We explore the use of air trenches to achieve compact high efficiency 90° waveguide bends and beamsplitters for waveguide material systems that have low refractive index and low refractive index contrast between the core and clad materials. For a single air interface, simulation results show that the optical efficiency of a waveguide bend can be increased from 78.4% to 99.2% by simply decreasing the bend angle from 90° to 60°. This can be explained by the angular spectrum of the waveguide mode optical field. For 90° bends we use a micro-genetic algorithm (µGA) with a 2-D finite difference time domain (FDTD) method to rigorously design high efficiency waveguide bends composed of multiple air trenches. Simulation results show an optical efficiency of 97.2% for an optimized bend composed of three air trenches. Similarly, a single air trench can be designed to function as a 90° beamsplitter with 98.5% total efficiency.
©2003 Optical Society of America
Much attention has been focused on reducing the size of waveguide components to enable higher levels of integration in planar lightwave circuits (PLCs). Waveguide bends and splitters are particularly critical elements that heavily influence overall component size. Low refractive index contrast waveguides in low index materials such as silica typically have a minimum bend radius of multiple millimeters to several centimeters, which creates an ultimate limit to device size reduction. An obvious approach to shrink the size of such components is to use waveguides with high refractive index contrast such that the minimum bend radius is much smaller. Examples for the 1.3 µm and 1.5 µm wavelength regions include core materials with large refractive index such as Si or poly-Si embedded in a low index cladding of SiO2 [1,2]. High refractive index contrast in such systems permits total internal reflection (TIR) to confine light to the waveguide even when the bend radius is reduced to the order of a few microns. Ninety degree bends can be decreased still further in size through the use of resonant cavities [3,4] or corner mirrors with either one or two segments . However, for a given level of interface roughness between the core and clad materials, high refractive index contrast waveguides give rise to more scattering loss than those with low refractive index contrast. This may well impose an eventual limit on the degree of device integration that can be achieved in PLCs based on high index contrast waveguides.
An alternate approach is to use materials with a large refractive index in configurations that lead to low refractive index contrast and hence weak optical confinement. Examples include SiGe on Si  and SOI ridge waveguides [5–7]. In such cases vertically etched faces can be used as mirrors to realize sharp waveguide bends [8–11]. Alternatively, an isolation trench on the outside edge of the bend can be used in conjunction with an offset between the bend waveguide segment and the straight input and output segments to dramatically reduce the bend radius .
However, most commercially available PLCs are based on very low loss silica waveguides, which have a low refractive index as well as low refractive index contrast between the core and clad. Moreover, polymer waveguides , which typically have similar low refractive index and low index contrast, continue to receive attention as a potentially attractive candidate for PLCs. One proposed solution for silica materials is to etch regions that define a high index contrast bend with tapers on both ends to couple light into and out of the bend region . Overall sizes of such structures are expected to be on the order of 100 µm or greater. In this paper we propose an alternate approach that permits the realization of significantly smaller bends and beamsplitters for 90° geometries in low index and low index contrast materials systems such as silica and polymers. Our approach is based on multiple etched trenches with planar interfaces. We have used a micro-genetic algorithm (µGA) with a 2-D finite difference time domain (FDTD) method to rigorously design high efficiency waveguide bends with multiple trenches that operate in TIR. The use of multiple trenches permits improved performance compared to a single air interface by reflecting optical field components in the waveguide mode’s angular spectrum that do not undergo TIR. We show an example in which this strategy leads to an optical bend efficiency of 97.2% for a 90° bend compared to 78.4% for a single air interface. We likewise show that a single trench that operates through frustrated TIR  can be used as a high efficiency beamsplitter.
In Section 2 we discuss bends made with a single air interface, and show how decreasing the bend angle (defined in Fig. 1) from 90° to 60° permits an increase in the optical bend efficiency from 78.4% to 99.2%, which is explained using the angular spectrum of the optical field of the waveguide mode. In Section 3 we evaluate multiple air trenches to create 90° waveguide bends. In Section 4 we show how a single air trench can form a 90° beamsplitter, followed in Section 5 by a brief summary.
2. Single interface TIR mirrors
To begin our discussion we first consider the situation shown in Fig. 1 in which a 90° waveguide bend is formed with an air interface oriented at 45° to the input and output waveguide sections. The core and clad refractive indices are assumed to be 1.500 and 1.465, respectively, and the waveguide width is 2 µm. For light with a free space wavelength, λ0, of 1.55 µm, the waveguide supports a single transverse mode. We assume TE polarized light (electric field normal to the plane of the figure). A 2-D finite difference time domain (FDTD) method  with perfectly matched layer (PML) boundary conditions  is used to simulate this and all subsequent structures. In each case the Yi cell size is λ0/80 and the simulation source launches the fundamental mode of the waveguide into the input waveguide.
Figure 2 shows an image plot of the magnitude squared of the time average electric field of light propagating through a structure similar to that shown in Fig. 1, except that the output waveguide has been shifted 0.7 µm in the +z-direction to account for the Goos-Hanchen shift. Note that most of the light is reflected by the air interface through total internal reflection into the output waveguide. The bend efficiency, which we define as the ratio of the optical power in the output waveguide (calculated with the Poynting vector) to the incident optical power, is 78.4%. Note that some of the light is transmitted through the air interface, which reduces the bend efficiency.
The reason behind this result can be understood qualitatively by analyzing the angular spectrum of the guided mode (see Ref.  for a description of the method). The mode profile and its angular spectrum are shown in Fig. 3. The kx=0 plane wave component of the angular spectrum corresponds to an incidence angle of 45 degrees at the air interface. For the waveguide mode effective index, which is 1.485, the critical angle for TIR is 42.3°. (Note that each plane wave component actually experiences a slightly different effective index so the critical angle for each is marginally different.) As seen in the figure, some fraction of the angular spectrum plane wave components are incident on the air interface at angles less than the critical angle. Hence some portion of these plane wave components are transmitted through the interface, which is what we observe in Fig. 2.
To verify that this is the case, consider an 80° bend as shown in Fig. 4(a). In this situation the kx=0 plane wave component is incident at an angle of 50° on the air interface. Fewer plane wave components are incident at less than the critical angle and thus are totally internally reflected so that the bend efficiency, 93.4%, is greater compared to the 90° bend. Likewise, as shown in Fig. 4(b), the situation is improved when the bend angle is further decreased to 60° (i.e., the incidence angle for the kx=0 component is increased to 60°). In this case the bend efficiency is 99.2%. Due to their high bend efficiency, single air interface bends with angles less than 90° appear to be attractive components for PLC size reduction.
3. Multiple air-trench 90° bends
In this section we consider a method to improve the bend efficiency for 90° bends. Specifically, we propose the use of multiple air trenches in which the first air interface reflects much of the incident energy through TIR and the rest of the interfaces in the stack act similar to a Bragg mirror to reflect that portion of the angular spectrum that does not undergo TIR. However, the design of such trenches is not entirely straightforward since the Bragg mirror function must operate over a range of angular spectrum plane wave components. Moreover, it must also compensate for frustrated TIR, which occurs due to the finite trench thickness. We have therefore used a rigorous design tool based on the micro-genetic algorithm (µGA) and FDTD  to achieve optimal designs for 90° waveguide bends based on one, two, and three air trenches. The parameters varied in the µGA optimization are the individual trench length and thickness, the separation between the trenches, and the overall position of the trench structure relative to the waveguide.
Simulation results for the three cases are shown in Fig. 5 with trench parameters and bend efficiencies given in Table 1. In the case of a single air trench the maximum bend efficiency achieved by our µGA-FDTD design tool was 85.7%, which is substantially greater than the result for a single air interface. As seen in Fig. 5(a), some fraction of the incident light is still transmitted through the 1.28 µm thick air trench. This can be understood as follows. The addition of the second air interface increases the reflection of angular spectrum plane wave components that do not undergo TIR, but at the cost of permitting frustrated TIR to occur for the other plane wave components. As is apparent from our results, a design can be realized in which the former effect more than compensates for the latter.
As seen in Fig. 5(b), the addition of a second air trench can significantly reduce the amount of light that leaks through the structure such that the bend efficiency is increased to 94.8%. Note that in this case the air trenches have comparable thicknesses (0.81 µm compared to 0.90 µm). A third air trench, shown in Fig. 5(c), increases the bend efficiency to 97.2%. In this design the thicknesses of the air trenches are significantly different (1.05 µm, 0.52 µm, and 0.70 µm, respectively), which illustrates the utility of our µGA-FDTD design tool in finding nonintuitive solutions to rigorous electromagnetic design problems. We have found that the addition of further layers does not significantly improve the bend efficiency.
An additional attractive feature of multiple air trench 90° bends is illustrated in Fig. 6 in which the bend efficiency is shown as a function of the wavelength of the guided mode. In each case, the bend efficiency is only weakly dependent on wavelength. For example, the bend efficiency for the 3-layer air trench is greater than 95% over the wavelength range 1.43 µm to 1.61 µm.
4. Air-trench beamsplitter
As shown in Fig. 7, a waveguide beamsplitter can be formed with a single air trench that is designed to work by frustrated TIR. Our µGA-FDTD design tool was used to optimize the position of the trench as well as its thickness and length. The result is a trench that is 10.5 µm long and 0.39 µm thick shifted by 0.38 µm along the +x-axis. This design transmits 48.8% of the incident light and reflects 49.7% for a total efficiency of 98.5%. As seen in Fig. 7(a), the positional shift of the trench permits the design to account for the Goos-Hanchen shift of the reflected light.
The spectral dependence of the reflected and transmitted light is shown in Fig. 8. While the air-trench beamsplitter exhibits somewhat greater wavelength sensitivity than air-trench 90° bends, the efficiency for reflected and transmitted light stays within 40%–60% over the entire wavelength range of 1.3 µm to 1.8 µm shown in the figure.
In summary, we have shown that high efficiency waveguide bends and beamplitters can be realized in waveguide systems having both low refractive index and low index contrast through the use of air interfaces and air trenches. In the case of a single air interface, the highest efficiencies are achieved for bend angles less than 90°. For multiple air trench 90° bends, µGA-designed air trenches can lead to high efficiency waveguide bends that are broadband in their performance. Single air trenches that operate through frustrated TIR can be used to form compact 90° beamsplitters. Important next steps include determining the polarization dependence of single air interface and multiple air trench bends, examining the sensitivity of device performance to air trench dimensional variations, evaluating out of plane scattering loss, and experimental demonstration.
This work was supported in part by DARPA Grant N66001-01-1-8938 and National Science Foundation Grant EPS-0091853.
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