1. Introduction

Optical fibers are the backbone of modern telecommunication networks. These fibers have also been applied in beam delivery for medicine, diagnostics, sensing and imaging. There have been significant works on fiber-optic design and performance. One of the most notable works is employing the periodic array of microscopic air holes in cladding that run along the entire fiber length to form new types of optical fibers [1]. Such fibers are known as photonic crystal fibers (PCFs) because periodic dielectric structures are called photonic crystals [2]. PCFs provide unique properties such as the generation of a single mode broadband optical supercontinuum, flattened dispersion, endlessly single mode, large birefringence, and so on [3–5]. These properties stem from the large contrast of the refractive index and the two dimensional nature caused by the periodic array of air holes in the cladding region.

Quasiperiodic structures, quasicrystals, are unique structures having long-range order but no periodicity. It has been found that quasiperiodic structures can give rise to unusual phenomena and properties that have not been observed in periodic structures [6–7]. For example, an introduction of 12-fold symmetric quasicrystal of air holes in a dielectric slab with low refractive index creates photonic band gaps, the frequency ranges in which light propagation is completely prohibited, while introducing periodic arrays of air holes in the dielectric slab does not [6]. Other quasiperiodic structures such as Penrose tiling and octagonal tiling have been also introduced to improve the performances of optical devices such as high Q cavity lasers and waveguides [8–12].

 

Fig. 1. Schematic cross section of a six-fold symmetric photonic quasicrystal fiber and elementary units of the quasicrystal (dotted lines). Black circles denote air holes. a is the distance between neighboring air holes and d is the diameter of air holes.

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Introducing quasiperiodic array of microscopic air holes in optical fibers could give rise to unique properties that were not found in the fibers having periodic array of air holes, PCFs. However, the properties of fibers with quasiperiodic arrays of air holes in cladding, photonic quasicrystal fibers (PQFs), have rarely been investigated. To study how the quasiperiodic arrangement of air holes in cladding affects optical properties of fibers, it is reasonable to compare optical properties of a PQF and those of a PCF, where their symmetry are the same but their arrangements of air holes are different.

In this paper, we investigated properties of the six-fold symmetric PQF and compared them with those of a triangular PCF having six-fold symmetry. The PQF exhibits larger cutoff ratio for endlessly single mode operation than that of a triangular PCF and almost zero ultra-flattened chromatic dispersion (0 ± 0.05 ps nm^-1 km^-1) in the wide wavelength range from 1.49 to 1.68 μm. The dispersion value is much less than those of the proposed dispersion-flattened PCFs. The nearly zero ultra-flattened dispersion property can be useful in the generation of flat and wideband supercontinuum and wavelength division multiplex communications systems.

2. Results and Discussion

Schematic cross section of six-fold symmetric PQF is shown in Fig. 1. Black circles denote air holes and elementary units of the quasicrystal are denoted by white lines. a is the distance between neighboring air holes and d is the diameter of air holes. The air holes in cladding decrease the average refractive index in the cladding region and confine light in the silica core. Thus, the light guidance in the PQF can be explained by the total internal reflection that describes well the guiding of light in PCFs with solid cores [3]. It has been demonstrated that the PCFs can support endlessly single mode that the fundamental mode can propagate in the silica core for all wavelengths [5]. PQFs are also expected to support endlessly single modes, like PCFs.

To verify the endlessly single mode operations of PQFs, we need to parameterize the optical properties of PQFs in terms of the V parameter that is characterized by the core radius r, the core index n_c, and the cladding index n_cl. The V parameter in the step index fiber is given by $V\left(\lambda \right)=\left(\frac{2\mathrm{\pi r}}{\lambda }\right)\sqrt{n_c^2-n_{\mathrm{cl}}^2}.$. However, the equation is not valid for a PQF because r, n_c, and n_cl are not clearly defined in a PQF. This problem can be solved by introducing a modified V parameter for a PCF given by $V_{\mathrm{PCF}}\left(\lambda \right)=\left(\frac{2\pi }{\lambda }\right)\Lambda \sqrt{n_{\mathrm{eff}.c}^2\left(\lambda \right)-n_{\mathrm{eff}.\mathrm{cl}}^2\left(\lambda \right)},$, where Λ is the pitch of air-holes, n_eff.c (λ) is the core index associated with the effective index of the fundamental mode confined in the silica core, and n_eff.cl (λ) is the effective index of the mode which distributes over the cladding with a periodic array of air holes [13]. For the six-fold symmetric PQF, The field distribution of the second order mode is strongly affected by six holes which are the nearest to the core, like the triangular lattice PCF. Therefore the single mode condition is identical to that of the triangular lattice PCF formed by removing the central air hole, even though their structures are different. The value of V parameter for the lowest second-order mode is given by

From the value, the single mode condition is found to be V_PQF (λ) < π.

 

Fig. 2. V parameters of the six-fold symmetric PQF, V_PQF, for various d/a. The values of d/a are inserted as insets and assigned to different colors. V_PQF is normalized to π. The dashed line denotes the cutoff value below which a single mode is allowed and circles denote the cutoff wavelength of the second-order mode.

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Figure 2 shows the numerical results of V_PQF (λ) for various d/a. The values of d/a are inserted into Fig. 2 as insets and assigned to different colors. V_PQF (λ) is normalized to π. We employed the vectorial plane wave expansion method to calculate n_eff.c and n_eff.cl for various d/a [14]. The material refractive indices are chosen to be 1.0 for air holes and 1.444 for the silica. The dashed line denotes the cutoff value below which a single mode is allowed and circles denote the cutoff wavelength of the second-order mode. One can see that a single mode is allowed for all wavelengths when d/a is less than 0.525, which is the cutoff ratio for the endlessly single mode operation. Previously, d/a values of 0.406 and 0.442 were reported as cutoff ratios for a triangular lattice PCF and a square lattice PCF, respectively [13, 15]. The cutoff ratio of six-fold symmetric PQF is larger than those of the PCFs. The large cutoff ratio could be of benefit to design and fabricate the endlessly single mode fiber.

One of important properties of optical fibers is group velocity dispersion, in short, dispersion. Especially, flattened dispersion is necessary in communication systems such as systems supporting ultrashort pulse propagation and wavelength division multiplexing communication systems because uniform response at different wavelengths is required to improve the performance of the communication systems. The flattened dispersion behavior has been often achieved by dispersion-shift fibers which are typically W fibers. However, the dispersion behavior is approximately flat in a rather narrow range. Recently, it has been demonstrated that PCFs can exhibit flattened dispersion over a wide range of wavelength [16, 17, 18]. The characteristics of flattened dispersion strongly depend on the diameters and the pitch of air holes.

The dispersion of PQF, D (λ), can be approximately expressed by sum of geometrical dispersion and material dispersion [16],

where n_g (λ) is the geometrical modal index and n _m (λ) is the refractive index of silica. Note that the dependence of n_g (λ) on the wavelength λ arises from the geometric configuration because n_g(λ) is calculated under the assumption that the refractive index of silica is 1.444. We investigated the dependence of D (λ) on the structural parameters of d and a in the six-fold symmetric PQF. We employed the full vectorial beam propagation method to calculate D _g (λ) [19] and D _m (λ) is approximated by the Sellmeier equation [20]. The dependence of D _g (λ) on a and d/a is found to be very similar to that of D _g (λ) on Λ and d/Λ in the triangular lattice PCF [16].

To achieve zero ultra-flattened dispersion over a wide range of wavelength, D _g (λ) should completely compensate for D _m (λ) in a wide range of wavelength. Since the above expression of D (λ) can be rewritten as D (λ) = D _g (λ) - (- D _m (λ)), we compared D _g (λ) and - D _m (λ) by varying finely the values of d and a and investigated the dispersion behavior. It was a formidable task to find the structural parameters of the PQF exhibiting the ultra-flattened dispersion in a wide range of optical communication wavelengths. Figure 3 shows the dispersion behavior of the PQF for d/a = 0.31 and a = 2.41 μm in the range of wavelength from 1.0 to 1.9 μm. D(λ), D _g(λ), and -D _m(λ) are denote by red line, blue line, and black line, respectively. One can see that the curves of D _g(λ) and -D _m(λ) are almost overlapped in the range of wavelength from 1.49 to 1.68μm, resulting in the nearly zero ultra-flattened dispersion behavior in the range of wavelength.

 

Fig. 3. Dispersion behavior of the six-fold symmetric PQF for d /a=0.31 and a = 2.41 μm in the range of wavelength from 1.0 to 1.9 μm. D(λ), D _g(λ), and -D _m(λ) are denoted by red line, blue line, and black line, respectively.

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To show the nearly zero ultra-flattened dispersion behavior in detail, D(λ) of the six-fold symmetric PQF (red line) and a triangular PCF (blue line) are plotted in the range of wavelength 1.3 to 1.8 mm in Fig. 4. The grey region denotes the nearly zero ultra-flattened dispersion of 0 ± 0.05 ps nm^-1 km^-1 in the range from 1.49 to 1.68 μm (~ 200 nm) including S, C, L, and U bands of optical communication bands. Detailed dispersion behavior is shown in the inset. Moreover, the dispersion slope in the range of wavelength from 1.53 to 1.63 μm is from zero to a maximum absolute value of 6 × 10^-4 ps nm^-2 km^-1 . Thus, even though the symmetries of the PQF and the PCF are the same, the different arrangement of air holes in the second layer which influence on the optical properties of the fiber can improve the dispersion properties of the fiber. The six-fold symmetric PQF exhibiting almost zero ultra-flattened dispersion and negligible third-order dispersion are expected to be very useful for wavelength division multiplex communications systems and flat and ultra-wideband supercontinuum generation. It would be worth comparing dispersion of the six-fold symmetric PQF with that of the optimized ultra-flattened triangular PCF composed of air holes of different sizes in cladding. The value of dispersion and the third-order dispersion (the slope of dispersion) of the PQF are less than those of the optimized ultra-flattened triangular PCFs, 0 ± 0.1 ps nm^-1 km^-1 in 50 nm bandwidth and 3 × 10^-3 ps nm^-2 km^-1, respectively [18]. In addition, the structure of the PQF is much simpler than that of the optimized ultra-flattened triangular PCF. The chirped air layer structures of the optimized PCF require more precise fabrication process because flattened dispersion of the optimized PCF is sensitive to the tolerance of the sizes and positions of the air holes [18].

 

Fig. 4. Dispersion of the six-fold symmetric PQF (red line) and a triangular lattice PCF (blue line) in the range from 1.3 to 1.8 μm for d/a = 0.31 and a = 2.41 μm. The gray region denotes the almost zero ultra-flattened dispersion (0 ± 0.05 ps nm^-1 km^-1) in the range from 1.49 to 1.68 μm. The detailed dispersion behavior is shown in the inset.

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It should be stressed that the proposed six-fold symmetric PQF is just one example showing that introducing quasicrystal structures in optical fibers could significantly improve the performance of optical fibers. The optical properties of PQFs having other quasicrystals such as Penrose tiling and octagonal tiling of air holes in cladding will be interesting.

3. Conclusions

We investigated the endlessly single mode condition and dispersion of the six-fold symmetric PQF. The PQF exhibits larger cutoff ratio for the endlessly single mode operation than those of triangular and square PCFs and almost zero ultra-flattened chromatic dispersion, 0 ± 0.05 ps nm^-1 km^-1, in the range of wavelength from 1490 to 1680 nm. Introducing quasicrystal structures in optical fibers can improve the performance of fibers significantly. The PQF could be useful for flat and ultra-wideband supercontinuum generation and wavelength division multiplexing communication systems.