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  • Editor: Alan E. Willner
  • Vol. 37, Iss. 22 — Nov. 15, 2012
  • pp: 4802–4804
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Subpixel smoothing finite-difference time-domain method for material interface between dielectric and dispersive media

Jinjie Liu, Moysey Brio, and Jerome V. Moloney  »View Author Affiliations


Optics Letters, Vol. 37, Issue 22, pp. 4802-4804 (2012)
http://dx.doi.org/10.1364/OL.37.004802


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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

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

3

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

]. 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

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

6

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

] is commonly used. A recently proposed subpixel smoothing technique in [7

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]

] has achieved second-order accuracy by using an inverse dielectric tensor formula, and this method has been extended to anisotropic media [8

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

] by employing a recently proposed stable FDTD scheme in anisotropic media [9

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

]. Another second-order EP algorithm was introduced in [10

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

]. In [11

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

], 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.

Now, we derive a numerical algorithm based on the CR to solve the following local constitutive equation:
E=ϵ01ϵ˜1D,
(1)
where ϵ0 is the permittivity in vacuum and ϵ˜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 ϵ1 and ϵ2. The unit normal vector is given by N=(cosϕ,sinϕ) and the unit tangential vector is T=(sinϕ,cosϕ), where ϕ is the angle between the interface and the vertical axis.

Fig. 1. Electric fields near the material interface. Ex and Ey are the electric fields in Cartesian coordinates. EN and ET correspond to the electric fields perpendicular and tangential to the material interface, respectively.

First, we show that on the interface between two dielectrics, our method is exactly the subpixel smoothing method proposed in [7

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]

]. Using coordinate transformation, we have:
(ENET)=(cosϕsinϕsinϕcosϕ)(ExEy)=R(ExEy),
(2)
where ET and EN represent the electric fields in the tangential and normal directions, respectively, and R is the rotation matrix. Similar formulas can be derived for electric fluxes DT and DN.

When computing effective permittivities in the constitutive Eq. (1), we use the arithmetic mean for ET and the harmonic mean for EN, then we obtain:
(ENET)=1ϵ0(ϵ100ϵ1)(DNDT),
(3)
where · denotes the volume average. After CR, Eq. (3) becomes
(ExEy)=1ϵ0R1(ϵ100ϵ1)R(DxDy).
(4)
When we let P=NTN be the projection matrix where N is the unit normal vector, we see that Eq. (4) is equivalent to Eq. (1) if we let
ϵ˜1=Pϵ1+(1P)ϵ1,
(5)
where ϵ˜1 is the inverse permittivity tensor for the subpixel smoothing algorithm [7

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]

]. Therefore, we see that the CR results in the same formulation as the subpixel smoothing method for a dielectric interface.

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:
  • 1. Compute DN and DT using CR:
    (DNDT)=R(DxDy);
    (6)
  • 2. Compute EN and ET by solving the constitutive Eq. (3) in the rotated coordinates; and
  • 3. Compute Ex and Ey using inverse rotation:
    (ExEy)=R1(ENET).
    (7)

In Step 2, Eq. (3) can be solved as two separated equations:
EN=1ϵ0ϵ1DN,
(8)
and
ET=1ϵ0ϵ1DT.
(9)

To 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

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

]. As shown in Fig. 1, we assume that ϵ1 is a constant (dielectric material) and ϵ2 represents a Lorentz dispersive medium with dielectric function given by
ϵ2(ω)=ϵαωp2ω2iγωωp2.
(10)

In Eq. (9), we first compute volume averaged permittivity:
ϵ=βϵ2(ω)+(1β)ϵ1,
(11)
where β represents the ratio of the shaded area of the Fig. 1 to the whole area of the figure. After some simple calculation, we get:
ϵ=ϵ^α^ω^p2ω2iγ^ωω^p2,
(12)
where ϵ^=βϵ+(1β)ϵ1, α^=βα, ω^p=ωp, and γ^=γ.

By introducing the polarization PT, we have:
DT=ϵ0ϵET=ϵ0(ϵ^ET+PT),
(13)
where
PT=α^ω^p2ω2iγ^ωω^p2ET.
(14)
Equation (14) is a Lorentz model and can be solved by the auxiliary differential equation algorithm [3

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

]:
2PTt2+γ^PTt+ω^p2PT=α^ω^p2ET.
(15)

For the normal component of the electric field in Eq. (8), we have the harmonic averaging
ϵ1=β1ϵ2(ω)+(1β)1ϵ1,
(16)
which leads to
1ϵ1=ϵ˘α˘ω˘p2ω2iωγ˘ω˘p2,
(17)
where ϵ˘=ϵϵ1/ξ, α˘=αβϵ12/ξ2, ω˘p2=ωp2+ωp2(1β)α/ξ, and ξ=βϵ1+(1β)ϵ.

The SF subpixel method proposed in [11

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

] 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 ϵ1 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 ω 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 (EN, ET, DN, and DT) 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 L2 norm of the global error of the field intensity and is defined as
En2Ee2=i,j(|En(i,j)|2|Ee(i,j)|2|Ee(i,j)|2)2,
(20)
where En(i,j) and Ee(i,j) are the numerical and exact (Mie) solutions at grid point (i,j), respectively. The subpixel smoothing method consistently achieves smaller error in comparison to the staircased FDTD results. We also see that, although our method does not achieve second-order accuracy, it does significantly improve the accuracy. Table 1 shows a comparison of CPU time and memory usage for the three methods. Compared to the SF-subpixel smoothing, our method appears to have better accuracy when the numerical grid resolution is low and it saves about 10% of memory and about 25% of CPU time.

Fig. 2. Electric fields intensity along the x axis passing through the center of a cylinder (radius is 1 μm). The incident TE plane wave (wavelength 2.5 μm) propagates in x direction.
Fig. 3. Relative error versus resolution for a cylinder illuminated by the TE plane wave.

Table 1. Comparison of CPU Time and Memory Usage

table-icon
View This Table

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]

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(050.1755) Diffraction and gratings : Computational electromagnetic methods

ToC Category:
Diffraction and Gratings

History
Original Manuscript: September 28, 2012
Revised Manuscript: October 16, 2012
Manuscript Accepted: October 16, 2012
Published: November 15, 2012

Virtual Issues
December 10, 2012 Spotlight on Optics

Citation
Jinjie Liu, Moysey Brio, and Jerome V. Moloney, "Subpixel smoothing finite-difference time-domain method for material interface between dielectric and dispersive media," Opt. Lett. 37, 4802-4804 (2012)
http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-37-22-4802


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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]

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