1. Introduction

Chromatic dispersion of optical fibers is one of the key parameters causing optical pulse broadening in an optical communication system. The dispersion is usually compensated by use of dispersion compensating fibers (DCFs). The typical design of DCFs is based on the dual-core structure of an inner concentric core and an outer ring core that supports two supermodes [1,2]. By introducing the outer ring core, it is possible to obtain a large negative dispersion coefficient ranging from several hundreds to thousands ps/nm/km. In the last few years, a photonic crystal fiber (PCF) has been receiving increasing attention because of its novel optical characteristics [3–5]. In particular, a PCF can control dispersion over a wide wavelength range and can be expected to provide a novel DCF. Dual-core structures have been employed in PCFs to get much large negative dispersion around 1.55 μm [6–12]. The inner cores of dual-core PCFs have been pure silica, and the outer cores have been made by reducing the diameters of the air holes in one layer of cladding. It is well known that PCFs are very useful for chromatic dispersion compensation without Ge doping [9].

It has been found that quasi-periodic structures can give rise to unusual phenomena and properties that have not been observed in periodic structures [13,14]. For example, the introduction of 12-fold symmetric quasi-structure of air holes in a dielectric slab with low refractive index creates photonic band gaps while introducing periodic arrays of air holes in the dielectric slab does not [14]. Other quasi-periodic 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 [15–19]. Recently, it has been reported that introducing quasi-periodic structures of microscopic air holes in optical fibers can give rise to a unique dispersion property such as almost zero ultra-flattened chromatic dispersion [12]. An optical fiber with a quasi-periodic array of air holes in cladding was named a photonic quasi-crystal fiber (PQF). It may be expected that introducing dual-core structure into a PQF makes a PQF a DCF with unusual dispersion compensating properties.

In this paper, we show that a dual-core PQF with a six-fold symmetric quasi-periodic array can exhibit a large negative chromatic dispersion value over optical communication band. Moreover, the dependence of the effective indices of the inner and the outer cores, and chromatic dispersion on the structure parameters shows that the negative chromatic dispersion property of the dual-core PQF is quite different from that of a dual-core triangular PCF with a six-fold symmetry. The difference comes from the strong confinement of out-core modes due to the circular-like-quasi-periodic air hole arrangement.

2. Results and discussion

The schematic cross section of a dual-core PQF with a six-fold symmetry is shown in Fig. 1 . The inner core is made of pure silica which is surrounded by the inner cladding consisting of the first and the second layer of air holes of the diameter D ₁. The outer core is the third layer of air holes of the diameter D ₂. The diameter of air holes in outer cladding and the distance between the centers of neighboring air holes are denoted D ₀ and Λ, respectively.

 

Fig. 1 Schematic of the cross section of a dual-core PQF with a six-fold symmetry

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Figure 2 (a) shows the effective refractive index of the dual-core PQF as a function of wavelength, where Λ = 1.8μm, D ₀/Λ = 0.53, D ₁/Λ = 0.65, and D ₂/Λ = 0.4. The parameters for the inner and outer cores were so chosen that the respective modes are phase-matched close to 1.5μm. In Fig. 2 (a), the dashed (solid) line shows the effective index curve of the guiding mode of a single core PQF with an inner (outer) core only. Both the plane wave expansion method with preconditioned conjugate-gradient minimization of the block Rayleigh quotient and 3D full-vectorial beam propagation method were employed in calculating the effective indices. The circles result from the combined effect of the inner and outer cores. At the phase matching wavelength, λ_p, where the combined effective index shows the kink, the fundamental mode suddenly moves from the inner core to the outer core. Also, it is observed that the lines for two individual cores seem to act as the asymptotes to the combined line on each side of the kink. This means that the most of fundamental mode is just like being confined in the outer core for a wavelength longer than λ_p and does not experience any effect from the inner core and vice versa [6–8]^,

 

Fig. 2 Variation of the dielectric refractive index with wavelength for PQF (a) and PCF (b). A dashed (solid) curve corresponds to a fiber with inner (outer) core only. Circles result from the combined effect of the inner and outer cores.

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Figure 2(b) shows the effective refractive index of a dual-core triangular PCF with the same structural parameters of Fig. 2(a). The dashed (solid) line also shows the effective index curve of the guiding mode of a single-core triangular PCF with an inner (outer) core only. It is worth noting that the effective refractive index of the inner core guiding mode for the dual core-PCF is nearly equal to that for the dual-core PQF. This is because the triangular air-hole arrangements of the first cladding layers to surround the inner cores of the PQF and the PCF are the same. The effective refractive index of the outer core mode for the dual-core PCF is lower than that of the dual-core PQF and thus λ_p of the dual-core PCF, 1.86 μm, is longer than that of the dual-core PQF. This means that the outer core mode of the dual-core PQF is more tightly confined in the core than that of the dual-core PCF. The strong confinement of the outer core mode of the PQF comes from the circular-like arrangement of air holes in the cladding layer to surround the outer core.

Figure 3 shows field distributions of the fundamental modes of the dual-core PCF ((a)-(c)) and the dual-core PQF ((d)-(f)) when λ < λ_p (λ = 1.6 μm (a) and λ = 1.2 μm (d)), λ = λ_p (λ = 1.86 μm (b) and 1.5 μm (e)), and λ > λ_p ((λ = 2.1 μm (c) and (λ = 1.8 μm (f)). The mode profiles were simulated by the plane wave expansion method with the preconditioned conjugate-gradient minimization. One can see that the field distributions of inner core modes of the PQF and the PCF are nearly the same. The mode shapes are hexagonal. The field distribution of the outer core mode of the PQF is different from that of the PCF. The shape of the outer core mode of the PQF is circular, while the shape of the outer core mode of the PCF is hexagonal. One can see that the outer core mode of the PQF is tightly confined in the core, while the outer core mode of the PCF penetrates into cladding layers, as mentioned. The tightly confined outer core mode has a smaller overlap with the inner core mode than for the PCF. This means that the effective index mismatch between the inner and the outer modes of the PQF is larger than that of the PCF, as seen in Fig. 2.

 

Fig. 3 Field distributions of the fundamental modes of the dual-core PCF ((a)-(c)) and the dual-core PQF ((d)-(f)) when λ < λ_p (λ = 1.6 μm (a) and λ = 1.2 μm (d)), λ = λ_p (λ = 1.86 μm (b) and 1.5 μm (e)), and λ > λ_p ((λ = 2.1 μm (c) and (λ = 1.8 μm (f)).

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The chromatic dispersion can be directly calculated from the effective index of the fundamental mode. The dispersion parameter is written as

 

Fig. 4 The chromatic dispersion of the dual-core PQF (solid line) and the dual-core PCF (dashed line).

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The dispersion of the dual-core PQF exhibits a sharp deep curve and a large negative dispersion value (~-2500 ps/km/nm) at λ_p of 1.5 μm. The full width half maximum is about 20 nm from 1490 to 1510 nm and thus the PQF can compensate dispersion of a typical single mode fiber in S-band. Of course, the proposed PQF can be designed to compensate dispersion of a single mode fiber in C-band (see Fig. 5(a) ). Equation (1) indicates that the sharpness and the dispersion value are determined by the difference between the slopes of the effective refractive indices of the inner and the outer core modes around λ_p. For avoiding confusion, λ_p for the dual-core PQF (PCF) is denoted by λ_{p PQF} (λ_{p PCF}). For the dual-core PQF, a sharp deep curve and a large negative dispersion value at λ_{p PQF}, implies the abrupt change of the slopes around λ_{p PQF}. The amount of slope change at λ_{p PQF} is about 0.015. The value is approximately fifteen times larger than that at λ_{p PCF}. The origin of the abrupt slope change can be physically understood from the mode profiles in shown Fig. 3. First, for the dual-core PCF, the slope change of the effective refractive indices around λ_{p PCF} is gentle because both the inner core mode and the outer core mode have the hexagonal shape. For the dual-core PQF, the quasi-periodic arrangement of air holes gives rise to the circular-like outer core mode. The abrupt slope change of the effective refractive indices around λ_{p PQF} caused by the mode change from triangular shape to circular shape around λ_{p PQF} Thus, an absolute value of negative dispersion of the dual-core PQF at λ_{p PQF} is expected to be usually larger than that of a dual-core PCF at λ_{p PCF} .

 

Fig. 5 The dependences of D_{p PQF} and λ_{p PQF} on Λ (a), and the dependences of D_{p PCF} and λ_{p PCF} on Λ (b) for Λ = 1.8μm, D ₀/Λ = 0.53, and D ₂/Λ = 0.4.

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We investigated a negative dispersion value at λ_{p PQF} (λ_{p PCF}), D _{p PQF} (D _{p PCF}), by varying the lattice constant Λ from 1.2 to 2.0 μm with keeping D ₀/Λ = 0.53, D ₁/Λ = 0.65, and D ₂/Λ = 0.4. Figure 5 shows the dependences of D _{p PQF} and λ_{p PQF} on Λ (a), and the dependences of D _{p PCF} and λ_{p PCF} on Λ (b). One can see that an absolute value of D _{p PQF} is always larger than that of D _{p PCF} in the wavelength range. As Λ increases, λ_{p PQF} and λ_{p PCF} are shifted toward a longer wavelength. This is a phenomenon commonly found in dual-core compensating fibers [20]. When Λ increases, the sizes of the inner and the outer cores increase and thus the effective indices of the inner and outer core modes increase and their slopes become gentle. The increase of the effective indices results in the strong mode confinement in the cores. The strong confinement of the inner core mode dominantly contributes to the shift of the phase matching to a longer wavelength.

An interesting phenomenon is that the absolute value of D _{p PQF} increases as Λ increases, while the absolute value of D _{p PCF} decreases. As mentioned, D _{p PQF} (D _{p PCF}) is determined by the difference between the slopes of the effective indices of the inner and the outer code modes around λ_{p PQF} (λ_{p PCF}). For the PCF, the slope of the effective index for the outer core mode gets similar to that of the effective index for the inner core mode as Λ increases and thus the difference between the slopes of the effective indices (D _{p PCF}) decreases as Λ increases. This property hinders realizing a large core PCF to reduce coupling loss to a conventional single mode fiber. For the PQF, however, the slope of the effective index for the outer core mode gets gentler than that of the effective index for the inner core mode as Λ increases and thus, in contrast of the PCF, the difference between the slopes of the effective indices of the PQF (D _{p PCF}) increases as Λ increases. This is due to the fact that the effective index for the outer core mode is strongly affected by the change of the core size because the circular-like outer core mode is tightly confined in the core. The property of the outer core mode caused by the quasi-periodic arrangement will be useful to realize a large core PQF to reduce coupling loss of a conventional single mode fiber.

Geometrical imperfection of a PQF can reduce its performance. The PQF can be fabricated by the complex fabrication method such as sol-gel methods [21], not a conventional stacking method [22], because the quasi-periodic arrangement cannot be realized by the stacking method. The sol-gel method usually gives well-arranged air holes. However, due to the surface tension, diameters of air holes of outer cladding layers can be smaller than an original diameter after the drawing process. To see dependence of dispersion of the PQF on the diameters, we calculated dispersion of the PQF by varying the diameters of the second and the third cladding layers of the outer core (see Fig. 1). We observed that dispersion of the PQF with the diameters of 0.53, 0.5, and 0.48 Λ are not sensitive to the diameters, even though λ_{p PQF} shifts slightly to a short wavelength. This is because the nearest neighbor holes of the outer core are crucial in determining properties of the outer core mode.

For a conventional DCF and a dispersion compensating PCF, there is generally a trade-off between large dispersion and mode area [9]. However, the proposed dual-core PQF can provide both large negative dispersion and large mode area. Thus the dual-core PQF can be useful in implementing long-haul data transmission systems without additional non-linear penalties.

3. Conclusions

We investigated dispersion properties of dual-core photonic quasi-crystal fiber with a six-fold symmetry. The absolute value of negative dispersion for the dual-core photonic quasi-crystal fiber increases as the distance between the nearest holes increases, while absolute value of negative dispersion for the dual-core photonic crystal fiber decreases. The circular-like outer core caused by the quasi-periodic arrangement causes the opposite dispersion behavior. The unusual dispersion property of the dual-core PQF can be useful in realizing a large mode area dispersion compensating fiber to reduce the coupling loss to a conventional single mode fiber.