What Are Hollow-Core Fibers?

This is a continuation from the previous tutorial - introduction to miniature and micro-optics.

 

1. Introduction

The history of the development of optical fibers has been largely determined by various constraints stemming from the materials used for the fiber core.

For example, in the case of long-haul telecom applications, it is essential to have very low losses at the wavelength of a reliable and commercially available laser.

Because silica has very low losses over wavelengths ranging from the visible to the near infrared (IR), which coincide with the operating wavelengths of a number of readily available lasers, it became the material of choice for the fiber core; most of the optical power travels through the core, so the properties of the core material determine the loss properties of such a fiber.

Furthermore, the carrier wavelength in telecom applications has migrated from 1.31 to 1.55 μm, to match the wavelength for which the loss of the (mostly) silica core material is minimized.

For other important applications, such as surgery, sensing, and industrial welding, it is often desirable to operate at longer wavelengths (>2 μm) where the losses of silica are prohibitively high. Consequently, many other potential core materials that have moderate losses at mid-IR wavelengths have been explored and used, including chalcogenide glasses, heavy-metal fluoride glasses, and polycrystalline materials.

However, instead of looking for a fiber core material with the best properties for a particular application or wavelength, there is another approach: one can make a fiber that guides light within a hollow core.

In hollow-core fibers, the cladding is designed to act as a ‘‘mirror,’’ reflecting light incident on it back into the core. In contrast to the solid-core fibers, the vast majority of optical power now travels through air, whose optical properties are dramatically different than the optical properties of any solid material.

Because of the small evanescent tails of the guided light, which penetrate into the cladding, the optical properties of the materials of the cladding still have an influence on the overall fiber transmission characteristics, but these optical properties become much less of a constraint.

In particular, the overall losses or nonlinearities of the fiber can now be orders of magnitude smaller than the intrinsic losses and nonlinearities of the solid materials used to manufacture the fiber.

It has been predicted, for example, that hollow-core silica fibers could someday lead to even lower transmission losses than solid-core silica fibers, whose losses—0.2 dB/km at \(\lambda\) = 1.55 μm—are already extremely low.

Similarly, the high-peak-power material breakdown can now be shifted to much stronger peak intensities. Moreover, because optical properties of the materials are not as much of a constraint any more, it becomes possible to use many materials with poor optical properties but attractive other properties (e.g., thermal or mechanical properties) to implement optical fibers.

This tutorial discusses the properties, applications, and manufacture of hollow-core fibers: Section 2 provides a discussion of their transmission properties, Section 3 discusses their applications, Section 4 discusses various ways hollow-core fibers are being manufactured, and Section 5 presents brief concluding remarks.

The remainder of the current section provides a short review of the principles of operation and history of solid-core fibers, followed by a description of the principles of operation and history of each of the main kinds of hollow-core fibers.

 

1.1. Wave-Guiding by Total Internal Reflection

Total internal reflection (TIR) fibers are the most commonly used optical fibers. For example, almost all fibers used for telecom applications are of the TIR type. The principle of operation of a TIR fiber is explained in Fig. 10.1.

 

 

The fact that TIR can be used to confine and guide light through objects of substantial length but small cross-sectional area was realized and experimented with as early as the nineteenth century; even some applications were proposed and were being explored at that time.

In 1956, Larry Curtis made the first successful fiber with a glass core and a glass cladding; similar fibers have been used in endoscopes to look inside the human body ever since. In 1965, C. K. Kao and G. Hockham published an analysis showing that an optical fiber could be a suitable medium for long-distance communication if the losses could be brought down below 20 dB/km. They proceeded to show that a highly purified glass should easily have losses below this limit. Finally, in 1970, four employees of Corning Glass Works—R. Maurer, D. Keck, P. Schultz, and F. Zimar— demonstrated an optical fiber with losses of 17 dB/km, thereby starting the era of optical fiber communications.

Interestingly enough, there even exist hollow-core fibers (\(n_\text{core}=1\)) that operate based on TIR; for certain frequency ranges, there are materials (e.g., sapphire at \(\lambda\) = 10.6 μm) that have an index of refraction \(n\lt1\) and can, therefore, serve as a suitable cladding. However, the range of applicability of such fibers is limited because these materials turn out to have other highly undesirable properties (e.g., loss, stiffness).

 

1.2. Wave-Guiding by Reflection Off a Conducting Boundary 

This class of waveguides has a hollow core surrounded by a conducting boundary that acts as a mirror, reflecting the guided light back toward the core center, thereby providing confinement for light.

Conductors can have good, but not perfect, reflection properties; more than 90% reflection for normal incidence is not unusual. Nevertheless, when the angle of incidence of the ray onto the conductor is far from normal and the core is large, the number of reflections per meter of fiber length can be quite small, so the losses can often be acceptably small.

Often, the properties of the fiber can be further improved by depositing one or more layers of dielectric onto the conducting surface. Interference behavior associated with the dielectric layers improves the guiding properties of the fiber.

Between the 1930s and the 1970s, Bell Labs had a substantial research program aimed at developing a hollow-core waveguide to transport microwave (GHz) electromagnetic signals for long-distance communications.

It developed a 60-mm-diameter hollow-core tube, the inside of which was coated with copper. The range of operational frequencies was aimed to be 40–110 GHz, with losses less than 1 dB/km.

Interestingly enough, one of the major obstacles that precluded it from deploying the cable was economic; the price of deployment was substantial, so only deployment for large markets could pay off, yet there were very few markets at the time that could make anywhere close to the full use of the 274-Mbps capacity of such a cable.

In the 1970s, Corning developed its optical silica fiber, whose properties were highly superior to the Bell Labs concept, so the interest in microwave transmission through metal waveguides for long-distance communication subsided.

The first optical-frequency hollow-core metallic waveguides were built in 1980. These rectangular aluminum waveguides could guide approximately 1 kW of \(\lambda\) = 10.6-μm light (from a CO₂ laser) with losses as low as approximately 1 dB/m.

Their rectangular shape prevented them from being uniformly bent, but since then, many circular designs have also been explored. Such fibers can today guide fairly high powers (10–1000 W), anywhere from visible wavelengths all the way to \(\lambda\) > 10.6 μm, with losses as low as 0.1 dB/m.

 

1.3. Wave-Guiding by Photonic Band-Gaps

In addition to TIR and reflection from a conducting boundary, there is yet another physical mechanism that produces a highly reflecting surface: the photonic band-gap. Photonic band-gap crystals are artificially created materials in which the index of refraction varies periodically between high-index values and low-index values.

When light of wavelengths comparable to the periodicity tries to propagate inside such a material, it experiences very strong scattering: it cannot propagate and is reflected out of it.

In more precise terms, in photonic crystals, a so-called photonic band-gap opens for certain frequency regimes; Maxwell’s equations do not have propagating solutions for the frequencies inside the photonic band-gap, so light with those frequencies is prohibited from propagation through the material. The material, therefore, acts as a perfect mirror.

Note that it is the periodicity length scale of the photonic crystal that determines the frequencies for which the photonic band-gaps occur; by adjusting the periodicity (while using the same constituent materials), one can select the frequency range in which the photonic crystal will act as a reflector.

The main advantage of using a photonic-crystal mirror (compared to a conducting-surface mirror) lies in the fact that photonic crystals can be implemented from dielectric materials, which can often have dramatically lower losses than any conductor (especially in the optical regime). As a result, reflection from a photonic-crystal mirror can incur much less absorption than reflection from any conducting-surface mirror.

Hollow-core photonic band-gap fibers have photonic band-gap crystals surrounding their cores, which act as mirrors so that light is confined to propagate within the core.

The two main kinds of photonic band-gap fibers are those that use one-dimensional (1D) photonic band-gaps and those that use two-dimensional (2D) photonic band-gaps; both kinds are described in the subsections that follow.

 

1.3.1. Wave-Guiding by 1D Photonic Band-gaps

A 1D photonic band-gap fiber uses a 1D periodic photonic crystal (periodicity is only in a single direction) as its mirror; an example of such a fiber is shown in Fig. 10.2.

 

 

The first such fiber, called a Bragg fiber (not to be confused with a fiber Bragg grating, which is periodic in the axial instead of the radial direction), was proposed by Yeh et al.. Since then, many authors have explored Bragg fibers.

One particular type of Bragg fiber, called an OmniGuide fiber, has an omnidirectional reflector surrounding its core. An omnidirectional reflector uses a 1D photonic band-gap to reflect light (within a certain frequency range) perfectly, except for material absorption, for all polarizations and angles of incidence.

Needless to say, such a perfect mirror acts as an excellent mechanism to confine light to the core. OmniGuide fibers have been built for many frequency regimes, extending from IR (\(\lambda\) > 10 μm), all the way down to ultraviolet (UV) (\(\lambda\) < 350 nm). In Fig. 10.2, we show an OmniGuide fiber designed to guide CO₂ laser light (\(\lambda\) = 10.6μm); it has losses less than 1 dB/m.

 

1.3.2. Wave-Guiding by 2D Photonic Band-gaps

The confining mechanism of a 2D photonic band-gap fiber is a mirror implemented by a 2D periodic photonic crystal, like the one shown in Fig. 10.3.

 

 

Photonic-crystal fibers (PCFs) were pioneered by P. Russell and have since been investigated by many researchers around the globe (such fibers can even be purchased from Crystal Fibre A/S).

Such fibers enable many important applications (to be discussed in greater detail in Section 3), but it is worth noting here that they are predicted to have lower losses than solid-core silica telecom fibers. The current minimum measured transmission loss in such fibers is 1.7 dB/km at \(\lambda\) = 1.5 μm.

 

2. Light Transmission in Hollow-Core Fiber

Hollow-core fibers guide light by means of a reflective cladding. Because the index of refraction of the hollow core is smaller than that of the cladding materials, the guiding mechanism cannot be based on TIR, as is the case for traditional optical fibers. Instead, three major types of reflective cladding are used: (1) a metal tube with optional dielectric coating, (2) a multilayer dielectric Bragg mirror (i.e., a 1D photonic crystal), or (3) a 2D photonic crystal. These are schematically depicted in Fig. 10.4.

 

 

The following subsections describe transmission in fibers that use these different reflection mechanisms for the cladding. We first, however, introduce a few physical quantities that are common to the electromagnetic fields of any axially uniform waveguide, whatever its confinement mechanism.

We consider monochromatic light of angular frequency \(\omega\), with a time dependence of the form \(e^{-i\omega{t}}\). The translational invariance of the waveguide along its axis (which we take to be the \(z\)-axis) suggests that the electromagnetic fields of the waveguide will have a \(z\)-dependence of the form \(e^{i\beta{z}}\), where \(\beta\) is the as-yet undetermined axial wave vector (also called the propagation constant).

Thus, at frequency \(\omega\), we expect that an electromagnetic field in the waveguide can be written as a plane wave along the \(z\)-direction times a field profile in the transverse \((x,y)\) plane:

\[\tag{10.1a}\mathbf{E}(x,y,z,t)=\mathbf{E}(x,y)e^{i(\beta{z}-\omega{t})}\]

\[\tag{10.1b}\mathbf{B}(x,y,z,t)=\mathbf{B}(x,y)e^{i(\beta{z}-\omega{t})}\]

It is well known that Eqs. (10.1a) and (10.1b) are valid solutions to Maxwell’s equations, and represent fields confined to the waveguide core, only for certain discrete values of the axial wave vector. These discrete solutions are known as the guided modes of the waveguide and, of course, depend on the particular distribution of the dielectric profile \(\epsilon(x,y)/\epsilon_0=n^2(x,y)\) in the cross-section. Except in certain simple cases, they must be evaluated numerically. For a given mode, the axial wave vector varies with the angular frequency: \(\beta=\beta(\omega)\). This functional dependence is known as the dispersion relation for the mode.

Although the evaluation of the dispersion relations for a particular fiber cross-section must, in general, be performed numerically, one can still draw important conclusions about the nature of a waveguide mode simply based on the values of \(\omega\), \beta\), and the refractive indices of the materials that make up the core and the cladding.

The transverse wave vector in a material with index of refraction n is given by \(k_t^2=(n\omega/c)^2-\beta^2\). Depending on whether \(\beta\) is smaller or larger than \(n\omega/c\), the transverse wave vector \(k_t\) will take real or imaginary values, thus determining whether light can or cannot propagate transversely through that material. The line that separates these two regimes is given by \(\beta=n\omega/c\) and is usually called the light line of a material with index of refraction \(n\).

Based on this simple argument, in Fig. 10.5, we compare the guiding mechanisms in a traditional index-guiding optical fiber, a hollow metal waveguide, and a generic photonic band-gap fiber.

 

 

The light lines of relevant materials are shown as dashed lines, and a schematic dispersion relation of a typical guided mode confined to the core is shown as a black curve. Regions of the (\(\omega\), \(\beta\)) diagram where modes can propagate through the cladding (so-called radiation modes) are shaded in dark gray.

Light gray regions correspond to (\(\omega\), \(\beta\)) pairs for which light can propagate through the core material, but not through the cladding, thus allowing for core-confined guided modes.

In all three cases, core guided modes can only exist above the light line corresponding to the index of refraction in the core. The differences between the three cases stem from the specific properties of the claddings.

For the traditional index-guiding fiber, core guided modes are situated between the light lines corresponding to the core and cladding materials and can only exist if the core has a refractive index larger than that of the cladding.

In the case of the hollow metal waveguide, light is not allowed in the cladding for any pair (\(\omega\), \(\beta\)), which means that core guided modes can exist in the entire region above the light line of vacuum. Finally, for a photonic band-gap fiber, light can be guided in the core at (\(\omega\), \(\beta\)) values that are above the light line of vacuum and inside a photonic band-gap of the 1D or 2D photonic-crystal cladding. 

 

2.1. Hollow Metal Waveguides

Perhaps the simplest method for guiding light in a hollow core is by enclosing the core with a highly reflective metal. The metal acts as a mirror, so that fields from the core incident on the metal are reflected back into the core, providing the confinement mechanism.

For a perfect metal, the evaluation of the transverse dependence of the mode fields from Maxwell’s equations is fairly straightforward and is covered in most standard textbooks.

When the interior of the waveguide consists of a single homogeneous dielectric material, the mode fields can be separated into two polarizations: TE (transverse electric) and TM (transverse magnetic) with vanishing axial components of the electric and magnetic fields, respectively.

The analysis of these modes is particularly simple for the case of a hollow metal waveguide with a circular cross-section.

For TM mode fields, the allowed values of the axial wave vector at the frequency \(\omega\) are

\[\tag{10.2a}\beta_{mn}^2=\frac{\omega^2}{c^2}-\frac{x_{mn}^2}{R^2}\]

where \(m\) is an index (the ‘‘angular momentum’’) denoting the angular dependence \(e^{im\theta}\) of the mode, \(n\) is an index denoting the radial dependence of the mode, \(R\) is the core radius, and \(x_{mn}\) is the \(n^\text{th}\) root of the Bessel function \(J_m(x)\).

For TE modes, the allowed axial wave vectors are

\[\tag{10.2b}\beta_{mn}^2=\frac{\omega^2}{c^2}-\frac{y_{mn}^2}{R^2}\]

where \(y_{mn}\) is the \(n^\text{th}\) root of \(dJ_m(y)=dy\). Equations (10.2a) and (10.2b) form the dispersion relations for the hollow metal fiber.

Each mode possesses a cutoff frequency given by \(\omega_c=c\frac{x_{mn}}{R}\) for the TM polarization and \(\omega_c=c\frac{y_{mn}}{R}\) for the TE polarization.

Below the cutoff frequency, the axial wave vector becomes imaginary and the mode decays exponentially instead of propagating; it is ‘‘cutoff.’’ However, it remains a valid solution to Maxwell’s equations and is needed, for example, in the modal decomposition of an arbitrary field in the waveguide. At a frequency below its cutoff frequency, a mode is said to be evanescent.

Note that the sequences of Bessel function roots \(x_{mn}\),\(y_{mn}\) are increasing and unbounded with index \(n\). Thus, at a given frequency, only a finite number of propagating modes exist for each angular index \(m\).

Indeed, the smallest Bessel function root is \(y_{11}\approx1.84\), associated via Eq. (10.2b) with the TE\(_{11}\) mode. For frequencies \(\omega\lt{c}\frac{y_{11}}{R}\), no propagating modes exist for the hollow metal waveguide.

In contrast, TIR fibers always have at least one propagating mode for any frequency. As the core radius \(R\) increases, the number of propagating modes at a given frequency will also increase.

For a perfect metal, propagating modes are loss free. Actual metals, of course, are not perfect; they have finite conductivities, so a propagating mode will penetrate to a small extent into the metal and thereby become lossy.

To reduce this loss, dielectric layers may be placed on top of the metal layer. By proper choice of the dielectric layer thicknesses, the reflected waves established at the dielectric layer interfaces will interfere destructively with the transmitted waves, reducing the amplitude of the mode field in the vicinity of the lossy metal and reducing the loss of the mode.

This process, which is only operational within a limited frequency range, essentially places a dielectric mirror in front of the metal mirror. Of course, with a sufficient number of properly chosen dielectric layers, a 1D photonic band-gap can be formed, which eliminates the need for the metal layer entirely.

 

2.2. Wave-Guiding in Bragg and OmniGuide Fibers

Bragg and OmniGuide fibers confine light to the hollow core with a mirror-like multilayer cladding, made up of concentric cylindrical rings having alternating low and high indices of refraction. In certain frequency ranges, waves reflected from the multiple dielectric interfaces interfere destructively with the transmitted waves, prohibiting transmission of light through the cladding and providing a confinement mechanism for light in the hollow core. A more detailed understanding of this mechanism can be obtained by analyzing a planar dielectric mirror formed from the same pattern of alternating high- and low-index layers.

Consider a planar Bragg mirror with alternating layers having indices of refraction \(n_1=2.7\), \(n_2=1.6\) and thicknesses \(d_1=0.33a\), \(d_2=0.67a\), where \(a=d_1+d_2\) is the thickness of one pair of high- and low-index layers, which we will refer to as a bilayer. A schematic view of this structure is shown in the inset of Fig. 10.6.

 

 

We again consider monochromatic light of angular frequency \(\omega\). The translational invariance of this system in the \(z\)-direction (the direction indicated by the \(\beta\) vector in the inset), like the waveguide situation considered earlier, suggests that the electromagnetic fields of this system will have a \(z\)-dependence \(e^{i\beta{z}}\).

For some \(\omega\) and \(\beta\) pairs, light can propagate through the mirror; for others, light decays exponentially in the mirror. The \(\omega-\beta\) plane can be divided into regions called bands, corresponding to propagation through the mirror, and regions called band-gaps, corresponding to exponential decay through the mirror.

Light incident from the outside (which will also have a time and \(z\) dependence of \(e^{i(\beta{z}-\omega{t})}\) on a half-infinite mirror will be partially transmitted for the \(\omega\), \(\beta\) pairs outside of the band-gaps of the mirror and will be 100% reflected for those \(\omega\), \(\beta\) pairs within the band-gaps of the mirror.

We plot the band structure of the mirror in the left panel of Fig. 10.6. Band-gaps (the unshaded regions in the figure) appear in frequency ranges that are approximately equidistant (e.g., at \(\beta=0\), corresponding to normal incidence on the mirror, they are centered at 0.25, 0.5, and 0.75).

The fundamental band-gap also includes a frequency range of omnidirectional reflectivity (shown in light gray between dashed horizontal lines); for frequencies in this range, no propagating modes with \(\omega\gt{c\beta}\) exist in the cladding, so light incident from air onto the mirror at any angle of incidence must be totally reflected.

An OmniGuide fiber with a diameter much larger than the bilayer thickness a has a transmission spectrum closely related to the band structure of the planar mirror. This can be seen in the right panel of Fig. 10.6, where we plot the measured transmission through a fiber with core radius of 250 μm and cladding layers made of As₂Se₃ (high index, \(n_2\) =1.68 at \(\lambda\) = 10.6 μm, \(d_1\) = 1 μm) and polyether sulfone (low index \(n_2\) = 1.68 at \(\lambda\) = 10.6 μm, \(d_2\) = 2 μm).

Note that the transmission peaks for a large core fiber will correspond to the portion of the planar band-gap close to the light line of air (delimited by the black dots in the figure). Modes with dispersion relations in this region have mostly grazing angles of incidence with the mirror (i.e., axial wave vectors close to \(\omega/c\)) and will have the lowest losses.

Note that the axial wave vectors in this figure are plotted in units of (2p=a), while the angular frequency is plotted in units of (\(2\pi{c}/a\)). The motivation for using dimensionless axes lies in the scale-invariance of Maxwell’s equations: ignoring material dispersion, given a wave-guiding structure at one wavelength, a wave-guiding structure with identical properties at another wavelength can be obtained by scaling all dimensions by the change in wavelength. Hence, there is no need (except for discussions of a specific application or where material dispersion is critical) to tie the results plotted to any particular wavelength or bilayer thickness.

A cylindrical OmniGuide fiber is obtained conceptually by wrapping a periodic dielectric mirror around a hollow core. Even though the optical properties of the dielectric mirror are somewhat modified by this change of geometry, the band structure of the planar mirror still largely determines the wave-guiding properties of the OmniGuide fiber.

This can be understood by noting that at a large distance from the origin, the curvature of the layers is very small. Thus, an OmniGuide fiber with an infinite number of bilayers will prevent light from escaping the core region for those frequency–axial wave vector pairs that lie within a band-gap of the planar dielectric mirror. This confinement property supports guided modes within the fiber core and, sometimes, surface modes that have the majority of the field at the interface between the core and the cladding.

For an analysis of the mode structure of hollow-core OmniGuide fibers, we focus on a fiber with a small core radius, \(R=3a\), such that the number of guided modes in the fundamental band-gap is not exceedingly large.

It is instructive to compare and contrast the modes of the OmniGuide fiber with those of a much simpler structure: a hollow metal waveguide having the same internal diameter (Fig. 10.7).

 

 

Both waveguides confine light in the core by a reflective cladding, but the simplicity of the hollow metal waveguide allows for analytical solutions. Rearranging Eqs. (10.2a) and (10.2b), the bands have the form \(\omega^2=\omega_c^2+c^2\beta^2\), where \(\omega_c\) is the cutoff frequency.

Starting from low frequencies, the modes of the metal waveguide are in order TE\(_{11}\), TM\(_{01}\), TE\(_{21}\), TE\(_{01}\) and TM\(_{11}\) (degenerate modes), TE\(_{31}\), and so on. The first of the two subscript indices is the angular index \(m\) of the mode and the second is the radial index \(n\), introduced in Section 2.1.

As can be seen in Fig. 10.7, the OmniGuide fiber supports guided core modes in the band-gap that have dispersion relations very similar to those of the metal waveguide. The intensity patterns of the modes of the two waveguides are also similar in appearance. For the OmniGuide fiber, however, only modes with angular index \(m=0\) maintain their pure transverse electric or transverse magnetic character, while all other modes become hybrid (e.g., TE\(_{11}\) becomes HE\(_{11}\)). Also, as a result of the frequency-dependent phase shift upon reflection from the dielectric mirror, the OmniGuide modes are pulled towards the mid-gap frequency when compared to their metallic counterparts. Finally, the degeneracy of the TE\(_{01}\) and TM\(_{11}\) modes of the metal waveguide is lifted in the all-dielectric fiber.

Two important modes of an OmniGuide fiber, HE\(_{11}\) and TE\(_{01}\), are analyzed in more detail in Fig. 10.8.

 

 

The lowest guided mode, HE\(_{11}\), is a doubly degenerate mode that resembles the fundamental mode of an index-guiding fiber. Its electric field distribution becomes very close to being linearly polarized for large-core diameters, so this mode couples very well with a linearly polarized input laser beam. The flux distribution for the HE\(_{11}\) mode shown in Fig. 10.8(c) is somewhat elongated in the direction parallel to the axis of polarization of the mode.

Note, however, that for fibers with larger cores (radius larger than \(30a\) or so), this asymmetry becomes much less pronounced and the flux distribution becomes almost azimuthally symmetric. The TE\(_{01}\) mode is a nondegenerate azimuthally symmetric mode for which the electric field is always oriented in the azimuthal direction, as can be seen in Fig. 10.8b.

This property, together with the boundary condition at the interface between the core and the dielectric mirror, results in a very small overlap of the field of the TE\(_{01}\) mode with the cladding, as can be seen in Fig. 10.8d. As we will see in the following section, this reduction in field overlap with the cladding results in advantageous loss properties for the mode.

 

2.3. Loss Mechanisms in OmniGuide Fibers

After studying the band structure and basic mode properties of the Omni-Guide fiber, we now focus on the various loss mechanisms that arise in this type of hollow-core fiber.

 

2.3.1. Effect of a Finite Mirror: Radiation Loss

With an infinite number of layers, the cladding of an OmniGuide fiber forms a perfect reflector. Any realistic OmniGuide fiber, of course, will only contain a finite number of layers in its cladding, which results in an intrinsic radiation leakage loss.

Fortunately, this radiation loss, which comes from the tiny exponential tails of the field profiles in the outermost cladding, scales exponentially with the number of cladding layers and can easily be made negligible when dielectric contrast is high.

 

 

More quantitatively, Fig. 10.9 shows the radiation loss spectrum of several modes in an OmniGuide fiber with a core radius of \(80a\) and with a cladding composed of 25 layers with alternating indices of refraction \(n_\text{H}/n_\text{L}=2.7/1.6\) and alternating thicknesses \(d_\text{H}/d_\text{L}=0.33a/0.67a\). The units of dB/m on the vertical axis correspond to the specific choice \(a\) = 3.2 μm, appropriate for a fiber with a desired operating wavelength of 10.6 μm. More generally, the units on the vertical axis will scale inversely with wavelength as the bilayer thickness is scaled for operation at a different wavelength.

Each of the loss curves has a characteristic shape; the losses are at a minimum near the middle of the bandgap, where the exponential confinement of the field is strongest, and increase towards the gap edges. The TE\(_{01}\) mode (and any TE\(_{0n}\) mode) is different from the others; because this mode is purely TE polarized, it sees the larger TE band-gap of the corresponding planar mirror and hence has larger bandwidth and smaller radiation loss. The other modes (the TM modes and the HE and EH hybrid modes) have a TM component and, hence, are limited by the smaller TM gap of the planar mirror, with consequently smaller bandwidth and larger radiation loss.

Regardless of the mode polarization, however, the radiation loss decreases exponentially as one increases the number of layers, simply because the fields are exponentially attenuated within the multilayer cladding. This is apparent in Fig. 10.10, which shows the loss at the roughly mid-gap frequency as a function of the number of layers.

The TE-mode losses decrease by about a factor of four per bilayer at this index contrast (\(n_\text{H}/n_\text{L}=2.7/1.6\)), while the other modes (because of the smaller TM band-gap of the planar mirror) decrease by about a factor of two per bilayer. By 25 layers (12 bilayers), the HE\(_{11}\) mode has radiation losses less than 0.01 dB/m for these parameters, and in practice radiative losses are not the limiting factor for high-contrast material systems where sufficiently many layers are easily achievable.

 

 

In fact, the index contrast need not be so large to achieve negligible radiation loss. In Fig. 10.11, we show the number of layers required to obtain 0.01 dB/m losses for this core radius (\(80a\)) and wavelength (10.6 μm), as a function of the index difference \(\Delta{n}\) between the high-index and low-index material, for a fixed low-index material with \(n\) = 1.6. (The previous graphs correspond to \(\Delta{n}\) = 1.1.)

For each index contrast, we look at the glancing-angle mid-gap frequency \(a/\lambda=(\tilde{n}_\text{hi}+\tilde{n}_\text{lo})/4\tilde{n}_\text{hi}\tilde{n}_\text{lo}\), where \(\tilde{n}=\sqrt{n^2-1}\) (this gives roughly the minimum attenuation). As can be seen from the plot, even an index contrast of \(\Delta{n}\) = 0.4 is sufficient to require only 60 layers for HE\(_{11}\) and 25 layers for TE\(_{01}\). On the other hand, a 1% contrast as in doped silica fibers would require hundreds or even thousands of layers.

 

 

The computation of the radiation losses is straightforward and can exploit any of several standard methods to implement open boundary conditions in electromagnetism. With finite-element or finite-difference methods that require a finite computational cell, one can use a perfectly matched layer (PML) to implement a reflectionless absorbing boundary to emulate radiative loss.

Alternatively, for multilayer systems, efficient transfer-matrix methods exist that allow one to directly solve for ‘‘leaky-mode’’ solutions; complex-\(\beta\) modes for a real \(\omega\) that satisfy the boundary condition of zero incoming waves from outside the fiber, where the imaginary part of \(\beta\) gives the radiation loss rate.

 

2.3.2. Material Absorption Losses

Of more common concern is loss due to material absorption, especially at wavelengths like 10.6 μm where transparent materials are unavailable. Hollow-core fibers, by design, greatly suppress such losses because most of the light is confined in the hollow core, but one must still evaluate the losses that result from the small portion of the mode that penetrates the cladding.

As is discussed later in this tutorial, such losses decrease rapidly with the core radius and quickly become several orders of magnitude smaller than the bulk absorption rates of the cladding materials.

Although a variety of computational methods allow one to evaluate directly the electromagnetic modes in absorbing materials, modeled by a complex refractive index at a given wavelength, more general insights can be obtained by perturbative methods.

In particular for all dielectric materials considered here, the imaginary part of the refractive index is much smaller than the real part (<1% even for the polymer at 10.6 μm), in which case the following formula is essentially exact:

\[\text{absorption loss}=\sum_{\text{materials}}\text{(material bulk absorption loss)}\cdot\text{(fraction of }\epsilon|E|^2\text{ in material)/}(v_g/c)\]

where (\(v_g/c\)) is the group velocity in units of \(c\) (usually ~ 1.0 for large-core hollow fibers). For example, if the clad materials have a bulk absorption loss of 1000 dB/m and a mode of interest has 99.9% of its electric-field energy in the core and a velocity approximately \(c\), then the waveguide’s absorption loss for that mode will be only 1 dB/m.

In fact, for the large-core fibers used to transmit 10.6-μm light, many modes have much more than 99.9% of their energy in the core and the bulk absorption losses can be suppressed by four or five orders of magnitude.

The effect of material absorption on transmission loss in an Omni-Guide fiber is graphically demonstrated in Fig. 10.12, which compares the measured transmission spectrum of an OmniGuide fiber with the material absorption of the low-index component of the cladding (the polymer polyether sulfone [PES]). Dips in the transmission spectrum clearly correspond with peaks in the material absorption. The material absorption of the high-index component of the cladding is negligible relative to the low-index material.

 

 

Because the absorption loss arises from the small exponential tail of the field in the innermost mirror layers, it is not surprising that this loss is a rather sensitive function of the thicknesses of these layers.

Small changes to their thicknesses, while having very little effect on the field structure in the core, can introduce a substantial change to the small exponential tail that determines the absorption loss.

This is particularly evident in an OmniGuide fiber where the high-index and low-index materials have vastly different bulk absorption losses: a small change to the thickness of the first layer (assumed to be of a high-index, low-loss material), for example, can push substantially more of the exponential tail into the next layer (assumed to be of a low-index, high-loss material), sharply raising the loss.

This effect should be mode dependent; a change that pushes more of the tail into the high-loss material for one mode may have the opposite effect on the field distribution and loss for a different mode.

Figure 10.13 shows the dependence of absorption loss on the first layer thickness for several modes of an OmniGuide fiber with indices \(n_\text{H}=2.7+1.94\times10^{-6}i\), \(n_\text{L}=1.6+7.77\times10^{-3}i\), and layer thicknesses \(d_\text{H}=1.05\) μm, \(d_\text{L}=2.10\) μm. The imaginary parts of the indices of refraction correspond to bulk absorption losses of 10 dB/m and 40,000 dB/m, respectively, at the operating wavelength of 10.6 μm. The core radius of the fiber is 250 μm. The horizontal axis gives the ratio of the first layer thickness to \(d_\text{H}\), the thickness of all other high-index layers.

 

 

The dependence of mode loss on the thicknesses of the innermost layers in general, and the first layer in particular, strongly suggests that the structure of an OmniGuide fiber be optimized for particular applications.

For example, in applications where the HE\(_{11}\) mode is the desired operating mode, the Omni-Guide fiber described earlier should be constructed with a first-layer thickness of about one-third the thickness of the other high-index layers (which is also close to optimal for the TM\(_{01}\) and EH\(_{11}\) modes). Alternatively, if the TE\(_{01}\) mode is the desired operating mode, all high-index layers should have the same thickness.

Figure 10.14 shows the absorption loss suppression (absorption loss/bulk absorption) as a function of wavelength (and dimensionless frequency) for several modes in an OmniGuide fiber with a core radius of \(80a\) (~ 24 wavelengths).

The thickness of the first layer (high index) is taken to be half the thickness of the other high-index layers. As for the radiative loss, the TE\(_{01}\) mode has the lowest absorption loss and the widest bandwidth; this is not only because of the larger TE band-gap in the corresponding planar mirror but also because of boundary-condition considerations discussed later in this tutorial. Even the HE\(_{11}\) fundamental mode, however, suppresses absorption losses by five orders of magnitude for this core radius.

 

 

This absorption-loss suppression is a dimensionless quantity; it is invariant as the whole structure is rescaled with wavelength. It is also almost independent of the number of layers, because the exponential field decay in the cladding means that almost all of the contribution to absorption loss comes from the first few layers.

It does depend on two other quantities, however: the index contrast and the core radius. One would expect, for example, that the larger the index contrast, the stronger the confinement of light in the core and the greater the absorption suppression.

The dependence on index contrast \(\Delta{n}\) for a fixed low-index material (\(n\) = 1.6), however, turns out to be fairly weak, as shown in Fig. 10.15; halving the index contrast increases absorption loss by less than a factor of two. Therefore, the use of more lossy materials to obtain greater index contrast at some point ceases to be advantageous. Index contrast is more important in determining bandwidth and radiation loss, as described in the previous section.

 

 

Another way to reduce absorption loss is to increase the core radius \(R\), and there is a general argument that the absorption losses will decrease as \(1/R^3\). This is demonstrated directly in Fig. 10.16, which shows the scaling of the absorption loss versus core radius at the mid-gap frequency.

 

 

The origin of this \(1/R^3\) behavior, which has been observed in many other systems including hollow metallic waveguides, can be seen in an analysis of the field-energy fraction in the cladding.

Since the depth of the field penetration into the cladding is determined by the band-gap and is independent of \(R\), a simple surface-area/volume analysis would suggest that the fraction of the field in the cladding and, hence, the absorption, scales as \(1/R\).

This neglects the boundary conditions on the field, however; similar to a metallic waveguide, the TE\(_{01}\) mode’s electric field (azimuthally polarized) goes approximately to zero at the boundary of the core.

Because of this, its \(\mathbf{E}\) amplitude in the cladding scales as the slope \(d\mathbf{E}/dr\) at \(R\), which goes as \(1/R\) for a given mode pattern. This gives an additional \(1/R^2\) factor in the \(\epsilon|\mathbf{E}|^2\) energy in the cladding and, thus, the \(1/R^3\) behavior of the absorption loss.

The explanation of the \(1/R^3\) behavior for the non-TE modes, for which the analogous metallic boundary condition does not force a node in the field at \(R\) even for metal waveguides, is more subtle.

In the large-\(R\) limit, the penetration of the field into the cladding becomes negligible compared to \(R\) or to the transverse wave vector \(\sqrt{\omega^2/c^2-\beta^2}\). One can, therefore, employ a scalar approximation of the field in the interior (where the index is uniform) and simply impose zero boundary conditions at \(R\) instead of the finite penetration (this is related to the LP mode categorization that applies for doped-silica fibers with low-index contrast and is mirrored by a similar scalar approximation that applies in the high-frequency limit for holey silica fibers).

Given this scalar approximation, the \(1/R^3\) behavior follows as for the TE mode. Because this boundary condition only applies asymptotically, however, the TE mode tends to have an advantage in the strength of its field confinement. Also, for smaller \(R\), one can see that the TM and HE mode losses in Fig. 10.15 begin to flatten; the losses are starting to change from \(1/R^3\) scaling to the \(1/R\) scaling that one expects in the absence of zero-field boundary conditions.

In fact, similar \(1/R^3\) scaling occurs for any other loss phenomenon associated with the penetration of the field into the cladding, including radiation loss and surface-roughness scattering, for exactly the same reasons.

 

2.3.3. Loss Discrimination and Single-Mode Behavior

A waveguide with a large hollow core, whether metallic or dielectric, is intrinsically multimode. However, in practice the observed behavior is typically that all of the power is guided in one or a few modes.

The reason for this can be seen in the absorption loss spectra of Fig. 10.14: different modes have different penetrations into the cladding and consequently greatly differing losses. This causes the high-loss (high-order) modes to be filtered out, with the effective number of modes being determined by the ratio of the intermodal coupling to the differential loss rates.

In early work on microwave communications via metallic waveguides, precisely this loss discrimination was employed to preferentially guide light in the low-loss TE\(_{01}\) mode.

 

2.3.4. Bending Loss

The wide utility of fiber waveguides derives partly from their ability to bend and guide optical energy to positions that are not in the direct line of sight of the optical source. However, the bending introduces coupling between the low-order and low-loss operating mode (usually the HE\(_{11}\) or TE\(_{01}\) mode) and higher order, higher loss modes are supported by the waveguide. Thus, the bend causes additional attenuation of the guided field.

It is also possible to couple energy into reflected fields, but this generally requires bends with extremely small radii of curvature that cannot be implemented without breaking the fiber. Coupled-mode theory provides the simplest and most efficient framework for evaluating the additional attenuation associated with a bend.

Maxwell’s equations are expressed in terms of a Serret–Frenet coordinate system based on the curve defined by the axis of the fiber. These equations can be separated into a portion that has the same form as the equations of a straight waveguide and a perturbation, proportional to the inverse of the radius of curvature of the bend.

The field in this perturbed waveguide is represented as an infinite series of modes (in general containing both forward and backward propagating components) of the straight waveguide with amplitudes that depend on arc length along the fiber axis.

Following standard manipulations, the orthogonality of the modes is employed to derive an infinite set of coupled ordinary differential equations for the arc length–dependent mode amplitudes.

In practice, coupling to higher order modes decays rapidly with mode order, so this infinite system can be truncated to obtain a computationally tractable problem. A perfectly matched layer, tailored for the bent waveguide, can be used to simulate radiation loss.

A straight circular waveguide with a confinement mechanism that involves dielectric materials supports TE, TM, and hybrid (HE, EH) modes, where each hybrid mode can have two degenerate polarizations. For planar bends (i.e., where the axis of the fiber remains in a plane), the degenerate polarizations of the hybrid modes are not coupled by the bend.

One hybrid polarization, corresponding to almost linear polarization perpendicular to the plane of the bend, couples only to other hybrid modes with the same polarization and to the TE modes. The second hybrid polarization, corresponding to almost linear polarization in the plane of the bend, couples only to other hybrid modes with the same polarization and to the TM modes. We refer to these two polarization states as the ‘‘low-loss’’ and ‘‘high-loss’’ polarizations, respectively.

Figure 10.17 shows the theoretical local attenuation, defined as the derivative with respect to arc length of the axial power flow, of the electromagnetic field in an OmniGuide fiber as it propagates through a 90-degree circular bend with radius 10 cm.

The bend used in this coupled mode calculation is obtained by winding the fiber, modeled as a stiff beam, under tension around a cylindrical mandrel. The fiber cladding is composed of 40 alternating layers with indices of refraction \(n_\text{H}=2.7+1.94\times10^{-6}i\), \(n_\text{L}=1.7+7.77\times10^{-3}i\), corresponding to a chalcogenide glass and a polymer, respectively.

The imaginary parts of the indices correspond to bulk absorption losses of 10 dB/m and 40,000 dB/m, respectively. The high- and low-index layers have thicknesses 0.93 and 1.87 μm, but the first high-index layer has half the thickness of the other high-index layers. The core radius of the fiber is 250 μm, and the operating wavelength is 10.6 μm. An HE\(_{11}\) mode with the high-loss polarization (i.e., in the plane of the bend) is incident on the bend.

 

 

Initially, only the HE\(_{11}\) mode is present, so the local attenuation is constant and small. The fiber then undergoes a rapid transition from straight to bent, and the local attenuation rises rapidly. It then exhibits a number of peaks and valleys, corresponding to destructive and constructive interference of all the modes that get excited by the bend.

The fiber then makes a second transition from bent back to straight, and the loss drops quickly. Note that the loss level reached after the end of the bend is higher than the loss level before the bend; some of the higher order and higher loss modes excited by the bend continue to propagate, raising the local attenuation.

Figure 10.18 shows the predicted bend loss, obtained via coupled-mode theory, for an OmniGuide fiber following a 90-degree circular bend as a function of the radius of curvature of the bend.

The bend loss is defined to be the difference between the loss (in dB) after propagation through the bend and the loss after propagation through a straight fiber of the same total arc length. The structure of the fiber used in this calculation is the same as Fig. 10.17. The bend losses for an incident HE\(_{11}\) mode in the low-loss and high-loss polarizations are given by the curves marked with open triangles and open squares, respectively.

 

 

The strength of the coupling to higher order modes scales inversely with the bend radius. However, the arc length of the bend scales directly with the bend radius so the total bend loss, which scales as the product of these two effects, should be roughly independent of bend radius, at least for relatively small-radius bends that couple many modes together.

At higher bend radii (in this example, more than ~100 cm), where very few modes are coupled by the bend (i.e., where second-order perturbation theory in \(1/R_\text{Bend}\) becomes applicable), the coupling to higher order modes scales with the inverse of the square of the bend radius, so the bend loss becomes proportional to \(1/R_\text{Bend}\).

The bend loss for the high-loss polarization generally follows this two-regime dependence on bend radius. The bend loss for the low-loss polarization also follows this same general behavior, although some deviation is observed for small bend radii (in this example, less than ~10 cm).

The theoretical bend attenuation evaluated with coupled-mode theory for a hollow metal waveguide, with a single-layer dielectric coating, is shown in Fig. 10.19, along with measured results. The fiber core radius is 125 μm and the incident field has the low-loss polarization (i.e., almost perpendicular to the plane of the bend).

Note that bend attenuation is obtained by dividing the bend loss by the arc length of the bend; with the removal of the effect of the arc length, the bend attenuation scales with the inverse of the bend radius. The generally good agreement between theory and measurement attests to the validity of the coupled-mode analysis.

 

 

Figure 10.20 displays the variation of bend loss with core radius for an OmniGuide fiber with the same cladding as in Figs. 10.17 and 10.18. The bend radius and wavelength used in this calculation are fixed at 10 cm and 10.6 μm, respectively, while the bend angle is kept at 90 degrees.

Two competing factors determine the dependence of bend loss with core radius. As the core radius increases, the losses of all modes propagating in the bend decrease (like \(1/R^3_\text{core}\) as discussed in Section 2.3.2), which tends to reduce the bend loss.

However, the strength of the coupling to higher order modes increases with core radius, which tends to increase the bend loss. The net effect is shown in Fig. 10.20; the bend loss initially increases with core radius but eventually levels off.

 

 

2.4. Wave-Guiding in 2D Photonic-Crystal Fiber

Another form of periodic cladding that supports band-gaps and can, therefore, serve to guide modes in a hollow core is a 2D periodic arrangement of materials with a significant index contrast.

In particular, the most common realization of such a 2D crystal is a triangular lattice of air holes in silica glass, which was first demonstrated to guide light in a hollow core by Knight et al..

Such ‘‘holey’’ fibers have the advantage that they are only fabricated from a single material, conventional silica; the main disadvantage is the lack of cylindrical symmetry, which allows modes to couple to one another more easily and, as is described later in this chapter, makes large-core holey fibers more susceptible to problems with surface states.

Conceptually, the analysis of holey PCFs follows the same outline as in Fig. 10.21. First, we evaluate the band diagram of the infinite cladding, plotting all possible propagating states as a function of frequency \(\omega\) and axial wave vector \(\beta\); this band diagram is called the ‘‘light cone’’ of the crystal (although it is only a cone for a homogeneous material).

Then, if we find gaps—regions in which there are no solutions that propagate in the infinite cladding—above the air light line, we can use the photonic crystal as a reflective cladding to guide light in a hollow core. This analysis is more complicated than for Bragg fibers because the computations are inherently 2D instead of 1D but can be performed in a few minutes on a workstation, for example, by a plane-wave–expansion method.

 

 

In particular, we consider the structure shown in the inset of Fig. 10.21, a triangular lattice (with period \(a\)) of air holes (radius \(0.47a\)) in silica (\(n=1.45\)), similar to fabricated structures.

The light cone and band-gaps of this infinite holey cladding are shown in Fig. 10.21. In addition to the space below the crystal’s light cone (corresponding to conventional TIR guiding), there are finger-like gaps extending to the left, most of which open monotonically as b goes to infinity.

These are the band-gaps, and it can be shown that they always appear for sufficiently large \(\beta\), regardless of the hole radius or the index contrast.

However, most of these gaps are useless for guiding in a hollow core; only the gaps that lie above the air light line (\(\omega=c\beta\)) are applicable. Below the air light line, the field would decay exponentially in the core, rather than propagate. Therefore, we focus on the region of the first (fundamental) gap that lies above the air light line, and in particular the region outlined by the dashed lines in Fig. 10.21.

Given the band-gap of the perfect cladding, we expect that it will support confinement of guided modes in a hollow core, which we form by removing from the crystal any dielectric within a radius \(1.2a\) or \(1.4a\). The resulting structures are shown in the insets of Fig. 10.22.

The former (core ‘‘radius’’ \(1.2a\)) is similar to a structure that was fabricated by Smith et al.. The latter (core ‘‘radius’’ \(1.4a\)) differs not only by being larger, but also by having a different termination of the surrounding crystal, which has beneficial effects predicted by West et al..

 

 

Figure 10.22 plots the dispersion relations of the guided modes in these two structures. It is remarkable that the larger core structure has fewer modes, in contrast to the behavior of TIR, OmniGuide, and hollow metal fibers, and this is due to the influence of the crystal termination—what part of the crystal lies at the core boundary.

For certain crystal terminations, it is known that the crystal supports surface states; these are modes that lie within the band-gap but below the air light line and are thus localized around the boundary of the crystal rather than within the core. Such states below the air light line are visible in Fig. 10.22 (top).

These states continue as they cross the air light line and influence the other guided modes, and the result is that the radius \(1.2a\) core exhibits a large number of guided modes that cross one another.

In contrast, by changing the termination, in Fig. 10.22 (bottom), we see that all of the surface modes disappear, and only a small number of modes remain above the air light line: two linearly polarized modes and four nearly degenerate modes of different symmetry (similar to an LP\(_{11}\) mode).

The intensity patterns for some of these modes are shown in Fig. 10.23. At the top right, note the surface-localized pattern of the surface state lying below the light line. Even the other two modes of the \(1.2a\) radius core at top left and middle, which lie above the air light line, have a significant fraction of their energy localized near the cladding; only about 50% of their energy is localized within the \(1.2a\)-radius core, corresponding to a factor of two suppression of, for example, cladding absorption loss, and a strong sensitivity to surface roughness.

Moreover, where the guided mode crosses over modes of other symmetry in Fig. 10.22, any symmetry-breaking perturbation can couple the modes and increase losses. This depends sensitively on the surface geometry, however. If we consider the fundamental mode in the \(1.4a\) radius core, at bottom left, it is almost entirely localized in the air; 95% of its energy is within the \(1.4a\) radius core, corresponding to more than an order of magnitude suppression of cladding absorption loss and much less sensitivity to surface effects.

A different symmetry mode in the same \(1.4a\) core is shown at the bottom right of Fig. 10.23, resembling the TE\(_{01}\) mode of a cylindrical metallic waveguide.

 

 

To further reduce losses, for example to match the \(10^5\) suppression of cladding absorption for the multilayer fibers of the previous sections, and more importantly to reduce scattering from surface roughness, one would have to increase core radius. Similar \(1/R-1/R^3\) scaling laws should apply.

However, as core radius is increased, the number of surface states also increases, unless care is taken with the crystal termination. In this way, Mangan et al. were able to reduce the losses from 13 to 1.7 dB/km by increasing the core radius, but because the number of surface states increased, the contiguous bandwidth (determined by the frequency separation of the surface-state intersections with the fundamental guided mode) was also decreased.

All hybrid modes of a circular dielectric waveguide, including the fundamental HE\(_{11}\) mode, have two orthogonal polarizations that travel at the same speed (i.e., are degenerate), but at different speeds when the circular symmetry is broken by imperfections. This phenomenon produces birefringence and polarization mode dispersion (PMD).

A very similar phenomenon occurs for PCFs, but the analysis of the symmetry is much more complicated because of the sixfold symmetry of the fiber structure. In particular, it might seem impossible that one would have pairs of degenerate modes, because the structure does not have 90-degree rotational symmetry; one cannot simply rotate a mode by 90 degrees and get an independent mode solution.

Nevertheless, pairs of degenerate modes are precisely what one gets with sixfold symmetry (and never trios or sextuples of degenerate modes). This result follows from group representation theory; technically, the structure has the \(C_\text{6v}\) symmetry group, and there are six possible representations of this group that modes can fall into, one of which is the doubly degenerate ‘‘two orthogonal polarizations’’ case.

In simple terms, one degenerate mode is the average of 60- and 120-degree rotations of the other mode. We will not go into more detail here but will simply note that the existence of these doubly degenerate modes means that PMD and birefringence occur when the symmetry of a PCF is broken; indeed, these effects occur much more strongly because the index contrast is so much larger than in traditional dopedsilica fiber.

Most experimental PCF work has relied on such doubly degenerate modes, because they are the fundamental modes of the fiber. Alternatively, it may be possible to employ a higher order nondegenerate operating mode analogous to the TE\(_{01}\) mode of Bragg fibers or metal waveguides.

 

3.  Applications of Hollow-Core Fibers

3.1. Hollow-Core Fibers for Medical Applications

Optical fibers are used in medicine for a diverse range of applications, from sensing and diagnostics to therapeutics. The main advantage of hollow-core fibers in medicine is their ability to transmit wavelengths for which traditional solid-core fibers are not transparent.

For example, traditional silica-based fibers cannot transmit wavelengths above approximately 2.1 μm because the material absorption of silica becomes too large. These longer wavelengths, such as the Er:YAG laser wavelength at 2.94 μm and the CO₂ laser wavelength at 10.6 μm, have significant clinical advantages over the shorter wavelengths accessible to silica-based fiber.

The interaction of light and the different components of biological tissue encountered in medical applications is determined mainly by the absorption of the light by these components. For wavelengths above 2 μm, the absorption of water, the main constituent of biological tissue, rises sharply.

This strong absorption of laser light by water can be used for very efficient cutting and ablation of soft tissue. The operating wavelengths of CO₂ lasers (10.6 μm) and Er:YAG lasers (2.94 μm) offer particularly strong water absorption and are, therefore, especially well suited for precise cutting and ablation; the strong absorption prevents the laser energy from penetrating tissue much beyond the point of application.

In contrast, wavelengths less than 2 μm penetrate as much as 1 cm beyond the intended target. CO₂ lasers, in particular, are reliable and commercially available and have been used in medicine for more than 30 years.

In addition, the CO₂ laser wavelength offers excellent coagulation capability to stop bleeding during laser-based surgery, a capability not shared by the Er:YAG laser wavelength. This unique combination of precise cutting, limited penetration beyond the point of application, and coagulation is found only with the CO₂ laser wavelength, making it the optimal wavelength for many surgical procedures.

In spite of the advantages associated with CO₂ lasers, their use in medicine has been relatively limited because of the lack of a flexible medium to transmit the laser power to a target within the human body. Without a flexible delivery medium, the use of the CO₂ laser was confined to procedures in which a direct line of sight could be established between the laser and the target, primarily in dermatology and ear, nose, and throat (ENT) surgery with a rigid laryngoscope.

Thus, since 1976, various academic and commercial groups have attempted to develop alternative waveguides and fibers that would allow the use of CO₂ lasers in a much wider variety of medical applications. For the most part, these attempts have been met with only limited success.

Products were developed that used either complicated and bulky articulated arms or restricted line-of-sight micromanipulators. Some commercial hollow metallic waveguides were developed (e.g., by Sharplan, Luxar, and Clinicon), mainly as an articulated-arm substitute. Some solid-core fibers were also developed, based on materials that are relatively transparent in the mid-IR, but these fibers proved either too lossy or too brittle.

Hollow-core fibers using an omnidirectionally reflective 1D photonic bandgap to confine light have been developed and used successfully in a number of minimally invasive procedures with CO₂ lasers.

These OmniGuide fibers, as can be seen in Fig. 10.24, are flexible enough to be introduced into the body through flexible endoscopes, delivering CO₂ laser radiation to regions not accessible by such lasers before.

In one particularly interesting example, an OmniGuide fiber was used through the working channel of a flexible endoscope to successfully remove a large malignant tumor from the bronchus intermedius of a patient with lung cancer.

 

3.2. Potential Telecom Applications

Various oxide glasses have material absorption losses below 100 dB/km and indices of refraction near 1.6 at the typical telecom wavelengths. When combined with the high-index glass formed from As₂Se₃, a very low loss OmniGuide fiber can be produced.

Indeed, with approximately 20 bilayers and the proper choice of the hollow-core radius, the theoretical loss for the TE\(_{01}\) mode is below 0.01 dB/km. Losses this low would reduce the number of relatively expensive amplifier modules needed in long-haul systems, changing the fundamental economics of this market. In addition, nonlinear effects in the hollow core would be greatly reduced, allowing for higher launch power and dense wavelength division multiplexing without the usual concerns raised by four-wave mixing.

Johnson et al. proposed that a relatively large hollow-core radius be employed in OmniGuide fiber for telecom applications to achieve the benefits of extremely low loss and nonlinearity. Of course, an OmniGuide fiber with this core size is multimoded but, because of large differential mode attenuation, behaves as essentially single moded, with the lowest loss mode serving as the operating mode.

The lowest loss mode for OmniGuide fiber is the TE\(_{01}\) mode; the use of a large core radius, therefore, necessitates the use of this mode as the operating mode. The TE\(_{01}\) mode, unlike the doubly degenerate HE\(_{11}\) mode, has the advantage of being free from PMD but does not couple at all to the TEM\(_{00}\) field output from conventional telecom lasers. Thus, its use in telecom systems will require the coincident development of efficient HE\(_{11}\rightarrow\) TE\(_{01}\) mode converters. In addition, this proposed telecom fiber would need to have very low surface roughness to prevent scattering of power from the operating mode to higher order lossy modes.

Two-dimensional, hollow-core PCF also offers the possibility of extremely low loss and nonlinearity. In addition, because the cladding matrix of this fiber is low-loss silica glass, there is no need for the use of a large core to achieve the desired loss level; true single-mode behavior can be obtained, with the doubly degenerate HE\(_{11}\)-like operating mode coupling efficiently to the output from typical telecom lasers.

On the other hand, as discussed in Section 2.4, this fiber supports lossy surface modes, which couple easily to the operating mode and provide an additional loss mechanism. In addition, the double degeneracy of the operating mode will produce PMD in the presence of perturbations that split the degeneracy.

The complexity of the 2D photonic-crystal mirror, with attendant difficulty in maintaining the desired structure during manufacture, enhances the likelihood of the existence of perturbations that both induce coupling between the operating and surface modes and split the degeneracy of the operating mode, giving rise to PMD.

The wealth of parameters that can be modified to fine-tune the confinement mechanism in OmniGuide fiber, combined with the virtual absence of material dispersion, suggest that OmniGuide fiber can be tailored to have certain desirable dispersion properties.

Engeness et al., for example, describe a simple modification to the multilayer cladding structure of an OmniGuide fiber to obtain a dispersion-compensating fiber (DCF) with a theoretical dispersion/loss figure of merit up to five times larger than conventional DCFs. Similar design freedom in 2D PCF has also been exploited to provide theoretically high-performance DCF.

 

3.3. Hollow-Core Fibers as Gas Cells

Hollow-core fibers also find application as gas cells. Interaction of light with gases leads to a variety of complex and important phenomena of both academic and commercial interest in fields such as nonlinear optics, chemical sensing, quantum optics, and frequency measurement.

Because of the low density of gaseous media, maximization of interactions with light is often a challenge. Typically, the signature strength of the phenomena being explored increases with the intensity of the probing beam and with the interaction length. To enhance intensity in a conventional gas cell, one can focus the laser beam more tightly. Unfortunately, that leads to a smaller diffraction length, which in turn implies shorter interaction lengths.

The use of hollow-core fibers as gas cells eliminates these restrictions on intensities and interaction lengths. If the laser beam is confined as a guided mode of a hollow-core fiber, diffraction is prevented, and one can explore very small beam profiles and, therefore, very high intensities. Moreover, simply for practical considerations, lengths of conventional gas cells are typically limited to less than approximately 1 m.

In contrast, the lengths of gas cells implemented via hollow-core fibers are much less limited in that respect; the only fundamental limitation is the transmission loss of the fiber. For example, losses of approximately 0.3 dB/m allow for propagation lengths of about 10 m. In addition, as we have seen in the previous sections, hollow-core fibers can often be bent into loops of fairly small radii, thereby enabling very compact gas cells that nevertheless have very long interaction lengths.

In chemical sensing applications, hollow-core fibers offer significant benefits over conventional gas cells because they require a drastically smaller volume of the sample gas. Moreover, the response time is faster since such smaller volumes fill up faster.

For example, Charlton et al. used a photonic band-gap hollow-core fiber to detect ethyl chloride at concentration levels of 30 ppb with a sample volume of 1.5 ml, and a response time of 8 seconds, representing an increase in sensitivity by almost three orders of magnitude over conventional gas cells.

Hollow-core fibers have also enabled many impressive accomplishments in nonlinear optics. For example, stimulated Raman scattering has been observed in hollow-core PCFs filled with hydrogen gas, with up to 92% quantum efficiency and threshold pulse energies six orders of magnitude lower than what was previously reported. In addition, slow-light effects have been observed in acetylene-filled hollow-core PCFs; similar fibers filled with acetylene have also been used to implement frequency locking of diode lasers.

 

3.4. Applications of Hollow-Core Fibers for Remote Sensing

Hollow-core optical fibers can alternatively be used as passive elements to deliver radiation from a source being sensed or studied, to the remote sensor, which then analyzes the radiation. Applications of interest include temperature sensing (analysis of black-body radiation) and chemical sensing.

This kind of remote sensing can be useful in hazardous environments (e.g., battlefields) or when the point of sensing is difficult to reach (e.g., inside a machine). It is also useful when many points within an environment need to be analyzed often, but not continuously (bottoms of oceans, industrial plants); in that case, the number of sensors that needs to be deployed is significantly reduced compared to the situation in which one sensor is placed at each point that needs monitoring.

The motivation to use hollow-core fibers is that they can often transmit broader bandwidths and access wavelength regimes that would be difficult to explore using any other solid-core fiber (e.g., for chemical sensing, the 3- to 20-μm regime is of particular interest).

Applications of interest include environmental sensing (pollution, etc.), homeland security, process monitoring, and biomedical sensing (e.g., breath analysis for detection of asthma).

 

3.5. Industrial Applications

CO₂ lasers are used in many industrial applications such as cutting, welding, and marking. It is a challenge, however, to bring the laser beam from the source to the working area, as the path is usually obstructed by other equipment.

The use of articulated arms is the only method for beam delivery, even though these bulky systems require a large working space and use mirrors that require constant maintenance and alignment.

Early efforts at using solid-core fibers to deliver high-power CO₂ beams have suffered from thermal damage, particularly at the air–fiber interfaces at the input or the output end of the fiber.

Some of the promising solid-core fibers also suffered from short lifetimes because of degradation of the guiding material when carrying high-power laser beams. To overcome these problems, researchers started to investigate high-power CO₂ laser delivery systems based on hollow-core fibers.

Initial attempts based on rectangular and circular metal-coated hollow-core fibers succeeded in transmitting up to 3 kW of laser power. However, these power levels were achieved with fibers having large core radii, which exhibited poor output mode quality when subjected to movements and bends.

Most of the industrial applications require well-defined, low-order output modes so that the cut or mark is clean and sharp. The only way to achieve this type of modal quality with hollow-core fibers is to have a smaller core size that will filter the higher order modes and transmit only the fundamental mode. Of course, as the core size is reduced, the losses of the fiber increase and the power capacity decreases.

These problems limited the utilization of hollow-core fibers to lower power industrial applications, like marking or cutting paper or plastic products, applications that require powers in the range of 1–100 W, which can be delivered with the small core fibers. The flow of gas through the fiber core and the use of a water-circulating jacket are often employed to improve the power handling capacity of these fibers, but they still find little use in industrial processing.

An interesting solution to the problem of fiber-motion–induced mode mixing in a marking application was investigated by Harrington, in cooperation with Domino Laser Corporation. The method involved using eight computer-controlled lasers coupled to eight stationary fibers to form the desired mark.

 

4. Hollow-Core Fiber Manufacturing

4.1. OmniGuide Fiber Manufacturing

Because of their flexibility, low cost, and relative ease of manufacture, polymer fibers are widely used in applications that do not involve the optical properties of the fiber (e.g., textile fabrics).

On the other hand, structures with refined abilities to control and manipulate light, such as Bragg mirrors, are costly to produce and have been mostly limited to planar geometries. OmniGuide fibers, however, combine the best properties of polymer fibers with the unique optical properties of Bragg mirrors.

Because OmniGuide fibers consist of approximately 98% polymer, they are extremely flexible, yet they have the required optical properties to guide high-power laser light.

The two materials that are used to fabricate the omnidirectional reflective cladding of OmniGuide fibers, the chalcogenide glass As₂Se₃ and the polymer PES, are chosen for their unique properties; they have very different indices of refraction critical for the efficient formation of a 1D omnidirectional photonic band-gap, and they can be thermally co-drawn into layered structures without cracking or delamination.

The preform/draw process used to manufacture OmniGuide fibers, similar to the process used for silica optical fiber manufacturing, allows for high-volume production with good geometrical control. For typical applications, a proximal- end connector (for ease of attachment to a laser) and distal tip are added to the OmniGuide fiber.

 

4.1.1. OmniGuide Fiber Manufacture: Preform Construction and Draw

The manufacturing processes of OmniGuide fiber are shown schematically in Fig. 10.25. This manufacturing technique enables easy tuning of the operational band-gap of the OmniGuide fiber, allowing it to transmit light of almost any desired wavelength.

For example, the technique described here has been successfully used to manufacture OmniGuide fibers operational at the CO₂ (\(\lambda\) = 10.6 μm) and Er:YAG (\(\lambda\) = 2.94 μm) laser wavelengths, using the same raw materials.

 

 

The high-index material, the chalcogenide glass As₂Se₃, is synthesized by melting the glass components inside a clean, evacuated, fused silica tube at 600\(^\circ\)C. The chemical uniformity of the glass is ensured by rocking the entire furnace assembly while at the melt temperature, thoroughly mixing the glass components.

A multilayer preform is produced by first evaporating a film of As₂Se₃ glass onto both sides of a sheet of the polymer PES. The coated sheet of polymer is then rolled into a cylinder, adding more PES sheets to form an outer cladding for mechanical strength.

Heating under vacuum then consolidates the rolled structure. This method allows the production of hollow-core preforms with more than 40 layers. Depending on the desired operating wavelength range, the PES sheet thickness varies from 25 to 50 μm and the deposited As₂Se₃ film thickness is 12–25 μm. These layer thicknesses can be scaled up as desired, although for thicker glass layer deposition, care must be taken to prevent cracking during the rolling process.

After consolidation, the preform is drawn into fiber. At this step, the drawdown ratio can be adjusted to control the layer thicknesses in the resulting fiber and thereby produce fibers that transmit at a desired wavelength. A single preform of approximately 30-cm length can yield more than 100 m of CO₂ fiber (\(\lambda\) = 10.6 μm) or 300 m of Er:YAG fiber (\(\lambda\) = 2.94 μm). Continuous monitoring of the outer diameter, as well as band-gap and inner diameter measurements performed on sample fibers during the draw, is used to ensure that the fiber has the desired optical properties.

Another preform/draw technique was successfully used to fabricate Bragglike hollow-core fibers. These fibers are composed of concentric cylindrical silica rings separated by nanometer-scale support bridges.

Because these support bridges are so small, the region between concentric silica rings behaves like a low-index layer (i.e., a layer of air). These alternating air–silica rings form a 1D photonic band-gap.

Although these fibers are useful for transmitting wavelengths below 2 μm, the use of silica prevents their application in the mid-IR. The use of a glass transparent in the mid-IR should eliminate this limitation, but the formation and preservation of the thin support bridges with such a glass would remain a challenge.

 

4.1.2. OmniGuide Fiber: Proximal End

For easy connection to a laser source, a standard ST connector is added to the proximal end of the OmniGuide fiber. The ferrule inside diameter (ID) in the connector is chosen to be slightly smaller than the fiber ID to protect the endface of the fiber (which is not reflective) from the incident laser beam.

The connector is attached to the outer surface of the fiber using a high thermal-conductivity epoxy for improved heat transfer. For CO₂ fibers, a short hollow zirconia tube is used as the ferrule to protect the fiber input and reduce the heat generated in the fiber near the coupling plane by filtering out higher order modes. For similar purposes, copper tubes are used to couple into Er:YAG fibers.

 

4.1.3. OmniGuide Fiber: Distal End Termination

For surgical applications, where the fiber will encounter a semi-aquatic environment, precautions are necessary to prevent liquids from entering the hollow core. For CO₂ fibers, continuous helium flow through the hollow core is used to solve this problem. To avoid damage to the fiber tip that results from clogging when in contact with tissue, a short stainless-steel tube is attached to the end of the fiber, with side holes to aid in gas flow venting.

However, not all applications allow for gas flow through the core. For example, in Er:YAG laser endoscopic lithotripsy (kidney stone destruction), which takes place in the closed aqueous environment of the urinary tract, gas flow cannot be used. In such cases, the distal end needs to be sealed off so liquid and debris do not penetrate into the fiber core. Distal-end pieces sealed with a low-OH silica window have been used successfully for this purpose. In either case, tips are attached to the outer diameter of the fiber with epoxy.

 

4.2. Techniques Used in the Manufacture of Other Hollow-Core Fibers

4.2.1. Hollow-Core Metal-Coated Fibers

There are many approaches used in the manufacture of hollow-core metal fibers. Early attempts to make rectangular metal waveguides by separating the metal strips with a metal or plastic spacer were successful; the resulting waveguides were able to transmit CO₂ laser powers on the order of kilowatts.

The use of an additional dielectric layer coating to lower the metal waveguide losses was also used with rectangular waveguides and was combined with new fabrication techniques to reduce the core size for better mode quality.

However, because the core was rectangular, these guides suffered from increased losses due to twists and had limited flexibility. Also, the core sizes achieved (1 x 1 mm) were still not small enough for good beam quality.

Circular metallic guides, which do not have the problem associated with twisting, were pioneered by Miyagi et al.. The fabrication of this type of fiber typically starts with an aluminum or glass mandrel, which is then coated by a dielectric (Ge, AsSe, or ZnS) and a metal film (Ag, Cu, or Au). Finally, a thick nickel layer is deposited on top of the metal layer. The mandrel is etched away to leave a hollow core.

Bhardwaj et al. used a different technique, starting with an extruded Ag tube with hollow core. The dielectric layer inside the tube, which is typically AgBr or AgCl, is deposited inside the core using liquid- or gas-phase reaction with Ag. The fibers fabricated with this technique suffered from surface roughness that increased the losses, despite polishing of the Ag core with an acid solution.

Alaluf et al. used a different technique, starting with polyethylene and Teflon tubing, to obtain flexible and low-cost fibers. They coated the inside surface of the tube with Ag and used wet or liquid chemistry to convert some of the Ag to AgI to form the dielectric layer.

George and Harrington improved the surface roughness on this type of fibers using polycarbonate or similar tubing that has better surface quality. Harrington also worked with silica glass tubing to improve the surface roughness.

In this technique, the starting Ag film needs to be thick to form both the conducting metal boundary and the desired AgI film after reaction with iodine. However, the Ag surface roughness also increases with film thickness. In addition to this roughness/thickness problem, it is also difficult to achieve a uniform film thickness over long lengths of fiber.

Miyagi et al. used a different approach to produce a dielectric film over the metal that does not depend on subtraction of the metal layer. They used liquidphase techniques to deposit polymer films on a thin Ag layer. Because they did not need a thick Ag film to start with, they were able to obtain much less surface roughness, lowering the losses of the fiber. However, the use of a relatively lossy polymer layer (17,000 dB/m at \(\lambda\) = 10.6 μm) limits the power handling capacity of this fiber.

Two important drawbacks are associated with all these techniques. The first is that the manufacturing process is not readily scalable. The metallic hollow-core fibers have to be processed individually to have a metallic and dielectric layer inside the small bore tubing, in contrast to a preform/draw process, which is easily scaled up simply by increasing the size of the preform.

Indeed, the preform/draw process associated with silica fiber manufacture has been scaled up from preforms yielding 12 km to preforms yielding more than 2000 km. The second drawback associated with the manufacture of hollow metal waveguides, with or without dielectric coatings, is the inability to easily reduce inhomogeneities or geometric non-uniformities formed during the macroscopic processing.

With a preform/draw process, non-uniformities introduced during the macroscopic preform production are greatly elongated during the subsequent draw process, decreasing their impact on the optical performance of the resulting fiber.

 

4.2.2. Hollow-Core 2D Photonic-Crystal Fibers

Hollow-core 2D PCFs are manufactured by drawing bundles of silica capillary tubes, where the hollow core is formed by removing some tubes from the center.

The bundle surrounding the hollow core forms the 2D photonic band-gap, which enables the guiding of light in the core. Fibers were successfully fabricated with this technique, but with the limitations on the operational wavelength imposed by the material absorption of silica (little transmission above 2 μm).

There have been efforts to manufacture similar structures using glasses transparent to mid-IR wavelengths, but preserving the uniformity of the structure during draw, which is necessary to keep losses low, is a challenge.

 

5. Conclusions

Hollow-core fibers offer a number of significant advantages over traditional solid-core fibers. They greatly ease the constraints—absorption, nonlinearity, material dispersion—associated with propagation through the core material and, by proper choice of cladding materials and geometry, are capable of guiding radiation of almost any wavelength.

Of particular interest for both commercial and academic applications are hollow-core photonic band-gap fibers, which, in different implementations, may be capable of exceeding the performance characteristics of silica-based solid-core fiber in the near-IR and are capable of the flexible low-loss transmission of mid- to far-IR radiation.

This latter capability is already opening up exciting new frontiers for minimally invasive laser surgery. Although the measured performance of these fibers has not yet approached theoretical limits, steady improvements in manufacturing technology and fiber design have reduced this gap and suggest that these fibers will play an increasingly important role in wave-guiding applications.

 

The next tutorial discusses about linear pulse propagation.