## An efficient iterative algorithm for computation of scattering from dielectric objects |

Optics Express, Vol. 19, Issue 4, pp. 3304-3315 (2011)

http://dx.doi.org/10.1364/OE.19.003304

Acrobat PDF (984 KB)

### Abstract

We have developed an efficient iterative algorithm for electromagnetic scattering of arbitrary but relatively smooth dielectric objects. The algorithm iteratively adapts the equivalent surface currents until the electromagnetic fields inside and outside the dielectric objects match the boundary conditions. Theoretical convergence is analyzed for two examples that solve scattering of plane waves incident upon air/dielectric slabs of semi-infinite and finite thicknesses. We applied the iterative algorithm for simulation of sinusoidally-perturbed dielectric slab on one side and the method converged for such unsmooth surfaces. We next simulated the shift in radiation pattern of a 6-inch dielectric lens for different offsets of the feed antenna on the focal plane. The result is compared to that of the Geometrical Optics (GO).

© 2011 Optical Society of America

## 1. Introduction

2. J. L. Volakis, L. C. Kempel, and A. Chatterjee, *Finite Element Method Electromagnetics* (IEEE Computer Society Press, 1998). [CrossRef]

3. R. F. Harrington, *Field Computation by Moment Methods* (Wiley IEEE Press, 1993). [CrossRef]

*M*

^{3}(we have assumed the number of mesh nodes

*M*=

_{x}*M*=

_{y}*M*=

_{z}*M*to be equal for all 3 dimensions). MOM is better in terms of the number of unknowns since it deals with boundary surface of the object instead of the volume of the object. However, MOM requires the inverse of the impedance matrix, which takes a computational complexity of

*N*

^{3}for

*N*unknowns. The required memory scales as

*N*

^{2}. Although Multi-level Fast Multipole Method (MLFMM) can reduce the computational complexity to

*N*log

*N*[4], but its computational procedure is too complicated and still not so efficient for some ultra-large-scale object simulation. Other boundary element methods can only simulate some particular geometries efficiently [5

5. C. M. Kelso, P. D. Flammer, J. A. DeSanto, and R. T. Collins, “Integral equations applied to wave propagation in two dimensions: modeling the tip of a near-field scanning optical microscope,” J. Opt. Soc. Am. A **18**(8), 1993–2001 (2001). [CrossRef]

8. X. An and Z. Q. Lu, “An efficient finite element-boundary integral method solving electromagnetic scattering problems,” Microwave Opt. Technol. Lett. **51**(9), 2065–2071 (2009). [CrossRef]

9. N. Gopalsami, S. Liao, E. R. Koehl, T. W. Elmer, A. Heifetz, H.-T. Chien, and A. C. Raptis, “Passive millimeter wave imaging and spectroscopy system for terrestrial remote sensing,” Proc. SPIE **7670**, 767003 (2010). [CrossRef]

## 2. Equivalent Surface Current Model for Scattering Problem

**E**,

**H**) are the total fields expressed by the sum of the incident (

**E**

*,*

_{i}**H**

*) and scattered fields (*

_{i}**E**

*,*

_{sca}**H**

*),*

_{sca}**A**and

**F**are given by, and from which we can obtain the scattered electric and magnetic fields [10], with the following definitions, and

## 3. The Iterative Algorithm

**J**

*and*

_{s}**M**

*for the scattering problem. Fig. 1 shows that the total field of the scattering problem can be viewed as the sum of the incident and the scattered fields. One way to obtain the equivalent surface currents*

_{s}**J**

*and*

_{s}**M**

*is [12]: the scattered fields*

_{s}**E**

*and*

_{sca}**H**

*outside the object are obtained when these surface currents radiate in a homogeneous medium*

_{sca}*ɛ*

_{0}; while the negative scattered fields inside the object

**E**

*and*

_{o}**H**

*are obtained when the these surface currents −*

_{o}**J**

*and −*

_{s}**M**

*radiate in a homogeneous medium*

_{s}*ɛ*. In this way,

_{r}**J**

*and*

_{s}**M**

*can be obtained as [12]*

_{s}**J**

*,*

_{s}**M**

*) through approximate incident fields*

_{s}**E**

*and*

_{i}**H**

*; then scattered fields (*

_{i}**E**

_{sca}_{,}

*,*

_{k}**H**

_{sca}_{,}

*) of the*

_{k}*k*

^{th}iteration are calculated as follows, where

**J**

_{s,k},

**M**

_{s,k}) of the

*k*

^{th}iteration according to Eq. (8),

- Make the initial guess of the total field (
**E**_{0},**H**_{0}): although there is no specific requirement of the initial guess, it is preferable to use reasonable value so that the iteration converges fast. The simplest guess is to begin with the incident field (**E**,_{i}**H**);_{i} - Update the equivalent surface currents (
**M**_{s,k},**J**_{s,k}) of the*k*^{th}iteration according to Eq. (10). - Update the scattered fields on both sides (inside and outside) of the dielectric object.
- If the correction is small compared to some criterion, the iterative algorithm converges and go to step 6) below; otherwise, repeat step 2) to step 4) until the algorithm converges.
- Calculate the far field pattern.

## 4. Convergence Analysis of the Iterative Algorithm

### 4.1. General Analysis

*k*

^{th}iterative equivalent surface currents

**M**

_{s,k},

**J**

_{s,k}can be expressed using the previous ones, where

*k*→ ∞,

_{1,2}are smaller than unit 1, and the result is,

### 4.2. Semi-infinite dielectric slab

**E**

*=*

_{i}*x̂E*

_{i,x};

**H**

*=*

_{i}*ŷH*

_{i,x}) incident upon an air-dielectric (

*ɛ*) interface, let us denote the total field of the (

_{r}*k*− 1)

^{th}iteration as (

**E**

_{k}_{−1},

**H**

_{k}_{−1}) = (

*x̂E*

_{k−1},

*ŷH*

_{k−1}). The equivalent surface currents are (we choose the surface normal to be

*n̂*= −

*ẑ*), Then the scattered field is given by,

*α*̿ =

*β*̿ =

*θ*̿ =

*γ*̿ = 0, so or as expected. Interestingly, for infinite dielectric slab, only one iteration is required to obtain the result.

### 4.3. Finite dielectric slab

*n̂*

^{−}= −

*n̂*

^{+}= −

*ẑ*(− and + denote the left and right sides respectively). Similarly, let us denote the total field of the (

*k*− 1)

^{th}iteration as follows

*k*and

*k*are the wave vectors of the air and the dielectric.

_{r}**C̿**are, where

^{−jkd}and 𝒫

*= exp*

_{r}^{−jkrd}. Since |

*r*𝒫

*| = |*

_{r}*r*𝒫| ≤ 1, the iterative algorithm converges according to Eq. (15).

## 5. Numerical Results

### 5.1. Dielectric slab with sinusoidal shape on one side

*w*= 5

*λ*= 10 mm in the simulation. The sinusoidal shape is given by

### 5.2. Dielectric lens simulation

13. J. P. Thakur, W.-G. Kim, and Y.-H. Kim, “Large aperture low aberration aspheric dielectric lens antenna for W-band quasi-optics,” PIER **103**, 57–65 (2010). [CrossRef]

15. A. V. Boriskin, G. Godi, R. Sauleau, and A. I. Nosich, “Small hemielliptic dielectric lens antenna analysis in 2-D: boundary integral equations versus geometrical and physical optics,” *IEEE Trans. Antennas Propag.*56, 485–492 (2008). [CrossRef]

*x*is the antenna offset.

_{off}*R*= 3.4″ is the radius of the lens and

*h*= 1.8″ is the thickness of the lens. The focal length

*F*is given by the lens maker’s formula, with

*n*= 1.5 for our simulation.

*E*

_{//}of the electric field and the tangential component

*H*

_{//}of the magnetic field inside and outside the dielectric surface after the algorithm runs for 7 iterations. The corresponding radiation patterns are shown in Fig. 8, together with the GO results for comparison. The GO result is obtained by using Snell’s law and Fresnel’s law on the plane surface of the lens to obtain the field inside the lens; the obtained field then is propagated to the convex surface of the lens, where again the Snell’s law and Fresnel’s law are applied to obtain the transmitted field outside the lens.

## 6. Discussion

*L*=

_{x}*L*=

_{y}*L*=

_{z}*L*, which also means the numbers of mesh cells are the same

*M*=

_{x}*M*=

_{y}*M*=

_{z}*M*. We also assumed that the boundary surface of the object is proportional to the square of dimension, i.e.,

*S*∝

*L*

^{2}(e.g., a simple cube). It is straight forward that the memory for FDTD and FEM is at the order of 𝒪(

*M*

^{3}) [1, 2

2. J. L. Volakis, L. C. Kempel, and A. Chatterjee, *Finite Element Method Electromagnetics* (IEEE Computer Society Press, 1998). [CrossRef]

*M*

^{3}×

*M*) = 𝒪(

*M*

^{4}) since it takes ∝

*M*time step to propagate a pulse through the scattering object of length

*L*[1]. The computation time for FEM using an iterative solver can be as low as 𝒪(

*M*

^{3}) [2

2. J. L. Volakis, L. C. Kempel, and A. Chatterjee, *Finite Element Method Electromagnetics* (IEEE Computer Society Press, 1998). [CrossRef]

*M*times as much as that of the FEM. This is because FDTD can give 𝒪(

*M*) frequency points after the time domain has been transformed to the frequency domain using FFT, while FEM only gives one frequency point. The memory for MOM-MLFMM can achieve the order of 𝒪(

*M*

^{2}log

*M*

^{2}) and the computation time using MLFMM scales as 𝒪(

*M*

^{2}log

*M*

^{2}) [4]. At last, the memory for our algorithm is 𝒪(

*M*

^{2}). The computation time is 𝒪(

*M*

^{2}log

*M*

^{2}) when fast field propagation method like FFT-based method [11

11. S. Liao and R. J. Vernon, “A fast algorithm for computation of electromagnetic wave propagation in half-space,” IEEE Trans. Antennas Propag. **57**(7), 2068–2075 (2009). [CrossRef]

*L*=

_{x}*L*= 127

_{y}*λ*,

*L*= 11

_{z}*λ*. In FDTD and FEM, assuming the discretization size Δ

*L*= 1/10

*λ*, we have,

*M*=

_{x}*M*= 1270,

_{y}*M*= 110. The memory for both FDTD and FEM is 𝒪(

_{z}*M*) = 𝒪(1.8 × 10

_{x}M_{y}M_{z}^{8}). The computation time for FDTD and FEM are 𝒪(

*M*×

_{x}M_{y}M_{z}*max*(

*M*,

_{x}*M*,

_{y}*M*)) = 𝒪(2.3 × 10

_{z}^{11}) and 𝒪(

*M*) = 𝒪(1.8 × 10

_{x}M_{y}M_{z}^{8}) respectively. The memory and computation time for MOM-MLFMM are the same, i.e., 𝒪(

*M*log (

_{x}M_{y}*M*)) = 𝒪(2.3 × 10

_{x}M_{y}^{7}). The memory and computation time for our algorithm are 𝒪(

*M*) = 𝒪(1.6 × 10

_{x}M_{y}^{6}) and 𝒪(

*M*log(

_{x}M_{y}*M*)) = 𝒪(2.3 × 10

_{x}M_{y}^{7}) respectively. In example 2, we used the following discretization:

*M*=

_{x}*M*= 760,

_{y}*M*= 230 for Δ

_{z}*L*= 1/10

*λ*. The result is shown in Table 1. Obviously, our algorithm is more efficient in memory requirement than or at least as efficient as other methods used here for comparison. In terms of computation time, our algorithm is at least as efficient as MOM-MLFMM (only a few iterations are needed: 7 in both of our examples) and is much simpler to be implemented.

**M**

*,*

_{s}**J**

*) in Eq. (10), we have implicitly used the geometric optics approximation, which means our algorithm is more efficient for relatively smooth surface, i.e., the smoother the surface, the fewer the iterations it takes to converge. The rule of thumb of this is that good convergence could be obtained if each of the principal radius of curvature of the object surface is greater than a few wavelengths [16*

_{s}16. S. Liao and R. J. Vernon, “On the image approximation for electromagnetic wave propagation and PEC scattering in cylindrical harmonics,” Prog. Electromagn. Res. **66**, 65–88 (2006). [CrossRef]

## 7. Conclusion

## References and links

1. | A. Taflove and S. C. Hagness, |

2. | J. L. Volakis, L. C. Kempel, and A. Chatterjee, |

3. | R. F. Harrington, |

4. | W. C. Chew, J. M. Jin, E. Michielssen, and J. Song, |

5. | C. M. Kelso, P. D. Flammer, J. A. DeSanto, and R. T. Collins, “Integral equations applied to wave propagation in two dimensions: modeling the tip of a near-field scanning optical microscope,” J. Opt. Soc. Am. A |

6. | Q. H. Liu, Y. Lin, J. Liu, J. H. Lee, and E. Simsek, “A 3-D spectral integral method (SIM) for surface integral equations,” IEEE Microw. Wirel. Compon. Lett. |

7. | M. Y. Xia, C. H. Chan, S. Q. Li, B. Zhang, and L. Tsang, “An efficient algorithm for electromagnetic scattering from rough surfaces using a single integral equation and multilevel sparse-matrix canonical-grid method,” IEEE Trans. Antennas Propag. |

8. | X. An and Z. Q. Lu, “An efficient finite element-boundary integral method solving electromagnetic scattering problems,” Microwave Opt. Technol. Lett. |

9. | N. Gopalsami, S. Liao, E. R. Koehl, T. W. Elmer, A. Heifetz, H.-T. Chien, and A. C. Raptis, “Passive millimeter wave imaging and spectroscopy system for terrestrial remote sensing,” Proc. SPIE |

10. | C. A. Balanis, |

11. | S. Liao and R. J. Vernon, “A fast algorithm for computation of electromagnetic wave propagation in half-space,” IEEE Trans. Antennas Propag. |

12. | S. B. Sorensen and K. Pontoppidan, Lens analysis methods for quasioptical systems, in The 2nd European Conference on Antennas and Propagation (EuCAP 2007), Edinburgh, UK, 11–16 Nov. 2007. |

13. | J. P. Thakur, W.-G. Kim, and Y.-H. Kim, “Large aperture low aberration aspheric dielectric lens antenna for W-band quasi-optics,” PIER |

14. | Z. X. Wang and W. B. Dou, “Full-wave analysis of monopulse dielectric lens antennas at W-band,” Int. J. Infrared Millim. Waves |

15. | A. V. Boriskin, G. Godi, R. Sauleau, and A. I. Nosich, “Small hemielliptic dielectric lens antenna analysis in 2-D: boundary integral equations versus geometrical and physical optics,” |

16. | S. Liao and R. J. Vernon, “On the image approximation for electromagnetic wave propagation and PEC scattering in cylindrical harmonics,” Prog. Electromagn. Res. |

**OCIS Codes**

(080.0080) Geometric optics : Geometric optics

(290.0290) Scattering : Scattering

**ToC Category:**

Scattering

**History**

Original Manuscript: November 22, 2010

Revised Manuscript: January 3, 2011

Manuscript Accepted: January 12, 2011

Published: February 4, 2011

**Citation**

Shaolin Liao, N. Gopalsami, A. Venugopal, A. Heifetz, and A. C. Raptis, "An efficient iterative algorithm for computation of scattering from dielectric objects," Opt. Express **19**, 3304-3315 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-4-3304

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### References

- A. Taflove, and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd. ed. (Artech House, 2005).
- J. L. Volakis, L. C. Kempel, and A. Chatterjee, Finite ElementMethod Electromagnetics (IEEE Computer Society Press, 1998). [CrossRef]
- R. F. Harrington, Field Computation by Moment Methods (Wiley IEEE Press, 1993). [CrossRef]
- W. C. Chew, J. M. Jin, E. Michielssen, and J. Song, Fast and Efficient Algorithm in Computational Electromagnetics (Artech House Publisher, 2001).
- C. M. Kelso, P. D. Flammer, J. A. DeSanto, and R. T. Collins, “Integral equations applied to wave propagation in two dimensions: modeling the tip of a near-field scanning optical microscope,” J. Opt. Soc. Am. A 18(8), 1993–2001 (2001). [CrossRef]
- Q. H. Liu, Y. Lin, J. Liu, J. H. Lee, and E. Simsek, “A 3-D spectral integral method (SIM) for surface integral equations,” IEEE Microw. Wirel. Compon. Lett. 19(2), 62–64 (2009). [CrossRef]
- M. Y. Xia, C. H. Chan, S. Q. Li, B. Zhang, and L. Tsang, “An efficient algorithm for electromagnetic scattering from rough surfaces using a single integral equation and multilevel sparse-matrix canonical-grid method,” IEEE Trans. Antenn. Propag. 51(6), 1142–1149 (2003). [CrossRef]
- X. An, and Z. Q. Lu, “An efficient finite element-boundary integral method solving electromagnetic scattering problems,” Microw. Opt. Technol. Lett. 51(9), 2065–2071 (2009). [CrossRef]
- N. Gopalsami, S. Liao, E. R. Koehl, T. W. Elmer, A. Heifetz, H.-T. Chien, and A. C. Raptis, “Passive millimeter wave imaging and spectroscopy system for terrestrial remote sensing,” Proc. SPIE 7670, 767003 (2010). [CrossRef]
- C. A. Balanis, Advanced Engineering Electromagnetics, (John Wiley & Sons, 1989).
- S. Liao, and R. J. Vernon, “A fast algorithm for computation of electromagnetic wave propagation in half-space,” IEEE Trans. Antenn. Propag. 57(7), 2068–2075 (2009). [CrossRef]
- S. B. Sorensen, and K. Pontoppidan, Lens analysis methods for quasioptical systems, in The 2nd European Conference on Antennas and Propagation (EuCAP 2007), Edinburgh, UK, 11–16 Nov. 2007.
- J. P. Thakur, W.-G. Kim, and Y.-H. Kim, “Large aperture low aberration aspheric dielectric lens antenna for W-band quasi-optics,” PIER 103, 57–65 (2010). [CrossRef]
- Z. X. Wang, and W. B. Dou, “Full-wave analysis of monopulse dielectric lens antennas atW-band,” Int. J. Infrared Millim. Waves 31, 151–161 (2010).
- A. V. Boriskin, G. Godi, R. Sauleau, and A. I. Nosich, “Small hemielliptic dielectric lens antenna analysis in 2-D: boundary integral equations versus geometrical and physical optics,” IEEE Trans. Antenn. Propag. 56, 485–492 (2008). [CrossRef]
- S. Liao, and R. J. Vernon, “On the image approximation for electromagnetic wave propagation and PEC scattering in cylindrical harmonics,” Prog. Electromagn. Res. 66, 65–88 (2006). [CrossRef]

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