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Virtual Journal for Biomedical Optics

Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editor: Gregory W. Faris
  • Vol. 3, Iss. 1 — Jan. 29, 2008

Optics InfoBase > Virtual Journal for Biomedical Optics > Volume 3 > Issue 1 > Page 17577

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Design of 7 and 19 cells core air-guiding photonic crystal fibers for low-loss, wide bandwidth and dispersion controlled operation

R. Amezcua-Correa, N. G. Broderick, M. N. Petrovich, F. Poletti, and D. J. Richardson  »View Author Affiliations


Optics Express, Vol. 15, Issue 26, pp. 17577-17586 (2007)
http://dx.doi.org/10.1364/OE.15.017577


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Abstract

We study the modal properties of feasible hollow-core photonic bandgap fibers (HC-PBGFs) with cores formed by omitting either 7 or 19 central unit-cells. Firstly, we analyze fibers with thin core surrounds and demonstrate that even for large cores the proposed structures are optimum for broad-band transmission. We compare these optimized structures with fibers which incorporate antiresonant core surrounds which are known to have low-loss. Trade-offs between loss and useful bandwidth are presented. Finally, we study the effects that small modifications to the core surround have on the fiber’s group velocity dispersion, showing the possibility of engineering the dispersion in hollow-core photonic bandgap fibers.

© 2007 Optical Society of America

1. Introduction

Photonic crystal fibers (PCFs) are usually single material fibers with air-holes that define their transverse profile. These fibers have already demonstrated several advantages over conventional silica-doped fibers due to their novel optical properties such as broadband single mode guidance, widely engineerable dispersion, nonlinear properties, mode area tailoring, and photonic bandgap guidance [1

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

]. Photonic band gap fibers are a special category of PCFs which exploit photonic bandgaps of the cladding structure, rather than total internal reflection, as the mechanism of light guidance. In hollow-core photonic bandgap fibres light can be predominantly guided in an air-core [1

P. St. 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. St.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]

]. These fibers have opened-up unique and technologically important fiber applications, especially for high power delivery, low nonlinearity and transmission beyond silica’s own transparency window [3

D. G. Ouzounov, F. R. Ahmad, D. Muller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, “Generation of megawatt optical solitons in hollow-core photonic band-gap fibers,” Science 301, 1702–17044 (2003). [CrossRef] [PubMed]

, 4

P. J. Roberts, F. Couny, T. A. Birks, J. C. Knight, P. St.J. Russell, B. J. Mangan, H. Sabert, D. P. Willliams, and L. Farr, “Achieving low loss and low nonlinearity in hollow-core photonic crystal fibers,” in Proc. CLEO2005 (Baltimore, 2005), paper CWA7.

, 5

J. D. Shephard, W. N. MacPherson, R. R.J. Maier, J. D.C. Jones, M. Mohebbi, A. K. George, P. J. Roberts, and J. C. Knight, “Single-mode mid-IR guidance in a hollow-core photonic crystal fiber,” Opt. Express 13, 7139–7144 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-18-7139. [CrossRef] [PubMed]

]. In addition, they are ideal for nonlinear-optics and sensors since there can be a big overlap between the optical guided modes and gases, liquids or particles introduced into the air core [1

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

, 6

F. Benabid, F. Couny, J. C. Knight, T. A. Birks, and P. St. J. Russell, “Stimulated Raman Scattering in Hydrogen-Filled Hollow-Core Photonic Crystal Fiber,” Science 298, 399–402 (2002). [CrossRef] [PubMed]

, 7

S. Fevrier, P. Viale, M. Lelek, F. Louradour, J. L. Auguste, P. Roy, and J. M. Blondy, “Singlemode low-index liquid core holey fibre,” in Proc. ECOC2005 (Glasgow, 2005), paper Tu1.4.3.

].

The fundamental limiting factor for loss in hollow-core PBGFs is light scattering from surface imperfections at the air/glass interfaces [8

P. J. Roberts, F. Couny, H. Sabert, B. J. Mangan, D. P. Willliams, L. Farr, M. W. Mason, A. Tomlinson, T. A. Birks, J. C. Knight, and P. St.J. Russell, “Ultimate low loss of hollow-core photonic crystal fibres,” Opt. Express 13, 236–244 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-236. [CrossRef] [PubMed]

]. The surfaces roughness in these fibers is mainly due to surface capillary waves frozen during the fiber drawing process, making it unfeasible for this loss mechanism to be completely eliminated [8

P. J. Roberts, F. Couny, H. Sabert, B. J. Mangan, D. P. Willliams, L. Farr, M. W. Mason, A. Tomlinson, T. A. Birks, J. C. Knight, and P. St.J. Russell, “Ultimate low loss of hollow-core photonic crystal fibres,” Opt. Express 13, 236–244 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-236. [CrossRef] [PubMed]

]. Thus if low-loss fibers are to be fabricated, mechanisms to reduce the light intensity at the interfaces must be investigated. Roberts et al showed that silica core surrounds which thickness satisfies an antiresonant condition, repel the optical field at the core/cladding interface producing low-loss fibers [4

P. J. Roberts, F. Couny, T. A. Birks, J. C. Knight, P. St.J. Russell, B. J. Mangan, H. Sabert, D. P. Willliams, and L. Farr, “Achieving low loss and low nonlinearity in hollow-core photonic crystal fibers,” in Proc. CLEO2005 (Baltimore, 2005), paper CWA7.

, 9

P. J. Roberts, D. P. Willliams, B. J. Mangan, H. Sabert, F. Couny, W. J. Wadsworth, T. A. Birks, J. C. Knight, and P. St.J. Russell, “Realizing low loss air core photonic crystal fibers by exploiting an antiresonant core surround,” Opt. Express 13, 8277–8285 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-20-8277. [CrossRef] [PubMed]

]. Antiresonant core surrounds incorporating elliptical features have also been proposed [10

P. J. Roberts, D. P. Williams, H. Sabert, B. J. Mangan, D. M. Bird, T. A. Birks, J. C. Knight, and P. St. J. Russell, “Design of low-loss and highly birefringent hollow-core photonic crystal fiber,” Opt. Express 14, 7329–7341 (2006), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-14-16-7329. [CrossRef] [PubMed]

]. However, the main drawback of these designs is that the reduction of loss comes with an increase in the number of undesirable surface modes (SMs). At wavelengths where surface modes anticross with air-guided modes the fiber’s loss drastically increases, and consequently low-loss transmission is only achievable over narrow wavelength ranges [10

P. J. Roberts, D. P. Williams, H. Sabert, B. J. Mangan, D. M. Bird, T. A. Birks, J. C. Knight, and P. St. J. Russell, “Design of low-loss and highly birefringent hollow-core photonic crystal fiber,” Opt. Express 14, 7329–7341 (2006), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-14-16-7329. [CrossRef] [PubMed]

–13

R. Amezcua-Correa, N. G. Broderick, M. N. Petrovich, F. Poletti1, and D. J. Richardson, “Optimizing the usable bandwidth and loss through core design in realistic hollow-core photonic bandgap fibers,” Opt. Express 14, 7974–7985 (2006), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-14-17-7974. [CrossRef] [PubMed]

].

Recently, we have numerically studied high air-filling fraction HC-PBGFs whose core is formed by omitting the 7 central unit cells and identified feasible core surrounds that maximize the wavelength range over which the “fundamental” air-guided mode is free from anticrossings with surface modes, and showed that these fiber designs also offer low loss [13

R. Amezcua-Correa, N. G. Broderick, M. N. Petrovich, F. Poletti1, and D. J. Richardson, “Optimizing the usable bandwidth and loss through core design in realistic hollow-core photonic bandgap fibers,” Opt. Express 14, 7974–7985 (2006), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-14-17-7974. [CrossRef] [PubMed]

]. The proposed designs incorporate thin silica walls surrounding the core, in these structures the width of the silica ring is around half the thickness of the thin silica struts of the cladding. A fiber based on these designs has been fabricated and presented a wide transmission spectra free from interactions with surface modes in the center of the bandgap [17

R. Amezcua-Correa, M. N. Petrovich, N. G. Broderick, D. J. Richardson, T. Delmonte, M. A. Watson, and E. J. O’Driscoll, “Broadband infrared transmission in a hollow-core photonic bandgap fibre free of surface modes,” in Proc. ECOC2006(Cannes, 2006), paper We4.4.4.

]. Previously, Digonnet et al [18

M. J. F. Digonnet, H. K. Kim, J. Shin, S. Fan, and G. S. Kino, “Simple geometric criterion to predict the existence of surface modes in air-core photonic band-gap fibers,” Opt. Express 12, 1864–1872 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1864. [CrossRef] [PubMed]

, 19

H. K. Kim, J. Shin, S. Fan, M. J. F. Digonnet, and G. S. Kino, “Designing air-core photonic-bandgap fibers free of surface modes,” IEEE J. Quantum Electron. 40, 551–556 (2004). [CrossRef]

] proposed that fibers with silica struts pointing into the core do not support surface modes, and it was demonstrated that even fibers with large cores can be designed for not supporting surface modes. However, these fiber designs are difficult to fabricate with the current technology. The need of designing feasible fibers with large cores and free from surface modes comes from the fact that enlarging the core is a straightforward way of reducing the overlap of the optical field with the glass structure and therefore the scattering loss. Within this paper, we extend our previous work to examine 19-cell core hollow-core fibers and show that the design regime identified previously for 7-cell core fibers can be extended to larger core defects, leading to novel wide bandwidth low-loss fibers.

The transmission band of HC-PBGFs always presents wavelength ranges of normal and anomalous dispersion, due to this and their nonlinear properties, there has been a growing interest in using HC-PBGFs for re-compressing pulses in chirped pulse amplification systems [25

J. Limpert, T. Schreiber, S. Nolte, H. Zellmer, and A. Tunnermann, “All fiber chirped-pulse amplification system based on compression in air-guiding photonic bandgap fiber,” Opt. Express 11, 3332–3337 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=oe-11-24-3332. [CrossRef] [PubMed]

, 26

C. J. S. de Matos and J. R. Taylor, “Chirped pulse Raman amplification with compression in air-core photonic bandgap fiber,” Opt. Express 13, 2828–2834 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-8-2828. [CrossRef] [PubMed]

]. In the second part of this paper we analyze ways of modifying the dispersion of HC-PBGFs and show that is possible to control the fiber dispersion over a reasonable wavelength range without incurring in a large loss penalty.

2. Design of wide bandwidth low-loss HC-PBGFs

Cross sections of the analyzed 7-cell and 19-cell core air-guiding photonic bandgap fibers are shown in Fig. 1(a)(b) respectively. These structures are idealized representations of fibers that can be manufactured by the stack-and-drawmethod, omitting the central capillaries to create the core. The cladding is a triangular arrangement of hexagonal holes with rounded corners [20

N. A Mortensen and M. D. Nielsen, “Modeling of realistic cladding structures in air-core photonic band-gap fibers,” Opt. Lett. 29, 349–351 (2004). [CrossRef] [PubMed]

], with parameters chosen to match those of the low-loss fibers studied in [12

K. Saitoh, N. A. Mortensen, and M. Koshiba, “Air-core photonic band-gap fibers: the impact of surface modes,” Opt. Express 12, 394–400 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-394. [CrossRef] [PubMed]

, 21

F. Poletti, N. G. R. Broderick, D. J. Richardson, and T. M. Monro, “The effect of core asymmetries on the polarization properties of hollow core photonic bandgap fibers,” Opt. Express 13, 9115–9124 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-22-9115. [CrossRef] [PubMed]

]: relative hole size d/Λ=0.98, curvature at the corners dc =0.44Λ, and distance from hole center to hole center Λ=4.7 µm. The cladding’s air-filling fraction is 94.2%. The silica ring surrounding the hollow-core is a dodecagon of thickness tr , with corners rounded using circles of radius rc =1.2Λ, and dp =0.2Λ. Normalized values of tr with respect to the thickness of the thin silica struts in the cladding are considered: T=tr /(Λ-d).

Fig. 1. Cross section of the analyzed HC-PBGFs: (a) 7-cell core, (b) 19-cell core. (c) Structural parameters: d/Λ=0.98, dc /Λ=0.44, dp /Λ=0.22, tr /(Λ-d)=1, and Λ=4.7µm.

For the 7 and 19-cells core fibers shown in Fig. 1(a)(b), the thickness of the core surround was varied in the range 0.15≤T≤4.5. Thicknesses at which the core surround becomes antiresonant can be approximated by considering it as an annular tube. In this case, antiresonance takes place for

tr= (2 j+1)λ 4 ns2−1,
(1)

where λ is the wavelength, ns is the glass index, and j=0,1,2,… For the fibers studied here, antiresonance at wavelengths within the bandgap is achieved for T in the range 3–4.5. It is important to mention that the analysis of fibers incorporating antiresonant core surrounds is not the objective of this paper and can be found elsewhere [8

P. J. Roberts, F. Couny, H. Sabert, B. J. Mangan, D. P. Willliams, L. Farr, M. W. Mason, A. Tomlinson, T. A. Birks, J. C. Knight, and P. St.J. Russell, “Ultimate low loss of hollow-core photonic crystal fibres,” Opt. Express 13, 236–244 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-236. [CrossRef] [PubMed]

, 9

P. J. Roberts, D. P. Willliams, B. J. Mangan, H. Sabert, F. Couny, W. J. Wadsworth, T. A. Birks, J. C. Knight, and P. St.J. Russell, “Realizing low loss air core photonic crystal fibers by exploiting an antiresonant core surround,” Opt. Express 13, 8277–8285 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-20-8277. [CrossRef] [PubMed]

, 14

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]

, 15

N. M. Litchinitser, S. C. Dunn, B. Usner, B. J. Eggleton, T. P. White, R. C. McPhedran, and C. Martijn 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]

, 16

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

]. In this section we are interested in identifying fiber designs with wide transmission windows. However, designs incorporating antiresonant core surrounds have been included in the calculations as a reference for low-loss fibers. For the different fibers structures, full-vector solutions of the optical modes within the bandgap and near the bandgap edges are obtained using a finite element method (FEM). Material dispersion was not considered and the refractive index of silica and air are ns =1.45 and na =1, respectively. Since this paper focuses on the modal properties of the fundamental air-guided mode (HE11-like) and surface modes of compatible symmetry, we have reduced the computational time and excluded some irrelevant modes by considering only one quarter of the structure [24

R. Guobin, W. Zhi, L. Shuqin, and J. Shuisheng, “Mode classification and degeneracy in photonic crystal fibers,” Opt. Express 11, 1310–1321 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-11-1310. [CrossRef] [PubMed]

]. The normalized interface field intensity of the fundamental air-guided mode at the glass air/interfaces represents a relative measure of the scattering loss due to surface roughness [8

P. J. Roberts, F. Couny, H. Sabert, B. J. Mangan, D. P. Willliams, L. Farr, M. W. Mason, A. Tomlinson, T. A. Birks, J. C. Knight, and P. St.J. Russell, “Ultimate low loss of hollow-core photonic crystal fibres,” Opt. Express 13, 236–244 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-236. [CrossRef] [PubMed]

, 9

P. J. Roberts, D. P. Willliams, B. J. Mangan, H. Sabert, F. Couny, W. J. Wadsworth, T. A. Birks, J. C. Knight, and P. St.J. Russell, “Realizing low loss air core photonic crystal fibers by exploiting an antiresonant core surround,” Opt. Express 13, 8277–8285 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-20-8277. [CrossRef] [PubMed]

], and can be defined as follows:

F= ( ε0 μ0) 1/2 ∮holeperimetersdl∣E∣2 ∫ cross− sectiondA∣ E× H*∣· ẑ,
(2)

where E and H are the electric and magnetic fields of the fundamental mode and ẑ is the unit vector along the fiber.

The calculated normalized interface field intensity of the fundamental air-guided mode for the various fibers is presented as a function of the normalized ring thickness and wavelength in Fig. 2. The contour map in Fig. 2(a) corresponds to fibers with a 7-cell core, while Fig. 2(b) is for a larger 19-cell core. In these maps, each horizontal line represents a different fiber design. The large values of FΛ at wavelengths within the bandgap, diagonal yellow lines, are due to anticrossings between the fundamental air-guided mode and surface modes. At wavelengths near the anticrossing, the fundamental core mode couples to a particular surface mode and its overlap with the core surround increases and therefore the associated scattering loss also increases. Dark blue, i.e. FΛ ≤1, corresponds to regions where the fundamental air-guided mode is tightly confined in the air-core and represents low loss fibers.

Fig. 2. Normalized interface field intensity of the fundamental air-guided mode vs. wavelength for different values of the normalized ring thickness T. For the (a) 7-cell core fiber and (b) 19-cell core fiber. Note that the color scale is the same for both maps

From the maps in Fig. 2 we can see that the dependence of the transmission properties in T is very similar for both core sizes. However, the 19 cell fiber have lower values FΛ (deeper blue zones) and supports more surface modes. Fig. 2 clearly indicates that for both core sizes, fibers with core surrounds of normalized thickness T~0.5 maximize the wavelength range over which the fundamental core mode is free from anticrossings with surface modes, and provide a good confinement of the mode. In contrast, for other ring thicknesses one or multiple surface modes are supported and the useful bandwidth is reduced accordingly. Fig. 2 also shows that fibers with thick antiresonant core surrounds, T in the range 3.5–4.5, present multiple surface modes and low-loss transmission is achieved only over narrow wavelength ranges. However, fibers incorporating antiresonant core surrounds should show lower loss and nonlinearity since the interface field intensity is minimized.

In order to obtain an estimation of the fiber’s operational bandwidth, we set FΛ<1 to be the condition for maximum allowed “scattering loss” and define the operational bandwidth to be the maximum continuous wavelength range for which this condition is satisfied within the bandgap. Using this definition, the calculated operational bandwidth as a function of T is shown in Fig. 3(a)(b) for 7 and 19-cell core respectively. From these results we can observe that for both core sizes, designs with core rings of thickness in the range of approximately 0.4≤T ≤0.7 are optimal for broadband transmission providing a wide low-loss usable bandwidth of more that 360 nm, which corresponds to ~23% of the central wavelength of the bandgap. The low-loss wavelength range for fibers with T=0.5 is 404 nm and 380 nm for 7 and 19-cell core respectively.

After identifying designs that provide wide band transmission we compare their performance in terms of loss and mode confinement against antiresonant designs. In order to do this, for each ring thickness the minimum value of FΛ and maximum value of the power fraction in the air-core were obtained, and are shown in Fig. 4(a)(b) as a function of T for 7 and 19-cells core fibers respectively. These plots clearly point out the advantages of antiresonant core surrounds. However, we found that the normalized interface field intensity and the power fraction in the air-core, also present a minimum/maximum for thin ring designs. The minimum FΛ value achieved by the 7-cell core fiber is FΛ=0.1, when T=3.8. This value is about 2.6 times lower than for a fiber with T=0.5. For the larger core fiber (19-cells) Fig. 4(b), the antiresonant regime becomes more important and the lowest value of FΛ=0.02 is obtained when T=4.25, which is 3.3 times lower than for a fiber with T=0.5. Comparing the normalized interface field intensity of fibers with T=0.5 of different core sizes we found that FΛ is typically about 3.2 times lower for the 19-cell fiber, implying that the loss will be reduced by a similar factor [8

P. J. Roberts, F. Couny, H. Sabert, B. J. Mangan, D. P. Willliams, L. Farr, M. W. Mason, A. Tomlinson, T. A. Birks, J. C. Knight, and P. St.J. Russell, “Ultimate low loss of hollow-core photonic crystal fibres,” Opt. Express 13, 236–244 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-236. [CrossRef] [PubMed]

], leading to novel wide bandwidth low-loss fibers.

Fig. 3. Calculated useful bandwidth vs. normalized ring thickness, (a) for the 7-cell core, and (b) 19-cell core fibers.
Fig. 4. Maximum of the power fraction in the core (solid) and minimum of the normalized field intensity F (dashed) vs. normalized ring thickness, (a) for the 7-cell core, and (b) 19-cell core fibers.

3. Group velocity dispersion control in hollow-core fibers

3.1. Group velocity dispersion vs. ring thickness

We have investigated a strategy to control the group velocity dispersion (GVD) in HC-PBGFs via small changes to the geometrical parameters of the core. This study focuses on fibers with thin core surrounds because they have low loss and because we believe that it would be easier to more precisely control the fiber properties since they present less surface modes. For the following simulations material dispersion was directly included in silica’s refractive index via Sellmeier equations. Fig. 5(a) illustrates the calculated group velocity dispersion of the fundamental air-guided mode as a function of wavelength for the 7-cell core fiber for different thicknesses of the silica ring: T varying in the range 0.45–0.65. Although the changes to the core structure are rather small, tens of nanometers-scale, the effect on the GVD is quite noticeable. This is in good agreement with earlier reports of important changes in the fiber’s nonlinearity [27

C. J. Hensley, D. G. Ouzounov, A. L. Gaeta, N. Venkataraman, M. T. Gallagher, and K. W. Koch, “Silica-glass contribution to the effective nonlinearity of hollow-core photonic band-gap fibers,” Opt. Express 15, 3507–3512 (2007), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-15-6-3507. [CrossRef] [PubMed]

] and birefringence [21

F. Poletti, N. G. R. Broderick, D. J. Richardson, and T. M. Monro, “The effect of core asymmetries on the polarization properties of hollow core photonic bandgap fibers,” Opt. Express 13, 9115–9124 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-22-9115. [CrossRef] [PubMed]

] due to small alterations of the core boundary. Particularly interesting is that as the thickness of the core surround increases, the zero-GVD wavelength (λ 0) can be precisely shifted over ~100 nm wavelength range: λ0=1.412 nm, 1.427 nm, 1.454 nm, 1.488 nm, and 1.523 nm for T=0.45, 0.5, 0.55, 0.6, and 0.65 respectively. Moreover, as λ 0 is shifted towards longer wavelengths, larger normal dispersion values are obtained for wavelengths on the short wavelength side of λ0. The origin of the change in the GVD can be attributed to the interaction of the fundamental air-guided mode and a surface mode that moves into the bandgap and anticrosses with the fundamental air-guided mode when the ring thickness is increased, Fig. 2(a). This anticrossing can be easily recognized in the effective mode index against wavelength plot shown in Fig. 5(c). Here, solid lines correspond to the fundamental air-guided mode, while dashed lines indicate surface modes of compatible symmetry, and the shaded region represents the bandgap. The fiber design with T=0.45 (red), does not support surface modes guided near the short wavelength edge of the bandgap. In clear contrast, when T=0.65 (black) one surface mode is guided near the short wavelength bandgap edge. In this case, the surface mode and the fundamental core mode anticross, repealing each other and present a curve in their mode indices. Consequently, producing large dispersion values near the anticrossing wavelength and the zero-GVD wavelength to be shifted towards the center of the bandgap.

As modifying the fiber’s GVD properties involves controlling the position of surface modes within the bandgap, it is important to evaluate the loss cost of the process. Fig. 5(b) shows the normalized interface field intensity of the fundamental air-guided mode against wavelength for different thicknesses of the core surround. Black circles indicate λ 0 for each fiber design. We found that the normalized interface interface field intensity at the zero-GVD wavelength for the fiber of T=0.65 is only 1.23 times larger than that for the design with T=0.45. Implying that shifting λ0 from 1.412 nm to 1.523 nm can be achieved at the expense of ~23% increase in loss [8

P. J. Roberts, F. Couny, H. Sabert, B. J. Mangan, D. P. Willliams, L. Farr, M. W. Mason, A. Tomlinson, T. A. Birks, J. C. Knight, and P. St.J. Russell, “Ultimate low loss of hollow-core photonic crystal fibres,” Opt. Express 13, 236–244 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-236. [CrossRef] [PubMed]

]. Furthermore, we have calculated the wavelength range over which the fibers have normal dispersion and the condition FΛ<1 is satisfied. The calculated normal dispersion wavelength ranges are: 40 nm, 55 nm, 82 nm, 113 nm, and 111 nm for T=0.45, 0.5, 0.55, 0.6, and 0.65 respectively. It is important to mention that the zero-GVD wavelength can be progressively shifter towards longer wavelengths as T is increased still further. However, this will imply an increase on the fiber loss and a reduction of the transmission bandwidth, see Fig. 4(a) and Fig. 3(a).

3.2. Group velocity dispersion vs. refractive index of the core surround

An alternative way of modifying the fiber’s transmission properties is by increasing or decreasing the refractive index of the glass region surrounding the core. It is reasonable to expected that by doing this, surface modes will be more affected than air-guided modes since these modes tightly confine energy in the core surround. Consequently, in this way it is possible to increase or decrease the mode index of surface modes and thus move their position inside the bandgap. In this section we study the effect that changes in the refractive index of the core surround have on the group velocity dispersion of HC-PBGs. Fig. 6(a) shows the cross section of the analyzed fiber: red represents the glass region for which we varied the refractive index, black corresponds to pure silica and white to air. A stack of capillaries that could be used to fabricate

this kind of fiber is presented in Fig. 6(b). For the following simulations we have calculated the refractive index of silica ns using Sellmeier equations, and the refractive index of the silica ring (red region) is obtained by increasing silica’s refractive index: ng =ns (1+I).

Fig. 5. (a) Group velocity dispersion, and (b) normalized interface field intensity of the fundamental mode for a 7-cell core fiber with T=0.45, 0.5, 0.55, 0.6, and 0.65. The black circles in (b) indicate the zero-GVD wavelength for each design. (c) Calculated mode indices vs. wavelength for T=0.45 (red) and 0.65 (black). Solid lines correspond to the fundamental air-guided mode, while dashed lines correspond to surface modes.
Fig. 6. (a) Schematic cross section of fiber with core surround of modified refractive index. (b) Stack of capillaries that can be used to fabricate the proposed fiber. Red represents glass of different refractive index, black is for silica and white air.

For the 7-cell core fiber, while keeping the core thickness constant T=0.5, the refractive index of the core surround is increased up to 5% in 1% intervals. Fig. 7(a) shows the effect of this perturbation on the fiber’s GVD. Similarly to the previous case, as the refractive index of the core surround increases, the zero-GVD wavelength is shifted towards long wavelengths. An increase in the refractive index of 5% produces a shift in λ 0 of 100 nm from: 1.427 nm to 1.527 nm. In addition, larger normal dispersion values are obtained for wavelengths on the short wavelength side of λ0. Surprisingly, although the changes seen in the fiber’s GVD are comparable to what can be achieved by increasing the thickness of the ring, Fig. 7(b) shows that in this case shifting the zero-GVD wavelength 100 nm can be achieved with less increase in loss. The normalized interface interface field intensity at the zero-GVD wavelength when the refractive index of the core surround is increased by 5% is only 1.055 times larger than that for the original fiber, see black circles in Fig. 7(b). Implying that a shift of 100 nm in λ 0 will only increase the loss at the fiber’s zero-GVD by ~5.5% [8

P. J. Roberts, F. Couny, H. Sabert, B. J. Mangan, D. P. Willliams, L. Farr, M. W. Mason, A. Tomlinson, T. A. Birks, J. C. Knight, and P. St.J. Russell, “Ultimate low loss of hollow-core photonic crystal fibres,” Opt. Express 13, 236–244 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-236. [CrossRef] [PubMed]

]. We believe that the increase in refractive index (5%) needed to shift λ 0 100 nm, is quite big and due to the different thermal properties of silica and doped silica this design might be difficult to fabricate. However, we have showed that the use of doped elements in HC-PBGFs can be used to control the fiber’s GVD properties with out drastically affecting the loss. Finally, the GVD in the 19-cell core fiber is similar to the GVD in the 7-cell fiber but with λ0 shifted to even shorter wavelengths and a smaller value of anomalous dispersion in the middle of the bandgap. And the strategies for GVD control presented here are applicable also in that case.

Fig. 7. (a) Group velocity dispersion, and (b) normalized interface field intensity of the fundamental mode for a 7-cell core fiber with T=0.5, for different increments ring’s refractive index. The black circles in (b) indicate the zero-GVD wavelength for each design.

4. Conclusions

Previously we showed that 7-cell air-guiding PBGFs with a thin core wall were optimal for maximising the useable bandwidth while maintain a reasonably low loss[13

R. Amezcua-Correa, N. G. Broderick, M. N. Petrovich, F. Poletti1, and D. J. Richardson, “Optimizing the usable bandwidth and loss through core design in realistic hollow-core photonic bandgap fibers,” Opt. Express 14, 7974–7985 (2006), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-14-17-7974. [CrossRef] [PubMed]

] by reducing the impact of surface modes. In this paper we have extended this work to the case of 19 cell fibres and showed that this design regime is robust and the principles developed for the 7-cell fibres apply here as well. In particular the useful bandwidth is maximized if the thickness of the silica ring surrounding the core is about half the thickness of the thin struts of the cladding. For the cladding configuration analyzed here, the confining ability provided by fibers with antiresonant core surrounds is superior than thin core surrounds and should thus be preferred for applications requiring low-loss and low-nonlinearity over narrow wavelength ranges. However, for applications requiring relatively low-loss over a wide wavelength range, a fiber featuring thin core walls and 19-cell core is to be preferred. We found that for core surrounds of normalized thickness T=0.5, the 19-cell core fiber typically has ~3.2 times lower loss than the 7-cell core fiber over a comparable bandwidth, leading to novel wide bandwidth low-loss fibers. Furthermore, since these designs are free of surface modes across a wide wavelength range, we believe they are more insensitive to fabrication errors and it should be possible to fabricate them.

Simulations presented here show that it is possible to control the fiber’s group velocity dispersion by slightly modifying the properties of the glass ring around the core without incurring large loss penalties. Two strategies are analyzed: the first one consist on modifying the thickness of the core surround, and the second by changing its refractive index. Particularly interesting is the possibility of shifting the zero dispersion wavelength over ~25% of the bandgap width. Future work will explore several alternative mechanisms of tailoring the dispersion in hollow-core photonic bandgap fibers.

Acknowledgments

R. Amezcua-Correa acknowledges financial support from the Mexican council for science and technology (CONACyT).

References and links

1.

P. St. 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. St.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.

D. G. Ouzounov, F. R. Ahmad, D. Muller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, “Generation of megawatt optical solitons in hollow-core photonic band-gap fibers,” Science 301, 1702–17044 (2003). [CrossRef] [PubMed]

4.

P. J. Roberts, F. Couny, T. A. Birks, J. C. Knight, P. St.J. Russell, B. J. Mangan, H. Sabert, D. P. Willliams, and L. Farr, “Achieving low loss and low nonlinearity in hollow-core photonic crystal fibers,” in Proc. CLEO2005 (Baltimore, 2005), paper CWA7.

5.

J. D. Shephard, W. N. MacPherson, R. R.J. Maier, J. D.C. Jones, M. Mohebbi, A. K. George, P. J. Roberts, and J. C. Knight, “Single-mode mid-IR guidance in a hollow-core photonic crystal fiber,” Opt. Express 13, 7139–7144 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-18-7139. [CrossRef] [PubMed]

6.

F. Benabid, F. Couny, J. C. Knight, T. A. Birks, and P. St. J. Russell, “Stimulated Raman Scattering in Hydrogen-Filled Hollow-Core Photonic Crystal Fiber,” Science 298, 399–402 (2002). [CrossRef] [PubMed]

7.

S. Fevrier, P. Viale, M. Lelek, F. Louradour, J. L. Auguste, P. Roy, and J. M. Blondy, “Singlemode low-index liquid core holey fibre,” in Proc. ECOC2005 (Glasgow, 2005), paper Tu1.4.3.

8.

P. J. Roberts, F. Couny, H. Sabert, B. J. Mangan, D. P. Willliams, L. Farr, M. W. Mason, A. Tomlinson, T. A. Birks, J. C. Knight, and P. St.J. Russell, “Ultimate low loss of hollow-core photonic crystal fibres,” Opt. Express 13, 236–244 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-236. [CrossRef] [PubMed]

9.

P. J. Roberts, D. P. Willliams, B. J. Mangan, H. Sabert, F. Couny, W. J. Wadsworth, T. A. Birks, J. C. Knight, and P. St.J. Russell, “Realizing low loss air core photonic crystal fibers by exploiting an antiresonant core surround,” Opt. Express 13, 8277–8285 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-20-8277. [CrossRef] [PubMed]

10.

P. J. Roberts, D. P. Williams, H. Sabert, B. J. Mangan, D. M. Bird, T. A. Birks, J. C. Knight, and P. St. J. Russell, “Design of low-loss and highly birefringent hollow-core photonic crystal fiber,” Opt. Express 14, 7329–7341 (2006), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-14-16-7329. [CrossRef] [PubMed]

11.

J. A. West, C. M. Smith, N. F. Borrelli, D. C. Allan, and K. W. Koch, “Surface modes in air-core photonic band-gap fibers,” Opt. Express 12, 1485–1496 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1485. [CrossRef] [PubMed]

12.

K. Saitoh, N. A. Mortensen, and M. Koshiba, “Air-core photonic band-gap fibers: the impact of surface modes,” Opt. Express 12, 394–400 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-394. [CrossRef] [PubMed]

13.

R. Amezcua-Correa, N. G. Broderick, M. N. Petrovich, F. Poletti1, and D. J. Richardson, “Optimizing the usable bandwidth and loss through core design in realistic hollow-core photonic bandgap fibers,” Opt. Express 14, 7974–7985 (2006), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-14-17-7974. [CrossRef] [PubMed]

14.

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]

15.

N. M. Litchinitser, S. C. Dunn, B. Usner, B. J. Eggleton, T. P. White, R. C. McPhedran, and C. Martijn 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]

16.

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

17.

R. Amezcua-Correa, M. N. Petrovich, N. G. Broderick, D. J. Richardson, T. Delmonte, M. A. Watson, and E. J. O’Driscoll, “Broadband infrared transmission in a hollow-core photonic bandgap fibre free of surface modes,” in Proc. ECOC2006(Cannes, 2006), paper We4.4.4.

18.

M. J. F. Digonnet, H. K. Kim, J. Shin, S. Fan, and G. S. Kino, “Simple geometric criterion to predict the existence of surface modes in air-core photonic band-gap fibers,” Opt. Express 12, 1864–1872 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1864. [CrossRef] [PubMed]

19.

H. K. Kim, J. Shin, S. Fan, M. J. F. Digonnet, and G. S. Kino, “Designing air-core photonic-bandgap fibers free of surface modes,” IEEE J. Quantum Electron. 40, 551–556 (2004). [CrossRef]

20.

N. A Mortensen and M. D. Nielsen, “Modeling of realistic cladding structures in air-core photonic band-gap fibers,” Opt. Lett. 29, 349–351 (2004). [CrossRef] [PubMed]

21.

F. Poletti, N. G. R. Broderick, D. J. Richardson, and T. M. Monro, “The effect of core asymmetries on the polarization properties of hollow core photonic bandgap fibers,” Opt. Express 13, 9115–9124 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-22-9115. [CrossRef] [PubMed]

22.

C. J. S. de Matos, J. R. Taylor, T. P. Hansen, K. P. Hansen, and J. Broeng, “All-fiber chirped pulse amplification using highly-dispersive air-core photonic bandgap fiber,” Opt. Express 11, 2832–2837 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2832. [CrossRef] [PubMed]

23.

J. Loegsgaard, N. A. Mortensen, J. Riishede, and A. Bjarklev, “Material effects in airguiding photonic bandgap fibers,” J. Opt. Soc. Am. B 20, 2046–2051 (2003). [CrossRef]

24.

R. Guobin, W. Zhi, L. Shuqin, and J. Shuisheng, “Mode classification and degeneracy in photonic crystal fibers,” Opt. Express 11, 1310–1321 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-11-1310. [CrossRef] [PubMed]

25.

J. Limpert, T. Schreiber, S. Nolte, H. Zellmer, and A. Tunnermann, “All fiber chirped-pulse amplification system based on compression in air-guiding photonic bandgap fiber,” Opt. Express 11, 3332–3337 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=oe-11-24-3332. [CrossRef] [PubMed]

26.

C. J. S. de Matos and J. R. Taylor, “Chirped pulse Raman amplification with compression in air-core photonic bandgap fiber,” Opt. Express 13, 2828–2834 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-8-2828. [CrossRef] [PubMed]

27.

C. J. Hensley, D. G. Ouzounov, A. L. Gaeta, N. Venkataraman, M. T. Gallagher, and K. W. Koch, “Silica-glass contribution to the effective nonlinearity of hollow-core photonic band-gap fibers,” Opt. Express 15, 3507–3512 (2007), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-15-6-3507. [CrossRef] [PubMed]

28.

* Please note that Rodrigo Amezcua-Correa is now at the Centre for Photonics and Photonic Materials, Department of Physics, University of Bath, Bath BA2 7AY, UK

OCIS Codes
(060.2280) Fiber optics and optical communications : Fiber design and fabrication
(060.2400) Fiber optics and optical communications : Fiber properties

ToC Category:
Photonic Crystal Fibers

History
Original Manuscript: June 26, 2007
Revised Manuscript: September 30, 2007
Manuscript Accepted: October 5, 2007
Published: December 11, 2007

Virtual Issues
Vol. 3, Iss. 1 Virtual Journal for Biomedical Optics

Citation
R. Amezcua-Correa, N. G. Broderick, M. N. Petrovich, F. Poletti, and D. J. Richardson, "Design of 7 and 19 cells core air-guiding photonic crystal fibers for low-loss, wide bandwidth and dispersion controlled operation," Opt. Express 15, 17577-17586 (2007)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-26-17577


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References

  1. P. St. 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. St.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. D. G. Ouzounov, F. R. Ahmad, D. Muller, N. Venkataraman, M. T. Gallagher, M. G. Thomas, J. Silcox, K. W. Koch, and A. L. Gaeta, "Generation of megawatt optical solitons in hollow-core photonic band-gap fibers," Science 301, 1702-1704 (2003). [CrossRef] [PubMed]
  4. P. J. Roberts, F. Couny, T. A. Birks, J. C. Knight, P. St. J. Russell, B. J. Mangan, H. Sabert, D. P. Willliams, and L. Farr, "Achieving low loss and low nonlinearity in hollow-core photonic crystal fibers," in Proc. CLEO2005 (Baltimore, 2005), paper CWA7.
  5. J. D. Shephard, W. N. MacPherson, R. R. J. Maier, J. D. C. Jones, M. Mohebbi, A. K. George, P. J. Roberts, and J. C. Knight, "Single-mode mid-IR guidance in a hollow-core photonic crystal fiber," Opt. Express 13, 7139-7144 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-18-7139. [CrossRef] [PubMed]
  6. F. Benabid, F. Couny, J. C. Knight, T. A. Birks, and P. St. J. Russell, "Stimulated Raman Scattering in Hydrogen-Filled Hollow-Core Photonic Crystal Fiber," Science 298, 399-402 (2002). [CrossRef] [PubMed]
  7. S. Fevrier, P. Viale, M. Lelek, F. Louradour, J. L. Auguste, P. Roy, J. M. Blondy, "Singlemode low-index liquid core holey fibre," in Proc. ECOC2005 (Glasgow, 2005), paper Tu1.4.3.
  8. P. J. Roberts, F. Couny, H. Sabert, B. J. Mangan, D. P. Willliams, L. Farr, M. W. Mason A. Tomlinson, T. A. Birks, J. C. Knight, and P. St.J. Russell, "Ultimate low loss of hollow-core photonic crystal fibres," Opt. Express 13, 236-244 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-1-236. [CrossRef] [PubMed]
  9. P. J. Roberts, D. P. Willliams, B. J. Mangan, H. Sabert, F. Couny, W. J. Wadsworth, T. A. Birks, J. C. Knight, and P. St. J. Russell, "Realizing low loss air core photonic crystal fibers by exploiting an antiresonant core surround," Opt. Express 13,8277-8285 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-20-8277. [CrossRef] [PubMed]
  10. P. J. Roberts, D. P. Williams, H. Sabert, B. J. Mangan, D. M. Bird, T. A. Birks, J. C. Knight, and P. St. J. Russell, "Design of low-loss and highly birefringent hollow-core photonic crystal fiber," Opt. Express 14,7329-7341 (2006), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-14-16-7329. [CrossRef] [PubMed]
  11. J. A. West, C. M. Smith, N. F. Borrelli, D. C. Allan, and K. W. Koch, "Surface modes in air-core photonic band-gap fibers," Opt. Express 12,1485-1496 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-8-1485. [CrossRef] [PubMed]
  12. K. Saitoh, N. A. Mortensen, and M. Koshiba, "Air-core photonic band-gap fibers: the impact of surface modes," Opt. Express 12,394-400 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-394. [CrossRef] [PubMed]
  13. R. Amezcua-Correa, N. G. Broderick, M. N. Petrovich, F. Poletti1, D. J. Richardson, "Optimizing the usable bandwidth and loss through core design in realistic hollow-core photonic bandgap fibers," Opt. Express 14, 7974-7985 (2006), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-14-17-7974. [CrossRef] [PubMed]
  14. 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]
  15. N. M. Litchinitser, S. C. Dunn, B. Usner, B. J. Eggleton, T. P. White, R. C. McPhedran, and C. Martijn 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]
  16. P. White, R. C. McPhedran, C. Martijnde Sterke, N. M. Litchinitser, and B. J. Eggleton, "Resonance and scattering in microstructured optical fibers," Opt. Lett. 27, 1977-1979 (2002). [CrossRef]
  17. R. Amezcua-Correa, M. N. Petrovich, N. G. Broderick, D. J. Richardson, T. Delmonte, M. A. Watson, and E. J. O’Driscoll, "Broadband infrared transmission in a hollow-core photonic bandgap fibre free of surface modes," in Proc. ECOC2006 (Cannes, 2006), paper We4.4.4.
  18. M. J. F. Digonnet, H. K. Kim, J. Shin, S. Fan, and G. S. Kino, "Simple geometric criterion to predict the existence of surface modes in air-core photonic band-gap fibers," Opt. Express 12, 1864-1872 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-9-1864. [CrossRef] [PubMed]
  19. H. K. Kim, J. Shin, S. Fan, M. J. F. Digonnet, and G. S. Kino, "Designing air-core photonic-bandgap fibers free of surface modes," IEEE J. Quantum Electron. 40, 551-556 (2004). [CrossRef]
  20. N. A Mortensen and M. D. Nielsen, "Modeling of realistic cladding structures in air-core photonic band-gap fibers," Opt. Lett. 29, 349-351 (2004). [CrossRef] [PubMed]
  21. F. Poletti, N. G. R. Broderick, D. J. Richardson, and T. M. Monro, "The effect of core asymmetries on the polarization properties of hollow core photonic bandgap fibers," Opt. Express 13, 9115-9124 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-22-9115. [CrossRef] [PubMed]
  22. C. J. S. de Matos, J. R. Taylor, T. P. Hansen, K. P. Hansen, and J. Broeng, "All-fiber chirped pulse amplification using highly-dispersive air-core photonic bandgap fiber," Opt. Express 11, 2832-2837 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-22-2832. [CrossRef] [PubMed]
  23. J. Loegsgaard, N. A. Mortensen, J. Riishede, and A. Bjarklev, "Material effects in airguiding photonic bandgap fibers," J. Opt. Soc. Am. B 20, 2046-2051 (2003). [CrossRef]
  24. R. Guobin, W. Zhi, L. Shuqin, and J. Shuisheng, "Mode classification and degeneracy in photonic crystal fibers," Opt. Express 11, 1310-1321 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-11-1310 [CrossRef] [PubMed]
  25. J. Limpert, T. Schreiber, S. Nolte, H. Zellmer, and A. Tunnermann, "All fiber chirped-pulse amplification system based on compression in air-guiding photonic bandgap fiber," Opt. Express 11, 3332-3337 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=oe-11-24-3332. [CrossRef] [PubMed]
  26. C. J. S. de Matos, and J. R. Taylor, "Chirped pulse Raman amplification with compression in air-core photonic bandgap fiber," Opt. Express 13,2828-2834 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-8-2828. [CrossRef] [PubMed]
  27. C. J. Hensley, D. G. Ouzounov, A. L. Gaeta, N. Venkataraman, M. T. Gallagher, and K. W. Koch, "Silica-glass contribution to the effective nonlinearity of hollow-core photonic band-gap fibers," Opt. Express 15, 3507-3512 (2007), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-15-6-3507. [CrossRef] [PubMed]

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