1. Introduction

Solid core photonic bandgap fibers (SC-PBGFs) [1–6] are optical fibers with a low index core surrounded by a microstructure of discrete high index regions [Fig. 1(b)] or concentric rings of high and low index glass [7]; in this work, we concentrate exclusively on the former geometry, specifically the case where the high index regions are cylinders. One may treat the microstructure as a photonic bandgap medium (hence the name), and in the limit of an infinite, periodic microstructure, the modal properties of SC-PBGFs can be modeled using Bloch’s theorem and plane wave expansions [2,3,8]. However, it has been shown that there is a more intuitive model of SC-PBGFs as a cylindrical analog of planar antiresonant reflecting optical waveguides (ARROWs) [9] [Fig. 1(a)], where the high index cylinders play the role of the antiresonant layer and the backward scattering of light from the cylinders provide mode confinement in the low index core (see Fig. 2) [10–12]; in the literature, SC-PBGFs are thus sometimes called ARROW microstructured optical fibers (ARROW-MOFs) or ARROW photonic crystal fibers (ARROW-PCFs).

 

Fig. 1. Schematic of (a) single layer planar ARROW with antiresonant mode in low index core, (b) SC-PGBF index profile, and (c) transmission loss through a SC-PBGF. Vertical lines correspond to cutoff frequencies of the vector modes of the high index cylinders: red=TE/TM_0,p,HE_2,p, blue=HE_1,p+1, and green=EH_1,p;HE_3,p

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The ARROW model presumes that the spectral properties of SC-PBGFs are dominated by the scattering properties of a single high index cylinder at oblique incidence, and that therefore neither periodicity nor multiple layers are necessary conditions to achieve mode confinement [13]. Furthermore, just as the high index layer of a planar ARROW may be treated as a waveguide in its own right, the modal structure of which is closely related to its resonances and antiresonances in the ARROW geometry [14, 15], so too the modes of a single cylinder in the cladding of the SC-PBGF geometry are linked to the cylinder scattering resonances [16] (see also [8, 10, 12, 17–20]). The scattering resonances of the cylinder are excited when the incident field (a plane wave originating in the low index region) is phase-matched to and has strong overlap with a leaky cylinder mode; in the SC-PBGF geometry, where the plane wave is at grazing incidence to the cylinder surface, these conditions are optimally satisfied for cylinder modes with low azimuthal order, at frequencies very near to the modal cutoff. One can thus understand many of the properties of SC-PBGFs (e.g. spectral position of the transmission bands, band structure, mode field patterns) in terms of the modes of a cylinder [8, 12, 17]. An example of this is shown schematically in Fig. 1(c), where we plot the modal loss in the first six transmission bands of a SC-PBGF. The vertical lines, which correspond to the cutoff frequencies of cylinder modes with low azimuthal order and are given by analytic formulae, determine the edges of the SC-PBGF transmission bands.

However, since SC-PBGFs are two dimensional structures, there are many ways in which they differ from one dimensional ARROWs. For example, the low frequency transmission bands of multi-layer planar ARROWs depend on the periodicity of the layers and so confinement in the low frequency bands of SC-PBGFs was presumed to arise from Bragg-like effects as well [10–12]; it was later demonstrated that in the cylindrical geometry, these are still strongly dominated by the scattering resonances and antiresonances of the single cylinder [8, 17, 18]. This is not to say, however, that the cylinder spacing is unimportant. In fact, as a number of recent papers have shown, the ratio between the cylinder diameter d and the pitch of the cladding microstructure Λ has a strong impact on a number of SC-PBGF properties, such as bend loss [19] and the influence of cylinder modes with high azimuthal order [8, 20]. This is explained in terms of the field distributions of different types of cylinder modes and the degree to which they couple to adjacent cylinders.

One may then ask, what happens in the limiting case where d/Λ is small and the coupling between cylinders is weak? This is where we expect the strongest analogy between SC-PBGFs and the single layer ARROW, in the sense that the SC-PBGF spectrum should truly mimic the scattering spectrum of a single cylinder at oblique incidence, just as the single layer ARROW spectrum mimics the reflection spectrum from a thin dielectric slab. Following the treatment in Ref. [13] (rederived in modified form in Sec. 2), we show that the loss spectra of SC-PBGFs with low d/Λ do in fact behave like the scattering spectrum of a single cylinder at all frequencies. More interestingly, as described in Secs. 3 and 4, we find that the single cylinder scattering spectrum is more complicated than previously thought, and in particular, for glancing incidence the scattering resonances behave like Fano resonances. We derive a semi-analytic formula to explain the origin of this phenomenon. Finally, in Secs. 5 and 6, we show how this behavior can be used to optimize the core design of SC-PBGFs with an eye towards mitigating the modal loss in the lowest frequency band of SC-PBGFs with moderate d/Λ.

2. Scattering formalism

The scattering of an electromagnetic plane wave by a dielectric cylinder at conical incidence is a well-studied problem, originally solved over 50 years ago [21], and we present only the basics of the formalism; we generally follow the treatment given in Ref. [22]. Consider a plane wave scattered by an infinite dielectric cylinder as in Fig. 2, where ϑ₀ is the angle of conical incidence [23]. In this geometry, all fields can be extracted from the E _z and H _z components of E and H, which in turn are a sum of the incoming and scattered fields

where β=n _low k ₀ cosϑ₀, k ₀=2π/λ and ω=ck ₀. A similar expression holds for H _z . The incoming and scattered fields can be expressed as Fourier Bessel series

where f denotes either E or H, k _⊥=n _low k ₀ sinϑ₀, J _q and $H_q^{\left(1\right)}$ are the q ^th order Bessel function and Hankel function of the first kind, respectively, and $A_q^f$ and $B_q^f$ are complex-valued expansion coefficients. The cylindrical harmonics in Eq. (2) are chosen to satisfy the incoming and outgoing wave conditions of the incident and scattered fields. The internal fields have the same form as Eq. (2a). The A _q coefficients are determined by the condition that ${\psi }_i^f$ is a plane wave

where Z₀ is the vacuum impedance, φ ₀ is the angle of incidence with respect to the origin of the coordinate axes, which we set to zero, and δ ₀ defines polarization angle of the incoming field (for φ ₀=0, δ ₀ is the angle between E and the y-axis in Fig. 2), which we set to zero as well; that is, the cylinder axis always lies in the plane defined by H and k (H polarization [16]), though our scattering calculations give nearly identical results for arbitrary polarization. The $B_q^f$ coefficients, which determine the scattered field, are found by applying boundary conditions at the cylinder surface. For simplicity, we define $B_q^H$ to include the Z ₀ terms which typically appear in the expressions for the scattering cross section given below.

 

Fig. 2. Geometry of plane wave scattering by a dielectric cylinder at conical incidence. The variable φ is defined such that regions of forward or backward scattering are given by cos(φ)>0 or cos(φ)<0, respectively. The polarization is chosen such that the cylinder axis lies in the plane defined by H and k. In the related SC-PBGF geometry, the region below the cylinder corresponds to the SC-PBGF core.

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One way to interpret the field representation of Eq. (2b) is that the field around the cylinder can be described as a superposition of its leaky modes, and so each $B_q^f$ term is related to the strength of the cylinder modes excited by the scattering process (the excitation of guided modes, of course, is forbidden). For glancing incidence, we excite the modes just below cutoff, where the radiation field is described as a leaky mode [24]. Since we are dealing with E _z and H _z , each $B_q^f$ corresponds to the excitation of a leaky HE_q,p or EH_q,p mode (TE_0,p or TM_0,p for q=0) which, depending on the mode type, has q+1 or q-1 azimuthal nodes in the major transverse field components [25].

The incident and scattered power are related through the scattering cross section σ_sc defined by P _sc=σ_sc I _i , where P _sc is the power scattered by the cylinder in the far field and I _i is the power density of the incoming wave. For scattering from a cylinder, σ_sc has units of length normalized to the cylinder radius. Note that even though σ_sc is a far field quantity, surprisingly it does strongly correlate with some features of the SC-PBGF spectra [13]. In our fiber geometry, we need the backward scattered power, since this gives us the SC-PBGF mode confinement (see Fig. 2, where the region below the cylinder corresponds to the SC-PBGF core), but σ_sc gives the total scattering independent of direction. That is to say, strong scattering is a necessary but insufficient condition to achieve mode confinement in SC-PBGFs; the scattered light must be redirected back into the defect core. Thus we must also consider the differential scattering cross section σ^d(φ), the power per unit irradiance scattered into a unit arc for a given angular direction, with σ_sc=${\int }_0^{2\pi }$ σ^d(φ)dφ. It is straightforward to show that

with

We then use σ^d(φ) to obtain the asymmetry parameter g, the average cosine of the scattering angle

g>0 means more forward scattering and g<0 means more backward scattering.

3. Single scatterer and SC-PBGF loss spectrum

In Fig. 3, we plot σ_sc and g for ϑ₀=0.025 rad≈1.43°. We plot the data in this and all subsequent figures as a function of the normalized frequency V=πd($n_{\text{high}}^2$-$n_{\text{low}}^2$)^1/2/λ so that the band edges or scattering resonances always line up along the same x-axis points independent of d or index contrast; in this paper we set n _high=1.65, n _low=1.45, and d=0.2µm. In Fig. 3, the red vertical lines correspond to TE_0,p and HE_2,p cutoff frequencies (for this polarization, $B_0^E$ is always zero, so the TM_0,p modes are never excited) and the blue vertical lines correspond to the HE_1,p+1=EH_1,p cutoff frequencies. The results shown are obtained using three Fourier-Bessel orders, which is all that are required for convergence for this value of ϑ₀ (we have checked up to nine orders); if we use four orders, we do find some additional very fine features associated with the HE_3,p modes, though they are not detectable on the scale shown in Fig. 3, so we do not include the green vertical lines associated with this resonance as in Figs. 1(c). For larger incident angles, we find resonances associated with higher order modes, and more Fourier-Bessel orders are required. We note that in terms of the transverse fields, the blue lines correspond to monopole resonances of the cylinder, where the scattered field is azimuthally symmetric and essentially in phase with the incident field, so σ_sc is effectively zero. That is, these resonances more or less behave like the resonances of the planar slab in that the high index region becomes transparent. The red lines, however, correspond to dipole resonances of the cylinder, and so σ_sc increases. Since the index contrast is sufficiently large to remove the degeneracy of the individual vector modes, we do observe some splitting between the TE_0,p and HE_2,p modes, though the splitting is very small as the frequency increases.

 

Fig. 3. Im(n _eff) versus V for the first 6 bands of a SC-PBGF with d/Λ=0.15 (upper panel), scattering cross section σ_sc (middle panel) and asymmetry factor g (lower panel) for ϑ₀=0.025 rad; the scattering data are calculated using 3 Fourier-Bessel orders. Red and blue vertical lines correspond to TE_0,p/HE_2,p and HE_1,p+1 cutoff frequencies, respectively.

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The top panel of Fig. 3 shows simulation results for a three ring SC-PBGF with d/Λ=0.15. We plot Im(n _eff), which is proportional to the modal loss [loss (dB/length) ~54.575×Im(n _eff)/λ], where we use the multipole method to simulate the mode index [26–28]. We note that this spectrum is quite different from what has been previously reported for SC-PBGFs with moderate d/Λ(d/Λ~0.4-0.7) [8, 13, 17, 18], shown schematically in Fig. 1(c), where the confinement is poor in the low order transmission bands and the lowest loss point of a given band occurs near its center. When d/Λ is small, only the even order bands have the expected quasi-parabolic shape, whereas the odd order bands are concave-up and appear to transmit better at the band edges than at the band centers. Also, the loss in the odd order bands is always lower as compared to the even order bands.

These features are linked to the behavior of g and σ_sc, where low Im(n _eff) should correlate with strong backwards scattering (high σ_sc and negative g). For the frequency ranges corresponding to the even order SC-PBGF transmission bands, g is larger (more forward scattering) and concave up, and σ_sc is lower relative to the adjacent odd bands (weaker overall scattering); this combination of g and σ_sc correlates with high loss, concave up bands in the SC-PBGF geometry. In the frequency ranges corresponding to the odd order bands, σ_sc is larger and g is lower relative to the adjacent even bands, and g is concave down, leading to the unexpected result that in the odd order SC-PBGF bands, the transmission loss is lower at the transmission band edge as compared to the band center.

4. Single scatterer Fano resonance

The above arguments link the peculiarities of confinement loss of low d/Λ SC-PBGFs to the scattering properties of a single infinite cylinder, but they beg the question why the cylinder has asymmetric scattering resonances under shallow angle illumination. By comparison, near normal incidence, dielectric cylinders scatter like spheres, and the resonances are smooth and roughly symmetric [16, 22]. To explain this difference, we note that if we use three Fourier-Bessel orders in the scattering calculation and write their complex coefficients as $B_q^f=b_q^f{e}^{i{\varphi }_q^f}$, then using Eqs. (4)–(6) we obtain

(again, $B_0^E$=0 for this polarization, and since φ ₀=0, we have $B_q^H$ =$B_{\mathit{-}q}^H$ and $B_q^E$ =-$B_{\mathit{-}q}^E$ ). The dominant term in Eq. (7) is the one which describes the interference between $B_0^H$ and $B_1^H$ (first term in the numerator). In Fig. 4 we plot the evolution of $B_0^H$ and $B_1^H$ in the complex plane as a function of V; the right panel shows the projection of these terms on the real line and the bottom panel shows the projection on the complex plane. The loops where the phase is rapidly varying correspond to resonances of the individual coefficients. As expected, the $B_0^H$ loops correspond to the TE_0,p cutoff frequencies and the $B_1^H$ loops correspond to the HE_1,p+1 cutoffs ($B_1^H$ actually shows two resonances; the second loop corresponds to the EH_1,p cutoff. Both loops widen with respect to the V axis as V decreases such that they merge into one resonance for the first $B_1^H$ resonance near V=3.83).

Figure 5 shows the sines of the individual phase terms of Eq. (7), and these are almost identical (indeed, the red curve is almost completely obscured by the green curve). This indicates that $B_2^H$ and $B_0^H$ are always out of phase with the $B_1^H$ term by the same amount and thus in phase with each other. The phase flips shown in Fig. 5 are what give rise to the asymmetric resonances of g in Fig. 3, and by extension, the corners in the odd order bands of the SC-PBGF spectrum. Returning to our interpretation of the $B_q^f$ coefficients in terms of the leaky cylinder modes, the asymmetric resonances can be viewed as the combination of a resonance associated with one leaky mode approaching cutoff and its interference with another, off-resonance leaky mode; as shown in Fig. 4, the resonant and off-resonant modes alternate between TE_0,p=HE_2,p and HE_1,p+1/EH_1,p.

 

Fig. 4. 3-D line plots of $B_0^H$ (red) and $B_1^H$ (blue) as a function of V, with projections on the real axis (right panel) and complex plane (bottom panel) shown.

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For example, let us consider the lowest frequency resonance loop in Fig 4, which occurs for $B_0^H$ (TE_0,1 cutoff). Below the loop, the phase difference between $B_0^H$ and $B_1^H$ is ~-π/3 (cf. Fig. 5, sin(${\varphi }_0^H$-${\varphi }_1^H$)~-0.8), which yields strong backward scattering. As $B_0^H$ goes through its loop, $B_1^H$ is off-resonance and has a relatively constant phase. The $B_0^H$ term does not complete a full loop and changes in phase by ~π. The interference between $B_0^H$ and $B_1^H$ on the high frequency side of the loop now yields strong forward scattering. At the next resonance loop near V=3.83, the roles of $B_0^H$ and $B_1^H$ are reversed. This type of asymmetric resonance arising from a term with rapidly varying phase interfering with another term with slowly varying phase is known as a Fano resonance and was first described in the context of atomic resonances interfering with a background ionization potential [29], though such resonances have been more recently described in the context of optical physics [30–32].

 

Fig. 5. Sine of the phase-dependent terms which contribute to g using same simulation parameters as in Fig. 3 and 4.

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We note in Fig. 5, there are actually two phase flips near V=5. In terms of the SC-PBGF spectrum, the first one accounts for the low loss “horn” at the high frequency edge of the 3^rd transmission band, and the second occurs at the transition between the 3^rd and 4^th transmission bands. In Fig. 4, however, these correspond to the regions around resonances of $B_0^H$ (specifically the TE_0,2 and resonance), where there is a single loop and therefore only one Fano resonance. In fact, only the second phase flip, the one corresponding to the transition between transmission bands, is a Fano resonance of the type described above. The first phase flip, the one responsible for the “horn”, occurs in the non-resonant section of $B_0^H$ just below the loop, and it arises from $B_0^H$ passing through the origin of the complex plane. In a sense, this can also be viewed as a Fano resonance in that it results from a very rapid variation of ${\varphi }_0^H$ (it discontinuously jumps from 3π/2 to π/2, which is why this feature is so sharp in Fig. 5), but it is not obviously linked to any of the modal properties of the cylinder. We find similar behavior near V=8.5, corresponding to the high frequency edge of the 5^th band. If we look at the projection of the scattering coefficients on the real axis in Fig 4, we see that at high frequency, $B_0^H$ crosses the origin within the loops rather than in the off-resonance sections, and so we expect that the “horns” at the high frequency edge of the odd order SC-PBGF transmission bands should disappear for large V. $B_1^H$ always crosses the origin within the loops, at least for the frequency range shown here.

One may ask how the low d/Λ SC-PBGF can be interpreted in terms of the bandgap model of SC-PBGFs. In this model, the cladding is treated as an infinite lattice which supports bands of Bloch modes, which in turn are described as coupled states of the cylinder modes, and guidance in the defect core is only allowed in the gaps between these bands of Bloch modes [2, 3, 8, 19]. In Ref. [19], the authors showed much can learned about the guidance characteristics of SC-PBGFs by considering the Bloch modes which define the bandgap edge. In their particular example, d/Λ=0.44 and fiber had low index contrast, so the Bloch modes were described in terms of the scalar modes of the cylinder and classified using the LP_q,p designation, where the subscripts denote the number of azimuthal and radial nodes in the transverse mode field, as opposed to the notation we use for the vector modes in terms of the longitudinal mode field. The authors found that for their geometry, the Bloch modes which define the bandgap edge alternate between LP_1,p-derived bands and LP_0,p+1-derived bands, depending on whether the associated transmission band is of odd or even order. The LP_0,p+1-derived bands are composed of much more strongly coupled states than the LP_1,p bands [8, 19], and this was shown to have a very strong impact on the bend loss behavior of the fiber [19]. In the present case, we relate the Bloch modes at gap edges of the low d/Λ SC-PBGF to the scattered field distribution at antiresonant frequencies. If we look at the projection of the scattering coefficients on the real line in Fig. 4, we see that in the off-resonance sections (away from the loops), $B_1^H$ is by far the dominant scattering coefficient regardless of whether these sections correspond to an even order or an odd order band (the $B_2^f$ coefficients, which are not shown, are even smaller than the $B_0^H$ coefficient and so we need not consider the behavior these terms). We thus expect that off-resonance, the scattered field inside and around the cylinder is primarily HE_1,p+1-like (or LP_0,p+1-like in the scalar nomenclature). Our multipole simulations in the SC-PBGF geometry do show that near the cladding cylinders, the field distribution of the SC-PBGF core mode always behaves like a leaky LP_0,p+1 mode when d\Λ is small. That is, the band gap edges are associated with leaky LP_0,p+1-like Bloch modes for all transmission bands because the cylinders are sufficiently far from each other that their LP_1,p modes cannot easily couple to their nearest neighbors. We still observe different behavior for odd and even order bands, as in Ref. [19], but here it depends on whether the gap edge represents a LP_0,p+1-derived Bloch mode near cutoff (even bands) or far from cutoff (odd bands).

 

Fig. 6. Im(n _eff) versus V for the first 6 bands of a 3 ring SC-PBGF with (a) d/Λ=0.6, (b) d/Λ=0.4, (c) d/Λ=0.2, and (d) d/Λ=0.15.

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5. Low d/Λ vs. moderate d/Λ

As a practical fiber design, low d/Λ SC-PBGFs are expected to have very high bend loss compared to SC-PBGFs with more densely packed cylinders. Furthermore, given that one would have to measure transmission through many meters of fiber in order to even begin to observe the Fano resonances described above (or longer lengths if the microstructure has more than three rings of cylinders), it is unclear what advantage these fibers provide over more conventional SC-PBGF designs, or for that matter, whether the analysis of the previous three sections contains any useful insights.

However, as we show in this section, there are cases for which SC-PBGFs with moderate d/Λ are not optimal, and our understanding of the low d/Λ may provide some practical benefits. In Fig. 6, we plot Im(n _eff) in the first six transmission bands of SC-PBGFs with varying d/Λ. Again, we see the behavior associated with Fano resonances described above in Fig. 6(d). When d/Λ=0.2 [Fig. 6(c)], the even order bands are essentially unchanged and the odd order still exhibit “horns”, at least in the 3^rd and 5^th bands, though these are somewhat broadened. The 1^st band, while clearly showing some structure associated with the Fano resonances of Fig. 6(d), has lost its horns, indicating the effect of coupling between the cladding cylinders at low frequency. When d/Λ increases to 0.4 [Fig. 6(b)], the Fano resonances are completely gone, and cylinders are sufficiently coupled to each other that the analysis of Secs. 3 and 4 is no longer adequate to explain the shape of the SC-PBGF transmission bands. The confinement in the first and second bands is poor, consistent with earlier results [8, 17, 18], but the modal loss in the third and fifth bands is clearly enhanced compared to the even order bands. This can be explained in terms of the weaker coupling between the LP_1,p-like cladding states as compared to to the LP_0,p+1-like cladding states as described in Ref. [19]. The d/Λ=0.4 spectrum also has discontinuities in the 3^rd–5^th bands associated with EH_2,p/HE_4,p, EH_3,p/HE_5,p, and EH_4,p/HE_6,p scattering resonances (these correspond to avoided crossings in the real part of n _eff [20]). When d/Λ is further increased to 0.6 [Fig. 6(a)], the cylinders are strongly coupled in all of the transmission bands, so we no longer observe qualitative differences between even and odd order bands. The higher order resonances also become wider and stronger and they appear in the 6^th transmission band as well. We note that in Fig. 6, the core is always a single missing cylinder defect, so when d/Λ increases, the core size decreases, as can be seen from the insets. However, the fact that the core becomes small relative to the wavelength for a fixed V has little bearing on the trends we have described. This can be seen by comparing Figs. 6(b), 6(c), and 8(a), where the loss spectra of Figs. 6(b) and 8(a), which correspond to structures having the same d/Λ, are qualitatively very similar, even though the core of the latter is nearly double the size of the former, whereas the spectra of Figs. 6(c) and 8(a) (similar core size, different d/Λ) are very different (also see Ref. [20]).

For moderate-to-high d/Λ, the propagation loss in the lowest frequency band is clearly very poor, and so the experimental papers in SC-PBGFs typically operate in the 3^rd band or higher [1, 2, 5, 19]. As suggested in Ref. [17], however, it may be advantageous to operate in this band since the cylinders support fewer resonances there, and so the propagation loss should be more robust to non-uniformities in the size and shape of the cylinders in the fiber cross section. As we show in the next section, we can employ the Fano resonances to modify the design of moderate d/Λ fibers and achieve low propagation loss in the 1^st band.

6. Fano-enhanced core

The Fano resonances described in Sec. 4 depend on a particular phase relationship between the individual scattering coefficients of the cylinder. We expect that when there are multiple cylinders, each with its own set of interdependent scattering coefficients, this phase relationship no longer necessarily holds and the Fano resonances disappear, even for glancing incidence. Of course if the cylinders are sufficiently far apart from each other, we do observe Fano resonances, as evidenced from the low d/Λ SC-PBGF spectra shown above. In order to give an idea for how the cylinder spacing affects the scattering spectrum, in Fig. 7 we show animations of σ_sc and g for scattering from three cylinders where d/Λ is varied from 0.1 to 0.8 (the general method for calculating the scattering spectrum from multiple cylinders at conical incidence is described in Ref. [33]). The purpose of these animations is to highlight the general trends that describe how the scattering spectrum for multiple cylinders changes as the cylinder spacing is changed. Our specific choice of three cylinders arranged in a particular way is somewhat arbitrary and is in no way meant to give a comprehensive description of all the scattering features for multiple cylinders.

In Fig. 7, the incident angle is fixed at ϑ₀=0.025 rad, and the cylinders are oriented as shown in the inset, where the transverse component of the incident plane wave travels from left to right. We have found that some features of the scattering spectra are orientation dependent (for example, the “undulation” of the spectra at low d/Λ, or the relative strengths of the resonances), though this dependence weakens as d/Λ increases, and as expected, the frequencies of the resonant features in the scattering spectra are independent of orientation.

 

Fig. 7. Animations of (a) scattering cross section σ_sc (b) and asymmetry factor g for three cylinders with d/Λ varying from 0.1 to 0.8 and ϑ₀=0.025 rad is constant. Vertical lines indicate modal cutoff frequencies of the single cylinder. Figure shown in text corresponds to d/Λ=0.4. [Media 1] [Media 2]

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The vertical lines indicate the cutoff frequencies of various vector modes of the single cylinder; the dashed lines correspond to modes of low azimuthal order (blue=HE_1,p+1, red=TE_0,p/HE_2,p, green=EH_1,p/HE_3,p) and the dotted lines correspond to modes of high azimuthal order (purple=EH_2,p/HE_4,p, orange=EH_3,p/HE_5,p, dark cyan=EH_4,p/HE_6,p, pink=EH_5,p/HE_7,p, gray=EH_6,p/HE_8,p); most of the high azimuthal order resonances appear only in the animations when the cylinders are very close together. We note that there is a very low frequency resonance (V~0.5-1.5) which does not line up with any of the vertical lines and has no analog in the single cylinder scattering spectrum. This resonance is associated with the low index region between the cylinders [34], and so its frequency shifts as the cylinder spacing is reduced. The distribution of the transverse scattered field at the resonance frequency looks like a hybrid TE_0,1/HE_2,1 mode localized in the low index region bounded by the cylinders. Surprisingly, this resonance is quite strong and narrow even when the cylinders are far apart, where one would not expect the low index region between the cylinders to be accurately approximated as a closed object.

As the cylinders move closer together, resonances associated with cylinder modes of increasing azimuthal order (which do correspond to the colored vertical lines) become apparent in the scattering spectrum, consistent with our observations in Fig. 6. Also, the low order modes couple to each other, and so where we once observed isolated resonances associated with a given mode of a cylinder, we now find multiple resonances associated with the supermodes of the three-cylinder object. As d/Λ increases, these supermode resonances split further apart from each other and cover an increasingly wide spectral bandwidth, which correlates with the high loss region between transmission bands in the SC-PBGF. We thus expect that for SC-PBGFs with large d/Λ, the spectral regions of high loss between the transmission bands are broadened. The number of supermodes equals the number of cylinders, so as more cylinders are added, these supermodes form continuous bands as described in Refs. [8, 19].

 

Fig. 8. Im(n _eff) versus V for the first 6 bands of a SC-PBGF with d/Λ=0.4 and different types of defect cores. The discontinuity at the center of the 5^th band in (b) is due to the fact that the numerical precision of the simulation does not allow one to find values of Im(n _eff) below 10^-14. The core designs in (c) and (d) show Fano resonance-like corners at the edges of the odd order bands and improved confinement in the 1^st order band.

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We note that Fig. 7 does exhibit Fano-like resonances for moderate d/Λ, even though no such behavior exists for the analogous structures in Fig. 6. We posit that for the multiple cylinder scattering spectrum, the appearance of Fano resonances depends not only on the cylinder spacing, but also on the number of cylinders - more cylinders means more coupling terms, and the likelihood that phases of the scattering coefficients add up to enhanced backscattering is reduced. That is, we may be able to generate Fano resonances in the spectrum of a SC-PBGF with moderate d/Λ by isolating the cylinders which surround the core from their nearest neighbors. The basic idea is shown in Fig. 8, where we plot Im(n _eff) simulated for SC-PBGFs with different core designs and constant d/Λ=0.4. In Figs. 8(a) (7-defect core) and (b) (19-defect core), the spectra are qualitatively very similar to the results we obtain in Fig. 6(b), though the overall losses are lower. In Figs. 8(c) and (d) (13-defect core), however, where the cylinders closest to the fiber center only have three or two nearest neighbors, respectively, rather than five nearest neighbors, we do observe “horns” in the odd order bands, similar to the Fano resonances described above. This effect is stronger in Fig. 8(d), where the cylinders nearest to the fiber center have only two nearest neighbors, and per our previous arguments, fewer nearest neighbors should increase the likelihood of Fano-like behavior. We also note that for largest core fibers [Figs. 8(b)–(d)], we find discontinuities along the low frequency edge of the 1^st band, which we argue are associated with the resonance of the low index region described in Fig. 7.

The core designs of Fig. 8(c) and (d) actually do not appear to offer much of an advantage in terms of the modal loss in 2^nd–6^th bands. In the 1^st band, however, they do. This is shown more clearly in Fig. 9. The best results (lowest loss over the widest bandwidth) are obtained for the core design of Fig. 8(c) (black line), where the modal loss in the 1^st band is three orders of magnitude lower than what we obtain for the single defect core (magenta line). We also note that this design gives nearly a factor of 10 lower loss than the 19-defect core fiber (blue line), even though the latter has a bigger core, and for the data shown here, the 19-defect core fiber incorporates a 72-cylinder microstructure whereas the Fano-enhanced design of Fig. 8(c) has only 48 cylinders.

 

Fig. 9. Im(n _eff) versus V for the first order band of a SC-PBGF with d/Λ=0.4 and different types of defect cores.

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7. Conclusion

We have numerically modeled the modal loss properties of SC-PBGFs with low d/Λ, which are expected to strongly correlate with the scattering properties of a single cylinder. We unexpectedly found that these fibers achieve lowest loss at the edges of the odd order transmission bands. We have shown that for glancing incidence, the scattering resonances of a single cylinder are in fact Fano resonances, which explains the unusual behavior of the SC-PBGF spectrum. We have further explained the origin of the Fano resonances in terms of the interference between the monopole and dipole components of the transverse scattered field. Finally, we have applied our analysis to design SC-PBGFs with moderate d/Λ having low loss in the lowest frequency transmission band.