1. Introduction

Microstructured optical fibers (MOFs), optical fibers with airholes running along the axial direction, have been the subject of intense research over the past decade. Photonic crystal fibers (PCFs) [1], MOFs in which the microstructure is periodic, have been of particular interest. PCFs with an airhole microstructure can guide by coherent Bragg scattering or total internal reflection, depending on whether they have an air core [2, 3], or a solid core [4]. Silica core PCFs with high index inclusions [5, 6] differ from the two former types of PCF in that many of their transmission properties can be explained in terms of the resonant properties of a single high index inclusion.

 

Fig. 1. (a) Schematic of ARROW-PCF geometry and (b) schematic of 2-D ARROW geometry with anti-resonant guided mode profile.

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High index inclusion PCFs, shown schematically in Fig. 1(a), can support modes confined to the low index core. Unlike solid core PCFs with air inclusions, which have broadband transmission, these fibers have discrete bands of high and low transmission. This is typically a signature of photonic band gap guidance, such as in air core PCFs. However, a simpler model [7] explains these spectral features in terms of anti-resonant reflecting optical waveguiding (ARROW), originally proposed for planar geometries [8, 9]. In the ARROW model, light is confined through anti-resonant scattering off the high index cylinders. It fundamentally differs from the band gap model in that the wavelengths of the transmission bands depend only on the cylinders’ index and diameter, but not on their geometric arrangement [10].

Previously [7, 11], we assumed that at long wavelengths, around the first anti-resonance of the cylinders, the ARROW model is not valid because the high index region cannot support a transverse standing wave. Also, since the fields of the long wavelength modes extend far into the high index microstructure and the width of the transmission bands appeared sensitive to the pitch of the microstructure, it was asserted that guidance at long wavelengths is due to a band gap effect, analogous to the air core fibers. However, these criteria are derived only for a 1-D planar model, and, as shown below, the ARROW-PCF model is in fact consistent with both the experimental and numerical data at long wavelengths. There have been two previous experimental investigations [5, 6] of PCFs with high index inclusions, both for geometries well within the short wavelength regime, and neither of which linked with theory in detail. In this paper we experimentally and numerically investigate transmission spectra for such structures at long wavelengths. In Section 2 we provide a review of the ARROW-PCF model, which is approximate but analytic. We then present our experimental results in Section 3, and in Section 4 we compare these with multipole and beam propagation simulations. Additional modeling of structures with nonuniformities and asymmetries are presented in the appendix.

2. ARROW model

We first consider the 1-D analog of the ARROW-PCF structure. This geometry, shown in Fig. 1(b), where z is the propagation direction, and where we include only the first ring of the high index microstructure, can be described as a low index core bounded on either side by 1-D Fabry-Pérot (F-P) resonators formed by the high index material. The resonances and anti-resonances of these inclusions are obtained from the phase relation (2π/λ)n_highdcos(θ ₂)=(m+σ)π, where λ is wavelength, m>0 is an integer, σ=0 or ½, and all other variables are defined as in Fig. 1(b). When this equation is satisfied for σ=0 (resonance), the high index layers are transparent [12] and there is no guidance in the core. At anti-resonance, σ=½, we get an enhanced reflection back to the core given by R _A-R=4R/(1+R)², where R is the Fresnel reflectance at a single planar interface. The schematic mode in Fig. 1(b) corresponds to the anti-resonant case. In the transmission spectrum these resonant effects manifest as wide pass bands with narrow high loss dips at the F-P resonances [7]. At long wavelengths, the diffraction angle θ in Fig. 1(b) increases, and so R, and hence R _A-R, decrease and the confinement in the core deteriorates. Note that for structures with more than one high index layer on either side of the core, the low index layers also support resonances and anti-resonances. These are less relevant when we extend our discussion to the 2-D cylindrical case and so their effect is ignored here. Also, due to the Brewster angle effect for TM polarized light at planar interfaces, 1-D ARROW waveguides often support only TE-like modes, whereas TM-like modes are strongly attenuated [8]. These strong polarization effects do not apply in the 2-D cylindrical case.

We extend the ARROW model to the 2-D cylindrical case. Litchinitser et al. [7] showed that the resonant and anti-resonant wavelengths can be approximated by

where the role of σ is reversed from the 1-D case, so σ=½ corresponds to resonance and σ=0 corresponds to anti-resonance. Equation (1) is independent of the period of the microstructure, consistent with the earlier discussion. It is derived by relating the resonances associated with scattering of a plane wave off a high index cylinder to the cutoffs of the guided modes of that cylinder. When light is incident on a dielectric cylinder at a glancing angle (θ ₁→90° in Fig. 1(b)), the condition for resonance, where the wave is not scattered, is equivalent to the cutoff of the highest order guided mode [13]. That is, in the PCF geometry, the analog for F-P resonance is modal cutoff [7]. If all cylinders are identical, the microstructure is transparent at the resonance wavelengths and no ARROW mode exists there. As we deviate from glancing incidence (e.g., at long wavelengths), the scattering resonances still exist but they broaden and do not exactly coincide with modal cutoff wavelengths. For an incident angle of ~70°, roughly corresponding to the largest deviation from glancing incidence in our experiments, the resonant scattering features are still well-defined, and are close enough to the modal cutoffs for our approximate model.

Figure 2 shows the mode structure of a single cylinder and the anti-guided modes of an ARROW-PCF. The solid black line indicates n_silica. The red curves underneath the line correspond to the real part of the effective index n_eff of the ARROW modes, while the blue curves above the line correspond to n_eff of the modes of a single high index cylinder. The vertical lines indicate the resonance wavelengths from Eq. (1). The ARROW modes were obtained using the multipole method [14,15] and the high index cylinder modes werecalculated directly from the Maxwell equations using a transfer matrix method. In the calculations, n_high corresponds to a commercial index fluid from Cargille Laboratories [16] used in one of ourexperiments with n_D=1.78, where nD is the index measured at 589.3 nm, n_low=n_silica, d=0.9 µm, and Λ=1.31 µm, where d and Λ are defined as in Fig. 1(a). When the dispersion in n_high and n_low are ignored, the resonances and anti-resonances are equally spaced in frequency. In fact, n_high is quite dispersive (e.g., for the n_D=1.78 fluid, the index varies from 1.778 at 600 nm to 1.734 at 1700 nm), so Eq. (1) must be solved self-consistently.

 

Fig. 2. Effective index n_eff versus λ for anti-guided modes of ARROW-PCF (red) and modes of a single high index cylinder (blue). Black line corresponds to n_silica. Regions of high loss for ARROW modes correspond to modal cutoffs of the high index cylinder modes. Resonances predicted by Eq. (1) are indicated by the vertical dashed lines.

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Equation (1) accurately predicts the regions where the ARROW mode does not exist in Fig. 2. In this example, the index contrast is quite large, and so approximate degeneracies of the cutoffs which occur for many modes of low index contrast structures do not occur here. It was recently shown [17] that, using a scalar treatment, the cutoff points of LP_νµ-type modes for ν=0,1 generally lead to spectrally wide regions of high loss whereas for ν>1 cutoff points, the high loss regions become spectrally narrow and for ν>4, the cutoff does not even contribute to loss in the ARROW mode. The polarization splitting due to the high index contrast is not expected to have an effect on the loss spectra (e.g., near the m=3 line in Fig. 2, TE₀₂/TM₀₂, and HE₂₂ cutoffs equally contribute to the resonant loss, even though their exact cutoff points vary by ~7 nm, since the high loss region is wider than this). The dispersion curves of the high index cylinder modes track along the n_silica line near cutoff, so it is difficult to see the exact cutoff points in Fig. 2, but we have confirmed that the cutoffs of the ν<3 modes agree with Eq. (1) to within a few percent.

3. Experiment

Figure 3(a) shows an SEM micrograph of the core region of the all-silica fiber, manufactured by Crystal Fibre A/S, which was used in the experiments. The fiber has an average hole diameter of d=0.90 µm, pitch of Λ=1.31 µm, core diameter of 1.7 µm, and 10 rings of holes surrounding the core. Figure 3(b) shows a schematic of the experiment. A short length (~4–5 cm) of fiber was cleaved on both sides and mounted between two XYZ microblock stages. Light was launched into the PCF and collected at the output by butt coupling two cleaved ends of high NA fiber (UHNA4 from Nufern, NA=0.36, core diameter=2.5 µm), each of which was spliced to a connectorized length of standard single mode fiber (SSMF); the collected light was measured using an optical spectrum analyzer (Agilent 86140B). Two different light sources were used: an incandescent white light source (ANDO AQ4303C) was used to characterize transmission over the 600–1250 nm bandwidth, and a broadband infrared source consisting of four edge-emitting LEDs (Agilent EELED 83437A) was used for the 1250–1700 nm bandwidth. We investigated two different ARROW-PCF structures using the same fiber but with different index fluids. Transmission through the PCF was first measured without high index material in the holes, and then a short length (~1–2 mm) of high index fluid (Cargille Laboratories refractive index fluid n_D=1.78 or 1.60) was infused into the fiber and the spectrum remeasured.

 

Fig. 3. (a) SEM micrograph of fiber used, and (b) block diagram of experiment

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The fluid-filled region within the MOF did not begin abruptly, but rather there were sections of non-uniform filling at either end of the fluid region. An example is shown by the optical microscope image in Fig. 4. The fluid profiles were measured by taking a series of images with an optical microscope at 500x magnification. For the n_D=1.78 sample, there was a uniformly filled region of 1.46 mm, bounded on either side by regions of axially varying non-uniform filling 0.25 mm and 0.73 mm in length, respectively, giving a total sample length of 2.44 mm. For the n=1.60 sample, the uniform region was 0.36 mm and bounded on either side by 0.33 mm long non-uniform sections, giving a total sample length of 1.02 mm.

 

Fig. 4. Optical microscope image showing axial variations in the relative position and lengths of the fluid plugs

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Figure 5 shows the measured transmission relative to that of the unfilled fiber. The gray line represents the noise floor of the OSA. For both samples, the fluid region drew back from the fiber endface by >100 µm, so any light which did not couple from the high NA fiber into the core mode of the airhole PCF leaked out before it reached the fluid. That is, Fig. 5 shows the combined effects of coupling loss between the core modes of the filled and unfilled fiber and the propagation loss in the filled fiber. The resonances and anti-resonances predicted by Eq. (1) are indicated by dashed and dotted vertical lines, respectively, and with the exception of the m=3 anti-resonance in Fig. 5(b), they line up quite well with the experimental data. Note that the loss is quite large for the m=1 transmission band, consistent with the discussion in Section 2.

 

Fig. 5. Measured transmission spectra for (a) n_D=1.60 and (b) n_D=1.78. Resonances and anti-resonances predicted by Eq. (1) are indicated by dashed and dotted lines, respectively. In (a), the m=2 anti-resonance lies just to the right of the graph at λ=592 nm (506.75 THz); in (b) the m=1 anti-resonance lies just to the left of the graph at λ=1735 nm (172.91 THz). Results from multipole and beam propagation simulations (see Sec. 4) are also shown.

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4. Discussion

The analysis from Section 2 explains where the ARROW modes do and do not exist and qualitatively indicates why they become lossier at long wavelengths, but does not give any further insight into what was measured experimentally. There are two loss mechanisms in our ARROW-PCF structure. The anti-guided modes of ARROWs are inherently leaky, and so there is a propagation loss given by the imaginary part of n_eff. However, since the fluid regions of the fiber are bounded on either side by regions with empty airholes, there are also coupling losses when light passes from one region to another. As mentioned in Section 3, coupling losses between the high NA input fiber and the PCF were normalized out of the transmission measurement. The material loss of the fluids used in the experiments is much larger than that of silica, but still quite low for our sample lengths (<0.5 dB/cm [16]). Inclusion of the material loss in the simulations had a negligible effect on the results.

We use multipole [14,15] and beam propagation [18] simulations to model the propagation and coupling losses. This section describes simulations of an idealized structure which is axially uniform and transversely symmetric. The effects of the non-uniform filling of the airholes (Fig. 4), and of nonuniformities in the fiber cross section (Fig 3(a)) are weak, as demonstrated in the appendix. In all simulations, we ignored the effects of the air/silica interface at the outer fiber cladding by using either an infinite silica matrix in the multipole calculations, or transparent boundary conditions in beam propagation.

Multipole simulations predict the pure propagation loss of the ARROW mode. In terms of the coupling loss, when the core mode of the empty fiber reaches the fluid region, it can couple either to the ARROW mode or to any number of supermodes of the high index microstructure, which are essentially linear combinations of the individual modes of each high index cylinder. At short wavelengths, higher order ARROW modes also exist, but these are leakier than the fundamental ARROW mode and more difficult to excite, and so we ignore their effect. Coupling losses were obtained using the beam propagation method. In these simulations, the launch field was the fundamental mode of the empty fiber, which was obtained using the multipole method, and we monitored the overlap P(z) of the field in the fluid filled fiber with the launch field as a function of propagation distance, with

where E _launch and E(z) are the major components of the launch field and field after distance z, respectively, and the integral is evaluated over the transverse simulation window. We ignore higher order modes in the unfilled fiber, as the almost all power is in the fundamental mode.

 

Fig. 6. Transmission spectra of beam propagation simulations for 10 µm (red) and 500 µm (black) of propagation for (a) n_D=1.60 and (b) n_D=1.78.

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Figure 5 compares the measured transmission spectra with results from multipole and beam propagation simulations, where the multipole data assume a propagation length of 1.02 mm and 2.44 mm in Fig. 5(a) and Fig. 5(b), respectively, and the beam propagation data assume a propagation length of 0.5 mm. Attempts to simulate longer lengths with beam propagation led to numerical instabilities. The multipole method predicts essentially zero loss in the m=2 and m=3 bands. The beam propagation method is closer to experiment in these bands since it includes coupling losses. This is shown in Fig. 6, where we plot the transmission of the ARROW-PCF after propagating 10 µm and 500 µm. The transmission in the center of the m=2 and m=3 bands does not change much, consistent with the multipole result that the propagation loss is small over these lengths.

The behavior in the m=1 band is somewhat different. The propagation loss of the ARROW mode predicted by the multipole method is very large, considering the short propagation length, especially in the n_D=1.78 case. Here, the mode fields extend throughout the microstructure. According to the multipole calculation of a 4 ring structure with n_D=1.78, the peak of the m=1 band occurs at at λ=1430 nm and Im(n_eff)=2.85×10^-3, whereas for a 10 ring structure, the peak shifts to at λ=1620 nm and Im(n_eff)=3.18×10^-4, decreasing by only a factor of 9. That is, adding six rings to the ARROW-PCF structure does little to improve the propagation loss, indicating that the anti-resonant reflectance is weak. Contrast this with the m=2 band where for 4 rings, the peak is at λ=864 nm, whereas for the 10 ring structure the peak shifts to at λ=874 nm and the propagation loss improves by a factor of 7,000. There is also a large difference in the effective area (A_eff) of the modes in the m=1 and higher order bands: in the 10 ring structure, A_eff=3.1 µm² and 2.0 µm² at the transmission peak of the m=2 and m=3 bands, respectively, whereas at the peak of the m=1 band, A_eff=15.0 µm², consistent with poor mode confinement. For n_D=1.60, the mode in the m=1 band is not nearly so leaky because the wavelength is shorter relative to the core, increasing the incident angle and thus the Fresnel coefficient, enhancing R _A-R as per the argument in Section 2.

Since A_eff is much larger in the m=1 band than in the higher order bands, the coupling losses between the empty and fluid filled regions of the fiber are also much greater. A_eff of the empty fiber mode only changes from 1.6 µm² at at λ=600 nm to 3.0 µm² at λ=1700 nm, so while this mode is well-matched to the ARROW mode in the higher order bands, in the m=1 band it also excites supermodes of the high index microstructure, many of which have field components in the fiber core. This effect can be seen in Fig. 7, which shows the same data as in Fig. 6, but as a contour plot of P(z) versus wavelength and propagation distance. The oscillations indicate mode beating, and suggest that many modes were excited at the launch. The beat length is short compared to the propagation length (~7–10 µm in Fig. 7(b)), indicating a large difference in effective index between the modes. In Fig. 7(b), the beat length is consistent with beating between ARROW modes with n_eff ~1.4 and high index modes with n_eff ~1.5–1.6. Since these supermodes are guided and can (weakly) couple back to the core at the end of the fluid region, they may explain why the measured transmission in Fig. 5(b) actually exceeds the propagation loss given by the multipole results.

 

Fig. 7. Contour plots of P(z) (dB scale) versus wavelength and distance in a 10 ring ARROW-PCF with (a) n_D=1.60 and (b) n_D=1.78. Black regions indicate P(z) <-75 dB

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Fig. 8. Infrared image of light coupled out of the core for (a) an unfilled fiber and (b) a fluid filled fiber with n_D=1.78.

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The existence of coupling to the supermodes was verified experimentally. Light from an erbium doped fiber ASE source at 1550 nm was launched into the empty fiber end, and the output face was imaged through a 16X microscope objective lens onto a vidicon infrared camera. Fig. 8(a) shows the imaged output in the fiber without fluid, while Fig. 8(b) shows the output when the fluid with n_D=1.78 is within 30 µm of the output face. In Fig. 8(a) the fiber is illuminated from the side so that the outer cladding can be seen and the input power is reduced so that the high-intensity core mode does not saturate the photodetector. In Fig. 8(b), there is no side illumination; all light originated in the PCF core. Here the light is distributed over the entire high index microstructure, indicating strong coupling between the core mode of the empty fiber and the supermodes of the fluid region.

Finally, the losses predicted by the beam propagation simulations in Fig. 6 differ from the experimentally measured loss in the m=1 band by approximately 10 dB for n_D=1.60 and by 10–20 dB for n_D=1.78. Given that the experimental lengths are longer than the simulated structures, this is an underestimation of the real discrepancy. One possible reason for this difference, aside from numerical error, is that the effect of the finite outer cladding not included in the simulations. Any light escaping the microstructure is lost in beam propagation, but it can be totally internally reflected at the cladding-air interface and may couple back into the core. This process is weak, but not inconsistent with measured losses of 30–35 dB. For the m=2 and m=3 bands, simulation and experiment agree quite well.

5. Conclusion

We have experimentally measured the transmission through PCF with high-index inclusions, for the lowest order bands. We also modeled this structure using the multipole and beam propagation methods. We show that the ARROW model is consistent with both experiment and numerical results, even at long wavelengths. The measured loss in the m>1 bands primarily arise from coupling loss between the fluid-filled and empty regions of the PCF, whereas in the m=1 band it is due to a combination of coupling loss and propagation loss.

Appendix — nonuniformity and asymmetry

Here we consider independently the effects of variations in the relative lengths of the fluid plugs and disorder in the size and position of the airholes. All simulations were done using beam propagation. For the axially varying fluid profile simulation, the structure we modeled approximates the measured fluid profile described in Sec. 3. Since the propagation lengths are long, we simulated a 4 ring structure, both to reduce simulation time and to stave off the onset of numerical instability. The results are shown in Fig. 9. Here, the mode beating in the m>1 bands is weaker than in Fig. 7. This is reasonable, as there are regions of hundreds of micrometers where the transverse index profile is changing and so there is no well-defined mode spectrum for the whole structure. The axial variations have little effect on the loss in the m>1 bands. This is reasonable because the uniformly filled fluid region represents a worst case scenario for loss; the air-silica interfaces which occur when some of the holes are not filled actually provide better light confinement than the fluid-silica interface. By contrast, in the m=1 band we initially see a broad, low loss region which gradually becomes lossier and narrower as more fluid-filled holes appear. For n_D=1.60, some of the light propagates to the end of the structure, whereas for n_D=1.78, everything is lost, mainly because the propagation length is more than twice as long. Even so, in both cases the loss is less than the equivalent structure with uniform filling. Though these effects cannot be readily discerned in the experimental spectra, these results show that asymmetry in the fluid plug length and position should not strongly contribute to the measured loss.

 

Fig. 9. Contour plots of P(z) (dB scale) as a function of wavelength and distance in an axially varying four ring ARROW-PCF with (a) n_D=1.60, uniform fluid length=0.36 mm and (b) n_D=1.78, uniform fluid length=1.46 mm. Black regions indicate P(z) <-75 dB

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Finally, we investigate the effect of structural disorder in the fiber microstructure. Using the digitized image of the fiber cross-section in Fig. 3(a), we generated an index profile for use in the beam propagation simulations which matches the actual fiber. Figure 10 shows P(z=500 µm) for the symmetric structure and the SEM-based structure together with the experimental results. The structural disorder narrows all wavelength bands. We attribute this primarily to disorder in the hole size, since it directly affects the resonance wavelengths. For the symmetric structure, wavelengths near the resonances are very weakly guided, and so any perturbations in the hole size which shifts a local resonance towards a transmission band leads to greater propagation loss at the band edge. The converse statement is also true, i.e. shifting the resonance away from the band will mitigate loss at the band edge, but a chain is only as strong as its weakest link, so only narrowing is observed. Since A_eff is large at the band edge [17], perturbations in the hole size also affect coupling loss, and this likely explains why the measured transmission bands are much narrower than the multipole result in Fig. 5. The narrowing in the beam propagation results improves the agreement with experiment in the m=2 band for n_D=1.60, but actually worsens the agreement in all other cases. This may to due to systemic errors in the SEM measurement, since variations of a few percent in the hole diameter lead to variations of tens of nanometers in the resonances.

 

Fig. 10. Experimental transmission spectra and beam propagation simulations assuming transverse symmetry or using an SEM-based index profile for (a) n_D=1.6 and (b) n_D=1.78.

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Acknowledgments

The authors thank Audrey Lobo for her contributions to Fig. 2 and Adam Sikorski for his invaluable help with the SEM sample preparation. This work was produced with the assistance of the Australian Research Council under the ARC Centres of Excellence program. CUDOS (the Centre for Ultrahigh bandwidth Devices for Optical Systems) is an ARC Centre of Excellence

References and links

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

2. R.F. Cregan, B.J. Mangan, J.C. Knight, T.A. Birks, P.S.J. Russell, P.J. Roberts, and D.C. Allan, “Single-mode photonic band gap guidance of light in air,” Science 285, 1537–1539 (1999). [CrossRef]   [PubMed]  

3. C.M. Smith, N. Venkataraman, M.T. Gallagher, D. Muller, J.A. West, N.F. Borelli, and K.W. Koch, “Low-loss hollow-core silica/air photonic bandgap fibre” Nature 424, 657–659 (2003). [CrossRef]   [PubMed]  

4. A. Bjarklev, J. Broeng, and A.S. Bjarklev, Photonic Crystal Fibers (Kluwer Academic Publishers, Boston, 2000).

5. R.T. Bise, R.S. Windeler, K.S. Kranz, C. Kerbage, B.J. Eggleton, and D.J. Trevor, “Tunable photonic band gap fiber,” in Optical Fiber Communications Conference, Postconference Edition, vol. 70 of OSA Trends in Optics and Photonics Series Technical Digest (Optical Society of America, Washington D.C.2002) pp. 4f66–468.

6. T.T. Larsen, A. Bjarklev, and D.S. Hermann, “Optical devices based on liquid crystal photonics bandgap fibers,” Opt. Express 11, 2589–2596 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-20-2589 [CrossRef]   [PubMed]  

7. 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), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-10-1243 [CrossRef]   [PubMed]  

8. 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]  

9. T. Baba and Y. Kokubun, “Dispersion and Radiation Loss Characteristics of Antiresonant Reflecting Optical Waveguides - Numerical Results and Analytical Expressions,” IEEE J. Quantum Electron. 28, 1689–1700 (1992). [CrossRef]  

10. T.P. White, R.C. McPhedran, C.M. de Sterke, N.M. Litchinitser, and B.J. Eggleton, “Resonance and scattering in microstructured optical fibers,” Opt. Lett. 27, 1977–1979 (2002). [CrossRef]  

11. 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), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1320 [CrossRef]   [PubMed]  

12. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon Press Ltd., Oxford, 1980), Sec. 7.6

13. A.C. Lind and J.M. Greenberg, “Electromagnetic Scattering by Obliquely Oriented Cylinders,” J. Appl. Phys. 37, 3195–3203 (1966). [CrossRef]  

14. T.P. White, B.T. Kuhlmey, R.C. McPhedran, D. Maystre, G. Renversez, C.M. de Sterke, and L.C. Botten, “Multipole method for microstructured optical fibers. I. Formulation,” J. Opt. Soc. Am. B 19, 2322–2330 (2002). [CrossRef]  

15. B.T. Kuhlmey, T.P. White, D. Maystre, G. Renversez, L.C. Botten, C.M. de Sterke, and R.C. McPhedran, “Multipole method for microstructured optical fibers. II. Implementation and results,” J. Opt. Soc. Am. B 19, 2331–2340 (2002). [CrossRef]  

16. Cargille Laboratories, Inc., Cedar Grove, NJ, USA, Data sheets: Refractive Index Liquid Series A, n_D=1.60, Refractive Index Liquid Series M, n_D=1.78

17. J. Laegsgaard, “Gap formation and guided modes in photonic bandgap fibres with high-index rods,” J. Opt. A 6, 798–804 (2004). [CrossRef]  

18. R. Scarmozzino, A. Gopinath, R. Pregla, and S. Helfert, “Numerical techniques for modeling guided-wave photonic devices,” IEEE J. Sel. Top. Quantum Electron. 6, 150–162 (2000). [CrossRef]