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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 22 — Oct. 27, 2008
  • pp: 17935–17951
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Bandgaps and antiresonances in integrated-ARROWs and Bragg fibers; a simple model

Kristopher J. Rowland, Shahraam Afshar V., and Tanya M. Monro  »View Author Affiliations


Optics Express, Vol. 16, Issue 22, pp. 17935-17951 (2008)
http://dx.doi.org/10.1364/OE.16.017935


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Abstract

We consider the spectral properties of dielectric waveguides with low refractive index cores and binary layered claddings, such as Bragg fibers and integrated-ARROWs. We show that the full, nontrivial, 2-D spectrum of Bloch bands (hence bandgaps) of such claddings correspond, in structure and topology, to the dispersion properties of both constituent layer types; quantitatively demonstrating an intimate relationship between the bandgap and antiresonance guidance mechanisms. The dispersion functions of these layers, and the interactions thereof, thus form what we coin the Stratified Planar Anti-Resonant Reflecting OpticalWaveguide (SPARROW) model, capable of quantitative, analytic, descriptions of many nontrivial bandgap and antiresonance properties. The SPARROW model is useful for the spectral analysis and design of Bragg fibers and integrated-ARROWs with cores of arbitrary refractive index (equal to or less than the lowest cladding index). Both waveguide types are of interest for sensing and microfluidic applications due to their natural ability to guide light within low-index cores, permitting low-loss guidance within a large range of gases and liquids. A liquid-core Bragg fiber is discussed as an example, demonstrating the applicability of the SPARROW model to realistic and important waveguide designs.

© 2008 Optical Society of America

1. Introduction

This work focuses on two primary themes: the reconciliation of the bandgap and antiresonance guidance mechanisms for dielectric waveguides with cores of low refractive index and binary layered claddings (such as Fig. 1); and, from this, the construction of a simple analytic model for the analysis of the full, nontrivial, 2-D bandgap spectra of such cladding structures.

Fig. 1. Schematic representations of the fiber, stack and slab geometries discussed here. Left: An arbitrary Bragg fiber geometry. All parameters defined within. Refractive indices take any value such that n 1>n 0n core≥1. Right: The equivalent planar slab representation of the Bragg-cladding with the equivalent isolated constituent layers shown below. The vector diagram represents the decomposition of an incident ray’s wavevector.

Considering the first theme, fibers with a binary-layered (Bragg) cladding are well known for their ability to confine light to cores with refractive indices equal to or lower than either of the cladding indices [1

1. P. Yeh, A. Yariv , and C. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. 67, 423–438 (1977). [CrossRef]

8

8. T. Katagiri, Y. Matsuura, and M. Miyagi, “All-solid single-mode bragg fibers for compact fiber devices,” J. Lightwave Technol. 24, 4314–4318 (2006). [CrossRef]

]. Analogous types of planar waveguides known as integrated Antiresonant Reflecting Optical Waveguides (integrated-ARROWs) exhibit similar low-index confinement behaviour [9

9. J. L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11, 416–423 (1993). [CrossRef]

13

13. H. Schmidt and A. R. Hawkins, “Optofluidic waveguides: I. Concepts and implementations,” Microfluid. Nanofluid. 4, 3–16 (2008). [CrossRef] [PubMed]

] but are typically treated as distinct to Bragg waveguides and their associated bandgap guidance mechanism. The cladding of a Bragg waveguide is usually considered as a 1-D photonic crystal such that the behaviour of the resultant Bloch modes dictates whether light can or cannot couple to the cladding [1

1. P. Yeh, A. Yariv , and C. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. 67, 423–438 (1977). [CrossRef]

, 14

14. Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998). [CrossRef] [PubMed]

]; core modes exist only for wavelengths and propagation constants that fall within the Bloch modes’ forbidden regions (bandgaps). ARROW guidance, on the other hand, is attributed to the antiresonance of light with the individual cladding layers [9

9. J. L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11, 416–423 (1993). [CrossRef]

, 15

15. M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures,” Appl. Phys. Lett. 49, 13–15 (1986). [CrossRef]

]; a particular layer will preferentially guide light at its resonant frequencies such that the transverse component of the light interferes constructively with itself for each round trip. Thus, light sufficiently far from the cladding resonances will be confined to the core due to restricted coupling (antiresonance) with the cladding layers themselves.

Here we suggest that the distinction between Bragg-cladding waveguides and integrated-ARROWs is somewhat artificial, with the only difference being semantic: whether the cladding is periodic (multiple unit cells: ‘Bragg’) or not (single unit cell: ‘ARROW’). Indeed, with hollow-core integrated-ARROWs now being considered with more than one binary cladding unit cell [10

10. D. Yin, J.P. Barber, A. R. Hawkins, and H. Schmidt, “Waveguide loss optimization in hollow-core ARROW waveguides,” Opt. Express 13, 9331–9336 (2005). [CrossRef] [PubMed]

12

12. F. Poli, M. Foroni, A. Cucinotta, and S. Selleri, “Spectral behavior of integrated antiresonant reflecting hollow-core waveguides,” J. Lightwave Technol. 25, 2599–2604 (2007). [CrossRef]

], this distinction ceases to exist. Thus, since these two waveguide structures are essentially the same, it is natural to expect a strong relationship between the bandgap and antiresonance guidance mechanisms themselves. We explore this idea within by analysing the two approaches and directly comparing the results.

It is useful here to spell out an important misnomer regarding the use of the term ‘ARROW’. As it was initially conceived, the model used to describe ARROWs typically assumed that one of the cladding layer refractive indices was much greater than the core and remaining cladding index [15

15. M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures,” Appl. Phys. Lett. 49, 13–15 (1986). [CrossRef]

]. As such, many ARROW designs have a core index equal to the lowest of the cladding indices [15

15. M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures,” Appl. Phys. Lett. 49, 13–15 (1986). [CrossRef]

21

21. N. M. Litchinitser, S. C. Dunn, P. E. Steinvurzel, B. J. Eggleton, T. P. White, R. C. McPhedran, and C. M. de Sterke, “Application of an ARROW model for designing tunable photonic devices,” Opt. Express 12, 1540–1550 (2004). [CrossRef] [PubMed]

]; we will call these level-core waveguides. Indeed, this line of thought saw the ARROWmodel successfully applied to (non-layered cladding) photonic crystal fibers (PCFs) [20

20. N. M. Litchinitser, S. C. Dunn, B. Usner, B. J. Eggleton, T. P. White, R. C. McPhedran, and C. M. de Sterke, “Resonances in microstructured optical waveguides,” Opt. Express 11, 1243–1251 (2003). [CrossRef] [PubMed]

, 21

21. N. M. Litchinitser, S. C. Dunn, P. E. Steinvurzel, B. J. Eggleton, T. P. White, R. C. McPhedran, and C. M. de Sterke, “Application of an ARROW model for designing tunable photonic devices,” Opt. Express 12, 1540–1550 (2004). [CrossRef] [PubMed]

] and has spawned much interest in what have been termed ‘ARROWfibers’: PCFs of a low-index substrate with a cladding of high-index rods (typically on, but not restricted to, a hexagonal lattice). It is implicitly assumed that such fibers have a core refractive index equal to the surrounding low-index substrate, just like the early (level-core) ARROWs; rightly so, since it is structurally the only possibility for a 2-D lattice based cladding as opposed to a layered one. For level-core waveguides, the predominant antiresonance behaviour comes from the high-index inclusions alone, with little dependence on their spacing [19

19. N. M. Litchinitser, A. K. Abeeluck, C. Headley, and B. J. Eggleton, “Antiresonant reflecting photonic crystal optical waveguides,” Opt. Lett. 27, 1592–1594 (2002). [CrossRef]

]. However, the general definition of an ARROW by Archambault et al. [9

9. J. L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11, 416–423 (1993). [CrossRef]

] dictates that the core index can have any value up to that of the lowest cladding index; we will call these depressed-core waveguides. All cladding layer resonances in depressed-core waveguides influence the spectral behaviour of the core modes [9

9. J. L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11, 416–423 (1993). [CrossRef]

], not just the high-index layer resonances. Indeed, it is the Archambault-ARROW model that is considered for most current work on integrated hollow-core ARROWs [10

10. D. Yin, J.P. Barber, A. R. Hawkins, and H. Schmidt, “Waveguide loss optimization in hollow-core ARROW waveguides,” Opt. Express 13, 9331–9336 (2005). [CrossRef] [PubMed]

, 11

11. D. Yin, H. Schmidt, J. P. Barber, E. J. Lunt, and A. R. Hawkins, “Optical characterisation of arch-shaped ARROW waveguides with liquid cores,” Opt. Express 13, 10564–10570 (2005). [CrossRef] [PubMed]

]. We believe this disparity between the use of the general Archambault-ARROW model and its restricted, Duguay-ARROW-based [15

15. M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures,” Appl. Phys. Lett. 49, 13–15 (1986). [CrossRef]

], form as applied to PCFs (the ‘ARROW-fibers’) [18

18. A. K. Abeeluck, N. M. Litchinitser, C. Headley, and B. J. Eggleton, “Analysis of spectral characteristics of photonic bandgap waveguides,” Opt. Express 10, 1320–1333 (2002). [PubMed]

21

21. N. M. Litchinitser, S. C. Dunn, P. E. Steinvurzel, B. J. Eggleton, T. P. White, R. C. McPhedran, and C. M. de Sterke, “Application of an ARROW model for designing tunable photonic devices,” Opt. Express 12, 1540–1550 (2004). [CrossRef] [PubMed]

] is responsible for the apparent bifurcation of the use of the ARROW principle in the integrated-waveguide and fiber fields. This is most clearly demonstrated (to the best of the Authors’ knowledge) by the absence of antiresonance analyses for hollow-core Bragg fibers and the absence of a Bloch analysis for hollow-core integrated-ARROWs; the two waveguide structures being fundamentally similar (depressed core, binary layered cladding), as discussed. The work presented here clearly demonstrates how both the antiresonance and bandgap principles are applicable to any depressed-core waveguide with a binary stratified cladding, particularly for fibers and integrated waveguides respectively.

Within, we develop a generalised version of the Archambault-ARROW model and use it to demonstrate an intimate relationship between the bandgap and antiresonance guidance mechanisms associated with stratified cladding waveguides. We coin this new model the Stratified Planar Anti-Resonant Reflecting OpticalWaveguide (SPARROW) model to distinguish it from both the original Archambault-ARROW model itself [9

9. J. L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11, 416–423 (1993). [CrossRef]

] and, in particular, the more restrictive level-core ARROW and (non-stratified) PCF applications based on the Duguay-ARROW model [15

15. M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures,” Appl. Phys. Lett. 49, 13–15 (1986). [CrossRef]

, 19

19. N. M. Litchinitser, A. K. Abeeluck, C. Headley, and B. J. Eggleton, “Antiresonant reflecting photonic crystal optical waveguides,” Opt. Lett. 27, 1592–1594 (2002). [CrossRef]

21

21. N. M. Litchinitser, S. C. Dunn, P. E. Steinvurzel, B. J. Eggleton, T. P. White, R. C. McPhedran, and C. M. de Sterke, “Application of an ARROW model for designing tunable photonic devices,” Opt. Express 12, 1540–1550 (2004). [CrossRef] [PubMed]

]. Indeed, as will be shown later (Section 4.3), the Duguay-ARROWmodel is a special case of the SPARROW model (found by enforcing a level-core index profile). Further, we will demonstrate how all (both high- and low-index) cladding layer resonances must be considered together for the most general waveguide analysis and design, rather than considering them separately as is typically done via the Archambault-ARROWmodel [9

9. J. L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11, 416–423 (1993). [CrossRef]

, 10

10. D. Yin, J.P. Barber, A. R. Hawkins, and H. Schmidt, “Waveguide loss optimization in hollow-core ARROW waveguides,” Opt. Express 13, 9331–9336 (2005). [CrossRef] [PubMed]

, 22

22. D. Yin, H. Schmidt, J. P. Barber, and A. R. Hawkins, “Integrated ARROW waveguides with hollow cores,” Opt. Express 12, 2710–2715 (2004). [CrossRef] [PubMed]

].

Here we examine the resonances of the individual cladding layers, via the SPARROWmodel, to present an analytical physical description of these nontrivial bandgap properties. We demonstrate how our model can determine a number of nontrivial properties with simple, fully analytic, expressions. Such tools are indispensable for the understanding and design of depressed-core layered-cladding waveguides, whether they be fibers (multi-dielectric, such as the aforementioned hollow-core Bragg fibers [3

3. S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. Engeness, M. Soljačić, S. Jacobs, J. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express 9, 748–779 (2001). [CrossRef] [PubMed]

, 4

4. 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]

, 6

6. K. Kuriki, O. Shapira, S. D. Hart, G. Benoit, Y. Kuriki, J. F. Viens, M. Bayindir, J. D. Joannopoulos, and Y. Fink, “Hollow multilayer photonic bandgap fibers for NIR applications,” Opt. Express 12, 1510–1517 (2004). [CrossRef] [PubMed]

], or single-material Bragg fibers [5

5. G. Vienne, Y. Xu, C. Jakobsen, H. Deyerl, J. Jensen, T. Sørensen, T. Hansen, Y. Huang, M. Terrel, R. Lee, N. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and Amnon Yariv, “Ultra-large bandwidth hollow-core guiding in all-silica Bragg fibers with nano-supports,” Opt. Express 12, 3500–3508 (2004). [CrossRef] [PubMed]

, 23

23. K. J. Rowland, S. Afshar V., and T. M. Monro, “Novel low-loss bandgaps in all-silica Bragg fibers,” J. Lightwave Technol. 26, 43–51 (2008). [CrossRef]

, 24

24. F. Poli, M. Foroni, D. Giovanelli, A. Cucinotta, S. Selleri, J. B. Jensen, J. Læsgaard, A. Bjarklev, G. Vienne, C. Jakobsen, and J. Broeng, “Silica bridge impact on hollow-core Bragg fiber transmission properties,” Proceedings OFC 2007 (Anaheim, California), paper OML8 (2007).

]) or integrated-ARROWs [9

9. J. L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11, 416–423 (1993). [CrossRef]

13

13. H. Schmidt and A. R. Hawkins, “Optofluidic waveguides: I. Concepts and implementations,” Microfluid. Nanofluid. 4, 3–16 (2008). [CrossRef] [PubMed]

]. We thus expect that the SPARROW model will be particularly useful for the design of layered-cladding waveguides with gas or liquid cores for applications in sensing, microfluidics, and novel nonlinear waveguides.

2. Background theory

2.1. Bragg stack Bloch analysis

It is well known that the bandgap properties of the cladding of a Bragg fiber can be well approximated by those of a planar Bragg stack [1

1. P. Yeh, A. Yariv , and C. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. 67, 423–438 (1977). [CrossRef]

, 3

3. S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. Engeness, M. Soljačić, S. Jacobs, J. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express 9, 748–779 (2001). [CrossRef] [PubMed]

6

6. K. Kuriki, O. Shapira, S. D. Hart, G. Benoit, Y. Kuriki, J. F. Viens, M. Bayindir, J. D. Joannopoulos, and Y. Fink, “Hollow multilayer photonic bandgap fibers for NIR applications,” Opt. Express 12, 1510–1517 (2004). [CrossRef] [PubMed]

, 23

23. K. J. Rowland, S. Afshar V., and T. M. Monro, “Novel low-loss bandgaps in all-silica Bragg fibers,” J. Lightwave Technol. 26, 43–51 (2008). [CrossRef]

]. Consider a Bragg fiber with cladding layer refractive indices n 1 and n 0 and core refractive index n core, such that n 1>n 0n core≥1 (a ‘depressed-core’ waveguide as defined above), with layer thicknesses t 1 and t 0 and core thickness t core, Fig. 1.

Two matrix recursion relations, describing transverse electric (TE) or transverse magnetic (TM) waves, can be derived via the Bloch-Floquet theorem, relating plane waves in the stack to an incident plane wave [1

1. P. Yeh, A. Yariv , and C. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. 67, 423–438 (1977). [CrossRef]

]. The resultant 2×2 Bloch-wave eigenvalue equation has only one unique matrix element, A TE,TM(k,ñ) [1

1. P. Yeh, A. Yariv , and C. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. 67, 423–438 (1977). [CrossRef]

], which is a function only of the layer parameters (ni and ti) and wave parameters (k and ñ). Only solutions satisfying the condition [1

1. P. Yeh, A. Yariv , and C. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. 67, 423–438 (1977). [CrossRef]

]:

Re{ATE,TM(k,n˜)}<1,
(1)

correspond to (non-evanescent) waves allowed to propagate perpendicular to the Bragg stack (the x-direction in Fig. 1). The evaluation of Inequality 1 for a particular layer configuration is straightforward since k and ñ are the only free parameters. The regions in the 2-dimensional space (k, ñ) satisfying Ineq. (1) are allowed bands (waves allowed to propagate in the cladding) for either TE or TM waves (in Fig. 2, black represents where the TE and TM bands overlap, so for ñ<n 0 all TE bands are black since the TM bands are larger, thus both black and grey). For ñ>n 0 the TE and TM bands diverge [1

1. P. Yeh, A. Yariv , and C. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. 67, 423–438 (1977). [CrossRef]

], Fig. 2. The regions in between the allowed bands are the bandgaps. Thus, any core mode of a depressed-core waveguide (1≤n coren 0) must have both k and ñ within a bandgap. All figures within plot over (Λ/λ,ñ)=(Λk/2π,ñ) so that both axes are unitless, naturally representing the scale-invariance of the system.

Fig. 2. Left: A bandgap map generated via the Bloch theorem (Section 2.1) for a Bragg fiber cladding like that considered in [4]: t 1=0.27µm, t 0=0.9µm, n 1=2.8 and n 0=1.55. Color scheme (for ñ<n 0): as described in text, Section 2.1; black for TE bands, black and grey for TM bands (⇒ white for TM bandgaps, white and grey for TE bandgaps). Solid blue line: the n 0-light-line. Dotted line: the Brewster line, ñ=n B. Right: A plot of all the SPARROW curves (dispersion curves of the equivalent isolated layers), via Eq. (6), for the same cladding and bandgap domain. Magenta: high-index (n 1) layer, n˜m1 (k). Cyan: low-index (n 0) layer, ñ m0 (k).

From Fig. 1, glancing incidence (θ=π/2) produces β=kiñ=ni, referred to as the ‘ni-light-line’. For waveguides of sufficiently large core radius, core-bound modes of low order exhibit dispersion curves ñ(k) close to those of plane waves of glancing incidence, hence ñ(k)≈ni. Core modes of the waveguides considered here will thus typically only exist where the n core-light line intercepts the relevant bandgap; namely, TE modes within TE gaps and TM and HE modes within TM gaps (since HE modes contain components of both TE and TM rays [27

27. J. A. Buck, Fundamentals of Optical Fibers (2nd ed., John Wiley & Sons., Inc, 2004).

]).

As demonstrated in Fig. 2, the TM bandgaps close up at a specific value of nB=n1n0n12+n02 due to the well known Brewster phenomenon [28

28. M. Born and E. Wolf, Principles of Optics (7th ed., Cambridge University Press, 2002).

30

30. A. Argyros, “Guided modes and loss in Bragg fibres,” Opt. Express 10, 1411–1417 (2002). [PubMed]

], where TM rays satisfying the Brewster condition are totally transmitted thus preventing the existence of a bandgap at that specific value of ñ [30

30. A. Argyros, “Guided modes and loss in Bragg fibres,” Opt. Express 10, 1411–1417 (2002). [PubMed]

]. Thus, for ñ<n 0, the TM gaps always lie within the TE gaps [23

23. K. J. Rowland, S. Afshar V., and T. M. Monro, “Novel low-loss bandgaps in all-silica Bragg fibers,” J. Lightwave Technol. 26, 43–51 (2008). [CrossRef]

]. The closure of the TE gaps close to n B in the case of Fig. 2 is coincidental (discussed further in Sections 3 and 4.1).

2.2. ARROW models

Originally discussed in the context of planar waveguides with a single high-index cladding layer in 1986 [15

15. M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures,” Appl. Phys. Lett. 49, 13–15 (1986). [CrossRef]

], the Anti-Resonant Reflecting Optical Waveguide (ARROW) model of Duguay et al. demonstrated how light could be confined to a low-index core by inhibited coupling (antiresonance) with a high-index cladding layer. In 1994, this model was later generalised to arbitrary numbers of cladding layers with arbitrary refractive indices by Archambault et al. [9

9. J. L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11, 416–423 (1993). [CrossRef]

]. The only restriction for core-guidance in the Archambault-ARROW model was that the core must have a refractive index equal to or less than the lowest of the cladding indices: n core≤min{ni}. By considering the transverse phase accumulated by a propagating ray (Fig. 1) per round trip (including that from traversing the core, via the V-parameter) and equating it to 2π (resonance), the Archambault-ARROW model determines the wavelengths at which the i th cladding layer will be resonant with the guided light [9

9. J. L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11, 416–423 (1993). [CrossRef]

, 22

22. D. Yin, H. Schmidt, J. P. Barber, and A. R. Hawkins, “Integrated ARROW waveguides with hollow cores,” Opt. Express 12, 2710–2715 (2004). [CrossRef] [PubMed]

]:

λmi=2timini2ncore2+(Uλ2πtcore)2,
(2)

From Eq. (2), the wavelengths producing antiresonance with individual cladding layers are found between the cladding layer resonance points by considering half-integer layer mode orders: mimi+½[9

9. J. L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11, 416–423 (1993). [CrossRef]

]. Using this approach, the Archambault-ARROWmodel has recently been applied successfully to depressed-core gas- and liquid-filled hollow-core integrated-ARROW waveguides [10

10. D. Yin, J.P. Barber, A. R. Hawkins, and H. Schmidt, “Waveguide loss optimization in hollow-core ARROW waveguides,” Opt. Express 13, 9331–9336 (2005). [CrossRef] [PubMed]

, 11

11. D. Yin, H. Schmidt, J. P. Barber, E. J. Lunt, and A. R. Hawkins, “Optical characterisation of arch-shaped ARROW waveguides with liquid cores,” Opt. Express 13, 10564–10570 (2005). [CrossRef] [PubMed]

, 22

22. D. Yin, H. Schmidt, J. P. Barber, and A. R. Hawkins, “Integrated ARROW waveguides with hollow cores,” Opt. Express 12, 2710–2715 (2004). [CrossRef] [PubMed]

] but tends to be used to achieve antiresonance with each cladding layer individually and only for mi=1 (e.g. [22

22. D. Yin, H. Schmidt, J. P. Barber, and A. R. Hawkins, “Integrated ARROW waveguides with hollow cores,” Opt. Express 12, 2710–2715 (2004). [CrossRef] [PubMed]

]). While this approach is sufficient for designing a guidance regime, the tuning of each layer separately is unnecessarily restrictive (and potentially produces a higher confinement loss than desired by bringing the resonances of both layer types, rather than just one, close to the k of interest). As will be shown in Section 4.2, a general analysis requires the resonances of all layer types to be considered together, leading to our definition of the general antiresonance point kc which is more suited for general analyses of arbitrary cladding configurations and strictly necessary for arbitrary ñ and mi.

By substituting U π/2 (planar core with p=0) in Eq. (2), the Duguay-ARROW resonance condition [15

15. M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures,” Appl. Phys. Lett. 49, 13–15 (1986). [CrossRef]

] is derived (where only one, high-index, cladding layer type is considered, making the i label redundant). This substitution is equivalent to assuming the core-bound rays only make glancing incidence with the cladding layer [15

15. M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures,” Appl. Phys. Lett. 49, 13–15 (1986). [CrossRef]

]. The Duguay-ARROW model has been successfully applied to different types of level-core (n core=n 0<n 1) ARROWs (e.g. [15

15. M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures,” Appl. Phys. Lett. 49, 13–15 (1986). [CrossRef]

17

17. T. Baba and Y. Kokubun, “Dispersion and radiations loss characteristics of antiresonant reflecting optical waveguides - numerical results and analytical expressions,” J. Quantum Electron. 28, 1689–1700 (1992). [CrossRef]

]). More recently, the Duguay-ARROWmodel has also been employed by Litchinitser et al. [19

19. N. M. Litchinitser, A. K. Abeeluck, C. Headley, and B. J. Eggleton, “Antiresonant reflecting photonic crystal optical waveguides,” Opt. Lett. 27, 1592–1594 (2002). [CrossRef]

21

21. N. M. Litchinitser, S. C. Dunn, P. E. Steinvurzel, B. J. Eggleton, T. P. White, R. C. McPhedran, and C. M. de Sterke, “Application of an ARROW model for designing tunable photonic devices,” Opt. Express 12, 1540–1550 (2004). [CrossRef] [PubMed]

] and Abeeluck et al. [18

18. A. K. Abeeluck, N. M. Litchinitser, C. Headley, and B. J. Eggleton, “Analysis of spectral characteristics of photonic bandgap waveguides,” Opt. Express 10, 1320–1333 (2002). [PubMed]

] to describe antiresonance guidance in photonic crystal fibers (PCFs) in the large-core regime, including level-core Bragg fibers [19

19. N. M. Litchinitser, A. K. Abeeluck, C. Headley, and B. J. Eggleton, “Antiresonant reflecting photonic crystal optical waveguides,” Opt. Lett. 27, 1592–1594 (2002). [CrossRef]

], and, in particular, PCFs with a cladding of high-index rods [20

20. N. M. Litchinitser, S. C. Dunn, B. Usner, B. J. Eggleton, T. P. White, R. C. McPhedran, and C. M. de Sterke, “Resonances in microstructured optical waveguides,” Opt. Express 11, 1243–1251 (2003). [CrossRef] [PubMed]

, 21

21. N. M. Litchinitser, S. C. Dunn, P. E. Steinvurzel, B. J. Eggleton, T. P. White, R. C. McPhedran, and C. M. de Sterke, “Application of an ARROW model for designing tunable photonic devices,” Opt. Express 12, 1540–1550 (2004). [CrossRef] [PubMed]

]. For stratified claddings, the large-core limit reduces the cladding layer resonance condition to [19

19. N. M. Litchinitser, A. K. Abeeluck, C. Headley, and B. J. Eggleton, “Antiresonant reflecting photonic crystal optical waveguides,” Opt. Lett. 27, 1592–1594 (2002). [CrossRef]

]:

λm=2t1mn12n02.
(3)

While level-core Bragg fibers have been discussed in the context of the Duguay-ARROW model [18

18. A. K. Abeeluck, N. M. Litchinitser, C. Headley, and B. J. Eggleton, “Analysis of spectral characteristics of photonic bandgap waveguides,” Opt. Express 10, 1320–1333 (2002). [PubMed]

, 19

19. N. M. Litchinitser, A. K. Abeeluck, C. Headley, and B. J. Eggleton, “Antiresonant reflecting photonic crystal optical waveguides,” Opt. Lett. 27, 1592–1594 (2002). [CrossRef]

], the more general Archambault-ARROW model appears to be absent from analyses of solid-cladding hollow-core (n 0>n core=1) Bragg fibers, where Bloch-wave analysis (as in Section 2.1) is more common [1

1. P. Yeh, A. Yariv , and C. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. 67, 423–438 (1977). [CrossRef]

, 3

3. S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. Engeness, M. Soljačić, S. Jacobs, J. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express 9, 748–779 (2001). [CrossRef] [PubMed]

6

6. K. Kuriki, O. Shapira, S. D. Hart, G. Benoit, Y. Kuriki, J. F. Viens, M. Bayindir, J. D. Joannopoulos, and Y. Fink, “Hollow multilayer photonic bandgap fibers for NIR applications,” Opt. Express 12, 1510–1517 (2004). [CrossRef] [PubMed]

, 23

23. K. J. Rowland, S. Afshar V., and T. M. Monro, “Novel low-loss bandgaps in all-silica Bragg fibers,” J. Lightwave Technol. 26, 43–51 (2008). [CrossRef]

]. Likewise, the analysis of multilayer hollow-core integrated-ARROWs [9

9. J. L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11, 416–423 (1993). [CrossRef]

13

13. H. Schmidt and A. R. Hawkins, “Optofluidic waveguides: I. Concepts and implementations,” Microfluid. Nanofluid. 4, 3–16 (2008). [CrossRef] [PubMed]

] appears to primarily use the Archambault-ARROW model, foregoing a Bloch-wave analysis. By demonstrating the intimate relationship between the bandgap and antiresonance guidance mechanisms here, we hope that the artificial bifurcation of these techniques is overcome.

2.3. Dispersion analysis of slab waveguides below the light-line

Consider now the layers that constitute a Bragg stack, each in isolation (represented in the bottom-right of Fig. 1). Let the isolated slab and its surrounding cladding have refractive indices n a and n b, respectively. There are thus two types of slab waveguide to consider: one with a high slab index n a=n 1 with a lower cladding index n b=n 0(n a>n b), and one with a low slab index n a=n 0 with a high cladding index n b=n 1(n a<n b); n 1>n 0 as before (Fig. 1). For this analysis, the same wave-vector decomposition as in Fig. 1 is used, with the wavevector ki=k a incident at angle θa to the normal within the dielectric of refractive index n a [the Bragg stack schematic in Fig. 2, though, is replaced by a homogeneous region with the equivalent vector diagram for a transmitted ray (ab)].

Since we are interested in slab modes with effective mode indices ñ a below the n 0-light-line (ñ a<n 0), all slab modes are inherently leaky [27

27. J. A. Buck, Fundamentals of Optical Fibers (2nd ed., John Wiley & Sons., Inc, 2004).

]. This is because for the former (high-index) slab ñ a=β 1/k=n 1 sinθ a<n 0, so θ a<θ c=sin-1(n b/n a) (the critical angle) and no total internal reflection occurs, thus relying on regular (lossy) reflection. In other words, the modes must exist below the bound mode cut-off [27

27. J. A. Buck, Fundamentals of Optical Fibers (2nd ed., John Wiley & Sons., Inc, 2004).

]. The latter (low-index) slab relies on inherently lossy Fresnel reflection for all θ a since n a<n b. The loss characteristics are considered here only insofar as they aid the phase analysis via the Fresnel reflection coefficients [27

27. J. A. Buck, Fundamentals of Optical Fibers (2nd ed., John Wiley & Sons., Inc, 2004).

, 28

28. M. Born and E. Wolf, Principles of Optics (7th ed., Cambridge University Press, 2002).

]:

ΓTE=kaxkbxkax+kbx,ΓTM=nb2kaxna2kbxnb2kaxna2kbx,
(4)

which relate the incident and reflected field amplitudes, A TE,TM and ATE,TM respectively, by A TE,TMTE,TM ATE,TM.

As discussed, for ñ a<n 0 guidance, Snell’s law implies the high-index (na>nb) slab must have θ a<θc. Since cos(θi)=kix/ki and 0≤θa<θ cπ/2 (ℤ0≤θ bπ/2) then cos(θ a)>cos(θ b) and thus k ax>k bx. Equation (4) then implies ΓTE,TM∈ℝ+ so that sign(A TE,TM)=sign(ATE,TM) and no phase shift occurs upon reflection. However, for the low-index (n a<n b) slab, θa>θb with no critical angle threshold, so similar reasoning implies k ax<k bx. Thus ΓTE,TM∈ℝ-, meaning sign(A TE,TM)=-sign(ATE,TM) such that a π phase shift occurs upon reflection at the interface. Note that the TE and TM modes are thus degenerate for ñ<n 0 due to their equivalence in this phase analysis (ΓTE,TM produce the same reflective phase conditions for both na>nb and na<nb for ña<n 0 guidance).

The slab waveguides will only support modes, leaky or otherwise, if the accumulated transverse phase for one round-trip of the slab (traversing the slab twice upon two reflections from the interfaces) is an integer multiple of 2π. For both slabs, the transverse phase accumulated by traversing the slab region once is k ax t a. The forms of the low- and high-index slabs’ phase relations thus differ only in their reflection terms. Equating the cumulative phase shifts to m2π (m∈ℤ+), a dispersion relation for each waveguide is derived [27

27. J. A. Buck, Fundamentals of Optical Fibers (2nd ed., John Wiley & Sons., Inc, 2004).

, 32

32. P. K. Tien, “Light waves in thin films and integrated optics,” Appl. Opt. 10, 2395–2413 (1971). [CrossRef] [PubMed]

]:

kaxta={mπforna>nbandm(m+1)πforna<nbandm+
(5)

where m=0 is obviously not allowed for the high-index slab, implying that the m=0 bound mode has no leaky counterpart [27

27. J. A. Buck, Fundamentals of Optical Fibers (2nd ed., John Wiley & Sons., Inc, 2004).

]. By rearranging the phase relations and setting a→1 and b→0 for the high-index (n a=n 1>n b=n 0) slab and a→0 and b→1 for the low-index (n a=n 0<n b=n 1) slab, we find a unified dispersion relation:

n˜mi=[ni2(miπtik)2]12,mi
(6)

such that m 1=m and m 0=m+1. Groups of dispersion curves for a range of mode orders are plotted in Fig. 2 (right) and subsequently in Figs. 3, 4 and 5.

Here it is convenient to define that m 1=0 refers to the ñ-axis (k=0) and m 0=0 to the n 0-light-line (ñ=n 0). It is easily shown that, while not representative of physical modes, these definitions still satisfy Eqs. 5 and 6. Hereon the ‘SPARROW curves’ will refer to both the physical slab dispersion curves (m 1,0∈ℕN) and these m 1,0=0 lines, unless otherwise specified. The lower limit line ñ=0 is also important but its inclusion in this set is not required, as will soon be evident.

Note that Eq. (6) is truly analytic since the non-analytic Goos-Haenchen phase shift [32

32. P. K. Tien, “Light waves in thin films and integrated optics,” Appl. Opt. 10, 2395–2413 (1971). [CrossRef] [PubMed]

] that appears in the bound-mode (n˜m1>n0) solution of the high-index slab does not appear in these phase relations due to the nature of the reflective phase shifts for n˜mi<n0. Also note how Eq. (6) depends only on ni, implying that, below the n 0-light-line, the dispersion properties of each slab depend only on the slab refractive index, not that of the medium surrounding it.

Equation (6) can be arranged to give the k values of resonances of order mi for arbitrary ñ as:

kmi=miπti[ni2n˜2]12,
(7)

where, once expressed in wavelength, it is obvious that the forms of the large-core Duguay-ARROW model (Eq. (3)) and SPARROW model (Eq. (7)) are identical save for two important differences: the SPARROW model is valid for all ñn 0 and depends explicitly on t 0.

3. Formal SPARROWmodel definition

Figure 3 (left) is produced by overlaying the slab curves (Fig. 2, right) upon the corresponding Bragg stack’s bandgap map (Fig. 2, left). A clear and striking similarity between the two plots is thus revealed: each high- and low-index slab dispersion curve corresponds to a band of the bandgap spectrum. Also, the gaps completely close at the intersection points of the high and low refractive index curves ( nm1 and nm0, discussed further in Section 4.1). This is because n˜m1=n˜m0 implies optimal coupling between the two layer types so that light can easily propagate through the cladding, precluding the generation of a bandgap.

Fig. 3. Left: The bandgap map of Fig. 2 with the cladding layer dispersion (SPARROW) curves (Fig. 2) overlayed. Right: The same configuration but with the high-index layer’s thickness decreased: t 1=0.27µm→0.18µm. The bandgap topology dramatically changes between the two cases (new bandgaps are created). Using the nomenclature and analyses of Section 4.5: t 1=0.27µm produces N 1=2 and N 2=4 whereas t 1=0.18µm produces N 1=3 and N 2=6.

We thus define the Stratified Planar Anti-Resonant Reflecting Optical Waveguide (SPARROW) model as the use of the slab waveguides’ dispersion curves (and the properties derived from them, Section 4) to quantitatively describe the resonant features of the associated bandgap spectrum. The SPARROWmodel thus gives direct physical insight into how the thicknesses and refractive indices of the constituent high- and low-index layers affect the resonant properties of the cladding. Our model is thus similar to the Archambault-ARROWmodel [9

9. J. L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11, 416–423 (1993). [CrossRef]

] except that the latter inseparably couples the core properties to the resonance analysis. Instead, the SPARROW model considers the effective mode index alone, with no allusion to the properties of the core itself. In this way, we have separated the cladding resonances from the core properties, giving two benefits: the ability to easily describe cladding resonances on a bandgap-style (Λ/λ,ñ) plot (e.g. Figs. 2,3,4 and 5); and the freedom to consider core modes of arbitrary ñ. The latter point requires that, for the SPARROWmodel to accurately predict core-mode spectral features, sensible values for a core mode’s ñ must be provided independently. For large-core waveguides, this is trivial, since most low-order modes will lie very close to the n core-light-line (demonstrated in Section 3.1). However, it has also been shown that, at least for air-core Bragg fibers, it is possible to infer the (real part of the) core modes’ ñ solely from the core geometry [23

23. K. J. Rowland, S. Afshar V., and T. M. Monro, “Novel low-loss bandgaps in all-silica Bragg fibers,” J. Lightwave Technol. 26, 43–51 (2008). [CrossRef]

, 30

30. A. Argyros, “Guided modes and loss in Bragg fibres,” Opt. Express 10, 1411–1417 (2002). [PubMed]

], including higher-order modes and small core (~λ) waveguides.

Note that between any two adjacent slab dispersion curves, a bandgap region exists in either the ñ or k dimension, meaning that the discrete bandgaps within the stack bandgap spectrum are each enclosed by a subset of the SPARROW curves. We thus define the concept of a bounding region for any particular bandgap: the (k, ñ) region enclosed by the curves surrounding a particular bandgap. This behaviour implies the nontrivial discrete bandgap spectrum of the stratified cladding is replicated in position and topology by the equivalent SPARROW curves (including the physical limits k=0, ñ=0 and ñ=n 0, as discussed). Consequences and applications of this are discussed in Section 4.

3.1. Confirmation via FEM analysis

The validity of the SPARROW model can be verified directly by calculating the confinement loss spectrum of a particular Bragg fiber configuration in which both high- and low-index resonances are evident. We model a realistically achievable configuration: a fabricable (as demonstrated by Temelkuran et al. [4

4. 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]

]) solid-cladding hollow-core Bragg fiber which is filled with a liquid of refractive index 1.45 (achievable with index-matching liquids or various oils). The cladding layers of [4

4. 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]

] were made from As 2 Se 3 chalcogenide glass (n≈2.8) and the polymer polyethersulphone (PES, n≈1.55) with thicknesses of 270nm and 900nm respectively. n core=1.45 is chosen here simply because it produces modes revealing interesting structure in the bandgap spectrum. We choose a smaller core diameter (20µm) than the cited fabricated fiber (≈700µm), and fewer rings (4 pairs of layers instead of 9), due to numerical restrictions of the method employed (the relevant discretisations are stored in finite computer memory), but note that the bandgap behaviour would be very similar between the two structures regardless (like the 700µm core, a 20µm core also produces modes close to the n core-light-line, Fig. 4).

Figure 4 shows the bandgap and CL spectra for the discussed fiber, demonstrating how the supported TE01 mode exists only within the cladding bandgaps, with ñ having no solutions within the allowed bands, as expected. Note that the core-mode dispersion curve lies relatively close to the n core-light-line for this t core.

Fig. 4. Top: A portion of the SPARROWcurves and bandgap map from Fig. 2. Green line: the TE01 mode’s Re{ñ} from the FEM calculation of the equivalent Bragg fiber, with a core of size t core=20µm. Red dashed line: the n core-light-line (ñ=1.45). Red circles: positions of the general antiresonance point kc (Section 4.2) on the n core-light-line. The gaps are labelled using the nomenclature of Section 4.1. The limit value Λ/λ=3 corresponds to λ=390nm in this case. Bottom: The associated CL spectrum. Cyan and magenta dashed lines: low- and high-index SPARROW resonances on the n core-light-line, respectively (corresponding to circles of the same color in the top plot). Red lines: positions of the general antiresonance points from the top plot (red circles).

4. Further analysis of the SPARROWmodel

4.1. Curve intersections and gap nomenclature

The intersection points of the high- and low-index layer dispersion curves (Eq. (6) for i={1,0}) can be found by equating either ñ or k. The intersection point of arbitrary n˜m1 (k) (high-index) and n˜m0 (k) (low-index) curves is found to be:

P(m1,m0)=(k,n˜)n˜m1=n˜m0=(π1n12n02(m12t12m02t02),n12n02η21η2),
(8)

where η=m1t0m0t1.. These points thus form the corners of the aforementioned bounding regions for a particular gap, which we shall call the bounding points.

Since the n˜mi (k) curves monotonically approach asymptotes at ni ( n˜mini as k→∞), all gaps will have a maximal bounding point: the intersection point whose k and ñ values are larger than those of all other bounding points for that gap, being the top-right bounding point when represented on a (Λ/λ, ñ) plot. Since the maximal bounding point exists for all gaps for all cladding configurations, it may be used to define a consistent nomenclature for arbitrary bandgap spectra. Here we adopt the convention that each gap is referred to by the orders of the bounding curves producing the maximal bounding point. Explicitly, an arbitrary gap is labelled as the 〈m 1,m 0〉 gap. Some examples: the lowest order (fundamental) gap is the 〈m 1,m 0〉=〈1,0〉 gap (using the Section 2.3 definition that m 0=0 corresponds to the n0-light-line); any gap bound above by the ñ=n 0 line is an 〈m 1,0〉 gap; and so on. The gaps shown in Figs. 4 and 5 are labelled using this convention. It should be noted that this nomenclature is different (and more general) than that previously proposed by the authors in [23

23. K. J. Rowland, S. Afshar V., and T. M. Monro, “Novel low-loss bandgaps in all-silica Bragg fibers,” J. Lightwave Technol. 26, 43–51 (2008). [CrossRef]

]; while that particular nomenclature was suitable for the particular fiber parameters examined, the system introduced here is more general and thus suitable for all possible regimes.

Using this nomenclature, we can now quantify that the portions of the 〈m 1,m 0〉 gap that close up within the 0<ñ<n 0 region will do so at the intersection points P(m 1,m 0) and P(m 1-1,m 0+1), should these points exist within the domain for the given gap. If these points don’t exist, the gap must then be terminated by the ñ=0 or n 0 lines, leaving it open (seen explicitly in Figs. 3 and 5). Incidentally, we believe that this gap closure behaviour helps explain the mode suppression phenomena observed in [35

35. J. Li and K. S. Chiang, “Disappearance of modes in planar Bragg waveguides,” Opt. Lett. 32, 2369–2371 (2007). [CrossRef] [PubMed]

].

4.2. The general antiresonance point

From Eq. (7) we define the general antiresonance point kc, at arbitrary ñ=ñ′, as the arithmetic mean of the bounding curve values kmi(n˜) either side of the gap (the central k value):

kc(n˜)=kmp(n˜)+kmq(n˜)2,
(9)

where p and q refer to the adjacent bounding curve types in the k-dimension (i.e. p,q∈{1,0}) and mq and mp to their order. Note that the curves adjacent (in the k-dimension) to kc will change as ñ′ varies within the bounding region. For example, ñ′ in the middle section of the 〈m 1,m 0〉 bounding region (i.e. between P(m 1-1,m 0) and P(m 1,m 0+1) in the ñ-dimension) implies kc is between curves with either mp=m 1 and mq=m 1-1 or mp=m 0 and mq=m 0+1 (depending on the type of gap), due to the monotonicity of the curves. For ñ′ in the top section of the bounding region (above the middle section but below P(m 1,m 0)), kc will be in between curves with mp=m 1 and mq=m 0, whereas for ñ′ near the bottom of the gap (below the middle section but above P(m11,m0+1)) kc is between the mp=m 1-1 and mq=m 0+1 curves.

Figure 4 demonstrates how kc naturally predicts the approximate position of lowest CL for a core mode of the fiber discussed in Section 3.1. The reason CL reaches a minimum near kc is that, at that point, the guided wave is maximally antiresonant with the pair of slabs producing the bounding dispersion curves (mp and mq); as one or the other bounding curve is approached, the wave becomes more resonant with the associated slab, thus allowing greater coupling from the core to the cladding. This is why the resonances of all layer types must be considered together, rather than separately, as discussed earlier, and is inherent in the definition of kc. The general antiresonance point is thus a powerful tool for waveguide design and analysis, allowing immediate determination of the approximate point of minimum CL at arbitrary ñ within an arbitrary bandgap.

Fig. 5. SPARROW model curves overlayed upon a portion of the cladding Bandgap map of the Bragg fiber examined in [5] and [23]: t 1=0.37µm, t 0=4.1µm, n 1=1.45 and n 0=1. The color scheme is the same as for Figs. 2 and 3. The bandgaps are labelled according to the nomenclature introduced in Section 4.1. Red circles: intersection points (P, Section 4.1) of the SPARROWcurves. Magenta circles: ñ=n 0 resonances (Eqs. 3 and 7). Green dashed curves: a specific bandgap’s half-order curves, the intersection of which form P c (Section 4.4), the green circle. The chosen gap is of order 〈m 1,m 0〉=〈3,1〉 with P c=(k c,ñ c)≈(5.34898µm -1,0.976641).

Note that, due to the analyticity of Eq. (9), the derivatives of kc with respect to all waveguide parameters (∂kc/∂ni,∂kc/∂ti, etc.) can be easily derived, but are omitted here for brevity. These derivatives are ideal for the direct calculation of fabrication tolerances in waveguide design, or sensitivities to core materials for sensing, for example.

4.3. Special cases

There are two important special cases of the SPARROW model: ñ=n 0 and ñ=0. It is easily shown that the former actually reduces to the large-core limit of the Duguay-ARROW model, since by setting i=1 and n˜m1=n0 in Eq. (6) we derive Eq. (3). These ñ=n 0 resonances are shown in Fig. 5 (magenta circles). Thus, the SPARROW model also explains why the Duguay-ARROW model in the large-core regime (Litchinitser et al. [20

20. N. M. Litchinitser, S. C. Dunn, B. Usner, B. J. Eggleton, T. P. White, R. C. McPhedran, and C. M. de Sterke, “Resonances in microstructured optical waveguides,” Opt. Express 11, 1243–1251 (2003). [CrossRef] [PubMed]

]) is typically independent of the thickness of the low-index region: all orders of the low-index slab dispersion curves (n˜m0) have an asymptote at ñ=n 0 and hence never intercept it; their resonant features can never appear on the n 0-light-line (the region of applicability of Eq. (3)).

The other special case, ñ=0 (the zero-line), is derived in much the same way. However, in this case, both high- and low-index curves intercept the zero-line. Setting ñi=0 we derive:

λmi=2nitimi
(10)

where i∈{1,0}. ñ=0 corresponds to rays normally incident to the layered cladding, similar to high-order modes within hollow-core solid-cladding Bragg fibers [30

30. A. Argyros, “Guided modes and loss in Bragg fibres,” Opt. Express 10, 1411–1417 (2002). [PubMed]

] which lie within 0<ñ≤1.

While useful in themselves, these two special cases reveal important information about the structure and topology of the Bandgap map, discussed in more detail in Section 4.5.

4.4. The central gap point

The concept of a central gap point can now be introduced. Note that the slab dispersion curves with half-orders (12,32,52,) fall mid-way between adjacent integer-order curves for both i={1,0} (when expressed in terms of k and ñ). An example is shown in Fig. 5 (green dotted curves). Thus, we define the central gap point as the intersection point of the half-order curves within a particular gap. In the nomenclature defined in Section 4.1, the curves producing the central gap point for the 〈m 1,m 0〉 gap are the m1=m112 and m0=m0+12 curves, so that the position of the gap’s center is given a modified form of Eq. (8) where m1m112 and m0m0+12, namely:

Pc=(kc,n˜c)=P(m112,m0+12)=(π1n12n02[(m112t1)2(m0+12t0)2],n12n02ηc21ηc2),
(11)

where ηc=(m112)t0(m0+12)t1. In fact it can be shown that Pc is entirely commensurate with the general antiresonance point (kc) of Eq. (9) such that:

kc(n˜c)=kc.
(12)

The proof for this is omitted for brevity, but only requires some careful manipulation and a demonstration that ñc always lies between the middle two intersection points P(m1+1,m0) and P(m1,m01) so that kc is always bound on either side by either the mp=m 1 and mq=m 1-1 curves or the mp=m 0 and mq=m 0+1 curves (as discussed in Section 4.2). An example based on the fiber cladding examined in [5

5. G. Vienne, Y. Xu, C. Jakobsen, H. Deyerl, J. Jensen, T. Sørensen, T. Hansen, Y. Huang, M. Terrel, R. Lee, N. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and Amnon Yariv, “Ultra-large bandwidth hollow-core guiding in all-silica Bragg fibers with nano-supports,” Opt. Express 12, 3500–3508 (2004). [CrossRef] [PubMed]

, 23

23. K. J. Rowland, S. Afshar V., and T. M. Monro, “Novel low-loss bandgaps in all-silica Bragg fibers,” J. Lightwave Technol. 26, 43–51 (2008). [CrossRef]

] is shown in Fig. 5 (green circle), where Pc=P(52,32)(5.34898μm1,0.976641) corresponding to the 〈3,1〉 gap.

Since the gaps bound by the ñ=n 0 and/or ñ=0 lines are open (above and below, respectively), this definition of the center point is insufficient when Pc reaches these lines, or where Pc simply doesn’t exist (as is the case for all 〈m 1,0〉 gaps). In these cases, we use the general antiresonance point (Eq. (9)) to define Pc=(kc(ñ′),ñ′) where ñ′=n 0 or 0 as appropriate. Thanks to Eq. (12), this gap center formalism is thus consistent for all bandgaps.

Also, similar to the discussion of Section 4.2, the derivatives of Pc with respect to all cladding parameters (∂kc/∂ni, ∂kc/∂ti, ∂ñc/∂ni and ∂ñc/∂ti) can also be easily derived, and would also be ideal for similar applications.

4.5. Bandgap topology and the bounding region

Here we derive another powerful feature of the SPARROW model. Equation (8) can be used to determine the number of bandgaps (or more precisely, bounding regions) that exist between a pair of adjacent high-index slab curves, n m1 (k), which have orders m 1-1 (left curve in the k-dimension) and m 1 (right curve). We define Dm1 as the domain enclosed by this curve pair and the ñ=n 0 and ñ=0 lines. We focus on the TE bandgaps here, since the TM gaps have the same topology save for the simple gap closure induced by the Brewster effect (Section 2.1, Figs. 2 and 3).

It is easy to see (e.g. Figs. 2 and 3) that the number of bounding regions, hence TE gaps, within Dm1 is one greater than the number of intersection points made by the low-index curves [n m0(k)] with the rightmost bounding curve [n m1(k)], excluding the n 0-light-line (m 0=0). To show this analytically, we must enforce upon the intersection point expression (Eq. (8)) the physical condition: P(m 1,m 0)∈ℝ2. By enforcing k∈ℝ, the square-root requires (m 1/t 1)2-(m 0/t 0)2>0⇒η>1 (where η is defined with Eq. (8)). This can be used to find an upper limit m 0<m 1(t 0/t 1), but a more strict limit is found by enforcing the second physical condition ñ∈ℝ: since η>1, 1-η 2<1 so that the numerator (within the square-root) must also be negative: n 2 1-n 2 0 η 2<1⇒n 1/n 0<η. This last inequality gives the most strict range physically imposed on m 0, namely:

m0<m1n0t0n1t1.
(13)

Thus, the maximum permissible order of an m 0-curve within Dm1 is:

m0max=floor{m1n0t0n1t1},
(14)

which is also the number of m 0-curves within Dm1(excluding the ñ={0,n 0} lines). The number of TE bandgaps within Dm1 is thus m max 0+1; the ‘+1’ accounting for the ever present 〈m 1,0⎚ gaps, bound above by the n 0-light-line, whose maximal bounding point P(m1,0) doesn’t contribute to m max 0 by definition, as discussed.

There is an exception to this analysis: where a maximal bounding point lies on the zero-line (ñ=0) such that the point exists but the associated gap does not (the bounding region becomes singular). In this case, the number of gaps within Dm1 is exactly equal to the number of intersection points on n˜m1 (k) (including the ñ=0 bounding point). Quantitatively, the condition for this behaviour can be deduced from the SPARROW model’s ñ=0 special case (Eq. (10)) by setting λm1=λm0, demonstrating that the above inequality (Eq. (13)) becomes an equality (i.e. the floor function of Eq. (14) becomes redundant).

We can thus express the total number of gaps bound within Dm1 as:

Nm1={m0max+1whenm0max<m1t0n0t1n1m0maxwhenm0max=m1t0n0t1n1
(15)

which depends only upon the cladding parameters and just one order parameter, m 1 (required to define Dm1).

The only difference in topology for the TM gaps over the TE gaps is that the Brewster-induced gap closure increases the number of gaps within Dm1 by 1 (i.e.Nm1+1), the position of the extra closure being ñ=n B (Section 2.1). The only exception is when n B coincides with a TE gap closure point, P(m1,m0), in which case the associated TM gap’s bounding region becomes singular (similar to the Pc at ñ=0 case above) and the number of gaps within Dm1 is identical to the TE case (i.e. Nm1).

The above expressions for the intersections points (Eq. (8)) and the number of gaps within a given domain Dm1 (Eq. (15)) explicitly define the topology for any given bandgap spectrum: the number of gaps (in a finite domain) and how they join together. Figure 3 gives an explicit example of how a bandgap spectrum topology can change with varying cladding parameters.

5. Concluding remarks

In this work we have demonstrated an intimate relationship between the bandgap and antiresonance pictures of light confinement in binary-layered-cladding waveguides on and below the low-cladding-index light-line, implying that Bragg fibers and integrated-ARROWs guide by fundamentally the same principles. This was done by developing an antiresonance model, the SPARROW model, which describes the resonances of an arbitrary waveguide’s cladding layers, independent of the core properties. The SPARROW model is a generalisation of the Archambault-ARROW model [9

9. J. L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11, 416–423 (1993). [CrossRef]

] in that we consider only the resonant properties of the cladding layers and not that of the core, permitting the direct comparison of the layers’ dispersive behaviour with the associated Bloch-wave bandgap spectra. Foremost, the model demonstrates that the cladding layer dispersion curves replicate the nontrivial structure and topology of the analogous 2-D Bloch-mode bandgap spectrum. By exploiting this, the model is also capable of quantitatively and analytically describing nontrivial features of such spectra. Among the most important features of the model derived were: a consistent nomenclature for arbitrary bandgap spectra (〈m 1,m 0〉); the approximate position of lowest core-mode confinement loss of any gap via the general antiresonance point (kc); the precise closure points of a given gap (P(m1,m0) and/or P(m11,m0+1)); the center of a gap in both ñ- and k-dimensions via the central gap point (Pc); and the number of bandgaps within a specific domain (e.g. Nm1 for TE), thus the bandgap spectrum topology; all via simple analytic expressions.

The SPARROWmodel is thus a powerful and simple tool for the spectral analysis and design of layered cladding dielectric waveguides with core refractive indices equal to or less than the lowest cladding index. Integrated-ARROWs have recently been demonstrated as ideal hollow-core waveguides for sensing and microfluidics [13

13. H. Schmidt and A. R. Hawkins, “Optofluidic waveguides: I. Concepts and implementations,” Microfluid. Nanofluid. 4, 3–16 (2008). [CrossRef] [PubMed]

]. However, our analysis of a liquid-core solid-cladding Bragg fiber implies that fibers have similar promise. These principles are also useful for achieving guidance in nonlinear liquids and gases, important for the development of novel nonlinear waveguides.

Acknowledgments

We acknowledge the Defence Science and Technology Organisation (DSTO) Australia for supporting work in the Centre of Expertise in Photonics. This work was supported under the Australian Research Council’s Discovery Projects funding scheme (project number LP0776947).

References and links

1.

P. Yeh, A. Yariv , and C. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. 67, 423–438 (1977). [CrossRef]

2.

Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, “Guiding optical light in air using an all-dielectric structure,” J. Lightwave Technol. 19, 2039–2041 (1999). [CrossRef]

3.

S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. Engeness, M. Soljačić, S. Jacobs, J. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express 9, 748–779 (2001). [CrossRef] [PubMed]

4.

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]

5.

G. Vienne, Y. Xu, C. Jakobsen, H. Deyerl, J. Jensen, T. Sørensen, T. Hansen, Y. Huang, M. Terrel, R. Lee, N. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and Amnon Yariv, “Ultra-large bandwidth hollow-core guiding in all-silica Bragg fibers with nano-supports,” Opt. Express 12, 3500–3508 (2004). [CrossRef] [PubMed]

6.

K. Kuriki, O. Shapira, S. D. Hart, G. Benoit, Y. Kuriki, J. F. Viens, M. Bayindir, J. D. Joannopoulos, and Y. Fink, “Hollow multilayer photonic bandgap fibers for NIR applications,” Opt. Express 12, 1510–1517 (2004). [CrossRef] [PubMed]

7.

T. Katagiri, Y. Matsuura, and M. Miyagi, “Photonic bandgap fiber with a silica core and multilayer dielectric cladding,” Opt. Lett. 29, 557–559 (2004). [CrossRef] [PubMed]

8.

T. Katagiri, Y. Matsuura, and M. Miyagi, “All-solid single-mode bragg fibers for compact fiber devices,” J. Lightwave Technol. 24, 4314–4318 (2006). [CrossRef]

9.

J. L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11, 416–423 (1993). [CrossRef]

10.

D. Yin, J.P. Barber, A. R. Hawkins, and H. Schmidt, “Waveguide loss optimization in hollow-core ARROW waveguides,” Opt. Express 13, 9331–9336 (2005). [CrossRef] [PubMed]

11.

D. Yin, H. Schmidt, J. P. Barber, E. J. Lunt, and A. R. Hawkins, “Optical characterisation of arch-shaped ARROW waveguides with liquid cores,” Opt. Express 13, 10564–10570 (2005). [CrossRef] [PubMed]

12.

F. Poli, M. Foroni, A. Cucinotta, and S. Selleri, “Spectral behavior of integrated antiresonant reflecting hollow-core waveguides,” J. Lightwave Technol. 25, 2599–2604 (2007). [CrossRef]

13.

H. Schmidt and A. R. Hawkins, “Optofluidic waveguides: I. Concepts and implementations,” Microfluid. Nanofluid. 4, 3–16 (2008). [CrossRef] [PubMed]

14.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998). [CrossRef] [PubMed]

15.

M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, “Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures,” Appl. Phys. Lett. 49, 13–15 (1986). [CrossRef]

16.

T. Baba and Y. Kokubun, “High efficiency light coupling from antiresonant reflecting optical waveguide to integrated photodetector using an antireflecting layer,” Appl. Opt. 29, 2781–2792 (1990). [CrossRef] [PubMed]

17.

T. Baba and Y. Kokubun, “Dispersion and radiations loss characteristics of antiresonant reflecting optical waveguides - numerical results and analytical expressions,” J. Quantum Electron. 28, 1689–1700 (1992). [CrossRef]

18.

A. K. Abeeluck, N. M. Litchinitser, C. Headley, and B. J. Eggleton, “Analysis of spectral characteristics of photonic bandgap waveguides,” Opt. Express 10, 1320–1333 (2002). [PubMed]

19.

N. M. Litchinitser, A. K. Abeeluck, C. Headley, and B. J. Eggleton, “Antiresonant reflecting photonic crystal optical waveguides,” Opt. Lett. 27, 1592–1594 (2002). [CrossRef]

20.

N. M. Litchinitser, S. C. Dunn, B. Usner, B. J. Eggleton, T. P. White, R. C. McPhedran, and C. M. de Sterke, “Resonances in microstructured optical waveguides,” Opt. Express 11, 1243–1251 (2003). [CrossRef] [PubMed]

21.

N. M. Litchinitser, S. C. Dunn, P. E. Steinvurzel, B. J. Eggleton, T. P. White, R. C. McPhedran, and C. M. de Sterke, “Application of an ARROW model for designing tunable photonic devices,” Opt. Express 12, 1540–1550 (2004). [CrossRef] [PubMed]

22.

D. Yin, H. Schmidt, J. P. Barber, and A. R. Hawkins, “Integrated ARROW waveguides with hollow cores,” Opt. Express 12, 2710–2715 (2004). [CrossRef] [PubMed]

23.

K. J. Rowland, S. Afshar V., and T. M. Monro, “Novel low-loss bandgaps in all-silica Bragg fibers,” J. Lightwave Technol. 26, 43–51 (2008). [CrossRef]

24.

F. Poli, M. Foroni, D. Giovanelli, A. Cucinotta, S. Selleri, J. B. Jensen, J. Læsgaard, A. Bjarklev, G. Vienne, C. Jakobsen, and J. Broeng, “Silica bridge impact on hollow-core Bragg fiber transmission properties,” Proceedings OFC 2007 (Anaheim, California), paper OML8 (2007).

25.

T. A. Birks, G. J. Pearce, and D. M. Bird, “Approximate band structure calculation for photonic bandgap fibres,” Opt. Express 14, 9483–9490 (2006). [CrossRef] [PubMed]

26.

J. J. Hu, G. Ren, P. Shum, X. Yu, G. Wang, and C. Lu, “Analytical method for band structure calculation of photonic crystal fibers filled with liquid,” Opt. Express 16, 6668–6674 (2008). [CrossRef] [PubMed]

27.

J. A. Buck, Fundamentals of Optical Fibers (2nd ed., John Wiley & Sons., Inc, 2004).

28.

M. Born and E. Wolf, Principles of Optics (7th ed., Cambridge University Press, 2002).

29.

I. M. Bassett and A. Argyros, “Elimination of polarization degeneracy in round waveguides,” Opt. Express 10, 1342–1346 (2002). [PubMed]

30.

A. Argyros, “Guided modes and loss in Bragg fibres,” Opt. Express 10, 1411–1417 (2002). [PubMed]

31.

J. P. Barber, D. B. Conkey, M. M. Smith, J. R. Lee, B. A. Peeni, Z. A. George, A. R. Hawkins, D. Yin, and H. Schmidt, “Hollow waveguides on planar substrates with selectable geometry cores,” Proceedings CLEO 2005 (San José, California), paper CTuD4 (2005).

32.

P. K. Tien, “Light waves in thin films and integrated optics,” Appl. Opt. 10, 2395–2413 (1971). [CrossRef] [PubMed]

33.

P. Viale, S. Février, F Gérôme, and H. Vilard, “Confinement loss computation in photonic crystal fibres using a novel perfectly matched layer design,” Proceedings of the COMSOL Multiphysics user’s conference (Paris, 2005).

34.

V. Finazzi, T. M. Monro, and D. J. Richardson, “Small-core silica holey fibers: nonlinearity and confinement loss trade-offs,” J. Opt. Soc. Am. B 20, 1427–1436 (2003). [CrossRef]

35.

J. Li and K. S. Chiang, “Disappearance of modes in planar Bragg waveguides,” Opt. Lett. 32, 2369–2371 (2007). [CrossRef] [PubMed]

OCIS Codes
(060.2280) Fiber optics and optical communications : Fiber design and fabrication
(130.0130) Integrated optics : Integrated optics
(230.4170) Optical devices : Multilayers
(230.7370) Optical devices : Waveguides
(060.4005) Fiber optics and optical communications : Microstructured fibers

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: August 29, 2008
Revised Manuscript: October 15, 2008
Manuscript Accepted: October 19, 2008
Published: October 21, 2008

Virtual Issues
Vol. 3, Iss. 12 Virtual Journal for Biomedical Optics

Citation
Kristopher J. Rowland, Shahraam Afshar V., and Tanya M. Monro, "Bandgaps and antiresonances in integrated-ARROWs and Bragg fibers; a simple model," Opt. Express 16, 17935-17951 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-22-17935


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References

  1. P. Yeh, A. Yariv and C. Hong, "Electromagnetic propagation in periodic stratified media. I. General theory," J. Opt. Soc. Am. 67, 423-438 (1977). [CrossRef]
  2. Y. Fink, D. J. Ripin, S. Fan, C. Chen, J. D. Joannopoulos, and E. L. Thomas, "Guiding optical light in air using an all-dielectric structure," J. Lightwave Technol. 19, 2039-2041 (1999). [CrossRef]
  3. S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. Engeness, M. Solja¡ci’c, S. Jacobs, J. Joannopoulos, and Y. Fink, "Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers," Opt. Express 9, 748-779 (2001). [CrossRef] [PubMed]
  4. 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]
  5. G. Vienne, Y. Xu, C. Jakobsen, H. Deyerl, J. Jensen, T. Sørensen, T. Hansen, Y. Huang, M. Terrel, R. Lee, N. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and A. Yariv, "Ultra-large bandwidth hollow-core guiding in all-silica Bragg fibers with nano-supports," Opt. Express 12, 3500-3508 (2004). [CrossRef] [PubMed]
  6. K. Kuriki, O. Shapira, S. D. Hart, G. Benoit, Y. Kuriki, J. F. Viens, M. Bayindir, J. D. Joannopoulos, and Y. Fink, "Hollow multilayer photonic bandgap fibers for NIR applications," Opt. Express 12, 1510-1517 (2004). [CrossRef] [PubMed]
  7. T. Katagiri, Y. Matsuura, and M. Miyagi, "Photonic bandgap fiber with a silica core and multilayer dielectric cladding," Opt. Lett. 29, 557-559 (2004). [CrossRef] [PubMed]
  8. T. Katagiri, Y. Matsuura, and M. Miyagi, "All-solid single-mode bragg fibers for compact fiber devices," J. Lightwave Technol. 24, 4314-4318 (2006). [CrossRef]
  9. J. L. Archambault, R. J. Black, S. Lacroix, and J. Bures, "Loss calculations for antiresonant waveguides," J. Lightwave Technol. 11, 416-423 (1993). [CrossRef]
  10. D. Yin, J. P. Barber, A. R. Hawkins, and H. Schmidt, "Waveguide loss optimization in hollow-core ARROW waveguides," Opt. Express 13, 9331-9336 (2005). [CrossRef] [PubMed]
  11. D. Yin, H. Schmidt, J. P. Barber, E. J. Lunt, and A. R. Hawkins, "Optical characterisation of arch-shaped ARROW waveguides with liquid cores," Opt. Express 13, 10564-10570 (2005). [CrossRef] [PubMed]
  12. F. Poli, M. Foroni, A. Cucinotta, and S. Selleri, "Spectral behavior of integrated antiresonant reflecting hollowcore waveguides," J. Lightwave Technol. 25, 2599-2604 (2007). [CrossRef]
  13. H. Schmidt and A. R. Hawkins, "Optofluidic waveguides: I. Concepts and implementations," Microfluid. Nanofluid. 4, 3-16 (2008). [CrossRef] [PubMed]
  14. Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, "A dielectric omnidirectional reflector," Science 282, 1679-1682 (1998). [CrossRef] [PubMed]
  15. M. A. Duguay, Y. Kokubun, T. L. Koch, and L. Pfeiffer, "Antiresonant reflecting optical waveguides in SiO2-Si multilayer structures," Appl. Phys. Lett. 49, 13-15 (1986). [CrossRef]
  16. T. Baba and Y. Kokubun, "High efficiency light coupling from antiresonant reflecting optical waveguide to integrated photodetector using an antireflecting layer," Appl. Opt. 29, 2781-2792 (1990). [CrossRef] [PubMed]
  17. T. Baba and Y. Kokubun, "Dispersion and radiations loss characteristics of antiresonant reflecting optical waveguides - numerical results and analytical expressions," J. Quantum Electron. 28, 1689-1700 (1992). [CrossRef]
  18. A. K. Abeeluck, N. M. Litchinitser, C. Headley, and B. J. Eggleton, "Analysis of spectral characteristics of photonic bandgap waveguides," Opt. Express 10, 1320-1333 (2002). [PubMed]
  19. N. M. Litchinitser, A. K. Abeeluck, C. Headley, and B. J. Eggleton, "Antiresonant reflecting photonic crystal optical waveguides," Opt. Lett. 27, 1592-1594 (2002). [CrossRef]
  20. N. M. Litchinitser, S. C. Dunn, B. Usner, B. J. Eggleton, T. P. White, R. C. McPhedran, and C. M. de Sterke, "Resonances in microstructured optical waveguides," Opt. Express 11, 1243-1251 (2003). [CrossRef] [PubMed]
  21. N. M. Litchinitser, S. C. Dunn, P. E. Steinvurzel, B. J. Eggleton, T. P. White, R. C. McPhedran, and C. M. de Sterke, "Application of an ARROW model for designing tunable photonic devices," Opt. Express 12, 1540-1550 (2004). [CrossRef] [PubMed]
  22. D. Yin, H. Schmidt, J. P. Barber, and A. R. Hawkins, "Integrated ARROW waveguides with hollow cores," Opt. Express 12, 2710-2715 (2004). [CrossRef] [PubMed]
  23. K. J. Rowland, S. Afshar V., and T. M. Monro, "Novel low-loss bandgaps in all-silica Bragg fibers," J. Lightwave Technol. 26, 43-51 (2008). [CrossRef]
  24. F. Poli, M. Foroni, D. Giovanelli, A. Cucinotta, S. Selleri, J. B. Jensen, J. Læsgaard, A. Bjarklev, G. Vienne, C. Jakobsen, and J. Broeng, "Silica bridge impact on hollow-core Bragg fiber transmission properties," Proceedings OFC 2007 (Anaheim, California), paper OML8 (2007).
  25. T. A. Birks, G. J. Pearce, and D. M. Bird, "Approximate band structure calculation for photonic bandgap fibres," Opt. Express 14, 9483-9490 (2006). [CrossRef] [PubMed]
  26. J. J. Hu, G. Ren, P. Shum, X. Yu, G. Wang, and C. Lu, "Analytical method for band structure calculation of photonic crystal fibers filled with liquid," Opt. Express 16, 6668-6674 (2008). [CrossRef] [PubMed]
  27. J. A. Buck, Fundamentals of Optical Fibers, 2nd ed., ( John Wiley & Sons, Inc., 2004).
  28. M. Born and E. Wolf, Principles of Optics, 7th ed., (Cambridge University Press, 2002).
  29. I. M. Bassett and A. Argyros, "Elimination of polarization degeneracy in round waveguides," Opt. Express 10, 1342-1346 (2002). [PubMed]
  30. A. Argyros, "Guided modes and loss in Bragg fibres," Opt. Express 10, 1411-1417 (2002). [PubMed]
  31. J. P. Barber, D. B. Conkey, M. M. Smith, J. R. Lee, B. A. Peeni, Z. A. George, A. R. Hawkins, D. Yin, and H. Schmidt, "Hollow waveguides on planar substrates with selectable geometry cores," Proceedings CLEO 2005 (San Jos’e, California), paper CTuD4 (2005).
  32. P. K. Tien, "Light waves in thin films and integrated optics," Appl. Opt. 10, 2395-2413 (1971). [CrossRef] [PubMed]
  33. P. Viale, S. Fevrier, F Gerome, and H. Vilard, "Confinement loss computation in photonic crystal fibres using a novel perfectly matched layer design," Proceedings of the COMSOL Multiphysics user’s conference (Paris, 2005).
  34. V. Finazzi, T. M. Monro, and D. J. Richardson, "Small-core silica holey fibers: nonlinearity and confinement loss trade-offs," J. Opt. Soc. Am. B 20, 1427-1436 (2003). [CrossRef]
  35. J. Li and K. S. Chiang, "Disappearance of modes in planar Bragg waveguides," Opt. Lett. 32, 2369-2371 (2007). [CrossRef] [PubMed]

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