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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 13 — Jun. 18, 2012
  • pp: 14350–14361
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Extra loss due to Fano resonances in inhibited coupling fibers based on a lattice of tubes

L. Vincetti and V. Setti  »View Author Affiliations


Optics Express, Vol. 20, Issue 13, pp. 14350-14361 (2012)
http://dx.doi.org/10.1364/OE.20.014350


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Abstract

Confinement loss of inhibited coupling fibers with a cladding composed of a lattice of tubes of various shapes is theoretically and numerically investigated. Both solid core and hollow core are taken into account. It is shown that in case of polygonal shaped tubes, confinement loss is affected by extra loss due to Fano resonances between core modes and cladding modes with high spatial dependence. This explains why hollow core Kagome fibers exhibit much higher confinement loss with respect to tube lattice fibers and why hypocycloid core cladding interfaces significantly reduce fiber loss. Moreover it is shown that tube deformations, due for example to fabrication process, affect fiber performances. A relationship between the number of polygon sides and the spectral position of the extra loss is found. This suggests general guide lines for the design and fabrication of fibers free of Fano resonance in the spectral range of interest.

© 2012 OSA

1. Introduction

Inhibited Coupling Fibers (ICFs) have been extensively studied in recent years [1

1. F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and photonic guidance of multi-octave optical-frequency combs,” Science 318, 1118–1121 (2007). [CrossRef] [PubMed]

9

9. S. Février, F. Gérôme, A. Labruyère, B. Beaudou, G. Humbert, and J. L. Auguste, “Ultraviolet guiding hollow-core photonic crystal fiber,” Opt. Lett. 34, 2888–2890 (2009). [CrossRef] [PubMed]

]. Their confinement mechanism is based on the inhibition of the coupling between the core and the cladding modes. When the coupling is high, then cladding becomes almost transparent to the electromagnetic wave, causing high confinement loss (CL). On the other hand, when the coupling is weak, the electromagnetic wave can propagate along the core with low loss [1

1. F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and photonic guidance of multi-octave optical-frequency combs,” Science 318, 1118–1121 (2007). [CrossRef] [PubMed]

,2

2. A. Argyros and J. Pla, “Hollow-core polymer fibers with a kagome lattice: potential for transmission in the infrared,” Opt. Express 15, 7713–7719 (2007). [CrossRef] [PubMed]

]. ICFs exhibit better performance than photonic band gap fibers in terms of low loss bandwidth and dispersion, but they suffer from higher loss. Reduction of loss in ICFs is thus a key point in the development of this kind of fibers. The microstructured cladding of ICFs is often composed of a regular arrangement of rods or tubes immersed in a background material. Tubes allow to obtain lower CL, larger transmission windows (TWs), and reduced bend sensitivity than rods [3

3. T. Grujic, B. T. Kuhlmey, A. Argyros, S. Coen, and C. M. de Sterke, “Solid-core fiber with ultra-wide bandwidth transmission window due to inhibited coupling,” Opt. Express 18, 25556–25566 (2010). [CrossRef] [PubMed]

5

5. A. F. Kosolapov, A. D. Pryamikov, A. S. Biriukov, V. S. Shiryaev, M. S. Astapovich, G. E. Snopatin, V. G. Plotnichenko, M. F. Churbanov, and E. M. Dianov, “Demonstration of CO2-laser power delivery through chalcogenide-glass fiber with negativecurvature hollow core,” Opt. Express 19, 25723–25728 (2011). [CrossRef]

]. For this reason, only Tube Lattice Fibers (TLF) will be considered in the following. Background material can be either air or a different dielectric material. In the first case, the fibers are known as Hollow Core TLFs (HC-TLFs), while in the second one as Solid Core TLFs (SC-TLFs). Kagome Fibers (KFs) belong to the group of HC-TLF since their microstructured cladding can be seen as composed of a regular arrangement of tubes with hexagonal shape [10

10. L. Vincetti and V. Setti, “Confinement loss in kagome and tube lattice fibers: comparison and analysis,” J. Light-wave Technol. 30, 1470–1474 (2012). [CrossRef]

]. KFs suffer from high confinement loss compared to HC-TLFs with circular tubes (Circular TLFs - CTLFs) [10

10. L. Vincetti and V. Setti, “Confinement loss in kagome and tube lattice fibers: comparison and analysis,” J. Light-wave Technol. 30, 1470–1474 (2012). [CrossRef]

, 11

11. L. Vincetti, V. Setti, and M. Zoboli, “Confinement loss of tube lattice and kagome fibers,” in Specialty Optical Fibers (SOF)Toronto, Canada (2011).

]. Numerical and experimental works have shown that this high confinement loss is connected to the shape of the core-cladding interface [12

12. Y. Y. Wang, N. V. Wheeler, F. Couny, P. J. Roberts, and F. Benabid, “Low loss broadband transmission in hypocycloid-core kagome hollow-core photonics crystal fiber,” Opt. Lett. 36, 669–671 (2011). [CrossRef] [PubMed]

], and to the presence or absence of struts around the core [13

13. S. Février, B. Beaudou, and P. Viale, “Understanding origin of loss in large pitch hollow-core photonic crystal fibers and their design simplification,” Opt. Express 18, 5142–5150 (2010). [CrossRef] [PubMed]

]. The polygonal shape of the tubes composing the cladding of TLFs with both HC and SC can also be due to the drawing step of the fabrication process [2

2. A. Argyros and J. Pla, “Hollow-core polymer fibers with a kagome lattice: potential for transmission in the infrared,” Opt. Express 15, 7713–7719 (2007). [CrossRef] [PubMed]

,14

14. 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, 6291–6296 (2006). [CrossRef] [PubMed]

,15

15. X. Jiang, T. G. Euser, A. Abdolvand, F. Babic, F. Tani, N. Y. Joly, J. C. Travers, and P. St. J. Russell, “Single-mode hollow-core photonic crystal fiber made from soft glass,” Opt. Express 19, 15438–15444 (2011). [CrossRef] [PubMed]

]. Recently, it has been numerically shown that in HC-TLFs with polygonal tubes (Polygonal TLFs - PTLFs), loss reduces by increasing the number of sides of the polygons, and that CTLFs exhibit the best performance [10

10. L. Vincetti and V. Setti, “Confinement loss in kagome and tube lattice fibers: comparison and analysis,” J. Light-wave Technol. 30, 1470–1474 (2012). [CrossRef]

]. The confinement loss in HC-TLFs seems thus to depend on the shape of the struts composing the core boundary. However simple numerical results are not enough to fully understand the physical origin of this dependence and what are the fiber geometrical parameters that affect it.

In this paper it will be shown that a similar coupling mechanism also appears in TLFs when the tubes composing the cladding are changed from a circular shape (CTLFs) to a polygonal one (PTLFs). By applying the coupled mode theory to PTLFs, a simple analytical formula connecting the extra loss spectral regions to the number of sides is obtained. The higher the number of the sides of the tubes, the higher the frequencies where the extra losses appear. Circular shaped cladding tubes represent thus the best case for the confinement loss spectrum, while KFs composed of hexagonal tubes are affected by extra loss over the whole spectrum. This is very important both from a design and manufacturing point of view.

The paper is organized as follows. Section 2 shows the structure of both SC-TLFs and HC-TLFs considered in the analysis and it gives a brief overview of the confinement mechanism of TLFs. In the section 3 the relationship between extra loss and the tube shape is outlined through numerical results. The section 4 is devoted to the development of the analytical model based on the coupled mode theory. The model is validated in the section 5 through numerical results and comparison with experimental results reported in the literature. Conclusions follow.

2. Outline of the waveguiding mechanism in CTLFs

Figures 1(a) and 1(b) show the transverse cross sections that are typically used in CTLFs with HC and SC respectively [3

3. T. Grujic, B. T. Kuhlmey, A. Argyros, S. Coen, and C. M. de Sterke, “Solid-core fiber with ultra-wide bandwidth transmission window due to inhibited coupling,” Opt. Express 18, 25556–25566 (2010). [CrossRef] [PubMed]

, 6

6. J. Lu, C. Yu, H. Chang, H. Chen, Y. Li, C. Pan, and C. Sun, “Terahertz air-core microstructure fiber,” Appl. Phys. Lett. 92, 064105 (2009). [CrossRef]

, 14

14. 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, 6291–6296 (2006). [CrossRef] [PubMed]

, 16

16. L. Vincetti and V. Setti, “Waveguiding mechanism in tube lattice fibers,” Opt. Express 18, 23133–23146 (2010). [CrossRef] [PubMed]

]. Claddings are composed of a regular arrangement of circular tubes with external radius Rcext, thickness t and refractive index n1, separated by a pitch Λ and immersed in a uniform dielectric background with lower refractive index n2. The cross section of a stand alone circular tube is reported in Fig. 1(c). In the HC-TLFs, the background material is air (n2 = 1). The tubes must be in contact with each other and thus the tube radius Rcext and the pitch Λ are bounded each other. Kagome fibers can be seen as a particular case of these fibers in which tubes have an hexagonal shape [10

10. L. Vincetti and V. Setti, “Confinement loss in kagome and tube lattice fibers: comparison and analysis,” J. Light-wave Technol. 30, 1470–1474 (2012). [CrossRef]

]. In the SC-TLFs, the background material has a refraction index n2 > 1. In this case, mechanical stability of the fiber is not a problem, thus Λ can have any desired value. Typically, for HC-TLFs also an external jacket that surrounds the cladding tubes is added in order to enhance the mechanical stability of the fiber structure. Its refractive index is here assumed to be equal to that of the cladding tubes for simplicity.

Fig. 1 (a)–(b) Cross sections of a HC-TLF and a SC-TLF respectively, with circular tubes in the cladding. (c)–(d) Cross sections of a standalone circular and a polygonal tube fiber. They also represent the tubes composing the cladding of CTLFs and PTLFs, respectively. White and gray regions represent low refractive index n1 background material and high index n2 one respectively.

TLFs support three different kind of modes: core modes, which confine the major part of their electromagnetic power inside the core region; cladding hole modes (airy modes in HC-TLFs [2

2. A. Argyros and J. Pla, “Hollow-core polymer fibers with a kagome lattice: potential for transmission in the infrared,” Opt. Express 15, 7713–7719 (2007). [CrossRef] [PubMed]

], [16

16. L. Vincetti and V. Setti, “Waveguiding mechanism in tube lattice fibers,” Opt. Express 18, 23133–23146 (2010). [CrossRef] [PubMed]

]), which are confined in the inner part of cladding tubes; cladding ring modes, that concentrate their power mainly in the high index rings of the cladding tubes [16

16. L. Vincetti and V. Setti, “Waveguiding mechanism in tube lattice fibers,” Opt. Express 18, 23133–23146 (2010). [CrossRef] [PubMed]

]. The first two kinds of modes have neff < n2, whereas the latter has n2neff < n1. For the confinement mechanism purposes, only the core modes and the cladding ring modes can be considered. Cladding ring modes can be described in terms of a combination of the ring modes of the single tubes composing the cladding [3

3. T. Grujic, B. T. Kuhlmey, A. Argyros, S. Coen, and C. M. de Sterke, “Solid-core fiber with ultra-wide bandwidth transmission window due to inhibited coupling,” Opt. Express 18, 25556–25566 (2010). [CrossRef] [PubMed]

, 16

16. L. Vincetti and V. Setti, “Waveguiding mechanism in tube lattice fibers,” Opt. Express 18, 23133–23146 (2010). [CrossRef] [PubMed]

] as shown in Figs. 2(a) and 2(b). In general, ring modes can be HEξ,γri, EHξ,γri, TE0,γri and TM0,γri modes. The subscripts ξ and γ, refer to the number of periods along the circumference direction of the tube and to the number of maxima/minima in the radial direction, respectively. In the SC-TLF usually (n2n1)/n2 << 1 and ring modes can be described in terms of LPξ,γri modes which are composed of HEξ+1,γri, EHξ1,γri modes if ξ ≥ 2. This approximation is no longer valid in HC-TLF because the step index is much higher. In this paper, for sake of simplicity, cladding ring modes are named after the ring modes composing them.

Fig. 2 (a)–(b) Eϕ field component for the cladding ring modes HE1,2ri and LP11,2ri respectively. (c) Top: comparison of the confinement loss spectra of a circular (black dots) and a 12-sided polygonal (red triangles) tube fibers. Geometrical and physical properties of fibers are described through the paper. Middle: cutoff frequencies for the tube modes of the circular tube fiber. Dotted lines highlight the tubes modes which cause Fano resonances in tube fiber, according to Eq. (2). Solid lines highlight the cladding modes which defines the boundaries of the Fano resonances regions according to Eq. (8) with μ̄ = 3. Red, green, and blue colors are used for the cases m = 1 and m = 2, respectively. Bottom: comparison of the confinement loss performance of SF-TLFs with circular (black dots) and 12-sided polygonal (red triangles) tubes in the cladding.

3. Fano resonances in polygonal TFs and TLFs

3.1. Polygonal tube fibers

In the SC-TF, being n2n1 << 1, the coupling strength is much lower and hybrid modes degenerate into linearly polarized modes: LPξ,γriHEξ+1,γri,EHξ1,γri. Due to the weakness of the coupling, only LPmN,γri modes give rise to non-negligible Fano resonances, because they are composed of hybrid modes both satisfying Eq. (2). In LPmN1,γri and LPmN+1,γri modes, only one of the two hybrid modes satisfy Eq. (2), thus the resonances are much lower and they are not visible with the scale used in Fig. 2(c). A detail of the resonance with the LP12,1ri mode is reported on the top of Fig. 3(a). Figure 3(b) clearly shows the hybridization between the core mode HE1,1co and the ring LP12,1 mode. In the middle of Fig. 2(c), the cut-off normalized frequencies of the ring modes HEξ,γri and EHξ,γri with γ = 1, 2 of a CTF are reported. Vertical dotted lines correspond to cut off of the HEmN+1,γri and EHmN1,γri modes (LPmN,γ) satisfying Eq. (2) with m = 1 (red lines), m = 2 (green lines), and m = 3 (blue lines). They coincide with Fano resonances in the confinement loss spectrum of the PTF.

Fig. 3 (a) Zoom of Fig. 2(c) for F = [0.39, 0.75]. In the middle the degeneration of the HEξ+1,γ and EHξ−1,γ modes composing the LPξ,γ modes is highlighted. (b) z-component of the Poynting vector of the guided mode at F = 0.569373 of the PTF (top) and PTLF (bottom) both with N = 24.

3.2. Polygonal tube lattice fibers

4. Analytical model

In order to investigate which cladding ring modes are able to couple to the core one, coupled mode theory is used [24

24. S. L. Chuang, “A coupled mode formulation by reciprocity and a variational principle,” J. Lightwave Technol. 5, 5–15 (1987). [CrossRef]

, 25

25. W. P. Huang and J. Mu, “Complex coupled-mode theory for optical waveguides,” Opt. Express 17, 19134–19152 (2009). [CrossRef]

]. A PTLF with polygonal tubes in the cladding can be seen as the perturbed version of a CTLF. An example with N = 6 is shown in Fig. 4(a). A similar approach has already been successfully applied to investigate this kind of resonances in PTFs [17

17. L. Vincetti and V. Setti, “Fano resonances in polygonal tube fibers,” J. Lightwave Technol. 30, 31–37 (2012). [CrossRef]

].

Fig. 4 (a) Example of perturbation function for a PTLF. z-component of the Poynting vector of the fundamental mode on log scale is also shown. The inset shows the perturbation function for a generic cladding tube, with the local reference system centered at its center. (b) Electric field components of the fundamental mode along the six innermost tubes of the TLF; different colors refer to different tubes.

In TLFs, the overlap between ring and core modes is quite low and the coupling coefficient can be written as [17

17. L. Vincetti and V. Setti, “Fano resonances in polygonal tube fibers,” J. Lightwave Technol. 30, 31–37 (2012). [CrossRef]

, 24

24. S. L. Chuang, “A coupled mode formulation by reciprocity and a variational principle,” J. Lightwave Technol. 5, 5–15 (1987). [CrossRef]

]:
K˜co,ri=πf2SΔε(E¯tcoE¯triεεc+ΔεEzcoEzri)dS,
(3)
where S is the whole transverse plane, and the subscripts t and z indicate the transverse and longitudinal components of the electric field, respectively; εc is the dielectric permittivity of the CTLF, and Δε is the perturbation function shown in Fig. 4(a).

In order to find the conditions which guarantee that the integral (Eq. (3)) is non-zero, the Nt tubes composing the cladding are indexed by i and their centers are specified by vector C̄i (see Fig. 4(a)). A local cylindrical coordinate system (ri, ϕi) is introduced at the center of each cylinder. The perturbation function can thus be expressed as:
Δε(r¯)=i=1NtΔε˜(r¯C¯i);
(4)
Δε̃ is the perturbation of a PTF centered in the origin of the reference system (r, ϕ) and it is periodic along ϕ with period 2π/N.

Due to the vectorial nature of the Eq. (3), its development gives rise to three similar integrals, one for each field component. By substituting Eq. (4) in each one of these integrals, it yields:
K˜co,rix=(1)δx,zπf2i=1NtAiΔε˜(ε˜cε˜c+Δε˜δx,zExcoExri)ridϕidri
(5)
where x = {r,ϕ,z}, δx,z is the Kronecker index, ε̃c is the permittivity of a CTF centered in the origin, and Ai is the surface where Δε̃(r̄ − C̄i) ≠ 0. Since cladding ring modes can be described as a composition of the modes of a CTF, in each integral of the series the field components of the ring modes can be expressed as [17

17. L. Vincetti and V. Setti, “Fano resonances in polygonal tube fibers,” J. Lightwave Technol. 30, 31–37 (2012). [CrossRef]

]:
Exri(ri,ϕi)=Rx1ri(ri)cos(ξϕi)+Rx2ri(ri)sin(ξϕi),
(6)
which are periodic functions with period 2π/ξ along ϕi. For the core modes, this is true in the TFs, but not in the TLFs as shown in Fig. 4(a), where the z component of the Poynting vector of the fundamental mode is reported on log scale. Despite that, since Δε̃(r̄ − C̄i) ≠ 0 only near the circular tube interface, Exco can be locally described by means of a Fourier-Bessel series [26

26. B. T. Kuhlmey, K. Pathmanandavel, and R. C. McPhedran, “Multipole analysis of photonic crystal fibers with coated inclusions,” Opt. Express 14, 10851–10864 (2006). [CrossRef] [PubMed]

]:
Exco(ri,ϕi)=μ=0μ¯AxμRx1co,μ(ri)cos(μϕi)+BxμRx2co,μ(ri)sin(μϕi),
(7)
where μ̄ is the integer at which the Fourier-Bessel series can be truncated without loss accuracy. As reported in Fig. 4(b), Ēco components are slowly varying along ϕi with i = 1,.., Nt. This allows to consider very few terms in the series (Eq. (7)). By introducing Eq. (7) into Eq. (5), for each tube in the cladding there are μ̄ + 1 integrals equal to those of a PTF. For a PTF with N sides, a core mode with an azimuthal index μ̃ couples with cladding modes with azimuthal index such that |mNξ| = μ̃ [17

17. L. Vincetti and V. Setti, “Fano resonances in polygonal tube fibers,” J. Lightwave Technol. 30, 31–37 (2012). [CrossRef]

]. In the TLF, all the harmonics of Eq. (7) must be considered and the condition on the ring modes becomes:
|mNξ|μ¯.
(8)
In Fig. 2(c) solid lines correspond to cut-off of the HEmN+1,γri and EHmN1,γri modes (LPmN,γ) satisfying Eq. (8) with μ̄ = 3 and m = 1 (red lines), m = 2 (green lines), and m = 3 (blue lines). They coincide with the edges of the high loss regions in the confinement loss spectrum of the PTLF.

In short, in TFs the core modes are described on the perturbation domain by only one sinusoidal function along the azimuthal direction. On the contrary, in the TLFs, the core modes do not exhibit a periodic trend on the perturbation. Despite that they can be described in terms of a series of periodic functions, each of which gives a non zero term to the integral (Eq. (3)). This justify the increment of the number of the resonances shown in Fig. 3(a).

4.1. Conditions to have a resonance free spectral region

The resonance having the lowest frequency Fq is the most important. It determines the spectral region [0 : Fq] without Fano resonances, where confinement loss is very close to that of a CTLF [10

10. L. Vincetti and V. Setti, “Confinement loss in kagome and tube lattice fibers: comparison and analysis,” J. Light-wave Technol. 30, 1470–1474 (2012). [CrossRef]

, 17

17. L. Vincetti and V. Setti, “Fano resonances in polygonal tube fibers,” J. Lightwave Technol. 30, 31–37 (2012). [CrossRef]

]. The first resonance is due to the HENμ¯,1ri mode. Cutoff frequencies of the modes of a TF are reported in the middle of Fig. 2(c). They are analytically known [27

27. M. M. Z. Kharadly and J. E. Lewis, “Properties of dielectric-tube waveguides,” Proc. IEEE 116, 214–224 (1969).

] and give a good approximation of those of the cladding ring modes. For every given Fq, it always exists a number of sides N which guarantee that resonance free spectral region. When ξ >> 1, the relationship between Fq and ξ is well approximated by:
ξ=aFb.
(9)
Both the a and the b parameters depend on the geometrical and physical parameters of the cladding tubes. By combining Eq. (9) and (8) it yields:
Fq=N+bμ¯a,
(10)
thus the number of the sides N which guarantee the absence of Fano resonances in the spectral region [0 : Fq] is:
N=aFqb+μ¯,
(11)
where ⌈x⌉ denotes the nearest integer value that is bigger than x.

5. Numerical validation

5.1. HC-TLF

The HC-TLF here considered is shown in Fig. 1(a) with Rcext=5μm, t = 500nm, n1 = 1.45 and n2 = 1. A single layer of tubes surrounds the HC obtained by removing the seven innermost ones. The parameters of the Eq. (9) are a = 31.8 and b = 8. The polygons in the cladding are oriented in such a way to contact each other only on vertices, as happens in the KFs [12

12. Y. Y. Wang, N. V. Wheeler, F. Couny, P. J. Roberts, and F. Benabid, “Low loss broadband transmission in hypocycloid-core kagome hollow-core photonics crystal fiber,” Opt. Lett. 36, 669–671 (2011). [CrossRef] [PubMed]

, 13

13. S. Février, B. Beaudou, and P. Viale, “Understanding origin of loss in large pitch hollow-core photonic crystal fibers and their design simplification,” Opt. Express 18, 5142–5150 (2010). [CrossRef] [PubMed]

].

In Fig. 5(a) a polygonal HC-TLF with N = 6 is considered. It corresponds to a Kagome fiber with pitch Λ = 9.5μm and strut thickness t = 500nm. Colored rectangles on the top of the graphs represent the cutoff regions of the rings modes that satisfy Eq. (8) with μ̄ = 3. Different colors correspond to different values of the m parameter. Confinement loss is compared to that of a HC-CTLF. CLs coincide only in the high loss spectral regions corresponding to resonances with low azimuthal dependence ring modes [10

10. L. Vincetti and V. Setti, “Confinement loss in kagome and tube lattice fibers: comparison and analysis,” J. Light-wave Technol. 30, 1470–1474 (2012). [CrossRef]

, 16

16. L. Vincetti and V. Setti, “Waveguiding mechanism in tube lattice fibers,” Opt. Express 18, 23133–23146 (2010). [CrossRef] [PubMed]

]. In the low loss regions, the Fano resonances due to hexagonal shape of the tubes composing the cladding cause a significant worsening of the confinement loss with respect to the circular case. Since Fq = 0.24 and the bands of m = 1,2,3 are partially overlapped, the Fano resonances cover the whole spectrum. Figure 5(b) shows the case of an HC-TLF with N = 12. The change from N = 6 to N = 12 shifts the bands toward higher frequencies, but Fq = 0.53 is still too low and the bands are still partially overlapped, thus there is not a significant improvement. By further increasing the sides up to N = 24 (Fig. 5(c)) an improvement of the confinement loss is obtained in the first and in the second transmission window. In fact, being Fq = 0.91, in the first transmission window there is a wide frequency range where confinement loss coincides with that of CTLF. Moreover, the bands with m = 1 and with m = 2 are no longer overlapped in the second transmission window creating a frequency range where confinement loss coincides with the circular case. For higher frequencies, there are still a lot of resonant modes that worsen the performance of the fiber. In order to widen the Fano resonance free region, polygons with a higher number of sides must be considered. From Eq. (11) by choosing Fq = 2, it yields N ≥ 62. Actually N = 66 is required in order to satisfy geometrical constraints about the polygon vertices. Figure 5(d) shows the confinement loss performance for such fiber. As expected, the confinement loss performance of the 66-sided HC-PTLF remain very close to those of the HC-CTLF in the first two transmission windows.

Fig. 5 Comparison of the confinement loss performances between a HC-TLF with circular (black dots) and N-sided polygonal HC-TLF (red triangles), with N = 6 (a), N = 12 (b), N = 24 (c), N = 66 (d). Rectangles on the top of the graphs represent the cutoff regions for the rings modes that satisfy Eq. (8) with μ̄ = 3. Different colors are used for different values of the m parameter. In (a) only m ≤ 4 has been considered for clearness.

Finally notice that the model and these results are in agreement with the results reported in [12

12. Y. Y. Wang, N. V. Wheeler, F. Couny, P. J. Roberts, and F. Benabid, “Low loss broadband transmission in hypocycloid-core kagome hollow-core photonics crystal fiber,” Opt. Lett. 36, 669–671 (2011). [CrossRef] [PubMed]

] where the shape of the tubes facing towards the HC has been changed into a rounded one, obtaining a reduction of the CL.

5.2. SC-TLF

The model that have been developed through this paper is very useful to predict the effect of the manufacturing imperfections for both SC and HC-TLFs. For this purpose a SC-CTLF with n1 = 1.47288, n2 = 1.457, t = 540nm and Rcext=5.4μm is taken into account [14

14. 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, 6291–6296 (2006). [CrossRef] [PubMed]

]. As it can be seen in the scanning electron micrograph reported in [14

14. 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, 6291–6296 (2006). [CrossRef] [PubMed]

], due to the manufacturing process, cladding tubes are actually hexagons with rounded corners. Rounded corners do not affect the spectral position of the Fano resonances but reduce their bandwidth [17

17. L. Vincetti and V. Setti, “Fano resonances in polygonal tube fibers,” J. Lightwave Technol. 30, 31–37 (2012). [CrossRef]

]. A smoothing parameter s = 1–2a/L, as shown in Fig. 6(a), is defined for the polygons of the cladding. In the simulations, s = 0.75 has been used. Figure 6(b) compares the confinement loss performance of the SC-PTLF with rounded hexagons with those of a SC-CTLF. Yellow regions represent the spectral regions with high transmission loss obtained experimentally in [14

14. 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, 6291–6296 (2006). [CrossRef] [PubMed]

]. For the SC-CTLF, the resonance intensities with LPξ,1ri quickly decrease as ξ increases and the fiber does not exhibit high loss peaks for F > 0.40. On the contrary, the SC-PTLF exhibits all high loss peaks experimentally observed, showing that they are due to the hexagonal shape of the cladding elements. This is further confirmed by Fig. 6(c) which shows the modes found in simulations at F = 0.305 and F = 0.47. They agree with near field images reported in [14

14. 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, 6291–6296 (2006). [CrossRef] [PubMed]

]. The differences of the spectral position of the loss peaks between numerical results and experimental ones are due to the fact that it was not possible to make a numerical prototype exactly like the fiber used in the experiment. For example, in the simulations the tube thickness was considered constant, whereas in the fabricated fiber the cross section of each tube is actually a ring of closely-spaced rods, spaced by around 0.6μm, but the reported image does not allow to determine exactly their shape and size. A better agreement between numerical and experimental results can be obtained only by a higher resolution scanning micrograph image of the fiber cross section.

Fig. 6 (a) Rounding scheme of the polygon vertex: L is the length of the polygon side and a is the distance of the rounding point from the center of the side. (b) Comparison of the confinement loss between a SC-TLF with circles (black dots) or rounded hexagons (red triangles) in the cladding. Resonant rings modes are highlighted on the top of the figure. Yellow regions represent the high confinement loss regions reported in [14]. (c) Hybridization between core mode and ring modes LP4,1ri (A) and LP8,1ri (B) computed at F = 0.305 and F = 0.47, respectively.

6. Conclusions

References and links

1.

F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and photonic guidance of multi-octave optical-frequency combs,” Science 318, 1118–1121 (2007). [CrossRef] [PubMed]

2.

A. Argyros and J. Pla, “Hollow-core polymer fibers with a kagome lattice: potential for transmission in the infrared,” Opt. Express 15, 7713–7719 (2007). [CrossRef] [PubMed]

3.

T. Grujic, B. T. Kuhlmey, A. Argyros, S. Coen, and C. M. de Sterke, “Solid-core fiber with ultra-wide bandwidth transmission window due to inhibited coupling,” Opt. Express 18, 25556–25566 (2010). [CrossRef] [PubMed]

4.

A. D. Pryamikov, A. S. Biriukov, A. F. Kosolapov, V. G. Plotnichenko, S. L. Semjov, and E. M. Dianov, “Demonstration of a waveguide regime for a silica hollow-core microstructured optical fiber with a negatice curvature of the core boundary in the spectral region >3.5μm,” Opt. Express 19, 1441–1448 (2011). [CrossRef] [PubMed]

5.

A. F. Kosolapov, A. D. Pryamikov, A. S. Biriukov, V. S. Shiryaev, M. S. Astapovich, G. E. Snopatin, V. G. Plotnichenko, M. F. Churbanov, and E. M. Dianov, “Demonstration of CO2-laser power delivery through chalcogenide-glass fiber with negativecurvature hollow core,” Opt. Express 19, 25723–25728 (2011). [CrossRef]

6.

J. Lu, C. Yu, H. Chang, H. Chen, Y. Li, C. Pan, and C. Sun, “Terahertz air-core microstructure fiber,” Appl. Phys. Lett. 92, 064105 (2009). [CrossRef]

7.

J. C. Travers, W. Chang, J. Nold, N. Y. Joly, and P. St. Russell, “Ultrafast nonlinear optics in gas-filled hollow-core photonic crystal fibers,” J. Opt. Soc. Am. B 28, A11–A26 (2011). [CrossRef]

8.

J. Anthony, R. Leonhardt, S. G. Leon-Saval, and A. Argyros, “THz propagation in kagome hollow-core microstructured fibers,” Opt. Express 19, 18470–18478 (2011). [CrossRef] [PubMed]

9.

S. Février, F. Gérôme, A. Labruyère, B. Beaudou, G. Humbert, and J. L. Auguste, “Ultraviolet guiding hollow-core photonic crystal fiber,” Opt. Lett. 34, 2888–2890 (2009). [CrossRef] [PubMed]

10.

L. Vincetti and V. Setti, “Confinement loss in kagome and tube lattice fibers: comparison and analysis,” J. Light-wave Technol. 30, 1470–1474 (2012). [CrossRef]

11.

L. Vincetti, V. Setti, and M. Zoboli, “Confinement loss of tube lattice and kagome fibers,” in Specialty Optical Fibers (SOF)Toronto, Canada (2011).

12.

Y. Y. Wang, N. V. Wheeler, F. Couny, P. J. Roberts, and F. Benabid, “Low loss broadband transmission in hypocycloid-core kagome hollow-core photonics crystal fiber,” Opt. Lett. 36, 669–671 (2011). [CrossRef] [PubMed]

13.

S. Février, B. Beaudou, and P. Viale, “Understanding origin of loss in large pitch hollow-core photonic crystal fibers and their design simplification,” Opt. Express 18, 5142–5150 (2010). [CrossRef] [PubMed]

14.

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, 6291–6296 (2006). [CrossRef] [PubMed]

15.

X. Jiang, T. G. Euser, A. Abdolvand, F. Babic, F. Tani, N. Y. Joly, J. C. Travers, and P. St. J. Russell, “Single-mode hollow-core photonic crystal fiber made from soft glass,” Opt. Express 19, 15438–15444 (2011). [CrossRef] [PubMed]

16.

L. Vincetti and V. Setti, “Waveguiding mechanism in tube lattice fibers,” Opt. Express 18, 23133–23146 (2010). [CrossRef] [PubMed]

17.

L. Vincetti and V. Setti, “Fano resonances in polygonal tube fibers,” J. Lightwave Technol. 30, 31–37 (2012). [CrossRef]

18.

U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 24, 1866–1878 (1961). [CrossRef]

19.

S. Glasberg, A. Sharon, D. Rosenblatt, and A. A. Friesem, “Spectral shifts and line-shapes asymmetries in the resonant response of grating waveguide structures,” Opt. Commun. 145, 291–299 (1998). [CrossRef]

20.

S. S. Wang, R. Magnusson, J. Bagby, and M. G. Moharam, “Guided-mode resonances in planar dielectric-layer diffraction gratings,” J. Opt. Soc. Am. A 7, 1470–1474 (1990). [CrossRef]

21.

S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the fano resonance in optical resonators,” J. Opt. Soc. Am. A 20, 569–572 (2002). [CrossRef]

22.

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

23.

S. Selleri, L. Vincetti, A. Cucinotta, and M. Zoboli, “Complex FEM modal solver of optical waveguides with PML boundary conditions,” Opt. Quantum Electron. 33, 359–371 (2001). [CrossRef]

24.

S. L. Chuang, “A coupled mode formulation by reciprocity and a variational principle,” J. Lightwave Technol. 5, 5–15 (1987). [CrossRef]

25.

W. P. Huang and J. Mu, “Complex coupled-mode theory for optical waveguides,” Opt. Express 17, 19134–19152 (2009). [CrossRef]

26.

B. T. Kuhlmey, K. Pathmanandavel, and R. C. McPhedran, “Multipole analysis of photonic crystal fibers with coated inclusions,” Opt. Express 14, 10851–10864 (2006). [CrossRef] [PubMed]

27.

M. M. Z. Kharadly and J. E. Lewis, “Properties of dielectric-tube waveguides,” Proc. IEEE 116, 214–224 (1969).

OCIS Codes
(060.2400) Fiber optics and optical communications : Fiber properties
(060.4005) Fiber optics and optical communications : Microstructured fibers
(060.5295) Fiber optics and optical communications : Photonic crystal fibers

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: March 29, 2012
Revised Manuscript: May 17, 2012
Manuscript Accepted: May 30, 2012
Published: June 12, 2012

Citation
L. Vincetti and V. Setti, "Extra loss due to Fano resonances in inhibited coupling fibers based on a lattice of tubes," Opt. Express 20, 14350-14361 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-13-14350


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References

  1. F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and photonic guidance of multi-octave optical-frequency combs,” Science318, 1118–1121 (2007). [CrossRef] [PubMed]
  2. A. Argyros and J. Pla, “Hollow-core polymer fibers with a kagome lattice: potential for transmission in the infrared,” Opt. Express15, 7713–7719 (2007). [CrossRef] [PubMed]
  3. T. Grujic, B. T. Kuhlmey, A. Argyros, S. Coen, and C. M. de Sterke, “Solid-core fiber with ultra-wide bandwidth transmission window due to inhibited coupling,” Opt. Express18, 25556–25566 (2010). [CrossRef] [PubMed]
  4. A. D. Pryamikov, A. S. Biriukov, A. F. Kosolapov, V. G. Plotnichenko, S. L. Semjov, and E. M. Dianov, “Demonstration of a waveguide regime for a silica hollow-core microstructured optical fiber with a negatice curvature of the core boundary in the spectral region >3.5μm,” Opt. Express19, 1441–1448 (2011). [CrossRef] [PubMed]
  5. A. F. Kosolapov, A. D. Pryamikov, A. S. Biriukov, V. S. Shiryaev, M. S. Astapovich, G. E. Snopatin, V. G. Plotnichenko, M. F. Churbanov, and E. M. Dianov, “Demonstration of CO2-laser power delivery through chalcogenide-glass fiber with negativecurvature hollow core,” Opt. Express19, 25723–25728 (2011). [CrossRef]
  6. J. Lu, C. Yu, H. Chang, H. Chen, Y. Li, C. Pan, and C. Sun, “Terahertz air-core microstructure fiber,” Appl. Phys. Lett.92, 064105 (2009). [CrossRef]
  7. J. C. Travers, W. Chang, J. Nold, N. Y. Joly, and P. St. Russell, “Ultrafast nonlinear optics in gas-filled hollow-core photonic crystal fibers,” J. Opt. Soc. Am. B28, A11–A26 (2011). [CrossRef]
  8. J. Anthony, R. Leonhardt, S. G. Leon-Saval, and A. Argyros, “THz propagation in kagome hollow-core microstructured fibers,” Opt. Express19, 18470–18478 (2011). [CrossRef] [PubMed]
  9. S. Février, F. Gérôme, A. Labruyère, B. Beaudou, G. Humbert, and J. L. Auguste, “Ultraviolet guiding hollow-core photonic crystal fiber,” Opt. Lett.34, 2888–2890 (2009). [CrossRef] [PubMed]
  10. L. Vincetti and V. Setti, “Confinement loss in kagome and tube lattice fibers: comparison and analysis,” J. Light-wave Technol.30, 1470–1474 (2012). [CrossRef]
  11. L. Vincetti, V. Setti, and M. Zoboli, “Confinement loss of tube lattice and kagome fibers,” in Specialty Optical Fibers (SOF)Toronto, Canada (2011).
  12. Y. Y. Wang, N. V. Wheeler, F. Couny, P. J. Roberts, and F. Benabid, “Low loss broadband transmission in hypocycloid-core kagome hollow-core photonics crystal fiber,” Opt. Lett.36, 669–671 (2011). [CrossRef] [PubMed]
  13. S. Février, B. Beaudou, and P. Viale, “Understanding origin of loss in large pitch hollow-core photonic crystal fibers and their design simplification,” Opt. Express18, 5142–5150 (2010). [CrossRef] [PubMed]
  14. 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. Express14, 6291–6296 (2006). [CrossRef] [PubMed]
  15. X. Jiang, T. G. Euser, A. Abdolvand, F. Babic, F. Tani, N. Y. Joly, J. C. Travers, and P. St. J. Russell, “Single-mode hollow-core photonic crystal fiber made from soft glass,” Opt. Express19, 15438–15444 (2011). [CrossRef] [PubMed]
  16. L. Vincetti and V. Setti, “Waveguiding mechanism in tube lattice fibers,” Opt. Express18, 23133–23146 (2010). [CrossRef] [PubMed]
  17. L. Vincetti and V. Setti, “Fano resonances in polygonal tube fibers,” J. Lightwave Technol.30, 31–37 (2012). [CrossRef]
  18. U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev.24, 1866–1878 (1961). [CrossRef]
  19. S. Glasberg, A. Sharon, D. Rosenblatt, and A. A. Friesem, “Spectral shifts and line-shapes asymmetries in the resonant response of grating waveguide structures,” Opt. Commun.145, 291–299 (1998). [CrossRef]
  20. S. S. Wang, R. Magnusson, J. Bagby, and M. G. Moharam, “Guided-mode resonances in planar dielectric-layer diffraction gratings,” J. Opt. Soc. Am. A7, 1470–1474 (1990). [CrossRef]
  21. S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the fano resonance in optical resonators,” J. Opt. Soc. Am. A20, 569–572 (2002). [CrossRef]
  22. S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B65, 235112 (2002). [CrossRef]
  23. S. Selleri, L. Vincetti, A. Cucinotta, and M. Zoboli, “Complex FEM modal solver of optical waveguides with PML boundary conditions,” Opt. Quantum Electron.33, 359–371 (2001). [CrossRef]
  24. S. L. Chuang, “A coupled mode formulation by reciprocity and a variational principle,” J. Lightwave Technol.5, 5–15 (1987). [CrossRef]
  25. W. P. Huang and J. Mu, “Complex coupled-mode theory for optical waveguides,” Opt. Express17, 19134–19152 (2009). [CrossRef]
  26. B. T. Kuhlmey, K. Pathmanandavel, and R. C. McPhedran, “Multipole analysis of photonic crystal fibers with coated inclusions,” Opt. Express14, 10851–10864 (2006). [CrossRef] [PubMed]
  27. M. M. Z. Kharadly and J. E. Lewis, “Properties of dielectric-tube waveguides,” Proc. IEEE116, 214–224 (1969).

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