1. Introduction

Waveguides based on photonic crystals provide unique properties that are not available in conventional TIR (total internal reflection) guided waveguides. One of the most important property is that their dispersive nature makes very large group indices achievable and tunable, which is attractive for the miniaturization and integration of many types of photonic devices [1]. In this sense, the concept of coupled resonator optical waveguide(CROW) [2] can be used for especially large and controllable dispersion [3, 4]. Combined with the linear defect waveguide, the CROW may function as delay lines, dispersion compensators, and nonlinear components. In this paper, we consider a CROW as a dispersive component in a photonic integrated circuit based on a linear defect waveguide. The linear defect waveguide also exhibits very low group velocity around the band edge which is applicable for dispersive components. It requires, however, the hole radii or the lattice constant of the component to be different from that of the other parts. This causes additional complexity in both the fabrication and the design of such devices. In addition, the predicted loss of suspended membrane linear defect photonic crystal waveguides increases at the band edge. Finally, unlike linear defect waveguides, both positive and negative dispersion can be obtained using the CROW structure.

 

Fig. 1. (a) Computational domain of linear defect waveguide in a triangular lattice photonic crystal. The radius of air holes is 0.3(r/a) and the refractive index of background is 3.4. (b) Eigenvalue plot of the structure (a).

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In this paper, we first calculate the group velocity and its dispersion for the linear defect waveguide. Then we introduce air hole defects in the waveguide channel to form a CROW structure. We take the size of these holes as a design parameter to speed up or slow down the wave packet in the waveguide. The dependence of the group velocity and the dispersion on the hole size is investigated and compared to the linear defect waveguide. Lastly, pulse propagation simulations are performed for the explicit illustration of the dispersion in these waveguides.

2. Group velocity and wavelength dispersion

We used the vector finite element method to calculate the dispersion relations of the guided modes of the photonic crystal waveguides. The FEM is proven to be versatile in modeling microwave components with inhomogeneous and complex structures. Considering that it is rather cumbersome to describe complex and fine structures using point sampling scheme, the FEM is especially advantageous in this sense because it is based on the element-wise formulation. Combined with the mesh generator and linear algebra software, the eigenvalue formulation of the FEM provides an accurate and efficient way of characterizing photonic crystal structures. The details can be found in our previous work in Ref. [6].

We consider a two-dimensional triangular lattice of air holes in a dielectric background. This is a waveguide that is uniform in the vertical direction. In practice, this structure is achieved by a sufficiently large height compared to the lattice constant or weakly guiding waveguide in the vertical direction in which case the effective index is close to that of bulk material.We start from the single line defect waveguide with the dispersion relation shown in Fig. 1. Figure 1(a) shows the computational domain for the finite element calculations and the eigenvalues are plotted in Fig. 1(b). The radius of of air holes is 0.3(r/a) and the refractive index of the background is 3.4 which corresponds to that of Si or GaAs around the wavelength of 1550 nm. The working variable is the electric field and odd and even modes are calculated separately with boundary conditions of the perfect electric conductor(PEC) and the perfect magnetic conductor(PMC), respectively [6]. From the band structure of odd and even guided modes drawn as solid lines, group velocities are simply the slope of the curve and can be calculated by taking derivatives of the dispersion relation of each mode. Then the group velocity dispersion can be calculated by taking derivatives of the group velocities with respect to the wavelength.

 

Fig. 2. (a) Group velocities and (b) their dispersion plots for the odd and even guided modes plotted as solid lines in Fig. 1(b).

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Figure 2 shows the group velocities and the group velocity dispersion of the waveguide modes with respect to the normalized wavelength. Around both the band edges of the odd mode, there are wavelengths at which the group velocities are zero due to anti-crossing which is a one-dimensional characteristic. The upper one is, however, outside the bandgap and may not be useful for practical consideration. The odd mode has zero dispersion at 3.84a and the even mode has two zero dispersion wavelengths at 4.067a and 4.097a. The bandwidth over which the dispersion is below 1 (ps/a/mm) is 0.15a(3.78a–3.93a) and increases to 0.622a(3.64a–4.262a) for the less strict constraint of the dispersion 10 (ps/a/mm) for the odd mode. The even mode bandwidth for which the group velocity dispersion is less than 10 (ps/a/mm) is smaller than that of odd mode by a factor of more than 40. Compared to the odd mode, the even mode is more dispersive and there is a zero group velocity point at β≈0.3(a/2π). This difference can be attributed to their different guiding mechanisms, index and gap guiding, for the odd and even modes, respectively [7]. We believe that the even mode is more difficult to excite than the odd mode because it is not the fundamental mode and the multi-valued group velocity and the group velocity dispersion makes it complicated for the practical applications. For this reason, we will focus on the odd modes of the CROW even though the even mode has more favorable dispersion features. Note that the material dispersion is not included in our model and only the waveguide dispersion is considered.

 

Fig. 3. (a) Computational domain of the CROWin a triangular lattice photonic crystal. The radius of air holes is 0.3(r/a) and the index of background is 3.4. (b) Eigenvalue plot of the structure (a) for different radii of a hole in the linear defect channel.

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The computational domain for the CROW and the corresponding photonic band structures are shown in Fig. 3(a) and (b), respectively. Air holes, marked as yellow arcs in Fig. 3(a), are inserted at every other a along the waveguide channel to form coupled resonators so the period of the CROW is 2a. The radii of air holes, r_d’s, are varied from 0 to 0.4 in steps of 0.1(r/a) and the corresponding band structures are shown in Fig. 3(b). When r_d=0, it corresponds to the linear defect waveguide as in Fig. 1(a) and the band structure is the same except that it folds back with respect to the center of the Brillouin zone. For non-zero radii of defect holes, the mode gap forms at the Brillouin zone edge and becomes larger as r_d increases. For r_d=0.3 and 0.4, the folded bands move out of the bandgap and are not shown in the picture. All folded bands are air bands and will not be considered hereafter. In the same manner as was done in the analysis of a linear defect waveguide, numerical differentiation is performed for the bands drawn as solid lines in Fig. 3(b), and the group velocity and the group velocity dispersion are shown in Fig. 4. From Fig. 4, the maximum group velocity decreases up to r_d=0.3 as r_d increases. This implies that the coupling coefficient does not change significantly for r_d>0.3. Further increase of r_d may lead mainly to the decrease in the wavelength of the mode. From the wavelength dispersion plot in Fig. 4(b), the wavelength of zero dispersion, which corresponds to the maximum point in Fig. 4(a), moves to the shorter wavelength. For a given operating wavelength, proper choice of r_d gives the desired group velocity difference from the linear defect waveguide. When considering the coupling between the linear defect waveguide and the CROW, the reflection at the transition junction is desired to be minimized by an adiabatic transition between the modes or by specially designing the interface region. A slow change of the hole size makes the momentum and group velocity of the guided mode gradually convert to one from the other. This is possible only over the wavelength region in which the bandwidth of the mode with different r_d overlaps. For example, from the Fig. 4(a), for the normalized wavelength of 4 which is outside the bandwidth of the CROW with r_d=0.1 and 0.2, the adiabatic coupling cannot be achieved. On the other hand, for the normalized wavelength of 4.2, the mode of the linear defect waveguide can be gradually converted to that of the CROW and the group velocity is reduced from 0.25 to 0.05. The wavelength dispersion of the group velocity is also an important factor for the propagation of short pulses. As shown in Fig. 4, the wavelength dispersion of the CROW is much larger than that of the linear defect waveguide. Pulses passing through the CROW section suffer distortion depending on the spectral width of the pulse. For the dispersion compensating element, the CROW’s with r_d=0.1 and 0.2 provide both positive(normal) and negative(anomalous) values in the bandwidth of the unfolded mode of the linear defect waveguide. Let’s consider a specific example of a system designed at 1550 nm or lattice constant a=405 nm. Then the wavelength range where D<1 (ps/a/mm)=0.00247 (ps/nm/mm) is from 1531 nm to 1592 nm and becomes wider from 1474 to 1726 for D<10 (ps/a/mm)=0.0247 (ps/nm/mm). For the linear defect waveguide carrying 100 GHz signal in this wavelength range, the signal distortion may be tolerable if the length of the waveguide satisfies $l\ll \frac{T}{D\Delta \lambda }$ , that is, l≪4000 mm and l≪400 mm for D=1 and 10 (ps/nm/mm), respectively. If we incorporate the delay line in the waveguide, a restriction on the total waveguide length will be raised. For the CROW with r_d=0.3a at the wavelength of 4.2a, D≈2730 (ps/a/mm) so the length is limited such that l≪1.45 mm.

 

Fig. 4. (a) Group velocities and (b) their dispersion plots for different radii of holes.

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To verify the argument in a more explicit way, we performed a two-dimensional finite-difference time domain (FDTD) simulation for the structure shown in Fig. 5(a), which consists of three parts. The left part is a 14a-long homogeneous dielectric medium with a refractive index of 3.4. The middle part is a single-line defect photonic crystal waveguide created in the Γ-K direction of a triangular lattice of air holes of which radius is 0.3a. The length of the linear defect waveguide in the propagation direction is 25a. This section is a bridge to the CROW to the right. The CROW is a slow wave structure where air holes with radii of 0.2a are created at every other lattice positions. The whole geometry is surrounded with 14-layer perfectly matched layer (PML) on all four sides. A source is positioned 2a from the PML and 10a from the photonic crystal waveguide. A 16,000Δt(=a/40/c) Blackman source is excited with a center frequency of 0.226(a/λ). The choice of Blackman envelope over conventional Gaussian envelope is to achieve reasonably small signal bandwidth in a shorter time period, which reduces the total simulation load.

Figure 5(b) shows the amplitude of the magnetic field H_z at different probe locations. Each time sequence labeled as A, B and C corresponds to the value of H_z at the probe A, B and C, respectively. At probe A, which is positioned at the same location as the center of the source, the reflection due to the first interface is almost overshadowed by the input pulse except for its tail which can be identified at around 15,000Δt of the first probe. Its small magnitude does not imply a weak reflection; however it is the result of incoherent propagation in the homogeneous dielectric medium. We don’t intend to suggest that a naive setup can be used for input coupling, rather we found that exciting the photonic crystal defect mode in the homogeneous region can reduce the energy dissipation due to parasitic radiation fields. The other ripple at 25,000Δt is the result of the reflection at the linear defect waveguide and the CROW interface. It can be also identified by probe B, which is 10a away from the first interface and 30a away from the second. The reflectance at the second interface is estimated at 50% for this structure and is attributed to the impedance mismatch of two types of waveguides. A third probe C is located at 46a away from the start of the CROW. Higher group velocity dispersion increases the pulse width in time, as is evident from the magnitude plot of H_z. Also as a comparison, we plot the propagation of an identical pulse in the structure where the CROW is replaced with a photonic crystal single-line defect waveguide, as labeled C* in Fig. 5(b). It is not hard to see that the pulse propagates much slower in the CROW. The time it takes for the pulse to reach its maximal envelop at probe C is 31,000 and 25,000Δt for the CROW and linear defect waveguide, respectively. A 46a-long CROW has resulted in a pulse propagation delay of 6000Δt.

 

Fig. 5. (a) Computational domain for the FDTD simulation. (b) Time sequences of H_z at different probe locations A, B and C. The reference plot labeled C* is that of the linear defect waveguide.

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From a practical point of view, the reflection at the interfaces between the linear defect waveguide and CROW is not desirable and may be reduced by optimizing the interface or adiabatically changing the hole sizes of the CROW. We have not pursued this here since we are only interested in the dispersive properties of such a structure in this study. Also, the significant amount of dispersion observed in the simulation is due to the narrow pulse width with a bandwidth of approximately 15 THz. This bandwidth was chosen to facilitate the FDTD simulations. Larger pulse width can be expected to give less dispersion but increase the computational time.

3. Conclusion

With a linear defect photonic crystal waveguide as a base guiding structure, a coupled resonator optical waveguide provides a way of controlling the group velocity and the group velocity dispersion by varying the radii of air holes in the linear defect. We performed the eigenvalue analysis using the FEM to derive the group velocity and dispersion as a function of the wavelength. We also conducted the simulation of pulse propagation using the FDTD and the results show the reasonable agreement with those of the FEM.