A locally one-dimensional finite difference time domain method for the analysis of a periodic structure at oblique incidence

Authors


Abstract

[1] An implicit finite difference time domain (FDTD) method is developed to analyze periodic structures at oblique incidence. The split-field technique is applied to the locally one-dimensional (LOD) FDTD method with the periodic boundary condition. In addition, the dispersion control parameters are introduced to reduce the numerical dispersion error. The effectiveness of the present method is investigated through the analysis of a photonic band gap structure. It is shown that the computational time is reduced to ≅16% of that of the explicit FDTD method with acceptable results being maintained. As an application, a broadband mirror consisting of a subwavelength grating is analyzed and discussed.

1. Introduction

[2] The finite difference time domain (FDTD) method [Taflove and Hagness, 2000] with the periodic boundary condition (PBC) is one of the most popular numerical methods for the efficient analysis of periodic structures. Several techniques for the analysis at oblique incidence have been proposed and applied to the periodic FDTD method [Aminian and Rahmat-Samii, 2006; Taflove and Hagness, 2000; Roden et al., 1998]. It should be noted, however, that the time step size (Δt) of the conventional FDTD method suffers from the Courant-Friedich-Levy (CFL) condition. To remove this, Wang et al. [2005] have applied the alternating direction implicit (ADI) scheme [Namiki, 1999; Zhen et al., 2000] to the periodic FDTD method, in which the cyclic matrix problem resulting from the implicit scheme is solved using the Sherman-Morrison formula [Thomas, 1995]. The implicit scheme allows us to use a large Δt beyond ΔtCFL that is the upper limit determined with the CFL condition. The periodic ADI-FDTD method has also been extended to the analysis at oblique incidence [Mao et al., 2009].

[3] As an alternative to the ADI-FDTD method, the unconditionally stable locally one-dimensional (LOD) FDTD method [Nascimento et al., 2006; Shibayama et al., 2005, 2006; Tan, 2007] has been proposed. The algorithm of the LOD-FDTD method is quite simple compared to the ADI-FDTD method, maintaining the comparable accuracy. Recently, we have introduced the PBC into the LOD-FDTD method and confirmed the effectiveness of the LOD technique [Shibayama et al., 2009]. It should be noted, however, that the formulation has been limited to the case at normal incidence.

[4] From this viewpoint, we have begun to investigate the possibility of deriving the periodic LOD-FDTD method for the analysis at oblique incidence [Shibayama et al., 2010; Wakabayashi et al., 2010]. In this paper, we systematically formulate the LOD-FDTD method at oblique incidence. The two-dimensional problems for obliquely incident transverse electric (TE) and transverse magnetic (TM) waves are treated with the aid of the split-field technique [Roden et al., 1998]. In the light of the advantage of the implicit FDTD method, we introduce the dispersion control parameters to reduce the dispersion error caused by a large Δt [Li et al., 2007; Shibayama et al., 2009]. After the formulation, we analyze a photonic band gap (PBG) structure to confirm the effectiveness of the present algorithm. With the control parameters, the numerical results of the periodic LOD-FDTD method for 8ΔtCFL fairly compare with those of the explicit periodic FDTD method. Thus, the computational time is reduced to ≅16% of that of the explicit FDTD method. Using the derived algorithm, we analyze a broadband mirror consisting of a subwavelength grating (SWG) [Mateus et al., 2004a, 2004b; Wakabayashi et al., 2009].

2. Numerical Method

2.1. PBC

[5] Since the infinite periodicity assumed in this paper lies only in the x direction, we impose the PBC on the two vertical edges of the unit cell shown in Figure 1a. The relationships between the fields at the edges for the TE case are expressed as

equation image
equation image
equation image

where xp is the periodicity, kx = k0 sin θ, k0 is the free space wave number, and θ is the angle of incidence. Note that (1)–(3) include the phase shift resulting in the time gradient. For removing this phase shift, we introduce the field transformation technique [Taflove and Hagness, 2000]:

equation image
equation image

where η0 is the intrinsic impedance of free space. Therefore, the relationships in (1)–(3) are transformed into

equation image
equation image
equation image

respectively.

Figure 1.

(a) Periodic boundary condition. (b) Arrangement of the field components.

2.2. TE Case

[6] Substituting (4) and (5) into Maxwell's equations for the two-dimensional TE case gives

equation image
equation image
equation image

where α and β are the dispersion control parameters, as will be defined in 2.4. c is the velocity of light in free space, ɛr and μr are the relative permittivity and permeability. In this paper, μr is fixed to be unity. There are time derivatives on both sides of (10) and (11). To eliminate the time derivatives on the right-hand sides, the split-field technique [Roden et al., 1998] is used:

equation image
equation image

We substitute (12) and (13) into the left-hand sides of (10) and (11), respectively, and have

equation image
equation image
equation image

We employ the Crank-Nicolson scheme to remove the CFL condition, and apply the LOD scheme to the resultant equation (see Appendix A). Thus, we derive the following equations:

equation image
equation image
equation image

for the first step and

equation image
equation image
equation image

for the second step, where the field components with the asterisk are the intermediate fields. Recall that the PBC is only applied to the first step.

2.2.1. First Step

[7] It should be noted that there are the four unknown components (Qza*, Pya*, Qz*, Py*) in the first step. Thus, these equations cannot be solved directly. To resolve this problem, we reduce these unknown components.

[8] According to (13), we can transform the Pya components in (19) into the Py and Qz components:

equation image

With the aid of (12), the Qz components in (23) are also transformed into the Py and Qza components. Then, we discretize the resultant equation and obtain

equation image

where γx = equation image − sin2 θ and

equation image

The four unknown components in the first step are reduced to the two unknown components (Qza*, Py*). Note that the field components in the third and fourth terms of (24) are not collocated in the Yee mesh shown in Figure 1b. Therefore, spatial averaging is needed for these field components.

[9] Furthermore, we eliminate Qza* by substituting (18) into (24), resulting in the following equation to be solved implicitly:

equation image

where

equation image

When the edges of the unit cell shown in Figure 1a are treated, the relationships of (6)–(8) are applied to (25). As a result, we obtain the following matrix:

equation image

where equation image is the known components of (25), equation image is the unknown components, and

equation image

Unfortunately, [M] is a cyclic matrix not solvable with the Thomas algorithm for a tridiagonal system of linear equations. To efficiently solve (26) with the Thomas algorithm, we take the following approach.

[10] The cyclic matrix [M] can be rewritten as

equation image

where

equation image

equation image = [−1, 0, …, 0, −1]T, and equation image = [C, 0, …, 0, A]T. Here, we resort to the Sherman-Morrison formula [Thomas, 1995], leading to

equation image

For the solution of equation image with (28), the following two auxiliary equations are introduced:

equation image
equation image

Once equation image and equation image are available, we obtain equation image in (26) as

equation image

where

equation image

Fortunately, (29) and (30) are tridiagonal systems of linear equations, to which the Thomas algorithm is applicable. Since equation image is constant, equation image of (30) is solved once and stored throughout the analysis. Following the procedure mentioned above, we implicitly solve (26), and then explicitly solve (18).

2.2.2. Second Step

[11] In the second step, we similarly transform the Pya components in (22) into the Py and Qza components. Note that the terms with the Qza components in the resultant equation are eliminated because of the relation in (21). Therefore, we obtain the following equation:

equation image

where γz = equation image − sin2 θ and

equation image

[12] We substitute (20) into (32), and obtain the following equation:

equation image

Note that we do not assume the infinite periodicity in the z direction. Therefore, we can implicitly solve (33) without the Sherman-Morrison formula, and then explicitly solve (20). To avoid the reflection from the horizontal edges of the unit cell, we place absorbing regions as shown in Figure 1a. The fields of the Qx, Qza, and Py components in the absorbing regions are multiplied by the absorber function [Saijonmaa and Yevick, 1983].

2.3. TM Case

[13] So far, we have discussed the formulation for the TE case. A similar approach can also be used for the formulation for the TM case. The application of the Crank-Nicolson scheme and the LOD scheme to Maxwell's equations for the TM case gives

equation image
equation image
equation image

for the first step and

equation image
equation image
equation image

for the second step. Equations (36) and (39) are transformed into

equation image

and

equation image

respectively, which correspond to (24) and (32), respectively, in the TE case. Equations (40) and (41) are solved in the same way as that for the TE case.

2.4. Dispersion Control Parameters

[14] To reduce the dispersion error caused by a large Δt, we introduce the dispersion control parameters. The parameters are derived from the dispersion characteristic equation [Li et al., 2007]. Although the dispersion characteristic equation becomes a function of equation image, θ treated in this paper is relatively small. Therefore, we approximately express the dispersion control parameters as those at normal incidence. With a material refractive index n (=equation image), the parameters are expressed as

equation image

where kz = k0 sin ϕ, kx = k0 cos ϕ (note that n is not taken into account in [Li et al., 2007] since the free space is considered). α and β are derived with ϕ = 90° and 0°, respectively. It is pointed out by Shibayama et al. [2009] that, for sufficiently small Δx and Δz, (42) is simplified to

equation image

Notice that knowledge regarding n is not required in (43). In other words, the control parameters can be used without paying attention to the background index, provided that spatial sampling widths are sufficiently small. When α and β are set to be unity, the present scheme reduces to the conventional method.

2.5. Validation of the Present Method

[15] To confirm the validity of the present algorithm, we analyze the PBG structure shown in Figure 2. The configuration is the same as that treated by Roden et al. [1998]. The radius of the dielectric rod is a = 2 mm and its dielectric constant is ɛr = 4.2 (surrounded by air). The rods are arranged in a square lattice such that the center-to-center separation distance b is equal to the unit cell width where b is taken to be 9 mm. A Gaussian pulse plane wave is launched toward the direction defined by θ from the incidence plane, where θ is set to be 20°. The numerical parameters are chosen to be Δx = Δz = 0.25 mm.

Figure 2.

Configuration of a PBG.

[16] Figure 3 shows the transmission as a function of frequency obtained from the LOD-FDTD method without the dispersion control parameters (α = β = 1). For reference, the transmission from the explicit FDTD method is also included for Δt = ΔtCFL. It is seen that the transmission for the LOD-FDTD method with a large Δt slightly deviates from that of the explicit FDTD method in the high frequency region, which stems from the numerical dispersion error.

Figure 3.

Transmission as a function of frequency.

[17] To achieve a more accurate result, we employ the dispersion control parameters. α and β are determined by (43) at 15 GHz. Figure 4 depicts the transmission with (43). It is found for 4ΔtCFL that the transmission is almost superimposed on the explicit counterpart. In addition, the result for 8ΔtCFL is found to be acceptable, although a slight deviation is seen in the high frequency region.

Figure 4.

Transmission as a function of frequency with the dispersion control parameters.

[18] Each computational time is indicated in Figure 5, where the results are normalized to that for the explicit FDTD method. The computational time is reduced to ≅33% and 16% for 4ΔtCFL and 8ΔtCFL, respectively.

Figure 5.

Computational time.

[19] We also investigate memory storage. Fortunately, the computational memory of the present method is comparable with that of the explicit FDTD method, although the implicit scheme generally needs a lot of memory storage compared with that for the explicit scheme. This is because Maxwell's equations with the field transformation technique require that both the electric and magnetic fields be computed at each time step. To meet this requirement, for the explicit FDTD method, we need to employ a dual time grid. Therefore, the computational memory is increased. In contrast, for the present method, we do not need to employ the dual time grid because the discretization with respect to time is based on the Crank-Nicolson scheme, that is, both the electric and magnetic fields are placed at each time step. As a result, the computational memory becomes equivalent to that of the explicit FDTD method.

3. Broadband Mirror Consisting of an SWG

3.1. Configuration

[20] As an application of the present algorithm, we analyze the broadband mirror consisting of the SWG shown in Figure 6. Following the configuration treated by Mateus et al. [2004a], the parameters are taken to be ng = nsub = 3.48 (Poly-Si), nL = 1.47 (SiO2), Tg = 0.46 μm, TL = 0.9 μm, and Λ = 0.7 μm. The fill factor (w/Λ) is set to be ≅0.75. The numerical parameters are fixed to be Δx = Δz = 0.01 μm and Δt = 4ΔtCFL. The TM wave is incident toward the direction defined by θ.

Figure 6.

Configuration of an SWG.

3.2. Reflection Characteristics

[21] Figure 7 shows the reflectivity as a function of λ. It is found at normal incidence that a reflectivity of more than 99% is obtained over a wavelength range of 1.34 μm to 1.81 μm. This broadband reflectivity is explained by the guided mode resonance (GMR) [Magnusson and Shokoon-Saremi, 2008; Sang et al., 2009], under which the fields propagate in opposite directions along the dielectric grating. However, when the angle of incidence is varied from θ = 0°, this symmetry is broken [Sang et al., 2009], with the result that a reduction in the reflectivity occurs in the middle of the high-reflection region. Consequently, the high-reflection region is separated into two regions. For example, a reflectivity of more than 99% is obtained over wavelength ranges of 1.32 μm to 1.42 μm and of 1.74 μm to 1.87 μm for θ = 10°.

Figure 7.

Reflectivity as a function of λ (Tg = 0.46 μm, TL = 0.9 μm, Λ = 0.7 μm, w/Λ ≅ 0.75).

[22] The reflectivity as a joint function of λ and θ is shown in Figure 8. The region, in which the reduction in the reflectivity occurs, broadens as the angle of incidence is increased, although the high-reflection regions are maintained around λ = 1.4 μm and 1.8 μm. Note that the reflectivity is reduced around λ = 1.55 μm. Therefore we modify the SWG structure in order to restore the high-reflection region around λ = 1.55 μm.

Figure 8.

Reflectivity as a joint function of λ and θ.

3.3. Modified Structures

[23] Since the GMR is mainly determined by Λ and Tg, we modify these parameters. The reflectivity as a joint function of λ and Λ is shown in Figure 9. It is seen that, as Λ is increased, the high-reflection regions shift toward longer wavelengths. For Λ = 0.62 μm, we obtain the high-reflection regions around λ = 1.30 μm and 1.55 μm. Figure 10 shows the contour plot of the reflectivity as a joint function of λ and Tg for Λ = 0.62 μm. It is revealed that wide reflection bands with high reflectivities are observed for Tg = 0.38 μm. From this view point, we newly choose the configuration parameters as Λ = 0.62 μm and Tg = 0.38 μm.

Figure 9.

Reflectivity as a joint function of λ and Λ.

Figure 10.

Reflectivity as a joint function of λ and Tg for Λ = 0.62 μm.

[24] Figure 11 shows the reflection characteristics of the modified structure. For comparison, the results obtained with the original structure are again plotted. It is observed that the modified structure offers improvement in the reflectivity around λ = 1.55 μm. A reflectivity of more than 99% is obtained over wavelength ranges of 1.23 μm to 1.34 μm and of 1.53 μm to 1.63 μm.

Figure 11.

Reflection characteristics of a modified structure (θ = 10°, Tg = 0.38 μm, TL = 0.9 μm, Λ = 0.62 μm, w/Λ ≅ 0.75).

[25] We finally refer to the results obtained from another modified structure, which are shown in Figure 12. The configuration parameters are chosen to be Tg = 0.46 μm, TL = 0.85 μm, Λ = 0.66 μm, and w/Λ ≅ 0.52. Although the high-reflection region around 1.3 μm is disappeared, a reflectivity of more than 99% is observed over a relatively wide wavelength range of 1.44 μm to 1.65 μm.

Figure 12.

Reflection characteristics of another modified structure (θ = 10°, Tg = 0.46 μm, TL = 0.85 μm, Λ = 0.66 μm, w/Λ ≅ 0.52).

4. Conclusion

[26] We have extended the LOD-FDTD method with the periodic boundary condition to the analysis at oblique incidence. The field transformation and the split-field techniques are used to treat the oblique incidence case. The dispersion control parameters are employed to allow a large time step size. After the formulation, we analyze the photonic band gap structure that has often been employed as a benchmark test. With the acceptable results, the computational time for 8ΔtCFL is reduced to ≅16% of that of the explicit FDTD method. As an application of the present method, the broadband mirror consisting of the subwavelength grating is analyzed. At oblique incidence, the high-reflection regions are maintained around λ = 1.4 μm and 1.8 μm, although the reduction in the reflectivity occurs around λ = 1.55 μm. To restore the high-reflection region around λ = 1.55 μm, we change the subwavelength grating structure into a smaller one. As a result, the high-reflection regions shift toward shorter wavelengths. A reflectivity of more than 99% is observed over wavelength ranges of 1.23 μm to 1.34 μm and of 1.53 μm to 1.63 μm.

Appendix A

[27] Maxwell's equations for the two-dimensional TE case, to which the split-field technique is applied, are expressed as

equation image

where equation image = [Qx, Qza, Qz, Pya, Py]T, [I] denotes the unit matrix, and

equation image

[28] To remove the CFL condition, we apply the Crank-Nicolson scheme to (A1) and have

equation image

where [A′] and [B′] are defined as

equation image
equation image

Application of the LOD scheme to (A2) gives

equation image

for the first step and

equation image

for the second step. Taking (A3) and (A4) into account, we can represent (A5) and (A6) as

equation image
equation image

We obtain (17)–(22) from the first, second, and fourth rows of (A7) and of (A8).

Acknowledgments

[29] This work was supported in part by MEXT, Grant-in-Aid for Scientific Research (c) (22560350).

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