Plane Wave Expansion Method and Reduced Bloch Mode Expansion Technique MAXWELL’S EQUATIONS IN FOURIER SPACE Real-Space THE UNIVERSITY OF TEXAS AT EL PASO Pioneering 21 st Century Electromagnetics and Photonics 0 0 0 0 0 0 y z r x x z r y y x r z y z r x x z r y y x r z H H k E y z H H k E z x H H k E x y E E k H y z E E k H z x E E k H x y , , , , 0 , , , , , , , 0 , , , , , , , 0 , , y pqr z pqr z pqr y pqr p pq qr r x pqr p q r z pqr x pqr x pqr z pqr p pq qr r y pqr p q r x pqr y pqr y pqr x pqr p pq qr r z jk U jk U k a S jk U jk U k a S jk U jk U k a S , , , , , 0 , , , , , , , 0 , , , , , , pqr p q r y pqr z pqr z pqr y pqr p pq qr r x pqr p q r z pqr x pqr x pqr z pqr p pq qr r y pqr p q r x pqr y pqr yp jk S jk S k b U jk S jk S k b U jk S jk , 0 , , , qr x pqr p pq qr r z pqr p q r S k b U 0 0 0 0 0 0 y z z y r x z x x z r y x y y x r z y z z y r x z x x z r y x y y x r z jk jk jk jk jk jk Ku Ku εs Ku Ku εs Ku Ku εs Ks Ks μu Ks Ks μu Ks Ks μu Fourier Space Matrix Form K Matrices ,1,1,1 ,1,1,2 , , , i i i iPQR k k k K Plane Wave ,1,1,1 ,1,1,1 ,1,1,2 ,1,1,2 , , , , , , i i i i i i iPQR iPQR U S U S U S u s The plane wave expansion method (PWEM) provides a highly efficient numerical solution to Maxwell’s equations for devices with low to moderate dielectric contrast. It expands the field into a set of plane waves and converts Maxwell’s equations to matrix form by assigning each plane wave a complex amplitude. The final matrix equation can be solved using any number of standard eigen-value THE CONVOLUTION MATRICES r ε 3D PWEM FORMULATION 2D PWEM FORMULATION 1 1 2 0 x r x y r y z r z k K μK KμK s εs 1 1 2 0 x r x y r y z r z k K εK KεK u μu E Mode H Mode 1 1 2 2 2 1 1 1 1 2 1 1 2 0 1 1 2 2 1 2 1 1 2 1 2 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ • • • • ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ • • • • r r r r r r r r P P P P k P P P P K ε PK K ε PK μP μP u u u u K ε PK K ε PK μP μP ˆ polarization vectors orthogonal to K i P BAND DIAGRAMS ISOFREQUENCY CONTOURS REDUCED BLOCH MODE EXPANSION Step 1: Calculate the eigen-vector matrices at the key points of symmetry 1 2 3 1 2 3 N N Γ X Γ Γ Γ Γ X X X X V V v v v v v v v v 1 2 3 N M M M M M V v v v v Step 2: Construct Bloch mode 1 2 3 1 2 3 N N Γ X Γ Γ Γ Γ X X X X V V v v v v v v v v 1 2 3 N M M M M M V v v v v 1 1 1 M M M Γ Γ X X M M U v v v v v v GramSchmidt U U Step 3: Calculate eigen-value problem using standard PWEM Ax λBx Step 4: Calculate and solve a reduced eigen- value problem. H H A U AU B U BU , Ax λBx Vλ Step 5: If needed, the eigen-vectors can be transformed back to the plane wave basis. H V UVU Effect of the number of spatial harmonics. M. I. Hussein, “Reduced Bloch mode expansion for periodic media band structure calculations,” Proc. Roy. Soc. Lond. Ser. A465, 2825–2848 (2009).