Periodic Structures in FDTD

Transcription

1 EE 5303 Electromagnetic Analsis Using Finite Difference Time Domain Lecture #19 Periodic Structures in FDTD Lecture 19 These notes ma contain coprighted material obtained under fair use rules. Distribution of these materials is strictl prohibited Slide 1 Lecture Outline Review of Lecture 18 Periodic Structures Periodic Boundar Conditions in FDTD Electromagnetic Band Calculation using FDTD Lecture 19 Slide 2 1

2 Review of Lecture 18 Lecture 19 Slide 3 Methods for Incorporating Metals Easier Implementation More Accurate Simulation Etreme Dielectric Constant Easiest because no modification to the code is necessar, but it does not account for loss. Perfect Electric Conductor Requires minimal modification to the code, but does not account for loss. Requires greater modification to the formulation of the update equations. It can account for loss, but cannot account for frequenc dependence. Lorentz Drude Model Requires a much more complicated formulation and implementation, but it can account for loss and frequenc dependence. Lecture 19 Slide 4 2

3 Placing Metals on a 2D Grid Ez Mode For the E z mode, the electric field is alwas tangential to metal interfaces and few problems eist modeling metallic structures. Hz Mode For the H z mode, the electric field can be polarized perpendicular to metal interfaces. This is problematic and it is best to place metals with the outermost fields being tangential to the interfaces. Bad placement of metals Good placement of metals E Hz Mode Hz E Lecture 19 Slide 5 Drawbacks of Uniform Grids Uniform grids are the easiest to implement, but do not conform well to arbitrar structures and ehibit high anisotropic dispersion. Anisotropic Dispersion (see Lecture 10) Staircase Approimation (see Lecture 18) Lecture 19 Slide 6 3

4 Alternative Grid Schemes Heagonal Grids Nonuniform Grids Curvilinear Grids Nonorthogonal Grids Irregular Unstructured Grids Bodies of Revolution Lecture 19 Slide 7 Periodic Structures Lecture 19 Slide 8 4

5 Eamples of Periodic Electromagnetic Devices Diffraction Gratings Waveguides Band Gap Materials Metamaterials Antennas Frequenc Selective Surfaces Slow Wave Devices Lecture 19 Slide 9 The Bravais Lattices and Seven Crstal Sstems Lecture 19 Slide 10 5

6 Two Dimensional Bravais Lattices Oblique Rectangular Rhombic Heagonal Square Lecture 19 Slide 11 Primitive Lattice Vectors Ais vectors most intuitivel define the shape and orientation of the unit cell. The cannot uniquel describe all 14 Bravais lattices. Translation vectors connect adjacent points in the lattice and can uniquel describe all 14 Bravais lattices. The are less intuitive to interpret. Primitive lattice vectors are the smallest possible vectors that still describe the unit cell. Lecture 19 Slide 12 6

7 Non Primitive Lattice Vectors Almost alwas, the label lattice vector refers to the translation vectors, not the ais vectors. A translation vector is an vector that connects two points in a lattice. The must be an integer combination of the primitive translation vectors. t pt qt rt pqr p, 2, 1,0,1,2, q, 2, 1,0,1,2, r, 2, 1,0,1,2, Primitive translation vector Non primitive translation vector Lecture 19 Slide 13 Fields in Periodic Structures Waves in periodic structures take on the same periodicit as their host. k inc Lecture 19 Slide 14 7

8 The Bloch Theorem The field inside a periodic structure takes on the same smmetr and periodicit of that structure according to the Bloch theorem. E r j r Ar e Overall field Amplitude envelope with same periodicit and smmetr as the device. Plane wave like phase tilt term E r A r Lecture 19 Slide 15 Bloch Waves Wave normall incident on a periodic structure. Wave incident at 45 on the same periodic structure. Lecture 19 Slide 16 8

9 Mathematical Description of Periodicit A structure is periodic if its material properties repeat. Given the lattice vectors, the periodicit is epressed as r tpqr r tpqr pt1qt2 rt3 Recall that it is the amplitude of the Bloch wave that has the same periodicit as the structure the wave is in. Therefore, A r t A r t pt qt rt pqr pqr For a device that is periodic onl along one direction, these relations reduce to p,, z,, z,,,, A p z A z more compact notation p A p A Lecture 19 Slide 17 Periodic Boundar Conditions in FDTD Lecture 19 Slide 18 9

10 Generalized Periodic Boundar Condition For a device that is periodic along with period, Bloch s theorem can be written as E r j j Ar e e Periodicit requires that the amplitude also have period. A A m m,, 2, 1,0,1,2,, The phase tilt term, however, can have an period along the ais. We can derive an equation that describes the periodic boundar condition (PBC) from the Bloch theorem. E E E A A E e e e j j j j j e j e e E E e j Lecture 19 Slide 19 Periodic Boundar Condition in the Time Domain The generalized PBC in the frequenc domain was derived from the Bloch theorem to be j E, E, e Recall the following propert of the Fourier transform 0, j t, g tt0 G e It follows that the generalized PBC in the time domain is E, t E, t c 0 sin 0 value from the future 0 value from the past Lecture 19 Slide 20 10

11 Generalized PBC At Normal Incidence At normal incidence, =0. The generalized PBC reduces to, E, t E t At the low boundar, this is E z t 0, j N, j E z t At the high boundar, this is E z t N 1, j 1, j E z t E z 0, j N, j E t z t N 1, j 1, j Ez E t z t Lecture 19 Slide 21 Generalized PBC At Oblique Incidence At oblique incidence, 0. The generalized PBC remains, E, t E t At the low boundar, this is 0, j N, j z t z t E E At the high boundar, this is E N 1, j 1, j z E t z t We can alwas compute the electric field at the past time of t b storing a record of the fields at the boundar and interpolating. How do we calculate the E field at a future time? 0, j N, j z t z t E E N 1, j 1, j Ez E t z t Lecture 19 Slide 22 11

12 Conclusions Time domain methods have serious problems when the following conditions eist simultaneousl Periodic boundar condition Oblique incidence Pulsed source Good solutions eist when an one of these conditions can be removed. One limited solution eists when all of these conditions eist at the same time. Angled update method Lecture 19 Slide 23 Case #1: No Periodic Boundar In this case, we are modeling scattering from a finite size device. Onl PMLs are needed and nothing else. Lecture 19 Slide 24 12

13 Case #2: No Oblique Incidence In this case, we can use the standard PBC we alread discussed. PML Periodic Boundar Periodic Boundar PML Lecture 19 Slide 25 Case #3: Pure Frequenc Source When a device can be modeled with a single frequenc, it becomes possible to incorporate a generalized periodic boundar condition. We lose the wideband capabilit of FDTD, but retain all other benefits. Sine Cosine Method e j cos jsin Re Ez A Ez B jez D e Im Ez C Ez B jez D e jk jk A B C D Re H B H A jh C e Im H D H A jh C e jk jk Lecture 19 Slide 26 13

14 Case #4: All Conditions (1 of 4) There is no known general solution. There does eist several limited solutions. Method #1: Multiple Unit Cell Method PBC requires past time values Error from ABC Grid serves as a record of the fields at A PBC would require future values. Lecture 19 Slide 27 Case #4: All Conditions (2 of 4) Method #2: Angled Update Method (1 of 2) T 5 T 4 T 4 T 3 T 3 T 3 T 2 T 2 T 2 T 2 T 1 T 1 T 1 T 1 T 1 Envision a grid where all the field components eist at T=1. We could then update all the field components up to some slanted boundar. Again, we could update all the field components up to some slanted boundar that is awa from the first slanted boundar. And again We have now built a time gradient into the grid. Lecture 19 Slide 28 14

15 Case #4: All Conditions (3 of 4) Method #2: Angled Update Method (2 of 2) This boundar requires fields from future time values. The right side of grid contains field values at future time steps. This boundar requires fields from past time values. The left side of the grid contains field values at previous time steps. From here, we iterate over the whole grid ver much like the standard FDTD algorithm. The difference is that we store the boundar fields for a few iterations from which we interpolate the field at whatever time value is needed. Lecture 19 Slide 29 ma ma 45 for 2D 35 for 3D Case #4: All Conditions (4 of 4) Field Transformation Technique Larger angles possible than for angled update method Difficult to implement Stabilit is an issue See Tet, pp Split Field Method Difficult to implement Stabilit is an issue See Tet, pp Lecture 19 Slide 30 15

16 Electromagnetic Band Calculation using FDTD Lecture 19 Slide 31 Band Diagrams Band diagrams relate frequenc to the direction and period of a Bloch wave propagating inside a periodic structure. The indicate which frequencies support waves with a given direction and wavelength in that periodic structure. Lecture 19 Slide 32 16

17 Computation of Band Diagrams Band diagrams are an efficient, but incomplete, means of characterizing the electromagnetic properties of a periodic structure. Given the unit cell and Bloch wave vector, Mawell s equations are solved to identif the frequencies of all the Bloch waves with that vector. The band diagram is constructed b repeating this computation over a range of Bloch wave vectors. Lecture 19 Slide 33 Animation of Construction of a Band Diagram Lecture 19 Slide 34 17

18 Benefits and Drawbacks of FDTD for Band Calculations Benefits Wideband Can account for dispersion (ver unique) Ecellent for large unit cells Ecellent for structures with metals or high dielectric contrast Drawbacks A large number of iterations is needed to accuratel identif bands Difficult to distinguish bands when the are in close proimit (degenerate modes) No guarantee that all modes are ecited and/or detected Lecture 19 Slide 35 Revised PBC for Band Calculations The generalized periodic boundar condition for a 3D periodic structure is E r t E r e j tpqr pqr Recall that periodic boundar conditions are incorporated into the curl equations. For band calculations, this is C C C z z z z in,, k i,1, k i, jk, i, jk, 1 i, jk, i, j, Nz i, j1, k i, jk, z i, j,1 i, jk, E CE i, j, Nz E E E E E E E E E z z z z ze i, j,1 E i, j, k E i1, j, ke i, j, k E N, j, k i, j, k1 E z z i, j, k E jke CE z z E E E E in,, k E E E E CEz 1,, i, jk, N, j, k 1, j, k i, j, k i, j 1, k i, j, k i 1, j, k i, j, k i,1, k i, j, k Ez e e e z j j jzz C C C H H H H z * z z z * i,1, k i, j, k i, N, k Hi, j, khi, j, k1 H,,1 i, j, kh i j i, j1, k i, j, kz i, j, Nz H CH z z z * z H H H H H H Hi, j, k H N z z * * Hi, j, k HN, j, k Hi, j, khi, j 1, k H H i, j, k H i 1, j, k H CHz * z z i, j,1 i, j, k z i, j, Nz i, j, k i1, j, k 1, j, k i, j, k i, j, k1 H CH 1, jk, i, 1, k Hz, jk, i, j, k i, N, k e * e * e * z j j jzz Lecture 19 Slide 36 18

19 Implementation of PBC in MATLAB % Calculate Phase Across Grid phi = ep(-1i*b*s); phi = ep(-1i*b*s); e e j j C C 1, jk, Hz i,1, k Hz H H H H H * i, j, k N, j, k i, j, k i, j1, k H i, j, k i1, j, k H H * i, j, k i, N, k % Compute CHz CHz(1,1) = (H(1,1) - conj(phi)*h(n,1))/d... - (H(1,1) - conj(phi)*h(1,n))/d; for n = 2 : N CHz(n,1) = (H(n,1) - H(n-1,1))/d... - (H(n,1) - conj(phi)*h(n,n))/d; end for n = 2 : N CHz(1,n) = (H(1,n) - conj(phi)*h(n,n))/d... - (H(1,n) - H(1,n-1))/d; for n = 2 : N CHz(n,n) = (H(n,n) - H(n-1,n))/d... - (H(n,n) - H(n,n-1))/d; end end Lecture 19 Slide 37 Implications of Revised PBC FDTD for band calculations is ver much like the sin cosine method Field values in the grid will be comple. This is not a problem for MATLAB and FORTRAN Other languages require separate grids for real and imaginar components (i.e. sin cosine method) While FDTD for band calculations is wideband, the PBCs are possible because the field is periodic and the spatial period of the wave is fied. Lecture 19 Slide 38 19

20 The Source For band calculations in FDTD, we use simple dipole sources that are randoml polarized and randoml distributed throughout the unit cell. It is best to avoid locations that are obvious smmetr points. This is done to ensure that all possible modes are ecited. Dipole source Lecture 19 Slide 39 Record Points Similarl, we record the response at multiple record points that are distributed randoml throughout the lattice like the source points. Record point Lecture 19 Slide 40 20

21 Identifing the Bloch Mode Frequencies First, we FFT the record arras to calculate the power spectral densit recorded at each record point. p PSD FFT 2 E t p source # p z Second, we add the power spectral densities from all the recorded points. PSD PSD p p Third, frequencies corresponding to Bloch modes are identified as sharp peaks in the overall PSD. indicates an eigen frequenc Lecture 19 Slide 41 Animation of Calculating Eigen Frequencies An arbitrar Bloch wave vector was chosen for this simulation. Peaks are identified in the PSD b the red circles. A larger number of iterations is needed before the peaks can be identified accuratel. Poor, or unluck, selection of source and record points can fail to ecite or detect certain bands. r 0.35a r Lecture 19 Slide 42 r 2 r 2 r1 a r a 21

22 Calculating the Fields of the Bloch Modes Step 1: Run a simulation and identif the eigen frequencies of the modes ou are interested in. Step 2: Run a second simulation and calculate the stead state field at each eigen frequenc of interest at each point throughout the grid. Lecture 19 Slide 43 Procedure for Calculating Band Diagrams Build unit cell on a grid Iterate over a list of Bloch wave vectors Initialize random sources and record points Run FDTD and record fields at record points Compute overall PSD Identif eigen frequencies Plot the eigen frequencies as a function of the Bloch wave vector You have now produced an electromagnetic band diagram using FDTD! Lecture 19 Slide 44 22

23 Eample 3D Simulation w 0.1a 2.34 r Lecture 19 Slide 45 Animation of Band Diagram Construction Using FDTD Eact bands calculated using the plane wave epansion method. Eigen Frequencies Lecture 19 Slide 46 23