## Abstract

Various bending waveguides with different bending angles in photonic crystal are proposed and studied. Analyses show that the bending waveguides can provide a high transmission for the whole frequencies within the photonic band gap. Numerical simulations confirm that high transmissions over 98.5% can be achieved for the proposed waveguides.

©2006 Optical Society of America

## 1. Introduction

Waveguide acts as an important element in photonic integrated circuit (PIC). Usually, dielectric waveguides have relatively large dimensions. The reason is that when dielectric waveguides are used as interconnections between optical components in PICs, they require a big radius of curvature to prevent suffering large bending loss. Photonic crystal (PC)-based waveguides, different from the dielectric waveguides, provide strong confinement and flexible control to lightwaves. One of the most remarkable applications of the PC waveguides is to transmit lightwaves through a sharp bend with low bending loss. [1–3]. Therefore, the PC waveguides have inspired great interest in designing [4,5] and fabricating [6] waveguides with bends. However, un-negligible bending loss still arises due to the reflections at the sharp bending corner, especially when bending angle is larger than 90°. Consequently, studies have been focused on the optimization at the bending corner to lessen the reflection loss. Prominent solutions include directly increasing the lengths of the bends [7], incorporating a taper structure into the bending corner [8], removing unit cells from the bending corner [9], and varying the radius of dielectric rods at the bending corner [10]. Still there remains a problem that most of the previous work dealt with cases of small bending angles, generally less than or equal to 90°. Although Tokushima *et al*. reported the 1.55 µm lightwave propagating through a 120° sharp bent waveguide [11], the bending loss at the un-optimized corner cannot be neglected. Moreover, in most of the previous work, high transmission through the bending waveguide can only be achieved for lightwave at a certain frequency, rather than for all the frequencies within the photonic band gap (PBG) (as an exception, the solution in reference [10] offers a transmission over 98% for the entire PBG). Until recently, a PC waveguide with bending angle of 135° was studied [12]. The PC model was of square lattice of dielectric rods in the air, and high transmission was achieved through the PC waveguide formed by coupled defect cavities. Luan *et al*. investigated the transmission characteristics of a finite periodic dielectric waveguide that can provide bending transmission for lightwaves [13]. But due to the confinement mechanism (total internal reflection), this waveguide cannot ensure very high transmission and high compactness at the same time. In this letter we present a study on line-defect bending waveguides in PC, which ensures bending transmission of lightwaves in various bending angles with high transmission for all the frequencies within the PBG.

## 2. Models of arc bends

The considered PC model is consisted of dielectric rods of square lattice with a dielectric constant of *ε*_{r}
=12.1 in air. The radius of the rods is *r*=0.18*a*, where *a* is the lattice constant. In this PC structure, a band gap opens for TM mode in the frequency ranging from *ω*
_{0}=*a/λ*=0.31 to 0.44, where λ is the wavelength in air. Waveguides are formed by introducing a single line-defect into the PC. For simplicity of numerical simulations, three layers of crystals are chosen as confinement of lightwaves in the waveguides, which are adequate to prevent the losses caused by the leakage of light. To form a line-defect bending waveguide, the design of the bending corner becomes crucial because the geometry of the rods at the bending corner will greatly affect the transmission performance. Thus a 90° line-defect bending waveguide in PC is formed with an arc bending corner, and the geometry is shown in Fig. 1(a).

For comparison, one of the bending waveguides studied in reference [7] is shown in Fig. 1(b). This bending waveguide has a beeline bend with a bending length close to that of Fig. 1(a). Compared with the Fig. 1(b), the dielectric rods at the bending corner of the Fig. 1(a) are rearranged on arcs with an incremental change of *a* in radius from the arc center denoted by *O*. The rearranged rods on the arcs are axially symmetrical regarding the bisector of the 90° centre angle. In addition, to preserve the PBG for crystal layers in the arc bend, the rods on the arcs are also with a center-to-center space of *a*, with an exception that rods located beside bisector have a space a little larger or smaller than *a* to ensure the symmetry of the bend.

## 3. Simulations and analyses of arc bends

To evaluate the performance of the proposed bending waveguide, a numerical simulation is run with the finite-difference time-domain (FDTD) method. The FDTD simulation is run in a domain where a perfectly matched layers surrounding is used as the boundary conditions to absorb the outgoing waves. Distribution of steady-state electric field for *ω*
_{0}=0.375(*a*/λ) (center frequency of the PBG) is obtained as shown in Fig. 2. The FDTD simulated results are captured at the bend of the bending waveguides. Also, by calculating the output power, the reflection loss spectrum is obtained as shown in Fig. 3, together with the result for the bending waveguide in Fig. 1(b) for comparison.

It can be seen from Fig. 3 that for the whole frequency region in the PBG, the reflection loss of the proposed bending waveguide is generally lower than that of the compared bending waveguide. This performance of low reflection loss can be explained both qualitatively and quantitively. The layers of the proposed 90° arc bend are bent bit by bit and so the proposed bending waveguide can be regarded as a gradually bending waveguide, rather than a sharply bending one. This forms a very smooth bending corner where the lightwaves can hardly suffer any sharp bending when traveling through the bend. Figures 4(a), 4(b) and 4(c) show the profiles of steady-state electric field at the frequency of 0.35(*a*/λ) obtained by the FDTD method for lightwaves in the straight waveguide, the arc bend of the proposed waveguide, and the beeline bend of the compared waveguide, respectively. It can be seen that the profiles of electric field in Figs. 4(a) and 4(b) are nearly the same, while the profile in Fig. 4(c) is distinct as reflections and resonances can also be observed. The accordance of profiles of the electric field shows that the propagating status of the guided lightwaves in the arc bend is much closer to that in the straight waveguide, compared with the case in the beeline bend. This leads to a very smooth changing environment for propagating lightwaves in the proposed bending waveguide, and thus provides lower reflection loss and higher transmission.

To explain the oscillatory behavior of the transmittance spectrum through the bends like Fig. 1(b), reflection coefficient *R* is applied in the model of reference [7] and can be expressed as,

where *k*
_{0} and *k*_{b}
are the wave vectors of the guided modes in the line-defect straight waveguides and the bends, respectively, and *L* is the length of the bend. The hindrance to directly apply Eq. (1) to explain the low reflection loss in the proposed bend is that no real “modes” exist in the arc bend due to its non-periodic nature. However, as the length of the arc bend is very small compared with the straight waveguide, the shape of the field structure nearly does not change in the bend. Thus we consider the arc bend as a contorted beeline bend in which the mode propagating with wave vector *k*
_{0} is scattered into a “quasi-mode” with wave vector *k’*_{b}
. Under this consideration, Eq. (1) is applied for taking an attempt to quantitively explain the low reflection loss in the proposed bend. For the waveguide in Fig. 1(b), *L*=3.33√2 *a*≈4.709*a* while for the proposed waveguide *L*’=(3π/2)*a*≈4.712*a*. Difference between the *L* and *L*’ is very small. So the difference of the reflection coefficients *R* for the waveguide in Fig. 1(b) and *R*’ for the bending waveguide in Fig. 1(a) is mainly determined by the difference of (${{k}_{\mathit{0}}}^{\mathit{2}}$${{\mathit{-}k}_{b}}^{\mathit{2}}$
) and (${{k}_{\mathit{0}}}^{\mathit{2}}$${{\mathit{-}k\mathit{\u2019}}_{b}}^{\mathit{2}}$
). The dispersion curves of the wave vectors versus the lightwave frequency *ω*
_{0} can be obtained by taking a calculation with plane wave expansion (PWE) method in the defects of the PC [14]. Dispersion curves of *k*
_{0} and *k*_{b}
can be easily calculated by taking the PWE calculation in a super-cell as the structures of the straight waveguide and the beeline bend are both periodic. Dispersion curve of *k’*_{b}
cannot be obtained by calculating a super-cell as the structure is not periodic in the arc bend. However, by directly taking a PWE calculation in the arc bend domain, we can still get an approximate curve of *k’*_{b}
as a summing result of the different parts in the non-periodic structure of the bend. For comparison, the dispersion curves of *k*_{0}*, k*_{b}
and *k’*_{b}
are shown in Fig. 5. It can be seen clearly that for the whole frequencies within the PBG, *k*
_{0} is much closer to *k’*_{b}
, so (${{k}_{\mathit{0}}}^{\mathit{2}}$${{\mathit{-}k\mathit{\u2019}}_{b}}^{\mathit{2}}$
) is much smaller than (${{k}_{\mathit{0}}}^{\mathit{2}}$${{\mathit{-}k}_{b}}^{\mathit{2}}$
). This leads to *R*<*R*’ and also explains why *R*’ is very small for the entire PBG rather than at certain frequencies.

## 4. Simulations and analyses of different arc bending waveguides

By analogy with the design of the 90° line-defect bending waveguide, bending waveguides with bending angles of 45°, 135° and 180° are formed. The geometries of the arc bending corners are shown in Figs. 6(a), 7(a), and 8(a), respectively. The distributions of the steady-state electric field at *ω*
_{0}=0.375(*a*/λ) are also obtained by the FDTD simulations, and the results are shown in Figs. 6(b), 7(b), and 8(b), respectively.

In Figs. 6(a), 7(a), and 8(a), arrangements of the dielectric rods at the arc bending corners are analogies of that in Fig. 1(a). To evaluate their transmission performances, power transmittance spectrums are obtained by calculating the output power in the waveguides and scanning the frequencies from 0.31(*a*/λ) to 0.44(*a*/λ). The results are shown in Fig. 9.

It can be seen from Fig. 9 that in the three bending waveguides, the power transmittances can reach above 98.5% for all frequencies in the PBG. This is attributed to the arc bending corners’ ability to transmit lightwaves with very low reflection loss. For further evaluation, we also form two “S-shape” line-defect bending waveguides by cascading two 180° bends and by cascading a 180° bend with a 135° bend, respectively. The FDTD simulated distributions of steady-state electric field at *ω*
_{0}=0.375(*a*/λ) and the power transmittance spectrums are shown in Figs. 10 and 11, respectively. High transmissions over 98% can still be achieved after traveling through the two consecutive arc bending corners for the whole frequencies within the PBG.

The performance of the line-defect bending waveguides will also be affected by the bending radius *R*_{b}
. Generally, the power transmissions will increase with the increase of bending radius. For all the aforementioned bending waveguides, the bending radius is *R*_{b}
=3*a*. To discuss the affection of *R*
_{b} on the transmission performance, the power transmittance spectrum for 90° line-defect bending waveguides are calculated with the bending radius of 2*a*, 3*a* and 4*a*, respectively and the results are shown in Fig. 12. It can be seen from Fig. 12 that the power transmission increases with the increase of bending radius. Transmittance for *R*
_{b}=2*a* is much lower than that for *R*
_{b}=3*a* and *R*
_{b}=4*a*. This is due to the sharp change of the propagation environment in the bending waveguide with *R*
_{b}=2*a*. Moreover, the bending waveguide with *R*
_{b}=4*a* provides a bit higher transmission than that with *R*
_{b}=3*a*. However, for applications, the bending waveguide with *R*
_{b}=3*a* is more suitable for optical interconnections because it provides higher compactness as well as acceptable low loss.

## 5. Conclusion

We proposed and studied PC-based line-defect bending waveguides with various bending angles. High transmission through the bending waveguides can be obtained for nearly all frequencies in the PBG. The performances are analyzed and discussed. Numerical simulation confirms that high transmissions over 98.5% can be achieved. It is expected that these PC-based line-defect bending waveguides can find some applications in PICs as optical interconnections. It should be pointed out that the proposed method disturbs the PC lattice at the bending corners. But according to the calculation of dispersion curves in the bend, this disturbance has little impact on the PBG of the crystal layers in the arc bend and so the disturbance on the PC lattice can be neglected. It should also be pointed out that the discussed structures and performed simulations in this work are two-dimensional. However, the main results should still be valid for a three-dimensional structure since the loss through the third dimension is not due to the bending.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China (Nos. 90401008, 60577001), the Research Fund for the Doctoral Program of Higher Education (No. 20040558009), the Key Project of Chinese Ministry of Education (No. 104144), and the Program for New Century Excellent Talents in University (No. NECT-04-0796).

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