Modal characteristics of hollow-core photonic-crystal fibers with elliptical veins are studied by use of a recently proposed numerical method. The dynamic behavior of bandgap guided modes, as the wavelength and aspect ratio are varied, is shown to include zero-crossings of the birefringence, polarization dependent radiation losses, and deformation of the fundamental mode.
© 2005 Optical Society of America
In Photonic-crystal fibers (PCFs), the photonic crystal cladding exhibits a photonic bandgap in the neighborhood of the propagating wave, which is thus confined to propagate in a central core [1, 2]. The most common cladding consists of circular air veins, arranged in a hexagonal lattice, producing a waveguide with six-fold rotational and mirror symmetry (C 6v symmetry) . These structures have been investigated extensively in recent years [4, 5, 6], however, with the advent of efficient numerical methods, interest is shifting towards less symmetrical structures. As explained in , C 6v symmetry implies the existence of doubly-degenerate pairs of modes, that share the same propagation constant, β, and free-space wavelength, λ, similarly to the fundamental HE11 modes in a conventional step-index fiber. When this symmetry is perturbed by bends and twists of the fiber, or by manufacturing imperfections, the degeneracy is lifted and the real parts of the effective indices, n eff ≜ λβ/2π, of the degenerate modes separate by an amount termed modal birefringence. These perturbations couple the modes that propagate at slightly different phase velocities, with the consequence that the polarization of light becomes unpredictable after a short propagation. When control over the polarization of light is crucial, high birefringence (up to about 5 × 10-4) may be induced on purpose in conventional fibers by a number of techniques , and this reduces the coupling between the once degenerate modes. A number of recent studies indicate that PCFs with a preferred direction in their geometry could exhibit birefringence of about an order of magnitude higher than that obtained with conventional techniques [8, 9]. Also, polarization dependent radiation losses, which give rise to an imaginary part of the effective index, even assuming lossless materials, could be exploited to make single-polarization fibers .
In anisotropic crystals, such as ZnO and CdS, zero-crossings of the birefringence vs. wavelength curves were used to explain the existence of polariton absorption lines , and we shall show similar birefringence zero-crossings in birefringent hollow-core PCFs. Similar results have also been reported recently in .
2. Numerical method
The numerical method used in this study is a recently proposed source-model technique (SMT) [13, 14], which is adequate for the analysis of general dielectric waveguide geometries. The PCFs we analyzed have elliptical veins, the likes of which have been successfully manufactured recently , and are depicted in Fig 1. Solid-core versions of these elliptical photonic-crystal fibers (EPCFs), as they have been tentatively dubbed , have been analyzed recently by an adjustable boundary condition Fourier decomposition method (ABC-FDM) , and a differential multipole method (DMM) . Before proceeding to discuss hollow-core EPCFs, we present results obtained with the SMT for solid-core PCFs, and compare them with these published data.
2.1. Validation by comparison with previously published data
In Table 1, the effective indices of the fundamental modes that are approximately polarized in the x direction in the center of the PCF, are shown for a few solid-core EPCFs. The effective indices calculated by the ABC-FDM and DMM are quoted from Table 3 in , and the geometry considered is shown in Fig. 1(a). Harmonic exp(jωt) time dependence is assumed throughout. The aspect ratio (vertical to horizontal axes ratio), is denoted by η, and takes on the values shown in the first column of the table. The excellent agreement in Re(n eff) reassures us of the correctness of our SMT implementation. The ABC-FDM results for Im(n eff) appear to be less accurate, as the agreement between the DMM and SMT results is very good.
3. Hollow-core EPCFs
While solid-core EPCFs have been studied by a number of authors, hollow-core EPCFs have hardly received any attention at all. In , a one-ring hollow-core EPCF was analyzed by a super-cell model which, however, could not take into account the radiation losses. For the modes studied in , the losses are anticipated to be relatively small, because they are guided by total internal reflection, so a super-cell analysis should be reasonable.
In this paper, we shall focus on modes that are guided by the photonic bandgap effect and not by total internal reflection as in the previously analyzed cases. In general, because bandgap guidance is due to multiple reflections throughout the cladding, it is expected that the differences between the previously degenerate modes would be greater than when the guidance mechanism was total internal reflection. The numerical results generally support this intuition, although further research would be required to form a comprehensive picture of the modal characteristics of these devices.
3.1. Modal Dynamics
To find bandgap-guided modes that carry most of their energy in the hollow core, we focused on a small portion of the complex n eff plane in the neighborhood of the light line, Re(n eff) = 1. We began with a PCF with circular veins, for which we calculated the bandgap for out-of-plane propagation by a plane-wave expansion method  (see Fig. 2), and then gradually decremented the aspect ratio. As in the solid-core example, the semimajor axis was kept fixed (see Fig. 1(b)).
We use McIsaac’s  symmetry-based mode numbering, where a mode that is approximately x-polarized in the PCF center is referred to as a p = 3 mode, and its counterpart, approximately y-polarized, is referred to as a p = 4 mode. The complex effective indices that correspond to modes of both polarizations are shown in Fig. 3, for decreasing aspect ratio, η. In the topmost panels the PCF has circular veins, and accordingly, most of the p = 3 modes have degenerate counterparts of symmetry class p = 4. There are some exceptions however, for example, the p = 4 mode at n eff = 0.975 - 0.01j; these are non-degenerate modes. McIsaac’s symmetry-based mode classification has a certain drawback, in that non-degenerate modes (referred to as p = 8) are also p = 4 modes, and non-degenerate p = 7 modes are also p = 3 modes. A different symmetry-based mode classification that circumvents this problem was proposed in , but we have stayed with the more widespread classification. These non-degenerate modes should maintain their polarization very well when propagating, but they may be difficult to excite because they are not the fundamental degenerate pair, which is shown in Figs. 4(a) and (b). Apart from that, the non-degenerate modes have zero light intensity at the PCF axis. We therefore focus only on p = 3 and p = 4 modes in this paper.
The most easily discernable feature of Fig. 3 is the rightward movement of all of the dots as the aspect ratio decreases. A possible explanation for this is that the air-filling fraction decreases with the decreasing aspect ratio, causing the effective indices to approach their upper limit. Although the aspect ratio in Fig. 3 is varied very slightly from one panel to the next, the movement of some of the modes is quite dynamic. If the aspect ratio were to be decremented in larger steps, it may become hard to make a correspondence between the modes of the EPCF and the modes of the circular-veins PCF. The similarities of the modal fields are generally not enough for this purpose as the modes are not spatially orthogonal . When many modes are present, it would be natural to define the modal birefringence as the difference in effective indices of two modes that correspond to a degenerate pair in the circular-veins PCF. Therefore, from the starting position in the topmost panel, we traced the movement of the effective indices of the fundamental degenerate pair, as the aspect ratio of the veins is decreased and they turn elliptical. This is shown in Fig. 5, where the dotted lines connect points of equal aspect ratio. Clearly, the p = 4 mode is affected much more moderately than the p = 3 mode by the deformation of the veins. Indeed, in Fig. 3 the p = 4 mode is seen to be the one with fewest losses left of the light line (modes right of the light line usually propagate mostly outside the core, and thus would not be excited by adequate coupling). In this sense, the p = 4 mode would seem to be the fundamental of the EPCF, although conceivably, at other aspect ratios a different mode may become the fundamental. It is also interesting to note that the losses of the p = 3 mode reach a maximum near η = 0.85 and then start to decrease again. In the p = 3 AVI file, an enhancement of the field near the boundary of the outermost veins can be observed around this aspect ratio, which may explain this maximum. We stopped the tracking process at η = 0.8 since both modes were well beyond the light line, and we wanted to focus on bandgap guidance of the air-core modes
3.2. Adiabatic deformation of the fundamental mode
Each curve in Fig. 5 corresponds to a single mode in the sense that any two points on the curve can be connected by a continuous deformation of the structure. The mode profile, however, changes shape substantially along the curve. In Fig. 4, the intensity of light, normalized to unit overall intensity in the cross-section, is shown for a number of modes along the p = 3 curve of the fundamental pair. The mode begins strongly concentrated in the core, but as the veins become more elliptical it spreads over the cross-section. Eventually, the propagation constant crosses the light line, and the mode becomes evanescent in the air and confined mostly to the dielectric, as seen in Fig. 4(f). When the PCF is deformed, a different mode may become the fundamental, but if the PCF starts out as circular and is then deformed adiabatically, the fields in the PCF cross-section would actually follow the sequence shown in Fig. 4.
3.3. Wavelength dependence of the radiation losses and the birefringence
In Fig. 6(a), the wavelength dependence of Im(n eff) of the mode that is the fundamental mode in the circular-veins PCF is shown for three aspect ratios, including the circular case. For most of the wavelength range, the p = 3 mode is more lossy than the p = 4 mode, however, at the shorter wavelengths this order is reversed. When one more ring of veins is added, as shown in Fig. 6(b), the losses are reduced approximately by a factor of two. Here too, the p = 4 mode is less leaky than the p = 3 mode in the longer wavelengths, and the order is reversed at shorter wavelengths. This is quite an interesting feature, because it implies that the dominant polarization is determined by whether the wavelength is to the right or to the left of the intersection of the p = 3 and p = 4 curves.
The wavelength dependence of Re(n eff) and the birefringence are shown, respectively, in Figs. 6(c) and (e), for three aspect ratios, including the circular case. When one more ring of veins is added, as shown in Fig. 6(d), Re(n eff) is almost unchanged. This is as it should be in a well-confined mode, since the added ring of veins is added in a region where the fields are small and thus perturb the effective index only be slightly. On the other hand, the birefringence, which is the difference of two such perturbed curves may be altered significantly. As shown in Fig. 6(f), higher birefringence is obtained in this case. Considering the very small deviation from a circular geometry, the birefringence is quite high. For comparison, birefringence of the same order of magnitude (7 × 10-3,15 × 10-3) was found in , for a solid-core PCF with two rings of veins of aspect ratio of 0.7. It is interesting to note the zero-crossings of the birefringence, implying an accidental degeneracy at a specific wavelength. At this wavelength the polarizations may be easily coupled, and this might be utilized to make very sharp optical filters, as was suggested in [21, 22], in the context of anisotropic crystals.
The modes shown above are too leaky to be of practical use for optical telecommunications, although they may be relevant at lower frequencies or in very small devices. Also, it is plausible that the modal characteristics that would be found in PCFs with a larger cladding would not be entirely different, because the extra rings of veins would be added in regions where the fields are small, at least for well-confined modes like the one in Figs. 4(a) and (b).
In conclusion, the modal characteristics of hollow-core EPCFs were shown to include a number of interesting features: the radiation losses may be polarization dependent, and the polarization which is less leaky depends on the wavelength. The less leaky mode, which may be considered the fundamental is rapidly deformed as the veins become elliptical. To define the birefringence of the various modes, a correspondence was made between the modes of the EPCF, and those of the circular-veins PCF. Zero-crossings of the birefringence curves were also shown. We hope that these dynamic modal characteristics will motivate further research into these devices.
The authors would like to thank Alon Ludwig for his assistance with the calculation of the bandgap for out-of-plane propagation.
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