Advances in Optical Technologies

Volume 2011, Article ID 265702, 6 pages

http://dx.doi.org/10.1155/2011/265702

## Nonstandard FDTD Simulation-Based Design of CROW Wavelength Splitters

^{1}Department of Computer Science, Graduate School of Systems and Information Engineering, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8573, Japan^{2}Department of Nanosystem Science, Graduate School of Nanobioscience, Yokohama City University, 22-2 Seto, Kanazawa-ku, Yokohama, Kanagawa 236-0027, Japan^{3}Optics and Electronics Laboratory, Fujikura Ltd., 1440 Mutsuzaki, Sakura, Chiba 285-8550, Japan^{4}Koga Soken Ltd., 344-6 Uchikoshi-machi, Hachioji, Tokyo 192-0082, Japan

Received 27 January 2011; Accepted 23 March 2011

Academic Editor: Augusto Belendez

Copyright © 2011 Naoki Okada et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The finite-difference time-domain (FDTD) algorithm has been used in simulation-based designs of many optical devices, but it fails to reproduce high-Q whispering gallery modes (WGMs). On the other hand, the nonstandard (NS) FDTD algorithm can accurately compute WGMs and can be used to make simulation-based designs of WGM devices. Wavelength splitters using the coupled resonator optical waveguides (CROWs) based on WGM couplings have recently attracted attention because they are potentially ultracompact. In this paper, we design a CROW wavelength splitter using NS FDTD simulations and demonstrate high interchannel extinction ratios of over 20 dB.

#### 1. Introduction

The finite-difference time-domain (FDTD) algorithm [1] has been used in simulation-based designs of optical devices such as optical fibers. However, the FDTD algorithm has not been used to design optical disc and ring resonators based on whispering gallery modes (WGMs) in the past, although these are essential building blocks of integrated optical circuits. This is due to failure of the conventional FDTD algorithm to reproduce the high-Q WGM resonances [2–4]. Instead the discontinuous Galerkin time-domain (DGTD) method [5] which can accurately calculate these resonances with lower memory consumption has been used, but its computational overhead is significantly higher than FDTD [6]. Recently, the nonstandard (NS) FDTD algorithm [7, 8] has been successfully used for high-accuracy WGM simulations with lower memory consumption and computational overhead [9, 10]. This is because high-accuracy difference operators are derived by optimizing to monochromatic wave propagation in the NS-FDTD algorithm and its temporal-spatial difference errors are considerably reduced by comparison with the conventional FDTD. Details of the NS-FDTD derivation are given in [8, 9]. Thus, the NS-FDTD algorithm can be used to make simulation-based designs of WGM devices.

On the other hand, the coupled resonator optical waveguide (CROW) [11, 12] has attracted much attention in recent years because it combines characteristics of both resonator and waveguide and is potentially ultracompact. In particular, the optical wavelength splitter using the CROW [13, 14] is very small in comparison with conventional splitters. For example, the microring resonator-based splitter [15] has high interchannel extinction ratios (IERs), but its size is much larger than a wavelength (a few dozen *μ*m) because the resonator employs total internal reflection with designs based on geometric optics theory. Other splitters using an arrayed waveguide grating [16] and multimode interferometer [17] are also large and have just barely enough IERs because they are not resonators. In contrast, the CROW splitter has high IERs and is potentially ultracompact (on the order of a wavelength) because the CROW is based on WGM couplings between contiguous disk resonators in the Mie regime.

In this paper, we design two CROW wavelength splitters based on the NS-FDTD simulation and demonstrate its high IERs of over 20 dB.

#### 2. Coupled Resonator Optical Waveguide

The coupled resonator optical waveguide (CROW) consisting of microdisks propagates energy with low loss by coupling whispering gallery mode (WGM) resonators in the Mie regime. As shown in Figure 1, the microdisks can be arranged to form any path. Thus, we can flexibly design optical devices.

The resonance wavelength and disk size are calculated using the WGM resonance condition. The resonance condition in geometric optics theory is given by [18]

where is the disk radius, is the wavelength outside the disk, is the specific refractive index of the disk, and is the mode number (integer). These high mode WGMs are widely used in many optical devices such as microring resonators, but they are much larger than a wavelength because high mode resonators employ total internal reflection.

On the other hand, in the Mie regime, the size of WGM resonators is of the order of a wavelength and the resonance conditions are given by the Mie theory [19, 20]. In the transverse electric (TE) mode in which the electric field is perpendicular on the media interfaces, the resonance condition is

where , is the Bessel function of the first kind, and is the Hankel function of the first kind. In the transverse magnetic (TM) mode in which the electric field is parallel to the media interfaces, the resonance condition is

We can numerically compute the appropriate disk size for a given wavelength and refractive index using (2) and (3). Example resonance conditions are listed in Table 1, where we choose the widely used specific refractive index between silicon (Si) inside the disk and silicon dioxide (SiO_{2}) outside.

#### 3. Optical Wavelength Splitter

In the CROW, disks of different size (tuned to specific wavelengths) can be used to select the components of a multiplexed input signal and function as a wavelength splitter. The interchannel extinction ratio is high because the disk resonators are strongly excited and coupled only at specific wavelengths. This CROW splitter is much smaller than conventional devices since the resonator sizes are comparable to a wavelength. Moreover, the CROW splitter can be used in multiplex polarization communication because the resonance conditions of (2) and (3) are polarization dependent unlike conventional devices.

We design two CROW splitters. The first device separates wavelength components from a disk resonator which is concurrently excited by two different wavelengths as shown in Figure 2. The WGMs of wavelength and are excited in a disk . Disks and are excitable only by and , respectively. We empirically computed the appropriate gap spacings between disks, (). If is too small, both the desired wavelength and undesired one propagate. But if is too large, the desired wavelength is also suppressed.

The second device extracts arbitrary wavelengths from an optical waveguide as shown in Figure 3. A multiplexed input signal of wavelengths and propagates in the optical waveguide. Disks and are excitable only by and , respectively.

#### 4. Simulation and Validation

We calculated the electromagnetic wave propagation in two devices (Figures 2 and 3) using the nonstandard (NS) finite-difference time-domain (FDTD) algorithm in the TE mode and demonstrated the interchannel extinction ratios (IERs) and effects of manufacturing errors. The computational domain is terminated by the NS perfectly matched layer absorbing boundary condition [21]. In the TM mode, similar devices and functions can be computed.

First we demonstrated that whispering gallery modes (WGMs) are accurately simulated by the NS-FDTD algorithm. Example parameters are listed in Table 2. In the TE mode, we calculated the scattered intensity of the electric field due to an infinite plane wave using the NS-FDTD algorithm and compared the result with the Mie theory in Figure 4. Figures 4(a) and 4(b) visualize scattered intensity distributions of analytical and numerical results at steady state (about 1000 wave periods). Figure 4(c) shows angular scattered intensity distributions on a circular contour of radius () around the disk center. As shown in Figure 4, the NS-FDTD algorithm accurately computes the WGM.

Before simulating the splitter shown in Figure 2, we must select the point source position because resonance intensity is sensitive to source position. We simulated WGMs in a disk using different source positions with parameters listed in Table 2. Figure 5 shows the average total intensity over the surface of a disk at steady state (about 2000 wave periods) for different source positions. As shown in Figure 5, if the source position is too far from the surface or too close, waves cross the surface are radiated outside the disk. For the 8 mode resonance in Figure 5, the strongest resonance is obtained at , but the optimal source position depends on the resonance mode. In low modes, we found that the optimal source position is shifted from (at 6 mode) to (at 12 mode). In the following simulations, we chose because both and modes are excited in disk (Figure 2).

We simulated the device shown in Figure 2 for 2^{20} time steps (about 2400 wave periods) and observed the IER on ports set at resonance-enhanced points outside disks and . We gradually damp the source amplitude after 2^{19} time steps to reduce spurious frequency components in the Fourier transform. We used the simulation parameters listed in Table 3. These wavelengths are widely used in optical communication. Figures 6(a) and 6(b) show the computed spectra on ports and , respectively. As shown in Figure 6, both IERs between nm and nm are over 20 dB (26.3 dB on port ; 29.1 dB on port ).

We similarly simulated the device shown in Figure 3 for 2^{20} time steps (gradually damping the source after 2^{19} time steps) and observed IERs on ports set at resonance-enhanced points outside disks and . We used the simulation parameters listed in Table 4. Figures 7(a) and 7(b) show the computed spectra on the port and , respectively. As shown in Figure 7, both IERs between nm and nm are over 20 dB (21.5 dB on port ; 24.5 dB on port ).

Finally, we examined effects of manufacturing errors for a practical realization. The radius error and refractive index error are produced during the manufacturing process and generate the shift of the resonance wavelength . In the TE and TM resonance conditions, we found that is approximated by

where , , and are the wavelength, radius, and refractive index on the mode resonance (, ). The relationship between and is derived from , where the resonance condition is maintained. The relationship between and is more complex and is numerically approximated. Figure 8 shows the distribution using example parameters listed in Table 2. The latest lithography gives nm and thus nm () as shown in Figure 8. The desired resonance wavelength () is obtained by adjusting the refractive index of the silicon, which depends on the temperature [22].

#### 5. Conclusion

Based on the nonstandard (NS) finite-difference time-domain (FDTD) simulations, we designed two ultracompact optical wavelength splitters using the coupled resonator optical waveguide (CROW) and demonstrated high interchannel extinction ratios (IERs) of over 20 dB. The first device in Figure 2 separates wavelengths from a disk resonator and also can function as a source separator. The second device in Figure 3 extracts arbitrary wavelengths from an optical waveguide and can theoretically split any multimode signal. These splitters can be used in multiplex polarization communication, since low mode whispering gallery modes are polarization dependent. In some applications polarization independence is desirable, but this could be achieved by using two disks excited in each mode.

The NS-FDTD algorithm has much lower computational overhead than the discontinuous Galerkin time-domain method and is much simpler to program. In this paper, we presented the NS-FDTD simulation-based design as a prototype tool for the design of more advanced devices such as photonic-crystal circuits. In future work we will try to optimize the gap spacing and obtain a higher IER, within such technological constraints as semiconductor lithography and wavelength dispersion.

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