# Nonstandard FDTD Simulation-Based Design of CROW Wavelength Splitters

Authors: Naoki Okada; James B. Cole; Shigeki Yamada; Kensuke Ogawa; Yoshifumi Katayama
Journal: Advances in Optical Technologies
 (2011)

---

## 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.


---

## Body

## 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 μm) 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 Figure1, the microdisks can be arranged to form any path. Thus, we can flexibly design optical devices.Figure 1
Flexible layout in the CROW. The gray regions represent microdisks, and black regions show whispering gallery modes in each disk.The resonance wavelength and disk size are calculated using the WGM resonance condition. The resonance condition in geometric optics theory is given by [18](1)2πr=lλn(r≫λn),wherer is the disk radius, λ is the wavelength outside the disk, n is the specific refractive index of the disk, and ℓ 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(2)Jl-2(nkr)Jl-1(nkr)=nHl-2(1)(kr)Hl-1(1)(kr)-(l-1)(n2-1)nkr,wherek=2π/λ, Jℓ is the Bessel function of the first kind, and Hℓ(1) 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(3)Jl-1(nkr)Jl(nkr)=1nHl-1(1)(kr)Hl(1)(kr).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 (SiO2) outside.Table 1
Example resonance conditions. The specific refractive index between Si and SiO2 is n=3.5/1.45.
TE modeWavelengthλRadiusrModeℓ1350 nm953.72 nm81400 nm1096.58 nm91450 nm1245.55 nm101500 nm1400.82 nm111550 nm1562.53 nm12TM modeWavelengthλRadiusrModeℓ1350 nm655.02 nm51400 nm790.51 nm61450 nm931.76 nm71500 nm1078.97 nm81550 nm1232.32 nm9


## 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 Figure2. The WGMs of wavelength λA and λB are excited in a disk O. Disks A and B are excitable only by λA and λB, respectively. We empirically computed the appropriate gap spacings between disks, Δ=0.2r (r=diskradius). If Δ is too small, both the desired wavelength and undesired one propagate. But if Δ is too large, the desired wavelength is also suppressed.Figure 2
Wavelength splitter for disk resonators. Gray regions represent microdisks with excited WGMs.rO, rA, and rB are the radii of disks O, A, B, respectively. ΔOA is the gap spacing between disks O and A (ΔOO, ΔOB, etc., analogously defined). Source is a superposition of wavelengths λA and λB. WGMs of λA and λB are excited in disks O. Disks A and B are excitable only by λA and λB, respectively.The second device extracts arbitrary wavelengths from an optical waveguide as shown in Figure3. A multiplexed input signal of wavelengths λA and λB propagates in the optical waveguide. Disks A and B are excitable only by λA and λB, respectively.Figure 3
Wavelength splitter for multimode signals in an optical waveguide. Circular gray regions represent microdisks with excited WGMs. Linear gray regions represent an optical waveguide.rA and rB are the radii of disks A and B, respectively. ΔAA is the gap spacing between disks A (ΔBB analogously defined). Source is a superposition of wavelengths λA and λB. Disks A and B are excitable only by λA and λB, respectively.


## 4. Simulation and Validation

We calculated the electromagnetic wave propagation in two devices (Figures2 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 Table2. 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 1.1r (r=diskradius) around the disk center. As shown in Figure 4, the NS-FDTD algorithm accurately computes the WGM.Table 2
Example parameters used to simulateℓ=8 mode WGM in the TE mode.
Wavelength1550 nmDisk radius1095 nmGrid spacing5 nmRefractive index3.5/1.45 (Si/SiO2)Computational space4μm × 4 μmScattered intensities of electric fields at steady state, TE mode. (a) Analytical solution. (b) Numerical result by the NS-FDTD algorithm. (c) Angular intensity distributions of (a) and (b) on a circular contour of radius1.1r (r= disk radius) around the disk center.
(a)(b)(c)Before simulating the splitter shown in Figure2, 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 sourceposition=0.75r, but the optimal source position depends on the resonance mode. In low modes, we found that the optimal source position is shifted from 0.7r (at 6 mode) to 0.8r (at 12 mode). In the following simulations, we chose sourceposition=0.775r because both ℓ=9 and ℓ=11 modes are excited in disk O (Figure 2).Figure 5
Effect of source positions on anL=8 WGM resonance intensity. The source position is shifted from the center =0.0r to the surface =1.0r (r=diskradius). The intensity is an average over the surface of a disk at steady state.We simulated the device shown in Figure2 for 220 time steps (about 2400 wave periods) and observed the IER on ports set at resonance-enhanced points outside disks A and B. We gradually damp the source amplitude after 219 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 A and B, respectively. As shown in Figure 6, both IERs between λA=1300 nm and λB=1550 nm are over 20 dB (26.3 dB on port A; 29.1 dB on port B).Table 3
Computational parameters and resonance modes for the wavelength splitter shown in Figure2. θA and θB are integral multiples of π/ℓ rad (ℓ=modenumber in the disk O).
Computational parametersλA1300 nmλB1550 nmrO1214.08 nmrA1310.52 nmrB1562.54 nmθA3π/11 radθB3π/9 radGrid spacing5 nmRefractive index3.5/1.45 (Si/SiO2)Computational space14μm × 12 mResonance modesPortλAλBO11 mode9 modeA12 modeunexcitedBunexcited12 modeWavelength spectra from the simulation of the wavelength splitter of Figure2. The observation ports are set at resonance-enhanced points outside disks A and B. (a) Spectrum on the port A. Interchannel extinction ratios = 26.3 dB. (b) Spectrum on the port B. Interchannel extinction ratio = 29.1 dB.
(a)(b)We similarly simulated the device shown in Figure3 for 220 time steps (gradually damping the source after 219 time steps) and observed IERs on ports set at resonance-enhanced points outside disks A and B. We used the simulation parameters listed in Table 4. Figures 7(a) and 7(b) show the computed spectra on the port A and B, respectively. As shown in Figure 7, both IERs between λA=1400 nm and λB=1550 nm are over 20 dB (21.5 dB on port A; 24.5 dB on port B).Table 4
Computational parameters and resonance modes in the wavelength splitter shown in Figure3.
Computational parametersλA1400 nmλB1550 nmrA1307.43 nmrB1447.51 nmGrid spacing5 nmRefractive index3.5/1.45 (Si/SiO2)Computational space12μm × 12 μmResonance modesPortλAλBA11 modeUnexcitedBUnexcited11 modeWavelength spectra from the simulation of the wavelength splitter of Figure3. The observation ports are set at resonance-enhanced points outside disks A and B. (a) Spectrum on port A. Interchannel extinction ratio = 21.5 dB. (b) Spectrum on port B. Interchannel extinction ratio = 24.5 dB.
(a)(b)Finally, we examined effects of manufacturing errors for a practical realization. The radius errorΔr and refractive index error Δn are produced during the manufacturing process and generate the shift of the resonance wavelength Δλ. In the TE and TM resonance conditions, we found that Δλ is approximated by(4)Δλ≅λlrlΔr+λl+1nlΔn,whereλℓ, rℓ, and nℓ are the wavelength, radius, and refractive index on the ℓ mode resonance (rℓ→rℓ+Δr, nℓ→nℓ+Δn⇒λℓ→λℓ+Δλ). The relationship between Δλ and Δr is derived from 2πrℓ/λℓ=2π(rℓ+Δr)/(λℓ+Δλ), where the resonance condition is maintained. The relationship between Δλ and Δn is more complex and is numerically approximated. Figure 8 shows the Δλ distribution using example parameters listed in Table 2. The latest lithography gives Δr≅5 nm and thus Δλ≅7 nm (Δn=0) as shown in Figure 8. The desired resonance wavelength (Δλ=0) is obtained by adjusting the refractive index of the silicon, which depends on the temperature [22].Figure 8
Effects of manufacturing errors.Δr is the radius error. Δn is the refractive index error. Δλ is the shift of the resonance wavelength.


## 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 Figure2 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.



---

*Source: 265702-2011-06-01.xml*