## Abstract

Topology optimization has been used to design intersections in two-dimensional photonic crystal slab waveguides. We have experimentally confirmed that the optimized intersection displays high-transmittance with low-crosstalk for the straightforward beam-propagation line.

©2006 Optical Society of America

## 1. Introduction

Two-dimensional (2D) photonic crystal (PC) slab waveguides (WGs) are attractive for novel applications in a miniaturized photonic integrated circuit (PIC) [1, 2]. We have already demonstrated a symmetric Mach-Zehnder type all-optical switch based on the 2DPC WG (PC-SMZ) [3]. To construct a compact PIC on a single chip, a 2DPC WG intersection is an important element for optical wiring in the PIC as well as bend, branch and directional coupler. Figure 1 shows an example of the WG intersection applied for monolithic integration of control pulse (CP) and signal pulse (SP) WGs in the PC-SMZ, where wide and flat bandwidth, high-transmittance and low-crosstalk are required for different wavelengths, i.e., λ_{1} for the SP and λ_{2} for the CP. The separation between *λ*
_{1} and *λ*
_{2} is typically 20 nm at 1.3 µm for utilizing quantum dots as a nonlinear material [4].

So far, the eliminations of crosstalk in 2DPC WG intersections were investigated to consider coupling of the four branches of the intersection in terms of a resonant cavity with the proper symmetry at the center [5–7]. In such designs, however, useful bandwidths are narrow (e. g. ~10 nm with crosstalk as low as-10 to-45 dB) due to a trade-off between the bandwidth and the transmittance determined by the Q-factor of the cavity. The trade-off can be avoided by optimizing the intersection design [8].

Recently, a new design method called topology optimization (TO) originating from mechanics [9] has been developed and proven effective for drastic improvement of bandwidths and transmittances in the 2DPC WGs such as bends and branches [10–12]. The TO is a gradient-based optimization method that creates optimized designs with no restrictions on resulting topologies and can thus be used to create designs with previously unattainable properties [13–15]. It has been more recently shown that the TO method is effective for the WG intersection design [16]. In this paper, we present the numerical results in detail together with the experimental results on the WG intersections with broad bandwidth, high-transmittance and low-crosstalk by using the TO method.

## 2. TO design of 2DPC WG intersection

We consider a 2DPC with a hexagonal lattice (lattice constant *a*) of air holes (radius, *r*=0.3*a*) in a dielectric substrate (effective index, *n*=2.95) [17]. It has a photonic bandgap for a TE mode (in-plane electronic field) in the normalized frequency range from 0.238 to 0.319 *c*/*a*, where *c* is the velocity of the light in a vacuum. An intersection is formed by a line-defect WG composed of one missing row of air holes, as shown in Fig. 2(a). An even guided mode below the light-line is supported for the frequency range from 0.249 to 0.274 *c*/*a*.

The optimization algorithm is based on a 2D finite-element frequency-domain solver. The full computational model consists of the domain shown in Fig. 2(b) as well as additional perfectly matching layers to eliminate the unwanted reflections from the input and output WG ports. Each unit cell is discretized using 14×12 four-noded quadrilateral elements. The solver is used repeatedly in an iterative scheme, in which the material distribution is updated every iteration based on analytical sensitivity analysis and use of a mathematical programming tool. The optimization method is described in detail in [9, 14]. A TO procedure is executed to maximize transmittance in a straightforward line (a-d) by modifying the refractive index distribution in the design domain indicated by a green area shown in Fig. 2(b). We use a technique based on active sets in which we fix a number of target frequencies in the desired frequency interval. During the optimization, these target frequencies are repeatedly changed, according to the most critical frequencies with lowest transmission. The critical frequencies are found every 25th iteration by performing a fast frequency sweep [18]. To avoid a leaky-mode and hence low-transmittance range, target frequencies are restricted to the frequency range for a non-leaky guided mode which is limited by a light-line in the band diagram for an air-bridge type 2DPC slab structure.

We use continuous design variables *x _{e}* (0≤

*x*≤1) to allow for utilizing a gradient-based optimization strategy. The refractive index is represented by applying a linear interpolation in

_{e}*x*for each finite element within the design domain. Thus the existence of elements in the final design with values between zero (

_{e}*n*

_{air}) and one (

*n*

_{dielctric}), so-called gray elements, is undesirable for fabrication. To remedy this problem, we use an implicit variant by introducing the penalty σ

_{pen}as an extra damping term for elements within the design domain. The penalty σ

_{pen}is set to be proportional to the damping matrix and the penalty function p=4 α

*x*(1-

_{e}*x*), where α is a real and positive constant. As shown in Fig. 3(h), the value of p is large in the middle range. Therefore the gray elements will be un-economical since they absorb energy and will be forced toward either zero or one by applying the parameter α.

_{e}Figures 3(a)–(g) show the snapshots of the design during the optimization process and Figs. 4(a)–(g) show the transmittance spectra for each design. The value of the parameter α is doubled for each 200th iteration from 0.01 to 0.32. Many gray elements appear in the designs shown in Fig. 3(b) and (c), though they result in the high-transmittance with low-crosstalk (>-40 dB~-50 dB). The gray elements gradually disappear with increasing α. The final optimized design shown in Fig. 3(g) is feasible for fabrication and exhibits high transmittance with low crosstalk.

Figures 5(a) and (b) show profiles of a steady-state magnetic-field at the frequency 0.26 *c*/*a* for the standard (Fig. 3(a)) and TO (Fig. 3(g)) designs, respectively. The light wave smoothly passes along the optimized intersection, while it is scattered around the standard intersection. The calculated transmittance spectra with straightforward line (a-d) and two crosstalk lines (a-b and a-c) for each design are shown in Figs. 5(c) and (d). Quite different from the standard design, TO design results in high-transmittance and low-crosstalk (-20dB~-30dB) in a wide frequency range. This means that the crosstalk is suppressed sufficiently for two different wavelengths of 1,300 nm for λ_{1} and 1,280 nm for λ_{2} for designing the PC-SMZ. For these reasons, we can say that the TO method provides us with advantages for the 2DPC intersection design.

## 3. Fabrication and measurements

The sample was fabricated in an epitaxial hetero-structure grown by molecular beam epitaxy. A 250-nm-thick GaAs core layer was deposited on top of a 2-µm-thick Al_{0.6}Ga_{0.4}As cladding/sacrificial layer on the GaAs substrate. The air-bridge WG was fabricated using high-resolution electron-beam lithography, dry etching, and selective wet-etching techniques [19]. The lattice constant and hole diameter are typically 360 and 210 nm, respectively.

The optical properties were characterized by observing transmittance spectra for the TE polarization over a broad-wavelength region from 850 to 1,600 nm [20]. Figures 6(a) and (b) show plan-view photographs observed by an IR-vidicon camera for an input optical beam wavelength of 1,370 nm. Input ports are indicated by arrows. It is found that the standard sample shows a bright spot even at the crosstalk port c in Fig. 6(a), while the TO sample shows no crosstalk at the port c in Fig. 6(b). Figures 6(c) and (d) show measured transmittance spectra for the standard- and TO-design samples, respectively. For reference, transmittance spectra of the straight 2DPC WG are inserted by dotted line curves as well as plan-view SEM photographs of the used samples. In the lower photograph, uniquely shaped air-hole patterns near the crossing spot are actually fabricated patterns corresponding to the calculated TO-designs, as shown in Fig. 5(b). A transmittance spectrum for the straightforward beam (a-d) in the standard sample is degraded seriously due to a large crosstalk (a-c), while the TO-design sample exhibits excellent transmittance characteristics: The peak transmittance (a-d) is more than 8-dB higher than the crosstalk level (a-c) and comparable to that for the straight WG. Moreover, a measured 3-dB bandwidth of ~60 nm is comparable to those (~70 nm) for the designed value and for the straight WG. The slight band-narrowing (~10 nm), appearing in the vicinity of the band edge (1,380~1,410 nm), is partly due to the structure imperfections which are sensitive to the low group velocity and partly due to the band-narrowing in designing the bend WG as compared to the straight WG. Improvement of this problem is currently under investigation by means of topology optimized design of the bend WG as well as optimization of the precise fabrication process.

## 4. Conclusions

We experimentally demonstrated the intersection of the 2DPC WG designed by the TO method and verified the effectiveness by achieving high transmittance with low crosstalk for the straightforward beam line. The results encourage us to develop the ultra-fast all-optical devices with high performances.

## Acknowledgments

The authors would like to thank Prof. O. Sigmund and Dr. J. S. Jensen (Technical University of Denmark) for the instructive suggestions and support, and Prof. M. Kristensen (University of Arhus) and Prof. P. I. Borel (Technical University of Denmark) for helpful suggestions. This work was supported in part by the New Energy and Industrial Technology Development Organization (NEDO) projects.

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