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Tbilisi State University (LAE) Method of Auxiliary Sources for Optical Nano Structures K.Tavzarashvili(1), G.Ghvedashvili(1), D. Kakulia(1), D.Karkashadze(1,2), (1)Laboratory of Applied Electrodynamics, Tbilisi State University, Georgia (2) EMCoS, EM Consulting and Software, Ltd, Tbilisi, Georgia, e-mail: [email protected] 5/7/2017 David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011 1 Tbilisi State University (LAE) The Content and Purpose • Conventional interpretation of MAS applied to the solution of electromagnetic scattering and propagation problems. • The general recommendations for the solution of these problems. • An application of the method to specific problems for the single body and a set of bodies of various material filling. • The application areas of MAS, its advantages and benefits 5/7/2017 David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011 2 Tbilisi State University (LAE) Mathematical Background of the Method of Auxiliary Sources (MAS) • • • 1. 2. 3. 4. 5. 6. 7. 8. The name “MAS”, currently used, did not appear at once. The authors themselves adhered to the names: “The Method of Generalized Fourier Series” [1-5] “The Method of Expansion in Terms of Metaharmonic Functions” [6] “The Method of Expansion by Fundamental Solutions [7,8] V.D. Kupradze, M.A. Aleksidze: On one approximate method for solving boundary problems. The BULLETIN of the Georgian Academy of Sciences. 30(1963)5, 529-536 (in Russian). V.D. Kupradze, M.A. Aleksidze: The method of functional equations for approximate solution of some boundary problems. Journal of Appl. Math. and Math. Physics. 4(1964)4, 683-715 (in Russian). V.D. Kupradze, M.A. Aleksidze: The method of functional equations for approximate solution of some boundary problems. Journal of Appl. Math. and Math. Physics. 4(1964)4, 683-715 (in Russian). V.D. Kupradze: Potential methods in the theory of elasticity. Fizmatizdat, Moscow 1963, 1-472 (in Russian, English translation available, reprinted in Jerusalem, 1965). V.D. Kupradze: On the one method of approximate solution of the boundary problems of mathematical physics. Journal of Appl. Math. and Math. Physics. 4(1964) 6, 1118 (in Russian). I.N. Vekua: On the completeness of the system of metaharmonic functions. Reports of the Acad. of Sciences of USSR. 90(1953)5, 715-717 (in Russian). V.D. Kupradze, T.G. Gegelia, M.O. Bashaleishvili, T.V. Burchuladze: Three dimensional problems of the theory of elastisity. Nauka, Moscow 1976, 1-664) (in Russian). M.A. Aleksidze: Fundamental functions in approximate solutions of the boundary problems, Nauka, Moscow 1991, 1-352 (in Russian). “A common idea of these works is a basic theorem of completeness in L2(S) … of infinite set of particular solutions, generated by the chosen fundamental or other singular solutions… ”[7] of wave equation (D.K.) 5/7/2017 David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011 3 Tbilisi State University (LAE) Electromagnetic Scattering and Propagation as the Boundary Problem The main goal of problem is to find vectors of secondary electromagnetic field in each bounded, simply connected domains m mM0 , Em , Bm , Dm , H m M m 0 confining with the set of smooth, closed surfaces S mn M (interfaces between neighbouring m and n domains), while the primary field m ,n 0 E 0 , H 0 , D 0 , B 0 is given. In corresponding domain secondary or primary electromagnetic field should satisfy: a) Maxwell’s equation; ˆ ( Em , Bm ) ; b) some type of constitutive relation among field vectors – Dm Fˆ ( Em , Bm ), H m c) boundary conditions Lˆmn ( Em , H m , En , H n ) 0 on S mn M m ,n 0 surfaces; d) in unbounded free space Γ0 field vectors should satisfy the radiation condition. 5/7/2017 David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011 4 Tbilisi State University (LAE) Method of Auxiliary Sources (MAS) (simple 2D case) ΔE scat z ( x, y ) k E 2 scat z ( x, y ) 0, P( x, y ) 0 r 0 n n 1 , r n0 S 0 2) Fundamental solutions of Helmholtz equation: Ezscat ( xs , ys ) Ezinc ( xs , ys ) 0 n(r rn0 ) k 2 n(r rn0 ) (r rn0 ) y ( E inc , H inc ) 1) Everywhere dense points on the surface S0 - n(kRn0 ) ( E scat , H scat ) i (1) H 0 (kRn0 ), Rn0 ( x xn0 ) 2 ( y yn0 ) 2 4 3) Construction of the set of fundamental solutions of wave equation with radiation centers on surface S0: 0 n ( rs , rn ) n 1 W n n1 n (rs , rn0 )n1 x Γ Scatterer - S Γ0 Auxiliary surface – S0 Theorem: It can be shown [*], that for an arbitrary smooth surface S (in the Lyapunov sense) one can always find the auxiliary surface S0 such that the constructed set of functions is complete and linearly independent on S in the functional space L2. *M.A. Aleksidze: “Fundamental functions in approximate solutions of the boundary problems”. Nauka, Moscow 1991, 1-352 (in Russian). 5/7/2017 David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011 5 r v M m m 1 V ; Rm = rs rmv Tbilisi State University (LAE) Method of Auxiliary Sources (MAS) (simple 2D case) Any continuous function on S can be expanded in terms of the first N functions of the given set of fundamental solutions: N E zscat ( xs , y s ) an H 0(1) k ( xs xn0 ) 2 ( ys yn0 ) 2 n 1 Properties of this set of fundamental solutions guarantee existence of corresponding coefficients providing the best in L2(S) mean-square approximation of any continuous function on S: (xs ,ys) ( xn0 , yn0 ) 5/7/2017 Corresponding discrepancy: S an nN1 N ( ) E inc ( x , y ) a H (1) k ( x x 0 ) 2 ( y y 0 ) 2 s s n 0 s n s n S z n 1 2 inc E ( x , y ) dS z s s S when N ( ) , then 0 David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011 2 1 2 dS ; 6 Tbilisi State University (LAE) Computational Procedures for Determination MAS Unknown Coefficients • • • • 5/7/2017 Gram-Schmidt Orthogonalization approach; MAS-MoM-Galerkin approach; Method of Colocation; ….. David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011 7 Tbilisi State University (LAE) Gram-Schmidt Orthogonalization Approach M Gm ( Rm ) m m ( rs )m 1 , L m ( Rm ) = I ( Rn ) M m 1 Orthogonalization procedure 1 ( s ) G1 ( s ), n=1 2 ( s ) G2 ( s ) 1 ( s ) A21 , From Fourier Theorem - The best expansion (in L2) of given vector function H(xs,ys,zs) on surface S: 3 ( s ) G3 ( s ) 1 ( s ) A31 2 ( s ) A32 , . . . . M H ( xs , ys , zs ) m ( s ) bm m 1 m ( s ) Gm ( s ) k ( s ) Amk ; m 1 H ( s) s m 1 k 1 2 M m . . . . ( s ) bm ds min( L2 ) Scalar product definition: 0, m k T T ds m k s m k Fm , m k n 1 Amn F Cnm AnkT Fk Amk , ( Amn 0, m n ); k 1 1 n G T m S Gk ds Cmk , G T k Gk ds Fk ; S bq Fq1 q H V.D. Kupradze: “On the one method of approximate solution of the boundary problems of mathematical physics”. Journal of Appl. Math. and Math. Physics. 4(1964) 6, 1118 (in Russian). 5/7/2017 David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011 8 Tbilisi State University (LAE) MAS-MoM-Galerkin Approach Tn n 3 ln nn Tn 4 n n 2 rs rn n O 1 MoM-like representation of current on the auxiliary surface ln 2 A n , N n J (rs ) I n f n rs , f n (rs ) ln n 1 2 A n , n 0 , Triangulated auxiliary surface Triangulated surface of the scatterer if rs Tn if rs Tn if rs Tn Current basis function and testing function 5/7/2017 David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011 9 Tbilisi State University (LAE) MAS+Collocation Approach (simple 2D case) r 0 N n n 1 , r n0 S 0 N a Z n 1 n nm rm Mm1 , rm S Vm , (m 1,2,..., M ) Z nm H 0(1) k ( xm xn0 ) 2 ( ym yn0 ) 2 Log10(ε) -0.30 -2.24 -4.18 Vm E ( xm , y m ) inc z -6.12 (xm ,ym) ( xn0 , yn0 ) n=3 4 -8.06 5 6 7 kd -10.00 0.50 1.80 3.10 4.40 5.70 7.00 Estimation of accuracy of solution d 2 E S inc z N ( xs , y s ) an H n 1 (1) 0 E k inc z ( xs x ) ( y s y ) 0 2 n 2 0 2 n 2 dS ; ( xs , y s ) dS S 5/7/2017 David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011 10 Tbilisi State University (LAE) Method of Auxiliary Sources (problems) Problem 1: unstability of linear system of algebraic equation -0.21 Colocation points (xm ,ym) Log10(ε) -0.30 b) a) n -0.87 -2.24 5 6 7 Auxiliary Sources ( xn0 , yn0 ) Log10(ε) -1.53 -4.18 -2.19 -6.12 -2.85 -8.06 n=3 d n k0 -3.51 0.50 kd 1.80 3.10 4.40 5.70 7.00 an Z nm Vm , (m 1,2,..., M ) n 1 (a n 1 n 1.80 kd 3.10 4.40 5.70 7.00 Accuracy of solutions versus relative distance kd N N -10.00 0.50 4 5 6 7 an ) ( Z nm Z nm ) Vm Vm N m am an n 1 Z nm Vm , (m 1,2,..., M ) Z11 Z11 Tikhonov-Arsenin → αm= 10-8 ÷10-10 αm - regularization parameter Hadamard : for Z nm 1012 and Vm 1012 an an A.N. Tikhonov, V.Ya. Arsenin: “The methods for solving non-correct problems”. Nauka, Moscow 1986, 1-288 5/7/2017 David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011 11 Tbilisi State University (LAE) Method of Auxiliary Sources (problems) Problem 2: scattered fields main singularities (SFMS) E tot E inc E scat E tot H tot 0 From Uniqueness Theorem The regular, in whole space, solution of Maxwell’s equation satisfying the radiation condition at infinity should be identically zero(*). PEC Scattered wave fields (both scalar and vector), which are continuously extended inside the scatterer's domain certainly has irregular points (singularities - SFMS). d q h q R Auxiliari charges (unstable) Auxiliari charges (stabile) (r , ) 1 q q 2 2 2 40 r d 2rd cos r ( R d ) 2r ( R d ) cos d R2 Rq ; q Rh Rh V.D. Kupradze(*): “The main problems in the mathematical theory of diffraction”. GROL. Leningrad, Moscow 1935,1-112. 5/7/2017 David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011 12 Tbilisi State University (LAE) Method of Auxiliary Sources (problems) • Problem 2: scattered fields main singularities (SFMS - conformal mapping procedure) 1.08 0.65 y 1.270.54 y y D1 L2 0.760.22 D1 0.22 0.25-0.11 -0.22 -0.25-0.43 -0.76-0.76 L1 S -1.08 -2.16 L2 L2 D2 -0.65 L1 -1.29 -0.43 0.43 1.29 x 2.16 D2 S -1.27-1.08 -1.38 -2.16 -0.83 -1.53 -0.29 -0.89 0.26 -0.26 0.81 0.37 x x 1.001.36 Image lines of primary source - L1 and L2; “Auxiliary Sources” ( monopoles) surrounding the areas of SFMS concentration, imitating the radiation from the images L1 and L2; The lines of other possible distributions of auxiliary sources; Some optimal distribution of collocation points on the main surface S. D.Karkashadze: "On Status of Main Singularities in 3D Scattering Problems". Proceedings of VIth International Seminar/Workshop on Direct and Inverse Problems of Electromagnetic and Acoustic Wave Theory (DIPED-2001), Lviv, Ukraine, September 18-20, 2001, pp. 81-84. http://www.ewh.ieee.org 5/7/2017 David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011 13 Tbilisi State University (LAE) Method of Auxiliary Sources (problems) • Problem 3: Most general, linear form of constitutive relation (Bi-Isotropic medium) Maxwell’s equations I . E iB; II . B 0; III . H iD; IV . D 0; Constitutive Relations 1 D E iB, H iE B Isotropic magnetodielectrics: Chiral medium: Tellegen medium: = = 0; = 0; =- 0; Wave equation any field 2U k 2U ( ) U 0, U any fildvectors vectors Material parameters Wave impedances r r, k r, ˆ , , r, r k r, r, r, ˆ , r ˆ , 0 r, 0 , 0 2 0 2 ( ) 4 2 1 , r r Wave numbers k ˆ k 0 r • • 5/7/2017 0 k 2 ( )2 r, , k k 4 2 , k k0 r r A.Lakhtakia: “Beltrami fields in chiral media”. World Sci. Publ. Co., Singapore 1994, 1-536 I.V.Lindell, A.H.Sihvola, A.A.Tretyakov, and A.J.Viitanen: “Electromagnetic waves in chiral and BiIsotropic media”. Artech House, Boston, London 1994, 1-332.. David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011 14 Tbilisi State University (LAE) Method of Auxiliary Sources (problems) • Problem 3: Fundamental Solution for Bi-Isotropic medium Spinor basis r v i u v i u l Spinor of electromagnetic field F r E iη r H F F E iη H Maxwell’s equation in the Majorana-Dirak form Fundamental Solution for Bi-Isotropic medium ˆ 0 F kF 0 Gn Green function Green function matrix Gnr ˆ Gn 0 Fn r , Gˆ n n 1r , Gˆ n n kˆ Gˆ n n k Gn r , r, n G ρ,ρ n H 0(1)(k r , n ) - 2D case r , rn 1 4 ik r , r rn e r rn - 3D case R.Penrose, W.Ringler: “Spinors and space-time”, vol. 1. Cambridge University Press, Cambridge (eng.), 1986 5/7/2017 David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011 15 Tbilisi State University (LAE) General Concept of MAS Problem formulation for most general form of constitutive relations Definition of fundamental or other singular solutions of the wave equation Geometry Analyzing for Scattered Field Main Singularities (SFMS) Finding the best placement and type of the fundamental solutions (AS) Definition of numerical method for evaluating the amplitudes of the auxiliary sources Method of collocation, method of moments (MOM), ortogonalization Deriving the system of linear algebraic equations Post processing stage 5/7/2017 Computation of amplitudes of AS Data processing and visualization David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011 16 Tbilisi State University (LAE) MAS Applicatin to Some Particular Problems • MAS approach for electromagnetic scattering on the Bi-Isotropic bodies; • MAS simulation of wave propogation in Double-Negative medium; • MAS and MMP simulations of Finite Photonic Crystal (PhC) based devices; • MAS approach for electromagnetic scattering on Double-Periodic structures; • MAS Approach for Band Structure Calculation and eigenvalue search problem; 5/7/2017 David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011 17 Tbilisi State University (LAE) Electromagnetic Scattering Upon the Chiral Bodies F.G. Bogdanov, D.D. Karkashadze, R.S. Zaridze. “The Method of Auxiliary Sources in Electromagnetic Scattering Problems”. North-Holland, Mechanics and Mathematical Methods, A Series of Handbooks. First Series: Computational Methods in Mechanics. Vol.4, Generalized Multipole Techniques for Electromagnetic and Light Scattering, Chapter 7, pp. 143-172, 1999. Fundamental solutin: Boundary problem for spinor field: ˆ 0 F kF Wˆ F(r ) s = f(r s ), M (r s ) S r r Required solution N N (N) anr 0 F (r ) aˆ n Fn (r , rn ), r D, aˆ n 0 an n 1 n 1 r , ˆ ˆ 1 ˆ ˆ Fn Gn n r , Gn n k Gn n ; k r ik r r G v i u 0 n , G r , r r 1 e Gˆ n , 0 G i 4 r r r r , ˆk k 0 r n 0 k v 2 ( )2 r, , k k0 , k k 4 2 u r r Scattered electric field reconstracted from spinor fields E sc (r ) Escr Esc r N N r r an Fn (r , rn ) an Fn (r , rn ) r r n1 n 1 Spherical shape chirolens Constitutive relations E ztot Ez Ezr D E iB, H iE 1B 0.3 , 3.0, 1.389, 0 120 0 k 0 400, heght d 0.6, thickness t 0.125 x 5/7/2017 David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011 18 Tbilisi State University (LAE) Plane Wave Excitation on Ciral Sphere Log a2 a 0.2 mm; 0 (5.0 i 0.1); ( 0 2 / ) 1 ; 110.8 nr Im(n) 77.9 44.9 12.0 -20.9 -53.9 150 3.22 fGHz GHz Re(n) 180 210 GHz 240 270 300 n 2.58 Re(n) Scattering cross-section versus frequency. MAS approach: γ=0; γ=0.0001; T-matrix approach: γ=0.0001 • • 1.94 1.30 0.66 A.Lakhtakia, V.K.Varadan, V.V.Varadan: “Scattering and absorption Im(n) characteristics of lossy dielectric, chiral, nonspherical objects”. Appl. Opt. 0.02 GHz 300 150 180 210 240 270 24(1985) 23, 4146-4154. F.G.Bogdanov and D.D.Karkashadze: “Conventional MAS in the problems of Refractive index versus frequancy for Right-hand and Left-hand components electromagnetic scattering by the bodies of complex materials”. Proc. of the 3-rd Workshop on Electromagnetic and Light Scattering, Bremen 1998,133-140. 5/7/2017 David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011 19 Tbilisi State University (LAE) Kirchhoff-Kotler formula + MAS for Field Reconstruction in DoubleNegative Medium (wave front reversal approach) k k0 1 1 y r =1; r=1 r =−1; r=−1 y r =1; r=1 r =−1; r=−1 x x Gaussian beam λ λ Gaussian beam illuminated transparent object The total electric field Ez component reconstruction from space x<0 to spece x>0. Total electric field Ez component was reconstructed from EM field tangential components on the YOZ plane. Electric field distribution from a Gaussian beam source left x<0 (actual) and MAS predicted electric field distribution in a virtual DNM half space right x>0 (reconstructed from EM field tangential components on YOZ plane). David Karkashadze, Juan Pablo Fernandez, and Fridon Shubitidze: “Scatterer localization using a left-handed medium”. OPTICS EXPRESS 9906, (C) 2009 OSA, 8 June 2009 / Vol. 17, No. 12 5/7/2017 David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011 20 Tbilisi State University (LAE) MAS and Multiple Multipole Method (MMP) Simulations of Finite Photonic Crystal (PhC) Based Devices • E. Moreno, D. Erni, Ch. Hafner: “Modeling of discontinuities in photonic crystal waveguides with the multiple multipole method”. Phys. Rev. E 66, 036618, 2002 • D. Karkashadze, R. Zaridze, A. Bijamov, Ch. Hafner, J. Smajic, D. Erni: “Reflection compensation scheme for the efficient and accurate computation of waveguide discontinuities in photonic crystals”. Applied Computational Electromagnetics Society Journal, Vol. 19, No. 1a, March 2004, pp. 10-21 • D. Karkashadze, R. Zaridze, A. Bijamov, Ch. Hafner, J. Smajic, D. Erni: “MAS and MMP Simulations of Photonic Crystal Devices”. Extended Papers of Progress In Electromagnetic Research Symposium (PIERS-2004). March 28-31, 2004, Pisa, Italy. pp. 29-32 5/7/2017 David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011 21 Tbilisi State University (LAE) “Filtering T-junction” Design (MMP, MAS) IWGAInput DFDI DFDI IWGARight IWGALeft TM-polarization. Lattice constant a=1μm , base rods permittivity ε=11.4, base rods radii r=0.18a f1=1.038 1014Hz Comparison (without optimization) MMP simulation Case MAS simulation R (%) Tl (%) Tr (%) (%) R (%) Tl (%) Tr (%) (%) Left 35.37 63.38 0.41 99.16 36.38 63.71 0.42 100.51 Right 36.51 0.11 63.24 99.86 36.02 0.11 63.76 99.89 Results of Filtering T_Junction optimization f2=1.230 1014Hz 5/7/2017 f1=1.038 1014 Hz f1=1.230 1014 Hz Rup=0.73% Rup=0.76% Tright=0.16% Tright=97.77% Tleft=98.59% Tleft=0.88% David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011 22 Tbilisi State University (LAE) Coupling a Slab with a PhC Waveguide: fa/c=0.38 (MMP, MAS) Optimizing Inclusion ro=0.15R Crossing waveguide operating with different frequency in each channels Before optimization: SWR_WGMAS= 1.72; SWR_WGMMP=1.66 After optimization : SWR_WG = 1.08 hD_WG = 3.379; hPhC_WG = 1.644; 5/7/2017 David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011 23 Tbilisi State University (LAE) 3D Double-Periodic Green Functions 2 2 2 exp ik x nd y nd z x y x, y, z exp(iknd x cos x ikmd y cos y ) 2 2 n m ik x nd x y nd y z 2 x, y , z 2 dxd y k xq k cos x k zqp Poisson Transformation 1 exp(ik xq x ik yp ik zpq z ) ik pq p q ik z zpq 2 2 q, k yp k cos y p, h p dx dy h 2 k 2 , if h k and k k p xq yp p xq , 2 2 i k xq hp ,if hp k xq and k k yp 2 k 2 k yp , p 0, 1, 2,... , q 0, 1, 2,... k zqp i hp2 k xq2 , if k k yp D. Kakulia, K. Tavzarashvili, G. Ghvedashvili, D. Karkashadze, and Ch. Hafner: “The Method of Auxiliary Sources Approach to Modeling of Electromagnetic Field Scattering on Two-Dimensional Periodic Structures”. Journal of Computational and Theoretical Nanoscience, Vol. 8, 1–10, 2011 5/7/2017 David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011 24 Tbilisi State University (LAE) Oblique Incident of Plane Wave on Bi-Periodic Array of Dielectric Spheres 1 Reflection coefficient Г z ES EP k y 2a d x d 0.8 20 0 0.6 s-pol. 0.4 p-pol. 0.2 0 0.6 0.7 0.8 0.9 1 Relative period d/λ a) Problem geometry (radius of spheres - a, period – d, incident angle θ=20º, φ=0º, permittivity of dielectric spheres ε=3.0, a/d=0.4); b) Transmission coefficient versus relative period. The solid curve is for p-polarization and the dotted curve is for s-polarization. Comparison of MAS approach with results presented in • M. Inoue: “Enhancement of local field by two-dimensional array of dielectric spheres placed on the substrate”. PHYSICAL REWIEV B, vol. 36 #5, 15 August 1987, p 2852-2862. 5/7/2017 David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011 25 Tbilisi State University (LAE) The Test for Convergence of MAS Results 1.0 Transmission Coefficient Т E 0.8 H N=2352 N=1200 0.6 k 0.4 X direction 0.2 a) 0.0 0.6 b) 0.7 0.8 0.9 Relative period dx / λ 1.0 Dielectric layer with hexagonally bi-periodic dents: a) problem geometry and incidents plane wave orientation; b) transmission coefficient versus relative period. 5/7/2017 David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011 26 Tbilisi State University (LAE) MAS and MMP Approach for Band Structure Calculation y Non periodic expansions dx D0 Sout dy D1 1 , 1 x k? Sin Periodic expansions Unit cell Ez 2D bi-periodic structure with arbitrary shaped dielectric scatterers and unit cell with distribution of auxiliary sources (left); Periodic Green’s functions for different angles14of incidence a) =00, b) =450 (right). x 10 1 5 4.5 0.8 4 3.5 0.6 fa/c fa/c 3 0.4 2.5 ky 2 Μ 1.5 0.2 1 TM TE 0 x*dx Γ Χ kx 0.5 0 2/a a Band structure for a perfect PhC made of dielectric rods (left); Band structure for a perfect PhC made of silver wires (right). K. Tavzarashvili, Ch. Hafner, Cui Xudong, Ruediger Vahldieck, D. Kakulia, G. Ghvedashvili and D. Karkashadze, “Method of Auxiliary Sources nd Model-Based Parameter Estimation for the Computation of Periodic Structures”, Journal of Computational and Theoretical Nanoscience Vol.4, 1–8, 2007 5/7/2017 David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011 27 Tbilisi State University (LAE) Conclusion The procedure of solution of scattering problems by the method of auxiliary sources (MAS) needs preliminary considerations of various problems. Correct solution of these problems can radically influence efficiency of MAS. 5/7/2017 David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011 28 Tbilisi State University (LAE) Thank you for attention [email protected] 5/7/2017 David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011 29