Finite-Difference Time-Domain Simulation of Light Propagation in 2D Periodic and Quasi-Periodic Photonic Structures

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1 JNS 3 (213) Fiite-Differece Time-Domai Simulatio of Light Propagatio i 2D Periodic ad Quasi-Periodic Photoic Structures N. Dadashadeh a,b*, O.G. Romaov b a Islamic Aad Uiversity of Hadishahr, Hadishahr, Ira b Belarusia State Uiversity, Neavisimostiave. 4, 223 Misk, Belarus Article history: Received 13/7/213 Accepted 11/11/213 Published olie 1/12/213 Keywords: photoic crystals umerical modelig Mawell s equatios FDTD method *Correspodig author: address: oushidadashadeh@yahoo.com Phoe: Fa: Abstract Ultra-short pulse is a promisig techology for achievig ultrahigh data rate trasmissio which is required to follow the icreased demad of data trasport over a optical commuicatio system. Therefore, the propagatio of such type of pulses ad the effects that it may suffer durig its trasmissio through a optical waveguide has received a great deal of attetio i the recet years. We provide a overview of recet theoretical developmets i a umerical modelig of Mawell's equatios to aalye the propagatio of short laser pulses i photoic structures. The process of short light pulse propagatio through 2D periodic ad quasi-periodic photoic structures is simulated based o Fiite-Differece Time-Domai calculatios of Mawell s equatios. 213 JNS All rights reserved 1. Itroductio The fiite-differece time-domai method (FDTD) is a very powerful techique for umerical aalysis of Mawell s equatios [1]. Due to its accuracy, the FDTD method is widely used for simulatio of light propagatio i optical waveguides, scatterig media or photoic crystals [2, 3]. Although the first studies of optical couplig processes i waveguide arrays were performed i early 196 s [4, 5], it was oly recetly that highcotrast dielectric elemets became available which made possible creatio of photoic crystal structures [6]. Recetly, discretied light propagatio i photoic lattices has attracted a lot of iterest [7-11]. I this work, we have used the FDTD method to aalye the processes of light propagatio i dielectric media with differet microstructures embedded i them. I particular, the tasks of the light pulses propagatio i two-dimesioal (2D) coupled-waveguides, periodic ad quasi-periodic structures have bee eamied umerically. The cases of differet polariatio state of

2 36 N. Dadashadeh et al./ JNS 3 (213) electromagetic waves (TE-, ad TM-modes) have bee studied, ad the space distributios of the amplitudes for electric ad magetic vectors, as well as for the eergy desity of the electromagetic field have bee calculated. The ifluece of the parameters of the light pulses (wavelegth, duratio of pulse, trasverse sie of the beam) ad the parameters of the medium (the values of refractive ide of dielectric layers, its geometrical sie ad mutual arragemet) o trasmissio characteristics of the electromagetic field have bee aalyed. As a result, the mai features of eergy trasitio modes betwee two coupled waveguides ( pedulum mode ), ad discrete diffractio i the system of several coupled waveguides have bee studied thoroughly. I additio, the features of the short light pulses propagatio i quasi-periodic photoic structures with a labyrithie distributio of the refractive ide have bee aalyed based o the previously developed techique [12]. 2. Theoretical model Light propagatio: Fiite-differece time-domai (FDTD) method The fiite-differece time-domai techique or FDTD-method is a widely used techique that umerically solves Mawell s equatios with a high accuracy, etailig a cosiderable computig time [1]. This method allows trackig the spatial periodic evolutio of the electromagetic field iside of the eviromet with a arbitrary distributio of the dielectric coductivity. I this sectio, we show that with the usig of FDTD method we ca aalye the propagatio of short laser pulses i periodic ad quasi-periodic structures. Let s cosider Mawell s equatios for a optical medium H E, (1) t E, y, H, (2) t These equatios ca be decomposed ito the three coordiate compoets to obtai a set of si differetial equatios: H 1 E E y t y H y 1 E E t, (3), (4) H 1 Ey E t y E 1 H H y t, y, y Ey 1 H H t, y, E 1 H y H t, y, y, (5), (6), (7). (8) From ow o, a 2D cofiguratio will be cosidered, i such a way that the electrical permittivity is reduced to ε = ε (, y). With this situatio, the differetial equatios for TE mode are: H 1 E, (9) t y H y 1 E t E 1 H y H t, y y, (1). (11) The fiite differece approimatio of equatio sets (9) (11) implies a set of equatios that ca be solved eplicitly [2]. Accordig to FDTD theory, these differetial fuctios are segregated i both space ad time, so that they ca be calculated as: H 1/2 i, j 1 / 2 t, (12), 1 / 2, 1, 1/2 y H i j E i j E i j

3 N. Dadashadeh et al./ JNS 3 (213) /2 y, (13) 1 / 2, 1,, H i 1 / 2, j t H i j E i j E i j 1 1/2 y,, E i j E i j t 1 1/2 1/2 H y i 1 / 2, j H y i 1 / 2, j i, j t 1 1/2 1/2 H i j H i j i, j y I these equatios, 1 / 2, 1 / 2 ad (14) y are the spatial mesh steps alog the coordiates ad y, respectively, t is the time step. The followig separatio for the desired fuctios is used here:,,,,, F i j F i jy t F y t. O the other had, the equatios for TM mode are: E 1 H t, y y Ey 1 H t, y H 1 Ey E t y, (15), (16). (17) Agai, after separatio of these fuctios accordig to FDTD algorithm, we have: 1 1/ 2, 1/ 2, E i j E i j 1/2 t 1 H i 1/ 2, j 1 / 2 1/2 i, j y H i 1/ 2, j 1 / 2, (18) 1 y, 1/ 2, 1/ 2 E i j E i j y 1/2 t 1 H i 1/ 2, j 1/ 2 1/2 i, j H i 1/ 2, j 1 / 2, (19) 1 / 2, 1 / 2 1 / 2, 1 / 2 1/2 1/2 H i j H i j 1 Ey i 1, j 1 / 2 Ey i, j 1 / 2 t 1 E i 1 / 2, j 1 E i 1 / 2, j y (2). 3. Results ad discussio Numerical modelig of the problems for light beam propagatio i the media with microstructures of refractive ide has bee performed for the followig cases: 1) propagatio of light pulses i the system of two coupled waveguides, 2) propagatio of radiatio i the system of N coupled waveguides, ad 3) propagatio of radiatio i photoic structures with labyrithie distributio of refractive ide. I the umeric modelig, it is assumed that the source of the electromagetic field i the form of a quasimoochromatic wave with time pulse ad spatial profile of Gaussia-like shape is started from the left border of the computatio domai. The time evolutio of spatial distributio of compoets of electric ad magetic fields, as well as the eergy desity has bee calculated, ad the results are preseted i Figs. 1 3.

4 362 N. Dadashadeh et al./ JNS 3 (213) The mode of eergy couplig of two closely positioed plaar waveguides is demostrated i Fig. 1. This task has bee cosidered for the case, whe the wavelegth of icidet electromagetic wave λ = 1μm, pulse duratio τ p =2 fs, diameter of waveguides d 1 = d 2 = 1μm, the distace betwee waveguides d = 1μm. Plaar waveguides are formed by the medium with refractive ide 2 =1.5 (Fig. 1, a, dark area) embedded ito the medium with refractive ide 1 =1.45 (white area). The TEpolariatio of radiatio has bee cosidered. Aswe ca see, iitially the light beam is focused to the bottom waveguide, ad propagates alog it (Fig. 1, b). After passig so-called couplig distace, the eergy of the light pulse is almost completely trasferred to the upper waveguide (Fig. 1, c) due to spatial itersectio of trasverse waveguide modes. The reverse process is takig place i doubled couplig distace (Fig. 1, d) providig the realiatio of pedulum mode for eergy echage betwee the two waveguides. The typical results of umerical modelig for the mode of discrete diffractio i a system of N coupled waveguides are show i Fig. 2 for TEpolariatio of icidet optical radiatio. The wavelegth of icidet electromagetic wave λ = 1μm, pulse duratio τ p =2 fs. The system of Plaar waveguides (Fig. 2, a) are formed by the mediums with refractive ide 1 =1.45 ad 2 =1.5. The width of each waveguides d 1 = 1μm, ad the distace betwee waveguides are d = 1μm. Iitially localied i the cetral waveguide (Fig. 2, b) the light pulse the is divided ito two spatially symmetrical pulses localied i eighbor waveguides (Fig. 2, c). Net divisio of pulses Fig. 1.Two coupled waveguides structure (a), ad spatial distributio of the eergy desity of the electromagetic field (b d). t = 126 fs (b), 42 fs (c), 1 fs (d). λ = 1μm, τ p =2 fs, 1 = 1.45, 2 = 1.5. (Fig. 2, d) leads to eergy trasfer to the outer waveguides, while the coheret elimiatio of light pulse i the cetral waveguide takes place. The aalysis results of the quasi-moochromatic light pulses propagatio kietics i labyrithie like photoic structures are preseted i Fig. 3. The labyrithie structures (Fig. 3, a) are characteried by the presece of short-distace ad the abseceof log-distace order. Therefore, these structures ca be cosidered as trasiet structures betwee photoic crystals, comple-structured waveguides, a b c d

5 N. Dadashadeh et al./ JNS 3 (213) a a b b c c d e Fig. 2. coupled waveguides structure (a), ad spatial distributio of the eergy desity of the electromagetic field (b d). t = 126 fs (b), 42 fs (c), 1 fs (d). λ = 1μm, τ p =2 fs, 1 = 1.45, 2 = 1.5. ad disordered systems. Based o the umerical eperimets performed, it is show that thedistributio of the eergy desity of the electromagetic field is characteried by a comple brachig structure (Fig. 3, b e). The structures applicatio possibilities are aalyed with referece to icrease the optical iformatio storage time ad localiatio of the electromagetic field eergy i the photoic structure [12]. d Fig. 3. Labyrithie photoic structure (a), ad spatial distributio of the eergy desity of the electromagetic field. (b e). t = 25 fs (b), 5 fs (c), 75 fs (d), 1 fs (e). λ = 1μm, τ p =1 fs, 1 = 1, 2 = 2; the computatioal domai 1 λ 1 λ. 4. Coclusio This article provides a overview of theoretical aalysis that has bee performed i order to evaluate the light propagatio i 2D photoic structures. We show that the FDTD method is well suited for studyig of the pulse propagatio i 2D photoic structures, ad a very useful tool for ivestigatig oliear pulse propagatio ad its iteractio with the media. Refereces [1] A. Taflove, S.C. Hagess, Computatioal Electrodyamics: the Fiite-Differece Time- Domai Method, 2, Norwood, MA: Artech. [2] K. Kawao, T. Kitoh, Itroductio to Optical Waveguide Aalysis: Solvig Mawell s Equatios

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