A Combination of BIM and BEM for efficiently analyzing optical. elements

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1 A Combination of BIM and BEM for effiiently analyzing optial elements SUN Fang, LIU Juan, ZHANG Guo-ting, HU Chuan-fei, SU Xiaoxing Institute of Lightwave Tehnology, hool of eletronis and information engineering, Beijing Jiaotong University, Beijing, 00044, P. R. China. ABSTRACT We propose the revised boundary integral method (RBIM) that ombines the boundary integral method (BIM) and the boundary element method (BEM) together. It is well nown that the boundary integral equations are ast into matrix form for ease of omputer implementation, and the points on the diagonal line of the matrix present the superposition of the observation and the soure points. The points are alled singularity points whih an ause the big error bar. Thus, we onsider replaing the BIM by the BEM at the diagonal line, omparing the numerial results by using the RBIM, the BIM, the BEM, and the analytial method, and find the error bar aused by the RBIM is smaller than that of the BIM. It indiates that the RBIM is not only faster than the BEM, but also it is preiser than BIM. Key words: boundary integral method (BIM), boundary element method (BEM), revised boundary integral method (RBIM). INTRODUCTION In lassial diffration theory, various alar integral methods have been used widely to analyze the properties of traditional optial elements whose feature sizes are always muh larger than the wavelength of light and to alulate the diffrated fields of the diffrative struture [, ]. However, with the development of refinements in mirofabriation methods suh as mirophotolithography, optoeletroni devies, and biologial sensors and so on, various optial elements whose feature sizes are always smaller than the wavelength of light are fabriated. In this ase, the alar integral methods beome inappliability beause of the prominent attering effets [3, 4]. To overome this limitation, people onsider the vetor-based analysis of nano-optial elements by using rigorous boundary integral method (BIM) [5]. It is well nown that BIM requires less omputational effort, alulate fast and are onsequently the preferred integral approah. However, omparing with the boundary element method (BEM), we find the preision of the BIM is inferior. In order to obtain a faster and preiser method to analyze the optial elements, we ompare the numerial results alulated by the BIM and the BEM, analyze the reason of impreision BIM, and find the error bar produed by the diagonal values of the matrix in BIM. It is found that the error bar of the BIM is muh higher than that of BEM. Based on the analysis, we introdue a new method whih ombines the BIM and the BEM together. In this presentation, we name the ombined method as revised boundary integral method (RBIM). Thus, The RBIM is used the BEM to replae the BIM at the diagonal line to ahieve high auray. In this presentation, we present the numerial analysis of a two-dimensional attering system, ompare the numerial results by using the BIM, the BEM, the RBIM and the analytial method, and find the error bar aused by the RBIM smaller than the BIM. It indiates that the RBIM is not International Symposium on Photoeletroni Detetion and Imaging 007: Optoeletroni System Design, Manufaturing, and Testing, edited by Liwei Zhou Pro. of SPIE Vol. 664, 664, (008) X/08/$8 doi: 0.7/.790 Pro. of SPIE Vol Downloaded From: on 0/8/06 Ter of Use:

2 only faster than the BEM, but also it is preiser than BIM. It is believed that the RBIM is a useful method for effiiently analyzing optial elements.. THE REVISED BOUNDARY INTEGRAL METHODS (RBIM) Regioni ( C Regi( s,) Fig..The hemati view of the two-dimentional attering system In this presentation, we assume a arbitrary attering struture in Fig.. we ompute the intensity distribution of magneti field of the points on the boundary of the attering with the BIM, the BEM and the RBIM to ompare their error bar. In order to introdue the RBIM, we must now the BIM at first. The Two-dimensional attering problem onsisting of a atter with its surfae ontour C and its outward normal n uv is displayed in Fig.. The wave equation read as follows: [6] () r ( r) 0, r V ( ) ( ) ( ) + = r + r = f r, r V () Where i ( π ni)/ λ0 spae. V and = (i =,) is wave number in region i. r represents position vetor of a random point in the V denote region and region, respetively. ( r) stands for the E () r for the inoming field of TE polarized and H ( r ) for the TM polarized (in this present, we apply TM polarized).the boundary integral equation an z be derived by applying Green s Seond Identity, whih an be written as follows: [6] Where ( ) i i 0 i i ( r ) G ( r, r ) ε ( r) = [ (, ) ( ), G r r r ] dl r V n n ( r) G ( r, r) in ( r) = [ G( r, r) ( r) ] dl+ ( r), r V n n ε uv uv G r, r = H ( r r ) is Green funtion, and z H is the zero-order Hanel funtion of the seond ind. 0 () r denotes the observation point on the boundary C. in ( r ) stands the inident field. Pro. of SPIE Vol Downloaded From: on 0/8/06 Ter of Use:

3 Beause the resonane is very big, when the wavelength of inident light is almost equal with the size of the struturem, the error bar may be very large,. Now we an redue the resonane error by the way of mae the al field dissoiate illumination field and attering field. The equation () an be written as: θ ( r ) G ( r, r ) ( ) ε (, ) π ( r) ( ) G r r r dl n n in θ in ( r) in G ( r, r) ( r) ε G r r ( r) dl r V π n n ( r) G ( r, r) + ( ) 0 dl = r V n n = ( ) (, ) (3) θ ( r ) ε G ( r, r ) r π Then we now the diretization of equation (3) as: θ ( ) G () mn m () mn n () m εg n ln π n= n n n m θ G = ( ) G l m=,,.. (4) θm l = 0, m=,,.. π in() mn m in() mn n in() m ε n n π n= n n n m () mn in() mn n () G m + εg n n n= n n n m Where denotes the number of the points on the boundary, m denotes the other points on the boundary. The ontour C is divided into piees and l denotes the length of eah piee. n In order to revise the BIM, we an give the equation by the method of BEM whih uses interpretation method at first, then we an use the value on the diagonal line of the matrix produed by the method of BEM replae the value by the BIM. The equation (5) an be written as: Pro. of SPIE Vol Downloaded From: on 0/8/06 Ter of Use:

4 () n () () m m () m () [ ] G [ ] m N N N3 ε G Jdξ N N N3 Jdξ π n = = n () () + + n in() in() in() m θ m in() m G in() ( ) m { [ N 3] [ 3] π ε ξ ξ n = = n in() in() + + n () n () () m m in() m G () m ε ξ ξ π n = = n () () + + n θ ( ) {,,,, } = θ n, N, N G J d N, N, N J d } m =,,..+ [ N N N3] G J d [ N N N3] J d m,, {,,,, } = 0, = Where N, N, N 3 denote primary funtion of two orders interpolating funtion. N N N 3 (5) = ξ ( ξ )/ = ξ There are = ξξ ( + )/ piees on the boundary and denotes the points in the piee. Eah piee inludes three points, so there are + points. 3. NUMERICAL SIMULATIONS AND RESULTS In this presentation, we analyze the error bar and estimate the alulation time by the BIM, the BEM and the RBIM. In order to mae the ompute easy, we adopt the struture of the irular ylinder. The parameters of the struture are as follows: r=000nm, λ =000nm. The materials in the Region and Region are air ( ε = ε =.0 ). At first, we ompare the field of the error bar on the bond of the irle between numerial results by the three inds of methods and analytial results in Fig.. (a) the amplitude of the points on the bond of the irle by the method of BIM. (b) the amplitude of the points on the bond of the irle by the method of BEM. Then we ompare derivation of the field of the error bar on the bond of the irle between numerial results by the three inds of the methods and analytial results in Fig.3. (a)the amplitude of the points on the bond of the irle by the method of BIM. (b) the amplitude of the points on the bond of the irle by the method of BEM. In Fig.3.we find the error bar aused by the method of BIM is larger than the method of BEM. Furthermore, we ompare the field of the error bar outside the irle between analytial results and numerial results by the BIM method and the RBIM mehod in Fig.4.(a),(b), respetively. We ompare the field of the error bar in the irle between analytial results and numerial results by the BIM method and the RBIM method in Fig.5.(a),(b), respetively. And in Fig.4.and in Fig.5.we find that the error bars aused by the BIM are larger than the BEM. However, the time is taen by the BEM is longer than the BIM. In order to obtain a faster and preise method to analyze the optial elements, we onsider introduing a new method (revised boundary integral method, RBIM) whih ombines the BIM and the BEM together. Moreover, we ompare the error bar between numerial results with the RBEM and analytial results in Fig.. () the field amplitude of the points on the bond of the irle by the method of RBIM. And in Fig.3.() the derivation amplitude of the points on the bond of the irle by the method of RBIM. It Pro. of SPIE Vol Downloaded From: on 0/8/06 Ter of Use:

5 indiates that the error bar of the derivation field on the bond aused by the BIM is big, beause the bla and magenta urve isn t oherene. But the urves in the Fig.3.(b), () produed by the BEM, RBIM, respetively, oherent with the urves produed by the analytial value though there is onussion. At the same time, Based on the analysis of Fig.4.we find the error bar aused by the RBIM is about %, but the error bar aused by the BIM is about 8%. In Fig.5.we find the error bar aused by the RBIM is smaller than the BIM. In addition, we analyze the time of the three methods taen, and we find the BEM is longer than the BIM and RBIM. Also, the time of the RBIM is almost equal with the BIM. So the RBIM is not only faster than the BEM, but also it is preiser than BIM. epfliidu? Opfl!IdLU (RBIM) () Fig.. The amplitude distributions on the boundary obtained by the BIM, the BEM, the RBIM, and the analytial method (a) the amplitude distribution by the BIM. (b) the amplitude distribution by the BEM. () the amplitude distribution by the RBIM. Blue and red stand for the real part of analytial value and alulation value, respetively. Bla and magenta stand for the image part of analytial value and alulation value, respetively. Pro. of SPIE Vol Downloaded From: on 0/8/06 Ter of Use:

6 (DIM) o 'CO '93 SO the sampling points (B EM) the sampling points Fig.3. The derivation values on the boundary by the BIM, the BEM, the RBIM, and the analytial method (a)the derivation amplitude distribution by the BIM. (b) the derivation amplitude distribution by the BEM. () the derivation amplitude distribution by the RBIM. Blue and red stand for the real part of analytial value and alulation value, respetively. Moreover, bla and magenta stand for the image part of analytial and alulation solution, respetively (DIM) the sampling points (R3) 00 lou 00 the sampling points (b) Fig.4. The amplitude distribution ( r ) alulated by the BIM, the RBIM, and the analytial method outside the irle (a) the amplitude distribution by the BIM (b) the amplitude distribution by the RBIM. Blue line and red line stand for the alulated and analytial results, respetively. Pro. of SPIE Vol Downloaded From: on 0/8/06 Ter of Use:

7 (RIM) o so the saniplino points (a) o 30 (RBIM) 0 3 U the saniplino points (b) Fig.5. The amplitude distribution ( r ) alulated by the BIM, the RBIM, and the analytial method in the irle (a) the amplitude distribution by the BIM (b) the amplitude distribution by the RBIM. Blue line and red line stand for the alulated and analytial results, respetively. 4. CONCLUSIONS In summary, we introdue a revised boundary integral method (RBIM) that ombines the BIM and the BEM together. We now the boundary integral equations are ast into matrix form for ease of omputer implementation, and the points on the diagonal line of the matrix present the superposition of the observation and the soure points. The points are alled singularity points whih an ause the big error bar. Thus, we onsider using the BEM to replae the BIM at the diagonal line. In this presentation, we ompare the numerial results alulated by the RBIM with that of the analysis results, and find the error bar aused by the RBIM is smaller than the BIM. It indiates that the RBIM is not only faster than the BEM, but also it is preiser than BIM. We believe that the RBIM is a useful method for effiiently analyzing and design optial elements in the future. ACKNOWLEDGEMENTS This wor was supported by the National Basi Researh Program of China (Grant No. 006CB3090), and the foundation of Beijing Jiaotong University (Grant No. 006XM005). REFERENCES. A. Ishimaru, Eletromagneti Wave Propagation, Radiation, and Sattering (Prentie-Hall, Englewood Cliffs, N.J. Pro. of SPIE Vol Downloaded From: on 0/8/06 Ter of Use:

8 99), Chap. 6.. C. J. Bouwamp, Diffration theory Rep. Prog. Phys. 7 (954), page D. A. Pommet, M. G. Moharam and E. B. Grann Limits of Salar Diffration Theory for Diffrative Phase Elements J. Opt. So. Am. A. (994), page J. N. Mait. Understanding Diffrative Opti Design in the Salar Domain J. Opt. So. Am. A. (995), page Dennis W.Prather,Mar S.Mirotzni and Joseph N.Mait Boundary integral methods applied to the analysis of diffrative optial elements J. Opt. So. Am. A. (997), page Nagayoshi Morita et al., Integral equation methods for eletromagnetis (99), page Pro. of SPIE Vol Downloaded From: on 0/8/06 Ter of Use: