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
Electromagnetic scattering of dielectric objects of arbitrary shape and size is difficult to simulate with reasonable speed and accuracy. There exist many computational methods such as Finite Difference Time Domain (FDTD) , Finite Element (FEM) , and Method of Moments (MoM)  which can provide accurate results, but the computation times become prohibitive for electrically large objects. For example, FDTD and FEM require generation of dense computational mesh of scattering object volume, with problem freedom or unknowns scaling as M3(we have assumed the number of mesh nodes Mx = My = Mz = 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 N3 for N unknowns. The required memory scales as N2. Although Multi-level Fast Multipole Method (MLFMM) can reduce the computational complexity to N log N , 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 [5–8]. Finally, methods such as Physical Optics (PO) and Geometric Optics (GO) are generally faster but less accurate.
In this paper, we describe a new iterative algorithm based on finding the equivalent surface currents (both magnetic and electrical) on the dielectric surface. Fast and accurate convergence of the iterative algorithm is theoretically validated for plane wave incidence upon dielectric slabs of semi-infinite and finite thicknesses. We then applied the iterative algorithm on two examples: 1) sinusoidally-perturbed dielectric slab on one side and 2) a 6-inch dielectric lens with different offsets that we used for a passive millimeter wave radiometer , both at the frequency of 150 GHz. Also, the radiation patterns for different offsets of the lens with respect to a horn antenna were obtained and compared with the GO results.
2. Equivalent Surface Current Model for Scattering Problem
Fig. 1 shows the simulation based on equivalent surface currents on an arbitrary dielctric surface. The equivalent surface currents are given by,
Now the potentials A and F are given by,10],
The calculation of the electromagnetic fields in Eq. (6) and Eq. (7) can be done efficiently through the Taylor-FFT algorithm developed by the author in . The problem now reduces to finding a way to obtain the equivalent surface currents (Ms, Js), which is the main subject of this paper.
3. The Iterative Algorithm
The iterative algorithm aims at obtaining the equivalent surface currents Js and Ms 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 Js and Ms is : the scattered fields Esca and Hsca outside the object are obtained when these surface currents radiate in a homogeneous medium ɛ0; while the negative scattered fields inside the object Eo and Ho are obtained when the these surface currents −Js and −Ms radiate in a homogeneous medium ɛr. In this way, Js and Ms can be obtained as 
The algorithm begins with initial guess of (Js, Ms) through approximate incident fields Ei and Hi; then scattered fields (Esca,k, Hsca,k) of the kth iteration are calculated as follows,Eq. (8),
The iteration continues until it converges for a given criterion. Fig. 2 shows the detailed procedures for the iterative algorithm, which are summarized below,
- Make the initial guess of the total field (E0, H0): 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 (Ei, Hi);
- Update the equivalent surface currents (Ms,k, Js,k) of the kth iteration according to Eq. (10).
- Update the scattered fields on both sides (inside and outside) of the dielectric object.
- Correct the total field at kth iteration (Ek, Hk) according to Eq. (2).
- 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
Before we show the numerical results for the lens simulation, let us first give a general analysis of the convergence of the iterative algorithm and apply it to two basic examples: i.e., air/dielectric interface and dielectric slab with finite thickness. Both examples assume plane wave incidence.
4.1. General Analysis
Following Eq. (10), for plane wave incidence of arbitrary number of dielectric slabs, the iterative algorithm follows the correction procedure shown in Fig. 3, i.e., the kth iterative equivalent surface currents Ms,k, Js,k can be expressed using the previous ones,
From Eq. (11) we can obtain the convergence of the iterative algorithm by letting k → ∞,
Eq. (13) can be solved by expressing the matrix in its eigenvalue-eigenvector form,
Eq. (13) converges if all the diagonal elements in matrices Λ̿1,2 are smaller than unit 1, and the result is,
4.2. Semi-infinite dielectric slab
For a plane wave (assuming Ei = x̂Ei,x; Hi = ŷHi,x) incident upon an air-dielectric (ɛr) interface, let us denote the total field of the (k − 1)th iteration as (Ek−1, Hk−1) = (x̂Ek−1, ŷHk−1). The equivalent surface currents are (we choose the surface normal to be n̂ = −ẑ),
The updated equivalent surface currents are given by Eq. (11) and we find that α̿ = β̿ = θ̿ = γ̿ = 0, so
4.3. Finite dielectric slab
Now let us extend the problem to the finite dielectric slab. In this case, we have surface currents on both left and right planes, with surface normals of 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 . The equivalent surface currents are,
From Eq. (10), we have,
The eigenvalues of matrix C̿ are,Eq. (15).
5. Numerical Results
We applied the iterative algorithm on two examples: 1) Gaussian beam propagation through a dielectric slab with sinusoidal shape on one side and 2) dielectric lens simulation with different offsets of feed horn on the focal plane.
5.1. Dielectric slab with sinusoidal shape on one side
Fig. 4 shows the schematic of the simulation: a Gaussian beam source is 1” away from the dielectric slab,
Fig. 5 shows the convergence of tangential fields along x-direction on the sinusoidal surface of the dielectric slab after 7 iterations. As can be seen from the Fig. 5, the accuracy is up to −120 dB, good enough for most applications.
5.2. Dielectric lens simulation
Lens simulation is important to obtain accurate radiation pattern [13–15]. We apply the iterative algorithm on the simulation of a 6-inch dielectric lens at the frequency of 150 GHz for feed antenna at different offsets on the focal plane (see Fig. 6). The antenna pattern is approximated by a Gaussian beam as,
The simulated lens is a thin spherical lens given by,
Fig. 7 shows the tangential component 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.
We compared the efficiency of our algorithm in terms of the memory costs and computational time to different computational electromagnetics methods, with result shown in Table 1. Without loss of generality, we chose methods of FDTD, FEM and MOM using MLFMM (MOM-MLFMM) for comparison. In Table 1, we have assumed the dimension sizes are equal in all 3 dimension, Lx = Ly = Lz = L, which also means the numbers of mesh cells are the same Mx = My = Mz = M. We also assumed that the boundary surface of the object is proportional to the square of dimension, i.e., S ∝ L2 (e.g., a simple cube). It is straight forward that the memory for FDTD and FEM is at the order of 𝒪(M3) [1, 2]. The computation time for FDTD scales as 𝒪(M3 × M) = 𝒪(M4) since it takes ∝ M time step to propagate a pulse through the scattering object of length L . The computation time for FEM using an iterative solver can be as low as 𝒪(M3) . It is noted that the FDTD computation time is 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(M2 log M2) and the computation time using MLFMM scales as 𝒪(M2 log M2) . At last, the memory for our algorithm is 𝒪(M2). The computation time is 𝒪(M2 log M2) when fast field propagation method like FFT-based method  is used. Now let’s apply the above analysis to our two examples. In example 1, Lx = Ly = 127λ, Lz = 11λ. In FDTD and FEM, assuming the discretization size ΔL = 1/10λ, we have, Mx = My = 1270, Mz = 110. The memory for both FDTD and FEM is 𝒪(MxMyMz) = 𝒪(1.8 × 108). The computation time for FDTD and FEM are 𝒪(MxMyMz × max (Mx, My, Mz)) = 𝒪(2.3 × 1011) and 𝒪(MxMyMz) = 𝒪(1.8 × 108) respectively. The memory and computation time for MOM-MLFMM are the same, i.e., 𝒪(MxMy log (MxMy)) = 𝒪(2.3 × 107). The memory and computation time for our algorithm are 𝒪(MxMy) = 𝒪(1.6 × 106) and 𝒪(MxMy log(MxMy)) = 𝒪(2.3 × 107) respectively. In example 2, we used the following discretization: Mx = My = 760, Mz = 230 for ΔL = 1/10λ. 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.
In updating the surface currents (Ms, Js) 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 .
We have developed an efficient iterative algorithm to simulate the electromagnetic scattering by dielectric objects of different shapes, including a dielectric lens. The new algorithm iteratively adapts the equivalent surface currents to match the boundary conditions inside and outside the dielectric objects. The iterative algorithm converges after just a few iterations. Theoretical convergence analysis has been done for plane wave incidence upon dielectric slab. The algorithm has been applied to two examples, i.e., dielectric slab with perturbation on one side and a 6-inch dielectric lens with different offsets of a feed antenna on the focal plane. A comparison has been made among our algorithm and other commonly used computational electromagnetics methods, which shows that our algorithm is more efficient or at least as efficient as the state-of-the-art method like MOM-MLFMM and our algorithm is straightforward and simpler to be implemented.
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