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FDTD Solutions Numerical Methods Outline Application area review Ray optics vs. wave optics Finite Difference Time Domain method : Yee cell : Uniform, graded, and conformal meshing : Obtaining frequency domain results from a time domain method Boundary conditions Coherence and polarization in FDTD Accuracy and convergence testing © 2012 Lumerical Solutions, Inc. 1 Application area overview Our products can accurately simulate many technologies Photonic crystals Bandstructure Plasmonics CMOS Image sensors Nanoparticles Solar cells Resonators LED/OLEDs Grating devices Lithography Metamaterials Waveguides © 2012 Lumerical Solutions, Inc. Wave optics vs. ray tracing Question: What features are common among these applications? (When do you need to use FDTD Solutions?) Answer: Feature sizes are on the order of the wavelength. © 2012 Lumerical Solutions, Inc. 2 Wave optics vs. ray tracing Source = 0.55 um 4 um © 2012 Lumerical Solutions, Inc. Wave optics vs. ray tracing Incident light R=100% 45 n=1.5 n=1 Snell’s Law gives c: 41.8 © 2012 Lumerical Solutions, Inc. 3 Wave optics vs. ray tracing = 0.4 um = 4 um 20 um rayvswave_700THz.mpg © 2012 Lumerical Solutions, Inc. rayvswave_70THz.mpg Wave optics vs. ray tracing Conclusion: You need FDTD Solutions when feature sizes are on the order of a wavelength © 2012 Lumerical Solutions, Inc. 4 Overview of FDTD method TOPICS Maxwell equations Yee cell Time domain technique Fourier transform Sources and Monitors Computational requirements 2D vs. 3D Advantages of the FDTD method © 2012 Lumerical Solutions, Inc. Maxwell’s equations Describe the behavior of both the electric and magnetic fields, as well as their interactions with matter. Name Differential form Integral form Gauss’ law Gauss' law for magnetism (absence of magnetic monopoles): Faraday’s law of induction: Ampère’s law (with Maxwell's extension): © 2012 Lumerical Solutions, Inc. 5 Maxwell’s equations Meaning Symbol SI Unit of Measure electric field volt per meter magnetic field also called the auxiliary field ampere per meter electric displacement field also called the electric flux density coulomb per square meter magnetic flux density also called the magnetic induction also called the magnetic field tesla, or equivalently, weber per square meter free electric charge density, not including dipole charges bound in a material coulomb per cubic meter free current density, not including polarization or magnetization currents bound in a material ampere per square meter © 2012 Lumerical Solutions, Inc. Wave Optics – Free space plane wave In vacuum, without charges (=0) or currents (J=0) Maxwell’s equations have a simple solution in terms of traveling sinusoidal plane waves The electric and magnetic field directions are orthogonal to one another and the direction of travel k The E, H fields are in phase, traveling at the speed c H © 2012 Lumerical Solutions, Inc. 6 Wave Optics - Simple materials In linear materials, the D and B fields are related to E and H by: D E B H where: ε is the electrical permittivity of the material, and μ is the permeability of the material In FDTD Solutions, we typically deal with the electrical permittivity only μ=μ0 is the permeability of the free space © 2012 Lumerical Solutions, Inc. How the FDTD method works E and H are discrete in time E (t ) E nt ( n 1 ) t H (t ) H 2 The basic FDTD time-stepping relation: t E n 1 E n H n 1 2 t H n 3 2 H n 1 2 E n 1 E0 H1 2 E1 H3 2 … 2nd order accurate in time: error ~ t2 © 2012 Lumerical Solutions, Inc. 7 Maxwell equations on a mesh Yee cell E and H are discrete in space © 2012 Lumerical Solutions, Inc. The Yee cell Spatially stagger the vector components of the E-field and H-field about rectangular unit cells of a Cartesian computational grid. Z Yee cell Ez Hx Ey Hy (x,y,z) Ex Y Hz X 2nd order accurate in space Kane Yee (1966). "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media". Antennas and Propagation, IEEE Transactions on 14: 302– 307. © 2012 Lumerical Solutions, Inc. 8 How are dielectric properties treated? (r ) The true structure The meshed structure i , j ,k Discrete mesh Y X Conformal mesh technology © 2012 Lumerical Solutions, Inc. Conformal mesh technology Interfaces are a problem for Maxwell’s equations on a discrete mesh : The fields can be discontinuous at interfaces : The positions of the interface cannot be resolved to better than dx : Staircasing effects Solutions : Graded mesh (reduce mesh size near interfaces) : Conformal mesh technology : Combination of both © 2012 Lumerical Solutions, Inc. 9 Conformal mesh technology Finite difference methods cannot resolve interface positions or layer thicknesses to better than the mesh size © 2012 Lumerical Solutions, Inc. Conformal mesh technology Staircasing can lead to unwanted effects © 2012 Lumerical Solutions, Inc. 10 Conformal mesh technology Conformal mesh technology uses an integral solution to Maxwell’s equations near interfaces : Lumerical’s approach can handle arbitrary dispersive media : More advanced than well known approaches such as the Yu-Mittra model for PEC © 2012 Lumerical Solutions, Inc. Example: Yu-Mittra approach for PEC C Bz E dl t C E dl C1 C1 Ex Ey Bz Ey y PEC Ex x © 2012 Lumerical Solutions, Inc. 11 Conformal mesh technology Application Simulation mesh Best Solution Multilayer Conformal mesh can allow you to use the default simulation mesh. Mie scattering We combine conformal meshing with graded meshing, but the mesh is not as fine as with staircasing. Waveguide couplers Conformal mesh can allow you to use the default simulation mesh. © 2012 Lumerical Solutions, Inc. Conformal mesh technology There are 3 variants used : Conformal variant 0 (default) • Conformal mesh applied to all materials except metals and PEC (Perfect Electrical Conductor) • Metals are materials with real() < 1 • This is the best setting without doing convergence testing : Conformal variant 1 • Conformal mesh applied to all interfaces : Conformal variant 2 • Yu-Mittra model for PEC applied to PEC and metals © 2012 Lumerical Solutions, Inc. 12 Conformal mesh technology Simple rule of thumb : Use the default conformal variant 0 Possible exceptions : If the simulation diverges • Test using staircasing : If you are studying plasmonic effects • Consider using conformal variant 1 • Do some careful convergence testing and read the page at http://docs.lumerical.com/en/fdtd/user_guide_testing_conver gence.html © 2012 Lumerical Solutions, Inc. Computational Resource Requirements How do the required computational resources scale with grid size? 3D 2D Memory Requirements ~ (/dx)3 ~ (/dx)2 Simulation Time ~ (/dx)4 ~ (/dx)3 dx=/10 © 2012 Lumerical Solutions, Inc. 13 Controlling the mesh We provide a mesh accuracy slider that ranges from 1-8 The mesh algorithm then targets a minimum Note that =0/n, where n is the refractive index Generally, /dx=10 (mesh accuracy 2) is considered reasonable for many FDTD simulations and /dx ~ 20 (mesh accuracy 4,5) is considered very high accuracy It is still often worth running at /dx=6 (mesh accuracy 1) for initial simulations The default is /dx=10 (mesh accuracy 2) We will see later why /dx = 1/(k dx) is the right quantity to consider The meshing can be fully customized : /dx = 6, 10, 14, 18, 22, 26, 30, 34 : : Typically, we use mesh override regions to force a particular mesh in a given region, which we recommend We can fully customize the mesh algorithm details if desired but this is not recommended © 2012 Lumerical Solutions, Inc. Tip Use a coarse mesh for simulations : Memory scales as 1/dx3 : Simulation time scales as 1/dx4 © 2012 Lumerical Solutions, Inc. 14 2D vs 3D FDTD simulations can be run in 2D or 3D 3D 3D 2D: 2D: Structure Structure is is infinite infinite in in zz direction direction Z Y X Y Z X © 2012 Lumerical Solutions, Inc. 2D vs 3D 2D assumes E H 0 z z z We get perfect separation of Maxwell’s equations into two independent sets of equations : Transverse Electric (TE) : involves only Ex, Ey, Hz : Transverse Magnetic (TM) : involves only Ez, Hx, Hy The terms “TE” and “TM” are no longer used in FDTD Solutions. Use the blue arrows of sources to determine the Electric field polarization. © 2012 Lumerical Solutions, Inc. 15 FDTD is a time domain technique! The simulation is run to solve Maxwell’s equations in time to obtain E(t) and H(t) Most users want to know the field as a function of wavelength, E(), or equivalently frequency, E(w) The steady state, continuous wave (CW) field E(w) is calculated from E(t) by Fourier transform during the simulation. TSim iwt E (w ) e E (t )dt 0 See section on Units and Normalization of Reference Manual for more details: http://www.lumerical.com/fdtd_online_help/ref_fdtd_units_units_and_normalization.php © 2012 Lumerical Solutions, Inc. FDTD is a time domain technique! Normalize E(w) to the source spectrum and we can obtain the impulse response of the system! T 1 Sim iwt Eimpulse(w ) e E (t )dt s(w ) 0 Eimpulse is a response of the system : It is independent of the source pulse used : It is the CW, or monochromatic response Ideally s(w)=1, meaning that s(t) is a delta function : In practice, we use a very short pulse © 2012 Lumerical Solutions, Inc. 16 Advantages of the FDTD method Advantages Few inherent approximations = accurate A very general technique that can deal with many types of problems Arbitrarily complex geometries One simulation gives broadband results © 2012 Lumerical Solutions, Inc. Example, ring resonator through n=2.915 4.4 m ~1.55m drop © 2012 Lumerical Solutions, Inc. 17 Example: waveguide ring resonator time 1 Fourier Transform ring.mpg 2 © 2012 Lumerical Solutions, Inc. Example: waveguide ring resonator through drop © 2012 Lumerical Solutions, Inc. 18 Boundary conditions PML • Absorbs incident fields • Use PML when the fields are meant to propagate away from the simulation region Metal • Perfect metal boundary • 100% reflection, 0% absorption Periodic • For periodic structures • The structure AND fields must be periodic • Typically used with the plane wave source © 2012 Lumerical Solutions, Inc. Symmetric/Anti-Symmetric boundaries Symmetric/Anti-Symmetric • Can reduce memory/time for symmetric simulations • Both structure AND fields must be symmetric © 2012 Lumerical Solutions, Inc. 19 Symmetric/Anti-Symmetric boundaries Non zero components of the electric and magnetic fields at symmetric/anti-symmetric boundaries © 2012 Lumerical Solutions, Inc. Symmetric/Anti-Symmetric boundaries How the different electric and mangetic components behave under different symmetry conditions © 2012 Lumerical Solutions, Inc. 20 Symmetric/Anti-Symmetric boundaries Symmetry/Anti-Symmetry can be used for periodic structures : Consider file from far field section, farfield4.fsp © 2012 Lumerical Solutions, Inc. Symmetric/Anti-Symmetric boundaries Symmetry/Anti-Symmetry can be used for periodic structures : Image the near and far fields at 1.3 microns © 2012 Lumerical Solutions, Inc. 21 Symmetric/Anti-Symmetric boundaries Symmetry/Anti-Symmetry can even be used for periodic structures : We can get the same results but the simulation runs faster : Far field projections can still be done : The data is automatically “unfolded” so we see the full image GUI and script results Actual simulation © 2012 Lumerical Solutions, Inc. Bloch boundary conditions Bloch • For periodic structures (similar to periodic) • Use Bloch when the plane wave source is at nonnormal incidence • Use Bloch for bandstructure simulations. • Bloch boundaries conditions ensure that E(x+a) = exp(ika)*E(x) a is the simulation span k is the Bloch vector © 2012 Lumerical Solutions, Inc. 22 Bloch boundary conditions • Periodic boundaries are just a special case of Bloch boundaries (k=0)! • Bloch BC requires 2x memory and simulation time • When using Bloch boundaries for non-normal plane waves, you must check the following © 2012 Lumerical Solutions, Inc. Bloch boundary conditions Consider the difference between correct and incorrect k settings for a plane wave in free space Correct usr_bloch_movie_2.mpg Incorrect usr_bloch_movie_3.mpg © 2012 Lumerical Solutions, Inc. 23 Bloch boundary conditions When the setting are correct, we can study periodic structure illuminated by plane waves at angles : We can calculate far field projections • Assume periodicity the same as with periodic boundary conditions : We can calculate grating order efficiencies, the same as with periodic boundary conditions : We must be cautious about the PML performance when the angle of incidence is steep • Sometimes, we need to increase the number of layers of PML to get accurate results © 2012 Lumerical Solutions, Inc. Coherence Temporal incoherence : The phase, j, of the light shifts randomly over time, on a time scale tc E (t ) E0 cos(wt j (t )) 2 T 10 15 s w t c 10 11 s : Even without random phase shifts, if the light is not monochromatic, it is incoherent t c f 1 : In either case, the coherence length of the system is often much longer than any standard simulation time (tc >> T) • It is not efficiency in general to directly model incoherence • It is not possible to perform near to far field projections of incoherent results in the near field © 2012 Lumerical Solutions, Inc. 24 Coherence Temporal incoherence : The frequency domain monitors of FDTD Solutions calculate the monochromatic response of the system • There are some advanced features to directly extract incoherent results where the value of f ~ 1/tc can be specified, see http://docs.lumerical.com/en/fdtd/user_guide_spectral_averaging.html for details : In general, the best approach is to calculate the monochromatic (or CW) response first, then calculate the incoherent result with 2 2 E (w0 ) W (w ) E (w ) dw • Where W(w) is the spectrum of the physical source used © 2012 Lumerical Solutions, Inc. Coherence Temporal incoherence example : Reflection of 50nm of silver on 500nm of silicon CW or monochromatic response © 2012 Lumerical Solutions, Inc. 25 Coherence From one simulation, we can calculate the incoherent results at many wavelengths 2 2 E (w0 ) W (w ) E (w ) dw © 2012 Lumerical Solutions, Inc. Coherence Spatial incoherence can be simulated using the ergodic principle + Each ensemble consists of dipoles with randomized phase, amplitude, position, orientation and pulse time A large number of ensembles must be averaged : There is a statistical error associated that decreases with increased number of ensembles Typically 50 to 100 simulations is a minimum requirement and more may be required : +……+ See Chan, Soljačić, and Joannopoulos, “Direct calculation of thermal emission for threedimensionally periodic photonic crystal slabs” http://pre.aps.org/abstract/PRE/v74/i3/e036615 for discussion It is often erroneously assumed that one simulation is enough : This may or may not be true, but it depends on the details of what is being recorded © 2012 Lumerical Solutions, Inc. 26 Coherence Spatial coherence : The same results can be reconstructed from spatially coherent results • There is no statistical error • The total number of simulations is typically less than is required by ensemble averaging : Example for an ensemble of incoherent dipole emitters © 2012 Lumerical Solutions, Inc. Coherence Incoherent sources are dealt with from the coherent results. For example, consider 2 dipoles log(|E(w)|2) Time domain dipole_coherent.mpg © 2012 Lumerical Solutions, Inc. 27 Coherence Incoherent dipoles, 2 simulations Simulation 1 log(|E1|2+|E2|2) log(|E1|2) dipole_incoherent1.mpg Simulation 2 log(|E2|2) log(|E1+E2|2) dipole_incoherent1.mpg © 2012 Lumerical Solutions, Inc. Polarization FDTD simulations have well defined polarization Unpolarized results are obtained by adding the results of 2 orthogonal polarization simulations incoherently ½( |E1|2 + |E2|2 ) = <|E|2>unpolarized © 2012 Lumerical Solutions, Inc. 28 Dipoles Incoherent, isotropic dipoles require the sum of 3 orthogonal polarization states, summed incoherently 2 E = 1/3 { 2 E px + 2 E py + 2 E pz } The above theory greatly simplifies LED/OLED simulations © 2012 Lumerical Solutions, Inc. Coherence We often have to find incoherent results : : : : : Temporal incoherence Spatial incoherence Unpolarized light Anisotropic dipole emitters and more... It is generally most efficient to reconstruct the incoherent results from the coherent results : There is no statistical error : The total number of simulations is typically less than a brute force approach © 2012 Lumerical Solutions, Inc. 29 Accuracy and convergence testing Dispersion relation in FDTD (in free space) 2 k y y 1 1 k x x 1 wt 1 k z z ct sin 2 x sin 2 y sin 2 z sin 2 2 2 t 0 When x 0 y 0 z 0 we have w c k x2 k y2 k z2 ck Typical target for accuracy might be Courant stability limit t 2 k x x 2 ~ 0.3 10 2 20 x x c 3 © 2012 Lumerical Solutions, Inc. Accuracy and convergence testing In reality, many factor beyond the mesh size affect the accuracy of FDTD For example : The proximity of the PML : PML reflections : The multi-coefficient model fit • How well do you know n,k for your materials? What is the experimental error? Do you expect the same n,k as other authors? : and more... Please read http://docs.lumerical.com/en/fdtd/user_guide_testing_convergence.html for a detailed list and steps on doing convergence testing Do not waste time making the mesh size really small without considering the other factors : And some of them can get worse as the mesh size gets small! © 2012 Lumerical Solutions, Inc. 30 Accuracy and convergence testing With care and computing power, almost any accuracy can be achieved The more important limit for real devices is often how well you know the geometry and the material properties (n,k) © 2012 Lumerical Solutions, Inc. Review and tips What mesh size should I use? : “mesh accuracy” of 1 or 2 for initial setup (faster) : Use “mesh accuracy” of 2-4 for final simulations : “mesh accuracy” 5-8 is almost never necessary • Use mesh overrides instead for most applications How long a simulation time should I use? : Start with longer simulations times and let the “auto-shutoff” feature find out when you can stop the simulation : Check with point time monitors How do I check the memory requirements? © 2012 Lumerical Solutions, Inc. 31 Review and tips What are some tricks for speeding up FDTD Solutions, and reducing the memory requirements? Avoid simulating homogeneous regions with no structure Use symmetry where possible Use periodicity where possible Use a coarse mesh (use a refined mesh for final simulations) : : : : Use far field projections instead Gain factors of 2, 4 or 8 Gain factors of 100s or 1000s Start with “mesh accuracy” of 1 instead of 2 • • • : : gives 8 times faster simulation 5 times less memory within 10-20% accuracy in general User mesh accuracy of 2-4 for final simulation Use mesh override regions for local regions of fine mesh Reduce amount of data collected : : : Down sample monitors spatially Record fewer frequency points in frequency monitors Record only the necessary field components © 2012 Lumerical Solutions, Inc. Review and tips Questions and answers… © 2012 Lumerical Solutions, Inc. 32 Getting help Technical Support : Email: [email protected] : Online help: docs.lumerical.com/en/fdtd/knowledge_base.html • Many examples, user guide, full text search, getting started, reference guide, installation manuals : Phone: +1-604-733-9006 and press 2 for support Sales information: [email protected] Find an authorized sales representative for your region: : www.lumerical.com and select Contact Us © 2012 Lumerical Solutions, Inc. 33