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