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We investigate photonic band-gap (PBG) profiles of a modified honeycomb lattice structure and we identify the structural parameters that possess the largest band-gap. By incorporating the identified profile into the cladding, the wavelength dependence of the dispersion properties and confinement losses of air-guiding modified honeycomb PBG fibers (PBGFs) is investigated through a full-vector modal solver based on finite element method. In particular, we find that broadband effectively single-mode operation from 1450 nm to 1850 nm can be achieved using a modified honeycomb PBGF with a defected core realized by removing 7 air holes.
©2006 Optical Society of America
1. Introduction
In recent years, photonic band-gap fibers (PBGFs) have attracted much attention because of their unusual properties of guiding light. In PBGFs though, the cladding is usually made of a periodic arrangement of air with n 1 = 1.0, and silica with n 2 = 1.45, so the light can propagate along the fiber length in the low-index defected core region, because photonic band-gap (PBG) exists in the cladding region for the localized guided mode with a longitudinal component of wave vector despite the small refractive index contrast of silica and air [1]. In particular, when the defect is formed by removing several air holes, which is called hollow core or air-guiding fiber, it is possible to guide light in air [2–9]. For designing PBGFs, not only the core structure but also the cladding structure is very important. When large band-gaps are obtained in the cladding, the fiber can possess desirable properties such as a low confinement loss and a broadband transmission range. It is a necessary requirement for air-guiding PBGFs that a photonic crystal cladding structure exhibits PBGs covering the β/k values equal or less than one (where β is the propagation constant and k is the wavenumber in free space). Generally, a triangular lattice for the cladding structure is used for realizing air-guiding PBGFs because of its large band-gaps. Honeycomb structures for air-guiding PBGFs with large band-gaps have also been proposed [8, 9] and have gathered much attention. In addition, a modified honeycomb structure has been investigated by Broeng et al. [10], however it has been evaluated only how the presences of interstitial air holes affect the band-gap properties [10–12]. More recently, Chen et al. suggested the possibility that the modified honeycomb structure produces completely different large band-gaps beyond the realm of triangular and honeycomb lattices as structural parameters are adjusted [13]. In [14] and [15], the authors calculated a modal dispersion curve and confinement loss properties for modified honeycomb-type air-guiding PBGFs with large core diameters, which result in suffering from multi-mode transmission. However, the possibility of realizing single-mode air-guiding PBGFs operating in a wide wavelength range with low confinement losses has not been investigated so far.
Fig. 1. Modified honeycomb lattice, where d is the diameter of air holes in the basic honeycomb lattice, dc is the diameter of extra air-holes in the center of unit cell, Λ is the distance between adjacent air holes, and n 1 = 1.0 and n 2 = 1.45 are the refractive indices of air and silica, respectively.
In this paper, and to the best of our knowledge for the first time, we investigate PBG profiles of a modified honeycomb lattice and we identify the structural parameters which possess the largest band-gap. This band-gap size is, surprisingly, as large as that of a regular triangular lattice corresponding to d/Λ = 0.98 (where d is the diameter of air holes and Λ is the distance between adjacent air holes). By incorporating the identified structural parameters into the fiber cladding, dispersion properties and confinement losses of air-guiding modified honeycomb PBGFs with a defected core realized by removing 7 and 13 air holes as a function of wavelength are investigated through a full-vector modal solver based on finite-element method (FEM) [16]. In particular, we find that broadband effectively single-mode operation from 1450 nm to 1850 nm can be achieved using a modified honeycomb PBGF with a defected core realized by removing 7 air holes.
2. PBG profiles of a modified honeycomb lattice
Figure 1 shows a modified honeycomb lattice as proposed in Ref. [13], where d is the diameter of air holes in the basic honeycomb lattice, dc is the diameter of additional air holes in the center of unit cell, Λ is the distance between adjacent air holes, and n 1 = 1.0 and n 2 = 1.45 are the refractive indices of air and silica, respectively. To design a realistic cladding structure, both d + dc < 2Λ and dc < √3Λ are necessary conditions. If the first condition is not satisfied, the air holes with diameters dc and adjacent air holes with diameters d intersect between each other. If the second condition is not satisfied, the air holes with diameters dc intersect with the adjacent air holes of the same diameters.
First, in Figs. 2(a) and 2(b), we show the PBG diagrams calculated for a modified honeycomb lattice at βΛ = 6.0 and 11.0, respectively, as an example, where d/Λ = 0.60 and dc/Λ = 1.30. The dispersion curves are calculated through FEM for an infinite periodic lattice of the cladding air-holes. The shaded regions represent the complete PBGs. For βΛ = 6.0, we can see that the region between band-6 and band-7 is large in Fig. 2(a), while for βΛ = 11.0, the regions between band-15 and band-16 as well as between band-6 and band-7 is large in Fig. 2(b). Figure 2(c) shows the βΛ-dependence of the band-gaps for the modified honeycomb lattice with d/AΛ = 0.60 and dc/Λ = 1.30. In general, for air-guiding PBGFs, dispersion curve of the fundamental core mode appear near the air line in the PBG. To estimate the available transmission band, it is very useful to consider the spans of the air line over the PBG regions.
Fig. 2. PBG diagrams for a modified honeycomb lattice at (a) βΛ = 6.0 and (b) βΛ = 11.0, where d/Λ = 0.60 and dc/Λ = 1.30. The shaded regions represent complete PBGs. (c) PBG diagram in k-β space for a modified honeycomb lattice with d/Λ = 0.60 and dc/Λ = 1.30.
Fig. 3. The spans of the air line over the PBG regions calculated for all the realistic structural parameters. (a) d/Λ = 0.30 to 0.60, (b) d/Λ = 0.65 to 0.85, and (c) d/Λ = 0.90 to 0.98. The solid red line corresponds to the triangular lattice (dc/Λ = d/Λ) and the solid cyan line represents the upper limit of the size of dc/Λ (dc/Λ = 1.96 - d/Λ). Lower order gaps (band 6–7) are represented by blue curves and the others are represented by green curves.
Figures 3(a)–3(c) show these spans systematically calculated for all the realistic structural parameters. These maps are very important to design an actual fiber cladding. We calculate the band-gap size for the structural parameters d/Λ = 0.30 to 0.98 and dc/Λ = d/Λ to 1.96 -d/Λ, because it has been reported that when dc < d, the cladding structure does not exhibit wide PBGs [13]. The red solid line corresponds to triangular lattice structure (dc/Λ = d/Λ) and the solid cyan line represents the maximum possible size of dc/Λ (dc/Λ = 1.96 - d/Λ). When dc is increased, several band-gaps can be enlarged. When d/Λ is small, band 12–13 is dominant, whereas when d/Λ is between 0.5 and 0.7, band 6–7 is dominant. In addition, when d/Λ becomes larger than 0.9, band 12–13 becomes dominant again. In order to compare each band-gap size quantitatively in the same wavelength range, we can use the following simple equation:
where k 1Λ and k 2Λ are the normalized wavenumbers at the band-gap edges obtained from Fig. 3 for the target structure. The value of w × 2λ0 represents the PBG wavelength range crossing the air-line, where λ0 stands for the central wavelength in the PBG. Using Eq. (1), we can find which structural parameters result in the largest band-gap, and by done so, we have derived the optimized structural parameters as: d/Λ = 0.60 and dc/Λ = 1.36, thus optimizing the 6–7 band-gap. When the cladding structure is realized based on these parameters, W×2λ0 ≈ 400 nm is obtained (provided λ0 = 1.55 μm), and this value is equivalent to that of a triangular lattice, with structural parameter of d/Λ = 0.98.
Fig. 4. Cross sections of the proposed modified honeycomb PBGFs with (a) the defected core realized by removing 13 air holes and (b) the defected core realized by removing 7 air holes.
3. Dispersion and confinement loss properties
Figures 4(a) and 4(b) show the cross sections of the two defected shapes, which are named type-A and type-B, respectively. The defected core of type-A in Fig. 4(a) is realized by removing the central unit cell and six air holes surrounding it and smoothing the resulting core edges, and type-B in Fig. 4(b) is made by removing the central unit cell. Considering the optimum structural parameters, d/Λ = 0.60 and dc/Λ = 1.36 as were obtained in Section 2 for the cladding structure, we evaluate the dispersion characteristics and confinement losses for these air-guiding modified honeycomb PBGFs. Because the defected core surface does not intersect the silica material where bulk mode has a high intensity, no surface modes exist in such core types [7].
Figure 5 shows the effective refractive indices of guided modes as a function of wavelength for the type-A PBGF. The black solid curves represent the PBG region provided that the number of cladding rings is infinite. The fundamental mode (red curve) exists inside the band-gap region and the lower and higher cutoff wavelengths are λ/Λ = 0.787 and λ/Λ = 1.03, respectively. Due to the large core diameter, higher order modes co-exist inside the band-gap region. Setting the hole pitch Λ = 1.747 μm, this fiber operates around 1.55 μm wavelength range. Figure 6 shows the surface plots for the x-component of the electric field distribution |Ex|, for (a) the x-polarized HE11 mode and (b) the TE01 mode of type-A PBGF with six cell rings at a wavelength of 1.55 μm, where Λ = 1.747 μm, d/Λ = 0.60, and dc/Λ = 1.36. The second-order mode is well confined in the air-core region as well as the fundamental mode. For actual PBGFs, due to the finite number of cladding rings, the power of guided modes attenuates in some degree. The confinement losses of the modes can be evaluated from the following equation [5]:
where Im[β] stands for the imaginary part of β. Figure 7 shows the wavelength dependence of the confinement losses for the modified honeycomb PBGF with the defected core of type-A with six and ten cell rings. The confinement losses of the guided modes decrease with increasing the number of rings as expected. Unfortunately, because the confinement losses of the higher-order modes as well as the fundamental modes are relatively low, the fiber with the defect of type-A suffers from multi-mode operation over a wide wavelength range. Especially as a single-mode fiber, there is a limitation of the wavelength range for which only the fundamental mode is confined, whereas the higher-order modes are sufficiently leaky [3].
Fig. 5. Dispersion curves for the type-A PBGF with d/Λ = 0.60 and dc/Λ = 1.36. The black solid curves represent PBG region. The fundamental mode is shown in red curve, the second-order mode is shown in blue curve, and the third-order mode is shown in green curve.
Fig. 6. Surface plots for the x-component of the electric field distribution |Ex| for (a) the x-polarized HE11 and (b) the TE01 modes for type-A PBGF with d/Λ = 0.60, dc/Λ = 1.36, and Λ = 1.747 μm at a wavelength of 1.55 μm.
Fig. 7. Confinement losses of the type-A PBGF with d/Λ = 0.60, dc/Λ = 1.36, and Λ = 1.747 μm for (a) six cell rings and (b) ten cell rings. Red, blue, and green curves correspond to the confinement losses for the fundamental, second-order, and third-order modes, respectively.
In order to realize effectively single-moded air-guiding PBGF, we consider type-B PBGF with smaller air-core region as shown in Fig. 4(b). Figure 8 shows modal dispersion curves as a function of wavelength for type-B PBGF. The black solid curves represent PBG region. The red and blue curves correspond to the effective indices for the fundamental and second-order modes, respectively. The lower and higher cutoff wavelengths of the fundamental mode are λ/Λ = 0.807 and λ/Λ = 1.10, respectively. Because the core diameter is small compared to that of type A, dispersion curves of the air-core modes shift downward from the position of air line. As the core size decreases, we can expect a single-mode operation [7]. In spite of this expectation, the second-order mode appears near the band-gap edge. Figure 9 shows the surface plots for the x-component of the electric field distribution |Ex|, for (a) the x-polarized HE11 mode and (b) the TE01 mode with six cell rings at a wavelength of 1.55 μm, where Λ = 1.747 μm, d/Λ = 0.60, and dc/Λ = 1.36. The fundamental mode is well confined to the air-core region, while the confinement of the second-order mode is weak and the confinement loss is very large. Figure 10 shows the wavelength dependence of the confinement losses of the type-B PBGF with six and ten cell rings. The confinement loss of the second-order mode is about 2-orders of magnitude larger than that of the fundamental mode for a six-rings fiber, and 4-orders of magnitude larger for a ten-rings fiber. So, as a conclusion type-B PBGF with ten cell rings and structural parameters of Λ = 1.747 μm, d/Λ = 0.60, and dc/Λ = 1.36 has low-losses and can operate as an effectively single-mode air-guiding PBGF from 1450 nm to 1850 nm.
Although the purpose of this paper was the theoretical investigation of the propagation properties of air-guiding PBGFs, at this point we would like to comment on the feasibility of our proposed PBGF. Perhaps at this stage the fabrication technologies may not be appropriate for constructing our proposed modified honeycomb crystal lattice. We believe however that in the near future with more advanced fabrication techniques the construction of this PBGF will become possible.
Another concern is the effect of possible surface tension to the core shape, associated with the effective single-mode operation. Even under such scenario, the fiber may suffer from surface modes resulting in reducing the operating wavelength in some degree, the confinement losses between fundamental and higher order modes will be kept in a relative high difference, making our fiber not to lose its effectively single-mode operation.
Fig. 8. Dispersion curves for the type-B PBGF with d/Λ = 0.60 and dc/Λ = 1.36. The black solid curves represent PBG region. The fundamental mode is shown in red curve and the second-order mode is shown in blue curve.
Fig. 9. Surface plots for the x-component of the electric field distribution |Ex| for (a) the x-polarized HE11 and (b) the TE01 modes for type-B PBGF with d/Λ = 0.60, dc/Λ = 1.36, and Λ = 1.747 μm at a wavelength of 1.55 μm.
Fig. 10. Confinement losses of the type-B PBGF with d/Λ = 0.60, dc/Λ = 1.36, and Λ = 1.747 μm for (a) six cell rings and (b) ten cell rings. Red and blue curves correspond to the confinement losses for the fundamental and second-order modes, respectively.
4. Conclusions
We have investigated the PBG profiles in a modified honeycomb lattice structure by varying the structural parameters d/Λ and dc/Λ, in order to identify the optimum parameters that possess the largest band-gap. Using the obtained optimized structural parameters, an air-guiding PBGF based on a modified honeycomb lattice has been demonstrated. We have analyzed the dispersion characteristics and confinement losses in two types of defected air-cores. The fundamental mode is well confined to the air-core region in both core types. In particular we showed that an air-guiding modified honeycomb PBGF with a defected core realized by removing 7 air holes can operate as an effectively single-mode fiber with low confinement losses, over a wide wavelength range. According to our calculation based on full-vector FEM, no surface modes exist in the proposed PBGFs.
Acknowledgments
The authors would like to acknowledge stimulating discussions with Dr. N. J. Florous from Hokkaido University.
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