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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 6 — Mar. 12, 2012
  • pp: 6746–6760
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Broadband bandgap guidance and mode filtering in radially hybrid photonic crystal fiber

Yacoub Ould-Agha, Aurelie Bétourné, Olivier Vanvincq, Géraud Bouwmans, and Yves Quiquempois  »View Author Affiliations


Optics Express, Vol. 20, Issue 6, pp. 6746-6760 (2012)
http://dx.doi.org/10.1364/OE.20.006746


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Abstract

Hybrid Photonic Crystal Fibers with a first ring of high index inclusions are studied and compared to both standard air-hole fibers and all solid photonic bandgap fibers. In such new fibers a bandgap-like core mode exists over a wide spectral range and exhibits confinement losses ten orders of magnitude smaller than those of the corresponding all-solid fiber. This particular fiber supports also a core mode guided by modified total internal reflection at long enough wavelengths. The origin and properties of these two kinds of modes are discussed in details. Such a design can also act as a mode filter (as compared to the standard air-hole structure) and could also be used to ease phase matching conditions for nonlinear optics.

© 2012 OSA

1. Introduction

Photonic Crystal Fibers (PCF), also called microstructured fibers, have been widely studied in recent years because of their original optical properties [1

1. J. C. Knight, “Photonic crystal fibres,” Nature (London) 424, 857–851 (2003). [CrossRef]

3

3. P. Russell, “Photonic-crystal fibers,” J. Lightwave Technol. 24, 4729–4749 (2006). [CrossRef]

]. Two guiding mechanisms are usually invoked to explain why light can be efficiently confined to the PCF core: either index-guiding or bandgap-guiding.

In the former, the core refractive index (ncore) is larger than the refractive index that can be associated to the microstructured cladding (nFSM). The mechanism responsible for light guiding is then similar to the one of conventional fibers and it is thus usually referred as the Modified Total Internal Reflection (MTIR) guiding mechanism. The cladding of such fibers is typically made of a periodic array of air-holes embedded within the material background. The linear optical properties of these fibers can be derived from the theory developed for standard step index fibers, having in mind that nFSM depends strongly on the wavelength. Many original optical effects have been emphasized in the literature: endlessly single-mode behavior [4

4. T. A. Birks, J. C. Knight, and P. St. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997). [CrossRef] [PubMed]

], an enlarge chromatic dispersion control [5

5. G. Renversez, B. Kuhlmey, and R. McPhedran, “Dispersion management with microstructured optical fibers: ultraflattened chromatic dispersion with low losses,” Opt. Lett. 28, 989–991 (1997). [CrossRef]

], supercontinuum generation [6

6. J. M. Dudley and J. R. Taylork, Supercontinuum Generation in Optical Fibers (Cambridge University Press, 2010). [CrossRef]

], high numerical aperture fibers [7

7. G. Bouwmans, R. M. Percival, W. J. Wadsworth, J. C. Knight, and P. St. J. Russel, “High-power Er:Yb fiber laser with very high numerical aperture pump-cladding waveguide,” Appl. Phys. Lett. 83, 817–818 (2003). [CrossRef]

], etc.

On the opposite, the second type of fibers have a core refractive index smaller than nFSM. The efficient confinement of the light into the fiber core is often explained by the existence of photonic bandgaps for the periodical cladding structure [1

1. J. C. Knight, “Photonic crystal fibres,” Nature (London) 424, 857–851 (2003). [CrossRef]

3

3. P. Russell, “Photonic-crystal fibers,” J. Lightwave Technol. 24, 4729–4749 (2006). [CrossRef]

]. Such a confinement for finite spectral windows leads to very unusual dispersive and spectral properties. Among these fibers, All Solid Photonic BandGap Fibers (ASPBGF) are particularly interesting since they associate the advantages of solid fibers with the amazing optical properties of PBGFs [8

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

,9

9. 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]

]. For example these fibers have been already used to realize tunable bandpass filters [10

10. B. W. Liu, M. L. Hu, X. H. Fang, Y. F. Li, L. Chai, J. Y. Li, W. Chen, and C. Y. Wang, “Tunable bandpass filter with solid-core photonic bandgap fiber and bragg fiber,” IEEE Photon. Technol. Lett. 20, 581–583 (2008). [CrossRef]

], femtosecond mode locked Ytterbium laser [11

11. A. Isomaki and O. G. Okhotnikov, “Femtosecond soliton mode-locked laser based on ytterbium-doped photonic bandgap fiber,” Opt. Express 14, 9238–9243 (2006). [CrossRef] [PubMed]

] and laser / amplifier working in specific wavelength ranges [12

12. A. Wang, A. K. George, and J. C. Knight, “Three-level neodymium fiber laser incorporating photonic bandgap fiber,” Opt. Lett. 31, 1388–1390 (2006). [CrossRef] [PubMed]

14

14. C. B. Olausson, A. Shirakawa, M. Chen, J. K. Lyngso, J. Broeng, K. P. Hansen, A. Bjarklev, and K. Ueda, “167 W, power scalable ytterbium-doped photonic bandgap fiber amplifier at 1178nm,” Opt. Express 18, 16345–16352 (2010). [CrossRef] [PubMed]

]. If the cladding is made of high index inclusions embedded in a low index matrix (as most of the practical cases of all solid PBGFs), the high loss transmission regions can be simply estimated thanks to the elegant ARROW model [15

15. 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]

, 16

16. 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]

]. Note that more complicated structures (as the Kagome one [17

17. 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]

]) can also guide light in a low index core despite that the cladding structure has no bandgap: the core mode coexists then with cladding modes at the same wavelength and propagation constant. The confinement in such structure is still possible because of the low density of cladding modes and the weak interaction between them and the core mode [17

17. 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]

].

Naturally several research groups have developed the concept of so called hybrid fibers that mixes both index and bandgap guiding mechanisms. First, Cerqueria et al. have studied an air-hole PCF where one line of holes is substituted by germanium-doped high index inclusions [18

18. A. Cerqueira, F. Luan, C. M. B. Cordeiro, A. K. George, and J. C. Knight, “Hybrid photonic crystal fiber,” Opt. Express 14, 926–931 (2006). [CrossRef]

] while Xiao et al. [19

19. L. Xiao, W. Jin, and M. S. Demokan, “Photonic crystal fibers confining light by both index-guiding and bandgap-guiding: hybrid PCFs,” Opt. Express 15, 15637–15647 (2006). [CrossRef]

] have also studied the opposite structure (one line of air holes, all the other inclusions being high index inclusions). In both cases, the bandgap and MTIR guiding mechanism coexist for all the wavelengths, the fiber being hybrid as the confinement is obtained in one direction by one mechanism and by the other mechanism for the other directions. Such fibers have been used to study nonlinear interaction between two different bandgaps in the femtosecond regime [20

20. A. Cerqueira, C. M. B. Cordeiro, F. Biancalana, P. J. Roberts, H. E. Hernandez-Figueroa, and C. H. Brito Cruz, “Nonlinear interaction between two different photonic bandgaps of a hybrid photonic crystal fiber,” Opt. Lett. 33, 2080–2082 (2008). [CrossRef]

] or to obtain broadband single-polarization guidance [21

21. A. Cerqueira, D. G. Lona, I. de Oliveira, H. E. Hernandez-Figueroa, and H. L. Fragnito, “Broadband single-polarization guidance in hybrid photonic crystal fibers,” Opt. Lett. 36, 133–135 (2011). [CrossRef]

].

In 2007, J. Sun and C.C. Chan had investigated numerically a design in which liquid crystal is inserted into the holes of an air-silica PCF in order to obtain a fiber that guides for all wavelengths and in all directions by MTIR for one polarisation and by BG in the orthogonal polarization [22

22. J. Sun and C. C. Chan, “Hybrid guiding in liquid-crystal photonic crystal fibers,” J. Opt. Soc. Am. B 24, 2640–2646 (2007). [CrossRef]

].

During the same year, we proposed another hybrid PCF for which light is confined (in all directions and polarisations) by MTIR at long wavelengths and by the bandgap mechanism at short wavelengths [23

23. M. Perrin, Y. Quiquempois, G. Bouwmans, and M. Douay, “Coexistence of total internal reflexion and bandgap modes in solid core photonic bandgap fibre with intersticial air holes,” Opt. Express 15, 13783–13795 (2007). [CrossRef] [PubMed]

]. The principle behind this effect lies in the fact that the cladding index nFSM is greater than ncore except for wavelengths higher than a specific cut-off value. To enable this particular effect, the fiber cross-section can be made of either an array of high index rods surrounded by six air-holes (honey-comb structure) [24

24. A. Bétourné, G. Bouwmans, Y. Quiquempois, M. Perrin, and M. Douay, “Improvements of solid-core photonic bandgap fibers by means of interstitial air holes,” Opt. Lett. 32, 1719–1721 (2007). [CrossRef] [PubMed]

] or the negative snapshot (an air-hole surrounded by six high-index rods) [25

25. A. Bétourné, A. Kudlinski, G. Bouwmans, O. Vanvincq, A. Mussot, and Y. Quiquempois, “Control of super-continuum generation and soliton self-frequency shift in solid-core photonic bandgap fibers,” Opt. Lett. 34, 3083–3085 (2009). [CrossRef] [PubMed]

]. Note that honeycomb structures have been also studied in the past [26

26. J. Lægsgaard and A. Bjarklev, “Doped photonic bandgap fibers for short-wavelength nonlinear devices,” Opt. Lett. 28, 783–785 (2003). [CrossRef] [PubMed]

] but the condition of a high index defect core in order to ensure a guided core mode in the MTIR region was not then fulfilled [23

23. M. Perrin, Y. Quiquempois, G. Bouwmans, and M. Douay, “Coexistence of total internal reflexion and bandgap modes in solid core photonic bandgap fibre with intersticial air holes,” Opt. Express 15, 13783–13795 (2007). [CrossRef] [PubMed]

].

Firstly, we have outlined in Ref. [24

24. A. Bétourné, G. Bouwmans, Y. Quiquempois, M. Perrin, and M. Douay, “Improvements of solid-core photonic bandgap fibers by means of interstitial air holes,” Opt. Lett. 32, 1719–1721 (2007). [CrossRef] [PubMed]

] that our particular hybrid designs greatly enhance the linear optical properties as compared to the corresponding ASPBGF (reduction of several orders of magnitude of confinement and bending losses) and secondly, that phase matching conditions required for efficient frequency doubling or tripling can be fulfilled between fundamental like modes for well-designed hybrid fibers [27

27. A. Bétourné, Y. Quiquempois, G. Bouwmans, and M. Douay, “Design of a photonic crystal fiber for phase-matched frequency doubling or tripling,” Opt. Express 16, 14255–14262 (2008). [CrossRef] [PubMed]

]. More recently we have also demonstrated that this kind of hybrid fibers allows the soliton self frequency shift dynamic to be controlled [28

28. O. Vanvincq, A. Kudlinski, A. Betourne, Y. Quiquempois, and G. Bouwmans, “Extreme deceleration of the soliton self-frequency shift by the third-order dispersion in solid-core photonic bandgap fibers,” J. Opt. Soc. Am. B 27, 2328–2335 (2010). [CrossRef]

] and thus the supercontinuum spectral extension to be manipulated without significant power loss [25

25. A. Bétourné, A. Kudlinski, G. Bouwmans, O. Vanvincq, A. Mussot, and Y. Quiquempois, “Control of super-continuum generation and soliton self-frequency shift in solid-core photonic bandgap fibers,” Opt. Lett. 34, 3083–3085 (2009). [CrossRef] [PubMed]

].

In this paper, we propose a more simple design where only the first ring around the core exhibits inclusions of higher refractive index than the matrix, the other outer rings being composed by air-holes. A scheme of this fiber is shown in Fig. 1 (Fiber RH). Although no true bandgap can be defined in this particular fiber (only one ring of high index rods), we demonstrate that a bandgap-like mode can exist over a large spectral range, with relatively low loss and that the transmission window of this core mode is significantly extended as compared to true photonics band gap fibers and other hybrid fibers. Such a fiber shares also some of its amazing optical properties with the fiber of ref. [27

27. A. Bétourné, Y. Quiquempois, G. Bouwmans, and M. Douay, “Design of a photonic crystal fiber for phase-matched frequency doubling or tripling,” Opt. Express 16, 14255–14262 (2008). [CrossRef] [PubMed]

], but it is much easier to realize as free of interstitial holes: one can then use the usual and well controlled stack-and-draw technique since the first ring is composed by only 6 Ge-doped rods and the other rings are made of regular air holes as in conventional PCFs cladding. We demonstrate also that this first ring placed around the silica core act as a mode filter as compared to the equivalent air-hole fiber with a 7 defect core (see Fig. 1).

Fig. 1 Cross sections of the radially hybrid fiber (RH) and the corresponding standard all-solid PBGF (ASPBG) and the air silica 7 defects core (7D). Darker blue corresponds to higher refractive index.

The outline of this paper is the following: in section 2, we will describe the linear properties of fiber RH and point out the differences with those of the corresponding All Solid Photonic Bandgap Fiber (ASPBGF). Section 3 will be focused on the discussion and explanations of the original behavior of fiber RH. A comparison with a 7 defect core air-hole fiber will be exposed in section 4 before concluding.

2. General features of the radially hybrid PCF

The fiber under investigation is sketched in Fig. 1 (labelled Fiber RH). It consists of a standard PCF in which the first ring of air-holes is replaced by 6 high index rods. The corresponding ASPBGF is also shown in Fig. 1. For the sake of simplicity, and to have a general under-standing of such a hybrid fiber, material dispersion is not taken into account, so that dispersion curves can be easily rescaled. The background refractive index is fixed to 1.45. The pitch Λ of the hexagonal lattice is normalized to 1 μm. Each high index rods presents a parabolic refractive index profile with a maximum refractive index in its center equal to 1.48. Except when mentioned elsewhere, the air-holes diameter d is taken equal to the one of the high index rods fixed to 0.7 μm (d/Λ = 0.7). The total number of rings has been limited here to 4 for both structures in order to reduce calculation time. Obviously the Confinement Loss (CL) could be easily decreased by adding more rings without changing qualitatively the results presented below.

In order to point out the specific properties of the proposed hybrid design we have studied the optical linear properties of both fibers (Fiber RH and ASPBG). To do so, numerical simulations were performed using a commercial finite element software, the CL of the modes being obtained thanks to a circular anisotropic perfectly matched layer added to the ends of the computational domain [29

29. Y. Ould-Agha, F. Zolla, A. Nicolet, and S. Guenneau, “On the use of PML for the computation of leaky modes : an application to microstructured optical fibres,” Int. J. Computation Math. Elec. Electron. Engineer. 27, 95–109 (2008). [CrossRef]

]. We also calculated the Density Of States (DOS) associated to the cladding of the ASPBG fiber thanks to a plane wave expansion method similar to those of ref. [30

30. G. J. Pearce, T. D. Hedley, and D. M. Bird, “Adaptative curvilinear coordinates in a plane-wave solution of Maxwell’s equations in photonic crystals,” Phys. Rev. B 71, 195108 (2005). [CrossRef]

]. The result is plotted in Fig. 2, lighter regions corresponding to higher DOS.

Fig. 2 Bottom: Density Of States (DOS) for the all-solid PBGF. Lighter regions correspond to higher DOS. Blue curve: effective index of the fundamental core mode in the first BG, red curve: dispersion curve of the highest effective index Block mode. TOP: confinement losses of the fundamental core mode in dB/km.

The effective index of the Bloch mode with the highest value is emphasized in red and will be refereed here after as nFSM ASPBG. As mentioned before, this particular effective index is generally associated with the refractive index of the equivalent homogeneous cladding material [4

4. T. A. Birks, J. C. Knight, and P. St. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997). [CrossRef] [PubMed]

]. As expected, nFSM ASPBG is greater that the core index (green curve) for all λ since the cladding is made here of an array of high index inclusions: thus the ASPBG fiber, as it is well known, can not support core mode guided by the MTIR mechanism. We will show in the following that the situation is different in the case of our hybrid fiber.

The refractive index as well as the CL of the guided core mode in the first BandGap (BG) of the ASPBG fibre are plotted in blue on Fig. 2. Both curves have the typical shape associated to bandgap modes [9

9. 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]

] and the guided core mode exists mainly inside the BG of the infinite periodical structure. Note that the CL are here very high compared to published realisations as we work here in the 1st BG while using a large d/Λ ratio (and so a small ratio Λ/λ) and a reduced number of rings.

2.1. Bandgap-like guided core mode (BG-like mode)

In Fig. 3 are plotted the effective index and the CL for fiber RH (black lines). These numerical studies show that, in the 1st BG window (typically for λ/Λ in [0.2,0.5]), the hybrid fiber presents a guided core mode strongly analogous to the fundamental mode of the ASPBG fiber. Indeed the mode profiles of both fibers (2 first sets of snapshots on Fig. 4) as well as their effective indices (see Fig. 3 and its inset) are very similar. This mode will thus be refereed here after as the BG-like mode of the hybrid fiber. Note that for λ/Λ = 0.6 (where the DOS is not zero in Fig. 2) the intensity pattern for the ASPBG core mode begins to couple to cladding modes since the energy begins to locate itself in some high index inclusions (Fig. 4); for greater values of λ/Λ, this core mode disappears in favor of cladding modes (strong coupling).

Fig. 3 Bottom: In black, effective indices of the BG-like mode (solid line) and MTIR-like mode (dashed line) of the fiber RH. Green lines correspond to the effective indices of the 12 cladding modes that can be associated to LP01 supermodes of the 6 high index rods. Red curves represent the effective index of the fundamental space filling mode calculated for an infinite structure with the periodicity of the Fiber ASPBG cladding (top curve) and with the periodicity of the Fiber 7D cladding (bottom curve). The dispersion curve of the Fiber ASPBG fundamental mode is plotted in blue in the first bandgap (red hatched area), the inset being a zoom to facilitate the comparison between this mode and the BG-like mode of the fiber RH. Top: Corresponding core modes confinement losses.
Fig. 4 Bottom: Intensity patterns of the Fiber RH BG-like core mode at λ/Λ = 0.25 (A2), 0.4 (B2), 0.6 (C2), 1.25 (D2) and 1.5 (E2). Only a quarter of the cross-section is shown. Top: Corresponding intensity pattern of the Fiber ASPBG fundamental core mode.

However two main differences have to be pointed out : (i) the CL of this hybrid fiber mode is dramatically reduced (at least 10 orders of magnitude) as compared to fiber ASPBG, and (ii) the domain where the guided core mode exists is extended far beyond the long wavelength edge of the 1st BG (a core mode still exists at λ/Λ = 1.5 see Fig. 3 and Fig. 4 E2). Note that there is a numerical artefact in the CL for short wavelengths because the CL have reached the numerical lower limit estimated to about 5 × 10−8 dB/km (see Fig. 3).

2.2. Index-guided core mode (MTIR-like mode)

Fig. 5 Intensity patterns of the Fiber RH MTIR-like core mode below its cut-off at λ/Λ = 0.25 (A1) and 0.4 (B1) and above its cut-off at λ/Λ = 0.6 (C1), 1.25 (D1) and 1.5 (E1) (only a quarter of the cross-section is shown).

The CL of this MTIR-like mode, shown in Fig. 3 in dashed lines, increases with the wavelength. The level of losses is relatively low (compared to the BG mode of the ASPBG fiber) and reaches ≈ 1 dB/m at λ/Λ = 1.56. Note that this level of loss can be easily reduced by simply adding more rings of air holes.

2.3. Group velocity dispersion and effective area

To finalize this first description of the hybrid fiber properties, let’us describe briefly the chromatic Group Velocity Dispersions (GVD) and effective areas of the MTIR and BG-like modes plotted respectively in dashed and continuous lines in Fig. 6. The MTIR-like mode presents a relatively flat and anomalous dispersion between 0.8 and 1.8, the maximum value of about 80 ps/(nm.km) being reached at λ/Λ = 1.35, the zero GVD being obtained around λ/Λ = 0.53. Its effective area is also relatively flat between λ/Λ ∈ [0.5,1.8] with a mean value of about 5 μm2.

Fig. 6 Chromatic group velocity dispersions and effective areas for the two guided modes of the RH fiber. Dashed lines: MTIR mode, continuous line: BG core mode. The vertical red line indicates the cut-off wavelength of the MTIR-like mode.

3. Discussion

3.1. Models for predicting the transmission windows of PBGF

Before investigating further these two kinds of modes of our hybrid design, let’s recall that two methods are generally used to predict roughly the spectral transmission windows: ever by exploiting the bandgap diagram associated to the cladding or by using the Anti Resonant Reflection Optical Waveguide (ARROW) model.

In the latter, the transmission bands are deduced from the optical properties of the high index inclusions of the cladding considered as isolated waveguides [15

15. 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]

]. Such isolated waveguides present modes that are mainly confined in the high index regions until they reach their cut-off wavelength. At their cut-off, the modes not only spread significantly out in the background material but their effective indices, equal to the background one, are then very close to the ASPBGF core mode one. Both phenomena facilitate the coupling between the core mode and these cladding modes leading to high CL for the core mode. Thus high transmission regions are then assumed to be located only between cut-off wavelengths. Note that the ARROW model has been recently extended by Renversez et al. to take into account that the indices equality for the core and isolated rod modes appears just below the cut-off of these latter (ie with the leaky modes of the isolated rods) [16

16. 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]

]. This model is very handily as the cut-off wavelengths of the inclusions are usually well known. On the other hand, it gives only approximative results especially because the coupling between the rods are completely dismissed.

On the contrary, these coupling are taken into account in the band diagram approach. However this procedure requires to calculate numerically the modes of an infinite periodic material having the same periodicity than the fiber cladding. Bloch modes are then solutions of Maxwell equations. Regions in the 2D plane (neff,λ/Λ) where no Bloch mode can exist are defined as photonic bandgaps of the medium. In such regions, the light cannot escape from the core and is reflected back by the cladding. This model can be seen as a collective model where all the high index rods of the cladding are coupled to each other. For large enough effective indices, Bloch modes (allowed bands) are approximately linear combinations of same LPlm modes of high index inclusions with well defined relative phases imposed by the periodicity [23

23. M. Perrin, Y. Quiquempois, G. Bouwmans, and M. Douay, “Coexistence of total internal reflexion and bandgap modes in solid core photonic bandgap fibre with intersticial air holes,” Opt. Express 15, 13783–13795 (2007). [CrossRef] [PubMed]

, 31

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

, 32

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

]. For example, the supermodes of the first allowed band (see Fig. 2) are made of LP01 modes.

Nevertheless, numerical simulations taking into account the finite size of the cladding (eg FEM method) are needed to accurately predict the transmission windows of a real fiber, particularly for the 1st BG. Indeed, the ARROW model predicts that the first BG would be semi-infinite since LP01 mode has no cut-off. On the contrary, the model based on the BG diagram underestimates the transmission window width as it can be seen in Fig. 2 where the core mode continues to exist in a region where the DOS is different from zero. This last phenomenon can be explained as follow. The allowed bands in the 2D plane (neff,λ/Λ) form a continuum of cladding modes in the case of an infinite array but these bands are reduced to a finite number of supermodes in a real structure. In the example of the ASPBGF of Fig. 1 (60 high index inclusions), the 1st allowed band is constituted of 120 LP01 supermodes (the factor 2 coming from taking into account polarization effects). These allowed modes bands have smaller width than their continuous counterparts obtained for an infinite cladding. Thus, decreasing the number of high index rods will reduce the number of cladding modes and so one can expect an increase of the spectral width where the guided core mode can exist. Another reason to explain that the core mode can subsist beyond the bandgap of the infinite structure is that the interaction of the core mode with cladding modes can be null (for symmetry reasons) or very weak so that the effective index curve of the core mode simply crosses these particular cladding modes curves without impacting significantly on the core mode CL, effective area and GVD.

3.2. Cladding modes of the hybrid fiber

Given the radial structure of the Fiber RH, its cladding modes can be schematically divided into 2 main categories. The first one concerns cladding modes with their intensity localized mainly in the 6 high index rods : these modes can be interpreted in terms of supermodes as it is done for ASPBG fibers [31

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

,32

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

], the first 12 supermodes of the Fiber RH being discussed in more details hereafter. The modes of the second type have their intensity mainly localized in the air-silica region (ie in the 3 rings of air holes) and have a behaviour similar to the ones of a conventional air-silica PCF. Thus lots of these cladding modes can exist with their exact properties depending on the number of air-hole rings, their d/Λ ratio and their potential hybridation with core and 1st ring modes. However, to extract the essential information and to present a reasoning that holds whatever the air-holes cladding structure (see section 3.4 and 4), the first cladding mode of this second category will be approximated by the fundamental cladding mode of an infinite air-holes cladding nFSM AH (calculated with the PWE method and shown in Fig. 3). Note that this mode in the real fiber has an effective index close to nFSM AH if couplings with other modes are weak and if the number of rings is large enough.

Let us now focus on the cladding modes that can be associated to the high index rods. Because the hybrid fiber studied here contains only 6 high index inclusions (cf. Fig. 1), the LP01 supermodes band is made of only 12 cladding modes. The dispersion of these modes are plotted in green on Fig. 3. The modes gather in 4 groups: group I, II, III, and IV ranked with decreasing effective indices. The group I and IV are made of two degenerate modes with linear orthogonal polarizations (see Fig. 7). Groups II and III are both composed of 4 modes, two of them being strictly degenerate so that only 3 curves are distinguishable for each of these groups in Fig. 3.

Fig. 7 Electric field (norm) at λ/Λ = 0.25 of the 12 cladding modes identified by the group number, same number meaning the same effective index (degenerate modes).

Such behaviour can be understood using symmetry arguments similar to the ones used in conventional coupled modes theory [33

33. K. Otsuka, “Self-induced phase turbulence and chaotic itinerancy in coupled laser systems,” Phys. Rev. Lett. 65, 329–332 (1990). [CrossRef] [PubMed]

35

35. L. Jin, Z. Wang, Y. Liu, G. Kai, and X. Dong, “Ultraviolet-inscribed long period gratings in all-solid photonic bandgap fibers,” Opt. Express 16, 21119–21131 (2008). [CrossRef] [PubMed]

]. In our case, the cycle condition imposes that the polarisation direction can only change from one rod to the neighbouring one by a factor equal to Δϕm = 2πm/6 with m an integer between 0 and 5. More precisely the group I is made of the 2 modes for which the polarisation is identical for all rods (m=0) while the group IV relates to a π rotation of the polarisation between adjacent rods (m=3). The group II and III correspond respectively to a rotation of ±π/3 (m=1 and 5) and ±2π/3 (m=2 and 4).

At short wavelengths, these 12 modes have effective indices higher than the silica core one and their mode profiles present, as expected, high intensity within the 6 high index rods (cf. Fig. 7 obtained for λ/Λ = 0.25). As the wavelength increases, the light spreads more and more out of the Ge-doped rods especially once the modes effective indices are smaller than the silica core. Figure 8 illustrates such a behavior for λ/Λ = 1.0.

Fig. 8 Electric field (norm) at λ/Λ = 1 of the 12 cladding modes identified by the group number, same number meaning the same effective index (degenerate modes).

3.3. The MTIR-like mode of the hybrid fiber

The most interesting feature visible on Fig. 8 concerns modes of group I. Indeed this group exhibits an intensity profile very similar to a conventional LP01 centered in the low index fiber core. Thus the MTIR-like mode described in section 2.2 is in fact the group I cladding modes above the cut-off wavelength λ/Λ = 0.53 at which their effective indices cross the silica core one. Note that a similar behaviour has been reported in ref. [23

23. M. Perrin, Y. Quiquempois, G. Bouwmans, and M. Douay, “Coexistence of total internal reflexion and bandgap modes in solid core photonic bandgap fibre with intersticial air holes,” Opt. Express 15, 13783–13795 (2007). [CrossRef] [PubMed]

] concerning a fiber having a honeycomb structure. It was then attributed to the fact that the fiber could guide light by MTIR mechanism at long enough wavelengths because the nFSM of the periodical cladding was therefore smaller than the silica core index. In our present case, this simple picture may be not valid, since the fiber cladding is now more complex (and not just a periodical structure) with a first ring of high index rods and the other rings made of air-holes. Another analogy can be made with three layers standard step index fibers whose transverse index profile is reported in ref. [36

36. C. Zhao, R. Peng, Z. Tang, Y. Ye, L. Shen, and D. Fan, “Model fields and bending analyses of three-layer large flattened mode fibers,” Opt. Commun. 266, 175–180 (2006). [CrossRef]

]. These fibers have been studied to obtain large flattened mode for high power high beam quality fiber laser. In such fibers, the fundamental mode that is confined within the annular high index waveguide transforms itself above a cut-off wavelength into a Gaussian-like mode that extends within the whole central core. At the cut-off, the mode shape is flattened (regime of interest for Zhao et al [36

36. C. Zhao, R. Peng, Z. Tang, Y. Ye, L. Shen, and D. Fan, “Model fields and bending analyses of three-layer large flattened mode fibers,” Opt. Commun. 266, 175–180 (2006). [CrossRef]

]). Note that this particular behaviour is also observed in the case of our hybrid fiber at the cut-off wavelength λ/Λ = 0.53 where the intensity pattern of the highest cladding mode is flattened within the silica core. This behaviour however will not be discussed in details in the present paper for consistency. Note that fiber RH presents some advantages compared to the 3 layers standard step index fiber: the stack-and-draw technique used usually to realize such hybrid fibers leads to a more flexible choice in the opto-geometrical parameters since the index contrasts between the core, the high index rods and the microstructures cladding are usually better controlled.

3.4. The BG-like mode of the hybrid fiber

Let’s now focus on the BG-like core mode in fiber RH. As pointed out earlier, this mode behaves like the fundamental core mode of ASPBG fiber (λ/Λ ∈ [0.2; 0.5]), but extends far beyond the bandgap edge shown in Fig. 3. The first cladding modes that could limit the transmission band of this mode are the modes of group IV, their effective index curves intersecting the one of the BG-like mode at point C2 of Fig. 3 (λ/Λ = 0.6). However, due to symmetry reasons, the antisymmetric cladding modes of group IV cannot couple to the symmetric BG-like core mode [35

35. L. Jin, Z. Wang, Y. Liu, G. Kai, and X. Dong, “Ultraviolet-inscribed long period gratings in all-solid photonic bandgap fibers,” Opt. Express 16, 21119–21131 (2008). [CrossRef] [PubMed]

]. In this case, the BG-like core mode crosses these cladding modes without any interactions. This behaviour is confirmed by the GVD and effective area curves shown in Fig. 6 where no significant slope modification can be seen at λ/Λ = 0.6.

Even if it is not obvious in the effective index curve, this effect is depicted without any ambiguity in the GVD and the effective area curves shown in Fig. 6. Indeed, the slope of the curve showing the BG-like mode effective area as a function of wavelength changes for wavelength values greater than 1.5 μm and the GVD experience a huge unexpected decrease with a minimum about −1200 ps/nm.km at 1.58 μm.

To confirm this effect, two other air-hole diameters have been tested (d/Λ = 0.5 and d/Λ = 0.9), while keeping the diameter of the high index inclusions constant (d/Λ = 0.7). In Fig. 9 are plotted in red nFSM for the 3 structures together with the effective indices of the BG-like core modes in dark. At short wavelengths, the effective indices of these core modes converge to the same value while they diverge at high wavelengths. As can be seen, the BG-like mode effective index curves intersect nFSM curves for increasing wavelength values as the diameter of the air-holes increases (≈ 1.0 μm for d/Λ = 0.5, and ≈ 2.0 μm for d/Λ = 0.9).

Fig. 9 Effective index of the BG core modes for 3 values of dair/Λ (in black). Corresponding effective index of the fundamental space filling mode of the air-hole cladding (in red).

Figure 10 shows the corresponding effective areas of the BG-like core modes and their GVD. This graph confirms that the BG-like mode experiences anti-crossings with cladding modes having effective indices close to the nFSM AH of the air-hole cladding. Indeed near the point where the two effective index curves intersect, the effective area increases significantly while the GVD reaches a minimum. One can notice also that the minimal value of the GVD decreases as the air-hole diameter increases with a value as high as −1900 ps/nm.km for d/Λ = 0.9 at 1.9 μm.

Fig. 10 Group Velocity Dispersion of the BG core modes for three values of dair/Λ (bottom). Corresponding effective area (top).

The limiting factor for the BG-like core mode transmission is then rather the coupling with the FSM of the air-hole cladding than the coupling with the high index modes of the first cladding. As visible on Fig. 10, this factor can be simply modified by changing the d/Λ ratio of the air-holes. Note that the very large transmission bandwidth of this mode strongly suggests a poor sensitivity of the fiber to small fabrications errors.

4. Comparison with a 7 defect core fiber

In this section, we carry on discussing on the two kinds of Fiber RH core modes by comparing their linear properties with the ones of a 7 defect core fiber with the same air-holes cladding (fiber 7D in Fig. 1). This study was motivated by the following idea: at wavelengths λ large enough compared to the inclusions diameter d, the light will stop to resolve the high index inclusions before the air holes, simply because of the huge index contrast difference between these two kinds of inclusions. Thus at long enough wavelengths, we expect the fiber RH to share some similarities with the associated fiber 7D. The results of this study are summarized on Fig. 11: the effective indices and the confinement loss of the BG and MTIR-like core mode of fiber RH are plotted respectively in continuous black line and dashed black line whereas the same quantities for the LP01 and LP02 modes of fiber 7D are plotted in blue. Note that LP notation is not really appropriate here since the index contrast is high between air and silica and that the symmetry is C6v, but for clarity, we will continue with this notation.

Fig. 11 Comparison of effective indices for the core and cladding modes of the seven defects air-hole photonic crystal fiber with those of the Fiber RH (top). Corresponding confinement losses are plotted above. Only cladding modes II2 and III2 of the RH fiber are plotted for clarity.

However, the most interesting features about this comparison between fiber RH and fiber 7D appear in the short wavelengths regime (typically in the first bandgap window associated with fiber ASPBG). Indeed, as can be seen in Fig. 11, the first cladding modes of fiber RH (group I to IV) experience a kind of cut-off so that, in this spectral range, the BG-like mode can be considered as the fundamental core mode since it is the mode with the highest effective index having its intensity mainly localized within the core. On the contrary the LP02 mode of fiber 7D can not obviously be considered as the fundamental mode of this fiber. Thus the 6 high index rods of our hybrid design act as a core mode filter in the short wavelengths regime. Note that both MTIR and BG-like modes of fiber RH can thus be considered as fundamental mode in their respective wavelength domains.

5. Conclusions

For the first time to our best knowledge, we report on a new hybrid fiber structure composed by one ring of six high index rods, the other rings being made of air-holes. We demonstrate that this fiber supports a BG-like core mode similarly to the corresponding ASPBG fibers. However, this BG core mode exists over an extended spectral range, far beyond the conventional bandgap edge of ASPBG fibers (the transmission band is extended by at least 200%). The confinement losses are also drastically reduced (by at least 10 orders of magnitude). Moreover we have shown that this hybrid fiber can support an additional core mode for wavelengths greater than a cut-off value depending mainly on the high index rod parameters. This mode, called MTIR-like mode, is index guided by the holey cladding and its origin and properties have been clearly investigated. By comparing our hybrid design with a 7 defect core fiber, we have demonstrated that our fiber can also act like a mode filter. The unconventional and interesting waveguide dispersion properties arising from the existence of both MTIR and BG-like modes can be also used to ease phase matching conditions, the dispersion curves of these two modes behaving like those of the hybrid fiber with air-holes and high index rods arranged in a honey comb structure and previously studied in ref. [27

27. A. Bétourné, Y. Quiquempois, G. Bouwmans, and M. Douay, “Design of a photonic crystal fiber for phase-matched frequency doubling or tripling,” Opt. Express 16, 14255–14262 (2008). [CrossRef] [PubMed]

].

Acknowledgments

We acknowledge financial support from the Ministry of Higher Education and Research, the Nord-Pas de Calais Regional Council and the FEDER through the “Contrat de Projets Etat Region (CPER) 2007–2013”.

References and links

1.

J. C. Knight, “Photonic crystal fibres,” Nature (London) 424, 857–851 (2003). [CrossRef]

2.

A. Bjarklev, J. Broeng, and A. Sanchez Bjarklev, Photonic crystal fibres (Kluwer Academic Publishers, 2003). [CrossRef]

3.

P. Russell, “Photonic-crystal fibers,” J. Lightwave Technol. 24, 4729–4749 (2006). [CrossRef]

4.

T. A. Birks, J. C. Knight, and P. St. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997). [CrossRef] [PubMed]

5.

G. Renversez, B. Kuhlmey, and R. McPhedran, “Dispersion management with microstructured optical fibers: ultraflattened chromatic dispersion with low losses,” Opt. Lett. 28, 989–991 (1997). [CrossRef]

6.

J. M. Dudley and J. R. Taylork, Supercontinuum Generation in Optical Fibers (Cambridge University Press, 2010). [CrossRef]

7.

G. Bouwmans, R. M. Percival, W. J. Wadsworth, J. C. Knight, and P. St. J. Russel, “High-power Er:Yb fiber laser with very high numerical aperture pump-cladding waveguide,” Appl. Phys. Lett. 83, 817–818 (2003). [CrossRef]

8.

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

9.

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]

10.

B. W. Liu, M. L. Hu, X. H. Fang, Y. F. Li, L. Chai, J. Y. Li, W. Chen, and C. Y. Wang, “Tunable bandpass filter with solid-core photonic bandgap fiber and bragg fiber,” IEEE Photon. Technol. Lett. 20, 581–583 (2008). [CrossRef]

11.

A. Isomaki and O. G. Okhotnikov, “Femtosecond soliton mode-locked laser based on ytterbium-doped photonic bandgap fiber,” Opt. Express 14, 9238–9243 (2006). [CrossRef] [PubMed]

12.

A. Wang, A. K. George, and J. C. Knight, “Three-level neodymium fiber laser incorporating photonic bandgap fiber,” Opt. Lett. 31, 1388–1390 (2006). [CrossRef] [PubMed]

13.

V. Pureur, L. Bigot, G. Bouwmans, Y. Quiquempois, M. Douay, and Y. Jaouen, “Ytterbium-doped solid core photonic bandgap fiber for laser operation around 980 nm,” Appl. Phys. Lett. 92, 061113 (2008). [CrossRef]

14.

C. B. Olausson, A. Shirakawa, M. Chen, J. K. Lyngso, J. Broeng, K. P. Hansen, A. Bjarklev, and K. Ueda, “167 W, power scalable ytterbium-doped photonic bandgap fiber amplifier at 1178nm,” Opt. Express 18, 16345–16352 (2010). [CrossRef] [PubMed]

15.

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]

16.

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]

17.

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]

18.

A. Cerqueira, F. Luan, C. M. B. Cordeiro, A. K. George, and J. C. Knight, “Hybrid photonic crystal fiber,” Opt. Express 14, 926–931 (2006). [CrossRef]

19.

L. Xiao, W. Jin, and M. S. Demokan, “Photonic crystal fibers confining light by both index-guiding and bandgap-guiding: hybrid PCFs,” Opt. Express 15, 15637–15647 (2006). [CrossRef]

20.

A. Cerqueira, C. M. B. Cordeiro, F. Biancalana, P. J. Roberts, H. E. Hernandez-Figueroa, and C. H. Brito Cruz, “Nonlinear interaction between two different photonic bandgaps of a hybrid photonic crystal fiber,” Opt. Lett. 33, 2080–2082 (2008). [CrossRef]

21.

A. Cerqueira, D. G. Lona, I. de Oliveira, H. E. Hernandez-Figueroa, and H. L. Fragnito, “Broadband single-polarization guidance in hybrid photonic crystal fibers,” Opt. Lett. 36, 133–135 (2011). [CrossRef]

22.

J. Sun and C. C. Chan, “Hybrid guiding in liquid-crystal photonic crystal fibers,” J. Opt. Soc. Am. B 24, 2640–2646 (2007). [CrossRef]

23.

M. Perrin, Y. Quiquempois, G. Bouwmans, and M. Douay, “Coexistence of total internal reflexion and bandgap modes in solid core photonic bandgap fibre with intersticial air holes,” Opt. Express 15, 13783–13795 (2007). [CrossRef] [PubMed]

24.

A. Bétourné, G. Bouwmans, Y. Quiquempois, M. Perrin, and M. Douay, “Improvements of solid-core photonic bandgap fibers by means of interstitial air holes,” Opt. Lett. 32, 1719–1721 (2007). [CrossRef] [PubMed]

25.

A. Bétourné, A. Kudlinski, G. Bouwmans, O. Vanvincq, A. Mussot, and Y. Quiquempois, “Control of super-continuum generation and soliton self-frequency shift in solid-core photonic bandgap fibers,” Opt. Lett. 34, 3083–3085 (2009). [CrossRef] [PubMed]

26.

J. Lægsgaard and A. Bjarklev, “Doped photonic bandgap fibers for short-wavelength nonlinear devices,” Opt. Lett. 28, 783–785 (2003). [CrossRef] [PubMed]

27.

A. Bétourné, Y. Quiquempois, G. Bouwmans, and M. Douay, “Design of a photonic crystal fiber for phase-matched frequency doubling or tripling,” Opt. Express 16, 14255–14262 (2008). [CrossRef] [PubMed]

28.

O. Vanvincq, A. Kudlinski, A. Betourne, Y. Quiquempois, and G. Bouwmans, “Extreme deceleration of the soliton self-frequency shift by the third-order dispersion in solid-core photonic bandgap fibers,” J. Opt. Soc. Am. B 27, 2328–2335 (2010). [CrossRef]

29.

Y. Ould-Agha, F. Zolla, A. Nicolet, and S. Guenneau, “On the use of PML for the computation of leaky modes : an application to microstructured optical fibres,” Int. J. Computation Math. Elec. Electron. Engineer. 27, 95–109 (2008). [CrossRef]

30.

G. J. Pearce, T. D. Hedley, and D. M. Bird, “Adaptative curvilinear coordinates in a plane-wave solution of Maxwell’s equations in photonic crystals,” Phys. Rev. B 71, 195108 (2005). [CrossRef]

31.

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

32.

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

33.

K. Otsuka, “Self-induced phase turbulence and chaotic itinerancy in coupled laser systems,” Phys. Rev. Lett. 65, 329–332 (1990). [CrossRef] [PubMed]

34.

H. S. Huang and H. C. Chang, “Guided vector modes of equilateral three-core fibres,” Electron. Lett. 25, 55–56 (1988). [CrossRef]

35.

L. Jin, Z. Wang, Y. Liu, G. Kai, and X. Dong, “Ultraviolet-inscribed long period gratings in all-solid photonic bandgap fibers,” Opt. Express 16, 21119–21131 (2008). [CrossRef] [PubMed]

36.

C. Zhao, R. Peng, Z. Tang, Y. Ye, L. Shen, and D. Fan, “Model fields and bending analyses of three-layer large flattened mode fibers,” Opt. Commun. 266, 175–180 (2006). [CrossRef]

37.

H. C. Park, I. K. Hwang, D. I. Yeom, and B. Y. Kim, “Analyses of cladding modes in photonic crystal fiber,” Opt. Express 15, 15154–15160 (2007). [CrossRef] [PubMed]

38.

C. Chen, A. Laronche, G. Bouwmans, L. Bigot, Y. Quiquempois, and J. Albert, “Sensitivity of photonic crystal fiber modes to temperature, strain and external refractive index,” Opt. Express 16, 9645–9653 (2008). [CrossRef] [PubMed]

OCIS Codes
(060.2280) Fiber optics and optical communications : Fiber design and fabrication
(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers
(060.5295) Fiber optics and optical communications : Photonic crystal fibers

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: November 15, 2011
Revised Manuscript: February 11, 2012
Manuscript Accepted: February 25, 2012
Published: March 8, 2012

Citation
Yacoub Ould-Agha, Aurelie Bétourné, Olivier Vanvincq, Géraud Bouwmans, and Yves Quiquempois, "Broadband bandgap guidance and mode filtering in radially hybrid photonic crystal fiber," Opt. Express 20, 6746-6760 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-6-6746


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References

  1. J. C. Knight, “Photonic crystal fibres,” Nature (London)424, 857–851 (2003). [CrossRef]
  2. A. Bjarklev, J. Broeng, and A. Sanchez Bjarklev, Photonic crystal fibres (Kluwer Academic Publishers, 2003). [CrossRef]
  3. P. Russell, “Photonic-crystal fibers,” J. Lightwave Technol.24, 4729–4749 (2006). [CrossRef]
  4. T. A. Birks, J. C. Knight, and P. St. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett.22, 961–963 (1997). [CrossRef] [PubMed]
  5. G. Renversez, B. Kuhlmey, and R. McPhedran, “Dispersion management with microstructured optical fibers: ultraflattened chromatic dispersion with low losses,” Opt. Lett.28, 989–991 (1997). [CrossRef]
  6. J. M. Dudley and J. R. Taylork, Supercontinuum Generation in Optical Fibers (Cambridge University Press, 2010). [CrossRef]
  7. G. Bouwmans, R. M. Percival, W. J. Wadsworth, J. C. Knight, and P. St. J. Russel, “High-power Er:Yb fiber laser with very high numerical aperture pump-cladding waveguide,” Appl. Phys. Lett.83, 817–818 (2003). [CrossRef]
  8. F. Luan, A. K. George, T. D. Hedley, G. J. Pearce, D. M. Bird, J. C. Knight, and P. St. Russel, “All-solid photonic bandgap fiber,” Opt. Lett.29, 2369–2371 (2004). [CrossRef] [PubMed]
  9. 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. Express13, 8452–8459 (2005). [CrossRef] [PubMed]
  10. B. W. Liu, M. L. Hu, X. H. Fang, Y. F. Li, L. Chai, J. Y. Li, W. Chen, and C. Y. Wang, “Tunable bandpass filter with solid-core photonic bandgap fiber and bragg fiber,” IEEE Photon. Technol. Lett.20, 581–583 (2008). [CrossRef]
  11. A. Isomaki and O. G. Okhotnikov, “Femtosecond soliton mode-locked laser based on ytterbium-doped photonic bandgap fiber,” Opt. Express14, 9238–9243 (2006). [CrossRef] [PubMed]
  12. A. Wang, A. K. George, and J. C. Knight, “Three-level neodymium fiber laser incorporating photonic bandgap fiber,” Opt. Lett.31, 1388–1390 (2006). [CrossRef] [PubMed]
  13. V. Pureur, L. Bigot, G. Bouwmans, Y. Quiquempois, M. Douay, and Y. Jaouen, “Ytterbium-doped solid core photonic bandgap fiber for laser operation around 980 nm,” Appl. Phys. Lett.92, 061113 (2008). [CrossRef]
  14. C. B. Olausson, A. Shirakawa, M. Chen, J. K. Lyngso, J. Broeng, K. P. Hansen, A. Bjarklev, and K. Ueda, “167 W, power scalable ytterbium-doped photonic bandgap fiber amplifier at 1178nm,” Opt. Express18, 16345–16352 (2010). [CrossRef] [PubMed]
  15. 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]
  16. G. Renversez, P. Boyer, and A. Sagrini, “Antiresonant reflecting optical waveguide microstructured fibers revisited: a new analysis based on leaky mode coupling,” Opt. Express14, 5682–5687 (2006). [CrossRef] [PubMed]
  17. 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]
  18. A. Cerqueira, F. Luan, C. M. B. Cordeiro, A. K. George, and J. C. Knight, “Hybrid photonic crystal fiber,” Opt. Express14, 926–931 (2006). [CrossRef]
  19. L. Xiao, W. Jin, and M. S. Demokan, “Photonic crystal fibers confining light by both index-guiding and bandgap-guiding: hybrid PCFs,” Opt. Express15, 15637–15647 (2006). [CrossRef]
  20. A. Cerqueira, C. M. B. Cordeiro, F. Biancalana, P. J. Roberts, H. E. Hernandez-Figueroa, and C. H. Brito Cruz, “Nonlinear interaction between two different photonic bandgaps of a hybrid photonic crystal fiber,” Opt. Lett.33, 2080–2082 (2008). [CrossRef]
  21. A. Cerqueira, D. G. Lona, I. de Oliveira, H. E. Hernandez-Figueroa, and H. L. Fragnito, “Broadband single-polarization guidance in hybrid photonic crystal fibers,” Opt. Lett.36, 133–135 (2011). [CrossRef]
  22. J. Sun and C. C. Chan, “Hybrid guiding in liquid-crystal photonic crystal fibers,” J. Opt. Soc. Am. B24, 2640–2646 (2007). [CrossRef]
  23. M. Perrin, Y. Quiquempois, G. Bouwmans, and M. Douay, “Coexistence of total internal reflexion and bandgap modes in solid core photonic bandgap fibre with intersticial air holes,” Opt. Express15, 13783–13795 (2007). [CrossRef] [PubMed]
  24. A. Bétourné, G. Bouwmans, Y. Quiquempois, M. Perrin, and M. Douay, “Improvements of solid-core photonic bandgap fibers by means of interstitial air holes,” Opt. Lett.32, 1719–1721 (2007). [CrossRef] [PubMed]
  25. A. Bétourné, A. Kudlinski, G. Bouwmans, O. Vanvincq, A. Mussot, and Y. Quiquempois, “Control of super-continuum generation and soliton self-frequency shift in solid-core photonic bandgap fibers,” Opt. Lett.34, 3083–3085 (2009). [CrossRef] [PubMed]
  26. J. Lægsgaard and A. Bjarklev, “Doped photonic bandgap fibers for short-wavelength nonlinear devices,” Opt. Lett.28, 783–785 (2003). [CrossRef] [PubMed]
  27. A. Bétourné, Y. Quiquempois, G. Bouwmans, and M. Douay, “Design of a photonic crystal fiber for phase-matched frequency doubling or tripling,” Opt. Express16, 14255–14262 (2008). [CrossRef] [PubMed]
  28. O. Vanvincq, A. Kudlinski, A. Betourne, Y. Quiquempois, and G. Bouwmans, “Extreme deceleration of the soliton self-frequency shift by the third-order dispersion in solid-core photonic bandgap fibers,” J. Opt. Soc. Am. B27, 2328–2335 (2010). [CrossRef]
  29. Y. Ould-Agha, F. Zolla, A. Nicolet, and S. Guenneau, “On the use of PML for the computation of leaky modes : an application to microstructured optical fibres,” Int. J. Computation Math. Elec. Electron. Engineer.27, 95–109 (2008). [CrossRef]
  30. G. J. Pearce, T. D. Hedley, and D. M. Bird, “Adaptative curvilinear coordinates in a plane-wave solution of Maxwell’s equations in photonic crystals,” Phys. Rev. B71, 195108 (2005). [CrossRef]
  31. T. A. Birks, G. J. Pearce, and D. M. Bird, “Approximate band structure calculation for photonic bandgap fibres,” Opt. Express14, 9483–9490 (2006). [CrossRef] [PubMed]
  32. J. Laegsgaard, “Gap formation and guided modes in photonic bandgap fibres with high-index rods,” J. Opt. A: Pure Appl. Opt.6, 798–804 (2004). [CrossRef]
  33. K. Otsuka, “Self-induced phase turbulence and chaotic itinerancy in coupled laser systems,” Phys. Rev. Lett.65, 329–332 (1990). [CrossRef] [PubMed]
  34. H. S. Huang and H. C. Chang, “Guided vector modes of equilateral three-core fibres,” Electron. Lett.25, 55–56 (1988). [CrossRef]
  35. L. Jin, Z. Wang, Y. Liu, G. Kai, and X. Dong, “Ultraviolet-inscribed long period gratings in all-solid photonic bandgap fibers,” Opt. Express16, 21119–21131 (2008). [CrossRef] [PubMed]
  36. C. Zhao, R. Peng, Z. Tang, Y. Ye, L. Shen, and D. Fan, “Model fields and bending analyses of three-layer large flattened mode fibers,” Opt. Commun.266, 175–180 (2006). [CrossRef]
  37. H. C. Park, I. K. Hwang, D. I. Yeom, and B. Y. Kim, “Analyses of cladding modes in photonic crystal fiber,” Opt. Express15, 15154–15160 (2007). [CrossRef] [PubMed]
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