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Optics Express

  • Editor: Michael Duncan
  • Vol. 12, Iss. 1 — Jan. 12, 2004
  • pp: 104–116
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Antiguiding in microstructured optical fibers

M. Yan and P. Shum  »View Author Affiliations


Optics Express, Vol. 12, Issue 1, pp. 104-116 (2004)
http://dx.doi.org/10.1364/OPEX.12.000104


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Abstract

Antiguiding, as opposed to positive index-contrast guiding (or index-guiding), in microstructured air-silica optical fibers is shown to have a significant influence on the fiber’s transmission property, especially when perturbations exist near the defect core. Antiguided modes are numerically analyzed in such fibers by treating the finite periodic air-silica composite (including the central defect) as the core and outer bulk silica region as the cladding. Higher-order modes, which can couple energy from the fundamental mode in the presence of waveguide irregularities, are predicted to be responsible for high leakage loss of realistic holey fibers. The modal property of an equivalent simple step-index antiguide model is also analyzed. Results show that approximation from a composite core waveguide to a simple step-index fiber always neglects some important modal characteristics.

© 2004 Optical Society of America

1. Introduction

Microstructured optical fibers (MOFs) made of air and pure silica [1

1. 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]

] have attracted a significant amount of attention in recent years. Two mechanisms are identified as being responsible for waveguiding in such fibers: index-guiding [2

2. T. A. Birks, J. C. Knight, and P. St. J. Russel, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997). [CrossRef] [PubMed]

] and photonic bandgap (PBG) guiding [3

3. J. Broeng, S. E. Barkou, T. Søndergaard, and A. Bjarklev, “Analysis of air-guiding photonic bandgap fibers,” Opt. Lett. 25, 96–98 (2000). [CrossRef]

]. Index-guiding utilizes the positive index-contrast between the solid silica core and periodic silica-air cladding, whereas PBG-guiding utilizes the photonic forbidden bandgap to localize light in the core, or defect region. To date, most research has focused on index-guiding MOFs, owing to ease of fabrication and to the many revolutionary properties that they can provide. These properties include endlessly single-mode operation, tailored dispersion, very large or ultrasmall effective core area, and so on. [4

4. T. M. Monro and D. J. Richardson, “Holey optical fibres: Fundamental properties and device applications,” C. R. Physique 4, 175–186 (2003). [CrossRef]

]. PBG air-silica fibers, although they have many promising applications, have narrow working bandwidth, difficulty in fabrication, and other shortcomings. New materials need to be identified in order to make more practical PBG fibers [5

5. B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650–653 (2002). [CrossRef] [PubMed]

]. In this paper, we will put our emphasis on the index-guiding fibers.

Veff=2πaλnsilica2nSFM2.

a is effective core radius. We used a=0.525Λ. nSFM is index of the fundamental space filling mode (SFM) in a uniform 2D air-silica photonic crystal characterized by value d and Λ (hereafter, this 2D photonic crystal is referred as the PC) when wave is propagating along the air hole axis. This chosen a value will give rise to single-mode cut-off Veff valued at 2.18 [11

11. N. A. Mortensen, J. R. Folkenberg, M. D. Nielsen, and K. P. Hansen, “Modal cutoff and the V parameter in photonic crystal fibers,” Opt. Lett. 28, 1879–1881 (2003). [CrossRef] [PubMed]

]. The same figure also shows Veff curves for other fibers characterized by different d/Λ values. We may conclude from the plot that the fiber in Fig. 1(a) is endlessly single-mode.

† These two measured values, which are later used in our numerical study, have 4-decimal-point accuracy. It should be noted that current MOF fabrication technology has little control at such precision level.

Fig. 1. (a) A commercial large-mode-area single mode MOF. (b) Its Veff is plotted as a function of normalized frequency Λ/λ (red curve). (b) also gives Veff curves for other types of fibers characterized by different d/Λ values (indicated beside each curve).

However, recent experiments done by Eggleton et al. [13

13. B. J. Eggleton, P. S. Westbrook, R. S. Windeler, S. Spälter, and T. A. Strasser, “Grating resonances in air-silica microstructured optical fibers,” Opt. Lett. 24, 1460–1462 (1999). [CrossRef]

15

15. B. J. Eggleton, C. Kerbage, P. Westbrook, R. S. Windeler, and A. Hale, “Microstructured optical fiber devices,” Opt. Express 9, 698–713 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13- 698 [CrossRef] [PubMed]

] have shown that when a fiber Bragg grating (FBG) is written in the core region of such a “single-mode” MOF, the transmission spectrum shows several dips other than only one dip. Each of these dips suggests one mode that is supported by the fiber. By comparing the spectrum with that obtained using a FBG written in a conventional single-mode fiber, they conclude that the dips at short wavelength region are due to coupling from core mode to backward-propagating “cladding modes”. Field patterns of these modes are experimentally recorded by tuning input wavelength at those dip positions. In this paper, we will show that these “cladding modes” are not actually comparable to those in conventional single-mode fibers for the fibers’ structural difference. These higher order modes are, in principle, caused by the antiguiding nature of the MOF, which is in turn due to finite periodic air-silica cladding size.

2. Numerical analysis of antiguided leaky modes

The MOF under study is shown in Fig. 2(a). The fiber’s dimensional parameters are the same as those for Fig. 1(a). The central red-filled circle has a refractive index of n0, and a diameter d0 which is equal to all surrounding air holes’. This red region acts as a defect in the PC if n0≠1.0. Despite 5 rings of air-holes, we have included 18 more air-holes in the 6th ring in order to make the air-hole region more circular. Their inclusion does not change the waveguide’s C6v group symmetry [19

19. P. R. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides. I. Summary of results,” IEEE Trans. Microwave Theory Tech. MTT-23, 421–429 (1975). [CrossRef]

]. Unlike usual way of defining core and cladding, we now treat the periodic air-silica composite (including red circle region) as core, and the bulk silica (white region) as cladding, which extends to infinity. We have neglected outer silica-air interface. This simplification will not affect our final result greatly since any light wandering into bulk silica region will eventually leak out of the outer silica-air interface; it will not be coupled back to the red circle region due to the woods of air cylinders. This point has been experimentally proven by Eggleton et al. [13

13. B. J. Eggleton, P. S. Westbrook, R. S. Windeler, S. Spälter, and T. A. Strasser, “Grating resonances in air-silica microstructured optical fibers,” Opt. Lett. 24, 1460–1462 (1999). [CrossRef]

] where the transmission spectrum of a MOF FBG/LPG has not varied much when the MOF is immersed into index-matching liquid. Fig. 2(b) gives the zoom-in view of the fiber at the central defect region.

Fig. 2. (a) Index profile of the MOF under studied. Black circles represent air-holes. The central red circle has diameter d0=d and refractive index n 0. (b) Zoom-in plot at the central defect region, with unit cell of the PC highlighted in light-red. (c) Equivalent simple antiguide. Core area (grey) has a diameter of 67.8308µm and a refractive index n 1 of 1.4443. Red circle has diameter d 0’ and refractive index n 0’.

The MOF in Fig. 2(a) can be roughly approximated by a simple step-index (SSI) antiguide whose index profile is shown in Fig. 2(c). The refractive index value of its core (grey area) is given as the effective index of the longitudinally-propagating fundamental space-filling mode (SFM) in the PC at λ=1.55µm, i.e., n 1=nSFM=1.4443. This approximation is accurate only when the periodic air-silica composite in Fig. 2(a) functions the same as an infinitely-extended composite, which is not always true here, as later their supported modes will show. Also this approximation disregards the fine structure and symmetry of the composite core. Hence this simple model only qualitatively reflects the MOF’s property. Nevertheless, it can provide us physical insight of antiguiding nature in the MOF. Central red circle also acts as a defect with index n 0’=1.45 and diameter d 0’. d 0’ depends on n 0. When n 0=1.0, d 0’=0µm, and both waveguides have a uniform core (one is composite, the other is single-material). When n 0=1.45, d 0’=1.05Λ, and two waveguides both have a defect at the center: defect in the MOF is shown by the unit cell in Fig. 2(b); defect in the SSI antiguide [denoted in red in Fig. 2(c)] has the same area as that unit cell in the MOF.

For SSI antiguides, we choose analytical transfer-matrix method (a similar formulation for planar multilayer waveguide can be found in [18

18. J. Chilwell and I. Hodgkinson, “Thin films field-transfer matrix theory of planar multiplayer waveguides and reflection from prism-loaded waveguides,” J. Opt. Soc. Am. A 1, 742–753 (1984). [CrossRef]

]), whose implementation is also based on scalar wave equation.

2.1. Antiguiding of the MOF with a uniform composite core

First we let n 0=1.0, i.e., the red circle in Fig. 2(a) represents an air hole (hereafter, we refer this MOF as MOF-1). The corresponding parameters in Fig. 2(b) are n 0’=1.4443 and d 0’=0µm, which make the simple antiguide like a hollow optical waveguide. Modes of such waveguides can be calculated using a transfer-matrix method in cylindrical coordinates [18

18. J. Chilwell and I. Hodgkinson, “Thin films field-transfer matrix theory of planar multiplayer waveguides and reflection from prism-loaded waveguides,” J. Opt. Soc. Am. A 1, 742–753 (1984). [CrossRef]

]. LP01-, LP11-, LP21- and LP02-like modes at λ=1.55µm are plotted in Fig. 3(a), (b), (c) and (d), respectively. Their neff values and corresponding leakage losses are summarized in Table 1.

Fig. 3. E-field distributions for (a) LP01-like (b) LP11-like (c) LP21-like and (d) LP02-like mode. Solid-black curve is the real part of the mode filed and dotted-red curve is the imaginary part. 2D color plots are in accord with 1D radial plots, with extra azimuthal field variations. Only real part of the mode field is shown in 2D color plots, which is true for all subsequent figures. The same color scale is shared by all 2D plots presented in this paper.

Table 1. Effective indices and confinement losses for four modes in Fig. 3.

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Relatively large imaginary part in both neff values and mode field values suggest high leakage losses for these modes in such type of optical waveguides. This is also manifested by oscillating mode field tail at remote radial positions, which transports energy out from core region (mathematically, these leaky transverse mode profiles can be represented by Hankel functions with certain phase velocities along radial direction). The fundamental mode will only propagate about 5 centimeters before its power drops to 1 percent. For a SSI antiguide, though not shown here, it can be concluded that all modes are leaky. Fundamental mode always has the smallest loss. And the loss value decreases as: (1) index contrast between core and cladding increases; (2) operating wavelength decreases.

Upper plot in Fig. 4 gives the mode spectrum of MOF-1. The calculation is done by using scalar BPM. Correlation mode solver [20

20. M. D. Feit and J. A. Fleck, “Computation of mode properties in optical fiber waveguides by a propagating beam method,” Appl. Opt. 19, 1154–1164 (1980). [CrossRef] [PubMed]

] has been used to derive the propagation constant and mode field distribution for each leaky mode. The fiber length is chosen as 215 µm for all similar calculations, unless otherwise specified.

Under symmetrical launching condition, only LP0m-like modes are excited. Six dominant modes are given in Fig. 4. We observe that these modes can be grouped according to their corresponding SFMs in the PC. Mode-a resembles the fundamental SFM of the PC when light propagates along z direction. Its field envelope is close to a Gaussian curve, hence we call it LP01-like mode. Mode-b and mode-c are also derived from the fundamental SFM but with radial field nodes. They are LP02- and LP03-like modes, respectively. Mode-d, e and f are grouped under a higher-order SFM of the PC. They can be similarly treated as LP01-, LP02- and LP03-like modes under this SFM category. Effective indices and corresponding confinement losses are given in Table 2. It should be noted that loss for LP01-like mode in the higher-order SFM group (mode-d) is smaller than that in the fundamental SFM group (mode-a). The same thing happens for LP02- and LP03-like modes. Physically, such loss difference is due to the fact that high-order SFM experiences many more times of reflections in the PC than fundamental SFM. It can be concluded from Fig. 4 that: (1) the modes excited with symmetric launching condition can be either SFM-like with a Gaussian-shaped envelope, or with extra envelope nodes in the radial direction; (2) each mode excited has their field envelope exhibiting full C6v group symmetry suggested by MOF-1’s core [19

19. P. R. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides. I. Summary of results,” IEEE Trans. Microwave Theory Tech. MTT-23, 421–429 (1975). [CrossRef]

]; (3) each mode excited has their amplitude exhibiting the symmetry not only suggested by the composite core, but also suggested by the composite itself. It is not difficult to imagine that all modes’ envelopes here tend to exhibit C∞v symmetry if more air-hole rings are included in the composite core (the composite core can be made infinitely close to a circular shape as the composite goes finer).

Only mode-a and b are comparable to modes we derived using SSI antiguide approximation. The difference in neff values between mode-a in Fig. 4 and that in Fig. 3 is 0.00033-0.000002i. The difference in neff values between mode-b in Fig. 4 and mode-d in Fig. 3 is 0.00042-0.000018i. Both neff values and the mode field patterns show our approximation is generally accurate. SSI antiguide approximation neglects structural detail of MOF-1, hence it overlooks mode-d, e and f here. But if we use the effective index of the PC’s SFM represented by mode-d as n 1 during our SSI antiguide approximation, we will get similar neff values corresponding to mode-d, e and f here; mode-a, b and c will be overlooked instead.

Fig. 4. Upper figure shows the mode spectrum of MOF-1 excited by a rectangular-shaped launching field of width Λ. Launching position is at x=y=0µm. Six mode profiles show modes a, b, c, d, e, f denoted in the spectrum.

Table 2. Effective indices and confinement losses for six modes in Fig. 4.

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Mode-b and c here are comparable to those in Fig. 3. Two LP11-like modes have a neff difference of 0.00025-0.00000091i, and two LP21-like modes have a neff difference of 0.000396-0.0000278i.

Fig. 5. Upper figure shows the mode spectrum of MOF-1 excited by a rectangular-shaped launching field of width Λ. Launching position is at x=2.5×Λ, y=0µm. Six mode profiles show mode-b, c, d, e, f, g denoted in the spectrum. Mode-a is the same as that in Fig. 4.

Table 3. Effective indices and confinement losses for six modes in Fig. 5.

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2.2. Antiguiding of the MOF with a defective cmposite core

Fig. 6. E field distributions of: (a) LP01-like (b) LP11-like (c) LP02-like and (d) LP121-like mode.

Table 4. Effective indices and confinement losses for four modes in Fig. 6.

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Figure 7 gives the mode spectrum and two dominant modes (mode-a and e) when MOF-2 is excited with a symmetrically launched source. Their mode fields all fully support C6v group symmetry. Whereas mode-a can find its analogue in SSI antiguide model [Fig. 6(a)], mode-e has no equivalent as such. These excited modes do not belong to any SFM of the PC. Instead, they belong to the defect modes of the PC. Such modes only reflect the symmetry of the defect. Unless the mode is extending to bulk silica cladding region, symmetry of the composite waveguide core is not evident in these modes. Mode-b, c and d shown in the spectrum are comparatively too lossy to be captured by the correlation mode solver. But as we will show later, they are dominating modes when the excitation is launched asymmetrically. neff values for mode-a and b are 1.446889+1.633e-9i and 1.437310+7.426e-5i, corresponding to a loss of 1.3239e-4cm -1 and 6.0205 cm -1, respectively.

Fig. 7. Upper figure shows the mode spectrum of MOF-2 excited by a rectangular-shaped launching field of width Λ. Launching position is at x=0µm, y=0µm. Only mode-a and e are shown. Mode-b, c, d corresponding to mode-a, b, c in the next figure, where the excitation is asymmetric. Fiber length here is 214 µm.

Comparing mode-a and that shown in Fig. 6, the difference in their real part of neff values is 0.000768. Their mode fields are both Gaussian-like. Unlike MOF-1, we cannot get neff value of mode-e by just using another n o’ value for SSI antiguide approximation.

Fig. 8. Upper figure shows the mode spectrum of MOF-2 excited by a rectangular-shaped launching field of width Λ. Launching position is at x=2.5×Λ, y=0µm. Six mode profiles show mode-a, b, c, d, f, g denoted in the spectrum. Mode-e is the same as that in Fig. 7.

Table 5. Effective indices and confinement losses for six modes in Fig. 8.

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Together with two modes shown in Fig. 7, we can see that except mode-a in Fig. 7, all other modes which can exist in MOF-2 are extremely lossy. They can only propagate for about 1–4cm in this fiber before their power drops by 2 orders. So effectively, as far as optical communication system is concerned, MOF-2 is a single-mode fiber, and it remains single-mode for an extended wavelength range for reason explained in [2

2. T. A. Birks, J. C. Knight, and P. St. J. Russel, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997). [CrossRef] [PubMed]

]. Following convention, we call mode-a in Fig. 7 as fundamental mode for MOF-2. All other modes are called higher-order leaky modes.

3. Discussion on antiguided leaky modes in MOFs

Higher-order leaky modes in MOF-2 are not equivalent to cladding modes as present in conventional step-index single-mode fibers (SISMF). The difference between them can be understood from their origins and their influence on optical devices and optical transmission systems.

In SISMF, core mode is the fundamental mode fully supported by doped-core and pure-silica cladding, and cladding modes are caused by reflection at the air-clad interface (we can understand these modes by treating the whole fiber as guiding core and surrounding air as cladding). Due to positive index-contrast between the Germanium-doped core and pure silica cladding in SISMF, light tends to settle into the core mode. Cladding mode cannot be excited easily if the light source is intentionally coupled into the core region within an acceptance angle. Possible irregularities in fiber cable and interconnects, which act as mode scramblers, can redistribute energy from propagating core mode into cladding modes. Current SISMF fabrication technology allows little variation on core shape along axial direction. And state-of-the-art fiber splicers can splice fibers with negligible loss. So cladding modes in SISMF do not pose a big problem in current optical communication system. And though cladding modes may appear at launching end or some other irregularity sites, they can easily be stripped off by applying an outer layer of high-index jacket around the fiber.

The effect of cladding mode becomes significant when we are analyzing fiber-based devices like FBG. When FBG is written in the doped core, the grating acts as discontinuities in the waveguide. It will perturb and thus redistribute energy among all possible modes, including cladding modes. If any mode has the propagation constant that fulfills the phase matching condition of the grating [21

21. T. Erdogan, “Fiber Grating Spectra,” J. Lightwave Technol. 155, 1277–1294 (1997). [CrossRef]

], it will get reflected. Those modes which did not get reflected will have their energy coupled back into the core mode after a short distance. Energy loss caused by FBG reflection can be observed clearly as sudden dips in the grating’s transmission spectrum [15

15. B. J. Eggleton, C. Kerbage, P. Westbrook, R. S. Windeler, and A. Hale, “Microstructured optical fiber devices,” Opt. Express 9, 698–713 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13- 698 [CrossRef] [PubMed]

]. In most of the cases, we want to reflect only one channel of information (say at λ) by using one uniform FBG. λ here is the fundamental Bragg wavelength. Coupling from forward-propagating core mode to backward-propagating core mode happens at this Bragg wavelength. But if the cladding mode comes into play (especially when the index modulation in the FBG is big or the grating is tilted), channels in slightly shorter wavelength region (say at λ’) may also experience reflection. λ’ here is a higher-order Bragg wavelength. Coupling from forward-propagating core mode to backward-propagating cladding mode happens at this wavelength. Such cladding mode resonances in SISMF give us undesired losses in wavelength-division-multiplexing (WDM) transmission system. But such losses can always be eliminated by employing special-designed cladding-mode suppression fibers [22

22. L. Dong, L. Reekie, J. L Cruz, J. E. Caplen, and D. N. Payne, “Cladding mode suppression in fibre Bragg gratings using fibres with a depressed cladding,” ECOC, 1.53–56 (1996).

] for FBG fabrication.

Leaky modes in MOF-2 originate from the antiguiding nature between the composite air-silica core and pure-silica cladding. These modes are inherently leaky. Such type of optical fiber, unless rigorously fabricated, will experience relatively high transmission loss, since any imperfection near the center will excite many leaky modes. Most of these leaky modes have very small overlapping ratio with the fundamental mode, hence this energy redistribution is irreversible. This suggests that using fiber like MOF-2 as a long-distance optical signal transmission medium is not feasible. Increasing number of air-hole rings in design and improving precision technology during fabrication are effective ways to reduce such leakage loss.

When FBG is written in the lifted-index region in MOF-2, its transmission spectrum has several dips, each corresponding to a different mode whose propagation constant phase-matches with the FBG [15

15. B. J. Eggleton, C. Kerbage, P. Westbrook, R. S. Windeler, and A. Hale, “Microstructured optical fiber devices,” Opt. Express 9, 698–713 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13- 698 [CrossRef] [PubMed]

]. By tuning source wavelength at one of the dips, fundamental mode and higher-order leaky modes can be recorded. Whereas cladding mode resonances in SISMF can be eliminated by using special fibers during FBG fabrication, dips corresponding to higher-order leaky mode in MOF-2’s transmission spectrum cannot be easily mitigated (one possible way could be to increase number of air hole rings in MOF designs). In addition, due to short distances between fundamental Bragg resonance and those higher-order Bragg resonances [15

15. B. J. Eggleton, C. Kerbage, P. Westbrook, R. S. Windeler, and A. Hale, “Microstructured optical fiber devices,” Opt. Express 9, 698–713 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13- 698 [CrossRef] [PubMed]

], one FBG might reflect more than one channel in WDM systems (but with only one peak appearing on the reflection spectrum since higher-order modes are all leaky). That limits the usage of MOF in FBG devices, or may bring some new ways of applying FBG devices for aught we know.

Some people utilized the fact that all modes in an antiguide are leaky to make a vertical-cavity surface-emitting laser (VCSEL), which has very high output power but remains single-mode [23

23. D. Zhou and L. J. Mawst, “High power single-mode antiresonant reflecting optical waveguide-type vertical cavity surface emitting lasers,” IEEE J. Quantum Electron. 38, 1599–1606 (2002). [CrossRef]

]. The high power is achieved by a big core area, and single-modeness is achieved by making higher-order modes far lossier than the fundamental one (only fundamental mode survives after gain competition). So theoretically speaking, we can use structure of index profile like MOF-2 in VCSEL cavity design, but with reduced number of air-hole rings and an optimized dimension so as to increase the loss discrimination between fundamental mode and higher-order modes. Some preliminary work has recently been published in [24

24. D.-S. Song, S.-H. Kim, H.-G. Park, C.-K. Kim, and Y.-H. Lee, “Single-fundamental-mode photonic-crystal vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 80, 3901–3903 (2002). [CrossRef]

].

4. Conclusion

Acknowledgments

M.Yan would like to thank J. Zhang for discussion on FBG, H. Liu for discussion on VCSEL, and Dr. Le Binh for valuable comments on the manuscript. He also would like to thank the Singapore government for providing financial support for his first degree and current research work.

References and Links

1.

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]

2.

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B. J. Eggleton, P. S. Westbrook, R. S. Windeler, S. Spälter, and T. A. Strasser, “Grating resonances in air-silica microstructured optical fibers,” Opt. Lett. 24, 1460–1462 (1999). [CrossRef]

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C. Kerbage, B. J. Eggleton, P. Westbrook, and R. S. Windeler, “Experimental and scalar beam propagation analysis of an air-silica microstructure fiber,” Opt. Express 7, 113–122 (2000), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-3-113 [CrossRef] [PubMed]

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B. J. Eggleton, C. Kerbage, P. Westbrook, R. S. Windeler, and A. Hale, “Microstructured optical fiber devices,” Opt. Express 9, 698–713 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13- 698 [CrossRef] [PubMed]

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J. Chilwell and I. Hodgkinson, “Thin films field-transfer matrix theory of planar multiplayer waveguides and reflection from prism-loaded waveguides,” J. Opt. Soc. Am. A 1, 742–753 (1984). [CrossRef]

19.

P. R. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides. I. Summary of results,” IEEE Trans. Microwave Theory Tech. MTT-23, 421–429 (1975). [CrossRef]

20.

M. D. Feit and J. A. Fleck, “Computation of mode properties in optical fiber waveguides by a propagating beam method,” Appl. Opt. 19, 1154–1164 (1980). [CrossRef] [PubMed]

21.

T. Erdogan, “Fiber Grating Spectra,” J. Lightwave Technol. 155, 1277–1294 (1997). [CrossRef]

22.

L. Dong, L. Reekie, J. L Cruz, J. E. Caplen, and D. N. Payne, “Cladding mode suppression in fibre Bragg gratings using fibres with a depressed cladding,” ECOC, 1.53–56 (1996).

23.

D. Zhou and L. J. Mawst, “High power single-mode antiresonant reflecting optical waveguide-type vertical cavity surface emitting lasers,” IEEE J. Quantum Electron. 38, 1599–1606 (2002). [CrossRef]

24.

D.-S. Song, S.-H. Kim, H.-G. Park, C.-K. Kim, and Y.-H. Lee, “Single-fundamental-mode photonic-crystal vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 80, 3901–3903 (2002). [CrossRef]

OCIS Codes
(060.0060) Fiber optics and optical communications : Fiber optics and optical communications
(230.3990) Optical devices : Micro-optical devices
(230.7370) Optical devices : Waveguides

ToC Category:
Research Papers

History
Original Manuscript: October 27, 2003
Revised Manuscript: December 14, 2003
Published: January 12, 2004

Citation
M. Yan and P. Shum, "Antiguiding in microstructured optical fibers," Opt. Express 12, 104-116 (2004)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-1-104


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References

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  12. K. Saitoh and M. Koshiba, �??Full-vectorial imaginary-distance beam propagation method based on a finite element scheme: application to photonic crystal fibers,�?? IEEE J. Quantum Electron. 38, 927-933 (2002). [CrossRef]
  13. B. J. Eggleton, P. S. Westbrook, R. S. Windeler, S. Spälter, and T. A. Strasser, �??Grating resonances in airsilica microstructured optical fibers,�?? Opt. Lett. 24, 1460-1462 (1999). [CrossRef]
  14. C. Kerbage, B. J. Eggleton, P. Westbrook, and R. S. Windeler, "Experimental and scalar beam propagation analysis of an air-silica microstructure fiber," Opt. Express 7, 113-122 (2000), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-3-113<a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-3-113">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-3-113</a> [CrossRef] [PubMed]
  15. B. J. Eggleton, C. Kerbage, P. Westbrook, R. S. Windeler, and A. Hale, "Microstructured optical fiber devices," Opt. Express 9, 698-713 (2001), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13- 698<a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13- 698">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-13- 698</a> [CrossRef] [PubMed]
  16. E. A. Marcatili and R. A. Schmeltzer, "Hollow metallic and dielectric waveguides for long distance optical transmission and lasers," Bell Syst. Tech. J. 43, 1783-1809 (1964).
  17. G. R. Hadley, �??Transparent boundary condition for the beam propagation method,�?? J. Quantum Electron 28, 363-370 (1992). [CrossRef]
  18. J. Chilwell and I. Hodgkinson, �??Thin films field-transfer matrix theory of planar multiplayer waveguides and reflection from prism-loaded waveguides,�?? J. Opt. Soc. Am. A 1, 742�??753 (1984). [CrossRef]
  19. P. R. McIsaac, �??�??Symmetry-induced modal characteristics of uniform waveguides. I. Summary of results,�??�?? IEEE Trans. Microwave Theory Tech. MTT-23, 421�??429 (1975). [CrossRef]
  20. M. D. Feit and J. A. Fleck, �??Computation of mode properties in optical fiber waveguides by a propagating beam method,�?? Appl. Opt. 19, 1154-1164 (1980). [CrossRef] [PubMed]
  21. T. Erdogan, "Fiber Grating Spectra," J. Lightwave Technol. 155, 1277-1294 (1997). [CrossRef]
  22. L. Dong, L. Reekie, J. L Cruz, J. E. Caplen and D. N. Payne, �??Cladding mode suppression in fibre Bragg gratings using fibres with a depressed cladding,�?? ECOC, 1.53-56 (1996).
  23. D. Zhou and L. J. Mawst, �??High power single-mode antiresonant reflecting optical waveguide-type vertical cavity surface emitting lasers,�?? IEEE J. Quantum Electron. 38, 1599�??1606 (2002). [CrossRef]
  24. D. S. Song, S.-H. Kim, H.-G. Park, C.-K. Kim, and Y.-H. Lee, �??Single-fundamental-mode photonic-crystal vertical-cavity surface-emitting lasers,�?? Appl. Phys. Lett. 80, 3901-3903 (2002). [CrossRef]

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