1. Introduction

McIsaac, in two seminal papers in 1975, essentially solved the problem of the symmetry-induced characteristics of modes in a uniform waveguide [1, 2]. He showed that modes can be classified according to the irreducible representations of the symmetry group of the waveguide and that all modes are either non-degenerate or doubly degenerate. For each class of mode he gave expressions, based on a Fourier expansion in the azimuthal variable, for the form of the longitudinal electric and magnetic fields in a waveguide whose transverse structure has a point group symmetry of either C_M or C_Mv, for arbitrary M. His analysis also provided the minimum sector of the waveguide required to determine the complete electromagnetic field, and the boundary conditions satisfied by the longitudinal components on the edge of the minimum sector. This work came to prominence in the context of the development of microstructured optical fibres and, in particular, photonic crystal fibres with hexagonal symmetry (i.e. point group C6v). Examples of the use and development of McIsaac’s symmetry analysis include [3–14].

The motivation for revisiting this topic comes from the development of a new generation of anti-resonant microstructured fibres (often termed negative curvature fibres) that dispense with a translationally-periodic cladding structure but which demonstrate highly efficient guidance of light in a low refractive index core, e.g. air ([15, 16], and references therein). These fibres are not restricted to hexagonal symmetry, and examples have been published where the point group of the idealised transverse structure is C_7v [17–19], C_8v [20–23], or C_10v [22]. The aim of this work is to provide a complete analysis of the symmetries of the field components for all mode classes for fibres with a C_Mv point group. In doing this two aspects of modal symmetry not fully covered by McIsaac’s and subsequent work are clarified. One concerns how the connection between the symmetries of the electric and magnetic field components emerges from the underlying governing equations. The second is the precise relationship between the field components (transverse as well as longitudinal) in doubly degenerate modes. Although these points have been touched upon in a number of previous papers (details are given in the next section), it is anticipated that bringing this analysis together in a consistent and coherent form will be of use to the community.

2. Symmetry analysis

The analysis is based on a fibre that is uniform along its length (the z direction) and with a transverse structure defined by an isotropic dielectric function ϵ(r, θ). Cylindrical polar coordinates (r, θ, z) are used throughout because they fit naturally with the symmetries of the fibre. It is assumed that the transverse dielectric profile has a point group C_Mv. The symmetry operations that leave the structure invariant are rotations about the fibre axis by 2pπ/M, where p = 0, 1, 2,…, M − 1, and mirror lines in the plane of the fibre that are separated by angles p′π/M, where p′ = 0, 1,…, M − 1. For even values of M there are two distinct types of mirror lines, while for odd M all the mirrors are equivalent. For simplicity it is assumed that the x-axis, i.e. θ = 0, is a mirror line. In the case of a hollow-core, anti-resonant fibre whose cladding consists of a ring of thin-walled capillaries [15,16], M corresponds to the number of capillaries; for even values of M the distinct types of mirror lines are those passing through or between the capillaries respectively (see Fig. 1).

 

Fig. 1 Schematic diagram of the cross section of hollow-core, anti-resonant fibres consisting of thin-walled capillaries uniformly distributed around an outer jacket. Air regions are white and glass regions are shaded. (a) and (b) have point groups C_7v and C_8v respectively. Mirror lines are shown by dashed and dotted lines; in (a) all mirror lines are equivalent, while in (b) there are two distinct sets of mirror lines represented by the two line types.

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The analysis is divided into three parts. In the following subsection, the symmetries of a scalar wave equation are analysed and expressions for the generic form of solutions with different symmetry types (i.e. the different irreducible representations) are presented. These expressions are equivalent to those given by [1], except for a clarification of the form of the doubly degenerate solutions. The extension to vector modes is then discussed, first by demonstrating the symmetry connection between the e_z and h_z components and then including the full set of in-plane components. It is shown that each component of the electric and magnetic fields in a mode has the symmetry characteristics of a scalar wave, but different components have different irreducible representations. In the third subsection the relationship between the modes of a fibre with C_Mv symmetry and those of a cylindrically symmetric fibre is discussed.

2.1. Symmetry of solutions of a scalar wave equation

The eigenstates, ψ, of a two-dimensional scalar wave equation with an underlying C_Mv symmetry can be characterised by the way in which they are transformed under the elements of the point group. If an initial ψ_i(r, θ) is rotated anti-clockwise by an angle ϕ relative to fixed coordinates then it transforms to ψ_f (r, θ) = ψ_i(r, θ − ϕ). Similarly, for reflection in a mirror line at an angle ϕ, ψ_f (r, θ) = ψ_i(r, 2ϕ − θ). Each eigenstate transforms according to one of the irreducible representations of the point group, and character tables of the irreducible representations are given in many textbooks. A useful web resource that includes data for C_Mv groups up to M = 32 is [24]. The approach taken by McIsaac [1, 2] is followed here, in which explicit forms for the irreducible representations are constructed from basis functions of the form f_n(r) cos nθ and g_n(r) sin nθ, where f and g are arbitrary functions of r and only certain integers n are allowed in each irreducible representation. As discussed by Fini [9] (see also Ferrando et al. [7]) an exponential form can be used instead of cosine/sine functions; this can be advantageous for degenerate modes in terms of the minimum sector required for a full numerical solution. In terms of symmetry, the approaches are essentially equivalent. Most of the papers referred to in the introduction use McIsaac’s notation to label the different mode classes. A more standard Mulliken notation for irreducible representations is used here; this notation was also used, for example, by Aghaie et al. [12] and Wolff et al. [14].

All C_Mv point groups have non-degenerate A₁ and A₂ representations for which n is an integer multiple of M and which can be expressed in the form

Under rotation by 2pπ/M both of these transform as ψ_f = cos(pπ) ψ_i and under reflection in a mirror line at p′ π/M, B₁ states transform as ψ_f = cos(p′π) ψ_i and B₂ states transform as ψ_f = − cos(p′π)ψ_i.

For all C_Mv points groups with M > 2 there are doubly degenerate representations labelled by E_q, with q = 1, 2,…. The number of E_q representations increases with M; for even and odd values of M there are M/2 − 1 and (M − 1)/2 distinct representations respectively. Within each doubly degenerate pair there are two states that can be constructed from cosine and sine basis functions and which are denoted by E_qc and E_qs. These are expanded as

The matrices in Eqs. (4) and (5) are consistent with the characters of the symmetry elements for E_q irreducible representations [24], as are the transformations given above for states with A-and B-type symmetry.

To make the analysis more concrete, explicit forms for states of different symmetries for point groups C_7v and C_8v are given below; any eigenstate ψ will be expressible in one of these forms:

The patterns seen here can easily be extended to other values of M. In the expressions for the functions of E_q symmetry, the alternating pattern of signs in the E_qs expansion arises from the negative values of j in Eq. (3).

The expansions given in Eq. (3) for doubly degenerate modes are a little different from those of McIsaac [1] and other authors. McIsaac uses two semi-infinite sums rather than the j sums in Eq. (3) running from −∞ to ∞. More importantly, he does not highlight the fact that the radial terms in the E_qc and E_qs expansions must be related to one another, as seen in Eqs. (3), (7) and (8). This connection is implicit in previous work (e.g. [3, 7, 9]) but does not appear to have been brought out explicitly.

As well as the basic transformation properties of the states, two extensions are required for the analysis in subsection 2.2. First, the symmetries of the derivatives of states are considered. Differentiation with respect to r results in a function with the same irreducible representation as the original state, but differentiation with respect to θ changes the representation, as shown in the first line of Table 1. It is straightforward to show from Eqs. (1) and (2) that the non-degenerate representations, A₁/A₂ and B₁/B₂ transform into a function with the symmetry of the other representation in the pair, i.e. 1 ↔ 2. The doubly degenerate E_q representations Eq. (3) transform into a pair of functions that also have E_q symmetry. For an initial function ψ = α_c E_qc + α_s E_qs the differential with respect to θ can be written in the form ${\psi }^{\prime }={\alpha }_c^{\prime }{E}_{qc}^{\prime }+{\alpha }_s^{\prime }{E}_{qs}^{\prime }$, where $E_{qc}^{\prime }$ and $E_{qs}^{\prime }$ are a pair of functions that have the standard form given by Eq. (3) and the coefficients ${\alpha }_{c,s}^{\prime }$ are related to the original coefficients by

The transformation matrix here is included in the first line of Table 1.

Table 1. Effect of differentiation with respect to θ and multiplication by functions with A₁ and A₂ symmetry on states with given irreducible representations. The top row gives the original representation and the other rows the representation resulting from the given operation.

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The symmetry properties of the product of a function that has A₁ or A₂ symmetry with states of different irreducible representations are also required. These can be determined by direct multiplication of the standard forms given in Eq. (1) with each of Eqs. (1), (2) and (3), followed by identification of the symmetry type of the resulting expression. The results are shown in Table 1; multiplication by a function of A₁ symmetry does not change the representation, while multiplication by a function of A₂ symmetry changes the representation in an equivalent way to differentiation with respect to θ. The transformation matrices given for the E_q representation are to be interpreted in the same way as in Eq. (8).

2.2. Modal symmetry in fibres with point group C_Mv

To extend these properties of scalar solutions to the vector modes of a fibre, an analysis of the symmetries of the field components e_r(r, θ), e_θ (r, θ), e_z (r, θ), ${\tilde{h}}_r(r,\theta )$, ${\tilde{h}}_{\theta }(r,\theta )$ and ${\tilde{h}}_z(r,\theta )$ is required (a scaled magnetic field defined by $\tilde{\mathbf{h}}(r,\theta )={({\mu }_0/{ϵ}_0)}^{1/2}\mathbf{h}(r,\theta )$ is used to simplify the notation, and lower case letters are used for the fields to avoid any confusion between the electric field and the E symmetry type). All field components are assumed to have a z dependence of the form exp(iβz), where β is the propagation constant. The components are closely inter-related and any two of them are sufficient to generate the full set. For example, if e_z and ${\tilde{h}}_z$ are known, the remaining components are given by

These equations are well known in the analysis of circularly symmetric fibres where ϵ ≡ ϵ(r) [25], but they are also valid for a general dielectric profile ϵ(r, θ). Similarly, e_z and ${\tilde{h}}_z$ can be expressed in terms of the transverse field components as

Coupled wave equations that depend only on e_z and ${\tilde{h}}_z$ are derived by substituting Eq. (9) into Eq. (11), yielding

We first note that all the functions, other than the field components, that appear in Eqs. (9) and (12) have either A₁(ϵ,1/ϵ, ∂ϵ/∂r, k², 1/k²) or A₂(∂ϵ/∂θ) symmetry. These functions always appear as multipliers of the field terms, and the symmetry effect of such multiplications is given in Table 1. Where there are repeated products (for example, in the term 1/ϵ × ∂ϵ/∂θ × 1/k²) the symmetry is found by repeated application of Table 1 (thus the overall symmetry of the example product is A₂).

The key point in determining the allowed symmetries of vector modes is that each term in Eq. (12a) (and separately in Eq. (12b)) must transform under the same irreducible representation, so that the whole equation transforms consistently under rotations and reflections. For example, if e_z has A₁ symmetry then, from Table 1, ∂e_z /∂θ has A₂ symmetry. Every term in Eq. (12a) must then have A₁ symmetry (because the (∇² + k²)e_z, ∂e_z /∂r, and ∂e_z /∂θ terms do) and every term in Eq. (12b) must have A₂ symmetry (because the ∂e_z /∂r and ∂e_z /∂θ terms do). It follows that, in this case, ${\tilde{h}}_z$ must have A₂ symmetry to keep the symmetries of the two coupled equations consistent. The symmetries of the other field components can then be found from Eqs. (9). The result is given in the first line of Table 2, where this overall symmetry type is labelled as A⁽¹⁾. An equivalent analysis can be applied to the other non-degenerate solutions. By starting with the symmetry of e_z being A₂, B₁ and B₂ in turn, the symmetries of the other field components can be determined, with the resulting modal symmetries being labelled as A⁽²⁾, B⁽¹⁾ and B⁽²⁾ in Table 2. In each case, it can be seen that every field component has a symmetry that is characteristic of a solution of a scalar wave equation, but components with the given symmetries must be combined to create a complete vector mode.

Table 2. Irreducible representations of the components of the electric and magnetic fields for modes of A⁽¹⁾, A⁽²⁾, B⁽¹⁾, B⁽²⁾ and $E_q^{(1,2)}$) symmetry (see text for details).

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The doubly degenerate modes are treated in a similar way. All the terms in both Eqs. (12a) and (12b) must be expressible in the form α_c E_qc + α_s E_qs where E_qc and E_qs are functions of the type given by Eq. (3) and the constants α_c and α_s are the same for each term within the separate equations. This will ensure that both equations transform correctly under the symmetry elements of the point group. If it is assumed that e_z has this standard form with α_c = α₁ and α_s = α₂ then all the terms in Eq. (12a) must have these same constants multiplying the E_qc and E_qs parts of each term. The effects of the products and differentiations in each term can be determined by repeated application of the matrices given in Table 1. It is straightforward to show that Eqs. (12a) and (12b) have the correct overall form if and only if the constants in the expansion of ${\tilde{h}}_z$have the form α_c = α₂ and α_s = −α₁. It then follows from Eqs. (9), in the same way as for the non-degenerate modes, that e_r and ${\tilde{h}}_{\theta }$ have the same form as e_z, while e_θ and ${\tilde{h}}_r$ have the same form as ${\tilde{h}}_z$. This overall symmetry type is labelled as $E_q^{(1,2)}$ in Table 2, where the column vectors for each component are to be interpreted in the same way as in Eqs. (4) and (5). Orthogonal modes within the doubly degenerate pair for this overall symmetry type can be obtained by choosing particular values for the constants α. The final two lines of Table 2 give a pair of modes: $E_q^{(1)}$ with α₁ = 1 and α₂ = 0, and $E_q^{(2)}$ with α₁ = 0 and α₂ = 1. It should be emphasised that the signs given for the field components for $E_q^{(1)}$ and $E_q^{(2)}$ do not imply any particular sign relationship between different components; they give only the sign relationship between the two modes for a given component. For example, if e_z for modes 1 and 2 within a degenerate pair has a E_qc and E_qs form respectively, then ${\tilde{h}}_z$ for modes 1 and 2 will have E_qs and E_qc forms respectively, but with opposite signs.

Previous authors (e.g. [11]) have given expressions for transverse field components based on McIsaac’s expansions, but Table 2, combined with Eqs. (1), (2) and (3), provides a complete description of the symmetry for every mode class in fibres with a C_Mv point group.

2.3. Relationship with modes of a fibre of cylindrical symmetry

Modes in fibres with cylindrical symmetry (which corresponds to a point group C_∞v) are conventionally labelled as TE_0m, TM_0m, HE_nm and EH_nm, where n describes the azimuthal dependence of the field components and m labels the different modes for each n. All field components in a given mode vary as cos(nθ) or sin(nθ) [25] and thus there is a clear connection with the generic scalar functions given in Eqs. (1) to (3). The difference is that in C_∞v the sums over M are absent and only one term is present in each case. A₁ corresponds to n = 0 and fields with this symmetry are constant in θ. The A₂, B₁ and B₂ functions do not exist in C_∞v. Functions of the E_q type retain only the cos(qθ)/sin(qθ) term, and so q can be identified with the azimuthal label n in cylindrically symmetric fibres.

TE_0m modes have $e_z=e_r={\tilde{h}}_{\theta }=0$ and can therefore be identified with the general A⁽²⁾ mode in Table 2. Similarly, TM_0m modes have ${\tilde{h}}_z={\tilde{h}}_r=e_{\theta }=0$ and are of the general A⁽¹⁾ type. HE_nm and EH_nm modes consist of a degenerate pair with the e_z components in the two members of the pair having the form f(r) cos nθ and f(r) sin nθ, and the ${\tilde{h}}_z$ components −g(r) sin nθ and g(r) cos nθ, where f and g are functions of the radial variable [25]. Thus both HE and EH modes have the generic $E_q^{(1,2)}$ form of Table 2; there is no underlying symmetry difference between HE and EH solutions.

The modes of a hollow core fibre with point group C_Mv are closely related to those of a hollow tube with C_∞v symmetry, but the reduction in symmetry caused by the microstructure can lead to changes in the modal symmetries, including degeneracy breaking. This is well known for point group C_6v (e.g. [6, 12]) and in Table 3 the analysis is extended to point groups from C_1v to C_10v. The symmetry properties of the 12 lowest order modes, grouped by their value of n, are considered and the Table shows which generic symmetry type given in Table 2 each group falls into. For these modes, microstructured fibres with M ≥ 9 have essentially the same symmetry characteristics as a hollow tube, but for M = 8 and lower, there are differences. For example, in a hexagonal fibre with C_6v symmetry, the HE₃₁ mode is split into two non-degenerate modes, and the HE₄₁ mode is symmetrically equivalent to the group of modes with n = 2. In contrast, in a fibre with C_7v symmetry the HE₃₁ mode remains doubly degenerate, while HE₃₁ and HE₄₁ share the same symmetry type which is distinct from the modes with n = 2. Fibres with C_8v symmetry are similar to those with a C_7v point group except that the eight-fold rotation symmetry splits the HE₄₁ mode into two non-degenerate modes. It is straightforward to extend the entries in Table 3 both to higher order modes and to larger values of M.

Table 3. Symmetry types for the 12 lowest modes in hollow-core, microstructured fibres with point groups from C_1v to C_10v. The modes are grouped and labelled according to the equivalent modes in a cylindrically symmetric hollow tube. Where the degeneracy is broken, the symmetries of the two resulting modes are separated by a semi-colon.

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3. Conclusions

The key result of this paper is the description of the symmetries of all field components for each class of mode that exists in fibres with a C_Mv point group, as given in Table 2 and Eqs. (1) to (3). As mentioned in the Introduction, it is not claimed that this work is entirely new, or that previous work is incorrect. However by grounding the symmetry analysis in the governing equations for the longitudinal field components, Eqs. (12a) and (12b), and by using a matrix-based approach for the degenerate modes, a more complete and systematic picture of the symmetry emerges. With the growing interest in negative curvature fibres with relatively unfamiliar point groups, it is anticipated that Table 3 will provide a useful reference for the allowed symmetries of the lower order modes.