## Abstract

A directional coupler switch structure capable of short switching length and wide bandwidth is proposed. The switching length and bandwidth have a trade-off relationship in conventional directional coupler switches. Dispersion curves that avoid this trade-off are derived, and a two-dimensional photonic crystal structure that achieves these dispersion curves is presented. Numerical calculations show that the switching length of the proposed structure is 7.1% of that for the conventional structure, while the bandwidth is 2.17 times larger.

©2006 Optical Society of America

## 1. Introduction

Photonic crystals are optical materials with a spatially periodic dielectric constant and may provide a basis for ultrasmall optical integrated circuits, including low-threshold lasers, high-Q cavities, and add-drop filters [1, 2, 3]. Our group is currently developing an optical buffer device as shown in Fig. 1 [4] as one application of photonic crystals. The optical buffer consists of a directional coupler switch as an input and output port, and a ring-shaped waveguide for the storage of optical pulses. To fabricate the buffer device, it is necessary to develop a ring waveguide with very low loss and a DC switch with high extinction ratio, wide bandwidth, and small coupling and switching lengths. Low-loss straight and bend waveguides have been studied by many researcher [5, 6, 7]. For directional coupler, theoretical studies of basic directional coupler structure [8, 9] and improved structures e.g. a structure with shorter coupling length [10] and experimental study [11] have been done. However, it remains difficult to realize a DC switch that satisfies the four conditions above simultaneously. For example, a small coupling length conflicts with the need for a high extinction ratio, and the trade-off relationship between these two parameters has been demonstrated [4]. In the present paper, the trade-off between short switching length and wide bandwidth is examined, and a solution to the problem is proposed.

## 2. Trade-off between switching length and bandwidth

A directional coupler consisting of a pair of parallel waveguides has two eigenmodes, referred to as even and odd modes. The light confined to one of the parallel waveguides can be expressed by the addition of even and odd modes with an appropriate phase difference. When the phase is added to (2*n* + 1)*π* after traveling the coupling length, the light is transfered to the opposite side waveguide. This process is described by

where *k*
_{e} and *k*
_{o} are the wavenumbers of the even and odd modes, respectively, and *L*
_{c} is the coupling length. For switching operation, it is necessary to change the right-hand side of equation (1) to 2*mπ*. The directional coupler is separated into two parts, as shown in Fig. 2. In one part, the wavenumbers of the even and odd modes are fixed at *k*
_{e,fix} and *k*
_{o,fix}, while in the other part, the wavenumbers change from *k*
_{e,off} and *k*
_{o,off} before switching to *k*
_{e,on} and *k*
_{o,on} after switching. When these parameters are satisfied, i.e.,

the switching operation will be successful. Here, *L*
_{fix} is the length of the fixed parameter region, *L*
_{sw} is the length of the variable wavenumber region, and *m* and *n* are arbitrary integers. *L*
_{sw} is referred to as the switching length in this paper. From equations (2) and (3), the switching length can be expressed in two separate parts as

A directional coupler switch can also be constructed using a directional coupler with varying wavenumbers and with length equal to a common multiple of the coupling length *L*
_{c} and the switching length *L*
_{sw}. In both cases, it is necessary to reduce the length of the device in order to shorten the switching length. As the denominator of the right-hand side of the equation (4) increases, the switching length decreases. The switching operation here is performed by changing the refractive index of the material from *n*
_{off} to *n*
_{on}. Expressing the dispersion relation for even and odd modes by *ω*
_{e}(*k*, *n*) and *ω*
_{o}(*k* ,*n*), the total differential of dispersion is given by $\mathrm{d\omega}=\frac{\partial \omega}{\partial k}\mathrm{dk}+\frac{\partial \omega}{\partial n}\mathrm{dn}$. For a single frequency of light, *dω*= 0, and the differential of wavenumber is obtained as

Equation (4) can then be rewritten as

$$\phantom{\rule{1em}{0ex}}\simeq \frac{\left(2n+1\right)\pi}{\left(-\frac{1}{\frac{\partial {\omega}_{e}}{\partial k}}+\frac{1}{\frac{\partial {\omega}_{o}}{\partial k}}\right)\frac{\partial \omega}{\partial n}\mathrm{dn}}.$$

Here, it is assumed that the shifts of the dispersion curves toward higher frequencies due to a slight change in refractive index are almost equal (i.e.,$\frac{\partial {\omega}_{e}}{\partial n}\simeq \frac{\partial {\omega}_{o}}{\partial n}\simeq \frac{\partial \omega}{\partial n}$). It can thus be seen that a small switching length can be obtained by increasing the difference between the group velocities of the even and odd modes.

The bandwidth of the directional coupler is defined as the frequency range in which the phase difference at the output port is within the allowable range. The relationship between the allowable shift of the phase difference at the output ports Δ*ϕ* and the frequency fluctuation Δ*ω* satisfies the following expression for the “off” switching condition:

This equation is derived by replacing (2*n* + 1)*π*, *k*
_{e,off} and *k*
_{o,off} with (2*n* + 1)*π* + Δ*ϕ*, *k*
_{e,off} + Δ*k*
_{e,off}, and *k*
_{o,off} + Δ*k*
_{o,off} in equation (2), respectively. The relationship $\Delta k=\frac{1}{\frac{\mathrm{d\omega}}{\mathrm{dk}}}\Delta \omega $ is used here assuming a constant refractive index. From this equation, Δ*ω* becomes large when the difference between the group velocities of the even and odd modes is small. The allowable frequency fluctuation can be obtained for the “on” switch condition by a similar equation. The bandwidth for switching is determined as the common frequency region between the band-widths for the on and off switching conditions. It can thus be seen that the condition for wide bandwidth conflicts with the condition of small switching length for this device structure.

## 3. Structure allowing small switching length and wide bandwidth

A small switching length and wide bandwidth may be obtained simultaneously by deriving a structure that provides an appropriate dispersion curve. The dispersion curve required is shown in Fig. 3. In the figures, the solid dispersion curve is that for the initial state, and the dashed curve is that after the refractive index of the material has been changed, and red and blue denote the two eigenmodes. The first mode has a monotonically decreasing dispersion curve (red line in the figure), while the other has a curve consisting of three parts; two monotonically decreasing regions separated by a fixed frequency region. The monotonically decreasing regions of both modes have the same slope, resulting in a wide bandwidth except in the fixed frequency region. The operating frequency is therefore set between the fixed frequencies before and after switching. Upon switching, the change in the wavenumber of the eigenmode with fixed frequency is larger than that for the other eigenmode, resulting in a small switching length. Because the wavenumber of even mode at operating frequency changes drastically by existence of flat frequency region, the term of *k*
_{e,on} - *k*
_{e,off} in equation (4) cannot approximate by utilizing equation (5) in this case. It means that we should use equation (4) instead of equation (6) to express the switching length, and the small switching length can be realized even if the even and odd modes have same group velocity around the operating frequency. The fact that the both eigenmodes have same group velocity is important to avoid the intermodal dispersion between two eigenmodes which obstructs the short pulse propagation [12].

In this way, it is possible to realize a directional coupler switch with small switching length and wide bandwidth if a structure with such dispersion curves can be constructed.

## 4. Numerical Design of a novel directional coupler switch

The dispersion curves above can be realized using a two-dimensional photonic crystal structure. Figure 4 shows the dispersion curves and energy distribution for parallel waveguides in a two-dimensional photonic crystal arranged in a triangular lattice consisting of holes of uniform radius. The data were calculated by a two-dimensional plane-wave expansion method. The holes have a radius of 0.29*a*, where *a* is the lattice constants, and are arranged in a triangular lattice in a material with refractive index of 2.76 (i.e., a single-mode GaAs/air slab). Here, we consider the GaAs system because we will utilize InAs/GaAs quantum dots as nonlinear effect materials. The energy distributions of the even and odd modes are concentrated near the waveguide region for almost all wavelengths. However, the energy distribution of the even mode (marked by a circle in the figure) spreads into the central region between the waveguides. If the refractive index around the center of the waveguides is reduced, the dispersion of the even mode will be arrested and a fixed-frequency region will be obtained for the even mode.

The structure shown in figure 5(a) is introduced in order to reduce the refractive index in the central region between waveguides. The radius of the center holes is large so as to reduce the refractive index, and the position of the holes besides the waveguides is shifted towards the center of the structure. The dispersion curves for the modified structure are shown in Fig. 5(b). A flat even mode is obtained in the wavenumber region of 0.360–0.395. In the Fig. 4 and 5, we show only two eigenmodes. Actually, there are some modes which originated in a higher waveguide mode. These modes are omitted from the figures because the frequency of these modes are higher than the flat frequency of even mode that we pay attention to.

The switching length and bandwidth calculated for this structure are shown in Fig. 6(a), and those for the conventional structure are shown in Fig. 6(b). Here, the conventional structure means the directional coupler consisting of 2D photonic crystal in which air holes with uniform radius (0.29a) are arranged with exact triangular lattice. The calculations were performed using the same refractive indices for the fixed- and variable-parameter regions for the off switch condition, that is, *k*
_{e,fix} = *k*
_{e,off} and *k*
_{o,fix} = *k*
_{o,off} in equation (2). In the on condition, only the refractive index of the variable-parameter region was changed (+0.1%). The index change will be achieved by the InAs/GaAs quantum dots system [13]. The equations for *L*
_{sw} and *L*
_{fix} includes an optional integer (see eqs. (2) and (4)). The plotted switching lengths *L*
_{sw} represent the minimum value of eq. (4). *L*
_{fix}, which is necessary to calculate the bandwidth, is the minimum value satisfying eq. (2). This results in steps in the bandwidth curves. From the figure, the switching length of the proposed structure is extremely small in the frequency region 0.2905 – 0.2907. The lower frequency, 0.2905, is the flat frequency of the even mode in the initial state, and the upper frequency, 0.2907, is the flat frequency after the change in refractive index. The switching length therefore becomes small at the operating frequency (between the flat frequencies). The minimum switching length for the proposed structure is 12*a*, which is 7% of the minimum switching length of the conventional structure. In the figure, the frequency range for novel structure is higher than the range for the conventional structure. It is necessary to shift the frequency range of the dispersion of novel structure keeping the flat dispersion shape for matching the frequency range with the conventional structure. For this purpose, we calculated several modified structure, and we have grasped the tendency of deformation of dispersion curves by changing the hole size and positions. The results will be shown in another paper [14].

The bandwidths are shown in the same figure (dashed lines). The bandwidths for the off and on conditions are calculated independently using eq. (7). The bandwidth of the switch is smaller value between the bandwidth of switch off and on conditions. -Δ*ω*/*ω*Δ*ϕ* is shown for each *ω* in the figure. When the absolute value is large, the bandwidth is large. The bandwidth of the proposed structure becomes quite small at the flat frequencies of the even mode, but remains large between the flat frequencies. A small switching length and wide bandwidth can therefore be obtained simultaneously using this structure by setting the operating frequency between these flat frequencies. The switching lengths and bandwidths for various refractive index changes are listed in table 1. For each condition, the proposed structure has a smaller switching length and wider bandwidth than a conventional directional coupler switch with a simple triangular lattice.

## 5. Conclusion

In the development of directional coupler switches constructed using photonic crystal, representing key components of optical circuits, it has been found that the switching length has a trade-off relationship with device bandwidth in a simple parallel waveguide structure. To resolve this conflict, an improved structure was proposed in which special eigenmodes are used to allow a small switching length and wide bandwidth to be realized simultaneously. One of the modes exhibits monotonically decreasing (or increasing) dispersion, while the other exhibits three discrete regions; two regions of monotonically decreasing (or increasing) dispersion separated by a region with a flat frequency response. The trade-off between switching length and bandwidth was shown to be eased when the operating frequency is near the flat frequency region. Numerical calculations showed that a photonic crystal directional coupler switch having an even mode with flat frequency response can be constructed by deforming the radius and position of holes between the parallel waveguides. The switching length of the proposed structure is 12.3*a*, which is 7.1% of that for the conventional structure, and the bandwidth is 0.391%, which is 2.17 times larger than that for the conventional structure. This is achieved using a refractive index change for switching of 0.1%. The means for changing the operating frequency will be presented in a subsequent paper [14].

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**14. **N. Yamamoto, J. Sugisaka, S. H. Jeong, and K. Komori, *unpublished*.