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

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 o[r]

Original 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 xxx

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 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 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/ ).

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 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 Thefilling 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 2shows the first 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

Fig 1 Dimensions of (a) a bare glass DFB, and (b) an active layer coated DFB unit cell.

N.D Le, T Nguyen-Tran / Journal of Science: Advanced Materials and Devices xxx (xxxx) xxx 2

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 forL 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

(forL 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 neff ¼

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 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 ofFig 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 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 neff¼ 1:48 (higher than that of SiO2) 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.

order 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 SiO2slab 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/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/SiO2layer, 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 xxx (xxxx) xxx 4

3.2 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

t2¼ 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 the DFB structures with no active layer, whereas the bottom row is for the DFB structures covered with an active layer of thickness t 1 ¼ 120 nm.

3.4 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 structures with no active layer, whereas the bottom row consists 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 xxx (xxxx) xxx 6

Concerning 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 2 ¼ 600 nm; h ¼ 500 nm, FF ¼ 0.3 and varying t 1 from 150 nm to 250 nm.

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.

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

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

https://doi.org/10.1016/j.jsamd.2020.01.008 References

[1] B.E.A Saleh, M.C Teich, Fundamentals of Photonics, second ed., John Wiley & Sons, Inc, Hoboken, New Jersey, 2007

[2] J.J.D Joannopoulos, S Johnson, J.N.J Winn, R.R.D Meade, Photonic Crystals: Molding the Flow of Light, second ed., Princeton University Press, Princeton,

2008 https://doi.org/10.1063/1.1586781 [3] A Yariv, P Yeh, Photonics Optical Electronics in Modern Communications, sixth ed., Oxford University Press, New York, 2007

[4] P Brenner, M Stulz, D Kapp, T Abzieher, U.W Paetzold, A Quintilla, I.A Howard, H Kalt, U Lemmer, Highly stable solution processed metal-halide

Fig 9 (a) Reflectivity spectrum 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, and (b) a zoom in the central region with two modelled coupling modes (black lines).

N.D Le, T Nguyen-Tran / Journal of Science: Advanced Materials and Devices xxx (xxxx) xxx 8

perovskite lasers on nanoimprinted distributed feedback structures, Appl.

Phys Lett 109 (2016) 141106, https://doi.org/10.1063/1.4963893

[5] S Chen, K Roh, J Lee, W.K Chong, Y Lu, N Mathews, T.C Sum, A Nurmikko,

A photonic crystal laser from solution based organo-lead iodide perovskite

thin films, ACS Nano 10 (2016) 3959e3967, https://doi.org/10.1021/

acsnano.5b08153

[6] T Feng, T Hosoda, L Shterengas, G Kipshidze, A Stein, M Lu, G Belenky,

Laterally coupled distributed feedback type-I quantum well cascade diode

lasers emitting near 3.22mm, Appl Optic 56 (2017) H74eH80, https://doi.org/

10.1364/AO.56.000H74

[7] T Shindo, T Okumura, H Ito, T Koguchi, D Takahashi, Y Atsumi, J Kang,

R Osabe, T Amemiya, N Nishiyama, S Arai, Lateral-current-injection

distributed feedback laser with surface grating structure, IEEE J Sel Top.

Quant Electron 17 (2011) 1175e1182, https://doi.org/10.1109/

JSTQE.2011.2131636

[8] G.L Whitworth, J.R Harwell, D.N Miller, G.J Hedley, W Zhang, H.J Snaith,

G.A Turnbull, I.D.W Samuel, Nanoimprinted distributed feedback lasers of

solution processed hybrid perovskites, Optic Express 24 (2016) 23677, https://

doi.org/10.1364/oe.24.023677

[9] I Vurgaftman, J.R Meyer, Photonic-crystal distributed-feedback lasers, Appl.

Phys Lett 78 (2001) 1475e1477, https://doi.org/10.1063/1.1355670

[10] D.P Schinke, R.G Smith, E.G Spencer, M.F Galvin, Thin-film

distributed-feedback laser fabricated by ion milling, Appl Phys Lett 21 (1972)

494e496, https://doi.org/10.1063/1.1654232

[11] G Heliotis, R Xia, D.D.C Bradley, G.A Turnbull, I.D.W Samuel, P Andrew,

W.L Barnes, Two-dimensional distributed feedback lasers using a broadband,

red polyfluorene gain medium, J Appl Phys 96 (2004) 6959e6965, https://

doi.org/10.1063/1.1811374

[12] H Kogelnik, C.V Shank, CoupledWave theory of distributed feedback lasers,

J Appl Phys 43 (1972) 2327e2335, https://doi.org/10.1063/1.1661499

[13] S Wang, Principles of distributed feedback and distributed Bragg-Reflector

lasers, IEEE J Quant Electron 10 (1974) 413e427, https://doi.org/10.1109/

JQE.1974.1068152

[14] S Wang, S Sheem, Two-dimensional distributed-feedback lasers and their

applications, Appl Phys Lett 22 (1973) 460e462, https://doi.org/10.1063/

1.1654712

[15] S Li, G Witjaksono, S Macomber, D Botez, Analysis of surface-emitting

sec-ond-order distributed feedback lasers with central grating Phaseshift, IEEE J.

Sel Top Quant Electron 9 (2003) 1153e1165, https://doi.org/10.1109/

JSTQE.2003.819467

[16] G.B Morrison, D.T Cassidy, A Probability-amplitude transfer matrix model for

distributed-feedback laser structures, IEEE J Quant Electron 36 (2000)

633e640, https://doi.org/10.1109/3.845716

[17] N Susa, Threshold gain and gain-enhancement due to distributed-feedback in

two-dimensional photonic-crystal lasers, J Appl Phys 89 (2001) 815e823,

https://doi.org/10.1063/1.1332806

[18] L Mahler, A Tredicucci, F Beltram, C Walther, J Faist, H.E Beere, D.A Ritchie, D.S Wiersma, Quasi-periodic distributed feedback laser, Nat Photon 4 (2010) 165e169, https://doi.org/10.1038/nphoton.2009.285

[19] C Ge, M Lu, Y Tan, B.T Cunningham, Enhancement of pump efficiency of a visible wavelength organic distributed feedback laser by resonant optical pumping, Optic Express 19 (2011) 5086, https://doi.org/10.1364/ oe.19.005086

[20] N Pourdavoud, A Mayer, M Buchmüller, K Brinkmann, T H€ager, T Hu,

R Heiderhoff, I Shutsko, P G€orrn, Y Chen, H.C Scheer, T Riedl, Distributed feedback lasers based on MAPbBr3, Adv Mater Technol 3 (2018) 1700253,

https://doi.org/10.1002/admt.201700253 [21] M Saliba, S.M Wood, J.B Patel, P.K Nayak, J Huang, J.A Alexander-Webber,

B Wenger, S.D Stranks, M.T H€orantner, J.T.-W Wang, R.J Nicholas, L.M Herz, M.B Johnston, S.M Morris, H.J Snaith, M.K Riede, Structured organic-inorganic perovskite toward a distributed feedback laser, Adv Mater 28 (2016) 923e929, https://doi.org/10.1002/adma.201502608

[22] F Mathies, P Brenner, G Hernandez-Sosa, I.A Howard, U.W Paetzold,

U Lemmer, Inkjet-printed perovskite distributed feedback lasers, Optic Ex-press 26 (2018) A144eA152, https://doi.org/10.1364/oe.26.00a144 [23] Y Jia, R.A Kerner, A.J Grede, A.N Brigeman, B.P Rand, N.C Giebink, Diode-pumped organo-lead halide perovskite lasing in a metal-clad distributed feedback resonator, Nano Lett 16 (2016) 4624e4629, https://doi.org/10.1021/ acs.nanolett.6b01946

[24] Y Jia, R.A Kerner, A.J Grede, B.P Rand, N.C Giebink, Continuous-wave lasing

in an organic-inorganic lead halide perovskite semiconductor, Nat Photon 11 (2017) 784e788, https://doi.org/10.1038/s41566-017-0047-6

[25] A Gharajeh, R Haroldson, Z Li, J Moon, B Balachandran, W Hu, A Zakhidov,

Q Gu, Continuous-wave operation in directly patterned perovskite distrib-uted feedback light source at room temperature, Opt Lett 43 (2018) 611e614, https://doi.org/10.1364/ol.43.000611

[26] L Zhang, R Gogna, W Burg, E Tutuc, H Deng, Photonic-crystal exciton-polaritons in monolayer semiconductors, Nat Commun 9 (2018) 713,

https://doi.org/10.1038/s41467-018-03188-x [27] M Polyanskiy, Refractive Index Database, 2019 https://refractiveindex.info/ (Accessed 3 January 2019).

[28] S Johnson, J Joannopoulos, Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis, Optic Express 8 (2001) 173e190,

https://doi.org/10.1364/OE.8.000173 [29] M.G Moharam, E.B Grann, D.A Pommet, Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,

J Opt Soc Am A 12 (1995) 1068e1076, https://doi.org/10.1364/ JOSAA.12.001068

[30] V Liu, S Fan, S4: a free electromagnetic solver for layered periodic structures, Comput Phys Commun 183 (2012) 2233e2244, https://doi.org/10.1016/ j.cpc.2012.04.026