Numerical Methods in Analysis of Photonic Crystals Integrated

Numerical Methods in Analysis of Photonic Crystals Integrated Photonics Laboratory School of Electrical Engineering Sharif University of Technology

Photonic Crystals Team n Faculty n n n Bizhan Rashidian Rahim Faez Farzad Akbari Sina Khorasani Khashayar Mehrany Students & Graduates n n n Special Acknowledgements n n n © Copyright 2005 Sharif University of Technology Alireza Dabirian Amir Hossein Atabaki Amir Hosseini Meysamreza Chamanzar Mohammad Ali Mahmoodzadeh Keyhan Kobravi Sadjad Jahanbakht Maryam Safari 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Outline n n n Overview Wave propagation in periodic media Band structure in one- and multidimensional structrues Brillouin Zones n 2 D & 3 D Lattices n n n Numerical methods Summing Up & Conclusions © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Outline n Numerical methods n Frequency Domain Methods n Plane Wave Expansion (PWE) n Finite-Difference Frequency Domain (FDFD) n Wannier Function Method (WFM) n Finite-Element Method (FEM) n Multiple-Multipole Method (MMP) n Time Domain Methods n Finite-Difference Time-Domain (FDTD) n Finite-Elements in Time-Domain (FETD) © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Periodic Medium n Typical layered periodic medium in 1 D a n No transmission for some frequencies © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Periodic Medium n Typical layered periodic medium in 1 D a n Non-zero transmission for some frequencies © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Periodic Medium n Zero-transmission depend on Dielectric and loss constants n Wavelength n Angle of incidence n Period n Number of dielectrics in each period n Thickness of each individual layer n n These windows are referred to as Photonic Band Gaps (PBGs). © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Bloch Waves n n How can an optical wave propagate in periodic media? Bloch-Floquet Theorem: © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Band Structure Light Cone (n=1) n n For simple dielectric, The dispersion equation is: © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005 .

Band Structure n Periodic Medium © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Band Structure n Periodic Medium PBG #2 PBG #1 © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Brillouin Zones n BZ # 1 © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Brillouin Zones n BZ #2 © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Brillouin Zones n Irreducible BZ © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

2 D Photonic Crystals n In higher dimensions we typically have n ai , i=1, 2, are primitive lattice vectors A(r) is either of magnetic or electric vector fields n © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

2 D Photonic Crystals n Square Lattice (Top View) a 1 a 2 © Copyright 2005 Sharif University of Technology Host Medium (Air, Si, Ga. As, etc. ) Holes/Rods (Air, Al 2 O 3, etc. ) 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

2 D Photonic Crystals n Square Lattice (Top View) Unit Cell 90 o rotation leaves the photonic crystal unchanged Symmetry © Copyright 2005 Sharif University of Technology Four-fold 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

2 D Photonic Crystals n Triangular Lattice (Top View) a 1 Host Medium (Air, Si, Ga. As, etc. ) Holes/Rods (Air, Al 2 O 3, etc. ) a 2 © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

2 D Photonic Crystals n Triangular Lattice (Top View) Unit Cell 60 o rotation leaves the photonic crystal unchanged Unit Cell © Copyright 2005 Sharif University of Technology Six-fold Symmetry 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

2 D Photonic Crystals n 2 D Photonic Crystals in Cylindrical Geometry (Top View) © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

2 D Photonic Crystals n Planar 2 D Photonic Crystal (Air Holes in Si) Lupu et. al. , Opt. Express 12, 5690 (2004) © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

2 D Photonic Crystals n 2 D Photonic Crystal Slabs Takayama et. al. , Appl. Phys. Lett. 87, 061107 (2005) © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

2 D Photonic Crystals n Five-fold symmetry ? © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

2 D Photonic Crystals n n n Five-fold symmetry ? Yes ! No exact periodicity No Bloch waves allowed 72 o Quasi-crystals http: //members. shaw. ca/quadibloc/math/penint. htm © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

2 D Photonic Crystals 12 -fold Symmetric Quasi-crystal with Photonic Band Gap Zoorob et. al. , Nature 404, 740 (2000) © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

3 D Photonic Crystals n Yablonovite Yablonovitch et. al. , Phys. Rev. Lett. 67, 2295 (1991) © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

3 D Photonic Crystals n Interwoven Helical Structure Toader & John, Science 292, 1133 (2001) © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

3 D Photonic Crystals n Interwoven Helical Structure Seet et. al. , Advanced Mat. 17, 541 (2005) © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

2 D Band Structure n Square Lattice n The Band Structure shown is typical © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

2 D Band Structure n BZ #1 © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

2 D Band Structure n BZ #2 © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

2 D Band Structure n Irreducible BZ © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

2 D Band Structure n Irreducible BZ © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

2 D Band Structure n G-C-M-G Path on the Irreducible BZ PBG #1 © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Numerical Methods in Photonic Crystals Frequency-Domain Methods © Copyright 2005 Sharif University of Technology Time-Domain Methods 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Numerical Methods Frequency Domain Methods Plane Wave Expansion (PWE) Finite-Difference Frequency Domain (FDFD) Wannier Function Method (WFM) Finite-Element Method (FEM) Multiple-Multipole Method (MMP) © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Numerical Methods Time Domain Methods Finite-Difference Time-Domain (FDTD) Finite-Elements in Time-Domain (FETD) © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Plane Wave Expansion n For simplicity we take the E-polarization: © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Plane Wave Expansion n Using Bloch Theorem we have n Hence © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Plane Wave Expansion n Using Discrete Fourier Expansion we have n Here , and are Inverse Lattice Vectors. © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Plane Wave Expansion n Inverse Lattice Vectors in 2 D are given by n Finally, the eigenvalue equation for © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005 is

Finite-Difference Frequency Domain n Maxwell’s equations and are discretized for Epolarization as © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Finite-Difference Frequency Domain n Thus which can be combined as © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Finite-Difference Frequency Domain n Field plots of y-fundamental mode of Holey fiber Zhu & Brown, Opt. Express 10, 853 (2002) © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Wannier Function Method n Orthogonality of E-polarizations reads: n n is the eigenfrequency number. Wannier functions are defined as (R is a lattice site): n © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Wannier Function Method n Wannier function satisfy the properties n Orthogonality n Translation n Reconstrucion n Real-valued for Inversion Symmetric Media © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Wannier Function Method n Maximally localized Wannier functions Busch et. al. , J. Phys. : Condens. Matter 15, R 1233 (2003) © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Finite-Element Method n Wave equation in frequency-domain is: n Taking the inner product by W and integration using Green’s theorem gives © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Finite-Element Method n n Boundary term vanishes on PEC, PMC. Using the interpolating approximating elements: © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Finite-Element Method n For band-structure computations we have © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Multiple-Multipole Method n 2 D region is divided into K sub-domains and expanded as © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Multiple-Multipole Method n n n Solution is approximated by a superposition of a Bessel expansion and K- Hankel expansions of sufficiently high order, each centered at a different local center. Field matching is done on collocation points 6 continuity conditions for field components: 4 tangential components of E, H n 2 normal components of D, B n © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Multiple-Multipole Method 6 Boundary Conditions Overdetermination More Uniform Error Distribution Less Error Where More Collocation Points Are Present © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Finite-Difference Time-Domain n Fields are discretized similar to FDFD grid. © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Finite-Difference Time-Domain © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Finite-Difference Time-Domain a L © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Finite-Elements in Time-Domain n n Time-dependent wave equation is transformed for dispersive media as Time-integration is done by Newmark-b scheme, which is unconditionally stable when : © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Finite-Elements in Time-Domain n n Slowly-varying envelope (BPM) formulation: Ritz-Galerkin FEM + Newmark-b method: © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Summing Up-PWE + Simple to code + Good results for the few lowest order bands (n<5) - Serious convergence problems for moderate number of plane waves - Poor computational complexity O(N 4) - Full matrices, hence eigenvalue extraction problems - Handles non-dispersive media © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Summing Up-FDFD + Efficient + Handles disperive structures + Handles lossy media + Sparse Matrices + Accurate - Performance not evaluated for large propagation problems © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Summing Up-WFM + Mathematically perfect + Uses natural properties of periodic media + The most efficient after MMP, both in 2 D and 3 D + Sparse Matrices + Handles large-scale Photonic Crystal circuits easily + Small number of bands for expansions are required (<10) + Handles guided propagating and leaky modes + Accurate in the vicinity of band edges + Handles degenerate bands - Difficult to construct the Wannier functions - Difficult coding © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Summing Up-FEM + Sparse matrices + Convergent and stable results + Reasonable computational effort for medium scale structures - Needs mesh generation - Difficult coding © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Summing Up-MMP + The most efficient existing method in frequency domain + Handles dispersive media + Ability to accurately compute near field distributions such as surface plasmons + Uniform error distribution - Difficulty with selection of collocation points (Nonautomatic) - Mathematically questionable in 3 D, limited to 2 D © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Summing Up-FDTD + Very versatile + The most widely used method + Conditionally stable and convergent + Best suited for distributed processing - Needs to be modified for non-orthogonal lattices - Computationally intensive - Not accurate for very low frequencies and misses high frequencies in pulse propagation - Produces spurious peaks in the spectrum - Some difficulty with dispersive/lossy media © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Summing Up-FETD + Sparse matrices + Suitable for parallel processing + Can be unconditionally stable + Convergent - Needs mesh generation - Difficult implementation - Some difficulty with dispersive/lossy media © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Conclusions n Best frequency domain methods n 2 D : n Plane Wave Expansion (only for band computation of non-defective structures) n 2 D/3 D : n Wannier Functions Method n Finite-Difference Frequency-Domain n Best time-domain methods 2 D : Finite-Elements in Time-Domain n 3 D : Finite-Difference Time-Domain n © Copyright 2005 Sharif University of Technology 1 st Workshop on Photonic Crystals Mashad, Iran, September 2005

Thanks for your attention !