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Simulation of coupling optical modes in 1D photonic crystals for optoelectronic applications
Journal Pre-proof Simulation of coupling optical modes in 1D photonic crystals for optoelectronic applications Ngoc Duc Le, Thuat Nguyen-Tran PII:
S2468-2179(20)30008-3
DOI:
https://doi.org/10.1016/j.jsamd.2020.01.008
Reference:
JSAMD 269
To appear in:
Journal of Science: Advanced Materials and Devices
Received Date: 9 April 2019 Revised Date:
30 January 2020
Accepted Date: 30 January 2020
Please cite this article as: N.D. Le, T. Nguyen-Tran, Simulation of coupling optical modes in 1D photonic crystals for optoelectronic applications, Journal of Science: Advanced Materials and Devices, https:// doi.org/10.1016/j.jsamd.2020.01.008. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Publishing services by Elsevier B.V. on behalf of Vietnam National University, Hanoi.
Simulation of coupling optical modes in 1D photonic crystals for optoelectronic applications Ngoc Duc Le1,2, Thuat Nguyen-Tran1,* 1
Nano and Energy Center, VNU University of Science, 334 Nguyen Trai, Thanh Xuan, Hanoi,
Vietnam 2
Department of Advanced Materials Science and Nanotechnology, University of Science and
Technology of Hanoi, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam Keywords: 1D photonic crystal, DFB structure, angle – resolved reflectivity, photonic band diagram, coupling waves
Corresponding author:
Address:
* [email protected]
Nano and Energy Center VNU University of Science Room 503, 5th floor, T2 building 334 Nguyen Trai street, Thanh Xuan, Hanoi, Vietnam Mol: +84 985 516 980, Tel: +84 435 406 136, Fax: +84 435 406 137
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
±, ,
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 anticrossing, 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.
Keywords: 1D photonic crystal, DFB structure, angle – resolved reflectivity, photonic band diagram, coupling waves
1
1.
Introduction Photonic crystals are periodic dielectric modulation media where light propagates in a
particular behavior [1,2]. In the simplest understanding, this behavior can be characterized by waves which are propagating in opposite directions and coupled by the reflection 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) photonic crystals with a discontinuous dielectric modulation, known as distributed feedback (DFB) structures, are often made of layered media, where there is at least one layer with periodic variation of the refractive index[4–11]. The light propagation in these low-dimensional photonic crystals can be considered as bound optical modes in the corresponding layers. Depending on the relative effective refractive index of the guided layer with respect to the underneath and the overlying layers, there may be leaky or confined waveguided modes. The term “leaky” is used here to describe the optical modes in a waveguide whose refractive index is smaller than that of one of the cladding layers, whereas the term “confined” is used for a waveguide whose refractive index is higher than that of the cladding layers. Both modes, “leaky” and “confined”, correspond to a phase matching condition of light reflection in a waveguide. By convention, they are called wave-guided modes, and they are interesting subjects for mathematical as well as technological points of views [1–3]. DFB structures play important roles in optoelectronics, especially 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 direction [12– 14]. 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 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 structures 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 [20–25], as well as in light-matter coupling phenomena [26]. In this paper, we present a simulation and modelization study of the photonic modes of a 1D DFB photonic crystal. The work was carried out on the bare DFB structures as well as 2
the same structures 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 the master 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 = 1.46) on a silicon substrate (refractive index ns = 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 = 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 Λ, 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 Λ. 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 reflectivity spectra were computed by implementing the rigorous coupled wave analysis (RCWA) method (also called Fourier modal method - FMM) [29] by using an open source package named Stanford stratified structure solver (S4) [30]. After computing the 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 periodic direction of the DFB structures. 3.
Results and Discussion
3.1.
Influence of the period of the DFB structures
3
Fig. 2 shows the first 15 lowest photonic modes, in the first Brillouin zone (BZ), of the DFB structures with fixed parameters t2 = 600 nm, h = 500 nm, FF = 0.3 while the period Λ varies from 250 nm to 1000 nm. We can observe that, when the period Λ 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 Λ = 250 nm, at 2.75 eV for Λ = 500 nm, at 2.5 eV for Λ = 800 nm, and at 2.0 eV for Λ = 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 between 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 Λ 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]: =
ℎ 2
DFB
eff
DFB
∈ℕ
where h is the Planck constant, c is the velocity of light in vacuum and refractive index of the DFB structure, and
DFB
eff
is the effective
is a natural number representing the order of
DFB modes. From the above equation, it is therefore quite trivial that when the period 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 Λ = 250 nm, at 2.5 eV for Λ = 500 nm, at 2.0 eV for Λ = 800 nm, and at 1.75 eV for Λ = 1000 nm. Since the energy levels the 15th lowest bands correlate 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 4
the lowest crossing at the center of the first BZ (kx = 0) between the two straight DFB modes (second order DFB modes
DFB
= 2), we find that the energy value equals to 3.45 eV (for Λ
= 250 nm), 1.79 eV (for Λ = 500 nm), 1.15 eV (for Λ = 800 nm) and 0.94 eV (for Λ = 1000 nm) for the structures with no active layer. This energy level equals to 3.08 eV (for Λ = 250 nm), 1.70 eV (for Λ = 500 nm), 1.10 eV (for Λ = 800 nm), and 0.89 eV (for Λ = 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 active layer
eff
eff
= 1.38, and with an
= 1.48 (higher than the refractive index of SiO2). This value of effective
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 are similar 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 Λ = 250 nm, and 1.80 eV for Λ = 500 nm, 1.18 eV for Λ = 800 nm, and 0.97 eV for Λ = 1000 nm. This is consistent with the photonic band diagrams in those regions, and the deduced average effective refractive index is eff
= 1.38. For the structures with an active layer, this energy level is around 2.88 eV for Λ =
250 nm, 1.58 eV for Λ = 500 nm, 1.05 eV for Λ = 800 nm, and 0.86 eV for Λ = 1000 nm. This corresponds to an average effective refractive index of
eff
= 1.48.
In addition, from the reflectivity spectra calculated by S4, for the DFB structures without the active layer, we can observe two families of the straight DFB modes for each value of Λ. The inclined angles of the two straight DFB families are different, corresponding to different effective refractive indices. For Λ = 500 nm, we observe that the first family of the DFB modes, corresponding to a low inclined angle (top row, second column from the left of Fig. 3), are wave-guided modes in the SiO2 layer with a refractive index of
eff
= 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 5
comb periodic structure between air and SiO2, with a refractive index of
eff
= 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
eff
= 1.48 (higher than that of SiO2). In order to identify all DFB modes
encountered in this study, we propose to use (i) (ii)
±, ,
with following suggested rules:
stands for transverse electric modes. The sign “±” is for indicating propagation direction, “−” is for indicating waves from the left to the right and “+” for the wave propagating in the opposite direction.
(iii)
is for indicating from which BZ the waves come. central BZ,
= −1 for waves from the left BZ,
= 0 for waves in the
= −2 for waves from
the second left BZ. The sign “+” is for waves from BZs on the right. (iv)
represents the order of conventional planar wave-guided modes, taking value 0, 1, 2, …
(v)
represents the nature of the planar waveguide, depending on the DFB structure in consideration. We note that the bare DFB structure),
= 1 for the top layer (the periodic SiO2/air for
= 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 corresponding 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 Fig. S2 and Fig. 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 6
medium/SiO2 layer, which we call the third family, would be correlated to the confined waveguided modes. This is in agreement with the strong bright contrast observed in the reflectivity spectra. We can see also that as the period Λ increases, the second family of DFB modes, corresponding to the wave-guided modes in the periodic air/SiO2 layer, is less pronounced. 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 in this study, the silicon medium, having a refractive index higher than that of the waveguide on top of it, makes the wave-guided modes “leaky”, and thus exhibiting dark contrast. With the coated active layer, only wave-guided modes in the periodic active medium/SiO2 layer structures, known as the third family modes, are the nonleaky ones. These play an important role in this study because they are related to both the appearance of the active layer and the periodicity of the DFB structures, and they are related to non-leaky wave-guided modes. When comparing the photonic band diagrams obtained by using the MPB package with the DFB modes calculated by using S4, we see that the agreement is more significant for larger periods. Another point is that the parabolic modes, which are similar for both the two calculation methods, are not of the same nature with the DFB modes, and we cannot use the above-mentioned rules for them. Looking at the coupling between these optical modes we can identify the following anti-crossings: (i) the ones between the DFB modes originating from different BZs; (ii) the quite weak ones between the DFB modes of the SiO2 slab (the first family); (iii) the quite strong ones between the DFB modes of the periodic air/SiO2 (the second family) is, (iv) the very strong one between the DFB modes of the periodic SiO2/active medium (the third family); (v) the strong one between the DFB and the parabolic modes; (vi) the ones between the DFB modes of the different families, and (vii) anti-crossing decreases generally as Λ increases. An analytical coupling model between the DFB modes of the third family will be carried out in a subsequent part of this study in order to get further understanding about these coupling modes. 3.2.
Influence of the thickness of the SiO2 layer 7
In Fig. 4, we can see that the reflectivity contrast is improved in the presence of the active layer. The same effect is observed in Fig. 3 because of the conditions of the refractive indices which are favorable for the wave-guided modes in the periodic active medium/SiO2 layer. We also see that although the thickness t2 of the SiO2 slab varies, in both DFB structures with no active layer and with an active layer, the inclined angles of the straight DFB modes of the first and the second families remain almost the same (except for the third family of the DFB modes). Rhombus shapes formed by crossing between those DFB modes are quite visible. For the parabolic modes, there are alternatively bright and dark fringes in the reflectivity spectra. For the DFB modes, in the structures with no active layer, we observe two families of DFB modes corresponding to two different propagation layers, which are the SiO2 slab and the periodic SiO2/air layer. By considering only one family of the DFB modes in Fig. 5, such as the first family, there are three (3) other “sub-modes” (single dark contrast line with the same inclined angle) which are quite equally distributed in energy, for t2 = 600 nm. As mentioned before, these submodes are conventional optical guided modes in a wave-guide (the same number X, same direction and the same BZ but the different n, illustrated in Fig. S2a). The number of submodes is increased to four (4) for t2 = 900 nm, and to approximately seven (7) for t2 = 1800 nm. It is easy to see that when the thickness of a waveguide increases, the number of guided modes increases. This suggests that guided modes in the SiO2 slab have a certain correlation with the vertical interference parabolic modes, since they both represent standing waves in the vertical direction. In addition, anti-crossing features are visible for the first lowest energy crossing point (of the straight DFB modes coming from the adjacent BZ from the left and from the right of the third family), at 1.5 eV, at the center of the central BZ. The opening energy gap is relatively the same for all values of t2. The anti-crossing feature is also observed for all DFB modes, as well as between the parabolic modes and the DFB modes. There is also a very bright Dirac cone feature at kx = 0 and 2 eV, resulted from the coupling between the DFB modes of the first family in a very bright parabolic mode. This feature appears on all three reflectivity spectra for the structures covered with an active layer, whose position remains almost unchanged with a large variation range of the value of t2. 3.3.
Influence of the height of the comb In Fig. 5, we can clearly observe that the contrast of the reflectivity spectra is enhanced
for the structures coated with an active layer. The energy levels of all photonic bands shift slightly to the lower energy region in the DFB structures coated with an active layer compared to the modes in the corresponding bare DFB structures with no active layer. For the parabolic 8
modes, the contrast is changed with increasing h. For h = 200 nm, there are four (4) principal dark fringes, at the energy range from 0 to 3 eV. For h = 600 nm, in addition to these four (4) principal fringes, we can observe more secondary fringes with less contrast. For h = 1000 nm, the number of secondary less-contrast fringes is increased. We suggest that the principal fringes are due to the reflection of the full SiO2 layer, and that the secondary fringes are due to the reflection of the combed layer. For the structure with no active layer, there are also two families of DFB modes corresponding to two different values of the inclined angle. The second family is barely observed for h = 200 nm. When h = 1000 nm, we can observe two submodes of the second family whereas the number of submodes of the first family is the same as for h = 200 nm. This observation confirms the fact that the second family of the DFB modes represents the guided modes in the periodic air/SiO2 combed layer, and that the first family are modes in the SiO2 slab. For the structures covered with an active layer, there appears a third family of DFB modes whose features stay almost unchanged with increasing h. It suggests that the height of the comb does not cause a significant impact on the third family. For all DFB modes, the anti-crossing features are visible, especially for the first lowest energy coupling mode at the central BZ, between two DFB modes of the third family, at around 1.5 eV. The energy gap of this coupling mode is the same as h increases. 3.4.
Influence of the filling factor As it can be seen in Fig. 6, the contrast of the reflectivity spectra is enhanced in the
samples with the presence of an active layer. The number of the parabolic fringes of the parabolic modes increases slightly as FF increases. This is a direct consequence of the fact that the effective refractive index of the DFB structure
eff
increases slightly with the filling
factor. For the DFB structures with no active layer, by observing the DFB modes, for FF = 0.3, we can recognize two different families of DFB modes, which correspond to two different inclined angles. For FF = 0.6, the first family is quite the same, but the second family is quite different. For FF = 0.9, we can hardly find the second family, which corresponds to the higher inclined angles and is related to the propagating waves in the periodic air/SiO2 structure. Since FF = 0.9, the full SiO2 slab looks to be extended in thickness, thus the first family is not the same as for FF = 0.3. In the DFB structures with an active layer, there appears a third family, which are guided modes in the periodic SiO2/active layer. As FF increases, this third family changes gradually. Concerning the anti-crossing features, for FF = 0.3, the opening energy gap of the lowest energy coupling of the DFB modes of the third family, at the central BZ, is quite high, and is clearly seen. For FF = 0.6, the opening energy gap of the lowest energy coupling mode 9
becomes rather complicated by the involvement of the DFB modes of the second family. For FF = 0.9, the energy gap of the lowest energy coupling mode reappears, accompanied by the fact that the contrast of all DFB modes becomes less important. 3.5.
Influence of the thickness of the active layer As it is seen in Fig. 7, when the thickness t1 of the active layer increases from 150 nm to
250 nm, all photonic bands shift slightly to the lower energy region. The parabolic modes stay almost unchanged. More precisely, by considering the first and the second order parabolic modes, their energy levels at the center of the first BZ are at 0.66 eV and 0.77 eV for t1 = 150 nm; 0.62 eV and 0.73 eV for t2 = 200 nm; and 0.59 eV and 0.69 eV for t2 = 250 nm, respectively. For the DFB modes, we can observe the first and the third families of the DFB modes as well as the DFB submodes of the same waveguide layer. However, it is more difficult to observe the second family as t1 increases. For the anti-crossing features, the energy gap of the lowest energy coupling modes of the third family increases from 0.034 eV (t1 = 150 nm) to 0.063 eV (t1 = 200 nm), and to 0.076 eV (t1 = 250 nm), respectively. This suggests that we can tune separately this Rabi splitting energy by varying t1. 3.6.
Comparison between the two methods of computation Fig. 8 shows in a same graph the photonic band diagram (dash grey, green, blue, pink
and brown lines, calculated by the MPB software package) on the background being the reflectivity spectrum (calculated by the S4 package) of the same DFB structure with the parameters of Λ = 500 nm, t2 = 600 nm, h = 400 nm, FF = 0.2, and t1 = 120 nm. There are slight differences between the results calculated by both methods. Some of the grey dash lines of the photonic band diagram do not appear in the reflectivity spectrum. Nevertheless, we find that the green band corresponds to the thin “reversed V – shaped” dispersion curve (A) and the pink band corresponds to the “V – shaped” bright dispersion curve (B), despite the difference of approximately 0.2 eV in terms of energy. The bright dispersion curve (C) also forms with the dispersion curve (B) a rhombus, similarly to the blue band forming a rhombus with the pink one. In addition, the brown band corresponds to the dark dispersion curve (D). The difference between the two computation results can be explained by two reasons. First, different bands have different density of states, so bands with a low density of states may become hidden into the background of the reflectivity spectrum, and cannot be observed as neither bright or dark dispersion curves. Second, the computational mechanism is different from one package to the other: while the MPB solves the eigenvalue problem of the master equation in the basis of plane waves, the S4 uses the rigorous coupled wave analysis method 10
and gives more pronounced coupled photon modes by taking into account the coupling between the photonic modes and the mixing between the bands.
3.7.
Modeling of coupling waves Fig. 9a shows the reflectivity spectrum of the chosen DFB structure with Λ = 500 nm, t2
= 600 nm, h = 400 nm, FF = 0.2, t1 = 120 nm, for a coupling optical modes analysis. Fig. 9b is a zoom in a region where the coupling is very strong for two DFB modes of the third family, originating from both the left and the right BZ. The detailed analysis of the coupling between these two modes can be found in the supporting information. The essential issues are the followings: (i) in the uncoupled regime, these two modes are straight, their group velocities have the same magnitude but are of different signs, they represent the propagating waves in a planar waveguide with constant refractive index; (ii) the interaction between these two modes is represented by an interacting second order Hamiltonian; (iii) the coupling constants U appears in the non-diagonal terms. Diagonalizing the interacting Hamiltonian in order to obtain the eigenvalues and eigenstates is a textbook procedure. The fitting was performed by tracing the eigenvalues in the same reflectivity spectrum in order to obtain the parameters, such as
",#
(energy at $% = 0), &' (group velocity) and U (coupling constant). The
black lines in Fig. 9b represent the eigenvalues of the coupling states after diagonalizing the interacting Hamiltonian. The best fit, illustrated by a good agreement between the modelled eigenvalues and the reflectivity spectrum in Fig. 9b, gave
",#
= 1.542 eV, &' = 0.633
eV.μm -.
and U = 0.032 eV. A similar model can be carried out for fitting the anti-crossing between two wave-guided modes in the vicinity of $% = ±//Λ, thus a full dispersion relation of coupling wave-guided modes can be obtained. From this fitting result, we suggest that there exists a procedure for fitting a reflectivity spectrum obtained by the RCWA simulation of a DFB structure. First, the planar wave guided modes are calculated, based on a conventional waveguide calculation, thus obtaining the dispersion curves of the
2,3
modes. Second, by
folding these modes in the different BZs, the dispersion curves of the uncoupled
±,456 2,3
modes are obtained. Note that the dispersion curves of the uncoupled modes can be modelled
by the straight lines in the vicinity of $% = 0 as done in this paper. Finally, by constructing the interacting higher order Hamiltonian with the non-diagonal coupling constants, and then by diagonalizing this Hamiltonian, the dispersion relations of coupling
11
±,456 2,3
modes can be
obtained. These exact dispersions curves would play a central role in applying the DFB structures for optoelectronics and light-matter coupling. 4.
Conclusion To conclude, we have simulated photonic TE modes in bare 1D photonic crystal
distributed feedback structures, and in those coated with an active layer. General trends about the variation of the photonic bands were studied. Different DFB modes corresponding to different wave guided modes have been identified, and are fully indexed by using
±,456 . 2,3
With the increasing period of these structures, the energy of the photonic modes becomes significantly lowered, combined with the fact that the anti-crossing feature is stronger for smaller periods. The thickness of the SiO2 layer separating the photonic crystal and the substrate causes the change of the parabolic modes, and affects one family of the DFB modes corresponding to one value of X, that we call the first family relating to wave-guided modes in the SiO2 slab. When increasing the height of the combs, the energy level of all photonic modes becomes lower, and the DFB modes corresponding to another value of X, that we call the second family relating to wave-guided modes in the SiO2/air layer, are affected. The filling factor and the thickness of the active coating layer have a large influence on the DFB guided modes on the periodic SiO2/active medium layer. The active layer is found to play an essential role in having non-leaky photonic modes. The difference between the photonic band diagrams and the reflectivity spectra calculated by two different software packages using two different methods expresses the difference between the way they solve the coupling modes. Finally, a simple coupling model between the two non-leaky modes is presented, in a good agreement with the RCWA simulation. Even by the fact of using constant refractive indices for all materials, which limits the correctness of the simulation results in a small range of the low energy region, these results could be further improved by using the dispersive materials for the high energy in comparison with analytical and numerical approaches, and confirmed by experimental works. This result paves a way for us to tune the photonic modes by changing the geometry of the structures in order to obtain desirable photonic modes for future optoelectronic applications. Acknowledgements We are grateful to Dr. Hai-Son Nguyen, Dr. Anh T. Le, Dr. Ha Q. Duong and Dr. Cam T.H. Hoang for helpful discussions. References [1]
B.E.A. Saleh, M.C. Teich, Fundamentals of Photonics, 2nd ed., John Wiley & Sons, 12
Inc, Hoboken, New Jersey, 2007. [2]
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15
List of captions Fig. 1. Dimensions of (a) a bare glass DFB, and (b) an active layer coated DFB unit cell. Fig. 2. Photonic band diagrams of the first 15 lowest modes of the DFB structures with t2 = 600 nm, h = 500 nm, FF = 0.3 and varying period Λ. The top row shows the photonic band diagrams of the DFB structures with no active layer, whereas the bottom row shows the photonic band diagrams of the DFB structures covered with an active layer of thickness t1 = 120 nm. Fig. 3. Reflectivity spectra of the DFB structures with t2 = 600 nm, h = 500 nm, FF = 0.3 and varying Λ, from 250 nm to 1000 nm, on the same energy and wave vector scale as shown in Fig. 2. The top row represents the reflectivity spectra of the DFB structures with no active layer, whereas the bottom row shows the reflectivity spectra of the DFB structures covered with an active layer of thickness t1 = 120 nm. Fig. 4. Reflectivity spectra of the DFB structures with Λ = 500 nm, h = 400 nm, FF = 0.2, and varying t2 from 600 nm to 1800 nm. The top row represents the reflectivity spectra of the DFB structures with no active layer, whereas the bottom row is for the DFB structures covered with an active layer of thickness t1 = 120 nm. Fig. 5. Reflectivity spectra of DFB structures with Λ = 500 nm, t2 = 600 nm, FF = 0.2, and varying h from 200 nm to 1000 nm. The top row represents the reflectivity spectra of the DFB structures with no active layer, whereas the bottom row shows the reflectivity spectra of the DFB structures covered with an active layer of thickness t1 = 120 nm. Fig. 6. Reflectivity spectra of the DFB structures with Λ = 500 nm, t2 = 600 nm, h = 500 nm, and varying FF from 0.3 to 0.9. The top row represents the reflectivity spectra of the DFB structures with no active layer, whereas the bottom row consists of the DFB structures covered with an active layer of thickness t1 = 120 nm. Fig. 7. Reflectivity spectra of the DFB structures with Λ = 500 nm; t2 = 600 nm; h = 500 nm, FF = 0.3 and varying t1 from 150 nm to 250 nm. Fig. 8. Comparison between the photonic modes calculated by MPB and the reflectivity spectrum calculated by S4 of the DFB structure with Λ = 500 nm; t2 = 600 nm; h = 400 nm; FF = 0.2; t1 = 120 nm.
16
Fig. 9. (a) Reflectivity spectrum of the DFB structure with Λ = 500 nm, t2 = 600 nm, h = 400 nm, FF = 0.2, t1 = 120 nm, and (b) a zoom in the central region with two modelled coupling modes (black lines).
17
Supplementary information 1. Reflectivity and folded wave-guided modes
Fig. S1. k-space reflectivity spectra for DFB structure with Λ = 500 nm, t2 = 600 nm, h = 500 nm, FF = 0.3 of (left) a bare DFB structure and (right) a DFB structure covered with an active layer of thickness t1 = 120 nm. For the bare DFB structure (left), two families of DFB modes in the central BZ, with different inclined angles, are highlighted (blue and pink lines). The extrapolated straight lines of these two families meet at the centers of two adjacent BZs (the left BZ and the right BZ). Bright lines are light lines in air, starting from two adjacent BZs. For the DFB structure covered with an active layer, a third family of DFB modes appears (black lines), with lowest inclined angle corresponding to highest refractive index. This third family should be wave-guided modes of the periodic active medium/SiO2 layer.
(a)
(b)
Fig. S2. (a) Simplified line diagram of uncoupling (no anti-crossing) folded waveguided modes (from ±,456 first adjacent BZs to the center BZ) in a DFB structures with nomenclature in this paper. 2,3 Different mode colors, with different inclined angles, represent different waveguides (green modes are from the waveguide number 1, blue modes are from the waveguide number 2, violet modes are from the waveguide number 3). (b) A typical reflectivity spectrum in k-space of an active layer covered DFB structure. Two families of straight modes can be easily observed (with conventional Bragg order denoted by m and m'). The energy level of the second order crossing of the first family (m = mDFB = 2) is smaller than that of the second family (m' = mDFB' = 2), thus implying that the effective refractive ±,456 , we can index for the first family is higher than that of the second. By using the notation 2,3 identify two lowest energy modes (first family), denoted as 18
7,89 ",#
and
8,79 ",# .
We suppose that X=3
is the third planar waveguide layer (the periodic active medium/SiO2 as shown in Fig. S3 below) counting from the top air to the bottom silicon.
Fig. S3. Simplified planar layers of (a) bare and (b) active layer covered DFB structures for waveguide consideration. Air and the substrate, being semi-infinite media, are not considered as waveguides. ±,456 Brighter color represents lower effective index of refraction. In the nomenclature , for the 2,3 bare DFB structure (a) X=1 for the periodic air/SiO2 layer, X=2 for the SiO2 slab; for the active layer covered DFB structure (b) X=1 for the periodic air/active medium layer, X=2 for the periodic air/SiO2 layer, X=3 for the periodic active medium/SiO2 layer, and X=4 for the SiO2 slab. Waveguided optical modes in the bare structure are leaky, wheresas there are non-leaky waveguided optical modes in the active-layer-covered structure (guided modes in the periodic active/air and active/SiO2 layers).
Fig. S4. Reflectivity spectra of a DFB structure on a silicon substrate (left column) and of a free standing DFB structure (right column – without Si substrate). On the top row reflectivity, we can observe that dark leaky wave-guided modes (of the DFB structure on a silicon substrate) are replaced by bright non-leaky wave-guided modes (of the free standing DFB structure). On the bottom row 19
reflectivity, when coated by an active layer, the additional bright modes (so called the third DFB family modes) are the same for the DFB structure on a silicon substrate and for the free standing DFB structure. They are wave-guided modes in the periodic active medium/SiO2 layer. 2. Coupling waveguides By using indices proposed in this paper, DFB modes of the third family are:
±,456 , 2,3
two states, represented in the bra-ket notation, of 8,79 ",#
|1; for the “uncoupling” mode
7,89 ",#
|2; for the “uncoupling” mode
The interacting Hamilton matrix between these two above modes is <==
8,79 ",#
>
$
>
7,89 ",#
$
?
where $ = $@ is the wavevector along the x direction 8,79 ",#
$ =
8,79 ",#
=
7,89 ",#
$ =
8,79 ",# 7,89 ",#
7,89 ",#
=
− &A $ is the energy of the state |1; + &A $ is the energy of the state |2; ",#
is the energy at $ = 0
&A is the group velocity of these two DFB guided modes U is the coupling constant between two modes By diagonalizing the above Hamiltonian, we obtain the following states (by neglecting a normalization constant) o
9
|−; = D&A $ − E$ - &A- + > - F ∙ |1; + |2; with an eigenvalue C
8
$ =
"
− E$ - &A- + > -,
"
+ E$ - &A- + > -,
corresponding to the hyperboloic lower branch of the coupling modes. o
9
|+; = |1; + D&A $ + E$ - &A- + > - F ∙ |2; with an eigenvalue C
7
$ =
corresponding to the hyperboloic upper branch of the coupling modes. The Rabi splitting energy between these two coupled eigenstates is H = 2> at $ = 0
By tracing 8 $ and 7 $ of the two coupling modes I–K and |+; in the refelectivity spectrum obtained by S4 of the same DFB structure, we obtained Figure 13b.
20
HIGHLIGHTS • • • •
Throughout investigation of photonic modes with various geometrical parameters of 1D photonic crystals The active layer plays an essential role in having non-leaky optical modes Strong anticrossing between two opposite direction non-leaky photonic modes Enhancement of anticrossing between photonic modes when decreasing the period and increasing the thickness of the perovskite layer
The authors declare no conflict of interest. On behalf of the authors NGUYEN-TRAN Thuat
S2468-2179(20)30008-3
DOI:
https://doi.org/10.1016/j.jsamd.2020.01.008
Reference:
JSAMD 269
To appear in:
Journal of Science: Advanced Materials and Devices
Received Date: 9 April 2019 Revised Date:
30 January 2020
Accepted Date: 30 January 2020
Please cite this article as: N.D. Le, T. Nguyen-Tran, Simulation of coupling optical modes in 1D photonic crystals for optoelectronic applications, Journal of Science: Advanced Materials and Devices, https:// doi.org/10.1016/j.jsamd.2020.01.008. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Publishing services by Elsevier B.V. on behalf of Vietnam National University, Hanoi.
Simulation of coupling optical modes in 1D photonic crystals for optoelectronic applications Ngoc Duc Le1,2, Thuat Nguyen-Tran1,* 1
Nano and Energy Center, VNU University of Science, 334 Nguyen Trai, Thanh Xuan, Hanoi,
Vietnam 2
Department of Advanced Materials Science and Nanotechnology, University of Science and
Technology of Hanoi, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Cau Giay, Hanoi, Vietnam Keywords: 1D photonic crystal, DFB structure, angle – resolved reflectivity, photonic band diagram, coupling waves
Corresponding author:
Address:
* [email protected]
Nano and Energy Center VNU University of Science Room 503, 5th floor, T2 building 334 Nguyen Trai street, Thanh Xuan, Hanoi, Vietnam Mol: +84 985 516 980, Tel: +84 435 406 136, Fax: +84 435 406 137
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
±, ,
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 anticrossing, 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.
Keywords: 1D photonic crystal, DFB structure, angle – resolved reflectivity, photonic band diagram, coupling waves
1
1.
Introduction Photonic crystals are periodic dielectric modulation media where light propagates in a
particular behavior [1,2]. In the simplest understanding, this behavior can be characterized by waves which are propagating in opposite directions and coupled by the reflection 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) photonic crystals with a discontinuous dielectric modulation, known as distributed feedback (DFB) structures, are often made of layered media, where there is at least one layer with periodic variation of the refractive index[4–11]. The light propagation in these low-dimensional photonic crystals can be considered as bound optical modes in the corresponding layers. Depending on the relative effective refractive index of the guided layer with respect to the underneath and the overlying layers, there may be leaky or confined waveguided modes. The term “leaky” is used here to describe the optical modes in a waveguide whose refractive index is smaller than that of one of the cladding layers, whereas the term “confined” is used for a waveguide whose refractive index is higher than that of the cladding layers. Both modes, “leaky” and “confined”, correspond to a phase matching condition of light reflection in a waveguide. By convention, they are called wave-guided modes, and they are interesting subjects for mathematical as well as technological points of views [1–3]. DFB structures play important roles in optoelectronics, especially 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 direction [12– 14]. 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 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 structures 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 [20–25], as well as in light-matter coupling phenomena [26]. In this paper, we present a simulation and modelization study of the photonic modes of a 1D DFB photonic crystal. The work was carried out on the bare DFB structures as well as 2
the same structures 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 the master 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 = 1.46) on a silicon substrate (refractive index ns = 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 = 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 Λ, 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 Λ. 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 reflectivity spectra were computed by implementing the rigorous coupled wave analysis (RCWA) method (also called Fourier modal method - FMM) [29] by using an open source package named Stanford stratified structure solver (S4) [30]. After computing the 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 periodic direction of the DFB structures. 3.
Results and Discussion
3.1.
Influence of the period of the DFB structures
3
Fig. 2 shows the first 15 lowest photonic modes, in the first Brillouin zone (BZ), of the DFB structures with fixed parameters t2 = 600 nm, h = 500 nm, FF = 0.3 while the period Λ varies from 250 nm to 1000 nm. We can observe that, when the period Λ 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 Λ = 250 nm, at 2.75 eV for Λ = 500 nm, at 2.5 eV for Λ = 800 nm, and at 2.0 eV for Λ = 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 between 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 Λ 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]: =
ℎ 2
DFB
eff
DFB
∈ℕ
where h is the Planck constant, c is the velocity of light in vacuum and refractive index of the DFB structure, and
DFB
eff
is the effective
is a natural number representing the order of
DFB modes. From the above equation, it is therefore quite trivial that when the period 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 Λ = 250 nm, at 2.5 eV for Λ = 500 nm, at 2.0 eV for Λ = 800 nm, and at 1.75 eV for Λ = 1000 nm. Since the energy levels the 15th lowest bands correlate 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 4
the lowest crossing at the center of the first BZ (kx = 0) between the two straight DFB modes (second order DFB modes
DFB
= 2), we find that the energy value equals to 3.45 eV (for Λ
= 250 nm), 1.79 eV (for Λ = 500 nm), 1.15 eV (for Λ = 800 nm) and 0.94 eV (for Λ = 1000 nm) for the structures with no active layer. This energy level equals to 3.08 eV (for Λ = 250 nm), 1.70 eV (for Λ = 500 nm), 1.10 eV (for Λ = 800 nm), and 0.89 eV (for Λ = 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 active layer
eff
eff
= 1.38, and with an
= 1.48 (higher than the refractive index of SiO2). This value of effective
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 are similar 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 Λ = 250 nm, and 1.80 eV for Λ = 500 nm, 1.18 eV for Λ = 800 nm, and 0.97 eV for Λ = 1000 nm. This is consistent with the photonic band diagrams in those regions, and the deduced average effective refractive index is eff
= 1.38. For the structures with an active layer, this energy level is around 2.88 eV for Λ =
250 nm, 1.58 eV for Λ = 500 nm, 1.05 eV for Λ = 800 nm, and 0.86 eV for Λ = 1000 nm. This corresponds to an average effective refractive index of
eff
= 1.48.
In addition, from the reflectivity spectra calculated by S4, for the DFB structures without the active layer, we can observe two families of the straight DFB modes for each value of Λ. The inclined angles of the two straight DFB families are different, corresponding to different effective refractive indices. For Λ = 500 nm, we observe that the first family of the DFB modes, corresponding to a low inclined angle (top row, second column from the left of Fig. 3), are wave-guided modes in the SiO2 layer with a refractive index of
eff
= 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 5
comb periodic structure between air and SiO2, with a refractive index of
eff
= 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
eff
= 1.48 (higher than that of SiO2). In order to identify all DFB modes
encountered in this study, we propose to use (i) (ii)
±, ,
with following suggested rules:
stands for transverse electric modes. The sign “±” is for indicating propagation direction, “−” is for indicating waves from the left to the right and “+” for the wave propagating in the opposite direction.
(iii)
is for indicating from which BZ the waves come. central BZ,
= −1 for waves from the left BZ,
= 0 for waves in the
= −2 for waves from
the second left BZ. The sign “+” is for waves from BZs on the right. (iv)
represents the order of conventional planar wave-guided modes, taking value 0, 1, 2, …
(v)
represents the nature of the planar waveguide, depending on the DFB structure in consideration. We note that the bare DFB structure),
= 1 for the top layer (the periodic SiO2/air for
= 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 corresponding 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 Fig. S2 and Fig. 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 6
medium/SiO2 layer, which we call the third family, would be correlated to the confined waveguided modes. This is in agreement with the strong bright contrast observed in the reflectivity spectra. We can see also that as the period Λ increases, the second family of DFB modes, corresponding to the wave-guided modes in the periodic air/SiO2 layer, is less pronounced. 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 in this study, the silicon medium, having a refractive index higher than that of the waveguide on top of it, makes the wave-guided modes “leaky”, and thus exhibiting dark contrast. With the coated active layer, only wave-guided modes in the periodic active medium/SiO2 layer structures, known as the third family modes, are the nonleaky ones. These play an important role in this study because they are related to both the appearance of the active layer and the periodicity of the DFB structures, and they are related to non-leaky wave-guided modes. When comparing the photonic band diagrams obtained by using the MPB package with the DFB modes calculated by using S4, we see that the agreement is more significant for larger periods. Another point is that the parabolic modes, which are similar for both the two calculation methods, are not of the same nature with the DFB modes, and we cannot use the above-mentioned rules for them. Looking at the coupling between these optical modes we can identify the following anti-crossings: (i) the ones between the DFB modes originating from different BZs; (ii) the quite weak ones between the DFB modes of the SiO2 slab (the first family); (iii) the quite strong ones between the DFB modes of the periodic air/SiO2 (the second family) is, (iv) the very strong one between the DFB modes of the periodic SiO2/active medium (the third family); (v) the strong one between the DFB and the parabolic modes; (vi) the ones between the DFB modes of the different families, and (vii) anti-crossing decreases generally as Λ increases. An analytical coupling model between the DFB modes of the third family will be carried out in a subsequent part of this study in order to get further understanding about these coupling modes. 3.2.
Influence of the thickness of the SiO2 layer 7
In Fig. 4, we can see that the reflectivity contrast is improved in the presence of the active layer. The same effect is observed in Fig. 3 because of the conditions of the refractive indices which are favorable for the wave-guided modes in the periodic active medium/SiO2 layer. We also see that although the thickness t2 of the SiO2 slab varies, in both DFB structures with no active layer and with an active layer, the inclined angles of the straight DFB modes of the first and the second families remain almost the same (except for the third family of the DFB modes). Rhombus shapes formed by crossing between those DFB modes are quite visible. For the parabolic modes, there are alternatively bright and dark fringes in the reflectivity spectra. For the DFB modes, in the structures with no active layer, we observe two families of DFB modes corresponding to two different propagation layers, which are the SiO2 slab and the periodic SiO2/air layer. By considering only one family of the DFB modes in Fig. 5, such as the first family, there are three (3) other “sub-modes” (single dark contrast line with the same inclined angle) which are quite equally distributed in energy, for t2 = 600 nm. As mentioned before, these submodes are conventional optical guided modes in a wave-guide (the same number X, same direction and the same BZ but the different n, illustrated in Fig. S2a). The number of submodes is increased to four (4) for t2 = 900 nm, and to approximately seven (7) for t2 = 1800 nm. It is easy to see that when the thickness of a waveguide increases, the number of guided modes increases. This suggests that guided modes in the SiO2 slab have a certain correlation with the vertical interference parabolic modes, since they both represent standing waves in the vertical direction. In addition, anti-crossing features are visible for the first lowest energy crossing point (of the straight DFB modes coming from the adjacent BZ from the left and from the right of the third family), at 1.5 eV, at the center of the central BZ. The opening energy gap is relatively the same for all values of t2. The anti-crossing feature is also observed for all DFB modes, as well as between the parabolic modes and the DFB modes. There is also a very bright Dirac cone feature at kx = 0 and 2 eV, resulted from the coupling between the DFB modes of the first family in a very bright parabolic mode. This feature appears on all three reflectivity spectra for the structures covered with an active layer, whose position remains almost unchanged with a large variation range of the value of t2. 3.3.
Influence of the height of the comb In Fig. 5, we can clearly observe that the contrast of the reflectivity spectra is enhanced
for the structures coated with an active layer. The energy levels of all photonic bands shift slightly to the lower energy region in the DFB structures coated with an active layer compared to the modes in the corresponding bare DFB structures with no active layer. For the parabolic 8
modes, the contrast is changed with increasing h. For h = 200 nm, there are four (4) principal dark fringes, at the energy range from 0 to 3 eV. For h = 600 nm, in addition to these four (4) principal fringes, we can observe more secondary fringes with less contrast. For h = 1000 nm, the number of secondary less-contrast fringes is increased. We suggest that the principal fringes are due to the reflection of the full SiO2 layer, and that the secondary fringes are due to the reflection of the combed layer. For the structure with no active layer, there are also two families of DFB modes corresponding to two different values of the inclined angle. The second family is barely observed for h = 200 nm. When h = 1000 nm, we can observe two submodes of the second family whereas the number of submodes of the first family is the same as for h = 200 nm. This observation confirms the fact that the second family of the DFB modes represents the guided modes in the periodic air/SiO2 combed layer, and that the first family are modes in the SiO2 slab. For the structures covered with an active layer, there appears a third family of DFB modes whose features stay almost unchanged with increasing h. It suggests that the height of the comb does not cause a significant impact on the third family. For all DFB modes, the anti-crossing features are visible, especially for the first lowest energy coupling mode at the central BZ, between two DFB modes of the third family, at around 1.5 eV. The energy gap of this coupling mode is the same as h increases. 3.4.
Influence of the filling factor As it can be seen in Fig. 6, the contrast of the reflectivity spectra is enhanced in the
samples with the presence of an active layer. The number of the parabolic fringes of the parabolic modes increases slightly as FF increases. This is a direct consequence of the fact that the effective refractive index of the DFB structure
eff
increases slightly with the filling
factor. For the DFB structures with no active layer, by observing the DFB modes, for FF = 0.3, we can recognize two different families of DFB modes, which correspond to two different inclined angles. For FF = 0.6, the first family is quite the same, but the second family is quite different. For FF = 0.9, we can hardly find the second family, which corresponds to the higher inclined angles and is related to the propagating waves in the periodic air/SiO2 structure. Since FF = 0.9, the full SiO2 slab looks to be extended in thickness, thus the first family is not the same as for FF = 0.3. In the DFB structures with an active layer, there appears a third family, which are guided modes in the periodic SiO2/active layer. As FF increases, this third family changes gradually. Concerning the anti-crossing features, for FF = 0.3, the opening energy gap of the lowest energy coupling of the DFB modes of the third family, at the central BZ, is quite high, and is clearly seen. For FF = 0.6, the opening energy gap of the lowest energy coupling mode 9
becomes rather complicated by the involvement of the DFB modes of the second family. For FF = 0.9, the energy gap of the lowest energy coupling mode reappears, accompanied by the fact that the contrast of all DFB modes becomes less important. 3.5.
Influence of the thickness of the active layer As it is seen in Fig. 7, when the thickness t1 of the active layer increases from 150 nm to
250 nm, all photonic bands shift slightly to the lower energy region. The parabolic modes stay almost unchanged. More precisely, by considering the first and the second order parabolic modes, their energy levels at the center of the first BZ are at 0.66 eV and 0.77 eV for t1 = 150 nm; 0.62 eV and 0.73 eV for t2 = 200 nm; and 0.59 eV and 0.69 eV for t2 = 250 nm, respectively. For the DFB modes, we can observe the first and the third families of the DFB modes as well as the DFB submodes of the same waveguide layer. However, it is more difficult to observe the second family as t1 increases. For the anti-crossing features, the energy gap of the lowest energy coupling modes of the third family increases from 0.034 eV (t1 = 150 nm) to 0.063 eV (t1 = 200 nm), and to 0.076 eV (t1 = 250 nm), respectively. This suggests that we can tune separately this Rabi splitting energy by varying t1. 3.6.
Comparison between the two methods of computation Fig. 8 shows in a same graph the photonic band diagram (dash grey, green, blue, pink
and brown lines, calculated by the MPB software package) on the background being the reflectivity spectrum (calculated by the S4 package) of the same DFB structure with the parameters of Λ = 500 nm, t2 = 600 nm, h = 400 nm, FF = 0.2, and t1 = 120 nm. There are slight differences between the results calculated by both methods. Some of the grey dash lines of the photonic band diagram do not appear in the reflectivity spectrum. Nevertheless, we find that the green band corresponds to the thin “reversed V – shaped” dispersion curve (A) and the pink band corresponds to the “V – shaped” bright dispersion curve (B), despite the difference of approximately 0.2 eV in terms of energy. The bright dispersion curve (C) also forms with the dispersion curve (B) a rhombus, similarly to the blue band forming a rhombus with the pink one. In addition, the brown band corresponds to the dark dispersion curve (D). The difference between the two computation results can be explained by two reasons. First, different bands have different density of states, so bands with a low density of states may become hidden into the background of the reflectivity spectrum, and cannot be observed as neither bright or dark dispersion curves. Second, the computational mechanism is different from one package to the other: while the MPB solves the eigenvalue problem of the master equation in the basis of plane waves, the S4 uses the rigorous coupled wave analysis method 10
and gives more pronounced coupled photon modes by taking into account the coupling between the photonic modes and the mixing between the bands.
3.7.
Modeling of coupling waves Fig. 9a shows the reflectivity spectrum of the chosen DFB structure with Λ = 500 nm, t2
= 600 nm, h = 400 nm, FF = 0.2, t1 = 120 nm, for a coupling optical modes analysis. Fig. 9b is a zoom in a region where the coupling is very strong for two DFB modes of the third family, originating from both the left and the right BZ. The detailed analysis of the coupling between these two modes can be found in the supporting information. The essential issues are the followings: (i) in the uncoupled regime, these two modes are straight, their group velocities have the same magnitude but are of different signs, they represent the propagating waves in a planar waveguide with constant refractive index; (ii) the interaction between these two modes is represented by an interacting second order Hamiltonian; (iii) the coupling constants U appears in the non-diagonal terms. Diagonalizing the interacting Hamiltonian in order to obtain the eigenvalues and eigenstates is a textbook procedure. The fitting was performed by tracing the eigenvalues in the same reflectivity spectrum in order to obtain the parameters, such as
",#
(energy at $% = 0), &' (group velocity) and U (coupling constant). The
black lines in Fig. 9b represent the eigenvalues of the coupling states after diagonalizing the interacting Hamiltonian. The best fit, illustrated by a good agreement between the modelled eigenvalues and the reflectivity spectrum in Fig. 9b, gave
",#
= 1.542 eV, &' = 0.633
eV.μm -.
and U = 0.032 eV. A similar model can be carried out for fitting the anti-crossing between two wave-guided modes in the vicinity of $% = ±//Λ, thus a full dispersion relation of coupling wave-guided modes can be obtained. From this fitting result, we suggest that there exists a procedure for fitting a reflectivity spectrum obtained by the RCWA simulation of a DFB structure. First, the planar wave guided modes are calculated, based on a conventional waveguide calculation, thus obtaining the dispersion curves of the
2,3
modes. Second, by
folding these modes in the different BZs, the dispersion curves of the uncoupled
±,456 2,3
modes are obtained. Note that the dispersion curves of the uncoupled modes can be modelled
by the straight lines in the vicinity of $% = 0 as done in this paper. Finally, by constructing the interacting higher order Hamiltonian with the non-diagonal coupling constants, and then by diagonalizing this Hamiltonian, the dispersion relations of coupling
11
±,456 2,3
modes can be
obtained. These exact dispersions curves would play a central role in applying the DFB structures for optoelectronics and light-matter coupling. 4.
Conclusion To conclude, we have simulated photonic TE modes in bare 1D photonic crystal
distributed feedback structures, and in those coated with an active layer. General trends about the variation of the photonic bands were studied. Different DFB modes corresponding to different wave guided modes have been identified, and are fully indexed by using
±,456 . 2,3
With the increasing period of these structures, the energy of the photonic modes becomes significantly lowered, combined with the fact that the anti-crossing feature is stronger for smaller periods. The thickness of the SiO2 layer separating the photonic crystal and the substrate causes the change of the parabolic modes, and affects one family of the DFB modes corresponding to one value of X, that we call the first family relating to wave-guided modes in the SiO2 slab. When increasing the height of the combs, the energy level of all photonic modes becomes lower, and the DFB modes corresponding to another value of X, that we call the second family relating to wave-guided modes in the SiO2/air layer, are affected. The filling factor and the thickness of the active coating layer have a large influence on the DFB guided modes on the periodic SiO2/active medium layer. The active layer is found to play an essential role in having non-leaky photonic modes. The difference between the photonic band diagrams and the reflectivity spectra calculated by two different software packages using two different methods expresses the difference between the way they solve the coupling modes. Finally, a simple coupling model between the two non-leaky modes is presented, in a good agreement with the RCWA simulation. Even by the fact of using constant refractive indices for all materials, which limits the correctness of the simulation results in a small range of the low energy region, these results could be further improved by using the dispersive materials for the high energy in comparison with analytical and numerical approaches, and confirmed by experimental works. This result paves a way for us to tune the photonic modes by changing the geometry of the structures in order to obtain desirable photonic modes for future optoelectronic applications. Acknowledgements We are grateful to Dr. Hai-Son Nguyen, Dr. Anh T. Le, Dr. Ha Q. Duong and Dr. Cam T.H. Hoang for helpful discussions. References [1]
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15
List of captions Fig. 1. Dimensions of (a) a bare glass DFB, and (b) an active layer coated DFB unit cell. Fig. 2. Photonic band diagrams of the first 15 lowest modes of the DFB structures with t2 = 600 nm, h = 500 nm, FF = 0.3 and varying period Λ. The top row shows the photonic band diagrams of the DFB structures with no active layer, whereas the bottom row shows the photonic band diagrams of the DFB structures covered with an active layer of thickness t1 = 120 nm. Fig. 3. Reflectivity spectra of the DFB structures with t2 = 600 nm, h = 500 nm, FF = 0.3 and varying Λ, from 250 nm to 1000 nm, on the same energy and wave vector scale as shown in Fig. 2. The top row represents the reflectivity spectra of the DFB structures with no active layer, whereas the bottom row shows the reflectivity spectra of the DFB structures covered with an active layer of thickness t1 = 120 nm. Fig. 4. Reflectivity spectra of the DFB structures with Λ = 500 nm, h = 400 nm, FF = 0.2, and varying t2 from 600 nm to 1800 nm. The top row represents the reflectivity spectra of the DFB structures with no active layer, whereas the bottom row is for the DFB structures covered with an active layer of thickness t1 = 120 nm. Fig. 5. Reflectivity spectra of DFB structures with Λ = 500 nm, t2 = 600 nm, FF = 0.2, and varying h from 200 nm to 1000 nm. The top row represents the reflectivity spectra of the DFB structures with no active layer, whereas the bottom row shows the reflectivity spectra of the DFB structures covered with an active layer of thickness t1 = 120 nm. Fig. 6. Reflectivity spectra of the DFB structures with Λ = 500 nm, t2 = 600 nm, h = 500 nm, and varying FF from 0.3 to 0.9. The top row represents the reflectivity spectra of the DFB structures with no active layer, whereas the bottom row consists of the DFB structures covered with an active layer of thickness t1 = 120 nm. Fig. 7. Reflectivity spectra of the DFB structures with Λ = 500 nm; t2 = 600 nm; h = 500 nm, FF = 0.3 and varying t1 from 150 nm to 250 nm. Fig. 8. Comparison between the photonic modes calculated by MPB and the reflectivity spectrum calculated by S4 of the DFB structure with Λ = 500 nm; t2 = 600 nm; h = 400 nm; FF = 0.2; t1 = 120 nm.
16
Fig. 9. (a) Reflectivity spectrum of the DFB structure with Λ = 500 nm, t2 = 600 nm, h = 400 nm, FF = 0.2, t1 = 120 nm, and (b) a zoom in the central region with two modelled coupling modes (black lines).
17
Supplementary information 1. Reflectivity and folded wave-guided modes
Fig. S1. k-space reflectivity spectra for DFB structure with Λ = 500 nm, t2 = 600 nm, h = 500 nm, FF = 0.3 of (left) a bare DFB structure and (right) a DFB structure covered with an active layer of thickness t1 = 120 nm. For the bare DFB structure (left), two families of DFB modes in the central BZ, with different inclined angles, are highlighted (blue and pink lines). The extrapolated straight lines of these two families meet at the centers of two adjacent BZs (the left BZ and the right BZ). Bright lines are light lines in air, starting from two adjacent BZs. For the DFB structure covered with an active layer, a third family of DFB modes appears (black lines), with lowest inclined angle corresponding to highest refractive index. This third family should be wave-guided modes of the periodic active medium/SiO2 layer.
(a)
(b)
Fig. S2. (a) Simplified line diagram of uncoupling (no anti-crossing) folded waveguided modes (from ±,456 first adjacent BZs to the center BZ) in a DFB structures with nomenclature in this paper. 2,3 Different mode colors, with different inclined angles, represent different waveguides (green modes are from the waveguide number 1, blue modes are from the waveguide number 2, violet modes are from the waveguide number 3). (b) A typical reflectivity spectrum in k-space of an active layer covered DFB structure. Two families of straight modes can be easily observed (with conventional Bragg order denoted by m and m'). The energy level of the second order crossing of the first family (m = mDFB = 2) is smaller than that of the second family (m' = mDFB' = 2), thus implying that the effective refractive ±,456 , we can index for the first family is higher than that of the second. By using the notation 2,3 identify two lowest energy modes (first family), denoted as 18
7,89 ",#
and
8,79 ",# .
We suppose that X=3
is the third planar waveguide layer (the periodic active medium/SiO2 as shown in Fig. S3 below) counting from the top air to the bottom silicon.
Fig. S3. Simplified planar layers of (a) bare and (b) active layer covered DFB structures for waveguide consideration. Air and the substrate, being semi-infinite media, are not considered as waveguides. ±,456 Brighter color represents lower effective index of refraction. In the nomenclature , for the 2,3 bare DFB structure (a) X=1 for the periodic air/SiO2 layer, X=2 for the SiO2 slab; for the active layer covered DFB structure (b) X=1 for the periodic air/active medium layer, X=2 for the periodic air/SiO2 layer, X=3 for the periodic active medium/SiO2 layer, and X=4 for the SiO2 slab. Waveguided optical modes in the bare structure are leaky, wheresas there are non-leaky waveguided optical modes in the active-layer-covered structure (guided modes in the periodic active/air and active/SiO2 layers).
Fig. S4. Reflectivity spectra of a DFB structure on a silicon substrate (left column) and of a free standing DFB structure (right column – without Si substrate). On the top row reflectivity, we can observe that dark leaky wave-guided modes (of the DFB structure on a silicon substrate) are replaced by bright non-leaky wave-guided modes (of the free standing DFB structure). On the bottom row 19
reflectivity, when coated by an active layer, the additional bright modes (so called the third DFB family modes) are the same for the DFB structure on a silicon substrate and for the free standing DFB structure. They are wave-guided modes in the periodic active medium/SiO2 layer. 2. Coupling waveguides By using indices proposed in this paper, DFB modes of the third family are:
±,456 , 2,3
two states, represented in the bra-ket notation, of 8,79 ",#
|1; for the “uncoupling” mode
7,89 ",#
|2; for the “uncoupling” mode
The interacting Hamilton matrix between these two above modes is <==
8,79 ",#
>
$
>
7,89 ",#
$
?
where $ = $@ is the wavevector along the x direction 8,79 ",#
$ =
8,79 ",#
=
7,89 ",#
$ =
8,79 ",# 7,89 ",#
7,89 ",#
=
− &A $ is the energy of the state |1; + &A $ is the energy of the state |2; ",#
is the energy at $ = 0
&A is the group velocity of these two DFB guided modes U is the coupling constant between two modes By diagonalizing the above Hamiltonian, we obtain the following states (by neglecting a normalization constant) o
9
|−; = D&A $ − E$ - &A- + > - F ∙ |1; + |2; with an eigenvalue C
8
$ =
"
− E$ - &A- + > -,
"
+ E$ - &A- + > -,
corresponding to the hyperboloic lower branch of the coupling modes. o
9
|+; = |1; + D&A $ + E$ - &A- + > - F ∙ |2; with an eigenvalue C
7
$ =
corresponding to the hyperboloic upper branch of the coupling modes. The Rabi splitting energy between these two coupled eigenstates is H = 2> at $ = 0
By tracing 8 $ and 7 $ of the two coupling modes I–K and |+; in the refelectivity spectrum obtained by S4 of the same DFB structure, we obtained Figure 13b.
20
HIGHLIGHTS • • • •
Throughout investigation of photonic modes with various geometrical parameters of 1D photonic crystals The active layer plays an essential role in having non-leaky optical modes Strong anticrossing between two opposite direction non-leaky photonic modes Enhancement of anticrossing between photonic modes when decreasing the period and increasing the thickness of the perovskite layer
The authors declare no conflict of interest. On behalf of the authors NGUYEN-TRAN Thuat