1. Introduction

Photonic crystals (PC) are periodic dielectric structures made for controlling light [1, 2]. The propagating eigenmodes of the electromagnetic (EM) wave in PC are the Bloch waves [3], and their dispersion relations form the photonic band structure. The passbands and bandgaps are the frequency ranges allowing and forbidding wave propagation, respectively. Utilizing the bandgap effect, photonic crystal waveguides (PCW) can be designed to guide light [4]. High transmission is crucial for applications in optical integrated circuit, which is hard to achieve if using conventional waveguides (CWG) [5, 6]. Although high transmission is the main advantage of PCW, it still has some disadvantages. For example, the guiding path of a PCW is restricted by the geometry of the PC, thus it cannot be bent arbitrarily.

Recently, a new type of optical waveguide named coupled resonator optical waveguide (CROW) has been proposed [7, 8]. A CROW consists of an array of coupled resonators with high quality factor (Q-factor), and the guiding of light is due to photon hopping along the successive resonators. The optical behaviors of CROW have been analyzed extensively using the tight-binding approximation (TBA) method [7, 9]. Usually a resonator is just a cavity embedded in a PC background, thus it is still subject to the geometric restriction of PC.

In this paper, we study the transmission properties of an alternative kind of waveguides that use index guiding (total internal refraction) mechanism [10, 11]. The waveguide is formed by aligning a one-dimensional array of dielectric rods in air, and we name it the periodic dielectric waveguide (PDWG). The band structure of the guided modes is calculated by using a finite-difference time-domain (FDTD) method proposed by M. Qiu et.al. [12]. When operates at a low frequency the guided mode has a small Bloch wavevector, and the behavior of PDWG is just like that of the conventional dielectric waveguide. However, if we increase the frequency such that the eigenmode acquires a Bloch wavevector close enough to the boundary of the Brillouin zone, the behavior of PDWGs become very similar to that of CROW.

The transmissions for the waveguides with different bend angles, bend radii, and at different frequencies are calculated using the multiple scattering method [14]. Since the guiding path of a PDWG can be bent arbitrarily, the geometric restriction of a PCW can be avoided. Furthermore, when the frequency is chosen appropriately, high transmission is observed even if the bends in the PDWG have radius of curvature as small as three wavelengths. These characteristics indicate that PDWG is a practical device at least operating in the microwave frequency regime, and it might also become useful as an optical component in photonic circuits in the future.

2. System description and numerical results

For simplicity in this paper we consider only the TM (E-polarization) modes , in which E = E ẑ , and ẑ is the direction of the cylinder axis. We begin with the consideration of an x-propagating mode.

According to Bloch’s Theorem [3], the eigenmode takes the form

 

Fig. 1. (a) Band structure (the first and second bands) of the TM (E-polarization) modes of the PDWG. The part of dispersion curve inside the shaded region corresponds to the extended modes, which cannot be used to guide light. Inset also shows the a×9a sized supercell, which is used in calculating the band structure. The radius of the dielectric cylinder is r = 0.2a, and its dielectric constant is ε = 11.56. (b) The group velocity derived from the dispersion curve of the first band.

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where k = k x̂ is the Bloch wavevector (-π/a < k ≤ π/a), and r = (x,y) is the position vector. The vector function U(r) = U(x,y)ẑ satisfies the periodic boundary condition along x

and it is localized about y = 0

Here a is the lattice spacing between two successive cylinders.

To calculate the dispersion relation of the x-propagating eigenmodes we adopt a finite-difference time-domain (FDTD) scheme proposed in Ref. [12]. In this approach we choose a supercell of size a × 9a (See Fig. 1(a)). The Bloch boundary condition Eq.(2) is imposed in the x direction, whereas the perfectly matched layers (PML) are in the |y| ≥ 4.5a regions. The dielectric constant of the cylinders is ε = 11.56 (GaAs), and the radius of the cylinders is r = 0.2a. The calculated results for the first and second bands are shown in Fig. 1(a). The fact that the first band curve is below the light line indicates that the modes in this band is extended in the x direction and localized in the y direction, thus they are indeed guided modes. Since the first band is wider than the second, it provides a wider frequency range for guiding light. In addition, since for a first band mode the local field pattern around each cylinder is nearly the monopole type, and the second band does not overlap in frequency with the first band, thus the waveguide is single mode for the first band. Hereafter we study only the first band modes.

From the dispersion relation, the group velocity v_g = dω/dk can be derived, which is the pulse-propagation velocity for a narrow-bandwidth pulse. As is shown in Fig. 1(b), the group velocity goes to zero as k approaches π/a. The slow group velocity behavior near the band edge can be understood as follows. A mode with Bloch wavevector k≈π/a is almost a standing wave [1], thus the field energy density (energy per unit length along x)

is strongly localized around each cylinder, with a very small energy flow (energy flowing through YZ plane per unit time)

Since in a periodic system the group velocity v_g of a Bloch mode is equal to the energy velocity [2], defined as

thus we expect that the group velocity becomes very slow near the band edge. The negative curvature of the band curve (d ² ω/dk ² < 0) there implies that photon becomes massive, with a negative effective mass m_eff = α(d ² ω/dk ²)^-1. Here α is a nonzero constant. In that region light is guided via the photon hopping mechanism [13]. This “massive photon effect” explains both the slow light behavior and the dispersive character of the guided pulse [8]. Given a frequency ω, we calculate the electromagnetic field using multiple scattering method [14], which is a frequency domain method for dealing with the two dimensional system that contains a finite number of cylindrical scatterers. The idea of this method is that the “incident field” and “scattered field” outside a cylinder as well as the “interior field” inside it can all be written as sums of Bessel and Hankel function series of integer orders. By using an addition theorem of Hankel functions and considering the boundary condition for every cylinder, coupled equations for the coefficients in front of the cylindrical functions can be written down. Solving these linear equations then give us these coefficients and the total field can be determined. To learn the details, please refer to Ref. [14]. In this paper, a two dimensional point source is assumed, which is a dipole oscillating along the third direction (ẑ) with frequency ω. The source is located one lattice spacing apart from the first cylinder of the waveguide.

As a test, we take the frequency to be ω = 0.25(2πc/a) and calculate the E field for a long enough (120 cylinders) straight PDWG. A snapshot of the calculated E-field (the real part of the complex amplitude) is shown in Fig. 2(a). As one can see, the field energy is indeed localized in the y direction. According to the dispersion relation, the Bloch wavevector corresponding to this frequency is k = 0.4(2π/a)x̂, having a mode wavelength λ = 2π/k = 2.5a. However, the x dependence of the field pattern we obtained does not seem like a sine curve of wavelength 2.5a. The reason for this is that the amplitude function U(x) is not a constant but a periodic function of period a, and 2.5a is not large enough when compared to a, so the field pattern becomes more complicated. To check if it is indeed the k = 0.4(2π/a)x̂ mode been excited, we make the following transformation. We define a new “effective wavenumber” k′ as

then we have

 

Fig. 2. (a) The snapshot of E-field pattern (the real part of the complex amplitude) for the mode with frequency ω = 0.25(2πc/a) and Bloch wavevector k = 0.4(2π/a)x̂. (b) The field pattern in (a), when multiplied by the factor (-1)ⁿ and evaluated at the center of the nth cylinder, fits to a sinusoidal wave. The wavelength is about 10a.

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That is, the value of Re [(-1)ⁿ E(na,0)] (the field evaluated at the nth cylinder center, times (-1)ⁿ) follows a sine curve. For the present case, we get |k′| = 0.1(2π/a), and the corresponding new effective wavelength is λ′ = 10a. Figure 2(b) shows the field pattern in (a), multiplied by the factor (-1)ⁿ and evaluated at the center of the nth cylinder. Obviously, the modified field pattern now fits to a sinusoidal wave of wavelength 10a.

We now study the transmission of PDWGs. In order to define the transmission, for an N-cylinder PDWG we insert two planes of width 6a (between y = ±3a) at x = 20.5a and x = (N - 0.5)a to evaluate the input power P_i and output power P_o. The transmission is then defined as T = P_o/P_i (See Fig. 3(a1)). Three cases of transmission calculation for straight PDWGs are shown in Fig. 3(a2), they correspond to L = 40a, 81a, and 100a. Here L = (N - 1)a stands for the length of the PDWG. The low frequency part of the these cases are slightly different, which is caused by the finite size effect. As we increase the frequency, the transmission curves merge, which implies they approach the result of infinite long PDWG. In this plot we also observe that at high frequency the transmission is slightly lower than that at low frequency, with some fluctuations in between. We believe this result is caused by the numerical restrictions of the method we used, such as the cross section planes are finite, and the sum of infinite Bessel series are replaced by a sum of only finite number of terms.

We now study the influence of bend on the transmission. The bend region is defined as the arc between the two dashed radii, as shown in Fig. 3(b1). In this region the distance between two successive cylinders is still a, and the radius of curvature of the bend is R. The bend angle is given by θ = (N_b-1)δ, where N_b is the number of cylinders in the bend, and δ = 2sin^-1 (a/2R) is the angle corresponding to one lattice spacing. Beyond this region are the input and output parts of the waveguide. In the following simulations about one single bend the we assume the input and output region contains 21 and 2 cylinders, respectively.

 

Fig. 3. The transmission properties of PDWGs. (a1) A typical straight PDWG. Two planes of width 6a inserted at x = 19.5a and x = (N - 1.5)a for an N-cylinder PDWG are used for evaluating the input power P_i and output power P_o. The transmission is defined as T =P_o/P_i. (a2) Transmission as function of frequency for three straight PDWGs of different lengths. (b1) A typical 90° bent PDWG. (b2) Transmission as function of frequency for four cases of different radii. One lattice period in the bend region for the R = 57.3a,19.1a,11.5a,5.7a waveguides correspond to δ = 1°,3°,5°,10°, respectively. The total number of cylinders of the four waveguides are 114, 54, 42, and 33, respectively.

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Fig. 4. The S shaped PDWG. The bend angle is θ, and the total length is 79a (80 cylinders). The S shaped section of the waveguide contains 38 cylinders. The radius of curvature is R = 11.5a, corresponding to δ = 5°.

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Fig. 5. Transmission properties of the S shaped PDWGs. (a) Transmission as function of frequency for four different bend angles. (b) Transmission as function of bend angle for four different frequencies.

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We first examine the effect of varying the bend radius R. Four cases are studied and the results are shown in Fig. 3(b2). One lattice spacing in the bend region for the R = 57.3a, 19.1a, 11.5a, 5.7a waveguides correspond to δ = 1°,3°,5°, and 10°. The total number of cylinders N of the four waveguides are 114, 54, 42, and 33, respectively. These results reveal that the transmission in the low frequency region can be dramatically reduced when a bend is present, whereas the bend effect diminishes if the frequency becomes high enough. At frequency ω = 0.25(2π/a) the calculated transmissions for the four cases all exceed 90%. In the following simulations for the θ ≠ 90° cases we always choose R = 11.5a, corresponding to δ = 5°. With this choice the transmission can keep higher than 0.85 if the frequency ω is chosen to be larger than 0.2(2πc/a).

We now turn to study the transmission of the S shaped PDWG with a pair of bends. Figure 4 shows the structure. The two bends are opposite in orientation and have the same size and bend angle θ. Such a structure is chosen to replace the one-bend PDWG so as to prevent possible numerical errors due to non-parallel cross section planes when we evaluate the energy flows. The S shaped PDWG contains five sections: the input region, the first bend, the connection part, the second bend, and the output region. We fix the number of cylinders of the whole PDWG to be N = 80. Both the input and output parts contain 21 cylinders. For each bend we choose δ = 5° and R = 11.5a, and it contains N_b = θ/δ + 1 cylinders. The remaining N - 2N_b - 2 × 21 = 38 - 2θ/δ cylinders are in the connection region.

Figure 5 shows the simulation results. In Fig. 5(a), four cases with different bend angles are chosen, and we plot the transmission as functions of frequency. In the region with reduced frequency below 0.174, large bend angle always gives low transmission, as expected. On the other hand, the transmission curves merge together in the high frequency region and give a transmission higher than 90% if the reduced frequency is above 0.23. In between these curves have oscillatory behavior, which we believe is caused by the finite size effect of the connection part.

 

Fig. 6. Three types of bent waveguides. (a1) A straight waveguide without any bend. (a2) An S-shaped waveguide formed by combining two 45°-bent waveguides. (a3) A U-shaped waveguide formed by combining two 90°-bent waveguides. Transmission in these waveguides is higher than 90% when the working frequency is chosen as ω = 0.25(2πc/a)

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In Fig. 5(b), four different reduced frequencies are chosen and the transmissions are plotted as functions of bend angles. When the bend angle is small, the bend effect has not yet been revealed, so the transmission curves look just like that for the straight PDWG, and low frequency implies high transmission, as observed in the θ < 10° region. When the bend angle is larger than 10°, the bend effect becomes important, and high frequency leads to high transmission. There are still some oscillatory behaviors, especially for the low frequency cases. This can be understood intuitively by the fact that the long wavelength propagating modes are more sensitive to the left-right asymmetry caused by bending the PDWG to one side. The results for the two cases with reduced frequencies a/λ = 0.225 and 0.25 are almost the same, which imply the transmission behaviors become stationary at high frequency region, consistent with the results in Fig. 5(a).

To illustrate the guiding ability of PDWG we plot in Fig. 6 the field patterns for three types of bent PDWG. The reduced frequency is chosen as 0.25. Figure 6(a1) is the result for a straight waveguide without any bend and Fig. 6(a2) is for an S shaped waveguide formed by combining two 45°-bent waveguides. Finally, the result for a U-shaped waveguide formed by combining two 90°-bent waveguides is shown in Fig. 6(a3).

To guide light in a conventional dielectric waveguide (with low index contrast) through a bend, the bend radius R must be large enough in order to reduce the loss. Typically the bend region has at least a size of millimeter order. For a PDWG, if the working wavelength is λ = 1.55μm, corresponding to a/λ = 0.25, then the bend radius R = 11.5a < 3λ can be reduced to 5μm, much smaller than that of the conventional dielectric waveguide. Furthermore, since it can be bent arbitrarily, it can avoid the geometric restriction that must be obeyed by the PCW.

One may expect that a CWG can guide light as good as PDWG does if high enough index contrast is used. To examine this possibility we further study the transmission characteristics of high index constarat (ε = 11.56) CWGs using FDTD method. The width and length of the CWG are chosen as 0.4a and 40a, respectively. The simulation results are compared with the results of a PDWG of the same size, shown in Fig. 7. The source is a point source located one lattice spacing apart from the insertion edge of the waveguides. In Fig. 7(a) the effective index of the waveguides, defined as [15]

 

Fig. 7. Comparison of the transmission and reflection characteristics between PDWG and CWG. (a) The effective index. (b) The reflection (insertion loss) for the straight waveguides. (c) Transmission as functions of frequency for 90° bent waveguides of two different radii

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are ploted, and in Fig. 7(b), the insertion losses (reflection) for straight CWG and PDWG are compared. The reflection is defined as

where P_inc is the incident power when the waveguide is absent (evaluated by integrating the normal component of the Poynting vector over a cross section plane 0.5a from the source), and P_out is the output power when the waveguide is present. From Fig. 7(a), we find that when the reduced frequency is lower than 0.23, the effective index of PDWG is smaller than that of CWG, which is consistent with the lower reflection in Fig. 7(b). The oscillatory behavior of the reflection curve for the PDWG might be caused by the discrete structure of the PDWG. As the frequency increases further, the effective index of PDWG grows to a very high value, corresponding to the very slow group velocity of the high frequency (near the band edge) guided modes in the PDWG. In Fig. 7(c), the transmission as functions of bend radius for the two kinds of waveguides in the 90° bent situation are compared. We find that if the reduced frequency is chosen to be higher than 0.18, than for any case with R ≥ 11.5a, the guiding ability of PDWG is better than CWG. We also find that if the bend radius is too small, for instance, R = 5.7a, the guiding ability of CWG becomes better than that of PDWG. For practical applications, the bend radius should be larger than 11.5a, and the operating (reduced) frequency should be chosen between 0.18 and 0.23.

3. Conclusion

In conclusion, we have studied the transmission properties of PDWG numerically. Effects of bend radius, bend angle, and frequency on the transmission are considered. The numerical results show that high transmission can be preserved if the bend radius is larger than 3λ and the frequency is chosen high enough. PDWG has the advantage that it can be bent arbitrarily in a small region, which might be helpful to make a more compact optical circuits. At the same time it can avoid the geometric restriction that must be obeyed by a photonic crystal waveguide.