Finite-Difference Time-Domain Method (FDTD)
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1 Computatioal Photoics Semiar 05 Fiite-Differece Time-Domai Method (FDTD) Lear how to implemet a D versio of the FDTD method ted the code to 3D problems
2 D FDTD: Yee Grid for & Compoets Chagig of ide otatio to iteger idices t().5 t() (i) 3 (i) start at =: start at = ( =0, =0) : t t i i i i i 0i 0i t i i 3 i 0 j i +½ t t j i i i i i 0i 0i t i i i i 0
3 3 D FDTD: Source Separable source: j i = A Δt + Spatial distributio: j t = 0 i Carrier e πifδt(+ Τ) velope: A(Δt + Τ ) Τ e πifδt + Τ j t = 0 ቚ i
4 temporal ide Computatioal Photoics, Prof. Thomas Pertsch, Abbe School of Photoics, FSU Jea 4 D FDTD: Laout of the Matlab Arras boudar values N N N N N.5 N N N N N N N N N N iitial values ε j ε j ε j.5.5 ε N j N ε N j N ε N jn N- N-.5 N- N-0.5 N spatial ide i
5 5 3D FDTD: Yee-Grid Ceter of the cube is i the ceter of the coordiate sstem ( ), j () ε, j, j (j) (i) Grid sie is determied b the permittivit distributio: sie ε = [N, N, N ]
6 6 3D FDTD: lectric Field Compoets, j ε, j, j () (j) Permittivit must be iterpolated: i0.5, j, i, j, i, j0.5, i, j, i, j, 0.5 i, j, (i) , 0.5, 0.5, 0.5, t i j i j i0.5, j, 0.5 i0.5, j, i 0.5, j, j i0.5, j, i0.5, j, 0 i0.5, j, j t, 0.5, 0.5, 0.5, 0.5 i j i j i0.5, j0.5, i0.5, j0.5, 0.5 i, j 0.5, i, j0.5, i, j 0.5, 0i, j0.5, ,, ,, 0.5 t i j i j i, j0.5, 0.5 i, j0.5, j 0. 5 i, j,
7 7 3D FDTD: Magetic Field Compoets, j (), j ε, j (j) (i) t, 0.5,, 0.5, i j i j i, j, i, j 0.5, 0.5 i, j0.5, t,, 0.5,, 0.5 i j i j i0.5, j, i0.5, j, i 0.5, j, 0.5 i0.5, j, ,, 0.5,, t i j i j i, j0.5, i, j0.5, i 0.5, j 0.5, i0.5, j0.5, 0
8 3D FDTD: lectric Field Compoets Chage Ide Notatio to Iteger Idices 8, j, ε () Reamig of fractioal idices: i0.5 i j0.5 j 0.5, j, ε ε, j, ε (i) (j) Reamig of iterpolated permittivit: i0.5, j, i, j0.5, 0.5,,,, t i j i j j 0 j t,,,, i j i j i, j,,, 0 i j,,,, t i j i j i, j, j 0
9 3D FDTD: Magetic Field Compoets Chage Ide Notatio to Iteger Idices 9, j, ε () Reamig of fractioal idices: i0.5 i j0.5 j 0.5 ε, j, ε, j, ε (j) (i) t,, i j i, j, 0 t,, i j 0,,,, t i j i j i, j, 0
10 0 3D FDTD: Arra Sies ad Boudar Coditios Permittivit grid ad output grid: sie ε = [N, N, N ] Fields: Tagetial -fields ad ormal -fields are stored at iteger idices : N N grid poits Normal -fields ad tagetial -field are stored at fractioal idices.5: N 0.5 N grid poits Arra sies: : N, N, N ; : N, N, N ; : N, N, N ; : N, N, N ; : N, N, N ; : N, N, N ; PC boudar coditios: At the boudaries the tagetial -fields ad the ormal -fields are set to ero ad are ot updated
11 3D FDTD: Arra Sies ad Boudar Coditios PC boudar coditios: At the boudaries the tagetial -fields ad the ormal -fields are set to ero ad are ot updated :,, : = 0 :, N, : = 0 :,, = 0 :, :, N = 0, :, : = 0 N, :, : = 0, :, : = 0 N, :, : = 0 :,, = 0 :, :, N = 0 :,, : = 0 :, N, : = 0, :, : = 0 N, :, : = 0 :,, : = 0 :, N, : = 0 :, :, = 0 :, :, N = 0
12 3D FDTD: Time Steppig Update of the lectric Field, j, ε, j, ε ε, j, ε () (j) Seperable source: 0.5 i t 0.5 ( 0) j A t e j t 0.5 i t 0.5 ( 0) j A t e j t 0.5 i t 0.5 ( 0) j A t e j t (i),,,, t i j i j j 0 i : N j : N : N j t,,,, i j i j i, j,,, 0 i j i : N j : N : N,,,, t i j i j i, j, j 0 Tagetial -fields at boudar are ot updated! i : N j : N : N
13 3D FDTD: Time Steppig Update of the Magetic Field 3, j, ε () ε, j, ε, j, ε (j) (i) t,, i j i, j, 0 i : N j : N : N t,, i j 0 i : N j : N : N,,,, t i j i j i, j, 0 Normal -fields at boudar are ot updated! i : N j : N : N
14 4 3D FDTD: Iterpolatio of Output For postprocessig purposes it is desirable to have all fields o a commo grid i space ad time fields must be iterpolated (e.g. to the iteger grid where ε is give), j (), j ε, j (j) (i) Field Iterpolated Aes Field Iterpolated Aes,, t,, t,, t
15 5 3D FDTD: Iterpolatio of Output For postprocessig purposes it is desirable to have all fields o a commo grid i space ad time fields must be iterpolated (e.g. to ε-grid) out,, i j i, j, out,, i j i, j, out,, i j, j ε, j, j () (j) 8 out 8 out 8 i, j, i, j, i, j, out i, j, i, j, i, j, i, j, i, j, i, j, i, j, i, j, (i) i, j, i, j, i, j,
16 6 3D FDTD: Iterpolatio of Output What about missig values at the boudaries?.g.: Iterpolatio of out (, :, : ) requires out 0, :, : Iterpolatio of out (:,, : ) requires out :, 0, : Iterpolatio of out (:, :, ) requires out :, :, 0 At the PC boudar the followig mirror smmetries hold: = +, = + = + +, = Missig values behid the boudar ca be obtaied b duplicatig the values i frot of the boudar PC 0.5
17 Useful Matlab fuctios: roud, cat, meshgrid, drawow, subplot, imagesc Useful costats: c = m/s, µ 0 = 4π 0 7 Vs/Am, ε 0 = /(c µ 0 ) As/Vm 7 Tas I: Implemetatio of the D FDTD method Phsical problem: Simulate the propagatio of a ultrashort pulse i a dispersio-free dielectric medium ε = See what happes whe the pulse hits the iterface betwee two differet dielectric media with permittivities ε = ad ε = 4, the iterface should be located at a distace of 4.5 µm i positive directio from the ceter of the computatioal domai citatio: Pulsed source with frequec f = 500T (red light) delta-shaped spatial profile j t = 0, = j 0 δ 0 with j 0 = A/m located at the ceter of the computatioal domai at 0 = 0 Gaussia temporal evelope A t = ep( (t t 0 ) /τ ) with τ = fs ad t 0 = 3τ Simulatio grid: Spatial widow sie of W = 8 µm with discretiatio Δ = 5 m ad metallic walls ( = 0 at the boudaries) Simulatio time spa T = 60fs with discretiatio Δt = Δ /(c) Output: (, t) ad (, t) at ever time step iterpolated to the iteger grid both i space ad time
18 8 Tas I: Implemetatio of the D FDTD method Please iclude relevat plots of the fields (e.g. sapshots a certai time steps, time traces) i our report but do ot iclude or submit video files!
19 9 Tas I: Implemetatio of the D FDTD method fuctio [,,,t] = fdtd_d(eps_rel, d, time_spa,... source_frequec, source_positio,... source_pulse_legth) % fuctio [,,,t] = fdtd_d(eps_rel, grid_sie, time_spa,... % source_frequec, source_positio,... % source_pulse_legth) % Computes the temporal evolutio of a pulsed ecitatio usig the % D FDTD method. The temporal ceter of the pulse is placed at a % simulatio time of 3*source_pulse_legth. The origi =0 is i the % ceter of the computatioal domai. All quatities have to be % specified i SI uits. % Argumets: % eps_rel : rel. permittivit distributio withi the % computatioal domai (vector) % d : spacig of the simulatio grid % (scalar, please esure d <= lambda/0) % time_spa : time spa of simulatio (scalar) % source_frequec : frequec of curret source (scalar) % source_positio : spatial positio of curret source (scalar) % source_pulse_legth : temporal width of Gaussia evelope of % the source (scalar) % Returs: % : compoet of (,t) (matri, each colum correspods to oe % time step) % : compoet of (,t) (matri, each colum correspods to oe % time step) % : spatial coordiates of the field output (vector) % t : time of the field output (vector)
20 0 Tas I: Implemetatio of the D FDTD method You ca use the provided aimatio fuctio to watch a movie of the fields fuctio fig = fdtd_d_aimatio(, t,,, _iterface, step,... fps, fileame) % fig = fdtd_d_aimatio(, t,,, _iterface, step, fps, fileame) % % Creates a aimatio of the D FDTD fields. % % Argumets: % : Spatial coordiates (vector) % t : Time (vector) % : field to aimate (matri, each colum correspods % to oe time step) % : field to aimate (matri, each colum correspods % to oe time step) % _iterface : Positio of the iterface (scalar) % step : Time step betwee frames (scalar) % fps : Frames per secod (iteger, default: 5) % fileame : Fileame of the video. A pg image of the % last frame is also save to the fileame with the % etesio replaced b '.pg'. If empt, o files % are saved to dis. (strig, default: '') % Returs: % fig : Figure of the aimatio.
21 Tas II: Implemetatio of the 3D FDTD method Phsical problem: Ivestigate the radiatio characteristics of a pulsed lie curret with a Gaussia spatial evelope j,,, t = j 0 ep πift ep t t 0 + ep Simulatio grid: Spatial domai sie of grid poits with a step sie of Δ = Δ = Δ = 30 m PC boudar coditios Simulatio time spa T = 0 fs with discretiatio Δt = Δ /(c) Specif all iput quatities (ε r, j r, j r ad j r ) o the same cetered iteger grid ad iterpolate the quatities to the required shifted grids withi the implemetatio citatio: Pulsed curret source with amplitude j 0 = A/m, frequec f=500 T (red light), temporal width τ = fs ad offset t 0 = 3τ ad spatial width w = Δ Output: ad i the -plae cetered i the middle alog the -directio at ever 4th time step iterpolated to the iteger grid i space ad time Useful Matlab fuctios: mod, cat τ w e
22 Tas II: Implemetatio of the 3D FDTD method Please iclude relevat plots of the fields (e.g. sapshots at t = T) i our report but do ot iclude or submit video files!
23 3 Tas II: Implemetatio of the 3D FDTD method fuctio [F, t] = fdtd_3d(eps_rel,dr,time_spa,freq,tau,j,j,j,... field_compoet,_id,output_step) % fuctio [F, t] = fdtd_3d(eps_rel,dr,time_spa,freq,tau,j,j,j,... % field_compoet,_id,output_step) % Computes the temporal evolutio of a pulsed spatiall eteded curret % source usig the 3D FDTD method. Returs -slices of the selected % field at the give -positio ever output_step time steps. The pulse % is cetered at a simulatio time of 3*tau. All quatities have to be % specified i SI uits. % Argumets: % eps_rel : rel. permittivit distributio withi the % computatioal domai (3D arra) % dr : grid spacig % (scalar, please esure dr<=lambda/0) % time_spa : time spa of simulatio (scalar) % freq : ceter frequec of the curret source (scalar) % tau : temporal width of Gaussia evelope of % the source (scalar) % j, j, j : spatial desit profile of the curret source % (3D arras) % field_compoet : field compoet which is stored % (oe of e, e, e, h, h, h ) % _ide : -positio of the field output (iteger) % output_step : umber of time steps betwee field outputs % (iteger) % Returs: % F : -slices of the selected field compoet at the % -positio specified b _id stored ever output_step % time steps (3D arra, time varies alog the last dimesio) % t : time of the field output (vector)
24 4 Tas II: Implemetatio of the 3D FDTD method You ca use the provided aimatio fuctio to watch a movie of the fields fuctio fig = fdtd_3d_aimatio(,, t, F, titlestr, cb_label,... rel_color_rage, fps, fileame) % fig = fdtd_3d_aimatio(,, t, F, titlestr, cb_label,... % rel_color_rage, fps, fileame) % % Creates a aimatio of a 3D FDTD field. % % Argumets: %, : Coordiate aes (vectors) % t : Time (vector) % F : Slices of the field to aimate (3d-arra, % the time ais is assumed to be the last ais % of the arra) % titlestr : Plot title (strig) % cb_label : Colorbar label (strig) % rel_color_rage : Rage of the colormap relative to the full scale % of the field magitude (scalar) % fps : Frames per secod (iteger, default: 5) % fileame : Fileame of the video. A pg image of the % last frame is also save to the fileame with the % etesio replaced p '.pg'. If empt, o files % are saved to dis. (strig, default: '') % Returs: % fig : Figure of the aimatio.
25 5 omewor 4 (Jue 5, 08) Solve tass I & II. Prepare a short report about our solutio with represetative figures ad a short discussio of our results The source code ad the report must be submitted via to teachig-aooptics@ui-jea.de b Frida (Jue 5, 08), 3 AM (sharp!) The subject lie of the should have the followig format: [famil ame]; [give ames]; [studet id]: CPho8 - solutio to the assigmet of semiar [semiar o.] The report should be a pdf file All source code files should be gathered i a sigle ip archive (o rar, tar, 7, g or a other compressio format!) O Jue 5, the solutios of the tass will be available olie at the lectures homepage >>> Computatioal Photoics. You are epected to solve the tas ourself ad a declaratio of idepedet wor must be siged b ever studet at the ed of the semester.
Finite-Difference Time-Domain Method (FDTD)
Computatioal Photoics, Summer Term 04, Abbe School of Photoics, FSU Jea, Prof. Thomas Pertsch Computatioal Photoics Semiar 05, 0 Jue 04 Fiite-Differece Time-Domai Method (FDTD) Lear how to implemet a D
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