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Method of Auxiliary Sources for Optical Nano Structures

This workshop presentation explores the Method of Auxiliary Sources (MAS) applied in solving electromagnetic scattering and propagation problems for optical nano structures. It discusses the mathematical background, application areas, and advantages of MAS, along with computational procedures. The presentation also delves into the concept of auxiliary surfaces and the construction of fundamental solutions for wave equations. Emphasizing the completeness and linear independence of the functions in L2 functional space, it demonstrates how continuous functions on surfaces can be expanded using MAS for precise approximations, ensuring the best mean-square approximation. Theoretical frameworks and practical applications of MAS are examined in the context of solving boundary problems for electromagnetic fields in complex domains.

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Method of Auxiliary Sources for Optical Nano Structures

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  1. 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: david.karkashadze@emcos.ge David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011

  2. 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 David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011

  3. Mathematical Background of the Method of Auxiliary Sources (MAS) 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 chosenfundamental or other singular solutions… ”[7] of wave equation (D.K.) David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011

  4. Electromagnetic Scattering and Propagation as the Boundary Problem The main goal of problem isto find vectors of secondary electromagnetic field in each bounded, simply connected domains confining with the set of smooth, closed surfaces (interfaces between neighbouring m and n domains), while the primary field is given. In corresponding domain secondary or primary electromagnetic field shouldsatisfy: a) Maxwell’s equation; b) some type of constitutive relation among field vectors – c) boundary conditions on surfaces; d) in unbounded free space Γ0 field vectors should satisfy the radiation condition. David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011

  5. y x Γ Γ0 Auxiliary surface – S0 Scatterer - S Method of Auxiliary Sources (MAS) (simple 2D case) 1) Everywhere dense points on the surface S0 - 2) Fundamental solutions of Helmholtz equation: 3) Construction of the set of fundamental solutions of wave equation with radiation centers on 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). David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011

  6. (xs ,ys) S 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: 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: Corresponding discrepancy: David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011

  7. Computational Procedures for Determination MAS Unknown Coefficients • Gram-SchmidtOrthogonalizationapproach; • MAS-MoM-Galerkin approach; • Method of Colocation; • ….. David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011

  8. Gram-SchmidtOrthogonalization Approach Orthogonalization procedure From Fourier Theorem - The best expansion (in L2) of given vector function H(xs,ys,zs) on surface S: Scalar product definition: 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). David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011

  9. O 1 2 3 4 MAS-MoM-Galerkin Approach MoM-like representation of current on the auxiliary surface Triangulated auxiliary surface Triangulated surface of the scatterer Current basis function and testing function David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011

  10. Log10(ε) -0.30 (xm ,ym) -2.24 -4.18 -6.12 nl=3 4 -8.06 5 6 7 kd -10.00 0.50 1.80 3.10 4.40 5.70 7.00 d MAS+Collocation Approach (simple 2D case) Estimation of accuracy of solution David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011

  11. Colocation points (xm ,ym) Auxiliary Sources Log10(ε) Log10(ε) -0.30 -0.21 nl -2.24 -0.87 5 6 d 7 -4.18 -1.53 -6.12 -2.19 nl=3 4 -8.06 -2.85 5 6 7 kd kd -10.00 -3.51 0.50 1.80 3.10 4.40 5.70 7.00 0.50 1.80 3.10 4.40 5.70 7.00 Method of Auxiliary Sources (problems) Problem 1:unstability of linear system of algebraic equation b) a) Accuracy of solutions versus relative distance kd Tikhonov-Arsenin → αm= 10-8 ÷10-10 αm - regularization parameter A.N. Tikhonov, V.Ya. Arsenin: “The methods for solving non-correct problems”. Nauka, Moscow 1986, 1-288 David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011

  12. From Uniqueness Theorem The regular, in whole space, solution of Maxwell’s equation satisfying the radiation condition at infinity should be identically zero(*). PEC h d Scattered wave fields (both scalar and vector), which are continuously extended inside the scatterer's domain certainly has irregular points (singularities - SFMS). q R Auxiliari charges (stabile) Auxiliari charges (unstable) Method of Auxiliary Sources (problems) Problem 2:scattered fields main singularities (SFMS) V.D. Kupradze(*): “The main problems in the mathematical theory of diffraction”. GROL. Leningrad, Moscow 1935,1-112. David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011

  13. y y 1.27 1.08 D1 L2 y 0.76 0.65 0.54 D1 0.25 0.22 0.22 L1 -0.25 -0.22 - 0.11 L2 L2 D2 -0.65 -0.76 - 0.43 D2 L1 • S S x x -1.27 -1.08 - 0.76 0.43 1.29 2.16 -2.16 -1.29 -0.43 0.26 1.36 -1.38 -0.83 -0.29 0.81 • Image lines of primary source-L1andL2; • “Auxiliary Sources” ( monopoles) surrounding the areas of SFMS concentration, imitating the • radiation from the imagesL1andL2; • The lines of other possible distributions of auxiliary sources; • Some optimal distribution of collocation points on the main surfaceS. x - 1.08 - 2.16 - 1.53 - 0.89 - 0.26 0.37 1.00 Method of Auxiliary Sources (problems) • Problem 2:scattered fields main singularities (SFMS - conformal mapping procedure) 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 David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011

  14. Wave equation Maxwell’s equations any field vectors Constitutive Relations Isotropic magnetodielectrics: =  = 0; Chiral medium: =   0; Tellegen medium:=-  0; Material parameters Wave impedances Wave numbers Method of Auxiliary Sources(problems) • Problem 3:Most general, linear form of constitutive relation (Bi-Isotropic medium) • 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 Bi-Isotropic media”. Artech House, Boston, London 1994, 1-332.. David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011

  15. Spinor basis Spinor of electromagnetic field Green function Green function matrix - 2D case - 3D case Method of Auxiliary Sources (problems) • Problem 3:Fundamental Solution for Bi-Isotropic medium Fundamental Solution for Bi-Isotropic medium Maxwell’s equation in the Majorana-Dirak form R.Penrose, W.Ringler: “Spinors and space-time”, vol. 1. Cambridge University Press, Cambridge (eng.), 1986 David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011

  16. Problem formulation for most general form of constitutive relations Definition of fundamental or other singular solutions of the wave equation Finding the best placement and type of the fundamental solutions (AS) Geometry Analyzing for Scattered Field Main Singularities (SFMS) Definition of numerical method for evaluating the amplitudes of the auxiliary sources Deriving the system of linear algebraic equations Method of collocation, method of moments (MOM), ortogonalization Computation of amplitudes of AS Post processing stage Data processing and visualization General Concept of MAS David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011

  17. 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; MAS Applicatin to SomeParticular Problems David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011

  18. Boundary problem for spinor field: Required solution Constitutive relations Scattered electric field reconstracted from spinor fields x 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: Spherical shape chirolens David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011

  19. Im(n) 110.8 3.22 77.9 2.58 44.9 1.94 Re(n) 12.0 1.30 -20.9 0.66 GHz -53.9 f GHz GHz 150 180 210 240 270 300 0.02 300 150 180 210 240 270 Re(n) Scattering cross-section versus frequency. MAS approach:γ=0; γ=0.0001; T-matrix approach:γ=0.0001 Im(n) Plane Wave Excitation on Ciral Sphere GHz • A.Lakhtakia, V.K.Varadan, V.V.Varadan: “Scattering and absorption characteristics of lossy dielectric, chiral, nonspherical objects”. Appl. Opt. 24(1985) 23, 4146-4154. • F.G.Bogdanov and D.D.Karkashadze:“Conventional MAS in the problems of electromagnetic scattering by the bodies of complex materials”. Proc. of the 3-rd Workshop on Electromagnetic and Light Scattering,Bremen 1998,133-140. Refractive index versus frequancy for Right-hand and Left-hand components David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011

  20. 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). Gaussian beam illuminated transparent object The total electric field Ez component reconstruction from space x<0 to specex>0. Total electric field Ez component was reconstructed from EM field tangential components on the YOZ plane. Kirchhoff-Kotler formula + MAS for Field Reconstruction in Double-Negative Medium (wave front reversal approach) 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 David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011

  21. 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 David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011

  22. IWGAInput DFDI DFDI IWGARight IWGALeft Results of Filtering T_Junction optimization f1=1.038 1014 Hz f1=1.230 1014 Hz Rup=0.73% Rup=0.76% TM-polarization. Lattice constant a=1μm , base rods permittivity ε=11.4, base rods radii r=0.18a Tright=0.16% Tright=97.77% Case MMP simulation MAS simulation Tleft=98.59% Tleft=0.88% R (%) Tl (%) Tr (%)  (%) R (%) Tl (%) Tr (%)  (%) 100.51 Left 35.37 63.38 0.41 99.16 36.38 63.71 0.42 Right 36.51 0.11 63.24 99.86 36.02 0.11 63.76 99.89 “Filtering T-junction” Design(MMP, MAS) Comparison (without optimization) f1=1.038 1014Hz f2=1.230 1014Hz David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011

  23. Optimizing Inclusion ro=0.15R Before optimization: SWR_WGMAS= 1.72; SWR_WGMMP=1.66 After optimization : SWR_WG = 1.08 hD_WG = 3.379; hPhC_WG = 1.644; Coupling a Slab with a PhC Waveguide: fa/c=0.38 (MMP, MAS) Crossing waveguide operating with different frequency in each channels David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011

  24. Poisson Transformation ikzpq 3D Double-Periodic Green Functions 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 ofComputational and Theoretical Nanoscience, Vol. 8, 1–10, 2011 David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011

  25. Oblique Incident of Plane Wave on Bi-Periodic Array of Dielectric Spheres • a) Problem geometry (radius of spheres - a, period –d,incident angleθ=20º, φ=0º, permittivity • of dielectric spheresε=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. David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011

  26. N=2352 X direction N=1200 E H k 1.0 0.8 0.6 Transmission CoefficientТ 0.4 0.2 0.0 0.6 0.7 0.8 0.9 1.0 Relative period dx/ λ The Test for Convergence of MAS Results b) a) Dielectric layer with hexagonally bi-periodic dents: a) problem geometry and incidents plane wave orientation; b) transmission coefficient versus relative period. David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011

  27. y Non periodic expansions d S x D out 0 d y S in D q 1 e , m 1 1 x k Periodic expansions ? Unit cell E z ky Μ Γ Χ kx 2/a MAS and MMP Approach for Band Structure Calculation 2D bi-periodic structure with arbitrary shaped dielectric scatterers and unit cell with distribution of auxiliary sources(left); Periodic Green’s functions for different angles of incidence a) =00, b) =450 (right). 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 David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011

  28. Conclusion The procedure of solution of scattering problems by the method of auxiliarysources (MAS) needs preliminary considerations of various problems. Correct solution of these problems can radically influence efficiency of MAS. David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011

  29. Thank you for attention david.karkashadze@emcos.ge David Karkashadze - Workshop 2011 on Numerical Methods for Optical Nano Structures - Zürich 2011

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