A novel silica index guiding holey fiber (IGHF) design is proposed utilizing a new defect structure that is composed of an elliptic high index ring structure and an elliptic air-hole at the center with triangular lattice structure. The proposed IGHF showed unique modal properties such as uniform and high birefringence over a wide spectral range and single polarization single mode (SPSM) guidance along with a flat negative chromatic dispersion. Optical waveguide properties were numerically analyzed using the plane wave expansion method in terms of mode intensity distribution, modal birefringence, chromatic dispersion for the new defect structural parameters.
©2005 Optical Society of America
Index guiding holey fibers (IGHFs) consisted of periodic distribution of air holes in silica cross section along with a central defect, have been intensively studied in recent years due to their novel optical properties such as endlessly single-mode operation [1,2], anomalous group-velocity dispersion in the visible region , high non-linearity  to name a few. In conventional IGHFs, the core-guidance of the optical signal is provided by a solid silica defect core surrounded by periodic air-hole arrays in the cladding. Since the core refractive index is higher than the effective cladding index, which is an average of air holes and background silica, light signal can be guided by the total internal reflection along the silica defect core similar to conventional optical fibers. Recently, due to the large index contrast of IGHF compared to the conventional fiber, high birefringence (HB) IGHFs have been reported by breaking the circular symmetry implementing asymmetric defect structures such as dissimilar air hole diameters along the two orthogonal axes [5–7], and asymmetric core design . Modal birefringence in these HB IGHFs has been predicted to have values an order of higher than that of the conventional HB fibers . The polarization and dispersive properties of elliptical hole IGHFs, as shown in Fig. 1(b), have been investigated with plane wave expansion method by Steel et al. [10–11], where single polarization single mode (SPSM) guidance has been predicted at a long wavelength regime. Broeng et al. have introduced a new defect design at the center by placing an elliptic air hole whose dimension is smaller than those of holes in the cladding as represented in Fig. 1(c) in order to further enhance the birefringence . To demonstrate the feasibility of elliptical hole IGHF, Issa et al. have reported fabrication of IGHFs with uniformly oriented elliptical holes and high ellipticity [28–29], where they used a technique that relies on hole deformation by asymmetric stress during the fiber drawing process.
In recent years, the authors have introduced a new type of optical fiber, hollow optical fiber (HOF), with a triple layered structure, a central air hole, germanosilicate ring core, and silica cladding for the versatile photonic device applications [13–15]. Birefringece in elliptic HOF has been also investigated .
In this paper, we present a novel defect design based on HOF structure embedded in IGHF, for the first time to the best knowledge of the authors, and its structure is schematically shown in Fig. 1(a). Based on the already established fabrication techniques of HOFs and the elliptical hole IGHFs, the proposed structure is believed to be attainable with a high feasibility.
In the Fig. 1(a), the defect consists of germanosilicate high index elliptic ring surrounding the elliptic air hole inside, which can endow a new degree of freedom in defect engineering for IGHFs such as hole diameter, (Dx, Dy), ring width (Wring_x, Wring_y), and the ring index (Δ), to control the birefringence and chromatic dispersion simultaneously. We numerically investigated the effects of the proposed defect designs on the bound mode intensity distributions, dispersion relations, the birefringence characteristics, and chromatic dispersion property.
2. Defect parameters and analysis of optical properties
2.1 Impacts of defect parameters over modal birefringence
In the prior elliptic hole IGHFs, the major design parameters were the air hole diameter D, its pitch Λ, and the ellipticity η (the ratio of major to minor axis for the air holes) as in Fig. 1(b) introduced by Steel et al [10,11] and one more defect parameter Dc in Fig. 1(c) introduced by Broeng et al . For the sake of convenience, we will assume the regime η > 1 with the major axis oriented along the y-direction. In contrast to these prior structures, the optical properties of the proposed IGHF with new defect design can be further tailored by three more defect parameters; the width of germanosilicate ring (Wring_x and Wring_y), the diameter of central elliptic hole (Dcx and Dcy) and the index difference between high index doped ring and pure silica Δ = nGe-doped-nsilica as illustrated in Fig. 1(a), which can provide flexible and versatile defect engineering.
We will focus on IGHF design as in Fig. 1(e) for higher birefringence due to the large contrast in the refractive index between the ring core and the air hole. We theoretically analyzed the optical properties of the proposed IGHF along with detailed comparisons with prior structures. The impacts of defect parameters will be also investigated in terms of mode intensity distribution, modal birefringence and its chromatic dispersion property utilizing plane wave expansion method [16–18]. In the following discussions, we will use a convention such that Wring = Wring_y and Dc = Dcy, for the sake of convenience along with ellipticity η = Dy/Dx = Dcy/Dcx = Wring_y/Wring_x.
The calculation accuracy in the plane wave expansion method is influenced by several parameters such as the number of plane waves, the size and shape of supercell, and the tolerance . In our calculations, the optimum parameters were chosen such as; supercell size of 7Λ×4√3Λ as a rectangular shape, the tolenrance of 10-7 and the resolution of 256, which corresponds to the number of plane waves of (7Λ×4√3Λ) × (7×256). The error due to the finite number of the plane waves in calculation of the birefringence was to be less than 5% [16, 19].
To illustrate the mode field distributions in the proposed elliptic hole IGHF with new defect design, we show two fundamental modes with orthogonal polarizations at wavelength λ= 1.31μm, parallel to the major axis (y-axis) in Fig. 2(a), and minor axis (x-axis) in Fig. 2(b), where Λ = 2.2 μm, D/Λ = 0.7, Dc/Λ=0.2, Wring = 2Λ, Δ= 0.013 and ellipticity, η= 4. The effective indices of y-polarization and x-polarization mode are 1.4214 and 1.4150 with a birefringence B = - = 6.4×10-3, respectively. The y-polarization mode is more bound to the central core region compared to x polarization, which can be directly inferred from the effective indices of two polarization modes in reference to the effective cladding index nclad = 1.4144. From the intensity distribution of the modes along the major and minor axes, we could also find that the bound modes are linearly polarized and transverse, similar to prior polarization maintaining fibers. In elliptical hollow optical fiber (E-HOF), high birefringence was found to originate from the boundary conditions for electromagnetic fields at the elliptical air and silica ring core interface .
The electric field intensity distribution directly modifies the effective index of the guided mode. Similar patterns were also observed in our studies with prominent intensity suppression around the hole in x polarization as shown in Fig. 2(b) to result in a lower effective index than the y polarization mode. Due to this polarization dependent disparity in the intensity distribution near the central elliptic hole defect, the difference of effective index between the slow axis and fast axis can be enhanced by optimizing defect parameters, Dc , along with hollow ring defects to further tailor polarization properties.
To compare the polarization mode dispersion of the proposed IGHF with newly introduced defect design (Fig. 1(d), (e)) with the prior elliptic hole IGHF (Fig. 1(b), (c)) the effective indices of the core-guided modes and cladding were calculated as a function of wavelength. The results are summarized in Fig. 3. In calculations, we have assumed the same structural parameters, Λ=2.2μm, D/Λ=0.7, and η=4. The open and solid symbols represent the x, and y polarization modes, respectively. As indicated the electric field intensity distribution in Fig. 2, the y-polarization modes consistently showed a higher effective index than the x-polarization modes..
The effect of the presence of high index ring (Wring≠0) on the dispersion relation is shown in Fig. 3(a) and those of the central elliptic hole (Dc≠0) are shown in Fig. 3(b). The high index ring defect along the minor axis was found to increase birefringence as indicated by solid and open triangle symbols in Fig. 3(a). It is also noteworthy that the cut-off wavelength of x-polarization mode shifted significantly to longer wavelength by more than 50nm. With the high index ring defects, the single polarization single mode (SPSM) operation can be achieved in the wavelength range over 1.7μm and below this cut-off the fiber retains a high birefringence. When the central elliptic hole is furthermore introduced (Dc=0.2Λ), the birefringence was found to further enhanced as indicated by solid and open star symbols in Fig. 3(b). The cut-off wavelength for SPSM condition was found to be shifted to 1.42 μm, which could be of high importance for optical communication and sensing in C, L band.
Based on calculations of the effective indices in Fig. 3, the birefriengence, Δn=∣nx-ny, ∣, was plotted as a function of wavelength in Fig. 4. With the proposed defect structure consisted of both high index rings along the minor axis and central elliptic hole, it is found that the IGHF showed high and uniform birefringence over a very broad wavelength range, λ=1.15 to λ=1.42μm, as shown in solid star symbols in Fig. 4. The similar behavior could be also achieved with a lower birefringence for the case only with central elliptic hole by optimizing the defect parameters, Dc and ellipticity η, which is represented by solid circles.
In prior HB IGHFs, birefringence was found to increase monochromatically with increasing wavelength [10–12]. In contrast, we could obtain a high (~ 6.4×10-3) and uniform birefringence with small deviation less than 10-5 over a wide wavelength range of 270nm in our proposed defect design, which has not been ever reported to the best knowledge of the authors. Note that the wavelength range from 1.15 to 1.42 μm, covers O, S-band for telecommunications and conventional polarimetric sensor optical window centered around 1300nm, which makes the proposed fiber highly potential in optical device and sensor applications. Especially with current development of broadband superluminscent LED, this uniform birefringence can find various applications in polarimetric sensory systems [23, 24]. Major requirements in elliptical hole IGHF for high birefringence have been, in general, confined to large diameter and higher ellipticity of the holes in the cladding . The proposed elliptic hole IGHF with new defect structure as in Fig. 1(e) can provide significantly enhanced flexibility to control the guiding properties such as modal birefringence and chromatic dispersion, which could not be obtained in prior arts.
Firstly we analyzed the impacts of the new defect parameters over the modal birefringence, whose results are summarized in Fig. 5. Birefringence for various defect parameters, ellipticity, η, Wring and Δ were theoretically calculated assuming the fiber structure as in Fig. 1(e), and the results are shown in Fig. 5(a), (b) and (c), respectively. As the ellipticity increases from η=1 to η=4, the birefringence gradually increases by a factor of six to assume the same behavior indicated by the solid stars symbols in Fig. 4. For the case of η=1, the asymmetry in the fiber is only introduced by two hollow ring defects along the minor axis, where the birefringence reached only to 10-4 with monotonic increase with wavelength. The cut-off wavelength for SPSM operation was also found to affected by the ellipticity, and the higher ellipticity resulted in the shorter cut-off wavelength. Birefringent and SPSM region, divided by the cut-off, would find their unique applications and the ability to control the cutoff wavelength obtained in the proposed IGHF structure would be of very high importance in both optical communications and polarimetric sensory systems.
For other parameters, Wring and Δ, we found that the cut-off wavelength did not change, fixed at 1.42 μm, yet only the maginitude of birefringence increased for larger Wring and higher Δ as shown in Fig. 5(b) and (c), respectively. It is observed that these two parameters did not significantly change the spectral uniformity of birefringence over the wavelength range. It is also noteworthy the hollow index ring with depressed refractive index, Δ<0, could also work as a defect as good as raised refractive index case, Δ>0. Depressed refractive index can be realized by doping Fluorine instead GeO2.
Regarding the last defect parameter, Dc, we found that it is playing a critical role on the modal birefringence. Broeng et al  have given a brief qualitative explanation on its role such that smaller hole diameter could induce a higher birefringence than the larger diameter. In our analysis we could find a quantitative value ranges that can alter the modal birefringence. For the optimal case for η=4 as shown in Fig. 4, it is noted that SPSM guidance was obtained in a very wide range λ>λcutoff=760nm if Dc>0.2Λ, irrespective of Wring and Δ. In this case modal birefringence, Δn=∣ nx-ny ∣, is no longer defined. In order to achieve birefringence, therefore, it was required to have Dc<0.2Λ and the analysis showed the maximum birefringence for Dc=0.2Λ. Note this specific optimal condition for Dc has never been reported thus far to the best knowledge of the authors.
From these parametric analysis for defect structure, we could find that uniformity in birefringence is affected by the diameter of central elliptic hole, Dc and ellipticity, η, and the magnitude of birefringence is related to Wring and Δ. Note that these parmetric studies directly shows higher flexibility in birefringence control than prior reports where only the ellipticity and air-hole size in cladding worked as the only structural defect parameters .
2.2 Impacts of defect parameters over chromatic dispersion
Chromatic dispersion control has been another area of key interest in IGHF that drew extensive attentions. Compared with conventional solid silica core fibers, IGHFs posses waveguide design parameters such as hole diameter, D, and the pitch, Λ, with greater flexibility dispersion design to suit the required application. For examples, high anomalous dispersion  in the near-visible and broad band flat dispersion [20–22] have been one of notable developments. Our proposed IGHF structure can further extend these ideas with larger flexibility provided by defect parameters, Dc, Wring, Δ, and η. The chromatic dispersion of the lowest order mode with y polarization was calculated for each of fiber types in Fig.1, and their spectral distributions are shown in Fig. 6. Here we have assumed the following structural parameters; Λ=2.2μm, D/Λ=0.7, η = 4, Wring = λ, Dc= 0.2λ, and Δ = 0.013. The diameter of central hole defect was found to play an important role in chromatics dispersion as in modal birefringence. For the case Dc=0, we observed a negative dispersion slope in S, C, L band, while for Dc=0.2Λ the dispersion slope changed its sign to positive in the optical windows.
In the case of Dc=0 as in Fig. 1(b) and (d), monotonically decreasing dispersion was obtained over S, C, L band in the range of ±3ps/km.nm2 with dispersion slope range of -0.02ps/km.nm2 at 1.55μm as shown in the upper plots in Fig. 6, which is comparable to prior reports with circular holes [10–11, 20–22]. It is noted that the proposed hollow ring defect can significantly alter the dispersion slope to enable detailed dispersion tailoring. In the case of Dc=0.2Λ as in Fig. 1(c) and (e), we could obtain a negative and flat dispersion of D ~-5.5ps/km.nm over a wide wavelength range of 1.42 to 1.65μm and very low dispersion slope less than 0.001ps/km.nm2 at λ=1.55μm as shown in the lower plots of Fig. 6.
Considering the single mode single polarization operation in these conditions as illustrated in Fig. 3(b), the negative flat dispersion,in addition, will make this fiber almost ideal transmission medium for high capacity WDM systems. Impacts of the other defect parameters such as ellipticity η, Wring, and Δ over the dispersion and its slope, have been numerically investigated and the results are summarized in Fig. 7.
As the ellipticity change from η=1 to 4, the overall dispersion value monotonically decrease from +36.5 to -5.07ps/km.nm at 1.55μm, changing the dispersion characteristics from anomalous to normal in S, C, and L band as shown in Fig. 7(a). The proposed fiber with η=3, showed interesting characteristics with a small non-zero dispersion value of 4.06ps/km.nm along with a very low negative dispersion slope of -0.001ps/km.nm2 at λ= 1.55μm. For higher ellipticity η=4, the dispersion properties change their signs such that dispersion and its slope are -5.07ps/km.nm and 0.002ps/km.nm2 , respectively at 1.55μm.
Fixing the ellipticity η=4, we then further investigated the impacts of hollow ring width, Wring, and the results are summarized in Fig. 7(b), As Wring varies from l.0Λto 2.0Λ, the dispersion slope monotonically increases and flat-negative dispersion was attainable only near Wring=Λ. Having η=4, and Wring=Λ, the impacts of the relative index difference, Δ, was also analyzed and the results are shown in Fig. 7(c). As Δ decreases from 0.02 to -0.015, both the chromatic dispersion and its slope decrease. Especially for the depressed refractive index ring defects with Δ=-0.015, dispersion and its slope are -5.57ps/km.nm and 0.0007ps/km.nm2, respectively. Note that in this case of Δ=-0.015, the dispersion is extremely flat over entire C, and L band, which can be directly utilized in WDM transmission dispersion control [26–27].
From these numerical analyses, we confirmed the potential and flexibility of the proposed defect structures in IGHF such that they could provide unique waveguide properties to obtain uniform high birefringence over the wide wavelength region, and a negative flat dispersion over the single polarization single mode operation region, simultaneously.
We have demonstrated a novel type of uniform birefringent and negative dispersion flattened IGHF introducing new defect design based on hollow index ring along with central hole defect. The impacts of the defect structure parameters, such as ring width, Wring, ring refractive index, Δ, ellipticity, η, for the hollow ring defect and central hole defect diameter, Dc, over modal birefringence and chromatic dispersion have been numerically analyzed. Optimizing the defect parameters, we have obtained uniform and high birefringence of 6.4×10-3 over the wavelength range 1.15 to 1.42μm for Λ=2.2μm, D/Λ=0.7, η=4, Wring=2Λ, Dc=0.2Λ, and Δ=0.013. This uniform birefringence can be of high importance for broadband WDM and polarimetric sensory systems. We have also obtained a negative and flat dispersion of -5.5ps/km.nm for the wavelength range 1.42 ~ 1.65μm, including S, C, and L-band, along with a very low dispersion slope of ~0.001ps/km.nm2 at the single polarization-single mode operation condition, which makes the proposed fiber an almost ideal transmission medium for high capacity WDM systems.
This work was partially supported by KOSEF through UFON, and ERC program, by MOE-BK21.
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