## Abstract

An analytic expression is obtained for the confinement loss of model anti-resonant fibres consisting of concentric regions of air and glass. Hankel functions in the regions surrounding the air core are approximated by their asymptotic form; apart from this, results are correct to leading order in the small parameter 1/(*k*_{0}*r _{c}*), where

*r*is the core radius and

_{c}*k*

_{0}the free space wavenumber. The results extend and generalise previous solutions for propagation in a hollow glass tube and a thin-walled capillary. Comparison with exact numerical calculations shows that the analytic expression provides an accurate description of the loss, including its dependence on the mode, the core radius and the widths of the surrounding glass and air regions. The relevance of the results to the recent generation of hollow-core, anti-resonant photonic crystal fibres is discussed.

© 2017 Optical Society of America

## 1. Introduction

There has been considerable recent interest in a new generation of hollow-core fibres with broad transmission windows and low loss, and their potential applications for high power delivery and guidance into the mid-infrared (see, for example, [1–7] and references therein). The basic mechanism of the guidance is qualitatively well understood in terms of anti-resonant reflection from the glass and air elements surrounding the core [3, 6, 8]. Another important structural feature is the shape of the core wall, and a negative curvature has been shown to have a substantial impact in reducing the attenuation [1,7,9,10].

Large-scale finite element calculations have been widely used to study the attenuation of realistic hollow-core, anti-resonant fibre structures. Although good agreement with experiment is found, such calculations do not always provide an understanding of the fundamental physics of the guidance. With this in mind, several groups have used model fibre structures, for which calculations can be performed rapidly, to provide insight into different aspects of the guidance mechanism. These model fibres consist of an air core, whose radius is large compared to the free space wavelength, surrounded by concentric glass and air regions. The widths of the cladding layers can be chosen to provide the conditions for anti-resonant reflection and therefore a reduced attenuation. Fig. 1 shows model fibres with an increasing number, *N*, of anti-resonant layers; for example, for *N* = 2, the air core is surrounded by an anti-resonant glass and an anti-resonant air region, with the outermost glass region extending to infinity. The structures shown in Fig. 1 are the focus of this paper, and examples of their previous use include: models of guidance in kagome fibres [11, 12] (both of these papers use the *N* = 1 and *N* = 3 models of Fig. 1); optimisation of the attenuation of hollow anti-resonant fibres (using *N* = 1 and *N* = 2) [13]; anti-resonant guidance in capillaries (using *N* = 1) [14]; analysis of a double anti-resonant fibre (using *N* = 1 and *N* = 2) [15]; analysis of hexagram fibres (using *N* = 1 and *N* = 2) [16]; use as a reference structure in the analysis of nodeless anti-resonant fibre designs (using *N* = 2) [3].

As well as being efficient for numerical calculations, the model fibres of Fig. 1 are also analytically tractable. The *N* = 0 case was analysed in the classic paper by Marcatili & Schmeltzer [17] and their result that the attenuation in a thick-walled capillary scales as ${\lambda}_{0}^{2}/{r}_{c}^{3}$ (where *λ*_{0} is the free space wavelength and *r _{c}* the radius of the air core) has been widely used. Miyagi & Nishida [18] considered the

*N*= 1 case and found a ${\lambda}_{0}^{3}/{r}_{c}^{4}$ scaling of the attenuation, although they give few details of the derivation and the approximations made. Archambault et al. [19] presented expressions for the attenuation of TE and TM modes in anti-resonant waveguides that include the

*N*= 1 and

*N*= 2 structures of Fig. 1, again without detailing the approximations made. They also made an ansatz for how their results can be extended to HE modes by averaging the attenuation of TE and TM solutions. Poletti et al. [13] showed that numerical results for the attenuation of the

*N*= 2 structure have a ${\lambda}_{0}^{4}/{r}_{c}^{5}$ dependence, and although it reasonable to assume that these scalings hold to

*N*= 2 and above [3], this has not explicitly been shown. The model air-glass structures of Fig. 1 are a special case of the more general class of Bragg fibres. Yeh, Yariv & Marom [20] set out a matrix-based theory of Bragg fibres, using a Bessel function basis, and this has been extensively used in both numerical calculations and theoretical analysis (for example [21–24]). These papers use an asymptotic approximation of the Bessel functions in the outer regions of the cladding, and make further approximations in the matrix formulation relevant to low index contrast Bragg fibres.

In this paper, analytic expressions are derived for the core-guided leaky modes of the model fibres shown in Fig. 1 (Hu & Menyuk [25] provide an excellent review of the theoretical basis of leaky modes). The analysis makes use of the fact that hollow fibres with a large core are characterised by a large parameter *D* = 2*πr _{c}/λ*

_{0}=

*k*

_{0}

*r*, which can be thought of either as a normalised radius or a normalised wavevector. As in previous work, an asymptotic approximation of Bessel functions is used but, apart from this, the results are correct to leading order in 1/

_{c}*D*. All modes are considered on the same footing and results for HE and EH modes are obtained directly without making further assumptions.

The paper is organised as follows. In section 2 the analytic expression for the confinement loss is given, together with an outline of its derivation, emphasising the features that are specific to anti-resonant fibres with an air core that is large relative to the free space wavelength. A comparison with exact numerical results is presented in section 3. In section 4 the relevance of the results for practical fibre designs is discussed, including the dependence of the loss on the mode, dielectric constant, core radius and the number and width of anti-resonant structural elements. Details of the derivation are given in the Appendix (section 5).

## 2. Analytic expression for the confinement loss

The main result of the paper is that the confinement loss (in dB) incurred in propagation through a distance of one wavelength *λ*_{0} for the model fibres shown in Fig. 1 is given to leading order in 1/*D* by

*N*more clearly; the factor of 8.686 (= 20/ ln 10) arises from converting the imaginary part of the propagation constant to a loss in dB.

*x*

_{0}is a dimensionless quantity that is related to the transverse wavevector in the core. For TE

_{0m}and TM

_{0m}modes,

*x*

_{0}is the

*m*

^{th}(non-trivial) zero of the Bessel function

*J*

_{1}, for HE

*modes it is the*

_{nm}*m*

^{th}zero of

*J*

_{n−1}, and for EH

*modes it is the*

_{nm}*m*

^{th}zero of

*J*

_{n+1}(see Eqs. (41)–(43)).

*∊*is the relative dielectric constant of the glass regions (the dielectric constant of air is assumed to be 1), and because the focus of the paper is on the basic anti-resonant guidance properties of the multi-layered structures, material loss is neglected, and

*∊*is assumed to be real. The

*∊*-dependent factors for TE and TM modes are the same as those found by [17–19] and Eq. (1) shows that the

*∊*-dependence of EH and HE modes is given by the average of the TE and TM terms. This generalises the results of [17] and [18] but, as discussed in section 5.5, there are corrections to this simple average for model structures with

*N*= 2 and above. These correction terms are neglected in Eq. (1) and the justification for doing this is also given in section 5.5.

As has been previously found [3, 13, 17–19] the loss given by Eq. (1) scales as (*λ*_{0}/*r _{c}*)

^{N+3}. The recent generation of hollow-core, anti-resonant fibres typically has

*r*

_{c}/λ_{0}of order 10 to 20, and the exponential decrease of attenuation with the number of anti-resonant layers,

*N*, accounts, at least in part, for the low loss of these fibres. The final product term in Eq. (1) describes the dependence of the loss on the widths of the anti-resonant glass and air layers (for

*N*= 0 the product term is set to one, and Eq. (1) reduces to the expression for the hollow glass tube given by [17]).

*ϕ*are phase angles corresponding to each successive layer in the structure, and for glass and air layers are given by (see Eqs. (29) and (32))

_{i}*i*

^{th}layer. The loss is minimised for

*ϕ*=

_{i}*π*/2, 3

*π*/2, . . . and this corresponds to the layer being perfectly anti-resonant. It follows from Eq. (2) that the thickness of an anti-resonant air layer depends on the mode (via

*x*

_{0}) but is independent of

*λ*

_{0}; conversely, the width of an anti-resonant glass layer depends on

*λ*

_{0}but is independent of the mode [13, 19]. Equation (1) shows that, in terms of anti-resonance, the individual layers of a multi-layered structure act independently of one another. It also shows that, due to the 1/ sin

^{2}

*ϕ*dependence, the loss is relatively insensitive to the precise width of the air and glass regions, provided they are fairly close to anti-resonance.

The loss given by Eq. (1) diverges as any of the layers tends towards a resonant thickness, where sin *ϕ _{i}* → 0. This is a consequence of retaining only the leading term in 1/

*D*, which provides the dominant contribution in the physically most important regime, close to anti-resonance. As shown in sections 5.4 and 5.5, for

*N*> 0 Eq. (1) is derived by retaining only the highest order terms in

*D*in both the numerator and denominator of an expression, so that the resulting loss is correct to leading order in 1/

*D*. As sin

*ϕ*for any layer tends towards zero, terms in the denominator with a lower power of

_{i}*D*will start to become dominant, and this suppresses the divergence.

A full derivation of Eq. (1) is given in the Appendix (section 5). A wave matching methodology, based on that of Yeh, Yariv and Marom [20], is presented in section 5.1, leading to a matrix formulation that can be used to find exact, numerical values of the propagation constant and hence the attenuation of leaky modes. The analysis exploits the cylindrical symmetry of the model structures, and solutions are characterised by an integer *n* that describes the angular variation of the fields. TE and TM modes correspond to *n* = 0 solutions and for these modes the analysis is considerably simplified because the 4 × 4 matrices of the general formulation reduce to 2 × 2 blocks. A parameterisation appropriate for large-core hollow fibres is introduced in section 5.2 and two key approximations are discussed that enable an analytic solution to be developed. First, it is assumed that *D* is large, both in absolute terms and in comparison with *x*_{0}. Second, outside the core region, Hankel functions are approximated by their asymptotic forms that are valid for large arguments. This is a good approximation for the glass layers but is more questionable for air regions. The magnitude of the error introduced by this approximation is shown to be of order *n/x*_{0}; this is the primary source of error in Eq. (1) and will be discussed further in the next section, where the results of Eq. (1) are compared with exact numerical calculations. Based on these approximations, transfer matrices are derived that form the basis of the subsequent analysis.

The *N* = 0 case is solved in section 5.3 and the results of Marcatili & Schmeltzer [17] are shown to arise from the leading order term in powers of 1/*D*. In section 5.4 TE and TM modes are analysed for model structures with *N* > 0. With the simplified matrix algebra for these cases it is shown that Eqs. (1a) and (1b) can be derived for an arbitrary number of layers *N*. A similar approach is used in section 5.5 to analyse HE and EH solutions but, in this case, it has not been possible to solve for all *N*. Eq. (1) is shown to hold (without the correction factors referred to above) up to *N* = 4, that is, for all the model structures shown in Fig. 1.

## 3. Comparison with numerical results

Table 1 gives numerical results for the confinement loss for *r _{c}/λ*

_{0}= 10, 15 and 20, (i.e.

*D*= 62.8, 94.2, 125.7), together with those obtained from Eq. (1). Numerical results are calculated using the method described in sections 5.1 and 5.2 with no approximations in the matrices or Hankel functions; values of the loss are essentially exact and are correct to the number of decimal places given. The selected values of

*r*

_{c}/λ_{0}have been chosen as being typical of experimental fibre designs. Values of the loss in Table 1 are scaled by a factor of (

*λ*

_{0}/

*r*)

_{c}^{N+3}to make comparisons between different structures and modes more clear. The 7 lowest order modes are included in the Table, plus TE

_{02}, as this mode has a lower attenuation than neighbouring higher order modes. The value of

*x*

_{0}for each mode is also given; equivalent results for

*x*(see section 5.2) obtained from numerical calculations are less than 0.2% different from

*x*

_{0}in all cases. For all the results in Table 1 the structure is chosen such that the phase angle

*ϕ*=

_{i}*π*/2 for all layers

*i*; from Eq. (2) the widths of glass and air layers are then

*r*

_{c}/λ_{0}, and any variation in the exact numerical results provides an indication of the accuracy of the (

*λ*

_{0}/

*r*)

_{c}^{N+3}scaling. It can be seen from Table 1 that the scaled numerical results for loss do not vary significantly with

*r*

_{c}/λ_{0}. The maximum percentage difference between results for

*r*

_{c}/λ_{0}= 10 and 20 is just 1.5%, occurring for the HE

_{12}and TE

_{02}modes for

*N*= 4, and most of the differences are significantly less than 1%. This indicates that the (

*λ*

_{0}/

*r*)

_{c}^{N+3}dependence of Eq. (1) is robust and that this scaling holds accurately in the exact case.

For *N* = 0 the numerical and analytic values of the loss given in Table 1 are very close to one another (numerical results are given only for *r _{c}/λ*

_{0}= 15 because differences for other values of

*r*

_{c}/λ_{0}are insignificant). This agreement is to be expected because, as discussed in section 5.2, for

*N*= 0 the argument of the Hankel function in the outermost glass region is large and Eq. (26) provides an excellent approximation. Larger differences between numerical and analytic results are found for the structures with higher values of

*N*. Table 1 gives the percentage error between the analytic expression and exact results for

*r*

_{c}/λ_{0}= 15, and it is found that the difference varies from 1% to 37% for the selected modes. The only additional approximation made for structures with

*N*> 0 relates to the form of the Hankel functions in air regions, and the variations seen in the Table are essentially a reflection of this approximation. There is a general trend that the percentage error between numerical and analytic results increases with

*N*for each mode. Within each structure, modes with

*n*= 0 (TE

_{01}, TM

_{01}, TE

_{02}) generally show the smallest error, and it is interesting to note that the analytic expression is most accurate for TE

_{02}, where the Hankel functions in the outer air regions will be better approximated as an exponential. Modes with

*n*= 1 (HE

_{11}, EH

_{11}, HE

_{12}) are the next most accurately described by Eq. (1), and again it is noted that the higher order mode HE

_{12}is the most accurate of this set. The largest difference between numerical and analytic values of the scaled loss is found for HE

_{21}and HE

_{31}, which have

*n*= 2 and 3 respectively and the largest values of (

*n/x*

_{0}). Despite some of the percentage differences being large, the key point of Table 1 is that the analytic expressions provide a remarkably good description of how the confinement loss varies across structures and modes. For example, the actual loss in one wavelength of HE

_{12}for

*N*= 1 and

*r*

_{c}/λ_{0}= 10, and TE

_{01}for

*N*= 4 and

*r*

_{c}/λ_{0}= 20, differ by nearly 7 orders of magnitude, and this difference is accurately captured by the analytic expression.

Although the loss given by Eq. (1) is minimised for the anti-resonant layer thicknesses given by Eq. (3), it is possible in principle that these widths are not optimal in the exact case, since Eq. (3) is derived following the approximation of the Hankel functions in Eq. (26). To test this, the loss given by full numerical calculations has been minimised as a function of the layer widths. Results are shown in Table 2 for *r _{c}/λ*

_{0}= 15. The percentage change of each optimal layer thickness, with respect to that given by Eq. (3), is shown for each mode and structure. It is found that the variation is small, with the exact optimal thickness typically being within 5% of the basic anti-resonant width. The reduction of the loss relative to that given in Table 1 is also small: in most cases the difference is less than 1%, with the largest reduction of 2.5% being for the HE

_{31}mode for

*N*= 4.

The 1/ sin^{2} *ϕ* dependence of the loss predicted by Eq. (1) has also been tested against full numerical calculations. Fig. 2 shows contour plots of the scaled loss of the HE_{11} and TE_{01} modes for *N* = 2 as a function of the thickness of the glass and air layers (i.e. g1 and a2). It can be seen that Eq. (1) provides an accurate description of the loss away from anti-resonance, with the discrepancy near the centre of the plots being a reflection of the small difference between the exact and analytic results at anti-resonance. The 1/ sin^{2} *ϕ _{i}* factors in Eq. (1) are symmetrical with respect to the anti-resonant condition, as can be seen in the ‘approx’ plots in Fig. 2. The numerical results are not exactly symmetrical, but this asymmetry becomes noticeable only when the scaled width is less than 0.5 or greater than 1.5. Fig. 3 shows plots of the scaled loss for the same modes and for

*N*= 4. Each of the four layer widths are varied separately, with the other layers held at their anti-resonant thickness (i.e.

*ϕ*=

_{i}*π*/2). Results given by Eq. (1) are the same for all four layers, because the variation of layer width is shown as a proportion of the anti-resonant thickness. The numerical results show a small difference between the variations of the different layers, but the overall picture is that Eq. (1) provides a very good approximation of the exact results away from anti-resonance.

## 4. Discussion and conclusions

An analytic expression has been derived for the confinement loss of model fibres consisting of concentric layers of glass and air surrounding an air core whose radius is large compared to the free space wavelength. Results for TE and TM modes apply to an arbitrary number of layers, while for HE and EH modes the loss has been calculated only up to four complete layers. However, it is likely that Eq. (1) holds for *N* > 4. The loss has been calculated to leading order in the small parameter 1/*D*, and the main source of error is in the approximation of Hankel functions by an exponential form. Comparison with exact numerical results shows that the analytic expression provides a good description of the attenuation, perhaps surprisingly good given the rather crude nature of the approximation Eq. (26) for air regions. For model structures with *N* ≥ 2 there are correction factors to the result that the loss for HE and EH modes is given by the average of TE- and TM-like expressions. However, the size of these correction terms is of a similar order to the error introduced by the Hankel function approximation, and it is therefore reasonable to drop them, given that they do not systematically improve the agreement with exact numerical results.

The analytic result Eq. (1) can be used to understand the variation of loss with mode, *r _{c}/λ*

_{0}and structure seen in Table 1 and Figs. 2 and 3. The largest effect is the (

*λ*

_{0}/

*r*)

_{c}^{N+3}scaling, but the other terms in Eq. (1) are also important. For TE modes, the

*∊*dependent factor is less than one provided

*∊*> 2 and, with the chosen value of

*∊*= 2.25, it acts to decrease the loss with increasing

*N*. For all other modes the

*∊*dependent term is unfavourable in terms of loss and it increases markedly with

*N*. Interestingly, the

*∊*dependent term for TM, HE and EH modes has a minimum value for

*∊*≈ 2 (it is exactly 2 for the TM function

*∊*/ √(

*∊*− 1)), which means that silica glass is close to being optimal in reducing the attenuation of these modes (including the fundamental HE

_{11}mode). The pattern of loss values seen in Table 1 arises from a combination of these

*∊*dependent factors with the mode-dependent (

*x*

_{0}/2

*π*)

^{N+2}term.

*x*

_{0}/2

*π*is less than 1 for all the modes considered except TE

_{02}, and so this factor tends to reduce the attenuation as

*N*increases, with this effect being most significant for the HE

_{11}mode. The overall competition between the

*∊*and

*x*

_{0}dependent terms means that there is a marked discrimination between the low order modes, with HE

_{11}and TE

_{01}having a loss substantially lower (an order of magnitude lower for

*N*≥ 3) than that of the other modes.

Equation (3) shows that the optimum thickness of air layers depends on the mode (via *x*_{0}) and it follows that if the air layers are exactly at anti-resonance for one mode, they will be non-optimal for modes with a different value of *x*_{0}. The 1/ sin^{2} *ϕ _{i}* dependence in Eq. (1) means that the loss is relatively insensitive to the precise width of the air and glass regions, and an individual layer can be as much as 50% thinner or thicker than the anti-resonant width and the loss increases only by a factor of 2. This creates the opportunity for enhancing the discrimination between modes. For example, the anti-resonant widths

*w*

^{(a)}for HE

_{11}and TE

_{01}are 0.653

*r*and 0.410

_{c}*r*respectively. If an air layer has a width of 0.820

_{c}*r*, the TE

_{c}_{01}mode will be resonant and strongly attenuated, while the loss of the HE

_{11}mode is increased by only 18%. This feature has been exploited in the design of real fibres that are essentially single-moded [5,26].

The model structures have features in common with practical anti-resonant-guiding fibres, but they are obviously highly simplified (and, for air-glass structures, unphysical for *N* ≥ 2). Nevertheless, it is anticipated that the results will be useful in providing an insight into realistic fibre designs, including the dependence of the loss on *r _{c}/λ*

_{0}, mode,

*∊*and the width of anti-resonant elements of the structure. For example, it would be possible to compare the variation of the loss of different modes of realistic fibres, as a function of

*r*

_{c}/λ_{0}and

*∊*, with those predicted by Eq. (1). This could be achieved with numerical calculations of the attenuation, and it would be interesting to see if realistic fibres can be described by a value of

*N*, and whether or not this value is consistent across the variation of different parameters. Previous calculations [3,27] have touched on some aspects of this. For example, Poletti [3] found for the fundamental mode of a nested nodeless anti-resonant fibre that the attenuation scales as ${\lambda}_{0}^{7}/{r}_{c}^{8}$, which indicates an

*N*= 5 behaviour. At first sight, this nested fibre would appear to be most closely analogous to the model

*N*= 4 structure, which implies that the space between the cladding rings and the outer jacket is providing additional anti-resonant reflection. A full, systematic study would provide an understanding of which aspects of the loss are or are not captured by the model structures, from which it may be possible to improve realistic fibre designs.

An interesting open question is the physics underlying the reduction of the attenuation seen with negative curvature of the core wall. In some respects, the model fibres considered here would appear to be optimal, with every element of the structure being anti-resonant and with no direct connection between the glass regions and the outer glass jacket. Any coupling between core modes and high-order cladding modes is also strictly zero by symmetry. However, finite element calculations indicate that realistic fibre designs with negative curvature have a lower attenuation than the most closely-matched model structure [3, 7, 9]. A useful extension of the calculations presented here would be to consider perturbation in the shape of the core wall to see if this can provide insight into the effect of negative curvature.

Finally, it would also be informative to include material loss via a complex dielectric constant, and to calculate the real as well as the imaginary part of the propagation constant. These extensions are straightforward for the *N* = 0 case [17] but are more problematic for *N* > 0. This is because the analysis in sections 5.4 and 5.5 depends on a careful separation of real and imaginary parts, taking into account powers of *D*, followed by cancellations that occur in the expressions for the imaginary part of the propagation constant. However, it is anticipated that analytic progress can be made with both the real part of the propagation constant and a complex dielectric constant, and this will be returned to in a later paper.

## 5. Appendix: Derivation of Eq. (1)

## 5.1. Wave matching

A cylindrically symmetric structure is considered that is uniform along its length (the *z* direction), consisting of concentric regions of glass, with relative dielectric constant *∊* (assumed to be real), and air, with *∊* = 1. The innermost, core region is air, the outermost region is assumed to extend to infinity, and each structure is labelled by an integer *N* that refers to the number of complete, finite layers it contains (Fig. 1).

The analysis is based on that of Yeh, Yariv and Marom [20] (hereafter referred to as YYM). Cylindrical polar coordinates (*r*, *θ*, *z*) are used and all field components have a *z* dependence of the form exp(*iβz*), where *β* is the propagation constant. The basic variables are the *z* components of the electric and magnetic fields:

*n*is an integer, ${H}_{n}^{(1)}$ and ${H}_{n}^{(2)}$ are Hankel functions of the first and second kind respectively, and

*a*,

*b*,

*c*,

*d*,

*ϕ*and

*ψ*are constants. To simplify the notation a scaled magnetic field defined by

**H̃**(

*r*,

*θ*) = (

*μ*

_{0}/

*∊*

_{0})

^{1/2}

**H**(

*r*,

*θ*) is used. The transverse wavevector

*k*is given by where

*k*

_{0}is the free space wavevector. Hankel functions are used in Eq. (4) rather than the Bessel functions

*J*and

_{n}*Y*used by YYM because, for the leaky modes considered here, they are a more natural basis for waves in the outermost region. All the other field components can be expressed in terms of

_{n}*E*and

_{z}*H̃*as:

_{z}Following YYM, standard electromagnetic boundary conditions are applied to derive the relationship between the coefficients in one layer (*a*_{1}, *b*_{1}, *c*_{1}, *d*_{1}) to those in the adjacent layer in the outward direction (*a*_{2}, *b*_{2}, *c*_{2}, *d*_{2}). The boundary conditions are that *E _{z}*,

*H̃*,

_{z}*E*and

_{θ}*H̃*must match across each boundary, and applying this to a boundary at

_{θ}*r*=

*R*leads to an equation of the form

*M*is a 4 × 4 matrix:

*H′*is the derivative of the Hankel function. Equation (8) is essentially the same as in YYM, with the four rows of the matrix corresponding to matching

_{n}*E*,

_{z}*H̃*,

_{θ}*H̃*and

_{z}*E*respectively. As discussed in YYM, Eq. (8) refers to solutions with

_{θ}*nβ/k*

^{2}

*R*factors in Eq. (8) become −

*nβ/k*

^{2}

*R*.

By using Eq. (7) at each boundary the coefficients in the outermost region (*a _{f}*,

*b*,

_{f}*c*,

_{f}*df*) of the structure can be expressed in terms of those in the core (

*a*,

_{c}*b*,

_{c}*c*,

_{c}*d*) by:

_{c}*r*(

_{i}*i*= 1, . . . ,

*N*) is the outer radius of the

*i*

^{th}complete layer in the concentric structure (counting from the centre outwards).

*k*and

_{c}*k*are the transverse wavevectors in the core and outermost regions respectively, and

_{f}*k*is the transverse wavevector of layer

_{i}*i*. As shown by YYM, the inverse matrices

*M*

^{−1}(

*k*,

*R*) can readily be constructed, and examples of inverse matrices will be given later. It follows from Eqs. (4a) and (4b) that modes that are finite at the origin must have and that solutions with only outwardly propagating waves at infinity satisfy Equations (12) and (13) are sufficient, along with Eq. (11), to determine the propagation constants

*β*of the allowed solutions. In a general case, if the refractive index of the core is greater than that of the outermost region, guided modes will exist for which

*β*is real (assuming lossless materials). If the refractive index of the core is equal to or less than that of the outermost region (as is the case for the structures considered here), there will be no bound states;

*β*will be complex even if the materials are lossless and the solutions represent leaky modes.

The matrix algebra is considerably simplified for solutions with *n* = 0 because Eq. (8) becomes block diagonal. The upper left block represents TM modes for which *H̃ _{z}* =

*E*= 0 and the lower right block represents TE modes for which

_{θ}*E*=

_{z}*H̃*= 0.

_{θ}## 5.2. Approximation for large-core PCF

The focus of the paper is on the leaky modes of hollow core fibres whose core diameter is large compared to the free space wavelength. It follows that *β* will be close to *k*_{0} and that the transverse wavevector in air regions (including the core) will be small relative to *k*_{0}. To reflect this the system is parameterised using dimensionless quantities *x* and *D* defined by:

*x*and

*D*. The propagation constant becomes and for a general dielectric constant

*∊*:

*∊*= 1 these become

*N*. The system is characterised by

*∊*and the set of dimensionless radii

*r*, while

_{i}/r_{c}*D*defines the core radius in terms of the wavelength. Equations (12) and (13) are used in Eq. (11) to search for allowed values of

*x*for any given value of

*n*, and the (complex) propagation constant then follows from Eq. (15).

For the fibres considered here, *r _{c}/λ*

_{0}will be of order 10 to 20, and therefore

*D*will be large, typically of order 50 to 100. As will be shown,

*x*is of order unity, typically between 2 and 6 for the lowest order modes. The fact that

*D*is large, both in absolute terms and in comparison with

*x*, is exploited by retaining only the largest order terms in

*D*in deriving an approximate expression for the loss of the leaky modes.

The *x*^{2}/*D*^{2} terms in Eqs. (16) and (17) are dropped as these are small relative to one and to (*∊* − 1) for glass-air structures. Equation (8) and its inverse then become:

*g*indicates that these matrices refer to glass regions. The equivalent expressions for air are:

In YYM, the *M* and *M*^{−1} matrices for a boundary at a given radius are combined, and an explicit expression for *M*^{−1}(*k*_{2}, *R*) × *M*(*k*_{1}, *R*) is given. Instead of this, *M* and *M*^{−1} are combined to provide a transfer matrix across a given layer of the structure:

*z*is a dummy argument)

*i*and −

*i*respectively. An indication of the validity of these approximations can be obtained by considering the standard recurrence relations for Bessel functions: ${{H}_{n}^{(m)}}^{\prime}(z)=\frac{1}{2}\left({H}_{n-1}^{(m)}(z)-{H}_{n+1}^{(m)}(z)\right)$; ${{H}_{n}^{(m)}}^{\prime}(z)={H}_{n-1}^{(m)}(z)-(n/z){H}_{n+1}^{(m)}(z)$; ${{H}_{n}^{(m)}}^{\prime}(z)=-{{H}_{n+1}^{(m)}}^{\prime}(z)+(n/z){H}_{n}^{(m)}(z)$ (where

*m*= 1 or 2). Only the first of these expressions, in conjunction with Eqs. (26a) and (26b), gives ${{H}_{n}^{(1)}}^{\prime}/{H}_{n}^{(1)}=i$ and ${{H}_{n}^{(2)}}^{\prime}/{H}_{n}^{(2)}=-i$; the other two forms differ by a term of order

*n/z*. It follows from Eq. (20) that for glass layers the error in approximating the Hankel functions will be small: of order 1/

*D*. However, Eq. (23) indicates that the approximation is less robust for air layers, and can be expected to introduce errors of order

*n/x*.

Equations (26a) and (26b) are substituted into Eqs. (18), (19), (21) and (22), and Eq. (24) is used to derive the transfer matrix for a layer with inner and outer radii of *R*_{1} and *R*_{2} respectively. For a glass layer the transfer matrix becomes

*t*

_{1}is given by

*R*=

*R*

_{2}−

*R*

_{1}. The equivalent result for an air layer is

## 5.3. N = 0 case

In this section the leaky modes of a hollow glass tube are analysed, where it is assumed that the outer glass region extends to infinity. The results for this structure, labelled *N* = 0 in Fig. 1, provide the basis for the multi-layered structures considered in the following two subsections. Expressions for the attenuation were first derived by Marcatili & Schmeltzer [17].

For *N* = 0, Eq. (25) reduces to

*p*is a constant to be determined) and multiplying by Eq. (21) evaluated at

*r*gives

_{c}*v̱*

_{1}and

*v̱*

_{2}have been introduced for use in section 5.5. The condition for allowed solutions follows from Eq. (13); the second and fourth rows of Eq. (19), evaluated at

*r*, are multiplied into Eq. (36) and setting the result equal to zero yields

_{c}*n*= 0 case, for which Eqs. (37a) and (37b) decouple into separate equations for the TM and TE modes respectively.

Equation (37) has a clear separation of its real and imaginary parts in terms of the large parameter *D*, and it is therefore natural to look for solutions in the form of a power series in 1/*D*:

*x*

_{0}and

*p*

_{0}. For

*n*= 0,

*p*

_{0}is undetermined and it follows that solutions must satisfy

*x*

_{0}values are determined by the zeroes of

*J*

_{1}. For

*n*≠ 0 it can be seen by inspection that solutions of Eq. (40) exist only for

*p*

_{0}= 1 and

*p*

_{0}= −1. For

*p*

_{0}= 1, solutions are defined by

*p*

_{0}= −1 by

*n*= 1, HE and EH solutions have

*x*

_{0}determined by the zeroes of

*J*

_{0}and

*J*

_{2}respectively. As pointed out in Eq. (10), there is a second category of solutions for which the

*n*/

*x*terms in Eq. (37) become −

*n*/

*x*. In this case, the zeroth order

*p*

_{0}= −1 solutions are degenerate with those of Eq. (42) and

*p*

_{0}= 1 solutions are degenerate with those of Eq. (43).

The attenuation of the leaky modes is found by considering the 1/*D* terms of the characteristic equations Eq. (37). These give

*J″*terms, and Eq. (42) or (43) is used to substitute for

_{n}*J′*(

_{n}*x*

_{0}) for

*p*

_{0}= 1 or

*p*

_{0}= −1 respectively. For

*n*= 0 the equations separate and, to leading order in 1/

*D*, the imaginary part of

*x*is given by where

*ν*is

*ν*

_{TE}for TE solutions and

*ν*

_{TM}for TM solutions. For

*n*≠ 0 the leading order term in

*δx*is

*n*, but

*n*is indirectly included via the value of

*x*

_{0}.

The attenuation follows directly from Eq. (15). To first order in *δx* the imaginary part of *β* is given by

*λ*

_{0}(to convert to loss in dB incurred in propagation through one wavelength) these expressions give the

*N*= 0 result of Eq. (1). As shown by Marcatili & Schmeltzer [17], the dielectric constant dependence of the attenuation of modes with

*n*≠ 0 is given by the average of those for the TE and TM solutions. This indicates that the attenuation of HE and EH modes can be considered in terms of TE and TM components; the value of

*p*

_{0}of 1 or −1 indicates an equal admixture, leading to an average of the TE-like and TM-like attenuation coefficients. It is shown in section 5.5 that this average extends to multi-layered structures, but with correction factors for

*N*≥ 2 (see also [19] and [24]).

## 5.4. TE and TM solutions for N > 0

For TE and TM solutions the matrices in Eq. (25) reduce to separate 2 × 2 blocks. To derive the characteristic equation, the vector

(which is the upper part of*v̱*

_{1}or the lower part of

*v̱*

_{2}in Eq. (36)) is multiplied by alternating transfer matrices for glass layers

*ν*is either

*ν*

_{TE}or

*ν*

_{TM}, as given by Eq. (38). The √(

*R*

_{1}/

*R*

_{2}) factors in Eqs. (28) and (30) have been dropped because an overall multiplying factor does not affect the characteristic equation. The final step in forming the characteristic equation is to multiply by the second (for TM modes) or fourth row (for TE modes) of

*M*

^{−1}(

*k*,

_{f}*r*) and set the result equal to zero. If the number of complete layers

_{N}*N*is even, the outermost region is glass and the relevant vector is (−

*iν*, 1). For

*N*odd, the outermost region is air and the vector is (−

*iD*/

*x*, 1).

If a generic product of *T* matrices in Eq. (25) is written as

*x*. It is instructive first to consider the order of magnitude in powers of

*D*of the terms in the

*T*matrix product and the left-most vector in Eq. (53). An increasing number of complete layers gives

*ϕ*factors for the relevant number of layers. This term is always multiplied by the highest power of

*D*for a given

*N*and, because it contains no cos

*ϕ*factors, it remains non-zero if all the layers are exactly at anti-resonance. To form the characteristic equation, the resultant vectors in Eq. (54) are multiplied into Eq. (49). It follows that the largest terms in powers of

*D*will arise from the second elements of the resultant vectors in Eq. (54). In turn, this means that the largest terms in the characteristic equation will be imaginary for

*N*odd and real for

*N*even. To derive an expression for the attenuation the real and imaginary parts of the characteristic equation are separated, and this difference between odd and even values of

*N*needs to be taken into account.

If *N* is odd the characteristic equation becomes

*D*larger than those on the right. It was shown in the previous subsection that for the thick-walled capillary substitution of a power series leads directly to the leading order terms of the solution. For multi-layered structures this method is not effective because of cancellations that eliminate higher order terms in

*D*. An alternative perturbative approach is therefore used and a zeroth order solution is defined by setting the right hand side of Eq. (55) to zero, giving The solution has been written as

*x̃*

_{0}to distinguish it from the solution of Eq. (41) for

*N*= 0.

*x*is written as

*x*=

*x̃*

_{0}+

*δx*and expansion of Eq. (55) in

*δx*gives

*J′*

_{0}. For the final step, note that (

*PS*−

*QR*) is the determinant of

*T*, where

*T*is given by Eq. (52). The determinants of

*T*and

_{g}*T*in Eqs. (50) and (51) are both 1 and it follows that the determinant of any product of these matrices is also 1. Although the individual elements in

_{a}*T*can include high powers of

*D*, taking the determinant always leads to a cancellation of terms and a result that is of order unity. This cancellation is critical in the dramatic decrease of the loss as the number of layers is increased.

The largest term on the left hand side of Eq. (57) is (*D*^{2}*QJ′*_{0})* ^{′}* and therefore, to leading order in

*D*,

*δx*becomes

*x̃*

_{0}. The right hand side of Eq. (56) has a factor 1/

*D*which means that the difference between

*x̃*

_{0}and

*x*

_{0}(the solution of Eq. (41)) is of order 1/

*D*or smaller.

*x̃*

_{0}can therefore be replaced with

*x*

_{0}in Eq. (59) and the result will still be correct to leading order in 1/

*D*. Noting that

*J′*

_{0}(

*x*

_{0}) = 0 and

*J″*

_{0}(

*x*

_{0}) = −

*J*

_{0}(

*x*

_{0}), the imaginary part of

*δx*becomes

The analysis is similar for even values of *N*. In this case the characteristic equation is

*D*larger than those on the right. By following the same approach as for

*N*odd, the leading term for the imaginary part of

*δx*becomes

The final step is to determine the largest factors in *Q* and *S* for *N* odd and even respectively. By multiplying out the *T* matrices the leading order terms are found to be

*x*=

*x*

_{0}. With a proof by induction it is straightforward to extend these results to an arbitrary number of layers

*N*and, by using Eq. (47), Eq. (63) leads directly to Eqs. (1a) and (1b).

## 5.5. HE and EH solutions for N > 0

A similar approach is used for solutions when *n* ≠ 0. In this case the algebra becomes long-winded and the simplicity provided by the determinant in Eq. (58) is absent. Only an outline of the approach used is therefore provided, together with the results obtained for structures with *N* from 1 to 4. The characteristic equations can be expressed as

*v̱*

_{1}and

*v̱*

_{2}are defined in Eq. (36),

*T̳*represents the product of

*T*matrices in Eq. (25), and the

*w̱*are vectors derived from the second and fourth rows of the

_{i}*M*

^{−1}(

*k*,

_{f}*r*) matrix. The separation of the

_{N}*w̱*vectors into real and imaginary parts simplifies the analysis, and is analogous to the separation made in Eqs. (55) and (61). For an even number of complete layers, the outermost region of the fibre structure is glass and the

*w̱*are given by

_{i}*w̱*

_{1}and

*w̱*

_{2}the relevant elements of the

*M*

^{−1}matrix have been multiplied by

*∊*. The equivalent expressions for

*N*odd, where the outermost region is air, are

Elimination of *p* from Eqs. (64a) and (64b) yields

*q*are given by The grouping of terms in Eq. (67) reflects the order of magnitude of the different

_{ij}*q*products in powers of

_{ij}*D*. The largest terms in all the products on the right hand side of Eq. (67) are one power of

*D*lower than the largest terms on the left hand side. For

*N*even, the largest terms in (

*q*

_{11}

*q*

_{32}−

*q*

_{12}

*q*

_{31}) exceed those in (

*q*

_{21}

*q*

_{42}−

*q*

_{22}

*q*

_{41}) by a factor of

*D*

^{2}. For

*N*odd, the largest factors in the two bracketed terms on the left hand side of Eq. (67) have equal powers of

*D*. However, if the

*n*= 0 limit is taken, then the largest terms of (

*q*

_{21}

*q*

_{42}−

*q*

_{22}

*q*

_{41}) are

*D*

^{2}greater than those in (

*q*

_{11}

*q*

_{32}−

*q*

_{12}

*q*

_{31}). For

*N*even the zeroth order

*x̃*

_{0}is therefore taken to be the solution of

*δx*becomes

*N*odd are

*x̃*

_{0}differ from

*x*

_{0}(given by Eqs. (42) and (43)) by terms of order 1/

*D*.

To evaluate the right hand side of Eqs. (70) and (72) note that each of the bracketed combinations of the *q _{ij}* in Eq. (67) can be expanded in terms proportional to

*J′*

_{n}^{2}(

*x*),

*J′*(

_{n}*x*)

*J*(

_{n}*x*) and ${J}_{n}^{2}(x)$, where the Bessel functions arise from the

*v̱*

_{1}and

*v̱*

_{2}vectors in Eq. (64). An expression for

*J′*

_{n}^{2}(

*x̃*

_{0}) can be obtained from Eq. (69) or (71) and substituted into Eq. (70) or (72) respectively. There is a substantial cancellation of terms in the resulting expressions for the numerators of Eqs. (70) and (72), analogous to the cancellation provided by the determinant for TE and TM solutions. As for the

*n*= 0 solutions, it is important to make an exact substitution of

*J′*

_{l}^{2}(

*x̃*

_{0}) at this stage (the same method works if ${J}_{l}^{2}({\tilde{x}}_{0})$ is substituted) to ensure that the correct highest order terms are found. Once the highest order terms are established,

*x̃*

_{0}can be replaced with

*x*

_{0}from Eq. (42) or (43). This is equivalent to what was done for the TE and TM solutions and the result for

*δx*remains correct to leading order in 1/

*D*.

The leading order terms for the imaginary part of *δx* for structures with *N* from 1 to 4, and *n* ≠ 0 are

*x*

_{0}. The connection with the results for the thick-walled capillary Eq. (46) and the TE and TM solutions for multi-layered structures Eq. (63) is clear. The attenuation for modes with

*n*≠ 0 is essentially the average of the equivalent TE and TM expressions; however there are corrections to this result for

*N*≥ 2. The additional factors in Eq. (73) are given by

*f*

_{2}and

*f*

_{4}in Eq. (75) is + for HE modes and − for EH modes.

Inspection of Eqs. (74) to (76) shows that the factors before the brackets in *C*_{2} to *C*_{4} differ from unity by terms of order (*n/x*_{0})^{2}. Although the bracketed terms in Eq. (74) differ from unity by order (*n/x*_{0}), note that both *f*_{2} and *f*_{4} are zero at the exact anti-resonance condition, and that the factors dependent on *∊* will also act to reduce the magnitude of this difference. The net effect of the *C*_{2} to *C*_{4} terms will therefore be to alter the value of the confinement loss by a factor between order (*n/x*_{0}) and (*n/x*_{0})^{2}. Given that the magnitude of (*n/x*_{0}) is not particularly small for the lowest order modes, the correction factors can be expected to have a significant effect on the loss. However, as discussed in section 5.2, the approximation of the Hankel functions that underlies all of the analysis is also expected to introduce errors of order (*n/x*_{0}). Extensive tests have been carried out to compare the results of Eq. (73), with and without the correction factors Eq. (74), with exact numerical calculations for a range of structures with different values of *N*. Although the correction factors are found to act in the right direction, they often over-compensate, leaving the resulting loss further from the exact value. Given that the correction factors do not provide a systematic improvement in the agreement between the analytic expressions and the full numerical results, a pragmatic approach in favour of simplicity is taken and the *C*_{2} to *C*_{4} terms are dropped in the final expression Eq. (1).

## Acknowledgments

I would like to thank William Wadsworth and Tim Birks for helpful discussions and a critical reading of the manuscript.

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