遇。PCFs 已经在许多领域得到应用，包括紧凑型 SC 光源、频率计量的频率组合系统、受限的光或生物化学和微流控（利用中空通道[148]）、生物光子学和生物医学设备、医学成像、天文学、粒子输送、高功率光纤激光器、制造业中高功率激光光纤输送以及气体光纤设备。未来十年应该会看到这些应用中的许多成熟为商业产品。

APPENDIX 附录

The analysis in [15] to estimate

in a triangular lattice of air holes leads to the following equation:
[15]中的分析用于估计空气孔三角晶格中的

，导致以下方程式：

where

(needed to ensure the correct value of

in the long-wavelength limit), and

. The leading root of (13), which is evaluated for

and

, yields the polynomial fit (7).
其中

（需要确保在长波长极限下

的正确值），而

。对（13）的主要根，对

和

进行评估，得到多项式拟合（7）。

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Manuscript received July 7, 2006; revised August 8, 2006.
收到手稿日期为 2006 年 7 月 7 日；修订日期为 2006 年 8 月 8 日。

The author is with the Max-Planck Research Group for Optics, Information, and Photonics, University of Erlangen-Nuremberg, 91058 Erlangen, Germany (e-mail: russell@optik.uni-erlangen.de).
作者隶属于德国埃尔朗根-纽伦堡大学光学、信息和光子学马克斯·普朗克研究小组（电子邮件：russell@optik.uni-erlangen.de）。

Digital Object Identifier 10.1109/JLT.2006.885258
数字对象标识符 10.1109/JLT.2006.885258

, where and are, respectively, the core and cladding refractive indices.
，其中和分别是核心和包层的折射率。

Photonic-Crystal Fibers 光子晶体光纤

Abstract 摘要

LIST OF ACRONYMS 首字母缩略词列表

I. Introduction I. 介绍

A. Outline of the Article文章概要

II. BRIEF HISTORY 二、简史

A. PCFs

B. Bragg Fibers B. 布拉格光纤

III. Fabrication TechniquesIII. 制造技术

A. Design Approach A. 设计方法

IV. Modeling And AnalysisIV. 建模与分析

A. Maxwell's Equations 麦克斯韦方程组

B. Scalar Approximation B. 标量近似

C. Numerical Techniques C. 数值技术

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

A. Maximum Refractive IndexA. 最大折射率

B. Transverse Effective WavelengthB. 横向有效波长

C.

VI. Characteristics OF GUIDANCEVI. 引导特性

A. Resonance and AntiresonanceA. 共振和反共振

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

C. Negative Core-Cladding Index DifferenceC. 负芯-包层折射率差

D. Birefringence D. 双折射

E.

F. Attenuation MechanismsF. 衰减机制

G. Kerr Nonlinearities G. Kerr 非线性

VII. AppliCATIONS 七、应用

A. High-Power and-Energy TransmissionA. 高功率和高能量传输

B. Fiber Lasers and AmplifiersB. 光纤激光器和放大器

C. Intrafiber Devices-Cutting and JoiningC. 光纤内部器件-切割和连接

D. Kerr-Related Nonlinear EffectsD. Kerr 相关的非线性效应

E. Brillouin Scattering 布里渊散射

F. Gas-Based Nonlinear Optics基于气体的非线性光学

G. Telecommunications G. 电信

H. Laser Tweezers in Hollow-Core PCFH. 空芯 PCF 中的激光光镊

I. Optical Sensors I. 光学传感器

VIII. Final Remarks 第八部分 总结

APPENDIX 附录

REFERENCES 参考文献

A. Outline of the Article
文章概要

III. Fabrication Techniques
III. 制造技术

IV. Modeling And Analysis
IV. 建模与分析

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

A. Maximum Refractive Index
A. 最大折射率

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

VI. Characteristics OF GUIDANCE
VI. 引导特性

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

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

C. Negative Core-Cladding Index Difference
C. 负芯-包层折射率差

F. Attenuation Mechanisms
F. 衰减机制

A. High-Power and-Energy Transmission
A. 高功率和高能量传输

B. Fiber Lasers and Amplifiers
B. 光纤激光器和放大器

C. Intrafiber Devices-Cutting and Joining
C. 光纤内部器件-切割和连接

D. Kerr-Related Nonlinear Effects
D. Kerr 相关的非线性效应

F. Gas-Based Nonlinear Optics
基于气体的非线性光学

H. Laser Tweezers in Hollow-Core PCF
H. 空芯 PCF 中的激光光镊

VIII. Final Remarks 第八部分总结