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Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 19 — Sep. 18, 2006
  • pp: 8797–8811
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Single scatterer Fano resonances in solid core photonic band gap fibers

P. Steinvurzel, C. Martijn de Sterke, M. J. Steel, B. T. Kuhlmey, and B. J. Eggleton  »View Author Affiliations


Optics Express, Vol. 14, Issue 19, pp. 8797-8811 (2006)
http://dx.doi.org/10.1364/OE.14.008797


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Abstract

Solid core photonic bandgap fibers (SC-PBGFs) consisting of an array of high index cylinders in a low index background and a low index defect core have been treated as a cylindrical analog of the planar anti-resonant reflecting optical waveguide (ARROW). We consider a limiting case of this model in which the cylinders in the SC-PBGF cladding are widely spaced apart, so that the SC-PBGF modal loss characteristics should resemble the antiresonant scattering properties of a single cylinder. We find that for glancing incidence, the single cylinder scattering resonances are Fano resonances, and these Fano resonances do in fact appear in the loss spectra of SC-PBGFs. We apply our analysis to enhance the core design of SC-PBGFs, specifically with an eye towards improving the mode confinement in the fundamental bandgap.

© 2006 Optical Society of America

1. Introduction

Fig. 1. Schematic of (a) single layer planar ARROW with antiresonant mode in low index core, (b) SC-PGBF index profile, and (c) transmission loss through a SC-PBGF. Vertical lines correspond to cutoff frequencies of the vector modes of the high index cylinders: red=TE/TM0,p,HE2,p, blue=HE1,p+1, and green=EH1,p;HE3,p

However, since SC-PBGFs are two dimensional structures, there are many ways in which they differ from one dimensional ARROWs. For example, the low frequency transmission bands of multi-layer planar ARROWs depend on the periodicity of the layers and so confinement in the low frequency bands of SC-PBGFs was presumed to arise from Bragg-like effects as well [10

10. 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]

12

12. 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]

]; it was later demonstrated that in the cylindrical geometry, these are still strongly dominated by the scattering resonances and antiresonances of the single cylinder [8

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

, 17

17. A. Argyros, T. A. Birks, S. G. Leon-Saval, C. M. B. Cordeiro, and P. St. J. Russell, “Guidance properties of low-contrast photonic bandgap fibres,” Opt. Express 13, 2503–2511 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-7-2503. [CrossRef] [PubMed]

, 18

18. P. Steinvurzel, B. T. Kuhlmey, T. P. White, M. J. Steel, C. M. de Sterke, and B. J. Eggleton, “Long wavelength anti-resonant guidance in high index inclusion microstructured fibers,” Opt. Express 12, 5424–5433 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5424. [CrossRef] [PubMed]

]. This is not to say, however, that the cylinder spacing is unimportant. In fact, as a number of recent papers have shown, the ratio between the cylinder diameter d and the pitch of the cladding microstructure Λ has a strong impact on a number of SC-PBGF properties, such as bend loss [19

19. T. A. Birks, F. Luan, G. J. Pearce, A. Wang, J. C. Knight, and D. M. Bird, “Bend loss in all-solid bandgap fibres,” Opt. Express 14, 5688–5698 (2006), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-14-12-5688. [CrossRef] [PubMed]

] and the influence of cylinder modes with high azimuthal order [8

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

, 20

20. G. Renversez, P. Boyer, and A. Sagrini, “Antiresonant reflecting optical waveguide microstructured fibers revisited: a new analysis based on leaky mode coupling,” Opt. Express 14, 5682–5687 (2006), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-14-12-5682. [CrossRef] [PubMed]

]. This is explained in terms of the field distributions of different types of cylinder modes and the degree to which they couple to adjacent cylinders.

2. Scattering formalism

The scattering of an electromagnetic plane wave by a dielectric cylinder at conical incidence is a well-studied problem, originally solved over 50 years ago [21

21. J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Canadian J. Phys. 33, 189–195 (1955). [CrossRef]

], and we present only the basics of the formalism; we generally follow the treatment given in Ref. [22

22. C. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1998). [CrossRef]

]. Consider a plane wave scattered by an infinite dielectric cylinder as in Fig. 2, where ϑ0 is the angle of conical incidence [23

23. Note that we define ϑ0 such that it approaches zero for near glancing incidence, which we feel is the natural choice in the current context; in most of the literature on scattering by cylinders [16,21,22], however, the conicity angle approaches π/2 for glancing incidence, and our definition is equivalent to π/2-α in the earlier references.

]. In this geometry, all fields can be extracted from the E z and H z components of E and H, which in turn are a sum of the incoming and scattered fields

Ez(r,φ,z,t)=[ψiE(r,φ)+ψsE(r,φ)]ei(βzωt)+c.c.,
(1)

where β=n low k 0 cosϑ0, k 0=2π/λ and ω=ck 0. A similar expression holds for H z . The incoming and scattered fields can be expressed as Fourier Bessel series

ψif(r,φ)=q=AqfJq(kr)eiqφ,
(2a)
ψsf(r,φ)=q=BqfHq(1)(kr)eiqφ,
(2b)

where f denotes either E or H, k =n low k 0 sinϑ0, J q and Hq(1) are the q th order Bessel function and Hankel function of the first kind, respectively, and Aqf and Bqf are complex-valued expansion coefficients. The cylindrical harmonics in Eq. (2) are chosen to satisfy the incoming and outgoing wave conditions of the incident and scattered fields. The internal fields have the same form as Eq. (2a). The A q coefficients are determined by the condition that ψif is a plane wave

AqE=E0sin(δ0)sin(ϑ0)eiq(π2φ0),
(3a)
AqH=nlowE0Z0cos(δ0)sin(ϑ0)eiq(π2φ0),
(3b)

where Z0 is the vacuum impedance, φ 0 is the angle of incidence with respect to the origin of the coordinate axes, which we set to zero, and δ 0 defines polarization angle of the incoming field (for φ 0=0, δ 0 is the angle between E and the y-axis in Fig. 2), which we set to zero as well; that is, the cylinder axis always lies in the plane defined by H and k (H polarization [16

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

]), though our scattering calculations give nearly identical results for arbitrary polarization. The Bqf coefficients, which determine the scattered field, are found by applying boundary conditions at the cylinder surface. For simplicity, we define BqH to include the Z 0 terms which typically appear in the expressions for the scattering cross section given below.

Fig. 2. Geometry of plane wave scattering by a dielectric cylinder at conical incidence. The variable φ is defined such that regions of forward or backward scattering are given by cos(φ)>0 or cos(φ)<0, respectively. The polarization is chosen such that the cylinder axis lies in the plane defined by H and k. In the related SC-PBGF geometry, the region below the cylinder corresponds to the SC-PBGF core.

σd(φ)=2πE02ksin2ϑ0(gE(φ)2+gH(φ)2),
(4)

with

gf=q=Bqfei[q(φπ2)π4].
(5)

We then use σd(φ) to obtain the asymmetry parameter g, the average cosine of the scattering angle

g=1σsc02πσd(φ)cosφdφ;
(6)

g>0 means more forward scattering and g<0 means more backward scattering.

3. Single scatterer and SC-PBGF loss spectrum

In Fig. 3, we plot σsc and g for ϑ0=0.025 rad≈1.43°. We plot the data in this and all subsequent figures as a function of the normalized frequency V=πd(nhigh2-nlow2)1/2/λ so that the band edges or scattering resonances always line up along the same x-axis points independent of d or index contrast; in this paper we set n high=1.65, n low=1.45, and d=0.2µm. In Fig. 3, the red vertical lines correspond to TE0,p and HE2,p cutoff frequencies (for this polarization, B0E is always zero, so the TM0,p modes are never excited) and the blue vertical lines correspond to the HE1,p+1=EH1,p cutoff frequencies. The results shown are obtained using three Fourier-Bessel orders, which is all that are required for convergence for this value of ϑ0 (we have checked up to nine orders); if we use four orders, we do find some additional very fine features associated with the HE3,p modes, though they are not detectable on the scale shown in Fig. 3, so we do not include the green vertical lines associated with this resonance as in Figs. 1(c). For larger incident angles, we find resonances associated with higher order modes, and more Fourier-Bessel orders are required. We note that in terms of the transverse fields, the blue lines correspond to monopole resonances of the cylinder, where the scattered field is azimuthally symmetric and essentially in phase with the incident field, so σsc is effectively zero. That is, these resonances more or less behave like the resonances of the planar slab in that the high index region becomes transparent. The red lines, however, correspond to dipole resonances of the cylinder, and so σsc increases. Since the index contrast is sufficiently large to remove the degeneracy of the individual vector modes, we do observe some splitting between the TE0,p and HE2,p modes, though the splitting is very small as the frequency increases.

Fig. 3. Im(n eff) versus V for the first 6 bands of a SC-PBGF with d/Λ=0.15 (upper panel), scattering cross section σsc (middle panel) and asymmetry factor g (lower panel) for ϑ0=0.025 rad; the scattering data are calculated using 3 Fourier-Bessel orders. Red and blue vertical lines correspond to TE0,p/HE2,p and HE1,p+1 cutoff frequencies, respectively.

These features are linked to the behavior of g and σsc, where low Im(n eff) should correlate with strong backwards scattering (high σsc and negative g). For the frequency ranges corresponding to the even order SC-PBGF transmission bands, g is larger (more forward scattering) and concave up, and σsc is lower relative to the adjacent odd bands (weaker overall scattering); this combination of g and σsc correlates with high loss, concave up bands in the SC-PBGF geometry. In the frequency ranges corresponding to the odd order bands, σsc is larger and g is lower relative to the adjacent even bands, and g is concave down, leading to the unexpected result that in the odd order SC-PBGF bands, the transmission loss is lower at the transmission band edge as compared to the band center.

4. Single scatterer Fano resonance

g=b0Hb1Hsin(ϕ0Hϕ1H)+b1Hb2Hsin(ϕ1Hϕ2H)+b1Eb2Esin(ϕ1Eϕ2E)12(b0H)2+(b1H)2+(b2H)2+(b1E)2+(b2E)2,
(7)

(again, B0E=0 for this polarization, and since φ 0=0, we have BqH =BqH and BqE =-BqE ). The dominant term in Eq. (7) is the one which describes the interference between B0H and B1H (first term in the numerator). In Fig. 4 we plot the evolution of B0H and B1H in the complex plane as a function of V; the right panel shows the projection of these terms on the real line and the bottom panel shows the projection on the complex plane. The loops where the phase is rapidly varying correspond to resonances of the individual coefficients. As expected, the B0H loops correspond to the TE0,p cutoff frequencies and the B1H loops correspond to the HE1,p+1 cutoffs (B1H actually shows two resonances; the second loop corresponds to the EH1,p cutoff. Both loops widen with respect to the V axis as V decreases such that they merge into one resonance for the first B1H resonance near V=3.83).

Figure 5 shows the sines of the individual phase terms of Eq. (7), and these are almost identical (indeed, the red curve is almost completely obscured by the green curve). This indicates that B2H and B0H are always out of phase with the B1H term by the same amount and thus in phase with each other. The phase flips shown in Fig. 5 are what give rise to the asymmetric resonances of g in Fig. 3, and by extension, the corners in the odd order bands of the SC-PBGF spectrum. Returning to our interpretation of the Bqf coefficients in terms of the leaky cylinder modes, the asymmetric resonances can be viewed as the combination of a resonance associated with one leaky mode approaching cutoff and its interference with another, off-resonance leaky mode; as shown in Fig. 4, the resonant and off-resonant modes alternate between TE0,p=HE2,p and HE1,p+1/EH1,p.

Fig. 4. 3-D line plots of B0H (red) and B1H (blue) as a function of V, with projections on the real axis (right panel) and complex plane (bottom panel) shown.

Fig. 5. Sine of the phase-dependent terms which contribute to g using same simulation parameters as in Fig. 3 and 4.

One may ask how the low d/Λ SC-PBGF can be interpreted in terms of the bandgap model of SC-PBGFs. In this model, the cladding is treated as an infinite lattice which supports bands of Bloch modes, which in turn are described as coupled states of the cylinder modes, and guidance in the defect core is only allowed in the gaps between these bands of Bloch modes [2

2. T. T. Larsen, A. Bjarklev, D. S. Hermann, and J. Broeng, “Optical devices based on liquid crystal photonic bandgap fibres,” Opt. Express 11, 2589–2896 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-20-2589. [CrossRef] [PubMed]

, 3

3. F. Luan, A. K. George, T. D. Hedley, G. J. Pearce, D. M. Bird, J. C. Knight, and P. St. J. Russell, “All-solid photonic bandgap fiber,” Opt. Lett. 29, 2369–2371 (2004). [CrossRef] [PubMed]

, 8

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

, 19

19. T. A. Birks, F. Luan, G. J. Pearce, A. Wang, J. C. Knight, and D. M. Bird, “Bend loss in all-solid bandgap fibres,” Opt. Express 14, 5688–5698 (2006), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-14-12-5688. [CrossRef] [PubMed]

]. In Ref. [19

19. T. A. Birks, F. Luan, G. J. Pearce, A. Wang, J. C. Knight, and D. M. Bird, “Bend loss in all-solid bandgap fibres,” Opt. Express 14, 5688–5698 (2006), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-14-12-5688. [CrossRef] [PubMed]

], the authors showed much can learned about the guidance characteristics of SC-PBGFs by considering the Bloch modes which define the bandgap edge. In their particular example, d/Λ=0.44 and fiber had low index contrast, so the Bloch modes were described in terms of the scalar modes of the cylinder and classified using the LPq,p designation, where the subscripts denote the number of azimuthal and radial nodes in the transverse mode field, as opposed to the notation we use for the vector modes in terms of the longitudinal mode field. The authors found that for their geometry, the Bloch modes which define the bandgap edge alternate between LP1,p-derived bands and LP0,p+1-derived bands, depending on whether the associated transmission band is of odd or even order. The LP0,p+1-derived bands are composed of much more strongly coupled states than the LP1,p bands [8

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

, 19

19. T. A. Birks, F. Luan, G. J. Pearce, A. Wang, J. C. Knight, and D. M. Bird, “Bend loss in all-solid bandgap fibres,” Opt. Express 14, 5688–5698 (2006), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-14-12-5688. [CrossRef] [PubMed]

], and this was shown to have a very strong impact on the bend loss behavior of the fiber [19

19. T. A. Birks, F. Luan, G. J. Pearce, A. Wang, J. C. Knight, and D. M. Bird, “Bend loss in all-solid bandgap fibres,” Opt. Express 14, 5688–5698 (2006), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-14-12-5688. [CrossRef] [PubMed]

]. In the present case, we relate the Bloch modes at gap edges of the low d/Λ SC-PBGF to the scattered field distribution at antiresonant frequencies. If we look at the projection of the scattering coefficients on the real line in Fig. 4, we see that in the off-resonance sections (away from the loops), B1H is by far the dominant scattering coefficient regardless of whether these sections correspond to an even order or an odd order band (the B2f coefficients, which are not shown, are even smaller than the B0H coefficient and so we need not consider the behavior these terms). We thus expect that off-resonance, the scattered field inside and around the cylinder is primarily HE1,p+1-like (or LP0,p+1-like in the scalar nomenclature). Our multipole simulations in the SC-PBGF geometry do show that near the cladding cylinders, the field distribution of the SC-PBGF core mode always behaves like a leaky LP0,p+1 mode when d\Λ is small. That is, the band gap edges are associated with leaky LP0,p+1-like Bloch modes for all transmission bands because the cylinders are sufficiently far from each other that their LP1,p modes cannot easily couple to their nearest neighbors. We still observe different behavior for odd and even order bands, as in Ref. [19

19. T. A. Birks, F. Luan, G. J. Pearce, A. Wang, J. C. Knight, and D. M. Bird, “Bend loss in all-solid bandgap fibres,” Opt. Express 14, 5688–5698 (2006), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-14-12-5688. [CrossRef] [PubMed]

], but here it depends on whether the gap edge represents a LP0,p+1-derived Bloch mode near cutoff (even bands) or far from cutoff (odd bands).

Fig. 6. Im(n eff) versus V for the first 6 bands of a 3 ring SC-PBGF with (a) d/Λ=0.6, (b) d/Λ=0.4, (c) d/Λ=0.2, and (d) d/Λ=0.15.

5. Low d/Λ vs. moderate d/Λ

6. Fano-enhanced core

In Fig. 7, the incident angle is fixed at ϑ0=0.025 rad, and the cylinders are oriented as shown in the inset, where the transverse component of the incident plane wave travels from left to right. We have found that some features of the scattering spectra are orientation dependent (for example, the “undulation” of the spectra at low d/Λ, or the relative strengths of the resonances), though this dependence weakens as d/Λ increases, and as expected, the frequencies of the resonant features in the scattering spectra are independent of orientation.

Fig. 7. Animations of (a) scattering cross section σsc (b) and asymmetry factor g for three cylinders with d/Λ varying from 0.1 to 0.8 and ϑ0=0.025 rad is constant. Vertical lines indicate modal cutoff frequencies of the single cylinder. Figure shown in text corresponds to d/Λ=0.4. [Media 1] [Media 2]

The vertical lines indicate the cutoff frequencies of various vector modes of the single cylinder; the dashed lines correspond to modes of low azimuthal order (blue=HE1,p+1, red=TE0,p/HE2,p, green=EH1,p/HE3,p) and the dotted lines correspond to modes of high azimuthal order (purple=EH2,p/HE4,p, orange=EH3,p/HE5,p, dark cyan=EH4,p/HE6,p, pink=EH5,p/HE7,p, gray=EH6,p/HE8,p); most of the high azimuthal order resonances appear only in the animations when the cylinders are very close together. We note that there is a very low frequency resonance (V~0.5-1.5) which does not line up with any of the vertical lines and has no analog in the single cylinder scattering spectrum. This resonance is associated with the low index region between the cylinders [34

34. T. A. Birks, D. M. Bird, T. D. Hedley, J. M. Pottage, and P. St. J. Russell, “Scaling laws and vector effects in bandgap-guiding fibers,” Opt. Express 12, 69–74 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-69. [CrossRef] [PubMed]

], and so its frequency shifts as the cylinder spacing is reduced. The distribution of the transverse scattered field at the resonance frequency looks like a hybrid TE0,1/HE2,1 mode localized in the low index region bounded by the cylinders. Surprisingly, this resonance is quite strong and narrow even when the cylinders are far apart, where one would not expect the low index region between the cylinders to be accurately approximated as a closed object.

As the cylinders move closer together, resonances associated with cylinder modes of increasing azimuthal order (which do correspond to the colored vertical lines) become apparent in the scattering spectrum, consistent with our observations in Fig. 6. Also, the low order modes couple to each other, and so where we once observed isolated resonances associated with a given mode of a cylinder, we now find multiple resonances associated with the supermodes of the three-cylinder object. As d/Λ increases, these supermode resonances split further apart from each other and cover an increasingly wide spectral bandwidth, which correlates with the high loss region between transmission bands in the SC-PBGF. We thus expect that for SC-PBGFs with large d/Λ, the spectral regions of high loss between the transmission bands are broadened. The number of supermodes equals the number of cylinders, so as more cylinders are added, these supermodes form continuous bands as described in Refs. [8

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

, 19

19. T. A. Birks, F. Luan, G. J. Pearce, A. Wang, J. C. Knight, and D. M. Bird, “Bend loss in all-solid bandgap fibres,” Opt. Express 14, 5688–5698 (2006), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-14-12-5688. [CrossRef] [PubMed]

].

Fig. 8. Im(n eff) versus V for the first 6 bands of a SC-PBGF with d/Λ=0.4 and different types of defect cores. The discontinuity at the center of the 5th band in (b) is due to the fact that the numerical precision of the simulation does not allow one to find values of Im(n eff) below 10-14. The core designs in (c) and (d) show Fano resonance-like corners at the edges of the odd order bands and improved confinement in the 1st order band.

Fig. 9. Im(n eff) versus V for the first order band of a SC-PBGF with d/Λ=0.4 and different types of defect cores.

7. Conclusion

Acknowledgements

The authors thank R. C. McPhedran and S. Tomljenovic-Hanic for many useful discussions. B.T. Kuhlmey acknowledges financial support from the Australian Research Council under Australian Postdoctoral Fellowship Project No. DP0665032. This work was produced with the assistance of the Australian Research Council under the ARC Centres of Excellence program. CUDOS is an ARC Centre of Excellence.

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A. Argyros, T. A. Birks, S. G. Leon-Saval, C. M. B. Cordeiro, and P. St. J. Russell, “Guidance properties of low-contrast photonic bandgap fibres,” Opt. Express 13, 2503–2511 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-7-2503. [CrossRef] [PubMed]

18.

P. Steinvurzel, B. T. Kuhlmey, T. P. White, M. J. Steel, C. M. de Sterke, and B. J. Eggleton, “Long wavelength anti-resonant guidance in high index inclusion microstructured fibers,” Opt. Express 12, 5424–5433 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5424. [CrossRef] [PubMed]

19.

T. A. Birks, F. Luan, G. J. Pearce, A. Wang, J. C. Knight, and D. M. Bird, “Bend loss in all-solid bandgap fibres,” Opt. Express 14, 5688–5698 (2006), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-14-12-5688. [CrossRef] [PubMed]

20.

G. Renversez, P. Boyer, and A. Sagrini, “Antiresonant reflecting optical waveguide microstructured fibers revisited: a new analysis based on leaky mode coupling,” Opt. Express 14, 5682–5687 (2006), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-14-12-5682. [CrossRef] [PubMed]

21.

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Canadian J. Phys. 33, 189–195 (1955). [CrossRef]

22.

C. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1998). [CrossRef]

23.

Note that we define ϑ0 such that it approaches zero for near glancing incidence, which we feel is the natural choice in the current context; in most of the literature on scattering by cylinders [16,21,22], however, the conicity angle approaches π/2 for glancing incidence, and our definition is equivalent to π/2-α in the earlier references.

24.

A. W. Snyder and J.D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).

25.

E. Snitzer, “Cylindrical Dielectric Waveguide Modes,” J. Opt. Soc. Am. 51, 491–498 (1961). [CrossRef]

26.

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]

27.

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]

28.

http://www.physics.usyd.edu.au/cudos/mofsoftware

29.

U. Fano, “Effects Of Configuration Interaction On Intensities And Phase Shifts,” Phys. Rev. 124, 1866–1878 (1961). [CrossRef]

30.

R. V. Andaloro, H. J. Simon, and R. T. Deck, “Temporal pulse reshaping with surface waves,” Appl. Opt. 33, 6340–6347 (1994). [CrossRef] [PubMed]

31.

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002). [CrossRef]

32.

A. E. Miroshnichenko, S. F. Mingaleev, S. Flach, and Yu. S. Kivshar, “Nonlinear Fano resonance and bistable wave transmission,” Phys. Rev. E 71, 036626 (2005). [CrossRef]

33.

E. Centeno and D. Felbacq, “Rigorous vector diffraction of electromagnetic waves by bidimensional photonic crystals,” J. Opt. Soc. Am. A 17, 320–327 (2000). [CrossRef]

34.

T. A. Birks, D. M. Bird, T. D. Hedley, J. M. Pottage, and P. St. J. Russell, “Scaling laws and vector effects in bandgap-guiding fibers,” Opt. Express 12, 69–74 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-69. [CrossRef] [PubMed]

OCIS Codes
(060.2310) Fiber optics and optical communications : Fiber optics
(060.2400) Fiber optics and optical communications : Fiber properties
(290.0290) Scattering : Scattering
(290.1350) Scattering : Backscattering

ToC Category:
Photonic Crystal Fibers

History
Original Manuscript: August 7, 2006
Manuscript Accepted: August 30, 2006
Published: September 18, 2006

Citation
P. Steinvurzel, C. Martijn de Sterke, M. J. Steel, B. T. Kuhlmey, and B. J. Eggleton, "Single scatterer Fano resonances in solid core photonic band gap fibers," Opt. Express 14, 8797-8811 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-19-8797


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References

  1. R. T. Bise, R. S. Windeler, K. S. Kranz, C. Kerbage, B. J. Eggleton, and D. J. Trevor, "Tunable photonic bandgap fiber," in Optical Fiber Communications Conference, Post Conference Ed., Vol. 70 of OSA Trends in Optics and Photonics Series Technical Digest (Optical Society of America, Washington, D. C., 2002), 466-468.
  2. T. T. Larsen, A. Bjarklev, D. S. Hermann, and J. Broeng, "Optical devices based on liquid crystal photonic bandgap fibres," Opt. Express 11, 2589-2596 (2003). [CrossRef] [PubMed]
  3. F. Luan, A. K. George, T. D. Hedley, G. J. Pearce, D. M. Bird, J. C. Knight, and P. St. J. Russell, "All-solid photonic bandgap fiber," Opt. Lett. 29, 2369-2371 (2004). [CrossRef] [PubMed]
  4. A. Argyros, T. A. Birks, S. G. Leon-Saval, C. M. B. Cordeiro, F. Luan, and P. St. J. Russell, "Photonic bandgap with an index step of one percent," Opt. Express 13, 309-314 (2005). [CrossRef] [PubMed]
  5. G. Bouwmans, L. Bigot, Y. Quiquempois, F. Lopez, L. Provino, and M. Douay, "Fabrication and characterization of an all-solid 2D photonic bandgap fiber with a low-loss region (< 20 dB/km) around 1550 nm," Opt. Express 13, 8452-8459 (2005). [CrossRef] [PubMed]
  6. J. M. Stone, G. J. Pearce, F. Luan, T. A. Birks, J. C. Knight, A. K. George, and D. M. Bird, "An improved photonic bandgap fiber based on an array of rings," Opt. Express 14, 6291-6295 (2006). [CrossRef] [PubMed]
  7. F. Brechet, P. Roy, J. Marcou, and D. Pagnoux, "Singlemode propagation into depressed-core-index photonic bandgap fibre designed for zero-dispersion propagation at short wavelengths," Electron. Lett. 36, 514-515 (2000). [CrossRef]
  8. J. Lægsgaard, "Gap formation and guided modes in photonic bandgap fibres with high-index rods," J. Opt. A, Pure Appl. Opt. 6, 798-804 (2004). [CrossRef]
  9. 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]
  10. N. M. Litchinitser, A. K. Abeeluck, C. Headley, and B. J. Eggleton, "Antiresonant reflecting photonic crystal optical waveguides," Opt. Lett. 27, 1 592-1594 (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). [PubMed]
  12. 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]
  13. 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]
  14. J. Kubica, D. Uttamchandani, and B. Culshaw, "Modal propagation within ARROWwaveguides," Opt. Commun. 78, 133-136 (1990). [CrossRef]
  15. 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]
  16. A. C. Lind and J. M. Greenberg, "Electromagnetic scattering by obliquely Oriented Cylinders," J. Appl. Phys. 37, 3195-3203 (1966). [CrossRef]
  17. A. Argyros, T. A. Birks, S. G. Leon-Saval, C. M. B. Cordeiro, and P. St. J. Russell, "Guidance properties of low-contrast photonic bandgap fibres," Opt. Express 13, 2503-2511 (2005). [CrossRef] [PubMed]
  18. P. Steinvurzel, B. T. Kuhlmey, T. P. White, M. J. Steel, C. M. de Sterke, and B. J. Eggleton, "Long wavelength anti-resonant guidance in high index inclusion microstructured fibers," Opt. Express 12, 5424-5433 (2004). [CrossRef] [PubMed]
  19. T. A. Birks, F. Luan, G. J. Pearce, A. Wang, J. C. Knight, and D. M. Bird, "Bend loss in all-solid bandgap fibres," Opt. Express 14, 5688-5698 (2006). [CrossRef] [PubMed]
  20. G. Renversez, P. Boyer, and A. Sagrini, "Antiresonant reflecting optical waveguide microstructured fibers revisited: a new analysis based on leaky mode coupling," Opt. Express 14, 5682-5687 (2006). [CrossRef] [PubMed]
  21. J. R. Wait, "Scattering of a plane wave from a circular dielectric cylinder at oblique incidence," Canadian J. Phys. 33, 189-195 (1955). [CrossRef]
  22. C. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1998). [CrossRef]
  23. Note that we define ϑ0 such that it approaches zero for near glancing incidence, which we feel is the natural choice in the current context; in most of the literature on scattering by cylinders [16,21,22], however, the conicity angle approaches π=2 for glancing incidence, and our definition is equivalent to π=2¡α in the earlier references.
  24. A. W. Snyder and J.D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983).
  25. E. Snitzer, "Cylindrical Dielectric Waveguide Modes," J. Opt. Soc. Am. 51, 491-498 (1961). [CrossRef]
  26. 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]
  27. 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]
  28. http://www.physics.usyd.edu.au/cudos/mofsoftware
  29. U. Fano, "Effects Of Configuration Interaction On Intensities And Phase Shifts," Phys. Rev. 124, 1866-1878 (1961). [CrossRef]
  30. R. V. Andaloro, H. J. Simon, and R. T. Deck, "Temporal pulse reshaping with surface waves," Appl. Opt. 33, 6340-6347 (1994). [CrossRef] [PubMed]
  31. S. Fan, and J. D. Joannopoulos, "Analysis of guided resonances in photonic crystal slabs," Phys. Rev. B 65, 235112 (2002). [CrossRef]
  32. A. E. Miroshnichenko, S. F. Mingaleev, S. Flach, and Yu. S. Kivshar, "Nonlinear Fano resonance and bistable wave transmission," Phys. Rev. E 71, 036626 (2005). [CrossRef]
  33. E. Centeno and D. Felbacq, "Rigorous vector diffraction of electromagnetic waves by bidimensional photonic crystals," J. Opt. Soc. Am. A 17, 320-327 (2000). [CrossRef]
  34. T. A. Birks, D. M. Bird, T. D. Hedley, J. M. Pottage, and P. St. J. Russell, "Scaling laws and vector effects in bandgap-guiding fibers," Opt. Express 12, 69-74 (2004). [CrossRef] [PubMed]

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