computational analysis and diagonal preconditioning for the discrete dipole approximation on surface

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Contents lists available at ScienceDirect

Journal of Quantitative Spectroscopy &Radiative Transfer

Journal of Quantitative Spectroscopy & Radiative Transfer 110 (2009) 51–61

0022-40

doi:10.1

� Cor

E-m

journal homepage: www.elsevier.com/locate/jqsrt

Computational analysis and diagonal preconditioning for the discretedipole approximation on surface

Euiwon Bae, E. Daniel Hirleman �

School of Mechanical Engineering, West Lafayette, IN 47906, USA

a r t i c l e i n f o

Article history:

Received 12 February 2008

Received in revised form

1 October 2008

Accepted 2 October 2008

Keywords:

Scattering

Discrete dipole approximation

Preconditioning

73/$ - see front matter & 2008 Elsevier Ltd. A

016/j.jqsrt.2008.10.001

responding author. Tel.: +1765 494 5688.

ail address: [email protected] (E.D. Hirle

a b s t r a c t

The construction of a matrix for the discrete dipole approximation (DDA) on surface and

its relationship to an iterative solver is analyzed. It is shown that the spectral

characteristics of the DDA for free space and surface correlates to different convergence

characteristics. Compared with the free space DDA, when a surface is introduced, both

the dipole polarizability matrix and the reflection–interaction matrix contributes to the

diagonal/off-diagonal element, and solvability of the iterative method is related to

several physical parameters such as incident angle, polarization, and refractive indices.

Finally, we propose a diagonal preconditioning technique and show the effectiveness of

the preconditioned to a semiconductor pattern with isolated contaminant which is

assumed to be PSL, Si3N4, and Si. The result shows that when there is difference in the

refractive index, the diagonal preconditioning reduces the total computation time up to

27% for low refractive index cases. However the result shows limitation for the higher

refractive index cases.

& 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Light scattering is widely applied to identify certain physical characteristics of an individual object or systems of objectsince its fast, accurate, and non-invasive nature. In semiconductor manufacturing field, light scattering from a particle isused to inspect various different types of surface and subsurface defects and contaminants created from fabricating circuitstructures [1]. Therefore, predicting a scattering signature from contaminants or defects is important not only to identifythe physical information of defects but also to apply to designing an in-line surface inspection instrument. The discretedipole approximation (DDA) [2] uses cubic lattice of dipoles to fill an arbitrary-shaped geometry of a structure andcompute the scattering response. The power of the DDA method is its flexibility to model inhomogeneous features ofarbitrary shape using dipole polarizability tensor. The dipole polarizability is a function of volume and refractive index ofthe element material. Extensive studies have been performed by Draine and Goodman to conclude that lattice dispersionrelationship was superior to other theories in effectively representing the dipole polarizability [3]. Draine and Flatau [4]used the DDA method to write a code called DDSCAT which is applicable under certain range of refractive index and severalresearchers had extended this work to surface scattering of semiconductor field [5] and biological field [6,7].

The solution of DDA method is provided by an iterative method rather than direct matrix inversion due to the nature ofthe size and computational time. In general, the research on solving a matrix equation is performed by calculating thespectral characteristics or by finding a faster method of solution. Computational characteristics of the matrix construction

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E. Bae, E.D. Hirleman / Journal of Quantitative Spectroscopy & Radiative Transfer 110 (2009) 51–6152

is analyzed via analyzing the eigenvalues and relating them to the matrix invertibility [8,9]. The effect of different iterativealgorithms on DDA method on free space was analyzed by various authors and concluded that stabilized biconjugategradient method (BiCGSTAB) provided the best result in respective testing [10,11]. Similar analysis was conducted byNebeker on DDA on surface and chose the QMR method as the best method among the algorithm tested [12]. Manyapplications of scatterometry, however, in defect inspection system usually have circuit design pattern with a contaminantwith different material on surface. This situation provides us a chance of applying acceleration technique such aspreconditioning to improve the convergence of the iterative method with the presence of surface or inhomogeneousrefractive indices. Preconditioning method is a widely accepted method applied to many iterative solution methods [13,14].However, they require additional memory storage and variation in the computation algorithm to modify the original matrixformulation which may or may not reduce the total computation time which depends on the various input parameters [15].In this paper, we investigate the matrix construction of the DDA method on surface (DDSURF) and relate them to iterativesolver such that the convergence characteristic can be improved. The formulation of the elements of the total matrix isanalyzed along with different input parameters such as wavelength, incident angle, polarization, and refractive index.Based on the analysis, we will apply the diagonal preconditioner to one of representative dipole modeling of integratedcircuit pattern and present the effectiveness of using diagonal preconditioner in accelerating the convergence and reducingthe total computation time.

2. Theoretical background

2.1. Discrete dipole approximation method on surface

When there exist an incident field on a dipole array, the dipole moment Pi at ith dipole is related to the electromagneticfield shown as [1]

Pi ¼ aiEi, (1)

where Ei is the total electric field present at the radiating dipole and ai is the dipole polarizability tensor. The electric fieldincident on a dipole is a superposition of incident fields and fields from other dipoles. As shown in Fig. 1, the electric fieldincident on the receiving dipole also includes components from surface-reflected incident in the presence of a surface anddipole fields and total equation is rearranged as

1

aiPi � Edirect;i � Ereflected;i ¼ Einc;i, (2)

where Edirect,i is the direct interaction at ith dipole and Ereflection,i is the surface reflected interaction between dipoles whereSommerfeld integral are used to compute. When we represent all N individual dipoles with Eq. (2), we obtain a global3N�3N matrix equation

TP ¼ Einc, (3)

where T is the total matrix P is the dipole moment matrix of 3N�1, and Einc is the incident beam matrix of 3N�1. Solvingthe scattering problem involves computing the dipole moment at each dipole using an iterative method and a relativeresidue is defined as

rk ¼kEinc � TPk

k2

kEinck2, (4)

where rk is the residue at the kth iteration. A more rigorous mathematical derivation is available in Schmehl et al. [1].

Fig. 1. Types of dipole interaction and dipole model of sphere on surface.

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E. Bae, E.D. Hirleman / Journal of Quantitative Spectroscopy & Radiative Transfer 110 (2009) 51–61 53

To predict the far-field scattering, Eq. (3) must be solved to provide the internal dipole moment distribution in all thedipole location in all three direction. Since the dipole moment vector P is solved by an iterative method, investigating theformulation of the total matrix is helpful to understand the characteristics of the convergence and matrix construction.According to Refs. [16,17], the convergence rate of solving a conjugate gradient type problem with iterative methoddepends on the spectrum of the eigenvalues which is also represented by a condition number of a matrix. The totalinteraction matrix T is formulated as the sum of 3�3 submatrices Tij which is the summation of the following threeinteraction submatrices:

Dii ¼1

ai, (5)

Aij ¼k2

0

�0

XN

jai

Gij, (6)

Rij ¼XN

j¼1

Sij þk2

0

�0

k21 � k2

0

k21 þ k2

0

Gij

!, (7)

where Dij is submatrix from dipole polarizability tensor ai at ith location, Aij is the direct interaction, Rij is the reflectioninteraction, k0 and k1 are the wavevector for air and substrate, e0 is the permittivity, and Gij and Sij are dyadic Green’sfunction and Sommerfeld integral terms. With all the submatrices, global matrix T is constructed from following threeN�N matrices:

D ¼ diag1

a1x;

1

a1y;

1

a1iz; . . . ;

1

aNx;

1

aNy;

1

aNiz

� �, (8)

A ¼

o � � � A1N

..

.o ..

.

AN1 � � � o

2664

3775; R ¼

R1N � � � R1N

..

. ...

RN1 � � � RNN

2664

3775. (9)

Among various different models to describe the polarizability, Draine suggests that lattice dispersion equation (LDR) showsthe best accuracy in DDA method on |m�1|o1 [3]. The LDR is formulated as

ai ¼a0i

1þa0i

d3

� �½b1 þm2

i b2 þm2i b3S�ðk0dÞ2 �

2

3iðk0d3

Þ

� � , (10)

a0i ¼3d3

4pm2

i � 1

m2i þ 2

!, (11)

where d is the dipole spacing, mi is the refractive index and b1, b2, b3 are constant of �1.8915316, 0.1648469, and�1.7700004, respectively. S is a parameter which is defined as

S ¼X3

l¼1

ðalelÞ2, (12)

where al and el are propagation and polarization vector.Another contribution to the diagonal element is the reflection interaction matrix. The dyadic greens function and

Sommerfeld integral terms of Eq. (7) are formulated as

Rij ¼ �1

4p�0

r2I;ij;xIH

r � r2I;ij;yIH

f rI;ij;xrI;ij;x½IHr þ IH

f� rI;ij;xIVr

rI;ij;xrI;ij;x½IHr þ IH

f� r2I;ij;yIH

r � r2I;ij;xIH

f rI;ij;yIVr

�rI;ij;xIVr �rI;ij;yIV

r IVz

266664

377775

�k2

1 � k22

k21 þ k2

2

expðik0rI;ijÞ

4p�0rI;ij

�ðbI;ij þ gI;ijr2I;ij;xÞ �ðgI;ijrI;ij;xrI;ij;yÞ gI;ij rI;ij;xrI;ij;z

�ðgI;ijrI;ij;yrI;ij;xÞ �ðbI;ij þ gI;ij r2I;ij;yÞ gI;ijrI;ij;yrI;ij;z

�ðgI;ijrI;ij;zrI;ij;xÞ �ðgI;ijrI;ij;zrI;ij;yÞ bI;ij þ gI;ijr2I;ij;z

266664

377775 (13)

rI;ij;x ¼rI;ij;x

rI;ij; rI;ij;y ¼

rI;ij;y

rI;ij; rI;ij;z ¼

rI;ij;z

rI;ij, (14)

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bI;ij ¼ ½1� ðk0rI;ijÞ�2þ iðk0rI;ijÞ

�1�, (15)

gI;ij ¼ �½1� 3ðk0rI;ijÞ�2þ 3iðk0rI;ijÞ

�1�, (16)

rI;ij ¼ ½ðxi � xjÞ2þ ðyi � yjÞ

2þ ðzi þ zjÞ

2�1=2, (17)

were k1 and k2 is the wavevector, IHr ; I

Hf; I

Vr , and IV

z are the Sommerfeld integral terms. The detailed derivation is shown byScheml [1].

2.2. Matrix analysis

Computational electromagnetic codes which use iterative method for their solver show different convergencecharacteristics depending on the condition number of the matrix. The condition number, which is defined as the ratio of themaximum and minimum eigenvalue, is a measure of invertibility of a matrix and the inverse of this index is related tosensitivity of the matrix to round-off errors. To relate the matrix structure with this condition number, we analyze theeigenvalue spectrum of the DDA on surface. Fig. 2 presents the histogram of the eigenvalue spectrum for 8�8�8 dipolecube case when the model is in free space and on surface. The wavelength is set to 632.8 nm and the surface is assumed tobe Si with refractive index (3.88, 0.023). The results indicates that the magnitude of the eigenvalue spectrum is distributedbetween 0 to 2.6 and the surface case shows larger number of eigenvalues on the bracket close to 0. The actual conditionnumber of the total matrix T is 52.40 (free space) and 58.54 (surface) and take 16 and 22 iteration to converge, respectively.To provide the understanding of the eigenvalue spectrum and the physical parameters, we computed the similarhistograms when the refractive indices of the 5�5�5 dipole cube are varied from 1.5 to 2.5 as shown in Fig. 3. Theincrease of the refractive index results in a wider range of eigenvalue spectrum which results in a higher condition numberand thus requires more number of iterations to converge. The final condition numbers are 2.51, 5.37, and 11.75 and required5, 6, and 11 iterations to converge. This indicates that the presence of the surface and higher refractive index effectsnegatively on the computational aspect of the DDA method which deteriorates the condition number of the total matrix T.

Based on this reasoning, we define a parameter called diagonal dominance (DD) [16,17]:

DDi ¼diagjTiijPN

jaijTijj, (18)

which is a ratio of absolute value of diagonal element to sum of absolute value of off-diagonal elements for ith row of thetotal matrix T. This parameter provides the basis for the so-called diagonal dominance condition as shown which is asufficient condition for fast convergence if the problem converges. The extreme case of diagonal dominance would be theidentity matrix which has ratio of N.

Since both polarizability and reflection interaction matrices are functions of various input parameters, we havecomputed the relationship of diagonal dominance and refractive index for a single row (i ¼ 1) as a sample case of160�160�160 nm cube while changing three different wavelengths (632.8, 532, and 460 nm), two polarizations (P and S),and four incident angles (01, 301, 451, 601) with Si surface. As shown in Fig. 4, increasing the refractive index dramaticallyreduces the DD value which affects the rate of convergence. Fig. 4(a) shows the comparison for changing wavelengths withP polarization and 451 of incident angle which exhibits the decrease of DD when shorter wavelength is used. Fig. 4(b) showsthe number of iterations required for each case. Comparing with Fig. 4(a), as the DD value decreases, the iterative solverneeds more iteration to reach the target residue. Iteration for 632.8 nm wavelength shows the lowest number especiallywhen the refractive index is larger than 2. Fig. 4(c) and (d) shows the similar cases when the wavelength and incident angleare fixed to 532 nm 451 while polarization is changed. The change in polarization effects the polarization through Eq. (11)which shows less difference between changing polarization. This case also shows that more number of iteration is requiredto converge for computing cases with lower DD value. Fig. 4(e) and (f) shows the result for variation of incident angle. Thewavelength and polarization is fixed to 532 nm and p polarization. In this case, normal incident shows the best DD valueand requires less number of iteration while oblique incidence cases tends to lower the DD value and convergence rate startsto slow down as the refractive index increases above 2. The simulation results shows that there is good inverse correlationbetween DD and number of iterations, hence DD can be used as a simple (computable much easier than condition number)and rude estimate of condition number and the solvability of the DDA method.

2.3. Diagonal preconditioning

Based on the analysis of the matrix construction and its effect on number of iteration, we propose to apply a diagonalpreconditioning to current DDSURF. Diagonal preconditioning which is also called Jacobi preconditioning is the simplestand one of the effective preconditioning methods provided that the main diagonal elements of the total matrix are notidentical [18,19]. As stated before, physical parameters such as the presence of the surface, wavelength, andinhomogeneous refractive index results in variations in the diagonal and off-diagonal elements. In limited cases, the

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Fig. 2. Histogram of magnitude of eigenvalue spectrum for (a) free space and (b) surface case for 8�8�8 dipole cube on a silicon surface.


diagonal preconditioning changes condition number to accelerate the convergence of the iteration but does not affectthe DD.

The DDSURF applies a conjugate gradient type iterative solver called QMR method and it has been shown that two-termvariation of the three-term version showed better convergence characteristics [7,20]. The preconditioning is executed byconstructing an additional matrix M, which resembles the original matrix T. The advantage of the diagonal preconditioningis that their inverse is easily computed which is just an inverse of the total matrix element diag(T). Therefore, here we applythe diagonal preconditioning to 2-term QMR method as shown in Table 1. The preconditioning matrix M�1 is multiplied ineach required step to finally calculate the estimated polarization vector Pk at the kth iteration step. The relative residue is

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Fig. 3. Histogram of magnitude of eigenvalue spectrum for refractive index change of (a) n ¼ 1.5, (b) 2.0, and (c) 2.5 for 8�8�8 dipole cube on silicon

surface.


then computed via the same Eq. (4) to provide the true residual as the previous authors [20,21]. Since the diagonalpreconditioning is multiplied as a constant on each matrix rows, the diagonal preconditioning is performing avector–vector multiplication and the only requirement is to store them in additional memories, which is N�3. Therefore,the normal case and preconditioned case will have similar computational time per iteration since both cases have the samenumber of matrix–vector multiplications which is the most time-consuming portion of the iterative algorithm.

3. Result

The effectiveness of the diagonal preconditioning with QMR algorithm is proved through a real-life example ofsemiconductor manufacturing process. The computation is performed on 16 core Linux server with 64 GB of RAM. Theexample is an area 10 sample which is shown in Fig. 5. Fig. 5(a) shows the schematics of the 1.0� 0.8�0.28mm SiO2

pattern along with a contaminant modeled as an sphere particle with 0.482mm. Fig. 5(b) shows the corresponding dipolemodel with incident angle y. In this chapter, we assume that the particle which is designated as a contaminant has differentmaterial characteristics such as PSL, Si3N4, and Si. Theses materials are widely used or found as a calibration material or inthe semiconductor manufacturing process. Both polarization, P and S, are considered while the incident angle is fixed to451. Incident wavelength is changed to 632.8, 532, and 460 nm. Total number of dipole used to model and material constantused is tabulated in Table 2. The computation result is shown in Fig. 6 as a spectral hemispherical scattering of differentialscattering cross-section (DSC) for both P and S polarization. The figures display a strong forward scattering lobe at the 451while their shape and number lobe changes depending on the polarization and wavelength. For example, 532 nm

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Fig. 4. Relationship of diagonal dominance and iteration against refractive index for (a) and (b) different wavelengths, (c) and (d) polarizations, and

(e) and (f) incident angles.


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Fig. 5. Test sample of region 10 area 10 with SiO2 feature and spherical contaminant. (a) Schematic of the pattern and dimensions and (b) dipole model.

Table 2Number of dipole and material constant used for modeling region 10 area 10 defect model

Material l (nm) Total of dipoles Refractive index of particle Refractive index of surface

PSL 632.8 15,652 (1.58, 0.0) (3.88, 0.023)

532 26,676 (1.59, 0.0) (4.15, 0.035)

460 44,127 (1.61, 0.0) (4.58, 0.073)

Si3N4 632.8 8991 (2.02, 0.0) (3.88, 0.023)

532 16,070 (2.03, 0.0) (4.15, 0.035)

460 26,005 (2.05, 0.0) (4.58, 0.073)

Si 632.8 65,141 (3.88, 0.023) (3.88, 0.023)

532 135,160 (4.15, 0.035) (4.15, 0.035)

460 276,844 (4.58, 0.073) (4.58, 0.073)

Table 1Algorithm for two-term preconitioning QMR method [18,19]

Preconditioned two-term QMR algorithm

1 Compute r0 ¼ Einc � TP0

x0 ¼ d0 ¼ 0, c0 ¼ �0 ¼ 1; r1 ¼ kM�1r0k; v1 ¼ r0=r1

2 For k ¼ 1,2,y,Do

Compute dk ¼ vTk M�1vk

3 Compute xk ¼M�1vk � xk�1ðrk�1dk=�k�1Þ

4 Compute �k ¼ xTkTxk; bk ¼ �k=dk

vkþ1 ¼ Tpk � vkbk; rkþ1 ¼ kM�1vkþ1k

5 Compute

Wk ¼rkþ1

ck�1 jbk j; ck ¼

1ffiffiffiffiffiffiffiffiffi1þW2

k

p ; Zk ¼ �Zk�1rkc2

k

bkc2k�1

dk ¼ xkZk þ dk�1ðWk�1ckÞ2 ; Pk ¼ Pk�1 þ dk

6 vkþ1 ¼ vkþ1=rkþ1

7 Compute

rk ¼ kEinc � TPkk2=kEinck2

8 End Do


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Fig. 6. Hemispherical distribution of far-field scattering of DSC Fig. 5 for different wavelengths and polarizations.

Table 3Comparison of computational time for normal and preconditioned cases for both polarization of PSL and Si3N4

Material l (nm) Normal (N) Preconditioned (P) (N�P/N)*100 (%)

CPU time ITER TSC (mm2) CPU time ITER TSC (mm2) CPU time ITER TSC

P polarization

PSL 632.8 0.0534 21 0.7484 0.0544 21 0.7484 �1.98 0.00 0.00

532 0.1786 33 0.8916 0.1669 29 0.8912 6.59 12.12 0.04

460 0.3041 43 0.9711 0.2757 36 0.9689 9.35 16.28 0.23

Si3N4 632.8 0.0448 59 0.8610 0.0397 48 0.8611 11.40 18.64 �0.01

532 0.1345 106 0.9448 0.1184 82 0.9447 11.98 22.64 0.01

460 0.4652 171 0.8408 0.3651 129 0.8409 21.50 24.56 �0.01

S polarization

PSL 632.8 0.0608 25 2.1324 0.0551 21 2.1322 9.45 16.00 0.01

532 0.1906 34 1.5373 0.1774 31 1.5367 6.90 8.82 0.04

460 0.3364 46 1.4122 0.3024 41 1.4108 10.10 10.87 0.10

Si3N4 632.8 0.0497 60 2.0381 0.0373 46 2.0385 24.86 23.33 �0.02

532 0.1447 112 1.3831 0.1256 94 1.3830 13.16 16.07 0.01

460 0.4889 197 0.9359 0.3764 142 0.9362 23.01 27.92 �0.03

The final residue is set to 1.0e�6 and the unit of CPU time is hour.


wavelength with S polarization case reveals some backscattering while the rest of the case shows minimal backscatteringlight. The result also shows that S polarization incident beam is more sensitive to the surface irregularities (i.e., particlecontaminant) where the total scattering cross-section (TSC) is larger than the P polarization cases. The comparison of theperformance for low refractive index cases is tabulated at Table 3 with CPU time, number of iteration, and TSC values where

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Table 4Comparison of computational time for normal and preconditioned cases for both polarization of Si case

l (nm) Residue Normal (N) Preconditioned (P) (N�P/N)*100 (%)

CPU time ITER TSC (mm2) CPU time ITER TSC (mm2) CPU time ITER TSC

P polarization

632.8 1.0e�03 1.9732 239 0.7634 1.8453 219 0.7766 6.48 8.37 �1.73

1.0e�04 9.2724 1161 0.8094 8.0401 988 0.8104 13.29 14.90 �0.12

1.0e�05 19.8728 2500 0.8155 20.0117 2500 0.8135 �0.70 0.00 0.25

532 1.0e�03 16.5750 970 0.7111 10.7852 589 0.7179 34.93 39.28 �0.96

1.0e�04 42.4532 2500 0.7099 42.9393 2500 0.7095 �1.15 0.00 0.06

1.0e�05 – – – – – – –

460 1.0e�03 129.7136 2500 0.775 132.5608 2500 0.7553 �2.20 0.00 2.54

1.0e�04 – – – – – – – – –

1.0e�05 – – – – – – – – –

S polarization

632.8 1.0e�03 3.6357 449 1.7001 2.1881 252 1.7103 39.82 43.88 �0.60

1.0e�04 5.2982 659 1.7307 5.1707 634 1.7484 2.41 3.79 �1.02

1.0e�05 19.8728 2500 1.7359 20.0811 2500 1.7438 �1.05 0.00 �0.46

532 1.0e�03 22.9771 1351 1.4659 13.2367 714 1.4077 42.39 47.15 3.97

1.0e�04 42.3792 2500 1.5561 43.1477 2500 1.5050 �1.81 0.00 3.28

1.0e�05 – – – – – – –

460 1.0e�03 138.7397 2500 0.9581 19.0634 327 0.9629 86.26 86.92 �0.50

1.0e�04 – – – 143.2744 2500 0.9513 – – –

1.0e�05 – – – – – – – – –

The result is monitored via different target residues for comparison.


the computation is performed until the residue reaches below 1.0e�6. According to results, the diagonal preconditioner canreduce the number of iteration and total computational time upto 27% depending on physical parameters. However, for PSLwith 632.8 cases shows that when the number of iteration is the same and the preconditioner requires more time since thediagonal preconditioning step is designed with extra vector–vector multiplications. The difference of the TSC of both casesincreases with higher refractive index and the maximum is 0.23%.

However, for high refractive index material (Si), the preconditioned iteration solver shows limitation on the convergencecharacteristics. As shown in Table 4, the same results are tabulated with different residue criteria of 1.0e�3 to 1.0e�5. Thecombination of increased number of dipoles and higher refractive index result in longer CPU time to reach the targetresidue. For the Si case, we can only compare the normal and preconditioned case when both achieved the same targetresidue. With this assumption, the diagonal precodnitioner still outperformed the normal case up to 42% in CUP time andthe TSC difference are upto 4% which is worse than the low refractive index cases. When we impose stronger residue bothnormal and preconiditioned cases have difficulty in reaching the target residue as shown for 1.0e�4 and 1.0e�5 for 532 and460 nm cases. The result clearly shows that higher refractive index requires more iteration to converge to the designatedresidue.

4. Conclusion

Analysis of DDA method on surface are conducted specializing in matrix construction and their iteration characteristicsfor QMR method. The analysis shows that depending on the physical input parameter such as the presence of surface,incident angle, polarization, wavelength, and refractive index, their spectral characteristics changes along with a diagonaldominance variation. Among these parameters, the presence of surface and high refractive index deteriorates the spectralcharacteristics thus requires more number of iteration to converge. Based on the analysis, we have suggested a diagonalpreconditioning method which requires minimum computational requirement to execute. The suggested new algorithm isevaluated on real-life semiconductor pattern assuming that a contaminant particle present. The effectiveness of usingdiagonal preconditioning shows that maximum of 27% reduction in computational time for P and S polarization on testedcases for low refractive index. However, the result show limited performance for high refractive index cases.

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Reference

[1] Schmehl R, Nebeker BM, Hirleman ED. Discrete-dipole approximation for scattering by features on surface by means of a two-dimensional fastFourier transform technique. J Opt Soc of Am A 1997;14:3026–36.

[2] Purcell EM, Pennypacker CR. Scattering and absorption of light by nonspherical dielectric grains. Astrophys J 1973;186:705–14.[3] Draine BT, Goodman J. Beyond Clausius–Mossotti: wave propagation on a polarizable point Lattice and the discrete dipole approximation. Astrophys J

1993;405:685–97.[4] Draine BT, Flatau PJ. Discrete-dipole approximation for scattering calculations. J Opt Soc Am A 1994;11:1491–9.[5] Nebeker BM. Modeling of light scattering from features above and below surfaces using the discrete-dipole approximation. PhD thesis, Arizona State

University, USA, 1998.[6] Nebeker BM, Buckner B, Hirleman ED, Lathrop A, Bhunia AK. Identification and characterization of bacteria on surfaces using light scattering. Proc

SPIE 2001;4026:224–34.[7] Bae E, Hirleman ED. Extending the applicability of the discrete dipole approximation for multi-scale features on surface. JQSRT 2007;107:470–8.[8] Rahola J. On the eigenvalues of the volume integral operator of electromagnetic scattering. SIAM J Sci Comput 2000;21:1740–54.[9] Budko NV, Samokhin AB. Spectrum of the volume integral operator of electromagnetic scattering. SIAM J Sci Comput 2006;28:682–700.

[10] Flatau P. Improvements in the discrete-dipole approximation method of computing scattering and absorption. Opt Express 1997;22:1205–7.[11] Fan ZH, Wang DX, Chen RS, Yung EKN. The application of iterative solvers in discrete dipole approximation method for computing electromagnetic

scattering. Micro Opt Techn Let 2006;48:1741–6.[12] Nebeker BM, Starr GW, Hirleman ED. Evaluation of iteration methods used when modeling scattering from features on surfaces using the discrete-

dipole approximation. JQSRT 1998;60:493–500.[13] Saad Y. Iterative methods for sparse linear systems. Soc Ind Appl Math 2000.[14] Kolotilina LY. Explicit preconditioning of systems of linear algebraic equations. J Soviet Math 1988;43:2566–73.[15] Carpentieri B, Duff IS, Giraud L. Magolu monga Made M, CERFACS Technical Report TR/PA/01/35.[16] Lewis DW. Matrix theory. Singapore: World Scientific; 1991.[17] Golub GH, Van Loan CF. Matrix computation. Baltimore, MD: Johns Hopkins University Press; 1996.[18] Golub G, Ortega JM. Scientific computing: an introduction with parallel computing. New York: Academic Press; 1993.[19] Kim HJ, Kim HS, Choi K, Lee HB, Jung HK, Hahn SY. Cost-effective parallel preconditioner for network-based computing. IEEE Trans Mag

1997;33:1800–3.[20] Freund RW, Nachtigal NM. An implementation of the QMR method based on coupled two-term recurrences. SIAM J Sci Stat Comput 1994;15:313–37.[21] Chen FC. A preconditioned quasi-minimal residual method for electrically large random surface scattering of remote sensing. Int J Remote Sens

1999;20:2627–36.

computational analysis and diagonal preconditioning for the discrete dipole approximation on surface

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