1. Introduction

In the past several years photonic crystals have attracted much interest and shown many fascinating characteristics. Among others, its ability to interact with light on a wavelength-scale promises ultra-compact structures for integrated optical circuits. Many functional devices utilizing phonic crystals (PCs) have been proposed and are expected to play an important role in future optical circuit. Components being used in planar optical circuits include couplers, de/multiplexers, power-splitters, cavities, etc., and they can be also realized in 2-dimensional (2D) PC slab structures [1]. The devices on the 2D PC slab have many advantages such as relatively easy fabrication and convenient integration into conventional devices. Recently, well-established silicon-on-insulator (SOI) process facilitates the fabrication of 2D PC slab devices with much improved accuracy [2].

Of functional devices mentioned above, a power-splitter is one of the most indispensable components, however, the power-splitting function on photonic crystals was implemented, to our knowledge, with a typical Y-junction (or branch) structure, which has poor transmittance without structural tuning or conditions for zero reflection [4,5]. In addition, although the performance of Y-junctions can be improved by such tuning, the Y-branch, which is composed of the tuned Y-junction and bending at output port, still has difficulties that can not be easily addressed for practical applications, which originated from the mode-mismatching of Y-junction and the bending losses of output ports [1]. A group of researchers, though, experimentally showed almost perfect transmission in each output port of Y-junction in microwave region, but their design using line-defect and coupled-cavity waveguides is relatively difficult to fabricate on optical wavelength scale [3]. Focusing on a practical design for integrated photonic circuits and circumventing complex problems involved in Y-branch, we try to use coupling characteristics of photonic crystal line-defect waveguides (PCWGs), which have been previously analyzed and adopted for functional PC devices by many authors [6–10]. In this literature, a new power-splitting scheme is proposed, which is based on the coupling between guided modes supported by photonic crystal line-defect waveguides. For efficient power-splitting in the proposed scheme, it is important to keep the structure of device symmetric with respect to the input waveguide, which leads to a three-waveguide structure with the central waveguide used as an input port. In the use of proposed scheme, a 1×2 PC power-splitter is designed and it is reported that up to 47.6% transmittance per each output waveguide is achieved at the designed spectral range.

2. Design and numerical experiment

The power-splitter designed by the proposed scheme is depicted in Fig. 1. For simplicity, the system under consideration is 2-dimensinal and consists of an array of air-holes with a triangular lattice structure. Air-holes with a radius of r=0.3a are perforated in GaAs (n=3), where a and n are lattice constant and refractive-index, respectively. In this structure, band-gap opens for the frequency range of 0.2303–0.2666(a/λ) for H-polarization (magnetic field parallel to air-holes), where λ is the wavelength in free space. The PCWGs are formed in Γ-K direction by removing one row of air holes.

 

Fig. 1. Geometry of photonic crystal power-splitter and setup for the finite-difference time-domain computation. The device can be divided into two regions, as labeled on top. Black boxed areas indicate super-cells for the PWE calculation. White holes represent air-holes perforated in GaAs (n=3).

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The device designed here has the power-splitting mechanism analogous to that of three-waveguide directional couplers, and it can be divided into two regions according to their functions; coupling region and output region, as shown in Fig. 1. In the coupling region, the input field propagating through the PCWG in the middle will be identically coupled into two PCWGs at both sides, and then, in the output region, the coupled power will be transferred to the output waveguide A and B without coupling from one waveguide into another at the designed frequency range (in the vicinity of a/λ=0.258).

There are two requirements for this power-splitter to function at a large spectral range: i) in the coupling region, the wavelength-insensitive coupling should be supported without change in the coupling length, ii) in the output region, on the other hand, the coupling between the two output waveguides is prevented for balanced output power. We start analysis of these regions shown in Fig. 1 by inspecting dispersion characteristics for each region. The dispersion relations for the coupling region and output region are calculated by the plane wave expansion method (PWE) [11] and shown in Figs. 2(a) and (b), respectively. PWE computational super-cells for the two regions are designated by black boxed areas in Figure 1 and enlarged view of them are shown in insets of Figs. 2(a) and (b), respectively. As shown in the inset of Fig. 2(a), air holes between PCWGs have smaller radius of 0.2a to increase the coupling strength in the coupling region [12]. In Fig. 2(b), two guided modes are labeled ‘even’ and ‘odd’: they are distinguished into even and odd symmetry, with respect to the z-axis centered between the two PCWGs in the output region.

 

Fig. 2. (a) The dispersion relation and the computational super-cell (inset) for coupling region. (b) The dispersion relation and the computational super-cell (inset) for output region. Gray areas in (a) and (b) indicate the designed frequency range. Red lines in (a) designate the possible frequency range with a spectrally constant coupling length and the excitable modes. Boxed area in (b) indicates the region of anti-crossing between modes of different order. The two modes labeled ‘even’ and ‘odd’ in (b) have the corresponding symmetry.

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Fig. 3. PWE calculated patterns of magnetic field for the modes of interest. (a) H_y field profile of the 0^th mode. (b) Hy field profile of the 1^st mode. (c) H_y field profile of the 2^nd mode. PWE calculation points are marked on dispersion curves in Fig. 2(a).

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Dispersion curves in Fig. 2 clearly explain the coupling characteristics of guided modes, in accordance with above requirements. From Fig. 2(a) for the coupling region, a spectrally constant coupling length is expected for a certain frequency range (gray area), between modes that show almost parallel decrease in frequency (marked by red lines). However, for the output region shown in Fig. 2(b), the coupling between two excitable modes, which are labeled ‘even’ and ‘odd’, does not occur because the difference between the propagation constants is almost zero for the frequency range designated by gray area [13].

To confirm the analyses of dispersion relations presented above, the field patterns of excitable modes are identified. Shown in Figs. 3(a)–(c) is the y-component of magnetic field for the eigen-modes of interest in the coupling region. The field patterns are calculated at the normalized frequency of a/λ=0.258 by the PWE and the points at which the calculation is performed are marked on the dispersion relation for coupling region, as shown in Fig. 2(a). Figs. 3(a), (b), and (c) are labeled ‘0^th,’ ‘1^st,’ and ‘2^nd’, respectively, due to their resemblance to modal field patterns of planar multi-mode waveguides. There are several modes at the normalized frequency of a/λ=0.258. The only excited modes, however, are the 0^th mode and the 2^nd, since other modes except them have odd symmetry with respect to the propagation axis. Hence, the power-splitting mechanism is attributed to the superposition of the 0^th mode and the 2^nd, and so the two modes are taken into account in calculating the coupling length L, which is the minimum distance where input field is split into a two-folded image.

To calculate the coupling length, the normal mode theory is employed [14]. Total field, Ψ(x,z), in the coupling region can be expressed by the superposition of excited modes:

where c _m is the field excitation coefficient, ${\psi }_m(x,y)e^{-j{\beta }_{m}z}$ is localized Bloch wave function with propagation constant of β _m, and subscript m denotes order of mode. After propagating coupling length z=L, Ψ(x,L) should satisfy following condition so that a two-folded image can be made:

By inspecting Eq. (2), the expression for coupling length can be derived:

Propagation constants, β ₀ and β ₂ at a/λ=0.258, are read out from dispersion relation shown in Fig. 3(d) and coupling length, L=19.762846a, is calculated by substituting β₀=0.2682 and β₂=0.0.2953(2π/a) into Eq. (3). The coupling length is rounded off to 20a, because photonic crystals have discrete structure in size.

Now that coupling length L is obtained, we verify this by the finite-difference time-domain (FDTD) computation [15]. The setup for the FDTD computation is shown in Fig. 1 and the length of coupling region is 20a, as calculated above. Since a finite structure is considered here, the whole computational domain is surrounded by perfectly matched layers to absorb the outgoing waves. The Gaussian modulated pulse is launched at the entrance and the Poynting vector penetrating through the line detectors is integrated to calculate the output power A and B. To avoid the back reflection at entrance, the width of the input field is spatially adjusted so that FWHM (full width at half maximum) of the input pulse is set to 0.6×√3a. The grid size of FDTD computational domain is set to a/32.

3. Results and summary

Figure 4 shows the calculated output power normalized to the total input power. Spectrally flat transmittance 46–47.6% per each output is achieved around a/λ=0.258 (grayed area in Fig. 4). An abrupt dip between 0.259 and 0.260 in normalized frequency (a/λ) results from the anti-crossing between modes of different order (the boxed area in Fig. 2(b)). Papers on this poor transmittance have been reported previously [16,17]. For the normalized frequency of a/λ=0.258, amplitude of magnetic field calculated by the FDTD is shown in Fig. 5. The FDTD simulated propagation pattern of magnetic field is in very good agreement with the calculated coupling length from Eq. (3).

 

Fig. 4. Normalized output power after FDTD computation. Grayed area indicates spectrally flat coupling range.

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Fig. 5. Amplitude of H_y field calculated by the FDTD at a/λ=0.258. The coupling and output regions are separated by black vertical lines.

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In summary, we designed a power-splitter on 2D photonic crystal with hexagonal structure utilizing the coupling between guided modes supported by PCWGs, with spectrally flat coupling in mind. It achieves up to 47.6% transmission per each output waveguide at the designed frequency range. Further optimization for the real 3-dimensonal slab structure remains for future work.