1. Introduction

The symmetry properties of microstructured optical fibers (MOF) were essentially completely described by McIsaac [1], a quarter century before anyone thought to consider the question. Thinking in terms of conventional closed lossless waveguides consisting of a single core and cladding, McIsaac determined the group theoretical implications of the waveguide symmetry on both modal degeneracy and the modal field patterns.

This work revealed two critical results. Firstly, every mode of a waveguide is either non-degenerate or two-fold degenerate, and (barring accidental degeneracies) two-fold degeneracy only occurs for pairs of modes which individually exhibit less than the full symmetry of the waveguide. In addition, once a given waveguide geometry is identified as belonging to one of the symmetry point groups of the plane, all the modes of the waveguide can be classified into a finite number of classes by the azimuthal expansion of their fields. The possible symmetry groups are C _∞v (circular waveguide), C _∞ (helical waveguide), C_nv (discrete rotational symmetry of order n with a reflection axis) or C_n (discrete rotational symmetry of order n with no reflection axis). Notably, the standard circular fiber is an instance of group C _∞v, and the familiar hexagonal photonic crystal fiber is an instance of C _6v.

The advent of the modern MOF has given new relevance to this early work, since if we ignore the small effects of leakiness on the mode profiles, (which are strictly complex quantities in a MOF), the group theoretical treatment is unchanged. Hence, McIsaac’s work had implied that the fundamental modes of a perfect hexagonal MOF must be degenerate, long before the issue arose in experimental and numerical studies of MOF [2]. This fact, having been rediscovered and illustrated through a simple geometric argument [2, 3], has lately become a standard test for new numerical mode-solvers [4, 5].

A point that up to now has been largely unremarked, is that only rotational symmetry is required for degeneracy. An axis of reflection, while virtually ubiquitous in waveguide designs is unnecessary. For instance, all three “satellite” fibers shown in Fig. 1 support degenerate fundamental modes, even though only the first two support reflection symmetry [2]. This result can at first seem a little surprising, since it conflicts with the strong intuitive sense that reflection symmetry, if it exists, “should be doing something.” In this work, we attempt to identify what that “something” is. Note that we refer to the fibers in Fig. 1 frequently, and for convenience label them as fibers A_6v, B_6v and C₆ respectively, where the subscripts serve as a reminder of their symmetry groups. The fibers have hole diameter-pitch ratios of d/Λ=0.8 (large holes), and d_s/Λ=0.2 (small holes), and glass index n_co=1.45. For the three fibers, the small holes are aligned at angles of 0°, 30°, and 15° with respect to the large holes.

In this paper we study the modal properties and profiles of a number of MOF designs, with and without reflection symmetry. While there may be other effects, we concentrate on the impact of reflection symmetry on the “transversality” of modes, and particularly on the behavior of the MOF analogs of the TE₀ and TM₀ modes of the standard circular fiber. By transversality, we mean both the degree to which the longitudinal components are small, and the degree to which the transverse mode profiles resemble the profiles of truly transverse modes. Overall, we find that reflection symmetry drives the modes towards transversality. Finally, we provide an explanation for this behavior based on the interaction of symmetry constraints and energy minimization principles.

We begin by reviewing some symmetry properties of Maxwell’s equations for waveguide modes.

 

Fig. 1. Schematic of three “satellite” fibers with symmetry groups C _6v [(a) and (b)] and C ₆ [(c)]. All three fibers support degenerate modes, but only (a) and (b) possess reflection symmetry. The fibers are referred to in the text as A_6v, B_6v and C ₆.

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2. Field symmetries in modal solutions to Maxwell’s equations

Consider modal fields in cylindrical coordinates propagating along the z axis with propagation constant β and frequency ω.

where r _t=(x,y). Then Maxwell’s equations allow us to express the transverse parts of the field in terms of the longitudinal components:

where n(r _t) is the local refractive index, k ₀=ω/c and $Z_0=\sqrt{{\mu }_0⁄{\epsilon }_0}$. Thus, the longitudinal fields may be thought of us as sources for the transverse fields.

Assuming now that the index distribution n(r _t) is piecewise constant, we take the transverse divergence and curl of Eqs. (3) to find

Note that only one longitudinal component appears in each relation. Thus we obtain the result that in any homogenous region, a longitudinal component can only induce a transverse divergence upon its own field, and a transverse curl upon the opposite field. (See the appendix for the definitions of the transverse vector operators.)

Consider for example, the TE₀ mode of a standard circular fiber, (see Fig. 2). The fields have the following properties: e_z vanishes (by definition), and h_z has no azimuthal dependence. Then Eqs. (4) require that h _t is purely radially directed with ∇_t×h _t=0, and e _t is purely azimuthally directed with ∇_t·e _t=0, consistent with the profiles shown in Fig. 2(b-c).

 

Fig. 2. (a) h_z, (b) h _t and (c) e _t for the TE₀ mode of a circular fiber. The fields of the TM₀ mode are qualitatively similar with the roles of E and H reversed.

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3. Review of symmetry classes

As mentioned already, the work of McIsaac [1] classifies the modes of optical waveguides according to the symmetries both of the guides and the modes. While we will not attempt to review the full details of this classification, it will be useful to have a few of the basic results to hand. For each symmetry group of the waveguide, the modes are labeled with a class index p. Modes belonging to the same class share both the form of the expansion of their longitudinal fields and their degeneracy properties—each class either contains modes which are non-degenerate, or modes which are degenerate with exactly one of the modes in the complementary class. We will only be interested in the non-degenerate classes, and in fact only in the first two non-degenerate classes, labeled p=1 and p=2. The expansions of the longitudinal fields of these modes are summarized in Table 1. For C _∞v (circular fibers), classes p=1 and p=2 are just the TE and TM modes, which have no azimuthal dependence, and by definition, one vanishing component. Next in the table, the corresponding modes for C_nv waveguides (eg. standard MOF with rotational symmetry of order n) are no longer transverse–both e_z and h_z are non-zero. However, in the limit of large n, all the terms average to zero except the v=0 cosine term, and we can think of these modes as tending towards the C _∞v states.

For the groups without reflection symmetry the situation is diferent. For C _∞ guides the non-degenerate states are hybrid, and form a single class p=1. (An example is an optical fiber with a short-pitch helical metal wire around the core [6]). Such guides do not support any transverse states. The analogous finite groups C_n follow suit: the C_n guides only support a single class p=1 and even at large n, the expansions for these modes always contain both e_z and h_z components, since the v=0 terms in the expansion survive.

4. Comparison of circular fibers and MOFs

We now turn to the mode profiles of a range of MOF with and without reflection symmetry. Calculations were performed using the plane-wave expansion technique with the commercial tool BandSOLVE by RSoft Design Group [7]. Typically, the calculations used 2¹⁶ plane waves per polarization. Note that the truncation of the plane wave expansion to a finite set of waves necessarily imposes a smoothing on the boundaries so that strictly speaking the results shown are not for perfectly discrete dielectric interfaces. Of course, with sufficient plane waves, the method converges to the results of other treatments that use discrete interfaces [2], and this approximation does not affect the validity of our conclusions. Finally, we note that we always

Table 1. First two symmetry classes (all containing non-degenerate modes) for several symmetry groups.

View Table

assume perfect structures. So a hexagonal PCF is assumed to be a perfect fiber supporting six-fold rotational and reflection symmetry.

From the many studies of the modal properties of hexagonal photonic crystal fibers and other MOF [2, 8–11] it is by now well-known that their guided modes share many characteristics with the modes of the circular step-index fiber. Firstly, the fundamental mode is usually a degenerate pair of HE₁₁-like orthogonal states. (Concentric circular Bragg fibers, for which the fundamental mode is a genuine TE-state, are a notable exception which we do not consider.) Secondly, in multimode fibers, the lowest order modes appear in similar clusters of degenerate and almostde-generate states [11]. For example, Fig. 3 shows the first dozen or so modes for an air-silica circular step-index fiber and fiber A_6v in Fig. 1c. In both cases, the HE₁₁ fundamental pair is followed by a quartet of states: a degenerate HE₂₁ pair and the non-degenerate “TE₀” and “TM₀” states. Finally, the corresponding modes from each type of fiber have similar spatial profiles. For example, the HE₁₁ modes are approximately uniformly polarized with a single central peak in intensity. It is for these reasons, that mode labels such as HE₁₁ or TE₀, which strictly apply only to circular fibers, are frequently applied to the modes of MOFs.

On the other hand, there are also important differences in the modal properties. Certain HE or EH modes which are degenerate in circular fibers are split in a fiber of given discrete symmetry [11], since the symmetry requirements are relaxed. For example, the LP₃₁ or EH₂₁/HE₄₁ mode pair of circular fiber is split into separate states in a fiber with six-fold rotational symmetry.

5. Transverse profiles

In our numerical studies, we have found that there is little qualitative difference between the fundamental mode profiles of C_nv and C_n fibers. Both cases show the typical HE₁₁ pattern of an approximately uniformly polarized field with an intensity maximum at the core. Consequently, we will not examine the degenerate pair states further.

5.1. Quasi-transverse states

It is more revealing to examine the analogs of the TE and TM modes of the circular fiber, which are non-degenerate. Fig. 4 shows the lowest non-degenerate mode of fiber B_6v. The field distributions show significant similarities with the TE₀ mode of the circular fiber—the h _t field appears radial and the e _t field azimuthual. Indeed this type of mode, which formally belongs to the McIsaac class p=1 is customarily referred to as TE₀ in the MOF literature. The longitudinal h_z field is also similar to the circular fiber mode, but of course it possesses six-fold symmetry rather than the continuous rotational symmetry in Fig. 2. The most obvious difference with the step-index fiber is the appearance of the longitudinal e_z field which exhibits a six-fold set of oscillatory lobes circling the origin. One way to understand the appearance of a non-zero e_z is to consider it as a source for the transverse fields in Eq. (3). In order for the transverse fields to satisfy the boundary conditions at the dielectric interfaces, h _t and e _t can not be perfectly radial or azimuthal—both fields must possess a non-zero transverse curl and transverse divergence in the vicinity of the boundaries. This fact is illustrated in Fig. 5 which shows the quantity |∇_t×h _t| for the p=1 mode. For the same mode in the circular fiber, this quantity vanishes. From the discussion following Eqs. (4), the non-vanishing of this quantity implies a nonzero e_z. Thus the oscillatory e_z component near the dielectric interfaces provides the extra freedom to the transverse fields needed to satisfy the boundary conditions. Note that this is consistent with the C_nv line in Table 1 which permits both longitudinal fields to be nonzero. All research to date suggests that all MOF modes are hybrid—there are no truly TE or TM states.

 

Fig. 3. Mode spectra for (a) a glass-air circular step-index fiber (core diameter 5µm, n_co=1.45), and (b) fiber A_6v at l=1.55µm.

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Fig. 4. Mode profiles for the lowest p=1 mode of the fiber in Fig. 1b. (a) h_z, (b) e_z, (c) h _t and (d) e _t.

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Fig. 5. Boundary effects for the p=1 mode in fiber B6v: transverse curl of h _t. For the p=2 mode, the profile looks similar with the roles of the fields swapped.

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The other non-degenerate member of the first quartet of modes in Fig. 3 belongs to class p=2 which for the circular fiber is the TM₀ mode. For this mode, we find very similar profiles to those in Fig. 4 but with the roles of E and H reversed—e _t is largely radial, h _t is largely azimuthal, and h_z shows oscillatory behavior. Since the plots are similar to those for the p=1 mode with the labels reversed, we do not include them here. We have also calculated the field profiles for the same two modes in A_6v. For each mode and field, the profiles are very similar to those of the first fiber.

5.2. Removing reflection symmetry

We now consider the first few modes of the Fig. 1 fiber C₆ which does not have reflection symmetry. As with the previous fiber, the first six modes consist of two degenerate pairs, and two non-degenerate states, modes 2 and 5, both of which belong to the class p=1. Figure 6 shows the field components for mode 5. While there are similarities to the earlier profiles, these fields are clearly quite different in nature. (We show the second of the two p=1 modes, because it resembles the profiles in Fig. 4 somewhat more closely than does the first p=1 mode.) For the transverse fields, the distinction between radial and azimuthal fields is now completely lost; instead both fields have a spiral shape. The spiral character is also seen in the rather beautiful longitudinal field profiles. Moreover, whereas as in the C_6v mode in Fig. 4, h_z was essentially a central peak, while the “minor” field e_z had an oscillatory character for the C ₆ mode, both h_z and e_z have a mixed oscillatory and peaked character.

 

Fig. 6. Mode profiles for the second lowest p=1 mode of the fiber in Fig. 1c. (a) h_z, (b) e_z, (c) h _t and (d) e _t.

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6. Degree of transversality

We have now seen that while the mode profiles for both the C _6v and C ₆ fibers are similar in structure to the corresponding transverse modes of step-index circular fibers, those of the C _6v fibers are considerably closer to the circular fiber modes. However, since neither the C _6v or C ₆ fibers support genuinely transverse modes, it is necessary to determine whether the greater transverse character of the shape of the C _6v modes is reflected in the actual strength of the longitudinal components.

We define the degree of transversality of the electric and magnetic fields as the fraction of the energy in the longitudinal component:

 

Fig. 7. Transversality of first 6 modes for the three fibers in Fig. 1. The two fibers with reflection symmetry exhibit larger contrast between the major and minor longitudinal components.

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where the inner product represents integration over the whole modal field

Figure 7 shows the transversality of the electric and magnetic fields for the first six modes of the three fibers in Fig. 1. The modes shown are the two pairs of degenerate hybrid HE states (shown in gray), and the two non-degenerate states, (shown in red for quasi-TE and blue for quasi-TM). The curves are labeled with the mode and field to which they correspond. For example, the designation “TE, H” indicates the transversality f_z[H] for the magnetic field of the TE mode. For the two C _6v fibers, (Fig. 7a and b), we can clearly distinguish “major” components (TE, H, and TM, E) and “minor” components (TE, E, and TM, H) Note that the minor components are substantially smaller than the major components. Indeed, for fiber B_6v, the minor components are a factor of ten smaller than the major components and for both fibers, the minor components are approximately equal or smaller to the transversality of the fundamental HE₁₁ modes. On the other hand, for fiber C₆, (Fig.7c), the distinction between minor and major components is much less sharp, particularly at higher frequencies. Thus in this case, the TE/TM designation is truly inappropriate and we simply label the modes as two instances of the p=1 class.

 

Fig. 8. Examples of systematic fibers with C _3v and C ₃ symmetry. (a) and (b) h_z and e_z for C _3v. (c) and (d) h_z and e_z for C ₃.

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7. Systematic studies

The results shown so far are suggestive of a relation between the presence of reflection symmetry and the smallness of the minor longitudinal component. One objection to the above results however, is that for different positions of the “satellite” holes, the effective core diameter is different. Note for example, that the mode diameter indicated in the Fig. 6 is substantially larger than that in Fig. 4. Thus it is possible that the difference in transversality is impacted by the difference in mode areas and effective indices of the different modes.

To overcome this problem, we performed a large range of calculations on two other classes of fibers. Representative examples are shown in Fig. 8. The fibers with C_nv symmetry [Fig. 8(a-b)], consisted of concentric rings of holes. In the innermost ring, n holes are removed. The fibers with C_n symmetry [Fig. 8(c-d)] consisted of n spirals. Such fibers have not been seen in practice yet, but could be easily created using polymer materials [12]. The mode profiles for these two new classes of fiber have similar characteristics to those we saw earlier. Again, since the range of possibilities to explore is enormous, we concentrate on the TE₀ and TM₀ like profiles. The C_nv modes again exhibit many characteristics of circular fibers while the spiral fibers show much less distinction between TE and TM-like states.

The advantage of the new profiles is that we have sufficient degrees of freedom to manipulate the mode areas and effective indices as required. We proceeded in the following fashion. For rotational symmetry ranging from n=3 to n=7 we adjusted the spacing of the rings and holes in order to match the effective index and mode areas of corresponding modes to within 3–4%. This was done essentially by trial and error. The transversality of the modes as defined by Eqs. (5) was then calculated as before. We show results for the case of n=4 which are typical. For the concentric ring C _4v fiber, we chose a scale length Λ=4 and hole diameter d/Λ=3/9. The first ring of holes had a radius of 2.75Λ, and the radii of subsequent rings increased in steps of 0.75Λ. The holes were equally spaced around each ring. For the spiral C ₄ fiber, the hole sizes were the same and successive rotations of the spirals were approximately Λ apart. Fig. 9 shows the fraction of each field in both the TE and TM-like modes for the C ₄ and C _4v fibers. The curves labeled “major components” represent h_z for the TE-like mode and e_z for the TM-like mode. The “minor components” denote h_z for the TM-like mode and e_z for TE-like mode. Note that the major components are very similar for both the C ₄ and C _4v fibers. However, the minor components for the C ₄ fiber are a factor six or so larger than in the C _4v fiber. Once again, therefore the fiber with reflection symmetry clearly exhibits a much greater degree of transversality in its quasi-transverse modes.

 

Fig. 9. Degree of transversality for TE and TM-like modes in matched C ₄ and C _4v fibers. Crosses denote the C ₄ results.

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8. Discussion

The preference for transversality in fibers with reflection symmetry can be understood using the field expansions introduced in Table 1. The table indicates that for C_nv fibers, the expansion of the minor components is a pure sine series. It is easily seen that as a result, the minor field must vanish along the lines ϕ=2mπ/n for integer m. This is apparent in images such as Figs. 4b and 8b. Now, recall from the discussion following Eqs. (4) that if one longitudinal component vanishes, then the transverse fields have either a vanishing transverse divergence or transverse curl. Thus along the n nodal lines, the non-degenerate states in C_nv fibers recover the exact field structure of the plain circular fiber—locally, the transverse fields are purely radial or azimuthal. On the other hand for the C_n fibers, the minor longitudinal field is less restricted, consisting of a complex Fourier series. As can be seen from Figs. 6b and 8d, while not forbidden by the longitudinal expansion, nodal lines do not appear. The difference in transversality can then be understood in terms of a simple energy argument. In a variational formulation of the wave equation [13], eigenmodes act to minimize their energy by reducing field curvature as far as possible and maximizing the fraction of energy in high index regions. This is completely true in the scalar problem, and remains the dominant effect in the vector treatment. In our problem of waveguides with rotational symmetry, the group theory places essentially no constraints on the modes of the C_n fibers, which can then seek out an optimum lowest energy configuration. In the C_nv fibers however, the group theory constrains the field structure along the nodal lines. Away from the nodal lines, the field is free to adopt any structure but since the field must be pure radial/azimuthal along the nodal lines, there must be a significant energy penalty due to field curvature if it is not approximately radial/azimuthal everywhere. In fact, by this argument we can predict an increasing degree of transversality as the order of rotational symmetry, (and thus number of reflection lines increases), since the field is constrained in more locations. This is confirmed in Fig. 10 which shows the transversality (using a slightly different measure) as a function of rotational order for fibers with a single ring of circular holes, again matched to have similar mode areas and effective indices. A small value represents a highly transverse mode. The transverse fields are also shown for the C _3v and C _9v instances of this structure, with the latter clearly showing a more ideally transverse structure.

 

Fig. 10. (a) Transversality µ _TE=f_z[E]/f_z[H] and µ _TM=f_z[H]/f_z[E] as a function of rotational order for single ring fibers. Small µ indicates strong transversality. Also profiles of H _t for (b) n=3 and (c) n=9.

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The highly transverse field structure, while it seems the more “ideal” in fact represents a higher energy state, since fewer degrees of freedom are available to minimize the energy.

9. Conclusion

This study is an illustration of the complementary roles played by group theory and the full wave equation in problems with high symmetry. The group theory analysis, long since completed by McIsaac delineates the minimum properties that must be satisfied by the modal fields. Within this framework, the eigensolutions to the wave equation then seek a configuration that minimizes their energy exploiting all the available degrees of freedom.

In the present work, we have only begun to investigate the possible impacts of reflection symmetry. Future work could examine the extent to which the tendency towards transversality survives in higher order modes, and whether more subtle effects can be discerned from the degenerate HE-like states.