## Abstract

In this Letter, we have shown that the subpixel smoothing technique that eliminates the staircasing error in the finite-difference time-domain method can be extended to material interface between dielectric and dispersive media by local coordinate rotation. First, we show our method is equivalent to the subpixel smoothing method for dielectric interface, then we extend it to a more general case where dispersive/dielectric interface is present. Finally, we provide a numerical example on a scattering problem to demonstrate that we were able to significantly improve the accuracy.

© 2012 Optical Society of America

The finite-difference time-domain (FDTD) Maxwell solver is a successful method in computational electrodynamics [1–3]. When modeling a curved material interface, the standard FDTD technique uses the staircased approximation, which reduces the accuracy. To eliminating the staircasing error effective permittivity (EP) [4–6] is commonly used. A recently proposed subpixel smoothing technique in [7] has achieved second-order accuracy by using an inverse dielectric tensor formula, and this method has been extended to anisotropic media [8] by employing a recently proposed stable FDTD scheme in anisotropic media [9]. Another second-order EP algorithm was introduced in [10]. In [11], the subpixel smoothing was extended to dispersive and conductive media by using split fields (SFs), where the electric field is written as the sum of multiple auxiliary variables. As a result, the computational cost is increased because of the additional memory and computations required for solving for the auxiliary variables.

In this Letter, we present what we believe is a new way to extend the subpixel smoothing algorithm to material interfaces between dielectric and dispersive media by using local coordinate rotation (CR). First, we show that our method is equivalent to the subpixel smoothing method [7,8] for a dielectric interface, then we extend it to a more general case where a dispersive/dielectric interface is present. Like the previous proposed algorithm in [11], although our method does not achieve second-order accuracy, it does significantly improve the accuracy. The advantage of our method is its efficiency because it does not require the storage and computation of the SFs.

Now, we derive a numerical algorithm based on the CR to solve the following local constitutive equation:

where ${\u03f5}_{0}$ is the permittivity in vacuum and ${\tilde{\u03f5}}^{-1}$ is the inverse permittivity tensor. For simplicity, we illustrate the two-dimensional case. The extension to three-dimensional is straightforward. We apply a coordinate transformation from Cartesian coordinates $(x,y)$ to the rotated coordinates $(N,T)$, where $N$ and $T$ represent the normal and tangential directions on the material interface, respectively. Figure 1 shows an example of the interface between two materials with two different permittivities ${\u03f5}_{1}$ and ${\u03f5}_{2}$. The unit normal vector is given by $N=(\mathrm{cos}\text{\hspace{0.17em}}\varphi ,\mathrm{sin}\text{\hspace{0.17em}}\varphi )$ and the unit tangential vector is $T=(\mathrm{sin}\text{\hspace{0.17em}}\varphi ,-\mathrm{cos}\text{\hspace{0.17em}}\varphi )$, where $\varphi $ is the angle between the interface and the vertical axis.First, we show that on the interface between two dielectrics, our method is exactly the subpixel smoothing method proposed in [7]. Using coordinate transformation, we have:

When computing effective permittivities in the constitutive Eq. (1), we use the arithmetic mean for ${E}_{T}$ and the harmonic mean for ${E}_{N}$, then we obtain:

Second, we show how to extend the subpixel smoothing to dispersive/dielectric interfaces without using SFs. Based on Eq. (3), our method includes the following three steps:

- 2. Compute ${E}_{N}$ and ${E}_{T}$ by solving the constitutive Eq. (3) in the rotated coordinates; and

In Step 2, Eq. (3) can be solved as two separated equations:

andTo illustrate how to solve Eqs. (8) and (9) on the material interface between dispersive and dielectric media, we use the Lorentz dispersive model as an example. The formulation for Drude and Debye models can be derived in the same fashion. Our method also can be applied to other similar models such as the effective optical response model proposed in [12]. As shown in Fig. 1, we assume that ${\u03f5}_{1}$ is a constant (dielectric material) and ${\u03f5}_{2}$ represents a Lorentz dispersive medium with dielectric function given by

In Eq. (9), we first compute volume averaged permittivity:

where $\beta $ represents the ratio of the shaded area of the Fig. 1 to the whole area of the figure. After some simple calculation, we get:By introducing the polarization ${P}_{T}$, we have:

For the normal component of the electric field in Eq. (8), we have the harmonic averaging

Similar to the previous case, from Eq. (8), we get:

The SF subpixel method proposed in [11] consider a general interface between two dispersive media. If one side of the interface is a dielectric medium, it can be reduced to two auxiliary variables since $\u3008{\u03f5}^{-1}\u3009$ in Eq. (5) is given in Eq. (17). However, SFs are still required since it cannot be further simplified, unless you use a common denominator, which will result in higher order differential equations due to the higher order $\omega $ terms. In contrast, our method is a diagonal system so the constitutive equation can be solved separately and the field splitting is avoided. A limitation of our method is that it applies to the case where one side of the interface is a dispersive medium. For an interface between two dispersive media, SFs are still required. We would like to point out that the normal and tangential components of the electric fields and fluxes (${E}_{N}$, ${E}_{T}$, ${D}_{N}$, and ${D}_{T}$) in our algorithm serve as intermediate variables, so except for a few temporary variables, no additional computational memory is required.

We tested our method (the CR subpixel smoothing) for a scattering problem and compared it with the SF subpixel smoothing method, the standard FDTD method, and the Mie solution. In our numerical simulation, a cylindrical particle with refractive index $n=0.23+2.97i$ is modeled by the Lorentz model. Figure 2 shows the electric field intensity along the $x$ axis passing through the center of the cylinder. Comparison of the relative error versus the resolution is shown in Fig. 3. The relative error is the ${L}_{2}$ norm of the global error of the field intensity and is defined as

In conclusion, we present what we believe is a new way to extend the subpixel smoothing FDTD algorithm to the material interface between dielectric and dispersive media by using local CR. Because our method does not require additional storage of auxiliary variables, better efficiency has been achieved compared to the previously proposed subpixel smoothing algorithm for dispersive media.

This work is supported by the U.S. Air Force Office of Scientific Research under Grant Numbers FA9550-10-1-0127 and FA9550-10-1-0064. Jinjie Liu also would like to acknowledge support from NSF Grant HRD-1242067.

## References

**1. **K. S. Yee, IEEE Trans. Antennas Propag. **14**, 302 (1966). [CrossRef]

**2. **A. Taflove and M. E. Brodwin, IEEE Trans. Microwave Theory Tech. **23**, 623 (1975). [CrossRef]

**3. **A. Taflove and S. Hagness, *Computational Electrodynamics: The Finite-Difference Time-Domain Method*, 3rd Ed. (Artech House, 2005).

**4. **N. Kaneda, B. Houshmand, and T. Itoh, IEEE Trans. Microwave Theory Tech. **45**, 1645 (1997). [CrossRef]

**5. **S. Dey and R. Mittra, IEEE Trans. Microwave Theory Tech. **47**, 1737 (1999). [CrossRef]

**6. **A. Mohammadi, H. Nadgaran, and M. Agio, Opt. Express **13**, 10367 (2005). [CrossRef]

**7. **A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. W. Burr, Opt. Lett. **31**, 2972 (2006). [CrossRef]

**8. **A. F. Oskooi, C. Kottke, and S. G. Johnson, Opt. Lett. **34**, 2778 (2009). [CrossRef]

**9. **G. R. Werner and J. R. Cary, J. Comput. Phys. **226**, 1085 (2007). [CrossRef]

**10. **T. Hirono, Y. Yoshikuni, and T. Yamanaka, Appl. Opt. **49**, 1080 (2010). [CrossRef]

**11. **A. Deinega and I. Valuev, Opt. Lett. **32**, 3429 (2007). [CrossRef]

**12. **A. Deinega and S. John, Opt. Lett. **37**, 112 (2012). [CrossRef]