1. Introduction

Optical fibers with silica-air microstructures called photonic crystal fibers (PCFs) [1, 2] have attracted a considerable amount of attention recently, because of their unique properties that are not realized in conventional optical fibers. PCFs are divided into two different kinds of fibers. The first one, index-guiding PCF, guides light by total internal reflection between a solid core and a cladding region with multiple air-holes [3, 4]. On the other hand, the second one uses a perfectly periodic structure exhibiting a photonic bandgap (PBG) effect at the operating wavelength to guide light in a low index core-region [5, 6]. In this paper, we will focus on index-guiding PCFs, also called holey fibers (HFs).

HFs possess numerous unusual properties such as wide single-mode wavelength [4], bend-loss edge at short wavelength [4], extremely large [7] or small [8] effective-core-area at single-mode region, and anomalous group-velocity dispersion at visible and near-infrared wavelengths [9]. In a usual PCF, there is one defect in the central region and the light is guided along this defect. Recently, introducing adjacent two defects, two cores, into a PCF, it has been shown that it is possible to use the PCF as an optical fiber coupler [10–12]. These PCF couplers with adjacent two cores have possibility of realizing a multiplexer-demultiplexer (MUX-DEMUX).

In this paper, coupling characteristics of dual-core PCF couplers are evaluated by using a vector finite element method (FEM) [13, 14] and their application to a MUX-DEMUX based on PCF is investigated. The PCF couplers for 1.48/1.55-µm, 1.3/1.55-µm, 0.98/1.55-µm, and 0.85/1.55-µm wavelength MUX-DEMUX are designed and the beam propagation analysis [15] of the proposed PCF couplers is performed. It is shown from numerical results that it is possible to realize significantly shorter MUX-DEMUX PCFs, compared to conventional optical fiber couplers.

2. Guided mode analysis

We consider a PCF with multiple air holes in the cladding region surrounded by perfectly matched layers (PMLs) regions 1 to 8 with thicknesses t_i (i=1 to 4), the cross section of which is shown in Fig. 1, where x and y are the transverse directions, z is the propagation direction, PML regions 1, 2 and 3, 4 are faced with the x and y directions, respectively, regions 5 to 8 correspond to the four corners, and W_x and W_y are the computational window sizes along the x and y directions, respectively.

Using an anisotropic PML [13, 16], from Maxwell’s equations the following vectorial wave equation is derived:

with

where k ₀=2π/λ the free-space wavenumber, λ is the wavelength, E denotes the electric field, and n is the refractive index. The PML parameters s_x and s_y are given in Table 1, where the values of s_i (i=1 to 4) are complex as

where ρ is the distance from the beginning of PML. Attenuation of the field E in PML regions can be controlled by choosing the values of α_i appropriately.

When applying a full-vector FEM to HFs, a curvilinear hybrid edge/nodal element [17] as shown in Fig. 2 is very useful for avoiding spurious solutions and for accurately modeling curved boundaries of circular air holes. For the axial electric field, E_z, a nodal element with six variables, E _z1 to E _z6, is employed, while for the transverse fields, E_x and E_y, an edge element with eight variables, E _t1 to E _t8, is employed, resulting in significantly fast convergence of solutions [17].

Dividing the fiber cross section into curvilinear hybrid edge/nodal elements, and applying the standard finite element technique, we can obtain the following eigenvalue equation:

where β is the propagation constant, [K] and [M] are the finite element matrices [13], and {E} is the discretized electric-field vector consisting of the edge and nodal variables.

 

Fig. 1. Cross section of PCF surrounded by PMLs.

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Fig. 2. Curvilinear hybrid edge/nodal element.

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Table 1. PML parameters.

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3. Characteristics of PCF couplers

We consider PCF coupler structures as shown in Fig. 3, where Λ is the hole pitch, d is the hole diameter, and the background silica index is assumed to be 1.45. The separation between the centers of two cores, A and B, shown in Fig. 3(a) is √3λ and we call this PCF Type 1. On the other hand, the separation between the centers of A and B shown in Fig. 3(b) is 2λ and we call this PCF Type 2. Figure 4 shows the hole-pitch dependence of coupling length L for the two PCF couplers shown in Fig. 3, where d/Λ=0.5, operating wavelength λ=1.55 µm, and the red and the blue curves denote the coupling lengths for the x-polarized and the y-polarized modes, respectively. The coupling length L is obtained by using the propagation constants of even mode β_e and odd mode β_o as

The coupling length of Type 1 is shorter than that of Type 2 for the same pitch and the same hole diameter, because the distance between the centers of A and B for Type 1 is nearer than that for Type 2. Thus, we consider a coupler of Type 1 with shorter coupling length.

 

Fig. 3. Cross sections of PCF couplers of (a) Type 1 and (b) Type 2.

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Figure 5 shows the hole-pitch dependence of coupling lengths for couplers with d/Λ=0.3, 0.5 0.7, and 0.9, where the operating wavelength λ=1.55 µm, the red and the blue curves denote the coupling lengths for the x-polarized and the y-polarized modes, respectively. The coupling lengths of couplers with smaller hole-pitch and smaller value of d/λ becomes shorter. Moreover, the coupling length of x-polarized mode is shorter than that of y-polarized mode, because the two cores are placed in parallel to x-axis and the coupling of x-polarized mode is stronger than that of y-polarized mode. Although, in order to obtain shorter coupling lengths, strong coupling between cores is necessary, too strong coupling causes lower extinction ratios. Therefore, in what follows, we consider PCF couplers using y-polarized mode. The structures shown in Fig. 3 exhibit relatively small difference between the coupling length for the x- and y-polarized modes. The difference can be enhanced by introducing structural birefringence into the cores. High birefringence can be obtained by adjusting the size of the air holes around the two core regions, which gives rise to an enhanced difference in the coupling lengths for the two polarization modes [18].

Figure 6 shows the hole-pitch dependence of coupling lengths, where d/Λ=0.7 and the operating wavelength λ is taken as a parameter. The coupling length increases with decreasing the operating wavelength. A PCF coupler can separate two wavelengths λ₁ and λ₂, if it is in a bar-coupled state for one wavelength and a cross-coupled state for the other; i.e., the coupling length L _λ1 at the wavelength λ₁ and the coupling length L _λ2 at the wavelength λ₂ satisfy the relation

or

 

Fig. 4. Coupling lengths of PCF couplers with d/Λ=0.5.

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Fig. 5. Hole-pitch dependence of coupling lengths, taking the value of d/Λ as a parameter.

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Fig. 6. Hole-pitch dependence of coupling lengths for PCF coupler with d/Λ=0.7, taking the operating wavelength as a parameter.

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Figure 7 shows the d/dependence of coupling length dependence of coupling length ratios L _λ2/L _λ1 for couplers with hole-pitch dependence of coupling length=1.8, 2.0, and 2.2 µm. The one wavelength λ₁ is fixed at 1.55 µm and the another one λ₂ is 1.48 µm, 1.3 µm, 0.98 µm, and 0.85 µm. The wavelengths of 1.48 µm and 0.98 µm are used as pump lights in erbium-doped fiber amplifiers and the wavelength of 0.85 µm is used in short wavelength-range optical communication systems. Here, each coupler is called Coupler 1, 2, 3, or 4. From Fig. 7, we could say that the optimum coupling-length ratio is different for each coupler.

The structural parameters of Couplers 1 to 4 for 1.48/1.55-µm, 1.3/1.55-µm, 0.98/1.55-µm, and 0.85/1.55-µm wavelength MUX-DEMUX are shown in Table 2. The fiber length of Coupler 1 is longer than that of the others, because the difference between the coupling lengths at 1.48 µm and 1.55 µm is shorter. The coupling length ratios for Couplers 1 to 4 are L _1.55 : L ₁.48=9 : 10, L _1.55 : L _1.3=2 : 3, L _1.55 : L _0.98=1 : 2, and L _1.55 : L _0.85=1: 4, respectively, and the fiber lengths are 2284 µm, 712 µm, 481 µm, and 1178 µm, respectively. The fiber lengths are much shorter than conventional optical fiber couplers. In conventional optical fiber couplers the two cores are separated by a long distance and the coupling between the modes of the two cores is week, and so the fiber coupler length becomes long. On the other hand, in dual-core PCF couplers investigated here the distance between the two cores is √3Λ (Λ ≅ 2 µm), and so very short coupling length can be realized.

Finally, using a vector beam propagation method [15], we confirm that the four PCF couplers designed as Table 2 could separate two wavelengths of λ₁ and λ₂. The y-polarized fundamental modes at λ₁ and λ₂ are inputted into the core A in Fig. 3(a) and the beam propagation analysis is performed. Figure 8 shows the normalized power variation along the propagation distance in the bar port. In Couplers 1 to 4, the separation of two wavelengths of λ₁ and λ₂ is achieved at the propagation distance of 2284 µm, 712 µm, 481 µm, and 1178 µm, respectively. These results are in good agreement with the coupling lengths estimated by using (5), and Couplers 1 to 4 operate as MUX/DEMUX for 1.48/1.55-µm, 1.3/1.55-µm, 0.98/1.55-µm, and 0.85/1.55-µm wavelength, respectively. In Coupler 3, the field confinement is weak because of smaller value of d/Λ, and so, the extinction ratio of this coupler is lower, compared to the other couplers.

 

Fig. 7. d/Λ-dependence of coupling-length ratios, where λ₁=1.55 µm.

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Fig. 8. Demultiplexing characteristics of PCF couplers.

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Table 2. Parameters of photonic crystal fiber couplers, where λ₁=1.55 µm.

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4. Conclusion

Coupling characteristics of dual-core PCF couplers have been numerically investigated and MUX-DEMUX using the PCF couplers for 1.48/1.55-µm, 1.3/1.55-µm, 0.98/1.55-µm, and 0.85/1.55-µm wavelength have been proposed. From numerical results, it has been shown that it is possible to design significantly shorter MUX-DEMUX PCFs with coupling lengths of a few millimeters, compared to conventional optical fiber couplers with coupling lengths of several tens or even hundreds of millimeters.

It is important that how the light should be launched into one core efficiently at the input and collected from one core at the end of the PCF. These are problems to be considered in a future work.