Abstract

The photonic bandgaps (PBGs) of honeycomb photonic bandgap fibers (HPBFs) with and without interstitial air holes (IAHs) are numerically investigated. It is shown that the IAHs can increase the width of PBGs in HPBFs, and also that at the same moderate total air filling fraction, HPBFs with IAHs produce more uniform PBGs than those without IAHs. The bandgap behavior is qualitatively explained using the node-and-vein concept.

©2005 Optical Society of America

1. Introduction

In recent years, photonic crystal fibers (PCFs) [1] consisting of a number of periodically arrayed air holes in the cladding have attracted considerable interest due to their novel properties unattainable with conventional optical fibers [2] [3]. In particular, if a lower-index defect is introduced into the otherwise periodic cladding, light guidance can occur owning to the photonic bandgap (PBG) effect created by the periodic air holes, namely, within a frequency range light is forbidden to propagate in the periodic structure [4]. PCFs guiding light by the PBG effect are often referred to as photonic bandgap fibers (PBFs). The PBG effect is a fundamentally different guiding mechanism from modified total internal reflection of the index-guiding PCFs with a solid and higher-index core [1–3].

The first reported PBF was based on a honeycomb lattice [5] [6] and honeycomb photonic bandgap fibers (HPBFs) are still a subject of intensive research [7–10]. This is because HPBFs were found to produce large absolute out-of-plane PBGs [5] and can offer many degrees of freedom in modifying the fiber parameters [7–10]. In the search of large PBGs, it was found that the addition of interstitial air holes (IAHs), which result from the common stack-and-draw process used in PCF fabrication [11] and are schematically shown by the blue circles in Fig.1, can increase the bandgap width of HPBFs. However, the influence of IAHs on the PBGs was just touched upon in [5] and there still lacks a complete investigation of the effect of IAHs.

In this letter, an investigation and comparison of the bandgap behavior of HPBFs with and without IAHs are presented. We show that the bandgap width of HPBFs is enlarged by the IAHs and that at the same moderate air filling fraction (AFF) HPBFs with IAHs can shift the position of the largest PBGs and produce more uniform PBGs in comparison with HPBFs without IAHs. The node-and-vein concept [4] is used to explain the results qualitatively.

2. Results and discussion

2.1 Bandgap of an HPBF without IAHs

Figure 1 shows the schematic of a hexagonal unit cell of the HPBF under study. The red circles stand for the regular air holes of the unit cell, whose diameter is D. The distance of two nearest air holes is Λ. IAHs with a diameter of Dint, which are located midway between three closest air holes in an otherwise triangular lattice formed if an additional air hole of the same diameter as the regular air holes is put at the center of each hexagonal unit cell, are shown as blue circles and also form a honeycomb lattice rotated 300 relative to the original lattice.

 

Fig. 1. Schematic of an HPBF unit cell. Red circles represent the air holes of the HPBF with IAHs shown as blue circles. The diameters of the air holes and IAHs are D and Dint, respectively. The two yellow regions denote nodes, and a vein is shown in magenta.

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We first compute the PBGs for an HPBF without IAHs using the widely employed plane wave expansion method as described in Ref. 12. The AFF is taken to be f= 30% [1,5], corresponding to D/Λ ≈ 0.70 . The bandgap diagram is given in Fig. 2, where the results are expressed in normalized frequency kΛ (k is the free-space wave number) as a function of normalized propagation constant βΛ. As can be seen from the figure, four PBGs are found in the range of interest, but two of them are too narrow and hardly visible. Therefore, in the rest of the paper we concentrate on the two largest bandgaps, the primary and secondary gaps. These two gaps are found to appear between bands 2 and 3, and between bands 6 and 7, respectively.

 

Fig. 2. Bandgap diagram for an HPBF of AFF f = 30% . The first four bandgaps are shown as the blue regions. The red line is the air line, and the grey area is the region where no light propagation is allowed.

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2.2 Bandgap of an HPBF with IAHs

First, we examine how the PBGs of the HPBF in Section 2.1 are influenced by the IAHs. When IAHs with an AFF of f int = 5% are included to raise the total AFF of the fiber to 35%, no new gaps are found except that the relative gaps sizes are increased due to the higher AFF, as shown in Fig. 3, where the relative gaps size is defined as the gap width divided by the central frequency of the gap. This result is in agreement with earlier findings [1] [5].

When the AFF of the IAHs f int is further increased, new gaps start to appear but we concentrate only on the first two gaps described above. The variation of the relative sizes of the two gaps is illustrated in Fig. 4. It is seen that the relative sizes of the two gaps increase monotonically but there is little change in the positions where the two gaps are widest. The same behavior is observed for an HPBF when f = 50% (that is, when D/Λ ≈ 0.91) and the IAHs are increased, as demonstrated in Fig. 5. Hence, we find that the interstitial air holes are favorable for the creation of large PBGs in HPBFs.

 

Fig. 3. Comparison of the relative bandgap sizes between the HPBFs with and without IAHs. The AFF of the HPBF without IAHs is 30%, and the AFF of IAHs is 5%.

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Fig. 4. The relative sizes of the primary and secondary gaps for an HBGF of f = 30% as a function of the AFF of the IAHs.

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Fig. 5. The relative sizes of the primary and secondary gaps for an HBGF of f = 50% as a function of the AFF of the IAHs.

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2.3 Bandgap comparison of HPBFs with and without IAHs at the same AFFs

Then, HPBFs with and without IAHs at the same total AFFs are compared to see how the IAHs affect the bandgaps. The relative sizes of the primary and secondary gaps are shown in Figs. 6 and 7, respectively, when the total AFFs of the HPBFs with and without IAHs are the same. It is observed that the increase of the total AFFs results in larger gaps for both the primary and secondary gaps. Nevertheless, the behavior is obviously different for PBFs with and without PBFs at the same total AFF: PBFs without IAHs possess larger PBFs than those with IAHs, whereas the latter exhibit a longer and larger tail and hence a more uniform PBG. Thought not so pronounced, the same behavior is found for the case of HPBFs with a larger AFF of f = 50% , see Figs. 8 and 9.

 

Fig. 6. The relative size of the primary gap for HPBFs with and without IAHs for fixed total AFFs. The dotted lines show the case of HPBFs without IAHs, while the sold lines represent the case of HPBFs with IAHs.

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Fig. 7. The relative size of the secondary gap for HBGFs with and without IAHs for fixed total AFFs. The dotted lines show the case of HPBFs without IAHs, while the sold lines represent the case of HPBFs with IAHs.

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2.4 Explanation

We attribute the difference in the bandgap behavior between HPBFs with and without IAHs to the structural change induced by the IAHs. It is a well-know rule of thumb that absolute PBGs are favored in a photonic crystal with high-dielectric regions (nodes) connected by narrow veins [4]. The nodes and vein are marked yellow and magenta in Fig. 1, respectively. Using the node-and-vein concept [4], we see that the IAHs have two competing effects on the gap width: On the one hand, larger IAHs reduce the central portions of the unit cells, creating favorably the isolated nodes required for a PBG to appear. On the other hand, the IAHs on the two opposite sides of each vein are detrimental to the appearance of a PBG because they block the connection of the nodes. The former effect is, however, seen to be dominant, leading to a monotonic increase in the two gaps when f int is increased [see Figs. 4 and 5]. When the regular air holes in the HPBFs without IAHs are circumferentially enlarged to make the HPBFs have the same AFF as those with IAHs, they both isolate the nodes and narrow the veins and therefore create larger gaps compared to HPBFs with IAHs because the IAHs on the opposite sides of the veins have the negative effect of reducing the gap width by blocking the veins. This structural difference in the air hole distribution also leads to a shift in the position of the largest gaps, as can be seen in Figs. 6 and 7.

 

Fig. 8. The relative size of the primary gap for HBGFs with and without IAHs for fixed total AFFs. The dotted lines show the case of HPBFs without IAHs, while the sold lines represent the case of HPBFs with IAHs.

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Although HPBFs with IAHs possess a slightly smaller maximum gap than their counterparts without IAHs at the same AFF, they have larger gaps at large normalized propagation constants, demonstrating more uniform PBGs. As a consequence, the existence of the IAHs are beneficial in terms of robust bandgap guidance, since the uniform PBGs can help trap a defect mode at the gap center. When the AFFs are extremely large, the role played by the IAHs becomes minimal, as can be seen from the comparisons in Figs. 8 and 9, where the relative gap sizes are almost equal. This is primarily due to the fact that the AFFs are so large and the veins are so squeezed that the room occupied by the IAHs is too small to have as pronounced effects as when the AFF is moderate, e.g., when f = 30% .

 

Fig. 9. The relative size of the secondary gap for HBGFs with and without IAHs for fixed total AFFs. The dotted lines show the case of HPBFs without IAHs, while the sold lines represent the case of HPBFs with IAHs.

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3. Conclusion

We have numerically examined the bandgaps of honeycomb photonic bandgap fibers with and without interstitial air holes. It is found that the interstitial air holes can favorably increase the bandgap width and also that they can beneficially make the bandgaps more uniform even though at the same moderate air filling fraction the gaps of photonic bandgap fibers with interstitial air holes are slightly smaller than those of fibers without interstitial air holes.

Acknowledgments

This work was supported by the National Key Basic Research Special Foundation (NKBRSF) (Grant Nos. G1999075201 and 2003CB314904), the National Natural Science Foundation of China (Grant No. 60278003), and the National High Technology Development Program of China (Grant No. 2003AA311010).

References and links

1. A. Bjarklev, J. Broeng, and A.S. Bjarklev, Photonic Crystal Fibres (Kluwer Academic Publishers, Boston, 2003). [CrossRef]  

2. P. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003). [CrossRef]   [PubMed]  

3. J.C. Knight, “Photonic crystal fibres,” Nature 424, 847–851 (2003). [CrossRef]   [PubMed]  

4. J. D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).

5. J. Broeng, S.E. Barkou, A. Bjarklev, J.C. Knight, T.A. Birks, and P.St.J. Russell, “Highly increased photonic band gaps in silica/air structures,” Opt. Commun. 156, 240–244 (1998). [CrossRef]  

6. J.C. Knight, J. Broeng, T.A. Birks, and P.St.J. Russell, “Photonic band gap guidance in optical fibers,” Science 282, 1476–1478 (1998). [CrossRef]   [PubMed]  

7. M. Chen and R. Yu, “Analysis of photonic bandgaps in modified honeycomb structures,” IEEE Photonics Technol. Lett. 16, 819–821 (2004). [CrossRef]  

8. L. Zhang and C. Yang, “Photonic crystal fibers with squeezed hexagonal lattice,” Opt. Express 12, 2371–2376 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-11-2371 [CrossRef]   [PubMed]  

9. M. Yan, X. Yu, P. Shum, C. Lu, and Y. Zhu, “Honeycomb photonic bandgap fiber with a modified core design,” IEEE Photonics Technol. Lett. 16, 2051–2053 (2004). [CrossRef]  

10. M. Yan, P. Shum, and J. Hu, “Design of air-guiding honeycomb photonic bandgap fiber,” Opt. Lett. 30, 465–467 (2005). [CrossRef]   [PubMed]  

11. J.C. Knight, T.A. Birks, P.St.J. Russell, and D.M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21, 1547–1549 (1996). [CrossRef]   [PubMed]  

12. S. Guo and S. Albin, “Simple plane wave implementation for photonic crystal calculations,” Opt. Express 11, 167–175 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-2-167 [CrossRef]   [PubMed]  

References

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  1. A. Bjarklev, J. Broeng, and A.S. Bjarklev, Photonic Crystal Fibres (Kluwer Academic Publishers, Boston, 2003).
    [Crossref]
  2. P. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003).
    [Crossref] [PubMed]
  3. J.C. Knight, “Photonic crystal fibres,” Nature 424, 847–851 (2003).
    [Crossref] [PubMed]
  4. J. D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).
  5. J. Broeng, S.E. Barkou, A. Bjarklev, J.C. Knight, T.A. Birks, and P.St.J. Russell, “Highly increased photonic band gaps in silica/air structures,” Opt. Commun. 156, 240–244 (1998).
    [Crossref]
  6. J.C. Knight, J. Broeng, T.A. Birks, and P.St.J. Russell, “Photonic band gap guidance in optical fibers,” Science 282, 1476–1478 (1998).
    [Crossref] [PubMed]
  7. M. Chen and R. Yu, “Analysis of photonic bandgaps in modified honeycomb structures,” IEEE Photonics Technol. Lett. 16, 819–821 (2004).
    [Crossref]
  8. L. Zhang and C. Yang, “Photonic crystal fibers with squeezed hexagonal lattice,” Opt. Express 12, 2371–2376 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-11-2371
    [Crossref] [PubMed]
  9. M. Yan, X. Yu, P. Shum, C. Lu, and Y. Zhu, “Honeycomb photonic bandgap fiber with a modified core design,” IEEE Photonics Technol. Lett. 16, 2051–2053 (2004).
    [Crossref]
  10. M. Yan, P. Shum, and J. Hu, “Design of air-guiding honeycomb photonic bandgap fiber,” Opt. Lett. 30, 465–467 (2005).
    [Crossref] [PubMed]
  11. J.C. Knight, T.A. Birks, P.St.J. Russell, and D.M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21, 1547–1549 (1996).
    [Crossref] [PubMed]
  12. S. Guo and S. Albin, “Simple plane wave implementation for photonic crystal calculations,” Opt. Express 11, 167–175 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-2-167
    [Crossref] [PubMed]

2005 (1)

2004 (3)

M. Chen and R. Yu, “Analysis of photonic bandgaps in modified honeycomb structures,” IEEE Photonics Technol. Lett. 16, 819–821 (2004).
[Crossref]

L. Zhang and C. Yang, “Photonic crystal fibers with squeezed hexagonal lattice,” Opt. Express 12, 2371–2376 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-11-2371
[Crossref] [PubMed]

M. Yan, X. Yu, P. Shum, C. Lu, and Y. Zhu, “Honeycomb photonic bandgap fiber with a modified core design,” IEEE Photonics Technol. Lett. 16, 2051–2053 (2004).
[Crossref]

2003 (3)

1998 (2)

J. Broeng, S.E. Barkou, A. Bjarklev, J.C. Knight, T.A. Birks, and P.St.J. Russell, “Highly increased photonic band gaps in silica/air structures,” Opt. Commun. 156, 240–244 (1998).
[Crossref]

J.C. Knight, J. Broeng, T.A. Birks, and P.St.J. Russell, “Photonic band gap guidance in optical fibers,” Science 282, 1476–1478 (1998).
[Crossref] [PubMed]

1996 (1)

Albin, S.

Atkin, D.M.

Barkou, S.E.

J. Broeng, S.E. Barkou, A. Bjarklev, J.C. Knight, T.A. Birks, and P.St.J. Russell, “Highly increased photonic band gaps in silica/air structures,” Opt. Commun. 156, 240–244 (1998).
[Crossref]

Birks, T.A.

J. Broeng, S.E. Barkou, A. Bjarklev, J.C. Knight, T.A. Birks, and P.St.J. Russell, “Highly increased photonic band gaps in silica/air structures,” Opt. Commun. 156, 240–244 (1998).
[Crossref]

J.C. Knight, J. Broeng, T.A. Birks, and P.St.J. Russell, “Photonic band gap guidance in optical fibers,” Science 282, 1476–1478 (1998).
[Crossref] [PubMed]

J.C. Knight, T.A. Birks, P.St.J. Russell, and D.M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21, 1547–1549 (1996).
[Crossref] [PubMed]

Bjarklev, A.

J. Broeng, S.E. Barkou, A. Bjarklev, J.C. Knight, T.A. Birks, and P.St.J. Russell, “Highly increased photonic band gaps in silica/air structures,” Opt. Commun. 156, 240–244 (1998).
[Crossref]

A. Bjarklev, J. Broeng, and A.S. Bjarklev, Photonic Crystal Fibres (Kluwer Academic Publishers, Boston, 2003).
[Crossref]

Bjarklev, A.S.

A. Bjarklev, J. Broeng, and A.S. Bjarklev, Photonic Crystal Fibres (Kluwer Academic Publishers, Boston, 2003).
[Crossref]

Broeng, J.

J. Broeng, S.E. Barkou, A. Bjarklev, J.C. Knight, T.A. Birks, and P.St.J. Russell, “Highly increased photonic band gaps in silica/air structures,” Opt. Commun. 156, 240–244 (1998).
[Crossref]

J.C. Knight, J. Broeng, T.A. Birks, and P.St.J. Russell, “Photonic band gap guidance in optical fibers,” Science 282, 1476–1478 (1998).
[Crossref] [PubMed]

A. Bjarklev, J. Broeng, and A.S. Bjarklev, Photonic Crystal Fibres (Kluwer Academic Publishers, Boston, 2003).
[Crossref]

Chen, M.

M. Chen and R. Yu, “Analysis of photonic bandgaps in modified honeycomb structures,” IEEE Photonics Technol. Lett. 16, 819–821 (2004).
[Crossref]

Guo, S.

Hu, J.

Joannopoulos, J. D.

J. D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).

Knight, J.C.

J.C. Knight, “Photonic crystal fibres,” Nature 424, 847–851 (2003).
[Crossref] [PubMed]

J.C. Knight, J. Broeng, T.A. Birks, and P.St.J. Russell, “Photonic band gap guidance in optical fibers,” Science 282, 1476–1478 (1998).
[Crossref] [PubMed]

J. Broeng, S.E. Barkou, A. Bjarklev, J.C. Knight, T.A. Birks, and P.St.J. Russell, “Highly increased photonic band gaps in silica/air structures,” Opt. Commun. 156, 240–244 (1998).
[Crossref]

J.C. Knight, T.A. Birks, P.St.J. Russell, and D.M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21, 1547–1549 (1996).
[Crossref] [PubMed]

Lu, C.

M. Yan, X. Yu, P. Shum, C. Lu, and Y. Zhu, “Honeycomb photonic bandgap fiber with a modified core design,” IEEE Photonics Technol. Lett. 16, 2051–2053 (2004).
[Crossref]

Meade, R.D.

J. D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).

Russell, P.

P. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003).
[Crossref] [PubMed]

Russell, P.St.J.

J. Broeng, S.E. Barkou, A. Bjarklev, J.C. Knight, T.A. Birks, and P.St.J. Russell, “Highly increased photonic band gaps in silica/air structures,” Opt. Commun. 156, 240–244 (1998).
[Crossref]

J.C. Knight, J. Broeng, T.A. Birks, and P.St.J. Russell, “Photonic band gap guidance in optical fibers,” Science 282, 1476–1478 (1998).
[Crossref] [PubMed]

J.C. Knight, T.A. Birks, P.St.J. Russell, and D.M. Atkin, “All-silica single-mode optical fiber with photonic crystal cladding,” Opt. Lett. 21, 1547–1549 (1996).
[Crossref] [PubMed]

Shum, P.

M. Yan, P. Shum, and J. Hu, “Design of air-guiding honeycomb photonic bandgap fiber,” Opt. Lett. 30, 465–467 (2005).
[Crossref] [PubMed]

M. Yan, X. Yu, P. Shum, C. Lu, and Y. Zhu, “Honeycomb photonic bandgap fiber with a modified core design,” IEEE Photonics Technol. Lett. 16, 2051–2053 (2004).
[Crossref]

Winn, J.N.

J. D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).

Yan, M.

M. Yan, P. Shum, and J. Hu, “Design of air-guiding honeycomb photonic bandgap fiber,” Opt. Lett. 30, 465–467 (2005).
[Crossref] [PubMed]

M. Yan, X. Yu, P. Shum, C. Lu, and Y. Zhu, “Honeycomb photonic bandgap fiber with a modified core design,” IEEE Photonics Technol. Lett. 16, 2051–2053 (2004).
[Crossref]

Yang, C.

Yu, R.

M. Chen and R. Yu, “Analysis of photonic bandgaps in modified honeycomb structures,” IEEE Photonics Technol. Lett. 16, 819–821 (2004).
[Crossref]

Yu, X.

M. Yan, X. Yu, P. Shum, C. Lu, and Y. Zhu, “Honeycomb photonic bandgap fiber with a modified core design,” IEEE Photonics Technol. Lett. 16, 2051–2053 (2004).
[Crossref]

Zhang, L.

Zhu, Y.

M. Yan, X. Yu, P. Shum, C. Lu, and Y. Zhu, “Honeycomb photonic bandgap fiber with a modified core design,” IEEE Photonics Technol. Lett. 16, 2051–2053 (2004).
[Crossref]

IEEE Photonics Technol. Lett. (2)

M. Chen and R. Yu, “Analysis of photonic bandgaps in modified honeycomb structures,” IEEE Photonics Technol. Lett. 16, 819–821 (2004).
[Crossref]

M. Yan, X. Yu, P. Shum, C. Lu, and Y. Zhu, “Honeycomb photonic bandgap fiber with a modified core design,” IEEE Photonics Technol. Lett. 16, 2051–2053 (2004).
[Crossref]

Nature (1)

J.C. Knight, “Photonic crystal fibres,” Nature 424, 847–851 (2003).
[Crossref] [PubMed]

Opt. Commun. (1)

J. Broeng, S.E. Barkou, A. Bjarklev, J.C. Knight, T.A. Birks, and P.St.J. Russell, “Highly increased photonic band gaps in silica/air structures,” Opt. Commun. 156, 240–244 (1998).
[Crossref]

Opt. Express (2)

Opt. Lett. (2)

Science (2)

J.C. Knight, J. Broeng, T.A. Birks, and P.St.J. Russell, “Photonic band gap guidance in optical fibers,” Science 282, 1476–1478 (1998).
[Crossref] [PubMed]

P. Russell, “Photonic crystal fibers,” Science 299, 358–362 (2003).
[Crossref] [PubMed]

Other (2)

A. Bjarklev, J. Broeng, and A.S. Bjarklev, Photonic Crystal Fibres (Kluwer Academic Publishers, Boston, 2003).
[Crossref]

J. D. Joannopoulos, R.D. Meade, and J.N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).

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Figures (9)

Fig. 1.
Fig. 1. Schematic of an HPBF unit cell. Red circles represent the air holes of the HPBF with IAHs shown as blue circles. The diameters of the air holes and IAHs are D and Dint, respectively. The two yellow regions denote nodes, and a vein is shown in magenta.
Fig. 2.
Fig. 2. Bandgap diagram for an HPBF of AFF f = 30% . The first four bandgaps are shown as the blue regions. The red line is the air line, and the grey area is the region where no light propagation is allowed.
Fig. 3.
Fig. 3. Comparison of the relative bandgap sizes between the HPBFs with and without IAHs. The AFF of the HPBF without IAHs is 30%, and the AFF of IAHs is 5%.
Fig. 4.
Fig. 4. The relative sizes of the primary and secondary gaps for an HBGF of f = 30% as a function of the AFF of the IAHs.
Fig. 5.
Fig. 5. The relative sizes of the primary and secondary gaps for an HBGF of f = 50% as a function of the AFF of the IAHs.
Fig. 6.
Fig. 6. The relative size of the primary gap for HPBFs with and without IAHs for fixed total AFFs. The dotted lines show the case of HPBFs without IAHs, while the sold lines represent the case of HPBFs with IAHs.
Fig. 7.
Fig. 7. The relative size of the secondary gap for HBGFs with and without IAHs for fixed total AFFs. The dotted lines show the case of HPBFs without IAHs, while the sold lines represent the case of HPBFs with IAHs.
Fig. 8.
Fig. 8. The relative size of the primary gap for HBGFs with and without IAHs for fixed total AFFs. The dotted lines show the case of HPBFs without IAHs, while the sold lines represent the case of HPBFs with IAHs.
Fig. 9.
Fig. 9. The relative size of the secondary gap for HBGFs with and without IAHs for fixed total AFFs. The dotted lines show the case of HPBFs without IAHs, while the sold lines represent the case of HPBFs with IAHs.

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