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.
Trang 1Original Article
Simulation of coupling optical modes in 1D photonic crystals for
optoelectronic applications
Ngoc Duc Lea,b, Thuat Nguyen-Trana,*
a Nano and Energy Center, VNU University of Science, 334 Nguyen Trai, Thanh Xuan, Hanoi, Viet Nam
b 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, Viet Nam
a r t i c l e i n f o
Article history:
Received 9 April 2019
Received in revised form
30 January 2020
Accepted 30 January 2020
Available online 7 February 2020
Keywords:
1D photonic crystal
DFB structure
Angle e resolved reflectivity
Photonic band diagram
Coupling waves
a b s t r a c t
In this paper, one-dimensional photonic crystal distributed feedback structures were chosen for simu-lating 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±;mBZ
n;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 (http://creativecommons.org/licenses/by/4.0/)
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 re
flec-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)
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 [4e11] 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 wave-guided 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 [1e3]
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 orfirst 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).
Peer review under responsibility of Vietnam National University, Hanoi.
Contents lists available atScienceDirect Journal of Science: Advanced Materials and Devices
j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / j s a m d
https://doi.org/10.1016/j.jsamd.2020.01.008
2468-2179/© 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
Trang 2called 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 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 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 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 1shows 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 ns¼ 3.97) There are two types of DFB structures
for each simulation: (i) the bare structure (Fig 1a), and the active
layer (refractive index n1¼ 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 byL, 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
lengthL 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 re
flec-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 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 kxof 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
Fig 2 shows thefirst 15 lowest photonic modes, in the first Brillouin zone (BZ), of the DFB structures withfixed parameters
t2¼ 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 periodL 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 forL 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 bebe-tween 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 periodL 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]:
2
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
N.D Le, T Nguyen-Tran / Journal of Science: Advanced Materials and Devices 5 (2020) 142e150 143
Trang 3period 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
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 the
lowest crossing at the center of thefirst BZ (kx¼ 0) between the two
straight DFB modes (second order DFB modes mDFB¼ 2), we find
that the energy value equals to 3.45 eV (forL 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 (forL ¼ 250 nm), 1.70 eV (for L ¼ 500 nm),
1.10 eV (forL 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 neff¼
1:48 (higher than the refractive index of SiO2) This value of
effec-tive refraceffec-tive index shows that these DFB photonic modes would
be correlated to the light propagation in the SiO2layer (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 inFig 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 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 average 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 forL 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 ofL The inclined angles of the two straight DFB families are different, corresponding
to different effective refractive indices ForL ¼ 500 nm, we observe that thefirst family of the DFB modes, corresponding to a low in-clined angle (top row, second column from the left ofFig 3), are wave-guided modes in the SiO2layer with a refractive index of
neff¼ 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 ofFig 3), are wave-guided modes in the comb periodic structure between air and SiO2, with a refractive index of
neff¼ 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 thefirst 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 inFig S1 Thefirst family of the DFB modes are parallel to the LL in SiO2; and the second one lies between the LL in SiO2and 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 n ¼ 1:48 (higher than that of SiO) In
Fig 2 Photonic band diagrams of the first 15 lowest modes of the DFB structures with t 2 ¼ 600 nm, h ¼ 500 nm, FF ¼ 0.3 and varying period L 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
t 1 ¼ 120 nm.
Trang 4order to identify all DFB modes encountered in this study, we
propose to use TE±;mBZ
n ;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) mBZis for indicating from which BZ the waves come mBZ¼ 0
for waves in the central BZ, mBZ¼ 1 for waves from the left
BZ, mBZ¼ 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
on the DFB structure in consideration We note that X¼ 1 for
the top layer (the periodic SiO2/air for the bare DFB
struc-ture), X¼ 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/SiO2layer, 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/SiO2layer is higher than its above layer, which is
the air/SiO2layer, and its below layer, which is the SiO2slab Such
details are illustrated inFigs S2 and S3 A comparison between the
conventional DFB structure index mDFBand 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
periodL increases, the second family of DFB modes,
correspond-ing to the wave-guided modes in the periodic air/SiO 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 thefirst 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 non-leaky 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 be-tween the DFB modes of the SiO2slab (thefirst 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 L 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
Fig 3 Reflectivity spectra of the DFB structures with t 2 ¼ 600 nm, h ¼ 500 nm, FF ¼ 0.3 and varying L, 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 t 1 ¼ 120 nm.
N.D Le, T Nguyen-Tran / Journal of Science: Advanced Materials and Devices 5 (2020) 142e150 145
Trang 53.2 Influence of the thickness of the SiO2layer
InFig 4, we can see that the reflectivity contrast is improved in
the presence of the active layer The same effect is observed inFig 3
because of the conditions of the refractive indices which are
favorable for the wave-guided modes in the periodic active
me-dium/SiO2layer We also see that although the thickness t2of the
SiO2slab varies, in both DFB structures with no active layer and
with an active layer, the inclined angles of the straight DFB modes
of thefirst 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
para-bolic 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 SiO2slab and the
periodic SiO2/air layer By considering only one family of the DFB
modes inFig 5, such as thefirst 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 ¼ 600 nm As mentioned before, these submodes are
conven-tional 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
t2¼ 900 nm, and to approximately seven (7) for t2¼ 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 SiO2slab 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 thefirst 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 kx¼ 0 and 2 eV, resulted from the coupling between the
DFB modes of thefirst 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 un-changed with a large variation range of the value of t2
3.3 Influence of the height of the comb
InFig 5, we can clearly observe that the contrast of the re flec-tivity 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 modes, the contrast is changed with increasing h For h¼ 200 nm, there are four (4) principal dark fringes, at the energy range from 0 to 3 eV For h¼ 600 nm, in addition to these four (4) principal fringes, we can observe more secondary fringes with less contrast For
h ¼ 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 SiO2layer, 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 corre-sponding to two different values of the inclined angle The second family is barely observed for h¼ 200 nm When h ¼ 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
h¼ 200 nm This observation confirms the fact that the second family of the DFB modes represents the guided modes in the pe-riodic air/SiO2combed layer, and that thefirst family are modes in the SiO2slab 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
Fig 4 Reflectivity spectra of the DFB structures with L ¼ 500 nm, h ¼ 400 nm, FF ¼ 0.2, and varying t 2 from 600 nm to 1800 nm The top row represents the reflectivity spectra of
¼ 120 nm.
Trang 63.4 Influence of the filling factor
As it can be seen inFig 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
in-creases slightly as FF inin-creases This is a direct consequence of the
fact that the effective refractive index of the DFB structure neff
increases slightly with the filling factor For the DFB structures
with no active layer, by observing the DFB modes, for FF¼ 0.3, we
can recognize two different families of DFB modes, which
correspond to two different inclined angles For FF¼ 0.6, the first family is quite the same, but the second family is quite different For FF¼ 0.9, we can hardly find the second family, which corre-sponds to the higher inclined angles and is related to the propa-gating waves in the periodic air/SiO2structure Since FF¼ 0.9, the full SiO2 slab looks to be extended in thickness, thus the first family is not the same as for FF¼ 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
Fig 5 Reflectivity spectra of DFB structures with L ¼ 500 nm, t 2 ¼ 600 nm, FF ¼ 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 t 1 ¼ 120 nm.
Fig 6 Reflectivity spectra of the DFB structures with L ¼ 500 nm, t 2 ¼ 600 nm, h ¼ 500 nm, and varying FF from 0.3 to 0.9 The top row represents the reflectivity spectra of the DFB
¼ 120 nm.
N.D Le, T Nguyen-Tran / Journal of Science: Advanced Materials and Devices 5 (2020) 142e150 147
Trang 7Concerning the anti-crossing features, for FF¼ 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
FF¼ 0.6, the opening energy gap of the lowest energy coupling
mode becomes rather complicated by the involvement of the DFB
modes of the second family For FF¼ 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 inFig 7, when the thickness t1of 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
un-changed More precisely, by considering thefirst and the second
order parabolic modes, their energy levels at the center of thefirst
BZ are at 0.66 eV and 0.77 eV for t1¼ 150 nm; 0.62 eV and 0.73 eV
for t2 ¼ 200 nm; and 0.59 eV and 0.69 eV for t2 ¼ 250 nm,
respectively For the DFB modes, we can observe thefirst 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 (t1¼ 150 nm) to 0.063 eV (t1¼ 200 nm),
and to 0.076 eV (t1¼ 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 8shows 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
spec-trum (calculated by the S4package) of the same DFB structure with
the parameters ofL ¼ 500 nm, t2¼ 600 nm, h ¼ 400 nm, FF ¼ 0.2,
and t1¼ 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, wefind that the green band corresponds to the thin
“reversed V e shaped” dispersion curve (A) and the pink band
corresponds to the “V e 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 S4uses the rigorous coupled wave analysis method and gives more pronounced coupled photon modes by taking into ac-count 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 withL ¼ 500 nm, t2¼ 600 nm, h ¼ 400 nm, FF ¼ 0.2,
t1¼ 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 be-tween these two modes is represented by an interacting second
Fig 7 Reflectivity spectra of the DFB structures with L ¼ 500 nm; t ¼ 600 nm; h ¼ 500 nm, FF ¼ 0.3 and varying t
Fig 8 Comparison between the photonic modes calculated by MPB and the reflec-tivity spectrum calculated by S 4 of the DFB structure with L ¼ 500 nm; t 2 ¼ 600 nm;
h ¼ 400 nm; FF ¼ 0.2; t 1 ¼ 120 nm.
Trang 8order 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
pro-cedure Thefitting was performed by tracing the eigenvalues in the
same reflectivity spectrum in order to obtain the parameters, such
as E0;3 (energy at kx¼ 0), vg (group velocity) and U (coupling
constant) The black lines inFig 9b represent the eigenvalues of the
coupling states after diagonalizing the interacting Hamiltonian The
bestfit, illustrated by a good agreement between the modelled
eigenvalues and the reflectivity spectrum inFig 9b, gave E0;3¼
1:542 eV, vg¼ 0:633eV:mm
2p and U¼ 0.032 eV A similar model can be carried out forfitting the anti-crossing between two wave-guided
modes in the vicinity of kx¼ ±p=L, thus a full dispersion relation
of coupling wave-guided modes can be obtained From thisfitting
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 TEn;X modes Second, by folding these
modes in the different BZs, the dispersion curves of the uncoupled
TE±;mBZ
n ;X modes are obtained Note that the dispersion curves of the
uncoupled modes can be modelled by the straight lines in the
vi-cinity of kx¼ 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 TE±;mBZ
n;X 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
corre-sponding to different wave guided modes have been identified, and
are fully indexed by using TE±;mBZ
n;X 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 thefirst family
relating to wave-guided modes in the SiO2slab 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 Thefilling 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 cor-rectness 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 experi-mental 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
Declaration of Competing Interest The authors declare no conflict of interest
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
Appendix A Supplementary data Supplementary data to this article can be found online at
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