## Abstract

In this paper, we propose a device to bend light in non-channel planar photonic crystal (PhC) waveguides using the self-collimation phenomenon. The mode distribution in a non-channel planar PhC waveguide is investigated in detail in order to help understand the proposed bending mechanism. Three-dimensional finite-difference time-domain simulations show an over 80% bending efficiency for a 90° bend. As the first proposal for bending light in a non-channel planar PhC waveguide, the presented device enables the application of routing in non-channel planar PhC waveguides.

©2003 Optical Society of America

## 1. Introduction

Since first proposed by Yablonovitch [1] and John [2], Photonic Crystals (PhCs) have attracted tremendous interest due to their unique capability to modify photon interaction with host materials, e.g. propagation and localization. With proper parameter design, a PhC can exhibit a photonic band gap (PBG) [3], which is a frequency region in which no wave propagation is allowed. In this case, if one perturbs the periodic structure at one point, one can create a point defect or a micro-cavity [4]. Similarly, if one perturbs the periodic structure along one column or one row, one can create a line defect or a PhC waveguide [5].

An alterative property of PhCs is their ability to significantly modify the dispersion property of the host material. For example, while the equal-frequency contours (EFCs) of an un-patterned silicon slab are circular, the EFCs of a silicon slab with different periodic patterns can exhibit different shapes [6]. An EFC is the cross-section of a dispersion surface, which is a surface that characterizes the relationship between all allowed wave vectors in the structure and their corresponding frequencies. Recently, there has been a growing interest in engineering the dispersion property of PhCs for possible applications. One of the very interesting phenomena found in planar PhCs during these explorations is the self-collimation phenomenon [6, 7]. This behavior, in which incident waves with a certain angular range are naturally collimated along certain directions, promises a variety of applications, such as self guiding [8, 9], and spatial beam routing. In comparison with conventional dielectric waveguides or line defect PhC waveguides working within the photonic band gap [5], a dispersion-guided waveguide does not require a physical boundary for confinement. Moreover, it also releases the strict alignment requirements imposed by the coupling efficiency in the case of narrow waveguides. As such, it enables high in-plane coupling. However, further implementation of this type of non-channel planar PhC waveguides, especially in high-density photonic integrated circuits, is greatly hindered due to their inability to efficiently bend and redirect light.

To this end, in this paper, we propose a device to bend light in a non-channel planar PhC waveguides at the wavelength of 1.5 *µm*. We start with the investigation of the mode distribution in a non-channel planar PhC waveguide. Once the spatial mode distribution is identified, we propose a device capable of bending light by 90°. We further explore various implementations to optimize the bending efficiency of the device by using the three-dimensional (3D) finite-difference time-domain (FDTD) method [10].

## 2. Mode mechanism in non-channel planar photonic crystal waveguide

To understand the mode mechanism in a non-channel planar PhC waveguide, we first examine a straight waveguide. As an example, we consider a silicon slab patterned with a structure consisting of a square array of 2-dimensional (2D) air holes in the silicon slab suspended in air, as illustrated in Fig. 1(a). The lattice constant is *450 nm*, the diameter of air holes is *270 nm*, and the thickness of the silicon slab is *260 nm*. The refractive index of silicon is *3.5*. The 3D iterative plane wave method (PWM) [11] is used to obtain the EFCs of this structure. Interestingly, the EFC at the wavelength of *1.5 µm* has a square-like shape, as indicated by the dotted line in Fig. 1(b). The solid circle represents the cross section of the light cone at the same wavelength. From the figure, we can see that the EFC of the structure is outside the cross section of the light cone, which means all modes supported by the PhC at this wavelength are below the light cone so that they can be guided in the slab without vertical loss, which is very important for practical spatial beam routing.

In order to see the response of this structure to an incident beam with a wide angular spectrum, we first examine the phase propagation and the energy propagation when a plane wave with a wave vector *k _{i}* is incident on this structure from the free space. According to Snell’s law, the components of the incident wave vector,

*k*, the reflected wave vector,

_{i}*k*, and the refracted wave vector,

_{r}*k*along the interface between free space and the structure should be equal. In this way, one is able to determine the length and orientation of

_{t}*k*and

_{r}*k*. On the other hand, the group velocity is equal to the derivative of the dispersion function

_{t}*ω*(

*k*)[6]. In other words, energy propagates in the direction of the steepest ascent of the dispersion surface, which is perpendicular to the EFC. As mentioned above, the EFC of this structure at the wavelength

*1.5 µm*is square-like, in contrast to a circular shape of the EFC of free space at the same wavelength. This is to say that in free space, the energy and the phase propagate in the same direction while in dispersion-based PhC, they propagate in different directions, as indicated in Fig. 1(b). In this figure,

*k*,

_{i}*k*, and

_{r}*k*are phase propagation directions and

_{t}*ν*,

_{gi}*ν*, and

_{gr}*ν*are energy propagation directions for the incident wave, the reflected wave, and the refracted wave, respectively. Because some sections of the EFC are approximately flat, the energy in these sections will propagate within a very narrow angular range. Figure 2 is the plot of the incident angle versus the energy propagation angle in the PhC. All angles here are relative to the Γ-X direction, which is along the X direction of the lattice, as indicated in Fig. 3. Because of the symmetry, only the plot for the incident angle ranging from 0° to 45° is given. From the figure, one can see that for the incident angle up to 28°, the energy of the refracted wave propagates within the range of ±2°. Thus, if a Guassian beam with a wide angular spectrum is launched into the PhC along one lattice direction, say Γ-X1, after a certain propagation distance, the guided beam contains only a certain range of wave vectors with narrow energy propagation angles to Γ-X1. Other components with wider energy propagation angles would dissipate most of their energy within the lattice. Since self-guiding can take place only along directions which are perpendicular to the four approximately flat sides of the square EFC, only 90° bending is allowed in this structure. To turn a self guided wave along Γ-X1, which includes a range of continuous wave vectors, or phase propagation angles, into its perpendicular direction Γ-X2, one needs to convert all wave vectors into another group of wave vectors with narrow energy propagation angles to Γ-X2.

_{gt}## 3. Design of the bending device

To this point, we have discussed the mode mechanism of the self-guiding beam and we now discuss a method of beam bending in a non-channel planar PhC waveguide.

Unlike conventional waveguides with a physical boundary, in which beam bending is achieved by simply defining a curved physical boundary, a non-channel planar PhC waveguide has no physical boundary. As such, a distinct bending mechanism has to be introduced. Recall that in free space optics, a collimated optical beam is usually redirected to other directions with an optical mirror, to achieve the free space optical interconnection [12]. Therefore, we propose a similar approach in a non-channel planar PhC waveguide to realize beam bending, which is illustrated in Fig. 3. In doing so, we introduce a small rectangular air area that is etched away along Γ-*M* direction, which behaves as 45° mirror to achieve 90° light bending. However, unlike collimated beam propagation in free space or other homogeneous materials, geometrical optics cannot be used to determine the interaction of the optical beam and the mirror. Therefore, in this effort, we used the 3D FDTD method to evaluate the performance of this bending mechanism. In this implementation of the 3D FDTD method, Perfectly Matched Layer (PML) absorbing boundary conditions are applied in all three directions. In order to avoid the evanescent slab modes, for which PML is not sufficiently valid [13, 14], PMLs on the top and bottom of the slab are placed *520 nm* away from the slab. A Guassian beam with a width of *1um* and a height of *260 nm* is launched along one lattice direction at the interface between the PhC and the un-patterned silicon slab, as shown in Fig. 3. Figure 4 shows a horizontal cross section of the steady state amplitude of the *Hz* component on the central plane of PhC slab when *d*=*0.107a*, where *d* is defined as the distance between the nearest edge of air holes and the edge of the etched mirror surface, as shown in Fig. 3, where *a* is the lattice constant. In order to characterize the bending efficiency, a straight waveguide with the same propagation length is also analyzed using the 3D FDTD method. Then at the output end for both the bent structure and the straight structure, a cube with length (in propagation direction) of *5a*, width of *3a*, and height of *260 nm* is taken for calculating the average energy flow along the propagation direction. The bending efficiency is then defined as the ratio of the average output energy flow for the bent structure and that for the straight structure. In this way, a bending efficiency of *82.9%* is obtained in this case. However, the bending efficiency is very sensitive to the mirror location. To show this, we plot the bending efficiency versus the mirror location, *d*, in Fig. 5, where *d*<0 means the opposite direction of as indicated in Fig. 3. Since the radius of air holes is *0.6a*, thus, when -0.6*a*<*d*<0, the etch part overlaps with air holes and the mirror edge is corrugated. To this end, we have observed four mechanisms that influence the bending efficiency. The first one is the symmetry of the PhC. In other words, the mirror edge is right at the place to which the original unperturbed structure is symmetric. This mechanism favors the bending efficiency and the most favored place is *d*=(√2/2-0.6)*a*/2=0.0535*a*, which is right at the middle of two rows of air holes along Γ-M direction, as indicated by AB in Fig. 5. The second is the symmetry of the mirror in which the central lines of the input beam and the output beam cross onto the mirror, as indicated by the CD in Fig. 5. From Fig. 4, one can see that for the self- collimated mode, most of its energy concentrates in the high dielectric material between the rows of air holes along Γ-X. Therefore, it is favorable that the mirror is placed at the center of the air holes because most of the energy of the input beam is capable of being redirected without undergoing scattering and relocating, as indicated by the arrows in the inset of Fig. 5. The third mechanism is the scattering of the corrugated mirror. The more corrugated the mirror, the more scattering and, hence, the lower the bending efficiency. This mechanism works only when -0.6*a*<*d*<0 because only in this case, the mirror is corrugated. The fourth mechanism is the Fabry-Perot effect. When *d*>0, thereby forming a Fabry-Perot (FP) cavity. Because some photons can leak into the lattice along the FP cavity, we observe that the FP effect is more pronounced as d increases.

## 4. Experimental results

The bending device, shown in Fig. 3, is validated experimentally by fabricating the respective structure in the 260 nm thick device layer of a silicon-on-insulator substrate. The structure is defined by electron-beam lithography and subsequently etched in an inductively coupled plasma reactive ion etching system. The self-collimating PhC lattice consists of air holes with a lattice constant *a*=442 nm and radius *r*=0.29*a*. The mirror is incorporated in the self-collimating PhC by lithographically defining a rectangular region rotated 45° with respect to the rectangular PhC lattice. The mirror is 3*a* wide in order to ensure the light is laterally confined to the PhC region. Figure 6(a) shows a scanning electron micrograph of the fabricated routing structure with mirror position at 0.25*a*. As shown in Fig. 6(b) we employ a J-coupler focusing four layers into the PhC lattice to observe the lateral confinement of light within the dispersion guiding structure. Six functional devices with various mirror positions are fabricated on one sample in order to comparatively analyze the bending efficiencies. The mirrors are placed between *d*=-0.39*a* and *d*=0.25*a* away from the PhC lattice as depicted in Fig. 3c. Figure 6(b) shows a top-down image, from a near IR camera, of the light routed by the mirror loacted 0.25*a* from the lattice. The scattered light is easily observed because the underlying oxide layer remained intact, rendering the wave guided above the light cone. After removal of the oxide layer, the bending efficiency is calculated as shown in Fig. 5. The structures are analyzed at a wavelength of 1432 nm.

## 5. Conclusion

In summary, we present a mechanism to bend self-collimated beams in a non-channel PhC waveguide. As an alternative to dielectric waveguides or line defect PhC waveguides, this type of waveguide offers significant advantages in that it permits highly efficient in-plane coupling because it has no physical boundary. However, shortage of available bending mechanisms greatly limits its use in practical applications. It was for this reason that we introduce the proposed device and present simulation results along with experimental results.

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