Simulation of coupling optical modes in 1D photonic crystals for optoelectronic applications

In this paper, one-dimensional photonic crystal distributed feedback structures were chosen for simulating the photonic modes. The corresponding photonic bands were calculated by using a numerical method for solving the master equation, while the reflectivity spectra of the structures were simulated by using a rigorous coupled wave analysis method. By observing the variation of the photonic band diagram and the reflectivity spectrum versus different geometrical parameters, the variation of the photonic bands was detailedly studied. We observed two kinds of photonic modes: (i) the one related to the vertical structures, and (ii) the other related to the horizontal periodic structures. The detailed analysis of the optical modes was illustrated by proposing TE± n;;XmBZ for indexing all transverse electric modes. An active layer coated on the distributed feedback structures plays an essential role in having radiative non-leaky photonic modes. The coupling between these modes, giving to anti-crossing, was also identified both by simulation and by modelling. This study can pave a way for further modelling optical modes in distributed feedback structures, and for selecting a suitable one-dimensional photonic crystal for optoelectronic applications with a specific active semiconductor layer.

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1anoi and Article history: Accepted 30 January 2020 Available online 7 February 2020 In this paper, one-dimensional photonic crystal distributed feedback structures were chosen for simu- method for solving the master equation, while the reflectivity spectra of the structures were simulated where light propagates in a particular behavior [1,2]. In the simplest photonic crystals with a discontinuous dielectric modulation, layered media, where there is at least one layer with periodic the relative effective refractive index of the guided layer with ers, there may be m “leaky” is used e whose refractive ayers, whereas the refractive index is odes, “leaky” and condition of light are called wave- guided modes, and they are interesting subjects for mathematical DFB structures play important roles in optoelectronics, espe- cially for application in lasing. Because of the multiple periodic reflection of light in a DFB structure, an optical gain can be obtained when an active layer is introduced along the light propagation di- rection [12e14]. In a 1D or 2D DFB structure, lasing effects occur in both the periodic direction [12], often called wave-guided DFB modes or first order modes, and the perpendicular direction, often * Corresponding author. Nano and Energy Center, VNU University of Science, Room 503, 5th floor, T2 building, 334 Nguyen Trai street, Thanh Xuan, Hanoi, Viet Nam. Fax: þ84 435 406 137. E-mail address: thuatnt@vnu.edu.vn (T. Nguyen-Tran). Contents lists available at ScienceDirect Journal of Science: Advanc journal homepage: www.el Journal of Science: Advanced Materials and Devices 5 (2020) 142e150Peer review under responsibility of Vietnam National University, Hanoi.known as distributed feedback (DFB) structures, are often made of as well as technological points of views [1e3].understanding, this behavior can be characterized by waves which are propagating in opposite directions and coupled by the reflec- tion of light from the periodic interfaces between the media of different refractive indices [3]. This coupling in turn gives raise to anti-crossing between the optical modes, thus creating forbidden bands for light in photonic crystals. As a consequence, the light behaviors in photonic crystals are the same as that of electrons in a crystalline solid. One-dimensional (1D) or two-dimensional (2D) respect to the underneath and the overlying lay leaky or confined wave-guided modes. The ter here to describe the optical modes in a waveguid index is smaller than that of one of the cladding l term “confined” is used for a waveguide whose higher than that of the cladding layers. Both m “confined”, correspond to a phase matching reflection in a waveguide. By convention, theyPhotonic crystals are periodic dielectric modulation media bound optical modes in the corresponding layers. Depending onKeywords: 1D photonic crystal DFB structure Angle e resolved reflectivity Photonic band diagram Coupling waves 1. Introductionhttps://doi.org/10.1016/j.jsamd.2020.01.008 2468-2179/© 2020 The Authors. Publishing services b ( and the reflectivity spectrum versus different geometrical parameters, the variation of the photonic bands was detailedly studied. We observed two kinds of photonic modes: (i) the one related to the vertical structures, and (ii) the other related to the horizontal periodic structures. The detailed analysis of the optical modes was illustrated by proposing TE±;mBZn;X for indexing all transverse electric modes. An active layer coated on the distributed feedback structures plays an essential role in having radiative non-leaky photonic modes. The coupling between these modes, giving to anti-crossing, was also identified both by simulation and by modelling. This study can pave a way for further modelling optical modes in distributed feedback structures, and for selecting a suitable one-dimensional photonic crystal for optoelectronic applications with a specific active semiconductor layer. © 2020 The Authors. Publishing services by Elsevier B.V. on behalf of Vietnam National University, Hanoi. This is an open access article under the CC BY license ( variation of the refractive index [4e11]. The light propagation in these low-dimensional photonic crystals can be considered asReceived in revised form 30 January 2020 by using a rigorous coupled wave analysis method. By observing the variation of the photonic bandReceived 9 April 2019 lating the photonic modes. The corresponding photonic bands were calculated by using a numericalOriginal Article Simulation of coupling optical modes in optoelectronic applications Ngoc Duc Le a, b, Thuat Nguyen-Tran a, * a Nano and Energy Center, VNU University of Science, 334 Nguyen Trai, Thanh Xuan, H b Department of Advanced Materials Science and Nanotechnology, University of Science Technology, 18 Hoang Quoc Viet, Cau Giay, Hanoi, Viet Nam a r t i c l e i n f o a b s t r a c ty Elsevier B.V. on behalf of VietnamD photonic crystals for , Viet Nam Technology of Hanoi, Vietnam Academy of Science and ed Materials and Devices sevier .com/locate/ jsamdNational University, Hanoi. This is an open access article under the CC BY license called radiative DFB modes or higher order modes [15]. These modes are generally eigenmodes of the master equation of light in photonic crystals. Resolving the master equation is a very difficult task, thus in order to understand the DFB modes, the simulation by using the transfer matrix methods in periodic multilayered struc- tures can be performed [16,17]. The information obtained from the eigenmodes is essential for better applications in laser diodes based on the conventional III-V semiconductor compounds [7,18,19] or on the novel hybrid organic-inorganic semiconductors [20e25], as well as in light-matter coupling phenomena [26]. In this paper, we present a simulation andmodelization study of the photonic modes of a 1D DFB photonic crystal. The work was photonic band diagrams and the reflectivity spectra of each series as well as comparing between the bare and the active layer coated structures, general trends were drawn for possible strong coupling applications. Noting that the horizontal component kx of the wave vector refers to the component of the wave vector along the peri- odic direction of the DFB structures. 3. Results and discussion 3.1. Influence of the period of the DFB structures N.D. Le, T. Nguyen-Tran / Journal of Science: Advanced Materials and Devices 5 (2020) 142e150 143carried out on the bare DFB structures as well as the same struc- tures covered with an active layer. The presence of the active layer emphasizes the outlet of the DFB structures simulated here for future optoelectronic applications, where the active layer can be a semiconductor material. The simulation of the optical modes was carried out by using the rigorous coupled wave analysis (RCWA) method; and was compared with a two-wave coupling model as well as an attempt of eigenvalues calculation of themaster equation in the simplest manner. Results obtained in the paper could pave the way for using the DFB structures for lasing devices and light- matter interaction effects. 2. Simulation methods Fig. 1 shows the one-dimensional photonic crystal of a comb- like DFB structure studied here. A typical periodic structure is made of SiO2 (refractive index n2 ¼ 1.46) on a silicon substrate (refractive index ns ¼ 3.97). There are two types of DFB structures for each simulation: (i) the bare structure (Fig. 1a), and the active layer (refractive index n1 ¼ 2.16) coated structure (Fig. 1b). Noting that the active layer studied here is as simple as a dielectric one with the corresponding dielectric constant having no imaginary part. In the real situation, the active layer is a semiconductor one having an ability of emitting light into the DFB structure. All refractive indices were taken from the source in Ref [27]. The period of the photonic crystal is denoted by L, the thickness of the active layer is t1, the height of the comb is h, and the thickness of the SiO2 layer not including the comb is t2. The filling factor (FF) is the fraction between the width of a comb over the whole periodic length L. The variation of the photonic band diagram and of the reflectivity spectrum versus these geometrical parameters was observed by varying each parameter while keeping the others constant. Incident light was polarized in the transverse electrical (TE) mode. For the photonic band diagrams, we used the open source package called MIT photonic bands (MPB) [28]. The reflec- tivity spectra were computed by implementing the rigorous coupled wave analysis (RCWA) method (also called Fourier modal method e FMM) [29] by using an open source package named Stanford stratified structure solver (S4) [30]. After computing theFig. 1. Dimensions of (a) a bare glass DFB, andFig. 2 shows the first 15 lowest photonic modes, in the first Brillouin zone (BZ), of the DFB structures with fixed parameters t2 ¼ 600 nm, h ¼ 500 nm, FF ¼ 0.3 while the period L varies from 250 nm to 1000 nm. We can observe that, when the period L is increased, all the dispersion curves shift to the lower energy region both at the edge and at the center of the first BZ. For the DFB structures with no active layer, the energy of the 15th mode (the highest energy black curve) is located at around 4 eV for L ¼ 250 nm, at 2.75 eV for L ¼ 500 nm, at 2.5 eV for L ¼ 800 nm, and at 2.0 eV for L ¼ 1000 nm. Except the 1st order mode (the lowest energy black curve), there are two types of curve shapes for the remaining modes: (i) parabolic and (ii) straight lines. On one hand, the parabolic modes would come from the vertical reflection from several interfaces between the layers. In the subsequent parts of this paper, we call them vertical parabolic modes. On the other hand, the straight lines would be principally due to the straight dispersion curves of the light originating from the other BZs to the first BZ, to which we hereafter refer as DFB modes. Anti-crossing and gap-opening features are also clearly observed in Fig. 2 be- tween the parabolic and the straight photonic modes or between straight modes only. These anti-crossing features are due to the strong coupling between the parabolic modes and the DFB modes, or between the DFB modes themselves. As a consequence, the 15 lowest energy optical modes shown in the band diagram are the total number of the lowest energy DFB coupling with the parabolic photonic modes. When the period L increases, the energy the 15th optical modes decreases. This decrease is mainly due to the lowering in the energy of all DFB modes, whilst the energy levels of the parabolic modes stay rather constant. In the literature, the energy of the constructive interference modes from a DFB structure is given by [13]: E¼hc 2 mDFB neffL ðmDFB2NÞ where h is the Planck constant, c is the velocity of light in vacuum and neff is the effective refractive index of the DFB structure, and mDFB is a natural number representing the order of DFB modes. From the above equation, it is therefore quite trivial that when the(b) an active layer coated DFB unit cell. t2 ¼ s the dvanperiod increases, the energy of the optical modes shifts to the lower range. We observe also that the energy levels of the structures covered with an active layer are lower than the corresponding energy levels of the structures without the active layer. The energy of the 15th band is located at around 3.5 eV for L ¼ 250 nm, at 2.5 eV for L ¼ 500 nm, at 2.0 eV for L ¼ 800 nm, and at 1.75 eV for L ¼ 1000 nm. Since the energy levels the 15th lowest bands Fig. 2. Photonic band diagrams of the first 15 lowest modes of the DFB structures with band diagrams of the DFB structures with no active layer, whereas the bottom row show t1 ¼ 120 nm. N.D. Le, T. Nguyen-Tran / Journal of Science: A144correlate strongly with the energy of the DFB modes as shown by the above equation, when an active layer is coated, the effective refractive index of the DFB structure increases slightly, and thus lowering the energy level of the optical modes.When looking at the lowest crossing at the center of the first BZ (kx¼ 0) between the two straight DFB modes (second order DFB modes mDFB ¼ 2), we find that the energy value equals to 3.45 eV (for L ¼ 250 nm), 1.79 eV (for L ¼ 500 nm), 1.15 eV (for L ¼ 800 nm) and 0.94 eV (for L ¼ 1000 nm) for the structures with no active layer. This energy level equals to 3.08 eV (for L ¼ 250 nm), 1.70 eV (for L ¼ 500 nm), 1.10 eV (for L ¼ 800 nm), and 0.89 eV (for L ¼ 1000 nm) for the structures with the active layer. We deduce that the average effective refractive index of these second order DFB modes without the active layer is about neff ¼ 1:38, and with an active layer neff ¼ 1:48 (higher than the refractive index of SiO2). This value of effec- tive refractive index shows that these DFB photonic modes would be correlated to the light propagation in the SiO2 layer (in the bare DFB structures) or in the comb periodic SiO2/active medium layer (in the structures covered with an active layer). In the reflectivity spectra shown in Fig. 3, the modes under the light line (LL) in vacuum are not obtained [2]. This is represented as the triangular limit of the reflectivity spectra at low energy levels. The spectra, in fact, give the information of the modes whose energy levels are strictly higher than the LL. The modes which are strictly under the LL are guided inside the photonic crystal without being able to couple to the outside of it, so are not present in the reflectivity spectra. The spectra of the structures with an active layer are of higher contrast in comparison to the spectra of the structures without the active layer. The patterns aresimilar when comparing between the structures with and without the active layer: (i) parabolic modes, and (ii) straight DFB modes. For the structures without the active layer, we observe two lowest energy straight bands converging at the center of the BZ at the energy level around 3.44 eV for the structures with the period L ¼ 250 nm, and 1.80 eV for L ¼ 500 nm, 1.18 eV for L ¼ 800 nm, and 0.97 eV for L ¼ 1000 nm. This is consistent with the photonic band diagrams in those regions, and the deduced 600 nm, h ¼ 500 nm, FF ¼ 0.3 and varying period L. The top row shows the photonic photonic band diagrams of the DFB structures covered with an active layer of thickness ced Materials and Devices 5 (2020) 142e150average effective refractive index is neff ¼ 1:38. For the structures with an active layer, this energy level is around 2.88 eV for L ¼ 250 nm, 1.58 eV for L ¼ 500 nm, 1.05 eV for L ¼ 800 nm, and 0.86 eV for L ¼ 1000 nm. This corresponds to an average effective refractive index of neff ¼ 1:48. In addition, from the reflectivity spectra calculated by S4, for the DFB structures without the active layer, we can observe two fam- ilies of the straight DFB modes for each value of L. The inclined angles of the two straight DFB families are different, corresponding to different effective refractive indices. For L¼ 500 nm, we observe that the first family of the DFB modes, corresponding to a low in- clined angle (top row, second column from the left of Fig. 3), are wave-guided modes in the SiO2 layer with a refractive index of neff ¼ 1:38 (without the active layer). The second family of DFB modes, corresponding to a higher inclined angle (top row, second column from the left of Fig. 3), are wave-guided modes in the comb periodic structure between air and SiO2, with a refractive index of neff ¼ 1:1. Fig. S1 shows that these two families of DFB modes originate from the centers of the left and the right BZs (with respect to the first central BZ). Noting that these modes are correlated to wave-guided modes in a planar waveguide limited by the LL in vacuum, which is demonstrated by the black dash line in Fig. S1. The first family of the DFB modes are parallel to the LL in SiO2; and the second one lies between the LL in SiO2 and the LL in vacuum. In contrast, for the DFB structure covered with an active layer, there exists an additional third family of the DFB modes, corresponding to the lowest inclined angle which are wave-guided modes in the periodic structure between SiO2 and the active layer, with the highest refractive index of neff ¼ 1:48 (higher than that of SiO2). In .3 a wit dvanorder to identify all DFB modes encountered in this study, we propose to use TE±;mBZn;X with following suggested rules: (i) TE stands for transverse electric modes. (ii) The sign “±” is for indicating propagation direction, “” is for indicating waves from the left to the right and “þ” for the wave propagating in the opposite direction. (iii) mBZ is for indicating fromwhich BZ the waves come.mBZ ¼ 0 for waves in the central BZ,mBZ ¼ 1 for waves from the left BZ,mBZ ¼ 2 for waves from the second left BZ. The sign “þ” is for waves from BZs on the right. (iv) n represents the order of conventional planar wave-guided modes, taking value 0, 1, 2, (v) X represents the nature of the planar waveguide, depending Fig. 3. Reflectivity spectra of the DFB structures with t2 ¼ 600 nm, h ¼ 500 nm, FF ¼ 0 shown in Fig. 2. The top row represents the reflectivity spectra of the DFB structures structures covered with an active layer of thickness t1 ¼ 120 nm. N.D. Le, T. Nguyen-Tran / Journal of Science: Aon the DFB structure in consideration. We note that X ¼ 1 for the top layer (the periodic SiO2/air for the bare DFB struc- ture), X ¼ 2 for the second top layer (the SiO2 slab for the bare DFB structure), and so on. By using the above proposed notations, we can point out, in the reflectivity spectra, that “parallel” DFB modes are modes corre- sponding to the same BZ, same direction, same value of X, but different values of n. Modes with different inclined angles are modes in different waveguides. For a bare DFB structure, there are two types of waveguides, but for the DFB structures coated with an active layer, there may be up to four types, for example, the periodic air/active medium layer, the periodic air/SiO2 layer, the periodic active medium/SiO2 layer and the SiO2 slab. For a DFB structure with an active layer, the refractive index of the periodic active medium/SiO2 layer is higher than its above layer, which is the air/SiO2 layer, and its below layer, which is the SiO2 slab. Such details are illustrated in Figs. S2 and S3. A comparison between the conventional DFB structure index mDFB and the indices proposed in this paper is shown in Fig. S2b. As a result, the wave-guided modes in the periodic active medium/SiO2 layer, which we call the third family, would be correlated to the confined wave-guided modes. This is in agreement with the strong bright contrast observed in the reflectivity spectra. We can see also that as the period L increases, the second family of DFB modes, correspond- ing to the wave-guided modes in the periodic air/SiO2 layer, is lesspronounced. The DFB wave-guided modes in the periodic air/ active medium are also not present, may be due to the very thin thickness of the top active layer. For better understanding the dark contrast of the first and the second families of DFB modes, as well as the bright contrast of the third family of DFB modes, Fig. S4 shows a comparison of the reflectivity of a DFB structure on a silicon substrate and with that of the same structure without the silicon substrate. It is true that for the DFB structures without the silicon substrate, the effective refractive index is higher than that of the surrounding medium, thus favoring the confinement of wave-guided modes. As a consequence, these modes are bright on the reflectivity spectra. For the DFB structures simulated