1. Introduction

Photonic Bandgap Fibers (PBGFs) are an emerging class of microstructured optical fibers whose guiding properties rely on the existence of photonic bandgaps (PBGs) [1]. In air-silica PBGFs, the periodic arrangement of holes gives rise to photonic band gaps where light propagation is prohibited for well-defined ranges of optical frequencies within the cladding, and consequently light is highly confined within the core (i.e. an introduced defect). As PBGFs do not rely on total internal refection for guiding, these novel fibers do not need a high refractive index core to achieve light guidance, allowing the propagation to take place within an air core [2], resulting in a very small overlap of the guided mode with the fiber structure. The benefits of propagating light within a hollow-core arise from the lower optical attenuation, nonlinear coefficients and Rayleigh scattering of air, as compared to those of silica. Additionally silica PBGFs offer the possibility of guiding light at wavelengths that cannot be guided using conventional fibers made from the same materials [3].

Within the bandgap of a hollow-core photonic band gap fiber it is possible to distinguish two types of modes: air-guided modes which concentrate most of their energy in the core and surface modes (SMs) which localize energy at the core-cladding interface. It has been shown that the coupling of energy between the “fundamental” air-guided mode (FM) and surface modes due to typical imperfections along the fiber is a major source of loss in PBGFs [5, 6]. Moreover, these interactions can drastically reduce the fiber’s transmission bandwidth [4, 6], making the presence of surface modes the major limitation for the development of air-silica PBGFs with a broad transmission spectrum. Note that in these fibers the position and width of the bandgap is determined by the photonic crystal cladding which in this study is kept fixed. However the useful bandwidth is determined by the core shape and in particular whether or not surface modes are supported. Therefore, in order to increase the transmission spectrum of photonic band gap fibers it is necessary to design new core structures that do not support SMs or at least “push” them towards the bandgap edges.

The problem of minimizing the impact of surface modes has attracted much attention and several studies have been carried out in order to understand their nature and ways to eliminate them [6–12]. For example, it has been proposed that cores formed by terminating the periodic photonic crystal cladding with a perfectly circular hole can suppress the presence of surface modes if the core radius is chosen properly [7, 8]. Alternatively, it was suggested that if an appropriate circular silica ring is added at the core-cladding interface, the impact of surface modes can be reduced [9]. However these two core designs do not resemble real fibers and cannot be readily fabricated with the current technology. In contrast real fibers always have a non-circular silica ring at the core-cladding boundary, and similarly variations in the core size deform the ring of holes immediately surrounding it. Consequently a proper investigation resembling realistic structures of the critical core-cladding interface region is needed to successfully address the problems of surface modes [11, 12].

In the following sections we present a detailed study of idealized but realistic silica PBGFs structures, investigate the impact that key structural parameters have on the fiber’s transmission properties and identify a new design regime with a fundamental core mode robustly free of anticrossings with surface modes for all frequencies within the bandgap.

2. Fiber structure and modeling method

Since this study focuses on the design of realistic silica hollow-core PBGFs with a broad transmission spectrum, we first need to consider that the width of the optical bandgap is strongly correlated with the air filling fraction of the cladding. Due to the high air filling fraction required to obtain wide bandgaps, the cladding holes tend to become hexagonal with rounded corners and are described via two parameters [13]: the relative hole diameter, d/ʌ and the relative diameter of curvature at the corners, d_c/ʌ, as shown in Fig. 1. In our model, the core is defined by a non-circular ring of nearly constant thickness t_r at the boundary with the cladding as shown in Fig. 1(b). The corners of the pentagons surrounding the core are rounded using circles of relative diameter d_p/ʌ. This core-cladding boundary representation was adopted trying to accurately imitate scanning electron microscopy (SEM) images of our latest fabricated fibers, where the core was formed by removing the 7 central capillaries from the stacked preform.

 

Fig. 1. (a) Cross section of a modeled PBGF with d/ʌ = 0.95, d_c/ʌ = 0.55, d_p/ʌ = 0.317 and t_r/(ʌ-d) = 1, (b) geometric parameters used to define the structure.

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During the fabrication of PBGFs, while keeping the cladding structure fixed, it is possible to control the size of the air core (i.e. it can be compressed or expanded) and the thickness of the silica ring surrounding it (t_r). Hence our study focuses on the effects on the fiber’s transmission properties due to modification of these two parameters. The size of the core is determined by the parameter Rc as shown in Fig. 1(b). In order to accurately model realistic fibers of different core sizes, compressions and expansions of the core were modeled considering that only the first ring of holes is affected and the rest of the cladding is unchanged. This assumption was justified based on experimental observation, as shown by SEM micrographs of PBGFs with expanded and compressed cores in Fig. 2. In Fig. 2.(a) (compressed core) we see that distortion of the structure beyond the first ring is minimal. For the expanded core a certain degree of distortion is observed for the second ring of holes. However this distortion has little impact on the surface modes since they are tightly confined in the ring around the core, so this distortion do not change the conclusions of our study. Idealized representations of the fibers used for the simulations are presented in Fig. 2(b) (d). The modeled fibers strongly resemble our fabricated fibers suggesting that the designs obtained from this study are feasible. Note that changes on the thickness of the ring do not affect the parameter Rc since it is measured from the center of the fiber to the boundary of the first cladding hole. Normalized variations of Rc were considered by defining an expansion coefficient E and the thickness of the core boundary was normalized with respect to the thickness of the thin silica struts in the cladding:

where negative values of E correspond to fibers with compressed cores and positive values to enlarged cores.

 

Fig. 2. (a) and (c) SEM images of hollow-core PBGFs fabricated at our facilities with compressed and expanded cores respectively. (b) and (c) modeled structures that resemble the fibers, with parameters d/ʌ = 0.95, d_c/ʌ = 0.55, d_p/ʌ = 0.317, normalized ring thickness T = 1, and expansion coefficient of -6.33% for (b) and +6.33% for (d).

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A full-vector finite element method was used to accurately and efficiently solve Maxwell’s equations [14]. In these calculations the dielectric properties of silica are directly included in the model through the Sellmeier equation. Since in this study we are only interested in the fundamental air-guided mode and surface modes belonging to the same symmetry class (note that modes of different symmetry classes do not strongly interact), we have obtained all the required information and reduced the computation time by only modeling one quarter of the fiber. In previous studies[14] the accuracy of the FEM method was discussed and we ensured that the mesh size was sufficient to ensure convergence of the propagation constants.

3. Parameters used to evaluate the fibers quality

We initially modeled the 7-cell core PBGF with an un-optimized silica ring of normalized thickness T = 1, shown in Fig. 1(a). In this study we used d/ʌ = 0.95 and d_c/ʌ = 0.55, which according to detailed structural analysis performed by SEM on our previously fabricated fibers we believe are realistic targets for fabrication. A hole-to-hole distance ʌ = 4.3μm was chosen to obtain a bandgap centered at ≈ 2 μm (although the results presented here are general and can be transposed to other wavelengths by scaling the structure). The optical modes within the bandgap were calculated and are presented in Fig. 3. Blue curves in Fig. 3(a) show the effective index of the fundamental air-guided mode as a function of wavelength while red curves correspond to a surface mode of the same symmetry class, and the green region represents the photonic bandgap. This un-optimized fiber has an anticrossing between the fundamental mode and a surface mode within the bandgap at λ = 2.13 μm. Away from the avoided crossing point (i), the fundamental mode localizes most of the optical power in the hollow-core whilst the surface mode [i.e. point (iv)] is tightly confined in the glass region surrounding the core as clearly seen from the modal plots in the lower panel of Fig. 3. For wavelengths close to the anticrossing [i.e. point (ii)] the modes of the fiber cannot be differentiated as “true” air-guided modes or as “true” surface modes since at these wavelengths both modes concentrate energy in the hollow-core and in the silica ring [6, 10]. Hence, the impact of the modal interaction can clearly be seen as a dip in the percentage of power in the core of the “fundamental” core mode Fig. 3(b).

Since modes concentrating energy in glass can be easily coupled to extended modes that are lossy, an increase in the fiber’s transmission loss is always observed at wavelengths near to the avoided crossing [6]. Thus the fiber’s operational bandwidth is correspondingly reduced. Furthermore, at the bandgap edges [i.e. point (iii)] the fundamental mode expands into the cladding region reducing its fraction of power concentrated in the core. Therefore the fraction of core confined energy of the fundamental mode against wavelength provides a useful qualitative estimation of the fiber’s transmission spectrum.

Away from anticrossing wavelengths, the dominant factor of loss in PBGFs is the scattering loss due to roughness at the glass/air interfaces [15, 16]. Even though this loss is unlikely to be completely eliminated, it can be reduced by decreasing the fundamental core mode’s intensity at the air-glass interfaces. This loss is proportional to an overlap integral of the fundamental air-guided mode with the silica surfaces:

where E and H are the electric and magnetic fields of the fundamental mode and ẑ is the unit vector along the fiber. The normalized interface field intensity (factor F) of the fundamental core mode is plotted in black in Fig. 3(b). This graph clearly shows the strong inverse relation between F and the fraction of power in the core (blue curve). In the following sections we assess the quality of different fibers by using the energy confined by the fundamental mode in the core region and its factor F since these two parameters suffice to identify good designs, and since both factors critically depend on whether or not surface modes exist.

Summarizing, we aim to design core structures that suppress surface modes. By their elimination the fiber’s operational bandwidth is increased and the loss due to mode coupling mechanisms is reduced [6]. In addition, we are looking for designs that minimize F in order to reduce the dominant factor of loss at wavelengths where the interaction with surface modes is small [15, 16]. Finally we note that the fibers studied here support higher order core modes however these modes are not relevant to our study since they do not interact with the fundamental core mode.

4. Results

4.1. Dependance of the fiber’s properties on the thickness of the core boundary

A systematic investigation of the effects that modifications of the ring thickness have on the fiber’s transmission properties was carried out. Holding the core size constant (E = 0) the normalized boundary thickness was varied in the range 0.175 ≤ T ≤ 3.5 and for each fiber geometry we solved for the modes at wavelengths within the bandgap and near to the bandgap edges. This range was chosen to reflect the range of possible fibres that can be fabricated with current technologies. In order to assess the performance of the various structures we calculated the percentage of power in the core (PinC) and the factor F of the fundamental air-guided mode as previously described. The results are presented as a function of the normalized ring thickness T and of the wavelength in Fig. 4(a) (b). For the design map in Fig. 4(a), red (PinC ≥ 90%) denotes regions where the fundamental mode is tightly confined in the core, and yellow diagonal lines (PinC ≈ 60%) correspond to anticrossing points between the fundamental and surface modes (note that some lines appear dis-continuous in the graph due to the discrete number of T values considered); green regions (PinC ≈ 50%) correspond to wavelengths where the fundamental mode is outside the gap and highly expanded into the cladding, i.e. it coexists with cladding modes as shown in Fig. 3.(iii). The regions where the fundamental air-guided mode is no longer supported are presented in blue. Similarly, for the normalized interface field intensity map shown in Fig. 4(b), blue (Fʌ ≤ 1) corresponds to regions of high confinement of the fundamental mode, cyan lines (Fʌ ≈ 3) to anticrossing points, and red to regions where a fundamental air-guided mode is no longer supported.

 

Fig. 3. (a) Effective index of modes (blue for FM and red for SM). (b) Fraction of core-confined energy of the FM (blue) and factor F (black) of the FM vs. wavelength for a fiber with T = 1 and E = 0. Plots of the axial Poynting vector: (i), (ii), and (iii) are of the FM “far” from the anticrossing, at the anticrossing point and near the long wavelength bandgap edge respectively, and (iv) is for the SM.

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By studying the maps in Fig. 4, we immediately notice that the fiber with a core boundary of thickness T = 0.175 (the lowest value examined in this study) supports two surface modes that interact with the fundamental at λ =2.1μm and λ = 2.19μm. Similarly, for certain values of T multiple surface modes appear inside the bandgap in all cases reducing the operational bandwidth of the fiber. In clear contrast, Fig. 4(a) (b) show that there are two important design regions for which the wavelength range over which the fundamental air-guided mode is free from anticrossings with surface modes is maximized. The first one is for core surrounds of thickness T ≈ 0.5 and the second for T ≈ 1.15. However, fiber designs with T around 0.5 are better since they provide a higher confinement of the core mode and a smaller overlap of the fundamental mode with the glass-air interfaces.

 

Fig. 4. (a) Fraction of core-confined energy and (b) factor F of the fundamental core mode vs. normalized ring thickness and vs. wavelength, for fibers with E = 0.

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The evolution of the dispersion curves of the fundamental and surface modes as the thickness of the core boundary increases is presented in Fig. 5(a) and (b). Solid lines correspond to the effective index of the fundamental air-guided mode while dashed lines correspond to surface modes. Similarly, Fig. 5(c) shows the factor F of the fundamental core mode for fibers with T = 0.4, 0.5, 0.6 and 0.7. As can be observed from Fig. 5(a), the addition of silica at the core-cladding interface (modifying T from 0.175 to 0.4) produces an increase in the effective index of the surface modes. Thus the anticrossings are shifted towards longer wavelengths and the fiber’s operational bandwidth is increased. Additionally, the strength of the interactions is reduced. As T increases still further, the surface modes are progressively shifted towards the long wavelength side of the bandgap, see Fig. 5(b). For T = 0.5 only one anticrossing occurs inside the gap while the fundamental mode of a PBGF with T = 0.6 is free of anticrossings with surface modes for all wavelengths within the bandgap. Increasing the thickness still further results in a new surface mode appearing on the short wavelength side of the gap along with a corresponding decrease in the operational bandwidth of the fiber.

We believe that the appearance of this design region free of surface modes can be explained as follows. The termination of the cladding periodicity to create the air core without adding a silica ring (T = 0), generates surface modes with intensity maxima at the corners of the photonic crystal, as described in [7–9]. The profile of a surface mode localized at the corners of the photonic crystal is shown in Fig. 5(a), which corresponds to the mode located by an arrow. If a silica ring of thickness T = 0.7 is added, the structure is perturbed strongly enough to support a new type of surface mode, which concentrates energy at the thin silica struts of the core boundary, see Fig. 5(b). However, we found that between these two fiber designs there is an intermediate design regime that induces a perturbation that is too weak to support surface modes of the second type but strong enough to shift surface modes of the first type outside the bandgap.

 

Fig. 5. Dispersion curves of the fundamental mode (solid lines) and surface modes (dashed lines) for fibers with (a) T = 0.175 and 0.4, (b) T = 0.5, 0.6 and 0.7. Insets are mode profiles of the surface modes located by arrows. (c) Factor F of the FM vs. wavelength for fibers with T = 0.4, 0.5, 0.6 and 0.7

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The factor Fʌ of the fundamental mode of a fiber with T ≤ 0.6 is approximately equal to unity in the vicinity of the short wavelength edge of the bandgap (λ ≈ 1.88 μm), see Fig. 5(c) and Fig. 4(b). Furthermore it remains less than one across most of the bandgap when there are no surface modes. We set Fʌ < 1 to be the condition for the useful bandwidth of the fibers and define the operational bandwidth to be the maximum continuous wavelength range for which this condition is satisfied within the bandgap. Using this definition the operational bandwidth as function of the normalized core thickness is shown in Fig. 6. In Fig. 6(a) we have normalized the operational bandwidth with respect to the central central gap wavelength λ_c = 2.05 μm while in Fig. 6(b) we present the results normalized with respect to the bandgap width measured at the airline, equal to 330 nm. As expected, PBGF structures with rings of thickness in the range of approximately 0.45 ≤ T ≤ 0.65 are optimal for broadband transmission providing a wide operational bandwidth of more than 15% of the central gap wavelength. Fig. 6(b) shows that fibers in this new design regime have an operational bandwidth of over 90% of the gap width at the airline. The maximum operational bandwidth is 17% of λ_c (105 % of the bandgap width at the airline) for a fiber with a ring of T = 0.575. This new design regime drastically reduces the impact of surface modes as compared with a more conventional core design of T = 1 which has an operational bandwidth 9% of λ_c.

 

Fig. 6. (a) Operational bandwidth normalized with respect to the central gap wavelength λ_c = 2.05 μm, and (b) normalized with respect to the bandgap width measured at the airline, equal to 330 nm.

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Finally to conclude this analysis we have summarized the contour plots as follows: for each different ring thickness we found the wavelengths that had the minimum F factor (maximum power in the core) and have plotted this minimum (maximum) value. These results are plotted in Fig. 7 against the normalized ring thickness. We found three design regimes where the confinement of the fundamental mode and the factor F are maximized and minimized respectively. These correspond to the thin ring region centered round T = 0.5, an intermediate region around T = 1.15 and the antiresonant region studied by Roberts et al. [15, 16] around T = 2.5. We note that for this fiber the antiresonant region is not particularly noticeable but for other fiber designs it offers the possibility of reducing the loss -see Section 4.2 and [15, 16]. Of these three regimes which one is best depends on the applications of the fiber, however our new thin core design regime offers in all cases the widest operational bandwidth and should thus be preferred when fabricating fibers for broadband operation. Also for this particular core size the thin ring designs have the lowest loss (F factor) in the bandwidth and are thus the preferred targets for fabrication.

 

Fig. 7. (blue) Maximum of the core-confined energy and (black) minimum F factor of the fundamental core mode vs. normalized ring thickness.

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Fig. 8. (Movie 312 KB) The movie in (a) shows contour maps of the percentage of power in the core of the fundamental air-guided mode as the core is enlarged. Contour maps of fraction of power in the core of the FM for fibers with the (a) smallest and (b) largest core analyzed, E = ±6.33%.

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4.2. Dependance of the fiber’s properties on the size of the core

In this section we analyze the effects that compressing and enlarging the core have on the fiber’s transmission properties. To do so, the fiber’s core has been modified by scanning the expansion coefficient E from -6.33% to 6.33% in 1.583% intervals. For each core radius the thickness of the core boundary was varied in the range presented in the previous section. The movie in Fig. 8(a) shows contour maps of the percentage of power in the core of the fundamental air-guided mode as a function of E. As the core is enlarged, the fundamental core mode becomes more tightly confined in the central air region (red regions of the contour maps become darker as E increases). We found that the number and position of surface modes in fibers with thick core surrounds (T > 2) is strongly dependent on the size of the core. Conversely, surface modes supported by fibers with thin core surrounds (T < 0.6) do not highly depend on the core size. This behavior is more evident when comparing the contour maps for the fibers with the smallest (E = -6.33%) and largest (E = 6.33%) core analyzed, Fig. 8(a) (b) respectively (note that the lower parts of the maps remain almost the same while the upper parts present big differences).

The maximum fraction of core confined energy, minimum Fʌ and relative useful bandwidth of fibers with different core radius are presented as a function of the normalized ring thickness in Fig. 9(a),(b) and (c) respectively. Changes of the core radius do not drastically shift the position of the three design regimes that maximize the mode confinement and minimize F found for E = 0. Antiresonant core effects (T around 2.5) become noticeable for large cores, i.e. E = 6.33% and 3.16%. Despite the presence of surface modes, fibers with large cores and antiresonant rings have the lowest loss but they do not offer wide operational bandwidth. However, for these fibers the size of the core has a big impact in the minimum achievable value of F. In contrast, we found that the thin ring design regime (T around 0.5) is robust against the size of the hollow-core. This is important as it is difficult to have precise control of the size of the core during the fabrication process. Figure 9(c) shows that variations in E do not drastically affect the operational bandwidth of fibers in the thin ring design regime. For all the core sizes studied, thin ring designs offer the widest operational bandwidth and have low scattering loss. Indeed for small core sizes considered (E < 3.16%) thin ring designs have lower F values than designs with antiresonant core surrounds.

 

Fig. 9. (a) Maximum of the core-confined energy, (b) minimum Fʌ, and (c) operational bandwidth normalized with respect to the center of the bandgap λ_c = 2.05 μm vs. normalized core thickness for E = ±6.33%,±3.16% and 0.

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5. Conclusion

Through computers simulations we have demonstrated that by carefully selecting the thickness of the silica ring around the core and the size of the core it is possible to design realistic PBGFs with high air filling fractions with a fundamental air-guided mode free of anticrossings with surface modes at all wavelengths within the bandgap. The fibers have a silica ring surrounding the core of thickness around the half of the thickness of the thin cladding struts. These designs offer the widest operational bandwidth and should thus be preferred when fabricating fibers for broadband operation. Furthermore, we have demonstrated that our hollow-core PBGFs designs are robust in eliminating surface modes and variations in the size of the core do not drastically reduce their operational bandwidth. These results can be transposed to other wavelengths by scaling the structures.

We have studied several fibers with different cladding parameters demonstrating that our conclusions are independent of these parameters. By increasing the air filling fraction of the cladding the achievable usable bandwidth is further increased. Initial studies of fibers with a 19-cell core suggest that the design regime presented here also maximizes the useful bandwidth of these fibers. The results of this study will be presented in a subsequent paper.

Finally we are confident that it is possible to fabricate these designs because they strongly resemble the fibers fabricated at our facilities. We expect that the optimum thickness of the central silica ring (T ≈ 0.5) can be obtained simply by removing the 7 central capillaries from the stacked preform without the need to add an extra element to create the core.