1. Introduction

Hollow core photonic bandgap fibers (HC-PBGFs) open up unprecedented application opportunities in various areas that include data transmission with low latency and ultralow nonlinearity, mid-IR gas spectroscopy and high power laser delivery [1]. Exploiting these opportunities requires improvements in the optical properties of these fibers. Most notably, achieving lower losses and wider operational bandwidths is of paramount importance. To this end, it is critical to be able to understand the effect that the exact geometry profile of the fibers’ cross section has on their optical properties. In performing theoretical studies on HC-PBGFs, their cross-sections have often been approximated by idealized models which regard them as two-dimensionally periodic distributions of hexagonal shaped air holes embedded in a silica matrix. While these idealized models are useful in providing physical insight into the operation of these fibers, the cross-sections of fabricated fibers often differ considerably from these models and feature visible structural distortions. These distortions and small details in the geometry can have far-reaching implications in terms of fiber performance [2, 3]. Our recent studies on the impact of typical structural distortions present in fabricated HC-PBGFs have revealed for example that most distortions prompt a stronger overlap of the guided mode field with air-glass interfaces and result in stronger scattering from surface roughness and thus higher loss [4].

To further understand the role of the exact fiber cross-section and close the HC-PBGF design loop, it is important to perform numerical modeling and a theoretical assessment of the properties of fabricated samples. This verification step is critical in providing an explanation for the experimentally measurable fiber properties and in identifying routes to address potential design issues. If performed with high accuracy, it can also be a reliable means to obtain good estimates of fiber properties which are more challenging to measure experimentally.

Traditionally, the modelling of fabricated HC-PBGFs has been performed by reconstructing the refractive index distribution of the fiber from scanning electron microscope (SEM) images of their cross-sections and using this as input to a chosen mode solver [2, 3]. However, obtaining reliable and faithful results from this seemingly simple procedure has proved more challenging than one may expect. The standard procedure consists of converting the grayscale SEM image into a two-level (black and white) one and using edge detection routines to turn the boundaries of the air holes into smooth curves, usually splines. The challenge arises because the limited resolution of the SEM images prevents one from resolving simultaneously the entire microstructured cross-section (∼ 100μm) and at the same time the smallest scale features such as the thin silica struts in the cladding (typically ≤ 200nm). Furthermore, the metallic coating required to prevent charging effects and the hard to control tilt of the samples during SEM acquisition contribute to exaggerating the value of the strut thickness, resulting in the images always showing a lower air-filling fraction (and therefore a photonic bandgap centered at longer wavelengths) than in reality [2, 3]. Figure 1 shows an example of simulation results obtained using this approach. Edge detection was performed on the high quality SEM image of Fig. 1 and the resulting profile used as input to a finite element solver. It can be seen from the comparison between simulated and measured transmission that in addition to the simulated photonic bandgap being centered at longer wavelengths, the effects described above lead to the simulation predicting several surface modes (seen as the dips in the transmission) that are not observed experimentally, despite the fact that the simulated geometry seemingly replicates what is seen in the SEM image. It is this discrepancy that has stalled progress in simulating fabricated fibers and has caused researchers to rely on simulating ’idealized’ structures for qualitative comparison with experimental results.

 

Fig. 1 Example simulation performed on a high quality SEM image using edge detection. (a) shows the SEM image overlapped with the reconstructed geometry in red. (b) shows the measured short length transmission (over 10m) of the fiber in blue and the simulated fraction of power in the core in red. The many dips are due to the ‘artificial’ surface modes arising from a thicker core surround than in reality.

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A more recent approach to circumvent this problem has consisted of increasing the magnification of the SEM images, limiting the field of view to the core and to the first two rings of air holes surrounding it and thereby increasing the width of single glass struts to a few pixels. Although selected fibre properties were accurately matched with simulations, the technique is far too cumbersome as it requires manually measuring all the relevant structural features in the cross section and using them as input parameters to manually reconstruct the fiber geometry [5]. It further assumes that the fibre geometry preserves a six-fold symmetry which is broken in most, if not all fabricated fibers.

Here we propose a novel, fast and efficient method to faithfully reproduce HC-PBGF structures from SEM images. The problems posed by the image resolution and the metallic coating are circumvented by using fiber information such as average cladding strut thickness and glass volume in each cladding unit cell, which we obtain by conserving the glass from the fiber preform. Simulations of several fabricated HC-PBGFs show a good agreement with measured fiber loss and surface modes position within the photonic bandgap. The speed and hands-off approach of our method are very useful features that can help support the improvement of these fibers by providing timely and accurate feedback to fiber design.

2. Methods

2.1. Reconstructing the geometry from SEM images

Our aim is to extract from an SEM image a geometry profile that accurately matches the distortions of the fabricated HC-PBGF under study, while having less severe demands on the image quality or resolution. This is achieved by realizing that in high quality SEM images (e.g. free of charging effects, debris or tilt), the center positions of the air holes and those of the glass nodes on the images are accurate representations of their counterparts in the actual fiber, regardless of the image resolution and of the effects of the coating. With knowledge of these center positions, an accurate representation of the fiber can be generated provided that the amount of glass at each node and strut thicknesses are known. In practice, we implement this procedure in two steps as illustrated in Fig. 2.

 

Fig. 2 Geometry reconstruction from SEM image: (a) High quality starting SEM image. (b) After conversion to black and white, the holes center positions are detected. A dilation is applied to air holes to leave isolated glass nodes. Knowledge of strut thickness and node area allows to reconstruct the original air hole as detailed in the appendix. (c) Examples showing the overlap between the reconstructed structure in red and the original image.

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Firstly, we filter and convert the grayscale image into a binary black and white one. At this stage, each air hole can be identified and its center position accurately located by using simple image processing routines (e.g. we use the MATLAB image processing toolbox). We then apply a dilation to each air hole in such a way that all the air holes are merged together. This leaves the interstitial nodes as isolated objects, whose center positions can again be accurately located. Secondly, with knowledge of the position of air holes and interstitial nodes, a hexagon (or pentagon) is built around each air hole by finding the six (or five) closest nodes to its center. Each edge of the hexagon (pentagon) is then moved closer to the holes center by a distance corresponding to half the strut thickness t which must be determined first. The next step is to fillet each corner with an arc of radius r which is chosen so that the volume of glass in each unit cell is the same and conserved from the quantity in the preform from which the fiber is drawn. The calculations involved are as detailed in the appendix. A good estimate of the average strut thickness can be obtained from averaging measurements on highly magnified portions of the cross section, adjusted by the estimated coating thickness. However, we have relied on a mass-conservation model predicting the evolution of second-stage preforms into fibers to predict both the average thickness and the unit cell area in the fiber [6]. The freedom to create struts of desired thicknesses allows one to generate geometries that better match those of the fibers, avoiding the limitation posed by the finite SEM resolution. Although this simple approach reproduces the shape of the air holes in the cladding with high fidelity (see third image in Fig. 2(c)), it does not exactly replicate all the air holes boundaries in the SEM image (more obvious in the second image in Fig. 2(c)), which as argued above are affected by the metallic coating. However, with the nodes positions accurately detected and the correct thicknesses and glass volume in each unit cell, the regenerated profile reproduces the experimentally measured optical properties of the fiber, as shown in the several examples below.

To improve the match between reconstructed and actual fiber profile, we further impose that each silica strut’s thickness inversely proportional to its length. In other words, struts longer than average are made thinner while shorter struts are made thicker in proportion, such that each individual strut contains the same amount of glass. Particular attention needs to be paid to the struts on the core boundary. In the traditional stack and draw method, identical capillaries are stacked in the triangular arrangement to make the preform. In a first stage draw, this preform is drawn into second-stage preforms (canes) in which segments on the core boundary can vary significantly in length and sometimes thickness, and thus contain a different mass of glass than the average ones in the cladding. As it is safe to assume that no transverse material flow occurs during the second stage draw, these differences in glass volumes are transferred to the fibers, so that the amount of glass in the struts on the core boundary is in proportion to that in the cane. Estimating their thickness from these considerations was found to be particularly crucial because these struts directly determine if and where surface modes are supported within the bandgap. Fibers incorporating special features on the core boundary such as typically used in birefringent fibers [7] can also be simulated using our method, since these features can be automatically detected and incorporated into the reconstructed profile.

Note that because there are nodes beyond the outermost ring of air holes, this final ring cannot reproduced in the final geometry. Its omission results, in principle, in an overestimate of the confinement loss. However, the confinement loss in fibers with several rings of air holes is very low and the rule of thumb whereby it reduces by approximately an order of magnitude for each additional ring of air holes may be used to extrapolate the simulated fibers confinement loss [8]. For the fibers simulated and presented here which typically incorporated at least six rings of air holes, we have found such extrapolation unnecessary for the well-confined fundamental LP₀₁-like mode, as the simulated confinement loss with one less ring remains negligible at wavelengths within the photonic bandgap. Moreover, since the guided-mode field quickly decays away from the core, scattering from surface roughness and material absorption predominantly occur in the first two rings of air holes near the core defect [4]. Omitting the final ring of air holes therefore has no impact on the most important and useful fiber properties.

2.2. Modeling the loss

At telecommunication wavelengths where silica is highly transparent, loss in HC-PBGFs is known to be dominated by scattering from surface roughness [9, 10]. Beyond 2μm, the glass absorption becomes the dominant loss mechanism. In [10], we have formulated a treatment of roughness scattering in HC-PBGFs and derived expressions for estimating the scattering loss. Such calculations however, require the full knowledge of the roughness power spectral density (PSD) at all spatial frequencies. Since this is a statistical property that can be measured only over a restricted spatial frequency range, some ’free parameter’ must be unavoidably introduced. Reported roughness measurements in HC-PBGFs and similar fibers attribute the origin of roughness to frozen-in surface capillary waves (SCWs), the power spectral density of which is given as [9, 11, 12]:

With this in mind, we reconstructed the geometry profile of the fiber of Fig. 1(a) using the proposed method in section 2.1,and then employed the theory detailed in [10] to estimate the fibre loss. Figure 3 shows comparison between simulated and measured fibre loss. The modal properties were simulated using the commercial finite element solver COMSOL Multi-physics. By using T_g/γ = 1500K/J · m⁻² and κ_c = 1/100μm [9, 12], we obtained a loss value of 2.5dB/km at 1.5μm, which is in fairly good agreement with the 3.5dB/km measured via cutback, considering the uncertainties on the roughness spectrum discussed above.

 

Fig. 3 Modelling the loss contributions in fabricated HC-PBGFs. The blue curve is the cutback measurement for the fiber shown in Fig. 1(a). The black one is the total loss with the scattering contribution proportional to the interface field intensity. The red curve has the scattering contribution calculated with the theory of ref. [10]. The dotted green line shows the confinement loss contribution which remains very low at wavelengths within the bandgap.

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Simulating fiber loss using this theory is however very time-consuming, as the far-field scattered power needs to be evaluated on closely spaced points on a distant sphere. With various optimization and exploiting the scattered field symmetry, it took us an average of 70 minutes to calculate the loss for a single wavelength on a computer with 4-core processor and 24GB of memory.

To speed up the simulation process while still giving useful insight and accurate results, we propose an alternative simpler approach which overlooks the exact form of the roughness spectral density and adopts a simple uncorrelated scatterers assumption. This is equivalent to assuming that loss is proportional to the field intensity at the air-glass interfaces:

However, the assumption of uncorrelated scatterers results in incorrect loss scaling laws unless an appropriate wavelength dependence is incorporated into the scaling factor η. It is known from reported experiments [9, 15] and from our theoretical treatment that the frozen-in surface capillary waves-induced roughness in HC-PBGFs causes the scattering loss to decrease as $1/{\lambda }_c^3$ when HC-PBGFs are rigidly scaled for operation at different wavelengths. Under such rigid scaling, the central operating wavelength is proportional to the fiber dimensions, provided the material refractive index does not change significantly. Just as in hollow dielectric waveguides, the field near the core interface in HC-PBGFs decrease approximately as λ_c/R (with R being the core radius and λ_c the operating cental wavelength) [16]. It can be shown therefore from Eq. (4) that F changes as ${\lambda }_c^2/R^3$ ( ${\lambda }_c^2/R$from the numerator, and R² from the denominator). In other words, under rigid scaling, F scales as 1/λ_c. If we assume that the roughness present on the surfaces is not affected by the dimensions of the fiber, then η must be proportional to $1/{\lambda }_c^2$ for the loss of Eq. (3) to decrease as $1/{\lambda }_c^3$ when the fiber is rigidly scaled. It follows therefore that the scattering loss may be estimated for a fiber operating at λ_c as :

3. Modeling results

We have routinely used the procedure detailed above to assess the properties of various fabricated fiber samples. The primary interest of our work was to compare simulations with the measurable fiber properties, in particular attenuation and bandwidth. Every simulation also gave accurate estimates of fiber properties which are not routinely measured or are more challenging to evaluate experimentally such as dispersion, differential modal loss, mode field diameters, effective mode areas, etc. As discussed above, loss contributions from roughness scattering and leakage are taken into account.

3.1. Low-loss 19c HC-PBGF operating at 1.55μm

The first example we present here is the fiber which we have already briefly discussed above. Its cross-section is shown again in Fig. 4(a) with an overlap of the reconstructed geometry profile. In order to achieve broad operational bandwidth, we chose not to use a core tube when preparing the preform from which the fiber was drawn. Measurements of the strut thickness from SEM images indicated an average of 130 ± 20nm. But this is inflated by the metallic coating as explained above, so in the modelling we used instead the value of 110nm obtained from mass conservation. The unit cell area in the fiber, also predicted from the preform via mass conservation was 1.09μm².

 

Fig. 4 Transmission and loss measurement compared with simulation for a low-loss fiber made with no core tube (a) shows the overlap between the original SEM image and the reconstructed geometry. (b) is the fundamental mode profile at the lowest loss wavelength of 1.5μm. (c) Short length transmission measurement (measured over 10m) and simulated power in the core (d) Cutback loss measurement (from 350 to 10m) and simulated fundamental mode loss. Loss is computed as the sum of contributions from scattering and leakage. Here $\mathit{Loss}=\frac{1}{2}\left(\mathit{loss}({\mathit{LP}}_{01x})+\mathit{loss}({\mathit{LP}}_{01y})\right)$.

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Figure 4(b) shows an example of the simulated fundamental mode profile at a low loss wavelength within the bandgap. Figure 4(c) shows plots of measured short length (10m) transmission and simulated fraction of guided power in the core. The agreement between bandgap positions and widths is remarkable and is a good testimony of the accuracy of the method used to reproduce the fiber geometry. Shown in Fig. 4(d) is a comparison between measured and simulated fiber attenuation already discussed above. The simulated loss was taken as an average of both polarizations of the fundamental mode and the dominating scattering contribution was estimated from the simple scaling of the normalized interface field intensity as in Eq. (5). As can be appreciated, the simple method employed to estimate the loss yields a good agreement with the measured loss values.

In addition to loss and bandwidth for the fundamental mode, the accurate reproduction of the fiber’s cross-section allows us to study the dispersive properties of each of its guided modes, their differential loss, and in general to understand which structural features are responsible for the guidance of surface modes or cause an excess loss in the fibers. Figure 5 shows the effective index vs. wavelength trajectories for the first five mode groups across the photonic bandgap. We see at once that the reduction in transmission at both edges of the photonic bandgap in Fig. 4(c) is due to the presence of surface modes. The first group of surface modes near the short wavelength edge have their power located in the struts of the first ring of air holes which appear to be shorter and thicker than average. The second group, near the long wavelength edge, have their power concentrated in glass nodes in the vicinity of the core which are smaller in size than those in the cladding. Because these two groups of surface modes have been pushed towards the edges of the bandgap, the fiber still has a wide operational bandwidth.

 

Fig. 5 Effective index map and modal power distribution for the first 5 mode groups of the fiber shown in Fig. 4(a). The surface modes close to the bandgap edge are responsible for the drop in transmission. The number underneath each mode profile indicates the mode’s minimum total attenuation in dB/km acroos the photonic bandgap.

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The visible asymmetries of the fiber’s core cause some slight phase birefringence of the order of ∼ 5 × 10⁻⁶ between the two polarizations of the LP₀₁-like mode, and also introduce a differential loss of 0.2dB/km between them. These same distortions also cause higher order LP_mn-like modes with no circular symmetry (m > 0) to split into two groups of two nearly degenerate modes, LP_mna and LP_mnb, with increasing difference in effective indices and loss. Figure 6(a) compares simulated and measured differential modal group delay (DMGD) with respect to the fundamental mode. The experimental data was collected using a time-of-flight setup, by offset-launching linearly polarized light into the PBGF [17]. This predominantly excites one lobe of the higher order modes, and as a result, only the two polarizations of the LP_mna or LP_mnb could be excited and detected under the same launching conditions. As can be observed, DMGD in HC-PBGFs is several orders of magnitude higher than the values reported for the best low DMGD few-moded solid fibers, such as the ∼ −0.08ps/m reported by Grüner-Nielsen in [18]. Our simulated DMGD values agree well in general with experimental data, except for the LP₀₂-like mode for which the discrepancy is the highest. We believe that this is due to the extreme sensitivity of this mode to the finest details of the silica core surround.

 

Fig. 6 (a) Differential modal group delay across the C-band for the first 5 mode groups of the fiber shown in Fig. 4(a). The markers are measured data obtained from time-of flight experiments and were obtained only for one subgroup of each mode group. The solid lines represent the simulated values. (b) Simulated modal loss for each mode within the first five mode groups.

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Figure 6(b) shows a plot of the loss computed for each of the modes in the first five mode groups. Although differential modal loss is an important consideration, especially in fibers to be employed for mode multiplexed data transmission, assessing it experimentally is a challenging task that requires exciting and detecting individual modes. The ability to model each mode’s loss accurately therefore becomes paramount. As the mode order increases, the mode-field expands and overlaps more prominently with the rough surfaces, causing an increasingly higher scattering loss. It can be observed that the differential modal loss both within a mode group and between the mode groups is significant. The minimum loss for the LP₁₁, LP₂₁, LP₀₂ and LP₃₁ mode groups are on average 2.8, 6.3, 10, and 14.5 times higher than that of the fundamental mode respectively. This is another important consideration if HC-PBGFs are to be used in long-haul multimode data transmission because it requires designing amplifiers that can provide higher gains to higher order modes or suitable interspan mode scramblers.

3.2. HC-PBGF with surface modes, operating at 1.55μm

Some HC-PBGF preforms often deliberately include a central core tube. Despite the known disadvantages of introducing surface modes in the final fibers [19], the use of a core tube increases the yield from a given stack. We have shown previously that when all the core boundary struts are of equal length and thickness, the inclusion of a core tube with a thickness matched to that of the smaller capillaries in the preform should result in fibers with little loss or bandwidth penalty when compared to fibers without any core tube [6]. Figure 7(a) shows the cross-section of a fiber drawn from a preform assembled with such a core tube. This fiber has an average pitch and strut thickness of 5.5±0.4μm and 60±10nm. Despite its high air-filling-fraction, the fiber suffers from relatively high loss and supports three groups of surface modes which essentially fragment its transmission window. Figure 7(b) highlights the special attention one must pay to the air holes on the first ring near the core. These are made from identical capillaries as the rest of the cladding, but the presence of the large core defect results in a higher fraction of glass ending up in the core boundary struts.

 

Fig. 7 Comparison between simulations and experiments for a fiber produced from a pre-form assembled with a core tube. (a) Scanning electron micrograph of the fiber cross-section, (b) shows how struts in the cane relate to those in the fiber and explains why some of the core struts must be thicker. (c) Comparison between short length transmission (10m) and simulated power in the core and (d) between simulated and measured loss via cutback.

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In the example shown, the core boundary edge of the twelve pentagonal holes have the same thickness in the cane as the cladding struts but are approximately 1.7 times longer. These same struts end up in the fiber being about 1.5 times shorter than the cladding struts. A simple mass conservation argument implies that these struts are approximately 2.5 times thicker than the average cladding strut. The four remaining struts forming the pentagonal hole contain similar glass volume as those in the cladding. These simple considerations led to the very good agreement between simulations and measurements, as can be appreciated in the transmission and loss curves of Figs. 7(c) and 7(d).

The measured minimum loss for this fiber was 6.6dB/km, while the simulated value which we estimated using Eq. (5) was 5.6dB/km. We see once again that both the scattering dominated loss and the position of the surface modes are captured fairly well by the simulations. This offers some indication that the surface modes and loss peaks can in principle be eliminated, for example by addressing the distortion that results in these struts being shorter and by reducing the volume of glass they contain.

3.3. Fibers operating at other wavelengths

Due to the combination of scattering from roughness and the infrared absorption of silica, the lowest achievable loss in HC-PBGFs is predicted to occur around 2μm. This is well beyond the typical wavelength of 1.6μm after which silica starts to absorb significantly, and it arises because of the small overlap between the guided mode field and the absorbing glass. Consequently, HC-PBGFs offer exciting new possibilities in the mid-IR, despite silica having very high absorption in that region. We have fabricated several fibers operating both at 2μm and further in the mid-IR near 3.3μm, and used the technique described above to analyse their properties. These fibers were used in experiments demonstrating the feasibility of amplified WDM data transmission near 2μm [20] and high sensitivity gas sensing in the mid-IR [21, 22].

Figure 8(a) shows an SEM image of the 19c 2μm fiber, along with a comparison between measured and simulated transmission and loss in Figs. 8(b) and 8(c) respectively. Here again, our reconstruction method reproduces the position of the surface modes within the photonic bandgap. Two groups of surface modes are supported as a result of the uneven strut thickness around the core defect as discussed above, with one group close to the short wavelength edge of the photonic bandgap. The narrow low-loss region close to this edge and visible in the simulated loss curve is not observed in experiments as the fiber is cutback from 1.1km to 10m. We believe this may be due to small structural changes along the fiber length, particularly changes in strut thickness which would affect both the bandgap edge and the surface mode close to it. Using the interface field intensity to estimate the loss as described in Eq. (5), the simulated minimum loss value of 2.3dB/km is in good agreement with the measured minimum loss of 2.5dB/km. This suggests that our fabrication process results in fibers that possess similar roughness, and hence, the differences in loss between any two fibers can be explained simply by the differences between their cross-sections and how distortions present therein cause the guided mode to overlap with the scattering surfaces.

 

Fig. 8 Fibers operating near 2 and 3μm respectively, with and without core tube. (a) cross section of the fiber guiding at 2μm which was made with a core tube. (b) Corresponding measured transmission over 10m and simulated fraction of power in the core and (c) measured loss by cutback from 1.1km to 10m (blue curve) and simulated total loss (red). (d) Cross-section of the fiber guiding around 3.3μm, (e) corresponding transmission over 5m length (blue) and simulated power in the core (red). (d) Cutback loss measurement from 58 to 5m before purging the fiber of absorbing gas species (blue), measured loss after gas purging (green) and simulated loss in the absence (magenta) and presence (red) of material absorption.

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To model the fiber guiding in the mid-IR and whose cross-section is shown in Fig. 8(d), we extracted bulk silica glass absorption data from Humbach et al. [23] and directly included it in the FEM simulations as an imaginary part of the silica refractive index. Figures 8(e) and 8(f) show the simulated and measured transmission and loss respectively. The preform from which this fiber was fabricated did not feature a core tube, but mass conservation considerations as explained above results in the core edges of the pentagonal holes being thicker than half the cladding strut thickness. As can be appreciated again, the simulation correctly predicts the four distinct surface modes close to the short wavelength edge of the photonic bandgap, in good agreement with the experiment. Ignoring the difference in thicknesses would fail to accurately predict the surface modes. This shows how sensitive the properties of HC-PBGFs are to the smallest structural changes, especially in the vicinity of the core defect. The scattering loss estimated from the normalized interface field intensity only amounts to 0.2dB/km and is therefore negligible as the loss is dominated by absorption. Despite the bulk silica absorption being as high as 600dB/m at this operating wavelength, the fiber had a minimum loss of 0.05dB/m after it was purged of absorbing gas species [21]. This compares very well with the absorption dominated value of 0.06dB/m obtained from simulations.

4. Conclusion

We have presented a new approach to simulate the properties of fabricated HC-PBGFs from their SEM images. The new method retains the geometrical irregularities of the cross-section of fabricated fibers, whilst solving known problems of previous methods (i.e. limited SEM image resolution and unknown metallic coating thickness), through the use of fibre preform data and mass conservation. We have shown that this tool can achieve remarkable accuracy in predicting the experimental loss of a range of different fibers operating in both a scattering loss or in a glass absorption loss dominated regime. A simple straightforward way to estimate numerically the scattering loss of HC-PBGFs has also been proposed and validated against more rigorous but also more time consuming methods. Besides the loss, the proposed method has been shown to allow, for the very first time, an accurate reproduction of both the spectral and the spatial position of the surface modes supported by any fabricated fibre. The former is key to an accurate estimate of various wavelength dependent modal parameters such as group delay, differential loss and group velocity dispersion that can now be calculated effortlessly for all the air-guided modes in the fiber. The latter can enable the identification of the geometrical deformations that are responsible for surface modes and therefore detrimental to the useable overall bandwidth in the fiber. We believe that the proposed method will represent a powerful tool in the development of wider bandwidth, lower loss HC-PGFs.

Appendix

Here we give details of how the air holes boundaries can be reconstructed after nodes and holes center positions have been detected. The objective is to assign desired thicknesses to the silica struts and to find the fillet radii which result in a target amount of glass mass in each unit cell of the fibre. Figure 9 shows a portion of the cladding of a HC-PBGF and highlights three neighbouring nodes A, B and C which determine part of the boundary of the air hole. The procedure to construct the air hole boundary B’NJQC’ where the arc NJQ defines the fillet is as follows.

 

Fig. 9 Reconstruction of the air-hole boundary with information on node positions, strut thicknesses and node areas.

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First, we find the position of the point A′ such that the distance between the parallel segments AC and A′C′ and between AB and A′B′ are the desired half thicknesses t₁/2 and t₂/2 respectively. If u⃗ and v⃗ are the unit vectors along AC and AB respectively, then the coordinates of A′ can be found by solving:

(6)$$\begin{array}{lll}\overrightarrow{A{A}^{\prime }}\cdot \overrightarrow{u}\hfill & =\hfill & \mathit{AH}\hfill \\ \overrightarrow{A{A}^{\prime }}\cdot \overrightarrow{v}\hfill & =\hfill & \mathit{AI}\hfill \end{array}$$

Since AKA′L is a parallelogram, the triangles HA′L and KA′I are similar. If the angle $\widehat{\mathit{BAC}}=\alpha$, then $\widehat{IK{A}^{\prime }}=\widehat{HL{A}^{\prime }}=\pi -\alpha$ and $\widehat{K{A}^{\prime }L}=\widehat{H{A}^{\prime }L}=\alpha -\pi /2$. It follows that,

(7)$$\begin{array}{lll}\hfill A^{\prime }K& =\hfill & \mathit{AL}\frac{t_2}{2\text{cos}(\alpha -\pi /2)}\hfill \\ \hfill A^{\prime }L& =\hfill & \mathit{KA}=\frac{t_1}{2\text{cos}(\alpha -\pi /2)}\hfill \\ \hfill \mathit{IK}& =\hfill & \frac{t_2}{2}\text{tan}(\alpha -\pi /2)\hfill \\ \hfill \mathit{HL}& =\hfill & \frac{t_1}{2}\text{tan}(\alpha -\pi /2)\hfill \end{array}$$

Since AH = AL − HL = A′K − HL and AI = AK − IK = A′L − IK, Eq. (6) becomes:

(8)$$\begin{array}{lll}\overrightarrow{A{A}^{\prime }}\cdot \overrightarrow{u}\hfill & =\hfill & \frac{t_2}{2\text{cos}(\alpha -\pi /2)}-\frac{t_1}{2}\text{tan}(\alpha -\pi /2)\hfill \\ \overrightarrow{A{A}^{\prime }}\cdot \overrightarrow{v}\hfill & =\hfill & \frac{t_1}{2\text{cos}(\alpha -\pi /2)}-\frac{t_2}{2}\text{tan}(\alpha -\pi /2)\hfill \end{array}$$

Having found the coordinates of the point A′, we search for the fillet radius r to apply to the corner B′A′C′ so that the total area of glass enclosed in the unit cell equals the specified value obtained from mass conservation. If A_u is the unit cell area, then the sum of the six shaded areas (see Fig. 9) is

(9)$$A_t=A_u-\left[\text{area}\hspace{0.17em}\text{of}\hspace{0.17em}\text{hexagon}(ABC\dots )-\text{area}\hspace{0.17em}\text{of}\hspace{0.17em}\text{hexagon}(A^{\prime }{B}^{\prime }{C}^{\prime }\dots )\right]$$

We have assumed for simplicity that A_t is equally distributed among the corners of the hexagon, that is, the shaded area shown in Fig. 9 is 1/6A_t. Since the triangles A′QO and A′NO are identical, we deduce that A′N = A′Q = r tan((π − α)/2). The shaded area is therefore:

(10)$$\frac{1}{6}{A}_t=r^2\text{tan}\left((\pi -\alpha )/2\right)-\frac{1}{2}\frac{\pi -\alpha }{2}{r}^2.$$

Solving Eq. (10) for r yields the correct fillet radius to apply and thereby completes the solution to the problem.

Acknowledgments

The authors wish to acknowledge Brian Mangan for providing some of the core tubes used in fiber fabrication. This work was supported by the EU 7th Framework Programme under grant agreement 258033(MODE-GAP), and by the UK EPSRC through grants EP/I01196X/1 (HYPERHIGHWAY) and EP/H02607X/1(IMRC). The data for the plots in this paper can be found at http://dx.doi.org/10.5258/SOTON/377919.

References and links

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